{"id": "2602.00112v1", "paper_link": "http://arxiv.org/abs/2602.00112v1", "theorems_cnt": 1, "theorem": {"env_name": "proposition", "content": "\\label{Prop1}\nFor every  torse-forming vector field $V$ that satisfies (\\ref{TFVF}),  we have\n\\begin{equation}\\label{Theta}\n\t\t\\theta ={\\rm d}\\rho - f {\\rm e}^{-2 \\rho} \\omega.\n\t\\end{equation}", "start_pos": 5584, "end_pos": 5812, "label": "Prop1"}, "ref_dict": {"TFVF": "\\begin{equation}\\label{TFVF}\n\t\t\\nabla_X V = fX  + \\theta(X)V,\n\t\\end{equation}", "Eq110": "\\begin{eqnarray}\\label{Eq110}\ng(\\nabla_X V, V)= X g(V,V)-g(V , \\nabla_X V)  &\\Leftrightarrow &   g(\\nabla_X V, V) = \\frac{1}{2} X({\\rm e}^{2\\rho}) \\notag\\\\\n&\\Leftrightarrow & g(\\nabla_X V, V) = {\\rm e}^{2\\rho} X(\\rho).\n\\end{eqnarray}", "Eq100": "\\begin{eqnarray}\\label{Eq100}\ng(\\nabla_X V, V) &=& f g(X,V)  + \\theta(X)g(V,V) \\notag\\\\\n&=& f \\omega(X) + {\\rm e}^{2\\rho} \\theta(X).\n\\end{eqnarray}"}, "pre_theorem_intro_text_len": 3383, "pre_theorem_intro_text": "It is well known that generalizing mathematical concepts is a prerequisite for most researchers. It is also often an important means of developing mathematics and answering many questions that have occupied the minds of researchers in many fields. For example, the notion of a Ricci soliton is a natural generalization of an Einstein metric; at the same time, it is a stationary solution of a famous PDE for Riemannian metrics, known under the name of the Ricci flow equation. Without forgetting that Ricci soliton itself has had many generalizations made recently in many works.\n\nGenerally, a mathematical concept is initially defined by a simple expression that satisfies the need to include it. Then questions begin: What if we replace the constant with a function? What if we add a differential one-form? What happens if we add a certain term? ... And so, little by little, the simple expression begins to take on longer and more complex forms. \n\nIn principle, if these generalizations fail to answer any outstanding mathematical problems or fail to impact other concepts, then they can be considered an intellectual luxury. \n\nIn this paper we present a step in the reverse direction of generalization; as a subject of study, we will discuss the concept of the torse-forming vector field as a generalization of both the  torqued vector field, the concircular vector field, the concurrent vector field, the recurrent vector field, and the parallel vector field.\n\n Torse-forming vector fields  demand special attention due to their applications not only in relativity and cosmology but in theory of submanifolds also. On a Riemannian manifold $(M^n, g)$, a the torse-forming vector field is a vector field $V$ that satisfies\n\\begin{equation}\\label{TFVF}\n\t\t\\nabla_X V = fX  + \\theta(X)V,\n\t\\end{equation}\nfor any vector field $X$ on $M$ where $\\nabla$ is the Levi-Civita connection of $g$, $\\theta$ is a one-form and $f$ is a smooth function on $(M^n, g)$. The 1-form $\\theta$ is called the generating form and the function $f$ is called the conformal scalar.  If $\\omega$ is the corresponding  $1$-form of $V$ , i.e., $\\omega(X) = g(X,V)$ then for all $X, Y$ vector fields on $M$\n\\begin{equation}\\label{TFVF2}\n (\\nabla_X \\omega)Y = g(\\nabla_X V , Y)= fg(X,Y)+ \\theta(X)\\omega(Y).\n\t\\end{equation}\n Some special types of torse-forming vector fields have been considered in various studies. A torse-forming vector field $V$ is called: \n\\begin{itemize}\n\\item[1)] proper torse-forming if $f \\neq 0$ and the 1-form $\\theta$ is nowhere zero on a dense open subset of $M$.\n\\item[2)] torqued vector field if $V$ satisfying (\\ref{TFVF}) with $\\theta(V)=0$ (see \\cite{CBY2, CBY3}).\n\\item[3)] concircular vector field  if $\\theta$ is identically zero \\cite{YK2} .\n\\item[4)] concurrent vector field if $\\theta = 0$ and $f=1$.\n\\item[5)] reccurent vector field if $\\theta \\neq 0$ and $f=0$.\n\\item[6)] geodesic vector field if $\\nabla_V V=0$.\n\\item[7)] parallel vector field if $\\theta = 0$ and $f=0$.\n\\end{itemize}\n\n\\textbf{Agreement:} Through the rest of this paper, $(M,g)$ always denotes a Riemannian manifold.  $V$ denotes a proper torse-forming vector field on $M$ that satisfies (\\ref{TFVF}) with $\\Vert V \\Vert = {\\rm e}^{\\rho}$, and  $\\omega $  the  $1$-form corresponding to $V$, i.e.,  for all $X$ vector field on $M$,\n$$ \\omega(X)=g(V , X)\\qquad and \\qquad  \\omega(V)={\\rm e}^{2\\rho}.$$", "context": "It is well known that generalizing mathematical concepts is a prerequisite for most researchers. It is also often an important means of developing mathematics and answering many questions that have occupied the minds of researchers in many fields. For example, the notion of a Ricci soliton is a natural generalization of an Einstein metric; at the same time, it is a stationary solution of a famous PDE for Riemannian metrics, known under the name of the Ricci flow equation. Without forgetting that Ricci soliton itself has had many generalizations made recently in many works.\n\nGenerally, a mathematical concept is initially defined by a simple expression that satisfies the need to include it. Then questions begin: What if we replace the constant with a function? What if we add a differential one-form? What happens if we add a certain term? ... And so, little by little, the simple expression begins to take on longer and more complex forms.\n\nIn this paper we present a step in the reverse direction of generalization; as a subject of study, we will discuss the concept of the torse-forming vector field as a generalization of both the torqued vector field, the concircular vector field, the concurrent vector field, the recurrent vector field, and the parallel vector field.\n\nTorse-forming vector fields demand special attention due to their applications not only in relativity and cosmology but in theory of submanifolds also. On a Riemannian manifold $(M^n, g)$, a the torse-forming vector field is a vector field $V$ that satisfies\n\\begin{equation}\\label{TFVF}\n \\nabla_X V = fX + \\theta(X)V,\n \\end{equation}\nfor any vector field $X$ on $M$ where $\\nabla$ is the Levi-Civita connection of $g$, $\\theta$ is a one-form and $f$ is a smooth function on $(M^n, g)$. The 1-form $\\theta$ is called the generating form and the function $f$ is called the conformal scalar. If $\\omega$ is the corresponding $1$-form of $V$ , i.e., $\\omega(X) = g(X,V)$ then for all $X, Y$ vector fields on $M$\n\\begin{equation}\\label{TFVF2}\n (\\nabla_X \\omega)Y = g(\\nabla_X V , Y)= fg(X,Y)+ \\theta(X)\\omega(Y).\n \\end{equation}\n Some special types of torse-forming vector fields have been considered in various studies. A torse-forming vector field $V$ is called: \n\\begin{itemize}\n\\item[1)] proper torse-forming if $f \\neq 0$ and the 1-form $\\theta$ is nowhere zero on a dense open subset of $M$.\n\\item[2)] torqued vector field if $V$ satisfying (\\ref{TFVF}) with $\\theta(V)=0$ (see \\cite{CBY2, CBY3}).\n\\item[3)] concircular vector field if $\\theta$ is identically zero \\cite{YK2} .\n\\item[4)] concurrent vector field if $\\theta = 0$ and $f=1$.\n\\item[5)] reccurent vector field if $\\theta \\neq 0$ and $f=0$.\n\\item[6)] geodesic vector field if $\\nabla_V V=0$.\n\\item[7)] parallel vector field if $\\theta = 0$ and $f=0$.\n\\end{itemize}\n\n\\textbf{Agreement:} Through the rest of this paper, $(M,g)$ always denotes a Riemannian manifold. $V$ denotes a proper torse-forming vector field on $M$ that satisfies (\\ref{TFVF}) with $\\Vert V \\Vert = {\\rm e}^{\\rho}$, and $\\omega $ the $1$-form corresponding to $V$, i.e., for all $X$ vector field on $M$,\n$$ \\omega(X)=g(V , X)\\qquad and \\qquad \\omega(V)={\\rm e}^{2\\rho}.$$\n\n\\begin{equation}\\label{TFVF}\n\t\t\\nabla_X V = fX  + \\theta(X)V,\n\t\\end{equation}", "full_context": "It is well known that generalizing mathematical concepts is a prerequisite for most researchers. It is also often an important means of developing mathematics and answering many questions that have occupied the minds of researchers in many fields. For example, the notion of a Ricci soliton is a natural generalization of an Einstein metric; at the same time, it is a stationary solution of a famous PDE for Riemannian metrics, known under the name of the Ricci flow equation. Without forgetting that Ricci soliton itself has had many generalizations made recently in many works.\n\nGenerally, a mathematical concept is initially defined by a simple expression that satisfies the need to include it. Then questions begin: What if we replace the constant with a function? What if we add a differential one-form? What happens if we add a certain term? ... And so, little by little, the simple expression begins to take on longer and more complex forms.\n\nIn this paper we present a step in the reverse direction of generalization; as a subject of study, we will discuss the concept of the torse-forming vector field as a generalization of both the torqued vector field, the concircular vector field, the concurrent vector field, the recurrent vector field, and the parallel vector field.\n\nTorse-forming vector fields demand special attention due to their applications not only in relativity and cosmology but in theory of submanifolds also. On a Riemannian manifold $(M^n, g)$, a the torse-forming vector field is a vector field $V$ that satisfies\n\\begin{equation}\\label{TFVF}\n \\nabla_X V = fX + \\theta(X)V,\n \\end{equation}\nfor any vector field $X$ on $M$ where $\\nabla$ is the Levi-Civita connection of $g$, $\\theta$ is a one-form and $f$ is a smooth function on $(M^n, g)$. The 1-form $\\theta$ is called the generating form and the function $f$ is called the conformal scalar. If $\\omega$ is the corresponding $1$-form of $V$ , i.e., $\\omega(X) = g(X,V)$ then for all $X, Y$ vector fields on $M$\n\\begin{equation}\\label{TFVF2}\n (\\nabla_X \\omega)Y = g(\\nabla_X V , Y)= fg(X,Y)+ \\theta(X)\\omega(Y).\n \\end{equation}\n Some special types of torse-forming vector fields have been considered in various studies. A torse-forming vector field $V$ is called: \n\\begin{itemize}\n\\item[1)] proper torse-forming if $f \\neq 0$ and the 1-form $\\theta$ is nowhere zero on a dense open subset of $M$.\n\\item[2)] torqued vector field if $V$ satisfying (\\ref{TFVF}) with $\\theta(V)=0$ (see \\cite{CBY2, CBY3}).\n\\item[3)] concircular vector field if $\\theta$ is identically zero \\cite{YK2} .\n\\item[4)] concurrent vector field if $\\theta = 0$ and $f=1$.\n\\item[5)] reccurent vector field if $\\theta \\neq 0$ and $f=0$.\n\\item[6)] geodesic vector field if $\\nabla_V V=0$.\n\\item[7)] parallel vector field if $\\theta = 0$ and $f=0$.\n\\end{itemize}\n\n\\textbf{Agreement:} Through the rest of this paper, $(M,g)$ always denotes a Riemannian manifold. $V$ denotes a proper torse-forming vector field on $M$ that satisfies (\\ref{TFVF}) with $\\Vert V \\Vert = {\\rm e}^{\\rho}$, and $\\omega $ the $1$-form corresponding to $V$, i.e., for all $X$ vector field on $M$,\n$$ \\omega(X)=g(V , X)\\qquad and \\qquad \\omega(V)={\\rm e}^{2\\rho}.$$\n\n\\begin{equation}\\label{TFVF}\n\t\t\\nabla_X V = fX  + \\theta(X)V,\n\t\\end{equation}\n\nFor a conformal transformation, it is well known that the Levi-Civita connection $\\nabla$ and $\\tilde{\\nabla}$ associated with the metrics $g$ and $\\tilde{g}$ respectively are connected by\n\\begin{eqnarray}\\label{TransfConf}\n\\tilde{\\nabla}_{X}Y= \\nabla_{X}Y + X(\\sigma)Y + Y(\\sigma)X - g(X,Y){\\rm grad}\\sigma.\n \\end{eqnarray}\nBy using (\\ref{TFVF}) and Proposition \\ref{Prop1}, we have\n\\begin{eqnarray}\\label{TildNabV}\n\\tilde{\\nabla}_{X}V &=& \\nabla_{X}V + X(\\sigma)V + V(\\sigma)X - g(X,V){\\rm grad}\\sigma \\notag\\\\\n&=& f X + \\big( X(\\rho) - f {\\rm e}^{-2\\rho} \\omega(X) \\big)V + X(\\sigma)V + V(\\sigma)X - \\omega(X){\\rm grad}\\sigma \\notag\\\\\n&=& \\big(f + V(\\sigma) \\big) X + \\big( X(\\rho) + X(\\sigma) \\big)V - \\omega(X) \\big( f {\\rm e}^{-2\\rho} V +{\\rm grad}\\sigma \\big).\n\\end{eqnarray}\nIt is clear that $V$ is a torse-formming vector field on $(M,\\tilde{g})$ if and only if\n$${\\rm grad}\\sigma = -f {\\rm e}^{-2\\rho} V.$$\nAlso, under this condition, we get $ f + V(\\sigma)=0$ and (\\ref{TildNabV}) gives\n$$\\tilde{\\nabla}_{X}V = \\theta(X)V.$$\nConsequently, $V$ is recurrent with respect to $\\tilde{g}$. Hence, we conclude the following:\n\\begin{theorem}\\label{Th1}\nLet $V$ be a proper torse-formming vector field on $(M,g)$ which satisfying (\\ref{TFVF}). $V$ is recurrent with respect to $\\tilde{g}={\\rm e}^{2\\sigma}g$ where $\\sigma \\in \\mathcal{C}^{\\infty}(M)$ if and only if ${\\rm grad}\\sigma = -f {\\rm e}^{-2\\rho} V$.\n\\end{theorem}\n\\begin{example}\nIt is convenient to have some sort of hyperspherical coordinates on the unit sphere $\\mathbb{S}^3$ in analogy to the usual spherical coordinates on $\\mathbb{S}^2$. One such choice is to use $(x_1, x_2,x_3)$ where\n\\begin{equation}\\label{parametrisationS3}\n \\left\\{\n \\begin{array}{lll}\nx_1 = \\cos x_1,\\\\\nx_2 = \\sin x_1 \\cos x_2,\\\\\nx_3 = \\sin x_1 \\sin x_2 \\cos x_3,\\\\\nx_4 = \\sin x_1 \\sin x_2 \\sin x_3,\n \\end{array}\n \\right.\n\\end{equation}\nwhere $x_1$ and $x_2$ run over the range $0$ to $\\pi$, and $x_3$ runs over $0$ to $2 \\pi$.\n\nOne can define the Riemannian metric $g$ in these coordinates as follows:\n$$ g = d x_1^2 + \\sin^2 x_1 ( dx_2^2 + \\sin^2 x_2 \\;d x_3^2).$$\nwhere\n$$\\left\\lbrace e_1 = \\frac{\\partial}{\\partial x_1},\\quad e_2 = \\frac{1}{\\sin x_1}\\frac{\\partial}{\\partial x_2},\\quad\ne_3=\\frac{1}{\\sin x_1 \\sin x_2}\\frac{\\partial}{\\partial x_3}\\right\\rbrace ,$$\nis an orthonormal basis for $g$. By Kozsul's formula, the covariant derivatives of the basis elements are as follows:\n$$\\begin{tabular}{lll}\n$ \\nabla_{e_1} e_1 = 0,$ & $\\nabla_{e_1} e_2 =0,$ & $\\nabla_{e_1} e_3 = 0$, \\\\ \n$\\nabla_{e_2} e_1= \\cot x_1 e_2,$ & $\\nabla_{e_2} e_2= -\\cot x_1 e_1,$ & $\\nabla_{e_2} e_3 =0,$ \\\\ \n$\\nabla_{e_3} e_1= \\cot x_1 e_3,$& $\\nabla_{e_3} e_2 = \\frac{\\cot x_2 }{\\sin x_1} e_3,$ & $\\nabla_{e_3} e_3 = -\\cot x_1 e_1 - \\frac{\\cot x_2 }{\\sin x_1} e_2.$ \\\\ \n\\end{tabular} \n$$\nEasily, we can see that for $ V = h(x_3) \\sin x_1 e_1$ we have \n\\begin{equation}\\label{NabV1}\n \\left\\{\n \\begin{array}{lll}\n\\nabla_{e_1} V =h(x_3) \\cos x_1 e_1 ,\\\\\n\\nabla_{e_2} V = h(x_3) \\cos x_1 e_2,\\\\\n\\nabla_{e_3} V =\\frac{h'}{\\sin x_2 } e_1 + h(x_3) \\cos x_1 e_3,\n \\end{array}\n \\right.\n\\end{equation}\nwhere $h' = \\frac{\\partial h}{\\partial x_3}$. Then, for $\\theta = \\frac{h'}{h} dx_3$ we get\n$$\\nabla_{e_i} V = h(x_3) \\cos x_1 e_i + \\theta(e_i) V,$$\nhence, according to (\\ref{TFVF}), $V$ is a proper torse-forming vector field on $(\\mathbb{S}^3, g)$ with $f =h(x_3) \\cos x_1$ and $\\theta = \\frac{h'}{h} dx_3$.\n\n\\begin{proposition}\n Let $\\nabla$ and $\\hat{\\nabla} $ denote the Riemannian connections of $ g$ and $\\hat{g}$ respectively. Then, for all $X,Y $ vector fields on $M$, we have the relation:\n\\begin{eqnarray}\\label{NablaHat}\n\\hat{\\nabla}_{X}Y &=& \\nabla_X Y + X(\\sigma)Y + Y(\\sigma)X \\notag\\\\\n&-& g(X,Y){\\rm grad} \\sigma - \\omega(X)\\omega(Y){\\rm grad}(\\sigma +\\rho)\\notag\\\\\n&+& \\frac{1}{1+{\\rm e}^{2\\rho}} \\Big( \\big(f + V(\\sigma)\\big) g(X,Y) +X(\\rho) \\omega(Y) + \\omega(X)Y(\\rho)\\notag\\\\\n&&\\qquad\\qquad +\\big(V(\\rho)+V(\\sigma)-f {\\rm e}^{-2\\rho}\\big)\\omega(X)\\omega(Y)\\Big)V.\n \\end{eqnarray}\n\\end{proposition}\n\\begin{proof}\nUsing (\\ref{TransfConf}), we get\n\\begin{eqnarray}\\label{TransfDConf}\n\\hat{\\nabla}_{X}Y= \\overline{\\nabla}_{X}Y + X(\\sigma)Y + Y(\\sigma)X - \\overline{g}(X,Y)\\overline{{\\rm grad}}\\sigma.\n \\end{eqnarray}\nFor all $X$ vector field on $M$, we compute\n\\begin{eqnarray}\\label{Eq300}\n\\overline{g}(\\overline{{\\rm grad}}\\sigma, X) = X(\\sigma)\n &=& g({\\rm grad} \\sigma, X) \\notag\\\\\n&=& \\overline{g}( {\\rm grad} \\sigma, X) - V(\\sigma)\\omega(X) \\notag\\\\\n&=& \\overline{g}\\Big( {\\rm grad} \\sigma -\\frac{ V(\\sigma)}{1+ {\\rm e}^{2\\rho}} V,X\\Big),\n\\end{eqnarray} \nthen,\n\\begin{equation}\\label{Eq310}\n\\overline{{\\rm grad}}\\sigma = {\\rm grad} \\sigma -\\frac{ V(\\sigma)}{1+ {\\rm e}^{2\\rho}} V.\n\\end{equation}\nNow, by substituting (\\ref{NablaOverline}) and (\\ref{Eq310}) in (\\ref{TransfDConf}) taking into account $\\overline{g} = g + \\omega \\otimes \\omega$ we get our formula.\n\\end{proof}\nLet's replace $Y$ by $V$ in (\\ref{NablaHat}) and using (\\ref{TFVF}), we get\n\\begin{eqnarray}\\label{NablaHat2}\n\\hat{\\nabla}_{X}V &=& \\big(f + V(\\sigma)\\big) X + \\theta(X)V + X(\\sigma)V \\notag\\\\\n&-& (1+{\\rm e}^{2 \\rho})\\omega(X){\\rm grad} \\sigma - {\\rm e}^{2 \\rho}\\omega(X){\\rm grad}(\\rho)\\notag\\\\\n&+& \\frac{1}{1+{\\rm e}^{2\\rho}} \\Big( {\\rm e}^{2 \\rho}X(\\rho) +(1+{\\rm e}^{2 \\rho})\\big(V(\\rho)+V(\\sigma)\\big)\\omega(X)\\Big)V.\n \\end{eqnarray}\n Reorganize this equation using (\\ref{Theta}), we find\n\n\\begin{eqnarray}\\label{Eq400}\n\\hat{\\nabla}_{X}V &=& \\big(f + V(\\sigma)\\big) X \\notag\\\\\n&&+ \\Big( X(\\sigma) + \\big(V(\\rho)+V(\\sigma)- f {\\rm e}^{-2 \\rho}\\big)\\omega(X) + \\frac{1+2{\\rm e}^{2 \\rho}}{1+{\\rm e}^{2\\rho}} X(\\rho) \\Big)V \\notag\\\\\n&&- (1+{\\rm e}^{2 \\rho})\\omega(X){\\rm grad} \\sigma - {\\rm e}^{2 \\rho}\\omega(X){\\rm grad}(\\rho).\n \\end{eqnarray}\nFor $V$ to be a torse forming vector field with respect to $\\hat{g}$, it is sufficient for it to be\n\\begin{equation}\\label{Eq410}\n {\\rm grad} \\sigma= - \\frac{{\\rm e}^{2 \\rho}}{1+{\\rm e}^{2\\rho}}{\\rm grad}\\rho.\n\\end{equation}\nFrom this equation, we get\n\\begin{equation}\\label{Eq420}\nX(\\sigma) = - \\frac{{\\rm e}^{2 \\rho}}{1+{\\rm e}^{2\\rho}}X(\\rho)\\qquad and \\qquad V(\\sigma) = - \\frac{{\\rm e}^{2 \\rho}}{1+{\\rm e}^{2\\rho}}V(\\rho).\n\\end{equation}\nBy (\\ref{Eq410}) and (\\ref{Eq420}), (\\ref{Eq400}) becomes\n\\begin{equation}\\label{430}\n\\hat{\\nabla}_{X}V = \\Big(f - \\frac{{\\rm e}^{2 \\rho}}{1+{\\rm e}^{2\\rho}} \\Big) X + \\Big( X(\\rho) + \\Big( \\frac{V(\\rho)}{1+{\\rm e}^{2\\rho}} - f {\\rm e}^{-2 \\rho}\\Big)\\omega(X) \\Big)V.\n \\end{equation}\nHence, $V$ is a torse-forming with respect to $\\hat{g}$. Therfore,\nwe can extract two important cases:\\\\\n(1):\\quad For $f = \\frac{{\\rm e}^{2 \\rho}}{1+{\\rm e}^{2\\rho}}$ we get\n$ \\hat{\\nabla}_{X}V = X(\\rho)V$ this means that $V$ is reccurent on $(M, \\hat{g})$.\\\\\n(2):\\quad For $ X(\\rho) = \\Big( f {\\rm e}^{-2 \\rho} -\\frac{V(\\rho)}{1+{\\rm e}^{2\\rho}}\\Big)\\omega(X)$, by taking $X=V$ we get $ f =\\frac{1+2{\\rm e}^{2\\rho}}{1+{\\rm e}^{2\\rho}}V(\\rho)$ then we obtain $ \\hat{\\nabla}_{X}V = V(\\rho)X$.\nBy combining the arguments of the above discussion, we obtain the following theorem:\n\\begin{theorem}\nLet $V$ be a proper torse-forming vector field on $(M,g)$ with $V^{\\flat}=\\omega$ and $\\omega(V)={\\rm e}^{2\\rho}$. If\n\\begin{equation}\\label{Eq450}\n {\\rm grad} \\sigma= - \\frac{{\\rm e}^{2 \\rho}}{1+{\\rm e}^{2\\rho}}{\\rm grad}\\rho.\n\\end{equation}\nthen $V$ is a torse-forming with respect to $\\hat{g}$. Moreover,\\\\\n\\textbf{(i)}\\quad if $f = \\frac{{\\rm e}^{2 \\rho}}{1+{\\rm e}^{2\\rho}}$ then\n $V$ is reccurent vector field on $(M, \\hat{g})$.\\\\\n\\textbf{(ii)}\\quad if $ {\\rm d}\\rho = \\Big( f {\\rm e}^{-2 \\rho} -\\frac{V(\\rho)}{1+{\\rm e}^{2\\rho}}\\Big)\\omega$ then $V$ is a concircular vector field on $(M, \\hat{g})$.", "post_theorem_intro_text_len": 643, "post_theorem_intro_text": "\\begin{proof}\nUsing (\\ref{TFVF}), for all $X$ vector field on $M$, we have\n\\begin{eqnarray}\\label{Eq100}\ng(\\nabla_X V, V) &=& f g(X,V)  + \\theta(X)g(V,V) \\notag\\\\\n&=& f \\omega(X) + {\\rm e}^{2\\rho} \\theta(X).\n\\end{eqnarray}\nOn the other hand, we have\n\\begin{eqnarray}\\label{Eq110}\ng(\\nabla_X V, V)= X g(V,V)-g(V , \\nabla_X V)  &\\Leftrightarrow &   g(\\nabla_X V, V) = \\frac{1}{2} X({\\rm e}^{2\\rho}) \\notag\\\\\n&\\Leftrightarrow & g(\\nabla_X V, V) = {\\rm e}^{2\\rho} X(\\rho).\n\\end{eqnarray}\nThen, from (\\ref{Eq100}) and (\\ref{Eq110}) we get\n$$ \\theta(X) =  X(\\rho) - f {\\rm e}^{-2\\rho} \\omega(X).$$\nThis is what is required to be proven,\n\n\\end{proof}", "sketch": "From (\\ref{TFVF}), compute for any vector field $X$:\n\\begin{eqnarray}\n g(\\nabla_X V, V) &=& f\\, g(X,V)+\\theta(X)g(V,V)= f\\,\\omega(X)+\\mathrm{e}^{2\\rho}\\theta(X). \\tag{\\ref{Eq100}}\n\\end{eqnarray}\nOn the other hand, using $g(\\nabla_X V,V)=\\tfrac12 X(g(V,V))$ and $g(V,V)=\\mathrm{e}^{2\\rho}$ gives\n\\begin{eqnarray}\n g(\\nabla_X V, V)=\\frac12 X(\\mathrm{e}^{2\\rho})=\\mathrm{e}^{2\\rho}X(\\rho). \\tag{\\ref{Eq110}}\n\\end{eqnarray}\nEquating (\\ref{Eq100}) and (\\ref{Eq110}) yields\n$$\\theta(X)=X(\\rho)-f\\,\\mathrm{e}^{-2\\rho}\\omega(X),$$\ni.e. $\\theta=\\mathrm{d}\\rho-f\\,\\mathrm{e}^{-2\\rho}\\omega$, which is (\\ref{Theta}).", "expanded_sketch": "From\n\\begin{equation}\\label{TFVF}\n\t\t\\nabla_X V = fX  + \\theta(X)V,\n\t\\end{equation}\ncompute for any vector field $X$:\n\\begin{eqnarray}\\label{Eq100}\ng(\\nabla_X V, V) &=& f g(X,V)  + \\theta(X)g(V,V) \\\\notag\\\\\n&=& f \\omega(X) + {\\rm e}^{2\\rho} \\theta(X).\n\\end{eqnarray}\nOn the other hand, using $g(\\nabla_X V,V)=\\tfrac12 X(g(V,V))$ and $g(V,V)=\\mathrm{e}^{2\\rho}$ gives\n\\begin{eqnarray}\\label{Eq110}\ng(\\nabla_X V, V)= X g(V,V)-g(V , \\nabla_X V)  &\\Leftrightarrow &   g(\\nabla_X V, V) = \\\\frac{1}{2} X({\\rm e}^{2\\rho}) \\\\notag\\\\\n&\\Leftrightarrow & g(\\nabla_X V, V) = {\\rm e}^{2\\rho} X(\\rho).\n\\end{eqnarray}\nEquating the two displayed identities yields\n$$\\theta(X)=X(\\rho)-f\\,\\mathrm{e}^{-2\\rho}\\omega(X),$$\ni.e. $\\theta=\\mathrm{d}\\rho-f\\,\\mathrm{e}^{-2\\rho}\\omega$, which is (\\ref{Theta}).", "expanded_theorem": "\\label{Prop1}\nFor every  torse-forming vector field $V$ that satisfies\n\\begin{equation}\\label{TFVF}\n\t\t\\nabla_X V = fX  + \\theta(X)V,\n\t\\end{equation}\nwe have\n\\begin{equation}\\label{Theta}\n\t\t\\theta ={\\rm d}\\rho - f {\\rm e}^{-2 \\rho} \\omega.\n\t\\end{equation}", "theorem_type": ["Universal"], "mcq": {"question": "Let \\((M,g)\\) be a Riemannian manifold, and let \\(V\\) be a torse-forming vector field, meaning that there exist a smooth function \\(f\\) and a 1-form \\(\\theta\\) such that for every vector field \\(X\\), \\(\\nabla_X V = fX + \\theta(X)V\\). Let \\(\\omega = V^{\\flat}\\) be the metric dual 1-form of \\(V\\), so \\(\\omega(X)=g(V,X)\\), and write \\(\\|V\\|=e^{\\rho}\\) (equivalently, \\(\\omega(V)=e^{2\\rho}\\)). Which statement holds for every such vector field \\(V\\)?", "correct_choice": {"label": "A", "text": "The generating 1-form satisfies \\(\\theta = d\\rho - f e^{-2\\rho}\\,\\omega\\); equivalently, for every vector field \\(X\\), \\(\\theta(X)=X(\\rho)-f e^{-2\\rho}\\omega(X)\\)."}, "choices": [{"label": "B", "text": "The generating 1-form satisfies \\(\\theta = d\\rho + f e^{-2\\rho}\\,\\omega\\); equivalently, for every vector field \\(X\\), \\(\\theta(X)=X(\\rho)+f e^{-2\\rho}\\omega(X)\\)."}, {"label": "C", "text": "For every vector field \\(X\\) orthogonal to \\(V\\) (that is, \\(\\omega(X)=0\\)), one has \\(\\theta(X)=X(\\rho)\\)."}, {"label": "D", "text": "The generating 1-form satisfies \\(\\theta = d\\rho - f e^{-\\rho}\\,\\omega\\); equivalently, for every vector field \\(X\\), \\(\\theta(X)=X(\\rho)-f e^{-\\rho}\\omega(X)\\)."}, {"label": "E", "text": "There exists a smooth function \\(\\lambda\\) on \\(M\\), depending only on \\(\\rho\\), such that \\(\\theta = d\\rho - \\lambda\\,\\omega\\); equivalently, for every vector field \\(X\\), \\(\\theta(X)=X(\\rho)-\\lambda\\omega(X)\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "trace_identity", "tampered_component": "sign from equating the two expressions for g(\\nabla_XV,V)", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "trace_identity", "tampered_component": "dropped the full formula by restricting to the case \\omega(X)=0", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "trace_identity", "tampered_component": "power of e^\\rho arising from dividing by g(V,V)=e^{2\\rho}", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "trace_identity", "tampered_component": "specific dependence of the coefficient on f and e^{-2\\rho}", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives the definition of a torse-forming vector field and relevant notation, but it does not state or strongly telegraph the target identity. The correct formula for \\(\\theta\\) must still be derived."}, "TAS": {"score": 2, "justification": "This is not a mere restatement of the definition in the stem. The correct choice is a derived consequence relating \\(\\theta\\), \\(\\rho\\), \\(f\\), and \\(\\omega\\), and the options present genuinely competing conclusions."}, "GPS": {"score": 1, "justification": "The item requires a short but real derivation: compare \\(X(g(V,V))\\) with \\(2g(\\nabla_XV,V)\\) and solve for \\(\\theta(X)\\). This is more than recall, but the reasoning is fairly direct rather than deeply generative."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: sign error (B), wrong exponent (D), a weaker-but-true statement (C), and an overgeneralized existential form (E). These reflect realistic failure modes."}, "total_score": 7, "overall_assessment": "A strong MCQ: it avoids answer leakage, is not tautological, and uses well-designed distractors. The reasoning required is meaningful but relatively short, so the generative demand is moderate rather than high."}}
{"id": "2602.01138v1", "paper_link": "http://arxiv.org/abs/2602.01138v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{propagation of chaos}\nThe assumptions of Lemma \\ref{weak solution} still hold. Let $\\{X_{N,i}^\\varepsilon\\}_{1\\le i\\le N}$ and $\\{\\overline{X}_i^\\varepsilon\\}_{1\\le i\\le N}$ be the solutions to systems \\eqref{sde} and \\eqref{mfs}, respectively. Then \nfor any $m,N\\in \\mathbb{N}^+$ and parameters $0<\\theta<\\frac{1}{2},\\quad 0<\\alpha<\\frac{\\theta}{2},\\quad m>\\frac{\\theta+1}{1-2\\theta}$, there exist a constant $C(m,T)>0$ and a parameter $\\gamma,\\eta>0$ satisfying the bounds\n\\begin{align}\n&0<\\gamma<\\min\\big\\{\\frac{\\alpha}{3},\\frac{-2\\alpha +m(1-2\\theta)-1}{4m+4}\\big\\},\\label{gamma} \\\\\n&0<\\eta\\le\\min\\{\\theta-2\\alpha, -(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}\\label{eta}\n\\end{align}\nsuch that for all $0\\le t\\le T$,\n\\begin{align}\\label{Pe}\n\\mathbb{P}\\big(\\max_{i=1,\\cdots,N}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|> N^{-\\alpha}\\big)\\le C(m,T)N^{-\\eta}, \n\\end{align}   \nwhere the cut-off parameter satisfies\n$\\varepsilon\\sim N^{-\\gamma} $.", "start_pos": 15136, "end_pos": 16132, "label": "propagation of chaos"}, "ref_dict": {"sde": "\\begin{align}\\label{sde}\n\\begin{cases}\ndX_{N,i}^\\varepsilon(t) = \\Big(2\\exp\\Big(-\\frac{1}{N}\\sum_{j=1}^N\\Phi^\\varepsilon(X_{N,i}^\\varepsilon(t) - X_{N,j}^\\varepsilon(t))\\Big)+ 2\\Big)^{1/2}dB_i(t),\\\\\nX_{N,i}^\\varepsilon(0) =\\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}", "mfs": "\\begin{align}\\label{mfs}\n\\begin{cases}\nd\\overline{X}_i^\\varepsilon(t) = \\big(2\\exp\\big(-\\Phi^\\varepsilon * u^\\varepsilon( \\overline{X}_i^\\varepsilon,t)\\big)+2\\big)^{1/2} dB_i(t),\\\\\n\\overline{X}_i^\\varepsilon(0) = \\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}", "pde": "\\begin{cases}\\label{pde}\n\\partial_t u = \\Delta(e^{-v}u+ u), \\qquad &x\\in\\R^2,~ t>0,\\\\\n-\\Delta v + v =\\chi u, \\qquad &x\\in\\R^2,~ t>0,\\\\\nu(x,0) = u_0(x), \\qquad &x\\in\\R^2,\n\\end{cases}", "Pe": "\\begin{align}\\label{Pe}\n\\mathbb{P}\\big(\\max_{i=1,\\cdots,N}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|> N^{-\\alpha}\\big)\\le C(m,T)N^{-\\eta}, \n\\end{align}", "propagation of chaos": "\\begin{theorem}\\label{propagation of chaos}\nThe assumptions of Lemma \\ref{weak solution} still hold. Let $\\{X_{N,i}^\\varepsilon\\}_{1\\le i\\le N}$ and $\\{\\overline{X}_i^\\varepsilon\\}_{1\\le i\\le N}$ be the solutions to systems \\eqref{sde} and \\eqref{mfs}, respectively. Then \nfor any $m,N\\in \\mathbb{N}^+$ and parameters $0<\\theta<\\frac{1}{2},\\quad 0<\\alpha<\\frac{\\theta}{2},\\quad m>\\frac{\\theta+1}{1-2\\theta}$, there exist a constant $C(m,T)>0$ and a parameter $\\gamma,\\eta>0$ satisfying the bounds\n\\begin{align}\n&0<\\gamma<\\min\\big\\{\\frac{\\alpha}{3},\\frac{-2\\alpha +m(1-2\\theta)-1}{4m+4}\\big\\},\\label{gamma} \\\\\n&0<\\eta\\le\\min\\{\\theta-2\\alpha, -(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}\\label{eta}\n\\end{align}\nsuch that for all $0\\le t\\le T$,\n\\begin{align}\\label{Pe}\n\\mathbb{P}\\big(\\max_{i=1,\\cdots,N}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|> N^{-\\alpha}\\big)\\le C(m,T)N^{-\\eta}, \n\\end{align}   \nwhere the cut-off parameter satisfies\n$\\varepsilon\\sim N^{-\\gamma} $.\n\\end{theorem}", "pde1": "\\begin{cases}\\label{pde1}\n\\partial_t u = \\Delta(\\gamma(v)u),\\\\\n\\tau\\partial_t v-\\Delta v + v = u,\n\\end{cases}", "weak solution": "\\begin{lemma}\\label{weak solution}\n\\textup{(Existence of weak solution, \\cite[Theorem 2, Lemma 16, and Lemma 18]{BOL2026113712})}.\nLet $u_0\\ge0$ be a initial probability density that satisfies\n\\begin{eqnarray*}\n&u_0\\log u_0\\in L^1(\\R^2), \\quad \\nabla\\log u_0\\in W^{1,q}(\\R^2)~(q>2),\\\\\n&u_0\\in L^1(\\R^2,|x|^2dx)\\cap L^p(\\R^2)~(1\\le p\\le \\infty).   \n\\end{eqnarray*}\nFurthermore, assume $\\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\\|\\omega\\|_{L^4(\\R^2)}^4\\le c_*\\|\\omega\\|_{L^2(\\R^2)}^2\\|\\nabla\\omega\\|_{L^2(\\R^2)}^2$. Then for any $T>0$ and $t\\in[0,T]$, the problem \\eqref{rpde} possesses weak solutions $(u^\\varepsilon, v^\\varepsilon)$ in $\\R^2\\times(0,T)$, and there exists a time $T^*\\in(0,T)$ such that \n\\begin{eqnarray}\n&\\|u^\\varepsilon\\|_{{L^2(0,T;H^1(\\R^2))}\\cap L^\\infty(0,T;L^p(\\R^2))}\\le C,\\label{uniformweaksolution}\\\\\n&\\||x|^2u^\\varepsilon\\|_{L^\\infty(0, T; L^1(\\R^2))}\\le C,\\quad\\|\\nabla\\log u^\\varepsilon\\|_{L^\\infty(0, T^*; W^{1,q}(\\R^2))}\\le C.\\label{uniformweaksolution1}\n\\end{eqnarray}\nwhere $C$ is a constant independent of $\\varepsilon$. \n\\end{lemma}", "Ps111": "\\begin{align}\\label{Ps111}\n\\|u_{N,r}^{\\varepsilon}(t) - u^{\\varepsilon\\otimes r}(t)\\|_{L^\\infty(0,T^*;L^1(\\R^{2r}))} \\le C(r,m,T)\\varepsilon^{\\beta}, \n  \\end{align}", "rpde": "\\begin{cases}\\label{rpde}\n\\partial_t u^\\varepsilon = \\Delta(e^{-v^\\varepsilon}u^\\varepsilon+ u^\\varepsilon),\\qquad &x\\in\\R^2,~ t>0,\\\\\n-\\Delta v^\\varepsilon + v^\\varepsilon =\\chi u^\\varepsilon * j^\\varepsilon,\\qquad &x\\in\\R^2,~ t>0,\\\\\nu^\\varepsilon(x,0) = u_0*j^\\varepsilon(x), \\qquad &x\\in\\R^2.\n\\end{cases}"}, "pre_theorem_intro_text_len": 12702, "pre_theorem_intro_text": "In this work, we provide a rigorous derivation of the two-dimensional Keller–Segel-type system with signal-dependent sensitivity expressed as\n\\begin{align}\n\\begin{cases}\\label{pde}\n\\partial_t u = \\Delta(e^{-v}u+ u), \\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\n-\\Delta v + v =\\chi u, \\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\nu(x,0) = u_0(x), \\qquad &x\\in\\mathbb{R}^2,\n\\end{cases}\n\\end{align}\nwhere $u(x,t)$ denotes cell density and $v(x,t)$ represents the concentration of the chemical signal. The coefficient $\\chi>0$ quantifies the signal-dependent strength of chemotactic sensitivity. \n\nThe formulation of the model is grounded in a robust biological rationale. This model describes the formation of stripe patterns through a self-trapping mechanism. Referring\nto \\cite{fu2012stripe}, we know that this process, which has been extensively studied using synthetic biological experimental methods, involves Escherichia coli secreting the signaling molecule acyl-homoserine lactone (AHL). At low concentrations of AHL, the bacteria exhibit high motility, characterized by random motion driven by typical swimming and tumbling behavior with little external interference. However, as the AHL concentration increases, the behavior of the bacterial population changes significantly, eventually leading to a macroscopically static state.\nThe partial differential equation (PDE) \\eqref{pde} is derived from the general signal-dependent Keller–Segel model proposed \n\\begin{align}\n\\begin{cases}\\label{pde1}\n\\partial_t u = \\Delta(\\gamma(v)u),\\\\\n\\tau\\partial_t v-\\Delta v + v = u,\n\\end{cases}\n\\end{align}\nwhere $\\tau$ takes the value of $0$ or $1$. Here, $\\gamma(v)$ represents the signal-dependent motility, which satisfies $\\gamma'(v)\\le 0$. \nA substantial body of research has been devoted to the existence of solutions for the model \\eqref{pde1}, particularly in cases where the signal-dependent function $\\gamma(v)$ is subject to different conditions.\nIn 2017, under the assumptions of upper and lower bounds for both $\\gamma(v)$ and its derivative $\\gamma'(v)$, Tao and Winkler \\cite{tao2017effects} proved the existence of global classical solutions for the two-dimensional case, while demonstrating that the system admits only weak solutions in higher dimensions. Fujie and Jiang \\cite{fujie2020global} established that when $\\gamma(v)$ satisfies the following conditions:\n\\begin{equation*}\n0<\\gamma(v)\\in C^3[0,\\infty),\\quad \\gamma'(v)\\leq 0 \\text{ on } [0,\\infty),\\quad \\lim_{v\\to\\infty}\\gamma(v)=0,\n\\end{equation*}\nglobal classical solutions exist for arbitrary initial data. Their subsequent work \\cite{fujie2021boundedness} showed that if the function $\\gamma(v)$ satisfies the asymptotic condition $\\lim_{v\\to\\infty} e^{\\alpha v}\\gamma(v) = +\\infty$ for any $\\alpha > 0$, then there exists a globally bounded classical solution.\nFor the case $\\gamma(v) = e^{-\\alpha v}$ ($\\alpha > 0$), the authors in \\cite{fujie2020global,fujie2021comparison,jin2020critical} derived a critical mass threshold: solutions remain uniformly bounded when the initial cell mass is below this critical value. In contrast, studies in \\cite{burger2021delayed, fujie2020global,fujie2021comparison} indicate that solutions  blow up as time approaches infinity when the initial cell mass exceeds the critical mass. Nevertheless, it is impossible to blow-up in finite time, which is a difference from the Keller-Segel model. \n\nIn their seminal 2025 work on the model \\eqref{pde} (with $\\gamma(v)=e^{-v}+1$), the reference \\cite{BOL2026113712}\nrigorously established the well-posedness of solutions to the PDE \\eqref{pde} and gave error estimates between \\eqref{pde} and its regularized version. The discussion is already so comprehensive that we can cite it directly. They also conducted an in-depth study in which the convergences for the stochastic particle systems of \\eqref{pde} inspired us to refine their methodologies. \nOur focus lies particularly on the stochastic particle equations and the mean field limit corresponding to  \\eqref{pde}. By introducing the concept of stopping time (achieving convergence in probability in Section 2), we obtained results with faster convergence rates than those in \\cite{BOL2026113712}. \nIn Section 3, we establish the propagation of chaos in strong senses for stochastic differential equations (SDEs), (where we apply the relative entropy method and derive a higher convergence rate), which build a foundational bridge between the macroscopic and microscopic systems.\nIn the following, we propose that the SDE corresponding to \\eqref{pde} for $N\\in\\mathbb{N}$ interacting particles $\\{X_{N,i}^\\varepsilon(\\cdot)\\}_{1\\le i\\le N}$ in $\\mathbb{R}^2$ reads as\n\\begin{align}\\label{sde}\n\\begin{cases}\ndX_{N,i}^\\varepsilon(t) = \\Big(2\\exp\\Big(-\\frac{1}{N}\\sum_{j=1}^N\\Phi^\\varepsilon(X_{N,i}^\\varepsilon(t) - X_{N,j}^\\varepsilon(t))\\Big)+ 2\\Big)^{1/2}dB_i(t),\\\\\nX_{N,i}^\\varepsilon(0) =\\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}\nwhere $0<\\varepsilon<1$. Here, we consider a filtered probability space defined by $(\\Omega,\\mathcal{F}, (\\mathcal{F})_{t\\ge0}, \\mathbb{P})$ and introduce $\\{B_i\\}_{1\\le i\\le N}$, a collection of independent $\\mathcal{F}_t$-Brownian motions. The initial data $\\zeta_1,\\zeta_2,\\cdots,\\zeta_N$ are assumed to be random variables independent and identically distributed (i.i.d.) with the common probability density function $u_0$. The potential $\\Phi^\\varepsilon$ is given by\n$$\\Phi^\\varepsilon:=\\Phi*j^\\varepsilon,\\quad \\Phi:=\\chi \\tilde{\\Phi},\\quad j^\\varepsilon(x):=\\frac{1}{\\varepsilon^2}j(\\frac{x}{\\varepsilon}),$$\nwhere $\\chi>0$ describes the strength rate of the signals in the model \\eqref{pde} and $\\tilde{\\Phi}$ is the Yukawa potential \\cite{lieb2001analysis}, which is defined for any $\\mu > 0$, by \n\\begin{align*}\n&\\tilde{\\Phi}(x) = \\int_0^\\infty(4\\pi t)^{-1}\\exp\\Big\\{-\\frac{|x|^2}{4t}-\\mu^2t\\Big\\}dt.\n\\end{align*}\nBased on \\cite[Appendix B]{li2023optimal}, the following properties are observed\n\\begin{align}\\label{Phi}\n&\\|\\tilde{\\Phi}\\|_{L^p(\\mathbb{R}^2)}<\\infty,~\\|\\nabla\\tilde{\\Phi}\\|_{L^q(\\mathbb{R}^2)}<\\infty,\\qquad p\\in[1,\\infty),~q\\in[1,2).\n\\end{align}\nThrough calculation, the following results are obtained that for any constant $C$ independent of $\\varepsilon$,\n\\begin{align}\n&\\|\\Phi^\\varepsilon\\|_{W^{1,1}(\\mathbb{R}^2)}\\le\\|\\Phi\\|_{W^{1,1}(\\mathbb{R}^2)}\\|j^\\varepsilon\\|_{L^1(\\mathbb{R}^2)}\\le C,\\label{D1phi}\\\\\n&\\|\\Phi^\\varepsilon\\|_{W^{1,\\infty}(\\mathbb{R}^2)}\\le\\|\\Phi\\|_{W^{1,1}(\\mathbb{R}^2)}\\|j^\\varepsilon\\|_{L^\\infty(\\mathbb{R}^2)}\\le \\frac{C}{\\varepsilon^2},\\label{phi}\\\\\n&\\|D^2\\Phi^\\varepsilon\\|_{L^\\infty(\\mathbb{R}^2)}\\le \\|\\nabla\\Phi\\|_{L^1(\\mathbb{R}^2)}\\|\\nabla j^\\varepsilon\\|_{L^\\infty(\\mathbb{R}^2)}\\le \\frac{C}{\\varepsilon^3}\\label{D2phi}.\n\\end{align}\nThe rigorous derivation of PDEs from stochastic particle systems represents a fundamental challenge in mathematical physics and applied analysis. \nA particularly influential framework for such derivations is the theory of moderately interacting particles, introduced in the pioneering works of Oelschl\\\"ager  \\cite{oelschlager1985law, oelschlager1987fluctuation, oelschlager1989derivation},\nwhere the Law of Large Numbers for interacting diffusions was established, together with central-limit-type fluctuation results.\nThis approach, characterized by a specific scaling of the interaction potential that preserves nonlocality while weakening pointwise strength, has since been extended to various biological and physical models. For example, Stevens \\cite{stevens2000derivation} derived the Keller–Segel chemotaxis model from a moderately interacting system, overcoming the lack of ellipticity, a common assumption in earlier mean field limits, by introducing novel analytical techniques. \n\nIn recent years, significant progress has been made in handling singular interactions, such as Coulomb and Riesz potentials, which arise naturally in models of collective behavior. Lazarovici and Pickl \\cite{lazarovici2017mean} analyzed a particle system incorporating a regularized potential and random initial conditions. Serfaty \\cite{serfaty2020mean} developed the modulated energy method to treat Coulomb-type flows, providing a powerful tool for the analysis of the mean field without confinement. \nSubsequent studies on related models employing diverse regularization strategies have further expanded this line of inquiry, as documented in \\cite{boers2016mean,bolley2011stochastic,chen2017mean,chen2020combined}.\nCurrently, the relative entropy method, advanced by Jabin and Wang \\cite{jabin2017mean} and further applied by Bresch et al.\\cite{bresch2023mean} and Chen et al.\\cite{chen2025quantitative,chen2025mean}, has enabled quantitative estimates of convergence for singular attractive kernels. These techniques have been instrumental in the establishment of various modes of convergence, including convergence in the Wasserstein distance (Carrillo et al.\\cite{carrillo2019propagation}) and convergence in probability Pickl et al. \\cite{lazarovici2017mean,huang2020mean,huang2017error}, strong $L^1$ convergence (Chen et al. \\cite{chen2025quantitative} and \\cite{olivera2020quantitative,olivera2023quantitative}). More recently, the moderate interaction framework has been instrumental in the derivation of cross-diffusion models; see, for example, \\cite{carrillo2024interacting,chen2021rigorous, chen2019rigorous,li2024convergence}.\n\nTo study the relationship between the macroscopic system \\eqref{pde} and its microscopic system \\eqref{sde}, \nwe need to introduce an intermediate particle system (Mean-Field equation). A key object in the mean-field limit is the empirical measure, a random probability measure defined as\n$$\\mu^\\varepsilon_N(t)=\\frac{1}{N}\\sum_{i=1}^N\\delta_{X_{N,i}^\\varepsilon(t)},\\quad t>0,$$\nwhere $\\delta$ is the Dirac delta distribution. The work \\cite{olivera2021quantitative} shows that $\\mu^\\varepsilon_N(t)$ converges to the following PDE solution $u^\\varepsilon$. For fixed $\\varepsilon>0$, the particle system \\eqref{sde} propagates chaos in the many-particle limit $N\\to\\infty$ towards the non-linear SDE system:\n\\begin{align}\\label{mfs}\n\\begin{cases}\nd\\overline{X}_i^\\varepsilon(t) = \\big(2\\exp\\big(-\\Phi^\\varepsilon * u^\\varepsilon( \\overline{X}_i^\\varepsilon,t)\\big)+2\\big)^{1/2} dB_i(t),\\\\\n\\overline{X}_i^\\varepsilon(0) = \\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}\nwhere\n$u^\\varepsilon(\\cdot,t)$ is the density function of i.i.d. random processes\n$\\overline{X}_1^\\varepsilon(t),\\cdots, \\overline{X}_N^\\varepsilon(t)$. \nAnd the initial data $\\{\\zeta_i\\}_{i=1}^N$\nare subject to the same conditions as in \\eqref{sde}. Using It\\^{o}'s formula, the density function $u^\\varepsilon$ satisfies the so-called intermediate nonlocal problem, namely,\n\\begin{align}\n\\begin{cases}\\label{rpde}\n\\partial_t u^\\varepsilon = \\Delta(e^{-v^\\varepsilon}u^\\varepsilon+ u^\\varepsilon),\\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\n-\\Delta v^\\varepsilon + v^\\varepsilon =\\chi u^\\varepsilon * j^\\varepsilon,\\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\nu^\\varepsilon(x,0) = u_0*j^\\varepsilon(x), \\qquad &x\\in\\mathbb{R}^2.\n\\end{cases}\n\\end{align}\n\nIn this paper, our argument relies on the Lemma \\ref{weak solution} from \\cite{BOL2026113712}, which establishes the existence and uniqueness of the weak solution $u^\\varepsilon$ to the above system \\eqref{rpde} and provides the $L^\\infty$ estimate for $\\nabla\\log u^\\varepsilon$.\n\\begin{lemma}\\label{weak solution}\n\\textup{(Existence of weak solution, \\cite[Theorem 2, Lemma 16, and Lemma 18]{BOL2026113712})}.\nLet $u_0\\ge0$ be a initial probability density that satisfies\n\\begin{eqnarray*}\n&u_0\\log u_0\\in L^1(\\mathbb{R}^2), \\quad \\nabla\\log u_0\\in W^{1,q}(\\mathbb{R}^2)~(q>2),\\\\\n&u_0\\in L^1(\\mathbb{R}^2,|x|^2dx)\\cap L^p(\\mathbb{R}^2)~(1\\le p\\le \\infty).   \n\\end{eqnarray*}\nFurthermore, assume $\\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\\|\\omega\\|_{L^4(\\mathbb{R}^2)}^4\\le c_*\\|\\omega\\|_{L^2(\\mathbb{R}^2)}^2\\|\\nabla\\omega\\|_{L^2(\\mathbb{R}^2)}^2$. Then for any $T>0$ and $t\\in[0,T]$, the problem \\eqref{rpde} possesses weak solutions $(u^\\varepsilon, v^\\varepsilon)$ in $\\mathbb{R}^2\\times(0,T)$, and there exists a time $T^*\\in(0,T)$ such that \n\\begin{eqnarray}\n&\\|u^\\varepsilon\\|_{{L^2(0,T;H^1(\\mathbb{R}^2))}\\cap L^\\infty(0,T;L^p(\\mathbb{R}^2))}\\le C,\\label{uniformweaksolution}\\\\\n&\\||x|^2u^\\varepsilon\\|_{L^\\infty(0, T; L^1(\\mathbb{R}^2))}\\le C,\\quad\\|\\nabla\\log u^\\varepsilon\\|_{L^\\infty(0, T^*; W^{1,q}(\\mathbb{R}^2))}\\le C.\\label{uniformweaksolution1}\n\\end{eqnarray}\nwhere $C$ is a constant independent of $\\varepsilon$. \n\\end{lemma}\n\nUsing the above lemma, we obtain our first main result: convergence in probability of the mean-field limit under algebraic scaling.", "context": "In this work, we provide a rigorous derivation of the two-dimensional Keller–Segel-type system with signal-dependent sensitivity expressed as\n\\begin{align}\n\\begin{cases}\\label{pde}\n\\partial_t u = \\Delta(e^{-v}u+ u), \\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\n-\\Delta v + v =\\chi u, \\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\nu(x,0) = u_0(x), \\qquad &x\\in\\mathbb{R}^2,\n\\end{cases}\n\\end{align}\nwhere $u(x,t)$ denotes cell density and $v(x,t)$ represents the concentration of the chemical signal. The coefficient $\\chi>0$ quantifies the signal-dependent strength of chemotactic sensitivity.\n\nTo study the relationship between the macroscopic system \\eqref{pde} and its microscopic system \\eqref{sde}, \nwe need to introduce an intermediate particle system (Mean-Field equation). A key object in the mean-field limit is the empirical measure, a random probability measure defined as\n$$\\mu^\\varepsilon_N(t)=\\frac{1}{N}\\sum_{i=1}^N\\delta_{X_{N,i}^\\varepsilon(t)},\\quad t>0,$$\nwhere $\\delta$ is the Dirac delta distribution. The work \\cite{olivera2021quantitative} shows that $\\mu^\\varepsilon_N(t)$ converges to the following PDE solution $u^\\varepsilon$. For fixed $\\varepsilon>0$, the particle system \\eqref{sde} propagates chaos in the many-particle limit $N\\to\\infty$ towards the non-linear SDE system:\n\\begin{align}\\label{mfs}\n\\begin{cases}\nd\\overline{X}_i^\\varepsilon(t) = \\big(2\\exp\\big(-\\Phi^\\varepsilon * u^\\varepsilon( \\overline{X}_i^\\varepsilon,t)\\big)+2\\big)^{1/2} dB_i(t),\\\\\n\\overline{X}_i^\\varepsilon(0) = \\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}\nwhere\n$u^\\varepsilon(\\cdot,t)$ is the density function of i.i.d. random processes\n$\\overline{X}_1^\\varepsilon(t),\\cdots, \\overline{X}_N^\\varepsilon(t)$. \nAnd the initial data $\\{\\zeta_i\\}_{i=1}^N$\nare subject to the same conditions as in \\eqref{sde}. Using It\\^{o}'s formula, the density function $u^\\varepsilon$ satisfies the so-called intermediate nonlocal problem, namely,\n\\begin{align}\n\\begin{cases}\\label{rpde}\n\\partial_t u^\\varepsilon = \\Delta(e^{-v^\\varepsilon}u^\\varepsilon+ u^\\varepsilon),\\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\n-\\Delta v^\\varepsilon + v^\\varepsilon =\\chi u^\\varepsilon * j^\\varepsilon,\\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\nu^\\varepsilon(x,0) = u_0*j^\\varepsilon(x), \\qquad &x\\in\\mathbb{R}^2.\n\\end{cases}\n\\end{align}\n\nIn this paper, our argument relies on the Lemma \\ref{weak solution} from \\cite{BOL2026113712}, which establishes the existence and uniqueness of the weak solution $u^\\varepsilon$ to the above system \\eqref{rpde} and provides the $L^\\infty$ estimate for $\\nabla\\log u^\\varepsilon$.\n\\begin{lemma}\\label{weak solution}\n\\textup{(Existence of weak solution, \\cite[Theorem 2, Lemma 16, and Lemma 18]{BOL2026113712})}.\nLet $u_0\\ge0$ be a initial probability density that satisfies\n\\begin{eqnarray*}\n&u_0\\log u_0\\in L^1(\\mathbb{R}^2), \\quad \\nabla\\log u_0\\in W^{1,q}(\\mathbb{R}^2)~(q>2),\\\\\n&u_0\\in L^1(\\mathbb{R}^2,|x|^2dx)\\cap L^p(\\mathbb{R}^2)~(1\\le p\\le \\infty). \n\\end{eqnarray*}\nFurthermore, assume $\\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\\|\\omega\\|_{L^4(\\mathbb{R}^2)}^4\\le c_*\\|\\omega\\|_{L^2(\\mathbb{R}^2)}^2\\|\\nabla\\omega\\|_{L^2(\\mathbb{R}^2)}^2$. Then for any $T>0$ and $t\\in[0,T]$, the problem \\eqref{rpde} possesses weak solutions $(u^\\varepsilon, v^\\varepsilon)$ in $\\mathbb{R}^2\\times(0,T)$, and there exists a time $T^*\\in(0,T)$ such that \n\\begin{eqnarray}\n&\\|u^\\varepsilon\\|_{{L^2(0,T;H^1(\\mathbb{R}^2))}\\cap L^\\infty(0,T;L^p(\\mathbb{R}^2))}\\le C,\\label{uniformweaksolution}\\\\\n&\\||x|^2u^\\varepsilon\\|_{L^\\infty(0, T; L^1(\\mathbb{R}^2))}\\le C,\\quad\\|\\nabla\\log u^\\varepsilon\\|_{L^\\infty(0, T^*; W^{1,q}(\\mathbb{R}^2))}\\le C.\\label{uniformweaksolution1}\n\\end{eqnarray}\nwhere $C$ is a constant independent of $\\varepsilon$. \n\\end{lemma}\n\nUsing the above lemma, we obtain our first main result: convergence in probability of the mean-field limit under algebraic scaling.", "full_context": "In this work, we provide a rigorous derivation of the two-dimensional Keller–Segel-type system with signal-dependent sensitivity expressed as\n\\begin{align}\n\\begin{cases}\\label{pde}\n\\partial_t u = \\Delta(e^{-v}u+ u), \\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\n-\\Delta v + v =\\chi u, \\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\nu(x,0) = u_0(x), \\qquad &x\\in\\mathbb{R}^2,\n\\end{cases}\n\\end{align}\nwhere $u(x,t)$ denotes cell density and $v(x,t)$ represents the concentration of the chemical signal. The coefficient $\\chi>0$ quantifies the signal-dependent strength of chemotactic sensitivity.\n\nTo study the relationship between the macroscopic system \\eqref{pde} and its microscopic system \\eqref{sde}, \nwe need to introduce an intermediate particle system (Mean-Field equation). A key object in the mean-field limit is the empirical measure, a random probability measure defined as\n$$\\mu^\\varepsilon_N(t)=\\frac{1}{N}\\sum_{i=1}^N\\delta_{X_{N,i}^\\varepsilon(t)},\\quad t>0,$$\nwhere $\\delta$ is the Dirac delta distribution. The work \\cite{olivera2021quantitative} shows that $\\mu^\\varepsilon_N(t)$ converges to the following PDE solution $u^\\varepsilon$. For fixed $\\varepsilon>0$, the particle system \\eqref{sde} propagates chaos in the many-particle limit $N\\to\\infty$ towards the non-linear SDE system:\n\\begin{align}\\label{mfs}\n\\begin{cases}\nd\\overline{X}_i^\\varepsilon(t) = \\big(2\\exp\\big(-\\Phi^\\varepsilon * u^\\varepsilon( \\overline{X}_i^\\varepsilon,t)\\big)+2\\big)^{1/2} dB_i(t),\\\\\n\\overline{X}_i^\\varepsilon(0) = \\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}\nwhere\n$u^\\varepsilon(\\cdot,t)$ is the density function of i.i.d. random processes\n$\\overline{X}_1^\\varepsilon(t),\\cdots, \\overline{X}_N^\\varepsilon(t)$. \nAnd the initial data $\\{\\zeta_i\\}_{i=1}^N$\nare subject to the same conditions as in \\eqref{sde}. Using It\\^{o}'s formula, the density function $u^\\varepsilon$ satisfies the so-called intermediate nonlocal problem, namely,\n\\begin{align}\n\\begin{cases}\\label{rpde}\n\\partial_t u^\\varepsilon = \\Delta(e^{-v^\\varepsilon}u^\\varepsilon+ u^\\varepsilon),\\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\n-\\Delta v^\\varepsilon + v^\\varepsilon =\\chi u^\\varepsilon * j^\\varepsilon,\\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\nu^\\varepsilon(x,0) = u_0*j^\\varepsilon(x), \\qquad &x\\in\\mathbb{R}^2.\n\\end{cases}\n\\end{align}\n\nIn this paper, our argument relies on the Lemma \\ref{weak solution} from \\cite{BOL2026113712}, which establishes the existence and uniqueness of the weak solution $u^\\varepsilon$ to the above system \\eqref{rpde} and provides the $L^\\infty$ estimate for $\\nabla\\log u^\\varepsilon$.\n\\begin{lemma}\\label{weak solution}\n\\textup{(Existence of weak solution, \\cite[Theorem 2, Lemma 16, and Lemma 18]{BOL2026113712})}.\nLet $u_0\\ge0$ be a initial probability density that satisfies\n\\begin{eqnarray*}\n&u_0\\log u_0\\in L^1(\\mathbb{R}^2), \\quad \\nabla\\log u_0\\in W^{1,q}(\\mathbb{R}^2)~(q>2),\\\\\n&u_0\\in L^1(\\mathbb{R}^2,|x|^2dx)\\cap L^p(\\mathbb{R}^2)~(1\\le p\\le \\infty). \n\\end{eqnarray*}\nFurthermore, assume $\\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\\|\\omega\\|_{L^4(\\mathbb{R}^2)}^4\\le c_*\\|\\omega\\|_{L^2(\\mathbb{R}^2)}^2\\|\\nabla\\omega\\|_{L^2(\\mathbb{R}^2)}^2$. Then for any $T>0$ and $t\\in[0,T]$, the problem \\eqref{rpde} possesses weak solutions $(u^\\varepsilon, v^\\varepsilon)$ in $\\mathbb{R}^2\\times(0,T)$, and there exists a time $T^*\\in(0,T)$ such that \n\\begin{eqnarray}\n&\\|u^\\varepsilon\\|_{{L^2(0,T;H^1(\\mathbb{R}^2))}\\cap L^\\infty(0,T;L^p(\\mathbb{R}^2))}\\le C,\\label{uniformweaksolution}\\\\\n&\\||x|^2u^\\varepsilon\\|_{L^\\infty(0, T; L^1(\\mathbb{R}^2))}\\le C,\\quad\\|\\nabla\\log u^\\varepsilon\\|_{L^\\infty(0, T^*; W^{1,q}(\\mathbb{R}^2))}\\le C.\\label{uniformweaksolution1}\n\\end{eqnarray}\nwhere $C$ is a constant independent of $\\varepsilon$. \n\\end{lemma}\n\nUsing the above lemma, we obtain our first main result: convergence in probability of the mean-field limit under algebraic scaling.\n\nUsing the above lemma, we obtain our first main result: convergence in probability of the mean-field limit under algebraic scaling.\n\nThe proof of Theorem \\ref{propagation of chaos}, given in Section 2, follows the approach of \\cite{lazarovici2017mean}, adopting a superior algebraic scaling with respect to the maximum norm of the trajectory, which \\cite{BOL2026113712} only achieved results under logaritheoremic scaling. \nSince the diffusion coefficients in the stochastic models \\eqref{sde} and \\eqref{mfs} given the nonlinear dependence of the diffusion coefficients on inter-particle interactions, the use of the Burkholder-Davis-Gundy inequality becomes indispensable. The key ingredient of our proof is the introduction of stopping times, a technique that facilitates the application of the Law of Large Numbers.\n\nTherefore, plugging \\eqref{I12} and \\eqref{I11} into \\eqref{EI1i} gives\n\\begin{align}\\label{I1i}\n&\\E\\Big(N^{2\\alpha }\\int_0^{t\\land\\tau_\\alpha}\\max_{i=1,\\cdots,N}|I_{1,i}(s)|^2 ds\\Big)\\nonumber\\\\\n\\le\\ & C(T) (1+\\varepsilon^{-6}N^{-2\\alpha })\\int_0^{t\\land\\tau_\\alpha}\\E(S_\\alpha^2(s))ds + C(m, T)\\varepsilon^{-4m-4}N^{-m+1}.\n\\end{align}\nSubstituting \\eqref{I2i} and \\eqref{I1i} into \\eqref{ES} yields the inequality\n\\begin{align}\\label{initial ineq}\n\\E(S_\\alpha^2(t))\n\\le\\ & C(T) (1+\\varepsilon^{-6}N^{-2\\alpha })\\int_0^{t\\land\\tau_\\alpha}\\E(S_\\alpha^2(s))ds \\nonumber\\\\\n&+ C(m, T)(\\varepsilon^{-4m-4}N^{-m+1} +N^{(2\\alpha-\\theta)} + \\varepsilon^{-4m-4}N^{2\\alpha +m(2\\theta -1)+1}).\n\\end{align}\nWe now show that the terms multiplying the constants decay as a negative power of $N$. Recall the scaling $\\varepsilon\\sim N^{-\\gamma},\\gamma>0$. To bound the error terms, we require a positive exponent\n$\\eta>0$ such that the following bounds hold\n\\begin{align*}\n\\varepsilon^{-6}N^{-2\\alpha } = N^{6\\gamma -2\\alpha}\\le N^{0}&\\Longleftrightarrow\\gamma\\le\\frac{\\alpha}{3},\\\\\nN^{2\\alpha-\\theta}\\le N^{-\\eta}&\\Longleftrightarrow\\eta\\le\\theta-2\\alpha, \\\\\n\\varepsilon^{-4m-4}N^{-m+1} = N^{(4m+4)\\gamma -m+1}\\le N^{-\\eta}\n&\\Longleftrightarrow\\eta\\le-(4m+4)\\gamma+m-1,\\\\\n\\varepsilon^{-4m-4}N^{2\\alpha +m(2\\theta -1)+1}=N^{(4m+4)\\gamma+2\\alpha +m(2\\theta -1)+1}\\le N^{-\\eta}&\\Longleftrightarrow\\eta\\le-(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1.\n\\end{align*}\nThe condition $\\eta>0$ implies that $\\theta$, $\\alpha$, $m$, and $\\gamma$ need to be selected to meet the following relationships\n\\begin{align*}\n0<\\theta<\\frac{1}{2},\\quad 0<\\alpha<\\frac{\\theta}{2},\\quad m>\\frac{\\theta+1}{1-2\\theta},\\quad 0<\\gamma<\\frac{-2\\alpha+m(1-2\\theta)-1}{4m+4}. \n\\end{align*}\nThus choose $\\gamma$ satisfying\n$$\n0<\\gamma<\\min\\big\\{\\frac{\\alpha}{3},\\frac{-2\\alpha +m(1-2\\theta)-1}{4m+4}\\big\\},\n$$\nand $\\eta$ satisfying \n$$\n0<\\eta\\le\\min\\{\\theta-2\\alpha, -(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}.\n$$ \nThen the inequality \\eqref{initial ineq} simplifies the estimate\n\\begin{align}\\label{Sak}\n\\E(S_\\alpha^2(t))\n\\le\\ & C\\int_0^{t\\land\\tau_\\alpha}\\E(S_\\alpha^2(s))ds + C(m, T)N^{-\\eta}.\n\\end{align}\nUsing Gr\\\"{o}nwall's inequality for \\eqref{Sak}, we obtain for any $t\\in[0,T]$\n\\begin{align} \\label{Egronwall}\n\\E(S_\\alpha^2(t))\\le C(m,T)N^{-\\eta} (1+CTe^{CT}). \n\\end{align}\nBy inserting \\eqref{MK} into \\eqref{Egronwall}, we derive \n\\begin{align}\n\\mathbb{P}\\big(\\max_{i=1,\\cdots,N}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|> N^{-\\alpha}\\big)\\le C(m,T)N^{-\\eta} (1+CTe^{CT}).\n\\end{align}\nThe proof of this theorem is completed.\n\n\\setcounter{theorem}{1}\n\\begin{theorem}\\label{Propagation of chaos in the strong sense}\\textup{(Propagation of chaos in the strong sense).} \nUnder the assumptions of Lemma \\ref{weak solution}, let \n$u_{N}^\\varepsilon(t,x_1,\\cdots,x_N)$ and $u^{\\varepsilon\\otimes N}(t,x_1,\\cdots,x_N)$ be the solutions of systems \\eqref{lpde1} and \\eqref{lpde2}, respectively. For any \n$r\\in\\mathbb{N^+}$, denote by $u_{N,r}^\\varepsilon(t,x_1,\\cdots,x_r)$ and $u^{\\varepsilon\\otimes r}(t,x_1,\\cdots,x_r)$ their corresponding \n$r$-th marginal distributions. \nThen for parameters $0<\\theta<1/2$, $0 <\\alpha<\\frac{\\theta}{2}$, and $m\\in \\mathbb{N}^+$ satisfying $m \\ge \\frac{1+\\theta}{1-2\\theta}$, there exist a time $T^*\\in(0,T)$, a constant $C(m,r,T)>0$, and a parameter $\\beta>0$ satisfying the bound $1<\\beta\\le\\min\\{\\frac{2\\alpha}{\\gamma}-6,\\frac{\\eta}{\\gamma}-4\\}$ such that \n\\begin{align}\\label{Ps}\n\\|u_{N,r}^{\\varepsilon}(t) - u^{\\varepsilon\\otimes r}(t)\\|_{L^\\infty(0,T^*;L^1(\\R^{2r}))} \\le C(r,m,T)\\varepsilon^{\\beta}, \n \\end{align}\nwhere the cut-off parameter satisfies\n$\\varepsilon\\sim N^{-\\gamma} $,\nand the parameter $\\gamma$ defined by\n$$0<\\gamma<\\min\\{\\frac{2\\alpha}{7},\\frac{-2\\alpha + m(1-2\\theta)-1}{4m+4}\\},$$\nand the parameter $\\eta$ satisfies\n$$5\\gamma<\\eta<\\min\\{\\theta-2\\alpha,-(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}.$$ \n\\end{theorem}\n\nPlugging \\eqref{J1} and \\eqref{J2} into \\eqref{K21} produces the following\n\\begin{align}\nK_2\\leq &-\\int_{\\R^{2N}}\\frac{1}{N}\\sum_{i=1}^Nu_N^\\varepsilon\\Big[\\exp\\big(-\\frac{1}{N}\\sum_{j=1}^N\\Phi^\\varepsilon(x_i - x_j)\\big) \\Big|\\nabla_{x_i}\\log\\frac{u_N^\\varepsilon}{u^{\\varepsilon\\otimes N}}\\Big|^2\\Big]d_{x_1}\n\\cdots d_{x_N}\\nonumber\\\\\n&+ C(m, T)(N^{-2\\alpha}\\varepsilon^{-6} + (N^{-1} + N^{-\\eta})\\varepsilon^{-4}).\\label{K_2}\n\\end{align}\nCombining \\eqref{K1} and \\eqref{H1}, we derive\n\\begin{align}\n\\frac{d}{dt}\\mathcal{H}(u_N^\\varepsilon|u^{\\varepsilon\\otimes N})\n\\leq& -\\frac{1}{2}\\int_{\\R^{2N}}\\frac{1}{N}\\sum_{i=1}^Nu_N^\\varepsilon\\Big|\\nabla_{x_i}\\log\\frac{u_N^\\varepsilon}{u^{\\varepsilon\\otimes N}}\\Big|^2d_{x_1}\\cdots d_{x_N}\\nonumber\\\\\n&-\\frac{1}{2}\\int_{\\R^{2N}}\\frac{1}{N}\\sum_{i=1}^Nu_N^\\varepsilon\\Big[\\exp\\big(-\\frac{1}{N}\\sum_{j=1}^N\\Phi^\\varepsilon(x_i - x_j)\\big) \\Big|\\nabla_{x_i}\\log\\frac{u_N^\\varepsilon}{u^{\\varepsilon\\otimes N}}\\Big|^2\\Big]d_{x_1}\n\\cdots d_{x_N}\\nonumber\\\\\n& + C(m, T)(N^{-2\\alpha}\\varepsilon^{-6} + (N^{-1} + N^{-\\eta})\\varepsilon^{-4}).\\label{entropy}\n\\end{align}\nNoticing that the initial relative entropy is zero, it can be obtained from (\\ref{entropy}) that for any $0<t<T^*<T$, \n$$\n\\mathcal{H}(u_{N}^{\\varepsilon}|u^{\\varepsilon\\otimes N})(t)\\le C(m, T)(N^{-2\\alpha}\\varepsilon^{-6} + (N^{-1} + N^{-\\eta})\\varepsilon^{-4}).\n$$\nCombing \\eqref{relative entropy} and the C-K-P inequality \\eqref{CKP}, we have for $r\\in\\mathbb{N}$, \n\\begin{align}\n\\sup_{t\\in(0,T^*)}\\|u_{N,r}^{\\varepsilon}(t) - u^{\\varepsilon\\otimes r}(t)\\|_{L^1(\\R^{2r})} \n&\\le \\sup_{t\\in(0,T^*)}2\\mathcal{H}_r(u_{N,r}^{\\varepsilon}|u^{\\varepsilon\\otimes r})(t)\\nonumber\\\\\n&\\le \\sup_{t\\in(0,T^*)}4r\\mathcal{H}(u_{N}^{\\varepsilon}|u^{\\varepsilon\\otimes N})(t)\\nonumber\\\\\n&\\le C(m, T)(N^{-2\\alpha}\\varepsilon^{-6} + (N^{-1} + N^{-\\eta})\\varepsilon^{-4}).\n\\end{align}\nAccording to Theorem \\ref{propagation of chaos}, we know that \n\\begin{align}\n&0<\\eta\\le\\min\\{\\theta-2\\alpha,-(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\},\\label{eta1}\\\\\n&0<\\gamma\\le\\min\\{\\frac{\\alpha}{3},\\frac{-2\\alpha + m(1-2\\theta)-1}{4m+4}\\}. \\label{gamma1} \n\\end{align}\nBased on the previous assumption $\\varepsilon\\sim N^{-\\gamma}$, it holds that there exists $\\beta>0$ such that \n\\begin{align}\nN^{-2\\alpha}\\varepsilon^{-6} + (N^{-1} + N^{- \\eta})\\varepsilon^{-4}=\\varepsilon^\\frac{2\\alpha}{\\gamma}\\varepsilon^{-6}+(\\varepsilon^\\frac{1}{\\gamma}+\\varepsilon^\\frac{\\eta}{\\gamma})\\varepsilon^{-4}\\le \\varepsilon^\\beta, \n\\end{align}\nwhere $\\beta$ satisfies\n$$1<\\beta\\le\\min\\{\\frac{2\\alpha}{\\gamma}-6,\\frac{\\eta}{\\gamma}-4\\}.$$ \nHere we have used $\\eta<1$.\nThis further requires that $0<\\gamma<\\frac{2\\alpha}{7}$, $\\eta>5\\gamma$. Building on the assumptions in \\eqref{eta1} and \\eqref{gamma1}, we assume\n\\begin{align*}\n&0<\\gamma<\\min\\{\\frac{2\\alpha}{7},\\frac{-2\\alpha + m(1-2\\theta)-1}{4m+4}\\},\\\\\n&5\\gamma<\\eta<\\min\\{\\theta-2\\alpha,-(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}.\n\\end{align*} \nThen we obtain that for $0<t<T^*<T$, \n\\begin{align}\n\\sup_{t\\in(0,T^*)}\\|u_{N,r}^{\\varepsilon}(t) - u^{\\varepsilon\\otimes r}(t)\\|_{L^1(\\R^{2r})} \\le C(m,T)\\varepsilon^\\beta,\\quad \\beta>1.\n\\end{align}\nThus, we complete the proof of propagation of chaos. \n\\end{proof}", "post_theorem_intro_text_len": 5262, "post_theorem_intro_text": "\\begin{remark}\nTo see the value of $\\gamma$ and $\\eta$ more clearly, we provide a special case. When $m\\in\\mathbb{N}^+$, $ m\\ge\\frac{1+\\theta}{1-2\\theta}$ and $0<\\alpha<\\frac{\\theta}{2}<\\frac{1}{2}$, we take $\\alpha=0.1$, $\\theta=0.3$ and $m=4$ such that\n$0<\\gamma<\\min\\{\\frac{\\alpha}{3},\\frac{-2\\alpha +m(1-2\\theta)-1}{2(2+2m)}\\}=\\min\\{0.033,0.02\\}=0.02$. we may assume that $\\gamma=0.019$, then $0<\\eta\\le\\min\\{\\theta-2\\alpha,-2\\gamma(2+2m)-2\\alpha -m(2\\theta -1)-1\\}=\\min\\{0.1, 0.02\\}=0.02$. \n\\end{remark}\n\nThe  proof of Theorem \\ref{propagation of chaos}, given in Section 2, follows the approach of \\cite{lazarovici2017mean}, adopting a superior algebraic scaling with respect to the maximum norm of the trajectory, which \\cite{BOL2026113712} only achieved results under logaritheoremic scaling. \nSince the diffusion coefficients in the stochastic models \\eqref{sde} and \\eqref{mfs} given the nonlinear dependence of the diffusion coefficients on inter-particle interactions, the use of the Burkholder-Davis-Gundy inequality becomes indispensable. The key ingredient of our proof is the introduction of stopping times, a technique that facilitates the application of the Law of Large Numbers. \n\nBased on the uniform $L^\\infty(0,T^*;H^2(\\mathbb{R}^2))$ bound for $\\nabla\\log u_\\varepsilon$ in Lemma \\ref{weak solution} and the propagation of chaos established in Theorem \\ref{propagation of chaos}, our next main result addresses the strong $L^1$ convergence for the propagation of chaos. The proof is mainly presented in Section 3, employing the relative entropy method \\cite{jabin2018quantitative} as recently refined in \\cite{chen2025quantitative}.\n\n\\begin{theorem}\\label{Propagation of chaos in the strong sense}\\textup{(Propagation of chaos in the strong sense).} \nUnder the assumptions of Theorem \\ref{propagation of chaos}, let $r\\in \\mathbb{N^+}$, $u_{N,r}^\\varepsilon(t, x_1,\\cdots, x_r)$ be the $r$-th marginal density of the joint density $u_N^\\varepsilon(t, x_1,\\cdots, x_N)$ of $\\{X_{N,i}^\\varepsilon\\}_{1\\le i\\le N}$, and $u^{\\varepsilon \\otimes r}(t, x_1,\\cdots, x_r)$ be the tensor product of the solutions $u^{\\varepsilon}$ to the model \\eqref{rpde}. Then for parameters $0<\\theta<1/2$, $0 <\\alpha<\\frac{\\theta}{2}$, and $m\\in \\mathbb{N}^+$ satisfying $m \\ge \\frac{1+\\theta}{1-2\\theta}$, there exist a time $T^*\\in(0,T)$, a constant $C(m,r,T)>0$, and a parameter $\\beta>0$ satisfying the bound $1<\\beta\\le\\min\\{\\frac{2\\alpha}{\\gamma}-6,\\frac{\\eta}{\\gamma}-4\\}$ such that \n\\begin{align}\\label{Ps111}\n\\|u_{N,r}^{\\varepsilon}(t) - u^{\\varepsilon\\otimes r}(t)\\|_{L^\\infty(0,T^*;L^1(\\mathbb{R}^{2r}))} \\le C(r,m,T)\\varepsilon^{\\beta}, \n  \\end{align}\nwhere the cut-off parameter satisfies\n$\\varepsilon\\sim N^{-\\gamma} $,\nand the parameter $\\gamma$ defined by\n$$0<\\gamma<\\min\\{\\frac{2\\alpha}{7},\\frac{-2\\alpha + m(1-2\\theta)-1}{4m+4}\\},$$\nand the parameter $\\eta$ satisfies\n$$5\\gamma<\\eta<\\min\\{\\theta-2\\alpha,-(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}.$$ \n\\end{theorem}\n\\begin{remark}\nTo illustrate the feasible range of parameters, we give an explicit example. Choose $\\theta = 0.4,\\ \\alpha = 0.1$ and $m\\ge \\frac{1+\\theta}{1-2\\theta}=7$ such that $m=7$.\nThen we obtain\n$$0<\\gamma<\\min\\{\\frac{2\\alpha}{7},\\frac{-2\\alpha + m(1-2\\theta)-1}{4m+4}\\}=\\min\\{0.029,0.006\\}=0.006.$$\nTaking $\\gamma=0.0025$ gives\n$$0.0125=5\\gamma<\\eta<\\min\\{\\theta-2\\alpha,-(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}=\\min\\{0.2,0.12\\}=0.12.$$\nso we pick $\\eta=0.03$. Finally, $$1<\\beta\\le\\min\\{\\frac{2\\alpha}{\\gamma}-6,\\frac{\\eta}{\\gamma}-4\\}=\\min\\{74,8\\}=8$$\nwe take $\\beta = 8$.\nWith these choices, the propagation of chaos estimate becomes\n\\begin{align*}\n\\max_{i=1,\\cdots,N}\\mathbb{E}\\big(\\sup_{t\\in[0,T]}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|^2\\big)\\le C\\varepsilon^8, \n\\end{align*} \n\\end{remark}\nWe comment here that the convergence of the mean-field limit have already been discussed in \\cite{BOL2026113712}. It is therefore especially gratifying that our results accelerate the convergence rates  for the solutions to the SDE \\eqref{sde} and \\eqref{mfs} across different norms. Specifically, Chen et al. in \\cite[Theorem 3]{BOL2026113712} presented as\n\\begin{align*}\n\\max_{i=1,\\cdots,N}\\mathbb{E}\\big(\\sup_{t\\in[0,T]}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|^2\\big)\\le C\\varepsilon^2, \n\\end{align*}   \nwhere the cut-off parameter $\\varepsilon=(\\lambda\\log N)^{-\\frac{1}{4}}$. However, in \\eqref{Pe}, we prove algebraic convergence in contrast to logarithmic convergence established in \\cite{BOL2026113712}. This represents an  improvement in the convergence rate. Furthermore, in \\cite[Theorem 5]{BOL2026113712}, they obtained\n\\begin{align*}\n\\|u_{N,r}^{\\varepsilon}(t) - u^{\\varepsilon\\otimes r}(t)\\|_{L^\\infty(0,T^*;L^1(\\mathbb{R}^{2r}))}^2 \\le C(r,m,T)\\varepsilon.\n\\end{align*}\nBy contrast, our estimate in \\eqref{Ps111} yields a  faster convergence rate than that bound.\n\nThe article is organized as follows. In section 2, we establish the propagation of chaos, which corresponds to the convergence in probability of solutions to the stochastic differential equations \\eqref{sde} and \\eqref{mfs}. Section 3 further derive quantitative propagation of chaos result in the strong sense by applying the relative entropy method.", "sketch": "The proof of Theorem~\\ref{propagation of chaos} (given in Section 2) “follows the approach of \\cite{lazarovici2017mean},” and “adopt[s] a superior algebraic scaling with respect to the maximum norm of the trajectory” (contrasting with \\cite{BOL2026113712}, which “only achieved results under logarithmic scaling”). Because “the diffusion coefficients … [have] the nonlinear dependence … on inter-particle interactions,” the argument says “the use of the Burkholder-Davis-Gundy inequality becomes indispensable.” The “key ingredient” is “the introduction of stopping times,” which “facilitates the application of the Law of Large Numbers.”", "expanded_sketch": "The proof (given later) “follows the approach of Lazarovici and Pickl, \\emph{A mean field limit for the Vlasov--Poisson system} (2017),” and “adopt[s] a superior algebraic scaling with respect to the maximum norm of the trajectory” (contrasting with \\cite{BOL2026113712}, which “only achieved results under logarithmic scaling”). Because “the diffusion coefficients … [have] the nonlinear dependence … on inter-particle interactions,” the argument says “the use of the Burkholder-Davis-Gundy inequality becomes indispensable.” The “key ingredient” is “the introduction of stopping times,” which “facilitates the application of the Law of Large Numbers.”", "expanded_theorem": "\\label{propagation of chaos}\nThe assumptions of the following lemma still hold.\n\\begin{lemma}\\label{weak solution}\n\\textup{(Existence of weak solution, \\cite[Theorem 2, Lemma 16, and Lemma 18]{BOL2026113712})}.\nLet $u_0\\ge0$ be a initial probability density that satisfies\n\\begin{eqnarray*}\n&u_0\\log u_0\\in L^1(\\R^2), \\quad \\nabla\\log u_0\\in W^{1,q}(\\R^2)~(q>2),\\\\\n&u_0\\in L^1(\\R^2,|x|^2dx)\\cap L^p(\\R^2)~(1\\le p\\le \\infty).   \n\\end{eqnarray*}\nFurthermore, assume $\\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\\|\\omega\\|_{L^4(\\R^2)}^4\\le c_*\\|\\omega\\|_{L^2(\\R^2)}^2\\|\\nabla\\omega\\|_{L^2(\\R^2)}^2$. Then for any $T>0$ and $t\\in[0,T]$, the problem \\eqref{rpde} possesses weak solutions $(u^\\varepsilon, v^\\varepsilon)$ in $\\R^2\\times(0,T)$, and there exists a time $T^*\\in(0,T)$ such that \n\\begin{eqnarray}\n&\\|u^\\varepsilon\\|_{{L^2(0,T;H^1(\\R^2))}\\cap L^\\infty(0,T;L^p(\\R^2))}\\le C,\\label{uniformweaksolution}\\\\\n&\\||x|^2u^\\varepsilon\\|_{L^\\infty(0, T; L^1(\\R^2))}\\le C,\\quad\\|\\nabla\\log u^\\varepsilon\\|_{L^\\infty(0, T^*; W^{1,q}(\\R^2))}\\le C.\\label{uniformweaksolution1}\n\\end{eqnarray}\nwhere $C$ is a constant independent of $\\varepsilon$. \n\\end{lemma}\nLet $\\{X_{N,i}^\\varepsilon\\}_{1\\le i\\le N}$ and $\\{\\overline{X}_i^\\varepsilon\\}_{1\\le i\\le N}$ be the solutions to systems\n\\begin{align}\\label{sde}\n\\begin{cases}\ndX_{N,i}^\\varepsilon(t) = \\Big(2\\exp\\Big(-\\frac{1}{N}\\sum_{j=1}^N\\Phi^\\varepsilon(X_{N,i}^\\varepsilon(t) - X_{N,j}^\\varepsilon(t))\\Big)+ 2\\Big)^{1/2}dB_i(t),\\\\\nX_{N,i}^\\varepsilon(0) =\\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}\nand\n\\begin{align}\\label{mfs}\n\\begin{cases}\nd\\overline{X}_i^\\varepsilon(t) = \\big(2\\exp\\big(-\\Phi^\\varepsilon * u^\\varepsilon( \\overline{X}_i^\\varepsilon,t)\\big)+2\\big)^{1/2} dB_i(t),\\\\\n\\overline{X}_i^\\varepsilon(0) = \\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}\nrespectively. Then \nfor any $m,N\\in \\mathbb{N}^+$ and parameters $0<\\theta<\\frac{1}{2},\\quad 0<\\alpha<\\frac{\\theta}{2},\\quad m>\\frac{\\theta+1}{1-2\\theta}$, there exist a constant $C(m,T)>0$ and a parameter $\\gamma,\\eta>0$ satisfying the bounds\n\\begin{align}\n&0<\\gamma<\\min\\big\\{\\frac{\\alpha}{3},\\frac{-2\\alpha +m(1-2\\theta)-1}{4m+4}\\big\\},\\label{gamma} \\\\\n&0<\\eta\\le\\min\\{\\theta-2\\alpha, -(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}\\label{eta}\n\\end{align}\nsuch that for all $0\\le t\\le T$,\n\\begin{align}\\label{Pe}\n\\mathbb{P}\\big(\\max_{i=1,\\cdots,N}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|> N^{-\\alpha}\\big)\\le C(m,T)N^{-\\eta}, \n\\end{align}   \nwhere the cut-off parameter satisfies\n$\\varepsilon\\sim N^{-\\gamma} $.", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let $u_0\\ge 0$ be an initial probability density on $\\mathbb{R}^2$ such that\n$$u_0\\log u_0\\in L^1(\\mathbb{R}^2),\\qquad \\nabla\\log u_0\\in W^{1,q}(\\mathbb{R}^2)\\ \\text{for some }q>2,$$\nand\n$$u_0\\in L^1(\\mathbb{R}^2,|x|^2dx)\\cap L^p(\\mathbb{R}^2)\\qquad (1\\le p\\le \\infty).$$\nAssume also that $\\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo--Nirenberg inequality\n$$\\|\\omega\\|_{L^4(\\mathbb{R}^2)}^4\\le c_*\\|\\omega\\|_{L^2(\\mathbb{R}^2)}^2\\|\\nabla\\omega\\|_{L^2(\\mathbb{R}^2)}^2.$$ \nLet $(u^\\varepsilon,v^\\varepsilon)$ be the weak solution of the intermediate nonlocal problem\n$$\\partial_t u^\\varepsilon=\\Delta(e^{-v^\\varepsilon}u^\\varepsilon+u^\\varepsilon),\\qquad -\\Delta v^\\varepsilon+v^\\varepsilon=\\chi \\,u^\\varepsilon * j^\\varepsilon,\\qquad u^\\varepsilon(\\cdot,0)=u_0*j^\\varepsilon,$$\non $\\mathbb{R}^2\\times(0,T)$, where $*$ denotes convolution.\nFor $1\\le i\\le N$, let $X_{N,i}^\\varepsilon$ and $\\overline X_i^\\varepsilon$ solve, with the same Brownian motions $B_i$ and the same initial data $\\zeta_i$,\n$$dX_{N,i}^\\varepsilon(t)=\\Big(2\\exp\\Big(-\\frac1N\\sum_{j=1}^N\\Phi^\\varepsilon(X_{N,i}^\\varepsilon(t)-X_{N,j}^\\varepsilon(t))\\Big)+2\\Big)^{1/2}dB_i(t),$$\n$$d\\overline X_i^\\varepsilon(t)=\\Big(2\\exp\\big(-\\Phi^\\varepsilon*u^\\varepsilon(\\overline X_i^\\varepsilon,t)\\big)+2\\Big)^{1/2}dB_i(t).$$\nFix $m,N\\in\\mathbb N^+$ and parameters $0<\\theta<\\tfrac12$, $0<\\alpha<\\tfrac\\theta2$, and $m>\\frac{\\theta+1}{1-2\\theta}$. Under these assumptions, which quantitative estimate holds for the probability that the interacting particle system and the mean-field system differ by more than $N^{-\\alpha}$ in maximum norm at time $t$?", "correct_choice": {"label": "A", "text": "There exist a constant $C(m,T)>0$ and parameters $\\gamma,\\eta>0$ such that\n$$0<\\gamma<\\min\\Big\\{\\frac\\alpha3,\\frac{-2\\alpha+m(1-2\\theta)-1}{4m+4}\\Big\\},$$\n$$0<\\eta\\le \\min\\Big\\{\\theta-2\\alpha,\n-(4m+4)\\gamma-2\\alpha+m(1-2\\theta)-1\\Big\\},$$\nand, if the cutoff parameter satisfies $\\varepsilon\\sim N^{-\\gamma}$, then for every $t\\in[0,T]$,\n$$\\mathbb P\\Big(\\max_{i=1,\\dots,N}|X_{N,i}^\\varepsilon(t)-\\overline X_i^\\varepsilon(t)|>N^{-\\alpha}\\Big)\\le C(m,T)N^{-\\eta}.$$"}, "choices": [{"label": "B", "text": "There exist a constant $C(m,T)>0$ and parameters $\\gamma,\\eta>0$ such that\n$$0<\\gamma<\\min\\Big\\{\\frac\\alpha3,\\frac{-2\\alpha+m(1-2\\theta)-1}{4m+4}\\Big\\},$$\n$$0<\\eta\\le \\min\\Big\\{\\theta-2\\alpha,\n-(4m+4)\\gamma-2\\alpha+m(1-2\\theta)-1\\Big\\},$$\nand, if the cutoff parameter satisfies $\\varepsilon\\sim N^{-\\gamma}$, then for every $t\\in[0,T]$,\n$$\\mathbb P\\Big(\\max_{i=1,\\dots,N}|X_{N,i}^\\varepsilon(t)-\\overline X_i^\\varepsilon(t)|>N^{-\\alpha}\\Big)\\le C(m,T)N^{-2\\eta}.$$"}, {"label": "C", "text": "There exist a constant $C(m,T)>0$ and parameters $\\gamma,\\eta>0$ such that\n$$0<\\gamma<\\min\\Big\\{\\frac\\alpha3,\\frac{-2\\alpha+m(1-2\\theta)-1}{4m+4}\\Big\\},$$\n$$0<\\eta\\le \\min\\Big\\{\\theta-2\\alpha,\n-(4m+4)\\gamma-2\\alpha+m(1-2\\theta)-1\\Big\\},$$\nand, if the cutoff parameter satisfies $\\varepsilon\\sim N^{-\\gamma}$, then for every $t\\in[0,T]$,\n$$\\mathbb P\\Big(\\max_{i=1,\\dots,N}|X_{N,i}^\\varepsilon(t)-\\overline X_i^\\varepsilon(t)|>N^{-\\alpha}\\Big)\\le C(m,T).$$"}, {"label": "D", "text": "For every choice of parameters $\\gamma,\\eta>0$ satisfying\n$$0<\\gamma<\\min\\Big\\{\\frac\\alpha3,\\frac{-2\\alpha+m(1-2\\theta)-1}{4m+4}\\Big\\},$$\n$$0<\\eta\\le \\min\\Big\\{\\theta-2\\alpha,\n-(4m+4)\\gamma-2\\alpha+m(1-2\\theta)-1\\Big\\},$$\nthere exists a constant $C(m,T)>0$ such that, if the cutoff parameter satisfies $\\varepsilon\\sim N^{-\\gamma}$, then for every $t\\in[0,T]$,\n$$\\mathbb P\\Big(\\max_{i=1,\\dots,N}|X_{N,i}^\\varepsilon(t)-\\overline X_i^\\varepsilon(t)|>N^{-\\alpha}\\Big)\\le C(m,T)N^{-\\eta},$$\nwith the same constant $C(m,T)$ working uniformly for all such admissible pairs $(\\gamma,\\eta)$."}, {"label": "E", "text": "There exist a constant $C(m,T)>0$ and parameters $\\gamma,\\eta>0$ such that\n$$0<\\gamma\\le\\min\\Big\\{\\frac\\alpha3,\\frac{-2\\alpha+m(1-2\\theta)-1}{4m+4}\\Big\\},$$\n$$0<\\eta\\le \\min\\Big\\{\\theta-2\\alpha,\n-(4m+4)\\gamma-2\\alpha+m(1-2\\theta)-1\\Big\\},$$\nand, if the cutoff parameter satisfies $\\varepsilon\\sim N^{-\\gamma}$, then for every $t\\in[0,T]$,\n$$\\mathbb P\\Big(\\max_{i=1,\\dots,N}|X_{N,i}^\\varepsilon(t)-\\overline X_i^\\varepsilon(t)|>N^{-\\alpha}\\Big)\\le C(m,T)N^{-\\eta}.$$"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "algebraic decay rate from stopping-time/BDG argument", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the quantitative decay factor $N^{-\\eta}$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence of the constant on the admissible pair $(\\gamma,\\eta)$", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "boundary_range", "tampered_component": "strict upper bound on $\\gamma$ replaced by an attained endpoint", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct option directly. It gives hypotheses and asks which conclusion holds, without explicitly signaling the exact probability estimate or the precise quantifier structure."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: the correct choice is essentially the exact theorem conclusion under the listed assumptions. It is not a pure restatement in the stem, but it mainly tests recognition of the formal statement rather than deriving a new consequence."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish subtle changes in rates, strict vs. non-strict bounds, pointwise-in-time vs. supremum-in-time estimates, and explicit rate vs. mere convergence. However, this is more precision/recall of a known result than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong: they are mathematically close to the target statement and differ in realistic ways that reflect common failure modes, such as weakening the conclusion, changing the rate variable, altering endpoint admissibility, or strengthening time uniformity improperly."}, "total_score": 6, "overall_assessment": "A technically strong MCQ with little answer leakage and high-quality distractors, but it leans more toward theorem-statement recognition than genuine generative reasoning."}}
{"id": "2602.01138v1", "paper_link": "http://arxiv.org/abs/2602.01138v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{propagation of chaos}\nThe assumptions of Lemma \\ref{weak solution} still hold. Let $\\{X_{N,i}^\\varepsilon\\}_{1\\le i\\le N}$ and $\\{\\overline{X}_i^\\varepsilon\\}_{1\\le i\\le N}$ be the solutions to systems \\eqref{sde} and \\eqref{mfs}, respectively. Then \nfor any $m,N\\in \\mathbb{N}^+$ and parameters $0<\\theta<\\frac{1}{2},\\quad 0<\\alpha<\\frac{\\theta}{2},\\quad m>\\frac{\\theta+1}{1-2\\theta}$, there exist a constant $C(m,T)>0$ and a parameter $\\gamma,\\eta>0$ satisfying the bounds\n\\begin{align}\n&0<\\gamma<\\min\\big\\{\\frac{\\alpha}{3},\\frac{-2\\alpha +m(1-2\\theta)-1}{4m+4}\\big\\},\\label{gamma} \\\\\n&0<\\eta\\le\\min\\{\\theta-2\\alpha, -(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}\\label{eta}\n\\end{align}\nsuch that for all $0\\le t\\le T$,\n\\begin{align}\\label{Pe}\n\\mathbb{P}\\big(\\max_{i=1,\\cdots,N}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|> N^{-\\alpha}\\big)\\le C(m,T)N^{-\\eta}, \n\\end{align}   \nwhere the cut-off parameter satisfies\n$\\varepsilon\\sim N^{-\\gamma} $.", "start_pos": 15136, "end_pos": 16132, "label": "propagation of chaos"}, "ref_dict": {"sde": "\\begin{align}\\label{sde}\n\\begin{cases}\ndX_{N,i}^\\varepsilon(t) = \\Big(2\\exp\\Big(-\\frac{1}{N}\\sum_{j=1}^N\\Phi^\\varepsilon(X_{N,i}^\\varepsilon(t) - X_{N,j}^\\varepsilon(t))\\Big)+ 2\\Big)^{1/2}dB_i(t),\\\\\nX_{N,i}^\\varepsilon(0) =\\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}", "mfs": "\\begin{align}\\label{mfs}\n\\begin{cases}\nd\\overline{X}_i^\\varepsilon(t) = \\big(2\\exp\\big(-\\Phi^\\varepsilon * u^\\varepsilon( \\overline{X}_i^\\varepsilon,t)\\big)+2\\big)^{1/2} dB_i(t),\\\\\n\\overline{X}_i^\\varepsilon(0) = \\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}", "pde": "\\begin{cases}\\label{pde}\n\\partial_t u = \\Delta(e^{-v}u+ u), \\qquad &x\\in\\R^2,~ t>0,\\\\\n-\\Delta v + v =\\chi u, \\qquad &x\\in\\R^2,~ t>0,\\\\\nu(x,0) = u_0(x), \\qquad &x\\in\\R^2,\n\\end{cases}", "Pe": "\\begin{align}\\label{Pe}\n\\mathbb{P}\\big(\\max_{i=1,\\cdots,N}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|> N^{-\\alpha}\\big)\\le C(m,T)N^{-\\eta}, \n\\end{align}", "propagation of chaos": "\\begin{theorem}\\label{propagation of chaos}\nThe assumptions of Lemma \\ref{weak solution} still hold. Let $\\{X_{N,i}^\\varepsilon\\}_{1\\le i\\le N}$ and $\\{\\overline{X}_i^\\varepsilon\\}_{1\\le i\\le N}$ be the solutions to systems \\eqref{sde} and \\eqref{mfs}, respectively. Then \nfor any $m,N\\in \\mathbb{N}^+$ and parameters $0<\\theta<\\frac{1}{2},\\quad 0<\\alpha<\\frac{\\theta}{2},\\quad m>\\frac{\\theta+1}{1-2\\theta}$, there exist a constant $C(m,T)>0$ and a parameter $\\gamma,\\eta>0$ satisfying the bounds\n\\begin{align}\n&0<\\gamma<\\min\\big\\{\\frac{\\alpha}{3},\\frac{-2\\alpha +m(1-2\\theta)-1}{4m+4}\\big\\},\\label{gamma} \\\\\n&0<\\eta\\le\\min\\{\\theta-2\\alpha, -(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}\\label{eta}\n\\end{align}\nsuch that for all $0\\le t\\le T$,\n\\begin{align}\\label{Pe}\n\\mathbb{P}\\big(\\max_{i=1,\\cdots,N}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|> N^{-\\alpha}\\big)\\le C(m,T)N^{-\\eta}, \n\\end{align}   \nwhere the cut-off parameter satisfies\n$\\varepsilon\\sim N^{-\\gamma} $.\n\\end{theorem}", "pde1": "\\begin{cases}\\label{pde1}\n\\partial_t u = \\Delta(\\gamma(v)u),\\\\\n\\tau\\partial_t v-\\Delta v + v = u,\n\\end{cases}", "weak solution": "\\begin{lemma}\\label{weak solution}\n\\textup{(Existence of weak solution, \\cite[Theorem 2, Lemma 16, and Lemma 18]{BOL2026113712})}.\nLet $u_0\\ge0$ be a initial probability density that satisfies\n\\begin{eqnarray*}\n&u_0\\log u_0\\in L^1(\\R^2), \\quad \\nabla\\log u_0\\in W^{1,q}(\\R^2)~(q>2),\\\\\n&u_0\\in L^1(\\R^2,|x|^2dx)\\cap L^p(\\R^2)~(1\\le p\\le \\infty).   \n\\end{eqnarray*}\nFurthermore, assume $\\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\\|\\omega\\|_{L^4(\\R^2)}^4\\le c_*\\|\\omega\\|_{L^2(\\R^2)}^2\\|\\nabla\\omega\\|_{L^2(\\R^2)}^2$. Then for any $T>0$ and $t\\in[0,T]$, the problem \\eqref{rpde} possesses weak solutions $(u^\\varepsilon, v^\\varepsilon)$ in $\\R^2\\times(0,T)$, and there exists a time $T^*\\in(0,T)$ such that \n\\begin{eqnarray}\n&\\|u^\\varepsilon\\|_{{L^2(0,T;H^1(\\R^2))}\\cap L^\\infty(0,T;L^p(\\R^2))}\\le C,\\label{uniformweaksolution}\\\\\n&\\||x|^2u^\\varepsilon\\|_{L^\\infty(0, T; L^1(\\R^2))}\\le C,\\quad\\|\\nabla\\log u^\\varepsilon\\|_{L^\\infty(0, T^*; W^{1,q}(\\R^2))}\\le C.\\label{uniformweaksolution1}\n\\end{eqnarray}\nwhere $C$ is a constant independent of $\\varepsilon$. \n\\end{lemma}", "Ps111": "\\begin{align}\\label{Ps111}\n\\|u_{N,r}^{\\varepsilon}(t) - u^{\\varepsilon\\otimes r}(t)\\|_{L^\\infty(0,T^*;L^1(\\R^{2r}))} \\le C(r,m,T)\\varepsilon^{\\beta}, \n  \\end{align}", "rpde": "\\begin{cases}\\label{rpde}\n\\partial_t u^\\varepsilon = \\Delta(e^{-v^\\varepsilon}u^\\varepsilon+ u^\\varepsilon),\\qquad &x\\in\\R^2,~ t>0,\\\\\n-\\Delta v^\\varepsilon + v^\\varepsilon =\\chi u^\\varepsilon * j^\\varepsilon,\\qquad &x\\in\\R^2,~ t>0,\\\\\nu^\\varepsilon(x,0) = u_0*j^\\varepsilon(x), \\qquad &x\\in\\R^2.\n\\end{cases}"}, "pre_theorem_intro_text_len": 12702, "pre_theorem_intro_text": "In this work, we provide a rigorous derivation of the two-dimensional Keller–Segel-type system with signal-dependent sensitivity expressed as\n\\begin{align}\n\\begin{cases}\\label{pde}\n\\partial_t u = \\Delta(e^{-v}u+ u), \\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\n-\\Delta v + v =\\chi u, \\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\nu(x,0) = u_0(x), \\qquad &x\\in\\mathbb{R}^2,\n\\end{cases}\n\\end{align}\nwhere $u(x,t)$ denotes cell density and $v(x,t)$ represents the concentration of the chemical signal. The coefficient $\\chi>0$ quantifies the signal-dependent strength of chemotactic sensitivity. \n\nThe formulation of the model is grounded in a robust biological rationale. This model describes the formation of stripe patterns through a self-trapping mechanism. Referring\nto \\cite{fu2012stripe}, we know that this process, which has been extensively studied using synthetic biological experimental methods, involves Escherichia coli secreting the signaling molecule acyl-homoserine lactone (AHL). At low concentrations of AHL, the bacteria exhibit high motility, characterized by random motion driven by typical swimming and tumbling behavior with little external interference. However, as the AHL concentration increases, the behavior of the bacterial population changes significantly, eventually leading to a macroscopically static state.\nThe partial differential equation (PDE) \\eqref{pde} is derived from the general signal-dependent Keller–Segel model proposed \n\\begin{align}\n\\begin{cases}\\label{pde1}\n\\partial_t u = \\Delta(\\gamma(v)u),\\\\\n\\tau\\partial_t v-\\Delta v + v = u,\n\\end{cases}\n\\end{align}\nwhere $\\tau$ takes the value of $0$ or $1$. Here, $\\gamma(v)$ represents the signal-dependent motility, which satisfies $\\gamma'(v)\\le 0$. \nA substantial body of research has been devoted to the existence of solutions for the model \\eqref{pde1}, particularly in cases where the signal-dependent function $\\gamma(v)$ is subject to different conditions.\nIn 2017, under the assumptions of upper and lower bounds for both $\\gamma(v)$ and its derivative $\\gamma'(v)$, Tao and Winkler \\cite{tao2017effects} proved the existence of global classical solutions for the two-dimensional case, while demonstrating that the system admits only weak solutions in higher dimensions. Fujie and Jiang \\cite{fujie2020global} established that when $\\gamma(v)$ satisfies the following conditions:\n\\begin{equation*}\n0<\\gamma(v)\\in C^3[0,\\infty),\\quad \\gamma'(v)\\leq 0 \\text{ on } [0,\\infty),\\quad \\lim_{v\\to\\infty}\\gamma(v)=0,\n\\end{equation*}\nglobal classical solutions exist for arbitrary initial data. Their subsequent work \\cite{fujie2021boundedness} showed that if the function $\\gamma(v)$ satisfies the asymptotic condition $\\lim_{v\\to\\infty} e^{\\alpha v}\\gamma(v) = +\\infty$ for any $\\alpha > 0$, then there exists a globally bounded classical solution.\nFor the case $\\gamma(v) = e^{-\\alpha v}$ ($\\alpha > 0$), the authors in \\cite{fujie2020global,fujie2021comparison,jin2020critical} derived a critical mass threshold: solutions remain uniformly bounded when the initial cell mass is below this critical value. In contrast, studies in \\cite{burger2021delayed, fujie2020global,fujie2021comparison} indicate that solutions  blow up as time approaches infinity when the initial cell mass exceeds the critical mass. Nevertheless, it is impossible to blow-up in finite time, which is a difference from the Keller-Segel model. \n\nIn their seminal 2025 work on the model \\eqref{pde} (with $\\gamma(v)=e^{-v}+1$), the reference \\cite{BOL2026113712}\nrigorously established the well-posedness of solutions to the PDE \\eqref{pde} and gave error estimates between \\eqref{pde} and its regularized version. The discussion is already so comprehensive that we can cite it directly. They also conducted an in-depth study in which the convergences for the stochastic particle systems of \\eqref{pde} inspired us to refine their methodologies. \nOur focus lies particularly on the stochastic particle equations and the mean field limit corresponding to  \\eqref{pde}. By introducing the concept of stopping time (achieving convergence in probability in Section 2), we obtained results with faster convergence rates than those in \\cite{BOL2026113712}. \nIn Section 3, we establish the propagation of chaos in strong senses for stochastic differential equations (SDEs), (where we apply the relative entropy method and derive a higher convergence rate), which build a foundational bridge between the macroscopic and microscopic systems.\nIn the following, we propose that the SDE corresponding to \\eqref{pde} for $N\\in\\mathbb{N}$ interacting particles $\\{X_{N,i}^\\varepsilon(\\cdot)\\}_{1\\le i\\le N}$ in $\\mathbb{R}^2$ reads as\n\\begin{align}\\label{sde}\n\\begin{cases}\ndX_{N,i}^\\varepsilon(t) = \\Big(2\\exp\\Big(-\\frac{1}{N}\\sum_{j=1}^N\\Phi^\\varepsilon(X_{N,i}^\\varepsilon(t) - X_{N,j}^\\varepsilon(t))\\Big)+ 2\\Big)^{1/2}dB_i(t),\\\\\nX_{N,i}^\\varepsilon(0) =\\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}\nwhere $0<\\varepsilon<1$. Here, we consider a filtered probability space defined by $(\\Omega,\\mathcal{F}, (\\mathcal{F})_{t\\ge0}, \\mathbb{P})$ and introduce $\\{B_i\\}_{1\\le i\\le N}$, a collection of independent $\\mathcal{F}_t$-Brownian motions. The initial data $\\zeta_1,\\zeta_2,\\cdots,\\zeta_N$ are assumed to be random variables independent and identically distributed (i.i.d.) with the common probability density function $u_0$. The potential $\\Phi^\\varepsilon$ is given by\n$$\\Phi^\\varepsilon:=\\Phi*j^\\varepsilon,\\quad \\Phi:=\\chi \\tilde{\\Phi},\\quad j^\\varepsilon(x):=\\frac{1}{\\varepsilon^2}j(\\frac{x}{\\varepsilon}),$$\nwhere $\\chi>0$ describes the strength rate of the signals in the model \\eqref{pde} and $\\tilde{\\Phi}$ is the Yukawa potential \\cite{lieb2001analysis}, which is defined for any $\\mu > 0$, by \n\\begin{align*}\n&\\tilde{\\Phi}(x) = \\int_0^\\infty(4\\pi t)^{-1}\\exp\\Big\\{-\\frac{|x|^2}{4t}-\\mu^2t\\Big\\}dt.\n\\end{align*}\nBased on \\cite[Appendix B]{li2023optimal}, the following properties are observed\n\\begin{align}\\label{Phi}\n&\\|\\tilde{\\Phi}\\|_{L^p(\\mathbb{R}^2)}<\\infty,~\\|\\nabla\\tilde{\\Phi}\\|_{L^q(\\mathbb{R}^2)}<\\infty,\\qquad p\\in[1,\\infty),~q\\in[1,2).\n\\end{align}\nThrough calculation, the following results are obtained that for any constant $C$ independent of $\\varepsilon$,\n\\begin{align}\n&\\|\\Phi^\\varepsilon\\|_{W^{1,1}(\\mathbb{R}^2)}\\le\\|\\Phi\\|_{W^{1,1}(\\mathbb{R}^2)}\\|j^\\varepsilon\\|_{L^1(\\mathbb{R}^2)}\\le C,\\label{D1phi}\\\\\n&\\|\\Phi^\\varepsilon\\|_{W^{1,\\infty}(\\mathbb{R}^2)}\\le\\|\\Phi\\|_{W^{1,1}(\\mathbb{R}^2)}\\|j^\\varepsilon\\|_{L^\\infty(\\mathbb{R}^2)}\\le \\frac{C}{\\varepsilon^2},\\label{phi}\\\\\n&\\|D^2\\Phi^\\varepsilon\\|_{L^\\infty(\\mathbb{R}^2)}\\le \\|\\nabla\\Phi\\|_{L^1(\\mathbb{R}^2)}\\|\\nabla j^\\varepsilon\\|_{L^\\infty(\\mathbb{R}^2)}\\le \\frac{C}{\\varepsilon^3}\\label{D2phi}.\n\\end{align}\nThe rigorous derivation of PDEs from stochastic particle systems represents a fundamental challenge in mathematical physics and applied analysis. \nA particularly influential framework for such derivations is the theory of moderately interacting particles, introduced in the pioneering works of Oelschl\\\"ager  \\cite{oelschlager1985law, oelschlager1987fluctuation, oelschlager1989derivation},\nwhere the Law of Large Numbers for interacting diffusions was established, together with central-limit-type fluctuation results.\nThis approach, characterized by a specific scaling of the interaction potential that preserves nonlocality while weakening pointwise strength, has since been extended to various biological and physical models. For example, Stevens \\cite{stevens2000derivation} derived the Keller–Segel chemotaxis model from a moderately interacting system, overcoming the lack of ellipticity, a common assumption in earlier mean field limits, by introducing novel analytical techniques. \n\nIn recent years, significant progress has been made in handling singular interactions, such as Coulomb and Riesz potentials, which arise naturally in models of collective behavior. Lazarovici and Pickl \\cite{lazarovici2017mean} analyzed a particle system incorporating a regularized potential and random initial conditions. Serfaty \\cite{serfaty2020mean} developed the modulated energy method to treat Coulomb-type flows, providing a powerful tool for the analysis of the mean field without confinement. \nSubsequent studies on related models employing diverse regularization strategies have further expanded this line of inquiry, as documented in \\cite{boers2016mean,bolley2011stochastic,chen2017mean,chen2020combined}.\nCurrently, the relative entropy method, advanced by Jabin and Wang \\cite{jabin2017mean} and further applied by Bresch et al.\\cite{bresch2023mean} and Chen et al.\\cite{chen2025quantitative,chen2025mean}, has enabled quantitative estimates of convergence for singular attractive kernels. These techniques have been instrumental in the establishment of various modes of convergence, including convergence in the Wasserstein distance (Carrillo et al.\\cite{carrillo2019propagation}) and convergence in probability Pickl et al. \\cite{lazarovici2017mean,huang2020mean,huang2017error}, strong $L^1$ convergence (Chen et al. \\cite{chen2025quantitative} and \\cite{olivera2020quantitative,olivera2023quantitative}). More recently, the moderate interaction framework has been instrumental in the derivation of cross-diffusion models; see, for example, \\cite{carrillo2024interacting,chen2021rigorous, chen2019rigorous,li2024convergence}.\n\nTo study the relationship between the macroscopic system \\eqref{pde} and its microscopic system \\eqref{sde}, \nwe need to introduce an intermediate particle system (Mean-Field equation). A key object in the mean-field limit is the empirical measure, a random probability measure defined as\n$$\\mu^\\varepsilon_N(t)=\\frac{1}{N}\\sum_{i=1}^N\\delta_{X_{N,i}^\\varepsilon(t)},\\quad t>0,$$\nwhere $\\delta$ is the Dirac delta distribution. The work \\cite{olivera2021quantitative} shows that $\\mu^\\varepsilon_N(t)$ converges to the following PDE solution $u^\\varepsilon$. For fixed $\\varepsilon>0$, the particle system \\eqref{sde} propagates chaos in the many-particle limit $N\\to\\infty$ towards the non-linear SDE system:\n\\begin{align}\\label{mfs}\n\\begin{cases}\nd\\overline{X}_i^\\varepsilon(t) = \\big(2\\exp\\big(-\\Phi^\\varepsilon * u^\\varepsilon( \\overline{X}_i^\\varepsilon,t)\\big)+2\\big)^{1/2} dB_i(t),\\\\\n\\overline{X}_i^\\varepsilon(0) = \\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}\nwhere\n$u^\\varepsilon(\\cdot,t)$ is the density function of i.i.d. random processes\n$\\overline{X}_1^\\varepsilon(t),\\cdots, \\overline{X}_N^\\varepsilon(t)$. \nAnd the initial data $\\{\\zeta_i\\}_{i=1}^N$\nare subject to the same conditions as in \\eqref{sde}. Using It\\^{o}'s formula, the density function $u^\\varepsilon$ satisfies the so-called intermediate nonlocal problem, namely,\n\\begin{align}\n\\begin{cases}\\label{rpde}\n\\partial_t u^\\varepsilon = \\Delta(e^{-v^\\varepsilon}u^\\varepsilon+ u^\\varepsilon),\\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\n-\\Delta v^\\varepsilon + v^\\varepsilon =\\chi u^\\varepsilon * j^\\varepsilon,\\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\nu^\\varepsilon(x,0) = u_0*j^\\varepsilon(x), \\qquad &x\\in\\mathbb{R}^2.\n\\end{cases}\n\\end{align}\n\nIn this paper, our argument relies on the Lemma \\ref{weak solution} from \\cite{BOL2026113712}, which establishes the existence and uniqueness of the weak solution $u^\\varepsilon$ to the above system \\eqref{rpde} and provides the $L^\\infty$ estimate for $\\nabla\\log u^\\varepsilon$.\n\\begin{lemma}\\label{weak solution}\n\\textup{(Existence of weak solution, \\cite[Theorem 2, Lemma 16, and Lemma 18]{BOL2026113712})}.\nLet $u_0\\ge0$ be a initial probability density that satisfies\n\\begin{eqnarray*}\n&u_0\\log u_0\\in L^1(\\mathbb{R}^2), \\quad \\nabla\\log u_0\\in W^{1,q}(\\mathbb{R}^2)~(q>2),\\\\\n&u_0\\in L^1(\\mathbb{R}^2,|x|^2dx)\\cap L^p(\\mathbb{R}^2)~(1\\le p\\le \\infty).   \n\\end{eqnarray*}\nFurthermore, assume $\\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\\|\\omega\\|_{L^4(\\mathbb{R}^2)}^4\\le c_*\\|\\omega\\|_{L^2(\\mathbb{R}^2)}^2\\|\\nabla\\omega\\|_{L^2(\\mathbb{R}^2)}^2$. Then for any $T>0$ and $t\\in[0,T]$, the problem \\eqref{rpde} possesses weak solutions $(u^\\varepsilon, v^\\varepsilon)$ in $\\mathbb{R}^2\\times(0,T)$, and there exists a time $T^*\\in(0,T)$ such that \n\\begin{eqnarray}\n&\\|u^\\varepsilon\\|_{{L^2(0,T;H^1(\\mathbb{R}^2))}\\cap L^\\infty(0,T;L^p(\\mathbb{R}^2))}\\le C,\\label{uniformweaksolution}\\\\\n&\\||x|^2u^\\varepsilon\\|_{L^\\infty(0, T; L^1(\\mathbb{R}^2))}\\le C,\\quad\\|\\nabla\\log u^\\varepsilon\\|_{L^\\infty(0, T^*; W^{1,q}(\\mathbb{R}^2))}\\le C.\\label{uniformweaksolution1}\n\\end{eqnarray}\nwhere $C$ is a constant independent of $\\varepsilon$. \n\\end{lemma}\n\nUsing the above lemma, we obtain our first main result: convergence in probability of the mean-field limit under algebraic scaling.", "context": "In this work, we provide a rigorous derivation of the two-dimensional Keller–Segel-type system with signal-dependent sensitivity expressed as\n\\begin{align}\n\\begin{cases}\\label{pde}\n\\partial_t u = \\Delta(e^{-v}u+ u), \\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\n-\\Delta v + v =\\chi u, \\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\nu(x,0) = u_0(x), \\qquad &x\\in\\mathbb{R}^2,\n\\end{cases}\n\\end{align}\nwhere $u(x,t)$ denotes cell density and $v(x,t)$ represents the concentration of the chemical signal. The coefficient $\\chi>0$ quantifies the signal-dependent strength of chemotactic sensitivity.\n\nTo study the relationship between the macroscopic system \\eqref{pde} and its microscopic system \\eqref{sde}, \nwe need to introduce an intermediate particle system (Mean-Field equation). A key object in the mean-field limit is the empirical measure, a random probability measure defined as\n$$\\mu^\\varepsilon_N(t)=\\frac{1}{N}\\sum_{i=1}^N\\delta_{X_{N,i}^\\varepsilon(t)},\\quad t>0,$$\nwhere $\\delta$ is the Dirac delta distribution. The work \\cite{olivera2021quantitative} shows that $\\mu^\\varepsilon_N(t)$ converges to the following PDE solution $u^\\varepsilon$. For fixed $\\varepsilon>0$, the particle system \\eqref{sde} propagates chaos in the many-particle limit $N\\to\\infty$ towards the non-linear SDE system:\n\\begin{align}\\label{mfs}\n\\begin{cases}\nd\\overline{X}_i^\\varepsilon(t) = \\big(2\\exp\\big(-\\Phi^\\varepsilon * u^\\varepsilon( \\overline{X}_i^\\varepsilon,t)\\big)+2\\big)^{1/2} dB_i(t),\\\\\n\\overline{X}_i^\\varepsilon(0) = \\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}\nwhere\n$u^\\varepsilon(\\cdot,t)$ is the density function of i.i.d. random processes\n$\\overline{X}_1^\\varepsilon(t),\\cdots, \\overline{X}_N^\\varepsilon(t)$. \nAnd the initial data $\\{\\zeta_i\\}_{i=1}^N$\nare subject to the same conditions as in \\eqref{sde}. Using It\\^{o}'s formula, the density function $u^\\varepsilon$ satisfies the so-called intermediate nonlocal problem, namely,\n\\begin{align}\n\\begin{cases}\\label{rpde}\n\\partial_t u^\\varepsilon = \\Delta(e^{-v^\\varepsilon}u^\\varepsilon+ u^\\varepsilon),\\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\n-\\Delta v^\\varepsilon + v^\\varepsilon =\\chi u^\\varepsilon * j^\\varepsilon,\\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\nu^\\varepsilon(x,0) = u_0*j^\\varepsilon(x), \\qquad &x\\in\\mathbb{R}^2.\n\\end{cases}\n\\end{align}\n\nIn this paper, our argument relies on the Lemma \\ref{weak solution} from \\cite{BOL2026113712}, which establishes the existence and uniqueness of the weak solution $u^\\varepsilon$ to the above system \\eqref{rpde} and provides the $L^\\infty$ estimate for $\\nabla\\log u^\\varepsilon$.\n\\begin{lemma}\\label{weak solution}\n\\textup{(Existence of weak solution, \\cite[Theorem 2, Lemma 16, and Lemma 18]{BOL2026113712})}.\nLet $u_0\\ge0$ be a initial probability density that satisfies\n\\begin{eqnarray*}\n&u_0\\log u_0\\in L^1(\\mathbb{R}^2), \\quad \\nabla\\log u_0\\in W^{1,q}(\\mathbb{R}^2)~(q>2),\\\\\n&u_0\\in L^1(\\mathbb{R}^2,|x|^2dx)\\cap L^p(\\mathbb{R}^2)~(1\\le p\\le \\infty). \n\\end{eqnarray*}\nFurthermore, assume $\\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\\|\\omega\\|_{L^4(\\mathbb{R}^2)}^4\\le c_*\\|\\omega\\|_{L^2(\\mathbb{R}^2)}^2\\|\\nabla\\omega\\|_{L^2(\\mathbb{R}^2)}^2$. Then for any $T>0$ and $t\\in[0,T]$, the problem \\eqref{rpde} possesses weak solutions $(u^\\varepsilon, v^\\varepsilon)$ in $\\mathbb{R}^2\\times(0,T)$, and there exists a time $T^*\\in(0,T)$ such that \n\\begin{eqnarray}\n&\\|u^\\varepsilon\\|_{{L^2(0,T;H^1(\\mathbb{R}^2))}\\cap L^\\infty(0,T;L^p(\\mathbb{R}^2))}\\le C,\\label{uniformweaksolution}\\\\\n&\\||x|^2u^\\varepsilon\\|_{L^\\infty(0, T; L^1(\\mathbb{R}^2))}\\le C,\\quad\\|\\nabla\\log u^\\varepsilon\\|_{L^\\infty(0, T^*; W^{1,q}(\\mathbb{R}^2))}\\le C.\\label{uniformweaksolution1}\n\\end{eqnarray}\nwhere $C$ is a constant independent of $\\varepsilon$. \n\\end{lemma}\n\nUsing the above lemma, we obtain our first main result: convergence in probability of the mean-field limit under algebraic scaling.", "full_context": "In this work, we provide a rigorous derivation of the two-dimensional Keller–Segel-type system with signal-dependent sensitivity expressed as\n\\begin{align}\n\\begin{cases}\\label{pde}\n\\partial_t u = \\Delta(e^{-v}u+ u), \\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\n-\\Delta v + v =\\chi u, \\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\nu(x,0) = u_0(x), \\qquad &x\\in\\mathbb{R}^2,\n\\end{cases}\n\\end{align}\nwhere $u(x,t)$ denotes cell density and $v(x,t)$ represents the concentration of the chemical signal. The coefficient $\\chi>0$ quantifies the signal-dependent strength of chemotactic sensitivity.\n\nTo study the relationship between the macroscopic system \\eqref{pde} and its microscopic system \\eqref{sde}, \nwe need to introduce an intermediate particle system (Mean-Field equation). A key object in the mean-field limit is the empirical measure, a random probability measure defined as\n$$\\mu^\\varepsilon_N(t)=\\frac{1}{N}\\sum_{i=1}^N\\delta_{X_{N,i}^\\varepsilon(t)},\\quad t>0,$$\nwhere $\\delta$ is the Dirac delta distribution. The work \\cite{olivera2021quantitative} shows that $\\mu^\\varepsilon_N(t)$ converges to the following PDE solution $u^\\varepsilon$. For fixed $\\varepsilon>0$, the particle system \\eqref{sde} propagates chaos in the many-particle limit $N\\to\\infty$ towards the non-linear SDE system:\n\\begin{align}\\label{mfs}\n\\begin{cases}\nd\\overline{X}_i^\\varepsilon(t) = \\big(2\\exp\\big(-\\Phi^\\varepsilon * u^\\varepsilon( \\overline{X}_i^\\varepsilon,t)\\big)+2\\big)^{1/2} dB_i(t),\\\\\n\\overline{X}_i^\\varepsilon(0) = \\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}\nwhere\n$u^\\varepsilon(\\cdot,t)$ is the density function of i.i.d. random processes\n$\\overline{X}_1^\\varepsilon(t),\\cdots, \\overline{X}_N^\\varepsilon(t)$. \nAnd the initial data $\\{\\zeta_i\\}_{i=1}^N$\nare subject to the same conditions as in \\eqref{sde}. Using It\\^{o}'s formula, the density function $u^\\varepsilon$ satisfies the so-called intermediate nonlocal problem, namely,\n\\begin{align}\n\\begin{cases}\\label{rpde}\n\\partial_t u^\\varepsilon = \\Delta(e^{-v^\\varepsilon}u^\\varepsilon+ u^\\varepsilon),\\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\n-\\Delta v^\\varepsilon + v^\\varepsilon =\\chi u^\\varepsilon * j^\\varepsilon,\\qquad &x\\in\\mathbb{R}^2,~ t>0,\\\\\nu^\\varepsilon(x,0) = u_0*j^\\varepsilon(x), \\qquad &x\\in\\mathbb{R}^2.\n\\end{cases}\n\\end{align}\n\nIn this paper, our argument relies on the Lemma \\ref{weak solution} from \\cite{BOL2026113712}, which establishes the existence and uniqueness of the weak solution $u^\\varepsilon$ to the above system \\eqref{rpde} and provides the $L^\\infty$ estimate for $\\nabla\\log u^\\varepsilon$.\n\\begin{lemma}\\label{weak solution}\n\\textup{(Existence of weak solution, \\cite[Theorem 2, Lemma 16, and Lemma 18]{BOL2026113712})}.\nLet $u_0\\ge0$ be a initial probability density that satisfies\n\\begin{eqnarray*}\n&u_0\\log u_0\\in L^1(\\mathbb{R}^2), \\quad \\nabla\\log u_0\\in W^{1,q}(\\mathbb{R}^2)~(q>2),\\\\\n&u_0\\in L^1(\\mathbb{R}^2,|x|^2dx)\\cap L^p(\\mathbb{R}^2)~(1\\le p\\le \\infty). \n\\end{eqnarray*}\nFurthermore, assume $\\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\\|\\omega\\|_{L^4(\\mathbb{R}^2)}^4\\le c_*\\|\\omega\\|_{L^2(\\mathbb{R}^2)}^2\\|\\nabla\\omega\\|_{L^2(\\mathbb{R}^2)}^2$. Then for any $T>0$ and $t\\in[0,T]$, the problem \\eqref{rpde} possesses weak solutions $(u^\\varepsilon, v^\\varepsilon)$ in $\\mathbb{R}^2\\times(0,T)$, and there exists a time $T^*\\in(0,T)$ such that \n\\begin{eqnarray}\n&\\|u^\\varepsilon\\|_{{L^2(0,T;H^1(\\mathbb{R}^2))}\\cap L^\\infty(0,T;L^p(\\mathbb{R}^2))}\\le C,\\label{uniformweaksolution}\\\\\n&\\||x|^2u^\\varepsilon\\|_{L^\\infty(0, T; L^1(\\mathbb{R}^2))}\\le C,\\quad\\|\\nabla\\log u^\\varepsilon\\|_{L^\\infty(0, T^*; W^{1,q}(\\mathbb{R}^2))}\\le C.\\label{uniformweaksolution1}\n\\end{eqnarray}\nwhere $C$ is a constant independent of $\\varepsilon$. \n\\end{lemma}\n\nUsing the above lemma, we obtain our first main result: convergence in probability of the mean-field limit under algebraic scaling.\n\nUsing the above lemma, we obtain our first main result: convergence in probability of the mean-field limit under algebraic scaling.\n\nThe proof of Theorem \\ref{propagation of chaos}, given in Section 2, follows the approach of \\cite{lazarovici2017mean}, adopting a superior algebraic scaling with respect to the maximum norm of the trajectory, which \\cite{BOL2026113712} only achieved results under logaritheoremic scaling. \nSince the diffusion coefficients in the stochastic models \\eqref{sde} and \\eqref{mfs} given the nonlinear dependence of the diffusion coefficients on inter-particle interactions, the use of the Burkholder-Davis-Gundy inequality becomes indispensable. The key ingredient of our proof is the introduction of stopping times, a technique that facilitates the application of the Law of Large Numbers.\n\nTherefore, plugging \\eqref{I12} and \\eqref{I11} into \\eqref{EI1i} gives\n\\begin{align}\\label{I1i}\n&\\E\\Big(N^{2\\alpha }\\int_0^{t\\land\\tau_\\alpha}\\max_{i=1,\\cdots,N}|I_{1,i}(s)|^2 ds\\Big)\\nonumber\\\\\n\\le\\ & C(T) (1+\\varepsilon^{-6}N^{-2\\alpha })\\int_0^{t\\land\\tau_\\alpha}\\E(S_\\alpha^2(s))ds + C(m, T)\\varepsilon^{-4m-4}N^{-m+1}.\n\\end{align}\nSubstituting \\eqref{I2i} and \\eqref{I1i} into \\eqref{ES} yields the inequality\n\\begin{align}\\label{initial ineq}\n\\E(S_\\alpha^2(t))\n\\le\\ & C(T) (1+\\varepsilon^{-6}N^{-2\\alpha })\\int_0^{t\\land\\tau_\\alpha}\\E(S_\\alpha^2(s))ds \\nonumber\\\\\n&+ C(m, T)(\\varepsilon^{-4m-4}N^{-m+1} +N^{(2\\alpha-\\theta)} + \\varepsilon^{-4m-4}N^{2\\alpha +m(2\\theta -1)+1}).\n\\end{align}\nWe now show that the terms multiplying the constants decay as a negative power of $N$. Recall the scaling $\\varepsilon\\sim N^{-\\gamma},\\gamma>0$. To bound the error terms, we require a positive exponent\n$\\eta>0$ such that the following bounds hold\n\\begin{align*}\n\\varepsilon^{-6}N^{-2\\alpha } = N^{6\\gamma -2\\alpha}\\le N^{0}&\\Longleftrightarrow\\gamma\\le\\frac{\\alpha}{3},\\\\\nN^{2\\alpha-\\theta}\\le N^{-\\eta}&\\Longleftrightarrow\\eta\\le\\theta-2\\alpha, \\\\\n\\varepsilon^{-4m-4}N^{-m+1} = N^{(4m+4)\\gamma -m+1}\\le N^{-\\eta}\n&\\Longleftrightarrow\\eta\\le-(4m+4)\\gamma+m-1,\\\\\n\\varepsilon^{-4m-4}N^{2\\alpha +m(2\\theta -1)+1}=N^{(4m+4)\\gamma+2\\alpha +m(2\\theta -1)+1}\\le N^{-\\eta}&\\Longleftrightarrow\\eta\\le-(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1.\n\\end{align*}\nThe condition $\\eta>0$ implies that $\\theta$, $\\alpha$, $m$, and $\\gamma$ need to be selected to meet the following relationships\n\\begin{align*}\n0<\\theta<\\frac{1}{2},\\quad 0<\\alpha<\\frac{\\theta}{2},\\quad m>\\frac{\\theta+1}{1-2\\theta},\\quad 0<\\gamma<\\frac{-2\\alpha+m(1-2\\theta)-1}{4m+4}. \n\\end{align*}\nThus choose $\\gamma$ satisfying\n$$\n0<\\gamma<\\min\\big\\{\\frac{\\alpha}{3},\\frac{-2\\alpha +m(1-2\\theta)-1}{4m+4}\\big\\},\n$$\nand $\\eta$ satisfying \n$$\n0<\\eta\\le\\min\\{\\theta-2\\alpha, -(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}.\n$$ \nThen the inequality \\eqref{initial ineq} simplifies the estimate\n\\begin{align}\\label{Sak}\n\\E(S_\\alpha^2(t))\n\\le\\ & C\\int_0^{t\\land\\tau_\\alpha}\\E(S_\\alpha^2(s))ds + C(m, T)N^{-\\eta}.\n\\end{align}\nUsing Gr\\\"{o}nwall's inequality for \\eqref{Sak}, we obtain for any $t\\in[0,T]$\n\\begin{align} \\label{Egronwall}\n\\E(S_\\alpha^2(t))\\le C(m,T)N^{-\\eta} (1+CTe^{CT}). \n\\end{align}\nBy inserting \\eqref{MK} into \\eqref{Egronwall}, we derive \n\\begin{align}\n\\mathbb{P}\\big(\\max_{i=1,\\cdots,N}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|> N^{-\\alpha}\\big)\\le C(m,T)N^{-\\eta} (1+CTe^{CT}).\n\\end{align}\nThe proof of this theorem is completed.\n\n\\setcounter{theorem}{1}\n\\begin{theorem}\\label{Propagation of chaos in the strong sense}\\textup{(Propagation of chaos in the strong sense).} \nUnder the assumptions of Lemma \\ref{weak solution}, let \n$u_{N}^\\varepsilon(t,x_1,\\cdots,x_N)$ and $u^{\\varepsilon\\otimes N}(t,x_1,\\cdots,x_N)$ be the solutions of systems \\eqref{lpde1} and \\eqref{lpde2}, respectively. For any \n$r\\in\\mathbb{N^+}$, denote by $u_{N,r}^\\varepsilon(t,x_1,\\cdots,x_r)$ and $u^{\\varepsilon\\otimes r}(t,x_1,\\cdots,x_r)$ their corresponding \n$r$-th marginal distributions. \nThen for parameters $0<\\theta<1/2$, $0 <\\alpha<\\frac{\\theta}{2}$, and $m\\in \\mathbb{N}^+$ satisfying $m \\ge \\frac{1+\\theta}{1-2\\theta}$, there exist a time $T^*\\in(0,T)$, a constant $C(m,r,T)>0$, and a parameter $\\beta>0$ satisfying the bound $1<\\beta\\le\\min\\{\\frac{2\\alpha}{\\gamma}-6,\\frac{\\eta}{\\gamma}-4\\}$ such that \n\\begin{align}\\label{Ps}\n\\|u_{N,r}^{\\varepsilon}(t) - u^{\\varepsilon\\otimes r}(t)\\|_{L^\\infty(0,T^*;L^1(\\R^{2r}))} \\le C(r,m,T)\\varepsilon^{\\beta}, \n \\end{align}\nwhere the cut-off parameter satisfies\n$\\varepsilon\\sim N^{-\\gamma} $,\nand the parameter $\\gamma$ defined by\n$$0<\\gamma<\\min\\{\\frac{2\\alpha}{7},\\frac{-2\\alpha + m(1-2\\theta)-1}{4m+4}\\},$$\nand the parameter $\\eta$ satisfies\n$$5\\gamma<\\eta<\\min\\{\\theta-2\\alpha,-(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}.$$ \n\\end{theorem}\n\nPlugging \\eqref{J1} and \\eqref{J2} into \\eqref{K21} produces the following\n\\begin{align}\nK_2\\leq &-\\int_{\\R^{2N}}\\frac{1}{N}\\sum_{i=1}^Nu_N^\\varepsilon\\Big[\\exp\\big(-\\frac{1}{N}\\sum_{j=1}^N\\Phi^\\varepsilon(x_i - x_j)\\big) \\Big|\\nabla_{x_i}\\log\\frac{u_N^\\varepsilon}{u^{\\varepsilon\\otimes N}}\\Big|^2\\Big]d_{x_1}\n\\cdots d_{x_N}\\nonumber\\\\\n&+ C(m, T)(N^{-2\\alpha}\\varepsilon^{-6} + (N^{-1} + N^{-\\eta})\\varepsilon^{-4}).\\label{K_2}\n\\end{align}\nCombining \\eqref{K1} and \\eqref{H1}, we derive\n\\begin{align}\n\\frac{d}{dt}\\mathcal{H}(u_N^\\varepsilon|u^{\\varepsilon\\otimes N})\n\\leq& -\\frac{1}{2}\\int_{\\R^{2N}}\\frac{1}{N}\\sum_{i=1}^Nu_N^\\varepsilon\\Big|\\nabla_{x_i}\\log\\frac{u_N^\\varepsilon}{u^{\\varepsilon\\otimes N}}\\Big|^2d_{x_1}\\cdots d_{x_N}\\nonumber\\\\\n&-\\frac{1}{2}\\int_{\\R^{2N}}\\frac{1}{N}\\sum_{i=1}^Nu_N^\\varepsilon\\Big[\\exp\\big(-\\frac{1}{N}\\sum_{j=1}^N\\Phi^\\varepsilon(x_i - x_j)\\big) \\Big|\\nabla_{x_i}\\log\\frac{u_N^\\varepsilon}{u^{\\varepsilon\\otimes N}}\\Big|^2\\Big]d_{x_1}\n\\cdots d_{x_N}\\nonumber\\\\\n& + C(m, T)(N^{-2\\alpha}\\varepsilon^{-6} + (N^{-1} + N^{-\\eta})\\varepsilon^{-4}).\\label{entropy}\n\\end{align}\nNoticing that the initial relative entropy is zero, it can be obtained from (\\ref{entropy}) that for any $0<t<T^*<T$, \n$$\n\\mathcal{H}(u_{N}^{\\varepsilon}|u^{\\varepsilon\\otimes N})(t)\\le C(m, T)(N^{-2\\alpha}\\varepsilon^{-6} + (N^{-1} + N^{-\\eta})\\varepsilon^{-4}).\n$$\nCombing \\eqref{relative entropy} and the C-K-P inequality \\eqref{CKP}, we have for $r\\in\\mathbb{N}$, \n\\begin{align}\n\\sup_{t\\in(0,T^*)}\\|u_{N,r}^{\\varepsilon}(t) - u^{\\varepsilon\\otimes r}(t)\\|_{L^1(\\R^{2r})} \n&\\le \\sup_{t\\in(0,T^*)}2\\mathcal{H}_r(u_{N,r}^{\\varepsilon}|u^{\\varepsilon\\otimes r})(t)\\nonumber\\\\\n&\\le \\sup_{t\\in(0,T^*)}4r\\mathcal{H}(u_{N}^{\\varepsilon}|u^{\\varepsilon\\otimes N})(t)\\nonumber\\\\\n&\\le C(m, T)(N^{-2\\alpha}\\varepsilon^{-6} + (N^{-1} + N^{-\\eta})\\varepsilon^{-4}).\n\\end{align}\nAccording to Theorem \\ref{propagation of chaos}, we know that \n\\begin{align}\n&0<\\eta\\le\\min\\{\\theta-2\\alpha,-(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\},\\label{eta1}\\\\\n&0<\\gamma\\le\\min\\{\\frac{\\alpha}{3},\\frac{-2\\alpha + m(1-2\\theta)-1}{4m+4}\\}. \\label{gamma1} \n\\end{align}\nBased on the previous assumption $\\varepsilon\\sim N^{-\\gamma}$, it holds that there exists $\\beta>0$ such that \n\\begin{align}\nN^{-2\\alpha}\\varepsilon^{-6} + (N^{-1} + N^{- \\eta})\\varepsilon^{-4}=\\varepsilon^\\frac{2\\alpha}{\\gamma}\\varepsilon^{-6}+(\\varepsilon^\\frac{1}{\\gamma}+\\varepsilon^\\frac{\\eta}{\\gamma})\\varepsilon^{-4}\\le \\varepsilon^\\beta, \n\\end{align}\nwhere $\\beta$ satisfies\n$$1<\\beta\\le\\min\\{\\frac{2\\alpha}{\\gamma}-6,\\frac{\\eta}{\\gamma}-4\\}.$$ \nHere we have used $\\eta<1$.\nThis further requires that $0<\\gamma<\\frac{2\\alpha}{7}$, $\\eta>5\\gamma$. Building on the assumptions in \\eqref{eta1} and \\eqref{gamma1}, we assume\n\\begin{align*}\n&0<\\gamma<\\min\\{\\frac{2\\alpha}{7},\\frac{-2\\alpha + m(1-2\\theta)-1}{4m+4}\\},\\\\\n&5\\gamma<\\eta<\\min\\{\\theta-2\\alpha,-(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}.\n\\end{align*} \nThen we obtain that for $0<t<T^*<T$, \n\\begin{align}\n\\sup_{t\\in(0,T^*)}\\|u_{N,r}^{\\varepsilon}(t) - u^{\\varepsilon\\otimes r}(t)\\|_{L^1(\\R^{2r})} \\le C(m,T)\\varepsilon^\\beta,\\quad \\beta>1.\n\\end{align}\nThus, we complete the proof of propagation of chaos. \n\\end{proof}", "post_theorem_intro_text_len": 5262, "post_theorem_intro_text": "\\begin{remark}\nTo see the value of $\\gamma$ and $\\eta$ more clearly, we provide a special case. When $m\\in\\mathbb{N}^+$, $ m\\ge\\frac{1+\\theta}{1-2\\theta}$ and $0<\\alpha<\\frac{\\theta}{2}<\\frac{1}{2}$, we take $\\alpha=0.1$, $\\theta=0.3$ and $m=4$ such that\n$0<\\gamma<\\min\\{\\frac{\\alpha}{3},\\frac{-2\\alpha +m(1-2\\theta)-1}{2(2+2m)}\\}=\\min\\{0.033,0.02\\}=0.02$. we may assume that $\\gamma=0.019$, then $0<\\eta\\le\\min\\{\\theta-2\\alpha,-2\\gamma(2+2m)-2\\alpha -m(2\\theta -1)-1\\}=\\min\\{0.1, 0.02\\}=0.02$. \n\\end{remark}\n\nThe  proof of Theorem \\ref{propagation of chaos}, given in Section 2, follows the approach of \\cite{lazarovici2017mean}, adopting a superior algebraic scaling with respect to the maximum norm of the trajectory, which \\cite{BOL2026113712} only achieved results under logaritheoremic scaling. \nSince the diffusion coefficients in the stochastic models \\eqref{sde} and \\eqref{mfs} given the nonlinear dependence of the diffusion coefficients on inter-particle interactions, the use of the Burkholder-Davis-Gundy inequality becomes indispensable. The key ingredient of our proof is the introduction of stopping times, a technique that facilitates the application of the Law of Large Numbers. \n\nBased on the uniform $L^\\infty(0,T^*;H^2(\\mathbb{R}^2))$ bound for $\\nabla\\log u_\\varepsilon$ in Lemma \\ref{weak solution} and the propagation of chaos established in Theorem \\ref{propagation of chaos}, our next main result addresses the strong $L^1$ convergence for the propagation of chaos. The proof is mainly presented in Section 3, employing the relative entropy method \\cite{jabin2018quantitative} as recently refined in \\cite{chen2025quantitative}.\n\n\\begin{theorem}\\label{Propagation of chaos in the strong sense}\\textup{(Propagation of chaos in the strong sense).} \nUnder the assumptions of Theorem \\ref{propagation of chaos}, let $r\\in \\mathbb{N^+}$, $u_{N,r}^\\varepsilon(t, x_1,\\cdots, x_r)$ be the $r$-th marginal density of the joint density $u_N^\\varepsilon(t, x_1,\\cdots, x_N)$ of $\\{X_{N,i}^\\varepsilon\\}_{1\\le i\\le N}$, and $u^{\\varepsilon \\otimes r}(t, x_1,\\cdots, x_r)$ be the tensor product of the solutions $u^{\\varepsilon}$ to the model \\eqref{rpde}. Then for parameters $0<\\theta<1/2$, $0 <\\alpha<\\frac{\\theta}{2}$, and $m\\in \\mathbb{N}^+$ satisfying $m \\ge \\frac{1+\\theta}{1-2\\theta}$, there exist a time $T^*\\in(0,T)$, a constant $C(m,r,T)>0$, and a parameter $\\beta>0$ satisfying the bound $1<\\beta\\le\\min\\{\\frac{2\\alpha}{\\gamma}-6,\\frac{\\eta}{\\gamma}-4\\}$ such that \n\\begin{align}\\label{Ps111}\n\\|u_{N,r}^{\\varepsilon}(t) - u^{\\varepsilon\\otimes r}(t)\\|_{L^\\infty(0,T^*;L^1(\\mathbb{R}^{2r}))} \\le C(r,m,T)\\varepsilon^{\\beta}, \n  \\end{align}\nwhere the cut-off parameter satisfies\n$\\varepsilon\\sim N^{-\\gamma} $,\nand the parameter $\\gamma$ defined by\n$$0<\\gamma<\\min\\{\\frac{2\\alpha}{7},\\frac{-2\\alpha + m(1-2\\theta)-1}{4m+4}\\},$$\nand the parameter $\\eta$ satisfies\n$$5\\gamma<\\eta<\\min\\{\\theta-2\\alpha,-(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}.$$ \n\\end{theorem}\n\\begin{remark}\nTo illustrate the feasible range of parameters, we give an explicit example. Choose $\\theta = 0.4,\\ \\alpha = 0.1$ and $m\\ge \\frac{1+\\theta}{1-2\\theta}=7$ such that $m=7$.\nThen we obtain\n$$0<\\gamma<\\min\\{\\frac{2\\alpha}{7},\\frac{-2\\alpha + m(1-2\\theta)-1}{4m+4}\\}=\\min\\{0.029,0.006\\}=0.006.$$\nTaking $\\gamma=0.0025$ gives\n$$0.0125=5\\gamma<\\eta<\\min\\{\\theta-2\\alpha,-(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}=\\min\\{0.2,0.12\\}=0.12.$$\nso we pick $\\eta=0.03$. Finally, $$1<\\beta\\le\\min\\{\\frac{2\\alpha}{\\gamma}-6,\\frac{\\eta}{\\gamma}-4\\}=\\min\\{74,8\\}=8$$\nwe take $\\beta = 8$.\nWith these choices, the propagation of chaos estimate becomes\n\\begin{align*}\n\\max_{i=1,\\cdots,N}\\mathbb{E}\\big(\\sup_{t\\in[0,T]}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|^2\\big)\\le C\\varepsilon^8, \n\\end{align*} \n\\end{remark}\nWe comment here that the convergence of the mean-field limit have already been discussed in \\cite{BOL2026113712}. It is therefore especially gratifying that our results accelerate the convergence rates  for the solutions to the SDE \\eqref{sde} and \\eqref{mfs} across different norms. Specifically, Chen et al. in \\cite[Theorem 3]{BOL2026113712} presented as\n\\begin{align*}\n\\max_{i=1,\\cdots,N}\\mathbb{E}\\big(\\sup_{t\\in[0,T]}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|^2\\big)\\le C\\varepsilon^2, \n\\end{align*}   \nwhere the cut-off parameter $\\varepsilon=(\\lambda\\log N)^{-\\frac{1}{4}}$. However, in \\eqref{Pe}, we prove algebraic convergence in contrast to logarithmic convergence established in \\cite{BOL2026113712}. This represents an  improvement in the convergence rate. Furthermore, in \\cite[Theorem 5]{BOL2026113712}, they obtained\n\\begin{align*}\n\\|u_{N,r}^{\\varepsilon}(t) - u^{\\varepsilon\\otimes r}(t)\\|_{L^\\infty(0,T^*;L^1(\\mathbb{R}^{2r}))}^2 \\le C(r,m,T)\\varepsilon.\n\\end{align*}\nBy contrast, our estimate in \\eqref{Ps111} yields a  faster convergence rate than that bound.\n\nThe article is organized as follows. In section 2, we establish the propagation of chaos, which corresponds to the convergence in probability of solutions to the stochastic differential equations \\eqref{sde} and \\eqref{mfs}. Section 3 further derive quantitative propagation of chaos result in the strong sense by applying the relative entropy method.", "sketch": "The proof of Theorem~\\ref{propagation of chaos} (given in Section 2) “follows the approach of \\cite{lazarovici2017mean},” and “adopt[s] a superior algebraic scaling with respect to the maximum norm of the trajectory” (contrasting with \\cite{BOL2026113712}, which “only achieved results under logarithmic scaling”). Because “the diffusion coefficients … [have] the nonlinear dependence … on inter-particle interactions,” the argument says “the use of the Burkholder-Davis-Gundy inequality becomes indispensable.” The “key ingredient” is “the introduction of stopping times,” which “facilitates the application of the Law of Large Numbers.”", "expanded_sketch": "The proof (given later) “follows the approach of Lazarovici and Pickl, \\emph{A mean field limit for the Vlasov--Poisson system} (2017),” and “adopt[s] a superior algebraic scaling with respect to the maximum norm of the trajectory” (contrasting with \\cite{BOL2026113712}, which “only achieved results under logarithmic scaling”). Because “the diffusion coefficients … [have] the nonlinear dependence … on inter-particle interactions,” the argument says “the use of the Burkholder-Davis-Gundy inequality becomes indispensable.” The “key ingredient” is “the introduction of stopping times,” which “facilitates the application of the Law of Large Numbers.”", "expanded_theorem": "\\label{propagation of chaos}\nThe assumptions of the following lemma still hold.\n\\begin{lemma}\\label{weak solution}\n\\textup{(Existence of weak solution, \\cite[Theorem 2, Lemma 16, and Lemma 18]{BOL2026113712})}.\nLet $u_0\\ge0$ be a initial probability density that satisfies\n\\begin{eqnarray*}\n&u_0\\log u_0\\in L^1(\\R^2), \\quad \\nabla\\log u_0\\in W^{1,q}(\\R^2)~(q>2),\\\\\n&u_0\\in L^1(\\R^2,|x|^2dx)\\cap L^p(\\R^2)~(1\\le p\\le \\infty).   \n\\end{eqnarray*}\nFurthermore, assume $\\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo-Nirenberg inequality: $\\|\\omega\\|_{L^4(\\R^2)}^4\\le c_*\\|\\omega\\|_{L^2(\\R^2)}^2\\|\\nabla\\omega\\|_{L^2(\\R^2)}^2$. Then for any $T>0$ and $t\\in[0,T]$, the problem \\eqref{rpde} possesses weak solutions $(u^\\varepsilon, v^\\varepsilon)$ in $\\R^2\\times(0,T)$, and there exists a time $T^*\\in(0,T)$ such that \n\\begin{eqnarray}\n&\\|u^\\varepsilon\\|_{{L^2(0,T;H^1(\\R^2))}\\cap L^\\infty(0,T;L^p(\\R^2))}\\le C,\\label{uniformweaksolution}\\\\\n&\\||x|^2u^\\varepsilon\\|_{L^\\infty(0, T; L^1(\\R^2))}\\le C,\\quad\\|\\nabla\\log u^\\varepsilon\\|_{L^\\infty(0, T^*; W^{1,q}(\\R^2))}\\le C.\\label{uniformweaksolution1}\n\\end{eqnarray}\nwhere $C$ is a constant independent of $\\varepsilon$. \n\\end{lemma}\nLet $\\{X_{N,i}^\\varepsilon\\}_{1\\le i\\le N}$ and $\\{\\overline{X}_i^\\varepsilon\\}_{1\\le i\\le N}$ be the solutions to systems\n\\begin{align}\\label{sde}\n\\begin{cases}\ndX_{N,i}^\\varepsilon(t) = \\Big(2\\exp\\Big(-\\frac{1}{N}\\sum_{j=1}^N\\Phi^\\varepsilon(X_{N,i}^\\varepsilon(t) - X_{N,j}^\\varepsilon(t))\\Big)+ 2\\Big)^{1/2}dB_i(t),\\\\\nX_{N,i}^\\varepsilon(0) =\\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}\nand\n\\begin{align}\\label{mfs}\n\\begin{cases}\nd\\overline{X}_i^\\varepsilon(t) = \\big(2\\exp\\big(-\\Phi^\\varepsilon * u^\\varepsilon( \\overline{X}_i^\\varepsilon,t)\\big)+2\\big)^{1/2} dB_i(t),\\\\\n\\overline{X}_i^\\varepsilon(0) = \\zeta_i,\\qquad 1\\le i\\le N,\n\\end{cases}\n\\end{align}\nrespectively. Then \nfor any $m,N\\in \\mathbb{N}^+$ and parameters $0<\\theta<\\frac{1}{2},\\quad 0<\\alpha<\\frac{\\theta}{2},\\quad m>\\frac{\\theta+1}{1-2\\theta}$, there exist a constant $C(m,T)>0$ and a parameter $\\gamma,\\eta>0$ satisfying the bounds\n\\begin{align}\n&0<\\gamma<\\min\\big\\{\\frac{\\alpha}{3},\\frac{-2\\alpha +m(1-2\\theta)-1}{4m+4}\\big\\},\\label{gamma} \\\\\n&0<\\eta\\le\\min\\{\\theta-2\\alpha, -(4m+4)\\gamma-2\\alpha +m(1-2\\theta)-1\\}\\label{eta}\n\\end{align}\nsuch that for all $0\\le t\\le T$,\n\\begin{align}\\label{Pe}\n\\mathbb{P}\\big(\\max_{i=1,\\cdots,N}\\big|(X_{N,i}^\\varepsilon - \\overline{X}_i^\\varepsilon)(t)\\big|> N^{-\\alpha}\\big)\\le C(m,T)N^{-\\eta}, \n\\end{align}   \nwhere the cut-off parameter satisfies\n$\\varepsilon\\sim N^{-\\gamma} $.", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let $u_0\\ge 0$ be an initial probability density on $\\mathbb{R}^2$ such that\n$$u_0\\log u_0\\in L^1(\\mathbb{R}^2),\\qquad \\nabla\\log u_0\\in W^{1,q}(\\mathbb{R}^2)\\ \\text{for some }q>2,$$\nand\n$$u_0\\in L^1(\\mathbb{R}^2,|x|^2dx)\\cap L^p(\\mathbb{R}^2)\\qquad (1\\le p\\le \\infty).$$\nAssume also that $\\chi<4/c_*$, where $c_*$ is the optimal constant in the Gagliardo--Nirenberg inequality\n$$\\|\\omega\\|_{L^4(\\mathbb{R}^2)}^4\\le c_*\\|\\omega\\|_{L^2(\\mathbb{R}^2)}^2\\|\\nabla\\omega\\|_{L^2(\\mathbb{R}^2)}^2.$$ \nLet $(u^\\varepsilon,v^\\varepsilon)$ be the weak solution of the intermediate nonlocal problem\n$$\\partial_t u^\\varepsilon=\\Delta(e^{-v^\\varepsilon}u^\\varepsilon+u^\\varepsilon),\\qquad -\\Delta v^\\varepsilon+v^\\varepsilon=\\chi \\,u^\\varepsilon * j^\\varepsilon,\\qquad u^\\varepsilon(\\cdot,0)=u_0*j^\\varepsilon,$$\non $\\mathbb{R}^2\\times(0,T)$, where $*$ denotes convolution.\nFor $1\\le i\\le N$, let $X_{N,i}^\\varepsilon$ and $\\overline X_i^\\varepsilon$ solve, with the same Brownian motions $B_i$ and the same initial data $\\zeta_i$,\n$$dX_{N,i}^\\varepsilon(t)=\\Big(2\\exp\\Big(-\\frac1N\\sum_{j=1}^N\\Phi^\\varepsilon(X_{N,i}^\\varepsilon(t)-X_{N,j}^\\varepsilon(t))\\Big)+2\\Big)^{1/2}dB_i(t),$$\n$$d\\overline X_i^\\varepsilon(t)=\\Big(2\\exp\\big(-\\Phi^\\varepsilon*u^\\varepsilon(\\overline X_i^\\varepsilon,t)\\big)+2\\Big)^{1/2}dB_i(t).$$\nFix $m,N\\in\\mathbb N^+$ and parameters $0<\\theta<\\tfrac12$, $0<\\alpha<\\tfrac\\theta2$, and $m>\\frac{\\theta+1}{1-2\\theta}$. Under these assumptions, which quantitative estimate holds for the probability that the interacting particle system and the mean-field system differ by more than $N^{-\\alpha}$ in maximum norm at time $t$?", "correct_choice": {"label": "A", "text": "There exist a constant $C(m,T)>0$ and parameters $\\gamma,\\eta>0$ such that\n$$0<\\gamma<\\min\\Big\\{\\frac\\alpha3,\\frac{-2\\alpha+m(1-2\\theta)-1}{4m+4}\\Big\\},$$\n$$0<\\eta\\le \\min\\Big\\{\\theta-2\\alpha,\n-(4m+4)\\gamma-2\\alpha+m(1-2\\theta)-1\\Big\\},$$\nand, if the cutoff parameter satisfies $\\varepsilon\\sim N^{-\\gamma}$, then for every $t\\in[0,T]$,\n$$\\mathbb P\\Big(\\max_{i=1,\\dots,N}|X_{N,i}^\\varepsilon(t)-\\overline X_i^\\varepsilon(t)|>N^{-\\alpha}\\Big)\\le C(m,T)N^{-\\eta}.$$"}, "choices": [{"label": "B", "text": "There exist a constant $C(m,T)>0$ and parameters $\\gamma,\\eta>0$ such that\n$$0<\\gamma<\\min\\Big\\{\\frac\\alpha3,\\frac{-2\\alpha+m(1-2\\theta)-1}{4m+4}\\Big\\},$$\n$$0<\\eta\\le \\min\\Big\\{\\theta-2\\alpha,\n-(4m+4)\\gamma-2\\alpha+m(1-2\\theta)-1\\Big\\},$$\nand, if the cutoff parameter satisfies $\\varepsilon\\sim N^{-\\gamma}$, then for every $t\\in[0,T]$,\n$$\\mathbb P\\Big(\\max_{i=1,\\dots,N}|X_{N,i}^\\varepsilon(t)-\\overline X_i^\\varepsilon(t)|>N^{-\\alpha}\\Big)\\le C(m,T)N^{-2\\eta}.$$"}, {"label": "C", "text": "There exist a constant $C(m,T)>0$ and parameters $\\gamma,\\eta>0$ such that\n$$0<\\gamma<\\min\\Big\\{\\frac\\alpha3,\\frac{-2\\alpha+m(1-2\\theta)-1}{4m+4}\\Big\\},$$\n$$0<\\eta\\le \\min\\Big\\{\\theta-2\\alpha,\n-(4m+4)\\gamma-2\\alpha+m(1-2\\theta)-1\\Big\\},$$\nand, if the cutoff parameter satisfies $\\varepsilon\\sim N^{-\\gamma}$, then for every $t\\in[0,T]$,\n$$\\mathbb P\\Big(\\max_{i=1,\\dots,N}|X_{N,i}^\\varepsilon(t)-\\overline X_i^\\varepsilon(t)|>N^{-\\alpha}\\Big)\\le C(m,T).$$"}, {"label": "D", "text": "For every choice of parameters $\\gamma,\\eta>0$ satisfying\n$$0<\\gamma<\\min\\Big\\{\\frac\\alpha3,\\frac{-2\\alpha+m(1-2\\theta)-1}{4m+4}\\Big\\},$$\n$$0<\\eta\\le \\min\\Big\\{\\theta-2\\alpha,\n-(4m+4)\\gamma-2\\alpha+m(1-2\\theta)-1\\Big\\},$$\nthere exists a constant $C(m,T)>0$ such that, if the cutoff parameter satisfies $\\varepsilon\\sim N^{-\\gamma}$, then for every $t\\in[0,T]$,\n$$\\mathbb P\\Big(\\max_{i=1,\\dots,N}|X_{N,i}^\\varepsilon(t)-\\overline X_i^\\varepsilon(t)|>N^{-\\alpha}\\Big)\\le C(m,T)N^{-\\eta},$$\nwith the same constant $C(m,T)$ working uniformly for all such admissible pairs $(\\gamma,\\eta)$."}, {"label": "E", "text": "There exist a constant $C(m,T)>0$ and parameters $\\gamma,\\eta>0$ such that\n$$0<\\gamma\\le\\min\\Big\\{\\frac\\alpha3,\\frac{-2\\alpha+m(1-2\\theta)-1}{4m+4}\\Big\\},$$\n$$0<\\eta\\le \\min\\Big\\{\\theta-2\\alpha,\n-(4m+4)\\gamma-2\\alpha+m(1-2\\theta)-1\\Big\\},$$\nand, if the cutoff parameter satisfies $\\varepsilon\\sim N^{-\\gamma}$, then for every $t\\in[0,T]$,\n$$\\mathbb P\\Big(\\max_{i=1,\\dots,N}|X_{N,i}^\\varepsilon(t)-\\overline X_i^\\varepsilon(t)|>N^{-\\alpha}\\Big)\\le C(m,T)N^{-\\eta}.$$"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "algebraic decay rate from stopping-time/BDG argument", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the quantitative decay factor $N^{-\\eta}$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence of the constant on the admissible pair $(\\gamma,\\eta)$", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "boundary_range", "tampered_component": "strict upper bound on $\\gamma$ replaced by an attained endpoint", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the conclusion outright; it states hypotheses and asks for the resulting quantitative estimate. The correct answer is not explicitly leaked."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the hypotheses are given in full and the task is to pick the exact stated conclusion. It functions as a restatement of a proposition rather than a conceptually restructured problem."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in distinguishing subtle variants involving decay rate, endpoint admissibility, and quantifier dependence, but the item mainly tests precise recall/recognition of the theorem statement rather than genuine derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically close and plausible: one weakens the bound, one changes the decay exponent, one alters endpoint strictness, and one changes quantifier/constant dependence. These reflect realistic theorem-statement failure modes."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-recognition MCQ with strong distractors, but it is largely tautological and only moderately tests generative reasoning."}}
{"id": "2602.01248v1", "paper_link": "http://arxiv.org/abs/2602.01248v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "[Structural output from primitive cycles]\n\\label{thm:structural}\nAssume the standing assumptions of Remark~\\ref{rem:standing-assumptions},\nso that the scaling-limit kernel $K_L$ exists.\nThen there exists a nonnegative kernel $\\Phi\\in L^1(\\mathbb{R})$ such that:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\Phi\\in\\mathrm{PF}_\\infty$;\n\\item its bilateral Laplace transform $\\mathcal B\\Phi$ satisfies the reflection law\n$\\mathcal B\\Phi(s)=\\mathcal B\\Phi(\\tfrac12-s)$ on the common domain of absolute convergence;\n\\item $\\mathcal B\\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation\n\\[\n\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)},\n\\]\nwhere $\\Psi$ is Laguerre--P\\'olya and hence has only real zeros.\n\\end{enumerate}", "start_pos": 11842, "end_pos": 12581, "label": "thm:structural"}, "ref_dict": {"eq:generator": "\\begin{equation}\n\\label{eq:generator}\n(\\mathcal{L}_N f)(j)\n=\na_j\\,\\bigl(f(j+1)-f(j)\\bigr)\n\\;+\na_{j-1}\\,\\bigl(f(j-1)-f(j)\\bigr).\n\\end{equation}", "rem:translation-invariant": "\\begin{remark}[Translation-invariant specialization]\n\\label{rem:translation-invariant}\nFor the purposes of this paper we restrict to the translation-invariant case\n\\(\na_j\\equiv a>0\n\\)\n(independent of $j$), so that \\eqref{eq:generator} is the usual continuous-time simple\nrandom walk on $\\Z/N\\Z$ with jump rate $a$. In this case the diffusive scaling limit is\ngoverned by the one-dimensional heat equation with diffusion constant \\(D=a\\), and the\nFourier analysis on $\\Z/N\\Z$ leading to the theta-series form is completely explicit.\nExtensions to uniformly elliptic periodic environments, or to quantitative homogenization\nregimes, can also yield a uniform local CLT and the same scaling limit; we do not pursue\nthose generalizations here.\n\\end{remark}", "thm:main-assembled": "\\begin{theorem}[Main theorem (assembled)]\n\\label{thm:main-assembled}\nAssume the strong regime hypotheses of Section~\\ref{sec:completion-selfdual}, so that the scaling-limit kernel\n$K_L$ exists and the self-dual normalization can be fixed. Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item (\\emph{Unconditional structural output.})\nThere exists a nonnegative kernel $\\Phi\\in L^1(\\mathbb R)$ satisfying:\n\\begin{itemize}\n\\item $\\Phi\\in\\mathrm{PF}_\\infty$;\n\\item its bilateral Laplace transform $\\mathcal B\\Phi$ satisfies the reflection law\n$\\mathcal B\\Phi(s)=\\mathcal B\\Phi(\\tfrac12-s)$ on the common domain of absolute convergence;\n\\item $\\mathcal B\\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation\n$\\mathcal B\\Phi(s)=E(s)/\\Psi(s)$ with $\\Psi$ Laguerre--P\\'olya and hence having only real zeros.\n\\end{itemize}\n\\item (\\emph{Unconditional Archimedean Mellin identification.})\nThe Archimedean-completed kernel $\\widetilde K_{\\mathrm{arch}}$ has Mellin transform\n$F_{\\mathrm{arch}}(z)=\\Xi(2z)$ as in Theorem~\\ref{thm:arch-mellin-identification}.\n\\end{enumerate}\n\\end{theorem}", "rem:standing-assumptions": "\\begin{remark}[Standing assumptions and existence of the scaling limit]\n\\label{rem:standing-assumptions}\nWe work throughout in a regime where the scaling-limit trace $K_L(t)$ exists for every $L>0$ and $t>0$, with convergence locally uniform in $t$ on compact subsets of $(0,\\infty)$. In particular, under the translation-invariant specialization of Remark~\\ref{rem:translation-invariant} this limit can be obtained explicitly by Fourier analysis on $\\Z/N\\Z$. Under these assumptions the limit admits the explicit theta-series form stated in Theorem~\\ref{thm:theta-series-form-recall}. Analytic interchanges (limits, sums, differentiation, and Mellin/Laplace integrals) are justified by the domination estimates recorded in Appendix~\\ref{app:technical}.\n\\end{remark}", "def:arch-operator": "\\begin{definition}[Archimedean completion operator]\n\\label{def:arch-operator}\nFor a sufficiently smooth function $f:(0,\\infty)\\to\\R$, define\n\\begin{equation}\\label{eq:arch-operator}\n(\\mathcal A f)(t):=\\frac{d}{dt}\\Bigl(t^{3/2}\\frac{d}{dt}f(t)\\Bigr),\\qquad t>0.\n\\end{equation}\n\\end{definition}", "thm:SEK": "\\begin{theorem}[Schoenberg--Edrei--Karlin]\n\\label{thm:SEK}\nLet $\\Phi\\in L^1(\\mathbb R)\\cap \\mathrm{PF}_\\infty$ be nonnegative.\nThen its bilateral Laplace transform admits the representation\n\\begin{equation}\n\\label{eq:SEK-form}\n\\mathcal B\\Phi(s)\n=\n\\frac{E(s)}{\\Psi(s)},\n\\end{equation}\nwhere:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $E(s)$ is an entire function with no zeros,\n\\item $\\Psi(s)$ is an entire function of Laguerre--Pólya type,\n\\item all zeros of $\\Psi$ are real,\n\\item the representation is unique up to multiplication of $E$ and $\\Psi$ by the\nsame nonzero constant.\n\\end{enumerate}\n\\end{theorem}", "app:technical": "\\begin{remark}[What this proof does and does not use]\nWe do \\emph{not} use any Mellin transform, any identification with $\\Xi$, nor any\nanalytic continuation beyond the natural half-planes of convergence. The only external\ninputs are the product formula \\eqref{eq:invGamma-product} and the standard\nSchoenberg--Edrei--Karlin criterion linking Laguerre--P\\'olya denominators to\n$\\mathrm{PF}_\\infty$ kernels.\n\\end{remark}\n\n\\section{The bridge problem (open)}\n\\label{sec:future-directions}\n\nThe constructions developed in this paper naturally suggest a further analytic\nproblem that we do not address here. On the one hand, the primitive finite dynamics\nconsidered above give rise, in a canonical and entirely Archimedean manner, to a\nnonnegative integrable kernel $\\Phi$ whose bilateral Laplace transform admits a\nSchoenberg--Edrei--Karlin factorization $\\mathcal B\\Phi=E/\\Psi$, with $\\Psi$ belonging\nto the Laguerre--P\\'olya class\\cite{Levin1964,PolyaSchur1914}. On the other hand, the Archimedean completion procedure\nintroduced here produces, at a distinguished self-dual normalization, a Mellin\ntransform coinciding with the classical function $\\Xi(2z)$.\n\n\\smallskip\n\\noindent\\textbf{Sanity checks for any future identification.}\nAny proposed identification should, at minimum, respect (i) the functional symmetries arising from the canonical centering and self-dual normalization, and (ii) the intrinsic total-positivity/Laguerre--P\\'olya structure on the $\\Psi$-side\\cite{Schoenberg1951,Edrei1952,karlin1968total,Levin1964}.\n\nA natural question is whether these two analytic structures can be directly related.\nMore precisely, one may ask whether the Laguerre--P\\'olya datum $\\Psi$ arising from the\ntotal-positivity side can be identified with $\\Xi(2z)$ up to multiplication by a\nzero-free entire factor, or whether a suitable rigidity principle forces such an\nidentification once both objects are placed in a common analytic framework. Any such\nidentification would amount to a comparison of the zero divisors of two functions that\noriginate from a priori unrelated constructions.\n\nWe view this identification problem as a promising direction for future work. Its\nresolution would require additional analytic input beyond the scope of the present\npaper, and we therefore leave it open.\n\n\\bibliographystyle{plain}\n\\bibliography{citations}\n\n\\appendix\n\\section{Technical analytic justifications}\n\\label{app:technical}\n\nThis appendix collects several analytic estimates and justifications that are\nused implicitly or explicitly in the main text. None of the results here\nintroduce new objects or alter the logical structure of the argument; they serve\nonly to justify standard limiting procedures, termwise operations, and boundary\nmanipulations already invoked.\n\n\\subsection{Uniform local central limit bounds}\n\\label{app:ULCLT}\n\nIn Section~\\ref{sec:completion-selfdual} we appealed to a uniform local central limit theorem (ULCLT) to pass\nfrom the discrete primitive dynamics to the scaling-limit kernel $K_L$. We record\nhere a representative bound sufficient for all uses in the paper.\n\n\\begin{lemma}[Uniform local CLT estimate]\n\\label{lem:ULCLT}\nLet $(X_t^{(N)})_{t\\ge0}$ be the rescaled primitive process on the $N$-cycle with\neffective diffusion constant $D>0$, and let $p_t^{(N)}(j)$ denote its transition\nkernel. Then for every compact interval $[t_0,t_1]\\subset(0,\\infty)$ there exists\na constant $C=C(t_0,t_1)$ such that\n\\[\n\\sup_{t\\in[t_0,t_1]}\\sup_{j\\in\\mathbb Z}\n\\Bigl|\np_t^{(N)}(j)\n-\n\\frac{1}{\\sqrt{4\\pi Dt}}\\exp\\!\\Bigl(-\\frac{j^2}{4Dt}\\Bigr)\n\\Bigr|\n\\le\n\\frac{C}{N},\n\\]\nfor all sufficiently large $N$.\n\\end{lemma}\n\n\\begin{proof}[Sketch]\nThis is a standard consequence of Fourier analysis on the discrete torus together\nwith Taylor expansion of the dispersion relation near the origin. Uniformity in\n$t\\in[t_0,t_1]$ follows from compactness. Precise proofs may be found, for example,\nin classical treatments of random walks on finite groups.\n\\end{proof}\n\n\\begin{remark}\nLemma~\\ref{lem:ULCLT} is used only to justify dominated convergence and termwise\nlimits when passing to the theta-series representation of the completed kernel.\nNo rate sharper than $O(N^{-1})$ is required.\n\\end{remark}", "thm:arch-mellin-identification": "\\begin{theorem}[Archimedean Mellin identification]\n\\label{thm:arch-mellin-identification}\nAssume the self-dual scale $L^2=4\\pi D$, so that $\\widetilde K_{\\mathrm{arch}}=\\Theta$\nby Lemma~\\ref{lem:arch-equals-Theta-new}.\nDefine, for $z\\in\\mathbb C$,\n\\begin{equation}\n\\label{eq:Farch-def-new}\nF_{\\mathrm{arch}}(z)\n:=\n\\int_0^\\infty \\widetilde K_{\\mathrm{arch}}(t)\\,t^{\\frac34+iz}\\,\\frac{dt}{t}.\n\\end{equation}\nThen for all $z\\in\\mathbb C$,\n\\begin{equation}\n\\label{eq:Farch-equals-Xi}\nF_{\\mathrm{arch}}(z)=\\Xi(2z).\n\\end{equation}\n\\end{theorem}", "eq:Phi-def": "\\begin{equation}\n\\label{eq:Phi-def}\n\\Phi(x)\n:=\ne^{x/4}\\,\\widetilde K_{\\mathrm{sym}}(e^{x})\n=\ne^{-x/4}\\,\\widetilde K(e^{x}),\n\\qquad x\\in\\mathbb R.\n\\end{equation}", "def:half-density": "\\begin{definition}[Archimedean (half--density) completion]\n\\label{def:half-density}\nDefine the \\emph{Archimedean-normalized} (``half--density'') kernel by\n\\[\n\\widetilde K_{\\mathrm{sym}}(t):=t^{-1/2}\\,\\widetilde K(t),\\qquad t>0.\n\\]\n\\end{definition}", "tab:notation": "\\label{tab:notation}\n\\end{table}\n\n\\section{Acknowledgements}\nThe author would like to thank Krishnaswami Alladi, Tiziano Valentinuzzi and Kenneth Valpey for helpful discussions.\n\n\\section{Self-dual Ar", "tab:proof-map": "\\begin{tabular}{|p{0.10\\linewidth}|p{0.52\\linewidth}|p{0.30\\linewidth}|}\n\\hline\n\\textbf{Step} & \\textbf{Input $\\to$ Output} & \\textbf{Main tool} \\\\\n\\hline\n1 &\nlocal dynamics $\\to$ theta-series kernel $\\widetilde K$ &\nlift/periodize + ULCLT$^*$\n+ dominated convergence\n\\\\ \\hline\n2 &\n$\\widetilde K \\to$ self-dual normalization and Archimedean completion $\\mathcal A$ &\nJacobi inversion + completion operator\n\\\\ \\hline\n3 &\n$\\widetilde K \\to \\Phi \\to (\\Psi,\\ \\mathcal B\\Phi=E/\\Psi)$ &\ncomplete monotonicity $\\Rightarrow \\mathrm{PF}_\\infty$ +\nSchoenberg--Edrei--Karlin\n\\\\ \\hline\n4 &\n$\\mathcal A(K_L-1)\\to \\Theta \\to \\Xi$ (anchor) and open bridge problem &\nArchimedean completion + Mellin;\nComparison problem (open)\n\\\\ \\hline\n\\end{tabular}\n\n\\caption{Proof map (Steps~1--4). $^*$ULCLT = uniform local central limit theorem.}\n\\label{tab:proof-map}\n\\end{table}\n\nThe proof relies on a small number of canonical kernels and transforms; these are summarized in Table~\\ref{tab:notation} with precise definitions and references.\n\n\\begin{table}[h!]\n\\centering\n\\setlength{\\arrayrulewidth}{0.8pt}\n\\renewcommand{\\arraystretch}{1.25}\n\n\\begin{tabular}{|p{0.22\\linewidth}|p{0.70\\linewidth}|}\n\\hline\n\\textbf{Symbol} & \\textbf{Meaning} \\\\\n\\hline\n$K_L(t)$ &\nScaling-limit trace\n(Definition~\\ref{def:scaling-limit}).\n\\\\ \\hline\n$\\widetilde K(t)$ &\nRenormalized trace kernel\n(Theorem~\\ref{thm:theta-series-form-recall}).\n\\\\ \\hline\n$\\widetilde K_{\\mathrm{sym}}(t)=t^{-1/2}\\widetilde K(t)$ &\nSymmetric half-density kernel\n(Definition~\\ref{def:half-density}).\n\\\\ \\hline\n$\\Phi(x)$ &\nLogarithmic kernel\n$\\Phi(x)=e^{x/4}\\,\\widetilde K_{\\mathrm{sym}}(e^x)$\n(equation~\\eqref{eq:Phi-def}).\n\\\\ \\hline\n$\\mathcal B\\Phi(s)$ &\nBilateral Laplace transform\n$\\displaystyle \\mathcal B\\Phi(s)=\\int_{\\R}\\Phi(x)e^{-s x}\\,dx$\n(on $\\Re(s)>-\\tfrac14$).\n\\\\ \\hline\n$\\Psi(s)$ &\nLaguerre--P\\'olya entire function with only real zeros satisfying\n$\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)}$ on a strip\n(Theorem~\\ref{thm:SEK}).\n\\\\ \\hline\n$F(z)$ &\nFourier transform of $\\Phi$,\n$\\displaystyle F(z)=\\int_{\\R}\\Phi(x)e^{izx}\\,dx$;\nboundary value $F(z)=\\mathcal B\\Phi(-iz)$ where both sides converge.\n\\\\ \\hline\n$\\mathcal A$ &\nArchimedean completion operator\n(Definition~\\ref{def:arch-operator}).\n\\\\ \\hline\n$\\widetilde K_{\\mathrm{arch}}$ &\nCompleted Archimedean kernel\n$\\widetilde K_{\\mathrm{arch}}:=\\mathcal A(K_L-1)$\n(Definition~\\ref{def:arch-theta-kernel}).\n\\\\ \\hline\n$F_{\\mathrm{arch}}(z)$ &\nMellin transform\n$\\displaystyle F_{\\mathrm{arch}}(z)=\\int_0^{\\infty}\n\\widetilde K_{\\mathrm{arch}}(t)\\,\nt^{\\frac14+iz}\\frac{dt}{t}$;\nat the self-dual scale,\n$F_{\\mathrm{arch}}(z)=\\Xi(2z)$\n(Theorem~\\ref{thm:arch-mellin-identification}).\n\\\\ \\hline\n$\\Psi_c(z)$ &\nCentered Laguerre--P\\'olya function\n$\\Psi_c(z):=\\Psi(\\tfrac14+iz)$. A central open problem is to compare $\\Psi_c$ with the Mellin-side\nfunction $\\Xi(2z)$ up to multiplication by a zero-free entire factor (Section~\\ref{sec:future-directions}).\n\\\\ \\hline\n\\end{tabular}", "def:arch-theta-kernel": "\\begin{definition}[Archimedean-completed kernel]\n\\label{def:arch-theta-kernel}\nLet $K_L(t)$ be the scaling-limit trace. Define the Archimedean-completed kernel\n\\[\n\\widetilde K_{\\mathrm{arch}}(t) :=\\bigl(\\mathcal{A}(K_L-1)\\bigr)(t),\\qquad t>0.\n\\]\n\\end{definition}", "thm:theta-series-form-recall": "\\begin{theorem}[Theta-series form of the scaling-limit trace]\n\\label{thm:theta-series-form-recall}\nFor every $L>0$ and $t>0$,\n\\begin{equation}\\label{eq:KL-theta-recall}\nK_L(t)=\\sum_{n\\in\\Z}\\exp\\!\\Bigl(-\\frac{4\\pi^2D}{L^2}\\,n^2\\,t\\Bigr).\n\\end{equation}\n\\end{theorem}", "def:scaling-limit": "\\begin{definition}[Scaling--limit completed trace kernel]\n\\label{def:scaling-limit}\nFix a macroscopic length $L>0$ and choose a scaling parameter $s\\to\\infty$ with $N=N(s)$ such that $N/s\\to L$.\nDefine the scaling--limit trace\n\\[\nK_L(t):=\\lim_{s\\to\\infty} N(s)\\,p^{\\mathrm{cyc}}_{s^2 t}(0,0),\\qquad t>0,\n\\]\nand set the (scaling--limit) completed trace kernel\n\\[\n\\widetilde K_L(t):=K_L(t)-\\frac{L}{\\sqrt{4\\pi D t}}.\n\\]\nWhen $L$ is fixed we suppress the subscript and write $\\widetilde K$.\n\\end{definition}"}, "pre_theorem_intro_text_len": 9896, "pre_theorem_intro_text": "\\label{sec:background}\n\nAcross several foundational currents in mathematics, a shared goal is to\nrecover high-level objects from minimal local principles and universal symmetry\nrequirements, rather than from distinguished presentations. In the present setting, we ask\nwhether analytic structures that are usually introduced through arithmetic data can instead\nbe forced by locality, reversibility, symmetry, and scaling alone. We use the term\n``primitive'' in this sense: the input is only finite local dynamics, while the outputs are\ndetermined by the invariances that survive in the scaling limit. More specifically, the aim herein is to show that two classical analytic objects can arise canonically\nfrom primitive input: finite, local, reversible Markov dynamics on discrete cycles, introduced\nwith no reference to primes, Euler products, or spectral/arithmetic data. \n\nThe first output is a \\emph{total-positivity object}: a logarithmic kernel $\\Phi$ whose\nbilateral Laplace transform admits a canonical Schoenberg--Edrei--Karlin factorization~\\cite{Schoenberg1951,Edrei1952,karlin1968total}.\nEquivalently, the reciprocal transform is a Laguerre--P\\'olya entire function $\\Psi$ and\ntherefore has only real zeros. The second output is an \\emph{Archimedean completion object}:\nan explicit completion operator $\\mathcal A$ that restores Mellin self-duality at a\ncanonical normalization, producing a completed kernel whose Mellin transform coincides\n(within a natural domain) with the classical Riemann $\\Xi$-function.\n\nThese constructions are logically independent but share the same primitive source, and they\nmotivate a natural identification question left open here (see\nSection~\\ref{sec:future-directions}): to what extent can the Laguerre--P\\'olya datum $\\Psi$\nbe identified with the $\\Xi$-function after the normalizations are fixed?\n\nSeveral classical approaches emphasize representations of $\\Xi$ as an integral transform. One may express $\\Xi$ through Mellin transforms of theta functions, through Fourier transforms of rapidly decaying kernels, or through trace formulas that relate zeros to spectral data\\cite{Selberg1956,BerryKeating1999}. These representations highlight a recurring theme: analytic properties of $\\Xi$ often reflect structural features of an underlying kernel. In particular, positivity and symmetry at the level of kernels tend to propagate to strong constraints on the associated transforms.\n\nHeat kernels and theta series provide a natural language\\cite{Chandrasekharan1985,SteinShakarchi2003} for these ideas. The fundamental solution of the heat equation on the real line is Gaussian, and its periodization produces classical theta functions. Poisson summation expresses a precise duality between spatial periodization and frequency localization. These mechanisms are robust and do not depend on arithmetic input. They arise whenever one studies diffusion on spaces with discrete symmetries. In analytic number theory, theta series appear as generating functions encoding spectral or lattice data, and they often serve as intermediaries between local and global descriptions.\n\nIn this work, heat kernels play a central conceptual role. Rather than appearing as analytic tools applied to $\\zeta(s)$, they arise from elementary local dynamics. The guiding idea is that long-time diffusion captures universal behavior, while finite-volume effects introduce discrete structure. When these effects are combined through exact lift and periodization identities, theta series emerge canonically. This viewpoint places theta functions in a broader probabilistic and geometric setting, where they reflect general principles rather than special constructions.\n\nA second strand of background comes from the theory of total positivity\\cite{Pinkus2010,karlin1968total,GantmacherKrein2002}. Total positivity originated in work of Schoenberg and was developed extensively by Karlin and others. It concerns kernels whose minors of all orders are nonnegative, a property that implies strong variation-diminishing and rigidity phenomena. Within this theory, a distinguished role is played by P\\'olya frequency functions of infinite order. In the form used here (Schoenberg--Edrei--Karlin), the bilateral Laplace transform of a $\\mathrm{PF}_\\infty$ function extends meromorphically and its reciprocal is a Laguerre--P\\'olya entire function. Entire functions in the Laguerre--P\\'olya class\\cite{Levin1964,PolyaSchur1914} are limits of polynomials with only real zeros, and hence have only real zeros.\n\nClosely related to this viewpoint is the Hermite--Biehler theory\\cite{Krein1947,deBranges1968} of entire functions and its modern formulation within de Branges spaces\\cite{deBranges1968}. Hermite--Biehler theory provides criteria ensuring that an entire function has all its zeros on a line, typically the real axis, based on positivity properties of associated kernels or on analytic inequalities in the upper half-plane. De Branges' work showed that these ideas can be organized into a Hilbert space framework with powerful classification theorems. Although the present paper does not rely on the full machinery of de Branges spaces\\cite{deBranges1968}, the Hermite--Biehler perspective clarifies why positivity at the kernel level leads naturally to real-zero conclusions.\n\nWe now introduce the central object of the paper and state the main results.\n\n\\subsection{Primitive dynamical model}\n\\label{sec:model}\n\nFor each integer $N\\ge 1$ we consider a continuous-time, reversible, nearest-neighbor Markov process on the discrete cycle $\\mathbb{Z}/N\\mathbb{Z}$. The dynamics are specified by a collection of strictly positive conductances $\\{a_j\\}_{j\\in\\mathbb{Z}/N\\mathbb{Z}}$, where $a_j$ is associated with the undirected edge between $j$ and $j+1$ (indices taken modulo $N$). The generator $\\mathcal{L}_N$ acts on functions $f:\\mathbb{Z}/N\\mathbb{Z}\\to\\mathbb{R}$ by\n\\begin{equation}\n\\label{eq:generator}\n(\\mathcal{L}_N f)(j)\n=\na_j\\,\\bigl(f(j+1)-f(j)\\bigr)\n\\;+\na_{j-1}\\,\\bigl(f(j-1)-f(j)\\bigr).\n\\end{equation}\n\n\\begin{remark}[Translation-invariant specialization]\n\\label{rem:translation-invariant}\nFor the purposes of this paper we restrict to the translation-invariant case\n\\(\na_j\\equiv a>0\n\\)\n(independent of $j$), so that \\eqref{eq:generator} is the usual continuous-time simple\nrandom walk on $\\mathbb{Z}/N\\mathbb{Z}$ with jump rate $a$. In this case the diffusive scaling limit is\ngoverned by the one-dimensional heat equation with diffusion constant \\(D=a\\), and the\nFourier analysis on $\\mathbb{Z}/N\\mathbb{Z}$ leading to the theta-series form is completely explicit.\nExtensions to uniformly elliptic periodic environments, or to quantitative homogenization\nregimes, can also yield a uniform local CLT and the same scaling limit; we do not pursue\nthose generalizations here.\n\\end{remark}\n\nWe write $p^{\\mathrm{cyc}}_t(j,k)$ for the associated heat kernel,\n\\(p^{\\mathrm{cyc}}_t(j,k)=(e^{t\\mathcal{L}_N}\\mathbf{1}_{\\{k\\}})(j)\\), and we denote by $D>0$ the macroscopic diffusion constant appearing in the Gaussian scaling limit established later.\n\nAs described above, the trace of the heat kernel associated with the finite dynamics captures global information about the system but contains a universal singular contribution reflecting diffusive behavior. This singular term is independent of the fine structure of the dynamics and depends only on the macroscopic scaling. To isolate the genuinely structural content, we subtract this term in a canonical way. The resulting object is the completed trace kernel.\n\n\\begin{definition}[Scaling--limit completed trace kernel]\n\\label{def:scaling-limit}\nFix a macroscopic length $L>0$ and choose a scaling parameter $s\\to\\infty$ with $N=N(s)$ such that $N/s\\to L$.\nDefine the scaling--limit trace\n\\[\nK_L(t):=\\lim_{s\\to\\infty} N(s)\\,p^{\\mathrm{cyc}}_{s^2 t}(0,0),\\qquad t>0,\n\\]\nand set the (scaling--limit) completed trace kernel\n\\[\n\\widetilde K_L(t):=K_L(t)-\\frac{L}{\\sqrt{4\\pi D t}}.\n\\]\nWhen $L$ is fixed we suppress the subscript and write $\\widetilde K$.\n\\end{definition}\n\n\\begin{definition}[Archimedean (half--density) completion]\n\\label{def:half-density}\nDefine the \\emph{Archimedean-normalized} (``half--density'') kernel by\n\\[\n\\widetilde K_{\\mathrm{sym}}(t):=t^{-1/2}\\,\\widetilde K(t),\\qquad t>0.\n\\]\n\\end{definition}\n\nThe existence of the scaling limit in Definition~\\ref{def:scaling-limit}, together with the basic domination estimates needed to justify termwise limits, follows from the standing assumptions recorded below; for convenience we collect the relevant analytic justifications in Appendix~\\ref{app:technical}.\n\n\\begin{remark}[Standing assumptions and existence of the scaling limit]\n\\label{rem:standing-assumptions}\nWe work throughout in a regime where the scaling-limit trace $K_L(t)$ exists for every $L>0$ and $t>0$, with convergence locally uniform in $t$ on compact subsets of $(0,\\infty)$. In particular, under the translation-invariant specialization of Remark~\\ref{rem:translation-invariant} this limit can be obtained explicitly by Fourier analysis on $\\mathbb{Z}/N\\mathbb{Z}$. Under these assumptions the limit admits the explicit theta-series form stated in Theorem~\\ref{thm:theta-series-form-recall}. Analytic interchanges (limits, sums, differentiation, and Mellin/Laplace integrals) are justified by the domination estimates recorded in Appendix~\\ref{app:technical}.\n\\end{remark}\n\n\\medskip\n\n\\noindent\nIn addition to the total-positivity output (producing a canonical Laguerre--P\\'olya function) and\nthe Archimedean completion output (producing the classical theta/Mellin representation of $\\Xi$),\nwe isolate a single remaining analytic identification problem: relate the Laguerre--P\\'olya datum\n$\\Psi$ to the Mellin-side function $\\Xi(2\\cdot)$ up to multiplication by a zero-free entire factor.\nWe treat this as an open ``bridge'' problem and discuss it in Section~\\ref{sec:future-directions}.", "context": "The first output is a \\emph{total-positivity object}: a logarithmic kernel $\\Phi$ whose\nbilateral Laplace transform admits a canonical Schoenberg--Edrei--Karlin factorization~\\cite{Schoenberg1951,Edrei1952,karlin1968total}.\nEquivalently, the reciprocal transform is a Laguerre--P\\'olya entire function $\\Psi$ and\ntherefore has only real zeros. The second output is an \\emph{Archimedean completion object}:\nan explicit completion operator $\\mathcal A$ that restores Mellin self-duality at a\ncanonical normalization, producing a completed kernel whose Mellin transform coincides\n(within a natural domain) with the classical Riemann $\\Xi$-function.\n\nA second strand of background comes from the theory of total positivity\\cite{Pinkus2010,karlin1968total,GantmacherKrein2002}. Total positivity originated in work of Schoenberg and was developed extensively by Karlin and others. It concerns kernels whose minors of all orders are nonnegative, a property that implies strong variation-diminishing and rigidity phenomena. Within this theory, a distinguished role is played by P\\'olya frequency functions of infinite order. In the form used here (Schoenberg--Edrei--Karlin), the bilateral Laplace transform of a $\\mathrm{PF}_\\infty$ function extends meromorphically and its reciprocal is a Laguerre--P\\'olya entire function. Entire functions in the Laguerre--P\\'olya class\\cite{Levin1964,PolyaSchur1914} are limits of polynomials with only real zeros, and hence have only real zeros.\n\n\\begin{definition}[Scaling--limit completed trace kernel]\n\\label{def:scaling-limit}\nFix a macroscopic length $L>0$ and choose a scaling parameter $s\\to\\infty$ with $N=N(s)$ such that $N/s\\to L$.\nDefine the scaling--limit trace\n\\[\nK_L(t):=\\lim_{s\\to\\infty} N(s)\\,p^{\\mathrm{cyc}}_{s^2 t}(0,0),\\qquad t>0,\n\\]\nand set the (scaling--limit) completed trace kernel\n\\[\n\\widetilde K_L(t):=K_L(t)-\\frac{L}{\\sqrt{4\\pi D t}}.\n\\]\nWhen $L$ is fixed we suppress the subscript and write $\\widetilde K$.\n\\end{definition}\n\n\\begin{remark}[Standing assumptions and existence of the scaling limit]\n\\label{rem:standing-assumptions}\nWe work throughout in a regime where the scaling-limit trace $K_L(t)$ exists for every $L>0$ and $t>0$, with convergence locally uniform in $t$ on compact subsets of $(0,\\infty)$. In particular, under the translation-invariant specialization of Remark~\\ref{rem:translation-invariant} this limit can be obtained explicitly by Fourier analysis on $\\mathbb{Z}/N\\mathbb{Z}$. Under these assumptions the limit admits the explicit theta-series form stated in Theorem~\\ref{thm:theta-series-form-recall}. Analytic interchanges (limits, sums, differentiation, and Mellin/Laplace integrals) are justified by the domination estimates recorded in Appendix~\\ref{app:technical}.\n\\end{remark}\n\n\\medskip\n\n\\noindent\nIn addition to the total-positivity output (producing a canonical Laguerre--P\\'olya function) and\nthe Archimedean completion output (producing the classical theta/Mellin representation of $\\Xi$),\nwe isolate a single remaining analytic identification problem: relate the Laguerre--P\\'olya datum\n$\\Psi$ to the Mellin-side function $\\Xi(2\\cdot)$ up to multiplication by a zero-free entire factor.\nWe treat this as an open ``bridge'' problem and discuss it in Section~\\ref{sec:future-directions}.", "full_context": "The first output is a \\emph{total-positivity object}: a logarithmic kernel $\\Phi$ whose\nbilateral Laplace transform admits a canonical Schoenberg--Edrei--Karlin factorization~\\cite{Schoenberg1951,Edrei1952,karlin1968total}.\nEquivalently, the reciprocal transform is a Laguerre--P\\'olya entire function $\\Psi$ and\ntherefore has only real zeros. The second output is an \\emph{Archimedean completion object}:\nan explicit completion operator $\\mathcal A$ that restores Mellin self-duality at a\ncanonical normalization, producing a completed kernel whose Mellin transform coincides\n(within a natural domain) with the classical Riemann $\\Xi$-function.\n\nA second strand of background comes from the theory of total positivity\\cite{Pinkus2010,karlin1968total,GantmacherKrein2002}. Total positivity originated in work of Schoenberg and was developed extensively by Karlin and others. It concerns kernels whose minors of all orders are nonnegative, a property that implies strong variation-diminishing and rigidity phenomena. Within this theory, a distinguished role is played by P\\'olya frequency functions of infinite order. In the form used here (Schoenberg--Edrei--Karlin), the bilateral Laplace transform of a $\\mathrm{PF}_\\infty$ function extends meromorphically and its reciprocal is a Laguerre--P\\'olya entire function. Entire functions in the Laguerre--P\\'olya class\\cite{Levin1964,PolyaSchur1914} are limits of polynomials with only real zeros, and hence have only real zeros.\n\n\\begin{definition}[Scaling--limit completed trace kernel]\n\\label{def:scaling-limit}\nFix a macroscopic length $L>0$ and choose a scaling parameter $s\\to\\infty$ with $N=N(s)$ such that $N/s\\to L$.\nDefine the scaling--limit trace\n\\[\nK_L(t):=\\lim_{s\\to\\infty} N(s)\\,p^{\\mathrm{cyc}}_{s^2 t}(0,0),\\qquad t>0,\n\\]\nand set the (scaling--limit) completed trace kernel\n\\[\n\\widetilde K_L(t):=K_L(t)-\\frac{L}{\\sqrt{4\\pi D t}}.\n\\]\nWhen $L$ is fixed we suppress the subscript and write $\\widetilde K$.\n\\end{definition}\n\n\\begin{remark}[Standing assumptions and existence of the scaling limit]\n\\label{rem:standing-assumptions}\nWe work throughout in a regime where the scaling-limit trace $K_L(t)$ exists for every $L>0$ and $t>0$, with convergence locally uniform in $t$ on compact subsets of $(0,\\infty)$. In particular, under the translation-invariant specialization of Remark~\\ref{rem:translation-invariant} this limit can be obtained explicitly by Fourier analysis on $\\mathbb{Z}/N\\mathbb{Z}$. Under these assumptions the limit admits the explicit theta-series form stated in Theorem~\\ref{thm:theta-series-form-recall}. Analytic interchanges (limits, sums, differentiation, and Mellin/Laplace integrals) are justified by the domination estimates recorded in Appendix~\\ref{app:technical}.\n\\end{remark}\n\n\\medskip\n\n\\noindent\nIn addition to the total-positivity output (producing a canonical Laguerre--P\\'olya function) and\nthe Archimedean completion output (producing the classical theta/Mellin representation of $\\Xi$),\nwe isolate a single remaining analytic identification problem: relate the Laguerre--P\\'olya datum\n$\\Psi$ to the Mellin-side function $\\Xi(2\\cdot)$ up to multiplication by a zero-free entire factor.\nWe treat this as an open ``bridge'' problem and discuss it in Section~\\ref{sec:future-directions}.\n\n\\begin{abstract}\nStarting from finite, local, reversible Markov dynamics on discrete cycles, we construct a\nscaling-limit renormalized trace kernel admitting an exact theta-series representation.\nThe construction is entirely Archimedean and uses no Euler products, primes, or\narithmetic spectral input.\nFrom this limit we define a logarithmic kernel $\\Phi$ and prove that it lies in the\nP\\'olya frequency class $\\mathrm{PF}_\\infty$, yielding via the Schoenberg--Edrei--Karlin\nclassification a canonical Laguerre--P\\'olya function $\\Psi$.\nIndependently, we introduce an Archimedean completion operator and show that, at a\nself-dual normalization, the completed kernel coincides with the classical theta kernel,\nwhose Mellin transform is the Riemann $\\Xi$-function.\nWe isolate a single remaining analytic problem relating $\\Psi$ to $\\Xi(2\\cdot)$.\n\\end{abstract}\n\n\\noindent\nIn addition to the total-positivity output (producing a canonical Laguerre--P\\'olya function) and\nthe Archimedean completion output (producing the classical theta/Mellin representation of $\\Xi$),\nwe isolate a single remaining analytic identification problem: relate the Laguerre--P\\'olya datum\n$\\Psi$ to the Mellin-side function $\\Xi(2\\cdot)$ up to multiplication by a zero-free entire factor.\nWe treat this as an open ``bridge'' problem and discuss it in Section~\\ref{sec:future-directions}.\n\n\\begin{theorem}[Archimedean Mellin identification]\n\\label{thm:arch-overview}\nAt the self-dual scale fixed in Section~\\ref{sec:completion-selfdual}, the Archimedean-completed\nkernel $\\widetilde K_{\\mathrm{arch}}:=\\mathcal A(K_L-1)$ has Mellin transform\n\\[\nF_{\\mathrm{arch}}(z)=\\Xi(2z)\n\\]\nas in Theorem~\\ref{thm:arch-mellin-identification}.\n\\end{theorem}\n\n\\begin{tabular}{|p{0.22\\linewidth}|p{0.70\\linewidth}|}\n\\hline\n\\textbf{Symbol} & \\textbf{Meaning} \\\\\n\\hline\n$K_L(t)$ &\nScaling-limit trace\n(Definition~\\ref{def:scaling-limit}).\n\\\\ \\hline\n$\\widetilde K(t)$ &\nRenormalized trace kernel\n(Theorem~\\ref{thm:theta-series-form-recall}).\n\\\\ \\hline\n$\\widetilde K_{\\mathrm{sym}}(t)=t^{-1/2}\\widetilde K(t)$ &\nSymmetric half-density kernel\n(Definition~\\ref{def:half-density}).\n\\\\ \\hline\n$\\Phi(x)$ &\nLogarithmic kernel\n$\\Phi(x)=e^{x/4}\\,\\widetilde K_{\\mathrm{sym}}(e^x)$\n(equation~\\eqref{eq:Phi-def}).\n\\\\ \\hline\n$\\mathcal B\\Phi(s)$ &\nBilateral Laplace transform\n$\\displaystyle \\mathcal B\\Phi(s)=\\int_{\\R}\\Phi(x)e^{-s x}\\,dx$\n(on $\\Re(s)>-\\tfrac14$).\n\\\\ \\hline\n$\\Psi(s)$ &\nLaguerre--P\\'olya entire function with only real zeros satisfying\n$\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)}$ on a strip\n(Theorem~\\ref{thm:SEK}).\n\\\\ \\hline\n$F(z)$ &\nFourier transform of $\\Phi$,\n$\\displaystyle F(z)=\\int_{\\R}\\Phi(x)e^{izx}\\,dx$;\nboundary value $F(z)=\\mathcal B\\Phi(-iz)$ where both sides converge.\n\\\\ \\hline\n$\\mathcal A$ &\nArchimedean completion operator\n(Definition~\\ref{def:arch-operator}).\n\\\\ \\hline\n$\\widetilde K_{\\mathrm{arch}}$ &\nCompleted Archimedean kernel\n$\\widetilde K_{\\mathrm{arch}}:=\\mathcal A(K_L-1)$\n(Definition~\\ref{def:arch-theta-kernel}).\n\\\\ \\hline\n$F_{\\mathrm{arch}}(z)$ &\nMellin transform\n$\\displaystyle F_{\\mathrm{arch}}(z)=\\int_0^{\\infty}\n\\widetilde K_{\\mathrm{arch}}(t)\\,\nt^{\\frac14+iz}\\frac{dt}{t}$;\nat the self-dual scale,\n$F_{\\mathrm{arch}}(z)=\\Xi(2z)$\n(Theorem~\\ref{thm:arch-mellin-identification}).\n\\\\ \\hline\n$\\Psi_c(z)$ &\nCentered Laguerre--P\\'olya function\n$\\Psi_c(z):=\\Psi(\\tfrac14+iz)$. A central open problem is to compare $\\Psi_c$ with the Mellin-side\nfunction $\\Xi(2z)$ up to multiplication by a zero-free entire factor (Section~\\ref{sec:future-directions}).\n\\\\ \\hline\n\\end{tabular}\n\n\\begin{proposition}[Reflection law for the Step-2 transform]\n\\label{prop:bilaplace-reflection}\nAssume \\eqref{eq:Ksym-selfdual}. Then for every $s\\in\\mathbb C$ with\n\\[\n-\\tfrac14<\\Re(s)<\\tfrac34\n\\]\n(one precisely needs absolute convergence of both sides), one has the exact identity\n\\begin{equation}\n\\label{eq:bilaplace-reflection}\n\\mathcal B\\Phi(s)=\\mathcal B\\Phi\\!\\left(\\tfrac12-s\\right).\n\\end{equation}\nConsequently, for all $t\\in\\mathbb R$ one has\n\\begin{equation}\n\\label{eq:bilaplace-real-on-centerline}\n\\mathcal B\\Phi\\!\\left(\\tfrac14+it\\right)\\in\\mathbb R.\n\\end{equation}\n\\end{proposition}\n\n\\begin{theorem}[Schoenberg--Edrei--Karlin]\n\\label{thm:SEK}\nLet $\\Phi\\in L^1(\\mathbb R)\\cap \\mathrm{PF}_\\infty$ be nonnegative.\nThen its bilateral Laplace transform admits the representation\n\\begin{equation}\n\\label{eq:SEK-form}\n\\mathcal B\\Phi(s)\n=\n\\frac{E(s)}{\\Psi(s)},\n\\end{equation}\nwhere:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $E(s)$ is an entire function with no zeros,\n\\item $\\Psi(s)$ is an entire function of Laguerre--Pólya type,\n\\item all zeros of $\\Psi$ are real,\n\\item the representation is unique up to multiplication of $E$ and $\\Psi$ by the\nsame nonzero constant.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{theorem}[Main theorem (assembled)]\n\\label{thm:main-assembled}\nAssume the strong regime hypotheses of Section~\\ref{sec:completion-selfdual}, so that the scaling-limit kernel\n$K_L$ exists and the self-dual normalization can be fixed. Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item (\\emph{Unconditional structural output.})\nThere exists a nonnegative kernel $\\Phi\\in L^1(\\mathbb R)$ satisfying:\n\\begin{itemize}\n\\item $\\Phi\\in\\mathrm{PF}_\\infty$;\n\\item its bilateral Laplace transform $\\mathcal B\\Phi$ satisfies the reflection law\n$\\mathcal B\\Phi(s)=\\mathcal B\\Phi(\\tfrac12-s)$ on the common domain of absolute convergence;\n\\item $\\mathcal B\\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation\n$\\mathcal B\\Phi(s)=E(s)/\\Psi(s)$ with $\\Psi$ Laguerre--P\\'olya and hence having only real zeros.\n\\end{itemize}\n\\item (\\emph{Unconditional Archimedean Mellin identification.})\nThe Archimedean-completed kernel $\\widetilde K_{\\mathrm{arch}}$ has Mellin transform\n$F_{\\mathrm{arch}}(z)=\\Xi(2z)$ as in Theorem~\\ref{thm:arch-mellin-identification}.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nWe compute the bilateral Laplace transform on the half-plane $\\Re(s)>0$:\n\\[\n\\mathcal B\\phi_u(s)\n=\n\\int_{\\mathbb R} e^{-u e^{-x}}e^{-s x}\\,dx.\n\\]\nWith the change of variables $y=e^{-x}$ (so $y\\in(0,\\infty)$ and $dx=-dy/y$), this becomes\n\\[\n\\mathcal B\\phi_u(s)\n=\n\\int_0^\\infty e^{-u y} y^{s-1}\\,dy\n=\nu^{-s}\\Gamma(s),\n\\qquad \\Re(s)>0.\n\\]\nNow recall the classical Weierstrass product for the reciprocal Gamma function:\n\\begin{equation}\n\\label{eq:invGamma-product}\n\\frac{1}{\\Gamma(s)}\n=\ns\\,e^{\\gamma s}\\prod_{n=1}^\\infty\\Bigl(1+\\frac{s}{n}\\Bigr)e^{-s/n},\n\\end{equation}\nwhere $\\gamma$ is Euler's constant. The right-hand side is an entire function whose\nzeros are exactly the nonpositive integers, all real and simple, and whose product\nrepresentation is of Laguerre--P\\'olya type. In particular,\n\\[\n\\Psi(s):=\\frac{1}{\\Gamma(s)}\n\\quad\\text{is Laguerre--P\\'olya,}\n\\qquad\nE(s):=u^{-s}=e^{-s\\log u}\n\\quad\\text{is entire and zero-free.}\n\\]\nThus on $\\Re(s)>0$ we have the representation\n\\[\n\\mathcal B\\phi_u(s)=\\frac{E(s)}{\\Psi(s)}.\n\\]", "post_theorem_intro_text_len": 4911, "post_theorem_intro_text": "\\begin{theorem}[Archimedean Mellin identification]\n\\label{thm:arch-overview}\nAt the self-dual scale fixed in Section~\\ref{sec:completion-selfdual}, the Archimedean-completed\nkernel $\\widetilde K_{\\mathrm{arch}}:=\\mathcal A(K_L-1)$ has Mellin transform\n\\[\nF_{\\mathrm{arch}}(z)=\\Xi(2z)\n\\]\nas in Theorem~\\ref{thm:arch-mellin-identification}.\n\\end{theorem}\n\nThe remainder of the paper constructs the kernel $\\Phi$, proves Theorem~\\ref{thm:main-assembled}, and isolates each logical seam explicitly.\n\n\\medskip\n\nThe paper is organized as follows. Section~\\ref{sec:completion-selfdual} fixes the self-dual Archimedean normalization and records the resulting theta-series form of the scaling-limit trace. Section~\\ref{sec:log-reflection} constructs the logarithmic kernel $\\Phi$ and establishes the exact reflection symmetry for its bilateral Laplace transform. Section~\\ref{sec:total-positivity} proves total positivity and derives the Schoenberg--Edrei--Karlin factorization $\\mathcal B\\Phi=E/\\Psi$. Section~\\ref{sec:arch-mellin-xi} identifies the Archimedean-completed Mellin transform with $\\Xi(2\\cdot)$ and records the Archimedean Mellin identification with $\\Xi(2\\cdot)$, while isolating the remaining identification problem (Section~\\ref{sec:future-directions}). Section~\\ref{sec:completion-ledger} assembles the argument and isolates the remaining seam. Section~\\ref{sec:tp-details} provides a detailed proof of total positivity. Appendix~\\ref{app:technical} collects analytic justifications for termwise operations and boundary terms.\n\n\\subsection{Map of the proof}\\label{subsec:map-proof}\nThe argument is organized into four steps, each producing a new object from the previous one.\nFor quick reference we summarize the pipeline in Table~\\ref{tab:proof-map}.\n\n\\begin{table}[h!]\n\\centering\n\\setlength{\\arrayrulewidth}{0.8pt}\n\\renewcommand{\\arraystretch}{1.25}\n\n\\begin{tabular}{|p{0.10\\linewidth}|p{0.52\\linewidth}|p{0.30\\linewidth}|}\n\\hline\n\\textbf{Step} & \\textbf{Input $\\to$ Output} & \\textbf{Main tool} \\\\\n\\hline\n1 &\nlocal dynamics $\\to$ theta-series kernel $\\widetilde K$ &\nlift/periodize + ULCLT$^*$\n+ dominated convergence\n\\\\ \\hline\n2 &\n$\\widetilde K \\to$ self-dual normalization and Archimedean completion $\\mathcal A$ &\nJacobi inversion + completion operator\n\\\\ \\hline\n3 &\n$\\widetilde K \\to \\Phi \\to (\\Psi,\\ \\mathcal B\\Phi=E/\\Psi)$ &\ncomplete monotonicity $\\Rightarrow \\mathrm{PF}_\\infty$ +\nSchoenberg--Edrei--Karlin\n\\\\ \\hline\n4 &\n$\\mathcal A(K_L-1)\\to \\Theta \\to \\Xi$ (anchor) and open bridge problem &\nArchimedean completion + Mellin;\nComparison problem (open)\n\\\\ \\hline\n\\end{tabular}\n\n\\caption{Proof map (Steps~1--4). $^*$ULCLT = uniform local central limit theorem.}\n\\label{tab:proof-map}\n\\end{table}\n\nThe proof relies on a small number of canonical kernels and transforms; these are summarized in Table~\\ref{tab:notation} with precise definitions and references.\n\n\\begin{table}[h!]\n\\centering\n\\setlength{\\arrayrulewidth}{0.8pt}\n\\renewcommand{\\arraystretch}{1.25}\n\n\\begin{tabular}{|p{0.22\\linewidth}|p{0.70\\linewidth}|}\n\\hline\n\\textbf{Symbol} & \\textbf{Meaning} \\\\\n\\hline\n$K_L(t)$ &\nScaling-limit trace\n(Definition~\\ref{def:scaling-limit}).\n\\\\ \\hline\n$\\widetilde K(t)$ &\nRenormalized trace kernel\n(Theorem~\\ref{thm:theta-series-form-recall}).\n\\\\ \\hline\n$\\widetilde K_{\\mathrm{sym}}(t)=t^{-1/2}\\widetilde K(t)$ &\nSymmetric half-density kernel\n(Definition~\\ref{def:half-density}).\n\\\\ \\hline\n$\\Phi(x)$ &\nLogarithmic kernel\n$\\Phi(x)=e^{x/4}\\,\\widetilde K_{\\mathrm{sym}}(e^x)$\n(equation~\\eqref{eq:Phi-def}).\n\\\\ \\hline\n$\\mathcal B\\Phi(s)$ &\nBilateral Laplace transform\n$\\displaystyle \\mathcal B\\Phi(s)=\\int_{\\mathbb{R}}\\Phi(x)e^{-s x}\\,dx$\n(on $\\Re(s)>-\\tfrac14$).\n\\\\ \\hline\n$\\Psi(s)$ &\nLaguerre--P\\'olya entire function with only real zeros satisfying\n$\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)}$ on a strip\n(Theorem~\\ref{thm:SEK}).\n\\\\ \\hline\n$F(z)$ &\nFourier transform of $\\Phi$,\n$\\displaystyle F(z)=\\int_{\\mathbb{R}}\\Phi(x)e^{izx}\\,dx$;\nboundary value $F(z)=\\mathcal B\\Phi(-iz)$ where both sides converge.\n\\\\ \\hline\n$\\mathcal A$ &\nArchimedean completion operator\n(Definition~\\ref{def:arch-operator}).\n\\\\ \\hline\n$\\widetilde K_{\\mathrm{arch}}$ &\nCompleted Archimedean kernel\n$\\widetilde K_{\\mathrm{arch}}:=\\mathcal A(K_L-1)$\n(Definition~\\ref{def:arch-theta-kernel}).\n\\\\ \\hline\n$F_{\\mathrm{arch}}(z)$ &\nMellin transform\n$\\displaystyle F_{\\mathrm{arch}}(z)=\\int_0^{\\infty}\n\\widetilde K_{\\mathrm{arch}}(t)\\,\nt^{\\frac14+iz}\\frac{dt}{t}$;\nat the self-dual scale,\n$F_{\\mathrm{arch}}(z)=\\Xi(2z)$\n(Theorem~\\ref{thm:arch-mellin-identification}).\n\\\\ \\hline\n$\\Psi_c(z)$ &\nCentered Laguerre--P\\'olya function\n$\\Psi_c(z):=\\Psi(\\tfrac14+iz)$. A central open problem is to compare $\\Psi_c$ with the Mellin-side\nfunction $\\Xi(2z)$ up to multiplication by a zero-free entire factor (Section~\\ref{sec:future-directions}).\n\\\\ \\hline\n\\end{tabular}\n\n\\caption{Notation used throughout the proof.}\n\\label{tab:notation}\n\\end{table}", "sketch": "The introduction gives a proof map rather than a detailed argument. It says the paper will “constructs the kernel $\\Phi$” and establishes the three outputs of Theorem~\\ref{thm:structural} by isolating “each logical seam explicitly,” with the following pipeline:\n\n1. **Local dynamics $\\to$ theta-series kernel $\\widetilde K$** using “lift/periodize + ULCLT$^*$ + dominated convergence.”\n\n2. **$\\widetilde K \\to$ self-dual normalization and Archimedean completion $\\mathcal A$** via “Jacobi inversion + completion operator.”\n\n3. **$\\widetilde K \\to \\Phi \\to (\\Psi,\\ \\mathcal B\\Phi=E/\\Psi)$**: construct the “logarithmic kernel”\n\\[\\Phi(x)=e^{x/4}\\,\\widetilde K_{\\mathrm{sym}}(e^x),\\]\nthen prove “complete monotonicity $\\Rightarrow \\mathrm{PF}_\\infty$ + Schoenberg--Edrei--Karlin,” yielding total positivity and the “Schoenberg--Edrei--Karlin factorization $\\mathcal B\\Phi=E/\\Psi$,” with $\\Psi$ in Laguerre--P\\'olya.\n\nAdditionally, it states that Section~\\ref{sec:log-reflection} “establishes the exact reflection symmetry for its bilateral Laplace transform,” i.e. the reflection law for $\\mathcal B\\Phi$.\n\n4. **Archimedean Mellin anchor**: “$\\mathcal A(K_L-1)\\to \\Theta \\to \\Xi$ (anchor) and open bridge problem” via “Archimedean completion + Mellin; Comparison problem (open).”", "expanded_sketch": "The introduction gives a proof map rather than a detailed argument. It says the paper will “constructs the kernel $\\Phi$” and establishes the three outputs of the main theorem by isolating “each logical seam explicitly,” with the following pipeline:\n\n1. **Local dynamics $\\to$ theta-series kernel $\\widetilde K$** using “lift/periodize + ULCLT$^*$ + dominated convergence.”\n\n2. **$\\widetilde K \\to$ self-dual normalization and Archimedean completion $\\mathcal A$** via “Jacobi inversion + completion operator.”\n\n3. **$\\widetilde K \\to \\Phi \\to (\\Psi,\\ \\mathcal B\\Phi=E/\\Psi)$**: construct the “logarithmic kernel”\n\\[\\Phi(x)=e^{x/4}\\,\\widetilde K_{\\mathrm{sym}}(e^x),\\]\nthen prove “complete monotonicity $\\Rightarrow \\mathrm{PF}_\\infty$ + Schoenberg--Edrei--Karlin,” yielding total positivity and the “Schoenberg--Edrei--Karlin factorization $\\mathcal B\\Phi=E/\\Psi$,” with $\\Psi$ in Laguerre--P\\'olya.\n\nAdditionally, it states that later the paper “establishes the exact reflection symmetry for its bilateral Laplace transform,” i.e. the reflection law for $\\mathcal B\\Phi$.\n\n4. **Archimedean Mellin anchor**: “$\\mathcal A(K_L-1)\\to \\Theta \\to \\Xi$ (anchor) and open bridge problem” via “Archimedean completion + Mellin; Comparison problem (open).”", "expanded_theorem": "[Structural output from primitive cycles]\n\\label{thm:structural}\n\\begin{remark}[Standing assumptions and existence of the scaling limit]\n\\label{rem:standing-assumptions}\nWe work throughout in a regime where the scaling-limit trace $K_L(t)$ exists for every $L>0$ and $t>0$, with convergence locally uniform in $t$ on compact subsets of $(0,\\infty)$. In particular, under the translation-invariant specialization of Remark~\\ref{rem:translation-invariant} this limit can be obtained explicitly by Fourier analysis on $\\Z/N\\Z$. Under these assumptions the limit admits the explicit theta-series form stated in Theorem~\\ref{thm:theta-series-form-recall}. Analytic interchanges (limits, sums, differentiation, and Mellin/Laplace integrals) are justified by the domination estimates recorded in Appendix~\\ref{app:technical}.\n\\end{remark}\nAssume the standing assumptions above, so that the scaling-limit kernel $K_L$ exists.\nThen there exists a nonnegative kernel $\\Phi\\in L^1(\\mathbb{R})$ such that:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\Phi\\in\\mathrm{PF}_\\infty$;\n\\item its bilateral Laplace transform $\\mathcal B\\Phi$ satisfies the reflection law\n$\\mathcal B\\Phi(s)=\\mathcal B\\Phi(\\tfrac12-s)$ on the common domain of absolute convergence;\n\\item $\\mathcal B\\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation\n\\[\n\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)},\n\\]\nwhere $\\Psi$ is Laguerre--P\\'olya and hence has only real zeros.\n\\end{enumerate}", "theorem_type": ["Existence", "Existential–Universal"], "mcq": {"question": "Fix a macroscopic length $L>0$ and define the scaling-limit trace by\n\\[\nK_L(t):=\\lim_{s\\to\\infty} N(s)\\,p^{\\mathrm{cyc}}_{s^2 t}(0,0),\\qquad t>0,\n\\]\nwhere $N=N(s)$ satisfies $N/s\\to L$. Assume that for every $L>0$ and $t>0$ this limit exists, with convergence locally uniform in $t$ on compact subsets of $(0,\\infty)$, so that the scaling-limit kernel $K_L$ exists. Let $\\mathcal B\\Phi(s):=\\int_{\\mathbb R} e^{-sx}\\Phi(x)\\,dx$ denote the bilateral Laplace transform on its domain of absolute convergence, and let $\\mathrm{PF}_\\infty$ denote the Pólya frequency class of infinite order. Which statement holds under these assumptions?", "correct_choice": {"label": "A", "text": "There exists a nonnegative kernel $\\Phi\\in L^1(\\mathbb R)$ such that $\\Phi\\in\\mathrm{PF}_\\infty$, its bilateral Laplace transform satisfies the reflection law $\\mathcal B\\Phi(s)=\\mathcal B\\Phi(\\tfrac12-s)$ on the common domain of absolute convergence, and $\\mathcal B\\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation\n\\[\n\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)},\n\\]\nwhere $\\Psi$ is a Laguerre--Pólya entire function and hence has only real zeros."}, "choices": [{"label": "B", "text": "There exists a nonnegative kernel $\\Phi\\in L^1(\\mathbb R)$ such that $\\Phi\\in\\mathrm{PF}_\\infty$, its bilateral Laplace transform satisfies the reflection law $\\mathcal B\\Phi(s)=\\mathcal B\\Phi(-s)$ on the common domain of absolute convergence, and $\\mathcal B\\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation\n\\[\n\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)},\n\\]\nwhere $\\Psi$ is a Laguerre--Pólya entire function and hence has only real zeros."}, {"label": "C", "text": "There exists a nonnegative kernel $\\Phi\\in L^1(\\mathbb R)$ such that $\\Phi\\in\\mathrm{PF}_\\infty$ and $\\mathcal B\\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation\n\\[\n\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)},\n\\]\nwhere $\\Psi$ is a Laguerre--Pólya entire function and hence has only real zeros."}, {"label": "D", "text": "There exists a nonnegative kernel $\\Phi\\in L^1(\\mathbb R)$ such that $\\Phi\\in\\mathrm{PF}_\\infty$, its bilateral Laplace transform satisfies the reflection law $\\mathcal B\\Phi(s)=\\mathcal B\\Phi(\\tfrac12-s)$ for all $s\\in\\mathbb C$, and $\\mathcal B\\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation\n\\[\n\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)},\n\\]\nwhere $\\Psi$ is a Laguerre--Pólya entire function and hence has only real zeros."}, {"label": "E", "text": "There exists a nonnegative kernel $\\Phi\\in L^1(\\mathbb R)$ such that its bilateral Laplace transform satisfies the reflection law $\\mathcal B\\Phi(s)=\\mathcal B\\Phi(\\tfrac12-s)$ on the common domain of absolute convergence, and $\\mathcal B\\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation\n\\[\n\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)},\n\\]\nwhere $\\Psi$ is a Laguerre--Pólya entire function and hence has only real zeros, with $\\Phi$ therefore necessarily in \\mathrm{PF}_\\infty."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "reflection-center-at-1/2", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped reflection law clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "domain-of-absolute-convergence restriction", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "direction of SEK implication versus PF_infty hypothesis", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the correct option. It introduces terminology used in the answers, but the key distinctions among the choices—existential vs. universal quantification, whether PF_infty is included, and the precise domain of the reflection law—are not leaked."}, "TAS": {"score": 0, "justification": "This is very close to a theorem-statement recognition item: the stem gives the hypotheses and asks which existence statement holds, with the correct choice essentially reproducing the intended conclusion. It tests recall of the exact theorem formulation more than independent conclusion-drawing."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the student must distinguish subtle logical variations: stronger regularity, weaker omission of a clause, and a universal quantifier trap. However, the task is still largely to identify the exact theorem conclusion rather than generate a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are plausible and well targeted: one is too strong (entire extension/all s), one is weaker but incomplete, one incorrectly strengthens existential to universal, and one omits the PF_infty conclusion. These reflect realistic mathematical failure modes."}, "total_score": 5, "overall_assessment": "Technically well-constructed with strong distractors and little answer leakage, but it is mainly a theorem-recall item rather than a genuinely generative reasoning question."}}
{"id": "2602.01248v1", "paper_link": "http://arxiv.org/abs/2602.01248v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "[Structural output from primitive cycles]\n\\label{thm:structural}\nAssume the standing assumptions of Remark~\\ref{rem:standing-assumptions},\nso that the scaling-limit kernel $K_L$ exists.\nThen there exists a nonnegative kernel $\\Phi\\in L^1(\\mathbb{R})$ such that:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\Phi\\in\\mathrm{PF}_\\infty$;\n\\item its bilateral Laplace transform $\\mathcal B\\Phi$ satisfies the reflection law\n$\\mathcal B\\Phi(s)=\\mathcal B\\Phi(\\tfrac12-s)$ on the common domain of absolute convergence;\n\\item $\\mathcal B\\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation\n\\[\n\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)},\n\\]\nwhere $\\Psi$ is Laguerre--P\\'olya and hence has only real zeros.\n\\end{enumerate}", "start_pos": 11842, "end_pos": 12581, "label": "thm:structural"}, "ref_dict": {"eq:generator": "\\begin{equation}\n\\label{eq:generator}\n(\\mathcal{L}_N f)(j)\n=\na_j\\,\\bigl(f(j+1)-f(j)\\bigr)\n\\;+\na_{j-1}\\,\\bigl(f(j-1)-f(j)\\bigr).\n\\end{equation}", "rem:translation-invariant": "\\begin{remark}[Translation-invariant specialization]\n\\label{rem:translation-invariant}\nFor the purposes of this paper we restrict to the translation-invariant case\n\\(\na_j\\equiv a>0\n\\)\n(independent of $j$), so that \\eqref{eq:generator} is the usual continuous-time simple\nrandom walk on $\\Z/N\\Z$ with jump rate $a$. In this case the diffusive scaling limit is\ngoverned by the one-dimensional heat equation with diffusion constant \\(D=a\\), and the\nFourier analysis on $\\Z/N\\Z$ leading to the theta-series form is completely explicit.\nExtensions to uniformly elliptic periodic environments, or to quantitative homogenization\nregimes, can also yield a uniform local CLT and the same scaling limit; we do not pursue\nthose generalizations here.\n\\end{remark}", "thm:main-assembled": "\\begin{theorem}[Main theorem (assembled)]\n\\label{thm:main-assembled}\nAssume the strong regime hypotheses of Section~\\ref{sec:completion-selfdual}, so that the scaling-limit kernel\n$K_L$ exists and the self-dual normalization can be fixed. Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item (\\emph{Unconditional structural output.})\nThere exists a nonnegative kernel $\\Phi\\in L^1(\\mathbb R)$ satisfying:\n\\begin{itemize}\n\\item $\\Phi\\in\\mathrm{PF}_\\infty$;\n\\item its bilateral Laplace transform $\\mathcal B\\Phi$ satisfies the reflection law\n$\\mathcal B\\Phi(s)=\\mathcal B\\Phi(\\tfrac12-s)$ on the common domain of absolute convergence;\n\\item $\\mathcal B\\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation\n$\\mathcal B\\Phi(s)=E(s)/\\Psi(s)$ with $\\Psi$ Laguerre--P\\'olya and hence having only real zeros.\n\\end{itemize}\n\\item (\\emph{Unconditional Archimedean Mellin identification.})\nThe Archimedean-completed kernel $\\widetilde K_{\\mathrm{arch}}$ has Mellin transform\n$F_{\\mathrm{arch}}(z)=\\Xi(2z)$ as in Theorem~\\ref{thm:arch-mellin-identification}.\n\\end{enumerate}\n\\end{theorem}", "rem:standing-assumptions": "\\begin{remark}[Standing assumptions and existence of the scaling limit]\n\\label{rem:standing-assumptions}\nWe work throughout in a regime where the scaling-limit trace $K_L(t)$ exists for every $L>0$ and $t>0$, with convergence locally uniform in $t$ on compact subsets of $(0,\\infty)$. In particular, under the translation-invariant specialization of Remark~\\ref{rem:translation-invariant} this limit can be obtained explicitly by Fourier analysis on $\\Z/N\\Z$. Under these assumptions the limit admits the explicit theta-series form stated in Theorem~\\ref{thm:theta-series-form-recall}. Analytic interchanges (limits, sums, differentiation, and Mellin/Laplace integrals) are justified by the domination estimates recorded in Appendix~\\ref{app:technical}.\n\\end{remark}", "def:arch-operator": "\\begin{definition}[Archimedean completion operator]\n\\label{def:arch-operator}\nFor a sufficiently smooth function $f:(0,\\infty)\\to\\R$, define\n\\begin{equation}\\label{eq:arch-operator}\n(\\mathcal A f)(t):=\\frac{d}{dt}\\Bigl(t^{3/2}\\frac{d}{dt}f(t)\\Bigr),\\qquad t>0.\n\\end{equation}\n\\end{definition}", "thm:SEK": "\\begin{theorem}[Schoenberg--Edrei--Karlin]\n\\label{thm:SEK}\nLet $\\Phi\\in L^1(\\mathbb R)\\cap \\mathrm{PF}_\\infty$ be nonnegative.\nThen its bilateral Laplace transform admits the representation\n\\begin{equation}\n\\label{eq:SEK-form}\n\\mathcal B\\Phi(s)\n=\n\\frac{E(s)}{\\Psi(s)},\n\\end{equation}\nwhere:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $E(s)$ is an entire function with no zeros,\n\\item $\\Psi(s)$ is an entire function of Laguerre--Pólya type,\n\\item all zeros of $\\Psi$ are real,\n\\item the representation is unique up to multiplication of $E$ and $\\Psi$ by the\nsame nonzero constant.\n\\end{enumerate}\n\\end{theorem}", "app:technical": "\\begin{remark}[What this proof does and does not use]\nWe do \\emph{not} use any Mellin transform, any identification with $\\Xi$, nor any\nanalytic continuation beyond the natural half-planes of convergence. The only external\ninputs are the product formula \\eqref{eq:invGamma-product} and the standard\nSchoenberg--Edrei--Karlin criterion linking Laguerre--P\\'olya denominators to\n$\\mathrm{PF}_\\infty$ kernels.\n\\end{remark}\n\n\\section{The bridge problem (open)}\n\\label{sec:future-directions}\n\nThe constructions developed in this paper naturally suggest a further analytic\nproblem that we do not address here. On the one hand, the primitive finite dynamics\nconsidered above give rise, in a canonical and entirely Archimedean manner, to a\nnonnegative integrable kernel $\\Phi$ whose bilateral Laplace transform admits a\nSchoenberg--Edrei--Karlin factorization $\\mathcal B\\Phi=E/\\Psi$, with $\\Psi$ belonging\nto the Laguerre--P\\'olya class\\cite{Levin1964,PolyaSchur1914}. On the other hand, the Archimedean completion procedure\nintroduced here produces, at a distinguished self-dual normalization, a Mellin\ntransform coinciding with the classical function $\\Xi(2z)$.\n\n\\smallskip\n\\noindent\\textbf{Sanity checks for any future identification.}\nAny proposed identification should, at minimum, respect (i) the functional symmetries arising from the canonical centering and self-dual normalization, and (ii) the intrinsic total-positivity/Laguerre--P\\'olya structure on the $\\Psi$-side\\cite{Schoenberg1951,Edrei1952,karlin1968total,Levin1964}.\n\nA natural question is whether these two analytic structures can be directly related.\nMore precisely, one may ask whether the Laguerre--P\\'olya datum $\\Psi$ arising from the\ntotal-positivity side can be identified with $\\Xi(2z)$ up to multiplication by a\nzero-free entire factor, or whether a suitable rigidity principle forces such an\nidentification once both objects are placed in a common analytic framework. Any such\nidentification would amount to a comparison of the zero divisors of two functions that\noriginate from a priori unrelated constructions.\n\nWe view this identification problem as a promising direction for future work. Its\nresolution would require additional analytic input beyond the scope of the present\npaper, and we therefore leave it open.\n\n\\bibliographystyle{plain}\n\\bibliography{citations}\n\n\\appendix\n\\section{Technical analytic justifications}\n\\label{app:technical}\n\nThis appendix collects several analytic estimates and justifications that are\nused implicitly or explicitly in the main text. None of the results here\nintroduce new objects or alter the logical structure of the argument; they serve\nonly to justify standard limiting procedures, termwise operations, and boundary\nmanipulations already invoked.\n\n\\subsection{Uniform local central limit bounds}\n\\label{app:ULCLT}\n\nIn Section~\\ref{sec:completion-selfdual} we appealed to a uniform local central limit theorem (ULCLT) to pass\nfrom the discrete primitive dynamics to the scaling-limit kernel $K_L$. We record\nhere a representative bound sufficient for all uses in the paper.\n\n\\begin{lemma}[Uniform local CLT estimate]\n\\label{lem:ULCLT}\nLet $(X_t^{(N)})_{t\\ge0}$ be the rescaled primitive process on the $N$-cycle with\neffective diffusion constant $D>0$, and let $p_t^{(N)}(j)$ denote its transition\nkernel. Then for every compact interval $[t_0,t_1]\\subset(0,\\infty)$ there exists\na constant $C=C(t_0,t_1)$ such that\n\\[\n\\sup_{t\\in[t_0,t_1]}\\sup_{j\\in\\mathbb Z}\n\\Bigl|\np_t^{(N)}(j)\n-\n\\frac{1}{\\sqrt{4\\pi Dt}}\\exp\\!\\Bigl(-\\frac{j^2}{4Dt}\\Bigr)\n\\Bigr|\n\\le\n\\frac{C}{N},\n\\]\nfor all sufficiently large $N$.\n\\end{lemma}\n\n\\begin{proof}[Sketch]\nThis is a standard consequence of Fourier analysis on the discrete torus together\nwith Taylor expansion of the dispersion relation near the origin. Uniformity in\n$t\\in[t_0,t_1]$ follows from compactness. Precise proofs may be found, for example,\nin classical treatments of random walks on finite groups.\n\\end{proof}\n\n\\begin{remark}\nLemma~\\ref{lem:ULCLT} is used only to justify dominated convergence and termwise\nlimits when passing to the theta-series representation of the completed kernel.\nNo rate sharper than $O(N^{-1})$ is required.\n\\end{remark}", "thm:arch-mellin-identification": "\\begin{theorem}[Archimedean Mellin identification]\n\\label{thm:arch-mellin-identification}\nAssume the self-dual scale $L^2=4\\pi D$, so that $\\widetilde K_{\\mathrm{arch}}=\\Theta$\nby Lemma~\\ref{lem:arch-equals-Theta-new}.\nDefine, for $z\\in\\mathbb C$,\n\\begin{equation}\n\\label{eq:Farch-def-new}\nF_{\\mathrm{arch}}(z)\n:=\n\\int_0^\\infty \\widetilde K_{\\mathrm{arch}}(t)\\,t^{\\frac34+iz}\\,\\frac{dt}{t}.\n\\end{equation}\nThen for all $z\\in\\mathbb C$,\n\\begin{equation}\n\\label{eq:Farch-equals-Xi}\nF_{\\mathrm{arch}}(z)=\\Xi(2z).\n\\end{equation}\n\\end{theorem}", "eq:Phi-def": "\\begin{equation}\n\\label{eq:Phi-def}\n\\Phi(x)\n:=\ne^{x/4}\\,\\widetilde K_{\\mathrm{sym}}(e^{x})\n=\ne^{-x/4}\\,\\widetilde K(e^{x}),\n\\qquad x\\in\\mathbb R.\n\\end{equation}", "def:half-density": "\\begin{definition}[Archimedean (half--density) completion]\n\\label{def:half-density}\nDefine the \\emph{Archimedean-normalized} (``half--density'') kernel by\n\\[\n\\widetilde K_{\\mathrm{sym}}(t):=t^{-1/2}\\,\\widetilde K(t),\\qquad t>0.\n\\]\n\\end{definition}", "tab:notation": "\\label{tab:notation}\n\\end{table}\n\n\\section{Acknowledgements}\nThe author would like to thank Krishnaswami Alladi, Tiziano Valentinuzzi and Kenneth Valpey for helpful discussions.\n\n\\section{Self-dual Ar", "tab:proof-map": "\\begin{tabular}{|p{0.10\\linewidth}|p{0.52\\linewidth}|p{0.30\\linewidth}|}\n\\hline\n\\textbf{Step} & \\textbf{Input $\\to$ Output} & \\textbf{Main tool} \\\\\n\\hline\n1 &\nlocal dynamics $\\to$ theta-series kernel $\\widetilde K$ &\nlift/periodize + ULCLT$^*$\n+ dominated convergence\n\\\\ \\hline\n2 &\n$\\widetilde K \\to$ self-dual normalization and Archimedean completion $\\mathcal A$ &\nJacobi inversion + completion operator\n\\\\ \\hline\n3 &\n$\\widetilde K \\to \\Phi \\to (\\Psi,\\ \\mathcal B\\Phi=E/\\Psi)$ &\ncomplete monotonicity $\\Rightarrow \\mathrm{PF}_\\infty$ +\nSchoenberg--Edrei--Karlin\n\\\\ \\hline\n4 &\n$\\mathcal A(K_L-1)\\to \\Theta \\to \\Xi$ (anchor) and open bridge problem &\nArchimedean completion + Mellin;\nComparison problem (open)\n\\\\ \\hline\n\\end{tabular}\n\n\\caption{Proof map (Steps~1--4). $^*$ULCLT = uniform local central limit theorem.}\n\\label{tab:proof-map}\n\\end{table}\n\nThe proof relies on a small number of canonical kernels and transforms; these are summarized in Table~\\ref{tab:notation} with precise definitions and references.\n\n\\begin{table}[h!]\n\\centering\n\\setlength{\\arrayrulewidth}{0.8pt}\n\\renewcommand{\\arraystretch}{1.25}\n\n\\begin{tabular}{|p{0.22\\linewidth}|p{0.70\\linewidth}|}\n\\hline\n\\textbf{Symbol} & \\textbf{Meaning} \\\\\n\\hline\n$K_L(t)$ &\nScaling-limit trace\n(Definition~\\ref{def:scaling-limit}).\n\\\\ \\hline\n$\\widetilde K(t)$ &\nRenormalized trace kernel\n(Theorem~\\ref{thm:theta-series-form-recall}).\n\\\\ \\hline\n$\\widetilde K_{\\mathrm{sym}}(t)=t^{-1/2}\\widetilde K(t)$ &\nSymmetric half-density kernel\n(Definition~\\ref{def:half-density}).\n\\\\ \\hline\n$\\Phi(x)$ &\nLogarithmic kernel\n$\\Phi(x)=e^{x/4}\\,\\widetilde K_{\\mathrm{sym}}(e^x)$\n(equation~\\eqref{eq:Phi-def}).\n\\\\ \\hline\n$\\mathcal B\\Phi(s)$ &\nBilateral Laplace transform\n$\\displaystyle \\mathcal B\\Phi(s)=\\int_{\\R}\\Phi(x)e^{-s x}\\,dx$\n(on $\\Re(s)>-\\tfrac14$).\n\\\\ \\hline\n$\\Psi(s)$ &\nLaguerre--P\\'olya entire function with only real zeros satisfying\n$\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)}$ on a strip\n(Theorem~\\ref{thm:SEK}).\n\\\\ \\hline\n$F(z)$ &\nFourier transform of $\\Phi$,\n$\\displaystyle F(z)=\\int_{\\R}\\Phi(x)e^{izx}\\,dx$;\nboundary value $F(z)=\\mathcal B\\Phi(-iz)$ where both sides converge.\n\\\\ \\hline\n$\\mathcal A$ &\nArchimedean completion operator\n(Definition~\\ref{def:arch-operator}).\n\\\\ \\hline\n$\\widetilde K_{\\mathrm{arch}}$ &\nCompleted Archimedean kernel\n$\\widetilde K_{\\mathrm{arch}}:=\\mathcal A(K_L-1)$\n(Definition~\\ref{def:arch-theta-kernel}).\n\\\\ \\hline\n$F_{\\mathrm{arch}}(z)$ &\nMellin transform\n$\\displaystyle F_{\\mathrm{arch}}(z)=\\int_0^{\\infty}\n\\widetilde K_{\\mathrm{arch}}(t)\\,\nt^{\\frac14+iz}\\frac{dt}{t}$;\nat the self-dual scale,\n$F_{\\mathrm{arch}}(z)=\\Xi(2z)$\n(Theorem~\\ref{thm:arch-mellin-identification}).\n\\\\ \\hline\n$\\Psi_c(z)$ &\nCentered Laguerre--P\\'olya function\n$\\Psi_c(z):=\\Psi(\\tfrac14+iz)$. A central open problem is to compare $\\Psi_c$ with the Mellin-side\nfunction $\\Xi(2z)$ up to multiplication by a zero-free entire factor (Section~\\ref{sec:future-directions}).\n\\\\ \\hline\n\\end{tabular}", "def:arch-theta-kernel": "\\begin{definition}[Archimedean-completed kernel]\n\\label{def:arch-theta-kernel}\nLet $K_L(t)$ be the scaling-limit trace. Define the Archimedean-completed kernel\n\\[\n\\widetilde K_{\\mathrm{arch}}(t) :=\\bigl(\\mathcal{A}(K_L-1)\\bigr)(t),\\qquad t>0.\n\\]\n\\end{definition}", "thm:theta-series-form-recall": "\\begin{theorem}[Theta-series form of the scaling-limit trace]\n\\label{thm:theta-series-form-recall}\nFor every $L>0$ and $t>0$,\n\\begin{equation}\\label{eq:KL-theta-recall}\nK_L(t)=\\sum_{n\\in\\Z}\\exp\\!\\Bigl(-\\frac{4\\pi^2D}{L^2}\\,n^2\\,t\\Bigr).\n\\end{equation}\n\\end{theorem}", "def:scaling-limit": "\\begin{definition}[Scaling--limit completed trace kernel]\n\\label{def:scaling-limit}\nFix a macroscopic length $L>0$ and choose a scaling parameter $s\\to\\infty$ with $N=N(s)$ such that $N/s\\to L$.\nDefine the scaling--limit trace\n\\[\nK_L(t):=\\lim_{s\\to\\infty} N(s)\\,p^{\\mathrm{cyc}}_{s^2 t}(0,0),\\qquad t>0,\n\\]\nand set the (scaling--limit) completed trace kernel\n\\[\n\\widetilde K_L(t):=K_L(t)-\\frac{L}{\\sqrt{4\\pi D t}}.\n\\]\nWhen $L$ is fixed we suppress the subscript and write $\\widetilde K$.\n\\end{definition}"}, "pre_theorem_intro_text_len": 9896, "pre_theorem_intro_text": "\\label{sec:background}\n\nAcross several foundational currents in mathematics, a shared goal is to\nrecover high-level objects from minimal local principles and universal symmetry\nrequirements, rather than from distinguished presentations. In the present setting, we ask\nwhether analytic structures that are usually introduced through arithmetic data can instead\nbe forced by locality, reversibility, symmetry, and scaling alone. We use the term\n``primitive'' in this sense: the input is only finite local dynamics, while the outputs are\ndetermined by the invariances that survive in the scaling limit. More specifically, the aim herein is to show that two classical analytic objects can arise canonically\nfrom primitive input: finite, local, reversible Markov dynamics on discrete cycles, introduced\nwith no reference to primes, Euler products, or spectral/arithmetic data. \n\nThe first output is a \\emph{total-positivity object}: a logarithmic kernel $\\Phi$ whose\nbilateral Laplace transform admits a canonical Schoenberg--Edrei--Karlin factorization~\\cite{Schoenberg1951,Edrei1952,karlin1968total}.\nEquivalently, the reciprocal transform is a Laguerre--P\\'olya entire function $\\Psi$ and\ntherefore has only real zeros. The second output is an \\emph{Archimedean completion object}:\nan explicit completion operator $\\mathcal A$ that restores Mellin self-duality at a\ncanonical normalization, producing a completed kernel whose Mellin transform coincides\n(within a natural domain) with the classical Riemann $\\Xi$-function.\n\nThese constructions are logically independent but share the same primitive source, and they\nmotivate a natural identification question left open here (see\nSection~\\ref{sec:future-directions}): to what extent can the Laguerre--P\\'olya datum $\\Psi$\nbe identified with the $\\Xi$-function after the normalizations are fixed?\n\nSeveral classical approaches emphasize representations of $\\Xi$ as an integral transform. One may express $\\Xi$ through Mellin transforms of theta functions, through Fourier transforms of rapidly decaying kernels, or through trace formulas that relate zeros to spectral data\\cite{Selberg1956,BerryKeating1999}. These representations highlight a recurring theme: analytic properties of $\\Xi$ often reflect structural features of an underlying kernel. In particular, positivity and symmetry at the level of kernels tend to propagate to strong constraints on the associated transforms.\n\nHeat kernels and theta series provide a natural language\\cite{Chandrasekharan1985,SteinShakarchi2003} for these ideas. The fundamental solution of the heat equation on the real line is Gaussian, and its periodization produces classical theta functions. Poisson summation expresses a precise duality between spatial periodization and frequency localization. These mechanisms are robust and do not depend on arithmetic input. They arise whenever one studies diffusion on spaces with discrete symmetries. In analytic number theory, theta series appear as generating functions encoding spectral or lattice data, and they often serve as intermediaries between local and global descriptions.\n\nIn this work, heat kernels play a central conceptual role. Rather than appearing as analytic tools applied to $\\zeta(s)$, they arise from elementary local dynamics. The guiding idea is that long-time diffusion captures universal behavior, while finite-volume effects introduce discrete structure. When these effects are combined through exact lift and periodization identities, theta series emerge canonically. This viewpoint places theta functions in a broader probabilistic and geometric setting, where they reflect general principles rather than special constructions.\n\nA second strand of background comes from the theory of total positivity\\cite{Pinkus2010,karlin1968total,GantmacherKrein2002}. Total positivity originated in work of Schoenberg and was developed extensively by Karlin and others. It concerns kernels whose minors of all orders are nonnegative, a property that implies strong variation-diminishing and rigidity phenomena. Within this theory, a distinguished role is played by P\\'olya frequency functions of infinite order. In the form used here (Schoenberg--Edrei--Karlin), the bilateral Laplace transform of a $\\mathrm{PF}_\\infty$ function extends meromorphically and its reciprocal is a Laguerre--P\\'olya entire function. Entire functions in the Laguerre--P\\'olya class\\cite{Levin1964,PolyaSchur1914} are limits of polynomials with only real zeros, and hence have only real zeros.\n\nClosely related to this viewpoint is the Hermite--Biehler theory\\cite{Krein1947,deBranges1968} of entire functions and its modern formulation within de Branges spaces\\cite{deBranges1968}. Hermite--Biehler theory provides criteria ensuring that an entire function has all its zeros on a line, typically the real axis, based on positivity properties of associated kernels or on analytic inequalities in the upper half-plane. De Branges' work showed that these ideas can be organized into a Hilbert space framework with powerful classification theorems. Although the present paper does not rely on the full machinery of de Branges spaces\\cite{deBranges1968}, the Hermite--Biehler perspective clarifies why positivity at the kernel level leads naturally to real-zero conclusions.\n\nWe now introduce the central object of the paper and state the main results.\n\n\\subsection{Primitive dynamical model}\n\\label{sec:model}\n\nFor each integer $N\\ge 1$ we consider a continuous-time, reversible, nearest-neighbor Markov process on the discrete cycle $\\mathbb{Z}/N\\mathbb{Z}$. The dynamics are specified by a collection of strictly positive conductances $\\{a_j\\}_{j\\in\\mathbb{Z}/N\\mathbb{Z}}$, where $a_j$ is associated with the undirected edge between $j$ and $j+1$ (indices taken modulo $N$). The generator $\\mathcal{L}_N$ acts on functions $f:\\mathbb{Z}/N\\mathbb{Z}\\to\\mathbb{R}$ by\n\\begin{equation}\n\\label{eq:generator}\n(\\mathcal{L}_N f)(j)\n=\na_j\\,\\bigl(f(j+1)-f(j)\\bigr)\n\\;+\na_{j-1}\\,\\bigl(f(j-1)-f(j)\\bigr).\n\\end{equation}\n\n\\begin{remark}[Translation-invariant specialization]\n\\label{rem:translation-invariant}\nFor the purposes of this paper we restrict to the translation-invariant case\n\\(\na_j\\equiv a>0\n\\)\n(independent of $j$), so that \\eqref{eq:generator} is the usual continuous-time simple\nrandom walk on $\\mathbb{Z}/N\\mathbb{Z}$ with jump rate $a$. In this case the diffusive scaling limit is\ngoverned by the one-dimensional heat equation with diffusion constant \\(D=a\\), and the\nFourier analysis on $\\mathbb{Z}/N\\mathbb{Z}$ leading to the theta-series form is completely explicit.\nExtensions to uniformly elliptic periodic environments, or to quantitative homogenization\nregimes, can also yield a uniform local CLT and the same scaling limit; we do not pursue\nthose generalizations here.\n\\end{remark}\n\nWe write $p^{\\mathrm{cyc}}_t(j,k)$ for the associated heat kernel,\n\\(p^{\\mathrm{cyc}}_t(j,k)=(e^{t\\mathcal{L}_N}\\mathbf{1}_{\\{k\\}})(j)\\), and we denote by $D>0$ the macroscopic diffusion constant appearing in the Gaussian scaling limit established later.\n\nAs described above, the trace of the heat kernel associated with the finite dynamics captures global information about the system but contains a universal singular contribution reflecting diffusive behavior. This singular term is independent of the fine structure of the dynamics and depends only on the macroscopic scaling. To isolate the genuinely structural content, we subtract this term in a canonical way. The resulting object is the completed trace kernel.\n\n\\begin{definition}[Scaling--limit completed trace kernel]\n\\label{def:scaling-limit}\nFix a macroscopic length $L>0$ and choose a scaling parameter $s\\to\\infty$ with $N=N(s)$ such that $N/s\\to L$.\nDefine the scaling--limit trace\n\\[\nK_L(t):=\\lim_{s\\to\\infty} N(s)\\,p^{\\mathrm{cyc}}_{s^2 t}(0,0),\\qquad t>0,\n\\]\nand set the (scaling--limit) completed trace kernel\n\\[\n\\widetilde K_L(t):=K_L(t)-\\frac{L}{\\sqrt{4\\pi D t}}.\n\\]\nWhen $L$ is fixed we suppress the subscript and write $\\widetilde K$.\n\\end{definition}\n\n\\begin{definition}[Archimedean (half--density) completion]\n\\label{def:half-density}\nDefine the \\emph{Archimedean-normalized} (``half--density'') kernel by\n\\[\n\\widetilde K_{\\mathrm{sym}}(t):=t^{-1/2}\\,\\widetilde K(t),\\qquad t>0.\n\\]\n\\end{definition}\n\nThe existence of the scaling limit in Definition~\\ref{def:scaling-limit}, together with the basic domination estimates needed to justify termwise limits, follows from the standing assumptions recorded below; for convenience we collect the relevant analytic justifications in Appendix~\\ref{app:technical}.\n\n\\begin{remark}[Standing assumptions and existence of the scaling limit]\n\\label{rem:standing-assumptions}\nWe work throughout in a regime where the scaling-limit trace $K_L(t)$ exists for every $L>0$ and $t>0$, with convergence locally uniform in $t$ on compact subsets of $(0,\\infty)$. In particular, under the translation-invariant specialization of Remark~\\ref{rem:translation-invariant} this limit can be obtained explicitly by Fourier analysis on $\\mathbb{Z}/N\\mathbb{Z}$. Under these assumptions the limit admits the explicit theta-series form stated in Theorem~\\ref{thm:theta-series-form-recall}. Analytic interchanges (limits, sums, differentiation, and Mellin/Laplace integrals) are justified by the domination estimates recorded in Appendix~\\ref{app:technical}.\n\\end{remark}\n\n\\medskip\n\n\\noindent\nIn addition to the total-positivity output (producing a canonical Laguerre--P\\'olya function) and\nthe Archimedean completion output (producing the classical theta/Mellin representation of $\\Xi$),\nwe isolate a single remaining analytic identification problem: relate the Laguerre--P\\'olya datum\n$\\Psi$ to the Mellin-side function $\\Xi(2\\cdot)$ up to multiplication by a zero-free entire factor.\nWe treat this as an open ``bridge'' problem and discuss it in Section~\\ref{sec:future-directions}.", "context": "The first output is a \\emph{total-positivity object}: a logarithmic kernel $\\Phi$ whose\nbilateral Laplace transform admits a canonical Schoenberg--Edrei--Karlin factorization~\\cite{Schoenberg1951,Edrei1952,karlin1968total}.\nEquivalently, the reciprocal transform is a Laguerre--P\\'olya entire function $\\Psi$ and\ntherefore has only real zeros. The second output is an \\emph{Archimedean completion object}:\nan explicit completion operator $\\mathcal A$ that restores Mellin self-duality at a\ncanonical normalization, producing a completed kernel whose Mellin transform coincides\n(within a natural domain) with the classical Riemann $\\Xi$-function.\n\nA second strand of background comes from the theory of total positivity\\cite{Pinkus2010,karlin1968total,GantmacherKrein2002}. Total positivity originated in work of Schoenberg and was developed extensively by Karlin and others. It concerns kernels whose minors of all orders are nonnegative, a property that implies strong variation-diminishing and rigidity phenomena. Within this theory, a distinguished role is played by P\\'olya frequency functions of infinite order. In the form used here (Schoenberg--Edrei--Karlin), the bilateral Laplace transform of a $\\mathrm{PF}_\\infty$ function extends meromorphically and its reciprocal is a Laguerre--P\\'olya entire function. Entire functions in the Laguerre--P\\'olya class\\cite{Levin1964,PolyaSchur1914} are limits of polynomials with only real zeros, and hence have only real zeros.\n\n\\begin{definition}[Scaling--limit completed trace kernel]\n\\label{def:scaling-limit}\nFix a macroscopic length $L>0$ and choose a scaling parameter $s\\to\\infty$ with $N=N(s)$ such that $N/s\\to L$.\nDefine the scaling--limit trace\n\\[\nK_L(t):=\\lim_{s\\to\\infty} N(s)\\,p^{\\mathrm{cyc}}_{s^2 t}(0,0),\\qquad t>0,\n\\]\nand set the (scaling--limit) completed trace kernel\n\\[\n\\widetilde K_L(t):=K_L(t)-\\frac{L}{\\sqrt{4\\pi D t}}.\n\\]\nWhen $L$ is fixed we suppress the subscript and write $\\widetilde K$.\n\\end{definition}\n\n\\begin{remark}[Standing assumptions and existence of the scaling limit]\n\\label{rem:standing-assumptions}\nWe work throughout in a regime where the scaling-limit trace $K_L(t)$ exists for every $L>0$ and $t>0$, with convergence locally uniform in $t$ on compact subsets of $(0,\\infty)$. In particular, under the translation-invariant specialization of Remark~\\ref{rem:translation-invariant} this limit can be obtained explicitly by Fourier analysis on $\\mathbb{Z}/N\\mathbb{Z}$. Under these assumptions the limit admits the explicit theta-series form stated in Theorem~\\ref{thm:theta-series-form-recall}. Analytic interchanges (limits, sums, differentiation, and Mellin/Laplace integrals) are justified by the domination estimates recorded in Appendix~\\ref{app:technical}.\n\\end{remark}\n\n\\medskip\n\n\\noindent\nIn addition to the total-positivity output (producing a canonical Laguerre--P\\'olya function) and\nthe Archimedean completion output (producing the classical theta/Mellin representation of $\\Xi$),\nwe isolate a single remaining analytic identification problem: relate the Laguerre--P\\'olya datum\n$\\Psi$ to the Mellin-side function $\\Xi(2\\cdot)$ up to multiplication by a zero-free entire factor.\nWe treat this as an open ``bridge'' problem and discuss it in Section~\\ref{sec:future-directions}.", "full_context": "The first output is a \\emph{total-positivity object}: a logarithmic kernel $\\Phi$ whose\nbilateral Laplace transform admits a canonical Schoenberg--Edrei--Karlin factorization~\\cite{Schoenberg1951,Edrei1952,karlin1968total}.\nEquivalently, the reciprocal transform is a Laguerre--P\\'olya entire function $\\Psi$ and\ntherefore has only real zeros. The second output is an \\emph{Archimedean completion object}:\nan explicit completion operator $\\mathcal A$ that restores Mellin self-duality at a\ncanonical normalization, producing a completed kernel whose Mellin transform coincides\n(within a natural domain) with the classical Riemann $\\Xi$-function.\n\nA second strand of background comes from the theory of total positivity\\cite{Pinkus2010,karlin1968total,GantmacherKrein2002}. Total positivity originated in work of Schoenberg and was developed extensively by Karlin and others. It concerns kernels whose minors of all orders are nonnegative, a property that implies strong variation-diminishing and rigidity phenomena. Within this theory, a distinguished role is played by P\\'olya frequency functions of infinite order. In the form used here (Schoenberg--Edrei--Karlin), the bilateral Laplace transform of a $\\mathrm{PF}_\\infty$ function extends meromorphically and its reciprocal is a Laguerre--P\\'olya entire function. Entire functions in the Laguerre--P\\'olya class\\cite{Levin1964,PolyaSchur1914} are limits of polynomials with only real zeros, and hence have only real zeros.\n\n\\begin{definition}[Scaling--limit completed trace kernel]\n\\label{def:scaling-limit}\nFix a macroscopic length $L>0$ and choose a scaling parameter $s\\to\\infty$ with $N=N(s)$ such that $N/s\\to L$.\nDefine the scaling--limit trace\n\\[\nK_L(t):=\\lim_{s\\to\\infty} N(s)\\,p^{\\mathrm{cyc}}_{s^2 t}(0,0),\\qquad t>0,\n\\]\nand set the (scaling--limit) completed trace kernel\n\\[\n\\widetilde K_L(t):=K_L(t)-\\frac{L}{\\sqrt{4\\pi D t}}.\n\\]\nWhen $L$ is fixed we suppress the subscript and write $\\widetilde K$.\n\\end{definition}\n\n\\begin{remark}[Standing assumptions and existence of the scaling limit]\n\\label{rem:standing-assumptions}\nWe work throughout in a regime where the scaling-limit trace $K_L(t)$ exists for every $L>0$ and $t>0$, with convergence locally uniform in $t$ on compact subsets of $(0,\\infty)$. In particular, under the translation-invariant specialization of Remark~\\ref{rem:translation-invariant} this limit can be obtained explicitly by Fourier analysis on $\\mathbb{Z}/N\\mathbb{Z}$. Under these assumptions the limit admits the explicit theta-series form stated in Theorem~\\ref{thm:theta-series-form-recall}. Analytic interchanges (limits, sums, differentiation, and Mellin/Laplace integrals) are justified by the domination estimates recorded in Appendix~\\ref{app:technical}.\n\\end{remark}\n\n\\medskip\n\n\\noindent\nIn addition to the total-positivity output (producing a canonical Laguerre--P\\'olya function) and\nthe Archimedean completion output (producing the classical theta/Mellin representation of $\\Xi$),\nwe isolate a single remaining analytic identification problem: relate the Laguerre--P\\'olya datum\n$\\Psi$ to the Mellin-side function $\\Xi(2\\cdot)$ up to multiplication by a zero-free entire factor.\nWe treat this as an open ``bridge'' problem and discuss it in Section~\\ref{sec:future-directions}.\n\n\\begin{abstract}\nStarting from finite, local, reversible Markov dynamics on discrete cycles, we construct a\nscaling-limit renormalized trace kernel admitting an exact theta-series representation.\nThe construction is entirely Archimedean and uses no Euler products, primes, or\narithmetic spectral input.\nFrom this limit we define a logarithmic kernel $\\Phi$ and prove that it lies in the\nP\\'olya frequency class $\\mathrm{PF}_\\infty$, yielding via the Schoenberg--Edrei--Karlin\nclassification a canonical Laguerre--P\\'olya function $\\Psi$.\nIndependently, we introduce an Archimedean completion operator and show that, at a\nself-dual normalization, the completed kernel coincides with the classical theta kernel,\nwhose Mellin transform is the Riemann $\\Xi$-function.\nWe isolate a single remaining analytic problem relating $\\Psi$ to $\\Xi(2\\cdot)$.\n\\end{abstract}\n\n\\noindent\nIn addition to the total-positivity output (producing a canonical Laguerre--P\\'olya function) and\nthe Archimedean completion output (producing the classical theta/Mellin representation of $\\Xi$),\nwe isolate a single remaining analytic identification problem: relate the Laguerre--P\\'olya datum\n$\\Psi$ to the Mellin-side function $\\Xi(2\\cdot)$ up to multiplication by a zero-free entire factor.\nWe treat this as an open ``bridge'' problem and discuss it in Section~\\ref{sec:future-directions}.\n\n\\begin{theorem}[Archimedean Mellin identification]\n\\label{thm:arch-overview}\nAt the self-dual scale fixed in Section~\\ref{sec:completion-selfdual}, the Archimedean-completed\nkernel $\\widetilde K_{\\mathrm{arch}}:=\\mathcal A(K_L-1)$ has Mellin transform\n\\[\nF_{\\mathrm{arch}}(z)=\\Xi(2z)\n\\]\nas in Theorem~\\ref{thm:arch-mellin-identification}.\n\\end{theorem}\n\n\\begin{tabular}{|p{0.22\\linewidth}|p{0.70\\linewidth}|}\n\\hline\n\\textbf{Symbol} & \\textbf{Meaning} \\\\\n\\hline\n$K_L(t)$ &\nScaling-limit trace\n(Definition~\\ref{def:scaling-limit}).\n\\\\ \\hline\n$\\widetilde K(t)$ &\nRenormalized trace kernel\n(Theorem~\\ref{thm:theta-series-form-recall}).\n\\\\ \\hline\n$\\widetilde K_{\\mathrm{sym}}(t)=t^{-1/2}\\widetilde K(t)$ &\nSymmetric half-density kernel\n(Definition~\\ref{def:half-density}).\n\\\\ \\hline\n$\\Phi(x)$ &\nLogarithmic kernel\n$\\Phi(x)=e^{x/4}\\,\\widetilde K_{\\mathrm{sym}}(e^x)$\n(equation~\\eqref{eq:Phi-def}).\n\\\\ \\hline\n$\\mathcal B\\Phi(s)$ &\nBilateral Laplace transform\n$\\displaystyle \\mathcal B\\Phi(s)=\\int_{\\R}\\Phi(x)e^{-s x}\\,dx$\n(on $\\Re(s)>-\\tfrac14$).\n\\\\ \\hline\n$\\Psi(s)$ &\nLaguerre--P\\'olya entire function with only real zeros satisfying\n$\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)}$ on a strip\n(Theorem~\\ref{thm:SEK}).\n\\\\ \\hline\n$F(z)$ &\nFourier transform of $\\Phi$,\n$\\displaystyle F(z)=\\int_{\\R}\\Phi(x)e^{izx}\\,dx$;\nboundary value $F(z)=\\mathcal B\\Phi(-iz)$ where both sides converge.\n\\\\ \\hline\n$\\mathcal A$ &\nArchimedean completion operator\n(Definition~\\ref{def:arch-operator}).\n\\\\ \\hline\n$\\widetilde K_{\\mathrm{arch}}$ &\nCompleted Archimedean kernel\n$\\widetilde K_{\\mathrm{arch}}:=\\mathcal A(K_L-1)$\n(Definition~\\ref{def:arch-theta-kernel}).\n\\\\ \\hline\n$F_{\\mathrm{arch}}(z)$ &\nMellin transform\n$\\displaystyle F_{\\mathrm{arch}}(z)=\\int_0^{\\infty}\n\\widetilde K_{\\mathrm{arch}}(t)\\,\nt^{\\frac14+iz}\\frac{dt}{t}$;\nat the self-dual scale,\n$F_{\\mathrm{arch}}(z)=\\Xi(2z)$\n(Theorem~\\ref{thm:arch-mellin-identification}).\n\\\\ \\hline\n$\\Psi_c(z)$ &\nCentered Laguerre--P\\'olya function\n$\\Psi_c(z):=\\Psi(\\tfrac14+iz)$. A central open problem is to compare $\\Psi_c$ with the Mellin-side\nfunction $\\Xi(2z)$ up to multiplication by a zero-free entire factor (Section~\\ref{sec:future-directions}).\n\\\\ \\hline\n\\end{tabular}\n\n\\begin{proposition}[Reflection law for the Step-2 transform]\n\\label{prop:bilaplace-reflection}\nAssume \\eqref{eq:Ksym-selfdual}. Then for every $s\\in\\mathbb C$ with\n\\[\n-\\tfrac14<\\Re(s)<\\tfrac34\n\\]\n(one precisely needs absolute convergence of both sides), one has the exact identity\n\\begin{equation}\n\\label{eq:bilaplace-reflection}\n\\mathcal B\\Phi(s)=\\mathcal B\\Phi\\!\\left(\\tfrac12-s\\right).\n\\end{equation}\nConsequently, for all $t\\in\\mathbb R$ one has\n\\begin{equation}\n\\label{eq:bilaplace-real-on-centerline}\n\\mathcal B\\Phi\\!\\left(\\tfrac14+it\\right)\\in\\mathbb R.\n\\end{equation}\n\\end{proposition}\n\n\\begin{theorem}[Schoenberg--Edrei--Karlin]\n\\label{thm:SEK}\nLet $\\Phi\\in L^1(\\mathbb R)\\cap \\mathrm{PF}_\\infty$ be nonnegative.\nThen its bilateral Laplace transform admits the representation\n\\begin{equation}\n\\label{eq:SEK-form}\n\\mathcal B\\Phi(s)\n=\n\\frac{E(s)}{\\Psi(s)},\n\\end{equation}\nwhere:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $E(s)$ is an entire function with no zeros,\n\\item $\\Psi(s)$ is an entire function of Laguerre--Pólya type,\n\\item all zeros of $\\Psi$ are real,\n\\item the representation is unique up to multiplication of $E$ and $\\Psi$ by the\nsame nonzero constant.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{theorem}[Main theorem (assembled)]\n\\label{thm:main-assembled}\nAssume the strong regime hypotheses of Section~\\ref{sec:completion-selfdual}, so that the scaling-limit kernel\n$K_L$ exists and the self-dual normalization can be fixed. Then:\n\\begin{enumerate}[label=(\\roman*)]\n\\item (\\emph{Unconditional structural output.})\nThere exists a nonnegative kernel $\\Phi\\in L^1(\\mathbb R)$ satisfying:\n\\begin{itemize}\n\\item $\\Phi\\in\\mathrm{PF}_\\infty$;\n\\item its bilateral Laplace transform $\\mathcal B\\Phi$ satisfies the reflection law\n$\\mathcal B\\Phi(s)=\\mathcal B\\Phi(\\tfrac12-s)$ on the common domain of absolute convergence;\n\\item $\\mathcal B\\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation\n$\\mathcal B\\Phi(s)=E(s)/\\Psi(s)$ with $\\Psi$ Laguerre--P\\'olya and hence having only real zeros.\n\\end{itemize}\n\\item (\\emph{Unconditional Archimedean Mellin identification.})\nThe Archimedean-completed kernel $\\widetilde K_{\\mathrm{arch}}$ has Mellin transform\n$F_{\\mathrm{arch}}(z)=\\Xi(2z)$ as in Theorem~\\ref{thm:arch-mellin-identification}.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nWe compute the bilateral Laplace transform on the half-plane $\\Re(s)>0$:\n\\[\n\\mathcal B\\phi_u(s)\n=\n\\int_{\\mathbb R} e^{-u e^{-x}}e^{-s x}\\,dx.\n\\]\nWith the change of variables $y=e^{-x}$ (so $y\\in(0,\\infty)$ and $dx=-dy/y$), this becomes\n\\[\n\\mathcal B\\phi_u(s)\n=\n\\int_0^\\infty e^{-u y} y^{s-1}\\,dy\n=\nu^{-s}\\Gamma(s),\n\\qquad \\Re(s)>0.\n\\]\nNow recall the classical Weierstrass product for the reciprocal Gamma function:\n\\begin{equation}\n\\label{eq:invGamma-product}\n\\frac{1}{\\Gamma(s)}\n=\ns\\,e^{\\gamma s}\\prod_{n=1}^\\infty\\Bigl(1+\\frac{s}{n}\\Bigr)e^{-s/n},\n\\end{equation}\nwhere $\\gamma$ is Euler's constant. The right-hand side is an entire function whose\nzeros are exactly the nonpositive integers, all real and simple, and whose product\nrepresentation is of Laguerre--P\\'olya type. In particular,\n\\[\n\\Psi(s):=\\frac{1}{\\Gamma(s)}\n\\quad\\text{is Laguerre--P\\'olya,}\n\\qquad\nE(s):=u^{-s}=e^{-s\\log u}\n\\quad\\text{is entire and zero-free.}\n\\]\nThus on $\\Re(s)>0$ we have the representation\n\\[\n\\mathcal B\\phi_u(s)=\\frac{E(s)}{\\Psi(s)}.\n\\]", "post_theorem_intro_text_len": 4911, "post_theorem_intro_text": "\\begin{theorem}[Archimedean Mellin identification]\n\\label{thm:arch-overview}\nAt the self-dual scale fixed in Section~\\ref{sec:completion-selfdual}, the Archimedean-completed\nkernel $\\widetilde K_{\\mathrm{arch}}:=\\mathcal A(K_L-1)$ has Mellin transform\n\\[\nF_{\\mathrm{arch}}(z)=\\Xi(2z)\n\\]\nas in Theorem~\\ref{thm:arch-mellin-identification}.\n\\end{theorem}\n\nThe remainder of the paper constructs the kernel $\\Phi$, proves Theorem~\\ref{thm:main-assembled}, and isolates each logical seam explicitly.\n\n\\medskip\n\nThe paper is organized as follows. Section~\\ref{sec:completion-selfdual} fixes the self-dual Archimedean normalization and records the resulting theta-series form of the scaling-limit trace. Section~\\ref{sec:log-reflection} constructs the logarithmic kernel $\\Phi$ and establishes the exact reflection symmetry for its bilateral Laplace transform. Section~\\ref{sec:total-positivity} proves total positivity and derives the Schoenberg--Edrei--Karlin factorization $\\mathcal B\\Phi=E/\\Psi$. Section~\\ref{sec:arch-mellin-xi} identifies the Archimedean-completed Mellin transform with $\\Xi(2\\cdot)$ and records the Archimedean Mellin identification with $\\Xi(2\\cdot)$, while isolating the remaining identification problem (Section~\\ref{sec:future-directions}). Section~\\ref{sec:completion-ledger} assembles the argument and isolates the remaining seam. Section~\\ref{sec:tp-details} provides a detailed proof of total positivity. Appendix~\\ref{app:technical} collects analytic justifications for termwise operations and boundary terms.\n\n\\subsection{Map of the proof}\\label{subsec:map-proof}\nThe argument is organized into four steps, each producing a new object from the previous one.\nFor quick reference we summarize the pipeline in Table~\\ref{tab:proof-map}.\n\n\\begin{table}[h!]\n\\centering\n\\setlength{\\arrayrulewidth}{0.8pt}\n\\renewcommand{\\arraystretch}{1.25}\n\n\\begin{tabular}{|p{0.10\\linewidth}|p{0.52\\linewidth}|p{0.30\\linewidth}|}\n\\hline\n\\textbf{Step} & \\textbf{Input $\\to$ Output} & \\textbf{Main tool} \\\\\n\\hline\n1 &\nlocal dynamics $\\to$ theta-series kernel $\\widetilde K$ &\nlift/periodize + ULCLT$^*$\n+ dominated convergence\n\\\\ \\hline\n2 &\n$\\widetilde K \\to$ self-dual normalization and Archimedean completion $\\mathcal A$ &\nJacobi inversion + completion operator\n\\\\ \\hline\n3 &\n$\\widetilde K \\to \\Phi \\to (\\Psi,\\ \\mathcal B\\Phi=E/\\Psi)$ &\ncomplete monotonicity $\\Rightarrow \\mathrm{PF}_\\infty$ +\nSchoenberg--Edrei--Karlin\n\\\\ \\hline\n4 &\n$\\mathcal A(K_L-1)\\to \\Theta \\to \\Xi$ (anchor) and open bridge problem &\nArchimedean completion + Mellin;\nComparison problem (open)\n\\\\ \\hline\n\\end{tabular}\n\n\\caption{Proof map (Steps~1--4). $^*$ULCLT = uniform local central limit theorem.}\n\\label{tab:proof-map}\n\\end{table}\n\nThe proof relies on a small number of canonical kernels and transforms; these are summarized in Table~\\ref{tab:notation} with precise definitions and references.\n\n\\begin{table}[h!]\n\\centering\n\\setlength{\\arrayrulewidth}{0.8pt}\n\\renewcommand{\\arraystretch}{1.25}\n\n\\begin{tabular}{|p{0.22\\linewidth}|p{0.70\\linewidth}|}\n\\hline\n\\textbf{Symbol} & \\textbf{Meaning} \\\\\n\\hline\n$K_L(t)$ &\nScaling-limit trace\n(Definition~\\ref{def:scaling-limit}).\n\\\\ \\hline\n$\\widetilde K(t)$ &\nRenormalized trace kernel\n(Theorem~\\ref{thm:theta-series-form-recall}).\n\\\\ \\hline\n$\\widetilde K_{\\mathrm{sym}}(t)=t^{-1/2}\\widetilde K(t)$ &\nSymmetric half-density kernel\n(Definition~\\ref{def:half-density}).\n\\\\ \\hline\n$\\Phi(x)$ &\nLogarithmic kernel\n$\\Phi(x)=e^{x/4}\\,\\widetilde K_{\\mathrm{sym}}(e^x)$\n(equation~\\eqref{eq:Phi-def}).\n\\\\ \\hline\n$\\mathcal B\\Phi(s)$ &\nBilateral Laplace transform\n$\\displaystyle \\mathcal B\\Phi(s)=\\int_{\\mathbb{R}}\\Phi(x)e^{-s x}\\,dx$\n(on $\\Re(s)>-\\tfrac14$).\n\\\\ \\hline\n$\\Psi(s)$ &\nLaguerre--P\\'olya entire function with only real zeros satisfying\n$\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)}$ on a strip\n(Theorem~\\ref{thm:SEK}).\n\\\\ \\hline\n$F(z)$ &\nFourier transform of $\\Phi$,\n$\\displaystyle F(z)=\\int_{\\mathbb{R}}\\Phi(x)e^{izx}\\,dx$;\nboundary value $F(z)=\\mathcal B\\Phi(-iz)$ where both sides converge.\n\\\\ \\hline\n$\\mathcal A$ &\nArchimedean completion operator\n(Definition~\\ref{def:arch-operator}).\n\\\\ \\hline\n$\\widetilde K_{\\mathrm{arch}}$ &\nCompleted Archimedean kernel\n$\\widetilde K_{\\mathrm{arch}}:=\\mathcal A(K_L-1)$\n(Definition~\\ref{def:arch-theta-kernel}).\n\\\\ \\hline\n$F_{\\mathrm{arch}}(z)$ &\nMellin transform\n$\\displaystyle F_{\\mathrm{arch}}(z)=\\int_0^{\\infty}\n\\widetilde K_{\\mathrm{arch}}(t)\\,\nt^{\\frac14+iz}\\frac{dt}{t}$;\nat the self-dual scale,\n$F_{\\mathrm{arch}}(z)=\\Xi(2z)$\n(Theorem~\\ref{thm:arch-mellin-identification}).\n\\\\ \\hline\n$\\Psi_c(z)$ &\nCentered Laguerre--P\\'olya function\n$\\Psi_c(z):=\\Psi(\\tfrac14+iz)$. A central open problem is to compare $\\Psi_c$ with the Mellin-side\nfunction $\\Xi(2z)$ up to multiplication by a zero-free entire factor (Section~\\ref{sec:future-directions}).\n\\\\ \\hline\n\\end{tabular}\n\n\\caption{Notation used throughout the proof.}\n\\label{tab:notation}\n\\end{table}", "sketch": "The introduction gives a proof map rather than a detailed argument. It says the paper will “constructs the kernel $\\Phi$” and establishes the three outputs of Theorem~\\ref{thm:structural} by isolating “each logical seam explicitly,” with the following pipeline:\n\n1. **Local dynamics $\\to$ theta-series kernel $\\widetilde K$** using “lift/periodize + ULCLT$^*$ + dominated convergence.”\n\n2. **$\\widetilde K \\to$ self-dual normalization and Archimedean completion $\\mathcal A$** via “Jacobi inversion + completion operator.”\n\n3. **$\\widetilde K \\to \\Phi \\to (\\Psi,\\ \\mathcal B\\Phi=E/\\Psi)$**: construct the “logarithmic kernel”\n\\[\\Phi(x)=e^{x/4}\\,\\widetilde K_{\\mathrm{sym}}(e^x),\\]\nthen prove “complete monotonicity $\\Rightarrow \\mathrm{PF}_\\infty$ + Schoenberg--Edrei--Karlin,” yielding total positivity and the “Schoenberg--Edrei--Karlin factorization $\\mathcal B\\Phi=E/\\Psi$,” with $\\Psi$ in Laguerre--P\\'olya.\n\nAdditionally, it states that Section~\\ref{sec:log-reflection} “establishes the exact reflection symmetry for its bilateral Laplace transform,” i.e. the reflection law for $\\mathcal B\\Phi$.\n\n4. **Archimedean Mellin anchor**: “$\\mathcal A(K_L-1)\\to \\Theta \\to \\Xi$ (anchor) and open bridge problem” via “Archimedean completion + Mellin; Comparison problem (open).”", "expanded_sketch": "The introduction gives a proof map rather than a detailed argument. It says the paper will “constructs the kernel $\\Phi$” and establishes the three outputs of the main theorem by isolating “each logical seam explicitly,” with the following pipeline:\n\n1. **Local dynamics $\\to$ theta-series kernel $\\widetilde K$** using “lift/periodize + ULCLT$^*$ + dominated convergence.”\n\n2. **$\\widetilde K \\to$ self-dual normalization and Archimedean completion $\\mathcal A$** via “Jacobi inversion + completion operator.”\n\n3. **$\\widetilde K \\to \\Phi \\to (\\Psi,\\ \\mathcal B\\Phi=E/\\Psi)$**: construct the “logarithmic kernel”\n\\[\\Phi(x)=e^{x/4}\\,\\widetilde K_{\\mathrm{sym}}(e^x),\\]\nthen prove “complete monotonicity $\\Rightarrow \\mathrm{PF}_\\infty$ + Schoenberg--Edrei--Karlin,” yielding total positivity and the “Schoenberg--Edrei--Karlin factorization $\\mathcal B\\Phi=E/\\Psi$,” with $\\Psi$ in Laguerre--P\\'olya.\n\nAdditionally, it states that later the paper “establishes the exact reflection symmetry for its bilateral Laplace transform,” i.e. the reflection law for $\\mathcal B\\Phi$.\n\n4. **Archimedean Mellin anchor**: “$\\mathcal A(K_L-1)\\to \\Theta \\to \\Xi$ (anchor) and open bridge problem” via “Archimedean completion + Mellin; Comparison problem (open).”", "expanded_theorem": "[Structural output from primitive cycles]\n\\label{thm:structural}\n\\begin{remark}[Standing assumptions and existence of the scaling limit]\n\\label{rem:standing-assumptions}\nWe work throughout in a regime where the scaling-limit trace $K_L(t)$ exists for every $L>0$ and $t>0$, with convergence locally uniform in $t$ on compact subsets of $(0,\\infty)$. In particular, under the translation-invariant specialization of Remark~\\ref{rem:translation-invariant} this limit can be obtained explicitly by Fourier analysis on $\\Z/N\\Z$. Under these assumptions the limit admits the explicit theta-series form stated in Theorem~\\ref{thm:theta-series-form-recall}. Analytic interchanges (limits, sums, differentiation, and Mellin/Laplace integrals) are justified by the domination estimates recorded in Appendix~\\ref{app:technical}.\n\\end{remark}\nAssume the standing assumptions above, so that the scaling-limit kernel $K_L$ exists.\nThen there exists a nonnegative kernel $\\Phi\\in L^1(\\mathbb{R})$ such that:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\Phi\\in\\mathrm{PF}_\\infty$;\n\\item its bilateral Laplace transform $\\mathcal B\\Phi$ satisfies the reflection law\n$\\mathcal B\\Phi(s)=\\mathcal B\\Phi(\\tfrac12-s)$ on the common domain of absolute convergence;\n\\item $\\mathcal B\\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation\n\\[\n\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)},\n\\]\nwhere $\\Psi$ is Laguerre--P\\'olya and hence has only real zeros.\n\\end{enumerate}", "theorem_type": ["Existence", "Existential–Universal"], "mcq": {"question": "Fix a macroscopic length $L>0$ and define the scaling-limit trace by\n\\[\nK_L(t):=\\lim_{s\\to\\infty} N(s)\\,p^{\\mathrm{cyc}}_{s^2 t}(0,0),\\qquad t>0,\n\\]\nwhere $N=N(s)$ satisfies $N/s\\to L$. Assume that for every $L>0$ and $t>0$ this limit exists, with convergence locally uniform in $t$ on compact subsets of $(0,\\infty)$, so that the scaling-limit kernel $K_L$ exists. Let $\\mathcal B\\Phi(s):=\\int_{\\mathbb R} e^{-sx}\\Phi(x)\\,dx$ denote the bilateral Laplace transform on its domain of absolute convergence, and let $\\mathrm{PF}_\\infty$ denote the Pólya frequency class of infinite order. Which statement holds under these assumptions?", "correct_choice": {"label": "A", "text": "There exists a nonnegative kernel $\\Phi\\in L^1(\\mathbb R)$ such that $\\Phi\\in\\mathrm{PF}_\\infty$, its bilateral Laplace transform satisfies the reflection law $\\mathcal B\\Phi(s)=\\mathcal B\\Phi(\\tfrac12-s)$ on the common domain of absolute convergence, and $\\mathcal B\\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation\n\\[\n\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)},\n\\]\nwhere $\\Psi$ is a Laguerre--Pólya entire function and hence has only real zeros."}, "choices": [{"label": "B", "text": "There exists a nonnegative kernel $\\Phi\\in L^1(\\mathbb R)$ such that $\\Phi\\in\\mathrm{PF}_\\infty$, its bilateral Laplace transform satisfies the reflection law $\\mathcal B\\Phi(s)=\\mathcal B\\Phi(-s)$ on the common domain of absolute convergence, and $\\mathcal B\\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation\n\\[\n\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)},\n\\]\nwhere $\\Psi$ is a Laguerre--Pólya entire function and hence has only real zeros."}, {"label": "C", "text": "There exists a nonnegative kernel $\\Phi\\in L^1(\\mathbb R)$ such that $\\Phi\\in\\mathrm{PF}_\\infty$ and $\\mathcal B\\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation\n\\[\n\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)},\n\\]\nwhere $\\Psi$ is a Laguerre--Pólya entire function and hence has only real zeros."}, {"label": "D", "text": "There exists a nonnegative kernel $\\Phi\\in L^1(\\mathbb R)$ such that $\\Phi\\in\\mathrm{PF}_\\infty$, its bilateral Laplace transform satisfies the reflection law $\\mathcal B\\Phi(s)=\\mathcal B\\Phi(\\tfrac12-s)$ for all $s\\in\\mathbb C$, and $\\mathcal B\\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation\n\\[\n\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)},\n\\]\nwhere $\\Psi$ is a Laguerre--Pólya entire function and hence has only real zeros."}, {"label": "E", "text": "There exists a nonnegative kernel $\\Phi\\in L^1(\\mathbb R)$ such that its bilateral Laplace transform satisfies the reflection law $\\mathcal B\\Phi(s)=\\mathcal B\\Phi(\\tfrac12-s)$ on the common domain of absolute convergence, and $\\mathcal B\\Phi$ admits a canonical Schoenberg--Edrei--Karlin representation\n\\[\n\\mathcal B\\Phi(s)=\\frac{E(s)}{\\Psi(s)},\n\\]\nwhere $\\Psi$ is a Laguerre--Pólya entire function and hence has only real zeros, with $\\Phi$ therefore necessarily in \\mathrm{PF}_\\infty."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "reflection-center-at-1/2", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped reflection law clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "domain-of-absolute-convergence restriction", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "direction of SEK implication versus PF_infty hypothesis", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option, nor does it contain a giveaway phrase uniquely matching choice A. It states hypotheses and asks for the valid conclusion."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem gives the assumptions and asks which theorem statement holds. The correct option is the theorem conclusion with only nearby perturbations in the distractors."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact valid conclusion from a weaker true statement (C) and subtle false strengthenings or alterations (B, D, E). However, it mostly tests precise theorem recognition rather than generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: wrong reflection center, omission of a necessary clause, unjustified extension of domain, and reversal/misuse of implication involving PF_infty."}, "total_score": 5, "overall_assessment": "Technically well-constructed with strong distractors and little answer leakage, but it is mainly a direct theorem-identification question rather than a genuinely generative reasoning task."}}
{"id": "2602.01380v1", "paper_link": "http://arxiv.org/abs/2602.01380v1", "theorems_cnt": 5, "theorem": {"env_name": "lemma", "content": "\\label{newlemma}\nLet $D$ be a square-free integer such that $\\operatorname{rank}_{\\mathbb Z}E_0^{D}(\\mathbb Q)=0$. Then $E_0(\\mathbb Q(\\sqrt{D}))=E_0(\\mathbb Q)$, except when $D=\\pm 2$.", "start_pos": 17703, "end_pos": 17885, "label": "newlemma"}, "ref_dict": {"pointsC4": "\\begin{center} \\caption{$C_4(\\QD)$}\\label{pointsC4}\n\\begin{tabular}{|c|c|c|c|}\n\\cline{2-4} \n \\multicolumn{1}{c|}{} & $D\\ne 2,3,6$ & $D=3$ & $D=2$  \\\\ \n\\hline\n$C_4$ & $\\begin{array}{c} (0,0)\\\\ (1,\\pm 4)  \\end{array}$ & $\\begin{array}{c}(-1,-2\\sqrt{3})\\\\ (7\\pm 4\\sqrt{3},0)\\end{array}$ & $\\begin{array}{c}\n(-3-2\\sqrt{2},\\pm(8+6\\sqrt{2}))\\\\ \n(-3+2\\sqrt{2},\\pm(8-6\\sqrt{2}))\\\\ \n\\end{array}$\\\\\n   \\hline\n\\end{tabular} \n\\end{center}", "growth": "\\begin{longtable}{|c|c|}\n\\caption{Torsion growth over quadratic fields}\\label{growth}\\\\\n\\hline \n$E$ & \\mbox{$D$ such that $E(\\Q)_{\\operatorname{tors}}\\ne \nE(\\QD)_{\\operatorname{tors}}$}\\\\\n\\hline \n\n$E_0$ & $-2$ \\\\\n\\hline\n$E_1$   &  $-1$, $\\pm 3$ \\\\\n\\hline\n$E^{-1}_1$     &  $-1$,$\\pm 3$\\\\\n\\hline \n$E_4$ & 2,3,6\\\\\n\\hline\n$E_6$ & $\\emptyset$ \\\\\n\\hline \n$E_6^{-1}$ & $-1$,$\\pm 3$\\\\\n\\hline\n\\end{longtable}", "Conj": "\\begin{conj}\\label{Conj}\nThere are infinitely many non-constant arithmetic progressions of five squares pro\\-perly defined over a quadratic extension $K_1$ (resp. $K_2$) of $\\Q(\\sqrt{-2})$ (resp. of $\\Q(\\sqrt{2}))$.\n\\begin{itemize}\n\\item[(i)] In the case $\\Q(\\sqrt{-2})$, any of them are, up to equivalence, of the form $\\left(a^2,b^2,-2c^2,-m d^2,-2e^2\\right)$, where $a,b,c,d,e,m\\in\\Z$ and $m>0$ is square-free. In particular, $K_1=\\Q(\\sqrt{-2},\\sqrt{-m})$. \n\\item[(ii)] In the case $\\Q(\\sqrt{2})$, any of them are, up to equivalence, of the form $\\left(2 a^2,b^2,2c^2,m,e^2\\right)$, where $a,b,c,e,m\\in\\Z$ and $m$ is square-free. In particular, $K_2=\\Q(\\sqrt{2},\\sqrt{m})$. \n\\end{itemize}\n\\end{conj}", "TableC": "\\begin{longtable}{|l|l|l|l|}\n\\caption{Curves} \\label{TableC} \\\\\n\\hline \n$C_0$ & $y^2 = x^4-2x^3+2x^2+2x+1$  &  \\href{https://www.lmfdb.org/EllipticCurve/Q/192a2/}{\\texttt{192.a2}} &  $E_0$\\\\\n\\hline \n$C_1$& $y^2 = (x+4)(x^2+4x+16)$  & \\href{https://www.lmfdb.org/EllipticCurve/Q/24a4/}{\\texttt{24.a5}} & $E_1$\\\\\n\\hline\n$C_2$& $y^2 = x(x^2+4x+16)$  & \\href{https://www.lmfdb.org/EllipticCurve/Q/48a4/}{\\texttt{48.a5}}&  $E^{-1}_1$\\\\\n\\hline \n$C_3$& $y^2 = x(x+4)(x^2+4x+16)$ &  \\href{https://www.lmfdb.org/EllipticCurve/Q/24a4/}{\\texttt{24.a5}}&  $E_1$\\\\\n\\hline \n$C_4$& $y^2 = x(x^2+14x+1)$ &  \\href{https://www.lmfdb.org/EllipticCurve/Q/24a3/}{\\texttt{24.a2}}&  $E_4$\\\\\n\\hline \n$C_5$ & $y^2 = x^4+4x^2+16$ &  \\href{https://www.lmfdb.org/EllipticCurve/Q/48a1/}{\\texttt{48.a4}} &  $E^{-1}_6$\\\\\n\\hline \n$C_6$ & $y^2 = x^4+14x^2+1$  &  \\href{https://www.lmfdb.org/EllipticCurve/Q/24a1/}{\\texttt{24.a4}} &  $E_6$\\\\\n\\hline\n\\end{longtable}", "points": "\\begin{center} \\caption{$C_k(\\QD)$ for $k=1,2,3,5,6$.}\\label{points}\n\\begin{tabular}{|c|c|c|c|c|}\n\\cline{2-5} \n \\multicolumn{1}{c|}{}\n & $D\\ne -1,\\pm 3$ & $D=-1$ & $D=-3$ & $D=3$  \\\\ \n\\hline\n$C_1$ & $\\begin{array}{c}(-4,0)\\\\ (0,\\pm 8) \\end{array}$& $(-4\\pm 4i,\\pm8)$ & $\\begin{array}{c}(-8,\\pm8\\sqrt{-3})\\\\(-2\\pm 2 \\sqrt{-3},0)\\end{array}$ & $\\begin{array}{c}\n(4-4\\sqrt{3},\\pm(24-16\\sqrt{3}))\\\\ \n(4+4 \\sqrt{3},\\pm(24+16\\sqrt{3}))\n\\end{array}$  \\\\\n\\hline\n$C_2$ & $(0,0)$ & $\\begin{array}{c}(\\pm 4i,\\pm 8i)\\\\(-4,\\pm 8i) \\end{array}$& $(-2\\pm 2\\sqrt{-3},0)$ & $(4,\\pm 8\\sqrt{3}) $ \\\\ \n\\hline\n$C_3$ & $\\begin{array}{c} (0,0)\\\\(-4,0) \\end{array}$& $(-2\\pm 2 i,\\pm 8i)$ & $\\begin{array}{c}(-2\\pm 2\\sqrt{-3},0)\\\\(-2,\\pm 4 \\sqrt{-3})\\end{array}$ & $(-2\\pm 2 \\sqrt{3},\\pm 8\\sqrt{3})$  \\\\\n\\hline\n$C_5$ & $(0,\\pm 4)$ & $(\\pm 2i,\\pm 4)$ & $(\\pm1\\pm\\sqrt{-3},0)$ & $(\\pm 2,\\pm 4\\sqrt{3})$  \\\\ \n\\hline\n$C_6$ & $\\begin{array}{c} (0,\\pm 1)\\\\ (\\pm 1,\\pm 4) \\end{array}$ & $-$ & $-$ & $- $ \\\\ \n\\hline\n\\end{tabular} \n\\end{center}"}, "pre_theorem_intro_text_len": 13083, "pre_theorem_intro_text": "Fermat stated that there are no non-constant arithmetic progressions of four squares over $\\mathbb{Q}$ in 1640 and Euler proved this in 1780. It is natural to ask for what would happen in number fields. Over quadratic fields, Xarles \\cite{art17} showed that there are no non-constant arithmetic progressions of six squares and Gonz\\'alez-Jim\\'enez and Xarles \\cite{GJX} characterized non-constant arithmetic progressions of five squares. Over cubic fields, Bremner and Siksek \\cite{art5} showed that there are no non-constant arithmetic progressions of five squares. \n\nIn this paper, we extend the results of  Xarles \\cite{art17} and of Gonz\\'alez-Jim\\'enez and Xarles \\cite{GJX}  to quadratic extensions of $\\mathbb Q(\\sqrt{D})$, where $D$ is a square-free integer satisfying certain conditions on the $D$-quadratic twist of the elliptic curves\n$$\nE_0\\,:\\, y^2 = x^3 - x^2 - 9x + 9 \\qquad \\mbox{and}\\qquad E_1\\,:\\, y^2 = x^3 - x^2 + x.\n$$\nFor any elliptic curve $E$ and integer $D$, we denote by $E^D$ the $D$-quadratic twist of $E$. \n\n{\nLet $n$ be a positive integer and $K$ be a field. Let $a_1,\\dots,a_n\\in K$ such that $(a_1^2,a_2^2,\\dots,a_n^2)$ is an arithmetic progression of length $n$. We say that this arithmetic progression is equivalent to $(s^2 a_1^2,s^2 a_2^2,\\dots,s^2a_n^2)$ for any $s\\in K^*$ and to the arithmetic progression $(a_n^2,\\dots,a_2^2,a_1^2)$. We say that $(a_1^2,a_2^2,\\dots,a_n^2)$ is properly defined over a number field $K$ if $a_1,\\dots,a_n\\in K$  and  $\\{a_1,\\dots,a_n\\}\\not\\subset F$ for any proper subfield $F$ of $K$.} Let $i=\\sqrt{-1}$. \n\n\\\n\nThe main theorems of the paper are the following.\n\n\\begin{thm}\\label{main}\nLet $D$ be a square-free integer, $D\\ne -1,\\pm 2,3$, such that $\\operatorname{rank}_{\\mathbb Z}E_1^{\\pm D}(\\mathbb Q)= 0$ and $K$ a quadratic extension of $\\Q(\\sqrt{D})$. Then\n\\begin{itemize}\n\\item[(A)] If $\\operatorname{rank}_{\\mathbb Z}E_0^{D}(\\mathbb Q)=0$, then there does not exist any non-constant arithmetic progression of five squares properly defined over $K$.\n\n\\item[(B)] If $\\operatorname{rank}_{\\mathbb Z}E_0^{D}(\\mathbb Q)\\ne 0$ and the class number of $\\Q(\\sqrt{D})$ is $1$, then a non-constant arithmetic progression of five squares properly defined over $K$ is, up to equivalence, of the form $\\left(a^2,b^2,c^2,\\alpha \\, d^2,e^2\\right)$, where $a,b,c,d,e,\\alpha \\in\\mathbb Q(\\sqrt{D})$ and $\\alpha$ is non-square.\n\\end{itemize}\n\\end{thm}\n\n\\begin{thm}\\label{maincor}\nThere exists a non-constant arithmetic progression of five squares properly defined over a quadratic extension $K$ of\n\\begin{itemize}\n    \\item $\\mathbb Q(i)$ if and only if $K=\\mathbb Q(i,\\sqrt{2})$. In this case, $(-2,-1, 0,1,2)$ is the unique, up to equivalence, non-constant arithmetic progression of five squares properly defined over $K$.\n     \\item $\\mathbb Q(\\sqrt{3})$ if and only if $K=\\mathbb Q(\\sqrt{3},\\sqrt{2})$. In this case, $(0,1,2,3,4)$ is the unique, up to equivalence, non-constant arithmetic progression of five squares properly defined over $K$.\n     \\item $\\mathbb Q(\\sqrt{2})$ if and only if $K=\\mathbb Q(\\sqrt{2},\\sqrt{3})$, $K=\\mathbb Q(\\sqrt{2},i)$ or $K=\\mathbb Q(\\sqrt{2},\\sqrt{\\alpha})$, for some squarefree $\\alpha \\in\\mathbb Q(\\sqrt{2})$. In the first case, $(0,1,2,3,4)$ is the unique, up to equivalence, non-constant arithmetic progression of five squares properly defined over $\\mathbb Q(\\sqrt{2},\\sqrt{3})$; and in the second case, $(-2,-1, 0,1,2)$ is the unique, up to equivalence, non-constant arithmetic progression of five squares properly defined over $\\mathbb Q(\\sqrt{2},i)$. In the last case, a non-constant arithmetic progression of five squares properly defined over $K$ is, up to equivalence, of the form $\\left(a^2,b^2,c^2,\\alpha \\, d^2,e^2\\right)$ where $a,b,c,d,e \\in\\mathbb Q(\\sqrt{2})$. \n     \\item $\\mathbb Q(\\sqrt{-2})$ if and only if $K=\\mathbb Q(\\sqrt{-2},i)$ or $K=\\mathbb Q(\\sqrt{-2},\\sqrt{\\alpha})$, for some squarefree $\\alpha \\in\\mathbb Q(\\sqrt{-2})$. In the first case, $(-2,-1, 0,1,2)$ is the unique, up to equivalence, non-constant arithmetic progression of five squares properly defined over $\\mathbb Q(\\sqrt{-2},i)$. In the second case, a non-constant arithmetic progression of five squares properly defined over $K$ is, up to equivalence, of the form $\\left(a^2,b^2,c^2,\\alpha \\, d^2,e^2\\right)$ where $a,b,c,d,e \\in\\mathbb Q(\\sqrt{-2})$.\n\\end{itemize}  \n\\end{thm}\n\\begin{remark}\nNote that $\\operatorname{rank}_{\\mathbb Z}E_1^{\\pm D}(\\mathbb Q)= 0$ and $\\operatorname{rank}_{\\mathbb Z}E_0^{D}(\\mathbb Q)= 0$ if $D\\in\\{-1,\\pm 2,3\\}$.\n\\end{remark}\n\n\\begin{thm}\\label{six}\nLet $D$ be a square-free integer such that \n$$\n\\operatorname{rank}_{\\mathbb Z}E_1^{\\pm D}(\\mathbb Q)= 0\\,,\n$$\nand $\\operatorname{rank}_{\\mathbb Z}E_0^{D}(\\mathbb Q)=0$ or if $\\operatorname{rank}_{\\mathbb Z}E_0^{D}(\\mathbb Q)\\ne 0$  the class number of $\\Q(\\sqrt{D})$ is $1$. Then there does not exist any non-constant arithmetic progression of six squares properly defined over any quadratic extension of $\\Q(\\sqrt{D})$. \n\\end{thm}\n\n\\begin{remark}\nThe following list shows all the square-free integers $D$, $|D|<200$, satisfying $\\operatorname{rank}_{\\mathbb Z}E_1^{\\pm D}(\\mathbb Q)= 0$ and  $\\operatorname{rank}_{\\mathbb Z}E_0^{D}(\\mathbb Q)= 0$\\,:\n$$\n-158, -123, -62, -51, -3, -2, -1, 2, 3, 7, 31, 51, 62, 79, 103, 110, 123, 127,\n151, 158, 194, 195, 199.\n$$\nIf we allow $\\operatorname{rank}_{\\mathbb Z}E_0^{D}(\\mathbb Q)\\ne 0$ and the class number of $\\Q(\\sqrt{D})$ is $1$ then $D\\in\\{-7, 38, 86\\}$. \n\\end{remark}\n\n\\begin{remark}\nThere exist square-free integers $D$ and non-constant arithmetic progressions of six squares properly defined over a quadratic extension of $\\Q(\\sqrt{D})$. For example, when $D=409$ then $$(7^2, 13^2, 17^2, (\\sqrt{409})^2, 23^2, (\\sqrt{649})^2)$$ is an arithmetic progression properly defined over $\\mathbb Q(\\sqrt{409},\\sqrt{649})$.\n\\end{remark}\n\n\\begin{conj}\\label{Conj}\nThere are infinitely many non-constant arithmetic progressions of five squares pro\\-perly defined over a quadratic extension $K_1$ (resp. $K_2$) of $\\mathbb Q(\\sqrt{-2})$ (resp. of $\\mathbb Q(\\sqrt{2}))$.\n\\begin{itemize}\n\\item[(i)] In the case $\\mathbb Q(\\sqrt{-2})$, any of them are, up to equivalence, of the form $\\left(a^2,b^2,-2c^2,-m d^2,-2e^2\\right)$, where $a,b,c,d,e,m\\in\\mathbb Z$ and $m>0$ is square-free. In particular, $K_1=\\mathbb Q(\\sqrt{-2},\\sqrt{-m})$. \n\\item[(ii)] In the case $\\mathbb Q(\\sqrt{2})$, any of them are, up to equivalence, of the form $\\left(2 a^2,b^2,2c^2,m,e^2\\right)$, where $a,b,c,e,m\\in\\mathbb Z$ and $m$ is square-free. In particular, $K_2=\\mathbb Q(\\sqrt{2},\\sqrt{m})$. \n\\end{itemize}\n\\end{conj}\n\nConjecture \\ref{Conj} is supported by the calculation in Section \\ref{sect_para}. For the cases $D\\in\\{-7, 38, 86\\}$, there is not a similar conjecture.\n\nOur approach is different from the approach in \\cite{GJX} and \\cite{art17}. We follow the approach in Mordell's paper \\cite{art12}, where he gave alternative proofs to the results of Aigner \\cite{art1} and Faddeev \\cite{art7} on the solution of the equation $x^4+y^4=1$ in quadratic and cubic number fields. Mordell's approach has the advantage that it is very concrete in calculation.\n\nAnother important ingredient  is the theory concerning the growth of the torsion subgroup of elliptic curves under base change. Upon assuming $\\operatorname{rank}_{\\mathbb Z}E_1^{\\pm D}(\\mathbb Q)= 0$, it becomes imperative to determine the torsion subgroup of certain elliptic curves defined over $\\mathbb Q$ over any quadratic field. The answer to this inquiry lies within the LMFDB online database  \\cite{lmfdb}. The theory of torsion growth has undergone extensive scrutiny in recent years. Particularly noteworthy references are \\cite{GJN1,GJN2} and \\cite{GJT1,GJT2,N} (for quadratic fields). \n\nIn the following subsection we include all the elliptic curves that are relevant to this article, together with the necessary information concerning rational points over quadratic fields. This material will be used in the proofs of the previous theorems.\n\n\\subsection*{Data of some elliptic curves}\nTable \\ref{TableC} shows the elliptic curves used in this article. For each curve \\( C_k \\), with \\( k = 0,1,\\dots,6 \\), the third column gives the corresponding label in the LMFDB database \\cite{lmfdb}, while the fourth column lists the corresponding \\(\\mathbb{Q}\\)-isomorphic elliptic curve.\n\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{longtable}{|l|l|l|l|}\n\\caption{Curves} \\label{TableC} \\\\\n\\hline \n$C_0$ & $y^2 = x^4-2x^3+2x^2+2x+1$  &  \\href{https://www.lmfdb.org/EllipticCurve/Q/192a2/}{\\texttt{192.a2}} &  $E_0$\\\\\n\\hline \n$C_1$& $y^2 = (x+4)(x^2+4x+16)$  & \\href{https://www.lmfdb.org/EllipticCurve/Q/24a4/}{\\texttt{24.a5}} & $E_1$\\\\\n\\hline\n$C_2$& $y^2 = x(x^2+4x+16)$  & \\href{https://www.lmfdb.org/EllipticCurve/Q/48a4/}{\\texttt{48.a5}}&  $E^{-1}_1$\\\\\n\\hline \n$C_3$& $y^2 = x(x+4)(x^2+4x+16)$ &  \\href{https://www.lmfdb.org/EllipticCurve/Q/24a4/}{\\texttt{24.a5}}&  $E_1$\\\\\n\\hline \n$C_4$& $y^2 = x(x^2+14x+1)$ &  \\href{https://www.lmfdb.org/EllipticCurve/Q/24a3/}{\\texttt{24.a2}}&  $E_4$\\\\\n\\hline \n$C_5$ & $y^2 = x^4+4x^2+16$ &  \\href{https://www.lmfdb.org/EllipticCurve/Q/48a1/}{\\texttt{48.a4}} &  $E^{-1}_6$\\\\\n\\hline \n$C_6$ & $y^2 = x^4+14x^2+1$  &  \\href{https://www.lmfdb.org/EllipticCurve/Q/24a1/}{\\texttt{24.a4}} &  $E_6$\\\\\n\\hline\n\\end{longtable}\n\\end{center}\nNote that the elliptic curves $E_1,E_4$ and $E_6$ belongs to the same $\\mathbb Q$-isogeny class.  In particular, for any integer $D$, the ranks of any $D$-quadratic twist coincide. In particular, we have $\\operatorname{rank}_{\\mathbb Z}C_k(\\mathbb Q)=0$ for $k=1,\\dots,6$. Let $E$ be an elliptic curve defined over $\\mathbb Q$. Using the known formula (see \\cite{Kra})\n$$\n\\operatorname{rank}_{\\mathbb Z}E(\\Q(\\sqrt{D}))=\\operatorname{rank}_{\\mathbb Z}E(\\mathbb Q)+\n\\operatorname{rank}_{\\mathbb Z}E^D(\\mathbb Q),\n$$\nwe obtain that $\\operatorname{rank}_{\\mathbb Z}E_1^{\\pm D}(\\mathbb Q)= 0$ is equivalent to  $\\operatorname{rank}_{\\mathbb Z}C_k(\\Q(\\sqrt{D}))=0$ for $k=1,\\dots,6$. Assuming $\\operatorname{rank}_{\\mathbb Z}E_1^{\\pm D}(\\mathbb Q)= 0$ for the square-free integer $D$, to compute $C_k(\\Q(\\sqrt{D}))$, for $k=1,\\dots,6$, we need to compute the torsion subgroup over $\\Q(\\sqrt{D})$. In our particular case, given an elliptic curve $E$ defined over $\\mathbb Q$, there are only a finite number of square-free integers $D$ such that $E(\\mathbb Q)_{\\operatorname{tors}}\\ne E(\\Q(\\sqrt{D}))_{\\operatorname{tors}}$. If the conductor of $E$ is less than $400000$, these square-free integers appear in the LMFDB database \\cite{lmfdb} and are listed in Table \\ref{growth}.\n\\begin{center}\n\\begin{longtable}{|c|c|}\n\\caption{Torsion growth over quadratic fields}\\label{growth}\\\\\n\\hline \n$E$ & \\mbox{$D$ such that $E(\\mathbb Q)_{\\operatorname{tors}}\\ne \nE(\\Q(\\sqrt{D}))_{\\operatorname{tors}}$}\\\\\n\\hline \n\n$E_0$ & $-2$ \\\\\n\\hline\n$E_1$   &  $-1$, $\\pm 3$ \\\\\n\\hline\n$E^{-1}_1$     &  $-1$,$\\pm 3$\\\\\n\\hline \n$E_4$ & 2,3,6\\\\\n\\hline\n$E_6$ & $\\emptyset$ \\\\\n\\hline \n$E_6^{-1}$ & $-1$,$\\pm 3$\\\\\n\\hline\n\\end{longtable}\n\\end{center}\nNote that for those elliptic curves and values of $D$, we have $\\operatorname{rank}_{\\mathbb Z}E^D(\\mathbb Q)=0$ except in the case $E^6_4$. Finally, for any square-free integer $D$, Tables \\ref{points} and \\ref{pointsC4}  show  $C_k(\\Q(\\sqrt{D}))$ for $k=1,\\dots,6$. When $D=-1,2,\\pm 3$, the points at the second column correspond to $C_k(\\mathbb Q)$ and they must be added to the corresponding set $C_k(\\Q(\\sqrt{D}))$.\n\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{table}[htbp] \n\\begin{center} \\caption{$C_k(\\Q(\\sqrt{D}))$ for $k=1,2,3,5,6$.}\\label{points}\n\\begin{tabular}{|c|c|c|c|c|}\n\\cline{2-5} \n \\multicolumn{1}{c|}{}\n & $D\\ne -1,\\pm 3$ & $D=-1$ & $D=-3$ & $D=3$  \\\\ \n\\hline\n$C_1$ & $\\begin{array}{c}(-4,0)\\\\ (0,\\pm 8) \\end{array}$& $(-4\\pm 4i,\\pm8)$ & $\\begin{array}{c}(-8,\\pm8\\sqrt{-3})\\\\(-2\\pm 2 \\sqrt{-3},0)\\end{array}$ & $\\begin{array}{c}\n(4-4\\sqrt{3},\\pm(24-16\\sqrt{3}))\\\\ \n(4+4 \\sqrt{3},\\pm(24+16\\sqrt{3}))\n\\end{array}$  \\\\\n\\hline\n$C_2$ & $(0,0)$ & $\\begin{array}{c}(\\pm 4i,\\pm 8i)\\\\(-4,\\pm 8i) \\end{array}$& $(-2\\pm 2\\sqrt{-3},0)$ & $(4,\\pm 8\\sqrt{3}) $ \\\\ \n\\hline\n$C_3$ & $\\begin{array}{c} (0,0)\\\\(-4,0) \\end{array}$& $(-2\\pm 2 i,\\pm 8i)$ & $\\begin{array}{c}(-2\\pm 2\\sqrt{-3},0)\\\\(-2,\\pm 4 \\sqrt{-3})\\end{array}$ & $(-2\\pm 2 \\sqrt{3},\\pm 8\\sqrt{3})$  \\\\\n\\hline\n$C_5$ & $(0,\\pm 4)$ & $(\\pm 2i,\\pm 4)$ & $(\\pm1\\pm\\sqrt{-3},0)$ & $(\\pm 2,\\pm 4\\sqrt{3})$  \\\\ \n\\hline\n$C_6$ & $\\begin{array}{c} (0,\\pm 1)\\\\ (\\pm 1,\\pm 4) \\end{array}$ & $-$ & $-$ & $- $ \\\\ \n\\hline\n\\end{tabular} \n\\end{center}\n\\end{table}\n\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{table}[htbp] \n\\begin{center} \\caption{$C_4(\\Q(\\sqrt{D}))$}\\label{pointsC4}\n\\begin{tabular}{|c|c|c|c|}\n\\cline{2-4} \n \\multicolumn{1}{c|}{} & $D\\ne 2,3,6$ & $D=3$ & $D=2$  \\\\ \n\\hline\n$C_4$ & $\\begin{array}{c} (0,0)\\\\ (1,\\pm 4)  \\end{array}$ & $\\begin{array}{c}(-1,-2\\sqrt{3})\\\\ (7\\pm 4\\sqrt{3},0)\\end{array}$ & $\\begin{array}{c}\n(-3-2\\sqrt{2},\\pm(8+6\\sqrt{2}))\\\\ \n(-3+2\\sqrt{2},\\pm(8-6\\sqrt{2}))\\\\ \n\\end{array}$\\\\\n   \\hline\n\\end{tabular} \n\\end{center}\n\\end{table}\n\n\\newpage", "context": "\\begin{thm}\\label{main}\nLet $D$ be a square-free integer, $D\\ne -1,\\pm 2,3$, such that $\\operatorname{rank}_{\\mathbb Z}E_1^{\\pm D}(\\mathbb Q)= 0$ and $K$ a quadratic extension of $\\Q(\\sqrt{D})$. Then\n\\begin{itemize}\n\\item[(A)] If $\\operatorname{rank}_{\\mathbb Z}E_0^{D}(\\mathbb Q)=0$, then there does not exist any non-constant arithmetic progression of five squares properly defined over $K$.\n\n\\begin{thm}\\label{six}\nLet $D$ be a square-free integer such that \n$$\n\\operatorname{rank}_{\\mathbb Z}E_1^{\\pm D}(\\mathbb Q)= 0\\,,\n$$\nand $\\operatorname{rank}_{\\mathbb Z}E_0^{D}(\\mathbb Q)=0$ or if $\\operatorname{rank}_{\\mathbb Z}E_0^{D}(\\mathbb Q)\\ne 0$ the class number of $\\Q(\\sqrt{D})$ is $1$. Then there does not exist any non-constant arithmetic progression of six squares properly defined over any quadratic extension of $\\Q(\\sqrt{D})$. \n\\end{thm}\n\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{longtable}{|l|l|l|l|}\n\\caption{Curves} \\label{TableC} \\\\\n\\hline \n$C_0$ & $y^2 = x^4-2x^3+2x^2+2x+1$ & \\href{https://www.lmfdb.org/EllipticCurve/Q/192a2/}{\\texttt{192.a2}} & $E_0$\\\\\n\\hline \n$C_1$& $y^2 = (x+4)(x^2+4x+16)$ & \\href{https://www.lmfdb.org/EllipticCurve/Q/24a4/}{\\texttt{24.a5}} & $E_1$\\\\\n\\hline\n$C_2$& $y^2 = x(x^2+4x+16)$ & \\href{https://www.lmfdb.org/EllipticCurve/Q/48a4/}{\\texttt{48.a5}}& $E^{-1}_1$\\\\\n\\hline \n$C_3$& $y^2 = x(x+4)(x^2+4x+16)$ & \\href{https://www.lmfdb.org/EllipticCurve/Q/24a4/}{\\texttt{24.a5}}& $E_1$\\\\\n\\hline \n$C_4$& $y^2 = x(x^2+14x+1)$ & \\href{https://www.lmfdb.org/EllipticCurve/Q/24a3/}{\\texttt{24.a2}}& $E_4$\\\\\n\\hline \n$C_5$ & $y^2 = x^4+4x^2+16$ & \\href{https://www.lmfdb.org/EllipticCurve/Q/48a1/}{\\texttt{48.a4}} & $E^{-1}_6$\\\\\n\\hline \n$C_6$ & $y^2 = x^4+14x^2+1$ & \\href{https://www.lmfdb.org/EllipticCurve/Q/24a1/}{\\texttt{24.a4}} & $E_6$\\\\\n\\hline\n\\end{longtable}\n\\end{center}\nNote that the elliptic curves $E_1,E_4$ and $E_6$ belongs to the same $\\mathbb Q$-isogeny class. In particular, for any integer $D$, the ranks of any $D$-quadratic twist coincide. In particular, we have $\\operatorname{rank}_{\\mathbb Z}C_k(\\mathbb Q)=0$ for $k=1,\\dots,6$. Let $E$ be an elliptic curve defined over $\\mathbb Q$. Using the known formula (see \\cite{Kra})\n$$\n\\operatorname{rank}_{\\mathbb Z}E(\\Q(\\sqrt{D}))=\\operatorname{rank}_{\\mathbb Z}E(\\mathbb Q)+\n\\operatorname{rank}_{\\mathbb Z}E^D(\\mathbb Q),\n$$\nwe obtain that $\\operatorname{rank}_{\\mathbb Z}E_1^{\\pm D}(\\mathbb Q)= 0$ is equivalent to $\\operatorname{rank}_{\\mathbb Z}C_k(\\Q(\\sqrt{D}))=0$ for $k=1,\\dots,6$. Assuming $\\operatorname{rank}_{\\mathbb Z}E_1^{\\pm D}(\\mathbb Q)= 0$ for the square-free integer $D$, to compute $C_k(\\Q(\\sqrt{D}))$, for $k=1,\\dots,6$, we need to compute the torsion subgroup over $\\Q(\\sqrt{D})$. In our particular case, given an elliptic curve $E$ defined over $\\mathbb Q$, there are only a finite number of square-free integers $D$ such that $E(\\mathbb Q)_{\\operatorname{tors}}\\ne E(\\Q(\\sqrt{D}))_{\\operatorname{tors}}$. If the conductor of $E$ is less than $400000$, these square-free integers appear in the LMFDB database \\cite{lmfdb} and are listed in Table \\ref{growth}.\n\\begin{center}\n\\begin{longtable}{|c|c|}\n\\caption{Torsion growth over quadratic fields}\\label{growth}\\\\\n\\hline \n$E$ & \\mbox{$D$ such that $E(\\mathbb Q)_{\\operatorname{tors}}\\ne \nE(\\Q(\\sqrt{D}))_{\\operatorname{tors}}$}\\\\\n\\hline\n\n$E_0$ & $-2$ \\\\\n\\hline\n$E_1$ & $-1$, $\\pm 3$ \\\\\n\\hline\n$E^{-1}_1$ & $-1$,$\\pm 3$\\\\\n\\hline \n$E_4$ & 2,3,6\\\\\n\\hline\n$E_6$ & $\\emptyset$ \\\\\n\\hline \n$E_6^{-1}$ & $-1$,$\\pm 3$\\\\\n\\hline\n\\end{longtable}\n\\end{center}\nNote that for those elliptic curves and values of $D$, we have $\\operatorname{rank}_{\\mathbb Z}E^D(\\mathbb Q)=0$ except in the case $E^6_4$. Finally, for any square-free integer $D$, Tables \\ref{points} and \\ref{pointsC4} show $C_k(\\Q(\\sqrt{D}))$ for $k=1,\\dots,6$. When $D=-1,2,\\pm 3$, the points at the second column correspond to $C_k(\\mathbb Q)$ and they must be added to the corresponding set $C_k(\\Q(\\sqrt{D}))$.\n\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{table}[htbp] \n\\begin{center} \\caption{$C_4(\\Q(\\sqrt{D}))$}\\label{pointsC4}\n\\begin{tabular}{|c|c|c|c|}\n\\cline{2-4} \n \\multicolumn{1}{c|}{} & $D\\ne 2,3,6$ & $D=3$ & $D=2$ \\\\ \n\\hline\n$C_4$ & $\\begin{array}{c} (0,0)\\\\ (1,\\pm 4) \\end{array}$ & $\\begin{array}{c}(-1,-2\\sqrt{3})\\\\ (7\\pm 4\\sqrt{3},0)\\end{array}$ & $\\begin{array}{c}\n(-3-2\\sqrt{2},\\pm(8+6\\sqrt{2}))\\\\ \n(-3+2\\sqrt{2},\\pm(8-6\\sqrt{2}))\\\\ \n\\end{array}$\\\\\n \\hline\n\\end{tabular} \n\\end{center}\n\\end{table}\n\n\\newpage", "full_context": "\\begin{thm}\\label{main}\nLet $D$ be a square-free integer, $D\\ne -1,\\pm 2,3$, such that $\\operatorname{rank}_{\\mathbb Z}E_1^{\\pm D}(\\mathbb Q)= 0$ and $K$ a quadratic extension of $\\Q(\\sqrt{D})$. Then\n\\begin{itemize}\n\\item[(A)] If $\\operatorname{rank}_{\\mathbb Z}E_0^{D}(\\mathbb Q)=0$, then there does not exist any non-constant arithmetic progression of five squares properly defined over $K$.\n\n\\begin{thm}\\label{six}\nLet $D$ be a square-free integer such that \n$$\n\\operatorname{rank}_{\\mathbb Z}E_1^{\\pm D}(\\mathbb Q)= 0\\,,\n$$\nand $\\operatorname{rank}_{\\mathbb Z}E_0^{D}(\\mathbb Q)=0$ or if $\\operatorname{rank}_{\\mathbb Z}E_0^{D}(\\mathbb Q)\\ne 0$ the class number of $\\Q(\\sqrt{D})$ is $1$. Then there does not exist any non-constant arithmetic progression of six squares properly defined over any quadratic extension of $\\Q(\\sqrt{D})$. \n\\end{thm}\n\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{longtable}{|l|l|l|l|}\n\\caption{Curves} \\label{TableC} \\\\\n\\hline \n$C_0$ & $y^2 = x^4-2x^3+2x^2+2x+1$ & \\href{https://www.lmfdb.org/EllipticCurve/Q/192a2/}{\\texttt{192.a2}} & $E_0$\\\\\n\\hline \n$C_1$& $y^2 = (x+4)(x^2+4x+16)$ & \\href{https://www.lmfdb.org/EllipticCurve/Q/24a4/}{\\texttt{24.a5}} & $E_1$\\\\\n\\hline\n$C_2$& $y^2 = x(x^2+4x+16)$ & \\href{https://www.lmfdb.org/EllipticCurve/Q/48a4/}{\\texttt{48.a5}}& $E^{-1}_1$\\\\\n\\hline \n$C_3$& $y^2 = x(x+4)(x^2+4x+16)$ & \\href{https://www.lmfdb.org/EllipticCurve/Q/24a4/}{\\texttt{24.a5}}& $E_1$\\\\\n\\hline \n$C_4$& $y^2 = x(x^2+14x+1)$ & \\href{https://www.lmfdb.org/EllipticCurve/Q/24a3/}{\\texttt{24.a2}}& $E_4$\\\\\n\\hline \n$C_5$ & $y^2 = x^4+4x^2+16$ & \\href{https://www.lmfdb.org/EllipticCurve/Q/48a1/}{\\texttt{48.a4}} & $E^{-1}_6$\\\\\n\\hline \n$C_6$ & $y^2 = x^4+14x^2+1$ & \\href{https://www.lmfdb.org/EllipticCurve/Q/24a1/}{\\texttt{24.a4}} & $E_6$\\\\\n\\hline\n\\end{longtable}\n\\end{center}\nNote that the elliptic curves $E_1,E_4$ and $E_6$ belongs to the same $\\mathbb Q$-isogeny class. In particular, for any integer $D$, the ranks of any $D$-quadratic twist coincide. In particular, we have $\\operatorname{rank}_{\\mathbb Z}C_k(\\mathbb Q)=0$ for $k=1,\\dots,6$. Let $E$ be an elliptic curve defined over $\\mathbb Q$. Using the known formula (see \\cite{Kra})\n$$\n\\operatorname{rank}_{\\mathbb Z}E(\\Q(\\sqrt{D}))=\\operatorname{rank}_{\\mathbb Z}E(\\mathbb Q)+\n\\operatorname{rank}_{\\mathbb Z}E^D(\\mathbb Q),\n$$\nwe obtain that $\\operatorname{rank}_{\\mathbb Z}E_1^{\\pm D}(\\mathbb Q)= 0$ is equivalent to $\\operatorname{rank}_{\\mathbb Z}C_k(\\Q(\\sqrt{D}))=0$ for $k=1,\\dots,6$. Assuming $\\operatorname{rank}_{\\mathbb Z}E_1^{\\pm D}(\\mathbb Q)= 0$ for the square-free integer $D$, to compute $C_k(\\Q(\\sqrt{D}))$, for $k=1,\\dots,6$, we need to compute the torsion subgroup over $\\Q(\\sqrt{D})$. In our particular case, given an elliptic curve $E$ defined over $\\mathbb Q$, there are only a finite number of square-free integers $D$ such that $E(\\mathbb Q)_{\\operatorname{tors}}\\ne E(\\Q(\\sqrt{D}))_{\\operatorname{tors}}$. If the conductor of $E$ is less than $400000$, these square-free integers appear in the LMFDB database \\cite{lmfdb} and are listed in Table \\ref{growth}.\n\\begin{center}\n\\begin{longtable}{|c|c|}\n\\caption{Torsion growth over quadratic fields}\\label{growth}\\\\\n\\hline \n$E$ & \\mbox{$D$ such that $E(\\mathbb Q)_{\\operatorname{tors}}\\ne \nE(\\Q(\\sqrt{D}))_{\\operatorname{tors}}$}\\\\\n\\hline\n\n$E_0$ & $-2$ \\\\\n\\hline\n$E_1$ & $-1$, $\\pm 3$ \\\\\n\\hline\n$E^{-1}_1$ & $-1$,$\\pm 3$\\\\\n\\hline \n$E_4$ & 2,3,6\\\\\n\\hline\n$E_6$ & $\\emptyset$ \\\\\n\\hline \n$E_6^{-1}$ & $-1$,$\\pm 3$\\\\\n\\hline\n\\end{longtable}\n\\end{center}\nNote that for those elliptic curves and values of $D$, we have $\\operatorname{rank}_{\\mathbb Z}E^D(\\mathbb Q)=0$ except in the case $E^6_4$. Finally, for any square-free integer $D$, Tables \\ref{points} and \\ref{pointsC4} show $C_k(\\Q(\\sqrt{D}))$ for $k=1,\\dots,6$. When $D=-1,2,\\pm 3$, the points at the second column correspond to $C_k(\\mathbb Q)$ and they must be added to the corresponding set $C_k(\\Q(\\sqrt{D}))$.\n\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{table}[htbp] \n\\begin{center} \\caption{$C_4(\\Q(\\sqrt{D}))$}\\label{pointsC4}\n\\begin{tabular}{|c|c|c|c|}\n\\cline{2-4} \n \\multicolumn{1}{c|}{} & $D\\ne 2,3,6$ & $D=3$ & $D=2$ \\\\ \n\\hline\n$C_4$ & $\\begin{array}{c} (0,0)\\\\ (1,\\pm 4) \\end{array}$ & $\\begin{array}{c}(-1,-2\\sqrt{3})\\\\ (7\\pm 4\\sqrt{3},0)\\end{array}$ & $\\begin{array}{c}\n(-3-2\\sqrt{2},\\pm(8+6\\sqrt{2}))\\\\ \n(-3+2\\sqrt{2},\\pm(8-6\\sqrt{2}))\\\\ \n\\end{array}$\\\\\n \\hline\n\\end{tabular} \n\\end{center}\n\\end{table}\n\n\\newpage\n\n\\newpage\n\n\\begin{proof}\nTable~\\ref{growth} shows that the torsion subgroup grows only when $D=-2$. Thus, for $D\\neq -2$, we have $E_0(\\Q(\\sqrt{D}))_{\\operatorname{tors}}=E_0(\\Q)_{\\operatorname{tors}}$. Therefore, if $D\\neq -2$ and $\\operatorname{rank}_{\\Z}E_0^{D}(\\Q)=0$, we deduce that $E_0(\\Q(\\sqrt{D}))$ is isomorphic to $E_0(\\Q)$. Let us prove that the only case where $E_0(\\Q(\\sqrt{D}))\\neq E_0(\\Q)$ under these assumptions occurs when $D=2$. Assume that there exists $P\\in E_0(\\Q)$ and $R\\in E_0(\\Q(\\sqrt{D}))\\setminus E_0(\\Q)$ such that $P=nR$ for some $n\\in \\Z$. Let $\\sigma\\in \\operatorname{Gal}(\\Q(\\sqrt{D})/\\Q)$ be the nontrivial automorphism. Then $P=n\\,\\sigma(R)$, and hence $n(R-\\sigma(R))=\\mathcal{O}$. This implies that $R-\\sigma(R)\\in E_0(\\Q)[n]$. Since $E_0(\\Q)_{\\operatorname{tors}}=E_0[2]$, we conclude that $n=2$.\n\n\\section{Setting}\nLet $D$ be a square-free integer and $K$ be a quadratic extension of $\\mathbb{Q}(\\sqrt{D})$. If $S\\subseteq K$, we denote by $S^2$ to the set of squares in $S$. Let $a, b, c, d, e \\in K$ be such that $\\left(a^{2}, b^{2}, c^{2}, d^{2}, e^{2}\\right)$ is an arithmetic progression. Then $[a: b: c: d: e]\\in V(K) \\subset \\mathbb{P}^4(K)$, where $V$ is the genus 5 curve defined by the system of equations\n$$\nV\\,\\,:\\,\\, \\{a^{2}+e^{2}=2 c^{2}\\,,\\,\\, a^{2}+c^{2}=2 b^{2}\\,,\\,\\,c^{2}+e^{2}=2 d^{2} \\}.\n$$\nEquation $a^2+e^2=2c^2$ has the parametrization\n\\begin{equation}\\label{eq2}\n[a: c: e]=\\left[t^{2}-2 t-1: t^{2}+1: t^{2}+2 t-1\\right] \n\\end{equation}\nwith $t \\in K$. Applying \\eqref{eq2} in the equations of $V$ gives, up to sign,\n\\begin{equation}\\label{eq3}\n\\left\\{\n\\begin{array}{l}\na = t^{2}-2 t-1, \\\\\nb= \\sqrt{t^{4}-2 t^{3}+2 t^{2}+2 t+1}, \\\\\nc= t^{2}+1, \\\\\nd= \\sqrt{t^{4}+2 t^{3}+2 t^{2}-2 t+1}, \\\\\ne= t^{2}+2 t-1.\n\\end{array}\n\\right.\n\\end{equation}\nReplacing $t$ by $-t$ in \\eqref{eq3} gives the arithmetic progression $(e^2,d^2,c^2,b^2,a^2)$, equivalent to $(a^2,b^2,c^2,d^2,e^2)$, and the constant arithmetic progressions correspond to the cases $t=0,\\pm 1$.\n\n\\begin{prop}\\label{prop}\nLet $D$ be a square-free integer and $K$ be a quadratic extension of $\\mathbb{Q}(\\sqrt{D})$. Let $a, b, c, d, e \\in K$ be such that $\\left(a^{2}, b^{2}, c^{2}, d^{2}, e^{2}\\right)$ is a non elementary arithmetic progression given by \\eqref{eq3}. If $\\operatorname{rank}_{\\Z}E_1^{\\pm D}(\\Q)= 0\\,,$ then\n\\begin{enumerate}\n\\item\\label{prop1} $r \\in \\QD$,\n\\item\\label{prop2} $s \\in \\QD$,\n\\item\\label{prop3} $t\\in\\QD$,\n\\item\\label{prop4} $t\\in\\Q$, if $D\\ne \\pm 2$ and $\\operatorname{rank}_{\\Z}E_0^{D}(\\Q)=0$.\n\\end{enumerate}\n\\end{prop}\n\n\\section{Proof of Proposition \\ref{prop}\\,\\eqref{prop4}}\\label{tinQ}\nLet {$D\\ne \\pm 2$} be a square-free integer satisfying $\\operatorname{rank}_{\\Z}E_1^{\\pm D}(\\Q)= 0$ and $\\operatorname{rank}_{\\Z}E_0^{D}(\\Q)=0$. Let \\( t \\in K \\setminus \\{0, \\pm 1, \\pm i, \\pm 1 \\pm \\sqrt{2}\\} \\) corresponding to a non-elementary arithmetic progression, as described in ~\\eqref{eq3}. We prove that $t \\in \\Q$. Since $[K:\\QD]=2$, there exists $\\alpha \\in K$ such that $\\alpha^2 \\in \\QD$ and $K=\\Q(\\sqrt{D},\\alpha)$. By \\eqref{eq6}, there exist $\\gamma_1,\\delta_1,\\gamma_2,\\delta_2\\in \\QD$ such that \n\\begin{equation}\\label{Section6.E1}\n\\begin{cases}\n t^{4}-2 t^{3}+2 t^{2}+2 t+1&\\!\\!\\!\\!=\\left(\\delta_{1}+\\gamma_{1}\\alpha\\right)^{2}, \\\\\n t^{4}+2 t^{3}+2 t^{2}-2 t+1&\\!\\!\\!\\!=\\left(\\delta_{2}+\\gamma_{2} \\alpha\\right)^{2}.\n\\end{cases}\n\\end{equation}\nSince $t \\in \\Q(\\sqrt{D})$ and $\\alpha \\notin \\QD$, we have $\\gamma_{1} \\delta_{1}=\\gamma_{2} \\delta_{2}=0$. There are four cases:\n\nLet us prove Theorem \\ref{main}:\n\\begin{itemize}\n\\item[(A)] Assume $\\operatorname{rank}_{\\Z}E_0^{D}(\\Q)=0$. Since $D\\ne \\pm 2$. By Proposition \\ref{prop} \\eqref{prop4}, we have $t\\in \\Q$. In this case we have \\( C_0(\\mathbb{Q}(\\sqrt{D})) = C_0(\\mathbb{Q}) \\). Hence, if the value of \\( G(t) \\) is a square over \\( \\mathbb{Q}(\\sqrt{D}) \\), it is already a square over \\( \\mathbb{Q} \\). Consequently, four out of the five terms of the arithmetic progression are defined over \\( \\mathbb{Q} \\), and therefore the full arithmetic progression of five terms is properly defined over a quadratic extension of \\( \\mathbb{Q} \\).\n\\item[(B)] Assume $\\operatorname{rank}_{\\Z}E_0^{D}(\\Q)\\ne 0$ and that the class number of $\\QD$ is $1$. By Proposition \\ref{prop} \\eqref{prop3}, we have $t\\in \\QD$. Since the ring of integers of $\\QD$ is a unique factorization domain, there exist $\\alpha,\\delta\\in \\Q(\\sqrt{D})$ such that $G(-t)=\\alpha\\,\\delta^2$. We can conclude that \\(\\alpha\\) is not a square in \\(\\mathbb{Q}(\\sqrt{D})\\), since otherwise there would exist a nontrivial arithmetic progression of five squares in \\(\\mathbb{Q}(\\sqrt{D})\\). However, this is impossible because \\cite[Proposition~5.2]{GJX} states that a necessary condition is that \\(\\operatorname{rank} E_1^{\\pm D}(\\mathbb{Q}) \\geq 2\\). Then the arithmetic progression $(a^2,b^2,c^2,d^2,e^2)$ is properly defined over $K=\\Q(\\sqrt{D},\\omega)$, where $\\omega^2=\\alpha$.\n\\end{itemize}\nNow, the proof of Theorem \\ref{maincor} assuming that $(a^2,b^2,c^2,d^2,e^2)$ is not an elementary arithmetic progression:\n\\begin{itemize}\n\\item $D\\in\\{-1,3\\}$: The proof is analogous to the case $(A)$ since $\\operatorname{rank}_{\\Z}E_0^{D}(\\Q)=0$ and {$D\\ne \\pm 2$}.\n\\item $D= \\pm 2$: In this case $\\operatorname{rank}_{\\Z}E_0^{D}(\\Q)=0$. By Proposition \\ref{prop} \\eqref{prop3} we have $t\\in \\Q(\\sqrt{D})$. The proof is analogous to the case $(B)$ since the class number of $\\Q(\\sqrt{D})$ is $1$.\n\\end{itemize}\n\n\\section{Parametrization in the cases $\\Q(\\sqrt{-2})$ and $\\Q(\\sqrt{2})$}\\label{sect_para}\nLet \\(D\\) be a square-free integer such that \\(\\operatorname{rank}_{\\mathbb{Z}} E_1^{\\pm D}(\\mathbb{Q}) = 0\\). We have proved that any non-elementary arithmetic progression of five squares over a quadratic extension of \\(\\QD\\) is equivalent to \n\\[\n\\bigl((t^{2}-2t-1)^2,\\, \\beta^2,\\, (t^{2}+1)^2,\\, \\alpha\\delta^2,\\, (t^{2}+2t-1)^2 \\bigr),\n\\]\nwhere \\(t, \\alpha, \\beta, \\delta \\in \\mathbb{Q}(\\sqrt{D})\\) satisfy \\(\\beta^2 = G(t)\\) and \\(\\alpha\\delta^2 = G(-t)\\). In particular, for any non-elementary arithmetic progression of five squares, there exists a point \\(R \\in E_0(\\mathbb{Q}(\\sqrt{D}))\\), and conversely. Note that this correspondence is not one-to-one. This construction recovers the one developed in \\cite{GJX} for the special case $D=1$. That is, for any \\(R \\in E_0(\\mathbb{Q})\\) we obtain an arithmetic progression of five squares over a quadratic field, except when \\(R \\in E_0(\\mathbb{Q})_{\\operatorname{tors}}\\), in which case the progression is trivial. Note that \\(\\operatorname{rank}_{\\mathbb{Z}} E_1^{\\pm 1}(\\mathbb{Q}) = 0\\). Let $T_1=(-3,0)$, $T_2=(1,0)$ and $P=(0,3)$, then \n$$\nE_0(\\Q)=\\langle T_1 \\rangle \\oplus \\langle T_2 \\rangle \\oplus \\langle P \\rangle.\n$$\nIf $n\\in\\Z$, $n\\ge 0$, the set of points $S_n=\\{n_1T_1+n_2T_2+m P \\,:\\, n_1,n_2\\in\\{0,1\\} \\,,\\,m\\in\\{n,-n-1\\}\\}$ corresponds to the same arithmetic progression of five squares, up to equivalence. For example, if $n=1$ the set $S_1$ corresponds to the arithmetic progression $(7^2, 13^2, 17^2, (\\sqrt{409})^2, 23^2)$.", "post_theorem_intro_text_len": 2725, "post_theorem_intro_text": "\\begin{proof}\nTable~\\ref{growth} shows that the torsion subgroup grows only when $D=-2$. Thus, for $D\\neq -2$, we have $E_0(\\mathbb Q(\\sqrt{D}))_{\\operatorname{tors}}=E_0(\\mathbb Q)_{\\operatorname{tors}}$. Therefore, if $D\\neq -2$ and $\\operatorname{rank}_{\\mathbb Z}E_0^{D}(\\mathbb Q)=0$, we deduce that $E_0(\\mathbb Q(\\sqrt{D}))$ is isomorphic to $E_0(\\mathbb Q)$. Let us prove that the only case where $E_0(\\mathbb Q(\\sqrt{D}))\\neq E_0(\\mathbb Q)$ under these assumptions occurs when $D=2$. Assume that there exists $P\\in E_0(\\mathbb Q)$ and $R\\in E_0(\\mathbb Q(\\sqrt{D}))\\setminus E_0(\\mathbb Q)$ such that $P=nR$ for some $n\\in \\mathbb Z$. Let $\\sigma\\in \\operatorname{Gal}(\\mathbb Q(\\sqrt{D})/\\mathbb Q)$ be the nontrivial automorphism. Then $P=n\\,\\sigma(R)$, and hence $n(R-\\sigma(R))=\\mathcal{O}$. This implies that $R-\\sigma(R)\\in E_0(\\mathbb Q)[n]$. Since $E_0(\\mathbb Q)_{\\operatorname{tors}}=E_0[2]$, we conclude that $n=2$. \n\nNow recall that $E_0(\\mathbb Q)=\\langle T_1 \\rangle \\oplus \\langle T_2 \\rangle \\oplus \\langle P_0 \\rangle$, where $T_1=(-3,0)$, $T_2=(1,0)$, and $P_0=(5,8)$. Thus any $P\\in E_0(\\mathbb Q)$ can be written as $ P=n_1T_1+n_2T_2+mP_0$ for some $n_1,n_2\\in \\{0,1\\}$ and $m\\in \\mathbb Z$. It is therefore enough to check the existence of such $R$ when $P=n_1T_1+n_2T_2+P_0$ for some $n_1,n_2\\in \\{0,1\\}$. Let us prove the above fact: suppose first that $m=2n$ for some $n\\neq 0$. Then we would obtain $2(R+nP_0)=n_1T_1+n_2T_2$,  which is impossible since the left-hand side has infinite order while the right-hand side has order  { dividing} $2$. Suppose instead that $m=2n+1$ for some $n\\neq 0$. Then we obtain $2(R+nP_0)=n_1T_1+n_2T_2+P_0$, which gives the desired relation.  \n\n{ Finally, one computes that the only solutions to $2 R=n_1T_1+n_2T_2+P_0$, $n_1,n_2\\in \\{0,1\\}$, are given by $n_1=n_2=0$ and $R=(1+2\\sqrt{2},-4)+T$ with $T\\in E[2]$.}  This completes the proof.\n\\end{proof}\n\n{\\bf Acknowledgments.} Nguyen Xuan Tho is also supported by Vietnam Institute for Advanced Study in Mathematics (VIASM) from April 2025 to May 2025. The author really appreciates the Institute for their help and funding. We would like to thank Xavier Xarles for pointing out that the elliptic curves $E_1,E_4$ and $E_6$ belongs to the same $\\mathbb Q$-isogeny class. The authors would like to thank the referees for their careful reading of the manuscript and for their valuable suggestions. In particular, we are grateful to the last referee, whose extensive and detailed comments greatly enhanced the overall quality of the paper. We are sincerely thankful for their contributions.\n\n\\\n\nAll computations in this paper are done in \\texttt{Magma} \\cite{Magma} and \\texttt{Mathematica} \\cite{mathematica}.", "sketch": "Table~\\ref{growth} is used to note that the torsion subgroup grows only when $D=-2$; hence for $D\\neq -2$ one has $E_0(\\mathbb Q(\\sqrt D))_{\\operatorname{tors}}=E_0(\\mathbb Q)_{\\operatorname{tors}}$, and with $\\operatorname{rank}_{\\mathbb Z}E_0^{D}(\\mathbb Q)=0$ this yields that $E_0(\\mathbb Q(\\sqrt D))$ is isomorphic to $E_0(\\mathbb Q)$.\n\nTo see when equality can fail under these assumptions, the argument assumes there exist $P\\in E_0(\\mathbb Q)$ and $R\\in E_0(\\mathbb Q(\\sqrt D))\\setminus E_0(\\mathbb Q)$ with $P=nR$. Let $\\sigma\\in \\operatorname{Gal}(\\mathbb Q(\\sqrt D)/\\mathbb Q)$ be nontrivial. Then $P=n\\,\\sigma(R)$, so $n(R-\\sigma(R))=\\mathcal O$, hence $R-\\sigma(R)\\in E_0(\\mathbb Q)[n]$. Since $E_0(\\mathbb Q)_{\\operatorname{tors}}=E_0[2]$, it follows that $n=2$.\n\nUsing $E_0(\\mathbb Q)=\\langle T_1\\rangle\\oplus\\langle T_2\\rangle\\oplus\\langle P_0\\rangle$ with $T_1=(-3,0)$, $T_2=(1,0)$, $P_0=(5,8)$, write $P=n_1T_1+n_2T_2+mP_0$ ($n_1,n_2\\in\\{0,1\\}$, $m\\in\\mathbb Z$). One reduces to checking the existence of $R$ only when $P=n_1T_1+n_2T_2+P_0$. Indeed, if $m=2n\\neq 0$ then $2(R+nP_0)=n_1T_1+n_2T_2$, impossible because the left side has infinite order while the right side has order dividing $2$; if $m=2n+1$ with $n\\neq 0$ then $2(R+nP_0)=n_1T_1+n_2T_2+P_0$, giving the desired relation.\n\nFinally, it is computed that the only solutions to $2R=n_1T_1+n_2T_2+P_0$ with $n_1,n_2\\in\\{0,1\\}$ occur for $n_1=n_2=0$ and $R=(1+2\\sqrt 2,-4)+T$ with $T\\in E[2]$, so the exceptional case is $D=2$ (together with the earlier torsion-growth exception $D=-2$).", "expanded_sketch": "\\begin{longtable}{|c|c|}\n\\caption{Torsion growth over quadratic fields}\\label{growth}\\\\\n\\hline \n$E$ & \\mbox{$D$ such that $E(\\Q)_{\\operatorname{tors}}\\ne \nE(\\QD)_{\\operatorname{tors}}$}\\\\\n\\hline \n\n$E_0$ & $-2$ \\\\\n\\hline\n$E_1$   &  $-1$, $\\pm 3$ \\\\\n\\hline\n$E^{-1}_1$     &  $-1$,$\\pm 3$\\\\\n\\hline \n$E_4$ & 2,3,6\\\\\n\\hline\n$E_6$ & $\\emptyset$ \\\\\n\\hline \n$E_6^{-1}$ & $-1$,$\\pm 3$\\\\\n\\hline\n\\end{longtable}\n\nThis table is used to note that for $E_0$ the torsion subgroup grows only when $D=-2$; hence for $D\\neq -2$ one has $E_0(\\mathbb Q(\\sqrt D))_{\\operatorname{tors}}=E_0(\\mathbb Q)_{\\operatorname{tors}}$, and with $\\operatorname{rank}_{\\mathbb Z}E_0^{D}(\\mathbb Q)=0$ this yields that $E_0(\\mathbb Q(\\sqrt D))$ is isomorphic to $E_0(\\mathbb Q)$.\n\nTo see when equality can fail under these assumptions, the argument assumes there exist $P\\in E_0(\\mathbb Q)$ and $R\\in E_0(\\mathbb Q(\\sqrt D))\\setminus E_0(\\mathbb Q)$ with $P=nR$. Let $\\sigma\\in \\operatorname{Gal}(\\mathbb Q(\\sqrt D)/\\mathbb Q)$ be nontrivial. Then $P=n\\,\\sigma(R)$, so $n(R-\\sigma(R))=\\mathcal O$, hence $R-\\sigma(R)\\in E_0(\\mathbb Q)[n]$. Since $E_0(\\mathbb Q)_{\\operatorname{tors}}=E_0[2]$, it follows that $n=2$.\n\nUsing $E_0(\\mathbb Q)=\\langle T_1\\rangle\\oplus\\langle T_2\\rangle\\oplus\\langle P_0\\rangle$ with $T_1=(-3,0)$, $T_2=(1,0)$, $P_0=(5,8)$, write $P=n_1T_1+n_2T_2+mP_0$ ($n_1,n_2\\in\\{0,1\\}$, $m\\in\\mathbb Z$). One reduces to checking the existence of $R$ only when $P=n_1T_1+n_2T_2+P_0$. Indeed, if $m=2n\\neq 0$ then $2(R+nP_0)=n_1T_1+n_2T_2$, impossible because the left side has infinite order while the right side has order dividing $2$; if $m=2n+1$ with $n\\neq 0$ then $2(R+nP_0)=n_1T_1+n_2T_2+P_0$, giving the desired relation.\n\nFinally, it is computed that the only solutions to $2R=n_1T_1+n_2T_2+P_0$ with $n_1,n_2\\in\\{0,1\\}$ occur for $n_1=n_2=0$ and $R=(1+2\\sqrt 2,-4)+T$ with $T\\in E[2]$, so the exceptional case is $D=2$ (together with the earlier torsion-growth exception $D=-2$).", "expanded_theorem": "\\label{newlemma}\nLet $D$ be a square-free integer such that $\\operatorname{rank}_{\\mathbb Z}E_0^{D}(\\mathbb Q)=0$. Then $E_0(\\mathbb Q(\\sqrt{D}))=E_0(\\mathbb Q)$, except when $D=\\pm 2$.", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let \\(E_0\\) denote the elliptic curve labeled \\(192.a2\\) in the context, and for a square-free integer \\(D\\), let \\(E_0^D\\) be the \\(D\\)-quadratic twist of \\(E_0\\). Which statement holds for every square-free integer \\(D\\) satisfying \\(\\operatorname{rank}_{\\mathbb Z} E_0^D(\\mathbb Q)=0\\)?", "correct_choice": {"label": "A", "text": "If \\(D\\neq \\pm 2\\), then the group of \\(\\mathbb Q(\\sqrt{D})\\)-rational points on \\(E_0\\) equals the group of \\(\\mathbb Q\\)-rational points: \\(E_0(\\mathbb Q(\\sqrt{D}))=E_0(\\mathbb Q)\\)."}, "choices": [{"label": "B", "text": "If \\(D\\neq -2\\), then the group of \\(\\mathbb Q(\\sqrt{D})\\)-rational points on \\(E_0\\) equals the group of \\(\\mathbb Q\\)-rational points: \\(E_0(\\mathbb Q(\\sqrt{D}))=E_0(\\mathbb Q)\\)."}, {"label": "C", "text": "If \\(D\\neq \\pm 2\\), then every \\(\\mathbb Q(\\sqrt{D})\\)-rational point on \\(E_0\\) is already \\(\\mathbb Q\\)-rational up to torsion; equivalently, the quotient group \\(E_0(\\mathbb Q(\\sqrt{D}))/E_0(\\mathbb Q)\\) is torsion."}, {"label": "D", "text": "There exists a finite set \\(S\\subset \\mathbb Z\\), independent of \\(D\\), such that whenever \\(D\\notin S\\) and \\(\\operatorname{rank}_{\\mathbb Z} E_0^D(\\mathbb Q)=0\\), one has \\(E_0(\\mathbb Q(\\sqrt{D}))_{\\operatorname{tors}}=E_0(\\mathbb Q)_{\\operatorname{tors}}\\), and hence \\(E_0(\\mathbb Q(\\sqrt{D}))=E_0(\\mathbb Q)\\)."}, {"label": "E", "text": "If \\(\\operatorname{rank}_{\\mathbb Z} E_0^D(\\mathbb Q)=0\\), then every point of \\(E_0(\\mathbb Q(\\sqrt{D}))\\) is twice a point of \\(E_0(\\mathbb Q)\\); in particular, \\(E_0(\\mathbb Q(\\sqrt{D}))=E_0(\\mathbb Q)\\) unless \\(D=\\pm 2\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "computational_check", "tampered_component": "exceptional_range_including_D=2", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped_equality_of_groups_to_torsion_quotient", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "uniform_exception_set_replacing_specific_exceptions", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "direction_of_2-divisibility_argument", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state or strongly hint at the exceptional values of D or the exact conclusion. It only gives the rank-0 hypothesis and asks for the resulting statement."}, "TAS": {"score": 1, "justification": "This is close to asking for the precise theorem statement attached to the named curve E_0, so it is largely recall of a specific result rather than a substantially reworked application."}, "GPS": {"score": 1, "justification": "There is some need to distinguish exact versus weaker or incorrect conclusions, especially around the exceptional cases and equality vs. isomorphism. However, option C is also true as a weaker consequence, so the item does not cleanly force selection by genuine generative reasoning."}, "DQS": {"score": 1, "justification": "Several distractors are plausible and reflect common errors: missing an exception, overgeneralizing to an unspecified finite set, or confusing equality with isomorphism. But C is not actually false; it is a weaker true statement, which weakens the distractor set in a single-answer MCQ."}, "total_score": 5, "overall_assessment": "Moderate quality: no answer leakage and some nuanced alternatives, but the item is mostly theorem-recall and is undermined by having a weaker true option among the distractors."}}
{"id": "2602.01431v1", "paper_link": "http://arxiv.org/abs/2602.01431v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\nLet $c \\neq 0$. \nSuppose ${\\omega d}/{c}=1+\\varepsilon$ for a sufficiently small absolute constant $\\varepsilon > 0$ and ${\\sigma}/{(c^{2}d)}>\\frac13$, the system (\\ref{main second}) admits a $C^2$ solitary wave solution with velocity $c$. Moreover, the solitary wave profile $\\eta$ has the asymptotic expansion\n\\begin{equation}\n\\eta(\\xi)=d\\, \\varepsilon\\, Q\\bigg(\\varepsilon^{\\frac{1}{2}}\\bigg(\\frac{\\sigma}{c^2d}-\\frac13\\bigg)^{-\\frac{1}{2}}\\frac{\\xi}{d}\\bigg)+O(\\varepsilon^2)\n\\end{equation}\nwhere $Q(\\bar x)=-3\\mathrm{sech}^2(\\bar x/2)$ solves the steady the KdV equation\n\\begin{equation}\nQ''=Q+\\frac{1}{2}Q^2.\n\\end{equation}", "start_pos": 9140, "end_pos": 9816, "label": "thm:main"}, "ref_dict": {"main second": "\\begin{equation}\\label{main second}\n\\begin{aligned}\n&\\phi_{xx}+\\phi_{yy}=0 && \\text{in } -d<y<\\eta(t,x),\\\\\n&\\phi_y=0 && \\text{on } y=-d,\\\\\n&\\eta_t=\\phi_y-(\\phi_x-\\omega \\eta)\\eta_x && \\text{on } y=\\eta(t,x),\\\\\n&\\phi_t+\\frac12(\\phi_x^2+\\phi_y^2)-\\omega\\eta\\,\\phi_x+\\omega\\psi\n-\\frac{\\sigma\\,\\eta_{xx}}{(1+\\eta_x^2)^{3/2}}=0 && \\text{on } y=\\eta(t,x).\n\\end{aligned}\n\\end{equation}", "eq:euler-freebdry": "\\begin{equation}\\label{eq:euler-freebdry}\n\\begin{aligned}\n&\\mathbf{u}_t+\\mathbf{u}\\cdot\\nabla \\mathbf{u}=-\\nabla p && \\text{in }\\mathcal{D}(t),\\\\\n&\\nabla\\cdot \\mathbf{u}=0,\\ \\mathrm{curl}\\,\\mathbf{u}=\\omega && \\text{in }\\mathcal{D}(t).\\\\\n\\end{aligned}\n\\end{equation}", "Young-Laplace": "\\begin{align}\\label{Young-Laplace}\np = \\sigma H\\quad \\text{on }\\partial\\mathcal{D}(t),\n\\end{align}"}, "pre_theorem_intro_text_len": 6008, "pre_theorem_intro_text": "The study of the motion of water waves is a classical question and one of the central\nproblems in fluid dynamics. Among the basic questions one can ask are the local\nwell-posedness of the Cauchy problem, the regularity of solutions and formation of\nsingularities, and the existence and stability of special solutions. In this paper, we consider two-dimensional pure capillary solitary water waves in finite depth with constant vorticity. More precisely, we study an inviscid, incompressible fluid separated from the air by a free surface, evolving under surface tension in the absence of gravity and with constant vorticity. We construct a family of solitary wave solutions in such setting, and, to the best of our knowledge, this fills a remaining gap in the finite-depth pure-capillary\nsolitary-wave theory in the presence of vorticity.\n\n\\subsection{Free-boundary Euler equations and capillary water waves with constant vorticity}\nLet the fluid occupy a time-dependent domain $\\mathcal{D}(t)\\subset\\mathbb{R}^2$ of finite depth,\nwith flat bottom $\\{y=-d\\}$ and free surface $\\partial\\mathcal{D}(t)$. We assume unit density\nand non-zero constant vorticity. The velocity field $\\mathbf{u}$ and pressure $p$ solve\nthe free-boundary incompressible Euler equations\n\\begin{equation}\\label{eq:euler-freebdry}\n\\begin{aligned}\n&\\mathbf{u}_t+\\mathbf{u}\\cdot\\nabla \\mathbf{u}=-\\nabla p && \\text{in }\\mathcal{D}(t),\\\\\n&\\nabla\\cdot \\mathbf{u}=0,\\ \\mathrm{curl}\\,\\mathbf{u}=\\omega && \\text{in }\\mathcal{D}(t).\\\\\n\\end{aligned}\n\\end{equation}\nIn addition, we have the kinematic boundary conditions\n\\begin{align}\\label{kinematic_top}\n\\mathcal{V}(\\partial\\mathcal{D}(t)) &= \\mathbf{u}\\cdot \\mathbf{n}\\,\\quad \\text{on }\\partial\\mathcal{D}(t),\\\\ \n\\label{kinematic_bottom}\n\\mathbf u\\cdot e_2 &= 0\\quad\\quad\\ \\ \\text{on }\\{y=-d\\},\n\\end{align}\nwhere $\\mathbf{n}$ denotes the outward unit normal to the free surface and\n$\\mathcal{V}(\\partial\\mathcal{D}(t))$ is its normal velocity. We also have the dynamic boundary condition\n\\begin{align}\\label{Young-Laplace}\np = \\sigma H\\quad \\text{on }\\partial\\mathcal{D}(t),\n\\end{align}\nwhere $H$ is the mean curvature of the interface, and $\\sigma > 0$ is the surface tension coefficient. \n\nThe system \\eqref{eq:euler-freebdry}--\\eqref{Young-Laplace} is often studied in the\nZakharov--Craig--Sulem framework, which we briefly recall below. A more detailed derivation can be seen in, for example, \\cite{Wahlen2007HamiltonianConstantVorticity}.  We assume that our free boundary is a graph $\\partial\\mathcal D(t)=\\{(x,y): y=\\eta(t,x)\\}$. Since the vorticity $\\mathrm{curl}\\,\\mathbf u = \\omega$ is a constant, the velocity can be written as a shear flow plus an irrotational perturbation as\n\\begin{equation}\\label{velocity}\n\\mathbf u = (-\\omega y,0)+\\nabla\\phi.\n\\end{equation}\nThen, in terms of the velocity potential $\\phi$, its harmonic conjugate $\\psi$, and the free boundary function $\\eta$, \\eqref{eq:euler-freebdry}--\\eqref{Young-Laplace} can be recast as\n\\begin{equation}\\label{main second}\n\\begin{aligned}\n&\\phi_{xx}+\\phi_{yy}=0 && \\text{in } -d<y<\\eta(t,x),\\\\\n&\\phi_y=0 && \\text{on } y=-d,\\\\\n&\\eta_t=\\phi_y-(\\phi_x-\\omega \\eta)\\eta_x && \\text{on } y=\\eta(t,x),\\\\\n&\\phi_t+\\frac12(\\phi_x^2+\\phi_y^2)-\\omega\\eta\\,\\phi_x+\\omega\\psi\n-\\frac{\\sigma\\,\\eta_{xx}}{(1+\\eta_x^2)^{3/2}}=0 && \\text{on } y=\\eta(t,x).\n\\end{aligned}\n\\end{equation}\n\n\\subsection{Previous work on solitary waves and our the main result}\nAmong the most studied special solutions of the water-wave system are solitary waves,\ni.e.\\ localized traveling disturbances propagating at constant speed over an otherwise flat\nsurface. The scientific interest in such waves dates back at least to the experiments of\nJ.\\ S.\\ Russell in 1844 \\cite{russell1844report}, whose observations anticipated key features\nof the modern theory. \n\nThe existence theory for solitary waves in two-dimensional irrotational water flows has a\nlong history. For finite-depth pure-gravity waves, the first rigorous constructions were\nobtained in the classical works of Friedrichs--Hyers \\cite{friedrichs1954existence}, Beale\n\\cite{thomas1977existence}, and Amick--Toland \\cite{amick1981periodic}. When both gravity\nand capillarity are present, small-amplitude solitary waves in finite depth were constructed\nusing a variety of techniques beginning with Kirchg\\\"assner \\cite{kirchgassner1988nonlinearly}\nand Amick--Kirchg\\\"assner \\cite{amick1989theory}, and extended in, for example,\n\\cite{buffoni1996plethora, iooss1992water, buffoni1999multiplicity, constantin2011nonlinear,\ngroves2004steady, groves2002dimension, groves2008fully, puaruau2005nonlinear, haragus2002finite,\ndeng2009three}. In infinite depth, solitary waves exist in the mixed gravity--capillary regime;\nsee \\cite{buffoni2004existence, buffoni2004existence2, groves2011existence, iooss1996capillary}.\n\nIn recent years, existence/non-existence theories in pure-gravity (infinite depth) and pure-capillary (finite/infinite depth) regimes have seen significant progress. Hur \\cite{hur2012no} proved nonexistence of two-dimensional irrotational pure-gravity solitary waves in deep water under a\ndecay assumption; Ifrim--Tataru \\cite{ifrim2020no} established complete nonexistence in deep water\nfor both pure-gravity and pure-capillary cases; and Ifrim--Pineau--Tataru--Taylor \\cite{ifrim2022no}\nproved that irrotational pure-capillary solitary waves do not exist in finite depth. See also \\cite{constantin2007rotational, groves2004steady, groves2007spatial, hur2008exact,\nhur2008symmetry} for steady waves with vorticity.\n\nIn this article, we prove the existence of finite-depth solitary waves for a remaining 2D case, namely the pure capillary case with nonzero constant vorticity. To describe a wave traveling at constant speed $c$, we introduce the moving-frame variable $\\xi:= x - ct$ and seek solutions of the form\n\\begin{equation}\\label{traveling_wave_ansatz}\n(\\phi(t,x,y),\\psi(t,x,y),\\eta(t,x))=(\\phi(\\xi,y),\\psi(\\xi,y),\\eta(\\xi)).\n\\end{equation}\n\nWe state our main theorem as follows.", "context": "The study of the motion of water waves is a classical question and one of the central\nproblems in fluid dynamics. Among the basic questions one can ask are the local\nwell-posedness of the Cauchy problem, the regularity of solutions and formation of\nsingularities, and the existence and stability of special solutions. In this paper, we consider two-dimensional pure capillary solitary water waves in finite depth with constant vorticity. More precisely, we study an inviscid, incompressible fluid separated from the air by a free surface, evolving under surface tension in the absence of gravity and with constant vorticity. We construct a family of solitary wave solutions in such setting, and, to the best of our knowledge, this fills a remaining gap in the finite-depth pure-capillary\nsolitary-wave theory in the presence of vorticity.\n\n\\subsection{Free-boundary Euler equations and capillary water waves with constant vorticity}\nLet the fluid occupy a time-dependent domain $\\mathcal{D}(t)\\subset\\mathbb{R}^2$ of finite depth,\nwith flat bottom $\\{y=-d\\}$ and free surface $\\partial\\mathcal{D}(t)$. We assume unit density\nand non-zero constant vorticity. The velocity field $\\mathbf{u}$ and pressure $p$ solve\nthe free-boundary incompressible Euler equations\n\\begin{equation}\\label{eq:euler-freebdry}\n\\begin{aligned}\n&\\mathbf{u}_t+\\mathbf{u}\\cdot\\nabla \\mathbf{u}=-\\nabla p && \\text{in }\\mathcal{D}(t),\\\\\n&\\nabla\\cdot \\mathbf{u}=0,\\ \\mathrm{curl}\\,\\mathbf{u}=\\omega && \\text{in }\\mathcal{D}(t).\\\\\n\\end{aligned}\n\\end{equation}\nIn addition, we have the kinematic boundary conditions\n\\begin{align}\\label{kinematic_top}\n\\mathcal{V}(\\partial\\mathcal{D}(t)) &= \\mathbf{u}\\cdot \\mathbf{n}\\,\\quad \\text{on }\\partial\\mathcal{D}(t),\\\\ \n\\label{kinematic_bottom}\n\\mathbf u\\cdot e_2 &= 0\\quad\\quad\\ \\ \\text{on }\\{y=-d\\},\n\\end{align}\nwhere $\\mathbf{n}$ denotes the outward unit normal to the free surface and\n$\\mathcal{V}(\\partial\\mathcal{D}(t))$ is its normal velocity. We also have the dynamic boundary condition\n\\begin{align}\\label{Young-Laplace}\np = \\sigma H\\quad \\text{on }\\partial\\mathcal{D}(t),\n\\end{align}\nwhere $H$ is the mean curvature of the interface, and $\\sigma > 0$ is the surface tension coefficient.\n\nThe system \\eqref{eq:euler-freebdry}--\\eqref{Young-Laplace} is often studied in the\nZakharov--Craig--Sulem framework, which we briefly recall below. A more detailed derivation can be seen in, for example, \\cite{Wahlen2007HamiltonianConstantVorticity}. We assume that our free boundary is a graph $\\partial\\mathcal D(t)=\\{(x,y): y=\\eta(t,x)\\}$. Since the vorticity $\\mathrm{curl}\\,\\mathbf u = \\omega$ is a constant, the velocity can be written as a shear flow plus an irrotational perturbation as\n\\begin{equation}\\label{velocity}\n\\mathbf u = (-\\omega y,0)+\\nabla\\phi.\n\\end{equation}\nThen, in terms of the velocity potential $\\phi$, its harmonic conjugate $\\psi$, and the free boundary function $\\eta$, \\eqref{eq:euler-freebdry}--\\eqref{Young-Laplace} can be recast as\n\\begin{equation}\\label{main second}\n\\begin{aligned}\n&\\phi_{xx}+\\phi_{yy}=0 && \\text{in } -d<y<\\eta(t,x),\\\\\n&\\phi_y=0 && \\text{on } y=-d,\\\\\n&\\eta_t=\\phi_y-(\\phi_x-\\omega \\eta)\\eta_x && \\text{on } y=\\eta(t,x),\\\\\n&\\phi_t+\\frac12(\\phi_x^2+\\phi_y^2)-\\omega\\eta\\,\\phi_x+\\omega\\psi\n-\\frac{\\sigma\\,\\eta_{xx}}{(1+\\eta_x^2)^{3/2}}=0 && \\text{on } y=\\eta(t,x).\n\\end{aligned}\n\\end{equation}\n\n\\subsection{Previous work on solitary waves and our the main result}\nAmong the most studied special solutions of the water-wave system are solitary waves,\ni.e.\\ localized traveling disturbances propagating at constant speed over an otherwise flat\nsurface. The scientific interest in such waves dates back at least to the experiments of\nJ.\\ S.\\ Russell in 1844 \\cite{russell1844report}, whose observations anticipated key features\nof the modern theory.\n\nIn this article, we prove the existence of finite-depth solitary waves for a remaining 2D case, namely the pure capillary case with nonzero constant vorticity. To describe a wave traveling at constant speed $c$, we introduce the moving-frame variable $\\xi:= x - ct$ and seek solutions of the form\n\\begin{equation}\\label{traveling_wave_ansatz}\n(\\phi(t,x,y),\\psi(t,x,y),\\eta(t,x))=(\\phi(\\xi,y),\\psi(\\xi,y),\\eta(\\xi)).\n\\end{equation}\n\nWe state our main theorem as follows.", "full_context": "The study of the motion of water waves is a classical question and one of the central\nproblems in fluid dynamics. Among the basic questions one can ask are the local\nwell-posedness of the Cauchy problem, the regularity of solutions and formation of\nsingularities, and the existence and stability of special solutions. In this paper, we consider two-dimensional pure capillary solitary water waves in finite depth with constant vorticity. More precisely, we study an inviscid, incompressible fluid separated from the air by a free surface, evolving under surface tension in the absence of gravity and with constant vorticity. We construct a family of solitary wave solutions in such setting, and, to the best of our knowledge, this fills a remaining gap in the finite-depth pure-capillary\nsolitary-wave theory in the presence of vorticity.\n\n\\subsection{Free-boundary Euler equations and capillary water waves with constant vorticity}\nLet the fluid occupy a time-dependent domain $\\mathcal{D}(t)\\subset\\mathbb{R}^2$ of finite depth,\nwith flat bottom $\\{y=-d\\}$ and free surface $\\partial\\mathcal{D}(t)$. We assume unit density\nand non-zero constant vorticity. The velocity field $\\mathbf{u}$ and pressure $p$ solve\nthe free-boundary incompressible Euler equations\n\\begin{equation}\\label{eq:euler-freebdry}\n\\begin{aligned}\n&\\mathbf{u}_t+\\mathbf{u}\\cdot\\nabla \\mathbf{u}=-\\nabla p && \\text{in }\\mathcal{D}(t),\\\\\n&\\nabla\\cdot \\mathbf{u}=0,\\ \\mathrm{curl}\\,\\mathbf{u}=\\omega && \\text{in }\\mathcal{D}(t).\\\\\n\\end{aligned}\n\\end{equation}\nIn addition, we have the kinematic boundary conditions\n\\begin{align}\\label{kinematic_top}\n\\mathcal{V}(\\partial\\mathcal{D}(t)) &= \\mathbf{u}\\cdot \\mathbf{n}\\,\\quad \\text{on }\\partial\\mathcal{D}(t),\\\\ \n\\label{kinematic_bottom}\n\\mathbf u\\cdot e_2 &= 0\\quad\\quad\\ \\ \\text{on }\\{y=-d\\},\n\\end{align}\nwhere $\\mathbf{n}$ denotes the outward unit normal to the free surface and\n$\\mathcal{V}(\\partial\\mathcal{D}(t))$ is its normal velocity. We also have the dynamic boundary condition\n\\begin{align}\\label{Young-Laplace}\np = \\sigma H\\quad \\text{on }\\partial\\mathcal{D}(t),\n\\end{align}\nwhere $H$ is the mean curvature of the interface, and $\\sigma > 0$ is the surface tension coefficient.\n\nThe system \\eqref{eq:euler-freebdry}--\\eqref{Young-Laplace} is often studied in the\nZakharov--Craig--Sulem framework, which we briefly recall below. A more detailed derivation can be seen in, for example, \\cite{Wahlen2007HamiltonianConstantVorticity}. We assume that our free boundary is a graph $\\partial\\mathcal D(t)=\\{(x,y): y=\\eta(t,x)\\}$. Since the vorticity $\\mathrm{curl}\\,\\mathbf u = \\omega$ is a constant, the velocity can be written as a shear flow plus an irrotational perturbation as\n\\begin{equation}\\label{velocity}\n\\mathbf u = (-\\omega y,0)+\\nabla\\phi.\n\\end{equation}\nThen, in terms of the velocity potential $\\phi$, its harmonic conjugate $\\psi$, and the free boundary function $\\eta$, \\eqref{eq:euler-freebdry}--\\eqref{Young-Laplace} can be recast as\n\\begin{equation}\\label{main second}\n\\begin{aligned}\n&\\phi_{xx}+\\phi_{yy}=0 && \\text{in } -d<y<\\eta(t,x),\\\\\n&\\phi_y=0 && \\text{on } y=-d,\\\\\n&\\eta_t=\\phi_y-(\\phi_x-\\omega \\eta)\\eta_x && \\text{on } y=\\eta(t,x),\\\\\n&\\phi_t+\\frac12(\\phi_x^2+\\phi_y^2)-\\omega\\eta\\,\\phi_x+\\omega\\psi\n-\\frac{\\sigma\\,\\eta_{xx}}{(1+\\eta_x^2)^{3/2}}=0 && \\text{on } y=\\eta(t,x).\n\\end{aligned}\n\\end{equation}\n\n\\subsection{Previous work on solitary waves and our the main result}\nAmong the most studied special solutions of the water-wave system are solitary waves,\ni.e.\\ localized traveling disturbances propagating at constant speed over an otherwise flat\nsurface. The scientific interest in such waves dates back at least to the experiments of\nJ.\\ S.\\ Russell in 1844 \\cite{russell1844report}, whose observations anticipated key features\nof the modern theory.\n\nIn this article, we prove the existence of finite-depth solitary waves for a remaining 2D case, namely the pure capillary case with nonzero constant vorticity. To describe a wave traveling at constant speed $c$, we introduce the moving-frame variable $\\xi:= x - ct$ and seek solutions of the form\n\\begin{equation}\\label{traveling_wave_ansatz}\n(\\phi(t,x,y),\\psi(t,x,y),\\eta(t,x))=(\\phi(\\xi,y),\\psi(\\xi,y),\\eta(\\xi)).\n\\end{equation}\n\nWe state our main theorem as follows.\n\nWe state our main theorem as follows.\n\n\\subsection{Main ideas and comparison with related work}\nOur analysis uses a spatial-dynamics and Hamiltonian formulation of the traveling-wave Euler system,\nfollowed by a nonlinear transformation that rectifies the free-boundary conditions and the symplectic\nstructure. This leads to an infinite-dimensional reversible Hamiltonian system which, along a\ndistinguished curve in parameter space, has a two-dimensional center subspace. We verify the resolvent\nbounds needed for Mielke's center-manifold theorem, obtain a finite-dimensional reduced canonical system,\nand identify a KdV-type normal form admitting a homoclinic orbit after a long-wave scaling and a cubic expansion of the reduced Hamiltonian. The persistence of this orbit yields solitary waves for the\noriginal free-boundary Euler equations.\n\n\\section{The derivation of Hamiltonian formulation}\\label{sec:formulation}\nThe goal of this section is to formulate the traveling wave problem of (\\ref{main second}) as a Hamiltonian system. As a first step, we use the moving-frame variable $\\xi:= x - ct$ and plug in ansatz as in (\\ref{traveling_wave_ansatz}) to the system \\eqref{main second}, obtaining\n\\begin{equation}\\label{main third}\n\\begin{aligned}\n&\\phi_{\\xi\\xi}+\\phi_{yy}=0 && \\text{in } -d<y<\\eta(\\xi),\\\\\n&\\phi_y=0 && \\text{on } y=-d,\\\\\n&-c\\,\\eta_{\\xi}=\\phi_y-(\\phi_{\\xi}-\\omega \\eta)\\eta_{\\xi} && \\text{on } y=\\eta(\\xi),\\\\\n&-c\\,\\phi_{\\xi}+\\frac12(\\phi_{\\xi}^2+\\phi_y^2)-\\omega\\eta\\,\\phi_{\\xi}+\\omega\\psi\n-\\frac{\\sigma\\,\\eta_{\\xi\\xi}}{(1+\\eta_{\\xi}^2)^{3/2}}=0 && \\text{on } y=\\eta(\\xi).\n\\end{aligned}\n\\end{equation}\nIt is convenient to nondimensionalize. Under the scalings\n\\[\n\\xi\\mapsto \\xi/d,\\quad y\\mapsto y/d,\\quad \\omega\\mapsto \\omega d/c,\\quad\n\\sigma\\mapsto \\sigma/(c^2 d),\n\\qquad\n\\phi\\mapsto \\phi/(cd),\\quad \\psi\\mapsto \\psi/(cd),\\quad \\eta\\mapsto \\eta/d,\n\\]\nwe may normalize $c=d=1$ in \\eqref{main third}. In this normalization, the third equation\nin \\eqref{main third} implies\n\\begin{equation}\\label{psi-eta=0 1}\n\\frac{d}{d\\xi}\\Bigl(\\psi(\\xi,\\eta(\\xi))-\\frac{\\omega}{2}\\eta(\\xi)^2-\\eta(\\xi)\\Bigr)=0.\n\\end{equation}\nand thus\n\\begin{equation}\\label{psi-eta=0 2}\n\\psi(\\xi,\\eta(\\xi))=\\frac{\\omega}{2}\\eta(\\xi)^2+\\eta(\\xi)+\\mathrm{const}.\n\\end{equation}\nFor solitary waves the profile decays, $\\eta(\\xi)\\to 0$ as $|\\xi|\\to\\infty$, and hence the\nconstant must be zero. Substituting \\eqref{psi-eta=0 2} into \\eqref{main third} yields\n\\begin{equation}\\label{main fourth}\n\\begin{aligned}\n&\\phi_{\\xi\\xi}+\\phi_{yy}=0 && \\text{in } -1<y<\\eta(\\xi),\\\\\n&\\phi_y=0 && \\text{on } y=-1,\\\\\n&-\\eta_{\\xi}=\\phi_y-(\\phi_{\\xi}-\\omega \\eta)\\eta_{\\xi} && \\text{on } y=\\eta(\\xi),\\\\\n&-\\phi_{\\xi}+\\frac12\\phi_{\\xi}^2+\\frac12\\phi_y^2-\\omega\\eta\\,\\phi_{\\xi}\n+\\omega\\eta+\\frac{\\omega^2}{2}\\eta^2\n-\\frac{\\sigma\\,\\eta_{\\xi\\xi}}{(1+\\eta_{\\xi}^2)^{3/2}}=0 && \\text{on } y=\\eta(\\xi).\n\\end{aligned}\n\\end{equation}\n\nNote that $\\tilde f_1^\\varepsilon$ transforms the boundary conditions \\eqref{BC first} into\n\\begin{align}\\label{105 boundary}\n\\Phi_y(x,0)=\\Phi_y(x,1)=0 .\n\\end{align}\nWe therefore introduce\n\\begin{align}\\label{domain of L}\n\\mathrm{dom}(\\mathbf{L})\n:=\\{(\\Phi,\\theta,Z,\\eta)\\in N_0:\\ \\Phi_y(0)=\\Phi_y(1)=0\\},\n\\end{align}\nequipped with the $\\|\\cdot\\|_{X^2}$ norm, and the operator $\\mathbf L$ will be defined shortly. Recalling \\eqref{def of u_H}, let $m\\in \\mathrm{dom}(\\mathbf{L})$ be sufficiently small in $\\|\\cdot\\|_{X^2}$.\nThen for every $v'_m\\in M_0$ we compute\n\\begin{align*}\nd\\tilde H^\\varepsilon_m[v'_m]\n&= d H^{\\omega_0+\\varepsilon_1,\\sigma_0+\\varepsilon_2}_{(f_1^\\varepsilon)^{-1}(m)}\n\\bigl[(d f_1^\\varepsilon)^{-1}_m[v'_m]\\bigr] \\\\\n&=\\Omega^{\\omega_0+\\varepsilon_1,\\sigma_0+\\varepsilon_2}_{(f_1^\\varepsilon)^{-1}(m)}\n\\Bigl[v_H^{\\omega_0+\\varepsilon_1,\\sigma_0+\\varepsilon_2}\\bigl((f_1^\\varepsilon)^{-1}(m)\\bigr),\n(d f_1^\\varepsilon)^{-1}_m[v'_m]\\Bigr] \\\\\n&=\\tilde\\Omega^\\varepsilon_m\\Bigl[\nd f_1^\\varepsilon\\bigl((f_1^\\varepsilon)^{-1}(m)\\bigr)\n\\,v_H^{\\omega_0+\\varepsilon_1,\\sigma_0+\\varepsilon_2}\\bigl((f_1^\\varepsilon)^{-1}(m)\\bigr),\n\\,v'_m\\Bigr].\n\\end{align*}\nIt follows that the Hamiltonian vector field associated with $(\\tilde H^\\varepsilon,\\tilde\\Omega^\\varepsilon)$ is\n\\begin{align}\\label{newvH}\n\\widetilde{v_H^\\varepsilon}(m)\n=\nd f_1^\\varepsilon\\bigl((f_1^\\varepsilon)^{-1}(m)\\bigr)\n\\Bigl[v_H^{\\omega_0+\\varepsilon_1,\\sigma_0+\\varepsilon_2}\\bigl((f_1^\\varepsilon)^{-1}(m)\\bigr)\\Bigr],\n\\end{align}\nfor all $m\\in \\mathrm{dom}(\\mathbf{L})$ with sufficiently small $\\|m\\|_{X^2}$. Using \\eqref{newvH}, a straightforward but tedious computation\n(see \\cite[Appendix~A]{buffoni1996plethora}) yields the explicit formula for\n$\\widetilde{v_H^\\varepsilon}=(\\widetilde{\\Phi_H^\\varepsilon},\\widetilde{\\theta_H^\\varepsilon},\n\\widetilde{Z_H^\\varepsilon},\\widetilde{\\eta_H^\\varepsilon})$.\nWriting $m=(\\Phi,\\theta,Z,\\eta)$, we obtain\n\\begin{equation}\\label{B'K equation 2}\n\\begin{aligned}\n\\widetilde{\\Phi_H^\\varepsilon}(m)\n&=\\frac{1}{\\eta+1}\\Bigl(\\theta_y+R\\bigl(2y\\Phi_y+Ry^2(\\theta_y-1)-\\Phi-\\Phi(1)+Z\\bigr)\\Bigr) \\\\\n&\\quad-\\frac{\\bar Z(1+R^2)^{3/2}}{\\sigma_0+\\varepsilon_2}\n\\left(\\theta_p-\\frac12\\left(y^2-\\frac13\\right)\\right) \\\\\n&\\quad+\\frac{(\\omega_0+\\varepsilon_1)R^2y^2(4y-3)}{6(\\eta+1)}\n+(\\omega_0+\\varepsilon_1)(\\eta+1)\\left(y-\\frac12\\right)\n-\\frac{\\omega_0+\\varepsilon_1}{\\eta+1}\\left(y-\\frac12\\right),\\\\\n\\widetilde{\\theta_H^\\varepsilon}(m)\n&=-\\frac{1}{\\eta+1}\\Bigl(\\Phi_y-(\\omega_0+\\varepsilon_1)R(1-y)y\\Bigr),\\\\\n\\widetilde{Z_H^\\varepsilon}(m)\n&=-\\frac{1}{\\eta+1}\\Bigl(\\theta_y(1)+R\\bigl(R(\\theta_y(1)-1)+Z\\bigr)\\Bigr)\n-\\frac{(\\omega_0+\\varepsilon_1)(\\eta+1)}{2}\n+\\frac{\\omega_0+\\varepsilon_1}{2(\\eta+1)},\\\\\n\\widetilde{\\eta_H^\\varepsilon}(m)\n&=R,\n\\end{aligned}\n\\end{equation}\nwhere\n\\[\n\\bar Z=\\frac{(1+R^2)\\,(\\theta_y(1)-1)^2}{2(\\eta+1)^2}-\\frac12.\n\\]\n\n\\subsubsection*{Step 6. Leading-order asymptotics for $\\eta$ in nondimensional variables}\nUsing (\\ref{eq:QPX-scaling}), we get\n\\begin{equation}\\label{eq:qPiEtae}\n\\begin{aligned}\n{q(x)}&=\\varepsilon_1\\,\\sqrt{\\sigma_0-\\tfrac13}\\, Q(\\bar x);\\\\\np(x) &= \\varepsilon_1^{3/2}P(\\bar x).\n\\end{aligned}\n\\end{equation}\nTherefore, we have $q=O(\\varepsilon_1)$ and $p=O(\\varepsilon_1^{3/2})$, and thus $|(q,p)|=O(\\varepsilon_1)$ and $|(q,p,\\varepsilon_1)|=O(\\varepsilon_1)$, and by \\eqref{eq:r-estimate} we obtain\n\\begin{equation}\\label{eq:r-order}\n\\|\\tilde r(v(x),\\varepsilon_1)\\|=O(\\varepsilon_1^2),\n\\qquad\\Rightarrow\\qquad\n\\Pi_\\eta(\\tilde r(v(x),\\varepsilon_1))=O(\\varepsilon_1^2).\n\\end{equation}\nCombining \\eqref{eq:eta-q-leading}, \\eqref{eq:qPiEtae}, and \\eqref{eq:r-order} yields the nondimensional leading-order expansion\n\\begin{equation}\\label{eq:eta-dimless-asympt}\n\\eta(x)=\\varepsilon_1\\,Q(\\bar x)+O(\\varepsilon_1^2),\n\\qquad\n\\bar x=\\sqrt{\\varepsilon_1}(\\sigma_0-\\tfrac13)^{-1/2}x,\n\\end{equation}\nwhere $Q$ is the stationary KdV soliton \\eqref{eq:homoclinic-QP}. Identifying $\\varepsilon_1=\\varepsilon$ via \\eqref{eq:eps-def} in the regime $\\omega_0=1$, we may rewrite \\eqref{eq:eta-dimless-asympt} as\n\\begin{equation}\\label{eq:eta-dimless-eps}\n\\eta(x)=\\varepsilon\\,Q(\\bar x)+O(\\varepsilon^2),\n\\qquad\n\\bar x=\\sqrt{\\varepsilon}\\,(\\sigma_0-\\tfrac13)^{-1/2}x,\n\\qquad\nQ(\\bar x)=-3\\,\\sech^2(\\bar x/2).\n\\end{equation}\n\nUsing $x=\\xi=\\xi_{\\mathrm{phys}}/d$ and \\eqref{eq:eta-phys-from-dimless}, together with $\\sigma_0=\\sigma=\\sigma_{\\mathrm{phys}}/(c^2 d)$ from \\eqref{eq:nondim-params}, we compute\n\\[\n\\bar x=\\sqrt{\\varepsilon}\\,(\\sigma_0-\\tfrac13)^{-1/2}x\n=\\sqrt{\\varepsilon}\\,\n\\Bigl(\\frac{\\sigma_{\\mathrm{phys}}}{c^2 d}-\\frac13\\Bigr)^{-1/2}\\,\n\\frac{\\xi_{\\mathrm{phys}}}{d}.\n\\]\nTherefore, the leading-order solitary-wave profile in physical variables is\n\\begin{equation}\\label{eq:eta-phys-asympt}\n\\eta_{\\mathrm{phys}}(\\xi_{\\mathrm{phys}})\n=\nd\\,\\varepsilon\\,\nQ\\!\\left(\\varepsilon^{\\frac{1}{2}}\\bigg(\\frac{\\sigma_{\\mathrm{phys}}}{c^2 d}-\\frac13\\bigg)^{-\\frac{1}{2}}\\frac{\\xi_{\\mathrm{phys}}}{d}\\right)\n+O(\\varepsilon^2),\n\\end{equation}\nas in Theorem \\ref{thm:main}.", "post_theorem_intro_text_len": 2744, "post_theorem_intro_text": "\\subsection{Main ideas and comparison with related work}\nOur analysis uses a spatial-dynamics and Hamiltonian formulation of the traveling-wave Euler system,\nfollowed by a nonlinear transformation that rectifies the free-boundary conditions and the symplectic\nstructure. This leads to an infinite-dimensional reversible Hamiltonian system which, along a\ndistinguished curve in parameter space, has a two-dimensional center subspace. We verify the resolvent\nbounds needed for Mielke's center-manifold theorem, obtain a finite-dimensional reduced canonical system,\nand identify a KdV-type normal form admitting a homoclinic orbit after a long-wave scaling and a cubic expansion of the reduced Hamiltonian. The persistence of this orbit yields solitary waves for the\noriginal free-boundary Euler equations.\n\nOur approach is complementary to Rowan--Wan \\cite{rowan2024two}, who constructed pure-capillary solitary waves in infinite depth via a Babenko-type boundary-integral equation; the infinite-depth geometry leads to a different nonlocal operator and dispersive structure. We also note that Kharif--Abid--Chen--Hsu \\cite{kharif2025nonlinear} derived amplitude equations predicting finite-depth capillary solitary waves with vorticity by formal asymptotics and matching arguments.\n\n\\subsection{Organization of the paper}\nThe rest of the paper is organized as follows. In Section~\\ref{sec:formulation} we introduce the traveling-wave\nreduction and the Hamiltonian/spatial-dynamics formulation. In Section~\\ref{sec:center} we carry out the center-manifold\nreduction and derive the reduced canonical system. In Section~\\ref{sec:normalform} we compute the normal form and prove\nthe existence of a homoclinic orbit, and we lift the construction to a solitary wave of the full Euler system.\n\n\\subsection*{Acknowledgement}\n(i)~T.-Y. Hsiao is supported by the European Union  ERC CONSOLIDATOR GRANT 2023 GUnDHam, Project Number: 101124921. T.-Y. Hsiao would like to express sincere gratitude to Vera Hur, Zhao Yang, and Alberto Maspero for their invaluable guidance and continuous encouragement. Special thanks are also due to Chongchun Zeng and Erik Wahlén for their insightful advice and generous help during the completion of this project; (ii)~Z. Liang was supported in part by NSF grant DMS-2153992 through his advisor during winter 2025 semester when part of this work was completed. Z. Liang would like to thank his advisor Sijue Wu for her continuous support, and would like to thank Zach Deiman, Noah Stevenson, and Yuchuan Yang for helpful discussions;  (iii)~G. To was supported by the Swedish Research Council (grant no. 2020-00440); (iv)~Y. Zhang is supported by  the European Union  ERC STARTING GRANT 2020 GeoSub, Project Number: 945655.", "sketch": "Our analysis uses a spatial-dynamics and Hamiltonian formulation of the traveling-wave Euler system, followed by a nonlinear transformation that rectifies the free-boundary conditions and the symplectic structure. This leads to an infinite-dimensional reversible Hamiltonian system which, along a distinguished curve in parameter space, has a two-dimensional center subspace. We verify the resolvent bounds needed for Mielke's center-manifold theorem, obtain a finite-dimensional reduced canonical system, and identify a KdV-type normal form admitting a homoclinic orbit after a long-wave scaling and a cubic expansion of the reduced Hamiltonian. The persistence of this orbit yields solitary waves for the original free-boundary Euler equations.", "expanded_sketch": "Our analysis uses a spatial-dynamics and Hamiltonian formulation of the traveling-wave Euler system, followed by a nonlinear transformation that rectifies the free-boundary conditions and the symplectic structure. This leads to an infinite-dimensional reversible Hamiltonian system which, along a distinguished curve in parameter space, has a two-dimensional center subspace. We verify the resolvent bounds needed for Mielke's center-manifold theorem, obtain a finite-dimensional reduced canonical system, and identify a KdV-type normal form admitting a homoclinic orbit after a long-wave scaling and a cubic expansion of the reduced Hamiltonian. The persistence of this orbit yields solitary waves for the original free-boundary Euler equations.", "expanded_theorem": "\\label{thm:main}\nLet $c \\neq 0$. \nSuppose ${\\omega d}/{c}=1+\\varepsilon$ for a sufficiently small absolute constant $\\varepsilon > 0$ and ${\\sigma}/{(c^{2}d)}>\\frac13$, the system\n\\begin{equation}\\label{main second}\n\\begin{aligned}\n&\\phi_{xx}+\\phi_{yy}=0 && \\text{in } -d<y<\\eta(t,x),\\\\\n&\\phi_y=0 && \\text{on } y=-d,\\\\\n&\\eta_t=\\phi_y-(\\phi_x-\\omega \\eta)\\eta_x && \\text{on } y=\\eta(t,x),\\\\\n&\\phi_t+\\frac12(\\phi_x^2+\\phi_y^2)-\\omega\\eta\\,\\phi_x+\\omega\\psi\n-\\frac{\\sigma\\,\\eta_{xx}}{(1+\\eta_x^2)^{3/2}}=0 && \\text{on } y=\\eta(t,x).\n\\end{aligned}\n\\end{equation}\nadmits a $C^2$ solitary wave solution with velocity $c$. Moreover, the solitary wave profile $\\eta$ has the asymptotic expansion\n\\begin{equation}\n\\eta(\\xi)=d\\, \\varepsilon\\, Q\\bigg(\\varepsilon^{\\frac{1}{2}}\\bigg(\\frac{\\sigma}{c^2d}-\\frac13\\bigg)^{-\\frac{1}{2}}\\frac{\\xi}{d}\\bigg)+O(\\varepsilon^2)\n\\end{equation}\nwhere $Q(\\bar x)=-3\\mathrm{sech}^2(\\bar x/2)$ solves the steady the KdV equation\n\\begin{equation}\nQ''=Q+\\frac{1}{2}Q^2.\n\\end{equation}", "theorem_type": ["Implication", "Existence"], "mcq": {"question": "Consider the two-dimensional pure-capillary water-wave system with constant vorticity in finite depth, where the fluid occupies the region \\(-d<y<\\eta(t,x)\\), the free surface is \\(y=\\eta(t,x)\\), \\(\\phi\\) is the velocity potential, and \\(\\psi\\) is the harmonic conjugate of \\(\\phi\\). Assume \\(c\\neq 0\\),\n\\[\n\\frac{\\omega d}{c}=1+\\varepsilon\n\\]\nfor a sufficiently small absolute constant \\(\\varepsilon>0\\), and\n\\[\n\\frac{\\sigma}{c^2d}>\\frac13.\n\\]\nSuppose \\((\\phi,\\psi,\\eta)\\) satisfies\n\\[\n\\begin{aligned}\n&\\phi_{xx}+\\phi_{yy}=0 && \\text{in } -d<y<\\eta(t,x),\\\\\n&\\phi_y=0 && \\text{on } y=-d,\\\\\n&\\eta_t=\\phi_y-(\\phi_x-\\omega \\eta)\\eta_x && \\text{on } y=\\eta(t,x),\\\\\n&\\phi_t+\\frac12(\\phi_x^2+\\phi_y^2)-\\omega\\eta\\,\\phi_x+\\omega\\psi\n-\\frac{\\sigma\\,\\eta_{xx}}{(1+\\eta_x^2)^{3/2}}=0 && \\text{on } y=\\eta(t,x).\n\\end{aligned}\n\\]\nA solitary wave with velocity \\(c\\) means a traveling solution of the form\n\\[\n(\\phi(t,x,y),\\psi(t,x,y),\\eta(t,x))=(\\phi(\\xi,y),\\psi(\\xi,y),\\eta(\\xi)),\\qquad \\xi=x-ct,\n\\]\nwith localized surface profile. Under these assumptions, which statement about the existence and leading-order profile of such a wave is valid?", "correct_choice": {"label": "A", "text": "The system admits a \\(C^2\\) solitary wave solution traveling with velocity \\(c\\). Moreover, its surface profile satisfies\n\\[\n\\eta(\\xi)=d\\,\\varepsilon\\,Q\\!\\left(\\varepsilon^{1/2}\\left(\\frac{\\sigma}{c^2d}-\\frac13\\right)^{-1/2}\\frac{\\xi}{d}\\right)+O(\\varepsilon^2),\n\\]\nwhere\n\\[\nQ(\\bar x)=-3\\,\\operatorname{sech}^2(\\bar x/2)\n\\]\nand \\(Q\\) solves the steady KdV equation\n\\[\nQ''=Q+\\frac12 Q^2.\n\\]"}, "choices": [{"label": "B", "text": "The system admits a \\(C^2\\) solitary wave solution traveling with velocity \\(c\\). Moreover, its surface profile satisfies\n\\[\n\\eta(\\xi)=d\\,\\varepsilon\\,Q\\!\\left(\\varepsilon^{1/2}\\left(\\frac{\\sigma}{c^2d}-\\frac13\\right)^{-1/2}\\frac{\\xi}{d}\\right)+O(\\varepsilon^2),\n\\]\nwhere\n\\[\nQ(\\bar x)=3\\,\\operatorname{sech}^2(\\bar x/2)\n\\]\nand \\(Q\\) solves the steady KdV equation\n\\[\nQ''=Q+\\frac12 Q^2.\n\\]"}, {"label": "C", "text": "The system admits a solitary wave solution traveling with velocity \\(c\\), and its surface profile has leading-order behavior\n\\[\n\\eta(\\xi)=d\\,\\varepsilon\\,Q\\!\\left(\\varepsilon^{1/2}\\left(\\frac{\\sigma}{c^2d}-\\frac13\\right)^{-1/2}\\frac{\\xi}{d}\\right)+o(\\varepsilon),\n\\]\nfor some localized profile \\(Q\\) solving\n\\[\nQ''=Q+\\frac12 Q^2.\n\\]"}, {"label": "D", "text": "The system admits a \\(C^2\\) solitary wave solution traveling with velocity \\(c\\) for every \\(\\sigma/(c^2d)>0\\). Moreover, its surface profile satisfies\n\\[\n\\eta(\\xi)=d\\,\\varepsilon\\,Q\\!\\left(\\varepsilon^{1/2}\\left(\\frac{\\sigma}{c^2d}-\\frac13\\right)^{-1/2}\\frac{\\xi}{d}\\right)+O(\\varepsilon^2),\n\\]\nwhere\n\\[\nQ(\\bar x)=-3\\,\\operatorname{sech}^2(\\bar x/2)\n\\]\nand \\(Q\\) solves the steady KdV equation\n\\[\nQ''=Q+\\frac12 Q^2.\n\\]"}, {"label": "E", "text": "For each sufficiently small \\(\\varepsilon>0\\), there exists an absolute constant \\(C>0\\), independent of \\(\\sigma\\), \\(c\\), and \\(d\\), such that whenever \\(\\omega d/c=1+\\varepsilon\\) and \\(\\sigma/(c^2d)>1/3\\), the system admits a \\(C^2\\) solitary wave solution traveling with velocity \\(c\\), and its surface profile satisfies\n\\[\n\\eta(\\xi)=d\\,\\varepsilon\\,Q\\!\\left(\\varepsilon^{1/2}\\left(\\frac{\\sigma}{c^2d}-\\frac13\\right)^{-1/2}\\frac{\\xi}{d}\\right)+R(\\xi),\n\\qquad \\|R\\|_{L^\\infty}\\le C\\varepsilon^2,\n\\]\nwhere\n\\[\nQ(\\bar x)=-3\\,\\operatorname{sech}^2(\\bar x/2)\n\\]\nand \\(Q\\) solves\n\\[\nQ''=Q+\\frac12 Q^2.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "homoclinic-orbit sign/profile selection", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "drops explicit formula for the unique leading soliton and weakens the remainder from O(\\varepsilon^2) to o(\\varepsilon)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "surface-tension threshold \\(\\sigma/(c^2d)>1/3\\)", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "non-uniform error bound promoted to a parameter-independent effective estimate", "template_used": "uniformity_effectivity"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It states the PDE system and assumptions, then asks which conclusion holds; the exact asymptotic conclusion is not given away in the prompt itself."}, "TAS": {"score": 1, "justification": "This is essentially a theorem-recall item: the hypotheses are presented and the student is asked to identify the matching conclusion. The altered choices introduce some comparison, but the task is still close to selecting the theorem statement rather than deriving a genuinely new consequence."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in distinguishing the precise asymptotic statement from nearby variants (sign error, stronger remainder, periodic vs solitary). However, the item is weakened by the presence of choice C, which is a weaker statement that is also true if A is true, so the question does not cleanly force identification of a uniquely strongest conclusion."}, "DQS": {"score": 1, "justification": "B, D, and E are reasonably targeted distractors based on plausible mathematical confusions. But C is not a proper distractor, since it is a weaker true consequence of A; this creates ambiguity and lowers distractor quality substantially."}, "total_score": 5, "overall_assessment": "Moderate-quality but flawed theorem-recall MCQ. It avoids direct answer leakage and includes some plausible near-miss options, but it is too close to restating a theorem and is ambiguous because one distractor is also true."}}
{"id": "2602.01471v4", "paper_link": "http://arxiv.org/abs/2602.01471v4", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm:main1}\n\t\tLet $n,s,k$ be positive integers, $n \\geq sk$, $k \\geq 1$.\n\t\tLet $\\mathcal{F} \\subseteq \\binom{[n]}{k}$ be a family with no $s$ pairwise disjoint sets. Then, $|\\mathcal{F}| \\leq \\max\\left\\{\\binom{n}{k}-\\binom{n-s+1}{k}, \\binom{sk-1}{k}\\right\\}$.", "start_pos": 5166, "end_pos": 5461, "label": "thm:main1"}, "ref_dict": {"lemma:1": "\\begin{lemma} \\label{lemma:1}\n\t\tLet $\\mathcal{F}$ be a family of subsets of $[n]$ and let $1 \\leq i, j \\leq n$. Then the following hold:\n\t\t\\begin{enumerate}[label=(\\roman*)]\n\t\t\t\\item For any $F \\in \\mathcal{F}$, $|\\mathcal{C}_{ij}(F)| = |F|$.\n\t\t\t\\item $|\\mathcal{C}_{ij}(\\mathcal{F})| = |\\mathcal{F}|$.\n\t\t\t\\item The matching number does not increase, i.e., $\\nu(\\mathcal{C}_{ij}(\\mathcal{F})) \\leq \\nu(\\mathcal{F})$.\n\t\t\\end{enumerate}\n\t\\end{lemma}", "lemma:2": "\\begin{lemma}\\label{lemma:2}\n\t\tLet $\\mathcal{F}$ be a trivial family on the ground set $[n]$. Then for any $1 \\leq i, j \\leq n$, $i\\neq j$, the family $\\mathcal{C}_{ij}(\\mathcal{F})$ is also a trivial family.\n\t\\end{lemma}"}, "pre_theorem_intro_text_len": 3571, "pre_theorem_intro_text": "A central problem in extremal combinatorics is to determine the maximum possible size of a set family subject to a given constraint. We consider families $\\mathcal{F} \\subseteq 2^{[n]}$ on the ground set $[n] = \\{1, \\dots, n\\}$. A \\emph{matching} in $\\mathcal{F}$ is a collection of pairwise disjoint members of $\\mathcal{F}$. The \\emph{matching number} of $\\mathcal{F}$, denoted $\\nu(\\mathcal{F})$, is the cardinality of a maximum matching.\n\n\tThis paper focuses on $k$-uniform families, $\\mathcal{F} \\subseteq \\binom{[n]}{k}$, that are $s$-matching-free, i.e., $\\nu(\\mathcal{F}) \\le s-1$. We denote the maximum possible size of such a family by $f(n,k,s)$. In 1965, Erd\\H{o}s \\cite{erdos1965} posed a conjecture for the exact value of $f(n,k,s)$, which has become a celebrated open problem in the field.\n\n\t\\begin{conjecture}[Erd\\H{o}s Matching Conjecture \\cite{erdos1971}] \\label{emc}\n\t\tFor $n \\ge sk$,\n\t\t\\[f(n,k,s) = \\max\\left\\{\\binom{sk-1}{k}, \\binom{n}{k}-\\binom{n-s+1}{k}\\right\\}.\\]\n\t\\end{conjecture}\n\n\tThe two expressions in the maximum correspond to two canonical extremal families.\n\t\\begin{enumerate}\n\t\t\\item The family $\\mathcal{G}^* = \\binom{[sk-1]}{k}$, which consists of all $k$-subsets of a fixed set of $sk-1$ elements.\n\t\t\\item The family $\\mathcal{F}^* = \\{F \\in \\binom{[n]}{k} : F \\cap S \\neq \\emptyset\\}$ for a fixed set $S \\subset [n]$ with $|S| = s-1$. This family consists of all $k$-subsets that intersect a fixed set of $s-1$ elements.\n\t\\end{enumerate}\n\tThe conjecture is trivially true for $n < sk$, as in this case $\\nu(\\mathcal{F}) \\le s-1$ holds for the entire family $\\binom{[n]}{k}$.\n\tThe conjecture has a rich history and has been verified in several important cases. For $s=2$, the problem is to find the largest intersecting family, and the conjecture follows from the Erd\\H{o}s-Ko-Rado theorem \\cite{erdos1961}. Kleitman \\cite{kleitman1968} proved the conjecture for the boundary case $n=sk$.\n\n\tSignificant progress has been made on the conjecture for sufficiently large $n$. Erd\\H{o}s \\cite{erdos1965} first established the conjecture for $n \\ge n_0(k,s)$. This bound was subsequently sharpened by Bollob\\'{a}s, Daykin, and Erd\\H{o}s \\cite{bollobas1976} to $n \\ge 2sk^3$, and later by Huang, Lo, and Sudakov \\cite{huang2012} to $n \\ge 3sk^2$.\n\n\tOther results have affirmed the conjecture for specific parameters or ranges. Frankl \\cite{frankl1987} proved the conjecture for $n \\geq (2s+1)k-s$ and also provided the general upper bound $f(n,k,s) \\leq (s-1)\\binom{n-1}{k-1}$. Frankl, R{\\\"{o}}dl, and Ruci{\\'{n}}ski \\cite{frankl2012} established the case $k=3$ for $n \\geq 4s$. More recently, Kolupaev and Kupavskii \\cite{kupavskii2023} proved the conjecture for $k \\geq 5$, $s > 100k^3$, and $sk \\leq n < s(k+100/k)$. For a comprehensive survey of recent results, see \\cite{frankl2022}.\n\n\tDespite this extensive body of work, the conjecture has remained open in its entirety. In this paper, we provide a complete proof.\n\n\t\\subsection{Our results}\n\tWe establish the conjecture for all $n \\ge sk$. Our proof is algorithmic and relies on a key distinction between trivial and non-trivial families.\n\n\t\\begin{definition}\n\t\tA family is said to be \\emph{trivial} if there exists at least one element $x \\in [n]$ that is not contained in any $F \\in \\mathcal{F}$. A family that is not trivial is \\emph{non-trivial}.\n\t\\end{definition}\n\n\tBy analyzing the behavior of families under shift operations, and paying close attention to whether intermediate families in our algorithm become trivial, we establish the following theorem in Section \\ref{sec:4}.", "context": "A central problem in extremal combinatorics is to determine the maximum possible size of a set family subject to a given constraint. We consider families $\\mathcal{F} \\subseteq 2^{[n]}$ on the ground set $[n] = \\{1, \\dots, n\\}$. A \\emph{matching} in $\\mathcal{F}$ is a collection of pairwise disjoint members of $\\mathcal{F}$. The \\emph{matching number} of $\\mathcal{F}$, denoted $\\nu(\\mathcal{F})$, is the cardinality of a maximum matching.\n\nThis paper focuses on $k$-uniform families, $\\mathcal{F} \\subseteq \\binom{[n]}{k}$, that are $s$-matching-free, i.e., $\\nu(\\mathcal{F}) \\le s-1$. We denote the maximum possible size of such a family by $f(n,k,s)$. In 1965, Erd\\H{o}s \\cite{erdos1965} posed a conjecture for the exact value of $f(n,k,s)$, which has become a celebrated open problem in the field.\n\n\\begin{conjecture}[Erd\\H{o}s Matching Conjecture \\cite{erdos1971}] \\label{emc}\n For $n \\ge sk$,\n \\[f(n,k,s) = \\max\\left\\{\\binom{sk-1}{k}, \\binom{n}{k}-\\binom{n-s+1}{k}\\right\\}.\\]\n \\end{conjecture}\n\nThe two expressions in the maximum correspond to two canonical extremal families.\n \\begin{enumerate}\n \\item The family $\\mathcal{G}^* = \\binom{[sk-1]}{k}$, which consists of all $k$-subsets of a fixed set of $sk-1$ elements.\n \\item The family $\\mathcal{F}^* = \\{F \\in \\binom{[n]}{k} : F \\cap S \\neq \\emptyset\\}$ for a fixed set $S \\subset [n]$ with $|S| = s-1$. This family consists of all $k$-subsets that intersect a fixed set of $s-1$ elements.\n \\end{enumerate}\n The conjecture is trivially true for $n < sk$, as in this case $\\nu(\\mathcal{F}) \\le s-1$ holds for the entire family $\\binom{[n]}{k}$.\n The conjecture has a rich history and has been verified in several important cases. For $s=2$, the problem is to find the largest intersecting family, and the conjecture follows from the Erd\\H{o}s-Ko-Rado theorem \\cite{erdos1961}. Kleitman \\cite{kleitman1968} proved the conjecture for the boundary case $n=sk$.\n\n\\begin{definition}\n A family is said to be \\emph{trivial} if there exists at least one element $x \\in [n]$ that is not contained in any $F \\in \\mathcal{F}$. A family that is not trivial is \\emph{non-trivial}.\n \\end{definition}\n\nBy analyzing the behavior of families under shift operations, and paying close attention to whether intermediate families in our algorithm become trivial, we establish the following theorem in Section \\ref{sec:4}.", "full_context": "A central problem in extremal combinatorics is to determine the maximum possible size of a set family subject to a given constraint. We consider families $\\mathcal{F} \\subseteq 2^{[n]}$ on the ground set $[n] = \\{1, \\dots, n\\}$. A \\emph{matching} in $\\mathcal{F}$ is a collection of pairwise disjoint members of $\\mathcal{F}$. The \\emph{matching number} of $\\mathcal{F}$, denoted $\\nu(\\mathcal{F})$, is the cardinality of a maximum matching.\n\nThis paper focuses on $k$-uniform families, $\\mathcal{F} \\subseteq \\binom{[n]}{k}$, that are $s$-matching-free, i.e., $\\nu(\\mathcal{F}) \\le s-1$. We denote the maximum possible size of such a family by $f(n,k,s)$. In 1965, Erd\\H{o}s \\cite{erdos1965} posed a conjecture for the exact value of $f(n,k,s)$, which has become a celebrated open problem in the field.\n\n\\begin{conjecture}[Erd\\H{o}s Matching Conjecture \\cite{erdos1971}] \\label{emc}\n For $n \\ge sk$,\n \\[f(n,k,s) = \\max\\left\\{\\binom{sk-1}{k}, \\binom{n}{k}-\\binom{n-s+1}{k}\\right\\}.\\]\n \\end{conjecture}\n\nThe two expressions in the maximum correspond to two canonical extremal families.\n \\begin{enumerate}\n \\item The family $\\mathcal{G}^* = \\binom{[sk-1]}{k}$, which consists of all $k$-subsets of a fixed set of $sk-1$ elements.\n \\item The family $\\mathcal{F}^* = \\{F \\in \\binom{[n]}{k} : F \\cap S \\neq \\emptyset\\}$ for a fixed set $S \\subset [n]$ with $|S| = s-1$. This family consists of all $k$-subsets that intersect a fixed set of $s-1$ elements.\n \\end{enumerate}\n The conjecture is trivially true for $n < sk$, as in this case $\\nu(\\mathcal{F}) \\le s-1$ holds for the entire family $\\binom{[n]}{k}$.\n The conjecture has a rich history and has been verified in several important cases. For $s=2$, the problem is to find the largest intersecting family, and the conjecture follows from the Erd\\H{o}s-Ko-Rado theorem \\cite{erdos1961}. Kleitman \\cite{kleitman1968} proved the conjecture for the boundary case $n=sk$.\n\n\\begin{definition}\n A family is said to be \\emph{trivial} if there exists at least one element $x \\in [n]$ that is not contained in any $F \\in \\mathcal{F}$. A family that is not trivial is \\emph{non-trivial}.\n \\end{definition}\n\nBy analyzing the behavior of families under shift operations, and paying close attention to whether intermediate families in our algorithm become trivial, we establish the following theorem in Section \\ref{sec:4}.\n\n\\newtheorem*{statement1}{Statement of Theorem 1}\n\\newtheorem*{statement2}{Statement of Lemma 2}\n\\begin{document}\n \\title{Erd\\H{o}s Matching (Conjecture) Theorem}\n \\author{Tapas Kumar Mishra \\footnote{Orcid: 0000-0002-9825-3828} \\\\\n Department of Computer Science and Engineering,\\\\\n National Institute of Technology, Rourkela\n 769008, India\\\\\n mishrat@nitrkl.ac.in\\\\\n https://mishra-tapas.github.io/}\n \\maketitle \\begin{abstract}\n Let $\\mathcal{F}$ be a family of $k$-sized subsets of $[n]$ that does not contain $s$ pairwise disjoint subsets.\n The Erd\\H{o}s Matching Conjecture, a celebrated and long-standing open problem in extremal combinatorics, asserts the maximum cardinality of $\\mathcal{F}$ is upper bounded by\n $\\max\\left\\{\\binom{sk-1}{k}, \\binom{n}{k}-\\allowbreak \\binom{n-s+1}{k}\\right\\}$.\n These two bounds correspond to the sizes of two canonical extremal families: one in which all subsets are contained within a ground set of $sk-1$ elements, and one in which every subset intersects a fixed set of $s-1$ elements.\n In this paper, we prove the conjecture.\n\n\\begin{conjecture}[Erd\\H{o}s Matching Conjecture \\cite{erdos1971}] \\label{emc}\n For $n \\ge sk$,\n \\[f(n,k,s) = \\max\\left\\{\\binom{sk-1}{k}, \\binom{n}{k}-\\binom{n-s+1}{k}\\right\\}.\\]\n \\end{conjecture}\n\nBy analyzing the behavior of families under shift operations, and paying close attention to whether intermediate families in our algorithm become trivial, we establish the following theorem in Section \\ref{sec:4}.\n\nThe key technical ingredients in our proof are the classical $(i,j)$ shift operator of Fr\\'{a}nkl. The precise effects of these operators on a family's matching number and its potential for triviality are detailed in Lemma \\ref{lemma:1}, \\ref{lemma:2}. This algorithmic framework, which tracks the properties of families through a sequence of shifts, can be extended to the non-uniform case as well.\n \\section{Shifting operator and supporting lemmas}\n \\label{sec:2}\n \\begin{definition}[Fr\\'{a}nkl's $(i,j)$ shift]\n Let $\\mathcal{F} \\subseteq 2^{[n]}$. For $1 \\leq i,j \\leq n$, define $\\mathcal{C}_{ij}(\\mathcal{F})= \\{\\mathcal{C}_{ij}(F): F \\in \\mathcal{F}\\}$, where\n \\begin{align*}\n \\mathcal{C}_{ij}(F) = \\begin{cases}\n (F \\setminus \\{j\\})\\cup \\{i\\} , \\text{ if $i \\not \\in F$, $j \\in F$, and $(F \\setminus \\{j\\})\\cup \\{i\\} \\not\\in \\mathcal{F}$},\\\\\n F, \\text{ otherwise.}\n \\end{cases}\n \\end{align*}\n \\end{definition}\n We have two lemmas on the effects of a $(i,j)$ shift on a family $\\mathcal{F}$.\n\n\\begin{lemma} \\label{lemma:1}\n Let $\\mathcal{F}$ be a family of subsets of $[n]$ and let $1 \\leq i, j \\leq n$. Then the following hold:\n \\begin{enumerate}[label=(\\roman*)]\n \\item For any $F \\in \\mathcal{F}$, $|\\mathcal{C}_{ij}(F)| = |F|$.\n \\item $|\\mathcal{C}_{ij}(\\mathcal{F})| = |\\mathcal{F}|$.\n \\item The matching number does not increase, i.e., $\\nu(\\mathcal{C}_{ij}(\\mathcal{F})) \\leq \\nu(\\mathcal{F})$.\n \\end{enumerate}\n \\end{lemma}\n \\begin{proof}\n The proof of (i) and (ii) are given in \\cite{frankl1987}. As (iii) is probably folklore, the proof is given in the Appendix \\ref{app:1}. \n \\end{proof}\n\n\\section{Proof of Theorem \\ref{thm:main1}}\n \\label{sec:4}\nFor a family $\\mathcal{F}$ and an element $x \\in [n]$, we denote the subfamily of sets containing $x$ as $\\mathcal{F}^x = \\{F \\in \\mathcal{F} : x \\in F\\}$. \n \\begin{statement1}\n Let $n,s,k$ be positive integers, $n \\geq sk$, $k \\geq 1$.\n Let $\\mathcal{F} \\subseteq \\binom{[n]}{k}$ be a family with no $s$ pairwise disjoint sets. Then, $|\\mathcal{F}| \\leq \\max\\left\\{\\binom{n}{k}-\\binom{n-s+1}{k}, \\binom{sk-1}{k}\\right\\}$.\n \\end{statement1}\n \\begin{proof}\n The family $\\mathcal{F}^* = \\{F \\in \\binom{[n]}{k} : F \\cap \\{1, \\dots, s-1\\} \\neq \\emptyset\\}$ satisfies the condition $\\nu(\\mathcal{F}^*) \\le s-1$ and has size exactly $\\binom{n}{k} - \\binom{n-s+1}{k}$. \n Let $\\mathcal{G}^*=\\binom{[sk-1]}{k}$.\n We will show that no family satisfying the hypothesis can be larger in cardinality than both $\\mathcal{F}^*$ and $\\mathcal{G}^*$. The proof uses an iterative algorithm that transforms any valid family $\\mathcal{F}$ into one where the desired structure is evident.\n\n\\begin{enumerate}\n \\item If $\\mathcal{F}$ is trivial, let $Y \\subseteq [n']$ be the largest set of vertices with no element of $Y$ being a member of any set in $\\mathcal{F}$ (i.e. $\\forall y \\in Y \\forall F \\in \\mathcal{F}$, $y \\not\\in F$). Set $\\mathcal{F}$ to be the non-trivial family on $[n']\\setminus Y$ and with a relabeling (that does not change $S$), assume the ground set is exactly $[n'-|Y|]$. Set $n'=n'-|Y|$. Note that this transformation preserves $|\\mathcal{F}|$ and $\\nu(\\mathcal{F})$.\n If $n' \\leq sk-1$, $\\mathcal{F} \\subseteq \\mathcal{G}^*$ and we terminate the algorithm.\n Otherwise, set $\\mathcal{F}^* = \\{F \\in \\binom{[n']}{k} : F \\cap \\{1, \\dots, s-1\\} \\neq \\emptyset\\}$.\n \\item If the current family $\\mathcal{F}$ is not a subfamily of $\\mathcal{F}^*$, then there must exist a set $A \\in \\mathcal{F}$ such that $A \\cap S = \\emptyset$.\n \\item There must also exist a set $B \\notin \\mathcal{F}$ such that $B \\cap S \\neq \\emptyset$. If no such $B$ exists, the algorithm terminates and proceeds to the ``Contradiction'' phase described in Section III.\n \\item Let $X = A \\cap B$. Define $A' = A \\setminus X = \\{a_1, \\dots, a_r\\}$ and $B' = B \\setminus X = \\{b_1, \\dots, b_r\\}$. By construction, all elements $a \\in A'$ are not in $S$. We can order the elements in $B'$ such that $b_1 \\in S$. The remaining elements in $B'$ and all elements in $A'$ are ordered arbitrarily.\n\nThe algorithm terminates under three conditions.\\\\\n \\textbf{Condition 1: $n' \\leq sk-1$}. In this case, $\\mathcal{F} \\subseteq \\mathcal{G}^*$ and $|\\mathcal{F}| \\leq \\binom{sk-1}{k}$.\\\\\n \\textbf{Condition 2: the current family $\\mathcal{F}$ is a subfamily of $\\mathcal{F}^*$}. In this case, $|\\mathcal{F}| \\leq \\binom{n'}{k}-\\binom{n'-s+1}{k} \\leq \\binom{n}{k}-\\binom{n-s+1}{k}$ and the bound follows.\\\\\n \\textbf{Condition 3: if the algorithm reaches a state where it can find a set $A \\in \\mathcal{F}$ with $A \\cap S = \\emptyset$, but it cannot find a set $B \\notin \\mathcal{F}$ with $B \\cap S \\neq \\emptyset$}. This implies that the current family $\\mathcal{F}$ contains \\textbf{all possible $k$-sets that intersect $S$}.\n We can now derive a contradiction. Let $A$ be the set in $\\mathcal{F}$ that is disjoint from $S$. We can construct $s-1$ new sets, $B_1, \\dots, B_{s-1}$, that are pairwise disjoint and also disjoint from $A$.\n \\begin{itemize}\n \\item Choose a set $X$ of size $(s-1)k$ from $[n']\\setminus A$\n such that $\\{1,\\ldots,s-1\\} \\subset X$.\n Partition $X$ into $s-1$ sets $B_1,\\ldots,B_{s-1}$\n with $B_i \\cap \\{1,\\ldots,s-1\\}=\\{i\\}$.\n \\end{itemize}\n This construction is possible because the condition $n' \\ge sk$ ensures there are enough elements available. Since each $B_i$ intersects $S$, and our family $\\mathcal{F}$ contains all such sets, it follows that $\\{B_1, \\dots, B_{s-1}\\} \\subset \\mathcal{F}$.\n The collection \\textbf{$\\{A, B_1, \\dots, B_{s-1}\\}$} is therefore a matching of size $s$ within $\\mathcal{F}$. This contradicts the hypothesis that $\\nu(\\mathcal{F}) \\le s-1$.\n\n\\begin{lemma} \\label{lemma:1}\n\t\tLet $\\mathcal{F}$ be a family of subsets of $[n]$ and let $1 \\leq i, j \\leq n$. Then the following hold:\n\t\t\\begin{enumerate}[label=(\\roman*)]\n\t\t\t\\item For any $F \\in \\mathcal{F}$, $|\\mathcal{C}_{ij}(F)| = |F|$.\n\t\t\t\\item $|\\mathcal{C}_{ij}(\\mathcal{F})| = |\\mathcal{F}|$.\n\t\t\t\\item The matching number does not increase, i.e., $\\nu(\\mathcal{C}_{ij}(\\mathcal{F})) \\leq \\nu(\\mathcal{F})$.\n\t\t\\end{enumerate}\n\t\\end{lemma}\n\n\\begin{lemma}\\label{lemma:2}\n\t\tLet $\\mathcal{F}$ be a trivial family on the ground set $[n]$. Then for any $1 \\leq i, j \\leq n$, $i\\neq j$, the family $\\mathcal{C}_{ij}(\\mathcal{F})$ is also a trivial family.\n\t\\end{lemma}", "post_theorem_intro_text_len": 399, "post_theorem_intro_text": "The key technical ingredients in our proof are the classical $(i,j)$ shift operator of Fr\\'{a}nkl. The precise effects of these operators on a family's matching number and its potential for triviality are detailed in Lemma \\ref{lemma:1}, \\ref{lemma:2}. This algorithmic framework, which tracks the properties of families through a sequence of shifts, can be extended to the non-uniform case as well.", "sketch": "The proof of Theorem~\\ref{thm:main1} uses as key technical ingredients the classical $(i,j)$ shift operator of Fr\\'{a}nkl. The effects of these operators on a family’s matching number and its potential for triviality are detailed in Lemma~\\ref{lemma:1} and Lemma~\\ref{lemma:2}. The argument is presented in an algorithmic framework that “tracks the properties of families through a sequence of shifts,” and this shifting framework can also be extended to the non-uniform case.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:main1}\n\t\tLet $n,s,k$ be positive integers, $n \\geq sk$, $k \\geq 1$.\n\t\tLet $\\mathcal{F} \\subseteq \\binom{[n]}{k}$ be a family with no $s$ pairwise disjoint sets. Then, $|\\mathcal{F}| \\leq \\max\\left\\{\\binom{n}{k}-\\binom{n-s+1}{k}, \\binom{sk-1}{k}\\right\\}$.", "theorem_type": ["Inequality or Bound", "Implication"], "mcq": {"question": "Let $n,s,k$ be positive integers with $n\\ge sk$ and $k\\ge 1$. Write $[n]=\\{1,2,\\dots,n\\}$, and let $\\binom{[n]}{k}$ denote the family of all $k$-element subsets of $[n]$. Suppose $\\mathcal F\\subseteq \\binom{[n]}{k}$ is a $k$-uniform family that contains no $s$ pairwise disjoint sets. Which of the following quantitative estimates holds for $|\\mathcal F|$?", "correct_choice": {"label": "A", "text": "$|\\mathcal F|\\le \\max\\left\\{\\binom{n}{k}-\\binom{n-s+1}{k},\\;\\binom{sk-1}{k}\\right\\}$."}, "choices": [{"label": "B", "text": "$|\\mathcal F|\\le \\max\\left\\{\\binom{n}{k}-\\binom{n-s}{k},\\;\\binom{sk-1}{k}\\right\\}$."}, {"label": "C", "text": "$|\\mathcal F|\\le \\binom{n}{k}-\\binom{n-s+1}{k}+\\binom{sk-1}{k}$."}, {"label": "D", "text": "$|\\mathcal F|\\le \\min\\left\\{\\binom{n}{k}-\\binom{n-s+1}{k},\\;\\binom{sk-1}{k}\\right\\}$."}, {"label": "E", "text": "$|\\mathcal F|\\le \\max\\left\\{\\binom{n}{k}-\\binom{n-s+1}{k},\\;\\binom{sk}{k}\\right\\}$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "extremal-count threshold in the intersecting-with-fixed-$(s-1)$-set term", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "replaced the sharp maximum of the two extremal bounds by their sum", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "aggregation of the two candidate extremal bounds", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "boundary case $sk-1$ replaced by $sk$ in the complete-support extremal term", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only the hypotheses and asks for the valid conclusion; it does not explicitly reveal the bound or strongly hint at the correct option."}, "TAS": {"score": 0, "justification": "This is essentially a direct statement of the extremal theorem under its standard hypotheses, with the correct answer being the theorem’s conclusion almost verbatim."}, "GPS": {"score": 1, "justification": "There is some pressure to distinguish the exact sharp bound from nearby variants (off-by-one, max vs. min, weaker true bound), but the item mainly tests theorem recall rather than genuine derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically meaningful and plausible: they include a subtle threshold shift, a weaker-but-true statement, a max/min confusion, and an incorrect extremal support size."}, "total_score": 5, "overall_assessment": "Strong distractor design and no answer leakage, but the question is mostly a theorem-recall item and is fairly tautological rather than generative."}}
{"id": "2602.01471v4", "paper_link": "http://arxiv.org/abs/2602.01471v4", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm:main1}\n\t\tLet $n,s,k$ be positive integers, $n \\geq sk$, $k \\geq 1$.\n\t\tLet $\\mathcal{F} \\subseteq \\binom{[n]}{k}$ be a family with no $s$ pairwise disjoint sets. Then, $|\\mathcal{F}| \\leq \\max\\left\\{\\binom{n}{k}-\\binom{n-s+1}{k}, \\binom{sk-1}{k}\\right\\}$.", "start_pos": 5166, "end_pos": 5461, "label": "thm:main1"}, "ref_dict": {"lemma:1": "\\begin{lemma} \\label{lemma:1}\n\t\tLet $\\mathcal{F}$ be a family of subsets of $[n]$ and let $1 \\leq i, j \\leq n$. Then the following hold:\n\t\t\\begin{enumerate}[label=(\\roman*)]\n\t\t\t\\item For any $F \\in \\mathcal{F}$, $|\\mathcal{C}_{ij}(F)| = |F|$.\n\t\t\t\\item $|\\mathcal{C}_{ij}(\\mathcal{F})| = |\\mathcal{F}|$.\n\t\t\t\\item The matching number does not increase, i.e., $\\nu(\\mathcal{C}_{ij}(\\mathcal{F})) \\leq \\nu(\\mathcal{F})$.\n\t\t\\end{enumerate}\n\t\\end{lemma}", "lemma:2": "\\begin{lemma}\\label{lemma:2}\n\t\tLet $\\mathcal{F}$ be a trivial family on the ground set $[n]$. Then for any $1 \\leq i, j \\leq n$, $i\\neq j$, the family $\\mathcal{C}_{ij}(\\mathcal{F})$ is also a trivial family.\n\t\\end{lemma}"}, "pre_theorem_intro_text_len": 3571, "pre_theorem_intro_text": "A central problem in extremal combinatorics is to determine the maximum possible size of a set family subject to a given constraint. We consider families $\\mathcal{F} \\subseteq 2^{[n]}$ on the ground set $[n] = \\{1, \\dots, n\\}$. A \\emph{matching} in $\\mathcal{F}$ is a collection of pairwise disjoint members of $\\mathcal{F}$. The \\emph{matching number} of $\\mathcal{F}$, denoted $\\nu(\\mathcal{F})$, is the cardinality of a maximum matching.\n\n\tThis paper focuses on $k$-uniform families, $\\mathcal{F} \\subseteq \\binom{[n]}{k}$, that are $s$-matching-free, i.e., $\\nu(\\mathcal{F}) \\le s-1$. We denote the maximum possible size of such a family by $f(n,k,s)$. In 1965, Erd\\H{o}s \\cite{erdos1965} posed a conjecture for the exact value of $f(n,k,s)$, which has become a celebrated open problem in the field.\n\n\t\\begin{conjecture}[Erd\\H{o}s Matching Conjecture \\cite{erdos1971}] \\label{emc}\n\t\tFor $n \\ge sk$,\n\t\t\\[f(n,k,s) = \\max\\left\\{\\binom{sk-1}{k}, \\binom{n}{k}-\\binom{n-s+1}{k}\\right\\}.\\]\n\t\\end{conjecture}\n\n\tThe two expressions in the maximum correspond to two canonical extremal families.\n\t\\begin{enumerate}\n\t\t\\item The family $\\mathcal{G}^* = \\binom{[sk-1]}{k}$, which consists of all $k$-subsets of a fixed set of $sk-1$ elements.\n\t\t\\item The family $\\mathcal{F}^* = \\{F \\in \\binom{[n]}{k} : F \\cap S \\neq \\emptyset\\}$ for a fixed set $S \\subset [n]$ with $|S| = s-1$. This family consists of all $k$-subsets that intersect a fixed set of $s-1$ elements.\n\t\\end{enumerate}\n\tThe conjecture is trivially true for $n < sk$, as in this case $\\nu(\\mathcal{F}) \\le s-1$ holds for the entire family $\\binom{[n]}{k}$.\n\tThe conjecture has a rich history and has been verified in several important cases. For $s=2$, the problem is to find the largest intersecting family, and the conjecture follows from the Erd\\H{o}s-Ko-Rado theorem \\cite{erdos1961}. Kleitman \\cite{kleitman1968} proved the conjecture for the boundary case $n=sk$.\n\n\tSignificant progress has been made on the conjecture for sufficiently large $n$. Erd\\H{o}s \\cite{erdos1965} first established the conjecture for $n \\ge n_0(k,s)$. This bound was subsequently sharpened by Bollob\\'{a}s, Daykin, and Erd\\H{o}s \\cite{bollobas1976} to $n \\ge 2sk^3$, and later by Huang, Lo, and Sudakov \\cite{huang2012} to $n \\ge 3sk^2$.\n\n\tOther results have affirmed the conjecture for specific parameters or ranges. Frankl \\cite{frankl1987} proved the conjecture for $n \\geq (2s+1)k-s$ and also provided the general upper bound $f(n,k,s) \\leq (s-1)\\binom{n-1}{k-1}$. Frankl, R{\\\"{o}}dl, and Ruci{\\'{n}}ski \\cite{frankl2012} established the case $k=3$ for $n \\geq 4s$. More recently, Kolupaev and Kupavskii \\cite{kupavskii2023} proved the conjecture for $k \\geq 5$, $s > 100k^3$, and $sk \\leq n < s(k+100/k)$. For a comprehensive survey of recent results, see \\cite{frankl2022}.\n\n\tDespite this extensive body of work, the conjecture has remained open in its entirety. In this paper, we provide a complete proof.\n\n\t\\subsection{Our results}\n\tWe establish the conjecture for all $n \\ge sk$. Our proof is algorithmic and relies on a key distinction between trivial and non-trivial families.\n\n\t\\begin{definition}\n\t\tA family is said to be \\emph{trivial} if there exists at least one element $x \\in [n]$ that is not contained in any $F \\in \\mathcal{F}$. A family that is not trivial is \\emph{non-trivial}.\n\t\\end{definition}\n\n\tBy analyzing the behavior of families under shift operations, and paying close attention to whether intermediate families in our algorithm become trivial, we establish the following theorem in Section \\ref{sec:4}.", "context": "A central problem in extremal combinatorics is to determine the maximum possible size of a set family subject to a given constraint. We consider families $\\mathcal{F} \\subseteq 2^{[n]}$ on the ground set $[n] = \\{1, \\dots, n\\}$. A \\emph{matching} in $\\mathcal{F}$ is a collection of pairwise disjoint members of $\\mathcal{F}$. The \\emph{matching number} of $\\mathcal{F}$, denoted $\\nu(\\mathcal{F})$, is the cardinality of a maximum matching.\n\nThis paper focuses on $k$-uniform families, $\\mathcal{F} \\subseteq \\binom{[n]}{k}$, that are $s$-matching-free, i.e., $\\nu(\\mathcal{F}) \\le s-1$. We denote the maximum possible size of such a family by $f(n,k,s)$. In 1965, Erd\\H{o}s \\cite{erdos1965} posed a conjecture for the exact value of $f(n,k,s)$, which has become a celebrated open problem in the field.\n\n\\begin{conjecture}[Erd\\H{o}s Matching Conjecture \\cite{erdos1971}] \\label{emc}\n For $n \\ge sk$,\n \\[f(n,k,s) = \\max\\left\\{\\binom{sk-1}{k}, \\binom{n}{k}-\\binom{n-s+1}{k}\\right\\}.\\]\n \\end{conjecture}\n\nThe two expressions in the maximum correspond to two canonical extremal families.\n \\begin{enumerate}\n \\item The family $\\mathcal{G}^* = \\binom{[sk-1]}{k}$, which consists of all $k$-subsets of a fixed set of $sk-1$ elements.\n \\item The family $\\mathcal{F}^* = \\{F \\in \\binom{[n]}{k} : F \\cap S \\neq \\emptyset\\}$ for a fixed set $S \\subset [n]$ with $|S| = s-1$. This family consists of all $k$-subsets that intersect a fixed set of $s-1$ elements.\n \\end{enumerate}\n The conjecture is trivially true for $n < sk$, as in this case $\\nu(\\mathcal{F}) \\le s-1$ holds for the entire family $\\binom{[n]}{k}$.\n The conjecture has a rich history and has been verified in several important cases. For $s=2$, the problem is to find the largest intersecting family, and the conjecture follows from the Erd\\H{o}s-Ko-Rado theorem \\cite{erdos1961}. Kleitman \\cite{kleitman1968} proved the conjecture for the boundary case $n=sk$.\n\n\\begin{definition}\n A family is said to be \\emph{trivial} if there exists at least one element $x \\in [n]$ that is not contained in any $F \\in \\mathcal{F}$. A family that is not trivial is \\emph{non-trivial}.\n \\end{definition}\n\nBy analyzing the behavior of families under shift operations, and paying close attention to whether intermediate families in our algorithm become trivial, we establish the following theorem in Section \\ref{sec:4}.", "full_context": "A central problem in extremal combinatorics is to determine the maximum possible size of a set family subject to a given constraint. We consider families $\\mathcal{F} \\subseteq 2^{[n]}$ on the ground set $[n] = \\{1, \\dots, n\\}$. A \\emph{matching} in $\\mathcal{F}$ is a collection of pairwise disjoint members of $\\mathcal{F}$. The \\emph{matching number} of $\\mathcal{F}$, denoted $\\nu(\\mathcal{F})$, is the cardinality of a maximum matching.\n\nThis paper focuses on $k$-uniform families, $\\mathcal{F} \\subseteq \\binom{[n]}{k}$, that are $s$-matching-free, i.e., $\\nu(\\mathcal{F}) \\le s-1$. We denote the maximum possible size of such a family by $f(n,k,s)$. In 1965, Erd\\H{o}s \\cite{erdos1965} posed a conjecture for the exact value of $f(n,k,s)$, which has become a celebrated open problem in the field.\n\n\\begin{conjecture}[Erd\\H{o}s Matching Conjecture \\cite{erdos1971}] \\label{emc}\n For $n \\ge sk$,\n \\[f(n,k,s) = \\max\\left\\{\\binom{sk-1}{k}, \\binom{n}{k}-\\binom{n-s+1}{k}\\right\\}.\\]\n \\end{conjecture}\n\nThe two expressions in the maximum correspond to two canonical extremal families.\n \\begin{enumerate}\n \\item The family $\\mathcal{G}^* = \\binom{[sk-1]}{k}$, which consists of all $k$-subsets of a fixed set of $sk-1$ elements.\n \\item The family $\\mathcal{F}^* = \\{F \\in \\binom{[n]}{k} : F \\cap S \\neq \\emptyset\\}$ for a fixed set $S \\subset [n]$ with $|S| = s-1$. This family consists of all $k$-subsets that intersect a fixed set of $s-1$ elements.\n \\end{enumerate}\n The conjecture is trivially true for $n < sk$, as in this case $\\nu(\\mathcal{F}) \\le s-1$ holds for the entire family $\\binom{[n]}{k}$.\n The conjecture has a rich history and has been verified in several important cases. For $s=2$, the problem is to find the largest intersecting family, and the conjecture follows from the Erd\\H{o}s-Ko-Rado theorem \\cite{erdos1961}. Kleitman \\cite{kleitman1968} proved the conjecture for the boundary case $n=sk$.\n\n\\begin{definition}\n A family is said to be \\emph{trivial} if there exists at least one element $x \\in [n]$ that is not contained in any $F \\in \\mathcal{F}$. A family that is not trivial is \\emph{non-trivial}.\n \\end{definition}\n\nBy analyzing the behavior of families under shift operations, and paying close attention to whether intermediate families in our algorithm become trivial, we establish the following theorem in Section \\ref{sec:4}.\n\n\\newtheorem*{statement1}{Statement of Theorem 1}\n\\newtheorem*{statement2}{Statement of Lemma 2}\n\\begin{document}\n \\title{Erd\\H{o}s Matching (Conjecture) Theorem}\n \\author{Tapas Kumar Mishra \\footnote{Orcid: 0000-0002-9825-3828} \\\\\n Department of Computer Science and Engineering,\\\\\n National Institute of Technology, Rourkela\n 769008, India\\\\\n mishrat@nitrkl.ac.in\\\\\n https://mishra-tapas.github.io/}\n \\maketitle \\begin{abstract}\n Let $\\mathcal{F}$ be a family of $k$-sized subsets of $[n]$ that does not contain $s$ pairwise disjoint subsets.\n The Erd\\H{o}s Matching Conjecture, a celebrated and long-standing open problem in extremal combinatorics, asserts the maximum cardinality of $\\mathcal{F}$ is upper bounded by\n $\\max\\left\\{\\binom{sk-1}{k}, \\binom{n}{k}-\\allowbreak \\binom{n-s+1}{k}\\right\\}$.\n These two bounds correspond to the sizes of two canonical extremal families: one in which all subsets are contained within a ground set of $sk-1$ elements, and one in which every subset intersects a fixed set of $s-1$ elements.\n In this paper, we prove the conjecture.\n\n\\begin{conjecture}[Erd\\H{o}s Matching Conjecture \\cite{erdos1971}] \\label{emc}\n For $n \\ge sk$,\n \\[f(n,k,s) = \\max\\left\\{\\binom{sk-1}{k}, \\binom{n}{k}-\\binom{n-s+1}{k}\\right\\}.\\]\n \\end{conjecture}\n\nBy analyzing the behavior of families under shift operations, and paying close attention to whether intermediate families in our algorithm become trivial, we establish the following theorem in Section \\ref{sec:4}.\n\nThe key technical ingredients in our proof are the classical $(i,j)$ shift operator of Fr\\'{a}nkl. The precise effects of these operators on a family's matching number and its potential for triviality are detailed in Lemma \\ref{lemma:1}, \\ref{lemma:2}. This algorithmic framework, which tracks the properties of families through a sequence of shifts, can be extended to the non-uniform case as well.\n \\section{Shifting operator and supporting lemmas}\n \\label{sec:2}\n \\begin{definition}[Fr\\'{a}nkl's $(i,j)$ shift]\n Let $\\mathcal{F} \\subseteq 2^{[n]}$. For $1 \\leq i,j \\leq n$, define $\\mathcal{C}_{ij}(\\mathcal{F})= \\{\\mathcal{C}_{ij}(F): F \\in \\mathcal{F}\\}$, where\n \\begin{align*}\n \\mathcal{C}_{ij}(F) = \\begin{cases}\n (F \\setminus \\{j\\})\\cup \\{i\\} , \\text{ if $i \\not \\in F$, $j \\in F$, and $(F \\setminus \\{j\\})\\cup \\{i\\} \\not\\in \\mathcal{F}$},\\\\\n F, \\text{ otherwise.}\n \\end{cases}\n \\end{align*}\n \\end{definition}\n We have two lemmas on the effects of a $(i,j)$ shift on a family $\\mathcal{F}$.\n\n\\begin{lemma} \\label{lemma:1}\n Let $\\mathcal{F}$ be a family of subsets of $[n]$ and let $1 \\leq i, j \\leq n$. Then the following hold:\n \\begin{enumerate}[label=(\\roman*)]\n \\item For any $F \\in \\mathcal{F}$, $|\\mathcal{C}_{ij}(F)| = |F|$.\n \\item $|\\mathcal{C}_{ij}(\\mathcal{F})| = |\\mathcal{F}|$.\n \\item The matching number does not increase, i.e., $\\nu(\\mathcal{C}_{ij}(\\mathcal{F})) \\leq \\nu(\\mathcal{F})$.\n \\end{enumerate}\n \\end{lemma}\n \\begin{proof}\n The proof of (i) and (ii) are given in \\cite{frankl1987}. As (iii) is probably folklore, the proof is given in the Appendix \\ref{app:1}. \n \\end{proof}\n\n\\section{Proof of Theorem \\ref{thm:main1}}\n \\label{sec:4}\nFor a family $\\mathcal{F}$ and an element $x \\in [n]$, we denote the subfamily of sets containing $x$ as $\\mathcal{F}^x = \\{F \\in \\mathcal{F} : x \\in F\\}$. \n \\begin{statement1}\n Let $n,s,k$ be positive integers, $n \\geq sk$, $k \\geq 1$.\n Let $\\mathcal{F} \\subseteq \\binom{[n]}{k}$ be a family with no $s$ pairwise disjoint sets. Then, $|\\mathcal{F}| \\leq \\max\\left\\{\\binom{n}{k}-\\binom{n-s+1}{k}, \\binom{sk-1}{k}\\right\\}$.\n \\end{statement1}\n \\begin{proof}\n The family $\\mathcal{F}^* = \\{F \\in \\binom{[n]}{k} : F \\cap \\{1, \\dots, s-1\\} \\neq \\emptyset\\}$ satisfies the condition $\\nu(\\mathcal{F}^*) \\le s-1$ and has size exactly $\\binom{n}{k} - \\binom{n-s+1}{k}$. \n Let $\\mathcal{G}^*=\\binom{[sk-1]}{k}$.\n We will show that no family satisfying the hypothesis can be larger in cardinality than both $\\mathcal{F}^*$ and $\\mathcal{G}^*$. The proof uses an iterative algorithm that transforms any valid family $\\mathcal{F}$ into one where the desired structure is evident.\n\n\\begin{enumerate}\n \\item If $\\mathcal{F}$ is trivial, let $Y \\subseteq [n']$ be the largest set of vertices with no element of $Y$ being a member of any set in $\\mathcal{F}$ (i.e. $\\forall y \\in Y \\forall F \\in \\mathcal{F}$, $y \\not\\in F$). Set $\\mathcal{F}$ to be the non-trivial family on $[n']\\setminus Y$ and with a relabeling (that does not change $S$), assume the ground set is exactly $[n'-|Y|]$. Set $n'=n'-|Y|$. Note that this transformation preserves $|\\mathcal{F}|$ and $\\nu(\\mathcal{F})$.\n If $n' \\leq sk-1$, $\\mathcal{F} \\subseteq \\mathcal{G}^*$ and we terminate the algorithm.\n Otherwise, set $\\mathcal{F}^* = \\{F \\in \\binom{[n']}{k} : F \\cap \\{1, \\dots, s-1\\} \\neq \\emptyset\\}$.\n \\item If the current family $\\mathcal{F}$ is not a subfamily of $\\mathcal{F}^*$, then there must exist a set $A \\in \\mathcal{F}$ such that $A \\cap S = \\emptyset$.\n \\item There must also exist a set $B \\notin \\mathcal{F}$ such that $B \\cap S \\neq \\emptyset$. If no such $B$ exists, the algorithm terminates and proceeds to the ``Contradiction'' phase described in Section III.\n \\item Let $X = A \\cap B$. Define $A' = A \\setminus X = \\{a_1, \\dots, a_r\\}$ and $B' = B \\setminus X = \\{b_1, \\dots, b_r\\}$. By construction, all elements $a \\in A'$ are not in $S$. We can order the elements in $B'$ such that $b_1 \\in S$. The remaining elements in $B'$ and all elements in $A'$ are ordered arbitrarily.\n\nThe algorithm terminates under three conditions.\\\\\n \\textbf{Condition 1: $n' \\leq sk-1$}. In this case, $\\mathcal{F} \\subseteq \\mathcal{G}^*$ and $|\\mathcal{F}| \\leq \\binom{sk-1}{k}$.\\\\\n \\textbf{Condition 2: the current family $\\mathcal{F}$ is a subfamily of $\\mathcal{F}^*$}. In this case, $|\\mathcal{F}| \\leq \\binom{n'}{k}-\\binom{n'-s+1}{k} \\leq \\binom{n}{k}-\\binom{n-s+1}{k}$ and the bound follows.\\\\\n \\textbf{Condition 3: if the algorithm reaches a state where it can find a set $A \\in \\mathcal{F}$ with $A \\cap S = \\emptyset$, but it cannot find a set $B \\notin \\mathcal{F}$ with $B \\cap S \\neq \\emptyset$}. This implies that the current family $\\mathcal{F}$ contains \\textbf{all possible $k$-sets that intersect $S$}.\n We can now derive a contradiction. Let $A$ be the set in $\\mathcal{F}$ that is disjoint from $S$. We can construct $s-1$ new sets, $B_1, \\dots, B_{s-1}$, that are pairwise disjoint and also disjoint from $A$.\n \\begin{itemize}\n \\item Choose a set $X$ of size $(s-1)k$ from $[n']\\setminus A$\n such that $\\{1,\\ldots,s-1\\} \\subset X$.\n Partition $X$ into $s-1$ sets $B_1,\\ldots,B_{s-1}$\n with $B_i \\cap \\{1,\\ldots,s-1\\}=\\{i\\}$.\n \\end{itemize}\n This construction is possible because the condition $n' \\ge sk$ ensures there are enough elements available. Since each $B_i$ intersects $S$, and our family $\\mathcal{F}$ contains all such sets, it follows that $\\{B_1, \\dots, B_{s-1}\\} \\subset \\mathcal{F}$.\n The collection \\textbf{$\\{A, B_1, \\dots, B_{s-1}\\}$} is therefore a matching of size $s$ within $\\mathcal{F}$. This contradicts the hypothesis that $\\nu(\\mathcal{F}) \\le s-1$.\n\n\\begin{lemma} \\label{lemma:1}\n\t\tLet $\\mathcal{F}$ be a family of subsets of $[n]$ and let $1 \\leq i, j \\leq n$. Then the following hold:\n\t\t\\begin{enumerate}[label=(\\roman*)]\n\t\t\t\\item For any $F \\in \\mathcal{F}$, $|\\mathcal{C}_{ij}(F)| = |F|$.\n\t\t\t\\item $|\\mathcal{C}_{ij}(\\mathcal{F})| = |\\mathcal{F}|$.\n\t\t\t\\item The matching number does not increase, i.e., $\\nu(\\mathcal{C}_{ij}(\\mathcal{F})) \\leq \\nu(\\mathcal{F})$.\n\t\t\\end{enumerate}\n\t\\end{lemma}\n\n\\begin{lemma}\\label{lemma:2}\n\t\tLet $\\mathcal{F}$ be a trivial family on the ground set $[n]$. Then for any $1 \\leq i, j \\leq n$, $i\\neq j$, the family $\\mathcal{C}_{ij}(\\mathcal{F})$ is also a trivial family.\n\t\\end{lemma}", "post_theorem_intro_text_len": 399, "post_theorem_intro_text": "The key technical ingredients in our proof are the classical $(i,j)$ shift operator of Fr\\'{a}nkl. The precise effects of these operators on a family's matching number and its potential for triviality are detailed in Lemma \\ref{lemma:1}, \\ref{lemma:2}. This algorithmic framework, which tracks the properties of families through a sequence of shifts, can be extended to the non-uniform case as well.", "sketch": "The proof of Theorem~\\ref{thm:main1} uses as key technical ingredients the classical $(i,j)$ shift operator of Fr\\'{a}nkl. The effects of these operators on a family’s matching number and its potential for triviality are detailed in Lemma~\\ref{lemma:1} and Lemma~\\ref{lemma:2}. The argument is presented in an algorithmic framework that “tracks the properties of families through a sequence of shifts,” and this shifting framework can also be extended to the non-uniform case.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:main1}\n\t\tLet $n,s,k$ be positive integers, $n \\geq sk$, $k \\geq 1$.\n\t\tLet $\\mathcal{F} \\subseteq \\binom{[n]}{k}$ be a family with no $s$ pairwise disjoint sets. Then, $|\\mathcal{F}| \\leq \\max\\left\\{\\binom{n}{k}-\\binom{n-s+1}{k}, \\binom{sk-1}{k}\\right\\}$.", "theorem_type": ["Inequality or Bound", "Implication"], "mcq": {"question": "Let $n,s,k$ be positive integers with $n\\ge sk$ and $k\\ge 1$. Write $[n]=\\{1,2,\\dots,n\\}$, and let $\\binom{[n]}{k}$ denote the family of all $k$-element subsets of $[n]$. Suppose $\\mathcal F\\subseteq \\binom{[n]}{k}$ is a $k$-uniform family that contains no $s$ pairwise disjoint sets. Which of the following quantitative estimates holds for $|\\mathcal F|$?", "correct_choice": {"label": "A", "text": "$|\\mathcal F|\\le \\max\\left\\{\\binom{n}{k}-\\binom{n-s+1}{k},\\;\\binom{sk-1}{k}\\right\\}$."}, "choices": [{"label": "B", "text": "$|\\mathcal F|\\le \\max\\left\\{\\binom{n}{k}-\\binom{n-s}{k},\\;\\binom{sk-1}{k}\\right\\}$."}, {"label": "C", "text": "$|\\mathcal F|\\le \\binom{n}{k}-\\binom{n-s+1}{k}+\\binom{sk-1}{k}$."}, {"label": "D", "text": "$|\\mathcal F|\\le \\min\\left\\{\\binom{n}{k}-\\binom{n-s+1}{k},\\;\\binom{sk-1}{k}\\right\\}$."}, {"label": "E", "text": "$|\\mathcal F|\\le \\max\\left\\{\\binom{n}{k}-\\binom{n-s+1}{k},\\;\\binom{sk}{k}\\right\\}$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "extremal-count threshold in the intersecting-with-fixed-$(s-1)$-set term", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "replaced the sharp maximum of the two extremal bounds by their sum", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "aggregation of the two candidate extremal bounds", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "boundary case $sk-1$ replaced by $sk$ in the complete-support extremal term", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct formula. It states the hypotheses and asks for the extremal estimate, without giving away the max-structure or the precise boundary terms."}, "TAS": {"score": 0, "justification": "This is essentially a direct restatement of a known extremal set theory result under its standard hypotheses. The task is mainly to recognize the theorem statement rather than derive or apply it in a new way."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in subtle but meaningful ways (max vs min, n-s vs n-s+1, sk-1 vs sk). However, the item primarily tests theorem recall/recognition rather than substantive generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong: they are mathematically nearby, distinct, and reflect realistic errors such as off-by-one mistakes, replacing a sharp maximum by a weaker sum, or using the wrong aggregation."}, "total_score": 5, "overall_assessment": "A solid recognition-style MCQ with high-quality distractors and no answer leakage, but it is largely tautological as a theorem-recall question and only moderately tests genuine reasoning."}}
{"id": "2602.01571v1", "paper_link": "http://arxiv.org/abs/2602.01571v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{theorem:main}\n    Let $f$ be a cuspidal modular form of weight $k$ on $\\SL_2(\\BZ)$. For any integers $l\\geq2$ and $d\\geq1$ and $dl>4$, one has\n    \\[\\sum_{n\\leq x}  \\lambda^l_{\\Sym^d f}(n)=xP_{d,l}(\\log x)+O\\left(x^{\\theta_{d,l}+\\epsilon}\\right).\\]\n    Here $P_{d,l}$ is a polynomial of degree $K_{0,d,l}-1$, where $K_{i,d,l}$ denote the Kostka number defined in sections \\ref{section:Weylmod} and \\ref{section:Kostka number}. The exponent $\\theta_{d, l}$ is given by\n    $$\\theta_{d,l}=1-1/\\left((d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14\\right).$$", "start_pos": 19116, "end_pos": 19700, "label": "theorem:main"}, "ref_dict": {"theorem:main": "\\begin{thm}\\label{theorem:main}\n    Let $f$ be a cuspidal modular form of weight $k$ on $\\SL_2(\\BZ)$. For any integers $l\\geq2$ and $d\\geq1$ and $dl>4$, one has\n    \\[\\sum_{n\\leq x}  \\lambda^l_{\\Sym^d f}(n)=xP_{d,l}(\\log x)+O\\left(x^{\\theta_{d,l}+\\epsilon}\\right).\\]\n    Here $P_{d,l}$ is a polynomial of degree $K_{0,d,l}-1$, where $K_{i,d,l}$ denote the Kostka number defined in sections \\ref{section:Weylmod} and \\ref{section:Kostka number}. The exponent $\\theta_{d, l}$ is given by\n    $$\\theta_{d,l}=1-1/\\left((d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14\\right).$$\n\\end{thm}", "equation:introduction4": "\\begin{equation}\\label{equation:introduction4}\n    \\sum_{n\\leq x}\\lambda^2_{\\Sym^d f}(n)=c_dx+O\\left(x^{\\tilde{\\theta}_d+\\epsilon}\\right),\n\\end{equation}", "equation:introduction3": "\\begin{equation} \\label{equation:introduction3}\n    \\sum_{n\\leq x}\\lambda^l_{\\Sym^2 f}(n)=xP_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right)\n\\end{equation}", "proposition:dirichletseriescomp": "\\begin{prop}\\label{proposition:dirichletseriescomp}\nThe following identities hold:\n\\begin{align*}\n   &D\\left(f^{\\otimes l},s\\right)=L\\left(f^{\\otimes l},s\\right)U_{f,l}(s),D\\left(\\left(\\Sym^d f\\right)^{\\otimes l},s\\right)=L\\left(\\left(\\Sym^df\\right)^{\\otimes l},s\\right)U_{\\Sym^d f,l}(s),\\\\ \n   &D\\left(f^{\\otimes l},\\chi,s\\right)=L\\left(f^{\\otimes l},\\chi,s\\right)U_{f,l,\\chi}(s),D\\left(\\left(\\Sym^d f\\right)^{\\otimes l},\\chi,s\\right)=L\\left(\\left(\\Sym^df\\right)^{\\otimes l},\\chi,s\\right)U_{\\Sym^d f,l,\\chi}(s).\n\\end{align*}\n       where each  $U_*(s)$ is an Euler product that is absolutely convergent for  $\\Re(s)>1/2$ and uniformly convergent in the region $\\Re(s)>1/2+\\epsilon$ for any $\\epsilon>0$.\n\\end{prop}", "equation:introduction2": "\\begin{equation}\\label{equation:introduction2}\n \\sum_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2)=xQ'_l(\\log x)+O\\left(x^{{\\theta}'_l+\\epsilon}\\right),\n\\end{equation}", "equation:introduction1": "\\begin{equation}\\label{equation:introduction1}\n  \\sum_{n\\leq x}  \\lambda^l_f(n)=xQ_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right),\n\\end{equation}", "corollary:Kostka": "\\begin{cor}\\label{corollary:Kostka}\n    \\begin{equation}\n        K_{i,d,l}=\\begin{cases}\n            0\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\  \\ \\ \\ \\ \\ \\ \\   \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\text{if $2\\nmid dl-i$}\\\\\n            \\sum\\limits_{j=0}^{\\lfloor(dl-i)/(2d+2)\\rfloor}(-1)^j\\binom{l}{j}\\binom\n        {(dl-i)/2-j(d+1)+l-2}{l-2}\n    \\ \\text{otherwise}\n        \\end{cases}.\n    \\end{equation}\n\\end{cor}"}, "pre_theorem_intro_text_len": 4707, "pre_theorem_intro_text": "Let $f$ be a cuspidal  eigenform of weight $k$ on $\\SL_2(\\BZ)$,  and let\n\\[f(z)=\\sum_{n=1}^\\infty \\lambda_f(n)n^{\\frac{k-1}{2}}q^n\\]\nbe its $q$-expansion. Estimates of $\\lambda_f(n)$ and their average behavior are fundamental problems in number theory. \n Estimating the sum\n$\\sum\\limits_{n\\leq x}\\lambda_f(n),$ has a long history, with contributions from\n \\cite{kloosterman1927asymptotische}, \\cite{rankin1983sums}, \\cite{weil1948some}, \\cite{wu2009power}, among others. The sum of higher power moments \n$\\sum\\limits_{n\\leq x}\\lambda_f^l(n)$\nhas also attracted considerable attention. If $l=2$, Rankin\\cite{rankin1939contributions} and Selberg\\cite{selberg1940bemerkungen} already studied this problem and recently Huang's work \\cite{huang2021rankin} provides improved estimates with\n\\[\\sum_{n\\leq x}\\lambda_f^2(n)=Cx+O\\left(x^{\\frac{3}{5}-\\frac{1}{560}+\\epsilon}\\right).\\]\n\nFor $l\\geq3$,  symmetric power $L$-functions of modular forms are required.  Kim and  Kim \\& Shahidi   \\cite{kim2003functoriality}, \\cite{kim2002cuspidality},\\cite{kim2002functorial} showed that  $\\Sym^l f$ is automorphic for $l\\leq 4$. Using these results,  the estimates for $l\\leq 8$  are derived in  \\cite{lau2011integral}, \\cite{lu2009average} and \\cite{lu2011higher}. Later Zhai\\cite{zhai2013average} considered the sum over sums of two squares\n\\[\\sum_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2),\\]\nand provided estimates for $l\\leq 8$.\n\nRecently, Newton \\& Thorne \\cite{newton2021symmetric1},\\cite{newton2021symmetric2} proved the automorphy and cuspidality of $\\Sym^l f$ for all $l\\geq1$. Using this celebrated work, Xu\\cite{xu2022general} derived a general formula for $\\sum\\limits_{n\\leq x}\\lambda_f^l(n)$ and $\\sum\\limits_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2)$ for every $l\\geq1$. This result was later refined by Liu\\cite{liu2023asymptotic} as\n\\begin{equation}\\label{equation:introduction1}\n  \\sum_{n\\leq x}  \\lambda^l_f(n)=xQ_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right),\n\\end{equation}\n\\begin{equation}\\label{equation:introduction2}\n \\sum_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2)=xQ'_l(\\log x)+O\\left(x^{{\\theta}'_l+\\epsilon}\\right),\n\\end{equation}\nwhere $Q_l$ and $Q'_l$ are polynomials of degree $\\binom{l}{m}-\\binom{l}{m-1}-1$  if $l=2m$ and equals 0 if $l$ is odd. Here $\\theta_l, {\\theta}'_l$ are explicit constants less than 1. \n\nAnother direction of research concerns the average behavior of  Fourier coefficients of symmetric powers of $f$. Several results have been obtained in this direction, but  they only focus on some special cases. For example,  in \\cite{fomenko2006identities} and \\cite{fomenko2008mean},  Fomenko proved asymptotic formulas for \n\\[\\sum_{n\\leq x}\\lambda_{\\Sym^2 f}(n)\\ \\text{and}\\ \\sum_{n\\leq x}\\lambda^2_{\\Sym^2 f}(n).\\]\nIn \\cite{he2019integral} and \\cite{luo2021asymptotics},  the summations \\[\\sum_{n\\leq x}\\lambda^l_{\\Sym^2 f}(n)\\ \\text{and}\\ \\sum_{n\\leq x}\\lambda^2_{\\Sym^d f}(n)\\]\nwith $d\\geq 2$ and $l\\leq 8$ were studied. Later  in \\cite{liu2023average}, Liu obtained the following results with better error terms:\n\\begin{equation} \\label{equation:introduction3}\n    \\sum_{n\\leq x}\\lambda^l_{\\Sym^2 f}(n)=xP_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right)\n\\end{equation}\nfor $2\\leq l\\leq 8$, and\n\\begin{equation}\\label{equation:introduction4}\n    \\sum_{n\\leq x}\\lambda^2_{\\Sym^d f}(n)=c_dx+O\\left(x^{\\tilde{\\theta}_d+\\epsilon}\\right),\n\\end{equation}\nfor any $d\\geq 2$.\nHere $P_l$ are polynomials with  explicitly computed degrees, $c_d$ is a constant and $\\theta_l, \\tilde{\\theta}_d$ are positive constants less than 1.\n\nExtending these results to arbitrary integers $d\\geq1$ and $l\\geq1$ presents significant challenges, as the associated Dirichlet series become increasingly complex. In this paper, we address this problem by studying the underlying Galois representations and applying combinatorial results concerning their decompositions. \nOur work generalizes the average formulas (\\ref{equation:introduction1}),(\\ref{equation:introduction2}),(\\ref{equation:introduction3}),(\\ref{equation:introduction4}) in the following aspects.\n\\begin{itemize}\n    \\item We obtain asymptotic formulas with improved error terms.\n    \\item We establish a universal asymptotic formula for $\\sum\\limits_{n\\leq x}\\lambda^l_{\\Sym^d f}(n)$ valid for any $d\\geq 1$  and $l\\geq 1$,which generalizes (\\ref{equation:introduction3}) and (\\ref{equation:introduction4}).\n        \\item We prove an average formula for sums over values of a binary quadratic form, namely $ \\sum\\limits_{Q(n_1,n_2)\\leq x}\\lambda^l_{\\Sym^d f}(Q(n_1,n_2))$ for any definite binary quadratic form $Q$ and any $d,l\\geq1$.\n\\end{itemize}\n\\subsection{Main theorems}\nWe now state our main results for general $d$ and $l$.", "context": "Let $f$ be a cuspidal eigenform of weight $k$ on $\\SL_2(\\BZ)$, and let\n\\[f(z)=\\sum_{n=1}^\\infty \\lambda_f(n)n^{\\frac{k-1}{2}}q^n\\]\nbe its $q$-expansion. Estimates of $\\lambda_f(n)$ and their average behavior are fundamental problems in number theory. \n Estimating the sum\n$\\sum\\limits_{n\\leq x}\\lambda_f(n),$ has a long history, with contributions from\n \\cite{kloosterman1927asymptotische}, \\cite{rankin1983sums}, \\cite{weil1948some}, \\cite{wu2009power}, among others. The sum of higher power moments \n$\\sum\\limits_{n\\leq x}\\lambda_f^l(n)$\nhas also attracted considerable attention. If $l=2$, Rankin\\cite{rankin1939contributions} and Selberg\\cite{selberg1940bemerkungen} already studied this problem and recently Huang's work \\cite{huang2021rankin} provides improved estimates with\n\\[\\sum_{n\\leq x}\\lambda_f^2(n)=Cx+O\\left(x^{\\frac{3}{5}-\\frac{1}{560}+\\epsilon}\\right).\\]\n\nFor $l\\geq3$, symmetric power $L$-functions of modular forms are required. Kim and Kim \\& Shahidi \\cite{kim2003functoriality}, \\cite{kim2002cuspidality},\\cite{kim2002functorial} showed that $\\Sym^l f$ is automorphic for $l\\leq 4$. Using these results, the estimates for $l\\leq 8$ are derived in \\cite{lau2011integral}, \\cite{lu2009average} and \\cite{lu2011higher}. Later Zhai\\cite{zhai2013average} considered the sum over sums of two squares\n\\[\\sum_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2),\\]\nand provided estimates for $l\\leq 8$.\n\nRecently, Newton \\& Thorne \\cite{newton2021symmetric1},\\cite{newton2021symmetric2} proved the automorphy and cuspidality of $\\Sym^l f$ for all $l\\geq1$. Using this celebrated work, Xu\\cite{xu2022general} derived a general formula for $\\sum\\limits_{n\\leq x}\\lambda_f^l(n)$ and $\\sum\\limits_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2)$ for every $l\\geq1$. This result was later refined by Liu\\cite{liu2023asymptotic} as\n\\begin{equation}\\label{equation:introduction1}\n \\sum_{n\\leq x} \\lambda^l_f(n)=xQ_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right),\n\\end{equation}\n\\begin{equation}\\label{equation:introduction2}\n \\sum_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2)=xQ'_l(\\log x)+O\\left(x^{{\\theta}'_l+\\epsilon}\\right),\n\\end{equation}\nwhere $Q_l$ and $Q'_l$ are polynomials of degree $\\binom{l}{m}-\\binom{l}{m-1}-1$ if $l=2m$ and equals 0 if $l$ is odd. Here $\\theta_l, {\\theta}'_l$ are explicit constants less than 1.\n\nAnother direction of research concerns the average behavior of Fourier coefficients of symmetric powers of $f$. Several results have been obtained in this direction, but they only focus on some special cases. For example, in \\cite{fomenko2006identities} and \\cite{fomenko2008mean}, Fomenko proved asymptotic formulas for \n\\[\\sum_{n\\leq x}\\lambda_{\\Sym^2 f}(n)\\ \\text{and}\\ \\sum_{n\\leq x}\\lambda^2_{\\Sym^2 f}(n).\\]\nIn \\cite{he2019integral} and \\cite{luo2021asymptotics}, the summations \\[\\sum_{n\\leq x}\\lambda^l_{\\Sym^2 f}(n)\\ \\text{and}\\ \\sum_{n\\leq x}\\lambda^2_{\\Sym^d f}(n)\\]\nwith $d\\geq 2$ and $l\\leq 8$ were studied. Later in \\cite{liu2023average}, Liu obtained the following results with better error terms:\n\\begin{equation} \\label{equation:introduction3}\n \\sum_{n\\leq x}\\lambda^l_{\\Sym^2 f}(n)=xP_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right)\n\\end{equation}\nfor $2\\leq l\\leq 8$, and\n\\begin{equation}\\label{equation:introduction4}\n \\sum_{n\\leq x}\\lambda^2_{\\Sym^d f}(n)=c_dx+O\\left(x^{\\tilde{\\theta}_d+\\epsilon}\\right),\n\\end{equation}\nfor any $d\\geq 2$.\nHere $P_l$ are polynomials with explicitly computed degrees, $c_d$ is a constant and $\\theta_l, \\tilde{\\theta}_d$ are positive constants less than 1.\n\nExtending these results to arbitrary integers $d\\geq1$ and $l\\geq1$ presents significant challenges, as the associated Dirichlet series become increasingly complex. In this paper, we address this problem by studying the underlying Galois representations and applying combinatorial results concerning their decompositions. \nOur work generalizes the average formulas (\\ref{equation:introduction1}),(\\ref{equation:introduction2}),(\\ref{equation:introduction3}),(\\ref{equation:introduction4}) in the following aspects.\n\\begin{itemize}\n \\item We obtain asymptotic formulas with improved error terms.\n \\item We establish a universal asymptotic formula for $\\sum\\limits_{n\\leq x}\\lambda^l_{\\Sym^d f}(n)$ valid for any $d\\geq 1$ and $l\\geq 1$,which generalizes (\\ref{equation:introduction3}) and (\\ref{equation:introduction4}).\n \\item We prove an average formula for sums over values of a binary quadratic form, namely $ \\sum\\limits_{Q(n_1,n_2)\\leq x}\\lambda^l_{\\Sym^d f}(Q(n_1,n_2))$ for any definite binary quadratic form $Q$ and any $d,l\\geq1$.\n\\end{itemize}\n\\subsection{Main theorems}\nWe now state our main results for general $d$ and $l$.\n\n\\begin{equation}\\label{equation:introduction1}\n  \\sum_{n\\leq x}  \\lambda^l_f(n)=xQ_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right),\n\\end{equation}\n\n\\begin{equation} \\label{equation:introduction3}\n    \\sum_{n\\leq x}\\lambda^l_{\\Sym^2 f}(n)=xP_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right)\n\\end{equation}\n\n\\begin{equation}\\label{equation:introduction4}\n    \\sum_{n\\leq x}\\lambda^2_{\\Sym^d f}(n)=c_dx+O\\left(x^{\\tilde{\\theta}_d+\\epsilon}\\right),\n\\end{equation}", "full_context": "Let $f$ be a cuspidal eigenform of weight $k$ on $\\SL_2(\\BZ)$, and let\n\\[f(z)=\\sum_{n=1}^\\infty \\lambda_f(n)n^{\\frac{k-1}{2}}q^n\\]\nbe its $q$-expansion. Estimates of $\\lambda_f(n)$ and their average behavior are fundamental problems in number theory. \n Estimating the sum\n$\\sum\\limits_{n\\leq x}\\lambda_f(n),$ has a long history, with contributions from\n \\cite{kloosterman1927asymptotische}, \\cite{rankin1983sums}, \\cite{weil1948some}, \\cite{wu2009power}, among others. The sum of higher power moments \n$\\sum\\limits_{n\\leq x}\\lambda_f^l(n)$\nhas also attracted considerable attention. If $l=2$, Rankin\\cite{rankin1939contributions} and Selberg\\cite{selberg1940bemerkungen} already studied this problem and recently Huang's work \\cite{huang2021rankin} provides improved estimates with\n\\[\\sum_{n\\leq x}\\lambda_f^2(n)=Cx+O\\left(x^{\\frac{3}{5}-\\frac{1}{560}+\\epsilon}\\right).\\]\n\nFor $l\\geq3$, symmetric power $L$-functions of modular forms are required. Kim and Kim \\& Shahidi \\cite{kim2003functoriality}, \\cite{kim2002cuspidality},\\cite{kim2002functorial} showed that $\\Sym^l f$ is automorphic for $l\\leq 4$. Using these results, the estimates for $l\\leq 8$ are derived in \\cite{lau2011integral}, \\cite{lu2009average} and \\cite{lu2011higher}. Later Zhai\\cite{zhai2013average} considered the sum over sums of two squares\n\\[\\sum_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2),\\]\nand provided estimates for $l\\leq 8$.\n\nRecently, Newton \\& Thorne \\cite{newton2021symmetric1},\\cite{newton2021symmetric2} proved the automorphy and cuspidality of $\\Sym^l f$ for all $l\\geq1$. Using this celebrated work, Xu\\cite{xu2022general} derived a general formula for $\\sum\\limits_{n\\leq x}\\lambda_f^l(n)$ and $\\sum\\limits_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2)$ for every $l\\geq1$. This result was later refined by Liu\\cite{liu2023asymptotic} as\n\\begin{equation}\\label{equation:introduction1}\n \\sum_{n\\leq x} \\lambda^l_f(n)=xQ_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right),\n\\end{equation}\n\\begin{equation}\\label{equation:introduction2}\n \\sum_{n_1^2+n_2^2\\leq x}\\lambda_f^l(n_1^2+n_2^2)=xQ'_l(\\log x)+O\\left(x^{{\\theta}'_l+\\epsilon}\\right),\n\\end{equation}\nwhere $Q_l$ and $Q'_l$ are polynomials of degree $\\binom{l}{m}-\\binom{l}{m-1}-1$ if $l=2m$ and equals 0 if $l$ is odd. Here $\\theta_l, {\\theta}'_l$ are explicit constants less than 1.\n\nAnother direction of research concerns the average behavior of Fourier coefficients of symmetric powers of $f$. Several results have been obtained in this direction, but they only focus on some special cases. For example, in \\cite{fomenko2006identities} and \\cite{fomenko2008mean}, Fomenko proved asymptotic formulas for \n\\[\\sum_{n\\leq x}\\lambda_{\\Sym^2 f}(n)\\ \\text{and}\\ \\sum_{n\\leq x}\\lambda^2_{\\Sym^2 f}(n).\\]\nIn \\cite{he2019integral} and \\cite{luo2021asymptotics}, the summations \\[\\sum_{n\\leq x}\\lambda^l_{\\Sym^2 f}(n)\\ \\text{and}\\ \\sum_{n\\leq x}\\lambda^2_{\\Sym^d f}(n)\\]\nwith $d\\geq 2$ and $l\\leq 8$ were studied. Later in \\cite{liu2023average}, Liu obtained the following results with better error terms:\n\\begin{equation} \\label{equation:introduction3}\n \\sum_{n\\leq x}\\lambda^l_{\\Sym^2 f}(n)=xP_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right)\n\\end{equation}\nfor $2\\leq l\\leq 8$, and\n\\begin{equation}\\label{equation:introduction4}\n \\sum_{n\\leq x}\\lambda^2_{\\Sym^d f}(n)=c_dx+O\\left(x^{\\tilde{\\theta}_d+\\epsilon}\\right),\n\\end{equation}\nfor any $d\\geq 2$.\nHere $P_l$ are polynomials with explicitly computed degrees, $c_d$ is a constant and $\\theta_l, \\tilde{\\theta}_d$ are positive constants less than 1.\n\nExtending these results to arbitrary integers $d\\geq1$ and $l\\geq1$ presents significant challenges, as the associated Dirichlet series become increasingly complex. In this paper, we address this problem by studying the underlying Galois representations and applying combinatorial results concerning their decompositions. \nOur work generalizes the average formulas (\\ref{equation:introduction1}),(\\ref{equation:introduction2}),(\\ref{equation:introduction3}),(\\ref{equation:introduction4}) in the following aspects.\n\\begin{itemize}\n \\item We obtain asymptotic formulas with improved error terms.\n \\item We establish a universal asymptotic formula for $\\sum\\limits_{n\\leq x}\\lambda^l_{\\Sym^d f}(n)$ valid for any $d\\geq 1$ and $l\\geq 1$,which generalizes (\\ref{equation:introduction3}) and (\\ref{equation:introduction4}).\n \\item We prove an average formula for sums over values of a binary quadratic form, namely $ \\sum\\limits_{Q(n_1,n_2)\\leq x}\\lambda^l_{\\Sym^d f}(Q(n_1,n_2))$ for any definite binary quadratic form $Q$ and any $d,l\\geq1$.\n\\end{itemize}\n\\subsection{Main theorems}\nWe now state our main results for general $d$ and $l$.\n\n\\begin{equation}\\label{equation:introduction1}\n  \\sum_{n\\leq x}  \\lambda^l_f(n)=xQ_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right),\n\\end{equation}\n\n\\begin{equation} \\label{equation:introduction3}\n    \\sum_{n\\leq x}\\lambda^l_{\\Sym^2 f}(n)=xP_l(\\log x)+O\\left(x^{\\theta_l+\\epsilon}\\right)\n\\end{equation}\n\n\\begin{equation}\\label{equation:introduction4}\n    \\sum_{n\\leq x}\\lambda^2_{\\Sym^d f}(n)=c_dx+O\\left(x^{\\tilde{\\theta}_d+\\epsilon}\\right),\n\\end{equation}\n\nThe estimation in the general case is new. And our estimates in the error terms are better than those in the work of \\cite{liu2023average} for small $d$ and $l$. We list the cases with $d=2,3\\leq l\\leq8$ and $l=2,3\\leq d\\leq8$ in the following tables.\n\nWe also have the following theorem for summation over binary quadratic forms.\n\\begin{thm}\\label{thm:binary}\n Let $f$ be a cuspidal modular form of weight $k$ on $\\SL_2(\\BZ)$. For any integers $l\\geq2,d\\geq1$ with $dl>4$ and any binary quadratic form $Q(n_1,n_2)$, one has\n \\[\\sum_{Q(n_1,n_2)\\leq x} \\lambda^l_{\\Sym^d f}(Q(n_1,n_2))=xP_{d,l,Q}(\\log x)+O\\left(x^{\\theta_{d,l,Q}+\\epsilon}\\right).\\]\n Here $P_{d,l,Q}$ is a polynomial of degree $K_{0,d,l}-1$. If the class number of $Q$ is not 1, then \n $$\\theta_{d,l,Q}=1-\\frac{3}{3(d+1)^l-K_{0,d,l}};$$ if the class number is 1, then \n $$\\theta_{d,l,Q}=1-1/\\left((d+1)^l-8K_{0,d,l}/21-2K_{1,d,l}/3-5K_{2,d,l}/7\\right).$$\n\\end{thm}\n\nWe now provide an alternative explicit description of these Kostka numbers. Let $K_{i,d,l}=K_{\\mu,\\lambda}$ (as defined above). By Proposition \\ref{prop:decomposition}, $K_{i,d,l}$ is the multiplicity of $\\Sym^i V$ in $\\left(\\Sym^d V\\right)^{\\otimes l}$ for a two dimensional representation $V$. We extend the definition of $K_{i,d,l}$ to all $i\\in\\BZ$ by setting $K_{i,d,l}=0$ for $i\\notin [0,dl]$, or when $i$ and $dl$ have different parity. From Proposition \\ref{prop:decomposition} (4), we derive a recursive formula for $K_{i,d,l}$:\n\\[K_{i,d,l}=K_{i-d,d,l-1}+K_{i-d+2,d,l-1}+\\cdots+K_{i+d,d,l-1}.\\]\nOn the other hand, consider the polynomial:\n\\[(1+x+x^2+\\cdots+x^d)^{l}=\\sum_{j=-\\infty}^{+\\infty}C_{j,d,l}x^j.\\]\nDefine $A_{i,d,l}=C_{\\lfloor\\frac{dl-i}{2}\\rfloor,d,l}-C_{\\lfloor\\frac{dl-i-1}{2}\\rfloor,d,l}.$\n\\begin{prop} We have \n $ K_{i,d,l}=A_{i,d,l}.$\n \\end{prop}\n\\begin{proof}\nWe only need to show that $A_{i,d,l}$ and $K_{i,d,l}$ satisfy the same recursive formula and boundary conditions.\n First, the coefficients $C_{i,d,l}$ satisfy the recursion:\n \\[C_{i,d,l}=C_{i-d,d,l-1}+C_{i-d+1,d,l-1}+\\cdots+C_{i,d,l-1}.\\]\n Thus\n \\begin{align*}\n A_{i,d,l}=&C_{\\lfloor\\frac{dl-i}{2}\\rfloor,d,l}-C_{\\lfloor\\frac{dl-i-1}{2}\\rfloor,d,l}\\\\\n =&\\sum_{t=j-d}^jC_{t,d,l-1}-\\sum_{t=j'-d}^{j'}C_{t,d,l-1}.\n \\end{align*}\n where $j=\\lfloor\\frac{dl-i}{2}\\rfloor, j'=\\lfloor\\frac{dl-i-1}{2}\\rfloor.$\n On the other hand,\n \\begin{align*}\n \\sum_{t=0}^dA_{i-d+2t,d,l-1}&=\\sum_{t=0}^d\\left(C_{\\lfloor\\frac{dl-d-(i-d+2t)}{2}\\rfloor,d,l-1}-C_{\\lfloor\\frac{dl-d-(i-d+2t)-1}{2}\\rfloor,d,l-1}\\right)\\\\\n &=\\sum_{t=0}^d\\left(C_{\\lfloor\\frac{dl-i-2t)}{2}\\rfloor,d,l-1}-C_{\\lfloor\\frac{dl-i-2t-1}{2}\\rfloor,d,l-1}\\right)\n \\end{align*}\n By telescoping, this sum equals $A_{i,d,l}$.\n Thus\n \\[A_{i,d,l}=A_{i-d,d,l-1}+A_{i-d+2,d,l-1}+\\cdots+A_{i+d,d,l-1}.\\]\n For the boundary condition $l=1$:\n \\begin{itemize}\n \\item $K_{d,d,1}=1$ and $K_{i,d,1}=0$ for $i\\neq d$\n \\item $A_{d,d,1}=C_{\\lfloor \\frac{d-d}{2}\\rfloor,d,1}-C_{\\lfloor \\frac{d-d-1}{2}\\rfloor,d,1}=C_{0,d,1}-C_{-1,d,1}=1-0=1$ and $A_{i,d,1}=0$ for $i\\neq d$.\n \\end{itemize}\n Thus $A_{i,d,d}$ and $K_{i,d,d}$ satisfy the same recursion and boundary conditions, so $A_{i,d,l}=K_{i,d,l}$.\n\\end{proof}\nFrom this proposition, the Kostka number $K_{i,d,l}$ can be expressed in terms of binomial coefficients. This gives the following formula.\n\\begin{cor}\\label{corollary:Kostka}\n \\begin{equation}\n K_{i,d,l}=\\begin{cases}\n 0\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\text{if $2\\nmid dl-i$}\\\\\n \\sum\\limits_{j=0}^{\\lfloor(dl-i)/(2d+2)\\rfloor}(-1)^j\\binom{l}{j}\\binom\n {(dl-i)/2-j(d+1)+l-2}{l-2}\n \\ \\text{otherwise}\n \\end{cases}.\n \\end{equation}\n\\end{cor}\n\\begin{proof}\nSince\n \\[(1+x+x^2+\\cdots+x^d)^{l}=(1-x^{d+1})^{l}(1-x)^{-l},\\]\n the coefficients $K_{i,d,l}=A_{i,d,l}$ can be computed as\n\\begin{align*}\n A_{i,d,l}=&\\sum_{j=0}^{\\lfloor(dl-i)/(2d+2)\\rfloor}(-1)^j\\binom{l}{j}\\binom{\\lfloor(dl-i)/2\\rfloor-j(d+1)+l-1}{l-1}\\\\\n &-\\sum_{j=0}^{\\lfloor(dl-i-1)/(2d+2)\\rfloor}(-1)^j\\begin{pmatrix}\n l\\\\j\n \\end{pmatrix}\\begin{pmatrix}\n \\lfloor(dl-i-1)/2\\rfloor-j(d+1)+l-1\\\\l-1\n \\end{pmatrix}.\n\\end{align*}\n\nNow let $f$ be a cuspidal eigenform on $\\SL_2(\\BZ)$. For any integer $d\\geq0$, the symmetric power lifting $\\Sym^d f$ is an automorphic form on $\\GL_{d+1}$. For a character $\\chi$ of $\\fh$, we define the associated $L$-function as follows. If $\\chi$ is non-trivial,\nlet\n\\[L\\left(\\Sym^d f, \\chi,s\\right):=\\sum_{n=1}^\\infty\\frac{\\lambda_{\\Sym^df}(n)a_\\chi(n)}{n^s}\\]\nbe the Rankin-Selberg $L$-function associated to $\\Sym^d f$ and $f_\\chi$. If $\\chi$ is trivial, define $$L\\left(\\Sym^d f, \\chi,s\\right)=L\\left(\\Sym^df,s\\right)L\\left(\\Sym^df,\\chi_F,s\\right).$$ Similarly, we can define the $L$-functions $L\\left(f^{\\otimes l},\\chi,s\\right)$ and $L\\left(\\left(\\Sym^d f\\right)^{\\otimes l},\\chi,s\\right)$ by incorporating the coefficients $a_\\chi(n)$. Leveraging the tensor product decompositions from previous sections, we obtain the following factorizations:\n \\begin{equation}\n L\\left(f^{\\otimes l},\\chi,s\\right)=\\prod_{i=0}^{[l/2]}L\\left(\\Sym^{l-2i}f,\\chi,s\\right)^{c_{l,i}},\n \\end{equation}\n \\begin{equation}\\label{equation:chilfactor}\n L\\left(\\left(\\Sym^d f\\right)^{\\otimes l},\\chi,s\\right)=\\prod_{i=0}^{dl}L\\left(\\Sym^i f,\\chi,s\\right)^{K_{i,d,l}}. \n \\end{equation}\n\\section{The estimations via Perron's formula}\n\\subsection{The Dirichlet series}\nLet $f$ be a cuspidal eigenform of weight $k$ on $\\SL_2(\\BZ)$, let $Q$ be a binary quadratic form and let $\\chi$ be a character of $\\fh$. We define the following Dirichlet series, which are central to our analysis:\n\\begin{align*}\n&D\\left(f^{\\otimes l},s\\right):=\\sum_{n=1}^\\infty \\frac{\\lambda_f^l(n)}{n^s},&D&\\left(\\left(\\Sym^df\\right)^{\\otimes l},s\\right):=\\sum_{n=1}^\\infty \\frac{\\lambda_{\\Sym^df}^l(n)}{n^s},\\\\ \n&D\\left(f^{\\otimes l},Q,s\\right):=\\sum_{n=1}^\\infty \\frac{\\lambda_f^l(n)r(n,Q)}{n^s},&D&\\left(\\left(\\Sym^d f\\right)^{\\otimes l},Q,s\\right):=\\sum_{n=1}^\\infty \\frac{\\lambda_{\\Sym^d f}^l(n)r(n,Q)}{n^s},\\\\\n&D\\left(f^{\\otimes l},\\chi,s\\right):=\\sum_{n=1}^\\infty \\frac{\\lambda_f^l(n)a_{\\chi}(n)}{n^s},&D&\\left(\\left(\\Sym^d f\\right)^{\\otimes l},\\chi,s\\right):=\\sum_{n=1}^\\infty \\frac{\\lambda_{\\Sym^d f}^l(n)a_\\chi(n)}{n^s}.\n\\end{align*}\nFrom the expression for $r(n,Q)$ given in (\\ref{equation:binarydecom}), we immediately obtain the relations\n\\begin{align*}\n D\\left(f^{\\otimes l},Q,s\\right)&=\\frac{w_D}{h(D)}\\sum_{\\chi}\\overline{\\chi(\\fa_Q)}D\\left(f^{\\otimes l},\\chi,s\\right),\\\\ \n D\\left(\\left(\\Sym^d f\\right)^{\\otimes l},Q,s\\right)&=\\frac{w_D}{h(D)}\\sum_{\\chi}\\overline{\\chi(\\fa_Q)}D\\left(\\left(\\Sym^d f\\right)^{\\otimes l},\\chi,s\\right).\n\\end{align*}\nThe connection between these Dirichlet series and the complete $L$-functions is provided by the following proposition, which is an analogue of Theorem \\ref{theorem:l-functions} for the twisted case.\n\nThe parameter $T$ is now chosen to balance the error terms.\n We first consider the case where the class number of $Q$ is one. In this case, only the trivial character contributes. Choosing\n$$T=x^{1/\\left((d+1)^l-\\frac{8}{21}K_{0,d,l}-\\frac{2}{3}K_{1,d,l}-\\frac{5}{7}K_{2,d,l}\\right)},$$\nwe obtain the asymptotic formula\n\\[\\sum_{n\\leq x}\\lambda^l_{\\Sym^df}(n)a_\\chi(n)=xP_{0,d,l}(\\log(x))+O\\left(x^{1-1/\\left((d+1)^l-\\frac{8}{21}K_{0,d,l}-\\frac{2}{3}K_{1,d,l}-\\frac{5}{7}K_{2,d,l}\\right)+\\epsilon}\\right).\\]\nNext assume the class number is greater than 1. The main term comes only from the trivial character, but the error terms include contributions from all characters. Choosing\n $$T=x^{\\frac{3}{3(d+1)^l-K_{0,d,l}}},$$ we obtain\n\\[\\sum_{n\\leq x}\\lambda^l_{\\Sym^df}(n)a_\\chi(n)=xP_{0,d,l}(\\log(x))+O\\left(x^{1-\\frac{3}{3(d+1)^l-K_{0,d,l}}+\\epsilon}\\right).\\]\nSumming over all characters $\\chi$, and using (\\ref{equation:binarydecom}) completes the proof of Theorem \\ref{thm:binary}. \n\\bibliography{bib/ref}\n\\bibliographystyle{plain}", "post_theorem_intro_text_len": 5762, "post_theorem_intro_text": "In Corollary \\ref{corollary:Kostka},  we  provide an explicit formula for the Kostka number $K_{i,d,l}$, which determines the degrees of the main terms and the exponents in the error terms. The formula is \n\\begin{align*}        K_{i,d,l}=\\begin{cases}\n            0\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\  \\ \\ \\ \\ \\ \\ \\   \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\text{if $2\\nmid dl-i$}\\\\\n            \\sum\\limits_{j=0}^{\\lfloor(dl-i)/(2d+2)\\rfloor}(-1)^j\\binom{l}{j}\\binom\n        {(dl-i)/2-j(d+1)+l-2}{l-2}\n    \\ \\text{otherwise}\n        \\end{cases}.\n    \\end{align*}\nNote that  if $dl$ is odd, then $K_{0,d,l}=0$ and $P_{d,l}$ is 0.\n\nThe estimation in the general case is new. And our estimates in the error terms are better than those in the work of \\cite{liu2023average} for small $d$ and $l$. We list the cases with $d=2,3\\leq l\\leq8$ and $l=2,3\\leq d\\leq8$ in the following tables.\n\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n$d=2, l=$ &  Exponents in \\cite{liu2023average} & $\\theta_{d, l}$ \\\\\n\\hline\n3 & 0.9193... & 0.918287938\n \\\\\n\\hline\n4 & 0.9737... & 0.973534972\n  \\\\\n\\hline\n5 & 0.99136... & 0.991304348\n\\\\\n\\hline\n6 & 0.99714... & 0.99713291\n \\\\\n\\hline\n7 & 0.9990558... & 0.999051362\n \\\\\n\\hline\n8 & 0.9996868... & 0.999685565\n  \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n$l=2, d=$ & Exponents in \\cite{liu2023average} & $\\theta_{d, l}$ \\\\\n\\hline\n3 & 0.866... & 0.865814696\n \\\\\n\\hline\n4 & 0.9166... & 0.916334661\n  \\\\\n\\hline\n5 & 0.9428... & 0.942701228\n \\\\\n\\hline\n6 & 0.9583... & 0.958250497\n \\\\\n\\hline\n7 & $0.96824\\ldots$ &0.968205905\n  \\\\\n\\hline\n8 & 0.97499... & 0.974970203\n \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\nWe also have the following theorem for summation over binary quadratic forms.\n\\begin{thm}\\label{thm:binary}\n      Let $f$ be a cuspidal modular form of weight $k$ on $\\SL_2(\\BZ)$. For any integers $l\\geq2,d\\geq1$ with $dl>4$ and any binary quadratic form $Q(n_1,n_2)$, one has\n       \\[\\sum_{Q(n_1,n_2)\\leq x}  \\lambda^l_{\\Sym^d f}(Q(n_1,n_2))=xP_{d,l,Q}(\\log x)+O\\left(x^{\\theta_{d,l,Q}+\\epsilon}\\right).\\]\n        Here $P_{d,l,Q}$ is a polynomial of degree $K_{0,d,l}-1$.  If the class number of $Q$ is not 1,  then \n        $$\\theta_{d,l,Q}=1-\\frac{3}{3(d+1)^l-K_{0,d,l}};$$ if the class number is 1, then \n        $$\\theta_{d,l,Q}=1-1/\\left((d+1)^l-8K_{0,d,l}/21-2K_{1,d,l}/3-5K_{2,d,l}/7\\right).$$\n\\end{thm}\n\n\\subsection{Proof Sketch }\nThe proofs of our theorems follow a framework that can be viewed as an $\\ell$-adic analogue of the methods for Artin representations developed in \\cite{Yang2024second} , \\cite{Yang2024first} and \\cite{Yang2026higher}.\n\nThe starting point is Perron's formula, which translates the estimation of the sum $\\sum\\limits_{n\\leq x}\\lambda^l_{\\Sym^d f}(n)$ into the problem of evaluating a contour integral of the associated Dirichlet series $\\sum\\limits_{n=1}^\\infty\\frac{\\lambda^l_{\\Sym^d f}(n)}{n^s}$. This Dirichlet series is intimately related via its Euler product to the $L$-function $L\\left(\\left(\\Sym^d f\\right)^{\\otimes l},s\\right)$, which corresponds to  the $\\ell$-adic representation $\\Sym^d(\\rho_f)^{\\otimes l}$ (see Proposition \\ref{proposition:dirichletseriescomp}).\n\nA key insight is the decomposition of this high-dimensional tensor representation into a direct sum of irreducible constituents, which are shown to be of the form $\\Sym^i \\rho_f$. Consequently, the $L$-function factors into a product of symmetric power $L$-functions:\n$$L\\left(\\left(\\Sym^d f\\right)^{\\otimes l},s\\right)=\\prod_{i=0}^{ld}L\\left(\\Sym^i f,s\\right)^{K_{i,d,l}}$$\nThe automorphy and entirety of these symmetric power $L$-functions, established by Newton and Thorne \\cite{newton2021symmetric1},\\cite{newton2021symmetric2}, are crucial here. The final asymptotic formulas are then derived by combining this factorization with established subconvexity bounds for the involved $L$-functions.\n\nFor sums over values of a binary quadratic form $Q$, the approach is similar but involves additional ingredients. The generating Dirichlet series becomes $\\sum\\limits_{n=1}^\\infty\\frac{\\lambda^l_{\\Sym^d f}(n)r(n,Q)}{n^s}$, where $r(n,Q)$ counts representations of $n$ by $Q$. Let $F=\\BQ(\\sqrt{D})$ be the corresponding imaginary quadratic field. When the class number $h(D)=1$, the series $\\sum\\limits_{n=1}^\\infty \\frac{r(n,Q)}{n^s}$ is essentially the Dedekind zeta function $\\zeta_F(s)$.  In general, it can be expressed as a finite sum of Hecke $L$-functions associated to characters of the class group of $F$. Each such $L$-function corresponds to a modular form $f_\\chi$\n of weight $1$. Thus, the analysis reduces to studying Rankin-Selberg type $L$-functions of the form $L\\left(\\left(\\Sym^d f\\right)^{\\otimes l}\\otimes\\rho_{f_\\chi},s\\right)$ which again admit a decomposition into products of simpler $L$-functions. The theorem is proved by applying subconvexity estimates to these components.\n\n\\begin{remark}\n    While this paper has been submitted and review, we noticed that recently Venkatasubbareddy submitted a paper\\cite{arXiv:2601.17079} working on a similar problem and obtained similar results as our Theorem \\ref{theorem:main}. But our work is different from \\cite{arXiv:2601.17079} in the following aspects. First we use a representation  point of view to deal with the Dirichlet series, this allows us to connect our work with combination. Such ideas are similar with our work \\cite{Yang2024first} and \\cite{Yang2026higher}. They can be used to deal with more complicated cases and play a key role in the following work \\cite{Yang2026second}. Next we also consider average results over binary quadratic forms, which are not considered in \\cite{arXiv:2601.17079}. Finally, the estimation for the error terms in the two works are different.\n\n\\end{remark}", "sketch": "The post-theorem introduction contains a \\subsection{Proof Sketch} explaining the strategy for Theorem~\\ref{theorem:main}:\n\n- **Perron's formula / contour integral reduction.** The “starting point is Perron's formula,” which converts estimating \\(\\sum_{n\\le x}\\lambda^l_{\\Sym^d f}(n)\\) into evaluating “a contour integral of the associated Dirichlet series” \\(\\sum_{n\\ge1}\\lambda^l_{\\Sym^d f}(n)n^{-s}\\).\n\n- **Relate the Dirichlet series to an \\(L\\)-function via Euler product.** This Dirichlet series is “intimately related via its Euler product” to \\(L((\\Sym^d f)^{\\otimes l},s)\\), corresponding to the \\(\\ell\\)-adic representation \\(\\Sym^d(\\rho_f)^{\\otimes l}\\) (cf. Proposition~\\ref{proposition:dirichletseriescomp}).\n\n- **Decompose the tensor representation and factor the \\(L\\)-function.** A “key insight” is decomposing the “high-dimensional tensor representation into a direct sum of irreducible constituents,” which are “of the form \\(\\Sym^i\\rho_f\\).” Hence\n  \\[\n  L\\bigl((\\Sym^d f)^{\\otimes l},s\\bigr)=\\prod_{i=0}^{ld} L(\\Sym^i f,s)^{K_{i,d,l}}.\n  \\]\n\n- **Use automorphy/entireness and subconvexity to get the asymptotic.** The “automorphy and entirety of these symmetric power \\(L\\)-functions,” due to Newton--Thorne, are “crucial,” and the “final asymptotic formulas are then derived by combining this factorization with established subconvexity bounds for the involved \\(L\\)-functions.”\n\n(An additional sketch is given for the binary quadratic form theorem \\(\\ref{thm:binary}\\): one studies \\(\\sum \\lambda^l_{\\Sym^d f}(n)r(n,Q)n^{-s}\\), expresses \\(\\sum r(n,Q)n^{-s}\\) via \\(\\zeta_F(s)\\) when \\(h(D)=1\\) or as a finite sum of Hecke \\(L\\)-functions in general, reducing to Rankin--Selberg type \\(L\\)-functions \\(L((\\Sym^d f)^{\\otimes l}\\otimes \\rho_{f_\\chi},s)\\), and then applies subconvexity to their decomposed components.)", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{theorem:main}\n    Let $f$ be a cuspidal modular form of weight $k$ on $\\SL_2(\\BZ)$. For any integers $l\\geq2$ and $d\\geq1$ and $dl>4$, one has\n    \\[\\sum_{n\\leq x}  \\lambda^l_{\\Sym^d f}(n)=xP_{d,l}(\\log x)+O\\left(x^{\\theta_{d,l}+\\epsilon}\\right).\\]\n    Here $P_{d,l}$ is a polynomial of degree $K_{0,d,l}-1$, where $K_{i,d,l}$ denote the Kostka number defined in sections \\ref{section:Weylmod} and \\ref{section:Kostka number}. The exponent $\\theta_{d, l}$ is given by\n    $$\\theta_{d,l}=1-1/\\left((d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14\\right).$$,", "theorem_type": ["Asymptotic or Limit", "Universal"], "mcq": {"question": "Let $f$ be a cuspidal modular form of weight $k$ on $\\SL_2(\\mathbb{Z})$. For each $d\\ge 1$, write the $d$th symmetric-power $L$-function in the form\n\\[\nL(\\Sym^d f,s)=\\sum_{n=1}^\\infty \\lambda_{\\Sym^d f}(n)n^{-s}.\n\\]\nLet $K_{i,d,l}$ denote the Kostka numbers attached to $(i,d,l)$, equivalently the multiplicities/exponents appearing in the factorization of the $l$-fold tensor power of $\\Sym^d$, such as\n\\[\nL\\big((\\Sym^d f)^{\\otimes l},s\\big)=\\prod_{i=0}^{dl}L(\\Sym^i f,s)^{K_{i,d,l}}.\n\\]\nFor integers $l\\ge 2$ and $d\\ge 1$ with $dl>4$, which statement holds for every such $f$, $d$, and $l$?", "correct_choice": {"label": "A", "text": "For every $\\varepsilon>0$,\n\\[\n\\sum_{n\\le x}\\lambda_{\\Sym^d f}(n)^l=xP_{d,l}(\\log x)+O\\bigl(x^{\\theta_{d,l}+\\varepsilon}\\bigr)\n\\]\nas $x\\to\\infty$, where $P_{d,l}$ is a polynomial of degree $K_{0,d,l}-1$ and\n\\[\n\\theta_{d,l}=1-\\frac{1}{(d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14}.\n\\]"}, "choices": [{"label": "B", "text": "For every $\\varepsilon>0$,\n\\[\n\\sum_{n\\le x}\\lambda_{\\Sym^d f}(n)^l=xP_{d,l}(\\log x)+O\\bigl(x^{\\theta_{d,l}+\\varepsilon}\\bigr)\n\\]\nas $x\\to\\infty$, where $P_{d,l}$ is a polynomial of degree $K_{0,d,l}$ and\n\\[\n\\theta_{d,l}=1-\\frac{1}{(d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14}.\n\\]"}, {"label": "C", "text": "For every $\\varepsilon>0$,\n\\[\n\\sum_{n\\le x}\\lambda_{\\Sym^d f}(n)^l=xP_{d,l}(\\log x)+O\\bigl(x^{\\theta_{d,l}+\\varepsilon}\\bigr)\n\\]\nas $x\\to\\infty$ for some polynomial $P_{d,l}$, where\n\\[\n\\theta_{d,l}=1-\\frac{1}{(d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14}.\n\\]"}, {"label": "D", "text": "There exists $\\varepsilon>0$ such that for every cuspidal modular form $f$ of weight $k$ on $\\SL_2(\\mathbb{Z})$ and every integers $l\\ge 2$, $d\\ge 1$ with $dl>4$,\n\\[\n\\sum_{n\\le x}\\lambda_{\\Sym^d f}(n)^l=xP_{d,l}(\\log x)+O\\bigl(x^{\\theta_{d,l}+\\varepsilon}\\bigr)\n\\]\nas $x\\to\\infty$, where $P_{d,l}$ is a polynomial of degree $K_{0,d,l}-1$ and\n\\[\n\\theta_{d,l}=1-\\frac{1}{(d+1)^l/2-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14}.\n\\]"}, {"label": "E", "text": "For every $\\varepsilon>0$,\n\\[\n\\sum_{n\\le x}\\lambda_{\\Sym^d f}(n)^l=xP_{d,l}(\\log x)+O\\bigl(x^{\\theta_{d,l}+\\varepsilon}\\bigr)\n\\]\nas $x\\to\\infty$, where $P_{d,l}$ is a polynomial of degree $K_{0,d,l}-1$ and\n\\[\n\\theta_{d,l}=1-\\frac{1}{(d+1)^l-4K_{0,d,l}/21-K_{1,d,l}/3-5K_{2,d,l}/14}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "degree_of_main_polynomial", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "explicit_degree_K0dl_minus_1", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "quantifier_order_on_epsilon", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "denominator_factor_one_half_in_theta", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct option explicitly or by an obvious cue. It sets up notation and asks for the valid theorem-level conclusion, but the exact degree, quantifier structure, and exponent formula must be identified from the choices."}, "TAS": {"score": 1, "justification": "The item is essentially a theorem-identification question: the correct answer is a precise asymptotic statement very close to a known result. It is not a pure verbatim restatement because the options introduce meaningful variants, but it still mainly tests recognition of the exact theorem statement."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in distinguishing the strongest valid statement from a weaker true one and from subtle false variants involving quantifiers, polynomial degree, and the exponent denominator. However, the problem relies much more on recall/discrimination than on generating a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one changes the polynomial degree by 1, one gives a weaker true statement, one alters quantifier order, and one perturbs the key denominator. These reflect realistic theorem-recall and logic mistakes."}, "total_score": 6, "overall_assessment": "A solid but theorem-recall-heavy MCQ: little answer leakage and strong distractors, but only moderate generative reasoning since success depends largely on recognizing the exact statement rather than deriving it."}}
{"id": "2602.02723v1", "paper_link": "http://arxiv.org/abs/2602.02723v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{maintheo}Let $(M, g)$ be a Lorentzian manifold of dimension at least $3$ and let $\\bf P$ be its pseudo-group of local conformal transformations, that is the collection \n of all conformal diffeomorphisms between open subsets of $M$. We assume $M$ locally conformally homogeneous, i.e.~$M$ is an orbit of $\\bf P$. Then one of the following possibilities holds:\n\n \\begin{enumerate}\n\n \\item $\\bf P$ preserves a Lorentzian metric in the $g$-conformal class.\n\n \\item $(M, g)$ is conformally flat. \n\n \\item $(M, g)$ is locally conformally equivalent to a plane wave that is isometrically locally homogeneous.  More precisely, there exists a homogeneous plane wave $X$, such that \n $(M, [g])$ is modeled on $(\\Conf(X), X)$.\n \\end{enumerate}", "start_pos": 11292, "end_pos": 12061, "label": "maintheo"}, "ref_dict": {"Penrose-prop": "\\begin{proposition}\\label{Penrose-prop}\nLet $g$ be a Lorentzian metric, and let $\\gamma$ be a null geodesic. Let $g_\\sigma := e^\\sigma g$ be a conformal change of $g$ (up to reparametrization, $\\gamma$ is also a null geodesic of $g_\\sigma$). Let $\\mathrm{PL}_g$ (resp.\\ $\\mathrm{PL}_{g_\\sigma}$) denote the Penrose limit of $g$ (resp.\\ of $g_\\sigma$) along $\\gamma$. Then there exists a conformal diffeomorphism\n$ \\mathrm{PL}_g \\to \\mathrm{PL}_{g_\\sigma}.$\n\\end{proposition}", "maintheo": "\\begin{theorem}\\label{maintheo}Let $(M, g)$ be a Lorentzian manifold of dimension at least $3$ and let $\\bf P$ be its pseudo-group of local conformal transformations, that is the collection \n of all conformal diffeomorphisms between open subsets of $M$. We assume $M$ locally conformally homogeneous, i.e.~$M$ is an orbit of $\\bf P$. Then one of the following possibilities holds:\n\n \\begin{enumerate}\n\n \\item $\\bf P$ preserves a Lorentzian metric in the $g$-conformal class.\n\n \\item $(M, g)$ is conformally flat. \n\n \\item $(M, g)$ is locally conformally equivalent to a plane wave that is isometrically locally homogeneous.  More precisely, there exists a homogeneous plane wave $X$, such that \n $(M, [g])$ is modeled on $(\\Conf(X), X)$.\n \\end{enumerate}\n \\end{theorem}", "Gromov's Theory": "\\label{Gromov's Theory}\n \\addtocontents{toc}{\\setcounter{tocdepth}{1}}\n\nFor references, see the foundational work of Gromov~\\cite{Gro}, the lecture notes by D'Ambra and Gromov~\\cite{DG}, and those by"}, "pre_theorem_intro_text_len": 2569, "pre_theorem_intro_text": "A fundamental problem in conformal geometry is to classify conformal manifolds with  \\textit{essential} conformal transformations, that is, global conformal transformations that do not preserve a metric in the conformal class. For Riemannian conformal manifolds the situation is very rigid: up to conformal equivalence only the round sphere and Euclidean space admit essential conformal transformations. This was conjectured by Lichnerowicz in the 1960's  and   proved in a series of papers by Ferrand and Obata (\\cite{Lelong-Ferrand71},  \\cite{Obata71}, \\cite{Ferrand96}, with contributions in \\cite{Alekseevskii72}). \n\nFor indefinite metrics, the situation is far more flexible: there exist pseudo-Riemannian metrics that are not conformally flat yet admit essential conformal transformations, both on non-compact manifolds and, as shown in \\cite{frances12}, on compact manifolds in signatures other than the Lorentzian one. Non-compact examples in Lorentzian signature consist of plane waves. These are manifolds for which the metric in dimension $n+2$  is locally given by\n\\begin{equation}\\label{pw}\ng=2\\,\\mathrm{d} t\\, \\mathrm{d} v + \\boldsymbol{x}^\\top Q(t) \\boldsymbol{x}\\ \\mathrm{d} t^2 +\\mathrm{d}\\boldsymbol{x}^\\top\\mathrm{d}\\boldsymbol{x},\n\\end{equation}\nwhere $\\boldsymbol{x}=(x^1, \\ldots , x^n)^\\top\\in \\mathbb{R}^n$, {$Q(t)$ is} a $t$-dependent symmetric $n\\times n$-matrix. If $Q$ is not a scalar matrix, this metric is not conformally flat.\nPlane waves for which this metric is defined on all of $\\mathbb{R}^{n+2}$, for example,   admit non-isometric homotheties with fixed points. Therefore these homotheties are essential (for this implication see for example \\cite[Proposition 2.5]{LeistnerTeisseire22}). \n\n\\bigskip\n\nRecently \nan important rigidity result for essential Lorentzian conformal structures in the  \\textit{conformally homogeneous} setting has been obtained  by Alekseevsky and Galaev\n in \\cite{AlekseevskyGalaev25}. They showed that if the manifold is simply connected and admits a transitive group of global conformal transformations that is essential, then the manifold is either conformally flat or the conformal class contains a plane wave metric that is isometrically homogeneous and geodesically complete. In the present article we will generalize this result in the following sense: we will drop the assumption that the manifold is simply connected and we will only assume that the manifold is  \\textit{locally conformally homogeneous}, i.e.~the local conformal transformations act transitively. We will prove the following result.", "context": "A fundamental problem in conformal geometry is to classify conformal manifolds with \\textit{essential} conformal transformations, that is, global conformal transformations that do not preserve a metric in the conformal class. For Riemannian conformal manifolds the situation is very rigid: up to conformal equivalence only the round sphere and Euclidean space admit essential conformal transformations. This was conjectured by Lichnerowicz in the 1960's and proved in a series of papers by Ferrand and Obata (\\cite{Lelong-Ferrand71}, \\cite{Obata71}, \\cite{Ferrand96}, with contributions in \\cite{Alekseevskii72}).\n\nFor indefinite metrics, the situation is far more flexible: there exist pseudo-Riemannian metrics that are not conformally flat yet admit essential conformal transformations, both on non-compact manifolds and, as shown in \\cite{frances12}, on compact manifolds in signatures other than the Lorentzian one. Non-compact examples in Lorentzian signature consist of plane waves. These are manifolds for which the metric in dimension $n+2$ is locally given by\n\\begin{equation}\\label{pw}\ng=2\\,\\mathrm{d} t\\, \\mathrm{d} v + \\boldsymbol{x}^\\top Q(t) \\boldsymbol{x}\\ \\mathrm{d} t^2 +\\mathrm{d}\\boldsymbol{x}^\\top\\mathrm{d}\\boldsymbol{x},\n\\end{equation}\nwhere $\\boldsymbol{x}=(x^1, \\ldots , x^n)^\\top\\in \\mathbb{R}^n$, {$Q(t)$ is} a $t$-dependent symmetric $n\\times n$-matrix. If $Q$ is not a scalar matrix, this metric is not conformally flat.\nPlane waves for which this metric is defined on all of $\\mathbb{R}^{n+2}$, for example, admit non-isometric homotheties with fixed points. Therefore these homotheties are essential (for this implication see for example \\cite[Proposition 2.5]{LeistnerTeisseire22}).\n\n\\bigskip\n\nRecently \nan important rigidity result for essential Lorentzian conformal structures in the \\textit{conformally homogeneous} setting has been obtained by Alekseevsky and Galaev\n in \\cite{AlekseevskyGalaev25}. They showed that if the manifold is simply connected and admits a transitive group of global conformal transformations that is essential, then the manifold is either conformally flat or the conformal class contains a plane wave metric that is isometrically homogeneous and geodesically complete. In the present article we will generalize this result in the following sense: we will drop the assumption that the manifold is simply connected and we will only assume that the manifold is \\textit{locally conformally homogeneous}, i.e.~the local conformal transformations act transitively. We will prove the following result.", "full_context": "A fundamental problem in conformal geometry is to classify conformal manifolds with \\textit{essential} conformal transformations, that is, global conformal transformations that do not preserve a metric in the conformal class. For Riemannian conformal manifolds the situation is very rigid: up to conformal equivalence only the round sphere and Euclidean space admit essential conformal transformations. This was conjectured by Lichnerowicz in the 1960's and proved in a series of papers by Ferrand and Obata (\\cite{Lelong-Ferrand71}, \\cite{Obata71}, \\cite{Ferrand96}, with contributions in \\cite{Alekseevskii72}).\n\nFor indefinite metrics, the situation is far more flexible: there exist pseudo-Riemannian metrics that are not conformally flat yet admit essential conformal transformations, both on non-compact manifolds and, as shown in \\cite{frances12}, on compact manifolds in signatures other than the Lorentzian one. Non-compact examples in Lorentzian signature consist of plane waves. These are manifolds for which the metric in dimension $n+2$ is locally given by\n\\begin{equation}\\label{pw}\ng=2\\,\\mathrm{d} t\\, \\mathrm{d} v + \\boldsymbol{x}^\\top Q(t) \\boldsymbol{x}\\ \\mathrm{d} t^2 +\\mathrm{d}\\boldsymbol{x}^\\top\\mathrm{d}\\boldsymbol{x},\n\\end{equation}\nwhere $\\boldsymbol{x}=(x^1, \\ldots , x^n)^\\top\\in \\mathbb{R}^n$, {$Q(t)$ is} a $t$-dependent symmetric $n\\times n$-matrix. If $Q$ is not a scalar matrix, this metric is not conformally flat.\nPlane waves for which this metric is defined on all of $\\mathbb{R}^{n+2}$, for example, admit non-isometric homotheties with fixed points. Therefore these homotheties are essential (for this implication see for example \\cite[Proposition 2.5]{LeistnerTeisseire22}).\n\n\\bigskip\n\nRecently \nan important rigidity result for essential Lorentzian conformal structures in the \\textit{conformally homogeneous} setting has been obtained by Alekseevsky and Galaev\n in \\cite{AlekseevskyGalaev25}. They showed that if the manifold is simply connected and admits a transitive group of global conformal transformations that is essential, then the manifold is either conformally flat or the conformal class contains a plane wave metric that is isometrically homogeneous and geodesically complete. In the present article we will generalize this result in the following sense: we will drop the assumption that the manifold is simply connected and we will only assume that the manifold is \\textit{locally conformally homogeneous}, i.e.~the local conformal transformations act transitively. We will prove the following result.\n\nRecently \nan important rigidity result for essential Lorentzian conformal structures in the \\textit{conformally homogeneous} setting has been obtained by Alekseevsky and Galaev\n in \\cite{AlekseevskyGalaev25}. They showed that if the manifold is simply connected and admits a transitive group of global conformal transformations that is essential, then the manifold is either conformally flat or the conformal class contains a plane wave metric that is isometrically homogeneous and geodesically complete. In the present article we will generalize this result in the following sense: we will drop the assumption that the manifold is simply connected and we will only assume that the manifold is \\textit{locally conformally homogeneous}, i.e.~the local conformal transformations act transitively. We will prove the following result.\n\nWhen the pseudo-group $\\mathbf{P}$ does not preserve any metric in the $g$-conformal class, we say that $(M,g)$ is \\textit{weakly essential}, as introduced in Section \\ref{Section: terminology}. This is a weaker condition than being essential.\n Our proof starts off similarly as in \\cite{AlekseevskyGalaev25} by noting that weak essentiality yields the existence of an element in the isotropy group that is not contained in $\\SO(T_pM)$.\nIn \\cite{AlekseevskyGalaev25}, simple connectedness and global conformal homogeneity ensure that the isotropy group is connected, which in turn provides a specific element in its Lie algebra (that does not lie in the isometry algebra). In our setting, however, we cannot directly conclude the existence of such an element. Therefore, we adopt a different approach. We work only with the local isotropy of local conformal transformations, but we use powerful tools from Gromov's theory of rigid transformations (see Appendix \\ref{Gromov's Theory}) and from algebraic groups, which allow us to reduce the problem to the Lie algebra level (see Section \\ref{Section: isotropy}), and in particular to obtain a hyperbolic element in the isotropy algebra that is not in $\\so(T_pM)$.\nSince we assume only local homogeneity, we can pass to the universal cover, where weak essentiality is preserved. Moreover, on a simply connected real-analytic manifold, local conformal vector fields extend to global ones. We may therefore work with the Lie algebra of conformal vector fields on the universal cover. Loosely speaking, by weakening the assumption of homogeneity to local homogeneity, we are able to overcome the assumption of simple connectedness. \nThen, in Section \\ref{Section: Heis algebra}, we use the decomposition of the Lie algebra of conformal vector fields induced by the hyperbolic element to derive the existence of a codimension-one lightlike foliation of \\textit{Heisenberg type}, i.e., with a transitive action of a Heisenberg algebra. \nOur approach, together with the new characterization of plane waves obtained in Section \\ref{Section: Determining a plane wave by Killing fields}, not only leads to the generalization in Theorem \\ref{maintheo}, but also provides a more conceptual approach and simplifies the proof in \\cite{AlekseevskyGalaev25} for the simply connected globally homogeneous case.\n\nIn the last section, we consider the consequence of our result for {\\bf Penrose limits} of locally conformally homogeneous Lorentzian manifolds. \nThe Penrose limit is a famous construction in gravitational physics, which shows that any Lorentzian manifold admits a plane wave spacetime as a limit, this limit being taken along any lightlike geodesic (see \\cite{Blau-PL} for instance). \nAs shown in Theorem \\ref{maintheo}, a weakly essential locally conformally homogeneous Lorentzian manifold of dimension $\\geq 3$ is either conformally flat or locally conformal to a homogeneous plane wave.\nA natural question is then whether this plane wave coincides with the Penrose limit of the Lorentzian manifold along some lightlike geodesic. A priori, the Penrose limit is a plane wave associated with a metric, rather than with a conformal structure.\nThe underlying question is then whether two conformally related metrics have conformally equivalent Penrose limits along some lightlike geodesic (note that conformally related metrics have the same unparameterized lightlike geodesics). It turns out that the answer is affirmative, so the Penrose limit is a conformal invariant, as shown in Proposition~\\ref{Penrose-prop}.\nAs a consequence, we obtain that the plane wave metric in the conformal class of a weakly essential locally conformally homogeneous Lorentzian manifold $(M,g)$ (as in item (3) of Theorem~\\ref{maintheo}) coincides with the Penrose limit of $(M,g)$ along some null geodesic.\n\\begin{corollary} \nLet $(M, g)$ be a non-conformally flat Lorentzian manifold of dimension at least $3$ which is locally conformally homogeneous. Assume that $(M,g)$ is weakly essential. Then, $(M,g)$ is locally conformally equivalent to its Penrose limit along some null geodesic. \n\\end{corollary}\nThis raises the question of whether a direct proof of this fact can be found, which we plan to investigate in future work.\n\n\\begin{proposition}\\label{distributionprop}\nLet $\\g$ be the algebra of conformal vector fields of $M$, and assume that $\\g$ acts transitively on $M$. \nFix a point \n$p\\in M$, and let $\\g_p$ be the isotropy subalgebra at $p$, with image $\\h_p$ under the isotropy representation $\\rho_p$. \nLet $V_p$ be a subspace of $T_pM$ that is invariant under $\\h_p \\subset \\co(T_pM)$. Then\n\\begin{enumerate}\n\\item There is a vector distribution $V$ with $V_p$ as fiber at $p$ that is invariant under $\\g$, i.e. $\\d\\psi_p(V_p)=V_{\\psi(p)}$ for all local conformal transformations $\\psi \\in \\mathbf{G}$ acting locally on $M$.\n\\item If there is a Lie algebra $\\l\\subset \\g$ with stabilizer algebra $\\l_p$ such that the isomorphism $T_pM\\simeq \\g/\\g_p$ restricts to an isomorphism $V_p\\simeq \\l/\\l_p$, then $V$ is involutive with fiber at $\\psi(p)$ spanned by $\\Ad_\\psi (\\l)|_{\\psi(p)}$, for $\\psi\\in \\mathbf{G}$ acting locally on $M$. \nIn particular, if $\\l$ is an ideal of $\\g$, then $V_q$ is spanned by $\\l|_q$ for each $q\\in M$.\n\\item If $\\mathbf{L}$ is the Lie group corresponding to $\\l$, then the leaves through $q=\\psi(p)$ are given \nby the orbits of $q$ under $\\psi \\mathbf L \\psi^{-1}$. \nIn particular, if $\\l$ is an ideal, then the integral manifold through $q$ is the orbit of $q$ under $\\mathbf{L}$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proposition}\\label{Prop. existence of foliation}\nLet $(M,g)$ be a Lorentzian manifold of dimension $n+2$, with a transitive Lie algebra $\\g$ of conformal vector fields, and stabilizer algebra $\\h$ at $p\\in M$.\nAssume that \n$\\h \\subset \\g^0 \\oplus \\g^{\\pm 1}$ and $\\g^0=\\R B \\oplus \\h^{00}$ with $B=\\mathrm{diag}(\\alpha+1,\\alpha-1,\\alpha,\\ldots, \\alpha)$. \nThen there is a metric $\\hat g$ in the conformal class of $g$ and \nan algebra of Killing vector fields $\\g_1$ for $\\hat g$ that acts locally transitively at $p$. Moreover, \n\\begin{enumerate}\n \\item If $\\alpha \\notin \\left\\{\\frac{1}{2}, 1, 2\\right\\}$, then $(M,g)$ is conformally flat.\n\n\\begin{corollary}\\label{Penrose-cor}\nLet $(M, g)$ be a non-conformally flat Lorentzian manifold of dimension at least $3$ which is locally conformally homogeneous. Assume that $(M,g)$ is weakly essential. Then, $(M,g)$ \n is locally conformally equivalent to its Penrose limit along some null geodesic. \n\\end{corollary}\nNote that the proof of Theorem~\\ref{maintheo} reveals more details than those stated in this corollary. The assumption implies that there is a null line bundle that is invariant under the local conformal vector fields, which is spanned by the parallel vector field of the plane wave, and that $(M,g)$ is locally conformally equivalent to the Penrose limit along all null geodesics that are \\textit{transversal} to this null line bundle.", "post_theorem_intro_text_len": 4292, "post_theorem_intro_text": "When the pseudo-group $\\mathbf{P}$ does not preserve any metric in the $g$-conformal class, we say that $(M,g)$ is \\textit{weakly essential}, as introduced in Section \\ref{Section: terminology}. This is a weaker condition than being essential.\n Our proof starts off similarly as in \\cite{AlekseevskyGalaev25} by  noting that weak essentiality yields the existence of an element in the isotropy group that is not contained in $\\mathbf{SO}(T_pM)$.\nIn \\cite{AlekseevskyGalaev25}, simple connectedness and global conformal homogeneity ensure that the isotropy group is connected, which in turn provides a specific element in its Lie algebra (that does not lie in the isometry algebra). In our setting, however, we cannot directly conclude the existence of such an element. Therefore, we adopt a different approach. We work only with the local isotropy of local conformal transformations, but we use powerful tools from Gromov's theory of rigid transformations (see Appendix \\ref{Gromov's Theory}) and from algebraic groups, which allow us to reduce the problem to the Lie algebra level (see Section \\ref{Section: isotropy}), and in particular to obtain a hyperbolic element in the isotropy algebra that is not in $\\mathfrak{so}(T_pM)$.\nSince we assume only local homogeneity, we can pass to the universal cover, where weak essentiality is preserved. Moreover, on a simply connected real-analytic manifold, local conformal vector fields extend to global ones. We may therefore work with the Lie algebra of conformal vector fields on the universal cover. Loosely speaking, by weakening the assumption of homogeneity to local homogeneity, we are able to overcome  the assumption of simple connectedness. \nThen, in Section \\ref{Section: Heis algebra}, we use the decomposition of the Lie algebra of conformal vector fields induced by the hyperbolic element to derive the existence of a codimension-one lightlike foliation of \\textit{Heisenberg type}, i.e., with a transitive action of a Heisenberg algebra.  \nOur approach, together with the new characterization of plane waves obtained in Section \\ref{Section: Determining a plane wave by Killing fields}, not only leads to the generalization in Theorem \\ref{maintheo}, but also provides a more conceptual approach and simplifies the proof in \\cite{AlekseevskyGalaev25} for the simply connected globally homogeneous case.\n\n \\bigskip\n\nIn the last section, we consider the consequence of our result for {\\bf Penrose limits} of locally conformally homogeneous Lorentzian manifolds. \nThe Penrose limit is a famous construction in gravitational physics, which shows that any Lorentzian manifold admits a  plane wave spacetime as a limit, this limit being taken along any lightlike geodesic (see \\cite{Blau-PL} for instance). \nAs shown in  Theorem \\ref{maintheo}, a weakly essential locally conformally homogeneous Lorentzian manifold of dimension $\\geq 3$ is either conformally flat or locally conformal to a homogeneous plane wave.\nA natural question is then whether this plane wave coincides with the Penrose limit of the Lorentzian manifold along some lightlike geodesic. A priori, the Penrose limit is a plane wave associated with a metric, rather than with a conformal structure.\nThe underlying question is then whether two conformally related metrics have conformally equivalent Penrose limits along some lightlike geodesic (note that conformally related metrics have the same unparameterized lightlike geodesics). It turns out that the answer is affirmative, so the Penrose limit is a conformal invariant, as shown  in Proposition~\\ref{Penrose-prop}.\nAs a consequence, we obtain that the plane wave metric in the conformal class of a weakly essential locally conformally homogeneous Lorentzian manifold $(M,g)$ (as in item (3) of  Theorem~\\ref{maintheo}) coincides with the Penrose limit of $(M,g)$ along some null geodesic.\n\\begin{corollary} \nLet $(M, g)$ be a non-conformally flat Lorentzian manifold of dimension at least $3$ which is locally conformally homogeneous. Assume that $(M,g)$ is weakly essential. Then, $(M,g)$  is locally conformally equivalent to its Penrose limit along some null geodesic.  \n\\end{corollary}\nThis raises the question of whether a direct proof of this fact can be found, which we plan to investigate in future work.", "sketch": "To prove Theorem~\\ref{maintheo}, the argument is described as follows.\n\n- Define the weakly essential case: when the pseudo-group $\\mathbf{P}$ does not preserve any metric in the $g$-conformal class.\n\n- As in \\cite{AlekseevskyGalaev25}, start by using weak essentiality to get “the existence of an element in the isotropy group that is not contained in $\\mathbf{SO}(T_pM)$.” The earlier approach relied on connectedness of the isotropy group to “provide a specific element in its Lie algebra,” but “in our setting … we cannot directly conclude the existence of such an element.”\n\n- Use a different approach: “work only with the local isotropy of local conformal transformations,” and apply “powerful tools from Gromov's theory of rigid transformations … and from algebraic groups,” which “allow us to reduce the problem to the Lie algebra level … and in particular to obtain a hyperbolic element in the isotropy algebra that is not in $\\mathfrak{so}(T_pM)$.”\n\n- Because only local homogeneity is assumed, “pass to the universal cover, where weak essentiality is preserved.” On a “simply connected real-analytic manifold, local conformal vector fields extend to global ones,” so one can “work with the Lie algebra of conformal vector fields on the universal cover.”\n\n- Then “use the decomposition of the Lie algebra of conformal vector fields induced by the hyperbolic element to derive the existence of a codimension-one lightlike foliation of \\textit{Heisenberg type}, i.e., with a transitive action of a Heisenberg algebra.”\n\n- Finally, combine this with “the new characterization of plane waves obtained in Section~\\ref{Section: Determining a plane wave by Killing fields},” which “leads to the generalization in Theorem~\\ref{maintheo}.”", "expanded_sketch": "To prove the main theorem, the argument is described as follows.\n\n- Define the weakly essential case: when the pseudo-group $\\mathbf{P}$ does not preserve any metric in the $g$-conformal class.\n\n- As in \\cite{AlekseevskyGalaev25}, start by using weak essentiality to get “the existence of an element in the isotropy group that is not contained in $\\mathbf{SO}(T_pM)$.” The earlier approach relied on connectedness of the isotropy group to “provide a specific element in its Lie algebra,” but “in our setting … we cannot directly conclude the existence of such an element.”\n\n- Use a different approach: “work only with the local isotropy of local conformal transformations,” and apply “powerful tools from Gromov's theory of rigid transformations … and from algebraic groups,” which “allow us to reduce the problem to the Lie algebra level … and in particular to obtain a hyperbolic element in the isotropy algebra that is not in $\\mathfrak{so}(T_pM)$.”\n\n- Because only local homogeneity is assumed, “pass to the universal cover, where weak essentiality is preserved.” On a “simply connected real-analytic manifold, local conformal vector fields extend to global ones,” so one can “work with the Lie algebra of conformal vector fields on the universal cover.”\n\n- Then “use the decomposition of the Lie algebra of conformal vector fields induced by the hyperbolic element to derive the existence of a codimension-one lightlike foliation of \\textit{Heisenberg type}, i.e., with a transitive action of a Heisenberg algebra.”\n\n- Finally, combine this with “the new characterization of plane waves obtained later,” which “leads to the generalization in the main theorem.”", "expanded_theorem": "\\label{maintheo}Let $(M, g)$ be a Lorentzian manifold of dimension at least $3$ and let $\\bf P$ be its pseudo-group of local conformal transformations, that is the collection \n of all conformal diffeomorphisms between open subsets of $M$. We assume $M$ locally conformally homogeneous, i.e.~$M$ is an orbit of $\\bf P$. Then one of the following possibilities holds:\n\n \\begin{enumerate}\n\n \\item $\\bf P$ preserves a Lorentzian metric in the $g$-conformal class.\n\n \\item $(M, g)$ is conformally flat. \n\n \\item $(M, g)$ is locally conformally equivalent to a plane wave that is isometrically locally homogeneous.  More precisely, there exists a homogeneous plane wave $X$, such that \n $(M, [g])$ is modeled on $(\\Conf(X), X)$.\n \\end{enumerate}", "theorem_type": ["Classification or Bijection", "Universal"], "mcq": {"question": "Which statement holds for every Lorentzian manifold $(M,g)$ of dimension at least $3$ whose pseudo-group $\\mathbf P$ of local conformal transformations acts transitively on $M$? Here $\\mathbf P$ means all conformal diffeomorphisms between open subsets of $M$, and “locally conformally homogeneous” means that $M$ is a single orbit of $\\mathbf P$. A plane wave is a Lorentzian manifold locally given in dimension $n+2$ by\n\\[\ng=2\\,dt\\,dv+\\boldsymbol{x}^\\top Q(t)\\boldsymbol{x}\\,dt^2+d\\boldsymbol{x}^\\top d\\boldsymbol{x},\n\\]\nwhere $\\boldsymbol{x}\\in\\mathbb R^n$ and $Q(t)$ is a symmetric $n\\times n$ matrix; “homogeneous plane wave” means such a plane wave is isometrically locally homogeneous.", "correct_choice": {"label": "A", "text": "At least one of the following holds: (1) $\\mathbf P$ preserves a Lorentzian metric in the conformal class $[g]$; or (2) $(M,g)$ is conformally flat; or (3) $(M,g)$ is locally conformally equivalent to an isometrically locally homogeneous plane wave. More precisely in case (3), there exists a homogeneous plane wave $X$ such that $(M,[g])$ is modeled on $(\\operatorname{Conf}(X),X)$."}, "choices": [{"label": "B", "text": "Exactly one of the following holds: (1) $\\mathbf P$ preserves a Lorentzian metric in the conformal class $[g]$; or (2) $(M,g)$ is conformally flat; or (3) $(M,g)$ is locally conformally equivalent to an isometrically locally homogeneous plane wave. More precisely in case (3), there exists a homogeneous plane wave $X$ such that $(M,[g])$ is modeled on $(\\operatorname{Conf}(X),X)$."}, {"label": "C", "text": "At least one of the following holds: (1) $\\mathbf P$ preserves a Lorentzian metric in the conformal class $[g]$; or (2) $(M,g)$ is conformally flat; or (3) $(M,g)$ is locally conformally equivalent to a plane wave."}, {"label": "D", "text": "At least one of the following holds: (1) after passing to the universal cover of $M$, the lifted pseudo-group $\\mathbf P$ preserves a Lorentzian metric in the lifted conformal class; or (2) $(M,g)$ is conformally flat; or (3) after passing to the universal cover, $(M,g)$ is locally conformally equivalent to an isometrically locally homogeneous plane wave. More precisely in case (3), there exists a homogeneous plane wave $X$ such that the universal cover of $(M,[g])$ is modeled on $(\\operatorname{Conf}(X),X)$."}, {"label": "E", "text": "If $\\mathbf P$ does not preserve any Lorentzian metric in the conformal class $[g]$, then $(M,g)$ is locally conformally equivalent to an isometrically locally homogeneous plane wave. More precisely, there exists a homogeneous plane wave $X$ such that $(M,[g])$ is modeled on $(\\operatorname{Conf}(X),X)$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "case_split", "tampered_component": "nonexclusive trichotomy", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped homogeneous/modeling refinement in the plane-wave alternative", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "conclusion only on universal cover instead of on M itself", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "omits conformally flat weakly essential branch", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the classification itself or uniquely signal choice A. It only sets up the setting and asks for the complete classification."}, "TAS": {"score": 1, "justification": "This is essentially a theorem-classification recall item: the correct option states the full trichotomy. However, it is not a pure tautology because the distractors are nearby variants that force attention to precise qualifiers such as local vs global, exactness, and isometric vs conformal homogeneity."}, "GPS": {"score": 1, "justification": "The item requires moderate reasoning to distinguish subtle mathematical differences among the options, but it mainly tests precise theorem recognition rather than genuinely generating a conclusion from given premises."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they exploit common failure modes such as weakening the statement, strengthening it improperly, confusing local and global conclusions, and conflating conformal with isometric homogeneity."}, "total_score": 6, "overall_assessment": "A solid, high-precision theorem-recognition MCQ with strong distractors and little answer leakage, though it leans more toward exact recall than deep generative reasoning."}}
{"id": "2602.02876v1", "paper_link": "http://arxiv.org/abs/2602.02876v1", "theorems_cnt": 3, "theorem": {"env_name": "observation", "content": "\\label{Chain}\nFor any graph $G$ with maximum degree $\\Delta$ and positive integer $k\\in[\\Delta]$,\n\\begin{center}\n$\\chi(G)=\\chi_{\\Delta}^{f}(G)\\leq \\ldots\\leq \\chi_{2}^{f}(G)\\leq \\chi_{1}^{f}(G)=\\chi(G^{2})$.\n\\end{center}", "start_pos": 10728, "end_pos": 10985, "label": "Chain"}, "ref_dict": {"ob:boundDelta": "\\begin{observation}\n\\label{ob:boundDelta}\nIf $G$ is a graph with maximum degree $\\Delta$, then $\\chi_{t}^{f}(G)\\le 1+\\Delta(1+\\big \\lfloor\\frac{\\Delta-1}{t}\\big\\rfloor)$.     \n\\end{observation}", "Chain": "\\begin{observation}\\label{Chain}\nFor any graph $G$ with maximum degree $\\Delta$ and positive integer $k\\in[\\Delta]$,\n\\begin{center}\n$\\chi(G)=\\chi_{\\Delta}^{f}(G)\\leq \\ldots\\leq \\chi_{2}^{f}(G)\\leq \\chi_{1}^{f}(G)=\\chi(G^{2})$.\n\\end{center}\n\\end{observation}"}, "pre_theorem_intro_text_len": 5701, "pre_theorem_intro_text": "A proper vertex coloring of a graph $G$ is $t$-\\textit{frugal} if no color appears more than $t$ times in the neighborhood of any vertex in $G$. The $t$-\\textit{frugal chromatic number} of $G$ is the minimum $k$ for which $G$ admits a $t$-frugal coloring with $k$ colors, and we denote it by $\\chi_{t}^{f}(G)$. Hind, Molloy and Reed~\\cite{HMR} initiated the concept of frugal coloring in 1997 proving that a graph $G$ with sufficiently large maximum degree $\\Delta$ admits a $({\\log}^8{\\Delta})$-frugal coloring with $\\Delta+1$ colors and mentioning an application for total colorings. The result was improved by Ndreca, Procacci and Scoppola~\\cite{nps} in 2012, while only a few other authors considered this type of coloring so far~\\cite{AEv,BMR,KM}; see also~\\cite{ch-2011} where its list version was considered. In this paper, we expand the consideration of frugal colorings by studying several aspects that are of relevance for coloring invariants. \n\nFrugal colorings are related to several known coloring invariants. A {\\em $2$-distance coloring} of a graph $G$ is a mapping $c:V(G)\\to \\{1,\\ldots,k\\}$ such that any two vertices at distance at most $2$ receive different colors. The minimum number of colors in such a coloring is the {\\em $2$-distance chromatic number} $\\chi_{2}(G)$ of $G$; see Kramer and Kramer~\\cite{kk0} for systematic treatment of $d$-distance coloring. Alternatively, $2$-distance coloring of $G$ coincides with the coloring of the square $G^2$ of $G$, where two vertices in $G^2$ are adjacent if they are adjacent or have a common neighbor in $G$, and so $\\chi_{2}(G)=\\chi(G^{2})$. Clearly, the condition of $2$-frugal coloring is weaker than that of $2$-distance coloring, thus $\\chi_{2}^{f}(G)\\le \\chi_2(G)$ for any graph $G$. A proper coloring of a graph $G$ in which the vertices of any two color classes induce a forest of paths is a {\\em linear coloring} of $G$, as introduced by Yuster~\\cite{yus}. Note that a linear coloring is a $2$-frugal coloring of $G$, but the converse is not necessarily true. Thus, for the resulting invariant ${\\rm lc}(G)$ of $G$, which is the minimum number of colors in a linear coloring in $G$, we get $\\chi_{2}^{f}(G)\\le {\\rm lc}(G)$. \n\nAnother concept related to frugal coloring arises from that of limited packing as introduced by Gallant et al.~\\cite{GGHR} and studied further in~\\cite{GZ,BS}. A set $B\\subseteq V(G)$ is a {\\em $t$-limited packing} if the closed neighborhood of each vertex in $G$ contains at most $t$ vertices in $B$. Note that when $t=1$, the resulting $1$-limited packing coincides with the concept of $2$-packing, as introduced by Meir and Moon~\\cite{MM} and studied in many papers concerning graph domination. Recently, a \\textit{$t$-limited packing partition} of $G$, which is a partition of $V(G)$ into $t$-limited packings, was considered and the following coloring invariant was introduced. \nThe \\textit{$t$-limited packing partition number}, denoted by $\\chi_{\\times t}(G)$, is the minimum cardinality of a $t$-limited packing partition in $G$; see~\\cite{AS,AL}. \nNote that when $t\\ge 2$, the partition is not necessarily a proper coloring. However, additionally imposing that the color classes in a $t$-limited packing partition are independent results precisely in a $t$-frugal coloring. In particular, $\\chi_{t}^{f}(G)\\ge \\chi_{\\times t}(G)$ holds for any graph $G$ and any positive integer $t$.\n\nThe behavior of a particular graph coloring is inherently related to the nature and structure of its color classes. On the other hand, the color classes of many types of graph colorings have been studied independently. For example, a large number of papers have been published about ``independent\", ``$2$-packing\", ``open packing\" and ``dissociation\" sets, which are color classes of the ``standard\", ``$2$-distance\", ``injective\" and ``defective\" colorings, respectively. Moreover, the color classes of frugal colorings are a variant of limited packings, notably, they are independent limited packings. Due to the fundamental role of these sets in the study of frugal colorings, we investigate them under the name of {\\em frugal independent sets}. More specifically, such a set is a {\\em $t$-frugal independent set} ($t$FI set for short), where $t$ is a positive integer. The maximum cardinality of a $t$FI set in $G$ will be denoted by $\\alpha_t^f(G)$, and called $t$-{\\em frugal independence number} of $G$.  \nIn line with the above arguments, we will investigate frugal independent sets along with frugal colorings both from computational and combinatorial points of view.\n\n\\subsection{Preliminaries}\n\nThroughout the paper, we consider $G$ as a simple graph with vertex set $V(G)$ and edge set $E(G)$. In addition, $G$ is finite unless explicitly stated otherwise. We use~\\cite{West} as a reference for terminology and notation which are not explicitly defined here. The (\\textit{open}) {\\em neighborhood} of a vertex $v$ is denoted by $N_{G}(v)$, and its {\\em closed neighborhood} is $N_{G}[v]=N_{G}(v)\\cup \\{v\\}$. The {\\em minimum} and {\\em maximum degrees} of $G$ are denoted by $\\delta(G)$ and $\\Delta(G)$, respectively. Given subsets $A,B\\subseteq V(G)$, let $[A,B]$ denote the set of all edges with one endvertex in $A$ and the other in $B$. For simplicity, we use the notation $[k]$ instead of $\\{1,\\ldots,k\\}$ for any positive integer $k$. By a $\\chi_{t}^{f}(G)$-coloring we mean a $t$-frugal coloring of $G$ with $\\chi_{t}^{f}(G)$ colors.\n\nSince $\\chi_{t}^{f}(G)=\\chi(G)$ for each integer $t\\ge \\Delta(G)$, where $\\chi(G)$ is the chromatic number of $G$,  we restrict our attention to the cases when $t\\in[\\Delta(G)-1]$. Moreover, the following inequality chain follows from the definitions.", "context": "A proper vertex coloring of a graph $G$ is $t$-\\textit{frugal} if no color appears more than $t$ times in the neighborhood of any vertex in $G$. The $t$-\\textit{frugal chromatic number} of $G$ is the minimum $k$ for which $G$ admits a $t$-frugal coloring with $k$ colors, and we denote it by $\\chi_{t}^{f}(G)$. Hind, Molloy and Reed~\\cite{HMR} initiated the concept of frugal coloring in 1997 proving that a graph $G$ with sufficiently large maximum degree $\\Delta$ admits a $({\\log}^8{\\Delta})$-frugal coloring with $\\Delta+1$ colors and mentioning an application for total colorings. The result was improved by Ndreca, Procacci and Scoppola~\\cite{nps} in 2012, while only a few other authors considered this type of coloring so far~\\cite{AEv,BMR,KM}; see also~\\cite{ch-2011} where its list version was considered. In this paper, we expand the consideration of frugal colorings by studying several aspects that are of relevance for coloring invariants.\n\nFrugal colorings are related to several known coloring invariants. A {\\em $2$-distance coloring} of a graph $G$ is a mapping $c:V(G)\\to \\{1,\\ldots,k\\}$ such that any two vertices at distance at most $2$ receive different colors. The minimum number of colors in such a coloring is the {\\em $2$-distance chromatic number} $\\chi_{2}(G)$ of $G$; see Kramer and Kramer~\\cite{kk0} for systematic treatment of $d$-distance coloring. Alternatively, $2$-distance coloring of $G$ coincides with the coloring of the square $G^2$ of $G$, where two vertices in $G^2$ are adjacent if they are adjacent or have a common neighbor in $G$, and so $\\chi_{2}(G)=\\chi(G^{2})$. Clearly, the condition of $2$-frugal coloring is weaker than that of $2$-distance coloring, thus $\\chi_{2}^{f}(G)\\le \\chi_2(G)$ for any graph $G$. A proper coloring of a graph $G$ in which the vertices of any two color classes induce a forest of paths is a {\\em linear coloring} of $G$, as introduced by Yuster~\\cite{yus}. Note that a linear coloring is a $2$-frugal coloring of $G$, but the converse is not necessarily true. Thus, for the resulting invariant ${\\rm lc}(G)$ of $G$, which is the minimum number of colors in a linear coloring in $G$, we get $\\chi_{2}^{f}(G)\\le {\\rm lc}(G)$.\n\nAnother concept related to frugal coloring arises from that of limited packing as introduced by Gallant et al.~\\cite{GGHR} and studied further in~\\cite{GZ,BS}. A set $B\\subseteq V(G)$ is a {\\em $t$-limited packing} if the closed neighborhood of each vertex in $G$ contains at most $t$ vertices in $B$. Note that when $t=1$, the resulting $1$-limited packing coincides with the concept of $2$-packing, as introduced by Meir and Moon~\\cite{MM} and studied in many papers concerning graph domination. Recently, a \\textit{$t$-limited packing partition} of $G$, which is a partition of $V(G)$ into $t$-limited packings, was considered and the following coloring invariant was introduced. \nThe \\textit{$t$-limited packing partition number}, denoted by $\\chi_{\\times t}(G)$, is the minimum cardinality of a $t$-limited packing partition in $G$; see~\\cite{AS,AL}. \nNote that when $t\\ge 2$, the partition is not necessarily a proper coloring. However, additionally imposing that the color classes in a $t$-limited packing partition are independent results precisely in a $t$-frugal coloring. In particular, $\\chi_{t}^{f}(G)\\ge \\chi_{\\times t}(G)$ holds for any graph $G$ and any positive integer $t$.\n\nThe behavior of a particular graph coloring is inherently related to the nature and structure of its color classes. On the other hand, the color classes of many types of graph colorings have been studied independently. For example, a large number of papers have been published about ``independent\", ``$2$-packing\", ``open packing\" and ``dissociation\" sets, which are color classes of the ``standard\", ``$2$-distance\", ``injective\" and ``defective\" colorings, respectively. Moreover, the color classes of frugal colorings are a variant of limited packings, notably, they are independent limited packings. Due to the fundamental role of these sets in the study of frugal colorings, we investigate them under the name of {\\em frugal independent sets}. More specifically, such a set is a {\\em $t$-frugal independent set} ($t$FI set for short), where $t$ is a positive integer. The maximum cardinality of a $t$FI set in $G$ will be denoted by $\\alpha_t^f(G)$, and called $t$-{\\em frugal independence number} of $G$. \nIn line with the above arguments, we will investigate frugal independent sets along with frugal colorings both from computational and combinatorial points of view.\n\nThroughout the paper, we consider $G$ as a simple graph with vertex set $V(G)$ and edge set $E(G)$. In addition, $G$ is finite unless explicitly stated otherwise. We use~\\cite{West} as a reference for terminology and notation which are not explicitly defined here. The (\\textit{open}) {\\em neighborhood} of a vertex $v$ is denoted by $N_{G}(v)$, and its {\\em closed neighborhood} is $N_{G}[v]=N_{G}(v)\\cup \\{v\\}$. The {\\em minimum} and {\\em maximum degrees} of $G$ are denoted by $\\delta(G)$ and $\\Delta(G)$, respectively. Given subsets $A,B\\subseteq V(G)$, let $[A,B]$ denote the set of all edges with one endvertex in $A$ and the other in $B$. For simplicity, we use the notation $[k]$ instead of $\\{1,\\ldots,k\\}$ for any positive integer $k$. By a $\\chi_{t}^{f}(G)$-coloring we mean a $t$-frugal coloring of $G$ with $\\chi_{t}^{f}(G)$ colors.\n\nSince $\\chi_{t}^{f}(G)=\\chi(G)$ for each integer $t\\ge \\Delta(G)$, where $\\chi(G)$ is the chromatic number of $G$, we restrict our attention to the cases when $t\\in[\\Delta(G)-1]$. Moreover, the following inequality chain follows from the definitions.", "full_context": "A proper vertex coloring of a graph $G$ is $t$-\\textit{frugal} if no color appears more than $t$ times in the neighborhood of any vertex in $G$. The $t$-\\textit{frugal chromatic number} of $G$ is the minimum $k$ for which $G$ admits a $t$-frugal coloring with $k$ colors, and we denote it by $\\chi_{t}^{f}(G)$. Hind, Molloy and Reed~\\cite{HMR} initiated the concept of frugal coloring in 1997 proving that a graph $G$ with sufficiently large maximum degree $\\Delta$ admits a $({\\log}^8{\\Delta})$-frugal coloring with $\\Delta+1$ colors and mentioning an application for total colorings. The result was improved by Ndreca, Procacci and Scoppola~\\cite{nps} in 2012, while only a few other authors considered this type of coloring so far~\\cite{AEv,BMR,KM}; see also~\\cite{ch-2011} where its list version was considered. In this paper, we expand the consideration of frugal colorings by studying several aspects that are of relevance for coloring invariants.\n\nFrugal colorings are related to several known coloring invariants. A {\\em $2$-distance coloring} of a graph $G$ is a mapping $c:V(G)\\to \\{1,\\ldots,k\\}$ such that any two vertices at distance at most $2$ receive different colors. The minimum number of colors in such a coloring is the {\\em $2$-distance chromatic number} $\\chi_{2}(G)$ of $G$; see Kramer and Kramer~\\cite{kk0} for systematic treatment of $d$-distance coloring. Alternatively, $2$-distance coloring of $G$ coincides with the coloring of the square $G^2$ of $G$, where two vertices in $G^2$ are adjacent if they are adjacent or have a common neighbor in $G$, and so $\\chi_{2}(G)=\\chi(G^{2})$. Clearly, the condition of $2$-frugal coloring is weaker than that of $2$-distance coloring, thus $\\chi_{2}^{f}(G)\\le \\chi_2(G)$ for any graph $G$. A proper coloring of a graph $G$ in which the vertices of any two color classes induce a forest of paths is a {\\em linear coloring} of $G$, as introduced by Yuster~\\cite{yus}. Note that a linear coloring is a $2$-frugal coloring of $G$, but the converse is not necessarily true. Thus, for the resulting invariant ${\\rm lc}(G)$ of $G$, which is the minimum number of colors in a linear coloring in $G$, we get $\\chi_{2}^{f}(G)\\le {\\rm lc}(G)$.\n\nAnother concept related to frugal coloring arises from that of limited packing as introduced by Gallant et al.~\\cite{GGHR} and studied further in~\\cite{GZ,BS}. A set $B\\subseteq V(G)$ is a {\\em $t$-limited packing} if the closed neighborhood of each vertex in $G$ contains at most $t$ vertices in $B$. Note that when $t=1$, the resulting $1$-limited packing coincides with the concept of $2$-packing, as introduced by Meir and Moon~\\cite{MM} and studied in many papers concerning graph domination. Recently, a \\textit{$t$-limited packing partition} of $G$, which is a partition of $V(G)$ into $t$-limited packings, was considered and the following coloring invariant was introduced. \nThe \\textit{$t$-limited packing partition number}, denoted by $\\chi_{\\times t}(G)$, is the minimum cardinality of a $t$-limited packing partition in $G$; see~\\cite{AS,AL}. \nNote that when $t\\ge 2$, the partition is not necessarily a proper coloring. However, additionally imposing that the color classes in a $t$-limited packing partition are independent results precisely in a $t$-frugal coloring. In particular, $\\chi_{t}^{f}(G)\\ge \\chi_{\\times t}(G)$ holds for any graph $G$ and any positive integer $t$.\n\nThe behavior of a particular graph coloring is inherently related to the nature and structure of its color classes. On the other hand, the color classes of many types of graph colorings have been studied independently. For example, a large number of papers have been published about ``independent\", ``$2$-packing\", ``open packing\" and ``dissociation\" sets, which are color classes of the ``standard\", ``$2$-distance\", ``injective\" and ``defective\" colorings, respectively. Moreover, the color classes of frugal colorings are a variant of limited packings, notably, they are independent limited packings. Due to the fundamental role of these sets in the study of frugal colorings, we investigate them under the name of {\\em frugal independent sets}. More specifically, such a set is a {\\em $t$-frugal independent set} ($t$FI set for short), where $t$ is a positive integer. The maximum cardinality of a $t$FI set in $G$ will be denoted by $\\alpha_t^f(G)$, and called $t$-{\\em frugal independence number} of $G$. \nIn line with the above arguments, we will investigate frugal independent sets along with frugal colorings both from computational and combinatorial points of view.\n\nThroughout the paper, we consider $G$ as a simple graph with vertex set $V(G)$ and edge set $E(G)$. In addition, $G$ is finite unless explicitly stated otherwise. We use~\\cite{West} as a reference for terminology and notation which are not explicitly defined here. The (\\textit{open}) {\\em neighborhood} of a vertex $v$ is denoted by $N_{G}(v)$, and its {\\em closed neighborhood} is $N_{G}[v]=N_{G}(v)\\cup \\{v\\}$. The {\\em minimum} and {\\em maximum degrees} of $G$ are denoted by $\\delta(G)$ and $\\Delta(G)$, respectively. Given subsets $A,B\\subseteq V(G)$, let $[A,B]$ denote the set of all edges with one endvertex in $A$ and the other in $B$. For simplicity, we use the notation $[k]$ instead of $\\{1,\\ldots,k\\}$ for any positive integer $k$. By a $\\chi_{t}^{f}(G)$-coloring we mean a $t$-frugal coloring of $G$ with $\\chi_{t}^{f}(G)$ colors.\n\nSince $\\chi_{t}^{f}(G)=\\chi(G)$ for each integer $t\\ge \\Delta(G)$, where $\\chi(G)$ is the chromatic number of $G$, we restrict our attention to the cases when $t\\in[\\Delta(G)-1]$. Moreover, the following inequality chain follows from the definitions.\n\nSince $\\chi_{t}^{f}(G)=\\chi(G)$ for each integer $t\\ge \\Delta(G)$, where $\\chi(G)$ is the chromatic number of $G$, we restrict our attention to the cases when $t\\in[\\Delta(G)-1]$. Moreover, the following inequality chain follows from the definitions.\n\nLet $\\mathcal{J}$ be the color classes of a $\\chi_{t}^{f}(G)$-coloring, and let $u$ be a vertex in $G$ of maximum degree. Let $u\\in J$, where $J\\in \\mathcal{J}$. By definition, $J$ is an independent set and $u$ has at most $t$ neighbors in each color class in $\\mathcal{J}\\setminus \\{J\\}$. This shows that\n\\begin{equation}\\label{Delta}\n\\Delta(G)=\\deg_{G}(u)=\\sum_{J'\\in \\mathcal{J}\\setminus \\{J\\}}|N_{G}(u)\\cap J'|\\leq t|\\mathcal{J}\\setminus \\{J\\}|=t\\fk(G)-t.\n\\end{equation}\nTherefore, $\\fk(G)\\geq \\lceil \\Delta(G)/t\\rceil+1$. The following lower bound on the $t$-frugal chromatic number follows directly from the latter inequality and Observation \\ref{Chain}.\n\n\\begin{observation}\n\\label{ob:boundDelta}\nIf $G$ is a graph with maximum degree $\\Delta$, then $\\chi_{t}^{f}(G)\\le 1+\\Delta(1+\\big \\lfloor\\frac{\\Delta-1}{t}\\big\\rfloor)$. \n\\end{observation}\nThe bound in Observation~\\ref{ob:boundDelta} can be sharp. For instance, when $t=2$ and $\\Delta=2$, the bound reads $\\chi_{2}^{f}(G)\\le 1+2(1+\\big \\lfloor\\frac{2-1}{2}\\big\\rfloor)=3$, and $\\chi_{2}^{f}(C_{2r+1})=3$ where $r\\in\\mathbb{N}$.\n\n\\begin{theorem}\\label{General1}\nIf $t$ is a positive integer and $G$ is a graph of size $m$, then\n\\begin{center}\n$\\chi_{t}^{f}(G)\\geq \\dfrac{1}{2}+\\sqrt{\\dfrac{1}{4}+\\dfrac{2m}{t\\alpha_{t}^{f}(G)}}$\n\\end{center}\nwith equality if and only if $G\\in \\Psi_{t}$.\n\\end{theorem}\n\\begin{proof}\nThe lower bound trivially holds for edgeless graphs. So, we assume that $\\chi_{t}^{f}(G)\\geq2$. Let $\\mathbb{B}=\\{B_{1},\\ldots,B_{\\chi_{t}^{f}(G)}\\}$ be a $\\chi_{t}^{f}(G)$-coloring. Without loss of generality, we may assume that $|B_{1}|\\leq \\ldots\\leq|B_{\\chi_{t}^{f}(G)}|$. Then, \n\\begin{equation}\\label{Ine}\n\\begin{array}{lcl}\nm&=&\\sum_{s=1}^{\\chi_{t}^{f}(G)-1}\\sum_{t=s+1}^{\\chi_{t}^{f}(G)}|[B_{s},B_{t}]|\\leq t\\sum_{s=1}^{\\chi_{t}^{f}(G)-1}|B_{s}|(\\chi_{t}^{f}(G)-s)\\vspace{1mm}\\\\\n&\\leq& t\\alpha_{t}^{f}(G)\\sum_{s=1}^{\\chi_{t}^{f}(G)-1}(\\chi_{t}^{f}(G)-s)=t\\alpha_{t}^{f}(G)\\big{(}\\dfrac{\\chi_{t}^{f}(G)(\\chi_{t}^{f}(G)-1)}{2}\\big{)}.\n\\end{array}\n\\end{equation}\nTherefore, $t\\alpha_{t}^{f}(G)\\chi_{t}^{f}(G)^{2}-t\\alpha_{t}^{f}(G)\\chi_{t}^{f}(G)-2m\\geq0$. Solving this inequality for $\\chi_{t}^{f}(G)$, we obtain the desired lower bound on $\\chi_{t}^{f}(G)$.\n\nNow, let us prove that any graph $G\\in \\Psi_{t}$ attains the lower bound. Let $X_{1},\\ldots,X_{r}$ be the partite sets of $G$. For any two distinct indices $i,j\\in[r]$, $|X_{i}|=|X_{j}|$ follows from the fact that every vertex in $X_{i}$ (resp. $X_{j}$) has precisely $t$ neighbors in $X_{j}$ (resp. $X_{i}$). \nBy the structure of $G$, each partite set $X_{t}$ is a $t$FI set. This shows that $|X_{t}|\\leq \\alpha_{t}^{f}(G)$ and that $\\chi_{t}^{f}(G)\\leq r$. Let $B$ be an $\\alpha_{t}^{f}(G)$-set. Suppose to the contrary that $|X_{t}|<\\alpha_{t}^{f}(G)$. This in particular implies that $B\\nsubseteq X_{t}$. Suppose that $X_{t}\\subseteq B$. In such a situation, the strict inequality $|X_{t}|<\\alpha_{t}^{f}(G)$ shows that $B$ contains a vertex $v$ from a partite set $X_{j}$ with $j\\neq t$. This contradicts the fact that $B$ is an independent set as $v$ has a neighbor in $X_{t}$ by the structure of $G$. Therefore, $X_{t}\\nsubseteq B$. Now set $Q=\\{(b,x):\\, b\\in B\\setminus X_{t}, x\\in X_{t}\\setminus B\\ \\mbox{and}\\ bx\\in E(G)\\}$. Since $B$ is a $t$FI set, every vertex in $X_{t}\\setminus B$ is adjacent to at most $t$ vertices in $B\\setminus X_{t}$. So, $|Q|\\leq t|X_{t}\\setminus B|=t|X_{t}|-t|B\\cap X_{t}|$. On the other hand, every vertex in $B\\setminus X_{t}$ has exactly $t$ neighbors in $X_{t}\\setminus B$ by the structure of $G$ and since $B$ is an independent set. This shows that $|Q|=t|B\\setminus X_{t}|=t|B|-t|B\\cap X_{t}|$. This, together with the last inequality, results in $|B|\\leq|X_{t}|$. Therefore, $\\alpha_{t}^{f}(G)=|X_{1}|=\\ldots=|X_{r}|=|V(G)|/r$. With this in mind, we have\n\\begin{center} \n$\\chi_{t}^{f}(G)\\geq \\dfrac{1}{2}+\\sqrt{\\dfrac{1}{4}+\\dfrac{2m}{t\\alpha_{t}^{f}(G)}}=\\dfrac{1}{2}+\\sqrt{\\dfrac{1}{4}+r^{2}-r}=r$.\n\\end{center}\nThis leads to the desired equality.\n\n\\begin{proposition}\\label{TF}\nFor any triangle-free graph $G$ of order $n$,\n\\begin{center}\n$\\chi_{2}^{f}(G)\\leq \\Big{\\lfloor}\\dfrac{n-\\alpha_2^{f}(G)+4}{2}\\Big{\\rfloor}$\n\\end{center}\nand this bound is sharp.\n\\end{proposition}\n\\begin{proof}\nIf $n\\leq \\alpha_2^{f}(G)+2$, then the upper bound is easily verified. So, we may assume that $n\\geq \\alpha_2^{f}(G)+3$. Let $B$ be an $\\alpha_2^{f}(G)$-set and set $G'=G[V(G)\\setminus B]$. Clearly, $G'$ is a triangle-free graph as well. With this in mind and since $|V(G')|\\geq3$, once can partition $V(G')$ into $t$ subsets $B_{1},\\ldots,B_{t}$ in such a way that\n\\begin{itemize}\n \\item $B_{i}$ consists of two nonadjacent vertices for each $i\\in[t-1]$, and\n \\item $|B_{t}|\\in \\{1,2\\}$.\n\\end{itemize}\nIf $|B_{t}|=1$ or $B_{t}$ consists of two nonadjacent vertices, then $\\{B,B_{1},\\ldots,B_{t}\\}$ is a $2$-frugal coloring of $G$. Therefore, $\\chi_{2}^{f}(G)\\leq2+(n-\\alpha_2^{f}(G)-1)/2=(n-\\alpha_2^{f}(G)+3)/2$ or $\\chi_{2}^{f}(G)\\leq1+\\big{(}n-\\alpha_2^{f}(G)\\big{)}/2=(n-\\alpha_2^{f}(G)+2)/2$ if $|B_{t}|=1$ or $|B_{t}|=2$, respectively.\n\n\\begin{theorem}\\label{Lexi}\nFor any graphs $G$ and $H$, \n\\begin{center}\n$\\chi_{2}^{f}(H)+\\Big{\\lceil}\\frac{\\Delta(G)|V(H)|}{2}\\Big{\\rceil}\\leq \\chi_{2}^{f}(G\\circ H)\\leq \\chi_{2}^{f}(G)|V(H)|$.\n\\end{center} \nThese bounds are sharp.\n\\end{theorem}\n\\begin{proof}\nLet $\\{B_{1},\\ldots,B_{\\chi_{2}^{f}(G)}\\}$ be a $\\chi_{2}^{f}(G)$-coloring and let $V(H)=\\{h_{1},\\ldots,h_{|V(H)|}\\}$. Let $f:V(G\\circ H)\\to [\\chi_{2}^{f}(G)|V(H)|]$ be defined by \n$f(g,h)=(t-1)n+j$, where $g\\in B_t$ and $h=h_j$. \nNote that $B_{t}\\times \\{h_{j}\\}$, for $t\\in[\\chi_{2}^{f}(G)]$ and $j\\in[|V(H)|]$, are the color classes of $f$. \nSince $B_{t}$ is an I$2$F set in $G$, the adjacency role of the lexicographic product graphs shows that every color class $B_{t}\\times \\{h_{j}\\}$ is an independent set in $G\\circ H$. Suppose that there exists a vertex $(g,h)\\in V(G\\circ H)$ adjacent to three vertices $(g_{1},h_{j})$, $(g_{2},h_{j})$ and $(g_{3},h_{j})$ in $B_{t}\\times \\{h_j\\}$ for some $t\\in[\\chi_{2}^{f}(G)]$ and $j\\in[|V(H)|]$. This in particular implies that $g_{1},g_{2},g_{3}\\in N_{G}[g]$. Since $B_{t}$ is an $2$FI set in $G$, we may assume that $g=g_{1}$ and $g_{2},g_{3}\\in N_{G}(g)$. This is impossible because $B_{t}$ is an independent set in $G$. Therefore, $B_{t}\\times \\{h_{j}\\}$ is an I$2$F set in $G\\circ H$ for each $t\\in[\\chi_{2}^{f}(G)]$ and $j\\in[|V(H)|]$. Thus, $\\chi_{2}^{f}(G\\circ H)\\leq|f\\big{(}V(G\\circ H)\\big{)}|=\\chi_{2}^{f}(G)|V(H)|$.", "post_theorem_intro_text_len": 5153, "post_theorem_intro_text": "Let $\\mathcal{J}$ be the color classes of a $\\chi_{t}^{f}(G)$-coloring, and let $u$ be a vertex in $G$ of maximum degree. Let $u\\in J$, where $J\\in \\mathcal{J}$. By definition, $J$ is an independent set and $u$ has at most $t$ neighbors in each color class in $\\mathcal{J}\\setminus \\{J\\}$. This shows that\n\\begin{equation}\\label{Delta}\n\\Delta(G)=\\deg_{G}(u)=\\sum_{J'\\in \\mathcal{J}\\setminus \\{J\\}}|N_{G}(u)\\cap J'|\\leq t|\\mathcal{J}\\setminus \\{J\\}|=t\\chi_{t}^{f}(G)-t.\n\\end{equation}\nTherefore, $\\chi_{t}^{f}(G)\\geq \\lceil \\Delta(G)/t\\rceil+1$. The following lower bound on the $t$-frugal chromatic number follows directly from the latter inequality and Observation \\ref{Chain}. \n\n\\begin{observation}\n\\label{ob:lowerboundfor2}\nIf $G$ is a graph and $t\\ge1$, then $\\chi_{t}^{f}(G)\\ge \\max\\{\\chi(G),\\big\\lceil\\frac{\\Delta(G)}{t}\\big\\rceil+1\\}.$   \n\\end{observation}\n\nConcerning upper bounds with respect to the maximum degree of a graph, we obtain the following observation, which immediately follows by using a greedy coloring algorithm.\n\n\\begin{observation}\n\\label{ob:boundDelta}\nIf $G$ is a graph with maximum degree $\\Delta$, then $\\chi_{t}^{f}(G)\\le 1+\\Delta(1+\\big \\lfloor\\frac{\\Delta-1}{t}\\big\\rfloor)$.     \n\\end{observation}\nThe bound in Observation~\\ref{ob:boundDelta} can be sharp. For instance, when $t=2$ and $\\Delta=2$, the bound reads $\\chi_{2}^{f}(G)\\le 1+2(1+\\big \\lfloor\\frac{2-1}{2}\\big\\rfloor)=3$, and $\\chi_{2}^{f}(C_{2r+1})=3$ where $r\\in\\mathbb{N}$.  \n\n\\subsection{Main results and organization of the paper}\n\nWe start with computational aspects of the two main invariants studied in this paper. As proved in~\\cite{BMR}, the decision version of the $t$-frugal chromatic number is NP-complete for any positive integer $t$. Hence, in Section~\\ref{sec:computational}, we concentrate on computational aspects with respect to the $t$-frugal independence number and prove that the decision version of $\\alpha_t^f(G)$ is NP-complete for any positive integer $t$ even if restricted to bipartite graphs $G$. In contrast, we present a linear-time algorithm for computing $\\alpha_t^f(T)$ in an arbitrary tree $T$ and for any $t$. \n\nIn Section~\\ref{sec:generalbounds}, we prove several general bounds involving one or both graph invariants. In particular, we prove that $\\chi_{t}^{f}(G)\\geq {1}/{2}+\\sqrt{{1}/{4}+{2m}/({t\\alpha_{t}^{f}(G))}}$ holds for any positive integer $t$ and any graph $G$ of size $m$, and we characterize the graphs that attain this bound. On the other hand, when $G$ is triangle-free, we prove an upper bound on $\\chi_{2}^{f}(G)$ expressed in terms of the order of $G$ and its $2$-frugal independence number. In addition, we prove an upper bound on the $2$-frugal independence number of an arbitrary graph $G$ expressed in terms of the order of $G$ and several other invariants that depend on its pendant vertices. In Section~\\ref{sec:subcubic} we focus on subcubic graphs, where our main result is that any graph $G$ with $\\Delta(G)=3$ satisfies $3\\le \\chi_{2}^{f}(G)\\le 5$. While the lower bound is trivial, the upper bound is proved by an extensive case analysis. If, in addition, $G$ is claw-free, then the equality $\\chi_{2}^{f}(G)=3$ holds.\n\nSection~\\ref{sec:NG} is devoted to Nordhaus-Gaddum results, where we again focus on the $t$-frugal chromatic number where $t=2$. We prove that\n$$\\dfrac{n}{2}+2\\le\\chi_{2}^{f}(G)+\\chi_{2}^{f}(\\overline{G})\\leq \\dfrac{3n}{2},$$\nwhere the lower bound holds for all graphs $G$ of order $n$ with the exception of $6$ sporadic graphs, while the upper holds for all graphs $G$. In addition, the upper bound holds with equality if and only if $n$ is even and $G\\in \\{K_{1,n-1},\\overline{K_{1,n-1}}\\}$.\n\nIn Section~\\ref{sec:classes}, we consider several graph classes and graph operations. For an arbitrary block graph $G$ we obtain the exact value of the $2$-frugal chromatic number, that is,\n $\\chi_{2}^{f}(G)=\\max\\big\\{\\omega,\\big\\lceil\\frac{\\Delta}{2}\\big\\rceil+1\\big\\}$, where $\\omega$ is the clique number and $\\Delta$ the maximum degree of $G$. If $G\\,\\square\\, H$ is the Cartesian product of graphs $G$ and $H$ we obtain the following lower and upper bound on its $2$-frugal chromatic number: $\\max\\{\\chi_{2}^{f}(G),\\chi_{2}^{f}(H)\\}\\le \\chi_{2}^{f}(G\\,\\square\\, H)\\le \\max\\{\\chi_2(G),\\chi_2(H)\\}$. The bounds are sharp and in some cases coincide. We also obtain sharp upper bounds for the $2$-frugal chromatic numbers in strong and direct products of two graphs. In addition, we bound the lexicographic product of two graphs as follows: $\\chi_{2}^{f}(H)+\\Big{\\lceil}\\frac{\\Delta(G)|V(H)|}{2}\\Big{\\rceil}\\leq \\chi_{2}^{f}(G\\circ H)\\leq \\chi_{2}^{f}(G)|V(H)|$, and provide examples of sharpness. We continue with the quest for classes of graphs that achieve the trivial lower bound for their $2$-frugal chromatic number, notably when $\\chi_{2}^{f}(G)=\\lceil\\Delta(G)/2\\rceil+1$. We prove that several well-known infinite lattices satisfy this equality. In addition, the equality holds for Cartesian products of several (infinite) paths as well as strong products of several (infinite) paths. We conclude the paper with some open problems and directions for future research.", "sketch": "Let $\\mathcal{J}$ be the color classes of a $\\chi_t^f(G)$-coloring, and let $u$ be a vertex of maximum degree with $u\\in J\\in\\mathcal{J}$. Since $J$ is independent and $u$ has at most $t$ neighbors in each other color class $J'\\in\\mathcal{J}\\setminus\\{J\\}$, one has\n\\[\n\\Delta(G)=\\deg_G(u)=\\sum_{J'\\in\\mathcal{J}\\setminus\\{J\\}} |N_G(u)\\cap J'|\\le t\\,|\\mathcal{J}\\setminus\\{J\\}|=t\\chi_t^f(G)-t.\n\\]\nTherefore $\\chi_t^f(G)\\ge \\lceil \\Delta(G)/t\\rceil+1$. The stated lower bound $\\chi_t^f(G)\\ge \\max\\{\\chi(G),\\lceil \\Delta(G)/t\\rceil+1\\}$ is said to follow directly from this inequality together with Observation~\\ref{Chain} (the chain $\\chi(G)=\\chi_\\Delta^f(G)\\le\\cdots\\le\\chi_1^f(G)=\\chi(G^2)$).\n\nFor an upper bound in terms of $\\Delta$, it is stated that one \"immediately\" obtains $\\chi_t^f(G)\\le 1+\\Delta\\bigl(1+\\lfloor(\\Delta-1)/t\\rfloor\\bigr)$ by using a greedy coloring algorithm.", "expanded_sketch": "Let $\\mathcal{J}$ be the color classes of a $\\chi_t^f(G)$-coloring, and let $u$ be a vertex of maximum degree with $u\\in J\\in\\mathcal{J}$. Since $J$ is independent and $u$ has at most $t$ neighbors in each other color class $J'\\in\\mathcal{J}\\setminus\\{J\\}$, one has\n\\[\n\\Delta(G)=\\deg_G(u)=\\sum_{J'\\in\\mathcal{J}\\setminus\\{J\\}} |N_G(u)\\cap J'|\\le t\\,|\\mathcal{J}\\setminus\\{J\\}|=t\\chi_t^f(G)-t.\n\\]\nTherefore $\\chi_t^f(G)\\ge \\lceil \\Delta(G)/t\\rceil+1$. The stated lower bound $\\chi_t^f(G)\\ge \\max\\{\\chi(G),\\lceil \\Delta(G)/t\\rceil+1\\}$ is said to follow directly from this inequality together with the relevant monotonicity chain used in establishing the main theorem (namely that $\\chi_t^f(G)$ is at least $\\chi(G)$).\n\nFor an upper bound in terms of $\\Delta$, it is stated that one \"immediately\" obtains $\\chi_t^f(G)\\le 1+\\Delta\\bigl(1+\\lfloor(\\Delta-1)/t\\rfloor\\bigr)$ by using a greedy coloring algorithm.,", "expanded_theorem": "\\label{Chain}\nFor any graph $G$ with maximum degree $\\Delta$ and positive integer $k\\in[\\Delta]$,\n\\begin{center}\n$\\chi(G)=\\chi_{\\Delta}^{f}(G)\\leq \\ldots\\leq \\chi_{2}^{f}(G)\\leq \\chi_{1}^{f}(G)=\\chi(G^{2})$.\n\\end{center}", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let $G$ be a finite simple graph with maximum degree $\\Delta$. For a positive integer $t$, a proper vertex coloring of $G$ is called $t$-frugal if no color appears more than $t$ times in the open neighborhood of any vertex of $G$, and $\\chi_t^f(G)$ denotes the minimum number of colors in a $t$-frugal coloring of $G$. Also, $G^2$ denotes the square of $G$, whose vertices are those of $G$ and where two vertices are adjacent whenever they are at distance at most $2$ in $G$, so $\\chi(G^2)$ is the $2$-distance chromatic number of $G$. Which statement holds for every such graph $G$?", "correct_choice": {"label": "A", "text": "$\\chi(G)=\\chi_{\\Delta}^{f}(G)\\le \\chi_{\\Delta-1}^{f}(G)\\le \\cdots \\le \\chi_{2}^{f}(G)\\le \\chi_{1}^{f}(G)=\\chi(G^{2}).$"}, "choices": [{"label": "B", "text": "$\\chi(G)=\\chi_{\\Delta}^{f}(G)\\le \\chi_{\\Delta-1}^{f}(G)\\le \\cdots \\le \\chi_{2}^{f}(G)\\le \\chi_{1}^{f}(G)\\le \\chi(G^{2}).$"}, {"label": "C", "text": "$\\chi(G)\\le \\chi_{\\Delta}^{f}(G)\\le \\chi_{\\Delta-1}^{f}(G)\\le \\cdots \\le \\chi_{2}^{f}(G)\\le \\chi_{1}^{f}(G)=\\chi(G^{2}).$"}, {"label": "D", "text": "$\\chi(G)=\\chi_{\\Delta}^{f}(G)\\ge \\chi_{\\Delta-1}^{f}(G)\\ge \\cdots \\ge \\chi_{2}^{f}(G)\\ge \\chi_{1}^{f}(G)=\\chi(G^{2}).$"}, {"label": "E", "text": "$\\chi(G)=\\chi_{\\Delta}^{f}(G)\\le \\chi_{\\Delta-1}^{f}(G)\\le \\cdots \\le \\chi_{2}^{f}(G)=\\chi_{1}^{f}(G)=\\chi(G^{2}).$"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "endpoint equality at t=1 replaced by mere upper bound", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the sharp equality \\chi(G)=\\chi_\\Delta^f(G) at the left endpoint", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "monotonicity direction in t", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "strictness/equality at the t=1 and t=2 end of the chain", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only definitions of t-frugal coloring, G^2, and the relevant invariants; it does not reveal the claimed monotonicity direction or the endpoint equalities. There is no direct answer leakage."}, "TAS": {"score": 1, "justification": "The item is close to asking for a standard theorem/proposition verbatim: monotonicity of χ_t^f in t together with the endpoint identities χ_Δ^f(G)=χ(G) and χ_1^f(G)=χ(G^2). It is not pure restatement because the choices vary the direction/strength of the claim, but it still mainly tests recall of the exact statement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to see why larger t weakens the constraint, why t=Δ recovers ordinary coloring, and why t=1 corresponds to coloring G^2. However, the task is undermined by the presence of choice C, which is also true as a weaker consequence, so the item does not cleanly force selection of a unique strongest conclusion."}, "DQS": {"score": 1, "justification": "Several distractors are mathematically plausible and target common errors: reversing monotonicity, incorrectly making inequalities strict, and confusing χ(G^2) with χ_2(G). But choice C is a genuinely true statement, not a distractor, which creates ambiguity and substantially weakens distractor quality."}, "total_score": 5, "overall_assessment": "Moderately good mathematical content, but flawed as a single-correct MCQ because it includes another true option (C). The stem avoids leakage, yet the question is close to theorem recall and does not cleanly test uniquely identifiable reasoning."}}
{"id": "2602.03027v2", "paper_link": "http://arxiv.org/abs/2602.03027v2", "theorems_cnt": 1, "theorem": {"env_name": "conjecture", "content": "\\label{conj:main}\nThe constant $8/\\pi^2$ admits the following GCF representation:\n\\begin{equation}\\label{eq:conjecture}\n\\frac{8}{\\pi^2} = 1 - \\cfrac{2 \\cdot 1^4 - 1^3}{7 - \\cfrac{2 \\cdot 2^4 - 2^3}{19 - \\cfrac{2 \\cdot 3^4 - 3^3}{37 - \\dots}}}\n\\end{equation}\nwhere the partial denominators are $b_n = 3n^2+3n+1$ and partial numerators are $a_n = -(2n^4-n^3)$.", "start_pos": 3657, "end_pos": 4050, "label": "conj:main"}, "ref_dict": {"conj:main": "\\begin{conjecture}\\label{conj:main}\nThe constant $8/\\pi^2$ admits the following GCF representation:\n\\begin{equation}\\label{eq:conjecture}\n\\frac{8}{\\pi^2} = 1 - \\cfrac{2 \\cdot 1^4 - 1^3}{7 - \\cfrac{2 \\cdot 2^4 - 2^3}{19 - \\cfrac{2 \\cdot 3^4 - 3^3}{37 - \\dots}}}\n\\end{equation}\nwhere the partial denominators are $b_n = 3n^2+3n+1$ and partial numerators are $a_n = -(2n^4-n^3)$.\n\\end{conjecture}"}, "pre_theorem_intro_text_len": 1017, "pre_theorem_intro_text": "The algorithmic generation of conjectures on fundamental constants, exemplified by the Ramanujan Machine \\cite{Raayoni2021}, has introduced new identities involving generalized continued fractions (GCFs). \nHistorically, these identities trace their lineage back to Srinivasa Ramanujan’s profound constructions in his \\textit{Notebooks}, particularly his series for $1/\\pi$. While Ramanujan’s classical identities typically involve quadratic or low-degree polynomial patterns, the conjectures surfaced by the Ramanujan Machine often exhibit a higher structural complexity. One such conjecture involves the constant $8/\\pi^2$, characterized by a polynomial architecture with quartic partial numerators. \nThis identity poses a significant challenge to classical limits due to its quartic degree, which exceeds the polynomial complexity of most historically known GCFs.\nIn this work, we provide a rigorous proof of this identity using the framework of operator factorization for variable-coefficient difference equations.", "context": "The algorithmic generation of conjectures on fundamental constants, exemplified by the Ramanujan Machine \\cite{Raayoni2021}, has introduced new identities involving generalized continued fractions (GCFs). \nHistorically, these identities trace their lineage back to Srinivasa Ramanujan’s profound constructions in his \\textit{Notebooks}, particularly his series for $1/\\pi$. While Ramanujan’s classical identities typically involve quadratic or low-degree polynomial patterns, the conjectures surfaced by the Ramanujan Machine often exhibit a higher structural complexity. One such conjecture involves the constant $8/\\pi^2$, characterized by a polynomial architecture with quartic partial numerators. \nThis identity poses a significant challenge to classical limits due to its quartic degree, which exceeds the polynomial complexity of most historically known GCFs.\nIn this work, we provide a rigorous proof of this identity using the framework of operator factorization for variable-coefficient difference equations.", "full_context": "The algorithmic generation of conjectures on fundamental constants, exemplified by the Ramanujan Machine \\cite{Raayoni2021}, has introduced new identities involving generalized continued fractions (GCFs). \nHistorically, these identities trace their lineage back to Srinivasa Ramanujan’s profound constructions in his \\textit{Notebooks}, particularly his series for $1/\\pi$. While Ramanujan’s classical identities typically involve quadratic or low-degree polynomial patterns, the conjectures surfaced by the Ramanujan Machine often exhibit a higher structural complexity. One such conjecture involves the constant $8/\\pi^2$, characterized by a polynomial architecture with quartic partial numerators. \nThis identity poses a significant challenge to classical limits due to its quartic degree, which exceeds the polynomial complexity of most historically known GCFs.\nIn this work, we provide a rigorous proof of this identity using the framework of operator factorization for variable-coefficient difference equations.\n\n\\section{Introduction}\nThe algorithmic generation of conjectures on fundamental constants, exemplified by the Ramanujan Machine \\cite{Raayoni2021}, has introduced new identities involving generalized continued fractions (GCFs). \nHistorically, these identities trace their lineage back to Srinivasa Ramanujan’s profound constructions in his \\textit{Notebooks}, particularly his series for $1/\\pi$. While Ramanujan’s classical identities typically involve quadratic or low-degree polynomial patterns, the conjectures surfaced by the Ramanujan Machine often exhibit a higher structural complexity. One such conjecture involves the constant $8/\\pi^2$, characterized by a polynomial architecture with quartic partial numerators. \nThis identity poses a significant challenge to classical limits due to its quartic degree, which exceeds the polynomial complexity of most historically known GCFs.\nIn this work, we provide a rigorous proof of this identity using the framework of operator factorization for variable-coefficient difference equations.\n\nConjecture~\\eqref{conj:main} presents a notable structural complexity where the complexity of the coefficients (quartic in $n$) seemingly contradicts the simplicity of the limit. In this paper, we provide a formal analytic regularization of this conjecture. By constructing a discrete Green's function analogue through operator factorization, we resolve the recurrence into a summation series, thereby proving the identity.\n\n\\section{Preliminaries}\nThe evaluation of GCFs of the form \n\\begin{equation}\nx = b_0 + \\K_{n=1}^{\\infty} \\left( \\frac{a_n}{b_n} \\right) = b_0 + \\cfrac{a_1}{b_1 + \\cfrac{a_2}{b_2 + \\dots}}\n\\end{equation}\nis fundamentally governed by the relationship between its partial coefficients and its convergents $A_n/B_n$.\n\n\\begin{proposition}\\label{prop:factor}\nFor the conjectured coefficients $b_n = 3n(n+1)+1$ and $a_n = -(2n-1)n^3$, the recurrence admits an exact reduction over $\\mathbb{N}^{+}$ with the auxiliary sequences:\n\\begin{equation}\nc_n = n^2, \\quad d_n = n(2n-1).\n\\end{equation}\n\\end{proposition}\n\n\\subsubsection{Series Representation of the Convergent Ratio}\nThe $n$-th convergent of the continued fraction, $x_n = A_n/B_n$, is formulated by the ratio of \\eqref{eq:Bn_sum_final} to \\eqref{eq:An_product_final}. \nThe common product terms cancel exactly, yielding:\n\\begin{equation}\\label{eq:xn_final}\nx_n = \\frac{A_n}{B_n} = \\left( \\sum_{k=0}^{n} u_k \\right)^{-1}, \\quad \\text{where } u_k = \\frac{\\prod_{j=1}^{k} c_j}{\\prod_{j=1}^{k+1} d_j}.\n\\end{equation}\nThis fundamental equivalence maps the convergence of the polynomial continued fraction to the partial sums of an infinite series. The proof of the conjecture~\\eqref{conj:main} is thereby reduced to the asymptotic evaluation of the sum $S = \\sum_{k=0}^{\\infty} u_k$.\n\nThe convergence of the GCF is predicated on the existence of the limit of the convergents $\\{x_n\\}$. According to~\\eqref{eq:xn_final}, the value of the GCF, denoted by $\\mathcal{K}$, is mapped to the reciprocal of an infinite series:\n\\begin{equation}\\label{eq:limit_mapping}\n\\mathcal{K} = \\lim_{n \\to \\infty} \\frac{A_n}{B_n} = \\left( \\sum_{k=0}^{\\infty} t_k \\right)^{-1}, \\quad \\text{where } t_k = \\frac{\\prod_{j=1}^{k} c_j}{\\prod_{j=1}^{k+1} d_j}.\n\\end{equation}\n\n\\begin{theorem}[Pincherle's Theorem~\\cite{Gautschi1967}]\nThe generalized continued fraction $b_0 + \\K_{n=1}^\\infty (a_n / b_n)$ converges to a finite value if and only if the associated second-order linear difference equation \n\\begin{equation}\\label{eq:rec_pincherle}\n y_n = b_n y_{n-1} + a_n y_{n-2}\n\\end{equation}\nadmits a minimal solution $\\{f_n\\}$. A non-trivial solution $\\{f_n\\}$ is minimal if, for any solution $\\{g_n\\}$ linearly independent of $\\{f_n\\}$, the following asymptotic condition is satisfied:\n\\begin{equation}\n \\lim_{n \\to \\infty} \\frac{f_n}{g_n} = 0.\n\\end{equation}\nIf such a solution exists, the GCF converges to $f_0/f_{-1}$.\n\\end{theorem}\n\nCrucially, from the perspective of Pincherle’s Theorem, the fact that $S = \\pi^2/8$ is a \\textbf{finite constant} ensures that $B_n$ grows at an asymptotically faster rate than $A_n$ in the sense of the solution hierarchy. Any solution not proportional to $\\{A_n\\}$ will eventually be dominated by the summation-induced growth. Thus, the existence of this convergent series confirms $\\{A_n\\}$ as the minimal solution, and the GCF converges to the reciprocal of the sum:\n\\begin{equation}\n \\mathcal{K} = S^{-1} = \\left( \\sum_{m=1}^{\\infty} \\frac{2^m}{m^2 \\binom{2m}{m}} \\right)^{-1} = \\frac{8}{\\pi^2}.\n\\end{equation}\nThe proof is complete.\n\\end{proof}\n\n\\begin{conjecture}\\label{conj:main}\nThe constant $8/\\pi^2$ admits the following GCF representation:\n\\begin{equation}\\label{eq:conjecture}\n\\frac{8}{\\pi^2} = 1 - \\cfrac{2 \\cdot 1^4 - 1^3}{7 - \\cfrac{2 \\cdot 2^4 - 2^3}{19 - \\cfrac{2 \\cdot 3^4 - 3^3}{37 - \\dots}}}\n\\end{equation}\nwhere the partial denominators are $b_n = 3n^2+3n+1$ and partial numerators are $a_n = -(2n^4-n^3)$.\n\\end{conjecture}", "post_theorem_intro_text_len": 423, "post_theorem_intro_text": "Conjecture~\\eqref{conj:main} presents a notable structural complexity where the complexity of the coefficients (quartic in $n$) seemingly contradicts the simplicity of the limit. In this paper, we provide a formal analytic regularization of this conjecture. By constructing a discrete Green's function analogue through operator factorization, we resolve the recurrence into a summation series, thereby proving the identity.", "sketch": "To prove Conjecture~\\eqref{conj:main}, the paper \"provide[s] a formal analytic regularization\" of the continued fraction. The argument proceeds by \"constructing a discrete Green's function analogue through operator factorization,\" and then using this construction to \"resolve the recurrence into a summation series,\" which \"thereby prov[es] the identity.\"", "expanded_sketch": "To prove the main conjecture, the paper \"provide[s] a formal analytic regularization\" of the continued fraction. The argument proceeds by \"constructing a discrete Green's function analogue through operator factorization,\" and then using this construction to \"resolve the recurrence into a summation series,\" which \"thereby prov[es] the identity.\"", "expanded_theorem": "\\label{conj:main}\nThe constant $8/\\pi^2$ admits the following GCF representation:\n\\begin{equation}\\label{eq:conjecture}\n\\frac{8}{\\pi^2} = 1 - \\cfrac{2 \\cdot 1^4 - 1^3}{7 - \\cfrac{2 \\cdot 2^4 - 2^3}{19 - \\cfrac{2 \\cdot 3^4 - 3^3}{37 - \\dots}}}\n\\end{equation}\nwhere the partial denominators are $b_n = 3n^2+3n+1$ and partial numerators are $a_n = -(2n^4-n^3)$.,", "theorem_type": ["Existence", "Algorithmic or Constructive"], "mcq": {"question": "Let\n\\[\n1+\\K_{n=1}^{\\infty}\\left(\\frac{a_n}{b_n}\\right)\n\\]\ndenote a generalized continued fraction, where\n\\[\na_n=-(2n^4-n^3)=-(2n-1)n^3,\\qquad b_n=3n^2+3n+1=3n(n+1)+1\\quad (n\\ge 1).\n\\]\nEquivalently, this is\n\\[\n1- \\cfrac{2\\cdot 1^4-1^3}{7-\\cfrac{2\\cdot 2^4-2^3}{19-\\cfrac{2\\cdot 3^4-3^3}{37-\\ddots}}}.\n\\]\nWhich explicit value does this generalized continued fraction represent?", "correct_choice": {"label": "A", "text": "It equals \\(\\dfrac{8}{\\pi^2}\\); that is,\n\\[\n\\frac{8}{\\pi^2}=1- \\cfrac{2\\cdot 1^4-1^3}{7-\\cfrac{2\\cdot 2^4-2^3}{19-\\cfrac{2\\cdot 3^4-3^3}{37-\\ddots}}}.\n\\]"}, "choices": [{"label": "B", "text": "It equals \\(\\dfrac{4}{\\pi^2}\\); that is,\n\\[\n\\frac{4}{\\pi^2}=1- \\cfrac{2\\cdot 1^4-1^3}{7-\\cfrac{2\\cdot 2^4-2^3}{19-\\cfrac{2\\cdot 3^4-3^3}{37-\\ddots}}}.\n\\]"}, {"label": "C", "text": "It equals the reciprocal of a finite positive constant; equivalently, the generalized continued fraction converges to some real number \\(L>0\\).\n\\[\n1- \\cfrac{2\\cdot 1^4-1^3}{7-\\cfrac{2\\cdot 2^4-2^3}{19-\\cfrac{2\\cdot 3^4-3^3}{37-\\ddots}}}=L.\n\\]"}, {"label": "D", "text": "It equals \\(\\dfrac{8}{\\pi}\\); that is,\n\\[\n\\frac{8}{\\pi}=1- \\cfrac{2\\cdot 1^4-1^3}{7-\\cfrac{2\\cdot 2^4-2^3}{19-\\cfrac{2\\cdot 3^4-3^3}{37-\\ddots}}}.\n\\]"}, {"label": "E", "text": "It equals \\(\\left(\\sum_{m=1}^{\\infty}\\dfrac{2^m}{m^2\\binom{2m}{m}}\\right)^{-1}\\), and this reciprocal sum is \\(\\dfrac{6}{\\pi^2}\\); that is,\n\\[\n1- \\cfrac{2\\cdot 1^4-1^3}{7-\\cfrac{2\\cdot 2^4-2^3}{19-\\cfrac{2\\cdot 3^4-3^3}{37-\\ddots}}}=\\left(\\sum_{m=1}^{\\infty}\\frac{2^m}{m^2\\binom{2m}{m}}\\right)^{-1}=\\frac{6}{\\pi^2}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "final_constant_after_series_regularization", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "explicit_identification_with_8_over_pi_squared", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "square_on_pi_from_summation_evaluation", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "exact_evaluation_of_resolved_series", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only the continued fraction data and asks for its value; it does not state or strongly hint at 8/pi^2. There is no explicit answer leakage."}, "TAS": {"score": 2, "justification": "The item is not a direct restatement of a conclusion already embedded in the stem. It asks the solver to identify an explicit evaluation among competing alternatives."}, "GPS": {"score": 2, "justification": "Determining the exact constant represented by this nontrivial generalized continued fraction requires substantial mathematical recognition or derivation, so the correct answer is not obvious from surface features alone."}, "DQS": {"score": 1, "justification": "Several distractors are plausible mathematical confusions (4/pi^2, 8/pi, and a related-looking reciprocal sum), but choice C is a weaker true statement rather than a genuinely false distractor, which weakens single-answer integrity."}, "total_score": 7, "overall_assessment": "A strong, non-leaky, reasoning-heavy MCQ, but its distractor set is weakened by the inclusion of a true-but-weaker option."}}
{"id": "2602.03179v1", "paper_link": "http://arxiv.org/abs/2602.03179v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm-finite.interp}\n    The following three statements are equivalent.\n    \\begin{enumerate}\n    \\item There exists a valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\dots,r$.\n    \\item There exists a quasi-monomial valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\dots,r$.\n    \\item The equality $\\vbI(\\a_1\\cdots\\a_r)=\\sum_{j=1}^r b_j$ holds.\n    \\end{enumerate}", "start_pos": 5782, "end_pos": 6192, "label": "thm-finite.interp"}, "ref_dict": {"thm-infinite.interp": "\\begin{theorem}\\label{thm-infinite.interp}\n    Additionally assume $R$ is local with the unique maximal ideal $\\m$. Let $(\\a_j)_{j\\in\\bbZ_+}$ be a countable sequence of nonzero ideals in $R$ such that $\\sqrt{\\sum_{j=1}^{\\infty} \\a_j}=\\m$, and let $(b_j)_{j\\in\\bbZ_+}$ be a sequence of positive real numbers. Then the following statements are equivalent:\n    \\begin{enumerate}\n        \\item There exists a valuation $v$ on $R$ centered at $\\m$ with $A(v)<\\infty$ such that $v(\\a_j)=b_j$ for all $j\\ge 1$;\n        \\item For every $r\\ge 1$,\n    \\[\\vb_{I_{\\blt}^{(r)}}\\big(\\a_1\\cdots\\a_r\\big)=\\sum_{j=1}^r b_j,\\]\n    and it holds that\n    \\[\\sup_{r\\ge 1}\\sup_{k\\in\\bbN}\\Big(\\lct^{\\a_1^k\\cdots\\a_r^k}\\big( I^{(r)}_{\\blt}\\big)-k\\sum_{j=1}^r b_j\\Big)<\\infty.\\]\n    \\end{enumerate}\n    Here $I^{(r)}_{\\blt}=\\sum_{j=1}^{r}\\big(\\frac{1}{b_j}\\cdot \\a_j\\big)_{\\blt}$.\n\\end{theorem}", "subsec-gr.fil": "\\begin{enumerate}\n        \\item There exists a valuation $v$ on $R$ centered at $\\m$ with $A(v)<\\infty$ such that $v(\\a_j)=b_j$ for all $j\\ge 1$;\n        \\item For every $r\\ge 1$,\n    \\[\\vb_{I_{\\blt}^{(r)}}\\big(\\a_1\\cdots\\a_r\\big)=\\sum_{j=1}^r b_j,\\]\n    and it holds that\n    \\[\\sup_{r\\ge 1}\\sup_{k\\in\\bbN}\\Big(\\lct^{\\a_1^k\\cdots\\a_r^k}\\big( I^{(r)}_{\\blt}\\big)-k\\sum_{j=1}^r b_j\\Big)<\\infty.\\]\n    \\end{enumerate}\n    Here $I^{(r)}_{\\blt}=\\sum_{j=1}^{r}\\big(\\frac{1}{b_j}\\cdot \\a_j\\big)_{\\blt}$.\n\\end{theorem}\n\n\\vspace{.1in} {\\bf Organization}. This paper is organized as follows. In \\Cref{sec-r.o}, we recall the notion of asymptotic Samuel function and quickly prove the implication (1) $\\Rightarrow$ (3) in \\Cref{thm-finite.interp}. In \\Cref{sec-f.v.i}, we prove \\Cref{thm-finite.interp} by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\\QM(Y,D)$ of quasi-monomial valuations. In \\Cref{sec-i.v.i}, we prove \\Cref{thm-infinite.interp} using valuation approximation, a strategy that more closely parallels the approach in \\cite{BGMY25}. Finally, we demonstrate how \\Cref{thm-finite.interp} recovers a result in \\cite{BGMY25} (which deals with the complex analytic case) and yields an analytic characterization, derived from \\Cref{thm-infinite.interp}, for the existence of infinite valuative interpolation.\n\n\\vspace{.1in} {\\em Acknowledgements}. The first-named author completed this work during a visit to the School of Mathematical Sciences at Peking University and is grateful for its hospitality and support. The second-named author was supported by National Key R\\&D Program of China 2021YFA1003100 and NSFC-12425101. The third-named author was supported by NSFC-12401099 and the Talent Fund of Beijing Jiaotong University 2024-004. The fourth-named author was supported by NSFC-12501106.\n\n\\section{Valuation and asymptotic Samuel function}\\label{sec-r.o}\n\nIn this section, $R$ is a noetherian integral domain.\n\n\\subsection{Valuation}\\label{subsec-val}\n\nRecall that a function $v\\colon R^*=R\\setminus\\{0\\} \\to \\bbR_{\\ge 0}$ is a (real) \\emph{valuation} if\n\\[v(xy)=v(x)+v(y) \\quad \\& \\quad v(x+y)\\ge \\min\\{v(x),v(y)\\},\\]\nfor all $x,y\\in R^*$. Set $v(0)=\\infty$. A valuation $v$ on $R$ can be extended uniquely to a valuation $v\\colon K\\to \\bbR\\cup\\{\\infty\\}$ on the field of fractions $K$ of $R$ by setting $v(x/y)=v(x)-v(y)$. Say $v\\le w$ for two valuations $v,w$ on $R$ if $v(x)\\le w(x)$ for all $x\\in R^*$. For any nonzero ideal $\\a$ of $R$, define $v(\\a)=\\inf_{x\\in \\a}v(x)$. Then it is easy to check: for nonzero ideals $\\a,\\b$ in $R$,\n\\[v(\\a\\b)=v(\\a)+v(\\b) \\quad \\& \\quad v(\\a+\\b)= \\min\\{v(\\a),v(\\b)\\}.\\]\nIn particular, we can see if the ideals $\\a_1,\\ldots,\\a_r$ in $R$ satisfies $\\sum_{j=1}^r\\a_j=R$, then\n\\[0=v(R)=v\\big(\\sum \\a_j\\big)=\\min v(\\a_j),\\]\nwhich implies $v(\\a_j)=0$ for some $j\\in\\{1,\\ldots,r\\}$.\n\n\\subsection{Graded sequence and filtration of ideals}\\label{subsec-gr.fil}\n\nA \\emph{graded sequence} of ideals $\\a_{\\blt}=(\\a_m)_{m\\in\\bbZ_{>0}}$ is a sequence of ideals in $R$ that satisfies $\\a_p\\cdot \\a_p\\subseteq \\a_{p+q}$ for all $p,q\\ge 1$. Put $\\a_0=R$. We always assume $\\a_m\\neq (0)$ for some $m\\ge 1$. For example, $v$ is a nontrivial valuation on $R$, then $\\a_{\\blt}^v$ is a graded sequence of ideals in $R$ defined by $\\a_{m}^v=\\{x\\in R \\mid v(x)\\ge m\\}$. \n\nNow let $v$ be a valuation on $R$, and $\\a_{\\blt}$ a graded sequence of ideals in $R$. Then $v(\\a_{p+q})\\le v(\\a_p)+v(\\a_q)$ for all $p,q\\ge 1$, which implies the limit\n\\[v(\\a_{\\blt})\\coloneqq \\lim_{m\\to \\infty, \\, \\a_m\\neq (0)}\\frac{v(\\a_m)}{m}=\\inf_{m\\ge 1}\\frac{v(\\a_m)}{m}\\]\nexists by Fekete's lemma. In particular, $v(\\a_{\\blt}^v)=1$ for any nontrivial valuation $v$ on $R$. A graded sequence $\\a_{\\blt}=\\{\\a_m\\}_{m\\in\\bbN}$ of ideals in $R$ is called a \\emph{filtration} if additionally it satisfies $\\a_m\\supseteq \\a_{m+1}$ for every $m\\in\\bbN$.\n\n\\subsection{Asymptotic Samuel function}\n\nThe classical asymptotic Samuel function (\\cite{Sa52}) has been generalized to the filtration case in \\cite{CP24}. Let us recall the constructions and some basic properties. Let $J_{\\blt}$ be a filtration of ideals in $R$. For every ideal $\\a$ in $R$, the \\emph{order} of $\\a$ with respect to $J_{\\blt}$ is defined by\n\\[\\vJ\\big(\\a\\big)\\coloneqq \\sup\\,\\{m\\in\\bbN \\mid \\a\\subseteq J_m\\}\\in\\bbN\\cup\\{\\infty\\}.\\]\nThen we can see $\\vJ(\\a^{p+q})\\ge \\vJ(\\a^p)+\\vJ(\\a^q)$ for every $p,q\\in\\bbN$. Therefore, it follows that\n\\[\\vb(\\a;J_{\\blt})\\coloneqq \\lim_{m\\to\\infty}\\frac{\\vJ(\\a^m)}{m}=\\sup_{m\\ge 1}\\frac{\\vJ(\\a^m)}{m}\\in \\bbR_{\\ge 0}\\cup \\{\\infty\\}\\]\nis well-defined by Fekete's lemma, called the \\emph{asymptotic Samuel function} of $J_{\\blt}$ (cf. \\cite[Theorem 3.4]{CP24}). We also denote $\\vb(\\a;J_{\\blt})\\coloneqq \\vbJ(\\a)$ in the present paper.\n\n\\begin{example}\\label{ex-asf}\n    If $J_{\\blt}=\\{J^m\\}_{m\\ge 0}$ for a nonzero ideal $J$ in $R$, then $\\vb(\\a;J_{\\blt})=\\ol{\\nu}_{J}(\\a)$, which is the classical asymptotic Samuel function (cf. \\cite{Sa52,HS06,LJT09}).\n\\end{example}\n\n\\begin{example}\n    If we take $J_{\\blt}=\\a_{\\blt}^v$ the filtration of valuation ideals associated to some valuation $v$ on $R$, then clearly $\\vb(\\a;J_{\\blt})=v(\\a)$.\n\\end{example}\n\nThe following lemma shows some basis properties of the asymptotic Samuel function, which are actually known in \\cite{CP24}. The proof below follows from the strategy of \\cite[Theorem 2]{Sa52}.\n\n\\begin{lemma}[{see \\cite[Proposition 3.11]{CP24}}]\\label{lem-mu.subadditivity}\n    Let $J_{\\blt}$ be a filtration of ideals in $R$. Then\n    \\begin{enumerate}\n    \\item $\\vb(\\a;J_{\\blt})\\ge \\vb(\\b;J_{\\blt})$, for ideals $\\a\\subseteq\\b$ in $R$;\n    \\item $\\vb(\\a^k;J_{\\blt})=k\\cdot\\vb(\\a;J_{\\blt})$, where $\\a$ is an ideal in $R$ and $k\\in\\bbZ_+$;\n    \\item $\\vb(\\a+\\b;J_{\\blt})= \\min\\{\\vb(\\a;J_{\\blt}), \\vb(\\b;J_{\\blt})\\}$, where $\\a,\\b$ are ideals in $R$;\n    \\item $\\vb(\\a\\b;J_{\\blt})\\ge \\vb(\\a;J_{\\blt}) +\\vb(\\b;J_{\\blt})$, where $\\a,\\b$ are ideals in $R$.\n    \\end{enumerate}", "subsec-val": "\\begin{enumerate}\n        \\item There exists a valuation $v$ on $R$ centered at $\\m$ with $A(v)<\\infty$ such that $v(\\a_j)=b_j$ for all $j\\ge 1$;\n        \\item For every $r\\ge 1$,\n    \\[\\vb_{I_{\\blt}^{(r)}}\\big(\\a_1\\cdots\\a_r\\big)=\\sum_{j=1}^r b_j,\\]\n    and it holds that\n    \\[\\sup_{r\\ge 1}\\sup_{k\\in\\bbN}\\Big(\\lct^{\\a_1^k\\cdots\\a_r^k}\\big( I^{(r)}_{\\blt}\\big)-k\\sum_{j=1}^r b_j\\Big)<\\infty.\\]\n    \\end{enumerate}\n    Here $I^{(r)}_{\\blt}=\\sum_{j=1}^{r}\\big(\\frac{1}{b_j}\\cdot \\a_j\\big)_{\\blt}$.\n\\end{theorem}\n\n\\vspace{.1in} {\\bf Organization}. This paper is organized as follows. In \\Cref{sec-r.o}, we recall the notion of asymptotic Samuel function and quickly prove the implication (1) $\\Rightarrow$ (3) in \\Cref{thm-finite.interp}. In \\Cref{sec-f.v.i}, we prove \\Cref{thm-finite.interp} by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\\QM(Y,D)$ of quasi-monomial valuations. In \\Cref{sec-i.v.i}, we prove \\Cref{thm-infinite.interp} using valuation approximation, a strategy that more closely parallels the approach in \\cite{BGMY25}. Finally, we demonstrate how \\Cref{thm-finite.interp} recovers a result in \\cite{BGMY25} (which deals with the complex analytic case) and yields an analytic characterization, derived from \\Cref{thm-infinite.interp}, for the existence of infinite valuative interpolation.\n\n\\vspace{.1in} {\\em Acknowledgements}. The first-named author completed this work during a visit to the School of Mathematical Sciences at Peking University and is grateful for its hospitality and support. The second-named author was supported by National Key R\\&D Program of China 2021YFA1003100 and NSFC-12425101. The third-named author was supported by NSFC-12401099 and the Talent Fund of Beijing Jiaotong University 2024-004. The fourth-named author was supported by NSFC-12501106.\n\n\\section{Valuation and asymptotic Samuel function}\\label{sec-r.o}\n\nIn this section, $R$ is a noetherian integral domain.\n\n\\subsection{Valuation}\\label{subsec-val}\n\nRecall that a function $v\\colon R^*=R\\setminus\\{0\\} \\to \\bbR_{\\ge 0}$ is a (real) \\emph{valuation} if\n\\[v(xy)=v(x)+v(y) \\quad \\& \\quad v(x+y)\\ge \\min\\{v(x),v(y)\\},\\]\nfor all $x,y\\in R^*$. Set $v(0)=\\infty$. A valuation $v$ on $R$ can be extended uniquely to a valuation $v\\colon K\\to \\bbR\\cup\\{\\infty\\}$ on the field of fractions $K$ of $R$ by setting $v(x/y)=v(x)-v(y)$. Say $v\\le w$ for two valuations $v,w$ on $R$ if $v(x)\\le w(x)$ for all $x\\in R^*$. For any nonzero ideal $\\a$ of $R$, define $v(\\a)=\\inf_{x\\in \\a}v(x)$. Then it is easy to check: for nonzero ideals $\\a,\\b$ in $R$,\n\\[v(\\a\\b)=v(\\a)+v(\\b) \\quad \\& \\quad v(\\a+\\b)= \\min\\{v(\\a),v(\\b)\\}.\\]\nIn particular, we can see if the ideals $\\a_1,\\ldots,\\a_r$ in $R$ satisfies $\\sum_{j=1}^r\\a_j=R$, then\n\\[0=v(R)=v\\big(\\sum \\a_j\\big)=\\min v(\\a_j),\\]\nwhich implies $v(\\a_j)=0$ for some $j\\in\\{1,\\ldots,r\\}$.\n\n\\subsection{Graded sequence and filtration of ideals}\\label{subsec-gr.fil}\n\nA \\emph{graded sequence} of ideals $\\a_{\\blt}=(\\a_m)_{m\\in\\bbZ_{>0}}$ is a sequence of ideals in $R$ that satisfies $\\a_p\\cdot \\a_p\\subseteq \\a_{p+q}$ for all $p,q\\ge 1$. Put $\\a_0=R$. We always assume $\\a_m\\neq (0)$ for some $m\\ge 1$. For example, $v$ is a nontrivial valuation on $R$, then $\\a_{\\blt}^v$ is a graded sequence of ideals in $R$ defined by $\\a_{m}^v=\\{x\\in R \\mid v(x)\\ge m\\}$. \n\nNow let $v$ be a valuation on $R$, and $\\a_{\\blt}$ a graded sequence of ideals in $R$. Then $v(\\a_{p+q})\\le v(\\a_p)+v(\\a_q)$ for all $p,q\\ge 1$, which implies the limit\n\\[v(\\a_{\\blt})\\coloneqq \\lim_{m\\to \\infty, \\, \\a_m\\neq (0)}\\frac{v(\\a_m)}{m}=\\inf_{m\\ge 1}\\frac{v(\\a_m)}{m}\\]\nexists by Fekete's lemma. In particular, $v(\\a_{\\blt}^v)=1$ for any nontrivial valuation $v$ on $R$. A graded sequence $\\a_{\\blt}=\\{\\a_m\\}_{m\\in\\bbN}$ of ideals in $R$ is called a \\emph{filtration} if additionally it satisfies $\\a_m\\supseteq \\a_{m+1}$ for every $m\\in\\bbN$.\n\n\\subsection{Asymptotic Samuel function}\n\nThe classical asymptotic Samuel function (\\cite{Sa52}) has been generalized to the filtration case in \\cite{CP24}. Let us recall the constructions and some basic properties. Let $J_{\\blt}$ be a filtration of ideals in $R$. For every ideal $\\a$ in $R$, the \\emph{order} of $\\a$ with respect to $J_{\\blt}$ is defined by\n\\[\\vJ\\big(\\a\\big)\\coloneqq \\sup\\,\\{m\\in\\bbN \\mid \\a\\subseteq J_m\\}\\in\\bbN\\cup\\{\\infty\\}.\\]\nThen we can see $\\vJ(\\a^{p+q})\\ge \\vJ(\\a^p)+\\vJ(\\a^q)$ for every $p,q\\in\\bbN$. Therefore, it follows that\n\\[\\vb(\\a;J_{\\blt})\\coloneqq \\lim_{m\\to\\infty}\\frac{\\vJ(\\a^m)}{m}=\\sup_{m\\ge 1}\\frac{\\vJ(\\a^m)}{m}\\in \\bbR_{\\ge 0}\\cup \\{\\infty\\}\\]\nis well-defined by Fekete's lemma, called the \\emph{asymptotic Samuel function} of $J_{\\blt}$ (cf. \\cite[Theorem 3.4]{CP24}). We also denote $\\vb(\\a;J_{\\blt})\\coloneqq \\vbJ(\\a)$ in the present paper.\n\n\\begin{example}\\label{ex-asf}\n    If $J_{\\blt}=\\{J^m\\}_{m\\ge 0}$ for a nonzero ideal $J$ in $R$, then $\\vb(\\a;J_{\\blt})=\\ol{\\nu}_{J}(\\a)$, which is the classical asymptotic Samuel function (cf. \\cite{Sa52,HS06,LJT09}).\n\\end{example}\n\n\\begin{example}\n    If we take $J_{\\blt}=\\a_{\\blt}^v$ the filtration of valuation ideals associated to some valuation $v$ on $R$, then clearly $\\vb(\\a;J_{\\blt})=v(\\a)$.\n\\end{example}\n\nThe following lemma shows some basis properties of the asymptotic Samuel function, which are actually known in \\cite{CP24}. The proof below follows from the strategy of \\cite[Theorem 2]{Sa52}.\n\n\\begin{lemma}[{see \\cite[Proposition 3.11]{CP24}}]\\label{lem-mu.subadditivity}\n    Let $J_{\\blt}$ be a filtration of ideals in $R$. Then\n    \\begin{enumerate}\n    \\item $\\vb(\\a;J_{\\blt})\\ge \\vb(\\b;J_{\\blt})$, for ideals $\\a\\subseteq\\b$ in $R$;\n    \\item $\\vb(\\a^k;J_{\\blt})=k\\cdot\\vb(\\a;J_{\\blt})$, where $\\a$ is an ideal in $R$ and $k\\in\\bbZ_+$;\n    \\item $\\vb(\\a+\\b;J_{\\blt})= \\min\\{\\vb(\\a;J_{\\blt}), \\vb(\\b;J_{\\blt})\\}$, where $\\a,\\b$ are ideals in $R$;\n    \\item $\\vb(\\a\\b;J_{\\blt})\\ge \\vb(\\a;J_{\\blt}) +\\vb(\\b;J_{\\blt})$, where $\\a,\\b$ are ideals in $R$.\n    \\end{enumerate}", "thm-finite.interp": "\\begin{theorem}\\label{thm-finite.interp}\n    The following three statements are equivalent.\n    \\begin{enumerate}\n    \\item There exists a valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\dots,r$.\n    \\item There exists a quasi-monomial valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\dots,r$.\n    \\item The equality $\\vbI(\\a_1\\cdots\\a_r)=\\sum_{j=1}^r b_j$ holds.\n    \\end{enumerate}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2845, "pre_theorem_intro_text": "Let $R$ be an excellent regular domain of equicharacteristic $0$ and of dimension $n$.\n\nLet $\\a_1,\\ldots,\\a_r$ be $r$ nonzero ideals of $R$, and let $b_1,\\ldots,b_r\\in\\mathbb{R}_{\\ge 0}$. We consider the following valuative interpolation problem which was considered by the authors in the complex analytic setting (e.g. \\cite[Theorem 1.2]{BGMY25} deals with the case where $R=\\O_{\\mathbb{C}^n,o}$ the ring of holomorphic germs):\n\n\\begin{question}\nCan one give a necessary and sufficient condition for the existence of a valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\ldots, r$?\n\\end{question}\n\nWe introduce some notations for explaining our results in the present paper. Suppose $\\mathfrak{a}$ is an ideal on $X$, and $\\lambda\\in\\bbR_{\\ge 0}\\cup\\{\\infty\\}$. Denote by $(\\lambda\\cdot\\mathfrak{a})_{\\bullet}=\\big((\\lambda\\cdot\\mathfrak{a})_m\\big)_{m\\in\\mathbb{N}}$ the \\emph{filtration} (cf. \\Cref{subsec-gr.fil}) of ideals such that each term:\n\\begin{equation*}\n    (\\lambda\\cdot\\mathfrak{a})_m=\n    \\begin{cases}\n        R & \\lambda=0 \\\\\n        \\mathfrak{a}^{\\lceil \\lambda m\\rceil} & 0<\\lambda<\\infty \\\\\n        0 & \\lambda=\\infty \\\\\n    \\end{cases}\n    \\quad \\quad m\\in\\mathbb{N}.\n\\end{equation*}\n\nIf $(\\b_1)_{\\bullet},\\ldots,(\\b_s)_{\\bullet}$ are filtrations of ideals in $R$, then the summation $\\mathfrak{c}_{\\bullet}\\coloneqq \\sum(\\b_i)_{\\bullet}$ of these filtrations is a filtration defined as:\n\\begin{equation*}\n        \\mathfrak{c}_{m}=\\sum_{\\substack{m_1+\\cdots+m_s=m \\\\ m_i\\in\\mathbb{N}}}(\\b_1)_{m_1}\\cdots(\\b_s)_{m_s}, \\quad m\\in\\mathbb{N}.\n\\end{equation*}\n\nNow we return to the valuative interpolation problem. For the ideals $\\a_1,\\ldots,\\a_r$ and non-negative real numbers $b_1,\\ldots,b_r$, if all of $b_j$ are zero, then we can directly choose the trivial valuation on $R$, so we only consider the case where at least one of them is nonzero. Hence, may assume $\\sum_{b_j>0}\\a_j\\neq R$, otherwise none of valuations $v$ on $R$ can satisfy $v(\\a_j)=b_j$ for all $j$ (cf. \\Cref{subsec-val}).\n\nConsider the filtration\n\\[I_{\\bullet}\\coloneqq \\Big(\\frac{1}{b_1}\\cdot \\a_{1}\\Big)_{\\bullet}+\\cdots+\\Big(\\frac{1}{b_r}\\cdot \\a_{r}\\Big)_{\\bullet}.\\]\nFollowing Cutkosky and Peaharaj \\cite[Definition 3.5]{CP24}, the \\emph{asymptotic Samuel function} associated with this filtration is defined as follows. For any ideal $\\mathfrak{a}$ of $R$, let\n\\[\\vbI(\\mathfrak{a})\\coloneqq \\lim_{k\\to\\infty}\\frac{\\vI\\big(\\mathfrak{a}^k\\big)}{k}\\in \\bbR_{\\ge 0}\\cup\\{\\infty\\},\\]\nwhere $\\vI\\big(\\mathfrak{a}\\big)\\coloneqq \\sup\\,\\{m\\in\\mathbb{N} \\mid \\mathfrak{a}\\subseteq I_m\\}$ denotes the \\emph{order function} of $I_{\\bullet}$. This construction generalizes the classical asymptotic Samuel function \\cite{Sa52} and serves as an algebraic analogue of the relative type introduced in complex analytic settings \\cite{Ras06} (cf. \\Cref{sec-c.a.c}).", "context": "Let $\\a_1,\\ldots,\\a_r$ be $r$ nonzero ideals of $R$, and let $b_1,\\ldots,b_r\\in\\mathbb{R}_{\\ge 0}$. We consider the following valuative interpolation problem which was considered by the authors in the complex analytic setting (e.g. \\cite[Theorem 1.2]{BGMY25} deals with the case where $R=\\O_{\\mathbb{C}^n,o}$ the ring of holomorphic germs):\n\n\\begin{question}\nCan one give a necessary and sufficient condition for the existence of a valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\ldots, r$?\n\\end{question}\n\nWe introduce some notations for explaining our results in the present paper. Suppose $\\mathfrak{a}$ is an ideal on $X$, and $\\lambda\\in\\bbR_{\\ge 0}\\cup\\{\\infty\\}$. Denote by $(\\lambda\\cdot\\mathfrak{a})_{\\bullet}=\\big((\\lambda\\cdot\\mathfrak{a})_m\\big)_{m\\in\\mathbb{N}}$ the \\emph{filtration} (cf. \\Cref{subsec-gr.fil}) of ideals such that each term:\n\\begin{equation*}\n (\\lambda\\cdot\\mathfrak{a})_m=\n \\begin{cases}\n R & \\lambda=0 \\\\\n \\mathfrak{a}^{\\lceil \\lambda m\\rceil} & 0<\\lambda<\\infty \\\\\n 0 & \\lambda=\\infty \\\\\n \\end{cases}\n \\quad \\quad m\\in\\mathbb{N}.\n\\end{equation*}\n\nIf $(\\b_1)_{\\bullet},\\ldots,(\\b_s)_{\\bullet}$ are filtrations of ideals in $R$, then the summation $\\mathfrak{c}_{\\bullet}\\coloneqq \\sum(\\b_i)_{\\bullet}$ of these filtrations is a filtration defined as:\n\\begin{equation*}\n \\mathfrak{c}_{m}=\\sum_{\\substack{m_1+\\cdots+m_s=m \\\\ m_i\\in\\mathbb{N}}}(\\b_1)_{m_1}\\cdots(\\b_s)_{m_s}, \\quad m\\in\\mathbb{N}.\n\\end{equation*}\n\nNow we return to the valuative interpolation problem. For the ideals $\\a_1,\\ldots,\\a_r$ and non-negative real numbers $b_1,\\ldots,b_r$, if all of $b_j$ are zero, then we can directly choose the trivial valuation on $R$, so we only consider the case where at least one of them is nonzero. Hence, may assume $\\sum_{b_j>0}\\a_j\\neq R$, otherwise none of valuations $v$ on $R$ can satisfy $v(\\a_j)=b_j$ for all $j$ (cf. \\Cref{subsec-val}).\n\nConsider the filtration\n\\[I_{\\bullet}\\coloneqq \\Big(\\frac{1}{b_1}\\cdot \\a_{1}\\Big)_{\\bullet}+\\cdots+\\Big(\\frac{1}{b_r}\\cdot \\a_{r}\\Big)_{\\bullet}.\\]\nFollowing Cutkosky and Peaharaj \\cite[Definition 3.5]{CP24}, the \\emph{asymptotic Samuel function} associated with this filtration is defined as follows. For any ideal $\\mathfrak{a}$ of $R$, let\n\\[\\vbI(\\mathfrak{a})\\coloneqq \\lim_{k\\to\\infty}\\frac{\\vI\\big(\\mathfrak{a}^k\\big)}{k}\\in \\bbR_{\\ge 0}\\cup\\{\\infty\\},\\]\nwhere $\\vI\\big(\\mathfrak{a}\\big)\\coloneqq \\sup\\,\\{m\\in\\mathbb{N} \\mid \\mathfrak{a}\\subseteq I_m\\}$ denotes the \\emph{order function} of $I_{\\bullet}$. This construction generalizes the classical asymptotic Samuel function \\cite{Sa52} and serves as an algebraic analogue of the relative type introduced in complex analytic settings \\cite{Ras06} (cf. \\Cref{sec-c.a.c}).", "full_context": "Let $\\a_1,\\ldots,\\a_r$ be $r$ nonzero ideals of $R$, and let $b_1,\\ldots,b_r\\in\\mathbb{R}_{\\ge 0}$. We consider the following valuative interpolation problem which was considered by the authors in the complex analytic setting (e.g. \\cite[Theorem 1.2]{BGMY25} deals with the case where $R=\\O_{\\mathbb{C}^n,o}$ the ring of holomorphic germs):\n\n\\begin{question}\nCan one give a necessary and sufficient condition for the existence of a valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\ldots, r$?\n\\end{question}\n\nWe introduce some notations for explaining our results in the present paper. Suppose $\\mathfrak{a}$ is an ideal on $X$, and $\\lambda\\in\\bbR_{\\ge 0}\\cup\\{\\infty\\}$. Denote by $(\\lambda\\cdot\\mathfrak{a})_{\\bullet}=\\big((\\lambda\\cdot\\mathfrak{a})_m\\big)_{m\\in\\mathbb{N}}$ the \\emph{filtration} (cf. \\Cref{subsec-gr.fil}) of ideals such that each term:\n\\begin{equation*}\n (\\lambda\\cdot\\mathfrak{a})_m=\n \\begin{cases}\n R & \\lambda=0 \\\\\n \\mathfrak{a}^{\\lceil \\lambda m\\rceil} & 0<\\lambda<\\infty \\\\\n 0 & \\lambda=\\infty \\\\\n \\end{cases}\n \\quad \\quad m\\in\\mathbb{N}.\n\\end{equation*}\n\nIf $(\\b_1)_{\\bullet},\\ldots,(\\b_s)_{\\bullet}$ are filtrations of ideals in $R$, then the summation $\\mathfrak{c}_{\\bullet}\\coloneqq \\sum(\\b_i)_{\\bullet}$ of these filtrations is a filtration defined as:\n\\begin{equation*}\n \\mathfrak{c}_{m}=\\sum_{\\substack{m_1+\\cdots+m_s=m \\\\ m_i\\in\\mathbb{N}}}(\\b_1)_{m_1}\\cdots(\\b_s)_{m_s}, \\quad m\\in\\mathbb{N}.\n\\end{equation*}\n\nNow we return to the valuative interpolation problem. For the ideals $\\a_1,\\ldots,\\a_r$ and non-negative real numbers $b_1,\\ldots,b_r$, if all of $b_j$ are zero, then we can directly choose the trivial valuation on $R$, so we only consider the case where at least one of them is nonzero. Hence, may assume $\\sum_{b_j>0}\\a_j\\neq R$, otherwise none of valuations $v$ on $R$ can satisfy $v(\\a_j)=b_j$ for all $j$ (cf. \\Cref{subsec-val}).\n\nConsider the filtration\n\\[I_{\\bullet}\\coloneqq \\Big(\\frac{1}{b_1}\\cdot \\a_{1}\\Big)_{\\bullet}+\\cdots+\\Big(\\frac{1}{b_r}\\cdot \\a_{r}\\Big)_{\\bullet}.\\]\nFollowing Cutkosky and Peaharaj \\cite[Definition 3.5]{CP24}, the \\emph{asymptotic Samuel function} associated with this filtration is defined as follows. For any ideal $\\mathfrak{a}$ of $R$, let\n\\[\\vbI(\\mathfrak{a})\\coloneqq \\lim_{k\\to\\infty}\\frac{\\vI\\big(\\mathfrak{a}^k\\big)}{k}\\in \\bbR_{\\ge 0}\\cup\\{\\infty\\},\\]\nwhere $\\vI\\big(\\mathfrak{a}\\big)\\coloneqq \\sup\\,\\{m\\in\\mathbb{N} \\mid \\mathfrak{a}\\subseteq I_m\\}$ denotes the \\emph{order function} of $I_{\\bullet}$. This construction generalizes the classical asymptotic Samuel function \\cite{Sa52} and serves as an algebraic analogue of the relative type introduced in complex analytic settings \\cite{Ras06} (cf. \\Cref{sec-c.a.c}).\n\nConsider the filtration\n\\[I_{\\blt}\\coloneqq \\Big(\\frac{1}{b_1}\\cdot \\a_{1}\\Big)_{\\blt}+\\cdots+\\Big(\\frac{1}{b_r}\\cdot \\a_{r}\\Big)_{\\blt}.\\]\nFollowing Cutkosky and Peaharaj \\cite[Definition 3.5]{CP24}, the \\emph{asymptotic Samuel function} associated with this filtration is defined as follows. For any ideal $\\a$ of $R$, let\n\\[\\vbI(\\a)\\coloneqq \\lim_{k\\to\\infty}\\frac{\\vI\\big(\\a^k\\big)}{k}\\in \\bbR_{\\ge 0}\\cup\\{\\infty\\},\\]\nwhere $\\vI\\big(\\a\\big)\\coloneqq \\sup\\,\\{m\\in\\bbN \\mid \\a\\subseteq I_m\\}$ denotes the \\emph{order function} of $I_{\\blt}$. This construction generalizes the classical asymptotic Samuel function \\cite{Sa52} and serves as an algebraic analogue of the relative type introduced in complex analytic settings \\cite{Ras06} (cf. \\Cref{sec-c.a.c}).\n\nIn fact, the equivalence of the first two statements, along with the equivalence of the existence of \\emph{Zhou valuations} (cf. \\cite{BGMY24,BGZ25}) satisfying the same interpolation conditions, has been proved in \\cite[Appendix A.1]{BGMY25}.\n\n\\begin{theorem}\\label{thm-infinite.interp}\n Additionally assume $R$ is local with the unique maximal ideal $\\m$. Let $(\\a_j)_{j\\in\\bbZ_+}$ be a countable sequence of nonzero ideals in $R$ such that $\\sqrt{\\sum_{j=1}^{\\infty} \\a_j}=\\m$, and let $(b_j)_{j\\in\\bbZ_+}$ be a sequence of positive real numbers. Then the following statements are equivalent:\n \\begin{enumerate}\n \\item There exists a valuation $v$ on $R$ centered at $\\m$ with $A(v)<\\infty$ such that $v(\\a_j)=b_j$ for all $j\\ge 1$;\n \\item For every $r\\ge 1$,\n \\[\\vb_{I_{\\blt}^{(r)}}\\big(\\a_1\\cdots\\a_r\\big)=\\sum_{j=1}^r b_j,\\]\n and it holds that\n \\[\\sup_{r\\ge 1}\\sup_{k\\in\\bbN}\\Big(\\lct^{\\a_1^k\\cdots\\a_r^k}\\big( I^{(r)}_{\\blt}\\big)-k\\sum_{j=1}^r b_j\\Big)<\\infty.\\]\n \\end{enumerate}\n Here $I^{(r)}_{\\blt}=\\sum_{j=1}^{r}\\big(\\frac{1}{b_j}\\cdot \\a_j\\big)_{\\blt}$.\n\\end{theorem}\n\n\\begin{proposition}\\label{prop-v(aj)=bj.implies.mu=bj-1}\n If $v$ is valuation on $R$ such that $v(\\a_j)\\ge b_j$ for $j=1,\\ldots, r$, then $\\vb(\\a_1\\cdots\\a_r;I_{\\blt})\\le \\sum_{j=1}^r v(\\a_j)$. Moreover, if $v(\\a_j) = b_j$ for all $j$, then $\\vb(\\a_1\\cdots\\a_r;I_{\\blt})=\\sum_{j=1}^r b_j$.\n\\end{proposition}\n\n\\begin{lemma}\\label{lem-finite.val.interp.exist.QM}\n Let $\\a_0$ be a nonzero ideal of $R$ with $\\vb(\\a_0;I_{\\blt})>0$. Suppose $(Y,D)$ is a log smooth pair over $X$ which gives a log resolution of $\\a_1\\cdots\\a_r$. Then there exists a quasi-monomial valuation $v\\in \\QM(Y,D)$ such that\n \\[\\vb(\\a_0;I_{\\blt})=\\frac{v(\\a_0)}{v(I_{\\blt})}.\\]\n\\end{lemma}\n\n\\begin{proposition}\\label{prop-val.interp.gr.seq}\n Assume $\\ord_{\\xi}(J_{\\blt})>0$ and $\\m_{\\xi}^p\\subseteq J_{q}$ for some $p,q\\in\\bbZ_+$. Let $\\a$ be a nonzero ideal of $R$ with $\\a\\subseteq\\m_{\\xi}$. Then the following two statements are equivalent:\n \\begin{enumerate}\n \\item There exists $v\\in\\Val_{X,\\xi}^{<\\infty}$ such that $v(J_{\\blt})=1$ and $v(\\a)=\\rho(\\a;J_{\\blt})$;\n \\item It holds that\n \\[M\\coloneqq \\sup_{k\\in\\bbN}\\Big(\\lct(\\a^k;J_{\\blt})-\\rho(\\a;J_{\\blt})\\cdot k\\Big)<\\infty.\\]\n \\end{enumerate} \n Moreover, the valuation $v$ in (1) can be chosen with $A(v)\\le M$ and $v(\\m_{\\xi})\\ge q/p$ if (2) holds.\n\\end{proposition}\n\n\\begin{lemma}\\label{lem-multiply.change.to.plus}\n Let $J_{\\blt}$ be a filtration of nonzero ideals in $R$ with $\\lct(J_{\\blt})<\\infty$, and let $\\a_1,\\ldots,\\a_r$ be nonzero ideals in $R$. Set $\\a=\\a_1\\cdots\\a_r$. Suppose there is a valuation $v\\in\\Val_X^{<\\infty}$ such that $v(\\a)=\\rho(\\a;J_{\\blt})$ and $v(J_{\\blt})=1$. Then $v(\\a_j)=\\rho(\\a_j;J_{\\blt})$ for $j=1,\\ldots,r$ if and only if $\\rho(\\a;J_{\\blt})=\\sum_{j=1}^r \\rho(\\a_j;J_{\\blt})$.\n\\end{lemma}\n\n\\begin{proposition}\\label{prop-gr.seq.infinite.compute}\n Let $J_{\\blt}$ be a filtration of ideals in $R$ such that $\\ord_{\\xi}(J_{\\blt})>0$ and $\\m_{\\xi}^p\\subseteq J_{q}$ for some $p,q\\in\\bbZ_+$. Assume $(\\a_j)_{j\\in\\bbZ_+}$ is a sequence of nonzero ideals of $R$ with $\\a_j\\subseteq\\m_{\\xi}$ for all $j\\ge 1$. Then the following two statements are equivalent:\n \\begin{enumerate}\n \\item There exists $v\\in\\Val_{X,\\xi}^{<\\infty}$ such that $v(J_{\\blt})=1$ and $v(\\a_j)=\\rho(\\a_j;J_{\\blt})$ for all $j\\ge 1$;\n \\item For every $r\\in\\bbZ_+$,\n \\[\\rho(\\a_1\\cdots\\a_r;J_{\\blt})=\\sum_{j=1}^r\\rho(\\a_j;J_{\\blt}),\\]\n and it holds that\n \\[\\sup_{r\\ge 1}\\sup_{k\\in\\bbN}\\Big(\\lct(\\a_1^k\\cdots\\a_r^k; J_{\\blt})-\\rho(\\a_1\\cdots\\a_r;J_{\\blt})\\cdot k\\Big)<\\infty.\\]\n \\end{enumerate}\n\\end{proposition}", "post_theorem_intro_text_len": 2651, "post_theorem_intro_text": "In fact, the equivalence of the first two statements, along with the equivalence of the existence of \\emph{Zhou valuations} (cf. \\cite{BGMY24,BGZ25}) satisfying the same interpolation conditions, has been proved in \\cite[Appendix A.1]{BGMY25}.\n\nIn addition, for the infinite valuative interpolation problem, we apply the method of valuation approximation to prove the following result in the present paper.\n\n\\begin{theorem}\\label{thm-infinite.interp}\n    Additionally assume $R$ is local with the unique maximal ideal $\\mathfrak{m}$. Let $(\\a_j)_{j\\in\\bbZ_+}$ be a countable sequence of nonzero ideals in $R$ such that $\\sqrt{\\sum_{j=1}^{\\infty} \\a_j}=\\mathfrak{m}$, and let $(b_j)_{j\\in\\bbZ_+}$ be a sequence of positive real numbers. Then the following statements are equivalent:\n    \\begin{enumerate}\n        \\item There exists a valuation $v$ on $R$ centered at $\\mathfrak{m}$ with $A(v)<\\infty$ such that $v(\\a_j)=b_j$ for all $j\\ge 1$;\n        \\item For every $r\\ge 1$,\n    \\[\\vb_{I_{\\bullet}^{(r)}}\\big(\\a_1\\cdots\\a_r\\big)=\\sum_{j=1}^r b_j,\\]\n    and it holds that\n    \\[\\sup_{r\\ge 1}\\sup_{k\\in\\mathbb{N}}\\Big(\\lct^{\\a_1^k\\cdots\\a_r^k}\\big( I^{(r)}_{\\bullet}\\big)-k\\sum_{j=1}^r b_j\\Big)<\\infty.\\]\n    \\end{enumerate}\n    Here $I^{(r)}_{\\bullet}=\\sum_{j=1}^{r}\\big(\\frac{1}{b_j}\\cdot \\a_j\\big)_{\\bullet}$.\n\\end{theorem}\n\n\\vspace{.1in} {\\bf Organization}. This paper is organized as follows. In \\Cref{sec-r.o}, we recall the notion of asymptotic Samuel function and quickly prove the implication (1) $\\Rightarrow$ (3) in \\Cref{thm-finite.interp}. In \\Cref{sec-f.v.i}, we prove \\Cref{thm-finite.interp} by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\\QM(Y,D)$ of quasi-monomial valuations. In \\Cref{sec-i.v.i}, we prove \\Cref{thm-infinite.interp} using valuation approximation, a strategy that more closely parallels the approach in \\cite{BGMY25}. Finally, we demonstrate how \\Cref{thm-finite.interp} recovers a result in \\cite{BGMY25} (which deals with the complex analytic case) and yields an analytic characterization, derived from \\Cref{thm-infinite.interp}, for the existence of infinite valuative interpolation.\n\n\\vspace{.1in} {\\em Acknowledgements}. The first-named author completed this work during a visit to the School of Mathematical Sciences at Peking University and is grateful for its hospitality and support. The second-named author was supported by National Key R\\&D Program of China 2021YFA1003100 and NSFC-12425101. The third-named author was supported by NSFC-12401099 and the Talent Fund of Beijing Jiaotong University 2024-004. The fourth-named author was supported by NSFC-12501106.", "sketch": "The post-theorem introduction only indicates where proofs appear and what methods are used: it says they “quickly prove the implication (1) $\\Rightarrow$ (3) in \\Cref{thm-finite.interp}” in \\Cref{sec-r.o}; they “prove \\Cref{thm-finite.interp} by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\\QM(Y,D)$ of quasi-monomial valuations” in \\Cref{sec-f.v.i}; and they “prove \\Cref{thm-infinite.interp} using valuation approximation.” No further proof outline for \\Cref{thm-finite.interp} is given beyond this organizational summary.", "expanded_sketch": "The post-theorem introduction only indicates where proofs appear and what methods are used: it says they “quickly prove the implication (1) $\\Rightarrow$ (3)” of the main theorem later; they “prove the main theorem by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\\QM(Y,D)$ of quasi-monomial valuations” later; and they “prove the following theorem using valuation approximation. \\begin{theorem}\\label{thm-infinite.interp}\n    Additionally assume $R$ is local with the unique maximal ideal $\\m$. Let $(\\a_j)_{j\\in\\bbZ_+}$ be a countable sequence of nonzero ideals in $R$ such that $\\sqrt{\\sum_{j=1}^{\\infty} \\a_j}=\\m$, and let $(b_j)_{j\\in\\bbZ_+}$ be a sequence of positive real numbers. Then the following statements are equivalent:\n    \\begin{enumerate}\n        \\item There exists a valuation $v$ on $R$ centered at $\\m$ with $A(v)<\\infty$ such that $v(\\a_j)=b_j$ for all $j\\ge 1$;\n        \\item For every $r\\ge 1$,\n    \\[\\vb_{I_{\\blt}^{(r)}}\\big(\\a_1\\cdots\\a_r\\big)=\\sum_{j=1}^r b_j,\\]\n    and it holds that\n    \\[\\sup_{r\\ge 1}\\sup_{k\\in\\bbN}\\Big(\\lct^{\\a_1^k\\cdots\\a_r^k}\\big( I^{(r)}_{\\blt}\\big)-k\\sum_{j=1}^r b_j\\Big)<\\infty.\\]\n    \\end{enumerate}\n    Here $I^{(r)}_{\\blt}=\\sum_{j=1}^{r}\\big(\\frac{1}{b_j}\\cdot \\a_j\\big)_{\\blt}$.\n\\end{theorem} No further proof outline for the main theorem is given beyond this organizational summary.", "expanded_theorem": "\\label{thm-finite.interp}\nThe following three statements are equivalent.\n\\begin{enumerate}\n\\item There exists a valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\dots,r$.\n\\item There exists a quasi-monomial valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\dots,r$.\n\\item The equality $\\vbI(\\a_1\\cdots\\a_r)=\\sum_{j=1}^r b_j$ holds.\n\\end{enumerate}", "theorem_type": ["Biconditional or Equivalence", "Existence"], "mcq": {"question": "Let \\(\\mathfrak a_1,\\ldots,\\mathfrak a_r\\) be nonzero ideals of a ring \\(R\\), and let \\(b_1,\\ldots,b_r\\in \\mathbb R_{\\ge 0}\\). For \\(\\lambda\\in \\mathbb R_{\\ge 0}\\cup\\{\\infty\\}\\) and an ideal \\(\\mathfrak a\\), define the filtration \\((\\lambda\\cdot \\mathfrak a)_\\bullet=((\\lambda\\cdot \\mathfrak a)_m)_{m\\in\\mathbb N}\\) by\n\\[\n(\\lambda\\cdot \\mathfrak a)_m=\n\\begin{cases}\nR,& \\lambda=0,\\\\\n\\mathfrak a^{\\lceil \\lambda m\\rceil},& 0<\\lambda<\\infty,\\\\\n0,& \\lambda=\\infty.\n\\end{cases}\n\\]\nIf \\((\\mathfrak b_1)_\\bullet,\\ldots,(\\mathfrak b_s)_\\bullet\\) are filtrations, their sum is the filtration \\(\\mathfrak c_\\bullet\\) with\n\\[\n\\mathfrak c_m=\\sum_{m_1+\\cdots+m_s=m}(\\mathfrak b_1)_{m_1}\\cdots(\\mathfrak b_s)_{m_s}.\n\\]\nSet\n\\[\nI_\\bullet:=\\Big(\\frac1{b_1}\\cdot \\mathfrak a_1\\Big)_\\bullet+\\cdots+\\Big(\\frac1{b_r}\\cdot \\mathfrak a_r\\Big)_\\bullet,\n\\]\nwhere \\(1/0\\) is interpreted as \\(\\infty\\). For any ideal \\(\\mathfrak a\\subseteq R\\), define the order function\n\\[\n\\nu_I(\\mathfrak a):=\\sup\\{m\\in\\mathbb N\\mid \\mathfrak a\\subseteq I_m\\},\n\\]\nand the associated asymptotic Samuel function\n\\[\n\\bar\\nu_I(\\mathfrak a):=\\lim_{k\\to\\infty}\\frac{\\nu_I(\\mathfrak a^k)}{k}.\n\\]\nWhich statement holds for these data?", "correct_choice": {"label": "A", "text": "The following three statements are equivalent: (i) there exists a valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); (ii) there exists a quasi-monomial valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); and (iii) the equality \\(\\bar\\nu_I(\\mathfrak a_1\\cdots \\mathfrak a_r)=\\sum_{j=1}^r b_j\\) holds."}, "choices": [{"label": "B", "text": "The following three statements are equivalent: (i) there exists a valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); (ii) there exists a quasi-monomial valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); and (iii) for every nonempty subset \\(J\\subseteq\\{1,\\dots,r\\}\\), one has \\(\\bar\\nu_I\\big(\\prod_{j\\in J}\\mathfrak a_j\\big)=\\sum_{j\\in J} b_j\\)."}, {"label": "C", "text": "If there exists a valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\), then \\(\\bar\\nu_I(\\mathfrak a_1\\cdots \\mathfrak a_r)=\\sum_{j=1}^r b_j\\)."}, {"label": "D", "text": "The following three statements are equivalent: (i) there exists a valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); (ii) there exists a quasi-monomial valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); and (iii) for every ideal \\(\\mathfrak a\\subseteq R\\), one has \\(\\bar\\nu_I(\\mathfrak a)=\\inf_{v}\\frac{v(\\mathfrak a)}{v(I_\\bullet)}\\), where the infimum runs over all valuations \\(v\\) on \\(R\\) with \\(v(I_\\bullet)>0\\)."}, {"label": "E", "text": "The following three statements are equivalent: (i) there exists a valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); (ii) there exists a quasi-monomial valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); and (iii) the inequality \\(\\bar\\nu_I(\\mathfrak a_1\\cdots \\mathfrak a_r)\\ge \\sum_{j=1}^r b_j\\) holds."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "single-product equality upgraded to all partial products", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped converse directions and quasi-monomial equivalence, retaining only (i) => (iii)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "extremal quasi-monomial valuation existence replaced by a global valuation-formula for every ideal", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "equality condition weakened to one-sided inequality", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option, and the definitions alone do not single out choice A. There is no obvious lexical or structural cue that leaks the answer."}, "TAS": {"score": 0, "justification": "The question is essentially a theorem-recall item: it asks which statement is equivalent to the valuation existence condition, and the correct option is the exact asymptotic equality appearing as the theorem’s companion condition. This makes it close to a direct restatement rather than an independently motivated problem."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the distractors are nearby variants: a stronger universalization, a weaker inequality, a coordinatewise-inequality confusion, and replacement of asymptotic order by raw order. However, success mainly depends on knowing the precise theorem statement rather than generating a new argument."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and reflect realistic failure modes: strengthening to all partial products, weakening equality to inequality, confusing exact interpolation with lower bounds, and replacing the asymptotic invariant with the non-asymptotic one."}, "total_score": 5, "overall_assessment": "A technically strong but theorem-recall-heavy MCQ. It avoids direct answer leakage and uses high-quality distractors, but it is largely tautological and only moderately tests generative reasoning."}}
{"id": "2602.03179v1", "paper_link": "http://arxiv.org/abs/2602.03179v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm-finite.interp}\n    The following three statements are equivalent.\n    \\begin{enumerate}\n    \\item There exists a valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\dots,r$.\n    \\item There exists a quasi-monomial valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\dots,r$.\n    \\item The equality $\\vbI(\\a_1\\cdots\\a_r)=\\sum_{j=1}^r b_j$ holds.\n    \\end{enumerate}", "start_pos": 5782, "end_pos": 6192, "label": "thm-finite.interp"}, "ref_dict": {"thm-infinite.interp": "\\begin{theorem}\\label{thm-infinite.interp}\n    Additionally assume $R$ is local with the unique maximal ideal $\\m$. Let $(\\a_j)_{j\\in\\bbZ_+}$ be a countable sequence of nonzero ideals in $R$ such that $\\sqrt{\\sum_{j=1}^{\\infty} \\a_j}=\\m$, and let $(b_j)_{j\\in\\bbZ_+}$ be a sequence of positive real numbers. Then the following statements are equivalent:\n    \\begin{enumerate}\n        \\item There exists a valuation $v$ on $R$ centered at $\\m$ with $A(v)<\\infty$ such that $v(\\a_j)=b_j$ for all $j\\ge 1$;\n        \\item For every $r\\ge 1$,\n    \\[\\vb_{I_{\\blt}^{(r)}}\\big(\\a_1\\cdots\\a_r\\big)=\\sum_{j=1}^r b_j,\\]\n    and it holds that\n    \\[\\sup_{r\\ge 1}\\sup_{k\\in\\bbN}\\Big(\\lct^{\\a_1^k\\cdots\\a_r^k}\\big( I^{(r)}_{\\blt}\\big)-k\\sum_{j=1}^r b_j\\Big)<\\infty.\\]\n    \\end{enumerate}\n    Here $I^{(r)}_{\\blt}=\\sum_{j=1}^{r}\\big(\\frac{1}{b_j}\\cdot \\a_j\\big)_{\\blt}$.\n\\end{theorem}", "subsec-gr.fil": "\\begin{enumerate}\n        \\item There exists a valuation $v$ on $R$ centered at $\\m$ with $A(v)<\\infty$ such that $v(\\a_j)=b_j$ for all $j\\ge 1$;\n        \\item For every $r\\ge 1$,\n    \\[\\vb_{I_{\\blt}^{(r)}}\\big(\\a_1\\cdots\\a_r\\big)=\\sum_{j=1}^r b_j,\\]\n    and it holds that\n    \\[\\sup_{r\\ge 1}\\sup_{k\\in\\bbN}\\Big(\\lct^{\\a_1^k\\cdots\\a_r^k}\\big( I^{(r)}_{\\blt}\\big)-k\\sum_{j=1}^r b_j\\Big)<\\infty.\\]\n    \\end{enumerate}\n    Here $I^{(r)}_{\\blt}=\\sum_{j=1}^{r}\\big(\\frac{1}{b_j}\\cdot \\a_j\\big)_{\\blt}$.\n\\end{theorem}\n\n\\vspace{.1in} {\\bf Organization}. This paper is organized as follows. In \\Cref{sec-r.o}, we recall the notion of asymptotic Samuel function and quickly prove the implication (1) $\\Rightarrow$ (3) in \\Cref{thm-finite.interp}. In \\Cref{sec-f.v.i}, we prove \\Cref{thm-finite.interp} by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\\QM(Y,D)$ of quasi-monomial valuations. In \\Cref{sec-i.v.i}, we prove \\Cref{thm-infinite.interp} using valuation approximation, a strategy that more closely parallels the approach in \\cite{BGMY25}. Finally, we demonstrate how \\Cref{thm-finite.interp} recovers a result in \\cite{BGMY25} (which deals with the complex analytic case) and yields an analytic characterization, derived from \\Cref{thm-infinite.interp}, for the existence of infinite valuative interpolation.\n\n\\vspace{.1in} {\\em Acknowledgements}. The first-named author completed this work during a visit to the School of Mathematical Sciences at Peking University and is grateful for its hospitality and support. The second-named author was supported by National Key R\\&D Program of China 2021YFA1003100 and NSFC-12425101. The third-named author was supported by NSFC-12401099 and the Talent Fund of Beijing Jiaotong University 2024-004. The fourth-named author was supported by NSFC-12501106.\n\n\\section{Valuation and asymptotic Samuel function}\\label{sec-r.o}\n\nIn this section, $R$ is a noetherian integral domain.\n\n\\subsection{Valuation}\\label{subsec-val}\n\nRecall that a function $v\\colon R^*=R\\setminus\\{0\\} \\to \\bbR_{\\ge 0}$ is a (real) \\emph{valuation} if\n\\[v(xy)=v(x)+v(y) \\quad \\& \\quad v(x+y)\\ge \\min\\{v(x),v(y)\\},\\]\nfor all $x,y\\in R^*$. Set $v(0)=\\infty$. A valuation $v$ on $R$ can be extended uniquely to a valuation $v\\colon K\\to \\bbR\\cup\\{\\infty\\}$ on the field of fractions $K$ of $R$ by setting $v(x/y)=v(x)-v(y)$. Say $v\\le w$ for two valuations $v,w$ on $R$ if $v(x)\\le w(x)$ for all $x\\in R^*$. For any nonzero ideal $\\a$ of $R$, define $v(\\a)=\\inf_{x\\in \\a}v(x)$. Then it is easy to check: for nonzero ideals $\\a,\\b$ in $R$,\n\\[v(\\a\\b)=v(\\a)+v(\\b) \\quad \\& \\quad v(\\a+\\b)= \\min\\{v(\\a),v(\\b)\\}.\\]\nIn particular, we can see if the ideals $\\a_1,\\ldots,\\a_r$ in $R$ satisfies $\\sum_{j=1}^r\\a_j=R$, then\n\\[0=v(R)=v\\big(\\sum \\a_j\\big)=\\min v(\\a_j),\\]\nwhich implies $v(\\a_j)=0$ for some $j\\in\\{1,\\ldots,r\\}$.\n\n\\subsection{Graded sequence and filtration of ideals}\\label{subsec-gr.fil}\n\nA \\emph{graded sequence} of ideals $\\a_{\\blt}=(\\a_m)_{m\\in\\bbZ_{>0}}$ is a sequence of ideals in $R$ that satisfies $\\a_p\\cdot \\a_p\\subseteq \\a_{p+q}$ for all $p,q\\ge 1$. Put $\\a_0=R$. We always assume $\\a_m\\neq (0)$ for some $m\\ge 1$. For example, $v$ is a nontrivial valuation on $R$, then $\\a_{\\blt}^v$ is a graded sequence of ideals in $R$ defined by $\\a_{m}^v=\\{x\\in R \\mid v(x)\\ge m\\}$. \n\nNow let $v$ be a valuation on $R$, and $\\a_{\\blt}$ a graded sequence of ideals in $R$. Then $v(\\a_{p+q})\\le v(\\a_p)+v(\\a_q)$ for all $p,q\\ge 1$, which implies the limit\n\\[v(\\a_{\\blt})\\coloneqq \\lim_{m\\to \\infty, \\, \\a_m\\neq (0)}\\frac{v(\\a_m)}{m}=\\inf_{m\\ge 1}\\frac{v(\\a_m)}{m}\\]\nexists by Fekete's lemma. In particular, $v(\\a_{\\blt}^v)=1$ for any nontrivial valuation $v$ on $R$. A graded sequence $\\a_{\\blt}=\\{\\a_m\\}_{m\\in\\bbN}$ of ideals in $R$ is called a \\emph{filtration} if additionally it satisfies $\\a_m\\supseteq \\a_{m+1}$ for every $m\\in\\bbN$.\n\n\\subsection{Asymptotic Samuel function}\n\nThe classical asymptotic Samuel function (\\cite{Sa52}) has been generalized to the filtration case in \\cite{CP24}. Let us recall the constructions and some basic properties. Let $J_{\\blt}$ be a filtration of ideals in $R$. For every ideal $\\a$ in $R$, the \\emph{order} of $\\a$ with respect to $J_{\\blt}$ is defined by\n\\[\\vJ\\big(\\a\\big)\\coloneqq \\sup\\,\\{m\\in\\bbN \\mid \\a\\subseteq J_m\\}\\in\\bbN\\cup\\{\\infty\\}.\\]\nThen we can see $\\vJ(\\a^{p+q})\\ge \\vJ(\\a^p)+\\vJ(\\a^q)$ for every $p,q\\in\\bbN$. Therefore, it follows that\n\\[\\vb(\\a;J_{\\blt})\\coloneqq \\lim_{m\\to\\infty}\\frac{\\vJ(\\a^m)}{m}=\\sup_{m\\ge 1}\\frac{\\vJ(\\a^m)}{m}\\in \\bbR_{\\ge 0}\\cup \\{\\infty\\}\\]\nis well-defined by Fekete's lemma, called the \\emph{asymptotic Samuel function} of $J_{\\blt}$ (cf. \\cite[Theorem 3.4]{CP24}). We also denote $\\vb(\\a;J_{\\blt})\\coloneqq \\vbJ(\\a)$ in the present paper.\n\n\\begin{example}\\label{ex-asf}\n    If $J_{\\blt}=\\{J^m\\}_{m\\ge 0}$ for a nonzero ideal $J$ in $R$, then $\\vb(\\a;J_{\\blt})=\\ol{\\nu}_{J}(\\a)$, which is the classical asymptotic Samuel function (cf. \\cite{Sa52,HS06,LJT09}).\n\\end{example}\n\n\\begin{example}\n    If we take $J_{\\blt}=\\a_{\\blt}^v$ the filtration of valuation ideals associated to some valuation $v$ on $R$, then clearly $\\vb(\\a;J_{\\blt})=v(\\a)$.\n\\end{example}\n\nThe following lemma shows some basis properties of the asymptotic Samuel function, which are actually known in \\cite{CP24}. The proof below follows from the strategy of \\cite[Theorem 2]{Sa52}.\n\n\\begin{lemma}[{see \\cite[Proposition 3.11]{CP24}}]\\label{lem-mu.subadditivity}\n    Let $J_{\\blt}$ be a filtration of ideals in $R$. Then\n    \\begin{enumerate}\n    \\item $\\vb(\\a;J_{\\blt})\\ge \\vb(\\b;J_{\\blt})$, for ideals $\\a\\subseteq\\b$ in $R$;\n    \\item $\\vb(\\a^k;J_{\\blt})=k\\cdot\\vb(\\a;J_{\\blt})$, where $\\a$ is an ideal in $R$ and $k\\in\\bbZ_+$;\n    \\item $\\vb(\\a+\\b;J_{\\blt})= \\min\\{\\vb(\\a;J_{\\blt}), \\vb(\\b;J_{\\blt})\\}$, where $\\a,\\b$ are ideals in $R$;\n    \\item $\\vb(\\a\\b;J_{\\blt})\\ge \\vb(\\a;J_{\\blt}) +\\vb(\\b;J_{\\blt})$, where $\\a,\\b$ are ideals in $R$.\n    \\end{enumerate}", "subsec-val": "\\begin{enumerate}\n        \\item There exists a valuation $v$ on $R$ centered at $\\m$ with $A(v)<\\infty$ such that $v(\\a_j)=b_j$ for all $j\\ge 1$;\n        \\item For every $r\\ge 1$,\n    \\[\\vb_{I_{\\blt}^{(r)}}\\big(\\a_1\\cdots\\a_r\\big)=\\sum_{j=1}^r b_j,\\]\n    and it holds that\n    \\[\\sup_{r\\ge 1}\\sup_{k\\in\\bbN}\\Big(\\lct^{\\a_1^k\\cdots\\a_r^k}\\big( I^{(r)}_{\\blt}\\big)-k\\sum_{j=1}^r b_j\\Big)<\\infty.\\]\n    \\end{enumerate}\n    Here $I^{(r)}_{\\blt}=\\sum_{j=1}^{r}\\big(\\frac{1}{b_j}\\cdot \\a_j\\big)_{\\blt}$.\n\\end{theorem}\n\n\\vspace{.1in} {\\bf Organization}. This paper is organized as follows. In \\Cref{sec-r.o}, we recall the notion of asymptotic Samuel function and quickly prove the implication (1) $\\Rightarrow$ (3) in \\Cref{thm-finite.interp}. In \\Cref{sec-f.v.i}, we prove \\Cref{thm-finite.interp} by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\\QM(Y,D)$ of quasi-monomial valuations. In \\Cref{sec-i.v.i}, we prove \\Cref{thm-infinite.interp} using valuation approximation, a strategy that more closely parallels the approach in \\cite{BGMY25}. Finally, we demonstrate how \\Cref{thm-finite.interp} recovers a result in \\cite{BGMY25} (which deals with the complex analytic case) and yields an analytic characterization, derived from \\Cref{thm-infinite.interp}, for the existence of infinite valuative interpolation.\n\n\\vspace{.1in} {\\em Acknowledgements}. The first-named author completed this work during a visit to the School of Mathematical Sciences at Peking University and is grateful for its hospitality and support. The second-named author was supported by National Key R\\&D Program of China 2021YFA1003100 and NSFC-12425101. The third-named author was supported by NSFC-12401099 and the Talent Fund of Beijing Jiaotong University 2024-004. The fourth-named author was supported by NSFC-12501106.\n\n\\section{Valuation and asymptotic Samuel function}\\label{sec-r.o}\n\nIn this section, $R$ is a noetherian integral domain.\n\n\\subsection{Valuation}\\label{subsec-val}\n\nRecall that a function $v\\colon R^*=R\\setminus\\{0\\} \\to \\bbR_{\\ge 0}$ is a (real) \\emph{valuation} if\n\\[v(xy)=v(x)+v(y) \\quad \\& \\quad v(x+y)\\ge \\min\\{v(x),v(y)\\},\\]\nfor all $x,y\\in R^*$. Set $v(0)=\\infty$. A valuation $v$ on $R$ can be extended uniquely to a valuation $v\\colon K\\to \\bbR\\cup\\{\\infty\\}$ on the field of fractions $K$ of $R$ by setting $v(x/y)=v(x)-v(y)$. Say $v\\le w$ for two valuations $v,w$ on $R$ if $v(x)\\le w(x)$ for all $x\\in R^*$. For any nonzero ideal $\\a$ of $R$, define $v(\\a)=\\inf_{x\\in \\a}v(x)$. Then it is easy to check: for nonzero ideals $\\a,\\b$ in $R$,\n\\[v(\\a\\b)=v(\\a)+v(\\b) \\quad \\& \\quad v(\\a+\\b)= \\min\\{v(\\a),v(\\b)\\}.\\]\nIn particular, we can see if the ideals $\\a_1,\\ldots,\\a_r$ in $R$ satisfies $\\sum_{j=1}^r\\a_j=R$, then\n\\[0=v(R)=v\\big(\\sum \\a_j\\big)=\\min v(\\a_j),\\]\nwhich implies $v(\\a_j)=0$ for some $j\\in\\{1,\\ldots,r\\}$.\n\n\\subsection{Graded sequence and filtration of ideals}\\label{subsec-gr.fil}\n\nA \\emph{graded sequence} of ideals $\\a_{\\blt}=(\\a_m)_{m\\in\\bbZ_{>0}}$ is a sequence of ideals in $R$ that satisfies $\\a_p\\cdot \\a_p\\subseteq \\a_{p+q}$ for all $p,q\\ge 1$. Put $\\a_0=R$. We always assume $\\a_m\\neq (0)$ for some $m\\ge 1$. For example, $v$ is a nontrivial valuation on $R$, then $\\a_{\\blt}^v$ is a graded sequence of ideals in $R$ defined by $\\a_{m}^v=\\{x\\in R \\mid v(x)\\ge m\\}$. \n\nNow let $v$ be a valuation on $R$, and $\\a_{\\blt}$ a graded sequence of ideals in $R$. Then $v(\\a_{p+q})\\le v(\\a_p)+v(\\a_q)$ for all $p,q\\ge 1$, which implies the limit\n\\[v(\\a_{\\blt})\\coloneqq \\lim_{m\\to \\infty, \\, \\a_m\\neq (0)}\\frac{v(\\a_m)}{m}=\\inf_{m\\ge 1}\\frac{v(\\a_m)}{m}\\]\nexists by Fekete's lemma. In particular, $v(\\a_{\\blt}^v)=1$ for any nontrivial valuation $v$ on $R$. A graded sequence $\\a_{\\blt}=\\{\\a_m\\}_{m\\in\\bbN}$ of ideals in $R$ is called a \\emph{filtration} if additionally it satisfies $\\a_m\\supseteq \\a_{m+1}$ for every $m\\in\\bbN$.\n\n\\subsection{Asymptotic Samuel function}\n\nThe classical asymptotic Samuel function (\\cite{Sa52}) has been generalized to the filtration case in \\cite{CP24}. Let us recall the constructions and some basic properties. Let $J_{\\blt}$ be a filtration of ideals in $R$. For every ideal $\\a$ in $R$, the \\emph{order} of $\\a$ with respect to $J_{\\blt}$ is defined by\n\\[\\vJ\\big(\\a\\big)\\coloneqq \\sup\\,\\{m\\in\\bbN \\mid \\a\\subseteq J_m\\}\\in\\bbN\\cup\\{\\infty\\}.\\]\nThen we can see $\\vJ(\\a^{p+q})\\ge \\vJ(\\a^p)+\\vJ(\\a^q)$ for every $p,q\\in\\bbN$. Therefore, it follows that\n\\[\\vb(\\a;J_{\\blt})\\coloneqq \\lim_{m\\to\\infty}\\frac{\\vJ(\\a^m)}{m}=\\sup_{m\\ge 1}\\frac{\\vJ(\\a^m)}{m}\\in \\bbR_{\\ge 0}\\cup \\{\\infty\\}\\]\nis well-defined by Fekete's lemma, called the \\emph{asymptotic Samuel function} of $J_{\\blt}$ (cf. \\cite[Theorem 3.4]{CP24}). We also denote $\\vb(\\a;J_{\\blt})\\coloneqq \\vbJ(\\a)$ in the present paper.\n\n\\begin{example}\\label{ex-asf}\n    If $J_{\\blt}=\\{J^m\\}_{m\\ge 0}$ for a nonzero ideal $J$ in $R$, then $\\vb(\\a;J_{\\blt})=\\ol{\\nu}_{J}(\\a)$, which is the classical asymptotic Samuel function (cf. \\cite{Sa52,HS06,LJT09}).\n\\end{example}\n\n\\begin{example}\n    If we take $J_{\\blt}=\\a_{\\blt}^v$ the filtration of valuation ideals associated to some valuation $v$ on $R$, then clearly $\\vb(\\a;J_{\\blt})=v(\\a)$.\n\\end{example}\n\nThe following lemma shows some basis properties of the asymptotic Samuel function, which are actually known in \\cite{CP24}. The proof below follows from the strategy of \\cite[Theorem 2]{Sa52}.\n\n\\begin{lemma}[{see \\cite[Proposition 3.11]{CP24}}]\\label{lem-mu.subadditivity}\n    Let $J_{\\blt}$ be a filtration of ideals in $R$. Then\n    \\begin{enumerate}\n    \\item $\\vb(\\a;J_{\\blt})\\ge \\vb(\\b;J_{\\blt})$, for ideals $\\a\\subseteq\\b$ in $R$;\n    \\item $\\vb(\\a^k;J_{\\blt})=k\\cdot\\vb(\\a;J_{\\blt})$, where $\\a$ is an ideal in $R$ and $k\\in\\bbZ_+$;\n    \\item $\\vb(\\a+\\b;J_{\\blt})= \\min\\{\\vb(\\a;J_{\\blt}), \\vb(\\b;J_{\\blt})\\}$, where $\\a,\\b$ are ideals in $R$;\n    \\item $\\vb(\\a\\b;J_{\\blt})\\ge \\vb(\\a;J_{\\blt}) +\\vb(\\b;J_{\\blt})$, where $\\a,\\b$ are ideals in $R$.\n    \\end{enumerate}", "thm-finite.interp": "\\begin{theorem}\\label{thm-finite.interp}\n    The following three statements are equivalent.\n    \\begin{enumerate}\n    \\item There exists a valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\dots,r$.\n    \\item There exists a quasi-monomial valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\dots,r$.\n    \\item The equality $\\vbI(\\a_1\\cdots\\a_r)=\\sum_{j=1}^r b_j$ holds.\n    \\end{enumerate}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2845, "pre_theorem_intro_text": "Let $R$ be an excellent regular domain of equicharacteristic $0$ and of dimension $n$.\n\nLet $\\a_1,\\ldots,\\a_r$ be $r$ nonzero ideals of $R$, and let $b_1,\\ldots,b_r\\in\\mathbb{R}_{\\ge 0}$. We consider the following valuative interpolation problem which was considered by the authors in the complex analytic setting (e.g. \\cite[Theorem 1.2]{BGMY25} deals with the case where $R=\\O_{\\mathbb{C}^n,o}$ the ring of holomorphic germs):\n\n\\begin{question}\nCan one give a necessary and sufficient condition for the existence of a valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\ldots, r$?\n\\end{question}\n\nWe introduce some notations for explaining our results in the present paper. Suppose $\\mathfrak{a}$ is an ideal on $X$, and $\\lambda\\in\\bbR_{\\ge 0}\\cup\\{\\infty\\}$. Denote by $(\\lambda\\cdot\\mathfrak{a})_{\\bullet}=\\big((\\lambda\\cdot\\mathfrak{a})_m\\big)_{m\\in\\mathbb{N}}$ the \\emph{filtration} (cf. \\Cref{subsec-gr.fil}) of ideals such that each term:\n\\begin{equation*}\n    (\\lambda\\cdot\\mathfrak{a})_m=\n    \\begin{cases}\n        R & \\lambda=0 \\\\\n        \\mathfrak{a}^{\\lceil \\lambda m\\rceil} & 0<\\lambda<\\infty \\\\\n        0 & \\lambda=\\infty \\\\\n    \\end{cases}\n    \\quad \\quad m\\in\\mathbb{N}.\n\\end{equation*}\n\nIf $(\\b_1)_{\\bullet},\\ldots,(\\b_s)_{\\bullet}$ are filtrations of ideals in $R$, then the summation $\\mathfrak{c}_{\\bullet}\\coloneqq \\sum(\\b_i)_{\\bullet}$ of these filtrations is a filtration defined as:\n\\begin{equation*}\n        \\mathfrak{c}_{m}=\\sum_{\\substack{m_1+\\cdots+m_s=m \\\\ m_i\\in\\mathbb{N}}}(\\b_1)_{m_1}\\cdots(\\b_s)_{m_s}, \\quad m\\in\\mathbb{N}.\n\\end{equation*}\n\nNow we return to the valuative interpolation problem. For the ideals $\\a_1,\\ldots,\\a_r$ and non-negative real numbers $b_1,\\ldots,b_r$, if all of $b_j$ are zero, then we can directly choose the trivial valuation on $R$, so we only consider the case where at least one of them is nonzero. Hence, may assume $\\sum_{b_j>0}\\a_j\\neq R$, otherwise none of valuations $v$ on $R$ can satisfy $v(\\a_j)=b_j$ for all $j$ (cf. \\Cref{subsec-val}).\n\nConsider the filtration\n\\[I_{\\bullet}\\coloneqq \\Big(\\frac{1}{b_1}\\cdot \\a_{1}\\Big)_{\\bullet}+\\cdots+\\Big(\\frac{1}{b_r}\\cdot \\a_{r}\\Big)_{\\bullet}.\\]\nFollowing Cutkosky and Peaharaj \\cite[Definition 3.5]{CP24}, the \\emph{asymptotic Samuel function} associated with this filtration is defined as follows. For any ideal $\\mathfrak{a}$ of $R$, let\n\\[\\vbI(\\mathfrak{a})\\coloneqq \\lim_{k\\to\\infty}\\frac{\\vI\\big(\\mathfrak{a}^k\\big)}{k}\\in \\bbR_{\\ge 0}\\cup\\{\\infty\\},\\]\nwhere $\\vI\\big(\\mathfrak{a}\\big)\\coloneqq \\sup\\,\\{m\\in\\mathbb{N} \\mid \\mathfrak{a}\\subseteq I_m\\}$ denotes the \\emph{order function} of $I_{\\bullet}$. This construction generalizes the classical asymptotic Samuel function \\cite{Sa52} and serves as an algebraic analogue of the relative type introduced in complex analytic settings \\cite{Ras06} (cf. \\Cref{sec-c.a.c}).", "context": "Let $\\a_1,\\ldots,\\a_r$ be $r$ nonzero ideals of $R$, and let $b_1,\\ldots,b_r\\in\\mathbb{R}_{\\ge 0}$. We consider the following valuative interpolation problem which was considered by the authors in the complex analytic setting (e.g. \\cite[Theorem 1.2]{BGMY25} deals with the case where $R=\\O_{\\mathbb{C}^n,o}$ the ring of holomorphic germs):\n\n\\begin{question}\nCan one give a necessary and sufficient condition for the existence of a valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\ldots, r$?\n\\end{question}\n\nWe introduce some notations for explaining our results in the present paper. Suppose $\\mathfrak{a}$ is an ideal on $X$, and $\\lambda\\in\\bbR_{\\ge 0}\\cup\\{\\infty\\}$. Denote by $(\\lambda\\cdot\\mathfrak{a})_{\\bullet}=\\big((\\lambda\\cdot\\mathfrak{a})_m\\big)_{m\\in\\mathbb{N}}$ the \\emph{filtration} (cf. \\Cref{subsec-gr.fil}) of ideals such that each term:\n\\begin{equation*}\n (\\lambda\\cdot\\mathfrak{a})_m=\n \\begin{cases}\n R & \\lambda=0 \\\\\n \\mathfrak{a}^{\\lceil \\lambda m\\rceil} & 0<\\lambda<\\infty \\\\\n 0 & \\lambda=\\infty \\\\\n \\end{cases}\n \\quad \\quad m\\in\\mathbb{N}.\n\\end{equation*}\n\nIf $(\\b_1)_{\\bullet},\\ldots,(\\b_s)_{\\bullet}$ are filtrations of ideals in $R$, then the summation $\\mathfrak{c}_{\\bullet}\\coloneqq \\sum(\\b_i)_{\\bullet}$ of these filtrations is a filtration defined as:\n\\begin{equation*}\n \\mathfrak{c}_{m}=\\sum_{\\substack{m_1+\\cdots+m_s=m \\\\ m_i\\in\\mathbb{N}}}(\\b_1)_{m_1}\\cdots(\\b_s)_{m_s}, \\quad m\\in\\mathbb{N}.\n\\end{equation*}\n\nNow we return to the valuative interpolation problem. For the ideals $\\a_1,\\ldots,\\a_r$ and non-negative real numbers $b_1,\\ldots,b_r$, if all of $b_j$ are zero, then we can directly choose the trivial valuation on $R$, so we only consider the case where at least one of them is nonzero. Hence, may assume $\\sum_{b_j>0}\\a_j\\neq R$, otherwise none of valuations $v$ on $R$ can satisfy $v(\\a_j)=b_j$ for all $j$ (cf. \\Cref{subsec-val}).\n\nConsider the filtration\n\\[I_{\\bullet}\\coloneqq \\Big(\\frac{1}{b_1}\\cdot \\a_{1}\\Big)_{\\bullet}+\\cdots+\\Big(\\frac{1}{b_r}\\cdot \\a_{r}\\Big)_{\\bullet}.\\]\nFollowing Cutkosky and Peaharaj \\cite[Definition 3.5]{CP24}, the \\emph{asymptotic Samuel function} associated with this filtration is defined as follows. For any ideal $\\mathfrak{a}$ of $R$, let\n\\[\\vbI(\\mathfrak{a})\\coloneqq \\lim_{k\\to\\infty}\\frac{\\vI\\big(\\mathfrak{a}^k\\big)}{k}\\in \\bbR_{\\ge 0}\\cup\\{\\infty\\},\\]\nwhere $\\vI\\big(\\mathfrak{a}\\big)\\coloneqq \\sup\\,\\{m\\in\\mathbb{N} \\mid \\mathfrak{a}\\subseteq I_m\\}$ denotes the \\emph{order function} of $I_{\\bullet}$. This construction generalizes the classical asymptotic Samuel function \\cite{Sa52} and serves as an algebraic analogue of the relative type introduced in complex analytic settings \\cite{Ras06} (cf. \\Cref{sec-c.a.c}).", "full_context": "Let $\\a_1,\\ldots,\\a_r$ be $r$ nonzero ideals of $R$, and let $b_1,\\ldots,b_r\\in\\mathbb{R}_{\\ge 0}$. We consider the following valuative interpolation problem which was considered by the authors in the complex analytic setting (e.g. \\cite[Theorem 1.2]{BGMY25} deals with the case where $R=\\O_{\\mathbb{C}^n,o}$ the ring of holomorphic germs):\n\n\\begin{question}\nCan one give a necessary and sufficient condition for the existence of a valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\ldots, r$?\n\\end{question}\n\nWe introduce some notations for explaining our results in the present paper. Suppose $\\mathfrak{a}$ is an ideal on $X$, and $\\lambda\\in\\bbR_{\\ge 0}\\cup\\{\\infty\\}$. Denote by $(\\lambda\\cdot\\mathfrak{a})_{\\bullet}=\\big((\\lambda\\cdot\\mathfrak{a})_m\\big)_{m\\in\\mathbb{N}}$ the \\emph{filtration} (cf. \\Cref{subsec-gr.fil}) of ideals such that each term:\n\\begin{equation*}\n (\\lambda\\cdot\\mathfrak{a})_m=\n \\begin{cases}\n R & \\lambda=0 \\\\\n \\mathfrak{a}^{\\lceil \\lambda m\\rceil} & 0<\\lambda<\\infty \\\\\n 0 & \\lambda=\\infty \\\\\n \\end{cases}\n \\quad \\quad m\\in\\mathbb{N}.\n\\end{equation*}\n\nIf $(\\b_1)_{\\bullet},\\ldots,(\\b_s)_{\\bullet}$ are filtrations of ideals in $R$, then the summation $\\mathfrak{c}_{\\bullet}\\coloneqq \\sum(\\b_i)_{\\bullet}$ of these filtrations is a filtration defined as:\n\\begin{equation*}\n \\mathfrak{c}_{m}=\\sum_{\\substack{m_1+\\cdots+m_s=m \\\\ m_i\\in\\mathbb{N}}}(\\b_1)_{m_1}\\cdots(\\b_s)_{m_s}, \\quad m\\in\\mathbb{N}.\n\\end{equation*}\n\nNow we return to the valuative interpolation problem. For the ideals $\\a_1,\\ldots,\\a_r$ and non-negative real numbers $b_1,\\ldots,b_r$, if all of $b_j$ are zero, then we can directly choose the trivial valuation on $R$, so we only consider the case where at least one of them is nonzero. Hence, may assume $\\sum_{b_j>0}\\a_j\\neq R$, otherwise none of valuations $v$ on $R$ can satisfy $v(\\a_j)=b_j$ for all $j$ (cf. \\Cref{subsec-val}).\n\nConsider the filtration\n\\[I_{\\bullet}\\coloneqq \\Big(\\frac{1}{b_1}\\cdot \\a_{1}\\Big)_{\\bullet}+\\cdots+\\Big(\\frac{1}{b_r}\\cdot \\a_{r}\\Big)_{\\bullet}.\\]\nFollowing Cutkosky and Peaharaj \\cite[Definition 3.5]{CP24}, the \\emph{asymptotic Samuel function} associated with this filtration is defined as follows. For any ideal $\\mathfrak{a}$ of $R$, let\n\\[\\vbI(\\mathfrak{a})\\coloneqq \\lim_{k\\to\\infty}\\frac{\\vI\\big(\\mathfrak{a}^k\\big)}{k}\\in \\bbR_{\\ge 0}\\cup\\{\\infty\\},\\]\nwhere $\\vI\\big(\\mathfrak{a}\\big)\\coloneqq \\sup\\,\\{m\\in\\mathbb{N} \\mid \\mathfrak{a}\\subseteq I_m\\}$ denotes the \\emph{order function} of $I_{\\bullet}$. This construction generalizes the classical asymptotic Samuel function \\cite{Sa52} and serves as an algebraic analogue of the relative type introduced in complex analytic settings \\cite{Ras06} (cf. \\Cref{sec-c.a.c}).\n\nConsider the filtration\n\\[I_{\\blt}\\coloneqq \\Big(\\frac{1}{b_1}\\cdot \\a_{1}\\Big)_{\\blt}+\\cdots+\\Big(\\frac{1}{b_r}\\cdot \\a_{r}\\Big)_{\\blt}.\\]\nFollowing Cutkosky and Peaharaj \\cite[Definition 3.5]{CP24}, the \\emph{asymptotic Samuel function} associated with this filtration is defined as follows. For any ideal $\\a$ of $R$, let\n\\[\\vbI(\\a)\\coloneqq \\lim_{k\\to\\infty}\\frac{\\vI\\big(\\a^k\\big)}{k}\\in \\bbR_{\\ge 0}\\cup\\{\\infty\\},\\]\nwhere $\\vI\\big(\\a\\big)\\coloneqq \\sup\\,\\{m\\in\\bbN \\mid \\a\\subseteq I_m\\}$ denotes the \\emph{order function} of $I_{\\blt}$. This construction generalizes the classical asymptotic Samuel function \\cite{Sa52} and serves as an algebraic analogue of the relative type introduced in complex analytic settings \\cite{Ras06} (cf. \\Cref{sec-c.a.c}).\n\nIn fact, the equivalence of the first two statements, along with the equivalence of the existence of \\emph{Zhou valuations} (cf. \\cite{BGMY24,BGZ25}) satisfying the same interpolation conditions, has been proved in \\cite[Appendix A.1]{BGMY25}.\n\n\\begin{theorem}\\label{thm-infinite.interp}\n Additionally assume $R$ is local with the unique maximal ideal $\\m$. Let $(\\a_j)_{j\\in\\bbZ_+}$ be a countable sequence of nonzero ideals in $R$ such that $\\sqrt{\\sum_{j=1}^{\\infty} \\a_j}=\\m$, and let $(b_j)_{j\\in\\bbZ_+}$ be a sequence of positive real numbers. Then the following statements are equivalent:\n \\begin{enumerate}\n \\item There exists a valuation $v$ on $R$ centered at $\\m$ with $A(v)<\\infty$ such that $v(\\a_j)=b_j$ for all $j\\ge 1$;\n \\item For every $r\\ge 1$,\n \\[\\vb_{I_{\\blt}^{(r)}}\\big(\\a_1\\cdots\\a_r\\big)=\\sum_{j=1}^r b_j,\\]\n and it holds that\n \\[\\sup_{r\\ge 1}\\sup_{k\\in\\bbN}\\Big(\\lct^{\\a_1^k\\cdots\\a_r^k}\\big( I^{(r)}_{\\blt}\\big)-k\\sum_{j=1}^r b_j\\Big)<\\infty.\\]\n \\end{enumerate}\n Here $I^{(r)}_{\\blt}=\\sum_{j=1}^{r}\\big(\\frac{1}{b_j}\\cdot \\a_j\\big)_{\\blt}$.\n\\end{theorem}\n\n\\begin{proposition}\\label{prop-v(aj)=bj.implies.mu=bj-1}\n If $v$ is valuation on $R$ such that $v(\\a_j)\\ge b_j$ for $j=1,\\ldots, r$, then $\\vb(\\a_1\\cdots\\a_r;I_{\\blt})\\le \\sum_{j=1}^r v(\\a_j)$. Moreover, if $v(\\a_j) = b_j$ for all $j$, then $\\vb(\\a_1\\cdots\\a_r;I_{\\blt})=\\sum_{j=1}^r b_j$.\n\\end{proposition}\n\n\\begin{lemma}\\label{lem-finite.val.interp.exist.QM}\n Let $\\a_0$ be a nonzero ideal of $R$ with $\\vb(\\a_0;I_{\\blt})>0$. Suppose $(Y,D)$ is a log smooth pair over $X$ which gives a log resolution of $\\a_1\\cdots\\a_r$. Then there exists a quasi-monomial valuation $v\\in \\QM(Y,D)$ such that\n \\[\\vb(\\a_0;I_{\\blt})=\\frac{v(\\a_0)}{v(I_{\\blt})}.\\]\n\\end{lemma}\n\n\\begin{proposition}\\label{prop-val.interp.gr.seq}\n Assume $\\ord_{\\xi}(J_{\\blt})>0$ and $\\m_{\\xi}^p\\subseteq J_{q}$ for some $p,q\\in\\bbZ_+$. Let $\\a$ be a nonzero ideal of $R$ with $\\a\\subseteq\\m_{\\xi}$. Then the following two statements are equivalent:\n \\begin{enumerate}\n \\item There exists $v\\in\\Val_{X,\\xi}^{<\\infty}$ such that $v(J_{\\blt})=1$ and $v(\\a)=\\rho(\\a;J_{\\blt})$;\n \\item It holds that\n \\[M\\coloneqq \\sup_{k\\in\\bbN}\\Big(\\lct(\\a^k;J_{\\blt})-\\rho(\\a;J_{\\blt})\\cdot k\\Big)<\\infty.\\]\n \\end{enumerate} \n Moreover, the valuation $v$ in (1) can be chosen with $A(v)\\le M$ and $v(\\m_{\\xi})\\ge q/p$ if (2) holds.\n\\end{proposition}\n\n\\begin{lemma}\\label{lem-multiply.change.to.plus}\n Let $J_{\\blt}$ be a filtration of nonzero ideals in $R$ with $\\lct(J_{\\blt})<\\infty$, and let $\\a_1,\\ldots,\\a_r$ be nonzero ideals in $R$. Set $\\a=\\a_1\\cdots\\a_r$. Suppose there is a valuation $v\\in\\Val_X^{<\\infty}$ such that $v(\\a)=\\rho(\\a;J_{\\blt})$ and $v(J_{\\blt})=1$. Then $v(\\a_j)=\\rho(\\a_j;J_{\\blt})$ for $j=1,\\ldots,r$ if and only if $\\rho(\\a;J_{\\blt})=\\sum_{j=1}^r \\rho(\\a_j;J_{\\blt})$.\n\\end{lemma}\n\n\\begin{proposition}\\label{prop-gr.seq.infinite.compute}\n Let $J_{\\blt}$ be a filtration of ideals in $R$ such that $\\ord_{\\xi}(J_{\\blt})>0$ and $\\m_{\\xi}^p\\subseteq J_{q}$ for some $p,q\\in\\bbZ_+$. Assume $(\\a_j)_{j\\in\\bbZ_+}$ is a sequence of nonzero ideals of $R$ with $\\a_j\\subseteq\\m_{\\xi}$ for all $j\\ge 1$. Then the following two statements are equivalent:\n \\begin{enumerate}\n \\item There exists $v\\in\\Val_{X,\\xi}^{<\\infty}$ such that $v(J_{\\blt})=1$ and $v(\\a_j)=\\rho(\\a_j;J_{\\blt})$ for all $j\\ge 1$;\n \\item For every $r\\in\\bbZ_+$,\n \\[\\rho(\\a_1\\cdots\\a_r;J_{\\blt})=\\sum_{j=1}^r\\rho(\\a_j;J_{\\blt}),\\]\n and it holds that\n \\[\\sup_{r\\ge 1}\\sup_{k\\in\\bbN}\\Big(\\lct(\\a_1^k\\cdots\\a_r^k; J_{\\blt})-\\rho(\\a_1\\cdots\\a_r;J_{\\blt})\\cdot k\\Big)<\\infty.\\]\n \\end{enumerate}\n\\end{proposition}", "post_theorem_intro_text_len": 2651, "post_theorem_intro_text": "In fact, the equivalence of the first two statements, along with the equivalence of the existence of \\emph{Zhou valuations} (cf. \\cite{BGMY24,BGZ25}) satisfying the same interpolation conditions, has been proved in \\cite[Appendix A.1]{BGMY25}.\n\nIn addition, for the infinite valuative interpolation problem, we apply the method of valuation approximation to prove the following result in the present paper.\n\n\\begin{theorem}\\label{thm-infinite.interp}\n    Additionally assume $R$ is local with the unique maximal ideal $\\mathfrak{m}$. Let $(\\a_j)_{j\\in\\bbZ_+}$ be a countable sequence of nonzero ideals in $R$ such that $\\sqrt{\\sum_{j=1}^{\\infty} \\a_j}=\\mathfrak{m}$, and let $(b_j)_{j\\in\\bbZ_+}$ be a sequence of positive real numbers. Then the following statements are equivalent:\n    \\begin{enumerate}\n        \\item There exists a valuation $v$ on $R$ centered at $\\mathfrak{m}$ with $A(v)<\\infty$ such that $v(\\a_j)=b_j$ for all $j\\ge 1$;\n        \\item For every $r\\ge 1$,\n    \\[\\vb_{I_{\\bullet}^{(r)}}\\big(\\a_1\\cdots\\a_r\\big)=\\sum_{j=1}^r b_j,\\]\n    and it holds that\n    \\[\\sup_{r\\ge 1}\\sup_{k\\in\\mathbb{N}}\\Big(\\lct^{\\a_1^k\\cdots\\a_r^k}\\big( I^{(r)}_{\\bullet}\\big)-k\\sum_{j=1}^r b_j\\Big)<\\infty.\\]\n    \\end{enumerate}\n    Here $I^{(r)}_{\\bullet}=\\sum_{j=1}^{r}\\big(\\frac{1}{b_j}\\cdot \\a_j\\big)_{\\bullet}$.\n\\end{theorem}\n\n\\vspace{.1in} {\\bf Organization}. This paper is organized as follows. In \\Cref{sec-r.o}, we recall the notion of asymptotic Samuel function and quickly prove the implication (1) $\\Rightarrow$ (3) in \\Cref{thm-finite.interp}. In \\Cref{sec-f.v.i}, we prove \\Cref{thm-finite.interp} by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\\QM(Y,D)$ of quasi-monomial valuations. In \\Cref{sec-i.v.i}, we prove \\Cref{thm-infinite.interp} using valuation approximation, a strategy that more closely parallels the approach in \\cite{BGMY25}. Finally, we demonstrate how \\Cref{thm-finite.interp} recovers a result in \\cite{BGMY25} (which deals with the complex analytic case) and yields an analytic characterization, derived from \\Cref{thm-infinite.interp}, for the existence of infinite valuative interpolation.\n\n\\vspace{.1in} {\\em Acknowledgements}. The first-named author completed this work during a visit to the School of Mathematical Sciences at Peking University and is grateful for its hospitality and support. The second-named author was supported by National Key R\\&D Program of China 2021YFA1003100 and NSFC-12425101. The third-named author was supported by NSFC-12401099 and the Talent Fund of Beijing Jiaotong University 2024-004. The fourth-named author was supported by NSFC-12501106.", "sketch": "The post-theorem introduction only indicates where proofs appear and what methods are used: it says they “quickly prove the implication (1) $\\Rightarrow$ (3) in \\Cref{thm-finite.interp}” in \\Cref{sec-r.o}; they “prove \\Cref{thm-finite.interp} by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\\QM(Y,D)$ of quasi-monomial valuations” in \\Cref{sec-f.v.i}; and they “prove \\Cref{thm-infinite.interp} using valuation approximation.” No further proof outline for \\Cref{thm-finite.interp} is given beyond this organizational summary.", "expanded_sketch": "The post-theorem introduction only indicates where proofs appear and what methods are used: it says they “quickly prove the implication (1) $\\Rightarrow$ (3)” of the main theorem later; they “prove the main theorem by studying the asymptotic behavior of jumping numbers and an extremal problem on a cone $\\QM(Y,D)$ of quasi-monomial valuations” later; and they “prove the following theorem using valuation approximation. \\begin{theorem}\\label{thm-infinite.interp}\n    Additionally assume $R$ is local with the unique maximal ideal $\\m$. Let $(\\a_j)_{j\\in\\bbZ_+}$ be a countable sequence of nonzero ideals in $R$ such that $\\sqrt{\\sum_{j=1}^{\\infty} \\a_j}=\\m$, and let $(b_j)_{j\\in\\bbZ_+}$ be a sequence of positive real numbers. Then the following statements are equivalent:\n    \\begin{enumerate}\n        \\item There exists a valuation $v$ on $R$ centered at $\\m$ with $A(v)<\\infty$ such that $v(\\a_j)=b_j$ for all $j\\ge 1$;\n        \\item For every $r\\ge 1$,\n    \\[\\vb_{I_{\\blt}^{(r)}}\\big(\\a_1\\cdots\\a_r\\big)=\\sum_{j=1}^r b_j,\\]\n    and it holds that\n    \\[\\sup_{r\\ge 1}\\sup_{k\\in\\bbN}\\Big(\\lct^{\\a_1^k\\cdots\\a_r^k}\\big( I^{(r)}_{\\blt}\\big)-k\\sum_{j=1}^r b_j\\Big)<\\infty.\\]\n    \\end{enumerate}\n    Here $I^{(r)}_{\\blt}=\\sum_{j=1}^{r}\\big(\\frac{1}{b_j}\\cdot \\a_j\\big)_{\\blt}$.\n\\end{theorem} No further proof outline for the main theorem is given beyond this organizational summary.", "expanded_theorem": "\\label{thm-finite.interp}\nThe following three statements are equivalent.\n\\begin{enumerate}\n\\item There exists a valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\dots,r$.\n\\item There exists a quasi-monomial valuation $v$ on $R$ such that $v(\\a_j)=b_j$ for $j=1,\\dots,r$.\n\\item The equality $\\vbI(\\a_1\\cdots\\a_r)=\\sum_{j=1}^r b_j$ holds.\n\\end{enumerate}", "theorem_type": ["Biconditional or Equivalence", "Existence"], "mcq": {"question": "Let \\(\\mathfrak a_1,\\ldots,\\mathfrak a_r\\) be nonzero ideals of a ring \\(R\\), and let \\(b_1,\\ldots,b_r\\in \\mathbb R_{\\ge 0}\\). For \\(\\lambda\\in \\mathbb R_{\\ge 0}\\cup\\{\\infty\\}\\) and an ideal \\(\\mathfrak a\\), define the filtration \\((\\lambda\\cdot \\mathfrak a)_\\bullet=((\\lambda\\cdot \\mathfrak a)_m)_{m\\in\\mathbb N}\\) by\n\\[\n(\\lambda\\cdot \\mathfrak a)_m=\n\\begin{cases}\nR,& \\lambda=0,\\\\\n\\mathfrak a^{\\lceil \\lambda m\\rceil},& 0<\\lambda<\\infty,\\\\\n0,& \\lambda=\\infty.\n\\end{cases}\n\\]\nIf \\((\\mathfrak b_1)_\\bullet,\\ldots,(\\mathfrak b_s)_\\bullet\\) are filtrations, their sum is the filtration \\(\\mathfrak c_\\bullet\\) with\n\\[\n\\mathfrak c_m=\\sum_{m_1+\\cdots+m_s=m}(\\mathfrak b_1)_{m_1}\\cdots(\\mathfrak b_s)_{m_s}.\n\\]\nSet\n\\[\nI_\\bullet:=\\Big(\\frac1{b_1}\\cdot \\mathfrak a_1\\Big)_\\bullet+\\cdots+\\Big(\\frac1{b_r}\\cdot \\mathfrak a_r\\Big)_\\bullet,\n\\]\nwhere \\(1/0\\) is interpreted as \\(\\infty\\). For any ideal \\(\\mathfrak a\\subseteq R\\), define the order function\n\\[\n\\nu_I(\\mathfrak a):=\\sup\\{m\\in\\mathbb N\\mid \\mathfrak a\\subseteq I_m\\},\n\\]\nand the associated asymptotic Samuel function\n\\[\n\\bar\\nu_I(\\mathfrak a):=\\lim_{k\\to\\infty}\\frac{\\nu_I(\\mathfrak a^k)}{k}.\n\\]\nWhich statement holds for these data?", "correct_choice": {"label": "A", "text": "The following three statements are equivalent: (i) there exists a valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); (ii) there exists a quasi-monomial valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); and (iii) the equality \\(\\bar\\nu_I(\\mathfrak a_1\\cdots \\mathfrak a_r)=\\sum_{j=1}^r b_j\\) holds."}, "choices": [{"label": "B", "text": "The following three statements are equivalent: (i) there exists a valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); (ii) there exists a quasi-monomial valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); and (iii) for every nonempty subset \\(J\\subseteq\\{1,\\dots,r\\}\\), one has \\(\\bar\\nu_I\\big(\\prod_{j\\in J}\\mathfrak a_j\\big)=\\sum_{j\\in J} b_j\\)."}, {"label": "C", "text": "If there exists a valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\), then \\(\\bar\\nu_I(\\mathfrak a_1\\cdots \\mathfrak a_r)=\\sum_{j=1}^r b_j\\)."}, {"label": "D", "text": "The following three statements are equivalent: (i) there exists a valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); (ii) there exists a quasi-monomial valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); and (iii) for every ideal \\(\\mathfrak a\\subseteq R\\), one has \\(\\bar\\nu_I(\\mathfrak a)=\\inf_{v}\\frac{v(\\mathfrak a)}{v(I_\\bullet)}\\), where the infimum runs over all valuations \\(v\\) on \\(R\\) with \\(v(I_\\bullet)>0\\)."}, {"label": "E", "text": "The following three statements are equivalent: (i) there exists a valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); (ii) there exists a quasi-monomial valuation \\(v\\) on \\(R\\) such that \\(v(\\mathfrak a_j)=b_j\\) for \\(j=1,\\dots,r\\); and (iii) the inequality \\(\\bar\\nu_I(\\mathfrak a_1\\cdots \\mathfrak a_r)\\ge \\sum_{j=1}^r b_j\\) holds."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "single-product equality upgraded to all partial products", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped converse directions and quasi-monomial equivalence, retaining only (i) => (iii)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "extremal quasi-monomial valuation existence replaced by a global valuation-formula for every ideal", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "equality condition weakened to one-sided inequality", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem provides only definitions and setup; it does not explicitly state the equivalence in the correct option or otherwise single out choice A. There are no obvious lexical cues that reveal the answer."}, "TAS": {"score": 1, "justification": "The item is still fairly close to theorem recognition: the correct option is essentially the precise theorem statement in this notation. However, the presence of several nearby variants means it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish exact equivalence from weaker, stronger, or misdirected alternatives (equality vs inequality, full equivalence vs one implication, single product vs all subsets). But the task mainly tests recognition of the exact theorem rather than substantial generative problem solving."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: overstrengthening the conclusion, weakening to a one-way implication, replacing equality by inequality, or swapping in an overly broad valuation formula. They are distinct and nontrivial."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with no answer leakage and strong distractors, but it leans more toward precise theorem recall than genuine generative reasoning."}}
{"id": "2602.03241v1", "paper_link": "http://arxiv.org/abs/2602.03241v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm:infinite-geometric-moduli-general}\n\tLet $(M, T^{1, 0} M)$ be a compact, connected, strictly pseudoconvex CR manifold of dimension $2 n + 1$\n\tsuch that \\Cref{keyassumption} holds\n\tand $\\pi_{1}(M)$ has infinite profinite completion.\n\tThen $\\# \\GModuli(\\wtM, T^{1, 0} \\wtM) = \\infty$.", "start_pos": 10514, "end_pos": 10836, "label": "thm:infinite-geometric-moduli-general"}, "ref_dict": {"subsection:boundary-of-Reinhardt-domain": "\\label{subsection:boundary-of-Reinhardt-domain}\n\nAssume that $n \\geq 2$.\nLet $M$ be the boundary of the bounded Reinhardt domain\n\\begin{equation}\n\t\\Omega\n\t\\coloneqq \\Set{ w = (w^{1}, \\dots , w^{n + 1}", "thm:infinite-geometric-moduli-general": "\\begin{theorem}\n\\label{thm:infinite-geometric-moduli-general}\n\tLet $(M, T^{1, 0} M)$ be a compact, connected, strictly pseudoconvex CR manifold of dimension $2 n + 1$\n\tsuch that \\Cref{keyassumption} holds\n\tand $\\pi_{1}(M)$ has infinite profinite completion.\n\tThen $\\# \\GModuli(\\wtM, T^{1, 0} \\wtM) = \\infty$.\n\\end{theorem}", "subsection:CR-manifolds": "\\begin{theorem}\n\\label{thm:infinite-geometric-moduli-complex-hyperbolic}\n\tThe complement $S^{2 n + 1} \\setminus S_{\\bbC}^{2 k + 1}$ satisfies\n\t$\\# \\GModuli(S^{2 n + 1} \\setminus S_{\\bbC}^{2 k + 1}) = \\infty$ if $0 \\leq k < (n - 2) / 2$.\n\\end{theorem}\n\nWe add some remarks on these theorems.\nLet $V$ be a real linear subspace in $\\bbC^{n + 1}$\nand set $\\Lambda \\coloneqq V \\cap S^{2 n + 1} \\subset S^{2 n + 1}$.\nIf $V$ is totally real or complex and $\\dim_{H} \\Lambda < n$,\nthen we proved that $\\# \\GModuli(S^{2 n + 1} \\setminus \\Lambda) = \\infty$.\nThe key point of the proof is that\nthere exists a torsion-free convex cocompact discrete subgroup $\\Gamma$ of $PU(n + 1, 1)$\nwhose limit set coincides with $\\Lambda$.\nThen the quotient of $S^{2 n + 1} \\setminus \\Lambda$ by $\\Gamma$ satisfies \\Cref{keyassumption}.\nOn the other hand,\nwhen $V$ is neither totally real nor complex,\nthere exist no discrete subgroups in $PU(n + 1, 1)$\nsuch that its limit set coincides with $\\Lambda$,\nand hence our argument does not apply.\nAs far as we know,\nthere are no existence results for complete contact forms\nwith constant Tanaka--Webster scalar curvature on $S^{2 n + 1} \\setminus \\Lambda$\n\nWe will also apply \\cref{thm:infinite-geometric-moduli-general}\nto circle bundles over compact \\Kahler manifolds (\\cref{thm:infinite-geometric-moduli-circle-bundle})\nand the boundary of a Reinhardt domain (\\cref{subsection:boundary-of-Reinhardt-domain}),\nwhich is neither spherical nor Sasakian.\n\nThis paper is organized as follows.\nIn \\cref{section:CR-geometry},\nwe review basic materials in CR geometry.\nIn \\cref{section:CR-Yamabe-problem},\nwe recall the CR Yamabe problem and summarize known existence results that motivate the present work.\nIn \\cref{section:complex-hyperbolic-geometry},\nwe collect several facts from complex hyperbolic geometry\nthat are needed to analyze complements of $\\bbR$- and $\\bbC$-spheres in the CR sphere.\nIn \\cref{section:infinite-tower},\nwe study an existence problem for infinite towers of finite connected coverings;\nthis constitutes the technical core of the paper.\nIn \\cref{section:proof-of-main-result},\nwe use this framework to prove \\cref{thm:infinite-geometric-moduli-general}.\nIn \\cref{section:complements-of-spheres},\nwe apply this theorem to complements of $\\bbR$- and $\\bbC$-spheres in the CR sphere.\nFinally, in \\cref{section:other-applications},\nwe discuss further applications to circle bundles over compact \\Kahler manifolds\nand to the boundary of a Reinhardt domain.\n\n\\section{CR geometry}\n\\label{section:CR-geometry}\n\n\\subsection{CR manifolds}\n\\label{subsection:CR-manifolds}\n\nLet $M$ be a smooth $(2 n + 1)$-dimensional manifold.\nA \\emph{CR structure} is a rank $n$ complex subbundle $T^{1, 0} M$\nof the complexified tangent bundle $T M \\otimes \\bbC$ such that\n\\begin{equation}\n\tT^{1, 0} M \\cap T^{0, 1} M = 0, \\qquad\n\t\\comm{\\Gamma(T^{1, 0} M)}{\\Gamma(T^{1, 0} M)} \\subset \\Gamma(T^{1, 0} M),\n\\end{equation}\nwhere $T^{0, 1} M$ is the complex conjugate of $T^{1, 0} M$ in $T M \\otimes \\bbC$.\nSet $H M = \\Re T^{1, 0} M$\nand let $J \\colon H M \\to H M$ be the unique complex structure on $H M$ \nsuch that\n\\begin{equation}\n\tT^{1, 0} M = \\Ker(J - \\sqrt{- 1} \\colon H M \\otimes \\bbC \\to H M \\otimes \\bbC).\n\\end{equation}\nA typical example of a CR manifold is a real hypersurface $M$ in an $(n + 1)$-dimensional complex manifold $X$;\nthis $M$ has the induced CR structure\n\\begin{equation}\n\tT^{1, 0} M\n\t\\coloneqq T^{1, 0} X |_{M} \\cap (T M \\otimes \\bbC).\n\\end{equation}\nIn particular,\nthe unit sphere\n\\begin{equation}\n\tS^{2 n + 1}\n\t\\coloneqq \\Set{z \\in \\bbC^{n + 1} | \\abs{z}^{2} = 1}\n\\end{equation}\nhas the \\emph{canonical} CR structure $T^{1, 0} S^{2 n + 1}$.\n\nLet $(M, T^{1, 0} M)$ and $(M^{\\prime}, T^{1, 0} M^{\\prime})$ be CR manifolds.\nA smooth map $f \\colon M \\to M^{\\prime}$ is called a \\emph{CR map}\nif $f_{\\ast} (T^{1, 0} M) \\subset T^{1, 0} M^{\\prime}$.\nIf $f$ is a diffeomorphism,\nthen we call $f$ a \\emph{CR diffeomorphism}.\nA CR manifold $(M, T^{1, 0} M)$ is said to be \\emph{spherical}\nif it is locally isomorphic to $(S^{2 n + 1}, T^{1, 0} S^{2 n + 1})$.\n\nA CR structure $T^{1, 0} M$ is said to be \\emph{strictly pseudoconvex}\nif there exists a smooth real-valued one-form $\\theta$ annihilating exactly $H M$\nsuch that the \\emph{Levi form}\n\\begin{equation}\n\tL_{\\theta}(V, W)\n\t\\coloneqq d \\theta(V, J W)\n\t\\qquad (V, W \\in H M)\n\\end{equation}\nis positive definite.\nWe call such a one-form a \\emph{contact form}.\nThe triple $(M, T^{1, 0} M, \\theta)$ is called a \\emph{pseudo-Hermitian manifold}.\nDenote by $T$ the \\emph{Reeb vector field} with respect to $\\theta$; \nthat is,\nthe unique vector field satisfying\n\\begin{equation}\n\t\\theta(T) = 1, \\qquad T \\contr d \\theta = 0.\n\\end{equation}\nLet $(Z_{\\alpha})$ be a local frame of $T^{1, 0} M$,\nand set $Z_{\\ovxa} = \\overline{Z_{\\alpha}}$.\nThen\n$(T, Z_{\\alpha}, Z_{\\ovxa})$ gives a local frame of $T M \\otimes \\bbC$,\ncalled an \\emph{admissible frame}.\nIts dual frame $(\\theta, \\theta^{\\alpha}, \\theta^{\\ovxa})$\nis called an \\emph{admissible coframe}.\nThe two-form $d \\theta$ is written as\n\\begin{equation}\n\td \\theta\n\t= \\sqrt{- 1} h_{\\alpha \\ovxb} \\theta^{\\alpha} \\wedge \\theta^{\\ovxb},\n\\end{equation}\nwhere $(h_{\\alpha \\ovxb})$ is a positive definite Hermitian matrix.\nWe use $h_{\\alpha \\ovxb}$ and its inverse $h^{\\alpha \\ovxb}$\nto raise and lower indices of tensors.\n\nThe flat model of pseudo-Hermitian manifolds is the \\emph{Heisenberg group} $\\bbH^{2 n + 1}$,\nthat is,\nthe Lie group with the underlying manifold $\\bbR \\times \\bbC^{n}$ and the multiplication\n\\begin{equation}\n\t(t, z) \\cdot (t^{\\prime}, z^{\\prime})\n\t\\coloneqq (t + t^{\\prime} + 2 \\Im (z \\cdot \\ovz^{\\prime}), z + z^{\\prime}).\n\\end{equation}\nFor $\\alpha = 1, \\dots , n$,\nwe introduce a left-invariant complex vector field $Z_{\\alpha}$ by\n\\begin{equation}\n\tZ_{\\alpha}\n\t\\coloneqq \\pdv{}{z^{\\alpha}} + \\sqrt{- 1} \\ovz^{\\alpha} \\pdv{}{t}.\n\\end{equation}\nThe canonical CR structure $T^{1, 0} \\bbH^{2 n + 1}$\nis spanned by $Z_{1}, \\dots , Z_{n}$.\nDefine a left-invariant one-form $\\theta$ on $\\bbH^{2 n + 1}$ by\n\\begin{equation}\n\t\\theta\n\t\\coloneqq d t + \\sqrt{-1} \\sum_{\\alpha = 1}^{n}\n\t\t(z^{\\alpha} d \\ovz^{\\alpha} - \\ovz^{\\alpha} d z^{\\alpha}).\n\\end{equation}\nThen $\\theta$ annihilates $T^{1, 0} \\bbH^{2 n + 1}$\nand the Levi form $L_{\\theta}$ satisfies\n$L_{\\theta}(Z_{\\alpha}, Z_{\\ovxb}) = 2 \\delta_{\\alpha \\beta}$;\nin particular,\n$(\\bbH^{2 n + 1}, T^{1, 0} \\bbH^{2 n + 1})$ is a strictly pseudoconvex CR manifold\nand $\\theta$ is a contact form on $\\bbH^{2 n + 1}$.\n\nA CR manifold $(M, T^{1, 0} M)$ is said to be \\emph{embeddable}\nif there exists a smooth embedding $F \\colon M \\to \\bbC^{N}$\nsuch that $F_{\\ast}(T^{1, 0} M) \\subset T^{1, 0} \\bbC^{N}$.\nIt is known that a compact strictly pseudoconvex CR manifold $(M, T^{1, 0} M)$\nof dimension at least five must be embeddable~\\cite{Boutet_de_Monvel1975}.\nOn the other hand,\nthere are many non-embeddable compact strictly pseudoconvex CR manifolds of dimension three;\nsee for example \\cite{Burns-Epstein1990-Embed}.\nWe say that $(M, T^{1, 0} M)$ is \\emph{universally embeddable}\nif every finite cover of $(M, T^{1, 0} M)$ is embeddable.\nThis condition is strictly stronger than the embeddability condition~\\cite{Case-Yang2025}.\n\nAssume that $M$ is connected and fix a contact form $\\theta$ on $M$.\nWe can also endow $M$ with the \\emph{Carnot--Carath\\'{e}odory distance} $d_{\\CC}$ on $M$ as follows.\nFor any $p, q \\in M$,\nwe can find a smooth path $c \\colon \\clcl{0}{1} \\to M$\nsuch that $c(0) = p$, $c(1) = q$, and $\\theta(c^{\\prime}(t)) = 0$.\nThen $d_{\\CC}(p, q)$ is the infimum of the length of such curves.\nDenote by $\\dim_{H} A$ the Hausdorff dimension of $A \\subset M$ with respect to $d_{\\CC}$.\nThe Carnot--Carath\\'{e}odory distance $d_{\\CC}$ induces the standard topology of $M$,\nbut the Hausdorff dimension $\\dim_{H} M$ of $M$ is equal to $2 n + 2$,\nwhich does not coincide with its topological dimension.\nSee \\cite{Gromov1996} for more details.\n\n\\subsection{Tanaka--Webster connection}\n\\label{subsection:TW-connection}\n\nA contact form $\\theta$ induces a canonical connection $\\nabla$ on $T M$,\ncalled the \\emph{Tanaka--Webster connection} with respect to $\\theta$.\nIt is defined by\n\\begin{equation}\n\t\\nabla T\n\t= 0,\n\t\\quad\n\t\\nabla Z_{\\alpha}\n\t= \\omega_{\\alpha} {}^{\\beta} Z_{\\beta},\n\t\\quad\n\t\\nabla Z_{\\ovxa}\n\t= \\omega_{\\ovxa} {}^{\\ovxb} Z_{\\ovxb},\n\\end{equation}\nwhere $\\omega_{\\ovxa} {}^{\\ovxb} = \\overline{\\omega_{\\alpha} {}^{\\beta}}$,\nwith the following structure equations:\n\\begin{gather}\n\\label{eq:str-eq-of-TW-conn1}\n\td \\theta^{\\beta}\n\t= \\theta^{\\alpha} \\wedge \\omega_{\\alpha} {}^{\\beta}\n\t+ A^{\\beta} {}_{\\ovxa} \\theta \\wedge \\theta^{\\ovxa}, \\\\\n\\label{eq:str-eq-of-TW-conn2}\n\td h_{\\alpha \\ovxb}\n\t= \\omega_{\\alpha} {}^{\\gamma} h_{\\gamma \\ovxb}\n\t\t+ h_{\\alpha \\ovxg} \\omega_{\\ovxb} {}^{\\ovxg}.\n\\end{gather}\nThe tensor $A_{\\alpha \\beta} = \\overline{A_{\\ovxa \\ovxb}}$\nis symmetric and is called the \\emph{Tanaka--Webster torsion}.\nThe \\emph{sub-Laplacian} $\\Delta_{b}$ is defined by\n\\begin{equation}\n\t\\Delta_{b} u\n\t\\coloneqq - \\nabla^{\\alpha} \\nabla_{\\alpha} u - \\nabla^{\\ovxb} \\nabla_{\\ovxb} u.\n\\end{equation}\n\nThe curvature form\n$\\Omega_{\\alpha} {}^{\\beta} \\coloneqq d \\omega_{\\alpha} {}^{\\beta}\n- \\omega_{\\alpha} {}^{\\gamma} \\wedge \\omega_{\\gamma} {}^{\\beta}$\nof the Tanaka--Webster connection satisfies\n\\begin{equation}\n\\label{eq:curvature-form-of-TW-connection}\n\t\\Omega_{\\alpha} {}^{\\beta}\n\t= R_{\\alpha} {}^{\\beta} {}_{\\rho \\ovxs} \\theta^{\\rho} \\wedge \\theta^{\\ovxs}\n\t\t\\qquad \\text{modulo $\\theta, \\theta^{\\rho} \\wedge \\theta^{\\gamma},\n\t\t\\theta^{\\ovxg} \\wedge \\theta^{\\ovxs}$}.\n\\end{equation}\nWe call the tensor $R_{\\alpha} {}^{\\beta} {}_{\\rho \\ovxs}$\nthe \\emph{Tanaka--Webster curvature}.\nThis tensor has the symmetry \n\\begin{equation}\n\tR_{\\alpha \\ovxb \\rho \\ovxs}\n\t= R_{\\rho \\ovxb \\alpha \\ovxs}\n\t= R_{\\alpha \\ovxs \\rho \\ovxb}.\n\\end{equation}\nContraction of indices gives the \\emph{Tanaka--Webster Ricci curvature}\n$\\Ric_{\\rho \\ovxs} = R_{\\alpha} {}^{\\alpha} {}_{\\rho \\ovxs}$\nand the \\emph{Tanaka--Webster scalar curvature}\n$R_{\\theta} = \\Ric_{\\rho} {}^{\\rho}$.\nWhen $n \\geq 2$,\nthe \\emph{Chern tensor} $S_{\\alpha \\ovxb \\rho \\ovxs}$ is the completely trace-free part\nof $R_{\\alpha \\ovxb \\rho \\ovxs}$;\nthis tensor is the CR analogue of the Weyl tensor in conformal geometry.\nIt is known that the Chern tensor vanishes identically\nif and only if $(M, T^{1, 0} M)$ is spherical~\\cite{Chern-Moser1974}.\n\nFix a contact form $\\theta$ on $M$.\nThen any contact form $\\whxth$ is of the form $u^{2 / n} \\theta$,\nwhere\n\\begin{equation}\n\tu \\in C^{\\infty}_{+}(M) \\coloneqq \\Set{u \\in C^{\\infty}(M) | u > 0}.\n\\end{equation}\nUnder this conformal change,\nthe Tanaka--Webster scalar curvature $R_{\\wtxth}$ with respect to $\\whxth$ satisfies\n\\begin{equation}\n\tR_{\\wtxth}\n\t= u^{- 1 - 2 / n} \\sbra*{(2 + 2 / n) \\Delta_{b} u + R_{\\theta} u};\n\\end{equation}\nsee \\cite{Jerison-Lee1987}*{p.\\ 174} for example.\n\n\\subsection{Sasakian manifolds}\n\\label{subsection:Sasakian-manifolds}\n\nSasakian manifolds constitute an important class of pseudo-Hermitian manifolds.\nSee~\\cite{Boyer-Galicki2008} for a comprehensive introduction.\n\nA \\emph{Sasakian manifold} is a pseudo-Hermitian manifold $(S, T^{1, 0}S, \\theta)$\nwith vanishing Tanaka--Webster torsion.\nThis condition is equivalent to the requirement that the Reeb vector field $T$ of $\\theta$\npreserves the CR structure $T^{1, 0} S$.\n\nA typical example of a Sasakian manifold\nis the circle bundle associated with a negative Hermitian holomorphic line bundle.\nLet $Y$ be an $n$-dimensional complex manifold\nand $(L, h)$ a Hermitian holomorphic line bundle over $Y$\nsuch that\n\\begin{equation}\n\t\\omega = - \\sqrt{- 1} \\Theta_{h} = d d^{c} \\log h\n\\end{equation}\nis a \\Kahler form on $Y$,\nwhere $d^{c} \\coloneqq (\\sqrt{- 1} / 2)(\\delb - \\partial)$.\nConsider the circle bundle\n\\begin{equation}\n\tS = \\Set{ v \\in L | h(v, v) = 1}\n\\end{equation}\nover $Y$,\nwhich is a real hypersurface in the total space of $L$\nand has the canonical CR structure $T^{1, 0} S$.\nThe one-form $\\theta = d^{c} \\log h |_{S}$ is a contact form on $S$\nand its Tanaka--Webster scalar curvature $R_{\\theta}$ coincides with $p^{\\ast} R_{\\omega}$,\nwhere $p \\colon S \\to Y$ is the projection\nand $R_{\\omega}$ is the scalar curvature of $\\omega$.\nMoreover,\nthe CR manifold $(S, T^{1, 0} S)$ is spherical if and only if\n$(Y, \\omega)$ is Bochner-flat.\nSee \\cite{Takeuchi2022-Chern}*{Section 2.3} for more details.\n\n\\subsection{Schoen's Theorem for CR automorphism group}\n\nThe CR sphere and the Heisenberg group play a distinguished role\nin CR geometry from the viewpoint of symmetry.\nTo explain this,\nwe begin with some definitions.\n\n\\begin{definition}\n\tLet $(M, T^{1, 0} M)$ be a strictly pseudoconvex CR manifold of dimension $2 n + 1$.\n\tThe \\emph{CR automorphism group} $\\Aut_{\\CR}(M, T^{1, 0} M)$\n\tconsists of all CR diffeomorphisms $\\varphi \\in \\Diff(M)$\n\tequipped with the compact-open topology.\n\tWe say that $\\Aut_{\\CR}(M, T^{1, 0} M)$ \\emph{acts properly} on $M$\n\tif $\\Set{\\varphi \\in \\Aut_{\\CR}(M, T^{1, 0} M) | \\varphi(K) \\cap K \\neq \\emptyset}$\n\tis relatively compact in $\\Aut_{\\CR}(M, T^{1, 0} M)$ for any compact set $K \\subset M$.\n\\end{definition}\n\nSchoen's theorem for CR automorphism group states that\n$\\Aut_{\\CR}(M, T^{1, 0} M)$ acts properly except for\n$(S^{2 n + 1}, T^{1, 0} S^{2 n + 1})$ or $(\\bbH^{2 n + 1}, T^{1, 0} \\bbH^{2 n + 1})$.\n\n\\begin{theorem}[\\cite{Schoen1995}*{Theorems 3.3' and 3.4'}]\n\\label{thm:Schoen's-theorem}\n\tLet $(M, T^{1, 0} M)$ be a strictly pseudoconvex CR manifold of dimension $2 n + 1$.\n\tIf the CR automorphism group $\\Aut_{\\CR}(M, T^{1, 0} M)$ of $(M, T^{1, 0} M)$ does not act properly on $M$,\n\tthen $(M, T^{1, 0} M)$ is CR diffeomorphic to\n\t$(S^{2 n + 1}, T^{1, 0} S^{2 n + 1})$ or $(\\bbH^{2 n + 1}, T^{1, 0} \\bbH^{2 n + 1})$.\n\\end{theorem}", "eq:singular-CR-Yamabe-problem": "\\begin{equation}\n\\label{eq:singular-CR-Yamabe-problem}\n\t\\begin{cases}\n\t\t(2 + 2 / n) \\Delta_{b} u + R_{\\theta} u = \\lambda u^{1 + 2 / n} \\text{\\ on $M \\setminus \\Lambda$}, \\\\\n\t\t\\lim_{x \\to \\Lambda} u(x) = \\infty,\n\t\\end{cases}\n\\end{equation}", "keyassumption": "\\begin{keyassumption}{$(\\blacklozenge)$}\n\\label{keyassumption}\n\tAssume that the CR Yamabe constant of $(M, T^{1, 0} M)$ is positive.\n\tIf $\\dim M = 3$,\n\tthen we further assume that $(M, T^{1, 0} M)$ is universally embeddable;\n\tsee \\cref{subsection:CR-manifolds} for definition.\n\tIf $\\dim M = 5$ and $(M, T^{1, 0} M)$ is spherical,\n\tthen we additionally assume that the developing map $\\Phi \\colon \\wtM \\to S^{5}$ is injective.\n\\end{keyassumption}", "thm:infinite-geometric-moduli-circle-bundle": "\\begin{theorem}\n\\label{thm:infinite-geometric-moduli-circle-bundle}\n\tIf the scalar curvature $R_{\\omega}$ of $\\omega$ is positive\n\tand $\\pi_{1}(Y)$ has infinite profinite completion,\n\tthen $\\# \\GModuli(\\wtS, T^{1, 0} \\wtS) = \\infty$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 8186, "pre_theorem_intro_text": "\\label{sec:Introduction}\n\nThe existence of complete metrics with constant scalar curvature\nis one of the most fundamental problems in Riemannian and conformal geometry.\nIn the compact case,\nthis problem, known as the Yamabe problem, was solved affirmatively;\nsee \\cite{Lee-Parker1987} and the references therein.\nA typical non-compact setting arises\nwhen one considers the complement of a closed subset in a compact manifold.\nIn this situation,\nthe problem is referred to as the singular Yamabe problem.\nIf the ambient manifold is the round sphere,\nthis problem has been studied by several authors~\\cites{Schoen1989,Mazzeo-Smale1991,Bettiol-Piccione-Santaro2016,Bettiol-Piccione2018}.\nRelated to our result,\nin joint work with Andrade, Piccione, and Wei~\\cite{Andrade-Case-Piccione-Wei2025-preprint},\nthe first author proved that\n$S^{n} \\setminus S^{k}$ admits infinitely many non-homothetic periodic conformally round metrics\nwith positive constant scalar curvature provided that $n \\geq 2 k + 3$.\n\nBased on the analogy between conformal and CR geometry,\nwe consider the existence problem for complete contact forms with constant Tanaka--Webster scalar curvature.\nWe first recall the status of this problem\non compact, connected, strictly pseudoconvex CR manifolds $(M, T^{1, 0} M)$ of dimension $2 n + 1$.\nThe \\emph{CR Yamabe functional} $\\frakF$ is defined by\n\\begin{equation}\n\\label{eq:CR-Yamabe-functional}\n\t\\frakF(\\theta)\n\t\\coloneqq \\Vol_{\\theta}(M)^{- n / (n + 1)} \\int_{M} R_{\\theta} \\, \\vol_{\\theta},\n\\end{equation}\nwhere $\\theta$ is a contact form on $M$,\n$R_{\\theta}$ is the Tanaka--Webster scalar curvature associated with $\\theta$,\n$\\vol_{\\theta} \\coloneqq \\theta \\wedge (d \\theta)^{n}$,\nand $\\Vol_{\\theta}(M) \\coloneqq \\int_{M} \\vol_{\\theta}$.\nThe infimum of this functional is called the \\emph{CR Yamabe constant} $Y(M, T^{1, 0} M)$.\nA critical point of this functional corresponds to a contact form with constant Tanaka--Webster scalar curvature.\nJerison and Lee~\\cites{Jerison-Lee1987,Jerison-Lee1988,Jerison-Lee1989} proved that\nthis functional admits a minimizer if $n \\geq 2$ and $(M, T^{1, 0} M)$ is not spherical,\nor if $(M, T^{1, 0} M)$ is CR diffeomorphic to the CR sphere.\nGamara and Yacoub~\\cites{Gamara-Yacoub2001,Gamara2001} established the existence of a critical point\nin the remaining case via the method of critical points at infinity.\nThe contact form constructed by their method is not necessarily a CR Yamabe minimizer.\nIn fact,\nCheng, Malchiodi, and Yang~\\cite{Cheng-Malchiodi-Yang2023} showed that the Rossi sphere,\nwhich is an example of a non-embeddable CR three-manifold,\nadmits no CR Yamabe minimizers.\nWe also remark that $(M, T^{1, 0} M)$ admits a CR Yamabe minimizer\nif $n = 1$ and $(M, T^{1, 0} M)$ is embeddable~\\cites{Cheng-Malchiodi-Yang2017,Takeuchi2020-Paneitz}.\n\nIn this paper,\nwe consider the non-compact case.\nLet $(M, T^{1, 0} M)$ be a (possibly non-compact) connected strictly pseudoconvex CR manifold.\nDenote by $\\CSC(M, T^{1, 0} M)$ the set of complete contact forms on $M$\nwith constant Tanaka--Webster scalar curvature.\nWe say that two contact forms $\\theta_{1}$ and $\\theta_{2}$ on $M$ are \\emph{homothetic}\nif there exist a CR diffeomorphism $\\varphi$ and a constant $c > 0$\nsuch that $\\varphi^{\\ast} \\theta_{1} = c \\theta_{2}$;\notherwise they are said to be \\emph{non-homothetic}.\nThe \\emph{geometric moduli space} of complete contact forms\nwith constant Tanaka--Webster scalar curvature on $(M, T^{1, 0} M)$ is the quotient\n\\begin{equation}\n\t\\GModuli(M, T^{1, 0} M)\n\t\\coloneqq \\CSC(M, T^{1, 0} M) / \\sim,\n\\end{equation}\nwhere $\\theta_{1} \\sim \\theta_{2}$ if $\\theta_{1}$ and $\\theta_{2}$ are homothetic.\nJerison and Lee proved that $\\# \\GModuli(M, T^{1, 0} M) = 1$\nif $M$ is compact and $Y(M, T^{1, 0} M) \\leq 0$~\\cite{Jerison-Lee1987}*{Theorem 7.1}\nor $(M, T^{1, 0} M) = (S^{2 n + 1}, T^{1, 0} S^{2 n + 1})$~\\cite{Jerison-Lee1988}*{Theorem A}.\nIt is also known that $\\# \\GModuli(M, T^{1, 0} M) = 1$\nif $M$ is compact and admits an Einstein contact form~\\cite{Wang2015}*{Theorem 4}.\n\nA typical non-compact situation arises when one considers the complement of a closed subset\nin a compact strictly pseudoconvex CR manifold.\nLet $(M, T^{1, 0} M)$ be a compact, connected, strictly pseudoconvex CR manifold of dimension $2 n + 1$,\nlet $\\theta$ be a contact form on $M$,\nand let $\\Lambda$ be a closed subset of $M$.\nOne seeks complete contact forms with constant Tanaka--Webster scalar curvature on $M \\setminus \\Lambda$;\nwe abbreviate the CR structure $T^{1, 0} (M \\setminus \\Lambda)$ for simplicity.\nThis leads to the \\emph{singular CR Yamabe equation}\n\\begin{equation}\n\\label{eq:singular-CR-Yamabe-problem}\n\t\\begin{cases}\n\t\t(2 + 2 / n) \\Delta_{b} u + R_{\\theta} u = \\lambda u^{1 + 2 / n} \\text{\\ on $M \\setminus \\Lambda$}, \\\\\n\t\t\\lim_{x \\to \\Lambda} u(x) = \\infty,\n\t\\end{cases}\n\\end{equation}\nfor some constant $\\lambda \\in \\bbR$ and $0 < u \\in C^{\\infty}(M \\setminus \\Lambda)$.\nThe condition $\\lim_{x \\to \\Lambda} u(x) = \\infty$ is necessary for the contact form\n$\\whxth = u^{2 / n} \\theta$ to be complete on $M \\setminus \\Lambda$.\nChanillo and Yang~\\cite{Chanillo-Yang2025-preprint}*{Theorem 2.1} proved that $\\dim_{H} \\Lambda \\leq n$\nif there exists a solution of \\cref{eq:singular-CR-Yamabe-problem} with $\\lambda \\geq 0$,\nwhere $\\dim_{H}$ is the Hausdorff dimension with respect to the Carnot--Carath\\'{e}odory distance;\nsee \\cref{subsection:CR-manifolds} for the definition.\nRegarding existence results,\nNayatani~\\cite{Nayatani1999}*{p.\\ 221} constructed a complete contact form $\\theta^{\\prime}$\non $S^{2 n + 1} \\setminus S_{\\bbC}^{2 k + 1}$ with $R_{\\theta'} = (n + 1) (n - 2 k - 2)$,\nwhere $S_{\\bbC}^{2 k + 1}$ is the $\\bbC$-sphere\n\\begin{equation}\n\tS_{\\bbC}^{2 k + 1}\n\t\\coloneqq \\Set{(z^{1}, \\dots , z^{k + 1}, 0, \\dots , 0) \\in S^{2 n + 1} | z^{i} \\in \\bbC}.\n\\end{equation}\nGuidi, Maalaoui, and Martino~\\cite{Guidi-Maalaoui-Martino2024}*{Theorem 1.1}\nconstructed the same contact form by a different method.\nThey also constructed an infinite family of complete contact forms\nwith constant Tanaka--Webster scalar curvature on $S^{2 n + 1} \\setminus S_{\\bbC}^{1}$\nby using a bifurcation method~\\cite{Guidi-Maalaoui-Martino2024}*{Theorem 1.3}.\nFurthermore,\nthey proved that $S^{2 n + 1} \\setminus \\tau(S_{\\bbC}^{1})$ admits a complete contact form\nwith constant Tanaka--Webster scalar curvature\nif $\\tau \\in \\Diff(S^{2 n + 1})$ satisfies certain conditions \\cite{Guidi-Maalaoui-Martino2024}*{Theorem 1.2}.\nAfeltra~\\cite{Afeltra2020}*{Theorem 1.1} constructed infinitely many $\\bbZ$-periodic\ncontact forms on $S^{2 n + 1} \\setminus \\{p, - p\\}$ via the Lyapunov--Schmidt method.\n\nIn this paper,\nwe establish the existence of infinitely many pairwise\nnon-homothetic complete contact forms with constant Tanaka--Webster scalar curvature\non certain non-compact strictly pseudoconvex CR manifolds.\nIn particular,\nwe prove that the geometric moduli space is infinite\non complements of $\\bbR$- and $\\bbC$-spheres in the CR sphere.\nFrom this perspective,\nour results can be viewed as CR analogues of the existence of\ninfinitely many non-homothetic complete constant scalar curvature metrics\nin the non-compact setting of conformal geometry.\n\nTo explain our result in the general setting,\nwe start with a compact, connected, strictly pseudoconvex CR manifold $(M, T^{1, 0} M)$ of dimension $2 n + 1$\nsatisfying the following assumption:\n\n\\begin{keyassumption}{$(\\blacklozenge)$}\n\\label{keyassumption}\n\tAssume that the CR Yamabe constant of $(M, T^{1, 0} M)$ is positive.\n\tIf $\\dim M = 3$,\n\tthen we further assume that $(M, T^{1, 0} M)$ is universally embeddable;\n\tsee \\cref{subsection:CR-manifolds} for definition.\n\tIf $\\dim M = 5$ and $(M, T^{1, 0} M)$ is spherical,\n\tthen we additionally assume that the developing map $\\Phi \\colon \\wtM \\to S^{5}$ is injective.\n\\end{keyassumption}\n\nThese additional assumptions in low dimensions are imposed\nto guarantee the existence of CR Yamabe minimizers on finite covers of $M$.\nIf the fundamental group $\\pi_{1}(M)$ of $M$ has infinitely many normal subgroups of finite index,\nthen there exist infinitely many pairwise non-homothetic periodic contact forms on $(\\wtM, T^{1, 0} \\wtM)$.", "context": "In this paper,\nwe consider the non-compact case.\nLet $(M, T^{1, 0} M)$ be a (possibly non-compact) connected strictly pseudoconvex CR manifold.\nDenote by $\\CSC(M, T^{1, 0} M)$ the set of complete contact forms on $M$\nwith constant Tanaka--Webster scalar curvature.\nWe say that two contact forms $\\theta_{1}$ and $\\theta_{2}$ on $M$ are \\emph{homothetic}\nif there exist a CR diffeomorphism $\\varphi$ and a constant $c > 0$\nsuch that $\\varphi^{\\ast} \\theta_{1} = c \\theta_{2}$;\notherwise they are said to be \\emph{non-homothetic}.\nThe \\emph{geometric moduli space} of complete contact forms\nwith constant Tanaka--Webster scalar curvature on $(M, T^{1, 0} M)$ is the quotient\n\\begin{equation}\n \\GModuli(M, T^{1, 0} M)\n \\coloneqq \\CSC(M, T^{1, 0} M) / \\sim,\n\\end{equation}\nwhere $\\theta_{1} \\sim \\theta_{2}$ if $\\theta_{1}$ and $\\theta_{2}$ are homothetic.\nJerison and Lee proved that $\\# \\GModuli(M, T^{1, 0} M) = 1$\nif $M$ is compact and $Y(M, T^{1, 0} M) \\leq 0$~\\cite{Jerison-Lee1987}*{Theorem 7.1}\nor $(M, T^{1, 0} M) = (S^{2 n + 1}, T^{1, 0} S^{2 n + 1})$~\\cite{Jerison-Lee1988}*{Theorem A}.\nIt is also known that $\\# \\GModuli(M, T^{1, 0} M) = 1$\nif $M$ is compact and admits an Einstein contact form~\\cite{Wang2015}*{Theorem 4}.\n\nA typical non-compact situation arises when one considers the complement of a closed subset\nin a compact strictly pseudoconvex CR manifold.\nLet $(M, T^{1, 0} M)$ be a compact, connected, strictly pseudoconvex CR manifold of dimension $2 n + 1$,\nlet $\\theta$ be a contact form on $M$,\nand let $\\Lambda$ be a closed subset of $M$.\nOne seeks complete contact forms with constant Tanaka--Webster scalar curvature on $M \\setminus \\Lambda$;\nwe abbreviate the CR structure $T^{1, 0} (M \\setminus \\Lambda)$ for simplicity.\nThis leads to the \\emph{singular CR Yamabe equation}\n\\begin{equation}\n\\label{eq:singular-CR-Yamabe-problem}\n \\begin{cases}\n (2 + 2 / n) \\Delta_{b} u + R_{\\theta} u = \\lambda u^{1 + 2 / n} \\text{\\ on $M \\setminus \\Lambda$}, \\\\\n \\lim_{x \\to \\Lambda} u(x) = \\infty,\n \\end{cases}\n\\end{equation}\nfor some constant $\\lambda \\in \\bbR$ and $0 < u \\in C^{\\infty}(M \\setminus \\Lambda)$.\nThe condition $\\lim_{x \\to \\Lambda} u(x) = \\infty$ is necessary for the contact form\n$\\whxth = u^{2 / n} \\theta$ to be complete on $M \\setminus \\Lambda$.\nChanillo and Yang~\\cite{Chanillo-Yang2025-preprint}*{Theorem 2.1} proved that $\\dim_{H} \\Lambda \\leq n$\nif there exists a solution of \\cref{eq:singular-CR-Yamabe-problem} with $\\lambda \\geq 0$,\nwhere $\\dim_{H}$ is the Hausdorff dimension with respect to the Carnot--Carath\\'{e}odory distance;\nsee \\cref{subsection:CR-manifolds} for the definition.\nRegarding existence results,\nNayatani~\\cite{Nayatani1999}*{p.\\ 221} constructed a complete contact form $\\theta^{\\prime}$\non $S^{2 n + 1} \\setminus S_{\\bbC}^{2 k + 1}$ with $R_{\\theta'} = (n + 1) (n - 2 k - 2)$,\nwhere $S_{\\bbC}^{2 k + 1}$ is the $\\bbC$-sphere\n\\begin{equation}\n S_{\\bbC}^{2 k + 1}\n \\coloneqq \\Set{(z^{1}, \\dots , z^{k + 1}, 0, \\dots , 0) \\in S^{2 n + 1} | z^{i} \\in \\bbC}.\n\\end{equation}\nGuidi, Maalaoui, and Martino~\\cite{Guidi-Maalaoui-Martino2024}*{Theorem 1.1}\nconstructed the same contact form by a different method.\nThey also constructed an infinite family of complete contact forms\nwith constant Tanaka--Webster scalar curvature on $S^{2 n + 1} \\setminus S_{\\bbC}^{1}$\nby using a bifurcation method~\\cite{Guidi-Maalaoui-Martino2024}*{Theorem 1.3}.\nFurthermore,\nthey proved that $S^{2 n + 1} \\setminus \\tau(S_{\\bbC}^{1})$ admits a complete contact form\nwith constant Tanaka--Webster scalar curvature\nif $\\tau \\in \\Diff(S^{2 n + 1})$ satisfies certain conditions \\cite{Guidi-Maalaoui-Martino2024}*{Theorem 1.2}.\nAfeltra~\\cite{Afeltra2020}*{Theorem 1.1} constructed infinitely many $\\bbZ$-periodic\ncontact forms on $S^{2 n + 1} \\setminus \\{p, - p\\}$ via the Lyapunov--Schmidt method.\n\nTo explain our result in the general setting,\nwe start with a compact, connected, strictly pseudoconvex CR manifold $(M, T^{1, 0} M)$ of dimension $2 n + 1$\nsatisfying the following assumption:\n\n\\begin{keyassumption}{$(\\blacklozenge)$}\n\\label{keyassumption}\n Assume that the CR Yamabe constant of $(M, T^{1, 0} M)$ is positive.\n If $\\dim M = 3$,\n then we further assume that $(M, T^{1, 0} M)$ is universally embeddable;\n see \\cref{subsection:CR-manifolds} for definition.\n If $\\dim M = 5$ and $(M, T^{1, 0} M)$ is spherical,\n then we additionally assume that the developing map $\\Phi \\colon \\wtM \\to S^{5}$ is injective.\n\\end{keyassumption}\n\nThese additional assumptions in low dimensions are imposed\nto guarantee the existence of CR Yamabe minimizers on finite covers of $M$.\nIf the fundamental group $\\pi_{1}(M)$ of $M$ has infinitely many normal subgroups of finite index,\nthen there exist infinitely many pairwise non-homothetic periodic contact forms on $(\\wtM, T^{1, 0} \\wtM)$.", "full_context": "In this paper,\nwe consider the non-compact case.\nLet $(M, T^{1, 0} M)$ be a (possibly non-compact) connected strictly pseudoconvex CR manifold.\nDenote by $\\CSC(M, T^{1, 0} M)$ the set of complete contact forms on $M$\nwith constant Tanaka--Webster scalar curvature.\nWe say that two contact forms $\\theta_{1}$ and $\\theta_{2}$ on $M$ are \\emph{homothetic}\nif there exist a CR diffeomorphism $\\varphi$ and a constant $c > 0$\nsuch that $\\varphi^{\\ast} \\theta_{1} = c \\theta_{2}$;\notherwise they are said to be \\emph{non-homothetic}.\nThe \\emph{geometric moduli space} of complete contact forms\nwith constant Tanaka--Webster scalar curvature on $(M, T^{1, 0} M)$ is the quotient\n\\begin{equation}\n \\GModuli(M, T^{1, 0} M)\n \\coloneqq \\CSC(M, T^{1, 0} M) / \\sim,\n\\end{equation}\nwhere $\\theta_{1} \\sim \\theta_{2}$ if $\\theta_{1}$ and $\\theta_{2}$ are homothetic.\nJerison and Lee proved that $\\# \\GModuli(M, T^{1, 0} M) = 1$\nif $M$ is compact and $Y(M, T^{1, 0} M) \\leq 0$~\\cite{Jerison-Lee1987}*{Theorem 7.1}\nor $(M, T^{1, 0} M) = (S^{2 n + 1}, T^{1, 0} S^{2 n + 1})$~\\cite{Jerison-Lee1988}*{Theorem A}.\nIt is also known that $\\# \\GModuli(M, T^{1, 0} M) = 1$\nif $M$ is compact and admits an Einstein contact form~\\cite{Wang2015}*{Theorem 4}.\n\nA typical non-compact situation arises when one considers the complement of a closed subset\nin a compact strictly pseudoconvex CR manifold.\nLet $(M, T^{1, 0} M)$ be a compact, connected, strictly pseudoconvex CR manifold of dimension $2 n + 1$,\nlet $\\theta$ be a contact form on $M$,\nand let $\\Lambda$ be a closed subset of $M$.\nOne seeks complete contact forms with constant Tanaka--Webster scalar curvature on $M \\setminus \\Lambda$;\nwe abbreviate the CR structure $T^{1, 0} (M \\setminus \\Lambda)$ for simplicity.\nThis leads to the \\emph{singular CR Yamabe equation}\n\\begin{equation}\n\\label{eq:singular-CR-Yamabe-problem}\n \\begin{cases}\n (2 + 2 / n) \\Delta_{b} u + R_{\\theta} u = \\lambda u^{1 + 2 / n} \\text{\\ on $M \\setminus \\Lambda$}, \\\\\n \\lim_{x \\to \\Lambda} u(x) = \\infty,\n \\end{cases}\n\\end{equation}\nfor some constant $\\lambda \\in \\bbR$ and $0 < u \\in C^{\\infty}(M \\setminus \\Lambda)$.\nThe condition $\\lim_{x \\to \\Lambda} u(x) = \\infty$ is necessary for the contact form\n$\\whxth = u^{2 / n} \\theta$ to be complete on $M \\setminus \\Lambda$.\nChanillo and Yang~\\cite{Chanillo-Yang2025-preprint}*{Theorem 2.1} proved that $\\dim_{H} \\Lambda \\leq n$\nif there exists a solution of \\cref{eq:singular-CR-Yamabe-problem} with $\\lambda \\geq 0$,\nwhere $\\dim_{H}$ is the Hausdorff dimension with respect to the Carnot--Carath\\'{e}odory distance;\nsee \\cref{subsection:CR-manifolds} for the definition.\nRegarding existence results,\nNayatani~\\cite{Nayatani1999}*{p.\\ 221} constructed a complete contact form $\\theta^{\\prime}$\non $S^{2 n + 1} \\setminus S_{\\bbC}^{2 k + 1}$ with $R_{\\theta'} = (n + 1) (n - 2 k - 2)$,\nwhere $S_{\\bbC}^{2 k + 1}$ is the $\\bbC$-sphere\n\\begin{equation}\n S_{\\bbC}^{2 k + 1}\n \\coloneqq \\Set{(z^{1}, \\dots , z^{k + 1}, 0, \\dots , 0) \\in S^{2 n + 1} | z^{i} \\in \\bbC}.\n\\end{equation}\nGuidi, Maalaoui, and Martino~\\cite{Guidi-Maalaoui-Martino2024}*{Theorem 1.1}\nconstructed the same contact form by a different method.\nThey also constructed an infinite family of complete contact forms\nwith constant Tanaka--Webster scalar curvature on $S^{2 n + 1} \\setminus S_{\\bbC}^{1}$\nby using a bifurcation method~\\cite{Guidi-Maalaoui-Martino2024}*{Theorem 1.3}.\nFurthermore,\nthey proved that $S^{2 n + 1} \\setminus \\tau(S_{\\bbC}^{1})$ admits a complete contact form\nwith constant Tanaka--Webster scalar curvature\nif $\\tau \\in \\Diff(S^{2 n + 1})$ satisfies certain conditions \\cite{Guidi-Maalaoui-Martino2024}*{Theorem 1.2}.\nAfeltra~\\cite{Afeltra2020}*{Theorem 1.1} constructed infinitely many $\\bbZ$-periodic\ncontact forms on $S^{2 n + 1} \\setminus \\{p, - p\\}$ via the Lyapunov--Schmidt method.\n\nTo explain our result in the general setting,\nwe start with a compact, connected, strictly pseudoconvex CR manifold $(M, T^{1, 0} M)$ of dimension $2 n + 1$\nsatisfying the following assumption:\n\n\\begin{keyassumption}{$(\\blacklozenge)$}\n\\label{keyassumption}\n Assume that the CR Yamabe constant of $(M, T^{1, 0} M)$ is positive.\n If $\\dim M = 3$,\n then we further assume that $(M, T^{1, 0} M)$ is universally embeddable;\n see \\cref{subsection:CR-manifolds} for definition.\n If $\\dim M = 5$ and $(M, T^{1, 0} M)$ is spherical,\n then we additionally assume that the developing map $\\Phi \\colon \\wtM \\to S^{5}$ is injective.\n\\end{keyassumption}\n\nThese additional assumptions in low dimensions are imposed\nto guarantee the existence of CR Yamabe minimizers on finite covers of $M$.\nIf the fundamental group $\\pi_{1}(M)$ of $M$ has infinitely many normal subgroups of finite index,\nthen there exist infinitely many pairwise non-homothetic periodic contact forms on $(\\wtM, T^{1, 0} \\wtM)$.\n\nThese additional assumptions in low dimensions are imposed\nto guarantee the existence of CR Yamabe minimizers on finite covers of $M$.\nIf the fundamental group $\\pi_{1}(M)$ of $M$ has infinitely many normal subgroups of finite index,\nthen there exist infinitely many pairwise non-homothetic periodic contact forms on $(\\wtM, T^{1, 0} \\wtM)$.\n\nThe novelty of our work lies in the fact that\nour results do not follow from existing constructions in the literature\nand rely on a different geometric mechanism.\nIn particular,\nour approach applies to examples that are not accessible by bifurcation or ODE-based methods,\nwhich have played a central role in previous studies.\n\nOur proof is similar to that of \\cite{Andrade-Case-Piccione-Wei2025-preprint}*{Theorem 3.3}.\nSince $\\pi_{1}(M)$ has infinite profinite completion,\nwe can construct a tower of finite connected coverings over $M$.\n\\Cref{keyassumption} implies that each such cover has a positive CR Yamabe minimizer.\nIts lift to the universal cover defines a periodic contact form\nwith positive constant Tanaka--Webster scalar curvature.\nSuppose to the contrary that $\\# \\GModuli(\\wtM, T^{1, 0} \\wtM) < \\infty$.\nThen the CR diffeomorphism group $\\Aut_{\\CR}(\\wtM, T^{1, 0} \\wtM)$ does not act properly.\nBy a result of Schoen~\\cite{Schoen1995},\nit follows that $(\\wtM, T^{1, 0} \\wtM)$ is CR diffeomorphic to the Heisenberg group.\nWe conclude from the characterization of CR automorphisms of the Heisenberg group that\n$Y(M, T^{1, 0} M) = 0$, a contradiction.\n\n\\begin{proposition}\n\\label{prop:existence-of-tower-implies-Heisenberg}\n Let $(M, T^{1, 0} M)$ be a compact, connected, strictly pseudoconvex CR manifold of dimension $2 n + 1$.\n Suppose there are an infinite tower $(\\pi_{j} \\colon M_{j} \\to M_{j - 1})_{j = 1}^{\\infty}$\n of finite connected coverings of degree $(m_{j})_{j = 1}^{\\infty}$ over $M$\n and a sequence $(\\theta_{j})_{j = 0}^{\\infty}$\n of unit volume CR Yamabe minimizers on $(M_{j}, T^{1, 0} M_{j})$\n such that for each $j \\in \\bbZ_{> 0}$,\n there exist $\\Phi_{j} \\in \\Aut_{\\CR}(\\wtM, T^{1, 0} \\wtM)$ and $c_{j} \\in \\bbR_{> 0}$\n satisfying $\\Phi_{j}^{\\ast} \\wtxp^{\\ast} \\theta_{0} = c_{j} \\wtxp_{j}^{\\ast} \\theta_{j}$.\n Then $Y(M, T^{1, 0} M) \\leq 0$.\n\\end{proposition}\n\n\\begin{proposition}\n\\label{prop:existence-of-non-homothetic}\n Let $(M, T^{1, 0} M)$ be a compact, connected, strictly pseudoconvex CR manifold of dimension $2 n + 1$\n such that \\Cref{keyassumption} holds and $\\pi_{1}(M)$ has infinite profinite completion.\n Then there exist a regular infinite tower $(\\pi_{j} \\colon M_{j} \\to M_{j - 1})_{j = 1}^{\\infty}$\n of finite connected coverings over $M$\n and a sequence $(\\theta_{j})_{j = 0}^{\\infty}$\n of unit volume CR Yamabe minimizers on $(M_{j}, T^{1, 0} M_{j})$\n such that for any $j \\in \\bbZ_{> 0}$,\n the contact forms $((\\Pi_{j}^{l})^{\\ast} \\theta_{l})_{l = 0}^{j}$ on $M_{j}$\n are pairwise non-homothetic\n and have positive constant Tanaka--Webster scalar curvature.\n\\end{proposition}\n\n\\begin{proof}[Proof of \\cref{thm:infinite-geometric-moduli-real-hyperbolic}]\n Since $S^{2 n + 1} \\setminus S_{\\bbR}^{k}$ is simply connected,\n $\\pi_{1}(M_{\\Gamma}) \\cong \\Gamma$\n and $\\wtM_{\\Gamma}$ is CR diffeomorphic to $S^{2 n + 1} \\setminus S_{\\bbR}^{k}$.\n In particular,\n the developing map of $M_{\\Gamma}$ is injective.\n We need to check that \\Cref{keyassumption} holds.\n It follows from $\\delta_{\\Gamma} < n$\n that $Y(M_{\\Gamma}, T^{1, 0} M_{\\Gamma}) > 0$.\n As we noted,\n the developing map of $M_{\\Gamma}$ is injective.\n Moreover,\n $M_{\\Gamma}$ is universally embeddable\n since any finite connected cover of $M_{\\Gamma}$ is of the form $M_{\\Gamma^{\\prime}}$\n for a subgroup $\\Gamma^{\\prime} \\subset \\Gamma$ of finite index.\n Furthermore,\n $\\pi_{1}(M_{\\Gamma})$ has infinite profinite completion\n since it is infinite and residually finite according to \\cref{prop:residually-finite-PU}.\n \\cref{thm:infinite-geometric-moduli-general} implies that\n $\\# \\GModuli(S^{2 n + 1} \\setminus S_{\\bbR}^{k}) = \\infty$.\n\\end{proof}\n\n\\begin{proof}[Proof of \\cref{thm:infinite-geometric-moduli-complex-hyperbolic}]\n Since $S^{2 n + 1} \\setminus S_{\\bbC}^{2 k + 1}$ is simply connected,\n $\\pi_{1}(M_{\\Gamma}) \\cong \\Gamma$\n and $\\wtM_{\\Gamma}$ is CR diffeomorphic to $S^{2 n + 1} \\setminus S_{\\bbC}^{2 k + 1}$.\n It follows from $\\delta_{\\Gamma} < n$\n that $Y(M_{\\Gamma}, T^{1, 0} M_{\\Gamma})$ is positive,\n which guarantees \\Cref{keyassumption}.\n Moreover,\n $\\pi_{1}(M_{\\Gamma})$ has infinite profinite completion\n since it is infinite and residually finite according to \\cref{prop:residually-finite-PU}.\n \\cref{thm:infinite-geometric-moduli-general} implies that\n $\\# \\GModuli(S^{2 n + 1} \\setminus S_{\\bbC}^{2 k + 1}) = \\infty$.\n\\end{proof}\n\nAssume that $n \\geq 2$.\nLet $M$ be the boundary of the bounded Reinhardt domain\n\\begin{equation}\n \\Omega\n \\coloneqq \\Set{ w = (w^{1}, \\dots , w^{n + 1}) \\in \\bbC^{n + 1} | \\rho(w) < 0 },\n\\end{equation}\nwhere\n\\begin{equation}\n \\rho(w) \\coloneqq \\sum_{j = 1}^{n + 1} (\\log \\abs{w^{j}})^{2} - 1.\n\\end{equation}\nThis $M$ has the contact form $\\theta \\coloneqq d^{c} \\rho |_{M}$.\nConsider the holomorphic map\n\\begin{equation}\n \\Psi \\colon \\bbC^{n + 1} \\to \\bbC^{n + 1}; \\qquad\n (z^{1}, \\dots , z^{n + 1}) \\mapsto (\\exp z^{1}, \\dots , \\exp z^{n + 1}).\n\\end{equation}\nThe pull-back $\\Psi^{*} \\rho$ coincides with\n\\begin{equation}\n \\wtxr(z)\n \\coloneqq \\sum_{j = 1}^{n + 1} (\\Re z^{j})^{2} - 1,\n\\end{equation}\nand the pre-image of $\\Omega$ by $\\Psi$ is the tube domain\n\\begin{equation}\n \\wtxco\n \\coloneqq \\Set{ z = (z^{1}, \\dots , z^{n + 1}) \\in \\bbC^{n + 1} |\n \\sum_{j = 1}^{n + 1} (\\Re z^{j})^{2} < 1 }.\n\\end{equation}\nThe holomorphic map $\\Psi$ induces a CR map $\\psi \\colon \\wtM \\coloneqq \\bdry \\wtxco \\to M$.\nNote that $\\psi \\colon \\wtM \\to M$ is the universal covering map.\nThe authors~\\cite{Case-Takeuchi2023}*{Section 8} proved that\n$R_{\\theta}$ is a positive constant and the Chern tensor is nowhere vanishing.\nIn particular,\n$Y(M, T^{1, 0} M) > 0$ and $(M, T^{1, 0} M)$ is not spherical,\nwhich implies that $(M, T^{1, 0} M)$ satisfies \\Cref{keyassumption}.\nThe fundamental group $\\pi_{1}(M)$ of $M$ is isomorphic to $\\bbZ^{n + 1}$,\nwhich has infinite profinite completion.\nTherefore \\cref{thm:infinite-geometric-moduli-general} implies $\\# \\GModuli(\\wtM, T^{1, 0} \\wtM) = \\infty$.\n\n\\begin{keyassumption}{$(\\blacklozenge)$}\n\\label{keyassumption}\n\tAssume that the CR Yamabe constant of $(M, T^{1, 0} M)$ is positive.\n\tIf $\\dim M = 3$,\n\tthen we further assume that $(M, T^{1, 0} M)$ is universally embeddable;\n\tsee \\cref{subsection:CR-manifolds} for definition.\n\tIf $\\dim M = 5$ and $(M, T^{1, 0} M)$ is spherical,\n\tthen we additionally assume that the developing map $\\Phi \\colon \\wtM \\to S^{5}$ is injective.\n\\end{keyassumption}\n\n\\begin{theorem}\n\\label{thm:infinite-geometric-moduli-general}\n\tLet $(M, T^{1, 0} M)$ be a compact, connected, strictly pseudoconvex CR manifold of dimension $2 n + 1$\n\tsuch that \\Cref{keyassumption} holds\n\tand $\\pi_{1}(M)$ has infinite profinite completion.\n\tThen $\\# \\GModuli(\\wtM, T^{1, 0} \\wtM) = \\infty$.\n\\end{theorem}", "post_theorem_intro_text_len": 4765, "post_theorem_intro_text": "The novelty of our work lies in the fact that\nour results do not follow from existing constructions in the literature\nand rely on a different geometric mechanism.\nIn particular,\nour approach applies to examples that are not accessible by bifurcation or ODE-based methods,\nwhich have played a central role in previous studies.\n\nOur proof is similar to that of \\cite{Andrade-Case-Piccione-Wei2025-preprint}*{Theorem 3.3}.\nSince $\\pi_{1}(M)$ has infinite profinite completion,\nwe can construct a tower of finite connected coverings over $M$.\n\\Cref{keyassumption} implies that each such cover has a positive CR Yamabe minimizer.\nIts lift to the universal cover defines a periodic contact form\nwith positive constant Tanaka--Webster scalar curvature.\nSuppose to the contrary that $\\# \\GModuli(\\wtM, T^{1, 0} \\wtM) < \\infty$.\nThen the CR diffeomorphism group $\\Aut_{\\CR}(\\wtM, T^{1, 0} \\wtM)$ does not act properly.\nBy a result of Schoen~\\cite{Schoen1995},\nit follows that $(\\wtM, T^{1, 0} \\wtM)$ is CR diffeomorphic to the Heisenberg group.\nWe conclude from the characterization of CR automorphisms of the Heisenberg group that\n$Y(M, T^{1, 0} M) = 0$, a contradiction.\n\nAs an application,\nconsider the complements of an $\\bbR$-sphere or a $\\bbC$-sphere in $S^{2 n + 1}$,\nwhere the $\\bbR$-sphere\n\\begin{equation}\n\tS_{\\bbR}^{k}\n\t\\coloneqq \\Set{(x^{1}, \\dots , x^{k + 1}, 0, \\dots , 0) \\in S^{2 n + 1} | x^{i} \\in \\bbR}\n\\end{equation}\nis the intersection of the totally real linear subspace $\\bbR^{k + 1} \\times \\{0\\}$ with $S^{2 n + 1}$.\nThere exists a torsion-free discrete subgroup $\\Gamma$ of $\\Aut_{\\CR}(S^{2 n + 1} \\setminus S_{\\bbR}^{k})$\nsuch that the quotient of $S^{2 n + 1} \\setminus S_{\\bbR}^{k}$ by $\\Gamma$\nsatisfies the assumptions of \\cref{thm:infinite-geometric-moduli-general}.\n\n\\begin{theorem}\n\\label{thm:infinite-geometric-moduli-real-hyperbolic}\n\tThe complement $S^{2 n + 1} \\setminus S_{\\bbR}^{k}$ satisfies\n\t$\\# \\GModuli(S^{2 n + 1} \\setminus S_{\\bbR}^{k}) = \\infty$ if $0 \\leq k \\leq n - 1$.\n\\end{theorem}\n\nSimilarly,\nwe consider the $\\bbC$-sphere $S_{\\bbC}^{2 k + 1}$,\nwhich is the intersection of the complex linear subspace $\\bbC^{k + 1} \\times \\{0\\}$ with $S^{2 n + 1}$.\nA similar argument to that for the $\\bbR$-sphere implies the following result.\n\n\\begin{theorem}\n\\label{thm:infinite-geometric-moduli-complex-hyperbolic}\n\tThe complement $S^{2 n + 1} \\setminus S_{\\bbC}^{2 k + 1}$ satisfies\n\t$\\# \\GModuli(S^{2 n + 1} \\setminus S_{\\bbC}^{2 k + 1}) = \\infty$ if $0 \\leq k < (n - 2) / 2$.\n\\end{theorem}\n\nWe add some remarks on these theorems.\nLet $V$ be a real linear subspace in $\\bbC^{n + 1}$\nand set $\\Lambda \\coloneqq V \\cap S^{2 n + 1} \\subset S^{2 n + 1}$.\nIf $V$ is totally real or complex and $\\dim_{H} \\Lambda < n$,\nthen we proved that $\\# \\GModuli(S^{2 n + 1} \\setminus \\Lambda) = \\infty$.\nThe key point of the proof is that\nthere exists a torsion-free convex cocompact discrete subgroup $\\Gamma$ of $PU(n + 1, 1)$\nwhose limit set coincides with $\\Lambda$.\nThen the quotient of $S^{2 n + 1} \\setminus \\Lambda$ by $\\Gamma$ satisfies \\Cref{keyassumption}.\nOn the other hand,\nwhen $V$ is neither totally real nor complex,\nthere exist no discrete subgroups in $PU(n + 1, 1)$\nsuch that its limit set coincides with $\\Lambda$,\nand hence our argument does not apply.\nAs far as we know,\nthere are no existence results for complete contact forms\nwith constant Tanaka--Webster scalar curvature on $S^{2 n + 1} \\setminus \\Lambda$\n\nWe will also apply \\cref{thm:infinite-geometric-moduli-general}\nto circle bundles over compact \\Kahler manifolds (\\cref{thm:infinite-geometric-moduli-circle-bundle})\nand the boundary of a Reinhardt domain (\\cref{subsection:boundary-of-Reinhardt-domain}),\nwhich is neither spherical nor Sasakian.\n\nThis paper is organized as follows.\nIn \\cref{section:CR-geometry},\nwe review basic materials in CR geometry.\nIn \\cref{section:CR-Yamabe-problem},\nwe recall the CR Yamabe problem and summarize known existence results that motivate the present work.\nIn \\cref{section:complex-hyperbolic-geometry},\nwe collect several facts from complex hyperbolic geometry\nthat are needed to analyze complements of $\\bbR$- and $\\bbC$-spheres in the CR sphere.\nIn \\cref{section:infinite-tower},\nwe study an existence problem for infinite towers of finite connected coverings;\nthis constitutes the technical core of the paper.\nIn \\cref{section:proof-of-main-result},\nwe use this framework to prove \\cref{thm:infinite-geometric-moduli-general}.\nIn \\cref{section:complements-of-spheres},\nwe apply this theorem to complements of $\\bbR$- and $\\bbC$-spheres in the CR sphere.\nFinally, in \\cref{section:other-applications},\nwe discuss further applications to circle bundles over compact \\Kahler manifolds\nand to the boundary of a Reinhardt domain.", "sketch": "The proof of Theorem~\\ref{thm:infinite-geometric-moduli-general} is described as being \"similar to that of \\cite{Andrade-Case-Piccione-Wei2025-preprint}*{Theorem 3.3}.\" Since $\\pi_{1}(M)$ has infinite profinite completion, \"we can construct a tower of finite connected coverings over $M$.\" The assumption \\Cref{keyassumption} \"implies that each such cover has a positive CR Yamabe minimizer,\" and \"[i]ts lift to the universal cover defines a periodic contact form with positive constant Tanaka--Webster scalar curvature.\"\n\nAssuming for contradiction that \"$\\# \\GModuli(\\wtM, T^{1, 0} \\wtM) < \\infty$,\" it follows that \"the CR diffeomorphism group $\\Aut_{\\CR}(\\wtM, T^{1, 0} \\wtM)$ does not act properly.\" By Schoen's result \\cite{Schoen1995}, \"it follows that $(\\wtM, T^{1, 0} \\wtM)$ is CR diffeomorphic to the Heisenberg group.\" Using \"the characterization of CR automorphisms of the Heisenberg group,\" they conclude that \"$Y(M, T^{1, 0} M) = 0$, a contradiction.\"", "expanded_sketch": "In establishing the main theorem, the proof is described as being “similar to that of Andrade–Case–Piccione–Wei, *Andrade-Case-Piccione-Wei2025-preprint* (2025), Theorem 3.3.” Since $\\pi_{1}(M)$ has infinite profinite completion, “we can construct a tower of finite connected coverings over $M$.”\n\nWe use the following assumption.\n\\begin{keyassumption}{$(\\blacklozenge)$}\n\\label{keyassumption}\n\tAssume that the CR Yamabe constant of $(M, T^{1, 0} M)$ is positive.\n\tIf $\\dim M = 3$,\n\tthen we further assume that $(M, T^{1, 0} M)$ is universally embeddable;\n\tsee \\cref{subsection:CR-manifolds} for definition.\n\tIf $\\dim M = 5$ and $(M, T^{1, 0} M)$ is spherical,\n\tthen we additionally assume that the developing map $\\Phi \\colon \\wtM \\to S^{5}$ is injective.\n\\end{keyassumption}\nThis assumption “implies that each such cover has a positive CR Yamabe minimizer,” and “[i]ts lift to the universal cover defines a periodic contact form with positive constant Tanaka–Webster scalar curvature.”\n\nAssuming for contradiction that “$\\# \\GModuli(\\wtM, T^{1, 0} \\wtM) < \\infty$,” it follows that “the CR diffeomorphism group $\\Aut_{\\CR}(\\wtM, T^{1, 0} \\wtM)$ does not act properly.” By Schoen’s result (\\cite{Schoen1995}), “it follows that $(\\wtM, T^{1, 0} \\wtM)$ is CR diffeomorphic to the Heisenberg group.” Using “the characterization of CR automorphisms of the Heisenberg group,” they conclude that “$Y(M, T^{1, 0} M) = 0$, a contradiction,” and this completes the proof of the main theorem.", "expanded_theorem": "\\label{thm:infinite-geometric-moduli-general}\n\tLet $(M, T^{1, 0} M)$ be a compact, connected, strictly pseudoconvex CR manifold of dimension $2 n + 1$\n\tsuch that the following key assumption holds.\n\n\\begin{keyassumption}{$(\\blacklozenge)$}\n\\label{keyassumption}\n\tAssume that the CR Yamabe constant of $(M, T^{1, 0} M)$ is positive.\n\tIf $\\dim M = 3$,\n\tthen we further assume that $(M, T^{1, 0} M)$ is universally embeddable;\n\tsee \\cref{subsection:CR-manifolds} for definition.\n\tIf $\\dim M = 5$ and $(M, T^{1, 0} M)$ is spherical,\n\tthen we additionally assume that the developing map $\\Phi \\colon \\wtM \\to S^{5}$ is injective.\n\\end{keyassumption}\n\n\tand $\\pi_{1}(M)$ has infinite profinite completion.\n\tThen $\\# \\GModuli(\\wtM, T^{1, 0} \\wtM) = \\infty$., ", "theorem_type": ["Implication", "Existence"], "mcq": {"question": "Let $(M,T^{1,0}M)$ be a compact, connected, strictly pseudoconvex CR manifold of dimension $2n+1$, and let $\\widetilde M$ denote its universal cover. Assume that the CR Yamabe constant of $(M,T^{1,0}M)$ is positive. In addition, if $\\dim M=3$, assume that $(M,T^{1,0}M)$ is universally embeddable; and if $\\dim M=5$ and $(M,T^{1,0}M)$ is spherical, assume that the developing map $\\Phi:\\widetilde M\\to S^5$ is injective. Also assume that the fundamental group $\\pi_1(M)$ has infinite profinite completion. For a CR manifold $X$, let $\\CSC(X,T^{1,0}X)$ be the set of complete contact forms on $X$ with constant Tanaka--Webster scalar curvature, and define the geometric moduli space\n\\[\n\\GModuli(X,T^{1,0}X):=\\CSC(X,T^{1,0}X)/\\sim,\n\\]\nwhere $\\theta_1\\sim\\theta_2$ means that there exist a CR diffeomorphism $\\varphi$ and a constant $c>0$ such that $\\varphi^*\\theta_1=c\\theta_2$. Which conclusion about $\\GModuli(\\widetilde M,T^{1,0}\\widetilde M)$ is valid under these hypotheses?", "correct_choice": {"label": "A", "text": "The geometric moduli space on the universal cover is infinite: \\[\\#\\GModuli(\\widetilde M,T^{1,0}\\widetilde M)=\\infty.\\] Equivalently, $\\widetilde M$ carries infinitely many pairwise non-homothetic complete contact forms with constant Tanaka--Webster scalar curvature."}, "choices": [{"label": "B", "text": "The geometric moduli space on the universal cover is finite and nonempty: \\[1\\le \\#\\GModuli(\\widetilde M,T^{1,0}\\widetilde M)<\\infty.\\] In particular, $\\widetilde M$ admits only finitely many homothety classes of complete contact forms with constant Tanaka--Webster scalar curvature."}, {"label": "C", "text": "The geometric moduli space on the universal cover is nonempty: \\[\\#\\GModuli(\\widetilde M,T^{1,0}\\widetilde M)\\ge 1.\\] Equivalently, $\\widetilde M$ carries at least one complete contact form with constant Tanaka--Webster scalar curvature."}, {"label": "D", "text": "Every complete contact form on $\\widetilde M$ with constant Tanaka--Webster scalar curvature is homothetic to every other one; equivalently, \\[\\#\\GModuli(\\widetilde M,T^{1,0}\\widetilde M)=1.\\]"}, {"label": "E", "text": "The same infinitude conclusion already holds on the base manifold itself: \\[\\#\\GModuli(M,T^{1,0}M)=\\infty.\\] Equivalently, $M$ carries infinitely many pairwise non-homothetic complete contact forms with constant Tanaka--Webster scalar curvature."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "outer_automorphism", "tampered_component": "contradiction-from-finite-moduli", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped infinitude of distinct homothety classes", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "outer_automorphism", "tampered_component": "properness-vs-rigidity conclusion", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "universal-cover target of periodic lifts", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the conclusion or quote the exact result. It lists hypotheses and asks which existence statement follows, so the correct answer is not leaked directly."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the stem states the hypotheses of a specific result and asks for its conclusion. The correct option is basically the theorem statement itself."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact theorem conclusion from weaker true statements and stronger false upgrades, but the item mainly tests recognition/recall rather than genuine mathematical generation or derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically structured: one is a weaker true statement, others impose finiteness, periodicity, or unjustified strengthening. These reflect realistic failure modes in theorem interpretation."}, "total_score": 5, "overall_assessment": "Well-constructed in terms of answer hiding and distractor design, but it is primarily a theorem-restatement question with only moderate reasoning demand."}}
{"id": "2602.03254v1", "paper_link": "http://arxiv.org/abs/2602.03254v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "\\label{open and discrete thm}\n    Let $\\Omega\\subset\\mathbb{R}^n$ be a domain, let $y_0\\in\\mathbb{R}^n$, and let $p, q \\in [1, \\infty]$. Suppose that\n    $f\\in W^{1,n}_{\\rm loc}(\\Omega,\\mathbb{R}^n)$ has a value of finite distortion at $y_0\\in\\mathbb{R}^n$ with data $(K,\\Sigma)$, where $K:\\Omega\\to[1,\\infty)$ and $\\Sigma:\\Omega\\to[0,\\infty)$ are measurable functions such that\n    \\begin{equation}\\label{K,Sigma condition}\n        K\\in L^p_{\\rm loc}(\\Omega),\n        \\quad{\\rm and}\\quad \n        \\frac{\\Sigma}{K}\\in L^{q}_{\\rm loc}(\\Omega).\n    \\end{equation}\n    If $p > n-1$ and $p^{-1} + q^{-1} < 1$, then either $f\\equiv y_0$ a.e.\\ in $\\Omega$, or the continuous representative of $f$ satisfies the following conditions:\n    \\begin{itemize}[noitemsep]\n        \\item $f^{-1}\\{y_0\\}$ is a discrete subset of $\\Omega$,\n        \\item at every $x_0\\in f^{-1}\\{y_0\\}$ the local index $i(x_0,f)$ is positive, and\n        \\item for every neighborhood $U$ of a point $x_0\\in f^{-1}\\{y_0\\}$, we have $y_0\\in  {\\rm int}\\ f(U)$.\n    \\end{itemize}", "start_pos": 14372, "end_pos": 15433, "label": "open and discrete thm"}, "ref_dict": {"open and discrete thm": "\\begin{thm}\\label{open and discrete thm}\n    Let $\\Omega\\subset\\mathbb{R}^n$ be a domain, let $y_0\\in\\mathbb{R}^n$, and let $p, q \\in [1, \\infty]$. Suppose that\n    $f\\in W^{1,n}_{\\rm loc}(\\Omega,\\mathbb{R}^n)$ has a value of finite distortion at $y_0\\in\\mathbb{R}^n$ with data $(K,\\Sigma)$, where $K:\\Omega\\to[1,\\infty)$ and $\\Sigma:\\Omega\\to[0,\\infty)$ are measurable functions such that\n    \\begin{equation}\\label{K,Sigma condition}\n        K\\in L^p_{\\rm loc}(\\Omega),\n        \\quad{\\rm and}\\quad \n        \\frac{\\Sigma}{K}\\in L^{q}_{\\rm loc}(\\Omega).\n    \\end{equation}\n    If $p > n-1$ and $p^{-1} + q^{-1} < 1$, then either $f\\equiv y_0$ a.e.\\ in $\\Omega$, or the continuous representative of $f$ satisfies the following conditions:\n    \\begin{itemize}[noitemsep]\n        \\item $f^{-1}\\{y_0\\}$ is a discrete subset of $\\Omega$,\n        \\item at every $x_0\\in f^{-1}\\{y_0\\}$ the local index $i(x_0,f)$ is positive, and\n        \\item for every neighborhood $U$ of a point $x_0\\in f^{-1}\\{y_0\\}$, we have $y_0\\in  {\\rm int}\\ f(U)$.\n    \\end{itemize}\n\\end{thm}", "K,Sigma-distortion": "\\begin{align}\\label{K,Sigma-distortion}\n        |Df(x)|^n\\le K(x)J_f(x)+\\Sigma(x)\n    \\end{align}", "thm:lusin_N": "\\begin{thm}\\label{thm:lusin_N}\n    Let $\\Omega\\subset\\mathbb{R}^n$ be a domain, and let $p,q\\in[1,\\infty]$.\n    Suppose that\n    $f\\in W^{1,n}_{\\loc}(\\Omega,\\mathbb{R}^n)$ satisfies the $(K,\\Sigma)$-distortion inequality with defect or has a value of finite distortion at a point $y_0 \\in \\R^n$ with data $(K, \\Sigma)$, where $K:\\Omega\\to[0,\\infty)$ and $\\Sigma:\\Omega\\to[0,\\infty)$ are measurable functions such that\n    \\begin{equation*}\n        K\\in L^p_{\\rm loc}(\\Omega) \\quad{\\rm and}\\quad\n        \\frac{\\Sigma}{K}\\in L^{q}_{\\rm loc}(\\Omega).\n    \\end{equation*}\n    If $p^{-1}+q^{-1}<1$, then $f$ satisfies the Lusin (N) -property.\n\\end{thm}", "K,Sigma-quasiregular": "\\begin{align}\\label{K,Sigma-quasiregular}\n        |Df(x)|^n\\le K(x)J_f(x)+\\Sigma(x)|f-y_0|^n\n    \\end{align}", "K,Sigma condition": "\\begin{equation}\\label{K,Sigma condition}\n        K\\in L^p_{\\rm loc}(\\Omega),\n        \\quad{\\rm and}\\quad \n        \\frac{\\Sigma}{K}\\in L^{q}_{\\rm loc}(\\Omega).\n    \\end{equation}", "ex:planar_counterexample": "\\begin{ex}\\label{ex:planar_counterexample}\n    We recall that in \\cite[Theorem 1.5]{dolevzalova2024mappings}, for some value of $r > 0$, a mapping $h \\colon \\B^2(0,r)\\setminus\\{0\\} \\to \\C$ was constructed with the following properties: $h \\in W^{1,2}(\\B^2(0, r), \\C)$, $\\abs{Dh}^2 \\le K J_h + \\Sigma$ with\n    \\[\n        K \\in L^1(\\B^2(0,r), [1, \\infty)) \\qquad \\text{and} \\qquad \\Sigma \\in L^\\infty(\\B^2(0, r)), \n    \\]\n    and $h$ is of the form\n    \\[\n        h(z) = \\abs{z} + i \\varphi(z)\n    \\]\n    where $\\varphi \\colon \\B^2(0, r) \\setminus \\{0\\} \\to [0, \\infty)$ is continuous with $\\lim_{z \\to 0} \\varphi(z) = \\infty$. We may assume that $r < \\pi$ by restricting $h$ if necessary.\n\n    We consider the mapping $f \\colon \\B^2(0,r) \\to \\C$ defined by $f(0) = 0$ and\n    \\[\n        f(z) = e^{ih(z)}, \\qquad \\text{ for } z \\in \\B^2(0, r) \\setminus \\{0\\}.\n    \\]\n    Since the complex exponential is $1$-Lipschitz on $(-\\infty, 0] \\times \\R$, which contains the image of $ih$, it follows that $f \\in W^{1,2}(\\B^2(0, r), \\C)$. Moreover, by the conformality of the complex exponential, we have $\\abs{Df(z)} = \\abs{f(z)} \\abs{Dh(z)}$ and $J_{f}(z) = \\abs{f(z)}^2 J_h(z)$; thus, we have\n    \\[\n        \\abs{Df(z)}^2 \\le K(z) J_f(z) + \\abs{f(z)}^2 \\Sigma(z)\n    \\]\n    for a.e.\\ $z \\in \\B^2(0, r)$, where $K \\in L^1(\\B^2(0, r))$ and $\\Sigma \\in L^\\infty(\\B^2(0,r))$. However, since $0 \\le \\Im(ih) < \\pi$ for all $z \\in \\B^2(0, r)$, the image of $f$ is contained in the closed upper half-plane; since $f(0) = 0$, the map $f$ hence cannot be sense-preserving at $0$, and does not map $\\B^2(0, r)$ into a neighborhood of $0$.\n\\end{ex}"}, "pre_theorem_intro_text_len": 6240, "pre_theorem_intro_text": "Let $\\Omega\\subset\\mathbb{R}^n$, $n \\ge 2$,  be a domain. A  mapping $f \\colon \\Omega\\to\\mathbb{R}^n$ belonging to the local Sobolev space  $W^{1,n}_{\\mathrm{loc}}(\\Omega,\\mathbb{R}^n)$ is said to have  \\emph{finite distortion} if there exists a measurable function $K \\colon \\Omega \\to [1,\\infty)$ such that\n\\begin{equation}\\label{eq: finite distortion}\n    |Df(x)|^n\\le K(x)J_f(x)\n\\end{equation}\nfor almost every $x\\in\\Omega$. Here, $\\left| Df(x) \\right|$ denotes the operator norm of the weak differential of  $f$ at $x$, and $J_f(x)$ its Jacobian determinant.\n\nThe special case in which $K$ is bounded gives rise to the class of {\\em mappings of bounded distortion}, also known as {\\em quasiregular mappings}.  A cornerstone of the theory of quasiregular maps is Reshetnyak’s theorem \\cite{reshetni_ak1989space,reshetnyak1968condition}, which asserts that such mappings are continuous, and either constant, or open, discrete, and sense preserving. This result places quasiregular mappings among the natural nonlinear analogues of holomorphic functions in higher dimensions.\n\nIn the case without a uniform bound on $K$, the continuity of $W^{1,n}_{\\mathrm{loc}}$-regular mappings of finite distortion was established by Gol'dstein and Vodop'yanov \\cite{vodop1976quasiconformal}, and later extended to larger Orlicz–Sobolev classes by Iwaniec, Koskela, and Onninen \\cite{iwaniec2001mappings}.  In addition to initiating a systematic study of mappings of finite distortion within geometric function theory, the paper \\cite{iwaniec2001mappings} also introduced a naming convention for a sequence of papers, which is also reflected in the title of the present paper.\n\nThe questions of openness and discreteness for mappings of finite distortion are considerably more subtle.  In the planar case, Iwaniec and \\v{S}ver\\'ak \\cite{iwaniec1993mappings} showed that every non-constant mapping  $f\\in W^{1,2}_{\\mathrm{loc}}(\\Omega,\\mathbb{R}^2)$ of finite distortion with $K\\in L^1_{\\mathrm{loc}}(\\Omega)$ is discrete and open, relying on techniques from the theory of Beltrami-type equations. Their work led to the conjecture that, in higher dimensions, non-constant mappings  $f\\in W^{1,n}_{\\mathrm{loc}}(\\Omega,\\mathbb{R}^2)$  with $K \\in L_{\\mathrm{loc}}^{n-1} (\\Omega )$ should enjoy the same properties. The exponent $n-1$ is critical, as shown by Ball’s example \\cite{ball1981global}, which produces, for every $p<n-1$, a Lipschitz mapping of finite distortion with $K\\in L^p(\\Omega)$ that is neither open nor discrete.\n\nIn dimensions $n \\ge 3$, Heinonen and Koskela \\cite{heinonen1993sobolev} proved that quasi-light mappings of finite distortion with $K\\in L^p_{\\mathrm{loc}} (\\Omega)$,  $p>n-1$,  are open and discrete. The quasi-lightness assumption was subsequently removed by Manfredi and Villamor \\cite{manfredi1995mappings}. Bj\\\"orn \\cite{bjorn2002mappings} further extended these results to distortion functions in suitable Orlicz spaces strictly containing $\\bigcup_{p>n-1}L^p_{\\mathrm{loc}}(\\Omega)$.  In the conjectured borderline case  $K \\in L_{\\mathrm{loc}}^{n-1}(\\Omega)$ Hencl and Mal\\'y \\cite{hencl2002mappings} established openness and discreteness under the additional assumption of quasi-lightness, while Hencl and Rajala~\\cite{henclrajala2013} later constructed counterexamples showing that this assumption cannot, in general, be removed when $n \\ge 3$, two decades after the corresponding positive planar result of Iwaniec and \\v{S}ver\\'ak.\n\n\\subsection{Quasiregular values and values of finite distortion}\n\nIn recent years, a theory of \\emph{quasiregular values} has emerged, providing single-value analogues of many fundamental results in the theory of quasiregular mappings. The origins of the theory trace back to a definition by Astala, Iwaniec, and Martin \\cite[Section 8.5]{iwaniec2001geometric}, where they formulate an $n$-dimensional counterpart to a class of mappings used in the solution to the planar Calder\\'on problem by Astala and P\\\"aiv\\\"arinta \\cite{astala2006calderon}. The term ``quasiregular value'' was later coined in a series of papers by Kangasniemi and Onninen \\cite{kangasniemi2022heterogeneous, kangasniemi2024correction, kangasniemi2025single, kangasniemi2024quasiregular, kangasniemi2024linear}, where the basic theory of these mappings was developed; see also the recent result \\cite{heikkila2025quasiregular} by Heikkil\\\"a and Kangasniemi.\n\nIn this article, we consider an analogous generalization of mappings of finite distortion. This condition first appeared in the work of Dole\\v{z}alov\\'a, Kangasniemi and Onninen \\cite{dolevzalova2024mappings}, though our choice of terminology corresponds to the more recent work \\cite{kangasniemi2025continuity}.\n\\begin{defn}\n    Let $\\Omega\\subset\\mathbb{R}^n$ be a domain, let $y_0\\in\\mathbb{R}^n$, let $K \\colon \\Omega \\to [1, \\infty)$ be measurable, and let $\\Sigma\\in L^{1}_\\mathrm{loc}(\\Omega)$. Suppose that $f\\in W^{1,n}_{\\rm loc}(\\Omega,\\mathbb{R}^n)$. We say that $f$ has a \\emph{value of finite distortion} at $y_0$ with data $(K, \\Sigma)$ if\n    \\begin{align}\\label{K,Sigma-quasiregular}\n        |Df(x)|^n\\le K(x)J_f(x)+\\Sigma(x)|f-y_0|^n\n    \\end{align}\n    for a.e.\\ $x\\in\\Omega$. Alternatively, we say that $f$ satisfies the \\emph{$(K, \\Sigma)$-distortion inequality with defect} if\n    \\begin{align}\\label{K,Sigma-distortion}\n        |Df(x)|^n\\le K(x)J_f(x)+\\Sigma(x)\n    \\end{align}\n    for a.e.\\ $x \\in \\Omega$.\n\\end{defn}\n\nIf $K \\in L^p_\\mathrm{loc}(\\Omega)$ and $\\Sigma/K \\in L^q_\\mathrm{loc}(\\Omega)$ with $p, q \\in [1, \\infty]$, then the solutions $f \\in W^{1,n}_{\\mathrm{loc}}(\\Omega,\\mathbb{R}^n)$ of \\eqref{K,Sigma-quasiregular} or \\eqref{K,Sigma-distortion} are continuous precisely in the cases where $p^{-1} + q^{-1} < 1$. Counterexamples in the complementary range  $p^{-1} + q^{-1} \\ge 1$ were constructed in \\cite{dolevzalova2024mappings}, while the sharp positive result for $p^{-1} + q^{-1} < 1$  was established recently by Kangasniemi and Onninen~ \\cite{kangasniemi2025continuity}.\n\nThe main result of the present article is a Reshetnyak-type theorem for mappings possessing a value of finite distortion. The corresponding result for quasiregular values was proved in \\cite[Theorem 1.2]{kangasniemi2025single}.", "context": "Let $\\Omega\\subset\\mathbb{R}^n$, $n \\ge 2$, be a domain. A mapping $f \\colon \\Omega\\to\\mathbb{R}^n$ belonging to the local Sobolev space $W^{1,n}_{\\mathrm{loc}}(\\Omega,\\mathbb{R}^n)$ is said to have \\emph{finite distortion} if there exists a measurable function $K \\colon \\Omega \\to [1,\\infty)$ such that\n\\begin{equation}\\label{eq: finite distortion}\n |Df(x)|^n\\le K(x)J_f(x)\n\\end{equation}\nfor almost every $x\\in\\Omega$. Here, $\\left| Df(x) \\right|$ denotes the operator norm of the weak differential of $f$ at $x$, and $J_f(x)$ its Jacobian determinant.\n\nThe questions of openness and discreteness for mappings of finite distortion are considerably more subtle. In the planar case, Iwaniec and \\v{S}ver\\'ak \\cite{iwaniec1993mappings} showed that every non-constant mapping $f\\in W^{1,2}_{\\mathrm{loc}}(\\Omega,\\mathbb{R}^2)$ of finite distortion with $K\\in L^1_{\\mathrm{loc}}(\\Omega)$ is discrete and open, relying on techniques from the theory of Beltrami-type equations. Their work led to the conjecture that, in higher dimensions, non-constant mappings $f\\in W^{1,n}_{\\mathrm{loc}}(\\Omega,\\mathbb{R}^2)$ with $K \\in L_{\\mathrm{loc}}^{n-1} (\\Omega )$ should enjoy the same properties. The exponent $n-1$ is critical, as shown by Ball’s example \\cite{ball1981global}, which produces, for every $p<n-1$, a Lipschitz mapping of finite distortion with $K\\in L^p(\\Omega)$ that is neither open nor discrete.\n\nIn dimensions $n \\ge 3$, Heinonen and Koskela \\cite{heinonen1993sobolev} proved that quasi-light mappings of finite distortion with $K\\in L^p_{\\mathrm{loc}} (\\Omega)$, $p>n-1$, are open and discrete. The quasi-lightness assumption was subsequently removed by Manfredi and Villamor \\cite{manfredi1995mappings}. Bj\\\"orn \\cite{bjorn2002mappings} further extended these results to distortion functions in suitable Orlicz spaces strictly containing $\\bigcup_{p>n-1}L^p_{\\mathrm{loc}}(\\Omega)$. In the conjectured borderline case $K \\in L_{\\mathrm{loc}}^{n-1}(\\Omega)$ Hencl and Mal\\'y \\cite{hencl2002mappings} established openness and discreteness under the additional assumption of quasi-lightness, while Hencl and Rajala~\\cite{henclrajala2013} later constructed counterexamples showing that this assumption cannot, in general, be removed when $n \\ge 3$, two decades after the corresponding positive planar result of Iwaniec and \\v{S}ver\\'ak.\n\nIn this article, we consider an analogous generalization of mappings of finite distortion. This condition first appeared in the work of Dole\\v{z}alov\\'a, Kangasniemi and Onninen \\cite{dolevzalova2024mappings}, though our choice of terminology corresponds to the more recent work \\cite{kangasniemi2025continuity}.\n\\begin{defn}\n Let $\\Omega\\subset\\mathbb{R}^n$ be a domain, let $y_0\\in\\mathbb{R}^n$, let $K \\colon \\Omega \\to [1, \\infty)$ be measurable, and let $\\Sigma\\in L^{1}_\\mathrm{loc}(\\Omega)$. Suppose that $f\\in W^{1,n}_{\\rm loc}(\\Omega,\\mathbb{R}^n)$. We say that $f$ has a \\emph{value of finite distortion} at $y_0$ with data $(K, \\Sigma)$ if\n \\begin{align}\\label{K,Sigma-quasiregular}\n |Df(x)|^n\\le K(x)J_f(x)+\\Sigma(x)|f-y_0|^n\n \\end{align}\n for a.e.\\ $x\\in\\Omega$. Alternatively, we say that $f$ satisfies the \\emph{$(K, \\Sigma)$-distortion inequality with defect} if\n \\begin{align}\\label{K,Sigma-distortion}\n |Df(x)|^n\\le K(x)J_f(x)+\\Sigma(x)\n \\end{align}\n for a.e.\\ $x \\in \\Omega$.\n\\end{defn}\n\nIf $K \\in L^p_\\mathrm{loc}(\\Omega)$ and $\\Sigma/K \\in L^q_\\mathrm{loc}(\\Omega)$ with $p, q \\in [1, \\infty]$, then the solutions $f \\in W^{1,n}_{\\mathrm{loc}}(\\Omega,\\mathbb{R}^n)$ of \\eqref{K,Sigma-quasiregular} or \\eqref{K,Sigma-distortion} are continuous precisely in the cases where $p^{-1} + q^{-1} < 1$. Counterexamples in the complementary range $p^{-1} + q^{-1} \\ge 1$ were constructed in \\cite{dolevzalova2024mappings}, while the sharp positive result for $p^{-1} + q^{-1} < 1$ was established recently by Kangasniemi and Onninen~ \\cite{kangasniemi2025continuity}.\n\nThe main result of the present article is a Reshetnyak-type theorem for mappings possessing a value of finite distortion. The corresponding result for quasiregular values was proved in \\cite[Theorem 1.2]{kangasniemi2025single}.\n\n\\begin{align}\\label{K,Sigma-distortion}\n        |Df(x)|^n\\le K(x)J_f(x)+\\Sigma(x)\n    \\end{align}\n\n\\begin{align}\\label{K,Sigma-quasiregular}\n        |Df(x)|^n\\le K(x)J_f(x)+\\Sigma(x)|f-y_0|^n\n    \\end{align}", "full_context": "Let $\\Omega\\subset\\mathbb{R}^n$, $n \\ge 2$, be a domain. A mapping $f \\colon \\Omega\\to\\mathbb{R}^n$ belonging to the local Sobolev space $W^{1,n}_{\\mathrm{loc}}(\\Omega,\\mathbb{R}^n)$ is said to have \\emph{finite distortion} if there exists a measurable function $K \\colon \\Omega \\to [1,\\infty)$ such that\n\\begin{equation}\\label{eq: finite distortion}\n |Df(x)|^n\\le K(x)J_f(x)\n\\end{equation}\nfor almost every $x\\in\\Omega$. Here, $\\left| Df(x) \\right|$ denotes the operator norm of the weak differential of $f$ at $x$, and $J_f(x)$ its Jacobian determinant.\n\nThe questions of openness and discreteness for mappings of finite distortion are considerably more subtle. In the planar case, Iwaniec and \\v{S}ver\\'ak \\cite{iwaniec1993mappings} showed that every non-constant mapping $f\\in W^{1,2}_{\\mathrm{loc}}(\\Omega,\\mathbb{R}^2)$ of finite distortion with $K\\in L^1_{\\mathrm{loc}}(\\Omega)$ is discrete and open, relying on techniques from the theory of Beltrami-type equations. Their work led to the conjecture that, in higher dimensions, non-constant mappings $f\\in W^{1,n}_{\\mathrm{loc}}(\\Omega,\\mathbb{R}^2)$ with $K \\in L_{\\mathrm{loc}}^{n-1} (\\Omega )$ should enjoy the same properties. The exponent $n-1$ is critical, as shown by Ball’s example \\cite{ball1981global}, which produces, for every $p<n-1$, a Lipschitz mapping of finite distortion with $K\\in L^p(\\Omega)$ that is neither open nor discrete.\n\nIn dimensions $n \\ge 3$, Heinonen and Koskela \\cite{heinonen1993sobolev} proved that quasi-light mappings of finite distortion with $K\\in L^p_{\\mathrm{loc}} (\\Omega)$, $p>n-1$, are open and discrete. The quasi-lightness assumption was subsequently removed by Manfredi and Villamor \\cite{manfredi1995mappings}. Bj\\\"orn \\cite{bjorn2002mappings} further extended these results to distortion functions in suitable Orlicz spaces strictly containing $\\bigcup_{p>n-1}L^p_{\\mathrm{loc}}(\\Omega)$. In the conjectured borderline case $K \\in L_{\\mathrm{loc}}^{n-1}(\\Omega)$ Hencl and Mal\\'y \\cite{hencl2002mappings} established openness and discreteness under the additional assumption of quasi-lightness, while Hencl and Rajala~\\cite{henclrajala2013} later constructed counterexamples showing that this assumption cannot, in general, be removed when $n \\ge 3$, two decades after the corresponding positive planar result of Iwaniec and \\v{S}ver\\'ak.\n\nIn this article, we consider an analogous generalization of mappings of finite distortion. This condition first appeared in the work of Dole\\v{z}alov\\'a, Kangasniemi and Onninen \\cite{dolevzalova2024mappings}, though our choice of terminology corresponds to the more recent work \\cite{kangasniemi2025continuity}.\n\\begin{defn}\n Let $\\Omega\\subset\\mathbb{R}^n$ be a domain, let $y_0\\in\\mathbb{R}^n$, let $K \\colon \\Omega \\to [1, \\infty)$ be measurable, and let $\\Sigma\\in L^{1}_\\mathrm{loc}(\\Omega)$. Suppose that $f\\in W^{1,n}_{\\rm loc}(\\Omega,\\mathbb{R}^n)$. We say that $f$ has a \\emph{value of finite distortion} at $y_0$ with data $(K, \\Sigma)$ if\n \\begin{align}\\label{K,Sigma-quasiregular}\n |Df(x)|^n\\le K(x)J_f(x)+\\Sigma(x)|f-y_0|^n\n \\end{align}\n for a.e.\\ $x\\in\\Omega$. Alternatively, we say that $f$ satisfies the \\emph{$(K, \\Sigma)$-distortion inequality with defect} if\n \\begin{align}\\label{K,Sigma-distortion}\n |Df(x)|^n\\le K(x)J_f(x)+\\Sigma(x)\n \\end{align}\n for a.e.\\ $x \\in \\Omega$.\n\\end{defn}\n\nIf $K \\in L^p_\\mathrm{loc}(\\Omega)$ and $\\Sigma/K \\in L^q_\\mathrm{loc}(\\Omega)$ with $p, q \\in [1, \\infty]$, then the solutions $f \\in W^{1,n}_{\\mathrm{loc}}(\\Omega,\\mathbb{R}^n)$ of \\eqref{K,Sigma-quasiregular} or \\eqref{K,Sigma-distortion} are continuous precisely in the cases where $p^{-1} + q^{-1} < 1$. Counterexamples in the complementary range $p^{-1} + q^{-1} \\ge 1$ were constructed in \\cite{dolevzalova2024mappings}, while the sharp positive result for $p^{-1} + q^{-1} < 1$ was established recently by Kangasniemi and Onninen~ \\cite{kangasniemi2025continuity}.\n\nThe main result of the present article is a Reshetnyak-type theorem for mappings possessing a value of finite distortion. The corresponding result for quasiregular values was proved in \\cite[Theorem 1.2]{kangasniemi2025single}.\n\n\\begin{align}\\label{K,Sigma-distortion}\n        |Df(x)|^n\\le K(x)J_f(x)+\\Sigma(x)\n    \\end{align}\n\n\\begin{align}\\label{K,Sigma-quasiregular}\n        |Df(x)|^n\\le K(x)J_f(x)+\\Sigma(x)|f-y_0|^n\n    \\end{align}\n\nThe main result of the present article is a Reshetnyak-type theorem for mappings possessing a value of finite distortion. The corresponding result for quasiregular values was proved in \\cite[Theorem 1.2]{kangasniemi2025single}.\n\nThe assumption $p^{-1} + q^{-1} < 1$ in Theorem \\ref{open and discrete thm} is essential. For one, solutions of \\eqref{K,Sigma-quasiregular} need not even admit a continuous representative otherwise. Moreover, even a continuous solution of \\eqref{K,Sigma-quasiregular} with $p^{-1} + q^{-1} \\ge 1$ may fail to be open and sense-preserving in $f^{-1}\\{y_0\\}$; see Example \\ref{ex:planar_counterexample}. We also note that when $p^{-1} + q^{-1} < 1$, it follows from \\eqref{K,Sigma condition} that $\\Sigma \\in L^r_\\loc(\\Omega)$ with $r = (p^{-1} + q^{-1})^{-1} > 1$. Such higher integrability of $\\Sigma$ is necessary, as demonstrated by the example given in \\cite[Example 8.1]{kangasniemi2025single}.\n\n\\begin{thm}\\label{thm:lusin_N}\n Let $\\Omega\\subset\\mathbb{R}^n$ be a domain, and let $p,q\\in[1,\\infty]$.\n Suppose that\n $f\\in W^{1,n}_{\\loc}(\\Omega,\\mathbb{R}^n)$ satisfies the $(K,\\Sigma)$-distortion inequality with defect or has a value of finite distortion at a point $y_0 \\in \\R^n$ with data $(K, \\Sigma)$, where $K:\\Omega\\to[0,\\infty)$ and $\\Sigma:\\Omega\\to[0,\\infty)$ are measurable functions such that\n \\begin{equation*}\n K\\in L^p_{\\rm loc}(\\Omega) \\quad{\\rm and}\\quad\n \\frac{\\Sigma}{K}\\in L^{q}_{\\rm loc}(\\Omega).\n \\end{equation*}\n If $p^{-1}+q^{-1}<1$, then $f$ satisfies the Lusin (N) -property.\n\\end{thm}\n\n\\begin{lem}\\label{Duk bounded lemma}\n Let $\\Omega\\subset\\mathbb{R}^n$ be a domain, let $y_0\\in\\mathbb{R}^n$, and let $p>n-1,q>1$ with $p^{-1}+q^{-1}<1$.\n Suppose that\n $f\\in W^{1,n}_{\\rm loc}(\\Omega,\\mathbb{R}^n)$ has a value of finite distortion at $y_0\\in\\mathbb{R}^n$ with data $(K, \\Sigma)$, where $K:\\Omega\\to[1,\\infty)$ and $\\Sigma:\\Omega\\to[0,\\infty)$ are measurable functions such that\n \\begin{equation}\n K\\in L^p_{\\rm loc}(\\Omega) \\quad{\\rm and}\\quad\n \\frac{\\Sigma}{K}\\in L^{q}_{\\rm loc}(\\Omega).\n \\end{equation} \n Let $u_k=\\min(\\log\\log|f-y_0|^{-1},k)$, then there exists a neighborhood $V$ of $y_0$ such that\n \\begin{equation}\\label{Duk uniformly bounded}\n \\int_D|\\nabla u_k|^\\beta\\le C_D<\\infty,\n \\end{equation}\n for $\\beta=\\frac{pn}{p+1}<n$, all $k\\in\\mathbb{Z}_{>0}$ and all domains $D$ compactly contained in $U = f^{-1}(V)$, where $C_D=C_D(U,D,n,p,K,\\Sigma)$ is independent of $k$.\n\\end{lem}\n\\begin{proof}\n We select $V = \\B^n(y_0, 1/e)$ and let $U=f^{-1} V$, which is open since $f$ is continuous by \\cite[Corollary 1.2]{kangasniemi2025continuity}. Let $D$ be a domain that is compactly contained in $U$, let $\\varepsilon > 0$, and select $\\eta\\in C^\\infty_0(U,[0,1])$ such that $\\eta\\equiv 1$ on $D$.\n\n\\begin{lem}\\label{totally disconnected}\n Let $\\Omega\\subset\\mathbb{R}^n$ be a domain, let $y_0\\in\\mathbb{R}^n$, and let $p>n-1,q>1$ with $p^{-1}+q^{-1}<1$.\n Suppose that $f\\in W^{1,n}_{\\rm loc}(\\Omega,\\mathbb{R}^n)$ has a value of finite distortion at $y_0\\in\\mathbb{R}^n$ with data $(K, \\Sigma)$, where $K:\\Omega\\to[1,\\infty)$ and $\\Sigma:\\Omega\\to[0,\\infty)$ are measurable functions such that\n \\begin{equation}\n K\\in L^p_{\\rm loc}(\\Omega) \\quad{\\rm and}\\quad\n \\frac{\\Sigma}{K}\\in L^{q}_{\\rm loc}(\\Omega).\n \\end{equation}\n Then either $f\\equiv y_0$ or $\\mathcal{H}^{1}(f^{-1}\\{y_0\\})=0$. In particular, in the later case $f^{-1}\\{y_0\\}$ is totally disconnected.\n\\end{lem}\n\n\\begin{lem}\\label{lem: Integral of J_g on annulus}\n Let $\\Omega\\subset\\mathbb{R}^n$ be a domain, let $y_0\\in\\mathbb{R}^n$, and let $p>n-1,q>1$ be with $p^{-1}+q^{-1}<1$.\n Suppose that\n $f\\in W^{1,n}_{\\rm loc}(\\Omega,\\mathbb{R}^n)$ has a value of finite distortion at $y_0\\in\\mathbb{R}^n$ with data $(K,\\Sigma)$, where $K:\\Omega\\to[1,\\infty)$ and $\\Sigma:\\Omega\\to[0,\\infty)$ are measurable functions such that\n \\begin{equation*}\n K\\in L^p(\\Omega) \\quad{\\rm and}\\quad\n \\frac{\\Sigma}{K}\\in L^{q}(\\Omega).\n \\end{equation*}\n Suppose that $U$ is a non-empty component of $f^{-1}\\mathbb{B}^n(y_0,\\varepsilon)$, where $\\varepsilon>0$ is small enough that $\\overline{U}\\subset\\Omega$ and $m_n(U)<c$ with $c=c(n,q,K,\\Sigma,\\Omega)>0$ given by Lemma \\ref{non negative lemma}. If ${\\rm deg}(f,U)=0$ and $g$ is defined as in \\eqref{spherical-log- formula}. Then we have $g\\in W^{1,\\beta}(U,\\mathbb{R}\\times\\mathbb{S}^{n-1})$ for some $p<n$, the integral of $J_g$ over $U$ vanishes, and for every $0<t<\\varepsilon$, we have\n \\begin{equation}\\label{Jacobian over annulus}\n \\int_{\\{x\\in U:\\ |f-y_0|<t\\}}J_g=0.\n \\end{equation}\n\\end{lem}\n\n\\begin{cor}\\label{cor: positive local index}\n Let $\\Omega\\subset\\mathbb{R}^n$ be a domain, let $y_0\\in\\mathbb{R}^n$, and let $p>n-1,q>1$ with $p^{-1}+q^{-1}<1$. Suppose that\n $f\\in W^{1,n}_{\\rm loc}(\\Omega,\\mathbb{R}^n)$ has a value of finite distortion at $y_0\\in\\mathbb{R}^n$ with data $(K,\\Sigma)$, where $K:\\Omega\\to[1,\\infty)$ and $\\Sigma:\\Omega\\to[0,\\infty)$ are measurable functions such that\n \\begin{equation*}\n K\\in L^p_{\\rm loc}(\\Omega) \\quad{\\rm and}\\quad\n \\frac{\\Sigma}{K}\\in L^{q}_{\\rm loc}(\\Omega).\n \\end{equation*}\n Then for every $x_0\\in f^{-1}\\{y_0\\}$, the local index $i(x_0,f)$ is well-defined and we have $i(x_0,f)>0$.\n\\end{cor}\n\nThe final property to be deduced is the local openness property.\n\\begin{lem}\\label{lem:open}\n Let $\\Omega\\subset\\mathbb{R}^n$ be a domain, let $y_0\\in\\mathbb{R}^n$, and let $p>n-1,q>1$ with $p^{-1}+q^{-1}<1$. Suppose that $f\\in W^{1,n}_{\\rm loc}(\\Omega,\\mathbb{R}^n)$ has a value of finite distortion at $y_0\\in\\mathbb{R}^n$ with data $(K,\\Sigma)$, where $K:\\Omega\\to[1,\\infty)$ and $\\Sigma:\\Omega\\to[0,\\infty)$ are measurable functions such that\n \\begin{equation*}\n K\\in L^p_{\\rm loc}(\\Omega) \\quad{\\rm and}\\quad\n \\frac{\\Sigma}{K}\\in L^{q}_{\\rm loc}(\\Omega).\n \\end{equation*}\n Then for every $x_0 \\in f^{-1}\\{y_0\\}$ and every neighborhood $V$ of $x_0$, we have $y_0\\in{\\rm int}\\ f(V)$.\n\\end{lem}", "post_theorem_intro_text_len": 6668, "post_theorem_intro_text": "The assumption $p^{-1} + q^{-1} < 1$ in Theorem \\ref{open and discrete thm} is essential. For one, solutions of \\eqref{K,Sigma-quasiregular} need not even admit a continuous representative otherwise. Moreover, even a continuous solution of \\eqref{K,Sigma-quasiregular} with $p^{-1} + q^{-1} \\ge 1$ may fail to be open and sense-preserving in $f^{-1}\\{y_0\\}$; see Example \\ref{ex:planar_counterexample}. We also note that when $p^{-1} + q^{-1} < 1$, it follows from \\eqref{K,Sigma condition} that  $\\Sigma \\in L^r_\\mathrm{loc}(\\Omega)$ with $r = (p^{-1} + q^{-1})^{-1} > 1$. Such higher integrability of $\\Sigma$ is necessary, as demonstrated by the example given in \\cite[Example 8.1]{kangasniemi2025single}. \n\nThe assumption that $K\\in L_{\\mathrm{loc}}^p (\\Omega)$ with $p>n-1$ is also necessary. For $n \\ge 3$, this is already required in the special case  $\\Sigma \\equiv 0$; see~\\cite{henclrajala2013}. For $n = 2$, although Reshetnyak's theorem remains valid for mappings of finite distortion when $p = 1$, the fundamental difference in our setting is that if $p = 1$, then there exists no $q \\in [1, \\infty]$ with $p^{-1} + q^{-1} < 1$; for a counterexample with $p = 1$ and $q = \\infty$, see again Example \\ref{ex:planar_counterexample}.\n\nFinally, one cannot expect Reshetnyak-type conclusions at the preimages of values  $y\\neq y_0$, nor for solutions of the defect inequality \\eqref{K,Sigma-distortion}. For instance, the mapping  $f(x)=(x_1,0,...,0)$, satisfies the $(K,1)$-distortion inequality with defect for any constant $K\\ge 1$, yet $f$ is neither\nconstant nor discrete or open.  Further counterexamples are given in  \\cite[Example 8.4]{kangasniemi2025single}, where a mapping $f \\colon \\mathbb{R}^n \\to \\mathbb{R}^n$ is constructed that has a $(1, \\Sigma)$-quasiregular value at $y_0 = 0$ with a constant $\\Sigma$, has no values of finite distortion with a locally integrable $\\Sigma$ at other $y \\in \\mathbb{R}^n \\setminus \\{0\\}$, and fails to be sense preserving at the preimages of any $y \\ne 0$.\n\n\\subsection{Methods and other notable results}\n\nThe proof of Theorem \\ref{open and discrete thm} follows a similar outline as that of \\cite[Theorem 1.2]{kangasniemi2025single}, which in turn inherits many of its steps from the original proofs for quasiregular maps and mappings of finite distortion. In particular, if $f$ is not identically $y_0$, we first show that $f^{-1}\\{y_0\\}$ is totally disconnected, by showing that the Hausdorff dimension of $f^{-1}\\{y_0\\}$ is less than 1. We then show a local positivity result for the topological degree of $f$, which in turn is used to derive all three parts of Theorem \\ref{open and discrete thm}.\n\nHowever, there are two parts in the proof where our approach differs significantly from that of \\cite{kangasniemi2025single}. The first such difference is centered around the Lusin (N) -property; recall that if $\\Omega\\subset\\mathbb{R}^n$ is a domain and $f:\\Omega\\to\\mathbb{R}^n$ is continuous, then $f$ satisfies the {\\em Lusin (N) -property} if, for every $m_n$-nullset $E\\subset\\Omega$, the image $f E$ is an $m_n$-nullset. The Lusin (N) -property holds for quasiregular mappings by a result of Reshetnyak, see e.g.\\ \\cite{reshetni_ak1989space}, and also for $W^{1,n}_\\mathrm{loc}$-regular mappings of finite distortion by a result of Kauhanen, Koskela and Mal\\'y \\cite{kauhanen2001mappings}.\n\nNotably, the proof of Theorem \\ref{open and discrete thm} relies on $f$ satisfying the Lusin (N) -property. When $K$ is constant, this follows from a higher integrability and/or H\\\"older regularity result for $f$. However, in the case with variable $K$, there are no available higher integrability results, and the modulus of continuity shown in \\cite{kangasniemi2025continuity} does not directly imply the Lusin (N) -property for $W^{1,n}$-maps; see the counterexamples of Koskela, Mal\\'y, and Z\\\"urcher \\cite{koskela2015luzin}, as well as the related positive result of Zapadinskaya \\cite{zapadinskaya2014holder}.\n\nFor this reason, the Lusin (N) -property for solutions of \\eqref{K,Sigma-quasiregular} and \\eqref{K,Sigma-distortion} ends up being a result of independent interest. We record this result here.\n\n\\begin{thm}\\label{thm:lusin_N}\n    Let $\\Omega\\subset\\mathbb{R}^n$ be a domain, and let $p,q\\in[1,\\infty]$.\n    Suppose that\n    $f\\in W^{1,n}_{\\mathrm{loc}}(\\Omega,\\mathbb{R}^n)$ satisfies the $(K,\\Sigma)$-distortion inequality with defect or has a value of finite distortion at a point $y_0 \\in \\mathbb{R}^n$ with data $(K, \\Sigma)$, where $K:\\Omega\\to[0,\\infty)$ and $\\Sigma:\\Omega\\to[0,\\infty)$ are measurable functions such that\n    \\begin{equation*}\n        K\\in L^p_{\\rm loc}(\\Omega) \\quad{\\rm and}\\quad\n        \\frac{\\Sigma}{K}\\in L^{q}_{\\rm loc}(\\Omega).\n    \\end{equation*}\n    If $p^{-1}+q^{-1}<1$, then $f$ satisfies the Lusin (N) -property.\n\\end{thm}\n\nThe proof of Theorem \\ref{thm:lusin_N} uses the key estimate \\cite[Theorem 1.6]{kangasniemi2025continuity} behind the continuity of $W^{1,n}$-solutions to \\eqref{K,Sigma-distortion}, along with ideas from the proof of the Lusin (N) -property for continuous pseudomonotone $W^{1,n}$-maps by Mal\\'y and Martio \\cite[Theorem A]{maly1995lusin}. We highlight that Theorem \\ref{thm:lusin_N} in fact uses \\cite[Theorem 1.6]{kangasniemi2025continuity} in a sharper manner than the main continuity result of its original paper. In particular, the continuity of $W^{1,n}$-solutions to \\eqref{K,Sigma-distortion} arises from them being almost weakly monotone, see \\cite[Definition 1.4]{kangasniemi2025continuity}, but the proof of Theorem \\ref{thm:lusin_N} does not work for arbitrary almost weakly monotone $W^{1,n}$-maps.\n\nThe other major difference between the proofs of Theorem \\ref{open and discrete thm} and \\cite[Theorem 1.2]{kangasniemi2025single} occurs when we deduce that the topological degree of $f$ cannot vanish locally. This follows by showing that if it does vanish, then one obtains a continuity result for $\\log \\lvert f - y_0 \\rvert$ which leads to a contradiction. The original version \\cite[Lemma 5.5]{kangasniemi2025single} of this step uses technical estimates from \\cite[Section 6]{kangasniemi2022heterogeneous}, but this approach is critically reliant on $K$ being constant.\n\nInstead, we obtain this continuity result by again using the recent new techniques from \\cite{kangasniemi2025continuity}. Notably, we cannot directly use the main result of \\cite{kangasniemi2025continuity}, since when $K$ is non-constant, it is not clear to us whether $\\log \\lvert f - y_0 \\rvert$ is in a $W^{1,n}$-space in this case. However, with some adjustments, the underlying ideas are still sufficient to obtain the desired result.", "sketch": "The proof of Theorem \\ref{open and discrete thm} is said to follow “a similar outline” as \\cite[Theorem 1.2]{kangasniemi2025single} (and ultimately the classical arguments for quasiregular maps and mappings of finite distortion). If $f$ is not identically $y_0$, the argument proceeds as follows:\n\n1. First show that $f^{-1}\\{y_0\\}$ is totally disconnected “by showing that the Hausdorff dimension of $f^{-1}\\{y_0\\}$ is less than 1.”\n\n2. Then prove “a local positivity result for the topological degree of $f$,” and use this positivity “to derive all three parts of Theorem \\ref{open and discrete thm}.”\n\nTwo proof steps are highlighted as substantially different from \\cite{kangasniemi2025single}:\n\n- A key input is that “the proof of Theorem \\ref{open and discrete thm} relies on $f$ satisfying the Lusin (N)-property.” Since for variable $K$ there are “no available higher integrability results” and the known modulus of continuity “does not directly imply the Lusin (N)-property,” the paper proves Lusin (N) separately (Theorem \\ref{thm:lusin_N}).\n\n- To show “that the topological degree of $f$ cannot vanish locally,” they argue that “if it does vanish, then one obtains a continuity result for $\\log \\lvert f-y_0\\rvert$ which leads to a contradiction.” Because the earlier approach depended critically on $K$ being constant, they instead “obtain this continuity result by again using the recent new techniques from \\cite{kangasniemi2025continuity},” with adjustments since for non-constant $K$ it is unclear whether “$\\log\\lvert f-y_0\\rvert$ is in a $W^{1,n}$-space.”", "expanded_sketch": "The proof of Theorem \\ref{open and discrete thm} is said to follow “a similar outline” as \\cite[Theorem 1.2]{kangasniemi2025single} (and ultimately the classical arguments for quasiregular maps and mappings of finite distortion). If $f$ is not identically $y_0$, the argument proceeds as follows:\n\n1. First show that $f^{-1}\\{y_0\\}$ is totally disconnected “by showing that the Hausdorff dimension of $f^{-1}\\{y_0\\}$ is less than 1.”\n\n2. Then prove “a local positivity result for the topological degree of $f$,” and use this positivity to establish the main theorem.\n\nTwo proof steps are highlighted as substantially different from \\cite{kangasniemi2025single}:\n\n- A key input is that in establishing the main theorem one needs that $f$ satisfies the Lusin (N)-property. Since for variable $K$ there are “no available higher integrability results” and the known modulus of continuity “does not directly imply the Lusin (N)-property,” the paper proves Lusin (N) separately. We first prove the following theorem.\n\n\\begin{thm}\\label{thm:lusin_N}\n    Let $\\Omega\\subset\\mathbb{R}^n$ be a domain, and let $p,q\\in[1,\\infty]$.\n    Suppose that\n    $f\\in W^{1,n}_{\\loc}(\\Omega,\\mathbb{R}^n)$ satisfies the $(K,\\Sigma)$-distortion inequality with defect or has a value of finite distortion at a point $y_0 \\in \\R^n$ with data $(K, \\Sigma)$, where $K:\\Omega\\to[0,\\infty)$ and $\\Sigma:\\Omega\\to[0,\\infty)$ are measurable functions such that\n    \\begin{equation*}\n        K\\in L^p_{\\rm loc}(\\Omega) \\quad{\\rm and}\\quad\n        \\frac{\\Sigma}{K}\\in L^{q}_{\\rm loc}(\\Omega).\n    \\end{equation*}\n    If $p^{-1}+q^{-1}<1$, then $f$ satisfies the Lusin (N) -property.\n\\end{thm}\n\n- To show “that the topological degree of $f$ cannot vanish locally,” they argue that “if it does vanish, then one obtains a continuity result for $\\log \\lvert f-y_0\\rvert$ which leads to a contradiction.” Because the earlier approach depended critically on $K$ being constant, they instead “obtain this continuity result by again using the recent new techniques from \\cite{kangasniemi2025continuity},” with adjustments since for non-constant $K$ it is unclear whether “$\\log\\lvert f-y_0\\rvert$ is in a $W^{1,n}$-space.”", "expanded_theorem": "\\label{open and discrete thm}\n    Let $\\Omega\\subset\\mathbb{R}^n$ be a domain, let $y_0\\in\\mathbb{R}^n$, and let $p, q \\in [1, \\infty]$. Suppose that\n    $f\\in W^{1,n}_{\\rm loc}(\\Omega,\\mathbb{R}^n)$ has a value of finite distortion at $y_0\\in\\mathbb{R}^n$ with data $(K,\\Sigma)$, where $K:\\Omega\\to[1,\\infty)$ and $\\Sigma:\\Omega\\to[0,\\infty)$ are measurable functions such that\n    \\begin{equation}\\label{K,Sigma condition}\n        K\\in L^p_{\\rm loc}(\\Omega),\n        \\quad{\\rm and}\\quad \n        \\frac{\\Sigma}{K}\\in L^{q}_{\\rm loc}(\\Omega).\n    \\end{equation}\n    If $p > n-1$ and $p^{-1} + q^{-1} < 1$, then either $f\\equiv y_0$ a.e.\\ in $\\Omega$, or the continuous representative of $f$ satisfies the following conditions:\n    \\begin{itemize}[noitemsep]\n        \\item $f^{-1}\\{y_0\\}$ is a discrete subset of $\\Omega$,\n        \\item at every $x_0\\in f^{-1}\\{y_0\\}$ the local index $i(x_0,f)$ is positive, and\n        \\item for every neighborhood $U$ of a point $x_0\\in f^{-1}\\{y_0\\}$, we have $y_0\\in  {\\rm int}\\ f(U)$.\n    \\end{itemize}", "theorem_type": "unknown", "mcq": {"question": "Let \\(\\Omega\\subset \\mathbb{R}^n\\) be a domain, let \\(y_0\\in\\mathbb{R}^n\\), and let \\(p,q\\in[1,\\infty]\\). Suppose \\(f\\in W^{1,n}_{\\mathrm{loc}}(\\Omega,\\mathbb{R}^n)\\) has a value of finite distortion at \\(y_0\\) with data \\((K,\\Sigma)\\), meaning that for measurable functions \\(K:\\Omega\\to[1,\\infty)\\) and \\(\\Sigma:\\Omega\\to[0,\\infty)\\),\n\\[\n|Df(x)|^n\\le K(x)J_f(x)+\\Sigma(x)|f(x)-y_0|^n\n\\]\nfor almost every \\(x\\in\\Omega\\), where \\(|Df(x)|\\) is the operator norm of the weak differential and \\(J_f(x)\\) is the Jacobian determinant. Assume also that\n\\[\nK\\in L^p_{\\mathrm{loc}}(\\Omega),\\qquad \\frac{\\Sigma}{K}\\in L^q_{\\mathrm{loc}}(\\Omega),\n\\]\nand that \\(p>n-1\\) and \\(p^{-1}+q^{-1}<1\\). Which statement holds for \\(f\\)?", "correct_choice": {"label": "A", "text": "Either \\(f\\equiv y_0\\) almost everywhere in \\(\\Omega\\), or \\(f\\) has a continuous representative such that \\(f^{-1}\\{y_0\\}\\) is a discrete subset of \\(\\Omega\\), the local index \\(i(x_0,f)\\) is positive for every \\(x_0\\in f^{-1}\\{y_0\\}\\), and for every neighborhood \\(U\\) of any point \\(x_0\\in f^{-1}\\{y_0\\}\\), one has \\(y_0\\in \\operatorname{int} f(U)\\)."}, "choices": [{"label": "B", "text": "Either \\(f\\equiv y_0\\) almost everywhere in \\(\\Omega\\), or \\(f\\) has a continuous representative such that \\(f^{-1}\\{y_0\\}\\) is a discrete subset of \\(\\Omega\\), the local index \\(i(x_0,f)\\) is nonnegative for every \\(x_0\\in f^{-1}\\{y_0\\}\\), and for every neighborhood \\(U\\) of any point \\(x_0\\in f^{-1}\\{y_0\\}\\), one has \\(y_0\\in \\partial f(U)\\cup \\operatorname{int} f(U)\\)."}, {"label": "C", "text": "Either \\(f\\equiv y_0\\) almost everywhere in \\(\\Omega\\), or \\(f\\) has a continuous representative such that \\(f^{-1}\\{y_0\\}\\) is a discrete subset of \\(\\Omega\\)."}, {"label": "D", "text": "Either \\(f\\equiv y_0\\) almost everywhere in \\(\\Omega\\), or \\(f\\) has a continuous representative such that \\(f^{-1}\\{y_0\\}\\) is a discrete subset of \\(\\Omega\\), the local index \\(i(x_0,f)\\) is positive for every \\(x_0\\in f^{-1}\\{y_0\\}\\), and there exists a neighborhood \\(U_0\\) of each point \\(x_0\\in f^{-1}\\{y_0\\}\\) for which \\(y_0\\in \\operatorname{int} f(U_0)\\)."}, {"label": "E", "text": "Either \\(f\\equiv y_0\\) almost everywhere in \\(\\Omega\\), or \\(f\\) has a continuous representative such that \\(f^{-1}\\{y_0\\}\\) has Hausdorff \\(1\\)-measure zero in \\(\\Omega\\), the local index \\(i(x_0,f)\\) is positive for every \\(x_0\\in f^{-1}\\{y_0\\}\\), and for every neighborhood \\(U\\) of any point \\(x_0\\in f^{-1}\\{y_0\\}\\), one has \\(y_0\\in \\operatorname{int} f(U)\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "strict positivity of local degree/opening conclusion", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped positivity of local index and local openness conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "quantifier on local openness at preimages of y_0", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "replacing discreteness by merely \\(\\mathcal{H}^1\\)-null preimage", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem states only the hypotheses and asks for the resulting conclusion; it does not explicitly reveal the correct option. There is no direct answer leakage beyond indicating the general theorem setting."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem gives the exact hypotheses of a specific result and asks which conclusion holds. The correct answer is basically the theorem statement itself rather than a genuinely derived consequence in a new context."}, "GPS": {"score": 1, "justification": "Some reasoning is needed because the options differ in subtle ways (positivity vs nonnegativity of index, discreteness vs weaker size conditions, universal vs existential neighborhood claims). However, the task is still mostly recognition of the exact theorem rather than substantial generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: weakening the conclusion, altering quantifiers, replacing positivity by nonnegativity, and confusing discreteness with measure-theoretic smallness. They are distinct and well aligned with common theorem-misremembering errors."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-recognition MCQ with strong distractors and no major answer leakage, but it is largely tautological and only moderately tests genuine reasoning."}}
{"id": "2602.03577v1", "paper_link": "http://arxiv.org/abs/2602.03577v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{main I}\n    Suppose $\\Gamma$ is a finite simplicial graph and $\\{G_{v}\\}_{v\\in V(\\Gamma)}$ is a collection of weakly Haagerup groups with $\\Lambda_{WH}(G_v)=1$, for each $v$. Then the group $G(\\Gamma)$ has the weak Haagerup property with $\\Lambda_{WH}(G(\\Gamma))=1$.", "start_pos": 11251, "end_pos": 11553, "label": "main I"}, "ref_dict": {"definition WH": "\\begin{definition}\\label{definition WH}\n    $G$ is \\textit{weakly Haagerup} if there is a sequence of maps $\\phi_n:G\\rightarrow\\mathbb{C}$ with \n    \\begin{enumerate}\n        \\item $\\phi_n$ vanishes at infinity, for each $n\\in \\mathbb{N}$,\n        \\item $\\{\\phi_n\\}$ is an approximate identity i.e. $\\lim_{n\\rightarrow\\infty}\\phi_n(g)=1$ for all $g\\in G$,\n         \\item there exists a constant $B>0$ such that $||\\phi_n||_{B_2}\\leq B$ for each $n\\in \\mathbb{N}$.\n    \\end{enumerate}\n        The weak Haagerup constant, denoted $\\Lambda_{WH}$ for $G$ is the infimum of all such $B>0$ for which there is a sequence $\\{\\phi_n\\}_n$ satisfying the above properties.\n\\end{definition}", "definition completely bounded": "\\begin{definition}\\label{definition completely bounded}\n    A function $\\phi:G\\rightarrow \\mathbb{C}$ is called \\textit{completely bounded} if there is a $B>0$ such that the operator-norm of $M_{n,\\phi}$, defined above is bounded by $B$, for each $n\\in\\mathbb{N}$. The $B_2$-norm (alternately the completely bounded norm or the cb-norm) of $\\phi$ is\n    $$||\\phi||_{B_2}:=\\sup_{n\\in\\mathbb{N}} ||M_{n,\\phi}||$$\n\\end{definition}", "main I": "\\begin{theorem}\\label{main I}\n    Suppose $\\Gamma$ is a finite simplicial graph and $\\{G_{v}\\}_{v\\in V(\\Gamma)}$ is a collection of weakly Haagerup groups with $\\Lambda_{WH}(G_v)=1$, for each $v$. Then the group $G(\\Gamma)$ has the weak Haagerup property with $\\Lambda_{WH}(G(\\Gamma))=1$.\n\\end{theorem}", "graph product definition": "\\begin{array}{rl}\\label{lemma cb equivalent computation}\n         \\langle Exp_o(\\frac{R(x)}{\\sqrt{n}}),Exp_o(\\frac{R(y)}{\\sqrt{n}}) \\rangle & =  e^{-\\frac{||R(x)-R(y)||^2}{n}}\\\\\n         \\\\\n        \\langle Exp_o(\\frac{S(x)}{\\sqrt{n}}),Exp_o(-\\frac{S(y)}{\\sqrt{n}}) \\rangle & = e^{-\\frac{||S(x)+S(y)||^2}{n}}\n    \\end{array}\n\\end{equation}\n\n\\medskip\n\nHence we have the analogue of lemma \\ref{cb equivalent} for the functions $e^{-\\frac{\\phi}{n}}$ for groups $G$ with $\\Lambda_{WH}=1$. Observe that the first kernel in equation \\ref{equivalent of weak haagerup} is CND. Lemma \\ref{schoenberg}, tells us that the kernel $G\\times G \\ni (x,y)\\mapsto e^{-||R(x)-R(y)||^2}$ is PD and therefore the $B_2$-norm of the kernel is 1. A simple calculation gives us $||Exp_o(\\frac{S(x)}{\\sqrt{n}})||, ||Exp_o(-\\frac{S(y)}{\\sqrt{n}})||$ equal to 1; therefore from lemma \\ref{cb equivalent} and equation \\ref{lemma cb equivalent computation} we can say that the other part of the kernel $(x,y)\\mapsto e^{-\\frac{\\phi(y^{-1}x)}{n}}$ also has bounded $B_2$-norm i.e.: \n$$||(x,y)\\mapsto e^{-\\frac{||S(x)+S(y)||^2}{n}}||_{B_2}\\leq 1$$\n\n\\subsection{Graph Product of Groups}\\label{graph product definition}\n\n Let $\\Gamma= (V(\\Gamma),E(\\Gamma))$ be a finite simplicial graph, which has no loops and no multiple edges. The \\textit{graph product} $G(\\Gamma)$ of a collection of groups $\\{G_v:v\\in V(\\Gamma)\\}$, indexed by the vertices of $\\Gamma$, is a group generated by the groups $G_v$'s where two elements from $G_v$ and $G_w$ will commute if and only if there is an edge $[v,w]\\in E(\\Gamma)$ \\cite{green1990graph}. \n\n \\medskip\n\n \\begin{definition}\n    Let $\\{G_v\\}_{v\\in V(\\Gamma)}$ be a collection of groups indexed by $V(\\Gamma)$.  The graph product $G(\\Gamma)$ of this collection is the quotient of the free product $*_{v\\in V(\\Gamma)}G_v$ by the normal subgroup generated by  $\\{g_vg_wg_{v}^{-1}g_{w}^{-1}:g_v\\in G_v,g_w\\in G_w ~\\text{and}~[v,w]\\in E(\\Gamma)\\}$ i.e. $$G(\\Gamma)=\\frac{ *_{v\\in V(\\Gamma)}G_v} { \\langle\\langle \\{[G_v,G_w]:[v,w]\\in E(\\Gamma)\\}\\rangle\\rangle}$$\n\n        \\begin{figure}[ht!]\n                \\centering\n                \\parbox{5.5cm}{\n                \\begin{center}\n                \\begin{tikzpicture}[scale=0.33]                 \\draw [fill] (1,0) circle [radius=5pt] node[below]{$G_1$};\n                \\draw [fill] (5,0) circle [radius=5pt] node[below]{$G_2$};\n                \\draw [fill] (6,3.5) circle [radius=5pt] node[right]{$G_3$};\n                \\draw [fill] (0,3.5) circle [radius=5pt] node[left]{$G_4$};\n                \\draw [fill] (3,6) circle [radius=5pt] node[above]{$G_5$};\n                \\end{tikzpicture}\n                \\end{center}\n                \\caption{$\\Gamma$ is disconnected\\\\ \\centering{$G(\\Gamma)=*_{i=1}^5G_i$}}\n                \\label{fig:2A}}\n                \\qquad\n                \\begin{minipage}{5.5cm}\n                \\begin{center}\n                \\begin{tikzpicture}[scale=0.33]                 \\draw [fill] (1,0) circle [radius=5pt] node[below]{$G_1$};\n                \\draw [fill] (5,0) circle [radius=5pt] node[below]{$G_2$};\n                \\draw [fill] (6,3.5) circle [radius=5pt] node[right]{$G_3$};gluing\n                \\draw [fill] (0,3.5) circle [radius=5pt] node[left]{$G_4$};\n                \\draw [fill] (3,6) circle [radius=5pt] node[above]{$G_5$};\n                \\draw (1,0) -- (5,0) -- (6,3.5) -- (0,3.5) -- (1,0) -- (3,6) -- (5,0) -- (0,3.5) -- (3,6) -- (6,3.5) -- (1,0);\n                \\end{tikzpicture}\n                \\end{center}\n                \\caption{$\\Gamma$ is complete\\\\\\centering{$G(\\Gamma)=\\oplus_{i=1}^5G_i$}}\n                \\label{fig:2B}\n                \\end{minipage}\n        \\end{figure} \n\n \\end{definition}\n\n Note that the graph product of groups over a graph $\\Gamma$ with $n$ vertices and no edges, is the free product of the vertex groups, and if $\\Gamma$ is a complete graph on $n$ vertices, the corresponding graph product turns out to be the direct product of the vertex groups, see the figures \\ref{fig:2A}, \\ref{fig:2B}. Considering these two cases as the two extremes (zero edges - all edges), graph product in general can be seen as an interpolation between the free product and direct product of the corresponding vertex groups. \n\n\\medskip\n\nSuppose $G= \\ast_v G_v$ is free product of groups, then any $g \\in G$ can be written as $g= g_1g_2\\cdots g_n$, where each $g_i\\in G_{v_i}\\setminus\\{e\\}$ and no two consecutive $g_i$'s will come from the same group. This expression for $g$ is unique in free product of groups, and is called \\textit{normal form} of $g$ \\cite{lyndon1977combinatorial},\\cite{serre2002trees}. \n\n\\medskip\n\nSimilarly, if $g\\in G(\\Gamma)$ is an element in a graph product of groups, $g$ can also be expressed as $g=g_1g_2\\dots g_n$, where $g_i\\in G_{v_i}\\setminus \\{e\\}$ and consecutive elements come from two different vertex groups. But the presence of relators $\\{[G_v,G_w]\\}_{[v,w]\\in E(\\Gamma)}$ in a general graph product $G(\\Gamma)$, prevents a unique normal form expression for an element $g$ in $G(\\Gamma)$. Instead each element has a well defined reduced form, which is essentially a normal form up to the `shuffles' facilitated by the relators. \n\n\\medskip\n\n By a \\textit{shuffle} on an expression $g=g_1g_2\\cdots g_{i-1}g_ig_{i+1}\\cdots g_n$ we mean that if $[v_i,v_{i+1}]\\in E(\\Gamma)$, then $g$ is also expressed as $g=g_1g_2\\cdots g_{i-1}g_{i+1}g_i\\cdots g_n$. A decomposition of $g$ as above is called \\textit{reduced} if no shuffles, followed by multiplication of two consecutive elements (whenever possible), reduces the number of letters present in the decomposition.  \n\n\\medskip\n\nGreen proved in her thesis (Theorem 3.9, \\cite{green1990graph}) that for each element $g$ in a graph product $G(\\Gamma)$, the set of letters appearing in a reduced expression is unique. Therefore, the number of elements in each expression of $g$ is the same, and this induces a ``reduced length\" metric on the graph product $G(\\Gamma)$, given by $d_r(g,h):=|h^{-1}g|_r$, where $|g|_r$ denotes the number of letters appearing in a reduced expression of $g$.  Note that it is the normal length in the case of free products of groups.\n\n\\medskip\n\nLet $v$ be a vertex of $\\Gamma$. The \\textit{star of} $v$, denoted by $st(v)$, is defined as the collection of vertices which are at a distance $\\leq 1$ from $v$ in $\\Gamma$, i.e. $st(v)=\\{w\\in \\Gamma:[v,w]\\in E(\\Gamma)\\}\\cup \\{v\\}$. Let $\\Gamma_{st(v)}\\subseteq \\Gamma$ be the maximum sub-graph in $\\Gamma$ having vertex set $st(v)$. Define $G(st(v)):=G(\\Gamma_{st(v)})$. \n\n\\medskip\n\nNotice that, if $g_1g_2\\cdots g_n$ is a reduced expression and for $1\\leq i<j\\leq n$, $g_i,g_j$ are from same vertex group $G_v$, then there has to be some $k$ with $i<k<j$ such that $g_k\\notin G(st(v))$. Otherwise it would contradict the definition of reduced expression.\n\n\\medskip\n\nBy a reduced form of an element $g\\in G(\\Gamma)$ we mean a class of reduced decompositions of $g$. The ``$d$-tail'' of $g$ is the collection of  strings of letters that appear in the last $d$-length portion of elements from the above class. \n\\begin{lemma} (see lemma 2.5 in \\cite{reckwerdt2017weak}\\label{d tail lemma})\n    Suppose that $g,h \\in G(\\Gamma)$ and $d = |h^{-1}g|_r$. Then each term of a reduced form of $h^{-1}g$ is either a term from the $d$-tail of $g$ or $h$, or an amalgamation of terms from the $d$-tails of $g$ and $h$.\n\\end{lemma}\n\nTherefore $d$-tails of $g$ in general can contain more than $d$ letters (in free product case it is exactly $d$).The following lemma, which is obvious from discussions here, has been used in the estimates of the main theorem.\n\\begin{lemma}\\cite{reckwerdt2017weak}\\label{maximum terms}\n    Suppose $M$ is maximum such that there is a complete sub-graph in $\\Gamma$ with $M$ number of vertices. Then the maximum number of letters in $d$-tail of any element of $G(\\Gamma)$ is $dM$.\n\\end{lemma}\n\n\\subsection{Construction of Cube Complex from Graph Products}\\label{construction of wall space}\n\nIn order to prove the weak Haagerup property, our goal is to construct an approximate identity $\\{\\phi_n\\}$ such that, \n\\begin{itemize}\n    \\item each $\\phi_n$ vanishes at infinity,\n    \\item and the $B_2$-norm of each $\\phi_n$ is uniformly bounded.\n\\end{itemize}\n\nIn section \\ref{weak haagerup property for graph product}, we produce such functions by gluing similar functions $\\phi_{n,v}$'s coming from each of the vertex group $G_v$'s along a certain `base space' $X$. In both the cases \\cite{bozejko1993weakly} and \\cite{reckwerdt2017weak}, in order to guarantee the uniform boundedness of the approximate identities in the $B_2$-norm, the corresponding prescriptions required scaling by a suitable PD factor. We too will need a similar positive definite scaling in order to keep $||\\phi_n||_{B_2}$'s uniformly bounded.\n\n\\medskip\n\nSimilar to \\cite{reckwerdt2017weak}, the base space in our case is a CAT(0)-cube complex canonically associated to the graph product $G(\\Gamma)$, and the distance function on a CAT(0)-cube complex was proven to be CND by Niblo and Reeves \\cite{niblo1997groups}. We obtain a PD scaling factor using this CND distance function following lemma \\ref{schoenberg}. In this subsection we will briefly discuss the CAT(0)-cube complex structure associated to a graph product $G(\\Gamma)$.\n\n\\medskip\n\\medskip\n\n A \\textit{cube complex} is a combinatorial object, obtained by gluing euclidean cubes of different dimensions, by isometries along faces of lower dimensions. The euclidean cubes induce a metric on the cube complex. A metric space $X$ is called `CAT(0)', if it satisfies the CAT(0)-comparison property i.e. every geodesic triangle in $X$ is as \\textit{thin} as its euclidean comparison triangle. By a \\textit{CAT(0)-cube complex} we mean a cube complex which satisfies the CAT(0) property. For details and other equivalent definitions see \\cite{gromov1987hyperbolic},\\cite{bestvina2014geometric}. \n\n \\medskip\n\n One can also endow a space $X$ with a CAT(0)-cube complex structure if it admits a certain ``\\textit{wall space}'' structure, given by the (combinatorial) intersection data of the hyperplanes partitioning $X$ into two halves, called \\textit{walls}. Informally speaking, a wall space is a space with a family of intersecting walls. \n\n \\begin{definition}(Wall Space)\n    A wall space structure on $X$ is given by a pair $(X,\\mathcal{W})$, where the collection $\\mathcal{W}\\subseteq \\mathcal{P}(X)~\\text{(the power set of $X$)}$ is considered as the space of \\textit{half-spaces}, such that\n\\begin{enumerate}\n    \\item $\\mathcal{W}$ is closed under taking complement i.e. $h\\in \\mathcal{W}\\iff h^c\\in \\mathcal{W}$ \n    \\item a \\textit{wall}, denoted simply by $h$, is defined to be the pair $(h,h^c)$, which is uniquely determined by the element $h\\in \\mathcal{W}$  \n    \\item two points $x,y\\in X$ are \\textit{separated} by a wall $h$ if $x\\in h$ and $y \\in h^c$ \n    \\item  two walls $u$ and $v$ are meant to be \\textit{crossing} each other if all the four sets $u\\cap v, u\\cap v^c, u^c\\cap v, u^c\\cap v^c$ are non-empty.\n\\end{enumerate}\n\\end{definition}\n\nIn \\cite{chatterji2005wall}, Chatterji and Niblo showed that if a group $G$ acts on a wall space $(X,\\mathcal{W})$, respecting the wall structure (i.e. sending walls to walls) given by $\\mathcal{W}$, then there is a dual CAT(0)-cube complex $\\hat{X}_{_\\mathcal{W}}$ associated to $(X,\\mathcal{W})$, on which $G$ acts suitably.\n\n\\begin{theorem} \\cite{chatterji2005wall}\\label{walls}\n    Suppose $G$ is a group acting on a wall space $X$, and $\\mathcal{W}$ is the set of walls with an upper bound on the number of mutually crossing walls. Then there is a finite dimensional $CAT(0)$ cube complex $\\hat{X}_{_\\mathcal{W}}$, on which $G$ admits an isometric action.\n\\end{theorem}\n\n\\begin{remark}\n    Although Chatterji and Niblo proved the above theorem in the proper setting (i.e. the group acting properly on the wall space), the theorem holds true both with or without the properness assumption.\n\\end{remark}\n\n\\subsubsection{Wall Space structure on Graph Products}\\label{Wall Space structure on Graph Product}\n Let $v$ be a vertex in $\\Gamma$. Define $W_v\\subset G(\\Gamma)$ to be the collection of those elements which have a reduced form that starts with elements from $G_v$ i.e. $$W_v=\\{g \\in G(\\Gamma):g=g_1g_2\\cdots g_m~\\text{is a reduced form and}~g_1\\in G_v\\}.$$\n\n\\begin{figure}[ht!]\n    \\centering\n       \\begin{tikzpicture}[scale= 0.5]\n            \\path [draw=none,fill=gray, fill opacity = 0.2,rotate=30] (0.5,-0.15) ellipse (4cm and 2.5cm);\n            \\draw  (0,0) -- (60:3) -- (3,0) -- cycle;\n            \\draw (0,0) -- (-1.5,-1.5);\n             \\draw (3,0) -- (4.5,-1.5);\n             \\draw (1.5,5.19/2) -- (1.5,5.19/2+2.12);\n             \\draw  (1.5,5.19/2+2.12) --  (3,5.19+2.12) --  (0,5.19+2.12) --(1.5,5.19/2+2.12);\n             \\draw  (1.5-4.5,-8.81+5.19/2+2.12) --  (-4.5+3,5.19+2.12-8.81) --  (0-4.5,5.19+2.12-8.81) --(1.5-4.5,5.19/2+2.12-8.81);\n             \\draw  (1.5+4.5,5.19/2+2.12-8.81) --  (3+4.5,5.19+2.12-8.81) --  (4.5,5.19+2.12-8.81) --(1.5+4.5,5.19/2+2.12-8.81);\n             \\draw (3,5.19+2.12)--(3+4.5,5.19+2.12-8.81);\n             \\draw (1.5+4.5,5.19/2+2.12-8.81)--(1.5-4.5,-8.81+5.19/2+2.12);\n             \\draw (0-4.5,5.19+2.12-8.81)--(0,5.19+2.12);\n              \\node at (0,-0.5) {$v$};\n              \\draw [fill] (0,0) circle [radius=3pt];\n              \\node at (-4,4) {Some graph $\\Gamma$};\n              \\draw [fill] (-1.5,-1.5) circle [radius=3pt];\n              \\node at (-1.3,-2) {$v_1$};\n             \\draw [fill] (3,0) circle [radius=3pt];\n              \\node at (3,-0.5) {$v_2$};\n            \\draw [fill] (1.5,2.598) circle [radius=3pt];\n              \\node at (2,2.5) {$v_3$};\n        \\end{tikzpicture}\n    \\caption{$G_v,G_{v_1},G_{v_2},G_{v_3}$ generates $G(\\Gamma_{st(v)})$  in $G(\\Gamma)$}\n    \\label{fig:3}\n\\end{figure}\n\n Observe that $W_v$ is a wall in $G(\\Gamma)$ (since $(W_v,W_v^c)$ forms a partition of $G(\\Gamma)$). In the figure \\ref{fig:3}, the star of $v$ is the sub-graph $\\Gamma_{st(v)}$ of $\\Gamma$, consisting of vertices $\\{v,v_1,v_2,v_3\\}$ along with the concerned edges and $W_v$ is exactly $G(\\Gamma_{st(v)})$. Notice that $W_v^c$ is the set of elements whose reduced words start with element from the groups associated to vertices lying outside the shaded region.\n\n \\medskip\n\n For any $g\\in G(\\Gamma)$, the sets $gW_v$ and $gW_v^c$ is naturally defined by left multiplication. One can see that the set $gW_v$ contains all the elements $k\\in G(\\Gamma)$ whose $g^{-1}$ translate has a reduced form starting with an element from $G_v$. Further it is not hard to check that $gW_v^c=(gW_v)^c$. Therefore $gW_v$ is also a wall of $G(\\Gamma)$. Let $\\mathcal{W}$ be the collection of walls given by \n \\begin{equation}\\label{definition of walls}\n    \\mathcal{W}:=\\{gW_v:g\\in G(\\Gamma), v\\in \\Gamma\\}    \n \\end{equation}\n\nIn \\cite{reckwerdt2017weak}, it was showed that  $\\mathcal{W}$ has an upper bound on the number of mutually crossing walls, thus one have a finite dimensional dual $CAT(0)$-cube complex $\\hat{X}_{_\\mathcal{W}}$, as in theorem \\ref{walls}. In the following we denote $\\hat{X}_{_\\mathcal{W}}$ simply by $X$. The vertex set $V(X)$ is the collection of all (left)-cosets of $G(st(v))$'s in $G(\\Gamma)$, for each $v\\in \\Gamma$, on which $G$ admits a canonical action.  Moreover there is an $x_0\\in X$ such that for all $ g,h\\in G(\\Gamma)$ (lemma 3.5, \\cite{reckwerdt2017weak}) \n\\begin{equation}\\label{CAT}\n    d_X(gx_0,hx_0)=2|h^{-1}g|_r\n\\end{equation}\n\n\\medskip\n\nThe distance metric on a finite dimensional $CAT(0)$-cube complex is of CND type \\cite{niblo1997groups}. The equation \\ref{CAT} tells us the reduced metric on $G(\\Gamma)$ is proportional to the combinatorial metric of the dual CAT(0)-cube complex, obtained from the above wall structure on $G(\\Gamma)$.  Hence the reduced length function $g\\mapsto |g|_r$ on $G(\\Gamma)$ is a CND function.\n\n\\section{Weak Haagerup property for Graph Product}\\label{weak haagerup property for graph product}\n\nLet $\\Gamma$ be a finite simplicial graph, and $\\{G_v\\}_{v\\in V(\\Gamma)}$ be a collection of groups having weak Haagerup property with weak Haagerup constant $\\Lambda_{WH}(G_v)=1$. For each vertex $v\\in V(\\Gamma)$, we are given a weak Haagerup data $(G_v,\\phi_v,R_v,S_v,\\mathcal{H}_v)$. In this section we will combine this entire vertex data $\\{(G_v,\\phi_v,R_v,S_v,\\mathcal{H}_v)\\}_{v \\in V(\\Gamma)}$ to prove weak Haagerup property for the graph product $G(\\Gamma)$ of the groups $\\{G_v\\}_{v\\in V(\\Gamma)}$. \n\n\\medskip\n\nFrom the above data, approximate identity on $G_v$ is given by $\\{e^{-\\frac{\\phi_v}{n}}\\}$. We scale $e^{-\\frac{\\phi_v}{n}}$ suitably to get a function $\\psi_{n,v}$, in order to define 1 at the corresponding identity of the groups $G_v$, for $v \\in V(\\Gamma)$. Following \\cite{bozejko1993weakly}, we will define an approximate identity on $G(\\Gamma)$ by combining $\\psi_{n,v}$'s.\n\n\\medskip\n\nConsider $\\psi_{n,v}:G_v \\rightarrow \\mathbb{R}$ given by $g \\mapsto e^{\\frac{\\phi_v(1_v)}{n}}e^{-\\frac{\\phi_v(g)}{n}}$. Observe that $\\{\\psi_{n,v}\\}$ is also an approximate identity of the same type as $e^{-\\frac{\\phi_v}{n}}$, and the $B_2$-norm: $||\\psi_{n,v}||_{B_2}\\leq e^{\\frac{\\phi_v(1_v)}{n}}$. Therefore for any $x,y \\in G_v$ we have,\n\\begin{equation}\\label{group invariance}\n    \\psi_{n,v}(y^{-1}x)=e^{-\\frac{-\\phi_v(1_v)+\\phi_v(y^{-1}x)}{n}}=e^{-\\frac{\\big[-\\phi_v(1_v)+||R_v(x)-R_v(y)||^2+||S_v(x)+S_v(y)||^2\\big]}{n}}\n\\end{equation}\n\n\\medskip\n\nFurther from the discussion of section \\ref{Exponential Hilbert space}, one can write the $\\psi_{n,v}(y^{-1}x)$ as an inner-product between two vectors in a Hilbert space (see \\ref{lemma cb equivalent computation}):\n\n\\begin{multline*}\n    \\psi_{n,v}(y^{-1}x)=\\langle Exp_o(\\frac{R_v(x)}{\\sqrt{n}}) \\otimes e^{\\frac{\\phi_v(1_v)}{2n}} Exp_o(\\frac{S_v(x)}{\\sqrt{n}}),\n    \\\\ Exp_o(\\frac{R_v(y)}{\\sqrt{n}}) \\otimes e^{\\frac{\\phi_v(1_v)}{2n}} Exp_o(-\\frac{S_v(y)}{\\sqrt{n}})\\rangle\n\\end{multline*} \nwhere the vector norm of each component is bounded by $\\sqrt{e^{\\frac{\\phi_v(1_v)}{n}}}$. \nFor simplicity of notation we write \n\n\\begin{equation*}\n    \\alpha_{n,v}(x) := \\underbrace{Exp_o(\\frac{R_v(x)}{\\sqrt{n}})}_{\\mbox{$\\alpha_{n,v,R_v}(x)$}} ~\\otimes~  \\underbrace{e^{\\frac{\\phi_v(1_v)}{2n}} Exp_o(\\frac{S_v(x)}{\\sqrt{n}})}_{\\mbox{$\\alpha_{n,v,S_v}(x)$}}\n\\end{equation*}\nand \n\\begin{equation*}\n    \\beta_{n,v}(x) :=  \\underbrace{Exp_o(\\frac{R_v(x)}{\\sqrt{n}})}_{\\mbox{$\\beta_{n,v,R_v}(x)$}} ~\\otimes~  \\underbrace{e^{\\frac{\\phi_v(1_v)}{2n}} Exp_o(\\frac{-S_v(x)}{\\sqrt{n}})}_{\\mbox{$\\beta_{n,v,S_v}(x)$}}\n\\end{equation*}\nwhere $\\alpha_{n,v}$ and $\\beta_{n,v}$ both are maps from $G_v$ to the Hilbert space $Exp(\\mathcal{H}_v)\\otimes Exp(\\mathcal{H}_v)$. Note that here $\\alpha_{n,v,R_v}=\\beta_{n,v,R_v}$.\n\n\\medskip\n\nLet $X$ be the finite dimensional CAT(0)-cube complex underlying the graph product $G(\\Gamma)$ described in the section \\ref{construction of wall space}. In \\cite{das2023stability}, the authors provided a way to combine $\\{R_v\\}$ along $X$. Here, we briefly recall that construction.\n\n\\medskip\n\nThe vertex set $V(X)$ is the collection of cosets of $G(st(v))$, for all $v\\in V(\\Gamma)$ i.e. $V(X)=\\{G(\\Gamma)/G(st(v))\\}_{v\\in V(\\Gamma)}$. Consider the Hilbert space $\\mathcal{H}_1:=\\oplus_{t\\in X} \\mathcal{H}_t$, where $\\mathcal{H}_t$ is $\\mathcal{H}_v$ if $t$ is a coset $gG(st(v))$. A non-trivial $\\gamma\\in G(\\Gamma)$ has a reduced form  $\\gamma=\\gamma_1\\gamma_2\\ldots \\gamma_m$, where $\\gamma_i\\in G_{v_i}\\leq G(st(v_i))$ for $i=1,2,\\dots,m$. Define $R_\\Gamma:G(\\Gamma)\\rightarrow \\mathcal{H}_1$ by:\n\\begin{equation}\\label{definition of R_Gamma}\n    R_\\Gamma(\\gamma):=\n        \\begin{cases}\n            \\oplus_{i=1}^m R_{v_i}(\\gamma_i)_{\\gamma_1\\gamma_2\\ldots \\gamma_{i-1}G(st(v_i))}, ~~\\text{if}~\\gamma\\neq 1_{G(\\Gamma)}\\\\\n            0_{\\mathcal{H}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\text{if}~\\gamma=1_{G(\\Gamma)}\\\\\n        \\end{cases}\n\\end{equation}\nThen we consider the map from $G(\\Gamma)$ to the exponential Hilbert space $Exp(\\mathcal{H}_1)$, given by $$ \\gamma \\mapsto Exp_o(\\frac{R_\\Gamma(\\gamma)}{\\sqrt{n}})$$\nsatisfying the following property: for any $\\gamma,\\eta\\in G(\\Gamma)$\n\n\\begin{equation}\\label{schoenberg in graph product}\n    \\langle Exp_o(\\frac{R_\\Gamma(\\gamma)}{\\sqrt{n}}),Exp_o(\\frac{R_\\Gamma(\\eta)}{\\sqrt{n}})\\rangle=e^{-\\frac{||R_\\Gamma(\\gamma)-R_\\Gamma(\\eta)||^2}{n}}.\n\\end{equation}\n\n\\medskip\n\nLet us choose $\\epsilon>0$. For each $v \\in V(\\Gamma)$, $G_v$ is weakly Haagerup with $\\Lambda_{WH}(G_v)=1$. From the definition \\ref{definition WH} and the above defined approximate identity $\\psi_{n,v}$ on $G_v$, we can choose $n$ sufficiently large so that\n$||\\psi_{n,v}||_{B_2}\\leq 1+\\epsilon$. By abuse of notation, we denote that tail of the above sequence to be $\\phi_{n,v}$ for which the $B_2$-norm is less than $1+\\epsilon$ for each $n$. More precisely for any $x \\in G_v$ we have $||\\alpha_{n,v,R_v}(x)||=1=||\\beta_{n,v,R_v}(x)||$. Hence\n\\begin{equation*}                   \n||\\psi_{n,v}||_{B_2}=||\\alpha_{n,v,S_v}||_\\infty\\cdot ||\\beta_{n,v,S_v}||_\\infty\\leq e^{\\frac{\\phi_v(1_v)}{n}}\\leq 1+\\epsilon \n\\end{equation*}\nand this choice of $\\epsilon$ and $n$ also gives:\n\\begin{equation}\\label{S_v estimate}\n\\text{for any }~ x \\in G_v\n    \\begin{cases}\n    ||\\alpha_{n,v,S_v}(x)|| \\leq \\sqrt{1+\\epsilon}\\\\\n    ||\\beta_{n,v,S_v}(x)|| \\leq \\sqrt{1+\\epsilon}\n    \\end{cases}\n\\end{equation}\n\n\\begin{remark}\\label{kernel at x,x}\n    Note that for each $x\\in G_v$ we have $\\langle\\alpha_{n,v,S_v}(x),\\beta_{n,v,S_v}(x)\\rangle=1$.\n\\end{remark}\n\nGiven the choice of $ \\epsilon >0$, we consider the tuple $$\\Sigma_{v,\\epsilon}:=(G_v,\\{\\psi_{n,v}\\},\\{\\alpha_{n,v}\\},\\{\\beta_{n,v}\\},Exp(\\mathcal{H}_v)\\otimes Exp(\\mathcal{H}_v))$$ as an $\\epsilon$-perturbed weak Haagerup data for $G_v$. In order to prove weak Haagerup  property for the graph product, we produce a similar $\\epsilon$-data on $G(\\Gamma)$, denoted $\\Sigma_{_{\\Gamma,\\epsilon}}$, obtained from the entire vertex data $\\{\\Sigma_{v,\\epsilon}\\}_{v\\in V(\\Gamma)}$. Finally, we let the parameter $\\epsilon$ tend to zero, in order to obtain $\\Lambda_{WH}(G(\\Gamma))=1$. \n\n\\medskip\n\nSo far we have managed to combine the functions $\\alpha_{n,v,R_v}$ and $\\beta_{n,v,R_v}$ over all $v \\in V(\\Gamma)$ into the corresponding functions for $G(\\Gamma)$ (see equation \\ref{schoenberg in graph product}). To construct a desired approximate identity on $G(\\Gamma)$, what remains is to glue the functions $\\alpha_{n,v,S_v}$ and $\\beta_{n,v,S_v}$ for all $v \\in V(\\Gamma)$. We break up the remainder of the proof into the following three parts.\n\n\\medskip\n\n\\begin{enumerate}\n    \\item[\\textit{Step-I}:]  We construct a family of kernels $\\psi_{n,\\Gamma,d}$ (not on the entire $G(\\Gamma)\\times G(\\Gamma)$, but) restricted to the $d$-spheres of each element of $G(\\Gamma)$, with respect to the reduced length. Finally, the kernels $\\psi_{n,\\Gamma,d}$ will be glued together over varying $d\\geq 0$, obtaining a family of kernels $\\{\\psi_{n,\\Gamma}\\}_{_{n\\in \\mathbb{N}}}$ on the whole group $G(\\Gamma)$.\n    \\medskip\n    \\item[\\textit{Step-II}:] The $B_2$-norms of the above kernels $\\psi_{n,\\Gamma}$ are estimated (see equation \\ref{1st B_2 estimate}) by comparing their distances from suitable PD kernels $\\sigma_{n,\\Gamma}$ on $G(\\Gamma)$ (constructed in \\ref{positive deinite comparison}).\n    \\item[\\textit{Step-III}:] We prove the kernels $\\psi_{n,\\Gamma}$ to be $G(\\Gamma)$-invariant, and thus providing a desired approximate identity $\\{\\phi_{n,\\Gamma}\\}$ on $G(\\Gamma)$.\n\\end{enumerate}\n\n\\medskip\n\n\\subsection{Construction of kernels}\\label{section I}\n\nFirst we consider an $\\epsilon$-perturbed average of the vectors $\\alpha_{n,v,S_v}(x)$ and $\\beta_{n,v,S_v}(x)$, so that the resulting vector lies very close to the other two vectors, i.e. for any $x \\in G_v$, define $$avg_{n,v,S_v}(x):=\\frac{\\alpha_{n,v,S_v}(x)+\\beta_{n,v,S_v}(x)}{2+\\sqrt{2\\epsilon}}.$$ Now extend the Hilbert space $Exp(\\mathcal{H}_v)$ to $\\widehat{Exp(\\mathcal{H}_v)}:=Exp(\\mathcal{H}_v)\\oplus \\mathbb{C}^2\\oplus \\mathbb{C}^2$ and consider the map $\\theta_{n,v,S_v}:G_v \\rightarrow \\widehat{Exp(\\mathcal{H}_v)}$ given by \n\\begin{align*}\n    x\\mapsto\n        \\begin{cases}\n            \\Bigg(avg_{n,v,S_v}(x),\\begin{pmatrix}\n                D_{n,v,S_v}(x)\\\\D_{n,v,S_v}(x)\n            \\end{pmatrix},\\begin{pmatrix}\n                0\\\\0\n            \\end{pmatrix}\\Bigg), ~\\text{if}~x\\neq 1_v\\\\ \n             \\\\\n                \\Bigg(avg_{n,v,S_v}(1_v),\\begin{pmatrix}\n                0\\\\0\n            \\end{pmatrix},\\begin{pmatrix}\n                D_{n,v,S_v}(1_v)\\\\D_{n,v,S_v}(1_v)\n            \\end{pmatrix} \\Bigg), ~\\text{if}~ x=1_v\n        \\end{cases}\n\\end{align*}\nwhere $D_{n,v,S_v}(x)=\\sqrt{\\frac{1-||avg_{n,v,S_v}(x)||^2}{2}} \\in \\mathbb{C}$.\n\n\\begin{remark}\n    Note that from the definition of $\\theta_{n,v,S_v}$ it follows that for any $x\\in G_v$, $||\\theta_{n,v,S_v}(x)||=1$.\n\\end{remark}\n\nLet $X$ be the finite dimensional CAT(0)-cube complex on which $G(\\Gamma)$ acts, (see \\ref{construction of wall space}). We know that $V(X)=\\cup_{v \\in \\Gamma}\\{G(\\Gamma)/G(st(v))\\}$. We consider the larger Hilbert space $$\\mathcal{H}_2=\\otimes_{t\\in V(X)}\\widehat{Exp(\\mathcal{H}_t)}$$  where $\\mathcal{H}_t:=\\mathcal{H}_v$, if $t=kG(st(v))$ for some $v$, and the corresponding vacuum vector be $\\Bigg(avg_{n,v,S_v}(1_v),\\begin{pmatrix}\n                0\\\\0\n            \\end{pmatrix},\\begin{pmatrix}\n                D_{n,v,S_v}(1_v)\\\\D_{n,v,S_v}(1_v)\n            \\end{pmatrix} \\Bigg)$. The reason for taking these extra $\\mathbb{C}^2$ components is to absorb some ``errors'' accumulated in the process (see expressions in \\ref{error manipulation} and afterwards). \n\n\\medskip \n\nLet $\\gamma=\\gamma_1\\gamma_2\\cdots\\gamma_m$ be a reduced form for the element $\\gamma\\in G(\\Gamma)$, and the collection of ordered $d$-tuples $(\\gamma_{m-d+1},\\cdots,\\gamma_m)$'s appearing in the last $d$-terms of any reduced form of $\\gamma$ is the $d$-tail of $\\gamma$ (see section \\ref{graph product definition}). For each $d\\in \\mathbb{N}$, we define $\\alpha_{n,\\Gamma,d} $ and $ \\beta_{n,\\Gamma,d}:G(\\Gamma)\\rightarrow \\mathcal{H}_2$, given by\n\n\\begin{equation}\\label{alpha}\n\\alpha_{n,\\Gamma,d}(\\gamma)=\\otimes_{i=1}^m \n\\begin{cases}\n\\theta_{n,\\ast,S_\\ast}(\\gamma_i)_{\\gamma_1\\cdots\\gamma_{i-1}G(st(v_i))}, ~~\\qquad\\qquad\\qquad\\qquad \\text{if}~\\gamma_i\\notin d \\text{-tail of}~\\gamma,\\\\\n\\\\\n\\Bigg(\\alpha_{n,\\ast,S_\\ast}(\\gamma_i),\\begin{pmatrix}\n             C^{\\alpha_{n,\\ast,S_\\ast}}(\\gamma_i,\\gamma_i)\\\\0\n         \\end{pmatrix},\\begin{pmatrix}\n             C^{\\alpha_{n,\\ast,S_\\ast}}(\\gamma_i,1_\\ast)\\\\0\n         \\end{pmatrix}\\Bigg)_{\\gamma_1\\cdots\\gamma_{i-1}G(st(v_i))},\\\\\n         \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\text{if}~\\gamma_i\\in d \\text{-tail of}~\\gamma\n\\end{cases}\n\\end{equation}\n\n\\smallskip\n\n\\begin{equation}\\label{beta}\n\\beta_{n,\\Gamma,d}(\\gamma)=\\otimes_{i=1}^m\n\\begin{cases}\n\\theta_{n,\\ast,S_\\ast}(\\gamma_i)_{\\gamma_1\\cdots\\gamma_{i-1}G(st(v_i))}, ~~\\qquad\\qquad\\qquad\\qquad \\text{if}~\\gamma_i\\notin d \\text{-tail of}~\\gamma,\\\\\n\\\\\n\\Bigg(\\beta_{n,\\ast,S_\\ast}(\\gamma_i),\n        \\begin{pmatrix}\n             0\\\\C^{\\beta_{n,\\ast,S_\\ast}}(\\gamma_i,\\gamma_i)\n         \\end{pmatrix},\n\n         \\begin{pmatrix}\n             0\\\\C^{\\beta_{n,\\ast,S_\\ast}}(\\gamma_i,1_\\ast)\n         \\end{pmatrix}\\Bigg)_{\\gamma_1\\cdots\\gamma_{i-1}G(st(v_i))},\\\\\n         \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\text{if}~\\gamma_i\\in d \\text{-tail of}~\\gamma\n\\end{cases}\n\\end{equation}\n\n\\noindent where $\\ast$ stands for that $v \\in \\Gamma$, such that $\\gamma_i \\in G_v$ and the expressions for $ C^{\\alpha_{n,v,S_v}}(x,y)$ and $C^{\\beta_{n,v,S_v}}(x,y)$ are given by: for all $x,y \\in G_v$, \n\n \\begin{align*}\n    C^{\\alpha_{n,v,S_v}}(x,y) &:=\\frac{\\langle\\alpha_{n,v,S_v}(x),\\beta_{n,v,S_v}(y)-avg_{n,v,S_v}(y)\\rangle}{D_{n,v,S_v}(y)}\\\\\n      C^{\\beta_{n,v,S_v}}(x,y) &:=\\frac{\\langle\\alpha_{n,v,S_v}(y)-avg_{n,v,S_v}(y),\\beta_{n,v,S_v}(x)\\rangle}{D_{n,v,S_v}(y)}\n \\end{align*}\n\nNow we prove the well-definedness of the above maps. Suppose there is an edge between $v_i$ and $v_{i+1}$ vertices of $\\Gamma$. Then we have \n$$\\gamma=\\gamma_1\\gamma_2\\cdots\\gamma_{i-1}\\gamma_i\\gamma_{i+1}\\cdots\\gamma_m=\\gamma_1\\cdots \\gamma_{i-1}\\gamma_{i+1}\\gamma_i\\gamma_{i+2}\\cdots \\gamma_m$$\nChecking the well-definedness, then reduces to validation of the following two equations:\n    \\begin{align*}\n        \\gamma_1\\cdots\\gamma_{i-1}G(st(v_i))=\\gamma_1\\cdots\\gamma_{i-1}\\gamma_{i+1}G(st(v_i)) \\\\\n        \\gamma_1\\cdots\\gamma_{i}G(st(v_{i+1}))=\\gamma_1\\cdots \\gamma_{i-1}G(st(v_{i+1}))\n    \\end{align*}\nThis is true due to the fact that $[G_{v_i},G_{v_{i+1}}]=1$ in $G(\\Gamma)$. Since the expression of the reduced form for $\\gamma$ is unique up to a finite number of shuffles between two consecutive factors whenever they commute. So the maps $\\alpha_{n,\\Gamma,d}$ and $\\beta_{n,\\Gamma,d}$ are well-defined. \n\n\\medskip\n\nTherefore we can define a kernel \n\\begin{equation*}\n    \\begin{array}{rl}\n       \\psi_{n,\\Gamma,d}:G(\\Gamma)\\times G(\\Gamma)  &\\rightarrow \\mathbb{C}  \\\\\n       (\\gamma,\\eta)  & \\mapsto \\langle\\alpha_{n,\\Gamma,d}(\\gamma),\\beta_{n,\\Gamma,d}(\\eta)\\rangle\n    \\end{array}"}, "pre_theorem_intro_text_len": 7287, "pre_theorem_intro_text": "A group is called \\textit{amenable} if it admits a sequence of functions, converging point-wise to the constant function one (called \\textit{approximate identities}), which are finitely supported, and positive definite. Amenable groups form a fairly large class which is stable under taking direct products, quotients and subgroups. Finite rank free groups are not amenable.\n\n\\medskip\n\nIn \\cite{haagerup1978example}, Haagerup showed existence of an approximate identity $\\{ \\phi_{n} \\}$ on the free group $\\mathbb{F}_n$, such that they are positive definite, vanishing at infinity (with respect to word metric). Akemann-Walter \\cite{akemann1981unbounded} and Choda \\cite{choda1983group} called a group to have the \\textit{Haagerup property} if it admits a similar approximate identity as in \\cite{haagerup1978example}. Clearly amenable groups have Haagerup property. Fundamental group of hyperbolic surfaces, CAT(0)-cubulated groups, groups acting properly on a tree and many others have this property. Cherix et. al \\cite{cherix2001groups} provides a good survey on this topic.\n\n\\medskip\n\nOne other way amenability can be generalized is called \\textit{weak amenability}, where $G$ is required to have approximate identities consisting of finitely supported functions which are, uniformly bounded in the `completely bounded'-norm ($B_2$-norm, see definition \\ref{definition completely bounded}). Since a positive definite map $\\phi$ on $G$ has bounded cb-norm, this notion generalizes amenability. There is a constant, canonically associated to a weakly amenable group $G$, obtained as the infimum of the cb-norms of these approximate identities, called the Cowling-Haagerup constant of $G$ and is denoted by $\\Lambda_{CH}(G)$. Clearly, every amenable group has $\\Lambda_{CH}=1$. Most groups mentioned above were shown to be weakly amenable with $\\Lambda_{CH}=1$(\\cite{haagerup1978example}, \\cite{ozawa2008weak}, \\cite{mizuta2008bozejko}, \\cite{guentner2010weak}). For an example of a group with $\\Lambda_{CH}>1$, we mention a uniform lattice $\\Gamma$ in $Sp(1,n)$, where the $\\Lambda_{CH}(\\Gamma)=2n-1$. This group satisfies the Kazhdan's property (T), which can be seen as a strong negation of the Haagerup property (see \\cite{bekka2008kazhdan},\\cite{cowling1989completely}).\n\n \\medskip\n\nClearly, groups like $Sp(n,1)$ (or, lattices in them) do not satisfy the Haagerup property. It is also observed that \\textit{most} weakly amenable groups with $\\Lambda_{CH}=1$ happens to be groups with the Haagerup property and the famous conjecture by Cowling in this regard stated: $G$ is weakly amenable with $\\Lambda_{CH}=1$ if and only if $G$ has Haagerup property (section 1.3.1, \\cite{cherix2001groups}). Ozawa and Popa constructed examples of a non-weakly amenable groups in \\cite{ozawa2010class}, which was later shown to be Haagerup by Cornulier et. al \\cite{cornulier2012proper}. The other direction of the conjecture remains open.\n\n\\begin{figure}[ht!]\n    \\centering\n         \\begin{tikzpicture}[scale=0.55]\n            \\scope\n            \\draw (-3.5,-3.5) rectangle (2.5,3);\n            \\draw (-5,-5) rectangle (5,5);\n            \\draw (-7,-7) rectangle (7,7);\n            \\endscope\n\n            \\scope\n            \\draw[black!30!white] (-0.5,-2.75) rectangle (2,2.25);\n            \\draw[dashed] (-1.5,-4.3) rectangle (4,4.3);\n            \\draw[dashed] (-2,-6) rectangle (6.5,6.5);\n            \\endscope\n\n            \\draw (-2.75,-0.5)  node [rotate=270,text=black,scale=0.75] {Haagerup groups};\n            \\draw (-4.25,0)  node [rotate=270,text=black,scale=0.75] {weakly Haagerup groups with $\\Lambda_{WH}=1$};\n            \\draw (-6,0)  node [rotate=270,text=black,scale=0.75] {weakly Haagerup groups};\n            \\draw (0.75,-0.25)  node [rotate=90,text=black,scale=0.75] {Amenable groups};\n            \\draw (3.25,0)  node [rotate=90,text=black,scale=0.75] {weakly amenable groups with $\\Lambda_{CH}=1$};\n            \\draw (5.75,0)  node [rotate=90,text=black,scale=0.75] {weakly amenable groups};\n        \\end{tikzpicture}\n    \\label{fig:1}\n\\end{figure}\n\nA further generalisation called the \\textit{weak Haagerup Property} interpolates between the Haagerup property and weak amenability. Introduced by Knudby \\cite{knudby2014semigroups} (see definition \\ref{definition WH}), a group is called weakly Haagerup if there is an approximate identity of vanishing at infinity maps, uniformly bounded in $B_2$-norm, on the group. A group $G$ with the weak Haagerup property also comes equipped with a constant $\\Lambda_{WH}(G)$, defined similar to $\\Lambda_{CH}$ and it is evident that $\\Lambda_{WH}(G)\\leq \\Lambda_{CH}(G)$, for any $G$. An example of a group not having weak Haagerup property is given in \\cite{haagerup2015weak}.\n\n\\medskip\n\nBoth amenability and weak amenability are preserved under direct products. In \\cite{bozejko1993weakly},\nBozejko and Picardello proved that free product of amenable groups is weakly amenable. Whereas it is still an open question whether free product of any two weakly amenable groups is weakly amenable or not. Though as a special case Ricard and Xu had proved that free product of two weakly amenable group with $\\Lambda_{CH}=1$ also has weak amenability \\cite{ricard2006khintchine}. In general, Haagerup property of groups also proved to be stable under taking direct products, free products and amalgamated free products (over finite subgroups) \\cite{cherix2001groups}. Knudby in \\cite{knudby2016weak} showed that direct product of two weak Haagerup groups is again weakly Haagerup. In this paper we wish to study stability of the weak Haagerup property under certain group construction (\\textit{graph products}).\n\n\\medskip\n\nThe \\textit{graph products of groups} $G(\\Gamma)$, defined by Green in her thesis \\cite{green1990graph}, is a novel way to combine a collection $\\{G_v\\}_{v\\in V(\\Gamma)}$ of groups parameterised along the vertex set $V(\\Gamma)$ of a finite graph $\\Gamma$. Depending on the nature of the given (finite) graph, a graph product of groups can also be thought of as an interpolation between the free product (which corresponds to a graph without an edge) and the direct product (which corresponds to the complete graph on a given set of vertices) of groups (for details see section \\ref{graph product definition}).      \n\n\\medskip\n\nFor a finite simplicial graph $\\Gamma$, and a collection $\\{G_v\\}_{v\\in V(\\Gamma)}$ of weakly amenable groups with $\\Lambda_{CH}(G_{v})=1$ for each vertex $v\\in V(\\Gamma)$, it was shown by Reckwerdt in \\cite{reckwerdt2017weak}, that the graph product $G(\\Gamma)$ is weakly amenable. Stability of Haagerup property under graph product was proved by the present authors in \\cite{das2023stability}, (see also \\cite{antolin2013haagerup}). The main result of this paper establishes stability of the weak Haagerup property under graph product. Note that a group with Haagerup property has $\\Lambda_{WH}=1$, and a weakly amenable group $G$ with $\\Lambda_{CH}(G)=1$ is also weakly Haagerup with $\\Lambda_{WH}(G)=1$. Therefore, following Knudby's program in \\cite{knudby2014semigroups}, proving a weakly Haagerup group with $\\Lambda_{WH}=1$ to be Haagerup settles the remaining part of the Cowling's conjecture. The main result of this paper can be seen as a supportive evidence for the conjecture.", "context": "One other way amenability can be generalized is called \\textit{weak amenability}, where $G$ is required to have approximate identities consisting of finitely supported functions which are, uniformly bounded in the `completely bounded'-norm ($B_2$-norm, see definition \\ref{definition completely bounded}). Since a positive definite map $\\phi$ on $G$ has bounded cb-norm, this notion generalizes amenability. There is a constant, canonically associated to a weakly amenable group $G$, obtained as the infimum of the cb-norms of these approximate identities, called the Cowling-Haagerup constant of $G$ and is denoted by $\\Lambda_{CH}(G)$. Clearly, every amenable group has $\\Lambda_{CH}=1$. Most groups mentioned above were shown to be weakly amenable with $\\Lambda_{CH}=1$(\\cite{haagerup1978example}, \\cite{ozawa2008weak}, \\cite{mizuta2008bozejko}, \\cite{guentner2010weak}). For an example of a group with $\\Lambda_{CH}>1$, we mention a uniform lattice $\\Gamma$ in $Sp(1,n)$, where the $\\Lambda_{CH}(\\Gamma)=2n-1$. This group satisfies the Kazhdan's property (T), which can be seen as a strong negation of the Haagerup property (see \\cite{bekka2008kazhdan},\\cite{cowling1989completely}).\n\nA further generalisation called the \\textit{weak Haagerup Property} interpolates between the Haagerup property and weak amenability. Introduced by Knudby \\cite{knudby2014semigroups} (see definition \\ref{definition WH}), a group is called weakly Haagerup if there is an approximate identity of vanishing at infinity maps, uniformly bounded in $B_2$-norm, on the group. A group $G$ with the weak Haagerup property also comes equipped with a constant $\\Lambda_{WH}(G)$, defined similar to $\\Lambda_{CH}$ and it is evident that $\\Lambda_{WH}(G)\\leq \\Lambda_{CH}(G)$, for any $G$. An example of a group not having weak Haagerup property is given in \\cite{haagerup2015weak}.\n\nBoth amenability and weak amenability are preserved under direct products. In \\cite{bozejko1993weakly},\nBozejko and Picardello proved that free product of amenable groups is weakly amenable. Whereas it is still an open question whether free product of any two weakly amenable groups is weakly amenable or not. Though as a special case Ricard and Xu had proved that free product of two weakly amenable group with $\\Lambda_{CH}=1$ also has weak amenability \\cite{ricard2006khintchine}. In general, Haagerup property of groups also proved to be stable under taking direct products, free products and amalgamated free products (over finite subgroups) \\cite{cherix2001groups}. Knudby in \\cite{knudby2016weak} showed that direct product of two weak Haagerup groups is again weakly Haagerup. In this paper we wish to study stability of the weak Haagerup property under certain group construction (\\textit{graph products}).\n\nThe \\textit{graph products of groups} $G(\\Gamma)$, defined by Green in her thesis \\cite{green1990graph}, is a novel way to combine a collection $\\{G_v\\}_{v\\in V(\\Gamma)}$ of groups parameterised along the vertex set $V(\\Gamma)$ of a finite graph $\\Gamma$. Depending on the nature of the given (finite) graph, a graph product of groups can also be thought of as an interpolation between the free product (which corresponds to a graph without an edge) and the direct product (which corresponds to the complete graph on a given set of vertices) of groups (for details see section \\ref{graph product definition}).\n\n\\medskip\n\nFor a finite simplicial graph $\\Gamma$, and a collection $\\{G_v\\}_{v\\in V(\\Gamma)}$ of weakly amenable groups with $\\Lambda_{CH}(G_{v})=1$ for each vertex $v\\in V(\\Gamma)$, it was shown by Reckwerdt in \\cite{reckwerdt2017weak}, that the graph product $G(\\Gamma)$ is weakly amenable. Stability of Haagerup property under graph product was proved by the present authors in \\cite{das2023stability}, (see also \\cite{antolin2013haagerup}). The main result of this paper establishes stability of the weak Haagerup property under graph product. Note that a group with Haagerup property has $\\Lambda_{WH}=1$, and a weakly amenable group $G$ with $\\Lambda_{CH}(G)=1$ is also weakly Haagerup with $\\Lambda_{WH}(G)=1$. Therefore, following Knudby's program in \\cite{knudby2014semigroups}, proving a weakly Haagerup group with $\\Lambda_{WH}=1$ to be Haagerup settles the remaining part of the Cowling's conjecture. The main result of this paper can be seen as a supportive evidence for the conjecture.", "full_context": "One other way amenability can be generalized is called \\textit{weak amenability}, where $G$ is required to have approximate identities consisting of finitely supported functions which are, uniformly bounded in the `completely bounded'-norm ($B_2$-norm, see definition \\ref{definition completely bounded}). Since a positive definite map $\\phi$ on $G$ has bounded cb-norm, this notion generalizes amenability. There is a constant, canonically associated to a weakly amenable group $G$, obtained as the infimum of the cb-norms of these approximate identities, called the Cowling-Haagerup constant of $G$ and is denoted by $\\Lambda_{CH}(G)$. Clearly, every amenable group has $\\Lambda_{CH}=1$. Most groups mentioned above were shown to be weakly amenable with $\\Lambda_{CH}=1$(\\cite{haagerup1978example}, \\cite{ozawa2008weak}, \\cite{mizuta2008bozejko}, \\cite{guentner2010weak}). For an example of a group with $\\Lambda_{CH}>1$, we mention a uniform lattice $\\Gamma$ in $Sp(1,n)$, where the $\\Lambda_{CH}(\\Gamma)=2n-1$. This group satisfies the Kazhdan's property (T), which can be seen as a strong negation of the Haagerup property (see \\cite{bekka2008kazhdan},\\cite{cowling1989completely}).\n\nA further generalisation called the \\textit{weak Haagerup Property} interpolates between the Haagerup property and weak amenability. Introduced by Knudby \\cite{knudby2014semigroups} (see definition \\ref{definition WH}), a group is called weakly Haagerup if there is an approximate identity of vanishing at infinity maps, uniformly bounded in $B_2$-norm, on the group. A group $G$ with the weak Haagerup property also comes equipped with a constant $\\Lambda_{WH}(G)$, defined similar to $\\Lambda_{CH}$ and it is evident that $\\Lambda_{WH}(G)\\leq \\Lambda_{CH}(G)$, for any $G$. An example of a group not having weak Haagerup property is given in \\cite{haagerup2015weak}.\n\nBoth amenability and weak amenability are preserved under direct products. In \\cite{bozejko1993weakly},\nBozejko and Picardello proved that free product of amenable groups is weakly amenable. Whereas it is still an open question whether free product of any two weakly amenable groups is weakly amenable or not. Though as a special case Ricard and Xu had proved that free product of two weakly amenable group with $\\Lambda_{CH}=1$ also has weak amenability \\cite{ricard2006khintchine}. In general, Haagerup property of groups also proved to be stable under taking direct products, free products and amalgamated free products (over finite subgroups) \\cite{cherix2001groups}. Knudby in \\cite{knudby2016weak} showed that direct product of two weak Haagerup groups is again weakly Haagerup. In this paper we wish to study stability of the weak Haagerup property under certain group construction (\\textit{graph products}).\n\nThe \\textit{graph products of groups} $G(\\Gamma)$, defined by Green in her thesis \\cite{green1990graph}, is a novel way to combine a collection $\\{G_v\\}_{v\\in V(\\Gamma)}$ of groups parameterised along the vertex set $V(\\Gamma)$ of a finite graph $\\Gamma$. Depending on the nature of the given (finite) graph, a graph product of groups can also be thought of as an interpolation between the free product (which corresponds to a graph without an edge) and the direct product (which corresponds to the complete graph on a given set of vertices) of groups (for details see section \\ref{graph product definition}).\n\n\\medskip\n\nFor a finite simplicial graph $\\Gamma$, and a collection $\\{G_v\\}_{v\\in V(\\Gamma)}$ of weakly amenable groups with $\\Lambda_{CH}(G_{v})=1$ for each vertex $v\\in V(\\Gamma)$, it was shown by Reckwerdt in \\cite{reckwerdt2017weak}, that the graph product $G(\\Gamma)$ is weakly amenable. Stability of Haagerup property under graph product was proved by the present authors in \\cite{das2023stability}, (see also \\cite{antolin2013haagerup}). The main result of this paper establishes stability of the weak Haagerup property under graph product. Note that a group with Haagerup property has $\\Lambda_{WH}=1$, and a weakly amenable group $G$ with $\\Lambda_{CH}(G)=1$ is also weakly Haagerup with $\\Lambda_{WH}(G)=1$. Therefore, following Knudby's program in \\cite{knudby2014semigroups}, proving a weakly Haagerup group with $\\Lambda_{WH}=1$ to be Haagerup settles the remaining part of the Cowling's conjecture. The main result of this paper can be seen as a supportive evidence for the conjecture.\n\nOne other way amenability can be generalized is called \\textit{weak amenability}, where $G$ is required to have approximate identities consisting of finitely supported functions which are, uniformly bounded in the `completely bounded'-norm ($B_2$-norm, see definition \\ref{definition completely bounded}). Since a positive definite map $\\phi$ on $G$ has bounded cb-norm, this notion generalizes amenability. There is a constant, canonically associated to a weakly amenable group $G$, obtained as the infimum of the cb-norms of these approximate identities, called the Cowling-Haagerup constant of $G$ and is denoted by $\\Lambda_{CH}(G)$. Clearly, every amenable group has $\\Lambda_{CH}=1$. Most groups mentioned above were shown to be weakly amenable with $\\Lambda_{CH}=1$(\\cite{haagerup1978example}, \\cite{ozawa2008weak}, \\cite{mizuta2008bozejko}, \\cite{guentner2010weak}). For an example of a group with $\\Lambda_{CH}>1$, we mention a uniform lattice $\\Gamma$ in $Sp(1,n)$, where the $\\Lambda_{CH}(\\Gamma)=2n-1$. This group satisfies the Kazhdan's property (T), which can be seen as a strong negation of the Haagerup property (see \\cite{bekka2008kazhdan},\\cite{cowling1989completely}).\n\nFor a finite simplicial graph $\\Gamma$, and a collection $\\{G_v\\}_{v\\in V(\\Gamma)}$ of weakly amenable groups with $\\Lambda_{CH}(G_{v})=1$ for each vertex $v\\in V(\\Gamma)$, it was shown by Reckwerdt in \\cite{reckwerdt2017weak}, that the graph product $G(\\Gamma)$ is weakly amenable. Stability of Haagerup property under graph product was proved by the present authors in \\cite{das2023stability}, (see also \\cite{antolin2013haagerup}). The main result of this paper establishes stability of the weak Haagerup property under graph product. Note that a group with Haagerup property has $\\Lambda_{WH}=1$, and a weakly amenable group $G$ with $\\Lambda_{CH}(G)=1$ is also weakly Haagerup with $\\Lambda_{WH}(G)=1$. Therefore, following Knudby's program in \\cite{knudby2014semigroups}, proving a weakly Haagerup group with $\\Lambda_{WH}=1$ to be Haagerup settles the remaining part of the Cowling's conjecture. The main result of this paper can be seen as a supportive evidence for the conjecture.\n\nThe graph products are in some sense generalization of free products of groups, so as a consequence of the above theorem we have the following as well.\n\n\\begin{corollary}\\label{main II}\n Suppose $A$ and $B$ are two weakly Haagerup groups with $\\Lambda_{WH}(G_v)=1$. Then the group $G=A\\ast B$ has the weak Haagerup property.\n\\end{corollary}\n\nFollowing Knudby \\cite{knudby2014semigroups}, a necessary and sufficient condition for a group $G$ to have weak Haagerup property with $\\Lambda_{WH}=1$, is the existence of a proper, symmetric function $\\phi$ on $G$, which can be expressed as sum of two kernels on the group, i.e. for any $x,y \\in G$,\n\\begin{equation}\n \\phi(y^{-1}x)=\\rho(x,y)+\\tau(x,y) \n\\end{equation}\nwhere $\\rho$ is proper, conditionally negative definite and $\\tau$ is bounded positive definite. So, one way to prove theorem \\ref{main I} is to come up with a proper function $\\phi$ as above, on $G(\\Gamma)$. Since on each $G_v$, we have a proper function $\\phi_v$ which is a sum of $\\rho_v$ and $\\tau_v$ as above, it would be reasonable to ask if we can combine the individual kernels $\\rho_v$'s into a $\\rho$ and $\\tau_v$'s to a $\\tau$, in order to obtain a `$\\phi=\\rho +\\tau$' on $G(\\Gamma)$. Upto a modification the sum of the conditionally negative definite kernels on $G_v$'s define a kernel of the same type on $G(\\Gamma)$. A similar approach was followed in \\cite{das2023stability} to combine conditionally negative definite kernels $\\rho_v$'s on Haagerup vertex groups $G_v$ to obtain a conditionally negative definite kernel on $G(\\Gamma)$, proving it to have the Haagerup property. But here, we were unable to integrate the bounded positive definite kernels $\\tau_v$'s to get a similar kernel $\\tau$ on $G(\\Gamma)$, and hence unable to obtain the desired $\\phi=\\rho+\\tau$. We prove theorem \\ref{main I} following a general line of arguments found in \\cite{bozejko1993weakly,ricard2006khintchine,reckwerdt2017weak}. Somewhat similar strategies were followed also in proposition 12.3.5 of \\cite{brown2008textrm} and the proof of the main result of \\cite{ozawa2008weak}.\n\nLet $\\Gamma$ be a finite simplicial graph, and $\\{G_v\\}_{v\\in V(\\Gamma)}$ be a collection of groups having weak Haagerup property with weak Haagerup constant $\\Lambda_{WH}(G_v)=1$. For each vertex $v\\in V(\\Gamma)$, we are given a weak Haagerup data $(G_v,\\phi_v,R_v,S_v,\\mathcal{H}_v)$. In this section we will combine this entire vertex data $\\{(G_v,\\phi_v,R_v,S_v,\\mathcal{H}_v)\\}_{v \\in V(\\Gamma)}$ to prove weak Haagerup property for the graph product $G(\\Gamma)$ of the groups $\\{G_v\\}_{v\\in V(\\Gamma)}$.\n\n\\begin{theorem}\\label{main theorem}\n Suppose $\\Gamma$ is a finite graph and for each vertex $v \\in V(\\Gamma)$, the associated group $G_v$ has the weak Haagerup property with $\\Lambda_{WH}(G_v)=1$. Then $G(\\Gamma)$ has weak Haagerup property. Moreover $\\Lambda_{WH}(G(\\Gamma))=1$.\n\\end{theorem}\n\n\\begin{corollary}\\label{main corollary}\n Suppose $A$ and $B$ are two weakly Haagerup groups such that $\\Lambda_{WH}(A)=1=\\Lambda_{WH}(B)$, then $G=A \\ast B$ has weak Haagerup property with $\\Lambda_{WH}(G)=1$.\n\\end{corollary}\n\n\\begin{definition}\\label{definition completely bounded}\n    A function $\\phi:G\\rightarrow \\mathbb{C}$ is called \\textit{completely bounded} if there is a $B>0$ such that the operator-norm of $M_{n,\\phi}$, defined above is bounded by $B$, for each $n\\in\\mathbb{N}$. The $B_2$-norm (alternately the completely bounded norm or the cb-norm) of $\\phi$ is\n    $$||\\phi||_{B_2}:=\\sup_{n\\in\\mathbb{N}} ||M_{n,\\phi}||$$\n\\end{definition}\n\n\\begin{theorem}\\label{main I}\n    Suppose $\\Gamma$ is a finite simplicial graph and $\\{G_{v}\\}_{v\\in V(\\Gamma)}$ is a collection of weakly Haagerup groups with $\\Lambda_{WH}(G_v)=1$, for each $v$. Then the group $G(\\Gamma)$ has the weak Haagerup property with $\\Lambda_{WH}(G(\\Gamma))=1$.\n\\end{theorem}", "post_theorem_intro_text_len": 2096, "post_theorem_intro_text": "The graph products are in some sense generalization of free products of groups, so as a consequence of the above theorem we have the following as well.\n\n\\begin{corollary}\\label{main II}\n    Suppose $A$ and $B$ are two weakly Haagerup groups with $\\Lambda_{WH}(G_v)=1$. Then the group $G=A\\ast B$ has the weak Haagerup property.\n\\end{corollary}\n\nFollowing Knudby \\cite{knudby2014semigroups}, a necessary and sufficient condition for a group $G$ to have weak Haagerup property with $\\Lambda_{WH}=1$, is the existence of a proper, symmetric function $\\phi$ on $G$, which can be expressed as sum of two kernels on the group, i.e. for any $x,y \\in G$,\n\\begin{equation}\n    \\phi(y^{-1}x)=\\rho(x,y)+\\tau(x,y) \n\\end{equation}\nwhere $\\rho$ is proper, conditionally negative definite and $\\tau$ is bounded positive definite. So, one way to prove theorem \\ref{main I} is to come up with a proper function $\\phi$ as above, on $G(\\Gamma)$. Since on each $G_v$, we have a proper function $\\phi_v$ which is a sum of $\\rho_v$ and $\\tau_v$ as above, it would be reasonable to ask if we can combine the individual kernels $\\rho_v$'s into a $\\rho$ and $\\tau_v$'s to a $\\tau$, in order to obtain a `$\\phi=\\rho +\\tau$' on $G(\\Gamma)$. Upto a modification the sum of the conditionally negative definite kernels on $G_v$'s define a kernel of the same type on $G(\\Gamma)$. A similar approach was followed in \\cite{das2023stability} to combine conditionally negative definite kernels $\\rho_v$'s on Haagerup vertex groups $G_v$ to obtain a conditionally negative definite kernel on $G(\\Gamma)$, proving it to have the Haagerup property. But here, we were unable to integrate the bounded positive definite kernels $\\tau_v$'s to get a similar kernel $\\tau$ on $G(\\Gamma)$, and hence unable to obtain the desired $\\phi=\\rho+\\tau$. We prove theorem \\ref{main I} following a general line of arguments found in \\cite{bozejko1993weakly,ricard2006khintchine,reckwerdt2017weak}. Somewhat similar strategies were followed also in proposition 12.3.5 of \\cite{brown2008textrm} and the proof of the main result of \\cite{ozawa2008weak}.", "sketch": "Following Knudby \\cite{knudby2014semigroups}, a necessary and sufficient condition for a group $G$ to have weak Haagerup property with $\\Lambda_{WH}=1$ is the existence of a proper, symmetric function $\\phi$ on $G$ of the form\n\\[\n\\phi(y^{-1}x)=\\rho(x,y)+\\tau(x,y),\n\\]\nwhere $\\rho$ is proper, conditionally negative definite and $\\tau$ is bounded positive definite. Thus, one way to prove Theorem~\\ref{main I} is to \"come up with a proper function $\\phi$ as above\" on $G(\\Gamma)$. Since each vertex group $G_v$ has such a decomposition $\\phi_v=\\rho_v+\\tau_v$, it is \"reasonable to ask\" whether one can combine the kernels $\\rho_v$ into a global $\\rho$ and the $\\tau_v$ into a global $\\tau$ to obtain $\\phi=\\rho+\\tau$ on $G(\\Gamma)$. The authors note that, \"up to a modification,\" the sum of the conditionally negative definite kernels $\\rho_v$ does define a kernel of the same type on $G(\\Gamma)$ (as in \\cite{das2023stability} for the Haagerup property), but they were \"unable to integrate the bounded positive definite kernels $\\tau_v$'s\" to get a similar bounded positive definite kernel $\\tau$ on $G(\\Gamma)$, hence unable to obtain $\\phi=\\rho+\\tau$ by this direct combination. Instead, they \"prove Theorem~\\ref{main I} following a general line of arguments\" from \\cite{bozejko1993weakly,ricard2006khintchine,reckwerdt2017weak}, with similar strategies also appearing in Proposition 12.3.5 of \\cite{brown2008textrm} and the main result of \\cite{ozawa2008weak}.", "expanded_sketch": "Following Knudby Knudby, semigroups (2014), a necessary and sufficient condition for a group $G$ to have weak Haagerup property with $\\Lambda_{WH}=1$ is the existence of a proper, symmetric function $\\phi$ on $G$ of the form\n\\[\n\\phi(y^{-1}x)=\\rho(x,y)+\\tau(x,y),\n\\]\nwhere $\\rho$ is proper, conditionally negative definite and $\\tau$ is bounded positive definite. Thus, one way to prove the main theorem is to \"come up with a proper function $\\phi$ as above\" on $G(\\Gamma)$. Since each vertex group $G_v$ has such a decomposition $\\phi_v=\\rho_v+\\tau_v$, it is \"reasonable to ask\" whether one can combine the kernels $\\rho_v$ into a global $\\rho$ and the $\\tau_v$ into a global $\\tau$ to obtain $\\phi=\\rho+\\tau$ on $G(\\Gamma)$. The authors note that, \"up to a modification,\" the sum of the conditionally negative definite kernels $\\rho_v$ does define a kernel of the same type on $G(\\Gamma)$ (as in Das, stability (2023) for the Haagerup property), but they were \"unable to integrate the bounded positive definite kernels $\\tau_v$'s\" to get a similar bounded positive definite kernel $\\tau$ on $G(\\Gamma)$, hence unable to obtain $\\phi=\\rho+\\tau$ by this direct combination. Instead, they establish the main theorem following a general line of arguments from Bo\\.{z}ejko, weakly (1993), Ricard, khintchine (2006), and Reckwerdt, weak (2017), with similar strategies also appearing in Proposition 12.3.5 of Brown, textrm (2008) and the main result of Ozawa, weak (2008).", "expanded_theorem": "\\label{main I}\n    Suppose $\\Gamma$ is a finite simplicial graph and $\\{G_{v}\\}_{v\\in V(\\Gamma)}$ is a collection of weakly Haagerup groups with $\\Lambda_{WH}(G_v)=1$, for each $v$. Then the group $G(\\Gamma)$ has the weak Haagerup property with $\\Lambda_{WH}(G(\\Gamma))=1$.", "theorem_type": ["Implication"], "mcq": {"question": "Let \\(\\Gamma\\) be a finite simplicial graph with vertex set \\(V(\\Gamma)\\), and let \\(\\{G_v\\}_{v\\in V(\\Gamma)}\\) be a family of groups. Define the graph product \\(G(\\Gamma)\\) to be the quotient of the free product \\(\\ast_{v\\in V(\\Gamma)} G_v\\) obtained by imposing the relations that elements of \\(G_u\\) commute with elements of \\(G_v\\) whenever \\(u\\) and \\(v\\) are adjacent vertices of \\(\\Gamma\\). Assume that for every vertex \\(v\\), the group \\(G_v\\) has the weak Haagerup property, meaning it admits an approximate identity consisting of functions vanishing at infinity whose \\(B_2\\)-norms are uniformly bounded, and that its weak Haagerup constant \\(\\Lambda_{WH}(G_v)\\) (the infimum of such uniform \\(B_2\\)-bounds) satisfies \\(\\Lambda_{WH}(G_v)=1\\). Under these hypotheses, which conclusion about \\(G(\\Gamma)\\) holds?", "correct_choice": {"label": "A", "text": "The graph product \\(G(\\Gamma)\\) has the weak Haagerup property, and its weak Haagerup constant is \\(\\Lambda_{WH}(G(\\Gamma))=1\\)."}, "choices": [{"label": "B", "text": "The graph product \\(G(\\Gamma)\\) has the weak Haagerup property, and its weak Haagerup constant satisfies \\(\\Lambda_{WH}(G(\\Gamma))\\leq |V(\\Gamma)|\\)."}, {"label": "C", "text": "The graph product \\(G(\\Gamma)\\) has the weak Haagerup property."}, {"label": "D", "text": "The graph product \\(G(\\Gamma)\\) has the weak Haagerup property provided one additionally assumes that the vertex groups admit weak Haagerup approximate identities with a uniform \\(B_2\\)-bound independent of \\(v\\)."}, {"label": "E", "text": "The graph product \\(G(\\Gamma)\\) has the weak Haagerup property, and its weak Haagerup constant is given by \\(\\Lambda_{WH}(G(\\Gamma))=\\prod_{v\\in V(\\Gamma)} \\Lambda_{WH}(G_v)\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "exact_constant_1_replaced_by_vertex_dependent_bound", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "drops_the_conclusion_\\(\\Lambda_{WH}(G(\\Gamma))=1\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "adds_extra_uniformity_hypothesis_across_vertices", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "multiplicative_constant_formula_for_graph_products", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the conclusion explicitly and does not single out the exact answer. It provides hypotheses and asks for the resulting property of the graph product."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the hypotheses are stated in full and the correct option is the theorem’s conclusion with no meaningful reformulation."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because one must distinguish the exact conclusion from weaker or altered variants, especially between A and the weaker true statement C. However, the item mainly tests recognition of the theorem rather than genuine derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically relevant and distinct: a weaker true conclusion (C), an unnecessary extra hypothesis (D), and plausible but incorrect quantitative claims (B, E). These align with common failure modes about sharp constants and closure properties."}, "total_score": 5, "overall_assessment": "A solid theorem-recognition MCQ with good distractors and no direct answer leakage, but it is largely tautological and only moderately tests generative reasoning."}}
{"id": "2602.03768v1", "paper_link": "http://arxiv.org/abs/2602.03768v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{thm;global}\nFor $(u_0,v_0)\\in L_+^1(\\mathbb R^2) \\times L^1_+(\\mathbb R^2)\\cap \\dot{H}^1(\\mathbb R^2)$,\nlet $(u,v)$ be the solution to \\eqref{eqn;KS} on $(0,T)\\times\\mathbb R^2$.\nSuppose that $\\|u_0\\|_1=8\\pi$.\nThen, the solution to \\eqref{eqn;KS} exists globally in time.", "start_pos": 14650, "end_pos": 14921, "label": "thm;global"}, "ref_dict": {"thm;global": "\\begin{thm}\\label{thm;global}\nFor $(u_0,v_0)\\in L_+^1(\\R^2) \\times L^1_+(\\R^2)\\cap \\dot{H}^1(\\R^2)$,\nlet $(u,v)$ be the solution to \\eqref{eqn;KS} on $(0,T)\\times\\R^2$.\nSuppose that $\\|u_0\\|_1=8\\pi$.\nThen, the solution to \\eqref{eqn;KS} exists globally in time.\n\\end{thm}", "eqn;KS": "\\begin{aligned}\n&\\pt_t u =\\Delta u- \\N\\cd \\left( u \\N v \\right),\n& t>0,\\,~ &x\\in\\R^2,\n\\\\\n&\\pt_t v = \\Delta v - \\lambda v +u,\n&t>0,\\, ~&x\\in\\R^2,\n\\\\\n&(u,v)(0,x)=(u_{0},v_0)(x),\n&\\, &x\\in\\R^2\n\\end{aligned}\n\\right.\n\\label{eqn;KS}\n\\end{equation}\nwith a constant $\\lambda\\ge0$, where $u_0,v_0\\ge 0$ on $\\R^2$ and\n$u_0,v_0\\not\\equiv0$.\nThe Keller--Segel system~\\eqref{eqn;KS} is a fundamental mathematical model\nof chemotaxis, describing chemotactic aggregation in the\ncellular slime mold {\\it Dictyostelium discoideum} during its life cycle \\cite{KeSe,Pa}.\nIn this model, $u=u(t,x)$ and $v=v(t,x)$ denote the densities of cells and\nthe chemoattractant, respectively.\nThe parameter $\\lambda$ represents the degradation rate of the chemical.\nCells migrate toward regions of higher concentrations of a chemical substance\nsecreted by the cells themselves.\nFrom a biological viewpoint, it is natural to assume that the initial data\nare nonnegative functions.\nFrom a mathematical viewpoint, cell aggregation is interpreted as the\nblowup of solutions at $t=T$ in the sense that\n$\\lim_{t\\to T}\\|u(t)\\|_\\infty=\\infty.$\n\nOne of the central mathematical features of such systems is the phenomenon of critical\nmass.\nFor positive sufficiently regular solutions $(u,v)$ to~\\eqref{eqn;KS}, the first\ncomponent $u$ satisfies the mass conservation law \n$$\\|u(t)\\|_1=\\|u_0\\|_1\\quad \\text{for}~~~t>0.$$\nMoreover, in two space dimensions, the global behavior of solutions is governed\nby the size of the initial mass of $u$.\nIn particular, there exists a threshold value such that solutions with initial\nmass below this threshold exist globally in time, whereas solutions with\ninitial mass above the threshold may blow up in finite time.\nThis critical-mass phenomenon has attracted considerable attention in the\nliterature.\n\nThe aim of this paper is to establish the global behavior of solutions to\n\\eqref{eqn;KS} whose initial mass is exactly equal to this threshold value.\n\n\\vspace{3mm}\nThe second equation in \\eqref{eqn;KS} takes into account that cells are producing the chemoattractant themselves \nwhile this is diffusing into the environment. \nSince the chemoattractant attains its equilibrium on a time scale much quicker\nthan that of the cells, the simplified parabolic-elliptic system has also been\ninvestigated \\cite{JaLu,Na95}:\n\\begin{align*}\n\\left\\{\n\\begin{aligned}\n\\partial_t u &= \\Delta u - \\nabla\\cdot (u\\nabla v),\n& t>0,\\ & x\\in\\R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\R^2.\n\\end{aligned}", "prop;LWP": "\\begin{prop}[local-in-time solution]\\label{prop;LWP}\n\tLet $4/3<p<2$ and $q=p/(p-1)$.\n\tFor  $(u_0,v_0)\\in L^1(\\R^2)\\times  \\dot{H}^{1}(\\R^2)$,\n\tthere exist $T\\in (0,\\infty]$ and a unique mild solution $(u,v)$ to~\\eqref{eqn;KS} on $[0,T)$ in the sense of the definition.\n\tBesides, the solution $(u,v)$ has higher regularity on $(0,T)$ and satisfies the system~\\eqref{eqn;KS} in a classical sense on $(0,T)\\times\\R^2$.\n\tIn addition,\n\t\\begin{equation*}\n\t\\text{either $T=\\infty$ or $T<\\infty$ and } \\lim_{t\\to T} \\left\\{ \\|u(t)\\|_{r} + \\|\\nabla v(t)\\|_{r+1}\\right\\} = \\infty,~~ 1<r\\le \\infty. \n\t\\eqntag\n\t\\label{eqn;bucrit}\n\t\\end{equation*}\t\n\tMoreover, if $u_0,v_0 \\ge 0$ then $u,v> 0$ in $(0,T)\\times\\R^2$ and\n\t\\begin{equation*}\n\t\\|u(t)\\|_1=\\|u_0\\|_1, \\qquad t\\in [0,T).\t\n\t\\end{equation*} \n\\end{prop}", "eqn;PKS": "\\begin{aligned}\n\\partial_t u &= \\Delta u - \\nabla\\cdot (u\\nabla v),\n& t>0,\\ & x\\in\\R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\R^2.\n\\end{aligned}\n\\right.\n\\eqntag\n\\label{eqn;PKS}\n\\end{align*}\nThe system \\eqref{eqn;PKS} is also related to models of gravitational interaction\nof particles \\cite{Bi-Na94,Wo}.\nIn~\\eqref{eqn;PKS}, the second equation can be written as\n$v = (-\\Delta+\\lambda)^{-1}u.$\nAs a consequence, the system~\\eqref{eqn;PKS} can be reduced to a single equation\nfor~$u$.\nOwing to this elliptic structure, the parabolic-elliptic system~\\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.\nIn particular, numerous works have been devoted to the critical-mass\nphenomenon.\nIndeed, the solution with $\\|u_0\\|_1\\le 8\\pi$ exists globally in time~\\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},\nwhile the solution may blow up in finite time if $\\|u_0\\|_1>8\\pi$~\\cite{DoPe,Ko-Su,Ku-Og03,Wei}.\nFor the corresponding Cauchy--Neumann problem in bounded domains,\nsupplemented with homogeneous Neumann boundary conditions, see \\cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.\n\n\\vspace{3mm}\nUnlike the parabolic-elliptic system \\eqref{eqn;PKS}, the fully parabolic system\n\\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches\ndeveloped for \\eqref{eqn;PKS} are no longer applicable.\nThe global existence of solutions with sub-critical mass\n$\\|u_0\\|_{1}<8\\pi$ has nevertheless been established by combining\nLyapunov functionals with the Trudinger--Moser type inequality and its optimal\nconstant.\nMore precisely, the following results are known:\n\\begin{enumerate}\n\t\\item[(i)]\n\tIf $\\|u_0\\|_{1}<8\\pi$, then the corresponding solution to\n\t\\eqref{eqn;KS} exists globally in time \\cite{CaCo08,Mi13,NaOg11}.\n\n\t\\item[(ii)]\n\tIf $\\|u_0\\|_{1}>8\\pi$ and $(u_0,v_0)$ is radially symmetric, then there\n\texists a radially symmetric solution that blows up in finite time\n\t\\cite{Mi20,Mi20-SIAM}.\n\n\t\\item[(iii)]\n\tIf $\\|u_0\\|_{1}=8\\pi$ and $(u_0,v_0)$ is either radially symmetric or\n\tsatisfies the additional moment condition\n\t$u_0\\ln(1+|x|^2)\\in L^1(\\R^2)$, then the corresponding solution exists globally\n\tin time.\n\tIn contrast, for general initial data, the global behavior in the critical case\n\tremains delicate, and the solution either exists globally or blows up on the plane \\cite{CaCo08,Mi13}.\n\\end{enumerate}\nAs for positive forward self-similar solutions to \\eqref{eqn;KS}, refer to \\cite{BiCoDo}.\nFor the corresponding Cauchy--Neumann problem in bounded domains, we refer to\n\\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.\n\nAs mentioned above, although several partial results are available, no complete global existence result has been obtained\nfor solutions with critical mass and general initial data without any\nsymmetry or moment assumptions.\nOne of the main difficulties lies in controlling the behavior of solutions at~$|x|\\to\\infty$.\nMoreover, in the critical mass case, no global existence result is currently\nknown  for the corresponding Cauchy--Neumann problem to~\\eqref{eqn;KS} in bounded domains.\n\nIn this paper, we establish the global existence of solutions with critical mass\nfor general initial data $(u_0,v_0)$.\nTo this end, we first recall the definition of solutions.\n\n\\vspace{2mm}\n\\begin{def*}\nLet $4/3<p<2$ and $q=p/(p-1)$.\nGiven $T\\in (0,\\infty]$ and  $(u_0,v_0) \\in L^1(\\R^2)\\times \\dot{H}^{1}(\\R^2)$,\nwe say that a pair of functions $(u,v)$ defined on $[0,T)\\times \\R^2$ is a mild solution to~\\eqref{eqn;KS} on $[0,T)$ if\n\\begin{enumerate}\n\t\\item $u\\in C\\big([0,T); L^1(\\R^2)\\big)\\cap C\\left( (0,T); L^p(\\mathbb{R}^2) \\right)$;\n\t\\item $v\\in C\\big([0,T); \\dot{H}^{1}(\\R^2)\\big)\\cap C\\left( (0,T); \\dot{W}^{1,q}(\\mathbb{R}^2) \\right)$;\n\t\\item $\\dsp\\sup_{0<t<T} t^{1-\\frac1p}\\|u(t)\\|_{p}+\\dsp\\sup_{0<t<T}t^{\\frac12-\\frac1q}\\|\\nabla v(t)\\|_{q}$;\n\t\\item $(u,v)$ satisfies the integral formulations\n\t\\begin{align*}\n\t&u(t)=e^{t\\Del}u_0-\\int_0^t \\N\\cd e^{(t-s)\\Del} \\big(u(s)\\N v(s)\\big) \\,\\d s,\n\t\\\\\n\t&v(t)= e^{t(\\Del-\\lm)} v_0+\\int_0^t e^{(t-s)(\\Del-\\lm)} u(s) \\,\\d s\n\t\\end{align*}\n\tfor $0<t<T$, where $e^{t\\Del}$ is the heat semigroup given by\n\t\\begin{align*}\n\te^{t\\Del} f(x) := G_t*f(x)=\\int_{ \\R^2}G_t(x-y) f(y)\\,\\d y,\n\t\\quad\n\tG_t(x)=\\frac{1}{4\\pi t} \\exp\\left(-\\frac{|x|^2}{4 t}\\right)\n\t\\end{align*}\n\tfor $x\\in\\R^2$ and $f \\in L^p(\\R^2)$.\n\\end{enumerate}\n\\end{def*}\nThe Lebesgue spaces $L^p(\\R^2)$ are equipped with the norm $\\|\\cdot\\|_{L^p(\\R^2)}$,\nwhich is abbreviated as $\\|\\cdot\\|_p$ when there is no ambiguity.\nWe define the homogeneous Sobolev spaces by\n\\[\n\\dot{W}^{1,p}(\\R^2)\n:= \\overline{C_0^\\infty(\\R^2)}^{\\|\\nabla (\\cdot)\\|_{L^p(\\R^2)}}\n\\]\nand $\\dot{H}^1(\\R^2):=\\dot{W}^{1,2}(\\R^2)$ when $p=2$.\n$L_+^1(\\R^2):=\\{f \\in L^1(\\R^2); f\\ge0~~ \\text{and}~~ f\\not\\equiv0~~ \\text{on $\\R^2$} \\}$.\n\n\\vspace{5mm}\nThe starting point for our analysis is the local well-posedness for \\eqref{eqn;KS} and the solution~$(u,v)$ based on the above definition satisfies~\\eqref{eqn;KS} in a classical sense on $(0,T)$, see Proposition~\\ref{prop;LWP} in Section~\\ref{sect;LWP} below.\n\nWe are now in a position to state our main result in this paper.\nMain result in this paper reads as follows.\n\\vspace{2mm}\n\\begin{thm}\\label{thm;global}\nFor $(u_0,v_0)\\in L_+^1(\\R^2) \\times L^1_+(\\R^2)\\cap \\dot{H}^1(\\R^2)$,\nlet $(u,v)$ be the solution to \\eqref{eqn;KS} on $(0,T)\\times\\R^2$.\nSuppose that $\\|u_0\\|_1=8\\pi$.\nThen, the solution to \\eqref{eqn;KS} exists globally in time.\n\\end{thm}\n\\vspace{2mm}\n\n\\begin{rem}\nIn Theorem~\\ref{thm;global}, the initial data are only required to satisfy\n\\[\n(u_0,v_0)\\in L_+^1(\\R^2)\\times\\bigl(L_+^1(\\R^2)\\cap \\dot H^1(\\R^2)\\bigr)\\quad \\text{with}~~\\|u_0\\|_1=8\\pi.\n\\]\nIn particular, no additional symmetry or moment assumptions are imposed.\n\\end{rem}\n\\vspace{2mm}\nAs mentioned above, the global existence of solutions with critical mass\nwithout any symmetry or moment assumptions remains delicate.\nIn~\\cite[Theorem~1.2]{Mi13}, it is shown that, for general initial data with\ncritical mass, solutions either exist globally in time or blow up on the plane,\nby means of a contradiction argument.\nIndeed, although in the sub-critical case regularity estimates can be obtained\nby combining (modified) Lyapunov functionals with the Trudinger--Moser\ninequality, such estimates are no longer directly available in the critical case due to the lack of regularity of solutions,\nsee~\\eqref{eqn;afford} below.\nThis difficulty prevents the direct extension of classical entropy methods.\n\nTo overcome this difficulty, we introduce a reconstructed Lyapunov functional.\nThe main novelty of this work lies in the construction of a refined Lyapunov\nfunctional specifically adapted to the whole space setting, which allows us\nto control the behavior of solutions at $|x|\\to\\infty$ and to derive\nregularity estimates even in the critical mass regime, without\nimposing any symmetry or moment assumptions on the initial data.\nAs a consequence, we establish global-in-time existence for general initial\ndata at the critical mass, a result that was previously out of reach by existing\nmethods, see Subsection~\\ref{sect;modified-Lyapunov} for details.\nMoreover, the present approach is expected to be applicable to a broad class\nof chemotaxis  systems in the whole space setting.\n\n\\vspace{3mm}\nThe remainder of this paper is organized as follows.\nIn Section~\\ref{sect;preliminary}, we recall several preliminary lemmas needed\nto prove the main result.\nSection~\\ref{sect;LWP} is devoted to the local well-posedness of~\\eqref{eqn;KS}, based on the above definition of solutions.\nFinally, in Section~\\ref{sect;global}, we present the proof of\nTheorem~\\ref{thm;global}.\n\n\\vspace{5mm}\n\\section{Preliminary}\\label{sect;preliminary}\nIn this section, we collect some tools to show our main result.\n\\begin{lem}\\label{lem;L2andL3}\n\tFor any $f\\in H^1(\\R^2)$ and nonnegative cut-off function $\\phi\\in C^\\infty_0(\\R^2)$,\n\t\\begin{align*}\n\t\\int_{\\R^2}|f|^2\\phi\\dx\\le\\,&2\\left(\\int_{ \\{|f|>1\\}\\cap \\{\\supp\\phi\\} }|f|\\dx\\right)\\left(\\int_{ \\{|f|>1\\} }\\frac{|\\nabla f|^2}{1+|f|} \\phi\\dx\\right)\n\t\\\\\n\t&+4\\left(\\int_{\\R^2}|f \\nabla \\phi^{\\frac12}|\\dx\\right)^2+4\\left(\\int_{\\R^2}|f|\\phi\\dx\\right)\n\t\\end{align*}\n\tand for any $\\ep>0$\n\t\\begin{align*}\n\t\\int_{\\R^2}|f|^3\\phi\\dx\n\t\\le\\,&\\ep \\left(\\int_{  \\{\\supp\\phi\\} }(1+|f|)\\ln(1+|f|)\\dx\\right)\\left(\\int_{\\R^2 }|\\nabla f|^2\\phi\\dx\\right)\n\t\\\\\n\t&+C\\left(\\int_{\\R^2}|f^{\\frac32} \\nabla \\phi^{\\frac12}|\\dx\\right)^2+C_\\ep\\left(\\int_{\\R^2}|f|\\phi\\dx\\right),\n\t\\end{align*}\n\twhere the constant $C_\\ep\\to \\infty$ as $\\ep\\to0$.\n\\end{lem}\nThe proof of Lemma \\ref{lem;L2andL3} can be found in \\cite[Lemma 2.2]{NaOg16}.\n\n\\begin{lem}\\label{lem;Lp_Lq_heat}\n\tFor $1 \\le q \\le p \\le \\infty$,\n\tlet $f\\in L^q (\\R^n)$ and let $\\al$ be a multi-index.\n\tThen it follows that\n\t\\begin{equation*}\n\t\\| \\pt_x^{\\al} e^{t\\Del} f \\|_p  \n\t\\le C t^{-\\frac{n}{2} \\left(\\frac{1}{q} - \\frac{1}{p} \\right) - \\frac{|\\al|}{2}}   \\|f\\|_q\n\t\\end{equation*}\n\tfor all $t>0$.\n\\end{lem}\n\nThe proof is immediately obtained by use of Young's inequality and the \nconvolution expression of the heat evolution\nby the heat kernel, see for instance \\cite{GiGiSa}.\n\\begin{lem}\\label{lem;v-Lpbound}\nFor $v_0\\in L^1(\\R^2)\\cap \\dot H^1(\\R^2)$ and $f\\in L^\\infty \\left(0,\\infty;  L^1(\\R^2) \\right)$,\nlet $v$ be the solution to $\\partial_t v =\\Delta v -\\lambda v +f$  on $(0,\\infty)\\times\\R^2$ with the initial data $v_0$ and $\\lambda\\ge0$. Then, for any $1\\le p<\\infty$, \n\\begin{align*}\n\\| v(t)\\|_p\n\\le\\,&\n\\left\\{\n\\begin{aligned}\n&\\|v_0\\|_p+\\frac1p\\|f\\|_{L^\\infty\\left(0,\\infty; L^1(\\R^2)\\right)}  t^{\\frac1p} &\\text{if $\\lambda=0$},\n\\\\\n&\\|v_0\\|_p+\\lambda^{-\\frac1p} \\|f\\|_{L^\\infty\\left(0,\\infty; L^1(\\R^2)\\right)}  \\Gamma\\left(\\frac1p\\right)&\\text{if $\\lambda>0$}.\n\\end{aligned}", "eqn;afford": "\\begin{pr}{Proposition \\ref{prop;LWP}}\nBy virtue of Lemmas~\\ref{lem;fixed-point} and~\\ref{lem;continuous-depend},\nwe have established the existence of a unique mild solution to~\\eqref{eqn;KS} defined on a maximal time interval $[0,T)$ with~$T\\le\\infty$. As for the regularity of the solutions, we use the standard iteration argument with respect to the derivative. Define $|\\N|^\\al f(x):=\\F^{-1}[ |\\xi|^\\al \\F f(\\xi)\\,](x)$  for $x\\in\\R^2$ and $\\al>0$, where $\\F$ denotes the Fourier transform. Let $\\tau\\in (0,T)$ and $t\\in (0,\\tau)$. Recalling that, for $4/3<p<2$,\n\\begin{equation*}\nU_{p,0}(\\tau) := \\sup_{0< s<\\tau}s^{1-\\frac1p}\\|u(s)\\|_{p}<+\\infty, \\quad V_{q,0}(\\tau) := \\sup_{0< s<\\tau}s^{\\frac12-\\frac1q}\\|\\nabla v(s)\\|_{q}<+\\infty,\n\\end{equation*}\nby Lemma~\\ref{lem;fixed-point}, we may argue as in the proof of~\\eqref{eqn;L-4/3} to obtain\n\\begin{align*}\n&t^{1-\\frac1p+\\frac{\\al}{2}} \\| |\\N|^{\\al} u(t) \\|_{p}\n\\\\\n\\le\\,& t^{1-\\frac1p+\\frac{\\al}{2}} \\| |\\N|^\\al e^{t\\Del}u_0\\|_{p} + C U_{p,0}(\\tau) V_{q,0}(\\tau) t^{1-\\frac1p+\\frac{\\al}{2}}  \\int_0^t  (t-s)^{-1+\\frac1{p}-\\frac{1+\\al}2} s^{-1+\\frac{1}p} s^{-\\frac12+\\frac1q}\\,\\d s\n\\\\\n\\le\\,&C\\|u_0\\|_1 + C  U_{p,0}(\\tau) V_{q,0}(\\tau) \\mathsf{B}\\left(\\frac1p-\\frac{1+\\al}2,\\frac12\\right),\n\\end{align*}\nwhere $4/3<p<2$ and\n\\begin{align*}\n\\al <\\frac2p-1<\\frac12,\n\\end{align*}\nwhich implies that\n\\begin{align*}\nU_{p.\\al}(\\tau):=\\sup_{0< t<\\tau} t^{1-\\frac1p+\\frac{\\al}{2}} \\| |\\N|^{\\al} u(t)\\|_{p} <+\\infty,\n\\quad\n\\al<\\frac2p-1.\n\\end{align*}\nThis property yields additional regularity estimates for $v$. Indeed, for $\\bt\\ge \\al$, \n\\begin{align*}\n\\| |\\N|^{1+\\beta} v(t) \\|_{q}\n\\le\\,&C t^{-\\frac12+\\frac1q-\\frac\\bt2} \\| \\nabla v_0\\|_{2} + C U_{p,\\al}(\\tau) \\int_0^t (t-s)^{-\\left(\\frac1p-\\frac1q\\right)-\\frac{1+\\bt-\\al}{2} } s^{-1+\\frac1p-\\frac\\al2}\\,\\d s \n\\\\\n=\\,&C  t^{-\\frac12+\\frac1q-\\frac\\bt2} \\| \\nabla v_0\\|_{2} \n+C U_{p,\\al}(\\tau)  t^{-\\frac12+\\frac1q-\\frac\\bt2} \\mathsf{B}\\left(\\frac12-\\frac1p+\\frac1q-\\frac{\\bt-\\al}{2}, \\frac1p-\\frac{\\al}{2}\\right),\n\\end{align*}\nprovided that\n\\begin{align*}\n\\al\\le \\bt<\\al+3-\\frac4p<2-\\frac2p<1,\n\\end{align*}\nfrom which it follows that \n\\begin{align*}\nV_{q,\\bt}(\\tau) := \\sup_{0< t<\\tau} t^{\\frac12-\\frac1q+\\frac\\bt2} \\| |\\N|^{1+\\bt} v(t) \\|_{q}<+\\infty,\n\\quad  \\bt<\\al+3-\\frac4p\\in (\\al,1).\n\\end{align*}\nThe previous analysis leads us to a higher regularity estimate for $u$. Indeed, let $\\th\\ge\\al$. Since $\\al\\le\\bt\\le\\gm$,\n\\begin{align*}\n& \\| |\\N|^{\\th}u(t)\\|_{p} \\\\\n\\le\\,&C t^{-\\left(1-\\frac1p\\right)-\\frac\\th2}\\|u_0\\|_1\n\\\\\n&+C\\int_0^t(t-s)^{-\\left(1-\\frac1p\\right)-\\frac{1+\\th-\\al}{2}} \\left(\\||\\N|^{\\al} u(s)\\|_{p}\\|\\nabla v(s)\\|_q+\\|u(s)\\|_{p}\\| |\\N|^{1+\\al} v(s)\\|_{q}\\right)\\,\\d s\n\\\\\n\\le\\,&Ct^{-\\left(1-\\frac1p\\right)-\\frac\\th2}\\|u_0\\|_1\n\\\\\n&+ C \\big(U_{p,\\alpha}(\\tau) V_{q,0}(\\tau) + U_{p,0}(\\tau) V_{q,\\al}(\\tau)\\big) \n\\int_0^t(t-s)^{-\\left(1-\\frac1p\\right)-\\frac{1+\\th-\\al}{2}} s^{-\\frac32+\\frac1p+\\frac1q-\\frac{\\al}{2}}\\,\\d s\n\\\\\n=\\,&Ct^{-\\left(1-\\frac1p\\right)-\\frac\\th2}\\|u_0\\|_1\n\\\\\n& + C \\big(U_{p,\\al}(\\tau) V_{q,0}(\\tau) + U_{p,\\al}(\\tau) V_{q,0}(\\tau)\\big) t^{-\\left(1-\\frac1p\\right)-\\frac\\th2} \\mathsf{B}\\left(\\frac1p-\\frac{1+\\th-\\al}{2},\\frac12-\\frac\\al2\\right),\n\\end{align*}\nwhere\n\\begin{align*}\n\\al \\le \\th<\\al+\\frac2p-1<1,\\quad\n\\al<\\frac2p-1.\n\\end{align*}\nHence, \n\\begin{align*}\nU_{p,\\th}(\\tau) = \\sup_{0< t< \\tau}t^{1-\\frac1p+\\frac{\\th}{2}}\\||\\N|^\\th u(t)\\|_{p}<+\\infty,\n\\quad\n\\al \\le \\th<\\al+\\frac2p-1<1,\\quad\n\\al<\\frac2p-1.\n\\end{align*}\nIterating further the above arguments, we obtain that the solution possesses higher regularity. In fact, using the uniqueness result established in Lemma~\\ref{lem;continuous-depend}, we may solve the problem~\\eqref{eqn;KS} with the initial data $(u(t_0),v(t_0))$ for some $t_0\\in (0,T)$. We thus eventually show that for $s>0$, $u(t)\\in L^1(\\R^2)\\cap W^{s,p}(\\R^2)$,\n$v(t)\\in  \\dot{H}^{1}(\\R^2)\\cap W^{s,q}(\\R^2)$\nfor $t_0<t<T$, \nand satisfies the system~\\eqref{eqn;KS} in a  classical sense on $(0,T)\\times\\R^2$.\nAlso, classical arguments based on the proof of Lemma~\\ref{lem;fixed-point} lead to the blowup criterion~\\eqref{eqn;bucrit}.\n\nLet  $(u_0,v_0)$ be nonnegative initial conditions with $u_0,v_0\\not\\equiv0$ and\n$\\{u_{0,k}\\}_{k\\in\\Nt }$ and\n$\\{v_{0,k}\\}_{k\\in\\Nt }$ be sequences\nof nonnegative functions in $C_0^{\\infty}(\\R^2)$ converging to $u_0$ in $L^{1}(\\R^2)$ and $v_0$ in $\\dot{H}^{1}(\\R^2)$  as $k\\to\\infty$, respectively. Then, for each $k\\in\\Nt$, we can construct the unique, smooth and integrable solution\nto~\\eqref{eqn;KS} corresponding to the initial data $(u_{0,k},v_{0,k})$.\nBy the parabolic regularity theory,\nthe solutions $(u_k,v_k)$ belong to $C^{\\infty}( [0,T)\\times \\R^2)$\nand are positive on $(0,T)\\times\\R^2$. In addition, by the continuous dependence on the initial data established in Lemma~\\ref{lem;continuous-depend}, we have the convergences $u_k\\to u$ in $C\\big([0,T); L^1(\\R^2)\\big)$ and $v_k\\to v$ in $C\\big([0,T); \\dot{H}^{1}(\\R^2)\\big)$ as $ k\\to \\infty$,\nwhere $(u,v)$ is the solution to~\\eqref{eqn;KS} with initial data $(u_0,v_0)$. Hence, one can find subsequences of the approximated solutions\nwhich converge to $(u,v)$ almost everywhere in $(0,T)\\times\\R^2$, which implies that $u$ and $v$ are positive on $(0,T)\\times\\R^2$.\nMoreover the mass conservation law also holds by integrating $u$ over $\\R^2$.\n\\end{pr}\n\n\\vspace{5mm}\n\\section{Global existence of solutions with critical mass}\\label{sect;global}\n\nIn this section, we prove the global existence of solutions to \\eqref{eqn;KS}\nwith critical mass, as stated in Theorem~\\ref{thm;global}.\nTo this end, we divide the proof into several steps.\n\nIn Subsection~\\ref{sect;modified-Lyapunov}, we introduce a suitably reconstructed\nLyapunov functional and derive regularity estimates for the associated\ndissipative terms.\nIn Subsection~\\ref{sect;exterior}, we establish various regularity estimates\nin exterior domains by using the estimates obtained in\nSubsection~\\ref{sect;modified-Lyapunov}.\nThanks to the results in the previous subsections, we then derive interior\nregularity estimates in Subsection~\\ref{sect;interior}.\nFinally, in Subsection~\\ref{sect;main}, we complete the proof of\nTheorem~\\ref{thm;global} by combining the exterior and interior estimates.\n\nThroughout this section, we denote by $(u,v)$ the solution to~\\eqref{eqn;KS}\nwith maximal existence time $T\\le\\infty$.\nIn the following, we derive a priori estimates on $[t_0,\\tau]\\cap(0,T)$ for arbitrary $t_0,\\tau$ with $t_0<\\tau$, which will allow us to conclude global existence by a contradiction argument.\nMoreover, $C$ denotes a generic positive constant depending only on $\\lambda$\nand the initial data~$(u_0,v_0)$, which may vary from line to line.\nThe dependence of $C$ on additional parameters will be indicated explicitly.\n\n\\subsection{Refined Lyapunov  functional}\\label{sect;modified-Lyapunov}\n\nWe here introduce a suitably reconstructed Lyapunov functional which is non-increasing along the evolution of solutions, and derive regularity estimates associated with the corresponding dissipation mechanism. These estimates further yield refined regularity properties for solutions to \\eqref{eqn;KS}.\n\nThe existence of a Lyapunov functional is known to play a fundamental role in the analysis of the long-time behavior of solutions, including global existence and finite-time blowup phenomena. In the whole space setting, however, the use of Lyapunov-based methods in~\\eqref{eqn;KS} faces intrinsic difficulties due to \n the necessity of controlling the behavior as $|x|\\to\\infty$. This issue becomes particularly delicate at the critical mass.\n\nFirst of all, it is worth mentioning that the system \\eqref{eqn;KS} has a usual Lyapunov functional~$\\L(t)$ as follows:\n\\begin{align*}\n\\frac{\\d}{\\d t}\\L(t)+ \\D(t) =0,\n\\eqntag\n\\label{eqn;Lyapunov}\n\\end{align*}\nwhere\n\\begin{align*}\n\\L(t):=\\int_{\\R^2}u\\ln u\\dx-\\int_{\\R^2}uv\\dx+\\frac12\\|\\nabla v\\|_2^2+\\frac\\lambda2\\|v\\|_2^2\n\\end{align*}\nand the dissipative term $\\D(t)$ is given by\n\\begin{align*}\n\\D(t):=\\int_{\\R^2} u \\left|\\nabla(\\ln u -v)\\right|^2\\dx+\\|\\partial_t v\\|_2^2.\n\\end{align*}\nThe upper bound for $\\L(t)$ follows from the monotonicity \\eqref{eqn;Lyapunov} and \nthe lower bound for that can be obtained by the optimal constraint of the initial mass $\\|u_0\\|_1$ deriving from the Trudinger--Moser type inequality and its best possible constant, see \\cite{Na-Se-Yo} and also \\cite{CaCo08,Mi13,NaOg11} for instance.\nOn the other hand,\nin order to ensure the well-definedness of a usual entropy $\\int_{\\R^2}u\\ln u \\dx$ on the whole space,\nat least the logarithmic moment assumption for the initial data $u_0\\ln(1+|x|^2)\\in L^1(\\R^2)$ is required to control the behavior of solutions as $|x|\\to\\infty$, that is,\n\\begin{align*}\n-\\int_{\\R^2}u(\\ln u)_-\\dx \\le\\,C\\int_{\\R^2}u\\ln(1+|x|^2)\\dx,\n\\end{align*}\nwhere $(\\ln u)_-:=\\min\\{\\ln u,0\\}$, according to \\cite[Lemma 2.4]{CaCo08}.\nLater, to get rid of any moment assumptions,\nthe modified entropy is adapted by Nagai \\cite{Na11};\n\\begin{align*}\n\\int_{\\R^2}(1+u)\\ln(1+u)\\dx,\n\\end{align*}\nas well as the following identity for the modified Lyapunov functional $\\L_m(t)$ is introduced:\n\\begin{align*}\n\\frac{\\d}{\\d t} \\L_m(t) + \\D_m(t)=\\frac14\\|\\nabla v(t)\\|_2^2,\n\\eqntag\n\\label{eqn;usualmodifiedfunctiona}\n\\end{align*}\nwhere\n\\begin{align*}\n\\L_m(t):=\\int_{\\R^2}(1+u)\\ln(1+u)\\dx-\\int_{\\R^2}uv\\dx+\\frac12\\|\\nabla v\\|_2^2+\\frac\\lambda2\\|v\\|_2^2\n\\end{align*}\nand\n\\begin{align*}\n\\D_m(t):=\\int_{\\R^2}u\\left|\\nabla(\\ln(1+u)-v)\\right|^2\\dx+\\int_{\\R^2}\\left|\\nabla\\left(\\ln(1+u)-\\frac12v\\right)\\right|^2\\dx+\\|\\partial_tv\\|_2^2.\n\\end{align*}\nThe functional $\\L_m(t)$ is no longer the Lyapunov functional, however, it is still useful for proving \nthe global existence of the solution to \\eqref{eqn;KS} with sub-critical mass. Indeed, it follows\nfrom the lower bound for $\\L_m(t)$ that\nfor any $\\al\\in[0,1)$\n\\begin{align*}\n\\al\\int_{\\R^2}(1+u)\\ln(1+u)\\dx+\\frac12\\left[1-\\frac{\\|u_0\\|_1}{8\\pi(1-\\al)}\\right]&\\|\\nabla v\\|_2^2\n+\\int_0^t \\D_m(s)\\,\\d s\n\\\\\n\\le\\,&\\L_m(0)+C(\\tau,\\alpha)+\\frac14\\int_0^t\\|\\nabla v\\|_2^2\\,\\d s\n\\eqntag\n\\label{eqn;afford}\n\\end{align*}\nfor $t\\in[0,\\tau]\\cap(0,T)$ with any $\\tau>0$ and some $C(\\tau,\\alpha)>0$ (cf.~\\cite{CaCo08,Mi13}). Hence, if $\\|u_0\\|<8\\pi$, then all terms on the left hand side of \\eqref{eqn;afford} are positive by choosing $\\al>0$ sufficiently small which depends on $\\|u_0\\|_1$, and Gronwall's inequality implies that\n$\\nabla v \\in L^2((0,T)\\times \\R^2)$, \nso that we also obtain the bound for the modified entropy $\\int_{\\R^2}(1+u)\\ln (1+u)\\dx$. \nNevertheless, for the critical mass $\\|u_0\\|_1=8\\pi$, this necessarily leads to $\\alpha=0$, as a result, \n\\begin{align*}\n\\int_0^t \\D_m(s)\\,\\d s\\le\\,&\\L_m(0)+C(\\tau)+\\frac14\\int_0^t\\|\\nabla v\\|_2^2\\,\\d s,\n\\eqntag\n\\label{eqn;lack-lyapunov}\n\\end{align*}\nwhich is useless and fails to obtain even the estimates for $\\D_m(t)$ due to the lack of the regularity of solutions.\nTherefore, we introduce a reconstructed Lyapunov functional $\\F_m(t)$ so as to show regularity estimates corresponding to the dissipative terms. Let $\\F_m(t)$ be the functional for solutions to \\eqref{eqn;KS} defined as\n\\begin{align*}\n\\F_m(t):=\\L_m(t)\n+\\int_{\\R^2}\\ln(1+u)\\,\\mathrm{d}x-\\int_{\\R^2}v\\,\\d x.\n\\eqntag\n\\label{eqn;modifiedLyapunov}\n\\end{align*}\nThen, the following functional differential inequality holds true:\n\\begin{prop}\\label{prop;energy-est}\nLet $(u,v)$ be the solution to \\eqref{eqn;KS}. Then, the functional $\\F_m(t)$ defined in~\\eqref{eqn;modifiedLyapunov} satisfies the following identity:\n\t\\begin{align*}\n\t\\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2\n\t\\le\\,&\n\t\\lambda\\int_{\\R^2}v\\,\\d x.\n\t\\end{align*}\n\tTherefore, if $\\lambda=0$ then\n\t\\begin{align*}\n\t\\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2 \\le\\,0.\n\t\\end{align*}\n\tIf $\\lambda>0$ then \n\t\\begin{align*}\n\t\\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2 \\le\\,\\|v_0\\|_1+\\|u_0\\|_1.\n\t\\end{align*}\n\\end{prop}\n\\vspace{5mm}\nIt is worth emphasizing that $\\F_m(t)$ is non-increasing in time when $\\lambda=0$ as well as\nunlike the identity \\eqref{eqn;usualmodifiedfunctiona} based on the usual modified functional $\\mathcal{L}_m(t)$, the error term appearing on the right-hand side of Proposition~\\ref{prop;energy-est} can be easily controlled by initial data.\nThis allows us to show the regularity estimates for the dissipative term associated with $\\F_m(t)$, see Proposition~\\ref{prop;bound-energy} below.\n\\begin{lem}\\label{lem;ModifiedLF}\n\tSuppose assumptions as in Theorem \\ref{thm;global}. Then,\n\\begin{align*}\n\\frac{\\d}{\\d t}\\F_m(t)+ \\widetilde{\\D}(t)=-\\int_{\\R^2}\\partial_t v \\frac{u}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\mathrm{d}x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x,\n\\end{align*}\nwhere\n$\\F_m(t)$ is the modified functional defined in \\eqref{eqn;modifiedLyapunov} and the dissipative term $\\widetilde{\\D}(t)$ is given by\n\\begin{align*}\n\\widetilde{\\D}(t):=\\int_{\\R^2}u\\left|\\nabla \\left(\\ln(1+u)-v\\right)\\right|^2\\,\\d x+\\|\\partial_t v\\|_2^2.\n\\end{align*}\n\\end{lem}\n\n\\begin{pr}{Lemma \\ref{lem;ModifiedLF}}\nThe following computations are already well-known:\n\\begin{align*}\n\\frac{\\d}{\\d t} \\int_{\\R^2}(1+u)\\ln (1+u)\\,\\d x=\\,&\n-\\int_{\\R^2}(1+u)|\\nabla\\ln(1+u)|^2\\,\\d x\n+\\int_{\\R^2}u\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x\n\\end{align*}\nas well as\n\\begin{align*}\n-\\frac{\\d}{\\d t}\\int_{\\R^2}uv\\,\\d x=\\,&\\int_{\\R^2}(1+u)\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x-\\int_{\\R^2}u|\\nabla v|^2\\,\\d x\n\\\\\n&-\\|\\partial_tv\\|_2^2-\\frac12\\frac{\\d}{\\d t}\\|\\nabla v\\|_2^2-\\frac\\lambda2\\frac{\\d}{\\d t}\\|v\\|_2^2,\n\\end{align*}\nso that\n\\begin{align*}\n&\\frac{\\d}{\\d t}\\left[\\int_{\\R^2}(1+u)\\ln(1+u)\\,\\d x-\\int_{\\R^2}uv\\,\\d x+\\frac12\\left(\\|\\nabla v\\|_2^2+\\lambda\\|v\\|_2^2\\right)\\right]\n\\\\\n&+\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\|\\partial_tv\\|_2^2\n\\\\\n=\\,&-\\|\\nabla\\ln(1+u)\\|_2^2+\\int_{\\R^2}\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x.\n\\end{align*}\nNext,\n\\begin{align*}\n\\frac{\\d}{\\d t}\\int_{\\R^2}\\ln(1+u)\\,\\d x=\\,&\\int_{\\R^2}\\frac{1}{1+u} \\left[\\Delta u -\\nabla\\cdot( u\\nabla v)\\right]\\,\\mathrm{d}x\n\\\\\n=\\,&-\\int_{\\R^2}\\nabla\\left(\\frac{1}{1+u}\\right)\\cdot \\nabla u\\,\\d x+\\int_{\\R^2}u\\nabla\\left(\\frac{1}{1+u}\\right)\\cdot \\nabla v\\,\\d x\n\\\\\n=\\,&\\int_{\\R^2}|\\nabla \\ln(1+u)|^2\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x\n\\\\\n=\\,&\\int_{\\R^2}|\\nabla \\ln(1+u)|^2\\,\\d x-\\int_{\\R^2}\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x\n\\\\\n&+\\int_{\\R^2}\\frac{1}{1+u}\\nabla\\ln(1+u)\\cdot\\nabla v \\,\\d x.\n\\end{align*}\nHence, combining the above computations implies that\n\\begin{align*}\n&\\frac{\\d}{\\d t}\\left[\\int_{\\R^2}(1+u)\\ln(1+u)\\,\\d x-\\int_{\\R^2}uv\\,\\d x+\\frac12\\left(\\|\\nabla v\\|_2^2+\\lambda\\|v\\|_2^2\\right)+\\int_{\\R^2}\\ln(1+u)\\,\\d x\\right]\n\\\\\n&+\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\|\\partial_tv\\|_2^2\n\\\\\n=\\,&\\int_{\\R^2}\\frac{1}{(1+u)^2}\\nabla u\\cdot\\nabla v \\,\\d x.\n\\end{align*}\nSince\n\\begin{align*}\n\\int_{\\R^2}\\frac{1}{(1+u)^2}\\nabla u\\cdot\\nabla v \\,\\d x=\\,&-\\int_{\\R^2}\\nabla\\left(\\frac{1}{1+u}\\right)\\cdot \\nabla v\\,\\d x\n\\\\\n=\\,&\\int_{\\R^2}\\frac{1}{1+u}\\Delta v\\,\\d x\n\\\\\n=\\,&\\int_{\\R^2}\\frac{\\partial_t v}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\d x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x\n\\\\\n=\\,&\\frac{\\d}{\\d t}\\int_{\\R^2}v\\,\\d x-\\int_{\\R^2}\\partial_t v \\frac{u}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\d x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x,\n\\end{align*}\nwe end up with\n\\begin{align*}\n&\\frac{\\d}{\\d t}\\F_m(t)+\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\|\\partial_tv\\|_2^2\n\\\\\n=\\,&-\\int_{\\R^2}\\partial_t v \\frac{u}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\d x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x,\n\\end{align*}\nas desired.\n\\end{pr}"}, "pre_theorem_intro_text_len": 8115, "pre_theorem_intro_text": "We study  the Cauchy problem for the parabolic-parabolic Keller--Segel system in~$\\mathbb R^2$\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n&\\pt_t u =\\Delta u- \\nabla\\cdot \\left( u \\nabla v \\right),\n& t>0,\\,~ &x\\in\\mathbb R^2,\n\\\\\n&\\pt_t v = \\Delta v - \\lambda v +u,\n&t>0,\\, ~&x\\in\\mathbb R^2,\n\\\\\n&(u,v)(0,x)=(u_{0},v_0)(x),\n&\\, &x\\in\\mathbb R^2\n\\end{aligned}\n\\right.\n\\label{eqn;KS}\n\\end{equation}\nwith a constant $\\lambda\\ge0$, where $u_0,v_0\\ge 0$ on $\\mathbb R^2$ and\n$u_0,v_0\\not\\equiv0$.\nThe Keller--Segel system~\\eqref{eqn;KS} is a fundamental mathematical model\nof chemotaxis, describing chemotactic aggregation in the\ncellular slime mold {\\it Dictyostelium discoideum} during its life cycle \\cite{KeSe,Pa}.\nIn this model, $u=u(t,x)$ and $v=v(t,x)$ denote the densities of cells and\nthe chemoattractant, respectively.\nThe parameter $\\lambda$ represents the degradation rate of the chemical.\nCells migrate toward regions of higher concentrations of a chemical substance\nsecreted by the cells themselves.\nFrom a biological viewpoint, it is natural to assume that the initial data\nare nonnegative functions.\nFrom a mathematical viewpoint, cell aggregation is interpreted as the\nblowup of solutions at $t=T$ in the sense that\n$\\lim_{t\\to T}\\|u(t)\\|_\\infty=\\infty.$\n\nOne of the central mathematical features of such systems is the phenomenon of critical\nmass.\nFor positive sufficiently regular solutions $(u,v)$ to~\\eqref{eqn;KS}, the first\ncomponent $u$ satisfies the mass conservation law \n$$\\|u(t)\\|_1=\\|u_0\\|_1\\quad \\text{for}~~~t>0.$$\nMoreover, in two space dimensions, the global behavior of solutions is governed\nby the size of the initial mass of $u$.\nIn particular, there exists a threshold value such that solutions with initial\nmass below this threshold exist globally in time, whereas solutions with\ninitial mass above the threshold may blow up in finite time.\nThis critical-mass phenomenon has attracted considerable attention in the\nliterature.\n\nThe aim of this paper is to establish the global behavior of solutions to\n\\eqref{eqn;KS} whose initial mass is exactly equal to this threshold value.\n\n\\vspace{3mm}\nThe second equation in \\eqref{eqn;KS} takes into account that cells are producing the chemoattractant themselves \nwhile this is diffusing into the environment. \nSince the chemoattractant attains its equilibrium on a time scale much quicker\nthan that of the cells, the simplified parabolic-elliptic system has also been\ninvestigated \\cite{JaLu,Na95}:\n\\begin{align*}\n\\left\\{\n\\begin{aligned}\n\\partial_t u &= \\Delta u - \\nabla\\cdot (u\\nabla v),\n& t>0,\\ & x\\in\\mathbb R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\mathbb R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\mathbb R^2.\n\\end{aligned}\n\\right.\n\\addtocounter{equation}{1}\\tag{\\theequation}\n\\label{eqn;PKS}\n\\end{align*}\nThe system \\eqref{eqn;PKS} is also related to models of gravitational interaction\nof particles \\cite{Bi-Na94,Wo}.\nIn~\\eqref{eqn;PKS}, the second equation can be written as\n$v = (-\\Delta+\\lambda)^{-1}u.$\nAs a consequence, the system~\\eqref{eqn;PKS} can be reduced to a single equation\nfor~$u$.\nOwing to this elliptic structure, the parabolic-elliptic system~\\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.\nIn particular, numerous works have been devoted to the critical-mass\nphenomenon.\nIndeed, the solution with $\\|u_0\\|_1\\le 8\\pi$ exists globally in time~\\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},\nwhile the solution may blow up in finite time if $\\|u_0\\|_1>8\\pi$~\\cite{DoPe,Ko-Su,Ku-Og03,Wei}.\nFor the corresponding Cauchy--Neumann problem in bounded domains,\nsupplemented with homogeneous Neumann boundary conditions, see \\cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.\n\n\\vspace{3mm}\nUnlike the parabolic-elliptic system \\eqref{eqn;PKS}, the fully parabolic system\n\\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches\ndeveloped for \\eqref{eqn;PKS} are no longer applicable.\nThe global existence of solutions with sub-critical mass\n$\\|u_0\\|_{1}<8\\pi$ has nevertheless been established by combining\nLyapunov functionals with the Trudinger--Moser type inequality and its optimal\nconstant.\nMore precisely, the following results are known:\n\\begin{enumerate}\n\t\\item[(i)]\n\tIf $\\|u_0\\|_{1}<8\\pi$, then the corresponding solution to\n\t\\eqref{eqn;KS} exists globally in time \\cite{CaCo08,Mi13,NaOg11}.\n\n\t\\item[(ii)]\n\tIf $\\|u_0\\|_{1}>8\\pi$ and $(u_0,v_0)$ is radially symmetric, then there\n\texists a radially symmetric solution that blows up in finite time\n\t\\cite{Mi20,Mi20-SIAM}.\n\n\t\\item[(iii)]\n\tIf $\\|u_0\\|_{1}=8\\pi$ and $(u_0,v_0)$ is either radially symmetric or\n\tsatisfies the additional moment condition\n\t$u_0\\ln(1+|x|^2)\\in L^1(\\mathbb R^2)$, then the corresponding solution exists globally\n\tin time.\n\tIn contrast, for general initial data, the global behavior in the critical case\n\tremains delicate, and the solution either exists globally or blows up on the plane \\cite{CaCo08,Mi13}.\n\\end{enumerate}\nAs for positive forward self-similar solutions to \\eqref{eqn;KS}, refer to \\cite{BiCoDo}.\nFor the corresponding Cauchy--Neumann problem in bounded domains, we refer to\n\\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.\n\nAs mentioned above, although several partial results are available, no complete global existence result has been obtained\nfor solutions with critical mass and general initial data without any\nsymmetry or moment assumptions.\nOne of the main difficulties lies in controlling the behavior of solutions at~$|x|\\to\\infty$.\nMoreover, in the critical mass case, no global existence result is currently\nknown  for the corresponding Cauchy--Neumann problem to~\\eqref{eqn;KS} in bounded domains.\n\nIn this paper, we establish the global existence of solutions with critical mass\nfor general initial data $(u_0,v_0)$.\nTo this end, we first recall the definition of solutions.\n\n\\vspace{2mm}\n\\begin{def*}\nLet $4/3<p<2$ and $q=p/(p-1)$.\nGiven $T\\in (0,\\infty]$ and  $(u_0,v_0) \\in L^1(\\mathbb R^2)\\times \\dot{H}^{1}(\\mathbb R^2)$,\nwe say that a pair of functions $(u,v)$ defined on $[0,T)\\times \\mathbb R^2$ is a mild solution to~\\eqref{eqn;KS} on $[0,T)$ if\n\\begin{enumerate}\n\t\\item $u\\in C\\big([0,T); L^1(\\mathbb R^2)\\big)\\cap C\\left( (0,T); L^p(\\mathbb{R}^2) \\right)$;\n\t\\item $v\\in C\\big([0,T); \\dot{H}^{1}(\\mathbb R^2)\\big)\\cap C\\left( (0,T); \\dot{W}^{1,q}(\\mathbb{R}^2) \\right)$;\n\t\\item $\\displaystyle\\sup_{0<t<T} t^{1-\\frac1p}\\|u(t)\\|_{p}+\\displaystyle\\sup_{0<t<T}t^{\\frac12-\\frac1q}\\|\\nabla v(t)\\|_{q}$;\n\t\\item $(u,v)$ satisfies the integral formulations\n\t\\begin{align*}\n\t&u(t)=e^{t\\Delta}u_0-\\int_0^t \\nabla\\cdot e^{(t-s)\\Delta} \\big(u(s)\\nabla v(s)\\big) \\,\\mathrm{d} s,\n\t\\\\\n\t&v(t)= e^{t(\\Delta-\\lambda)} v_0+\\int_0^t e^{(t-s)(\\Delta-\\lambda)} u(s) \\,\\mathrm{d} s\n\t\\end{align*}\n\tfor $0<t<T$, where $e^{t\\Delta}$ is the heat semigroup given by\n\t\\begin{align*}\n\te^{t\\Delta} f(x) := G_t*f(x)=\\int_{ \\mathbb R^2}G_t(x-y) f(y)\\,\\mathrm{d} y,\n\t\\quad\n\tG_t(x)=\\frac{1}{4\\pi t} \\exp\\left(-\\frac{|x|^2}{4 t}\\right)\n\t\\end{align*}\n\tfor $x\\in\\mathbb R^2$ and $f \\in L^p(\\mathbb R^2)$.\n\\end{enumerate}\n\\end{def*}\nThe Lebesgue spaces $L^p(\\mathbb R^2)$ are equipped with the norm $\\|\\cdot\\|_{L^p(\\mathbb R^2)}$,\nwhich is abbreviated as $\\|\\cdot\\|_p$ when there is no ambiguity.\nWe define the homogeneous Sobolev spaces by\n\\[\n\\dot{W}^{1,p}(\\mathbb R^2)\n:= \\overline{C_0^\\infty(\\mathbb R^2)}^{\\|\\nabla (\\cdot)\\|_{L^p(\\mathbb R^2)}}\n\\]\nand $\\dot{H}^1(\\mathbb R^2):=\\dot{W}^{1,2}(\\mathbb R^2)$ when $p=2$.\n$L_+^1(\\mathbb R^2):=\\{f \\in L^1(\\mathbb R^2); f\\ge0~~ \\text{and}~~ f\\not\\equiv0~~ \\text{on $\\mathbb R^2$} \\}$.\n\n\\vspace{5mm}\nThe starting point for our analysis is the local well-posedness for \\eqref{eqn;KS} and the solution~$(u,v)$ based on the above definition satisfies~\\eqref{eqn;KS} in a classical sense on $(0,T)$, see Proposition~\\ref{prop;LWP} in Section~\\ref{sect;LWP} below.\n\nWe are now in a position to state our main result in this paper.\nMain result in this paper reads as follows.\n\\vspace{2mm}", "context": "\\vspace{3mm}\nThe second equation in \\eqref{eqn;KS} takes into account that cells are producing the chemoattractant themselves \nwhile this is diffusing into the environment. \nSince the chemoattractant attains its equilibrium on a time scale much quicker\nthan that of the cells, the simplified parabolic-elliptic system has also been\ninvestigated \\cite{JaLu,Na95}:\n\\begin{align*}\n\\left\\{\n\\begin{aligned}\n\\partial_t u &= \\Delta u - \\nabla\\cdot (u\\nabla v),\n& t>0,\\ & x\\in\\mathbb R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\mathbb R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\mathbb R^2.\n\\end{aligned}\n\\right.\n\\addtocounter{equation}{1}\\tag{\\theequation}\n\\label{eqn;PKS}\n\\end{align*}\nThe system \\eqref{eqn;PKS} is also related to models of gravitational interaction\nof particles \\cite{Bi-Na94,Wo}.\nIn~\\eqref{eqn;PKS}, the second equation can be written as\n$v = (-\\Delta+\\lambda)^{-1}u.$\nAs a consequence, the system~\\eqref{eqn;PKS} can be reduced to a single equation\nfor~$u$.\nOwing to this elliptic structure, the parabolic-elliptic system~\\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.\nIn particular, numerous works have been devoted to the critical-mass\nphenomenon.\nIndeed, the solution with $\\|u_0\\|_1\\le 8\\pi$ exists globally in time~\\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},\nwhile the solution may blow up in finite time if $\\|u_0\\|_1>8\\pi$~\\cite{DoPe,Ko-Su,Ku-Og03,Wei}.\nFor the corresponding Cauchy--Neumann problem in bounded domains,\nsupplemented with homogeneous Neumann boundary conditions, see \\cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.\n\n\\vspace{3mm}\nUnlike the parabolic-elliptic system \\eqref{eqn;PKS}, the fully parabolic system\n\\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches\ndeveloped for \\eqref{eqn;PKS} are no longer applicable.\nThe global existence of solutions with sub-critical mass\n$\\|u_0\\|_{1}<8\\pi$ has nevertheless been established by combining\nLyapunov functionals with the Trudinger--Moser type inequality and its optimal\nconstant.\nMore precisely, the following results are known:\n\\begin{enumerate}\n \\item[(i)]\n If $\\|u_0\\|_{1}<8\\pi$, then the corresponding solution to\n \\eqref{eqn;KS} exists globally in time \\cite{CaCo08,Mi13,NaOg11}.\n\n\\item[(iii)]\n If $\\|u_0\\|_{1}=8\\pi$ and $(u_0,v_0)$ is either radially symmetric or\n satisfies the additional moment condition\n $u_0\\ln(1+|x|^2)\\in L^1(\\mathbb R^2)$, then the corresponding solution exists globally\n in time.\n In contrast, for general initial data, the global behavior in the critical case\n remains delicate, and the solution either exists globally or blows up on the plane \\cite{CaCo08,Mi13}.\n\\end{enumerate}\nAs for positive forward self-similar solutions to \\eqref{eqn;KS}, refer to \\cite{BiCoDo}.\nFor the corresponding Cauchy--Neumann problem in bounded domains, we refer to\n\\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.\n\n\\vspace{2mm}\n\\begin{def*}\nLet $4/3<p<2$ and $q=p/(p-1)$.\nGiven $T\\in (0,\\infty]$ and $(u_0,v_0) \\in L^1(\\mathbb R^2)\\times \\dot{H}^{1}(\\mathbb R^2)$,\nwe say that a pair of functions $(u,v)$ defined on $[0,T)\\times \\mathbb R^2$ is a mild solution to~\\eqref{eqn;KS} on $[0,T)$ if\n\\begin{enumerate}\n \\item $u\\in C\\big([0,T); L^1(\\mathbb R^2)\\big)\\cap C\\left( (0,T); L^p(\\mathbb{R}^2) \\right)$;\n \\item $v\\in C\\big([0,T); \\dot{H}^{1}(\\mathbb R^2)\\big)\\cap C\\left( (0,T); \\dot{W}^{1,q}(\\mathbb{R}^2) \\right)$;\n \\item $\\displaystyle\\sup_{0<t<T} t^{1-\\frac1p}\\|u(t)\\|_{p}+\\displaystyle\\sup_{0<t<T}t^{\\frac12-\\frac1q}\\|\\nabla v(t)\\|_{q}$;\n \\item $(u,v)$ satisfies the integral formulations\n \\begin{align*}\n &u(t)=e^{t\\Delta}u_0-\\int_0^t \\nabla\\cdot e^{(t-s)\\Delta} \\big(u(s)\\nabla v(s)\\big) \\,\\mathrm{d} s,\n \\\\\n &v(t)= e^{t(\\Delta-\\lambda)} v_0+\\int_0^t e^{(t-s)(\\Delta-\\lambda)} u(s) \\,\\mathrm{d} s\n \\end{align*}\n for $0<t<T$, where $e^{t\\Delta}$ is the heat semigroup given by\n \\begin{align*}\n e^{t\\Delta} f(x) := G_t*f(x)=\\int_{ \\mathbb R^2}G_t(x-y) f(y)\\,\\mathrm{d} y,\n \\quad\n G_t(x)=\\frac{1}{4\\pi t} \\exp\\left(-\\frac{|x|^2}{4 t}\\right)\n \\end{align*}\n for $x\\in\\mathbb R^2$ and $f \\in L^p(\\mathbb R^2)$.\n\\end{enumerate}\n\\end{def*}\nThe Lebesgue spaces $L^p(\\mathbb R^2)$ are equipped with the norm $\\|\\cdot\\|_{L^p(\\mathbb R^2)}$,\nwhich is abbreviated as $\\|\\cdot\\|_p$ when there is no ambiguity.\nWe define the homogeneous Sobolev spaces by\n\\[\n\\dot{W}^{1,p}(\\mathbb R^2)\n:= \\overline{C_0^\\infty(\\mathbb R^2)}^{\\|\\nabla (\\cdot)\\|_{L^p(\\mathbb R^2)}}\n\\]\nand $\\dot{H}^1(\\mathbb R^2):=\\dot{W}^{1,2}(\\mathbb R^2)$ when $p=2$.\n$L_+^1(\\mathbb R^2):=\\{f \\in L^1(\\mathbb R^2); f\\ge0~~ \\text{and}~~ f\\not\\equiv0~~ \\text{on $\\mathbb R^2$} \\}$.\n\n\\vspace{5mm}\nThe starting point for our analysis is the local well-posedness for \\eqref{eqn;KS} and the solution~$(u,v)$ based on the above definition satisfies~\\eqref{eqn;KS} in a classical sense on $(0,T)$, see Proposition~\\ref{prop;LWP} in Section~\\ref{sect;LWP} below.\n\nWe are now in a position to state our main result in this paper.\nMain result in this paper reads as follows.\n\\vspace{2mm}", "full_context": "\\vspace{3mm}\nThe second equation in \\eqref{eqn;KS} takes into account that cells are producing the chemoattractant themselves \nwhile this is diffusing into the environment. \nSince the chemoattractant attains its equilibrium on a time scale much quicker\nthan that of the cells, the simplified parabolic-elliptic system has also been\ninvestigated \\cite{JaLu,Na95}:\n\\begin{align*}\n\\left\\{\n\\begin{aligned}\n\\partial_t u &= \\Delta u - \\nabla\\cdot (u\\nabla v),\n& t>0,\\ & x\\in\\mathbb R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\mathbb R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\mathbb R^2.\n\\end{aligned}\n\\right.\n\\addtocounter{equation}{1}\\tag{\\theequation}\n\\label{eqn;PKS}\n\\end{align*}\nThe system \\eqref{eqn;PKS} is also related to models of gravitational interaction\nof particles \\cite{Bi-Na94,Wo}.\nIn~\\eqref{eqn;PKS}, the second equation can be written as\n$v = (-\\Delta+\\lambda)^{-1}u.$\nAs a consequence, the system~\\eqref{eqn;PKS} can be reduced to a single equation\nfor~$u$.\nOwing to this elliptic structure, the parabolic-elliptic system~\\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.\nIn particular, numerous works have been devoted to the critical-mass\nphenomenon.\nIndeed, the solution with $\\|u_0\\|_1\\le 8\\pi$ exists globally in time~\\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},\nwhile the solution may blow up in finite time if $\\|u_0\\|_1>8\\pi$~\\cite{DoPe,Ko-Su,Ku-Og03,Wei}.\nFor the corresponding Cauchy--Neumann problem in bounded domains,\nsupplemented with homogeneous Neumann boundary conditions, see \\cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.\n\n\\vspace{3mm}\nUnlike the parabolic-elliptic system \\eqref{eqn;PKS}, the fully parabolic system\n\\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches\ndeveloped for \\eqref{eqn;PKS} are no longer applicable.\nThe global existence of solutions with sub-critical mass\n$\\|u_0\\|_{1}<8\\pi$ has nevertheless been established by combining\nLyapunov functionals with the Trudinger--Moser type inequality and its optimal\nconstant.\nMore precisely, the following results are known:\n\\begin{enumerate}\n \\item[(i)]\n If $\\|u_0\\|_{1}<8\\pi$, then the corresponding solution to\n \\eqref{eqn;KS} exists globally in time \\cite{CaCo08,Mi13,NaOg11}.\n\n\\item[(iii)]\n If $\\|u_0\\|_{1}=8\\pi$ and $(u_0,v_0)$ is either radially symmetric or\n satisfies the additional moment condition\n $u_0\\ln(1+|x|^2)\\in L^1(\\mathbb R^2)$, then the corresponding solution exists globally\n in time.\n In contrast, for general initial data, the global behavior in the critical case\n remains delicate, and the solution either exists globally or blows up on the plane \\cite{CaCo08,Mi13}.\n\\end{enumerate}\nAs for positive forward self-similar solutions to \\eqref{eqn;KS}, refer to \\cite{BiCoDo}.\nFor the corresponding Cauchy--Neumann problem in bounded domains, we refer to\n\\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.\n\n\\vspace{2mm}\n\\begin{def*}\nLet $4/3<p<2$ and $q=p/(p-1)$.\nGiven $T\\in (0,\\infty]$ and $(u_0,v_0) \\in L^1(\\mathbb R^2)\\times \\dot{H}^{1}(\\mathbb R^2)$,\nwe say that a pair of functions $(u,v)$ defined on $[0,T)\\times \\mathbb R^2$ is a mild solution to~\\eqref{eqn;KS} on $[0,T)$ if\n\\begin{enumerate}\n \\item $u\\in C\\big([0,T); L^1(\\mathbb R^2)\\big)\\cap C\\left( (0,T); L^p(\\mathbb{R}^2) \\right)$;\n \\item $v\\in C\\big([0,T); \\dot{H}^{1}(\\mathbb R^2)\\big)\\cap C\\left( (0,T); \\dot{W}^{1,q}(\\mathbb{R}^2) \\right)$;\n \\item $\\displaystyle\\sup_{0<t<T} t^{1-\\frac1p}\\|u(t)\\|_{p}+\\displaystyle\\sup_{0<t<T}t^{\\frac12-\\frac1q}\\|\\nabla v(t)\\|_{q}$;\n \\item $(u,v)$ satisfies the integral formulations\n \\begin{align*}\n &u(t)=e^{t\\Delta}u_0-\\int_0^t \\nabla\\cdot e^{(t-s)\\Delta} \\big(u(s)\\nabla v(s)\\big) \\,\\mathrm{d} s,\n \\\\\n &v(t)= e^{t(\\Delta-\\lambda)} v_0+\\int_0^t e^{(t-s)(\\Delta-\\lambda)} u(s) \\,\\mathrm{d} s\n \\end{align*}\n for $0<t<T$, where $e^{t\\Delta}$ is the heat semigroup given by\n \\begin{align*}\n e^{t\\Delta} f(x) := G_t*f(x)=\\int_{ \\mathbb R^2}G_t(x-y) f(y)\\,\\mathrm{d} y,\n \\quad\n G_t(x)=\\frac{1}{4\\pi t} \\exp\\left(-\\frac{|x|^2}{4 t}\\right)\n \\end{align*}\n for $x\\in\\mathbb R^2$ and $f \\in L^p(\\mathbb R^2)$.\n\\end{enumerate}\n\\end{def*}\nThe Lebesgue spaces $L^p(\\mathbb R^2)$ are equipped with the norm $\\|\\cdot\\|_{L^p(\\mathbb R^2)}$,\nwhich is abbreviated as $\\|\\cdot\\|_p$ when there is no ambiguity.\nWe define the homogeneous Sobolev spaces by\n\\[\n\\dot{W}^{1,p}(\\mathbb R^2)\n:= \\overline{C_0^\\infty(\\mathbb R^2)}^{\\|\\nabla (\\cdot)\\|_{L^p(\\mathbb R^2)}}\n\\]\nand $\\dot{H}^1(\\mathbb R^2):=\\dot{W}^{1,2}(\\mathbb R^2)$ when $p=2$.\n$L_+^1(\\mathbb R^2):=\\{f \\in L^1(\\mathbb R^2); f\\ge0~~ \\text{and}~~ f\\not\\equiv0~~ \\text{on $\\mathbb R^2$} \\}$.\n\n\\vspace{5mm}\nThe starting point for our analysis is the local well-posedness for \\eqref{eqn;KS} and the solution~$(u,v)$ based on the above definition satisfies~\\eqref{eqn;KS} in a classical sense on $(0,T)$, see Proposition~\\ref{prop;LWP} in Section~\\ref{sect;LWP} below.\n\nWe are now in a position to state our main result in this paper.\nMain result in this paper reads as follows.\n\\vspace{2mm}\n\n\\vspace{5mm}\nThe starting point for our analysis is the local well-posedness for \\eqref{eqn;KS} and the solution~$(u,v)$ based on the above definition satisfies~\\eqref{eqn;KS} in a classical sense on $(0,T)$, see Proposition~\\ref{prop;LWP} in Section~\\ref{sect;LWP} below.\n\nFirst of all, it is worth mentioning that the system \\eqref{eqn;KS} has a usual Lyapunov functional~$\\L(t)$ as follows:\n\\begin{align*}\n\\frac{\\d}{\\d t}\\L(t)+ \\D(t) =0,\n\\eqntag\n\\label{eqn;Lyapunov}\n\\end{align*}\nwhere\n\\begin{align*}\n\\L(t):=\\int_{\\R^2}u\\ln u\\dx-\\int_{\\R^2}uv\\dx+\\frac12\\|\\nabla v\\|_2^2+\\frac\\lambda2\\|v\\|_2^2\n\\end{align*}\nand the dissipative term $\\D(t)$ is given by\n\\begin{align*}\n\\D(t):=\\int_{\\R^2} u \\left|\\nabla(\\ln u -v)\\right|^2\\dx+\\|\\partial_t v\\|_2^2.\n\\end{align*}\nThe upper bound for $\\L(t)$ follows from the monotonicity \\eqref{eqn;Lyapunov} and \nthe lower bound for that can be obtained by the optimal constraint of the initial mass $\\|u_0\\|_1$ deriving from the Trudinger--Moser type inequality and its best possible constant, see \\cite{Na-Se-Yo} and also \\cite{CaCo08,Mi13,NaOg11} for instance.\nOn the other hand,\nin order to ensure the well-definedness of a usual entropy $\\int_{\\R^2}u\\ln u \\dx$ on the whole space,\nat least the logarithmic moment assumption for the initial data $u_0\\ln(1+|x|^2)\\in L^1(\\R^2)$ is required to control the behavior of solutions as $|x|\\to\\infty$, that is,\n\\begin{align*}\n-\\int_{\\R^2}u(\\ln u)_-\\dx \\le\\,C\\int_{\\R^2}u\\ln(1+|x|^2)\\dx,\n\\end{align*}\nwhere $(\\ln u)_-:=\\min\\{\\ln u,0\\}$, according to \\cite[Lemma 2.4]{CaCo08}.\nLater, to get rid of any moment assumptions,\nthe modified entropy is adapted by Nagai \\cite{Na11};\n\\begin{align*}\n\\int_{\\R^2}(1+u)\\ln(1+u)\\dx,\n\\end{align*}\nas well as the following identity for the modified Lyapunov functional $\\L_m(t)$ is introduced:\n\\begin{align*}\n\\frac{\\d}{\\d t} \\L_m(t) + \\D_m(t)=\\frac14\\|\\nabla v(t)\\|_2^2,\n\\eqntag\n\\label{eqn;usualmodifiedfunctiona}\n\\end{align*}\nwhere\n\\begin{align*}\n\\L_m(t):=\\int_{\\R^2}(1+u)\\ln(1+u)\\dx-\\int_{\\R^2}uv\\dx+\\frac12\\|\\nabla v\\|_2^2+\\frac\\lambda2\\|v\\|_2^2\n\\end{align*}\nand\n\\begin{align*}\n\\D_m(t):=\\int_{\\R^2}u\\left|\\nabla(\\ln(1+u)-v)\\right|^2\\dx+\\int_{\\R^2}\\left|\\nabla\\left(\\ln(1+u)-\\frac12v\\right)\\right|^2\\dx+\\|\\partial_tv\\|_2^2.\n\\end{align*}\nThe functional $\\L_m(t)$ is no longer the Lyapunov functional, however, it is still useful for proving \nthe global existence of the solution to \\eqref{eqn;KS} with sub-critical mass. Indeed, it follows\nfrom the lower bound for $\\L_m(t)$ that\nfor any $\\al\\in[0,1)$\n\\begin{align*}\n\\al\\int_{\\R^2}(1+u)\\ln(1+u)\\dx+\\frac12\\left[1-\\frac{\\|u_0\\|_1}{8\\pi(1-\\al)}\\right]&\\|\\nabla v\\|_2^2\n+\\int_0^t \\D_m(s)\\,\\d s\n\\\\\n\\le\\,&\\L_m(0)+C(\\tau,\\alpha)+\\frac14\\int_0^t\\|\\nabla v\\|_2^2\\,\\d s\n\\eqntag\n\\label{eqn;afford}\n\\end{align*}\nfor $t\\in[0,\\tau]\\cap(0,T)$ with any $\\tau>0$ and some $C(\\tau,\\alpha)>0$ (cf.~\\cite{CaCo08,Mi13}). Hence, if $\\|u_0\\|<8\\pi$, then all terms on the left hand side of \\eqref{eqn;afford} are positive by choosing $\\al>0$ sufficiently small which depends on $\\|u_0\\|_1$, and Gronwall's inequality implies that\n$\\nabla v \\in L^2((0,T)\\times \\R^2)$, \nso that we also obtain the bound for the modified entropy $\\int_{\\R^2}(1+u)\\ln (1+u)\\dx$. \nNevertheless, for the critical mass $\\|u_0\\|_1=8\\pi$, this necessarily leads to $\\alpha=0$, as a result, \n\\begin{align*}\n\\int_0^t \\D_m(s)\\,\\d s\\le\\,&\\L_m(0)+C(\\tau)+\\frac14\\int_0^t\\|\\nabla v\\|_2^2\\,\\d s,\n\\eqntag\n\\label{eqn;lack-lyapunov}\n\\end{align*}\nwhich is useless and fails to obtain even the estimates for $\\D_m(t)$ due to the lack of the regularity of solutions.\nTherefore, we introduce a reconstructed Lyapunov functional $\\F_m(t)$ so as to show regularity estimates corresponding to the dissipative terms. Let $\\F_m(t)$ be the functional for solutions to \\eqref{eqn;KS} defined as\n\\begin{align*}\n\\F_m(t):=\\L_m(t)\n+\\int_{\\R^2}\\ln(1+u)\\,\\mathrm{d}x-\\int_{\\R^2}v\\,\\d x.\n\\eqntag\n\\label{eqn;modifiedLyapunov}\n\\end{align*}\nThen, the following functional differential inequality holds true:\n\\begin{prop}\\label{prop;energy-est}\nLet $(u,v)$ be the solution to \\eqref{eqn;KS}. Then, the functional $\\F_m(t)$ defined in~\\eqref{eqn;modifiedLyapunov} satisfies the following identity:\n \\begin{align*}\n \\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2\n \\le\\,&\n \\lambda\\int_{\\R^2}v\\,\\d x.\n \\end{align*}\n Therefore, if $\\lambda=0$ then\n \\begin{align*}\n \\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2 \\le\\,0.\n \\end{align*}\n If $\\lambda>0$ then \n \\begin{align*}\n \\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2 \\le\\,\\|v_0\\|_1+\\|u_0\\|_1.\n \\end{align*}\n\\end{prop}\n\\vspace{5mm}\nIt is worth emphasizing that $\\F_m(t)$ is non-increasing in time when $\\lambda=0$ as well as\nunlike the identity \\eqref{eqn;usualmodifiedfunctiona} based on the usual modified functional $\\mathcal{L}_m(t)$, the error term appearing on the right-hand side of Proposition~\\ref{prop;energy-est} can be easily controlled by initial data.\nThis allows us to show the regularity estimates for the dissipative term associated with $\\F_m(t)$, see Proposition~\\ref{prop;bound-energy} below.\n\\begin{lem}\\label{lem;ModifiedLF}\n Suppose assumptions as in Theorem \\ref{thm;global}. Then,\n\\begin{align*}\n\\frac{\\d}{\\d t}\\F_m(t)+ \\widetilde{\\D}(t)=-\\int_{\\R^2}\\partial_t v \\frac{u}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\mathrm{d}x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x,\n\\end{align*}\nwhere\n$\\F_m(t)$ is the modified functional defined in \\eqref{eqn;modifiedLyapunov} and the dissipative term $\\widetilde{\\D}(t)$ is given by\n\\begin{align*}\n\\widetilde{\\D}(t):=\\int_{\\R^2}u\\left|\\nabla \\left(\\ln(1+u)-v\\right)\\right|^2\\,\\d x+\\|\\partial_t v\\|_2^2.\n\\end{align*}\n\\end{lem}\n\n\\vspace{5mm}\n\\begin{pr}{Theorem \\ref{thm;global}}\nLet $R$ be taken as in Lemma \\ref{lem;L^2-exterior}.\nBy Lemma \\ref{lem;LlogL-bound-exterior},\n\\begin{align*}\n\\int_{ |x|>2R } (1+u) \\ln(1+u) \\dx\\le\\, C(t_0,\\tau,R).\n\\end{align*}\nOn the other hand, Lemma \\ref{prop;interior-bdd-entropy} gives\n\\begin{align*}\n\\int_{ \\R^2}(u\\ln u)\\psi_R^2\\dx\\le\\, C(t_0,\\tau,R)\n\\end{align*}\nfor $t\\in[t_0,\\tau]\\cap(0,T)$.\nNow, according to \\cite[Lemma 2.3]{NaOg16},\n\\begin{align*}\n\\int_{ \\Omega} (1+u)\\ln(1+u)\\dx\\le\\,2\\int_{ \\Omega} u |\\ln u | \\dx+(2\\ln 2)\\int_{ \\Omega} u\\dx,\n\\end{align*}\nwhere $\\Omega$ is a measurable set in $\\R^2$,\n so that since $\\supp\\psi_R\\subset \\{x; |x|\\le 16 R\\}$\n \\begin{align*}\n \\int_{ |x|<8R }(1+u)\\ln(1+u)\\dx\\le\\,&2\\int_{ |x|<8R }u|\\ln u| \\dx+(2\\ln 2)\\|u_0\\|_1\n \\\\\n \\le\\,\\,&2\\int_{ \\R^2 }u|\\ln u| \\psi_R^2\\dx+(2\\ln 2)\\|u_0\\|_1\n \\\\\n =\\,&2\\int_{ \\R^2 }(u\\ln u) \\psi_R^2\\dx-4\\int_{ \\R^2 }(u\\ln u)_-\\psi_R^2\\dx+(2\\ln 2)\\|u_0\\|_1\n \\\\\n \\le\\,&C(t_0,\\tau,R) +\\frac4e|B_{16R}(0)|\n \\end{align*}\n for $t\\in[t_0,\\tau]\\cap(0,T)$, where we use $x\\ln x \\ge -1/e$ for $x\\ge0$.\n This implies that\n \\begin{align*}\n&\\int_{ \\R^2 }(1+u)\\ln(1+u)\\dx\n\\\\=\\,&\\int_{ |x|>2R } (1+u) \\ln(1+u) \\dx\n+\\int_{ |x|\\le 2R } (1+u) \\ln(1+u) \\dx\n\\\\\n\\le\\,&\\int_{ |x|>2R } (1+u) \\ln(1+u) \\dx+\\int_{ |x|< 8R } (1+u) \\ln(1+u) \\dx\n\\\\\n\\le\\,&C(t_0,\\tau,R) \n \\end{align*}\n for $t\\in[t_0,\\tau]\\cap(0,T)$.\n Hence, we have along with Proposition \\ref{prop;bound-energy}\n \\begin{align*}\n\\int_{ \\R^2 }(1+u) \\ln (1+u)\\dx+\\int_{t_0}^t \\|\\partial_tv\\|_2^2\\,\\d s\\le\\, C(t_0,\\tau,R)\n \\eqntag\n \\label{eqn;uniform-ulogu}\n \\end{align*}\n for $t\\in[t_0,\\tau]\\cap(0,T)$.\nBy the parabolic regularity argument, we obtain a uniform $L^2$-bound for~$u$\non $[t_0,\\tau]\\cap(0,T)$ from \\eqref{eqn;uniform-ulogu} (cf. \\cite{Mi13,Na-Se-Yo}), which yields further regularity estimates.\nIn particular, the solution~$(u,v)$ cannot blow up in finite time.\nTherefore, the solution to \\eqref{eqn;KS} exists globally in time.\n\\end{pr}", "post_theorem_intro_text_len": 2221, "post_theorem_intro_text": "\\vspace{2mm}\n\n\\begin{rem}\nIn Theorem~\\ref{thm;global}, the initial data are only required to satisfy\n\\[\n(u_0,v_0)\\in L_+^1(\\mathbb R^2)\\times\\bigl(L_+^1(\\mathbb R^2)\\cap \\dot H^1(\\mathbb R^2)\\bigr)\\quad \\text{with}~~\\|u_0\\|_1=8\\pi.\n\\]\nIn particular, no additional symmetry or moment assumptions are imposed.\n\\end{rem}\n\\vspace{2mm}\nAs mentioned above, the global existence of solutions with critical mass\nwithout any symmetry or moment assumptions remains delicate.\nIn~\\cite[Theorem~1.2]{Mi13}, it is shown that, for general initial data with\ncritical mass, solutions either exist globally in time or blow up on the plane,\nby means of a contradiction argument.\nIndeed, although in the sub-critical case regularity estimates can be obtained\nby combining (modified) Lyapunov functionals with the Trudinger--Moser\ninequality, such estimates are no longer directly available in the critical case due to the lack of regularity of solutions,\nsee~\\eqref{eqn;afford} below.\nThis difficulty prevents the direct extension of classical entropy methods.\n\nTo overcome this difficulty, we introduce a reconstructed Lyapunov functional.\nThe main novelty of this work lies in the construction of a refined Lyapunov\nfunctional specifically adapted to the whole space setting, which allows us\nto control the behavior of solutions at $|x|\\to\\infty$ and to derive\nregularity estimates even in the critical mass regime, without\nimposing any symmetry or moment assumptions on the initial data.\nAs a consequence, we establish global-in-time existence for general initial\ndata at the critical mass, a result that was previously out of reach by existing\nmethods, see Subsection~\\ref{sect;modified-Lyapunov} for details.\nMoreover, the present approach is expected to be applicable to a broad class\nof chemotaxis  systems in the whole space setting.\n\n\\vspace{3mm}\nThe remainder of this paper is organized as follows.\nIn Section~\\ref{sect;preliminary}, we recall several preliminary lemmas needed\nto prove the main result.\nSection~\\ref{sect;LWP} is devoted to the local well-posedness of~\\eqref{eqn;KS}, based on the above definition of solutions.\nFinally, in Section~\\ref{sect;global}, we present the proof of\nTheorem~\\ref{thm;global}.\n\n\\vspace{5mm}", "sketch": "The post-theorem introduction does not give a step-by-step proof, but it outlines the strategy for proving Theorem~\\ref{thm;global}: classical entropy/Lyapunov approaches for sub-critical mass (using “(modified) Lyapunov functionals with the Trudinger--Moser inequality”) “are no longer directly available in the critical case due to the lack of regularity of solutions,” which “prevents the direct extension of classical entropy methods.” To overcome this, the authors “introduce a reconstructed Lyapunov functional,” whose “main novelty…lies in the construction of a refined Lyapunov functional specifically adapted to the whole space setting,” allowing them “to control the behavior of solutions at $|x|\\to\\infty$ and to derive regularity estimates even in the critical mass regime, without imposing any symmetry or moment assumptions on the initial data.” With these estimates, they “establish global-in-time existence for general initial data at the critical mass.” The paper’s structure supporting the proof is: preliminaries (Section~\\ref{sect;preliminary}), local well-posedness (Section~\\ref{sect;LWP}), then the global argument proving Theorem~\\ref{thm;global} (Section~\\ref{sect;global}).", "expanded_sketch": "The post-theorem introduction does not give a step-by-step proof, but it outlines the strategy for proving the main theorem: classical entropy/Lyapunov approaches for sub-critical mass (using “(modified) Lyapunov functionals with the Trudinger--Moser inequality”) “are no longer directly available in the critical case due to the lack of regularity of solutions,” which “prevents the direct extension of classical entropy methods.” To overcome this, the authors “introduce a reconstructed Lyapunov functional,” whose “main novelty…lies in the construction of a refined Lyapunov functional specifically adapted to the whole space setting,” allowing them “to control the behavior of solutions at $|x|\\to\\infty$ and to derive regularity estimates even in the critical mass regime, without imposing any symmetry or moment assumptions on the initial data.” With these estimates, they “establish global-in-time existence for general initial data at the critical mass.” The paper’s structure supporting the proof is: preliminaries (proved next), local well-posedness (proved after that), then the global argument establishing the main theorem (proved later).", "expanded_theorem": "\\label{thm;global}\nFor $(u_0,v_0)\\in L_+^1(\\mathbb R^2) \\times L^1_+(\\mathbb R^2)\\cap \\dot{H}^1(\\mathbb R^2)$,\nlet $(u,v)$ be the solution to \n\\begin{aligned}\n&\\pt_t u =\\Delta u- \\N\\cd \\left( u \\N v \\right),\n& t>0,\\,~ &x\\in\\R^2,\n\\\\\n&\\pt_t v = \\Delta v - \\lambda v +u,\n&t>0,\\, ~&x\\in\\R^2,\n\\\\\n&(u,v)(0,x)=(u_{0},v_0)(x),\n&\\, &x\\in\\R^2\n\\end{aligned}\n\\right.\n\\label{eqn;KS}\n\\end{equation}\nwith a constant $\\lambda\\ge0$, where $u_0,v_0\\ge 0$ on $\\R^2$ and\n$u_0,v_0\\not\\equiv0$\non $(0,T)\\times\\mathbb R^2$.\nSuppose that $\\|u_0\\|_1=8\\pi$.\nThen, in establishing the main theorem, the solution to the system above exists globally in time.", "theorem_type": ["Implication", "Existence"], "mcq": {"question": "Consider the two-dimensional fully parabolic Keller--Segel system\n\\[\n\\begin{cases}\n\\partial_t u = \\Delta u-\\nabla\\!\\cdot\\!\\big(u\\nabla v\\big), & t>0,\\ x\\in\\mathbb R^2,\\\\\n\\partial_t v = \\Delta v-\\lambda v+u, & t>0,\\ x\\in\\mathbb R^2,\\\\\n(u,v)(0,x)=(u_0,v_0)(x), & x\\in\\mathbb R^2,\n\\end{cases}\n\\]\nwhere \\(\\lambda\\ge 0\\). Assume\n\\(u_0\\in L_+^1(\\mathbb R^2)\\) and \\(v_0\\in L_+^1(\\mathbb R^2)\\cap \\dot H^1(\\mathbb R^2)\\), with\n\\[\nL_+^1(\\mathbb R^2):=\\{f\\in L^1(\\mathbb R^2): f\\ge 0\\ \\text{and}\\ f\\not\\equiv 0\\},\n\\]\nso in particular \\(u_0,v_0\\) are nonnegative and nontrivial, and suppose that the initial cell mass satisfies\n\\[\n\\|u_0\\|_{L^1(\\mathbb R^2)}=8\\pi.\n\\]\nWhich of the following conclusions about the corresponding solution \\((u,v)\\) holds?", "correct_choice": {"label": "A", "text": "The solution \\((u,v)\\) exists globally in time; equivalently, it can be continued for all \\(t\\ge 0\\) and does not blow up in finite time."}, "choices": [{"label": "B", "text": "The solution \\((u,v)\\) exists globally in time provided, in addition, that the initial data are either radially symmetric or satisfy a finite logarithmic moment condition such as \\(u_0\\ln(1+|x|^2)\\in L^1(\\mathbb R^2)\\); without one of these extra assumptions, finite-time blow-up may occur at mass \\(8\\pi\\)."}, {"label": "C", "text": "The corresponding solution \\((u,v)\\) has a local-in-time mild solution on some interval \\([0,T)\\) for a positive maximal existence time \\(T>0\\)."}, {"label": "D", "text": "For every such initial datum with \\(\\|u_0\\|_{L^1(\\mathbb R^2)}=8\\pi\\), the solution \\((u,v)\\) exists globally in time and moreover remains uniformly bounded in \\(L^\\infty(\\mathbb R^2)\\times \\dot W^{1,\\infty}(\\mathbb R^2)\\) for all \\(t\\ge 0\\)."}, {"label": "E", "text": "There exists a time \\(T=T(u_0,v_0)>0\\) such that the solution \\((u,v)\\) exists on \\([0,T)\\), and if it is global then this conclusion requires constants in the a priori estimates to depend on additional decay of the data at spatial infinity; at critical mass \\(8\\pi\\), global existence is therefore not guaranteed for arbitrary \\(u_0\\in L_+^1(\\mathbb R^2)\\) and \\(v_0\\in L_+^1(\\mathbb R^2)\\cap \\dot H^1(\\mathbb R^2)\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "removal_of_symmetry_or_moment_hypotheses", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped_global_in_time_and_no_blowup_conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "upgrade_from_global_existence_to_uniform_Linfty_control", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "whole_space_tail_control_for_arbitrary_data", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct conclusion. It states the PDE, hypotheses, and critical-mass condition, but the answer still depends on knowing or reasoning about the exact existence theorem."}, "TAS": {"score": 1, "justification": "The item is close to theorem recall: it essentially asks which existence conclusion matches the stated hypotheses. However, it is not a pure restatement because the options include weaker, stronger, and conditionally modified alternatives."}, "GPS": {"score": 1, "justification": "The question requires some discrimination among nearby claims (local vs global existence, extra decay assumptions, stronger boundedness conclusions), but it mainly tests recognition of the correct theorem rather than substantial generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic confusions: requiring extra symmetry/decay assumptions, settling for only local existence, or overstrengthening the conclusion to uniform L^infty bounds."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with good distractors and no major answer leakage, though it leans more toward recall of a specific result than deep generative reasoning."}}
{"id": "2602.03768v1", "paper_link": "http://arxiv.org/abs/2602.03768v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{thm;global}\nFor $(u_0,v_0)\\in L_+^1(\\mathbb R^2) \\times L^1_+(\\mathbb R^2)\\cap \\dot{H}^1(\\mathbb R^2)$,\nlet $(u,v)$ be the solution to \\eqref{eqn;KS} on $(0,T)\\times\\mathbb R^2$.\nSuppose that $\\|u_0\\|_1=8\\pi$.\nThen, the solution to \\eqref{eqn;KS} exists globally in time.", "start_pos": 14650, "end_pos": 14921, "label": "thm;global"}, "ref_dict": {"thm;global": "\\begin{thm}\\label{thm;global}\nFor $(u_0,v_0)\\in L_+^1(\\R^2) \\times L^1_+(\\R^2)\\cap \\dot{H}^1(\\R^2)$,\nlet $(u,v)$ be the solution to \\eqref{eqn;KS} on $(0,T)\\times\\R^2$.\nSuppose that $\\|u_0\\|_1=8\\pi$.\nThen, the solution to \\eqref{eqn;KS} exists globally in time.\n\\end{thm}", "eqn;KS": "\\begin{aligned}\n&\\pt_t u =\\Delta u- \\N\\cd \\left( u \\N v \\right),\n& t>0,\\,~ &x\\in\\R^2,\n\\\\\n&\\pt_t v = \\Delta v - \\lambda v +u,\n&t>0,\\, ~&x\\in\\R^2,\n\\\\\n&(u,v)(0,x)=(u_{0},v_0)(x),\n&\\, &x\\in\\R^2\n\\end{aligned}\n\\right.\n\\label{eqn;KS}\n\\end{equation}\nwith a constant $\\lambda\\ge0$, where $u_0,v_0\\ge 0$ on $\\R^2$ and\n$u_0,v_0\\not\\equiv0$.\nThe Keller--Segel system~\\eqref{eqn;KS} is a fundamental mathematical model\nof chemotaxis, describing chemotactic aggregation in the\ncellular slime mold {\\it Dictyostelium discoideum} during its life cycle \\cite{KeSe,Pa}.\nIn this model, $u=u(t,x)$ and $v=v(t,x)$ denote the densities of cells and\nthe chemoattractant, respectively.\nThe parameter $\\lambda$ represents the degradation rate of the chemical.\nCells migrate toward regions of higher concentrations of a chemical substance\nsecreted by the cells themselves.\nFrom a biological viewpoint, it is natural to assume that the initial data\nare nonnegative functions.\nFrom a mathematical viewpoint, cell aggregation is interpreted as the\nblowup of solutions at $t=T$ in the sense that\n$\\lim_{t\\to T}\\|u(t)\\|_\\infty=\\infty.$\n\nOne of the central mathematical features of such systems is the phenomenon of critical\nmass.\nFor positive sufficiently regular solutions $(u,v)$ to~\\eqref{eqn;KS}, the first\ncomponent $u$ satisfies the mass conservation law \n$$\\|u(t)\\|_1=\\|u_0\\|_1\\quad \\text{for}~~~t>0.$$\nMoreover, in two space dimensions, the global behavior of solutions is governed\nby the size of the initial mass of $u$.\nIn particular, there exists a threshold value such that solutions with initial\nmass below this threshold exist globally in time, whereas solutions with\ninitial mass above the threshold may blow up in finite time.\nThis critical-mass phenomenon has attracted considerable attention in the\nliterature.\n\nThe aim of this paper is to establish the global behavior of solutions to\n\\eqref{eqn;KS} whose initial mass is exactly equal to this threshold value.\n\n\\vspace{3mm}\nThe second equation in \\eqref{eqn;KS} takes into account that cells are producing the chemoattractant themselves \nwhile this is diffusing into the environment. \nSince the chemoattractant attains its equilibrium on a time scale much quicker\nthan that of the cells, the simplified parabolic-elliptic system has also been\ninvestigated \\cite{JaLu,Na95}:\n\\begin{align*}\n\\left\\{\n\\begin{aligned}\n\\partial_t u &= \\Delta u - \\nabla\\cdot (u\\nabla v),\n& t>0,\\ & x\\in\\R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\R^2.\n\\end{aligned}", "prop;LWP": "\\begin{prop}[local-in-time solution]\\label{prop;LWP}\n\tLet $4/3<p<2$ and $q=p/(p-1)$.\n\tFor  $(u_0,v_0)\\in L^1(\\R^2)\\times  \\dot{H}^{1}(\\R^2)$,\n\tthere exist $T\\in (0,\\infty]$ and a unique mild solution $(u,v)$ to~\\eqref{eqn;KS} on $[0,T)$ in the sense of the definition.\n\tBesides, the solution $(u,v)$ has higher regularity on $(0,T)$ and satisfies the system~\\eqref{eqn;KS} in a classical sense on $(0,T)\\times\\R^2$.\n\tIn addition,\n\t\\begin{equation*}\n\t\\text{either $T=\\infty$ or $T<\\infty$ and } \\lim_{t\\to T} \\left\\{ \\|u(t)\\|_{r} + \\|\\nabla v(t)\\|_{r+1}\\right\\} = \\infty,~~ 1<r\\le \\infty. \n\t\\eqntag\n\t\\label{eqn;bucrit}\n\t\\end{equation*}\t\n\tMoreover, if $u_0,v_0 \\ge 0$ then $u,v> 0$ in $(0,T)\\times\\R^2$ and\n\t\\begin{equation*}\n\t\\|u(t)\\|_1=\\|u_0\\|_1, \\qquad t\\in [0,T).\t\n\t\\end{equation*} \n\\end{prop}", "eqn;PKS": "\\begin{aligned}\n\\partial_t u &= \\Delta u - \\nabla\\cdot (u\\nabla v),\n& t>0,\\ & x\\in\\R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\R^2.\n\\end{aligned}\n\\right.\n\\eqntag\n\\label{eqn;PKS}\n\\end{align*}\nThe system \\eqref{eqn;PKS} is also related to models of gravitational interaction\nof particles \\cite{Bi-Na94,Wo}.\nIn~\\eqref{eqn;PKS}, the second equation can be written as\n$v = (-\\Delta+\\lambda)^{-1}u.$\nAs a consequence, the system~\\eqref{eqn;PKS} can be reduced to a single equation\nfor~$u$.\nOwing to this elliptic structure, the parabolic-elliptic system~\\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.\nIn particular, numerous works have been devoted to the critical-mass\nphenomenon.\nIndeed, the solution with $\\|u_0\\|_1\\le 8\\pi$ exists globally in time~\\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},\nwhile the solution may blow up in finite time if $\\|u_0\\|_1>8\\pi$~\\cite{DoPe,Ko-Su,Ku-Og03,Wei}.\nFor the corresponding Cauchy--Neumann problem in bounded domains,\nsupplemented with homogeneous Neumann boundary conditions, see \\cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.\n\n\\vspace{3mm}\nUnlike the parabolic-elliptic system \\eqref{eqn;PKS}, the fully parabolic system\n\\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches\ndeveloped for \\eqref{eqn;PKS} are no longer applicable.\nThe global existence of solutions with sub-critical mass\n$\\|u_0\\|_{1}<8\\pi$ has nevertheless been established by combining\nLyapunov functionals with the Trudinger--Moser type inequality and its optimal\nconstant.\nMore precisely, the following results are known:\n\\begin{enumerate}\n\t\\item[(i)]\n\tIf $\\|u_0\\|_{1}<8\\pi$, then the corresponding solution to\n\t\\eqref{eqn;KS} exists globally in time \\cite{CaCo08,Mi13,NaOg11}.\n\n\t\\item[(ii)]\n\tIf $\\|u_0\\|_{1}>8\\pi$ and $(u_0,v_0)$ is radially symmetric, then there\n\texists a radially symmetric solution that blows up in finite time\n\t\\cite{Mi20,Mi20-SIAM}.\n\n\t\\item[(iii)]\n\tIf $\\|u_0\\|_{1}=8\\pi$ and $(u_0,v_0)$ is either radially symmetric or\n\tsatisfies the additional moment condition\n\t$u_0\\ln(1+|x|^2)\\in L^1(\\R^2)$, then the corresponding solution exists globally\n\tin time.\n\tIn contrast, for general initial data, the global behavior in the critical case\n\tremains delicate, and the solution either exists globally or blows up on the plane \\cite{CaCo08,Mi13}.\n\\end{enumerate}\nAs for positive forward self-similar solutions to \\eqref{eqn;KS}, refer to \\cite{BiCoDo}.\nFor the corresponding Cauchy--Neumann problem in bounded domains, we refer to\n\\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.\n\nAs mentioned above, although several partial results are available, no complete global existence result has been obtained\nfor solutions with critical mass and general initial data without any\nsymmetry or moment assumptions.\nOne of the main difficulties lies in controlling the behavior of solutions at~$|x|\\to\\infty$.\nMoreover, in the critical mass case, no global existence result is currently\nknown  for the corresponding Cauchy--Neumann problem to~\\eqref{eqn;KS} in bounded domains.\n\nIn this paper, we establish the global existence of solutions with critical mass\nfor general initial data $(u_0,v_0)$.\nTo this end, we first recall the definition of solutions.\n\n\\vspace{2mm}\n\\begin{def*}\nLet $4/3<p<2$ and $q=p/(p-1)$.\nGiven $T\\in (0,\\infty]$ and  $(u_0,v_0) \\in L^1(\\R^2)\\times \\dot{H}^{1}(\\R^2)$,\nwe say that a pair of functions $(u,v)$ defined on $[0,T)\\times \\R^2$ is a mild solution to~\\eqref{eqn;KS} on $[0,T)$ if\n\\begin{enumerate}\n\t\\item $u\\in C\\big([0,T); L^1(\\R^2)\\big)\\cap C\\left( (0,T); L^p(\\mathbb{R}^2) \\right)$;\n\t\\item $v\\in C\\big([0,T); \\dot{H}^{1}(\\R^2)\\big)\\cap C\\left( (0,T); \\dot{W}^{1,q}(\\mathbb{R}^2) \\right)$;\n\t\\item $\\dsp\\sup_{0<t<T} t^{1-\\frac1p}\\|u(t)\\|_{p}+\\dsp\\sup_{0<t<T}t^{\\frac12-\\frac1q}\\|\\nabla v(t)\\|_{q}$;\n\t\\item $(u,v)$ satisfies the integral formulations\n\t\\begin{align*}\n\t&u(t)=e^{t\\Del}u_0-\\int_0^t \\N\\cd e^{(t-s)\\Del} \\big(u(s)\\N v(s)\\big) \\,\\d s,\n\t\\\\\n\t&v(t)= e^{t(\\Del-\\lm)} v_0+\\int_0^t e^{(t-s)(\\Del-\\lm)} u(s) \\,\\d s\n\t\\end{align*}\n\tfor $0<t<T$, where $e^{t\\Del}$ is the heat semigroup given by\n\t\\begin{align*}\n\te^{t\\Del} f(x) := G_t*f(x)=\\int_{ \\R^2}G_t(x-y) f(y)\\,\\d y,\n\t\\quad\n\tG_t(x)=\\frac{1}{4\\pi t} \\exp\\left(-\\frac{|x|^2}{4 t}\\right)\n\t\\end{align*}\n\tfor $x\\in\\R^2$ and $f \\in L^p(\\R^2)$.\n\\end{enumerate}\n\\end{def*}\nThe Lebesgue spaces $L^p(\\R^2)$ are equipped with the norm $\\|\\cdot\\|_{L^p(\\R^2)}$,\nwhich is abbreviated as $\\|\\cdot\\|_p$ when there is no ambiguity.\nWe define the homogeneous Sobolev spaces by\n\\[\n\\dot{W}^{1,p}(\\R^2)\n:= \\overline{C_0^\\infty(\\R^2)}^{\\|\\nabla (\\cdot)\\|_{L^p(\\R^2)}}\n\\]\nand $\\dot{H}^1(\\R^2):=\\dot{W}^{1,2}(\\R^2)$ when $p=2$.\n$L_+^1(\\R^2):=\\{f \\in L^1(\\R^2); f\\ge0~~ \\text{and}~~ f\\not\\equiv0~~ \\text{on $\\R^2$} \\}$.\n\n\\vspace{5mm}\nThe starting point for our analysis is the local well-posedness for \\eqref{eqn;KS} and the solution~$(u,v)$ based on the above definition satisfies~\\eqref{eqn;KS} in a classical sense on $(0,T)$, see Proposition~\\ref{prop;LWP} in Section~\\ref{sect;LWP} below.\n\nWe are now in a position to state our main result in this paper.\nMain result in this paper reads as follows.\n\\vspace{2mm}\n\\begin{thm}\\label{thm;global}\nFor $(u_0,v_0)\\in L_+^1(\\R^2) \\times L^1_+(\\R^2)\\cap \\dot{H}^1(\\R^2)$,\nlet $(u,v)$ be the solution to \\eqref{eqn;KS} on $(0,T)\\times\\R^2$.\nSuppose that $\\|u_0\\|_1=8\\pi$.\nThen, the solution to \\eqref{eqn;KS} exists globally in time.\n\\end{thm}\n\\vspace{2mm}\n\n\\begin{rem}\nIn Theorem~\\ref{thm;global}, the initial data are only required to satisfy\n\\[\n(u_0,v_0)\\in L_+^1(\\R^2)\\times\\bigl(L_+^1(\\R^2)\\cap \\dot H^1(\\R^2)\\bigr)\\quad \\text{with}~~\\|u_0\\|_1=8\\pi.\n\\]\nIn particular, no additional symmetry or moment assumptions are imposed.\n\\end{rem}\n\\vspace{2mm}\nAs mentioned above, the global existence of solutions with critical mass\nwithout any symmetry or moment assumptions remains delicate.\nIn~\\cite[Theorem~1.2]{Mi13}, it is shown that, for general initial data with\ncritical mass, solutions either exist globally in time or blow up on the plane,\nby means of a contradiction argument.\nIndeed, although in the sub-critical case regularity estimates can be obtained\nby combining (modified) Lyapunov functionals with the Trudinger--Moser\ninequality, such estimates are no longer directly available in the critical case due to the lack of regularity of solutions,\nsee~\\eqref{eqn;afford} below.\nThis difficulty prevents the direct extension of classical entropy methods.\n\nTo overcome this difficulty, we introduce a reconstructed Lyapunov functional.\nThe main novelty of this work lies in the construction of a refined Lyapunov\nfunctional specifically adapted to the whole space setting, which allows us\nto control the behavior of solutions at $|x|\\to\\infty$ and to derive\nregularity estimates even in the critical mass regime, without\nimposing any symmetry or moment assumptions on the initial data.\nAs a consequence, we establish global-in-time existence for general initial\ndata at the critical mass, a result that was previously out of reach by existing\nmethods, see Subsection~\\ref{sect;modified-Lyapunov} for details.\nMoreover, the present approach is expected to be applicable to a broad class\nof chemotaxis  systems in the whole space setting.\n\n\\vspace{3mm}\nThe remainder of this paper is organized as follows.\nIn Section~\\ref{sect;preliminary}, we recall several preliminary lemmas needed\nto prove the main result.\nSection~\\ref{sect;LWP} is devoted to the local well-posedness of~\\eqref{eqn;KS}, based on the above definition of solutions.\nFinally, in Section~\\ref{sect;global}, we present the proof of\nTheorem~\\ref{thm;global}.\n\n\\vspace{5mm}\n\\section{Preliminary}\\label{sect;preliminary}\nIn this section, we collect some tools to show our main result.\n\\begin{lem}\\label{lem;L2andL3}\n\tFor any $f\\in H^1(\\R^2)$ and nonnegative cut-off function $\\phi\\in C^\\infty_0(\\R^2)$,\n\t\\begin{align*}\n\t\\int_{\\R^2}|f|^2\\phi\\dx\\le\\,&2\\left(\\int_{ \\{|f|>1\\}\\cap \\{\\supp\\phi\\} }|f|\\dx\\right)\\left(\\int_{ \\{|f|>1\\} }\\frac{|\\nabla f|^2}{1+|f|} \\phi\\dx\\right)\n\t\\\\\n\t&+4\\left(\\int_{\\R^2}|f \\nabla \\phi^{\\frac12}|\\dx\\right)^2+4\\left(\\int_{\\R^2}|f|\\phi\\dx\\right)\n\t\\end{align*}\n\tand for any $\\ep>0$\n\t\\begin{align*}\n\t\\int_{\\R^2}|f|^3\\phi\\dx\n\t\\le\\,&\\ep \\left(\\int_{  \\{\\supp\\phi\\} }(1+|f|)\\ln(1+|f|)\\dx\\right)\\left(\\int_{\\R^2 }|\\nabla f|^2\\phi\\dx\\right)\n\t\\\\\n\t&+C\\left(\\int_{\\R^2}|f^{\\frac32} \\nabla \\phi^{\\frac12}|\\dx\\right)^2+C_\\ep\\left(\\int_{\\R^2}|f|\\phi\\dx\\right),\n\t\\end{align*}\n\twhere the constant $C_\\ep\\to \\infty$ as $\\ep\\to0$.\n\\end{lem}\nThe proof of Lemma \\ref{lem;L2andL3} can be found in \\cite[Lemma 2.2]{NaOg16}.\n\n\\begin{lem}\\label{lem;Lp_Lq_heat}\n\tFor $1 \\le q \\le p \\le \\infty$,\n\tlet $f\\in L^q (\\R^n)$ and let $\\al$ be a multi-index.\n\tThen it follows that\n\t\\begin{equation*}\n\t\\| \\pt_x^{\\al} e^{t\\Del} f \\|_p  \n\t\\le C t^{-\\frac{n}{2} \\left(\\frac{1}{q} - \\frac{1}{p} \\right) - \\frac{|\\al|}{2}}   \\|f\\|_q\n\t\\end{equation*}\n\tfor all $t>0$.\n\\end{lem}\n\nThe proof is immediately obtained by use of Young's inequality and the \nconvolution expression of the heat evolution\nby the heat kernel, see for instance \\cite{GiGiSa}.\n\\begin{lem}\\label{lem;v-Lpbound}\nFor $v_0\\in L^1(\\R^2)\\cap \\dot H^1(\\R^2)$ and $f\\in L^\\infty \\left(0,\\infty;  L^1(\\R^2) \\right)$,\nlet $v$ be the solution to $\\partial_t v =\\Delta v -\\lambda v +f$  on $(0,\\infty)\\times\\R^2$ with the initial data $v_0$ and $\\lambda\\ge0$. Then, for any $1\\le p<\\infty$, \n\\begin{align*}\n\\| v(t)\\|_p\n\\le\\,&\n\\left\\{\n\\begin{aligned}\n&\\|v_0\\|_p+\\frac1p\\|f\\|_{L^\\infty\\left(0,\\infty; L^1(\\R^2)\\right)}  t^{\\frac1p} &\\text{if $\\lambda=0$},\n\\\\\n&\\|v_0\\|_p+\\lambda^{-\\frac1p} \\|f\\|_{L^\\infty\\left(0,\\infty; L^1(\\R^2)\\right)}  \\Gamma\\left(\\frac1p\\right)&\\text{if $\\lambda>0$}.\n\\end{aligned}", "eqn;afford": "\\begin{pr}{Proposition \\ref{prop;LWP}}\nBy virtue of Lemmas~\\ref{lem;fixed-point} and~\\ref{lem;continuous-depend},\nwe have established the existence of a unique mild solution to~\\eqref{eqn;KS} defined on a maximal time interval $[0,T)$ with~$T\\le\\infty$. As for the regularity of the solutions, we use the standard iteration argument with respect to the derivative. Define $|\\N|^\\al f(x):=\\F^{-1}[ |\\xi|^\\al \\F f(\\xi)\\,](x)$  for $x\\in\\R^2$ and $\\al>0$, where $\\F$ denotes the Fourier transform. Let $\\tau\\in (0,T)$ and $t\\in (0,\\tau)$. Recalling that, for $4/3<p<2$,\n\\begin{equation*}\nU_{p,0}(\\tau) := \\sup_{0< s<\\tau}s^{1-\\frac1p}\\|u(s)\\|_{p}<+\\infty, \\quad V_{q,0}(\\tau) := \\sup_{0< s<\\tau}s^{\\frac12-\\frac1q}\\|\\nabla v(s)\\|_{q}<+\\infty,\n\\end{equation*}\nby Lemma~\\ref{lem;fixed-point}, we may argue as in the proof of~\\eqref{eqn;L-4/3} to obtain\n\\begin{align*}\n&t^{1-\\frac1p+\\frac{\\al}{2}} \\| |\\N|^{\\al} u(t) \\|_{p}\n\\\\\n\\le\\,& t^{1-\\frac1p+\\frac{\\al}{2}} \\| |\\N|^\\al e^{t\\Del}u_0\\|_{p} + C U_{p,0}(\\tau) V_{q,0}(\\tau) t^{1-\\frac1p+\\frac{\\al}{2}}  \\int_0^t  (t-s)^{-1+\\frac1{p}-\\frac{1+\\al}2} s^{-1+\\frac{1}p} s^{-\\frac12+\\frac1q}\\,\\d s\n\\\\\n\\le\\,&C\\|u_0\\|_1 + C  U_{p,0}(\\tau) V_{q,0}(\\tau) \\mathsf{B}\\left(\\frac1p-\\frac{1+\\al}2,\\frac12\\right),\n\\end{align*}\nwhere $4/3<p<2$ and\n\\begin{align*}\n\\al <\\frac2p-1<\\frac12,\n\\end{align*}\nwhich implies that\n\\begin{align*}\nU_{p.\\al}(\\tau):=\\sup_{0< t<\\tau} t^{1-\\frac1p+\\frac{\\al}{2}} \\| |\\N|^{\\al} u(t)\\|_{p} <+\\infty,\n\\quad\n\\al<\\frac2p-1.\n\\end{align*}\nThis property yields additional regularity estimates for $v$. Indeed, for $\\bt\\ge \\al$, \n\\begin{align*}\n\\| |\\N|^{1+\\beta} v(t) \\|_{q}\n\\le\\,&C t^{-\\frac12+\\frac1q-\\frac\\bt2} \\| \\nabla v_0\\|_{2} + C U_{p,\\al}(\\tau) \\int_0^t (t-s)^{-\\left(\\frac1p-\\frac1q\\right)-\\frac{1+\\bt-\\al}{2} } s^{-1+\\frac1p-\\frac\\al2}\\,\\d s \n\\\\\n=\\,&C  t^{-\\frac12+\\frac1q-\\frac\\bt2} \\| \\nabla v_0\\|_{2} \n+C U_{p,\\al}(\\tau)  t^{-\\frac12+\\frac1q-\\frac\\bt2} \\mathsf{B}\\left(\\frac12-\\frac1p+\\frac1q-\\frac{\\bt-\\al}{2}, \\frac1p-\\frac{\\al}{2}\\right),\n\\end{align*}\nprovided that\n\\begin{align*}\n\\al\\le \\bt<\\al+3-\\frac4p<2-\\frac2p<1,\n\\end{align*}\nfrom which it follows that \n\\begin{align*}\nV_{q,\\bt}(\\tau) := \\sup_{0< t<\\tau} t^{\\frac12-\\frac1q+\\frac\\bt2} \\| |\\N|^{1+\\bt} v(t) \\|_{q}<+\\infty,\n\\quad  \\bt<\\al+3-\\frac4p\\in (\\al,1).\n\\end{align*}\nThe previous analysis leads us to a higher regularity estimate for $u$. Indeed, let $\\th\\ge\\al$. Since $\\al\\le\\bt\\le\\gm$,\n\\begin{align*}\n& \\| |\\N|^{\\th}u(t)\\|_{p} \\\\\n\\le\\,&C t^{-\\left(1-\\frac1p\\right)-\\frac\\th2}\\|u_0\\|_1\n\\\\\n&+C\\int_0^t(t-s)^{-\\left(1-\\frac1p\\right)-\\frac{1+\\th-\\al}{2}} \\left(\\||\\N|^{\\al} u(s)\\|_{p}\\|\\nabla v(s)\\|_q+\\|u(s)\\|_{p}\\| |\\N|^{1+\\al} v(s)\\|_{q}\\right)\\,\\d s\n\\\\\n\\le\\,&Ct^{-\\left(1-\\frac1p\\right)-\\frac\\th2}\\|u_0\\|_1\n\\\\\n&+ C \\big(U_{p,\\alpha}(\\tau) V_{q,0}(\\tau) + U_{p,0}(\\tau) V_{q,\\al}(\\tau)\\big) \n\\int_0^t(t-s)^{-\\left(1-\\frac1p\\right)-\\frac{1+\\th-\\al}{2}} s^{-\\frac32+\\frac1p+\\frac1q-\\frac{\\al}{2}}\\,\\d s\n\\\\\n=\\,&Ct^{-\\left(1-\\frac1p\\right)-\\frac\\th2}\\|u_0\\|_1\n\\\\\n& + C \\big(U_{p,\\al}(\\tau) V_{q,0}(\\tau) + U_{p,\\al}(\\tau) V_{q,0}(\\tau)\\big) t^{-\\left(1-\\frac1p\\right)-\\frac\\th2} \\mathsf{B}\\left(\\frac1p-\\frac{1+\\th-\\al}{2},\\frac12-\\frac\\al2\\right),\n\\end{align*}\nwhere\n\\begin{align*}\n\\al \\le \\th<\\al+\\frac2p-1<1,\\quad\n\\al<\\frac2p-1.\n\\end{align*}\nHence, \n\\begin{align*}\nU_{p,\\th}(\\tau) = \\sup_{0< t< \\tau}t^{1-\\frac1p+\\frac{\\th}{2}}\\||\\N|^\\th u(t)\\|_{p}<+\\infty,\n\\quad\n\\al \\le \\th<\\al+\\frac2p-1<1,\\quad\n\\al<\\frac2p-1.\n\\end{align*}\nIterating further the above arguments, we obtain that the solution possesses higher regularity. In fact, using the uniqueness result established in Lemma~\\ref{lem;continuous-depend}, we may solve the problem~\\eqref{eqn;KS} with the initial data $(u(t_0),v(t_0))$ for some $t_0\\in (0,T)$. We thus eventually show that for $s>0$, $u(t)\\in L^1(\\R^2)\\cap W^{s,p}(\\R^2)$,\n$v(t)\\in  \\dot{H}^{1}(\\R^2)\\cap W^{s,q}(\\R^2)$\nfor $t_0<t<T$, \nand satisfies the system~\\eqref{eqn;KS} in a  classical sense on $(0,T)\\times\\R^2$.\nAlso, classical arguments based on the proof of Lemma~\\ref{lem;fixed-point} lead to the blowup criterion~\\eqref{eqn;bucrit}.\n\nLet  $(u_0,v_0)$ be nonnegative initial conditions with $u_0,v_0\\not\\equiv0$ and\n$\\{u_{0,k}\\}_{k\\in\\Nt }$ and\n$\\{v_{0,k}\\}_{k\\in\\Nt }$ be sequences\nof nonnegative functions in $C_0^{\\infty}(\\R^2)$ converging to $u_0$ in $L^{1}(\\R^2)$ and $v_0$ in $\\dot{H}^{1}(\\R^2)$  as $k\\to\\infty$, respectively. Then, for each $k\\in\\Nt$, we can construct the unique, smooth and integrable solution\nto~\\eqref{eqn;KS} corresponding to the initial data $(u_{0,k},v_{0,k})$.\nBy the parabolic regularity theory,\nthe solutions $(u_k,v_k)$ belong to $C^{\\infty}( [0,T)\\times \\R^2)$\nand are positive on $(0,T)\\times\\R^2$. In addition, by the continuous dependence on the initial data established in Lemma~\\ref{lem;continuous-depend}, we have the convergences $u_k\\to u$ in $C\\big([0,T); L^1(\\R^2)\\big)$ and $v_k\\to v$ in $C\\big([0,T); \\dot{H}^{1}(\\R^2)\\big)$ as $ k\\to \\infty$,\nwhere $(u,v)$ is the solution to~\\eqref{eqn;KS} with initial data $(u_0,v_0)$. Hence, one can find subsequences of the approximated solutions\nwhich converge to $(u,v)$ almost everywhere in $(0,T)\\times\\R^2$, which implies that $u$ and $v$ are positive on $(0,T)\\times\\R^2$.\nMoreover the mass conservation law also holds by integrating $u$ over $\\R^2$.\n\\end{pr}\n\n\\vspace{5mm}\n\\section{Global existence of solutions with critical mass}\\label{sect;global}\n\nIn this section, we prove the global existence of solutions to \\eqref{eqn;KS}\nwith critical mass, as stated in Theorem~\\ref{thm;global}.\nTo this end, we divide the proof into several steps.\n\nIn Subsection~\\ref{sect;modified-Lyapunov}, we introduce a suitably reconstructed\nLyapunov functional and derive regularity estimates for the associated\ndissipative terms.\nIn Subsection~\\ref{sect;exterior}, we establish various regularity estimates\nin exterior domains by using the estimates obtained in\nSubsection~\\ref{sect;modified-Lyapunov}.\nThanks to the results in the previous subsections, we then derive interior\nregularity estimates in Subsection~\\ref{sect;interior}.\nFinally, in Subsection~\\ref{sect;main}, we complete the proof of\nTheorem~\\ref{thm;global} by combining the exterior and interior estimates.\n\nThroughout this section, we denote by $(u,v)$ the solution to~\\eqref{eqn;KS}\nwith maximal existence time $T\\le\\infty$.\nIn the following, we derive a priori estimates on $[t_0,\\tau]\\cap(0,T)$ for arbitrary $t_0,\\tau$ with $t_0<\\tau$, which will allow us to conclude global existence by a contradiction argument.\nMoreover, $C$ denotes a generic positive constant depending only on $\\lambda$\nand the initial data~$(u_0,v_0)$, which may vary from line to line.\nThe dependence of $C$ on additional parameters will be indicated explicitly.\n\n\\subsection{Refined Lyapunov  functional}\\label{sect;modified-Lyapunov}\n\nWe here introduce a suitably reconstructed Lyapunov functional which is non-increasing along the evolution of solutions, and derive regularity estimates associated with the corresponding dissipation mechanism. These estimates further yield refined regularity properties for solutions to \\eqref{eqn;KS}.\n\nThe existence of a Lyapunov functional is known to play a fundamental role in the analysis of the long-time behavior of solutions, including global existence and finite-time blowup phenomena. In the whole space setting, however, the use of Lyapunov-based methods in~\\eqref{eqn;KS} faces intrinsic difficulties due to \n the necessity of controlling the behavior as $|x|\\to\\infty$. This issue becomes particularly delicate at the critical mass.\n\nFirst of all, it is worth mentioning that the system \\eqref{eqn;KS} has a usual Lyapunov functional~$\\L(t)$ as follows:\n\\begin{align*}\n\\frac{\\d}{\\d t}\\L(t)+ \\D(t) =0,\n\\eqntag\n\\label{eqn;Lyapunov}\n\\end{align*}\nwhere\n\\begin{align*}\n\\L(t):=\\int_{\\R^2}u\\ln u\\dx-\\int_{\\R^2}uv\\dx+\\frac12\\|\\nabla v\\|_2^2+\\frac\\lambda2\\|v\\|_2^2\n\\end{align*}\nand the dissipative term $\\D(t)$ is given by\n\\begin{align*}\n\\D(t):=\\int_{\\R^2} u \\left|\\nabla(\\ln u -v)\\right|^2\\dx+\\|\\partial_t v\\|_2^2.\n\\end{align*}\nThe upper bound for $\\L(t)$ follows from the monotonicity \\eqref{eqn;Lyapunov} and \nthe lower bound for that can be obtained by the optimal constraint of the initial mass $\\|u_0\\|_1$ deriving from the Trudinger--Moser type inequality and its best possible constant, see \\cite{Na-Se-Yo} and also \\cite{CaCo08,Mi13,NaOg11} for instance.\nOn the other hand,\nin order to ensure the well-definedness of a usual entropy $\\int_{\\R^2}u\\ln u \\dx$ on the whole space,\nat least the logarithmic moment assumption for the initial data $u_0\\ln(1+|x|^2)\\in L^1(\\R^2)$ is required to control the behavior of solutions as $|x|\\to\\infty$, that is,\n\\begin{align*}\n-\\int_{\\R^2}u(\\ln u)_-\\dx \\le\\,C\\int_{\\R^2}u\\ln(1+|x|^2)\\dx,\n\\end{align*}\nwhere $(\\ln u)_-:=\\min\\{\\ln u,0\\}$, according to \\cite[Lemma 2.4]{CaCo08}.\nLater, to get rid of any moment assumptions,\nthe modified entropy is adapted by Nagai \\cite{Na11};\n\\begin{align*}\n\\int_{\\R^2}(1+u)\\ln(1+u)\\dx,\n\\end{align*}\nas well as the following identity for the modified Lyapunov functional $\\L_m(t)$ is introduced:\n\\begin{align*}\n\\frac{\\d}{\\d t} \\L_m(t) + \\D_m(t)=\\frac14\\|\\nabla v(t)\\|_2^2,\n\\eqntag\n\\label{eqn;usualmodifiedfunctiona}\n\\end{align*}\nwhere\n\\begin{align*}\n\\L_m(t):=\\int_{\\R^2}(1+u)\\ln(1+u)\\dx-\\int_{\\R^2}uv\\dx+\\frac12\\|\\nabla v\\|_2^2+\\frac\\lambda2\\|v\\|_2^2\n\\end{align*}\nand\n\\begin{align*}\n\\D_m(t):=\\int_{\\R^2}u\\left|\\nabla(\\ln(1+u)-v)\\right|^2\\dx+\\int_{\\R^2}\\left|\\nabla\\left(\\ln(1+u)-\\frac12v\\right)\\right|^2\\dx+\\|\\partial_tv\\|_2^2.\n\\end{align*}\nThe functional $\\L_m(t)$ is no longer the Lyapunov functional, however, it is still useful for proving \nthe global existence of the solution to \\eqref{eqn;KS} with sub-critical mass. Indeed, it follows\nfrom the lower bound for $\\L_m(t)$ that\nfor any $\\al\\in[0,1)$\n\\begin{align*}\n\\al\\int_{\\R^2}(1+u)\\ln(1+u)\\dx+\\frac12\\left[1-\\frac{\\|u_0\\|_1}{8\\pi(1-\\al)}\\right]&\\|\\nabla v\\|_2^2\n+\\int_0^t \\D_m(s)\\,\\d s\n\\\\\n\\le\\,&\\L_m(0)+C(\\tau,\\alpha)+\\frac14\\int_0^t\\|\\nabla v\\|_2^2\\,\\d s\n\\eqntag\n\\label{eqn;afford}\n\\end{align*}\nfor $t\\in[0,\\tau]\\cap(0,T)$ with any $\\tau>0$ and some $C(\\tau,\\alpha)>0$ (cf.~\\cite{CaCo08,Mi13}). Hence, if $\\|u_0\\|<8\\pi$, then all terms on the left hand side of \\eqref{eqn;afford} are positive by choosing $\\al>0$ sufficiently small which depends on $\\|u_0\\|_1$, and Gronwall's inequality implies that\n$\\nabla v \\in L^2((0,T)\\times \\R^2)$, \nso that we also obtain the bound for the modified entropy $\\int_{\\R^2}(1+u)\\ln (1+u)\\dx$. \nNevertheless, for the critical mass $\\|u_0\\|_1=8\\pi$, this necessarily leads to $\\alpha=0$, as a result, \n\\begin{align*}\n\\int_0^t \\D_m(s)\\,\\d s\\le\\,&\\L_m(0)+C(\\tau)+\\frac14\\int_0^t\\|\\nabla v\\|_2^2\\,\\d s,\n\\eqntag\n\\label{eqn;lack-lyapunov}\n\\end{align*}\nwhich is useless and fails to obtain even the estimates for $\\D_m(t)$ due to the lack of the regularity of solutions.\nTherefore, we introduce a reconstructed Lyapunov functional $\\F_m(t)$ so as to show regularity estimates corresponding to the dissipative terms. Let $\\F_m(t)$ be the functional for solutions to \\eqref{eqn;KS} defined as\n\\begin{align*}\n\\F_m(t):=\\L_m(t)\n+\\int_{\\R^2}\\ln(1+u)\\,\\mathrm{d}x-\\int_{\\R^2}v\\,\\d x.\n\\eqntag\n\\label{eqn;modifiedLyapunov}\n\\end{align*}\nThen, the following functional differential inequality holds true:\n\\begin{prop}\\label{prop;energy-est}\nLet $(u,v)$ be the solution to \\eqref{eqn;KS}. Then, the functional $\\F_m(t)$ defined in~\\eqref{eqn;modifiedLyapunov} satisfies the following identity:\n\t\\begin{align*}\n\t\\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2\n\t\\le\\,&\n\t\\lambda\\int_{\\R^2}v\\,\\d x.\n\t\\end{align*}\n\tTherefore, if $\\lambda=0$ then\n\t\\begin{align*}\n\t\\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2 \\le\\,0.\n\t\\end{align*}\n\tIf $\\lambda>0$ then \n\t\\begin{align*}\n\t\\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2 \\le\\,\\|v_0\\|_1+\\|u_0\\|_1.\n\t\\end{align*}\n\\end{prop}\n\\vspace{5mm}\nIt is worth emphasizing that $\\F_m(t)$ is non-increasing in time when $\\lambda=0$ as well as\nunlike the identity \\eqref{eqn;usualmodifiedfunctiona} based on the usual modified functional $\\mathcal{L}_m(t)$, the error term appearing on the right-hand side of Proposition~\\ref{prop;energy-est} can be easily controlled by initial data.\nThis allows us to show the regularity estimates for the dissipative term associated with $\\F_m(t)$, see Proposition~\\ref{prop;bound-energy} below.\n\\begin{lem}\\label{lem;ModifiedLF}\n\tSuppose assumptions as in Theorem \\ref{thm;global}. Then,\n\\begin{align*}\n\\frac{\\d}{\\d t}\\F_m(t)+ \\widetilde{\\D}(t)=-\\int_{\\R^2}\\partial_t v \\frac{u}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\mathrm{d}x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x,\n\\end{align*}\nwhere\n$\\F_m(t)$ is the modified functional defined in \\eqref{eqn;modifiedLyapunov} and the dissipative term $\\widetilde{\\D}(t)$ is given by\n\\begin{align*}\n\\widetilde{\\D}(t):=\\int_{\\R^2}u\\left|\\nabla \\left(\\ln(1+u)-v\\right)\\right|^2\\,\\d x+\\|\\partial_t v\\|_2^2.\n\\end{align*}\n\\end{lem}\n\n\\begin{pr}{Lemma \\ref{lem;ModifiedLF}}\nThe following computations are already well-known:\n\\begin{align*}\n\\frac{\\d}{\\d t} \\int_{\\R^2}(1+u)\\ln (1+u)\\,\\d x=\\,&\n-\\int_{\\R^2}(1+u)|\\nabla\\ln(1+u)|^2\\,\\d x\n+\\int_{\\R^2}u\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x\n\\end{align*}\nas well as\n\\begin{align*}\n-\\frac{\\d}{\\d t}\\int_{\\R^2}uv\\,\\d x=\\,&\\int_{\\R^2}(1+u)\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x-\\int_{\\R^2}u|\\nabla v|^2\\,\\d x\n\\\\\n&-\\|\\partial_tv\\|_2^2-\\frac12\\frac{\\d}{\\d t}\\|\\nabla v\\|_2^2-\\frac\\lambda2\\frac{\\d}{\\d t}\\|v\\|_2^2,\n\\end{align*}\nso that\n\\begin{align*}\n&\\frac{\\d}{\\d t}\\left[\\int_{\\R^2}(1+u)\\ln(1+u)\\,\\d x-\\int_{\\R^2}uv\\,\\d x+\\frac12\\left(\\|\\nabla v\\|_2^2+\\lambda\\|v\\|_2^2\\right)\\right]\n\\\\\n&+\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\|\\partial_tv\\|_2^2\n\\\\\n=\\,&-\\|\\nabla\\ln(1+u)\\|_2^2+\\int_{\\R^2}\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x.\n\\end{align*}\nNext,\n\\begin{align*}\n\\frac{\\d}{\\d t}\\int_{\\R^2}\\ln(1+u)\\,\\d x=\\,&\\int_{\\R^2}\\frac{1}{1+u} \\left[\\Delta u -\\nabla\\cdot( u\\nabla v)\\right]\\,\\mathrm{d}x\n\\\\\n=\\,&-\\int_{\\R^2}\\nabla\\left(\\frac{1}{1+u}\\right)\\cdot \\nabla u\\,\\d x+\\int_{\\R^2}u\\nabla\\left(\\frac{1}{1+u}\\right)\\cdot \\nabla v\\,\\d x\n\\\\\n=\\,&\\int_{\\R^2}|\\nabla \\ln(1+u)|^2\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x\n\\\\\n=\\,&\\int_{\\R^2}|\\nabla \\ln(1+u)|^2\\,\\d x-\\int_{\\R^2}\\nabla\\ln(1+u)\\cdot\\nabla v\\,\\d x\n\\\\\n&+\\int_{\\R^2}\\frac{1}{1+u}\\nabla\\ln(1+u)\\cdot\\nabla v \\,\\d x.\n\\end{align*}\nHence, combining the above computations implies that\n\\begin{align*}\n&\\frac{\\d}{\\d t}\\left[\\int_{\\R^2}(1+u)\\ln(1+u)\\,\\d x-\\int_{\\R^2}uv\\,\\d x+\\frac12\\left(\\|\\nabla v\\|_2^2+\\lambda\\|v\\|_2^2\\right)+\\int_{\\R^2}\\ln(1+u)\\,\\d x\\right]\n\\\\\n&+\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\|\\partial_tv\\|_2^2\n\\\\\n=\\,&\\int_{\\R^2}\\frac{1}{(1+u)^2}\\nabla u\\cdot\\nabla v \\,\\d x.\n\\end{align*}\nSince\n\\begin{align*}\n\\int_{\\R^2}\\frac{1}{(1+u)^2}\\nabla u\\cdot\\nabla v \\,\\d x=\\,&-\\int_{\\R^2}\\nabla\\left(\\frac{1}{1+u}\\right)\\cdot \\nabla v\\,\\d x\n\\\\\n=\\,&\\int_{\\R^2}\\frac{1}{1+u}\\Delta v\\,\\d x\n\\\\\n=\\,&\\int_{\\R^2}\\frac{\\partial_t v}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\d x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x\n\\\\\n=\\,&\\frac{\\d}{\\d t}\\int_{\\R^2}v\\,\\d x-\\int_{\\R^2}\\partial_t v \\frac{u}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\d x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x,\n\\end{align*}\nwe end up with\n\\begin{align*}\n&\\frac{\\d}{\\d t}\\F_m(t)+\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\|\\partial_tv\\|_2^2\n\\\\\n=\\,&-\\int_{\\R^2}\\partial_t v \\frac{u}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\d x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x,\n\\end{align*}\nas desired.\n\\end{pr}"}, "pre_theorem_intro_text_len": 8115, "pre_theorem_intro_text": "We study  the Cauchy problem for the parabolic-parabolic Keller--Segel system in~$\\mathbb R^2$\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n&\\pt_t u =\\Delta u- \\nabla\\cdot \\left( u \\nabla v \\right),\n& t>0,\\,~ &x\\in\\mathbb R^2,\n\\\\\n&\\pt_t v = \\Delta v - \\lambda v +u,\n&t>0,\\, ~&x\\in\\mathbb R^2,\n\\\\\n&(u,v)(0,x)=(u_{0},v_0)(x),\n&\\, &x\\in\\mathbb R^2\n\\end{aligned}\n\\right.\n\\label{eqn;KS}\n\\end{equation}\nwith a constant $\\lambda\\ge0$, where $u_0,v_0\\ge 0$ on $\\mathbb R^2$ and\n$u_0,v_0\\not\\equiv0$.\nThe Keller--Segel system~\\eqref{eqn;KS} is a fundamental mathematical model\nof chemotaxis, describing chemotactic aggregation in the\ncellular slime mold {\\it Dictyostelium discoideum} during its life cycle \\cite{KeSe,Pa}.\nIn this model, $u=u(t,x)$ and $v=v(t,x)$ denote the densities of cells and\nthe chemoattractant, respectively.\nThe parameter $\\lambda$ represents the degradation rate of the chemical.\nCells migrate toward regions of higher concentrations of a chemical substance\nsecreted by the cells themselves.\nFrom a biological viewpoint, it is natural to assume that the initial data\nare nonnegative functions.\nFrom a mathematical viewpoint, cell aggregation is interpreted as the\nblowup of solutions at $t=T$ in the sense that\n$\\lim_{t\\to T}\\|u(t)\\|_\\infty=\\infty.$\n\nOne of the central mathematical features of such systems is the phenomenon of critical\nmass.\nFor positive sufficiently regular solutions $(u,v)$ to~\\eqref{eqn;KS}, the first\ncomponent $u$ satisfies the mass conservation law \n$$\\|u(t)\\|_1=\\|u_0\\|_1\\quad \\text{for}~~~t>0.$$\nMoreover, in two space dimensions, the global behavior of solutions is governed\nby the size of the initial mass of $u$.\nIn particular, there exists a threshold value such that solutions with initial\nmass below this threshold exist globally in time, whereas solutions with\ninitial mass above the threshold may blow up in finite time.\nThis critical-mass phenomenon has attracted considerable attention in the\nliterature.\n\nThe aim of this paper is to establish the global behavior of solutions to\n\\eqref{eqn;KS} whose initial mass is exactly equal to this threshold value.\n\n\\vspace{3mm}\nThe second equation in \\eqref{eqn;KS} takes into account that cells are producing the chemoattractant themselves \nwhile this is diffusing into the environment. \nSince the chemoattractant attains its equilibrium on a time scale much quicker\nthan that of the cells, the simplified parabolic-elliptic system has also been\ninvestigated \\cite{JaLu,Na95}:\n\\begin{align*}\n\\left\\{\n\\begin{aligned}\n\\partial_t u &= \\Delta u - \\nabla\\cdot (u\\nabla v),\n& t>0,\\ & x\\in\\mathbb R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\mathbb R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\mathbb R^2.\n\\end{aligned}\n\\right.\n\\addtocounter{equation}{1}\\tag{\\theequation}\n\\label{eqn;PKS}\n\\end{align*}\nThe system \\eqref{eqn;PKS} is also related to models of gravitational interaction\nof particles \\cite{Bi-Na94,Wo}.\nIn~\\eqref{eqn;PKS}, the second equation can be written as\n$v = (-\\Delta+\\lambda)^{-1}u.$\nAs a consequence, the system~\\eqref{eqn;PKS} can be reduced to a single equation\nfor~$u$.\nOwing to this elliptic structure, the parabolic-elliptic system~\\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.\nIn particular, numerous works have been devoted to the critical-mass\nphenomenon.\nIndeed, the solution with $\\|u_0\\|_1\\le 8\\pi$ exists globally in time~\\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},\nwhile the solution may blow up in finite time if $\\|u_0\\|_1>8\\pi$~\\cite{DoPe,Ko-Su,Ku-Og03,Wei}.\nFor the corresponding Cauchy--Neumann problem in bounded domains,\nsupplemented with homogeneous Neumann boundary conditions, see \\cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.\n\n\\vspace{3mm}\nUnlike the parabolic-elliptic system \\eqref{eqn;PKS}, the fully parabolic system\n\\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches\ndeveloped for \\eqref{eqn;PKS} are no longer applicable.\nThe global existence of solutions with sub-critical mass\n$\\|u_0\\|_{1}<8\\pi$ has nevertheless been established by combining\nLyapunov functionals with the Trudinger--Moser type inequality and its optimal\nconstant.\nMore precisely, the following results are known:\n\\begin{enumerate}\n\t\\item[(i)]\n\tIf $\\|u_0\\|_{1}<8\\pi$, then the corresponding solution to\n\t\\eqref{eqn;KS} exists globally in time \\cite{CaCo08,Mi13,NaOg11}.\n\n\t\\item[(ii)]\n\tIf $\\|u_0\\|_{1}>8\\pi$ and $(u_0,v_0)$ is radially symmetric, then there\n\texists a radially symmetric solution that blows up in finite time\n\t\\cite{Mi20,Mi20-SIAM}.\n\n\t\\item[(iii)]\n\tIf $\\|u_0\\|_{1}=8\\pi$ and $(u_0,v_0)$ is either radially symmetric or\n\tsatisfies the additional moment condition\n\t$u_0\\ln(1+|x|^2)\\in L^1(\\mathbb R^2)$, then the corresponding solution exists globally\n\tin time.\n\tIn contrast, for general initial data, the global behavior in the critical case\n\tremains delicate, and the solution either exists globally or blows up on the plane \\cite{CaCo08,Mi13}.\n\\end{enumerate}\nAs for positive forward self-similar solutions to \\eqref{eqn;KS}, refer to \\cite{BiCoDo}.\nFor the corresponding Cauchy--Neumann problem in bounded domains, we refer to\n\\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.\n\nAs mentioned above, although several partial results are available, no complete global existence result has been obtained\nfor solutions with critical mass and general initial data without any\nsymmetry or moment assumptions.\nOne of the main difficulties lies in controlling the behavior of solutions at~$|x|\\to\\infty$.\nMoreover, in the critical mass case, no global existence result is currently\nknown  for the corresponding Cauchy--Neumann problem to~\\eqref{eqn;KS} in bounded domains.\n\nIn this paper, we establish the global existence of solutions with critical mass\nfor general initial data $(u_0,v_0)$.\nTo this end, we first recall the definition of solutions.\n\n\\vspace{2mm}\n\\begin{def*}\nLet $4/3<p<2$ and $q=p/(p-1)$.\nGiven $T\\in (0,\\infty]$ and  $(u_0,v_0) \\in L^1(\\mathbb R^2)\\times \\dot{H}^{1}(\\mathbb R^2)$,\nwe say that a pair of functions $(u,v)$ defined on $[0,T)\\times \\mathbb R^2$ is a mild solution to~\\eqref{eqn;KS} on $[0,T)$ if\n\\begin{enumerate}\n\t\\item $u\\in C\\big([0,T); L^1(\\mathbb R^2)\\big)\\cap C\\left( (0,T); L^p(\\mathbb{R}^2) \\right)$;\n\t\\item $v\\in C\\big([0,T); \\dot{H}^{1}(\\mathbb R^2)\\big)\\cap C\\left( (0,T); \\dot{W}^{1,q}(\\mathbb{R}^2) \\right)$;\n\t\\item $\\displaystyle\\sup_{0<t<T} t^{1-\\frac1p}\\|u(t)\\|_{p}+\\displaystyle\\sup_{0<t<T}t^{\\frac12-\\frac1q}\\|\\nabla v(t)\\|_{q}$;\n\t\\item $(u,v)$ satisfies the integral formulations\n\t\\begin{align*}\n\t&u(t)=e^{t\\Delta}u_0-\\int_0^t \\nabla\\cdot e^{(t-s)\\Delta} \\big(u(s)\\nabla v(s)\\big) \\,\\mathrm{d} s,\n\t\\\\\n\t&v(t)= e^{t(\\Delta-\\lambda)} v_0+\\int_0^t e^{(t-s)(\\Delta-\\lambda)} u(s) \\,\\mathrm{d} s\n\t\\end{align*}\n\tfor $0<t<T$, where $e^{t\\Delta}$ is the heat semigroup given by\n\t\\begin{align*}\n\te^{t\\Delta} f(x) := G_t*f(x)=\\int_{ \\mathbb R^2}G_t(x-y) f(y)\\,\\mathrm{d} y,\n\t\\quad\n\tG_t(x)=\\frac{1}{4\\pi t} \\exp\\left(-\\frac{|x|^2}{4 t}\\right)\n\t\\end{align*}\n\tfor $x\\in\\mathbb R^2$ and $f \\in L^p(\\mathbb R^2)$.\n\\end{enumerate}\n\\end{def*}\nThe Lebesgue spaces $L^p(\\mathbb R^2)$ are equipped with the norm $\\|\\cdot\\|_{L^p(\\mathbb R^2)}$,\nwhich is abbreviated as $\\|\\cdot\\|_p$ when there is no ambiguity.\nWe define the homogeneous Sobolev spaces by\n\\[\n\\dot{W}^{1,p}(\\mathbb R^2)\n:= \\overline{C_0^\\infty(\\mathbb R^2)}^{\\|\\nabla (\\cdot)\\|_{L^p(\\mathbb R^2)}}\n\\]\nand $\\dot{H}^1(\\mathbb R^2):=\\dot{W}^{1,2}(\\mathbb R^2)$ when $p=2$.\n$L_+^1(\\mathbb R^2):=\\{f \\in L^1(\\mathbb R^2); f\\ge0~~ \\text{and}~~ f\\not\\equiv0~~ \\text{on $\\mathbb R^2$} \\}$.\n\n\\vspace{5mm}\nThe starting point for our analysis is the local well-posedness for \\eqref{eqn;KS} and the solution~$(u,v)$ based on the above definition satisfies~\\eqref{eqn;KS} in a classical sense on $(0,T)$, see Proposition~\\ref{prop;LWP} in Section~\\ref{sect;LWP} below.\n\nWe are now in a position to state our main result in this paper.\nMain result in this paper reads as follows.\n\\vspace{2mm}", "context": "\\vspace{3mm}\nThe second equation in \\eqref{eqn;KS} takes into account that cells are producing the chemoattractant themselves \nwhile this is diffusing into the environment. \nSince the chemoattractant attains its equilibrium on a time scale much quicker\nthan that of the cells, the simplified parabolic-elliptic system has also been\ninvestigated \\cite{JaLu,Na95}:\n\\begin{align*}\n\\left\\{\n\\begin{aligned}\n\\partial_t u &= \\Delta u - \\nabla\\cdot (u\\nabla v),\n& t>0,\\ & x\\in\\mathbb R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\mathbb R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\mathbb R^2.\n\\end{aligned}\n\\right.\n\\addtocounter{equation}{1}\\tag{\\theequation}\n\\label{eqn;PKS}\n\\end{align*}\nThe system \\eqref{eqn;PKS} is also related to models of gravitational interaction\nof particles \\cite{Bi-Na94,Wo}.\nIn~\\eqref{eqn;PKS}, the second equation can be written as\n$v = (-\\Delta+\\lambda)^{-1}u.$\nAs a consequence, the system~\\eqref{eqn;PKS} can be reduced to a single equation\nfor~$u$.\nOwing to this elliptic structure, the parabolic-elliptic system~\\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.\nIn particular, numerous works have been devoted to the critical-mass\nphenomenon.\nIndeed, the solution with $\\|u_0\\|_1\\le 8\\pi$ exists globally in time~\\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},\nwhile the solution may blow up in finite time if $\\|u_0\\|_1>8\\pi$~\\cite{DoPe,Ko-Su,Ku-Og03,Wei}.\nFor the corresponding Cauchy--Neumann problem in bounded domains,\nsupplemented with homogeneous Neumann boundary conditions, see \\cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.\n\n\\vspace{3mm}\nUnlike the parabolic-elliptic system \\eqref{eqn;PKS}, the fully parabolic system\n\\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches\ndeveloped for \\eqref{eqn;PKS} are no longer applicable.\nThe global existence of solutions with sub-critical mass\n$\\|u_0\\|_{1}<8\\pi$ has nevertheless been established by combining\nLyapunov functionals with the Trudinger--Moser type inequality and its optimal\nconstant.\nMore precisely, the following results are known:\n\\begin{enumerate}\n \\item[(i)]\n If $\\|u_0\\|_{1}<8\\pi$, then the corresponding solution to\n \\eqref{eqn;KS} exists globally in time \\cite{CaCo08,Mi13,NaOg11}.\n\n\\item[(iii)]\n If $\\|u_0\\|_{1}=8\\pi$ and $(u_0,v_0)$ is either radially symmetric or\n satisfies the additional moment condition\n $u_0\\ln(1+|x|^2)\\in L^1(\\mathbb R^2)$, then the corresponding solution exists globally\n in time.\n In contrast, for general initial data, the global behavior in the critical case\n remains delicate, and the solution either exists globally or blows up on the plane \\cite{CaCo08,Mi13}.\n\\end{enumerate}\nAs for positive forward self-similar solutions to \\eqref{eqn;KS}, refer to \\cite{BiCoDo}.\nFor the corresponding Cauchy--Neumann problem in bounded domains, we refer to\n\\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.\n\n\\vspace{2mm}\n\\begin{def*}\nLet $4/3<p<2$ and $q=p/(p-1)$.\nGiven $T\\in (0,\\infty]$ and $(u_0,v_0) \\in L^1(\\mathbb R^2)\\times \\dot{H}^{1}(\\mathbb R^2)$,\nwe say that a pair of functions $(u,v)$ defined on $[0,T)\\times \\mathbb R^2$ is a mild solution to~\\eqref{eqn;KS} on $[0,T)$ if\n\\begin{enumerate}\n \\item $u\\in C\\big([0,T); L^1(\\mathbb R^2)\\big)\\cap C\\left( (0,T); L^p(\\mathbb{R}^2) \\right)$;\n \\item $v\\in C\\big([0,T); \\dot{H}^{1}(\\mathbb R^2)\\big)\\cap C\\left( (0,T); \\dot{W}^{1,q}(\\mathbb{R}^2) \\right)$;\n \\item $\\displaystyle\\sup_{0<t<T} t^{1-\\frac1p}\\|u(t)\\|_{p}+\\displaystyle\\sup_{0<t<T}t^{\\frac12-\\frac1q}\\|\\nabla v(t)\\|_{q}$;\n \\item $(u,v)$ satisfies the integral formulations\n \\begin{align*}\n &u(t)=e^{t\\Delta}u_0-\\int_0^t \\nabla\\cdot e^{(t-s)\\Delta} \\big(u(s)\\nabla v(s)\\big) \\,\\mathrm{d} s,\n \\\\\n &v(t)= e^{t(\\Delta-\\lambda)} v_0+\\int_0^t e^{(t-s)(\\Delta-\\lambda)} u(s) \\,\\mathrm{d} s\n \\end{align*}\n for $0<t<T$, where $e^{t\\Delta}$ is the heat semigroup given by\n \\begin{align*}\n e^{t\\Delta} f(x) := G_t*f(x)=\\int_{ \\mathbb R^2}G_t(x-y) f(y)\\,\\mathrm{d} y,\n \\quad\n G_t(x)=\\frac{1}{4\\pi t} \\exp\\left(-\\frac{|x|^2}{4 t}\\right)\n \\end{align*}\n for $x\\in\\mathbb R^2$ and $f \\in L^p(\\mathbb R^2)$.\n\\end{enumerate}\n\\end{def*}\nThe Lebesgue spaces $L^p(\\mathbb R^2)$ are equipped with the norm $\\|\\cdot\\|_{L^p(\\mathbb R^2)}$,\nwhich is abbreviated as $\\|\\cdot\\|_p$ when there is no ambiguity.\nWe define the homogeneous Sobolev spaces by\n\\[\n\\dot{W}^{1,p}(\\mathbb R^2)\n:= \\overline{C_0^\\infty(\\mathbb R^2)}^{\\|\\nabla (\\cdot)\\|_{L^p(\\mathbb R^2)}}\n\\]\nand $\\dot{H}^1(\\mathbb R^2):=\\dot{W}^{1,2}(\\mathbb R^2)$ when $p=2$.\n$L_+^1(\\mathbb R^2):=\\{f \\in L^1(\\mathbb R^2); f\\ge0~~ \\text{and}~~ f\\not\\equiv0~~ \\text{on $\\mathbb R^2$} \\}$.\n\n\\vspace{5mm}\nThe starting point for our analysis is the local well-posedness for \\eqref{eqn;KS} and the solution~$(u,v)$ based on the above definition satisfies~\\eqref{eqn;KS} in a classical sense on $(0,T)$, see Proposition~\\ref{prop;LWP} in Section~\\ref{sect;LWP} below.\n\nWe are now in a position to state our main result in this paper.\nMain result in this paper reads as follows.\n\\vspace{2mm}", "full_context": "\\vspace{3mm}\nThe second equation in \\eqref{eqn;KS} takes into account that cells are producing the chemoattractant themselves \nwhile this is diffusing into the environment. \nSince the chemoattractant attains its equilibrium on a time scale much quicker\nthan that of the cells, the simplified parabolic-elliptic system has also been\ninvestigated \\cite{JaLu,Na95}:\n\\begin{align*}\n\\left\\{\n\\begin{aligned}\n\\partial_t u &= \\Delta u - \\nabla\\cdot (u\\nabla v),\n& t>0,\\ & x\\in\\mathbb R^2,\\\\\n0 &= \\Delta v - \\lambda v + u,\n& t>0,\\ & x\\in\\mathbb R^2,\n\\\\\n&u(0,x)=u_{0}(x),\n&\\, &x\\in\\mathbb R^2.\n\\end{aligned}\n\\right.\n\\addtocounter{equation}{1}\\tag{\\theequation}\n\\label{eqn;PKS}\n\\end{align*}\nThe system \\eqref{eqn;PKS} is also related to models of gravitational interaction\nof particles \\cite{Bi-Na94,Wo}.\nIn~\\eqref{eqn;PKS}, the second equation can be written as\n$v = (-\\Delta+\\lambda)^{-1}u.$\nAs a consequence, the system~\\eqref{eqn;PKS} can be reduced to a single equation\nfor~$u$.\nOwing to this elliptic structure, the parabolic-elliptic system~\\eqref{eqn;PKS} is more amenable to analysis than the fully parabolic system~\\eqref{eqn;KS}, and has therefore been extensively studied and is now well understood in many aspects.\nIn particular, numerous works have been devoted to the critical-mass\nphenomenon.\nIndeed, the solution with $\\|u_0\\|_1\\le 8\\pi$ exists globally in time~\\cite{BiKaLaNa06,Bl-Ca-Ca,Bl-Ca-Ma,BlDoPe,DoPe,Lo-Na-Ya1,Lo-Na-Ya2,Na11,NaOg11,NaOg16,Na-Se,Wei},\nwhile the solution may blow up in finite time if $\\|u_0\\|_1>8\\pi$~\\cite{DoPe,Ko-Su,Ku-Og03,Wei}.\nFor the corresponding Cauchy--Neumann problem in bounded domains,\nsupplemented with homogeneous Neumann boundary conditions, see \\cite{BiKaLaNa06disc,Bi-Na94,Ga-Za98,Na95,Na01} for instance.\n\n\\vspace{3mm}\nUnlike the parabolic-elliptic system \\eqref{eqn;PKS}, the fully parabolic system\n\\eqref{eqn;KS} is a strongly coupled parabolic system, and many approaches\ndeveloped for \\eqref{eqn;PKS} are no longer applicable.\nThe global existence of solutions with sub-critical mass\n$\\|u_0\\|_{1}<8\\pi$ has nevertheless been established by combining\nLyapunov functionals with the Trudinger--Moser type inequality and its optimal\nconstant.\nMore precisely, the following results are known:\n\\begin{enumerate}\n \\item[(i)]\n If $\\|u_0\\|_{1}<8\\pi$, then the corresponding solution to\n \\eqref{eqn;KS} exists globally in time \\cite{CaCo08,Mi13,NaOg11}.\n\n\\item[(iii)]\n If $\\|u_0\\|_{1}=8\\pi$ and $(u_0,v_0)$ is either radially symmetric or\n satisfies the additional moment condition\n $u_0\\ln(1+|x|^2)\\in L^1(\\mathbb R^2)$, then the corresponding solution exists globally\n in time.\n In contrast, for general initial data, the global behavior in the critical case\n remains delicate, and the solution either exists globally or blows up on the plane \\cite{CaCo08,Mi13}.\n\\end{enumerate}\nAs for positive forward self-similar solutions to \\eqref{eqn;KS}, refer to \\cite{BiCoDo}.\nFor the corresponding Cauchy--Neumann problem in bounded domains, we refer to\n\\cite{Bi98,He-Ve,HoWa,Na-Se-Yo}, for instance.\n\n\\vspace{2mm}\n\\begin{def*}\nLet $4/3<p<2$ and $q=p/(p-1)$.\nGiven $T\\in (0,\\infty]$ and $(u_0,v_0) \\in L^1(\\mathbb R^2)\\times \\dot{H}^{1}(\\mathbb R^2)$,\nwe say that a pair of functions $(u,v)$ defined on $[0,T)\\times \\mathbb R^2$ is a mild solution to~\\eqref{eqn;KS} on $[0,T)$ if\n\\begin{enumerate}\n \\item $u\\in C\\big([0,T); L^1(\\mathbb R^2)\\big)\\cap C\\left( (0,T); L^p(\\mathbb{R}^2) \\right)$;\n \\item $v\\in C\\big([0,T); \\dot{H}^{1}(\\mathbb R^2)\\big)\\cap C\\left( (0,T); \\dot{W}^{1,q}(\\mathbb{R}^2) \\right)$;\n \\item $\\displaystyle\\sup_{0<t<T} t^{1-\\frac1p}\\|u(t)\\|_{p}+\\displaystyle\\sup_{0<t<T}t^{\\frac12-\\frac1q}\\|\\nabla v(t)\\|_{q}$;\n \\item $(u,v)$ satisfies the integral formulations\n \\begin{align*}\n &u(t)=e^{t\\Delta}u_0-\\int_0^t \\nabla\\cdot e^{(t-s)\\Delta} \\big(u(s)\\nabla v(s)\\big) \\,\\mathrm{d} s,\n \\\\\n &v(t)= e^{t(\\Delta-\\lambda)} v_0+\\int_0^t e^{(t-s)(\\Delta-\\lambda)} u(s) \\,\\mathrm{d} s\n \\end{align*}\n for $0<t<T$, where $e^{t\\Delta}$ is the heat semigroup given by\n \\begin{align*}\n e^{t\\Delta} f(x) := G_t*f(x)=\\int_{ \\mathbb R^2}G_t(x-y) f(y)\\,\\mathrm{d} y,\n \\quad\n G_t(x)=\\frac{1}{4\\pi t} \\exp\\left(-\\frac{|x|^2}{4 t}\\right)\n \\end{align*}\n for $x\\in\\mathbb R^2$ and $f \\in L^p(\\mathbb R^2)$.\n\\end{enumerate}\n\\end{def*}\nThe Lebesgue spaces $L^p(\\mathbb R^2)$ are equipped with the norm $\\|\\cdot\\|_{L^p(\\mathbb R^2)}$,\nwhich is abbreviated as $\\|\\cdot\\|_p$ when there is no ambiguity.\nWe define the homogeneous Sobolev spaces by\n\\[\n\\dot{W}^{1,p}(\\mathbb R^2)\n:= \\overline{C_0^\\infty(\\mathbb R^2)}^{\\|\\nabla (\\cdot)\\|_{L^p(\\mathbb R^2)}}\n\\]\nand $\\dot{H}^1(\\mathbb R^2):=\\dot{W}^{1,2}(\\mathbb R^2)$ when $p=2$.\n$L_+^1(\\mathbb R^2):=\\{f \\in L^1(\\mathbb R^2); f\\ge0~~ \\text{and}~~ f\\not\\equiv0~~ \\text{on $\\mathbb R^2$} \\}$.\n\n\\vspace{5mm}\nThe starting point for our analysis is the local well-posedness for \\eqref{eqn;KS} and the solution~$(u,v)$ based on the above definition satisfies~\\eqref{eqn;KS} in a classical sense on $(0,T)$, see Proposition~\\ref{prop;LWP} in Section~\\ref{sect;LWP} below.\n\nWe are now in a position to state our main result in this paper.\nMain result in this paper reads as follows.\n\\vspace{2mm}\n\n\\vspace{5mm}\nThe starting point for our analysis is the local well-posedness for \\eqref{eqn;KS} and the solution~$(u,v)$ based on the above definition satisfies~\\eqref{eqn;KS} in a classical sense on $(0,T)$, see Proposition~\\ref{prop;LWP} in Section~\\ref{sect;LWP} below.\n\nFirst of all, it is worth mentioning that the system \\eqref{eqn;KS} has a usual Lyapunov functional~$\\L(t)$ as follows:\n\\begin{align*}\n\\frac{\\d}{\\d t}\\L(t)+ \\D(t) =0,\n\\eqntag\n\\label{eqn;Lyapunov}\n\\end{align*}\nwhere\n\\begin{align*}\n\\L(t):=\\int_{\\R^2}u\\ln u\\dx-\\int_{\\R^2}uv\\dx+\\frac12\\|\\nabla v\\|_2^2+\\frac\\lambda2\\|v\\|_2^2\n\\end{align*}\nand the dissipative term $\\D(t)$ is given by\n\\begin{align*}\n\\D(t):=\\int_{\\R^2} u \\left|\\nabla(\\ln u -v)\\right|^2\\dx+\\|\\partial_t v\\|_2^2.\n\\end{align*}\nThe upper bound for $\\L(t)$ follows from the monotonicity \\eqref{eqn;Lyapunov} and \nthe lower bound for that can be obtained by the optimal constraint of the initial mass $\\|u_0\\|_1$ deriving from the Trudinger--Moser type inequality and its best possible constant, see \\cite{Na-Se-Yo} and also \\cite{CaCo08,Mi13,NaOg11} for instance.\nOn the other hand,\nin order to ensure the well-definedness of a usual entropy $\\int_{\\R^2}u\\ln u \\dx$ on the whole space,\nat least the logarithmic moment assumption for the initial data $u_0\\ln(1+|x|^2)\\in L^1(\\R^2)$ is required to control the behavior of solutions as $|x|\\to\\infty$, that is,\n\\begin{align*}\n-\\int_{\\R^2}u(\\ln u)_-\\dx \\le\\,C\\int_{\\R^2}u\\ln(1+|x|^2)\\dx,\n\\end{align*}\nwhere $(\\ln u)_-:=\\min\\{\\ln u,0\\}$, according to \\cite[Lemma 2.4]{CaCo08}.\nLater, to get rid of any moment assumptions,\nthe modified entropy is adapted by Nagai \\cite{Na11};\n\\begin{align*}\n\\int_{\\R^2}(1+u)\\ln(1+u)\\dx,\n\\end{align*}\nas well as the following identity for the modified Lyapunov functional $\\L_m(t)$ is introduced:\n\\begin{align*}\n\\frac{\\d}{\\d t} \\L_m(t) + \\D_m(t)=\\frac14\\|\\nabla v(t)\\|_2^2,\n\\eqntag\n\\label{eqn;usualmodifiedfunctiona}\n\\end{align*}\nwhere\n\\begin{align*}\n\\L_m(t):=\\int_{\\R^2}(1+u)\\ln(1+u)\\dx-\\int_{\\R^2}uv\\dx+\\frac12\\|\\nabla v\\|_2^2+\\frac\\lambda2\\|v\\|_2^2\n\\end{align*}\nand\n\\begin{align*}\n\\D_m(t):=\\int_{\\R^2}u\\left|\\nabla(\\ln(1+u)-v)\\right|^2\\dx+\\int_{\\R^2}\\left|\\nabla\\left(\\ln(1+u)-\\frac12v\\right)\\right|^2\\dx+\\|\\partial_tv\\|_2^2.\n\\end{align*}\nThe functional $\\L_m(t)$ is no longer the Lyapunov functional, however, it is still useful for proving \nthe global existence of the solution to \\eqref{eqn;KS} with sub-critical mass. Indeed, it follows\nfrom the lower bound for $\\L_m(t)$ that\nfor any $\\al\\in[0,1)$\n\\begin{align*}\n\\al\\int_{\\R^2}(1+u)\\ln(1+u)\\dx+\\frac12\\left[1-\\frac{\\|u_0\\|_1}{8\\pi(1-\\al)}\\right]&\\|\\nabla v\\|_2^2\n+\\int_0^t \\D_m(s)\\,\\d s\n\\\\\n\\le\\,&\\L_m(0)+C(\\tau,\\alpha)+\\frac14\\int_0^t\\|\\nabla v\\|_2^2\\,\\d s\n\\eqntag\n\\label{eqn;afford}\n\\end{align*}\nfor $t\\in[0,\\tau]\\cap(0,T)$ with any $\\tau>0$ and some $C(\\tau,\\alpha)>0$ (cf.~\\cite{CaCo08,Mi13}). Hence, if $\\|u_0\\|<8\\pi$, then all terms on the left hand side of \\eqref{eqn;afford} are positive by choosing $\\al>0$ sufficiently small which depends on $\\|u_0\\|_1$, and Gronwall's inequality implies that\n$\\nabla v \\in L^2((0,T)\\times \\R^2)$, \nso that we also obtain the bound for the modified entropy $\\int_{\\R^2}(1+u)\\ln (1+u)\\dx$. \nNevertheless, for the critical mass $\\|u_0\\|_1=8\\pi$, this necessarily leads to $\\alpha=0$, as a result, \n\\begin{align*}\n\\int_0^t \\D_m(s)\\,\\d s\\le\\,&\\L_m(0)+C(\\tau)+\\frac14\\int_0^t\\|\\nabla v\\|_2^2\\,\\d s,\n\\eqntag\n\\label{eqn;lack-lyapunov}\n\\end{align*}\nwhich is useless and fails to obtain even the estimates for $\\D_m(t)$ due to the lack of the regularity of solutions.\nTherefore, we introduce a reconstructed Lyapunov functional $\\F_m(t)$ so as to show regularity estimates corresponding to the dissipative terms. Let $\\F_m(t)$ be the functional for solutions to \\eqref{eqn;KS} defined as\n\\begin{align*}\n\\F_m(t):=\\L_m(t)\n+\\int_{\\R^2}\\ln(1+u)\\,\\mathrm{d}x-\\int_{\\R^2}v\\,\\d x.\n\\eqntag\n\\label{eqn;modifiedLyapunov}\n\\end{align*}\nThen, the following functional differential inequality holds true:\n\\begin{prop}\\label{prop;energy-est}\nLet $(u,v)$ be the solution to \\eqref{eqn;KS}. Then, the functional $\\F_m(t)$ defined in~\\eqref{eqn;modifiedLyapunov} satisfies the following identity:\n \\begin{align*}\n \\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2\n \\le\\,&\n \\lambda\\int_{\\R^2}v\\,\\d x.\n \\end{align*}\n Therefore, if $\\lambda=0$ then\n \\begin{align*}\n \\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2 \\le\\,0.\n \\end{align*}\n If $\\lambda>0$ then \n \\begin{align*}\n \\frac{\\d}{\\d t}\\F_m(t) +\\int_{\\R^2}u\\left|\\nabla\\left(\\ln(1+u)-v\\right)\\right|^2\\,\\mathrm{d}x+\\frac12\\|\\partial_tv\\|_2^2 \\le\\,\\|v_0\\|_1+\\|u_0\\|_1.\n \\end{align*}\n\\end{prop}\n\\vspace{5mm}\nIt is worth emphasizing that $\\F_m(t)$ is non-increasing in time when $\\lambda=0$ as well as\nunlike the identity \\eqref{eqn;usualmodifiedfunctiona} based on the usual modified functional $\\mathcal{L}_m(t)$, the error term appearing on the right-hand side of Proposition~\\ref{prop;energy-est} can be easily controlled by initial data.\nThis allows us to show the regularity estimates for the dissipative term associated with $\\F_m(t)$, see Proposition~\\ref{prop;bound-energy} below.\n\\begin{lem}\\label{lem;ModifiedLF}\n Suppose assumptions as in Theorem \\ref{thm;global}. Then,\n\\begin{align*}\n\\frac{\\d}{\\d t}\\F_m(t)+ \\widetilde{\\D}(t)=-\\int_{\\R^2}\\partial_t v \\frac{u}{1+u}\\,\\d x-\\int_{\\R^2}\\frac{u}{1+u}\\,\\mathrm{d}x+\\lambda\\int_{\\R^2}\\frac{v}{1+u}\\,\\d x,\n\\end{align*}\nwhere\n$\\F_m(t)$ is the modified functional defined in \\eqref{eqn;modifiedLyapunov} and the dissipative term $\\widetilde{\\D}(t)$ is given by\n\\begin{align*}\n\\widetilde{\\D}(t):=\\int_{\\R^2}u\\left|\\nabla \\left(\\ln(1+u)-v\\right)\\right|^2\\,\\d x+\\|\\partial_t v\\|_2^2.\n\\end{align*}\n\\end{lem}\n\n\\vspace{5mm}\n\\begin{pr}{Theorem \\ref{thm;global}}\nLet $R$ be taken as in Lemma \\ref{lem;L^2-exterior}.\nBy Lemma \\ref{lem;LlogL-bound-exterior},\n\\begin{align*}\n\\int_{ |x|>2R } (1+u) \\ln(1+u) \\dx\\le\\, C(t_0,\\tau,R).\n\\end{align*}\nOn the other hand, Lemma \\ref{prop;interior-bdd-entropy} gives\n\\begin{align*}\n\\int_{ \\R^2}(u\\ln u)\\psi_R^2\\dx\\le\\, C(t_0,\\tau,R)\n\\end{align*}\nfor $t\\in[t_0,\\tau]\\cap(0,T)$.\nNow, according to \\cite[Lemma 2.3]{NaOg16},\n\\begin{align*}\n\\int_{ \\Omega} (1+u)\\ln(1+u)\\dx\\le\\,2\\int_{ \\Omega} u |\\ln u | \\dx+(2\\ln 2)\\int_{ \\Omega} u\\dx,\n\\end{align*}\nwhere $\\Omega$ is a measurable set in $\\R^2$,\n so that since $\\supp\\psi_R\\subset \\{x; |x|\\le 16 R\\}$\n \\begin{align*}\n \\int_{ |x|<8R }(1+u)\\ln(1+u)\\dx\\le\\,&2\\int_{ |x|<8R }u|\\ln u| \\dx+(2\\ln 2)\\|u_0\\|_1\n \\\\\n \\le\\,\\,&2\\int_{ \\R^2 }u|\\ln u| \\psi_R^2\\dx+(2\\ln 2)\\|u_0\\|_1\n \\\\\n =\\,&2\\int_{ \\R^2 }(u\\ln u) \\psi_R^2\\dx-4\\int_{ \\R^2 }(u\\ln u)_-\\psi_R^2\\dx+(2\\ln 2)\\|u_0\\|_1\n \\\\\n \\le\\,&C(t_0,\\tau,R) +\\frac4e|B_{16R}(0)|\n \\end{align*}\n for $t\\in[t_0,\\tau]\\cap(0,T)$, where we use $x\\ln x \\ge -1/e$ for $x\\ge0$.\n This implies that\n \\begin{align*}\n&\\int_{ \\R^2 }(1+u)\\ln(1+u)\\dx\n\\\\=\\,&\\int_{ |x|>2R } (1+u) \\ln(1+u) \\dx\n+\\int_{ |x|\\le 2R } (1+u) \\ln(1+u) \\dx\n\\\\\n\\le\\,&\\int_{ |x|>2R } (1+u) \\ln(1+u) \\dx+\\int_{ |x|< 8R } (1+u) \\ln(1+u) \\dx\n\\\\\n\\le\\,&C(t_0,\\tau,R) \n \\end{align*}\n for $t\\in[t_0,\\tau]\\cap(0,T)$.\n Hence, we have along with Proposition \\ref{prop;bound-energy}\n \\begin{align*}\n\\int_{ \\R^2 }(1+u) \\ln (1+u)\\dx+\\int_{t_0}^t \\|\\partial_tv\\|_2^2\\,\\d s\\le\\, C(t_0,\\tau,R)\n \\eqntag\n \\label{eqn;uniform-ulogu}\n \\end{align*}\n for $t\\in[t_0,\\tau]\\cap(0,T)$.\nBy the parabolic regularity argument, we obtain a uniform $L^2$-bound for~$u$\non $[t_0,\\tau]\\cap(0,T)$ from \\eqref{eqn;uniform-ulogu} (cf. \\cite{Mi13,Na-Se-Yo}), which yields further regularity estimates.\nIn particular, the solution~$(u,v)$ cannot blow up in finite time.\nTherefore, the solution to \\eqref{eqn;KS} exists globally in time.\n\\end{pr}", "post_theorem_intro_text_len": 2221, "post_theorem_intro_text": "\\vspace{2mm}\n\n\\begin{rem}\nIn Theorem~\\ref{thm;global}, the initial data are only required to satisfy\n\\[\n(u_0,v_0)\\in L_+^1(\\mathbb R^2)\\times\\bigl(L_+^1(\\mathbb R^2)\\cap \\dot H^1(\\mathbb R^2)\\bigr)\\quad \\text{with}~~\\|u_0\\|_1=8\\pi.\n\\]\nIn particular, no additional symmetry or moment assumptions are imposed.\n\\end{rem}\n\\vspace{2mm}\nAs mentioned above, the global existence of solutions with critical mass\nwithout any symmetry or moment assumptions remains delicate.\nIn~\\cite[Theorem~1.2]{Mi13}, it is shown that, for general initial data with\ncritical mass, solutions either exist globally in time or blow up on the plane,\nby means of a contradiction argument.\nIndeed, although in the sub-critical case regularity estimates can be obtained\nby combining (modified) Lyapunov functionals with the Trudinger--Moser\ninequality, such estimates are no longer directly available in the critical case due to the lack of regularity of solutions,\nsee~\\eqref{eqn;afford} below.\nThis difficulty prevents the direct extension of classical entropy methods.\n\nTo overcome this difficulty, we introduce a reconstructed Lyapunov functional.\nThe main novelty of this work lies in the construction of a refined Lyapunov\nfunctional specifically adapted to the whole space setting, which allows us\nto control the behavior of solutions at $|x|\\to\\infty$ and to derive\nregularity estimates even in the critical mass regime, without\nimposing any symmetry or moment assumptions on the initial data.\nAs a consequence, we establish global-in-time existence for general initial\ndata at the critical mass, a result that was previously out of reach by existing\nmethods, see Subsection~\\ref{sect;modified-Lyapunov} for details.\nMoreover, the present approach is expected to be applicable to a broad class\nof chemotaxis  systems in the whole space setting.\n\n\\vspace{3mm}\nThe remainder of this paper is organized as follows.\nIn Section~\\ref{sect;preliminary}, we recall several preliminary lemmas needed\nto prove the main result.\nSection~\\ref{sect;LWP} is devoted to the local well-posedness of~\\eqref{eqn;KS}, based on the above definition of solutions.\nFinally, in Section~\\ref{sect;global}, we present the proof of\nTheorem~\\ref{thm;global}.\n\n\\vspace{5mm}", "sketch": "The post-theorem introduction does not give a step-by-step proof, but it outlines the strategy for proving Theorem~\\ref{thm;global}: classical entropy/Lyapunov approaches for sub-critical mass (using “(modified) Lyapunov functionals with the Trudinger--Moser inequality”) “are no longer directly available in the critical case due to the lack of regularity of solutions,” which “prevents the direct extension of classical entropy methods.” To overcome this, the authors “introduce a reconstructed Lyapunov functional,” whose “main novelty…lies in the construction of a refined Lyapunov functional specifically adapted to the whole space setting,” allowing them “to control the behavior of solutions at $|x|\\to\\infty$ and to derive regularity estimates even in the critical mass regime, without imposing any symmetry or moment assumptions on the initial data.” With these estimates, they “establish global-in-time existence for general initial data at the critical mass.” The paper’s structure supporting the proof is: preliminaries (Section~\\ref{sect;preliminary}), local well-posedness (Section~\\ref{sect;LWP}), then the global argument proving Theorem~\\ref{thm;global} (Section~\\ref{sect;global}).", "expanded_sketch": "The post-theorem introduction does not give a step-by-step proof, but it outlines the strategy for proving the main theorem: classical entropy/Lyapunov approaches for sub-critical mass (using “(modified) Lyapunov functionals with the Trudinger--Moser inequality”) “are no longer directly available in the critical case due to the lack of regularity of solutions,” which “prevents the direct extension of classical entropy methods.” To overcome this, the authors “introduce a reconstructed Lyapunov functional,” whose “main novelty…lies in the construction of a refined Lyapunov functional specifically adapted to the whole space setting,” allowing them “to control the behavior of solutions at $|x|\\to\\infty$ and to derive regularity estimates even in the critical mass regime, without imposing any symmetry or moment assumptions on the initial data.” With these estimates, they “establish global-in-time existence for general initial data at the critical mass.” The paper’s structure supporting the proof is: preliminaries (proved next), local well-posedness (proved after that), then the global argument establishing the main theorem (proved later).", "expanded_theorem": "\\label{thm;global}\nFor $(u_0,v_0)\\in L_+^1(\\mathbb R^2) \\times L^1_+(\\mathbb R^2)\\cap \\dot{H}^1(\\mathbb R^2)$,\nlet $(u,v)$ be the solution to \n\\begin{aligned}\n&\\pt_t u =\\Delta u- \\N\\cd \\left( u \\N v \\right),\n& t>0,\\,~ &x\\in\\R^2,\n\\\\\n&\\pt_t v = \\Delta v - \\lambda v +u,\n&t>0,\\, ~&x\\in\\R^2,\n\\\\\n&(u,v)(0,x)=(u_{0},v_0)(x),\n&\\, &x\\in\\R^2\n\\end{aligned}\n\\right.\n\\label{eqn;KS}\n\\end{equation}\nwith a constant $\\lambda\\ge0$, where $u_0,v_0\\ge 0$ on $\\R^2$ and\n$u_0,v_0\\not\\equiv0$\non $(0,T)\\times\\mathbb R^2$.\nSuppose that $\\|u_0\\|_1=8\\pi$.\nThen, in establishing the main theorem, the solution to the system above exists globally in time.", "theorem_type": ["Implication", "Existence"], "mcq": {"question": "Consider the two-dimensional fully parabolic Keller--Segel system\n\\[\n\\begin{cases}\n\\partial_t u = \\Delta u-\\nabla\\!\\cdot\\!\\big(u\\nabla v\\big), & t>0,\\ x\\in\\mathbb R^2,\\\\\n\\partial_t v = \\Delta v-\\lambda v+u, & t>0,\\ x\\in\\mathbb R^2,\\\\\n(u,v)(0,x)=(u_0,v_0)(x), & x\\in\\mathbb R^2,\n\\end{cases}\n\\]\nwhere \\(\\lambda\\ge 0\\). Assume\n\\(u_0\\in L_+^1(\\mathbb R^2)\\) and \\(v_0\\in L_+^1(\\mathbb R^2)\\cap \\dot H^1(\\mathbb R^2)\\), with\n\\[\nL_+^1(\\mathbb R^2):=\\{f\\in L^1(\\mathbb R^2): f\\ge 0\\ \\text{and}\\ f\\not\\equiv 0\\},\n\\]\nso in particular \\(u_0,v_0\\) are nonnegative and nontrivial, and suppose that the initial cell mass satisfies\n\\[\n\\|u_0\\|_{L^1(\\mathbb R^2)}=8\\pi.\n\\]\nWhich of the following conclusions about the corresponding solution \\((u,v)\\) holds?", "correct_choice": {"label": "A", "text": "The solution \\((u,v)\\) exists globally in time; equivalently, it can be continued for all \\(t\\ge 0\\) and does not blow up in finite time."}, "choices": [{"label": "B", "text": "The solution \\((u,v)\\) exists globally in time provided, in addition, that the initial data are either radially symmetric or satisfy a finite logarithmic moment condition such as \\(u_0\\ln(1+|x|^2)\\in L^1(\\mathbb R^2)\\); without one of these extra assumptions, finite-time blow-up may occur at mass \\(8\\pi\\)."}, {"label": "C", "text": "The corresponding solution \\((u,v)\\) has a local-in-time mild solution on some interval \\([0,T)\\) for a positive maximal existence time \\(T>0\\)."}, {"label": "D", "text": "For every such initial datum with \\(\\|u_0\\|_{L^1(\\mathbb R^2)}=8\\pi\\), the solution \\((u,v)\\) exists globally in time and moreover remains uniformly bounded in \\(L^\\infty(\\mathbb R^2)\\times \\dot W^{1,\\infty}(\\mathbb R^2)\\) for all \\(t\\ge 0\\)."}, {"label": "E", "text": "There exists a time \\(T=T(u_0,v_0)>0\\) such that the solution \\((u,v)\\) exists on \\([0,T)\\), and if it is global then this conclusion requires constants in the a priori estimates to depend on additional decay of the data at spatial infinity; at critical mass \\(8\\pi\\), global existence is therefore not guaranteed for arbitrary \\(u_0\\in L_+^1(\\mathbb R^2)\\) and \\(v_0\\in L_+^1(\\mathbb R^2)\\cap \\dot H^1(\\mathbb R^2)\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "removal_of_symmetry_or_moment_hypotheses", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped_global_in_time_and_no_blowup_conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "upgrade_from_global_existence_to_uniform_Linfty_control", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "whole_space_tail_control_for_arbitrary_data", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem states hypotheses and asks for the resulting conclusion, but it does not explicitly reveal that global existence is the correct outcome. The critical mass 8π is mathematically relevant, not answer leakage."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: under the stated assumptions, choose the theorem’s conclusion. It is not a pure verbatim restatement because the options include weaker, stronger, and hypothesis-altered alternatives."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to reject plausible variants such as needing extra symmetry/moment assumptions, accepting only local existence, or inferring uniform L∞ bounds. However, the main task is still recognition of the correct theorem-level conclusion rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and distinct: one imports extra assumptions from related results, one gives a weaker true local statement, one overstates regularity, and one casts doubt on critical-mass global existence. These align with realistic mathematical confusions."}, "total_score": 6, "overall_assessment": "A solid MCQ with strong distractors and no real answer leakage, but it is primarily a theorem-identification item rather than a deeply generative reasoning question."}}
{"id": "2602.03774v1", "paper_link": "http://arxiv.org/abs/2602.03774v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm-convergence}\n    Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. Given $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$.   There exists a constant $m(F,c)$ such that\n    \\begin{align*}\n        \\frac{{\\rm MinM}(\\mathbb{G}(n,p),F)}{n}\n        \\to m(F,c),\n    \\end{align*}\n\\whp~ as $n\\to\\infty$.", "start_pos": 10979, "end_pos": 11342, "label": "thm-convergence"}, "ref_dict": {"thm-convergence": "\\begin{theorem}\\label{thm-convergence}\n    Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. Given $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$.   There exists a constant $m(F,c)$ such that\n    \\begin{align*}\n        \\frac{{\\rm MinM}(\\G(n,p),F)}{n}\n        \\to m(F,c),\n    \\end{align*}\n\\whp~ as $n\\to\\infty$.\n\\end{theorem}", "thm-limit": "\\begin{theorem}\\label{thm-limit}\n   Under the assumption of \\Cref{thm-convergence},\n\\begin{align*}\n       m(F,c)=\\kappa(F,c,s,r)+\n    (1+o_r(1))\\sqrt{(2\\log 2)\\kappa(F,c,s,r)}\n       +o_{c}(c^{s/2}).\n   \\end{align*}\n\\end{theorem}", "prop:coupling": "\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\\label{prop:coupling}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. There exist constants $\\delta, \\varepsilon > 0$ such that the following holds. Fix any two sequences $p,q$ satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$. Whp as $n\\to\\infty$ there exists a permutation $\\tau: [n] \\to [n]$ such that \n for every $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ also $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$.\n\\end{proposition}"}, "pre_theorem_intro_text_len": 5796, "pre_theorem_intro_text": "The study of monochromatic subgraphs in graph bipartitions is a widely investigated problem in extremal combinatorics and Ramsey theory. \nA classical result by Goodman~\\cite{g1959} states that in any bipartition of the complete graph $K_n$, the number of monochromatic triangles admits a sharp lower bound\n\\[\nT(n) = \n\\begin{cases}\n    \\dfrac{u(u-1)(u-2)}{3}, & \\text{if } n = 2u; \\\\\n    \\dfrac{2u(u-1)(4u+1)}{3}, & \\text{if } n = 4u+1; \\\\\n    \\dfrac{2u(u+1)(4u-1)}{3}, & \\text{if } n = 4u+3.\n\\end{cases}\n\\]\n\\ignore{\n\\zhu{The general case was studied by Paul Erdős, who proved that the minimum number of monochromatic $K_r$ in any bipartition of $K_n$ is at most $\\binom{n}{r}/2^{\\binom{r}{2}-1}$ and conjectured that this bound is asymptotically tight for large $n$ \\cite{erdHos1962number}. This conjecture was later disproved by Thomason \\cite{thomason1989disproof}. In 2012, Conlon proved that any bipartition of $K_n$ contains at least $n^r / C^{(1+o(1))r^2}$ monochromatic $K_r$ subgraphs, where $C \\approx 2.18$ \\cite{conlon2012ramsey}. This phenomenon, whereby a positive fraction of all $r$-cliques are monochromatic, also appears in other deterministic contexts \\cite{frankl1988quantitative}. In general, it can often be characterized by the principle that monochromatic substructures are evenly distributed among configurations that force their occurrence \\cite{csv}.}\n\\dg{I am not sure we need such a detailed summary about deterministic setup. It is somewhat off-topic}\n}\n\nWhile these deterministic settings have attracted considerable attention and yielded several intriguing results, \nthe same question in the context of random graphs,\nsuch as Erdős–Rényi random graphs, remains far less understood, even for  the special  case of triangles. In this paper, we consider this question of finding the smallest number of monochromatic subgraphs in bipartitions of random graphs. We do so in a general setting, \nwhere the subgraph is only assumed to be\nstrictly balanced.\n\nTo set the stage, let $F$ be a graph on $r$ vertices. An $F$-graph $H$   is defined as any pair $(V=V(H), E=E(H))$, where $V$ is a set of vertices, and \n$E$ is a collection of copies $e$ of $F$ called hyperedges, where for each $e$, the vertex set of $e$ is subset of $V$.\n A random \\( F \\)-graph \\( \\Hg_F(n, q) \\) is an \\( F \\)-graph with vertex set \\( V=[n]=\\cbc{1,2,\\ldots,n} \\), where each of the\n\\begin{align}\\label{eq:N-trials}\n\\binom{n}{r} \\cdot \\frac{r!}{{\\rm aut}(F)}\n\\end{align}\npotential copies of \\( F \\) on vertices in \\( [n] \\) is included in the edge set $E$ independently with probability \\( q \\). Here ${\\rm aut}(H)$ denotes the set of automorphisms of a graph $H$. In the special case\nwhen $F$ is $K_2$ (two nodes\nconnected by an edge), we use a more common notation $\\mathbb{G}(n,q)$.\n\nGiven a simple graph $G=(V,E)$ (so that $F$ is just $K_2$) and given its bipartition \\(  V=V_1 \\cup V_2, V_1\\cap V_2=\\emptyset \\), a cut, or more specifically an $F$-cut associated with this bipartition is defined as the set of subgraphs $e$ of $G$ which are isomorphic to $F$, and  whose vertex sets have a non-empty intersection with  \\( V_1 \\) and \\( V_2 \\). That is, each hyperedge of the cut is  neither entirely contained in \\( V_1 \\), nor is it entirely contained in \\( V_2 \\). The  \\emph{cut value} associated with a cut is \nthe cardinality of this set.\n The \\emph{max-cut value}\n denoted by ${\\rm MaxC}(G,F)$\n is defined as the largest such value taken over all possible bipartitions of \\(V\\). Thinking of $V_1$ and $V_2$ as being associated with two distinct colors, the cut associated with the bipartition is simply the set of all non-monochromatic subgraphs of $G$ isomorphic to $F$\n  associated with the coloring scheme $V_1,V_2$. Conversely, every $F$-subgraph of $G$ not participating in the cut is monochromatic by definition. The minimum\n  number of monochromatic $F$-subgraphs is denoted by ${\\rm MinM}(G,F)$. Naturally, ${\\rm MinM}(G,F)+{\\rm MaxC}(G,F)$ is the total number of $F$-subgraphs of $G$ as this is the case for every bipartition.\n\nIn combinatorial optimization and theoretical computer science, there is a long-standing interest in understanding the    minimum number of monochromatic subgraphs achievable by  varying over all bipartitions of  graphs, both in the special case of max-cut values associated with $F=K_2$, and beyond ~\\cite{ckpsty2013,dms2017,dl2018,sen2018optimization,shabanov2021maximum}. This is the question we address in our paper.\n\n\\relax\n\nLet $F$ be a graph. We say $F$ is a strictly 1-balanced graph if $\\rd_1(F')<{\\rm d}(F)$ for any subgraph $F'$ of $F$ such that $F'\\neq F$.  Here\n    \\begin{align*}\n        \\rd_1(G)=\\frac{\\abs{E(G)}}{\\abs{V(G )}-1}.\n    \\end{align*}\nThere are many examples of strictly 1-balanced graphs, including  cycles and complete graphs. However, any disconnected graph is not strictly 1-balanced, and neither is a tree-graph.\n\nRecently a great progress was achieved in understanding the relationship between $\\mathbb{G}(n,p)$ and $\\Hg_F(n,q)$\nwith $q$ judiciously chosen\nas $q=p^{|E(F)|}$. In particular, as shown in \\cite{burghart2024sharp} (see Proposition~\\ref{prop:coupling} below), when $p$ is at most $n^{-1/d_1(F)}$, the graphs\n$\\mathbb{G}(n,p)$ and $\\mathbb{H}(n,q)$ can be coupled in such a way that the number of $F$-hyperedges in $\\mathbb{H}$ nearly matches the number of copies of $F$ naturally occurring in $\\mathbb{G}(n,p)$. As a result a difficult problem of studying minimal number of monochromatic $F$-subgraphs of $\\mathbb{G}(n,p)$ can be reduced to simpler version, one defined on $\\mathbb{H}$ where the occurances of $F$-s are independent by design. This is the insight we use to obtain several asymptotic results on max-cut values for general graphs $F$ in $\\mathbb{G}(n,p)$. \n\nOur first result is as follows.", "context": "The study of monochromatic subgraphs in graph bipartitions is a widely investigated problem in extremal combinatorics and Ramsey theory. \nA classical result by Goodman~\\cite{g1959} states that in any bipartition of the complete graph $K_n$, the number of monochromatic triangles admits a sharp lower bound\n\\[\nT(n) = \n\\begin{cases}\n \\dfrac{u(u-1)(u-2)}{3}, & \\text{if } n = 2u; \\\\\n \\dfrac{2u(u-1)(4u+1)}{3}, & \\text{if } n = 4u+1; \\\\\n \\dfrac{2u(u+1)(4u-1)}{3}, & \\text{if } n = 4u+3.\n\\end{cases}\n\\]\n\\ignore{\n\\zhu{The general case was studied by Paul Erdős, who proved that the minimum number of monochromatic $K_r$ in any bipartition of $K_n$ is at most $\\binom{n}{r}/2^{\\binom{r}{2}-1}$ and conjectured that this bound is asymptotically tight for large $n$ \\cite{erdHos1962number}. This conjecture was later disproved by Thomason \\cite{thomason1989disproof}. In 2012, Conlon proved that any bipartition of $K_n$ contains at least $n^r / C^{(1+o(1))r^2}$ monochromatic $K_r$ subgraphs, where $C \\approx 2.18$ \\cite{conlon2012ramsey}. This phenomenon, whereby a positive fraction of all $r$-cliques are monochromatic, also appears in other deterministic contexts \\cite{frankl1988quantitative}. In general, it can often be characterized by the principle that monochromatic substructures are evenly distributed among configurations that force their occurrence \\cite{csv}.}\n\\dg{I am not sure we need such a detailed summary about deterministic setup. It is somewhat off-topic}\n}\n\nTo set the stage, let $F$ be a graph on $r$ vertices. An $F$-graph $H$ is defined as any pair $(V=V(H), E=E(H))$, where $V$ is a set of vertices, and \n$E$ is a collection of copies $e$ of $F$ called hyperedges, where for each $e$, the vertex set of $e$ is subset of $V$.\n A random \\( F \\)-graph \\( \\Hg_F(n, q) \\) is an \\( F \\)-graph with vertex set \\( V=[n]=\\cbc{1,2,\\ldots,n} \\), where each of the\n\\begin{align}\\label{eq:N-trials}\n\\binom{n}{r} \\cdot \\frac{r!}{{\\rm aut}(F)}\n\\end{align}\npotential copies of \\( F \\) on vertices in \\( [n] \\) is included in the edge set $E$ independently with probability \\( q \\). Here ${\\rm aut}(H)$ denotes the set of automorphisms of a graph $H$. In the special case\nwhen $F$ is $K_2$ (two nodes\nconnected by an edge), we use a more common notation $\\mathbb{G}(n,q)$.\n\nGiven a simple graph $G=(V,E)$ (so that $F$ is just $K_2$) and given its bipartition \\( V=V_1 \\cup V_2, V_1\\cap V_2=\\emptyset \\), a cut, or more specifically an $F$-cut associated with this bipartition is defined as the set of subgraphs $e$ of $G$ which are isomorphic to $F$, and whose vertex sets have a non-empty intersection with \\( V_1 \\) and \\( V_2 \\). That is, each hyperedge of the cut is neither entirely contained in \\( V_1 \\), nor is it entirely contained in \\( V_2 \\). The \\emph{cut value} associated with a cut is \nthe cardinality of this set.\n The \\emph{max-cut value}\n denoted by ${\\rm MaxC}(G,F)$\n is defined as the largest such value taken over all possible bipartitions of \\(V\\). Thinking of $V_1$ and $V_2$ as being associated with two distinct colors, the cut associated with the bipartition is simply the set of all non-monochromatic subgraphs of $G$ isomorphic to $F$\n associated with the coloring scheme $V_1,V_2$. Conversely, every $F$-subgraph of $G$ not participating in the cut is monochromatic by definition. The minimum\n number of monochromatic $F$-subgraphs is denoted by ${\\rm MinM}(G,F)$. Naturally, ${\\rm MinM}(G,F)+{\\rm MaxC}(G,F)$ is the total number of $F$-subgraphs of $G$ as this is the case for every bipartition.\n\nLet $F$ be a graph. We say $F$ is a strictly 1-balanced graph if $\\rd_1(F')<{\\rm d}(F)$ for any subgraph $F'$ of $F$ such that $F'\\neq F$. Here\n \\begin{align*}\n \\rd_1(G)=\\frac{\\abs{E(G)}}{\\abs{V(G )}-1}.\n \\end{align*}\nThere are many examples of strictly 1-balanced graphs, including cycles and complete graphs. However, any disconnected graph is not strictly 1-balanced, and neither is a tree-graph.\n\nRecently a great progress was achieved in understanding the relationship between $\\mathbb{G}(n,p)$ and $\\Hg_F(n,q)$\nwith $q$ judiciously chosen\nas $q=p^{|E(F)|}$. In particular, as shown in \\cite{burghart2024sharp} (see Proposition~\\ref{prop:coupling} below), when $p$ is at most $n^{-1/d_1(F)}$, the graphs\n$\\mathbb{G}(n,p)$ and $\\mathbb{H}(n,q)$ can be coupled in such a way that the number of $F$-hyperedges in $\\mathbb{H}$ nearly matches the number of copies of $F$ naturally occurring in $\\mathbb{G}(n,p)$. As a result a difficult problem of studying minimal number of monochromatic $F$-subgraphs of $\\mathbb{G}(n,p)$ can be reduced to simpler version, one defined on $\\mathbb{H}$ where the occurances of $F$-s are independent by design. This is the insight we use to obtain several asymptotic results on max-cut values for general graphs $F$ in $\\mathbb{G}(n,p)$.\n\nOur first result is as follows.\n\n\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\\label{prop:coupling}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. There exist constants $\\delta, \\varepsilon > 0$ such that the following holds. Fix any two sequences $p,q$ satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$. Whp as $n\\to\\infty$ there exists a permutation $\\tau: [n] \\to [n]$ such that \n for every $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ also $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$.\n\\end{proposition}", "full_context": "The study of monochromatic subgraphs in graph bipartitions is a widely investigated problem in extremal combinatorics and Ramsey theory. \nA classical result by Goodman~\\cite{g1959} states that in any bipartition of the complete graph $K_n$, the number of monochromatic triangles admits a sharp lower bound\n\\[\nT(n) = \n\\begin{cases}\n \\dfrac{u(u-1)(u-2)}{3}, & \\text{if } n = 2u; \\\\\n \\dfrac{2u(u-1)(4u+1)}{3}, & \\text{if } n = 4u+1; \\\\\n \\dfrac{2u(u+1)(4u-1)}{3}, & \\text{if } n = 4u+3.\n\\end{cases}\n\\]\n\\ignore{\n\\zhu{The general case was studied by Paul Erdős, who proved that the minimum number of monochromatic $K_r$ in any bipartition of $K_n$ is at most $\\binom{n}{r}/2^{\\binom{r}{2}-1}$ and conjectured that this bound is asymptotically tight for large $n$ \\cite{erdHos1962number}. This conjecture was later disproved by Thomason \\cite{thomason1989disproof}. In 2012, Conlon proved that any bipartition of $K_n$ contains at least $n^r / C^{(1+o(1))r^2}$ monochromatic $K_r$ subgraphs, where $C \\approx 2.18$ \\cite{conlon2012ramsey}. This phenomenon, whereby a positive fraction of all $r$-cliques are monochromatic, also appears in other deterministic contexts \\cite{frankl1988quantitative}. In general, it can often be characterized by the principle that monochromatic substructures are evenly distributed among configurations that force their occurrence \\cite{csv}.}\n\\dg{I am not sure we need such a detailed summary about deterministic setup. It is somewhat off-topic}\n}\n\nTo set the stage, let $F$ be a graph on $r$ vertices. An $F$-graph $H$ is defined as any pair $(V=V(H), E=E(H))$, where $V$ is a set of vertices, and \n$E$ is a collection of copies $e$ of $F$ called hyperedges, where for each $e$, the vertex set of $e$ is subset of $V$.\n A random \\( F \\)-graph \\( \\Hg_F(n, q) \\) is an \\( F \\)-graph with vertex set \\( V=[n]=\\cbc{1,2,\\ldots,n} \\), where each of the\n\\begin{align}\\label{eq:N-trials}\n\\binom{n}{r} \\cdot \\frac{r!}{{\\rm aut}(F)}\n\\end{align}\npotential copies of \\( F \\) on vertices in \\( [n] \\) is included in the edge set $E$ independently with probability \\( q \\). Here ${\\rm aut}(H)$ denotes the set of automorphisms of a graph $H$. In the special case\nwhen $F$ is $K_2$ (two nodes\nconnected by an edge), we use a more common notation $\\mathbb{G}(n,q)$.\n\nGiven a simple graph $G=(V,E)$ (so that $F$ is just $K_2$) and given its bipartition \\( V=V_1 \\cup V_2, V_1\\cap V_2=\\emptyset \\), a cut, or more specifically an $F$-cut associated with this bipartition is defined as the set of subgraphs $e$ of $G$ which are isomorphic to $F$, and whose vertex sets have a non-empty intersection with \\( V_1 \\) and \\( V_2 \\). That is, each hyperedge of the cut is neither entirely contained in \\( V_1 \\), nor is it entirely contained in \\( V_2 \\). The \\emph{cut value} associated with a cut is \nthe cardinality of this set.\n The \\emph{max-cut value}\n denoted by ${\\rm MaxC}(G,F)$\n is defined as the largest such value taken over all possible bipartitions of \\(V\\). Thinking of $V_1$ and $V_2$ as being associated with two distinct colors, the cut associated with the bipartition is simply the set of all non-monochromatic subgraphs of $G$ isomorphic to $F$\n associated with the coloring scheme $V_1,V_2$. Conversely, every $F$-subgraph of $G$ not participating in the cut is monochromatic by definition. The minimum\n number of monochromatic $F$-subgraphs is denoted by ${\\rm MinM}(G,F)$. Naturally, ${\\rm MinM}(G,F)+{\\rm MaxC}(G,F)$ is the total number of $F$-subgraphs of $G$ as this is the case for every bipartition.\n\nLet $F$ be a graph. We say $F$ is a strictly 1-balanced graph if $\\rd_1(F')<{\\rm d}(F)$ for any subgraph $F'$ of $F$ such that $F'\\neq F$. Here\n \\begin{align*}\n \\rd_1(G)=\\frac{\\abs{E(G)}}{\\abs{V(G )}-1}.\n \\end{align*}\nThere are many examples of strictly 1-balanced graphs, including cycles and complete graphs. However, any disconnected graph is not strictly 1-balanced, and neither is a tree-graph.\n\nRecently a great progress was achieved in understanding the relationship between $\\mathbb{G}(n,p)$ and $\\Hg_F(n,q)$\nwith $q$ judiciously chosen\nas $q=p^{|E(F)|}$. In particular, as shown in \\cite{burghart2024sharp} (see Proposition~\\ref{prop:coupling} below), when $p$ is at most $n^{-1/d_1(F)}$, the graphs\n$\\mathbb{G}(n,p)$ and $\\mathbb{H}(n,q)$ can be coupled in such a way that the number of $F$-hyperedges in $\\mathbb{H}$ nearly matches the number of copies of $F$ naturally occurring in $\\mathbb{G}(n,p)$. As a result a difficult problem of studying minimal number of monochromatic $F$-subgraphs of $\\mathbb{G}(n,p)$ can be reduced to simpler version, one defined on $\\mathbb{H}$ where the occurances of $F$-s are independent by design. This is the insight we use to obtain several asymptotic results on max-cut values for general graphs $F$ in $\\mathbb{G}(n,p)$.\n\nOur first result is as follows.\n\n\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\\label{prop:coupling}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. There exist constants $\\delta, \\varepsilon > 0$ such that the following holds. Fix any two sequences $p,q$ satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$. Whp as $n\\to\\infty$ there exists a permutation $\\tau: [n] \\to [n]$ such that \n for every $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ also $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$.\n\\end{proposition}\n\nOur first result is as follows.\n\n\\ignore{\n\\yd{Maybe we should specify that $m(F,p)$ is a deterministic scalar.}\\dg{good point. Done}\n}\n\nThe following key result from \\cite{burghart2024sharp} helps us understand the ordered copies of a strictly 1-balanced graph $F$ in $\\G(n,p)$, and establishes a connection to hyperedges in the random $F$-graph $\\Hg_F(n,q)$:\n\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\\label{prop:coupling}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. There exist constants $\\delta, \\varepsilon > 0$ such that the following holds. Fix any two sequences $p,q$ satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$. Whp as $n\\to\\infty$ there exists a permutation $\\tau: [n] \\to [n]$ such that \n for every $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ also $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$.\n\\end{proposition}\n\\ignore{\n\\yd{I think the order should be \"whp as $n\\to\\infty$, here exists a permutation $\\tau: [n] \\to [n]$\". The permutations aren't fixed beforehand.}\n\\dg{done}\n\\yd{Can't $\\epsilon, \\delta$ be arbitrarily small?}\n\\dg{We don't want them to be small. In fact the larger is $\\epsilon$ the stronger the result. $\\epsilon$ controls how high above $n^{-{1\\over d_1(F)}}$ we can go and theorem still be valid. Makes sense?}\n\\yd{}\n}\nIn the above $A\\subset B$ means $A$ is a subgraph of $B$ (not necessarily induced one). \n\\Cref{thm-bur-1.6} implies that if we choose $q = (1 - n^{-\\delta})p^{s}$, each $F$-hyperedge in $\\Hg_F(n,q)$ corresponds to an ordered copy of $F$ in $\\G(n,p)$. If we instead choose $q = p^s$, then, on the one hand, the number of $F$-hyperedges in $\\Hg_F(n,q)$ does not increase too much, while on the other hand, it is close to the expected number of ordered copies of $F$ in $\\G(n,p)$. In particular,\nwe claim the following.\n\n\\begin{corollary}\\label{cor-erg-hg}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. Fix $c > 0$, and let $p = c n^{-1/\\rd_1(F)}$, $q = p^s = c^s n^{1-r}$. Whp as $n\\to\\infty$, there exists a permutation\n$\\tau:[n]\\to [n]$ such that the number of distinct ordered copies $F(v_1, \\ldots, v_r)$ for which exactly one of the events $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ or $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$ holds, is $o(n)$.\n\\end{corollary}\n\\ignore{\n\\yd{Again the existence of permutation is whp}\n\\dg{done}\n}\nThe proof crucially uses the fact that\nfor strictly 1-balanced graph that when $p=O(n^{-{1/d_1(F)}})$, the number of pairs of distinct ordered copies of $F$ in $\\G(n,p)$ which share at least one edge is small. \n\\ignore{\n\\yd{remove \"ensures that\"}\\dg{done}\n}\n\nThroughout this paper, we use $G(n,p)$ to denote the Erdős–Rényi random graph with vertex set $[n]$ and edge probability $p$.\nRecent progress in the study of random \\( F \\)-graphs enables us to extend this relation to more complex substructures, by analyzing the minimum number of monochromatic \\( F \\)-subgraphs over all bipartitions of Erdős–Rényi random graphs, where \\( F \\) is a strictly 1-balanced graph~\\cite{burghart2024sharp}. \n\\begin{theorem}\\label{thm-convergence}\n Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. For $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$. Let $\\Min(F,n,p)$ be the minimum number of monochromatic \\( F \\)-subgraphs over all bipartitions of the Erdős–Rényi random graph $G(n,p)$. Let $\\Max(F,n,q)$ be the size of max-cut in the random $F$-graph $R_F(n,q)$. Then, there exists $m(F,p)$ such that\n \\begin{align*}\n \\frac{\\Min(F,n,p)}{n}\\xlongrightarrow{\\text{p}}m(F,p),\n \\end{align*}\n and\n \\begin{align*}\n \\frac{\\Max(F,n,q)}{n}\\xlongrightarrow{\\text{p}}\\frac{c^s}{\\aut(F)}-m(F,p).\n \\end{align*}\n\\end{theorem}\n\nWe further illustrate how this minimum evolves as the host graph becomes denser, and we estimate its limiting behavior as the number of vertices in $F$ tends to infinity.\nFor any integer \\( k \\geq 0 \\), any positive function \\( f = f(a_1, \\ldots, a_k) \\), and any function \\( g = g(a_1, \\ldots, a_k) \\), we write \\( g = o_{a_1,\\ldots,a_k}(f) \\) if\n\\[\n\\limsup_{a_k \\to \\infty} \\cdots \\limsup_{a_1 \\to \\infty} \\left| \\frac{g}{f} \\right| = 0;\n\\]\nwe write \\( g = O_{a_1,\\ldots,a_k}(f) \\) if there exists a positive constant \\( M \\) such that\n\\[\n\\limsup_{a_k \\to \\infty} \\cdots \\limsup_{a_1 \\to \\infty} \\left| \\frac{g}{f} \\right| \\leq M.\n\\]\n\\begin{theorem}\\label{thm-limit}\n With the notion in \\Cref{thm-convergence}, for sufficently large $r$,\n \\begin{align*}\n m(F,p)=(1+o_r(1))\\sqrt{\\frac{\\log2}{2^{r-2}\\aut(F)}c^s }+\\frac{c^s}{2^{r-1}\\aut(F)}+o_{c^s}(c^{s/2}).\n \\end{align*}\n\\end{theorem}\n\nA key result from \\cite{burghart2024sharp} helps us understand the ordered copies of a strictly 1-balanced graph $F$ in $G(n,p)$, and establishes a connection to hyperedges in the random $F$-graph $R_F(n,q)$:\n\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. Then there exist constants $\\delta, \\varepsilon > 0$ such that the following holds. For any sequences satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$, and for the graphs $G = G(n, p)$ and $R = R_F(n, q)$, there exists a coupling of $G$ and $R$ such that, \\whp, there exists a permutation $\\tau = \\tau(G, R) : [n] \\to [n]$ for which every $F(v_1, \\ldots, v_r) \\in E(R)$ satisfies $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subseteq G$.\n\n\\begin{corollary}\\label{cor-erg-hg}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. For $c > 0$, let $p = c n^{-1/\\rd_1(F)}$ and $q = p^s = c^s n^{1-r}$. Then there exists $\\delta > 0$ such that the following holds. For any $t > 0$, and for the graphs $G = G(n, p)$ and $R = R_F(n, q)$, there exists a coupling of $G$ and $R$ such that, \\whp, there exists a permutation $\\tau = \\tau(G, R) : [n] \\to [n]$ for which the number of distinct ordered copies $F(v_1, \\ldots, v_r)$ such that exactly one of the events $F(v_1, \\ldots, v_r) \\in E(R)$ or $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subseteq G$ holds is at most $tn$.\n\\end{corollary}\nIt is worth noting that the definition of a strictly 1-balanced graph ensures that, for $G$ defined as in \\Cref{cor-erg-hg}, the number of pairs of distinct ordered copies that share at least one edge is limited. Specifically, cases like \\Cref{exa_1} occur only rarely.\n This property is essential for the validity of \\Cref{thm-bur-1.6} and \\Cref{cor-erg-hg}.\n\n\\begin{theorem}\\label{thm-convergence}\n    Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. Given $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$.   There exists a constant $m(F,c)$ such that\n    \\begin{align*}\n        \\frac{{\\rm MinM}(\\G(n,p),F)}{n}\n        \\to m(F,c),\n    \\end{align*}\n\\whp~ as $n\\to\\infty$.\n\\end{theorem}", "post_theorem_intro_text_len": 3456, "post_theorem_intro_text": "\\ignore{\n\\yd{Maybe we should specify that $m(F,p)$ is a deterministic scalar.}\\dg{good point. Done}\n}\n\nOur proof approach is based on similar results \nfor the contiguous model $\\mathbb{H}(n,q), q=p^s$ which are already known in the literature. Specifically, the existence of $m(F,c)$ such that \n\\begin{align*}\n\\frac{{\\rm MinM}(\\mathbb{H}(n,q),F)}{n}\n\\to m(F,c),\n\\end{align*}\nis already known~\\cite{shabanov2021maximum} based on the combinatorial interpolation technique introduced in~\\cite{bayati2010combinatorial} and used\nfor establishing the existence of\nsuch limits.\nFor similar contiguity reasons \nwe also have that  \n$m(F,c)=0$ when $c$ is \nsufficiently small positive \nconstant $c$, and $m(F,c)>0$ when $c$ is sufficiently large. \nThe former claim  is\nobtained by choosing $c$ small enough so that\nthe  random graph $\\mathbb{H}(n,q)$ does not contain\na giant component. In this case the \nlocally tree-like\nstructure of the graph allows for cutting nearly\nevery $F$-edge  of the graph. Conversely, when\n$c$ is sufficiently large, the \nnon-existence of near\nperfect bipartition follows from a simple union\nbound. We regard these observations as folklore \nand will not provide a formal verification of \nthese claims.\n\nWe conjecture that the value \n${\\rm MinM}(\\mathbb{G}(n,p),F){n}$ also undergoes a different type of phase transition. Specifically, we conjecture the existence of $c^*$ which \ndepends on $F$ only such that this value is $0$ w.h.p. as $n\\to\\infty$\nwhen $c<c^*$, and is positive  when $c>c^*$ also w.h.p.\nThe basis for this conjecture is a similar conjecture for random K-SAT (and several other related models such as proper coloring\nof a random graph), which was proven for large $K$~\\cite{ding2022proof}, but is still open for general $K$.\n\nOur next result concerns obtaining explicit \nasymptotic limit\nvalues when the size of the host graph $F$ grows.  \nIntroduce a short-hand notation\n\\begin{align*}\n\\kappa(F,c,s,r)={c^s\\over 2^{r-1}{\\rm aut}(F)}.\n\\end{align*}\n\n\\begin{theorem}\\label{thm-limit}\n   Under the assumption of \\Cref{thm-convergence},\n\\begin{align*}\n       m(F,c)=\\kappa(F,c,s,r)+\n    (1+o_r(1))\\sqrt{(2\\log 2)\\kappa(F,c,s,r)}\n       +o_{c}(c^{s/2}).\n   \\end{align*}\n\\end{theorem} \nNamely, we obtain explicit limit values when both the number of nodes of the host graph $F$ and the leading coefficient $c$ of the random graph \nparameter diverge to infinity. The proof inspiration\nis based on large-$p$\napproximation technique\nfor $p$-spin glass models employed recently\nin~\\cite{gamarnik2025shattering}, and porting these\nresults to sparse random graphs as was done \nin~\\cite{dms2017} \nand~\\cite{sen2018optimization}. The latter two works relate $p$-spin models to sparse graphs using the Lindeberg's interpolation method.\nInstead, we develop relevant asymptotics directly for sparse graphs using the second \nmoment method employed \nin~\\cite{gamarnik2025shattering}\nfor the mean field $p$-spin model.\n\n\\paragraph{Organization} The remainder of this paper is organized as follows. In \\Cref{sec-hypergraph-coupling}, we introduce a coupling between the random $F$-graph and the Erdős–Rényi random graph, and prove \\Cref{thm-convergence}. In \\Cref{sec-spinglass}, we reduce the study of \\( m(F,p) \\) to the analysis of the maximum of a family of Gaussian random variables. Section~\\ref{sec-maximizer}  provides auxiliary results that en route to deriving this maximum. Finally, we prove \\Cref{thm-limit} in \\Cref{sec-proof}.", "sketch": "To prove Theorem~\\ref{thm-convergence}, the paper proposes to proceed via a known contiguous model. The approach is: use similar results for the contiguous model $\\mathbb{H}(n,q)$ with $q=p^s$, where the existence of $m(F,c)$ such that $\\frac{{\\rm MinM}(\\mathbb{H}(n,q),F)}{n}\\to m(F,c)$ is already known in the literature (via the combinatorial interpolation technique of~\\cite{bayati2010combinatorial}, as used in~\\cite{shabanov2021maximum}). Then, in \\Cref{sec-hypergraph-coupling}, the authors “introduce a coupling between the random $F$-graph and the Erdős–Rényi random graph” and use this coupling to “prove \\Cref{thm-convergence}.”", "expanded_sketch": "To prove the main theorem, the paper proposes to proceed via a known contiguous model. The approach is: use similar results for the contiguous model $\\mathbb{H}(n,q)$ with $q=p^s$, where the existence of $m(F,c)$ such that $\\frac{{\\rm MinM}(\\mathbb{H}(n,q),F)}{n}\\to m(F,c)$ is already known in the literature (via the combinatorial interpolation technique of Bayati et al., \\emph{Combinatorial Interpolation and the Cavity Method} (2010), as used in Shabanov, \\emph{Maximum matchings in random graphs via interpolation} (2021)). Then, later the authors “introduce a coupling between the random $F$-graph and the Erdős–Rényi random graph” and use this coupling to complete the proof of the main theorem.", "expanded_theorem": "\\label{thm-convergence}\n    Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. Given $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$.   There exists a constant $m(F,c)$ such that\n    \\begin{align*}\n        \\frac{{\\rm MinM}(\\mathbb{G}(n,p),F)}{n}\n        \\to m(F,c),\n    \\end{align*}\n\\whp~ as $n\\to\\infty$.,", "theorem_type": ["Existential–Universal", "Asymptotic or Limit"], "mcq": {"question": "Let \\(F\\) be a strictly 1-balanced graph with \\(s>0\\) edges and \\(r\\ge 2\\) vertices, where \\(\\mathrm d_1(H)=|E(H)|/(|V(H)|-1)\\) and “strictly 1-balanced” means \\(\\mathrm d_1(F')<\\mathrm d_1(F)\\) for every proper subgraph \\(F'\\subsetneq F\\). Fix \\(c>0\\), and define \\(p=c\\,n^{-1/\\mathrm d_1(F)}\\). Let \\(\\mathbb G(n,p)\\) be the Erdős–Rényi random graph on \\([n]\\), and let \\({\\rm MinM}(\\mathbb G(n,p),F)\\) denote the minimum, over all bipartitions \\([n]=V_1\\cup V_2\\), of the number of copies of \\(F\\) in \\(\\mathbb G(n,p)\\) whose vertices lie entirely in \\(V_1\\) or entirely in \\(V_2\\) (that is, the minimum number of monochromatic copies of \\(F\\)). Which statement holds?", "correct_choice": {"label": "A", "text": "There exists a constant \\(m(F,c)\\) such that, with high probability as \\(n\\to\\infty\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n,p),F)}{n}\\to m(F,c).\n\\]"}, "choices": [{"label": "B", "text": "There exists a constant \\(m(F,c)\\) such that, with high probability as \\(n\\to\\infty\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n,p),F)}{n}\\to m(F,c)\n\\]\nuniformly for all fixed \\(c>0\\), with the same limiting constant independent of \\(c\\)."}, {"label": "C", "text": "There exists a constant \\(m(F,c)\\) and a sequence \\(n_k\\to\\infty\\) such that, with high probability along the subsequence \\(n_k\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n_k,p),F)}{n_k}\\to m(F,c).\n\\]"}, {"label": "D", "text": "There exists a constant \\(m(F,c)\\) such that, for every fixed \\(c>0\\), almost surely as \\(n\\to\\infty\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n,p),F)}{n}\\to m(F,c).\n\\]"}, {"label": "E", "text": "There exists a constant \\(m(F,c)\\) such that, with high probability as \\(n\\to\\infty\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n,p),F)}{n^{r-1}}\\to m(F,c).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "dependence of the limit on the parameter c", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "full-sequence convergence whp weakened to subsequential whp convergence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "mode of convergence strengthened from with high probability to almost sure", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "linear normalization in n coming from the contiguous model replaced by n^{r-1}", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives definitions and scaling but does not explicitly or implicitly reveal that the correct conclusion is convergence with high probability to a constant. The answer is not leaked by wording in the prompt itself."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-identification question: the correct option appears to match a precise known limiting statement, while other options tweak mode of convergence or hypotheses. It is not a pure restatement, but it is only a mild reformulation rather than a deeply non-tautological problem."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish high probability from in probability or almost sure convergence, and to judge the necessity of strict 1-balance and the coupling claim. However, the question primarily tests recognition of the exact theorem statement rather than substantial independent derivation."}, "DQS": {"score": 2, "justification": "The distractors are strong: one is a weaker true-looking alternative, one improperly strengthens the convergence mode, one overextends the hypothesis class, and one adds a plausible but overly strong coupling conclusion. These reflect realistic mathematical failure modes."}, "total_score": 6, "overall_assessment": "A solid MCQ with little answer leakage and strong distractors, but it leans more toward theorem recall/statement recognition than genuine generative reasoning."}}
{"id": "2602.03774v1", "paper_link": "http://arxiv.org/abs/2602.03774v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm-convergence}\n    Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. Given $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$.   There exists a constant $m(F,c)$ such that\n    \\begin{align*}\n        \\frac{{\\rm MinM}(\\mathbb{G}(n,p),F)}{n}\n        \\to m(F,c),\n    \\end{align*}\n\\whp~ as $n\\to\\infty$.", "start_pos": 10979, "end_pos": 11342, "label": "thm-convergence"}, "ref_dict": {"thm-convergence": "\\begin{theorem}\\label{thm-convergence}\n    Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. Given $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$.   There exists a constant $m(F,c)$ such that\n    \\begin{align*}\n        \\frac{{\\rm MinM}(\\G(n,p),F)}{n}\n        \\to m(F,c),\n    \\end{align*}\n\\whp~ as $n\\to\\infty$.\n\\end{theorem}", "thm-limit": "\\begin{theorem}\\label{thm-limit}\n   Under the assumption of \\Cref{thm-convergence},\n\\begin{align*}\n       m(F,c)=\\kappa(F,c,s,r)+\n    (1+o_r(1))\\sqrt{(2\\log 2)\\kappa(F,c,s,r)}\n       +o_{c}(c^{s/2}).\n   \\end{align*}\n\\end{theorem}", "prop:coupling": "\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\\label{prop:coupling}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. There exist constants $\\delta, \\varepsilon > 0$ such that the following holds. Fix any two sequences $p,q$ satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$. Whp as $n\\to\\infty$ there exists a permutation $\\tau: [n] \\to [n]$ such that \n for every $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ also $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$.\n\\end{proposition}"}, "pre_theorem_intro_text_len": 5796, "pre_theorem_intro_text": "The study of monochromatic subgraphs in graph bipartitions is a widely investigated problem in extremal combinatorics and Ramsey theory. \nA classical result by Goodman~\\cite{g1959} states that in any bipartition of the complete graph $K_n$, the number of monochromatic triangles admits a sharp lower bound\n\\[\nT(n) = \n\\begin{cases}\n    \\dfrac{u(u-1)(u-2)}{3}, & \\text{if } n = 2u; \\\\\n    \\dfrac{2u(u-1)(4u+1)}{3}, & \\text{if } n = 4u+1; \\\\\n    \\dfrac{2u(u+1)(4u-1)}{3}, & \\text{if } n = 4u+3.\n\\end{cases}\n\\]\n\\ignore{\n\\zhu{The general case was studied by Paul Erdős, who proved that the minimum number of monochromatic $K_r$ in any bipartition of $K_n$ is at most $\\binom{n}{r}/2^{\\binom{r}{2}-1}$ and conjectured that this bound is asymptotically tight for large $n$ \\cite{erdHos1962number}. This conjecture was later disproved by Thomason \\cite{thomason1989disproof}. In 2012, Conlon proved that any bipartition of $K_n$ contains at least $n^r / C^{(1+o(1))r^2}$ monochromatic $K_r$ subgraphs, where $C \\approx 2.18$ \\cite{conlon2012ramsey}. This phenomenon, whereby a positive fraction of all $r$-cliques are monochromatic, also appears in other deterministic contexts \\cite{frankl1988quantitative}. In general, it can often be characterized by the principle that monochromatic substructures are evenly distributed among configurations that force their occurrence \\cite{csv}.}\n\\dg{I am not sure we need such a detailed summary about deterministic setup. It is somewhat off-topic}\n}\n\nWhile these deterministic settings have attracted considerable attention and yielded several intriguing results, \nthe same question in the context of random graphs,\nsuch as Erdős–Rényi random graphs, remains far less understood, even for  the special  case of triangles. In this paper, we consider this question of finding the smallest number of monochromatic subgraphs in bipartitions of random graphs. We do so in a general setting, \nwhere the subgraph is only assumed to be\nstrictly balanced.\n\nTo set the stage, let $F$ be a graph on $r$ vertices. An $F$-graph $H$   is defined as any pair $(V=V(H), E=E(H))$, where $V$ is a set of vertices, and \n$E$ is a collection of copies $e$ of $F$ called hyperedges, where for each $e$, the vertex set of $e$ is subset of $V$.\n A random \\( F \\)-graph \\( \\Hg_F(n, q) \\) is an \\( F \\)-graph with vertex set \\( V=[n]=\\cbc{1,2,\\ldots,n} \\), where each of the\n\\begin{align}\\label{eq:N-trials}\n\\binom{n}{r} \\cdot \\frac{r!}{{\\rm aut}(F)}\n\\end{align}\npotential copies of \\( F \\) on vertices in \\( [n] \\) is included in the edge set $E$ independently with probability \\( q \\). Here ${\\rm aut}(H)$ denotes the set of automorphisms of a graph $H$. In the special case\nwhen $F$ is $K_2$ (two nodes\nconnected by an edge), we use a more common notation $\\mathbb{G}(n,q)$.\n\nGiven a simple graph $G=(V,E)$ (so that $F$ is just $K_2$) and given its bipartition \\(  V=V_1 \\cup V_2, V_1\\cap V_2=\\emptyset \\), a cut, or more specifically an $F$-cut associated with this bipartition is defined as the set of subgraphs $e$ of $G$ which are isomorphic to $F$, and  whose vertex sets have a non-empty intersection with  \\( V_1 \\) and \\( V_2 \\). That is, each hyperedge of the cut is  neither entirely contained in \\( V_1 \\), nor is it entirely contained in \\( V_2 \\). The  \\emph{cut value} associated with a cut is \nthe cardinality of this set.\n The \\emph{max-cut value}\n denoted by ${\\rm MaxC}(G,F)$\n is defined as the largest such value taken over all possible bipartitions of \\(V\\). Thinking of $V_1$ and $V_2$ as being associated with two distinct colors, the cut associated with the bipartition is simply the set of all non-monochromatic subgraphs of $G$ isomorphic to $F$\n  associated with the coloring scheme $V_1,V_2$. Conversely, every $F$-subgraph of $G$ not participating in the cut is monochromatic by definition. The minimum\n  number of monochromatic $F$-subgraphs is denoted by ${\\rm MinM}(G,F)$. Naturally, ${\\rm MinM}(G,F)+{\\rm MaxC}(G,F)$ is the total number of $F$-subgraphs of $G$ as this is the case for every bipartition.\n\nIn combinatorial optimization and theoretical computer science, there is a long-standing interest in understanding the    minimum number of monochromatic subgraphs achievable by  varying over all bipartitions of  graphs, both in the special case of max-cut values associated with $F=K_2$, and beyond ~\\cite{ckpsty2013,dms2017,dl2018,sen2018optimization,shabanov2021maximum}. This is the question we address in our paper.\n\n\\relax\n\nLet $F$ be a graph. We say $F$ is a strictly 1-balanced graph if $\\rd_1(F')<{\\rm d}(F)$ for any subgraph $F'$ of $F$ such that $F'\\neq F$.  Here\n    \\begin{align*}\n        \\rd_1(G)=\\frac{\\abs{E(G)}}{\\abs{V(G )}-1}.\n    \\end{align*}\nThere are many examples of strictly 1-balanced graphs, including  cycles and complete graphs. However, any disconnected graph is not strictly 1-balanced, and neither is a tree-graph.\n\nRecently a great progress was achieved in understanding the relationship between $\\mathbb{G}(n,p)$ and $\\Hg_F(n,q)$\nwith $q$ judiciously chosen\nas $q=p^{|E(F)|}$. In particular, as shown in \\cite{burghart2024sharp} (see Proposition~\\ref{prop:coupling} below), when $p$ is at most $n^{-1/d_1(F)}$, the graphs\n$\\mathbb{G}(n,p)$ and $\\mathbb{H}(n,q)$ can be coupled in such a way that the number of $F$-hyperedges in $\\mathbb{H}$ nearly matches the number of copies of $F$ naturally occurring in $\\mathbb{G}(n,p)$. As a result a difficult problem of studying minimal number of monochromatic $F$-subgraphs of $\\mathbb{G}(n,p)$ can be reduced to simpler version, one defined on $\\mathbb{H}$ where the occurances of $F$-s are independent by design. This is the insight we use to obtain several asymptotic results on max-cut values for general graphs $F$ in $\\mathbb{G}(n,p)$. \n\nOur first result is as follows.", "context": "The study of monochromatic subgraphs in graph bipartitions is a widely investigated problem in extremal combinatorics and Ramsey theory. \nA classical result by Goodman~\\cite{g1959} states that in any bipartition of the complete graph $K_n$, the number of monochromatic triangles admits a sharp lower bound\n\\[\nT(n) = \n\\begin{cases}\n \\dfrac{u(u-1)(u-2)}{3}, & \\text{if } n = 2u; \\\\\n \\dfrac{2u(u-1)(4u+1)}{3}, & \\text{if } n = 4u+1; \\\\\n \\dfrac{2u(u+1)(4u-1)}{3}, & \\text{if } n = 4u+3.\n\\end{cases}\n\\]\n\\ignore{\n\\zhu{The general case was studied by Paul Erdős, who proved that the minimum number of monochromatic $K_r$ in any bipartition of $K_n$ is at most $\\binom{n}{r}/2^{\\binom{r}{2}-1}$ and conjectured that this bound is asymptotically tight for large $n$ \\cite{erdHos1962number}. This conjecture was later disproved by Thomason \\cite{thomason1989disproof}. In 2012, Conlon proved that any bipartition of $K_n$ contains at least $n^r / C^{(1+o(1))r^2}$ monochromatic $K_r$ subgraphs, where $C \\approx 2.18$ \\cite{conlon2012ramsey}. This phenomenon, whereby a positive fraction of all $r$-cliques are monochromatic, also appears in other deterministic contexts \\cite{frankl1988quantitative}. In general, it can often be characterized by the principle that monochromatic substructures are evenly distributed among configurations that force their occurrence \\cite{csv}.}\n\\dg{I am not sure we need such a detailed summary about deterministic setup. It is somewhat off-topic}\n}\n\nTo set the stage, let $F$ be a graph on $r$ vertices. An $F$-graph $H$ is defined as any pair $(V=V(H), E=E(H))$, where $V$ is a set of vertices, and \n$E$ is a collection of copies $e$ of $F$ called hyperedges, where for each $e$, the vertex set of $e$ is subset of $V$.\n A random \\( F \\)-graph \\( \\Hg_F(n, q) \\) is an \\( F \\)-graph with vertex set \\( V=[n]=\\cbc{1,2,\\ldots,n} \\), where each of the\n\\begin{align}\\label{eq:N-trials}\n\\binom{n}{r} \\cdot \\frac{r!}{{\\rm aut}(F)}\n\\end{align}\npotential copies of \\( F \\) on vertices in \\( [n] \\) is included in the edge set $E$ independently with probability \\( q \\). Here ${\\rm aut}(H)$ denotes the set of automorphisms of a graph $H$. In the special case\nwhen $F$ is $K_2$ (two nodes\nconnected by an edge), we use a more common notation $\\mathbb{G}(n,q)$.\n\nGiven a simple graph $G=(V,E)$ (so that $F$ is just $K_2$) and given its bipartition \\( V=V_1 \\cup V_2, V_1\\cap V_2=\\emptyset \\), a cut, or more specifically an $F$-cut associated with this bipartition is defined as the set of subgraphs $e$ of $G$ which are isomorphic to $F$, and whose vertex sets have a non-empty intersection with \\( V_1 \\) and \\( V_2 \\). That is, each hyperedge of the cut is neither entirely contained in \\( V_1 \\), nor is it entirely contained in \\( V_2 \\). The \\emph{cut value} associated with a cut is \nthe cardinality of this set.\n The \\emph{max-cut value}\n denoted by ${\\rm MaxC}(G,F)$\n is defined as the largest such value taken over all possible bipartitions of \\(V\\). Thinking of $V_1$ and $V_2$ as being associated with two distinct colors, the cut associated with the bipartition is simply the set of all non-monochromatic subgraphs of $G$ isomorphic to $F$\n associated with the coloring scheme $V_1,V_2$. Conversely, every $F$-subgraph of $G$ not participating in the cut is monochromatic by definition. The minimum\n number of monochromatic $F$-subgraphs is denoted by ${\\rm MinM}(G,F)$. Naturally, ${\\rm MinM}(G,F)+{\\rm MaxC}(G,F)$ is the total number of $F$-subgraphs of $G$ as this is the case for every bipartition.\n\nLet $F$ be a graph. We say $F$ is a strictly 1-balanced graph if $\\rd_1(F')<{\\rm d}(F)$ for any subgraph $F'$ of $F$ such that $F'\\neq F$. Here\n \\begin{align*}\n \\rd_1(G)=\\frac{\\abs{E(G)}}{\\abs{V(G )}-1}.\n \\end{align*}\nThere are many examples of strictly 1-balanced graphs, including cycles and complete graphs. However, any disconnected graph is not strictly 1-balanced, and neither is a tree-graph.\n\nRecently a great progress was achieved in understanding the relationship between $\\mathbb{G}(n,p)$ and $\\Hg_F(n,q)$\nwith $q$ judiciously chosen\nas $q=p^{|E(F)|}$. In particular, as shown in \\cite{burghart2024sharp} (see Proposition~\\ref{prop:coupling} below), when $p$ is at most $n^{-1/d_1(F)}$, the graphs\n$\\mathbb{G}(n,p)$ and $\\mathbb{H}(n,q)$ can be coupled in such a way that the number of $F$-hyperedges in $\\mathbb{H}$ nearly matches the number of copies of $F$ naturally occurring in $\\mathbb{G}(n,p)$. As a result a difficult problem of studying minimal number of monochromatic $F$-subgraphs of $\\mathbb{G}(n,p)$ can be reduced to simpler version, one defined on $\\mathbb{H}$ where the occurances of $F$-s are independent by design. This is the insight we use to obtain several asymptotic results on max-cut values for general graphs $F$ in $\\mathbb{G}(n,p)$.\n\nOur first result is as follows.\n\n\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\\label{prop:coupling}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. There exist constants $\\delta, \\varepsilon > 0$ such that the following holds. Fix any two sequences $p,q$ satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$. Whp as $n\\to\\infty$ there exists a permutation $\\tau: [n] \\to [n]$ such that \n for every $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ also $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$.\n\\end{proposition}", "full_context": "The study of monochromatic subgraphs in graph bipartitions is a widely investigated problem in extremal combinatorics and Ramsey theory. \nA classical result by Goodman~\\cite{g1959} states that in any bipartition of the complete graph $K_n$, the number of monochromatic triangles admits a sharp lower bound\n\\[\nT(n) = \n\\begin{cases}\n \\dfrac{u(u-1)(u-2)}{3}, & \\text{if } n = 2u; \\\\\n \\dfrac{2u(u-1)(4u+1)}{3}, & \\text{if } n = 4u+1; \\\\\n \\dfrac{2u(u+1)(4u-1)}{3}, & \\text{if } n = 4u+3.\n\\end{cases}\n\\]\n\\ignore{\n\\zhu{The general case was studied by Paul Erdős, who proved that the minimum number of monochromatic $K_r$ in any bipartition of $K_n$ is at most $\\binom{n}{r}/2^{\\binom{r}{2}-1}$ and conjectured that this bound is asymptotically tight for large $n$ \\cite{erdHos1962number}. This conjecture was later disproved by Thomason \\cite{thomason1989disproof}. In 2012, Conlon proved that any bipartition of $K_n$ contains at least $n^r / C^{(1+o(1))r^2}$ monochromatic $K_r$ subgraphs, where $C \\approx 2.18$ \\cite{conlon2012ramsey}. This phenomenon, whereby a positive fraction of all $r$-cliques are monochromatic, also appears in other deterministic contexts \\cite{frankl1988quantitative}. In general, it can often be characterized by the principle that monochromatic substructures are evenly distributed among configurations that force their occurrence \\cite{csv}.}\n\\dg{I am not sure we need such a detailed summary about deterministic setup. It is somewhat off-topic}\n}\n\nTo set the stage, let $F$ be a graph on $r$ vertices. An $F$-graph $H$ is defined as any pair $(V=V(H), E=E(H))$, where $V$ is a set of vertices, and \n$E$ is a collection of copies $e$ of $F$ called hyperedges, where for each $e$, the vertex set of $e$ is subset of $V$.\n A random \\( F \\)-graph \\( \\Hg_F(n, q) \\) is an \\( F \\)-graph with vertex set \\( V=[n]=\\cbc{1,2,\\ldots,n} \\), where each of the\n\\begin{align}\\label{eq:N-trials}\n\\binom{n}{r} \\cdot \\frac{r!}{{\\rm aut}(F)}\n\\end{align}\npotential copies of \\( F \\) on vertices in \\( [n] \\) is included in the edge set $E$ independently with probability \\( q \\). Here ${\\rm aut}(H)$ denotes the set of automorphisms of a graph $H$. In the special case\nwhen $F$ is $K_2$ (two nodes\nconnected by an edge), we use a more common notation $\\mathbb{G}(n,q)$.\n\nGiven a simple graph $G=(V,E)$ (so that $F$ is just $K_2$) and given its bipartition \\( V=V_1 \\cup V_2, V_1\\cap V_2=\\emptyset \\), a cut, or more specifically an $F$-cut associated with this bipartition is defined as the set of subgraphs $e$ of $G$ which are isomorphic to $F$, and whose vertex sets have a non-empty intersection with \\( V_1 \\) and \\( V_2 \\). That is, each hyperedge of the cut is neither entirely contained in \\( V_1 \\), nor is it entirely contained in \\( V_2 \\). The \\emph{cut value} associated with a cut is \nthe cardinality of this set.\n The \\emph{max-cut value}\n denoted by ${\\rm MaxC}(G,F)$\n is defined as the largest such value taken over all possible bipartitions of \\(V\\). Thinking of $V_1$ and $V_2$ as being associated with two distinct colors, the cut associated with the bipartition is simply the set of all non-monochromatic subgraphs of $G$ isomorphic to $F$\n associated with the coloring scheme $V_1,V_2$. Conversely, every $F$-subgraph of $G$ not participating in the cut is monochromatic by definition. The minimum\n number of monochromatic $F$-subgraphs is denoted by ${\\rm MinM}(G,F)$. Naturally, ${\\rm MinM}(G,F)+{\\rm MaxC}(G,F)$ is the total number of $F$-subgraphs of $G$ as this is the case for every bipartition.\n\nLet $F$ be a graph. We say $F$ is a strictly 1-balanced graph if $\\rd_1(F')<{\\rm d}(F)$ for any subgraph $F'$ of $F$ such that $F'\\neq F$. Here\n \\begin{align*}\n \\rd_1(G)=\\frac{\\abs{E(G)}}{\\abs{V(G )}-1}.\n \\end{align*}\nThere are many examples of strictly 1-balanced graphs, including cycles and complete graphs. However, any disconnected graph is not strictly 1-balanced, and neither is a tree-graph.\n\nRecently a great progress was achieved in understanding the relationship between $\\mathbb{G}(n,p)$ and $\\Hg_F(n,q)$\nwith $q$ judiciously chosen\nas $q=p^{|E(F)|}$. In particular, as shown in \\cite{burghart2024sharp} (see Proposition~\\ref{prop:coupling} below), when $p$ is at most $n^{-1/d_1(F)}$, the graphs\n$\\mathbb{G}(n,p)$ and $\\mathbb{H}(n,q)$ can be coupled in such a way that the number of $F$-hyperedges in $\\mathbb{H}$ nearly matches the number of copies of $F$ naturally occurring in $\\mathbb{G}(n,p)$. As a result a difficult problem of studying minimal number of monochromatic $F$-subgraphs of $\\mathbb{G}(n,p)$ can be reduced to simpler version, one defined on $\\mathbb{H}$ where the occurances of $F$-s are independent by design. This is the insight we use to obtain several asymptotic results on max-cut values for general graphs $F$ in $\\mathbb{G}(n,p)$.\n\nOur first result is as follows.\n\n\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\\label{prop:coupling}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. There exist constants $\\delta, \\varepsilon > 0$ such that the following holds. Fix any two sequences $p,q$ satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$. Whp as $n\\to\\infty$ there exists a permutation $\\tau: [n] \\to [n]$ such that \n for every $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ also $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$.\n\\end{proposition}\n\nOur first result is as follows.\n\n\\ignore{\n\\yd{Maybe we should specify that $m(F,p)$ is a deterministic scalar.}\\dg{good point. Done}\n}\n\nThe following key result from \\cite{burghart2024sharp} helps us understand the ordered copies of a strictly 1-balanced graph $F$ in $\\G(n,p)$, and establishes a connection to hyperedges in the random $F$-graph $\\Hg_F(n,q)$:\n\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\\label{prop:coupling}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. There exist constants $\\delta, \\varepsilon > 0$ such that the following holds. Fix any two sequences $p,q$ satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$. Whp as $n\\to\\infty$ there exists a permutation $\\tau: [n] \\to [n]$ such that \n for every $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ also $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$.\n\\end{proposition}\n\\ignore{\n\\yd{I think the order should be \"whp as $n\\to\\infty$, here exists a permutation $\\tau: [n] \\to [n]$\". The permutations aren't fixed beforehand.}\n\\dg{done}\n\\yd{Can't $\\epsilon, \\delta$ be arbitrarily small?}\n\\dg{We don't want them to be small. In fact the larger is $\\epsilon$ the stronger the result. $\\epsilon$ controls how high above $n^{-{1\\over d_1(F)}}$ we can go and theorem still be valid. Makes sense?}\n\\yd{}\n}\nIn the above $A\\subset B$ means $A$ is a subgraph of $B$ (not necessarily induced one). \n\\Cref{thm-bur-1.6} implies that if we choose $q = (1 - n^{-\\delta})p^{s}$, each $F$-hyperedge in $\\Hg_F(n,q)$ corresponds to an ordered copy of $F$ in $\\G(n,p)$. If we instead choose $q = p^s$, then, on the one hand, the number of $F$-hyperedges in $\\Hg_F(n,q)$ does not increase too much, while on the other hand, it is close to the expected number of ordered copies of $F$ in $\\G(n,p)$. In particular,\nwe claim the following.\n\n\\begin{corollary}\\label{cor-erg-hg}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. Fix $c > 0$, and let $p = c n^{-1/\\rd_1(F)}$, $q = p^s = c^s n^{1-r}$. Whp as $n\\to\\infty$, there exists a permutation\n$\\tau:[n]\\to [n]$ such that the number of distinct ordered copies $F(v_1, \\ldots, v_r)$ for which exactly one of the events $F(v_1, \\ldots, v_r) \\in E(\\Hg_F(n,q))$ or $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subset \\G(n,p)$ holds, is $o(n)$.\n\\end{corollary}\n\\ignore{\n\\yd{Again the existence of permutation is whp}\n\\dg{done}\n}\nThe proof crucially uses the fact that\nfor strictly 1-balanced graph that when $p=O(n^{-{1/d_1(F)}})$, the number of pairs of distinct ordered copies of $F$ in $\\G(n,p)$ which share at least one edge is small. \n\\ignore{\n\\yd{remove \"ensures that\"}\\dg{done}\n}\n\nThroughout this paper, we use $G(n,p)$ to denote the Erdős–Rényi random graph with vertex set $[n]$ and edge probability $p$.\nRecent progress in the study of random \\( F \\)-graphs enables us to extend this relation to more complex substructures, by analyzing the minimum number of monochromatic \\( F \\)-subgraphs over all bipartitions of Erdős–Rényi random graphs, where \\( F \\) is a strictly 1-balanced graph~\\cite{burghart2024sharp}. \n\\begin{theorem}\\label{thm-convergence}\n Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. For $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$. Let $\\Min(F,n,p)$ be the minimum number of monochromatic \\( F \\)-subgraphs over all bipartitions of the Erdős–Rényi random graph $G(n,p)$. Let $\\Max(F,n,q)$ be the size of max-cut in the random $F$-graph $R_F(n,q)$. Then, there exists $m(F,p)$ such that\n \\begin{align*}\n \\frac{\\Min(F,n,p)}{n}\\xlongrightarrow{\\text{p}}m(F,p),\n \\end{align*}\n and\n \\begin{align*}\n \\frac{\\Max(F,n,q)}{n}\\xlongrightarrow{\\text{p}}\\frac{c^s}{\\aut(F)}-m(F,p).\n \\end{align*}\n\\end{theorem}\n\nWe further illustrate how this minimum evolves as the host graph becomes denser, and we estimate its limiting behavior as the number of vertices in $F$ tends to infinity.\nFor any integer \\( k \\geq 0 \\), any positive function \\( f = f(a_1, \\ldots, a_k) \\), and any function \\( g = g(a_1, \\ldots, a_k) \\), we write \\( g = o_{a_1,\\ldots,a_k}(f) \\) if\n\\[\n\\limsup_{a_k \\to \\infty} \\cdots \\limsup_{a_1 \\to \\infty} \\left| \\frac{g}{f} \\right| = 0;\n\\]\nwe write \\( g = O_{a_1,\\ldots,a_k}(f) \\) if there exists a positive constant \\( M \\) such that\n\\[\n\\limsup_{a_k \\to \\infty} \\cdots \\limsup_{a_1 \\to \\infty} \\left| \\frac{g}{f} \\right| \\leq M.\n\\]\n\\begin{theorem}\\label{thm-limit}\n With the notion in \\Cref{thm-convergence}, for sufficently large $r$,\n \\begin{align*}\n m(F,p)=(1+o_r(1))\\sqrt{\\frac{\\log2}{2^{r-2}\\aut(F)}c^s }+\\frac{c^s}{2^{r-1}\\aut(F)}+o_{c^s}(c^{s/2}).\n \\end{align*}\n\\end{theorem}\n\nA key result from \\cite{burghart2024sharp} helps us understand the ordered copies of a strictly 1-balanced graph $F$ in $G(n,p)$, and establishes a connection to hyperedges in the random $F$-graph $R_F(n,q)$:\n\\begin{proposition}[{\\cite[Theorem 1.6]{burghart2024sharp}}]\\label{thm-bur-1.6}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. Then there exist constants $\\delta, \\varepsilon > 0$ such that the following holds. For any sequences satisfying $p \\leq n^{-1/\\rd_1(F) + \\varepsilon}$ and $q \\leq (1 - n^{-\\delta})p^{s}$, and for the graphs $G = G(n, p)$ and $R = R_F(n, q)$, there exists a coupling of $G$ and $R$ such that, \\whp, there exists a permutation $\\tau = \\tau(G, R) : [n] \\to [n]$ for which every $F(v_1, \\ldots, v_r) \\in E(R)$ satisfies $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subseteq G$.\n\n\\begin{corollary}\\label{cor-erg-hg}\nLet $F$ be a strictly 1-balanced graph with $s > 0$ edges on $r \\geq 2$ vertices. For $c > 0$, let $p = c n^{-1/\\rd_1(F)}$ and $q = p^s = c^s n^{1-r}$. Then there exists $\\delta > 0$ such that the following holds. For any $t > 0$, and for the graphs $G = G(n, p)$ and $R = R_F(n, q)$, there exists a coupling of $G$ and $R$ such that, \\whp, there exists a permutation $\\tau = \\tau(G, R) : [n] \\to [n]$ for which the number of distinct ordered copies $F(v_1, \\ldots, v_r)$ such that exactly one of the events $F(v_1, \\ldots, v_r) \\in E(R)$ or $F(\\tau(v_1), \\ldots, \\tau(v_r)) \\subseteq G$ holds is at most $tn$.\n\\end{corollary}\nIt is worth noting that the definition of a strictly 1-balanced graph ensures that, for $G$ defined as in \\Cref{cor-erg-hg}, the number of pairs of distinct ordered copies that share at least one edge is limited. Specifically, cases like \\Cref{exa_1} occur only rarely.\n This property is essential for the validity of \\Cref{thm-bur-1.6} and \\Cref{cor-erg-hg}.\n\n\\begin{theorem}\\label{thm-convergence}\n    Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. Given $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$.   There exists a constant $m(F,c)$ such that\n    \\begin{align*}\n        \\frac{{\\rm MinM}(\\G(n,p),F)}{n}\n        \\to m(F,c),\n    \\end{align*}\n\\whp~ as $n\\to\\infty$.\n\\end{theorem}", "post_theorem_intro_text_len": 3456, "post_theorem_intro_text": "\\ignore{\n\\yd{Maybe we should specify that $m(F,p)$ is a deterministic scalar.}\\dg{good point. Done}\n}\n\nOur proof approach is based on similar results \nfor the contiguous model $\\mathbb{H}(n,q), q=p^s$ which are already known in the literature. Specifically, the existence of $m(F,c)$ such that \n\\begin{align*}\n\\frac{{\\rm MinM}(\\mathbb{H}(n,q),F)}{n}\n\\to m(F,c),\n\\end{align*}\nis already known~\\cite{shabanov2021maximum} based on the combinatorial interpolation technique introduced in~\\cite{bayati2010combinatorial} and used\nfor establishing the existence of\nsuch limits.\nFor similar contiguity reasons \nwe also have that  \n$m(F,c)=0$ when $c$ is \nsufficiently small positive \nconstant $c$, and $m(F,c)>0$ when $c$ is sufficiently large. \nThe former claim  is\nobtained by choosing $c$ small enough so that\nthe  random graph $\\mathbb{H}(n,q)$ does not contain\na giant component. In this case the \nlocally tree-like\nstructure of the graph allows for cutting nearly\nevery $F$-edge  of the graph. Conversely, when\n$c$ is sufficiently large, the \nnon-existence of near\nperfect bipartition follows from a simple union\nbound. We regard these observations as folklore \nand will not provide a formal verification of \nthese claims.\n\nWe conjecture that the value \n${\\rm MinM}(\\mathbb{G}(n,p),F){n}$ also undergoes a different type of phase transition. Specifically, we conjecture the existence of $c^*$ which \ndepends on $F$ only such that this value is $0$ w.h.p. as $n\\to\\infty$\nwhen $c<c^*$, and is positive  when $c>c^*$ also w.h.p.\nThe basis for this conjecture is a similar conjecture for random K-SAT (and several other related models such as proper coloring\nof a random graph), which was proven for large $K$~\\cite{ding2022proof}, but is still open for general $K$.\n\nOur next result concerns obtaining explicit \nasymptotic limit\nvalues when the size of the host graph $F$ grows.  \nIntroduce a short-hand notation\n\\begin{align*}\n\\kappa(F,c,s,r)={c^s\\over 2^{r-1}{\\rm aut}(F)}.\n\\end{align*}\n\n\\begin{theorem}\\label{thm-limit}\n   Under the assumption of \\Cref{thm-convergence},\n\\begin{align*}\n       m(F,c)=\\kappa(F,c,s,r)+\n    (1+o_r(1))\\sqrt{(2\\log 2)\\kappa(F,c,s,r)}\n       +o_{c}(c^{s/2}).\n   \\end{align*}\n\\end{theorem} \nNamely, we obtain explicit limit values when both the number of nodes of the host graph $F$ and the leading coefficient $c$ of the random graph \nparameter diverge to infinity. The proof inspiration\nis based on large-$p$\napproximation technique\nfor $p$-spin glass models employed recently\nin~\\cite{gamarnik2025shattering}, and porting these\nresults to sparse random graphs as was done \nin~\\cite{dms2017} \nand~\\cite{sen2018optimization}. The latter two works relate $p$-spin models to sparse graphs using the Lindeberg's interpolation method.\nInstead, we develop relevant asymptotics directly for sparse graphs using the second \nmoment method employed \nin~\\cite{gamarnik2025shattering}\nfor the mean field $p$-spin model.\n\n\\paragraph{Organization} The remainder of this paper is organized as follows. In \\Cref{sec-hypergraph-coupling}, we introduce a coupling between the random $F$-graph and the Erdős–Rényi random graph, and prove \\Cref{thm-convergence}. In \\Cref{sec-spinglass}, we reduce the study of \\( m(F,p) \\) to the analysis of the maximum of a family of Gaussian random variables. Section~\\ref{sec-maximizer}  provides auxiliary results that en route to deriving this maximum. Finally, we prove \\Cref{thm-limit} in \\Cref{sec-proof}.", "sketch": "To prove Theorem~\\ref{thm-convergence}, the paper proposes to proceed via a known contiguous model. The approach is: use similar results for the contiguous model $\\mathbb{H}(n,q)$ with $q=p^s$, where the existence of $m(F,c)$ such that $\\frac{{\\rm MinM}(\\mathbb{H}(n,q),F)}{n}\\to m(F,c)$ is already known in the literature (via the combinatorial interpolation technique of~\\cite{bayati2010combinatorial}, as used in~\\cite{shabanov2021maximum}). Then, in \\Cref{sec-hypergraph-coupling}, the authors “introduce a coupling between the random $F$-graph and the Erdős–Rényi random graph” and use this coupling to “prove \\Cref{thm-convergence}.”", "expanded_sketch": "To prove the main theorem, the paper proposes to proceed via a known contiguous model. The approach is: use similar results for the contiguous model $\\mathbb{H}(n,q)$ with $q=p^s$, where the existence of $m(F,c)$ such that $\\frac{{\\rm MinM}(\\mathbb{H}(n,q),F)}{n}\\to m(F,c)$ is already known in the literature (via the combinatorial interpolation technique of Bayati et al., \\emph{Combinatorial Interpolation and the Cavity Method} (2010), as used in Shabanov, \\emph{Maximum matchings in random graphs via interpolation} (2021)). Then, later the authors “introduce a coupling between the random $F$-graph and the Erdős–Rényi random graph” and use this coupling to complete the proof of the main theorem.", "expanded_theorem": "\\label{thm-convergence}\n    Let $F$ be a strictly 1-balanced graph with $s>0$ edges on $r \\geq 2$ vertices. Given $c>0$, let $p= cn^{-1/\\rd_1(F)}$ and $q=p^s=c^sn^{1-r}$.   There exists a constant $m(F,c)$ such that\n    \\begin{align*}\n        \\frac{{\\rm MinM}(\\mathbb{G}(n,p),F)}{n}\n        \\to m(F,c),\n    \\end{align*}\n\\whp~ as $n\\to\\infty$.,", "theorem_type": ["Existential–Universal", "Asymptotic or Limit"], "mcq": {"question": "Let \\(F\\) be a strictly 1-balanced graph with \\(s>0\\) edges and \\(r\\ge 2\\) vertices, where \\(\\mathrm d_1(H)=|E(H)|/(|V(H)|-1)\\) and “strictly 1-balanced” means \\(\\mathrm d_1(F')<\\mathrm d_1(F)\\) for every proper subgraph \\(F'\\subsetneq F\\). Fix \\(c>0\\), and define \\(p=c\\,n^{-1/\\mathrm d_1(F)}\\). Let \\(\\mathbb G(n,p)\\) be the Erdős–Rényi random graph on \\([n]\\), and let \\({\\rm MinM}(\\mathbb G(n,p),F)\\) denote the minimum, over all bipartitions \\([n]=V_1\\cup V_2\\), of the number of copies of \\(F\\) in \\(\\mathbb G(n,p)\\) whose vertices lie entirely in \\(V_1\\) or entirely in \\(V_2\\) (that is, the minimum number of monochromatic copies of \\(F\\)). Which statement holds?", "correct_choice": {"label": "A", "text": "There exists a constant \\(m(F,c)\\) such that, with high probability as \\(n\\to\\infty\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n,p),F)}{n}\\to m(F,c).\n\\]"}, "choices": [{"label": "B", "text": "There exists a constant \\(m(F,c)\\) such that, with high probability as \\(n\\to\\infty\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n,p),F)}{n}\\to m(F,c)\n\\]\nuniformly for all fixed \\(c>0\\), with the same limiting constant independent of \\(c\\)."}, {"label": "C", "text": "There exists a constant \\(m(F,c)\\) and a sequence \\(n_k\\to\\infty\\) such that, with high probability along the subsequence \\(n_k\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n_k,p),F)}{n_k}\\to m(F,c).\n\\]"}, {"label": "D", "text": "There exists a constant \\(m(F,c)\\) such that, for every fixed \\(c>0\\), almost surely as \\(n\\to\\infty\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n,p),F)}{n}\\to m(F,c).\n\\]"}, {"label": "E", "text": "There exists a constant \\(m(F,c)\\) such that, with high probability as \\(n\\to\\infty\\),\n\\[\n\\frac{{\\rm MinM}(\\mathbb G(n,p),F)}{n^{r-1}}\\to m(F,c).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "dependence of the limit on the parameter c", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "full-sequence convergence whp weakened to subsequential whp convergence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "mode of convergence strengthened from with high probability to almost sure", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "linear normalization in n coming from the contiguous model replaced by n^{r-1}", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the setup and notation but does not explicitly state the limiting conclusion. The correct answer is not leaked directly; one must distinguish among several nearby asymptotic formulations."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-identification question: one option is essentially the exact theorem statement, while the others are quantifier/convergence/normalization variants. So it is not fully tautological, but it is still largely a restatement-discrimination task rather than a fresh conclusion derived from premises."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the distractors differ in subtle but meaningful ways (dependence on c, subsequence vs full sequence, whp vs almost sure, and scaling by n vs n^{r-1}). However, the problem mainly tests recognition of the correct theorem-strength statement rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong: they are plausible, mathematically distinct, and target common failure modes such as overstrengthening convergence, weakening to subsequences, removing parameter dependence, or using an incorrect normalization."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no direct answer leakage, but it leans more toward recall/statement recognition than genuine generative reasoning."}}
{"id": "2602.03807v1", "paper_link": "http://arxiv.org/abs/2602.03807v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{theo:main}\nFor every non-orientable regular map $\\M$ of type $\\{p,q\\}$ where at least one of $p$ or $q$ is odd, there exists an unstable fully transitive $2$-orbit map $\\M^\\omega$, that covers $\\M$. Moreover, for all $n>3$ there exist $2^{n-3}$ non-isomorphic unstable $2$-orbit fully transitive $n$-maniplexes, whose $3$-faces are isomorphic to $\\M^\\omega$.", "start_pos": 10978, "end_pos": 11373, "label": "theo:main"}, "ref_dict": {"theo:main": "\\begin{theorem}\n\\label{theo:main}\nFor every non-orientable regular map $\\M$ of type $\\{p,q\\}$ where at least one of $p$ or $q$ is odd, there exists an unstable fully transitive $2$-orbit map $\\M^\\omega$, that covers $\\M$. Moreover, for all $n>3$ there exist $2^{n-3}$ non-isomorphic unstable $2$-orbit fully transitive $n$-maniplexes, whose $3$-faces are isomorphic to $\\M^\\omega$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 4819, "pre_theorem_intro_text": "Every non-bipartite graph $\\Gamma$ admits a connected {\\em canonical double cover} $\\widetilde{\\Gamma}$, isomorphic to the direct graph product $\\Gamma \\times K_2$. The group of automorphisms of $\\widetilde{\\Gamma}$ contains a subgroup isomorphic to $\\hbox{\\rm Aut}(\\Gamma) \\times \\ZZ_2$, but may be, in general, significantly larger. We say $\\Gamma$ is stable precisely when $\\hbox{\\rm Aut}(\\widetilde{\\Gamma}) \\cong \\hbox{\\rm Aut}(\\Gamma) \\times \\ZZ_2$, and say it is unstable otherwise.\nStability in graphs was first studied in \\cite{Marusic1989} and have since received considerable attention (see for instance, \\cite{BlasDoubleCovers,ademirdorde,QIN2019154,Surowski,Surowski2001Stability,WILSON2008359}).\n\nThe problem of stability extends naturally to other types of combinatorial objects, that can ultimately be regarded as graphs. For instance, in  \\cite{gareth} Jones considers stability in maps on surfaces. Typically, a map is defined as the embedding of a connected graph on a compact surface with no boundary, with the property that the embedded graph divides the surface into simply connected regions. Every map uniquely determines an edge-coloured graph, called its {\\em flag-graph}, that completely encodes its structure, allowing us to define maps in pure graph-theoretical terms. What is more, a map $\\widetilde{M}$ is the canonical double cover of a map $\\M$, in the topological sense, if and only if the flag graph of $\\widetilde{M}$ is the canonical double cover of the flag graph of $\\M$, in the graph theoretical sense. That is, the problem of stability in maps connects naturally to stability in graphs. There is however, a caveat. While there is a correspondence between the automorphism of a map and those of its flag graph, this correspondence is, in general, not one-to-one. The automorphisms of a map $\\M$ correspond precisely to the edge-colour preserving automorphism of its flag graph, and only to those. As a consequence of this, the group of automorphism of a map, when seen as a flag-graph, acts semiregularly on its nodes, which are usually called {\\em flags} in this context. This implies, as Jones remarks in \\cite{gareth}, that every regular (maximally symmetric) non-orientable map is stable. Therefore, the problem of stability takes a somewhat different flavour in the setting of maps, as stable maps are very easily produced. A natural and more interesting question is whether weaker symmetry conditions also imply stability, which is partially answered in \\cite{gareth}. The degree of symmetry of a map is usually measured by the number of flag-orbits of its automorphism group. Maps whose automorphism group has exactly $k$ distinct flag-orbits are called $k$-orbit maps, with $1$-orbit maps (which we call {\\em regular} here) being by far the most studied class. \nJones proves, among other things, that unstable edge-transitive maps exist. However, with the exception of a small degenerate map, having (one face, one vertex and two edges). all the maps constructed in \\cite{gareth} have $4$ or more orbits on flags. The question whether $2$-orbit maps exist is left unanswered.\n\nIn this paper, we consider this question in the much broader setting of maniplexes (\\cite{wilson2012maniplexes}). A maniplex of rank $n$ is a properly $n$-edge-coloured $n$-valent connected graph that generalises the notion of a flag-graph. In particular, a maniplex of rank $3$ is the combinatorial (graph-theoretical) equivalent of a map on a surface, in the sense that the flag-graph of a map is a $3$-maniplex, and every $3$-maniplex arises in this way. In short, maniplexes can be regarded as higher-dimension analogues of maps on surfaces (or as graph-theoretical generalisations of abstract polytopes \\cite{mcmullen2002abstract}, \\cite{garza2018polytopality}). \nNaturally, many of the notions (such as symmetry or orientability) of the theory of maps extend to maniplexes of higher ranks in an intuitive way. We call the nodes of a maniplex flags and we define a maniplex automorphism as an edge-colour preserving graph automorphism. \nWe say a maniplex is regular (often also called reflexible) if its automorphism group is transitive on the flags. \nJust as with maps, every regular non-orientable maniplex is automatically stable and thus the problem of finding highly symmetric (but non-regular) unstable maniplexes is natural. In this paper, we prove that for rank $n\\geq3$, there exist infinitely many $2$-orbit $n$-maniplexes that are unstable. Moreover, the maniplexes constructed are fully-transitive, meaning that their group of automorphisms is transitive on the set of faces of rank $i$ for every $i<n$. Precise definitions of the terms appearing in these paragraphs are provided in \\Cref{sec:preliminaries} and throughout the paper. We now state the main theorem of this paper.", "context": "Every non-bipartite graph $\\Gamma$ admits a connected {\\em canonical double cover} $\\widetilde{\\Gamma}$, isomorphic to the direct graph product $\\Gamma \\times K_2$. The group of automorphisms of $\\widetilde{\\Gamma}$ contains a subgroup isomorphic to $\\hbox{\\rm Aut}(\\Gamma) \\times \\ZZ_2$, but may be, in general, significantly larger. We say $\\Gamma$ is stable precisely when $\\hbox{\\rm Aut}(\\widetilde{\\Gamma}) \\cong \\hbox{\\rm Aut}(\\Gamma) \\times \\ZZ_2$, and say it is unstable otherwise.\nStability in graphs was first studied in \\cite{Marusic1989} and have since received considerable attention (see for instance, \\cite{BlasDoubleCovers,ademirdorde,QIN2019154,Surowski,Surowski2001Stability,WILSON2008359}).\n\nThe problem of stability extends naturally to other types of combinatorial objects, that can ultimately be regarded as graphs. For instance, in \\cite{gareth} Jones considers stability in maps on surfaces. Typically, a map is defined as the embedding of a connected graph on a compact surface with no boundary, with the property that the embedded graph divides the surface into simply connected regions. Every map uniquely determines an edge-coloured graph, called its {\\em flag-graph}, that completely encodes its structure, allowing us to define maps in pure graph-theoretical terms. What is more, a map $\\widetilde{M}$ is the canonical double cover of a map $\\M$, in the topological sense, if and only if the flag graph of $\\widetilde{M}$ is the canonical double cover of the flag graph of $\\M$, in the graph theoretical sense. That is, the problem of stability in maps connects naturally to stability in graphs. There is however, a caveat. While there is a correspondence between the automorphism of a map and those of its flag graph, this correspondence is, in general, not one-to-one. The automorphisms of a map $\\M$ correspond precisely to the edge-colour preserving automorphism of its flag graph, and only to those. As a consequence of this, the group of automorphism of a map, when seen as a flag-graph, acts semiregularly on its nodes, which are usually called {\\em flags} in this context. This implies, as Jones remarks in \\cite{gareth}, that every regular (maximally symmetric) non-orientable map is stable. Therefore, the problem of stability takes a somewhat different flavour in the setting of maps, as stable maps are very easily produced. A natural and more interesting question is whether weaker symmetry conditions also imply stability, which is partially answered in \\cite{gareth}. The degree of symmetry of a map is usually measured by the number of flag-orbits of its automorphism group. Maps whose automorphism group has exactly $k$ distinct flag-orbits are called $k$-orbit maps, with $1$-orbit maps (which we call {\\em regular} here) being by far the most studied class. \nJones proves, among other things, that unstable edge-transitive maps exist. However, with the exception of a small degenerate map, having (one face, one vertex and two edges). all the maps constructed in \\cite{gareth} have $4$ or more orbits on flags. The question whether $2$-orbit maps exist is left unanswered.\n\nIn this paper, we consider this question in the much broader setting of maniplexes (\\cite{wilson2012maniplexes}). A maniplex of rank $n$ is a properly $n$-edge-coloured $n$-valent connected graph that generalises the notion of a flag-graph. In particular, a maniplex of rank $3$ is the combinatorial (graph-theoretical) equivalent of a map on a surface, in the sense that the flag-graph of a map is a $3$-maniplex, and every $3$-maniplex arises in this way. In short, maniplexes can be regarded as higher-dimension analogues of maps on surfaces (or as graph-theoretical generalisations of abstract polytopes \\cite{mcmullen2002abstract}, \\cite{garza2018polytopality}). \nNaturally, many of the notions (such as symmetry or orientability) of the theory of maps extend to maniplexes of higher ranks in an intuitive way. We call the nodes of a maniplex flags and we define a maniplex automorphism as an edge-colour preserving graph automorphism. \nWe say a maniplex is regular (often also called reflexible) if its automorphism group is transitive on the flags. \nJust as with maps, every regular non-orientable maniplex is automatically stable and thus the problem of finding highly symmetric (but non-regular) unstable maniplexes is natural. In this paper, we prove that for rank $n\\geq3$, there exist infinitely many $2$-orbit $n$-maniplexes that are unstable. Moreover, the maniplexes constructed are fully-transitive, meaning that their group of automorphisms is transitive on the set of faces of rank $i$ for every $i<n$. Precise definitions of the terms appearing in these paragraphs are provided in \\Cref{sec:preliminaries} and throughout the paper. We now state the main theorem of this paper.", "full_context": "Every non-bipartite graph $\\Gamma$ admits a connected {\\em canonical double cover} $\\widetilde{\\Gamma}$, isomorphic to the direct graph product $\\Gamma \\times K_2$. The group of automorphisms of $\\widetilde{\\Gamma}$ contains a subgroup isomorphic to $\\hbox{\\rm Aut}(\\Gamma) \\times \\ZZ_2$, but may be, in general, significantly larger. We say $\\Gamma$ is stable precisely when $\\hbox{\\rm Aut}(\\widetilde{\\Gamma}) \\cong \\hbox{\\rm Aut}(\\Gamma) \\times \\ZZ_2$, and say it is unstable otherwise.\nStability in graphs was first studied in \\cite{Marusic1989} and have since received considerable attention (see for instance, \\cite{BlasDoubleCovers,ademirdorde,QIN2019154,Surowski,Surowski2001Stability,WILSON2008359}).\n\nThe problem of stability extends naturally to other types of combinatorial objects, that can ultimately be regarded as graphs. For instance, in \\cite{gareth} Jones considers stability in maps on surfaces. Typically, a map is defined as the embedding of a connected graph on a compact surface with no boundary, with the property that the embedded graph divides the surface into simply connected regions. Every map uniquely determines an edge-coloured graph, called its {\\em flag-graph}, that completely encodes its structure, allowing us to define maps in pure graph-theoretical terms. What is more, a map $\\widetilde{M}$ is the canonical double cover of a map $\\M$, in the topological sense, if and only if the flag graph of $\\widetilde{M}$ is the canonical double cover of the flag graph of $\\M$, in the graph theoretical sense. That is, the problem of stability in maps connects naturally to stability in graphs. There is however, a caveat. While there is a correspondence between the automorphism of a map and those of its flag graph, this correspondence is, in general, not one-to-one. The automorphisms of a map $\\M$ correspond precisely to the edge-colour preserving automorphism of its flag graph, and only to those. As a consequence of this, the group of automorphism of a map, when seen as a flag-graph, acts semiregularly on its nodes, which are usually called {\\em flags} in this context. This implies, as Jones remarks in \\cite{gareth}, that every regular (maximally symmetric) non-orientable map is stable. Therefore, the problem of stability takes a somewhat different flavour in the setting of maps, as stable maps are very easily produced. A natural and more interesting question is whether weaker symmetry conditions also imply stability, which is partially answered in \\cite{gareth}. The degree of symmetry of a map is usually measured by the number of flag-orbits of its automorphism group. Maps whose automorphism group has exactly $k$ distinct flag-orbits are called $k$-orbit maps, with $1$-orbit maps (which we call {\\em regular} here) being by far the most studied class. \nJones proves, among other things, that unstable edge-transitive maps exist. However, with the exception of a small degenerate map, having (one face, one vertex and two edges). all the maps constructed in \\cite{gareth} have $4$ or more orbits on flags. The question whether $2$-orbit maps exist is left unanswered.\n\nIn this paper, we consider this question in the much broader setting of maniplexes (\\cite{wilson2012maniplexes}). A maniplex of rank $n$ is a properly $n$-edge-coloured $n$-valent connected graph that generalises the notion of a flag-graph. In particular, a maniplex of rank $3$ is the combinatorial (graph-theoretical) equivalent of a map on a surface, in the sense that the flag-graph of a map is a $3$-maniplex, and every $3$-maniplex arises in this way. In short, maniplexes can be regarded as higher-dimension analogues of maps on surfaces (or as graph-theoretical generalisations of abstract polytopes \\cite{mcmullen2002abstract}, \\cite{garza2018polytopality}). \nNaturally, many of the notions (such as symmetry or orientability) of the theory of maps extend to maniplexes of higher ranks in an intuitive way. We call the nodes of a maniplex flags and we define a maniplex automorphism as an edge-colour preserving graph automorphism. \nWe say a maniplex is regular (often also called reflexible) if its automorphism group is transitive on the flags. \nJust as with maps, every regular non-orientable maniplex is automatically stable and thus the problem of finding highly symmetric (but non-regular) unstable maniplexes is natural. In this paper, we prove that for rank $n\\geq3$, there exist infinitely many $2$-orbit $n$-maniplexes that are unstable. Moreover, the maniplexes constructed are fully-transitive, meaning that their group of automorphisms is transitive on the set of faces of rank $i$ for every $i<n$. Precise definitions of the terms appearing in these paragraphs are provided in \\Cref{sec:preliminaries} and throughout the paper. We now state the main theorem of this paper.\n\nIn this paper, we consider this question in the much broader setting of maniplexes (\\cite{wilson2012maniplexes}). A maniplex of rank $n$ is a properly $n$-edge-coloured $n$-valent connected graph that generalises the notion of a flag-graph. In particular, a maniplex of rank $3$ is the combinatorial (graph-theoretical) equivalent of a map on a surface, in the sense that the flag-graph of a map is a $3$-maniplex, and every $3$-maniplex arises in this way. In short, maniplexes can be regarded as higher-dimension analogues of maps on surfaces (or as graph-theoretical generalisations of abstract polytopes \\cite{mcmullen2002abstract}, \\cite{garza2018polytopality}). \nNaturally, many of the notions (such as symmetry or orientability) of the theory of maps extend to maniplexes of higher ranks in an intuitive way. We call the nodes of a maniplex flags and we define a maniplex automorphism as an edge-colour preserving graph automorphism. \nWe say a maniplex is regular (often also called reflexible) if its automorphism group is transitive on the flags. \nJust as with maps, every regular non-orientable maniplex is automatically stable and thus the problem of finding highly symmetric (but non-regular) unstable maniplexes is natural. In this paper, we prove that for rank $n\\geq3$, there exist infinitely many $2$-orbit $n$-maniplexes that are unstable. Moreover, the maniplexes constructed are fully-transitive, meaning that their group of automorphisms is transitive on the set of faces of rank $i$ for every $i<n$. Precise definitions of the terms appearing in these paragraphs are provided in \\Cref{sec:preliminaries} and throughout the paper. We now state the main theorem of this paper.\n\nThe basis for our construction consists of two operations: the cross-cover operation for graphs, first defined in \\cite{WILSON2008359}, and the colour-coded extensions for maniplexes (see \\cite{twistop}), which we define here in graph-theoretical terms.\n\nFor $i \\in \\nset$ we say $\\M$ is $i$-face-transitive if $\\aut(\\M)$ acts transitively on the set of $i$-faces of $\\M$. If $\\M$ is $i$-face-transitive for all $i \\in \\nset$ then we say $\\M$ is \\emph{fully transitive}. Note that information about the $i$-face-transitivity of $\\M$ is encoded in its symmetry-type graph. Indeed, the number of orbits of $\\Aut(\\M)$ on $i$-faces is equal to the number of connected components of the multigraph resulting from deleting all edges and semi-edges of colour $i$ from $\\stg(\\M)$. Regular maniplexes are fully-transitive, although not all fully-transitive maniplexes are regular. Figure \\ref{fig:2orbstgs} shows the seven possible symmetry-type graphs for a $2$-orbit $3$-maniplex. Observe that those on the top row correspond to fully-transitive maniplexes, while those on the bottom correspond to maniplexes that are not $i$-face-transitive for some $i$. Thus, we can think of a $2$-orbit maniplex of the first kind as being 'more symmetric' than those of the second kind.\n\nNow, consider the graph $Q_\\ell$ whose vertices are the facets of $\\M$, where two vertices are adjacent if the corresponding facets are joined by an $n$-edge in $\\M$ (that is, if there in an $n$-edge having one endpoint in the first facet, and the other in the second facet). The graph $Q_\\ell$ is isomorphic to the $\\ell$-cube graph and every automorphism of $\\M$ (which permutes its facets) induces a permutation of the vertices of $Q_\\ell$. Moreover, since an automorphism of $\\M$ maps $n$-edges to $n$-edges, the induced permutation on the vertices of $Q_\\ell$ must map adjacent vertices to adjacent vertices. In other words, every automorphism of $\\M$ induces a graph automorphism of $Q_\\ell$. Let $G = \\Aut(\\M)$ and let $G'\\leq\\Aut(Q_\\ell)$ be the group of automorphisms of $Q_\\ell$ induced by $G$. Now consider the group $H = \\langle \\tau_1,\\ldots,\\tau_\\ell \\rangle$ and observe that $H$ acts not only transitively, but regularly, on the set of facets of $\\M$. It follows that the group $H'$ of automorphisms of $Q_\\ell$ induced by $H$ acts regularly on the vertices of $Q_\\ell$. Moreover, one can see that the set of all pairs of antipodal vertices of $Q_\\ell$ is a block system for $H'$ (indeed, $H'$ acts on the main diagonals of the $\\ell$-cube). Since this is also a block system for the full automorphism group of $Q_\\ell$, and $H' \\leq G' \\leq \\Aut(Q_\\ell)$, we see that it is a block system for $G'$ too. This block system corresponds to the partition into chromatic classes (of the facets of $\\M$) with the antipodal colouring $\\C^\\times$ of $\\M$, and thus, this partition must be a block system for $\\Aut(\\M)$ (acting on facets). We conclude that the antipodal colouring is $\\Aut(\\M)$-invariant.\n\\end{proof}\n\n\\Cref{prop:pares} tells us that it suffices to find a map $\\M$ with a weight function $\\omega$ such that $\\M^\\omega$ is a non-orientable maniplex (which must necessarily be unstable by \\Cref{theo:cross}), to show that there are unstable maniplexes of every rank $n \\geq 3$. Finding such a pair $(\\M,\\omega)$ is in fact not difficult, as we will see in the following pages. However, the maniplexes obtained by extending $\\M$ and $\\omega$ may, in principle, be of any symmetry type. In fact, they may very well completely lack symmetry. Since we are interested in maniplexes that are both unstable and very symmetric, extra conditions need to be imposed when considering a pair $(\\M,\\omega)$. This motivates the following definition.\n\n\\begin{lemma}\n\\label{lem:nicecover}\nLet $\\M$ be a regular $n$-maniplex and let $\\omega\\colon \\M \\to \\ZZ_k$ be a weight function for $\\M$. If $(\\M,\\omega)$ is a proper pair and $k=4$, \nthen $\\M^\\omega$ is an unstable $2$-orbit maniplex. \n\\end{lemma}\n\nTheorem \\ref{theo:main} now follows from Proposition \\ref{prop: goodweight} and Lemmas \\ref{lem:niceext}, \\ref{lem:nicecover} and \\ref{lem:stgcover}. Indeed, given a non-orientable regular map $\\M$ of type $\\{p,q\\}$, where $p$ or $q$ is odd, the pair $(\\M,\\vartheta)$ is a proper pair by \\Cref{prop: goodweight}. The cross-cover of $(\\M,\\vartheta)$ is an unstable $2$-orbit maniplex by \\Cref{lem:nicecover}, and it has symmetry-type $2^3_{\\{1\\}}$ by \\Cref{lem:stgcover}, making it fully-transitive.", "post_theorem_intro_text_len": 1046, "post_theorem_intro_text": "The basis for our construction consists of two operations: the cross-cover operation for graphs, first defined in \\cite{WILSON2008359}, and the colour-coded extensions for maniplexes (see \\cite{twistop}), which we define here in graph-theoretical terms.\n\nThe paper is structured as follows. In \\Cref{sec:preliminaries}, we give basic definitions pertaining to graphs and maniplexes. In \\Cref{sec:cross} we introduce cross-covers of graph (and maniplexes) and prove a few auxiliary results relating to their connectivity and symmetry.\nIn \\Cref{sec:ext} we discuss colour-coded extensions and define two well-known ways of extending maniplexes that will be useful for our purposes. \nIn \\Cref{sec:extweight} we show how the notions of cross-covers and colour-coded extensions interweave nicely, and we can use a combination of both operations to produce unstable maniplexes with desirable symmetry properties. \nIn particular, we prove the two propositions that will be the key to the proof of \\Cref{theo:main}, which we complete in \\Cref{sec:proof}.", "sketch": "The proof of \\Cref{theo:main} is based on combining two operations: the \"cross-cover\" operation for graphs and \"colour-coded extensions\" for maniplexes (defined in graph-theoretical terms). The argument develops auxiliary results on connectivity and symmetry for cross-covers, then introduces two standard extension methods, and then shows in \\Cref{sec:extweight} how cross-covers and colour-coded extensions \"interweave\" so that a combination of both \"produce unstable maniplexes with desirable symmetry properties\". In particular, the proof relies on \"two propositions\" proved in \\Cref{sec:extweight} that are \"key to the proof of \\Cref{theo:main}\", with the theorem then completed in \\Cref{sec:proof}.", "expanded_sketch": "In establishing the main theorem, the argument is based on combining two operations: the “cross-cover” operation for graphs and “colour-coded extensions” for maniplexes (defined in graph-theoretical terms). The argument develops auxiliary results on connectivity and symmetry for cross-covers, then introduces two standard extension methods, and then shows in \\Cref{sec:extweight} how cross-covers and colour-coded extensions “interweave” so that a combination of both “produce unstable maniplexes with desirable symmetry properties”. In particular, the proof relies on “two propositions” proved in \\Cref{sec:extweight} that are key to the proof of the main theorem, with the main theorem then completed in \\Cref{sec:proof}.", "expanded_theorem": "\\label{theo:main}\nFor every non-orientable regular map $\\M$ of type $\\{p,q\\}$ where at least one of $p$ or $q$ is odd, there exists an unstable fully transitive $2$-orbit map $\\M^\\omega$, that covers $\\M$. Moreover, for all $n>3$ there exist $2^{n-3}$ non-isomorphic unstable $2$-orbit fully transitive $n$-maniplexes, whose $3$-faces are isomorphic to $\\M^\\omega$.", "theorem_type": ["Universal–Existential", "Existence"], "mcq": {"question": "Let \\(\\mathcal M\\) be a non-orientable regular map of type \\(\\{p,q\\}\\), meaning that every face has length \\(p\\), every vertex has valence \\(q\\), and the automorphism group of \\(\\mathcal M\\) is transitive on flags. Assume that at least one of \\(p\\) or \\(q\\) is odd. A map or maniplex is called \\(2\\)-orbit if its automorphism group has exactly two flag-orbits, and fully transitive if its automorphism group is transitive on faces of every rank \\(i<n\\). It is called unstable if the automorphism group of its canonical double cover is larger than the expected subgroup \\(\\operatorname{Aut}(X)\\times \\mathbb Z_2\\). Which statement holds under these assumptions?", "correct_choice": {"label": "A", "text": "There exists an unstable fully transitive \\(2\\)-orbit map \\(\\mathcal M^{\\omega}\\) that covers \\(\\mathcal M\\). Moreover, for every \\(n>3\\), there exist \\(2^{n-3}\\) pairwise non-isomorphic unstable fully transitive \\(2\\)-orbit \\(n\\)-maniplexes whose \\(3\\)-faces are isomorphic to \\(\\mathcal M^{\\omega}\\)."}, "choices": [{"label": "B", "text": "There exists an unstable fully transitive \\(2\\)-orbit map \\(\\mathcal M^{\\omega}\\) that covers \\(\\mathcal M\\). Moreover, for every \\(n\\ge 3\\), there exist \\(2^{n-2}\\) pairwise non-isomorphic unstable fully transitive \\(2\\)-orbit \\(n\\)-maniplexes whose \\(3\\)-faces are isomorphic to \\(\\mathcal M^{\\omega}\\)."}, {"label": "C", "text": "There exists an unstable \\(2\\)-orbit map \\(\\mathcal M^{\\omega}\\) that covers \\(\\mathcal M\\). Moreover, for every \\(n>3\\), there exist \\(2^{n-3}\\) pairwise non-isomorphic unstable \\(2\\)-orbit \\(n\\)-maniplexes whose \\(3\\)-faces are isomorphic to \\(\\mathcal M^{\\omega}\\)."}, {"label": "D", "text": "There exists an unstable fully transitive \\(2\\)-orbit map \\(\\mathcal M^{\\omega}\\) that covers \\(\\mathcal M\\). Moreover, for every \\(n>3\\), for each such \\(n\\) there exists at least one unstable fully transitive \\(2\\)-orbit \\(n\\)-maniplex whose \\(3\\)-faces are isomorphic to \\(\\mathcal M^{\\omega}\\), and these maniplexes may be chosen canonically from \\(\\mathcal M\\), independently of the choice of \\(\\omega\\)."}, {"label": "E", "text": "There exists a fully transitive \\(2\\)-orbit map \\(\\mathcal M^{\\omega}\\) that covers \\(\\mathcal M\\) and is stable. Moreover, for every \\(n>3\\), there exist \\(2^{n-3}\\) pairwise non-isomorphic fully transitive \\(2\\)-orbit \\(n\\)-maniplexes whose \\(3\\)-faces are isomorphic to \\(\\mathcal M^{\\omega}\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "dimension range and enumeration count", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "full transitivity conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence on the chosen weight function \\(\\omega\\) and exact counting conclusion", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "cross-cover produces instability rather than stability", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives definitions and hypotheses but does not explicitly reveal the correct existence claim, the instability condition, or the exact count \\(2^{n-3}\\). There is no direct answer leakage."}, "TAS": {"score": 0, "justification": "The item is essentially asking for the exact theorem conclusion under the stated assumptions. This makes it very close to a direct restatement rather than a conceptually reworked application question."}, "GPS": {"score": 1, "justification": "The distractors differ by subtle quantifiers, counts, and stability assumptions, so some careful comparison is needed. However, the task mainly tests recall/recognition of the theorem statement rather than genuine mathematical generation or derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically structured: they vary in exact multiplicity, range of \\(n\\), stability versus instability, and omission of full transitivity. These align with realistic theorem-misreading failure modes."}, "total_score": 5, "overall_assessment": "A well-constructed recognition item with strong distractors and no leakage, but it is largely a theorem-recall question rather than a non-tautological test of generative reasoning."}}
{"id": "2602.03830v1", "paper_link": "http://arxiv.org/abs/2602.03830v1", "theorems_cnt": 1, "theorem": {"env_name": "thm", "content": "\\label{thm:main}\nLet $G_{1}$,~$G_{2}$, \\dots,~$G_{k}$ be a sequence of almost simple groups. Let $S$ be the socle of $G_1.$ Let\n\\[\nW = G_{k} \\operatorname{wr} G_{k-1} \\operatorname{wr} \\dots \\operatorname{wr} G_{1}\n\\]\nbe the iterated wreath product constructed via the regular action of each factor. Set\n\\[\nA = (G_{k}/G_{k}') \\times (G_{k-1}/G_{k-1}') \\times \\dots \\times (G_{2}/G_{2}').\n\\]\nThen\n\\[\nd(W)=\\max_p(d(A\\times G_1), d(A)+1, d_p(A)+2),\n\\]\nwhere $p$ ranges over the set of prime numbers dividing both $\\mathopen{|}A\\mathclose{|}$ and $\\mathopen{|}S\\mathclose{|}$.", "start_pos": 6724, "end_pos": 7241, "label": "thm:main"}, "ref_dict": {"lem:keylemmaregular-abelian": "\\begin{lem}\\label{lem:keylemmaregular-abelian}\nLet $S$ be the socle of $G_1$. The minimum number of generators of the wreath product $A\\wr G_1$ is given by\n\\[\nd(A\\wr G)=\\max_q(d(A\\times G), d(A)+1, d_q(A)+2),\n\\]\nwhere $q$ ranges over the set of prime numbers dividing $\\order{A}$ and $\\order{S}$.\n\\end{lem}", "lem:regularinduction": "\\begin{lem}\\label{lem:regularinduction}\n  Let $W$~and~$A$ be as defined above. Then\n  \\begin{equation}\n    d(W) = d(A \\wr G_{1}).\n    \\label{eq:regularreduction}\n  \\end{equation}\n\\end{lem}", "thm:main": "\\begin{thm}\\label{thm:main}\nLet $G_{1}$,~$G_{2}$, \\dots,~$G_{k}$ be a sequence of almost simple groups. Let $S$ be the socle of $G_1.$ Let\n\\[\nW = G_{k} \\wr G_{k-1} \\wr \\dots \\wr G_{1}\n\\]\nbe the iterated wreath product constructed via the regular action of each factor. Set\n\\[\nA = (G_{k}/G_{k}') \\times (G_{k-1}/G_{k-1}') \\times \\dots \\times (G_{2}/G_{2}').\n\\]\nThen\n\\[\nd(W)=\\max_p(d(A\\times G_1), d(A)+1, d_p(A)+2),\n\\]\nwhere $p$ ranges over the set of prime numbers dividing both $\\order{A}$ and $\\order{S}$.\n\\end{thm}"}, "pre_theorem_intro_text_len": 3858, "pre_theorem_intro_text": "A classic theme of group theory research is the generation property of groups. The structure of a group can be described using its generators. There have been numerous research on generators of specific groups. For example, it is delivered in undergraduate group theory courses that symmetric groups can be generated with two elements. Using the Classification of Finite Simple Groups, it is proved that every finite non-abelian simple group is $2$\\nobreakdash- generated.\n\nOne of the interesting generation problems is the growth rate of the minimum number of generators of the direct product $G^n$ where $G$ is a finite group. Wiegold shows in \\cite{Wiegold} that the number of required generators grows linearly when $G$ is an imperfect group and it grows logarithmically when $G$ is perfect. Apart from direct products, another way to construct groups is via wreath products. In this paper, we will investigate the minimum number of generators of iterated wreath products.\n\nBhattacharjee \\cite{Bhatt} investigates the generation property of iterated wreath product of alternating groups constructed via the natural action. She proves that the probability that such products can be generated by two random elements is positive. This shows that iterated wreath products of alternating groups are $2$\\nobreakdash- generated. Quick \\cite{MRQ} generalises this result and shows that the same result stills holds for iterated wreath products of non-abelian simple groups constructed from any faithful transitive action of the factors. Bondarenko \\cite{Bond} considers  iterated wreath products of transitive permutation groups with uniformly bounded number of generators. He establishes that the number of generators of the product involving $G_1, \\dots, G_n$ is bounded if and only if the number of generators of the abelian direct product $G_1/G_1'\\times G_2/G_2'\\times\\dots\\times G_n/G_n'$ is bounded as well.\n\nRecently, the author of this paper \\cite{LuQuick} considers the generators required for the iterated wreath products of symmetric groups, alternating groups or cyclic groups. In particular, it is established that the minimum number of generators of iterated wreath products of symmetric groups and alternating groups is the maximum of $2$ and the number of symmetric groups involved. The method used relies on Lucchini's work \\cite{Lu97}. Lucchini determines the number of generators required for the wreath product $A\\operatorname{wr} S_n$ of an abelian group $A$ constructed via the natural action of the symmetric group of degree $n$. This is then adapted for the wreath product $A\\operatorname{wr} A_n$ constructed via the natural action of the alternating group of degree $n$ in \\cite{LuQuick}. In \\cite{Lu97}, Lucchini also establishes the minimum number of generators for the wreath product $A\\operatorname{wr} G$ constructed via the regular action of a simple group $G$. We will use this result as part of our argument and generalise it to apply to the case that $G$ is an almost simple group.\n\nIn this article, we will establish the precise number of generators required for an iterated wreath product constructed from the regular action of almost simple groups. If we take the factors involved to be simple groups, then we recover the observation that iterated wreath products constructed via the regular action of non-abelian simple groups are $2$\\nobreakdash- generated.\n\nTo state the theorem, we recall some basic notations. For a group $G$, we write $d(G)$ for the minimum number of generators for $G$. For a finite abelian group $A$ and a prime number $p\\mid\\mathopen{|}A\\mathclose{|}$, we write $d_p(A)$ for the minimum number of generators for the Sylow-$p$ subgroup of $A$. Because of the Fundamental Theorem of Finite Abelian Groups, $d(A)=\\max_{p\\mid\\mathopen{|}A\\mathclose{|}}d_p(A)$. We will establish the following theorem.", "context": "A classic theme of group theory research is the generation property of groups. The structure of a group can be described using its generators. There have been numerous research on generators of specific groups. For example, it is delivered in undergraduate group theory courses that symmetric groups can be generated with two elements. Using the Classification of Finite Simple Groups, it is proved that every finite non-abelian simple group is $2$\\nobreakdash- generated.\n\nOne of the interesting generation problems is the growth rate of the minimum number of generators of the direct product $G^n$ where $G$ is a finite group. Wiegold shows in \\cite{Wiegold} that the number of required generators grows linearly when $G$ is an imperfect group and it grows logarithmically when $G$ is perfect. Apart from direct products, another way to construct groups is via wreath products. In this paper, we will investigate the minimum number of generators of iterated wreath products.\n\nBhattacharjee \\cite{Bhatt} investigates the generation property of iterated wreath product of alternating groups constructed via the natural action. She proves that the probability that such products can be generated by two random elements is positive. This shows that iterated wreath products of alternating groups are $2$\\nobreakdash- generated. Quick \\cite{MRQ} generalises this result and shows that the same result stills holds for iterated wreath products of non-abelian simple groups constructed from any faithful transitive action of the factors. Bondarenko \\cite{Bond} considers iterated wreath products of transitive permutation groups with uniformly bounded number of generators. He establishes that the number of generators of the product involving $G_1, \\dots, G_n$ is bounded if and only if the number of generators of the abelian direct product $G_1/G_1'\\times G_2/G_2'\\times\\dots\\times G_n/G_n'$ is bounded as well.\n\nRecently, the author of this paper \\cite{LuQuick} considers the generators required for the iterated wreath products of symmetric groups, alternating groups or cyclic groups. In particular, it is established that the minimum number of generators of iterated wreath products of symmetric groups and alternating groups is the maximum of $2$ and the number of symmetric groups involved. The method used relies on Lucchini's work \\cite{Lu97}. Lucchini determines the number of generators required for the wreath product $A\\operatorname{wr} S_n$ of an abelian group $A$ constructed via the natural action of the symmetric group of degree $n$. This is then adapted for the wreath product $A\\operatorname{wr} A_n$ constructed via the natural action of the alternating group of degree $n$ in \\cite{LuQuick}. In \\cite{Lu97}, Lucchini also establishes the minimum number of generators for the wreath product $A\\operatorname{wr} G$ constructed via the regular action of a simple group $G$. We will use this result as part of our argument and generalise it to apply to the case that $G$ is an almost simple group.\n\nIn this article, we will establish the precise number of generators required for an iterated wreath product constructed from the regular action of almost simple groups. If we take the factors involved to be simple groups, then we recover the observation that iterated wreath products constructed via the regular action of non-abelian simple groups are $2$\\nobreakdash- generated.\n\nTo state the theorem, we recall some basic notations. For a group $G$, we write $d(G)$ for the minimum number of generators for $G$. For a finite abelian group $A$ and a prime number $p\\mid\\mathopen{|}A\\mathclose{|}$, we write $d_p(A)$ for the minimum number of generators for the Sylow-$p$ subgroup of $A$. Because of the Fundamental Theorem of Finite Abelian Groups, $d(A)=\\max_{p\\mid\\mathopen{|}A\\mathclose{|}}d_p(A)$. We will establish the following theorem.", "full_context": "A classic theme of group theory research is the generation property of groups. The structure of a group can be described using its generators. There have been numerous research on generators of specific groups. For example, it is delivered in undergraduate group theory courses that symmetric groups can be generated with two elements. Using the Classification of Finite Simple Groups, it is proved that every finite non-abelian simple group is $2$\\nobreakdash- generated.\n\nOne of the interesting generation problems is the growth rate of the minimum number of generators of the direct product $G^n$ where $G$ is a finite group. Wiegold shows in \\cite{Wiegold} that the number of required generators grows linearly when $G$ is an imperfect group and it grows logarithmically when $G$ is perfect. Apart from direct products, another way to construct groups is via wreath products. In this paper, we will investigate the minimum number of generators of iterated wreath products.\n\nBhattacharjee \\cite{Bhatt} investigates the generation property of iterated wreath product of alternating groups constructed via the natural action. She proves that the probability that such products can be generated by two random elements is positive. This shows that iterated wreath products of alternating groups are $2$\\nobreakdash- generated. Quick \\cite{MRQ} generalises this result and shows that the same result stills holds for iterated wreath products of non-abelian simple groups constructed from any faithful transitive action of the factors. Bondarenko \\cite{Bond} considers iterated wreath products of transitive permutation groups with uniformly bounded number of generators. He establishes that the number of generators of the product involving $G_1, \\dots, G_n$ is bounded if and only if the number of generators of the abelian direct product $G_1/G_1'\\times G_2/G_2'\\times\\dots\\times G_n/G_n'$ is bounded as well.\n\nRecently, the author of this paper \\cite{LuQuick} considers the generators required for the iterated wreath products of symmetric groups, alternating groups or cyclic groups. In particular, it is established that the minimum number of generators of iterated wreath products of symmetric groups and alternating groups is the maximum of $2$ and the number of symmetric groups involved. The method used relies on Lucchini's work \\cite{Lu97}. Lucchini determines the number of generators required for the wreath product $A\\operatorname{wr} S_n$ of an abelian group $A$ constructed via the natural action of the symmetric group of degree $n$. This is then adapted for the wreath product $A\\operatorname{wr} A_n$ constructed via the natural action of the alternating group of degree $n$ in \\cite{LuQuick}. In \\cite{Lu97}, Lucchini also establishes the minimum number of generators for the wreath product $A\\operatorname{wr} G$ constructed via the regular action of a simple group $G$. We will use this result as part of our argument and generalise it to apply to the case that $G$ is an almost simple group.\n\nIn this article, we will establish the precise number of generators required for an iterated wreath product constructed from the regular action of almost simple groups. If we take the factors involved to be simple groups, then we recover the observation that iterated wreath products constructed via the regular action of non-abelian simple groups are $2$\\nobreakdash- generated.\n\nTo state the theorem, we recall some basic notations. For a group $G$, we write $d(G)$ for the minimum number of generators for $G$. For a finite abelian group $A$ and a prime number $p\\mid\\mathopen{|}A\\mathclose{|}$, we write $d_p(A)$ for the minimum number of generators for the Sylow-$p$ subgroup of $A$. Because of the Fundamental Theorem of Finite Abelian Groups, $d(A)=\\max_{p\\mid\\mathopen{|}A\\mathclose{|}}d_p(A)$. We will establish the following theorem.\n\n\\begin{abstract}\n Let $G_{1}$,~$G_{2}$, \\dots\\ be a sequence of almost simple groups and construct a sequence~$(W_{i})$ of wreath products via $W_{1} =\n G_{1}$ and, for each $i >1$, $W_{i+1} = G_{i+1} \\wr W_{i}$ via\n the regular action of each $G_i$. We determine the minimum\n number~$d(W_{i})$ of generators required for each wreath product in\n this sequence.\n\\end{abstract}\n\nBhattacharjee \\cite{Bhatt} investigates the generation property of iterated wreath product of alternating groups constructed via the natural action. She proves that the probability that such products can be generated by two random elements is positive. This shows that iterated wreath products of alternating groups are $2$\\nbd generated. Quick \\cite{MRQ} generalises this result and shows that the same result stills holds for iterated wreath products of non-abelian simple groups constructed from any faithful transitive action of the factors. Bondarenko \\cite{Bond} considers iterated wreath products of transitive permutation groups with uniformly bounded number of generators. He establishes that the number of generators of the product involving $G_1, \\dots, G_n$ is bounded if and only if the number of generators of the abelian direct product $G_1/G_1'\\times G_2/G_2'\\times\\dots\\times G_n/G_n'$ is bounded as well.\n\nIn this article, we will establish the precise number of generators required for an iterated wreath product constructed from the regular action of almost simple groups. If we take the factors involved to be simple groups, then we recover the observation that iterated wreath products constructed via the regular action of non-abelian simple groups are $2$\\nbd generated.\n\nWe finish this introduction by giving the structure of this paper. In Section~\\ref{sec:pre}, we recall basic definition that we will use. We will also introduce the results that our conclusions depend on, particularly the ones from \\cite{Lu97}. In Section~\\ref{sec:AwrG}, using the results in~\\cite{Lu97} and also extending them to cover a more general case, we will determine the minimum number of generators of the wreath product of an abelian group by an almost simple group constructed via the regular action. We will then provide the main proof of the theorem in the final section. We will prove it by showing that $d(W)=d(A\\wr G_1)$ and determining $d(A\\wr G_1)$, where $W, A, G_1$ are as defined in Theorem~\\ref{thm:main} (see Lemmas~\\ref{lem:keylemmaregular-abelian} and \\ref{lem:regularinduction} respectively).\n\n\\begin{lem}\\label{lem:LuStam}\n\\begin{enumerate}\n\\item\\label{i:Lucchini-s}{\\normalfont (Lucchini~{\\cite[1.2]{Lu97}})}\n Let $M$~be an irreducible $G$\\nbd module for a finite group~$G$.\n Then\n \\[\n s_{G}(M) = \\delta_{G}(M) + \\dim_{\\End{M}} \\Cohom{G/\\Cent{G}{M},M}.\n \\]\n\\item\\label{i:H1<rGM}{\\normalfont (Lucchini~{\\cite[1.3]{Lu97}})}\nLet $M$ be an irreducible $G$\\nbd module. Then\n\\[\n\\dim_{\\End{M}}H^1(G/C_G(M),M)<r_G(M).\n\\]\n\\item\\label{i:Stam}{\\normalfont (Stammbach~{\\cite[Theorem A]{Stammbach}})}\nGiven a prime integer $p$, a finite group $G$ is $p$-soluble if and only if $\\Cohom{G/C_G(M),M}=0$ for all irreducible $\\Field{p}G$\\nbd modules $M$.\n\\item\\label{i:augmentationgenerator}{\\normalfont (Lucchini~{\\cite[Proposition 1]{Lu97}})}\nLet $H$ be a finite group and $G$ be a transitive permutation group of degree $n$, then\n\\begin{align*}\nd(I_{H\\wr G})=\\max\\left(d(I_{(H/H')\\wr G}), \\left\\lfloor\\frac{d(I_H)-2}{n}\\right\\rfloor+2\\right).\n\\end{align*}\n\\item\\label{i:soluble}{\\normalfont (Lucchini~{\\cite[Theorem 2]{Lu97}})}\nIf $H$ is a finite soluble group and $G$ is a transitive permutation group of degree $n$, then\n\\[\nd(H\\wr G)=\\max\\left(d\\left(\\frac{H}{H'}\\wr G\\right),\\left\\lfloor\\frac{d(H)-2}{n}\\right\\rfloor+2\\right).\n\\]\n\\item\\label{i:Lucchini-AwrG}{\\normalfont (Lucchini~{\\cite[Theorem 4]{Lu97}})}\nLet $A\\wr G$ be the wreath product of a finite abelian group $A$ by a finite group $G$ constructed via the regular action of $G$. Set \n\\[\n\\rho_p=\\max_Mh_G(M)+d_p(A)\n\\]\nwhere $M$ ranges over the set of non-trivial irreducible $\\Field{p}G$\\nbd modules, with $\\rho_p=0$ if every irreducible $\\Field{p}G$\\nbd modules is trivial. Then\n\\[\nd(A\\wr G)=\\max_{p\\mid\\order{A}}(d(A\\times G),\\rho_p).\n\\]\n\\end{enumerate}\n\\end{lem}\n\n\\begin{lem}\\label{lem:AwrGLietype}\nLet $S$ be a simple group of Lie type and $G$ be an almost simple group with socle $S$. Let $A$ be a finite abelian group and set $W=A\\wr G$ to be the wreath product of the abelian group by the almost simple group with respect to the regular action. Then\n\\[\nd(W)=\\max_q(d(A\\times G), d(A)+1, d_q(A)+2),\n\\]\nwhere $q$ ranges over the set of the prime numbers dividing $\\order{A}$ and $\\order{S}$.\n\\end{lem}\n\\begin{prf}\nUsing Lemma~\\ref{lem:LuStam}\\ref{i:Lucchini-AwrG}, it suffices to show that\n\\begin{equation}\\label{eq:maxsigma=dq}\n\\max_{p\\mid\\order{A}}\\rho_p=\\max_q(d(A)+1, d_q(A)+2),\n\\end{equation}\nwhere $q$ ranges over the set of the prime numbers dividing $\\order{A}$ and $\\order{S}$ and\n\\[\n\\rho_p=\\max_Mh_G(M)+d_p(A).\n\\]\nLemma~\\ref{lem:maxhG} tells us that $\\rho_p\\geq1+d_p(A) $ for all $p\\mid\\order{A}$. Hence $\\rho_p\\geq 1+d(A)$. Lemma~\\ref{lem:maxhG} also shows that, for any prime number $p$ dividing $\\order{A}$ and $\\order{S}$, $\\rho_p=2+d_p(A)$. Thus \n\\[\n\\max_{p\\mid\\order{A}}\\rho_p\\geq\\max_{p\\mid\\order{A}, p\\mid\\order{S}}\\rho_p=\\max_{p\\mid\\order{A}, p\\mid\\order{S}} 2+d_p(A).\n\\]\nThen \n\\[\n\\max_{p\\mid\\order{A}}\\rho_p\\geq\\max_q(d(A)+1, d_q(A)+2).\n\\]\n\nWe will now establish the main theorem. Accordingly, let $k\\geq2$, and let $G_1, G_2,\\dots, G_k$ be almost simple groups. Define \n\\[\nW=G_k\\wr G_{k-1}\\wr\\dots\\wr G_1\n\\]\nto be the wreath product constructed via the regular action of each factor. We also define\n\\[\nH=G_k\\wr G_{k-1}\\wr \\dots\\wr G_2\n\\]\nand set $A=H/H'$ to be the abelianization of $H$; that is,\n\\[\nA=(G_k/G_k')\\times (G_{k-1}/G_{k-1}')\\times\\dots\\times (G_{2}/G_{2}').\n\\]\n\n\\begin{lem}\\label{lem:keylemmaregular-abelian}\nLet $S$ be the socle of $G_1$. The minimum number of generators of the wreath product $A\\wr G_1$ is given by\n\\[\nd(A\\wr G)=\\max_q(d(A\\times G), d(A)+1, d_q(A)+2),\n\\]\nwhere $q$ ranges over the set of prime numbers dividing $\\order{A}$ and $\\order{S}$.\n\\end{lem}\n\\begin{prf}\nThe case when $S$ is a simple group of Lie type is given in Lemma~\\ref{lem:AwrGLietype}. If $S$ is an alternating group, then $\\Out(S)$ is abelian. According to~\\cite[Table 1]{Atlas}, $\\Out(S)$ is also abelian when $S$ is one of the sporadic groups. It follows that the abelian chief factors of $G$ are all trivial in the case that $S$ is an alternating simple group or one of the sporadic simple groups. Then for any non-trivial $\\Field{p}G$\\nbd module $M$, $\\delta_G(M)=0$. In this case, the formula is a direct consequence of~\\cite[Corollary 5]{Lu97}.\n\\end{prf}\n\n\\begin{lem}\\label{lem:regularinduction}\n  Let $W$~and~$A$ be as defined above. Then\n  \\begin{equation}\n    d(W) = d(A \\wr G_{1}).\n    \\label{eq:regularreduction}\n  \\end{equation}\n\\end{lem}\n\n\\begin{thm}\\label{thm:main}\nLet $G_{1}$,~$G_{2}$, \\dots,~$G_{k}$ be a sequence of almost simple groups. Let $S$ be the socle of $G_1.$ Let\n\\[\nW = G_{k} \\wr G_{k-1} \\wr \\dots \\wr G_{1}\n\\]\nbe the iterated wreath product constructed via the regular action of each factor. Set\n\\[\nA = (G_{k}/G_{k}') \\times (G_{k-1}/G_{k-1}') \\times \\dots \\times (G_{2}/G_{2}').\n\\]\nThen\n\\[\nd(W)=\\max_p(d(A\\times G_1), d(A)+1, d_p(A)+2),\n\\]\nwhere $p$ ranges over the set of prime numbers dividing both $\\order{A}$ and $\\order{S}$.\n\\end{thm}", "post_theorem_intro_text_len": 934, "post_theorem_intro_text": "We finish this introduction by giving the structure of this paper. In Section~\\ref{sec:pre}, we recall basic definition that we will use. We will also introduce the results that our conclusions depend on, particularly the ones from \\cite{Lu97}. In Section~\\ref{sec:AwrG}, using the results in~\\cite{Lu97} and also extending them to cover a more general case, we will determine the minimum number of generators of the wreath product of an abelian group by an almost simple group constructed via the regular action. We will then provide the main proof of the theorem in the final section. We will prove it by showing that $d(W)=d(A\\operatorname{wr} G_1)$ and determining $d(A\\operatorname{wr} G_1)$, where $W, A, G_1$ are as defined in Theorem~\\ref{thm:main} (see Lemmas~\\ref{lem:keylemmaregular-abelian} and \\ref{lem:regularinduction} respectively).\n\n\\paragraph{Acknowledgements:} The author is funded by the China\nScholarship Council.", "sketch": "To prove Theorem~\\ref{thm:main}, the paper \"determin[es] the minimum number of generators of the wreath product of an abelian group by an almost simple group constructed via the regular action\" (using results in~\\cite{Lu97} and extending them). The \"main proof\" is then given by \"showing that $d(W)=d(A\\operatorname{wr} G_1)$ and determining $d(A\\operatorname{wr} G_1)$, where $W, A, G_1$ are as defined in Theorem~\\ref{thm:main}\" (see Lemmas~\\ref{lem:keylemmaregular-abelian} and~\\ref{lem:regularinduction}).", "expanded_sketch": "To prove the main theorem, the paper determin[es] the minimum number of generators of the wreath product of an abelian group by an almost simple group constructed via the regular action (using results in~\\cite{Lu97} and extending them). The main proof is then given by showing that $d(W)=d(A\\operatorname{wr} G_1)$ and determining $d(A\\operatorname{wr} G_1)$, where $W, A, G_1$ are as defined in the statement of the main theorem. We first prove the following lemma.\n\n\\begin{lem}\\label{lem:regularinduction}\n  Let $W$~and~$A$ be as defined above. Then\n  \\begin{equation}\n    d(W) = d(A \\wr G_{1}).\n    \\label{eq:regularreduction}\n  \\end{equation}\n\\end{lem}\n\nNext we determine $d(A\\wr G_1)$ via the following lemma.\n\n\\begin{lem}\\label{lem:keylemmaregular-abelian}\nLet $S$ be the socle of $G_1$. The minimum number of generators of the wreath product $A\\wr G_1$ is given by\n\\[\nd(A\\wr G)=\\max_q(d(A\\times G), d(A)+1, d_q(A)+2),\n\\]\nwhere $q$ ranges over the set of prime numbers dividing $\\order{A}$ and $\\order{S}$.\n\\end{lem}", "expanded_theorem": "\\label{thm:main}\nLet $G_{1}$,~$G_{2}$, \\dots,~$G_{k}$ be a sequence of almost simple groups. Let $S$ be the socle of $G_1.$ Let\n\\[\nW = G_{k} \\operatorname{wr} G_{k-1} \\operatorname{wr} \\dots \\operatorname{wr} G_{1}\n\\]\nbe the iterated wreath product constructed via the regular action of each factor. Set\n\\[\nA = (G_{k}/G_{k}') \\times (G_{k-1}/G_{k-1}') \\times \\dots \\times (G_{2}/G_{2}').\n\\]\nIn establishing the main theorem, we show that\n\\[\nd(W)=\\max_p(d(A\\times G_1), d(A)+1, d_p(A)+2),\n\\]\nwhere $p$ ranges over the set of prime numbers dividing both $\\mathopen{|}A\\mathclose{|}$ and $\\mathopen{|}S\\mathclose{|}$.,", "theorem_type": ["Equality or Bound", "Universal"], "mcq": {"question": "For a finite group $X$, let $d(X)$ denote the minimum number of generators of $X$. For a finite abelian group $B$ and a prime $p\\mid |B|$, let $d_p(B)$ denote the minimum number of generators of a Sylow-$p$ subgroup of $B$. Let $G_1, G_2, \\dots, G_k$ be a sequence of almost simple groups, let $S$ be the socle of $G_1$, form the iterated wreath product\n\\[\nW = G_k \\operatorname{wr} G_{k-1} \\operatorname{wr} \\dots \\operatorname{wr} G_1\n\\]\nusing the regular action of each factor, and set\n\\[\nA=(G_k/G_k')\\times (G_{k-1}/G_{k-1}')\\times \\dots \\times (G_2/G_2').\n\\]\nWhich statement holds for every such choice of groups?", "correct_choice": {"label": "A", "text": "\\[\nd(W)=\\max_p\\bigl(d(A\\times G_1),\\ d(A)+1,\\ d_p(A)+2\\bigr),\n\\]\nwhere $p$ ranges over the prime numbers dividing both $|A|$ and $|S|$."}, "choices": [{"label": "B", "text": "\\[\nd(W)=\\max_p\\bigl(d(A\\times G_1),\\ d(A)+1,\\ d_p(A)+2\\bigr),\n\\]\nwhere $p$ ranges over all prime numbers dividing $|A|$."}, {"label": "C", "text": "\\[\nd(W)\\ge \\max_p\\bigl(d(A\\times G_1),\\ d(A)+1,\\ d_p(A)+2\\bigr),\n\\]\nwhere $p$ ranges over the prime numbers dividing both $|A|$ and $|S|$."}, {"label": "D", "text": "\\[\nd(W)=\\max_p\\bigl(d(A\\times G_1),\\ d(A)+1,\\ d_p(A)+1\\bigr),\n\\]\nwhere $p$ ranges over the prime numbers dividing both $|A|$ and $|S|$."}, {"label": "E", "text": "\\[\nd(W)=\\max_p\\bigl(d(A\\times S),\\ d(A)+1,\\ d_p(A)+2\\bigr),\n\\]\nwhere $p$ ranges over the prime numbers dividing both $|A|$ and $|S|$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "prime range restricted to divisors of both |A| and |S|", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "exact equality replaced by the implied lower bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "the load-bearing +2 term in d_p(A)+2", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "reduction to A\\wr G_1; replacing G_1 by its socle S inside d(\\,\\cdot\\,\\times\\,\\cdot\\, )", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the objects and asks for the correct formula, but it does not explicitly reveal the answer or give away the distinctive features of choice A."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the question asks directly which statement holds for d(W), and the correct option is the exact theorem statement rather than a conclusion derived from a new setup."}, "GPS": {"score": 1, "justification": "There is some reasoning/precision required because the distractors alter subtle components of the formula (prime range, equality vs. inequality, additive constant), but the task is mainly recognition or recall rather than genuine generation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic errors: overbroad prime indexing, weakening equality to a lower bound, changing +2 to +1, or omitting extra wreath-product contributions."}, "total_score": 5, "overall_assessment": "A technically well-constructed recall question with strong distractors, but it is largely a direct restatement of a known result and only moderately tests reasoning."}}
{"id": "2602.03830v1", "paper_link": "http://arxiv.org/abs/2602.03830v1", "theorems_cnt": 1, "theorem": {"env_name": "thm", "content": "\\label{thm:main}\nLet $G_{1}$,~$G_{2}$, \\dots,~$G_{k}$ be a sequence of almost simple groups. Let $S$ be the socle of $G_1.$ Let\n\\[\nW = G_{k} \\operatorname{wr} G_{k-1} \\operatorname{wr} \\dots \\operatorname{wr} G_{1}\n\\]\nbe the iterated wreath product constructed via the regular action of each factor. Set\n\\[\nA = (G_{k}/G_{k}') \\times (G_{k-1}/G_{k-1}') \\times \\dots \\times (G_{2}/G_{2}').\n\\]\nThen\n\\[\nd(W)=\\max_p(d(A\\times G_1), d(A)+1, d_p(A)+2),\n\\]\nwhere $p$ ranges over the set of prime numbers dividing both $\\mathopen{|}A\\mathclose{|}$ and $\\mathopen{|}S\\mathclose{|}$.", "start_pos": 6724, "end_pos": 7241, "label": "thm:main"}, "ref_dict": {"lem:keylemmaregular-abelian": "\\begin{lem}\\label{lem:keylemmaregular-abelian}\nLet $S$ be the socle of $G_1$. The minimum number of generators of the wreath product $A\\wr G_1$ is given by\n\\[\nd(A\\wr G)=\\max_q(d(A\\times G), d(A)+1, d_q(A)+2),\n\\]\nwhere $q$ ranges over the set of prime numbers dividing $\\order{A}$ and $\\order{S}$.\n\\end{lem}", "lem:regularinduction": "\\begin{lem}\\label{lem:regularinduction}\n  Let $W$~and~$A$ be as defined above. Then\n  \\begin{equation}\n    d(W) = d(A \\wr G_{1}).\n    \\label{eq:regularreduction}\n  \\end{equation}\n\\end{lem}", "thm:main": "\\begin{thm}\\label{thm:main}\nLet $G_{1}$,~$G_{2}$, \\dots,~$G_{k}$ be a sequence of almost simple groups. Let $S$ be the socle of $G_1.$ Let\n\\[\nW = G_{k} \\wr G_{k-1} \\wr \\dots \\wr G_{1}\n\\]\nbe the iterated wreath product constructed via the regular action of each factor. Set\n\\[\nA = (G_{k}/G_{k}') \\times (G_{k-1}/G_{k-1}') \\times \\dots \\times (G_{2}/G_{2}').\n\\]\nThen\n\\[\nd(W)=\\max_p(d(A\\times G_1), d(A)+1, d_p(A)+2),\n\\]\nwhere $p$ ranges over the set of prime numbers dividing both $\\order{A}$ and $\\order{S}$.\n\\end{thm}"}, "pre_theorem_intro_text_len": 3858, "pre_theorem_intro_text": "A classic theme of group theory research is the generation property of groups. The structure of a group can be described using its generators. There have been numerous research on generators of specific groups. For example, it is delivered in undergraduate group theory courses that symmetric groups can be generated with two elements. Using the Classification of Finite Simple Groups, it is proved that every finite non-abelian simple group is $2$\\nobreakdash- generated.\n\nOne of the interesting generation problems is the growth rate of the minimum number of generators of the direct product $G^n$ where $G$ is a finite group. Wiegold shows in \\cite{Wiegold} that the number of required generators grows linearly when $G$ is an imperfect group and it grows logarithmically when $G$ is perfect. Apart from direct products, another way to construct groups is via wreath products. In this paper, we will investigate the minimum number of generators of iterated wreath products.\n\nBhattacharjee \\cite{Bhatt} investigates the generation property of iterated wreath product of alternating groups constructed via the natural action. She proves that the probability that such products can be generated by two random elements is positive. This shows that iterated wreath products of alternating groups are $2$\\nobreakdash- generated. Quick \\cite{MRQ} generalises this result and shows that the same result stills holds for iterated wreath products of non-abelian simple groups constructed from any faithful transitive action of the factors. Bondarenko \\cite{Bond} considers  iterated wreath products of transitive permutation groups with uniformly bounded number of generators. He establishes that the number of generators of the product involving $G_1, \\dots, G_n$ is bounded if and only if the number of generators of the abelian direct product $G_1/G_1'\\times G_2/G_2'\\times\\dots\\times G_n/G_n'$ is bounded as well.\n\nRecently, the author of this paper \\cite{LuQuick} considers the generators required for the iterated wreath products of symmetric groups, alternating groups or cyclic groups. In particular, it is established that the minimum number of generators of iterated wreath products of symmetric groups and alternating groups is the maximum of $2$ and the number of symmetric groups involved. The method used relies on Lucchini's work \\cite{Lu97}. Lucchini determines the number of generators required for the wreath product $A\\operatorname{wr} S_n$ of an abelian group $A$ constructed via the natural action of the symmetric group of degree $n$. This is then adapted for the wreath product $A\\operatorname{wr} A_n$ constructed via the natural action of the alternating group of degree $n$ in \\cite{LuQuick}. In \\cite{Lu97}, Lucchini also establishes the minimum number of generators for the wreath product $A\\operatorname{wr} G$ constructed via the regular action of a simple group $G$. We will use this result as part of our argument and generalise it to apply to the case that $G$ is an almost simple group.\n\nIn this article, we will establish the precise number of generators required for an iterated wreath product constructed from the regular action of almost simple groups. If we take the factors involved to be simple groups, then we recover the observation that iterated wreath products constructed via the regular action of non-abelian simple groups are $2$\\nobreakdash- generated.\n\nTo state the theorem, we recall some basic notations. For a group $G$, we write $d(G)$ for the minimum number of generators for $G$. For a finite abelian group $A$ and a prime number $p\\mid\\mathopen{|}A\\mathclose{|}$, we write $d_p(A)$ for the minimum number of generators for the Sylow-$p$ subgroup of $A$. Because of the Fundamental Theorem of Finite Abelian Groups, $d(A)=\\max_{p\\mid\\mathopen{|}A\\mathclose{|}}d_p(A)$. We will establish the following theorem.", "context": "A classic theme of group theory research is the generation property of groups. The structure of a group can be described using its generators. There have been numerous research on generators of specific groups. For example, it is delivered in undergraduate group theory courses that symmetric groups can be generated with two elements. Using the Classification of Finite Simple Groups, it is proved that every finite non-abelian simple group is $2$\\nobreakdash- generated.\n\nOne of the interesting generation problems is the growth rate of the minimum number of generators of the direct product $G^n$ where $G$ is a finite group. Wiegold shows in \\cite{Wiegold} that the number of required generators grows linearly when $G$ is an imperfect group and it grows logarithmically when $G$ is perfect. Apart from direct products, another way to construct groups is via wreath products. In this paper, we will investigate the minimum number of generators of iterated wreath products.\n\nBhattacharjee \\cite{Bhatt} investigates the generation property of iterated wreath product of alternating groups constructed via the natural action. She proves that the probability that such products can be generated by two random elements is positive. This shows that iterated wreath products of alternating groups are $2$\\nobreakdash- generated. Quick \\cite{MRQ} generalises this result and shows that the same result stills holds for iterated wreath products of non-abelian simple groups constructed from any faithful transitive action of the factors. Bondarenko \\cite{Bond} considers iterated wreath products of transitive permutation groups with uniformly bounded number of generators. He establishes that the number of generators of the product involving $G_1, \\dots, G_n$ is bounded if and only if the number of generators of the abelian direct product $G_1/G_1'\\times G_2/G_2'\\times\\dots\\times G_n/G_n'$ is bounded as well.\n\nRecently, the author of this paper \\cite{LuQuick} considers the generators required for the iterated wreath products of symmetric groups, alternating groups or cyclic groups. In particular, it is established that the minimum number of generators of iterated wreath products of symmetric groups and alternating groups is the maximum of $2$ and the number of symmetric groups involved. The method used relies on Lucchini's work \\cite{Lu97}. Lucchini determines the number of generators required for the wreath product $A\\operatorname{wr} S_n$ of an abelian group $A$ constructed via the natural action of the symmetric group of degree $n$. This is then adapted for the wreath product $A\\operatorname{wr} A_n$ constructed via the natural action of the alternating group of degree $n$ in \\cite{LuQuick}. In \\cite{Lu97}, Lucchini also establishes the minimum number of generators for the wreath product $A\\operatorname{wr} G$ constructed via the regular action of a simple group $G$. We will use this result as part of our argument and generalise it to apply to the case that $G$ is an almost simple group.\n\nIn this article, we will establish the precise number of generators required for an iterated wreath product constructed from the regular action of almost simple groups. If we take the factors involved to be simple groups, then we recover the observation that iterated wreath products constructed via the regular action of non-abelian simple groups are $2$\\nobreakdash- generated.\n\nTo state the theorem, we recall some basic notations. For a group $G$, we write $d(G)$ for the minimum number of generators for $G$. For a finite abelian group $A$ and a prime number $p\\mid\\mathopen{|}A\\mathclose{|}$, we write $d_p(A)$ for the minimum number of generators for the Sylow-$p$ subgroup of $A$. Because of the Fundamental Theorem of Finite Abelian Groups, $d(A)=\\max_{p\\mid\\mathopen{|}A\\mathclose{|}}d_p(A)$. We will establish the following theorem.", "full_context": "A classic theme of group theory research is the generation property of groups. The structure of a group can be described using its generators. There have been numerous research on generators of specific groups. For example, it is delivered in undergraduate group theory courses that symmetric groups can be generated with two elements. Using the Classification of Finite Simple Groups, it is proved that every finite non-abelian simple group is $2$\\nobreakdash- generated.\n\nOne of the interesting generation problems is the growth rate of the minimum number of generators of the direct product $G^n$ where $G$ is a finite group. Wiegold shows in \\cite{Wiegold} that the number of required generators grows linearly when $G$ is an imperfect group and it grows logarithmically when $G$ is perfect. Apart from direct products, another way to construct groups is via wreath products. In this paper, we will investigate the minimum number of generators of iterated wreath products.\n\nBhattacharjee \\cite{Bhatt} investigates the generation property of iterated wreath product of alternating groups constructed via the natural action. She proves that the probability that such products can be generated by two random elements is positive. This shows that iterated wreath products of alternating groups are $2$\\nobreakdash- generated. Quick \\cite{MRQ} generalises this result and shows that the same result stills holds for iterated wreath products of non-abelian simple groups constructed from any faithful transitive action of the factors. Bondarenko \\cite{Bond} considers iterated wreath products of transitive permutation groups with uniformly bounded number of generators. He establishes that the number of generators of the product involving $G_1, \\dots, G_n$ is bounded if and only if the number of generators of the abelian direct product $G_1/G_1'\\times G_2/G_2'\\times\\dots\\times G_n/G_n'$ is bounded as well.\n\nRecently, the author of this paper \\cite{LuQuick} considers the generators required for the iterated wreath products of symmetric groups, alternating groups or cyclic groups. In particular, it is established that the minimum number of generators of iterated wreath products of symmetric groups and alternating groups is the maximum of $2$ and the number of symmetric groups involved. The method used relies on Lucchini's work \\cite{Lu97}. Lucchini determines the number of generators required for the wreath product $A\\operatorname{wr} S_n$ of an abelian group $A$ constructed via the natural action of the symmetric group of degree $n$. This is then adapted for the wreath product $A\\operatorname{wr} A_n$ constructed via the natural action of the alternating group of degree $n$ in \\cite{LuQuick}. In \\cite{Lu97}, Lucchini also establishes the minimum number of generators for the wreath product $A\\operatorname{wr} G$ constructed via the regular action of a simple group $G$. We will use this result as part of our argument and generalise it to apply to the case that $G$ is an almost simple group.\n\nIn this article, we will establish the precise number of generators required for an iterated wreath product constructed from the regular action of almost simple groups. If we take the factors involved to be simple groups, then we recover the observation that iterated wreath products constructed via the regular action of non-abelian simple groups are $2$\\nobreakdash- generated.\n\nTo state the theorem, we recall some basic notations. For a group $G$, we write $d(G)$ for the minimum number of generators for $G$. For a finite abelian group $A$ and a prime number $p\\mid\\mathopen{|}A\\mathclose{|}$, we write $d_p(A)$ for the minimum number of generators for the Sylow-$p$ subgroup of $A$. Because of the Fundamental Theorem of Finite Abelian Groups, $d(A)=\\max_{p\\mid\\mathopen{|}A\\mathclose{|}}d_p(A)$. We will establish the following theorem.\n\n\\begin{abstract}\n Let $G_{1}$,~$G_{2}$, \\dots\\ be a sequence of almost simple groups and construct a sequence~$(W_{i})$ of wreath products via $W_{1} =\n G_{1}$ and, for each $i >1$, $W_{i+1} = G_{i+1} \\wr W_{i}$ via\n the regular action of each $G_i$. We determine the minimum\n number~$d(W_{i})$ of generators required for each wreath product in\n this sequence.\n\\end{abstract}\n\nBhattacharjee \\cite{Bhatt} investigates the generation property of iterated wreath product of alternating groups constructed via the natural action. She proves that the probability that such products can be generated by two random elements is positive. This shows that iterated wreath products of alternating groups are $2$\\nbd generated. Quick \\cite{MRQ} generalises this result and shows that the same result stills holds for iterated wreath products of non-abelian simple groups constructed from any faithful transitive action of the factors. Bondarenko \\cite{Bond} considers iterated wreath products of transitive permutation groups with uniformly bounded number of generators. He establishes that the number of generators of the product involving $G_1, \\dots, G_n$ is bounded if and only if the number of generators of the abelian direct product $G_1/G_1'\\times G_2/G_2'\\times\\dots\\times G_n/G_n'$ is bounded as well.\n\nIn this article, we will establish the precise number of generators required for an iterated wreath product constructed from the regular action of almost simple groups. If we take the factors involved to be simple groups, then we recover the observation that iterated wreath products constructed via the regular action of non-abelian simple groups are $2$\\nbd generated.\n\nWe finish this introduction by giving the structure of this paper. In Section~\\ref{sec:pre}, we recall basic definition that we will use. We will also introduce the results that our conclusions depend on, particularly the ones from \\cite{Lu97}. In Section~\\ref{sec:AwrG}, using the results in~\\cite{Lu97} and also extending them to cover a more general case, we will determine the minimum number of generators of the wreath product of an abelian group by an almost simple group constructed via the regular action. We will then provide the main proof of the theorem in the final section. We will prove it by showing that $d(W)=d(A\\wr G_1)$ and determining $d(A\\wr G_1)$, where $W, A, G_1$ are as defined in Theorem~\\ref{thm:main} (see Lemmas~\\ref{lem:keylemmaregular-abelian} and \\ref{lem:regularinduction} respectively).\n\n\\begin{lem}\\label{lem:LuStam}\n\\begin{enumerate}\n\\item\\label{i:Lucchini-s}{\\normalfont (Lucchini~{\\cite[1.2]{Lu97}})}\n Let $M$~be an irreducible $G$\\nbd module for a finite group~$G$.\n Then\n \\[\n s_{G}(M) = \\delta_{G}(M) + \\dim_{\\End{M}} \\Cohom{G/\\Cent{G}{M},M}.\n \\]\n\\item\\label{i:H1<rGM}{\\normalfont (Lucchini~{\\cite[1.3]{Lu97}})}\nLet $M$ be an irreducible $G$\\nbd module. Then\n\\[\n\\dim_{\\End{M}}H^1(G/C_G(M),M)<r_G(M).\n\\]\n\\item\\label{i:Stam}{\\normalfont (Stammbach~{\\cite[Theorem A]{Stammbach}})}\nGiven a prime integer $p$, a finite group $G$ is $p$-soluble if and only if $\\Cohom{G/C_G(M),M}=0$ for all irreducible $\\Field{p}G$\\nbd modules $M$.\n\\item\\label{i:augmentationgenerator}{\\normalfont (Lucchini~{\\cite[Proposition 1]{Lu97}})}\nLet $H$ be a finite group and $G$ be a transitive permutation group of degree $n$, then\n\\begin{align*}\nd(I_{H\\wr G})=\\max\\left(d(I_{(H/H')\\wr G}), \\left\\lfloor\\frac{d(I_H)-2}{n}\\right\\rfloor+2\\right).\n\\end{align*}\n\\item\\label{i:soluble}{\\normalfont (Lucchini~{\\cite[Theorem 2]{Lu97}})}\nIf $H$ is a finite soluble group and $G$ is a transitive permutation group of degree $n$, then\n\\[\nd(H\\wr G)=\\max\\left(d\\left(\\frac{H}{H'}\\wr G\\right),\\left\\lfloor\\frac{d(H)-2}{n}\\right\\rfloor+2\\right).\n\\]\n\\item\\label{i:Lucchini-AwrG}{\\normalfont (Lucchini~{\\cite[Theorem 4]{Lu97}})}\nLet $A\\wr G$ be the wreath product of a finite abelian group $A$ by a finite group $G$ constructed via the regular action of $G$. Set \n\\[\n\\rho_p=\\max_Mh_G(M)+d_p(A)\n\\]\nwhere $M$ ranges over the set of non-trivial irreducible $\\Field{p}G$\\nbd modules, with $\\rho_p=0$ if every irreducible $\\Field{p}G$\\nbd modules is trivial. Then\n\\[\nd(A\\wr G)=\\max_{p\\mid\\order{A}}(d(A\\times G),\\rho_p).\n\\]\n\\end{enumerate}\n\\end{lem}\n\n\\begin{lem}\\label{lem:AwrGLietype}\nLet $S$ be a simple group of Lie type and $G$ be an almost simple group with socle $S$. Let $A$ be a finite abelian group and set $W=A\\wr G$ to be the wreath product of the abelian group by the almost simple group with respect to the regular action. Then\n\\[\nd(W)=\\max_q(d(A\\times G), d(A)+1, d_q(A)+2),\n\\]\nwhere $q$ ranges over the set of the prime numbers dividing $\\order{A}$ and $\\order{S}$.\n\\end{lem}\n\\begin{prf}\nUsing Lemma~\\ref{lem:LuStam}\\ref{i:Lucchini-AwrG}, it suffices to show that\n\\begin{equation}\\label{eq:maxsigma=dq}\n\\max_{p\\mid\\order{A}}\\rho_p=\\max_q(d(A)+1, d_q(A)+2),\n\\end{equation}\nwhere $q$ ranges over the set of the prime numbers dividing $\\order{A}$ and $\\order{S}$ and\n\\[\n\\rho_p=\\max_Mh_G(M)+d_p(A).\n\\]\nLemma~\\ref{lem:maxhG} tells us that $\\rho_p\\geq1+d_p(A) $ for all $p\\mid\\order{A}$. Hence $\\rho_p\\geq 1+d(A)$. Lemma~\\ref{lem:maxhG} also shows that, for any prime number $p$ dividing $\\order{A}$ and $\\order{S}$, $\\rho_p=2+d_p(A)$. Thus \n\\[\n\\max_{p\\mid\\order{A}}\\rho_p\\geq\\max_{p\\mid\\order{A}, p\\mid\\order{S}}\\rho_p=\\max_{p\\mid\\order{A}, p\\mid\\order{S}} 2+d_p(A).\n\\]\nThen \n\\[\n\\max_{p\\mid\\order{A}}\\rho_p\\geq\\max_q(d(A)+1, d_q(A)+2).\n\\]\n\nWe will now establish the main theorem. Accordingly, let $k\\geq2$, and let $G_1, G_2,\\dots, G_k$ be almost simple groups. Define \n\\[\nW=G_k\\wr G_{k-1}\\wr\\dots\\wr G_1\n\\]\nto be the wreath product constructed via the regular action of each factor. We also define\n\\[\nH=G_k\\wr G_{k-1}\\wr \\dots\\wr G_2\n\\]\nand set $A=H/H'$ to be the abelianization of $H$; that is,\n\\[\nA=(G_k/G_k')\\times (G_{k-1}/G_{k-1}')\\times\\dots\\times (G_{2}/G_{2}').\n\\]\n\n\\begin{lem}\\label{lem:keylemmaregular-abelian}\nLet $S$ be the socle of $G_1$. The minimum number of generators of the wreath product $A\\wr G_1$ is given by\n\\[\nd(A\\wr G)=\\max_q(d(A\\times G), d(A)+1, d_q(A)+2),\n\\]\nwhere $q$ ranges over the set of prime numbers dividing $\\order{A}$ and $\\order{S}$.\n\\end{lem}\n\\begin{prf}\nThe case when $S$ is a simple group of Lie type is given in Lemma~\\ref{lem:AwrGLietype}. If $S$ is an alternating group, then $\\Out(S)$ is abelian. According to~\\cite[Table 1]{Atlas}, $\\Out(S)$ is also abelian when $S$ is one of the sporadic groups. It follows that the abelian chief factors of $G$ are all trivial in the case that $S$ is an alternating simple group or one of the sporadic simple groups. Then for any non-trivial $\\Field{p}G$\\nbd module $M$, $\\delta_G(M)=0$. In this case, the formula is a direct consequence of~\\cite[Corollary 5]{Lu97}.\n\\end{prf}\n\n\\begin{lem}\\label{lem:regularinduction}\n  Let $W$~and~$A$ be as defined above. Then\n  \\begin{equation}\n    d(W) = d(A \\wr G_{1}).\n    \\label{eq:regularreduction}\n  \\end{equation}\n\\end{lem}\n\n\\begin{thm}\\label{thm:main}\nLet $G_{1}$,~$G_{2}$, \\dots,~$G_{k}$ be a sequence of almost simple groups. Let $S$ be the socle of $G_1.$ Let\n\\[\nW = G_{k} \\wr G_{k-1} \\wr \\dots \\wr G_{1}\n\\]\nbe the iterated wreath product constructed via the regular action of each factor. Set\n\\[\nA = (G_{k}/G_{k}') \\times (G_{k-1}/G_{k-1}') \\times \\dots \\times (G_{2}/G_{2}').\n\\]\nThen\n\\[\nd(W)=\\max_p(d(A\\times G_1), d(A)+1, d_p(A)+2),\n\\]\nwhere $p$ ranges over the set of prime numbers dividing both $\\order{A}$ and $\\order{S}$.\n\\end{thm}", "post_theorem_intro_text_len": 934, "post_theorem_intro_text": "We finish this introduction by giving the structure of this paper. In Section~\\ref{sec:pre}, we recall basic definition that we will use. We will also introduce the results that our conclusions depend on, particularly the ones from \\cite{Lu97}. In Section~\\ref{sec:AwrG}, using the results in~\\cite{Lu97} and also extending them to cover a more general case, we will determine the minimum number of generators of the wreath product of an abelian group by an almost simple group constructed via the regular action. We will then provide the main proof of the theorem in the final section. We will prove it by showing that $d(W)=d(A\\operatorname{wr} G_1)$ and determining $d(A\\operatorname{wr} G_1)$, where $W, A, G_1$ are as defined in Theorem~\\ref{thm:main} (see Lemmas~\\ref{lem:keylemmaregular-abelian} and \\ref{lem:regularinduction} respectively).\n\n\\paragraph{Acknowledgements:} The author is funded by the China\nScholarship Council.", "sketch": "To prove Theorem~\\ref{thm:main}, the paper \"determin[es] the minimum number of generators of the wreath product of an abelian group by an almost simple group constructed via the regular action\" (using results in~\\cite{Lu97} and extending them). The \"main proof\" is then given by \"showing that $d(W)=d(A\\operatorname{wr} G_1)$ and determining $d(A\\operatorname{wr} G_1)$, where $W, A, G_1$ are as defined in Theorem~\\ref{thm:main}\" (see Lemmas~\\ref{lem:keylemmaregular-abelian} and~\\ref{lem:regularinduction}).", "expanded_sketch": "To prove the main theorem, the paper determin[es] the minimum number of generators of the wreath product of an abelian group by an almost simple group constructed via the regular action (using results in~\\cite{Lu97} and extending them). The main proof is then given by showing that $d(W)=d(A\\operatorname{wr} G_1)$ and determining $d(A\\operatorname{wr} G_1)$, where $W, A, G_1$ are as defined in the statement of the main theorem. We first prove the following lemma.\n\n\\begin{lem}\\label{lem:regularinduction}\n  Let $W$~and~$A$ be as defined above. Then\n  \\begin{equation}\n    d(W) = d(A \\wr G_{1}).\n    \\label{eq:regularreduction}\n  \\end{equation}\n\\end{lem}\n\nNext we determine $d(A\\wr G_1)$ via the following lemma.\n\n\\begin{lem}\\label{lem:keylemmaregular-abelian}\nLet $S$ be the socle of $G_1$. The minimum number of generators of the wreath product $A\\wr G_1$ is given by\n\\[\nd(A\\wr G)=\\max_q(d(A\\times G), d(A)+1, d_q(A)+2),\n\\]\nwhere $q$ ranges over the set of prime numbers dividing $\\order{A}$ and $\\order{S}$.\n\\end{lem}", "expanded_theorem": "\\label{thm:main}\nLet $G_{1}$,~$G_{2}$, \\dots,~$G_{k}$ be a sequence of almost simple groups. Let $S$ be the socle of $G_1.$ Let\n\\[\nW = G_{k} \\operatorname{wr} G_{k-1} \\operatorname{wr} \\dots \\operatorname{wr} G_{1}\n\\]\nbe the iterated wreath product constructed via the regular action of each factor. Set\n\\[\nA = (G_{k}/G_{k}') \\times (G_{k-1}/G_{k-1}') \\times \\dots \\times (G_{2}/G_{2}').\n\\]\nIn establishing the main theorem, we show that\n\\[\nd(W)=\\max_p(d(A\\times G_1), d(A)+1, d_p(A)+2),\n\\]\nwhere $p$ ranges over the set of prime numbers dividing both $\\mathopen{|}A\\mathclose{|}$ and $\\mathopen{|}S\\mathclose{|}$.,", "theorem_type": ["Equality or Bound", "Universal"], "mcq": {"question": "For a finite group $X$, let $d(X)$ denote the minimum number of generators of $X$. For a finite abelian group $B$ and a prime $p\\mid |B|$, let $d_p(B)$ denote the minimum number of generators of a Sylow-$p$ subgroup of $B$. Let $G_1, G_2, \\dots, G_k$ be a sequence of almost simple groups, let $S$ be the socle of $G_1$, form the iterated wreath product\n\\[\nW = G_k \\operatorname{wr} G_{k-1} \\operatorname{wr} \\dots \\operatorname{wr} G_1\n\\]\nusing the regular action of each factor, and set\n\\[\nA=(G_k/G_k')\\times (G_{k-1}/G_{k-1}')\\times \\dots \\times (G_2/G_2').\n\\]\nWhich statement holds for every such choice of groups?", "correct_choice": {"label": "A", "text": "\\[\nd(W)=\\max_p\\bigl(d(A\\times G_1),\\ d(A)+1,\\ d_p(A)+2\\bigr),\n\\]\nwhere $p$ ranges over the prime numbers dividing both $|A|$ and $|S|$."}, "choices": [{"label": "B", "text": "\\[\nd(W)=\\max_p\\bigl(d(A\\times G_1),\\ d(A)+1,\\ d_p(A)+2\\bigr),\n\\]\nwhere $p$ ranges over all prime numbers dividing $|A|$."}, {"label": "C", "text": "\\[\nd(W)\\ge \\max_p\\bigl(d(A\\times G_1),\\ d(A)+1,\\ d_p(A)+2\\bigr),\n\\]\nwhere $p$ ranges over the prime numbers dividing both $|A|$ and $|S|$."}, {"label": "D", "text": "\\[\nd(W)=\\max_p\\bigl(d(A\\times G_1),\\ d(A)+1,\\ d_p(A)+1\\bigr),\n\\]\nwhere $p$ ranges over the prime numbers dividing both $|A|$ and $|S|$."}, {"label": "E", "text": "\\[\nd(W)=\\max_p\\bigl(d(A\\times S),\\ d(A)+1,\\ d_p(A)+2\\bigr),\n\\]\nwhere $p$ ranges over the prime numbers dividing both $|A|$ and $|S|$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "prime range restricted to divisors of both |A| and |S|", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "exact equality replaced by the implied lower bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "the load-bearing +2 term in d_p(A)+2", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "reduction to A\\wr G_1; replacing G_1 by its socle S inside d(\\,\\cdot\\,\\times\\,\\cdot\\, )", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct formula or uniquely cue choice A. It only sets up notation and asks which universal statement is true."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-identification item: the correct option states the exact target formula, and the task is to recognize the precise theorem rather than derive a new conclusion from the setup."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to compare subtle variants (prime range, equality vs. lower bound, +2 vs. +1, G_1 vs. S), but the item mainly tests recall/recognition of the exact statement rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful. They target realistic failure modes: over-broad prime range, replacing equality by a weaker true inequality, weakening the critical +2 term, and substituting S for G_1."}, "total_score": 5, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and little answer leakage, but it is largely tautological and only moderately tests genuine reasoning."}}
{"id": "2602.04034v1", "paper_link": "http://arxiv.org/abs/2602.04034v1", "theorems_cnt": 1, "theorem": {"env_name": "conjecture", "content": "\\label{conjecture:main}\nLet $\\alg{A}$ and $\\alg{B}$ be finite modules. Then, there are only finitely many clonoids from $\\alg{A}$ to $\\alg{B}$, if and only if $\\alg{A}$ and $\\alg{B}$ are of coprime order.", "start_pos": 10644, "end_pos": 10872, "label": "conjecture:main"}, "ref_dict": {"theorem:main": "\\begin{theorem} \\label{theorem:main}\nLet $\\F$ be a finite field, $k \\in \\N$, and let $\\algB$ be an $\\algR$-module, such that $|F|$ is invertible in $\\algR$. Then $\\clodO_{\\F^k,\\algB}$ is uniformly generated by $k$-ary $(\\F^k,\\algB)$-minors.\n\\end{theorem}", "theorem:unigen": "\\begin{theorem} \\label{theorem:unigen}\nLet $\\cloA$, $\\cloB$ be clones, $n \\geq 1$, and let $\\cloA_{const} = \\Clo(\\cloA \\cup \\{a\\}_{a\\in A})$ be the clone generated by $\\cloA$ and all constant operations over its set. Let $\\clodC$ a clonoid from $\\cloA_{const}$\nto $\\cloB$. Then the following are equivalent:\n\\begin{enumerate}\n\\item\\label{ite:th_unigen1} $\\clodC^{(n+1)}$ is $(n,\\cloA,\\cloB)$-UG\n\\item\\label{ite:th_unigen2} $\\clodC^{(k)}$ is $(n,\\cloA,\\cloB)$-UG for some $k>n$\n\\item\\label{ite:th_unigen3} $\\clodC$ is $(n,\\cloA,\\cloB)$-UG\n\\item\\label{ite:th_unigen4} $\\forall k,l \\in \\N$, every partial operation $I \\colon \\clodC^{(k)} \\to \\clodC^{(l)}$ \nthat is $(\\cloA,\\cloB)$-UR is $(n,\\cloA,\\cloB)$-UR.\n\\end{enumerate}\n\\end{theorem}", "conjecture:main": "\\begin{conjecture} \\label{conjecture:main}\nLet $\\algA$ and $\\algB$ be finite modules. Then, there are only finitely many clonoids from $\\algA$ to $\\algB$, if and only if $\\algA$ and $\\algB$ are of coprime order.\n\\end{conjecture}", "theorem:clonoidlattice": "\\begin{theorem} \\label{theorem:clonoidlattice}\nThe lattice of clonoids from $\\F^k$ to $\\algB$ is isomorphic to $\\prod_{i=0}^k \\Sub(\\algM_{i,k}(\\F,\\algB))$.\n\nMoreover, for $j\\leq k$, the sublattice $\\prod_{i=0}^j \\Sub(\\algM_{i,k}(\\F,\\algB))$ corresponds to the clonoids consisting of functions that are generated by their $j$-ary minors.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 4708, "pre_theorem_intro_text": "A \\emph{clone} is a set of finitary operations on a given set that contains all projections and is closed under functional composition. Clones are a fundamental object in universal algebra, as they describe the set of term operations of any algebraic structure. Thus, starting with Post's classification of clones on a two-element set \\cite{post-lattice}, there has been a rich history of structural results on finite algebras based on clone theory, see e.g. \\cite{szendrei-clones}. On the other hand, clones theory has also several applications in theoretical computer science. One of the most prominent examples is the study of polymorphism clones to determining the complexity of fixed template constraint satisfaction problems (see e.g. the survey \\cite{BKW-polymorphisms}), which lead to Bulatov's and Zhuk's independent proofs of the CSP dichotomy theorem \\cite{bulatov-dichotomy}, \\cite{zhuk-dichotomy-short,zhuk-dichotomy}.\n\nIn recent years, generalizations of clones such as \\emph{minions} and \\emph{clonoids} have received increasing attention. Minions are sets of finitary operations from a set $A$ to a (possibly different) set $B$ that are closed under permutation and identification of variables, as well as the addition of dummy variables. Research on minions gained much traction due to their application on (promise) constraint satisfaction, see e.g. the survey \\cite{BBKO-PCSP}. A \\emph{clonoid} from an algebra $\\alg{A}$ to an algebra $\\alg{B}$ is a minion that is closed under (pre-)composition with operations of $\\alg{A}$ and (post-)composition with the operations of $\\alg{B}$. While the term `clonoid' was first introduced by Aichinger and Mayr in~\\cite{AM-eqclasses}, the concept has appeared under various names earlier, e.g. in \\cite{ CF-clonoids}. \n\nDue to their weak structure, it is does not seem to be feasible to explicitly describe all minions between two given finite sets. A first hurdle is their sheer number: it was shown in \\cite{pippenger-minions} that there are already continuum many minions between two 2-element sets. However, depending on the expressiveness of the algebras $\\alg{A}$ and $\\alg{B}$, much more can be said about the clonoids between $\\alg{A}$ and $\\alg{B}$. Sparks showed in \\cite{sparks-clonoids}, that there are only finitely many clonoids from $\\alg{A}$ to $\\alg{B}$ if $\\alg{B}$ has a near-unanimity term, and at most countably many, if $\\alg{B}$ has an edge-term. However, also a rich source algebra $\\alg{A}$ can lead to finiteness results: As showed in \\cite{LS-discriminator}, every clonoids from an algebra $\\alg{A}$ with a discriminator term, to an algebra $\\alg{B}$ on the same set, gives rise to only finitely many clonoids.\n\nWe would also like to highlight a series of papers by Lehtonen and coauthors \\cite{LN-minorsclique,LS-discriminator, CL-stability, lehtonen-majorityclonoids, lehtonen-NUclonoids}, which culminated in \\cite{lehtonen-Booleanclonoids} to a complete characterization of all two-element algebras $(\\alg{A},\\alg{B})$, for which there are at most countable clonoids, together with a complete classification of clonoids in all such cases.\n\nIn this paper, we are interested in classifying clonoids, for which $\\alg{B}$ is a module. Such clonoids often appear naturally when studying the term operations of Mal'cev algebras with a central congruence; this connection was recently made more explicit by Peter Mayr's definition of the \\emph{difference clonoid}, see e.g. \\cite{kompatscher-SMP2nil}. In particular, 2-nilpotent algebras give rise to clonoids between modules $\\alg{A}$ and $\\alg{B}$.\n\nAs an example of an application let us let us mention the classification of extensions of square-free groups in \\cite{AM-Zpqextensions} and later \\cite{fioravanti-groupexpansions}, which heavily relied on the classification of certain clonoids.\n\nThere is already a series of classification results in the literature. The clonoids between groups of prime order $\\Z_p$ and $\\Z_q$ were completely classified by Kreinecker \\cite{kreinecker-zpclonoids} (in the case that $p=q$) and the first author, for $p \\neq q$ \\cite{fioravanti-clonoids}. Interestingly, in the latter case, there are only finitely many clonoids that are all generated by their unary functions (while for $p=q$ one obtains infinitely many). \n\nIn subsequent generalizations of these results, it was also always observed that there are only finitely many clonoids between certain modules $\\alg{A}$ and $\\alg{B}$ of coprime order - the most recent, and most general such finiteness result is by Mayr and Wynne \\cite{MW-clonoidsmodules}, and applies to all cases where $\\alg{A}$ has a distributive lattice of submodules. This leads us to the following Conjecture:", "context": "In recent years, generalizations of clones such as \\emph{minions} and \\emph{clonoids} have received increasing attention. Minions are sets of finitary operations from a set $A$ to a (possibly different) set $B$ that are closed under permutation and identification of variables, as well as the addition of dummy variables. Research on minions gained much traction due to their application on (promise) constraint satisfaction, see e.g. the survey \\cite{BBKO-PCSP}. A \\emph{clonoid} from an algebra $\\alg{A}$ to an algebra $\\alg{B}$ is a minion that is closed under (pre-)composition with operations of $\\alg{A}$ and (post-)composition with the operations of $\\alg{B}$. While the term `clonoid' was first introduced by Aichinger and Mayr in~\\cite{AM-eqclasses}, the concept has appeared under various names earlier, e.g. in \\cite{ CF-clonoids}.\n\nDue to their weak structure, it is does not seem to be feasible to explicitly describe all minions between two given finite sets. A first hurdle is their sheer number: it was shown in \\cite{pippenger-minions} that there are already continuum many minions between two 2-element sets. However, depending on the expressiveness of the algebras $\\alg{A}$ and $\\alg{B}$, much more can be said about the clonoids between $\\alg{A}$ and $\\alg{B}$. Sparks showed in \\cite{sparks-clonoids}, that there are only finitely many clonoids from $\\alg{A}$ to $\\alg{B}$ if $\\alg{B}$ has a near-unanimity term, and at most countably many, if $\\alg{B}$ has an edge-term. However, also a rich source algebra $\\alg{A}$ can lead to finiteness results: As showed in \\cite{LS-discriminator}, every clonoids from an algebra $\\alg{A}$ with a discriminator term, to an algebra $\\alg{B}$ on the same set, gives rise to only finitely many clonoids.\n\nWe would also like to highlight a series of papers by Lehtonen and coauthors \\cite{LN-minorsclique,LS-discriminator, CL-stability, lehtonen-majorityclonoids, lehtonen-NUclonoids}, which culminated in \\cite{lehtonen-Booleanclonoids} to a complete characterization of all two-element algebras $(\\alg{A},\\alg{B})$, for which there are at most countable clonoids, together with a complete classification of clonoids in all such cases.\n\nIn this paper, we are interested in classifying clonoids, for which $\\alg{B}$ is a module. Such clonoids often appear naturally when studying the term operations of Mal'cev algebras with a central congruence; this connection was recently made more explicit by Peter Mayr's definition of the \\emph{difference clonoid}, see e.g. \\cite{kompatscher-SMP2nil}. In particular, 2-nilpotent algebras give rise to clonoids between modules $\\alg{A}$ and $\\alg{B}$.\n\nThere is already a series of classification results in the literature. The clonoids between groups of prime order $\\Z_p$ and $\\Z_q$ were completely classified by Kreinecker \\cite{kreinecker-zpclonoids} (in the case that $p=q$) and the first author, for $p \\neq q$ \\cite{fioravanti-clonoids}. Interestingly, in the latter case, there are only finitely many clonoids that are all generated by their unary functions (while for $p=q$ one obtains infinitely many).\n\nIn subsequent generalizations of these results, it was also always observed that there are only finitely many clonoids between certain modules $\\alg{A}$ and $\\alg{B}$ of coprime order - the most recent, and most general such finiteness result is by Mayr and Wynne \\cite{MW-clonoidsmodules}, and applies to all cases where $\\alg{A}$ has a distributive lattice of submodules. This leads us to the following Conjecture:", "full_context": "In recent years, generalizations of clones such as \\emph{minions} and \\emph{clonoids} have received increasing attention. Minions are sets of finitary operations from a set $A$ to a (possibly different) set $B$ that are closed under permutation and identification of variables, as well as the addition of dummy variables. Research on minions gained much traction due to their application on (promise) constraint satisfaction, see e.g. the survey \\cite{BBKO-PCSP}. A \\emph{clonoid} from an algebra $\\alg{A}$ to an algebra $\\alg{B}$ is a minion that is closed under (pre-)composition with operations of $\\alg{A}$ and (post-)composition with the operations of $\\alg{B}$. While the term `clonoid' was first introduced by Aichinger and Mayr in~\\cite{AM-eqclasses}, the concept has appeared under various names earlier, e.g. in \\cite{ CF-clonoids}.\n\nDue to their weak structure, it is does not seem to be feasible to explicitly describe all minions between two given finite sets. A first hurdle is their sheer number: it was shown in \\cite{pippenger-minions} that there are already continuum many minions between two 2-element sets. However, depending on the expressiveness of the algebras $\\alg{A}$ and $\\alg{B}$, much more can be said about the clonoids between $\\alg{A}$ and $\\alg{B}$. Sparks showed in \\cite{sparks-clonoids}, that there are only finitely many clonoids from $\\alg{A}$ to $\\alg{B}$ if $\\alg{B}$ has a near-unanimity term, and at most countably many, if $\\alg{B}$ has an edge-term. However, also a rich source algebra $\\alg{A}$ can lead to finiteness results: As showed in \\cite{LS-discriminator}, every clonoids from an algebra $\\alg{A}$ with a discriminator term, to an algebra $\\alg{B}$ on the same set, gives rise to only finitely many clonoids.\n\nWe would also like to highlight a series of papers by Lehtonen and coauthors \\cite{LN-minorsclique,LS-discriminator, CL-stability, lehtonen-majorityclonoids, lehtonen-NUclonoids}, which culminated in \\cite{lehtonen-Booleanclonoids} to a complete characterization of all two-element algebras $(\\alg{A},\\alg{B})$, for which there are at most countable clonoids, together with a complete classification of clonoids in all such cases.\n\nIn this paper, we are interested in classifying clonoids, for which $\\alg{B}$ is a module. Such clonoids often appear naturally when studying the term operations of Mal'cev algebras with a central congruence; this connection was recently made more explicit by Peter Mayr's definition of the \\emph{difference clonoid}, see e.g. \\cite{kompatscher-SMP2nil}. In particular, 2-nilpotent algebras give rise to clonoids between modules $\\alg{A}$ and $\\alg{B}$.\n\nThere is already a series of classification results in the literature. The clonoids between groups of prime order $\\Z_p$ and $\\Z_q$ were completely classified by Kreinecker \\cite{kreinecker-zpclonoids} (in the case that $p=q$) and the first author, for $p \\neq q$ \\cite{fioravanti-clonoids}. Interestingly, in the latter case, there are only finitely many clonoids that are all generated by their unary functions (while for $p=q$ one obtains infinitely many).\n\nIn subsequent generalizations of these results, it was also always observed that there are only finitely many clonoids between certain modules $\\alg{A}$ and $\\alg{B}$ of coprime order - the most recent, and most general such finiteness result is by Mayr and Wynne \\cite{MW-clonoidsmodules}, and applies to all cases where $\\alg{A}$ has a distributive lattice of submodules. This leads us to the following Conjecture:\n\n\\begin{abstract}\nClonoids are sets of finitary operations between two algebraic structures that are closed under composition with their term operations on both sides. We conjecture that, for finite modules $\\algA$ and $\\algB$ there are only finitely many clonoids from $\\algA$ to $\\algB$ if and only if $\\algA$, $\\algB$ are of coprime order.\n\nWe confirm this conjecture for a broad class of modules $\\algA$. In particular we show that, if $\\algA$ is a finite $k$-dimensional vector space, then every clonoid from $\\alg A$ to a coprime module $\\algB$ is generated by its $k$-ary functions (and arity $k-1$ does not suffice). In order to prove this results, we investigate `uniform generation by $(\\algA,\\algB)$-minors', a general criterion, which we show to apply to several other existing classifications results. Based on our analysis, we further prove that the subpower membership problem of certain 2-nilpotent Mal'cev algebras is solvable in polynomial time.\n\\end{abstract}\n\nIn subsequent generalizations of these results, it was also always observed that there are only finitely many clonoids between certain modules $\\algA$ and $\\algB$ of coprime order - the most recent, and most general such finiteness result is by Mayr and Wynne \\cite{MW-clonoidsmodules}, and applies to all cases where $\\algA$ has a distributive lattice of submodules. This leads us to the following Conjecture:\n\nWe remark that the ``only if direction'' follows from \\cite[Theorem $1.3$]{MW-clonoidsmodules}, where the authors proved that for modules with a common divisor, one can always construct an infinite ascending chain of clonoids.\n\nThe first main contribution of our paper is to confirm Conjecture \\ref{conjecture:main} if $\\algA$ is a finite vector space. In the case where $\\algB$ is coprime to the vector space $\\algA$, we are moreover able to give an explicit description of the lattice of all clonoids (Theorem \\ref{theorem:clonoidlattice}). In particular, this answers \\cite[Question 1.2]{MW-clonoidsmodules}, which asked for a classification of clonoids, if $\\algA$ is the group $(\\Z_p)^2$, for some prime $p$. By combining our result with \\cite{MW-clonoidsmodules}, we are further able to confirm Conjecture \\ref{conjecture:main} for all modules $\\algA =\\F_1^{k_1}\\times \\cdots \\times \\F_n^{k_n} \\times \\alg D$ as $\\F_1\\times \\cdots \\times\\F_n \\times \\algR$-module, where $\\alg D$ is a distributive $\\algR$-module, and all $\\F_i$ are fields.\n\n\\begin{corollary} \\label{corollary:main2}\nLet $\\F^k$ be the $k$-dimensional vector space over a finite field $\\F$ and let $\\algB$ be a finite module of coprime order. Then\n\\begin{enumerate}\n\\item there are only finitely many clonoids from $\\F^k$ to $\\algB$.\n\\item every clonoid from $\\F^k$ to $\\algB$ is finitely related by an at most $|F|^{k\\times k}$-ary relation.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{corollary} \\label{corollary:mainresultproducts}\nLet $\\algA =\\F_1^{k_1}\\times \\cdots \\times \\F_n^{k_n} \\times \\alg D$ as $\\F_1\\times \\cdots \\times\\F_n \\times \\algR$-module, where all $\\F_i$ are finite fields, and $\\alg D$ is a finite distributive $\\algR$-module. Let $l$ be the nilpotence degree of the Jacobson radical of $\\algR$, and let $k = \\max(k_1,\\ldots,k_n,l)$. Further let $\\algB$ be an $\\algS$-module, such that $|A|$ is invertible in $\\algS$. Then\n\\begin{enumerate}\n\\item every clonoid from $\\alg A$ to $\\algB$ is generated by its $k$-ary part.\n\\item If $\\algB$ is finite then there are only finitely many clonoids from $\\algA$ to $\\algB$.\n\\end{enumerate}\n\\end{corollary}\n\nIn this paper, we studied clonoids between finite modules. We proved that there are only finitely many clonoids from a finite vector space $\\algA$ to a coprime module $\\algB$, which is also in accordance to Conjecture \\ref{conjecture:main}.\n\n\\begin{conjecture} \\label{conjecture:main}\nLet $\\algA$ and $\\algB$ be finite modules. Then, there are only finitely many clonoids from $\\algA$ to $\\algB$, if and only if $\\algA$ and $\\algB$ are of coprime order.\n\\end{conjecture}\n\n\\begin{theorem} \\label{theorem:clonoidlattice}\nThe lattice of clonoids from $\\F^k$ to $\\algB$ is isomorphic to $\\prod_{i=0}^k \\Sub(\\algM_{i,k}(\\F,\\algB))$.\n\nMoreover, for $j\\leq k$, the sublattice $\\prod_{i=0}^j \\Sub(\\algM_{i,k}(\\F,\\algB))$ corresponds to the clonoids consisting of functions that are generated by their $j$-ary minors.\n\\end{theorem}", "post_theorem_intro_text_len": 4549, "post_theorem_intro_text": "We remark that the ``only if direction'' follows from \\cite[Theorem $1.3$]{MW-clonoidsmodules}, where the authors proved that for modules with a common divisor, one can always construct an infinite ascending chain of clonoids.\n\n The first main contribution of our paper is to confirm Conjecture \\ref{conjecture:main} if $\\alg{A}$ is a finite vector space. In the case where $\\alg{B}$ is coprime to the vector space $\\alg{A}$, we are moreover able to give an explicit description of the lattice of all clonoids (Theorem \\ref{theorem:clonoidlattice}). In particular, this answers \\cite[Question 1.2]{MW-clonoidsmodules}, which asked for a classification of clonoids, if $\\alg{A}$ is the group $(\\Z_p)^2$, for some prime $p$. By combining our result with \\cite{MW-clonoidsmodules}, we are further able to confirm Conjecture \\ref{conjecture:main} for all modules $\\alg{A} =\\F_1^{k_1}\\times \\cdots \\times \\F_n^{k_n} \\times \\alg D$ as $\\F_1\\times \\cdots \\times\\F_n \\times \\alg{R}$-module, where $\\alg D$ is a distributive $\\alg{R}$-module, and all $\\F_i$ are fields.\n\nOur classification is based on proving that, in the case of a vector space $\\alg{A} = {\\mathbf F}^k$ and coprime $\\alg{B}$, the set of all operations from $A$ to $B$ is \\emph{uniformly generated} by $k$-ary minors. While this technique was already hinted at in \\cite{MW-clonoidsmodules}, we are going to discuss it in great detail in Section \\ref{sec:unifgen}, and argue that many of the existing clonoid classifications follow the same pattern. We furthermore show that the arity $k$ is optimal, i.e., not every clonoid from ${\\mathbf F}^k$ to $\\alg{B}$ is generated by their $k-1$-ary functions. This coincides with a lower bound on the arity of generators of the \\emph{full} clonoid between modules that we derive in Section \\ref{sec:lowerbound}, and was already pointed out in the Bachelor thesis by Jan Van\\v{e}\\v{c}ek \\cite{vanecek-thesis}.\n\nLast, let us mention that, clonoids between affine algebras were also (sometimes implicitly) used to discuss some computational problems of Mal'cev algebras. The fact that some clonoids between coprime modules are generated by unary functions, which allows for a representation of their elements by nice normal forms, was used in \\cite{KKK-CEQV2nil} to prove that checking polynomial identities in a given 2-nilpotent algebra can always be done in polynomial time. In \\cite{mayr-VLloop} and \\cite{KompatscherMayr2026} such normal forms were similarly used to find finite equational basis for some 2-nilpotent loops.\n\nWhile we are not going to discuss such syntactic aspects of clonoids, we will apply our results to the \\emph{subpower membership problem} of certain algebras. The subpower membership problem (SMP($\\alg{A}$)), for a fixed finite algebra $\\alg{A}$, is the computational problem whether a given partial operation $f\\colon A^n \\to A$ can be extended to a term operation of $\\alg{A}$. In \\cite{IMMVW-subpowers} it was asked, whether the subpower membership problem is always polynomial time solvable for algebras with few subpowers. This question, however remains unsolved even in the Mal'cev case. The easiest examples that are not covered by existing tractability results \\cite{mayr-SMP, BMS-SMP}, are 2-nilpotent algebras, for which it was shown in \\cite{kompatscher-SMP2nil} that the problem is polynomial time equivalent to a similar `interpolation problem' for their clonoids. In Section \\ref{sect:SMP} we are going to use our results, to prove that the subpower membership problem of a big class of 2-nilpotent algebras is in P.\n\n\\subsection*{Organization of the paper}\n\nThe paper is organized as follows. In Section~\\ref{sec:prel} we introduce the necessary concepts from universal algebra and fix some notation. In Section~\\ref{sec:unifgen} we develop a theory of `uniform generation' and `uniform representation' by minors, a concept introduced in~\\cite{MW-clonoidsmodules}. In particular we provide a purely combinatorial criterion for all $(\\alg{A},\\alg{B})$-clonoids to be generated by their $k$-ary functions (Theorem \\ref{theorem:unigen}).\n\nSection~\\ref{sec:mainres} presents the main results of the paper, Theorem \\ref{theorem:main} and discusses some consequences. In Section \\ref{sect:SMP} we apply our results to the subpower membership problem. In Sections~\\ref{sec:lowerbound} we provide a lower bound on the arity of the generators of the full clonoid between finite modules, which is sharp in our setting. Section~\\ref{sec:conclusion} is dedicated to future work and possible applications of our results.", "sketch": "We remark that the ``only if direction'' follows from \\cite[Theorem $1.3$]{MW-clonoidsmodules}, where the authors proved that for modules with a common divisor, one can always construct an infinite ascending chain of clonoids.\n\nFor confirming Conjecture~\\ref{conjecture:main} when $\\alg{A}$ is a finite vector space and $\\alg{B}$ is coprime to $\\alg{A}$, the paper’s approach is described as follows: the classification is based on proving that, for a vector space $\\alg{A}={\\mathbf F}^k$ and coprime $\\alg{B}$, “the set of all operations from $A$ to $B$ is \\emph{uniformly generated} by $k$-ary minors.” The authors note this technique was “already hinted at in \\cite{MW-clonoidsmodules},” and they “discuss it in great detail in Section~\\ref{sec:unifgen},” including “a purely combinatorial criterion for all $(\\alg{A},\\alg{B})$-clonoids to be generated by their $k$-ary functions (Theorem~\\ref{theorem:unigen}).” They also establish an optimality statement for this method: “the arity $k$ is optimal, i.e., not every clonoid from ${\\mathbf F}^k$ to $\\alg{B}$ is generated by their $k-1$-ary functions,” and relate this to a “lower bound on the arity of generators of the \\emph{full} clonoid between modules” derived in Section~\\ref{sec:lowerbound}. Finally, they state that “by combining our result with \\cite{MW-clonoidsmodules}, we are further able to confirm Conjecture~\\ref{conjecture:main}” for broader classes of modules $\\alg{A}=\\F_1^{k_1}\\times\\cdots\\times\\F_n^{k_n}\\times\\alg D.", "expanded_sketch": "We remark that the ``only if direction'' follows from \\cite[Theorem $1.3$]{MW-clonoidsmodules}, where the authors proved that for modules with a common divisor, one can always construct an infinite ascending chain of clonoids.\n\nFor confirming the main conjecture when $\\alg{A}$ is a finite vector space and $\\alg{B}$ is coprime to $\\alg{A}$, the paper’s approach is described as follows: the classification is based on proving that, for a vector space $\\alg{A}={\\mathbf F}^k$ and coprime $\\alg{B}$, “the set of all operations from $A$ to $B$ is \\emph{uniformly generated} by $k$-ary minors.” The authors note this technique was “already hinted at in \\cite{MW-clonoidsmodules},” and they “discuss it in great detail” later, including “a purely combinatorial criterion for all $(\\alg{A},\\alg{B})$-clonoids to be generated by their $k$-ary functions.” We first recall this criterion.\n\n\\begin{theorem} \\label{theorem:unigen}\nLet $\\cloA$, $\\cloB$ be clones, $n \\geq 1$, and let $\\cloA_{const} = \\Clo(\\cloA \\cup \\{a\\}_{a\\in A})$ be the clone generated by $\\cloA$ and all constant operations over its set. Let $\\clodC$ a clonoid from $\\cloA_{const}$\nto $\\cloB$. Then the following are equivalent:\n\\begin{enumerate}\n\\item\\label{ite:th_unigen1} $\\clodC^{(n+1)}$ is $(n,\\cloA,\\cloB)$-UG\n\\item\\label{ite:th_unigen2} $\\clodC^{(k)}$ is $(n,\\cloA,\\cloB)$-UG for some $k>n$\n\\item\\label{ite:th_unigen3} $\\clodC$ is $(n,\\cloA,\\cloB)$-UG\n\\item\\label{ite:th_unigen4} $\\forall k,l \\in \\N$, every partial operation $I \\colon \\clodC^{(k)} \\to \\clodC^{(l)}$ \nthat is $(\\cloA,\\cloB)$-UR is $(n,\\cloA,\\cloB)$-UR.\n\\end{enumerate}\n\\end{theorem}\n\nThey also establish an optimality statement for this method: “the arity $k$ is optimal, i.e., not every clonoid from ${\\mathbf F}^k$ to $\\alg{B}$ is generated by their $k-1$-ary functions,” and relate this to a “lower bound on the arity of generators of the \\emph{full} clonoid between modules” derived later. Finally, they state that “by combining our result with \\cite{MW-clonoidsmodules}, we are further able to confirm the main conjecture” for broader classes of modules $\\alg{A}=\\F_1^{k_1}\\times\\cdots\\times\\F_n^{k_n}\\times\\alg D.", "expanded_theorem": "\\label{conjecture:main}\nLet $\\alg{A}$ and $\\alg{B}$ be finite modules. Then, there are only finitely many clonoids from $\\alg{A}$ to $\\alg{B}$, if and only if $\\alg{A}$ and $\\alg{B}$ are of coprime order.", "theorem_type": ["Biconditional or Equivalence", "Existence"], "mcq": {"question": "Let \\(\\alg A\\) and \\(\\alg B\\) be finite modules. A clonoid from \\(\\alg A\\) to \\(\\alg B\\) is a set of finitary functions from powers of \\(A\\) to \\(B\\) that is closed under taking minors (permuting variables, identifying variables, and adding dummy variables), as well as under precomposition with term operations of \\(\\alg A\\) and postcomposition with term operations of \\(\\alg B\\). Two finite modules are said to be of coprime order when \\(\\gcd(|A|,|B|)=1\\). Which statement holds about the number of clonoids from \\(\\alg A\\) to \\(\\alg B\\)?", "correct_choice": {"label": "A", "text": "There are only finitely many clonoids from \\(\\alg A\\) to \\(\\alg B\\) if and only if \\(\\alg A\\) and \\(\\alg B\\) are of coprime order."}, "choices": [{"label": "B", "text": "There are only finitely many clonoids from \\(\\alg A\\) to \\(\\alg B\\) whenever \\(\\alg A\\) and \\(\\alg B\\) have distinct orders."}, {"label": "C", "text": "If \\(\\alg A\\) and \\(\\alg B\\) are of coprime order, then there are only finitely many clonoids from \\(\\alg A\\) to \\(\\alg B\\)."}, {"label": "D", "text": "There are only finitely many clonoids from \\(\\alg A\\) to \\(\\alg B\\) if and only if every clonoid from \\(\\alg A\\) to \\(\\alg B\\) is generated by its unary functions."}, {"label": "E", "text": "There are only finitely many clonoids from \\(\\alg A\\) to \\(\\alg B\\) if and only if, for some integer \\(n\\ge 1\\) depending only on \\(|A|\\), every clonoid from \\(\\alg A\\) to \\(\\alg B\\) is generated by its \\(n\\)-ary functions."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "coprime-order hypothesis replaced by distinct-order condition", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped the converse direction from the iff statement", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "optimal k-ary generation confused with unary generation", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "uniform generation bound made to depend only on |A| and asserted equivalent to finiteness for all modules", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly or implicitly reveal the coprime-order criterion; it only asks for an equivalent characterization."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: it asks for the statement equivalent to a known finiteness condition, with the correct option restating the theorem almost verbatim."}, "GPS": {"score": 1, "justification": "The item requires some logical discrimination among nearby alternatives (weaker true condition, quantifier shift, overstrengthening), but it does not demand substantial mathematical generation beyond recalling the theorem."}, "DQS": {"score": 2, "justification": "The distractors are distinct and mathematically meaningful: one substitutes the wrong invariant, one gives a weaker consequence, one introduces a quantifier shift, and one adds an unjustified strengthening."}, "total_score": 5, "overall_assessment": "A solid recall-and-discrimination MCQ with good distractors and no answer leakage, but it is largely tautological as a theorem restatement rather than a genuine generative-reasoning question."}}
{"id": "2602.04034v1", "paper_link": "http://arxiv.org/abs/2602.04034v1", "theorems_cnt": 1, "theorem": {"env_name": "conjecture", "content": "\\label{conjecture:main}\nLet $\\alg{A}$ and $\\alg{B}$ be finite modules. Then, there are only finitely many clonoids from $\\alg{A}$ to $\\alg{B}$, if and only if $\\alg{A}$ and $\\alg{B}$ are of coprime order.", "start_pos": 10644, "end_pos": 10872, "label": "conjecture:main"}, "ref_dict": {"theorem:main": "\\begin{theorem} \\label{theorem:main}\nLet $\\F$ be a finite field, $k \\in \\N$, and let $\\algB$ be an $\\algR$-module, such that $|F|$ is invertible in $\\algR$. Then $\\clodO_{\\F^k,\\algB}$ is uniformly generated by $k$-ary $(\\F^k,\\algB)$-minors.\n\\end{theorem}", "theorem:unigen": "\\begin{theorem} \\label{theorem:unigen}\nLet $\\cloA$, $\\cloB$ be clones, $n \\geq 1$, and let $\\cloA_{const} = \\Clo(\\cloA \\cup \\{a\\}_{a\\in A})$ be the clone generated by $\\cloA$ and all constant operations over its set. Let $\\clodC$ a clonoid from $\\cloA_{const}$\nto $\\cloB$. Then the following are equivalent:\n\\begin{enumerate}\n\\item\\label{ite:th_unigen1} $\\clodC^{(n+1)}$ is $(n,\\cloA,\\cloB)$-UG\n\\item\\label{ite:th_unigen2} $\\clodC^{(k)}$ is $(n,\\cloA,\\cloB)$-UG for some $k>n$\n\\item\\label{ite:th_unigen3} $\\clodC$ is $(n,\\cloA,\\cloB)$-UG\n\\item\\label{ite:th_unigen4} $\\forall k,l \\in \\N$, every partial operation $I \\colon \\clodC^{(k)} \\to \\clodC^{(l)}$ \nthat is $(\\cloA,\\cloB)$-UR is $(n,\\cloA,\\cloB)$-UR.\n\\end{enumerate}\n\\end{theorem}", "conjecture:main": "\\begin{conjecture} \\label{conjecture:main}\nLet $\\algA$ and $\\algB$ be finite modules. Then, there are only finitely many clonoids from $\\algA$ to $\\algB$, if and only if $\\algA$ and $\\algB$ are of coprime order.\n\\end{conjecture}", "theorem:clonoidlattice": "\\begin{theorem} \\label{theorem:clonoidlattice}\nThe lattice of clonoids from $\\F^k$ to $\\algB$ is isomorphic to $\\prod_{i=0}^k \\Sub(\\algM_{i,k}(\\F,\\algB))$.\n\nMoreover, for $j\\leq k$, the sublattice $\\prod_{i=0}^j \\Sub(\\algM_{i,k}(\\F,\\algB))$ corresponds to the clonoids consisting of functions that are generated by their $j$-ary minors.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 4708, "pre_theorem_intro_text": "A \\emph{clone} is a set of finitary operations on a given set that contains all projections and is closed under functional composition. Clones are a fundamental object in universal algebra, as they describe the set of term operations of any algebraic structure. Thus, starting with Post's classification of clones on a two-element set \\cite{post-lattice}, there has been a rich history of structural results on finite algebras based on clone theory, see e.g. \\cite{szendrei-clones}. On the other hand, clones theory has also several applications in theoretical computer science. One of the most prominent examples is the study of polymorphism clones to determining the complexity of fixed template constraint satisfaction problems (see e.g. the survey \\cite{BKW-polymorphisms}), which lead to Bulatov's and Zhuk's independent proofs of the CSP dichotomy theorem \\cite{bulatov-dichotomy}, \\cite{zhuk-dichotomy-short,zhuk-dichotomy}.\n\nIn recent years, generalizations of clones such as \\emph{minions} and \\emph{clonoids} have received increasing attention. Minions are sets of finitary operations from a set $A$ to a (possibly different) set $B$ that are closed under permutation and identification of variables, as well as the addition of dummy variables. Research on minions gained much traction due to their application on (promise) constraint satisfaction, see e.g. the survey \\cite{BBKO-PCSP}. A \\emph{clonoid} from an algebra $\\alg{A}$ to an algebra $\\alg{B}$ is a minion that is closed under (pre-)composition with operations of $\\alg{A}$ and (post-)composition with the operations of $\\alg{B}$. While the term `clonoid' was first introduced by Aichinger and Mayr in~\\cite{AM-eqclasses}, the concept has appeared under various names earlier, e.g. in \\cite{ CF-clonoids}. \n\nDue to their weak structure, it is does not seem to be feasible to explicitly describe all minions between two given finite sets. A first hurdle is their sheer number: it was shown in \\cite{pippenger-minions} that there are already continuum many minions between two 2-element sets. However, depending on the expressiveness of the algebras $\\alg{A}$ and $\\alg{B}$, much more can be said about the clonoids between $\\alg{A}$ and $\\alg{B}$. Sparks showed in \\cite{sparks-clonoids}, that there are only finitely many clonoids from $\\alg{A}$ to $\\alg{B}$ if $\\alg{B}$ has a near-unanimity term, and at most countably many, if $\\alg{B}$ has an edge-term. However, also a rich source algebra $\\alg{A}$ can lead to finiteness results: As showed in \\cite{LS-discriminator}, every clonoids from an algebra $\\alg{A}$ with a discriminator term, to an algebra $\\alg{B}$ on the same set, gives rise to only finitely many clonoids.\n\nWe would also like to highlight a series of papers by Lehtonen and coauthors \\cite{LN-minorsclique,LS-discriminator, CL-stability, lehtonen-majorityclonoids, lehtonen-NUclonoids}, which culminated in \\cite{lehtonen-Booleanclonoids} to a complete characterization of all two-element algebras $(\\alg{A},\\alg{B})$, for which there are at most countable clonoids, together with a complete classification of clonoids in all such cases.\n\nIn this paper, we are interested in classifying clonoids, for which $\\alg{B}$ is a module. Such clonoids often appear naturally when studying the term operations of Mal'cev algebras with a central congruence; this connection was recently made more explicit by Peter Mayr's definition of the \\emph{difference clonoid}, see e.g. \\cite{kompatscher-SMP2nil}. In particular, 2-nilpotent algebras give rise to clonoids between modules $\\alg{A}$ and $\\alg{B}$.\n\nAs an example of an application let us let us mention the classification of extensions of square-free groups in \\cite{AM-Zpqextensions} and later \\cite{fioravanti-groupexpansions}, which heavily relied on the classification of certain clonoids.\n\nThere is already a series of classification results in the literature. The clonoids between groups of prime order $\\Z_p$ and $\\Z_q$ were completely classified by Kreinecker \\cite{kreinecker-zpclonoids} (in the case that $p=q$) and the first author, for $p \\neq q$ \\cite{fioravanti-clonoids}. Interestingly, in the latter case, there are only finitely many clonoids that are all generated by their unary functions (while for $p=q$ one obtains infinitely many). \n\nIn subsequent generalizations of these results, it was also always observed that there are only finitely many clonoids between certain modules $\\alg{A}$ and $\\alg{B}$ of coprime order - the most recent, and most general such finiteness result is by Mayr and Wynne \\cite{MW-clonoidsmodules}, and applies to all cases where $\\alg{A}$ has a distributive lattice of submodules. This leads us to the following Conjecture:", "context": "In recent years, generalizations of clones such as \\emph{minions} and \\emph{clonoids} have received increasing attention. Minions are sets of finitary operations from a set $A$ to a (possibly different) set $B$ that are closed under permutation and identification of variables, as well as the addition of dummy variables. Research on minions gained much traction due to their application on (promise) constraint satisfaction, see e.g. the survey \\cite{BBKO-PCSP}. A \\emph{clonoid} from an algebra $\\alg{A}$ to an algebra $\\alg{B}$ is a minion that is closed under (pre-)composition with operations of $\\alg{A}$ and (post-)composition with the operations of $\\alg{B}$. While the term `clonoid' was first introduced by Aichinger and Mayr in~\\cite{AM-eqclasses}, the concept has appeared under various names earlier, e.g. in \\cite{ CF-clonoids}.\n\nDue to their weak structure, it is does not seem to be feasible to explicitly describe all minions between two given finite sets. A first hurdle is their sheer number: it was shown in \\cite{pippenger-minions} that there are already continuum many minions between two 2-element sets. However, depending on the expressiveness of the algebras $\\alg{A}$ and $\\alg{B}$, much more can be said about the clonoids between $\\alg{A}$ and $\\alg{B}$. Sparks showed in \\cite{sparks-clonoids}, that there are only finitely many clonoids from $\\alg{A}$ to $\\alg{B}$ if $\\alg{B}$ has a near-unanimity term, and at most countably many, if $\\alg{B}$ has an edge-term. However, also a rich source algebra $\\alg{A}$ can lead to finiteness results: As showed in \\cite{LS-discriminator}, every clonoids from an algebra $\\alg{A}$ with a discriminator term, to an algebra $\\alg{B}$ on the same set, gives rise to only finitely many clonoids.\n\nWe would also like to highlight a series of papers by Lehtonen and coauthors \\cite{LN-minorsclique,LS-discriminator, CL-stability, lehtonen-majorityclonoids, lehtonen-NUclonoids}, which culminated in \\cite{lehtonen-Booleanclonoids} to a complete characterization of all two-element algebras $(\\alg{A},\\alg{B})$, for which there are at most countable clonoids, together with a complete classification of clonoids in all such cases.\n\nIn this paper, we are interested in classifying clonoids, for which $\\alg{B}$ is a module. Such clonoids often appear naturally when studying the term operations of Mal'cev algebras with a central congruence; this connection was recently made more explicit by Peter Mayr's definition of the \\emph{difference clonoid}, see e.g. \\cite{kompatscher-SMP2nil}. In particular, 2-nilpotent algebras give rise to clonoids between modules $\\alg{A}$ and $\\alg{B}$.\n\nThere is already a series of classification results in the literature. The clonoids between groups of prime order $\\Z_p$ and $\\Z_q$ were completely classified by Kreinecker \\cite{kreinecker-zpclonoids} (in the case that $p=q$) and the first author, for $p \\neq q$ \\cite{fioravanti-clonoids}. Interestingly, in the latter case, there are only finitely many clonoids that are all generated by their unary functions (while for $p=q$ one obtains infinitely many).\n\nIn subsequent generalizations of these results, it was also always observed that there are only finitely many clonoids between certain modules $\\alg{A}$ and $\\alg{B}$ of coprime order - the most recent, and most general such finiteness result is by Mayr and Wynne \\cite{MW-clonoidsmodules}, and applies to all cases where $\\alg{A}$ has a distributive lattice of submodules. This leads us to the following Conjecture:", "full_context": "In recent years, generalizations of clones such as \\emph{minions} and \\emph{clonoids} have received increasing attention. Minions are sets of finitary operations from a set $A$ to a (possibly different) set $B$ that are closed under permutation and identification of variables, as well as the addition of dummy variables. Research on minions gained much traction due to their application on (promise) constraint satisfaction, see e.g. the survey \\cite{BBKO-PCSP}. A \\emph{clonoid} from an algebra $\\alg{A}$ to an algebra $\\alg{B}$ is a minion that is closed under (pre-)composition with operations of $\\alg{A}$ and (post-)composition with the operations of $\\alg{B}$. While the term `clonoid' was first introduced by Aichinger and Mayr in~\\cite{AM-eqclasses}, the concept has appeared under various names earlier, e.g. in \\cite{ CF-clonoids}.\n\nDue to their weak structure, it is does not seem to be feasible to explicitly describe all minions between two given finite sets. A first hurdle is their sheer number: it was shown in \\cite{pippenger-minions} that there are already continuum many minions between two 2-element sets. However, depending on the expressiveness of the algebras $\\alg{A}$ and $\\alg{B}$, much more can be said about the clonoids between $\\alg{A}$ and $\\alg{B}$. Sparks showed in \\cite{sparks-clonoids}, that there are only finitely many clonoids from $\\alg{A}$ to $\\alg{B}$ if $\\alg{B}$ has a near-unanimity term, and at most countably many, if $\\alg{B}$ has an edge-term. However, also a rich source algebra $\\alg{A}$ can lead to finiteness results: As showed in \\cite{LS-discriminator}, every clonoids from an algebra $\\alg{A}$ with a discriminator term, to an algebra $\\alg{B}$ on the same set, gives rise to only finitely many clonoids.\n\nWe would also like to highlight a series of papers by Lehtonen and coauthors \\cite{LN-minorsclique,LS-discriminator, CL-stability, lehtonen-majorityclonoids, lehtonen-NUclonoids}, which culminated in \\cite{lehtonen-Booleanclonoids} to a complete characterization of all two-element algebras $(\\alg{A},\\alg{B})$, for which there are at most countable clonoids, together with a complete classification of clonoids in all such cases.\n\nIn this paper, we are interested in classifying clonoids, for which $\\alg{B}$ is a module. Such clonoids often appear naturally when studying the term operations of Mal'cev algebras with a central congruence; this connection was recently made more explicit by Peter Mayr's definition of the \\emph{difference clonoid}, see e.g. \\cite{kompatscher-SMP2nil}. In particular, 2-nilpotent algebras give rise to clonoids between modules $\\alg{A}$ and $\\alg{B}$.\n\nThere is already a series of classification results in the literature. The clonoids between groups of prime order $\\Z_p$ and $\\Z_q$ were completely classified by Kreinecker \\cite{kreinecker-zpclonoids} (in the case that $p=q$) and the first author, for $p \\neq q$ \\cite{fioravanti-clonoids}. Interestingly, in the latter case, there are only finitely many clonoids that are all generated by their unary functions (while for $p=q$ one obtains infinitely many).\n\nIn subsequent generalizations of these results, it was also always observed that there are only finitely many clonoids between certain modules $\\alg{A}$ and $\\alg{B}$ of coprime order - the most recent, and most general such finiteness result is by Mayr and Wynne \\cite{MW-clonoidsmodules}, and applies to all cases where $\\alg{A}$ has a distributive lattice of submodules. This leads us to the following Conjecture:\n\n\\begin{abstract}\nClonoids are sets of finitary operations between two algebraic structures that are closed under composition with their term operations on both sides. We conjecture that, for finite modules $\\algA$ and $\\algB$ there are only finitely many clonoids from $\\algA$ to $\\algB$ if and only if $\\algA$, $\\algB$ are of coprime order.\n\nWe confirm this conjecture for a broad class of modules $\\algA$. In particular we show that, if $\\algA$ is a finite $k$-dimensional vector space, then every clonoid from $\\alg A$ to a coprime module $\\algB$ is generated by its $k$-ary functions (and arity $k-1$ does not suffice). In order to prove this results, we investigate `uniform generation by $(\\algA,\\algB)$-minors', a general criterion, which we show to apply to several other existing classifications results. Based on our analysis, we further prove that the subpower membership problem of certain 2-nilpotent Mal'cev algebras is solvable in polynomial time.\n\\end{abstract}\n\nIn subsequent generalizations of these results, it was also always observed that there are only finitely many clonoids between certain modules $\\algA$ and $\\algB$ of coprime order - the most recent, and most general such finiteness result is by Mayr and Wynne \\cite{MW-clonoidsmodules}, and applies to all cases where $\\algA$ has a distributive lattice of submodules. This leads us to the following Conjecture:\n\nWe remark that the ``only if direction'' follows from \\cite[Theorem $1.3$]{MW-clonoidsmodules}, where the authors proved that for modules with a common divisor, one can always construct an infinite ascending chain of clonoids.\n\nThe first main contribution of our paper is to confirm Conjecture \\ref{conjecture:main} if $\\algA$ is a finite vector space. In the case where $\\algB$ is coprime to the vector space $\\algA$, we are moreover able to give an explicit description of the lattice of all clonoids (Theorem \\ref{theorem:clonoidlattice}). In particular, this answers \\cite[Question 1.2]{MW-clonoidsmodules}, which asked for a classification of clonoids, if $\\algA$ is the group $(\\Z_p)^2$, for some prime $p$. By combining our result with \\cite{MW-clonoidsmodules}, we are further able to confirm Conjecture \\ref{conjecture:main} for all modules $\\algA =\\F_1^{k_1}\\times \\cdots \\times \\F_n^{k_n} \\times \\alg D$ as $\\F_1\\times \\cdots \\times\\F_n \\times \\algR$-module, where $\\alg D$ is a distributive $\\algR$-module, and all $\\F_i$ are fields.\n\n\\begin{corollary} \\label{corollary:main2}\nLet $\\F^k$ be the $k$-dimensional vector space over a finite field $\\F$ and let $\\algB$ be a finite module of coprime order. Then\n\\begin{enumerate}\n\\item there are only finitely many clonoids from $\\F^k$ to $\\algB$.\n\\item every clonoid from $\\F^k$ to $\\algB$ is finitely related by an at most $|F|^{k\\times k}$-ary relation.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{corollary} \\label{corollary:mainresultproducts}\nLet $\\algA =\\F_1^{k_1}\\times \\cdots \\times \\F_n^{k_n} \\times \\alg D$ as $\\F_1\\times \\cdots \\times\\F_n \\times \\algR$-module, where all $\\F_i$ are finite fields, and $\\alg D$ is a finite distributive $\\algR$-module. Let $l$ be the nilpotence degree of the Jacobson radical of $\\algR$, and let $k = \\max(k_1,\\ldots,k_n,l)$. Further let $\\algB$ be an $\\algS$-module, such that $|A|$ is invertible in $\\algS$. Then\n\\begin{enumerate}\n\\item every clonoid from $\\alg A$ to $\\algB$ is generated by its $k$-ary part.\n\\item If $\\algB$ is finite then there are only finitely many clonoids from $\\algA$ to $\\algB$.\n\\end{enumerate}\n\\end{corollary}\n\nIn this paper, we studied clonoids between finite modules. We proved that there are only finitely many clonoids from a finite vector space $\\algA$ to a coprime module $\\algB$, which is also in accordance to Conjecture \\ref{conjecture:main}.\n\n\\begin{conjecture} \\label{conjecture:main}\nLet $\\algA$ and $\\algB$ be finite modules. Then, there are only finitely many clonoids from $\\algA$ to $\\algB$, if and only if $\\algA$ and $\\algB$ are of coprime order.\n\\end{conjecture}\n\n\\begin{theorem} \\label{theorem:clonoidlattice}\nThe lattice of clonoids from $\\F^k$ to $\\algB$ is isomorphic to $\\prod_{i=0}^k \\Sub(\\algM_{i,k}(\\F,\\algB))$.\n\nMoreover, for $j\\leq k$, the sublattice $\\prod_{i=0}^j \\Sub(\\algM_{i,k}(\\F,\\algB))$ corresponds to the clonoids consisting of functions that are generated by their $j$-ary minors.\n\\end{theorem}", "post_theorem_intro_text_len": 4549, "post_theorem_intro_text": "We remark that the ``only if direction'' follows from \\cite[Theorem $1.3$]{MW-clonoidsmodules}, where the authors proved that for modules with a common divisor, one can always construct an infinite ascending chain of clonoids.\n\n The first main contribution of our paper is to confirm Conjecture \\ref{conjecture:main} if $\\alg{A}$ is a finite vector space. In the case where $\\alg{B}$ is coprime to the vector space $\\alg{A}$, we are moreover able to give an explicit description of the lattice of all clonoids (Theorem \\ref{theorem:clonoidlattice}). In particular, this answers \\cite[Question 1.2]{MW-clonoidsmodules}, which asked for a classification of clonoids, if $\\alg{A}$ is the group $(\\Z_p)^2$, for some prime $p$. By combining our result with \\cite{MW-clonoidsmodules}, we are further able to confirm Conjecture \\ref{conjecture:main} for all modules $\\alg{A} =\\F_1^{k_1}\\times \\cdots \\times \\F_n^{k_n} \\times \\alg D$ as $\\F_1\\times \\cdots \\times\\F_n \\times \\alg{R}$-module, where $\\alg D$ is a distributive $\\alg{R}$-module, and all $\\F_i$ are fields.\n\nOur classification is based on proving that, in the case of a vector space $\\alg{A} = {\\mathbf F}^k$ and coprime $\\alg{B}$, the set of all operations from $A$ to $B$ is \\emph{uniformly generated} by $k$-ary minors. While this technique was already hinted at in \\cite{MW-clonoidsmodules}, we are going to discuss it in great detail in Section \\ref{sec:unifgen}, and argue that many of the existing clonoid classifications follow the same pattern. We furthermore show that the arity $k$ is optimal, i.e., not every clonoid from ${\\mathbf F}^k$ to $\\alg{B}$ is generated by their $k-1$-ary functions. This coincides with a lower bound on the arity of generators of the \\emph{full} clonoid between modules that we derive in Section \\ref{sec:lowerbound}, and was already pointed out in the Bachelor thesis by Jan Van\\v{e}\\v{c}ek \\cite{vanecek-thesis}.\n\nLast, let us mention that, clonoids between affine algebras were also (sometimes implicitly) used to discuss some computational problems of Mal'cev algebras. The fact that some clonoids between coprime modules are generated by unary functions, which allows for a representation of their elements by nice normal forms, was used in \\cite{KKK-CEQV2nil} to prove that checking polynomial identities in a given 2-nilpotent algebra can always be done in polynomial time. In \\cite{mayr-VLloop} and \\cite{KompatscherMayr2026} such normal forms were similarly used to find finite equational basis for some 2-nilpotent loops.\n\nWhile we are not going to discuss such syntactic aspects of clonoids, we will apply our results to the \\emph{subpower membership problem} of certain algebras. The subpower membership problem (SMP($\\alg{A}$)), for a fixed finite algebra $\\alg{A}$, is the computational problem whether a given partial operation $f\\colon A^n \\to A$ can be extended to a term operation of $\\alg{A}$. In \\cite{IMMVW-subpowers} it was asked, whether the subpower membership problem is always polynomial time solvable for algebras with few subpowers. This question, however remains unsolved even in the Mal'cev case. The easiest examples that are not covered by existing tractability results \\cite{mayr-SMP, BMS-SMP}, are 2-nilpotent algebras, for which it was shown in \\cite{kompatscher-SMP2nil} that the problem is polynomial time equivalent to a similar `interpolation problem' for their clonoids. In Section \\ref{sect:SMP} we are going to use our results, to prove that the subpower membership problem of a big class of 2-nilpotent algebras is in P.\n\n\\subsection*{Organization of the paper}\n\nThe paper is organized as follows. In Section~\\ref{sec:prel} we introduce the necessary concepts from universal algebra and fix some notation. In Section~\\ref{sec:unifgen} we develop a theory of `uniform generation' and `uniform representation' by minors, a concept introduced in~\\cite{MW-clonoidsmodules}. In particular we provide a purely combinatorial criterion for all $(\\alg{A},\\alg{B})$-clonoids to be generated by their $k$-ary functions (Theorem \\ref{theorem:unigen}).\n\nSection~\\ref{sec:mainres} presents the main results of the paper, Theorem \\ref{theorem:main} and discusses some consequences. In Section \\ref{sect:SMP} we apply our results to the subpower membership problem. In Sections~\\ref{sec:lowerbound} we provide a lower bound on the arity of the generators of the full clonoid between finite modules, which is sharp in our setting. Section~\\ref{sec:conclusion} is dedicated to future work and possible applications of our results.", "sketch": "We remark that the ``only if direction'' follows from \\cite[Theorem $1.3$]{MW-clonoidsmodules}, where the authors proved that for modules with a common divisor, one can always construct an infinite ascending chain of clonoids.\n\nFor confirming Conjecture~\\ref{conjecture:main} when $\\alg{A}$ is a finite vector space and $\\alg{B}$ is coprime to $\\alg{A}$, the paper’s approach is described as follows: the classification is based on proving that, for a vector space $\\alg{A}={\\mathbf F}^k$ and coprime $\\alg{B}$, “the set of all operations from $A$ to $B$ is \\emph{uniformly generated} by $k$-ary minors.” The authors note this technique was “already hinted at in \\cite{MW-clonoidsmodules},” and they “discuss it in great detail in Section~\\ref{sec:unifgen},” including “a purely combinatorial criterion for all $(\\alg{A},\\alg{B})$-clonoids to be generated by their $k$-ary functions (Theorem~\\ref{theorem:unigen}).” They also establish an optimality statement for this method: “the arity $k$ is optimal, i.e., not every clonoid from ${\\mathbf F}^k$ to $\\alg{B}$ is generated by their $k-1$-ary functions,” and relate this to a “lower bound on the arity of generators of the \\emph{full} clonoid between modules” derived in Section~\\ref{sec:lowerbound}. Finally, they state that “by combining our result with \\cite{MW-clonoidsmodules}, we are further able to confirm Conjecture~\\ref{conjecture:main}” for broader classes of modules $\\alg{A}=\\F_1^{k_1}\\times\\cdots\\times\\F_n^{k_n}\\times\\alg D.", "expanded_sketch": "We remark that the ``only if direction'' follows from \\cite[Theorem $1.3$]{MW-clonoidsmodules}, where the authors proved that for modules with a common divisor, one can always construct an infinite ascending chain of clonoids.\n\nFor confirming the main conjecture when $\\alg{A}$ is a finite vector space and $\\alg{B}$ is coprime to $\\alg{A}$, the paper’s approach is described as follows: the classification is based on proving that, for a vector space $\\alg{A}={\\mathbf F}^k$ and coprime $\\alg{B}$, “the set of all operations from $A$ to $B$ is \\emph{uniformly generated} by $k$-ary minors.” The authors note this technique was “already hinted at in \\cite{MW-clonoidsmodules},” and they “discuss it in great detail” later, including “a purely combinatorial criterion for all $(\\alg{A},\\alg{B})$-clonoids to be generated by their $k$-ary functions.” We first recall this criterion.\n\n\\begin{theorem} \\label{theorem:unigen}\nLet $\\cloA$, $\\cloB$ be clones, $n \\geq 1$, and let $\\cloA_{const} = \\Clo(\\cloA \\cup \\{a\\}_{a\\in A})$ be the clone generated by $\\cloA$ and all constant operations over its set. Let $\\clodC$ a clonoid from $\\cloA_{const}$\nto $\\cloB$. Then the following are equivalent:\n\\begin{enumerate}\n\\item\\label{ite:th_unigen1} $\\clodC^{(n+1)}$ is $(n,\\cloA,\\cloB)$-UG\n\\item\\label{ite:th_unigen2} $\\clodC^{(k)}$ is $(n,\\cloA,\\cloB)$-UG for some $k>n$\n\\item\\label{ite:th_unigen3} $\\clodC$ is $(n,\\cloA,\\cloB)$-UG\n\\item\\label{ite:th_unigen4} $\\forall k,l \\in \\N$, every partial operation $I \\colon \\clodC^{(k)} \\to \\clodC^{(l)}$ \nthat is $(\\cloA,\\cloB)$-UR is $(n,\\cloA,\\cloB)$-UR.\n\\end{enumerate}\n\\end{theorem}\n\nThey also establish an optimality statement for this method: “the arity $k$ is optimal, i.e., not every clonoid from ${\\mathbf F}^k$ to $\\alg{B}$ is generated by their $k-1$-ary functions,” and relate this to a “lower bound on the arity of generators of the \\emph{full} clonoid between modules” derived later. Finally, they state that “by combining our result with \\cite{MW-clonoidsmodules}, we are further able to confirm the main conjecture” for broader classes of modules $\\alg{A}=\\F_1^{k_1}\\times\\cdots\\times\\F_n^{k_n}\\times\\alg D.", "expanded_theorem": "\\label{conjecture:main}\nLet $\\alg{A}$ and $\\alg{B}$ be finite modules. Then, there are only finitely many clonoids from $\\alg{A}$ to $\\alg{B}$, if and only if $\\alg{A}$ and $\\alg{B}$ are of coprime order.", "theorem_type": ["Biconditional or Equivalence", "Existence"], "mcq": {"question": "Let \\(\\alg A\\) and \\(\\alg B\\) be finite modules. A clonoid from \\(\\alg A\\) to \\(\\alg B\\) is a set of finitary functions from powers of \\(A\\) to \\(B\\) that is closed under taking minors (permuting variables, identifying variables, and adding dummy variables), as well as under precomposition with term operations of \\(\\alg A\\) and postcomposition with term operations of \\(\\alg B\\). Two finite modules are said to be of coprime order when \\(\\gcd(|A|,|B|)=1\\). Which statement holds about the number of clonoids from \\(\\alg A\\) to \\(\\alg B\\)?", "correct_choice": {"label": "A", "text": "There are only finitely many clonoids from \\(\\alg A\\) to \\(\\alg B\\) if and only if \\(\\alg A\\) and \\(\\alg B\\) are of coprime order."}, "choices": [{"label": "B", "text": "There are only finitely many clonoids from \\(\\alg A\\) to \\(\\alg B\\) whenever \\(\\alg A\\) and \\(\\alg B\\) have distinct orders."}, {"label": "C", "text": "If \\(\\alg A\\) and \\(\\alg B\\) are of coprime order, then there are only finitely many clonoids from \\(\\alg A\\) to \\(\\alg B\\)."}, {"label": "D", "text": "There are only finitely many clonoids from \\(\\alg A\\) to \\(\\alg B\\) if and only if every clonoid from \\(\\alg A\\) to \\(\\alg B\\) is generated by its unary functions."}, {"label": "E", "text": "There are only finitely many clonoids from \\(\\alg A\\) to \\(\\alg B\\) if and only if, for some integer \\(n\\ge 1\\) depending only on \\(|A|\\), every clonoid from \\(\\alg A\\) to \\(\\alg B\\) is generated by its \\(n\\)-ary functions."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "coprime-order hypothesis replaced by distinct-order condition", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped the converse direction from the iff statement", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "optimal k-ary generation confused with unary generation", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "uniform generation bound made to depend only on |A| and asserted equivalent to finiteness for all modules", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem provides definitions and asks for the correct structural statement, but it does not explicitly reveal the coprime-order criterion or otherwise give away choice A."}, "TAS": {"score": 1, "justification": "The item is largely theorem-recognition: the correct option states the main characterization directly rather than requiring application in a new setting. Still, the alternatives introduce competing formulations, so it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the full iff statement from the weaker true one-way implication in C and from plausible but false variants. However, the question mainly tests recall of the theorem rather than substantial generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: B confuses coprime with distinct order, C is a weaker true statement, and D/E exploit realistic misunderstandings about generation by low-arity functions and quantifier dependence."}, "total_score": 6, "overall_assessment": "A solid theorem-recognition MCQ with no answer leakage and strong distractors, but it primarily tests recall rather than deeper generative mathematical reasoning."}}
{"id": "2602.04664v1", "paper_link": "http://arxiv.org/abs/2602.04664v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{main theorem k+1-gon-bad}\n\tFor each $\\epsilon>0$ and integer $N > 0$, there exists a constant $C_{\\epsilon, N}$ such that for any positive numbers $\\tau_{1},\\tau_{2},\\ldots,\\tau_{k}$, we have\n\t\\[\n\t\t\\sum_{\\lambda_{i_{k+1}}\\geq (1+\\epsilon)(\\tau_1+\\tau_2+\\dots+\\tau_k)} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\leq C_{\\epsilon, N}(\\tau_1+\\tau_2+\\dots+\\tau_k)^{-N},\n\t\\]\n\twhere $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$.", "start_pos": 10297, "end_pos": 10791, "label": "main theorem k+1-gon-bad"}, "ref_dict": {"main theorem k+1-good": "\\begin{theorem}\\label{main theorem k+1-good}\nLet $\\rho$ be a Schwartz function on $\\R^{k+1}$ with $\\int \\rho = 1$. Assume that\n\\[\n\t\\supp \\hat \\rho \\subset (-\\inj M, \\inj M)^{k+1}.\n\\]\nLet $\\Gamma$ be a $k$-good cone. Then for $\\tau = (\\tau_1, \\tau_2, \\dots, \\tau_{k+1}) \\in \\Gamma$,\n\\[\n\t\\rho * \\mu(\\tau) = (2\\pi)^{-kn} \\vol M \\vol F^{-1}(\\tau) + O(|\\tau|^{k(n-1)-2}),\n\\]\nwhere the constants implicit in the big-$O$ notation depend on $M$, $\\Gamma$, and $\\rho$ but not on $\\tau$.\n\\end{theorem}", "harmonic expansion": "\\begin{equation}\\label{harmonic expansion}\n\te_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}", "4": "\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{4}\n\\caption{An example of $\\tau\\notin\\Gamma_g$.}\n\\label{4}\n\\end{figure}", "main theorem k+1-gon-bad": "\\begin{theorem}\\label{main theorem k+1-gon-bad}\n\tFor each $\\epsilon>0$ and integer $N > 0$, there exists a constant $C_{\\epsilon, N}$ such that for any positive numbers $\\tau_{1},\\tau_{2},\\ldots,\\tau_{k}$, we have\n\t\\[\n\t\t\\sum_{\\lambda_{i_{k+1}}\\geq (1+\\epsilon)(\\tau_1+\\tau_2+\\dots+\\tau_k)} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\leq C_{\\epsilon, N}(\\tau_1+\\tau_2+\\dots+\\tau_k)^{-N},\n\t\\]\n\twhere $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$.\n\\end{theorem}", "2": "\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{2}\n\\caption{A pentagon with side lengths $a_1, \\ldots, a_5$.}\n\\label{2}\n\\includegraphics[width=0.6\\textwidth]{3}\n\\caption{A self-intersecting pentagon with the same side lengths.}\n\\label{3}\n\\end{figure}", "single sums": "\\begin{equation}\\label{single sums}\n\t\\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}", "rewrite harmonic expansion": "\\begin{equation}\\label{rewrite harmonic expansion}\n\t\\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}", "Scaling of F": "\\begin{proposition}\n [Scaling of $F$]\\label{Scaling of F} Let $\\Gamma$ be a $k$-good cone and fix $\\tau \\in \\Gamma$. Then $F^{-1}(\\tau) \\neq \\emptyset$ and $\\tau$ is a regular value of $F$, so $F^{-1}(\\tau)$ is a smooth compact submanifold of codimension $k+1$. Let\n\\[\nd:=k n-(k+1)=k(n-1)-1\n\\]\nbe its dimension. Then $0<\\operatorname{vol} F^{-1}(\\tau)<\\infty$ and, for every $r>0$,\n$$\n\\operatorname{vol} F^{-1}(r \\tau)=r^d \\operatorname{vol} F^{-1}(\\tau).\n$$\n\\end{proposition}", "3": "\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{2}\n\\caption{A pentagon with side lengths $a_1, \\ldots, a_5$.}\n\\label{2}\n\\includegraphics[width=0.6\\textwidth]{3}\n\\caption{A self-intersecting pentagon with the same side lengths.}\n\\label{3}\n\\end{figure}"}, "pre_theorem_intro_text_len": 7505, "pre_theorem_intro_text": "\\subsection{Background}\n\nLet $M$ be a compact Riemannian manifold without boundary, and $e_1, e_2, \\ldots$ be an orthonormal basis of Laplace-Beltrami eigenfunctions with\n\\[\n\t\\Delta_g e_j = -\\lambda_j^2 e_j,\n\\]\nwhere $\\lambda_j$ is the frequency of $e_j$.\n\nWe are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \\dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion\n\\begin{equation}\\label{harmonic expansion}\n\te_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\nThe classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \\dots e_{i_k}$. If we set $\\lambda_{i_1}=\\lambda_{i_2}=\\dots =\\lambda_{i_k}=\\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\\lambda_j}$. For this problem, Sogge provided a complete answer for $k\\geq 1$, see \\cite{MR930395}. \nHowever, for general eigenvalues $\\lambda_{i_1}, \\lambda_{i_2}, \\ldots, \\lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \\eqref{harmonic expansion} as follows:\n\\begin{equation}\\label{rewrite harmonic expansion}\n\t\\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\nThe decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \\eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}}$ do the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$ exhibit decay, and how is their $\\ell^2$-mass distributed as these frequencies vary?\n\nWhen $k=2$, as in many spectral problems, this question is closely connected to number theory in the automorphic setting. Sarnak showed that for fixed $j$ the coefficients $\\langle e_j^2, e_\\ell \\rangle$ decay exponentially in $\\lambda_\\ell$ \\cite{MR1277052}. Bernstein, Reznikov, Kr\\\"otz, and Stanton obtained the optimal estimate \\cite{MR1715328, MR2081437}. For the more general coefficients $\\langle e_i e_j, e_\\ell \\rangle$, a classical decay estimate in terms of the eigenvalues was given by Lu, Sogge, and Steinerberger \\cite{MR3997636}. More recently, the first author improved their estimate \\cite{MR4376455}. Given $\\epsilon > 0$ and an integer $N \\geq 1$, there exists a constant $C_{\\epsilon,N}$ for which\n\\begin{equation}\\label{Emmett conclusion}\n\t\\sum_{\\lambda_\\ell \\geq (2+\\epsilon)\\lambda} |\\langle e_i e_j, e_\\ell \\rangle|^2 \\leq C_{\\epsilon, N} \\lambda^{-N} \\qquad \\text{for all $i$, $j$ with $\\lambda_i, \\lambda_j \\leq \\lambda$.}\n\\end{equation}\n\nFurthermore, he proved that the $(2+\\epsilon)\\lambda$ factor is nearly optimal. He also obtained an asymptotic expansion for the non-rapidly decaying part \\cite{MR4376455}. Building on this work, we study the multi-product of $(k+1)$ eigenfunctions and obtain analogous results. \nOne of the most striking features of \\cite{MR4376455} is that it connects triple products of eigenfunctions to geometric configurations, namely triangles. We extend this perspective to general $k \\geq 2$, where the corresponding configurations are naturally described by polygons.\n\n\\subsection{Polygons}\n\nBased on \\eqref{rewrite harmonic expansion}, we study the sums\n\\begin{equation}\\label{single sums}\n\t\\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}\nfor given eigenvalues $a_1, a_2, \\dots, a_{k+1}$. These sums are independent of the choice of eigenbasis; hence their behavior is determined solely by the underlying manifold and its metric.\n\nThe main point of the present paper is to show that the sum \\eqref{single sums} can be estimated by counting configurations of $(k+1)$-gons with side lengths $a_1, a_2, \\dots, a_{k+1}$. To state our results, we first introduce the following definition concerning the side lengths of a $(k+1)$-gon.\n\n\\begin{definition}\\label{k+1-good and k+1-gon-bad}\nA vector $\\tau = (\\tau_1, \\ldots, \\tau_{k+1}) \\in (0,\\infty)^{k+1}$ is called $k$-good if it satisfies the following two conditions.\n\\begin{enumerate}\n\\item the $(k+1)$-gon inequality: for every $1 \\leq j \\leq k+1$,\n\\begin{equation}\\label{2.1}\n\\tau_j < \\sum_{\\ell \\neq j} \\tau_{\\ell}.\n\\end{equation}\n\\item the nondegeneracy condition: for every sign vector $\\varepsilon = (\\varepsilon_1, \\ldots, \\varepsilon_{k+1}) \\in \\{ \\pm 1\\}^{k+1}$,\n\\begin{equation}\\label{2.2}\n|\\varepsilon \\cdot \\tau| = \\left|\\sum_{j=1}^{k+1} \\varepsilon_j \\tau_j\\right| > 0.\n\\end{equation}\n\\end{enumerate}\nWe denote the set of $k$-good vectors by $\\Gamma_g$.\n\nA vector $\\tau = (\\tau_1, \\ldots, \\tau_{k+1}) \\in (0,\\infty)^{k+1}$ is called $k$-bad if there exists $j$, $1 \\leq j \\leq k+1$, such that\n\\begin{equation}\\label{2.3}\n\\tau_j > \\sum_{\\ell \\neq j} \\tau_{\\ell}.\n\\end{equation}\nWe denote the set of $k$-bad vectors by $\\Gamma_b$.\n\nA closed cone $\\Gamma \\subset (0,\\infty)^{k+1}$ is called $k$-good if $\\Gamma \\subset \\Gamma_g$, and it is called $k$-bad if $\\Gamma \\subset \\Gamma_b$.\n\\end{definition}\n\nThe set of $\\tau$ satisfying condition (1) consists of side-length data that can be realized by a $(k+1)$-gon in the plane, whereas the set of $k$-bad vectors contains no such points. The set of $k$-good vectors is obtained from the set of $\\tau$ satisfying condition (1) by removing those $\\tau$ for which a linear combination $|\\varepsilon \\cdot \\tau|$ vanishes for some $\\varepsilon \\in \\{ \\pm 1\\}^{k+1}$. For example, when $k=3$, the vector $(2,2,2,2)$ satisfies the $(k+1)$-gon inequality, but $(2,2,2,2)\\notin \\Gamma_g$ since it fails the nondegeneracy condition. Concretely, consider the degenerate quadrilateral in the plane formed by the vectors $(2,0), (-2,0), (2,0), (-2,0)$. Each side has length $2$, but the polygon is collinear and has zero area. In this case, $\\tau \\notin \\Gamma_g$, as illustrated in Figure \\ref{4}. When $k=2$, the nondegeneracy condition is automatic once one assumes condition (1), due to the rigidity of triangles. This is the primary reason it was not clear whether triple-product results could be generalized to higher multi-products, where such rigidity is absent. Our nondegeneracy condition is designed to fill this gap. As we will see in the proof, it ensures that several crucial steps go through and yields clean asymptotics.\n\n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{4}\n\\caption{An example of $\\tau\\notin\\Gamma_g$.}\n\\label{4}\n\\end{figure}\nIn what follows, we establish a connection between \\eqref{single sums} and the count of configurations of $(k+1)$-gons using the spectral projection operator $\\chi_\\lambda$ and the theory of Fourier integral operators. Our argument also shows that \\eqref{single sums} decays rapidly in the regime where $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ fails to satisfy the $(k+1)$-gon inequality.\n\n\\subsection{Main results}\nOur first theorem shows that the rapidly decaying contribution to \\eqref{single sums} comes from frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.", "context": "We are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \\dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion\n\\begin{equation}\\label{harmonic expansion}\n e_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\nThe classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \\dots e_{i_k}$. If we set $\\lambda_{i_1}=\\lambda_{i_2}=\\dots =\\lambda_{i_k}=\\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\\lambda_j}$. For this problem, Sogge provided a complete answer for $k\\geq 1$, see \\cite{MR930395}. \nHowever, for general eigenvalues $\\lambda_{i_1}, \\lambda_{i_2}, \\ldots, \\lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \\eqref{harmonic expansion} as follows:\n\\begin{equation}\\label{rewrite harmonic expansion}\n \\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\nThe decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \\eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}}$ do the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$ exhibit decay, and how is their $\\ell^2$-mass distributed as these frequencies vary?\n\nWhen $k=2$, as in many spectral problems, this question is closely connected to number theory in the automorphic setting. Sarnak showed that for fixed $j$ the coefficients $\\langle e_j^2, e_\\ell \\rangle$ decay exponentially in $\\lambda_\\ell$ \\cite{MR1277052}. Bernstein, Reznikov, Kr\\\"otz, and Stanton obtained the optimal estimate \\cite{MR1715328, MR2081437}. For the more general coefficients $\\langle e_i e_j, e_\\ell \\rangle$, a classical decay estimate in terms of the eigenvalues was given by Lu, Sogge, and Steinerberger \\cite{MR3997636}. More recently, the first author improved their estimate \\cite{MR4376455}. Given $\\epsilon > 0$ and an integer $N \\geq 1$, there exists a constant $C_{\\epsilon,N}$ for which\n\\begin{equation}\\label{Emmett conclusion}\n \\sum_{\\lambda_\\ell \\geq (2+\\epsilon)\\lambda} |\\langle e_i e_j, e_\\ell \\rangle|^2 \\leq C_{\\epsilon, N} \\lambda^{-N} \\qquad \\text{for all $i$, $j$ with $\\lambda_i, \\lambda_j \\leq \\lambda$.}\n\\end{equation}\n\nBased on \\eqref{rewrite harmonic expansion}, we study the sums\n\\begin{equation}\\label{single sums}\n \\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}\nfor given eigenvalues $a_1, a_2, \\dots, a_{k+1}$. These sums are independent of the choice of eigenbasis; hence their behavior is determined solely by the underlying manifold and its metric.\n\nA vector $\\tau = (\\tau_1, \\ldots, \\tau_{k+1}) \\in (0,\\infty)^{k+1}$ is called $k$-bad if there exists $j$, $1 \\leq j \\leq k+1$, such that\n\\begin{equation}\\label{2.3}\n\\tau_j > \\sum_{\\ell \\neq j} \\tau_{\\ell}.\n\\end{equation}\nWe denote the set of $k$-bad vectors by $\\Gamma_b$.\n\n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{4}\n\\caption{An example of $\\tau\\notin\\Gamma_g$.}\n\\label{4}\n\\end{figure}\nIn what follows, we establish a connection between \\eqref{single sums} and the count of configurations of $(k+1)$-gons using the spectral projection operator $\\chi_\\lambda$ and the theory of Fourier integral operators. Our argument also shows that \\eqref{single sums} decays rapidly in the regime where $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ fails to satisfy the $(k+1)$-gon inequality.\n\n\\subsection{Main results}\nOur first theorem shows that the rapidly decaying contribution to \\eqref{single sums} comes from frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.\n\n\\begin{equation}\\label{harmonic expansion}\n\te_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\n\n\\begin{equation}\\label{rewrite harmonic expansion}\n\t\\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\n\n\\begin{equation}\\label{single sums}\n\t\\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}", "full_context": "We are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \\dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion\n\\begin{equation}\\label{harmonic expansion}\n e_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\nThe classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \\dots e_{i_k}$. If we set $\\lambda_{i_1}=\\lambda_{i_2}=\\dots =\\lambda_{i_k}=\\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\\lambda_j}$. For this problem, Sogge provided a complete answer for $k\\geq 1$, see \\cite{MR930395}. \nHowever, for general eigenvalues $\\lambda_{i_1}, \\lambda_{i_2}, \\ldots, \\lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \\eqref{harmonic expansion} as follows:\n\\begin{equation}\\label{rewrite harmonic expansion}\n \\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\nThe decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \\eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}}$ do the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$ exhibit decay, and how is their $\\ell^2$-mass distributed as these frequencies vary?\n\nWhen $k=2$, as in many spectral problems, this question is closely connected to number theory in the automorphic setting. Sarnak showed that for fixed $j$ the coefficients $\\langle e_j^2, e_\\ell \\rangle$ decay exponentially in $\\lambda_\\ell$ \\cite{MR1277052}. Bernstein, Reznikov, Kr\\\"otz, and Stanton obtained the optimal estimate \\cite{MR1715328, MR2081437}. For the more general coefficients $\\langle e_i e_j, e_\\ell \\rangle$, a classical decay estimate in terms of the eigenvalues was given by Lu, Sogge, and Steinerberger \\cite{MR3997636}. More recently, the first author improved their estimate \\cite{MR4376455}. Given $\\epsilon > 0$ and an integer $N \\geq 1$, there exists a constant $C_{\\epsilon,N}$ for which\n\\begin{equation}\\label{Emmett conclusion}\n \\sum_{\\lambda_\\ell \\geq (2+\\epsilon)\\lambda} |\\langle e_i e_j, e_\\ell \\rangle|^2 \\leq C_{\\epsilon, N} \\lambda^{-N} \\qquad \\text{for all $i$, $j$ with $\\lambda_i, \\lambda_j \\leq \\lambda$.}\n\\end{equation}\n\nBased on \\eqref{rewrite harmonic expansion}, we study the sums\n\\begin{equation}\\label{single sums}\n \\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}\nfor given eigenvalues $a_1, a_2, \\dots, a_{k+1}$. These sums are independent of the choice of eigenbasis; hence their behavior is determined solely by the underlying manifold and its metric.\n\nA vector $\\tau = (\\tau_1, \\ldots, \\tau_{k+1}) \\in (0,\\infty)^{k+1}$ is called $k$-bad if there exists $j$, $1 \\leq j \\leq k+1$, such that\n\\begin{equation}\\label{2.3}\n\\tau_j > \\sum_{\\ell \\neq j} \\tau_{\\ell}.\n\\end{equation}\nWe denote the set of $k$-bad vectors by $\\Gamma_b$.\n\n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{4}\n\\caption{An example of $\\tau\\notin\\Gamma_g$.}\n\\label{4}\n\\end{figure}\nIn what follows, we establish a connection between \\eqref{single sums} and the count of configurations of $(k+1)$-gons using the spectral projection operator $\\chi_\\lambda$ and the theory of Fourier integral operators. Our argument also shows that \\eqref{single sums} decays rapidly in the regime where $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ fails to satisfy the $(k+1)$-gon inequality.\n\n\\subsection{Main results}\nOur first theorem shows that the rapidly decaying contribution to \\eqref{single sums} comes from frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.\n\n\\begin{equation}\\label{harmonic expansion}\n\te_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\n\n\\begin{equation}\\label{rewrite harmonic expansion}\n\t\\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\n\n\\begin{equation}\\label{single sums}\n\t\\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}\n\nWe are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \\dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion\n\\begin{equation}\\label{harmonic expansion}\n e_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\nThe classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \\dots e_{i_k}$. If we set $\\lambda_{i_1}=\\lambda_{i_2}=\\dots =\\lambda_{i_k}=\\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\\lambda_j}$. For this problem, Sogge provided a complete answer for $k\\geq 1$, see \\cite{MR930395}. \nHowever, for general eigenvalues $\\lambda_{i_1}, \\lambda_{i_2}, \\ldots, \\lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \\eqref{harmonic expansion} as follows:\n\\begin{equation}\\label{rewrite harmonic expansion}\n \\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\nThe decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \\eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}}$ do the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$ exhibit decay, and how is their $\\ell^2$-mass distributed as these frequencies vary?\n\n\\subsection{Main results}\nOur first theorem shows that the rapidly decaying contribution to \\eqref{single sums} comes from frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.\n\nThe hypothesis $\\lambda_{i_{k+1}} \\geq (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$ forces the $(k+1)$-tuple $(\\tau_1,\\ldots,\\tau_k,\\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others. In particular, any $(\\lambda_{i_1},\\ldots,\\lambda_{i_k},\\lambda_{i_{k+1}})$ with $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$ is also $k$-bad.\n\nFor clarity, we do not organize the left-hand side by summing over all tuples $(\\lambda_{i_1},\\ldots,\\lambda_{i_{k+1}})$ lying in a specified region of $(0,\\infty)^{k+1}$. Instead, we fix upper bounds $\\tau_j$ for $\\lambda_{i_j}$, $1 \\leq j \\leq k$, and then estimate the tail sum over those $i_{k+1}$ with $\\lambda_{i_{k+1}} \\geq (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$. The proof uses the method of non-stationary phase.\nWe record a flat-torus example in Section \\ref{sec:flat-torus-example} showing that the frequency threshold in Theorem \\ref{main theorem k+1-gon-bad} is essentially sharp.\n\nOur second and main theorem concerns the contribution from $k$-good vectors $\\tau \\in (0,\\infty)^{k+1}$.\n Following \\cite{MR4376455}, we introduce the joint spectral measure\n\\begin{equation}\\label{joint spectral measure}\n \\mu = \\sum_{i_1,i_2,\\dots,i_{k+1}} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\, \\delta_{(\\lambda_{i_1},\\lambda_{i_2},\\dots,\\lambda_{i_{k+1}})}\n\\end{equation}\non $\\R^{k+1}$, where $\\delta$ denotes the Dirac measure. Thus $\\mu$ is a weighted sum of point masses, with weights given by the squared magnitudes of the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$. Our second theorem describes the mass of $\\mu$ carried by frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ lying in $k$-good cones.\n\nNext we verify that $\\tau$ is a regular value of $F$. Fix $\\xi=(\\xi_1,\\ldots,\\xi_k)\\in F^{-1}(\\tau)$ and set\n\\[\n\\omega_j=\\frac{\\xi_j}{|\\xi_j|}\\in S^{n-1}\\ (1\\le j\\le k),\\qquad\n\\omega_{k+1}=\\frac{\\xi_1+\\cdots+\\xi_k}{|\\xi_1+\\cdots+\\xi_k|}\\in S^{n-1}.\n\\]\nFor $h=(h_1,\\ldots,h_k)\\in\\R^{kn}$ we compute\n\\[\nd(|\\xi_j|)[h]=\\langle \\omega_j,h_j\\rangle,\\qquad\nd(|\\xi_1+\\cdots+\\xi_k|)[h]=\\Bigl\\langle \\omega_{k+1},\\sum_{j=1}^k h_j\\Bigr\\rangle.\n\\]\nIf $\\mathrm{rank}(dF_\\xi)<k+1$, these $k+1$ linear functionals are linearly dependent, so there exist\n$(c_1,\\ldots,c_k,c_{k+1})\\neq 0$ such that for all $h$,\n\\[\n\\sum_{j=1}^k c_j\\langle \\omega_j,h_j\\rangle\n+c_{k+1}\\Bigl\\langle \\omega_{k+1},\\sum_{j=1}^k h_j\\Bigr\\rangle=0.\n\\]\nEquivalently,\n\\[\n\\sum_{j=1}^k \\langle c_j\\omega_j+c_{k+1}\\omega_{k+1},\\ h_j\\rangle=0\n\\quad\\text{for all }h_1,\\ldots,h_k,\n\\]\nhence $c_j\\omega_j+c_{k+1}\\omega_{k+1}=0$ for every $j$. In particular $c_{k+1}\\neq 0$, and\n$\\omega_j=\\varepsilon_j\\,\\omega_{k+1}$ for some $\\varepsilon_j\\in\\{\\pm1\\}$. Therefore\n\\[\n\\xi_1+\\cdots+\\xi_k=\\sum_{j=1}^k |\\xi_j|\\omega_j\n=\\Bigl(\\sum_{j=1}^k \\varepsilon_j|\\xi_j|\\Bigr)\\omega_{k+1},\n\\]\nand taking norms gives\n\\[\n\\tau_{k+1}=|\\xi_1+\\cdots+\\xi_k|\n=\\Bigl|\\sum_{j=1}^k \\varepsilon_j|\\xi_j|\\Bigr|\n=\\Bigl|\\sum_{j=1}^k \\varepsilon_j\\,\\tau_j\\Bigr|.\n\\]\nLet $s:=\\sum_{j=1}^k \\varepsilon_j\\,\\tau_j$ and choose $\\varepsilon_{k+1}\\in\\{\\pm1\\}$ so that\n$\\varepsilon_{k+1}\\tau_{k+1}+s=0$. Then the sign vector\n\\[\n\\varepsilon=(\\varepsilon_1,\\ldots,\\varepsilon_k,\\varepsilon_{k+1})\\in\\{\\pm1\\}^{k+1}\n\\]\nsatisfies $\\varepsilon\\cdot\\tau=0$, contradicting the nondegeneracy condition \\eqref{2.2}.\nHence $\\mathrm{rank}(dF_\\xi)=k+1$ for all $\\xi\\in F^{-1}(\\tau)$, so $\\tau$ is a regular value.\n\n\\begin{lemma}\\label{lem:kplus1-eigenfunction-product}\nLet $\\epsilon>0$ and set\n\\[\nT_m := (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)+m ,\\qquad m\\in\\N.\n\\]\nAssume that $\\lambda_{i_j}\\le \\tau_j$ for $1\\le j\\le k$ and that\n\\[\n\\lambda_{i_{k+1}}\\in [T_m,T_m+1].\n\\]\nThen for every integer $N\\ge 0$ there exists a constant $C_{\\epsilon,N}$ such that\n\\[\n\\bigl|\\langle e_{i_1} e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|\n\\le C_{\\epsilon,N}\\, T_m^{-N}.\n\\]\n\\end{lemma}\n\n\\begin{proof}[Proof of Theorem \\ref{main theorem k+1-gon-bad}]\nDecompose the tail into unit spectral windows:\n\\begin{equation*}\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)}\n\\bigl|\\langle e_{i_1}\\cdots e_{i_k}, e_{i_{k+1}}\\rangle\\bigr|^2\n=\\sum_{m=0}^\\infty \\sum_{\\lambda_{i_{k+1}}\\in[T_m,T_m+1]}\n\\bigl|\\langle e_{i_1}\\cdots e_{i_k}, e_{i_{k+1}}\\rangle\\bigr|^2 \n\\end{equation*}\nBy the Weyl law,\n\\[\n\\#\\{\\lambda_{i_{k+1}}\\in[T_m,T_m+1]\\}\\ \\lesssim\\ T_m^{n-1}.\n\\]\nLemma \\ref{lem:kplus1-eigenfunction-product} gives\n\\[\n\\sup_{\\lambda_{i_{k+1}}\\in[T_m,T_m+1]}\n\\bigl|\\langle e_{i_1}\\cdots e_{i_k}, e_{i_{k+1}}\\rangle\\bigr|^2\n\\ \\lesssim_{\\epsilon,N}\\ T_m^{-2N}.\n\\]\nHence\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)}\n\\bigl|\\langle e_{i_1}\\cdots e_{i_k}, e_{i_{k+1}}\\rangle\\bigr|^2\n\\ \\lesssim_{\\epsilon,N}\\ \\sum_{m=0}^\\infty T_m^{\\,n-1-2N}.\n\\]\nGiven $N'>0$, choose $N$ so that $2N-(n-1) > N'$. Then the last sum is bounded by\n$C_{\\epsilon,N'}(\\tau_1+\\cdots+\\tau_k)^{-N'}$, which proves the theorem.\n\\end{proof}", "post_theorem_intro_text_len": 6544, "post_theorem_intro_text": "The hypothesis $\\lambda_{i_{k+1}} \\geq (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$ forces the $(k+1)$-tuple $(\\tau_1,\\ldots,\\tau_k,\\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others. In particular, any $(\\lambda_{i_1},\\ldots,\\lambda_{i_k},\\lambda_{i_{k+1}})$ with $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$ is also $k$-bad.\n\nFor clarity, we do not organize the left-hand side by summing over all tuples $(\\lambda_{i_1},\\ldots,\\lambda_{i_{k+1}})$ lying in a specified region of $(0,\\infty)^{k+1}$. Instead, we fix upper bounds $\\tau_j$ for $\\lambda_{i_j}$, $1 \\leq j \\leq k$, and then estimate the tail sum over those $i_{k+1}$ with $\\lambda_{i_{k+1}} \\geq (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$. The proof uses the method of non-stationary phase.\nWe record a flat-torus example in Section \\ref{sec:flat-torus-example} showing that the frequency threshold in Theorem \\ref{main theorem k+1-gon-bad} is essentially sharp.\n\nOur second and main theorem concerns the contribution from $k$-good vectors $\\tau \\in (0,\\infty)^{k+1}$.\n Following \\cite{MR4376455}, we introduce the joint spectral measure\n\\begin{equation}\\label{joint spectral measure}\n\t\\mu = \\sum_{i_1,i_2,\\dots,i_{k+1}} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\, \\delta_{(\\lambda_{i_1},\\lambda_{i_2},\\dots,\\lambda_{i_{k+1}})}\n\\end{equation}\non ${\\mathbb R}^{k+1}$, where $\\delta$ denotes the Dirac measure. Thus $\\mu$ is a weighted sum of point masses, with weights given by the squared magnitudes of the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$. Our second theorem describes the mass of $\\mu$ carried by frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ lying in $k$-good cones.\n\nBefore stating the result, we introduce some terminology by recalling the definition and basic properties of the Leray density. Let $F : {\\mathbb R}^n \\to {\\mathbb R}^d$ be a smooth function, and let $y_0 \\in {\\mathbb R}^d$ be such that the differential $dF$ has full rank at every point of the level set $F^{-1}(y_0)$. By the inverse function theorem, $F^{-1}(y_0)$ is a smooth $(n-d)$-dimensional submanifold. Choose local coordinates $(z,z')$ in a neighborhood of a point of $F^{-1}(y_0)$ so that $z$ parametrizes $F^{-1}(y_0)$ and $\\det(\\partial F/\\partial z') \\neq 0$. The Leray density on $F^{-1}(y_0)$ is defined by\n\\begin{equation}\\label{Leray density definition}\n\td_F = \\left|\\det \\frac{\\partial F}{\\partial z'}\\right|^{-1} \\, dz,\n\\end{equation}\nand is independent of the choice of complementary coordinates $z'$. The corresponding Leray volume is\n\\[\n\t\\operatorname{vol} F^{-1}(y_0) = \\int_{F^{-1}(y_0)} d_F.\n\\]\n\nGiven a continuous function $f$ defined on a neighborhood of $F^{-1}(y_0)$, the integral of $f$ over $F^{-1}(y_0)$ with respect to the Leray measure can be expressed using the Dirac distribution as\n\\[\n\t\\int_{F^{-1}(y_0)} f \\, d_F = \\int_{{\\mathbb R}^n} f(x) \\delta(F(x) - y_0) \\, dx.\n\\]\n\nFor the purposes of this paper, we set $F : {\\mathbb R}^{kn} \\to {\\mathbb R}^{k+1}$ by\n\\begin{equation}\\label{def F}\n\tF(\\xi_1,\\xi_2,\\dots,\\xi_k) = \\bigl(|\\xi_1|,|\\xi_2|,\\dots,|\\xi_k|,\\ |\\xi_1 + \\cdots + \\xi_k|\\bigr).\n\\end{equation}\nThen the level set $F^{-1}(\\tau)$ consists of $k$-tuples $(\\xi_1,\\xi_2,\\dots,\\xi_k)$ satisfying\n\\[\n|\\xi_j|=\\tau_j \\ \\ (1\\le j\\le k)\n\\qquad \\text{and} \\qquad\n|\\xi_1+\\cdots+\\xi_k|=\\tau_{k+1}.\n\\]\nSuch a $k$-tuple determines a closed $(k+1)$-gon in ${\\mathbb R}^n$ with side lengths $\\tau_1,\\tau_2,\\dots,\\tau_{k+1}$, where the final side is the closing vector $-(\\xi_1+\\cdots+\\xi_k)$.\nThe next theorem provides a description of the concentration of $\\mu$ in $k$-good cones.\n\n\\begin{theorem}\\label{main theorem k+1-good}\nLet $\\rho$ be a Schwartz function on ${\\mathbb R}^{k+1}$ with $\\int \\rho = 1$. Assume that\n\\[\n\t\\operatorname{supp} \\hat \\rho \\subset (-\\operatorname{inj} M, \\operatorname{inj} M)^{k+1}.\n\\]\nLet $\\Gamma$ be a $k$-good cone. Then for $\\tau = (\\tau_1, \\tau_2, \\dots, \\tau_{k+1}) \\in \\Gamma$,\n\\[\n\t\\rho * \\mu(\\tau) = (2\\pi)^{-kn} \\operatorname{vol} M \\operatorname{vol} F^{-1}(\\tau) + O(|\\tau|^{k(n-1)-2}),\n\\]\nwhere the constants implicit in the big-$O$ notation depend on $M$, $\\Gamma$, and $\\rho$ but not on $\\tau$.\n\\end{theorem}\n\nSince $\\rho$ is rapidly decaying and $\\operatorname{supp} \\hat \\rho$ is contained in $(-\\operatorname{inj} M, \\operatorname{inj} M)^{k+1}$, the convolution $\\rho * \\mu$ provides a smoothed version of $\\mu$ that captures the mass of $\\mu$ near $\\tau$. In particular, Theorem \\ref{main theorem k+1-good} gives an asymptotic description of this smoothed local mass for $\\tau \\in \\Gamma$. Next, we determine the growth rate of $\\operatorname{vol} F^{-1}(\\tau)$ as $|\\tau| \\to \\infty$. When $k=2$, this quantity is related to the area of the corresponding triangle \\cite{MR4376455}. For $k>2$, the shape of a $(k+1)$-gon with fixed side lengths is not uniquely determined (Figures \\ref{2} and \\ref{3}), so it is not sufficient to consider only its planar area. We prove in Proposition \\ref{Scaling of F} that, under our assumptions, $\\operatorname{vol} F^{-1}(\\tau)$ is of order $|\\tau|^{k(n-1)-1}$, so the remainder term $O(|\\tau|^{k(n-1)-2})$ is indeed lower order. As in \\cite{MR4376455}, the leading term can be interpreted in terms of the volume of the configuration space of $(k+1)$-gons with side lengths prescribed by $\\tau$.\n\\subsection*{Organization}\nIn Section~2 we analyze the configuration space that appears in the main term of Theorem~\\ref{main theorem k+1-good}, and in particular prove the scaling law for the relevant fiber volume. Section~3 contains the proof of Theorem~\\ref{main theorem k+1-gon-bad}, establishing rapid decay of the high-frequency tail in the regime where the $(k+1)$-gon inequality fails, via spectral projectors and a non-stationary phase argument. Section~4 proves Theorem~\\ref{main theorem k+1-good} by expressing the Fourier transform of the joint spectral measure in terms of Fourier integral operators and computing the associated symbolic data using clean composition and half-density calculus. Finally, Section~\\ref{sec:flat-torus-example} presents a flat-torus example showing that the frequency cutoff in Theorem~\\ref{main theorem k+1-gon-bad} is essentially sharp.\n\n\\subsection*{Acknowledgement} This project is supported by the National Key R\\&D Program of China under Grant No. 2022YFA1007200, the Natural Science Foundation of China under Grant No. 12571107, and the Zhejiang Provincial Natural Science Foundation of China under Grant No. LR25A010001.", "sketch": "The introduction explains that the condition $\\lambda_{i_{k+1}} \\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$ “forces the $(k+1)$-tuple $(\\tau_1,\\ldots,\\tau_k,\\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others,” hence any $(\\lambda_{i_1},\\ldots,\\lambda_{i_k},\\lambda_{i_{k+1}})$ with $\\lambda_{i_j}\\le \\tau_j$ is also $k$-bad.\n\nRather than summing over a region in $(0,\\infty)^{k+1}$, the argument is organized by “fix[ing] upper bounds $\\tau_j$ for $\\lambda_{i_j}$, $1\\le j\\le k$, and then estimat[ing] the tail sum over those $i_{k+1}$ with $\\lambda_{i_{k+1}} \\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$.” The proof of Theorem~\\ref{main theorem k+1-gon-bad} then “uses the method of non-stationary phase,” and (per the Organization paragraph) establishes “rapid decay of the high-frequency tail in the regime where the $(k+1)$-gon inequality fails, via spectral projectors and a non-stationary phase argument.” A flat-torus example is recorded showing the frequency threshold is “essentially sharp.”", "expanded_sketch": "The introduction explains that the condition $\\lambda_{i_{k+1}} \\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$ “forces the $(k+1)$-tuple $(\\tau_1,\\ldots,\\tau_k,\\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others,” hence any $(\\lambda_{i_1},\\ldots,\\lambda_{i_k},\\lambda_{i_{k+1}})$ with $\\lambda_{i_j}\\le \\tau_j$ is also $k$-bad.\n\nRather than summing over a region in $(0,\\infty)^{k+1}$, the argument is organized by “fix[ing] upper bounds $\\tau_j$ for $\\lambda_{i_j}$, $1\\le j\\le k$, and then estimat[ing] the tail sum over those $i_{k+1}$ with $\\lambda_{i_{k+1}} \\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$.” To prove the main theorem, the proof then “uses the method of non-stationary phase,” and (per the Organization paragraph) establishes “rapid decay of the high-frequency tail in the regime where the $(k+1)$-gon inequality fails, via spectral projectors and a non-stationary phase argument.” A flat-torus example is recorded showing the frequency threshold is “essentially sharp.”", "expanded_theorem": "\\label{main theorem k+1-gon-bad}\n\tFor each $\\epsilon>0$ and integer $N > 0$, there exists a constant $C_{\\epsilon, N}$ such that for any positive numbers $\\tau_{1},\\tau_{2},\\ldots,\\tau_{k}$, we have\n\t\\[\n\t\t\\sum_{\\lambda_{i_{k+1}}\\geq (1+\\epsilon)(\\tau_1+\\tau_2+\\dots+\\tau_k)} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\leq C_{\\epsilon, N}(\\tau_1+\\tau_2+\\dots+\\tau_k)^{-N},\n\t\\]\n\twhere $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$.", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let \\(\\{e_i\\}\\) be an \\(L^2\\)-orthonormal family of eigenfunctions with corresponding eigenvalues (frequencies) \\(\\lambda_i\\), and write \\(\\langle f,g\\rangle\\) for the \\(L^2\\) inner product. Fix an integer \\(k\\ge 1\\). For positive numbers \\(\\tau_1,\\dots,\\tau_k\\), suppose the indices \\(i_1,\\dots,i_k\\) satisfy \\(\\lambda_{i_j}\\le \\tau_j\\) for each \\(1\\le j\\le k\\). Under these assumptions, which quantitative estimate holds for the tail of the harmonic expansion coefficients of the product \\(e_{i_1}e_{i_2}\\cdots e_{i_k}\\)?", "correct_choice": {"label": "A", "text": "For every \\(\\epsilon>0\\) and every integer \\(N>0\\), there exists a constant \\(C_{\\epsilon,N}\\) such that\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\tau_2+\\cdots+\\tau_k)} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon,N}(\\tau_1+\\tau_2+\\cdots+\\tau_k)^{-N}.\n\\]"}, "choices": [{"label": "B", "text": "For every \\(\\epsilon>0\\) and every integer \\(N>0\\), there exists a constant \\(C_{\\epsilon,N}\\) such that\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (\\tau_1+\\tau_2+\\cdots+\\tau_k)+\\epsilon} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon,N}(\\tau_1+\\tau_2+\\cdots+\\tau_k)^{-N}.\n\\]"}, {"label": "C", "text": "For every \\(\\epsilon>0\\), there exists a constant \\(C_{\\epsilon}\\) such that\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\tau_2+\\cdots+\\tau_k)} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon}.\n\\]"}, {"label": "D", "text": "There exists an integer \\(N>0\\) such that for every \\(\\epsilon>0\\) there is a constant \\(C_{\\epsilon,N}\\) with\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\tau_2+\\cdots+\\tau_k)} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon,N}(\\tau_1+\\tau_2+\\cdots+\\tau_k)^{-N}\n\\]\nfor all positive \\(\\tau_1,\\dots,\\tau_k\\) and all indices \\(i_1,\\dots,i_k\\) with \\(\\lambda_{i_j}\\le \\tau_j\\)."}, {"label": "E", "text": "For every \\(\\epsilon>0\\) and every integer \\(N>0\\), there exists a constant \\(C_{\\epsilon,N}\\) such that\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)\\max\\{\\tau_1,\\tau_2,\\dots,\\tau_k\\}} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon,N}(\\tau_1+\\tau_2+\\cdots+\\tau_k)^{-N}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "non-stationary phase", "tampered_component": "multiplicative tail threshold replaced by additive offset", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped rapid-decay power saving in \\((\\tau_1+\\cdots+\\tau_k)\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "quantifier order on N weakened from arbitrary N to existence of one N", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "sum-of-sides bad-region threshold replaced by max-side threshold", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It asks for the valid uniform decay statement without embedding the key threshold, decay rate, or quantifier structure of the correct answer."}, "TAS": {"score": 1, "justification": "The item is largely theorem-recognition: it asks which precise statement holds, and one option is essentially the target theorem. However, it is not purely tautological because the alternatives vary in meaningful ways (threshold type, decay strength, tail norm, and quantifiers)."}, "GPS": {"score": 1, "justification": "Answering requires moderate reasoning about logical strength and mathematical form: distinguishing multiplicative versus additive thresholds, rapid decay versus mere boundedness, and sum versus supremum. Still, it mainly tests recall/recognition of the theorem statement rather than generating a conclusion from premises."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically plausible. They reflect realistic failure modes: weakening the decay, altering quantifier order, replacing an \\(\\ell^2\\)-tail bound by a pointwise supremum, or using an incorrect boundary threshold."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with high-quality distractors and no answer leakage, but it leans more toward precise statement recognition than genuinely generative reasoning."}}
{"id": "2602.04664v1", "paper_link": "http://arxiv.org/abs/2602.04664v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{main theorem k+1-gon-bad}\n\tFor each $\\epsilon>0$ and integer $N > 0$, there exists a constant $C_{\\epsilon, N}$ such that for any positive numbers $\\tau_{1},\\tau_{2},\\ldots,\\tau_{k}$, we have\n\t\\[\n\t\t\\sum_{\\lambda_{i_{k+1}}\\geq (1+\\epsilon)(\\tau_1+\\tau_2+\\dots+\\tau_k)} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\leq C_{\\epsilon, N}(\\tau_1+\\tau_2+\\dots+\\tau_k)^{-N},\n\t\\]\n\twhere $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$.", "start_pos": 10297, "end_pos": 10791, "label": "main theorem k+1-gon-bad"}, "ref_dict": {"main theorem k+1-good": "\\begin{theorem}\\label{main theorem k+1-good}\nLet $\\rho$ be a Schwartz function on $\\R^{k+1}$ with $\\int \\rho = 1$. Assume that\n\\[\n\t\\supp \\hat \\rho \\subset (-\\inj M, \\inj M)^{k+1}.\n\\]\nLet $\\Gamma$ be a $k$-good cone. Then for $\\tau = (\\tau_1, \\tau_2, \\dots, \\tau_{k+1}) \\in \\Gamma$,\n\\[\n\t\\rho * \\mu(\\tau) = (2\\pi)^{-kn} \\vol M \\vol F^{-1}(\\tau) + O(|\\tau|^{k(n-1)-2}),\n\\]\nwhere the constants implicit in the big-$O$ notation depend on $M$, $\\Gamma$, and $\\rho$ but not on $\\tau$.\n\\end{theorem}", "harmonic expansion": "\\begin{equation}\\label{harmonic expansion}\n\te_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}", "4": "\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{4}\n\\caption{An example of $\\tau\\notin\\Gamma_g$.}\n\\label{4}\n\\end{figure}", "main theorem k+1-gon-bad": "\\begin{theorem}\\label{main theorem k+1-gon-bad}\n\tFor each $\\epsilon>0$ and integer $N > 0$, there exists a constant $C_{\\epsilon, N}$ such that for any positive numbers $\\tau_{1},\\tau_{2},\\ldots,\\tau_{k}$, we have\n\t\\[\n\t\t\\sum_{\\lambda_{i_{k+1}}\\geq (1+\\epsilon)(\\tau_1+\\tau_2+\\dots+\\tau_k)} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\leq C_{\\epsilon, N}(\\tau_1+\\tau_2+\\dots+\\tau_k)^{-N},\n\t\\]\n\twhere $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$.\n\\end{theorem}", "2": "\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{2}\n\\caption{A pentagon with side lengths $a_1, \\ldots, a_5$.}\n\\label{2}\n\\includegraphics[width=0.6\\textwidth]{3}\n\\caption{A self-intersecting pentagon with the same side lengths.}\n\\label{3}\n\\end{figure}", "single sums": "\\begin{equation}\\label{single sums}\n\t\\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}", "rewrite harmonic expansion": "\\begin{equation}\\label{rewrite harmonic expansion}\n\t\\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}", "Scaling of F": "\\begin{proposition}\n [Scaling of $F$]\\label{Scaling of F} Let $\\Gamma$ be a $k$-good cone and fix $\\tau \\in \\Gamma$. Then $F^{-1}(\\tau) \\neq \\emptyset$ and $\\tau$ is a regular value of $F$, so $F^{-1}(\\tau)$ is a smooth compact submanifold of codimension $k+1$. Let\n\\[\nd:=k n-(k+1)=k(n-1)-1\n\\]\nbe its dimension. Then $0<\\operatorname{vol} F^{-1}(\\tau)<\\infty$ and, for every $r>0$,\n$$\n\\operatorname{vol} F^{-1}(r \\tau)=r^d \\operatorname{vol} F^{-1}(\\tau).\n$$\n\\end{proposition}", "3": "\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{2}\n\\caption{A pentagon with side lengths $a_1, \\ldots, a_5$.}\n\\label{2}\n\\includegraphics[width=0.6\\textwidth]{3}\n\\caption{A self-intersecting pentagon with the same side lengths.}\n\\label{3}\n\\end{figure}"}, "pre_theorem_intro_text_len": 7505, "pre_theorem_intro_text": "\\subsection{Background}\n\nLet $M$ be a compact Riemannian manifold without boundary, and $e_1, e_2, \\ldots$ be an orthonormal basis of Laplace-Beltrami eigenfunctions with\n\\[\n\t\\Delta_g e_j = -\\lambda_j^2 e_j,\n\\]\nwhere $\\lambda_j$ is the frequency of $e_j$.\n\nWe are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \\dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion\n\\begin{equation}\\label{harmonic expansion}\n\te_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\nThe classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \\dots e_{i_k}$. If we set $\\lambda_{i_1}=\\lambda_{i_2}=\\dots =\\lambda_{i_k}=\\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\\lambda_j}$. For this problem, Sogge provided a complete answer for $k\\geq 1$, see \\cite{MR930395}. \nHowever, for general eigenvalues $\\lambda_{i_1}, \\lambda_{i_2}, \\ldots, \\lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \\eqref{harmonic expansion} as follows:\n\\begin{equation}\\label{rewrite harmonic expansion}\n\t\\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\nThe decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \\eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}}$ do the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$ exhibit decay, and how is their $\\ell^2$-mass distributed as these frequencies vary?\n\nWhen $k=2$, as in many spectral problems, this question is closely connected to number theory in the automorphic setting. Sarnak showed that for fixed $j$ the coefficients $\\langle e_j^2, e_\\ell \\rangle$ decay exponentially in $\\lambda_\\ell$ \\cite{MR1277052}. Bernstein, Reznikov, Kr\\\"otz, and Stanton obtained the optimal estimate \\cite{MR1715328, MR2081437}. For the more general coefficients $\\langle e_i e_j, e_\\ell \\rangle$, a classical decay estimate in terms of the eigenvalues was given by Lu, Sogge, and Steinerberger \\cite{MR3997636}. More recently, the first author improved their estimate \\cite{MR4376455}. Given $\\epsilon > 0$ and an integer $N \\geq 1$, there exists a constant $C_{\\epsilon,N}$ for which\n\\begin{equation}\\label{Emmett conclusion}\n\t\\sum_{\\lambda_\\ell \\geq (2+\\epsilon)\\lambda} |\\langle e_i e_j, e_\\ell \\rangle|^2 \\leq C_{\\epsilon, N} \\lambda^{-N} \\qquad \\text{for all $i$, $j$ with $\\lambda_i, \\lambda_j \\leq \\lambda$.}\n\\end{equation}\n\nFurthermore, he proved that the $(2+\\epsilon)\\lambda$ factor is nearly optimal. He also obtained an asymptotic expansion for the non-rapidly decaying part \\cite{MR4376455}. Building on this work, we study the multi-product of $(k+1)$ eigenfunctions and obtain analogous results. \nOne of the most striking features of \\cite{MR4376455} is that it connects triple products of eigenfunctions to geometric configurations, namely triangles. We extend this perspective to general $k \\geq 2$, where the corresponding configurations are naturally described by polygons.\n\n\\subsection{Polygons}\n\nBased on \\eqref{rewrite harmonic expansion}, we study the sums\n\\begin{equation}\\label{single sums}\n\t\\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}\nfor given eigenvalues $a_1, a_2, \\dots, a_{k+1}$. These sums are independent of the choice of eigenbasis; hence their behavior is determined solely by the underlying manifold and its metric.\n\nThe main point of the present paper is to show that the sum \\eqref{single sums} can be estimated by counting configurations of $(k+1)$-gons with side lengths $a_1, a_2, \\dots, a_{k+1}$. To state our results, we first introduce the following definition concerning the side lengths of a $(k+1)$-gon.\n\n\\begin{definition}\\label{k+1-good and k+1-gon-bad}\nA vector $\\tau = (\\tau_1, \\ldots, \\tau_{k+1}) \\in (0,\\infty)^{k+1}$ is called $k$-good if it satisfies the following two conditions.\n\\begin{enumerate}\n\\item the $(k+1)$-gon inequality: for every $1 \\leq j \\leq k+1$,\n\\begin{equation}\\label{2.1}\n\\tau_j < \\sum_{\\ell \\neq j} \\tau_{\\ell}.\n\\end{equation}\n\\item the nondegeneracy condition: for every sign vector $\\varepsilon = (\\varepsilon_1, \\ldots, \\varepsilon_{k+1}) \\in \\{ \\pm 1\\}^{k+1}$,\n\\begin{equation}\\label{2.2}\n|\\varepsilon \\cdot \\tau| = \\left|\\sum_{j=1}^{k+1} \\varepsilon_j \\tau_j\\right| > 0.\n\\end{equation}\n\\end{enumerate}\nWe denote the set of $k$-good vectors by $\\Gamma_g$.\n\nA vector $\\tau = (\\tau_1, \\ldots, \\tau_{k+1}) \\in (0,\\infty)^{k+1}$ is called $k$-bad if there exists $j$, $1 \\leq j \\leq k+1$, such that\n\\begin{equation}\\label{2.3}\n\\tau_j > \\sum_{\\ell \\neq j} \\tau_{\\ell}.\n\\end{equation}\nWe denote the set of $k$-bad vectors by $\\Gamma_b$.\n\nA closed cone $\\Gamma \\subset (0,\\infty)^{k+1}$ is called $k$-good if $\\Gamma \\subset \\Gamma_g$, and it is called $k$-bad if $\\Gamma \\subset \\Gamma_b$.\n\\end{definition}\n\nThe set of $\\tau$ satisfying condition (1) consists of side-length data that can be realized by a $(k+1)$-gon in the plane, whereas the set of $k$-bad vectors contains no such points. The set of $k$-good vectors is obtained from the set of $\\tau$ satisfying condition (1) by removing those $\\tau$ for which a linear combination $|\\varepsilon \\cdot \\tau|$ vanishes for some $\\varepsilon \\in \\{ \\pm 1\\}^{k+1}$. For example, when $k=3$, the vector $(2,2,2,2)$ satisfies the $(k+1)$-gon inequality, but $(2,2,2,2)\\notin \\Gamma_g$ since it fails the nondegeneracy condition. Concretely, consider the degenerate quadrilateral in the plane formed by the vectors $(2,0), (-2,0), (2,0), (-2,0)$. Each side has length $2$, but the polygon is collinear and has zero area. In this case, $\\tau \\notin \\Gamma_g$, as illustrated in Figure \\ref{4}. When $k=2$, the nondegeneracy condition is automatic once one assumes condition (1), due to the rigidity of triangles. This is the primary reason it was not clear whether triple-product results could be generalized to higher multi-products, where such rigidity is absent. Our nondegeneracy condition is designed to fill this gap. As we will see in the proof, it ensures that several crucial steps go through and yields clean asymptotics.\n\n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{4}\n\\caption{An example of $\\tau\\notin\\Gamma_g$.}\n\\label{4}\n\\end{figure}\nIn what follows, we establish a connection between \\eqref{single sums} and the count of configurations of $(k+1)$-gons using the spectral projection operator $\\chi_\\lambda$ and the theory of Fourier integral operators. Our argument also shows that \\eqref{single sums} decays rapidly in the regime where $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ fails to satisfy the $(k+1)$-gon inequality.\n\n\\subsection{Main results}\nOur first theorem shows that the rapidly decaying contribution to \\eqref{single sums} comes from frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.", "context": "We are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \\dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion\n\\begin{equation}\\label{harmonic expansion}\n e_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\nThe classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \\dots e_{i_k}$. If we set $\\lambda_{i_1}=\\lambda_{i_2}=\\dots =\\lambda_{i_k}=\\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\\lambda_j}$. For this problem, Sogge provided a complete answer for $k\\geq 1$, see \\cite{MR930395}. \nHowever, for general eigenvalues $\\lambda_{i_1}, \\lambda_{i_2}, \\ldots, \\lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \\eqref{harmonic expansion} as follows:\n\\begin{equation}\\label{rewrite harmonic expansion}\n \\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\nThe decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \\eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}}$ do the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$ exhibit decay, and how is their $\\ell^2$-mass distributed as these frequencies vary?\n\nWhen $k=2$, as in many spectral problems, this question is closely connected to number theory in the automorphic setting. Sarnak showed that for fixed $j$ the coefficients $\\langle e_j^2, e_\\ell \\rangle$ decay exponentially in $\\lambda_\\ell$ \\cite{MR1277052}. Bernstein, Reznikov, Kr\\\"otz, and Stanton obtained the optimal estimate \\cite{MR1715328, MR2081437}. For the more general coefficients $\\langle e_i e_j, e_\\ell \\rangle$, a classical decay estimate in terms of the eigenvalues was given by Lu, Sogge, and Steinerberger \\cite{MR3997636}. More recently, the first author improved their estimate \\cite{MR4376455}. Given $\\epsilon > 0$ and an integer $N \\geq 1$, there exists a constant $C_{\\epsilon,N}$ for which\n\\begin{equation}\\label{Emmett conclusion}\n \\sum_{\\lambda_\\ell \\geq (2+\\epsilon)\\lambda} |\\langle e_i e_j, e_\\ell \\rangle|^2 \\leq C_{\\epsilon, N} \\lambda^{-N} \\qquad \\text{for all $i$, $j$ with $\\lambda_i, \\lambda_j \\leq \\lambda$.}\n\\end{equation}\n\nBased on \\eqref{rewrite harmonic expansion}, we study the sums\n\\begin{equation}\\label{single sums}\n \\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}\nfor given eigenvalues $a_1, a_2, \\dots, a_{k+1}$. These sums are independent of the choice of eigenbasis; hence their behavior is determined solely by the underlying manifold and its metric.\n\nA vector $\\tau = (\\tau_1, \\ldots, \\tau_{k+1}) \\in (0,\\infty)^{k+1}$ is called $k$-bad if there exists $j$, $1 \\leq j \\leq k+1$, such that\n\\begin{equation}\\label{2.3}\n\\tau_j > \\sum_{\\ell \\neq j} \\tau_{\\ell}.\n\\end{equation}\nWe denote the set of $k$-bad vectors by $\\Gamma_b$.\n\n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{4}\n\\caption{An example of $\\tau\\notin\\Gamma_g$.}\n\\label{4}\n\\end{figure}\nIn what follows, we establish a connection between \\eqref{single sums} and the count of configurations of $(k+1)$-gons using the spectral projection operator $\\chi_\\lambda$ and the theory of Fourier integral operators. Our argument also shows that \\eqref{single sums} decays rapidly in the regime where $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ fails to satisfy the $(k+1)$-gon inequality.\n\n\\subsection{Main results}\nOur first theorem shows that the rapidly decaying contribution to \\eqref{single sums} comes from frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.\n\n\\begin{equation}\\label{harmonic expansion}\n\te_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\n\n\\begin{equation}\\label{rewrite harmonic expansion}\n\t\\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\n\n\\begin{equation}\\label{single sums}\n\t\\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}", "full_context": "We are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \\dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion\n\\begin{equation}\\label{harmonic expansion}\n e_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\nThe classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \\dots e_{i_k}$. If we set $\\lambda_{i_1}=\\lambda_{i_2}=\\dots =\\lambda_{i_k}=\\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\\lambda_j}$. For this problem, Sogge provided a complete answer for $k\\geq 1$, see \\cite{MR930395}. \nHowever, for general eigenvalues $\\lambda_{i_1}, \\lambda_{i_2}, \\ldots, \\lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \\eqref{harmonic expansion} as follows:\n\\begin{equation}\\label{rewrite harmonic expansion}\n \\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\nThe decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \\eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}}$ do the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$ exhibit decay, and how is their $\\ell^2$-mass distributed as these frequencies vary?\n\nWhen $k=2$, as in many spectral problems, this question is closely connected to number theory in the automorphic setting. Sarnak showed that for fixed $j$ the coefficients $\\langle e_j^2, e_\\ell \\rangle$ decay exponentially in $\\lambda_\\ell$ \\cite{MR1277052}. Bernstein, Reznikov, Kr\\\"otz, and Stanton obtained the optimal estimate \\cite{MR1715328, MR2081437}. For the more general coefficients $\\langle e_i e_j, e_\\ell \\rangle$, a classical decay estimate in terms of the eigenvalues was given by Lu, Sogge, and Steinerberger \\cite{MR3997636}. More recently, the first author improved their estimate \\cite{MR4376455}. Given $\\epsilon > 0$ and an integer $N \\geq 1$, there exists a constant $C_{\\epsilon,N}$ for which\n\\begin{equation}\\label{Emmett conclusion}\n \\sum_{\\lambda_\\ell \\geq (2+\\epsilon)\\lambda} |\\langle e_i e_j, e_\\ell \\rangle|^2 \\leq C_{\\epsilon, N} \\lambda^{-N} \\qquad \\text{for all $i$, $j$ with $\\lambda_i, \\lambda_j \\leq \\lambda$.}\n\\end{equation}\n\nBased on \\eqref{rewrite harmonic expansion}, we study the sums\n\\begin{equation}\\label{single sums}\n \\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}\nfor given eigenvalues $a_1, a_2, \\dots, a_{k+1}$. These sums are independent of the choice of eigenbasis; hence their behavior is determined solely by the underlying manifold and its metric.\n\nA vector $\\tau = (\\tau_1, \\ldots, \\tau_{k+1}) \\in (0,\\infty)^{k+1}$ is called $k$-bad if there exists $j$, $1 \\leq j \\leq k+1$, such that\n\\begin{equation}\\label{2.3}\n\\tau_j > \\sum_{\\ell \\neq j} \\tau_{\\ell}.\n\\end{equation}\nWe denote the set of $k$-bad vectors by $\\Gamma_b$.\n\n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{4}\n\\caption{An example of $\\tau\\notin\\Gamma_g$.}\n\\label{4}\n\\end{figure}\nIn what follows, we establish a connection between \\eqref{single sums} and the count of configurations of $(k+1)$-gons using the spectral projection operator $\\chi_\\lambda$ and the theory of Fourier integral operators. Our argument also shows that \\eqref{single sums} decays rapidly in the regime where $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ fails to satisfy the $(k+1)$-gon inequality.\n\n\\subsection{Main results}\nOur first theorem shows that the rapidly decaying contribution to \\eqref{single sums} comes from frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.\n\n\\begin{equation}\\label{harmonic expansion}\n\te_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\n\n\\begin{equation}\\label{rewrite harmonic expansion}\n\t\\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\n\n\\begin{equation}\\label{single sums}\n\t\\sum_{(\\lambda_{i_1}, \\lambda_{i_2}, \\dots ,\\lambda_{i_{k+1}}) = (a_1, a_2, \\dots ,a_{k+1})} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2\n\\end{equation}\n\nWe are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \\dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion\n\\begin{equation}\\label{harmonic expansion}\n e_{i_1} e_{i_2} \\dots e_{i_k} = \\sum_{i_{k+1}} \\langle e_{i_1} e_{i_2} \\dots e_{i_k}, e_{i_{k+1}} \\rangle e_{i_{k+1}}.\n\\end{equation}\nThe classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \\dots e_{i_k}$. If we set $\\lambda_{i_1}=\\lambda_{i_2}=\\dots =\\lambda_{i_k}=\\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\\lambda_j}$. For this problem, Sogge provided a complete answer for $k\\geq 1$, see \\cite{MR930395}. \nHowever, for general eigenvalues $\\lambda_{i_1}, \\lambda_{i_2}, \\ldots, \\lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \\eqref{harmonic expansion} as follows:\n\\begin{equation}\\label{rewrite harmonic expansion}\n \\|e_{i_1} e_{i_2} \\cdots e_{i_k}\\|_{L^2}^2 = \\sum_{i_{k+1}\\in\\mathbb Z^+} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2.\n\\end{equation}\nThe decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \\eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}}$ do the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$ exhibit decay, and how is their $\\ell^2$-mass distributed as these frequencies vary?\n\n\\subsection{Main results}\nOur first theorem shows that the rapidly decaying contribution to \\eqref{single sums} comes from frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.\n\nThe hypothesis $\\lambda_{i_{k+1}} \\geq (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$ forces the $(k+1)$-tuple $(\\tau_1,\\ldots,\\tau_k,\\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others. In particular, any $(\\lambda_{i_1},\\ldots,\\lambda_{i_k},\\lambda_{i_{k+1}})$ with $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$ is also $k$-bad.\n\nFor clarity, we do not organize the left-hand side by summing over all tuples $(\\lambda_{i_1},\\ldots,\\lambda_{i_{k+1}})$ lying in a specified region of $(0,\\infty)^{k+1}$. Instead, we fix upper bounds $\\tau_j$ for $\\lambda_{i_j}$, $1 \\leq j \\leq k$, and then estimate the tail sum over those $i_{k+1}$ with $\\lambda_{i_{k+1}} \\geq (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$. The proof uses the method of non-stationary phase.\nWe record a flat-torus example in Section \\ref{sec:flat-torus-example} showing that the frequency threshold in Theorem \\ref{main theorem k+1-gon-bad} is essentially sharp.\n\nOur second and main theorem concerns the contribution from $k$-good vectors $\\tau \\in (0,\\infty)^{k+1}$.\n Following \\cite{MR4376455}, we introduce the joint spectral measure\n\\begin{equation}\\label{joint spectral measure}\n \\mu = \\sum_{i_1,i_2,\\dots,i_{k+1}} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\, \\delta_{(\\lambda_{i_1},\\lambda_{i_2},\\dots,\\lambda_{i_{k+1}})}\n\\end{equation}\non $\\R^{k+1}$, where $\\delta$ denotes the Dirac measure. Thus $\\mu$ is a weighted sum of point masses, with weights given by the squared magnitudes of the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$. Our second theorem describes the mass of $\\mu$ carried by frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ lying in $k$-good cones.\n\nNext we verify that $\\tau$ is a regular value of $F$. Fix $\\xi=(\\xi_1,\\ldots,\\xi_k)\\in F^{-1}(\\tau)$ and set\n\\[\n\\omega_j=\\frac{\\xi_j}{|\\xi_j|}\\in S^{n-1}\\ (1\\le j\\le k),\\qquad\n\\omega_{k+1}=\\frac{\\xi_1+\\cdots+\\xi_k}{|\\xi_1+\\cdots+\\xi_k|}\\in S^{n-1}.\n\\]\nFor $h=(h_1,\\ldots,h_k)\\in\\R^{kn}$ we compute\n\\[\nd(|\\xi_j|)[h]=\\langle \\omega_j,h_j\\rangle,\\qquad\nd(|\\xi_1+\\cdots+\\xi_k|)[h]=\\Bigl\\langle \\omega_{k+1},\\sum_{j=1}^k h_j\\Bigr\\rangle.\n\\]\nIf $\\mathrm{rank}(dF_\\xi)<k+1$, these $k+1$ linear functionals are linearly dependent, so there exist\n$(c_1,\\ldots,c_k,c_{k+1})\\neq 0$ such that for all $h$,\n\\[\n\\sum_{j=1}^k c_j\\langle \\omega_j,h_j\\rangle\n+c_{k+1}\\Bigl\\langle \\omega_{k+1},\\sum_{j=1}^k h_j\\Bigr\\rangle=0.\n\\]\nEquivalently,\n\\[\n\\sum_{j=1}^k \\langle c_j\\omega_j+c_{k+1}\\omega_{k+1},\\ h_j\\rangle=0\n\\quad\\text{for all }h_1,\\ldots,h_k,\n\\]\nhence $c_j\\omega_j+c_{k+1}\\omega_{k+1}=0$ for every $j$. In particular $c_{k+1}\\neq 0$, and\n$\\omega_j=\\varepsilon_j\\,\\omega_{k+1}$ for some $\\varepsilon_j\\in\\{\\pm1\\}$. Therefore\n\\[\n\\xi_1+\\cdots+\\xi_k=\\sum_{j=1}^k |\\xi_j|\\omega_j\n=\\Bigl(\\sum_{j=1}^k \\varepsilon_j|\\xi_j|\\Bigr)\\omega_{k+1},\n\\]\nand taking norms gives\n\\[\n\\tau_{k+1}=|\\xi_1+\\cdots+\\xi_k|\n=\\Bigl|\\sum_{j=1}^k \\varepsilon_j|\\xi_j|\\Bigr|\n=\\Bigl|\\sum_{j=1}^k \\varepsilon_j\\,\\tau_j\\Bigr|.\n\\]\nLet $s:=\\sum_{j=1}^k \\varepsilon_j\\,\\tau_j$ and choose $\\varepsilon_{k+1}\\in\\{\\pm1\\}$ so that\n$\\varepsilon_{k+1}\\tau_{k+1}+s=0$. Then the sign vector\n\\[\n\\varepsilon=(\\varepsilon_1,\\ldots,\\varepsilon_k,\\varepsilon_{k+1})\\in\\{\\pm1\\}^{k+1}\n\\]\nsatisfies $\\varepsilon\\cdot\\tau=0$, contradicting the nondegeneracy condition \\eqref{2.2}.\nHence $\\mathrm{rank}(dF_\\xi)=k+1$ for all $\\xi\\in F^{-1}(\\tau)$, so $\\tau$ is a regular value.\n\n\\begin{lemma}\\label{lem:kplus1-eigenfunction-product}\nLet $\\epsilon>0$ and set\n\\[\nT_m := (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)+m ,\\qquad m\\in\\N.\n\\]\nAssume that $\\lambda_{i_j}\\le \\tau_j$ for $1\\le j\\le k$ and that\n\\[\n\\lambda_{i_{k+1}}\\in [T_m,T_m+1].\n\\]\nThen for every integer $N\\ge 0$ there exists a constant $C_{\\epsilon,N}$ such that\n\\[\n\\bigl|\\langle e_{i_1} e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|\n\\le C_{\\epsilon,N}\\, T_m^{-N}.\n\\]\n\\end{lemma}\n\n\\begin{proof}[Proof of Theorem \\ref{main theorem k+1-gon-bad}]\nDecompose the tail into unit spectral windows:\n\\begin{equation*}\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)}\n\\bigl|\\langle e_{i_1}\\cdots e_{i_k}, e_{i_{k+1}}\\rangle\\bigr|^2\n=\\sum_{m=0}^\\infty \\sum_{\\lambda_{i_{k+1}}\\in[T_m,T_m+1]}\n\\bigl|\\langle e_{i_1}\\cdots e_{i_k}, e_{i_{k+1}}\\rangle\\bigr|^2 \n\\end{equation*}\nBy the Weyl law,\n\\[\n\\#\\{\\lambda_{i_{k+1}}\\in[T_m,T_m+1]\\}\\ \\lesssim\\ T_m^{n-1}.\n\\]\nLemma \\ref{lem:kplus1-eigenfunction-product} gives\n\\[\n\\sup_{\\lambda_{i_{k+1}}\\in[T_m,T_m+1]}\n\\bigl|\\langle e_{i_1}\\cdots e_{i_k}, e_{i_{k+1}}\\rangle\\bigr|^2\n\\ \\lesssim_{\\epsilon,N}\\ T_m^{-2N}.\n\\]\nHence\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)}\n\\bigl|\\langle e_{i_1}\\cdots e_{i_k}, e_{i_{k+1}}\\rangle\\bigr|^2\n\\ \\lesssim_{\\epsilon,N}\\ \\sum_{m=0}^\\infty T_m^{\\,n-1-2N}.\n\\]\nGiven $N'>0$, choose $N$ so that $2N-(n-1) > N'$. Then the last sum is bounded by\n$C_{\\epsilon,N'}(\\tau_1+\\cdots+\\tau_k)^{-N'}$, which proves the theorem.\n\\end{proof}", "post_theorem_intro_text_len": 6544, "post_theorem_intro_text": "The hypothesis $\\lambda_{i_{k+1}} \\geq (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$ forces the $(k+1)$-tuple $(\\tau_1,\\ldots,\\tau_k,\\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others. In particular, any $(\\lambda_{i_1},\\ldots,\\lambda_{i_k},\\lambda_{i_{k+1}})$ with $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$ is also $k$-bad.\n\nFor clarity, we do not organize the left-hand side by summing over all tuples $(\\lambda_{i_1},\\ldots,\\lambda_{i_{k+1}})$ lying in a specified region of $(0,\\infty)^{k+1}$. Instead, we fix upper bounds $\\tau_j$ for $\\lambda_{i_j}$, $1 \\leq j \\leq k$, and then estimate the tail sum over those $i_{k+1}$ with $\\lambda_{i_{k+1}} \\geq (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$. The proof uses the method of non-stationary phase.\nWe record a flat-torus example in Section \\ref{sec:flat-torus-example} showing that the frequency threshold in Theorem \\ref{main theorem k+1-gon-bad} is essentially sharp.\n\nOur second and main theorem concerns the contribution from $k$-good vectors $\\tau \\in (0,\\infty)^{k+1}$.\n Following \\cite{MR4376455}, we introduce the joint spectral measure\n\\begin{equation}\\label{joint spectral measure}\n\t\\mu = \\sum_{i_1,i_2,\\dots,i_{k+1}} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\, \\delta_{(\\lambda_{i_1},\\lambda_{i_2},\\dots,\\lambda_{i_{k+1}})}\n\\end{equation}\non ${\\mathbb R}^{k+1}$, where $\\delta$ denotes the Dirac measure. Thus $\\mu$ is a weighted sum of point masses, with weights given by the squared magnitudes of the coefficients $\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle$. Our second theorem describes the mass of $\\mu$ carried by frequencies $(\\lambda_{i_1}, \\lambda_{i_2}, \\dots, \\lambda_{i_{k+1}})$ lying in $k$-good cones.\n\nBefore stating the result, we introduce some terminology by recalling the definition and basic properties of the Leray density. Let $F : {\\mathbb R}^n \\to {\\mathbb R}^d$ be a smooth function, and let $y_0 \\in {\\mathbb R}^d$ be such that the differential $dF$ has full rank at every point of the level set $F^{-1}(y_0)$. By the inverse function theorem, $F^{-1}(y_0)$ is a smooth $(n-d)$-dimensional submanifold. Choose local coordinates $(z,z')$ in a neighborhood of a point of $F^{-1}(y_0)$ so that $z$ parametrizes $F^{-1}(y_0)$ and $\\det(\\partial F/\\partial z') \\neq 0$. The Leray density on $F^{-1}(y_0)$ is defined by\n\\begin{equation}\\label{Leray density definition}\n\td_F = \\left|\\det \\frac{\\partial F}{\\partial z'}\\right|^{-1} \\, dz,\n\\end{equation}\nand is independent of the choice of complementary coordinates $z'$. The corresponding Leray volume is\n\\[\n\t\\operatorname{vol} F^{-1}(y_0) = \\int_{F^{-1}(y_0)} d_F.\n\\]\n\nGiven a continuous function $f$ defined on a neighborhood of $F^{-1}(y_0)$, the integral of $f$ over $F^{-1}(y_0)$ with respect to the Leray measure can be expressed using the Dirac distribution as\n\\[\n\t\\int_{F^{-1}(y_0)} f \\, d_F = \\int_{{\\mathbb R}^n} f(x) \\delta(F(x) - y_0) \\, dx.\n\\]\n\nFor the purposes of this paper, we set $F : {\\mathbb R}^{kn} \\to {\\mathbb R}^{k+1}$ by\n\\begin{equation}\\label{def F}\n\tF(\\xi_1,\\xi_2,\\dots,\\xi_k) = \\bigl(|\\xi_1|,|\\xi_2|,\\dots,|\\xi_k|,\\ |\\xi_1 + \\cdots + \\xi_k|\\bigr).\n\\end{equation}\nThen the level set $F^{-1}(\\tau)$ consists of $k$-tuples $(\\xi_1,\\xi_2,\\dots,\\xi_k)$ satisfying\n\\[\n|\\xi_j|=\\tau_j \\ \\ (1\\le j\\le k)\n\\qquad \\text{and} \\qquad\n|\\xi_1+\\cdots+\\xi_k|=\\tau_{k+1}.\n\\]\nSuch a $k$-tuple determines a closed $(k+1)$-gon in ${\\mathbb R}^n$ with side lengths $\\tau_1,\\tau_2,\\dots,\\tau_{k+1}$, where the final side is the closing vector $-(\\xi_1+\\cdots+\\xi_k)$.\nThe next theorem provides a description of the concentration of $\\mu$ in $k$-good cones.\n\n\\begin{theorem}\\label{main theorem k+1-good}\nLet $\\rho$ be a Schwartz function on ${\\mathbb R}^{k+1}$ with $\\int \\rho = 1$. Assume that\n\\[\n\t\\operatorname{supp} \\hat \\rho \\subset (-\\operatorname{inj} M, \\operatorname{inj} M)^{k+1}.\n\\]\nLet $\\Gamma$ be a $k$-good cone. Then for $\\tau = (\\tau_1, \\tau_2, \\dots, \\tau_{k+1}) \\in \\Gamma$,\n\\[\n\t\\rho * \\mu(\\tau) = (2\\pi)^{-kn} \\operatorname{vol} M \\operatorname{vol} F^{-1}(\\tau) + O(|\\tau|^{k(n-1)-2}),\n\\]\nwhere the constants implicit in the big-$O$ notation depend on $M$, $\\Gamma$, and $\\rho$ but not on $\\tau$.\n\\end{theorem}\n\nSince $\\rho$ is rapidly decaying and $\\operatorname{supp} \\hat \\rho$ is contained in $(-\\operatorname{inj} M, \\operatorname{inj} M)^{k+1}$, the convolution $\\rho * \\mu$ provides a smoothed version of $\\mu$ that captures the mass of $\\mu$ near $\\tau$. In particular, Theorem \\ref{main theorem k+1-good} gives an asymptotic description of this smoothed local mass for $\\tau \\in \\Gamma$. Next, we determine the growth rate of $\\operatorname{vol} F^{-1}(\\tau)$ as $|\\tau| \\to \\infty$. When $k=2$, this quantity is related to the area of the corresponding triangle \\cite{MR4376455}. For $k>2$, the shape of a $(k+1)$-gon with fixed side lengths is not uniquely determined (Figures \\ref{2} and \\ref{3}), so it is not sufficient to consider only its planar area. We prove in Proposition \\ref{Scaling of F} that, under our assumptions, $\\operatorname{vol} F^{-1}(\\tau)$ is of order $|\\tau|^{k(n-1)-1}$, so the remainder term $O(|\\tau|^{k(n-1)-2})$ is indeed lower order. As in \\cite{MR4376455}, the leading term can be interpreted in terms of the volume of the configuration space of $(k+1)$-gons with side lengths prescribed by $\\tau$.\n\\subsection*{Organization}\nIn Section~2 we analyze the configuration space that appears in the main term of Theorem~\\ref{main theorem k+1-good}, and in particular prove the scaling law for the relevant fiber volume. Section~3 contains the proof of Theorem~\\ref{main theorem k+1-gon-bad}, establishing rapid decay of the high-frequency tail in the regime where the $(k+1)$-gon inequality fails, via spectral projectors and a non-stationary phase argument. Section~4 proves Theorem~\\ref{main theorem k+1-good} by expressing the Fourier transform of the joint spectral measure in terms of Fourier integral operators and computing the associated symbolic data using clean composition and half-density calculus. Finally, Section~\\ref{sec:flat-torus-example} presents a flat-torus example showing that the frequency cutoff in Theorem~\\ref{main theorem k+1-gon-bad} is essentially sharp.\n\n\\subsection*{Acknowledgement} This project is supported by the National Key R\\&D Program of China under Grant No. 2022YFA1007200, the Natural Science Foundation of China under Grant No. 12571107, and the Zhejiang Provincial Natural Science Foundation of China under Grant No. LR25A010001.", "sketch": "The introduction explains that the condition $\\lambda_{i_{k+1}} \\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$ “forces the $(k+1)$-tuple $(\\tau_1,\\ldots,\\tau_k,\\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others,” hence any $(\\lambda_{i_1},\\ldots,\\lambda_{i_k},\\lambda_{i_{k+1}})$ with $\\lambda_{i_j}\\le \\tau_j$ is also $k$-bad.\n\nRather than summing over a region in $(0,\\infty)^{k+1}$, the argument is organized by “fix[ing] upper bounds $\\tau_j$ for $\\lambda_{i_j}$, $1\\le j\\le k$, and then estimat[ing] the tail sum over those $i_{k+1}$ with $\\lambda_{i_{k+1}} \\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$.” The proof of Theorem~\\ref{main theorem k+1-gon-bad} then “uses the method of non-stationary phase,” and (per the Organization paragraph) establishes “rapid decay of the high-frequency tail in the regime where the $(k+1)$-gon inequality fails, via spectral projectors and a non-stationary phase argument.” A flat-torus example is recorded showing the frequency threshold is “essentially sharp.”", "expanded_sketch": "The introduction explains that the condition $\\lambda_{i_{k+1}} \\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$ “forces the $(k+1)$-tuple $(\\tau_1,\\ldots,\\tau_k,\\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others,” hence any $(\\lambda_{i_1},\\ldots,\\lambda_{i_k},\\lambda_{i_{k+1}})$ with $\\lambda_{i_j}\\le \\tau_j$ is also $k$-bad.\n\nRather than summing over a region in $(0,\\infty)^{k+1}$, the argument is organized by “fix[ing] upper bounds $\\tau_j$ for $\\lambda_{i_j}$, $1\\le j\\le k$, and then estimat[ing] the tail sum over those $i_{k+1}$ with $\\lambda_{i_{k+1}} \\ge (1+\\epsilon)(\\tau_1+\\cdots+\\tau_k)$.” To prove the main theorem, the proof then “uses the method of non-stationary phase,” and (per the Organization paragraph) establishes “rapid decay of the high-frequency tail in the regime where the $(k+1)$-gon inequality fails, via spectral projectors and a non-stationary phase argument.” A flat-torus example is recorded showing the frequency threshold is “essentially sharp.”", "expanded_theorem": "\\label{main theorem k+1-gon-bad}\n\tFor each $\\epsilon>0$ and integer $N > 0$, there exists a constant $C_{\\epsilon, N}$ such that for any positive numbers $\\tau_{1},\\tau_{2},\\ldots,\\tau_{k}$, we have\n\t\\[\n\t\t\\sum_{\\lambda_{i_{k+1}}\\geq (1+\\epsilon)(\\tau_1+\\tau_2+\\dots+\\tau_k)} \\bigl|\\langle e_{i_1} e_{i_2} \\cdots e_{i_k}, e_{i_{k+1}} \\rangle\\bigr|^2 \\leq C_{\\epsilon, N}(\\tau_1+\\tau_2+\\dots+\\tau_k)^{-N},\n\t\\]\n\twhere $\\lambda_{i_j} \\leq \\tau_j$ for $1 \\leq j \\leq k$.", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let \\(\\{e_i\\}\\) be an \\(L^2\\)-orthonormal family of eigenfunctions with corresponding eigenvalues (frequencies) \\(\\lambda_i\\), and write \\(\\langle f,g\\rangle\\) for the \\(L^2\\) inner product. Fix an integer \\(k\\ge 1\\). For positive numbers \\(\\tau_1,\\dots,\\tau_k\\), suppose the indices \\(i_1,\\dots,i_k\\) satisfy \\(\\lambda_{i_j}\\le \\tau_j\\) for each \\(1\\le j\\le k\\). Under these assumptions, which quantitative estimate holds for the tail of the harmonic expansion coefficients of the product \\(e_{i_1}e_{i_2}\\cdots e_{i_k}\\)?", "correct_choice": {"label": "A", "text": "For every \\(\\epsilon>0\\) and every integer \\(N>0\\), there exists a constant \\(C_{\\epsilon,N}\\) such that\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\tau_2+\\cdots+\\tau_k)} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon,N}(\\tau_1+\\tau_2+\\cdots+\\tau_k)^{-N}.\n\\]"}, "choices": [{"label": "B", "text": "For every \\(\\epsilon>0\\) and every integer \\(N>0\\), there exists a constant \\(C_{\\epsilon,N}\\) such that\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (\\tau_1+\\tau_2+\\cdots+\\tau_k)+\\epsilon} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon,N}(\\tau_1+\\tau_2+\\cdots+\\tau_k)^{-N}.\n\\]"}, {"label": "C", "text": "For every \\(\\epsilon>0\\), there exists a constant \\(C_{\\epsilon}\\) such that\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\tau_2+\\cdots+\\tau_k)} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon}.\n\\]"}, {"label": "D", "text": "There exists an integer \\(N>0\\) such that for every \\(\\epsilon>0\\) there is a constant \\(C_{\\epsilon,N}\\) with\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)(\\tau_1+\\tau_2+\\cdots+\\tau_k)} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon,N}(\\tau_1+\\tau_2+\\cdots+\\tau_k)^{-N}\n\\]\nfor all positive \\(\\tau_1,\\dots,\\tau_k\\) and all indices \\(i_1,\\dots,i_k\\) with \\(\\lambda_{i_j}\\le \\tau_j\\)."}, {"label": "E", "text": "For every \\(\\epsilon>0\\) and every integer \\(N>0\\), there exists a constant \\(C_{\\epsilon,N}\\) such that\n\\[\n\\sum_{\\lambda_{i_{k+1}}\\ge (1+\\epsilon)\\max\\{\\tau_1,\\tau_2,\\dots,\\tau_k\\}} \\bigl|\\langle e_{i_1}e_{i_2}\\cdots e_{i_k},\\, e_{i_{k+1}}\\rangle\\bigr|^2\n\\le C_{\\epsilon,N}(\\tau_1+\\tau_2+\\cdots+\\tau_k)^{-N}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "non-stationary phase", "tampered_component": "multiplicative tail threshold replaced by additive offset", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped rapid-decay power saving in \\((\\tau_1+\\cdots+\\tau_k)\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "quantifier order on N weakened from arbitrary N to existence of one N", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "sum-of-sides bad-region threshold replaced by max-side threshold", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the exact estimate or its key features such as the multiplicative threshold \u001d(1+\u001b)\u001d and rapid decay for every \u001dN\u001d. It only asks which estimate holds, so there is no explicit answer leakage."}, "TAS": {"score": 1, "justification": "The item is very close to theorem recall: it presents the full setup and asks for the correct quantitative conclusion. However, the options differ in meaningful ways (threshold scale, power decay, quantifier order), so it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "Selecting the correct choice requires some reasoning about which formulation is strongest and correctly quantified, but it mainly tests recognition of the theorem's exact statement rather than substantial derivation or synthesis."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: additive versus multiplicative cutoff, loss of rapid decay, weakened quantifiers, and replacing the sum by a max. These align with realistic failure modes and are clearly distinct."}, "total_score": 6, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and little answer leakage, though it leans more toward precise recall than deep generative reasoning."}}
{"id": "2602.04777v1", "paper_link": "http://arxiv.org/abs/2602.04777v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm:main_asymmetric}\n\t\tLet $k >\\frac 1 2 \\alpha_N$, where $\\alpha_i$ is defined by \\eqref{def:alpha_i} for $i=1,\\dots, N$, and assume that $\\Sigma$ is a $k$-symmetric Riemann surface with smooth boundary. Suppose the potential functions $V_1, \\dots, V_N$ are $\\fR_k$-invariant, and \n\t\t\\[\n\t\t\\Sigma_0:=\\left\\{x\\in \\Sigma: \\fR_{k}^i(x)=x \\text{ for any } i=1,2,3,\\dots\\right\\}. \n\t\t\\]\n\t\tThen for any $m$ distinct points   $\\xi^*_1, \\dots, \\xi^*_m $ in  $ \\Sigma_0$, there exists a family of solutions $\\bu_\\varepsilon = (u_{1,\\varepsilon}, \\dots, u_{N,\\varepsilon})$ to the  Toda system \\eqref{eq:toda_mfN} such that:\n\t\t\\begin{itemize}\n\t\t\t\\item[i)] $\\bu_\\varepsilon$ blows up exactly at the points $\\xi^*_1, \\dots, \\xi^*_m$ as $\\varepsilon \\to 0$;\n\t\t\t\\item [ii)]the parameters $\\rho^\\varepsilon = (\\rho_1^\\varepsilon, \\dots, \\rho_N^\\varepsilon)$ satisfy\n\t\t\t\\[\n\t\t\t\\rho^\\varepsilon \\to (2\\alpha_1 \\pi m,\\dots,  2\\alpha_N \\pi m);\n\t\t\t\\]\n\t\t\t\\item[iii)] for each $i = 1, \\dots, N$, the following  weak-$*$ convergence holds:\n\t\t\t\\[\n\t\t\t\\rho_i^\\varepsilon \\frac{V_i e^{u_{i,\\varepsilon}}}{\\int_\\Sigma V_i e^{u_{i,\\varepsilon}}  dv_g \\, } dv_g   \\stackrel{*}{\\rightharpoonup}  \\sum_{j=1}^m 2\\alpha_i \\pi \\delta_{\\xi^*_j},  \\text{ as } \\varepsilon \\to 0,\n\t\t\t\\]\n\t\t\\end{itemize}\n\t\twhere $\\delta_{x}$ is the Dirac measure on $\\Sigma$ concentrated at the point $x.$", "start_pos": 18076, "end_pos": 19459, "label": "thm:main_asymmetric"}, "ref_dict": {"eq:toda_mfN": "\\begin{equation}\\label{eq:toda_mfN}\n\t\t\\begin{cases}\n\t\t\t-\\Delta_g u_1 = \\sum_{j=1}^N a_{ij}\\rho_j\\left( \\frac{V_je^{u_j}}{\\int_{\\Sigma} V_je^{u_j} dv_g}-\\frac 1 {|\\Sigma|_g} \\right)    & \\text{ in } \\intsigma\\\\\n\t\t\t\\partial_{\\nu_g} u_1=\\dots= \\partial_{\\nu_g} u_N= 0 & \\text{ on } \\partial\\Sigma\n\t\t\\end{cases},\n\t\\end{equation}", "def:alpha_i": "\\begin{equation}\n\t\t\\label{def:alpha_i}\n\t\t\\begin{split}\n\t\t\t\\alpha_i&=2i \n\t\t\t\\text{ and }\n\t\t\t\\alpha_N= 2-2(N-1)a_{N N-1}=\\begin{cases}\n\t\t\t\t2 N & \\text{ for }  \\mathbf{A}_N\\text{ or } \\mathbf{B}_N \\\\\n\t\t\t\t4N-2 & \\text{ for } \\mathbf{C}_N\\\\\n\t\t\t\t8& \\text{ for } \\mathbf{G}_2\n\t\t\t\\end{cases},\n\t\t\\end{split} \n\t\\end{equation}", "thm:main_asymmetric": "\\begin{theorem}\\label{thm:main_asymmetric}\n\t\tLet $k >\\frac 1 2 \\alpha_N$, where $\\alpha_i$ is defined by \\eqref{def:alpha_i} for $i=1,\\dots, N$, and assume that $\\Sigma$ is a $k$-symmetric Riemann surface with smooth boundary. Suppose the potential functions $V_1, \\dots, V_N$ are $\\fR_k$-invariant, and \n\t\t\\[\n\t\t\\Sigma_0:=\\left\\{x\\in \\Sigma: \\fR_{k}^i(x)=x \\text{ for any } i=1,2,3,\\dots\\right\\}. \n\t\t\\]\n\t\tThen for any $m$ distinct points   $\\xi^*_1, \\dots, \\xi^*_m $ in  $ \\Sigma_0$, there exists a family of solutions $\\bu_\\varepsilon = (u_{1,\\varepsilon}, \\dots, u_{N,\\varepsilon})$ to the  Toda system \\eqref{eq:toda_mfN} such that:\n\t\t\\begin{itemize}\n\t\t\t\\item[i)] $\\bu_\\varepsilon$ blows up exactly at the points $\\xi^*_1, \\dots, \\xi^*_m$ as $\\varepsilon \\to 0$;\n\t\t\t\\item [ii)]the parameters $\\rho^\\varepsilon = (\\rho_1^\\varepsilon, \\dots, \\rho_N^\\varepsilon)$ satisfy\n\t\t\t\\[\n\t\t\t\\rho^\\varepsilon \\to (2\\alpha_1 \\pi m,\\dots,  2\\alpha_N \\pi m);\n\t\t\t\\]\n\t\t\t\\item[iii)] for each $i = 1, \\dots, N$, the following  weak-$*$ convergence holds:\n\t\t\t\\[\n\t\t\t\\rho_i^\\varepsilon \\frac{V_i e^{u_{i,\\varepsilon}}}{\\int_\\Sigma V_i e^{u_{i,\\varepsilon}}  dv_g \\, } dv_g   \\stackrel{*}{\\rightharpoonup}  \\sum_{j=1}^m 2\\alpha_i \\pi \\delta_{\\xi^*_j},  \\text{ as } \\varepsilon \\to 0,\n\t\t\t\\]\n\t\t\\end{itemize}\n\t\twhere $\\delta_{x}$ is the Dirac measure on $\\Sigma$ concentrated at the point $x.$\n\t\\end{theorem}", "eq:A_2_Toda": "\\begin{equation}\\label{eq:A_2_Toda}\n\t\t\\left\\{\n\t\t\\begin{aligned}\n\t\t\t-\\Delta_g u_1 &= 2\\rho_1\\left( \\frac{V_1 e^{u_1}}{\\int_{\\Sigma} V_1 e^{u_1} \\, dv_g} - 1\\right) - \\rho_2\\left( \\frac{V_2 e^{u_2}}{\\int_{\\Sigma} V_2 e^{u_2} \\, dv_g} - 1\\right) & &\\text{ in } \\intsigma, \\\\\n\t\t\t-\\Delta_g u_2 &= 2\\rho_2\\left( \\frac{V_2 e^{u_2}}{\\int_{\\Sigma} V_2 e^{u_2} \\, dv_g} - 1\\right) - \\rho_1\\left( \\frac{V_1 e^{u_1}}{\\int_{\\Sigma} V_1 e^{u_1} \\, dv_g} - 1\\right) && \\text{ in } \\intsigma, \\\\\n\t\t\t\\partial_{\\nu_g} u_1 &= \\partial_{\\nu_g} u_2 = 0 & &\\text{ on } \\partial \\Sigma.\n\t\t\\end{aligned}\n\t\t\\right.\n\t\\end{equation}"}, "pre_theorem_intro_text_len": 11833, "pre_theorem_intro_text": "\\label{sec:1}\n\tGiven a Riemann surface $\n\t(\\Sigma,g)$ with smooth boundary \n\t$\\partial \\Sigma$ (possible to be empty) and a positive integer $N\\geq 2$, we consider the following  Toda system equipped with homogeneous Neumann boundary conditions\n\t\\begin{equation}\\label{eq:toda_mfN}\n\t\t\\begin{cases}\n\t\t\t-\\Delta_g u_1 = \\sum_{j=1}^N a_{ij}\\rho_j\\left( \\frac{V_je^{u_j}}{\\int_{\\Sigma} V_je^{u_j} dv_g}-\\frac 1 {|\\Sigma|_g} \\right)    & \\text{ in } {\\mathring\\Si}\\\\\n\t\t\t\\partial_{\\nu_g} u_1=\\dots= \\partial_{\\nu_g} u_N= 0 & \\text{ on } \\partial\\Sigma\n\t\t\\end{cases},\n\t\\end{equation}\n\twhere \n\t${\\mathring\\Si}=\\Sigma\\setminus\\partial\\Sigma$ is the interior of $\\Sigma$, $\\nu_g$  is the outward unit normal vector on $\\partial\\Sigma$, $\\Delta_g$ is the Laplace-Beltrami operator, $dv_g$  is the Riemannian volume element, $|\\Sigma|_g=\\int_{\\Sigma} dv_g$ denotes the total volume of $\\Sigma$, \t$\\rho_i$ is a non-negative parameter,  $V_i:\\Sigma\\rightarrow \\mathbb{R}$ is smooth positive functions, and the matrix \n\t$(a_{ij})$ is one of the Cartan matrices of a general simple Lie algebra with rank $N$. \tFor simplicity, we normalize the volume of $\\Sigma$, i.e., $|\\Sigma|_g =1$.\n\n\tIn this paper, we consider the case where the Cartan matrix \n\t$(a_{ij})$ takes one of the following forms:\n\t\\[\n\t\\mathbf{A}_N =\n\t\\begin{pmatrix}\n\t\t2 & -1 & 0 & \\dots & 0 \\\\\n\t\t-1 & 2 & -1 & \\dots & 0 \\\\\n\t\t\\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\\\\n\t\t0 & \\dots & -1 & 2 & -1 \\\\\n\t\t0 & \\dots & 0 & -1 & 2\n\t\\end{pmatrix},\n\t\\quad\n\t\\mathbf{B}_N =\n\t\\begin{pmatrix}\n\t\t2 & -1 & 0 & \\dots & 0 \\\\\n\t\t-1 & 2 & -1 & \\dots & 0 \\\\\n\t\t\\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\\\\n\t\t0 & \\dots & -1 & 2 & -2 \\\\\n\t\t0 & \\dots & 0 & -1 & 2\n\t\\end{pmatrix},\n\t\\]\n\t\\[\n\t\\mathbf{C}_N =\\left(\n\t\\begin{matrix}\n\t\t2 & -1 & 0 & \\dots & 0 \\\\\n\t\t-1 & 2 & -1 & \\dots & 0 \\\\\n\t\t\\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\\\\n\t\t0 & \\dots & -1 & 2 & -1 \\\\\n\t\t0 & \\dots & 0 & -2 & 2\n\t\\end{matrix}\\right), \\text{ and }\n\t\\mathbf{G}_2:= \\begin{pmatrix}\n\t\t2&-1\\\\\n\t\t-3&2\n\t\\end{pmatrix}.\\]\n\tAmong these, the system \\eqref{eq:toda_mfN} associated with the Cartan matrix $\\mathbf A_N$ is usually referred to as the $SU(N+1)$ Toda system. \n\tMoreover, the arguments developed in this paper can be extended, with only minor modifications, to Toda systems associated with more general Cartan matrices. \n\tFor simplicity of presentation, we restrict ourselves to the Cartan matrices considered above.\n\n\tToda system \\eqref{eq:toda_mfN} has been extensively studied over the past few decades due to its deep connections with various areas in geometry and physics. In differential geometry, it is related to the theory of harmonic maps and holomorphic curves into complex projective spaces $\\mathbb{C}\\mathbb{P}^N$ (see~\\cite{guest1997harmonic,bolton1988conformal,doliwa1997holomorphic,chern1987harmonic,bolton1997geometrical} and the references therein). In particular, when $\\Sigma=\\mathbb{S}^2$, the solution space of the $SU(3)$ Toda system coincides with the space of  holomorphic curves of $\\mathbb{S}^2$ into $\\mathbb{C}\\mathbb{P}^2$~\\cite{LWY2021Cla}. In mathematical physics,  Toda systems arise naturally in the study of non-abelian Chern–Simons gauge theories (see~\\cite{dunne1991self,dunne1995self,nolasco1999double,nolasco2000vortex,yang1999relativistic,yang2001solitons}, and the reference therein).\n\n\tSince the solution space of  \\eqref{eq:toda_mfN} is invariant under the addition of constants, we consider weak solutions in the subspace of the $N$-fold product of the Sobolev space $H^1(\\Sigma)$ with zero average. Specifically, we define\n\t$$\n\t\\mathcal{H} := \\underbrace{\\oH \\times \\dots \\times \\oH}_{N \\text{ times }},\n\t$$\n\twhere\n\t$\n\t\\oH := \\left\\{ u \\in H^1(\\Sigma) : \\int_\\Sigma u \\, dv_g = 0 \\right\\}.\n\t$\n\n\tToda system has been widely studied in planar domains with Dirichlet boundary conditions and on closed Riemann surfaces. \n\tThese studies focus on various aspects of the system, including existence, uniqueness, and blow-up behavior. For existence results, we refer the reader to~\\cite{jostandwang2001,jost_lin_wang2006,li2005solutions,battaglia_jevnikar_ruiz2015,Malchiodi2007SomeER,jevnikar_kallel_Malchiodi2015} and references therein. For uniqueness and non-degeneracy results, see~\\cite{Bartolucci2023Non,LWY2021Cla}, and for blow-up analysis, we refer to~\\cite{jost_lin_wang2006,LWZ2016Local,LWZ2025Cla,lee_degree_2018,musso_new_2016,D'aprile_asymmetric_2016}, among others.\n\n\tThe  Toda system under Neumann boundary conditions on Riemann surfaces has been much less studied. In~\\cite{zhu2011solutions}, X. B. Zhu constructs solutions to the $SU(3)$ Toda system as minimizers of the corresponding Euler-Lagrange functional in the case $\\rho_i = 2\\pi$ for all $i = 1, 2$.  \n\n\tLet the local limit mass $\\sigma_i(x')$ be defined by\n\t\\[\n\t\\sigma_i(x') = \\lim_{n \\to \\infty} \\lim_{r \\to 0} \\int_{\\substack{x \\in \\Sigma, d_g(x,x') < r}} \\frac{\\rho_i^n V_i e^{u_i^n}}{\\int_\\Sigma V_i e^{u_i^n} dv_g} \\, dv_g,\n\t\\]\n\tfor $i = 1, \\ldots, N$, where $d_g(\\cdot, \\cdot)$ denotes the geodesic distance on $\\Sigma$ with respect to the metric $g$.\n\n\tWe are particularly interested in   the $SU(3)$ Toda system:\n\t\\begin{equation}\\label{eq:A_2_Toda}\n\t\t\\left\\{\n\t\t\\begin{aligned}\n\t\t\t-\\Delta_g u_1 &= 2\\rho_1\\left( \\frac{V_1 e^{u_1}}{\\int_{\\Sigma} V_1 e^{u_1} \\, dv_g} - 1\\right) - \\rho_2\\left( \\frac{V_2 e^{u_2}}{\\int_{\\Sigma} V_2 e^{u_2} \\, dv_g} - 1\\right) & &\\text{ in } {\\mathring\\Si}, \\\\\n\t\t\t-\\Delta_g u_2 &= 2\\rho_2\\left( \\frac{V_2 e^{u_2}}{\\int_{\\Sigma} V_2 e^{u_2} \\, dv_g} - 1\\right) - \\rho_1\\left( \\frac{V_1 e^{u_1}}{\\int_{\\Sigma} V_1 e^{u_1} \\, dv_g} - 1\\right) && \\text{ in } {\\mathring\\Si}, \\\\\n\t\t\t\\partial_{\\nu_g} u_1 &= \\partial_{\\nu_g} u_2 = 0 & &\\text{ on } \\partial \\Sigma.\n\t\t\\end{aligned}\n\t\t\\right.\n\t\\end{equation}\n\tBy the classical blow-up analysis, for any family of blow-up solutions $u^n := (u_1^n, u_2^n)$ to the $SU(3)$ Toda system with parameters $\\rho^n := (\\rho_1^n, \\rho_2^n) \\to (\\rho_1,\\rho_2)$, the set of blow-up points is given by\n\t\\[\n\t\\mathcal{S} := \\Big \\{ x \\in \\Sigma : \\exists x_n \\to x \\text{ such that } \\max_i \\Big(u_i^n(x_n) -\\int_{\\Sigma} V_i e^{u_i^n} dv_g\\Big)  \\to \\infty \\Big\\}.\n\t\\]\n\tThis set is finite and consists of points with local limit masses $(\\sigma_1(x), \\sigma_2(x))$ that take  one of the following values:\n\t\\[\n\t\\Big\\{\n\t\\Big( 0, \\frac{1}{2} \\varrho(x) \\Big), \\Big( \\frac{1}{2} \\varrho(x), 0 \\Big), \\Big( \\frac{1}{2} \\varrho(x), \\varrho(x) \\Big), \\Big( \\varrho(x), \\frac{1}{2} \\varrho(x) \\Big), \\Big( \\varrho(x), \\varrho(x) \\Big)\n\t\\Big\\},\n\t\\]\n\tfor $x \\in \\mathcal{S}$, where $\\varrho(x) = 8\\pi$ if $x \\in \\Sigma$ and $\\varrho(x) = 4\\pi$ if $x \\in \\partial \\Sigma$.\n\n\tThe above result was established in~\\cite{jostandwang2001,jost_lin_wang2006} for interior blow-up points. The boundary case is expected to follow by a similar blow-up analysis, with half-mass contributions arising from boundary effects. Since this lies outside the scope of the present paper, we omit further details.\n\n\tAccording to the value of the local limit masses, we classify the blow-up phenomena of the $SU(3)$ Toda system around the blow-up points into the following three scenarios:\n\t\\begin{itemize}\n\t\t\\item \\textbf{Partial blow-ups}: $\\left(0, \\frac{1}{2} \\varrho(x)\\right), \\left(\\frac{1}{2} \\varrho(x), 0\\right)$;\n\t\t\\item \\textbf{Asymmetric blow-ups}: $ \\left(\\frac{1}{2} \\varrho(x), \\varrho(x)\\right), \\left(\\varrho(x), \\frac{1}{2} \\varrho(x)\\right)$;\n\t\t\\item \\textbf{Full blow-ups}: $\\left(\\varrho(x), \\varrho(x)\\right)$.\n\t\\end{itemize}\n\n\tNext, we introduce works related to the existence of blow-up solutions for the $SU(3)$ Toda system. \n\n\tPartial blow-up solutions, under a  non-degeneracy condition, have been constructed in various settings: on planar domains under Dirichlet boundary condition in \\cite{DAprile_Pistoia_Ruiz2015}, on closed surfaces in \\cite{lee_degree_2018}, and on Riemann surfaces with boundary under Neumann boundary condition   in \\cite{ABH2024Partial}.\n\n\tFor asymmetric blow-ups, W. Ao and L. Wang in \\cite{ao2014new} introduce a family of blow-up solutions for Toda systems with Dirichlet boundary conditions on a unit ball centered at the origin, exhibiting a single blow-up point at the center. Musso, Pistoia, and J. Wei introduce a so-called ``$k$-symmetric'' property for the planar domain  and construct blow-up solutions for a $SU(N+1)$ Toda system, applying singular perturbation methods in \\cite{musso_new_2016}.  Meanwhile, D'Aprile, Pistoia, and Ruiz obtain a similar result to $SU(3)$ Toda system for planar domains with Dirichlet boundary conditions as $\\rho_1 \\to 8\\pi$, while keeping the fixed parameter $\\rho_2 \\in (4\\pi, 8\\pi)$ in \\cite{D'aprile_asymmetric_2016}. The ``$k$-symmetric'' condition is a technical assumption introduced to ensure non-degeneracy of the limiting linearized problem.\n\n\tFor full blow-ups, the special case where $V_1 = V_2$, $u_1 = u_2$, and $\\rho_1 = \\rho_2$ reduces the system to a mean field equation. Blow-up solutions for this equation have been constructed in \\cite{baraket_construction_1997, Esposito2005} for bounded domains in $\\mathbb{R}^2$, in \\cite{Bartolucci2020, Esposito2014singular, figueroa2022bubbling} for closed Riemann surfaces, and in \\cite{HBA2024} for Riemann  surfaces with boundary.  For the general case, the full blow-up scenario remains largely unresolved. C.-S. Lin, J. Wei, and C. Zhao   analyzed fully blow-up solutions of the $SU(3)$ Toda system  and derived  necessary conditions for their existence, highlighting the substantial difficulties in constructing such solutions in \\cite{LWZ12}. Further progress on fully blowup  solutions has been made in subsequent works, see for instance \\cite{Ao16, LWZ2016Local,Zhang20} and the references therein.\n\n\tTo extend the results in \\cite{musso_new_2016, D'aprile_asymmetric_2016} to the case of Toda systems of rank $N$ equipped with homogeneous Neumann boundary conditions on Riemann surfaces with boundary, this paper aims to construct blow-up solutions of the  Toda system \\eqref{eq:toda_mfN} with $\\rho_i \\to 2\\alpha_i \\pi m$  for $i = 1, \\dots, N$ on ``$k$-symmetric'' surfaces with $k > \\frac{1}{2} \\alpha_N $ via singular perturbation methods, where \n\t\\[ \\alpha_i=2i \\text{ for }i=1,\\dots, N-1, \\text{ and }\\alpha_N= \\begin{cases}\n\t\t2 N & \\text{ for }  \\mathbf{A}_N\\text{ or } \\mathbf{B}_N \\\\\n\t\t4N-2 & \\text{ for } \\mathbf{C}_N\\\\\n\t\t8 & \\text{ for } \\mathbf{G}_2\n\t\\end{cases}.\\]\n\tSee  \\eqref{def:alpha_i}  in Section \\ref{sec:app} for more details.\n\n\tWe denote by $O(3)$,  \n\tthe orthogonal group of degree \n\t$3$ acting on  $\\mathbb{R}^3$. \n\tFor any $k \\in \\mathbb{N}_+$, we define\n\t\\begin{equation*}\n\t\t\\fR_{k} =\n\t\t\\begin{pmatrix}\n\t\t\t\\cos (2\\pi/k) & \\sin (2\\pi/k) & 0 \\\\\n\t\t\t-\\sin(2\\pi/k) & \\cos (2\\pi/k) & 0 \\\\\n\t\t\t0 & 0 & 1\n\t\t\\end{pmatrix}\n\t\t\\in O(3),\n\t\\end{equation*}\n\twhich represents a rotation by angle $\\frac{2\\pi} k$ around the $z$-axis.  \n\n\t\\begin{definition}\\label{def:surface}\n\t\tLet $\\Sigma$ be a Riemann surface. For an integer $k\\geq 1$, we say $\\Sigma$ is {\\it $k$-symmetric} if \n\t\t$\\Sigma$ can be embedded in $\\mathbb{R}^3$ and is invariant under $\\fR_{k}$. \n\t\tWe say $x$ is a {\\it $k$-symmetric center} of $\\Sigma$ if $x\\in \\Sigma_0:=\\left\\{x\\in \\Sigma: \\fR_{k}^i(x)=x \\text{ for any } i=1,2,3,\\dots\\right\\}$.\n\t\\end{definition}\n\n\t\\begin{definition}\n\t\tLet $\\Sigma$ be a $k$-symmetric surface. A function $f: \\Sigma \\rightarrow \\mathbb{R}$ is said to be \\emph{$\\fR_k$-invariant} if \n\t\t$\n\t\tf(x) = f(\\fR_k x)  \\text{ for all } x \\in \\Sigma.\n\t\t$\n\t\\end{definition}\n\n\tFor technical reasons, we assume throughout that $\\Sigma$ is a $k$-symmetric surface with $k > \\frac 1 2 \\alpha_N$, which is given by \\eqref{def:alpha_i} in Section \\ref{sec:app}. Due to the smoothness of the boundary, it follows that \n\t$\n\t\\Sigma_0 \\cap \\partial \\Sigma = \\emptyset.\n\t$\n\n\tWe are now ready to state the main result of this paper.", "context": "\\label{sec:1}\n Given a Riemann surface $\n (\\Sigma,g)$ with smooth boundary \n $\\partial \\Sigma$ (possible to be empty) and a positive integer $N\\geq 2$, we consider the following Toda system equipped with homogeneous Neumann boundary conditions\n \\begin{equation}\\label{eq:toda_mfN}\n \\begin{cases}\n -\\Delta_g u_1 = \\sum_{j=1}^N a_{ij}\\rho_j\\left( \\frac{V_je^{u_j}}{\\int_{\\Sigma} V_je^{u_j} dv_g}-\\frac 1 {|\\Sigma|_g} \\right) & \\text{ in } {\\mathring\\Si}\\\\\n \\partial_{\\nu_g} u_1=\\dots= \\partial_{\\nu_g} u_N= 0 & \\text{ on } \\partial\\Sigma\n \\end{cases},\n \\end{equation}\n where \n ${\\mathring\\Si}=\\Sigma\\setminus\\partial\\Sigma$ is the interior of $\\Sigma$, $\\nu_g$ is the outward unit normal vector on $\\partial\\Sigma$, $\\Delta_g$ is the Laplace-Beltrami operator, $dv_g$ is the Riemannian volume element, $|\\Sigma|_g=\\int_{\\Sigma} dv_g$ denotes the total volume of $\\Sigma$, $\\rho_i$ is a non-negative parameter, $V_i:\\Sigma\\rightarrow \\mathbb{R}$ is smooth positive functions, and the matrix \n $(a_{ij})$ is one of the Cartan matrices of a general simple Lie algebra with rank $N$. For simplicity, we normalize the volume of $\\Sigma$, i.e., $|\\Sigma|_g =1$.\n\nWe are particularly interested in the $SU(3)$ Toda system:\n \\begin{equation}\\label{eq:A_2_Toda}\n \\left\\{\n \\begin{aligned}\n -\\Delta_g u_1 &= 2\\rho_1\\left( \\frac{V_1 e^{u_1}}{\\int_{\\Sigma} V_1 e^{u_1} \\, dv_g} - 1\\right) - \\rho_2\\left( \\frac{V_2 e^{u_2}}{\\int_{\\Sigma} V_2 e^{u_2} \\, dv_g} - 1\\right) & &\\text{ in } {\\mathring\\Si}, \\\\\n -\\Delta_g u_2 &= 2\\rho_2\\left( \\frac{V_2 e^{u_2}}{\\int_{\\Sigma} V_2 e^{u_2} \\, dv_g} - 1\\right) - \\rho_1\\left( \\frac{V_1 e^{u_1}}{\\int_{\\Sigma} V_1 e^{u_1} \\, dv_g} - 1\\right) && \\text{ in } {\\mathring\\Si}, \\\\\n \\partial_{\\nu_g} u_1 &= \\partial_{\\nu_g} u_2 = 0 & &\\text{ on } \\partial \\Sigma.\n \\end{aligned}\n \\right.\n \\end{equation}\n By the classical blow-up analysis, for any family of blow-up solutions $u^n := (u_1^n, u_2^n)$ to the $SU(3)$ Toda system with parameters $\\rho^n := (\\rho_1^n, \\rho_2^n) \\to (\\rho_1,\\rho_2)$, the set of blow-up points is given by\n \\[\n \\mathcal{S} := \\Big \\{ x \\in \\Sigma : \\exists x_n \\to x \\text{ such that } \\max_i \\Big(u_i^n(x_n) -\\int_{\\Sigma} V_i e^{u_i^n} dv_g\\Big) \\to \\infty \\Big\\}.\n \\]\n This set is finite and consists of points with local limit masses $(\\sigma_1(x), \\sigma_2(x))$ that take one of the following values:\n \\[\n \\Big\\{\n \\Big( 0, \\frac{1}{2} \\varrho(x) \\Big), \\Big( \\frac{1}{2} \\varrho(x), 0 \\Big), \\Big( \\frac{1}{2} \\varrho(x), \\varrho(x) \\Big), \\Big( \\varrho(x), \\frac{1}{2} \\varrho(x) \\Big), \\Big( \\varrho(x), \\varrho(x) \\Big)\n \\Big\\},\n \\]\n for $x \\in \\mathcal{S}$, where $\\varrho(x) = 8\\pi$ if $x \\in \\Sigma$ and $\\varrho(x) = 4\\pi$ if $x \\in \\partial \\Sigma$.\n\nTo extend the results in \\cite{musso_new_2016, D'aprile_asymmetric_2016} to the case of Toda systems of rank $N$ equipped with homogeneous Neumann boundary conditions on Riemann surfaces with boundary, this paper aims to construct blow-up solutions of the Toda system \\eqref{eq:toda_mfN} with $\\rho_i \\to 2\\alpha_i \\pi m$ for $i = 1, \\dots, N$ on ``$k$-symmetric'' surfaces with $k > \\frac{1}{2} \\alpha_N $ via singular perturbation methods, where \n \\[ \\alpha_i=2i \\text{ for }i=1,\\dots, N-1, \\text{ and }\\alpha_N= \\begin{cases}\n 2 N & \\text{ for } \\mathbf{A}_N\\text{ or } \\mathbf{B}_N \\\\\n 4N-2 & \\text{ for } \\mathbf{C}_N\\\\\n 8 & \\text{ for } \\mathbf{G}_2\n \\end{cases}.\\]\n See \\eqref{def:alpha_i} in Section \\ref{sec:app} for more details.\n\n\\begin{definition}\\label{def:surface}\n Let $\\Sigma$ be a Riemann surface. For an integer $k\\geq 1$, we say $\\Sigma$ is {\\it $k$-symmetric} if \n $\\Sigma$ can be embedded in $\\mathbb{R}^3$ and is invariant under $\\fR_{k}$. \n We say $x$ is a {\\it $k$-symmetric center} of $\\Sigma$ if $x\\in \\Sigma_0:=\\left\\{x\\in \\Sigma: \\fR_{k}^i(x)=x \\text{ for any } i=1,2,3,\\dots\\right\\}$.\n \\end{definition}\n\nFor technical reasons, we assume throughout that $\\Sigma$ is a $k$-symmetric surface with $k > \\frac 1 2 \\alpha_N$, which is given by \\eqref{def:alpha_i} in Section \\ref{sec:app}. Due to the smoothness of the boundary, it follows that \n $\n \\Sigma_0 \\cap \\partial \\Sigma = \\emptyset.\n $\n\nWe are now ready to state the main result of this paper.\n\n\\begin{equation}\n\t\t\\label{def:alpha_i}\n\t\t\\begin{split}\n\t\t\t\\alpha_i&=2i \n\t\t\t\\text{ and }\n\t\t\t\\alpha_N= 2-2(N-1)a_{N N-1}=\\begin{cases}\n\t\t\t\t2 N & \\text{ for }  \\mathbf{A}_N\\text{ or } \\mathbf{B}_N \\\\\n\t\t\t\t4N-2 & \\text{ for } \\mathbf{C}_N\\\\\n\t\t\t\t8& \\text{ for } \\mathbf{G}_2\n\t\t\t\\end{cases},\n\t\t\\end{split} \n\t\\end{equation}\n\n\\begin{equation}\\label{eq:toda_mfN}\n\t\t\\begin{cases}\n\t\t\t-\\Delta_g u_1 = \\sum_{j=1}^N a_{ij}\\rho_j\\left( \\frac{V_je^{u_j}}{\\int_{\\Sigma} V_je^{u_j} dv_g}-\\frac 1 {|\\Sigma|_g} \\right)    & \\text{ in } \\intsigma\\\\\n\t\t\t\\partial_{\\nu_g} u_1=\\dots= \\partial_{\\nu_g} u_N= 0 & \\text{ on } \\partial\\Sigma\n\t\t\\end{cases},\n\t\\end{equation}", "full_context": "\\label{sec:1}\n Given a Riemann surface $\n (\\Sigma,g)$ with smooth boundary \n $\\partial \\Sigma$ (possible to be empty) and a positive integer $N\\geq 2$, we consider the following Toda system equipped with homogeneous Neumann boundary conditions\n \\begin{equation}\\label{eq:toda_mfN}\n \\begin{cases}\n -\\Delta_g u_1 = \\sum_{j=1}^N a_{ij}\\rho_j\\left( \\frac{V_je^{u_j}}{\\int_{\\Sigma} V_je^{u_j} dv_g}-\\frac 1 {|\\Sigma|_g} \\right) & \\text{ in } {\\mathring\\Si}\\\\\n \\partial_{\\nu_g} u_1=\\dots= \\partial_{\\nu_g} u_N= 0 & \\text{ on } \\partial\\Sigma\n \\end{cases},\n \\end{equation}\n where \n ${\\mathring\\Si}=\\Sigma\\setminus\\partial\\Sigma$ is the interior of $\\Sigma$, $\\nu_g$ is the outward unit normal vector on $\\partial\\Sigma$, $\\Delta_g$ is the Laplace-Beltrami operator, $dv_g$ is the Riemannian volume element, $|\\Sigma|_g=\\int_{\\Sigma} dv_g$ denotes the total volume of $\\Sigma$, $\\rho_i$ is a non-negative parameter, $V_i:\\Sigma\\rightarrow \\mathbb{R}$ is smooth positive functions, and the matrix \n $(a_{ij})$ is one of the Cartan matrices of a general simple Lie algebra with rank $N$. For simplicity, we normalize the volume of $\\Sigma$, i.e., $|\\Sigma|_g =1$.\n\nWe are particularly interested in the $SU(3)$ Toda system:\n \\begin{equation}\\label{eq:A_2_Toda}\n \\left\\{\n \\begin{aligned}\n -\\Delta_g u_1 &= 2\\rho_1\\left( \\frac{V_1 e^{u_1}}{\\int_{\\Sigma} V_1 e^{u_1} \\, dv_g} - 1\\right) - \\rho_2\\left( \\frac{V_2 e^{u_2}}{\\int_{\\Sigma} V_2 e^{u_2} \\, dv_g} - 1\\right) & &\\text{ in } {\\mathring\\Si}, \\\\\n -\\Delta_g u_2 &= 2\\rho_2\\left( \\frac{V_2 e^{u_2}}{\\int_{\\Sigma} V_2 e^{u_2} \\, dv_g} - 1\\right) - \\rho_1\\left( \\frac{V_1 e^{u_1}}{\\int_{\\Sigma} V_1 e^{u_1} \\, dv_g} - 1\\right) && \\text{ in } {\\mathring\\Si}, \\\\\n \\partial_{\\nu_g} u_1 &= \\partial_{\\nu_g} u_2 = 0 & &\\text{ on } \\partial \\Sigma.\n \\end{aligned}\n \\right.\n \\end{equation}\n By the classical blow-up analysis, for any family of blow-up solutions $u^n := (u_1^n, u_2^n)$ to the $SU(3)$ Toda system with parameters $\\rho^n := (\\rho_1^n, \\rho_2^n) \\to (\\rho_1,\\rho_2)$, the set of blow-up points is given by\n \\[\n \\mathcal{S} := \\Big \\{ x \\in \\Sigma : \\exists x_n \\to x \\text{ such that } \\max_i \\Big(u_i^n(x_n) -\\int_{\\Sigma} V_i e^{u_i^n} dv_g\\Big) \\to \\infty \\Big\\}.\n \\]\n This set is finite and consists of points with local limit masses $(\\sigma_1(x), \\sigma_2(x))$ that take one of the following values:\n \\[\n \\Big\\{\n \\Big( 0, \\frac{1}{2} \\varrho(x) \\Big), \\Big( \\frac{1}{2} \\varrho(x), 0 \\Big), \\Big( \\frac{1}{2} \\varrho(x), \\varrho(x) \\Big), \\Big( \\varrho(x), \\frac{1}{2} \\varrho(x) \\Big), \\Big( \\varrho(x), \\varrho(x) \\Big)\n \\Big\\},\n \\]\n for $x \\in \\mathcal{S}$, where $\\varrho(x) = 8\\pi$ if $x \\in \\Sigma$ and $\\varrho(x) = 4\\pi$ if $x \\in \\partial \\Sigma$.\n\nTo extend the results in \\cite{musso_new_2016, D'aprile_asymmetric_2016} to the case of Toda systems of rank $N$ equipped with homogeneous Neumann boundary conditions on Riemann surfaces with boundary, this paper aims to construct blow-up solutions of the Toda system \\eqref{eq:toda_mfN} with $\\rho_i \\to 2\\alpha_i \\pi m$ for $i = 1, \\dots, N$ on ``$k$-symmetric'' surfaces with $k > \\frac{1}{2} \\alpha_N $ via singular perturbation methods, where \n \\[ \\alpha_i=2i \\text{ for }i=1,\\dots, N-1, \\text{ and }\\alpha_N= \\begin{cases}\n 2 N & \\text{ for } \\mathbf{A}_N\\text{ or } \\mathbf{B}_N \\\\\n 4N-2 & \\text{ for } \\mathbf{C}_N\\\\\n 8 & \\text{ for } \\mathbf{G}_2\n \\end{cases}.\\]\n See \\eqref{def:alpha_i} in Section \\ref{sec:app} for more details.\n\n\\begin{definition}\\label{def:surface}\n Let $\\Sigma$ be a Riemann surface. For an integer $k\\geq 1$, we say $\\Sigma$ is {\\it $k$-symmetric} if \n $\\Sigma$ can be embedded in $\\mathbb{R}^3$ and is invariant under $\\fR_{k}$. \n We say $x$ is a {\\it $k$-symmetric center} of $\\Sigma$ if $x\\in \\Sigma_0:=\\left\\{x\\in \\Sigma: \\fR_{k}^i(x)=x \\text{ for any } i=1,2,3,\\dots\\right\\}$.\n \\end{definition}\n\nFor technical reasons, we assume throughout that $\\Sigma$ is a $k$-symmetric surface with $k > \\frac 1 2 \\alpha_N$, which is given by \\eqref{def:alpha_i} in Section \\ref{sec:app}. Due to the smoothness of the boundary, it follows that \n $\n \\Sigma_0 \\cap \\partial \\Sigma = \\emptyset.\n $\n\nWe are now ready to state the main result of this paper.\n\n\\begin{equation}\n\t\t\\label{def:alpha_i}\n\t\t\\begin{split}\n\t\t\t\\alpha_i&=2i \n\t\t\t\\text{ and }\n\t\t\t\\alpha_N= 2-2(N-1)a_{N N-1}=\\begin{cases}\n\t\t\t\t2 N & \\text{ for }  \\mathbf{A}_N\\text{ or } \\mathbf{B}_N \\\\\n\t\t\t\t4N-2 & \\text{ for } \\mathbf{C}_N\\\\\n\t\t\t\t8& \\text{ for } \\mathbf{G}_2\n\t\t\t\\end{cases},\n\t\t\\end{split} \n\t\\end{equation}\n\n\\begin{equation}\\label{eq:toda_mfN}\n\t\t\\begin{cases}\n\t\t\t-\\Delta_g u_1 = \\sum_{j=1}^N a_{ij}\\rho_j\\left( \\frac{V_je^{u_j}}{\\int_{\\Sigma} V_je^{u_j} dv_g}-\\frac 1 {|\\Sigma|_g} \\right)    & \\text{ in } \\intsigma\\\\\n\t\t\t\\partial_{\\nu_g} u_1=\\dots= \\partial_{\\nu_g} u_N= 0 & \\text{ on } \\partial\\Sigma\n\t\t\\end{cases},\n\t\\end{equation}\n\n\\section{Introduction}\\label{sec:1}\n Given a Riemann surface $\n (\\Sigma,g)$ with smooth boundary \n $\\partial \\Sigma$ (possible to be empty) and a positive integer $N\\geq 2$, we consider the following Toda system equipped with homogeneous Neumann boundary conditions\n \\begin{equation}\\label{eq:toda_mfN}\n \\begin{cases}\n -\\Delta_g u_1 = \\sum_{j=1}^N a_{ij}\\rho_j\\left( \\frac{V_je^{u_j}}{\\int_{\\Sigma} V_je^{u_j} dv_g}-\\frac 1 {|\\Sigma|_g} \\right) & \\text{ in } \\intsigma\\\\\n \\partial_{\\nu_g} u_1=\\dots= \\partial_{\\nu_g} u_N= 0 & \\text{ on } \\partial\\Sigma\n \\end{cases},\n \\end{equation}\n where \n $\\intsigma=\\Sigma\\setminus\\partial\\Sigma$ is the interior of $\\Sigma$, $\\nu_g$ is the outward unit normal vector on $\\partial\\Sigma$, $\\Delta_g$ is the Laplace-Beltrami operator, $dv_g$ is the Riemannian volume element, $|\\Sigma|_g=\\int_{\\Sigma} dv_g$ denotes the total volume of $\\Sigma$, $\\rho_i$ is a non-negative parameter, $V_i:\\Sigma\\rightarrow \\R$ is smooth positive functions, and the matrix \n $(a_{ij})$ is one of the Cartan matrices of a general simple Lie algebra with rank $N$. For simplicity, we normalize the volume of $\\Sigma$, i.e., $|\\Sigma|_g =1$.\n\nFor technical reasons, we assume throughout that $\\Sigma$ is a $k$-symmetric surface with $k > \\frac 1 2 \\alpha_N$, which is given by \\eqref{def:alpha_i} in Section \\ref{sec:app}. Due to the smoothness of the boundary, it follows that \n $\n \\Sigma_0 \\cap \\partial \\Sigma = \\emptyset.\n $\n\n\\noindent\n As an immediate consequence, we obtain the following result for the $SU(3)$ Toda system.\n \\begin{corollary}\n Let $k > 2$, and suppose that $\\Sigma$ is a $k$-symmetric Riemann surface, and the potentials $V_1$ and $ V_2$ are $\\fR_k$-invariant. Assume $\\Sigma_0 \\neq \\emptyset$ and $m \\leq \\# \\Sigma_0$, where \n $\\# \\Sigma_0$\n denotes the cardinality of $\\Sigma_0$. Then, there exists a family of solutions to the system \\eqref{eq:A_2_Toda} with $(\\rho_1^{\\varepsilon}, \\rho_2^\\varepsilon) \\to (4\\pi m, 8\\pi m)$ or $(8\\pi m, 4\\pi m)$ that blows up at exactly $m$ distinct points in $\\Sigma_0$, exhibiting asymmetric blow-ups.\n \\end{corollary}\n\nWe consider the following linear operator associated with the problem~\\eqref{eq:todaN}: \n \\begin{equation}\n \\label{eq:linear_op_a} \\cL_{\\xi,\\varepsilon}(\\bphi):=( L^1_{\\xi,\\varepsilon}(\\bphi),\\dots, L^N_{\\xi,\\varepsilon}(\\bphi) ),\n \\end{equation}\n where for any $i=1, \\dots, N $\n \\begin{equation*}\n \\begin{split}\n L^i_{\\xi,\\varepsilon}(\\bphi):&= -\\Delta_g\\phi_i -\\sum_{j=1}^m\\left( \\chi_j e^{-\\varphi_j} |y_{\\xi_j}|^{\\alpha_i-2} e^{U^i_j} \\phi_i-\\overline{\\chi_j e^{-\\varphi_j} |y_{\\xi_j}|^{\\alpha_i-2} e^{U^i_j} \\phi_i}\\right)\\\\\n &- \\sum^N_{\\sumi\n } \\sum_{j=1}^m\\frac{a_{ii'}}2\\left( \\chi_j e^{-\\varphi_j} |y_{\\xi_j}|^{\\alpha_{i'}-2} e^{U^{i'}_j} \\phi_{i'}-\\overline{\\chi_j e^{-\\varphi_j} |y_{\\xi_j}|^{\\alpha_{i'}-2} e^{U^{i'}_j} \\phi_{i'}}\\right).\n \\end{split}\n \\end{equation*}\n The key lemma is the non-degeneracy of the linear operator $\\cL_{\\xi,\\varepsilon}$. Formally, for $i=1,\\dots, N, j=1,\\dots,m$, we can derive the local limit operator of $L^i_{\\xi,\\varepsilon}$ is \n $$-\\Delta \\phi -2\\alpha_i^2 \\frac{|y|^{\\alpha_i-2}}{(1+|y|^{\\alpha_i})^2}\\phi$$ \n by a proper scaling around $\\xi_j$ on an isothermal chart (refer to Lemma~\\ref{lem:invertible_as}). Del Pino et al. in \\cite{DelPino2012Nondegeneracy} prove that the kernel space is generated by (in polar coordinate $(r,\\theta)$) \n \\[\\phi^0(r,\\theta):=\\frac{1-|r|^{\\alpha_i}}{1+|r|^{\\alpha_i}}, \\, \\phi^1(r,\\theta):=\\frac{|r|^{\\frac{\\alpha_i}{2}}}{1+|r|^{\\alpha_i}}\\cos \\frac{\\alpha_i}{2}\\theta , \\, \\phi^2(r,\\theta):=\\frac{|r|^{\\frac{\\alpha_i}{2}}}{1+|r|^{\\alpha_i}} \\sin \\frac{\\alpha_i}{2} \\theta. \\] \n The $k$-symmetric condition of $\\cH_{k}$ with $k>\\frac {\\alpha_{N}} 2$ excludes $\\phi^1$ and $\\phi^2$. To obtain the invertibility of the linearized operator $\\cL_{\\xi,\\varepsilon}$, we need to lay out $\\phi^0$ by introducing a family of test functions in $\\oH$. \n Let $z_i(y)=\\frac {1-|y|^{\\alpha_i}}{ 1+|y|^{\\alpha_i}}.$\n It is easy to see that \n $$-\\Delta z_i(y)= 2\\alpha_i^2 \\frac{|y|^{\\alpha_i-2}}{(1+|y|^{\\alpha_i})^2}z_i(y), y\\in\\R^2. $$\n For any $j=1,\\dots,m$, we define \n \\[ Z_{ij}(x)=\\left\\{ \\begin{array}{ll}\n z_i\\left(\\frac{|y_{\\xi_j}(x)| } {\\delta_{i,j}} \\right), & x\\in U(\\xi_j)\\\\\n 0, & x\\in \\Sigma\\setminus U(\\xi_j)\n \\end{array}\\right.. \\]\n Then, we project $Z_{ij}$ into the space $\\oH$ by following equations: \n \\begin{equation}\n \\label{eq:proj_Z}\n \\left\\{\\begin{array}{ll}\n -\\Delta_gPZ_{ij}= \\chi_j e^{-\\varphi_j} |y_{\\xi_j}|^{\\alpha_i-2} e^{U^i_j}Z_{ij}-\\overline{\\chi_j e^{-\\varphi_j} |y_{\\xi_j}|^{\\alpha_i-2} e^{U^i_j}Z_{ij}} &\\text{ in } \\intsigma\\\\\n \\partial_{\\nu_g} PZ_{ij}=0 &\\text{ on } \\partial\\Sigma\\\\\n \\int_{\\Sigma} PZ_{ij} dv_g=0&\n \\end{array}\\right..\n \\end{equation}\n We define that \n $$\\fL^{0}_{k}:= \\left\\{ h=(h_1,\\dots, h_N): \\int_{\\Sigma} h_i dv_g =0\\text{ and } h_i \\text{ is }\\fR_{k}\\text{ invariant} \\right\\}. $$\n \\begin{lemma}\n \\label{lem:invertible_as}\n For any $p>1$, \n there exist $\\varepsilon_0 > 0$ and $C > 0$ such that for any $\\varepsilon \\in (0, \\varepsilon_0)$, $\\bh=(h_1,\\dots, h_N) \\in (L^p(\\Sigma))^N\\cap\\fL^0_{k}$ there exists $\\bphi=(\\phi_1,\\dots, \\phi_N) \\in (W^{2,p}(\\Sigma))^N \\cap \\cH_{k}$ as a unique solution of \n \\begin{equation}\n \\label{eq:linear_key_a} \\left\\{ \\begin{array}{ll}\n \\cL_{\\xi,\\varepsilon}(\\bphi)=\\bh & \\text{ in }\\intsigma\\\\\n \\partial_{\\nu_g} \\bphi =0 & \\text{ on }\\partial\\Sigma\\\\\n \\int_{\\Sigma}\\bphi dv_g=0 \n \\end{array}\\right.,\n \\end{equation}\n satisfying that \n $$\n \\|\\bphi\\| \\leq C |\\log \\varepsilon| \\|\\bh\\|_p,\n $$\n where $\\|\\bphi\\|:= \\sqrt{\\sum_i \\|\\phi_i\\|^2} \\text{ and } \\|\\bh\\|_p=\\sum_i \\|h_i\\|_{p}$.\n \\end{lemma}\n\n\\begin{proof}\n We prove it by contradiction. Suppose Lemma~\\ref{lem:invertible_as} fails, i.e. there exist $p>1$ and a sequence of $\\varepsilon_n\\rightarrow 0$ and $\\bh_n:=(h_{1,n},\\dots, h_{N,n})\\in (L^p(\\Sigma))^N\\cap \\fL^0_{k}$ and $\\bphi_n:=(\\phi_{1,n},\\dots, \\phi_{N,n})\\in (W^{2,p}(\\Sigma))^N\\cap \\cH_{k}$ solves ~\\eqref{eq:linear_key_a} for $\\bh_n$ satisfying \n \\[ \\|\\bphi_n\\|=1\\text{ and } |\\log \\varepsilon_n|\\|\\bh_n\\|_p:=|\\log\\varepsilon_n|\\sum_{i=1}^N \\|h_{i,n}\\|_{p}\\rightarrow 0, \\]\n as $n\\rightarrow +\\infty.$\n For simplicity, we still use the notations $\\phi_i,$ $ h_i,$ $\\varepsilon$ instead of $\\phi_{i,n},$ $h_{i,n},$ $ \\varepsilon_n$ for $i=1,\\dots, N$. \n We define that for $i=1,\\dots, N, j=1,\\dots,m$\n \\[ \\tilde{\\phi}_{ij}(y)= \\left\\{\\begin{aligned}\n & \\chi\\left( \\frac { \\delta_{i,j}|y|}{r_0}\\right) \\phi_i\\circ y_{\\xi_j}^{-1}(\\delta_{i,j} y),& &y \\in \\Omega_{ij}:= \\frac{1}{\\delta_{i,j}} B^{\\xi_j}\\\\\n & 0& & y\\in\\R^2\\setminus\\Omega_{ij}\n \\end{aligned}\\right.. \\]\n Then we consider the following spaces for $\\alpha\\geq 2 $ and $\\xi\\in\\Sigma$,\n $$ \\rL^{\\alpha}:=\\Big\\{ u: \\Big\\| \\frac{|y|^{\\frac{\\alpha-2}{2}}}{1+|y|^{\\alpha}} u \\Big\\|_{2} <+\\infty\\Big\\}$$\n and \n $$ \\rH^{\\alpha}:=\\Big\\{u: \\|\\nabla u\\|_{2}+\\Big\\| \\frac{|y|^{\\frac{\\alpha-2}{2}}}{1+|y|^{\\alpha}} u \\Big\\|_{2}<+\\infty \\Big\\}.$$\n \\begin{itemize}\n \\item[{\\bf Step 1.}]\\label{item:step1_a} {\\it $\\tilde{\\phi}_{ij}\\rightarrow c_{ij} \\frac{1-|y|^{\\alpha_i}}{1+|y|^{\\alpha_i}}$ as $\\varepsilon\\rightarrow 0$ for some $c_{ij}\\in\\R$, which is weakly in $\\rH^{\\alpha_i}$ and strongly in $\\rL^{\\alpha_i}$.}\n \\end{itemize}\n We fix an arbitrary $l=1,\\cdots, N$. \n The Sobolev inequality and H\\\"{o}lder's inequality yield that\n \\begin{equation}\\label{eq:h_phi}\n \\Big| \\int_{\\Sigma} h_i \\phi_l\\Big|\\leq \\|\\bh\\|_p \\|\\phi_l\\|_{p'}\\leq \\|\\bh\\|_p \\|\\phi_l\\|=o\\Big(|\\log\\varepsilon|^{-1}\\Big)\\rightarrow 0,\n \\end{equation} \n where $p, p'>1$ with $\\frac 1 p+\\frac 1 {p'}=1. $ \n Moreover, $|\\la \\phi_i, \\phi_l\\ra|\\leq C \\|\\phi_i\\|\\|\\phi_l\\|=\\cO(1)$.", "post_theorem_intro_text_len": 3679, "post_theorem_intro_text": "\\noindent\n\tAs an immediate consequence, we obtain the following result for the $SU(3)$ Toda system.\n\t\\begin{corollary}\n\t\tLet $k > 2$, and suppose that $\\Sigma$ is a $k$-symmetric Riemann surface, and the potentials $V_1$ and $ V_2$ are $\\fR_k$-invariant. Assume $\\Sigma_0 \\neq \\emptyset$ and $m \\leq \\# \\Sigma_0$, where \n\t\t$\\# \\Sigma_0$\n\t\tdenotes the cardinality of $\\Sigma_0$. Then,  there exists a family of solutions to the system \\eqref{eq:A_2_Toda} with $(\\rho_1^{\\varepsilon}, \\rho_2^\\varepsilon) \\to (4\\pi m, 8\\pi m)$ or $(8\\pi m, 4\\pi m)$ that blows up at exactly $m$ distinct points in $\\Sigma_0$, exhibiting asymmetric blow-ups.\n\t\\end{corollary}\n\n\t\\begin{remark}\n\t\tThe $k$-symmetry assumption is restrictive, but it includes two standard geometric models: the sphere $\\mathbb{S}^2$ and the upper hemisphere $\\mathbb{S}^2_+$. \n\t\tOur result applies to compact surfaces with boundary (hence including $\\mathbb{S}^2_+$), and the method also allows the case $\\partial\\Sigma=\\emptyset$ (thus covering closed surfaces such as $\\mathbb{S}^2$).\n\n\t\tFor these two model cases, the blow-up locations of the solutions constructed in Theorem~\\ref{thm:main_asymmetric} can be described more explicitly. \n\t\t\\begin{itemize}\n\t\t\t\\item Let $\\Sigma=\\mathbb{S}^2$ and assume that $V_1$ and $V_2$ are constants.\n\t\t\tFor any $\\xi\\in\\mathbb{S}^2$, there exists a family of blow-up solutions to the $SU(3)$ Toda system~\\eqref{eq:toda_mfN} as $(\\rho_1^\\varepsilon,\\rho_2^\\varepsilon)\\to(4\\pi,8\\pi)$ or $(8\\pi,4\\pi)$, blowing up at $\\xi$.\n\t\t\tMoreover, there exists a family of blow-up solutions as $(\\rho_1^\\varepsilon,\\rho_2^\\varepsilon)\\to(8\\pi,16\\pi)$ or $(16\\pi,8\\pi)$, blowing up at the antipodal pair $\\{\\xi,-\\xi\\}$, where $-\\xi$ denotes the antipodal point of $\\xi$.\n\n\t\t\t\\item Let $\\Sigma=\\mathbb{S}^2_+$ and assume that $V_1$ and $V_2$ are $\\mathfrak{R}_3$-invariant.\n\t\t\tThen there exists a family of blow-up solutions to the $SU(3)$ Toda system~\\eqref{eq:toda_mfN} as $(\\rho_1^\\varepsilon,\\rho_2^\\varepsilon)\\to(4\\pi,8\\pi)$ or $(8\\pi,4\\pi)$, blowing up at the north pole $\\xi=(0,0,1)$.\n\t\t\\end{itemize}\n\n\t\\end{remark}\n\n\tFollowing the approach in \\cite{D'aprile_asymmetric_2016, musso_new_2016}, we construct blow-up solutions of \\eqref{eq:toda_mfN} via singular perturbation methods that combine variational methods with a Lyapunov–Schmidt reduction. One key difference in our setting is that we study the Toda system on surfaces, requiring the use of isothermal coordinates to pull back standard bubbles from the plane. Moreover, the Neumann boundary condition poses additional challenges, as the maximum principle is no longer available. Instead, we employ Green representation formulas and $L^p$-estimates to analyze the projected bubbles.\n\n\tTo perform the finite-dimensional reduction, we compute the kernel of the limiting linearized operator. Due to the higher-order blow-up we consider, the kernel contains non-radial elements that do not align with the tangent space of the approximation manifold. To overcome this, we introduce $k$-symmetry on the surface, inspired by \\cite{musso_new_2016}, which addresses invertibility in a $k$-symmetric subspace and implies that  the finite-dimensional part is trivial.\n\n\tThe main analytical difficulty lies in dealing with component interactions and the non-symmetry of the Cartan matrices $\\mathbf{B}_N, \\mathbf{C}_N$ and $\\mathbf{G}_2$. We resolve this using localized estimates in annular regions and treating the last component separately. The resulting solutions exhibit blow-up at $k$-symmetric centers which are interior points only; boundary blow-up remains out of reach due to degeneracy in our setting, and is left for future study.", "sketch": "Following the approach in \\cite{D'aprile_asymmetric_2016, musso_new_2016}, the proof of Theorem~\\ref{thm:main_asymmetric} proceeds by constructing blow-up solutions of \\eqref{eq:toda_mfN} via “singular perturbation methods that combine variational methods with a Lyapunov–Schmidt reduction.” Since the Toda system is on surfaces, one “require[s] the use of isothermal coordinates to pull back standard bubbles from the plane.” Because of the Neumann boundary condition, “the maximum principle is no longer available,” so the analysis uses “Green representation formulas and $L^p$-estimates to analyze the projected bubbles.”\n\nFor the finite-dimensional reduction, one “compute[s] the kernel of the limiting linearized operator.” In the “higher-order blow-up” regime, “the kernel contains non-radial elements that do not align with the tangent space of the approximation manifold”; to restore invertibility one “introduce[s] $k$-symmetry on the surface … which addresses invertibility in a $k$-symmetric subspace and implies that the finite-dimensional part is trivial.”\n\nThe “main analytical difficulty” is “component interactions and the non-symmetry of the Cartan matrices $\\mathbf{B}_N, \\mathbf{C}_N$ and $\\mathbf{G}_2$,” handled by “localized estimates in annular regions” and by “treating the last component separately.” The construction yields solutions “blow[ing] up at $k$-symmetric centers which are interior points only”; “boundary blow-up remains out of reach due to degeneracy … and is left for future study.”", "expanded_sketch": "Following the approach in D'aprile_asymmetric_2016 and musso_new_2016, the proof proceeds by constructing blow-up solutions of\n\\begin{equation}\\label{eq:toda_mfN}\n\t\t\\begin{cases}\n\t\t\t-\\Delta_g u_1 = \\sum_{j=1}^N a_{ij}\\rho_j\\left( \\frac{V_je^{u_j}}{\\int_{\\Sigma} V_je^{u_j} dv_g}-\\frac 1 {|\\Sigma|_g} \\right)    & \\text{ in } \\intsigma\\\\\n\t\t\t\\partial_{\\nu_g} u_1=\\dots= \\partial_{\\nu_g} u_N= 0 & \\text{ on } \\partial\\Sigma\n\t\t\\end{cases},\n\t\\end{equation}\nvia “singular perturbation methods that combine variational methods with a Lyapunov–Schmidt reduction.” Since the Toda system is on surfaces, one “require[s] the use of isothermal coordinates to pull back standard bubbles from the plane.” Because of the Neumann boundary condition, “the maximum principle is no longer available,” so the analysis uses “Green representation formulas and $L^p$-estimates to analyze the projected bubbles.”\n\nFor the finite-dimensional reduction, one “compute[s] the kernel of the limiting linearized operator.” In the “higher-order blow-up” regime, “the kernel contains non-radial elements that do not align with the tangent space of the approximation manifold”; to restore invertibility one “introduce[s] $k$-symmetry on the surface … which addresses invertibility in a $k$-symmetric subspace and implies that the finite-dimensional part is trivial.”\n\nThe “main analytical difficulty” is “component interactions and the non-symmetry of the Cartan matrices $\\mathbf{B}_N, \\mathbf{C}_N$ and $\\mathbf{G}_2$,” handled by “localized estimates in annular regions” and by “treating the last component separately.” The construction yields solutions “blow[ing] up at $k$-symmetric centers which are interior points only”; “boundary blow-up remains out of reach due to degeneracy … and is left for future study.”\n\nThis completes the proof of the main theorem.", "expanded_theorem": "\\label{thm:main_asymmetric}\n\t\tLet $k >\\frac 1 2 \\alpha_N$, where $\\alpha_i$ is defined by\n\t\t\\begin{equation}\n\t\t\\label{def:alpha_i}\n\t\t\\begin{split}\n\t\t\t\\alpha_i&=2i \n\t\t\t\\text{ and }\n\t\t\t\\alpha_N= 2-2(N-1)a_{N N-1}=\\begin{cases}\n\t\t\t\t2 N & \\text{ for }  \\mathbf{A}_N\\text{ or } \\mathbf{B}_N \\\\\n\t\t\t\t4N-2 & \\text{ for } \\mathbf{C}_N\\\\\n\t\t\t\t8& \\text{ for } \\mathbf{G}_2\n\t\t\t\\end{cases},\n\t\t\\end{split} \n\t\\end{equation}\n\t\tfor $i=1,\\dots, N$, and assume that $\\Sigma$ is a $k$-symmetric Riemann surface with smooth boundary. Suppose the potential functions $V_1, \\dots, V_N$ are $\\fR_k$-invariant, and \n\t\t\\[\n\t\t\\Sigma_0:=\\left\\{x\\in \\Sigma: \\fR_{k}^i(x)=x \\text{ for any } i=1,2,3,\\dots\\right\\}. \n\t\t\\]\n\t\tThen for any $m$ distinct points   $\\xi^*_1, \\dots, \\xi^*_m $ in  $ \\Sigma_0$, there exists a family of solutions $\\bu_\\varepsilon = (u_{1,\\varepsilon}, \\dots, u_{N,\\varepsilon})$ to the  Toda system\n\t\t\\begin{equation}\\label{eq:toda_mfN}\n\t\t\\begin{cases}\n\t\t\t-\\Delta_g u_1 = \\sum_{j=1}^N a_{ij}\\rho_j\\left( \\frac{V_je^{u_j}}{\\int_{\\Sigma} V_je^{u_j} dv_g}-\\frac 1 {|\\Sigma|_g} \\right)    & \\text{ in } \\intsigma\\\\\n\t\t\t\\partial_{\\nu_g} u_1=\\dots= \\partial_{\\nu_g} u_N= 0 & \\text{ on } \\partial\\Sigma\n\t\t\\end{cases},\n\t\\end{equation}\n\t\tsuch that:\n\t\t\\begin{itemize}\n\t\t\t\\item[i)] $\\bu_\\varepsilon$ blows up exactly at the points $\\xi^*_1, \\dots, \\xi^*_m$ as $\\varepsilon \\to 0$;\n\t\t\t\\item [ii)]the parameters $\\rho^\\varepsilon = (\\rho_1^\\varepsilon, \\dots, \\rho_N^\\varepsilon)$ satisfy\n\t\t\t\\[\n\t\t\t\\rho^\\varepsilon \\to (2\\alpha_1 \\pi m,\\dots,  2\\alpha_N \\pi m);\n\t\t\t\\]\n\t\t\t\\item[iii)] for each $i = 1, \\dots, N$, the following  weak-$*$ convergence holds:\n\t\t\t\\[\n\t\t\t\\rho_i^\\varepsilon \\frac{V_i e^{u_{i,\\varepsilon}}}{\\int_\\Sigma V_i e^{u_{i,\\varepsilon}}  dv_g \\, } dv_g   \\stackrel{*}{\\rightharpoonup}  \\sum_{j=1}^m 2\\alpha_i \\pi \\delta_{\\xi^*_j},  \\text{ as } \\varepsilon \\to 0,\n\t\t\t\\]\n\t\t\\end{itemize}\n\t\twhere $\\delta_{x}$ is the Dirac measure on $\\Sigma$ concentrated at the point $x.$,", "theorem_type": ["Universal–Existential", "Existence"], "mcq": {"question": "Let \\((a_{ij})\\) be a Cartan matrix of rank \\(N\\), and let \\((\\Sigma,g)\\) be a \\(k\\)-symmetric Riemann surface with smooth boundary, meaning that \\(\\Sigma\\) can be embedded in \\(\\mathbb{R}^3\\) and is invariant under a rotation \\(\\mathcal R_k\\). Define the set of \\(k\\)-symmetric centers by\n\\[\n\\Sigma_0:=\\{x\\in\\Sigma: \\mathcal R_k^i(x)=x \\text{ for every } i=1,2,3,\\dots\\}.\n\\]\nAssume \\(k>\\tfrac12\\alpha_N\\), where \\(\\alpha_i=2i\\) for \\(i=1,\\dots,N-1\\), and\n\\[\n\\alpha_N=2-2(N-1)a_{N,N-1}=\\begin{cases}\n2N & \\text{for } \\mathbf A_N \\text{ or } \\mathbf B_N,\\\\\n4N-2 & \\text{for } \\mathbf C_N,\\\\\n8 & \\text{for } \\mathbf G_2.\n\\end{cases}\n\\]\nSuppose the positive smooth potential functions \\(V_1,\\dots,V_N\\) are \\(\\mathcal R_k\\)-invariant. For any \\(m\\) distinct points \\(\\xi_1^*,\\dots,\\xi_m^*\\in\\Sigma_0\\), consider the Toda system with homogeneous Neumann boundary conditions\n\\[\n-\\Delta_g u_i=\\sum_{j=1}^N a_{ij}\\rho_j\\left(\\frac{V_j e^{u_j}}{\\int_{\\Sigma} V_j e^{u_j}\\,dv_g}-\\frac1{|\\Sigma|_g}\\right)\\quad\\text{in }\\mathring\\Sigma,\\qquad i=1,\\dots,N,\n\\]\nwith\n\\[\n\\partial_{\\nu_g}u_1=\\cdots=\\partial_{\\nu_g}u_N=0\\quad\\text{on }\\partial\\Sigma.\n\\]\nWhich statement holds under these assumptions?", "correct_choice": {"label": "A", "text": "There exists a family of solutions \\(\\mathbf u_\\varepsilon=(u_{1,\\varepsilon},\\dots,u_{N,\\varepsilon})\\) of the above Toda system, with parameters \\(\\rho^\\varepsilon=(\\rho_1^\\varepsilon,\\dots,\\rho_N^\\varepsilon)\\), such that as \\(\\varepsilon\\to0\\): (i) \\(\\mathbf u_\\varepsilon\\) blows up exactly at the points \\(\\xi_1^*,\\dots,\\xi_m^*\\); (ii) \\(\\rho^\\varepsilon\\to(2\\alpha_1\\pi m,\\dots,2\\alpha_N\\pi m)\\); and (iii) for every \\(i=1,\\dots,N\\),\n\\[\n\\rho_i^\\varepsilon\\frac{V_i e^{u_{i,\\varepsilon}}}{\\int_\\Sigma V_i e^{u_{i,\\varepsilon}}\\,dv_g}\\,dv_g \\stackrel{*}{\\rightharpoonup} \\sum_{j=1}^m 2\\alpha_i\\pi\\,\\delta_{\\xi_j^*}\n\\quad\\text{as }\\varepsilon\\to0,\n\\]\nwhere \\(\\delta_{\\xi_j^*}\\) is the Dirac measure at \\(\\xi_j^*\\)."}, "choices": [{"label": "B", "text": "There exists a family of solutions \\(\\mathbf u_\\varepsilon=(u_{1,\\varepsilon},\\dots,u_{N,\\varepsilon})\\) of the above Toda system, with parameters \\(\\rho^\\varepsilon=(\\rho_1^\\varepsilon,\\dots,\\rho_N^\\varepsilon)\\), such that as \\(\\varepsilon\\to0\\): (i) \\(\\mathbf u_\\varepsilon\\) blows up exactly at the points \\(\\xi_1^*,\\dots,\\xi_m^*\\); (ii) \\(\\rho^\\varepsilon\\to(2\\alpha_1\\pi m,\\dots,2\\alpha_N\\pi m)\\); and (iii) for every \\(i=1,\\dots,N\\),\n\\[\n\\rho_i^\\varepsilon\\frac{V_i e^{u_{i,\\varepsilon}}}{\\int_\\Sigma V_i e^{u_{i,\\varepsilon}}\\,dv_g}\\,dv_g \\stackrel{*}{\\rightharpoonup} \\sum_{j=1}^m \\alpha_i\\pi\\,\\delta_{\\xi_j^*}\n\\quad\\text{as }\\varepsilon\\to0,\n\\]\nwhere \\(\\delta_{\\xi_j^*}\\) is the Dirac measure at \\(\\xi_j^*\\)."}, {"label": "C", "text": "There exists a family of solutions \\(\\mathbf u_\\varepsilon=(u_{1,\\varepsilon},\\dots,u_{N,\\varepsilon})\\) of the above Toda system, with parameters \\(\\rho^\\varepsilon=(\\rho_1^\\varepsilon,\\dots,\\rho_N^\\varepsilon)\\), such that as \\(\\varepsilon\\to0\\): (i) \\(\\mathbf u_\\varepsilon\\) blows up exactly at the points \\(\\xi_1^*,\\dots,\\xi_m^*\\); and (ii) \\(\\rho^\\varepsilon\\to(2\\alpha_1\\pi m,\\dots,2\\alpha_N\\pi m)\\)."}, {"label": "D", "text": "There exists a family of solutions \\(\\mathbf u_\\varepsilon=(u_{1,\\varepsilon},\\dots,u_{N,\\varepsilon})\\) of the above Toda system for any \\(m\\) distinct points \\(\\xi_1^*,\\dots,\\xi_m^*\\in\\Sigma\\), with parameters \\(\\rho^\\varepsilon=(\\rho_1^\\varepsilon,\\dots,\\rho_N^\\varepsilon)\\), such that as \\(\\varepsilon\\to0\\): (i) \\(\\mathbf u_\\varepsilon\\) blows up exactly at the points \\(\\xi_1^*,\\dots,\\xi_m^*\\); (ii) \\(\\rho^\\varepsilon\\to(2\\alpha_1\\pi m,\\dots,2\\alpha_N\\pi m)\\); and (iii) for every \\(i=1,\\dots,N\\),\n\\[\n\\rho_i^\\varepsilon\\frac{V_i e^{u_{i,\\varepsilon}}}{\\int_\\Sigma V_i e^{u_{i,\\varepsilon}}\\,dv_g}\\,dv_g \\stackrel{*}{\\rightharpoonup} \\sum_{j=1}^m 2\\alpha_i\\pi\\,\\delta_{\\xi_j^*}\n\\quad\\text{as }\\varepsilon\\to0.\n\\]"}, {"label": "E", "text": "There exists a family of solutions \\(\\mathbf u_\\varepsilon=(u_{1,\\varepsilon},\\dots,u_{N,\\varepsilon})\\) of the above Toda system, with parameters \\(\\rho^\\varepsilon=(\\rho_1^\\varepsilon,\\dots,\\rho_N^\\varepsilon)\\), such that as \\(\\varepsilon\\to0\\): (i) \\(\\mathbf u_\\varepsilon\\) blows up exactly at the points \\(\\xi_1^*,\\dots,\\xi_m^*\\); (ii) \\(\\rho^\\varepsilon=(2\\alpha_1\\pi m,\\dots,2\\alpha_N\\pi m)\\) for all sufficiently small \\(\\varepsilon\\); and (iii) for every \\(i=1,\\dots,N\\),\n\\[\n\\rho_i^\\varepsilon\\frac{V_i e^{u_{i,\\varepsilon}}}{\\int_\\Sigma V_i e^{u_{i,\\varepsilon}}\\,dv_g}\\,dv_g \\stackrel{*}{\\rightharpoonup} \\sum_{j=1}^m 2\\alpha_i\\pi\\,\\delta_{\\xi_j^*}\n\\quad\\text{as }\\varepsilon\\to0.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "exact local mass coefficient at each blow-up point", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "weak-* convergence of the normalized measures", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "restriction to k-symmetric centers in \\Sigma_0 (interior fixed points only)", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "asymptotic convergence of parameters versus eventual exact equality", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives hypotheses and asks for the resulting existence theorem, but it does not explicitly reveal the correct conclusion. There is no direct statement of the exact blow-up, parameter limit, and weak-* convergence package."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem reproduces the full hypotheses of a specific result and the correct option restates its conclusion almost verbatim. The task is mainly to recognize the theorem statement rather than infer among genuinely different mathematical outcomes."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to reject subtle variants involving boundary blow-up points, loss of exactness, quantifier dependence, and the missing factor of m. However, the item mostly tests precise recall/discrimination of theorem wording rather than substantial generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one weakens the conclusion, one alters admissible blow-up locations, one introduces an unjustified uniformity in parameters, and one omits the factor m. These reflect realistic failure modes and are clearly distinct."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-recognition MCQ with strong distractors and no obvious answer leakage, but it is highly close to a direct restatement of the source theorem and only moderately tests genuine reasoning."}}
{"id": "2602.05203v1", "paper_link": "http://arxiv.org/abs/2602.05203v1", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "\\label{EPS} Assume that $p=\\frac{2n}{n-2k}$ and $n\\geq2k+2$.\n    Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality \n    for  the inequality \\eqref{BPS}. Furthermore, the extremal function $f_0$ must be radially symmetric and monotone decreasing about some point $P\\in \\mathbb{B}^n$, that is, $f_0$ is a constant on the geodesic sphere centered at $P\\in \\mathbb{B}^n$ and radially decreasing about the geodesic distance from P.", "start_pos": 12151, "end_pos": 12655, "label": "EPS"}, "ref_dict": {"BSobolev": "\\begin{equation}\\label{BSobolev}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV\\geq S_{n,k}\\left(\\int_{\\mathbb{B}^n}|f|^{\\frac{2n}{n-2k}}dV\\right)^{\\frac{n-2k}{n}},\n\\end{equation}", "sublambdaBN": "\\begin{equation}\\label{sublambdaBN}\nP_{k}\\left(f\\right)-\\alpha f=|f|^{p-2}f.\n\\end{equation}", "HHSM": "\\begin{equation}\\label{HHSM}\n\\int_{\\mathbb{R}^{n}_{+}}|\\nabla^{k}u|^2dx-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{R}^{n}_{+}}\\frac{u^2}{x_1^{2k}}dx\\geq C_{n,k,p} \\left(\\int_{\\mathbb{R}^{n}_{+}}x_1^\\gamma|u|^{p}dx\\right)^{\\frac{2}{p}}.\n\\end{equation}", "AHSM": "\\begin{equation}\\label{AHSM}\n\\int_{\\mathbb{R}^{n}_{+}}|\\nabla^{k}u|^2dx-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{R}^{n}_{+}}\\frac{u^2}{x_1^{2k}}dx\\geq C_{n,k,\\frac{2n}{n-2k}} \\left(\\int_{\\mathbb{R}^{n}_{+}}|u|^{\\frac{2n}{n-2k}}dx\\right)^{\\frac{n-2k}{n}}.\n\\end{equation}", "E-subHPS": "\\begin{theorem}\\label{E-subHPS}\n  Assume that $2<p<\\frac{2n}{n-2k}$ and $n>2k$. Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality in the inequality \\eqref{BPS},\n\\end{theorem}", "Preliminaries": "\\label{Preliminaries}\n\nIn this section, we recall some facts on the hyperbolic space. There are several models of hyperbolic space, such as  the Poincar\\'e half space model and the Poincar\\'e ball mod", "EPS": "\\begin{theorem}\\label{EPS} Assume that $p=\\frac{2n}{n-2k}$ and $n\\geq2k+2$.\n    Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality \n    for  the inequality \\eqref{BPS}. Furthermore, the extremal function $f_0$ must be radially symmetric and monotone decreasing about some point $P\\in \\mathbb{B}^n$, that is, $f_0$ is a constant on the geodesic sphere centered at $P\\in \\mathbb{B}^n$ and radially decreasing about the geodesic distance from P. \n\\end{theorem}", "BPS": "\\begin{equation}\\label{BPS}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\geq C_{n,k,p}\\left(\\int_{\\mathbb{B}^n}|f|^{p}dV\\right)^{\\frac{2}{p}},\n\\end{equation}"}, "pre_theorem_intro_text_len": 7570, "pre_theorem_intro_text": "The classical Hardy-Sobolev-Maz'ya (HSM) inequality provides a refinement of both the  Sobolev and the Hardy inequalities on the upper half space $\\mathbb{R}^n_+$. It reads as follows: for $n>2k$, $2<p\\leq \\frac{2n}{n-2k}$ and $\\gamma=\\frac{\\left(n-2k\\right)p}{2}-n$, there exists some positive constant $C_{n,k,p}$ such that for each $u\\in C_{c}^{\\infty}\\left(\\mathbb{R}^{n}_{+}\\right)$, there holds\n\\begin{equation}\\label{HHSM}\n\\int_{\\mathbb{R}^{n}_{+}}|\\nabla^{k}u|^2dx-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{R}^{n}_{+}}\\frac{u^2}{x_1^{2k}}dx\\geq C_{n,k,p} \\left(\\int_{\\mathbb{R}^{n}_{+}}x_1^\\gamma|u|^{p}dx\\right)^{\\frac{2}{p}}.\n\\end{equation}\n\nThe first order case ($k=1$) was established by Maz'ya \\cite{Mazya}, while the higher order case was proved by Lu and Yang \\cite{LY19}. In particular, when $p=\\frac{2n}{n-2k}$ (corresponding to $\\gamma=0$), inequality \\eqref{HHSM} is referred to as the critical HSM inequality:\n\\begin{equation}\\label{AHSM}\n\\int_{\\mathbb{R}^{n}_{+}}|\\nabla^{k}u|^2dx-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{R}^{n}_{+}}\\frac{u^2}{x_1^{2k}}dx\\geq C_{n,k,\\frac{2n}{n-2k}} \\left(\\int_{\\mathbb{R}^{n}_{+}}|u|^{\\frac{2n}{n-2k}}dx\\right)^{\\frac{n-2k}{n}}.\n\\end{equation}\n\n Lu and Yang \\cite{LY19} developed Fourier analysis  techniques on the hyperbolic space which is a noncompact complete Riemannian manifold to establish the HSM inequalities for higher order derivatives on half spaces. (see also Flynn et al \\cite{FlynnLuYang} and \\cite{LuYang-AIM} for the other cases of symmetric spaces of rank one.)  In fact, they derived the following  Poincar\\'e-Sobolev inequalities for the GJMS operators on hyperbolic spaces  $\\mathbb{B}^n$: for $n>2k$ and $2<p\\leq \\frac{2n}{n-2k}$, there exists some constant $C_{n,k,p}$ such that for each $f\\in C_{c}^{\\infty}\\left(\\mathbb{B}^n\\right)$, there holds\n\n\\begin{equation}\\label{BPS}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\geq C_{n,k,p}\\left(\\int_{\\mathbb{B}^n}|f|^{p}dV\\right)^{\\frac{2}{p}},\n\\end{equation}\nwhere $P_k$ is the GJMS operator on the hyperbolic space  $\\mathbb{B}^n$. The precise definition of this operator, along with relevant background on hyperbolic space, is provided in Section \\ref{Preliminaries}.\nBy establishing the equivalence of the Poincar\\'e-Sobolev inequalities on $\\mathbb{B}^n$ and the HSM inequality on half spaces, they derived the higher order HSM inequality. This approach provides a technically powerful route to the HSM inequality, allowing the higher order HSM inequality to be obtained in a unified and indirect manner via its hyperbolic counterpart.\nMotivated by this equivalence, in this article,  we first study the existence of the extremal functions for the Poincar\\'e-Sobolev inequalities on the hyperbolic space   $\\mathbb{B}^n$ and use this framework to establish the existence of extremal functions for the corresponding HSM inequalities.\n\\medskip\n\nThe $k$-th order Sobolev inequality on the hyperbolic space $\\mathbb{B}^n$ reads as follows:\n\\begin{equation}\\label{BSobolev}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV\\geq S_{n,k}\\left(\\int_{\\mathbb{B}^n}|f|^{\\frac{2n}{n-2k}}dV\\right)^{\\frac{n-2k}{n}},\n\\end{equation}\nwhere $f\\in C_{c}^{\\infty}\\left(\\mathbb{B}^n\\right)$, $1\\leq k <\\frac{n}{2}$, $S_{n,k}$ is the sharp constant of $k$-th order Sobolev  inequality on $\\mathbb{R}^n$.\n(see Hebey \\cite{Hebey} and Liu \\cite{Liu}). In \\cite{LY19}, Lu and Yang  gave an alternative way to prove \\eqref{BSobolev}. In fact, let $g(x)=\\big(\\frac{2}{1-|x|^2}\\big)^{\\frac{n}{2}-k}f(x)$, using the conformal transformation laws (see Lemma 5.1 of \\cite{LY19}) $$P_k(f)=\\big(\\frac{2}{1-|x|^2}\\big)^{-\\frac{n}{2}-k}(-\\Delta)^k\\big((\\frac{2}{1-|x|^2})^{\\frac{n}{2}-k}f(x)\\big),$$\nwe can see that the $k$-th order Sobolev inequality on the hyperbolic space $\\mathbb{B}^n$ is equivalent to the following $k$-th order  Sobolev inequality on the unit ball of $\\mathbb{R}^n$:\n\\begin{equation}\\label{BallSobolev}\n\\int_{\\mathbb{B}^n}(-\\Delta)^k g\\cdot gdx\\geq S_{n,k}\\left(\\int_{\\mathbb{B}^n}|g|^{\\frac{2n}{n-2k}}dx\\right)^{\\frac{n-2k}{n}}.\n\\end{equation}\nObviously, the $k$-th order Sobolev inequality on the unit ball of $\\mathbb{R}^n$ has no extremals using translation, dilation and classification of extremals of Sobolev inequality in $\\mathbb{R}^n$ (see Talenti \\cite{Talenti} and Aubin \\cite{Aubin} and also Proposition 1.43 of \\cite{Willem} for the first order, and  Lieb \\cite{Lieb} forthe  high-order case). Hence the $k$-th order Sobolev inequality on the hyperbolic space $\\mathbb{B}^n$ does not admit any extremal function.\n\\medskip\n\nOn the other hand, we have the following $k$-th Poincar\\'e inequality on the hyperbolic space $\\mathbb{B}^n$:\n\\begin{equation}\\label{Bpoincare}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV\\geq \\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\left(\\int_{\\mathbb{B}^n}|f|^{2}dV\\right),\n\\end{equation}\nwhere $f\\in C_{c}^{\\infty}\\left(\\mathbb{B}^n\\right)$, and the constant $\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}$ is sharp. We need to note that the sharp constants cannot be attained.    Combining $k$-th order Sobolev inequality and $k$-th order Poincar\\'e inequality on the hyperbolic space $\\mathbb{B}^n$, there must exist some refinement of both of these two kinds of sharp inequalities. The refinement is simply demonstrated by the high-order Poincare-Sobolev inequality on the hyperbolic space \\eqref{BPS}. However, the problem for the existence of extremal of high-order Poincare-Sobolev inequality on the hyperbolic space still remains open. \n\nFor the first order critical HSM inequality, Hebey \\cite{Hebey} showed that the best constant $C_{n,1,\\frac{2n}{n-2}}$ is strictly smaller than the Sobolev constant $S_{n,1}$ for $n\\geq 4$. Tertikas and Tintarev  \\cite{TT} further proved that the existence of the extremal function for $n\\geq 4$. On the other hand, Benguria, Frank and Loss \\cite{BFL} demonstrated that the best constant $C_{n,1,\\frac{2n}{n-2}}$ is the same as the Sobolev constant $S_{n,1}$ and the best constant $C_{n,1,\\frac{2n}{n-2}}$ is not achieved when  $n=3$. For the high order critical case, Lu and Yang    proved in \\cite {LY22}  that the best constant $C_{n,k,\\frac{2n}{n-k}}$ for the kth-order HSM inequality coincides with the $k$-th Sobolev contant $S_{n,k}$ if $n=2k+1$ and the best constant $C_{n,k,\\frac{2n}{n-k}}$ is strictly smaller than $k$-th Sobolev contant $S_{n,k}$ if $n\\geq 2k+2$. However, the attainability of the best constant $C_{n,k,\\frac{2n}{n-2k}}$ remains open for $n\\ge 2k+2$ when $k\\ge 2$. In \\cite{TT}, the authors employed a clever transformation that reduces the HSM inequality to a weighted Sobolev inequality on the upper half-space, and by applying the concentration compactness principle, they established the existence of extremal functions in the first order case $k=1$. Due to the increased complexity of higher order operators, such a transformation is no longer available in the higher order setting.\n\\medskip\n\nOur first main result concerns the existence of extremal functions for the critical higher order Poincar\\'e-Sobolev inequality.\nWe define the Hilbert space $H\\left(\\mathbb{B}^n\\right)$ by the completion of $C_c^{\\infty}\\left(\\mathbb{B}^n\\right)$ under the norm\n$$\\|f\\|_{H\\left(\\mathbb{B}^n\\right)}=\\left(\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\right)^{\\frac{1}{2}}.$$\n\nIt is noted that $\\|f\\|_{H\\left(\\mathbb{B}^n\\right)}$ is indeed a norm by using the Poincare-Sobolev inequality.", "context": "The classical Hardy-Sobolev-Maz'ya (HSM) inequality provides a refinement of both the Sobolev and the Hardy inequalities on the upper half space $\\mathbb{R}^n_+$. It reads as follows: for $n>2k$, $2<p\\leq \\frac{2n}{n-2k}$ and $\\gamma=\\frac{\\left(n-2k\\right)p}{2}-n$, there exists some positive constant $C_{n,k,p}$ such that for each $u\\in C_{c}^{\\infty}\\left(\\mathbb{R}^{n}_{+}\\right)$, there holds\n\\begin{equation}\\label{HHSM}\n\\int_{\\mathbb{R}^{n}_{+}}|\\nabla^{k}u|^2dx-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{R}^{n}_{+}}\\frac{u^2}{x_1^{2k}}dx\\geq C_{n,k,p} \\left(\\int_{\\mathbb{R}^{n}_{+}}x_1^\\gamma|u|^{p}dx\\right)^{\\frac{2}{p}}.\n\\end{equation}\n\nLu and Yang \\cite{LY19} developed Fourier analysis techniques on the hyperbolic space which is a noncompact complete Riemannian manifold to establish the HSM inequalities for higher order derivatives on half spaces. (see also Flynn et al \\cite{FlynnLuYang} and \\cite{LuYang-AIM} for the other cases of symmetric spaces of rank one.) In fact, they derived the following Poincar\\'e-Sobolev inequalities for the GJMS operators on hyperbolic spaces $\\mathbb{B}^n$: for $n>2k$ and $2<p\\leq \\frac{2n}{n-2k}$, there exists some constant $C_{n,k,p}$ such that for each $f\\in C_{c}^{\\infty}\\left(\\mathbb{B}^n\\right)$, there holds\n\nThe $k$-th order Sobolev inequality on the hyperbolic space $\\mathbb{B}^n$ reads as follows:\n\\begin{equation}\\label{BSobolev}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV\\geq S_{n,k}\\left(\\int_{\\mathbb{B}^n}|f|^{\\frac{2n}{n-2k}}dV\\right)^{\\frac{n-2k}{n}},\n\\end{equation}\nwhere $f\\in C_{c}^{\\infty}\\left(\\mathbb{B}^n\\right)$, $1\\leq k <\\frac{n}{2}$, $S_{n,k}$ is the sharp constant of $k$-th order Sobolev inequality on $\\mathbb{R}^n$.\n(see Hebey \\cite{Hebey} and Liu \\cite{Liu}). In \\cite{LY19}, Lu and Yang gave an alternative way to prove \\eqref{BSobolev}. In fact, let $g(x)=\\big(\\frac{2}{1-|x|^2}\\big)^{\\frac{n}{2}-k}f(x)$, using the conformal transformation laws (see Lemma 5.1 of \\cite{LY19}) $$P_k(f)=\\big(\\frac{2}{1-|x|^2}\\big)^{-\\frac{n}{2}-k}(-\\Delta)^k\\big((\\frac{2}{1-|x|^2})^{\\frac{n}{2}-k}f(x)\\big),$$\nwe can see that the $k$-th order Sobolev inequality on the hyperbolic space $\\mathbb{B}^n$ is equivalent to the following $k$-th order Sobolev inequality on the unit ball of $\\mathbb{R}^n$:\n\\begin{equation}\\label{BallSobolev}\n\\int_{\\mathbb{B}^n}(-\\Delta)^k g\\cdot gdx\\geq S_{n,k}\\left(\\int_{\\mathbb{B}^n}|g|^{\\frac{2n}{n-2k}}dx\\right)^{\\frac{n-2k}{n}}.\n\\end{equation}\nObviously, the $k$-th order Sobolev inequality on the unit ball of $\\mathbb{R}^n$ has no extremals using translation, dilation and classification of extremals of Sobolev inequality in $\\mathbb{R}^n$ (see Talenti \\cite{Talenti} and Aubin \\cite{Aubin} and also Proposition 1.43 of \\cite{Willem} for the first order, and Lieb \\cite{Lieb} forthe high-order case). Hence the $k$-th order Sobolev inequality on the hyperbolic space $\\mathbb{B}^n$ does not admit any extremal function.\n\\medskip\n\nOn the other hand, we have the following $k$-th Poincar\\'e inequality on the hyperbolic space $\\mathbb{B}^n$:\n\\begin{equation}\\label{Bpoincare}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV\\geq \\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\left(\\int_{\\mathbb{B}^n}|f|^{2}dV\\right),\n\\end{equation}\nwhere $f\\in C_{c}^{\\infty}\\left(\\mathbb{B}^n\\right)$, and the constant $\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}$ is sharp. We need to note that the sharp constants cannot be attained. Combining $k$-th order Sobolev inequality and $k$-th order Poincar\\'e inequality on the hyperbolic space $\\mathbb{B}^n$, there must exist some refinement of both of these two kinds of sharp inequalities. The refinement is simply demonstrated by the high-order Poincare-Sobolev inequality on the hyperbolic space \\eqref{BPS}. However, the problem for the existence of extremal of high-order Poincare-Sobolev inequality on the hyperbolic space still remains open.\n\nOur first main result concerns the existence of extremal functions for the critical higher order Poincar\\'e-Sobolev inequality.\nWe define the Hilbert space $H\\left(\\mathbb{B}^n\\right)$ by the completion of $C_c^{\\infty}\\left(\\mathbb{B}^n\\right)$ under the norm\n$$\\|f\\|_{H\\left(\\mathbb{B}^n\\right)}=\\left(\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\right)^{\\frac{1}{2}}.$$\n\nIt is noted that $\\|f\\|_{H\\left(\\mathbb{B}^n\\right)}$ is indeed a norm by using the Poincare-Sobolev inequality.\n\n\\begin{equation}\\label{BPS}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\geq C_{n,k,p}\\left(\\int_{\\mathbb{B}^n}|f|^{p}dV\\right)^{\\frac{2}{p}},\n\\end{equation}", "full_context": "The classical Hardy-Sobolev-Maz'ya (HSM) inequality provides a refinement of both the Sobolev and the Hardy inequalities on the upper half space $\\mathbb{R}^n_+$. It reads as follows: for $n>2k$, $2<p\\leq \\frac{2n}{n-2k}$ and $\\gamma=\\frac{\\left(n-2k\\right)p}{2}-n$, there exists some positive constant $C_{n,k,p}$ such that for each $u\\in C_{c}^{\\infty}\\left(\\mathbb{R}^{n}_{+}\\right)$, there holds\n\\begin{equation}\\label{HHSM}\n\\int_{\\mathbb{R}^{n}_{+}}|\\nabla^{k}u|^2dx-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{R}^{n}_{+}}\\frac{u^2}{x_1^{2k}}dx\\geq C_{n,k,p} \\left(\\int_{\\mathbb{R}^{n}_{+}}x_1^\\gamma|u|^{p}dx\\right)^{\\frac{2}{p}}.\n\\end{equation}\n\nLu and Yang \\cite{LY19} developed Fourier analysis techniques on the hyperbolic space which is a noncompact complete Riemannian manifold to establish the HSM inequalities for higher order derivatives on half spaces. (see also Flynn et al \\cite{FlynnLuYang} and \\cite{LuYang-AIM} for the other cases of symmetric spaces of rank one.) In fact, they derived the following Poincar\\'e-Sobolev inequalities for the GJMS operators on hyperbolic spaces $\\mathbb{B}^n$: for $n>2k$ and $2<p\\leq \\frac{2n}{n-2k}$, there exists some constant $C_{n,k,p}$ such that for each $f\\in C_{c}^{\\infty}\\left(\\mathbb{B}^n\\right)$, there holds\n\nThe $k$-th order Sobolev inequality on the hyperbolic space $\\mathbb{B}^n$ reads as follows:\n\\begin{equation}\\label{BSobolev}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV\\geq S_{n,k}\\left(\\int_{\\mathbb{B}^n}|f|^{\\frac{2n}{n-2k}}dV\\right)^{\\frac{n-2k}{n}},\n\\end{equation}\nwhere $f\\in C_{c}^{\\infty}\\left(\\mathbb{B}^n\\right)$, $1\\leq k <\\frac{n}{2}$, $S_{n,k}$ is the sharp constant of $k$-th order Sobolev inequality on $\\mathbb{R}^n$.\n(see Hebey \\cite{Hebey} and Liu \\cite{Liu}). In \\cite{LY19}, Lu and Yang gave an alternative way to prove \\eqref{BSobolev}. In fact, let $g(x)=\\big(\\frac{2}{1-|x|^2}\\big)^{\\frac{n}{2}-k}f(x)$, using the conformal transformation laws (see Lemma 5.1 of \\cite{LY19}) $$P_k(f)=\\big(\\frac{2}{1-|x|^2}\\big)^{-\\frac{n}{2}-k}(-\\Delta)^k\\big((\\frac{2}{1-|x|^2})^{\\frac{n}{2}-k}f(x)\\big),$$\nwe can see that the $k$-th order Sobolev inequality on the hyperbolic space $\\mathbb{B}^n$ is equivalent to the following $k$-th order Sobolev inequality on the unit ball of $\\mathbb{R}^n$:\n\\begin{equation}\\label{BallSobolev}\n\\int_{\\mathbb{B}^n}(-\\Delta)^k g\\cdot gdx\\geq S_{n,k}\\left(\\int_{\\mathbb{B}^n}|g|^{\\frac{2n}{n-2k}}dx\\right)^{\\frac{n-2k}{n}}.\n\\end{equation}\nObviously, the $k$-th order Sobolev inequality on the unit ball of $\\mathbb{R}^n$ has no extremals using translation, dilation and classification of extremals of Sobolev inequality in $\\mathbb{R}^n$ (see Talenti \\cite{Talenti} and Aubin \\cite{Aubin} and also Proposition 1.43 of \\cite{Willem} for the first order, and Lieb \\cite{Lieb} forthe high-order case). Hence the $k$-th order Sobolev inequality on the hyperbolic space $\\mathbb{B}^n$ does not admit any extremal function.\n\\medskip\n\nOn the other hand, we have the following $k$-th Poincar\\'e inequality on the hyperbolic space $\\mathbb{B}^n$:\n\\begin{equation}\\label{Bpoincare}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV\\geq \\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\left(\\int_{\\mathbb{B}^n}|f|^{2}dV\\right),\n\\end{equation}\nwhere $f\\in C_{c}^{\\infty}\\left(\\mathbb{B}^n\\right)$, and the constant $\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}$ is sharp. We need to note that the sharp constants cannot be attained. Combining $k$-th order Sobolev inequality and $k$-th order Poincar\\'e inequality on the hyperbolic space $\\mathbb{B}^n$, there must exist some refinement of both of these two kinds of sharp inequalities. The refinement is simply demonstrated by the high-order Poincare-Sobolev inequality on the hyperbolic space \\eqref{BPS}. However, the problem for the existence of extremal of high-order Poincare-Sobolev inequality on the hyperbolic space still remains open.\n\nOur first main result concerns the existence of extremal functions for the critical higher order Poincar\\'e-Sobolev inequality.\nWe define the Hilbert space $H\\left(\\mathbb{B}^n\\right)$ by the completion of $C_c^{\\infty}\\left(\\mathbb{B}^n\\right)$ under the norm\n$$\\|f\\|_{H\\left(\\mathbb{B}^n\\right)}=\\left(\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\right)^{\\frac{1}{2}}.$$\n\nIt is noted that $\\|f\\|_{H\\left(\\mathbb{B}^n\\right)}$ is indeed a norm by using the Poincare-Sobolev inequality.\n\n\\begin{equation}\\label{BPS}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\geq C_{n,k,p}\\left(\\int_{\\mathbb{B}^n}|f|^{p}dV\\right)^{\\frac{2}{p}},\n\\end{equation}\n\nIt is noted that $\\|f\\|_{H\\left(\\mathbb{B}^n\\right)}$ is indeed a norm by using the Poincare-Sobolev inequality.\n\n\\begin{theorem}\\label{EHMS}\n Assume $n\\geq2k+2$. Then there is a positive function $f_0\\in \\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)$ which achieves the equality in inequality\n \\eqref{AHSM},\nwhere $\\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)$ is the completion of $C_c^{\\infty}\\left(\\mathbb{R}^{n}_{+}\\right)$ under the norm\n$$\\|u\\|_{\\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)}=\\left(\\int_{\\mathbb{R}^{n}_{+}}|\\nabla^{k}u|^2dx-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{R}^{n}_{+}}\\frac{u^2}{x_1^{2k}}dx\\right)^{\\frac{1}{2}}.$$\n\n\\begin{theorem}\\label{E-subHPS}\n Assume that $2<p<\\frac{2n}{n-2k}$ and $n>2k$. Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality in the inequality \\eqref{BPS},\n\\end{theorem}\n\n\\begin{theorem}\\label{E-subHHSM}\nAssume that $2<p<\\frac{2n}{n-2k}$ and $n>2k$. \n Then there is a positive function $f_0\\in \\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)$ which achieves the equality in inequality \\eqref{HHSM}.\n\n\\begin{corollary}\\label{EL}\nLet $\\alpha=\\prod\\limits_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}$. If either $2k+2\\leq n$, $p=\\frac{2n}{n-2k}$ or $2k<n$, $2<p<\\frac{2n}{n-2k}$, then there exists at least one positive solution u in $H\\left(\\mathbb{B}^n\\right)$ to the equation \\eqref{sublambdaBN}, which is radially symmetry and monotone decreasing about some point $P\\in \\mathbb{B}^n$.\n\\end{corollary}\n\nFrom Lemma \\ref{kernel-decreasing} (presented later), we know that $G^{-1}_k\\left(x,y\\right)$ is symmetric and decreasing about the distance function $\\rho\\left(T_{x}\\left(y\\right)\\right)$. From the Symmetrization Lemma \\ref{becknersymm} in \\cite{Beckner93}, we can pick up radially decreasing maximizing sequence $\\{h_m\\}$ for the Hardy-Littlewood-Sobolev inequality \\eqref{BHLS}. By the direct equivalence of the inequality \\eqref{BHLS} and \\eqref{G2}, we know that $\\{h_m\\}$ is also the radially decreasing maximizing sequence for inequality \\eqref{G2}, that is, \n$$\\int_{\\mathbb{B}^n}|h_m|^{\\frac{2n}{n+2k}}dV=1,\\ \\ \\lim\\limits_{m\\rightarrow +\\infty} \\int_{\\mathbb{B}^n}|G_k^{-\\frac{1}{2}}\\left(h_m\\right)|^{2}dV=C_{n,k,\\frac{2n}{n-2k}}^{-1}.$$ \nIn the following Lemma \\ref{relationshipsequence}, we proved the relationship between the extremizing sequence of \\eqref{G1} and the extremizing sequence of \\eqref{G2}. Pick $f_m=G_{k}^{-1}\\left(h_m\\right)$, then $f_m\\left(x\\right)$ is a radially decreasing extremizing sequence for inequality \\eqref{G1}. Up to some constants, we have \n$$\\lim_{m\\rightarrow\\infty}\\|f_m\\|^2_{H(\\mathbb{B}^n)}=C_{n,k,\\frac{2n}{n-2k}},\\ \\ \\ \\int_{\\mathbb{B}^n}|f_m|^{\\frac{2n}{n-2k}}dV=1.$$\n\\end{proof}\n\n\\section{The proof of Theorem \\ref{E-subHPS}}\nWe now start to prove that the high-order subcritical Poincar\\'e-Sobolev inequality on Hyperbolic space $\\mathbb{B}^n$:\n$$\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\geq C_{n,k,q}\\left(\\int_{\\mathbb{B}^n}|f|^{q}dV\\right)^{\\frac{2}{q}}$$\nfor $2<q<\\frac{2n}{n-2k}$\nadmits an extremal function. Assume that $\\{f_m\\}$ is a minimizing sequence for the subcritical Poincar\\'e-Sobolev inequality on hyperbolic space $\\mathbb{B}^n$, that is\n$$\\int_{\\mathbb{B}^n}|f_m|^{q}dV=1,\\ \\ \\lim\\limits_{m\\rightarrow +\\infty}\\left(\\int_{\\mathbb{B}^n}P_k\\left(f_m\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f_m|^2dV\\right)=C_{n,k,q}.$$\nSimilar to the proof of the existence of Poincar\\'e-Sobolev inequality on hyperbolic space $\\mathbb{B}^n$ (Theorem \\ref{EPS}), by Green representative formula and Riesz rearrangement inequality, we can assume that the minimizing sequence $\\{f_m\\}$ is radially decreasing about origin.\nIn order to prove the existence of extremal function, we only need to prove that\n$f_m\\rightarrow f_0$ strongly in $L^{q}\\left(\\mathbb{B}^n\\right)$. Since $f_m$ is radially decreasing, by Helly theorem, we know that there exists $f_0\\in L^{p}\\left(\\mathbb{B}^n\\right)$ such that $f_m$ converges to $f_0$ for almost $x\\in \\mathbb{B}^n$. Through Poincar\\'e-Sobolev inequality, we see that\n$f_m$ is also bounded in $L^{\\frac{2n}{n-2k}}\\left(\\mathbb{B}^n\\right)$. This together with the Vitali-convergence theorem yields that\n$$\\lim\\limits_{R\\rightarrow 1}\\lim\\limits_{m\\rightarrow +\\infty}\\int_{B^n\\left(0,R\\right)}|f_m|^qdV=\\lim\\limits_{R\\rightarrow 1}\\int_{B^n\\left(0,R\\right)}|f_0|^qdV=\\int_{\\mathbb{B}^n}|f_0|^qdV.$$\nIn order to prove that $f_m$ strongly converges to $f_0$ in $L^q\\left(\\mathbb{B}^n\\right)$, it remains to show that\n$$\\lim\\limits_{R\\rightarrow 1}\\lim\\limits_{m\\rightarrow +\\infty}\\int_{\\mathbb{B}^n\\setminus B^n\\left(0,R\\right)}|f_m|^qdV=0,$$\nthat is,\n$$\\lim\\limits_{R\\rightarrow 1}\\lim\\limits_{m\\rightarrow +\\infty}\\int_{B^n\\setminus B^n\\left(0,R\\right)}|f_m|^q\\left(\\frac{2}{1-|x|^2}\\right)^ndx=0.$$\nFrom the previous radial Lemma \\ref{radial}, we see that there exists $D_{n,k,q_0}>0$ such that\n$f_m\\left(R\\right)\\leq \\left(1-R\\right)^{\\frac{n-1}{q_0}}$, where $q_0$ is chosen as $q_0<q$. Then it is not difficult to check that\n\\begin{equation}\\begin{split}\n&\\lim\\limits_{R\\rightarrow 1}\\lim\\limits_{m\\rightarrow +\\infty}\\int_{B^n\\setminus B^n\\left(0,R\\right)}|f_m|^q\\left(\\frac{2}{1-|x|^2}\\right)^ndx\\\\\n\\leq &D_{n,k,q_0}\\lim\\limits_{R\\rightarrow 1}\\lim\\limits_{m\\rightarrow +\\infty}\\int_{B^n\\setminus B^n\\left(0,R\\right)}\\left(1-R\\right)^{\\frac{q\\left(n-1\\right)}{q_0}}\\left(\\frac{2}{1-|x|^2}\\right)^ndx=0.\n\\end{split}\\end{equation}", "post_theorem_intro_text_len": 6158, "post_theorem_intro_text": "The proof relies on developing the Lions' type of the concentration-compactness principle \\cite{Lions841, Lions1} for differential inequalities on the hyperbolic space. The concentration-compactness principle for differential inequalities can be divided into the first and the second differential concentration-compactness. One of the key points is to prove the minimizing sequence $\\{u_k\\}$ for higher order Poincar\\'e-Sobolev inequality is tight. In order to achieve this, we must exclude the vanishing  and dichotomy phenomenon. The loss of invariance of dilation causes much challenge to exclude the vanishing phenomenon. In order to overcome these difficulties, we need to find the radially minimizing sequence which decays at infinity of hyperbolic space and help to exclude the vanishing phenomenon. However, P\\\"{o}lya-Szeg\\\"{o} rearrangement inequality is invalid for high-order differential inequality. Applying the equivalence of Poincar\\'e-Sobolev inequality and Hardy-Littlewood-Sobolev inequality on the hyperbolic space, Riesz's rearrangement inequality on hyperbolic space and Green's representative formula, one can find the radially minimizing sequence for Poincar\\'e-Sobolev inequality. To exclude  the dichotomy, one cannot follow the same line of Lions' proof in \\cite{Lions1}. In fact, recall the proof of excluding dichotomy for minimizing sequence $w_k$ of $L^p$ Sobolev inequality involving high order derivatives: $$I=\\inf\\{\\int_{\\mathbb{R}^n}|\\nabla^{m}w|^pdx,\\ \\ \\int_{\\mathbb{R}^n}|w|^{\\frac{np}{n-mp}}dx=1\\}.$$ Lions constructed a suitable concentration-function $$\\rho_k=\\sum_{j=0}^{m}|D^jw_k|^{q_j},\\  q_j=\\frac{np}{n-(m-j)p}$$\nand prove that for any $\\epsilon>0$, there exists $y_k\\in \\mathbb{R}^n$ and $R_{\\epsilon}>0$ such that $\\int_{|x-y_k|\\geq R}\\rho_k(x)dx<\\epsilon$ for any $R>R_{\\epsilon}$. This is achieved by scaling cut-off function and \n$$\\int_{R_0<|x-y_k|<R_k}\\sum_{j=0}^{m-1}|D^jw_k|^{q_j}dx=o_k(1).$$ However, for higher order Poincar\\'e-Sobolev inequality in hyperbolic space, it is impossible to introduce the replacement of concentration-function $\\rho_k$ of $\\mathbb{R}^n$. Moreover, the loss of scaling cut-off function can also add extra difficulty. In order to overcome this difficulty, we found a quantitative relationship between the minimizing sequences for the Poincare-Sobolev inequalities and those for the  equivalent integral inequalities (It is differen from the equivalence of extremals), allowing us to apply the concentration-compactness techniques for equivalent integral inequalities to rule out dichotomy of the minimizing sequences for Poincar\\'e-Sobolev inequality. Once we prove that the minimizing sequence is compact, we can follow the standard second Lions' concentration-compactness procedure to exclude the minimizing sequence converging $\\delta$-measure using $C_{n,k,\\frac{2n}{n-k}}<S_{n,k}$.\n\n\\medskip\n\nThrough the equivalence, combining Theorem \\ref{EPS}, we immediately obtain the existence of the extremal function for the  high order critical Hardy-Sobolev-Maz'ya inequality \\eqref{AHSM} on the upper half space.\n\n\\begin{theorem}\\label{EHMS}\n Assume $n\\geq2k+2$.  Then there is a positive function $f_0\\in \\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)$ which achieves the equality in inequality\n \\eqref{AHSM},\nwhere $\\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)$ is the completion of $C_c^{\\infty}\\left(\\mathbb{R}^{n}_{+}\\right)$ under the norm\n$$\\|u\\|_{\\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)}=\\left(\\int_{\\mathbb{R}^{n}_{+}}|\\nabla^{k}u|^2dx-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{R}^{n}_{+}}\\frac{u^2}{x_1^{2k}}dx\\right)^{\\frac{1}{2}}.$$\n\n\\end{theorem}\n\nFor $n=2k+1$, according to Lemma 4.3 of \\cite{LY22}, we can deduce the nonexistence of the extremal function for critical inequality \\eqref{BPS} and \\eqref{HHSM}. Different from the critical case, the subcritical inequalities have an extremal function for all $n>2k$. We have the following results:\n\n\\begin{theorem}\\label{E-subHPS}\n  Assume that $2<p<\\frac{2n}{n-2k}$ and $n>2k$. Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality in the inequality \\eqref{BPS},\n\\end{theorem}\n\n\\begin{theorem}\\label{E-subHHSM}\nAssume that $2<p<\\frac{2n}{n-2k}$ and $n>2k$. \n Then there is a positive function $f_0\\in \\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)$ which achieves the equality in   inequality \\eqref{HHSM}.\n\n\\end{theorem}\n\nThe Poincar\\'e-Sobolev inequalities  are highly linked with the  Brezis-Nirenberg problem \\cite{Brezis} on the entire hyperbolic space $\\mathbb{B}^n$:\n\\begin{equation}\\label{sublambdaBN}\nP_{k}\\left(f\\right)-\\alpha f=|f|^{p-2}f.\n\\end{equation}\nFor $k=1$, Mancini and Sandeep \\cite{MS} showed that the entire solution exists either when  $n\\geq4$, $p=\\frac{2n}{n-2}$, $0<\\alpha\\leq\\frac{1}{4}$ or when $n\\geq3$, $1<p<\\frac{2n}{n-2}$ and $\\alpha\\leq\\frac{1}{4}$. Bhakta et al. \\cite{BGKM25} proved the quantitative stability for the equation \\eqref{sublambdaBN} when $k=1$. \n For $k\\geq 2$, Li et al. \\cite{LLY} established the existence of the nontrivial solution  when $n\\geq 2k+2$, $p=\\frac{2n}{n-2k}$, $\\alpha<\\prod\\limits_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}$. It is not hard to calculate that the corresponding Euler-Lagrange equation of the inequality \\eqref{BPS} is the  equation \\eqref{sublambdaBN} with $\\alpha=\\prod\\limits_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}$, $2<p\\leq\\frac{2n}{n-2k}$.  A straightforward corollary of Theorem \\ref{EPS} and Theorem \\ref{E-subHPS} is as follows:\n\n\\begin{corollary}\\label{EL}\nLet $\\alpha=\\prod\\limits_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}$. If either $2k+2\\leq n$, $p=\\frac{2n}{n-2k}$ or $2k<n$, $2<p<\\frac{2n}{n-2k}$,  then there exists at least one positive solution u in $H\\left(\\mathbb{B}^n\\right)$ to the equation \\eqref{sublambdaBN},  which is radially symmetry  and monotone decreasing about some point $P\\in \\mathbb{B}^n$.\n\\end{corollary}\n\n\\begin{remark}\n   It is the first time to derive the existence of positive solutions for high-order Brezis-Nirenberg problem for $\\alpha=\\prod\\limits_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}$ on the whole hyperbolic space.\n\\end{remark}", "sketch": "The proof of Theorem~\\ref{EPS} is sketched as follows.\n\n1. **Concentration--compactness framework on hyperbolic space.** The proof “relies on developing the Lions' type of the concentration-compactness principle … for differential inequalities on the hyperbolic space,” split into “the first and the second differential concentration-compactness.”\n\n2. **Tightness of a minimizing sequence and exclusion of vanishing/dichotomy.** A “key point is to prove the minimizing sequence $\\{u_k\\}$ for higher order Poincar\\'e-Sobolev inequality is tight,” which requires excluding “the vanishing and dichotomy phenomenon.” The “loss of invariance of dilation causes much challenge to exclude the vanishing phenomenon.”\n\n3. **Radially minimizing sequence to rule out vanishing (via integral-equation equivalence and rearrangement tools).** To overcome the failure of direct rearrangement (“P\\\"{o}lya-Szeg\\\"{o} rearrangement inequality is invalid for high-order differential inequality”), the sketch uses “the equivalence of Poincar\\'e-Sobolev inequality and Hardy-Littlewood-Sobolev inequality on the hyperbolic space, Riesz's rearrangement inequality on hyperbolic space and Green's representative formula,” to “find the radially minimizing sequence for Poincar\\'e-Sobolev inequality” which “decays at infinity of hyperbolic space” and “help[s] to exclude the vanishing phenomenon.”\n\n4. **Ruling out dichotomy via a quantitative link to equivalent integral inequalities.** Since “to exclude the dichotomy, one cannot follow the same line of Lions' proof,” and it is “impossible to introduce the replacement of concentration-function $\\rho_k$ of $\\mathbb{R}^n$” (and the “loss of scaling cut-off function” adds difficulty), the approach is: “we found a quantitative relationship between the minimizing sequences for the Poincare-Sobolev inequalities and those for the equivalent integral inequalities … allowing us to apply the concentration-compactness techniques for equivalent integral inequalities to rule out dichotomy of the minimizing sequences for Poincar\\'e-Sobolev inequality.”\n\n5. **Compactness and exclusion of concentration to a $\\delta$-measure (second Lions procedure).** “Once we prove that the minimizing sequence is compact, we can follow the standard second Lions' concentration-compactness procedure to exclude the minimizing sequence converging $\\delta$-measure using $C_{n,k,\\frac{2n}{n-k}}<S_{n,k}$.”\n\nThese steps yield existence of an extremal, and the sketch indicates that the construction produces a “radially minimizing sequence,” matching the stated radial symmetry/monotone decrease conclusion in Theorem~\\ref{EPS}.", "expanded_sketch": "The proof of the main theorem is sketched as follows.\n\n1. **Concentration--compactness framework on hyperbolic space.** The proof “relies on developing the Lions' type of the concentration-compactness principle … for differential inequalities on the hyperbolic space,” split into “the first and the second differential concentration-compactness.”\n\n2. **Tightness of a minimizing sequence and exclusion of vanishing/dichotomy.** A “key point is to prove the minimizing sequence $\\{u_k\\}$ for higher order Poincar\\'e-Sobolev inequality is tight,” which requires excluding “the vanishing and dichotomy phenomenon.” The “loss of invariance of dilation causes much challenge to exclude the vanishing phenomenon.”\n\n3. **Radially minimizing sequence to rule out vanishing (via integral-equation equivalence and rearrangement tools).** To overcome the failure of direct rearrangement (“P\\\"{o}lya-Szeg\\\"{o} rearrangement inequality is invalid for high-order differential inequality”), the sketch uses “the equivalence of Poincar\\'e-Sobolev inequality and Hardy-Littlewood-Sobolev inequality on the hyperbolic space, Riesz's rearrangement inequality on hyperbolic space and Green's representative formula,” to “find the radially minimizing sequence for Poincar\\'e-Sobolev inequality” which “decays at infinity of hyperbolic space” and “help[s] to exclude the vanishing phenomenon.”\n\n4. **Ruling out dichotomy via a quantitative link to equivalent integral inequalities.** Since “to exclude the dichotomy, one cannot follow the same line of Lions' proof,” and it is “impossible to introduce the replacement of concentration-function $\\rho_k$ of $\\mathbb{R}^n$” (and the “loss of scaling cut-off function” adds difficulty), the approach is: “we found a quantitative relationship between the minimizing sequences for the Poincare-Sobolev inequalities and those for the equivalent integral inequalities … allowing us to apply the concentration-compactness techniques for equivalent integral inequalities to rule out dichotomy of the minimizing sequences for Poincar\\'e-Sobolev inequality.”\n\n5. **Compactness and exclusion of concentration to a $\\delta$-measure (second Lions procedure).** “Once we prove that the minimizing sequence is compact, we can follow the standard second Lions' concentration-compactness procedure to exclude the minimizing sequence converging $\\delta$-measure using $C_{n,k,\\frac{2n}{n-k}}<S_{n,k}$.”\n\nThese steps yield existence of an extremal, and the sketch indicates that the construction produces a “radially minimizing sequence,” matching the stated radial symmetry/monotone decrease conclusion in the main theorem.", "expanded_theorem": "\\label{EPS} Assume that $p=\\frac{2n}{n-2k}$ and $n\\geq2k+2$.\nThen there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves equality in the inequality\n\\begin{equation}\\label{BPS}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\geq C_{n,k,p}\\left(\\int_{\\mathbb{B}^n}|f|^{p}dV\\right)^{\\frac{2}{p}},\n\\end{equation}\nFurthermore, the extremal function $f_0$ must be radially symmetric and monotone decreasing about some point $P\\in \\mathbb{B}^n$, that is, $f_0$ is a constant on the geodesic sphere centered at $P\\in \\mathbb{B}^n$ and radially decreasing about the geodesic distance from $P$.", "theorem_type": ["Existence", "Universal"], "mcq": {"question": "Let $p=\\frac{2n}{n-2k}$ and assume $n\\ge 2k+2$. On the hyperbolic space $\\mathbb{B}^n$ with volume element $dV$, let $P_k$ denote the $k$th-order GJMS operator, and let $H(\\mathbb{B}^n)$ be the completion of $C_c^{\\infty}(\\mathbb{B}^n)$ under the norm\n$$\n\\|f\\|_{H(\\mathbb{B}^n)}=\\left(\\int_{\\mathbb{B}^n}P_k(f)f\\,dV-\\prod_{i=1}^{k}\\frac{(2i-1)^2}{4}\\int_{\\mathbb{B}^n}|f|^2\\,dV\\right)^{1/2}.\n$$\nWhich statement holds for the critical higher-order Poincar\\'e-Sobolev inequality\n$$\n\\int_{\\mathbb{B}^n}P_k(f)f\\,dV-\\prod_{i=1}^{k}\\frac{(2i-1)^2}{4}\\int_{\\mathbb{B}^n}|f|^2\\,dV\\ge C_{n,k,p}\\left(\\int_{\\mathbb{B}^n}|f|^{p}\\,dV\\right)^{2/p}?\n$$", "correct_choice": {"label": "A", "text": "There exists a positive function $f_0\\in H(\\mathbb{B}^n)$ that attains equality in this inequality. Moreover, this extremal function is radially symmetric and monotone decreasing about some point $P\\in\\mathbb{B}^n$; equivalently, $f_0$ is constant on each geodesic sphere centered at $P$ and decreases with the geodesic distance from $P$."}, "choices": [{"label": "B", "text": "There exists a positive function $f_0\\in H(\\mathbb{B}^n)$ that attains equality in this inequality. Moreover, every extremal function is radially symmetric and monotone decreasing about the origin; equivalently, $f_0$ is constant on each geodesic sphere centered at $0\\in\\mathbb{B}^n$ and decreases with the geodesic distance from the origin."}, {"label": "C", "text": "There exists a positive function $f_0\\in H(\\mathbb{B}^n)$ that attains equality in this inequality."}, {"label": "D", "text": "For every minimizing sequence $\\{f_m\\}\\subset H(\\mathbb{B}^n)$ for this inequality, there exists a point $P\\in\\mathbb{B}^n$ such that, after passing to a subsequence, $f_m$ converges strongly in $H(\\mathbb{B}^n)$ to a positive function $f_0$ attaining equality. Moreover, $f_0$ is radially symmetric and monotone decreasing about $P$."}, {"label": "E", "text": "There exists a positive function $f_0\\in H(\\mathbb{B}^n)$ that attains equality in this inequality, and the best constant satisfies $C_{n,k,p}=S_{n,k}$. Moreover, the extremal function is radially symmetric and monotone decreasing about some point $P\\in\\mathbb{B}^n$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "center_of_radial_symmetry", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "radial_symmetry_and_monotonicity_clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "concentration_compactness", "tampered_component": "compactness_of_every_minimizing_sequence_in_H", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "strict_inequality_Cnkp_less_than_Snk", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem states the setting, assumptions, and inequality, but it does not reveal the existence of an extremal or its symmetry properties. The correct conclusion is not explicitly or trivially encoded in the wording."}, "TAS": {"score": 1, "justification": "This is essentially a theorem-conclusion identification question: the hypotheses are given and the student must pick the exact existence statement that follows. However, it is not a pure tautology because the options vary in strength, symmetry center, compactness claim, and sharp constant."}, "GPS": {"score": 1, "justification": "The item requires moderate reasoning or precise theorem recall to distinguish the exact valid conclusion from weaker, stronger, or subtly altered variants. Still, it mainly tests recognition of the theorem rather than generating a new argument."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target common failure modes: over-specifying symmetry about the origin, giving only a weaker true statement, asserting unjustified compactness of minimizing sequences, and confusing the hyperbolic sharp constant with the Euclidean Sobolev constant."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and strong distractors. Its main limitation is that it leans more toward precise theorem recall than genuinely generative mathematical reasoning."}}
{"id": "2602.05203v1", "paper_link": "http://arxiv.org/abs/2602.05203v1", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "\\label{EPS} Assume that $p=\\frac{2n}{n-2k}$ and $n\\geq2k+2$.\n    Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality \n    for  the inequality \\eqref{BPS}. Furthermore, the extremal function $f_0$ must be radially symmetric and monotone decreasing about some point $P\\in \\mathbb{B}^n$, that is, $f_0$ is a constant on the geodesic sphere centered at $P\\in \\mathbb{B}^n$ and radially decreasing about the geodesic distance from P.", "start_pos": 12151, "end_pos": 12655, "label": "EPS"}, "ref_dict": {"BSobolev": "\\begin{equation}\\label{BSobolev}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV\\geq S_{n,k}\\left(\\int_{\\mathbb{B}^n}|f|^{\\frac{2n}{n-2k}}dV\\right)^{\\frac{n-2k}{n}},\n\\end{equation}", "sublambdaBN": "\\begin{equation}\\label{sublambdaBN}\nP_{k}\\left(f\\right)-\\alpha f=|f|^{p-2}f.\n\\end{equation}", "HHSM": "\\begin{equation}\\label{HHSM}\n\\int_{\\mathbb{R}^{n}_{+}}|\\nabla^{k}u|^2dx-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{R}^{n}_{+}}\\frac{u^2}{x_1^{2k}}dx\\geq C_{n,k,p} \\left(\\int_{\\mathbb{R}^{n}_{+}}x_1^\\gamma|u|^{p}dx\\right)^{\\frac{2}{p}}.\n\\end{equation}", "AHSM": "\\begin{equation}\\label{AHSM}\n\\int_{\\mathbb{R}^{n}_{+}}|\\nabla^{k}u|^2dx-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{R}^{n}_{+}}\\frac{u^2}{x_1^{2k}}dx\\geq C_{n,k,\\frac{2n}{n-2k}} \\left(\\int_{\\mathbb{R}^{n}_{+}}|u|^{\\frac{2n}{n-2k}}dx\\right)^{\\frac{n-2k}{n}}.\n\\end{equation}", "E-subHPS": "\\begin{theorem}\\label{E-subHPS}\n  Assume that $2<p<\\frac{2n}{n-2k}$ and $n>2k$. Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality in the inequality \\eqref{BPS},\n\\end{theorem}", "Preliminaries": "\\label{Preliminaries}\n\nIn this section, we recall some facts on the hyperbolic space. There are several models of hyperbolic space, such as  the Poincar\\'e half space model and the Poincar\\'e ball mod", "EPS": "\\begin{theorem}\\label{EPS} Assume that $p=\\frac{2n}{n-2k}$ and $n\\geq2k+2$.\n    Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality \n    for  the inequality \\eqref{BPS}. Furthermore, the extremal function $f_0$ must be radially symmetric and monotone decreasing about some point $P\\in \\mathbb{B}^n$, that is, $f_0$ is a constant on the geodesic sphere centered at $P\\in \\mathbb{B}^n$ and radially decreasing about the geodesic distance from P. \n\\end{theorem}", "BPS": "\\begin{equation}\\label{BPS}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\geq C_{n,k,p}\\left(\\int_{\\mathbb{B}^n}|f|^{p}dV\\right)^{\\frac{2}{p}},\n\\end{equation}"}, "pre_theorem_intro_text_len": 7570, "pre_theorem_intro_text": "The classical Hardy-Sobolev-Maz'ya (HSM) inequality provides a refinement of both the  Sobolev and the Hardy inequalities on the upper half space $\\mathbb{R}^n_+$. It reads as follows: for $n>2k$, $2<p\\leq \\frac{2n}{n-2k}$ and $\\gamma=\\frac{\\left(n-2k\\right)p}{2}-n$, there exists some positive constant $C_{n,k,p}$ such that for each $u\\in C_{c}^{\\infty}\\left(\\mathbb{R}^{n}_{+}\\right)$, there holds\n\\begin{equation}\\label{HHSM}\n\\int_{\\mathbb{R}^{n}_{+}}|\\nabla^{k}u|^2dx-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{R}^{n}_{+}}\\frac{u^2}{x_1^{2k}}dx\\geq C_{n,k,p} \\left(\\int_{\\mathbb{R}^{n}_{+}}x_1^\\gamma|u|^{p}dx\\right)^{\\frac{2}{p}}.\n\\end{equation}\n\nThe first order case ($k=1$) was established by Maz'ya \\cite{Mazya}, while the higher order case was proved by Lu and Yang \\cite{LY19}. In particular, when $p=\\frac{2n}{n-2k}$ (corresponding to $\\gamma=0$), inequality \\eqref{HHSM} is referred to as the critical HSM inequality:\n\\begin{equation}\\label{AHSM}\n\\int_{\\mathbb{R}^{n}_{+}}|\\nabla^{k}u|^2dx-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{R}^{n}_{+}}\\frac{u^2}{x_1^{2k}}dx\\geq C_{n,k,\\frac{2n}{n-2k}} \\left(\\int_{\\mathbb{R}^{n}_{+}}|u|^{\\frac{2n}{n-2k}}dx\\right)^{\\frac{n-2k}{n}}.\n\\end{equation}\n\n Lu and Yang \\cite{LY19} developed Fourier analysis  techniques on the hyperbolic space which is a noncompact complete Riemannian manifold to establish the HSM inequalities for higher order derivatives on half spaces. (see also Flynn et al \\cite{FlynnLuYang} and \\cite{LuYang-AIM} for the other cases of symmetric spaces of rank one.)  In fact, they derived the following  Poincar\\'e-Sobolev inequalities for the GJMS operators on hyperbolic spaces  $\\mathbb{B}^n$: for $n>2k$ and $2<p\\leq \\frac{2n}{n-2k}$, there exists some constant $C_{n,k,p}$ such that for each $f\\in C_{c}^{\\infty}\\left(\\mathbb{B}^n\\right)$, there holds\n\n\\begin{equation}\\label{BPS}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\geq C_{n,k,p}\\left(\\int_{\\mathbb{B}^n}|f|^{p}dV\\right)^{\\frac{2}{p}},\n\\end{equation}\nwhere $P_k$ is the GJMS operator on the hyperbolic space  $\\mathbb{B}^n$. The precise definition of this operator, along with relevant background on hyperbolic space, is provided in Section \\ref{Preliminaries}.\nBy establishing the equivalence of the Poincar\\'e-Sobolev inequalities on $\\mathbb{B}^n$ and the HSM inequality on half spaces, they derived the higher order HSM inequality. This approach provides a technically powerful route to the HSM inequality, allowing the higher order HSM inequality to be obtained in a unified and indirect manner via its hyperbolic counterpart.\nMotivated by this equivalence, in this article,  we first study the existence of the extremal functions for the Poincar\\'e-Sobolev inequalities on the hyperbolic space   $\\mathbb{B}^n$ and use this framework to establish the existence of extremal functions for the corresponding HSM inequalities.\n\\medskip\n\nThe $k$-th order Sobolev inequality on the hyperbolic space $\\mathbb{B}^n$ reads as follows:\n\\begin{equation}\\label{BSobolev}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV\\geq S_{n,k}\\left(\\int_{\\mathbb{B}^n}|f|^{\\frac{2n}{n-2k}}dV\\right)^{\\frac{n-2k}{n}},\n\\end{equation}\nwhere $f\\in C_{c}^{\\infty}\\left(\\mathbb{B}^n\\right)$, $1\\leq k <\\frac{n}{2}$, $S_{n,k}$ is the sharp constant of $k$-th order Sobolev  inequality on $\\mathbb{R}^n$.\n(see Hebey \\cite{Hebey} and Liu \\cite{Liu}). In \\cite{LY19}, Lu and Yang  gave an alternative way to prove \\eqref{BSobolev}. In fact, let $g(x)=\\big(\\frac{2}{1-|x|^2}\\big)^{\\frac{n}{2}-k}f(x)$, using the conformal transformation laws (see Lemma 5.1 of \\cite{LY19}) $$P_k(f)=\\big(\\frac{2}{1-|x|^2}\\big)^{-\\frac{n}{2}-k}(-\\Delta)^k\\big((\\frac{2}{1-|x|^2})^{\\frac{n}{2}-k}f(x)\\big),$$\nwe can see that the $k$-th order Sobolev inequality on the hyperbolic space $\\mathbb{B}^n$ is equivalent to the following $k$-th order  Sobolev inequality on the unit ball of $\\mathbb{R}^n$:\n\\begin{equation}\\label{BallSobolev}\n\\int_{\\mathbb{B}^n}(-\\Delta)^k g\\cdot gdx\\geq S_{n,k}\\left(\\int_{\\mathbb{B}^n}|g|^{\\frac{2n}{n-2k}}dx\\right)^{\\frac{n-2k}{n}}.\n\\end{equation}\nObviously, the $k$-th order Sobolev inequality on the unit ball of $\\mathbb{R}^n$ has no extremals using translation, dilation and classification of extremals of Sobolev inequality in $\\mathbb{R}^n$ (see Talenti \\cite{Talenti} and Aubin \\cite{Aubin} and also Proposition 1.43 of \\cite{Willem} for the first order, and  Lieb \\cite{Lieb} forthe  high-order case). Hence the $k$-th order Sobolev inequality on the hyperbolic space $\\mathbb{B}^n$ does not admit any extremal function.\n\\medskip\n\nOn the other hand, we have the following $k$-th Poincar\\'e inequality on the hyperbolic space $\\mathbb{B}^n$:\n\\begin{equation}\\label{Bpoincare}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV\\geq \\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\left(\\int_{\\mathbb{B}^n}|f|^{2}dV\\right),\n\\end{equation}\nwhere $f\\in C_{c}^{\\infty}\\left(\\mathbb{B}^n\\right)$, and the constant $\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}$ is sharp. We need to note that the sharp constants cannot be attained.    Combining $k$-th order Sobolev inequality and $k$-th order Poincar\\'e inequality on the hyperbolic space $\\mathbb{B}^n$, there must exist some refinement of both of these two kinds of sharp inequalities. The refinement is simply demonstrated by the high-order Poincare-Sobolev inequality on the hyperbolic space \\eqref{BPS}. However, the problem for the existence of extremal of high-order Poincare-Sobolev inequality on the hyperbolic space still remains open. \n\nFor the first order critical HSM inequality, Hebey \\cite{Hebey} showed that the best constant $C_{n,1,\\frac{2n}{n-2}}$ is strictly smaller than the Sobolev constant $S_{n,1}$ for $n\\geq 4$. Tertikas and Tintarev  \\cite{TT} further proved that the existence of the extremal function for $n\\geq 4$. On the other hand, Benguria, Frank and Loss \\cite{BFL} demonstrated that the best constant $C_{n,1,\\frac{2n}{n-2}}$ is the same as the Sobolev constant $S_{n,1}$ and the best constant $C_{n,1,\\frac{2n}{n-2}}$ is not achieved when  $n=3$. For the high order critical case, Lu and Yang    proved in \\cite {LY22}  that the best constant $C_{n,k,\\frac{2n}{n-k}}$ for the kth-order HSM inequality coincides with the $k$-th Sobolev contant $S_{n,k}$ if $n=2k+1$ and the best constant $C_{n,k,\\frac{2n}{n-k}}$ is strictly smaller than $k$-th Sobolev contant $S_{n,k}$ if $n\\geq 2k+2$. However, the attainability of the best constant $C_{n,k,\\frac{2n}{n-2k}}$ remains open for $n\\ge 2k+2$ when $k\\ge 2$. In \\cite{TT}, the authors employed a clever transformation that reduces the HSM inequality to a weighted Sobolev inequality on the upper half-space, and by applying the concentration compactness principle, they established the existence of extremal functions in the first order case $k=1$. Due to the increased complexity of higher order operators, such a transformation is no longer available in the higher order setting.\n\\medskip\n\nOur first main result concerns the existence of extremal functions for the critical higher order Poincar\\'e-Sobolev inequality.\nWe define the Hilbert space $H\\left(\\mathbb{B}^n\\right)$ by the completion of $C_c^{\\infty}\\left(\\mathbb{B}^n\\right)$ under the norm\n$$\\|f\\|_{H\\left(\\mathbb{B}^n\\right)}=\\left(\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\right)^{\\frac{1}{2}}.$$\n\nIt is noted that $\\|f\\|_{H\\left(\\mathbb{B}^n\\right)}$ is indeed a norm by using the Poincare-Sobolev inequality.", "context": "The classical Hardy-Sobolev-Maz'ya (HSM) inequality provides a refinement of both the Sobolev and the Hardy inequalities on the upper half space $\\mathbb{R}^n_+$. It reads as follows: for $n>2k$, $2<p\\leq \\frac{2n}{n-2k}$ and $\\gamma=\\frac{\\left(n-2k\\right)p}{2}-n$, there exists some positive constant $C_{n,k,p}$ such that for each $u\\in C_{c}^{\\infty}\\left(\\mathbb{R}^{n}_{+}\\right)$, there holds\n\\begin{equation}\\label{HHSM}\n\\int_{\\mathbb{R}^{n}_{+}}|\\nabla^{k}u|^2dx-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{R}^{n}_{+}}\\frac{u^2}{x_1^{2k}}dx\\geq C_{n,k,p} \\left(\\int_{\\mathbb{R}^{n}_{+}}x_1^\\gamma|u|^{p}dx\\right)^{\\frac{2}{p}}.\n\\end{equation}\n\nLu and Yang \\cite{LY19} developed Fourier analysis techniques on the hyperbolic space which is a noncompact complete Riemannian manifold to establish the HSM inequalities for higher order derivatives on half spaces. (see also Flynn et al \\cite{FlynnLuYang} and \\cite{LuYang-AIM} for the other cases of symmetric spaces of rank one.) In fact, they derived the following Poincar\\'e-Sobolev inequalities for the GJMS operators on hyperbolic spaces $\\mathbb{B}^n$: for $n>2k$ and $2<p\\leq \\frac{2n}{n-2k}$, there exists some constant $C_{n,k,p}$ such that for each $f\\in C_{c}^{\\infty}\\left(\\mathbb{B}^n\\right)$, there holds\n\nThe $k$-th order Sobolev inequality on the hyperbolic space $\\mathbb{B}^n$ reads as follows:\n\\begin{equation}\\label{BSobolev}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV\\geq S_{n,k}\\left(\\int_{\\mathbb{B}^n}|f|^{\\frac{2n}{n-2k}}dV\\right)^{\\frac{n-2k}{n}},\n\\end{equation}\nwhere $f\\in C_{c}^{\\infty}\\left(\\mathbb{B}^n\\right)$, $1\\leq k <\\frac{n}{2}$, $S_{n,k}$ is the sharp constant of $k$-th order Sobolev inequality on $\\mathbb{R}^n$.\n(see Hebey \\cite{Hebey} and Liu \\cite{Liu}). In \\cite{LY19}, Lu and Yang gave an alternative way to prove \\eqref{BSobolev}. In fact, let $g(x)=\\big(\\frac{2}{1-|x|^2}\\big)^{\\frac{n}{2}-k}f(x)$, using the conformal transformation laws (see Lemma 5.1 of \\cite{LY19}) $$P_k(f)=\\big(\\frac{2}{1-|x|^2}\\big)^{-\\frac{n}{2}-k}(-\\Delta)^k\\big((\\frac{2}{1-|x|^2})^{\\frac{n}{2}-k}f(x)\\big),$$\nwe can see that the $k$-th order Sobolev inequality on the hyperbolic space $\\mathbb{B}^n$ is equivalent to the following $k$-th order Sobolev inequality on the unit ball of $\\mathbb{R}^n$:\n\\begin{equation}\\label{BallSobolev}\n\\int_{\\mathbb{B}^n}(-\\Delta)^k g\\cdot gdx\\geq S_{n,k}\\left(\\int_{\\mathbb{B}^n}|g|^{\\frac{2n}{n-2k}}dx\\right)^{\\frac{n-2k}{n}}.\n\\end{equation}\nObviously, the $k$-th order Sobolev inequality on the unit ball of $\\mathbb{R}^n$ has no extremals using translation, dilation and classification of extremals of Sobolev inequality in $\\mathbb{R}^n$ (see Talenti \\cite{Talenti} and Aubin \\cite{Aubin} and also Proposition 1.43 of \\cite{Willem} for the first order, and Lieb \\cite{Lieb} forthe high-order case). Hence the $k$-th order Sobolev inequality on the hyperbolic space $\\mathbb{B}^n$ does not admit any extremal function.\n\\medskip\n\nOn the other hand, we have the following $k$-th Poincar\\'e inequality on the hyperbolic space $\\mathbb{B}^n$:\n\\begin{equation}\\label{Bpoincare}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV\\geq \\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\left(\\int_{\\mathbb{B}^n}|f|^{2}dV\\right),\n\\end{equation}\nwhere $f\\in C_{c}^{\\infty}\\left(\\mathbb{B}^n\\right)$, and the constant $\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}$ is sharp. We need to note that the sharp constants cannot be attained. Combining $k$-th order Sobolev inequality and $k$-th order Poincar\\'e inequality on the hyperbolic space $\\mathbb{B}^n$, there must exist some refinement of both of these two kinds of sharp inequalities. The refinement is simply demonstrated by the high-order Poincare-Sobolev inequality on the hyperbolic space \\eqref{BPS}. However, the problem for the existence of extremal of high-order Poincare-Sobolev inequality on the hyperbolic space still remains open.\n\nOur first main result concerns the existence of extremal functions for the critical higher order Poincar\\'e-Sobolev inequality.\nWe define the Hilbert space $H\\left(\\mathbb{B}^n\\right)$ by the completion of $C_c^{\\infty}\\left(\\mathbb{B}^n\\right)$ under the norm\n$$\\|f\\|_{H\\left(\\mathbb{B}^n\\right)}=\\left(\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\right)^{\\frac{1}{2}}.$$\n\nIt is noted that $\\|f\\|_{H\\left(\\mathbb{B}^n\\right)}$ is indeed a norm by using the Poincare-Sobolev inequality.\n\n\\begin{equation}\\label{BPS}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\geq C_{n,k,p}\\left(\\int_{\\mathbb{B}^n}|f|^{p}dV\\right)^{\\frac{2}{p}},\n\\end{equation}", "full_context": "The classical Hardy-Sobolev-Maz'ya (HSM) inequality provides a refinement of both the Sobolev and the Hardy inequalities on the upper half space $\\mathbb{R}^n_+$. It reads as follows: for $n>2k$, $2<p\\leq \\frac{2n}{n-2k}$ and $\\gamma=\\frac{\\left(n-2k\\right)p}{2}-n$, there exists some positive constant $C_{n,k,p}$ such that for each $u\\in C_{c}^{\\infty}\\left(\\mathbb{R}^{n}_{+}\\right)$, there holds\n\\begin{equation}\\label{HHSM}\n\\int_{\\mathbb{R}^{n}_{+}}|\\nabla^{k}u|^2dx-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{R}^{n}_{+}}\\frac{u^2}{x_1^{2k}}dx\\geq C_{n,k,p} \\left(\\int_{\\mathbb{R}^{n}_{+}}x_1^\\gamma|u|^{p}dx\\right)^{\\frac{2}{p}}.\n\\end{equation}\n\nLu and Yang \\cite{LY19} developed Fourier analysis techniques on the hyperbolic space which is a noncompact complete Riemannian manifold to establish the HSM inequalities for higher order derivatives on half spaces. (see also Flynn et al \\cite{FlynnLuYang} and \\cite{LuYang-AIM} for the other cases of symmetric spaces of rank one.) In fact, they derived the following Poincar\\'e-Sobolev inequalities for the GJMS operators on hyperbolic spaces $\\mathbb{B}^n$: for $n>2k$ and $2<p\\leq \\frac{2n}{n-2k}$, there exists some constant $C_{n,k,p}$ such that for each $f\\in C_{c}^{\\infty}\\left(\\mathbb{B}^n\\right)$, there holds\n\nThe $k$-th order Sobolev inequality on the hyperbolic space $\\mathbb{B}^n$ reads as follows:\n\\begin{equation}\\label{BSobolev}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV\\geq S_{n,k}\\left(\\int_{\\mathbb{B}^n}|f|^{\\frac{2n}{n-2k}}dV\\right)^{\\frac{n-2k}{n}},\n\\end{equation}\nwhere $f\\in C_{c}^{\\infty}\\left(\\mathbb{B}^n\\right)$, $1\\leq k <\\frac{n}{2}$, $S_{n,k}$ is the sharp constant of $k$-th order Sobolev inequality on $\\mathbb{R}^n$.\n(see Hebey \\cite{Hebey} and Liu \\cite{Liu}). In \\cite{LY19}, Lu and Yang gave an alternative way to prove \\eqref{BSobolev}. In fact, let $g(x)=\\big(\\frac{2}{1-|x|^2}\\big)^{\\frac{n}{2}-k}f(x)$, using the conformal transformation laws (see Lemma 5.1 of \\cite{LY19}) $$P_k(f)=\\big(\\frac{2}{1-|x|^2}\\big)^{-\\frac{n}{2}-k}(-\\Delta)^k\\big((\\frac{2}{1-|x|^2})^{\\frac{n}{2}-k}f(x)\\big),$$\nwe can see that the $k$-th order Sobolev inequality on the hyperbolic space $\\mathbb{B}^n$ is equivalent to the following $k$-th order Sobolev inequality on the unit ball of $\\mathbb{R}^n$:\n\\begin{equation}\\label{BallSobolev}\n\\int_{\\mathbb{B}^n}(-\\Delta)^k g\\cdot gdx\\geq S_{n,k}\\left(\\int_{\\mathbb{B}^n}|g|^{\\frac{2n}{n-2k}}dx\\right)^{\\frac{n-2k}{n}}.\n\\end{equation}\nObviously, the $k$-th order Sobolev inequality on the unit ball of $\\mathbb{R}^n$ has no extremals using translation, dilation and classification of extremals of Sobolev inequality in $\\mathbb{R}^n$ (see Talenti \\cite{Talenti} and Aubin \\cite{Aubin} and also Proposition 1.43 of \\cite{Willem} for the first order, and Lieb \\cite{Lieb} forthe high-order case). Hence the $k$-th order Sobolev inequality on the hyperbolic space $\\mathbb{B}^n$ does not admit any extremal function.\n\\medskip\n\nOn the other hand, we have the following $k$-th Poincar\\'e inequality on the hyperbolic space $\\mathbb{B}^n$:\n\\begin{equation}\\label{Bpoincare}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV\\geq \\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\left(\\int_{\\mathbb{B}^n}|f|^{2}dV\\right),\n\\end{equation}\nwhere $f\\in C_{c}^{\\infty}\\left(\\mathbb{B}^n\\right)$, and the constant $\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}$ is sharp. We need to note that the sharp constants cannot be attained. Combining $k$-th order Sobolev inequality and $k$-th order Poincar\\'e inequality on the hyperbolic space $\\mathbb{B}^n$, there must exist some refinement of both of these two kinds of sharp inequalities. The refinement is simply demonstrated by the high-order Poincare-Sobolev inequality on the hyperbolic space \\eqref{BPS}. However, the problem for the existence of extremal of high-order Poincare-Sobolev inequality on the hyperbolic space still remains open.\n\nOur first main result concerns the existence of extremal functions for the critical higher order Poincar\\'e-Sobolev inequality.\nWe define the Hilbert space $H\\left(\\mathbb{B}^n\\right)$ by the completion of $C_c^{\\infty}\\left(\\mathbb{B}^n\\right)$ under the norm\n$$\\|f\\|_{H\\left(\\mathbb{B}^n\\right)}=\\left(\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\right)^{\\frac{1}{2}}.$$\n\nIt is noted that $\\|f\\|_{H\\left(\\mathbb{B}^n\\right)}$ is indeed a norm by using the Poincare-Sobolev inequality.\n\n\\begin{equation}\\label{BPS}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\geq C_{n,k,p}\\left(\\int_{\\mathbb{B}^n}|f|^{p}dV\\right)^{\\frac{2}{p}},\n\\end{equation}\n\nIt is noted that $\\|f\\|_{H\\left(\\mathbb{B}^n\\right)}$ is indeed a norm by using the Poincare-Sobolev inequality.\n\n\\begin{theorem}\\label{EHMS}\n Assume $n\\geq2k+2$. Then there is a positive function $f_0\\in \\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)$ which achieves the equality in inequality\n \\eqref{AHSM},\nwhere $\\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)$ is the completion of $C_c^{\\infty}\\left(\\mathbb{R}^{n}_{+}\\right)$ under the norm\n$$\\|u\\|_{\\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)}=\\left(\\int_{\\mathbb{R}^{n}_{+}}|\\nabla^{k}u|^2dx-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{R}^{n}_{+}}\\frac{u^2}{x_1^{2k}}dx\\right)^{\\frac{1}{2}}.$$\n\n\\begin{theorem}\\label{E-subHPS}\n Assume that $2<p<\\frac{2n}{n-2k}$ and $n>2k$. Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality in the inequality \\eqref{BPS},\n\\end{theorem}\n\n\\begin{theorem}\\label{E-subHHSM}\nAssume that $2<p<\\frac{2n}{n-2k}$ and $n>2k$. \n Then there is a positive function $f_0\\in \\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)$ which achieves the equality in inequality \\eqref{HHSM}.\n\n\\begin{corollary}\\label{EL}\nLet $\\alpha=\\prod\\limits_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}$. If either $2k+2\\leq n$, $p=\\frac{2n}{n-2k}$ or $2k<n$, $2<p<\\frac{2n}{n-2k}$, then there exists at least one positive solution u in $H\\left(\\mathbb{B}^n\\right)$ to the equation \\eqref{sublambdaBN}, which is radially symmetry and monotone decreasing about some point $P\\in \\mathbb{B}^n$.\n\\end{corollary}\n\nFrom Lemma \\ref{kernel-decreasing} (presented later), we know that $G^{-1}_k\\left(x,y\\right)$ is symmetric and decreasing about the distance function $\\rho\\left(T_{x}\\left(y\\right)\\right)$. From the Symmetrization Lemma \\ref{becknersymm} in \\cite{Beckner93}, we can pick up radially decreasing maximizing sequence $\\{h_m\\}$ for the Hardy-Littlewood-Sobolev inequality \\eqref{BHLS}. By the direct equivalence of the inequality \\eqref{BHLS} and \\eqref{G2}, we know that $\\{h_m\\}$ is also the radially decreasing maximizing sequence for inequality \\eqref{G2}, that is, \n$$\\int_{\\mathbb{B}^n}|h_m|^{\\frac{2n}{n+2k}}dV=1,\\ \\ \\lim\\limits_{m\\rightarrow +\\infty} \\int_{\\mathbb{B}^n}|G_k^{-\\frac{1}{2}}\\left(h_m\\right)|^{2}dV=C_{n,k,\\frac{2n}{n-2k}}^{-1}.$$ \nIn the following Lemma \\ref{relationshipsequence}, we proved the relationship between the extremizing sequence of \\eqref{G1} and the extremizing sequence of \\eqref{G2}. Pick $f_m=G_{k}^{-1}\\left(h_m\\right)$, then $f_m\\left(x\\right)$ is a radially decreasing extremizing sequence for inequality \\eqref{G1}. Up to some constants, we have \n$$\\lim_{m\\rightarrow\\infty}\\|f_m\\|^2_{H(\\mathbb{B}^n)}=C_{n,k,\\frac{2n}{n-2k}},\\ \\ \\ \\int_{\\mathbb{B}^n}|f_m|^{\\frac{2n}{n-2k}}dV=1.$$\n\\end{proof}\n\n\\section{The proof of Theorem \\ref{E-subHPS}}\nWe now start to prove that the high-order subcritical Poincar\\'e-Sobolev inequality on Hyperbolic space $\\mathbb{B}^n$:\n$$\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\geq C_{n,k,q}\\left(\\int_{\\mathbb{B}^n}|f|^{q}dV\\right)^{\\frac{2}{q}}$$\nfor $2<q<\\frac{2n}{n-2k}$\nadmits an extremal function. Assume that $\\{f_m\\}$ is a minimizing sequence for the subcritical Poincar\\'e-Sobolev inequality on hyperbolic space $\\mathbb{B}^n$, that is\n$$\\int_{\\mathbb{B}^n}|f_m|^{q}dV=1,\\ \\ \\lim\\limits_{m\\rightarrow +\\infty}\\left(\\int_{\\mathbb{B}^n}P_k\\left(f_m\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f_m|^2dV\\right)=C_{n,k,q}.$$\nSimilar to the proof of the existence of Poincar\\'e-Sobolev inequality on hyperbolic space $\\mathbb{B}^n$ (Theorem \\ref{EPS}), by Green representative formula and Riesz rearrangement inequality, we can assume that the minimizing sequence $\\{f_m\\}$ is radially decreasing about origin.\nIn order to prove the existence of extremal function, we only need to prove that\n$f_m\\rightarrow f_0$ strongly in $L^{q}\\left(\\mathbb{B}^n\\right)$. Since $f_m$ is radially decreasing, by Helly theorem, we know that there exists $f_0\\in L^{p}\\left(\\mathbb{B}^n\\right)$ such that $f_m$ converges to $f_0$ for almost $x\\in \\mathbb{B}^n$. Through Poincar\\'e-Sobolev inequality, we see that\n$f_m$ is also bounded in $L^{\\frac{2n}{n-2k}}\\left(\\mathbb{B}^n\\right)$. This together with the Vitali-convergence theorem yields that\n$$\\lim\\limits_{R\\rightarrow 1}\\lim\\limits_{m\\rightarrow +\\infty}\\int_{B^n\\left(0,R\\right)}|f_m|^qdV=\\lim\\limits_{R\\rightarrow 1}\\int_{B^n\\left(0,R\\right)}|f_0|^qdV=\\int_{\\mathbb{B}^n}|f_0|^qdV.$$\nIn order to prove that $f_m$ strongly converges to $f_0$ in $L^q\\left(\\mathbb{B}^n\\right)$, it remains to show that\n$$\\lim\\limits_{R\\rightarrow 1}\\lim\\limits_{m\\rightarrow +\\infty}\\int_{\\mathbb{B}^n\\setminus B^n\\left(0,R\\right)}|f_m|^qdV=0,$$\nthat is,\n$$\\lim\\limits_{R\\rightarrow 1}\\lim\\limits_{m\\rightarrow +\\infty}\\int_{B^n\\setminus B^n\\left(0,R\\right)}|f_m|^q\\left(\\frac{2}{1-|x|^2}\\right)^ndx=0.$$\nFrom the previous radial Lemma \\ref{radial}, we see that there exists $D_{n,k,q_0}>0$ such that\n$f_m\\left(R\\right)\\leq \\left(1-R\\right)^{\\frac{n-1}{q_0}}$, where $q_0$ is chosen as $q_0<q$. Then it is not difficult to check that\n\\begin{equation}\\begin{split}\n&\\lim\\limits_{R\\rightarrow 1}\\lim\\limits_{m\\rightarrow +\\infty}\\int_{B^n\\setminus B^n\\left(0,R\\right)}|f_m|^q\\left(\\frac{2}{1-|x|^2}\\right)^ndx\\\\\n\\leq &D_{n,k,q_0}\\lim\\limits_{R\\rightarrow 1}\\lim\\limits_{m\\rightarrow +\\infty}\\int_{B^n\\setminus B^n\\left(0,R\\right)}\\left(1-R\\right)^{\\frac{q\\left(n-1\\right)}{q_0}}\\left(\\frac{2}{1-|x|^2}\\right)^ndx=0.\n\\end{split}\\end{equation}", "post_theorem_intro_text_len": 6158, "post_theorem_intro_text": "The proof relies on developing the Lions' type of the concentration-compactness principle \\cite{Lions841, Lions1} for differential inequalities on the hyperbolic space. The concentration-compactness principle for differential inequalities can be divided into the first and the second differential concentration-compactness. One of the key points is to prove the minimizing sequence $\\{u_k\\}$ for higher order Poincar\\'e-Sobolev inequality is tight. In order to achieve this, we must exclude the vanishing  and dichotomy phenomenon. The loss of invariance of dilation causes much challenge to exclude the vanishing phenomenon. In order to overcome these difficulties, we need to find the radially minimizing sequence which decays at infinity of hyperbolic space and help to exclude the vanishing phenomenon. However, P\\\"{o}lya-Szeg\\\"{o} rearrangement inequality is invalid for high-order differential inequality. Applying the equivalence of Poincar\\'e-Sobolev inequality and Hardy-Littlewood-Sobolev inequality on the hyperbolic space, Riesz's rearrangement inequality on hyperbolic space and Green's representative formula, one can find the radially minimizing sequence for Poincar\\'e-Sobolev inequality. To exclude  the dichotomy, one cannot follow the same line of Lions' proof in \\cite{Lions1}. In fact, recall the proof of excluding dichotomy for minimizing sequence $w_k$ of $L^p$ Sobolev inequality involving high order derivatives: $$I=\\inf\\{\\int_{\\mathbb{R}^n}|\\nabla^{m}w|^pdx,\\ \\ \\int_{\\mathbb{R}^n}|w|^{\\frac{np}{n-mp}}dx=1\\}.$$ Lions constructed a suitable concentration-function $$\\rho_k=\\sum_{j=0}^{m}|D^jw_k|^{q_j},\\  q_j=\\frac{np}{n-(m-j)p}$$\nand prove that for any $\\epsilon>0$, there exists $y_k\\in \\mathbb{R}^n$ and $R_{\\epsilon}>0$ such that $\\int_{|x-y_k|\\geq R}\\rho_k(x)dx<\\epsilon$ for any $R>R_{\\epsilon}$. This is achieved by scaling cut-off function and \n$$\\int_{R_0<|x-y_k|<R_k}\\sum_{j=0}^{m-1}|D^jw_k|^{q_j}dx=o_k(1).$$ However, for higher order Poincar\\'e-Sobolev inequality in hyperbolic space, it is impossible to introduce the replacement of concentration-function $\\rho_k$ of $\\mathbb{R}^n$. Moreover, the loss of scaling cut-off function can also add extra difficulty. In order to overcome this difficulty, we found a quantitative relationship between the minimizing sequences for the Poincare-Sobolev inequalities and those for the  equivalent integral inequalities (It is differen from the equivalence of extremals), allowing us to apply the concentration-compactness techniques for equivalent integral inequalities to rule out dichotomy of the minimizing sequences for Poincar\\'e-Sobolev inequality. Once we prove that the minimizing sequence is compact, we can follow the standard second Lions' concentration-compactness procedure to exclude the minimizing sequence converging $\\delta$-measure using $C_{n,k,\\frac{2n}{n-k}}<S_{n,k}$.\n\n\\medskip\n\nThrough the equivalence, combining Theorem \\ref{EPS}, we immediately obtain the existence of the extremal function for the  high order critical Hardy-Sobolev-Maz'ya inequality \\eqref{AHSM} on the upper half space.\n\n\\begin{theorem}\\label{EHMS}\n Assume $n\\geq2k+2$.  Then there is a positive function $f_0\\in \\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)$ which achieves the equality in inequality\n \\eqref{AHSM},\nwhere $\\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)$ is the completion of $C_c^{\\infty}\\left(\\mathbb{R}^{n}_{+}\\right)$ under the norm\n$$\\|u\\|_{\\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)}=\\left(\\int_{\\mathbb{R}^{n}_{+}}|\\nabla^{k}u|^2dx-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{R}^{n}_{+}}\\frac{u^2}{x_1^{2k}}dx\\right)^{\\frac{1}{2}}.$$\n\n\\end{theorem}\n\nFor $n=2k+1$, according to Lemma 4.3 of \\cite{LY22}, we can deduce the nonexistence of the extremal function for critical inequality \\eqref{BPS} and \\eqref{HHSM}. Different from the critical case, the subcritical inequalities have an extremal function for all $n>2k$. We have the following results:\n\n\\begin{theorem}\\label{E-subHPS}\n  Assume that $2<p<\\frac{2n}{n-2k}$ and $n>2k$. Then there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves the equality in the inequality \\eqref{BPS},\n\\end{theorem}\n\n\\begin{theorem}\\label{E-subHHSM}\nAssume that $2<p<\\frac{2n}{n-2k}$ and $n>2k$. \n Then there is a positive function $f_0\\in \\widetilde{H}\\left(\\mathbb{R}^{n}_{+}\\right)$ which achieves the equality in   inequality \\eqref{HHSM}.\n\n\\end{theorem}\n\nThe Poincar\\'e-Sobolev inequalities  are highly linked with the  Brezis-Nirenberg problem \\cite{Brezis} on the entire hyperbolic space $\\mathbb{B}^n$:\n\\begin{equation}\\label{sublambdaBN}\nP_{k}\\left(f\\right)-\\alpha f=|f|^{p-2}f.\n\\end{equation}\nFor $k=1$, Mancini and Sandeep \\cite{MS} showed that the entire solution exists either when  $n\\geq4$, $p=\\frac{2n}{n-2}$, $0<\\alpha\\leq\\frac{1}{4}$ or when $n\\geq3$, $1<p<\\frac{2n}{n-2}$ and $\\alpha\\leq\\frac{1}{4}$. Bhakta et al. \\cite{BGKM25} proved the quantitative stability for the equation \\eqref{sublambdaBN} when $k=1$. \n For $k\\geq 2$, Li et al. \\cite{LLY} established the existence of the nontrivial solution  when $n\\geq 2k+2$, $p=\\frac{2n}{n-2k}$, $\\alpha<\\prod\\limits_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}$. It is not hard to calculate that the corresponding Euler-Lagrange equation of the inequality \\eqref{BPS} is the  equation \\eqref{sublambdaBN} with $\\alpha=\\prod\\limits_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}$, $2<p\\leq\\frac{2n}{n-2k}$.  A straightforward corollary of Theorem \\ref{EPS} and Theorem \\ref{E-subHPS} is as follows:\n\n\\begin{corollary}\\label{EL}\nLet $\\alpha=\\prod\\limits_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}$. If either $2k+2\\leq n$, $p=\\frac{2n}{n-2k}$ or $2k<n$, $2<p<\\frac{2n}{n-2k}$,  then there exists at least one positive solution u in $H\\left(\\mathbb{B}^n\\right)$ to the equation \\eqref{sublambdaBN},  which is radially symmetry  and monotone decreasing about some point $P\\in \\mathbb{B}^n$.\n\\end{corollary}\n\n\\begin{remark}\n   It is the first time to derive the existence of positive solutions for high-order Brezis-Nirenberg problem for $\\alpha=\\prod\\limits_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}$ on the whole hyperbolic space.\n\\end{remark}", "sketch": "The proof of Theorem~\\ref{EPS} is sketched as follows.\n\n1. **Concentration--compactness framework on hyperbolic space.** The proof “relies on developing the Lions' type of the concentration-compactness principle … for differential inequalities on the hyperbolic space,” split into “the first and the second differential concentration-compactness.”\n\n2. **Tightness of a minimizing sequence and exclusion of vanishing/dichotomy.** A “key point is to prove the minimizing sequence $\\{u_k\\}$ for higher order Poincar\\'e-Sobolev inequality is tight,” which requires excluding “the vanishing and dichotomy phenomenon.” The “loss of invariance of dilation causes much challenge to exclude the vanishing phenomenon.”\n\n3. **Radially minimizing sequence to rule out vanishing (via integral-equation equivalence and rearrangement tools).** To overcome the failure of direct rearrangement (“P\\\"{o}lya-Szeg\\\"{o} rearrangement inequality is invalid for high-order differential inequality”), the sketch uses “the equivalence of Poincar\\'e-Sobolev inequality and Hardy-Littlewood-Sobolev inequality on the hyperbolic space, Riesz's rearrangement inequality on hyperbolic space and Green's representative formula,” to “find the radially minimizing sequence for Poincar\\'e-Sobolev inequality” which “decays at infinity of hyperbolic space” and “help[s] to exclude the vanishing phenomenon.”\n\n4. **Ruling out dichotomy via a quantitative link to equivalent integral inequalities.** Since “to exclude the dichotomy, one cannot follow the same line of Lions' proof,” and it is “impossible to introduce the replacement of concentration-function $\\rho_k$ of $\\mathbb{R}^n$” (and the “loss of scaling cut-off function” adds difficulty), the approach is: “we found a quantitative relationship between the minimizing sequences for the Poincare-Sobolev inequalities and those for the equivalent integral inequalities … allowing us to apply the concentration-compactness techniques for equivalent integral inequalities to rule out dichotomy of the minimizing sequences for Poincar\\'e-Sobolev inequality.”\n\n5. **Compactness and exclusion of concentration to a $\\delta$-measure (second Lions procedure).** “Once we prove that the minimizing sequence is compact, we can follow the standard second Lions' concentration-compactness procedure to exclude the minimizing sequence converging $\\delta$-measure using $C_{n,k,\\frac{2n}{n-k}}<S_{n,k}$.”\n\nThese steps yield existence of an extremal, and the sketch indicates that the construction produces a “radially minimizing sequence,” matching the stated radial symmetry/monotone decrease conclusion in Theorem~\\ref{EPS}.", "expanded_sketch": "The proof of the main theorem is sketched as follows.\n\n1. **Concentration--compactness framework on hyperbolic space.** The proof “relies on developing the Lions' type of the concentration-compactness principle … for differential inequalities on the hyperbolic space,” split into “the first and the second differential concentration-compactness.”\n\n2. **Tightness of a minimizing sequence and exclusion of vanishing/dichotomy.** A “key point is to prove the minimizing sequence $\\{u_k\\}$ for higher order Poincar\\'e-Sobolev inequality is tight,” which requires excluding “the vanishing and dichotomy phenomenon.” The “loss of invariance of dilation causes much challenge to exclude the vanishing phenomenon.”\n\n3. **Radially minimizing sequence to rule out vanishing (via integral-equation equivalence and rearrangement tools).** To overcome the failure of direct rearrangement (“P\\\"{o}lya-Szeg\\\"{o} rearrangement inequality is invalid for high-order differential inequality”), the sketch uses “the equivalence of Poincar\\'e-Sobolev inequality and Hardy-Littlewood-Sobolev inequality on the hyperbolic space, Riesz's rearrangement inequality on hyperbolic space and Green's representative formula,” to “find the radially minimizing sequence for Poincar\\'e-Sobolev inequality” which “decays at infinity of hyperbolic space” and “help[s] to exclude the vanishing phenomenon.”\n\n4. **Ruling out dichotomy via a quantitative link to equivalent integral inequalities.** Since “to exclude the dichotomy, one cannot follow the same line of Lions' proof,” and it is “impossible to introduce the replacement of concentration-function $\\rho_k$ of $\\mathbb{R}^n$” (and the “loss of scaling cut-off function” adds difficulty), the approach is: “we found a quantitative relationship between the minimizing sequences for the Poincare-Sobolev inequalities and those for the equivalent integral inequalities … allowing us to apply the concentration-compactness techniques for equivalent integral inequalities to rule out dichotomy of the minimizing sequences for Poincar\\'e-Sobolev inequality.”\n\n5. **Compactness and exclusion of concentration to a $\\delta$-measure (second Lions procedure).** “Once we prove that the minimizing sequence is compact, we can follow the standard second Lions' concentration-compactness procedure to exclude the minimizing sequence converging $\\delta$-measure using $C_{n,k,\\frac{2n}{n-k}}<S_{n,k}$.”\n\nThese steps yield existence of an extremal, and the sketch indicates that the construction produces a “radially minimizing sequence,” matching the stated radial symmetry/monotone decrease conclusion in the main theorem.", "expanded_theorem": "\\label{EPS} Assume that $p=\\frac{2n}{n-2k}$ and $n\\geq2k+2$.\nThen there is a positive function $f_0\\in H\\left(\\mathbb{B}^n\\right)$ which achieves equality in the inequality\n\\begin{equation}\\label{BPS}\n\\int_{\\mathbb{B}^n}P_k\\left(f\\right)fdV-\\prod_{i=1}^{k}\\frac{\\left(2i-1\\right)^2}{4}\\int_{\\mathbb{B}^n}|f|^2dV\\geq C_{n,k,p}\\left(\\int_{\\mathbb{B}^n}|f|^{p}dV\\right)^{\\frac{2}{p}},\n\\end{equation}\nFurthermore, the extremal function $f_0$ must be radially symmetric and monotone decreasing about some point $P\\in \\mathbb{B}^n$, that is, $f_0$ is a constant on the geodesic sphere centered at $P\\in \\mathbb{B}^n$ and radially decreasing about the geodesic distance from $P$.", "theorem_type": ["Existence", "Universal"], "mcq": {"question": "Let $p=\\frac{2n}{n-2k}$ and assume $n\\ge 2k+2$. On the hyperbolic space $\\mathbb{B}^n$ with volume element $dV$, let $P_k$ denote the $k$th-order GJMS operator, and let $H(\\mathbb{B}^n)$ be the completion of $C_c^{\\infty}(\\mathbb{B}^n)$ under the norm\n$$\n\\|f\\|_{H(\\mathbb{B}^n)}=\\left(\\int_{\\mathbb{B}^n}P_k(f)f\\,dV-\\prod_{i=1}^{k}\\frac{(2i-1)^2}{4}\\int_{\\mathbb{B}^n}|f|^2\\,dV\\right)^{1/2}.\n$$\nWhich statement holds for the critical higher-order Poincar\\'e-Sobolev inequality\n$$\n\\int_{\\mathbb{B}^n}P_k(f)f\\,dV-\\prod_{i=1}^{k}\\frac{(2i-1)^2}{4}\\int_{\\mathbb{B}^n}|f|^2\\,dV\\ge C_{n,k,p}\\left(\\int_{\\mathbb{B}^n}|f|^{p}\\,dV\\right)^{2/p}?\n$$", "correct_choice": {"label": "A", "text": "There exists a positive function $f_0\\in H(\\mathbb{B}^n)$ that attains equality in this inequality. Moreover, this extremal function is radially symmetric and monotone decreasing about some point $P\\in\\mathbb{B}^n$; equivalently, $f_0$ is constant on each geodesic sphere centered at $P$ and decreases with the geodesic distance from $P$."}, "choices": [{"label": "B", "text": "There exists a positive function $f_0\\in H(\\mathbb{B}^n)$ that attains equality in this inequality. Moreover, every extremal function is radially symmetric and monotone decreasing about the origin; equivalently, $f_0$ is constant on each geodesic sphere centered at $0\\in\\mathbb{B}^n$ and decreases with the geodesic distance from the origin."}, {"label": "C", "text": "There exists a positive function $f_0\\in H(\\mathbb{B}^n)$ that attains equality in this inequality."}, {"label": "D", "text": "For every minimizing sequence $\\{f_m\\}\\subset H(\\mathbb{B}^n)$ for this inequality, there exists a point $P\\in\\mathbb{B}^n$ such that, after passing to a subsequence, $f_m$ converges strongly in $H(\\mathbb{B}^n)$ to a positive function $f_0$ attaining equality. Moreover, $f_0$ is radially symmetric and monotone decreasing about $P$."}, {"label": "E", "text": "There exists a positive function $f_0\\in H(\\mathbb{B}^n)$ that attains equality in this inequality, and the best constant satisfies $C_{n,k,p}=S_{n,k}$. Moreover, the extremal function is radially symmetric and monotone decreasing about some point $P\\in\\mathbb{B}^n$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "center_of_radial_symmetry", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "radial_symmetry_and_monotonicity_clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "concentration_compactness", "tampered_component": "compactness_of_every_minimizing_sequence_in_H", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "strict_inequality_Cnkp_less_than_Snk", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only the setup and asks which conclusion is valid; it does not explicitly state or strongly hint at the correct option."}, "TAS": {"score": 1, "justification": "This is close to a theorem-identification question: the correct choice essentially states the theorem's conclusion, though the alternatives introduce meaningful variations in symmetry center, compactness, and sharp constant claims."}, "GPS": {"score": 1, "justification": "Selecting the correct answer requires comparing nuanced mathematical claims, but it mainly tests recognition of the exact theorem statement rather than deriving a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one fixes the symmetry center at the origin, one gives a weaker true statement, one overstates compactness of minimizing sequences, and one tampers with the sharp constant. These reflect realistic failure modes."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it is still fairly theorem-recall-driven rather than strongly generative."}}
{"id": "2602.05254v1", "paper_link": "http://arxiv.org/abs/2602.05254v1", "theorems_cnt": 2, "theorem": {"env_name": "Lemma", "content": "\\label{lem:lower}\nIf there exists a complete capset in $\\F_3^n$ with $N$ points, then\n$N(N+1)/2 \\ge 3^n$.", "start_pos": 8194, "end_pos": 8325, "label": "lem:lower"}, "ref_dict": {"cor:upper": "\\begin{Corollary}\n\\label{cor:upper}\nFor every $n$, there exists a complete capset in $\\F_3^n$ of size $O(\\sqrt{3^n})$. \\end{Corollary}", "lem:lower": "\\begin{Lemma}\n\\label{lem:lower}\nIf there exists a complete capset in $\\F_3^n$ with $N$ points, then\n$N(N+1)/2 \\ge 3^n$.\n\\end{Lemma}"}, "pre_theorem_intro_text_len": 1938, "pre_theorem_intro_text": "A capset is a subset $C \\subset \\F_3^n$ with no three points on a line. A question that has attracted a lot of interest (see e.g. \\cite{Tao},\\cite{Matrix}) is the following: What is the largest size of a capset in $\\F_3^n$ as a function of $n$? If we denote this value by $a(n)$ then it can be shown (see \\cite[Proposition 2.2]{Tyrrell}) that $c=\\lim a(n)^{1/n}$ exists. Moreover, it is known that $2.2202 \\le c$ (\\cite{Nature}) and $c \\le 2.756$ (\\cite{Jordan}). The exact value of $a(n)$ is known for $n \\le 6$ (\\cite{capsetKnownSizes1, capetKnownSizes2, capsetKnownSizes3, capsetKnownSizes4}). It is also known that $a(8)\\geq 512$ (\\cite{Nature}).\n\n\\begin{Definition}\n    A \\emph{capset} is a subset of $\\mathbb{F}_3^n$ containing no three distinct points on a line. A capset is \\emph{complete} if it is not a subset of a larger capset. \n    \\end{Definition}\n\nThe main purpose of this paper is to exhibit some constructions of complete capsets and, in particular, construct the smallest known complete capsets.\n\nIf $q=3^m$ is a power of $3$, choosing a basis for $\\F_q/\\F_3$ induces an\nidentification of $\\F_q^d$ with $\\F_3^{dm}$ for any integer $d$. We will\ndefine some subsets of $\\F_q^d$ by algebraic equations and view them as\ncapsets in $\\F_3^{dm}$ to study their properties.\n\nWe can also consider subsets of $\\F_q^d$ with no three points on an $\\F_q$-line.\nWe call such a set a cap (to distinguish it from a capset under the above \nidentification). It is clear that a cap is a capset but not conversely.\nWe can also consider the analogous notion of complete caps. A cap $C$ is complete as a cap if $C$ is not contained in a larger cap. That is, for every\npoint $P$ in the complement of $C$, there exists an $\\F_q$-line through $P$\nwhich meets $C$ in two points. We remark that a cap can be complete as a cap and yet not be complete as a capset. For example, a conic in the plane $\\F_q^2, q >3$. This follows from the lemma below.", "context": "A capset is a subset $C \\subset \\F_3^n$ with no three points on a line. A question that has attracted a lot of interest (see e.g. \\cite{Tao},\\cite{Matrix}) is the following: What is the largest size of a capset in $\\F_3^n$ as a function of $n$? If we denote this value by $a(n)$ then it can be shown (see \\cite[Proposition 2.2]{Tyrrell}) that $c=\\lim a(n)^{1/n}$ exists. Moreover, it is known that $2.2202 \\le c$ (\\cite{Nature}) and $c \\le 2.756$ (\\cite{Jordan}). The exact value of $a(n)$ is known for $n \\le 6$ (\\cite{capsetKnownSizes1, capetKnownSizes2, capsetKnownSizes3, capsetKnownSizes4}). It is also known that $a(8)\\geq 512$ (\\cite{Nature}).\n\n\\begin{Definition}\n A \\emph{capset} is a subset of $\\mathbb{F}_3^n$ containing no three distinct points on a line. A capset is \\emph{complete} if it is not a subset of a larger capset. \n \\end{Definition}\n\nThe main purpose of this paper is to exhibit some constructions of complete capsets and, in particular, construct the smallest known complete capsets.\n\nIf $q=3^m$ is a power of $3$, choosing a basis for $\\F_q/\\F_3$ induces an\nidentification of $\\F_q^d$ with $\\F_3^{dm}$ for any integer $d$. We will\ndefine some subsets of $\\F_q^d$ by algebraic equations and view them as\ncapsets in $\\F_3^{dm}$ to study their properties.\n\nWe can also consider subsets of $\\F_q^d$ with no three points on an $\\F_q$-line.\nWe call such a set a cap (to distinguish it from a capset under the above \nidentification). It is clear that a cap is a capset but not conversely.\nWe can also consider the analogous notion of complete caps. A cap $C$ is complete as a cap if $C$ is not contained in a larger cap. That is, for every\npoint $P$ in the complement of $C$, there exists an $\\F_q$-line through $P$\nwhich meets $C$ in two points. We remark that a cap can be complete as a cap and yet not be complete as a capset. For example, a conic in the plane $\\F_q^2, q >3$. This follows from the lemma below.", "full_context": "A capset is a subset $C \\subset \\F_3^n$ with no three points on a line. A question that has attracted a lot of interest (see e.g. \\cite{Tao},\\cite{Matrix}) is the following: What is the largest size of a capset in $\\F_3^n$ as a function of $n$? If we denote this value by $a(n)$ then it can be shown (see \\cite[Proposition 2.2]{Tyrrell}) that $c=\\lim a(n)^{1/n}$ exists. Moreover, it is known that $2.2202 \\le c$ (\\cite{Nature}) and $c \\le 2.756$ (\\cite{Jordan}). The exact value of $a(n)$ is known for $n \\le 6$ (\\cite{capsetKnownSizes1, capetKnownSizes2, capsetKnownSizes3, capsetKnownSizes4}). It is also known that $a(8)\\geq 512$ (\\cite{Nature}).\n\n\\begin{Definition}\n A \\emph{capset} is a subset of $\\mathbb{F}_3^n$ containing no three distinct points on a line. A capset is \\emph{complete} if it is not a subset of a larger capset. \n \\end{Definition}\n\nThe main purpose of this paper is to exhibit some constructions of complete capsets and, in particular, construct the smallest known complete capsets.\n\nIf $q=3^m$ is a power of $3$, choosing a basis for $\\F_q/\\F_3$ induces an\nidentification of $\\F_q^d$ with $\\F_3^{dm}$ for any integer $d$. We will\ndefine some subsets of $\\F_q^d$ by algebraic equations and view them as\ncapsets in $\\F_3^{dm}$ to study their properties.\n\nWe can also consider subsets of $\\F_q^d$ with no three points on an $\\F_q$-line.\nWe call such a set a cap (to distinguish it from a capset under the above \nidentification). It is clear that a cap is a capset but not conversely.\nWe can also consider the analogous notion of complete caps. A cap $C$ is complete as a cap if $C$ is not contained in a larger cap. That is, for every\npoint $P$ in the complement of $C$, there exists an $\\F_q$-line through $P$\nwhich meets $C$ in two points. We remark that a cap can be complete as a cap and yet not be complete as a capset. For example, a conic in the plane $\\F_q^2, q >3$. This follows from the lemma below.\n\n\\begin{abstract}\nCapsets are subsets of $\\F_3^n$ with no three points on a line and a capset is complete if it is not a subset of a larger capset. We study some new constructions of capsets via algebraic equations over extensions of $\\F_3$. In particular we construct the smallest known complete capsets with size proportional to the best known lower bound.\n\\end{abstract}\n\nA capset is a subset $C \\subset \\F_3^n$ with no three points on a line. A question that has attracted a lot of interest (see e.g. \\cite{Tao},\\cite{Matrix}) is the following: What is the largest size of a capset in $\\F_3^n$ as a function of $n$? If we denote this value by $a(n)$ then it can be shown (see \\cite[Proposition 2.2]{Tyrrell}) that $c=\\lim a(n)^{1/n}$ exists. Moreover, it is known that $2.2202 \\le c$ (\\cite{Nature}) and $c \\le 2.756$ (\\cite{Jordan}). The exact value of $a(n)$ is known for $n \\le 6$ (\\cite{capsetKnownSizes1, capetKnownSizes2, capsetKnownSizes3, capsetKnownSizes4}). It is also known that $a(8)\\geq 512$ (\\cite{Nature}).\n\nWe can also consider subsets of $\\F_q^d$ with no three points on an $\\F_q$-line.\nWe call such a set a cap (to distinguish it from a capset under the above \nidentification). It is clear that a cap is a capset but not conversely.\nWe can also consider the analogous notion of complete caps. A cap $C$ is complete as a cap if $C$ is not contained in a larger cap. That is, for every\npoint $P$ in the complement of $C$, there exists an $\\F_q$-line through $P$\nwhich meets $C$ in two points. We remark that a cap can be complete as a cap and yet not be complete as a capset. For example, a conic in the plane $\\F_q^2, q >3$. This follows from the lemma below.\n\n\\begin{proof}\nThe $\\F_3$ lines through the $N(N-1)/2$ pairs of points of the capset have their third point in the complement of the capset, so if $N(N-1)/2 < 3^n -N$, there is a point in the complement not on any line through two points in the capset, hence the capset is not complete. \n\\end{proof}\nIn \\cite{Csajbok}, subsets of $\\F_p^n$ with $p$ prime no three points in arithmetic progression are studied. This condition is equivalent to the capset condition when $p=3$. They prove a lower bound similar, but slightly weaker, to Lemma \\ref{lem:lower} in this more general situation. They also ask if this lower bound is best possible up to a multiplicative constant (loc. cit., Problem 5.1). We give a positive answer to this question in the case $p=3$ (see Corollary \\ref{cor:upper}).\n\n\\begin{Corollary}\n\\label{cor:upper}\nFor every $n$, there exists a complete capset in $\\F_3^n$ of size $O(\\sqrt{3^n})$. \\end{Corollary}\n\n\\begin{proof}\nAssume first that $n=2m$ even and start with the construction given by Theorem \\ref{thm:two-conics} of a capset of size $2(3^m -1) = O(\\sqrt{3^n})$ in $\\F_{3^m}^2$. If $m$ is odd, this is already complete.\nWhen $m$ is even, we can only add points of the form $(0,b), b$ non-square. So we need a complete capset in $\\F_{3^m}$ comprised of non-squares. Viewing $\\F_{3^m}$ as $\\F_{3^{m/2}}(\\sqrt{d})$ and selecting $\\lambda$ a non-square in $\\F_{3^m}$, we can take elements of the form $\\lambda(\\pm 1+x\\sqrt{d})^2, x \\in \\F_{3^{m/2}}^*$ which will be a capset as in the theorem, up to an affine transformation, since $(\\pm 1+x\\sqrt{d})^2 = (1+dx^2) \\pm 2x\\sqrt{d}$. Iterating this construction gives the result for $n$ even.\n\nTo construct small complete capsets in $\\F_{3^n}$ when $n$ is odd, write $n=2m+1$, take the complete capset $C \\subset \\F_{3^{2m}}$ constructed above and consider $C \\times \\{0,1\\} \\subset \\F_{3^n}$. It is straightfoward that this is a capset. To see it is complete, consider a pointi $P$ in its complement with last coordinate $c \\in \\F_3$ and let $P_0 \\in \\F_{3^{2m}}$ be the point corresponding to the other coordinates. If $c \\in \\{0,1\\}$, then $P_0 \\not\\in C$ and there is a line in the subspace of points whose last coordinate is $c$ through $P$ meeting $C \\times \\{c\\}$ in two points. If $c=2$ and $P_0 \\in C$, the vertical line through $P$ meets the capset in two points. If $c=2$ and $P_0 \\not\\in C$, there is a line through $P_0$ meeting $C$ in $Q_0,Q_1$, say. Then $(Q_0,0), (Q_1,1)$ are in $C \\times \\{0,1\\}$ and are collinear with $P$.\n\\end{proof}\n\n\\begin{Theorem}\n\\label{thm:quadric}\nLet $\\lambda$ be a non-square in $\\F_q, q=3^m$ and $Q \\subset \\F_q^3$ be given by\n$$Q = \\{(x,y,x^2-\\lambda y^2) \\mid x,y \\in \\F_q\\}.$$\nThen $Q$ is a complete capset.\n\\end{Theorem}\n\n\\begin{Lemma}\n\\label{lem:lower}\nIf there exists a complete capset in $\\F_3^n$ with $N$ points, then\n$N(N+1)/2 \\ge 3^n$.\n\\end{Lemma}", "post_theorem_intro_text_len": 1006, "post_theorem_intro_text": "\\begin{proof}\nThe $\\F_3$ lines through the $N(N-1)/2$ pairs of points of the capset have their third point in the complement of the capset, so if $N(N-1)/2 < 3^n -N$, there is a point in the complement not on any line through two points in the capset, hence the capset is not complete. \n\\end{proof}\nIn \\cite{Csajbok}, subsets of $\\F_p^n$ with $p$ prime no three points in arithmetic progression are studied. This condition is equivalent to the capset condition when $p=3$. They prove a lower bound similar, but slightly weaker, to Lemma \\ref{lem:lower} in this more general situation. They also ask if this lower bound is best possible up to a multiplicative constant (loc. cit., Problem 5.1). We give a positive answer to this question in the case $p=3$ (see Corollary \\ref{cor:upper}).\n\n\\subsection*{Acknowledgements}\nThe second author was supported by the Marsden Fund administered by the Royal\nSociety of New Zealand. He would also like to thank J. Sheekey and G. Van de Voorde for helpful discussions.", "sketch": "The argument is a counting/completeness contradiction: the $\\F_3$-lines through the $N(N-1)/2$ pairs of points of the capset have their third point in the complement. Thus these lines can cover at most $N(N-1)/2$ points of the complement. If $N(N-1)/2 < 3^n-N$, then there is a point in the complement not lying on any line through two capset points, so it cannot be added to form a 3-term progression; hence the capset is not complete. Therefore completeness forces $N(N-1)/2 \\ge 3^n-N$, i.e. $N(N+1)/2\\ge 3^n$.", "expanded_sketch": "The argument is a counting/completeness contradiction: the $\\F_3$-lines through the $N(N-1)/2$ pairs of points of the capset have their third point in the complement. Thus these lines can cover at most $N(N-1)/2$ points of the complement. If $N(N-1)/2 < 3^n-N$, then there is a point in the complement not lying on any line through two capset points, so it cannot be added to form a 3-term progression; hence the capset is not complete. Therefore completeness forces $N(N-1)/2 \\ge 3^n-N$, i.e. $N(N+1)/2\\ge 3^n$.", "expanded_theorem": "\\label{lem:lower}\nIf there exists a complete capset in $\\F_3^n$ with $N$ points, then\n$N(N+1)/2 \\ge 3^n$.", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let \\(C \\subset \\mathbb{F}_3^n\\) be a complete capset, where a capset means a subset of \\(\\mathbb{F}_3^n\\) containing no three distinct points on a line, and complete means that \\(C\\) is not contained in any larger capset. If \\(|C|=N\\), which quantitative estimate must hold?", "correct_choice": {"label": "A", "text": "\\(\\dfrac{N(N+1)}{2} \\ge 3^n\\)."}, "choices": [{"label": "B", "text": "\\(\\dfrac{N(N-1)}{2} \\ge 3^n\\)."}, {"label": "C", "text": "\\(\\dfrac{N(N+1)}{2} \\ge 3^n-N\\)."}, {"label": "D", "text": "\\(\\dfrac{N(N-1)}{2} \\ge 3^n-N+1\\)."}, {"label": "E", "text": "For every point \\(P\\in \\mathbb{F}_3^n\\setminus C\\), there exists a unique line through \\(P\\) meeting \\(C\\) in two points; consequently \\(\\dfrac{N(N-1)}{2} \\ge 3^n-N\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "counting_estimate", "tampered_component": "final algebraic rearrangement from complement count to total count", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "dropped the final rewrite using \\(3^n-N\\) to obtain a weaker bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "strict-vs-nonstrict boundary in the contradiction inequality", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "existence of a covering line for each complement point replaced by uniqueness", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the target inequality or give away the answer directly. It provides definitions of capset and completeness, but the conclusion still must be derived by a counting argument."}, "TAS": {"score": 2, "justification": "This is not a mere restatement of a theorem in the stem. The solver must translate completeness into a covering property for points outside the capset and then count via pairs of points in C."}, "GPS": {"score": 1, "justification": "The intended solution requires moderate reasoning: infer that each outside point lies on a line determined by two points of C, then compare the complement size with the number of unordered pairs. However, the presence of an algebraically equivalent option weakens the item's ability to test genuine generative reasoning cleanly."}, "DQS": {"score": 0, "justification": "The distractors are flawed overall because choice C is actually algebraically equivalent to the marked correct choice A: N(N-1)/2 >= 3^n - N rearranges to N(N+1)/2 >= 3^n. This creates multiple correct answers. Although B and D reflect plausible mistakes, the item is invalid as written."}, "total_score": 5, "overall_assessment": "Good stem with no answer leakage and a genuinely non-tautological counting task, but the MCQ is seriously compromised by an equivalent 'distractor,' so it does not function as a valid single-answer question."}}
{"id": "2602.05303v1", "paper_link": "http://arxiv.org/abs/2602.05303v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{thm1}\n   Let \\( x: \\Sigma \\rightarrow \\mathbb{S}^2 \\times \\mathbb{S}^2 \\) be a minimal-Willmore surface. Then either \\( x \\) is a special complex curve given by a slice or a diagonal; or, up to an isometry, is contained in a totally geodesic submanifold $\\mathbb{S}^2 \\times \\mathbb{S}^1$, and can be described by a solution of the sinh-Gordon equation in one variable.", "start_pos": 4463, "end_pos": 4860, "label": "thm1"}, "ref_dict": {"eq-mwill": "\\begin{equation}\\label{eq-mwill}\n    e^{it}\\cosh^3(v-w)(v+w)_{\\bar{z}} = \\cosh^3(v+w)(v-w)_z.\n    \\end{equation}", "rk-mini": "\\begin{remark}\\label{rk-mini}If $w=0$, then it follows from \\eqref{eq-vw} that $v$ only depends on the variable $u_1$, and satisfies \n\\begin{equation}\\label{eq-ellip}\n\\frac{d^2 v}{d u_1 ^2}=-2\\sinh(2v).\n\\end{equation}\nThe solution to this ordinary differential equation can be expressed using an elliptic function, \n\\[\nv = \\operatorname{arcsinh}\\Bigl( k \\; \\mathrm{sn}\\bigl( 2u_1 + \\delta,\\; i k \\bigr) \\Bigr),\n\\]\nwhere $\\delta$ and $k>0$ are integration constants, and $\\mathrm{sn}(\\,\\cdot,\\,ik)$ denotes the Jacobian elliptic sine function with  modulus $i k$.  \nApplying the classical Jacobi imaginary modulus transformation, the solution also admits the equivalent form \n\\[\nv = \\operatorname{arcsinh}\\Bigl( \\frac{k}{\\sqrt{1+k^2}} \\; \\mathrm{sn}\\bigl( \\sqrt{1+k^2}\\,(2u_1 + \\delta),\\; \\frac{k}{\\sqrt{1+k^2}} \\bigr) \\Bigr).\\]\n\nGeometrically, given a solution of~\\eqref{eq-ellip}, one first obtains a minimal surface $\\psi:\\mathbb{C}\\rightarrow \\mathbb{S}^3$ with induced metric $e^{2v}|dz|^2$ \nand Hopf differential $\\frac i 2 dz\\otimes dz$. \nSuch a surface should be necessarily homogeneous or of cohomogeneity one and hence belongs to the $T_{m,k,a}$ family constructed and classified by Hsiang and Lawson in \\cite{Hsiang-Lawson}. This family contains infinitely many closed minimal surfaces, whose topology is either a torus or a Klein bottle. Denote by $N$ the Gauss map of such a surface. Then the map \\emph{(}see the proof of Corollary $1$ in \\cite{Urbano}\\emph{)}\n\\[\n\\phi(z)=(V_\\psi, e^{2i u_2}):\\mathbb{C}\\rightarrow \\mathbb{S}^2\\times\\mathbb{S}^1\n\\]\ndefines a minimal surface in $\\mathbb{S}^2\\times\\mathbb{S}^1$, where\n\\[\nV_\\psi=\\frac{1}{\\sqrt{2}}(-2ie^{-2v}\\psi_{z}\\wedge\\psi_{\\overline{z}}+\\psi\\wedge N):\\mathbb{C}\\rightarrow \\mathbb{S}^2\\subset \\Lambda^2\\mathbb{R}^4. \n\\]\nApart from the obvious slices, these surfaces constitute all minimal-Willmore surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^1$. \n\\end{remark}", "thm1": "\\begin{thm}\\label{thm1}\n   Let \\( x: \\Sigma \\rightarrow \\mathbb{S}^2 \\times \\mathbb{S}^2 \\) be a minimal-Willmore surface. Then either \\( x \\) is a special complex curve given by a slice or a diagonal; or, up to an isometry, is contained in a totally geodesic submanifold $\\mathbb{S}^2 \\times \\mathbb{S}^1$, and can be described by a solution of the sinh-Gordon equation in one variable.\n\\end{thm}"}, "pre_theorem_intro_text_len": 2863, "pre_theorem_intro_text": "For a closed surface $x:\\Sigma\\rightarrow (N^n,g)$ in a Riemannian manifold, the squared $L^2$-norm of the trace-free second fundamental form is a fundamental conformal invariant; that is, it is preserved under conformal changes of the metric. This functional, referred to as the conformally invariant Willmore functional \\cite{Michelat,Mondino}, will henceforth be called simply the Willmore functional. The Willmore functional is equivalent, up to a topological invariant, to the integral   \n$$\\int_{\\Sigma}(|\\vec{H}|^2+K_{1212})dA,$$\nwhere $\\vec{H}$ is the mean curvature vector and $K_{1212}$ denotes the sectional curvature of $(N^n,g)$ restricted to the tangent plane of the surface. In what follows, we will denote this integral by $\\mathcal{W}$ and also refer to it as the Willmore functional. This functional exhibits notable links with other fundamental quantities, including, among others, the renormalized area functional within the AdS/CFT correspondence \\cite{Alexakis, Graham}. Surfaces that satisfy the Euler-Lagrange equation of this functional are called {\\em Willmore surfaces}. \n\nWhen the ambient space is a real space form, both the Willmore functional and Willmore surfaces have been extensively investigated. Notable advances include the resolution of the Willmore conjecture in $\\mathbb{S}^3$, originally proposed by Willmore in \\cite{Willmore} and proved by Marques and Neves in \\cite{Marques}.   Regarding the construction and classification of Willmore surfaces, we refer to \\cite{Bryant, Ejiri, WD, MWW} and references  therein. A basic fact is that every minimal surface in a real space form is  automatically Willmore. \nWhen the ambient space is a non-space form Riemannian manifold, the Euler–Lagrange equation of the Willmore functional has been derived independently by several groups of geometers from different viewpoints; see, for example, \\cite{Hu-Li, Mondino, Pedit, Wang-Xie}. Owing to the presence of ambient curvature terms in the Euler–Lagrange equation, not every minimal surface is Willmore in a non-space form Riemannian manifold. In \\cite{Montiel-Urbano}, Montiel and Urbano proved that the only surfaces in $\\mathbb{CP}^2$ that are both minimal and Willmore are the superminimal surfaces of positive spin, i.e., complex curves and minimal Lagrangian surfaces. Recently, the last two authors of  this paper have generalized this result to the self-dual K\\\"ahler-Einstein surfaces (see Proposition 2.3 in \\cite{Wang-Xie}). Note that $\\mathbb{S}^2 \\times \\mathbb{S}^2$ (equipped with the standard product metric and complex structure), being neither self-dual nor anti-self-dual, is another canonical example of a K\\\"ahler-Einstein surface. This motivates the study of which minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ are Willmore. We refer to a surface as minimal-Willmore if it is both minimal and Willmore.", "context": "For a closed surface $x:\\Sigma\\rightarrow (N^n,g)$ in a Riemannian manifold, the squared $L^2$-norm of the trace-free second fundamental form is a fundamental conformal invariant; that is, it is preserved under conformal changes of the metric. This functional, referred to as the conformally invariant Willmore functional \\cite{Michelat,Mondino}, will henceforth be called simply the Willmore functional. The Willmore functional is equivalent, up to a topological invariant, to the integral \n$$\\int_{\\Sigma}(|\\vec{H}|^2+K_{1212})dA,$$\nwhere $\\vec{H}$ is the mean curvature vector and $K_{1212}$ denotes the sectional curvature of $(N^n,g)$ restricted to the tangent plane of the surface. In what follows, we will denote this integral by $\\mathcal{W}$ and also refer to it as the Willmore functional. This functional exhibits notable links with other fundamental quantities, including, among others, the renormalized area functional within the AdS/CFT correspondence \\cite{Alexakis, Graham}. Surfaces that satisfy the Euler-Lagrange equation of this functional are called {\\em Willmore surfaces}.\n\nWhen the ambient space is a real space form, both the Willmore functional and Willmore surfaces have been extensively investigated. Notable advances include the resolution of the Willmore conjecture in $\\mathbb{S}^3$, originally proposed by Willmore in \\cite{Willmore} and proved by Marques and Neves in \\cite{Marques}. Regarding the construction and classification of Willmore surfaces, we refer to \\cite{Bryant, Ejiri, WD, MWW} and references therein. A basic fact is that every minimal surface in a real space form is automatically Willmore. \nWhen the ambient space is a non-space form Riemannian manifold, the Euler–Lagrange equation of the Willmore functional has been derived independently by several groups of geometers from different viewpoints; see, for example, \\cite{Hu-Li, Mondino, Pedit, Wang-Xie}. Owing to the presence of ambient curvature terms in the Euler–Lagrange equation, not every minimal surface is Willmore in a non-space form Riemannian manifold. In \\cite{Montiel-Urbano}, Montiel and Urbano proved that the only surfaces in $\\mathbb{CP}^2$ that are both minimal and Willmore are the superminimal surfaces of positive spin, i.e., complex curves and minimal Lagrangian surfaces. Recently, the last two authors of this paper have generalized this result to the self-dual K\\\"ahler-Einstein surfaces (see Proposition 2.3 in \\cite{Wang-Xie}). Note that $\\mathbb{S}^2 \\times \\mathbb{S}^2$ (equipped with the standard product metric and complex structure), being neither self-dual nor anti-self-dual, is another canonical example of a K\\\"ahler-Einstein surface. This motivates the study of which minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ are Willmore. We refer to a surface as minimal-Willmore if it is both minimal and Willmore.", "full_context": "For a closed surface $x:\\Sigma\\rightarrow (N^n,g)$ in a Riemannian manifold, the squared $L^2$-norm of the trace-free second fundamental form is a fundamental conformal invariant; that is, it is preserved under conformal changes of the metric. This functional, referred to as the conformally invariant Willmore functional \\cite{Michelat,Mondino}, will henceforth be called simply the Willmore functional. The Willmore functional is equivalent, up to a topological invariant, to the integral \n$$\\int_{\\Sigma}(|\\vec{H}|^2+K_{1212})dA,$$\nwhere $\\vec{H}$ is the mean curvature vector and $K_{1212}$ denotes the sectional curvature of $(N^n,g)$ restricted to the tangent plane of the surface. In what follows, we will denote this integral by $\\mathcal{W}$ and also refer to it as the Willmore functional. This functional exhibits notable links with other fundamental quantities, including, among others, the renormalized area functional within the AdS/CFT correspondence \\cite{Alexakis, Graham}. Surfaces that satisfy the Euler-Lagrange equation of this functional are called {\\em Willmore surfaces}.\n\nWhen the ambient space is a real space form, both the Willmore functional and Willmore surfaces have been extensively investigated. Notable advances include the resolution of the Willmore conjecture in $\\mathbb{S}^3$, originally proposed by Willmore in \\cite{Willmore} and proved by Marques and Neves in \\cite{Marques}. Regarding the construction and classification of Willmore surfaces, we refer to \\cite{Bryant, Ejiri, WD, MWW} and references therein. A basic fact is that every minimal surface in a real space form is automatically Willmore. \nWhen the ambient space is a non-space form Riemannian manifold, the Euler–Lagrange equation of the Willmore functional has been derived independently by several groups of geometers from different viewpoints; see, for example, \\cite{Hu-Li, Mondino, Pedit, Wang-Xie}. Owing to the presence of ambient curvature terms in the Euler–Lagrange equation, not every minimal surface is Willmore in a non-space form Riemannian manifold. In \\cite{Montiel-Urbano}, Montiel and Urbano proved that the only surfaces in $\\mathbb{CP}^2$ that are both minimal and Willmore are the superminimal surfaces of positive spin, i.e., complex curves and minimal Lagrangian surfaces. Recently, the last two authors of this paper have generalized this result to the self-dual K\\\"ahler-Einstein surfaces (see Proposition 2.3 in \\cite{Wang-Xie}). Note that $\\mathbb{S}^2 \\times \\mathbb{S}^2$ (equipped with the standard product metric and complex structure), being neither self-dual nor anti-self-dual, is another canonical example of a K\\\"ahler-Einstein surface. This motivates the study of which minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ are Willmore. We refer to a surface as minimal-Willmore if it is both minimal and Willmore.\n\n\\section{Introduction}\nFor a closed surface $x:\\Sigma\\rightarrow (N^n,g)$ in a Riemannian manifold, the squared $L^2$-norm of the trace-free second fundamental form is a fundamental conformal invariant; that is, it is preserved under conformal changes of the metric. This functional, referred to as the conformally invariant Willmore functional \\cite{Michelat,Mondino}, will henceforth be called simply the Willmore functional. The Willmore functional is equivalent, up to a topological invariant, to the integral \n$$\\int_{\\Sigma}(|\\vec{H}|^2+K_{1212})dA,$$\nwhere $\\vec{H}$ is the mean curvature vector and $K_{1212}$ denotes the sectional curvature of $(N^n,g)$ restricted to the tangent plane of the surface. In what follows, we will denote this integral by $\\mathcal{W}$ and also refer to it as the Willmore functional. This functional exhibits notable links with other fundamental quantities, including, among others, the renormalized area functional within the AdS/CFT correspondence \\cite{Alexakis, Graham}. Surfaces that satisfy the Euler-Lagrange equation of this functional are called {\\em Willmore surfaces}.\n\nIn their work \\cite{Urbano}, Tobarro and Urbano gave a beautiful local characterization of minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ that possess no complex points. Their description is formulated in terms of a pair of solutions $v$ and $w$ to the sinh‑Gordon equation \n$$v_{z\\bar z}+\\frac{1}{2}\\sinh (2v)=0.$$\nWithin this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w$, see \\eqref{eq-mwill}. By analyzing this coupled system, we establish the theorem above.\n\n\\begin{example}\\label{ex-slice} For any point $p\\in\\mathbb{S}^2$, the corresponding slices \n\\[\n\\mathbb{S}^2\\times\\{p\\}=\\{(x,p)\\in\\mathbb{S}^2\\times\\mathbb{S}^2 \\mid x\\in\\mathbb{S}^2\\},\n\\]\n\\[\n\\{p\\}\\times\\mathbb{S}^2=\\{(p,x)\\in\\mathbb{S}^2\\times\\mathbb{S}^2 \\mid x\\in\\mathbb{S}^2\\}.\n\\]\nare totally geodesic with the first one satisfying $C_1=C_2=1$ and the second one satisfying $C_1=-C_2=1$. That is, the slices are complex with respect to both $J_1$ and $J_2$.\n\\end{example}\n\\begin{example}\\label{ex-diagonal} The diagonal \n$$D\\triangleq\\{(x,x)\\in\\mathbb{S}^2\\times\\mathbb{S}^2 \\mid x\\in\\mathbb{S}^2\\}$$\nis totally geodesic and satisfies $C_1=1$ and $C_2=0$, which means that $D$ is complex with respect to $J_1$ and Lagrangian with respect to $J_2$.\n\\end{example}\n\\begin{example}\\label{ex-Clifford}\n The torus defined by \n $$T\\triangleq\\{(x,y)\\in\\mathbb{S}^2\\times\\mathbb{S}^2 \\mid x_1=y_1=0\\}$$ \n is totally geodesic and satisfies $C_1=C_2=0$, which means that $T$ is Lagrangian for both $J_1$ and $J_2$. This surface is known as the Clifford torus in $\\mathbb{S}^2 \\times \\mathbb{S}^2$, characterized by the flatness of both its tangent and normal bundles. \n\\end{example}\n\\begin{example} \nLet $\\Sigma=\\mathbb{C}/\\Lambda$ be a torus generated by the lattice $\\Lambda=\\{m+n\\tau \\mid m,n\\in\\mathbb{Z}\\}$, where $\\tau$ is a complex number with $\\operatorname{Im}\\tau>0$. Let $\\wp:\\Sigma\\rightarrow\\mathbb{S}^2$ be the Weierstrass $\\wp$-function with a second-order pole at the origin, and $z_0\\in\\Sigma$ be a point at which $\\wp$ is not ramified. Then $x=\\big(\\wp(z),\\wp(z-z_0)\\big):\\Sigma\\rightarrow(\\mathbb{S}^2\\times\\mathbb{S}^2)$ is a holomorphic embedding with the complex structure $J_1$. Such minimal tori are referred to as Weierstrass tori.\n\\end{example}\n\n\\begin{proposition}\\label{pro-mw}\n Let $x:\\Sigma \\rightarrow \\mathbb{S}^2\\times \\mathbb{S}^2$ be a complex curve or minimal Lagrangian surface. Then $x$ is Willmore if and only if it is one of the surfaces given in Example~\\ref{ex-slice} $\\sim$ Example~\\ref{ex-Clifford}. \n\\end{proposition}\n\\begin{proof}\nWithout loss of generality, we assume that $x$ is complex or minimal Lagrangian with respect to the complex structure $J_1$. Consequently, $C_1$ is a constant which must be either $0$ or $\\pm 1$. It follows from \\cite[Proposition 3]{Urbano} (see also \\cite{Chen}) that we only need to prove that $C_2$ is a constant.\n\nBy \\eqref{eq-C1z}, we have $f_1\\bar{\\gamma_1}=0$. Substituting this into \\eqref{eq-mW} yields $\\bar{f_1}{\\gamma_2}=0$, which, by \\eqref{eq-C2z}, implies that $C_2$ is holomorphic. Since $C_2$ is real-valued, it must be constant. \n\\end{proof}\n\\begin{theorem}\\label{thm-mw}\n Let $x:\\Sigma \\rightarrow \\mathbb{S}^2\\times \\mathbb{S}^2$ be a minimal surface without complex points. Then $x$ is Willmore if and only if up to an isometry it is a minimal-Willmore surface in $\\mathbb{S}^2\\times\\mathbb{S}^1$. \n\\end{theorem}\n\nLet $x:\\Sigma \\rightarrow \\mathbb{S}^2\\times \\mathbb{S}^2$ be a minimal immersion of a simply connected surface without complex points and $z$ be a complex coordinate such that the Hopf differential $\\Theta(z)\\triangleq\\frac{\\langle J_1 x_z, J_2 x_z\\rangle}{2}dz^2$ (which is holomorphic) satisfies $\\Theta(z)=dz^2$. \n Then the functions \n $$v\\triangleq\\frac{1}{2}\\ln \\sqrt{\\frac{(1+C_1)(1+C_2)}{(1-C_1)(1-C_2)}},~~~w\\triangleq\\frac{1}{2}\\ln \\sqrt{\\frac{(1-C_1)(1+C_2)}{(1+C_1)(1-C_2)}}$$\n satisfy the sinh-Gordon equation, i.e., \n $$v_{z\\bar z}+\\frac{1}{2}\\sinh (2v)=0,~~~w_{z\\bar z}+\\frac{1}{2}\\sinh (2w)=0.$$\n Conversely, given two solutions $v, w:\\mathbb{C}\\rightarrow \\mathbb{R}$ of the sinh-Gordon equation, one can construct a 1-parameter family of minimal immersions $X_t: \\mathbb{S}^2 \\times \\mathbb{S}^2$ by taking the following quantities as the fundamental data, \n$$\\sigma=\\frac{1}{2}\\ln\\big(4\\cosh(v+w)\\cosh(v-w)\\big),~~~ \\rho=\\left(\\ln \\sqrt{\\frac{\\cosh(v+w)}{\\cosh(v-w)}}\\right)_{\\!\\!z},~~~$$\n$$C_1=\\tanh(v-w),~~~C_2=\\tanh(v+w),$$\n$$\\gamma_1=\\sqrt{2}e^{\\frac{it}{2}}\\sqrt{\\frac{\\cosh(v+w)}{\\cosh(v-w)}},~~~\\gamma_2=\\sqrt{2}e^{\\frac{it}{2}}\\sqrt{\\frac{\\cosh(v-w)}{\\cosh(v+w)}},~~~f_1=-i \\gamma_1 (v-w)_z,~~~f_2=-i\\gamma_2(v+w)_z.$$\nGeometrically, this construction originates from the Gauss maps of two minimal surfaces in \\( \\mathbb{S}^3 \\), whose induced metrics are \\( e^{2v} |dz|^2 \\) and \\( e^{2w} |dw|^2 \\), respectively, and which share the same Hopf differential \\( \\theta(z) = \\frac{i}{2} dz^2 \\). For further details, we refer to Section~5 of \\cite{Urbano}.\n\nTherefore, we establish the claim that $p=\\pm q$, which means either $v=0$ or $w=0$. It then follows from Proposition 4 in \\cite{Urbano} that $x$ is non-full in $\\mathbb{S}^2 \\times \\mathbb{S}^2$. This means it lies in a totally geodesic hypersurface of the ambient space, which, up to an isometry, is an open subset of $\\mathbb{S}^2 \\times \\mathbb{S}^1$. \n\\end{proof}\n\\begin{remark}\\label{rk-mini}If $w=0$, then it follows from \\eqref{eq-vw} that $v$ only depends on the variable $u_1$, and satisfies \n\\begin{equation}\\label{eq-ellip}\n\\frac{d^2 v}{d u_1 ^2}=-2\\sinh(2v).\n\\end{equation}\nThe solution to this ordinary differential equation can be expressed using an elliptic function, \n\\[\nv = \\operatorname{arcsinh}\\Bigl( k \\; \\mathrm{sn}\\bigl( 2u_1 + \\delta,\\; i k \\bigr) \\Bigr),\n\\]\nwhere $\\delta$ and $k>0$ are integration constants, and $\\mathrm{sn}(\\,\\cdot,\\,ik)$ denotes the Jacobian elliptic sine function with modulus $i k$. \nApplying the classical Jacobi imaginary modulus transformation, the solution also admits the equivalent form \n\\[\nv = \\operatorname{arcsinh}\\Bigl( \\frac{k}{\\sqrt{1+k^2}} \\; \\mathrm{sn}\\bigl( \\sqrt{1+k^2}\\,(2u_1 + \\delta),\\; \\frac{k}{\\sqrt{1+k^2}} \\bigr) \\Bigr).\\]", "post_theorem_intro_text_len": 2347, "post_theorem_intro_text": "\\noindent In other words, when the ambient space is $\\mathbb{S}^2\\times\\mathbb{S}^2$,  within the minimal class the Willmore condition turns out to be  rigid: it forces special holomorphicity, or reduces the problem to a totally geodesic hypersurface. \n Geometrically, the last class of minimal surfaces in Theorem~\\ref{thm1} arises from a certain Gauss-map construction applied to a cohomogeneity {one} minimal surface in $\\mathbb{S}^3\\subset\\mathbb{R}^4$, see Remark~\\ref{rk-mini}. \n\nIn their work \\cite{Urbano}, Tobarro and Urbano gave a beautiful local characterization of minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ that possess no complex points. Their description is formulated in terms of a pair of solutions $v$ and $w$ to the sinh‑Gordon equation \n$$v_{z\\bar z}+\\frac{1}{2}\\sinh (2v)=0.$$\nWithin this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w$, see \\eqref{eq-mwill}. By analyzing this coupled system, we establish the theorem above. \n\nExcept for seeking Willmore surfaces among minimal surfaces, another natural approach is to use the product structure of  $\\mathbb{S}^2\\times\\mathbb{S}^2$ to produce Willmore surfaces. \n\\begin{thm}\\label{thm2}\n   All Willmore surfaces of product type in $\\mathbb{S}^2 \\times \\mathbb{S}^2$ are exhausted by products of an elastic curve in $\\mathbb{S}^2$ with a great circle. \n\\end{thm}\n\nNote that both of these two classes of examples yield Willmore surfaces that are non-linearly full in $\\mathbb{S}^2 \\times \\mathbb{S}^2$. No linearly full examples are currently known. We point out that constructing explicit examples, or even addressing the existence problem, in non-space-form Riemannian manifolds is particularly challenging; see the efforts in \\cite{Modino,Ikoma,Michelat} and the references therein. \n\nThe paper is organized as follows. {Section~\\ref{sec2}  begins with the  preliminaries on  $\\mathbb{S}^2\\times \\mathbb{S}^2$ and on surfaces within it. In Section~\\ref{sec3}, we first derive the fundamental equations for surfaces in $\\mathbb{S}^2\\times \\mathbb{S}^2$, and subsequently compute the corresponding Willmore equation. Section~\\ref{sec4} is devoted to the classification of minimal-Willmore surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$. Finally, in Section~\\ref{sec5}, we classify Willmore surfaces of product type.", "sketch": "Using the local characterization of Tobarro--Urbano \\cite{Urbano} for minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ with no complex points, the surface is described \"in terms of a pair of solutions $v$ and $w$ to the sinh-Gordon equation\" $$v_{z\\bar z}+\\tfrac12\\sinh(2v)=0.$$ \"Within this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w$, see \\eqref{eq-mwill}.\" The proof then proceeds by \"analyzing this coupled system\" (the sinh-Gordon equation together with the additional PDE imposed by the Willmore condition), which yields the classification stated in Theorem~\\ref{thm1}.", "expanded_sketch": "Using the local characterization of Tobarro--Urbano \\cite{Urbano} for minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ with no complex points, the surface is described \"in terms of a pair of solutions $v$ and $w$ to the sinh-Gordon equation\" $$v_{z\\bar z}+\\tfrac12\\sinh(2v)=0.$$ \"Within this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w,\" namely\n\\begin{equation}\\label{eq-mwill}\n    e^{it}\\cosh^3(v-w)(v+w)_{\\bar{z}} = \\cosh^3(v+w)(v-w)_z.\n    \\end{equation}\nThe proof then proceeds by \"analyzing this coupled system\" (the sinh-Gordon equation together with the additional PDE imposed by the Willmore condition). To prove the main theorem, one shows that this analysis yields the stated classification.", "expanded_theorem": "\\label{thm1}\n   Let \\( x: \\Sigma \\rightarrow \\mathbb{S}^2 \\times \\mathbb{S}^2 \\) be a minimal-Willmore surface. Then either \\( x \\) is a special complex curve given by a slice or a diagonal; or, up to an isometry, is contained in a totally geodesic submanifold $\\mathbb{S}^2 \\times \\mathbb{S}^1$, and can be described by a solution of the sinh-Gordon equation in one variable.", "theorem_type": ["Classification or Bijection", "Universal"], "mcq": {"question": "A minimal-Willmore surface in \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\) means a surface \\(x:\\Sigma\\to \\mathbb{S}^2\\times\\mathbb{S}^2\\) that is both minimal and Willmore, where a Willmore surface satisfies the Euler-Lagrange equation of the Willmore functional \\(\\mathcal W=\\int_\\Sigma (|\\vec H|^2+K_{1212})\\,dA\\). A slice is a surface of the form \\(\\mathbb{S}^2\\times\\{p\\}\\) or \\(\\{p\\}\\times\\mathbb{S}^2\\), and a diagonal means a diagonal copy of \\(\\mathbb{S}^2\\) in \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\). Which statement holds for every minimal-Willmore surface \\(x:\\Sigma\\to \\mathbb{S}^2\\times\\mathbb{S}^2\\)?", "correct_choice": {"label": "A", "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\), the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\), and the surface is described by a solution of the sinh-Gordon equation depending on a single variable."}, "choices": [{"label": "B", "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\), the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\), and the surface is described by a pair of independent solutions of the sinh-Gordon equation in two variables."}, {"label": "C", "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\), the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\)."}, {"label": "D", "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\), and every such non-complex minimal-Willmore surface is described by a solution of the sinh-Gordon equation depending on two variables."}, {"label": "E", "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\), the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\), and the surface is described by a solution of the sinh-Gordon equation, with no restriction that the solution depend on only one variable."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "collapse from coupled system to one-variable reduction", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "geometric_construction", "tampered_component": "single-variable sinh-Gordon description", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "one-variable dependence conclusion", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "restriction to one-variable solutions", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem names the setting and asks for the complete classification, but it does not reveal the specific conclusion. The correct answer is not stated or strongly hinted beyond the topic itself."}, "TAS": {"score": 1, "justification": "This is still very close to a direct theorem-recall question: it asks for the complete classification of minimal-Willmore surfaces. However, the presence of subtly varied alternatives means it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "Selecting the correct option requires moderate reasoning/discrimination, especially distinguishing the complete statement from a weaker true version and from overgeneralized variants. Still, the task is mainly recognition of the theorem rather than genuine derivation."}, "DQS": {"score": 2, "justification": "The distractors are strong: one is weaker but true, others are plausible overstatements or property confusions, and they target realistic mathematical mistakes about completeness, generality, and PDE structure."}, "total_score": 6, "overall_assessment": "A solid theorem-classification MCQ with excellent distractors and no answer leakage, but it remains largely a recall/recognition item rather than a strongly generative reasoning question."}}
{"id": "2602.05303v1", "paper_link": "http://arxiv.org/abs/2602.05303v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{thm1}\n   Let \\( x: \\Sigma \\rightarrow \\mathbb{S}^2 \\times \\mathbb{S}^2 \\) be a minimal-Willmore surface. Then either \\( x \\) is a special complex curve given by a slice or a diagonal; or, up to an isometry, is contained in a totally geodesic submanifold $\\mathbb{S}^2 \\times \\mathbb{S}^1$, and can be described by a solution of the sinh-Gordon equation in one variable.", "start_pos": 4463, "end_pos": 4860, "label": "thm1"}, "ref_dict": {"eq-mwill": "\\begin{equation}\\label{eq-mwill}\n    e^{it}\\cosh^3(v-w)(v+w)_{\\bar{z}} = \\cosh^3(v+w)(v-w)_z.\n    \\end{equation}", "rk-mini": "\\begin{remark}\\label{rk-mini}If $w=0$, then it follows from \\eqref{eq-vw} that $v$ only depends on the variable $u_1$, and satisfies \n\\begin{equation}\\label{eq-ellip}\n\\frac{d^2 v}{d u_1 ^2}=-2\\sinh(2v).\n\\end{equation}\nThe solution to this ordinary differential equation can be expressed using an elliptic function, \n\\[\nv = \\operatorname{arcsinh}\\Bigl( k \\; \\mathrm{sn}\\bigl( 2u_1 + \\delta,\\; i k \\bigr) \\Bigr),\n\\]\nwhere $\\delta$ and $k>0$ are integration constants, and $\\mathrm{sn}(\\,\\cdot,\\,ik)$ denotes the Jacobian elliptic sine function with  modulus $i k$.  \nApplying the classical Jacobi imaginary modulus transformation, the solution also admits the equivalent form \n\\[\nv = \\operatorname{arcsinh}\\Bigl( \\frac{k}{\\sqrt{1+k^2}} \\; \\mathrm{sn}\\bigl( \\sqrt{1+k^2}\\,(2u_1 + \\delta),\\; \\frac{k}{\\sqrt{1+k^2}} \\bigr) \\Bigr).\\]\n\nGeometrically, given a solution of~\\eqref{eq-ellip}, one first obtains a minimal surface $\\psi:\\mathbb{C}\\rightarrow \\mathbb{S}^3$ with induced metric $e^{2v}|dz|^2$ \nand Hopf differential $\\frac i 2 dz\\otimes dz$. \nSuch a surface should be necessarily homogeneous or of cohomogeneity one and hence belongs to the $T_{m,k,a}$ family constructed and classified by Hsiang and Lawson in \\cite{Hsiang-Lawson}. This family contains infinitely many closed minimal surfaces, whose topology is either a torus or a Klein bottle. Denote by $N$ the Gauss map of such a surface. Then the map \\emph{(}see the proof of Corollary $1$ in \\cite{Urbano}\\emph{)}\n\\[\n\\phi(z)=(V_\\psi, e^{2i u_2}):\\mathbb{C}\\rightarrow \\mathbb{S}^2\\times\\mathbb{S}^1\n\\]\ndefines a minimal surface in $\\mathbb{S}^2\\times\\mathbb{S}^1$, where\n\\[\nV_\\psi=\\frac{1}{\\sqrt{2}}(-2ie^{-2v}\\psi_{z}\\wedge\\psi_{\\overline{z}}+\\psi\\wedge N):\\mathbb{C}\\rightarrow \\mathbb{S}^2\\subset \\Lambda^2\\mathbb{R}^4. \n\\]\nApart from the obvious slices, these surfaces constitute all minimal-Willmore surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^1$. \n\\end{remark}", "thm1": "\\begin{thm}\\label{thm1}\n   Let \\( x: \\Sigma \\rightarrow \\mathbb{S}^2 \\times \\mathbb{S}^2 \\) be a minimal-Willmore surface. Then either \\( x \\) is a special complex curve given by a slice or a diagonal; or, up to an isometry, is contained in a totally geodesic submanifold $\\mathbb{S}^2 \\times \\mathbb{S}^1$, and can be described by a solution of the sinh-Gordon equation in one variable.\n\\end{thm}"}, "pre_theorem_intro_text_len": 2863, "pre_theorem_intro_text": "For a closed surface $x:\\Sigma\\rightarrow (N^n,g)$ in a Riemannian manifold, the squared $L^2$-norm of the trace-free second fundamental form is a fundamental conformal invariant; that is, it is preserved under conformal changes of the metric. This functional, referred to as the conformally invariant Willmore functional \\cite{Michelat,Mondino}, will henceforth be called simply the Willmore functional. The Willmore functional is equivalent, up to a topological invariant, to the integral   \n$$\\int_{\\Sigma}(|\\vec{H}|^2+K_{1212})dA,$$\nwhere $\\vec{H}$ is the mean curvature vector and $K_{1212}$ denotes the sectional curvature of $(N^n,g)$ restricted to the tangent plane of the surface. In what follows, we will denote this integral by $\\mathcal{W}$ and also refer to it as the Willmore functional. This functional exhibits notable links with other fundamental quantities, including, among others, the renormalized area functional within the AdS/CFT correspondence \\cite{Alexakis, Graham}. Surfaces that satisfy the Euler-Lagrange equation of this functional are called {\\em Willmore surfaces}. \n\nWhen the ambient space is a real space form, both the Willmore functional and Willmore surfaces have been extensively investigated. Notable advances include the resolution of the Willmore conjecture in $\\mathbb{S}^3$, originally proposed by Willmore in \\cite{Willmore} and proved by Marques and Neves in \\cite{Marques}.   Regarding the construction and classification of Willmore surfaces, we refer to \\cite{Bryant, Ejiri, WD, MWW} and references  therein. A basic fact is that every minimal surface in a real space form is  automatically Willmore. \nWhen the ambient space is a non-space form Riemannian manifold, the Euler–Lagrange equation of the Willmore functional has been derived independently by several groups of geometers from different viewpoints; see, for example, \\cite{Hu-Li, Mondino, Pedit, Wang-Xie}. Owing to the presence of ambient curvature terms in the Euler–Lagrange equation, not every minimal surface is Willmore in a non-space form Riemannian manifold. In \\cite{Montiel-Urbano}, Montiel and Urbano proved that the only surfaces in $\\mathbb{CP}^2$ that are both minimal and Willmore are the superminimal surfaces of positive spin, i.e., complex curves and minimal Lagrangian surfaces. Recently, the last two authors of  this paper have generalized this result to the self-dual K\\\"ahler-Einstein surfaces (see Proposition 2.3 in \\cite{Wang-Xie}). Note that $\\mathbb{S}^2 \\times \\mathbb{S}^2$ (equipped with the standard product metric and complex structure), being neither self-dual nor anti-self-dual, is another canonical example of a K\\\"ahler-Einstein surface. This motivates the study of which minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ are Willmore. We refer to a surface as minimal-Willmore if it is both minimal and Willmore.", "context": "For a closed surface $x:\\Sigma\\rightarrow (N^n,g)$ in a Riemannian manifold, the squared $L^2$-norm of the trace-free second fundamental form is a fundamental conformal invariant; that is, it is preserved under conformal changes of the metric. This functional, referred to as the conformally invariant Willmore functional \\cite{Michelat,Mondino}, will henceforth be called simply the Willmore functional. The Willmore functional is equivalent, up to a topological invariant, to the integral \n$$\\int_{\\Sigma}(|\\vec{H}|^2+K_{1212})dA,$$\nwhere $\\vec{H}$ is the mean curvature vector and $K_{1212}$ denotes the sectional curvature of $(N^n,g)$ restricted to the tangent plane of the surface. In what follows, we will denote this integral by $\\mathcal{W}$ and also refer to it as the Willmore functional. This functional exhibits notable links with other fundamental quantities, including, among others, the renormalized area functional within the AdS/CFT correspondence \\cite{Alexakis, Graham}. Surfaces that satisfy the Euler-Lagrange equation of this functional are called {\\em Willmore surfaces}.\n\nWhen the ambient space is a real space form, both the Willmore functional and Willmore surfaces have been extensively investigated. Notable advances include the resolution of the Willmore conjecture in $\\mathbb{S}^3$, originally proposed by Willmore in \\cite{Willmore} and proved by Marques and Neves in \\cite{Marques}. Regarding the construction and classification of Willmore surfaces, we refer to \\cite{Bryant, Ejiri, WD, MWW} and references therein. A basic fact is that every minimal surface in a real space form is automatically Willmore. \nWhen the ambient space is a non-space form Riemannian manifold, the Euler–Lagrange equation of the Willmore functional has been derived independently by several groups of geometers from different viewpoints; see, for example, \\cite{Hu-Li, Mondino, Pedit, Wang-Xie}. Owing to the presence of ambient curvature terms in the Euler–Lagrange equation, not every minimal surface is Willmore in a non-space form Riemannian manifold. In \\cite{Montiel-Urbano}, Montiel and Urbano proved that the only surfaces in $\\mathbb{CP}^2$ that are both minimal and Willmore are the superminimal surfaces of positive spin, i.e., complex curves and minimal Lagrangian surfaces. Recently, the last two authors of this paper have generalized this result to the self-dual K\\\"ahler-Einstein surfaces (see Proposition 2.3 in \\cite{Wang-Xie}). Note that $\\mathbb{S}^2 \\times \\mathbb{S}^2$ (equipped with the standard product metric and complex structure), being neither self-dual nor anti-self-dual, is another canonical example of a K\\\"ahler-Einstein surface. This motivates the study of which minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ are Willmore. We refer to a surface as minimal-Willmore if it is both minimal and Willmore.", "full_context": "For a closed surface $x:\\Sigma\\rightarrow (N^n,g)$ in a Riemannian manifold, the squared $L^2$-norm of the trace-free second fundamental form is a fundamental conformal invariant; that is, it is preserved under conformal changes of the metric. This functional, referred to as the conformally invariant Willmore functional \\cite{Michelat,Mondino}, will henceforth be called simply the Willmore functional. The Willmore functional is equivalent, up to a topological invariant, to the integral \n$$\\int_{\\Sigma}(|\\vec{H}|^2+K_{1212})dA,$$\nwhere $\\vec{H}$ is the mean curvature vector and $K_{1212}$ denotes the sectional curvature of $(N^n,g)$ restricted to the tangent plane of the surface. In what follows, we will denote this integral by $\\mathcal{W}$ and also refer to it as the Willmore functional. This functional exhibits notable links with other fundamental quantities, including, among others, the renormalized area functional within the AdS/CFT correspondence \\cite{Alexakis, Graham}. Surfaces that satisfy the Euler-Lagrange equation of this functional are called {\\em Willmore surfaces}.\n\nWhen the ambient space is a real space form, both the Willmore functional and Willmore surfaces have been extensively investigated. Notable advances include the resolution of the Willmore conjecture in $\\mathbb{S}^3$, originally proposed by Willmore in \\cite{Willmore} and proved by Marques and Neves in \\cite{Marques}. Regarding the construction and classification of Willmore surfaces, we refer to \\cite{Bryant, Ejiri, WD, MWW} and references therein. A basic fact is that every minimal surface in a real space form is automatically Willmore. \nWhen the ambient space is a non-space form Riemannian manifold, the Euler–Lagrange equation of the Willmore functional has been derived independently by several groups of geometers from different viewpoints; see, for example, \\cite{Hu-Li, Mondino, Pedit, Wang-Xie}. Owing to the presence of ambient curvature terms in the Euler–Lagrange equation, not every minimal surface is Willmore in a non-space form Riemannian manifold. In \\cite{Montiel-Urbano}, Montiel and Urbano proved that the only surfaces in $\\mathbb{CP}^2$ that are both minimal and Willmore are the superminimal surfaces of positive spin, i.e., complex curves and minimal Lagrangian surfaces. Recently, the last two authors of this paper have generalized this result to the self-dual K\\\"ahler-Einstein surfaces (see Proposition 2.3 in \\cite{Wang-Xie}). Note that $\\mathbb{S}^2 \\times \\mathbb{S}^2$ (equipped with the standard product metric and complex structure), being neither self-dual nor anti-self-dual, is another canonical example of a K\\\"ahler-Einstein surface. This motivates the study of which minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ are Willmore. We refer to a surface as minimal-Willmore if it is both minimal and Willmore.\n\n\\section{Introduction}\nFor a closed surface $x:\\Sigma\\rightarrow (N^n,g)$ in a Riemannian manifold, the squared $L^2$-norm of the trace-free second fundamental form is a fundamental conformal invariant; that is, it is preserved under conformal changes of the metric. This functional, referred to as the conformally invariant Willmore functional \\cite{Michelat,Mondino}, will henceforth be called simply the Willmore functional. The Willmore functional is equivalent, up to a topological invariant, to the integral \n$$\\int_{\\Sigma}(|\\vec{H}|^2+K_{1212})dA,$$\nwhere $\\vec{H}$ is the mean curvature vector and $K_{1212}$ denotes the sectional curvature of $(N^n,g)$ restricted to the tangent plane of the surface. In what follows, we will denote this integral by $\\mathcal{W}$ and also refer to it as the Willmore functional. This functional exhibits notable links with other fundamental quantities, including, among others, the renormalized area functional within the AdS/CFT correspondence \\cite{Alexakis, Graham}. Surfaces that satisfy the Euler-Lagrange equation of this functional are called {\\em Willmore surfaces}.\n\nIn their work \\cite{Urbano}, Tobarro and Urbano gave a beautiful local characterization of minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ that possess no complex points. Their description is formulated in terms of a pair of solutions $v$ and $w$ to the sinh‑Gordon equation \n$$v_{z\\bar z}+\\frac{1}{2}\\sinh (2v)=0.$$\nWithin this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w$, see \\eqref{eq-mwill}. By analyzing this coupled system, we establish the theorem above.\n\n\\begin{example}\\label{ex-slice} For any point $p\\in\\mathbb{S}^2$, the corresponding slices \n\\[\n\\mathbb{S}^2\\times\\{p\\}=\\{(x,p)\\in\\mathbb{S}^2\\times\\mathbb{S}^2 \\mid x\\in\\mathbb{S}^2\\},\n\\]\n\\[\n\\{p\\}\\times\\mathbb{S}^2=\\{(p,x)\\in\\mathbb{S}^2\\times\\mathbb{S}^2 \\mid x\\in\\mathbb{S}^2\\}.\n\\]\nare totally geodesic with the first one satisfying $C_1=C_2=1$ and the second one satisfying $C_1=-C_2=1$. That is, the slices are complex with respect to both $J_1$ and $J_2$.\n\\end{example}\n\\begin{example}\\label{ex-diagonal} The diagonal \n$$D\\triangleq\\{(x,x)\\in\\mathbb{S}^2\\times\\mathbb{S}^2 \\mid x\\in\\mathbb{S}^2\\}$$\nis totally geodesic and satisfies $C_1=1$ and $C_2=0$, which means that $D$ is complex with respect to $J_1$ and Lagrangian with respect to $J_2$.\n\\end{example}\n\\begin{example}\\label{ex-Clifford}\n The torus defined by \n $$T\\triangleq\\{(x,y)\\in\\mathbb{S}^2\\times\\mathbb{S}^2 \\mid x_1=y_1=0\\}$$ \n is totally geodesic and satisfies $C_1=C_2=0$, which means that $T$ is Lagrangian for both $J_1$ and $J_2$. This surface is known as the Clifford torus in $\\mathbb{S}^2 \\times \\mathbb{S}^2$, characterized by the flatness of both its tangent and normal bundles. \n\\end{example}\n\\begin{example} \nLet $\\Sigma=\\mathbb{C}/\\Lambda$ be a torus generated by the lattice $\\Lambda=\\{m+n\\tau \\mid m,n\\in\\mathbb{Z}\\}$, where $\\tau$ is a complex number with $\\operatorname{Im}\\tau>0$. Let $\\wp:\\Sigma\\rightarrow\\mathbb{S}^2$ be the Weierstrass $\\wp$-function with a second-order pole at the origin, and $z_0\\in\\Sigma$ be a point at which $\\wp$ is not ramified. Then $x=\\big(\\wp(z),\\wp(z-z_0)\\big):\\Sigma\\rightarrow(\\mathbb{S}^2\\times\\mathbb{S}^2)$ is a holomorphic embedding with the complex structure $J_1$. Such minimal tori are referred to as Weierstrass tori.\n\\end{example}\n\n\\begin{proposition}\\label{pro-mw}\n Let $x:\\Sigma \\rightarrow \\mathbb{S}^2\\times \\mathbb{S}^2$ be a complex curve or minimal Lagrangian surface. Then $x$ is Willmore if and only if it is one of the surfaces given in Example~\\ref{ex-slice} $\\sim$ Example~\\ref{ex-Clifford}. \n\\end{proposition}\n\\begin{proof}\nWithout loss of generality, we assume that $x$ is complex or minimal Lagrangian with respect to the complex structure $J_1$. Consequently, $C_1$ is a constant which must be either $0$ or $\\pm 1$. It follows from \\cite[Proposition 3]{Urbano} (see also \\cite{Chen}) that we only need to prove that $C_2$ is a constant.\n\nBy \\eqref{eq-C1z}, we have $f_1\\bar{\\gamma_1}=0$. Substituting this into \\eqref{eq-mW} yields $\\bar{f_1}{\\gamma_2}=0$, which, by \\eqref{eq-C2z}, implies that $C_2$ is holomorphic. Since $C_2$ is real-valued, it must be constant. \n\\end{proof}\n\\begin{theorem}\\label{thm-mw}\n Let $x:\\Sigma \\rightarrow \\mathbb{S}^2\\times \\mathbb{S}^2$ be a minimal surface without complex points. Then $x$ is Willmore if and only if up to an isometry it is a minimal-Willmore surface in $\\mathbb{S}^2\\times\\mathbb{S}^1$. \n\\end{theorem}\n\nLet $x:\\Sigma \\rightarrow \\mathbb{S}^2\\times \\mathbb{S}^2$ be a minimal immersion of a simply connected surface without complex points and $z$ be a complex coordinate such that the Hopf differential $\\Theta(z)\\triangleq\\frac{\\langle J_1 x_z, J_2 x_z\\rangle}{2}dz^2$ (which is holomorphic) satisfies $\\Theta(z)=dz^2$. \n Then the functions \n $$v\\triangleq\\frac{1}{2}\\ln \\sqrt{\\frac{(1+C_1)(1+C_2)}{(1-C_1)(1-C_2)}},~~~w\\triangleq\\frac{1}{2}\\ln \\sqrt{\\frac{(1-C_1)(1+C_2)}{(1+C_1)(1-C_2)}}$$\n satisfy the sinh-Gordon equation, i.e., \n $$v_{z\\bar z}+\\frac{1}{2}\\sinh (2v)=0,~~~w_{z\\bar z}+\\frac{1}{2}\\sinh (2w)=0.$$\n Conversely, given two solutions $v, w:\\mathbb{C}\\rightarrow \\mathbb{R}$ of the sinh-Gordon equation, one can construct a 1-parameter family of minimal immersions $X_t: \\mathbb{S}^2 \\times \\mathbb{S}^2$ by taking the following quantities as the fundamental data, \n$$\\sigma=\\frac{1}{2}\\ln\\big(4\\cosh(v+w)\\cosh(v-w)\\big),~~~ \\rho=\\left(\\ln \\sqrt{\\frac{\\cosh(v+w)}{\\cosh(v-w)}}\\right)_{\\!\\!z},~~~$$\n$$C_1=\\tanh(v-w),~~~C_2=\\tanh(v+w),$$\n$$\\gamma_1=\\sqrt{2}e^{\\frac{it}{2}}\\sqrt{\\frac{\\cosh(v+w)}{\\cosh(v-w)}},~~~\\gamma_2=\\sqrt{2}e^{\\frac{it}{2}}\\sqrt{\\frac{\\cosh(v-w)}{\\cosh(v+w)}},~~~f_1=-i \\gamma_1 (v-w)_z,~~~f_2=-i\\gamma_2(v+w)_z.$$\nGeometrically, this construction originates from the Gauss maps of two minimal surfaces in \\( \\mathbb{S}^3 \\), whose induced metrics are \\( e^{2v} |dz|^2 \\) and \\( e^{2w} |dw|^2 \\), respectively, and which share the same Hopf differential \\( \\theta(z) = \\frac{i}{2} dz^2 \\). For further details, we refer to Section~5 of \\cite{Urbano}.\n\nTherefore, we establish the claim that $p=\\pm q$, which means either $v=0$ or $w=0$. It then follows from Proposition 4 in \\cite{Urbano} that $x$ is non-full in $\\mathbb{S}^2 \\times \\mathbb{S}^2$. This means it lies in a totally geodesic hypersurface of the ambient space, which, up to an isometry, is an open subset of $\\mathbb{S}^2 \\times \\mathbb{S}^1$. \n\\end{proof}\n\\begin{remark}\\label{rk-mini}If $w=0$, then it follows from \\eqref{eq-vw} that $v$ only depends on the variable $u_1$, and satisfies \n\\begin{equation}\\label{eq-ellip}\n\\frac{d^2 v}{d u_1 ^2}=-2\\sinh(2v).\n\\end{equation}\nThe solution to this ordinary differential equation can be expressed using an elliptic function, \n\\[\nv = \\operatorname{arcsinh}\\Bigl( k \\; \\mathrm{sn}\\bigl( 2u_1 + \\delta,\\; i k \\bigr) \\Bigr),\n\\]\nwhere $\\delta$ and $k>0$ are integration constants, and $\\mathrm{sn}(\\,\\cdot,\\,ik)$ denotes the Jacobian elliptic sine function with modulus $i k$. \nApplying the classical Jacobi imaginary modulus transformation, the solution also admits the equivalent form \n\\[\nv = \\operatorname{arcsinh}\\Bigl( \\frac{k}{\\sqrt{1+k^2}} \\; \\mathrm{sn}\\bigl( \\sqrt{1+k^2}\\,(2u_1 + \\delta),\\; \\frac{k}{\\sqrt{1+k^2}} \\bigr) \\Bigr).\\]", "post_theorem_intro_text_len": 2347, "post_theorem_intro_text": "\\noindent In other words, when the ambient space is $\\mathbb{S}^2\\times\\mathbb{S}^2$,  within the minimal class the Willmore condition turns out to be  rigid: it forces special holomorphicity, or reduces the problem to a totally geodesic hypersurface. \n Geometrically, the last class of minimal surfaces in Theorem~\\ref{thm1} arises from a certain Gauss-map construction applied to a cohomogeneity {one} minimal surface in $\\mathbb{S}^3\\subset\\mathbb{R}^4$, see Remark~\\ref{rk-mini}. \n\nIn their work \\cite{Urbano}, Tobarro and Urbano gave a beautiful local characterization of minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ that possess no complex points. Their description is formulated in terms of a pair of solutions $v$ and $w$ to the sinh‑Gordon equation \n$$v_{z\\bar z}+\\frac{1}{2}\\sinh (2v)=0.$$\nWithin this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w$, see \\eqref{eq-mwill}. By analyzing this coupled system, we establish the theorem above. \n\nExcept for seeking Willmore surfaces among minimal surfaces, another natural approach is to use the product structure of  $\\mathbb{S}^2\\times\\mathbb{S}^2$ to produce Willmore surfaces. \n\\begin{thm}\\label{thm2}\n   All Willmore surfaces of product type in $\\mathbb{S}^2 \\times \\mathbb{S}^2$ are exhausted by products of an elastic curve in $\\mathbb{S}^2$ with a great circle. \n\\end{thm}\n\nNote that both of these two classes of examples yield Willmore surfaces that are non-linearly full in $\\mathbb{S}^2 \\times \\mathbb{S}^2$. No linearly full examples are currently known. We point out that constructing explicit examples, or even addressing the existence problem, in non-space-form Riemannian manifolds is particularly challenging; see the efforts in \\cite{Modino,Ikoma,Michelat} and the references therein. \n\nThe paper is organized as follows. {Section~\\ref{sec2}  begins with the  preliminaries on  $\\mathbb{S}^2\\times \\mathbb{S}^2$ and on surfaces within it. In Section~\\ref{sec3}, we first derive the fundamental equations for surfaces in $\\mathbb{S}^2\\times \\mathbb{S}^2$, and subsequently compute the corresponding Willmore equation. Section~\\ref{sec4} is devoted to the classification of minimal-Willmore surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$. Finally, in Section~\\ref{sec5}, we classify Willmore surfaces of product type.", "sketch": "Using the local characterization of Tobarro--Urbano \\cite{Urbano} for minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ with no complex points, the surface is described \"in terms of a pair of solutions $v$ and $w$ to the sinh-Gordon equation\" $$v_{z\\bar z}+\\tfrac12\\sinh(2v)=0.$$ \"Within this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w$, see \\eqref{eq-mwill}.\" The proof then proceeds by \"analyzing this coupled system\" (the sinh-Gordon equation together with the additional PDE imposed by the Willmore condition), which yields the classification stated in Theorem~\\ref{thm1}.", "expanded_sketch": "Using the local characterization of Tobarro--Urbano \\cite{Urbano} for minimal surfaces in $\\mathbb{S}^2\\times\\mathbb{S}^2$ with no complex points, the surface is described \"in terms of a pair of solutions $v$ and $w$ to the sinh-Gordon equation\" $$v_{z\\bar z}+\\tfrac12\\sinh(2v)=0.$$ \"Within this framework, the Willmore condition introduces a new partial differential equation linking $v$ and $w,\" namely\n\\begin{equation}\\label{eq-mwill}\n    e^{it}\\cosh^3(v-w)(v+w)_{\\bar{z}} = \\cosh^3(v+w)(v-w)_z.\n    \\end{equation}\nThe proof then proceeds by \"analyzing this coupled system\" (the sinh-Gordon equation together with the additional PDE imposed by the Willmore condition). To prove the main theorem, one shows that this analysis yields the stated classification.", "expanded_theorem": "\\label{thm1}\n   Let \\( x: \\Sigma \\rightarrow \\mathbb{S}^2 \\times \\mathbb{S}^2 \\) be a minimal-Willmore surface. Then either \\( x \\) is a special complex curve given by a slice or a diagonal; or, up to an isometry, is contained in a totally geodesic submanifold $\\mathbb{S}^2 \\times \\mathbb{S}^1$, and can be described by a solution of the sinh-Gordon equation in one variable.", "theorem_type": ["Classification or Bijection", "Universal"], "mcq": {"question": "A minimal-Willmore surface in \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\) means a surface \\(x:\\Sigma\\to \\mathbb{S}^2\\times\\mathbb{S}^2\\) that is both minimal and Willmore, where a Willmore surface satisfies the Euler-Lagrange equation of the Willmore functional \\(\\mathcal W=\\int_\\Sigma (|\\vec H|^2+K_{1212})\\,dA\\). A slice is a surface of the form \\(\\mathbb{S}^2\\times\\{p\\}\\) or \\(\\{p\\}\\times\\mathbb{S}^2\\), and a diagonal means a diagonal copy of \\(\\mathbb{S}^2\\) in \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\). Which statement holds for every minimal-Willmore surface \\(x:\\Sigma\\to \\mathbb{S}^2\\times\\mathbb{S}^2\\)?", "correct_choice": {"label": "A", "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\), the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\), and the surface is described by a solution of the sinh-Gordon equation depending on a single variable."}, "choices": [{"label": "B", "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\), the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\), and the surface is described by a pair of independent solutions of the sinh-Gordon equation in two variables."}, {"label": "C", "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\), the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\)."}, {"label": "D", "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\), and every such non-complex minimal-Willmore surface is described by a solution of the sinh-Gordon equation depending on two variables."}, {"label": "E", "text": "Either \\(x\\) is a special complex curve, namely a slice or a diagonal; or, after applying an isometry of \\(\\mathbb{S}^2\\times\\mathbb{S}^2\\), the image of \\(x\\) is contained in a totally geodesic submanifold \\(\\mathbb{S}^2\\times\\mathbb{S}^1\\), and the surface is described by a solution of the sinh-Gordon equation, with no restriction that the solution depend on only one variable."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "collapse from coupled system to one-variable reduction", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "geometric_construction", "tampered_component": "single-variable sinh-Gordon description", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "one-variable dependence conclusion", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "restriction to one-variable solutions", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the setting and terminology but does not reveal the classification result or explicitly hint at the one-variable sinh-Gordon reduction. The correct answer is not leaked by the wording of the question itself."}, "TAS": {"score": 1, "justification": "The item is essentially a theorem-identification question: the correct option states a precise classification theorem rather than requiring derivation from the definitions in the stem. However, it is not a pure tautology because the distractors alter meaningful logical strength and technical details."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to choose the strongest true statement over weaker or overextended variants, especially regarding the one-variable sinh-Gordon condition. Still, this mainly tests precise recall/discrimination of a known result, not substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically structured: one is weaker but true, while others introduce realistic confusions about dimensional dependence or strength of the conclusion. They are distinct and align with common failure modes in reading classification theorems."}, "total_score": 6, "overall_assessment": "A reasonably strong theorem-discrimination MCQ with no answer leakage and good distractors, but it leans more toward recall of a classification result than genuine generative reasoning."}}
{"id": "2602.05643v1", "paper_link": "http://arxiv.org/abs/2602.05643v1", "theorems_cnt": 1, "theorem": {"env_name": "thmABC", "content": "\\label{thm:A}\n\tLet $\\Sigma$ be an $S$-integral reduction type with cuspidal part $\\Sigma^{\\csp}$ and let $\\mathfrak p \\not\\in S$ be a prime of $K$.\n\tIf $(a_1,\\ldots,a_{g+n-1}) \\in K_{\\mathfrak p}^{g+n-1}$ is contained in the kernel of the matrix $M(\\Sigma^{\\csp})$ defined in \\eqref{eq:MSigma}--\\eqref{eq:matrixD}, let $\\omega = \\omega(\\Sigma^{\\csp}) \\coloneqq a_1 \\omega_1 + \\dots + a_{g+n-1} \\omega_{g+n-1}$ and let $c = c(P_0,\\Sigma,\\omega)$ be the constant defined by\n\t\\eqref{eq:constants} and \\eqref{eq:b-lambda}.\n\tThen\n\t\\[\n\t\t\\int_{P_0}^P \\omega = c \\quad \\text{ for all } P \\in \\mathcal{Y}(\\cO_{K,S})_\\Sigma.\n\t\\]", "start_pos": 10558, "end_pos": 11179, "label": "thm:A"}, "ref_dict": {"eq:matrixD": "\\begin{equation}\n\t\t\\label{eq:matrixD}\n\t\tD(U)_{i,j} \\coloneqq \\sum_{\\substack{Q \\in |D|,\\\\ \\varphi\\colon k(Q) \\to \\Kpbar}} \\varphi(\\Res_{Q} \\omega_{g+j}) \\sum_{\\substack{\\text{$\\fq$ prime,}\\\\ \\text{$\\lambda|\\fq$ in $\\cO_{k(Q)}$}}} u_{i,\\lambda} \\log \\varphi(\\pi_{\\lambda}).\n\t\\end{equation}", "eq:t-ell": "\\begin{equation}\n\t\t\\label{eq:t-ell}\n\t\t\\sigma(F) = \\sum_{i=1}^s t_{i}(F) u_i \\qquad \\text{in $V_D$},\n\t\\end{equation}", "eq:F-and-Gi": "\\begin{equation}\n\t\t\\label{eq:F-and-Gi}\n\t\tF = \\sum_{i=1}^r x_i(F) G_i + \\div(f)\n\t\\end{equation}", "eq:f-of-Q": "\\begin{equation}\n\t\t\\label{eq:f-of-Q}\n\t\t(f(Q))_Q = \\prod_{i=1}^{k} e_{i}^{a_{i}(F)} \\cdot  \\left(\\prod_{\\text{$\\lambda$ prime of $\\cO_{k(Q)}$}} \\pi_{\\lambda}^{v_\\lambda(f(Q))}\\right)_Q \\quad \\text{in} \\quad \\left.\\left(\\prod_Q k(Q)^\\times_{\\bQ}\\right)\\right/K_{\\bQ}^\\times,\n\t\\end{equation}", "eq:chabauty-condition-over-number-field-intro": "\\begin{equation}\n\\label{eq:chabauty-condition-over-number-field-intro}\nr + \\# S + ([K : \\bQ]-1)n < g + \\rank \\cO_K^\\times + \\#|D| + n_2(D) - 1,\n\\end{equation}", "item:algo-units": "\\begin{enumerate}[label=(\\arabic*)]\n\t\\item \\label{item:algo-D-transversal-model}\n\tExtend $\\cX$ to a regular model over~$\\cO_K$. Ensure that the model is $D$-transversal over primes in~$S$ via blow-ups if necessary (see Section~\\ref{sec:redtypes}).\n\t\\item \\label{item:algo-intersection-numbers}\n\tFor each prime~$\\fq$ of bad reduction for~$\\cX$, determine the set of components and the intersection matrix of the mod-$\\fq$ fibre~$\\cX_{\\fq}$. \n\t\\item \\label{item:algo-mw-basis}\n\tFind divisors $G_i\\in \\Div^0(Y)$ generating a full rank subgroup of the Mordell--Weil group $J(K)$.\n\t\\item \\label{item:algo-correction-divisors}\n\tUsing \\cite[Lemma~2.7]{affchab1}, determine the vertical correction divisors $\\Phi_{\\fq}(G_i)$ for each prime $\\fq$ of $K$. This involves the Moore--Penrose pseudoinverse of the intersection matrix of the mod-$\\fq$ fibre computed in step~\\ref{item:algo-intersection-numbers}.\n\t\\item \\label{item:algo-log-differentials}\n\tFind a basis $\\omega_1,\\ldots,\\omega_{g+n-1}$ of the space of log differentials $\\rH^0(X,\\Omega^1(D))$ and compute their residues $\\Res_Q(\\omega_j) \\in k(Q)$ at all cusps $Q \\in |D|$.\n\t\\item \\label{item:algo-units}\n\tFor each $Q\\in |D|$, find $\\varepsilon_i \\in \\cO_{k(Q)}^\\times$ generating a full rank subgroup of $\\cO_{k(Q)}^\\times$.\n\t\\item \\label{item:algo-prime-ideal-generators}\n\tFor finitely many ``special'' primes $\\fq$ of $K$, determine generators $\\rho_{\\fq}$ of $\\fq$ in $K^\\times\\otimes {\\bQ}$.\n\tSimilarly, for $Q\\in |D|$ and finitely many ``special'' primes $\\lambda$ of $k(Q)$, determine generators $\\pi_{\\lambda}$ of $\\lambda$ in $k(Q)^\\times\\otimes {\\bQ}$.\n\t\\item \\label{item:algo-integrals}\n\tFind a method for calculating $\\fp$-adic integrals of log differentials on $X$.\n\t\\item \\label{item:algo-chabauty-function}\n\tFor each $S$-integral reduction type~$\\Sigma$, determine the matrix $M(\\Sigma^{\\csp})$ and find a non-trivial element $(a_1,\\ldots,a_{g+n-1})$ in the kernel. Compute the constant $c=c(P_0,\\Sigma,\\omega)$ for the  Chabauty differential $\\omega = \\sum_j a_j \\omega_j$.\n\t\\item \\label{item:algo-chabauty-locus}\n\tOn each residue disc, expand the $\\fp$-adic analytic function $P \\mapsto \\int_{P_0}^P \\omega$ into a power series and determine the locus where $\\int_{P_0}^P \\omega=c$.\n\\end{enumerate}", "item:algo-prime-ideal-generators": "\\begin{enumerate}[label=(\\arabic*)]\n\t\\item \\label{item:algo-D-transversal-model}\n\tExtend $\\cX$ to a regular model over~$\\cO_K$. Ensure that the model is $D$-transversal over primes in~$S$ via blow-ups if necessary (see Section~\\ref{sec:redtypes}).\n\t\\item \\label{item:algo-intersection-numbers}\n\tFor each prime~$\\fq$ of bad reduction for~$\\cX$, determine the set of components and the intersection matrix of the mod-$\\fq$ fibre~$\\cX_{\\fq}$. \n\t\\item \\label{item:algo-mw-basis}\n\tFind divisors $G_i\\in \\Div^0(Y)$ generating a full rank subgroup of the Mordell--Weil group $J(K)$.\n\t\\item \\label{item:algo-correction-divisors}\n\tUsing \\cite[Lemma~2.7]{affchab1}, determine the vertical correction divisors $\\Phi_{\\fq}(G_i)$ for each prime $\\fq$ of $K$. This involves the Moore--Penrose pseudoinverse of the intersection matrix of the mod-$\\fq$ fibre computed in step~\\ref{item:algo-intersection-numbers}.\n\t\\item \\label{item:algo-log-differentials}\n\tFind a basis $\\omega_1,\\ldots,\\omega_{g+n-1}$ of the space of log differentials $\\rH^0(X,\\Omega^1(D))$ and compute their residues $\\Res_Q(\\omega_j) \\in k(Q)$ at all cusps $Q \\in |D|$.\n\t\\item \\label{item:algo-units}\n\tFor each $Q\\in |D|$, find $\\varepsilon_i \\in \\cO_{k(Q)}^\\times$ generating a full rank subgroup of $\\cO_{k(Q)}^\\times$.\n\t\\item \\label{item:algo-prime-ideal-generators}\n\tFor finitely many ``special'' primes $\\fq$ of $K$, determine generators $\\rho_{\\fq}$ of $\\fq$ in $K^\\times\\otimes {\\bQ}$.\n\tSimilarly, for $Q\\in |D|$ and finitely many ``special'' primes $\\lambda$ of $k(Q)$, determine generators $\\pi_{\\lambda}$ of $\\lambda$ in $k(Q)^\\times\\otimes {\\bQ}$.\n\t\\item \\label{item:algo-integrals}\n\tFind a method for calculating $\\fp$-adic integrals of log differentials on $X$.\n\t\\item \\label{item:algo-chabauty-function}\n\tFor each $S$-integral reduction type~$\\Sigma$, determine the matrix $M(\\Sigma^{\\csp})$ and find a non-trivial element $(a_1,\\ldots,a_{g+n-1})$ in the kernel. Compute the constant $c=c(P_0,\\Sigma,\\omega)$ for the  Chabauty differential $\\omega = \\sum_j a_j \\omega_j$.\n\t\\item \\label{item:algo-chabauty-locus}\n\tOn each residue disc, expand the $\\fp$-adic analytic function $P \\mapsto \\int_{P_0}^P \\omega$ into a power series and determine the locus where $\\int_{P_0}^P \\omega=c$.\n\\end{enumerate}", "eq:MSigma": "\\begin{equation}\n\t\\label{eq:MSigma}\n\tM(U) \\coloneqq \\begin{pmatrix}\n\t\tA & B  \\\\\n\t\t0 & C  \\\\\n\t\t0 & D(U) \n\t\\end{pmatrix}\n\\end{equation}", "item:algo-integrals": "\\begin{enumerate}[label=(\\arabic*)]\n\t\\item \\label{item:algo-D-transversal-model}\n\tExtend $\\cX$ to a regular model over~$\\cO_K$. Ensure that the model is $D$-transversal over primes in~$S$ via blow-ups if necessary (see Section~\\ref{sec:redtypes}).\n\t\\item \\label{item:algo-intersection-numbers}\n\tFor each prime~$\\fq$ of bad reduction for~$\\cX$, determine the set of components and the intersection matrix of the mod-$\\fq$ fibre~$\\cX_{\\fq}$. \n\t\\item \\label{item:algo-mw-basis}\n\tFind divisors $G_i\\in \\Div^0(Y)$ generating a full rank subgroup of the Mordell--Weil group $J(K)$.\n\t\\item \\label{item:algo-correction-divisors}\n\tUsing \\cite[Lemma~2.7]{affchab1}, determine the vertical correction divisors $\\Phi_{\\fq}(G_i)$ for each prime $\\fq$ of $K$. This involves the Moore--Penrose pseudoinverse of the intersection matrix of the mod-$\\fq$ fibre computed in step~\\ref{item:algo-intersection-numbers}.\n\t\\item \\label{item:algo-log-differentials}\n\tFind a basis $\\omega_1,\\ldots,\\omega_{g+n-1}$ of the space of log differentials $\\rH^0(X,\\Omega^1(D))$ and compute their residues $\\Res_Q(\\omega_j) \\in k(Q)$ at all cusps $Q \\in |D|$.\n\t\\item \\label{item:algo-units}\n\tFor each $Q\\in |D|$, find $\\varepsilon_i \\in \\cO_{k(Q)}^\\times$ generating a full rank subgroup of $\\cO_{k(Q)}^\\times$.\n\t\\item \\label{item:algo-prime-ideal-generators}\n\tFor finitely many ``special'' primes $\\fq$ of $K$, determine generators $\\rho_{\\fq}$ of $\\fq$ in $K^\\times\\otimes {\\bQ}$.\n\tSimilarly, for $Q\\in |D|$ and finitely many ``special'' primes $\\lambda$ of $k(Q)$, determine generators $\\pi_{\\lambda}$ of $\\lambda$ in $k(Q)^\\times\\otimes {\\bQ}$.\n\t\\item \\label{item:algo-integrals}\n\tFind a method for calculating $\\fp$-adic integrals of log differentials on $X$.\n\t\\item \\label{item:algo-chabauty-function}\n\tFor each $S$-integral reduction type~$\\Sigma$, determine the matrix $M(\\Sigma^{\\csp})$ and find a non-trivial element $(a_1,\\ldots,a_{g+n-1})$ in the kernel. Compute the constant $c=c(P_0,\\Sigma,\\omega)$ for the  Chabauty differential $\\omega = \\sum_j a_j \\omega_j$.\n\t\\item \\label{item:algo-chabauty-locus}\n\tOn each residue disc, expand the $\\fp$-adic analytic function $P \\mapsto \\int_{P_0}^P \\omega$ into a power series and determine the locus where $\\int_{P_0}^P \\omega=c$.\n\\end{enumerate}", "item:algo-chabauty-function": "\\begin{enumerate}[label=(\\arabic*)]\n\t\\item \\label{item:algo-D-transversal-model}\n\tExtend $\\cX$ to a regular model over~$\\cO_K$. Ensure that the model is $D$-transversal over primes in~$S$ via blow-ups if necessary (see Section~\\ref{sec:redtypes}).\n\t\\item \\label{item:algo-intersection-numbers}\n\tFor each prime~$\\fq$ of bad reduction for~$\\cX$, determine the set of components and the intersection matrix of the mod-$\\fq$ fibre~$\\cX_{\\fq}$. \n\t\\item \\label{item:algo-mw-basis}\n\tFind divisors $G_i\\in \\Div^0(Y)$ generating a full rank subgroup of the Mordell--Weil group $J(K)$.\n\t\\item \\label{item:algo-correction-divisors}\n\tUsing \\cite[Lemma~2.7]{affchab1}, determine the vertical correction divisors $\\Phi_{\\fq}(G_i)$ for each prime $\\fq$ of $K$. This involves the Moore--Penrose pseudoinverse of the intersection matrix of the mod-$\\fq$ fibre computed in step~\\ref{item:algo-intersection-numbers}.\n\t\\item \\label{item:algo-log-differentials}\n\tFind a basis $\\omega_1,\\ldots,\\omega_{g+n-1}$ of the space of log differentials $\\rH^0(X,\\Omega^1(D))$ and compute their residues $\\Res_Q(\\omega_j) \\in k(Q)$ at all cusps $Q \\in |D|$.\n\t\\item \\label{item:algo-units}\n\tFor each $Q\\in |D|$, find $\\varepsilon_i \\in \\cO_{k(Q)}^\\times$ generating a full rank subgroup of $\\cO_{k(Q)}^\\times$.\n\t\\item \\label{item:algo-prime-ideal-generators}\n\tFor finitely many ``special'' primes $\\fq$ of $K$, determine generators $\\rho_{\\fq}$ of $\\fq$ in $K^\\times\\otimes {\\bQ}$.\n\tSimilarly, for $Q\\in |D|$ and finitely many ``special'' primes $\\lambda$ of $k(Q)$, determine generators $\\pi_{\\lambda}$ of $\\lambda$ in $k(Q)^\\times\\otimes {\\bQ}$.\n\t\\item \\label{item:algo-integrals}\n\tFind a method for calculating $\\fp$-adic integrals of log differentials on $X$.\n\t\\item \\label{item:algo-chabauty-function}\n\tFor each $S$-integral reduction type~$\\Sigma$, determine the matrix $M(\\Sigma^{\\csp})$ and find a non-trivial element $(a_1,\\ldots,a_{g+n-1})$ in the kernel. Compute the constant $c=c(P_0,\\Sigma,\\omega)$ for the  Chabauty differential $\\omega = \\sum_j a_j \\omega_j$.\n\t\\item \\label{item:algo-chabauty-locus}\n\tOn each residue disc, expand the $\\fp$-adic analytic function $P \\mapsto \\int_{P_0}^P \\omega$ into a power series and determine the locus where $\\int_{P_0}^P \\omega=c$.\n\\end{enumerate}", "item:algo-chabauty-locus": "\\begin{enumerate}[label=(\\arabic*)]\n\t\\item \\label{item:algo-D-transversal-model}\n\tExtend $\\cX$ to a regular model over~$\\cO_K$. Ensure that the model is $D$-transversal over primes in~$S$ via blow-ups if necessary (see Section~\\ref{sec:redtypes}).\n\t\\item \\label{item:algo-intersection-numbers}\n\tFor each prime~$\\fq$ of bad reduction for~$\\cX$, determine the set of components and the intersection matrix of the mod-$\\fq$ fibre~$\\cX_{\\fq}$. \n\t\\item \\label{item:algo-mw-basis}\n\tFind divisors $G_i\\in \\Div^0(Y)$ generating a full rank subgroup of the Mordell--Weil group $J(K)$.\n\t\\item \\label{item:algo-correction-divisors}\n\tUsing \\cite[Lemma~2.7]{affchab1}, determine the vertical correction divisors $\\Phi_{\\fq}(G_i)$ for each prime $\\fq$ of $K$. This involves the Moore--Penrose pseudoinverse of the intersection matrix of the mod-$\\fq$ fibre computed in step~\\ref{item:algo-intersection-numbers}.\n\t\\item \\label{item:algo-log-differentials}\n\tFind a basis $\\omega_1,\\ldots,\\omega_{g+n-1}$ of the space of log differentials $\\rH^0(X,\\Omega^1(D))$ and compute their residues $\\Res_Q(\\omega_j) \\in k(Q)$ at all cusps $Q \\in |D|$.\n\t\\item \\label{item:algo-units}\n\tFor each $Q\\in |D|$, find $\\varepsilon_i \\in \\cO_{k(Q)}^\\times$ generating a full rank subgroup of $\\cO_{k(Q)}^\\times$.\n\t\\item \\label{item:algo-prime-ideal-generators}\n\tFor finitely many ``special'' primes $\\fq$ of $K$, determine generators $\\rho_{\\fq}$ of $\\fq$ in $K^\\times\\otimes {\\bQ}$.\n\tSimilarly, for $Q\\in |D|$ and finitely many ``special'' primes $\\lambda$ of $k(Q)$, determine generators $\\pi_{\\lambda}$ of $\\lambda$ in $k(Q)^\\times\\otimes {\\bQ}$.\n\t\\item \\label{item:algo-integrals}\n\tFind a method for calculating $\\fp$-adic integrals of log differentials on $X$.\n\t\\item \\label{item:algo-chabauty-function}\n\tFor each $S$-integral reduction type~$\\Sigma$, determine the matrix $M(\\Sigma^{\\csp})$ and find a non-trivial element $(a_1,\\ldots,a_{g+n-1})$ in the kernel. Compute the constant $c=c(P_0,\\Sigma,\\omega)$ for the  Chabauty differential $\\omega = \\sum_j a_j \\omega_j$.\n\t\\item \\label{item:algo-chabauty-locus}\n\tOn each residue disc, expand the $\\fp$-adic analytic function $P \\mapsto \\int_{P_0}^P \\omega$ into a power series and determine the locus where $\\int_{P_0}^P \\omega=c$.\n\\end{enumerate}", "item:algo-log-differentials": "\\begin{enumerate}[label=(\\arabic*)]\n\t\\item \\label{item:algo-D-transversal-model}\n\tExtend $\\cX$ to a regular model over~$\\cO_K$. Ensure that the model is $D$-transversal over primes in~$S$ via blow-ups if necessary (see Section~\\ref{sec:redtypes}).\n\t\\item \\label{item:algo-intersection-numbers}\n\tFor each prime~$\\fq$ of bad reduction for~$\\cX$, determine the set of components and the intersection matrix of the mod-$\\fq$ fibre~$\\cX_{\\fq}$. \n\t\\item \\label{item:algo-mw-basis}\n\tFind divisors $G_i\\in \\Div^0(Y)$ generating a full rank subgroup of the Mordell--Weil group $J(K)$.\n\t\\item \\label{item:algo-correction-divisors}\n\tUsing \\cite[Lemma~2.7]{affchab1}, determine the vertical correction divisors $\\Phi_{\\fq}(G_i)$ for each prime $\\fq$ of $K$. This involves the Moore--Penrose pseudoinverse of the intersection matrix of the mod-$\\fq$ fibre computed in step~\\ref{item:algo-intersection-numbers}.\n\t\\item \\label{item:algo-log-differentials}\n\tFind a basis $\\omega_1,\\ldots,\\omega_{g+n-1}$ of the space of log differentials $\\rH^0(X,\\Omega^1(D))$ and compute their residues $\\Res_Q(\\omega_j) \\in k(Q)$ at all cusps $Q \\in |D|$.\n\t\\item \\label{item:algo-units}\n\tFor each $Q\\in |D|$, find $\\varepsilon_i \\in \\cO_{k(Q)}^\\times$ generating a full rank subgroup of $\\cO_{k(Q)}^\\times$.\n\t\\item \\label{item:algo-prime-ideal-generators}\n\tFor finitely many ``special'' primes $\\fq$ of $K$, determine generators $\\rho_{\\fq}$ of $\\fq$ in $K^\\times\\otimes {\\bQ}$.\n\tSimilarly, for $Q\\in |D|$ and finitely many ``special'' primes $\\lambda$ of $k(Q)$, determine generators $\\pi_{\\lambda}$ of $\\lambda$ in $k(Q)^\\times\\otimes {\\bQ}$.\n\t\\item \\label{item:algo-integrals}\n\tFind a method for calculating $\\fp$-adic integrals of log differentials on $X$.\n\t\\item \\label{item:algo-chabauty-function}\n\tFor each $S$-integral reduction type~$\\Sigma$, determine the matrix $M(\\Sigma^{\\csp})$ and find a non-trivial element $(a_1,\\ldots,a_{g+n-1})$ in the kernel. Compute the constant $c=c(P_0,\\Sigma,\\omega)$ for the  Chabauty differential $\\omega = \\sum_j a_j \\omega_j$.\n\t\\item \\label{item:algo-chabauty-locus}\n\tOn each residue disc, expand the $\\fp$-adic analytic function $P \\mapsto \\int_{P_0}^P \\omega$ into a power series and determine the locus where $\\int_{P_0}^P \\omega=c$.\n\\end{enumerate}", "eq:constants": "\\begin{equation}\n\t\\label{eq:constants}\n\tc = c(b, \\omega) \\coloneqq \\sum_{\\substack{Q \\in |D|,\\\\ \\varphi\\colon k(Q) \\to \\Kpbar}} \\varphi(\\Res_Q \\omega) \\sum_{\\textnormal{$\\lambda$ prime of $\\cO_{k(Q)}$}} b_{\\lambda} \\log \\varphi(\\pi_{\\lambda}).\n\t\\end{equation}", "item:algo-mw-basis": "\\begin{enumerate}[label=(\\arabic*)]\n\t\\item \\label{item:algo-D-transversal-model}\n\tExtend $\\cX$ to a regular model over~$\\cO_K$. Ensure that the model is $D$-transversal over primes in~$S$ via blow-ups if necessary (see Section~\\ref{sec:redtypes}).\n\t\\item \\label{item:algo-intersection-numbers}\n\tFor each prime~$\\fq$ of bad reduction for~$\\cX$, determine the set of components and the intersection matrix of the mod-$\\fq$ fibre~$\\cX_{\\fq}$. \n\t\\item \\label{item:algo-mw-basis}\n\tFind divisors $G_i\\in \\Div^0(Y)$ generating a full rank subgroup of the Mordell--Weil group $J(K)$.\n\t\\item \\label{item:algo-correction-divisors}\n\tUsing \\cite[Lemma~2.7]{affchab1}, determine the vertical correction divisors $\\Phi_{\\fq}(G_i)$ for each prime $\\fq$ of $K$. This involves the Moore--Penrose pseudoinverse of the intersection matrix of the mod-$\\fq$ fibre computed in step~\\ref{item:algo-intersection-numbers}.\n\t\\item \\label{item:algo-log-differentials}\n\tFind a basis $\\omega_1,\\ldots,\\omega_{g+n-1}$ of the space of log differentials $\\rH^0(X,\\Omega^1(D))$ and compute their residues $\\Res_Q(\\omega_j) \\in k(Q)$ at all cusps $Q \\in |D|$.\n\t\\item \\label{item:algo-units}\n\tFor each $Q\\in |D|$, find $\\varepsilon_i \\in \\cO_{k(Q)}^\\times$ generating a full rank subgroup of $\\cO_{k(Q)}^\\times$.\n\t\\item \\label{item:algo-prime-ideal-generators}\n\tFor finitely many ``special'' primes $\\fq$ of $K$, determine generators $\\rho_{\\fq}$ of $\\fq$ in $K^\\times\\otimes {\\bQ}$.\n\tSimilarly, for $Q\\in |D|$ and finitely many ``special'' primes $\\lambda$ of $k(Q)$, determine generators $\\pi_{\\lambda}$ of $\\lambda$ in $k(Q)^\\times\\otimes {\\bQ}$.\n\t\\item \\label{item:algo-integrals}\n\tFind a method for calculating $\\fp$-adic integrals of log differentials on $X$.\n\t\\item \\label{item:algo-chabauty-function}\n\tFor each $S$-integral reduction type~$\\Sigma$, determine the matrix $M(\\Sigma^{\\csp})$ and find a non-trivial element $(a_1,\\ldots,a_{g+n-1})$ in the kernel. Compute the constant $c=c(P_0,\\Sigma,\\omega)$ for the  Chabauty differential $\\omega = \\sum_j a_j \\omega_j$.\n\t\\item \\label{item:algo-chabauty-locus}\n\tOn each residue disc, expand the $\\fp$-adic analytic function $P \\mapsto \\int_{P_0}^P \\omega$ into a power series and determine the locus where $\\int_{P_0}^P \\omega=c$.\n\\end{enumerate}", "item:algo-D-transversal-model": "\\begin{enumerate}[label=(\\arabic*)]\n\t\\item \\label{item:algo-D-transversal-model}\n\tExtend $\\cX$ to a regular model over~$\\cO_K$. Ensure that the model is $D$-transversal over primes in~$S$ via blow-ups if necessary (see Section~\\ref{sec:redtypes}).\n\t\\item \\label{item:algo-intersection-numbers}\n\tFor each prime~$\\fq$ of bad reduction for~$\\cX$, determine the set of components and the intersection matrix of the mod-$\\fq$ fibre~$\\cX_{\\fq}$. \n\t\\item \\label{item:algo-mw-basis}\n\tFind divisors $G_i\\in \\Div^0(Y)$ generating a full rank subgroup of the Mordell--Weil group $J(K)$.\n\t\\item \\label{item:algo-correction-divisors}\n\tUsing \\cite[Lemma~2.7]{affchab1}, determine the vertical correction divisors $\\Phi_{\\fq}(G_i)$ for each prime $\\fq$ of $K$. This involves the Moore--Penrose pseudoinverse of the intersection matrix of the mod-$\\fq$ fibre computed in step~\\ref{item:algo-intersection-numbers}.\n\t\\item \\label{item:algo-log-differentials}\n\tFind a basis $\\omega_1,\\ldots,\\omega_{g+n-1}$ of the space of log differentials $\\rH^0(X,\\Omega^1(D))$ and compute their residues $\\Res_Q(\\omega_j) \\in k(Q)$ at all cusps $Q \\in |D|$.\n\t\\item \\label{item:algo-units}\n\tFor each $Q\\in |D|$, find $\\varepsilon_i \\in \\cO_{k(Q)}^\\times$ generating a full rank subgroup of $\\cO_{k(Q)}^\\times$.\n\t\\item \\label{item:algo-prime-ideal-generators}\n\tFor finitely many ``special'' primes $\\fq$ of $K$, determine generators $\\rho_{\\fq}$ of $\\fq$ in $K^\\times\\otimes {\\bQ}$.\n\tSimilarly, for $Q\\in |D|$ and finitely many ``special'' primes $\\lambda$ of $k(Q)$, determine generators $\\pi_{\\lambda}$ of $\\lambda$ in $k(Q)^\\times\\otimes {\\bQ}$.\n\t\\item \\label{item:algo-integrals}\n\tFind a method for calculating $\\fp$-adic integrals of log differentials on $X$.\n\t\\item \\label{item:algo-chabauty-function}\n\tFor each $S$-integral reduction type~$\\Sigma$, determine the matrix $M(\\Sigma^{\\csp})$ and find a non-trivial element $(a_1,\\ldots,a_{g+n-1})$ in the kernel. Compute the constant $c=c(P_0,\\Sigma,\\omega)$ for the  Chabauty differential $\\omega = \\sum_j a_j \\omega_j$.\n\t\\item \\label{item:algo-chabauty-locus}\n\tOn each residue disc, expand the $\\fp$-adic analytic function $P \\mapsto \\int_{P_0}^P \\omega$ into a power series and determine the locus where $\\int_{P_0}^P \\omega=c$.\n\\end{enumerate}", "item:algo-intersection-numbers": "\\begin{enumerate}[label=(\\arabic*)]\n\t\\item \\label{item:algo-D-transversal-model}\n\tExtend $\\cX$ to a regular model over~$\\cO_K$. Ensure that the model is $D$-transversal over primes in~$S$ via blow-ups if necessary (see Section~\\ref{sec:redtypes}).\n\t\\item \\label{item:algo-intersection-numbers}\n\tFor each prime~$\\fq$ of bad reduction for~$\\cX$, determine the set of components and the intersection matrix of the mod-$\\fq$ fibre~$\\cX_{\\fq}$. \n\t\\item \\label{item:algo-mw-basis}\n\tFind divisors $G_i\\in \\Div^0(Y)$ generating a full rank subgroup of the Mordell--Weil group $J(K)$.\n\t\\item \\label{item:algo-correction-divisors}\n\tUsing \\cite[Lemma~2.7]{affchab1}, determine the vertical correction divisors $\\Phi_{\\fq}(G_i)$ for each prime $\\fq$ of $K$. This involves the Moore--Penrose pseudoinverse of the intersection matrix of the mod-$\\fq$ fibre computed in step~\\ref{item:algo-intersection-numbers}.\n\t\\item \\label{item:algo-log-differentials}\n\tFind a basis $\\omega_1,\\ldots,\\omega_{g+n-1}$ of the space of log differentials $\\rH^0(X,\\Omega^1(D))$ and compute their residues $\\Res_Q(\\omega_j) \\in k(Q)$ at all cusps $Q \\in |D|$.\n\t\\item \\label{item:algo-units}\n\tFor each $Q\\in |D|$, find $\\varepsilon_i \\in \\cO_{k(Q)}^\\times$ generating a full rank subgroup of $\\cO_{k(Q)}^\\times$.\n\t\\item \\label{item:algo-prime-ideal-generators}\n\tFor finitely many ``special'' primes $\\fq$ of $K$, determine generators $\\rho_{\\fq}$ of $\\fq$ in $K^\\times\\otimes {\\bQ}$.\n\tSimilarly, for $Q\\in |D|$ and finitely many ``special'' primes $\\lambda$ of $k(Q)$, determine generators $\\pi_{\\lambda}$ of $\\lambda$ in $k(Q)^\\times\\otimes {\\bQ}$.\n\t\\item \\label{item:algo-integrals}\n\tFind a method for calculating $\\fp$-adic integrals of log differentials on $X$.\n\t\\item \\label{item:algo-chabauty-function}\n\tFor each $S$-integral reduction type~$\\Sigma$, determine the matrix $M(\\Sigma^{\\csp})$ and find a non-trivial element $(a_1,\\ldots,a_{g+n-1})$ in the kernel. Compute the constant $c=c(P_0,\\Sigma,\\omega)$ for the  Chabauty differential $\\omega = \\sum_j a_j \\omega_j$.\n\t\\item \\label{item:algo-chabauty-locus}\n\tOn each residue disc, expand the $\\fp$-adic analytic function $P \\mapsto \\int_{P_0}^P \\omega$ into a power series and determine the locus where $\\int_{P_0}^P \\omega=c$.\n\\end{enumerate}", "eq:b-lambda": "\\begin{equation}\n\\label{eq:b-lambda}\nb_{\\lambda} = -i_{\\lambda}(\\cP_0,\\tilde\\cQ) + i_{\\lambda}(\\Phi_{\\fq}(\\cpt(\\Sigma_{\\fq}) - \\cpt_{\\fq}(P_0)), \\tilde\\cQ),\n\\end{equation}", "thm:A": "\\begin{thmABC}\\label{thm:A}\n\tLet $\\Sigma$ be an $S$-integral reduction type with cuspidal part $\\Sigma^{\\csp}$ and let $\\fp \\not\\in S$ be a prime of $K$.\n\tIf $(a_1,\\ldots,a_{g+n-1}) \\in K_{\\fp}^{g+n-1}$ is contained in the kernel of the matrix $M(\\Sigma^{\\csp})$ defined in \\eqref{eq:MSigma}--\\eqref{eq:matrixD}, let $\\omega = \\omega(\\Sigma^{\\csp}) \\coloneqq a_1 \\omega_1 + \\dots + a_{g+n-1} \\omega_{g+n-1}$ and let $c = c(P_0,\\Sigma,\\omega)$ be the constant defined by\n\t\\eqref{eq:constants} and \\eqref{eq:b-lambda}.\n\tThen\n\t\\[\n\t\t\\int_{P_0}^P \\omega = c \\quad \\text{ for all } P \\in \\cY(\\cO_{K,S})_\\Sigma.\n\t\\]\n\\end{thmABC}"}, "pre_theorem_intro_text_len": 3794, "pre_theorem_intro_text": "\\label{sec: introduction}\n\nLet $K$ be a number field and $S$ a finite set of primes of $\\mathcal{O}_K$.\nLet $Y/K$ be a smooth affine curve and let $\\mathcal{Y}/\\cO_{K,S}$ be a\nregular model of $Y$ over the ring of $S$-integers of $K$.\nIf $Y$ is hyperbolic, the set of $S$-integral points $\\mathcal{Y}(\\cO_{K,S})$ is finite by the theorems of Siegel, Mahler, and Faltings.\nHowever, finding the points in $\\mathcal{Y}(\\cO_{K,S})$ remains a difficult open problem in general.\nIn \\cite{affchab1} we presented a new approach addressing this question for curves satisfying the hypothesis\n\\begin{equation}\n\\label{eq:chabauty-condition-over-number-field-intro}\nr + \\# S + ([K : \\mathbb Q]-1)n < g + \\rank \\cO_K^\\times + \\#|D| + n_2(D) - 1,\n\\end{equation}\nan inequality involving the genus $g$, the Mordell--Weil rank $r$, the cardinality of~$S$, the degree and unit rank of $K$, and invariants $n=\\# D(\\overline{K})$ and $n_2(D) = \\frac12\\# (D(\\mathbb C) \\smallsetminus  D(\\mathbb{R}))$ coming from the boundary $D$ of $Y$.\nOur method is an $S$-integral analogue of the method of Chabauty--Coleman \\cite{Cha41, Col85}.\nFor an auxiliary prime $\\mathfrak p \\not\\in S$, the set $\\mathcal{Y}(\\cO_{K,S})$ is cut out inside $\\mathcal{Y}(\\cO_{\\mathfrak p})$ by finitely many $\\mathfrak p$-adic integrals of \\emph{logarithmic differentials}, i.e.\\ meromorphic differentials on a compactification of~$Y$ with at worst simple poles at $D$.\nIn this article we present an algorithm to determine these log differentials and the zeros of their integrals in practice, thus computing a finite subset of $\\mathcal{Y}(\\cO_{\\mathfrak p})$ containing $\\mathcal{Y}(\\cO_{K,S})$.\n\n\\subsection{Main results}\n\nLet $Y/K$ be given as $X\\smallsetminus D$ where $X/K$ is a smooth projective curve and $D \\neq \\emptyset$ is a finite set of closed points called \\emph{cusps}.\nLet $S$ be a finite set of primes.\nLet~$\\mathcal{X}$ be a regular model of $X$ over the ring~$\\cO_{K,S}$ of $S$-integers, i.e., a regular, flat, projective $\\cO_{K,S}$-scheme with an isomorphism $\\cX_{K} \\cong X$ \\cite[Definition~10.1.1]{liu2006algebraic}.\nLet $\\cD$ be the closure of~$D$ in~$\\mathcal{X}$ and set $\\mathcal{Y} \\coloneqq \\mathcal{X} \\smallsetminus \\cD$. Assume that $Y(K) \\neq\\emptyset$ and fix a base point $P_0 \\in Y(K)$. \nWe use the following notation, which will be kept throughout this paper:\n\\begin{itemize}\n\t\\item $r \\coloneqq \\rank J(K)$ the Mordell--Weil rank of the Jacobian $J$ of~$X$;\n\t\\item $g$ the genus of~$X$;\n\t\\item $\\#|D| > 0$ the number of cusps;\n\t\\item $n \\coloneqq \\#D(\\overline{K})>0$ the number of geometric cusps;\t\\item $[K:\\mathbb Q]n = n_1(D) + 2n_2(D)$ with $n_1(D) \\coloneqq \\#D(\\mathbb{R})$ the number of real cusps and $n_2(D) = \\frac12\\# (D(\\mathbb C) \\smallsetminus  D(\\mathbb{R}))$ the number of conjugate pairs of complex cusps.\n\tHere $D$ is viewed as a scheme over $\\mathbb Q$ (rather than over $K$).\n\\end{itemize}\n\nWe partition the $S$-integral points\n\\[\n\t\\mathcal{Y}(\\cO_{K,S}) = \\coprod_{\\Sigma} \\mathcal{Y}(\\cO_{K,S})_{\\Sigma},\n\\]\nwhere $\\Sigma$ runs through the finitely many $S$-integral \\emph{reduction types} of $\\mathcal{Y}$, see \\Cref{sec:redtypes}.\nIn \\cite[Proposition 3.13]{affchab1} we showed that the Abel--Jacobi image of $\\mathcal{Y}(\\cO_{K,S})_{\\Sigma}$ inside the generalised Jacobian $J_Y(K)$ is contained in the Selmer set $\\Sel(P_0,\\Sigma)$, which is (empty or) a translate of a finitely generated abelian group of controllable rank and is defined using arithmetic intersection theory and the $D$-intersection map $\\sigma\\colon J_Y(K) \\to V_D$, see \\Cref{sec:affchab1}.\nLet $\\omega_1,\\ldots,\\omega_{g+n-1}$ be a basis of the space $\\mathrm H^0(X,\\Omega^1(D))$ of logarithmic differentials on $(X,D)$ with $\\omega_1,\\ldots,\\omega_g$ holomorphic. Assume $n \\geq 2$. \nOur main result is:", "context": "\\label{sec: introduction}\n\nLet $K$ be a number field and $S$ a finite set of primes of $\\mathcal{O}_K$.\nLet $Y/K$ be a smooth affine curve and let $\\mathcal{Y}/\\cO_{K,S}$ be a\nregular model of $Y$ over the ring of $S$-integers of $K$.\nIf $Y$ is hyperbolic, the set of $S$-integral points $\\mathcal{Y}(\\cO_{K,S})$ is finite by the theorems of Siegel, Mahler, and Faltings.\nHowever, finding the points in $\\mathcal{Y}(\\cO_{K,S})$ remains a difficult open problem in general.\nIn \\cite{affchab1} we presented a new approach addressing this question for curves satisfying the hypothesis\n\\begin{equation}\n\\label{eq:chabauty-condition-over-number-field-intro}\nr + \\# S + ([K : \\mathbb Q]-1)n < g + \\rank \\cO_K^\\times + \\#|D| + n_2(D) - 1,\n\\end{equation}\nan inequality involving the genus $g$, the Mordell--Weil rank $r$, the cardinality of~$S$, the degree and unit rank of $K$, and invariants $n=\\# D(\\overline{K})$ and $n_2(D) = \\frac12\\# (D(\\mathbb C) \\smallsetminus D(\\mathbb{R}))$ coming from the boundary $D$ of $Y$.\nOur method is an $S$-integral analogue of the method of Chabauty--Coleman \\cite{Cha41, Col85}.\nFor an auxiliary prime $\\mathfrak p \\not\\in S$, the set $\\mathcal{Y}(\\cO_{K,S})$ is cut out inside $\\mathcal{Y}(\\cO_{\\mathfrak p})$ by finitely many $\\mathfrak p$-adic integrals of \\emph{logarithmic differentials}, i.e.\\ meromorphic differentials on a compactification of~$Y$ with at worst simple poles at $D$.\nIn this article we present an algorithm to determine these log differentials and the zeros of their integrals in practice, thus computing a finite subset of $\\mathcal{Y}(\\cO_{\\mathfrak p})$ containing $\\mathcal{Y}(\\cO_{K,S})$.\n\nLet $Y/K$ be given as $X\\smallsetminus D$ where $X/K$ is a smooth projective curve and $D \\neq \\emptyset$ is a finite set of closed points called \\emph{cusps}.\nLet $S$ be a finite set of primes.\nLet~$\\mathcal{X}$ be a regular model of $X$ over the ring~$\\cO_{K,S}$ of $S$-integers, i.e., a regular, flat, projective $\\cO_{K,S}$-scheme with an isomorphism $\\cX_{K} \\cong X$ \\cite[Definition~10.1.1]{liu2006algebraic}.\nLet $\\cD$ be the closure of~$D$ in~$\\mathcal{X}$ and set $\\mathcal{Y} \\coloneqq \\mathcal{X} \\smallsetminus \\cD$. Assume that $Y(K) \\neq\\emptyset$ and fix a base point $P_0 \\in Y(K)$. \nWe use the following notation, which will be kept throughout this paper:\n\\begin{itemize}\n \\item $r \\coloneqq \\rank J(K)$ the Mordell--Weil rank of the Jacobian $J$ of~$X$;\n \\item $g$ the genus of~$X$;\n \\item $\\#|D| > 0$ the number of cusps;\n \\item $n \\coloneqq \\#D(\\overline{K})>0$ the number of geometric cusps; \\item $[K:\\mathbb Q]n = n_1(D) + 2n_2(D)$ with $n_1(D) \\coloneqq \\#D(\\mathbb{R})$ the number of real cusps and $n_2(D) = \\frac12\\# (D(\\mathbb C) \\smallsetminus D(\\mathbb{R}))$ the number of conjugate pairs of complex cusps.\n Here $D$ is viewed as a scheme over $\\mathbb Q$ (rather than over $K$).\n\\end{itemize}\n\nWe partition the $S$-integral points\n\\[\n \\mathcal{Y}(\\cO_{K,S}) = \\coprod_{\\Sigma} \\mathcal{Y}(\\cO_{K,S})_{\\Sigma},\n\\]\nwhere $\\Sigma$ runs through the finitely many $S$-integral \\emph{reduction types} of $\\mathcal{Y}$, see \\Cref{sec:redtypes}.\nIn \\cite[Proposition 3.13]{affchab1} we showed that the Abel--Jacobi image of $\\mathcal{Y}(\\cO_{K,S})_{\\Sigma}$ inside the generalised Jacobian $J_Y(K)$ is contained in the Selmer set $\\Sel(P_0,\\Sigma)$, which is (empty or) a translate of a finitely generated abelian group of controllable rank and is defined using arithmetic intersection theory and the $D$-intersection map $\\sigma\\colon J_Y(K) \\to V_D$, see \\Cref{sec:affchab1}.\nLet $\\omega_1,\\ldots,\\omega_{g+n-1}$ be a basis of the space $\\mathrm H^0(X,\\Omega^1(D))$ of logarithmic differentials on $(X,D)$ with $\\omega_1,\\ldots,\\omega_g$ holomorphic. Assume $n \\geq 2$. \nOur main result is:\n\n\\begin{equation}\n\t\\label{eq:MSigma}\n\tM(U) \\coloneqq \\begin{pmatrix}\n\t\tA & B  \\\\\n\t\t0 & C  \\\\\n\t\t0 & D(U) \n\t\\end{pmatrix}\n\\end{equation}\n\n\\begin{equation}\n\\label{eq:b-lambda}\nb_{\\lambda} = -i_{\\lambda}(\\cP_0,\\tilde\\cQ) + i_{\\lambda}(\\Phi_{\\fq}(\\cpt(\\Sigma_{\\fq}) - \\cpt_{\\fq}(P_0)), \\tilde\\cQ),\n\\end{equation}\n\n\\begin{equation}\n\t\\label{eq:constants}\n\tc = c(b, \\omega) \\coloneqq \\sum_{\\substack{Q \\in |D|,\\\\ \\varphi\\colon k(Q) \\to \\Kpbar}} \\varphi(\\Res_Q \\omega) \\sum_{\\textnormal{$\\lambda$ prime of $\\cO_{k(Q)}$}} b_{\\lambda} \\log \\varphi(\\pi_{\\lambda}).\n\t\\end{equation}\n\n\\begin{equation}\n\t\t\\label{eq:matrixD}\n\t\tD(U)_{i,j} \\coloneqq \\sum_{\\substack{Q \\in |D|,\\\\ \\varphi\\colon k(Q) \\to \\Kpbar}} \\varphi(\\Res_{Q} \\omega_{g+j}) \\sum_{\\substack{\\text{$\\fq$ prime,}\\\\ \\text{$\\lambda|\\fq$ in $\\cO_{k(Q)}$}}} u_{i,\\lambda} \\log \\varphi(\\pi_{\\lambda}).\n\t\\end{equation}", "full_context": "\\label{sec: introduction}\n\nLet $K$ be a number field and $S$ a finite set of primes of $\\mathcal{O}_K$.\nLet $Y/K$ be a smooth affine curve and let $\\mathcal{Y}/\\cO_{K,S}$ be a\nregular model of $Y$ over the ring of $S$-integers of $K$.\nIf $Y$ is hyperbolic, the set of $S$-integral points $\\mathcal{Y}(\\cO_{K,S})$ is finite by the theorems of Siegel, Mahler, and Faltings.\nHowever, finding the points in $\\mathcal{Y}(\\cO_{K,S})$ remains a difficult open problem in general.\nIn \\cite{affchab1} we presented a new approach addressing this question for curves satisfying the hypothesis\n\\begin{equation}\n\\label{eq:chabauty-condition-over-number-field-intro}\nr + \\# S + ([K : \\mathbb Q]-1)n < g + \\rank \\cO_K^\\times + \\#|D| + n_2(D) - 1,\n\\end{equation}\nan inequality involving the genus $g$, the Mordell--Weil rank $r$, the cardinality of~$S$, the degree and unit rank of $K$, and invariants $n=\\# D(\\overline{K})$ and $n_2(D) = \\frac12\\# (D(\\mathbb C) \\smallsetminus D(\\mathbb{R}))$ coming from the boundary $D$ of $Y$.\nOur method is an $S$-integral analogue of the method of Chabauty--Coleman \\cite{Cha41, Col85}.\nFor an auxiliary prime $\\mathfrak p \\not\\in S$, the set $\\mathcal{Y}(\\cO_{K,S})$ is cut out inside $\\mathcal{Y}(\\cO_{\\mathfrak p})$ by finitely many $\\mathfrak p$-adic integrals of \\emph{logarithmic differentials}, i.e.\\ meromorphic differentials on a compactification of~$Y$ with at worst simple poles at $D$.\nIn this article we present an algorithm to determine these log differentials and the zeros of their integrals in practice, thus computing a finite subset of $\\mathcal{Y}(\\cO_{\\mathfrak p})$ containing $\\mathcal{Y}(\\cO_{K,S})$.\n\nLet $Y/K$ be given as $X\\smallsetminus D$ where $X/K$ is a smooth projective curve and $D \\neq \\emptyset$ is a finite set of closed points called \\emph{cusps}.\nLet $S$ be a finite set of primes.\nLet~$\\mathcal{X}$ be a regular model of $X$ over the ring~$\\cO_{K,S}$ of $S$-integers, i.e., a regular, flat, projective $\\cO_{K,S}$-scheme with an isomorphism $\\cX_{K} \\cong X$ \\cite[Definition~10.1.1]{liu2006algebraic}.\nLet $\\cD$ be the closure of~$D$ in~$\\mathcal{X}$ and set $\\mathcal{Y} \\coloneqq \\mathcal{X} \\smallsetminus \\cD$. Assume that $Y(K) \\neq\\emptyset$ and fix a base point $P_0 \\in Y(K)$. \nWe use the following notation, which will be kept throughout this paper:\n\\begin{itemize}\n \\item $r \\coloneqq \\rank J(K)$ the Mordell--Weil rank of the Jacobian $J$ of~$X$;\n \\item $g$ the genus of~$X$;\n \\item $\\#|D| > 0$ the number of cusps;\n \\item $n \\coloneqq \\#D(\\overline{K})>0$ the number of geometric cusps; \\item $[K:\\mathbb Q]n = n_1(D) + 2n_2(D)$ with $n_1(D) \\coloneqq \\#D(\\mathbb{R})$ the number of real cusps and $n_2(D) = \\frac12\\# (D(\\mathbb C) \\smallsetminus D(\\mathbb{R}))$ the number of conjugate pairs of complex cusps.\n Here $D$ is viewed as a scheme over $\\mathbb Q$ (rather than over $K$).\n\\end{itemize}\n\nWe partition the $S$-integral points\n\\[\n \\mathcal{Y}(\\cO_{K,S}) = \\coprod_{\\Sigma} \\mathcal{Y}(\\cO_{K,S})_{\\Sigma},\n\\]\nwhere $\\Sigma$ runs through the finitely many $S$-integral \\emph{reduction types} of $\\mathcal{Y}$, see \\Cref{sec:redtypes}.\nIn \\cite[Proposition 3.13]{affchab1} we showed that the Abel--Jacobi image of $\\mathcal{Y}(\\cO_{K,S})_{\\Sigma}$ inside the generalised Jacobian $J_Y(K)$ is contained in the Selmer set $\\Sel(P_0,\\Sigma)$, which is (empty or) a translate of a finitely generated abelian group of controllable rank and is defined using arithmetic intersection theory and the $D$-intersection map $\\sigma\\colon J_Y(K) \\to V_D$, see \\Cref{sec:affchab1}.\nLet $\\omega_1,\\ldots,\\omega_{g+n-1}$ be a basis of the space $\\mathrm H^0(X,\\Omega^1(D))$ of logarithmic differentials on $(X,D)$ with $\\omega_1,\\ldots,\\omega_g$ holomorphic. Assume $n \\geq 2$. \nOur main result is:\n\n\\begin{equation}\n\t\\label{eq:MSigma}\n\tM(U) \\coloneqq \\begin{pmatrix}\n\t\tA & B  \\\\\n\t\t0 & C  \\\\\n\t\t0 & D(U) \n\t\\end{pmatrix}\n\\end{equation}\n\n\\begin{equation}\n\\label{eq:b-lambda}\nb_{\\lambda} = -i_{\\lambda}(\\cP_0,\\tilde\\cQ) + i_{\\lambda}(\\Phi_{\\fq}(\\cpt(\\Sigma_{\\fq}) - \\cpt_{\\fq}(P_0)), \\tilde\\cQ),\n\\end{equation}\n\n\\begin{equation}\n\t\\label{eq:constants}\n\tc = c(b, \\omega) \\coloneqq \\sum_{\\substack{Q \\in |D|,\\\\ \\varphi\\colon k(Q) \\to \\Kpbar}} \\varphi(\\Res_Q \\omega) \\sum_{\\textnormal{$\\lambda$ prime of $\\cO_{k(Q)}$}} b_{\\lambda} \\log \\varphi(\\pi_{\\lambda}).\n\t\\end{equation}\n\n\\begin{equation}\n\t\t\\label{eq:matrixD}\n\t\tD(U)_{i,j} \\coloneqq \\sum_{\\substack{Q \\in |D|,\\\\ \\varphi\\colon k(Q) \\to \\Kpbar}} \\varphi(\\Res_{Q} \\omega_{g+j}) \\sum_{\\substack{\\text{$\\fq$ prime,}\\\\ \\text{$\\lambda|\\fq$ in $\\cO_{k(Q)}$}}} u_{i,\\lambda} \\log \\varphi(\\pi_{\\lambda}).\n\t\\end{equation}\n\nThus $\\cY(\\cO_{K,S})_{\\Sigma}$ is contained in the set of zeros of the $\\fp$-adic analytic function $\\cY(\\cO_{\\fp}) \\to K_{\\fp}$ that sends $P$ to $\\int_{P_0}^P \\omega - c$.\nThe constant $c(P_0,\\Sigma,\\omega)$ and the entries of the matrix $M(\\Sigma^{\\csp})$ are given by explicit formulas in terms of $\\fp$-adic integrals of the basis differentials $\\omega_j$, the residues of the $\\omega_j$ at the cusps, the $\\fp$-adic logarithm of certain elements of the residue fields of the cusps, and intersection numbers on the regular model~$\\cX$.\n\n\\begin{thm}\n \\label{thm:explicit-function}\n If $(a_1,\\ldots,a_{g+n-1}) \\in K_{\\fp}^{g+n-1}$ is contained in the kernel of the matrix $M(U)$ defined in equations~\\eqref{eq:MSigma}--\\eqref{eq:matrixD}, let $\\omega = \\omega(U) \\coloneqq a_1 \\omega_1 + \\dots + a_{g+n-1} \\omega_{g+n-1}$.\n Define\n \\begin{equation}\n \\label{eq:constants}\n c = c(b, \\omega) \\coloneqq \\sum_{\\substack{Q \\in |D|,\\\\ \\varphi\\colon k(Q) \\to \\Kpbar}} \\varphi(\\Res_Q \\omega) \\sum_{\\textnormal{$\\lambda$ prime of $\\cO_{k(Q)}$}} b_{\\lambda} \\log \\varphi(\\pi_{\\lambda}).\n \\end{equation}\n Then\n \\[\n \\int_F \\omega = c \\quad \\text{ for all } F \\in b_0 + W.\n \\]\n\\end{thm}\n\nWe choose the base point $P_0 = (-1,1) \\in \\cY(\\bZ)$ and the auxiliary prime~$p = 7$.\nSetting $P_0 = (0,3)$ and $P_1 = (1,3)$, the classes of the divisors $G_i = P_i - P_0$ ($i = 1,2$) generate a subgroup of finite index in the Mordell--Weil group. The intersection numbers $i_{\\ell}(\\Psi_{\\ell}(G_i), \\cQ_+' - \\cQ_-')$ in~\\eqref{eq:hyp-M-entries} are all zero: for the horizontal part $i_{\\ell}(\\cP_i - \\cP_0, \\cQ_+' - \\cQ_-')$ this follows from $\\cP_0,\\cP_1,\\cP_2$ being integral points. The vertical divisors~$V$ in $\\cX_{\\ell}'$ we have $i_{\\ell}(V, \\cQ_+' - \\cQ_-') = 0$ since the points $Q_{+}$ and $Q_-$ reduce onto the same component of~$\\cX_{\\ell}'$. For $\\ell \\neq 3$ this is clear since~$\\cX'_{\\ell}$ has only one component; for $\\ell = 3$ this follows from the fact that the points at infinity reduce to smooth points of the mod-3 fibre of the non-regular model~$\\cX$, and since the desingularisation $\\cX' \\to \\cX$ is an isomorphism over the non-singular locus, they still reduce onto the same component of~$\\cX_3'$. We find that the matrix~$M$ is given by\n\\begin{equation*}\n M = \\begin{pmatrix}\n \\int_{P_0}^{P_1} \\omega_0 & \\int_{P_0}^{P_1} \\omega_1 & \\int_{P_0}^{P_1} \\omega_2 \\\\[1mm]\n \\int_{P_0}^{P_2} \\omega_0 & \\int_{P_0}^{P_2} \\omega_1 & \\int_{P_0}^{P_2} \\omega_2\n \\end{pmatrix},\n\\end{equation*}\nwithout the correction terms in the last column. A nontrivial element of the kernel is $(a_0,a_1,a_2) \\in \\bQ_7^3$ with\n\\begin{align*}\n a_0 &= 1 + O(7^8), \\\\\n a_1 &= 5 + 3\\cdot 7 + 3\\cdot 7^2 + 5\\cdot 7^3 + 3\\cdot 7^4 + 2\\cdot 7^6 + 2\\cdot 7^7 + O(7^8), \\\\\n a_2 &= 5 + 6\\cdot 7 + 6\\cdot 7^2 + 7^3 + 4\\cdot 7^4 + 6\\cdot 7^5 + 5\\cdot 7^6 + 3\\cdot 7^7 + O(7^8).\n\\end{align*}\nSetting $\\omega \\coloneqq a_0 \\omega_0 + a_1 \\omega_1 + a_2 \\omega_2$, the function $\\rho\\colon \\cY(\\bZ_p) \\to \\bQ_p$, $\\rho(P) = \\int_{P_0}^P \\omega$ takes a constant value $c(P_0,\\Sigma,\\omega)$ on all integral points of a fixed reduction type~$\\Sigma$. By the same argument as above, the intersection numbers in~\\eqref{eq:split-hyperelliptic-constants-final} vanish, so the constants $c(P_0,\\Sigma,\\omega)$ are all zero. The function~$\\rho$ thus vanishes on all of~$\\cY(\\bZ)$.\nWe can analyse the zeros of~$\\rho$ on each residue disc by computing the power series expansion in a uniformising parameter. For example, consider the point $P_0 = (-1,1)$. Its residue disc is parametrised by $Q(t) = (x(t), y(t))$ with\n\\begin{align*}\n x(t) = -1 + 7t,\\quad \n y(t) = \\sqrt{f(x(t))} = \\sum_{n=0}^\\infty \\binom{1/2}{n}(f(-1 + 7t)-1)^n \\end{align*}\nand in terms of the parameter~$t$, the function~$\\rho$ is given by\n\\begin{align*}\n \\rho(Q(t)) = \\sum_{j=0}^2 a_j \\int_{P_0}^{Q(t)} x(t)^j \\frac{\\rd x(t)}{y(t)} = (7 + 3\\cdot 7^2) t + 6\\cdot 7^2 \\cdot t^2 + O(7^3,t^3)\n\\end{align*}\nThe sequence of valuations of the coefficients starts $(\\infty, 1, 2, 3, 4, 6, 7, \\ldots)$, so by Strassmann's Theorem (see for example \\cite[Theorem~5.6.1]{gouvea}), the only zero of $\\rho(Q(t))$ in~$\\bZ_7$ is at $t = 0$. This shows that $P_0 = (-1,1)$ is the only integral point of~$\\cY$ in its residue disc. Each of the remaining residue discs contains one of the known integral points, and in each case one can show with Strassmann's Theorem that there are no other integral points in that disc, proving \\Cref{thm:example-split}. Sage code for computing the vanishing locus of the function~$\\rho$ in this example, for any auxiliary prime~$p$, can be found at \\url{https://github.com/martinluedtke/AffChab2}.\n\nChoose the auxiliary prime $p=7$, a primitive third root of unity $\\zeta_3\\in \\bQ_p$ and put $\\sqrt{-3}\\coloneqq 1+2\\zeta_3\\in\\bQ_p$.\n Then the matrix $M(\\Sigma^{\\csp})$ of \\eqref{eq:MSigma} is equal to\n \\begin{equation*}\n M(\\Sigma^{\\csp}) = \\begin{pmatrix}\n \\int_{P_0}^A \\omega_1 & \\int_{P_0}^A \\omega_2 - \\beta_2 & \\int_{P_0}^A \\omega_3 - \\beta_3\\\\\n 0 & D(\\Sigma^{\\csp})_2 & D(\\Sigma^{\\csp})_3\n \\end{pmatrix},\n \\end{equation*}\n where using \\eqref{eq:H} we have\n \\begin{align*}\n \\beta_2 &= \n(-1) \\left(i_{\\pi_2}(\\cA - \\cP_0, \\cQ_1) \\log(2) + i_{\\pi_3}(\\cA - \\cP_0, \\cQ_1) \\log(3)\\right) \\\\\n &\\qquad + (-\\zeta_3) i_{\\pi_{(\\sqrt{-3})}}(\\cA - \\cP_0, \\cQ_2) \\log(\\sqrt{-3}) + (-\\zeta_3^{-1}) i_{\\pi_{(\\sqrt{-3})}}(\\cA - \\cP_0, \\cQ_2) \\log(-\\sqrt{-3}) \\\\\n &= -\\log(2) - \\frac12 \\log(3)\n \\end{align*}\n because $A$ reduces onto $\\cQ_1$ modulo $2$ and onto $\\cQ_1 \\cap \\cQ_2$ modulo $3$; $P_0$ never reduces to a cusp; $i_{\\pi_\\bullet}(\\cA,\\cQ_i)=1$ in all cases; and $\\log(\\sqrt{-3})=\\log(3)/2$.\n A similar calculation shows $\\beta_3 = \\beta_2$.\n The entries $D(\\Sigma^{\\csp})_i$ are computed using \\eqref{eq:matrixD}:\n \\begin{align*}\n D(\\Sigma^{\\csp})_2 &= -\\zeta_3 \\log(2\\zeta_3 +23) - \\zeta_3^{-1} \\log(2\\zeta_3^{-1} + 23),\\\\\n D(\\Sigma^{\\csp})_3 &= -\\zeta_3^{-1} \\log(2\\zeta_3 +23) - \\zeta_3 \\log(2\\zeta_3^{-1} + 23).\n \\end{align*}\n We also calculate the integrals $\\int_{P_0}^A \\omega_j$ with the method outlined above resulting in the matrix\n \\begin{equation*}\n M(\\Sigma^{\\csp})= \\footnotesize{\n \\begin{pmatrix}\n 2\\cdot 7 + 5\\cdot 7^2 + 4 \\cdot 7^4 + 5 \\cdot 7^5 & 6\\cdot 7 + 7^2 + 3\\cdot 7^4 & 2\\cdot 7 + 6\\cdot 7^2 + 2\\cdot 7^3 \\\\\n 0 & 6\\cdot 7^2 + 2\\cdot 7^3 + 6\\cdot 7^4 & 7 + 4\\cdot 7^2 + 7^3 + 5\\cdot 7^4 + 4\\cdot 7^5\n \\end{pmatrix}\n }\n \\end{equation*}\n displayed with precision $O(7^6)$.\n The element\n \\[\n (a_1,a_2,a_3) = (1, 2 + 6\\cdot 7 + 2\\cdot 7^2 + 3\\cdot 7^3 + 4\\cdot 7^5, 2\\cdot 7 + 6\\cdot 7^2 + 2\\cdot 7^5) + O(7^6)\n \\]\n lies in the kernel of $M(\\Sigma^{\\csp})$, so the differential $\\omega = a_1\\omega_1 + a_2 \\omega_2 + a_3\\omega_3$ vanishes on all $\\{487\\}$-integral points of reduction type $\\Sigma$; note that the constant $c(P_0,\\Sigma,\\omega)$ is zero by \\eqref{eq:constants} and $b=0$ by (5) above.\n The point $(x,y)=(\\tfrac{216}{487},\\tfrac{438}{487})$ is a $\\{487\\}$-integral point on $\\cY$ which has reduction type $\\Sigma$.\n We checked numerically that indeed\n \\[\n \\int_{P_0}^P \\omega = O(7^{14}).\n \\]\n In order to fully determine the Chabauty locus $\\cY(\\bZ_7)_{S,\\Sigma}$, one now needs to determine the zeros of $P\\mapsto \\int_{P_0}^{P} \\omega$ on each residue disc.\n To give the full Chabauty locus $\\cY(\\bZ_7)_{S}$, one then has to similarly deal with the other $\\{487\\}$-integral reduction types.\n One can apply the same procedure for $S$-integral points for any singleton set $S$.\n The code for this example can be found on \\url{https://github.com/martinluedtke/AffChab2}.\n\\end{example}", "post_theorem_intro_text_len": 7422, "post_theorem_intro_text": "Thus $\\mathcal{Y}(\\cO_{K,S})_{\\Sigma}$ is contained in the set of zeros of the $\\mathfrak p$-adic analytic function $\\mathcal{Y}(\\cO_{\\mathfrak p}) \\to K_{\\mathfrak p}$ that sends $P$ to $\\int_{P_0}^P \\omega - c$.\nThe constant $c(P_0,\\Sigma,\\omega)$ and the entries of the matrix $M(\\Sigma^{\\csp})$ are given by explicit formulas in terms of $\\mathfrak p$-adic integrals of the basis differentials $\\omega_j$, the residues of the $\\omega_j$ at the cusps, the $\\mathfrak p$-adic logarithm of certain elements of the residue fields of the cusps, and intersection numbers on the regular model~$\\mathcal{X}$.\n\nTo prove \\Cref{thm:A}, let $\\log_{J_Y}\\colon J_Y(K_{\\mathfrak p}) \\to \\mathrm H^0(X_{K_{\\mathfrak p}},\\Omega^1(D))^\\vee$ be the logarithm of the $\\mathfrak p$-adic Lie group $J_Y(K_{\\mathfrak p})$, which we restrict to the direction $W$ of $\\Sel(P_0,\\Sigma)$.\nWe show that the matrix $M(\\Sigma^{\\csp})$ represents the dual map $\\log_{J_Y}^\\sharp \\colon \\mathrm H^0(X_{K_{\\mathfrak p}}, \\Omega^1(D)) \\to W^\\vee\\otimes_{\\mathbb Q} K_{\\mathfrak p}$ with respect to a certain explicit basis of $W$.\nThis basis is constructed in \\eqref{eq:F-and-Gi}, \\eqref{eq:f-of-Q} and \\eqref{eq:t-ell} and consists of dual bases of the Mordell--Weil group $J(K)$ of the Jacobian of $X$, of the unit groups at the cusps, and of a basis of $\\sigma(W)=U$.\nFor $F\\in W$ and $\\omega$ a logarithmic differential, the fundamental calculation of $\\log_{J_Y}^\\sharp(\\omega)(F) = \\int_F \\omega$ reduces to calculating $\\int_{\\mathrm{div}(f)} \\omega$ for a certain rational function~$f$.\nA $p$-adic residue theorem, proved in \\Cref{sec:integration}, expresses $\\int_{\\mathrm{div}(f)} \\omega$ in terms of the residues of $\\omega$ and the values of $f$ at the cusps.\nThe latter can be read off from $\\sigma(F)$, allowing us to conclude.\nThe main insight is how to combine the $D$-intersection map $\\sigma$ with $\\mathfrak p$-adic integration.\n\n\\subsection{Affine Chabauty algorithm}\n\\label{sec:algorithm}\n\n\\Cref{thm:A} is the central step in building an algorithm based on the Affine Chabauty method.\nThat is, on input an affine curve $Y=X\\smallsetminus D$, a finite set of primes $S$ satisfying \\eqref{eq:chabauty-condition-over-number-field-intro}, a regular model $\\mathcal{Y} = \\mathcal{X} \\smallsetminus \\cD$ over $\\cO_{K,S}$, and an auxiliary prime $\\mathfrak p \\not\\in S$, we want to determine a finite subset of $\\mathcal{Y}(\\cO_{\\mathfrak p})$ containing $\\mathcal{Y}(\\cO_{K,S})$.\nLet us give a sketch of how such an algorithm proceeds:\n\\begin{enumerate}[label=(\\arabic*)]\n\t\\item \\label{item:algo-D-transversal-model}\n\tExtend $\\mathcal{X}$ to a regular model over~$\\cO_K$. Ensure that the model is $D$-transversal over primes in~$S$ via blow-ups if necessary (see Section~\\ref{sec:redtypes}).\n\t\\item \\label{item:algo-intersection-numbers}\n\tFor each prime~$\\mathfrak q$ of bad reduction for~$\\mathcal{X}$, determine the set of components and the intersection matrix of the mod-$\\mathfrak q$ fibre~$\\cX_{\\mathfrak q}$. \n\t\\item \\label{item:algo-mw-basis}\n\tFind divisors $G_i\\in \\Div^0(Y)$ generating a full rank subgroup of the Mordell--Weil group $J(K)$.\n\t\\item \\label{item:algo-correction-divisors}\n\tUsing \\cite[Lemma~2.7]{affchab1}, determine the vertical correction divisors $\\Phi_{\\mathfrak q}(G_i)$ for each prime $\\mathfrak q$ of $K$. This involves the Moore--Penrose pseudoinverse of the intersection matrix of the mod-$\\mathfrak q$ fibre computed in step~\\ref{item:algo-intersection-numbers}.\n\t\\item \\label{item:algo-log-differentials}\n\tFind a basis $\\omega_1,\\ldots,\\omega_{g+n-1}$ of the space of log differentials $\\mathrm H^0(X,\\Omega^1(D))$ and compute their residues $\\Res_Q(\\omega_j) \\in k(Q)$ at all cusps $Q \\in |D|$.\n\t\\item \\label{item:algo-units}\n\tFor each $Q\\in |D|$, find $\\varepsilon_i \\in \\cO_{k(Q)}^\\times$ generating a full rank subgroup of $\\cO_{k(Q)}^\\times$.\n\t\\item \\label{item:algo-prime-ideal-generators}\n\tFor finitely many ``special'' primes $\\mathfrak q$ of $K$, determine generators $\\rho_{\\mathfrak q}$ of $\\mathfrak q$ in $K^\\times\\otimes {\\mathbb Q}$.\n\tSimilarly, for $Q\\in |D|$ and finitely many ``special'' primes $\\lambda$ of $k(Q)$, determine generators $\\pi_{\\lambda}$ of $\\lambda$ in $k(Q)^\\times\\otimes {\\mathbb Q}$.\n\t\\item \\label{item:algo-integrals}\n\tFind a method for calculating $\\mathfrak p$-adic integrals of log differentials on $X$.\n\t\\item \\label{item:algo-chabauty-function}\n\tFor each $S$-integral reduction type~$\\Sigma$, determine the matrix $M(\\Sigma^{\\csp})$ and find a non-trivial element $(a_1,\\ldots,a_{g+n-1})$ in the kernel. Compute the constant $c=c(P_0,\\Sigma,\\omega)$ for the  Chabauty differential $\\omega = \\sum_j a_j \\omega_j$.\n\t\\item \\label{item:algo-chabauty-locus}\n\tOn each residue disc, expand the $\\mathfrak p$-adic analytic function $P \\mapsto \\int_{P_0}^P \\omega$ into a power series and determine the locus where $\\int_{P_0}^P \\omega=c$.\n\\end{enumerate}\n\n\\Cref{thm:A} guarantees that the union of the sets obtained in step~\\ref{item:algo-chabauty-locus} over all $S$-integral reduction types~$\\Sigma$ is a finite subset of $\\mathcal{Y}(\\cO_{\\mathfrak p})$ containing $\\mathcal{Y}(\\cO_{K,S})$. The central step and main focus of this article is~\\ref{item:algo-chabauty-function}. The results of all prior steps enter in the computation of the matrices $M(\\Sigma^{\\csp})$ and the constants $c(P_0,\\Sigma,\\omega)$, as described in detail in Section~\\ref{sec:explicit}.\n\nIn individual examples, calculating intersection numbers \\ref{item:algo-intersection-numbers} and a basis of log differentials \\ref{item:algo-log-differentials} can be done by hand. For hyperelliptic curves and the superelliptic curves considered in Section~\\ref{sec:examples}, we are able to compute $\\mathfrak p$-adic integrals of logarithmic differentials \\ref{item:algo-integrals} by reducing it to integrals of the particular differential $x^g \\, \\mathrm d x/y$ on various hyperelliptic curves of genus~$g$, for which we can use existing code \\cite{BBK10, BB12}.\nThere are methods in Magma and Sagemath for computing regular models, Mordell--Weil bases, unit groups and generators of primes ideals, as required in steps \\ref{item:algo-D-transversal-model}, \\ref{item:algo-intersection-numbers}, \\ref{item:algo-mw-basis}, \\ref{item:algo-units}, \\ref{item:algo-prime-ideal-generators}.\n\n\\subsection{Structure of the paper}\n\n\\Cref{sec:affchab1} reviews the Affine Chabauty method developed in \\cite{affchab1}.\nWe define intersection numbers, the $D$-intersection map, reduction types, and the Selmer set.\nIn \\Cref{sec:integration} we prove the residue theorem for $p$-adic integrals of logarithmic differentials.\nIn \\Cref{sec:explicit} we define the matrix $M(\\Sigma^{\\csp})$ and prove the main result.\nWe illustrate our method with several examples in \\Cref{sec:examples}.\n\n\\subsection*{Acknowledgements}\n\nThe first author is supported by a Walter Benjamin Scholarship from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project number LE 5634/1, and also acknowledges support from the DFG through TRR 326 Geometry and Arithmetic of Uniformized Structures, project number 444845124. The second author is supported by a Minerva Fellowship of the Minerva Stiftung Gesellschaft für die Forschung mbH and also acknowledges support through a guest postdoc fellowship at the Max Planck Institute for Mathematics in Bonn and through an NWO Grant, project number VI.Vidi.192.106.", "sketch": "To prove \\Cref{thm:A}, consider the \\(\\mathfrak p\\)-adic Lie group logarithm \\(\\log_{J_Y}\\colon J_Y(K_{\\mathfrak p}) \\to \\mathrm H^0(X_{K_{\\mathfrak p}},\\Omega^1(D))^\\vee\\), restricted to the direction \\(W\\) of \\(\\Sel(P_0,\\Sigma)\\). Show that the matrix \\(M(\\Sigma^{\\csp})\\) represents the dual map \\(\\log_{J_Y}^\\sharp\\colon \\mathrm H^0(X_{K_{\\mathfrak p}}, \\Omega^1(D)) \\to W^\\vee\\otimes_{\\mathbb Q} K_{\\mathfrak p}\\) with respect to an explicit basis of \\(W\\). This basis (from \\eqref{eq:F-and-Gi}, \\eqref{eq:f-of-Q}, \\eqref{eq:t-ell}) consists of dual bases coming from the Mordell--Weil group \\(J(K)\\), unit groups at the cusps, and a basis of \\(\\sigma(W)=U\\). For \\(F\\in W\\) and a logarithmic differential \\(\\omega\\), the key computation \\(\\log_{J_Y}^\\sharp(\\omega)(F)=\\int_F\\omega\\) is reduced to computing \\(\\int_{\\mathrm{div}(f)}\\omega\\) for a suitable rational function \\(f\\). A \\(p\\)-adic residue theorem (\\Cref{sec:integration}) expresses \\(\\int_{\\mathrm{div}(f)}\\omega\\) in terms of residues of \\(\\omega\\) and values of \\(f\\) at the cusps; these values can be read off from \\(\\sigma(F)\\), which allows one to conclude. The stated relation \\(\\int_{P_0}^P\\omega=c\\) for \\(P\\in\\mathcal{Y}(\\cO_{K,S})_\\Sigma\\) then follows from taking \\((a_1,\\ldots,a_{g+n-1})\\) in the kernel of the representing matrix \\(M(\\Sigma^{\\csp})\\), i.e. combining the \\(D\\)-intersection map \\(\\sigma\\) with \\(\\mathfrak p\\)-adic integration.", "expanded_sketch": "To prove the main theorem, consider the \\(\\mathfrak p\\)-adic Lie group logarithm \\(\\log_{J_Y}\\colon J_Y(K_{\\mathfrak p}) \\to \\mathrm H^0(X_{K_{\\mathfrak p}},\\Omega^1(D))^\\vee\\), restricted to the direction \\(W\\) of \\(\\Sel(P_0,\\Sigma)\\). Show that the matrix \\(M(\\Sigma^{\\csp})\\) represents the dual map \\(\\log_{J_Y}^\\sharp\\colon \\mathrm H^0(X_{K_{\\mathfrak p}}, \\Omega^1(D)) \\to W^\\vee\\otimes_{\\mathbb Q} K_{\\mathfrak p}\\) with respect to an explicit basis of \\(W\\). This basis (from\n\\begin{equation}\n\t\t\\label{eq:F-and-Gi}\n\t\tF = \\sum_{i=1}^r x_i(F) G_i + \\div(f)\n\t\\end{equation}\n,\n\\begin{equation}\n\t\t\\label{eq:f-of-Q}\n\t\t(f(Q))_Q = \\prod_{i=1}^{k} e_{i}^{a_{i}(F)} \\cdot  \\left(\\prod_{\\text{$\\lambda$ prime of $\\cO_{k(Q)}$}} \\pi_{\\lambda}^{v_\\lambda(f(Q))}\\right)_Q \\quad \\text{in} \\quad \\left.\\left(\\prod_Q k(Q)^\\times_{\\bQ}\\right)\\right/K_{\\bQ}^\\times,\n\t\\end{equation}\n,\n\\begin{equation}\n\t\t\\label{eq:t-ell}\n\t\t\\sigma(F) = \\sum_{i=1}^s t_{i}(F) u_i \\qquad \\text{in $V_D$},\n\t\\end{equation}\n) consists of dual bases coming from the Mordell--Weil group \\(J(K)\\), unit groups at the cusps, and a basis of \\(\\sigma(W)=U\\). For \\(F\\in W\\) and a logarithmic differential \\(\\omega\\), the key computation \\(\\log_{J_Y}^\\sharp(\\omega)(F)=\\int_F\\omega\\) is reduced to computing \\(\\int_{\\mathrm{div}(f)}\\omega\\) for a suitable rational function \\(f\\). A \\(p\\)-adic residue theorem (We discuss this later.) expresses \\(\\int_{\\mathrm{div}(f)}\\omega\\) in terms of residues of \\(\\omega\\) and values of \\(f\\) at the cusps; these values can be read off from \\(\\sigma(F)\\), which allows one to conclude. The stated relation \\(\\int_{P_0}^P\\omega=c\\) for \\(P\\in\\mathcal{Y}(\\cO_{K,S})_\\Sigma\\) then follows from taking \\((a_1,\\ldots,a_{g+n-1})\\) in the kernel of the representing matrix \\(M(\\Sigma^{\\csp})\\), i.e. combining the \\(D\\)-intersection map \\(\\sigma\\) with \\(\\mathfrak p\\)-adic integration.", "expanded_theorem": "\\label{thm:A}\n\tLet $\\Sigma$ be an $S$-integral reduction type with cuspidal part $\\Sigma^{\\csp}$ and let $\\mathfrak p \\not\\in S$ be a prime of $K$.\n\tIf $(a_1,\\ldots,a_{g+n-1}) \\in K_{\\mathfrak p}^{g+n-1}$ is contained in the kernel of the matrix\n\t\\begin{equation}\n\t\\label{eq:MSigma}\n\tM(U) \\coloneqq \\begin{pmatrix}\n\t\tA & B  \\\\\n\t\t0 & C  \\\\\n\t\t0 & D(U) \n\t\\end{pmatrix}\n\\end{equation}\n\twith\n\t\\begin{equation}\n\t\t\\label{eq:matrixD}\n\t\tD(U)_{i,j} \\coloneqq \\sum_{\\substack{Q \\in |D|,\\\\ \\varphi\\colon k(Q) \\to \\Kpbar}} \\varphi(\\Res_{Q} \\omega_{g+j}) \\sum_{\\substack{\\text{$\\fq$ prime,}\\\\ \\text{$\\lambda|\\fq$ in $\\cO_{k(Q)}$}}} u_{i,\\lambda} \\log \\varphi(\\pi_{\\lambda}).\n\t\\end{equation}\n\t(evaluated at $U=\\Sigma^{\\csp}$), let $\\omega = \\omega(\\Sigma^{\\csp}) \\coloneqq a_1 \\omega_1 + \\dots + a_{g+n-1} \\omega_{g+n-1}$ and let $c = c(P_0,\\Sigma,\\omega)$ be the constant defined by\n\t\\begin{equation}\n\t\\label{eq:constants}\n\tc = c(b, \\omega) \\coloneqq \\sum_{\\substack{Q \\in |D|,\\\\ \\varphi\\colon k(Q) \\to \\Kpbar}} \\varphi(\\Res_Q \\omega) \\sum_{\\textnormal{$\\lambda$ prime of $\\cO_{k(Q)}$}} b_{\\lambda} \\log \\varphi(\\pi_{\\lambda}).\n\t\\end{equation}\n\tand\n\t\\begin{equation}\n\\label{eq:b-lambda}\nb_{\\lambda} = -i_{\\lambda}(\\cP_0,\\tilde\\cQ) + i_{\\lambda}(\\Phi_{\\fq}(\\cpt(\\Sigma_{\\fq}) - \\cpt_{\\fq}(P_0)), \\tilde\\cQ),\n\\end{equation}\n\tThen\n\t\\[\n\t\t\\int_{P_0}^P \\omega = c \\quad \\text{ for all } P \\in \\mathcal{Y}(\\cO_{K,S})_\\Sigma.\n\t\\],", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let $K$ be a number field, $S$ a finite set of primes of $\\mathcal O_K$, $Y=X\\setminus D$ with $X/K$ a smooth projective curve and $D\\neq\\varnothing$ a finite set of closed points, and let $\\mathcal X/\\mathcal O_{K,S}$ be a regular model with $\\mathcal Y=\\mathcal X\\setminus \\mathcal D$. Fix a base point $P_0\\in Y(K)$, write $g$ for the genus of $X$ and $n=\\#D(\\overline K)$, and choose a basis $\\omega_1,\\dots,\\omega_{g+n-1}$ of $H^0(X,\\Omega^1(D))$ (the logarithmic differentials on $(X,D)$). For a fixed $S$-integral reduction type $\\Sigma$, let $\\mathcal Y(\\mathcal O_{K,S})_\\Sigma$ denote the set of $S$-integral points of reduction type $\\Sigma$, and let $\\Sigma^{\\mathrm{csp}}$ be its cuspidal part. Let $\\mathfrak p\\notin S$ be a prime of $K$. Suppose $(a_1,\\ldots,a_{g+n-1})\\in K_{\\mathfrak p}^{g+n-1}$ lies in the kernel of the matrix\n\\[\nM(U)=\\begin{pmatrix}A&B\\\\0&C\\\\0&D(U)\\end{pmatrix}\n\\]\nevaluated at $U=\\Sigma^{\\mathrm{csp}}$, where\n\\[\nD(U)_{i,j}=\\sum_{\\substack{Q\\in |D|,\\\\ \\varphi:k(Q)\\to \\overline{K_{\\mathfrak p}}}} \\varphi(\\operatorname{Res}_Q\\omega_{g+j})\\sum_{\\substack{\\mathfrak q\\text{ prime},\\\\ \\lambda\\mid \\mathfrak q\\text{ in }\\mathcal O_{k(Q)}}} u_{i,\\lambda}\\log \\varphi(\\pi_\\lambda).\n\\]\nSet\n\\[\n\\omega=a_1\\omega_1+\\cdots+a_{g+n-1}\\omega_{g+n-1},\n\\]\nand define\n\\[\nc=c(P_0,\\Sigma,\\omega)=\\sum_{\\substack{Q\\in |D|,\\\\ \\varphi:k(Q)\\to \\overline{K_{\\mathfrak p}}}} \\varphi(\\operatorname{Res}_Q\\omega)\\sum_{\\lambda\\text{ prime of }\\mathcal O_{k(Q)}} b_\\lambda\\log \\varphi(\\pi_\\lambda),\n\\]\nwith\n\\[\nb_\\lambda=-i_\\lambda(\\mathcal P_0,\\widetilde{\\mathcal Q})+i_\\lambda\\bigl(\\Phi_{\\mathfrak q}(\\operatorname{cpt}(\\Sigma_{\\mathfrak q})-\\operatorname{cpt}_{\\mathfrak q}(P_0)),\\widetilde{\\mathcal Q}\\bigr).\n\\]\nWhich statement holds for every point $P\\in \\mathcal Y(\\mathcal O_{K,S})_\\Sigma$?", "correct_choice": {"label": "A", "text": "For every $P\\in \\mathcal Y(\\mathcal O_{K,S})_\\Sigma$, the $\\mathfrak p$-adic integral of $\\omega$ from $P_0$ to $P$ equals the constant $c$; equivalently, \\[\\int_{P_0}^P \\omega = c(P_0,\\Sigma,\\omega).\\]"}, "choices": [{"label": "B", "text": "For every $P\\in \\mathcal Y(\\mathcal O_{K,S})_\\Sigma$, the $\\mathfrak p$-adic integral of $\\omega$ from $P_0$ to $P$ vanishes; equivalently, \\[\\int_{P_0}^P \\omega = 0.\\]"}, {"label": "C", "text": "For every $P\\in \\mathcal Y(\\mathcal O_{K,S})_\\Sigma$, the value of the $\\mathfrak p$-adic integral $\\int_{P_0}^P \\omega$ is independent of the choice of $P$; equivalently, there exists a constant in $K_{\\mathfrak p}$, depending on $P_0$, $\\Sigma$, and $\\omega$, such that \\[\\int_{P_0}^P \\omega\\] is the same for all $P\\in \\mathcal Y(\\mathcal O_{K,S})_\\Sigma$. "}, {"label": "D", "text": "For every $P\\in \\mathcal Y(\\mathcal O_{K,S})$, the $\\mathfrak p$-adic integral of $\\omega$ from $P_0$ to $P$ equals the constant $c(P_0,\\Sigma,\\omega)$; equivalently, \\[\\int_{P_0}^P \\omega = c(P_0,\\Sigma,\\omega).\\]"}, {"label": "E", "text": "If $(a_1,\\ldots,a_{g+n-1})\\in K_{\\mathfrak p}^{g+n-1}$ lies in the kernel of $M(U)$ for some $U$, then for every $P\\in \\mathcal Y(\\mathcal O_{K,S})_\\Sigma$ one has \\[\\int_{P_0}^P \\omega = c(P_0,\\Sigma,\\omega),\\] regardless of whether $U=\\Sigma^{\\mathrm{csp}}$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "trace_identity", "tampered_component": "explicit nonzero constant c from residue computation", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "trace_identity", "tampered_component": "dropped identification of the common value with the explicit formula c(P_0,\\Sigma,\\omega)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence on the fixed reduction type \\Sigma", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "evaluation of the matrix at U=\\Sigma^{\\mathrm{csp}}", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the conclusion that the integral equals c(P_0,\\Sigma,\\omega). It introduces the relevant objects and hypotheses, but the correct answer is not directly leaked."}, "TAS": {"score": 0, "justification": "This is essentially a direct restatement of a theorem/proposition: under the kernel hypothesis for M(\\Sigma^{\\mathrm{csp}}), conclude that \\int_{P_0}^P \\omega equals the explicit constant c(P_0,\\Sigma,\\omega). The correct option reproduces that conclusion almost verbatim."}, "GPS": {"score": 1, "justification": "Some precision is needed to distinguish the exact theorem statement from nearby variants (weaker true statement C, overgeneralization D, and quantifier error E), but the item mainly tests recall of the theorem rather than substantial generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically meaningful and target realistic failure modes: forgetting the nonzero constant term (B), choosing a weaker but true conclusion (C), overextending beyond the fixed reduction type (D), and ignoring the dependence on evaluating at U=\\Sigma^{\\mathrm{csp}} (E)."}, "total_score": 5, "overall_assessment": "Technically well-constructed in its distractors, but it is largely a theorem-recall item rather than a genuinely non-tautological or generative reasoning question."}}
{"id": "2602.05652v1", "paper_link": "http://arxiv.org/abs/2602.05652v1", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "\\label{thm:maintheorem}\n    Let $L$ be a hyperbolic lattice of rank at least $46$. Then, the exceptional \n    lattice of $L$ is trivial.", "start_pos": 7399, "end_pos": 7565, "label": "thm:maintheorem"}, "ref_dict": {"rmk: cusp exists": "\\begin{remark} \\label{rmk: cusp exists}\n\\begin{enumerate}\n    \\item  Every hyperbolic lattice of rank $\\ge 20$ admits a cusp with infinite stabilizer. Indeed, every hyperbolic lattice of rank $\\ge 20$ has an infinite symmetry group (cf. \\cite{nikulin.rank.greater.than.five}), and therefore admits a cusp with infinite stabilizer by \\cite[Theorem~6.4.1]{nikulin.rank.greater.than.five}.\n\n    \\item Note that \\Cref{prop:exceptionalIntersection} fails if $L$ admits no cusps with infinite stabilizer. As an example, let $L$ a hyperbolic lattice of rank $3$ with $\\Aut(\\D_L)\\cong \\ZZ$ and no cusp with infinite stabilizer. Such lattices exist, and were classified in \\cite{brandhorst.mezzedimi.abelian}; for instance we can take $L=U(11)\\oplus A_1$. Then $\\Aut(\\D_L)$ is generated by a symmetry $f$ with infinite order and Salem degree $2$, and thus $f$ preserves a line in $L$. Hence $E(L)$ has rank $1$, while $E'$ is $L$ itself.\n\\end{enumerate}\n\\end{remark}", "thm:maintheorem": "\\begin{theorem}\\label{thm:maintheorem}\n    Let $L$ be a hyperbolic lattice of rank at least $46$. Then, the exceptional \n    lattice of $L$ is trivial. \n\\end{theorem}", "prop:structure": "\\begin{proposition}\\label{prop:structure}\n\tIf $L$ is a hyperbolic lattice of rank at least $6$, then there exist an overlattice \n    $L'$ of $L$, an integer $n$ and a negative-definite lattice $M$ such that\n    $$L'\\cong U\\oplus E_{8}^{n}\\oplus M,$$\n    with $\\rk M\\le 11$ and such that the root part of $M$ does not contain any $E_{8}$ or an ADE lattice of rank at least $9$.\n\\end{proposition}", "cor:finitelymany": "\\begin{corollary} \\label{cor:finitelymany}\n    Up to isometry, there are only finitely many hyperbolic lattices of rank $\\ge 3$ with non-trivial exceptional lattice.\n\\end{corollary}", "prop:exceptionalIntersection": "\\begin{proposition}\\label{prop:exceptionalIntersection}\nLet $L$ be a lattice such that $\\mathcal E_\\infty(L)$ is non-empty. \nThen the exceptional lattice satisfies\n\\begin{align*}\n    E(L)=\\bigcap _{e\\in \\mathcal E_{\\infty}(L)} (e^{\\perp})^{(2)}_{\\mathrm{pr}}.\n\\end{align*}\n\\end{proposition}"}, "pre_theorem_intro_text_len": 2437, "pre_theorem_intro_text": "\\subsection{Symmetries and exceptional lattice}\n\nLet $L$ be an even, integral hyperbolic lattice. The quotient $\\mathrm{Aut}(\\D_L)=\\mathrm{O}^+(L)/W(L)$ of the group of positive isometries of $L$ by the Weyl subgroup of reflections along $(-2)$-roots is called the \\emph{symmetry group} of $L$. Foundational work by Nikulin \\cite{nikulin.rank.greater.than.five,nikulin.rank.three} and Vinberg \\cite{vinberg.rank.four} in the 80s gives a complete classification of hyperbolic lattices with a finite symmetry group. This was recently generalized by Yu \\cite{yu.zero.entropy} (in the case of Picard lattices of K3 surfaces) and Brandhorst--Mezzedimi \\cite{brandhorst.mezzedimi.borcherds,brandhorst.mezzedimi.abelian}, who classified hyperbolic lattices with a \\emph{virtually abelian} symmetry group, that is, containing an abelian subgroup of finite index. The maximum rank of such a hyperbolic lattice is $26$, attained by the unimodular lattice ${\\rm II}_{1,25} \\coloneqq U\\oplus \\Lambda$, where $\\Lambda$ denotes the Leech lattice.\n\nIn order to capture the complexity of symmetry groups of general hyperbolic lattices, Nikulin introduced in \\cite{nikulin.elliptic.fibrations} the notion of \\emph{exceptional lattice}: If $L$ is a hyperbolic lattice, then the exceptional lattice $E(L)\\subseteq L$ is the sublattice generated by elements that have finite orbit under $\\mathrm{Aut}(\\D_L)$. Following \\cite[§4]{nikulin.elliptic.fibrations}, there are three possibilities for the exceptional lattice if it is non-trivial: \n\\begin{enumerate}\n    \\item $E(L)$ is a hyperbolic lattice. Then $E(L)=L$, and in particular $\\mathrm{Aut}(\\D_L)$ is finite.\n    \\item $E(L)$ is negative semidefinite. Then it contains an isotropic vector, and therefore $L$ is a \\emph{Borcherds lattice} (see \\cite[Definition~1.1]{brandhorst.mezzedimi.borcherds}). In particular $\\mathrm{Aut}(\\D_L)$ is virtually abelian.\n    \\item $E(L)$ is negative definite. \n\\end{enumerate}\n\nAs explained above, the lattices satisfying (1) or (2) have already been classified. Moreover, Nikulin proves in \\cite[Theorem~4.4]{nikulin.elliptic.fibrations} that there are only finitely many hyperbolic lattices of \\emph{fixed} rank $\\ge 3$ with non-trivial exceptional lattice. \nThe goal of this article is to make the next step towards a full classification also in case (3), by proving that $E(L)$ is in fact trivial in large enough rank. More precisely, we will show the following:", "context": "\\subsection{Symmetries and exceptional lattice}\n\nLet $L$ be an even, integral hyperbolic lattice. The quotient $\\mathrm{Aut}(\\D_L)=\\mathrm{O}^+(L)/W(L)$ of the group of positive isometries of $L$ by the Weyl subgroup of reflections along $(-2)$-roots is called the \\emph{symmetry group} of $L$. Foundational work by Nikulin \\cite{nikulin.rank.greater.than.five,nikulin.rank.three} and Vinberg \\cite{vinberg.rank.four} in the 80s gives a complete classification of hyperbolic lattices with a finite symmetry group. This was recently generalized by Yu \\cite{yu.zero.entropy} (in the case of Picard lattices of K3 surfaces) and Brandhorst--Mezzedimi \\cite{brandhorst.mezzedimi.borcherds,brandhorst.mezzedimi.abelian}, who classified hyperbolic lattices with a \\emph{virtually abelian} symmetry group, that is, containing an abelian subgroup of finite index. The maximum rank of such a hyperbolic lattice is $26$, attained by the unimodular lattice ${\\rm II}_{1,25} \\coloneqq U\\oplus \\Lambda$, where $\\Lambda$ denotes the Leech lattice.\n\nIn order to capture the complexity of symmetry groups of general hyperbolic lattices, Nikulin introduced in \\cite{nikulin.elliptic.fibrations} the notion of \\emph{exceptional lattice}: If $L$ is a hyperbolic lattice, then the exceptional lattice $E(L)\\subseteq L$ is the sublattice generated by elements that have finite orbit under $\\mathrm{Aut}(\\D_L)$. Following \\cite[§4]{nikulin.elliptic.fibrations}, there are three possibilities for the exceptional lattice if it is non-trivial: \n\\begin{enumerate}\n \\item $E(L)$ is a hyperbolic lattice. Then $E(L)=L$, and in particular $\\mathrm{Aut}(\\D_L)$ is finite.\n \\item $E(L)$ is negative semidefinite. Then it contains an isotropic vector, and therefore $L$ is a \\emph{Borcherds lattice} (see \\cite[Definition~1.1]{brandhorst.mezzedimi.borcherds}). In particular $\\mathrm{Aut}(\\D_L)$ is virtually abelian.\n \\item $E(L)$ is negative definite. \n\\end{enumerate}\n\nAs explained above, the lattices satisfying (1) or (2) have already been classified. Moreover, Nikulin proves in \\cite[Theorem~4.4]{nikulin.elliptic.fibrations} that there are only finitely many hyperbolic lattices of \\emph{fixed} rank $\\ge 3$ with non-trivial exceptional lattice. \nThe goal of this article is to make the next step towards a full classification also in case (3), by proving that $E(L)$ is in fact trivial in large enough rank. More precisely, we will show the following:", "full_context": "\\subsection{Symmetries and exceptional lattice}\n\nLet $L$ be an even, integral hyperbolic lattice. The quotient $\\mathrm{Aut}(\\D_L)=\\mathrm{O}^+(L)/W(L)$ of the group of positive isometries of $L$ by the Weyl subgroup of reflections along $(-2)$-roots is called the \\emph{symmetry group} of $L$. Foundational work by Nikulin \\cite{nikulin.rank.greater.than.five,nikulin.rank.three} and Vinberg \\cite{vinberg.rank.four} in the 80s gives a complete classification of hyperbolic lattices with a finite symmetry group. This was recently generalized by Yu \\cite{yu.zero.entropy} (in the case of Picard lattices of K3 surfaces) and Brandhorst--Mezzedimi \\cite{brandhorst.mezzedimi.borcherds,brandhorst.mezzedimi.abelian}, who classified hyperbolic lattices with a \\emph{virtually abelian} symmetry group, that is, containing an abelian subgroup of finite index. The maximum rank of such a hyperbolic lattice is $26$, attained by the unimodular lattice ${\\rm II}_{1,25} \\coloneqq U\\oplus \\Lambda$, where $\\Lambda$ denotes the Leech lattice.\n\nIn order to capture the complexity of symmetry groups of general hyperbolic lattices, Nikulin introduced in \\cite{nikulin.elliptic.fibrations} the notion of \\emph{exceptional lattice}: If $L$ is a hyperbolic lattice, then the exceptional lattice $E(L)\\subseteq L$ is the sublattice generated by elements that have finite orbit under $\\mathrm{Aut}(\\D_L)$. Following \\cite[§4]{nikulin.elliptic.fibrations}, there are three possibilities for the exceptional lattice if it is non-trivial: \n\\begin{enumerate}\n \\item $E(L)$ is a hyperbolic lattice. Then $E(L)=L$, and in particular $\\mathrm{Aut}(\\D_L)$ is finite.\n \\item $E(L)$ is negative semidefinite. Then it contains an isotropic vector, and therefore $L$ is a \\emph{Borcherds lattice} (see \\cite[Definition~1.1]{brandhorst.mezzedimi.borcherds}). In particular $\\mathrm{Aut}(\\D_L)$ is virtually abelian.\n \\item $E(L)$ is negative definite. \n\\end{enumerate}\n\nAs explained above, the lattices satisfying (1) or (2) have already been classified. Moreover, Nikulin proves in \\cite[Theorem~4.4]{nikulin.elliptic.fibrations} that there are only finitely many hyperbolic lattices of \\emph{fixed} rank $\\ge 3$ with non-trivial exceptional lattice. \nThe goal of this article is to make the next step towards a full classification also in case (3), by proving that $E(L)$ is in fact trivial in large enough rank. More precisely, we will show the following:\n\n\\begin{abstract}\nFor an even, integral hyperbolic lattice $L$, the symmetry group of $L$ is the quotient of the group of isometries of $L$ by the Weyl subgroup of $(-2)$-reflections. Following Nikulin, the exceptional lattice of $L$ is defined as the sublattice generated by elements that have finite orbit under the symmetry group of $L$. We prove that every hyperbolic lattice of rank at least $46$ has trivial exceptional lattice. In particular, every such lattice admits a symmetry of maximal Salem degree.\n\\end{abstract}\n\nAs explained above, the lattices satisfying (1) or (2) have already been classified. Moreover, Nikulin proves in \\cite[Theorem~4.4]{nikulin.elliptic.fibrations} that there are only finitely many hyperbolic lattices of \\emph{fixed} rank $\\ge 3$ with non-trivial exceptional lattice. \nThe goal of this article is to make the next step towards a full classification also in case (3), by proving that $E(L)$ is in fact trivial in large enough rank. More precisely, we will show the following:\n\nIn particular, the classification of hyperbolic lattices with non-trivial exceptional lattice becomes a finite problem:\n\n\\begin{corollary} \\label{cor:finitelymany}\n Up to isometry, there are only finitely many hyperbolic lattices of rank $\\ge 3$ with non-trivial exceptional lattice.\n\\end{corollary}\n\n\\begin{corollary}\nLet $L$ be a hyperbolic lattice of rank at least $46$. Then, the following hold:\n\\begin{enumerate}\n\\item There are no non-trivial $\\Aut(\\D_L)$-invariant $\\RR$-linear subspaces of $L_{\\mathbb{R}}$.\n\\item There exists an isometry $f \\in \\Aut(\\D_L)$ of maximal Salem degree.\n\\end{enumerate}\n\\end{corollary}\n\nNow let $L$ be a hyperbolic lattice of rank at least $46$. By taking an arbitrary maximal overlattice of $L$, we reduce to lattices of the form $L=U\\oplus E_8^n\\oplus M$, where $n\\ge 5$ and the rank of $M$ is at most $11$ (cf. \\Cref{prop:structure}). By using the isometry $U\\oplus E_8^3\\cong U\\oplus \\Lambda$, and the fact that the Leech lattice $\\Lambda$ has no $(-2)$-root, we deduce that the exceptional lattice $E(L)$ is contained in $M$. Since there are only finitely many possibilities for the root part of $M$, we are able to show in \\Cref{section:proof} that $E(L)=0$ for all of them, thus concluding the proof of \\Cref{thm:maintheorem}.\n\n\\begin{proposition} \\label{prop: big roots contain UE8}\nLet $R$ be an ADE lattice of rank $n \\geq 9$. Then, $U \\oplus R$ contains $U \\oplus E_8$.\n\\end{proposition}\n\\begin{proof}\nFrom the classification of ADE lattices, we have $R \\in \\{A_n,D_n\\}$, hence $\\ell(A_R) \\leq 2$ and even $\\ell(A_R) = 1$ unless $R = D_n$ with $n$ even (see \\cite[Table~1.1]{Eb13}).\nUsing \\cite[Corollary 1.13.5]{Ni80} and the inequality\n$$\n{\\rm rk}(U \\oplus R) \\geq 11 \\geq 9 + {\\ell}(A_R),\n$$\nwe conclude that $U \\oplus R \\cong E_8 \\oplus M$ for a hyperbolic lattice $M$ of rank $n -6 \\geq 3$ with $\\ell(A_M) = \\ell(A_R)$. By \\cite[Corollary 1.13.5]{Ni80}, we are done if $n - 6 \\geq 3 + \\ell(A_R)$. \nIt thus remains to consider the cases $R \\in \\{D_9,D_{10},A_9\\}$. A straightforward computation shows that the discriminant group of $D_9$ (resp. $D_{10}$, $A_9$) is isomorphic as a finite quadratic module to the discriminant group of $\\langle -4\\rangle$ (resp. $A_1\\oplus A_1$, $\\langle -10\\rangle$), see e.g. \\cite[Table~1]{gvirtz.mezzedimi}. Therefore $D_9$ (resp. $D_{10}$, $A_9$) is in the same genus of $E_8\\oplus \\langle -4\\rangle$ (resp. $E_8\\oplus A_1\\oplus A_1$, $E_8\\oplus \\langle -10\\rangle$), from which\n$$U\\oplus D_9\\cong U\\oplus E_8\\oplus \\langle -4\\rangle, \\quad U\\oplus D_{10} \\cong U\\oplus E_8\\oplus A_1\\oplus A_1, \\quad U\\oplus A_9\\cong U\\oplus E_8\\oplus \\langle -10\\rangle. $$ \n\\end{proof}\n\n\\begin{proposition}\\label{prop:structure}\n If $L$ is a hyperbolic lattice of rank at least $6$, then there exist an overlattice \n $L'$ of $L$, an integer $n$ and a negative-definite lattice $M$ such that\n $$L'\\cong U\\oplus E_{8}^{n}\\oplus M,$$\n with $\\rk M\\le 11$ and such that the root part of $M$ does not contain any $E_{8}$ or an ADE lattice of rank at least $9$.\n\\end{proposition}\n\n\\begin{proposition}\\label{prop:structure}\n\tIf $L$ is a hyperbolic lattice of rank at least $6$, then there exist an overlattice \n    $L'$ of $L$, an integer $n$ and a negative-definite lattice $M$ such that\n    $$L'\\cong U\\oplus E_{8}^{n}\\oplus M,$$\n    with $\\rk M\\le 11$ and such that the root part of $M$ does not contain any $E_{8}$ or an ADE lattice of rank at least $9$.\n\\end{proposition}\n\n\\begin{theorem}\\label{thm:maintheorem}\n    Let $L$ be a hyperbolic lattice of rank at least $46$. Then, the exceptional \n    lattice of $L$ is trivial. \n\\end{theorem}", "post_theorem_intro_text_len": 7024, "post_theorem_intro_text": "In particular, the classification of hyperbolic lattices with non-trivial exceptional lattice becomes a finite problem:\n\n\\begin{corollary} \\label{cor:finitelymany}\n    Up to isometry, there are only finitely many hyperbolic lattices of rank $\\ge 3$ with non-trivial exceptional lattice.\n\\end{corollary}\n\nObserve that in rank $2$, a hyperbolic lattice has non-trivial exceptional lattice if and only if it has finite symmetry group, or equivalently if and only if it represents $0$ or $-2$ (see \\cite[Corollary~3.4]{galluzzi.lombardo.peters}).\n\nGiven the finiteness result in \\Cref{cor:finitelymany}, it is natural to ask what is the largest rank of a hyperbolic lattice with non-trivial exceptional lattice. If $L\\coloneqq {\\rm II}_{1,25}$, then already Borcherds noted in \\cite{borcherds.leech} that $E(L)$ has rank $1$, and it is generated by the unique cusp (i.e. primitive, isotropic vector in the closure of the fundamental domain $\\D_L$) with infinite stabilizer. We do not know any example of larger rank. In the spirit of the classifications in \\cite{nikulin.rank.greater.than.five,nikulin.rank.three,vinberg.rank.four,brandhorst.mezzedimi.borcherds}, it would be interesting to have a complete classification of hyperbolic lattices with a non-trivial exceptional lattice, thus completing Nikulin's program in \\cite{nikulin.elliptic.fibrations}.\n\n\\subsection{Motivation from K3 surfaces and applications} \nIf $L = \\mathrm{Pic}(X)$ is the Picard lattice of a complex K3 surface $X$, then the classes of effective $(-2)$-curves in $L$ determine a choice of fundamental domain $\\D_L$ for the action of $W(L)$ on $L$ and the associated action\n\\begin{eqnarray*}\n\\mathrm{Aut}(X) & \\to & \\mathrm{Aut}(\\D_L) \\\\\ng & \\mapsto & g^*\n\\end{eqnarray*}\nhas finite kernel and cokernel. The \\emph{cusps} of $L$ that we consider in this article correspond to elliptic fibrations of $X$ and cusps with infinite stabilizer correspond to elliptic fibrations whose Jacobian has infinite Mordell--Weil group. If $X$ admits at least one elliptic fibration with infinite automorphism group, then the exceptional lattice is the primitive sublattice of $\\mathrm{Pic}(X)$ generated by all $(-2)$-curves that are contained in fibers of all such elliptic fibrations (cf. \\Cref{prop:exceptionalIntersection}).\n\nTo measure the complexity of an automorphism $g$, one considers the characteristic polynomial of $f=g^*$, which factors over $L_{\\mathbb{R}}$ as $S(x)C(x)$, where $S(x)$ is a Salem polynomial (or a quadratic polynomial or trivial), called the \\emph{Salem factor}, and $C(x)$ is a product of cyclotomic polynomials (see \\cite[Discussion before D\\'efinition 1.2]{cantat.dynamique}). The Salem factor is particularly interesting, since the logarithm of its largest root coincides with the topological entropy of $X$ \\cite{Gromov,Yomdin}. Thus, to measure the complexity of this number, we define the \\emph{Salem degree} of $f$ as the degree of $f$ and, since this degree is always even, we say that $f$ has \\emph{maximal Salem degree} if its Salem degree is $\\rk(L)$ if $\\rk(L)$ is even or $\\rk(L)-1$ if $\\rk(L)$ is odd.\n\nNow, observe that these definitions do not use the fact that there is an underlying K3 surface $X$ at all. In fact, even the choice of $\\D_L$, which is needed to consider $\\mathrm{Aut}(\\D_L)$ as a subgroup of $\\mathrm{O}^+(L)$, is irrelevant, since any two choices lead to conjugate subgroups.\nOur results on hyperbolic lattices can therefore be used to apply statements in the literature about automorphisms of K3 surfaces in terms of the exceptional lattice to the symmetry group $\\mathrm{Aut}(\\D_L)$. One such statement is the following, due to Yu \\cite[Theorem 6.2]{yu.elliptic.fibrations}. The original proof in the case of K3 surfaces of even Picard rank extends easily to the abstract setting and we can rephrase the result in our terminology as follows:\n\n\\begin{theorem}[Yu]\nLet $L$ be a hyperbolic lattice with trivial exceptional lattice $E(L)$. Then:\n\\begin{enumerate}\n    \\item The only $\\mathrm{Aut}(\\D_L)$-invariant $\\mathbb{R}$-linear subspaces of $L_\\mathbb{R}$ are $\\{0\\}$ and $L_\\mathbb{R}$.\n    \\item There exists a symmetry $f\\in \\mathrm{Aut}(\\D_L)$ of maximal Salem degree.\n\\end{enumerate}\nIf moreover $\\rk(L)\\ge 4$ is even, then the conditions $E(L)=0$, (1) and (2) are all equivalent.\n\\end{theorem}\n\nBy \\Cref{rmk: cusp exists}, there always exists a cusp of infinite stabilizer if ${\\rm rk}(L) \\geq 20$ and by our main \\Cref{thm:maintheorem}, the exceptional lattice is trivial as soon as ${\\rm rk}(L) \\geq 46$. We thus obtain the following application to Salem degrees:\n\n\\begin{corollary}\nLet $L$ be a hyperbolic lattice of rank at least $46$. Then, the following hold:\n\\begin{enumerate}\n\\item There are no non-trivial $\\mathrm{Aut}(\\D_L)$-invariant $\\mathbb{R}$-linear subspaces of $L_{\\mathbb{R}}$.\n\\item There exists an isometry $f \\in \\mathrm{Aut}(\\D_L)$ of maximal Salem degree.\n\\end{enumerate}\n\\end{corollary}\n\n\\subsection{Strategy of the proof and outline}\n\nThe starting point of the proof of \\Cref{thm:maintheorem} is an explicit description of the exceptional lattice of hyperbolic lattices, originally due to Nikulin \\cite[Theorem~4.2]{nikulin.elliptic.fibrations} (see also \\Cref{prop:exceptionalIntersection}). More precisely, it asserts that $E(L)$ is trivial as soon as every simple $(-2)$-root $r\\in L$ admits a cusp with infinite stabilizer $e\\in \\overline{\\mathcal{D}}_L$ such that $e.r>0$.\n\nNow let $L$ be a hyperbolic lattice of rank at least $46$. By taking an arbitrary maximal overlattice of $L$, we reduce to lattices of the form $L=U\\oplus E_8^n\\oplus M$, where $n\\ge 5$ and the rank of $M$ is at most $11$ (cf. \\Cref{prop:structure}). By using the isometry $U\\oplus E_8^3\\cong U\\oplus \\Lambda$, and the fact that the Leech lattice $\\Lambda$ has no $(-2)$-root, we deduce that the exceptional lattice $E(L)$ is contained in $M$. Since there are only finitely many possibilities for the root part of $M$, we are able to show in \\Cref{section:proof} that $E(L)=0$ for all of them, thus concluding the proof of \\Cref{thm:maintheorem}.\n\nGiven the nature of our approach, we expect the bound in \\Cref{thm:maintheorem} to be quite far from sharp. Moreover, it is probable that a (possibly computer assisted) refinement of our argument in \\Cref{section:proof} via extended Dynkin diagrams leads to a much better bound in \\Cref{thm:maintheorem}.\n\nWe briefly outline the structure of the paper. In Section 2 we recall basic properties of negative definite and hyperbolic lattices, and prove the structure result for maximal overlattices of hyperbolic lattices in \\Cref{prop:structure}. In Section 3 we review Nikulin's definition of the exceptional lattice, and the explicit description in \\Cref{prop:exceptionalIntersection}. Finally, in Section 4 we prove \\Cref{thm:maintheorem}.\n\n\\begin{ack}\nWe gratefully acknowledge the hospitality and support of the Max-Planck-Institut f\\\"ur Mathematik Bonn, where we started working on this article during an internship program.\n\\end{ack}", "sketch": "The proof of \\Cref{thm:maintheorem} starts from Nikulin’s explicit description of the exceptional lattice (\\cite[Theorem~4.2]{nikulin.elliptic.fibrations}, see also \\Cref{prop:exceptionalIntersection}): it implies that $E(L)$ is trivial “as soon as every simple $(-2)$-root $r\\in L$ admits a cusp with infinite stabilizer $e\\in \\overline{\\mathcal{D}}_L$ such that $e.r>0$.”\n\nFor a hyperbolic lattice $L$ of rank at least $46$, one “tak[es] an arbitrary maximal overlattice of $L$” to reduce to lattices of the form\n\\[\nL = U\\oplus E_8^n\\oplus M,\\quad n\\ge 5,\\ \\rk(M)\\le 11\n\\]\n(cf. \\Cref{prop:structure}). Using the isometry $U\\oplus E_8^3\\cong U\\oplus \\Lambda$ and that the Leech lattice $\\Lambda$ has no $(-2)$-root, one deduces that “the exceptional lattice $E(L)$ is contained in $M$.” Since there are “only finitely many possibilities for the root part of $M$,” the argument in \\Cref{section:proof} checks all cases and shows “that $E(L)=0$ for all of them,” which “conclud[es] the proof of \\Cref{thm:maintheorem}.”", "expanded_sketch": "To prove the main theorem, we start from Nikulin’s explicit description of the exceptional lattice (Nikulin, “Elliptic fibrations on K3 surfaces”, Theorem~4.2), see also the following proposition.\n\n\\begin{proposition}\\label{prop:exceptionalIntersection}\nLet $L$ be a lattice such that $\\mathcal E_\\infty(L)$ is non-empty. \nThen the exceptional lattice satisfies\n\\begin{align*}\n    E(L)=\\bigcap _{e\\in \\mathcal E_{\\infty}(L)} (e^{\\perp})^{(2)}_{\\mathrm{pr}}.\n\\end{align*}\n\\end{proposition}\n\nIt implies that $E(L)$ is trivial “as soon as every simple $(-2)$-root $r\\in L$ admits a cusp with infinite stabilizer $e\\in \\overline{\\mathcal{D}}_L$ such that $e.r>0$.”\n\nFor a hyperbolic lattice $L$ of rank at least $46$, one “tak[es] an arbitrary maximal overlattice of $L$” to reduce to lattices of the form\n\\[\nL = U\\oplus E_8^n\\oplus M,\\quad n\\ge 5,\\ \\rk(M)\\le 11\n\\]\nby the following proposition.\n\n\\begin{proposition}\\label{prop:structure}\n\\tIf $L$ is a hyperbolic lattice of rank at least $6$, then there exist an overlattice \n    $L'$ of $L$, an integer $n$ and a negative-definite lattice $M$ such that\n    $$L'\\cong U\\oplus E_{8}^{n}\\oplus M,$$\n    with $\\rk M\\le 11$ and such that the root part of $M$ does not contain any $E_{8}$ or an ADE lattice of rank at least $9$.\n\\end{proposition}\n\nUsing the isometry $U\\oplus E_8^3\\cong U\\oplus \\Lambda$ and that the Leech lattice $\\Lambda$ has no $(-2)$-root, one deduces that “the exceptional lattice $E(L)$ is contained in $M$.” Since there are “only finitely many possibilities for the root part of $M$,” the argument given later checks all cases and shows “that $E(L)=0$ for all of them,” which “conclud[es] the proof of the main theorem.”", "expanded_theorem": "\\label{thm:maintheorem}\n    Let $L$ be a hyperbolic lattice of rank at least $46$. Then, the exceptional \n    lattice of $L$ is trivial.,", "theorem_type": ["Universal"], "mcq": {"question": "Let $L$ be a hyperbolic lattice of rank at least $46$. Write $\\mathrm{Aut}(\\mathcal D_L)=\\mathrm{O}^+(L)/W(L)$ for its symmetry group, where $\\mathrm{O}^+(L)$ is the group of positive isometries of $L$ and $W(L)$ is the Weyl subgroup generated by reflections in $(-2)$-roots. The exceptional lattice $E(L)\\subseteq L$ is the sublattice generated by all elements of $L$ whose orbit under $\\mathrm{Aut}(\\mathcal D_L)$ is finite. Which statement holds for every such lattice $L$?", "correct_choice": {"label": "A", "text": "The exceptional lattice is trivial; equivalently, $E(L)=0$."}, "choices": [{"label": "B", "text": "The exceptional lattice is always negative definite of rank at most $11$; equivalently, after passing to a maximal overlattice one has $E(L)\\subseteq M$ for some negative-definite lattice $M$ with $\\operatorname{rk}(M)\\le 11$, and hence $E(L)\\neq L$."}, {"label": "C", "text": "The exceptional lattice contains no nonzero hyperbolic sublattice; equivalently, $E(L)$ is not hyperbolic."}, {"label": "D", "text": "There exists a universal constant $N\\le 46$ such that for every hyperbolic lattice $L$ of rank at least $N$, the exceptional lattice is contained in the negative-definite factor $M$ arising from some decomposition of a maximal overlattice as $U\\oplus E_8^n\\oplus M$ with $\\operatorname{rk}(M)\\le 11$, and therefore $E(L)$ is negative definite."}, {"label": "E", "text": "For every hyperbolic lattice $L$ of rank at least $46$, there exists a maximal overlattice $L'\\cong U\\oplus E_8^n\\oplus M$ with $n\\ge 5$ and $\\operatorname{rk}(M)\\le 11$ such that the exceptional lattice of $L$ coincides with the exceptional lattice of $L'$ and is equal to $M$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "containment_in_M_vs_triviality", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_triviality_kept_nonhyperbolic_consequence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "containment_in_M_implies_negative_definite", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "overlattice_transfer_of_exceptional_lattice", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the objects involved but does not explicitly or implicitly reveal that the exceptional lattice must be trivial. The correct conclusion is not embedded in the wording."}, "TAS": {"score": 0, "justification": "The item is essentially a direct theorem-recall question: under the stated rank hypothesis, the correct option states the theorem's conclusion almost verbatim as 'E(L)=0.'"}, "GPS": {"score": 1, "justification": "There is some pressure to distinguish the exact theorem from weaker true consequences and stronger false variants, but the task mainly tests recognition/recall rather than substantial mathematical generation or derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically sophisticated and plausible: one is a weaker true statement, others are natural overstrengthenings or transfer claims a student might incorrectly infer from the proof structure."}, "total_score": 5, "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it is largely a theorem-restatement item and only moderately tests genuine reasoning."}}
{"id": "2602.05697v1", "paper_link": "http://arxiv.org/abs/2602.05697v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm1}\n    Let $\\psi$ be a Hecke–Maass form on $X$ with spectral parameter $\\lambda\\gg1$. We have\n    \\begin{equation}\\label{i2}\n        \\| \\psi \\|_{L^6(X)} \\lesssim_\\varepsilon\n        \\lambda^{\\frac{1}{6}-\\frac{1}{144}+\\varepsilon}.\n    \\end{equation}", "start_pos": 25651, "end_pos": 25933, "label": "thm1"}, "ref_dict": {"eq: IS bound": "\\begin{align}\\label{eq: IS bound}\n    \\|\\psi\\|_{L^\\infty(X)} \\lesssim_\\epsilon\\lambda^{\\frac{5}{12}+\\epsilon},\n\\end{align}", "eq: HK L4": "\\begin{align}\\label{eq: HK L4}\n    \\| \\psi \\|_{L^4(X)}\\lesssim_\\epsilon \\la^{\\frac{3}{152}+\\epsilon}.\n\\end{align}", "i4": "\\begin{equation}\\label{i4}\n        \\|e_\\la\\|_{L^\\infty(M)}\\lesssim_\\e \\la^{\\frac12-\\delta_2+\\e}\n    \\end{equation}", "eq: Marshall L4": "\\begin{align}\\label{eq: Marshall L4}\n    \\| \\psi \\|_{L^4(X)}\\lesssim_\\epsilon \\la^{\\frac{1}{8}-\\frac{1}{56}+\\epsilon}.\n\\end{align}", "thm1": "\\begin{theorem}\\label{thm1}\n    Let $\\psi$ be a Hecke–Maass form on $X$ with spectral parameter $\\lambda\\gg1$. We have\n    \\begin{equation}\\label{i2}\n        \\| \\psi \\|_{L^6(X)} \\lesssim_\\epsilon\n        \\lambda^{\\frac{1}{6}-\\frac{1}{144}+\\epsilon}.\n    \\end{equation}\n\\end{theorem}", "prop: bound for I_A": "\\begin{proposition}\\label{prop: bound for I_A}\n    Let $\\phi\\in C^\\infty(X)$ be such that $\\|\\phi\\|_{L^{6/5}(X)}=1$. \n     \\begin{enumerate}\n         \\item For any $\\gamma\\in G$, $I_\\sA(\\lambda,\\phi,\\gamma)\\lesssim_\\epsilon\\lambda^{1/3+\\epsilon}$.\n         \\item Fix $0<\\epsilon_0\\ll\\delta_1$. If $d(\\gamma,e)\\lesssim 1$ and $d(g_0^{-1}\\gamma g_0,A)\\geq \\lambda^{-\\delta_1+2\\epsilon_0}$, then $I_\\sA(\\lambda,\\phi,\\gamma)\\lesssim_{\\epsilon_0,N} \\lambda^{-N}$ for any $N>0$.\n     \\end{enumerate}\n\\end{proposition}", "eq: HK L6": "\\begin{align}\\label{eq: HK L6}\n    \\| \\psi \\|_{L^6(X)}\\lesssim_\\epsilon \\la^{\\frac{26}{171}+\\epsilon}=\\la^{\\frac{1}{6}-\\frac{5}{342}+\\epsilon}.\n\\end{align}", "qnusymbol": "\\begin{equation}\\label{qnusymbol}\nA_\\nu^\\theta(x,\\xi)=\\phi(x) \\, a_j^\\theta(x,\\xi) \\, \\tilde\\beta\\bigl(p(x,\\xi)/\\la\\bigr),\n\\quad \\nu =\\theta j\\in \\theta \\cdot {\\mathbb Z}^{2}.\n\\end{equation}", "r1": "\\begin{equation}\\label{r1}\n\\|e_\\la\\|_{L^q(M)}\\lesssim \\la^{\\delta(q,n)}\\end{equation}", "1.2": "\\begin{equation}\\label{1.2}\n(\\Delta_g+\\la^2_j )e_{\\la_j}=0, \\quad\n\\text{and } \\quad \\int_M |e_{\\la_j}(x)|^2 \\, dx=1.\n\\end{equation}", "i3": "\\begin{equation}\\label{i3}\n        \\sup_\\nu\\|A_\\nu^{\\theta_0} e_\\la\\|_{L^6(M)}\\lesssim_\\e \\la^{\\frac16-a{\\delta_1}+\\e},\n    \\end{equation}", "thm2": "\\begin{theorem}\\label{thm2}\n    Let $(M,g)$ be a compact $2$-dimensional Riemannian manifold without boundary. Let $e_\\la$   be an $L^2$ normalized eigenfunction of  $-\\Delta_g$ with eigenvalue $\\la^2$, satisfying \\eqref{1.2} for some $\\la\\gg1$. Set $\\theta_0=\\la^{-\\delta_1}$ with $0<\\delta_1< \\frac12$, and let $A_\\nu^{\\theta_0}$ be the pseudo-differential operator defined in \\eqref{qnusymbol}. Assume that \n\\begin{equation}\\label{i3}\n        \\sup_\\nu\\|A_\\nu^{\\theta_0} e_\\la\\|_{L^6(M)}\\lesssim_\\e \\la^{\\frac16-a{\\delta_1}+\\e},\n    \\end{equation}\n    and \n        \\begin{equation}\\label{i4}\n        \\|e_\\la\\|_{L^\\infty(M)}\\lesssim_\\e \\la^{\\frac12-\\delta_2+\\e}\n    \\end{equation}\nfor some constant $a>0$ and $0<\\delta_2<\\frac{1}{2}$. Then \n \\begin{equation}\\label{i5}\n        \\|e_\\la\\|_{L^6(M)}\\lesssim_\\e \\la^{\\frac16-\\frac23a{\\delta_1}+\\e}+\\la^{\\frac23(-\\frac14+\\frac53\\delta_1)}+\\la^{\\frac16(1+\\delta_1-2\\delta_2)+\\e}.\n    \\end{equation}\n\\end{theorem}", "thm microlocal KN": "\\begin{theorem}\\label{thm microlocal KN}\n    Let $\\psi$ be a Hecke–Maass form on $X$ with spectral parameter $\\lambda\\gg1$. Set $\\theta_0=\\la^{-\\delta_1}$ with $0<\\delta_1<\\frac12$, and let $A_\\nu^{\\theta_0}$ be the pseudo-differential operator defined in  \\eqref{qnusymbol}. We have\n    \\begin{equation*}\n        \\sup_\\nu\\|A_\\nu^{\\theta_0} \\psi\\|_{L^6(X)}\\lesssim_\\e \\la^{\\frac16-\\frac{\\delta_1}{12}+\\e}.\n    \\end{equation*}\n\\end{theorem}", "prop: amplification inequality": "\\begin{proposition}\\label{prop: amplification inequality}\n    For any $\\phi\\in C^\\infty(X)$, we have\n    \\[\n    \\left|\\langle \\sA\\cT\\psi,\\phi\\rangle_X\\right|^2\\leq\\sum_{m,n\\leq N} |\\alpha_m  \\alpha_n | \\sum_{d|(m,n)} \\frac{d}{\\sqrt{mn}}\\sum_{\\gamma\\in R(mn/d^2)} |I_\\sA(\\la,\\phi,\\gamma)|\n    \\]\n\\end{proposition}", "eq: Marshall geodesic bound": "\\begin{align}\\label{eq: Marshall geodesic bound}\n    \\sup_{\\gamma\\in\\Pi} \\| \\psi|_\\gamma \\|_{L^2(\\gamma)}\\lesssim_\\epsilon \\lambda^{\\frac{3}{14}+\\epsilon}.\n\\end{align}"}, "pre_theorem_intro_text_len": 9150, "pre_theorem_intro_text": "The purpose of this paper is to obtain improved $L^q$-norm estimates for eigenfunctions on an arithmetic hyperbolic surface for the critical exponent $q=6$, which in turn also yield improved estimates for all $q>2$ by interpolation.\n\nLet $(M,g)$ be a  compact $n$-dimensional Riemannian manifold without boundary, \n$\\Delta_g$ be the\nLaplace-Beltrami operator associated with the metric $g$ on $M$. We let $0=\\la_0^2<\\lambda^2_1\\le \\lambda^2_2\\le \\cdots$ denote the\neigenvalues labeled with respect to the multiplicity of \n${-\\Delta_g}$\nand $e_{\\la_j}$ the associated $L^2$-normalized eigenfunctions, that is,\n\\begin{equation}\\label{1.2}\n(\\Delta_g+\\lambda^2_j )e_{\\la_j}=0, \\quad\n\\text{and } \\quad \\int_M |e_{\\la_j}(x)|^2 \\, dx=1.\n\\end{equation}\n\nIn Sogge~\\cite{sogge881} (see also Avakumovi\\'c \\cite{Avakumovic} and Levitan \\cite{Levitan} when $p = \\infty$), the universal bound for $q\\ge2$\n\\begin{equation}\\label{r1}\n\\|e_\\lambda\\|_{L^q(M)}\\lesssim \\lambda^{\\delta(q,n)}\\end{equation}\nwas obtained. Here $\\delta(q,n)$ is given by\n\\begin{equation*}\n            \\delta(q,n) = \\begin{dcases}\n                \\frac{n-1}{4} - \\frac{n-1}{2q} \\;&\\text{ for }\\;2\\leq q\\leq q_c,\\\\\n                \\frac{n-1}{2} - \\frac{n}{q} \\;\\;\\;\\;&\\text{ for }\\;q_c\\leq q\\leq\\infty,\n            \\end{dcases}\n            \\quad\\text{ and }\\quad q_c=\\frac{2(n+1)}{n-1}.\n\\end{equation*}\nThe result also applies to quasimodes that are linear combinations of eigenfunctions whose eigenvalues lie in the unit interval $[\\lambda,\\lambda+1]$. \nWhen $M$ is a compact locally symmetric space of higher rank, besides the Laplace-Beltrami operator, one can consider eigenfunctions of the full ring of differential operators on $M$. The general local bound was obtained by Sarnak \\cite{sarnak2004letter} (for $q=\\infty$) and Marshall \\cite{Mar16HigerRank} (for any $q\\ge2$ with some log losses at the corresponding critical exponents), which extends and improves the bound \\eqref{r1}. \n\nThe estimate \\eqref{r1} is sharp on the standard round sphere $\\mathbb{S}^n$ by taking $e_\\lambda$ to be the zonal functions or the highest weight spherical harmonics, which are point-focusing and geodesic-focusing eigenfunctions, respectively. Under additional geometric assumptions--such as nonpositive sectional curvature or on flat tori--one can improve \\eqref{r1}. See e.g., the work of Bourgain-Demeter \\cite[Section 2.3]{BoDe}, as well as  Germain-Myerson \\cite{germain2022bounds}, and the reference therein for related results on the torus. On compact manifolds with non-positive sectional curvature,\n$(\\log\\lambda)^{-\\frac12}$ improvements to \\eqref{r1} was obtained in B\\'erard \\cite{Berard} for $q=\\infty$, and by Hassell and Tacy \\cite{HassellTacy} for $q>q_c=\\tfrac{2(n+1)}{n-1}$. The same improvement also holds, more generally, for spectral projection operators onto the shorter interval $[\\lambda,\\lambda+(\\log\\lambda)^{-1}]$. These results were further generalized under weaker dynamical assumptions in the recent work of Canzani-Galkowski \\cite{CGGrowth}. \n\nIn the other direction, for $q\\le q_c$ similar improvements were obtained in Huang--Sogge \\cite{huang2024curvature}. More generally, they proved sharp quasimode estimates for logarithmic quasimodes, which were also used to characterize compact manifolds with constant curvature. See also Blair and Sogge \\cite{blair2018concerning,SBLog} and Blair, Huang and Sogge \\cite{blair2024improved} for related earlier work in this direction.\n\n\\subsection{\\texorpdfstring{$L^q$}{Lq}-norm bounds for Hecke-Maass forms}\nThe main result of the paper concerns improvements of \\eqref{r1} for arithmetic eigenfunctions on hyperbolic surfaces. We let $\\mathbb{H}$ be the upper half plane and $X=\\Gamma\\backslash\\mathbb{H}$ an arithmetic hyperbolic surface. Here $\\Gamma\\subset \\operatorname{SL}(2,\\mathbb{R})$ is taken to be the norm one units of a maximal order of an\nindefinite quaternion division algebra over $\\mathbb{Q}$ (see Section \\ref{sec: arith hyper sur} for the precise definitions) so that $X$ is compact. Let $\\psi$ be an $L^2$-normalized Hecke–Maass form on $X$. Namely, $\\int_X|\\psi(x)|^2dx=1$ and $\\psi$ is a joint eigenfunction of the Laplace-Beltrami operator $\\Delta_g$ and Hecke operators on $X$. Let $\\lambda$ be the spectral parameter of $\\psi$, so that $\\Delta \\psi+(\\frac14+\\lambda^2)\\psi=0$. As we are considering large-eigenvalue asymptotics, we will also assume that $\\lambda$ is real and $\\lambda\\gg1$. Iwaniec and Sarnak \\cite{IS95} made a Lindel\\\"of hypothesis type conjecture that $\\|\\psi\\|_{L^\\infty(X)}\\lesssim_\\varepsilon\\lambda^{\\varepsilon}$, which is strong and widely open. The conjecture implies that $\\|\\psi\\|_{L^q(X)}\\lesssim_\\varepsilon\\lambda^{\\varepsilon}$ for any $q>2$.\n\nFor these $X$ and $\\psi$, the first power-saving result was obtained by Iwaniec and Sarnak \\cite{IS95}. They showed that \n\\begin{align}\\label{eq: IS bound}\n    \\|\\psi\\|_{L^\\infty(X)} \\lesssim_\\varepsilon\\lambda^{\\frac{5}{12}+\\varepsilon},\n\\end{align}\nwhich is a $\\lambda^{\\frac1{12}}$ improvement over the uniform bound \\eqref{r1} for $n=2$ and $q=\\infty$. In two-dimensional cases, the critical exponent is $q_c=6$, by interpolation, \\eqref{eq: IS bound} implies an improvement for  $ \\| \\psi \\|_{L^q(X)}$ for all $q>6$. Their proof uses the technique known as arithmetic amplification. This approach has since been used by many authors to bound $L^\\infty$-norms of Hecke-Maass forms on other groups. For instance, see \\cite{BM13,Mar14,BHM16,BM16,BP16,BHM20} for results in the spectral aspect.\n\nFor $q<6$, Marshall \\cite{Mar16} studied the related $L^2$ geodesic restriction problem for the Hecke-Maass form $\\psi$ and obtained a similar $\\lambda$-power improvement over the uniform result of Burq,  G{\\'e}rard and Tzvetkov \\cite{BGTrestr}. We let $\\varPi$ denote the space of all unit-length geodesic segments on $X$. Marshall obtained the uniform bound\n\\begin{align}\\label{eq: Marshall geodesic bound}\n    \\sup_{\\gamma\\in\\varPi} \\| \\psi|_\\gamma \\|_{L^2(\\gamma)}\\lesssim_\\varepsilon \\lambda^{\\frac{3}{14}+\\varepsilon}.\n\\end{align}\nHe reduced $L^2$-norm bounds for $\\psi$ along $\\gamma\\in\\varPi$ to bounds for various\nFourier coefficients along $\\gamma$.\nHis improvements over the local bounds extend the technique of arithmetic amplification developed by Iwaniec-Sarnak.\nUsing a result of Blair and Sogge \\cite[Theorem 1.1]{blair2015refined}, the geodesic restriction bound \\eqref{eq: Marshall geodesic bound} implies that\n\\begin{align}\\label{eq: Marshall L4}\n    \\| \\psi \\|_{L^4(X)}\\lesssim_\\varepsilon \\lambda^{\\frac{1}{8}-\\frac{1}{56}+\\varepsilon}.\n\\end{align}\nMoreover, \\eqref{eq: Marshall L4} also implies an improvement over \\eqref{r1} for all $2<q<6$ by interpolation. See \\cite{hou2024restrictions,hou2025kakeya} for the extension of this method to improve bounds for Kakeya-Nikodym norms on other groups of rank one. For the higher rank group $\\operatorname{SL}(3,\\mathbb{R})$, \\eqref{eq: Marshall geodesic bound} was extended by Marshall to $L^2$ maximal flat restrictions of Hecke-Maass forms. It would be interesting to see whether this result could be used to prove a power saving for the global $L^p$-norms of $\\operatorname{SL}(3,\\mathbb{R})$ Hecke Maass forms for small $p$ as in Blair-Sogge \\cite{blair2015refined}.\n\nIt would be interesting to extend the above results and obtain a power saving over \\eqref{r1} for the critical exponent $q=6$. A much stronger $L^4$-norm bound than \\eqref{eq: Marshall L4} was obtained by Humphries and Khan \\cite[Theorem 1.4]{HK25}:\n\\begin{align}\\label{eq: HK L4}\n    \\| \\psi \\|_{L^4(X)}\\lesssim_\\varepsilon \\lambda^{\\frac{3}{152}+\\varepsilon}.\n\\end{align}\nThey applied Parseval’s identity to $|\\psi|^2$ and used the Watson–Ichino triple product formula to reduce bounds for $\\| \\psi \\|_{L^4(X)}$ to bounds for certain moments of $L$-functions. By interpolating with the $L^\\infty$-norm bound \\eqref{eq: IS bound} of Iwaniec-Sarnak, Humphries and Khan got\n\\begin{align}\\label{eq: HK L6}\n    \\| \\psi \\|_{L^6(X)}\\lesssim_\\varepsilon \\lambda^{\\frac{26}{171}+\\varepsilon}=\\lambda^{\\frac{1}{6}-\\frac{5}{342}+\\varepsilon}.\n\\end{align}\n\nOn the noncompact quotient $\\operatorname{SL}(2,\\mathbb{Z})\\backslash\\mathbb{H}$, for a Hecke-Maass cusp form $\\psi$ whose spectral parameter is $\\lambda$, both $L^\\infty$-norm bound \\eqref{eq: IS bound} and $L^4$-norm bound \\eqref{eq: HK L4} hold as well, so one still has the same $L^6$-norm bound \\eqref{eq: HK L6}. In \\cite{ki20234}, Ki showed an essentially sharp upper bound $\\|\\psi\\|_{L^4(\\operatorname{SL}(2,\\mathbb{Z})\\backslash\\mathbb{H})}\\lesssim_\\varepsilon\\lambda^{\\varepsilon}$, by using the\nFourier-Whittaker expansions of Hecke-Maass cusp forms. This result justifies Iwaniec-Sarnak's conjecture for $2<q\\leq4$, and therefore implies a better $L^6$-norm bound\n\\begin{align*}\n    \\| \\psi \\|_{L^6(\\operatorname{SL}(2,\\mathbb{Z})\\backslash\\mathbb{H})}\\lesssim_\\varepsilon \\lambda^{\\frac{1}{6}-\\frac{1}{36}+\\varepsilon}.\n\\end{align*}\n\nThe main result of the paper is the following sublocal $L^6$-norm bound in the \\textit{compact} setting, which is weaker than \\eqref{eq: HK L6} but is based on a completely different approach.", "context": "\\subsection{\\texorpdfstring{$L^q$}{Lq}-norm bounds for Hecke-Maass forms}\nThe main result of the paper concerns improvements of \\eqref{r1} for arithmetic eigenfunctions on hyperbolic surfaces. We let $\\mathbb{H}$ be the upper half plane and $X=\\Gamma\\backslash\\mathbb{H}$ an arithmetic hyperbolic surface. Here $\\Gamma\\subset \\operatorname{SL}(2,\\mathbb{R})$ is taken to be the norm one units of a maximal order of an\nindefinite quaternion division algebra over $\\mathbb{Q}$ (see Section \\ref{sec: arith hyper sur} for the precise definitions) so that $X$ is compact. Let $\\psi$ be an $L^2$-normalized Hecke–Maass form on $X$. Namely, $\\int_X|\\psi(x)|^2dx=1$ and $\\psi$ is a joint eigenfunction of the Laplace-Beltrami operator $\\Delta_g$ and Hecke operators on $X$. Let $\\lambda$ be the spectral parameter of $\\psi$, so that $\\Delta \\psi+(\\frac14+\\lambda^2)\\psi=0$. As we are considering large-eigenvalue asymptotics, we will also assume that $\\lambda$ is real and $\\lambda\\gg1$. Iwaniec and Sarnak \\cite{IS95} made a Lindel\\\"of hypothesis type conjecture that $\\|\\psi\\|_{L^\\infty(X)}\\lesssim_\\varepsilon\\lambda^{\\varepsilon}$, which is strong and widely open. The conjecture implies that $\\|\\psi\\|_{L^q(X)}\\lesssim_\\varepsilon\\lambda^{\\varepsilon}$ for any $q>2$.\n\nFor these $X$ and $\\psi$, the first power-saving result was obtained by Iwaniec and Sarnak \\cite{IS95}. They showed that \n\\begin{align}\\label{eq: IS bound}\n \\|\\psi\\|_{L^\\infty(X)} \\lesssim_\\varepsilon\\lambda^{\\frac{5}{12}+\\varepsilon},\n\\end{align}\nwhich is a $\\lambda^{\\frac1{12}}$ improvement over the uniform bound \\eqref{r1} for $n=2$ and $q=\\infty$. In two-dimensional cases, the critical exponent is $q_c=6$, by interpolation, \\eqref{eq: IS bound} implies an improvement for $ \\| \\psi \\|_{L^q(X)}$ for all $q>6$. Their proof uses the technique known as arithmetic amplification. This approach has since been used by many authors to bound $L^\\infty$-norms of Hecke-Maass forms on other groups. For instance, see \\cite{BM13,Mar14,BHM16,BM16,BP16,BHM20} for results in the spectral aspect.\n\nFor $q<6$, Marshall \\cite{Mar16} studied the related $L^2$ geodesic restriction problem for the Hecke-Maass form $\\psi$ and obtained a similar $\\lambda$-power improvement over the uniform result of Burq, G{\\'e}rard and Tzvetkov \\cite{BGTrestr}. We let $\\varPi$ denote the space of all unit-length geodesic segments on $X$. Marshall obtained the uniform bound\n\\begin{align}\\label{eq: Marshall geodesic bound}\n \\sup_{\\gamma\\in\\varPi} \\| \\psi|_\\gamma \\|_{L^2(\\gamma)}\\lesssim_\\varepsilon \\lambda^{\\frac{3}{14}+\\varepsilon}.\n\\end{align}\nHe reduced $L^2$-norm bounds for $\\psi$ along $\\gamma\\in\\varPi$ to bounds for various\nFourier coefficients along $\\gamma$.\nHis improvements over the local bounds extend the technique of arithmetic amplification developed by Iwaniec-Sarnak.\nUsing a result of Blair and Sogge \\cite[Theorem 1.1]{blair2015refined}, the geodesic restriction bound \\eqref{eq: Marshall geodesic bound} implies that\n\\begin{align}\\label{eq: Marshall L4}\n \\| \\psi \\|_{L^4(X)}\\lesssim_\\varepsilon \\lambda^{\\frac{1}{8}-\\frac{1}{56}+\\varepsilon}.\n\\end{align}\nMoreover, \\eqref{eq: Marshall L4} also implies an improvement over \\eqref{r1} for all $2<q<6$ by interpolation. See \\cite{hou2024restrictions,hou2025kakeya} for the extension of this method to improve bounds for Kakeya-Nikodym norms on other groups of rank one. For the higher rank group $\\operatorname{SL}(3,\\mathbb{R})$, \\eqref{eq: Marshall geodesic bound} was extended by Marshall to $L^2$ maximal flat restrictions of Hecke-Maass forms. It would be interesting to see whether this result could be used to prove a power saving for the global $L^p$-norms of $\\operatorname{SL}(3,\\mathbb{R})$ Hecke Maass forms for small $p$ as in Blair-Sogge \\cite{blair2015refined}.\n\nIt would be interesting to extend the above results and obtain a power saving over \\eqref{r1} for the critical exponent $q=6$. A much stronger $L^4$-norm bound than \\eqref{eq: Marshall L4} was obtained by Humphries and Khan \\cite[Theorem 1.4]{HK25}:\n\\begin{align}\\label{eq: HK L4}\n \\| \\psi \\|_{L^4(X)}\\lesssim_\\varepsilon \\lambda^{\\frac{3}{152}+\\varepsilon}.\n\\end{align}\nThey applied Parseval’s identity to $|\\psi|^2$ and used the Watson–Ichino triple product formula to reduce bounds for $\\| \\psi \\|_{L^4(X)}$ to bounds for certain moments of $L$-functions. By interpolating with the $L^\\infty$-norm bound \\eqref{eq: IS bound} of Iwaniec-Sarnak, Humphries and Khan got\n\\begin{align}\\label{eq: HK L6}\n \\| \\psi \\|_{L^6(X)}\\lesssim_\\varepsilon \\lambda^{\\frac{26}{171}+\\varepsilon}=\\lambda^{\\frac{1}{6}-\\frac{5}{342}+\\varepsilon}.\n\\end{align}\n\nOn the noncompact quotient $\\operatorname{SL}(2,\\mathbb{Z})\\backslash\\mathbb{H}$, for a Hecke-Maass cusp form $\\psi$ whose spectral parameter is $\\lambda$, both $L^\\infty$-norm bound \\eqref{eq: IS bound} and $L^4$-norm bound \\eqref{eq: HK L4} hold as well, so one still has the same $L^6$-norm bound \\eqref{eq: HK L6}. In \\cite{ki20234}, Ki showed an essentially sharp upper bound $\\|\\psi\\|_{L^4(\\operatorname{SL}(2,\\mathbb{Z})\\backslash\\mathbb{H})}\\lesssim_\\varepsilon\\lambda^{\\varepsilon}$, by using the\nFourier-Whittaker expansions of Hecke-Maass cusp forms. This result justifies Iwaniec-Sarnak's conjecture for $2<q\\leq4$, and therefore implies a better $L^6$-norm bound\n\\begin{align*}\n \\| \\psi \\|_{L^6(\\operatorname{SL}(2,\\mathbb{Z})\\backslash\\mathbb{H})}\\lesssim_\\varepsilon \\lambda^{\\frac{1}{6}-\\frac{1}{36}+\\varepsilon}.\n\\end{align*}\n\nThe main result of the paper is the following sublocal $L^6$-norm bound in the \\textit{compact} setting, which is weaker than \\eqref{eq: HK L6} but is based on a completely different approach.", "full_context": "\\subsection{\\texorpdfstring{$L^q$}{Lq}-norm bounds for Hecke-Maass forms}\nThe main result of the paper concerns improvements of \\eqref{r1} for arithmetic eigenfunctions on hyperbolic surfaces. We let $\\mathbb{H}$ be the upper half plane and $X=\\Gamma\\backslash\\mathbb{H}$ an arithmetic hyperbolic surface. Here $\\Gamma\\subset \\operatorname{SL}(2,\\mathbb{R})$ is taken to be the norm one units of a maximal order of an\nindefinite quaternion division algebra over $\\mathbb{Q}$ (see Section \\ref{sec: arith hyper sur} for the precise definitions) so that $X$ is compact. Let $\\psi$ be an $L^2$-normalized Hecke–Maass form on $X$. Namely, $\\int_X|\\psi(x)|^2dx=1$ and $\\psi$ is a joint eigenfunction of the Laplace-Beltrami operator $\\Delta_g$ and Hecke operators on $X$. Let $\\lambda$ be the spectral parameter of $\\psi$, so that $\\Delta \\psi+(\\frac14+\\lambda^2)\\psi=0$. As we are considering large-eigenvalue asymptotics, we will also assume that $\\lambda$ is real and $\\lambda\\gg1$. Iwaniec and Sarnak \\cite{IS95} made a Lindel\\\"of hypothesis type conjecture that $\\|\\psi\\|_{L^\\infty(X)}\\lesssim_\\varepsilon\\lambda^{\\varepsilon}$, which is strong and widely open. The conjecture implies that $\\|\\psi\\|_{L^q(X)}\\lesssim_\\varepsilon\\lambda^{\\varepsilon}$ for any $q>2$.\n\nFor these $X$ and $\\psi$, the first power-saving result was obtained by Iwaniec and Sarnak \\cite{IS95}. They showed that \n\\begin{align}\\label{eq: IS bound}\n \\|\\psi\\|_{L^\\infty(X)} \\lesssim_\\varepsilon\\lambda^{\\frac{5}{12}+\\varepsilon},\n\\end{align}\nwhich is a $\\lambda^{\\frac1{12}}$ improvement over the uniform bound \\eqref{r1} for $n=2$ and $q=\\infty$. In two-dimensional cases, the critical exponent is $q_c=6$, by interpolation, \\eqref{eq: IS bound} implies an improvement for $ \\| \\psi \\|_{L^q(X)}$ for all $q>6$. Their proof uses the technique known as arithmetic amplification. This approach has since been used by many authors to bound $L^\\infty$-norms of Hecke-Maass forms on other groups. For instance, see \\cite{BM13,Mar14,BHM16,BM16,BP16,BHM20} for results in the spectral aspect.\n\nFor $q<6$, Marshall \\cite{Mar16} studied the related $L^2$ geodesic restriction problem for the Hecke-Maass form $\\psi$ and obtained a similar $\\lambda$-power improvement over the uniform result of Burq, G{\\'e}rard and Tzvetkov \\cite{BGTrestr}. We let $\\varPi$ denote the space of all unit-length geodesic segments on $X$. Marshall obtained the uniform bound\n\\begin{align}\\label{eq: Marshall geodesic bound}\n \\sup_{\\gamma\\in\\varPi} \\| \\psi|_\\gamma \\|_{L^2(\\gamma)}\\lesssim_\\varepsilon \\lambda^{\\frac{3}{14}+\\varepsilon}.\n\\end{align}\nHe reduced $L^2$-norm bounds for $\\psi$ along $\\gamma\\in\\varPi$ to bounds for various\nFourier coefficients along $\\gamma$.\nHis improvements over the local bounds extend the technique of arithmetic amplification developed by Iwaniec-Sarnak.\nUsing a result of Blair and Sogge \\cite[Theorem 1.1]{blair2015refined}, the geodesic restriction bound \\eqref{eq: Marshall geodesic bound} implies that\n\\begin{align}\\label{eq: Marshall L4}\n \\| \\psi \\|_{L^4(X)}\\lesssim_\\varepsilon \\lambda^{\\frac{1}{8}-\\frac{1}{56}+\\varepsilon}.\n\\end{align}\nMoreover, \\eqref{eq: Marshall L4} also implies an improvement over \\eqref{r1} for all $2<q<6$ by interpolation. See \\cite{hou2024restrictions,hou2025kakeya} for the extension of this method to improve bounds for Kakeya-Nikodym norms on other groups of rank one. For the higher rank group $\\operatorname{SL}(3,\\mathbb{R})$, \\eqref{eq: Marshall geodesic bound} was extended by Marshall to $L^2$ maximal flat restrictions of Hecke-Maass forms. It would be interesting to see whether this result could be used to prove a power saving for the global $L^p$-norms of $\\operatorname{SL}(3,\\mathbb{R})$ Hecke Maass forms for small $p$ as in Blair-Sogge \\cite{blair2015refined}.\n\nIt would be interesting to extend the above results and obtain a power saving over \\eqref{r1} for the critical exponent $q=6$. A much stronger $L^4$-norm bound than \\eqref{eq: Marshall L4} was obtained by Humphries and Khan \\cite[Theorem 1.4]{HK25}:\n\\begin{align}\\label{eq: HK L4}\n \\| \\psi \\|_{L^4(X)}\\lesssim_\\varepsilon \\lambda^{\\frac{3}{152}+\\varepsilon}.\n\\end{align}\nThey applied Parseval’s identity to $|\\psi|^2$ and used the Watson–Ichino triple product formula to reduce bounds for $\\| \\psi \\|_{L^4(X)}$ to bounds for certain moments of $L$-functions. By interpolating with the $L^\\infty$-norm bound \\eqref{eq: IS bound} of Iwaniec-Sarnak, Humphries and Khan got\n\\begin{align}\\label{eq: HK L6}\n \\| \\psi \\|_{L^6(X)}\\lesssim_\\varepsilon \\lambda^{\\frac{26}{171}+\\varepsilon}=\\lambda^{\\frac{1}{6}-\\frac{5}{342}+\\varepsilon}.\n\\end{align}\n\nOn the noncompact quotient $\\operatorname{SL}(2,\\mathbb{Z})\\backslash\\mathbb{H}$, for a Hecke-Maass cusp form $\\psi$ whose spectral parameter is $\\lambda$, both $L^\\infty$-norm bound \\eqref{eq: IS bound} and $L^4$-norm bound \\eqref{eq: HK L4} hold as well, so one still has the same $L^6$-norm bound \\eqref{eq: HK L6}. In \\cite{ki20234}, Ki showed an essentially sharp upper bound $\\|\\psi\\|_{L^4(\\operatorname{SL}(2,\\mathbb{Z})\\backslash\\mathbb{H})}\\lesssim_\\varepsilon\\lambda^{\\varepsilon}$, by using the\nFourier-Whittaker expansions of Hecke-Maass cusp forms. This result justifies Iwaniec-Sarnak's conjecture for $2<q\\leq4$, and therefore implies a better $L^6$-norm bound\n\\begin{align*}\n \\| \\psi \\|_{L^6(\\operatorname{SL}(2,\\mathbb{Z})\\backslash\\mathbb{H})}\\lesssim_\\varepsilon \\lambda^{\\frac{1}{6}-\\frac{1}{36}+\\varepsilon}.\n\\end{align*}\n\nThe main result of the paper is the following sublocal $L^6$-norm bound in the \\textit{compact} setting, which is weaker than \\eqref{eq: HK L6} but is based on a completely different approach.\n\nFor these $X$ and $\\psi$, the first power-saving result was obtained by Iwaniec and Sarnak \\cite{IS95}. They showed that \n\\begin{align}\\label{eq: IS bound}\n \\|\\psi\\|_{L^\\infty(X)} \\lesssim_\\epsilon\\lambda^{\\frac{5}{12}+\\epsilon},\n\\end{align}\nwhich is a $\\la^{\\frac1{12}}$ improvement over the uniform bound \\eqref{r1} for $n=2$ and $q=\\infty$. In two-dimensional cases, the critical exponent is $q_c=6$, by interpolation, \\eqref{eq: IS bound} implies an improvement for $ \\| \\psi \\|_{L^q(X)}$ for all $q>6$. Their proof uses the technique known as arithmetic amplification. This approach has since been used by many authors to bound $L^\\infty$-norms of Hecke-Maass forms on other groups. For instance, see \\cite{BM13,Mar14,BHM16,BM16,BP16,BHM20} for results in the spectral aspect.\n\nOn the noncompact quotient $\\SL(2,\\BZ)\\backslash\\BH$, for a Hecke-Maass cusp form $\\psi$ whose spectral parameter is $\\la$, both $L^\\infty$-norm bound \\eqref{eq: IS bound} and $L^4$-norm bound \\eqref{eq: HK L4} hold as well, so one still has the same $L^6$-norm bound \\eqref{eq: HK L6}. In \\cite{ki20234}, Ki showed an essentially sharp upper bound $\\|\\psi\\|_{L^4(\\SL(2,\\BZ)\\backslash\\BH)}\\lesssim_\\epsilon\\la^{\\epsilon}$, by using the\nFourier-Whittaker expansions of Hecke-Maass cusp forms. This result justifies Iwaniec-Sarnak's conjecture for $2<q\\leq4$, and therefore implies a better $L^6$-norm bound\n\\begin{align*}\n \\| \\psi \\|_{L^6(\\SL(2,\\BZ)\\backslash\\BH)}\\lesssim_\\epsilon \\la^{\\frac{1}{6}-\\frac{1}{36}+\\epsilon}.\n\\end{align*}\n\n\\subsection{Proof strategy}\nA main step in the proof of Theorem~\\ref{thm1} is the following.\n\\begin{theorem}\\label{thm2}\n Let $(M,g)$ be a compact $2$-dimensional Riemannian manifold without boundary. Let $e_\\la$ be an $L^2$ normalized eigenfunction of $-\\Delta_g$ with eigenvalue $\\la^2$, satisfying \\eqref{1.2} for some $\\la\\gg1$. Set $\\theta_0=\\la^{-\\delta_1}$ with $0<\\delta_1< \\frac12$, and let $A_\\nu^{\\theta_0}$ be the pseudo-differential operator defined in \\eqref{qnusymbol}. Assume that \n\\begin{equation}\\label{i3}\n \\sup_\\nu\\|A_\\nu^{\\theta_0} e_\\la\\|_{L^6(M)}\\lesssim_\\e \\la^{\\frac16-a{\\delta_1}+\\e},\n \\end{equation}\n and \n \\begin{equation}\\label{i4}\n \\|e_\\la\\|_{L^\\infty(M)}\\lesssim_\\e \\la^{\\frac12-\\delta_2+\\e}\n \\end{equation}\nfor some constant $a>0$ and $0<\\delta_2<\\frac{1}{2}$. Then \n \\begin{equation}\\label{i5}\n \\|e_\\la\\|_{L^6(M)}\\lesssim_\\e \\la^{\\frac16-\\frac23a{\\delta_1}+\\e}+\\la^{\\frac23(-\\frac14+\\frac53\\delta_1)}+\\la^{\\frac16(1+\\delta_1-2\\delta_2)+\\e}.\n \\end{equation}\n\\end{theorem}\nFor each fixed $\\nu$, the operator $A_\\nu^{\\theta_0}$ is microlocally supported in a $\\theta_0$-neighborhood of a geodesic segment and is frequency-localized to directions near that geodesic; the precise definition will be given in Section \\ref{sec: microlocal decomposition}. The proof of Theorem~\\ref{thm2} follows a strategy similar to that of \\cite{huang2024curvature}, relying on bilinear techniques from harmonic analysis together with the improved $L^\\infty$ estimate \\eqref{i4}. We will prove Theorem~\\ref{thm2} in Section \\ref{sec: 2}. \nThe idea of using improved microlocal $L^6$-norm estimate to derive improved global $L^6$-norm estimate is inspired by the recent work \\cite{blair2024improved}.\n\n\\begin{theorem}\\label{thm microlocal KN}\n Let $\\psi$ be a Hecke–Maass form on $X$ with spectral parameter $\\lambda\\gg1$. Set $\\theta_0=\\la^{-\\delta_1}$ with $0<\\delta_1<\\frac12$, and let $A_\\nu^{\\theta_0}$ be the pseudo-differential operator defined in \\eqref{qnusymbol}. We have\n \\begin{equation*}\n \\sup_\\nu\\|A_\\nu^{\\theta_0} \\psi\\|_{L^6(X)}\\lesssim_\\e \\la^{\\frac16-\\frac{\\delta_1}{12}+\\e}.\n \\end{equation*}\n\\end{theorem}\n\nWe let $\\Delta_g$ be the Laplace-Beltrami operator on $\\BH$ and $X$, which is induced from the standard hyperbolic Riemannian metric. Let $\\psi\\in L^2(X)$ be a Hecke-Maass form that is an eigenfunction of $\\Delta_g$ and the operators $T_n$ with $(n,q)=1$. We let $\\lambda(n)$ be the Hecke eigenvalues of $\\psi$ and $\\lambda$ be its spectral parameter, so that\n\\begin{equation}\\label{def}\n \\begin{aligned}\n &T_n\\psi=\\lambda(n)\\psi,\\\\\n &\\Delta_g\\psi+(\\frac{1}{4}+\\lambda^2)\\psi=0. \n \\end{aligned}\n\\end{equation}\nWe assume that $\\| \\psi \\|_{L^2(X)}=1$ with respect to the hyperbolic volume on $X$ and assume that $\\lambda\\gg1$. Note that because $\\Delta_g$ and $T_n$ with $(n,q)=1$ are self-adjoint, we may assume that $\\psi$ is real-valued. For functions on $X$, we will also think of them as functions on $G$ that are left $\\Gamma$-invariant and right $K$-invariant.\n\n\\begin{proof}[Proof of Theorem \\ref{thm microlocal KN}]\n By Lemma \\ref{lem 3.10}, we may bound $\\| \\sA\\psi\\|_{L^6(X)}$ instead, and moreover by duality, it suffices to bound $\\langle\\sA\\psi,\\phi\\rangle_X$ uniformly for any $\\phi\\in C^\\infty(X)$ with $\\|\\phi\\|_{L^{6/5}(X)}=1$.\n By Proposition \\ref{prop: amplification inequality}, we have\n \\begin{align}\\label{eq: amplification inequality}\n \\left|\\langle \\sA\\cT\\psi,\\phi\\rangle_X\\right|^2\\leq\\sum_{m,n\\leq N} |\\alpha_m \\alpha_n | \\sum_{d|(m,n)} \\frac{d}{\\sqrt{mn}}\\sum_{\\gamma\\in R(mn/d^2)} |I_\\sA(\\la,\\phi,\\gamma )|.\n \\end{align}\n Fix $0<\\epsilon_0\\ll\\delta_1$. By assuming the support of $k_\\la$ and $\\Omega$ are sufficiently small, and by Proposition \\ref{prop: bound for I_A}, we only need to consider the terms in \\eqref{eq: amplification inequality} with $d(g_0^{-1}\\gamma g_0,e)\\leq1$ and $d(g_0^{-1}\\gamma g_0,A)\\leq\\lambda^{-\\delta_1+2\\epsilon_0}$. Lemma \\ref{lem: Hecke return} gives\n \\[\n M(g_0,n,\\lambda^{-\\delta_1 +2\\epsilon_0})\\ll_\\epsilon n^\\epsilon \\lambda^{\\epsilon_0+\\epsilon}(n\\lambda^{-\\delta_1/2}+1),\n \\]\n and so by the uniform estimate $|I_\\sA(\\la,\\phi,\\gamma )|\\lesssim_\\epsilon \\la^{1/3+\\epsilon}$ from Proposition \\ref{prop: bound for I_A} we have\n \\begin{align*}\n &\\sum_{m,n\\leq N} |\\alpha_m\\alpha_n| \\sum_{d|(m,n)} \\frac{d}{\\sqrt{mn}}\\sum_{\\gamma\\in R(mn/d^2)}\\left| I_{\\sA}(\\lambda, \\phi, \\gamma ) \\right|\\\\\n \\lesssim& \\sum_{m,n\\leq N} |\\alpha_m\\alpha_n| \\sum_{d|(m,n)} \\frac{d}{\\sqrt{mn}}\\lambda^{1/3+\\e} M(g_0,\\frac{mn}{d^2},\\lambda^{-\\delta_1+2\\epsilon_0})+O_{\\epsilon_0,A}(\\lambda^{-A})\\\\\n \\lesssim&_\\epsilon N^\\epsilon\\lambda^{\\epsilon_0+\\epsilon}\\sum_{m,n\\leq N} |\\alpha_m\\alpha_n| \\sum_{d|(m,n)} \\left(\\frac{\\sqrt{mn}}{d}\\lambda^{1/3-\\delta_1/2}+\\frac{d}{\\sqrt{mn}}\\lambda^{1/3}\\right) +O_{\\epsilon_0,A}(\\lambda^{-A}).\n \\end{align*}\n As in \\cite[p.310]{IS95}, we have\n \\begin{align*}\n \\sum_{m,n\\leq N} \\sum_{d|(m,n)}|\\alpha_m\\alpha_n|\\frac{\\sqrt{mn}}{d}\\lesssim_\\epsilon N^{1+\\epsilon} \\left( \\sum_{n\\leq N} |\\alpha_n|\\right)^2\n \\end{align*}\n and\n \\begin{align*}\n \\sum_{m,n\\leq N} \\sum_{d|(m,n)}|\\alpha_m\\alpha_n|\\frac{d}{\\sqrt{mn}}&=\\sum_{\\substack{m,n\\leq N\\\\(m,n)=1}} \\sum_{ml,nl\\leq N}\\sum_{d|l}|\\alpha_{ml}\\alpha_{nl}|\\frac{d}{l\\sqrt{mn}}\\\\\n &\\lesssim_\\epsilon N^\\epsilon\\sum_{ml,nl\\leq N}\\left( \\frac{|\\alpha_{ml}|^2}{n} + \\frac{|\\alpha_{nl}|^2}{m} \\right)\\\\\n &\\lesssim_\\epsilon N^\\epsilon \\sum_{n\\leq N} |\\alpha_n|^2.\n \\end{align*}\n Hence from \\eqref{eq: amplification inequality} we have\n \\begin{align*}\n \\left| \\langle \\sA\\cT\\psi,\\phi\\rangle_X \\right|^2 \\lesssim&_\\epsilon N^\\epsilon\\lambda^{\\epsilon_0+\\epsilon}\\left(N\\lambda^{1/3-\\delta_1/2}\\left( \\sum_{n\\leq N} |\\alpha_n|\\right)^2+\\lambda^{1/3}\\sum_{n\\leq N} |\\alpha_n|^2\\right) +O_{\\epsilon_0,A}(\\lambda^{-A}).\n \\end{align*}\n Combining this with \\eqref{eq: moment for alpha} and \\eqref{eq: lower bound for amplifier eigenvalue} and choosing $\\epsilon_0$ small gives\n \\begin{align*}\n N^{1-\\epsilon}\\left|\\langle \\sA\\psi,\\phi\\rangle_X \\right|^2\\lesssim&_\\epsilon N^\\epsilon\\lambda^{\\epsilon}\\left(N^2\\lambda^{1/3-\\delta_1/2}+N^{1/2}\\lambda^{1/3}\\right)+O_{\\epsilon,A}(\\lambda^{-A}),\n \\end{align*}\n which completes the proof by choosing $N=\\lambda^{\\delta_1/3}$.\n\\end{proof}", "post_theorem_intro_text_len": 7172, "post_theorem_intro_text": "\\subsection{Proof strategy}\nA main step in the proof of Theorem~\\ref{thm1} is the following.\n\\begin{theorem}\\label{thm2}\n    Let $(M,g)$ be a compact $2$-dimensional Riemannian manifold without boundary. Let $e_\\lambda$   be an $L^2$ normalized eigenfunction of  $-\\Delta_g$ with eigenvalue $\\lambda^2$, satisfying \\eqref{1.2} for some $\\lambda\\gg1$. Set $\\theta_0=\\lambda^{-\\delta_1}$ with $0<\\delta_1< \\frac12$, and let $A_\\nu^{\\theta_0}$ be the pseudo-differential operator defined in \\eqref{qnusymbol}. Assume that \n\\begin{equation}\\label{i3}\n        \\sup_\\nu\\|A_\\nu^{\\theta_0} e_\\lambda\\|_{L^6(M)}\\lesssim_\\varepsilon \\lambda^{\\frac16-a{\\delta_1}+\\varepsilon},\n    \\end{equation}\n    and \n        \\begin{equation}\\label{i4}\n        \\|e_\\lambda\\|_{L^\\infty(M)}\\lesssim_\\varepsilon \\lambda^{\\frac12-\\delta_2+\\varepsilon}\n    \\end{equation}\nfor some constant $a>0$ and $0<\\delta_2<\\frac{1}{2}$. Then \n \\begin{equation}\\label{i5}\n        \\|e_\\lambda\\|_{L^6(M)}\\lesssim_\\varepsilon \\lambda^{\\frac16-\\frac23a{\\delta_1}+\\varepsilon}+\\lambda^{\\frac23(-\\frac14+\\frac53\\delta_1)}+\\lambda^{\\frac16(1+\\delta_1-2\\delta_2)+\\varepsilon}.\n    \\end{equation}\n\\end{theorem}\nFor each fixed $\\nu$, the operator $A_\\nu^{\\theta_0}$ is microlocally supported in a $\\theta_0$-neighborhood of a geodesic segment and is frequency-localized to directions near that geodesic; the precise definition will be given in Section \\ref{sec: microlocal decomposition}. The proof of Theorem~\\ref{thm2} follows a strategy similar to that of \\cite{huang2024curvature}, relying on bilinear techniques from harmonic analysis together with the improved $L^\\infty$ estimate \\eqref{i4}. We will prove Theorem~\\ref{thm2} in Section \\ref{sec: 2}. \nThe idea of using improved microlocal $L^6$-norm estimate to derive improved global $L^6$-norm estimate is inspired by the recent work \\cite{blair2024improved}.\n\nNote that for a Hecke–Maass form $\\psi$ on $X$, as in Theorem \\ref{thm1}, the bound \\eqref{i4} holds with $\\delta_2=\\tfrac{1}{12}$ by Iwaniec-Sarnak's $L^\\infty$-norm bound \\eqref{eq: IS bound}. If, in addition, \\eqref{i3} holds for such $X$ and $\\psi$ with $a=\\frac1{12}$, then Theorem~\\ref{thm1} follows immediately by choosing $\\delta_1=\\frac{3}{2}\\delta_2=\\tfrac{1}{8}$.\n\n\\begin{theorem}\\label{thm microlocal KN}\n    Let $\\psi$ be a Hecke–Maass form on $X$ with spectral parameter $\\lambda\\gg1$. Set $\\theta_0=\\lambda^{-\\delta_1}$ with $0<\\delta_1<\\frac12$, and let $A_\\nu^{\\theta_0}$ be the pseudo-differential operator defined in  \\eqref{qnusymbol}. We have\n    \\begin{equation*}\n        \\sup_\\nu\\|A_\\nu^{\\theta_0} \\psi\\|_{L^6(X)}\\lesssim_\\varepsilon \\lambda^{\\frac16-\\frac{\\delta_1}{12}+\\varepsilon}.\n    \\end{equation*}\n\\end{theorem}\n\nWe will prove Theorem \\ref{thm microlocal KN} in Section \\ref{sec 3}. This theorem can be viewed as the $L^6$-version of Theorem 1.1 and Lemma 3.9 in \\cite{Mar16}, where Marshall used the amplification method to bound the $L^2$-mass of $\\psi$ restricted to a geodesic segment with frequency localized. We give the amplification inequality in Proposition \\ref{prop: amplification inequality} by integrating the amplified pretrace formula against a test function under the operator $A_\\nu^{\\theta_0}$. Let $\\gamma$ be the geodesic segment centered in $A_\\nu^{\\theta_0}$.\nSimilar to usual amplification arguments, we have a counting problem and an analytic problem. The counting problem is to estimate the number of times a Hecke operator maps $\\gamma$ back close to itself, which has been studied by Marshall \\cite[Section 3.2]{Mar16}. Using the microlocal support of $A_\\nu^{\\theta_0}$, we can reduce the analytic problem to several known results from \\cite{sogge881,Mar16,Mar16HigerRank}, which will be proved in Proposition \\ref{prop: bound for I_A}.\n\n\\subsection{Generalizations}\nIn Theorem \\ref{thm1}, the arithmetic assumption that $\\psi$ is a Hecke eigenfunction is only used to invoke the bound Iwaniec-Sarnak \\cite{IS95} and to prove Theorem \\ref{thm microlocal KN}, both of which rely on arithmetic amplification. Hence, the maximal order in the theorem can be replaced with an Eichler order more generally, as in \\cite{IS95}. Moreover, we use only unramified Hecke operators, and we may assume $\\psi$ to be an eigenfunction under even fewer Hecke operators as long as the amplification argument can be adapted.\n\nThe strategy developed in this paper can also be generalized to study the $L^{q_c}$-norm problems for Hecke-Maass forms on compact locally symmetric spaces of rank one, as long as the corresponding $L^\\infty$-norm problem and microlocal $L^{q_c}$ Kakeya-Nikodym problem can be solved by the arithmetic amplification technique; we will explore this direction in future work.\nTheorem \\ref{thm2} is expected to be generalized to Laplace-Beltrami eigenfunctions on compact Riemannian manifolds of any dimension, which allows us to control the $L^{q_c}$-norms by the $L^\\infty$-norms and the microlocal Kakeya-Nikodym norms. \nFor example, when the manifold is taken to be a compact arithmetic hyperbolic 3-manifold (the critical exponent is $q_c=4$), the corresponding $L^\\infty$-norm problem for Hecke-Maass forms is expected to be solved (see \\cite{BHM16} in the cases of noncompact arithmetic\nhyperbolic 3-manifolds with cusps). The corresponding microlocal $L^4$ Kakeya-Nikodym estimate should be established in a similar way as Theorem \\ref{thm microlocal KN}, by modifying the proof for the usual $L^2$ Kakeya-Nikodym estimate in \\cite{hou2024restrictions}.\n\n\\subsection{Notation}\nThroughout the paper, the notation $A\\lesssim B$ means that there is a positive constant $C$ such that $|A| \\leq C B$, and $A\\sim B$ means that $A\\lesssim B\\lesssim A$. We also use $A =O(B)$ to mean $A\\lesssim B$. The notation  $A\\gg B$ means there is a sufficiently large positive constant $C$ such that $A\\ge CB$, and similarly for $A\\ll B$.\n\nIf $f\\in L^1(\\mathbb{R}^n)$, the Fourier transform $\\hat{f}$ of $f$ in this paper is defined as\n\\begin{align*}\n    \\hat{f}(\\xi)=\\int_{\\mathbb{R}^n} f(x)e^{-i\\langle x,\\xi\\rangle} dx. \n\\end{align*}\nHere $x=(x_i),\\xi=(\\xi_i)\\in\\mathbb{R}^n$ and $\\langle x,\\xi\\rangle=\\sum_i x_i\\xi_i$.\nIf $f\\in\\mathcal{S}(\\mathbb{R}^n)$ is a Schwartz function, the Fourier inversion formula reads\n\\[\n    f(x) = (2\\pi)^{-n}\\int_{\\mathbb{R}^n}\\hat{f}(\\xi)e^{i\\langle x,\\xi\\rangle} d\\xi.\n\\]\nWe say a symbol $p(x,\\xi)$ is in the class $S^m_{\\rho,\\delta}$ if\\[\n|\\partial_x^\\beta\\partial^\\alpha_\\xi p(x,\\xi)|\\lesssim_{\\alpha,\\beta} \\langle\\xi\\rangle^{m-\\rho|\\alpha|+\\delta|\\beta|},\n\\]\nwhere $\\langle\\xi\\rangle=(1+|\\xi|^2)^{1/2}$ is the Japanese bracket.\nThe pseudo-differential operator $p(x,D)$ is defined by the integral\n\\[\np(x,D)f(x)=(2\\pi)^{-n}\\int p(x,\\xi)\\hat{f}(\\xi) e^{i\\langle x,\\xi\\rangle}d\\xi=(2\\pi)^{-n}\\iint p(x,\\xi)f(y) e^{i\\langle x-y,\\xi\\rangle}d\\xi dy.\n\\]\nWe say $P$ is the pseudo-differential operator with the compound symbol $p(x,y,\\xi)$ if\n\\[Pf(x)=(2\\pi)^{-n}\\iint p(x,y,\\xi)f(y) e^{i\\langle x-y,\\xi\\rangle}d\\xi dy.\\]\n\n\\subsection{Acknowledgements}\nThe authors would like to thank Simon Marshall for helpful comments.  The second author was supported in part by the Simons Foundation and NSF (DMS-2452860). \n\\bigskip", "sketch": "A stated proof strategy for Theorem~\\ref{thm1} is:\n\n- Reduce Theorem~\\ref{thm1} to Theorem~\\ref{thm2}: “A main step in the proof of Theorem~\\ref{thm1} is the following” theorem (Theorem~\\ref{thm2}), which turns (i) a microlocal $L^6$ bound of the form \\eqref{i3} for $A_\\nu^{\\theta_0}e_\\lambda$ and (ii) an improved $L^\\infty$ bound \\eqref{i4} into a global $L^6$ bound \\eqref{i5}. Here, “for each fixed $\\nu$, the operator $A_\\nu^{\\theta_0}$ is microlocally supported in a $\\theta_0$-neighborhood of a geodesic segment and is frequency-localized to directions near that geodesic.”\n\n- Establish Theorem~\\ref{thm2} by “a strategy similar to that of \\cite{huang2024curvature}, relying on bilinear techniques from harmonic analysis together with the improved $L^\\infty$ estimate \\eqref{i4}.”\n\n- For Hecke–Maass forms $\\psi$ on $X$ (as in Theorem~\\ref{thm1}), use that \\eqref{i4} holds “with $\\delta_2=\\tfrac{1}{12}$ by Iwaniec-Sarnak’s $L^\\infty$-norm bound.” Then prove the needed microlocal estimate \\eqref{i3} via Theorem~\\ref{thm microlocal KN}, which gives\n  \\[\\sup_\\nu\\|A_\\nu^{\\theta_0} \\psi\\|_{L^6(X)}\\lesssim_\\varepsilon \\lambda^{\\frac16-\\frac{\\delta_1}{12}+\\varepsilon},\\]\n  i.e. \\eqref{i3} “with $a=\\frac1{12}$.”\n\n- Conclude Theorem~\\ref{thm1} by parameter choice: “If, in addition, \\eqref{i3} holds for such $X$ and $\\psi$ with $a=\\frac1{12}$, then Theorem~\\ref{thm1} follows immediately by choosing $\\delta_1=\\frac{3}{2}\\delta_2=\\tfrac{1}{8}$.”\n\n- Proof approach for Theorem~\\ref{thm microlocal KN} (supplying \\eqref{i3}): it “can be viewed as the $L^6$-version” of Marshall’s microlocal restriction results; the authors “give the amplification inequality … by integrating the amplified pretrace formula against a test function under the operator $A_\\nu^{\\theta_0}$.” As in “usual amplification arguments,” there is “a counting problem” (how often a Hecke operator maps the relevant geodesic segment $\\gamma$ back close to itself) and “an analytic problem,” and “using the microlocal support of $A_\\nu^{\\theta_0}$, we can reduce the analytic problem to several known results from \\cite{sogge881,Mar16,Mar16HigerRank}.”", "expanded_sketch": "A stated proof strategy for the main theorem is:\n\n- Reduce the main theorem to the following auxiliary theorem.\n\n\\begin{theorem}\\label{thm2}\n    Let $(M,g)$ be a compact $2$-dimensional Riemannian manifold without boundary. Let $e_\\la$   be an $L^2$ normalized eigenfunction of  $-\\Delta_g$ with eigenvalue $\\la^2$, satisfying \\eqref{1.2} for some $\\la\\gg1$. Set $\\theta_0=\\la^{-\\delta_1}$ with $0<\\delta_1< \\frac12$, and let $A_\\nu^{\\theta_0}$ be the pseudo-differential operator defined in \\eqref{qnusymbol}. Assume that \n\\begin{equation}\\label{i3}\n        \\sup_\\nu\\|A_\\nu^{\\theta_0} e_\\la\\|_{L^6(M)}\\lesssim_\\e \\la^{\\frac16-a{\\delta_1}+\\e},\n    \\end{equation}\n    and \n        \\begin{equation}\\label{i4}\n        \\|e_\\la\\|_{L^\\infty(M)}\\lesssim_\\e \\la^{\\frac12-\\delta_2+\\e}\n    \\end{equation}\nfor some constant $a>0$ and $0<\\delta_2<\\frac{1}{2}$. Then \n \\begin{equation}\\label{i5}\n        \\|e_\\la\\|_{L^6(M)}\\lesssim_\\e \\la^{\\frac16-\\frac23a{\\delta_1}+\\e}+\\la^{\\frac23(-\\frac14+\\frac53\\delta_1)}+\\la^{\\frac16(1+\\delta_1-2\\delta_2)+\\e}.\n    \\end{equation}\n\\end{theorem}\n\nHere, “for each fixed $\\nu$, the operator $A_\\nu^{\\theta_0}$ is microlocally supported in a $\\theta_0$-neighborhood of a geodesic segment and is frequency-localized to directions near that geodesic.”\n\n- Establish the auxiliary theorem by “a strategy similar to that of \\cite{huang2024curvature}, relying on bilinear techniques from harmonic analysis together with the improved $L^\\infty$ estimate\n\\begin{equation}\\label{i4}\n        \\|e_\\la\\|_{L^\\infty(M)}\\lesssim_\\e \\la^{\\frac12-\\delta_2+\\e}\n    \\end{equation}.”\n\n- For Hecke–Maass forms $\\psi$ on $X$ (as in the main theorem), use that the preceding estimate holds “with $\\delta_2=\\tfrac{1}{12}$ by Iwaniec-Sarnak’s $L^\\infty$-norm bound.” Then prove the needed microlocal estimate\n\\begin{equation}\\label{i3}\n        \\sup_\\nu\\|A_\\nu^{\\theta_0} e_\\la\\|_{L^6(M)}\\lesssim_\\e \\la^{\\frac16-a{\\delta_1}+\\e},\n    \\end{equation}\nvia the following theorem.\n\n\\begin{theorem}\\label{thm microlocal KN}\n    Let $\\psi$ be a Hecke\u0013Maass form on $X$ with spectral parameter $\\lambda\\gg1$. Set $\\theta_0=\\la^{-\\delta_1}$ with $0<\\delta_1<\\frac12$, and let $A_\\nu^{\\theta_0}$ be the pseudo-differential operator defined in  \\eqref{qnusymbol}. We have\n    \\begin{equation*}\n        \\sup_\\nu\\|A_\\nu^{\\theta_0} \\psi\\|_{L^6(X)}\\lesssim_\\e \\la^{\\frac16-\\frac{\\delta_1}{12}+\\e}.\n    \\end{equation*}\n\\end{theorem}\n\nThis gives\n\\[\\sup_\\nu\\|A_\\nu^{\\theta_0} \\psi\\|_{L^6(X)}\\lesssim_\\varepsilon \\lambda^{\\frac16-\\frac{\\delta_1}{12}+\\varepsilon},\\]\nthat is, the microlocal bound above “with $a=\\frac1{12}$.”\n\n- Conclude the main theorem by parameter choice: “If, in addition, the microlocal bound above holds for such $X$ and $\\psi$ with $a=\\frac1{12}$, then the main theorem follows immediately by choosing $\\delta_1=\\frac{3}{2}\\delta_2=\\tfrac{1}{8}$.”\n\n- Proof approach for the preceding microlocal theorem (supplying the microlocal bound): it “can be viewed as the $L^6$-version” of Marshall’s microlocal restriction results; the authors “give the amplification inequality … by integrating the amplified pretrace formula against a test function under the operator $A_\\nu^{\\theta_0}$.” As in “usual amplification arguments,” there is “a counting problem” (how often a Hecke operator maps the relevant geodesic segment $\\gamma$ back close to itself) and “an analytic problem,” and “using the microlocal support of $A_\\nu^{\\theta_0}$, we can reduce the analytic problem to several known results from \\cite{sogge881,Mar16,Mar16HigerRank}.”", "expanded_theorem": "\\label{thm1}\n    Let $\\psi$ be a Hecke–Maass form on $X$ with spectral parameter $\\lambda\\gg1$. We have\n    \\begin{equation}\\label{i2}\n        \\| \\psi \\|_{L^6(X)} \\lesssim_\\varepsilon\n        \\lambda^{\\frac{1}{6}-\\frac{1}{144}+\\varepsilon}.\n    \\end{equation}", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let $X=\\Gamma\\backslash\\mathbb H$ be the compact arithmetic hyperbolic surface obtained from the upper half-plane $\\mathbb H$ by quotienting by a group $\\Gamma\\subset \\mathrm{SL}(2,\\mathbb R)$ coming from the norm-one units of a maximal order in an indefinite quaternion division algebra over $\\mathbb Q$. Let $\\psi$ be an $L^2$-normalized Hecke--Maass form on $X$, meaning that $\\int_X |\\psi(x)|^2\\,dx=1$ and $\\psi$ is a joint eigenfunction of the Laplace--Beltrami operator and the Hecke operators on $X$. If $\\psi$ has spectral parameter $\\lambda\\gg 1$, i.e. it satisfies\n\\[\n\\Delta \\psi + \\left(\\tfrac14+\\lambda^2\\right)\\psi=0,\n\\]\nwhich statement holds for every such $\\psi$?", "correct_choice": {"label": "A", "text": "For every $\\varepsilon>0$, one has\n\\[\n\\|\\psi\\|_{L^6(X)} \\lesssim_{\\varepsilon} \\lambda^{\\frac16-\\frac{1}{144}+\\varepsilon},\n\\]\nthat is, there exists a constant $C_{\\varepsilon}$ such that\n\\[\n\\|\\psi\\|_{L^6(X)} \\le C_{\\varepsilon}\\,\\lambda^{\\frac16-\\frac{1}{144}+\\varepsilon}.\n\\]"}, "choices": [{"label": "B", "text": "For every $\\varepsilon>0$, one has\n\\[\n\\|\\psi\\|_{L^6(X)} \\lesssim_{\\varepsilon} \\lambda^{\\frac16-\\frac{1}{72}+\\varepsilon},\n\\]\nthat is, there exists a constant $C_{\\varepsilon}$ such that\n\\[\n\\|\\psi\\|_{L^6(X)} \\le C_{\\varepsilon}\\,\\lambda^{\\frac16-\\frac{1}{72}+\\varepsilon}.\n\\]"}, {"label": "C", "text": "For every $\\varepsilon>0$, one has\n\\[\n\\|\\psi\\|_{L^6(X)} \\lesssim_{\\varepsilon} \\lambda^{\\frac16+\\varepsilon},\n\\]\nthat is, there exists a constant $C_{\\varepsilon}$ such that\n\\[\n\\|\\psi\\|_{L^6(X)} \\le C_{\\varepsilon}\\,\\lambda^{\\frac16+\\varepsilon}.\n\\]"}, {"label": "D", "text": "There exists $\\varepsilon_0>0$ such that one has\n\\[\n\\|\\psi\\|_{L^6(X)} \\lesssim \\lambda^{\\frac16-\\frac{1}{144}+\\varepsilon_0},\n\\]\nthat is, there exists a constant $C$ such that\n\\[\n\\|\\psi\\|_{L^6(X)} \\le C\\,\\lambda^{\\frac16-\\frac{1}{144}+\\varepsilon_0}.\n\\]"}, {"label": "E", "text": "For every $\\varepsilon>0$, one has\n\\[\n\\|\\psi\\|_{L^6(X)} \\lesssim_{\\varepsilon} \\lambda^{\\frac16-\\frac{1}{96}+\\varepsilon},\n\\]\nthat is, there exists a constant $C_{\\varepsilon}$ such that\n\\[\n\\|\\psi\\|_{L^6(X)} \\le C_{\\varepsilon}\\,\\lambda^{\\frac16-\\frac{1}{96}+\\varepsilon}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "final exponent from parameter choice $\\delta_1=1/8$", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the power-saving term $-1/144$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "quantifier on $\\varepsilon$ and dependence of implicit constant", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "miscomputed optimization of auxiliary exponents $a=1/12$, $\\delta_2=1/12$, $\\delta_1=1/8$", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only the geometric and spectral setup; it does not state or strongly hint at the specific $L^6$ bound or the exponent $1/6-1/144$. The correct answer is not leaked."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: it asks which precise bound holds for this standard setup. Although the options differ by subtle exponent and quantifier changes, the task is still largely a reformulation of a known result rather than an independently derived conclusion."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the choices differ in strength, sharpness, and quantifier dependence, so one must identify the strongest valid statement. However, success depends more on recalling the exact theorem than on generating a conclusion from the stem."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and well targeted: one is a weaker true-type bound, others are overly strong exponents, and one tampers with the quantifier/constant dependence. These reflect realistic failure modes."}, "total_score": 6, "overall_assessment": "A solid theorem-identification MCQ with no answer leakage and strong distractors, but it mainly tests precise recall/recognition of a known bound rather than deeper generative reasoning."}}
{"id": "2602.06332v1", "paper_link": "http://arxiv.org/abs/2602.06332v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm}\n\tLet $s>0, \\lambda>0, d=2,3$ and $T>0$. Then the data-to-solution map $(u_0,b_0)\\mapsto(u,b)$ for the equations \\eqref{mhd} is non-uniformly continuous from a bounded subset in $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C\\left([0,T],H^{s}(\\mathbb{R}^d)\\right)\\times C\\left([0,T],H^{s}(\\mathbb{R}^d)\\right)$.\n\n  More precisely, for any $\\gamma>0$ and arbitrary constant magnetic field ${\\bf B_0}\\in \\mathbb{R}^d$,  there exists two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$ such that\n\t\\begin{itemize}\n\t\t\\item the solutions satisfy\n\\begin{align*}\n\t\t\t(u^{\\pm 1,\\lambda}, b^{\\pm 1,\\lambda} - {\\bf B_0}) \\in C([0,T], H^{s}(\\mathbb{R}^{d})) \\times C([0,T], H^{s}(\\mathbb{R}^{d}));\n\\end{align*}\n\t\t\\item  the initial data satisfy\n\\begin{equation}\\label{nonuniform-initial-bound}\n\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\|u^{+1,\\lambda}(0,\\cdot)\\|_{H^{s}} + \\|b^{+1,\\lambda}(0,\\cdot) - {\\bf B_0}\\|_{H^{s}} \\leq \\gamma, \\\\[2mm]\n\t\t\t\\|u^{-1,\\lambda}(0,\\cdot)\\|_{H^{s}} + \\|b^{-1,\\lambda}(0,\\cdot) - {\\bf B_0}\\|_{H^{s}} \\leq \\gamma;\n\t\\end{array}\n\\end{equation}\n\t\t\\item and the non-uniform continuity is characterized by\n\t\t\\begin{itemize}\n\t\t\t\\item at initial time $t=0$,\n\t\t\t\\begin{align}\\label{nonuniform-initial-diff}\n\t\t\t\t\\lim_{\\lambda\\to\\infty} \\|u^{+1,\\lambda}(0,\\cdot) - u^{-1,\\lambda}(0,\\cdot)\\|_{H^{s}} = 0,\n\t\t\t\\end{align}\n\t\t\t\\item for evolution times $t>0$,\n\t\t\t\\begin{align}\\label{nonuniform-low-bound-sint}\n\t\t\t\t\\lim_{\\lambda \\to\\infty} \\|u^{+1,\\lambda}(t) - u^{-1,\\lambda}(t)\\|_{H^{s}} \\geq c\\gamma |\\sin t|,\n\t\t\t\\end{align}\n\t\t\\end{itemize}\n\t\twhere $c = c(s,d) > 0$ is a constant depending only on  $s$ and $d$.\n\t\\end{itemize}", "start_pos": 10028, "end_pos": 11702, "label": "thm"}, "ref_dict": {"1.2": "\\begin{align}\\label{1.2}\n\t\tu^{h,\\pm1,\\lambda}(x,t)=\\nabla^\\perp \\left(\\lambda^{-\\delta-s-1}\\phi\\left(\\dfrac{x_1}{\\lambda^\\delta}\\right)\\phi\\left(\\dfrac{x_2}{\\lambda^\\delta}\\right)\\sin (\\lambda x_2\\mp t)\\right).\n\t\\end{align}", "mhd-B": "\\begin{align}\\label{mhd-B}\n\t\\begin{cases}\n\t\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b+{\\bf B_0}\\cdot\\nabla b,\\\\[1mm]\n\t\t\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u+{\\bf B_0}\\cdot\\nabla u,  \\\\[1mm]\n\t\t\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\n\t\tu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}", "1.5": "\\begin{align}\\label{1.4}\n\tE^{\\pm1,\\lambda}&=\\partial_t u^{h,\\pm1,\\lambda}+u^{l,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{l,\\pm1,\\lambda},\\\\\\label{1.5}\n\tF^{\\pm1,\\lambda}&=u^{h,\\pm1,\\lambda}\\cdot\\nabla b^{\\pm1,\\lambda}-b^{\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda},\n\\end{align}", "mhd": "\\begin{align}\\label{mhd}\n\t\\begin{cases}\n\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u,  \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}", "thm": "\\begin{theorem}\\label{thm}\n\tLet $s>0, \\lambda>0, d=2,3$ and $T>0$. Then the data-to-solution map $(u_0,b_0)\\mapsto(u,b)$ for the equations \\eqref{mhd} is non-uniformly continuous from a bounded subset in $H^{s}(\\mr^d)\\times H^{s}(\\mr^d)$ into $C\\left([0,T],H^{s}(\\mr^d)\\right)\\times C\\left([0,T],H^{s}(\\mr^d)\\right)$.\n\n  More precisely, for any $\\gamma>0$ and arbitrary constant magnetic field ${\\bf B_0}\\in \\mr^d$,  there exists two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$ such that\n\t\\begin{itemize}\n\t\t\\item the solutions satisfy\n\\begin{align*}\n\t\t\t(u^{\\pm 1,\\lambda}, b^{\\pm 1,\\lambda} - {\\bf B_0}) \\in C([0,T], H^{s}(\\mathbb{R}^{d})) \\times C([0,T], H^{s}(\\mathbb{R}^{d}));\n\\end{align*}\n\t\t\\item  the initial data satisfy\n\\begin{equation}\\label{nonuniform-initial-bound}\n\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\|u^{+1,\\lambda}(0,\\cdot)\\|_{H^{s}} + \\|b^{+1,\\lambda}(0,\\cdot) - {\\bf B_0}\\|_{H^{s}} \\leq \\gamma, \\\\[2mm]\n\t\t\t\\|u^{-1,\\lambda}(0,\\cdot)\\|_{H^{s}} + \\|b^{-1,\\lambda}(0,\\cdot) - {\\bf B_0}\\|_{H^{s}} \\leq \\gamma;\n\t\\end{array}\n\\end{equation}\n\t\t\\item and the non-uniform continuity is characterized by\n\t\t\\begin{itemize}\n\t\t\t\\item at initial time $t=0$,\n\t\t\t\\begin{align}\\label{nonuniform-initial-diff}\n\t\t\t\t\\lim_{\\lambda\\to\\infty} \\|u^{+1,\\lambda}(0,\\cdot) - u^{-1,\\lambda}(0,\\cdot)\\|_{H^{s}} = 0,\n\t\t\t\\end{align}\n\t\t\t\\item for evolution times $t>0$,\n\t\t\t\\begin{align}\\label{nonuniform-low-bound-sint}\n\t\t\t\t\\lim_{\\lambda \\to\\infty} \\|u^{+1,\\lambda}(t) - u^{-1,\\lambda}(t)\\|_{H^{s}} \\geq c\\gamma |\\sin t|,\n\t\t\t\\end{align}\n\t\t\\end{itemize}\n\t\twhere $c = c(s,d) > 0$ is a constant depending only on  $s$ and $d$.\n\t\\end{itemize}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 7134, "pre_theorem_intro_text": "\\,\\,\\,\\,\\,\\,\\,\\, In this paper, we consider the Cauchy problem for the incompressible MHD equations with only magnetic diffusion (hereinafter called the resistive MHD equations):\n\\begin{align}\\label{mhd}\n\t\\begin{cases}\n\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u,  \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}\nwhere $u(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}^d$,\n$b(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}^d$,\nand $p(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}$\ndenote the velocity field, magnetic field, and pressure field, respectively,\nwith $x \\in \\mathbb{R}^d~(d=2,3)$ and $t>0$. Clearly, when $b\\equiv0$, \\eqref{mhd} reduces to the classical Euler equations.\n\nThe resistive MHD equations provide a fundamental framework for modeling key plasma phenomena where finite electrical resistivity plays a critical role, particularly in astrophysical magnetic reconnection processes that govern energy release in solar flares and magnetospheric activity, as well as in the geodynamo mechanisms responsible for generating and sustaining planetary magnetic fields \\cite{priest-2000,roberts1967introduction}.\n\n There have been a large number of mathematical studies on the  well-posedness theory for the incompressible MHD equations under various assumptions on velocity dissipation and magnetic diffusion.\nPrevious results on local well-posedness can be found in \\cite{sermange-1983,jiu-2006-local,li-local-nonresistive-2017-adv,chemin-local-nonresistive-2016-adv,fefferman-local-nonressitive-2014-jfa,fefferman-local-nonressitive-2017-arma}. The pioneering work of Bardos, Sulem and Sulem \\cite{full-not-bardos} and Lin and Zhang \\cite{lin-2014-GlobalSmallSolutions} on perturbation theory near constant background magnetic fields first revealed the stabilizing role of such equilibrium configurations in the MHD equations. Physical interpretations of this stabilization effect are discussed in \\cite{background-phy1,background-phy2}. These foundational results have inspired extensive research on global well-posedness and stability of the MHD equations, as documented in \\cite{lei-bkm-2009-dcds,full-1-duvaut-lions,sermange-1983,cao-wu-1-mix,cao-wu-2-mix,full-not-bardos,lin-2014-GlobalSmallSolutions,chen-2022-3dmhd-Diophant,deng-zhang-2018-decay,panGlobalClassicalSolutions2018,ren2014global,zhang-2016-non-resistive-global-jde,xie-2024-cvpde-dio,wei--global-resistive-2020-cmr,zhou-zhu-glbal-symmetry-jmp-2018,ye-global-resistive-2022-acta} and references therein.\n\n  In recent years, ill-posedness theory  for the incompressible MHD equations has also been invetigated. For the non-resistive MHD equations  which contain velocity dissipation but  no magnetic diffusion, Chen, Nie and Ye \\cite{chen-ye-sharp-ill-nonresistive-2024-jfa}  established sharp strong ill-posedness results that provide a striking contrast to the local well-posedness theory developed by Fefferman et al. \\cite{fefferman-local-nonressitive-2017-arma}. For the resistive case \\eqref{mhd}, Wu and Zhao \\cite{wu-zhao-mild-resistive-2023-IMRN}  obtained mild ill-posedness results near the background magnetic field $(1,0)$ in $\\mathbb{R}^2$.\n\nIt is noted that the concept of ill-posedness manifests rather strongly in many fundamental cases. As evidenced by \\cite{chen-ye-sharp-ill-nonresistive-2024-jfa,wu-zhao-mild-resistive-2023-IMRN} and other works of PDE systems, such behavior typically occurs in critical or supercritical function spaces, while  some problems remain unresolved to this day. These substantial difficulties have motivated researchers to consider relaxed notions of ill-posedness by examining weaker properties.\n For certain equations, the solution operator may exhibit non-uniform continuity properties under stronger topological frameworks, which provides meaningful insights into the refined continuity structure of solution mappings and can be seen a kind of instability of the solutions or ill-posedness the equations.\n\nIn this paper, we are concerned with the non-uniform continuity properties of the data-to-solution map for the resistive MHD equations. We begin by precisely defining the notion of non-uniform continuity as follows:\n\\begin{definition}\\label{define-nonuniform}\n\tLet $X$ be a Banach space, and consider the Cauchy problem:\n\t\\begin{align*}\n\t\t\\begin{cases}\n\t\t\t\\partial_t v = N(v), \\\\\n\t\t\tv(0) = v_0,\n\t\t\\end{cases}\n\t\\end{align*}\n\twhere $N$ is a (possibly nonlinear) differential operator. The \\textbf{data-to-solution map} $\\Phi_t \\colon X \\to X$ (for fixed $t > 0$) defined by $\\Phi_t(v_0) = v(t)$ is said to be \\textbf{non-uniformly continuous} on $X$ if the following holds:\n\n\t\\noindent\n\t\\textbf{Sequential Formulation:} \\\\\n\tFor every $t > 0$, there exists $\\epsilon_0 > 0$ and sequences $\\{v_{1,n}(0)\\}, \\{v_{2,n}(0)\\} \\subset X$ such that:\n\t\\begin{align*}\n\t\t\\lim_{n \\to \\infty} \\| v_{1,n}(0) - v_{2,n}(0) \\|_X = 0,\n\t\\end{align*}\n\tbut\n\t\\begin{align*}\n\t\t\\limsup_{n \\to \\infty} \t\\| \\Phi_t(v_{1,n}(0)) - \\Phi_t(v_{2,n}(0)) \\|_X \\geq \\epsilon_0.\n\t\\end{align*}\n\n\t\\noindent\n\t\\textbf{$\\delta$-$\\epsilon$ Formulation (Equivalent):} \\\\\n\tFor every $t > 0$, there exists $\\epsilon_0 > 0$ such that for any $\\delta > 0$, one can find initial data $v_1(0), v_2(0) \\in X$ satisfying:\n\t\\begin{align*}\n\t\t\\| v_1(0) - v_2(0) \\|_X < \\delta,\n\t\\end{align*}\n\tbut\n\t\\begin{align*}\n\t\t\\| \\Phi_t(v_1(0)) - \\Phi_t(v_2(0)) \\|_X \\geq \\epsilon_0.\n\t\\end{align*}\n\\end{definition}\n\nIn \\cite{nonuniform-2010-cmp-himonas}, Himonas and Misiołek first  proved non-uniform continuity of the data-to-solution map  on the  incompressible Euler equations in both $H^s(\\mathbb{R}^d)$ ($s > 0$) and $H^r(\\mathbb{T}^d)$ ($r \\in \\mathbb{R}$) with $d = 2,3$, which was later extended by Li and Bourgain \\cite{nonuniform-2019-cmp-bourgain-li} through Galilean boost techniques to the endpoint case $s \\geq 0$ in $H^s(\\mathbb{R}^d)$, where they further proved the stronger property of nowhere uniform continuity.\nFor the non-resistive MHD equations with only velocity dissipation, recent work by Li, Yin, and Zhu \\cite{nonuniforem-2023-adv-li-yin-zhu} demonstrated non-uniform continuity in $H^s(\\mathbb{R}^d)$ for $s > \\frac{d}{2}$.\n\n  We will prove in this paper the non-uniform continuity of the data-to-solution map for the resistive MHD equations in Sobolev spaces $H^s(\\mathbb{R}^d)$ for all $s>0$ and  $d=2,3$. In comparison with the work by Li, Yin, and Zhu \\cite{nonuniforem-2023-adv-li-yin-zhu} which is for non-resistive MHD equations with only velocity dissipation and in $H^s(\\mathbb{R}^d)$ for $s > \\frac{d}{2}$,  our results hold for the resistive MHD equations and  in  $H^s(\\mathbb{R}^d)$ for all $s>0$ and  $d=2,3$.  Moreover, our results permit the solution perturbation around an arbitrary constant background magnetic fields $\\mathbf{B_0} \\in \\mathbb{R}^d$, which shows that the strong magnetic field may provide the stabilization effect but no help for the uniform continuity of the data-to-solution map.\n\nOur main results can be stated as follows:", "context": "\\,\\,\\,\\,\\,\\,\\,\\, In this paper, we consider the Cauchy problem for the incompressible MHD equations with only magnetic diffusion (hereinafter called the resistive MHD equations):\n\\begin{align}\\label{mhd}\n \\begin{cases}\n \\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u, \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n \\end{cases}\n\\end{align}\nwhere $u(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}^d$,\n$b(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}^d$,\nand $p(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}$\ndenote the velocity field, magnetic field, and pressure field, respectively,\nwith $x \\in \\mathbb{R}^d~(d=2,3)$ and $t>0$. Clearly, when $b\\equiv0$, \\eqref{mhd} reduces to the classical Euler equations.\n\nIn this paper, we are concerned with the non-uniform continuity properties of the data-to-solution map for the resistive MHD equations. We begin by precisely defining the notion of non-uniform continuity as follows:\n\\begin{definition}\\label{define-nonuniform}\n Let $X$ be a Banach space, and consider the Cauchy problem:\n \\begin{align*}\n \\begin{cases}\n \\partial_t v = N(v), \\\\\n v(0) = v_0,\n \\end{cases}\n \\end{align*}\n where $N$ is a (possibly nonlinear) differential operator. The \\textbf{data-to-solution map} $\\Phi_t \\colon X \\to X$ (for fixed $t > 0$) defined by $\\Phi_t(v_0) = v(t)$ is said to be \\textbf{non-uniformly continuous} on $X$ if the following holds:\n\n\\noindent\n \\textbf{Sequential Formulation:} \\\\\n For every $t > 0$, there exists $\\epsilon_0 > 0$ and sequences $\\{v_{1,n}(0)\\}, \\{v_{2,n}(0)\\} \\subset X$ such that:\n \\begin{align*}\n \\lim_{n \\to \\infty} \\| v_{1,n}(0) - v_{2,n}(0) \\|_X = 0,\n \\end{align*}\n but\n \\begin{align*}\n \\limsup_{n \\to \\infty} \\| \\Phi_t(v_{1,n}(0)) - \\Phi_t(v_{2,n}(0)) \\|_X \\geq \\epsilon_0.\n \\end{align*}\n\n\\noindent\n \\textbf{$\\delta$-$\\epsilon$ Formulation (Equivalent):} \\\\\n For every $t > 0$, there exists $\\epsilon_0 > 0$ such that for any $\\delta > 0$, one can find initial data $v_1(0), v_2(0) \\in X$ satisfying:\n \\begin{align*}\n \\| v_1(0) - v_2(0) \\|_X < \\delta,\n \\end{align*}\n but\n \\begin{align*}\n \\| \\Phi_t(v_1(0)) - \\Phi_t(v_2(0)) \\|_X \\geq \\epsilon_0.\n \\end{align*}\n\\end{definition}\n\nWe will prove in this paper the non-uniform continuity of the data-to-solution map for the resistive MHD equations in Sobolev spaces $H^s(\\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. In comparison with the work by Li, Yin, and Zhu \\cite{nonuniforem-2023-adv-li-yin-zhu} which is for non-resistive MHD equations with only velocity dissipation and in $H^s(\\mathbb{R}^d)$ for $s > \\frac{d}{2}$, our results hold for the resistive MHD equations and in $H^s(\\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. Moreover, our results permit the solution perturbation around an arbitrary constant background magnetic fields $\\mathbf{B_0} \\in \\mathbb{R}^d$, which shows that the strong magnetic field may provide the stabilization effect but no help for the uniform continuity of the data-to-solution map.\n\nOur main results can be stated as follows:\n\n\\begin{align}\\label{mhd}\n\t\\begin{cases}\n\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u,  \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}", "full_context": "\\,\\,\\,\\,\\,\\,\\,\\, In this paper, we consider the Cauchy problem for the incompressible MHD equations with only magnetic diffusion (hereinafter called the resistive MHD equations):\n\\begin{align}\\label{mhd}\n \\begin{cases}\n \\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u, \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n \\end{cases}\n\\end{align}\nwhere $u(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}^d$,\n$b(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}^d$,\nand $p(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}$\ndenote the velocity field, magnetic field, and pressure field, respectively,\nwith $x \\in \\mathbb{R}^d~(d=2,3)$ and $t>0$. Clearly, when $b\\equiv0$, \\eqref{mhd} reduces to the classical Euler equations.\n\nIn this paper, we are concerned with the non-uniform continuity properties of the data-to-solution map for the resistive MHD equations. We begin by precisely defining the notion of non-uniform continuity as follows:\n\\begin{definition}\\label{define-nonuniform}\n Let $X$ be a Banach space, and consider the Cauchy problem:\n \\begin{align*}\n \\begin{cases}\n \\partial_t v = N(v), \\\\\n v(0) = v_0,\n \\end{cases}\n \\end{align*}\n where $N$ is a (possibly nonlinear) differential operator. The \\textbf{data-to-solution map} $\\Phi_t \\colon X \\to X$ (for fixed $t > 0$) defined by $\\Phi_t(v_0) = v(t)$ is said to be \\textbf{non-uniformly continuous} on $X$ if the following holds:\n\n\\noindent\n \\textbf{Sequential Formulation:} \\\\\n For every $t > 0$, there exists $\\epsilon_0 > 0$ and sequences $\\{v_{1,n}(0)\\}, \\{v_{2,n}(0)\\} \\subset X$ such that:\n \\begin{align*}\n \\lim_{n \\to \\infty} \\| v_{1,n}(0) - v_{2,n}(0) \\|_X = 0,\n \\end{align*}\n but\n \\begin{align*}\n \\limsup_{n \\to \\infty} \\| \\Phi_t(v_{1,n}(0)) - \\Phi_t(v_{2,n}(0)) \\|_X \\geq \\epsilon_0.\n \\end{align*}\n\n\\noindent\n \\textbf{$\\delta$-$\\epsilon$ Formulation (Equivalent):} \\\\\n For every $t > 0$, there exists $\\epsilon_0 > 0$ such that for any $\\delta > 0$, one can find initial data $v_1(0), v_2(0) \\in X$ satisfying:\n \\begin{align*}\n \\| v_1(0) - v_2(0) \\|_X < \\delta,\n \\end{align*}\n but\n \\begin{align*}\n \\| \\Phi_t(v_1(0)) - \\Phi_t(v_2(0)) \\|_X \\geq \\epsilon_0.\n \\end{align*}\n\\end{definition}\n\nWe will prove in this paper the non-uniform continuity of the data-to-solution map for the resistive MHD equations in Sobolev spaces $H^s(\\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. In comparison with the work by Li, Yin, and Zhu \\cite{nonuniforem-2023-adv-li-yin-zhu} which is for non-resistive MHD equations with only velocity dissipation and in $H^s(\\mathbb{R}^d)$ for $s > \\frac{d}{2}$, our results hold for the resistive MHD equations and in $H^s(\\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. Moreover, our results permit the solution perturbation around an arbitrary constant background magnetic fields $\\mathbf{B_0} \\in \\mathbb{R}^d$, which shows that the strong magnetic field may provide the stabilization effect but no help for the uniform continuity of the data-to-solution map.\n\nOur main results can be stated as follows:\n\n\\begin{align}\\label{mhd}\n\t\\begin{cases}\n\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u,  \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}\n\n\\section{Preliminaries}\\label{sec:2}\n\\subsection{Auxiliary Analysis Tools}\\label{sec:2.1}\nIn this subsection, we introduce two key lemmas needed later. The first one is\n\\begin{lemma}\\cite{nonuniform-2010-cmp-himonas}\\label{phi-cmp-2010}\n Let $\\sigma \\geq 0, \\delta\\geq 0, a\\in \\mr$ and $\\lambda\\gg 1$. For any Schwartz function $\\psi\\in \\mathcal{S}(\\mr)$, it holds that\n \\begin{align*}\n \\lambda^{\\delta/2}\\left\\|\\psi\\right\\|_{L^2(\\mathbb{R})}\\leq\\left\\|\\psi\\left(\\frac{\\cdot}{\\lambda^\\delta}\\right)\\right\\|_{H^\\sigma(\\mathbb{R})}\\leq\\lambda^{\\delta/2}\\left\\|\\psi\\right\\|_{H^\\sigma(\\mathbb{R})},\n \\end{align*}\nand \\begin{align}\\label{622}\n \\left\\|\\psi\\left(\\frac{\\cdot}{\\lambda^\\delta}\\right)\\cos(\\lambda\\cdot-a)\\right\\|_{H^\\sigma(\\mathbb{R})}\\simeq\\lambda^{\\sigma+\\delta/2}\\|\\psi\\|_{L^2(\\mathbb{R})}.\n\\end{align}\nMoreover, \\eqref{622} holds true if $\\cos(\\lambda \\cdot-a)$ is replaced by $\\sin(\\lambda \\cdot-a)$.\n\\end{lemma}\n\n\\subsection{Approximate Solutions Scheme}\\label{sec:3.1}\nWe begin by constructing two sequences of approximate solutions $b^{w,\\lambda}$ and $u^{w,\\lambda}$ with $w=\\pm 1$, where $b^{w,\\lambda}$ contains low frequencies only and $u^{w,\\lambda}$ contain both high and low frequencies. More precisely, we set\n\\begin{align}\\label{u-low+high}\n u^{w,\\lambda}(x,t)=u^{l,w,\\lambda}(x,t)+u^{h,w,\\lambda}(x,t).\n\\end{align}\nThe high-frequency components $u^{h,w,\\lambda}$ are constructed as\n\\begin{align}\\label{u-high}\n u^{h,w,\\lambda}(x,t)=\\begin{cases}\n \\nabla^\\perp \\phi^{h,w,\\lambda}(x,t)=(\\partial_{x_2}\\phi^{h,w,\\lambda}(x,t),-\\partial_{x_1}\\phi^{h,w,\\lambda}(x,t)), &~d=2,\\\\[3mm]\n \\left(\\nabla^\\perp \\phi^{h,w,\\lambda}(x,t),0 \\right)=(\\partial_{x_2}\\phi^{h,w,\\lambda}(x,t),-\\partial_{x_1}\\phi^{h,w,\\lambda}(x,t),0), &~d=3.\n \\end{cases}\n\\end{align}\nIn \\eqref{u-high}, the function $\\phi^{h,w,\\lambda}(x,t)$ is defined as\n\\begin{align}\\label{phi-high}\n \\phi^{h,w,\\lambda}(x,t)=\\begin{cases}\n \\lambda^{-s-\\delta-1}\\phi(\\dfrac{x_1}{\\lambda^\\delta})\\phi(\\dfrac{x_2}{\\lambda^\\delta})\\sin (\\lambda x_2-wt),&~d=2,\\\\[3mm]\n \\lambda^{-s-\\delta-1}\\phi(\\dfrac{x_1}{\\lambda^\\delta})\\phi(\\dfrac{x_2}{\\lambda^\\delta})\\sin (\\lambda x_2-wt)\\phi(x_3),&~d=3,\n \\end{cases}\n\\end{align}\nwhere $s > 0, \\lambda>0, w=\\pm 1$, the parameter $\\delta > 0$ is to be specified later and the smooth function $\\phi \\in C_c^\\infty(\\mathbb{R})$ satisfies $\\text{supp}\\,\\phi \\subset [-2,2]$ and $\\phi(x) \\equiv 1$ on $|x| < 1$.\n\nFollowing similar estimates established in \\eqref{u,b-low-t-local-bound} and \\eqref{u-high-bound}, we can obtain\n\\begin{align}\\label{4.5}\n \\|\\bar{u}^{l,w,\\lambda}(t)\\|_{H^m} + \\|\\bar{b}^{w,\\lambda}(t)\\|_{H^{m}}\n \\leq C\\lambda^{-1+\\delta},\n\\end{align}\nand\n\\begin{align}\\label{4.6}\n \\|\\bar{u}^{h,w,\\lambda}(t)\\|_{H^r}\\leq C\\lambda^{r-s},\\quad \\|\\bar{u}^{h,w,\\lambda}(t)\\|_{L^\\infty}\\leq C\\lambda^{-s-\\delta},\\quad \\|\\nabla \\bar{u}^{h,w,\\lambda}(t)\\|_{L^\\infty}\\leq C\\lambda^{-s+1-\\delta}\n\\end{align}\nfor any $0 \\leq t \\leq T\\leq 1$, where $[0,T]$ denotes the existence interval of the solution guaranteed by Lemma~\\ref{local-well-pri}.\n\\subsection{Error Estimates for the Approximate Solutions}\\label{sec:4.2}\nFollowing the approach in subsection \\ref{sec:3.2} and based on system \\eqref{mhd-B}, we construct a perturbed system as follows\n\\begin{align}\\label{4.9}\n \\begin{cases}\n \\partial_t \\bar{u}^{w,\\lambda}+\\bar{u}^{w,\\lambda} \\cdot \\nabla \\bar{u}^{w,\\lambda}+\\nabla p^{w,\\lambda}=\\bar{b}^{w,\\lambda}\\cdot \\nabla \\bar{b}^{w,\\lambda}+{\\bf B_0}\\cdot\\nabla \\bar{b}^{w,\\lambda}+E_1^{w,\\lambda},\\\\[1mm]\n \\partial_t \\bar{b}^{w,\\lambda}-\\Delta \\bar{b}^{w,\\lambda}+\\bar{u}^{w,\\lambda}\\cdot \\nabla \\bar{b}^{w,\\lambda}=\\bar{b}^{w,\\lambda}\\cdot\\nabla \\bar{u}^{w,\\lambda}+{\\bf B_0}\\cdot\\nabla \\bar{u}^{w,\\lambda}+F_1^{w,\\lambda}, \\\\[1mm]\n \\diver\\, \\bar{u}^{w,\\lambda}=\\diver\\, \\bar{b}^{w,\\lambda}=0,\n \\end{cases}\n\\end{align}\nwhere\n\\begin{align*}\n E_1^{w,\\lambda}&=\\partial_t \\bar{u}^{h,w,\\lambda}+\\bar{u}^{l,w,\\lambda}\\cdot\\nabla \\bar{u}^{h,w,\\lambda}+\\bar{u}^{h,w,\\lambda}\\cdot\\nabla \\bar{u}^{h,w,\\lambda}+\\bar{u}^{h,w,\\lambda}\\cdot\\nabla \\bar{u}^{l,w,\\lambda},\\\\\\nonumber\n F_1^{w,\\lambda}&=\\bar{u}^{h,w,\\lambda}\\cdot\\nabla \\bar{b}^{w,\\lambda}-\\bar{b}^{w,\\lambda}\\cdot\\nabla \\bar{u}^{h,w,\\lambda}-{\\bf B_0}\\cdot \\nabla \\bar{u}^{h,w,\\lambda}\\\\\n &\\define\\diver \\tilde{F}_1^{w,\\lambda}-{\\bf B_0}\\cdot \\nabla \\bar{u}^{h,w,\\lambda}.\n\\end{align*}\nThe perturbed system \\eqref{4.9} is analogous to \\eqref{mhd-low+high}. The directional derivative of the high-frequency component along the background magnetic field ${\\bf B_0} = (B_1, B_2)$ is\n\\begin{align*}\n {\\bf B_0} \\cdot \\nabla \\bar{u}^{h,w,\\lambda} = B_1\\partial_{x_1}\\bar{u}^{h,w,\\lambda} + B_2\\partial_{x_2}\\bar{u}^{h,w,\\lambda}.\n\\end{align*}\nRemarkably, we observe that the oscillatory part in $\\bar{u}^{h,w,\\lambda}$ satisfies\n\\begin{align*}\n {\\bf B_0} \\cdot \\nabla \\left[\\phi\\left(\\frac{B_1x_2 - B_2x_1}{\\lambda^\\delta}\\right) \\cos\\left(\\lambda(B_1x_2 - B_2x_1) - wt\\right)\\right] = 0,\n\\end{align*}\nwhich leads to the following simplified expression\n\\begin{align*}\n & {\\bf B_0}\\cdot \\nabla \\left(\\lambda^{-s-\\delta}\\phi\\left(\\dfrac{x_1}{\\lambda^\\delta}\\right)\n \\phi\\left(\\dfrac{B_1x_2-B_2x_1}{\\lambda^\\delta}\\right)\\cos \\left(\\lambda \\left(B_1x_2-B_2x_1\\right)-wt\\right)\\right)\\\\\n &=B_1\\lambda^{-s-2\\delta}\\phi'\\left(\\frac{x_1}{\\lambda^{\\delta}}\\right) \\phi\\left(\\dfrac{B_1x_2-B_2x_1}{\\lambda^\\delta}\\right)\\cos \\left(\\lambda \\left(B_1x_2-B_2x_1\\right)-wt\\right)\n\\end{align*}\nTherefore, following analogous arguments as \\eqref{4.3}, we can obtain\n\\begin{align}\\label{4.7}\n \\left\\| {\\bf B_0}\\cdot\\nabla \\bar{u}^{h,w,\\lambda}\\right\\|_{L^2}\\leq C\\lambda^{-s-\\delta}.\n\\end{align}\nCombining the estimates \\eqref{4.5}, \\eqref{4.6} with Lemma \\ref{E,F-error-estimate}, we deduce that\n\\begin{align}\\label{4.8}\n \\|E_1^{w,\\lambda}(t)\\|_{L^2}\\leq C\\lambda^{-\\sigma_{s,\\delta}},\\quad \\|\\tilde{F}_1^{w,\\lambda}(t)\\|_{L^2}\\leq C\\lambda^{-s-1},\n\\end{align}\nwhere \\begin{align*}\n \\sigma_{s,\\delta}=\\min\\{s+1-\\delta,2s-1+\\delta\\}.\n\\end{align*}\n\\subsection{Exact Solutions}\\label{sec:4.3}\nLet $(\\bar{u}_{w,\\lambda},\\bar{b}_{w,\\lambda})$ be the unique solution to system \\eqref{mhd-B} with the initial data $\\left(\\bar{u}^{w,\\lambda}(x,0),\\bar{b}^{w,\\lambda}(x,0)\\right)$, satisfying\n\\begin{align*}\n \\begin{cases}\n \\partial_t \\bar{u}_{w,\\lambda}+\\bar{u}_{w,\\lambda} \\cdot \\nabla \\bar{u}_{w,\\lambda}+\\nabla p_{w,\\lambda}=\\bar{b}_{w,\\lambda}\\cdot \\nabla \\bar{b}_{w,\\lambda}+{\\bf B_0}\\cdot \\nabla \\bar{b}_{w,\\lambda},\\\\[1mm]\n \\partial_t \\bar{b}_{w,\\lambda}-\\Delta \\bar{b}_{w,\\lambda}+\\bar{u}_{w,\\lambda}\\cdot \\nabla \\bar{b}_{w,\\lambda}=\\bar{b}_{w,\\lambda}\\cdot\\nabla \\bar{u}_{w,\\lambda}+{\\bf B_0}\\cdot \\nabla \\bar{u}_{w,\\lambda}, \\\\[1mm]\n \\diver\\, \\bar{u}_{w,\\lambda}=\\diver\\, \\bar{b}_{w,\\lambda}=0,\n \\end{cases}\n\\end{align*}\nIt follows from \\eqref{4.9} that\n\\begin{align*}\n \\begin{cases}\n \\partial_t \\bar{u}^{w,\\lambda}+\\bar{u}^{w,\\lambda} \\cdot \\nabla \\bar{u}^{w,\\lambda}+\\nabla p^{w,\\lambda}=\\bar{b}^{w,\\lambda}\\cdot \\nabla \\bar{b}^{w,\\lambda}+{\\bf B_0}\\cdot\\nabla \\bar{b}^{w,\\lambda}+E_1^{w,\\lambda},\\\\[1mm]\n \\partial_t \\bar{b}^{w,\\lambda}-\\Delta \\bar{b}^{w,\\lambda}+\\bar{u}^{w,\\lambda}\\cdot \\nabla \\bar{b}^{w,\\lambda}=\\bar{b}^{w,\\lambda}\\cdot\\nabla \\bar{u}^{w,\\lambda}+{\\bf B_0}\\cdot\\nabla \\bar{u}^{w,\\lambda}+\\diver \\tilde{F}_1^{w,\\lambda}-{\\bf B_0}\\cdot \\nabla \\bar{u}^{h,w,\\lambda}, \\\\[1mm]\n \\diver\\, \\bar{u}^{w,\\lambda}=\\diver\\, \\bar{b}^{w,\\lambda}=0.\n\\end{cases}\n\\end{align*}\nThe two systems share same initial conditions, which are\n\\begin{align*}\n \\bar{u}_{w,\\lambda}(x, 0)&=\\bar{u}^{w,\\lambda}(x,0)=\\nabla^\\perp\\left( \\bar{\\phi}^{l,w,\\lambda}(x)+\\bar{\\phi}^{h,w,\\lambda}(x,0)\\right),\\\\[1mm]\n \\bar{b}_{w,\\lambda}(x, 0)&=\\bar{b}^{w,\\lambda}(x,0)=\\nabla^\\perp\\bar{\\phi}^{l,w,\\lambda}(x)\n\\end{align*}", "post_theorem_intro_text_len": 6867, "post_theorem_intro_text": "\\begin{remark}\n\tThe stabilizing role of background magnetic fields in the MHD equations has been well studied in previous works \\cite{lin-2014-GlobalSmallSolutions,full-not-bardos}. Our analysis reveals that despite the stabilization effect of the  background magnetic fields, the data-to-solution map for the resistive MHD equations \\eqref{mhd} maintains its non-uniform continuity property for arbitrary non-zero constant fields ${\\bf B_0}\\neq {\\bf 0}$.\n\\end{remark}\n\nTo simplify the notations, we write the perturbed system as follows\n\\begin{align}\\label{mhd-B}\n\t\\begin{cases}\n\t\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b+{\\bf B_0}\\cdot\\nabla b,\\\\[1mm]\n\t\t\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u+{\\bf B_0}\\cdot\\nabla u,  \\\\[1mm]\n\t\t\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\n\t\tu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}\nwhere we denote $b - {\\bf B_0}$ by $b$ for convenience.\n\nWe now explain the main ideas of the proof of  Theorem \\ref{thm}. It is recalled that  Himonas and Misiołek  \\cite{nonuniform-2010-cmp-himonas} introduced a frequency decomposition to construct  approximate solutions of the two-dimensional incompressible Euler equations. Denote\n \\begin{align}\\label{1.1}\n\tu^{\\pm1,\\lambda}(x,t)=u^{l,\\pm1,\\lambda}(x,t)+u^{h,\\pm1,\\lambda}(x,t),\n\\end{align}\nwhere\n\\begin{itemize}\n\t\\item[(i)] the low-frequency components $u^{l,\\pm1,\\lambda}$ are  obtained by solving the Euler equations with  low-frequency initial data;\n\t\\item[(ii)] the high-frequency components $u^{h,\\pm1,\\lambda}$ are explicitly constructed by using  oscillatory profiles, with frequency parameter $\\lambda$ controlling the spatial concentration,\n\t\\begin{align}\\label{1.2}\n\t\tu^{h,\\pm1,\\lambda}(x,t)=\\nabla^\\perp \\left(\\lambda^{-\\delta-s-1}\\phi\\left(\\dfrac{x_1}{\\lambda^\\delta}\\right)\\phi\\left(\\dfrac{x_2}{\\lambda^\\delta}\\right)\\sin (\\lambda x_2\\mp t)\\right).\n\t\\end{align}\nHere, $\\lambda>0, \\max\\{1-s,0\\}<\\delta < 1$ and $\\phi\\in C_c^\\infty(\\mathbb{R})$ with $\\text{supp}\\,\\phi \\subset [-2,2]$ and $\\phi(x) \\equiv 1$ on $|x| < 1$.\n\\end{itemize}\n\nIt is required to modify the construction of the approximate solutions in the presence of the magnetic field. To the resistive MHD equations, due to the diffusive nature of the magnetic field evolution (versus the purely transport equations of the Euler equations ), we restrict high-frequency perturbations to the velocity field alone and  maintain low-frequency components for the magnetic field. More precisely, in the case $\\mathbf{B_0} = \\mathbf{0}$ (no background magnetic field) and $d=2$,  we set\n\\begin{itemize}\n\t\\item the low-frequency pairs $(u^{l,\\pm1,\\lambda}, b^{\\pm1,\\lambda})$ by solving the resistive MHD equations \\eqref{mhd} (equivalently, system \\eqref{mhd-B} with $\\mathbf{B_0} = \\mathbf{0}$) with low-frequency initial data;\n\t\\item the high-frequency velocity components $u^{h,\\pm1,\\lambda}$ by retain their profiles as in  \\eqref{1.2}.\n\\end{itemize}\n\nIn the three-dimensional case $(d = 3)$, we modify  the high-frequency velocity components as\n\t\\begin{align}\\label{1.3}\n\tu^{h,\\pm1,\\lambda}(x,t)=\\begin{pmatrix}\n\t\t\\partial_{x_1}\\\\[1mm]\n\t\t-\\partial_{x_2}\\\\[1mm]\n\t\t0\n\t\\end{pmatrix} \\left(\\lambda^{-\\delta-s-1}\\phi\\left(\\dfrac{x_1}{\\lambda^\\delta}\\right)\\phi\\left(\\dfrac{x_2}{\\lambda^\\delta}\\right)\\sin (\\lambda x_2\\mp t)\\phi(x_3)\\right).\n\\end{align}\n\nThe error terms induced by the  high-frequency  $u^{h,\\pm1,\\lambda}$   can be written as\n\\begin{align}\\label{1.4}\n\tE^{\\pm1,\\lambda}&=\\partial_t u^{h,\\pm1,\\lambda}+u^{l,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{l,\\pm1,\\lambda},\\\\\\label{1.5}\n\tF^{\\pm1,\\lambda}&=u^{h,\\pm1,\\lambda}\\cdot\\nabla b^{\\pm1,\\lambda}-b^{\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda},\n\\end{align}\nrespectively.\n\nDirect estimation of \\eqref{1.5} will lead to uncontrolled error terms. Our alternative approach consists of three key steps:\n\\begin{enumerate}\n\t\\item {\\bf Divergence reformulation:} We express $F^{\\pm1,\\lambda} =\\text{div}\\,\\, \\tilde{F}^{\\pm1,\\lambda}$ and establish estimates for $\\tilde{F}^{\\pm1,\\lambda}$;\n\t\\item {\\bf Integration by parts:} In $L^2$ inner product computations, we transfer the divergence operator to $b^{\\pm1,\\lambda}$ via integration by parts;\n\t\\item {\\bf Diffusion cancellation:} The resulting terms are precisely canceled by exploiting the magnetic diffusion term $\\Delta b^{\\pm1,\\lambda}$.\n\\end{enumerate}\n\nFor non-zero background fields $\\mathbf{B_0}=(B_1,B_2) \\neq \\mathbf{0}$, we introduce the coordinate transformation\n\\begin{align*}\n\tx_2 \\mapsto B_1x_2 - B_2x_1,\n\\end{align*}\nwhich maintains all norm estimates while achieving exact cancellation of the dominant high-frequency linear term $\\mathbf{B_0} \\cdot \\nabla u^{h,\\pm 1,\\lambda}$ appearing in the error analysis of $F^{\\pm1,\\lambda}$.\n\nA key observation is that the directional derivative exhibits a better decay:\n\\begin{align*}\n\t\\left\\|\\mathbf{B_0} \\cdot \\nabla u^{h,\\pm 1,\\lambda}\\right\\|_{L^2}\\leq C\\lambda^{-s-\\delta} \\ll \\left\\|u^{h,\\pm 1,\\lambda}\\right\\|_{L^2}\\leq C\\lambda^{-s}, \\text{~for~}\\lambda \\to \\infty,\n\\end{align*}\nwhere $\\delta>0$ represent the improved decay rate.\n\n\t The paper is organized as follows. section \\ref{sec:2} is the preliminary analytical framework, in which subsection \\ref{sec:2.1} is about key Lemmas and subsection \\ref{sec:2.2} is on the local well-posedness theory. The core  strategy of the proof is implemented in section \\ref{sec:3} (zero background field case, $\\mathbf{B_0} = \\mathbf{0}$) and section  \\ref{sec:4} (non-zero background field case, $\\mathbf{B_0} \\neq \\mathbf{0}$), following the approach as follows: construction of approximate solutions in subsections \\ref{sec:3.1} and \\ref{sec:4.1}, precise estimation of approximation errors in subsections \\ref{sec:3.2} and \\ref{sec:4.2}, rigorous construction of exact solutions in subsections \\ref{sec:3.3} and \\ref{sec:4.3}, finishing the proof of Theorem in the case $\\mathbf{B_0} = \\mathbf{0}$ and $\\mathbf{B_0} \\neq \\mathbf{0}$ in subsections \\ref{sec:3.4} and \\ref{sec:4.4}, respectively.\n\n{\\bf Notations:}\n\\begin{enumerate}\n\\item {\\bf Function Spaces:} Throughout this work, let $X$ denote a Banach space equipped with norm $\\|\\cdot\\|_X$. Since all function spaces considered here are defined on $\\mathbb{R}^d$ for $d = 2,3$, we will suppress the domain $\\mathbb{R}^d$ in our notation unless otherwise specified.\n\t\\item {\\bf Differential Operators:} For $x \\in \\mathbb{R}^2$, we define the perpendicular gradient operator as $\\nabla^\\perp := (\\partial_{x_2}, -\\partial_{x_1})$.\n\t\\item {\\bf Joint Norms:} Given functions $f, g \\in X(\\mathbb{R}^d)$, we define their joint norm by\n\\begin{align*}\n\t\\|f(\\cdot),g(\\cdot)\\|_{X(\\mathbb{R}^d)}=\t\\|f(\\cdot)\\|_{X(\\mathbb{R}^d)}+\t\\|g(\\cdot)\\|_{X(\\mathbb{R}^d)}\n\\end{align*}\nadopting this concise notation for simplicity.\n\\end{enumerate}", "sketch": "To prove Theorem~\\ref{thm}, the authors adapt the “frequency decomposition” method of Himonas–Misio\\l{}ek. They decompose the velocity as\n\\[\n u^{\\pm1,\\lambda}(x,t)=u^{l,\\pm1,\\lambda}(x,t)+u^{h,\\pm1,\\lambda}(x,t),\n\\]\nwhere (i) the low-frequency parts $u^{l,\\pm1,\\lambda}$ come from solving the (appropriate) equations with low-frequency initial data, and (ii) the high-frequency parts $u^{h,\\pm1,\\lambda}$ are “explicitly constructed by using oscillatory profiles,” with frequency parameter $\\lambda$ (in 2D, profile as in \\eqref{1.2}; in 3D, modified as in \\eqref{1.3}).\n\nFor resistive MHD, “due to the diffusive nature of the magnetic field evolution,” they “restrict high-frequency perturbations to the velocity field alone and maintain low-frequency components for the magnetic field”: the low-frequency pair $(u^{l,\\pm1,\\lambda}, b^{\\pm1,\\lambda})$ is obtained by solving the resistive MHD system with low-frequency initial data, while $u^{h,\\pm1,\\lambda}$ retains the oscillatory profile.\n\nThey then estimate the approximation errors induced by $u^{h,\\pm1,\\lambda}$, written as\n\\[\nE^{\\pm1,\\lambda}=\\partial_t u^{h,\\pm1,\\lambda}+u^{l,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{l,\\pm1,\\lambda},\n\\]\n\\[\nF^{\\pm1,\\lambda}=u^{h,\\pm1,\\lambda}\\cdot\\nabla b^{\\pm1,\\lambda}-b^{\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}.\n\\]\nSince “direct estimation of \\eqref{1.5} will lead to uncontrolled error terms,” their alternative consists of three steps: (1) “express $F^{\\pm1,\\lambda}=\\mathrm{div}\\,\\tilde F^{\\pm1,\\lambda}$” and estimate $\\tilde F^{\\pm1,\\lambda}$; (2) “transfer the divergence operator to $b^{\\pm1,\\lambda}$ via integration by parts” in $L^2$ inner products; (3) use “diffusion cancellation,” where “the resulting terms are precisely canceled by exploiting the magnetic diffusion term $\\Delta b^{\\pm1,\\lambda}$.”\n\nFor nonzero constant background fields $\\mathbf{B_0}\\neq\\mathbf{0}$, they “introduce the coordinate transformation $x_2\\mapsto B_1x_2-B_2x_1$,” which “maintains all norm estimates” and yields “exact cancellation of the dominant high-frequency linear term $\\mathbf{B_0}\\cdot\\nabla u^{h,\\pm1,\\lambda}$” in the error analysis of $F^{\\pm1,\\lambda}$. They also note the key decay bound\n\\[\n\\|\\mathbf{B_0}\\cdot\\nabla u^{h,\\pm1,\\lambda}\\|_{L^2}\\le C\\lambda^{-s-\\delta}\\ll \\|u^{h,\\pm1,\\lambda}\\|_{L^2}\\le C\\lambda^{-s}\\quad (\\lambda\\to\\infty),\n\\]\nwith $\\delta>0$ giving “improved decay.”\n\nFinally, they describe the proof strategy as: “construction of approximate solutions,” “precise estimation of approximation errors,” “rigorous construction of exact solutions,” and “finishing the proof” in the cases $\\mathbf{B_0}=\\mathbf{0}$ and $\\mathbf{B_0}\\neq\\mathbf{0}$.", "expanded_sketch": "To prove the main theorem, the authors adapt the “frequency decomposition” method of Himonas–Misio\\l{}ek. They decompose the velocity as\n\\[\n u^{\\pm1,\\lambda}(x,t)=u^{l,\\pm1,\\lambda}(x,t)+u^{h,\\pm1,\\lambda}(x,t),\n\\]\nwhere (i) the low-frequency parts $u^{l,\\pm1,\\lambda}$ come from solving the (appropriate) equations with low-frequency initial data, and (ii) the high-frequency parts $u^{h,\\pm1,\\lambda}$ are “explicitly constructed by using oscillatory profiles,” with frequency parameter $\\lambda$ (in 2D, profile as in\n\\begin{align}\\label{1.2}\n\t\tu^{h,\\pm1,\\lambda}(x,t)=\\nabla^\\perp \\left(\\lambda^{-\\delta-s-1}\\phi\\left(\\dfrac{x_1}{\\lambda^\\delta}\\right)\\phi\\left(\\dfrac{x_2}{\\lambda^\\delta}\\right)\\sin (\\lambda x_2\\mp t)\\right).\n\t\\end{align}\n; in 3D, modified as in \\eqref{1.3}).\n\nFor resistive MHD, “due to the diffusive nature of the magnetic field evolution,” they “restrict high-frequency perturbations to the velocity field alone and maintain low-frequency components for the magnetic field”: the low-frequency pair $(u^{l,\\pm1,\\lambda}, b^{\\pm1,\\lambda})$ is obtained by solving the resistive MHD system with low-frequency initial data, while $u^{h,\\pm1,\\lambda}$ retains the oscillatory profile.\n\nThey then estimate the approximation errors induced by $u^{h,\\pm1,\\lambda}$, written as\n\\begin{align}\\label{1.4}\n\tE^{\\pm1,\\lambda}&=\\partial_t u^{h,\\pm1,\\lambda}+u^{l,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{l,\\pm1,\\lambda},\\\\\\label{1.5}\n\tF^{\\pm1,\\lambda}&=u^{h,\\pm1,\\lambda}\\cdot\\nabla b^{\\pm1,\\lambda}-b^{\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda},\n\\end{align}\nSince “direct estimation of the equation above will lead to uncontrolled error terms,” their alternative consists of three steps: (1) “express $F^{\\pm1,\\lambda}=\\mathrm{div}\\,\\tilde F^{\\pm1,\\lambda}$” and estimate $\\tilde F^{\\pm1,\\lambda}$; (2) “transfer the divergence operator to $b^{\\pm1,\\lambda}$ via integration by parts” in $L^2$ inner products; (3) use “diffusion cancellation,” where “the resulting terms are precisely canceled by exploiting the magnetic diffusion term $\\Delta b^{\\pm1,\\lambda}$.”\n\nFor nonzero constant background fields $\\mathbf{B_0}\\neq\\mathbf{0}$, they “introduce the coordinate transformation $x_2\\mapsto B_1x_2-B_2x_1$,” which “maintains all norm estimates” and yields “exact cancellation of the dominant high-frequency linear term $\\mathbf{B_0}\\cdot\\nabla u^{h,\\pm1,\\lambda}$” in the error analysis of $F^{\\pm1,\\lambda}$. They also note the key decay bound\n\\[\n\\|\\mathbf{B_0}\\cdot\\nabla u^{h,\\pm1,\\lambda}\\|_{L^2}\\le C\\lambda^{-s-\\delta}\\ll \\|u^{h,\\pm1,\\lambda}\\|_{L^2}\\le C\\lambda^{-s}\\quad (\\lambda\\to\\infty),\n\\]\nwith $\\delta>0$ giving “improved decay.”\n\nFinally, they describe the proof strategy as: “construction of approximate solutions,” “precise estimation of approximation errors,” “rigorous construction of exact solutions,” and “finishing the proof” in the cases $\\mathbf{B_0}=\\mathbf{0}$ and $\\mathbf{B_0}\\neq\\mathbf{0}$.", "expanded_theorem": "\\label{thm}\n\tLet $s>0, \\lambda>0, d=2,3$ and $T>0$. Then the data-to-solution map $(u_0,b_0)\\mapsto(u,b)$ for the equations\n\\begin{align}\\label{mhd}\n\t\\begin{cases}\n\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u,  \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}\n is non-uniformly continuous from a bounded subset in $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C\\left([0,T],H^{s}(\\mathbb{R}^d)\\right)\\times C\\left([0,T],H^{s}(\\mathbb{R}^d)\\right)$.\n\n  More precisely, for any $\\gamma>0$ and arbitrary constant magnetic field ${\\bf B_0}\\in \\mathbb{R}^d$,  there exists two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$ such that\n\t\\begin{itemize}\n\t\t\\item the solutions satisfy\n\\begin{align*}\n\t\t\t(u^{\\pm 1,\\lambda}, b^{\\pm 1,\\lambda} - {\\bf B_0}) \\in C([0,T], H^{s}(\\mathbb{R}^{d})) \\times C([0,T], H^{s}(\\mathbb{R}^{d}));\n\\end{align*}\n\t\t\\item  the initial data satisfy\n\\begin{equation}\\label{nonuniform-initial-bound}\n\t\\begin{array}{l}\n\t\t\t\t\t\t\\|u^{+1,\\lambda}(0,\\cdot)\\|_{H^{s}} + \\|b^{+1,\\lambda}(0,\\cdot) - {\\bf B_0}\\|_{H^{s}} \\leq \\gamma, \\\\[2mm]\n\t\t\t\\|u^{-1,\\lambda}(0,\\cdot)\\|_{H^{s}} + \\|b^{-1,\\lambda}(0,\\cdot) - {\\bf B_0}\\|_{H^{s}} \\leq \\gamma;\n\t\\end{array}\n\\end{equation}\n\t\t\\item and the non-uniform continuity is characterized by\n\t\t\\begin{itemize}\n\t\t\t\\item at initial time $t=0$,\n\t\t\t\\begin{align}\\label{nonuniform-initial-diff}\n\t\t\t\t\\lim_{\\lambda\\to\\infty} \\|u^{+1,\\lambda}(0,\\cdot) - u^{-1,\\lambda}(0,\\cdot)\\|_{H^{s}} = 0,\n\t\t\t\\end{align}\n\t\t\t\\item for evolution times $t>0$,\n\t\t\t\\begin{align}\\label{nonuniform-low-bound-sint}\n\t\t\t\t\\lim_{\\lambda \\to\\infty} \\|u^{+1,\\lambda}(t) - u^{-1,\\lambda}(t)\\|_{H^{s}} \\geq c\\gamma |\\sin t|,\n\t\t\t\\end{align}\n\t\t\\end{itemize}\n\t\twhere $c = c(s,d) > 0$ is a constant depending only on  $s$ and $d$.\n\t\\end{itemize}", "theorem_type": ["Existential–Universal", "Asymptotic or Limit"], "mcq": {"question": "For the incompressible resistive MHD system on $\\mathbb{R}^d$ with $d\\in\\{2,3\\}$,\n\\[\n\\begin{cases}\n\\partial_t u+u\\cdot\\nabla u+\\nabla p=b\\cdot\\nabla b,\\\\\n\\partial_t b-\\Delta b+u\\cdot\\nabla b=b\\cdot\\nabla u,\\\\\n\\operatorname{div}u=\\operatorname{div}b=0,\\\\\nu(x,0)=u_0(x),\\quad b(x,0)=b_0(x),\n\\end{cases}\n\\]\nlet the data-to-solution map send initial data $(u_0,b_0)$ to the corresponding solution $(u,b)$. Fix $s>0$ and $T>0$, and view this map as acting from a bounded subset of $H^s(\\mathbb{R}^d)\\times H^s(\\mathbb{R}^d)$ into $C([0,T],H^s(\\mathbb{R}^d))\\times C([0,T],H^s(\\mathbb{R}^d))$. Which statement holds, including the explicit sequential form of non-uniform continuity around an arbitrary constant background magnetic field ${\\bf B}_0\\in\\mathbb{R}^d$?", "correct_choice": {"label": "A", "text": "The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C([0,T],H^{s}(\\mathbb{R}^d))\\times C([0,T],H^{s}(\\mathbb{R}^d))$. More precisely, for every $\\gamma>0$ and every constant magnetic field ${\\bf B}_0\\in\\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$, indexed by $\\lambda\\to\\infty$, such that\n\\[\n(u^{\\pm1,\\lambda},\\, b^{\\pm1,\\lambda}-{\\\\bf B}_0)\\in C([0,T],H^s(\\mathbb{R}^d))\\times C([0,T],H^s(\\mathbb{R}^d)),\n\\]\nwith initial bounds\n\\[\n\\|u^{+1,\\lambda}(0)\\|_{H^s}+\\|b^{+1,\\lambda}(0)-{\\bf B}_0\\|_{H^s}\\le \\gamma,\n\\qquad\n\\|u^{-1,\\lambda}(0)\\|_{H^s}+\\|b^{-1,\\lambda}(0)-{\\bf B}_0\\|_{H^s}\\le \\gamma,\n\\]\nand such that\n\\[\n\\lim_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(0)-u^{-1,\\lambda}(0)\\|_{H^s}=0,\n\\]\nwhile for every evolution time $t>0$ (in particular, $t\\in(0,T]$),\n\\[\n\\lim_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(t)-u^{-1,\\lambda}(t)\\|_{H^s}\\ge c(s,d)\\,\\gamma\\,|\\sin t|,\n\\]\nwhere $c(s,d)>0$ depends only on $s$ and $d$."}, "choices": [{"label": "B", "text": "The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C([0,T],H^{s}(\\mathbb{R}^d))\\times C([0,T],H^{s}(\\mathbb{R}^d))$. More precisely, for every $\\gamma>0$ and every constant magnetic field ${\\bf B}_0\\in\\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$, indexed by $\\lambda\\to\\infty$, such that\n\\[\n(u^{\\pm1,\\lambda},\\, b^{\\pm1,\\lambda}-{\\bf B}_0)\\in C([0,T],H^s(\\mathbb{R}^d))\\times C([0,T],H^s(\\mathbb{R}^d)),\n\\]\nwith the same initial bounds as in the theorem, and such that\n\\[\n\\lim_{\\lambda\\to\\infty}\\Big(\\|u^{+1,\\lambda}(0)-u^{-1,\\lambda}(0)\\|_{H^s}+\\|b^{+1,\\lambda}(0)-b^{-1,\\lambda}(0)\\|_{H^s}\\Big)=0,\n\\]\nwhile for every evolution time $t>0$,\n\\[\n\\lim_{\\lambda\\to\\infty}\\Big(\\|u^{+1,\\lambda}(t)-u^{-1,\\lambda}(t)\\|_{H^s}+\\|b^{+1,\\lambda}(t)-b^{-1,\\lambda}(t)\\|_{H^s}\\Big)\\ge c(s,d)\\,\\gamma\\,|\\sin t|,\n\\]\nwhere $c(s,d)>0$ depends only on $s$ and $d$."}, {"label": "C", "text": "The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C([0,T],H^{s}(\\mathbb{R}^d))\\times C([0,T],H^{s}(\\mathbb{R}^d))$. More precisely, for every $\\gamma>0$ and every constant magnetic field ${\\bf B}_0\\in\\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$, indexed by $\\lambda\\to\\infty$, such that\n\\[\n(u^{\\pm1,\\lambda},\\, b^{\\pm1,\\lambda}-{\\bf B}_0)\\in C([0,T],H^s(\\mathbb{R}^d))\\times C([0,T],H^s(\\mathbb{R}^d)),\n\\]\nwith initial bounds\n\\[\n\\|u^{+1,\\lambda}(0)\\|_{H^s}+\\|b^{+1,\\lambda}(0)-{\\bf B}_0\\|_{H^s}\\le \\gamma,\n\\qquad\n\\|u^{-1,\\lambda}(0)\\|_{H^s}+\\|b^{-1,\\lambda}(0)-{\\bf B}_0\\|_{H^s}\\le \\gamma,\n\\]\nand such that\n\\[\n\\lim_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(0)-u^{-1,\\lambda}(0)\\|_{H^s}=0.\n\\]\nIn particular, the data-to-solution map fails to be uniformly continuous on that bounded subset."}, {"label": "D", "text": "The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C([0,T],H^{s}(\\mathbb{R}^d))\\times C([0,T],H^{s}(\\mathbb{R}^d))$. More precisely, there exists a constant $c=c(s,d)>0$ such that for every $\\gamma>0$ one can choose two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$ with the property that for every constant magnetic field ${\\bf B}_0\\in\\mathbb{R}^d$,\n\\[\n(u^{\\pm1,\\lambda},\\, b^{\\pm1,\\lambda}-{\\bf B}_0)\\in C([0,T],H^s(\\mathbb{R}^d))\\times C([0,T],H^s(\\mathbb{R}^d)),\n\\]\nwith the same initial bounds, and such that\n\\[\n\\lim_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(0)-u^{-1,\\lambda}(0)\\|_{H^s}=0,\n\\]\nwhile for every evolution time $t>0$,\n\\[\n\\lim_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(t)-u^{-1,\\lambda}(t)\\|_{H^s}\\ge c\\,\\gamma\\,|\\sin t|.\n\\]"}, {"label": "E", "text": "The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C([0,T],H^{s}(\\mathbb{R}^d))\\times C([0,T],H^{s}(\\mathbb{R}^d))$. More precisely, for every $\\gamma>0$ and every constant magnetic field ${\\bf B}_0\\in\\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$, indexed by $\\lambda\\to\\infty$, such that\n\\[\n(u^{\\pm1,\\lambda},\\, b^{\\pm1,\\lambda}-{\\bf B}_0)\\in C([0,T],H^s(\\mathbb{R}^d))\\times C([0,T],H^s(\\mathbb{R}^d)),\n\\]\nwith the same initial bounds, and such that\n\\[\n\\lim_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(0)-u^{-1,\\lambda}(0)\\|_{H^s}=0,\n\\]\nwhile for every evolution time $t\\in[0,T]$,\n\\[\n\\liminf_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(t)-u^{-1,\\lambda}(t)\\|_{H^s}\\ge c(s,d)\\,\\gamma,\n\\]\nwhere $c(s,d)>0$ depends only on $s$ and $d$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "diffusion cancellation", "tampered_component": "velocity-only high-frequency instability promoted to joint $(u,b)$ separation", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the explicit positive-time lower bound $\\ge c\\gamma|\\sin t|$ and retained only failure of uniform continuity", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "quantifier order in dependence on ${\\bf B}_0$", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "time-dependent lower bound $|\\sin t|$ replaced by uniform-in-time positive separation on $[0,T]$", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct option explicitly or through obvious phrasing cues. It asks for an explicit asymptotic statement, and the learner must inspect quantifiers, norms, and which component exhibits separation."}, "TAS": {"score": 1, "justification": "This is largely a theorem-statement recognition item: the task is to identify the precise asymptotic formulation of a known non-uniform continuity result. The alternatives introduce meaningful variations, but the question still remains close to restating the theorem rather than applying it in a new setting."}, "GPS": {"score": 1, "justification": "Some reasoning is required to compare subtle differences among the options, especially quantifier order, strength of conclusion, and whether separation is asserted for u or b. However, the item mainly tests precise recall/discrimination rather than substantial generative mathematical reasoning."}, "DQS": {"score": 1, "justification": "Most distractors are plausible and target realistic failure modes: weakening liminf to limsup, changing quantifiers, or confusing velocity and magnetic-field instability. However, option C is a weaker statement that is still implied by the correct theorem, so it is not a clean false distractor and creates answer ambiguity."}, "total_score": 5, "overall_assessment": "A technically sophisticated but only moderately strong MCQ: it avoids answer leakage and uses plausible distractors, yet it is close to theorem recall and is weakened by ambiguity because one distractor is also true in a weaker form."}}
{"id": "2602.06332v1", "paper_link": "http://arxiv.org/abs/2602.06332v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm}\n\tLet $s>0, \\lambda>0, d=2,3$ and $T>0$. Then the data-to-solution map $(u_0,b_0)\\mapsto(u,b)$ for the equations \\eqref{mhd} is non-uniformly continuous from a bounded subset in $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C\\left([0,T],H^{s}(\\mathbb{R}^d)\\right)\\times C\\left([0,T],H^{s}(\\mathbb{R}^d)\\right)$.\n\n  More precisely, for any $\\gamma>0$ and arbitrary constant magnetic field ${\\bf B_0}\\in \\mathbb{R}^d$,  there exists two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$ such that\n\t\\begin{itemize}\n\t\t\\item the solutions satisfy\n\\begin{align*}\n\t\t\t(u^{\\pm 1,\\lambda}, b^{\\pm 1,\\lambda} - {\\bf B_0}) \\in C([0,T], H^{s}(\\mathbb{R}^{d})) \\times C([0,T], H^{s}(\\mathbb{R}^{d}));\n\\end{align*}\n\t\t\\item  the initial data satisfy\n\\begin{equation}\\label{nonuniform-initial-bound}\n\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\|u^{+1,\\lambda}(0,\\cdot)\\|_{H^{s}} + \\|b^{+1,\\lambda}(0,\\cdot) - {\\bf B_0}\\|_{H^{s}} \\leq \\gamma, \\\\[2mm]\n\t\t\t\\|u^{-1,\\lambda}(0,\\cdot)\\|_{H^{s}} + \\|b^{-1,\\lambda}(0,\\cdot) - {\\bf B_0}\\|_{H^{s}} \\leq \\gamma;\n\t\\end{array}\n\\end{equation}\n\t\t\\item and the non-uniform continuity is characterized by\n\t\t\\begin{itemize}\n\t\t\t\\item at initial time $t=0$,\n\t\t\t\\begin{align}\\label{nonuniform-initial-diff}\n\t\t\t\t\\lim_{\\lambda\\to\\infty} \\|u^{+1,\\lambda}(0,\\cdot) - u^{-1,\\lambda}(0,\\cdot)\\|_{H^{s}} = 0,\n\t\t\t\\end{align}\n\t\t\t\\item for evolution times $t>0$,\n\t\t\t\\begin{align}\\label{nonuniform-low-bound-sint}\n\t\t\t\t\\lim_{\\lambda \\to\\infty} \\|u^{+1,\\lambda}(t) - u^{-1,\\lambda}(t)\\|_{H^{s}} \\geq c\\gamma |\\sin t|,\n\t\t\t\\end{align}\n\t\t\\end{itemize}\n\t\twhere $c = c(s,d) > 0$ is a constant depending only on  $s$ and $d$.\n\t\\end{itemize}", "start_pos": 10028, "end_pos": 11702, "label": "thm"}, "ref_dict": {"1.2": "\\begin{align}\\label{1.2}\n\t\tu^{h,\\pm1,\\lambda}(x,t)=\\nabla^\\perp \\left(\\lambda^{-\\delta-s-1}\\phi\\left(\\dfrac{x_1}{\\lambda^\\delta}\\right)\\phi\\left(\\dfrac{x_2}{\\lambda^\\delta}\\right)\\sin (\\lambda x_2\\mp t)\\right).\n\t\\end{align}", "mhd-B": "\\begin{align}\\label{mhd-B}\n\t\\begin{cases}\n\t\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b+{\\bf B_0}\\cdot\\nabla b,\\\\[1mm]\n\t\t\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u+{\\bf B_0}\\cdot\\nabla u,  \\\\[1mm]\n\t\t\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\n\t\tu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}", "1.5": "\\begin{align}\\label{1.4}\n\tE^{\\pm1,\\lambda}&=\\partial_t u^{h,\\pm1,\\lambda}+u^{l,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{l,\\pm1,\\lambda},\\\\\\label{1.5}\n\tF^{\\pm1,\\lambda}&=u^{h,\\pm1,\\lambda}\\cdot\\nabla b^{\\pm1,\\lambda}-b^{\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda},\n\\end{align}", "mhd": "\\begin{align}\\label{mhd}\n\t\\begin{cases}\n\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u,  \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}", "thm": "\\begin{theorem}\\label{thm}\n\tLet $s>0, \\lambda>0, d=2,3$ and $T>0$. Then the data-to-solution map $(u_0,b_0)\\mapsto(u,b)$ for the equations \\eqref{mhd} is non-uniformly continuous from a bounded subset in $H^{s}(\\mr^d)\\times H^{s}(\\mr^d)$ into $C\\left([0,T],H^{s}(\\mr^d)\\right)\\times C\\left([0,T],H^{s}(\\mr^d)\\right)$.\n\n  More precisely, for any $\\gamma>0$ and arbitrary constant magnetic field ${\\bf B_0}\\in \\mr^d$,  there exists two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$ such that\n\t\\begin{itemize}\n\t\t\\item the solutions satisfy\n\\begin{align*}\n\t\t\t(u^{\\pm 1,\\lambda}, b^{\\pm 1,\\lambda} - {\\bf B_0}) \\in C([0,T], H^{s}(\\mathbb{R}^{d})) \\times C([0,T], H^{s}(\\mathbb{R}^{d}));\n\\end{align*}\n\t\t\\item  the initial data satisfy\n\\begin{equation}\\label{nonuniform-initial-bound}\n\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\|u^{+1,\\lambda}(0,\\cdot)\\|_{H^{s}} + \\|b^{+1,\\lambda}(0,\\cdot) - {\\bf B_0}\\|_{H^{s}} \\leq \\gamma, \\\\[2mm]\n\t\t\t\\|u^{-1,\\lambda}(0,\\cdot)\\|_{H^{s}} + \\|b^{-1,\\lambda}(0,\\cdot) - {\\bf B_0}\\|_{H^{s}} \\leq \\gamma;\n\t\\end{array}\n\\end{equation}\n\t\t\\item and the non-uniform continuity is characterized by\n\t\t\\begin{itemize}\n\t\t\t\\item at initial time $t=0$,\n\t\t\t\\begin{align}\\label{nonuniform-initial-diff}\n\t\t\t\t\\lim_{\\lambda\\to\\infty} \\|u^{+1,\\lambda}(0,\\cdot) - u^{-1,\\lambda}(0,\\cdot)\\|_{H^{s}} = 0,\n\t\t\t\\end{align}\n\t\t\t\\item for evolution times $t>0$,\n\t\t\t\\begin{align}\\label{nonuniform-low-bound-sint}\n\t\t\t\t\\lim_{\\lambda \\to\\infty} \\|u^{+1,\\lambda}(t) - u^{-1,\\lambda}(t)\\|_{H^{s}} \\geq c\\gamma |\\sin t|,\n\t\t\t\\end{align}\n\t\t\\end{itemize}\n\t\twhere $c = c(s,d) > 0$ is a constant depending only on  $s$ and $d$.\n\t\\end{itemize}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 7134, "pre_theorem_intro_text": "\\,\\,\\,\\,\\,\\,\\,\\, In this paper, we consider the Cauchy problem for the incompressible MHD equations with only magnetic diffusion (hereinafter called the resistive MHD equations):\n\\begin{align}\\label{mhd}\n\t\\begin{cases}\n\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u,  \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}\nwhere $u(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}^d$,\n$b(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}^d$,\nand $p(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}$\ndenote the velocity field, magnetic field, and pressure field, respectively,\nwith $x \\in \\mathbb{R}^d~(d=2,3)$ and $t>0$. Clearly, when $b\\equiv0$, \\eqref{mhd} reduces to the classical Euler equations.\n\nThe resistive MHD equations provide a fundamental framework for modeling key plasma phenomena where finite electrical resistivity plays a critical role, particularly in astrophysical magnetic reconnection processes that govern energy release in solar flares and magnetospheric activity, as well as in the geodynamo mechanisms responsible for generating and sustaining planetary magnetic fields \\cite{priest-2000,roberts1967introduction}.\n\n There have been a large number of mathematical studies on the  well-posedness theory for the incompressible MHD equations under various assumptions on velocity dissipation and magnetic diffusion.\nPrevious results on local well-posedness can be found in \\cite{sermange-1983,jiu-2006-local,li-local-nonresistive-2017-adv,chemin-local-nonresistive-2016-adv,fefferman-local-nonressitive-2014-jfa,fefferman-local-nonressitive-2017-arma}. The pioneering work of Bardos, Sulem and Sulem \\cite{full-not-bardos} and Lin and Zhang \\cite{lin-2014-GlobalSmallSolutions} on perturbation theory near constant background magnetic fields first revealed the stabilizing role of such equilibrium configurations in the MHD equations. Physical interpretations of this stabilization effect are discussed in \\cite{background-phy1,background-phy2}. These foundational results have inspired extensive research on global well-posedness and stability of the MHD equations, as documented in \\cite{lei-bkm-2009-dcds,full-1-duvaut-lions,sermange-1983,cao-wu-1-mix,cao-wu-2-mix,full-not-bardos,lin-2014-GlobalSmallSolutions,chen-2022-3dmhd-Diophant,deng-zhang-2018-decay,panGlobalClassicalSolutions2018,ren2014global,zhang-2016-non-resistive-global-jde,xie-2024-cvpde-dio,wei--global-resistive-2020-cmr,zhou-zhu-glbal-symmetry-jmp-2018,ye-global-resistive-2022-acta} and references therein.\n\n  In recent years, ill-posedness theory  for the incompressible MHD equations has also been invetigated. For the non-resistive MHD equations  which contain velocity dissipation but  no magnetic diffusion, Chen, Nie and Ye \\cite{chen-ye-sharp-ill-nonresistive-2024-jfa}  established sharp strong ill-posedness results that provide a striking contrast to the local well-posedness theory developed by Fefferman et al. \\cite{fefferman-local-nonressitive-2017-arma}. For the resistive case \\eqref{mhd}, Wu and Zhao \\cite{wu-zhao-mild-resistive-2023-IMRN}  obtained mild ill-posedness results near the background magnetic field $(1,0)$ in $\\mathbb{R}^2$.\n\nIt is noted that the concept of ill-posedness manifests rather strongly in many fundamental cases. As evidenced by \\cite{chen-ye-sharp-ill-nonresistive-2024-jfa,wu-zhao-mild-resistive-2023-IMRN} and other works of PDE systems, such behavior typically occurs in critical or supercritical function spaces, while  some problems remain unresolved to this day. These substantial difficulties have motivated researchers to consider relaxed notions of ill-posedness by examining weaker properties.\n For certain equations, the solution operator may exhibit non-uniform continuity properties under stronger topological frameworks, which provides meaningful insights into the refined continuity structure of solution mappings and can be seen a kind of instability of the solutions or ill-posedness the equations.\n\nIn this paper, we are concerned with the non-uniform continuity properties of the data-to-solution map for the resistive MHD equations. We begin by precisely defining the notion of non-uniform continuity as follows:\n\\begin{definition}\\label{define-nonuniform}\n\tLet $X$ be a Banach space, and consider the Cauchy problem:\n\t\\begin{align*}\n\t\t\\begin{cases}\n\t\t\t\\partial_t v = N(v), \\\\\n\t\t\tv(0) = v_0,\n\t\t\\end{cases}\n\t\\end{align*}\n\twhere $N$ is a (possibly nonlinear) differential operator. The \\textbf{data-to-solution map} $\\Phi_t \\colon X \\to X$ (for fixed $t > 0$) defined by $\\Phi_t(v_0) = v(t)$ is said to be \\textbf{non-uniformly continuous} on $X$ if the following holds:\n\n\t\\noindent\n\t\\textbf{Sequential Formulation:} \\\\\n\tFor every $t > 0$, there exists $\\epsilon_0 > 0$ and sequences $\\{v_{1,n}(0)\\}, \\{v_{2,n}(0)\\} \\subset X$ such that:\n\t\\begin{align*}\n\t\t\\lim_{n \\to \\infty} \\| v_{1,n}(0) - v_{2,n}(0) \\|_X = 0,\n\t\\end{align*}\n\tbut\n\t\\begin{align*}\n\t\t\\limsup_{n \\to \\infty} \t\\| \\Phi_t(v_{1,n}(0)) - \\Phi_t(v_{2,n}(0)) \\|_X \\geq \\epsilon_0.\n\t\\end{align*}\n\n\t\\noindent\n\t\\textbf{$\\delta$-$\\epsilon$ Formulation (Equivalent):} \\\\\n\tFor every $t > 0$, there exists $\\epsilon_0 > 0$ such that for any $\\delta > 0$, one can find initial data $v_1(0), v_2(0) \\in X$ satisfying:\n\t\\begin{align*}\n\t\t\\| v_1(0) - v_2(0) \\|_X < \\delta,\n\t\\end{align*}\n\tbut\n\t\\begin{align*}\n\t\t\\| \\Phi_t(v_1(0)) - \\Phi_t(v_2(0)) \\|_X \\geq \\epsilon_0.\n\t\\end{align*}\n\\end{definition}\n\nIn \\cite{nonuniform-2010-cmp-himonas}, Himonas and Misiołek first  proved non-uniform continuity of the data-to-solution map  on the  incompressible Euler equations in both $H^s(\\mathbb{R}^d)$ ($s > 0$) and $H^r(\\mathbb{T}^d)$ ($r \\in \\mathbb{R}$) with $d = 2,3$, which was later extended by Li and Bourgain \\cite{nonuniform-2019-cmp-bourgain-li} through Galilean boost techniques to the endpoint case $s \\geq 0$ in $H^s(\\mathbb{R}^d)$, where they further proved the stronger property of nowhere uniform continuity.\nFor the non-resistive MHD equations with only velocity dissipation, recent work by Li, Yin, and Zhu \\cite{nonuniforem-2023-adv-li-yin-zhu} demonstrated non-uniform continuity in $H^s(\\mathbb{R}^d)$ for $s > \\frac{d}{2}$.\n\n  We will prove in this paper the non-uniform continuity of the data-to-solution map for the resistive MHD equations in Sobolev spaces $H^s(\\mathbb{R}^d)$ for all $s>0$ and  $d=2,3$. In comparison with the work by Li, Yin, and Zhu \\cite{nonuniforem-2023-adv-li-yin-zhu} which is for non-resistive MHD equations with only velocity dissipation and in $H^s(\\mathbb{R}^d)$ for $s > \\frac{d}{2}$,  our results hold for the resistive MHD equations and  in  $H^s(\\mathbb{R}^d)$ for all $s>0$ and  $d=2,3$.  Moreover, our results permit the solution perturbation around an arbitrary constant background magnetic fields $\\mathbf{B_0} \\in \\mathbb{R}^d$, which shows that the strong magnetic field may provide the stabilization effect but no help for the uniform continuity of the data-to-solution map.\n\nOur main results can be stated as follows:", "context": "\\,\\,\\,\\,\\,\\,\\,\\, In this paper, we consider the Cauchy problem for the incompressible MHD equations with only magnetic diffusion (hereinafter called the resistive MHD equations):\n\\begin{align}\\label{mhd}\n \\begin{cases}\n \\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u, \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n \\end{cases}\n\\end{align}\nwhere $u(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}^d$,\n$b(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}^d$,\nand $p(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}$\ndenote the velocity field, magnetic field, and pressure field, respectively,\nwith $x \\in \\mathbb{R}^d~(d=2,3)$ and $t>0$. Clearly, when $b\\equiv0$, \\eqref{mhd} reduces to the classical Euler equations.\n\nIn this paper, we are concerned with the non-uniform continuity properties of the data-to-solution map for the resistive MHD equations. We begin by precisely defining the notion of non-uniform continuity as follows:\n\\begin{definition}\\label{define-nonuniform}\n Let $X$ be a Banach space, and consider the Cauchy problem:\n \\begin{align*}\n \\begin{cases}\n \\partial_t v = N(v), \\\\\n v(0) = v_0,\n \\end{cases}\n \\end{align*}\n where $N$ is a (possibly nonlinear) differential operator. The \\textbf{data-to-solution map} $\\Phi_t \\colon X \\to X$ (for fixed $t > 0$) defined by $\\Phi_t(v_0) = v(t)$ is said to be \\textbf{non-uniformly continuous} on $X$ if the following holds:\n\n\\noindent\n \\textbf{Sequential Formulation:} \\\\\n For every $t > 0$, there exists $\\epsilon_0 > 0$ and sequences $\\{v_{1,n}(0)\\}, \\{v_{2,n}(0)\\} \\subset X$ such that:\n \\begin{align*}\n \\lim_{n \\to \\infty} \\| v_{1,n}(0) - v_{2,n}(0) \\|_X = 0,\n \\end{align*}\n but\n \\begin{align*}\n \\limsup_{n \\to \\infty} \\| \\Phi_t(v_{1,n}(0)) - \\Phi_t(v_{2,n}(0)) \\|_X \\geq \\epsilon_0.\n \\end{align*}\n\n\\noindent\n \\textbf{$\\delta$-$\\epsilon$ Formulation (Equivalent):} \\\\\n For every $t > 0$, there exists $\\epsilon_0 > 0$ such that for any $\\delta > 0$, one can find initial data $v_1(0), v_2(0) \\in X$ satisfying:\n \\begin{align*}\n \\| v_1(0) - v_2(0) \\|_X < \\delta,\n \\end{align*}\n but\n \\begin{align*}\n \\| \\Phi_t(v_1(0)) - \\Phi_t(v_2(0)) \\|_X \\geq \\epsilon_0.\n \\end{align*}\n\\end{definition}\n\nWe will prove in this paper the non-uniform continuity of the data-to-solution map for the resistive MHD equations in Sobolev spaces $H^s(\\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. In comparison with the work by Li, Yin, and Zhu \\cite{nonuniforem-2023-adv-li-yin-zhu} which is for non-resistive MHD equations with only velocity dissipation and in $H^s(\\mathbb{R}^d)$ for $s > \\frac{d}{2}$, our results hold for the resistive MHD equations and in $H^s(\\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. Moreover, our results permit the solution perturbation around an arbitrary constant background magnetic fields $\\mathbf{B_0} \\in \\mathbb{R}^d$, which shows that the strong magnetic field may provide the stabilization effect but no help for the uniform continuity of the data-to-solution map.\n\nOur main results can be stated as follows:\n\n\\begin{align}\\label{mhd}\n\t\\begin{cases}\n\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u,  \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}", "full_context": "\\,\\,\\,\\,\\,\\,\\,\\, In this paper, we consider the Cauchy problem for the incompressible MHD equations with only magnetic diffusion (hereinafter called the resistive MHD equations):\n\\begin{align}\\label{mhd}\n \\begin{cases}\n \\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u, \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n \\end{cases}\n\\end{align}\nwhere $u(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}^d$,\n$b(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}^d$,\nand $p(x,t) \\colon \\mathbb{R}^d \\times \\mathbb{R}_+ \\to \\mathbb{R}$\ndenote the velocity field, magnetic field, and pressure field, respectively,\nwith $x \\in \\mathbb{R}^d~(d=2,3)$ and $t>0$. Clearly, when $b\\equiv0$, \\eqref{mhd} reduces to the classical Euler equations.\n\nIn this paper, we are concerned with the non-uniform continuity properties of the data-to-solution map for the resistive MHD equations. We begin by precisely defining the notion of non-uniform continuity as follows:\n\\begin{definition}\\label{define-nonuniform}\n Let $X$ be a Banach space, and consider the Cauchy problem:\n \\begin{align*}\n \\begin{cases}\n \\partial_t v = N(v), \\\\\n v(0) = v_0,\n \\end{cases}\n \\end{align*}\n where $N$ is a (possibly nonlinear) differential operator. The \\textbf{data-to-solution map} $\\Phi_t \\colon X \\to X$ (for fixed $t > 0$) defined by $\\Phi_t(v_0) = v(t)$ is said to be \\textbf{non-uniformly continuous} on $X$ if the following holds:\n\n\\noindent\n \\textbf{Sequential Formulation:} \\\\\n For every $t > 0$, there exists $\\epsilon_0 > 0$ and sequences $\\{v_{1,n}(0)\\}, \\{v_{2,n}(0)\\} \\subset X$ such that:\n \\begin{align*}\n \\lim_{n \\to \\infty} \\| v_{1,n}(0) - v_{2,n}(0) \\|_X = 0,\n \\end{align*}\n but\n \\begin{align*}\n \\limsup_{n \\to \\infty} \\| \\Phi_t(v_{1,n}(0)) - \\Phi_t(v_{2,n}(0)) \\|_X \\geq \\epsilon_0.\n \\end{align*}\n\n\\noindent\n \\textbf{$\\delta$-$\\epsilon$ Formulation (Equivalent):} \\\\\n For every $t > 0$, there exists $\\epsilon_0 > 0$ such that for any $\\delta > 0$, one can find initial data $v_1(0), v_2(0) \\in X$ satisfying:\n \\begin{align*}\n \\| v_1(0) - v_2(0) \\|_X < \\delta,\n \\end{align*}\n but\n \\begin{align*}\n \\| \\Phi_t(v_1(0)) - \\Phi_t(v_2(0)) \\|_X \\geq \\epsilon_0.\n \\end{align*}\n\\end{definition}\n\nWe will prove in this paper the non-uniform continuity of the data-to-solution map for the resistive MHD equations in Sobolev spaces $H^s(\\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. In comparison with the work by Li, Yin, and Zhu \\cite{nonuniforem-2023-adv-li-yin-zhu} which is for non-resistive MHD equations with only velocity dissipation and in $H^s(\\mathbb{R}^d)$ for $s > \\frac{d}{2}$, our results hold for the resistive MHD equations and in $H^s(\\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. Moreover, our results permit the solution perturbation around an arbitrary constant background magnetic fields $\\mathbf{B_0} \\in \\mathbb{R}^d$, which shows that the strong magnetic field may provide the stabilization effect but no help for the uniform continuity of the data-to-solution map.\n\nOur main results can be stated as follows:\n\n\\begin{align}\\label{mhd}\n\t\\begin{cases}\n\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u,  \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}\n\n\\section{Preliminaries}\\label{sec:2}\n\\subsection{Auxiliary Analysis Tools}\\label{sec:2.1}\nIn this subsection, we introduce two key lemmas needed later. The first one is\n\\begin{lemma}\\cite{nonuniform-2010-cmp-himonas}\\label{phi-cmp-2010}\n Let $\\sigma \\geq 0, \\delta\\geq 0, a\\in \\mr$ and $\\lambda\\gg 1$. For any Schwartz function $\\psi\\in \\mathcal{S}(\\mr)$, it holds that\n \\begin{align*}\n \\lambda^{\\delta/2}\\left\\|\\psi\\right\\|_{L^2(\\mathbb{R})}\\leq\\left\\|\\psi\\left(\\frac{\\cdot}{\\lambda^\\delta}\\right)\\right\\|_{H^\\sigma(\\mathbb{R})}\\leq\\lambda^{\\delta/2}\\left\\|\\psi\\right\\|_{H^\\sigma(\\mathbb{R})},\n \\end{align*}\nand \\begin{align}\\label{622}\n \\left\\|\\psi\\left(\\frac{\\cdot}{\\lambda^\\delta}\\right)\\cos(\\lambda\\cdot-a)\\right\\|_{H^\\sigma(\\mathbb{R})}\\simeq\\lambda^{\\sigma+\\delta/2}\\|\\psi\\|_{L^2(\\mathbb{R})}.\n\\end{align}\nMoreover, \\eqref{622} holds true if $\\cos(\\lambda \\cdot-a)$ is replaced by $\\sin(\\lambda \\cdot-a)$.\n\\end{lemma}\n\n\\subsection{Approximate Solutions Scheme}\\label{sec:3.1}\nWe begin by constructing two sequences of approximate solutions $b^{w,\\lambda}$ and $u^{w,\\lambda}$ with $w=\\pm 1$, where $b^{w,\\lambda}$ contains low frequencies only and $u^{w,\\lambda}$ contain both high and low frequencies. More precisely, we set\n\\begin{align}\\label{u-low+high}\n u^{w,\\lambda}(x,t)=u^{l,w,\\lambda}(x,t)+u^{h,w,\\lambda}(x,t).\n\\end{align}\nThe high-frequency components $u^{h,w,\\lambda}$ are constructed as\n\\begin{align}\\label{u-high}\n u^{h,w,\\lambda}(x,t)=\\begin{cases}\n \\nabla^\\perp \\phi^{h,w,\\lambda}(x,t)=(\\partial_{x_2}\\phi^{h,w,\\lambda}(x,t),-\\partial_{x_1}\\phi^{h,w,\\lambda}(x,t)), &~d=2,\\\\[3mm]\n \\left(\\nabla^\\perp \\phi^{h,w,\\lambda}(x,t),0 \\right)=(\\partial_{x_2}\\phi^{h,w,\\lambda}(x,t),-\\partial_{x_1}\\phi^{h,w,\\lambda}(x,t),0), &~d=3.\n \\end{cases}\n\\end{align}\nIn \\eqref{u-high}, the function $\\phi^{h,w,\\lambda}(x,t)$ is defined as\n\\begin{align}\\label{phi-high}\n \\phi^{h,w,\\lambda}(x,t)=\\begin{cases}\n \\lambda^{-s-\\delta-1}\\phi(\\dfrac{x_1}{\\lambda^\\delta})\\phi(\\dfrac{x_2}{\\lambda^\\delta})\\sin (\\lambda x_2-wt),&~d=2,\\\\[3mm]\n \\lambda^{-s-\\delta-1}\\phi(\\dfrac{x_1}{\\lambda^\\delta})\\phi(\\dfrac{x_2}{\\lambda^\\delta})\\sin (\\lambda x_2-wt)\\phi(x_3),&~d=3,\n \\end{cases}\n\\end{align}\nwhere $s > 0, \\lambda>0, w=\\pm 1$, the parameter $\\delta > 0$ is to be specified later and the smooth function $\\phi \\in C_c^\\infty(\\mathbb{R})$ satisfies $\\text{supp}\\,\\phi \\subset [-2,2]$ and $\\phi(x) \\equiv 1$ on $|x| < 1$.\n\nFollowing similar estimates established in \\eqref{u,b-low-t-local-bound} and \\eqref{u-high-bound}, we can obtain\n\\begin{align}\\label{4.5}\n \\|\\bar{u}^{l,w,\\lambda}(t)\\|_{H^m} + \\|\\bar{b}^{w,\\lambda}(t)\\|_{H^{m}}\n \\leq C\\lambda^{-1+\\delta},\n\\end{align}\nand\n\\begin{align}\\label{4.6}\n \\|\\bar{u}^{h,w,\\lambda}(t)\\|_{H^r}\\leq C\\lambda^{r-s},\\quad \\|\\bar{u}^{h,w,\\lambda}(t)\\|_{L^\\infty}\\leq C\\lambda^{-s-\\delta},\\quad \\|\\nabla \\bar{u}^{h,w,\\lambda}(t)\\|_{L^\\infty}\\leq C\\lambda^{-s+1-\\delta}\n\\end{align}\nfor any $0 \\leq t \\leq T\\leq 1$, where $[0,T]$ denotes the existence interval of the solution guaranteed by Lemma~\\ref{local-well-pri}.\n\\subsection{Error Estimates for the Approximate Solutions}\\label{sec:4.2}\nFollowing the approach in subsection \\ref{sec:3.2} and based on system \\eqref{mhd-B}, we construct a perturbed system as follows\n\\begin{align}\\label{4.9}\n \\begin{cases}\n \\partial_t \\bar{u}^{w,\\lambda}+\\bar{u}^{w,\\lambda} \\cdot \\nabla \\bar{u}^{w,\\lambda}+\\nabla p^{w,\\lambda}=\\bar{b}^{w,\\lambda}\\cdot \\nabla \\bar{b}^{w,\\lambda}+{\\bf B_0}\\cdot\\nabla \\bar{b}^{w,\\lambda}+E_1^{w,\\lambda},\\\\[1mm]\n \\partial_t \\bar{b}^{w,\\lambda}-\\Delta \\bar{b}^{w,\\lambda}+\\bar{u}^{w,\\lambda}\\cdot \\nabla \\bar{b}^{w,\\lambda}=\\bar{b}^{w,\\lambda}\\cdot\\nabla \\bar{u}^{w,\\lambda}+{\\bf B_0}\\cdot\\nabla \\bar{u}^{w,\\lambda}+F_1^{w,\\lambda}, \\\\[1mm]\n \\diver\\, \\bar{u}^{w,\\lambda}=\\diver\\, \\bar{b}^{w,\\lambda}=0,\n \\end{cases}\n\\end{align}\nwhere\n\\begin{align*}\n E_1^{w,\\lambda}&=\\partial_t \\bar{u}^{h,w,\\lambda}+\\bar{u}^{l,w,\\lambda}\\cdot\\nabla \\bar{u}^{h,w,\\lambda}+\\bar{u}^{h,w,\\lambda}\\cdot\\nabla \\bar{u}^{h,w,\\lambda}+\\bar{u}^{h,w,\\lambda}\\cdot\\nabla \\bar{u}^{l,w,\\lambda},\\\\\\nonumber\n F_1^{w,\\lambda}&=\\bar{u}^{h,w,\\lambda}\\cdot\\nabla \\bar{b}^{w,\\lambda}-\\bar{b}^{w,\\lambda}\\cdot\\nabla \\bar{u}^{h,w,\\lambda}-{\\bf B_0}\\cdot \\nabla \\bar{u}^{h,w,\\lambda}\\\\\n &\\define\\diver \\tilde{F}_1^{w,\\lambda}-{\\bf B_0}\\cdot \\nabla \\bar{u}^{h,w,\\lambda}.\n\\end{align*}\nThe perturbed system \\eqref{4.9} is analogous to \\eqref{mhd-low+high}. The directional derivative of the high-frequency component along the background magnetic field ${\\bf B_0} = (B_1, B_2)$ is\n\\begin{align*}\n {\\bf B_0} \\cdot \\nabla \\bar{u}^{h,w,\\lambda} = B_1\\partial_{x_1}\\bar{u}^{h,w,\\lambda} + B_2\\partial_{x_2}\\bar{u}^{h,w,\\lambda}.\n\\end{align*}\nRemarkably, we observe that the oscillatory part in $\\bar{u}^{h,w,\\lambda}$ satisfies\n\\begin{align*}\n {\\bf B_0} \\cdot \\nabla \\left[\\phi\\left(\\frac{B_1x_2 - B_2x_1}{\\lambda^\\delta}\\right) \\cos\\left(\\lambda(B_1x_2 - B_2x_1) - wt\\right)\\right] = 0,\n\\end{align*}\nwhich leads to the following simplified expression\n\\begin{align*}\n & {\\bf B_0}\\cdot \\nabla \\left(\\lambda^{-s-\\delta}\\phi\\left(\\dfrac{x_1}{\\lambda^\\delta}\\right)\n \\phi\\left(\\dfrac{B_1x_2-B_2x_1}{\\lambda^\\delta}\\right)\\cos \\left(\\lambda \\left(B_1x_2-B_2x_1\\right)-wt\\right)\\right)\\\\\n &=B_1\\lambda^{-s-2\\delta}\\phi'\\left(\\frac{x_1}{\\lambda^{\\delta}}\\right) \\phi\\left(\\dfrac{B_1x_2-B_2x_1}{\\lambda^\\delta}\\right)\\cos \\left(\\lambda \\left(B_1x_2-B_2x_1\\right)-wt\\right)\n\\end{align*}\nTherefore, following analogous arguments as \\eqref{4.3}, we can obtain\n\\begin{align}\\label{4.7}\n \\left\\| {\\bf B_0}\\cdot\\nabla \\bar{u}^{h,w,\\lambda}\\right\\|_{L^2}\\leq C\\lambda^{-s-\\delta}.\n\\end{align}\nCombining the estimates \\eqref{4.5}, \\eqref{4.6} with Lemma \\ref{E,F-error-estimate}, we deduce that\n\\begin{align}\\label{4.8}\n \\|E_1^{w,\\lambda}(t)\\|_{L^2}\\leq C\\lambda^{-\\sigma_{s,\\delta}},\\quad \\|\\tilde{F}_1^{w,\\lambda}(t)\\|_{L^2}\\leq C\\lambda^{-s-1},\n\\end{align}\nwhere \\begin{align*}\n \\sigma_{s,\\delta}=\\min\\{s+1-\\delta,2s-1+\\delta\\}.\n\\end{align*}\n\\subsection{Exact Solutions}\\label{sec:4.3}\nLet $(\\bar{u}_{w,\\lambda},\\bar{b}_{w,\\lambda})$ be the unique solution to system \\eqref{mhd-B} with the initial data $\\left(\\bar{u}^{w,\\lambda}(x,0),\\bar{b}^{w,\\lambda}(x,0)\\right)$, satisfying\n\\begin{align*}\n \\begin{cases}\n \\partial_t \\bar{u}_{w,\\lambda}+\\bar{u}_{w,\\lambda} \\cdot \\nabla \\bar{u}_{w,\\lambda}+\\nabla p_{w,\\lambda}=\\bar{b}_{w,\\lambda}\\cdot \\nabla \\bar{b}_{w,\\lambda}+{\\bf B_0}\\cdot \\nabla \\bar{b}_{w,\\lambda},\\\\[1mm]\n \\partial_t \\bar{b}_{w,\\lambda}-\\Delta \\bar{b}_{w,\\lambda}+\\bar{u}_{w,\\lambda}\\cdot \\nabla \\bar{b}_{w,\\lambda}=\\bar{b}_{w,\\lambda}\\cdot\\nabla \\bar{u}_{w,\\lambda}+{\\bf B_0}\\cdot \\nabla \\bar{u}_{w,\\lambda}, \\\\[1mm]\n \\diver\\, \\bar{u}_{w,\\lambda}=\\diver\\, \\bar{b}_{w,\\lambda}=0,\n \\end{cases}\n\\end{align*}\nIt follows from \\eqref{4.9} that\n\\begin{align*}\n \\begin{cases}\n \\partial_t \\bar{u}^{w,\\lambda}+\\bar{u}^{w,\\lambda} \\cdot \\nabla \\bar{u}^{w,\\lambda}+\\nabla p^{w,\\lambda}=\\bar{b}^{w,\\lambda}\\cdot \\nabla \\bar{b}^{w,\\lambda}+{\\bf B_0}\\cdot\\nabla \\bar{b}^{w,\\lambda}+E_1^{w,\\lambda},\\\\[1mm]\n \\partial_t \\bar{b}^{w,\\lambda}-\\Delta \\bar{b}^{w,\\lambda}+\\bar{u}^{w,\\lambda}\\cdot \\nabla \\bar{b}^{w,\\lambda}=\\bar{b}^{w,\\lambda}\\cdot\\nabla \\bar{u}^{w,\\lambda}+{\\bf B_0}\\cdot\\nabla \\bar{u}^{w,\\lambda}+\\diver \\tilde{F}_1^{w,\\lambda}-{\\bf B_0}\\cdot \\nabla \\bar{u}^{h,w,\\lambda}, \\\\[1mm]\n \\diver\\, \\bar{u}^{w,\\lambda}=\\diver\\, \\bar{b}^{w,\\lambda}=0.\n\\end{cases}\n\\end{align*}\nThe two systems share same initial conditions, which are\n\\begin{align*}\n \\bar{u}_{w,\\lambda}(x, 0)&=\\bar{u}^{w,\\lambda}(x,0)=\\nabla^\\perp\\left( \\bar{\\phi}^{l,w,\\lambda}(x)+\\bar{\\phi}^{h,w,\\lambda}(x,0)\\right),\\\\[1mm]\n \\bar{b}_{w,\\lambda}(x, 0)&=\\bar{b}^{w,\\lambda}(x,0)=\\nabla^\\perp\\bar{\\phi}^{l,w,\\lambda}(x)\n\\end{align*}", "post_theorem_intro_text_len": 6867, "post_theorem_intro_text": "\\begin{remark}\n\tThe stabilizing role of background magnetic fields in the MHD equations has been well studied in previous works \\cite{lin-2014-GlobalSmallSolutions,full-not-bardos}. Our analysis reveals that despite the stabilization effect of the  background magnetic fields, the data-to-solution map for the resistive MHD equations \\eqref{mhd} maintains its non-uniform continuity property for arbitrary non-zero constant fields ${\\bf B_0}\\neq {\\bf 0}$.\n\\end{remark}\n\nTo simplify the notations, we write the perturbed system as follows\n\\begin{align}\\label{mhd-B}\n\t\\begin{cases}\n\t\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b+{\\bf B_0}\\cdot\\nabla b,\\\\[1mm]\n\t\t\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u+{\\bf B_0}\\cdot\\nabla u,  \\\\[1mm]\n\t\t\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\n\t\tu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}\nwhere we denote $b - {\\bf B_0}$ by $b$ for convenience.\n\nWe now explain the main ideas of the proof of  Theorem \\ref{thm}. It is recalled that  Himonas and Misiołek  \\cite{nonuniform-2010-cmp-himonas} introduced a frequency decomposition to construct  approximate solutions of the two-dimensional incompressible Euler equations. Denote\n \\begin{align}\\label{1.1}\n\tu^{\\pm1,\\lambda}(x,t)=u^{l,\\pm1,\\lambda}(x,t)+u^{h,\\pm1,\\lambda}(x,t),\n\\end{align}\nwhere\n\\begin{itemize}\n\t\\item[(i)] the low-frequency components $u^{l,\\pm1,\\lambda}$ are  obtained by solving the Euler equations with  low-frequency initial data;\n\t\\item[(ii)] the high-frequency components $u^{h,\\pm1,\\lambda}$ are explicitly constructed by using  oscillatory profiles, with frequency parameter $\\lambda$ controlling the spatial concentration,\n\t\\begin{align}\\label{1.2}\n\t\tu^{h,\\pm1,\\lambda}(x,t)=\\nabla^\\perp \\left(\\lambda^{-\\delta-s-1}\\phi\\left(\\dfrac{x_1}{\\lambda^\\delta}\\right)\\phi\\left(\\dfrac{x_2}{\\lambda^\\delta}\\right)\\sin (\\lambda x_2\\mp t)\\right).\n\t\\end{align}\nHere, $\\lambda>0, \\max\\{1-s,0\\}<\\delta < 1$ and $\\phi\\in C_c^\\infty(\\mathbb{R})$ with $\\text{supp}\\,\\phi \\subset [-2,2]$ and $\\phi(x) \\equiv 1$ on $|x| < 1$.\n\\end{itemize}\n\nIt is required to modify the construction of the approximate solutions in the presence of the magnetic field. To the resistive MHD equations, due to the diffusive nature of the magnetic field evolution (versus the purely transport equations of the Euler equations ), we restrict high-frequency perturbations to the velocity field alone and  maintain low-frequency components for the magnetic field. More precisely, in the case $\\mathbf{B_0} = \\mathbf{0}$ (no background magnetic field) and $d=2$,  we set\n\\begin{itemize}\n\t\\item the low-frequency pairs $(u^{l,\\pm1,\\lambda}, b^{\\pm1,\\lambda})$ by solving the resistive MHD equations \\eqref{mhd} (equivalently, system \\eqref{mhd-B} with $\\mathbf{B_0} = \\mathbf{0}$) with low-frequency initial data;\n\t\\item the high-frequency velocity components $u^{h,\\pm1,\\lambda}$ by retain their profiles as in  \\eqref{1.2}.\n\\end{itemize}\n\nIn the three-dimensional case $(d = 3)$, we modify  the high-frequency velocity components as\n\t\\begin{align}\\label{1.3}\n\tu^{h,\\pm1,\\lambda}(x,t)=\\begin{pmatrix}\n\t\t\\partial_{x_1}\\\\[1mm]\n\t\t-\\partial_{x_2}\\\\[1mm]\n\t\t0\n\t\\end{pmatrix} \\left(\\lambda^{-\\delta-s-1}\\phi\\left(\\dfrac{x_1}{\\lambda^\\delta}\\right)\\phi\\left(\\dfrac{x_2}{\\lambda^\\delta}\\right)\\sin (\\lambda x_2\\mp t)\\phi(x_3)\\right).\n\\end{align}\n\nThe error terms induced by the  high-frequency  $u^{h,\\pm1,\\lambda}$   can be written as\n\\begin{align}\\label{1.4}\n\tE^{\\pm1,\\lambda}&=\\partial_t u^{h,\\pm1,\\lambda}+u^{l,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{l,\\pm1,\\lambda},\\\\\\label{1.5}\n\tF^{\\pm1,\\lambda}&=u^{h,\\pm1,\\lambda}\\cdot\\nabla b^{\\pm1,\\lambda}-b^{\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda},\n\\end{align}\nrespectively.\n\nDirect estimation of \\eqref{1.5} will lead to uncontrolled error terms. Our alternative approach consists of three key steps:\n\\begin{enumerate}\n\t\\item {\\bf Divergence reformulation:} We express $F^{\\pm1,\\lambda} =\\text{div}\\,\\, \\tilde{F}^{\\pm1,\\lambda}$ and establish estimates for $\\tilde{F}^{\\pm1,\\lambda}$;\n\t\\item {\\bf Integration by parts:} In $L^2$ inner product computations, we transfer the divergence operator to $b^{\\pm1,\\lambda}$ via integration by parts;\n\t\\item {\\bf Diffusion cancellation:} The resulting terms are precisely canceled by exploiting the magnetic diffusion term $\\Delta b^{\\pm1,\\lambda}$.\n\\end{enumerate}\n\nFor non-zero background fields $\\mathbf{B_0}=(B_1,B_2) \\neq \\mathbf{0}$, we introduce the coordinate transformation\n\\begin{align*}\n\tx_2 \\mapsto B_1x_2 - B_2x_1,\n\\end{align*}\nwhich maintains all norm estimates while achieving exact cancellation of the dominant high-frequency linear term $\\mathbf{B_0} \\cdot \\nabla u^{h,\\pm 1,\\lambda}$ appearing in the error analysis of $F^{\\pm1,\\lambda}$.\n\nA key observation is that the directional derivative exhibits a better decay:\n\\begin{align*}\n\t\\left\\|\\mathbf{B_0} \\cdot \\nabla u^{h,\\pm 1,\\lambda}\\right\\|_{L^2}\\leq C\\lambda^{-s-\\delta} \\ll \\left\\|u^{h,\\pm 1,\\lambda}\\right\\|_{L^2}\\leq C\\lambda^{-s}, \\text{~for~}\\lambda \\to \\infty,\n\\end{align*}\nwhere $\\delta>0$ represent the improved decay rate.\n\n\t The paper is organized as follows. section \\ref{sec:2} is the preliminary analytical framework, in which subsection \\ref{sec:2.1} is about key Lemmas and subsection \\ref{sec:2.2} is on the local well-posedness theory. The core  strategy of the proof is implemented in section \\ref{sec:3} (zero background field case, $\\mathbf{B_0} = \\mathbf{0}$) and section  \\ref{sec:4} (non-zero background field case, $\\mathbf{B_0} \\neq \\mathbf{0}$), following the approach as follows: construction of approximate solutions in subsections \\ref{sec:3.1} and \\ref{sec:4.1}, precise estimation of approximation errors in subsections \\ref{sec:3.2} and \\ref{sec:4.2}, rigorous construction of exact solutions in subsections \\ref{sec:3.3} and \\ref{sec:4.3}, finishing the proof of Theorem in the case $\\mathbf{B_0} = \\mathbf{0}$ and $\\mathbf{B_0} \\neq \\mathbf{0}$ in subsections \\ref{sec:3.4} and \\ref{sec:4.4}, respectively.\n\n{\\bf Notations:}\n\\begin{enumerate}\n\\item {\\bf Function Spaces:} Throughout this work, let $X$ denote a Banach space equipped with norm $\\|\\cdot\\|_X$. Since all function spaces considered here are defined on $\\mathbb{R}^d$ for $d = 2,3$, we will suppress the domain $\\mathbb{R}^d$ in our notation unless otherwise specified.\n\t\\item {\\bf Differential Operators:} For $x \\in \\mathbb{R}^2$, we define the perpendicular gradient operator as $\\nabla^\\perp := (\\partial_{x_2}, -\\partial_{x_1})$.\n\t\\item {\\bf Joint Norms:} Given functions $f, g \\in X(\\mathbb{R}^d)$, we define their joint norm by\n\\begin{align*}\n\t\\|f(\\cdot),g(\\cdot)\\|_{X(\\mathbb{R}^d)}=\t\\|f(\\cdot)\\|_{X(\\mathbb{R}^d)}+\t\\|g(\\cdot)\\|_{X(\\mathbb{R}^d)}\n\\end{align*}\nadopting this concise notation for simplicity.\n\\end{enumerate}", "sketch": "To prove Theorem~\\ref{thm}, the authors adapt the “frequency decomposition” method of Himonas–Misio\\l{}ek. They decompose the velocity as\n\\[\n u^{\\pm1,\\lambda}(x,t)=u^{l,\\pm1,\\lambda}(x,t)+u^{h,\\pm1,\\lambda}(x,t),\n\\]\nwhere (i) the low-frequency parts $u^{l,\\pm1,\\lambda}$ come from solving the (appropriate) equations with low-frequency initial data, and (ii) the high-frequency parts $u^{h,\\pm1,\\lambda}$ are “explicitly constructed by using oscillatory profiles,” with frequency parameter $\\lambda$ (in 2D, profile as in \\eqref{1.2}; in 3D, modified as in \\eqref{1.3}).\n\nFor resistive MHD, “due to the diffusive nature of the magnetic field evolution,” they “restrict high-frequency perturbations to the velocity field alone and maintain low-frequency components for the magnetic field”: the low-frequency pair $(u^{l,\\pm1,\\lambda}, b^{\\pm1,\\lambda})$ is obtained by solving the resistive MHD system with low-frequency initial data, while $u^{h,\\pm1,\\lambda}$ retains the oscillatory profile.\n\nThey then estimate the approximation errors induced by $u^{h,\\pm1,\\lambda}$, written as\n\\[\nE^{\\pm1,\\lambda}=\\partial_t u^{h,\\pm1,\\lambda}+u^{l,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{l,\\pm1,\\lambda},\n\\]\n\\[\nF^{\\pm1,\\lambda}=u^{h,\\pm1,\\lambda}\\cdot\\nabla b^{\\pm1,\\lambda}-b^{\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}.\n\\]\nSince “direct estimation of \\eqref{1.5} will lead to uncontrolled error terms,” their alternative consists of three steps: (1) “express $F^{\\pm1,\\lambda}=\\mathrm{div}\\,\\tilde F^{\\pm1,\\lambda}$” and estimate $\\tilde F^{\\pm1,\\lambda}$; (2) “transfer the divergence operator to $b^{\\pm1,\\lambda}$ via integration by parts” in $L^2$ inner products; (3) use “diffusion cancellation,” where “the resulting terms are precisely canceled by exploiting the magnetic diffusion term $\\Delta b^{\\pm1,\\lambda}$.”\n\nFor nonzero constant background fields $\\mathbf{B_0}\\neq\\mathbf{0}$, they “introduce the coordinate transformation $x_2\\mapsto B_1x_2-B_2x_1$,” which “maintains all norm estimates” and yields “exact cancellation of the dominant high-frequency linear term $\\mathbf{B_0}\\cdot\\nabla u^{h,\\pm1,\\lambda}$” in the error analysis of $F^{\\pm1,\\lambda}$. They also note the key decay bound\n\\[\n\\|\\mathbf{B_0}\\cdot\\nabla u^{h,\\pm1,\\lambda}\\|_{L^2}\\le C\\lambda^{-s-\\delta}\\ll \\|u^{h,\\pm1,\\lambda}\\|_{L^2}\\le C\\lambda^{-s}\\quad (\\lambda\\to\\infty),\n\\]\nwith $\\delta>0$ giving “improved decay.”\n\nFinally, they describe the proof strategy as: “construction of approximate solutions,” “precise estimation of approximation errors,” “rigorous construction of exact solutions,” and “finishing the proof” in the cases $\\mathbf{B_0}=\\mathbf{0}$ and $\\mathbf{B_0}\\neq\\mathbf{0}$.", "expanded_sketch": "To prove the main theorem, the authors adapt the “frequency decomposition” method of Himonas–Misio\\l{}ek. They decompose the velocity as\n\\[\n u^{\\pm1,\\lambda}(x,t)=u^{l,\\pm1,\\lambda}(x,t)+u^{h,\\pm1,\\lambda}(x,t),\n\\]\nwhere (i) the low-frequency parts $u^{l,\\pm1,\\lambda}$ come from solving the (appropriate) equations with low-frequency initial data, and (ii) the high-frequency parts $u^{h,\\pm1,\\lambda}$ are “explicitly constructed by using oscillatory profiles,” with frequency parameter $\\lambda$ (in 2D, profile as in\n\\begin{align}\\label{1.2}\n\t\tu^{h,\\pm1,\\lambda}(x,t)=\\nabla^\\perp \\left(\\lambda^{-\\delta-s-1}\\phi\\left(\\dfrac{x_1}{\\lambda^\\delta}\\right)\\phi\\left(\\dfrac{x_2}{\\lambda^\\delta}\\right)\\sin (\\lambda x_2\\mp t)\\right).\n\t\\end{align}\n; in 3D, modified as in \\eqref{1.3}).\n\nFor resistive MHD, “due to the diffusive nature of the magnetic field evolution,” they “restrict high-frequency perturbations to the velocity field alone and maintain low-frequency components for the magnetic field”: the low-frequency pair $(u^{l,\\pm1,\\lambda}, b^{\\pm1,\\lambda})$ is obtained by solving the resistive MHD system with low-frequency initial data, while $u^{h,\\pm1,\\lambda}$ retains the oscillatory profile.\n\nThey then estimate the approximation errors induced by $u^{h,\\pm1,\\lambda}$, written as\n\\begin{align}\\label{1.4}\n\tE^{\\pm1,\\lambda}&=\\partial_t u^{h,\\pm1,\\lambda}+u^{l,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda}+u^{h,\\pm1,\\lambda}\\cdot\\nabla u^{l,\\pm1,\\lambda},\\\\\\label{1.5}\n\tF^{\\pm1,\\lambda}&=u^{h,\\pm1,\\lambda}\\cdot\\nabla b^{\\pm1,\\lambda}-b^{\\pm1,\\lambda}\\cdot\\nabla u^{h,\\pm1,\\lambda},\n\\end{align}\nSince “direct estimation of the equation above will lead to uncontrolled error terms,” their alternative consists of three steps: (1) “express $F^{\\pm1,\\lambda}=\\mathrm{div}\\,\\tilde F^{\\pm1,\\lambda}$” and estimate $\\tilde F^{\\pm1,\\lambda}$; (2) “transfer the divergence operator to $b^{\\pm1,\\lambda}$ via integration by parts” in $L^2$ inner products; (3) use “diffusion cancellation,” where “the resulting terms are precisely canceled by exploiting the magnetic diffusion term $\\Delta b^{\\pm1,\\lambda}$.”\n\nFor nonzero constant background fields $\\mathbf{B_0}\\neq\\mathbf{0}$, they “introduce the coordinate transformation $x_2\\mapsto B_1x_2-B_2x_1$,” which “maintains all norm estimates” and yields “exact cancellation of the dominant high-frequency linear term $\\mathbf{B_0}\\cdot\\nabla u^{h,\\pm1,\\lambda}$” in the error analysis of $F^{\\pm1,\\lambda}$. They also note the key decay bound\n\\[\n\\|\\mathbf{B_0}\\cdot\\nabla u^{h,\\pm1,\\lambda}\\|_{L^2}\\le C\\lambda^{-s-\\delta}\\ll \\|u^{h,\\pm1,\\lambda}\\|_{L^2}\\le C\\lambda^{-s}\\quad (\\lambda\\to\\infty),\n\\]\nwith $\\delta>0$ giving “improved decay.”\n\nFinally, they describe the proof strategy as: “construction of approximate solutions,” “precise estimation of approximation errors,” “rigorous construction of exact solutions,” and “finishing the proof” in the cases $\\mathbf{B_0}=\\mathbf{0}$ and $\\mathbf{B_0}\\neq\\mathbf{0}$.", "expanded_theorem": "\\label{thm}\n\tLet $s>0, \\lambda>0, d=2,3$ and $T>0$. Then the data-to-solution map $(u_0,b_0)\\mapsto(u,b)$ for the equations\n\\begin{align}\\label{mhd}\n\t\\begin{cases}\n\t\\partial_t u+u \\cdot \\nabla u+\\nabla p=b\\cdot \\nabla b,\\\\[1mm]\n\\partial_t b-\\Delta b+u\\cdot \\nabla b=b\\cdot\\nabla u,  \\\\[1mm]\n\\text{div}\\, u=\\text{div}\\, b=0, \\\\[1mm]\\nu(x, 0)=u_0(x),\\,\\, b(x, 0)=b_0(x),\n\t\\end{cases}\n\\end{align}\n is non-uniformly continuous from a bounded subset in $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C\\left([0,T],H^{s}(\\mathbb{R}^d)\\right)\\times C\\left([0,T],H^{s}(\\mathbb{R}^d)\\right)$.\n\n  More precisely, for any $\\gamma>0$ and arbitrary constant magnetic field ${\\bf B_0}\\in \\mathbb{R}^d$,  there exists two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$ such that\n\t\\begin{itemize}\n\t\t\\item the solutions satisfy\n\\begin{align*}\n\t\t\t(u^{\\pm 1,\\lambda}, b^{\\pm 1,\\lambda} - {\\bf B_0}) \\in C([0,T], H^{s}(\\mathbb{R}^{d})) \\times C([0,T], H^{s}(\\mathbb{R}^{d}));\n\\end{align*}\n\t\t\\item  the initial data satisfy\n\\begin{equation}\\label{nonuniform-initial-bound}\n\t\\begin{array}{l}\n\t\t\t\t\t\t\\|u^{+1,\\lambda}(0,\\cdot)\\|_{H^{s}} + \\|b^{+1,\\lambda}(0,\\cdot) - {\\bf B_0}\\|_{H^{s}} \\leq \\gamma, \\\\[2mm]\n\t\t\t\\|u^{-1,\\lambda}(0,\\cdot)\\|_{H^{s}} + \\|b^{-1,\\lambda}(0,\\cdot) - {\\bf B_0}\\|_{H^{s}} \\leq \\gamma;\n\t\\end{array}\n\\end{equation}\n\t\t\\item and the non-uniform continuity is characterized by\n\t\t\\begin{itemize}\n\t\t\t\\item at initial time $t=0$,\n\t\t\t\\begin{align}\\label{nonuniform-initial-diff}\n\t\t\t\t\\lim_{\\lambda\\to\\infty} \\|u^{+1,\\lambda}(0,\\cdot) - u^{-1,\\lambda}(0,\\cdot)\\|_{H^{s}} = 0,\n\t\t\t\\end{align}\n\t\t\t\\item for evolution times $t>0$,\n\t\t\t\\begin{align}\\label{nonuniform-low-bound-sint}\n\t\t\t\t\\lim_{\\lambda \\to\\infty} \\|u^{+1,\\lambda}(t) - u^{-1,\\lambda}(t)\\|_{H^{s}} \\geq c\\gamma |\\sin t|,\n\t\t\t\\end{align}\n\t\t\\end{itemize}\n\t\twhere $c = c(s,d) > 0$ is a constant depending only on  $s$ and $d$.\n\t\\end{itemize}", "theorem_type": ["Existential–Universal", "Asymptotic or Limit"], "mcq": {"question": "For the incompressible resistive MHD system on $\\mathbb{R}^d$ with $d\\in\\{2,3\\}$,\n\\[\n\\begin{cases}\n\\partial_t u+u\\cdot\\nabla u+\\nabla p=b\\cdot\\nabla b,\\\\\n\\partial_t b-\\Delta b+u\\cdot\\nabla b=b\\cdot\\nabla u,\\\\\n\\operatorname{div}u=\\operatorname{div}b=0,\\\\\nu(x,0)=u_0(x),\\quad b(x,0)=b_0(x),\n\\end{cases}\n\\]\nlet the data-to-solution map send initial data $(u_0,b_0)$ to the corresponding solution $(u,b)$. Fix $s>0$ and $T>0$, and view this map as acting from a bounded subset of $H^s(\\mathbb{R}^d)\\times H^s(\\mathbb{R}^d)$ into $C([0,T],H^s(\\mathbb{R}^d))\\times C([0,T],H^s(\\mathbb{R}^d))$. Which statement holds, including the explicit sequential form of non-uniform continuity around an arbitrary constant background magnetic field ${\\bf B}_0\\in\\mathbb{R}^d$?", "correct_choice": {"label": "A", "text": "The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C([0,T],H^{s}(\\mathbb{R}^d))\\times C([0,T],H^{s}(\\mathbb{R}^d))$. More precisely, for every $\\gamma>0$ and every constant magnetic field ${\\bf B}_0\\in\\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$, indexed by $\\lambda\\to\\infty$, such that\n\\[\n(u^{\\pm1,\\lambda},\\, b^{\\pm1,\\lambda}-{\\\\bf B}_0)\\in C([0,T],H^s(\\mathbb{R}^d))\\times C([0,T],H^s(\\mathbb{R}^d)),\n\\]\nwith initial bounds\n\\[\n\\|u^{+1,\\lambda}(0)\\|_{H^s}+\\|b^{+1,\\lambda}(0)-{\\bf B}_0\\|_{H^s}\\le \\gamma,\n\\qquad\n\\|u^{-1,\\lambda}(0)\\|_{H^s}+\\|b^{-1,\\lambda}(0)-{\\bf B}_0\\|_{H^s}\\le \\gamma,\n\\]\nand such that\n\\[\n\\lim_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(0)-u^{-1,\\lambda}(0)\\|_{H^s}=0,\n\\]\nwhile for every evolution time $t>0$ (in particular, $t\\in(0,T]$),\n\\[\n\\lim_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(t)-u^{-1,\\lambda}(t)\\|_{H^s}\\ge c(s,d)\\,\\gamma\\,|\\sin t|,\n\\]\nwhere $c(s,d)>0$ depends only on $s$ and $d$."}, "choices": [{"label": "B", "text": "The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C([0,T],H^{s}(\\mathbb{R}^d))\\times C([0,T],H^{s}(\\mathbb{R}^d))$. More precisely, for every $\\gamma>0$ and every constant magnetic field ${\\bf B}_0\\in\\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$, indexed by $\\lambda\\to\\infty$, such that\n\\[\n(u^{\\pm1,\\lambda},\\, b^{\\pm1,\\lambda}-{\\bf B}_0)\\in C([0,T],H^s(\\mathbb{R}^d))\\times C([0,T],H^s(\\mathbb{R}^d)),\n\\]\nwith the same initial bounds as in the theorem, and such that\n\\[\n\\lim_{\\lambda\\to\\infty}\\Big(\\|u^{+1,\\lambda}(0)-u^{-1,\\lambda}(0)\\|_{H^s}+\\|b^{+1,\\lambda}(0)-b^{-1,\\lambda}(0)\\|_{H^s}\\Big)=0,\n\\]\nwhile for every evolution time $t>0$,\n\\[\n\\lim_{\\lambda\\to\\infty}\\Big(\\|u^{+1,\\lambda}(t)-u^{-1,\\lambda}(t)\\|_{H^s}+\\|b^{+1,\\lambda}(t)-b^{-1,\\lambda}(t)\\|_{H^s}\\Big)\\ge c(s,d)\\,\\gamma\\,|\\sin t|,\n\\]\nwhere $c(s,d)>0$ depends only on $s$ and $d$."}, {"label": "C", "text": "The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C([0,T],H^{s}(\\mathbb{R}^d))\\times C([0,T],H^{s}(\\mathbb{R}^d))$. More precisely, for every $\\gamma>0$ and every constant magnetic field ${\\bf B}_0\\in\\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$, indexed by $\\lambda\\to\\infty$, such that\n\\[\n(u^{\\pm1,\\lambda},\\, b^{\\pm1,\\lambda}-{\\bf B}_0)\\in C([0,T],H^s(\\mathbb{R}^d))\\times C([0,T],H^s(\\mathbb{R}^d)),\n\\]\nwith initial bounds\n\\[\n\\|u^{+1,\\lambda}(0)\\|_{H^s}+\\|b^{+1,\\lambda}(0)-{\\bf B}_0\\|_{H^s}\\le \\gamma,\n\\qquad\n\\|u^{-1,\\lambda}(0)\\|_{H^s}+\\|b^{-1,\\lambda}(0)-{\\bf B}_0\\|_{H^s}\\le \\gamma,\n\\]\nand such that\n\\[\n\\lim_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(0)-u^{-1,\\lambda}(0)\\|_{H^s}=0.\n\\]\nIn particular, the data-to-solution map fails to be uniformly continuous on that bounded subset."}, {"label": "D", "text": "The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C([0,T],H^{s}(\\mathbb{R}^d))\\times C([0,T],H^{s}(\\mathbb{R}^d))$. More precisely, there exists a constant $c=c(s,d)>0$ such that for every $\\gamma>0$ one can choose two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$ with the property that for every constant magnetic field ${\\bf B}_0\\in\\mathbb{R}^d$,\n\\[\n(u^{\\pm1,\\lambda},\\, b^{\\pm1,\\lambda}-{\\bf B}_0)\\in C([0,T],H^s(\\mathbb{R}^d))\\times C([0,T],H^s(\\mathbb{R}^d)),\n\\]\nwith the same initial bounds, and such that\n\\[\n\\lim_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(0)-u^{-1,\\lambda}(0)\\|_{H^s}=0,\n\\]\nwhile for every evolution time $t>0$,\n\\[\n\\lim_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(t)-u^{-1,\\lambda}(t)\\|_{H^s}\\ge c\\,\\gamma\\,|\\sin t|.\n\\]"}, {"label": "E", "text": "The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\\mathbb{R}^d)\\times H^{s}(\\mathbb{R}^d)$ into $C([0,T],H^{s}(\\mathbb{R}^d))\\times C([0,T],H^{s}(\\mathbb{R}^d))$. More precisely, for every $\\gamma>0$ and every constant magnetic field ${\\bf B}_0\\in\\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\\lambda},b^{+1,\\lambda})$ and $(u^{-1,\\lambda},b^{-1,\\lambda})$, indexed by $\\lambda\\to\\infty$, such that\n\\[\n(u^{\\pm1,\\lambda},\\, b^{\\pm1,\\lambda}-{\\bf B}_0)\\in C([0,T],H^s(\\mathbb{R}^d))\\times C([0,T],H^s(\\mathbb{R}^d)),\n\\]\nwith the same initial bounds, and such that\n\\[\n\\lim_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(0)-u^{-1,\\lambda}(0)\\|_{H^s}=0,\n\\]\nwhile for every evolution time $t\\in[0,T]$,\n\\[\n\\liminf_{\\lambda\\to\\infty}\\|u^{+1,\\lambda}(t)-u^{-1,\\lambda}(t)\\|_{H^s}\\ge c(s,d)\\,\\gamma,\n\\]\nwhere $c(s,d)>0$ depends only on $s$ and $d$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "diffusion cancellation", "tampered_component": "velocity-only high-frequency instability promoted to joint $(u,b)$ separation", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the explicit positive-time lower bound $\\ge c\\gamma|\\sin t|$ and retained only failure of uniform continuity", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "quantifier order in dependence on ${\\bf B}_0$", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "time-dependent lower bound $|\\sin t|$ replaced by uniform-in-time positive separation on $[0,T]$", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It specifies the PDE setting and asks for the valid statement, but the decisive details—velocity-only separation, the exact quantifiers in ${\\bf B}_0$, and the $c(s,d)\\gamma|\\sin t|$ lower bound—are not given away."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: it mainly asks for the precise statement of a known non-uniform continuity result. However, it is not a pure tautology, since the options differ in meaningful ways (quantifier order, weaker-vs-stronger conclusions, and which norms/sequences appear)."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact theorem from plausible variants, especially between the correct sharp statement and nearby strengthenings/weakenings. Still, the task is primarily recognition/recall of a theorem statement rather than genuine generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: strengthening to joint $(u,b)$ separation, dropping the explicit lower bound to a weaker true statement, altering quantifier dependence on ${\\bf B}_0$, and replacing the $|\\sin t|$ dependence by an unrealistically uniform-in-time gap."}, "total_score": 6, "overall_assessment": "A solid MCQ with strong distractors and little answer leakage, but it tests precise theorem recall more than generative reasoning."}}
{"id": "2602.06882v1", "paper_link": "http://arxiv.org/abs/2602.06882v1", "theorems_cnt": 5, "theorem": {"env_name": "maintheorem", "content": "\\label{mthm:CompAFCert} If a unital AF algebra has a c.e. presentation, then it also has a computable AF certificate.", "start_pos": 7148, "end_pos": 7303, "label": "mthm:CompAFCert"}, "ref_dict": {"cor:CompPresAF": "\\begin{corollary}\\label{cor:CompPresAF}\nEvery c.e. presentation of a unital AF algebra is computable.\n\\end{corollary}", "subsubsec:BackCompKThy": "\\begin{proposition}\\label{prop:PreimCeOpen}\nSuppose $\\boldA^\\#$ and $\\boldB^\\#$ are c.e. presentations of \\cstar-algebras, and let \n$U$ be a c.e. open set of $\\boldB^\\#$.  If $f$ is a computable map from \n$\\boldA^\\#$ to $\\boldB^\\#$, then $f^{-1}[U]$ is a c.e. open set of $\\boldA^\\#$. \nMoreover, it is possible to compute an $\\boldA^\\#$ index of $f^{-1}[U]$ from a $\\boldB^\\#$ index \nof $U$ and an $(\\boldA^\\#, \\boldB^\\#)$ index of $f$.\n\\end{proposition}\n\nSuppose $\\boldA^\\#$ is a c.e. presentation of a stably finite unital \\cstar-algebra. \nIn \\cite{UHFPaper}, it is shown that $\\unit_\\boldA$ is a computable point of \n$\\boldA^\\#$.  \nWe say that $e \\in \\N$ is a \\emph{unital index of $\\boldA^\\#$} if\n$e$ is a code of a pair $(e_0, e_1)$ that consists of an index of $\\boldA^\\#$ and \nan $\\boldA^\\#$ index of $\\unit_\\boldA$.  Our motivation for introducing such indexes\nis that even for the class of stably finite unital \\cstar-algebras, an $\\boldA^\\#$-index of $\\unit_\\boldA$ may not be computable from an index of \n$\\boldA^\\#$; see \\cite{mcnicholl2024evaluative}.\n\nWe now discuss prior developments regarding the computability of $K$-theory.\n\n\\subsubsection{$K$-theory}\\label{subsubsec:BackCompKThy}\n\nIn \\cite{UHFPaper}, it is shown that there is a \\emph{computable} functor from the \ncategory of c.e. presentations of unital \\cstar-algebras (where the morphisms are \ncomputable unital $\\star$-morphisms) to the category of c.e. presentations of abelian groups.\nThis functor is denoted $K_0$. \nThe functor is computable in the sense that it is possible to compute an index of $K_0(\\boldA^\\#)$ from \nan index of $\\boldA^\\#$.  and from an $(\\boldA^\\#, \\boldB^\\#)$ index of a computable $\\star$-homomorphism $\\phi$, it is possible to compute a $(K_0(\\boldA^\\#), K_0(\\boldB^\\#))$-index of $K_0(\\phi)$. \n\nThe application of computable $K$-theory to UHF algebras relied heavily on the \nconcept of computable weak stability, which will also support our proof of the Effective Glimm Lemma.\nHence, it is our next topic for review.\n\n\\subsubsection{Computable weak stability}\n\nThe concept of computable weak stability was introduced by Fox, Goldbring, and Hart \\cite\n{FoxGoldbringHart.2024+} based on the corresponding notion of weak stability in continuous \nmodel theory, which was introduced in \\cite{FarahEtAll.2021}.  \nIn \\cite{UHFPaper}, we presented computable weak stability in a way that illuminated \nits connection with c.e. closedness.  \nFor the sake of referencing some of its details, we recall this definition now. \n\n\\begin{definition}\\label{def:CompWklyStbl}\nSuppose that $\\vec{x} = (x_1, \\ldots, x_N)$ is a tuple of variables, $C_1, \\ldots, C_N \\in \\N$, \nand $p_1(\\vec{x}), \\ldots, p_M(\\vec{x})$ are rational $*$-polynomials.  Consider the set of relations\n$$\\mathcal{R} = \\{p_i(\\vec{x})=0 \\ : \\ i=1,\\ldots,M\\}\\cup\\{\\norm{x_j} \\leq C_j \\ : \\ j=1,\\ldots,N\\}$$.\n\\begin{enumerate}\n    \\item For a \\cstar-algebra $\\boldB$ and $w_1, \\ldots, w_N \\in \\boldB$, \n    let $\\mathcal{R}^{\\boldB}$ denote the quantity \n    $$\n    \\max(\\{\\norm{p_j(\\vec{w})}\\ :\\ j \\in \\{1, \\ldots, M\\}\\} \\cup \\{\\norm{w_j}- C_j\\ : j \\in \\{1, \\ldots, N\\}\\}$$\n    and write $\\boldB \\models \\mathcal{R}(\\vec{w})$ if \n    $p_j(\\vec{w}) = 0$ for all $j \\in \\{1, \\ldots, M\\}$ and $\\norm{w_j} \\leq C_j$ for all $j \\in \\{1, \\ldots, N\\}$.\n\n    \\item A function $g : \\N \\rightarrow \\N$ is a \\emph{modulus of weak stability} for \n    $\\mathcal{R}$ provided that, for every $k \\in \\N$, every \\cstar-algebra $\\boldB$, \n    and every $w_1, \\ldots, w_N \\in \\boldB$, if $\\mathcal{R}^\\boldB(\\vec{w}) < 2^{-g(k)}$, \n    then there exists $z_1, \\ldots, z_N \\in \\boldB$ so that \n    $\\max_j \\norm{w_j - z_j} < 2^{-k}$ and $\\mathbf{B} \\models \\mathcal{R}(\\vec{z})$.\n\n    \\item We say that $\\mathcal{R}$ is \\emph{weakly stable} if it has a modulus of weak stability and \\emph{computably weakly stable} if it has a computable modulus of weak stability.\n\\end{enumerate}\n\\end{definition}\n\nIn \\cite[Theorem 1.12]{UHFPaper} we showed that computably weakly stable relations define c.e. closed sets in c.e. presentations of \\cstar-algebras.\n\nWe have now completed our description of relevant prior developments in \nclassical and computable mathematics.  We now attend to some purely prelminary matters \nin these realms which will support our main efforts later.\n\n\\section{Preliminaries from classical mathematics}\\label{sec:PrlmClscl}\n\n\\subsection{Ordered abelian groups}\n\n\\subsubsection{Inductive limits}\n\nOur goal here is to arrive at a more concrete understanding of inductive limits \nof ordered abelian groups which will be valuable when developing their computability \ntheory. We begin with abelian groups.\n\n\\begin{lemma}\\label{lm:IndLimAbl}\nSuppose $(G_s, \\phi_s)_{s \\in \\N}$ is an inductive sequence of abelian groups and let \n$(G, (\\nu_s)_{s \\in \\N})$ be an inductive limit of $(G_s, \\phi_s)_{s \\in \\N}$.  Then:\n\\begin{enumerate}\n    \\item $G = \\bigcup_{s \\in \\N} \\ran(\\nu_s)$.\\label{lm:IndLimAbl:ran}\n\n    \\item For each $s \\in \\N$, $\\ker(\\nu_s) = \\bigcup_{k \\in \\N} \\ker(\\phi_{s, s+k})$.\\label{lm:IndLimAbl:ker}\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}[Proof sketch]\nWe first review a standard construction of \nan inductive limit of $(G_s, \\phi_s)_{s \\in \\N}$. For each $s \\in \\N$, let $\\rho_s : G_s \\rightarrow \\prod_n G_n$ be the homomorphism defined by \nsetting $\\rho_s(a) = (\\phi_{s,n}(a))_{n \\in \\N}$.\nWhen $f,g \\in \\prod_n G_n$, write $f \\sim g$ if there exists $k \\in \\N$\nso that $f(k+n) = g(k + n)$ for all $n \\in \\N$.  Thus, $\\sim$ is a congruence relation on \n$\\prod_n G_n$.  Let $\\pi : \\prod_n G_n \\rightarrow \\prod_n G_n/ \\sim$ denote the canonical epimorphism, \nand set $\\mu_s = \\pi \\circ \\rho_s$ for all $s \\in \\N$.\n\nDefine $H$ to be $\\bigcup_{s \\in \\N} \\ran(\\mu_s)$.  By construction, \n$\\ran(\\mu_s) \\subseteq \\ran(\\mu_{s+1})$.  Hence, $H$ is an abelian group.\nIt is well-known, and easy to verify, that $(H, (\\mu_n)_{n \\in \\N})$ is\nan inductive limit of $(G_s, \\phi_s)_{s \\in \\N}$.  Furthermore, \nfor each $s \\in \\N$, $\\ker(\\mu_s) = \\bigcup_{k \\in \\N} \\ker(\\phi_{s,s+k})$. \n\nLet $\\lambda$ be the reduction of $(G, (\\nu_s)_{s \\in \\N})$ to \n$(H, (\\mu_s)_{s \\in \\N})$.  \nBy Proposition \\ref{prop:IndLimUni}, $\\lambda$ is an isomorphism. \nSince $(H, (\\mu_s)_{s \\in \\N})$ satisfies (\\ref{lm:IndLimAbl:ran}) and \n(\\ref{lm:IndLimAbl:ker}), it follows that $(G, (\\nu_s)_{s \\in \\N})$ does as well.\n\\end{proof}\n\nWe now proceed to describe inductive limits of ordered abelian groups.\n\n\\begin{lemma}\\label{lm:IndLimOrdAbl}\nSuppose $(\\calG_s, \\phi_s)_{s \\in \\N}$ is an inductive sequence of ordered abelian groups.\nAssume $(\\calG, (\\nu_s)_{s \\in \\N})$ is an inductive upper limit of $(\\calG_s, \\phi_s)_{s \\in \\N}$. \nThen $(\\calG, (\\nu_s)_{s \\in \\N})$ is an inductive limit of $(\\calG_s, \\phi_s)_{s \\in \\N}$\nif and only if it satisfies the following:\n\\begin{enumerate}\n    \\item $\\calG^+ = \\bigcup_{s \\in \\N} \\nu_s[\\calG_s^+]$.\\label{lm:IndLimOrdGrp:ran}\n\n    \\item For each $s \\in \\N$, $\\ker(\\nu_s) = \\bigcup_{k \\in \\N} \\ker(\\phi_{s,s+k})$. \\label{lm:IndLimOrdGrp:ker}\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}  \nAssume $\\calG = (G, P)$ and $\\calG_s = (G_s, P_s)$ for all $s \\in \\N$. \nLet $(H, (\\mu_s)_{s \\in \\N})$ be an inductive limit of \n$(G_s, \\phi_s)_{s \\in \\N}$, and let $\\lambda$ be the reduction of \n$(H, (\\mu_s)_{s \\in \\N})$ to $(G, (\\nu_s))_{s \\in \\N}$.  \nLet $P' = \\bigcup_{s \\in \\N} \\mu_s[P_s]$ and set $\\calH = (H, P')$.  \nBy \\cite[Proposition 6.2.6]{Rordam.Larsen.Laustsen.2000}, \n$(\\calH, (\\mu_s)_{s \\in \\N})$ is an inductive limit of $(\\calG_s, \\phi_s)_{s \\in \\N}$.\nIt also follows that $\\lambda$ is positive and so \n$\\lambda$ is the reduction of $(\\calH, (\\mu_s)_{s \\in \\N})$ to $(\\calG, (\\nu_s)_{s \\in \\N})$.\nFurthermore, by construction, $\\calH^+ = \\bigcup_s \\mu_s[\\calG^+_s]$, and by \nLemma \\ref{lm:IndLimAbl}, $\\ker(\\mu_s) = \\bigcup_{k \\in \\N} \\ker(\\phi_{s,s+k})$ for all $s \\in \\N$.\n\nSuppose $(\\calG, (\\nu_s)_{s \\in \\N})$ is an inductive limit of \n$(\\calG_s, \\phi_s)_{s \\in \\N}$.  \nThen $\\lambda$ is an isomorphism.  Since $(\\calH, (\\mu_s)_{s \\in \\N})$ satisfies \n(\\ref{lm:IndLimOrdGrp:ran}) and (\\ref{lm:IndLimOrdGrp:ker}), \nit follows that $(\\calG, (\\nu_s)_{s \\in \\N})$ does as well.\n\nFinally, suppose $(\\calG, (\\nu_s)_{s \\in \\N})$ satisfies (\\ref{lm:IndLimOrdGrp:ran}) and (\\ref{lm:IndLimOrdGrp:ker}).  It follows from (\\ref{lm:IndLimOrdGrp:ran}) that $\\lambda$\nis surjective.  Let $h \\in \\ker(\\lambda)$.  Then, there exists $s$ and $g \\in \\calG_s$ so \nthat $\\mu_s(g) = h$.  Hence, $g \\in \\ker(\\nu_s)$, and so there exists $k \\in \\N$\nso that $g \\in \\ker(\\phi_{s,s+k})$.  \nThus, $h = \\mu_{s+k}(\\phi_{s,s+k}(g)) = \\zerovec_G$, and so $\\lambda$ is an isomorphism.\nHence, $(\\calG, (\\nu_s)_{s \\in \\N})$ is an inductive limit of $(\\calG_s, \\phi_s)_{s \\in \\N}$.\n\\end{proof}\n\n\\subsubsection{Unital dimension groups}\n\nUnital dimension groups are central to the $K$-theory of unital AF algebras, and \naccordingly we elaborate on our definition of certificate of dimensionality \nas follows.\n\n\\begin{definition}\\label{def:UntlCertDim}\nSuppose $(\\calG, u)$ is a unital dimension group.  We say that $(n_s, u_s \\nu_s, \\phi_s)_{s \\in \\N}$\nis a \\emph{unital certificate of dimensionality} of $(\\calG, u)$ if $u_s$ is an order unit of \n$\\Z^{n_s}$ and if \n$((\\calG, u), (\\nu_s)_{s \\in \\N})$ is an inductive limit of \n$((\\Z^{n_s}, u_s), \\phi_s)_{s \\in \\N}$ in the category of unital ordered abelian groups. \n\\end{definition}\n\nThese certificates will form the backbone of our proof of Main Theorem \\ref{mthm:ClsCePresDimGrp::Untl}.\n\n\\subsection{\\cstar-algebras}\n\n\\subsubsection{Inductive limits}\n\nAs in the previous section, we give a concrete description of \ninductive limits of \\cstar-algebras.\n\n\\begin{lemma}\\label{lm:IndLimAlg}\nSuppose $(\\boldA_s, \\phi_s)_{s \\in \\N}$ is an inductive sequence of \\cstar-algebras, \nand let $(\\boldA, (\\nu_s)_{s \\in \\N})$ be an inductive upper limit of \n$(\\boldA_s, \\phi_s)_{s \\in \\N}$.  Then, $(\\boldA, (\\nu_s)_{s \\in \\N})$ is an inductive limit of \n$(\\boldA_s, \\phi_s)_{s \\in \\N}$ if and only if it satisfies the following:\n\\begin{enumerate}\n    \\item $\\boldA = \\overline{\\bigcup_{s \\in \\N} \\ran(\\nu_s)}$.\\label{lm:IndLimAlg::ran}\n\n    \\item For every $s \\in \\boldA_s$, $\\norm{\\nu_s(a)} = \\lim_k \\norm{\\phi_{s, s + k}(a)}$.\\label{lm:IndLimAlg::norm}\n\n    \\item For every $s \\in \\N$, $\\ker(\\nu_s) = \\{a \\in A_s\\ : \\lim_k \\norm{\\phi_{s,s+k}(a)} = 0\\}$.\\label{lm:IndLimAlg::ker}\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nThe forward direction is \\cite[Proposition 6.2.4]{Rordam.Larsen.Laustsen.2000}.\n\nSuppose (\\ref{lm:IndLimAlg::ran}), (\\ref{lm:IndLimAlg::norm}), and (\\ref{lm:IndLimAlg::ker}) are satisfied.\nLet $(\\boldB, (\\mu_s)_{s \\in \\N})$ be an inductive upper limit of \n$(\\boldA_s, \\phi_s)_{s \\in \\N}$.  \n\nWe first demonstrate that $\\ker(\\nu_s) \\subseteq \\ker(\\mu_s)$.  \nLet $a \\in \\ker(\\nu_s)$.  \nThen, by (\\ref{lm:IndLimAlg::norm}), $\\lim_k \\norm{\\phi_{s, s+k}(a)} = 0$. \nSince every $\\star$-homomorphism is $1$-Lipschitz, \nwe have $\\lim_k \\norm{\\mu_{s + k}(\\phi_{s, s+k}(a))} = 0$.  However, since \n$(\\boldB, (\\mu_s)_{s \\in \\N})$ is an inductive upper limit of $(\\boldA_s, \\phi_s)_{s \\in \\N}$,\nit follows that $\\mu_{s+k}(\\phi_{s,s+k}(a)) = \\mu_s(a)$ and so $a \\in \\ker(\\mu_s)$.\n\nIt now follows that if $\\nu_s(a_0) = \\nu_{s + k}(a_1)$, then \n$\\mu_s(a_0) = \\mu_{s+k}(a_1)$.\n\nWe now construct a $\\star$-homomorphism $\\lambda'$ from $\\bigcup_{s \\in \\N} \\ran(\\nu_s)$ to $\\boldB$.  \nTo begin, let $a \\in \\ran(\\nu_s)$, and choose any $a' \\in \\boldA_s$ so that $\\nu_s(a') = a$. \nDefine $\\lambda'(a)$ to be $\\mu_s(a')$.  It follows from what has just been shown that \n$\\lambda'$ is well defined. Since each $\\nu_s$ and each $\\mu_s$ is a $\\star$-homomorphism,\nit also follows that $\\lambda'$ is a $\\star$-homomorphism. \n\nWe now claim that $\\lambda'$ is $1$-Lipschtiz.  Let $a \\in \\ran(\\nu_s)$, and suppose \n$\\nu_s(a') = a$.  By (\\ref{lm:IndLimAlg::norm}), $\\norm{\\nu_s(a')} = \\lim_k \\norm{\\phi_{s, s+k}(a')}$.\nSince each $\\mu_{s+k}$ is $1$-Lipschitz, it follows that \n$\\norm{\\nu_s(a')} \\geq \\lim_k \\norm{\\mu_{s + k}(\\phi_{s, s+k}(a'))}$. But, \n$\\mu_{s + k}(\\phi_{s, s + k}(a')) = \\mu_s(a') = \\lambda'(a)$.\n\nIt now follows that $\\lambda'$ has a unique continuous extension $\\lambda$ to \n$\\boldA$.  Furthermore, $\\lambda$ is a $\\star$-homomorphism and \nby construction $\\lambda \\circ \\nu_s = \\mu_s$ for all $s \\in \\N$.\n\nFinally, it directly follows from the definitions that any two reductions of \n$(\\boldA, (\\nu_s)_{s \\in \\N})$ to $(\\boldB, (\\mu_s)_{s \\in \\N})$ agree on \n$\\bigcup_{s \\in \\N} \\ran(\\nu_s)$ and hence must be identical.\n\\end{proof}\n\nAn immediate consequence of Lemma \\ref{lm:IndLimAlg} is the following connection\n between AF certificates and inductive limits.\n\n\\begin{corollary}\\label{cor:IndLimAFCert}\nIf $(F_j,\\psi_j)_{j \\in \\N}$ is an AF certificate of $\\boldA$, \nthen $(\\boldA, (\\psi_j)_{j \\in \\N})$ is an inductive limit of \n$(\\bigoplus_{n \\in F_j} M_n(\\C), (\\psi_{j + 1}^{-1} \\circ \\psi_j)_{j \\in \\N})$.\n\\end{corollary} \n\n\\subsubsection{Matricial systems}\n\nOur only item here is to present some terminology which will support \nour proof of the Effective Glimm Lemma.\n\n\\begin{definition}\\label{def:CompMat}\nSuppose $\\boldA$ is a \\cstar-algebra and let $n_1, \\ldots, n_m$ be positive integers.   \nAn array $(e^s_{i,j})_{s,i,j}$ is a \\emph{type-$(n_1, \\ldots, n_m)$ matricial system of $\\boldA$}\nif it satisfies the following: \n\\begin{enumerate}\n    \\item For each $s \\in \\{1, \\ldots, m\\}$, \n    $(e^s_{i,j})_{i,j}$ is an $n_s \\times n_s$ system of matrix units of $\\boldA$. \n\n    \\item When $s,s' \\in \\{1, \\ldots, m\\}$ are distinct, then $e^s_{i,j} e^{s'}_{i',j'} = 0$\n    for all $i,j \\in \\{1, \\ldots, n_s\\}$ and all $i',j' \\in \\{1, \\ldots, n_{s'}\\}$.\n\\end{enumerate}\nIf, in addition, $\\boldA$ is unital and $\\sum_{s,i} e^s_{i,i} = \\unit_\\boldA$, then we say that the system $(e^s_{i,j})_{s,i,j}$ is \n\\emph{unital}.\n\\end{definition}\n\nIf $\\boldF$ is a \\cstar-algebra that is generated by a matricial system $(e^s_{i,j})_{s,i,j}$, \nthen\nwe call $(e^s_{i,j})_{s,i,j}$ a \\emph{matricial generating system for $\\boldF$}.\nThus a \\cstar-algebra is finite-dimensional if and only if it has a matricial generating system.\n\n\\section{Preliminaries from effective mathematics}\\label{sec:PrlmEff}\n\nHerein, we present our frameworks for the computability of inductive limits \nin a computably indexed category and for the computability of ordered \nabelian groups.\n\n\\subsection{Inductive limits in computably indexed categories}\n\nWe now present a computable picture of the material discussed in \\ref{subsec:BackIndLimCat}.\nThus, we again assume $\\calC$ is a category with a zero object.  \nHowever, we assume we have fixed an \\emph{indexing} of $\\calC$.  This consists of \nan assignment of natural numbers to the objects of $\\calC$.\nThe number assigned to an object is referred to as one of its indexes. \nEach object has at least one index.  A number may not index more the one object.\nEach morphism is indexed by at least one triple of natural numbers.\nIf $(e_0, e_1,e)$ indexes a morphism $\\gamma$, then $e_0$ must index its \ndomain and $e_1$ must index its co-domain.\n\nWe also assume this indexing is \\emph{computable} in the sense that there is \nan algorithm that given an index $(e_0,e_1,e)$ of $\\phi : A \\rightarrow B$ and\nan index $(e_1, e_2,e')$ of $\\psi : B \\rightarrow C$\ncomputes an index of $\\psi \\circ \\phi$ of the form $(e_0, e_2, e'')$.\nFurthermore, we require that from an index $e$ of an object $A$ it is possible to \ncompute an index of its identity morphism.\n\nNow that the computable indexing is fixed, our remaining definitions are \nstraightforward modifications of those in Section \\ref{subsec:BackIndLimCat}.\nFix an inductive sequence $(A_n, \\phi_n)_{n \\in \\N}$ of $\\calC$.\nWe say that $(A_n, \\phi_n)_{n \\in \\N}$ is \\emph{computable} if there is an algorithm\nthat given $n \\in \\N$ computes indexes of $A_n$ and $\\phi_n$.\nNow suppose \n$(A, (\\nu_n)_{n \\in \\N})$ is an inductive upper limit of $(A_n, \\phi_n)_{n \\in \\N}$, \nand assume it is possible to compute an index of $\\nu_n$ from $n$.\nWe say that an upper inductive limit $(A, (\\nu_n)_{n \\in \\N})$ \nof $(A_n, \\phi_n)_{n \\in \\N}$ is \\emph{computable} with index $e \\in \\N$ if $e$ is the code of a pair $(e_0, e_1)$ so that \n$e_0$ is an index of $A$ and $e_1$ is an index of an algorithm that computes an index of \n$\\nu_n$ from $n$.\nWe say that $(A, (\\nu_n)_{n \\in \\N})$ is \\emph{a computable inductive limit} if \nit is a computable inductive upper limit and if it is possible to compute from an index of a computable inductive upper limit \n$(B, (\\mu_n)_{n \\in \\N})$ of $(A_n, \\phi_n)_{n \\in \\N}$ an index of the reduction from $(A, (\\nu_n)_{n \\in \\N})$\nto $(B, (\\mu_n)_{n \\in \\N})$.\n\n\\subsection{Abelian groups}\n\n\\begin{lemma}\\label{lm:IndLimPresAbl}\nSuppose $(G, (\\nu_s)_{s \\in \\N})$ is an inductive limit of \n$(G_s, \\phi_s)_{s \\in \\N}$ in the category of \nabelian groups. \n\\begin{enumerate}\n    \\item If $(G_s^\\#, \\phi_s)_{s\\in \\N}$ is an inductive sequence in the category \n    of c.e. presentations of abelian groups, and if \n    $(G^\\#, (\\nu_s)_{s \\in \\N})$ is a computable inductive upper limit \n    of this sequence, then $(G^\\#, (\\nu_s)_{s \\in \\N})$ is \n    a computable inductive limit of $(G^\\#, (\\nu_s)_{s \\in \\N})$ in the category \n    of c.e. presentations of abelian groups.\n\n    \\item If $(G_s^\\#, \\phi_s)_{s\\in \\N}$ is an inductive sequence in the category \n    of computable presentations of abelian groups, and if \n    $(G^\\#, (\\nu_s)_{s \\in \\N})$ is a computable inductive upper limit \n    of this sequence, then $(G^\\#, (\\nu_s)_{s \\in \\N})$ is \n    a computable inductive limit of $(G^\\#, (\\nu_s)_{s \\in \\N})$ in the category \n    of computable presentations of abelian groups.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nSuppose $(G_s^\\#, \\phi_s)_{s\\in \\N}$ is an inductive sequence in the category \n    of c.e. presentations of abelian groups, and assume \n    $(G^\\#, (\\nu_s)_{s \\in \\N})$ is a computable inductive upper limit \n    of this sequence.  \n    Let $(H^\\#, (\\mu_s)_{s \\in \\N}$ be a computable inductive upper limit of \n    $(G_s^\\#, \\phi_s)_{s \\in \\N}$ in the category of c.e. presentations of abelian groups.\n    Thus, $(H, (\\mu_s)_{s \\in \\N})$ is an inductive upper limit of \n    $(G_s, \\phi_s)_{s \\in \\N}$ in the category of abelian groups. \n    Hence, there is a unique reduction $\\lambda$ of  \n$(G, (\\nu_s)_{s \\in \\N})$ to $(H, (\\mu_s)_{s \\in \\N})$.  \n\nIt only remains to show that $\\lambda$ is a computable map from $G^\\#$ to $H^\\#$.  \nSuppose $w$ is a $G^\\#$ label of $g \\in G$.  Then, there exists $s_0 \\in \\N$ so that \n$g \\in \\ran(\\nu_{s_0})$ and so there is a $\\ran(\\nu_{s_0})^\\#$ label $w'$ of $g$. \nSince the inclusion map is a computable map from $\\ran(\\nu_s)^\\#$ to \n$G^\\#$ uniformly in $s$, $s_0$ and $w'$ can be found by a search procedure. \nThen, by a search procedure, we can compute a $G_{s_0}^\\#$ label $w''$ of a \n$g' \\in G_{s_0}$ so that $\\nu_{s_0}(g') = g$ (since $\\nu_s$ is a computable map \nfrom $G_s^\\#$ to $G^\\#$ uniformly in $s$).  Since $\\lambda(g) = \\mu_{s_0}(g')$, \nit follows that we can now compute an $H^\\#$ label of $\\lambda(g)$ from $w$.\n\nThe proof of the second part is almost identical.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:CompIndLimPresAbl}\n\\begin{enumerate}\n    \\item Every computable inductive sequence of c.e. presentations of abelian \n    groups has a computable inductive limit.\n\n    \\item Every computable inductive sequence of computable presentations of abelian \n    groups has a computable inductive limit.\n\\end{enumerate}\n\\end{proposition}", "cor:NotCompCatAF": "\\begin{corollary}\\label{cor:NotCompCatAF}\n\n\\\n\n\\begin{enumerate}\n    \\item The AF algebra $C(\\omega +1 )$ is not computably categorical.\n\n    \\item There is a unital dimension group that has two c.e. presentations that are \n    not computably isomorphic.\n\n    \\item There exist two computably presentable Bratteli diagrams that are\n    equivalent but not computably equivalent.\n\\end{enumerate}\n\\end{corollary}", "subsubsec:BackAF": "\\begin{theorem}[Classification of finite-dimensional algebras]\\label{thm:ClsFinDimAlg}\nA \\cstar-algebra $\\boldA$ is finite-dimensional if and only if \nthere is a finite multiset $F$ of positive integers so that \n$\\boldA$ is $\\star$-isomorphic to $\\bigoplus_{n \\in F} M_n(\\C)$.  \n\\end{theorem}\n\nSuppose $F$ is a finite multiset of positive integers and set $\\boldB = \\bigoplus_{n \\in F} M_n(\\C)$.  \nWhen $\\pi$ is the projection map associated with a summand \nof $\\boldB$, we associate $\\pi$ with an injection map $\\iota_\\pi :\\ran(\\pi) \\rightarrow \\boldB$ \nso that $\\pi \\circ \\iota_\\pi = \\Id_{\\ran(\\pi)}$. \nIn addition, we ensure that $\\pi \\circ \\iota_{\\pi'} = \\zerovec$ if $\\pi \\neq \\pi'$.\n\nWe now discuss the $K$-theory of finite-dimensional algebras.  \nLet $k = $ the cardinality of $F$.  \nThen, by the properties of the $K_0$ functor (see, e.g., \\cite[Section 3.2]{Rordam.Larsen.Laustsen.2000}), it is possible to construct from the isomorphisms and order units  described in \nSection \\ref{subsubsec:BackMtrx} an order unit \n$v_F$ for $\\Z^k$ and\na canonical isomorphism $\\zeta_F$ from $\\uK{\\boldB}$ to $(\\Z^k, v_F)$.\nMoreover, this construction provides a \ncanonical bijective mapping $\\xi_F$ from the projection maps of the \nsummands of $\\boldB$ to the generators of $\\Z^k$.\n\nSuppose $F'$ is a finite multiset of positive integers and set $\\boldC = \\bigoplus_{n \\in F'} M_n(\\C)$. \nLet $k' = $ the cardinality of $F'$.  \nSuppose $\\phi$ is a $\\star$-homomorphism from $\\boldB$ to $\\boldC$.  \nThe properties of the $K_0$ functor allow one to construct, from the maps \n$\\eta_{m,n}$ described in Section \\ref{subsubsec:BackMtrx}, \na canonical homomorphism $\\eta_{F,F'}(\\phi)$ from the ordered group $\\Z^k$ to \n$\\Z^{k'}$\nso that $\\zeta_{F'} \\circ \\phi  = \\eta_{F,F'}(\\phi)\\circ\\zeta_F$.\nIn addition, the construction of $\\eta_{F,F'}$ ensures that whenever \n$\\pi$ is the projection map for a summand of $\\boldB$ and $\\pi'$ is a projection \nmap for a summand of $\\boldC$, \n$\\pi_{\\xi_{F'}(\\pi')}(\\eta_{F,F'}(\\phi)(\\xi_F(\\pi))) = $\nthe multiplicity of $\\pi' \\circ \\phi \\circ \\iota_\\pi$.\nIt follows form the discussion in Section \\ref{subsubsec:BackMtrx}\nthat $\\eta_{F, F'}(\\phi)(v_F) \\leq v_{F'}$ and so \n$K_0(\\phi)(K_0(\\unit_\\boldB)) \\leq K_0(\\unit_\\boldC)$.  \nConversely, if $\\gamma : (K_0(\\boldB), K_0(\\boldB)^+) \\rightarrow (K_0(\\boldC), K_0(\\boldC)^+)$\nis a homomorphism and if $\\gamma(K_0(\\unit_\\boldB)) \\leq K_0(\\unit_\\boldC)$, then \nthere is a $\\star$-homomorphism $\\delta$ from $\\boldA$ to $\\boldB$ so \nthat $K_0(\\delta) = \\gamma$. \n\n\\subsubsection{AF algebras}\\label{subsubsec:BackAF}\nWe finally arrive at a formal treatment of AF algebras.\n\\begin{definition}\\label{def:AFCert}\nSuppose $\\boldA$ is a \\cstar-algebra.  \n An \\emph{AF certificate} of $\\boldA$ is a sequence \n$(F_j, \\psi_j)_{j \\in \\N}$ that satisfies the following:\n\\begin{enumerate}\n    \\item Each $F_j$ is a finite multiset of positive integers.\n\n    \\item For each $j \\in \\N$, $\\psi_j$ is a unital $\\star$-embedding of $\\bigoplus_{n \\in F_j} M_n(\\C)$\n    into $\\boldA$.\n\n    \\item For all $j$, $\\ran(\\psi_j) \\subseteq \\ran(\\psi_{j+1})$.\n\n    \\item $\\boldA = \\overline{\\bigcup_{j \\in \\N} \\ran(\\psi_j)}$.\n\\end{enumerate}\n\\end{definition}\n\nA \\cstar-algebra is an \\emph{AF algebra} if it has an AF certificate.  \nWe remark that this definition is not, but nevertheless is equivalent to, \nthe standard definition of an AF algebra as an inductive limit of \nfinite-dimensional algebras.  Our choice of this definition is motivated by forthcoming \ncomputabilty considerations.\n\nWe can now state Elliott's Theorem formally, a proof of which can be found in \n\\cite{Rordam.Larsen.Laustsen.2000}.\n\n\\begin{theorem}[Elliott's Theorem]\\label{thm:Elliott}\nSuppose $\\boldA$ and $\\boldB$ are AF algebras.  If \n$\\uK{\\boldA}$ and \n$\\uK{\\boldB}$ are isomorphic, \nthen $\\boldA$ and $\\boldB$ are  $\\star$-isomorphic.  \nMoreover, if $\\alpha$ is an isomorphism from $\\uK{\\boldA}$\nto $\\uK{\\boldB}$, \nthen there is a unital $\\star$-isomorphism $\\phi : \\boldA \\rightarrow \\boldB$\nso that $K_0(\\phi) = \\alpha$. \n\\end{theorem}", "mthm:CompAFCert": "\\begin{maintheorem}\\label{mthm:CompAFCert} If a unital AF algebra has a c.e. presentation, then it also has a computable AF certificate. \n\\end{maintheorem}", "thm:EffGlimm": "\\begin{theorem}[Effective Glimm Lemma]\\label{thm:EffGlimm}\nThere is a computable function $\\Delta:\\mathbb{N}^2\\to \\mathbb{N}$\nso that for every c.e.-closed $\\star$-subalgebra $\\boldB$ of $\\boldA^\\#$,\n all $n,k \\in \\N$, and every c.e.-closed $\\star$-subalgebra $\\boldF$ of $\\boldA^\\#$ for which $\\unit_\\boldA \\in \\boldF$, \nif $\\dim(\\boldF) = n$, and if $B(\\boldB; 2^{-\\Delta(n,k)})$ contains a matricial generating \nsystem of $\\boldF$, then there is a computable unitary $u$ of $\\boldA^\\#$ so that \n$\\norm{u - \\unit_\\boldA} < 2^{-k}$ and $u^*\\boldF u \\subseteq \\boldB$. \nFurthermore, an $\\boldA^\\#$ index of $u$ can be computed from an index of $\\boldA^\\#$ and $\\boldA^\\#$ indexes of $\\boldF$ and $\\boldB$. \n\\end{theorem}"}, "pre_theorem_intro_text_len": 3401, "pre_theorem_intro_text": "\\label{sec:Intro} \nIn our previous work \\cite{UHFPaper} with R. Miller, we initiated the effective study of $K$-theory for $\\mathrm{C}^*$-algebras.  More precisely, we established a computable functor which associated, to each computably enumerable (c.e.) presentation $\\mathbf{A}^\\#$ of a $\\mathrm{C}^*$-algebra $\\mathbf{A}$, a c.e. presentation $K_0(\\mathbf{A}^\\#)$ of its $K_0$ group $K_0(\\mathbf{A})$.  In addition, this functor maps \ncomputable $\\ast$-homomorphisms to computable group homomorphisms.  These statements are \nmade precise in Section \\ref{subsubsec:BackCompKThy} below.  \nA similar such computable functor was also defined for $K_1$ as well.  As a proof of concept, the special case of the $K_0$ functor restricted to \\emph{uniformly hyperfinite (UHF)} algebras was completely analyzed.\n\nIn this paper, we broaden our scope and consider our computable $K_0$ functor restricted to the class of \\emph{approximately finite-dimensional (AF)} algebras, which are the $\\mathrm{C}^*$-algebras isomorphic to those obtained as inductive limits of finite-dimensional $\\mathrm{C}^*$-algebras.  Every AF algebra $\\mathbf{A}$ has $K_1(\\mathbf{A}) = 0$, so the $K$-theory for AF algebras is entirely focused on $K_0$.  Since AF algebras are stably finite, their $K_0$ group has a natural ordering which renders it a (partially) ordered abelian group.  \nIn addition, for a c.e. presentation $\\mathbf{A}^\\#$, the positive cone of this group, \nwhich we denote $K_0(\\mathbf{A})^+$, is a c.e. set of the presentation $K_0(\\mathbf{A}^\\#)$; \naccordingly, we refer to $(K_0(\\mathbf{A}^\\#), K_0(\\mathbf{A})^+)$ as the \\emph{ordered presented group \nof $\\mathbf{A}^\\#$}.\nThe ordered abelian groups that arise as ordered $K_0$ groups of AF algebras are known as \\emph{dimension groups}.\nBy the Effros-Handelman-Shen Theorem, these are precisely the ordered abelian groups isomorphic to an inductive limit of \\emph{simplicial groups}, that is, groups of the form $\\mathbb{Z}^n$ with their canonical ordering.  By distinguishing the image $K_0(\\unit_\\mathbf{A})$ of the unit of $\\mathbf{A}$ in $K_0(\\mathbf{A})$, one views $K_0(\\mathbf{A})$ as a \\emph{unital (or scaled) group}, that is, as an ordered abelian group with a distinguished order unit.  Elliott's classification theorem \\cite{Elliott.1976} states that AF algebras are characterized, up to isomorphism, by their unital ordered $K_0$ groups.  In fact, the $K_0$ functor yields an equivalence of categories between the category of AF algebras, with approximate unitary equivalence classes of $*$-algebra homomorphisms as morphisms, and the category of unital dimension groups, with order-unit preserving positive group homomorphisms as morphisms.\n\nWhile $K_0$ groups provide an algebraic representation of AF algebras, Bratteli diagrams provide a combinatorial method of representing AF algebras, namely as \ninfinite multi-graphs arranged in a sequence of levels.  \nLabeled Bratteli diagrams assign numbers to the vertexes, and every AF algebra is completely described \nby a labeled Bratteli diagram.  \n\nOur first overarching goal is to classify the c.e. presentations of AF algebras.  \nTo this end, we introduce the concept of an \\emph{AF certificate} (defined in Section \\ref{subsubsec:BackAF}\nbelow).  These essentially describe an AF algebra as a specific inductive limit of finite dimensional \nalgebras.  Our first main result is the following.", "context": "\\label{sec:Intro} \nIn our previous work \\cite{UHFPaper} with R. Miller, we initiated the effective study of $K$-theory for $\\mathrm{C}^*$-algebras. More precisely, we established a computable functor which associated, to each computably enumerable (c.e.) presentation $\\mathbf{A}^\\#$ of a $\\mathrm{C}^*$-algebra $\\mathbf{A}$, a c.e. presentation $K_0(\\mathbf{A}^\\#)$ of its $K_0$ group $K_0(\\mathbf{A})$. In addition, this functor maps \ncomputable $\\ast$-homomorphisms to computable group homomorphisms. These statements are \nmade precise in Section \\ref{subsubsec:BackCompKThy} below. \nA similar such computable functor was also defined for $K_1$ as well. As a proof of concept, the special case of the $K_0$ functor restricted to \\emph{uniformly hyperfinite (UHF)} algebras was completely analyzed.\n\nIn this paper, we broaden our scope and consider our computable $K_0$ functor restricted to the class of \\emph{approximately finite-dimensional (AF)} algebras, which are the $\\mathrm{C}^*$-algebras isomorphic to those obtained as inductive limits of finite-dimensional $\\mathrm{C}^*$-algebras. Every AF algebra $\\mathbf{A}$ has $K_1(\\mathbf{A}) = 0$, so the $K$-theory for AF algebras is entirely focused on $K_0$. Since AF algebras are stably finite, their $K_0$ group has a natural ordering which renders it a (partially) ordered abelian group. \nIn addition, for a c.e. presentation $\\mathbf{A}^\\#$, the positive cone of this group, \nwhich we denote $K_0(\\mathbf{A})^+$, is a c.e. set of the presentation $K_0(\\mathbf{A}^\\#)$; \naccordingly, we refer to $(K_0(\\mathbf{A}^\\#), K_0(\\mathbf{A})^+)$ as the \\emph{ordered presented group \nof $\\mathbf{A}^\\#$}.\nThe ordered abelian groups that arise as ordered $K_0$ groups of AF algebras are known as \\emph{dimension groups}.\nBy the Effros-Handelman-Shen Theorem, these are precisely the ordered abelian groups isomorphic to an inductive limit of \\emph{simplicial groups}, that is, groups of the form $\\mathbb{Z}^n$ with their canonical ordering. By distinguishing the image $K_0(\\unit_\\mathbf{A})$ of the unit of $\\mathbf{A}$ in $K_0(\\mathbf{A})$, one views $K_0(\\mathbf{A})$ as a \\emph{unital (or scaled) group}, that is, as an ordered abelian group with a distinguished order unit. Elliott's classification theorem \\cite{Elliott.1976} states that AF algebras are characterized, up to isomorphism, by their unital ordered $K_0$ groups. In fact, the $K_0$ functor yields an equivalence of categories between the category of AF algebras, with approximate unitary equivalence classes of $*$-algebra homomorphisms as morphisms, and the category of unital dimension groups, with order-unit preserving positive group homomorphisms as morphisms.\n\nWhile $K_0$ groups provide an algebraic representation of AF algebras, Bratteli diagrams provide a combinatorial method of representing AF algebras, namely as \ninfinite multi-graphs arranged in a sequence of levels. \nLabeled Bratteli diagrams assign numbers to the vertexes, and every AF algebra is completely described \nby a labeled Bratteli diagram.\n\nOur first overarching goal is to classify the c.e. presentations of AF algebras. \nTo this end, we introduce the concept of an \\emph{AF certificate} (defined in Section \\ref{subsubsec:BackAF}\nbelow). These essentially describe an AF algebra as a specific inductive limit of finite dimensional \nalgebras. Our first main result is the following.", "full_context": "\\label{sec:Intro} \nIn our previous work \\cite{UHFPaper} with R. Miller, we initiated the effective study of $K$-theory for $\\mathrm{C}^*$-algebras. More precisely, we established a computable functor which associated, to each computably enumerable (c.e.) presentation $\\mathbf{A}^\\#$ of a $\\mathrm{C}^*$-algebra $\\mathbf{A}$, a c.e. presentation $K_0(\\mathbf{A}^\\#)$ of its $K_0$ group $K_0(\\mathbf{A})$. In addition, this functor maps \ncomputable $\\ast$-homomorphisms to computable group homomorphisms. These statements are \nmade precise in Section \\ref{subsubsec:BackCompKThy} below. \nA similar such computable functor was also defined for $K_1$ as well. As a proof of concept, the special case of the $K_0$ functor restricted to \\emph{uniformly hyperfinite (UHF)} algebras was completely analyzed.\n\nIn this paper, we broaden our scope and consider our computable $K_0$ functor restricted to the class of \\emph{approximately finite-dimensional (AF)} algebras, which are the $\\mathrm{C}^*$-algebras isomorphic to those obtained as inductive limits of finite-dimensional $\\mathrm{C}^*$-algebras. Every AF algebra $\\mathbf{A}$ has $K_1(\\mathbf{A}) = 0$, so the $K$-theory for AF algebras is entirely focused on $K_0$. Since AF algebras are stably finite, their $K_0$ group has a natural ordering which renders it a (partially) ordered abelian group. \nIn addition, for a c.e. presentation $\\mathbf{A}^\\#$, the positive cone of this group, \nwhich we denote $K_0(\\mathbf{A})^+$, is a c.e. set of the presentation $K_0(\\mathbf{A}^\\#)$; \naccordingly, we refer to $(K_0(\\mathbf{A}^\\#), K_0(\\mathbf{A})^+)$ as the \\emph{ordered presented group \nof $\\mathbf{A}^\\#$}.\nThe ordered abelian groups that arise as ordered $K_0$ groups of AF algebras are known as \\emph{dimension groups}.\nBy the Effros-Handelman-Shen Theorem, these are precisely the ordered abelian groups isomorphic to an inductive limit of \\emph{simplicial groups}, that is, groups of the form $\\mathbb{Z}^n$ with their canonical ordering. By distinguishing the image $K_0(\\unit_\\mathbf{A})$ of the unit of $\\mathbf{A}$ in $K_0(\\mathbf{A})$, one views $K_0(\\mathbf{A})$ as a \\emph{unital (or scaled) group}, that is, as an ordered abelian group with a distinguished order unit. Elliott's classification theorem \\cite{Elliott.1976} states that AF algebras are characterized, up to isomorphism, by their unital ordered $K_0$ groups. In fact, the $K_0$ functor yields an equivalence of categories between the category of AF algebras, with approximate unitary equivalence classes of $*$-algebra homomorphisms as morphisms, and the category of unital dimension groups, with order-unit preserving positive group homomorphisms as morphisms.\n\nWhile $K_0$ groups provide an algebraic representation of AF algebras, Bratteli diagrams provide a combinatorial method of representing AF algebras, namely as \ninfinite multi-graphs arranged in a sequence of levels. \nLabeled Bratteli diagrams assign numbers to the vertexes, and every AF algebra is completely described \nby a labeled Bratteli diagram.\n\nOur first overarching goal is to classify the c.e. presentations of AF algebras. \nTo this end, we introduce the concept of an \\emph{AF certificate} (defined in Section \\ref{subsubsec:BackAF}\nbelow). These essentially describe an AF algebra as a specific inductive limit of finite dimensional \nalgebras. Our first main result is the following.\n\n\\section{Introduction}\\label{sec:Intro} \nIn our previous work \\cite{UHFPaper} with R. Miller, we initiated the effective study of $K$-theory for \\cstar-algebras. More precisely, we established a computable functor which associated, to each computably enumerable (c.e.) presentation $\\boldA^\\#$ of a \\cstar-algebra $\\boldA$, a c.e. presentation $K_0(\\boldA^\\#)$ of its $K_0$ group $K_0(\\boldA)$. In addition, this functor maps \ncomputable $\\star$-homomorphisms to computable group homomorphisms. These statements are \nmade precise in Section \\ref{subsubsec:BackCompKThy} below. \nA similar such computable functor was also defined for $K_1$ as well. As a proof of concept, the special case of the $K_0$ functor restricted to \\emph{uniformly hyperfinite (UHF)} algebras was completely analyzed.\n\nOur first overarching goal is to classify the c.e. presentations of AF algebras. \nTo this end, we introduce the concept of an \\emph{AF certificate} (defined in Section \\ref{subsubsec:BackAF}\nbelow). These essentially describe an AF algebra as a specific inductive limit of finite dimensional \nalgebras. Our first main result is the following.\n\nIn classical mathematics, when an AF algebra is considered, a representation of it \nas an inductive limit of finite-dimensional algebras is assumed. However, a c.e. \npresentation of an AF algebra, which merely provides information about the \nnorm on a certain dense set, does not in and of itself readily provide such a description. \nRather, from the information provided by a c.e. presentation of an AF algebra $\\boldA$, one must algorithmically discern the finite-dimensional $\\star$-subalgebras of $\\boldA$ (which are not the same\nas the finitely generated subalgebras). Locating these $\\star$-subalgebras turns out to be no \nmean feat, and as an intermediate step we go to great lengths to first prove an effective \nversion of Glimm's Lemma (Theorem \\ref{thm:EffGlimm} below). We note that our proof of Main Theorem \\ref{mthm:CompAFCert} is uniform in that it provides an algorithm for producing AF certificates\nfrom c.e. presentations of AF algebras.\n\n\\item If $u$ is an order unit of $\\calG$, \n then there is a c.e. presentation $\\boldA^\\#$ of a\n unital AF algebra so that the presented unital ordered $K_0$ group of $\\boldA^\\#$\nis computably isomorphic to $(\\calG, u)^\\#$.\\label{mthm:ClsCePresDimGrp::Untl}\n\\end{enumerate}\n\\end{maintheorem}\n\n\\begin{maintheorem}[Classification of computably presentable unital AF algebras]\\label{mthm:ClsCompPrsntAF}\nSuppose $\\boldA$ is a unital AF algebra. Then, the following are equivalent.\n\\begin{enumerate}\n \\item $\\boldA$ has a c.e. presentation. \\label{mthm:ClsCompPrsntAF::CePres}\n\n\\item There is a computable inductive sequence of finite dimensional algebras whose inductive limit is $\\star$-isomorphic\n to $\\boldA$ and so that each bonding map is unital.\\label{mthm:ClsCompPrsntAF::CompIndSeqFinDimAlg}\n\n\\begin{maintheorem}\\label{mthm:CompIso}\nSuppose $\\boldA^\\#$ and $\\boldB^\\#$ are c.e. presentations of unital AF algebras. \nThen, the following are equivalent.\n\\begin{enumerate}\n \\item $\\boldA^\\#$ is computably $\\star$-isomorphic to $\\boldB^\\#$.\\label{mthm:CompIso::Pres}\n\n\\begin{definition}\\label{def:CompAFCert}\nSuppose $(F_j, \\psi_j)_{j \\in \\N}$ is an AF certificate for $\\boldA$, and assume $\\boldA^\\#$ is a \npresentation of $\\boldA$. We say that $(F_j, \\psi_j)_{j \\in \\N}$ is a \\emph{computable AF certificate of $\\boldA^\\#$} if $(F_j)_{j \\in \\N}$ is computable and if each $\\psi_j$ is a computable map\nfrom $\\bigoplus_{n \\in F_j}M_n(\\C)$\n to $\\boldA^\\#$.\n\\end{definition}", "post_theorem_intro_text_len": 6300, "post_theorem_intro_text": "In classical mathematics, when an AF algebra is considered, a representation of it \nas an inductive limit of finite-dimensional algebras is assumed.  However, a c.e. \npresentation of an AF algebra, which merely provides information about the \nnorm on a certain dense set, does not in and of itself readily provide such a description.  \nRather, from the information provided by a c.e. presentation of an AF algebra $\\mathbf{A}$, one must algorithmically discern the finite-dimensional $\\ast$-subalgebras of $\\mathbf{A}$ (which are not the same\nas the finitely generated subalgebras).  Locating these $\\ast$-subalgebras turns out to be no \nmean feat, and as an intermediate step we go to great lengths to first prove an effective \nversion of Glimm's Lemma (Theorem \\ref{thm:EffGlimm} below).  We note that our proof of Main Theorem \\ref{mthm:CompAFCert} is uniform in that it provides an algorithm for producing AF certificates\nfrom c.e. presentations of AF algebras.\n\nOur second main result, which will also contribute to our first overarching goal, \nis an effective version of the Effros-Handelman-Shen Theorem referred to above.\n\n\\begin{maintheorem}\\label{mthm:ClsCePresDimGrp} Suppose $\\mathcal{G}^\\#$ is a c.e. presentation of a dimension group.\n\\begin{enumerate}\n    \\item There is a c.e. presentation $\\mathbf{A}^\\#$ of an AF algebra so that \nthe presented ordered $K_0$ group of $\\mathbf{A}^\\#$\nis computably isomorphic to $\\mathcal{G}^\\#$.\\label{mthm:ClsCePresDimGrp::NotUntl}\n\n    \\item If $u$ is an order unit of $\\mathcal{G}$, \n    then there is a c.e. presentation $\\mathbf{A}^\\#$ of a\n    unital AF algebra so that the presented unital ordered $K_0$ group of $\\mathbf{A}^\\#$\nis computably isomorphic to $(\\mathcal{G}, u)^\\#$.\\label{mthm:ClsCePresDimGrp::Untl}\n\\end{enumerate}\n\\end{maintheorem}\n\nAgain, the proof is uniform.  With these two results in hand, we then \nachieve our goal of classifying the c.e. presentations of unital AF algebras.\n\n\\begin{maintheorem}[Classification of computably presentable unital AF algebras]\\label{mthm:ClsCompPrsntAF}\nSuppose $\\mathbf{A}$ is a unital AF algebra. Then, the following are equivalent.\n\\begin{enumerate}\n    \\item $\\mathbf{A}$ has a c.e. presentation. \\label{mthm:ClsCompPrsntAF::CePres}\n\n    \\item There is a computable inductive sequence     of finite dimensional algebras whose inductive limit is $\\ast$-isomorphic\n    to $\\mathbf{A}$ and so that each     bonding map is unital.\\label{mthm:ClsCompPrsntAF::CompIndSeqFinDimAlg}\n\n    \\item $\\mathbf{A}$ is computably presentatable. \\label{mthm:ClsCompPrsntAF::CompPres}\n\n    \\item The unital ordered $K_0$ group of $\\mathbf{A}$ is computably presentatable. \\label{mthm:ClsCompPrsntAF::CompPresUntlOrdAbl}\n\n    \\item $\\mathbf{A}$ has a computably presentable labeled Bratteli diagram.\\label{mthm:ClsCompPrsntAF::CompBrat}\n\\end{enumerate}\n\\end{maintheorem}\n\nOnce again, the proof is uniform.  Our next overarching goal is to strengthen this \nclassification result by showing that computably isomorphic objects \nmap to computably isomorphic objects.  This is the content of the following.\n\n\\begin{maintheorem}\\label{mthm:CompIso}\nSuppose $\\mathbf{A}^\\#$ and $\\mathbf{B}^\\#$ are c.e. presentations of unital AF algebras.  \nThen, the following are equivalent.\n\\begin{enumerate}\n    \\item $\\mathbf{A}^\\#$ is computably $\\ast$-isomorphic to $\\mathbf{B}^\\#$.\\label{mthm:CompIso::Pres}\n\n    \\item The presented unital ordered $K_0$ groups of $\\mathbf{A}^\\#$ and $\\mathbf{B}^\\#$ are\n    computably isomorphic.\\label{mthm:CompIso::Grp}\n\n    \\item Every Bratteli diagram of a computable AF certificate of \n    $\\mathbf{A}^\\#$ is computably equivalent to \n    every Brattelii diagram of a computable AF certificate of $\\mathbf{B}^\\#$. \\label{mthm:CompIso::Brat}\n\\end{enumerate}\n\\end{maintheorem}\n\nNot surprisingly, the proof is uniform.  Our results so far suggest that \nit should be possible to demonstrate an effective version of Elliott's \nequivalence of categories, and in fact it is:\n\n\\begin{maintheorem}\\label{mthm:CompEqvCat}\nThe following categories are computably equivalent:\n\\begin{enumerate}\n    \\item The category of c.e. presentations of unital AF algebras.\n\n    \\item The category of c.e. presentations of unital dimension groups.\n\\end{enumerate}\n\\end{maintheorem}\n\nWe formally define the terminology in this theorem in Section \\ref{sec:CompEqvCat} below; for now\nlet us say that computable equivalence means the required functors and natural isomorphisms \ncan be computed.  The proof is very much based on the uniformity of the proofs of the prior \ntheorems.\n\nOur main theorems have several interesting consequences.  For one, \nevery c.e. presentation of a unital AF algebra is computable (Corollary \\ref{cor:CompPresAF}). \nAlso, in contrast to UHF algebras, there is an AF algebra that is not computably categorical\n(Corollary \\ref{cor:NotCompCatAF}).  We can use this observation to prove \nsimilar statements for dimension groups and Bratteli diagrams. \nFinally, we use our results to determine the complexity of the index set and isomorphism \nproblems for AF and UHF algebras.\n\nThis paper brings together material from several areas of mathematics: computability theory, \ncategory theory, group theory, and the theory of $\\mathrm{C}^*$-algebras. \nIn order to make the presentation fairly self-contained and navigable, we have chosen \nthe following organization.  We first cover background material from these areas; this consists \nof information that is already in the literature but which we attempt to organize and summarize\nso that the reader need not constantly consult an array of other sources. \nWe divide this coverage into two components: the background from classical mathematics\n(Section \\ref{sec:BackClscl}) and from effective mathematics (Section \\ref{sec:BackEff}).  \nThe latter presents a picture of the \ncomputability theory of the topics covered in the former with a parallel organization. \nWe then attend to the development of preliminary matters from classical and effective mathematics \n(Sections \\ref{sec:PrlmClscl} and \\ref{sec:PrlmEff}). \nThe Effective Glimm Lemma is proven in Section \\ref{sec:EffGlimm}, and our main theorems are proven in Sections \\ref{sec:CompAFCert} through \\ref{sec:CompEqvCat}.\nSection \\ref{sec:IndxIso} contains our results on index sets", "sketch": "From a c.e. presentation of an AF algebra $\\mathbf{A}$ (which “merely provides information about the norm on a certain dense set”), one must “algorithmically discern the finite-dimensional $\\ast$-subalgebras of $\\mathbf{A}$ (which are not the same as the finitely generated subalgebras).” As an intermediate step toward proving Main Theorem~\\ref{mthm:CompAFCert}, the authors “go to great lengths to first prove an effective version of Glimm's Lemma (Theorem \\ref{thm:EffGlimm} below)” in order to help locate these $\\ast$-subalgebras. The proof of Main Theorem~\\ref{mthm:CompAFCert} is stated to be “uniform in that it provides an algorithm for producing AF certificates from c.e. presentations of AF algebras.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{mthm:CompAFCert} If a unital AF algebra has a c.e. presentation, then it also has a computable AF certificate.", "theorem_type": ["Implication"], "mcq": {"question": "Let \\(\\mathbf A\\) be a unital approximately finite-dimensional (AF) \\(C^*\\)-algebra, and suppose \\(\\mathbf A\\) has a computably enumerable (c.e.) presentation. An AF certificate for \\(\\mathbf A\\) is data \\((F_j,\\psi_j)_{j\\in\\mathbb N}\\) describing \\(\\mathbf A\\) as a specific inductive limit of finite-dimensional algebras, where each \\(\\psi_j\\) is a map from \\(\\bigoplus_{n\\in F_j} M_n(\\mathbb C)\\) into \\(\\mathbf A\\). A computable AF certificate of the given presentation \\(\\mathbf A^\\#\\) means such an AF certificate for which the sequence \\((F_j)_{j\\in\\mathbb N}\\) is computable and each \\(\\psi_j\\) is computable as a map into \\(\\mathbf A^\\#\\). Under these assumptions, which statement holds?", "correct_choice": {"label": "A", "text": "There exists a computable AF certificate for the c.e. presentation of \\(\\mathbf A\\); equivalently, every unital AF algebra with a c.e. presentation also has a computable AF certificate."}, "choices": [{"label": "B", "text": "There exists a single algorithm which, given any c.e. presentation of a unital AF algebra, computes a computable AF certificate and moreover computes each map \\(\\psi_j\\) uniformly from the presentation with no additional information."}, {"label": "C", "text": "There exists an AF certificate for the c.e. presentation of \\(\\mathbf A\\); equivalently, every unital AF algebra with a c.e. presentation has some AF certificate."}, {"label": "D", "text": "There exists a computable sequence \\((F_j)_{j\\in\\mathbb N}\\) of finite-dimensional building blocks for \\(\\mathbf A\\), but the maps \\(\\psi_j\\colon \\bigoplus_{n\\in F_j} M_n(\\mathbb C)\\to \\mathbf A^\\#\\) need not be computable."}, {"label": "E", "text": "There exists a computable AF certificate for the c.e. presentation of \\(\\mathbf A\\), and in addition the finite-dimensional \\(^*\\)-subalgebras appearing in the certificate can be effectively recovered as the finitely generated \\(^*\\)-subalgebras of \\(\\mathbf A^\\#\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "uniform algorithm from presentation", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped computability of the certificate", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "computability of each embedding map \\(\\psi_j\\)", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "identifying finite-dimensional \\(^*\\)-subalgebras with finitely generated \\(^*\\)-subalgebras", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the relevant notions but does not explicitly reveal that a computable AF certificate exists. The correct answer is not directly stated in the prompt."}, "TAS": {"score": 0, "justification": "Choice A is essentially the theorem statement being tested, with only minimal reformulation. The item mainly asks for recognition of the exact claimed conclusion."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact theorem from stronger uniform-effectivity claims and weaker existence claims, but the task is still largely theorem recall rather than genuine derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: one is a stronger uniformity claim, one is a weaker true-but-insufficient statement, one confuses computability of objects versus maps, and one adds an unjustified extra effectiveness claim."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-recognition MCQ with strong distractors, but it is close to a direct restatement of the target result and only moderately tests generative reasoning."}}
{"id": "2602.06897v1", "paper_link": "http://arxiv.org/abs/2602.06897v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "\\label{th:main}\n\\[\nN^{1/3} \\log N \\ll f_0(I(H_N)) \\ll N^{1/3} \\log N.\n\\]", "start_pos": 4664, "end_pos": 4757, "label": "th:main"}, "ref_dict": {"th:Andr": "\\begin{thm}\\label{th:Andr} For a lattice polytope $P \\subset \\RR^d$ with $\\vol P>0$\n\\[\nf_0(P) \\ll (\\vol P)^{\\frac {d-1}{d+1}}\n\\]\nwith the implied constant depending only on $d$.\n\\end{thm}", "th:main": "\\begin{thm}\\label{th:main}\n\\[\nN^{1/3} \\log N \\ll f_0(I(H_N)) \\ll N^{1/3} \\log N.\n\\]\n\\end{thm}", "eq:BalB": "\\begin{equation}\\label{eq:BalB}\n 0.33 r^{2/3} < f_0(P_r) < 5.55 r^{2/3}.   \n\\end{equation}"}, "pre_theorem_intro_text_len": 1410, "pre_theorem_intro_text": "Define $H_N=\\{(x,y)\\in \\mathbb{R}^2: xy\\ge N, x,y>0\\}$ which is an unbounded convex set and $N$ is a (large) positive integer. Its {\\sl integer convex hull} or {\\sl integer hull} for short is the convex hull of the integer (or lattice) points in $H_N$:\n\\[\nI(H_N)=\\conv(\\mathbb{Z}^2 \\cap H_N).\n\\]\nThis is bounded by the two halflines $\\{(x,1): x\\ge 1\\}$ and $\\{(1,y): y\\ge 1\\}$ and a convex lattice chain connecting $(1,N)$ with $(N,1)$. Our target is to determine the order of magnitude of the number of vertices of $I(H_N)$. The traditional notation for the number of vertices of a polytope $P\\subset \\mathbb{R}^d$ is $f_0(P)$, so we are interested in $f_0(I(H_N))$.\n\n\\medskip\nMotivation for this question comes from integer programming and, more generally, from the theory of geometry of numbers. The determination or estimation of $f_0(I(H_N))$ is considered in a recent paper by Alc\\'antara et al. \\cite{Santos}. They establish among other results that\n\\begin{equation}\\label{eq:Sant}\nN^{1/3} \\ll f_0(I(H_N)) \\ll N^{1/3} \\log N,\n\\end{equation}\nand ask what the right order of magnitude of $f_0(I(H_N))$ is. Here and throughout the paper the notation $f(N)\\ll g(N)$ means that there is a constant $C>0$ such that $0<f(N)<Cg(N)$ for every $N \\in \\mathbb{N}$. The notation $f(N)\\gg g(N)$ is analogous.  We will also use the $O(.)$ notation. Our main result establishes the order of magnitude of $f_0(I(H_N))$.", "context": "Define $H_N=\\{(x,y)\\in \\mathbb{R}^2: xy\\ge N, x,y>0\\}$ which is an unbounded convex set and $N$ is a (large) positive integer. Its {\\sl integer convex hull} or {\\sl integer hull} for short is the convex hull of the integer (or lattice) points in $H_N$:\n\\[\nI(H_N)=\\conv(\\mathbb{Z}^2 \\cap H_N).\n\\]\nThis is bounded by the two halflines $\\{(x,1): x\\ge 1\\}$ and $\\{(1,y): y\\ge 1\\}$ and a convex lattice chain connecting $(1,N)$ with $(N,1)$. Our target is to determine the order of magnitude of the number of vertices of $I(H_N)$. The traditional notation for the number of vertices of a polytope $P\\subset \\mathbb{R}^d$ is $f_0(P)$, so we are interested in $f_0(I(H_N))$.\n\n\\medskip\nMotivation for this question comes from integer programming and, more generally, from the theory of geometry of numbers. The determination or estimation of $f_0(I(H_N))$ is considered in a recent paper by Alc\\'antara et al. \\cite{Santos}. They establish among other results that\n\\begin{equation}\\label{eq:Sant}\nN^{1/3} \\ll f_0(I(H_N)) \\ll N^{1/3} \\log N,\n\\end{equation}\nand ask what the right order of magnitude of $f_0(I(H_N))$ is. Here and throughout the paper the notation $f(N)\\ll g(N)$ means that there is a constant $C>0$ such that $0<f(N)<Cg(N)$ for every $N \\in \\mathbb{N}$. The notation $f(N)\\gg g(N)$ is analogous. We will also use the $O(.)$ notation. Our main result establishes the order of magnitude of $f_0(I(H_N))$.", "full_context": "Define $H_N=\\{(x,y)\\in \\mathbb{R}^2: xy\\ge N, x,y>0\\}$ which is an unbounded convex set and $N$ is a (large) positive integer. Its {\\sl integer convex hull} or {\\sl integer hull} for short is the convex hull of the integer (or lattice) points in $H_N$:\n\\[\nI(H_N)=\\conv(\\mathbb{Z}^2 \\cap H_N).\n\\]\nThis is bounded by the two halflines $\\{(x,1): x\\ge 1\\}$ and $\\{(1,y): y\\ge 1\\}$ and a convex lattice chain connecting $(1,N)$ with $(N,1)$. Our target is to determine the order of magnitude of the number of vertices of $I(H_N)$. The traditional notation for the number of vertices of a polytope $P\\subset \\mathbb{R}^d$ is $f_0(P)$, so we are interested in $f_0(I(H_N))$.\n\n\\medskip\nMotivation for this question comes from integer programming and, more generally, from the theory of geometry of numbers. The determination or estimation of $f_0(I(H_N))$ is considered in a recent paper by Alc\\'antara et al. \\cite{Santos}. They establish among other results that\n\\begin{equation}\\label{eq:Sant}\nN^{1/3} \\ll f_0(I(H_N)) \\ll N^{1/3} \\log N,\n\\end{equation}\nand ask what the right order of magnitude of $f_0(I(H_N))$ is. Here and throughout the paper the notation $f(N)\\ll g(N)$ means that there is a constant $C>0$ such that $0<f(N)<Cg(N)$ for every $N \\in \\mathbb{N}$. The notation $f(N)\\gg g(N)$ is analogous. We will also use the $O(.)$ notation. Our main result establishes the order of magnitude of $f_0(I(H_N))$.\n\n\\medskip\nMotivation for this question comes from integer programming and, more generally, from the theory of geometry of numbers. The determination or estimation of $f_0(I(H_N))$ is considered in a recent paper by Alc\\'antara et al. \\cite{Santos}. They establish among other results that\n\\begin{equation}\\label{eq:Sant}\nN^{1/3} \\ll f_0(I(H_N)) \\ll N^{1/3} \\log N,\n\\end{equation}\nand ask what the right order of magnitude of $f_0(I(H_N))$ is. Here and throughout the paper the notation $f(N)\\ll g(N)$ means that there is a constant $C>0$ such that $0<f(N)<Cg(N)$ for every $N \\in \\NN$. The notation $f(N)\\gg g(N)$ is analogous. We will also use the $O(.)$ notation. Our main result establishes the order of magnitude of $f_0(I(H_N))$.\n\nA similar question is considered in our earlier paper \\cite{BalBar}. Writing $B^2$ for the Euclidean unit disk centered at the origin the set $P_r=I(rB^2)$ is a convex lattice polytope, the integer hull of $rB^2$. It is shown in \\cite{BalBar} that \n\\begin{equation}\\label{eq:BalB}\n 0.33 r^{2/3} < f_0(P_r) < 5.55 r^{2/3}. \n\\end{equation}\nMost likely the limit $r^{-2/3}f_0(P_r)$ exists. Balog and Deshoullier \\cite{BalD}\nproved that the average of $r^{-2/3}f_0(P_r)$ in the interval $[R,R+M]$ tends to an explicit constant 3.453.. as $R \\to \\infty$ and $M \\to\\infty$ (for instance $M=\\log R$). The upper bound in (\\ref{eq:BalB}) is easier and follows from Andrews theorem (Theorem~\\ref{th:Andr} below). The lower bound is more difficult and the method of its proof will be used here with suitable modifications. But the case of hyperbola is more involved because the curvature changes from $N^{-1/2}$ to $N^{-2}$ while it is constant for the disk.\n\n\\medskip\nWe observe that $I(H_N)$ has few, about $2N^{1/3}$ vertices outside a square $[1,N_2]^2$, where $N_2$ is about $N^{2/3}$, because we can actually list these vertices. They are\n\\[\n(x,y) = (k,\\lceil\\frac Nk\\rceil),\\quad 1\\leq k\\leq N^{1/3},\n\\]\nand by symmetry\n\\[\n(x,y) = (\\lceil\\frac Nk\\rceil,k),\\quad 1\\leq k\\leq N^{1/3}.\n\\]\nIndeed, the integer point $(1,N)$ is a vertex and $(k,\\lceil\\frac Nk\\rceil)$ is certainly a vertex whenever $(k-1,\\lceil\\frac N{k-1}\\rceil)$ is a vertex and\n\\[\n\\frac N{k+1} > 2\\lceil\\frac Nk\\rceil - \\lceil\\frac N{k-1}\\rceil. \n\\]\nThis inequality follows if $k\\leq N^{1/3}$. We mention that the above list of vertices is practically the lower bound of Alc\\'antara et al. ~\\cite{Santos}. For later use we point out that with\n\\begin{equation}\\label{eq:N_12}\n N_1=\\lfloor N^{1/3}\\rfloor =N^{1/3}+O(1),\\quad N_2=\\lceil \\frac N{N_1} \\rceil=N^{2/3}+O(N^{1/3}),\n\\end{equation}\nthe points $(N_1,N_2)$ and $(N_2,N_1)$ are vertices of $I(H_N)$.\nIntroducing the notation $Q_N=H_N\\cap [1,N_2]^2$ another form of Theorem~\\ref{th:main} is the following.\n\n\\begin{thm}\\label{th:main2}\n\\[\nN^{1/3} \\log N \\ll f_0(I(Q_N)) \\ll N^{1/3} \\log N.\n\\]\n\\end{thm}\n\n\\begin{thm}\\label{th:area}\n\\[ N^{1/3} \\log N \\ll A_N \\ll N^{1/3} \\log N.\\]\n\\end{thm}\n\n\\medskip\nOur Theorem~\\ref{th:main} claims that a positive proportion of the integer points in the above hyperbolic strip are actually vertices of $I(H_N)$. For technical reasons, we change to a somewhat narrower strip. In this way, we can miss a few vertices, but we can afford to get a lower bound. \nSet $\\Delta=\\frac12 N^{1/3}$, and write $H^1$ and $H^2$ for the hyperbolas with equations $xy=N$ and $xy=N+\\Delta$, respectively, and finally $S$ for the strip between $H^1$ and $H^2$. The above argument shows that \n\\begin{equation}\\label{eq:delta} \n|S \\cap \\ZZ^2| = \\Delta \\log N+O(N^{1/3}).\n\\end{equation}\n\n\\medskip\nThe number of non-vertices associated with the fixed primitive vector $(a,b)$ is then at most $M_1+M_2\\le 4\\sqrt{\\Delta/ab}$. Suppose $(a_i,b_i)\\in \\ZZ^2$ for $i=1,\\ldots,R$ is the set of primitive vectors (with distinct directions) that take part in the relations $z\\to v$ with $z \\in NV$ and $v \\in V$. Then $R\\le 2|V|$ as one vertex is used with at most two vectors $(a_i,b_i).$ At the same time \n\\begin{equation}\\label{eq:3root}\n|NV| \\le \\sum_{i=1}^R 4\\sqrt{\\frac{\\Delta}{a_ib_i}}=4\\sqrt{\\Delta}\\sum_{i=1}^R \\frac1{\\sqrt{a_ib_i}}.\n\\end{equation} \nThe last sum is the largest when we choose $R$ different primitive vectors with $a_ib_i$ as small as possible, so with $a_ib_i\\le w$ for the least possible $w\\in \\NN$. From {\\bf Fact ~\\ref{fa:divisors}} this $w$ satisfies $F(w-1)<R\\le F(w)$ and then\n\\[\nR=\\frac1{\\zeta(2)}w\\log w+O(w). \n\\]\nThis $w$ satisfies \n\\[ w\\ll \\frac R{\\log R}\\ll \\frac {|V|}{\\log |V|}. \\]\n\nLet $G_j$ be the set of inner normals $p$ with $x_p \\in I_j$. We are going to show that $ \\Area \\left(\\bigcup_{p\\in G_j}\\right) C^p \\ll N^{1/3}$ for every $j$. This will complete the proof as there are $O(\\log N)$ such intervals. For a fixed $p \\in G_j$, the cap $C^p$ has parameters $x_p,h_x,r_x$ and $\\rho_x$. The advantage of the intervals $I_j$ is that the radius of curvature $r_x\\asymp \\frac {x^3}N$ on $I_j$ is almost constant, and so is the slope $-\\frac N{x^2}$ of the tangent to $H^1$ at $x$:\n\\[\n2^{-3j}N \\ll r_x \\ll 2^{-3j+3}N \\mbox{ and } -2^{2j}N^{-1/3} \\le -\\frac N{x^2} \\le -2^{2j-2}N^{-1/3} \n\\]\nWe set $R_j:= 2^{-3j}N$ and define $\\Phi_j\\in (0,\\pi/4]$ via $\\tan \\Phi_j:= 2^{2j}N^{-1/3}$. So for all $x\\in I_j$\n\\begin{equation}\\label{eq:bounded}\nx\\asymp 2^{-j}N^{2/3}\\mbox{ and } r_x\\asymp R_j \\mbox{ and } \\frac N{x^2}\\asymp \\tan \\Phi_j \\mbox{ and }|I_j|=2^{-j}N^{2/3}.\n\\end{equation}\nWe distinguish three cases.\n\n\\begin{equation}\\label{eq:BalB}\n 0.33 r^{2/3} < f_0(P_r) < 5.55 r^{2/3}.   \n\\end{equation}\n\n\\begin{thm}\\label{th:Andr} For a lattice polytope $P \\subset \\RR^d$ with $\\vol P>0$\n\\[\nf_0(P) \\ll (\\vol P)^{\\frac {d-1}{d+1}}\n\\]\nwith the implied constant depending only on $d$.\n\\end{thm}\n\n\\begin{thm}\\label{th:main}\n\\[\nN^{1/3} \\log N \\ll f_0(I(H_N)) \\ll N^{1/3} \\log N.\n\\]\n\\end{thm}", "post_theorem_intro_text_len": 2750, "post_theorem_intro_text": "A similar question is considered in our earlier paper \\cite{BalBar}. Writing $B^2$ for the Euclidean unit disk centered at the origin the set $P_r=I(rB^2)$ is a convex lattice polytope, the integer hull of $rB^2$. It is shown in \\cite{BalBar} that \n\\begin{equation}\\label{eq:BalB}\n 0.33 r^{2/3} < f_0(P_r) < 5.55 r^{2/3}.   \n\\end{equation}\nMost likely the limit $r^{-2/3}f_0(P_r)$ exists. Balog and Deshoullier \\cite{BalD}\nproved that the average of $r^{-2/3}f_0(P_r)$ in the interval $[R,R+M]$ tends to an explicit constant 3.453.. as $R \\to \\infty$ and $M \\to\\infty$ (for instance $M=\\log R$). The upper bound in (\\ref{eq:BalB}) is easier and follows from Andrews theorem (Theorem~\\ref{th:Andr} below). The lower bound is more difficult and the method of its proof will be used here with suitable modifications. But the case of hyperbola is more involved because the curvature changes from $N^{-1/2}$ to $N^{-2}$ while it is constant for the disk.\n\n\\medskip\nWe observe that $I(H_N)$ has few, about $2N^{1/3}$ vertices outside a square $[1,N_2]^2$, where $N_2$ is about $N^{2/3}$, because we can actually list these vertices. They are\n\\[\n(x,y) = (k,\\lceil\\frac Nk\\rceil),\\quad 1\\leq k\\leq N^{1/3},\n\\]\nand by symmetry\n\\[\n(x,y) = (\\lceil\\frac Nk\\rceil,k),\\quad 1\\leq k\\leq N^{1/3}.\n\\]\nIndeed, the integer point $(1,N)$ is a vertex and $(k,\\lceil\\frac Nk\\rceil)$ is certainly a vertex whenever $(k-1,\\lceil\\frac N{k-1}\\rceil)$ is a vertex and\n\\[\n\\frac N{k+1} > 2\\lceil\\frac Nk\\rceil - \\lceil\\frac N{k-1}\\rceil. \n\\]\nThis inequality follows if $k\\leq N^{1/3}$. We mention that the above list of vertices is practically the lower bound of Alc\\'antara et al. ~\\cite{Santos}. For later use we point out that with\n\\begin{equation}\\label{eq:N_12}\n    N_1=\\lfloor N^{1/3}\\rfloor =N^{1/3}+O(1),\\quad N_2=\\lceil \\frac N{N_1} \\rceil=N^{2/3}+O(N^{1/3}),\n\\end{equation}\nthe points $(N_1,N_2)$ and $(N_2,N_1)$ are vertices of $I(H_N)$.\nIntroducing the notation $Q_N=H_N\\cap [1,N_2]^2$ another form of Theorem~\\ref{th:main} is the following.\n\n\\begin{thm}\\label{th:main2}\n\\[\nN^{1/3} \\log N \\ll f_0(I(Q_N)) \\ll N^{1/3} \\log N.\n\\]\n\\end{thm}\n\nOur second main result is about how well $I(H_N)$ approximates $H_N$ when the approximation is measured by the area $H_N$ missed by $I(H_N)$. But this area is infinite because $H_N \\setminus I(H_N)$ contains a large part of the set $\\{(x,y)\\in \\mathbb{R}^2: x\\in [0,\\infty), y\\in [0,1)\\}$. So it is better to consider $(H_N \\setminus I(H_N))\\cap [1,N]^2$. The area of this set is of order $N$; see the remark at the end of Section~\\ref{sec:prepa}. So we rather work with $Q_N$ again and define\n\\[\nA_N=\\Area(Q_N \\setminus I(Q_N)).\n\\]\n\n\\begin{thm}\\label{th:area}\n\\[ N^{1/3} \\log N \\ll A_N \\ll N^{1/3} \\log N.\\]\n\\end{thm}\n\n\\bigskip", "sketch": "The post-theorem introduction indicates how the proof of Theorem~\\ref{th:main} will proceed only at a high level: the authors note that for the disk case the upper bound “follows from Andrews theorem (Theorem~\\ref{th:Andr} below)”, while “the lower bound is more difficult and the method of its proof will be used here with suitable modifications.” They then explain an initial structural step for the hyperbola case: “we observe that $I(H_N)$ has few, about $2N^{1/3}$ vertices outside a square $[1,N_2]^2$, where $N_2$ is about $N^{2/3}$, because we can actually list these vertices,” namely $(x,y)=(k,\\lceil N/k\\rceil)$ and by symmetry $(\\lceil N/k\\rceil,k)$ for $1\\le k\\le N^{1/3}$. They justify these are vertices via an inductive condition and an inequality that “follows if $k\\le N^{1/3}$,” and record that with $N_1=\\lfloor N^{1/3}\\rfloor$ and $N_2=\\lceil N/N_1\\rceil$ the points $(N_1,N_2)$ and $(N_2,N_1)$ are vertices. Finally, they reformulate Theorem~\\ref{th:main} as the equivalent statement for $Q_N=H_N\\cap[1,N_2]^2$ (Theorem~\\ref{th:main2}), indicating the argument reduces to bounding $f_0(I(Q_N))$. They caution that “the case of hyperbola is more involved because the curvature changes from $N^{-1/2}$ to $N^{-2}$ while it is constant for the disk.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{th:main}\n\\[\nN^{1/3} \\log N \\ll f_0(I(H_N)) \\ll N^{1/3} \\log N.\n\\],", "theorem_type": ["Inequality or Bound"], "mcq": {"question": "Let\n\\[\nH_N=\\{(x,y)\\in\\mathbb{R}^2: xy\\ge N,\\ x,y>0\\},\n\\]\nwhere $N$ is a large positive integer. Its integer hull is\n\\[\nI(H_N)=\\operatorname{conv}(\\mathbb{Z}^2\\cap H_N),\n\\]\nand $f_0(P)$ denotes the number of vertices of a polytope $P$. Using the notation $f(N)\\ll g(N)$ to mean that $f(N)\\le Cg(N)$ for some absolute constant $C>0$, which quantitative estimate holds for the number of vertices of $I(H_N)$?", "correct_choice": {"label": "A", "text": "\\[\nN^{1/3}\\log N\\ll f_0\\bigl(I(H_N)\\bigr)\\ll N^{1/3}\\log N,\n\\]\nso $f_0(I(H_N))$ has order of magnitude $N^{1/3}\\log N$."}, "choices": [{"label": "B", "text": "\\[\nN^{1/3}\\ll f_0\\bigl(I(H_N)\\bigr)\\ll N^{1/3},\n\\]\nso $f_0(I(H_N))$ has order of magnitude $N^{1/3}$."}, {"label": "C", "text": "\\[\nN^{1/3}\\ll f_0\\bigl(I(H_N)\\bigr)\\ll N^{1/3}\\log N.\n\\]\nIn particular, the number of vertices grows between constant multiples of $N^{1/3}$ and $N^{1/3}\\log N$."}, {"label": "D", "text": "\\[\nN^{1/3}(\\log N)^2\\ll f_0\\bigl(I(H_N)\\bigr)\\ll N^{1/3}(\\log N)^2,\n\\]\nso $f_0(I(H_N))$ has order of magnitude $N^{1/3}(\\log N)^2$."}, {"label": "E", "text": "\\[\nN^{1/3}\\log N\\ll f_0\\bigl(I(H_N)\\bigr)\\ll N^{2/3}\\log N,\n\\]\nso one only gets a lower bound of order $N^{1/3}\\log N$ together with an upper bound of order $N^{2/3}\\log N$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "missing_log_factor", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "drop_matching_lower_bound_log_factor", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "extra_log_factor", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "asymmetric_upper_exponent", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct asymptotic estimate, nor does it contain strong hints favoring the matching upper/lower bound with a log factor."}, "TAS": {"score": 1, "justification": "This is essentially a theorem-recall question: it asks for the quantitative estimate for a specific object rather than requiring derivation from intermediate facts. The nearby alternatives make it slightly less tautological, but it remains close to a restatement of a known result."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in comparing closely related asymptotic forms and deciding whether the sharp order includes a log factor, but the item mainly tests recall of the exact result. Also, option C is a weaker statement that is still true, which weakens the demand for precise generative reasoning."}, "DQS": {"score": 1, "justification": "The distractors are mathematically plausible and target common asymptotic mistakes (missing log, extra log, wrong exponent). However, one distractor (C) appears to be a weaker true statement, making the options non-exclusive and reducing distractor quality in a single-correct MCQ."}, "total_score": 5, "overall_assessment": "Reasonably well-formed and free of answer leakage, with plausible asymptotic distractors, but it is close to theorem restatement and suffers from ambiguity because a weaker true option is included alongside the intended sharp answer."}}
{"id": "2602.06972v1", "paper_link": "http://arxiv.org/abs/2602.06972v1", "theorems_cnt": 1, "theorem": {"env_name": "thm", "content": "\\label{main}\nLet $S$ be a two-element ai-semiring, and let $n \\geq 2$ be an integer.\nThen the matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ is not a distributive lattice.", "start_pos": 12231, "end_pos": 12447, "label": "main"}, "ref_dict": {"main": "\\begin{thm}\\label{main}\nLet $S$ be a two-element ai-semiring, and let $n \\geq 2$ be an integer.\nThen the matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ is not a distributive lattice.\n\\end{thm}"}, "pre_theorem_intro_text_len": 5955, "pre_theorem_intro_text": "An \\emph{additively idempotent semiring} (abbreviated as ai-semiring)\nis an algebraic structure $(S, +, \\cdot)$ equipped with two binary operations $+$ and $\\cdot$ such that:\n\\begin{itemize}\n\\item the additive reduct $(S, +)$ is a commutative idempotent semigroup;\n\\item the multiplicative reduct $(S, \\cdot)$ is a semigroup;\n\\item the distributive laws hold:\n\\[\n(x+y)z \\approx xz + yz, \\qquad x(y+z) \\approx xy + xz.\n\\]\n\\end{itemize}\nThis class of algebras includes well-known examples such as the Kleene semiring of regular languages~\\cite{con},\nthe max-plus algebra~\\cite{aei}, the power semiring of a semigroup~\\cite{dgv24},\nthe endomorphism semiring of a semilattice~\\cite{dgv25},\nthe semiring of all binary relations on a set \\cite{dolinka2009}, and distributive lattices~\\cite{burris1981}.\nThese and other similar algebras have found significant applications in several branches of mathematics,\nsuch as algebraic geometry~\\cite{cc}, tropical geometry~\\cite{ms}, information science~\\cite{gl},\ntheoretical computer science~\\cite{go}, and soft constraint solving and programming~\\cite{bis2004}.\n\nLet $S$ be an ai-semiring. Define a binary relation $\\leq$ on $S$ by\n\\[\na\\leq b\\Leftrightarrow a+b=b.\n\\]\nThen $\\leq$ is a partial order on $S$, and $(S, \\leq)$ becomes an upper semilattice in which\nthe supremum of any elements $a$ and $b$ is $a+b$.\nConsequently, the additive reduct $(S, +)$ is uniquely determined by this semilattice order.\nTherefore, it is often convenient to represent the addition operation using the Hasse diagram of $(S, \\leq)$.\nMoreover, one can easily verify that this semilattice order is compatible with multiplication.\nFor this reason, an ai-semiring is often called a \\emph{semilattice-ordered semigroup}.\n\nLet $n\\geq 1$ be an integer,\nlet $S$ be an ai-semiring, and let $\\mathbf{M}_n(S)$ denote the set of all $n \\times n$ matrices over $S$.\nThen $\\mathbf{M}_n(S)$ forms an ai-semiring under the usual matrix addition and multiplication, that is,\n\\[\nA+B = [a_{ij}+b_{ij}]_{n\\times n}, \\quad A \\cdot B = \\left[ \\sum_{k=1}^n a_{ik} b_{kj} \\right]_{n\\times n}\n\\]\nfor all $A=[a_{ij}]_{n\\times n}, B=[b_{ij}]_{n\\times n}\\in \\mathbf{M}_n(S)$.\nIt is obvious that $S$ is isomorphic to $\\mathbf{M}_1(S)$.\nFor $n\\geq 2$, $S$ can be embedded into $\\mathbf{M}_n(S)$,\nsince the mapping\n\\[\n\\varphi\\colon S \\rightarrow \\mathbf{M}_n(S),\\quad a \\mapsto [a]_{n\\times n},\n\\]\nwhere $[a]_{n\\times n}$ denotes the constant matrix with all entries equal to $a$,\nis an embedding mapping.\nMoreover, if $S$ has an additive identity that is also a multiplicative zero,\nthen we can easily obtain the following infinite strict ascending chain\n\\[\nS\\hookrightarrow \\mathbf{M}_2(S)\\hookrightarrow \\mathbf{M}_{3}(S)\\hookrightarrow  \\mathbf{M}_{4}(S)\\hookrightarrow\\cdots.\n\\]\n\nLet $\\mathcal{V}$ be a class of algebras.\nThen $\\mathcal{V}$ is called a \\emph{variety} if it is closed under\ntaking subalgebras, homomorphic images, and arbitrary direct products.\nBy the celebrated Birkhoff's theorem \\cite{birkhoff1935}, $\\mathcal{V}$ is a variety if and only if\nit is an \\emph{equational class}, that is, the class of all algebras satisfying some set of identities.\n\nA variety is \\emph{finitely based} if it can be defined by a finite set of identities;\notherwise, it is \\emph{nonfinitely based}.\nAn algebra $A$ is finitely based or nonfinitely based if the variety $\\mathsf{V}(A)$\nit generates is finitely based or not.\nThe \\emph{finite basis problem} for a class of algebras,\none of the central problems in universal algebra,\nconcerns the classification of\nits members with respect to the property of being finitely based.\n\nA variety is \\emph{locally finite} if each of its finitely generated members is finite.\nA locally finite variety is \\emph{inherently nonfinitely based}\nif it is not contained in any finitely based locally finite variety.\nSince every finite algebra generates a locally finite variety,\nwe say that a finite algebra $A$ is inherently nonfinitely based if\nthe variety $\\mathsf{V}(A)$ is inherently nonfinitely based.\nIt follows immediately that every finite algebra whose variety contains\nan inherently nonfinitely based algebra is nonfinitely based.\nConsequently, every inherently nonfinitely based algebra must be nonfinitely based.\n\nOver the past two decades,\nthe finite basis problem for ai-semirings has attracted considerable attention,\nresulting in substantial\nprogress~\\cite{dol07, dolinka2009, dgv24, dgv25, gpz05, gv23, gv2501, gv2510, jrz, pas05, rz16, rzw, sr, volkov2024, yr25, yrzs, zrc}.\nIn particular, Dolinka~\\cite{dol07} constructed the first example of a nonfinitely based finite ai-semiring.\nPastijn et al.~\\cite{gpz05, pas05} proved that every ai-semiring satisfying the identity $x^2 \\approx x$ is finitely based;\nRen et al.~\\cite{rz16, rzw} later extended this to ai-semirings satisfying the identity $x^3 \\approx x$.\nShao and Ren~\\cite{sr} established that every algebra in the variety generated by all two-element ai-semirings is finitely based.\nSubsequently,\nJackson et al.~\\cite{jrz} and Zhao et al.~\\cite{zrc}\nprovided a complete classification of all three-element ai-semirings with respect to the finite basis property.\nMost recently,\nDolinka, Gusev and Volkov~\\cite{dgv25, gv2501} settled the finite basis problem for\nthe endomorphism semirings of finite semilattices.\n\nTo the best of our knowledge,\nthe study of the finite basis problem for matrix semirings was initiated by\nDolinka~\\cite{dolinka2009}, who considered the problem for $\\mathbf{M}_n(D_2)$ for $n\\geqslant 2$,\nwhere $D_2$ denotes the two-element distributive lattice.\nGusev and Volkov~\\cite{gv2510, volkov2024} are currently investigating\nthe same problem for upper triangular matrix semirings over $D_2$.\nMotivated by these works,\nwe undertake a systematic classification of matrix semirings  $\\mathbf{M}_n(S)$,\nwhere $S$ ranges over all two-element ai-semirings.\nThe main result of this paper provides the following complete characterization.", "context": "An \\emph{additively idempotent semiring} (abbreviated as ai-semiring)\nis an algebraic structure $(S, +, \\cdot)$ equipped with two binary operations $+$ and $\\cdot$ such that:\n\\begin{itemize}\n\\item the additive reduct $(S, +)$ is a commutative idempotent semigroup;\n\\item the multiplicative reduct $(S, \\cdot)$ is a semigroup;\n\\item the distributive laws hold:\n\\[\n(x+y)z \\approx xz + yz, \\qquad x(y+z) \\approx xy + xz.\n\\]\n\\end{itemize}\nThis class of algebras includes well-known examples such as the Kleene semiring of regular languages~\\cite{con},\nthe max-plus algebra~\\cite{aei}, the power semiring of a semigroup~\\cite{dgv24},\nthe endomorphism semiring of a semilattice~\\cite{dgv25},\nthe semiring of all binary relations on a set \\cite{dolinka2009}, and distributive lattices~\\cite{burris1981}.\nThese and other similar algebras have found significant applications in several branches of mathematics,\nsuch as algebraic geometry~\\cite{cc}, tropical geometry~\\cite{ms}, information science~\\cite{gl},\ntheoretical computer science~\\cite{go}, and soft constraint solving and programming~\\cite{bis2004}.\n\nLet $S$ be an ai-semiring. Define a binary relation $\\leq$ on $S$ by\n\\[\na\\leq b\\Leftrightarrow a+b=b.\n\\]\nThen $\\leq$ is a partial order on $S$, and $(S, \\leq)$ becomes an upper semilattice in which\nthe supremum of any elements $a$ and $b$ is $a+b$.\nConsequently, the additive reduct $(S, +)$ is uniquely determined by this semilattice order.\nTherefore, it is often convenient to represent the addition operation using the Hasse diagram of $(S, \\leq)$.\nMoreover, one can easily verify that this semilattice order is compatible with multiplication.\nFor this reason, an ai-semiring is often called a \\emph{semilattice-ordered semigroup}.\n\nLet $n\\geq 1$ be an integer,\nlet $S$ be an ai-semiring, and let $\\mathbf{M}_n(S)$ denote the set of all $n \\times n$ matrices over $S$.\nThen $\\mathbf{M}_n(S)$ forms an ai-semiring under the usual matrix addition and multiplication, that is,\n\\[\nA+B = [a_{ij}+b_{ij}]_{n\\times n}, \\quad A \\cdot B = \\left[ \\sum_{k=1}^n a_{ik} b_{kj} \\right]_{n\\times n}\n\\]\nfor all $A=[a_{ij}]_{n\\times n}, B=[b_{ij}]_{n\\times n}\\in \\mathbf{M}_n(S)$.\nIt is obvious that $S$ is isomorphic to $\\mathbf{M}_1(S)$.\nFor $n\\geq 2$, $S$ can be embedded into $\\mathbf{M}_n(S)$,\nsince the mapping\n\\[\n\\varphi\\colon S \\rightarrow \\mathbf{M}_n(S),\\quad a \\mapsto [a]_{n\\times n},\n\\]\nwhere $[a]_{n\\times n}$ denotes the constant matrix with all entries equal to $a$,\nis an embedding mapping.\nMoreover, if $S$ has an additive identity that is also a multiplicative zero,\nthen we can easily obtain the following infinite strict ascending chain\n\\[\nS\\hookrightarrow \\mathbf{M}_2(S)\\hookrightarrow \\mathbf{M}_{3}(S)\\hookrightarrow \\mathbf{M}_{4}(S)\\hookrightarrow\\cdots.\n\\]\n\nLet $\\mathcal{V}$ be a class of algebras.\nThen $\\mathcal{V}$ is called a \\emph{variety} if it is closed under\ntaking subalgebras, homomorphic images, and arbitrary direct products.\nBy the celebrated Birkhoff's theorem \\cite{birkhoff1935}, $\\mathcal{V}$ is a variety if and only if\nit is an \\emph{equational class}, that is, the class of all algebras satisfying some set of identities.\n\nOver the past two decades,\nthe finite basis problem for ai-semirings has attracted considerable attention,\nresulting in substantial\nprogress~\\cite{dol07, dolinka2009, dgv24, dgv25, gpz05, gv23, gv2501, gv2510, jrz, pas05, rz16, rzw, sr, volkov2024, yr25, yrzs, zrc}.\nIn particular, Dolinka~\\cite{dol07} constructed the first example of a nonfinitely based finite ai-semiring.\nPastijn et al.~\\cite{gpz05, pas05} proved that every ai-semiring satisfying the identity $x^2 \\approx x$ is finitely based;\nRen et al.~\\cite{rz16, rzw} later extended this to ai-semirings satisfying the identity $x^3 \\approx x$.\nShao and Ren~\\cite{sr} established that every algebra in the variety generated by all two-element ai-semirings is finitely based.\nSubsequently,\nJackson et al.~\\cite{jrz} and Zhao et al.~\\cite{zrc}\nprovided a complete classification of all three-element ai-semirings with respect to the finite basis property.\nMost recently,\nDolinka, Gusev and Volkov~\\cite{dgv25, gv2501} settled the finite basis problem for\nthe endomorphism semirings of finite semilattices.\n\nTo the best of our knowledge,\nthe study of the finite basis problem for matrix semirings was initiated by\nDolinka~\\cite{dolinka2009}, who considered the problem for $\\mathbf{M}_n(D_2)$ for $n\\geqslant 2$,\nwhere $D_2$ denotes the two-element distributive lattice.\nGusev and Volkov~\\cite{gv2510, volkov2024} are currently investigating\nthe same problem for upper triangular matrix semirings over $D_2$.\nMotivated by these works,\nwe undertake a systematic classification of matrix semirings $\\mathbf{M}_n(S)$,\nwhere $S$ ranges over all two-element ai-semirings.\nThe main result of this paper provides the following complete characterization.", "full_context": "An \\emph{additively idempotent semiring} (abbreviated as ai-semiring)\nis an algebraic structure $(S, +, \\cdot)$ equipped with two binary operations $+$ and $\\cdot$ such that:\n\\begin{itemize}\n\\item the additive reduct $(S, +)$ is a commutative idempotent semigroup;\n\\item the multiplicative reduct $(S, \\cdot)$ is a semigroup;\n\\item the distributive laws hold:\n\\[\n(x+y)z \\approx xz + yz, \\qquad x(y+z) \\approx xy + xz.\n\\]\n\\end{itemize}\nThis class of algebras includes well-known examples such as the Kleene semiring of regular languages~\\cite{con},\nthe max-plus algebra~\\cite{aei}, the power semiring of a semigroup~\\cite{dgv24},\nthe endomorphism semiring of a semilattice~\\cite{dgv25},\nthe semiring of all binary relations on a set \\cite{dolinka2009}, and distributive lattices~\\cite{burris1981}.\nThese and other similar algebras have found significant applications in several branches of mathematics,\nsuch as algebraic geometry~\\cite{cc}, tropical geometry~\\cite{ms}, information science~\\cite{gl},\ntheoretical computer science~\\cite{go}, and soft constraint solving and programming~\\cite{bis2004}.\n\nLet $S$ be an ai-semiring. Define a binary relation $\\leq$ on $S$ by\n\\[\na\\leq b\\Leftrightarrow a+b=b.\n\\]\nThen $\\leq$ is a partial order on $S$, and $(S, \\leq)$ becomes an upper semilattice in which\nthe supremum of any elements $a$ and $b$ is $a+b$.\nConsequently, the additive reduct $(S, +)$ is uniquely determined by this semilattice order.\nTherefore, it is often convenient to represent the addition operation using the Hasse diagram of $(S, \\leq)$.\nMoreover, one can easily verify that this semilattice order is compatible with multiplication.\nFor this reason, an ai-semiring is often called a \\emph{semilattice-ordered semigroup}.\n\nLet $n\\geq 1$ be an integer,\nlet $S$ be an ai-semiring, and let $\\mathbf{M}_n(S)$ denote the set of all $n \\times n$ matrices over $S$.\nThen $\\mathbf{M}_n(S)$ forms an ai-semiring under the usual matrix addition and multiplication, that is,\n\\[\nA+B = [a_{ij}+b_{ij}]_{n\\times n}, \\quad A \\cdot B = \\left[ \\sum_{k=1}^n a_{ik} b_{kj} \\right]_{n\\times n}\n\\]\nfor all $A=[a_{ij}]_{n\\times n}, B=[b_{ij}]_{n\\times n}\\in \\mathbf{M}_n(S)$.\nIt is obvious that $S$ is isomorphic to $\\mathbf{M}_1(S)$.\nFor $n\\geq 2$, $S$ can be embedded into $\\mathbf{M}_n(S)$,\nsince the mapping\n\\[\n\\varphi\\colon S \\rightarrow \\mathbf{M}_n(S),\\quad a \\mapsto [a]_{n\\times n},\n\\]\nwhere $[a]_{n\\times n}$ denotes the constant matrix with all entries equal to $a$,\nis an embedding mapping.\nMoreover, if $S$ has an additive identity that is also a multiplicative zero,\nthen we can easily obtain the following infinite strict ascending chain\n\\[\nS\\hookrightarrow \\mathbf{M}_2(S)\\hookrightarrow \\mathbf{M}_{3}(S)\\hookrightarrow \\mathbf{M}_{4}(S)\\hookrightarrow\\cdots.\n\\]\n\nLet $\\mathcal{V}$ be a class of algebras.\nThen $\\mathcal{V}$ is called a \\emph{variety} if it is closed under\ntaking subalgebras, homomorphic images, and arbitrary direct products.\nBy the celebrated Birkhoff's theorem \\cite{birkhoff1935}, $\\mathcal{V}$ is a variety if and only if\nit is an \\emph{equational class}, that is, the class of all algebras satisfying some set of identities.\n\nOver the past two decades,\nthe finite basis problem for ai-semirings has attracted considerable attention,\nresulting in substantial\nprogress~\\cite{dol07, dolinka2009, dgv24, dgv25, gpz05, gv23, gv2501, gv2510, jrz, pas05, rz16, rzw, sr, volkov2024, yr25, yrzs, zrc}.\nIn particular, Dolinka~\\cite{dol07} constructed the first example of a nonfinitely based finite ai-semiring.\nPastijn et al.~\\cite{gpz05, pas05} proved that every ai-semiring satisfying the identity $x^2 \\approx x$ is finitely based;\nRen et al.~\\cite{rz16, rzw} later extended this to ai-semirings satisfying the identity $x^3 \\approx x$.\nShao and Ren~\\cite{sr} established that every algebra in the variety generated by all two-element ai-semirings is finitely based.\nSubsequently,\nJackson et al.~\\cite{jrz} and Zhao et al.~\\cite{zrc}\nprovided a complete classification of all three-element ai-semirings with respect to the finite basis property.\nMost recently,\nDolinka, Gusev and Volkov~\\cite{dgv25, gv2501} settled the finite basis problem for\nthe endomorphism semirings of finite semilattices.\n\nTo the best of our knowledge,\nthe study of the finite basis problem for matrix semirings was initiated by\nDolinka~\\cite{dolinka2009}, who considered the problem for $\\mathbf{M}_n(D_2)$ for $n\\geqslant 2$,\nwhere $D_2$ denotes the two-element distributive lattice.\nGusev and Volkov~\\cite{gv2510, volkov2024} are currently investigating\nthe same problem for upper triangular matrix semirings over $D_2$.\nMotivated by these works,\nwe undertake a systematic classification of matrix semirings $\\mathbf{M}_n(S)$,\nwhere $S$ ranges over all two-element ai-semirings.\nThe main result of this paper provides the following complete characterization.\n\n\\begin{abstract}\nWe provide a complete classification of matrix semirings $\\mathbf{M}_n(S)$ over two-element additively idempotent semirings $S$\nwith respect to the finite basis property.\nOur main theorem shows that for every integer $n \\geq 2$,\nthe semiring $\\mathbf{M}_n(S)$ is finitely based\nif and only if $S$ is distinct from a distributive lattice.\n\\end{abstract}\n\nTo the best of our knowledge,\nthe study of the finite basis problem for matrix semirings was initiated by\nDolinka~\\cite{dolinka2009}, who considered the problem for $\\mathbf{M}_n(D_2)$ for $n\\ge 2$,\nwhere $D_2$ denotes the two-element distributive lattice.\nGusev and Volkov~\\cite{gv2510, volkov2024} are currently investigating\nthe same problem for upper triangular matrix semirings over $D_2$.\nMotivated by these works,\nwe undertake a systematic classification of matrix semirings $\\mathbf{M}_n(S)$,\nwhere $S$ ranges over all two-element ai-semirings.\nThe main result of this paper provides the following complete characterization.\n\nFor the proof,\nwe first note that the case where $S$ is a distributive lattice follows from Dolinka's work.\nSpecifically,\n\\cite[Theorem B]{dolinka2009} states that the semiring $\\mathcal{R}_2$ of all binary relations on a two-element set is inherently nonfinitely based.\nSince $\\mathcal{R}_2 $ is isomorphic to $\\mathbf{M}_2(D_2)$,\nit follows immediately that $\\mathbf{M}_2(D_2)$ inherits this property.\nFurthermore, because $\\mathbf{M}_n(D_2)$ contains a copy of $\\mathbf{M}_2(D_2)$ for all $n \\geq 2$,\nthe same property extends to $\\mathbf{M}_n(D_2)$; this result is explicitly stated in \\cite[Corollary 6.2]{dolinka2009}.\n(We note that $D_2$ is also denoted $\\mathbb{B}_2$ and called the two-element Boolean semiring; our notation follows \\cite{sr}.)\n\nTherefore, to establish Theorem~\\ref{main}, it remains to prove the converse:\nthat the matrix semiring $\\mathbf{M}_n(S)$ is finitely based\nfor every two-element ai-semiring $S$ distinct from a distributive lattice and every $n \\geq 2$.\nThe necessary preliminaries are collected in Section 2, and the proof is carried out in Sections 3 and 4.\n\n\\begin{pro}\\label{pro21}\nLet $n\\geq 2$ be an integer. Then\n$\\mathsf{V}(\\mathbf{M}_n(L_2))$ is the ai-semiring variety determined by the identities \\eqref{eq:T22} and\n\\begin{align}\n&xy\\approx xz. \\label{f1}\n\\end{align}\n\\end{pro}\n\\begin{proof}\nWe first show that $\\mathbf{M}_n(L_2)$ satisfies identities \\eqref{eq:T22} and \\eqref{f1}.\nLet $A = [a_{ij}]$, $B = [b_{ij}]$, $C = [c_{ij}]$ be arbitrary matrices in $\\mathbf{M}_n(L_2)$, where all entries belong to $L_2$.\nFor any indices $1 \\leq i, j \\leq n$, we have\n\\[\n(AB)_{ij} = \\sum_{k=1}^{n} a_{ik}b_{kj} \\stackrel{\\eqrefbasis{eq:L2}}=\n\\sum_{k=1}^{n} a_{ik} \\stackrel{\\eqrefbasis{eq:L2}}= \\sum_{k=1}^{n} a_{ik}c_{kj} = (AC)_{ij},\n\\]\nand\n\\begin{align*}\n(A^2)_{ij} = \\sum_{k=1}^{n} a_{ik}a_{kj}\n&= \\left( \\sum_{k=1}^{n} a_{ik}a_{kj} \\right) + a_{ij}a_{jj} \\\\\n&\\stackrel{\\eqrefbasis{eq:L2}}= \\left( \\sum_{k=1}^{n} a_{ik}a_{kj} \\right) + a_{ij} \\\\\n&= (A^2)_{ij} + A_{ij} = (A^2 + A)_{ij}.\n\\end{align*}\nThis shows that $AB = AC$ and $A^2 = A^2 + A$.\nHence $\\mathbf{M}_n(L_2)$ satisfies the identities \\eqref{eq:T22} and \\eqref{f1}.\n\n\\begin{pro}\\label{pro22}\nLet $n\\geq 2$ be an integer. Then\n$\\mathsf{V}(\\mathbf{M}_n(R_2))$ coincides with $\\mathsf{V}(S_{56})$, properly contains $\\mathsf{V}(R_2)$,\nand is the ai-semiring variety determined by the identities \\eqref{eq:T22} and\n\\begin{align}\n&xy\\approx zy. \\label{f3}\n\\end{align}\n\\end{pro}\n\\begin{proof}\nObserve that $R_2$ and $L_2$ have dual multiplications.\nThe remaining argument parallels those of Proposition~\\ref{pro21} and Remark~\\ref{rem1}.\n\\end{proof}\n\n\\begin{pro}\\label{pro233}\nLet $n\\geq 1$ be an integer. Then\n$\\mathsf{V}(\\mathbf{M}_n(N_2))$ is the ai-semiring variety determined by the identities \\eqref{eq:N21} and \\eqref{eq:N22}.\n\\end{pro}\n\\begin{proof}\nLet $A = [a_{ij}]$, $B = [b_{ij}]$, $C = [c_{ij}]$, $D = [d_{ij}]$ be arbitrary matrices in $\\mathbf{M}_n(N_2)$.\nFor any $1 \\leq i, j \\leq n$, we have\n\\[\n(AB)_{ij} = \\sum_{k=1}^{n} a_{ik}b_{kj} \\stackrel{\\eqrefbasis{eq:N21}}= \\sum_{k=1}^{n} c_{ik}d_{kj} = (CD)_{ij},\n\\]\nand\n\\begin{align*}\n(A + A^2)_{ij}\n&= A_{ij} + (A^2)_{ij}=a_{ij} + \\left(\\sum_{k=1}^n a_{ik}a_{kj}\\right)\\\\\n&\\stackrel{\\eqrefbasis{eq:N21}}= a_{ij} + a_{ij}a_{ij}\n\\stackrel{\\eqrefbasis{eq:N22}}= a_{ij}= A_{ij}.\n\\end{align*}\nThis shows that $AB = CD$ and $A=A+A^2$.\nHence $\\mathbf{M}_n(N_2)$ satisfies both identities \\eqref{eq:N21} and \\eqref{eq:N22}.\n\n\\section{Conclusion}\nWe have provided a complete classification of\nmatrix semirings $\\mathbf{M}_n(S)$ over two-element ai-semirings $S$\nwith respect to the finite basis property.\nFrom Remark~\\ref{rem1} and \\cite[Figure 1]{yr25},\nthe variety $\\mathsf{V}(\\mathbf{M}_n(L_2))$ has precisely $5$ subvarieties;\nthey form a distributive lattice and are all finitely based.\nThe same conclusion holds for the variety $\\mathsf{V}(\\mathbf{M}_n(R_2))$.\nBy \\cite[Theorem 1]{polin} together with Propositions~\\ref{pro233} and \\ref{pro23},\nthe varieties $\\mathsf{V}(\\mathbf{M}_n(N_2))$ and $\\mathsf{V}(\\mathbf{M}_n(T_2))$\nare both minimal nontrivial varieties.\nIn contrast, the subvariety lattice of the variety $\\mathsf{V}(\\mathbf{M}_n(M_2))$\nremains to be explicitly described.\n\n\\begin{thm}\\label{main}\nLet $S$ be a two-element ai-semiring, and let $n \\geq 2$ be an integer.\nThen the matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ is not a distributive lattice.\n\\end{thm}", "post_theorem_intro_text_len": 1096, "post_theorem_intro_text": "For the proof,\nwe first note that the case where $S$ is a distributive lattice follows from Dolinka's work.\nSpecifically,\n\\cite[Theorem B]{dolinka2009} states that the semiring $\\mathcal{R}_2$ of all binary relations on a two-element set is inherently nonfinitely based.\nSince $\\mathcal{R}_2 $ is isomorphic to $\\mathbf{M}_2(D_2)$,\nit follows immediately that $\\mathbf{M}_2(D_2)$ inherits this property.\nFurthermore, because $\\mathbf{M}_n(D_2)$ contains a copy of $\\mathbf{M}_2(D_2)$ for all $n \\geq 2$,\nthe same property extends to $\\mathbf{M}_n(D_2)$; this result is explicitly stated in \\cite[Corollary 6.2]{dolinka2009}.\n(We note that $D_2$ is also denoted $\\mathbb{B}_2$ and called the two-element Boolean semiring; our notation follows \\cite{sr}.)\n\nTherefore, to establish Theorem~\\ref{main}, it remains to prove the converse:\nthat the matrix semiring $\\mathbf{M}_n(S)$ is finitely based\nfor every two-element ai-semiring $S$ distinct from a distributive lattice and every $n \\geq 2$.\nThe necessary preliminaries are collected in Section 2, and the proof is carried out in Sections 3 and 4.", "sketch": "For the proof of Theorem~\\ref{main}: (1) In the case where $S$ is a distributive lattice, the result follows from Dolinka's work: \\cite[Theorem B]{dolinka2009} shows that the semiring $\\mathcal{R}_2$ of all binary relations on a two-element set is inherently nonfinitely based. Since $\\mathcal{R}_2\\cong \\mathbf{M}_2(D_2)$, it follows that $\\mathbf{M}_2(D_2)$ is inherently nonfinitely based; and because $\\mathbf{M}_n(D_2)$ contains a copy of $\\mathbf{M}_2(D_2)$ for all $n\\ge2$, the same property extends to $\\mathbf{M}_n(D_2)$ (explicitly \\cite[Corollary 6.2]{dolinka2009}). (2) Thus, to establish Theorem~\\ref{main} it remains to prove the converse: $\\mathbf{M}_n(S)$ is finitely based for every two-element ai-semiring $S$ distinct from a distributive lattice and every $n\\ge2$. The preliminaries are in Section~2 and the proof is carried out in Sections~3 and~4.", "expanded_sketch": "For the proof of the main theorem: (1) In the case where $S$ is a distributive lattice, the result follows from Dolinka's work: \\cite[Theorem B]{dolinka2009} shows that the semiring $\\mathcal{R}_2$ of all binary relations on a two-element set is inherently nonfinitely based. Since $\\mathcal{R}_2\\cong \\mathbf{M}_2(D_2)$, it follows that $\\mathbf{M}_2(D_2)$ is inherently nonfinitely based; and because $\\mathbf{M}_n(D_2)$ contains a copy of $\\mathbf{M}_2(D_2)$ for all $n\\ge2$, the same property extends to $\\mathbf{M}_n(D_2)$ (explicitly \\cite[Corollary 6.2]{dolinka2009}). (2) Thus, to establish the main theorem it remains to prove the converse: $\\mathbf{M}_n(S)$ is finitely based for every two-element ai-semiring $S$ distinct from a distributive lattice and every $n\\ge2$. The preliminaries are given next, and the proof is then carried out subsequently.", "expanded_theorem": "\\label{main}\nLet $S$ be a two-element ai-semiring, and let $n \\geq 2$ be an integer.\nThen the matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ is not a distributive lattice.", "theorem_type": ["Biconditional or Equivalence", "Universal"], "mcq": {"question": "An additively idempotent semiring (ai-semiring) is an algebra $(S,+,\\cdot)$ such that $(S,+)$ is a commutative idempotent semigroup, $(S,\\cdot)$ is a semigroup, and multiplication distributes over addition on both sides. Let $S$ be a two-element ai-semiring, and let $n\\ge 2$ be an integer. Write $\\mathbf{M}_n(S)$ for the ai-semiring of all $n\\times n$ matrices over $S$ with the usual matrix addition and multiplication. An algebra is called finitely based if the variety it generates is definable by finitely many identities. Which statement holds for every such $S$ and $n$?", "correct_choice": {"label": "A", "text": "The matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ is not a distributive lattice."}, "choices": [{"label": "B", "text": "The matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ is a distributive lattice."}, {"label": "C", "text": "If $S$ is not a distributive lattice, then the matrix semiring $\\mathbf{M}_n(S)$ is finitely based."}, {"label": "D", "text": "For every two-element ai-semiring $S$, the matrix semiring $\\mathbf{M}_n(S)$ is finitely based for all sufficiently large integers $n\\ge 2$."}, {"label": "E", "text": "The matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ itself is finitely based."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "distributive-lattice case direction", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the converse implication excluding distributive lattices", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "uniformity in n from the embedding of $\\mathbf{M}_2(D_2)$ into all $\\mathbf{M}_n(D_2)$ for $n\\ge2$", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "replacement of the matrix-semiring criterion by a criterion on the base semiring", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct criterion. It asks for an equivalent characterization, and the correct answer is not implied by the wording alone."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: the correct option is essentially the theorem statement itself. However, the presence of nearby stronger, weaker, and boundary-case variants makes it slightly more than a pure restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to reject the distractors by checking sufficiency vs. equivalence, the range restriction on n, and uniformity claims. Still, the question mainly tests precise recall of a characterization rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and well targeted: one weakens the biconditional, one overextends the range of n, one adds an unjustified uniform finite-basis claim, and one substitutes an unrelated finite-basis criterion on S itself."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it is primarily theorem recognition rather than a deeply generative reasoning task."}}
{"id": "2602.06972v1", "paper_link": "http://arxiv.org/abs/2602.06972v1", "theorems_cnt": 1, "theorem": {"env_name": "thm", "content": "\\label{main}\nLet $S$ be a two-element ai-semiring, and let $n \\geq 2$ be an integer.\nThen the matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ is not a distributive lattice.", "start_pos": 12231, "end_pos": 12447, "label": "main"}, "ref_dict": {"main": "\\begin{thm}\\label{main}\nLet $S$ be a two-element ai-semiring, and let $n \\geq 2$ be an integer.\nThen the matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ is not a distributive lattice.\n\\end{thm}"}, "pre_theorem_intro_text_len": 5955, "pre_theorem_intro_text": "An \\emph{additively idempotent semiring} (abbreviated as ai-semiring)\nis an algebraic structure $(S, +, \\cdot)$ equipped with two binary operations $+$ and $\\cdot$ such that:\n\\begin{itemize}\n\\item the additive reduct $(S, +)$ is a commutative idempotent semigroup;\n\\item the multiplicative reduct $(S, \\cdot)$ is a semigroup;\n\\item the distributive laws hold:\n\\[\n(x+y)z \\approx xz + yz, \\qquad x(y+z) \\approx xy + xz.\n\\]\n\\end{itemize}\nThis class of algebras includes well-known examples such as the Kleene semiring of regular languages~\\cite{con},\nthe max-plus algebra~\\cite{aei}, the power semiring of a semigroup~\\cite{dgv24},\nthe endomorphism semiring of a semilattice~\\cite{dgv25},\nthe semiring of all binary relations on a set \\cite{dolinka2009}, and distributive lattices~\\cite{burris1981}.\nThese and other similar algebras have found significant applications in several branches of mathematics,\nsuch as algebraic geometry~\\cite{cc}, tropical geometry~\\cite{ms}, information science~\\cite{gl},\ntheoretical computer science~\\cite{go}, and soft constraint solving and programming~\\cite{bis2004}.\n\nLet $S$ be an ai-semiring. Define a binary relation $\\leq$ on $S$ by\n\\[\na\\leq b\\Leftrightarrow a+b=b.\n\\]\nThen $\\leq$ is a partial order on $S$, and $(S, \\leq)$ becomes an upper semilattice in which\nthe supremum of any elements $a$ and $b$ is $a+b$.\nConsequently, the additive reduct $(S, +)$ is uniquely determined by this semilattice order.\nTherefore, it is often convenient to represent the addition operation using the Hasse diagram of $(S, \\leq)$.\nMoreover, one can easily verify that this semilattice order is compatible with multiplication.\nFor this reason, an ai-semiring is often called a \\emph{semilattice-ordered semigroup}.\n\nLet $n\\geq 1$ be an integer,\nlet $S$ be an ai-semiring, and let $\\mathbf{M}_n(S)$ denote the set of all $n \\times n$ matrices over $S$.\nThen $\\mathbf{M}_n(S)$ forms an ai-semiring under the usual matrix addition and multiplication, that is,\n\\[\nA+B = [a_{ij}+b_{ij}]_{n\\times n}, \\quad A \\cdot B = \\left[ \\sum_{k=1}^n a_{ik} b_{kj} \\right]_{n\\times n}\n\\]\nfor all $A=[a_{ij}]_{n\\times n}, B=[b_{ij}]_{n\\times n}\\in \\mathbf{M}_n(S)$.\nIt is obvious that $S$ is isomorphic to $\\mathbf{M}_1(S)$.\nFor $n\\geq 2$, $S$ can be embedded into $\\mathbf{M}_n(S)$,\nsince the mapping\n\\[\n\\varphi\\colon S \\rightarrow \\mathbf{M}_n(S),\\quad a \\mapsto [a]_{n\\times n},\n\\]\nwhere $[a]_{n\\times n}$ denotes the constant matrix with all entries equal to $a$,\nis an embedding mapping.\nMoreover, if $S$ has an additive identity that is also a multiplicative zero,\nthen we can easily obtain the following infinite strict ascending chain\n\\[\nS\\hookrightarrow \\mathbf{M}_2(S)\\hookrightarrow \\mathbf{M}_{3}(S)\\hookrightarrow  \\mathbf{M}_{4}(S)\\hookrightarrow\\cdots.\n\\]\n\nLet $\\mathcal{V}$ be a class of algebras.\nThen $\\mathcal{V}$ is called a \\emph{variety} if it is closed under\ntaking subalgebras, homomorphic images, and arbitrary direct products.\nBy the celebrated Birkhoff's theorem \\cite{birkhoff1935}, $\\mathcal{V}$ is a variety if and only if\nit is an \\emph{equational class}, that is, the class of all algebras satisfying some set of identities.\n\nA variety is \\emph{finitely based} if it can be defined by a finite set of identities;\notherwise, it is \\emph{nonfinitely based}.\nAn algebra $A$ is finitely based or nonfinitely based if the variety $\\mathsf{V}(A)$\nit generates is finitely based or not.\nThe \\emph{finite basis problem} for a class of algebras,\none of the central problems in universal algebra,\nconcerns the classification of\nits members with respect to the property of being finitely based.\n\nA variety is \\emph{locally finite} if each of its finitely generated members is finite.\nA locally finite variety is \\emph{inherently nonfinitely based}\nif it is not contained in any finitely based locally finite variety.\nSince every finite algebra generates a locally finite variety,\nwe say that a finite algebra $A$ is inherently nonfinitely based if\nthe variety $\\mathsf{V}(A)$ is inherently nonfinitely based.\nIt follows immediately that every finite algebra whose variety contains\nan inherently nonfinitely based algebra is nonfinitely based.\nConsequently, every inherently nonfinitely based algebra must be nonfinitely based.\n\nOver the past two decades,\nthe finite basis problem for ai-semirings has attracted considerable attention,\nresulting in substantial\nprogress~\\cite{dol07, dolinka2009, dgv24, dgv25, gpz05, gv23, gv2501, gv2510, jrz, pas05, rz16, rzw, sr, volkov2024, yr25, yrzs, zrc}.\nIn particular, Dolinka~\\cite{dol07} constructed the first example of a nonfinitely based finite ai-semiring.\nPastijn et al.~\\cite{gpz05, pas05} proved that every ai-semiring satisfying the identity $x^2 \\approx x$ is finitely based;\nRen et al.~\\cite{rz16, rzw} later extended this to ai-semirings satisfying the identity $x^3 \\approx x$.\nShao and Ren~\\cite{sr} established that every algebra in the variety generated by all two-element ai-semirings is finitely based.\nSubsequently,\nJackson et al.~\\cite{jrz} and Zhao et al.~\\cite{zrc}\nprovided a complete classification of all three-element ai-semirings with respect to the finite basis property.\nMost recently,\nDolinka, Gusev and Volkov~\\cite{dgv25, gv2501} settled the finite basis problem for\nthe endomorphism semirings of finite semilattices.\n\nTo the best of our knowledge,\nthe study of the finite basis problem for matrix semirings was initiated by\nDolinka~\\cite{dolinka2009}, who considered the problem for $\\mathbf{M}_n(D_2)$ for $n\\geqslant 2$,\nwhere $D_2$ denotes the two-element distributive lattice.\nGusev and Volkov~\\cite{gv2510, volkov2024} are currently investigating\nthe same problem for upper triangular matrix semirings over $D_2$.\nMotivated by these works,\nwe undertake a systematic classification of matrix semirings  $\\mathbf{M}_n(S)$,\nwhere $S$ ranges over all two-element ai-semirings.\nThe main result of this paper provides the following complete characterization.", "context": "An \\emph{additively idempotent semiring} (abbreviated as ai-semiring)\nis an algebraic structure $(S, +, \\cdot)$ equipped with two binary operations $+$ and $\\cdot$ such that:\n\\begin{itemize}\n\\item the additive reduct $(S, +)$ is a commutative idempotent semigroup;\n\\item the multiplicative reduct $(S, \\cdot)$ is a semigroup;\n\\item the distributive laws hold:\n\\[\n(x+y)z \\approx xz + yz, \\qquad x(y+z) \\approx xy + xz.\n\\]\n\\end{itemize}\nThis class of algebras includes well-known examples such as the Kleene semiring of regular languages~\\cite{con},\nthe max-plus algebra~\\cite{aei}, the power semiring of a semigroup~\\cite{dgv24},\nthe endomorphism semiring of a semilattice~\\cite{dgv25},\nthe semiring of all binary relations on a set \\cite{dolinka2009}, and distributive lattices~\\cite{burris1981}.\nThese and other similar algebras have found significant applications in several branches of mathematics,\nsuch as algebraic geometry~\\cite{cc}, tropical geometry~\\cite{ms}, information science~\\cite{gl},\ntheoretical computer science~\\cite{go}, and soft constraint solving and programming~\\cite{bis2004}.\n\nLet $S$ be an ai-semiring. Define a binary relation $\\leq$ on $S$ by\n\\[\na\\leq b\\Leftrightarrow a+b=b.\n\\]\nThen $\\leq$ is a partial order on $S$, and $(S, \\leq)$ becomes an upper semilattice in which\nthe supremum of any elements $a$ and $b$ is $a+b$.\nConsequently, the additive reduct $(S, +)$ is uniquely determined by this semilattice order.\nTherefore, it is often convenient to represent the addition operation using the Hasse diagram of $(S, \\leq)$.\nMoreover, one can easily verify that this semilattice order is compatible with multiplication.\nFor this reason, an ai-semiring is often called a \\emph{semilattice-ordered semigroup}.\n\nLet $n\\geq 1$ be an integer,\nlet $S$ be an ai-semiring, and let $\\mathbf{M}_n(S)$ denote the set of all $n \\times n$ matrices over $S$.\nThen $\\mathbf{M}_n(S)$ forms an ai-semiring under the usual matrix addition and multiplication, that is,\n\\[\nA+B = [a_{ij}+b_{ij}]_{n\\times n}, \\quad A \\cdot B = \\left[ \\sum_{k=1}^n a_{ik} b_{kj} \\right]_{n\\times n}\n\\]\nfor all $A=[a_{ij}]_{n\\times n}, B=[b_{ij}]_{n\\times n}\\in \\mathbf{M}_n(S)$.\nIt is obvious that $S$ is isomorphic to $\\mathbf{M}_1(S)$.\nFor $n\\geq 2$, $S$ can be embedded into $\\mathbf{M}_n(S)$,\nsince the mapping\n\\[\n\\varphi\\colon S \\rightarrow \\mathbf{M}_n(S),\\quad a \\mapsto [a]_{n\\times n},\n\\]\nwhere $[a]_{n\\times n}$ denotes the constant matrix with all entries equal to $a$,\nis an embedding mapping.\nMoreover, if $S$ has an additive identity that is also a multiplicative zero,\nthen we can easily obtain the following infinite strict ascending chain\n\\[\nS\\hookrightarrow \\mathbf{M}_2(S)\\hookrightarrow \\mathbf{M}_{3}(S)\\hookrightarrow \\mathbf{M}_{4}(S)\\hookrightarrow\\cdots.\n\\]\n\nLet $\\mathcal{V}$ be a class of algebras.\nThen $\\mathcal{V}$ is called a \\emph{variety} if it is closed under\ntaking subalgebras, homomorphic images, and arbitrary direct products.\nBy the celebrated Birkhoff's theorem \\cite{birkhoff1935}, $\\mathcal{V}$ is a variety if and only if\nit is an \\emph{equational class}, that is, the class of all algebras satisfying some set of identities.\n\nOver the past two decades,\nthe finite basis problem for ai-semirings has attracted considerable attention,\nresulting in substantial\nprogress~\\cite{dol07, dolinka2009, dgv24, dgv25, gpz05, gv23, gv2501, gv2510, jrz, pas05, rz16, rzw, sr, volkov2024, yr25, yrzs, zrc}.\nIn particular, Dolinka~\\cite{dol07} constructed the first example of a nonfinitely based finite ai-semiring.\nPastijn et al.~\\cite{gpz05, pas05} proved that every ai-semiring satisfying the identity $x^2 \\approx x$ is finitely based;\nRen et al.~\\cite{rz16, rzw} later extended this to ai-semirings satisfying the identity $x^3 \\approx x$.\nShao and Ren~\\cite{sr} established that every algebra in the variety generated by all two-element ai-semirings is finitely based.\nSubsequently,\nJackson et al.~\\cite{jrz} and Zhao et al.~\\cite{zrc}\nprovided a complete classification of all three-element ai-semirings with respect to the finite basis property.\nMost recently,\nDolinka, Gusev and Volkov~\\cite{dgv25, gv2501} settled the finite basis problem for\nthe endomorphism semirings of finite semilattices.\n\nTo the best of our knowledge,\nthe study of the finite basis problem for matrix semirings was initiated by\nDolinka~\\cite{dolinka2009}, who considered the problem for $\\mathbf{M}_n(D_2)$ for $n\\geqslant 2$,\nwhere $D_2$ denotes the two-element distributive lattice.\nGusev and Volkov~\\cite{gv2510, volkov2024} are currently investigating\nthe same problem for upper triangular matrix semirings over $D_2$.\nMotivated by these works,\nwe undertake a systematic classification of matrix semirings $\\mathbf{M}_n(S)$,\nwhere $S$ ranges over all two-element ai-semirings.\nThe main result of this paper provides the following complete characterization.", "full_context": "An \\emph{additively idempotent semiring} (abbreviated as ai-semiring)\nis an algebraic structure $(S, +, \\cdot)$ equipped with two binary operations $+$ and $\\cdot$ such that:\n\\begin{itemize}\n\\item the additive reduct $(S, +)$ is a commutative idempotent semigroup;\n\\item the multiplicative reduct $(S, \\cdot)$ is a semigroup;\n\\item the distributive laws hold:\n\\[\n(x+y)z \\approx xz + yz, \\qquad x(y+z) \\approx xy + xz.\n\\]\n\\end{itemize}\nThis class of algebras includes well-known examples such as the Kleene semiring of regular languages~\\cite{con},\nthe max-plus algebra~\\cite{aei}, the power semiring of a semigroup~\\cite{dgv24},\nthe endomorphism semiring of a semilattice~\\cite{dgv25},\nthe semiring of all binary relations on a set \\cite{dolinka2009}, and distributive lattices~\\cite{burris1981}.\nThese and other similar algebras have found significant applications in several branches of mathematics,\nsuch as algebraic geometry~\\cite{cc}, tropical geometry~\\cite{ms}, information science~\\cite{gl},\ntheoretical computer science~\\cite{go}, and soft constraint solving and programming~\\cite{bis2004}.\n\nLet $S$ be an ai-semiring. Define a binary relation $\\leq$ on $S$ by\n\\[\na\\leq b\\Leftrightarrow a+b=b.\n\\]\nThen $\\leq$ is a partial order on $S$, and $(S, \\leq)$ becomes an upper semilattice in which\nthe supremum of any elements $a$ and $b$ is $a+b$.\nConsequently, the additive reduct $(S, +)$ is uniquely determined by this semilattice order.\nTherefore, it is often convenient to represent the addition operation using the Hasse diagram of $(S, \\leq)$.\nMoreover, one can easily verify that this semilattice order is compatible with multiplication.\nFor this reason, an ai-semiring is often called a \\emph{semilattice-ordered semigroup}.\n\nLet $n\\geq 1$ be an integer,\nlet $S$ be an ai-semiring, and let $\\mathbf{M}_n(S)$ denote the set of all $n \\times n$ matrices over $S$.\nThen $\\mathbf{M}_n(S)$ forms an ai-semiring under the usual matrix addition and multiplication, that is,\n\\[\nA+B = [a_{ij}+b_{ij}]_{n\\times n}, \\quad A \\cdot B = \\left[ \\sum_{k=1}^n a_{ik} b_{kj} \\right]_{n\\times n}\n\\]\nfor all $A=[a_{ij}]_{n\\times n}, B=[b_{ij}]_{n\\times n}\\in \\mathbf{M}_n(S)$.\nIt is obvious that $S$ is isomorphic to $\\mathbf{M}_1(S)$.\nFor $n\\geq 2$, $S$ can be embedded into $\\mathbf{M}_n(S)$,\nsince the mapping\n\\[\n\\varphi\\colon S \\rightarrow \\mathbf{M}_n(S),\\quad a \\mapsto [a]_{n\\times n},\n\\]\nwhere $[a]_{n\\times n}$ denotes the constant matrix with all entries equal to $a$,\nis an embedding mapping.\nMoreover, if $S$ has an additive identity that is also a multiplicative zero,\nthen we can easily obtain the following infinite strict ascending chain\n\\[\nS\\hookrightarrow \\mathbf{M}_2(S)\\hookrightarrow \\mathbf{M}_{3}(S)\\hookrightarrow \\mathbf{M}_{4}(S)\\hookrightarrow\\cdots.\n\\]\n\nLet $\\mathcal{V}$ be a class of algebras.\nThen $\\mathcal{V}$ is called a \\emph{variety} if it is closed under\ntaking subalgebras, homomorphic images, and arbitrary direct products.\nBy the celebrated Birkhoff's theorem \\cite{birkhoff1935}, $\\mathcal{V}$ is a variety if and only if\nit is an \\emph{equational class}, that is, the class of all algebras satisfying some set of identities.\n\nOver the past two decades,\nthe finite basis problem for ai-semirings has attracted considerable attention,\nresulting in substantial\nprogress~\\cite{dol07, dolinka2009, dgv24, dgv25, gpz05, gv23, gv2501, gv2510, jrz, pas05, rz16, rzw, sr, volkov2024, yr25, yrzs, zrc}.\nIn particular, Dolinka~\\cite{dol07} constructed the first example of a nonfinitely based finite ai-semiring.\nPastijn et al.~\\cite{gpz05, pas05} proved that every ai-semiring satisfying the identity $x^2 \\approx x$ is finitely based;\nRen et al.~\\cite{rz16, rzw} later extended this to ai-semirings satisfying the identity $x^3 \\approx x$.\nShao and Ren~\\cite{sr} established that every algebra in the variety generated by all two-element ai-semirings is finitely based.\nSubsequently,\nJackson et al.~\\cite{jrz} and Zhao et al.~\\cite{zrc}\nprovided a complete classification of all three-element ai-semirings with respect to the finite basis property.\nMost recently,\nDolinka, Gusev and Volkov~\\cite{dgv25, gv2501} settled the finite basis problem for\nthe endomorphism semirings of finite semilattices.\n\nTo the best of our knowledge,\nthe study of the finite basis problem for matrix semirings was initiated by\nDolinka~\\cite{dolinka2009}, who considered the problem for $\\mathbf{M}_n(D_2)$ for $n\\geqslant 2$,\nwhere $D_2$ denotes the two-element distributive lattice.\nGusev and Volkov~\\cite{gv2510, volkov2024} are currently investigating\nthe same problem for upper triangular matrix semirings over $D_2$.\nMotivated by these works,\nwe undertake a systematic classification of matrix semirings $\\mathbf{M}_n(S)$,\nwhere $S$ ranges over all two-element ai-semirings.\nThe main result of this paper provides the following complete characterization.\n\n\\begin{abstract}\nWe provide a complete classification of matrix semirings $\\mathbf{M}_n(S)$ over two-element additively idempotent semirings $S$\nwith respect to the finite basis property.\nOur main theorem shows that for every integer $n \\geq 2$,\nthe semiring $\\mathbf{M}_n(S)$ is finitely based\nif and only if $S$ is distinct from a distributive lattice.\n\\end{abstract}\n\nTo the best of our knowledge,\nthe study of the finite basis problem for matrix semirings was initiated by\nDolinka~\\cite{dolinka2009}, who considered the problem for $\\mathbf{M}_n(D_2)$ for $n\\ge 2$,\nwhere $D_2$ denotes the two-element distributive lattice.\nGusev and Volkov~\\cite{gv2510, volkov2024} are currently investigating\nthe same problem for upper triangular matrix semirings over $D_2$.\nMotivated by these works,\nwe undertake a systematic classification of matrix semirings $\\mathbf{M}_n(S)$,\nwhere $S$ ranges over all two-element ai-semirings.\nThe main result of this paper provides the following complete characterization.\n\nFor the proof,\nwe first note that the case where $S$ is a distributive lattice follows from Dolinka's work.\nSpecifically,\n\\cite[Theorem B]{dolinka2009} states that the semiring $\\mathcal{R}_2$ of all binary relations on a two-element set is inherently nonfinitely based.\nSince $\\mathcal{R}_2 $ is isomorphic to $\\mathbf{M}_2(D_2)$,\nit follows immediately that $\\mathbf{M}_2(D_2)$ inherits this property.\nFurthermore, because $\\mathbf{M}_n(D_2)$ contains a copy of $\\mathbf{M}_2(D_2)$ for all $n \\geq 2$,\nthe same property extends to $\\mathbf{M}_n(D_2)$; this result is explicitly stated in \\cite[Corollary 6.2]{dolinka2009}.\n(We note that $D_2$ is also denoted $\\mathbb{B}_2$ and called the two-element Boolean semiring; our notation follows \\cite{sr}.)\n\nTherefore, to establish Theorem~\\ref{main}, it remains to prove the converse:\nthat the matrix semiring $\\mathbf{M}_n(S)$ is finitely based\nfor every two-element ai-semiring $S$ distinct from a distributive lattice and every $n \\geq 2$.\nThe necessary preliminaries are collected in Section 2, and the proof is carried out in Sections 3 and 4.\n\n\\begin{pro}\\label{pro21}\nLet $n\\geq 2$ be an integer. Then\n$\\mathsf{V}(\\mathbf{M}_n(L_2))$ is the ai-semiring variety determined by the identities \\eqref{eq:T22} and\n\\begin{align}\n&xy\\approx xz. \\label{f1}\n\\end{align}\n\\end{pro}\n\\begin{proof}\nWe first show that $\\mathbf{M}_n(L_2)$ satisfies identities \\eqref{eq:T22} and \\eqref{f1}.\nLet $A = [a_{ij}]$, $B = [b_{ij}]$, $C = [c_{ij}]$ be arbitrary matrices in $\\mathbf{M}_n(L_2)$, where all entries belong to $L_2$.\nFor any indices $1 \\leq i, j \\leq n$, we have\n\\[\n(AB)_{ij} = \\sum_{k=1}^{n} a_{ik}b_{kj} \\stackrel{\\eqrefbasis{eq:L2}}=\n\\sum_{k=1}^{n} a_{ik} \\stackrel{\\eqrefbasis{eq:L2}}= \\sum_{k=1}^{n} a_{ik}c_{kj} = (AC)_{ij},\n\\]\nand\n\\begin{align*}\n(A^2)_{ij} = \\sum_{k=1}^{n} a_{ik}a_{kj}\n&= \\left( \\sum_{k=1}^{n} a_{ik}a_{kj} \\right) + a_{ij}a_{jj} \\\\\n&\\stackrel{\\eqrefbasis{eq:L2}}= \\left( \\sum_{k=1}^{n} a_{ik}a_{kj} \\right) + a_{ij} \\\\\n&= (A^2)_{ij} + A_{ij} = (A^2 + A)_{ij}.\n\\end{align*}\nThis shows that $AB = AC$ and $A^2 = A^2 + A$.\nHence $\\mathbf{M}_n(L_2)$ satisfies the identities \\eqref{eq:T22} and \\eqref{f1}.\n\n\\begin{pro}\\label{pro22}\nLet $n\\geq 2$ be an integer. Then\n$\\mathsf{V}(\\mathbf{M}_n(R_2))$ coincides with $\\mathsf{V}(S_{56})$, properly contains $\\mathsf{V}(R_2)$,\nand is the ai-semiring variety determined by the identities \\eqref{eq:T22} and\n\\begin{align}\n&xy\\approx zy. \\label{f3}\n\\end{align}\n\\end{pro}\n\\begin{proof}\nObserve that $R_2$ and $L_2$ have dual multiplications.\nThe remaining argument parallels those of Proposition~\\ref{pro21} and Remark~\\ref{rem1}.\n\\end{proof}\n\n\\begin{pro}\\label{pro233}\nLet $n\\geq 1$ be an integer. Then\n$\\mathsf{V}(\\mathbf{M}_n(N_2))$ is the ai-semiring variety determined by the identities \\eqref{eq:N21} and \\eqref{eq:N22}.\n\\end{pro}\n\\begin{proof}\nLet $A = [a_{ij}]$, $B = [b_{ij}]$, $C = [c_{ij}]$, $D = [d_{ij}]$ be arbitrary matrices in $\\mathbf{M}_n(N_2)$.\nFor any $1 \\leq i, j \\leq n$, we have\n\\[\n(AB)_{ij} = \\sum_{k=1}^{n} a_{ik}b_{kj} \\stackrel{\\eqrefbasis{eq:N21}}= \\sum_{k=1}^{n} c_{ik}d_{kj} = (CD)_{ij},\n\\]\nand\n\\begin{align*}\n(A + A^2)_{ij}\n&= A_{ij} + (A^2)_{ij}=a_{ij} + \\left(\\sum_{k=1}^n a_{ik}a_{kj}\\right)\\\\\n&\\stackrel{\\eqrefbasis{eq:N21}}= a_{ij} + a_{ij}a_{ij}\n\\stackrel{\\eqrefbasis{eq:N22}}= a_{ij}= A_{ij}.\n\\end{align*}\nThis shows that $AB = CD$ and $A=A+A^2$.\nHence $\\mathbf{M}_n(N_2)$ satisfies both identities \\eqref{eq:N21} and \\eqref{eq:N22}.\n\n\\section{Conclusion}\nWe have provided a complete classification of\nmatrix semirings $\\mathbf{M}_n(S)$ over two-element ai-semirings $S$\nwith respect to the finite basis property.\nFrom Remark~\\ref{rem1} and \\cite[Figure 1]{yr25},\nthe variety $\\mathsf{V}(\\mathbf{M}_n(L_2))$ has precisely $5$ subvarieties;\nthey form a distributive lattice and are all finitely based.\nThe same conclusion holds for the variety $\\mathsf{V}(\\mathbf{M}_n(R_2))$.\nBy \\cite[Theorem 1]{polin} together with Propositions~\\ref{pro233} and \\ref{pro23},\nthe varieties $\\mathsf{V}(\\mathbf{M}_n(N_2))$ and $\\mathsf{V}(\\mathbf{M}_n(T_2))$\nare both minimal nontrivial varieties.\nIn contrast, the subvariety lattice of the variety $\\mathsf{V}(\\mathbf{M}_n(M_2))$\nremains to be explicitly described.\n\n\\begin{thm}\\label{main}\nLet $S$ be a two-element ai-semiring, and let $n \\geq 2$ be an integer.\nThen the matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ is not a distributive lattice.\n\\end{thm}", "post_theorem_intro_text_len": 1096, "post_theorem_intro_text": "For the proof,\nwe first note that the case where $S$ is a distributive lattice follows from Dolinka's work.\nSpecifically,\n\\cite[Theorem B]{dolinka2009} states that the semiring $\\mathcal{R}_2$ of all binary relations on a two-element set is inherently nonfinitely based.\nSince $\\mathcal{R}_2 $ is isomorphic to $\\mathbf{M}_2(D_2)$,\nit follows immediately that $\\mathbf{M}_2(D_2)$ inherits this property.\nFurthermore, because $\\mathbf{M}_n(D_2)$ contains a copy of $\\mathbf{M}_2(D_2)$ for all $n \\geq 2$,\nthe same property extends to $\\mathbf{M}_n(D_2)$; this result is explicitly stated in \\cite[Corollary 6.2]{dolinka2009}.\n(We note that $D_2$ is also denoted $\\mathbb{B}_2$ and called the two-element Boolean semiring; our notation follows \\cite{sr}.)\n\nTherefore, to establish Theorem~\\ref{main}, it remains to prove the converse:\nthat the matrix semiring $\\mathbf{M}_n(S)$ is finitely based\nfor every two-element ai-semiring $S$ distinct from a distributive lattice and every $n \\geq 2$.\nThe necessary preliminaries are collected in Section 2, and the proof is carried out in Sections 3 and 4.", "sketch": "For the proof of Theorem~\\ref{main}: (1) In the case where $S$ is a distributive lattice, the result follows from Dolinka's work: \\cite[Theorem B]{dolinka2009} shows that the semiring $\\mathcal{R}_2$ of all binary relations on a two-element set is inherently nonfinitely based. Since $\\mathcal{R}_2\\cong \\mathbf{M}_2(D_2)$, it follows that $\\mathbf{M}_2(D_2)$ is inherently nonfinitely based; and because $\\mathbf{M}_n(D_2)$ contains a copy of $\\mathbf{M}_2(D_2)$ for all $n\\ge2$, the same property extends to $\\mathbf{M}_n(D_2)$ (explicitly \\cite[Corollary 6.2]{dolinka2009}). (2) Thus, to establish Theorem~\\ref{main} it remains to prove the converse: $\\mathbf{M}_n(S)$ is finitely based for every two-element ai-semiring $S$ distinct from a distributive lattice and every $n\\ge2$. The preliminaries are in Section~2 and the proof is carried out in Sections~3 and~4.", "expanded_sketch": "For the proof of the main theorem: (1) In the case where $S$ is a distributive lattice, the result follows from Dolinka's work: \\cite[Theorem B]{dolinka2009} shows that the semiring $\\mathcal{R}_2$ of all binary relations on a two-element set is inherently nonfinitely based. Since $\\mathcal{R}_2\\cong \\mathbf{M}_2(D_2)$, it follows that $\\mathbf{M}_2(D_2)$ is inherently nonfinitely based; and because $\\mathbf{M}_n(D_2)$ contains a copy of $\\mathbf{M}_2(D_2)$ for all $n\\ge2$, the same property extends to $\\mathbf{M}_n(D_2)$ (explicitly \\cite[Corollary 6.2]{dolinka2009}). (2) Thus, to establish the main theorem it remains to prove the converse: $\\mathbf{M}_n(S)$ is finitely based for every two-element ai-semiring $S$ distinct from a distributive lattice and every $n\\ge2$. The preliminaries are given next, and the proof is then carried out subsequently.", "expanded_theorem": "\\label{main}\nLet $S$ be a two-element ai-semiring, and let $n \\geq 2$ be an integer.\nThen the matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ is not a distributive lattice.", "theorem_type": ["Biconditional or Equivalence", "Universal"], "mcq": {"question": "An additively idempotent semiring (ai-semiring) is an algebra $(S,+,\\cdot)$ such that $(S,+)$ is a commutative idempotent semigroup, $(S,\\cdot)$ is a semigroup, and multiplication distributes over addition on both sides. Let $S$ be a two-element ai-semiring, and let $n\\ge 2$ be an integer. Write $\\mathbf{M}_n(S)$ for the ai-semiring of all $n\\times n$ matrices over $S$ with the usual matrix addition and multiplication. An algebra is called finitely based if the variety it generates is definable by finitely many identities. Which statement holds for every such $S$ and $n$?", "correct_choice": {"label": "A", "text": "The matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ is not a distributive lattice."}, "choices": [{"label": "B", "text": "The matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ is a distributive lattice."}, {"label": "C", "text": "If $S$ is not a distributive lattice, then the matrix semiring $\\mathbf{M}_n(S)$ is finitely based."}, {"label": "D", "text": "For every two-element ai-semiring $S$, the matrix semiring $\\mathbf{M}_n(S)$ is finitely based for all sufficiently large integers $n\\ge 2$."}, {"label": "E", "text": "The matrix semiring $\\mathbf{M}_n(S)$ is finitely based if and only if $S$ itself is finitely based."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "distributive-lattice case direction", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the converse implication excluding distributive lattices", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "uniformity in n from the embedding of $\\mathbf{M}_2(D_2)$ into all $\\mathbf{M}_n(D_2)$ for $n\\ge2$", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "replacement of the matrix-semiring criterion by a criterion on the base semiring", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the objects and asks for a universal characterization, but it does not reveal or strongly hint at the correct criterion. The key discriminator involving distributive lattices appears only in the answer choices."}, "TAS": {"score": 0, "justification": "The correct option appears to be essentially the theorem statement itself: a precise if-and-only-if characterization of when \\(\\mathbf{M}_n(S)\\) is finitely based. The item mainly asks for recognition of the exact theorem rather than deriving a new consequence."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the student must distinguish the exact biconditional from a weaker true statement and several plausible overgeneralizations. However, the question primarily tests theorem recall/recognition rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and well targeted: one reverses the distributive-lattice condition, one gives a weaker one-way implication, one overgeneralizes in \\(n\\), and one confuses properties of the matrix semiring with those of the base semiring. These reflect realistic logical and structural failure modes."}, "total_score": 5, "overall_assessment": "A solid recognition-style theorem MCQ with strong distractors and no answer leakage, but it is close to a direct restatement of the result and only moderately tests reasoning."}}
{"id": "2602.07167v1", "paper_link": "http://arxiv.org/abs/2602.07167v1", "theorems_cnt": 2, "theorem": {"env_name": "proposition", "content": "[Intermittency] \\label{prop:intermittency}For any $p\\geq 1$ integer, we have \n\\begin{align*}\n   n \\exp\\left((p+\\frac{2p(p-1)}{n+2})\\tau \\right) \\leq  \\mathbb{E} |F_\\tau|^{2p}\\leq n^{p} \\exp\\left((p+\\frac{2p(p-1)}{n+2})\\tau \\right),\n\\end{align*}\nwhere $|F|^2:={\\rm tr} F^*F$ denotes the square of the Frobenius norm.", "start_pos": 6809, "end_pos": 7148, "label": "prop:intermittency"}, "ref_dict": {"2pthmoment": "\\begin{align}\n    \\E|F_\\tau|^{2p} \\sim_{n,p}\\footnotemark \\left(\\E^p |F_\\tau|^2\\right)^{1+\\frac{2(p-1)}{n+2}}. \\label{2pthmoment}\n\\end{align}", "brownian1": "\\begin{enumerate}[label=(B\\arabic*)]\n    \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n    \\item \\label{brownian2} $\\bbE[B_{\\tau} B_\\tau] =0$, and\n    \\item\\label{brownian3} $\\bbE[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n    \\end{enumerate}", "brownian3": "\\begin{enumerate}[label=(B\\arabic*)]\n    \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n    \\item \\label{brownian2} $\\bbE[B_{\\tau} B_\\tau] =0$, and\n    \\item\\label{brownian3} $\\bbE[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n    \\end{enumerate}", "prop:intermittency": "\\begin{proposition}[Intermittency] \\label{prop:intermittency}For any $p\\geq 1$ integer, we have \n\\begin{align*}\n   n \\exp\\left((p+\\frac{2p(p-1)}{n+2})\\tau \\right) \\leq  \\E |F_\\tau|^{2p}\\leq n^{p} \\exp\\left((p+\\frac{2p(p-1)}{n+2})\\tau \\right),\n\\end{align*}\nwhere $|F|^2:=\\tr F^*F$ denotes the square of the Frobenius norm.\n\\end{proposition}", "F": "\\begin{align}\\label{F}\n    dF=F_\\tau\\circ dB, \\quad F_{\\tau=0}=id ,\n\\end{align}", "prop:nontightness": "\\begin{proposition}[Non-tightness] \\label{prop:nontightness}\n    For $\\tau \\gg_n 1$\\footnote{Here $\\tau \\gg_n 1$ means that there is a possibly large constant $C(n)$ depending only on $n$ such that the statement is true for any $\\tau$ with $\\tau \\geq C(n)$.}, we have \t\\begin{align}\\label{prop02}\n\t\t\\E | F_{ \\tau } |^2 I \\Big( { \\textstyle \\frac{ 1 }{ n } } | F_{ \\tau } |^2 \\leq { \\textstyle \\frac{ 1 }{ e  } } \\big( { \\textstyle \\E \\frac{ 1 }{ n } } | F_{ \\tau } |^2 \\big)^{ \\frac{ n + 4 }{ n + 2 } } \\Big)\n\t\t\\leq \\Big( \\frac{ 1 }{ 2 } + \\frac{ C(n) }{ \\sqrt{\\tau} } \\Big) \\E| F_{ \\tau }  |^2  .\n\t\\end{align}\n where $I(A)$ is the indicator function of the event $A$.\t\n\\end{proposition}", "dd": "\\begin{align}\n    dX=b(X_t)dt+\\sqrt{2}dW \\label{dd}\n\\end{align}"}, "pre_theorem_intro_text_len": 1023, "pre_theorem_intro_text": "\\noindent\nWe are interested in the  process $\\{F_\\tau\\}_{\\tau \\geq 0}$ that solves the Stratonovich stochastic differential equation (SDE)\n\\begin{align}\\label{F}\n    dF=F_\\tau\\circ dB, \\quad F_{\\tau=0}=id ,\n\\end{align}\nwhere $\\{B_\\tau\\}_{\\tau \\geq 0}$ is a Brownian motion on ${\\frak{sl}}(n):=\\{{\\rm tr} B=0\\}$ that satisfies the following three assumptions:\n\\begin{enumerate}[label=(B\\arabic*)]\n    \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n    \\item \\label{brownian2} $\\mathbb E[B_{\\tau} B_\\tau] =0$, and\n    \\item\\label{brownian3} $\\mathbb E[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n    \\end{enumerate}\nSince the chain rule holds for Stratonovich SDEs, $F\\in \\textbf{SL}(n):=\\{\\text{det} F=1\\}$. In \\cite{MOW25}, it is proved that for $n\\geq 2$, there is a unique Brownian motion on $\\frak{sl}(n)$ that satisfies the Assumption \\ref{brownian1}-\\ref{brownian3}. We are interested in exploring the intermittent behavior of $F$ in \\eqref{F}. Our first result is on the moments:", "context": "\\noindent\nWe are interested in the process $\\{F_\\tau\\}_{\\tau \\geq 0}$ that solves the Stratonovich stochastic differential equation (SDE)\n\\begin{align}\\label{F}\n dF=F_\\tau\\circ dB, \\quad F_{\\tau=0}=id ,\n\\end{align}\nwhere $\\{B_\\tau\\}_{\\tau \\geq 0}$ is a Brownian motion on ${\\frak{sl}}(n):=\\{{\\rm tr} B=0\\}$ that satisfies the following three assumptions:\n\\begin{enumerate}[label=(B\\arabic*)]\n \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n \\item \\label{brownian2} $\\mathbb E[B_{\\tau} B_\\tau] =0$, and\n \\item\\label{brownian3} $\\mathbb E[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n \\end{enumerate}\nSince the chain rule holds for Stratonovich SDEs, $F\\in \\textbf{SL}(n):=\\{\\text{det} F=1\\}$. In \\cite{MOW25}, it is proved that for $n\\geq 2$, there is a unique Brownian motion on $\\frak{sl}(n)$ that satisfies the Assumption \\ref{brownian1}-\\ref{brownian3}. We are interested in exploring the intermittent behavior of $F$ in \\eqref{F}. Our first result is on the moments:\n\n\\begin{align}\\label{F}\n    dF=F_\\tau\\circ dB, \\quad F_{\\tau=0}=id ,\n\\end{align}\n\n\\begin{enumerate}[label=(B\\arabic*)]\n    \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n    \\item \\label{brownian2} $\\bbE[B_{\\tau} B_\\tau] =0$, and\n    \\item\\label{brownian3} $\\bbE[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n    \\end{enumerate}\n\n\\begin{enumerate}[label=(B\\arabic*)]\n    \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n    \\item \\label{brownian2} $\\bbE[B_{\\tau} B_\\tau] =0$, and\n    \\item\\label{brownian3} $\\bbE[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n    \\end{enumerate}", "full_context": "\\noindent\nWe are interested in the process $\\{F_\\tau\\}_{\\tau \\geq 0}$ that solves the Stratonovich stochastic differential equation (SDE)\n\\begin{align}\\label{F}\n dF=F_\\tau\\circ dB, \\quad F_{\\tau=0}=id ,\n\\end{align}\nwhere $\\{B_\\tau\\}_{\\tau \\geq 0}$ is a Brownian motion on ${\\frak{sl}}(n):=\\{{\\rm tr} B=0\\}$ that satisfies the following three assumptions:\n\\begin{enumerate}[label=(B\\arabic*)]\n \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n \\item \\label{brownian2} $\\mathbb E[B_{\\tau} B_\\tau] =0$, and\n \\item\\label{brownian3} $\\mathbb E[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n \\end{enumerate}\nSince the chain rule holds for Stratonovich SDEs, $F\\in \\textbf{SL}(n):=\\{\\text{det} F=1\\}$. In \\cite{MOW25}, it is proved that for $n\\geq 2$, there is a unique Brownian motion on $\\frak{sl}(n)$ that satisfies the Assumption \\ref{brownian1}-\\ref{brownian3}. We are interested in exploring the intermittent behavior of $F$ in \\eqref{F}. Our first result is on the moments:\n\n\\begin{align}\\label{F}\n    dF=F_\\tau\\circ dB, \\quad F_{\\tau=0}=id ,\n\\end{align}\n\n\\begin{enumerate}[label=(B\\arabic*)]\n    \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n    \\item \\label{brownian2} $\\bbE[B_{\\tau} B_\\tau] =0$, and\n    \\item\\label{brownian3} $\\bbE[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n    \\end{enumerate}\n\n\\begin{enumerate}[label=(B\\arabic*)]\n    \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n    \\item \\label{brownian2} $\\bbE[B_{\\tau} B_\\tau] =0$, and\n    \\item\\label{brownian3} $\\bbE[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n    \\end{enumerate}\n\n\\end{abstract}\n\\maketitle\n\n\\medskip \\noindent\nThe fact that $2p$-th moment scales with a rate much larger than the $p$-th power of the second moment amounts to strongly non-Gaussian and intermittent behavior. Proposition~\\ref{prop:intermittency} generalizes \\cite[Lemma 2]{MOW25} for $n=2$ to general $n\\ge 2$.\n\n\\medskip \\noindent\nWhile Proposition~\\ref{prop:intermittency} clearly expresses intermittency at the level of $F$, we are not able to transfer this higher-moment information to $\\nabla u$. This is because we can capture the proximity of $\\nabla u$ and $F$ only on the second moments. Hence we need to capture the shadow of the intermittency on the level of the second moment. In fact, it comes as a non-tightness result in the second moment in the sense that the extreme tails contain a substantial fraction of the second moment.\n \\begin{proposition}[Non-tightness] \\label{prop:nontightness}\n For $\\tau \\gg_n 1$\\footnote{Here $\\tau \\gg_n 1$ means that there is a possibly large constant $C(n)$ depending only on $n$ such that the statement is true for any $\\tau$ with $\\tau \\geq C(n)$.}, we have \\begin{align}\\label{prop02}\n \\E | F_{ \\tau } |^2 I \\Big( { \\textstyle \\frac{ 1 }{ n } } | F_{ \\tau } |^2 \\leq { \\textstyle \\frac{ 1 }{ e } } \\big( { \\textstyle \\E \\frac{ 1 }{ n } } | F_{ \\tau } |^2 \\big)^{ \\frac{ n + 4 }{ n + 2 } } \\Big)\n \\leq \\Big( \\frac{ 1 }{ 2 } + \\frac{ C(n) }{ \\sqrt{\\tau} } \\Big) \\E| F_{ \\tau } |^2 .\n \\end{align}\n where $I(A)$ is the indicator function of the event $A$. \n\\end{proposition}\n\\begin{remark}\\label{cor:nontightness} The information in Proposition~\\ref{prop:nontightness} can easily be upgraded to the lower bound as is done in \\cite[Theorem 3 \\& (118)]{MOW25}\n \\begin{align*}\n\\E |F_\\tau|^{2p} \\gtrsim_{n,p} \\bbE^{1+\\frac{n+4}{n+2}(p-1)} |F_\\tau|^{2},\n\\end{align*}\nwhich is weaker than \\eqref{2pthmoment} but not completely unrelated. This exponent is precisely the linearization of the quadratic exponent $p+\\frac{2p(p-1)}{n+2}$ around $p=1$. \n\\end{remark}\n\n\\subsection{Proof of Proposition~\\ref{prop:intermittency}}\nRecall from Lemma~\\ref{lem:odes}, we have for $p\\geq 1$ integer\n\\begin{align*}\n \\frac{ d\\bbE[\\tr G^p]}{d\\tau} =(p+\\frac{p(p-1)(n-2)}{\\alpha_n})\\bbE \\tr G_\\tau^p +\\frac{pn}{\\alpha_n}\\sum_{j=1}^{p-1}\\bbE \\tr G_\\tau^j \\tr G_\\tau^{p-j} \n\\end{align*}\n with initial condition $\\tr G_{\\tau=0}=\\tr id=n$. Using $\\tr G^j \\tr G^{p-j} \\geq \\tr G^p$ for $0\\leq j\\leq p$, we can bound the second term \n \\begin{align*}\n \\sum_{j=1}^{p-1}\\bbE \\tr G_\\tau^j \\tr G_\\tau^{p-j} \\geq (p-1) \\bbE \\tr G_\\tau^p\n\\end{align*}\nwhich leads to the differential inequality \n\\begin{align*}\n \\frac{ d\\bbE\\tr G^p}{d\\tau} \\geq (p+\\frac{p(p-1)(2n-2)}{\\alpha_n}) \\bbE\\tr G_\\tau^p\n\\end{align*}\nwhere $\\alpha_n$ is as defined \\eqref{alpha}. Hence we get \n\\begin{align*}\n \\bbE\\tr G_\\tau^p \\geq n e^{(p+\\frac{2p(p-1)}{n+2})\\tau} .\n\\end{align*}\nOn the other hand, recall again from Lemma~\\ref{lem:odes}, $\\bbE \\tr^p G_\\tau$ satisfies \n\\begin{align*}\n \\frac{ d\\bbE\\tr^p G}{d\\tau}& =(p-\\frac{2p(p-1)}{\\alpha_n}) \\bbE\\tr^p G_\\tau +\\frac{2p(p-1)n}{\\alpha_n} \\bbE\\tr^{p-2} G_\\tau\\tr G_\\tau^2 \n\\end{align*}\nwith initial condition $\\bbE\\tr^p G_{\\tau=0}=n^p$. This time using $\\tr G^2\\leq \\tr^2 G$, we get the differential inequality\n\\begin{align*}\n \\frac{d\\bbE \\tr^p G}{d\\tau}& \\leq (p+\\frac{p(p-1)(2n-2)}{\\alpha_n}) \\bbE\\tr^p G_\\tau \n\\end{align*}\nwhich leads to\n\\begin{align*}\n \\bbE\\tr^p G_\\tau \\leq n^pe^{(p+\\frac{2p(p-1)}{n+2})\\tau} .\n\\end{align*}\nHence we obtain \\eqref{lpestimates} which is a restatement of Proposition~\\ref{prop:intermittency}.\n\\qed\n\nFor parameters $ \\tau^* \\geq 0 $ and $ \\hat{\\sigma}^* $ that we will choose later we consider the terminal condition\n\\begin{align}\\label{tc}\n \\hat{\\zeta}( \\tau^*,\\hat{\\sigma})\\begin{cases}\n = 1, \\text{ if } \\hat{\\sigma} \\leq \\hat{\\sigma}^*,\\\\\n \\in [0,1] \\text{ for all } \\hat{\\sigma},\\\\\n =0, \\text{ if }\\hat{\\sigma}\\geq \\hat{\\sigma}^*+1,\n \\end{cases}\n \\text{ with } \\left| \\frac{\\partial^2 \\hat{\\zeta}( \\tau^* ,\\hat{\\sigma}) }{\\partial \\hat{\\sigma}^2}\\right| \\lesssim 1\n \\end{align}\n for the equation \\eqref{zetahat2} in Lemma~\\ref{lem:zeta}.\nIt is readily seen from \\eqref{tc} that we have\n \\begin{align}\n\\bbE \\hat R_{ \\tau^* } \\hat{\\zeta}(\\tau^*, \\hat R_{ \\tau^* }) \\stackrel{ (\\ref{tc}) }{ \\geq } \\bbE \\hat R_{ \\tau^* } I( \\hat R_{ \\tau^* } \\leq \\hat{R}^*) \n\\quad \\text{provided} \\quad \\hat\\sigma^* =: \\ln \\hat R^{ * }. \\label{ononehand}\n\\end{align}\nOur goal is to choose $ \\hat\\sigma^* $ large enough so that at initial time $ \\tau = 0 $ it holds\n\\begin{align}\\hat{\\zeta}( \\tau=0, \\hat\\sigma = 0 ) \\leq \\frac{1}{2} , \\label{initial}\n\\end{align}\nso that Lemma~\\ref{lem:zeta} implies\n\\begin{align}\n \\bbE \\hat R_{ \\tau^* } I( \\hat R_{ \\tau^* } \\leq \\hat{R}^*) - \\frac{ 1 }{ 2 }\n \\stackrel{ (\\ref{ononehand}) }{ \\leq } \\bbE \\hat{R}_{ \\tau^* } \\hat{\\zeta}( \\tau^*, \\hat R_{ \\tau^* }) - \\frac{ 1 }{ 2 }\n \\stackrel{ \\eqref{zetahatineq} \\& (\\ref{initial}) }{ \\lesssim_{ n } } \\int_0^{\\tau^*}d \\tau\\, e^{ - ( 1 - \\frac{ \\lambda_2 }{ 2 }) \\tau} \\sup_{\\hat \\sigma}\\left|\\frac{\\partial \\hat \\zeta}{\\partial\\hat \\sigma}+ \\frac{\\partial^2 \\hat \\zeta}{\\partial\\hat \\sigma^2} \\right| . \\label{zetahatineqb}\n\\end{align}\n\n\\noindent\nTo conclude the proof, it is left to show that the integral in \\eqref{zetahatineqb} is small. To this end, note that using the estimates for the terminal data \n\\begin{align*}\n \\sup_{\\hat\\sigma } \\left|\\frac{\\partial \\hat{\\zeta}( \\tau^*,\\hat{\\sigma})}{\\partial {\\hat\\sigma}}\\right|+\\left|\\frac{\\partial^2 \\hat{\\zeta}(\\tau^*,\\hat{\\sigma})}{\\partial^2 {\\hat \\sigma } }\\right| \n \\lesssim 1\n \\quad \\text{and} \\quad\n \\int d{\\hat{\\sigma}}\\left|\\frac{\\partial \\hat{\\zeta}( \\tau^*,\\hat{\\sigma})}{\\partial {\\hat\\sigma}}\\right| \n \\lesssim 1\n\\end{align*}\nwe learn (by convolution with the heat kernel) that \n\\begin{align*}\n \\sup_{ \\hat \\sigma } \\left|\\frac{\\partial \\hat{\\zeta} ( \\tau^* , \\hat\\sigma ) }{\\partial \\hat{\\sigma}}\\right| \\lesssim_{ n } \\frac{1}{ \\sqrt{1+ \\tau^*- \\tau}}, \\hskip 10pt \\sup_{ \\hat\\sigma } \\left|\\frac{\\partial^2 \\hat{\\zeta} ( \\tau^* , \\hat\\sigma ) }{\\partial \\hat{\\sigma}^2}\\right| \n \\lesssim_{ n } \\frac{1}{ 1+ \\tau^*- \\tau} .\n\\end{align*}\nHence, the integral term in \\eqref{zetahatineqb} is bounded by\n\\begin{align*} \\int_0^{\\tau*}d\\tau e^{ - ( 1 - \\frac{ \\lambda_2 }{ 2 } ) \\tau} \\sup_{\\hat \\sigma}\\left|\\frac{\\partial \\hat \\zeta}{\\partial\\hat \\sigma}+ \\frac{\\partial^2 \\hat \\zeta}{\\partial\\hat \\sigma^2} \\right| \\lesssim_{ n } \\int_0^{\\tau^*} d\\tau e^{-\\frac{4}{n-1} \\tau } \\frac{1}{\\sqrt{1+\\tau^*-\\tau}} \\lesssim_{ n } \\frac{ 1 }{\\sqrt{\\tau^*}},\n\\end{align*}\nand with some constant $ C = C ( n ) $ estimate \\eqref{zetahatineqb} turns into\n\\begin{align}\\label{ontheotherhand}\n \\bbE \\hat R_{ \\tau^* } I( \\hat R_{ \\tau^* } \\leq \\hat{R}^*) \\leq \\frac{ 1 }{ 2 } + \\frac{ C(n) }{ \\sqrt{ \\tau^* } }.\n\\end{align}\nFinally, undoing the change of variables we have\n\\begin{align*}\n \\hat R_{ \\tau^* }\n \\stackrel{ \\eqref{Rhat} }{ = } e^{ - \\tau^* } { \\textstyle \\frac{ 1 }{ n } } R_{ \\tau^* }\n \\stackrel{ \\ref{F3} , (\\ref{R}) \\& (\\ref{:G}) }{ = } \\frac{ | F_{ \\tau^* } |^2 }{ \\E | F_{ \\tau^* } |^2 } ,\n \\quad \n \\hat R^{ * }\n \\stackrel{ (\\ref{eqnHatRStar}) }{ \\leq } e^{ \\frac{ 2 }{ n + 2 } \\tau^* - 1 } \n \\stackrel{\\ref{F3} }{ = } \\frac{ 1 }{ e } \\Big( \\frac{ \\E | F_{ \\tau^* } |^2 }{ n } \\Big)^{ \\frac{ 2 }{ n + 2 } } ,\n\\end{align*}\nso that since $ \\frac{ 2 }{ n + 2 } + 1 = \\frac{ n + 4 }{ n + 2 } $ \n\\begin{align*}\n \\hat R_{ \\tau^* } I( \\hat R_{ \\tau^* } \\leq \\hat{R}^*)\n = \\frac{ | F_{ \\tau^* } |^2 }{ \\E | F_{ \\tau^* } |^2 } I \\Big( { \\textstyle \\frac{ 1 }{ n } } | F_{ \\tau^* } |^2 \\leq { \\textstyle \\frac{ 1 }{ e } } \\big( { \\textstyle \\frac{ \\E | F_{ \\tau^* } |^2 }{ n } } \\big)^{ \\frac{ n + 4 }{ n + 2 } } \\Big) .\n\\end{align*}\nThus \\eqref{ontheotherhand} turns into \\eqref{prop02}. \\qed", "post_theorem_intro_text_len": 6008, "post_theorem_intro_text": "\\noindent\nProposition~\\ref{prop:intermittency} implies that \n\\begin{align}\n    \\mathbb{E}|F_\\tau|^{2p} \\sim_{n,p}\\footnotemark \\left(\\mathbb{E}^p |F_\\tau|^2\\right)^{1+\\frac{2(p-1)}{n+2}}. \\label{2pthmoment}\n\\end{align}\n\\footnotetext{Here $A \\sim_{n,p} B$ means that there exist constants $c=c(n,p)$ and $C=C(n,p)$ depending only on $n$ and $p$ such that $cB\\leq A\\leq CB$.} \n\n\\medskip \\noindent\nThe fact that $2p$-th moment scales with a rate much larger than the $p$-th power of the second moment amounts to strongly non-Gaussian and intermittent behavior. Proposition~\\ref{prop:intermittency} generalizes \\cite[Lemma 2]{MOW25} for $n=2$ to general $n\\ge 2$.\n\n\\medskip \\noindent\nThe interest in studying the geometric Brownian motion comes from its\nintriguing relation to the drift-diffusion process \n\\begin{align}\n    dX=b(X_t)dt+\\sqrt{2}dW \\label{dd}\n\\end{align}\nwith $b$ being a divergence-free and time-independent vector field. One is interested in the case where $b$ is sampled from a stationary and isotropic Gaussian ensemble with the scaling \n \\begin{align}\n   \\label{b} b(\\mu \\cdot)=\\frac{1}{\\mu}b \\text{ in law for all } \\mu>0,\n\\end{align}\nin which convection and diffusion balance at every scale. After implementing an ultraviolet cutoff, w.l.o.g.\\ at scale 1, $b$ is fixed up to a single constant which we describe in terms of $\\mathbb{E}|b|^2=\\epsilon^2 \\frac{ n }{ 4 } $, where $\\epsilon \\ll 1$ can be interpreted as the P\\'{e}clet number. It is now well-known that the mean-square displacement displays super-diffusive behavior $\\frac{1}{2t}\\mathbb E|X_t|^2 \\approx \\lambda(t)$ for $t \\gg 1$, where \n\\begin{align*}\n\\lambda(t):=\\sqrt{1+\\frac{\\epsilon^2}{2} \\ln(1+t)}.\n\\end{align*}\nThis behavior was rigorously established with increasing precision in \\cite{TV12, CHT22, CMOW22,ABK24, MOW25}. \n\n\\medskip \\noindent\nThe relationship between the geometric Brownian motion process $F$ in \\eqref{F} and the drift-diffusion process in \\eqref{dd} was discovered in \\cite{MOW25} on the level of the expected position, $u(x,t)$, (w.r.t. $W_t$) of the process $\\{X_t\\}_{t\\geq 0}$ starting from $X_{t=0}=x$. More precisely, one obtains $\\nabla u(0,t) \\approx F_{\\tau(T)}$ in law on average over $t\\in (0,T)$, where $$\\tau(T):=\\ln \\lambda(T), $$ see \\cite[Theorem 2 \\& (19)]{MOW25}.\n\n\\medskip \\noindent\nOur result shows that the intermittency of the drift-diffusion equation depends less on topology (stream lines of the divergence-free $b$ are closed iff $n=2$) and more on geometry ($\\textbf{SL}(n)$ has less curvature as $n$ increases). \n\n\\medskip \\noindent\nWhile Proposition~\\ref{prop:intermittency} clearly expresses intermittency at the level of $F$, we are not able to transfer this higher-moment information to $\\nabla u$. This is because we can capture the proximity of $\\nabla u$ and $F$ only on the second moments. Hence we need to capture the shadow of the intermittency on the level of the second moment. In fact, it comes as a non-tightness result in the second moment in the sense that the extreme tails contain a substantial fraction of the second moment.\n \\begin{proposition}[Non-tightness] \\label{prop:nontightness}\n    For $\\tau \\gg_n 1$\\footnote{Here $\\tau \\gg_n 1$ means that there is a possibly large constant $C(n)$ depending only on $n$ such that the statement is true for any $\\tau$ with $\\tau \\geq C(n)$.}, we have \t\\begin{align}\\label{prop02}\n\t\t\\mathbb{E} | F_{ \\tau } |^2 I \\Big( { \\textstyle \\frac{ 1 }{ n } } | F_{ \\tau } |^2 \\leq { \\textstyle \\frac{ 1 }{ e  } } \\big( { \\textstyle \\mathbb{E} \\frac{ 1 }{ n } } | F_{ \\tau } |^2 \\big)^{ \\frac{ n + 4 }{ n + 2 } } \\Big)\n\t\t\\leq \\Big( \\frac{ 1 }{ 2 } + \\frac{ C(n) }{ \\sqrt{\\tau} } \\Big) \\mathbb{E}| F_{ \\tau }  |^2  .\n\t\\end{align}\n where $I(A)$ is the indicator function of the event $A$.\t\n\\end{proposition}\n\\begin{remark}\\label{cor:nontightness} The information in Proposition~\\ref{prop:nontightness} can easily be upgraded to the lower bound as is done in \\cite[Theorem 3 \\& (118)]{MOW25}\n \\begin{align*}\n\\mathbb{E}  |F_\\tau|^{2p} \\gtrsim_{n,p}  \\mathbb E^{1+\\frac{n+4}{n+2}(p-1)}  |F_\\tau|^{2},\n\\end{align*}\nwhich is weaker than \\eqref{2pthmoment} but not completely unrelated. This exponent is precisely the linearization of the quadratic exponent $p+\\frac{2p(p-1)}{n+2}$ around $p=1$. \n\\end{remark}\n\n\\noindent\nThe results in Propositions~\\ref{prop:intermittency} and ~\\ref{prop:nontightness} were previously proven in the case $n=2$ (see \\cite[Lemma 2, Lemma 4]{MOW25}). The argument relied on the fact that the law of $R=\\frac{1}{2}{\\rm tr} F^*F$ can be identified as the Itô evolution \\begin{align*}\n    dR= R_{\\tau} d\\tau +\\sqrt{R_\\tau^2-1} dw, \\quad R_{\\tau =0}=1,\n\\end{align*} \nwhere $\\{w_\\tau\\}_{\\tau\\geq 0}$ is a one-dimensional Brownian motion. In any dimension $n$, as a consequence of the $\\textbf{O}(n)$-invariance, it is not difficult to see that one obtains an evolution for the $(n-1)$ quantities ${\\rm tr} F^*F, {\\rm tr} (F^*F)^2,\\dots ,{\\rm tr} (F^*F)^{n-1}$. However, we are unable to leverage this evolution in order to characterize the evolution of $R$. In this paper, we take a different route: Relying on the observation that the quantities $\\mathbb{E} {\\rm tr} (F^*F)^2 $ and $\\mathbb{E}{\\rm tr}^2 F^*F $ satisfy a linear system of ODEs which is closed in any dimension $n$ and can be solved explicitly, we learn that $\\mathbb E{\\rm tr} (F^*F)^2$ can be well approximated by $\\mathbb E {\\rm tr}^2 F^* F$. This in turn allows us to approximately close the equation for $ R $ in dimensions $ n > 2 $.\n\n\\medskip\n \\noindent\nNote that $|F_\\tau|^2$ is close to a stochastic exponential, which is consistent with Proposition~\\ref{prop:intermittency} and \\ref{prop:nontightness}, namely $\\ln |F_\\tau|^2$ behaves like a Gaussian random variable with mean $(1-\\frac{2}{n+2})\\tau$ and variance $\\frac{4\\tau}{n+2}$. \n\n\\subsection*{Acknowledgment}\nWe thank Anna Wienhard and Corentin Le Bars for helpful discussions concerning random walks on groups. We thank Peter Morfe and Ofer Zeitouni for helpful discussions.", "sketch": "For $n=2$, the previously used argument (\\cite[Lemma 2, Lemma 4]{MOW25}) identified the law of $R:=\\frac{1}{2}{\\rm tr}\\,F^*F$ via the one-dimensional It\\^o SDE\n\\[\n dR=R_{\\tau}\\,d\\tau+\\sqrt{R_{\\tau}^2-1}\\,dw,\\qquad R_{\\tau=0}=1,\n\\]\nwhich yields the needed moment information.\n\nIn general dimension $n$, by $\\mathbf O(n)$-invariance “one obtains an evolution for the $(n-1)$ quantities ${\\rm tr}F^*F,\\,{\\rm tr}(F^*F)^2,\\dots,{\\rm tr}(F^*F)^{n-1}$,” but the authors state they are “unable to leverage this evolution in order to characterize the evolution of $R$.”\n\nInstead, they “take a different route”: they use that “$\\mathbb E\\,{\\rm tr}(F^*F)^2$ and $\\mathbb E\\,{\\rm tr}^2 F^*F$ satisfy a linear system of ODEs which is closed in any dimension $n$ and can be solved explicitly.” From this explicit solution, they “learn that $\\mathbb E\\,{\\rm tr}(F^*F)^2$ can be well approximated by $\\mathbb E\\,{\\rm tr}^2 F^*F$,” which “in turn allows us to approximately close the equation for $R$ in dimensions $n>2$.”", "expanded_sketch": "For $n=2$, the previously used argument (\\cite[Lemma 2, Lemma 4]{MOW25}) identified the law of $R:=\\frac{1}{2}{\\rm tr}\\,F^*F$ via the one-dimensional It\\^o SDE\n\\[\n dR=R_{\\tau}\\,d\\tau+\\sqrt{R_{\\tau}^2-1}\\,dw,\\qquad R_{\\tau=0}=1,\n\\]\nwhich yields the needed moment information.\n\nIn general dimension $n$, by $\\mathbf O(n)$-invariance “one obtains an evolution for the $(n-1)$ quantities ${\\rm tr}F^*F,\\,{\\rm tr}(F^*F)^2,\\dots,{\\rm tr}(F^*F)^{n-1}$,” but the authors state they are “unable to leverage this evolution in order to characterize the evolution of $R$.”\n\nInstead, they “take a different route”: they use that “$\\mathbb E\\,{\\rm tr}(F^*F)^2$ and $\\mathbb E\\,{\\rm tr}^2 F^*F$ satisfy a linear system of ODEs which is closed in any dimension $n$ and can be solved explicitly.” From this explicit solution, they “learn that $\\mathbb E\\,{\\rm tr}(F^*F)^2$ can be well approximated by $\\mathbb E\\,{\\rm tr}^2 F^*F$,” which “in turn allows us to approximately close the equation for $R$ in dimensions $n>2$.”,", "expanded_theorem": "[Intermittency] \\label{prop:intermittency}For any $p\\geq 1$ integer, we have \n\\begin{align*}\n   n \\exp\\left((p+\\frac{2p(p-1)}{n+2})\\tau \\right) \\leq  \\mathbb{E} |F_\\tau|^{2p}\\leq n^{p} \\exp\\left((p+\\frac{2p(p-1)}{n+2})\\tau \\right),\n\\end{align*}\nwhere $|F|^2:={\\rm tr} F^*F$ denotes the square of the Frobenius norm.", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let $n\\ge 2$, let $\\mathfrak{sl}(n)=\\{A:\\operatorname{tr}A=0\\}$, and let $\\{B_\\tau\\}_{\\tau\\ge 0}$ be a Brownian motion on $\\mathfrak{sl}(n)$ such that for every $O\\in \\mathbf O(n)=\\{O^*O=\\mathrm{id}\\}$ one has $OBO^{-1}=_{\\mathrm{law}}B$, and moreover $\\mathbb E[B_\\tau B_\\tau]=0$ and $\\mathbb E[B_\\tau^*B_\\tau]=\\tau\\,\\mathrm{id}$. Let $\\{F_\\tau\\}_{\\tau\\ge 0}$ solve the Stratonovich SDE\n\\[\n dF = F_\\tau\\circ dB,\\qquad F_{\\tau=0}=\\mathrm{id}.\n\\]\nIf $|F|^2:=\\operatorname{tr}(F^*F)$ denotes the square of the Frobenius norm, which statement holds for every integer $p\\ge 1$ and every $\\tau\\ge 0$?", "correct_choice": {"label": "A", "text": "\\[\n n\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right)\n \\le \\mathbb E|F_\\tau|^{2p}\n \\le n^p\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right).\n\\]"}, "choices": [{"label": "B", "text": "\\[\n n\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n}\\right)\\tau\\right)\n \\le \\mathbb E|F_\\tau|^{2p}\n \\le n^p\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n}\\right)\\tau\\right).\n\\]"}, {"label": "C", "text": "\\[\n \\mathbb E|F_\\tau|^{2p}\n \\le n^p\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right).\n\\]"}, {"label": "D", "text": "\\[\n \\text{For each integer }p\\ge 1\\text{ there exists }C_p\\in[1,\\infty)\\text{ such that for all }\\tau\\ge 0,\n\\quad C_p^{-1}\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right)\n \\le \\mathbb E|F_\\tau|^{2p}\n \\le C_p\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right).\n\\]"}, {"label": "E", "text": "\\[\n n\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right)\n \\le \\mathbb E|F_\\tau|^{2p}\n \\le n^p\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right)\n\\]\nfor every real number \\(p\\ge 1\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "explicit exponent denominator n+2", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped lower bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "explicit sharp prefactors n and n^p replaced by non-explicit constants depending on p", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "range of p extended from integers to all reals", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the target estimate or uniquely reveal the correct option. It only provides the setup and asks which bound holds."}, "TAS": {"score": 0, "justification": "This is very close to a direct theorem-recall item: the task is essentially to recognize the exact stated estimate among near-variants, rather than derive a new conclusion from the data."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in sharp constants, exponent structure, and strength of conclusion, including a weaker true statement. But the item mainly tests precise recall/recognition of the theorem rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one perturbs the denominator, one gives a weaker true statement, one alters quantifier dependence, and one inflates the upper exponential rate. These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "Well-constructed in terms of plausible distractors and lack of leakage, but it is primarily a theorem-recognition question and thus only moderately tests genuine reasoning."}}
{"id": "2602.07167v1", "paper_link": "http://arxiv.org/abs/2602.07167v1", "theorems_cnt": 2, "theorem": {"env_name": "proposition", "content": "[Intermittency] \\label{prop:intermittency}For any $p\\geq 1$ integer, we have \n\\begin{align*}\n   n \\exp\\left((p+\\frac{2p(p-1)}{n+2})\\tau \\right) \\leq  \\mathbb{E} |F_\\tau|^{2p}\\leq n^{p} \\exp\\left((p+\\frac{2p(p-1)}{n+2})\\tau \\right),\n\\end{align*}\nwhere $|F|^2:={\\rm tr} F^*F$ denotes the square of the Frobenius norm.", "start_pos": 6809, "end_pos": 7148, "label": "prop:intermittency"}, "ref_dict": {"2pthmoment": "\\begin{align}\n    \\E|F_\\tau|^{2p} \\sim_{n,p}\\footnotemark \\left(\\E^p |F_\\tau|^2\\right)^{1+\\frac{2(p-1)}{n+2}}. \\label{2pthmoment}\n\\end{align}", "brownian1": "\\begin{enumerate}[label=(B\\arabic*)]\n    \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n    \\item \\label{brownian2} $\\bbE[B_{\\tau} B_\\tau] =0$, and\n    \\item\\label{brownian3} $\\bbE[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n    \\end{enumerate}", "brownian3": "\\begin{enumerate}[label=(B\\arabic*)]\n    \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n    \\item \\label{brownian2} $\\bbE[B_{\\tau} B_\\tau] =0$, and\n    \\item\\label{brownian3} $\\bbE[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n    \\end{enumerate}", "prop:intermittency": "\\begin{proposition}[Intermittency] \\label{prop:intermittency}For any $p\\geq 1$ integer, we have \n\\begin{align*}\n   n \\exp\\left((p+\\frac{2p(p-1)}{n+2})\\tau \\right) \\leq  \\E |F_\\tau|^{2p}\\leq n^{p} \\exp\\left((p+\\frac{2p(p-1)}{n+2})\\tau \\right),\n\\end{align*}\nwhere $|F|^2:=\\tr F^*F$ denotes the square of the Frobenius norm.\n\\end{proposition}", "F": "\\begin{align}\\label{F}\n    dF=F_\\tau\\circ dB, \\quad F_{\\tau=0}=id ,\n\\end{align}", "prop:nontightness": "\\begin{proposition}[Non-tightness] \\label{prop:nontightness}\n    For $\\tau \\gg_n 1$\\footnote{Here $\\tau \\gg_n 1$ means that there is a possibly large constant $C(n)$ depending only on $n$ such that the statement is true for any $\\tau$ with $\\tau \\geq C(n)$.}, we have \t\\begin{align}\\label{prop02}\n\t\t\\E | F_{ \\tau } |^2 I \\Big( { \\textstyle \\frac{ 1 }{ n } } | F_{ \\tau } |^2 \\leq { \\textstyle \\frac{ 1 }{ e  } } \\big( { \\textstyle \\E \\frac{ 1 }{ n } } | F_{ \\tau } |^2 \\big)^{ \\frac{ n + 4 }{ n + 2 } } \\Big)\n\t\t\\leq \\Big( \\frac{ 1 }{ 2 } + \\frac{ C(n) }{ \\sqrt{\\tau} } \\Big) \\E| F_{ \\tau }  |^2  .\n\t\\end{align}\n where $I(A)$ is the indicator function of the event $A$.\t\n\\end{proposition}", "dd": "\\begin{align}\n    dX=b(X_t)dt+\\sqrt{2}dW \\label{dd}\n\\end{align}"}, "pre_theorem_intro_text_len": 1023, "pre_theorem_intro_text": "\\noindent\nWe are interested in the  process $\\{F_\\tau\\}_{\\tau \\geq 0}$ that solves the Stratonovich stochastic differential equation (SDE)\n\\begin{align}\\label{F}\n    dF=F_\\tau\\circ dB, \\quad F_{\\tau=0}=id ,\n\\end{align}\nwhere $\\{B_\\tau\\}_{\\tau \\geq 0}$ is a Brownian motion on ${\\frak{sl}}(n):=\\{{\\rm tr} B=0\\}$ that satisfies the following three assumptions:\n\\begin{enumerate}[label=(B\\arabic*)]\n    \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n    \\item \\label{brownian2} $\\mathbb E[B_{\\tau} B_\\tau] =0$, and\n    \\item\\label{brownian3} $\\mathbb E[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n    \\end{enumerate}\nSince the chain rule holds for Stratonovich SDEs, $F\\in \\textbf{SL}(n):=\\{\\text{det} F=1\\}$. In \\cite{MOW25}, it is proved that for $n\\geq 2$, there is a unique Brownian motion on $\\frak{sl}(n)$ that satisfies the Assumption \\ref{brownian1}-\\ref{brownian3}. We are interested in exploring the intermittent behavior of $F$ in \\eqref{F}. Our first result is on the moments:", "context": "\\noindent\nWe are interested in the process $\\{F_\\tau\\}_{\\tau \\geq 0}$ that solves the Stratonovich stochastic differential equation (SDE)\n\\begin{align}\\label{F}\n dF=F_\\tau\\circ dB, \\quad F_{\\tau=0}=id ,\n\\end{align}\nwhere $\\{B_\\tau\\}_{\\tau \\geq 0}$ is a Brownian motion on ${\\frak{sl}}(n):=\\{{\\rm tr} B=0\\}$ that satisfies the following three assumptions:\n\\begin{enumerate}[label=(B\\arabic*)]\n \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n \\item \\label{brownian2} $\\mathbb E[B_{\\tau} B_\\tau] =0$, and\n \\item\\label{brownian3} $\\mathbb E[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n \\end{enumerate}\nSince the chain rule holds for Stratonovich SDEs, $F\\in \\textbf{SL}(n):=\\{\\text{det} F=1\\}$. In \\cite{MOW25}, it is proved that for $n\\geq 2$, there is a unique Brownian motion on $\\frak{sl}(n)$ that satisfies the Assumption \\ref{brownian1}-\\ref{brownian3}. We are interested in exploring the intermittent behavior of $F$ in \\eqref{F}. Our first result is on the moments:\n\n\\begin{align}\\label{F}\n    dF=F_\\tau\\circ dB, \\quad F_{\\tau=0}=id ,\n\\end{align}\n\n\\begin{enumerate}[label=(B\\arabic*)]\n    \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n    \\item \\label{brownian2} $\\bbE[B_{\\tau} B_\\tau] =0$, and\n    \\item\\label{brownian3} $\\bbE[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n    \\end{enumerate}\n\n\\begin{enumerate}[label=(B\\arabic*)]\n    \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n    \\item \\label{brownian2} $\\bbE[B_{\\tau} B_\\tau] =0$, and\n    \\item\\label{brownian3} $\\bbE[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n    \\end{enumerate}", "full_context": "\\noindent\nWe are interested in the process $\\{F_\\tau\\}_{\\tau \\geq 0}$ that solves the Stratonovich stochastic differential equation (SDE)\n\\begin{align}\\label{F}\n dF=F_\\tau\\circ dB, \\quad F_{\\tau=0}=id ,\n\\end{align}\nwhere $\\{B_\\tau\\}_{\\tau \\geq 0}$ is a Brownian motion on ${\\frak{sl}}(n):=\\{{\\rm tr} B=0\\}$ that satisfies the following three assumptions:\n\\begin{enumerate}[label=(B\\arabic*)]\n \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n \\item \\label{brownian2} $\\mathbb E[B_{\\tau} B_\\tau] =0$, and\n \\item\\label{brownian3} $\\mathbb E[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n \\end{enumerate}\nSince the chain rule holds for Stratonovich SDEs, $F\\in \\textbf{SL}(n):=\\{\\text{det} F=1\\}$. In \\cite{MOW25}, it is proved that for $n\\geq 2$, there is a unique Brownian motion on $\\frak{sl}(n)$ that satisfies the Assumption \\ref{brownian1}-\\ref{brownian3}. We are interested in exploring the intermittent behavior of $F$ in \\eqref{F}. Our first result is on the moments:\n\n\\begin{align}\\label{F}\n    dF=F_\\tau\\circ dB, \\quad F_{\\tau=0}=id ,\n\\end{align}\n\n\\begin{enumerate}[label=(B\\arabic*)]\n    \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n    \\item \\label{brownian2} $\\bbE[B_{\\tau} B_\\tau] =0$, and\n    \\item\\label{brownian3} $\\bbE[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n    \\end{enumerate}\n\n\\begin{enumerate}[label=(B\\arabic*)]\n    \\item \\label{brownian1} $OB O^{-1}=_{\\text{law}}B$ for all $O\\in \\textbf{O}(n):=\\{O^*O=id\\}$,\n    \\item \\label{brownian2} $\\bbE[B_{\\tau} B_\\tau] =0$, and\n    \\item\\label{brownian3} $\\bbE[B_\\tau^* B_\\tau]=\\tau \\text{id}$. \n    \\end{enumerate}\n\n\\end{abstract}\n\\maketitle\n\n\\medskip \\noindent\nThe fact that $2p$-th moment scales with a rate much larger than the $p$-th power of the second moment amounts to strongly non-Gaussian and intermittent behavior. Proposition~\\ref{prop:intermittency} generalizes \\cite[Lemma 2]{MOW25} for $n=2$ to general $n\\ge 2$.\n\n\\medskip \\noindent\nWhile Proposition~\\ref{prop:intermittency} clearly expresses intermittency at the level of $F$, we are not able to transfer this higher-moment information to $\\nabla u$. This is because we can capture the proximity of $\\nabla u$ and $F$ only on the second moments. Hence we need to capture the shadow of the intermittency on the level of the second moment. In fact, it comes as a non-tightness result in the second moment in the sense that the extreme tails contain a substantial fraction of the second moment.\n \\begin{proposition}[Non-tightness] \\label{prop:nontightness}\n For $\\tau \\gg_n 1$\\footnote{Here $\\tau \\gg_n 1$ means that there is a possibly large constant $C(n)$ depending only on $n$ such that the statement is true for any $\\tau$ with $\\tau \\geq C(n)$.}, we have \\begin{align}\\label{prop02}\n \\E | F_{ \\tau } |^2 I \\Big( { \\textstyle \\frac{ 1 }{ n } } | F_{ \\tau } |^2 \\leq { \\textstyle \\frac{ 1 }{ e } } \\big( { \\textstyle \\E \\frac{ 1 }{ n } } | F_{ \\tau } |^2 \\big)^{ \\frac{ n + 4 }{ n + 2 } } \\Big)\n \\leq \\Big( \\frac{ 1 }{ 2 } + \\frac{ C(n) }{ \\sqrt{\\tau} } \\Big) \\E| F_{ \\tau } |^2 .\n \\end{align}\n where $I(A)$ is the indicator function of the event $A$. \n\\end{proposition}\n\\begin{remark}\\label{cor:nontightness} The information in Proposition~\\ref{prop:nontightness} can easily be upgraded to the lower bound as is done in \\cite[Theorem 3 \\& (118)]{MOW25}\n \\begin{align*}\n\\E |F_\\tau|^{2p} \\gtrsim_{n,p} \\bbE^{1+\\frac{n+4}{n+2}(p-1)} |F_\\tau|^{2},\n\\end{align*}\nwhich is weaker than \\eqref{2pthmoment} but not completely unrelated. This exponent is precisely the linearization of the quadratic exponent $p+\\frac{2p(p-1)}{n+2}$ around $p=1$. \n\\end{remark}\n\n\\subsection{Proof of Proposition~\\ref{prop:intermittency}}\nRecall from Lemma~\\ref{lem:odes}, we have for $p\\geq 1$ integer\n\\begin{align*}\n \\frac{ d\\bbE[\\tr G^p]}{d\\tau} =(p+\\frac{p(p-1)(n-2)}{\\alpha_n})\\bbE \\tr G_\\tau^p +\\frac{pn}{\\alpha_n}\\sum_{j=1}^{p-1}\\bbE \\tr G_\\tau^j \\tr G_\\tau^{p-j} \n\\end{align*}\n with initial condition $\\tr G_{\\tau=0}=\\tr id=n$. Using $\\tr G^j \\tr G^{p-j} \\geq \\tr G^p$ for $0\\leq j\\leq p$, we can bound the second term \n \\begin{align*}\n \\sum_{j=1}^{p-1}\\bbE \\tr G_\\tau^j \\tr G_\\tau^{p-j} \\geq (p-1) \\bbE \\tr G_\\tau^p\n\\end{align*}\nwhich leads to the differential inequality \n\\begin{align*}\n \\frac{ d\\bbE\\tr G^p}{d\\tau} \\geq (p+\\frac{p(p-1)(2n-2)}{\\alpha_n}) \\bbE\\tr G_\\tau^p\n\\end{align*}\nwhere $\\alpha_n$ is as defined \\eqref{alpha}. Hence we get \n\\begin{align*}\n \\bbE\\tr G_\\tau^p \\geq n e^{(p+\\frac{2p(p-1)}{n+2})\\tau} .\n\\end{align*}\nOn the other hand, recall again from Lemma~\\ref{lem:odes}, $\\bbE \\tr^p G_\\tau$ satisfies \n\\begin{align*}\n \\frac{ d\\bbE\\tr^p G}{d\\tau}& =(p-\\frac{2p(p-1)}{\\alpha_n}) \\bbE\\tr^p G_\\tau +\\frac{2p(p-1)n}{\\alpha_n} \\bbE\\tr^{p-2} G_\\tau\\tr G_\\tau^2 \n\\end{align*}\nwith initial condition $\\bbE\\tr^p G_{\\tau=0}=n^p$. This time using $\\tr G^2\\leq \\tr^2 G$, we get the differential inequality\n\\begin{align*}\n \\frac{d\\bbE \\tr^p G}{d\\tau}& \\leq (p+\\frac{p(p-1)(2n-2)}{\\alpha_n}) \\bbE\\tr^p G_\\tau \n\\end{align*}\nwhich leads to\n\\begin{align*}\n \\bbE\\tr^p G_\\tau \\leq n^pe^{(p+\\frac{2p(p-1)}{n+2})\\tau} .\n\\end{align*}\nHence we obtain \\eqref{lpestimates} which is a restatement of Proposition~\\ref{prop:intermittency}.\n\\qed\n\nFor parameters $ \\tau^* \\geq 0 $ and $ \\hat{\\sigma}^* $ that we will choose later we consider the terminal condition\n\\begin{align}\\label{tc}\n \\hat{\\zeta}( \\tau^*,\\hat{\\sigma})\\begin{cases}\n = 1, \\text{ if } \\hat{\\sigma} \\leq \\hat{\\sigma}^*,\\\\\n \\in [0,1] \\text{ for all } \\hat{\\sigma},\\\\\n =0, \\text{ if }\\hat{\\sigma}\\geq \\hat{\\sigma}^*+1,\n \\end{cases}\n \\text{ with } \\left| \\frac{\\partial^2 \\hat{\\zeta}( \\tau^* ,\\hat{\\sigma}) }{\\partial \\hat{\\sigma}^2}\\right| \\lesssim 1\n \\end{align}\n for the equation \\eqref{zetahat2} in Lemma~\\ref{lem:zeta}.\nIt is readily seen from \\eqref{tc} that we have\n \\begin{align}\n\\bbE \\hat R_{ \\tau^* } \\hat{\\zeta}(\\tau^*, \\hat R_{ \\tau^* }) \\stackrel{ (\\ref{tc}) }{ \\geq } \\bbE \\hat R_{ \\tau^* } I( \\hat R_{ \\tau^* } \\leq \\hat{R}^*) \n\\quad \\text{provided} \\quad \\hat\\sigma^* =: \\ln \\hat R^{ * }. \\label{ononehand}\n\\end{align}\nOur goal is to choose $ \\hat\\sigma^* $ large enough so that at initial time $ \\tau = 0 $ it holds\n\\begin{align}\\hat{\\zeta}( \\tau=0, \\hat\\sigma = 0 ) \\leq \\frac{1}{2} , \\label{initial}\n\\end{align}\nso that Lemma~\\ref{lem:zeta} implies\n\\begin{align}\n \\bbE \\hat R_{ \\tau^* } I( \\hat R_{ \\tau^* } \\leq \\hat{R}^*) - \\frac{ 1 }{ 2 }\n \\stackrel{ (\\ref{ononehand}) }{ \\leq } \\bbE \\hat{R}_{ \\tau^* } \\hat{\\zeta}( \\tau^*, \\hat R_{ \\tau^* }) - \\frac{ 1 }{ 2 }\n \\stackrel{ \\eqref{zetahatineq} \\& (\\ref{initial}) }{ \\lesssim_{ n } } \\int_0^{\\tau^*}d \\tau\\, e^{ - ( 1 - \\frac{ \\lambda_2 }{ 2 }) \\tau} \\sup_{\\hat \\sigma}\\left|\\frac{\\partial \\hat \\zeta}{\\partial\\hat \\sigma}+ \\frac{\\partial^2 \\hat \\zeta}{\\partial\\hat \\sigma^2} \\right| . \\label{zetahatineqb}\n\\end{align}\n\n\\noindent\nTo conclude the proof, it is left to show that the integral in \\eqref{zetahatineqb} is small. To this end, note that using the estimates for the terminal data \n\\begin{align*}\n \\sup_{\\hat\\sigma } \\left|\\frac{\\partial \\hat{\\zeta}( \\tau^*,\\hat{\\sigma})}{\\partial {\\hat\\sigma}}\\right|+\\left|\\frac{\\partial^2 \\hat{\\zeta}(\\tau^*,\\hat{\\sigma})}{\\partial^2 {\\hat \\sigma } }\\right| \n \\lesssim 1\n \\quad \\text{and} \\quad\n \\int d{\\hat{\\sigma}}\\left|\\frac{\\partial \\hat{\\zeta}( \\tau^*,\\hat{\\sigma})}{\\partial {\\hat\\sigma}}\\right| \n \\lesssim 1\n\\end{align*}\nwe learn (by convolution with the heat kernel) that \n\\begin{align*}\n \\sup_{ \\hat \\sigma } \\left|\\frac{\\partial \\hat{\\zeta} ( \\tau^* , \\hat\\sigma ) }{\\partial \\hat{\\sigma}}\\right| \\lesssim_{ n } \\frac{1}{ \\sqrt{1+ \\tau^*- \\tau}}, \\hskip 10pt \\sup_{ \\hat\\sigma } \\left|\\frac{\\partial^2 \\hat{\\zeta} ( \\tau^* , \\hat\\sigma ) }{\\partial \\hat{\\sigma}^2}\\right| \n \\lesssim_{ n } \\frac{1}{ 1+ \\tau^*- \\tau} .\n\\end{align*}\nHence, the integral term in \\eqref{zetahatineqb} is bounded by\n\\begin{align*} \\int_0^{\\tau*}d\\tau e^{ - ( 1 - \\frac{ \\lambda_2 }{ 2 } ) \\tau} \\sup_{\\hat \\sigma}\\left|\\frac{\\partial \\hat \\zeta}{\\partial\\hat \\sigma}+ \\frac{\\partial^2 \\hat \\zeta}{\\partial\\hat \\sigma^2} \\right| \\lesssim_{ n } \\int_0^{\\tau^*} d\\tau e^{-\\frac{4}{n-1} \\tau } \\frac{1}{\\sqrt{1+\\tau^*-\\tau}} \\lesssim_{ n } \\frac{ 1 }{\\sqrt{\\tau^*}},\n\\end{align*}\nand with some constant $ C = C ( n ) $ estimate \\eqref{zetahatineqb} turns into\n\\begin{align}\\label{ontheotherhand}\n \\bbE \\hat R_{ \\tau^* } I( \\hat R_{ \\tau^* } \\leq \\hat{R}^*) \\leq \\frac{ 1 }{ 2 } + \\frac{ C(n) }{ \\sqrt{ \\tau^* } }.\n\\end{align}\nFinally, undoing the change of variables we have\n\\begin{align*}\n \\hat R_{ \\tau^* }\n \\stackrel{ \\eqref{Rhat} }{ = } e^{ - \\tau^* } { \\textstyle \\frac{ 1 }{ n } } R_{ \\tau^* }\n \\stackrel{ \\ref{F3} , (\\ref{R}) \\& (\\ref{:G}) }{ = } \\frac{ | F_{ \\tau^* } |^2 }{ \\E | F_{ \\tau^* } |^2 } ,\n \\quad \n \\hat R^{ * }\n \\stackrel{ (\\ref{eqnHatRStar}) }{ \\leq } e^{ \\frac{ 2 }{ n + 2 } \\tau^* - 1 } \n \\stackrel{\\ref{F3} }{ = } \\frac{ 1 }{ e } \\Big( \\frac{ \\E | F_{ \\tau^* } |^2 }{ n } \\Big)^{ \\frac{ 2 }{ n + 2 } } ,\n\\end{align*}\nso that since $ \\frac{ 2 }{ n + 2 } + 1 = \\frac{ n + 4 }{ n + 2 } $ \n\\begin{align*}\n \\hat R_{ \\tau^* } I( \\hat R_{ \\tau^* } \\leq \\hat{R}^*)\n = \\frac{ | F_{ \\tau^* } |^2 }{ \\E | F_{ \\tau^* } |^2 } I \\Big( { \\textstyle \\frac{ 1 }{ n } } | F_{ \\tau^* } |^2 \\leq { \\textstyle \\frac{ 1 }{ e } } \\big( { \\textstyle \\frac{ \\E | F_{ \\tau^* } |^2 }{ n } } \\big)^{ \\frac{ n + 4 }{ n + 2 } } \\Big) .\n\\end{align*}\nThus \\eqref{ontheotherhand} turns into \\eqref{prop02}. \\qed", "post_theorem_intro_text_len": 6008, "post_theorem_intro_text": "\\noindent\nProposition~\\ref{prop:intermittency} implies that \n\\begin{align}\n    \\mathbb{E}|F_\\tau|^{2p} \\sim_{n,p}\\footnotemark \\left(\\mathbb{E}^p |F_\\tau|^2\\right)^{1+\\frac{2(p-1)}{n+2}}. \\label{2pthmoment}\n\\end{align}\n\\footnotetext{Here $A \\sim_{n,p} B$ means that there exist constants $c=c(n,p)$ and $C=C(n,p)$ depending only on $n$ and $p$ such that $cB\\leq A\\leq CB$.} \n\n\\medskip \\noindent\nThe fact that $2p$-th moment scales with a rate much larger than the $p$-th power of the second moment amounts to strongly non-Gaussian and intermittent behavior. Proposition~\\ref{prop:intermittency} generalizes \\cite[Lemma 2]{MOW25} for $n=2$ to general $n\\ge 2$.\n\n\\medskip \\noindent\nThe interest in studying the geometric Brownian motion comes from its\nintriguing relation to the drift-diffusion process \n\\begin{align}\n    dX=b(X_t)dt+\\sqrt{2}dW \\label{dd}\n\\end{align}\nwith $b$ being a divergence-free and time-independent vector field. One is interested in the case where $b$ is sampled from a stationary and isotropic Gaussian ensemble with the scaling \n \\begin{align}\n   \\label{b} b(\\mu \\cdot)=\\frac{1}{\\mu}b \\text{ in law for all } \\mu>0,\n\\end{align}\nin which convection and diffusion balance at every scale. After implementing an ultraviolet cutoff, w.l.o.g.\\ at scale 1, $b$ is fixed up to a single constant which we describe in terms of $\\mathbb{E}|b|^2=\\epsilon^2 \\frac{ n }{ 4 } $, where $\\epsilon \\ll 1$ can be interpreted as the P\\'{e}clet number. It is now well-known that the mean-square displacement displays super-diffusive behavior $\\frac{1}{2t}\\mathbb E|X_t|^2 \\approx \\lambda(t)$ for $t \\gg 1$, where \n\\begin{align*}\n\\lambda(t):=\\sqrt{1+\\frac{\\epsilon^2}{2} \\ln(1+t)}.\n\\end{align*}\nThis behavior was rigorously established with increasing precision in \\cite{TV12, CHT22, CMOW22,ABK24, MOW25}. \n\n\\medskip \\noindent\nThe relationship between the geometric Brownian motion process $F$ in \\eqref{F} and the drift-diffusion process in \\eqref{dd} was discovered in \\cite{MOW25} on the level of the expected position, $u(x,t)$, (w.r.t. $W_t$) of the process $\\{X_t\\}_{t\\geq 0}$ starting from $X_{t=0}=x$. More precisely, one obtains $\\nabla u(0,t) \\approx F_{\\tau(T)}$ in law on average over $t\\in (0,T)$, where $$\\tau(T):=\\ln \\lambda(T), $$ see \\cite[Theorem 2 \\& (19)]{MOW25}.\n\n\\medskip \\noindent\nOur result shows that the intermittency of the drift-diffusion equation depends less on topology (stream lines of the divergence-free $b$ are closed iff $n=2$) and more on geometry ($\\textbf{SL}(n)$ has less curvature as $n$ increases). \n\n\\medskip \\noindent\nWhile Proposition~\\ref{prop:intermittency} clearly expresses intermittency at the level of $F$, we are not able to transfer this higher-moment information to $\\nabla u$. This is because we can capture the proximity of $\\nabla u$ and $F$ only on the second moments. Hence we need to capture the shadow of the intermittency on the level of the second moment. In fact, it comes as a non-tightness result in the second moment in the sense that the extreme tails contain a substantial fraction of the second moment.\n \\begin{proposition}[Non-tightness] \\label{prop:nontightness}\n    For $\\tau \\gg_n 1$\\footnote{Here $\\tau \\gg_n 1$ means that there is a possibly large constant $C(n)$ depending only on $n$ such that the statement is true for any $\\tau$ with $\\tau \\geq C(n)$.}, we have \t\\begin{align}\\label{prop02}\n\t\t\\mathbb{E} | F_{ \\tau } |^2 I \\Big( { \\textstyle \\frac{ 1 }{ n } } | F_{ \\tau } |^2 \\leq { \\textstyle \\frac{ 1 }{ e  } } \\big( { \\textstyle \\mathbb{E} \\frac{ 1 }{ n } } | F_{ \\tau } |^2 \\big)^{ \\frac{ n + 4 }{ n + 2 } } \\Big)\n\t\t\\leq \\Big( \\frac{ 1 }{ 2 } + \\frac{ C(n) }{ \\sqrt{\\tau} } \\Big) \\mathbb{E}| F_{ \\tau }  |^2  .\n\t\\end{align}\n where $I(A)$ is the indicator function of the event $A$.\t\n\\end{proposition}\n\\begin{remark}\\label{cor:nontightness} The information in Proposition~\\ref{prop:nontightness} can easily be upgraded to the lower bound as is done in \\cite[Theorem 3 \\& (118)]{MOW25}\n \\begin{align*}\n\\mathbb{E}  |F_\\tau|^{2p} \\gtrsim_{n,p}  \\mathbb E^{1+\\frac{n+4}{n+2}(p-1)}  |F_\\tau|^{2},\n\\end{align*}\nwhich is weaker than \\eqref{2pthmoment} but not completely unrelated. This exponent is precisely the linearization of the quadratic exponent $p+\\frac{2p(p-1)}{n+2}$ around $p=1$. \n\\end{remark}\n\n\\noindent\nThe results in Propositions~\\ref{prop:intermittency} and ~\\ref{prop:nontightness} were previously proven in the case $n=2$ (see \\cite[Lemma 2, Lemma 4]{MOW25}). The argument relied on the fact that the law of $R=\\frac{1}{2}{\\rm tr} F^*F$ can be identified as the Itô evolution \\begin{align*}\n    dR= R_{\\tau} d\\tau +\\sqrt{R_\\tau^2-1} dw, \\quad R_{\\tau =0}=1,\n\\end{align*} \nwhere $\\{w_\\tau\\}_{\\tau\\geq 0}$ is a one-dimensional Brownian motion. In any dimension $n$, as a consequence of the $\\textbf{O}(n)$-invariance, it is not difficult to see that one obtains an evolution for the $(n-1)$ quantities ${\\rm tr} F^*F, {\\rm tr} (F^*F)^2,\\dots ,{\\rm tr} (F^*F)^{n-1}$. However, we are unable to leverage this evolution in order to characterize the evolution of $R$. In this paper, we take a different route: Relying on the observation that the quantities $\\mathbb{E} {\\rm tr} (F^*F)^2 $ and $\\mathbb{E}{\\rm tr}^2 F^*F $ satisfy a linear system of ODEs which is closed in any dimension $n$ and can be solved explicitly, we learn that $\\mathbb E{\\rm tr} (F^*F)^2$ can be well approximated by $\\mathbb E {\\rm tr}^2 F^* F$. This in turn allows us to approximately close the equation for $ R $ in dimensions $ n > 2 $.\n\n\\medskip\n \\noindent\nNote that $|F_\\tau|^2$ is close to a stochastic exponential, which is consistent with Proposition~\\ref{prop:intermittency} and \\ref{prop:nontightness}, namely $\\ln |F_\\tau|^2$ behaves like a Gaussian random variable with mean $(1-\\frac{2}{n+2})\\tau$ and variance $\\frac{4\\tau}{n+2}$. \n\n\\subsection*{Acknowledgment}\nWe thank Anna Wienhard and Corentin Le Bars for helpful discussions concerning random walks on groups. We thank Peter Morfe and Ofer Zeitouni for helpful discussions.", "sketch": "For $n=2$, the previously used argument (\\cite[Lemma 2, Lemma 4]{MOW25}) identified the law of $R:=\\frac{1}{2}{\\rm tr}\\,F^*F$ via the one-dimensional It\\^o SDE\n\\[\n dR=R_{\\tau}\\,d\\tau+\\sqrt{R_{\\tau}^2-1}\\,dw,\\qquad R_{\\tau=0}=1,\n\\]\nwhich yields the needed moment information.\n\nIn general dimension $n$, by $\\mathbf O(n)$-invariance “one obtains an evolution for the $(n-1)$ quantities ${\\rm tr}F^*F,\\,{\\rm tr}(F^*F)^2,\\dots,{\\rm tr}(F^*F)^{n-1}$,” but the authors state they are “unable to leverage this evolution in order to characterize the evolution of $R$.”\n\nInstead, they “take a different route”: they use that “$\\mathbb E\\,{\\rm tr}(F^*F)^2$ and $\\mathbb E\\,{\\rm tr}^2 F^*F$ satisfy a linear system of ODEs which is closed in any dimension $n$ and can be solved explicitly.” From this explicit solution, they “learn that $\\mathbb E\\,{\\rm tr}(F^*F)^2$ can be well approximated by $\\mathbb E\\,{\\rm tr}^2 F^*F$,” which “in turn allows us to approximately close the equation for $R$ in dimensions $n>2$.”", "expanded_sketch": "For $n=2$, the previously used argument (\\cite[Lemma 2, Lemma 4]{MOW25}) identified the law of $R:=\\frac{1}{2}{\\rm tr}\\,F^*F$ via the one-dimensional It\\^o SDE\n\\[\n dR=R_{\\tau}\\,d\\tau+\\sqrt{R_{\\tau}^2-1}\\,dw,\\qquad R_{\\tau=0}=1,\n\\]\nwhich yields the needed moment information.\n\nIn general dimension $n$, by $\\mathbf O(n)$-invariance “one obtains an evolution for the $(n-1)$ quantities ${\\rm tr}F^*F,\\,{\\rm tr}(F^*F)^2,\\dots,{\\rm tr}(F^*F)^{n-1}$,” but the authors state they are “unable to leverage this evolution in order to characterize the evolution of $R$.”\n\nInstead, they “take a different route”: they use that “$\\mathbb E\\,{\\rm tr}(F^*F)^2$ and $\\mathbb E\\,{\\rm tr}^2 F^*F$ satisfy a linear system of ODEs which is closed in any dimension $n$ and can be solved explicitly.” From this explicit solution, they “learn that $\\mathbb E\\,{\\rm tr}(F^*F)^2$ can be well approximated by $\\mathbb E\\,{\\rm tr}^2 F^*F$,” which “in turn allows us to approximately close the equation for $R$ in dimensions $n>2$.”,", "expanded_theorem": "[Intermittency] \\label{prop:intermittency}For any $p\\geq 1$ integer, we have \n\\begin{align*}\n   n \\exp\\left((p+\\frac{2p(p-1)}{n+2})\\tau \\right) \\leq  \\mathbb{E} |F_\\tau|^{2p}\\leq n^{p} \\exp\\left((p+\\frac{2p(p-1)}{n+2})\\tau \\right),\n\\end{align*}\nwhere $|F|^2:={\\rm tr} F^*F$ denotes the square of the Frobenius norm.", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let $n\\ge 2$, let $\\mathfrak{sl}(n)=\\{A:\\operatorname{tr}A=0\\}$, and let $\\{B_\\tau\\}_{\\tau\\ge 0}$ be a Brownian motion on $\\mathfrak{sl}(n)$ such that for every $O\\in \\mathbf O(n)=\\{O^*O=\\mathrm{id}\\}$ one has $OBO^{-1}=_{\\mathrm{law}}B$, and moreover $\\mathbb E[B_\\tau B_\\tau]=0$ and $\\mathbb E[B_\\tau^*B_\\tau]=\\tau\\,\\mathrm{id}$. Let $\\{F_\\tau\\}_{\\tau\\ge 0}$ solve the Stratonovich SDE\n\\[\n dF = F_\\tau\\circ dB,\\qquad F_{\\tau=0}=\\mathrm{id}.\n\\]\nIf $|F|^2:=\\operatorname{tr}(F^*F)$ denotes the square of the Frobenius norm, which statement holds for every integer $p\\ge 1$ and every $\\tau\\ge 0$?", "correct_choice": {"label": "A", "text": "\\[\n n\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right)\n \\le \\mathbb E|F_\\tau|^{2p}\n \\le n^p\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right).\n\\]"}, "choices": [{"label": "B", "text": "\\[\n n\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n}\\right)\\tau\\right)\n \\le \\mathbb E|F_\\tau|^{2p}\n \\le n^p\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n}\\right)\\tau\\right).\n\\]"}, {"label": "C", "text": "\\[\n \\mathbb E|F_\\tau|^{2p}\n \\le n^p\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right).\n\\]"}, {"label": "D", "text": "\\[\n \\text{For each integer }p\\ge 1\\text{ there exists }C_p\\in[1,\\infty)\\text{ such that for all }\\tau\\ge 0,\n\\quad C_p^{-1}\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right)\n \\le \\mathbb E|F_\\tau|^{2p}\n \\le C_p\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right).\n\\]"}, {"label": "E", "text": "\\[\n n\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right)\n \\le \\mathbb E|F_\\tau|^{2p}\n \\le n^p\\exp\\!\\left(\\left(p+\\frac{2p(p-1)}{n+2}\\right)\\tau\\right)\n\\]\nfor every real number \\(p\\ge 1\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "explicit exponent denominator n+2", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped lower bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "explicit sharp prefactors n and n^p replaced by non-explicit constants depending on p", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "range of p extended from integers to all reals", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only the stochastic setup and asks for the valid moment estimate; it does not explicitly state or strongly hint at the exact bound, the sharp denominator n+2, or the precise prefactors."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: the correct choice appears to be the exact target estimate under the stated hypotheses. However, it is not a pure restatement because the options include nearby variants that force comparison of strength, sharpness, and parameter range."}, "GPS": {"score": 2, "justification": "Selecting the correct answer requires substantial mathematical judgment: one must track the exact exponent, distinguish sharp two-sided bounds from weaker true statements, and notice the restriction to integer p. The answer is not obvious from surface cues."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically meaningful: one alters n+2 to n, one gives a weaker but still plausible one-sided bound, one replaces sharp constants by existential constants, and one improperly extends the range of p. These reflect realistic failure modes."}, "total_score": 7, "overall_assessment": "A high-quality MCQ with no answer leakage and excellent distractors; its main limitation is that it leans somewhat toward exact theorem recall rather than fully independent generative reasoning."}}
{"id": "2602.07171v1", "paper_link": "http://arxiv.org/abs/2602.07171v1", "theorems_cnt": 1, "theorem": {"env_name": "conjecture", "content": "[Noferini--Williams \\cite{MR4418964}]\\label{conj}\nFor $n\\ge1$ and $0\\le m<n$, $H(n,m)^{\\mathrm{ab}}\\cong\\mathbb{Z}^2$\nif and only if \n$n\\equiv0\\pmod6$ and \n$m\\equiv2\\pmod n.$", "start_pos": 6243, "end_pos": 6452, "label": "conj"}, "ref_dict": {"conj": "\\begin{conjecture}[Noferini--Williams \\cite{MR4418964}]\\label{conj}\nFor $n\\ge1$ and $0\\le m<n$, $H(n,m)^{\\mathrm{ab}}\\cong\\mathbb{Z}^2$\nif and only if \n$n\\equiv0\\pmod6$ and \n$m\\equiv2\\pmod n.$\n\\end{conjecture}", "eq:Gnmk": "\\begin{equation}\\label{eq:Gnmk}\nG_n(m,k)\n=\n\\langle x_0,\\dots,x_{n-1}\n\\mid\nx_i x_{i+m}=x_{i+k}\\ (0\\le i<n)\n\\rangle,\n\\qquad (\\text{indices mod } n).\n\\end{equation}", "tech": "\\label{tech}\nIn this section, we present the technical results that underpin the proof of Conjecture~\\ref{conj}. All notation is as in Section~\\ref{std}.\n\n\\medskip\nOur first result extends \\cite[Corol", "prelim": "\\label{prelim}\n\nThis section lays the foundation for the remainder of the paper by presenting the core ideas and setting the notation that will be used consistently throughout. All notations introduce"}, "pre_theorem_intro_text_len": 2776, "pre_theorem_intro_text": "Cyclically presented groups are finitely presented groups whose relations are invariant under cyclic permutation of generators. Since their introduction, they have provided a setting uniting combinatorial group theory, number theory, and low-dimensional topology. A key theme is expressing algebraic invariants in terms of arithmetic data from the presentation.\n\n\\medskip\nA particularly important family is given by the Fibonacci-type cyclically presented\ngroups\n\\begin{equation}\\label{eq:Gnmk}\nG_n(m,k)\n=\n\\langle x_0,\\dots,x_{n-1}\n\\mid\nx_i x_{i+m}=x_{i+k}\\ (0\\le i<n)\n\\rangle,\n\\qquad (\\text{indices mod } n).\n\\end{equation}\n\nThese groups were first discussed in \\cite{MR1634446,MR387419} (see also the survey \\cite{MR3010816}). They include the classical Fibonacci groups $F(2,n)=G_n(1,2)$\\cite{Conway1965}, the Sieradski groups $S(2,n)=G_n(2,1)$ \\cite{MR830041}, and the Gilbert-Howie groups $H(n,m)=G_n(m,1)$ \\cite{MR1332862}, as well as many generalisations. These groups have been studied from various perspectives, including finiteness and asphericity \\cite{MR2003626,MR2464807,MR1622427,MR1298588,MR1332862,MR2488144}, hyperbolicity \\cite{MR4223853,MR4243359,MR4424974}, perfectness \\cite{MR1700021,MR2661287} (see also \\cite{MR4315549}), and topology \\cite{MR1332862}.\n\n\\medskip\nA basic invariant of a finitely presented group is its abelianization.\nFor cyclically presented groups, this invariant is especially accessible: abelianization gives an integer circulant relation matrix, so its structure is determined by the Smith normal form of this circulant matrix.\nEquivalently, the \\(\\mathbb{Z}\\)-rank of the abelianization, that is, the minimal number of generators of its torsion-free part, is determined by the greatest common divisor of the \\emph{representer} polynomial (see Section~\\ref{prelim}) and $t^n-1$. For the Fibonacci-type presentation \\eqref{eq:Gnmk}, the representer polynomial is\n\\[\n1+t^m-t^k.\n\\]\nThe \\(\\mathbb{Z}\\)-rank of $G_n(m,k)^{\\mathrm{ab}}$ is then equal to\nthe degree of $\\gcd(1+t^m-t^k,t^n-1)$, that is, the number of common roots of\n$1+t^m-t^k$ and $t^n-1$, counted with multiplicity.\n\n\\medskip\n\nFor a generic choice of parameters $n$ and $m$, the abelianization of $H(n,m)$ is finite or of $\\mathbb{Z}$-rank 1. In particular, the cases with $\\mathbb{Z}$-rank 2 are exceptional. When the abelianization has $\\mathbb{Z}$-rank 2, it is necessary that $m \\equiv 2 \\pmod n$. This places these groups in close relation with the Sieradski groups, for which $m=2$. This motivates the problem of determining exactly when the abelianization is free and has $\\mathbb{Z}$-rank 2. Guided by extensive computations and structural results, Noferini and Williams proposed the following conjecture which characterizes exactly when this phenomenon occurs.", "context": "Cyclically presented groups are finitely presented groups whose relations are invariant under cyclic permutation of generators. Since their introduction, they have provided a setting uniting combinatorial group theory, number theory, and low-dimensional topology. A key theme is expressing algebraic invariants in terms of arithmetic data from the presentation.\n\n\\medskip\nA particularly important family is given by the Fibonacci-type cyclically presented\ngroups\n\\begin{equation}\\label{eq:Gnmk}\nG_n(m,k)\n=\n\\langle x_0,\\dots,x_{n-1}\n\\mid\nx_i x_{i+m}=x_{i+k}\\ (0\\le i<n)\n\\rangle,\n\\qquad (\\text{indices mod } n).\n\\end{equation}\n\nThese groups were first discussed in \\cite{MR1634446,MR387419} (see also the survey \\cite{MR3010816}). They include the classical Fibonacci groups $F(2,n)=G_n(1,2)$\\cite{Conway1965}, the Sieradski groups $S(2,n)=G_n(2,1)$ \\cite{MR830041}, and the Gilbert-Howie groups $H(n,m)=G_n(m,1)$ \\cite{MR1332862}, as well as many generalisations. These groups have been studied from various perspectives, including finiteness and asphericity \\cite{MR2003626,MR2464807,MR1622427,MR1298588,MR1332862,MR2488144}, hyperbolicity \\cite{MR4223853,MR4243359,MR4424974}, perfectness \\cite{MR1700021,MR2661287} (see also \\cite{MR4315549}), and topology \\cite{MR1332862}.\n\n\\medskip\nA basic invariant of a finitely presented group is its abelianization.\nFor cyclically presented groups, this invariant is especially accessible: abelianization gives an integer circulant relation matrix, so its structure is determined by the Smith normal form of this circulant matrix.\nEquivalently, the \\(\\mathbb{Z}\\)-rank of the abelianization, that is, the minimal number of generators of its torsion-free part, is determined by the greatest common divisor of the \\emph{representer} polynomial (see Section~\\ref{prelim}) and $t^n-1$. For the Fibonacci-type presentation \\eqref{eq:Gnmk}, the representer polynomial is\n\\[\n1+t^m-t^k.\n\\]\nThe \\(\\mathbb{Z}\\)-rank of $G_n(m,k)^{\\mathrm{ab}}$ is then equal to\nthe degree of $\\gcd(1+t^m-t^k,t^n-1)$, that is, the number of common roots of\n$1+t^m-t^k$ and $t^n-1$, counted with multiplicity.\n\n\\medskip\n\nFor a generic choice of parameters $n$ and $m$, the abelianization of $H(n,m)$ is finite or of $\\mathbb{Z}$-rank 1. In particular, the cases with $\\mathbb{Z}$-rank 2 are exceptional. When the abelianization has $\\mathbb{Z}$-rank 2, it is necessary that $m \\equiv 2 \\pmod n$. This places these groups in close relation with the Sieradski groups, for which $m=2$. This motivates the problem of determining exactly when the abelianization is free and has $\\mathbb{Z}$-rank 2. Guided by extensive computations and structural results, Noferini and Williams proposed the following conjecture which characterizes exactly when this phenomenon occurs.\n\n\\label{prelim}\n\nThis section lays the foundation for the remainder of the paper by presenting the core ideas and setting the notation that will be used consistently throughout. All notations introduce", "full_context": "Cyclically presented groups are finitely presented groups whose relations are invariant under cyclic permutation of generators. Since their introduction, they have provided a setting uniting combinatorial group theory, number theory, and low-dimensional topology. A key theme is expressing algebraic invariants in terms of arithmetic data from the presentation.\n\n\\medskip\nA particularly important family is given by the Fibonacci-type cyclically presented\ngroups\n\\begin{equation}\\label{eq:Gnmk}\nG_n(m,k)\n=\n\\langle x_0,\\dots,x_{n-1}\n\\mid\nx_i x_{i+m}=x_{i+k}\\ (0\\le i<n)\n\\rangle,\n\\qquad (\\text{indices mod } n).\n\\end{equation}\n\nThese groups were first discussed in \\cite{MR1634446,MR387419} (see also the survey \\cite{MR3010816}). They include the classical Fibonacci groups $F(2,n)=G_n(1,2)$\\cite{Conway1965}, the Sieradski groups $S(2,n)=G_n(2,1)$ \\cite{MR830041}, and the Gilbert-Howie groups $H(n,m)=G_n(m,1)$ \\cite{MR1332862}, as well as many generalisations. These groups have been studied from various perspectives, including finiteness and asphericity \\cite{MR2003626,MR2464807,MR1622427,MR1298588,MR1332862,MR2488144}, hyperbolicity \\cite{MR4223853,MR4243359,MR4424974}, perfectness \\cite{MR1700021,MR2661287} (see also \\cite{MR4315549}), and topology \\cite{MR1332862}.\n\n\\medskip\nA basic invariant of a finitely presented group is its abelianization.\nFor cyclically presented groups, this invariant is especially accessible: abelianization gives an integer circulant relation matrix, so its structure is determined by the Smith normal form of this circulant matrix.\nEquivalently, the \\(\\mathbb{Z}\\)-rank of the abelianization, that is, the minimal number of generators of its torsion-free part, is determined by the greatest common divisor of the \\emph{representer} polynomial (see Section~\\ref{prelim}) and $t^n-1$. For the Fibonacci-type presentation \\eqref{eq:Gnmk}, the representer polynomial is\n\\[\n1+t^m-t^k.\n\\]\nThe \\(\\mathbb{Z}\\)-rank of $G_n(m,k)^{\\mathrm{ab}}$ is then equal to\nthe degree of $\\gcd(1+t^m-t^k,t^n-1)$, that is, the number of common roots of\n$1+t^m-t^k$ and $t^n-1$, counted with multiplicity.\n\n\\medskip\n\nFor a generic choice of parameters $n$ and $m$, the abelianization of $H(n,m)$ is finite or of $\\mathbb{Z}$-rank 1. In particular, the cases with $\\mathbb{Z}$-rank 2 are exceptional. When the abelianization has $\\mathbb{Z}$-rank 2, it is necessary that $m \\equiv 2 \\pmod n$. This places these groups in close relation with the Sieradski groups, for which $m=2$. This motivates the problem of determining exactly when the abelianization is free and has $\\mathbb{Z}$-rank 2. Guided by extensive computations and structural results, Noferini and Williams proposed the following conjecture which characterizes exactly when this phenomenon occurs.\n\n\\label{prelim}\n\nThis section lays the foundation for the remainder of the paper by presenting the core ideas and setting the notation that will be used consistently throughout. All notations introduce\n\n\\begin{abstract}\n\\noindent\nThe Gilbert–Howie groups $H(n, m)$ form a notable subclass within the broader family of Fibonacci-type cyclically presented groups $G_n(m, k)$. Noferini and Williams conjectured that the abelianization $H(n, m)^{ab}$ is torsion-free with \\(\\mathbb{Z}\\)-rank $2$ if and only if $n \\equiv 0 \\pmod{6}$ and $m \\equiv 2 \\pmod{n}$. We confirm this conjecture by proving that \\[ \\operatorname{Res}(F, G) = \\pm 1, \\] where \\[ F = \\frac{1 + t^m - t}{\\Phi_6} \\quad \\text{and} \\quad G = \\frac{t^n - 1}{\\Phi_6}, \\] with $\\Phi_6$ denoting the sixth cyclotomic polynomial. The proof uses a minimality argument, reducing the general problem to three cases: $m = 2 + \\frac{n}{3}$, $m = 2 + \\frac{n}{2}$, and $m = 2 + \\frac{2n}{3}$.\nThese cases are then handled using polynomial resultant analysis and field-theoretic methods.\n As a consequence, we complete the classification of all $G_n(m, k)$ that arise as labelled oriented graph groups.\n\nFor a generic choice of parameters $n$ and $m$, the abelianization of $H(n,m)$ is finite or of $\\mathbb{Z}$-rank 1. In particular, the cases with $\\mathbb{Z}$-rank 2 are exceptional. When the abelianization has $\\mathbb{Z}$-rank 2, it is necessary that $m \\equiv 2 \\pmod n$. This places these groups in close relation with the Sieradski groups, for which $m=2$. This motivates the problem of determining exactly when the abelianization is free and has $\\mathbb{Z}$-rank 2. Guided by extensive computations and structural results, Noferini and Williams proposed the following conjecture which characterizes exactly when this phenomenon occurs.\n\nThe forward implication follows from cyclotomic roots of order six for the trinomial $1+t^m-t$ under the stated congruences (see \\cite[Lemma 4.2]{MR4418964}).\nThe converse is more difficult as one must rule out all other possible cyclotomic coincidences uniformly in $n$ and $m$. This brings us to the purpose of this paper, which is to prove Conjecture~\\ref{conj}.\n\n\\begin{proof}\nSuppose for contradiction that $\\operatorname{Res}(F,G)=1$ and that $v\\nmid u$. Then $v>1$. By \\cite[Lemma~4.3]{MR4418964}, one of the following holds:\n\\[\nv\\mid m-1,\\quad v\\mid m-2,\\quad v\\mid 1.\n\\]\nSince $\\gcd(v,m-2)=1$ and $v>1$, the only possibility is $v\\mid m-1$. Write $m=1+\\alpha v$ for some $\\alpha\\in\\mathbb{N}$. Since $6\\mid m-2$, it follows that $6\\mid\\alpha v- 1$. Set $\\alpha_0 = n/v$. Since $\\alpha_0$ is even, an argument similar to that used in the proof of Lemma~\\ref{lucas} shows that $t^v + 1$ divides $t^n - 1$. Since $\\gcd(6, v) = 1$, we have\n\\[\n\\gcd(t^v+1,\\Phi_6(t))=1.\n\\]\nHence $t^v+1$ divides $G(t)$, and so for some $S(t)\\in\\mathbb{Z}[t]$, we may write\n\\[\nG(t)=(t^v+1)S(t).\n\\]\nBy \\eqref{eq:res-mult}, we have\n\\[\n\\operatorname{Res}(F,G)\n=\\operatorname{Res}(F,t^v+1)\\operatorname{Res}(F,S).\n\\]\nSince $\\operatorname{Res}(F,G)=1$, it follows that\n\\begin{equation}\\label{eq:resultant-F}\n\\operatorname{Res}(F,t^v+1)=1.\n\\end{equation}\n\n\\begin{corollary}\\label{special}\nLet $n=2\\cdot 3^s$ with $s\\geq 2$. Then $H(n,2\\pm n/3)^{ab}\\not\\cong \\mathbb{Z}^2.$\n\\end{corollary}\n\\begin{proof}\nThis is exactly the situation where $f(t)=t^m-t+1$ with $m=2\\pm n/3\\pmod n$. Set $K=\\pm n/3\\pmod n$ and apply Lemma~\\ref{lem:v1_reduction} and Lemma~\\ref{grow}.\n\\end{proof}\n\n\\begin{lemma}\\label{minimality}\nUnder Assumptions~\\ref{std}, let $u=\\tau v$ for some $\\tau\\ge 1$. If $a\\ge b+2$ or $s\\ge r+2$ or $(a=b+1$ and $s=r+1)$, then there exists an integer $n_1$ with $n_1\\mid n$ and $n_1<n$, and an integer\n$m_1$ such that $2<m_1<n_1$, $m_1\\equiv m\\pmod{n_1}$, and $m_1\\equiv2\\pmod6$.\n\\end{lemma}\n\nIn this section, we give a proof of Conjecture~1.1. To begin, let us recall the setting. Our aim is to show that $H(n,m)^{ab}$ is isomorphic to $\\mathbb{Z}^2$ if and only if $n \\equiv 0 \\pmod{6}$ and $m \\equiv 2 \\pmod{n}$. By~ \\cite[Lemma 4.2]{MR4418964}, a necessary and sufficient condition for \\(\\mathbb{Z}\\)-rank to be~$2$ is $m \\equiv 2 \\pmod{6}$ and $n \\equiv 0 \\pmod{6}$. Hence, we write\n\\[\nn = 2^{a} 3^{s} v \\quad \\text{and} \\quad m = 2^{b} 3^{r} u,\n\\]\nwhere $\\gcd(uv,6)=1$ and $a,s,b,r \\geq 1$.\n\n\\medskip\nA direct computation shows that\n\\[\nH(18,14)^{\\mathrm{ab}} \\cong H(18,8)^{\\mathrm{ab}} \\cong \\mathbb{Z}^2 \\oplus \\mathbb{Z}/19\\mathbb{Z}, \\qquad\n H(12,8)^{\\mathrm{ab}} \\cong \\mathbb{Z}^2 \\oplus \\mathbb{Z}/5\\mathbb{Z}.\n\\]\nIn particular, all three groups have \\(\\mathbb{Z}\\)-rank~2 and nontrivial torsion.\nHence, while the conditions \\(n \\equiv 0 \\pmod{6}\\) and \\(m \\equiv 2 \\pmod{6}\\) are sufficient for the abelianization to have \\(\\mathbb{Z}\\)-rank~2, they are not sufficient to ensure that it is torsion-free.\n\n\\begin{conjecture}[Noferini--Williams \\cite{MR4418964}]\\label{conj}\nFor $n\\ge1$ and $0\\le m<n$, $H(n,m)^{\\mathrm{ab}}\\cong\\mathbb{Z}^2$\nif and only if \n$n\\equiv0\\pmod6$ and \n$m\\equiv2\\pmod n.$\n\\end{conjecture}", "post_theorem_intro_text_len": 1024, "post_theorem_intro_text": "The forward implication follows from cyclotomic roots of order six for the trinomial $1+t^m-t$ under the stated congruences (see \\cite[Lemma 4.2]{MR4418964}).\nThe converse is more difficult as one must rule out all other possible cyclotomic coincidences uniformly in $n$ and $m$. This brings us to the purpose of this paper, which is to prove Conjecture~\\ref{conj}. \n\n\\medskip\nOur method is arithmetic: the \\(\\mathbb{Z}\\)-rank~2, torsion-free condition reduces to $\\operatorname{Res}(F,G)=1,$ where\n\\[\nF=\\frac{1+t^m-t}{\\Phi_6(t)},\\quad\nG=\\frac{t^n-1}{\\Phi_6(t)}.\n\\]\nThe proof proceeds by a minimality argument. In Section~\\ref{tech}, we show that the conjecture holds for the groups $H(n, 2 \\pm n/3)$ and $H(n, 2 + n/2)$. We also derive strong necessary conditions on $n$ and $m$ that any minimal counterexample must satisfy. In Section \\ref{sec:mainproof}, we show that any such minimal counterexample must in fact belong to one of these families. Consequently, no minimal counterexample exists, and the conjecture follows.", "sketch": "The introduction outlines a proof of Conjecture~\\ref{conj} as follows. The forward implication is obtained by exhibiting “cyclotomic roots of order six for the trinomial $1+t^m-t$ under the stated congruences.” For the converse, the paper uses an arithmetic method: the “$\\mathbb{Z}$-rank~2, torsion-free condition reduces to $\\operatorname{Res}(F,G)=1$,” with\n\\[\nF=\\frac{1+t^m-t}{\\Phi_6(t)},\\quad\nG=\\frac{t^n-1}{\\Phi_6(t)}.\n\\]\nThe rest of the argument is “a minimality argument.” First (Section~\\ref{tech}) the conjecture is proved for the families $H(n, 2 \\pm n/3)$ and $H(n, 2 + n/2)$, and “strong necessary conditions on $n$ and $m$ that any minimal counterexample must satisfy” are derived. Then (Section~\\ref{sec:mainproof}) it is shown that any minimal counterexample “must in fact belong to one of these families,” so “no minimal counterexample exists,” yielding the conjecture.", "expanded_sketch": "The introduction outlines a proof of\n\\begin{conjecture}[Noferini--Williams \\cite{MR4418964}]\\label{conj}\nFor $n\\ge1$ and $0\\le m<n$, $H(n,m)^{\\mathrm{ab}}\\cong\\mathbb{Z}^2$\nif and only if \n$n\\equiv0\\pmod6$ and \n$m\\equiv2\\pmod n.$\n\\end{conjecture}\nas follows. The forward implication is obtained by exhibiting “cyclotomic roots of order six for the trinomial $1+t^m-t$ under the stated congruences.” For the converse, the paper uses an arithmetic method: the “$\\mathbb{Z}$-rank~2, torsion-free condition reduces to $\\operatorname{Res}(F,G)=1$,” with\n\\[\nF=\\frac{1+t^m-t}{\\Phi_6(t)},\\quad\nG=\\frac{t^n-1}{\\Phi_6(t)}.\n\\]\nThe rest of the argument is “a minimality argument.” First, we use the technical results underpinning the proof of the main theorem:\n\\label{tech}\nIn this section, we present the technical results that underpin the proof of Conjecture~\\ref{conj}. All notation is as in Section~\\ref{std}.\n\n\\medskip\nOur first result extends \\cite[Corol\nIn this part the conjecture is proved for the families $H(n, 2 \\pm n/3)$ and $H(n, 2 + n/2)$, and “strong necessary conditions on $n$ and $m$ that any minimal counterexample must satisfy” are derived. Then (Section~\\ref{sec:mainproof}) it is shown that any minimal counterexample “must in fact belong to one of these families,” so “no minimal counterexample exists,” yielding the conjecture.", "expanded_theorem": "[Noferini--Williams \\cite{MR4418964}]\\label{conj}\nFor $n\\ge1$ and $0\\le m<n$, $H(n,m)^{\\mathrm{ab}}\\cong\\mathbb{Z}^2$\nif and only if \n$n\\equiv0\\pmod6$ and \n$m\\equiv2\\pmod n.$", "theorem_type": ["Biconditional or Equivalence", "Universal"], "mcq": {"question": "For integers $n\\ge 1$ and $0\\le m<n$, let the Gilbert--Howie group be the cyclically presented group\n\\[\nH(n,m)=G_n(m,1)=\\langle x_0,\\dots,x_{n-1}\\mid x_i x_{i+m}=x_{i+1}\\ (0\\le i<n)\\rangle,\n\\]\nwhere indices are taken modulo $n$, and let $H(n,m)^{\\mathrm{ab}}$ denote its abelianization. Which statement holds for every such pair $(n,m)$?", "correct_choice": {"label": "A", "text": "$H(n,m)^{\\mathrm{ab}}\\cong \\mathbb{Z}^2$ if and only if $n\\equiv 0\\pmod 6$ and $m\\equiv 2\\pmod n$."}, "choices": [{"label": "B", "text": "$H(n,m)^{\\mathrm{ab}}\\cong \\mathbb{Z}^2$ if and only if $n\\equiv 0\\pmod 6$ and $m\\equiv 2\\pmod 6$."}, {"label": "C", "text": "If $H(n,m)^{\\mathrm{ab}}\\cong \\mathbb{Z}^2$, then $n\\equiv 0\\pmod 6$ and $m\\equiv 2\\pmod 6$."}, {"label": "D", "text": "If $n\\equiv 0\\pmod 6$ and $m\\equiv 2\\pmod n$, then $H(n,m)^{\\mathrm{ab}}$ has $\\mathbb{Z}$-rank $2$."}, {"label": "E", "text": "$H(n,m)^{\\mathrm{ab}}\\cong \\mathbb{Z}^2$ if and only if $n\\equiv 0\\pmod 6$ and $m\\equiv 2,\\ 2+\\frac{n}{3},\\ 2+\\frac{n}{2},\\ \\text{or }2+\\frac{2n}{3}\\pmod n$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "replace the sharp congruence $m\\equiv2\\pmod n$ by the weaker necessary condition $m\\equiv2\\pmod6$", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "characteristic", "tampered_component": "drop torsion-freeness/isomorphism to $\\mathbb{Z}^2$ down to the necessary congruence conclusion modulo $6$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "confuse 'isomorphic to $\\mathbb{Z}^2$' with merely 'having $\\mathbb{Z}$-rank $2$'", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "promote the three minimal-counterexample families $m=2+n/3,\\ 2+n/2,\\ 2+2n/3$ to actual positive cases", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem only defines the Gilbert--Howie group and asks for a universally valid statement; it does not reveal the key congruence condition or otherwise point directly to choice A."}, "TAS": {"score": 1, "justification": "The correct option is essentially a theorem-level characterization stated in finished form, so the item is somewhat close to theorem recall. However, the presence of nearby alternatives with weakened or altered conclusions prevents it from being a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "The question requires some reasoning to distinguish exact equivalence from weaker necessity, sufficiency-only claims, and rank-versus-isomorphism confusion. Still, success depends heavily on knowing the characterization rather than generating a substantial argument from first principles."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target real failure modes: replacing a sharp mod-n condition by mod 6, weakening iff to implication, confusing free abelian rank with isomorphism to Z^2, and overextending exceptional families."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise recall/recognition of a known characterization than deep generative reasoning."}}
{"id": "2602.07619v1", "paper_link": "http://arxiv.org/abs/2602.07619v1", "theorems_cnt": 7, "theorem": {"env_name": "lemma", "content": "\\label{lem:kqexist} Let $\\mathbb{F}$ be a field.\n Then there exists a uniform Kronecker quotient over $\\mathbb{F}$.", "start_pos": 6965, "end_pos": 7105, "label": "lem:kqexist"}, "ref_dict": {"def:kqdefineskd": "\\begin{definition}\n \\label{def:kqdefineskd} Let $\\oslash$ be a Kronecker quotient. The Kronecker difference $\\ominus$ induced by $\\oslash$ is the\n Kronecker difference given by\n \\begin{equation*}\n  M\\ominus B := (M-I_m\\otimes B)\\oslash I_n\n \\end{equation*}\n for all $M\\in\\mathbb{F}_{mn}$, $B\\in\\mathbb{F}_n$ and $m,n\\in\\mathbb{N}$.\n\\end{definition}", "rem:kqdefineskd": "\\begin{remark}\n \\label{rem:kqdefineskd} It is easily verified that this expression satisfies the condition for a\n Kronecker difference, i.e. for every $A\\in\\mathbb{F}_m$, $B\\in\\mathbb{F}_n$ and $m,n\\in\\mathbb{N}$,\n \\begin{align*}\n  (A\\oplus B)\\ominus B\n   &= (A\\otimes I_n + I_m\\otimes B)\\ominus B \\\\\n   &= (A\\otimes I_n + I_m\\otimes B - I_m\\otimes B)\\oslash I_n \\\\\n   &= (A\\otimes I_n)\\oslash I_n \\\\\n   &= A.\n \\end{align*}\n\\end{remark}", "def:kquniform": "\\begin{definition}\n \\label{def:kquniform} A Kronecker quotient $\\oslash$ is a uniform Kronecker quotient if\n \\begin{equation*}\n  (A\\otimes C)\\oslash B = A\\otimes (C\\oslash B)\n \\end{equation*}\n for all matrices $A\\in\\mathbb{F}_{m}$, $B\\in\\mathbb{F}_n$, $C\\in\\mathbb{F}_p\\otimes\\mathbb{F}_n$ and $m,n,p\\in\\mathbb{N}$,\n and\n \\begin{equation*}\n  (A+B)\\oslash C = (A\\oslash C)+(B\\oslash C), \\qquad\n  (kA)\\oslash C = k(A\\oslash C)\n \\end{equation*}\n for all matrices $A$, $B$ and $C$, and scalars $k\\in\\mathbb{F}$ whenever the operations are defined.\n\\end{definition}"}, "pre_theorem_intro_text_len": 4691, "pre_theorem_intro_text": "Let $\\mathbb{F}_m$ denote the vector space of $m\\times m$ matrices\nover the field $\\mathbb{F}$ with characteristic $\\chr(\\mathbb{F})$, where $m\\in\\mathbb{N}$. We will denote\nby $\\mathbb{F}_m\\otimes\\mathbb{F}_n=\\mathbb{F}_{mn}$ the tensor\nproduct of $\\mathbb{F}_m$ and $\\mathbb{F}_n$. By the matrix $E_{ij}$\nwe mean a matrix in $\\mathbb{F}_m$ with a 1 in row $i$ and column $j$ and\n0 in every other entry. In general, the order of the matrix $E_{ij}$ will\nbe clear from the context. The trace $\\tr(A)$ of a matrix $A\\in\\mathbb{F}_m$\nis the sum of the main diagonal entries of $A$.\n\n\\subsection{Kronecker products and Kronecker quotients}\n\nLet $A\\in\\mathbb{F}_m$ and $B\\in\\mathbb{F}_s$. The Kronecker product\n\\cite{graham81a,horn91,kronecker3}\n$A\\otimes B\\in\\mathbb{F}_{ms}$ is the $ms\\times ms$ matrix over $F$ with entries (here $:=$ denotes a definition)\n\\begin{equation}\n \\label{eq:kronent}\n (A\\otimes B)_{(i-1)s+p,(j-1)s+q} := (A)_{i,j}(B)_{p,q},\n\\end{equation}\nwhere\n $i,j\\in\\{1,\\ldots,m\\}$ and\n $p,q\\in\\{1,\\ldots,s\\}$.\nIn other words, we can write in block matrix form\n\\begin{equation*}\nA\\otimes B=\\begin{bmatrix}\n                    (A)_{1,1}B & (A)_{1,2}B & \\ldots & (A)_{1,n} B \\\\\n                    (A)_{2,1}B & (A)_{2,2}B & \\ldots & (A)_{2,n} B \\\\\n                    \\vdots     & \\vdots     & \\ddots & \\vdots      \\\\\n                    (A)_{m,1}B & (A)_{m,2}B & \\ldots & (A)_{m,n} B\n           \\end{bmatrix}.\n\\end{equation*}\nWe note that the Kronecker product may be viewed as a family of operations\non vector spaces of square matrices (indexed by the orders of the matrices involved), \nor as an operation on the graded vector space of all square matrices. In other words,\nthe Kronecker product is the family of operations\n$\\otimes:\\mathbb{F}_m\\times\\mathbb{F}_n\\to\\mathbb{F}_{mn}$ indexed by $m,n\\in\\mathbb{N}$.\n\nIn the same way that a Kronecker product is a family\nof operations on various matrix vector spaces (or on the graded vector space of\nall square matrices), a Kronecker quotient may be viewed as a family of operations\non various matrix vector spaces (or as a partially defined operation on the graded\nvector space of all square matrices). In other words, we consider the family\n$\\oslash:\\mathbb{F}_{mn}\\times\\mathbb{F}_n\\to\\mathbb{F}_m$ indexed by $m,n\\in\\mathbb{N}$\nsatisfying the following definition.\n\n\\begin{definition}[Kronecker quotient \\cite{kronquotient,leopardi05a}]\n \\label{def:kquotient} A (right) Kronecker quotient is an operation $\\oslash$ satisfying\n \\begin{equation*}\n  (A\\otimes B)\\oslash B = A\n \\end{equation*}\n for all matrices $A\\in\\mathbb{F}_{m}$ and $B\\in\\mathbb{F}_{n}$\n ($B$ non-zero) and $m,n\\in\\mathbb{N}$.\n\\end{definition}\n\nIn general, $(M\\oslash A)\\otimes A\\neq M$ for $mn\\times mn$ matrices $M$\nand $n\\times n$ non-zero matrices $A$. A detailed study of Kronecker quotients\nappears in \\cite{kronquotient}. Throughout this article, we will consider\nonly right Kronecker quotients and differences, analogous results hold for\nleft Kronecker quotients and differences in an obvious way. Uniform Kronecker\nquotients are of particular interest, since the family of operations are\nrelated to each other via algebraic equations. More precisely, there is a generating\noperation for the entire family in the algebraically obvious way:\n\n\\begin{definition}\n \\label{def:kquniform} A Kronecker quotient $\\oslash$ is a uniform Kronecker quotient if\n \\begin{equation*}\n  (A\\otimes C)\\oslash B = A\\otimes (C\\oslash B)\n \\end{equation*}\n for all matrices $A\\in\\mathbb{F}_{m}$, $B\\in\\mathbb{F}_n$, $C\\in\\mathbb{F}_p\\otimes\\mathbb{F}_n$ and $m,n,p\\in\\mathbb{N}$,\n and\n \\begin{equation*}\n  (A+B)\\oslash C = (A\\oslash C)+(B\\oslash C), \\qquad\n  (kA)\\oslash C = k(A\\oslash C)\n \\end{equation*}\n for all matrices $A$, $B$ and $C$, and scalars $k\\in\\mathbb{F}$ whenever the operations are defined.\n\\end{definition}\n\n\\begin{remark}\n An equivalent definition for uniform Kronecker quotients\n is as follows, by simply setting $p=1$ in Definition \\ref{def:kquniform}:\n A Kronecker quotient $\\oslash$ is a uniform Kronecker quotient if\n \\begin{math}\n  (A\\otimes C)\\oslash B = A\\otimes (C\\oslash B)\n \\end{math}\n for all matrices $A\\in\\mathbb{F}_{m}$, $B,C\\in\\mathbb{F}_n$ and $m,n\\in\\mathbb{N}$,\n and\n \\begin{math}\n  (A+B)\\oslash C = (A\\oslash C)+(B\\oslash C),\n \\end{math}\n \\begin{math}\n  (kA)\\oslash C = k(A\\oslash C)\n \\end{math}\n for all matrices $A$, $B$ and $C$, and scalars $k\\in\\mathbb{F}$ whenever the operations are defined.\n\\end{remark}\n\nIn \\cite{kronquotient}, examples of uniform Kronecker quotients over the real\nand complex numbers are given. The following lemma establishes the existence\nof a uniform Kronecker quotient over an arbitrary field $\\mathbb{F}$.", "context": "Let $\\mathbb{F}_m$ denote the vector space of $m\\times m$ matrices\nover the field $\\mathbb{F}$ with characteristic $\\chr(\\mathbb{F})$, where $m\\in\\mathbb{N}$. We will denote\nby $\\mathbb{F}_m\\otimes\\mathbb{F}_n=\\mathbb{F}_{mn}$ the tensor\nproduct of $\\mathbb{F}_m$ and $\\mathbb{F}_n$. By the matrix $E_{ij}$\nwe mean a matrix in $\\mathbb{F}_m$ with a 1 in row $i$ and column $j$ and\n0 in every other entry. In general, the order of the matrix $E_{ij}$ will\nbe clear from the context. The trace $\\tr(A)$ of a matrix $A\\in\\mathbb{F}_m$\nis the sum of the main diagonal entries of $A$.\n\nLet $A\\in\\mathbb{F}_m$ and $B\\in\\mathbb{F}_s$. The Kronecker product\n\\cite{graham81a,horn91,kronecker3}\n$A\\otimes B\\in\\mathbb{F}_{ms}$ is the $ms\\times ms$ matrix over $F$ with entries (here $:=$ denotes a definition)\n\\begin{equation}\n \\label{eq:kronent}\n (A\\otimes B)_{(i-1)s+p,(j-1)s+q} := (A)_{i,j}(B)_{p,q},\n\\end{equation}\nwhere\n $i,j\\in\\{1,\\ldots,m\\}$ and\n $p,q\\in\\{1,\\ldots,s\\}$.\nIn other words, we can write in block matrix form\n\\begin{equation*}\nA\\otimes B=\\begin{bmatrix}\n (A)_{1,1}B & (A)_{1,2}B & \\ldots & (A)_{1,n} B \\\\\n (A)_{2,1}B & (A)_{2,2}B & \\ldots & (A)_{2,n} B \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n (A)_{m,1}B & (A)_{m,2}B & \\ldots & (A)_{m,n} B\n \\end{bmatrix}.\n\\end{equation*}\nWe note that the Kronecker product may be viewed as a family of operations\non vector spaces of square matrices (indexed by the orders of the matrices involved), \nor as an operation on the graded vector space of all square matrices. In other words,\nthe Kronecker product is the family of operations\n$\\otimes:\\mathbb{F}_m\\times\\mathbb{F}_n\\to\\mathbb{F}_{mn}$ indexed by $m,n\\in\\mathbb{N}$.\n\n\\begin{definition}[Kronecker quotient \\cite{kronquotient,leopardi05a}]\n \\label{def:kquotient} A (right) Kronecker quotient is an operation $\\oslash$ satisfying\n \\begin{equation*}\n (A\\otimes B)\\oslash B = A\n \\end{equation*}\n for all matrices $A\\in\\mathbb{F}_{m}$ and $B\\in\\mathbb{F}_{n}$\n ($B$ non-zero) and $m,n\\in\\mathbb{N}$.\n\\end{definition}\n\n\\begin{definition}\n \\label{def:kquniform} A Kronecker quotient $\\oslash$ is a uniform Kronecker quotient if\n \\begin{equation*}\n (A\\otimes C)\\oslash B = A\\otimes (C\\oslash B)\n \\end{equation*}\n for all matrices $A\\in\\mathbb{F}_{m}$, $B\\in\\mathbb{F}_n$, $C\\in\\mathbb{F}_p\\otimes\\mathbb{F}_n$ and $m,n,p\\in\\mathbb{N}$,\n and\n \\begin{equation*}\n (A+B)\\oslash C = (A\\oslash C)+(B\\oslash C), \\qquad\n (kA)\\oslash C = k(A\\oslash C)\n \\end{equation*}\n for all matrices $A$, $B$ and $C$, and scalars $k\\in\\mathbb{F}$ whenever the operations are defined.\n\\end{definition}\n\n\\begin{remark}\n An equivalent definition for uniform Kronecker quotients\n is as follows, by simply setting $p=1$ in Definition \\ref{def:kquniform}:\n A Kronecker quotient $\\oslash$ is a uniform Kronecker quotient if\n \\begin{math}\n (A\\otimes C)\\oslash B = A\\otimes (C\\oslash B)\n \\end{math}\n for all matrices $A\\in\\mathbb{F}_{m}$, $B,C\\in\\mathbb{F}_n$ and $m,n\\in\\mathbb{N}$,\n and\n \\begin{math}\n (A+B)\\oslash C = (A\\oslash C)+(B\\oslash C),\n \\end{math}\n \\begin{math}\n (kA)\\oslash C = k(A\\oslash C)\n \\end{math}\n for all matrices $A$, $B$ and $C$, and scalars $k\\in\\mathbb{F}$ whenever the operations are defined.\n\\end{remark}\n\nIn \\cite{kronquotient}, examples of uniform Kronecker quotients over the real\nand complex numbers are given. The following lemma establishes the existence\nof a uniform Kronecker quotient over an arbitrary field $\\mathbb{F}$.\n\n\\begin{definition}\n \\label{def:kquniform} A Kronecker quotient $\\oslash$ is a uniform Kronecker quotient if\n \\begin{equation*}\n  (A\\otimes C)\\oslash B = A\\otimes (C\\oslash B)\n \\end{equation*}\n for all matrices $A\\in\\mathbb{F}_{m}$, $B\\in\\mathbb{F}_n$, $C\\in\\mathbb{F}_p\\otimes\\mathbb{F}_n$ and $m,n,p\\in\\mathbb{N}$,\n and\n \\begin{equation*}\n  (A+B)\\oslash C = (A\\oslash C)+(B\\oslash C), \\qquad\n  (kA)\\oslash C = k(A\\oslash C)\n \\end{equation*}\n for all matrices $A$, $B$ and $C$, and scalars $k\\in\\mathbb{F}$ whenever the operations are defined.\n\\end{definition}", "full_context": "Let $\\mathbb{F}_m$ denote the vector space of $m\\times m$ matrices\nover the field $\\mathbb{F}$ with characteristic $\\chr(\\mathbb{F})$, where $m\\in\\mathbb{N}$. We will denote\nby $\\mathbb{F}_m\\otimes\\mathbb{F}_n=\\mathbb{F}_{mn}$ the tensor\nproduct of $\\mathbb{F}_m$ and $\\mathbb{F}_n$. By the matrix $E_{ij}$\nwe mean a matrix in $\\mathbb{F}_m$ with a 1 in row $i$ and column $j$ and\n0 in every other entry. In general, the order of the matrix $E_{ij}$ will\nbe clear from the context. The trace $\\tr(A)$ of a matrix $A\\in\\mathbb{F}_m$\nis the sum of the main diagonal entries of $A$.\n\nLet $A\\in\\mathbb{F}_m$ and $B\\in\\mathbb{F}_s$. The Kronecker product\n\\cite{graham81a,horn91,kronecker3}\n$A\\otimes B\\in\\mathbb{F}_{ms}$ is the $ms\\times ms$ matrix over $F$ with entries (here $:=$ denotes a definition)\n\\begin{equation}\n \\label{eq:kronent}\n (A\\otimes B)_{(i-1)s+p,(j-1)s+q} := (A)_{i,j}(B)_{p,q},\n\\end{equation}\nwhere\n $i,j\\in\\{1,\\ldots,m\\}$ and\n $p,q\\in\\{1,\\ldots,s\\}$.\nIn other words, we can write in block matrix form\n\\begin{equation*}\nA\\otimes B=\\begin{bmatrix}\n (A)_{1,1}B & (A)_{1,2}B & \\ldots & (A)_{1,n} B \\\\\n (A)_{2,1}B & (A)_{2,2}B & \\ldots & (A)_{2,n} B \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n (A)_{m,1}B & (A)_{m,2}B & \\ldots & (A)_{m,n} B\n \\end{bmatrix}.\n\\end{equation*}\nWe note that the Kronecker product may be viewed as a family of operations\non vector spaces of square matrices (indexed by the orders of the matrices involved), \nor as an operation on the graded vector space of all square matrices. In other words,\nthe Kronecker product is the family of operations\n$\\otimes:\\mathbb{F}_m\\times\\mathbb{F}_n\\to\\mathbb{F}_{mn}$ indexed by $m,n\\in\\mathbb{N}$.\n\n\\begin{definition}[Kronecker quotient \\cite{kronquotient,leopardi05a}]\n \\label{def:kquotient} A (right) Kronecker quotient is an operation $\\oslash$ satisfying\n \\begin{equation*}\n (A\\otimes B)\\oslash B = A\n \\end{equation*}\n for all matrices $A\\in\\mathbb{F}_{m}$ and $B\\in\\mathbb{F}_{n}$\n ($B$ non-zero) and $m,n\\in\\mathbb{N}$.\n\\end{definition}\n\n\\begin{definition}\n \\label{def:kquniform} A Kronecker quotient $\\oslash$ is a uniform Kronecker quotient if\n \\begin{equation*}\n (A\\otimes C)\\oslash B = A\\otimes (C\\oslash B)\n \\end{equation*}\n for all matrices $A\\in\\mathbb{F}_{m}$, $B\\in\\mathbb{F}_n$, $C\\in\\mathbb{F}_p\\otimes\\mathbb{F}_n$ and $m,n,p\\in\\mathbb{N}$,\n and\n \\begin{equation*}\n (A+B)\\oslash C = (A\\oslash C)+(B\\oslash C), \\qquad\n (kA)\\oslash C = k(A\\oslash C)\n \\end{equation*}\n for all matrices $A$, $B$ and $C$, and scalars $k\\in\\mathbb{F}$ whenever the operations are defined.\n\\end{definition}\n\n\\begin{remark}\n An equivalent definition for uniform Kronecker quotients\n is as follows, by simply setting $p=1$ in Definition \\ref{def:kquniform}:\n A Kronecker quotient $\\oslash$ is a uniform Kronecker quotient if\n \\begin{math}\n (A\\otimes C)\\oslash B = A\\otimes (C\\oslash B)\n \\end{math}\n for all matrices $A\\in\\mathbb{F}_{m}$, $B,C\\in\\mathbb{F}_n$ and $m,n\\in\\mathbb{N}$,\n and\n \\begin{math}\n (A+B)\\oslash C = (A\\oslash C)+(B\\oslash C),\n \\end{math}\n \\begin{math}\n (kA)\\oslash C = k(A\\oslash C)\n \\end{math}\n for all matrices $A$, $B$ and $C$, and scalars $k\\in\\mathbb{F}$ whenever the operations are defined.\n\\end{remark}\n\nIn \\cite{kronquotient}, examples of uniform Kronecker quotients over the real\nand complex numbers are given. The following lemma establishes the existence\nof a uniform Kronecker quotient over an arbitrary field $\\mathbb{F}$.\n\n\\begin{definition}\n \\label{def:kquniform} A Kronecker quotient $\\oslash$ is a uniform Kronecker quotient if\n \\begin{equation*}\n  (A\\otimes C)\\oslash B = A\\otimes (C\\oslash B)\n \\end{equation*}\n for all matrices $A\\in\\mathbb{F}_{m}$, $B\\in\\mathbb{F}_n$, $C\\in\\mathbb{F}_p\\otimes\\mathbb{F}_n$ and $m,n,p\\in\\mathbb{N}$,\n and\n \\begin{equation*}\n  (A+B)\\oslash C = (A\\oslash C)+(B\\oslash C), \\qquad\n  (kA)\\oslash C = k(A\\oslash C)\n \\end{equation*}\n for all matrices $A$, $B$ and $C$, and scalars $k\\in\\mathbb{F}$ whenever the operations are defined.\n\\end{definition}\n\n\\begin{proof}\n Let $B,C\\in\\mathbb{F}_m$ with $C\\neq 0$. Then there exists\n an non-zero entry $(C)_{i,j}\\neq 0$ in $C$. Now let $f:\\mathbb{F}_m\\to\\mathbb{N}\\times\\mathbb{N}$\n be defined by $f(C)=(i,j)$ and $f(0_m)=(1,1)$, i.e. $f$ chooses a non-zero entry of $C$, unless\n $C$ is the zero matrix. Define $\\text{---}\\oslash C$\n as the linear extension of\n \\begin{equation*}\n (A\\otimes B)\\oslash C := A\\dfrac{(B)_{f(C)}}{(C)_{f(C)}}\n \\end{equation*}\n for all $A\\in\\mathbb{F}_s$, $s\\in\\mathbb{N}$. Then\n $\\oslash$ is a uniform Kronecker quotient by construction.\n\\end{proof}\n\n\\begin{definition}\n \\label{def:kqdefineskd} Let $\\oslash$ be a Kronecker quotient. The Kronecker difference $\\ominus$ induced by $\\oslash$ is the\n Kronecker difference given by\n \\begin{equation*}\n M\\ominus B := (M-I_m\\otimes B)\\oslash I_n\n \\end{equation*}\n for all $M\\in\\mathbb{F}_{mn}$, $B\\in\\mathbb{F}_n$ and $m,n\\in\\mathbb{N}$.\n\\end{definition}\n\n\\begin{proposition}\n \\label{prop:induceddiffprop} Let $\\oslash$ be a Kronecker quotient and let $\\ominus$\n be the induced Kronecker difference\n \\begin{equation*}\n X\\ominus B:=(X-I_m\\otimes B)\\oslash I_n\n \\end{equation*}\n for all $X\\in \\mathbb{F}_{mn}$ and $B\\in\\mathbb{F}_n$.\n \\begin{enumerate}[label=(\\alph*)]\n \\item\n If $(A\\oslash B)^T = A^T\\oslash B^T$\n then $(A\\ominus B)^T = A^T\\ominus B^T$.\\phantom{$\\dfrac1n$}\n \\item\n If $\\tr(A\\oslash B) = \\dfrac{\\tr(A)}{\\tr(B)}$\n then $\\tr(A\\ominus B) = \\dfrac1n(\\tr(A)-m\\tr(B))$.\n \\item\n If $\\oslash$ is uniform then $\\ominus$ is uniform.\\phantom{$\\dfrac1n$}\n \\item\n If $(kA)\\oslash B = k(A\\oslash B)$\n then $(kA)\\ominus(kB) = k(A\\ominus B)$.\\phantom{$\\dfrac1n$}\n \\item\n If $(A+C)\\oslash B = A\\oslash B+C\\oslash B$\n then $(A+C)\\ominus(B+D) = A\\ominus B+C\\ominus D$.\\phantom{$\\dfrac1n$}\n \\end{enumerate}\n\\end{proposition}\n\\begin{proof}\n The proof of (c) appears in Proposition \\ref{prop:ukqdefinesukd}, which also proves (d) and (e). The proofs of $(a)$ and $(b)$ each follow\n by straightforward calculation:\n \\begin{multline*}\n (A\\ominus B)^T = ((A-I_m\\otimes B)\\oslash I_n)^T = (A-I_m\\otimes B)^T\\oslash I_n \\\\\n = (A^T-I_m\\otimes B^T)\\oslash I_n = A^T\\ominus B^T,\n \\end{multline*}\n \\begin{multline*}\n \\tr(A\\ominus B) = \\tr((A-I_m\\otimes B)\\oslash I_n) \\\\\n = \\dfrac{\\tr(A-I_m\\otimes B)}n\n = \\dfrac1n(\\tr(A)-m\\tr(B)).\n \\qedhere\n \\end{multline*}\n\\end{proof}\n\n\\begin{lemma}\n \\label{lem:unicanon} Let $\\upsilon_n\\in\\mathbb{F}_n$, $n\\in\\mathbb{N}$ be a sequence of matrices with\n $\\tr(\\upsilon_n)=1$. Then the Kronecker difference defined below is uniform, where\n \\begin{equation*}\n A\\ominus B = \\tr_{12}(\\alpha_{m,n}^T(A\\otimes I_m-I_m\\otimes B\\otimes I_m)),\\qquad\n \\alpha_{m,n}:= \\sum_{i,j=1}^m E_{ij}\\otimes\\upsilon_n\\otimes E_{ji},\n \\end{equation*}\n for all $A\\in\\mathbb{F}_m\\otimes\\mathbb{F}_n$ and $B\\in\\mathbb{F}_n$, $m,n\\in\\mathbb{N}$.\n\\end{lemma}\n\n\\begin{theorem}\n \\label{thm:unicanon2} Let $\\ominus$ be a uniform Kronecker difference. Then there exists a sequence of matrices\n $\\upsilon_n\\in\\mathbb{F}_n$, $n\\in\\mathbb{N}$, with $\\tr(\\upsilon_n)=1$ and matrices\n $\\gamma_{m,n}\\in\\mathbb{F}_{m^2n}$, $m,n\\in\\mathbb{N}$, with $\\tr_{1}(\\gamma_{m,n})=0$ and $\\tr_{2}(\\gamma_{m,n})=0$ and\n \\begin{equation*}\n A\\ominus B = \\tr_{12}(\\alpha_{m,n}^T(A\\otimes I_m-I_m\\otimes B\\otimes I_m)),\\qquad\n \\alpha_{m,n}:= \\sum_{i,j=1}^m E_{ij}\\otimes\\upsilon_n\\otimes E_{ji} + \\gamma_{m,n},\n \\end{equation*}\n for all $A\\in\\mathbb{F}_m\\otimes\\mathbb{F}_n$ and $B\\in\\mathbb{F}_n$, $m,n\\in\\mathbb{N}$.\n\\end{theorem}\n\n\\begin{proof}\n By Remark \\ref{rmk:trivial}, for each $n\\in\\mathbb{N}$ there exists a matrix $\\upsilon_n\\in\\mathbb{F}_n$\n such that $\\tr(\\upsilon_n)=1$ and $A\\ominus B=\\tr(\\upsilon_n^T(A-B))$ for all $A,B\\in\\mathbb{F}_n$.\n Following the proof of Theorem \\ref{thm:canon2} (the choice of $E_{11}$ was arbitrary -- we\n required a matrix with unit trace), there exists $\\gamma_{m,n}\\in\\mathbb{F}_{m^2n}$ (dependent on $\\upsilon_n$) such that\n $\\tr_2(\\gamma_{m,n})=0$, and\n \\begin{equation*}\n A\\ominus B = \\tr_{12}(\\alpha_{m,n}^T(A\\otimes I_m-I_m\\otimes B\\otimes I_m)),\\qquad\n \\alpha_{m,n}:= \\sum_{i,j=1}^m E_{ij}\\otimes\\upsilon_n\\otimes E_{ji} + \\gamma_{m,n}\n \\end{equation*}\n for all $A\\in\\mathbb{F}_m\\otimes\\mathbb{F}_n$ and $B\\in\\mathbb{F}_n$, $m,n\\in\\mathbb{N}$.\n It remains to\n show that $\\tr_1(\\gamma_{m,n})=0$.\n First let us consider $B,C\\in\\mathbb{F}_n$.\n Since $\\ominus$ is uniform, and following the proof of Lemma \\ref{lem:unicanon},\n $(A\\oplus C)\\ominus B=A\\oplus (C\\ominus B)$ yields that (recalling that $\\tr_2(\\gamma_{m,n})=0$ and by use of Lemma \\ref{lem:blockpartial})\n \\begin{align*}\n \\lefteqn{\\tr_{12}(\\gamma_{m,n}^T(A\\otimes I_n\\otimes I_m+I_m\\otimes C\\otimes I_m-I_m\\otimes B\\otimes I_m))}\\qquad & \\\\\n &= \\Btr(\\tr_2(\\gamma_{m,n})^T(A\\otimes I_m))\n + \\Btr(\\tr_{1}(\\gamma_{m,n})^T ((C-B)\\otimes I_m)) \\\\\n &= \\Btr(\\tr_{1}(\\gamma_{m,n})^T ((C-B)\\otimes I_m)) \\\\\n &= 0,\n \\end{align*}\n for all $A\\in\\mathbb{F}_m$, $B,C\\in\\mathbb{F}_n$ and $m,n\\in\\mathbb{N}$. Writing $\\tr_{1}(\\gamma_{m,n})$\n in the form\n \\begin{equation*}\n \\tr_{1}(\\gamma_{m,n}) = \\sum_{i,j=1}^n E_{ij}\\otimes G_{ij}\n \\end{equation*}\n and setting $C-B=E_{kl}$ yields that\n \\begin{equation*}\n \\sum_{i,j=1}^n \\Btr((E_{ji}\\otimes G_{ij}^T)(E_{kl}\\otimes I_m))\n = \\sum_{j=1}^n \\Btr(E_{jl}\\otimes G_{kj}^T)\n = G_{kl}^T = 0.\n \\end{equation*}\n Since $j$ and $k$ were arbitrary, $\\tr_{1}(\\gamma_{m,n}) = 0$ for all $m,n\\in\\mathbb{N}.$\n \\def\\ignore{\\color{red}\n We require that\n \\begin{equation*}\n (A\\oplus C)\\ominus B= A\\oplus(C\\ominus B)=A\\oplus 0_p + 0_m\\oplus(C\\ominus B) \n \\end{equation*}\n for all $m,n,p\\in\\mathbb{N}$, $A\\in\\mathbb{F}_m$, $C\\in\\mathbb{F}_{p}\\otimes\\mathbb{F}_n$ and $B\\in\\mathbb{F}_n$.\n By Lemma \\ref{lem:unicanon} we need only satisfy\n \\begin{gather*}\n \\tr_{12}(\\gamma_{mp,n}^T(A\\otimes I_{pn}\\otimes I_{mp}+I_{m}\\otimes C\\otimes I_{mp}-I_{mp}\\otimes B\\otimes I_{mp})) \\\\\n = 0_m\\oplus(\\tr_{12}(\\gamma_{p,n}^T(C\\otimes I_p-I_p\\otimes B\\otimes I_p)))\n \\end{gather*}\n for all $A\\in\\mathbb{F}_m$, $B\\in\\mathbb{F}_n$ and\n $C\\in\\mathbb{F}_p\\otimes\\mathbb{F}_n$. Equivalently, since $\\tr_2(\\gamma_{mp,n})=0$ (over $\\mathbb{F}_{n}$) and $\\tr_1(\\gamma_{m,n})=0$ (over $\\mathbb{F}_m$),\n \\begin{equation}\n \\label{eq:unigamma}\n \\tr_{12}(\\gamma_{mp,n}^T(I_{m}\\otimes C-I_{mp}\\otimes B)\\otimes I_{mp})\n = I_m\\otimes(\\tr_{12}(\\gamma_{p,n}^T(C\\otimes I_p)))\n \\end{equation}\n for all $B\\in\\mathbb{F}_n$, and $C\\in\\mathbb{F}_p\\otimes\\mathbb{F}_n$.\n If $C=0$, we obtain for all $B\\in\\mathbb{F}_n$\n \\begin{equation}\n \\label{eq:unigammaB}\n \\tr_{12}(\\gamma_{mp,n}^T(I_{mp}\\otimes B\\otimes I_{mp})) = 0_{mp}.\n \\end{equation}\n Hence,\n \\begin{equation}\n \\label{eq:unigammaC}\n \\tr_{12}(\\gamma_{mp,n}^T(I_{m}\\otimes C\\otimes I_{mp}))\n = I_m\\otimes(\\tr_{12}(\\gamma_{p,n}^T(C\\otimes I_p)))\n \\end{equation}\n for all $C\\in\\mathbb{F}_p\\otimes\\mathbb{F}_n$.\n First, let us write $\\gamma_{mp,n}$ and $\\gamma_{p,n}$ in block form, namely\n \\begin{equation*}\n \\gamma_{p,n} = \\sum_{k,l=1}^{pn} E_{kl}\\otimes H_{kl},\\qquad\n \\gamma_{mp,n} = \\sum_{i,j=1}^{m}\\sum_{k,l=1}^{pn}E_{ij}\\otimes E_{kl}\\otimes H_{ij;kl}\n \\end{equation*}\n where each $H_{kl}\\in\\mathbb{F}_p$ and each $H_{ij;kl}\\in\\mathbb{F}_{m}\\otimes\\mathbb{F}_p$.\n It follows from \\eqref{eq:unigammaC} that\n \\begin{equation*}\n \\sum_{i,j=1}^m\\sum_{k,l=1}^{pn}\n \\tr(E_{ji})\\tr(E_{lk}C)H_{ij;kl}^T\n =\n \\sum_{k,l=1}^{pn}\\tr(E_{lk}C)I_m\\otimes H_{kl}^T\n \\end{equation*}\n for all $C\\in\\mathbb{F}_p\\otimes\\mathbb{F}_n$.\n If $C=E_{kl}\\in\\mathbb{F}_{pn}$, then\n \\begin{equation*}\n \\sum_{i=1}^m H_{ii;kl}^T= I_m\\otimes H_{kl}^T.\n \\end{equation*}\n In particular, if $p=1$ we obtain\n \\begin{equation*}\n \\tr_1(\\gamma_{m,n}) = \\gamma_{1,n}\\otimes I_m.\n \\end{equation*}\n Now \\eqref{eq:unigammaB} becomes\n \\begin{equation*}\n \\tr(\\gamma_{1,n}^TB)\\otimes I_{mp} = 0_{mp}\n \\end{equation*}\n for all $B\\in\\mathbb{F}_n$. Hence, $\\gamma_{1,n}=0$ and $\\tr_1(\\gamma_{m,n})=0$ for all $m,n\\in\\mathbb{N}$.\n }\n\\end{proof}", "post_theorem_intro_text_len": 6421, "post_theorem_intro_text": "\\begin{proof}\n Let $B,C\\in\\mathbb{F}_m$ with $C\\neq 0$. Then there exists\n an non-zero entry $(C)_{i,j}\\neq 0$ in $C$. Now let $f:\\mathbb{F}_m\\to\\mathbb{N}\\times\\mathbb{N}$\n be defined by $f(C)=(i,j)$ and $f(0_m)=(1,1)$, i.e. $f$ chooses a non-zero entry of $C$, unless\n $C$ is the zero matrix. Define $\\text{---}\\oslash C$\n as the linear extension of\n \\begin{equation*}\n  (A\\otimes B)\\oslash C := A\\dfrac{(B)_{f(C)}}{(C)_{f(C)}}\n \\end{equation*}\n for all $A\\in\\mathbb{F}_s$, $s\\in\\mathbb{N}$. Then\n $\\oslash$ is a uniform Kronecker quotient by construction.\n\\end{proof}\n\n\\subsection{Kronecker sums and Kronecker differences}\n\nThe Kronecker sum $A\\oplus B$ of an $m\\times m$ matrix $A$ and an\n$n\\times n$ matrix $B$ is given by \\cite[pp. 268]{horn91}\n\\begin{equation*}\n A\\oplus B = A\\otimes I_n + I_m\\otimes B.\n\\end{equation*}\nOver the real and complex numbers, we have\n\\begin{equation*}\n \\exp(A\\oplus B) = \\exp(A)\\otimes\\exp(B).\n\\end{equation*}\nThe Kronecker sum uniquely satisfies the above equation. The Kronecker sum\nalso arises in the study of matrix equations of the form\n\\begin{equation*}\n BX+XA^T = Y\n\\end{equation*}\nwhere $X$ is the subject of the equation. Taking the vectorization (vec)\non both sides of the equation, and using the fact that $\\vc(ABC)=(C^T\\otimes A)\\vc(B)$,\nyields \\cite[pp. 268]{horn91}\n\\begin{equation*}\n (A\\oplus B) \\vc(X) = \\vc(Y).\n\\end{equation*}\n\nThis article defines and studies Kronecker differences and their connection with Kronecker\nquotients \\cite{kronquotient}.\nSimilar to the definition of Kronecker quotients, we consider the family $\\ominus:\\mathbb{F}_{mn}\\times\\mathbb{F}_n\\to\\mathbb{F}_m$\nindexed by $m,n\\in\\mathbb{N}$ satisfying the following definition:\n\n\\begin{definition}\n \\label{def:kqdifference} A Kronecker difference is an operation $\\ominus$ satisfying\n \\begin{equation*}\n  (A\\oplus B)\\ominus B = A\n \\end{equation*}\n for all matrices $A\\in\\mathbb{F}_{m}$ and $B\\in\\mathbb{F}_{n}$,\n $m,n\\in\\mathbb{N}$.\n\\end{definition}\n\nEvery Kronecker quotient induces a Kronecker difference in the following way.\n\n\\begin{definition}\n \\label{def:kqdefineskd} Let $\\oslash$ be a Kronecker quotient. The Kronecker difference $\\ominus$ induced by $\\oslash$ is the\n Kronecker difference given by\n \\begin{equation*}\n  M\\ominus B := (M-I_m\\otimes B)\\oslash I_n\n \\end{equation*}\n for all $M\\in\\mathbb{F}_{mn}$, $B\\in\\mathbb{F}_n$ and $m,n\\in\\mathbb{N}$.\n\\end{definition}\n\n\\begin{remark}\n \\label{rem:kqdefineskd} It is easily verified that this expression satisfies the condition for a\n Kronecker difference, i.e. for every $A\\in\\mathbb{F}_m$, $B\\in\\mathbb{F}_n$ and $m,n\\in\\mathbb{N}$,\n \\begin{align*}\n  (A\\oplus B)\\ominus B\n   &= (A\\otimes I_n + I_m\\otimes B)\\ominus B \\\\\n   &= (A\\otimes I_n + I_m\\otimes B - I_m\\otimes B)\\oslash I_n \\\\\n   &= (A\\otimes I_n)\\oslash I_n \\\\\n   &= A.\n \\end{align*}\n\\end{remark}\n\n\\begin{remark}[Duality 1]\n \\label{rem:kddefineskq} Definition \\ref{def:kqdefineskd} has a weak dual, namely: Let $\\oslash$ be a Kronecker quotient such that\n \\begin{equation*}\n  M\\oslash B = (M(0_m\\oplus B)^{-1})\\ominus 0_n\n \\end{equation*}\n for all $M\\in\\mathbb{F}_{mn}$, invertible $B\\in\\mathbb{F}_n$ and $m,n\\in\\mathbb{N}$,\n where $\\ominus$ is a Kronecker difference. Indeed, for invertible $B$,\n $(0_m\\oplus B)^{-1} = (I_m\\otimes B)^{-1} = I_m\\otimes B^{-1}$ and\n \\begin{align*}\n  (A\\otimes B)\\oslash B\n   &= ((A\\otimes I_n)(I_m\\otimes B))\\oslash B \\\\\n   &= ((A\\otimes I_n)(I_m\\otimes B)(I_m\\otimes B)^{-1})\\ominus 0_n \\\\\n   &= (A\\oplus 0_n)\\ominus 0_n \\\\\n   &= A.\n \\end{align*}\n\\end{remark}\n\nThe notion of uniformity in Kronecker quotients has an analogue for Kronecker differences. \n\\begin{definition}\n \\label{def:kduniform} A Kronecker difference $\\ominus$ is a uniform Kronecker difference if\n \\begin{equation*}\n  (A\\oplus C)\\ominus B = A\\oplus (C\\ominus B)\n \\end{equation*}\n for all matrices $A\\in\\mathbb{F}_{m}$, $B\\in\\mathbb{F}_n$, $C\\in\\mathbb{F}_p\\otimes\\mathbb{F}_n$ and $m,n,p\\in\\mathbb{N}$,\n and\n \\begin{equation*}\n  (A+B)\\ominus (C+D) = (A\\ominus C)+(B\\ominus D), \\qquad\n  (kA)\\ominus (kC) = k(A\\ominus C)\n \\end{equation*}\n for all matrices $A$, $B$, $C$ and $D$, and scalars $k\\in\\mathbb{F}$ whenever the operations are defined.\n\\end{definition}\nOnce again, we immediately have a connection between uniform Kronecker quotients and uniform Kronecker differences.\n\n\\begin{proposition}\n \\label{prop:ukqdefinesukd} Let $\\oslash$ be a uniform Kronecker quotient.\n Then the induced Kronecker difference $\\ominus$ is uniform.\n\\end{proposition}\n\n\\begin{proof}\n Following the definition in Definition \\ref{def:kqdefineskd},  \\begin{equation*}\n  M\\ominus B := (M-I_m\\otimes B)\\oslash I_n\n \\end{equation*}\n for all $M\\in\\mathbb{F}_{mn}$, $B\\in\\mathbb{F}_n$ and $m,n\\in\\mathbb{N}$,\n where $\\oslash$ is a uniform Kronecker quotient.\n In Remark \\ref{rem:kqdefineskd} it was shown that $\\ominus$ is a Kronecker difference,\n it remains to show that this Kronecker difference is uniform.\n We have for all $A\\in\\mathbb{F}_{m}$, $B\\in\\mathbb{F}_n$, and $C\\in\\mathbb{F}_p\\otimes\\mathbb{F}_n$, \n \\begin{align*}\n  (A\\oplus C)\\ominus B\n   &= (A\\otimes I_p\\otimes I_n + I_m\\otimes C - I_m\\otimes I_p\\otimes B) \\oslash I_n \\\\\n   &= (A\\otimes I_p\\otimes I_n)\\oslash I_n + (I_m\\otimes C)\\oslash I_n - (I_m\\otimes I_p\\otimes B) \\oslash I_n \\\\\n   &= A\\otimes I_p + I_m\\otimes(C\\oslash I_n) - I_m\\otimes I_p\\otimes(B\\oslash I_n) \\\\\n   &= A\\otimes I_p + I_m\\otimes[(C\\oslash I_n) - I_p\\otimes (B\\oslash I_n)] \\\\\n   &= A\\otimes I_p + I_m\\otimes [(C-I_p\\otimes B)\\oslash I_n] \\\\\n   &= A\\otimes I_p + I_m\\otimes (C\\ominus B) \\\\\n   &= A\\oplus (C\\ominus B).\n \\end{align*}\n Finally, for all $A,B\\in\\mathbb{F}_{mn}$ and $C,D\\in\\mathbb{F}_n$,\n \\begin{align*}\n  (A+B)\\ominus (C+D)\n   &= (A+B-I_m\\otimes C-I_m\\otimes D)\\oslash I_n \\\\\n   &= (A-I_m\\otimes C+ B-I_m\\otimes D)\\oslash I_n \\\\\n   &= (A-I_m\\otimes C)\\oslash I_n + (B-I_m\\otimes D)\\oslash I_n \\\\\n   &= A\\ominus C + B\\ominus D\n \\end{align*}\n and for any scalar $k\\in\\mathbb{F}$,\n \\begin{align*}\n  (kA)\\ominus (kC)\n   &= (kA-I_m\\otimes(kC))\\oslash I_n \\\\\n   &= [k(A-I_m\\otimes C)]\\oslash I_n \\\\\n   &= k[(A-I_m\\otimes C)\\oslash I_n] \\\\\n   &= k(A\\ominus C). \\qedhere\n \\end{align*}\n\\end{proof}\n\nBefore studying Kronecker differences in more detail, we will need some additional definitions\nand technical results. \nThe presentations of these definitions and technicalities are left for Appendix \\ref{sec:orthogonality} and Appendix \\ref{sec:lemmata}.", "sketch": "Let $B,C\\in\\mathbb{F}_m$ with $C\\neq 0$. Choose a non-zero entry $(C)_{i,j}\\neq 0$ and define a selection map $f:\\mathbb{F}_m\\to\\mathbb{N}\\times\\mathbb{N}$ by $f(C)=(i,j)$ (and $f(0_m)=(1,1)$). Define $\\text{---}\\oslash C$ as the linear extension of\n\\[\n(A\\otimes B)\\oslash C := A\\dfrac{(B)_{f(C)}}{(C)_{f(C)}}\\quad\\text{for all }A\\in\\mathbb{F}_s.\n\\]\nThen “$\\oslash$ is a uniform Kronecker quotient by construction.”", "expanded_sketch": "Let $B,C\\in\\mathbb{F}_m$ with $C\\neq 0$. Choose a non-zero entry $(C)_{i,j}\\neq 0$ and define a selection map $f:\\mathbb{F}_m\\to\\mathbb{N}\\times\\mathbb{N}$ by $f(C)=(i,j)$ (and $f(0_m)=(1,1)$). Define $\\text{---}\\oslash C$ as the linear extension of\n\\[\n(A\\otimes B)\\oslash C := A\\dfrac{(B)_{f(C)}}{(C)_{f(C)}}\\quad\\text{for all }A\\in\\mathbb{F}_s.\n\\]\nThen $\\oslash$ is a uniform Kronecker quotient by construction.", "expanded_theorem": "\\label{lem:kqexist} Let $\\mathbb{F}$ be a field.\n Then there exists a uniform Kronecker quotient over $\\mathbb{F}$.", "theorem_type": ["Existence"], "mcq": {"question": "Let \\(\\mathbb F\\) be a field, and for each \\(m\\in\\mathbb N\\) let \\(\\mathbb F_m\\) denote the vector space of \\(m\\times m\\) matrices over \\(\\mathbb F\\). Let \\(\\otimes:\\mathbb F_m\\times\\mathbb F_n\\to\\mathbb F_{mn}\\) be the usual Kronecker product. A (right) Kronecker quotient is an operation \\(\\oslash\\) on square matrices over \\(\\mathbb F\\) such that for every \\(m,n\\in\\mathbb N\\), every \\(A\\in\\mathbb F_m\\), and every nonzero \\(B\\in\\mathbb F_n\\), one has \\((A\\otimes B)\\oslash B=A\\). It is called a uniform Kronecker quotient if, in addition, for all compatible sizes, all \\(A\\in\\mathbb F_m\\), \\(B\\in\\mathbb F_n\\), \\(C\\in\\mathbb F_p\\otimes\\mathbb F_n=\\mathbb F_{pn}\\), and all scalars \\(k\\in\\mathbb F\\),\n\\[\n(A\\otimes C)\\oslash B = A\\otimes (C\\oslash B),\n\\]\nand whenever defined,\n\\[\n(A_1+A_2)\\oslash C=(A_1\\oslash C)+(A_2\\oslash C),\\qquad (kA)\\oslash C=k(A\\oslash C).\n\\]\nUnder these assumptions, which existence statement holds?", "correct_choice": {"label": "A", "text": "There exists a uniform Kronecker quotient over \\(\\mathbb F\\); equivalently, there exists an operation \\(\\oslash\\) on square matrices over \\(\\mathbb F\\) satisfying \\((A\\otimes B)\\oslash B=A\\) for every \\(A\\in\\mathbb F_m\\) and every nonzero \\(B\\in\\mathbb F_n\\), and also satisfying the uniformity and linearity conditions above for all compatible matrix sizes."}, "choices": [{"label": "B", "text": "There exists a Kronecker quotient \\(\\oslash\\) on square matrices over \\(\\mathbb F\\) such that for every \\(m,n\\in\\mathbb N\\), every \\(A\\in\\mathbb F_m\\), and every nonzero \\(B\\in\\mathbb F_n\\), one has \\((A\\otimes B)\\oslash B=A\\); moreover, one may choose \\(\\oslash\\) so that for each fixed size \\(n\\) there is a single pair of indices \\((i_n,j_n)\\) with the property that for every nonzero \\(C\\in\\mathbb F_n\\), the quotient is obtained by reading the \\((i_n,j_n)\\)-entry of \\(C\\) and extending linearly, while still satisfying the stated uniformity and linearity conditions for all compatible sizes."}, {"label": "C", "text": "There exists a Kronecker quotient \\(\\oslash\\) over \\(\\mathbb F\\); that is, there exists an operation \\(\\oslash\\) on square matrices over \\(\\mathbb F\\) such that \\((A\\otimes B)\\oslash B=A\\) for every \\(A\\in\\mathbb F_m\\) and every nonzero \\(B\\in\\mathbb F_n\\)."}, {"label": "D", "text": "There exists an operation \\(\\oslash\\) on square matrices over \\(\\mathbb F\\) such that for every \\(m,n\\in\\mathbb N\\), every \\(A\\in\\mathbb F_m\\), and every nonzero \\(B\\in\\mathbb F_n\\), one has \\((A\\otimes B)\\oslash B=A\\), and such that the same operation is defined and satisfies the uniformity and linearity identities for all matrices \\(B\\in\\mathbb F_n\\), including \\(B=0\\)."}, {"label": "E", "text": "There exists a uniform Kronecker quotient over \\(\\mathbb F\\), and one may choose it so that for every nonzero \\(C\\in\\mathbb F_n\\) the map \\(M\\mapsto M\\oslash C\\) is given by a formula depending only on \\(C\\) and not on any auxiliary choice, namely by selecting a canonically determined nonzero entry of \\(C\\) and extending linearly, while satisfying \\((A\\otimes B)\\oslash B=A\\) for every \\(A\\in\\mathbb F_m\\) and every nonzero \\(B\\in\\mathbb F_n\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "dependence of the selected index on the matrix C itself", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "uniformity and linearity requirements", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "nonzero restriction on the divisor matrix", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "arbitrary choice function for a nonzero entry replaced by canonical choice", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the notions of Kronecker quotient and uniform Kronecker quotient but does not explicitly state that such an object exists. The correct answer is not directly leaked, though it is the most natural existential claim built from the definitions."}, "TAS": {"score": 1, "justification": "The correct option is very close to a direct existential restatement of the definition of a uniform Kronecker quotient. The presence of stronger and weaker alternatives prevents full tautology, but the item still closely mirrors the theorem statement."}, "GPS": {"score": 1, "justification": "There is some reasoning required to compare logical strength among the options, especially against the weaker true statement and the overstrengthened variants. However, the correct answer remains the most straightforward 'existence of the defined object' option, so generative pressure is only moderate."}, "DQS": {"score": 2, "justification": "The distractors are mathematically meaningful and distinct: one weakens the claim, others overstrengthen it via fixed indices, inclusion of the zero divisor, or canonical choice. These align well with common quantifier and boundary-case errors."}, "total_score": 6, "overall_assessment": "A reasonably solid MCQ with strong distractors and little direct answer leakage, but the correct option is still close to a theorem restatement, so it only moderately tests genuine generative reasoning."}}
{"id": "2602.07734v1", "paper_link": "http://arxiv.org/abs/2602.07734v1", "theorems_cnt": 2, "theorem": {"env_name": "theoremM", "content": "\\label{thmM:mp}\nLet $G/H$ be any simply connected, homogeneous space such that $T\\subset H\\subset G$ is a maximal torus, ${\\rm rank}\\ T=n$, and $G$ is a compact, connected, simple Lie group.\nThen one has \n\\[\nk(G/H)=2,3\\ \\text{or}\\ n(=\\frac{1}{2}\\dim G/H).\n\\]\nFurthermore, \n$k(G/H) = 3$ or $n$ if and only if $G/H$ is a maximal rank compact symmetric space with rank $>2$ other than $B_{n}/D_{i}\\times B_{n-i}$, $C_{n}/A_{n-1}\\times T^{1}$, $F_{4}/C_{3}\\times A_{1}$.", "start_pos": 5918, "end_pos": 6415, "label": "thmM:mp"}, "ref_dict": {"ex:origra": "\\begin{example}\\label{ex:origra}\nConsider the homogeneous manifold\n\\[\nGr^{+}(2,n)=SO(n)/SO(n-2)\\times SO(2)=Q_{n},\n\\]\nwith the natural effective $T$-action.\nHere, if $n=2r$ then $T=T^{r}/{\\pm 1}$, and if $n=2r+1$ then $T=T^{r}$.\nThis oriented Grassmanian is a connected double cover of the real Grassmanian $Gr(2,n)$.\nThis implies $Gr^{+}(2,3)=S^3$.\nIt is not difficult to show that $Gr^+(2,4)$ is homeomorphic to $S^2\\times S^2$.\n(Furthermore, $Gr^+(2,6)$ is homeomorphic to the complex Grassmanian $Gr_{\\C}(2,4)$.)\nThe orbit space is homeomorphic to $\\Sigma^{\\lfloor n/2\\rfloor} \\C P^{\\lfloor n/2\\rfloor-2}$ by~\\cite{S19}.\nIn particular, the orbit space is ${\\lfloor n/2\\rfloor}$-connected (where the suspension of an empty set is by definition a point).\nIn particular, it is contractible for $n=3,4$.\nThis homogeneous manifold has type $D_{r}/D_{r-1}\\times T^{1}$ for $n=2r$ and $B_{r}/B_{r-1}\\times T^1$ for $n=2r+1$, where $r\\geq 2$.\nHere, $B_1=A_1$, $D_1=T^{1}$, and $D_2=(A_1)^2$.\nSince $Gr^{+}(2,2r)=Q_{2r}$ and $Gr^{+}(2,2r+1)=Q_{2r+1}$, \nIt is known that $H^{odd}(Q_{n};\\mathbb{Z})=0$.\nTherefore, Theorem~\\ref{thm:AM} implies that the orbit space is $4$-acyclic for $n=2r$ and $3$-acyclic for $n=2r+1$.\nTherefore, our result agrees with previous known results, in particular, in dimensions $n=6,7,8$.\n\\end{example}", "ex:comgra": "\\begin{example}\\label{ex:comgra}\nConsider the homogeneous manifold\n\\[\nGr(2,5)=SU(5)/S(U(2)\\times U(3)),\n\\]\nwith the natural effective $T^{4}$-action.\nThis is the complex Grassmanian of two-planes in $\\C^5$.\nThe fixed-point data at the identity is a root system of type $A_{4}/A_{1}\\times A_{2}\\times T^{1}$.\nBy Theorems~\\ref{main} and Corollary~\\ref{cor:acyc},\nthis GKM-manifold has $4$-acyclic orbit space $Gr(2,5)/T^4$.\nOn the other hand, by~\\cite{BT} or~\\cite{S21},\n\\[\n\\widetilde{H}_{*}(Gr(2,5)/T^4;\\Z)\\simeq\n\\begin{cases}\n\\Z/2\\Z,\\ n=5,\\\\\n\\Z,\\ n=8,\\\\\n0,\\ \\mbox{else}.\n\\end{cases}\n\\]\nThis agrees with the above acyclicity claim.\n\\end{example}", "cor:acyc": "\\begin{corollary}\\label{cor:acyc}\nIf $G/H$ is a symmetric GKM$_{3}$ space (see Theorem~\\ref{thm:cart} and (1), (2) of Theorem~\\ref{main})\nwith the following coefficient $\\Bbbk$ for the reduced homology, then $\\widetilde{H}_{i}(T\\backslash G/H;\\Bbbk)=0$ for $i\\le 4$.\n\nIf $\\Bbbk=\\mathbb{Z}$, the the list is as follows:\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|} \\hline\nlabel & pair & $G$ & $H$ &  \\\\ \\hline\n   {\\rm AIII} & $(A_{n},A_{i}\\times A_{n-1-i}\\times T^{1})$ & $SU(n+1)$ & $S(U(i+1)\\times U(n-i))$ & $0\\le i\\le n-1$ \\\\\n   {\\rm BDI}    & $(D_{n}, D_{n-1}\\times T^{1})$ & $SO(2n)$ & $SO(2n-2)\\times SO(2)$ &    \\\\\n    {\\rm DIII} & $(D_{n}, A_{n-1}\\times T^{1})$ & $SO(2n)$ & $U(n)$ & \\\\\n    {\\rm CII} & $(C_{n},C_{i}\\times C_{n-i})$ & $Sp(n)$ & $Sp(i)\\times Sp(n-i)$ & $1\\le i\\le n-1$ \\\\ \n    {\\rm EIII} & $(E_{6}, D_{5}\\times T^{1})$ & $E_{6}$ & $(Spin(10)\\times T^{1})/\\mathbb{Z}_{4}$ & \\\\\n    {\\rm EVII} & $(E_{7},E_{6}\\times T^{1})$ & $E_{7}$ & $(E_{6}\\times T^{1})/\\mathbb{Z}_{3}$ & \\\\\n   {\\rm FII} & $(F_{4}, B_{4})$ & $F_{4}$ & $Spin(9)$ & \\\\\n   \\hline\n \\end{tabular}\n\\end{center}\n\nIf $\\Bbbk=\\mathbb{Z}[\\frac{1}{2}]$, the the list is as follows: \n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|} \n\\hline\nlabel & pair  & $G$  & $H$  &  \\\\ \\hline\n  {\\rm BDI}  & $(D_{n}, D_{i}\\times D_{n-i})$ & $SO(2n)$ & $SO(2i)\\times SO(2n-2i)$ & $i>2$,   $n-i>2$  \\\\\n    {\\rm EII} & $(E_{6}, A_{5}\\times A_{1})$ & $E_{6}$ & $(SU(6)\\times SU(2))/\\mathbb{Z}_{2}$ & \\\\\n      {\\rm EV} & $(E_{7},A_{7})$ & $E_{7}$ & $SU(8)/\\mathbb{Z}_{2}$ & \\\\\n      {\\rm EVI} & $(E_{7},D_{6}\\times A_{1})$ & $E_{7}$ & $(Spin(12)\\times SU(2))/\\mathbb{Z}_{2}$ & \\\\\n    {\\rm EIX} & $(E_{8},E_{7}\\times A_{1})$ & $E_{8}$ & $(E_{7}\\times SU(2))/\\mathbb{Z}_{2}$ & \\\\\n   \\hline\n \\end{tabular}\n \\end{center}\n\\end{corollary}", "split_computation": "\\begin{proposition}\n\\label{split_computation}\nAssume that $G/H$ is not a torus manifold. \nThen, there is the following equality for the splitting \\eqref{splitting}:\n\\begin{align*}\nk(G/H)=\\min \\{k(G_{i}/H_{i})\\ |\\ 1\\le i\\le r \\}.\n\\end{align*}\n\\end{proposition}", "main": "\\begin{theorem}\n\\label{main}\nAssume that $G$ is simply connected, simple Lie group and $H$ be its maximal rank \nsubgroup such that ${\\rm rank}(G)={\\rm rank}(H)=\\dim T=n$.\nThen, $k(G/H)=2,3,n$ and the following holds:\n\\begin{enumerate}\n\\item $k(G/H)=n={\\rm rank}(G)={\\rm rank}(H)$ if and only if $(G,H)$ is one of the following types:\n\\begin{description}\n\\item[Type $A_{n}$] $(A_{n},A_{n-1}\\times T^{1})$;\n\\item[Type $B_{n}$] $(B_{n}, D_{n})$.\n\\end{description}\n\\item $k(G/H)=3(\\not=n)$ if and only if \n$(G,H)$ is one of the pairs appered in Theorem \\ref{thm:cart} except for the above two cases, $(B_{n},D_{i}\\times B_{n-i})$, $(C_{n},A_{n-1}\\times T^{1})$ and $(F_{4}, C_{3}\\times A_{1})$.\n\\item $k(G/H)=2$, otherwise (all of the other cases). \n\\end{enumerate}\n\\end{theorem}", "lm:23dep": "\\begin{lemma}\\label{lm:23dep}\nLet $G$ be a simple Lie group of type $A$--$D$, and $H$ be\nits proper maximal connected (closed) maximal rank subgroup.\nThen one has the following:\n\\begin{enumerate}\n\\item $k(G/H)=2$ if and only if the weighted signed graph $\\Gamma(G,H)$ has a subgraph (with weighted vertices) appeared in Figure~\\ref{example_weighted_vertex}, where $i<j<k$.\n\\begin{figure}[H]\n\\begin{tikzpicture}\n\\begin{scope}[xscale=0.5, yscale=0.5]\n\nll(-2,-2)circle (5pt);\nll(-2,2)circle (5pt);\nll(2,-2)circle (5pt);\n\\draw[thick](-2,-2)--(-2,2);\n\\node[above] at (-2,2) {$w_{V}(i)$};\n\\draw[thick](2,-2)--(-2,2);\n\\node[below] at (-2,-2) {$w_{V}(j)$};\n\\draw[thick](-2,-2)--(2,-2);\n\\node[below] at (2,-2) {$w_{V}(k)$};\n\nll(5,2)circle (5pt);\nll(5,-2)circle (5pt);\nll(9,-2)circle (5pt);\n\\draw[dashed](5,-2)--(5,2);\n\\node[above] at (5,2) {$w_{V}(i)$};\n\\draw[dashed](9,-2)--(5,2);\n\\node[below] at (5,-2) {$w_{V}(j)$};\n\\draw[thick](9,-2)--(5,-2);\n\\node[below] at (9,-2) {$w_{V}(k)$};\n\nll(14,2)circle (5pt);\nll(14,-2)circle (5pt);\n\\draw[thick](14,-2)to[out=75,in=-75](14,2);\n\\draw[dashed](14,-2)to[out=105,in=-105](14,2);\n\\draw[thick](14,2.7) circle [x radius=0.7cm, y radius=0.7cm];\n\\node[above] at (14,3.4) {$w_{V}(i)$}; \n\\node[below] at (14,-2) {$w_{V}(j)$};\n\nll(19,2)circle (5pt);\nll(19,-2)circle (5pt);\n\\draw[dashed](19,-2)to[out=105,in=-105](19,2);\n\\draw[thick](19,2.7) circle [x radius=0.7cm, y radius=0.7cm];\n\\node[above] at (19,3.4) {$w_{V}(i)$};\n\\draw[thick](19,-2.7) circle [x radius=0.7cm, y radius=0.7cm];\n\\node[below] at (19,-3.4) {$w_{V}(j)$};\n\nll(23,2)circle (5pt);\nll(23,-2)circle (5pt);\n\\draw[thick](23,-2)to[out=75,in=-75](23,2);\n\\draw[thick](23,2.7) circle [x radius=0.7cm, y radius=0.7cm];\n\\node[above] at (23,3.4) {$w_{V}(i)$};\n\\draw[thick](23,-2.7) circle [x radius=0.7cm, y radius=0.7cm];\n\\node[below] at (23,-3.4) {$w_{V}(j)$};\n\n\\end{scope}\n\\end{tikzpicture}\n\\caption{$2$-independent subgraphs.} \n\\label{example_weighted_vertex}\n\\end{figure}\n\\item $k(G/H)=3$ if the weighted signed graph $\\Gamma(G,H)$ has neither of the subgraphs (with weighted vertices) depicted in Figure~\\ref{example_weighted_vertex} but has a weighted subgraph appeared in  Figure~\\ref{example_weighted_vertex2}, where $i<j<k<l$.\n\\begin{figure}[H]\n\\begin{tikzpicture}\n\\begin{scope}[xscale=0.5, yscale=0.5]\n\nll(-2,-2)circle (5pt);\nll(-2,2)circle (5pt);\nll(2,-2)circle (5pt);\nll(2,2)circle (5pt);\n\\draw[thick](-2,-2)--(-2,2)--(2,2)--(2,-2)--cycle;\n\\node[above] at (-2,2) {$w_{V}(i)$};\n\\node[below] at (-2,-2) {$w_{V}(j)$};\n\\node[below] at (2,-2) {$w_{V}(k)$};\n\\node[above] at (2,2) {$w_{V}(l)$};\n\n\\coordinate (A) at (5,2);\n\\coordinate (B) at (5,-2);\n\\coordinate (C) at (9,2);\nll(A)circle (5pt);\nll(B)circle (5pt);\nll(C)circle (5pt);\n\\draw[thick](A)to[out=240,in=120](B);\n\\node[above] at (A) {$w_{V}(i)$};\n\\draw[dashed](A)to[out=300,in=60](B);\n\\node[below] at (B) {$w_{V}(j)$};\n\\draw[thick](A)to[out=330,in=210](C);\n\\node[above] at (C) {$w_{V}(k)$};\n\\draw[dashed](A)to[out=30,in=150](C);\n\n\\coordinate (a) at (12,2);\n\\coordinate (b) at (12,-2);\n\\coordinate (c) at (16,-2);\n\\coordinate (d) at (16,2);\nll(a)circle (5pt);\nll(b)circle (5pt);\nll(c)circle (5pt);\nll(d)circle (5pt);\n\n\\draw[dashed] (a)--(b)--(d)--(c)--(a);\n\n\\node[above] at (a) {$w_{V}(i)$};\n\\node[below] at (b) {$w_{V}(j)$};\n\\node[below] at (c) {$w_{V}(k)$};\n\\node[above] at (d) {$w_{V}(l)$};\n\n\\end{scope}\n\\end{tikzpicture}\n\\caption{$3$-independent subgraphs.}\n\\label{example_weighted_vertex2}\n\\end{figure}\n\\end{enumerate}\n\\end{lemma}", "auxiliary lemma": "\\begin{lemma}\\label{auxiliary lemma}\nUnder the hypothesis in Theorem~\\ref{main}, \nif $k(G/H)\\ge 3$, then  $G/H$ is a symmetric space given in Theorem~\\ref{thm:cart}.\n\\end{lemma}", "Masuda-problem": "\\begin{probM}\n\\label{Masuda-problem}\nLet $M$ be an equivariantly formal (i.e., $H^{odd}(M)=0$) $2m$-dimensional GKM$_4$ manifold with a $T$-action.\nIs $M$ homeomorphic to a torus manifold, and is the given $T$-action obtained as the restriction of a half-dimensional torus $T^{m}$-action to a subtorus $T$?\n\\end{probM}", "thmM:mp": "\\begin{theoremM}\\label{thmM:mp}\nLet $G/H$ be any simply connected, homogeneous space such that $T\\subset H\\subset G$ is a maximal torus, ${\\rm rank}\\ T=n$, and $G$ is a compact, connected, simple Lie group.\nThen one has \n\\[\nk(G/H)=2,3\\ \\text{or}\\ n(=\\frac{1}{2}\\dim G/H).\n\\]\nFurthermore, \n$k(G/H) = 3$ or $n$ if and only if $G/H$ is a maximal rank compact symmetric space with rank $>2$ other than $B_{n}/D_{i}\\times B_{n-i}$, $C_{n}/A_{n-1}\\times T^{1}$, $F_{4}/C_{3}\\times A_{1}$.\n\\end{theoremM}"}, "pre_theorem_intro_text_len": 2190, "pre_theorem_intro_text": "\\label{sect:1}\n\n\\subsection{Motivation and Main theorem}\n\\label{sect:1.1}\n\nLet $M$ be a $2m$-dimensional, smooth $T$-manifold with a nonempty set of isolated fixed points $M^{T}$, where $T\\cong (S^{1})^{n}$ and we assume that the $T$-action is {\\it almost effective}, i.e., the kernel of the $T$-action is finite.\nThere is the following irreducible decomposition \n\\begin{align}\n\\label{cpx_on_p}\nT_{p}M\\cong \\bigoplus_{i=1}^{m}V_{\\alpha_{p,i}},\n\\end{align}\nof the isotropy representation at any fixed point $p\\in M^{T}$, where $V_{\\alpha_{p,i}}$ is the (complex) irreducible representation space of $T$, and $\\alpha_{p,i}\\in \\mathfrak{t}_{\\mathbb{Z}}^{*}/\\pm 1$ for $i=1,\\ldots, m$ is determined up to a sign.\nHere, $\\mathfrak{t}_{\\mathbb{Z}}^{*}\\cong \\Z^{n}$ denotes the character lattice of \nthe dual of the Lie algebra $\\mathfrak{t}^{*}$ of $T$.\nSet $S_{p}:=\\{\\alpha_{p,i}\\ |\\ i=1,\\ldots, m\\}\\subset \\mathbb{Z}^{n}$ and define the following numbers:\n\\begin{align}\n\\label{independency}\nk(p):=\\max\\{q\\ |\\ \\text{every $q$ elements in $S_{p}$ are linearly independent}\\},\\ \nk(M):=\\min\\limits_{p\\in M^{T}} k(p).\n\\end{align}\nIf $2\\le k(M)$, then this manifold is called a \\textit{GKM manifold}~\\cite{GKM},~\\cite{GZ}. \nA GKM manifold is called a \\textit{$j$-independent} or \\textit{GKM$_{j}$) manifold} if $k(M)\\geq j$ holds (see \\cite{AMS, GS}. In \\cite{AM}, this is called a {\\it $j$-generality}). \nIf $k(M)=m$ then $n=m$; in such case this manifold is called a {\\it torus manifold}~\\cite{MP},~\\cite{HaMa}.  \nMikiya Masuda has asked the following question to the first named author: \n\\begin{probM}\n\\label{Masuda-problem}\nLet $M$ be an equivariantly formal (i.e., $H^{odd}(M)=0$) $2m$-dimensional GKM$_4$ manifold with a $T$-action.\nIs $M$ homeomorphic to a torus manifold, and is the given $T$-action obtained as the restriction of a half-dimensional torus $T^{m}$-action to a subtorus $T$?\n\\end{probM}\n\nThe main result of the present paper is given as follows (see Theorem \\ref{main}), where by a {\\it maximal rank compact symmetric space} we mean a compact symmetric space $G/H$ that is a homogeneous space of maximal rank (also referred to as a {\\it class 1 symmetric space of inner type}).", "context": "\\label{sect:1}\n\n\\subsection{Motivation and Main theorem}\n\\label{sect:1.1}\n\nLet $M$ be a $2m$-dimensional, smooth $T$-manifold with a nonempty set of isolated fixed points $M^{T}$, where $T\\cong (S^{1})^{n}$ and we assume that the $T$-action is {\\it almost effective}, i.e., the kernel of the $T$-action is finite.\nThere is the following irreducible decomposition \n\\begin{align}\n\\label{cpx_on_p}\nT_{p}M\\cong \\bigoplus_{i=1}^{m}V_{\\alpha_{p,i}},\n\\end{align}\nof the isotropy representation at any fixed point $p\\in M^{T}$, where $V_{\\alpha_{p,i}}$ is the (complex) irreducible representation space of $T$, and $\\alpha_{p,i}\\in \\mathfrak{t}_{\\mathbb{Z}}^{*}/\\pm 1$ for $i=1,\\ldots, m$ is determined up to a sign.\nHere, $\\mathfrak{t}_{\\mathbb{Z}}^{*}\\cong \\Z^{n}$ denotes the character lattice of \nthe dual of the Lie algebra $\\mathfrak{t}^{*}$ of $T$.\nSet $S_{p}:=\\{\\alpha_{p,i}\\ |\\ i=1,\\ldots, m\\}\\subset \\mathbb{Z}^{n}$ and define the following numbers:\n\\begin{align}\n\\label{independency}\nk(p):=\\max\\{q\\ |\\ \\text{every $q$ elements in $S_{p}$ are linearly independent}\\},\\ \nk(M):=\\min\\limits_{p\\in M^{T}} k(p).\n\\end{align}\nIf $2\\le k(M)$, then this manifold is called a \\textit{GKM manifold}~\\cite{GKM},~\\cite{GZ}. \nA GKM manifold is called a \\textit{$j$-independent} or \\textit{GKM$_{j}$) manifold} if $k(M)\\geq j$ holds (see \\cite{AMS, GS}. In \\cite{AM}, this is called a {\\it $j$-generality}). \nIf $k(M)=m$ then $n=m$; in such case this manifold is called a {\\it torus manifold}~\\cite{MP},~\\cite{HaMa}. \nMikiya Masuda has asked the following question to the first named author: \n\\begin{probM}\n\\label{Masuda-problem}\nLet $M$ be an equivariantly formal (i.e., $H^{odd}(M)=0$) $2m$-dimensional GKM$_4$ manifold with a $T$-action.\nIs $M$ homeomorphic to a torus manifold, and is the given $T$-action obtained as the restriction of a half-dimensional torus $T^{m}$-action to a subtorus $T$?\n\\end{probM}\n\nThe main result of the present paper is given as follows (see Theorem \\ref{main}), where by a {\\it maximal rank compact symmetric space} we mean a compact symmetric space $G/H$ that is a homogeneous space of maximal rank (also referred to as a {\\it class 1 symmetric space of inner type}).\n\n\\begin{theorem}\n\\label{main}\nAssume that $G$ is simply connected, simple Lie group and $H$ be its maximal rank \nsubgroup such that ${\\rm rank}(G)={\\rm rank}(H)=\\dim T=n$.\nThen, $k(G/H)=2,3,n$ and the following holds:\n\\begin{enumerate}\n\\item $k(G/H)=n={\\rm rank}(G)={\\rm rank}(H)$ if and only if $(G,H)$ is one of the following types:\n\\begin{description}\n\\item[Type $A_{n}$] $(A_{n},A_{n-1}\\times T^{1})$;\n\\item[Type $B_{n}$] $(B_{n}, D_{n})$.\n\\end{description}\n\\item $k(G/H)=3(\\not=n)$ if and only if \n$(G,H)$ is one of the pairs appered in Theorem \\ref{thm:cart} except for the above two cases, $(B_{n},D_{i}\\times B_{n-i})$, $(C_{n},A_{n-1}\\times T^{1})$ and $(F_{4}, C_{3}\\times A_{1})$.\n\\item $k(G/H)=2$, otherwise (all of the other cases). \n\\end{enumerate}\n\\end{theorem}", "full_context": "\\label{sect:1}\n\n\\subsection{Motivation and Main theorem}\n\\label{sect:1.1}\n\nLet $M$ be a $2m$-dimensional, smooth $T$-manifold with a nonempty set of isolated fixed points $M^{T}$, where $T\\cong (S^{1})^{n}$ and we assume that the $T$-action is {\\it almost effective}, i.e., the kernel of the $T$-action is finite.\nThere is the following irreducible decomposition \n\\begin{align}\n\\label{cpx_on_p}\nT_{p}M\\cong \\bigoplus_{i=1}^{m}V_{\\alpha_{p,i}},\n\\end{align}\nof the isotropy representation at any fixed point $p\\in M^{T}$, where $V_{\\alpha_{p,i}}$ is the (complex) irreducible representation space of $T$, and $\\alpha_{p,i}\\in \\mathfrak{t}_{\\mathbb{Z}}^{*}/\\pm 1$ for $i=1,\\ldots, m$ is determined up to a sign.\nHere, $\\mathfrak{t}_{\\mathbb{Z}}^{*}\\cong \\Z^{n}$ denotes the character lattice of \nthe dual of the Lie algebra $\\mathfrak{t}^{*}$ of $T$.\nSet $S_{p}:=\\{\\alpha_{p,i}\\ |\\ i=1,\\ldots, m\\}\\subset \\mathbb{Z}^{n}$ and define the following numbers:\n\\begin{align}\n\\label{independency}\nk(p):=\\max\\{q\\ |\\ \\text{every $q$ elements in $S_{p}$ are linearly independent}\\},\\ \nk(M):=\\min\\limits_{p\\in M^{T}} k(p).\n\\end{align}\nIf $2\\le k(M)$, then this manifold is called a \\textit{GKM manifold}~\\cite{GKM},~\\cite{GZ}. \nA GKM manifold is called a \\textit{$j$-independent} or \\textit{GKM$_{j}$) manifold} if $k(M)\\geq j$ holds (see \\cite{AMS, GS}. In \\cite{AM}, this is called a {\\it $j$-generality}). \nIf $k(M)=m$ then $n=m$; in such case this manifold is called a {\\it torus manifold}~\\cite{MP},~\\cite{HaMa}. \nMikiya Masuda has asked the following question to the first named author: \n\\begin{probM}\n\\label{Masuda-problem}\nLet $M$ be an equivariantly formal (i.e., $H^{odd}(M)=0$) $2m$-dimensional GKM$_4$ manifold with a $T$-action.\nIs $M$ homeomorphic to a torus manifold, and is the given $T$-action obtained as the restriction of a half-dimensional torus $T^{m}$-action to a subtorus $T$?\n\\end{probM}\n\nThe main result of the present paper is given as follows (see Theorem \\ref{main}), where by a {\\it maximal rank compact symmetric space} we mean a compact symmetric space $G/H$ that is a homogeneous space of maximal rank (also referred to as a {\\it class 1 symmetric space of inner type}).\n\n\\begin{theorem}\n\\label{main}\nAssume that $G$ is simply connected, simple Lie group and $H$ be its maximal rank \nsubgroup such that ${\\rm rank}(G)={\\rm rank}(H)=\\dim T=n$.\nThen, $k(G/H)=2,3,n$ and the following holds:\n\\begin{enumerate}\n\\item $k(G/H)=n={\\rm rank}(G)={\\rm rank}(H)$ if and only if $(G,H)$ is one of the following types:\n\\begin{description}\n\\item[Type $A_{n}$] $(A_{n},A_{n-1}\\times T^{1})$;\n\\item[Type $B_{n}$] $(B_{n}, D_{n})$.\n\\end{description}\n\\item $k(G/H)=3(\\not=n)$ if and only if \n$(G,H)$ is one of the pairs appered in Theorem \\ref{thm:cart} except for the above two cases, $(B_{n},D_{i}\\times B_{n-i})$, $(C_{n},A_{n-1}\\times T^{1})$ and $(F_{4}, C_{3}\\times A_{1})$.\n\\item $k(G/H)=2$, otherwise (all of the other cases). \n\\end{enumerate}\n\\end{theorem}\n\n\\begin{abstract}\nLet $G/H$ be a simply connected homogeneous space of maximal rank. \nThen the maximal torus $T$-action on $G/H$ is a GKM manifold.\nWe call the $T$-action $j$-independent if any $i(\\leq j)$ pairwise distinct isotropy weights at a fixed point are linearly independent.\nUsing weighted graphs, we show that the maximal independence of $G/H$ is $2$, $3$ or $n=\\dim T$, and that the cases of $3$ or $n=\\dim T$ correspond to some symmetric spaces of rank $>2$.\nAs a corollary, \nusing the results of Ayzenberg and Masuda, the lower-degree reduced homology groups (with appropriate coefficients) of the orbit space $T\\backslash G/H$ vanish.\n\\end{abstract}\n\nLet $M$ be a $2m$-dimensional, smooth $T$-manifold with a nonempty set of isolated fixed points $M^{T}$, where $T\\cong (S^{1})^{n}$ and we assume that the $T$-action is {\\it almost effective}, i.e., the kernel of the $T$-action is finite.\nThere is the following irreducible decomposition \n\\begin{align}\n\\label{cpx_on_p}\nT_{p}M\\cong \\bigoplus_{i=1}^{m}V_{\\alpha_{p,i}},\n\\end{align}\nof the isotropy representation at any fixed point $p\\in M^{T}$, where $V_{\\alpha_{p,i}}$ is the (complex) irreducible representation space of $T$, and $\\alpha_{p,i}\\in \\mathfrak{t}_{\\mathbb{Z}}^{*}/\\pm 1$ for $i=1,\\ldots, m$ is determined up to a sign.\nHere, $\\mathfrak{t}_{\\mathbb{Z}}^{*}\\cong \\Z^{n}$ denotes the character lattice of \nthe dual of the Lie algebra $\\mathfrak{t}^{*}$ of $T$.\nSet $S_{p}:=\\{\\alpha_{p,i}\\ |\\ i=1,\\ldots, m\\}\\subset \\mathbb{Z}^{n}$ and define the following numbers:\n\\begin{align}\n\\label{independency}\nk(p):=\\max\\{q\\ |\\ \\text{every $q$ elements in $S_{p}$ are linearly independent}\\},\\ \nk(M):=\\min\\limits_{p\\in M^{T}} k(p).\n\\end{align}\nIf $2\\le k(M)$, then this manifold is called a \\textit{GKM manifold}~\\cite{GKM},~\\cite{GZ}. \nA GKM manifold is called a \\textit{$j$-independent} or \\textit{GKM$_{j}$) manifold} if $k(M)\\geq j$ holds (see \\cite{AMS, GS}. In \\cite{AM}, this is called a {\\it $j$-generality}). \nIf $k(M)=m$ then $n=m$; in such case this manifold is called a {\\it torus manifold}~\\cite{MP},~\\cite{HaMa}. \nMikiya Masuda has asked the following question to the first named author: \n\\begin{probM}\n\\label{Masuda-problem}\nLet $M$ be an equivariantly formal (i.e., $H^{odd}(M)=0$) $2m$-dimensional GKM$_4$ manifold with a $T$-action.\nIs $M$ homeomorphic to a torus manifold, and is the given $T$-action obtained as the restriction of a half-dimensional torus $T^{m}$-action to a subtorus $T$?\n\\end{probM}\n\nAs an application of Theorem~\\ref{thmM:mp}, together with the acyclicity results of~\\cite{AMS},~\\cite{AM}, we obtain vanishing (in degrees $\\leq 4$) for reduced homology with an appropriate coefficients of the orbit spaces for homogeneous GKM manifolds, see Corollary \\ref{cor:acyc}.\nOur results are consistent with those in~\\cite{S19},~\\cite{S21},~\\cite{BT}, while providing new insights, particularly for several symmetric spaces.\n\n\\subsection{The torus action on the homogeneous space $G/H$}\n\\label{sect:2.3}\nLet $G$ be a compact, connected, semi-simple Lie group, i.e., $\\pi_{1}(G)$ is finite, and \n$T$ be its maximal torus, and $H$ be a closed, connected subgroup of $G$ such that $T\\subseteq H\\subset G$, i.e., a {\\it maximal rank subgroup} of $G$.\nIt is known in this case that \nthe homogeneous space $G/H$ is a $2m$-dimensional, simply connected manifold satisfying $H^{odd}(G/H;\\mathbb{Q})=0$, see e.g.~\\cite{MiTo, Bu} and~\\cite[Theorem 2.4]{GHZ}.\nBelow we recall the description of the tangential weights of the $T$-action on $G/H$ by following these references.\n\n\\begin{proposition}\n\\label{k-independence}\nLet $G$ be a compact, connected semi-simple Lie group, $H$ be its maximal rank subgroup and $T\\subseteq H\\subset G$ be a maximal torus. \nThen $k(G/H)=k(gH)$ for any $gH\\in (G/H)^{T}$.\n\\end{proposition}\nThis proposition shows that if we compute the $k(G/H)$-independence on $\\Delta_{G}^{+}\\setminus \\Delta_{H}^{+}\\simeq \\Delta_{G,H}/\\{\\pm 1\\}$, then we have the $k(G/H)$-independence of all tangent spaces on the fixed points of $G/H$\n(Cf.~Example~\\ref{ex_depends_k(p)}).\nIn this paper, we determine the $k$-independence for all maximal rank homogeneous spaces $G/H$.\n\n\\begin{theorem}\n\\label{thm:cart}\nLet $G/H$ be a maximal rank compact symmetric spaces with rank $>2$.\nThen, the pair $(G,H)$ corresponds to one of the following list.\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|} \\hline\n label & pair & $G$ & $H$ & \\\\ \\hline\n {\\rm AIII} & $(A_{n},A_{i}\\times A_{n-1-i}\\times T^{1})$ & $SU(n+1)$ & $S(U(i+1)\\times U(n-i))$ & $0\\le i\\le n-1$ \\\\\n {\\rm BDI} & $(B_{n}, D_{i}\\times B_{n-i})$ & $SO(2n+1)$ & $SO(2i)\\times SO(2n-2i+1)$ & $1\\le i\\le n$ \\\\ \n & $(D_{n}, D_{i}\\times D_{n-i})$ & $SO(2n)$ & $SO(2i)\\times SO(2n-2i)$ & $1\\le i\\le n-1$ \\\\\n {\\rm DIII} & $(D_{n}, A_{n-1}\\times T^{1})$ & $SO(2n)$ & $U(n)$ & \\\\\n {\\rm CI} & $(C_{n}, A_{n-1}\\times T^{1})$ & $Sp(n)$ & $U(n)$ & \\\\\n {\\rm CII} & $(C_{n},C_{i}\\times C_{n-i})$ & $Sp(n)$ & $Sp(i)\\times Sp(n-i)$ & $1\\le i\\le n-1$ \\\\ \n {\\rm EII} & $(E_{6}, A_{5}\\times A_{1})$ & $E_{6}$ & $(SU(6)\\times SU(2))/\\mathbb{Z}_{2}$ & \\\\\n {\\rm EIII} & $(E_{6}, D_{5}\\times T^{1})$ & $E_{6}$ & $(Spin(10)\\times T^{1})/\\mathbb{Z}_{4}$ & \\\\\n {\\rm EV} & $(E_{7},A_{7})$ & $E_{7}$ & $SU(8)/\\mathbb{Z}_{2}$ & \\\\\n {\\rm EVI} & $(E_{7},D_{6}\\times A_{1})$ & $E_{7}$ & $(Spin(12)\\times SU(2))/\\mathbb{Z}_{2}$ & \\\\\n {\\rm EVII} & $(E_{7},E_{6}\\times T^{1})$ & $E_{7}$ & $(E_{6}\\times T^{1})/\\mathbb{Z}_{3}$ & \\\\\n {\\rm EVIII} & $(E_{8},D_{8})$ & $E_{8}$ & $Ss(16)$ & \\\\\n {\\rm EIX} & $(E_{8},E_{7}\\times A_{1})$ & $E_{8}$ & $(E_{7}\\times SU(2))/\\mathbb{Z}_{2}$ & \\\\\n {\\rm FI} & $(F_{4}, C_{3}\\times A_{1})$ & $F_{4}$ & $(Sp(3)\\times Sp(1))/\\mathbb{Z}_{2}$ & \\\\\n {\\rm FII} & $(F_{4}, B_{4})$ & $F_{4}$ & $Spin(9)$ & \\\\\n \\hline\n \\end{tabular}\n\\end{center}\n\n\\subsection{Main theorem}\n\\label{sect:3.1}\nIf we assume $G$ is simply connected, then the standard $T$-action on the simply connected homogeneous space $G/H$ is equivariantly diffeomorphic to the following product of the homogeneous spaces (see e.g.~\\cite[Section 2.2]{Ku10}):\n\\begin{align}\n\\label{splitting}\nG/H\\cong G_{1}/H_{1}\\times \\cdots \\times G_{r}/H_{r},\n\\end{align}\nwhere $G_{i}$ is a compact, simply connected simple Lie group, $H_{i}$ be its maximal rank subgroup and there is an isomorphism $T\\cong T_{1}\\times \\cdots \\times T_{r}$ such that $T_{i}\\subset H_{i}\\subset G_{i}$ is a maximal torus for $i=1,\\ldots, r$. \nMoreover, we can easily show the following proposition. (Recall that a torus manifold is equipped with a half-dimensional torus action.)\n\\begin{proposition}\n\\label{split_computation}\nAssume that $G/H$ is not a torus manifold. \nThen, there is the following equality for the splitting \\eqref{splitting}:\n\\begin{align*}\nk(G/H)=\\min \\{k(G_{i}/H_{i})\\ |\\ 1\\le i\\le r \\}.\n\\end{align*}\n\\end{proposition}\n\n\\begin{theorem}\n\\label{main}\nAssume that $G$ is simply connected, simple Lie group and $H$ be its maximal rank \nsubgroup such that ${\\rm rank}(G)={\\rm rank}(H)=\\dim T=n$.\nThen, $k(G/H)=2,3,n$ and the following holds:\n\\begin{enumerate}\n\\item $k(G/H)=n={\\rm rank}(G)={\\rm rank}(H)$ if and only if $(G,H)$ is one of the following types:\n\\begin{description}\n\\item[Type $A_{n}$] $(A_{n},A_{n-1}\\times T^{1})$;\n\\item[Type $B_{n}$] $(B_{n}, D_{n})$.\n\\end{description}\n\\item $k(G/H)=3(\\not=n)$ if and only if \n$(G,H)$ is one of the pairs appered in Theorem \\ref{thm:cart} except for the above two cases, $(B_{n},D_{i}\\times B_{n-i})$, $(C_{n},A_{n-1}\\times T^{1})$ and $(F_{4}, C_{3}\\times A_{1})$.\n\\item $k(G/H)=2$, otherwise (all of the other cases). \n\\end{enumerate}\n\\end{theorem}", "post_theorem_intro_text_len": 5118, "post_theorem_intro_text": "Note that by the classification of the homogeneous torus manifolds in \\cite{Ku09}, $k(G/H) =n$ if and only if $G/H$ is equivariantly diffeomorphic to $A_{n}/A_{n-1}\\times T^{1}(\\cong \\mathbb{C}P^{n})$ or $B_{n}/D_{n}(\\cong S^{2n})$.\nMoreover, the independence of $M_{1}\\times \\cdots \\times M_{r}$ coincides with \nthe minimal independence of $M_{i}$'s (if it is not a torus manifold, see Proposition~\\ref{split_computation}), and the independence of any connected homogeneous space $G/H$ is equal to that of its universal covering space $G_{1}/H_{1}\\times \\cdots \\times G_{r}/H_{r}$.\nTherefore, together with these facts, Theorem~\\ref{thmM:mp} answers Problem~\\ref{Masuda-problem} positively for the class of homogeneous GKM manifolds.\n\nAs an application of Theorem~\\ref{thmM:mp}, together with the acyclicity results of~\\cite{AMS},~\\cite{AM}, we obtain vanishing (in degrees $\\leq 4$) for reduced homology with an appropriate coefficients of the orbit spaces for homogeneous GKM manifolds, see Corollary \\ref{cor:acyc}.\nOur results are consistent with those in~\\cite{S19},~\\cite{S21},~\\cite{BT}, while providing new insights, particularly for several symmetric spaces.\n\n\\subsection{Sketch of the proof}\n\\label{sect:1.2}\n\nThe proof of Theorem~\\ref{thmM:mp} follows by computing $k(G/H)$ for every homogeneous GKM manifold $G/H$.\nHere, $k(G/H)=k(gH)$ for every fixed point $gH\\in (G/H)^{T}$ (see~\\cite{GHZ}).\nWe prove that $k(G/H)\\geq 3$ implies that $G/H$ is a maximal rank compact symmetric space with rank $>2$ using the well known characterization of symmetric spaces in terms of the Lie algebra $T_{eH} G/H$ (see Lemma~\\ref{auxiliary lemma}).\nFor maximal rank compact symmetric space $G/H$ such that $G$ is a classical Lie group (i.e., type $A$--$D$), the problem of finding maximal independence for $G/H$ reduces to the study of the matroid formed by the complement $\\Delta^{+}_{G,H}:=\\Delta^{+}_{G}\\setminus \\Delta^{+}_{H}$ of positive root systems.\nThe analysis of circuits for such a matroid is possible using {\\it weighted signed graphs}, which were introduced in~\\cite{Ha} and studied in~\\cite{Za} (also see \\cite{FGLP} and \\cite{FGGP}).\nWe apply this technique together with Borel-de Siebenthal theory~\\cite{BoSi} and Cartan's classification of symmetric spaces \\cite{C27} to compute $k(G/H)$ for all pairs $(G,H)$ except for $G$ being of type $F_{4},E_{6},E_{7},E_{8}$.\n(We remark that signed graphs under the name of Levi graphs or crystallographs were also used in~\\cite{Re} to study root subsystems.\nThe difference of our application for signed graphs is that we study \\textit{complements} of root subsystems.)\nIn the remaining cases, i.e. for $G$ being of type $F_{4},E_{6},E_{7},E_{8}$, the proof is a simple case study.\n\n\\subsection{Structure of the paper}\n\\label{sect:1.3}\n\nWe briefly outline the structure of the present paper.\nIn Section~\\ref{sec:prep}, we recall the isotropy representation for homogeneous GKM manifolds, Borel-de Siebenthal theory, Cartan's classification of symmetric spaces and homology vanishing of the orbit space for independent GKM manifolds. \nOne of the main contributions of this paper is to clarify the relationship between symmetric spaces and GKM manifolds from the viewpoint of independence.\nThe value of $k(G/H)$ for each homogeneous space $G/H$ is listed in Theorem~\\ref{main}. \nThis result implies $\\widetilde{H}_{i}(T\\backslash G/H;\\Bbbk)=0$ for all $i\\leq k(G/H)+1$, where $\\Bbbk$ is additionally specified (see Corollary~\\ref{cor:acyc} for symmetric GKM$_{3}$ cases).\nThis homology vanishing is compared with several previously known results in Examples~\\ref{ex:origra} and~\\ref{ex:comgra}.\nThe remainder of the paper is the proof of Theorem~\\ref{main}.\nIn Section~\\ref{sect:4}, linearly dependent triples are described in several cases in terms of weighted signed graphs (see Lemma~\\ref{lm:23dep}).\nIn Sections~\\ref{sect:5}--~\\ref{sect:7}, we analyse linear dependencies for root subsystems of classical types, using Borel-de Siebenthal theory and weighted signed graphs.\nIn Sections~\\ref{sect:8}--~\\ref{sect:9},\nthe remaining cases (when $G$ is of type $F_{4},E_{6},E_{7},E_{8}$) are studied. \n\n\\\n\n\\noindent {\\bf Acknowledgements:} The authors gratefully acknowledge funding of the Deutsche Forschungsgemeinschaft\n(DFG, German Research Foundation): Project number 561158824 (the second named author's Walter Benjamin Fellowship). \nWe thank M.~Masuda for many valuable discussions on toric manifolds without whom we would not be able to conduct this project, and to O.~Goertsches for the interest to our paper.\nWe also thank M.~Nakagawa for giving us a useful information on symmetric spaces.\nThe first author was partially supported by JSPS KAKENHI Grant Number 21K03262.\nHe would like to thank the Philipps University of Marburg for providing him with an excellent environment for research during his stay in August 2025.\nThe second named author is grateful to Okayama University of Science  and Osaka Metropolitan University for a very comfortable work atmosphere, where the initial part of the present work was performed during his research visits in 2023.", "sketch": "The proof of Theorem~\\ref{thmM:mp} follows by computing $k(G/H)$ for every homogeneous GKM manifold $G/H$. Here, $k(G/H)=k(gH)$ for every fixed point $gH\\in (G/H)^{T}$ (see~\\cite{GHZ}).\n\nOne shows that $k(G/H)\\geq 3$ implies that $G/H$ is a maximal rank compact symmetric space with rank $>2$, using the well known characterization of symmetric spaces in terms of the Lie algebra $T_{eH}G/H$ (see Lemma~\\ref{auxiliary lemma}).\n\nFor maximal rank compact symmetric spaces $G/H$ with $G$ a classical Lie group (type $A$--$D$), finding maximal independence reduces to studying the matroid formed by the complement of positive root systems $\\Delta^{+}_{G,H}:=\\Delta^{+}_{G}\\setminus \\Delta^{+}_{H}$. The analysis of circuits in this matroid is carried out using \\textit{weighted signed graphs} (as in~\\cite{Ha,Za}, also \\cite{FGLP,FGGP}). Applying this technique together with Borel-de Siebenthal theory~\\cite{BoSi} and Cartan's classification of symmetric spaces~\\cite{C27} yields the computation of $k(G/H)$ for all pairs $(G,H)$ except when $G$ is of type $F_{4},E_{6},E_{7},E_{8}$.\n\nIn the remaining cases ($G$ of type $F_{4},E_{6},E_{7},E_{8}$), the proof is described as a simple case study.", "expanded_sketch": "The proof of Theorem~\\ref{thmM:mp} follows by computing $k(G/H)$ for every homogeneous GKM manifold $G/H$. Here, $k(G/H)=k(gH)$ for every fixed point $gH\\in (G/H)^{T}$ (see GHZ).\n\nOne shows that $k(G/H)\\geq 3$ implies that $G/H$ is a maximal rank compact symmetric space with rank $>2$, using the well known characterization of symmetric spaces in terms of the Lie algebra $T_{eH}G/H$. We first prove the following lemma.\n\\begin{lemma}\\label{auxiliary lemma}\nUnder the hypothesis in Theorem~\\ref{main}, \nif $k(G/H)\\ge 3$, then  $G/H$ is a symmetric space given in Theorem~\\ref{thm:cart}.\n\\end{lemma}\n\nFor maximal rank compact symmetric spaces $G/H$ with $G$ a classical Lie group (type $A$--$D$), finding maximal independence reduces to studying the matroid formed by the complement of positive root systems $\\Delta^{+}_{G,H}:=\\Delta^{+}_{G}\\setminus \\Delta^{+}_{H}$. The analysis of circuits in this matroid is carried out using \\textit{weighted signed graphs} (as in Ha and Za, also FGLP and FGGP). Applying this technique together with Borel-de Siebenthal theory BoSi and Cartan's classification of symmetric spaces C27 yields the computation of $k(G/H)$ for all pairs $(G,H)$ except when $G$ is of type $F_{4},E_{6},E_{7},E_{8}$.\n\nIn the remaining cases ($G$ of type $F_{4},E_{6},E_{7},E_{8}$), the proof is described as a simple case study.,", "expanded_theorem": "\\label{thmM:mp}\nLet $G/H$ be any simply connected, homogeneous space such that $T\\subset H\\subset G$ is a maximal torus, ${\\rm rank}\\ T=n$, and $G$ is a compact, connected, simple Lie group.\nThen one has \n\\[\nk(G/H)=2,3\\ \\text{or}\\ n(=\\frac{1}{2}\\dim G/H).\n\\]\nFurthermore, \n$k(G/H) = 3$ or $n$ if and only if $G/H$ is a maximal rank compact symmetric space with rank $>2$ other than $B_{n}/D_{i}\\times B_{n-i}$, $C_{n}/A_{n-1}\\times T^{1}$, $F_{4}/C_{3}\\times A_{1}$.,", "theorem_type": ["Classification or Bijection", "Biconditional or Equivalence"], "mcq": {"question": "Let $G/H$ be a simply connected homogeneous space with $G$ a compact, connected, simple Lie group and $T\\subset H\\subset G$ a maximal torus of rank $n$. For the maximal $T$-action on $G/H$, define $k(G/H)$ as follows: at each $T$-fixed point $p$, let $k(p)$ be the largest integer $q$ such that every $q$ isotropy weights at $p$ are linearly independent, and set $k(G/H)=\\min_{p\\in (G/H)^T} k(p)$. Which statement gives the correct complete classification of the cases with $k(G/H)>2$?", "correct_choice": {"label": "A", "text": "The only possible values are $k(G/H)=2,3,$ or $n$ (with $n=\\tfrac12\\dim G/H$ in the case $k(G/H)=n$). Moreover, $k(G/H)=3$ or $k(G/H)=n$ if and only if $G/H$ is a maximal rank compact symmetric space of rank $>2$, excluding the spaces $B_{n}/(D_{i}\\times B_{n-i})$, $C_{n}/(A_{n-1}\\times T^{1})$, and $F_{4}/(C_{3}\\times A_{1})$; equivalently, all other cases have $k(G/H)=2$."}, "choices": [{"label": "B", "text": "The only possible values are $k(G/H)=2,3,$ or $n$. Moreover, $k(G/H)=3$ or $k(G/H)=n$ if and only if $G/H$ is a maximal rank compact symmetric space, excluding only the spaces $B_{n}/(D_{i}\\times B_{n-i})$, $C_{n}/(A_{n-1}\\times T^{1})$, and $F_{4}/(C_{3}\\times A_{1})$; equivalently, all other cases have $k(G/H)=2$."}, {"label": "C", "text": "If $k(G/H)>2$, then $G/H$ is a maximal rank compact symmetric space of rank $>2$ other than $B_{n}/(D_{i}\\times B_{n-i})$, $C_{n}/(A_{n-1}\\times T^{1})$, and $F_{4}/(C_{3}\\times A_{1})$."}, {"label": "D", "text": "The only possible values are $k(G/H)=2,3,$ or $n$ (with $n=\\tfrac12\\dim G/H$ in the case $k(G/H)=n$). Moreover, $k(G/H)=3$ if and only if $G/H$ is a maximal rank compact symmetric space of rank $>2$, excluding the spaces $B_{n}/(D_{i}\\times B_{n-i})$, $C_{n}/(A_{n-1}\\times T^{1})$, and $F_{4}/(C_{3}\\times A_{1})$; and $k(G/H)=n$ holds exactly for every remaining maximal rank compact symmetric space of rank $>2$."}, {"label": "E", "text": "The only possible values are $k(G/H)=2,3,$ or $n$ (with $n=\\tfrac12\\dim G/H$ in the case $k(G/H)=n$). Moreover, $k(G/H)=3$ or $k(G/H)=n$ if and only if $G/H$ is a compact symmetric space of rank $>2$, excluding the spaces $B_{n}/(D_{i}\\times B_{n-i})$, $C_{n}/(A_{n-1}\\times T^{1})$, and $F_{4}/(C_{3}\\times A_{1})$; equivalently, all other cases have $k(G/H)=2$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "rank_greater_than_2_requirement", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "characteristic", "tampered_component": "possible_values_statement_and_full_iff_classification", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "distinction_between_k_equals_3_and_k_equals_n_cases", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "maximal_rank_hypothesis", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state or strongly hint at the classification itself; it only defines k(G/H) and asks for the correct complete classification."}, "TAS": {"score": 0, "justification": "This is essentially a direct recall/restatement of a classification theorem. The correct option appears to reproduce the theorem almost verbatim rather than requiring application to a new situation."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in distinguishing subtle variants (rank > 2, maximal-rank hypothesis, weaker true statement, incorrect case split), but the task is still primarily recognition of the exact theorem statement rather than genuine generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one omits the rank > 2 condition, one is a weaker true statement, one incorrectly separates the k=3 and k=n cases, and one confuses compact symmetric with maximal-rank compact symmetric. These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A solid theorem-recall MCQ with strong distractors and no answer leakage, but it is largely tautological and tests recognition of a known classification more than generative reasoning."}}
{"id": "2602.07843v1", "paper_link": "http://arxiv.org/abs/2602.07843v1", "theorems_cnt": 3, "theorem": {"env_name": "lemma", "content": "\\label{lem:L2Green}\nLet $(M,g)$ be a compact connected $2$-dimensional Riemannian manifold without boundary, and let $G$ be the symmetric mean-zero Green function of $-\\Delta$.\nIt then holds that $G\\in L^2(M\\times M, dx\\otimes dx)$, and so\n\\[\n\\sigma^2 \\coloneqq \\int_{M}\\int_{M} G(x,y)^2\\,dx\\,dy <\\infty.\n\\]", "start_pos": 6355, "end_pos": 6692, "label": "lem:L2Green"}, "ref_dict": {"eq:GreenLogBound": "\\begin{equation}\\label{eq:GreenLogBound}\n|G(x,y)| \\le \\frac{1}{2\\pi}|\\log \\dist(x,y)|+C_0 \\le C\\bigl(1+|\\log \\dist(x,y)|\\bigr)\n\\end{equation}", "prob:Steinerberger": "\\begin{problem}[Steinerberger {\\cite[Problem 53]{SteOpen}}]\\label{prob:Steinerberger}\nIn dimension $d=2$, can one replace the remainder $\\sqrt{\\frac{\\log n}{n}}$ in \\eqref{eq:Steinerberger2D}\nby $\\frac{1}{\\sqrt n}$ while keeping the same off-diagonal Green term? In other terms, does there exist\n$C_\\M>0$ such that for all $n\\in\\mathbb N$ and all $x_1,\\dots,x_n\\in \\M$,\n\\[\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\le C_\\M\\left(\\frac1{\\sqrt n}\n+\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\\right)?\n\\]\n\\end{problem}", "eq:GreenMeanZero": "\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{\\M} G(x,y)\\,dy=0.\n\\end{equation}", "eq:Steinerberger2D": "\\begin{equation}\\label{eq:Steinerberger2D}\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_\\M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}", "thm:main": "\\begin{theorem}[The $\\sqrt{\\log n}$ remainder]\\label{thm:main}\nLet $G$ be the symmetric mean-zero Green function of $-\\Delta$.\nThere does not exist a constant $C_{\\M}>0$ such that for all $n\\in\\mathbb{N}$ and all $x_1,\\dots,x_n\\in \\M$,\n\\begin{equation}\\label{eq:NoUniversalO1sqrtN}\n\\W\\!\\left(\\frac1n\\sum_{i=1}^n\\delta_{x_i},\\, dx\\right)\n\\le\nC_{\\M}\\left(\\frac{1}{\\sqrt{n}}+\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\\right).\n\\end{equation}\nIn particular, the $\\sqrt{\\log n}$ factor in the two-dimensional inequality \\eqref{eq:Steinerberger2D} cannot be removed if the unrenormalized Green-energy term were to be preserved.\n\\end{theorem}", "def:green": "\\begin{definition}[Mean-zero Green function]\\label{def:green}\nA mean-zero Green function for $-\\Delta$ is a measurable function\n\\[\nG:\\M\\times \\M \\to \\mathbb{R}\\cup\\{+\\infty\\}\n\\]\nwith the following properties.\n\\begin{enumerate}\n\\item For each $x\\in \\M$, $y\\mapsto G(x,y)$ is locally integrable on $\\M$ and satisfies the normalization:\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{\\M} G(x,y)\\,dy=0.\n\\end{equation}\n\\item For each $f\\in C^\\infty(\\M)$ with $\\int_{\\M} f\\,dx=0$, the function\n\\[\nu(x)\\coloneqq \\int_{\\M} G(x,y)\\, f(y)\\,dy\n\\]\nis a weak solution of $-\\Delta u = f$ with $\\int_{\\M}u\\,dx=0$.\n\\end{enumerate}\n\\end{definition}"}, "pre_theorem_intro_text_len": 4896, "pre_theorem_intro_text": "Let $(M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary, $\\mathrm{d}_g$ is the induced distance, and let $G(x,y)$ denote the symmetric mean-zero Green function of the Laplacian (Definition \\ref{def:green}). We work with the normalized volume measure $dx=\\mathrm{vol}(M)^{-1}\\,\\mathrm{d}\\mathrm{vol}$. Whenever we integrate in the variable $y$, we write $dy$ for the same measure. For points $x_1,\\dots,x_n\\in M$, we define the empirical measure:\n\\[\n\\mu_n \\coloneqq \\frac1n\\sum_{i=1}^n \\delta_{x_i}.\n\\]\nSteinerberger \\cite{Ste21} showed that for $d\\ge 3$, the following Green--Wasserstein inequality holds:\n\\[\nW_2(\\mu_n,dx)\\ \\lesssim_{M}\\ n^{-1/d} + \\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}.\n\\]\nIn \\cite[Problem 53]{SteOpen}, he gives the following estimate:\n\\begin{equation}\\label{eq:Steinerberger2D}\nW_2\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}\nIn this note, we use\n\\[\n\\sum_{i\\neq j}(\\cdots)\\ \\coloneqq\\ \\sum_{\\substack{1\\le i,j\\le n\\\\ i\\neq j}}(\\cdots)\n\\]\nto denote the ordered sum over distinct indices. Notice that, by symmetry of $G$, it is twice the sum over $i<j$. Steinerberger asked whether the factor $\\sqrt{\\log n}$ in \\eqref{eq:Steinerberger2D} can be removed while keeping the same Green-energy term. The precise question is as follows.\n\n\\begin{problem}[Steinerberger {\\cite[Problem 53]{SteOpen}}]\\label{prob:Steinerberger}\nIn dimension $d=2$, can one replace the remainder $\\sqrt{\\frac{\\log n}{n}}$ in \\eqref{eq:Steinerberger2D}\nby $\\frac{1}{\\sqrt n}$ while keeping the same off-diagonal Green term? In other terms, does there exist\n$C_M>0$ such that for all $n\\in\\mathbb N$ and all $x_1,\\dots,x_n\\in M$,\n\\[\nW_2\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\le C_M\\left(\\frac1{\\sqrt n}\n+\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\\right)?\n\\]\n\\end{problem}\n\\noindent Our main result, Theorem \\ref{thm:main}, gives a negative answer to Problem \\ref{prob:Steinerberger}.\n\n\\begin{remark}[The diagonal]\\label{rem:diagonal}\nWe note that the Green function $G$ is smooth on $(M\\times M)\\setminus \\{(x,x):x\\inM\\}$ with a logarithmic singularity along the diagonal.\nIn a deterministic context, if $x_i=x_j$ for some $i\\neq j$, then the term $G(x_i,x_j)$ is not finite. It is then natural to interpret the Green term in \\eqref{eq:Steinerberger2D} and Problem \\ref{prob:Steinerberger} as $+\\infty$, in which case the inequality holds trivially.\nIn the random settings that we use in this note, collisions occur with probability $0$ because $dx$ is non-atomic.\n\\end{remark}\n\nWe now recall some definitions that will be useful in the main argument.\nFor Borel probability measures $\\mu,\\nu$ on $M$, let us denote by $\\Gamma(\\mu,\\nu)$ the set of couplings of $\\mu$ and $\\nu$ on $M\\times M$.\nThe quadratic Wasserstein distance is\n\\[\nW_2(\\mu,\\nu)\\coloneqq \\left(\\inf_{\\gamma\\in \\Gamma(\\mu,\\nu)} \\int_{M\\times M} \\mathrm{d}_g(x,y)^2\\,\\,\\mathrm{d}\\gamma(x,y)\\right)^{1/2}.\n\\]\nGiven that $M$ is compact, it follows that $W_2(\\mu,\\nu)$ is always finite and satisfies $W_2(\\mu,\\nu)\\le \\mathrm{diam}(M)$. Let $\\Delta$ be the Laplace--Beltrami operator on $(M,g)$, which is realized here as a self-adjoint operator on $L^2(M,dx)$ with domain $H^2(M)$. Again, since $M$ is compact and connected, $\\ker(-\\Delta)$ consists of the constant functions.\n\n\\begin{definition}[Mean-zero Green function]\\label{def:green}\nA mean-zero Green function for $-\\Delta$ is a measurable function\n\\[\nG:M\\times M \\to \\mathbb{R}\\cup\\{+\\infty\\}\n\\]\nwith the following properties.\n\\begin{enumerate}\n\\item For each $x\\in M$, $y\\mapsto G(x,y)$ is locally integrable on $M$ and satisfies the normalization:\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{M} G(x,y)\\,dy=0.\n\\end{equation}\n\\item For each $f\\in C^\\infty(M)$ with $\\int_{M} f\\,dx=0$, the function\n\\[\nu(x)\\coloneqq \\int_{M} G(x,y)\\, f(y)\\,dy\n\\]\nis a weak solution of $-\\Delta u = f$ with $\\int_{M}u\\,dx=0$.\n\\end{enumerate}\n\\end{definition}\n\nRecall that the uniqueness and existence of the mean-zero Green function, as a distribution kernel, are guaranteed. We can also choose it to be symmetric (for example, see \\cite{Aubin98} or \\cite{Rosenberg97}). We now fix a symmetric mean-zero Green function $G$, so that $G(x,y)=G(y,x)$ holds.\nBy symmetry and \\eqref{eq:GreenMeanZero}, we obtain:\n\\[\n\\int_{M}G(y,x)\\,dy=0\\qquad\\text{for every }x\\in M.\n\\]\n\n\\begin{remark}[Local singularity]\\label{rem:logsing}\nIn dimension $2$, the following classical local expansion (in geodesic normal coordinates) is true:\n\\[\nG(x,y)= -\\frac{1}{2\\pi}\\log \\mathrm{d}_g(x,y) + H(x,y).\n\\]\nNote that $H$ extends continuously to the diagonal and is smooth off the diagonal.\nIn particular, we have $G(x,\\cdot)\\in L^1(M,dx)$ for each fixed $x$.\n\\end{remark}", "context": "Let $(M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary, $\\mathrm{d}_g$ is the induced distance, and let $G(x,y)$ denote the symmetric mean-zero Green function of the Laplacian (Definition \\ref{def:green}). We work with the normalized volume measure $dx=\\mathrm{vol}(M)^{-1}\\,\\mathrm{d}\\mathrm{vol}$. Whenever we integrate in the variable $y$, we write $dy$ for the same measure. For points $x_1,\\dots,x_n\\in M$, we define the empirical measure:\n\\[\n\\mu_n \\coloneqq \\frac1n\\sum_{i=1}^n \\delta_{x_i}.\n\\]\nSteinerberger \\cite{Ste21} showed that for $d\\ge 3$, the following Green--Wasserstein inequality holds:\n\\[\nW_2(\\mu_n,dx)\\ \\lesssim_{M}\\ n^{-1/d} + \\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}.\n\\]\nIn \\cite[Problem 53]{SteOpen}, he gives the following estimate:\n\\begin{equation}\\label{eq:Steinerberger2D}\nW_2\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}\nIn this note, we use\n\\[\n\\sum_{i\\neq j}(\\cdots)\\ \\coloneqq\\ \\sum_{\\substack{1\\le i,j\\le n\\\\ i\\neq j}}(\\cdots)\n\\]\nto denote the ordered sum over distinct indices. Notice that, by symmetry of $G$, it is twice the sum over $i<j$. Steinerberger asked whether the factor $\\sqrt{\\log n}$ in \\eqref{eq:Steinerberger2D} can be removed while keeping the same Green-energy term. The precise question is as follows.\n\n\\begin{remark}[The diagonal]\\label{rem:diagonal}\nWe note that the Green function $G$ is smooth on $(M\\times M)\\setminus \\{(x,x):x\\inM\\}$ with a logarithmic singularity along the diagonal.\nIn a deterministic context, if $x_i=x_j$ for some $i\\neq j$, then the term $G(x_i,x_j)$ is not finite. It is then natural to interpret the Green term in \\eqref{eq:Steinerberger2D} and Problem \\ref{prob:Steinerberger} as $+\\infty$, in which case the inequality holds trivially.\nIn the random settings that we use in this note, collisions occur with probability $0$ because $dx$ is non-atomic.\n\\end{remark}\n\nWe now recall some definitions that will be useful in the main argument.\nFor Borel probability measures $\\mu,\\nu$ on $M$, let us denote by $\\Gamma(\\mu,\\nu)$ the set of couplings of $\\mu$ and $\\nu$ on $M\\times M$.\nThe quadratic Wasserstein distance is\n\\[\nW_2(\\mu,\\nu)\\coloneqq \\left(\\inf_{\\gamma\\in \\Gamma(\\mu,\\nu)} \\int_{M\\times M} \\mathrm{d}_g(x,y)^2\\,\\,\\mathrm{d}\\gamma(x,y)\\right)^{1/2}.\n\\]\nGiven that $M$ is compact, it follows that $W_2(\\mu,\\nu)$ is always finite and satisfies $W_2(\\mu,\\nu)\\le \\mathrm{diam}(M)$. Let $\\Delta$ be the Laplace--Beltrami operator on $(M,g)$, which is realized here as a self-adjoint operator on $L^2(M,dx)$ with domain $H^2(M)$. Again, since $M$ is compact and connected, $\\ker(-\\Delta)$ consists of the constant functions.\n\n\\begin{definition}[Mean-zero Green function]\\label{def:green}\nA mean-zero Green function for $-\\Delta$ is a measurable function\n\\[\nG:M\\times M \\to \\mathbb{R}\\cup\\{+\\infty\\}\n\\]\nwith the following properties.\n\\begin{enumerate}\n\\item For each $x\\in M$, $y\\mapsto G(x,y)$ is locally integrable on $M$ and satisfies the normalization:\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{M} G(x,y)\\,dy=0.\n\\end{equation}\n\\item For each $f\\in C^\\infty(M)$ with $\\int_{M} f\\,dx=0$, the function\n\\[\nu(x)\\coloneqq \\int_{M} G(x,y)\\, f(y)\\,dy\n\\]\nis a weak solution of $-\\Delta u = f$ with $\\int_{M}u\\,dx=0$.\n\\end{enumerate}\n\\end{definition}\n\nRecall that the uniqueness and existence of the mean-zero Green function, as a distribution kernel, are guaranteed. We can also choose it to be symmetric (for example, see \\cite{Aubin98} or \\cite{Rosenberg97}). We now fix a symmetric mean-zero Green function $G$, so that $G(x,y)=G(y,x)$ holds.\nBy symmetry and \\eqref{eq:GreenMeanZero}, we obtain:\n\\[\n\\int_{M}G(y,x)\\,dy=0\\qquad\\text{for every }x\\in M.\n\\]\n\n\\begin{remark}[Local singularity]\\label{rem:logsing}\nIn dimension $2$, the following classical local expansion (in geodesic normal coordinates) is true:\n\\[\nG(x,y)= -\\frac{1}{2\\pi}\\log \\mathrm{d}_g(x,y) + H(x,y).\n\\]\nNote that $H$ extends continuously to the diagonal and is smooth off the diagonal.\nIn particular, we have $G(x,\\cdot)\\in L^1(M,dx)$ for each fixed $x$.\n\\end{remark}\n\n\\begin{definition}[Mean-zero Green function]\\label{def:green}\nA mean-zero Green function for $-\\Delta$ is a measurable function\n\\[\nG:\\M\\times \\M \\to \\mathbb{R}\\cup\\{+\\infty\\}\n\\]\nwith the following properties.\n\\begin{enumerate}\n\\item For each $x\\in \\M$, $y\\mapsto G(x,y)$ is locally integrable on $\\M$ and satisfies the normalization:\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{\\M} G(x,y)\\,dy=0.\n\\end{equation}\n\\item For each $f\\in C^\\infty(\\M)$ with $\\int_{\\M} f\\,dx=0$, the function\n\\[\nu(x)\\coloneqq \\int_{\\M} G(x,y)\\, f(y)\\,dy\n\\]\nis a weak solution of $-\\Delta u = f$ with $\\int_{\\M}u\\,dx=0$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{\\M} G(x,y)\\,dy=0.\n\\end{equation}\n\n\\begin{equation}\\label{eq:Steinerberger2D}\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_\\M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}\n\n\\begin{problem}[Steinerberger {\\cite[Problem 53]{SteOpen}}]\\label{prob:Steinerberger}\nIn dimension $d=2$, can one replace the remainder $\\sqrt{\\frac{\\log n}{n}}$ in \\eqref{eq:Steinerberger2D}\nby $\\frac{1}{\\sqrt n}$ while keeping the same off-diagonal Green term? In other terms, does there exist\n$C_\\M>0$ such that for all $n\\in\\mathbb N$ and all $x_1,\\dots,x_n\\in \\M$,\n\\[\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\le C_\\M\\left(\\frac1{\\sqrt n}\n+\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\\right)?\n\\]\n\\end{problem}", "full_context": "Let $(M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary, $\\mathrm{d}_g$ is the induced distance, and let $G(x,y)$ denote the symmetric mean-zero Green function of the Laplacian (Definition \\ref{def:green}). We work with the normalized volume measure $dx=\\mathrm{vol}(M)^{-1}\\,\\mathrm{d}\\mathrm{vol}$. Whenever we integrate in the variable $y$, we write $dy$ for the same measure. For points $x_1,\\dots,x_n\\in M$, we define the empirical measure:\n\\[\n\\mu_n \\coloneqq \\frac1n\\sum_{i=1}^n \\delta_{x_i}.\n\\]\nSteinerberger \\cite{Ste21} showed that for $d\\ge 3$, the following Green--Wasserstein inequality holds:\n\\[\nW_2(\\mu_n,dx)\\ \\lesssim_{M}\\ n^{-1/d} + \\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}.\n\\]\nIn \\cite[Problem 53]{SteOpen}, he gives the following estimate:\n\\begin{equation}\\label{eq:Steinerberger2D}\nW_2\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}\nIn this note, we use\n\\[\n\\sum_{i\\neq j}(\\cdots)\\ \\coloneqq\\ \\sum_{\\substack{1\\le i,j\\le n\\\\ i\\neq j}}(\\cdots)\n\\]\nto denote the ordered sum over distinct indices. Notice that, by symmetry of $G$, it is twice the sum over $i<j$. Steinerberger asked whether the factor $\\sqrt{\\log n}$ in \\eqref{eq:Steinerberger2D} can be removed while keeping the same Green-energy term. The precise question is as follows.\n\n\\begin{remark}[The diagonal]\\label{rem:diagonal}\nWe note that the Green function $G$ is smooth on $(M\\times M)\\setminus \\{(x,x):x\\inM\\}$ with a logarithmic singularity along the diagonal.\nIn a deterministic context, if $x_i=x_j$ for some $i\\neq j$, then the term $G(x_i,x_j)$ is not finite. It is then natural to interpret the Green term in \\eqref{eq:Steinerberger2D} and Problem \\ref{prob:Steinerberger} as $+\\infty$, in which case the inequality holds trivially.\nIn the random settings that we use in this note, collisions occur with probability $0$ because $dx$ is non-atomic.\n\\end{remark}\n\nWe now recall some definitions that will be useful in the main argument.\nFor Borel probability measures $\\mu,\\nu$ on $M$, let us denote by $\\Gamma(\\mu,\\nu)$ the set of couplings of $\\mu$ and $\\nu$ on $M\\times M$.\nThe quadratic Wasserstein distance is\n\\[\nW_2(\\mu,\\nu)\\coloneqq \\left(\\inf_{\\gamma\\in \\Gamma(\\mu,\\nu)} \\int_{M\\times M} \\mathrm{d}_g(x,y)^2\\,\\,\\mathrm{d}\\gamma(x,y)\\right)^{1/2}.\n\\]\nGiven that $M$ is compact, it follows that $W_2(\\mu,\\nu)$ is always finite and satisfies $W_2(\\mu,\\nu)\\le \\mathrm{diam}(M)$. Let $\\Delta$ be the Laplace--Beltrami operator on $(M,g)$, which is realized here as a self-adjoint operator on $L^2(M,dx)$ with domain $H^2(M)$. Again, since $M$ is compact and connected, $\\ker(-\\Delta)$ consists of the constant functions.\n\n\\begin{definition}[Mean-zero Green function]\\label{def:green}\nA mean-zero Green function for $-\\Delta$ is a measurable function\n\\[\nG:M\\times M \\to \\mathbb{R}\\cup\\{+\\infty\\}\n\\]\nwith the following properties.\n\\begin{enumerate}\n\\item For each $x\\in M$, $y\\mapsto G(x,y)$ is locally integrable on $M$ and satisfies the normalization:\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{M} G(x,y)\\,dy=0.\n\\end{equation}\n\\item For each $f\\in C^\\infty(M)$ with $\\int_{M} f\\,dx=0$, the function\n\\[\nu(x)\\coloneqq \\int_{M} G(x,y)\\, f(y)\\,dy\n\\]\nis a weak solution of $-\\Delta u = f$ with $\\int_{M}u\\,dx=0$.\n\\end{enumerate}\n\\end{definition}\n\nRecall that the uniqueness and existence of the mean-zero Green function, as a distribution kernel, are guaranteed. We can also choose it to be symmetric (for example, see \\cite{Aubin98} or \\cite{Rosenberg97}). We now fix a symmetric mean-zero Green function $G$, so that $G(x,y)=G(y,x)$ holds.\nBy symmetry and \\eqref{eq:GreenMeanZero}, we obtain:\n\\[\n\\int_{M}G(y,x)\\,dy=0\\qquad\\text{for every }x\\in M.\n\\]\n\n\\begin{remark}[Local singularity]\\label{rem:logsing}\nIn dimension $2$, the following classical local expansion (in geodesic normal coordinates) is true:\n\\[\nG(x,y)= -\\frac{1}{2\\pi}\\log \\mathrm{d}_g(x,y) + H(x,y).\n\\]\nNote that $H$ extends continuously to the diagonal and is smooth off the diagonal.\nIn particular, we have $G(x,\\cdot)\\in L^1(M,dx)$ for each fixed $x$.\n\\end{remark}\n\n\\begin{definition}[Mean-zero Green function]\\label{def:green}\nA mean-zero Green function for $-\\Delta$ is a measurable function\n\\[\nG:\\M\\times \\M \\to \\mathbb{R}\\cup\\{+\\infty\\}\n\\]\nwith the following properties.\n\\begin{enumerate}\n\\item For each $x\\in \\M$, $y\\mapsto G(x,y)$ is locally integrable on $\\M$ and satisfies the normalization:\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{\\M} G(x,y)\\,dy=0.\n\\end{equation}\n\\item For each $f\\in C^\\infty(\\M)$ with $\\int_{\\M} f\\,dx=0$, the function\n\\[\nu(x)\\coloneqq \\int_{\\M} G(x,y)\\, f(y)\\,dy\n\\]\nis a weak solution of $-\\Delta u = f$ with $\\int_{\\M}u\\,dx=0$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{\\M} G(x,y)\\,dy=0.\n\\end{equation}\n\n\\begin{equation}\\label{eq:Steinerberger2D}\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_\\M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}\n\n\\begin{problem}[Steinerberger {\\cite[Problem 53]{SteOpen}}]\\label{prob:Steinerberger}\nIn dimension $d=2$, can one replace the remainder $\\sqrt{\\frac{\\log n}{n}}$ in \\eqref{eq:Steinerberger2D}\nby $\\frac{1}{\\sqrt n}$ while keeping the same off-diagonal Green term? In other terms, does there exist\n$C_\\M>0$ such that for all $n\\in\\mathbb N$ and all $x_1,\\dots,x_n\\in \\M$,\n\\[\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\le C_\\M\\left(\\frac1{\\sqrt n}\n+\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\\right)?\n\\]\n\\end{problem}\n\nLet $(\\M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary, $\\dist$ is the induced distance, and let $G(x,y)$ denote the symmetric mean-zero Green function of the Laplacian (Definition \\ref{def:green}). We work with the normalized volume measure $dx=\\mathrm{vol}(\\M)^{-1}\\dd\\mathrm{vol}$. Whenever we integrate in the variable $y$, we write $dy$ for the same measure. For points $x_1,\\dots,x_n\\in \\M$, we define the empirical measure:\n\\[\n\\mu_n \\coloneqq \\frac1n\\sum_{i=1}^n \\delta_{x_i}.\n\\]\nSteinerberger \\cite{Ste21} showed that for $d\\ge 3$, the following Green--Wasserstein inequality holds:\n\\[\n\\W(\\mu_n,dx)\\ \\lesssim_{\\M}\\ n^{-1/d} + \\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}.\n\\]\nIn \\cite[Problem 53]{SteOpen}, he gives the following estimate:\n\\begin{equation}\\label{eq:Steinerberger2D}\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_\\M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}\nIn this note, we use\n\\[\n\\sum_{i\\neq j}(\\cdots)\\ \\coloneqq\\ \\sum_{\\substack{1\\le i,j\\le n\\\\ i\\neq j}}(\\cdots)\n\\]\nto denote the ordered sum over distinct indices. Notice that, by symmetry of $G$, it is twice the sum over $i<j$. Steinerberger asked whether the factor $\\sqrt{\\log n}$ in \\eqref{eq:Steinerberger2D} can be removed while keeping the same Green-energy term. The precise question is as follows.\n\n\\begin{remark}[The diagonal]\\label{rem:diagonal}\nWe note that the Green function $G$ is smooth on $(\\M\\times \\M)\\setminus \\{(x,x):x\\in\\M\\}$ with a logarithmic singularity along the diagonal.\nIn a deterministic context, if $x_i=x_j$ for some $i\\neq j$, then the term $G(x_i,x_j)$ is not finite. It is then natural to interpret the Green term in \\eqref{eq:Steinerberger2D} and Problem \\ref{prob:Steinerberger} as $+\\infty$, in which case the inequality holds trivially.\nIn the random settings that we use in this note, collisions occur with probability $0$ because $dx$ is non-atomic.\n\\end{remark}\n\nWe now recall some definitions that will be useful in the main argument.\nFor Borel probability measures $\\mu,\\nu$ on $\\M$, let us denote by $\\Gamma(\\mu,\\nu)$ the set of couplings of $\\mu$ and $\\nu$ on $\\M\\times \\M$.\nThe quadratic Wasserstein distance is\n\\[\n\\W(\\mu,\\nu)\\coloneqq \\left(\\inf_{\\gamma\\in \\Gamma(\\mu,\\nu)} \\int_{\\M\\times \\M} \\dist(x,y)^2\\,\\dd\\gamma(x,y)\\right)^{1/2}.\n\\]\nGiven that $\\M$ is compact, it follows that $\\W(\\mu,\\nu)$ is always finite and satisfies $\\W(\\mu,\\nu)\\le \\mathrm{diam}(\\M)$. Let $\\Delta$ be the Laplace--Beltrami operator on $(\\M,g)$, which is realized here as a self-adjoint operator on $L^2(\\M,dx)$ with domain $H^2(\\M)$. Again, since $\\M$ is compact and connected, $\\ker(-\\Delta)$ consists of the constant functions.\n\n\\begin{definition}[Mean-zero Green function]\\label{def:green}\nA mean-zero Green function for $-\\Delta$ is a measurable function\n\\[\nG:\\M\\times \\M \\to \\mathbb{R}\\cup\\{+\\infty\\}\n\\]\nwith the following properties.\n\\begin{enumerate}\n\\item For each $x\\in \\M$, $y\\mapsto G(x,y)$ is locally integrable on $\\M$ and satisfies the normalization:\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{\\M} G(x,y)\\,dy=0.\n\\end{equation}\n\\item For each $f\\in C^\\infty(\\M)$ with $\\int_{\\M} f\\,dx=0$, the function\n\\[\nu(x)\\coloneqq \\int_{\\M} G(x,y)\\, f(y)\\,dy\n\\]\nis a weak solution of $-\\Delta u = f$ with $\\int_{\\M}u\\,dx=0$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark}[Local singularity]\\label{rem:logsing}\nIn dimension $2$, the following classical local expansion (in geodesic normal coordinates) is true:\n\\[\nG(x,y)= -\\frac{1}{2\\pi}\\log \\dist(x,y) + H(x,y).\n\\]\nNote that $H$ extends continuously to the diagonal and is smooth off the diagonal.\nIn particular, we have $G(x,\\cdot)\\in L^1(\\M,dx)$ for each fixed $x$.\n\\end{remark}\n\n\\begin{proof}\nRecall that $\\mathrm{vol}$ denotes the Riemannian volume measure on $(\\M,g)$ and $dx = \\mathrm{vol}(\\M)^{-1}\\dd\\mathrm{vol}$. Let $\\mathrm{inj}(\\M)>0$ be the injectivity radius of $(\\M,g)$ and set $r_0\\coloneqq \\mathrm{inj}(\\M)/4$. We now apply standard results on the singularity structure of the Green function in dimension $2$ (for instance, see \\cite{Aubin98}). In particular, the function\n\\[\nH(x,y)\\coloneqq G(x,y)+\\frac{1}{2\\pi}\\log \\dist(x,y),\n\\qquad x\\neq y,\n\\]\nextends continuously to the set $\\{(x,y)\\in \\M\\times\\M: \\dist(x,y)\\le 2r_0\\}$ (this also holds across the diagonal). Observe that this set is compact, so there exists $C_0>0$ such that\n\\[\n|H(x,y)|\\le C_0\\qquad\\text{whenever }\\dist(x,y)\\le 2r_0.\n\\]\nAs a result, for all $x\\neq y$ with $\\dist(x,y)<2r_0$, it follows that\n\\begin{equation}\\label{eq:GreenLogBound}\n|G(x,y)| \\le \\frac{1}{2\\pi}|\\log \\dist(x,y)|+C_0 \\le C\\bigl(1+|\\log \\dist(x,y)|\\bigr)\n\\end{equation}\nfor some constant $C>0$.\n\nWe now split $\\M\\times\\M$ into the near-diagonal region\n\\[\nD\\coloneqq \\{(x,y)\\in\\M\\times\\M: \\dist(x,y)<r_0\\}\n\\]\nand its complement $D^c$. Notice that $D^c$ is compact and is always a positive distance away from the diagonal, so $G$ is smooth on $D^c$ and therefore bounded there. We obtain:\n\\[\n\\int_{D^c} G(x,y)^2\\,dx\\,dy<\\infty.\n\\]\nLet us now analyze $D$. By \\eqref{eq:GreenLogBound}, we set\n\\[\nI(x)\\coloneqq \\int_{B_{r_0}(x)} \\bigl(1+|\\log \\dist(x,y)|\\bigr)^2\\,dy,\n\\]\nso that\n\\[\n\\int_{D} G(x,y)^2\\,dx\\,dy\n\\le C^2\\int_{\\M} I(x)\\,dx.\n\\]\nWe also fix $x\\in\\M$. Note that on the ball $B_{r_0}(x)$, we may use geodesic polar coordinates centered at $x$: $y=\\exp_x(\\rho\\theta)$ with $\\rho\\in(0,r_0)$ and $\\theta\\in\\mathbb{S}^1$. In these coordinates, the volume form has the following form:\n\\[\n\\dd\\mathrm{vol}(y)=J_x(\\rho,\\theta)\\,\\dd\\rho\\,\\dd\\theta,\n\\qquad\nJ_x(\\rho,\\theta)=\\rho\\,a_x(\\rho,\\theta),\n\\]\nwhere $a_x$ is smooth and satisfies $a_x(0,\\theta)=1$.\nIn particular, by compactness, there exists $C_1>0$ with:\n\\begin{equation}\\label{eq:JacobianBound}\nJ_x(\\rho,\\theta)\\le C_1\\rho\\qquad\\text{for all }x\\in\\M,\\ \\theta\\in\\mathbb{S}^1,\\ \\rho\\in(0,r_0)\n\\end{equation}\n(see, for instance, \\cite{Chavel84}).\nWe now apply the normalization of $dx$ and obtain:\n\\begin{align*}\n\\int_{B_{r_0}(x)} \\bigl(1+|\\log \\dist(x,y)|\\bigr)^2\\,dy\n&=\\frac{1}{\\mathrm{vol}(\\M)}\\int_{\\mathbb{S}^1}\\!\\!\\int_0^{r_0} \\bigl(1+|\\log \\rho|\\bigr)^2 J_x(\\rho,\\theta)\\,\\dd\\rho\\,\\dd\\theta\\\\\n&\\le \\frac{C_1}{\\mathrm{vol}(\\M)}\\int_{\\mathbb{S}^1}\\!\\!\\int_0^{r_0} \\bigl(1+|\\log \\rho|\\bigr)^2 \\rho\\,\\dd\\rho\\,\\dd\\theta.\n\\end{align*}\nThe right-hand side is finite since near $0$ the function $\\rho\\mapsto \\rho(\\log \\rho)^2$ is integrable. We apply the substitution $t=-\\log\\rho$ (so $\\rho=e^{-t}$ and $\\rho\\,\\dd\\rho=-e^{-2t}\\,\\dd t$) and get:\n\\[\n\\int_0^{r_0} \\bigl(1+|\\log \\rho|\\bigr)^2\\rho\\,\\dd\\rho\n=\\int_{\\infty}^{-\\log r_0} \\bigl(1+|t|\\bigr)^2(-e^{-2t})\\,\\dd t\n=\\int_{-\\log r_0}^{\\infty} \\bigl(1+|t|\\bigr)^2 e^{-2t}\\,\\dd t <\\infty.\n\\]\nIt follows that $\\int_{B_{r_0}(x)} (1+|\\log \\dist(x,y)|)^2\\,dy$ is bounded uniformly in $x$.\nIf we integrate over $x$, we arrive at $\\int_D G^2\\,dx\\,dy<\\infty$. Finally, we combine the estimates on $D$ and $D^c$. This finishes the proof of $\\sigma^2<\\infty$.\n\\end{proof}\n\n\\begin{theorem}[The $\\sqrt{\\log n}$ remainder]\\label{thm:main}\nLet $G$ be the symmetric mean-zero Green function of $-\\Delta$.\nThere does not exist a constant $C_{\\M}>0$ such that for all $n\\in\\mathbb{N}$ and all $x_1,\\dots,x_n\\in \\M$,\n\\begin{equation}\\label{eq:NoUniversalO1sqrtN}\n\\W\\!\\left(\\frac1n\\sum_{i=1}^n\\delta_{x_i},\\, dx\\right)\n\\le\nC_{\\M}\\left(\\frac{1}{\\sqrt{n}}+\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\\right).\n\\end{equation}\nIn particular, the $\\sqrt{\\log n}$ factor in the two-dimensional inequality \\eqref{eq:Steinerberger2D} cannot be removed if the unrenormalized Green-energy term were to be preserved.\n\\end{theorem}", "post_theorem_intro_text_len": 3405, "post_theorem_intro_text": "\\begin{proof}\nRecall that $\\mathrm{vol}$ denotes the Riemannian volume measure on $(M,g)$ and $dx = \\mathrm{vol}(M)^{-1}\\,\\mathrm{d}\\mathrm{vol}$. Let $\\mathrm{inj}(M)>0$ be the injectivity radius of $(M,g)$ and set $r_0\\coloneqq \\mathrm{inj}(M)/4$. We now apply standard results on the singularity structure of the Green function in dimension $2$ (for instance, see \\cite{Aubin98}). In particular, the function\n\\[\nH(x,y)\\coloneqq G(x,y)+\\frac{1}{2\\pi}\\log \\mathrm{d}_g(x,y),\n\\qquad x\\neq y,\n\\]\nextends continuously to the set $\\{(x,y)\\in M\\timesM: \\mathrm{d}_g(x,y)\\le 2r_0\\}$ (this also holds across the diagonal). Observe that this set is compact, so there exists $C_0>0$ such that\n\\[\n|H(x,y)|\\le C_0\\qquad\\text{whenever }\\mathrm{d}_g(x,y)\\le 2r_0.\n\\]\nAs a result, for all $x\\neq y$ with $\\mathrm{d}_g(x,y)<2r_0$, it follows that\n\\begin{equation}\\label{eq:GreenLogBound}\n|G(x,y)| \\le \\frac{1}{2\\pi}|\\log \\mathrm{d}_g(x,y)|+C_0 \\le C\\bigl(1+|\\log \\mathrm{d}_g(x,y)|\\bigr)\n\\end{equation}\nfor some constant $C>0$.\n\nWe now split $M\\timesM$ into the near-diagonal region\n\\[\nD\\coloneqq \\{(x,y)\\inM\\timesM: \\mathrm{d}_g(x,y)<r_0\\}\n\\]\nand its complement $D^c$. Notice that $D^c$ is compact and is always a positive distance away from the diagonal, so $G$ is smooth on $D^c$ and therefore bounded there. We obtain:\n\\[\n\\int_{D^c} G(x,y)^2\\,dx\\,dy<\\infty.\n\\]\nLet us now analyze $D$. By \\eqref{eq:GreenLogBound}, we set\n\\[\nI(x)\\coloneqq \\int_{B_{r_0}(x)} \\bigl(1+|\\log \\mathrm{d}_g(x,y)|\\bigr)^2\\,dy,\n\\]\nso that\n\\[\n\\int_{D} G(x,y)^2\\,dx\\,dy\n\\le C^2\\int_{M} I(x)\\,dx.\n\\]\nWe also fix $x\\inM$. Note that on the ball $B_{r_0}(x)$, we may use geodesic polar coordinates centered at $x$: $y=\\exp_x(\\rho\\theta)$ with $\\rho\\in(0,r_0)$ and $\\theta\\in\\mathbb{S}^1$. In these coordinates, the volume form has the following form:\n\\[\n\\,\\mathrm{d}\\mathrm{vol}(y)=J_x(\\rho,\\theta)\\,\\,\\mathrm{d}\\rho\\,\\,\\mathrm{d}\\theta,\n\\qquad\nJ_x(\\rho,\\theta)=\\rho\\,a_x(\\rho,\\theta),\n\\]\nwhere $a_x$ is smooth and satisfies $a_x(0,\\theta)=1$.\nIn particular, by compactness, there exists $C_1>0$ with:\n\\begin{equation}\\label{eq:JacobianBound}\nJ_x(\\rho,\\theta)\\le C_1\\rho\\qquad\\text{for all }x\\inM,\\ \\theta\\in\\mathbb{S}^1,\\ \\rho\\in(0,r_0)\n\\end{equation}\n(see, for instance, \\cite{Chavel84}).\nWe now apply the normalization of $dx$ and obtain:\n\\begin{align*}\n\\int_{B_{r_0}(x)} \\bigl(1+|\\log \\mathrm{d}_g(x,y)|\\bigr)^2\\,dy\n&=\\frac{1}{\\mathrm{vol}(M)}\\int_{\\mathbb{S}^1}\\!\\!\\int_0^{r_0} \\bigl(1+|\\log \\rho|\\bigr)^2 J_x(\\rho,\\theta)\\,\\,\\mathrm{d}\\rho\\,\\,\\mathrm{d}\\theta\\\\\n&\\le \\frac{C_1}{\\mathrm{vol}(M)}\\int_{\\mathbb{S}^1}\\!\\!\\int_0^{r_0} \\bigl(1+|\\log \\rho|\\bigr)^2 \\rho\\,\\,\\mathrm{d}\\rho\\,\\,\\mathrm{d}\\theta.\n\\end{align*}\nThe right-hand side is finite since near $0$ the function $\\rho\\mapsto \\rho(\\log \\rho)^2$ is integrable. We apply the substitution $t=-\\log\\rho$ (so $\\rho=e^{-t}$ and $\\rho\\,\\,\\mathrm{d}\\rho=-e^{-2t}\\,\\,\\mathrm{d} t$) and get:\n\\[\n\\int_0^{r_0} \\bigl(1+|\\log \\rho|\\bigr)^2\\rho\\,\\,\\mathrm{d}\\rho\n=\\int_{\\infty}^{-\\log r_0} \\bigl(1+|t|\\bigr)^2(-e^{-2t})\\,\\,\\mathrm{d} t\n=\\int_{-\\log r_0}^{\\infty} \\bigl(1+|t|\\bigr)^2 e^{-2t}\\,\\,\\mathrm{d} t <\\infty.\n\\]\nIt follows that $\\int_{B_{r_0}(x)} (1+|\\log \\mathrm{d}_g(x,y)|)^2\\,dy$ is bounded uniformly in $x$.\nIf we integrate over $x$, we arrive at $\\int_D G^2\\,dx\\,dy<\\infty$. Finally, we combine the estimates on $D$ and $D^c$. This finishes the proof of $\\sigma^2<\\infty$.\n\\end{proof}", "sketch": "Using the 2D Green function singularity, write\n\\[\nH(x,y)\\coloneqq G(x,y)+\\frac{1}{2\\pi}\\log \\mathrm{d}_g(x,y)\n\\]\nwhich “extends continuously” to a compact neighborhood of the diagonal, hence is bounded there. This gives the near-diagonal logarithmic bound\n\\[\n|G(x,y)|\\le C\\bigl(1+|\\log \\mathrm{d}_g(x,y)|\\bigr)\\qquad (\\mathrm{d}_g(x,y)<2r_0).\n\\]\nSplit \\(M\\times M\\) into the near-diagonal region \\(D=\\{\\mathrm{d}_g(x,y)<r_0\\}\\) and its complement \\(D^c\\). On \\(D^c\\), “\\(G\\) is smooth … and therefore bounded,” so \\(\\int_{D^c}G^2<\\infty\\).\n\nOn \\(D\\), use the bound above and reduce to showing finiteness of\n\\[\nI(x)\\coloneqq \\int_{B_{r_0}(x)}\\bigl(1+|\\log \\mathrm{d}_g(x,y)|\\bigr)^2\\,dy.\n\\]\nIn geodesic polar coordinates \\(y=\\exp_x(\\rho\\theta)\\), the Jacobian satisfies \\(J_x(\\rho,\\theta)=\\rho a_x(\\rho,\\theta)\\le C_1\\rho\\), hence\n\\[\nI(x)\\le \\frac{C_1}{\\mathrm{vol}(M)}\\int_{\\mathbb{S}^1}\\!\\!\\int_0^{r_0}(1+|\\log\\rho|)^2\\rho\\,d\\rho\\,d\\theta.\n\\]\nThe radial integral is finite because “near \\(0\\) the function \\(\\rho\\mapsto \\rho(\\log\\rho)^2\\) is integrable,” shown via the substitution \\(t=-\\log\\rho\\) giving an integrand \\((1+|t|)^2e^{-2t}\\). This yields a uniform bound on \\(I(x)\\), so integrating over \\(x\\) gives \\(\\int_D G^2<\\infty\\). Combining \\(D\\) and \\(D^c\\) proves \\(G\\in L^2(M\\times M)\\) and hence \\(\\sigma^2<\\infty\\).", "expanded_sketch": "Using the 2D Green function singularity, write\n\\[\nH(x,y)\\coloneqq G(x,y)+\\frac{1}{2\\pi}\\log \\mathrm{d}_g(x,y)\n\\]\nwhich extends continuously to a compact neighborhood of the diagonal, hence is bounded there. This gives the near-diagonal logarithmic bound\n\\[\n|G(x,y)|\\le C\\bigl(1+|\\log \\mathrm{d}_g(x,y)|\\bigr)\\qquad (\\mathrm{d}_g(x,y)<2r_0).\n\\]\nSplit \\(M\\times M\\) into the near-diagonal region \\(D=\\{\\mathrm{d}_g(x,y)<r_0\\}\\) and its complement \\(D^c\\). On \\(D^c\\), \\(G\\) is smooth and therefore bounded, so \\(\\int_{D^c}G^2<\\infty\\).\n\nOn \\(D\\), use the bound above and reduce to showing finiteness of\n\\[\nI(x)\\coloneqq \\int_{B_{r_0}(x)}\\bigl(1+|\\log \\mathrm{d}_g(x,y)|\\bigr)^2\\,dy.\n\\]\nIn geodesic polar coordinates \\(y=\\exp_x(\\rho\\theta)\\), the Jacobian satisfies \\(J_x(\\rho,\\theta)=\\rho a_x(\\rho,\\theta)\\le C_1\\rho\\), hence\n\\[\nI(x)\\le \\frac{C_1}{\\mathrm{vol}(M)}\\int_{\\mathbb{S}^1}\\!\\!\\int_0^{r_0}(1+|\\log\\rho|)^2\\rho\\,d\\rho\\,d\\theta.\n\\]\nThe radial integral is finite because near \\(0\\) the function \\(\\rho\\mapsto \\rho(\\log\\rho)^2\\) is integrable; via the substitution \\(t=-\\log\\rho\\) the integrand becomes \\((1+|t|)^2e^{-2t}\\). This yields a uniform bound on \\(I(x)\\), so integrating over \\(x\\) gives \\(\\int_D G^2<\\infty\\). Combining \\(D\\) and \\(D^c\\) proves \\(G\\in L^2(M\\times M)\\) and hence \\(\\sigma^2<\\infty\\).", "expanded_theorem": "\\label{lem:L2Green}\nLet $(M,g)$ be a compact connected $2$-dimensional Riemannian manifold without boundary, and let $G$ be the symmetric mean-zero Green function of $-\\Delta$.\nIt then holds that $G\\in L^2(M\\times M, dx\\otimes dx)$, and so\n\\[\n\\sigma^2 \\coloneqq \\int_{M}\\int_{M} G(x,y)^2\\,dx\\,dy <\\infty.\n\\]", "theorem_type": ["Universal", "Existence"], "mcq": {"question": "Let $(M,g)$ be a compact connected $2$-dimensional Riemannian manifold without boundary, let $dx=\\operatorname{vol}(M)^{-1}\\,d\\operatorname{vol}$ be the normalized volume measure, and let $G:M\\times M\\to \\mathbb{R}\\cup\\{+\\infty\\}$ be a symmetric mean-zero Green function for $-\\Delta$, meaning that $G(x,y)=G(y,x)$, for each fixed $x\\in M$ the map $y\\mapsto G(x,y)$ is locally integrable and satisfies $\\int_M G(x,y)\\,dy=0$, and for each $f\\in C^\\infty(M)$ with $\\int_M f\\,dx=0$, the function $u(x)=\\int_M G(x,y)f(y)\\,dy$ is a weak solution of $-\\Delta u=f$ with $\\int_M u\\,dx=0$. Which statement holds for every such $(M,g)$ and $G$?", "correct_choice": {"label": "A", "text": "The Green function is square-integrable on $M\\times M$ with respect to the product measure $dx\\otimes dx$; equivalently, $G\\in L^2(M\\times M,dx\\otimes dx)$ and therefore\n\\[\n\\sigma^2\\coloneqq \\int_M\\int_M G(x,y)^2\\,dx\\,dy<\\infty.\n\\]"}, "choices": [{"label": "B", "text": "The Green function is essentially bounded on $M\\times M$ with respect to the product measure $dx\\otimes dx$; equivalently, $G\\in L^\\infty(M\\times M,dx\\otimes dx)$ and therefore\n\\[\n\\int_M\\int_M G(x,y)^2\\,dx\\,dy<\\infty.\n\\]"}, {"label": "C", "text": "For each fixed $x\\in M$, the section $y\\mapsto G(x,y)$ belongs to $L^2(M,dy)$, and in particular\n\\[\n\\int_M G(x,y)^2\\,dy<\\infty\\qquad\\text{for every }x\\in M.\n\\]"}, {"label": "D", "text": "The Green function is square-integrable off the diagonal but not on all of $M\\times M$; more precisely, for every $\\varepsilon>0$ one has\n\\[\nG\\in L^2\\bigl(\\{(x,y)\\in M\\times M:\\ d_g(x,y)\\ge \\varepsilon\\},dx\\otimes dx\\bigr),\n\\]\nwhereas\n\\[\n\\int_M\\int_M G(x,y)^2\\,dx\\,dy=+\\infty.\n\\]"}, {"label": "E", "text": "For each fixed $x\\in M$, the map $y\\mapsto G(x,y)$ belongs to $L^p(M,dy)$ for every $p>2$, and consequently\n\\[\nG\\in L^p(M\\times M,dx\\otimes dx)\\qquad\\text{for every }p>2.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "logarithmic singularity near the diagonal is bounded only after subtracting the log term, not in absolute size", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "joint $L^2(M\\times M)$ conclusion weakened to fiberwise $L^2$ in the $y$-variable for each fixed $x$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "near-diagonal integrability of $\\rho(\\log\\rho)^2$", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "critical exponent from the logarithmic singularity: $L^2$ holds but no gain to all $p>2$", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only the setup and defining properties of the Green function; it does not reveal or strongly hint that square-integrability on M×M is the intended conclusion."}, "TAS": {"score": 2, "justification": "The item is not a bare restatement of a theorem in the stem. It asks the student to distinguish among several regularity/integrability claims (L1, L2, L∞, sectionwise bounds, divergence), so it involves competing conclusions."}, "GPS": {"score": 1, "justification": "A solver must reason about the 2D logarithmic singularity of the Green function and which integrability thresholds it satisfies, so some genuine mathematical reasoning is required. However, the pressure is weakened because choice C is also true, making the task less about uniquely generating the best conclusion."}, "DQS": {"score": 1, "justification": "B, D, and E are plausible and target common misconceptions about regularity and singularities. But C is a weaker true statement, so it is not a valid distractor under a single-answer format, which significantly harms the quality of the option set."}, "total_score": 6, "overall_assessment": "Conceptually strong and free of answer leakage, but the MCQ is flawed as a single-choice item because at least one non-key option (C) is also true. This ambiguity weakens both the distractor set and the assessment of reasoning."}}
{"id": "2602.07843v1", "paper_link": "http://arxiv.org/abs/2602.07843v1", "theorems_cnt": 3, "theorem": {"env_name": "lemma", "content": "\\label{lem:L2Green}\nLet $(M,g)$ be a compact connected $2$-dimensional Riemannian manifold without boundary, and let $G$ be the symmetric mean-zero Green function of $-\\Delta$.\nIt then holds that $G\\in L^2(M\\times M, dx\\otimes dx)$, and so\n\\[\n\\sigma^2 \\coloneqq \\int_{M}\\int_{M} G(x,y)^2\\,dx\\,dy <\\infty.\n\\]", "start_pos": 6355, "end_pos": 6692, "label": "lem:L2Green"}, "ref_dict": {"eq:GreenLogBound": "\\begin{equation}\\label{eq:GreenLogBound}\n|G(x,y)| \\le \\frac{1}{2\\pi}|\\log \\dist(x,y)|+C_0 \\le C\\bigl(1+|\\log \\dist(x,y)|\\bigr)\n\\end{equation}", "prob:Steinerberger": "\\begin{problem}[Steinerberger {\\cite[Problem 53]{SteOpen}}]\\label{prob:Steinerberger}\nIn dimension $d=2$, can one replace the remainder $\\sqrt{\\frac{\\log n}{n}}$ in \\eqref{eq:Steinerberger2D}\nby $\\frac{1}{\\sqrt n}$ while keeping the same off-diagonal Green term? In other terms, does there exist\n$C_\\M>0$ such that for all $n\\in\\mathbb N$ and all $x_1,\\dots,x_n\\in \\M$,\n\\[\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\le C_\\M\\left(\\frac1{\\sqrt n}\n+\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\\right)?\n\\]\n\\end{problem}", "eq:GreenMeanZero": "\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{\\M} G(x,y)\\,dy=0.\n\\end{equation}", "eq:Steinerberger2D": "\\begin{equation}\\label{eq:Steinerberger2D}\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_\\M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}", "thm:main": "\\begin{theorem}[The $\\sqrt{\\log n}$ remainder]\\label{thm:main}\nLet $G$ be the symmetric mean-zero Green function of $-\\Delta$.\nThere does not exist a constant $C_{\\M}>0$ such that for all $n\\in\\mathbb{N}$ and all $x_1,\\dots,x_n\\in \\M$,\n\\begin{equation}\\label{eq:NoUniversalO1sqrtN}\n\\W\\!\\left(\\frac1n\\sum_{i=1}^n\\delta_{x_i},\\, dx\\right)\n\\le\nC_{\\M}\\left(\\frac{1}{\\sqrt{n}}+\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\\right).\n\\end{equation}\nIn particular, the $\\sqrt{\\log n}$ factor in the two-dimensional inequality \\eqref{eq:Steinerberger2D} cannot be removed if the unrenormalized Green-energy term were to be preserved.\n\\end{theorem}", "def:green": "\\begin{definition}[Mean-zero Green function]\\label{def:green}\nA mean-zero Green function for $-\\Delta$ is a measurable function\n\\[\nG:\\M\\times \\M \\to \\mathbb{R}\\cup\\{+\\infty\\}\n\\]\nwith the following properties.\n\\begin{enumerate}\n\\item For each $x\\in \\M$, $y\\mapsto G(x,y)$ is locally integrable on $\\M$ and satisfies the normalization:\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{\\M} G(x,y)\\,dy=0.\n\\end{equation}\n\\item For each $f\\in C^\\infty(\\M)$ with $\\int_{\\M} f\\,dx=0$, the function\n\\[\nu(x)\\coloneqq \\int_{\\M} G(x,y)\\, f(y)\\,dy\n\\]\nis a weak solution of $-\\Delta u = f$ with $\\int_{\\M}u\\,dx=0$.\n\\end{enumerate}\n\\end{definition}"}, "pre_theorem_intro_text_len": 4896, "pre_theorem_intro_text": "Let $(M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary, $\\mathrm{d}_g$ is the induced distance, and let $G(x,y)$ denote the symmetric mean-zero Green function of the Laplacian (Definition \\ref{def:green}). We work with the normalized volume measure $dx=\\mathrm{vol}(M)^{-1}\\,\\mathrm{d}\\mathrm{vol}$. Whenever we integrate in the variable $y$, we write $dy$ for the same measure. For points $x_1,\\dots,x_n\\in M$, we define the empirical measure:\n\\[\n\\mu_n \\coloneqq \\frac1n\\sum_{i=1}^n \\delta_{x_i}.\n\\]\nSteinerberger \\cite{Ste21} showed that for $d\\ge 3$, the following Green--Wasserstein inequality holds:\n\\[\nW_2(\\mu_n,dx)\\ \\lesssim_{M}\\ n^{-1/d} + \\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}.\n\\]\nIn \\cite[Problem 53]{SteOpen}, he gives the following estimate:\n\\begin{equation}\\label{eq:Steinerberger2D}\nW_2\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}\nIn this note, we use\n\\[\n\\sum_{i\\neq j}(\\cdots)\\ \\coloneqq\\ \\sum_{\\substack{1\\le i,j\\le n\\\\ i\\neq j}}(\\cdots)\n\\]\nto denote the ordered sum over distinct indices. Notice that, by symmetry of $G$, it is twice the sum over $i<j$. Steinerberger asked whether the factor $\\sqrt{\\log n}$ in \\eqref{eq:Steinerberger2D} can be removed while keeping the same Green-energy term. The precise question is as follows.\n\n\\begin{problem}[Steinerberger {\\cite[Problem 53]{SteOpen}}]\\label{prob:Steinerberger}\nIn dimension $d=2$, can one replace the remainder $\\sqrt{\\frac{\\log n}{n}}$ in \\eqref{eq:Steinerberger2D}\nby $\\frac{1}{\\sqrt n}$ while keeping the same off-diagonal Green term? In other terms, does there exist\n$C_M>0$ such that for all $n\\in\\mathbb N$ and all $x_1,\\dots,x_n\\in M$,\n\\[\nW_2\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\le C_M\\left(\\frac1{\\sqrt n}\n+\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\\right)?\n\\]\n\\end{problem}\n\\noindent Our main result, Theorem \\ref{thm:main}, gives a negative answer to Problem \\ref{prob:Steinerberger}.\n\n\\begin{remark}[The diagonal]\\label{rem:diagonal}\nWe note that the Green function $G$ is smooth on $(M\\times M)\\setminus \\{(x,x):x\\inM\\}$ with a logarithmic singularity along the diagonal.\nIn a deterministic context, if $x_i=x_j$ for some $i\\neq j$, then the term $G(x_i,x_j)$ is not finite. It is then natural to interpret the Green term in \\eqref{eq:Steinerberger2D} and Problem \\ref{prob:Steinerberger} as $+\\infty$, in which case the inequality holds trivially.\nIn the random settings that we use in this note, collisions occur with probability $0$ because $dx$ is non-atomic.\n\\end{remark}\n\nWe now recall some definitions that will be useful in the main argument.\nFor Borel probability measures $\\mu,\\nu$ on $M$, let us denote by $\\Gamma(\\mu,\\nu)$ the set of couplings of $\\mu$ and $\\nu$ on $M\\times M$.\nThe quadratic Wasserstein distance is\n\\[\nW_2(\\mu,\\nu)\\coloneqq \\left(\\inf_{\\gamma\\in \\Gamma(\\mu,\\nu)} \\int_{M\\times M} \\mathrm{d}_g(x,y)^2\\,\\,\\mathrm{d}\\gamma(x,y)\\right)^{1/2}.\n\\]\nGiven that $M$ is compact, it follows that $W_2(\\mu,\\nu)$ is always finite and satisfies $W_2(\\mu,\\nu)\\le \\mathrm{diam}(M)$. Let $\\Delta$ be the Laplace--Beltrami operator on $(M,g)$, which is realized here as a self-adjoint operator on $L^2(M,dx)$ with domain $H^2(M)$. Again, since $M$ is compact and connected, $\\ker(-\\Delta)$ consists of the constant functions.\n\n\\begin{definition}[Mean-zero Green function]\\label{def:green}\nA mean-zero Green function for $-\\Delta$ is a measurable function\n\\[\nG:M\\times M \\to \\mathbb{R}\\cup\\{+\\infty\\}\n\\]\nwith the following properties.\n\\begin{enumerate}\n\\item For each $x\\in M$, $y\\mapsto G(x,y)$ is locally integrable on $M$ and satisfies the normalization:\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{M} G(x,y)\\,dy=0.\n\\end{equation}\n\\item For each $f\\in C^\\infty(M)$ with $\\int_{M} f\\,dx=0$, the function\n\\[\nu(x)\\coloneqq \\int_{M} G(x,y)\\, f(y)\\,dy\n\\]\nis a weak solution of $-\\Delta u = f$ with $\\int_{M}u\\,dx=0$.\n\\end{enumerate}\n\\end{definition}\n\nRecall that the uniqueness and existence of the mean-zero Green function, as a distribution kernel, are guaranteed. We can also choose it to be symmetric (for example, see \\cite{Aubin98} or \\cite{Rosenberg97}). We now fix a symmetric mean-zero Green function $G$, so that $G(x,y)=G(y,x)$ holds.\nBy symmetry and \\eqref{eq:GreenMeanZero}, we obtain:\n\\[\n\\int_{M}G(y,x)\\,dy=0\\qquad\\text{for every }x\\in M.\n\\]\n\n\\begin{remark}[Local singularity]\\label{rem:logsing}\nIn dimension $2$, the following classical local expansion (in geodesic normal coordinates) is true:\n\\[\nG(x,y)= -\\frac{1}{2\\pi}\\log \\mathrm{d}_g(x,y) + H(x,y).\n\\]\nNote that $H$ extends continuously to the diagonal and is smooth off the diagonal.\nIn particular, we have $G(x,\\cdot)\\in L^1(M,dx)$ for each fixed $x$.\n\\end{remark}", "context": "Let $(M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary, $\\mathrm{d}_g$ is the induced distance, and let $G(x,y)$ denote the symmetric mean-zero Green function of the Laplacian (Definition \\ref{def:green}). We work with the normalized volume measure $dx=\\mathrm{vol}(M)^{-1}\\,\\mathrm{d}\\mathrm{vol}$. Whenever we integrate in the variable $y$, we write $dy$ for the same measure. For points $x_1,\\dots,x_n\\in M$, we define the empirical measure:\n\\[\n\\mu_n \\coloneqq \\frac1n\\sum_{i=1}^n \\delta_{x_i}.\n\\]\nSteinerberger \\cite{Ste21} showed that for $d\\ge 3$, the following Green--Wasserstein inequality holds:\n\\[\nW_2(\\mu_n,dx)\\ \\lesssim_{M}\\ n^{-1/d} + \\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}.\n\\]\nIn \\cite[Problem 53]{SteOpen}, he gives the following estimate:\n\\begin{equation}\\label{eq:Steinerberger2D}\nW_2\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}\nIn this note, we use\n\\[\n\\sum_{i\\neq j}(\\cdots)\\ \\coloneqq\\ \\sum_{\\substack{1\\le i,j\\le n\\\\ i\\neq j}}(\\cdots)\n\\]\nto denote the ordered sum over distinct indices. Notice that, by symmetry of $G$, it is twice the sum over $i<j$. Steinerberger asked whether the factor $\\sqrt{\\log n}$ in \\eqref{eq:Steinerberger2D} can be removed while keeping the same Green-energy term. The precise question is as follows.\n\n\\begin{remark}[The diagonal]\\label{rem:diagonal}\nWe note that the Green function $G$ is smooth on $(M\\times M)\\setminus \\{(x,x):x\\inM\\}$ with a logarithmic singularity along the diagonal.\nIn a deterministic context, if $x_i=x_j$ for some $i\\neq j$, then the term $G(x_i,x_j)$ is not finite. It is then natural to interpret the Green term in \\eqref{eq:Steinerberger2D} and Problem \\ref{prob:Steinerberger} as $+\\infty$, in which case the inequality holds trivially.\nIn the random settings that we use in this note, collisions occur with probability $0$ because $dx$ is non-atomic.\n\\end{remark}\n\nWe now recall some definitions that will be useful in the main argument.\nFor Borel probability measures $\\mu,\\nu$ on $M$, let us denote by $\\Gamma(\\mu,\\nu)$ the set of couplings of $\\mu$ and $\\nu$ on $M\\times M$.\nThe quadratic Wasserstein distance is\n\\[\nW_2(\\mu,\\nu)\\coloneqq \\left(\\inf_{\\gamma\\in \\Gamma(\\mu,\\nu)} \\int_{M\\times M} \\mathrm{d}_g(x,y)^2\\,\\,\\mathrm{d}\\gamma(x,y)\\right)^{1/2}.\n\\]\nGiven that $M$ is compact, it follows that $W_2(\\mu,\\nu)$ is always finite and satisfies $W_2(\\mu,\\nu)\\le \\mathrm{diam}(M)$. Let $\\Delta$ be the Laplace--Beltrami operator on $(M,g)$, which is realized here as a self-adjoint operator on $L^2(M,dx)$ with domain $H^2(M)$. Again, since $M$ is compact and connected, $\\ker(-\\Delta)$ consists of the constant functions.\n\n\\begin{definition}[Mean-zero Green function]\\label{def:green}\nA mean-zero Green function for $-\\Delta$ is a measurable function\n\\[\nG:M\\times M \\to \\mathbb{R}\\cup\\{+\\infty\\}\n\\]\nwith the following properties.\n\\begin{enumerate}\n\\item For each $x\\in M$, $y\\mapsto G(x,y)$ is locally integrable on $M$ and satisfies the normalization:\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{M} G(x,y)\\,dy=0.\n\\end{equation}\n\\item For each $f\\in C^\\infty(M)$ with $\\int_{M} f\\,dx=0$, the function\n\\[\nu(x)\\coloneqq \\int_{M} G(x,y)\\, f(y)\\,dy\n\\]\nis a weak solution of $-\\Delta u = f$ with $\\int_{M}u\\,dx=0$.\n\\end{enumerate}\n\\end{definition}\n\nRecall that the uniqueness and existence of the mean-zero Green function, as a distribution kernel, are guaranteed. We can also choose it to be symmetric (for example, see \\cite{Aubin98} or \\cite{Rosenberg97}). We now fix a symmetric mean-zero Green function $G$, so that $G(x,y)=G(y,x)$ holds.\nBy symmetry and \\eqref{eq:GreenMeanZero}, we obtain:\n\\[\n\\int_{M}G(y,x)\\,dy=0\\qquad\\text{for every }x\\in M.\n\\]\n\n\\begin{remark}[Local singularity]\\label{rem:logsing}\nIn dimension $2$, the following classical local expansion (in geodesic normal coordinates) is true:\n\\[\nG(x,y)= -\\frac{1}{2\\pi}\\log \\mathrm{d}_g(x,y) + H(x,y).\n\\]\nNote that $H$ extends continuously to the diagonal and is smooth off the diagonal.\nIn particular, we have $G(x,\\cdot)\\in L^1(M,dx)$ for each fixed $x$.\n\\end{remark}\n\n\\begin{definition}[Mean-zero Green function]\\label{def:green}\nA mean-zero Green function for $-\\Delta$ is a measurable function\n\\[\nG:\\M\\times \\M \\to \\mathbb{R}\\cup\\{+\\infty\\}\n\\]\nwith the following properties.\n\\begin{enumerate}\n\\item For each $x\\in \\M$, $y\\mapsto G(x,y)$ is locally integrable on $\\M$ and satisfies the normalization:\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{\\M} G(x,y)\\,dy=0.\n\\end{equation}\n\\item For each $f\\in C^\\infty(\\M)$ with $\\int_{\\M} f\\,dx=0$, the function\n\\[\nu(x)\\coloneqq \\int_{\\M} G(x,y)\\, f(y)\\,dy\n\\]\nis a weak solution of $-\\Delta u = f$ with $\\int_{\\M}u\\,dx=0$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{\\M} G(x,y)\\,dy=0.\n\\end{equation}\n\n\\begin{equation}\\label{eq:Steinerberger2D}\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_\\M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}\n\n\\begin{problem}[Steinerberger {\\cite[Problem 53]{SteOpen}}]\\label{prob:Steinerberger}\nIn dimension $d=2$, can one replace the remainder $\\sqrt{\\frac{\\log n}{n}}$ in \\eqref{eq:Steinerberger2D}\nby $\\frac{1}{\\sqrt n}$ while keeping the same off-diagonal Green term? In other terms, does there exist\n$C_\\M>0$ such that for all $n\\in\\mathbb N$ and all $x_1,\\dots,x_n\\in \\M$,\n\\[\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\le C_\\M\\left(\\frac1{\\sqrt n}\n+\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\\right)?\n\\]\n\\end{problem}", "full_context": "Let $(M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary, $\\mathrm{d}_g$ is the induced distance, and let $G(x,y)$ denote the symmetric mean-zero Green function of the Laplacian (Definition \\ref{def:green}). We work with the normalized volume measure $dx=\\mathrm{vol}(M)^{-1}\\,\\mathrm{d}\\mathrm{vol}$. Whenever we integrate in the variable $y$, we write $dy$ for the same measure. For points $x_1,\\dots,x_n\\in M$, we define the empirical measure:\n\\[\n\\mu_n \\coloneqq \\frac1n\\sum_{i=1}^n \\delta_{x_i}.\n\\]\nSteinerberger \\cite{Ste21} showed that for $d\\ge 3$, the following Green--Wasserstein inequality holds:\n\\[\nW_2(\\mu_n,dx)\\ \\lesssim_{M}\\ n^{-1/d} + \\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}.\n\\]\nIn \\cite[Problem 53]{SteOpen}, he gives the following estimate:\n\\begin{equation}\\label{eq:Steinerberger2D}\nW_2\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}\nIn this note, we use\n\\[\n\\sum_{i\\neq j}(\\cdots)\\ \\coloneqq\\ \\sum_{\\substack{1\\le i,j\\le n\\\\ i\\neq j}}(\\cdots)\n\\]\nto denote the ordered sum over distinct indices. Notice that, by symmetry of $G$, it is twice the sum over $i<j$. Steinerberger asked whether the factor $\\sqrt{\\log n}$ in \\eqref{eq:Steinerberger2D} can be removed while keeping the same Green-energy term. The precise question is as follows.\n\n\\begin{remark}[The diagonal]\\label{rem:diagonal}\nWe note that the Green function $G$ is smooth on $(M\\times M)\\setminus \\{(x,x):x\\inM\\}$ with a logarithmic singularity along the diagonal.\nIn a deterministic context, if $x_i=x_j$ for some $i\\neq j$, then the term $G(x_i,x_j)$ is not finite. It is then natural to interpret the Green term in \\eqref{eq:Steinerberger2D} and Problem \\ref{prob:Steinerberger} as $+\\infty$, in which case the inequality holds trivially.\nIn the random settings that we use in this note, collisions occur with probability $0$ because $dx$ is non-atomic.\n\\end{remark}\n\nWe now recall some definitions that will be useful in the main argument.\nFor Borel probability measures $\\mu,\\nu$ on $M$, let us denote by $\\Gamma(\\mu,\\nu)$ the set of couplings of $\\mu$ and $\\nu$ on $M\\times M$.\nThe quadratic Wasserstein distance is\n\\[\nW_2(\\mu,\\nu)\\coloneqq \\left(\\inf_{\\gamma\\in \\Gamma(\\mu,\\nu)} \\int_{M\\times M} \\mathrm{d}_g(x,y)^2\\,\\,\\mathrm{d}\\gamma(x,y)\\right)^{1/2}.\n\\]\nGiven that $M$ is compact, it follows that $W_2(\\mu,\\nu)$ is always finite and satisfies $W_2(\\mu,\\nu)\\le \\mathrm{diam}(M)$. Let $\\Delta$ be the Laplace--Beltrami operator on $(M,g)$, which is realized here as a self-adjoint operator on $L^2(M,dx)$ with domain $H^2(M)$. Again, since $M$ is compact and connected, $\\ker(-\\Delta)$ consists of the constant functions.\n\n\\begin{definition}[Mean-zero Green function]\\label{def:green}\nA mean-zero Green function for $-\\Delta$ is a measurable function\n\\[\nG:M\\times M \\to \\mathbb{R}\\cup\\{+\\infty\\}\n\\]\nwith the following properties.\n\\begin{enumerate}\n\\item For each $x\\in M$, $y\\mapsto G(x,y)$ is locally integrable on $M$ and satisfies the normalization:\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{M} G(x,y)\\,dy=0.\n\\end{equation}\n\\item For each $f\\in C^\\infty(M)$ with $\\int_{M} f\\,dx=0$, the function\n\\[\nu(x)\\coloneqq \\int_{M} G(x,y)\\, f(y)\\,dy\n\\]\nis a weak solution of $-\\Delta u = f$ with $\\int_{M}u\\,dx=0$.\n\\end{enumerate}\n\\end{definition}\n\nRecall that the uniqueness and existence of the mean-zero Green function, as a distribution kernel, are guaranteed. We can also choose it to be symmetric (for example, see \\cite{Aubin98} or \\cite{Rosenberg97}). We now fix a symmetric mean-zero Green function $G$, so that $G(x,y)=G(y,x)$ holds.\nBy symmetry and \\eqref{eq:GreenMeanZero}, we obtain:\n\\[\n\\int_{M}G(y,x)\\,dy=0\\qquad\\text{for every }x\\in M.\n\\]\n\n\\begin{remark}[Local singularity]\\label{rem:logsing}\nIn dimension $2$, the following classical local expansion (in geodesic normal coordinates) is true:\n\\[\nG(x,y)= -\\frac{1}{2\\pi}\\log \\mathrm{d}_g(x,y) + H(x,y).\n\\]\nNote that $H$ extends continuously to the diagonal and is smooth off the diagonal.\nIn particular, we have $G(x,\\cdot)\\in L^1(M,dx)$ for each fixed $x$.\n\\end{remark}\n\n\\begin{definition}[Mean-zero Green function]\\label{def:green}\nA mean-zero Green function for $-\\Delta$ is a measurable function\n\\[\nG:\\M\\times \\M \\to \\mathbb{R}\\cup\\{+\\infty\\}\n\\]\nwith the following properties.\n\\begin{enumerate}\n\\item For each $x\\in \\M$, $y\\mapsto G(x,y)$ is locally integrable on $\\M$ and satisfies the normalization:\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{\\M} G(x,y)\\,dy=0.\n\\end{equation}\n\\item For each $f\\in C^\\infty(\\M)$ with $\\int_{\\M} f\\,dx=0$, the function\n\\[\nu(x)\\coloneqq \\int_{\\M} G(x,y)\\, f(y)\\,dy\n\\]\nis a weak solution of $-\\Delta u = f$ with $\\int_{\\M}u\\,dx=0$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{\\M} G(x,y)\\,dy=0.\n\\end{equation}\n\n\\begin{equation}\\label{eq:Steinerberger2D}\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_\\M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}\n\n\\begin{problem}[Steinerberger {\\cite[Problem 53]{SteOpen}}]\\label{prob:Steinerberger}\nIn dimension $d=2$, can one replace the remainder $\\sqrt{\\frac{\\log n}{n}}$ in \\eqref{eq:Steinerberger2D}\nby $\\frac{1}{\\sqrt n}$ while keeping the same off-diagonal Green term? In other terms, does there exist\n$C_\\M>0$ such that for all $n\\in\\mathbb N$ and all $x_1,\\dots,x_n\\in \\M$,\n\\[\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\le C_\\M\\left(\\frac1{\\sqrt n}\n+\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\\right)?\n\\]\n\\end{problem}\n\nLet $(\\M,g)$ be a compact connected two-dimensional Riemannian manifold without boundary, $\\dist$ is the induced distance, and let $G(x,y)$ denote the symmetric mean-zero Green function of the Laplacian (Definition \\ref{def:green}). We work with the normalized volume measure $dx=\\mathrm{vol}(\\M)^{-1}\\dd\\mathrm{vol}$. Whenever we integrate in the variable $y$, we write $dy$ for the same measure. For points $x_1,\\dots,x_n\\in \\M$, we define the empirical measure:\n\\[\n\\mu_n \\coloneqq \\frac1n\\sum_{i=1}^n \\delta_{x_i}.\n\\]\nSteinerberger \\cite{Ste21} showed that for $d\\ge 3$, the following Green--Wasserstein inequality holds:\n\\[\n\\W(\\mu_n,dx)\\ \\lesssim_{\\M}\\ n^{-1/d} + \\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}.\n\\]\nIn \\cite[Problem 53]{SteOpen}, he gives the following estimate:\n\\begin{equation}\\label{eq:Steinerberger2D}\n\\W\\!\\left(\\frac1n\\sum_{k=1}^n \\delta_{x_k},\\,dx\\right)\n\\lesssim_\\M\n\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\n+\n\\begin{cases}\n\\sqrt{\\frac{\\log n}{n}} & \\text{if } d=2,\\\\\nn^{-1/d} & \\text{if } d\\ge 3.\n\\end{cases}\n\\end{equation}\nIn this note, we use\n\\[\n\\sum_{i\\neq j}(\\cdots)\\ \\coloneqq\\ \\sum_{\\substack{1\\le i,j\\le n\\\\ i\\neq j}}(\\cdots)\n\\]\nto denote the ordered sum over distinct indices. Notice that, by symmetry of $G$, it is twice the sum over $i<j$. Steinerberger asked whether the factor $\\sqrt{\\log n}$ in \\eqref{eq:Steinerberger2D} can be removed while keeping the same Green-energy term. The precise question is as follows.\n\n\\begin{remark}[The diagonal]\\label{rem:diagonal}\nWe note that the Green function $G$ is smooth on $(\\M\\times \\M)\\setminus \\{(x,x):x\\in\\M\\}$ with a logarithmic singularity along the diagonal.\nIn a deterministic context, if $x_i=x_j$ for some $i\\neq j$, then the term $G(x_i,x_j)$ is not finite. It is then natural to interpret the Green term in \\eqref{eq:Steinerberger2D} and Problem \\ref{prob:Steinerberger} as $+\\infty$, in which case the inequality holds trivially.\nIn the random settings that we use in this note, collisions occur with probability $0$ because $dx$ is non-atomic.\n\\end{remark}\n\nWe now recall some definitions that will be useful in the main argument.\nFor Borel probability measures $\\mu,\\nu$ on $\\M$, let us denote by $\\Gamma(\\mu,\\nu)$ the set of couplings of $\\mu$ and $\\nu$ on $\\M\\times \\M$.\nThe quadratic Wasserstein distance is\n\\[\n\\W(\\mu,\\nu)\\coloneqq \\left(\\inf_{\\gamma\\in \\Gamma(\\mu,\\nu)} \\int_{\\M\\times \\M} \\dist(x,y)^2\\,\\dd\\gamma(x,y)\\right)^{1/2}.\n\\]\nGiven that $\\M$ is compact, it follows that $\\W(\\mu,\\nu)$ is always finite and satisfies $\\W(\\mu,\\nu)\\le \\mathrm{diam}(\\M)$. Let $\\Delta$ be the Laplace--Beltrami operator on $(\\M,g)$, which is realized here as a self-adjoint operator on $L^2(\\M,dx)$ with domain $H^2(\\M)$. Again, since $\\M$ is compact and connected, $\\ker(-\\Delta)$ consists of the constant functions.\n\n\\begin{definition}[Mean-zero Green function]\\label{def:green}\nA mean-zero Green function for $-\\Delta$ is a measurable function\n\\[\nG:\\M\\times \\M \\to \\mathbb{R}\\cup\\{+\\infty\\}\n\\]\nwith the following properties.\n\\begin{enumerate}\n\\item For each $x\\in \\M$, $y\\mapsto G(x,y)$ is locally integrable on $\\M$ and satisfies the normalization:\n\\begin{equation}\\label{eq:GreenMeanZero}\n\\int_{\\M} G(x,y)\\,dy=0.\n\\end{equation}\n\\item For each $f\\in C^\\infty(\\M)$ with $\\int_{\\M} f\\,dx=0$, the function\n\\[\nu(x)\\coloneqq \\int_{\\M} G(x,y)\\, f(y)\\,dy\n\\]\nis a weak solution of $-\\Delta u = f$ with $\\int_{\\M}u\\,dx=0$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark}[Local singularity]\\label{rem:logsing}\nIn dimension $2$, the following classical local expansion (in geodesic normal coordinates) is true:\n\\[\nG(x,y)= -\\frac{1}{2\\pi}\\log \\dist(x,y) + H(x,y).\n\\]\nNote that $H$ extends continuously to the diagonal and is smooth off the diagonal.\nIn particular, we have $G(x,\\cdot)\\in L^1(\\M,dx)$ for each fixed $x$.\n\\end{remark}\n\n\\begin{proof}\nRecall that $\\mathrm{vol}$ denotes the Riemannian volume measure on $(\\M,g)$ and $dx = \\mathrm{vol}(\\M)^{-1}\\dd\\mathrm{vol}$. Let $\\mathrm{inj}(\\M)>0$ be the injectivity radius of $(\\M,g)$ and set $r_0\\coloneqq \\mathrm{inj}(\\M)/4$. We now apply standard results on the singularity structure of the Green function in dimension $2$ (for instance, see \\cite{Aubin98}). In particular, the function\n\\[\nH(x,y)\\coloneqq G(x,y)+\\frac{1}{2\\pi}\\log \\dist(x,y),\n\\qquad x\\neq y,\n\\]\nextends continuously to the set $\\{(x,y)\\in \\M\\times\\M: \\dist(x,y)\\le 2r_0\\}$ (this also holds across the diagonal). Observe that this set is compact, so there exists $C_0>0$ such that\n\\[\n|H(x,y)|\\le C_0\\qquad\\text{whenever }\\dist(x,y)\\le 2r_0.\n\\]\nAs a result, for all $x\\neq y$ with $\\dist(x,y)<2r_0$, it follows that\n\\begin{equation}\\label{eq:GreenLogBound}\n|G(x,y)| \\le \\frac{1}{2\\pi}|\\log \\dist(x,y)|+C_0 \\le C\\bigl(1+|\\log \\dist(x,y)|\\bigr)\n\\end{equation}\nfor some constant $C>0$.\n\nWe now split $\\M\\times\\M$ into the near-diagonal region\n\\[\nD\\coloneqq \\{(x,y)\\in\\M\\times\\M: \\dist(x,y)<r_0\\}\n\\]\nand its complement $D^c$. Notice that $D^c$ is compact and is always a positive distance away from the diagonal, so $G$ is smooth on $D^c$ and therefore bounded there. We obtain:\n\\[\n\\int_{D^c} G(x,y)^2\\,dx\\,dy<\\infty.\n\\]\nLet us now analyze $D$. By \\eqref{eq:GreenLogBound}, we set\n\\[\nI(x)\\coloneqq \\int_{B_{r_0}(x)} \\bigl(1+|\\log \\dist(x,y)|\\bigr)^2\\,dy,\n\\]\nso that\n\\[\n\\int_{D} G(x,y)^2\\,dx\\,dy\n\\le C^2\\int_{\\M} I(x)\\,dx.\n\\]\nWe also fix $x\\in\\M$. Note that on the ball $B_{r_0}(x)$, we may use geodesic polar coordinates centered at $x$: $y=\\exp_x(\\rho\\theta)$ with $\\rho\\in(0,r_0)$ and $\\theta\\in\\mathbb{S}^1$. In these coordinates, the volume form has the following form:\n\\[\n\\dd\\mathrm{vol}(y)=J_x(\\rho,\\theta)\\,\\dd\\rho\\,\\dd\\theta,\n\\qquad\nJ_x(\\rho,\\theta)=\\rho\\,a_x(\\rho,\\theta),\n\\]\nwhere $a_x$ is smooth and satisfies $a_x(0,\\theta)=1$.\nIn particular, by compactness, there exists $C_1>0$ with:\n\\begin{equation}\\label{eq:JacobianBound}\nJ_x(\\rho,\\theta)\\le C_1\\rho\\qquad\\text{for all }x\\in\\M,\\ \\theta\\in\\mathbb{S}^1,\\ \\rho\\in(0,r_0)\n\\end{equation}\n(see, for instance, \\cite{Chavel84}).\nWe now apply the normalization of $dx$ and obtain:\n\\begin{align*}\n\\int_{B_{r_0}(x)} \\bigl(1+|\\log \\dist(x,y)|\\bigr)^2\\,dy\n&=\\frac{1}{\\mathrm{vol}(\\M)}\\int_{\\mathbb{S}^1}\\!\\!\\int_0^{r_0} \\bigl(1+|\\log \\rho|\\bigr)^2 J_x(\\rho,\\theta)\\,\\dd\\rho\\,\\dd\\theta\\\\\n&\\le \\frac{C_1}{\\mathrm{vol}(\\M)}\\int_{\\mathbb{S}^1}\\!\\!\\int_0^{r_0} \\bigl(1+|\\log \\rho|\\bigr)^2 \\rho\\,\\dd\\rho\\,\\dd\\theta.\n\\end{align*}\nThe right-hand side is finite since near $0$ the function $\\rho\\mapsto \\rho(\\log \\rho)^2$ is integrable. We apply the substitution $t=-\\log\\rho$ (so $\\rho=e^{-t}$ and $\\rho\\,\\dd\\rho=-e^{-2t}\\,\\dd t$) and get:\n\\[\n\\int_0^{r_0} \\bigl(1+|\\log \\rho|\\bigr)^2\\rho\\,\\dd\\rho\n=\\int_{\\infty}^{-\\log r_0} \\bigl(1+|t|\\bigr)^2(-e^{-2t})\\,\\dd t\n=\\int_{-\\log r_0}^{\\infty} \\bigl(1+|t|\\bigr)^2 e^{-2t}\\,\\dd t <\\infty.\n\\]\nIt follows that $\\int_{B_{r_0}(x)} (1+|\\log \\dist(x,y)|)^2\\,dy$ is bounded uniformly in $x$.\nIf we integrate over $x$, we arrive at $\\int_D G^2\\,dx\\,dy<\\infty$. Finally, we combine the estimates on $D$ and $D^c$. This finishes the proof of $\\sigma^2<\\infty$.\n\\end{proof}\n\n\\begin{theorem}[The $\\sqrt{\\log n}$ remainder]\\label{thm:main}\nLet $G$ be the symmetric mean-zero Green function of $-\\Delta$.\nThere does not exist a constant $C_{\\M}>0$ such that for all $n\\in\\mathbb{N}$ and all $x_1,\\dots,x_n\\in \\M$,\n\\begin{equation}\\label{eq:NoUniversalO1sqrtN}\n\\W\\!\\left(\\frac1n\\sum_{i=1}^n\\delta_{x_i},\\, dx\\right)\n\\le\nC_{\\M}\\left(\\frac{1}{\\sqrt{n}}+\\frac1n\\left|\\sum_{i\\neq j} G(x_i,x_j)\\right|^{1/2}\\right).\n\\end{equation}\nIn particular, the $\\sqrt{\\log n}$ factor in the two-dimensional inequality \\eqref{eq:Steinerberger2D} cannot be removed if the unrenormalized Green-energy term were to be preserved.\n\\end{theorem}", "post_theorem_intro_text_len": 3405, "post_theorem_intro_text": "\\begin{proof}\nRecall that $\\mathrm{vol}$ denotes the Riemannian volume measure on $(M,g)$ and $dx = \\mathrm{vol}(M)^{-1}\\,\\mathrm{d}\\mathrm{vol}$. Let $\\mathrm{inj}(M)>0$ be the injectivity radius of $(M,g)$ and set $r_0\\coloneqq \\mathrm{inj}(M)/4$. We now apply standard results on the singularity structure of the Green function in dimension $2$ (for instance, see \\cite{Aubin98}). In particular, the function\n\\[\nH(x,y)\\coloneqq G(x,y)+\\frac{1}{2\\pi}\\log \\mathrm{d}_g(x,y),\n\\qquad x\\neq y,\n\\]\nextends continuously to the set $\\{(x,y)\\in M\\timesM: \\mathrm{d}_g(x,y)\\le 2r_0\\}$ (this also holds across the diagonal). Observe that this set is compact, so there exists $C_0>0$ such that\n\\[\n|H(x,y)|\\le C_0\\qquad\\text{whenever }\\mathrm{d}_g(x,y)\\le 2r_0.\n\\]\nAs a result, for all $x\\neq y$ with $\\mathrm{d}_g(x,y)<2r_0$, it follows that\n\\begin{equation}\\label{eq:GreenLogBound}\n|G(x,y)| \\le \\frac{1}{2\\pi}|\\log \\mathrm{d}_g(x,y)|+C_0 \\le C\\bigl(1+|\\log \\mathrm{d}_g(x,y)|\\bigr)\n\\end{equation}\nfor some constant $C>0$.\n\nWe now split $M\\timesM$ into the near-diagonal region\n\\[\nD\\coloneqq \\{(x,y)\\inM\\timesM: \\mathrm{d}_g(x,y)<r_0\\}\n\\]\nand its complement $D^c$. Notice that $D^c$ is compact and is always a positive distance away from the diagonal, so $G$ is smooth on $D^c$ and therefore bounded there. We obtain:\n\\[\n\\int_{D^c} G(x,y)^2\\,dx\\,dy<\\infty.\n\\]\nLet us now analyze $D$. By \\eqref{eq:GreenLogBound}, we set\n\\[\nI(x)\\coloneqq \\int_{B_{r_0}(x)} \\bigl(1+|\\log \\mathrm{d}_g(x,y)|\\bigr)^2\\,dy,\n\\]\nso that\n\\[\n\\int_{D} G(x,y)^2\\,dx\\,dy\n\\le C^2\\int_{M} I(x)\\,dx.\n\\]\nWe also fix $x\\inM$. Note that on the ball $B_{r_0}(x)$, we may use geodesic polar coordinates centered at $x$: $y=\\exp_x(\\rho\\theta)$ with $\\rho\\in(0,r_0)$ and $\\theta\\in\\mathbb{S}^1$. In these coordinates, the volume form has the following form:\n\\[\n\\,\\mathrm{d}\\mathrm{vol}(y)=J_x(\\rho,\\theta)\\,\\,\\mathrm{d}\\rho\\,\\,\\mathrm{d}\\theta,\n\\qquad\nJ_x(\\rho,\\theta)=\\rho\\,a_x(\\rho,\\theta),\n\\]\nwhere $a_x$ is smooth and satisfies $a_x(0,\\theta)=1$.\nIn particular, by compactness, there exists $C_1>0$ with:\n\\begin{equation}\\label{eq:JacobianBound}\nJ_x(\\rho,\\theta)\\le C_1\\rho\\qquad\\text{for all }x\\inM,\\ \\theta\\in\\mathbb{S}^1,\\ \\rho\\in(0,r_0)\n\\end{equation}\n(see, for instance, \\cite{Chavel84}).\nWe now apply the normalization of $dx$ and obtain:\n\\begin{align*}\n\\int_{B_{r_0}(x)} \\bigl(1+|\\log \\mathrm{d}_g(x,y)|\\bigr)^2\\,dy\n&=\\frac{1}{\\mathrm{vol}(M)}\\int_{\\mathbb{S}^1}\\!\\!\\int_0^{r_0} \\bigl(1+|\\log \\rho|\\bigr)^2 J_x(\\rho,\\theta)\\,\\,\\mathrm{d}\\rho\\,\\,\\mathrm{d}\\theta\\\\\n&\\le \\frac{C_1}{\\mathrm{vol}(M)}\\int_{\\mathbb{S}^1}\\!\\!\\int_0^{r_0} \\bigl(1+|\\log \\rho|\\bigr)^2 \\rho\\,\\,\\mathrm{d}\\rho\\,\\,\\mathrm{d}\\theta.\n\\end{align*}\nThe right-hand side is finite since near $0$ the function $\\rho\\mapsto \\rho(\\log \\rho)^2$ is integrable. We apply the substitution $t=-\\log\\rho$ (so $\\rho=e^{-t}$ and $\\rho\\,\\,\\mathrm{d}\\rho=-e^{-2t}\\,\\,\\mathrm{d} t$) and get:\n\\[\n\\int_0^{r_0} \\bigl(1+|\\log \\rho|\\bigr)^2\\rho\\,\\,\\mathrm{d}\\rho\n=\\int_{\\infty}^{-\\log r_0} \\bigl(1+|t|\\bigr)^2(-e^{-2t})\\,\\,\\mathrm{d} t\n=\\int_{-\\log r_0}^{\\infty} \\bigl(1+|t|\\bigr)^2 e^{-2t}\\,\\,\\mathrm{d} t <\\infty.\n\\]\nIt follows that $\\int_{B_{r_0}(x)} (1+|\\log \\mathrm{d}_g(x,y)|)^2\\,dy$ is bounded uniformly in $x$.\nIf we integrate over $x$, we arrive at $\\int_D G^2\\,dx\\,dy<\\infty$. Finally, we combine the estimates on $D$ and $D^c$. This finishes the proof of $\\sigma^2<\\infty$.\n\\end{proof}", "sketch": "Using the 2D Green function singularity, write\n\\[\nH(x,y)\\coloneqq G(x,y)+\\frac{1}{2\\pi}\\log \\mathrm{d}_g(x,y)\n\\]\nwhich “extends continuously” to a compact neighborhood of the diagonal, hence is bounded there. This gives the near-diagonal logarithmic bound\n\\[\n|G(x,y)|\\le C\\bigl(1+|\\log \\mathrm{d}_g(x,y)|\\bigr)\\qquad (\\mathrm{d}_g(x,y)<2r_0).\n\\]\nSplit \\(M\\times M\\) into the near-diagonal region \\(D=\\{\\mathrm{d}_g(x,y)<r_0\\}\\) and its complement \\(D^c\\). On \\(D^c\\), “\\(G\\) is smooth … and therefore bounded,” so \\(\\int_{D^c}G^2<\\infty\\).\n\nOn \\(D\\), use the bound above and reduce to showing finiteness of\n\\[\nI(x)\\coloneqq \\int_{B_{r_0}(x)}\\bigl(1+|\\log \\mathrm{d}_g(x,y)|\\bigr)^2\\,dy.\n\\]\nIn geodesic polar coordinates \\(y=\\exp_x(\\rho\\theta)\\), the Jacobian satisfies \\(J_x(\\rho,\\theta)=\\rho a_x(\\rho,\\theta)\\le C_1\\rho\\), hence\n\\[\nI(x)\\le \\frac{C_1}{\\mathrm{vol}(M)}\\int_{\\mathbb{S}^1}\\!\\!\\int_0^{r_0}(1+|\\log\\rho|)^2\\rho\\,d\\rho\\,d\\theta.\n\\]\nThe radial integral is finite because “near \\(0\\) the function \\(\\rho\\mapsto \\rho(\\log\\rho)^2\\) is integrable,” shown via the substitution \\(t=-\\log\\rho\\) giving an integrand \\((1+|t|)^2e^{-2t}\\). This yields a uniform bound on \\(I(x)\\), so integrating over \\(x\\) gives \\(\\int_D G^2<\\infty\\). Combining \\(D\\) and \\(D^c\\) proves \\(G\\in L^2(M\\times M)\\) and hence \\(\\sigma^2<\\infty\\).", "expanded_sketch": "Using the 2D Green function singularity, write\n\\[\nH(x,y)\\coloneqq G(x,y)+\\frac{1}{2\\pi}\\log \\mathrm{d}_g(x,y)\n\\]\nwhich extends continuously to a compact neighborhood of the diagonal, hence is bounded there. This gives the near-diagonal logarithmic bound\n\\[\n|G(x,y)|\\le C\\bigl(1+|\\log \\mathrm{d}_g(x,y)|\\bigr)\\qquad (\\mathrm{d}_g(x,y)<2r_0).\n\\]\nSplit \\(M\\times M\\) into the near-diagonal region \\(D=\\{\\mathrm{d}_g(x,y)<r_0\\}\\) and its complement \\(D^c\\). On \\(D^c\\), \\(G\\) is smooth and therefore bounded, so \\(\\int_{D^c}G^2<\\infty\\).\n\nOn \\(D\\), use the bound above and reduce to showing finiteness of\n\\[\nI(x)\\coloneqq \\int_{B_{r_0}(x)}\\bigl(1+|\\log \\mathrm{d}_g(x,y)|\\bigr)^2\\,dy.\n\\]\nIn geodesic polar coordinates \\(y=\\exp_x(\\rho\\theta)\\), the Jacobian satisfies \\(J_x(\\rho,\\theta)=\\rho a_x(\\rho,\\theta)\\le C_1\\rho\\), hence\n\\[\nI(x)\\le \\frac{C_1}{\\mathrm{vol}(M)}\\int_{\\mathbb{S}^1}\\!\\!\\int_0^{r_0}(1+|\\log\\rho|)^2\\rho\\,d\\rho\\,d\\theta.\n\\]\nThe radial integral is finite because near \\(0\\) the function \\(\\rho\\mapsto \\rho(\\log\\rho)^2\\) is integrable; via the substitution \\(t=-\\log\\rho\\) the integrand becomes \\((1+|t|)^2e^{-2t}\\). This yields a uniform bound on \\(I(x)\\), so integrating over \\(x\\) gives \\(\\int_D G^2<\\infty\\). Combining \\(D\\) and \\(D^c\\) proves \\(G\\in L^2(M\\times M)\\) and hence \\(\\sigma^2<\\infty\\).", "expanded_theorem": "\\label{lem:L2Green}\nLet $(M,g)$ be a compact connected $2$-dimensional Riemannian manifold without boundary, and let $G$ be the symmetric mean-zero Green function of $-\\Delta$.\nIt then holds that $G\\in L^2(M\\times M, dx\\otimes dx)$, and so\n\\[\n\\sigma^2 \\coloneqq \\int_{M}\\int_{M} G(x,y)^2\\,dx\\,dy <\\infty.\n\\]", "theorem_type": ["Universal", "Existence"], "mcq": {"question": "Let $(M,g)$ be a compact connected $2$-dimensional Riemannian manifold without boundary, let $dx=\\operatorname{vol}(M)^{-1}\\,d\\operatorname{vol}$ be the normalized volume measure, and let $G:M\\times M\\to \\mathbb{R}\\cup\\{+\\infty\\}$ be a symmetric mean-zero Green function for $-\\Delta$, meaning that $G(x,y)=G(y,x)$, for each fixed $x\\in M$ the map $y\\mapsto G(x,y)$ is locally integrable and satisfies $\\int_M G(x,y)\\,dy=0$, and for each $f\\in C^\\infty(M)$ with $\\int_M f\\,dx=0$, the function $u(x)=\\int_M G(x,y)f(y)\\,dy$ is a weak solution of $-\\Delta u=f$ with $\\int_M u\\,dx=0$. Which statement holds for every such $(M,g)$ and $G$?", "correct_choice": {"label": "A", "text": "The Green function is square-integrable on $M\\times M$ with respect to the product measure $dx\\otimes dx$; equivalently, $G\\in L^2(M\\times M,dx\\otimes dx)$ and therefore\n\\[\n\\sigma^2\\coloneqq \\int_M\\int_M G(x,y)^2\\,dx\\,dy<\\infty.\n\\]"}, "choices": [{"label": "B", "text": "The Green function is essentially bounded on $M\\times M$ with respect to the product measure $dx\\otimes dx$; equivalently, $G\\in L^\\infty(M\\times M,dx\\otimes dx)$ and therefore\n\\[\n\\int_M\\int_M G(x,y)^2\\,dx\\,dy<\\infty.\n\\]"}, {"label": "C", "text": "For each fixed $x\\in M$, the section $y\\mapsto G(x,y)$ belongs to $L^2(M,dy)$, and in particular\n\\[\n\\int_M G(x,y)^2\\,dy<\\infty\\qquad\\text{for every }x\\in M.\n\\]"}, {"label": "D", "text": "The Green function is square-integrable off the diagonal but not on all of $M\\times M$; more precisely, for every $\\varepsilon>0$ one has\n\\[\nG\\in L^2\\bigl(\\{(x,y)\\in M\\times M:\\ d_g(x,y)\\ge \\varepsilon\\},dx\\otimes dx\\bigr),\n\\]\nwhereas\n\\[\n\\int_M\\int_M G(x,y)^2\\,dx\\,dy=+\\infty.\n\\]"}, {"label": "E", "text": "For each fixed $x\\in M$, the map $y\\mapsto G(x,y)$ belongs to $L^p(M,dy)$ for every $p>2$, and consequently\n\\[\nG\\in L^p(M\\times M,dx\\otimes dx)\\qquad\\text{for every }p>2.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "logarithmic singularity near the diagonal is bounded only after subtracting the log term, not in absolute size", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "joint $L^2(M\\times M)$ conclusion weakened to fiberwise $L^2$ in the $y$-variable for each fixed $x$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "near-diagonal integrability of $\\rho(\\log\\rho)^2$", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "critical exponent from the logarithmic singularity: $L^2$ holds but no gain to all $p>2$", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state or strongly hint at the correct integrability conclusion; it only defines the Green function and asks about a nontrivial property."}, "TAS": {"score": 2, "justification": "This is not a restatement of the definition. It asks for a substantive consequence about global integrability of the 2D Green kernel, which goes beyond the setup."}, "GPS": {"score": 1, "justification": "The intended solution requires reasoning from the logarithmic diagonal singularity in dimension 2 to decide which integrability claims are valid. However, the item's ambiguity weakens the generative demand because more than one option appears true."}, "DQS": {"score": 0, "justification": "The distractors are not functioning well in a single-answer format: option C is a weaker true statement, and option E also appears true for a 2D logarithmic singularity (finite L^p for every finite p). This creates multiple correct answers and undermines validity."}, "total_score": 5, "overall_assessment": "Conceptually strong and non-tautological, but flawed as an MCQ because the answer set is not uniquely keyed: at least C, and likely E, are also true alongside A."}}
{"id": "2602.07957v1", "paper_link": "http://arxiv.org/abs/2602.07957v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{main}\n        Let $f_{\\varepsilon}$  be of the form $f_{\\varepsilon} = M + \\varepsilon M g_{\\varepsilon} $, and $\\{f_{\\varepsilon}\\}$ is a family of solutions to \\eqref{longtime_Boltzmann} with $\\tau_\\varepsilon=\\varepsilon$ satisfying the assumptions \\eqref{conservation_of_mass},\\eqref{conservation_of_momentum}, \\eqref{conservation_of_energy} and \\eqref{entropy_inequality}.\n        Let $(\\rho_{\\varepsilon}, u_{\\varepsilon},\\theta_{\\varepsilon})$ be of the form $(\\rho_{\\varepsilon},u_{\\varepsilon},\\theta_{\\varepsilon}) = (1+ \\tilde{\\rho}_{\\varepsilon},\\varepsilon \\tilde{u}_{\\varepsilon},1+ \\varepsilon \\tilde{\\theta}_{\\varepsilon})$, and $\\{(\\rho_{\\varepsilon},u_{\\varepsilon},\\theta_{\\varepsilon})\\}$ is a family of solutions to \\eqref{longtime_CNS}.\n        Then we have, for $t \\geqslant 0$,\n        \\begin{equation}\\label{main_estimate}\n            \\frac{1}{{\\varepsilon}^2} H(f_{\\varepsilon}| M_{\\varepsilon})(t)  \\leqslant \\frac{1}{\\varepsilon^2}H(f^{\\mathrm{in}}_{\\varepsilon}| M^{\\mathrm{in}}_{\\varepsilon} ) + \\mathcal{O}(\\varepsilon),\n        \\end{equation}\n        \\begin{equation}\\label{entropy_dissipation_bound}\n           \\int_0^t \\int_{\\mathbb{T}^3_x}\\frac{1}{\\varepsilon^4}D(f_\\varepsilon)(s,x) - \\frac{1}{2} \\mu \\sigma( \\mathrm{u} ^b_\\varepsilon): \\sigma( \\mathrm{u} ^b_\\varepsilon) - \\frac{5}{2}\\kappa (\\nabla_x \\theta^b_{\\varepsilon})^2\\  dxds \\lesssim \\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}^{\\mathrm{in}}|M_{\\varepsilon}^{\\mathrm{in}}) + \\mathcal{O}(\\varepsilon),\n         \\end{equation}\n         \\begin{equation}\\label{energy_momentum_flux_bound}\n             \\int_0^t \\int_{\\mathbb{T}^3_x} \\frac{1}{2}\\mu \\sigma(\\tilde{ \\mathrm{u} }_{\\varepsilon}- \\mathrm{u} ^b_{\\varepsilon}):\\sigma(\\tilde{ \\mathrm{u} }_{\\varepsilon}- \\mathrm{u} ^b_{\\varepsilon}) + \\frac{5}{2}  \\kappa(\\nabla_x \\tilde{\\theta} -\\nabla_x \\theta^b_{\\varepsilon})^2 \\ dxds \\lesssim \\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}^{\\mathrm{in}}|M_{\\varepsilon}^{\\mathrm{in}}) + \\mathcal{O}(\\varepsilon),\n        \\end{equation}\n        where\n        \\begin{equation}\n            M_{\\varepsilon}^{\\mathrm{in}} = M(1+ \\varepsilon \\tilde\\rho^{\\mathrm{in}}, \\varepsilon \\tilde{u}^{\\mathrm{in}},1 + \\varepsilon \\tilde{\\theta}^{\\mathrm{in}}).\n        \\end{equation}", "start_pos": 29487, "end_pos": 31716, "label": "main"}, "ref_dict": {"longtime_Boltzmann": "\\begin{equation}\\label{longtime_Boltzmann} \\tag{BE}\n    \\left\\{\\begin{aligned}\n        &\\tau_{\\varepsilon} \\partial_t f_{\\varepsilon} + v\\mdot \\nabla_x f_{\\varepsilon} =\\frac{1}{\\varepsilon} C(f_{\\varepsilon},\\ f_{\\varepsilon}), \\quad x\\in \\mathbb{T}^3, \\quad v \\in \\mathbb{R}^3\\\\\n        &f_{\\varepsilon}(0,x) = f_{\\varepsilon}^{\\mathrm{in}} \\geqslant 0,\n    \\end{aligned}\\right.\n\\end{equation}", "conservation_of_mass": "\\begin{align}\n            &\\partial_t \\int_{\\mathbb{R}^3_v} f_{\\varepsilon} dv + \\nabla_x \\cdot  \\int_{\\mathbb{R}^3_v} f_{\\varepsilon}dv = 0, \\label{conservation_of_mass} \\\\\n            &\\partial_t \\int_{\\mathbb{R}^3_v} vf_{\\varepsilon} dv + \\nabla_x \\cdot \\int_{\\mathbb{R}^3_v} vf_{\\varepsilon}dv =0,\\label{conservation_of_momentum}\\\\\n            &\\partial_t \\int_{\\mathbb{R}^3_v} |v|^2 f_{\\varepsilon} dv + \\nabla_x \\cdot \\int_{\\mathbb{R}^3_v} |v|^2f_{\\varepsilon}dv=0.\\label{conservation_of_energy}\\\\\n        \\end{align}", "local_viscosity_and_heat_conductivity": "\\begin{equation}\\label{local_viscosity_and_heat_conductivity}\n        \\begin{aligned}\n            \\mu(\\rho,\\theta) = \\frac{1}{10} \\int_{\\mathbb{R}^3_v} A(V) :  \\hat{A}(V) \\mathcal{M} dv,\\\\\n            \\kappa(\\rho,\\theta) = \\frac{2}{15} \\int_{\\mathbb{R}^3_v} B(V) \\cdot \\hat{B}(V) \\mathcal{M} dv,\n        \\end{aligned}\n    \\end{equation}", "entropy_inequality": "\\begin{equation}\\label{entropy_inequality}\n        H(f_{\\varepsilon}| M)(t) + \\frac{1}{\\varepsilon^2}\\int_0^t \\int_{\\mathbb{T}^3_x} D(f_\\varepsilon)(s,x) \\ ds dx \\leqslant H(f^{\\mathrm{in}}_{\\varepsilon}| M ),\n    \\end{equation}", "SS_EDC": "\\begin{aligned}\n        \\frac{1}{{\\varepsilon}^2} &H(f_{\\varepsilon}| M_{\\varepsilon})(t) +\\int_0^t \\int_{\\mathbb{T}^3_x}\\frac{1}{\\varepsilon^4}D(f_\\varepsilon)(s,x) - \\frac{1}{2} \\mu \\sigma(u^b): \\sigma(u^b) - \\frac{5}{2}\\kappa (\\nabla_x \\theta^b)^2\\  dxds \\\\\n        &+\\int_0^t \\int_{\\mathbb{T}^3_x} \\frac{1}{2}\\mu \\sigma(\\tilde{u}-u^b):\\sigma(\\tilde{u}-u^b) + \\frac{5}{2}  \\kappa(\\nabla_x \\tilde{\\theta} -\\nabla_x \\theta^b)^2 \\ dxds\\\\\n        &\\lesssim \\frac{1}{\\varepsilon^2} H(f^{\\mathrm{in}}_{\\varepsilon}| M^{\\mathrm{in}}_{\\varepsilon}) + \\int^t_0 \\|(\\nabla_x \\tilde{u},\\nabla_x \\tilde{\\theta})\\|_{L^{\\infty}(dx)} \\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}|M_{\\varepsilon}) ds +\\tilde{R} + R_5+R_6+R_7.\n    \\end{aligned}\n\\end{equation}\n\n\\subsection{Entropy dissipation control}\\label{SS_EDC}\n We now deal with the term\n\\begin{equation}\\label{dissipation_control}\n    \\begin{aligned}\n        \\int_0^t \\int_{\\mathbb{T}^3_x}\\frac{1}{\\varepsilon^4}D(f_\\varepsilon)(s,x) - \\frac{1}{2} \\mu \\sigma(u^b): \\sigma(u^b) - \\frac{5}{2}\\kappa (\\nabla_x \\theta^b)^2\\  dxds ,\n    \\end{aligned}", "RE": "\\begin{equation}\\label{RE}\n        H(f|g) = \\int_{\\mathbb{T}^3_x}\\int_{\\mathbb{R}^3_v} f \\log(\\frac{f}{g}) - f + g \\  dxdv.\n    \\end{equation}", "SS_Conclusion": "\\begin{aligned}\n        R_{12}= \\int_0^t \\int_{\\mathbb{T}^3_x} \\left( \\langle v \\mdot \\nabla_x \\Loth g_{\\varepsilon} , \\hat{A}\\rangle + \\langle \\varepsilon \\partial_t g_{\\varepsilon} , \\hat{A}\\rangle\\right) :\\sigma(u^b) \\ dxds ,\\\\\n        R_{13}=\\int_0^t \\int_{\\mathbb{T}^3_x}  \\left(\\langle v \\mdot \\nabla_x \\Loth g_{\\varepsilon} , \\hat{B}\\rangle + \\langle \\varepsilon \\partial_t g_{\\varepsilon} , \\hat{B}\\rangle\\right) \\cdot \\nabla_x \\theta^b \\ dxds.\n    \\end{aligned}\n\\end{equation}\n\n\\subsection{Conclusion} \\label{SS_Conclusion}\nWe now complete the proof of Theorem \\ref{main}.\nNote that\n\\begin{equation}\\label{quadratic_term}\n    \\int_0^t \\int_{\\mathbb{T}^3_x} \\frac{1}{2}\\mu \\sigma(\\tilde{u}-u^b):\\sigma(\\tilde{u}-u^b) + \\frac{5}{2}  \\kappa(\\nabla_x \\tilde{\\theta} -\\nabla_x \\theta^b)^2 \\ dxds \\geqslant 0.\n\\end{equation}\nCombining with \\eqref{mid_result} and \\eqref{dissipation_control_final_result} we have\n\\begin{equation}\n    \\begin{aligned}\n        \\frac{1}{{\\varepsilon}^2} &H(f_{\\varepsilon}| M_{\\varepsilon})(t)  \\\\\n        &\\lesssim \\frac{1}{\\varepsilon^2} H(f^{\\mathrm{in}}_{\\varepsilon}| M^{\\mathrm{in}}_{\\varepsilon}) + \\int^t_0 \\|(\\nabla_x \\tilde{u},\\nabla_x \\tilde{\\theta})\\|_{L^{\\infty}(dx)} \\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}|M_{\\varepsilon}) ds \\\\\n        & +\\tilde{R}+ R_5+R_6+R_7 -R_{11}-R_{12} - 2R_{13}.\n    \\end{aligned}", "entropy_dissipation_bound": "\\begin{equation}\\label{entropy_dissipation_bound}\n           \\int_0^t \\int_{\\mathbb{T}^3_x}\\frac{1}{\\varepsilon^4}D(f_\\varepsilon)(s,x) - \\frac{1}{2} \\mu \\sigma(\\u^b_\\varepsilon): \\sigma(\\u^b_\\varepsilon) - \\frac{5}{2}\\kappa (\\nabla_x \\theta^b_{\\varepsilon})^2\\  dxds \\lesssim \\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}^{\\mathrm{in}}|M_{\\varepsilon}^{\\mathrm{in}}) + \\mathcal{O}(\\varepsilon),\n         \\end{equation}", "conservation_of_momentum": "\\begin{align}\n            &\\partial_t \\int_{\\mathbb{R}^3_v} f_{\\varepsilon} dv + \\nabla_x \\cdot  \\int_{\\mathbb{R}^3_v} f_{\\varepsilon}dv = 0, \\label{conservation_of_mass} \\\\\n            &\\partial_t \\int_{\\mathbb{R}^3_v} vf_{\\varepsilon} dv + \\nabla_x \\cdot \\int_{\\mathbb{R}^3_v} vf_{\\varepsilon}dv =0,\\label{conservation_of_momentum}\\\\\n            &\\partial_t \\int_{\\mathbb{R}^3_v} |v|^2 f_{\\varepsilon} dv + \\nabla_x \\cdot \\int_{\\mathbb{R}^3_v} |v|^2f_{\\varepsilon}dv=0.\\label{conservation_of_energy}\\\\\n        \\end{align}", "INSF": "\\begin{equation}\\label{INSF} \\tag{INSF}\n        \\left\\{\n            \\begin{aligned}\n                &\\partial_t \\u + \\u \\mdot \\nabla_x \\u + \\nabla_x p = \\mu \\Delta \\u, \\\\\n                &\\partial_t \\vartheta + \\u \\mdot \\nabla_x \\vartheta = \\kappa \\Delta \\vartheta,\\\\\n                &\\nabla_x \\cdot \\u =0,\n            \\end{aligned}\n        \\right.\n    \\end{equation}", "perturbed_Boltzmann": "\\begin{equation}\\label{perturbed_Boltzmann}\n    \\left\\{ \\begin{matrix}\n        \\varepsilon^2 \\partial_t g_{\\varepsilon} + \\varepsilon v\\mdot \\nabla_x g_{\\varepsilon} + \\mathcal{L}g_{\\varepsilon} = \\varepsilon Q(g_{\\varepsilon},g_{\\varepsilon}),\\\\\n        g_{\\varepsilon}(0,x,v) = g^{{in}}_{\\varepsilon},\n    \\end{matrix}\\right.\n\\end{equation}", "main_estimate": "\\begin{equation}\\label{main_estimate}\n            \\frac{1}{{\\varepsilon}^2} H(f_{\\varepsilon}| M_{\\varepsilon})(t)  \\leqslant \\frac{1}{\\varepsilon^2}H(f^{\\mathrm{in}}_{\\varepsilon}| M^{\\mathrm{in}}_{\\varepsilon} ) + \\mathcal{O}(\\varepsilon),\n        \\end{equation}", "conservation_of_energy": "\\begin{align}\n            &\\partial_t \\int_{\\mathbb{R}^3_v} f_{\\varepsilon} dv + \\nabla_x \\cdot  \\int_{\\mathbb{R}^3_v} f_{\\varepsilon}dv = 0, \\label{conservation_of_mass} \\\\\n            &\\partial_t \\int_{\\mathbb{R}^3_v} vf_{\\varepsilon} dv + \\nabla_x \\cdot \\int_{\\mathbb{R}^3_v} vf_{\\varepsilon}dv =0,\\label{conservation_of_momentum}\\\\\n            &\\partial_t \\int_{\\mathbb{R}^3_v} |v|^2 f_{\\varepsilon} dv + \\nabla_x \\cdot \\int_{\\mathbb{R}^3_v} |v|^2f_{\\varepsilon}dv=0.\\label{conservation_of_energy}\\\\\n        \\end{align}", "longtime_CNS": "\\begin{equation} \\label{longtime_CNS}\\tag{$\\text{CNS}_{\\varepsilon}$}\n        \\left\\{ \\begin{aligned}\n            &\\varepsilon\\partial_t \\rho_{\\varepsilon} + \\nabla_x (\\rho_{\\varepsilon} u_{\\varepsilon}) = 0,\\\\\n            &\\varepsilon\\partial_t u_{\\varepsilon} + u_{\\varepsilon} \\mdot \\nabla_x u_{\\varepsilon}  + \\frac{1}{\\rho_{\\varepsilon}} \\nabla_x (\\rho_{\\varepsilon} \\theta_{\\varepsilon})  = \\varepsilon \\frac{1}{\\rho_{\\varepsilon}} \\nabla_x [ \\mu_{\\varepsilon}(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\sigma(u_{\\varepsilon})],\\\\\n            &\\varepsilon \\partial_t \\theta_{\\varepsilon}  + u_{\\varepsilon} \\mdot \\nabla_x \\theta_{\\varepsilon} + \\frac{2}{3} \\theta_{\\varepsilon}  \\nabla_x \\mdot u_{\\varepsilon} = \\varepsilon  \\frac{1}{\\rho_{\\varepsilon}} \\frac{1}{3} \\mu_{\\varepsilon}(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\sigma(u_{\\varepsilon}):\\sigma(u_{\\varepsilon}) +  \\frac{1}{\\rho_{\\varepsilon}}\\frac{5}{3} \\varepsilon \\nabla_x \\cdot[\\kappa(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\theta_{\\varepsilon}],\\\\\n            &(\\rho_{\\varepsilon}(0,x),\\ u_{\\varepsilon}(0,x), \\ \\theta_{\\varepsilon}(0,x)) = (1+ \\varepsilon \\tilde{\\rho}^{\\mathrm{in}},\\ \\tilde{u}^{\\mathrm{in}},\\ 1+ \\varepsilon \\tilde{\\theta}^{\\mathrm{in}}).\n        \\end{aligned}\\right.\n    \\end{equation}", "g_moments": "\\begin{equation}\\label{g_moments}\n    \\begin{aligned}\n        \\rho^b_{\\varepsilon} = \\langle g_{\\varepsilon} \\rangle,\\\n        u^b_{\\varepsilon} =\\langle  vg_{\\varepsilon} \\rangle,\\\n        \\theta^b_{\\varepsilon} = \\langle  \\frac{|v|^2 -3}{3} g_{\\varepsilon} \\rangle.\n    \\end{aligned}\n\\end{equation}", "SS_REC": "\\begin{aligned}\n        R_6&= \\int^t_0 \\int_{\\mathbb{T}^3_x} (\\frac{1}{\\rho \\theta^2}-1 ) \\frac{5}{2} \\nabla_x \\mdot [\\kappa(\\rho,\\theta) \\nabla_x \\tilde{\\theta} ](\\theta^b -\\tilde{\\theta}) + (1-\\frac{1}{\\theta^2})\\kappa \\nabla_x \\theta^b \\cdot \\nabla_x \\tilde{\\theta}\\ dxds\\\\\n         &+ \\int^t_0 \\int_{\\mathbb{T}^3_x} \\frac{5}{2} (\\kappa - \\kappa(\\rho,\\theta))\\nabla_x \\tilde{\\theta}\\cdot (\\nabla_x \\tilde{\\theta}-\\nabla_x \\theta^b) \\ dxds.\n    \\end{aligned}\n\\end{equation}\n\n\\subsection{Relative entropy control}\\label{SS_REC}\nThe terms \\Rmnum{3} and \\Rmnum{4} are unsigned, however they can be controlled by the quadratic of $(\\rho^b-\\tilde{\\rho}^{\\varepsilon}, u^b - \\tilde{u}^{\\varepsilon},\\theta^b - \\tilde{\\theta}^{\\varepsilon})$ which  can be controlled by relative entropy $\\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}|M_{\\varepsilon})$.\nIn fact, we have the following two lemmas.\n\\begin{lemma} \\label{same_moments}\n    \\begin{equation}\n        \\begin{aligned}\n            H(f_{\\varepsilon}|M_{\\varepsilon}) = H(f_{\\varepsilon}| M_{f_\\varepsilon}) + H(M_{f_{\\varepsilon}}|M_{\\varepsilon}),\n        \\end{aligned}", "energy_momentum_flux_bound": "\\begin{equation}\\label{energy_momentum_flux_bound}\n             \\int_0^t \\int_{\\mathbb{T}^3_x} \\frac{1}{2}\\mu \\sigma(\\tilde{\\u}_{\\varepsilon}-\\u^b_{\\varepsilon}):\\sigma(\\tilde{\\u}_{\\varepsilon}-\\u^b_{\\varepsilon}) + \\frac{5}{2}  \\kappa(\\nabla_x \\tilde{\\theta} -\\nabla_x \\theta^b_{\\varepsilon})^2 \\ dxds \\lesssim \\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}^{\\mathrm{in}}|M_{\\varepsilon}^{\\mathrm{in}}) + \\mathcal{O}(\\varepsilon),\n        \\end{equation}", "short_CNSF": "\\begin{equation}\\label{short_CNSF} \\tag{CNS}\n    \\left\\{ \\begin{aligned}\n        &\\partial_t \\rho_{\\varepsilon} + \\nabla_x (\\rho_{\\varepsilon} u_{\\varepsilon}) = 0,\\\\\n        & \\rho_{\\varepsilon} (\\partial_t + u_{\\varepsilon} \\mdot \\nabla_x) u_{\\varepsilon}  + \\nabla_x ({\\rho_{\\varepsilon} \\theta_{\\varepsilon}}) = \\varepsilon \\nabla_x [ \\mu_{\\varepsilon}(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\sigma(u_{\\varepsilon})],\\\\\n        & \\tfrac{3}{2} \\rho_{\\varepsilon} (\\partial_t   + u_{\\varepsilon} \\mdot \\nabla_x )\\theta_{\\varepsilon} + \\rho_{\\varepsilon} \\theta_{\\varepsilon}  \\nabla_x \\mdot u_{\\varepsilon} = \\varepsilon  \\tfrac{1}{2} \\mu_{\\varepsilon}(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\sigma(u_{\\varepsilon}):\\sigma(u_{\\varepsilon}) + \\tfrac{5}{2} \\varepsilon \\nabla_x \\cdot[\\kappa(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\theta_{\\varepsilon}],\\\\\n        &(\\rho_{\\varepsilon}(0,x),\\ u_{\\varepsilon}(0,x), \\ \\theta_{\\varepsilon}(0,x)) =  (\\rho^{\\mathrm{in}},\\ u^{\\mathrm{in}},\\ \\theta^{\\mathrm{in}}).\n    \\end{aligned}\\right.\n\\end{equation}"}, "pre_theorem_intro_text_len": 26962, "pre_theorem_intro_text": "In this paper, we study, in the formal level, the asymptotic behavior of the following rescaled Boltzmann equation to the compressible Navier-Stokes system with small Mach number. To avoid the difficulty might be brought from the boundary, we consider the periodic domain $\\mathbb{T}^3$.\n\\begin{equation}\\label{longtime_Boltzmann} \\tag{BE}\n    \\left\\{\\begin{aligned}\n        &\\tau_{\\varepsilon} \\partial_t f_{\\varepsilon} + v \\!\\cdot\\! \\nabla_x f_{\\varepsilon} =\\frac{1}{\\varepsilon} C(f_{\\varepsilon},\\ f_{\\varepsilon}), \\quad x\\in \\mathbb{T}^3, \\quad v \\in \\mathbb{R}^3\\\\\n        &f_{\\varepsilon}(0,x) = f_{\\varepsilon}^{\\mathrm{in}} \\geqslant 0,\n    \\end{aligned}\\right.\n\\end{equation}\n where $f_{\\varepsilon} = f_{\\varepsilon}(t,x,v)$ denotes the number density of gas molecules at position $ x\\in \\mathbb{T}^3$ and time $t \\geqslant 0$. Moreover, $\\varepsilon$ denotes the Knudsen number which is the ratio of the mean free path and the macroscopic length. $\\tau_{\\varepsilon}$ denotes the Mach number which is the ratio of the bulk velocity to the sound speed.  The case $\\tau_{\\varepsilon} =1 $ is corresponding to the short time scale (or Euler time scale), and in this paper we shall focus on the case $\\tau_{\\varepsilon} = \\varepsilon $ which is corresponding to the longer time scale (or Navier-Stokes scale).\nThe smaller the Knudsen number is, the behavior of the Boltzmann equation is closer to the fluid equations. The limiting process when $\\varepsilon$ goes to zero is called hydrodynamic limit.\n\nThe operator $ C(f,g)$ is the collision operator which describe the binary elastic collision between particles and is defined as follows\n\\begin{equation}\n    \\begin{aligned}\n        C( f , g)  = \\frac{1}{2}\\iint_{\\mathbb{T}^3_x \\times \\mathbb{S}^2} B(v-v_1 , \\sigma) \\left( g^\\prime_1  f^\\prime + f^\\prime_1  g^\\prime -g_1 f -f_1 g   \\right) d\\sigma dv_1.\n    \\end{aligned}\n\\end{equation}\nHere and in the sequel we use the notation\n\\begin{equation}\n    f^{\\prime}_{1} = f(t,x,v^{\\prime}_1), \\ f^{\\prime} = f(t,x,v^{\\prime}),\\ f_{1} = f(t,x,v_1),\n\\end{equation}\nand for $\\sigma \\in \\mathbb{S}^2 $,\n\\begin{equation}\n    v^{\\prime} = \\frac{v+ v_1}{2}+ \\frac{|v-v_1|}{2} \\sigma,  \\ v^{\\prime}_1 = \\frac{v+ v_1}{2} - \\frac{|v-v_1|}{2} \\sigma.\n\\end{equation}\nThe function $B(v-v_1,\\sigma)$, called cross-section, is assumed to depend only on $v-v_1$ and $\\frac{v-v_1}{|v-v_1|} \\cdot \\sigma$.\n\nIt is well-known that the equilibrium of the Boltzmann collision operator $C$, i.e. the number density $\\mathcal{M}$ such that $C(\\mathcal{M}, \\mathcal{M})=0$, have a specific form, which is called the Maxwellian distributions:\n$$\\mathcal{M}(\\rho, u ,\\theta) :=\\frac{\\rho}{(2\\pi\\theta)^{\\frac{3}{2}}}\\exp(-\\frac{|v- \\mathrm{u} |^2}{2\\theta})\\,,$$\nwhere $(\\rho, \\mathrm{u} ,\\theta)$ denotes the fluid variables: density, bulk velocity and temperature. When $(\\rho,u,\\theta)$ are functions of $(x,t)$, $\\mathcal{M}(\\rho,  \\mathrm{u}  ,\\theta)$ is called the local Maxwellians, while when $(\\rho, \\mathrm{u} ,\\theta)$ are constant states, $\\mathcal{M}(\\rho,  \\mathrm{u}  ,\\theta)$ is called the global (or absolute) Maxwellians.\n\nIn different scales for time and fluctuations relative to Maxwellians, the fluid dynamics of Boltzmann equations are represented by different equations, such as compressible/incompressible Navier-Stokes or Euler equations. As formally derived in \\cite{BGL1}, when $\\tau_{\\varepsilon} = \\varepsilon $, this is the case where Mach number and Knudsen number both equals to $\\varepsilon$ (then the Reynold number $\\mathrm{Re}=O(1)$ from the von Kármán relation $\\mathrm{Re}=\\frac{\\mathrm{Ma}}{\\mathrm{Kn}}$ )\n and the incompressible Navier-Stokes limit can be expected. When considering small Mach number, it is natural to consider distributions  as perturbations about a given absolute Maxwellian such as $M = \\mathcal{M}(1,0,1)$. More specifically, the solutions to the equation \\eqref{longtime_Boltzmann} are sought in the form $f_{\\varepsilon} = M + \\varepsilon M g_{\\varepsilon}$.\nThen the equation \\eqref{longtime_Boltzmann} can be rewritten as\n\\begin{equation}\\label{perturbed_Boltzmann}\n    \\left\\{ \\begin{matrix}\n        \\varepsilon^2 \\partial_t g_{\\varepsilon} + \\varepsilon v \\!\\cdot\\! \\nabla_x g_{\\varepsilon} + \\mathcal{L}g_{\\varepsilon} = \\varepsilon Q(g_{\\varepsilon},g_{\\varepsilon}),\\\\\n        g_{\\varepsilon}(0,x,v) = g^{{in}}_{\\varepsilon},\n    \\end{matrix}\\right.\n\\end{equation}\nwith the linearized Boltzmann operator\n\\begin{equation}\n    \\mathcal{L}(g) := - \\frac{1}{M}(C(M,Mg) + C(Mg,M)),\n\\end{equation}\nand\n\\begin{equation}\n    \\begin{aligned}\n        &Q(f,g) := \\frac{1}{M}C(Mf,Mg).\\\\\n    \\end{aligned}\n\\end{equation}\nWe shall use the following notations in the sequel:\n    \\begin{equation}\n        \\begin{aligned}\n            &\\langle f,g \\rangle = \\int_{\\mathbb{R}^3_v} f g M\\, \\mathrm{d}  v, \\ \\ \\\n            \\langle f \\rangle = \\int_{\\mathbb{T}^3} f M \\, \\mathrm{d}  v,\\\\\n            &\\langle\\! \\langle  f  \\rangle\\! \\rangle := \\int_{\\mathbb{R}^3_v} \\int_{\\mathbb{R}^3_v}  \\int_{\\mathbb{S}^2}f M M _1 b \\, \\mathrm{d}  v  \\mathrm{d}  v_1  \\mathrm{d} \\sigma,\n        \\end{aligned}\n    \\end{equation}\n    and the standard inner product $({a} \\!\\  , {b}) = \\sum_{i=1}^5 a^ib^i$ in $\\mathbb{R}^5$.\n\n    It is well known that the linearized Boltzmann operator $\\mathcal{L}$ is self-adjoint on $L^2(Mdv)$ with inner product $\\langle f,g \\rangle $.\n    Moreover, the null space $\\mathcal{N}$ of $\\mathcal{L}$ is spanned by the set of collision invariants:\n    \\begin{equation}\n        \\begin{aligned}\n            \\mathcal{N} = \\text{Span} \\{ 1 , v, |v|^2\\}.\n        \\end{aligned}\n    \\end{equation}\n    We also let $\\mathcal{N}^{\\perp}$ to denote the orthogonal space of $\\mathcal{N}$ with respect to the inner product $\\langle \\cdot, \\cdot \\rangle$ defined above.\n    Furthermore, we use the notation $\\Lpre g$ and $\\Loth g$ to denote the projection of $g$ to $\\mathcal{N}$ and $\\mathcal{N}^{\\perp}$ respectively.\n\n    Formally, $g_{\\varepsilon}$ should converge to some $g$ which lies in the kernel of $\\mathcal{L}$. Specifically, $g$ should be of the form: $g = \\rho + v \\cdot  \\mathrm{u}   +( \\frac{1}{2} |v|^2 -\\frac{3}{2})\\theta$. Moreover,\n    the velocity $u$ is divergence free and the density and the temperature fluctuations $\\rho$ and $\\theta$ satisfy the Boussinesq relation\n    \\begin{equation}\n        \\nabla_x \\cdot  \\mathrm{u}  =0; \\ \\ \\rho + \\theta = 0.\n    \\end{equation}\n    Furthermore, $(\\rho, \\mathrm{u} , \\theta)$ should satisfy the incompressible Navier-Stokes-Fourier system\n    \\begin{equation}\\label{INSF} \\tag{INSF}\n        \\left\\{\n            \\begin{aligned}\n                &\\partial_t  \\mathrm{u}  +  \\mathrm{u}   \\!\\cdot\\! \\nabla_x  \\mathrm{u}  + \\nabla_x p = \\mu \\Delta  \\mathrm{u} , \\\\\n                &\\partial_t \\vartheta +  \\mathrm{u}   \\!\\cdot\\! \\nabla_x \\vartheta = \\kappa \\Delta \\vartheta,\\\\\n                &\\nabla_x \\cdot  \\mathrm{u}  =0,\n            \\end{aligned}\n        \\right.\n    \\end{equation}\n    where $\\vartheta = \\frac{3}{5} \\theta - \\frac{2}{5}\\rho$. The viscosity coefficient $\\mu$ and heat-conductivity coefficient $\\kappa$ are defined by $\\mu = \\mu(1,1)$ and $\\kappa = \\kappa(1,1)$ as in \\eqref{local_viscosity_and_heat_conductivity} by setting $(\\rho,\\theta) = (1,1)$. For more detailed derivation of the above incompressible Navier-Stokes system, see \\cite{BGL1}.\n\n    Various works have been contributed to justify this process since late 1970s. Among them, we shall emphasize the so-called BGL program started by Bardos, Golse, and Levermore since the late 1980s. They aimed to justify Leray's solutions to incompressible Navier-Stokes equations from Diperna-Lions renormalized solutions \\cites{BGL1,BGL2}.\nAfter that, Bardos, Golse, Levermore, Lions, Saint-Raymond, Masmoudi then made significant contributions (see e.g., \\cites{BGL_2000,Golse_Levermore_2002,Lions_Masmoudi_2001}). Finally Golse and Saint-Raymond obtained the first complete convergence result without any additional compactness assumption in \\cite{Golse_Raymond_2004} for cutoff Maxwellian collision kernel and in \\cite{Golse_Raymond_2005} for hard cutoff potentials. These results were later extended to include soft potentials by Levermore and Masmoudi \\cite{Levermore_Masmoudi_2010}, and to bounded domain by Jiang, Levermore, Masmoudi and Saint-Raymond (see e.g., \\cites{Jiang_Masmoudi_2017, Jiang_Levermore_Masmoudi_2010,Masmoudi_Raymond_2003}).\n\nAside from the uss of renormalized solutions of Boltzmann equation to deduce hydrodynamic limits, numerous works have also been based on the classical solution of Boltzmann equation. The first work of this type is that of Bardos and Ukai \\cite{Bardos_Ukai_1991}, in which they proved the global existence of classical solutions of \\eqref{perturbed_Boltzmann} for cutoff hard potentials, and established estimates uniform in $\\varepsilon$ for $0< \\varepsilon<1$. Later Briant, Merino-Aceituno, and Mouhot \\cites{Briant_2015,Briant_Merino-Aceituno_Mouhot_2019} used the semigroup approach to prove the incompressible Navier-Stokes limit on torus for cutoff kernels with hard potentials.\nFor a border class of collision kernels for both cutoff and non-cutoff cases, Jiang, Xu and Zhao proved the incompressible Navier-Stokes-Fourier limit in \\cite{Jiang_Xu_Zhao_2018}, using the non-isotropic norm developed in \\cites{AMUXY_2011,AMUXY_2012,AMUXY_2012_Anal_Appl} and equivalently in \\cite{Gressman_Strain}.\nThere are also many works based on the Hilbert expansion in the context of classical solutions.\nThis approach was started from Caflisch and Nishida's work on compressible Euler limit \\cites{Caflisch_1980,Nishida_1978}. Guo, Jiang, Jang also used this method combining with some nonlinear estimate to derive the  acoustic limit  \\cites{Guo_Jang_Jiang_2009,Guo_Jang_Jiang_2010,Jang_Jiang_2009}. Afterwards, this approach was used for incompressible Navier-Stokes limit in \\cites{Guo_CPAM_2006,Masi_Esposito_Lebowitz}. It was later extended to general initial data to include fast acoustic waves by Jiang and Xiong in \\cite{Jiang_Xiong}.\n\nIn \\cite{Golse_Levermore_2002}, Golse and Levermore made the following nice observation (in the {\\em Concluding Remarks} of \\cite{Golse_Levermore_2002}): they proposed the so-called {\\em compressible\n Stokes system}, which is the linearization about a homogeneous state of the compressible Navier-Stokes system. In the short time scale (i.e. $\\tau_\\varepsilon=1$), as $\\varepsilon\\rightarrow 0$, the compressible Stokes system converges to the acoustic system. While in the longer time scale, (i.e. $\\tau_\\varepsilon=\\varepsilon$), the limit is the incompressible Stokes equations. In this sense, the (compressible) acoustic system and the incompressible Stokes equations can be unified in one system, which is the compressible Stokes system. \n\nGolse and Levermore's proposal is on the linear case. Motivated by their idea, in this paper, we start from the scaled general compressible Navier-Stokes, i.e. the diffusion terms are order $O(\\varepsilon$. When the time scale $\\tau_\\varepsilon=1$, the limit will be the compressible Euler system. This is the inviscid limit, which is a well-known analytically hard problem. When the time scale $\\tau_\\varepsilon=\\varepsilon$, the limit will be the incompressible Navier-Stokes equations, which has been well-studied, as mentioned above. The main novelty of this paper is that we propose a new relative entropy approach, which can be analytically proved, as least in the framework of classical solutions. For the clarity of the presentation, we mainly focus on the formal calculation. However, we clearly state the the approach how the rigorous analytical proof could be followed from our formal analysis. The main advantage of this approach is that we could obtain the convergence rate to the incompressible Navier-Stokes equations. This could not be obtained from the previous compactness arguments. We state the main ideas of our appraoch in the following.   \n\nWhen $\\tau_{\\varepsilon}=1$, it has been shown in the formal level (see e.g., \\cite{BGL1}), using the Chapman-Enskog expansion, that the solution of \\eqref{longtime_Boltzmann} can be approximated by $f_{\\varepsilon} = M_{\\varepsilon} (1 + \\varepsilon g_{\\varepsilon} + \\varepsilon^2 w_{\\varepsilon})$,\nwhere $M_{\\varepsilon} = \\mathcal{M}(\\rho_{\\varepsilon}, u_{\\varepsilon},\\theta_{\\varepsilon})$ and $(\\rho_{\\varepsilon}, u_{\\varepsilon} ,\\theta_{\\varepsilon})$ satisfies the following compressible Navier-Stokes system with dissipation of the order $\\varepsilon$:\n\\begin{equation}\\label{short_CNSF} \\tag{CNS}\n    \\left\\{ \\begin{aligned}\n        &\\partial_t \\rho_{\\varepsilon} + \\nabla_x (\\rho_{\\varepsilon} u_{\\varepsilon}) = 0,\\\\\n        & \\rho_{\\varepsilon} (\\partial_t + u_{\\varepsilon}  \\!\\cdot\\! \\nabla_x) u_{\\varepsilon}  + \\nabla_x ({\\rho_{\\varepsilon} \\theta_{\\varepsilon}}) = \\varepsilon \\nabla_x [ \\mu_{\\varepsilon}(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\sigma(u_{\\varepsilon})],\\\\\n        & \\tfrac{3}{2} \\rho_{\\varepsilon} (\\partial_t   + u_{\\varepsilon}  \\!\\cdot\\! \\nabla_x )\\theta_{\\varepsilon} + \\rho_{\\varepsilon} \\theta_{\\varepsilon}  \\nabla_x  \\!\\cdot\\! u_{\\varepsilon} = \\varepsilon  \\tfrac{1}{2} \\mu_{\\varepsilon}(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\sigma(u_{\\varepsilon}):\\sigma(u_{\\varepsilon}) + \\tfrac{5}{2} \\varepsilon \\nabla_x \\cdot[\\kappa(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\theta_{\\varepsilon}],\\\\\n        &(\\rho_{\\varepsilon}(0,x),\\ u_{\\varepsilon}(0,x), \\ \\theta_{\\varepsilon}(0,x)) =  (\\rho^{\\mathrm{in}},\\ u^{\\mathrm{in}},\\ \\theta^{\\mathrm{in}}).\n    \\end{aligned}\\right.\n\\end{equation}\nHere $\\sigma(u)= \\nabla_x u + {\\nabla_x u}^T  - \\frac{2}{3}(\\nabla_x  \\!\\cdot\\! u) \\mathrm{I}$ is the stress tensor, \n    and the viscosity coefficient $\\mu_{\\varepsilon}(\\rho,\\theta)$ along with the heat-conductivity coefficient $\\kappa_{\\varepsilon}(\\rho,\\theta)$ are defined as follows\n    \\begin{equation}\\label{local_viscosity_and_heat_conductivity}\n        \\begin{aligned}\n            \\mu(\\rho,\\theta) = \\frac{1}{10} \\int_{\\mathbb{R}^3_v} A(V) :  \\hat{A}(V) \\mathcal{M} dv,\\\\\n            \\kappa(\\rho,\\theta) = \\frac{2}{15} \\int_{\\mathbb{R}^3_v} B(V) \\cdot \\hat{B}(V) \\mathcal{M} dv,\n        \\end{aligned}\n    \\end{equation}\n    where\n    \\begin{equation}\n        A(V) = V\\otimes V - \\frac{|V|^2}{3}I, \\qquad B(V) = V(\\frac{|V|^2}{2}- \\frac{5}{2}),\n    \\end{equation}\n    with $\\displaystyle V= \\frac{v-u}{\\sqrt{\\theta}}$.\n    Moreover, $\\hat{A} (V)$ and $\\hat{B}(V)$ are the unique solutions in $(\\text{Ker} \\mathcal{L}_{\\mathcal{M}}) ^{\\perp}$ of the following equations respectively\n    \\begin{equation}\n        \\mathcal{L}_{\\mathcal{M}} \\hat{A} (V) = A(V), \\quad  \\mathcal{L}_{\\mathcal{M}} \\hat{B}(V) = B(V),\n    \\end{equation}\n    with\n        \\begin{equation}\n        \\mathcal{L}_{\\mathcal{M}}(g) := - \\frac{1}{\\mathcal{M}}(C(\\mathcal{M},\\mathcal{M}g) + C(\\mathcal{M}g,\\mathcal{M})).\n    \\end{equation}\n\n    We note that the compressible Navier-Stokes system \\eqref{short_CNSF} is not a limit of the Boltzmann equation as the Knudsen number $\\varepsilon \\rightarrow 0$ but a second order approximation in the Chapman-Enskog expansion.\n    There are also a lot of results on this approximation, we refer to \\cites{BGL1,Kawashima_Matsumura_Nishida_1979,Duan_Liu_2021,Liu_Yang_Zhao}  and the references within.\n\n    When Mach number is small (when we set $\\tau_\\varepsilon = \\varepsilon$ as will be assumed throughout), it is expected that the slightly compressible fluids are close to incompressible fluids. \n    Then there is a natural relation between \\eqref{short_CNSF} and \\eqref{INSF} by studying the fluctuation of $(\\rho_\\varepsilon,u_\\varepsilon,\\theta_\\varepsilon)$ around $(1,0,1)$. More specifically, one may consider the following  compressible Navier-Stokes system with small Mach number:\n    \\begin{equation} \\label{longtime_CNS}\\tag{$\\text{CNS}_{\\varepsilon}$}\n        \\left\\{ \\begin{aligned}\n            &\\varepsilon\\partial_t \\rho_{\\varepsilon} + \\nabla_x (\\rho_{\\varepsilon} u_{\\varepsilon}) = 0,\\\\\n            &\\varepsilon\\partial_t u_{\\varepsilon} + u_{\\varepsilon}  \\!\\cdot\\! \\nabla_x u_{\\varepsilon}  + \\frac{1}{\\rho_{\\varepsilon}} \\nabla_x (\\rho_{\\varepsilon} \\theta_{\\varepsilon})  = \\varepsilon \\frac{1}{\\rho_{\\varepsilon}} \\nabla_x [ \\mu_{\\varepsilon}(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\sigma(u_{\\varepsilon})],\\\\\n            &\\varepsilon \\partial_t \\theta_{\\varepsilon}  + u_{\\varepsilon}  \\!\\cdot\\! \\nabla_x \\theta_{\\varepsilon} + \\frac{2}{3} \\theta_{\\varepsilon}  \\nabla_x  \\!\\cdot\\! u_{\\varepsilon} = \\varepsilon  \\frac{1}{\\rho_{\\varepsilon}} \\frac{1}{3} \\mu_{\\varepsilon}(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\sigma(u_{\\varepsilon}):\\sigma(u_{\\varepsilon}) +  \\frac{1}{\\rho_{\\varepsilon}}\\frac{5}{3} \\varepsilon \\nabla_x \\cdot[\\kappa(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\theta_{\\varepsilon}],\\\\\n            &(\\rho_{\\varepsilon}(0,x),\\ u_{\\varepsilon}(0,x), \\ \\theta_{\\varepsilon}(0,x)) = (1+ \\varepsilon \\tilde{\\rho}^{\\mathrm{in}},\\ \\tilde{u}^{\\mathrm{in}},\\ 1+ \\varepsilon \\tilde{\\theta}^{\\mathrm{in}}).\n        \\end{aligned}\\right.\n    \\end{equation}\n\n    Then the fluctuations $(\\tilde{\\rho}_{\\varepsilon}, \\tilde{u}_{\\varepsilon},\\tilde{\\theta}_{\\varepsilon})$ of the solutions of \\eqref{longtime_CNS}, which satisfying $ (\\rho_{\\varepsilon}, u_{\\varepsilon},\\theta_{\\varepsilon} )= (1+ \\varepsilon \\tilde{\\rho}_{\\varepsilon}, \\varepsilon \\tilde{u}, 1+ \\varepsilon \\tilde{\\theta}) $, should converge to the solutions of \\eqref{INSF} in some strong or weak sense (depending on the initial data and the domain along with proper boundary conditions).\n        This process is known as the low Mach number limit, and we refer to \\cites{Klainerman_Majda_1981,Klainerman_Majda_1982,Danchin_AJM,Alazard_2006,Ebin,MS_ARMA,DGLM,Masmoudi_2022,CGHJ} for more information.\n        We mention that when the domain is considered to be torus, and the initial data are ill-prepared(i.e. the initial data are not required to satisfy the incompressibility and Boussinesq relation), the convergence cannot be strong, due to the acoustic waves generated by initial compression, which will exist in the long time.\n\n        This phenomenon shall also arise when considering the incompressible Navier-Stokes-Fourier limit of the Boltzmann equation, which prevents the strong convergence of $g_{\\varepsilon}$ to $g= v \\cdot u + \\theta(\\frac{3}{2} |v|^2 -\\frac{5}{2}) $ if the kinetic part of initial data is well-prepared, but the fluid part of initial data is ill-prepared.\n        Therefore, when considering the incompressible Navier-Stokes limit of the Boltzmann equation on torus with ill-prepared initial data, we need also to characterize the behavior of acoustic waves in the meantime.\n\n        We deal with this problem by splitting the incompressible Navier-Stokes limit into two steps. The first step is the approximation of Boltzmann equation by compressible Navier-Stokes system with small Mach number, and the second step is the low Mach number limit of the compressible Navier-Stokes system.\n        Their relationship is shown in the following diagram. The low Mach number limit process has been discussed in our former paper \\cite{CGHJ}. In this paper we focus on the compressible Navier-Stokes approximation for the Boltzmann equation.\n        \\begin{figure}[H]\n            \\centering\n            \\begin{tikzpicture}[>=latex, node distance=2cm]\n                \\node [draw, rectangle]  (A) at (0,0) {Kinetic Level: };\n                \\node [draw, rectangle,text width=4cm, align=center] (A_1) at (5,0) {Boltzmann equation\\\\Kn=Ma=$\\varepsilon$};\n                \\node [draw, rectangle] (B) at (0,-3) {Fluid Level:} ;\n                \\node [draw, rectangle, text width=3.5cm, align=center]  (C) at(12,-3) { Compressible \\\\ Navier-Stokes system\\\\  Ma = $\\varepsilon$};\n                \\node [draw, rectangle, text width=3cm, align=center] (D)  at (5,-3) {Incompressible \\\\  Naiver-Stokes-Fourier system};\n\n                 \\draw[->] (A) -- (B);\n                 \\draw [->] (A_1) -- (C) node[midway, right,text width=4cm, align=center] {Hydrodynamic \\\\asymptotic\\\\ $\\varepsilon \\rightarrow 0$};\n                 \\draw [->] (A_1) -- (D) node[midway, left,text width=4cm, align=center] {Hydrodynamic \\\\limit\\\\ $\\varepsilon \\rightarrow 0$};\n                 \\draw [->] (C) -- (D)node[midway, above,text width=4cm, align=center] {Low Mach number}\n                                        node[midway, below,text width=4cm, align=center] {limit $\\varepsilon \\rightarrow 0$};\n\n            \\end{tikzpicture}\n            \\caption{Relationship between Boltzmann equation, incompressible Navier-Stokes-Fourier system and compressible Navier-Stokes system.}\n            \\label{fig:tikz_example}\n        \\end{figure}\n\n    As illustrated in \\eqref{short_CNSF}, the compressible Navier-Stokes-Fourier system derived from Boltzmann equation is {\\em not} a limit, since in this setting, the compressible Navier-Stokes-Fourier  system  depends on the Knudsen number itself. So it is not a {\\em limit}, but an {\\em asymptotic} problem. \n    We shall use the relative entropy to describe the compressible Navier-Stokes approximation of the Boltzmann equation. The relative entropy $H(f|g)$ for $f>0$ and $g>0$\n    is defined as\n    \\begin{equation}\\label{RE}\n        H(f|g) = \\int_{\\mathbb{T}^3_x}\\int_{\\mathbb{R}^3_v} f \\log(\\frac{f}{g}) - f + g \\  dxdv.\n    \\end{equation}\n    The key issue is that in this paper, the ``target'', i.e. $g$ in the relative entropy \\eqref{RE} must depends on the Kundsen number $\\varepsilon$. More specifically, we set $f$ in \\eqref{RE} as the solutions to the scaled Boltzmann equation $f_\\varepsilon$,  and $g$ as the local Maxwellian ($M_{\\varepsilon}$ defined below) governed by the solutions to the scaled compressible NSF. We calculate the evolution of this scaled relative entropy $\\frac{1}{{\\varepsilon}^2} H(f_{\\varepsilon}| M_{\\varepsilon})(t)$ and prove its stability, i.e. the evolution of the scaled relative entropy in any time $t>0$ can be controlled by its initial data. More importantly, we characterize its dissipation rate. It is the first quantitative convergence result for the compressible Navier-Stokes-Fourier system in the sense of relative entropy. This approach avoids any expansion method, such as Chapman-Enskog or Hilbert expansions. The main disadvantage of the expansion methods is it can only construct {\\em special} (in the format of expansions) solution of the original Boltzmann equation. On the other hand, the relative entropy method employed here is to directly measure the distance in the sense of relative entropy between {\\em any} solutions of the scaled Boltzmann equation and those of compressible NSF. Furthermore, the asymptotic behavior is quantitatively characterized by relative entropy and the relative entropy dissipation is explicit. This is the main novelty of this paper.\n\n    The idea of using the notion of relative entropy for this kind of problems comes from the notion of entropic convergence developed by Bardos, Golse and Levermore in \\cite{BGL2}, and on the other hand from Yau's derivation of the hydrodynamic limit of Ginzburg-Landau lattice model \\cite{Yau}.\n    Latter, Saint-Raymond use relative entropy to deduce the incompressible Euler limit for well-prepared initial data in \\cite{Saint-Raymond_2003} and for ill-prepared initial data in \\cite{Saint-Raymond_2009}.\n\n    We use the solution $(\\rho_{\\varepsilon},u_{\\varepsilon},\\theta_{\\varepsilon}) = (1+ \\varepsilon \\tilde{\\rho}_{\\varepsilon}, \\varepsilon \\tilde {u}_{\\varepsilon},1+ \\varepsilon \\tilde{\\theta}_{\\varepsilon})$ of \\eqref{longtime_CNS} to construct a local Maxwellian $M_{\\varepsilon} = \\mathcal{M}(\\rho_{\\varepsilon},u_{\\varepsilon},\\theta_{\\varepsilon}) = \\mathcal{M} (1+ \\varepsilon \\tilde{\\rho}_{\\varepsilon}, \\varepsilon \\tilde {u}_{\\varepsilon},1+ \\varepsilon \\tilde{\\theta}_{\\varepsilon}) $,\n    and we would like to use the relative entropy $H(f_{\\varepsilon}| M_{\\varepsilon})$ to measure the difference between the solution of the rescaled Boltzmann equation $\\eqref{longtime_Boltzmann}$ and the solution of the compressible Navier-Stokes system $\\eqref{longtime_CNS}$.\n    More specifically, we derive the time evolution of the relative entropy $\\frac{d}{dt} \\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}| M_{\\varepsilon})$, which leads to the following modulated entropy inequality\n    \\begin{equation}\n        \\begin{aligned}\n            \\tfrac{1}{{\\varepsilon}^2} &H(f_{\\varepsilon}| M_{\\varepsilon})(t) +\\int_0^t \\int_{\\mathbb{T}^3_x}\\left\\{\\tfrac{1}{\\varepsilon^4}D(f_\\varepsilon)(s,x) - \\tfrac{1}{2} \\mu \\sigma( \\mathrm{u} ^b_\\varepsilon): \\sigma( \\mathrm{u} ^b_\\varepsilon) - \\tfrac{5}{2}\\kappa (\\nabla_x \\theta^b_{\\varepsilon})^2\\right\\}\\,   \\mathrm{d}  x \\mathrm{d}  s \\\\\n            &+\\int_0^t \\int_{\\mathbb{T}^3_x} \\tfrac{1}{2}\\mu \\sigma(\\tilde{\\u_{\\varepsilon}}- \\mathrm{u} ^b_{\\varepsilon}):\\sigma(\\tilde{ \\mathrm{u} }_{\\varepsilon}- \\mathrm{u} ^b_{\\varepsilon}) + \\tfrac{5}{2}  \\kappa(\\nabla_x \\tilde{\\theta}_{\\varepsilon} -\\nabla_x \\theta^b_{\\varepsilon})^2 \\ dxds\\\\\n            &\\lesssim \\frac{1}{\\varepsilon^2} H(f^{\\mathrm{in}}_{\\varepsilon}| M^{\\mathrm{in}}_{\\varepsilon}) + \\mathcal{O}(\\varepsilon).\n        \\end{aligned}\n    \\end{equation}\n    Here and in the sequel $(\\rho^b_{\\varepsilon},  \\mathrm{u} ^b_{\\varepsilon},\\theta^b_{\\varepsilon})$ are moments of perturbation $g_{\\varepsilon}$ as defined in \\eqref{g_moments}.\n\n    Under the assumption of vanishing initial relative entropy\n    \\begin{equation}\n        \\frac{1}{\\varepsilon^2} H(f^{\\mathrm{in}}_{\\varepsilon}| M_{\\varepsilon}^{\\mathrm{in}}) \\rightarrow 0  \\ \\ \\text{as } \\varepsilon \\rightarrow 0,\n    \\end{equation}\n    for each $t > 0$, we obtain the following asymptotics:\n    \\begin{enumerate}[label=\\textbullet]\n        \\item Asymptotic of the entropy dissipation rate\n            \\begin{equation}\n               \\lim_{\\varepsilon \\rightarrow 0} \\left[\\int_0^t \\int_{\\mathbb{T}^3_x}\\frac{1}{\\varepsilon^4}D(f_\\varepsilon)(s,x) - \\frac{1}{2} \\mu \\sigma(u^b_{\\varepsilon}): \\sigma(u^b_{\\varepsilon}) - \\frac{5}{2}\\kappa (\\nabla_x \\theta^b_{\\varepsilon})^2\\  dxds \\right] = 0;\n            \\end{equation}\n        \\item Asymptotic of the momentum and energy flux\n            \\begin{equation}\n                \\lim_{\\varepsilon \\rightarrow 0} \\left[ \\int_0^t \\int_{\\mathbb{T}^3_x} \\frac{1}{2}\\mu \\sigma(\\tilde{ \\mathrm{u} }_{\\varepsilon}- \\mathrm{u} ^b_{\\varepsilon}):\\sigma(\\tilde{ \\mathrm{u} }_{\\varepsilon}- \\mathrm{u} ^b_{\\varepsilon}) + \\frac{5}{2}  \\kappa(\\nabla_x \\tilde{\\theta}_{\\varepsilon} -\\nabla_x \\theta^b_{\\varepsilon})^2 \\ dxds \\right]=0;\n            \\end{equation}\n        \\item Asymptotic of the relative entropy\n            \\begin{equation}\n                \\lim_{\\varepsilon \\rightarrow 0}\\frac{1}{{\\varepsilon}^2} H(f_{\\varepsilon}| M_{\\varepsilon})(t)= 0.\n            \\end{equation}\n    \\end{enumerate}\n   We state our main theorem as follows:", "context": "When Mach number is small (when we set $\\tau_\\varepsilon = \\varepsilon$ as will be assumed throughout), it is expected that the slightly compressible fluids are close to incompressible fluids. \n Then there is a natural relation between \\eqref{short_CNSF} and \\eqref{INSF} by studying the fluctuation of $(\\rho_\\varepsilon,u_\\varepsilon,\\theta_\\varepsilon)$ around $(1,0,1)$. More specifically, one may consider the following compressible Navier-Stokes system with small Mach number:\n \\begin{equation} \\label{longtime_CNS}\\tag{$\\text{CNS}_{\\varepsilon}$}\n \\left\\{ \\begin{aligned}\n &\\varepsilon\\partial_t \\rho_{\\varepsilon} + \\nabla_x (\\rho_{\\varepsilon} u_{\\varepsilon}) = 0,\\\\\n &\\varepsilon\\partial_t u_{\\varepsilon} + u_{\\varepsilon} \\!\\cdot\\! \\nabla_x u_{\\varepsilon} + \\frac{1}{\\rho_{\\varepsilon}} \\nabla_x (\\rho_{\\varepsilon} \\theta_{\\varepsilon}) = \\varepsilon \\frac{1}{\\rho_{\\varepsilon}} \\nabla_x [ \\mu_{\\varepsilon}(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\sigma(u_{\\varepsilon})],\\\\\n &\\varepsilon \\partial_t \\theta_{\\varepsilon} + u_{\\varepsilon} \\!\\cdot\\! \\nabla_x \\theta_{\\varepsilon} + \\frac{2}{3} \\theta_{\\varepsilon} \\nabla_x \\!\\cdot\\! u_{\\varepsilon} = \\varepsilon \\frac{1}{\\rho_{\\varepsilon}} \\frac{1}{3} \\mu_{\\varepsilon}(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\sigma(u_{\\varepsilon}):\\sigma(u_{\\varepsilon}) + \\frac{1}{\\rho_{\\varepsilon}}\\frac{5}{3} \\varepsilon \\nabla_x \\cdot[\\kappa(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\theta_{\\varepsilon}],\\\\\n &(\\rho_{\\varepsilon}(0,x),\\ u_{\\varepsilon}(0,x), \\ \\theta_{\\varepsilon}(0,x)) = (1+ \\varepsilon \\tilde{\\rho}^{\\mathrm{in}},\\ \\tilde{u}^{\\mathrm{in}},\\ 1+ \\varepsilon \\tilde{\\theta}^{\\mathrm{in}}).\n \\end{aligned}\\right.\n \\end{equation}\n\nWe use the solution $(\\rho_{\\varepsilon},u_{\\varepsilon},\\theta_{\\varepsilon}) = (1+ \\varepsilon \\tilde{\\rho}_{\\varepsilon}, \\varepsilon \\tilde {u}_{\\varepsilon},1+ \\varepsilon \\tilde{\\theta}_{\\varepsilon})$ of \\eqref{longtime_CNS} to construct a local Maxwellian $M_{\\varepsilon} = \\mathcal{M}(\\rho_{\\varepsilon},u_{\\varepsilon},\\theta_{\\varepsilon}) = \\mathcal{M} (1+ \\varepsilon \\tilde{\\rho}_{\\varepsilon}, \\varepsilon \\tilde {u}_{\\varepsilon},1+ \\varepsilon \\tilde{\\theta}_{\\varepsilon}) $,\n and we would like to use the relative entropy $H(f_{\\varepsilon}| M_{\\varepsilon})$ to measure the difference between the solution of the rescaled Boltzmann equation $\\eqref{longtime_Boltzmann}$ and the solution of the compressible Navier-Stokes system $\\eqref{longtime_CNS}$.\n More specifically, we derive the time evolution of the relative entropy $\\frac{d}{dt} \\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}| M_{\\varepsilon})$, which leads to the following modulated entropy inequality\n \\begin{equation}\n \\begin{aligned}\n \\tfrac{1}{{\\varepsilon}^2} &H(f_{\\varepsilon}| M_{\\varepsilon})(t) +\\int_0^t \\int_{\\mathbb{T}^3_x}\\left\\{\\tfrac{1}{\\varepsilon^4}D(f_\\varepsilon)(s,x) - \\tfrac{1}{2} \\mu \\sigma( \\mathrm{u} ^b_\\varepsilon): \\sigma( \\mathrm{u} ^b_\\varepsilon) - \\tfrac{5}{2}\\kappa (\\nabla_x \\theta^b_{\\varepsilon})^2\\right\\}\\, \\mathrm{d} x \\mathrm{d} s \\\\\n &+\\int_0^t \\int_{\\mathbb{T}^3_x} \\tfrac{1}{2}\\mu \\sigma(\\tilde{\\u_{\\varepsilon}}- \\mathrm{u} ^b_{\\varepsilon}):\\sigma(\\tilde{ \\mathrm{u} }_{\\varepsilon}- \\mathrm{u} ^b_{\\varepsilon}) + \\tfrac{5}{2} \\kappa(\\nabla_x \\tilde{\\theta}_{\\varepsilon} -\\nabla_x \\theta^b_{\\varepsilon})^2 \\ dxds\\\\\n &\\lesssim \\frac{1}{\\varepsilon^2} H(f^{\\mathrm{in}}_{\\varepsilon}| M^{\\mathrm{in}}_{\\varepsilon}) + \\mathcal{O}(\\varepsilon).\n \\end{aligned}\n \\end{equation}\n Here and in the sequel $(\\rho^b_{\\varepsilon}, \\mathrm{u} ^b_{\\varepsilon},\\theta^b_{\\varepsilon})$ are moments of perturbation $g_{\\varepsilon}$ as defined in \\eqref{g_moments}.\n\nUnder the assumption of vanishing initial relative entropy\n \\begin{equation}\n \\frac{1}{\\varepsilon^2} H(f^{\\mathrm{in}}_{\\varepsilon}| M_{\\varepsilon}^{\\mathrm{in}}) \\rightarrow 0 \\ \\ \\text{as } \\varepsilon \\rightarrow 0,\n \\end{equation}\n for each $t > 0$, we obtain the following asymptotics:\n \\begin{enumerate}[label=\\textbullet]\n \\item Asymptotic of the entropy dissipation rate\n \\begin{equation}\n \\lim_{\\varepsilon \\rightarrow 0} \\left[\\int_0^t \\int_{\\mathbb{T}^3_x}\\frac{1}{\\varepsilon^4}D(f_\\varepsilon)(s,x) - \\frac{1}{2} \\mu \\sigma(u^b_{\\varepsilon}): \\sigma(u^b_{\\varepsilon}) - \\frac{5}{2}\\kappa (\\nabla_x \\theta^b_{\\varepsilon})^2\\ dxds \\right] = 0;\n \\end{equation}\n \\item Asymptotic of the momentum and energy flux\n \\begin{equation}\n \\lim_{\\varepsilon \\rightarrow 0} \\left[ \\int_0^t \\int_{\\mathbb{T}^3_x} \\frac{1}{2}\\mu \\sigma(\\tilde{ \\mathrm{u} }_{\\varepsilon}- \\mathrm{u} ^b_{\\varepsilon}):\\sigma(\\tilde{ \\mathrm{u} }_{\\varepsilon}- \\mathrm{u} ^b_{\\varepsilon}) + \\frac{5}{2} \\kappa(\\nabla_x \\tilde{\\theta}_{\\varepsilon} -\\nabla_x \\theta^b_{\\varepsilon})^2 \\ dxds \\right]=0;\n \\end{equation}\n \\item Asymptotic of the relative entropy\n \\begin{equation}\n \\lim_{\\varepsilon \\rightarrow 0}\\frac{1}{{\\varepsilon}^2} H(f_{\\varepsilon}| M_{\\varepsilon})(t)= 0.\n \\end{equation}\n \\end{enumerate}\n We state our main theorem as follows:", "full_context": "When Mach number is small (when we set $\\tau_\\varepsilon = \\varepsilon$ as will be assumed throughout), it is expected that the slightly compressible fluids are close to incompressible fluids. \n Then there is a natural relation between \\eqref{short_CNSF} and \\eqref{INSF} by studying the fluctuation of $(\\rho_\\varepsilon,u_\\varepsilon,\\theta_\\varepsilon)$ around $(1,0,1)$. More specifically, one may consider the following compressible Navier-Stokes system with small Mach number:\n \\begin{equation} \\label{longtime_CNS}\\tag{$\\text{CNS}_{\\varepsilon}$}\n \\left\\{ \\begin{aligned}\n &\\varepsilon\\partial_t \\rho_{\\varepsilon} + \\nabla_x (\\rho_{\\varepsilon} u_{\\varepsilon}) = 0,\\\\\n &\\varepsilon\\partial_t u_{\\varepsilon} + u_{\\varepsilon} \\!\\cdot\\! \\nabla_x u_{\\varepsilon} + \\frac{1}{\\rho_{\\varepsilon}} \\nabla_x (\\rho_{\\varepsilon} \\theta_{\\varepsilon}) = \\varepsilon \\frac{1}{\\rho_{\\varepsilon}} \\nabla_x [ \\mu_{\\varepsilon}(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\sigma(u_{\\varepsilon})],\\\\\n &\\varepsilon \\partial_t \\theta_{\\varepsilon} + u_{\\varepsilon} \\!\\cdot\\! \\nabla_x \\theta_{\\varepsilon} + \\frac{2}{3} \\theta_{\\varepsilon} \\nabla_x \\!\\cdot\\! u_{\\varepsilon} = \\varepsilon \\frac{1}{\\rho_{\\varepsilon}} \\frac{1}{3} \\mu_{\\varepsilon}(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\sigma(u_{\\varepsilon}):\\sigma(u_{\\varepsilon}) + \\frac{1}{\\rho_{\\varepsilon}}\\frac{5}{3} \\varepsilon \\nabla_x \\cdot[\\kappa(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\theta_{\\varepsilon}],\\\\\n &(\\rho_{\\varepsilon}(0,x),\\ u_{\\varepsilon}(0,x), \\ \\theta_{\\varepsilon}(0,x)) = (1+ \\varepsilon \\tilde{\\rho}^{\\mathrm{in}},\\ \\tilde{u}^{\\mathrm{in}},\\ 1+ \\varepsilon \\tilde{\\theta}^{\\mathrm{in}}).\n \\end{aligned}\\right.\n \\end{equation}\n\nWe use the solution $(\\rho_{\\varepsilon},u_{\\varepsilon},\\theta_{\\varepsilon}) = (1+ \\varepsilon \\tilde{\\rho}_{\\varepsilon}, \\varepsilon \\tilde {u}_{\\varepsilon},1+ \\varepsilon \\tilde{\\theta}_{\\varepsilon})$ of \\eqref{longtime_CNS} to construct a local Maxwellian $M_{\\varepsilon} = \\mathcal{M}(\\rho_{\\varepsilon},u_{\\varepsilon},\\theta_{\\varepsilon}) = \\mathcal{M} (1+ \\varepsilon \\tilde{\\rho}_{\\varepsilon}, \\varepsilon \\tilde {u}_{\\varepsilon},1+ \\varepsilon \\tilde{\\theta}_{\\varepsilon}) $,\n and we would like to use the relative entropy $H(f_{\\varepsilon}| M_{\\varepsilon})$ to measure the difference between the solution of the rescaled Boltzmann equation $\\eqref{longtime_Boltzmann}$ and the solution of the compressible Navier-Stokes system $\\eqref{longtime_CNS}$.\n More specifically, we derive the time evolution of the relative entropy $\\frac{d}{dt} \\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}| M_{\\varepsilon})$, which leads to the following modulated entropy inequality\n \\begin{equation}\n \\begin{aligned}\n \\tfrac{1}{{\\varepsilon}^2} &H(f_{\\varepsilon}| M_{\\varepsilon})(t) +\\int_0^t \\int_{\\mathbb{T}^3_x}\\left\\{\\tfrac{1}{\\varepsilon^4}D(f_\\varepsilon)(s,x) - \\tfrac{1}{2} \\mu \\sigma( \\mathrm{u} ^b_\\varepsilon): \\sigma( \\mathrm{u} ^b_\\varepsilon) - \\tfrac{5}{2}\\kappa (\\nabla_x \\theta^b_{\\varepsilon})^2\\right\\}\\, \\mathrm{d} x \\mathrm{d} s \\\\\n &+\\int_0^t \\int_{\\mathbb{T}^3_x} \\tfrac{1}{2}\\mu \\sigma(\\tilde{\\u_{\\varepsilon}}- \\mathrm{u} ^b_{\\varepsilon}):\\sigma(\\tilde{ \\mathrm{u} }_{\\varepsilon}- \\mathrm{u} ^b_{\\varepsilon}) + \\tfrac{5}{2} \\kappa(\\nabla_x \\tilde{\\theta}_{\\varepsilon} -\\nabla_x \\theta^b_{\\varepsilon})^2 \\ dxds\\\\\n &\\lesssim \\frac{1}{\\varepsilon^2} H(f^{\\mathrm{in}}_{\\varepsilon}| M^{\\mathrm{in}}_{\\varepsilon}) + \\mathcal{O}(\\varepsilon).\n \\end{aligned}\n \\end{equation}\n Here and in the sequel $(\\rho^b_{\\varepsilon}, \\mathrm{u} ^b_{\\varepsilon},\\theta^b_{\\varepsilon})$ are moments of perturbation $g_{\\varepsilon}$ as defined in \\eqref{g_moments}.\n\nUnder the assumption of vanishing initial relative entropy\n \\begin{equation}\n \\frac{1}{\\varepsilon^2} H(f^{\\mathrm{in}}_{\\varepsilon}| M_{\\varepsilon}^{\\mathrm{in}}) \\rightarrow 0 \\ \\ \\text{as } \\varepsilon \\rightarrow 0,\n \\end{equation}\n for each $t > 0$, we obtain the following asymptotics:\n \\begin{enumerate}[label=\\textbullet]\n \\item Asymptotic of the entropy dissipation rate\n \\begin{equation}\n \\lim_{\\varepsilon \\rightarrow 0} \\left[\\int_0^t \\int_{\\mathbb{T}^3_x}\\frac{1}{\\varepsilon^4}D(f_\\varepsilon)(s,x) - \\frac{1}{2} \\mu \\sigma(u^b_{\\varepsilon}): \\sigma(u^b_{\\varepsilon}) - \\frac{5}{2}\\kappa (\\nabla_x \\theta^b_{\\varepsilon})^2\\ dxds \\right] = 0;\n \\end{equation}\n \\item Asymptotic of the momentum and energy flux\n \\begin{equation}\n \\lim_{\\varepsilon \\rightarrow 0} \\left[ \\int_0^t \\int_{\\mathbb{T}^3_x} \\frac{1}{2}\\mu \\sigma(\\tilde{ \\mathrm{u} }_{\\varepsilon}- \\mathrm{u} ^b_{\\varepsilon}):\\sigma(\\tilde{ \\mathrm{u} }_{\\varepsilon}- \\mathrm{u} ^b_{\\varepsilon}) + \\frac{5}{2} \\kappa(\\nabla_x \\tilde{\\theta}_{\\varepsilon} -\\nabla_x \\theta^b_{\\varepsilon})^2 \\ dxds \\right]=0;\n \\end{equation}\n \\item Asymptotic of the relative entropy\n \\begin{equation}\n \\lim_{\\varepsilon \\rightarrow 0}\\frac{1}{{\\varepsilon}^2} H(f_{\\varepsilon}| M_{\\varepsilon})(t)= 0.\n \\end{equation}\n \\end{enumerate}\n We state our main theorem as follows:\n\n\\begin{equation}\\label{temp_relative_entropy}\n \\begin{aligned}\n &\\frac{1}{{\\varepsilon}^2} H(f_{\\varepsilon}| M_{\\varepsilon})(t) +\\frac{1}{\\varepsilon^4}\\int_0^t \\int_{\\mathbb{T}^3_x} D(f_\\varepsilon)(s,x)\\ ds dx + \\tilde{R} \\\\\n &+ \\int^t_0 \\int_{\\mathbb{T}^3_x} \\nabla_x [ \\mu(\\rho,\\theta) \\sigma(\\tilde{u})] \\frac{1}{\\rho} \\frac{u^b-\\tilde{u}}{\\theta} - \\frac{1}{\\theta} \\mu \\sigma(u^b):\\nabla_x \\tilde{u} \\ dx ds \\\\\n & + \\int^t_0 \\int_{\\mathbb{T}^3_x} \\frac{1}{\\rho}\\left(\\frac{5}{2} \\nabla_x [\\kappa(\\rho,\\theta) \\nabla_x \\tilde{\\theta}]\\right) \\frac{1}{\\theta^2} (\\theta^b -\\tilde{\\theta}) -\\frac{1}{\\theta^2}\\frac{5}{2}\\kappa \\nabla_x \\theta^b \\cdot \\nabla_x \\tilde{\\theta} \\ dx ds \\\\\n &+ \\int^t_0 \\int_{\\mathbb{T}^3_x} \\frac{1}{\\theta} \\left[ (\\tilde{u}-u^b)\\otimes(\\tilde{u}-u^b) - \\frac{1}{3} (\\tilde{u} - u^b)^2 I \\right] : \\nabla_x \\tilde{u} \\ dx ds \\\\\n &+ \\int^t_0 \\int_{\\mathbb{T}^3_x} \\left[\\frac{5}{2} \\frac{1}{\\theta^2} (\\tilde{u}-u^b)(\\tilde{\\theta}-\\theta^b) \\right] \\cdot \\nabla_x \\tilde{\\theta} \\ dx ds \\\\\n &\\leqslant \\frac{1}{\\varepsilon^2} H(f^{\\mathrm{in}}_{\\varepsilon}| M^{\\mathrm{in}}_{\\varepsilon} ),\n \\end{aligned}\n \\end{equation}\nwhere\n\\begin{equation}\n \\begin{aligned}\n \\tilde{R} = R_1 + R_2 + R_3 + R_4 + \\int^t_0 \\int_{\\mathbb{T}^3_x} R_A : \\frac{\\nabla_x \\tilde{u}}{\\theta} + R_B \\cdot \\frac{\\nabla_x \\tilde{\\theta}}{\\theta^2} \\ dxds.\n \\end{aligned}\n\\end{equation}\nAnd we denote\n\\begin{align}\n & \\text{\\Rmnum{1}}=\\int^t_0 \\int_{\\mathbb{T}^3_x} \\nabla_x [ \\mu(\\rho,\\theta) \\sigma(\\tilde{u})] \\frac{1}{\\rho} \\frac{u^b-\\tilde{u}}{\\theta} - \\frac{1}{\\theta} \\mu \\sigma(u^b):\\nabla_x \\tilde{u} \\ dx ds \\label{viscous_term},\\\\\n & \\text{\\Rmnum{2}}= \\int^t_0 \\int_{\\mathbb{T}^3_x} \\frac{1}{\\rho}\\left(\\frac{5}{2} \\nabla_x [\\kappa(\\rho,\\theta) \\nabla_x \\tilde{\\theta}]\\right) \\frac{1}{\\theta^2} (\\theta^b -\\tilde{\\theta}) -\\frac{1}{\\theta^2}\\frac{5}{2}\\kappa \\nabla_x \\theta^b \\cdot \\nabla_x \\tilde{\\theta} \\ dx ds \\label{heat_conduct_term} ,\\\\\n & \\text{\\Rmnum{3}}= \\int^t_0 \\int_{\\mathbb{T}^3_x} \\frac{1}{\\theta} \\left[ (\\tilde{u}-u^b)\\otimes(\\tilde{u}-u^b) - \\frac{1}{3} (\\tilde{u} - u^b)^2 I \\right] : \\nabla_x \\tilde{u} \\ dx ds \\label{u_convection_term}, \\\\\n & \\text{\\Rmnum{4}}= \\int^t_0 \\int_{\\mathbb{T}^3_x} \\left[\\frac{5}{2} \\frac{1}{\\theta^2} (\\tilde{u}-u^b)(\\tilde{\\theta}-\\theta^b) \\right] \\cdot \\nabla_x \\tilde{\\theta} \\ dx ds \\label{theta_convection_term} .\\\\\n \\end{align}\nFor the term \\Rmnum{1}, by symmetry of $\\sigma(\\tilde{u})$ and integration by parts, we have\n\\begin{equation}\\label{integration_by_part_u}\n \\begin{aligned}\n \\int^t_0 \\int_{\\mathbb{T}^3_x} &\\nabla_x \\mdot [ \\mu(\\rho,\\theta) \\sigma(\\tilde{u})] \\mdot \\frac{1}{\\rho} \\frac{u^b-\\tilde{u}}{\\theta} - \\frac{1}{\\theta} \\mu \\sigma(u^b):\\nabla_x \\tilde{u} \\ dx ds\\\\\n &= \\int^t_0 \\int_{\\mathbb{T}^3_x}\\frac{1}{2} \\mu \\sigma(\\tilde{u}-u^b):\\sigma(\\tilde{u}-u^b) - \\frac{1}{2} \\mu \\sigma(u^b):\\sigma(u^b) \\ dxds + R_5,\n \\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n \\begin{aligned}\n R_5&= \\int^t_0 \\int_{\\mathbb{T}^3_x} \\nabla_x \\mdot [ \\mu(\\rho,\\theta) \\sigma(\\tilde{u})] \\mdot (\\frac{-\\varepsilon\\tilde{\\rho} -\\varepsilon \\tilde{\\theta} -\\varepsilon^2 \\tilde{\\rho}\\tilde{\\theta}}{\\rho \\theta})(u^b-\\tilde{u})+ (\\frac{\\varepsilon\\theta}{\\theta})\\mu \\sigma(u^b):\\nabla_x \\tilde{u} \\ dxds\\\\\n &+\\int^t_0 \\int_{\\mathbb{T}^3_x} \\frac{1}{2} \\left( \\mu(\\rho,\\theta) - \\mu \\right)\\sigma(\\tilde{u}):\\sigma(\\tilde{u}-u^b) \\ dxds.\n \\end{aligned}\n\\end{equation}\nFor the term II, same reasoning shows that\n\\begin{equation}\\label{integration_by_part_theta}\n \\begin{aligned}\n \\int^t_0 \\int_{\\mathbb{T}^3_x} \\frac{1}{\\rho}\\left(\\frac{5}{2} \\nabla_x \\mdot [\\kappa(\\rho,\\theta) \\nabla_x \\tilde{\\theta}]\\right) \\frac{1}{\\theta^2} (\\theta^b -\\tilde{\\theta}) -\\frac{1}{\\theta^2}\\frac{5}{2}\\kappa \\nabla_x \\theta^b \\cdot \\nabla_x \\tilde{\\theta} \\ dx ds \\\\\n = \\int^t_0 \\int_{\\mathbb{T}^3_x} \\frac{5}{2} \\kappa(\\nabla_x \\tilde{\\theta} -\\nabla_x \\theta^b)^2 -\\frac{5}{2}\\kappa( \\nabla_x \\theta^b )^2 \\ dxds + R_6,\n \\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n \\begin{aligned}\n R_6&= \\int^t_0 \\int_{\\mathbb{T}^3_x} (\\frac{1}{\\rho \\theta^2}-1 ) \\frac{5}{2} \\nabla_x \\mdot [\\kappa(\\rho,\\theta) \\nabla_x \\tilde{\\theta} ](\\theta^b -\\tilde{\\theta}) + (1-\\frac{1}{\\theta^2})\\kappa \\nabla_x \\theta^b \\cdot \\nabla_x \\tilde{\\theta}\\ dxds\\\\\n &+ \\int^t_0 \\int_{\\mathbb{T}^3_x} \\frac{5}{2} (\\kappa - \\kappa(\\rho,\\theta))\\nabla_x \\tilde{\\theta}\\cdot (\\nabla_x \\tilde{\\theta}-\\nabla_x \\theta^b) \\ dxds.\n \\end{aligned}\n\\end{equation}\n\n\\subsection{Conclusion} \\label{SS_Conclusion}\nWe now complete the proof of Theorem \\ref{main}.\nNote that\n\\begin{equation}\\label{quadratic_term}\n \\int_0^t \\int_{\\mathbb{T}^3_x} \\frac{1}{2}\\mu \\sigma(\\tilde{u}-u^b):\\sigma(\\tilde{u}-u^b) + \\frac{5}{2} \\kappa(\\nabla_x \\tilde{\\theta} -\\nabla_x \\theta^b)^2 \\ dxds \\geqslant 0.\n\\end{equation}\nCombining with \\eqref{mid_result} and \\eqref{dissipation_control_final_result} we have\n\\begin{equation}\n \\begin{aligned}\n \\frac{1}{{\\varepsilon}^2} &H(f_{\\varepsilon}| M_{\\varepsilon})(t) \\\\\n &\\lesssim \\frac{1}{\\varepsilon^2} H(f^{\\mathrm{in}}_{\\varepsilon}| M^{\\mathrm{in}}_{\\varepsilon}) + \\int^t_0 \\|(\\nabla_x \\tilde{u},\\nabla_x \\tilde{\\theta})\\|_{L^{\\infty}(dx)} \\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}|M_{\\varepsilon}) ds \\\\\n & +\\tilde{R}+ R_5+R_6+R_7 -R_{11}-R_{12} - 2R_{13}.\n \\end{aligned}\n\\end{equation}\nAll the remainder terms can be shown to tend to zero as $\\varepsilon$ tends to zero by using that $(\\rho^b_{\\varepsilon}, u^b_{\\varepsilon}, \\theta^b_{\\varepsilon})$, $(\\tilde{\\rho}_{\\varepsilon},\\tilde{u}_{\\varepsilon},\\tilde{\\theta}_{\\varepsilon})$, $g_{\\varepsilon}$ and $\\frac{1}{\\varepsilon} \\Loth g_{\\varepsilon}$\nare separately uniformly bounded in some proper functional spaces, and Grönwall's inequality then implies that\n\\begin{equation}\n \\begin{aligned}\n \\frac{1}{{\\varepsilon}^2} H(f_{\\varepsilon}| M_{\\varepsilon})(t) \\leqslant\n \\left( \\frac{1}{\\varepsilon^2} H(f^{\\mathrm{in}}_{\\varepsilon}| M^{\\mathrm{in}}_{\\varepsilon})+ \\mathcal{O}(\\varepsilon)\\right) \\exp \\left( C \\int_0^{\\infty} \\| (\\nabla_x \\tilde{u}, \\nabla_x \\tilde{\\theta}) \\|_{L^{\\infty}(dx)} dt \\right).\n \\end{aligned}\n\\end{equation}\nHence,\n\\begin{equation}\\label{entropy_control}\n \\begin{aligned}\n \\frac{1}{{\\varepsilon}^2} H(f_{\\varepsilon}| M_{\\varepsilon})(t) \\lesssim\n \\frac{1}{\\varepsilon^2} H(f^{\\mathrm{in}}_{\\varepsilon}| M^{\\mathrm{in}}_{\\varepsilon})+ \\mathcal{O}(\\varepsilon).\n \\end{aligned}\n\\end{equation}\nThis estimate together with \\eqref{mid_result}, \\eqref{quadratic_term} and the fact $H(f|g) \\geqslant 0$ leads to\n\\begin{equation}\\label{entropy_dissipation_bound_temp}\n \\int_0^t \\int_{\\mathbb{T}^3_x}\\frac{1}{\\varepsilon^4}D(f_\\varepsilon)(s,x) - \\frac{1}{2} \\mu \\sigma(u^b): \\sigma(u^b) - \\frac{5}{2}\\kappa (\\nabla_x \\theta^b)^2\\ dxds \\lesssim \\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}^{\\mathrm{in}}|M_{\\varepsilon}^{\\mathrm{in}}) + \\mathcal{O}(\\varepsilon).\n \\end{equation}\n Using \\eqref{mid_result}, \\eqref{entropy_control}, \\eqref{entropy_dissipation_bound_temp} and \\eqref{dissipation_control_final_result},we then have\n \\begin{equation}\n \\int_0^t \\int_{\\mathbb{T}^3_x} \\frac{1}{2}\\mu \\sigma(\\tilde{u}-u^b):\\sigma(\\tilde{u}-u^b) + \\frac{5}{2} \\kappa(\\nabla_x \\tilde{\\theta} -\\nabla_x \\theta^b)^2 \\ dxds \\lesssim \\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}^{\\mathrm{in}}|M_{\\varepsilon}^{\\mathrm{in}}) + \\mathcal{O}(\\varepsilon),\n\\end{equation}\nwhich completes the proof of Theorem \\ref{main}.", "post_theorem_intro_text_len": 2153, "post_theorem_intro_text": "\\begin{remark}\n  We remark that although the above theorem is in the formal level, its proof is indeed provide a clear way to the later rigorous proof. If we work in the framework of the classical solutions, all the corresponding existence results are available, for the Boltzmann equations, compressible and incompressible Navier-Stokes equations, at least near the constant states. Because the scalings themselves are near the constant states, so in this sense, the formal analysis is not far from the rigorous proof. \n\n    In fact, the estimates \\eqref{main_estimate}, \\eqref{entropy_dissipation_bound} and \\eqref{energy_momentum_flux_bound} hold for $0 \\leqslant t \\leqslant T^*$, where $T^*$ is determined by the minimal lifespan of the solutions to \\eqref{longtime_Boltzmann} and \\eqref{longtime_CNS}. In particular, if  for all $ \\varepsilon > 0$ both $f_\\varepsilon$ and $(\\rho_\\varepsilon, u_\\varepsilon, \\theta_\\varepsilon)$ exist globally in time, then these estimates also hold globally. Furthermore, the validity of the $\\mathcal{O}(\\varepsilon)$ residual term relies on uniform in $\\varepsilon$ estimates for the solutions. As we are working within a perturbation framework, such existence of global solutions of \\eqref{longtime_CNS} are provided in \\cites{Danchin_Inventiones,Chen-Gui-Jiang}, while the required uniform bounds for $(\\rho_\\varepsilon, u_{\\varepsilon}, \\theta_{\\varepsilon})$ can be found in \\cite{CGHJ}. Moreover, the global existence and uniform in $\\varepsilon$ estimates for the solution $f_{\\varepsilon}$ of \\eqref{longtime_Boltzmann} can be found in \\cite{Jiang_Xu_Zhao_2018}.\n\\end{remark}\n    In the next section, we shall prove, at a formal level, the above estimate by calculating the evolution of the relative entropy. In section \\ref{SS_REC}, we show that the unsigned terms will be controlled by the relative entropy itself. In section \\ref{SS_EDC}  we show that the negative terms will be controlled by the entropy dissipation.\n    And in section \\ref{SS_Conclusion} we conclude the estimate \\eqref{main_estimate}, \\eqref{entropy_dissipation_bound}, \\eqref{energy_momentum_flux_bound} by Grönwall's inequality.", "sketch": "To prove Theorem~\\ref{main} (\"at a formal level\"), the paper proceeds as follows:\n\\begin{itemize}\n\\item First, it proves the estimate by \"calculating the evolution of the relative entropy.\"\n\\item In Section~\\ref{SS_REC}, it shows that the \"unsigned terms will be controlled by the relative entropy itself.\"\n\\item In Section~\\ref{SS_EDC}, it shows that the \"negative terms will be controlled by the entropy dissipation.\"\n\\item Finally, in Section~\\ref{SS_Conclusion}, it concludes \\eqref{main_estimate}, \\eqref{entropy_dissipation_bound}, \\eqref{energy_momentum_flux_bound} \"by Gr\\\"onwall's inequality.\"\n\\end{itemize}\nIt is also remarked that the argument is \"in the formal level\" but \"provide[s] a clear way to the later rigorous proof,\" and that the validity of the $\\mathcal{O}(\\varepsilon)$ residual term \"relies on uniform in $\\varepsilon$ estimates for the solutions\" (with cited sources providing global existence and uniform bounds in the perturbative framework).", "expanded_sketch": "To prove the main theorem (\"at a formal level\"), the paper proceeds as follows:\n\\begin{itemize}\n\\item First, it proves the estimate by \"calculating the evolution of the relative entropy.\"\n\\item Next, in the part labeled\n\\begin{aligned}\n        R_6&= \\int^t_0 \\int_{\\mathbb{T}^3_x} (\\frac{1}{\\rho \\theta^2}-1 ) \\frac{5}{2} \\nabla_x \\mdot [\\kappa(\\rho,\\theta) \\nabla_x \\tilde{\\theta} ](\\theta^b -\\tilde{\\theta}) + (1-\\frac{1}{\\theta^2})\\kappa \\nabla_x \\theta^b \\cdot \\nabla_x \\tilde{\\theta}\\ dxds\\\\\n         &+ \\int^t_0 \\int_{\\mathbb{T}^3_x} \\frac{5}{2} (\\kappa - \\kappa(\\rho,\\theta))\\nabla_x \\tilde{\\theta}\\cdot (\\nabla_x \\tilde{\\theta}-\\nabla_x \\theta^b) \\ dxds.\n    \\end{aligned}\n\\end{equation}\n\n\\subsection{Relative entropy control}\\label{SS_REC}\nThe terms \\Rmnum{3} and \\Rmnum{4} are unsigned, however they can be controlled by the quadratic of $(\\rho^b-\\tilde{\\rho}^{\\varepsilon}, u^b - \\tilde{u}^{\\varepsilon},\\theta^b - \\tilde{\\theta}^{\\varepsilon})$ which  can be controlled by relative entropy $\\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}|M_{\\varepsilon})$.\nIn fact, we have the following two lemmas.\n\\begin{lemma} \\label{same_moments}\n    \\begin{equation}\n        \\begin{aligned}\n            H(f_{\\varepsilon}|M_{\\varepsilon}) = H(f_{\\varepsilon}| M_{f_\\varepsilon}) + H(M_{f_{\\varepsilon}}|M_{\\varepsilon}),\n        \\end{aligned}\n\\end{itemize}\n(so that the \"unsigned terms will be controlled by the relative entropy itself\").\n\\begin{itemize}\n\\item Then, in the part labeled\n\\begin{aligned}\n        \\frac{1}{{\\varepsilon}^2} &H(f_{\\varepsilon}| M_{\\varepsilon})(t) +\\int_0^t \\int_{\\mathbb{T}^3_x}\\frac{1}{\\varepsilon^4}D(f_\\varepsilon)(s,x) - \\frac{1}{2} \\mu \\sigma(u^b): \\sigma(u^b) - \\frac{5}{2}\\kappa (\\nabla_x \\theta^b)^2\\  dxds \\\\\n        &+\\int_0^t \\int_{\\mathbb{T}^3_x} \\frac{1}{2}\\mu \\sigma(\\tilde{u}-u^b):\\sigma(\\tilde{u}-u^b) + \\frac{5}{2}  \\kappa(\\nabla_x \\tilde{\\theta} -\\nabla_x \\theta^b)^2 \\ dxds\\\\\n        &\\lesssim \\frac{1}{\\varepsilon^2} H(f^{\\mathrm{in}}_{\\varepsilon}| M^{\\mathrm{in}}_{\\varepsilon}) + \\int^t_0 \\|(\\nabla_x \\tilde{u},\\nabla_x \\tilde{\\theta})\\|_{L^{\\infty}(dx)} \\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}|M_{\\varepsilon}) ds +\\tilde{R} + R_5+R_6+R_7.\n    \\end{aligned}\n\\end{equation}\n\n\\subsection{Entropy dissipation control}\\label{SS_EDC}\n We now deal with the term\n\\begin{equation}\\label{dissipation_control}\n    \\begin{aligned}\n        \\int_0^t \\int_{\\mathbb{T}^3_x}\\frac{1}{\\varepsilon^4}D(f_\\varepsilon)(s,x) - \\frac{1}{2} \\mu \\sigma(u^b): \\sigma(u^b) - \\frac{5}{2}\\kappa (\\nabla_x \\theta^b)^2\\  dxds ,\n    \\end{aligned}\n\\end{itemize}\n(it shows that the \"negative terms will be controlled by the entropy dissipation\").\n\\begin{itemize}\n\\item Finally, in the part labeled\n\\begin{aligned}\n        R_{12}= \\int_0^t \\int_{\\mathbb{T}^3_x} \\left( \\langle v \\mdot \\nabla_x \\Loth g_{\\varepsilon} , \\hat{A}\\rangle + \\langle \\varepsilon \\partial_t g_{\\varepsilon} , \\hat{A}\\rangle\\right) :\\sigma(u^b) \\ dxds ,\\\\\n        R_{13}=\\int_0^t \\int_{\\mathbb{T}^3_x}  \\left(\\langle v \\mdot \\nabla_x \\Loth g_{\\varepsilon} , \\hat{B}\\rangle + \\langle \\varepsilon \\partial_t g_{\\varepsilon} , \\hat{B}\\rangle\\right) \\cdot \\nabla_x \\theta^b \\ dxds.\n    \\end{aligned}\n\\end{equation}\n\n\\subsection{Conclusion} \\label{SS_Conclusion}\nWe now complete the proof of Theorem \\ref{main}.\nNote that\n\\begin{equation}\\label{quadratic_term}\n    \\int_0^t \\int_{\\mathbb{T}^3_x} \\frac{1}{2}\\mu \\sigma(\\tilde{u}-u^b):\\sigma(\\tilde{u}-u^b) + \\frac{5}{2}  \\kappa(\\nabla_x \\tilde{\\theta} -\\nabla_x \\theta^b)^2 \\ dxds \\geqslant 0.\n\\end{equation}\nCombining with \\eqref{mid_result} and \\eqref{dissipation_control_final_result} we have\n\\begin{equation}\n    \\begin{aligned}\n        \\frac{1}{{\\varepsilon}^2} &H(f_{\\varepsilon}| M_{\\varepsilon})(t)  \\\\\n        &\\lesssim \\frac{1}{\\varepsilon^2} H(f^{\\mathrm{in}}_{\\varepsilon}| M^{\\mathrm{in}}_{\\varepsilon}) + \\int^t_0 \\|(\\nabla_x \\tilde{u},\\nabla_x \\tilde{\\theta})\\|_{L^{\\infty}(dx)} \\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}|M_{\\varepsilon}) ds \\\\\n        & +\\tilde{R}+ R_5+R_6+R_7 -R_{11}-R_{12} - 2R_{13}.\n    \\end{aligned}\n\\end{equation}\n\nit concludes the bounds\n\\begin{equation}\\label{main_estimate}\n            \\frac{1}{{\\varepsilon}^2} H(f_{\\varepsilon}| M_{\\varepsilon})(t)  \\leqslant \\frac{1}{\\varepsilon^2}H(f^{\\mathrm{in}}_{\\varepsilon}| M^{\\mathrm{in}}_{\\varepsilon} ) + \\mathcal{O}(\\varepsilon),\n        \\end{equation}\n\\begin{equation}\\label{entropy_dissipation_bound}\n           \\int_0^t \\int_{\\mathbb{T}^3_x}\\frac{1}{\\varepsilon^4}D(f_\\varepsilon)(s,x) - \\frac{1}{2} \\mu \\sigma(\\u^b_\\varepsilon): \\sigma(\\u^b_\\varepsilon) - \\frac{5}{2}\\kappa (\\nabla_x \\theta^b_{\\varepsilon})^2\\  dxds \\lesssim \\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}^{\\mathrm{in}}|M_{\\varepsilon}^{\\mathrm{in}}) + \\mathcal{O}(\\varepsilon),\n         \\end{equation}\n\\begin{equation}\\label{energy_momentum_flux_bound}\n             \\int_0^t \\int_{\\mathbb{T}^3_x} \\frac{1}{2}\\mu \\sigma(\\tilde{\\u}_{\\varepsilon}-\\u^b_{\\varepsilon}):\\sigma(\\tilde{\\u}_{\\varepsilon}-\\u^b_{\\varepsilon}) + \\frac{5}{2}  \\kappa(\\nabla_x \\tilde{\\theta} -\\nabla_x \\theta^b_{\\varepsilon})^2 \\ dxds \\lesssim \\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}^{\\mathrm{in}}|M_{\\varepsilon}^{\\mathrm{in}}) + \\mathcal{O}(\\varepsilon),\n        \\end{equation}\n\"by Gr\\\"onwall's inequality.\"\n\\end{itemize}\nIt is also remarked that the argument is \"in the formal level\" but \"provide[s] a clear way to the later rigorous proof,\" and that the validity of the $\\mathcal{O}(\\varepsilon)$ residual term \"relies on uniform in $\\varepsilon$ estimates for the solutions\" (with cited sources providing global existence and uniform bounds in the perturbative framework).", "expanded_theorem": "\\label{main}\n        Let $f_{\\varepsilon}$  be of the form $f_{\\varepsilon} = M + \\varepsilon M g_{\\varepsilon} $, and $\\{f_{\\varepsilon}\\}$ is a family of solutions to\n        \\begin{equation}\\label{longtime_Boltzmann} \\tag{BE}\n    \\left\\{\\begin{aligned}\n        &\\tau_{\\varepsilon} \\partial_t f_{\\varepsilon} + v\\mdot \\nabla_x f_{\\varepsilon} =\\frac{1}{\\varepsilon} C(f_{\\varepsilon},\\ f_{\\varepsilon}), \\quad x\\in \\mathbb{T}^3, \\quad v \\in \\mathbb{R}^3\\\\\n        &f_{\\varepsilon}(0,x) = f_{\\varepsilon}^{\\mathrm{in}} \\geqslant 0,\n    \\end{aligned}\\right.\n\\end{equation}\n        with $\\tau_\\varepsilon=\\varepsilon$ satisfying the assumptions\n        \\begin{align}\n            &\\partial_t \\int_{\\mathbb{R}^3_v} f_{\\varepsilon} dv + \\nabla_x \\cdot  \\int_{\\mathbb{R}^3_v} f_{\\varepsilon}dv = 0, \\label{conservation_of_mass} \\\\\n            &\\partial_t \\int_{\\mathbb{R}^3_v} vf_{\\varepsilon} dv + \\nabla_x \\cdot \\int_{\\mathbb{R}^3_v} vf_{\\varepsilon}dv =0,\\label{conservation_of_momentum}\\\\\n            &\\partial_t \\int_{\\mathbb{R}^3_v} |v|^2 f_{\\varepsilon} dv + \\nabla_x \\cdot \\int_{\\mathbb{R}^3_v} |v|^2f_{\\varepsilon}dv=0.\\label{conservation_of_energy}\\\\\n        \\end{align}\n        and\n        \\begin{equation}\\label{entropy_inequality}\n        H(f_{\\varepsilon}| M)(t) + \\frac{1}{\\varepsilon^2}\\int_0^t \\int_{\\mathbb{T}^3_x} D(f_\\varepsilon)(s,x) \\ ds dx \\leqslant H(f^{\\mathrm{in}}_{\\varepsilon}| M ),\n    \\end{equation}\n        Let $(\\rho_{\\varepsilon}, u_{\\varepsilon},\\theta_{\\varepsilon})$ be of the form $(\\rho_{\\varepsilon},u_{\\varepsilon},\\theta_{\\varepsilon}) = (1+ \\tilde{\\rho}_{\\varepsilon},\\varepsilon \\tilde{u}_{\\varepsilon},1+ \\varepsilon \\tilde{\\theta}_{\\varepsilon})$, and $\\{(\\rho_{\\varepsilon},u_{\\varepsilon},\\theta_{\\varepsilon})\\}$ is a family of solutions to\n        \\begin{equation} \\label{longtime_CNS}\\tag{$\\text{CNS}_{\\varepsilon}$}\n        \\left\\{ \\begin{aligned}\n            &\\varepsilon\\partial_t \\rho_{\\varepsilon} + \\nabla_x (\\rho_{\\varepsilon} u_{\\varepsilon}) = 0,\\\\\n            &\\varepsilon\\partial_t u_{\\varepsilon} + u_{\\varepsilon} \\mdot \\nabla_x u_{\\varepsilon}  + \\frac{1}{\\rho_{\\varepsilon}} \\nabla_x (\\rho_{\\varepsilon} \\theta_{\\varepsilon})  = \\varepsilon \\frac{1}{\\rho_{\\varepsilon}} \\nabla_x [ \\mu_{\\varepsilon}(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\sigma(u_{\\varepsilon})],\\\\\n            &\\varepsilon \\partial_t \\theta_{\\varepsilon}  + u_{\\varepsilon} \\mdot \\nabla_x \\theta_{\\varepsilon} + \\frac{2}{3} \\theta_{\\varepsilon}  \\nabla_x \\mdot u_{\\varepsilon} = \\varepsilon  \\frac{1}{\\rho_{\\varepsilon}} \\frac{1}{3} \\mu_{\\varepsilon}(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\sigma(u_{\\varepsilon}):\\sigma(u_{\\varepsilon}) +  \\frac{1}{\\rho_{\\varepsilon}}\\frac{5}{3} \\varepsilon \\nabla_x \\cdot[\\kappa(\\rho_{\\varepsilon},\\theta_{\\varepsilon}) \\theta_{\\varepsilon}],\\\\\n            &(\\rho_{\\varepsilon}(0,x),\\ u_{\\varepsilon}(0,x), \\ \\theta_{\\varepsilon}(0,x)) = (1+ \\varepsilon \\tilde{\\rho}^{\\mathrm{in}},\\ \\tilde{u}^{\\mathrm{in}},\\ 1+ \\varepsilon \\tilde{\\theta}^{\\mathrm{in}}).\n        \\end{aligned}\\right.\n    \\end{equation}\n        Then we have, for $t \\geqslant 0$,\n        \\begin{equation}\\label{main_estimate}\n            \\frac{1}{{\\varepsilon}^2} H(f_{\\varepsilon}| M_{\\varepsilon})(t)  \\leqslant \\frac{1}{\\varepsilon^2}H(f^{\\mathrm{in}}_{\\varepsilon}| M^{\\mathrm{in}}_{\\varepsilon} ) + \\mathcal{O}(\\varepsilon),\n        \\end{equation}\n        \\begin{equation}\\label{entropy_dissipation_bound}\n           \\int_0^t \\int_{\\mathbb{T}^3_x}\\frac{1}{\\varepsilon^4}D(f_\\varepsilon)(s,x) - \\frac{1}{2} \\mu \\sigma( \\mathrm{u} ^b_\\varepsilon): \\sigma( \\mathrm{u} ^b_\\varepsilon) - \\frac{5}{2}\\kappa (\\nabla_x \\theta^b_{\\varepsilon})^2\\  dxds \\lesssim \\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}^{\\mathrm{in}}|M_{\\varepsilon}^{\\mathrm{in}}) + \\mathcal{O}(\\varepsilon),\n         \\end{equation}\n         \\begin{equation}\\label{energy_momentum_flux_bound}\n             \\int_0^t \\int_{\\mathbb{T}^3_x} \\frac{1}{2}\\mu \\sigma(\\tilde{ \\mathrm{u} }_{\\varepsilon}- \\mathrm{u} ^b_{\\varepsilon}):\\sigma(\\tilde{ \\mathrm{u} }_{\\varepsilon}- \\mathrm{u} ^b_{\\varepsilon}) + \\frac{5}{2}  \\kappa(\\nabla_x \\tilde{\\theta} -\\nabla_x \\theta^b_{\\varepsilon})^2 \\ dxds \\lesssim \\frac{1}{\\varepsilon^2} H(f_{\\varepsilon}^{\\mathrm{in}}|M_{\\varepsilon}^{\\mathrm{in}}) + \\mathcal{O}(\\varepsilon),\n        \\end{equation}\n        where\n        \\begin{equation}\n            M_{\\varepsilon}^{\\mathrm{in}} = M(1+ \\varepsilon \\tilde\\rho^{\\mathrm{in}}, \\varepsilon \\tilde{u}^{\\mathrm{in}},1 + \\varepsilon \\tilde{\\theta}^{\\mathrm{in}}).\n        \\end{equation}", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let $M$ be the reference Maxwellian, and suppose $f_{\\varepsilon}=M+\\varepsilon M g_{\\varepsilon}$ is a family of nonnegative solutions of the rescaled Boltzmann equation on $\\mathbb T^3_x\\times\\mathbb R^3_v$ with $\\tau_{\\varepsilon}=\\varepsilon$,\n\\[\n\\varepsilon\\,\\partial_t f_{\\varepsilon}+v\\cdot\\nabla_x f_{\\varepsilon}=\\frac1\\varepsilon C(f_{\\varepsilon},f_{\\varepsilon}),\\qquad f_{\\varepsilon}(0,x)=f_{\\varepsilon}^{\\mathrm{in}},\n\\]\nwhich satisfy the local conservation laws of mass, momentum, and energy, as well as the entropy inequality\n\\[\nH(f_{\\varepsilon}\\mid M)(t)+\\frac1{\\varepsilon^2}\\int_0^t\\!\\int_{\\mathbb T^3_x} D(f_{\\varepsilon})(s,x)\\,ds\\,dx\\le H(f_{\\varepsilon}^{\\mathrm{in}}\\mid M),\n\\]\nwhere $H(\\cdot\\mid\\cdot)$ is the relative entropy and $D(f_{\\varepsilon})$ is the Boltzmann entropy dissipation. Suppose also that $(\\rho_{\\varepsilon},u_{\\varepsilon},\\theta_{\\varepsilon})=(1+\\tilde\\rho_{\\varepsilon},\\varepsilon\\tilde u_{\\varepsilon},1+\\varepsilon\\tilde\\theta_{\\varepsilon})$ is a family of solutions of\n\\[\n\\left\\{\\begin{aligned}\n&\\varepsilon\\partial_t \\rho_{\\varepsilon}+\\nabla_x(\\rho_{\\varepsilon}u_{\\varepsilon})=0,\\\\\n&\\varepsilon\\partial_t u_{\\varepsilon}+u_{\\varepsilon}\\cdot\\nabla_xu_{\\varepsilon}+\\frac1{\\rho_{\\varepsilon}}\\nabla_x(\\rho_{\\varepsilon}\\theta_{\\varepsilon})=\\varepsilon\\frac1{\\rho_{\\varepsilon}}\\nabla_x[\\mu_{\\varepsilon}(\\rho_{\\varepsilon},\\theta_{\\varepsilon})\\sigma(u_{\\varepsilon})],\\\\\n&\\varepsilon\\partial_t\\theta_{\\varepsilon}+u_{\\varepsilon}\\cdot\\nabla_x\\theta_{\\varepsilon}+\\frac23\\theta_{\\varepsilon}\\nabla_x\\cdot u_{\\varepsilon}=\\varepsilon\\frac1{\\rho_{\\varepsilon}}\\frac13\\mu_{\\varepsilon}(\\rho_{\\varepsilon},\\theta_{\\varepsilon})\\sigma(u_{\\varepsilon}):\\sigma(u_{\\varepsilon})+\\frac1{\\rho_{\\varepsilon}}\\frac53\\varepsilon\\nabla_x\\cdot[\\kappa(\\rho_{\\varepsilon},\\theta_{\\varepsilon})\\theta_{\\varepsilon}],\\\\\n&(\\rho_{\\varepsilon}(0,x),u_{\\varepsilon}(0,x),\\theta_{\\varepsilon}(0,x))=(1+\\varepsilon\\tilde\\rho^{\\mathrm{in}},\\tilde u^{\\mathrm{in}},1+\\varepsilon\\tilde\\theta^{\\mathrm{in}}),\n\\end{aligned}\\right.\n\\]\nand define the local Maxwellians\n\\[\nM_{\\varepsilon}=\\mathcal M(\\rho_{\\varepsilon},u_{\\varepsilon},\\theta_{\\varepsilon}),\\qquad M_{\\varepsilon}^{\\mathrm{in}}=\\mathcal M(1+\\varepsilon\\tilde\\rho^{\\mathrm{in}},\\varepsilon\\tilde u^{\\mathrm{in}},1+\\varepsilon\\tilde\\theta^{\\mathrm{in}}).\n\\]\nLet $(\\rho^b_{\\varepsilon},u^b_{\\varepsilon},\\theta^b_{\\varepsilon})$ denote the density, momentum, and temperature moments associated with the perturbation $g_{\\varepsilon}$. Which statement holds for every such pair of families and every $t\\ge 0$?", "correct_choice": {"label": "A", "text": "For every $t\\ge 0$, all three estimates hold:\n\\[\n\\frac1{\\varepsilon^2}H(f_{\\varepsilon}\\mid M_{\\varepsilon})(t)\\le \\frac1{\\varepsilon^2}H(f_{\\varepsilon}^{\\mathrm{in}}\\mid M_{\\varepsilon}^{\\mathrm{in}})+\\mathcal O(\\varepsilon),\n\\]\n\\[\n\\int_0^t\\!\\int_{\\mathbb T^3_x}\\left(\\frac1{\\varepsilon^4}D(f_{\\varepsilon})(s,x)-\\frac12\\mu\\,\\sigma(u^b_{\\varepsilon}):\\sigma(u^b_{\\varepsilon})-\\frac52\\kappa\\,(\\nabla_x\\theta^b_{\\varepsilon})^2\\right)dx\\,ds\\lesssim \\frac1{\\varepsilon^2}H(f_{\\varepsilon}^{\\mathrm{in}}\\mid M_{\\varepsilon}^{\\mathrm{in}})+\\mathcal O(\\varepsilon),\n\\]\n\\[\n\\int_0^t\\!\\int_{\\mathbb T^3_x}\\left(\\frac12\\mu\\,\\sigma(\\tilde u_{\\varepsilon}-u^b_{\\varepsilon}):\\sigma(\\tilde u_{\\varepsilon}-u^b_{\\varepsilon})+\\frac52\\kappa\\,(\\nabla_x\\tilde\\theta_{\\varepsilon}-\\nabla_x\\theta^b_{\\varepsilon})^2\\right)dx\\,ds\\lesssim \\frac1{\\varepsilon^2}H(f_{\\varepsilon}^{\\mathrm{in}}\\mid M_{\\varepsilon}^{\\mathrm{in}})+\\mathcal O(\\varepsilon).\n\\]"}, "choices": [{"label": "B", "text": "For every $t\\ge 0$, all three estimates hold with the same right-hand side but with the dissipation term scaled only by $\\frac1{\\varepsilon^2}$, namely\n\\[\n\\frac1{\\varepsilon^2}H(f_{\\varepsilon}\\mid M_{\\varepsilon})(t)\\le \\frac1{\\varepsilon^2}H(f_{\\varepsilon}^{\\mathrm{in}}\\mid M_{\\varepsilon}^{\\mathrm{in}})+\\mathcal O(\\varepsilon),\n\\]\n\\[\n\\int_0^t\\!\\int_{\\mathbb T^3_x}\\left(\\frac1{\\varepsilon^2}D(f_{\\varepsilon})(s,x)-\\frac12\\mu\\,\\sigma(u^b_{\\varepsilon}):\\sigma(u^b_{\\varepsilon})-\\frac52\\kappa\\,(\\nabla_x\\theta^b_{\\varepsilon})^2\\right)dx\\,ds\\lesssim \\frac1{\\varepsilon^2}H(f_{\\varepsilon}^{\\mathrm{in}}\\mid M_{\\varepsilon}^{\\mathrm{in}})+\\mathcal O(\\varepsilon),\n\\]\n\\[\n\\int_0^t\\!\\int_{\\mathbb T^3_x}\\left(\\frac12\\mu\\,\\sigma(\\tilde u_{\\varepsilon}-u^b_{\\varepsilon}):\\sigma(\\tilde u_{\\varepsilon}-u^b_{\\varepsilon})+\\frac52\\kappa\\,(\\nabla_x\\tilde\\theta_{\\varepsilon}-\\nabla_x\\theta^b_{\\varepsilon})^2\\right)dx\\,ds\\lesssim \\frac1{\\varepsilon^2}H(f_{\\varepsilon}^{\\mathrm{in}}\\mid M_{\\varepsilon}^{\\mathrm{in}})+\\mathcal O(\\varepsilon).\n\\]"}, {"label": "C", "text": "For every $t\\ge 0$, the relative entropy estimate holds:\n\\[\n\\frac1{\\varepsilon^2}H(f_{\\varepsilon}\\mid M_{\\varepsilon})(t)\\le \\frac1{\\varepsilon^2}H(f_{\\varepsilon}^{\\mathrm{in}}\\mid M_{\\varepsilon}^{\\mathrm{in}})+\\mathcal O(\\varepsilon).\n\\]"}, {"label": "D", "text": "For every $t\\ge 0$, all three estimates hold with the same left-hand sides as above, but the error term can be taken uniformly negligible in the sense that\n\\[\n\\frac1{\\varepsilon^2}H(f_{\\varepsilon}\\mid M_{\\varepsilon})(t)\\le \\frac1{\\varepsilon^2}H(f_{\\varepsilon}^{\\mathrm{in}}\\mid M_{\\varepsilon}^{\\mathrm{in}}),\n\\]\n\\[\n\\int_0^t\\!\\int_{\\mathbb T^3_x}\\left(\\frac1{\\varepsilon^4}D(f_{\\varepsilon})(s,x)-\\frac12\\mu\\,\\sigma(u^b_{\\varepsilon}):\\sigma(u^b_{\\varepsilon})-\\frac52\\kappa\\,(\\nabla_x\\theta^b_{\\varepsilon})^2\\right)dx\\,ds\\lesssim \\frac1{\\varepsilon^2}H(f_{\\varepsilon}^{\\mathrm{in}}\\mid M_{\\varepsilon}^{\\mathrm{in}}),\n\\]\n\\[\n\\int_0^t\\!\\int_{\\mathbb T^3_x}\\left(\\frac12\\mu\\,\\sigma(\\tilde u_{\\varepsilon}-u^b_{\\varepsilon}):\\sigma(\\tilde u_{\\varepsilon}-u^b_{\\varepsilon})+\\frac52\\kappa\\,(\\nabla_x\\tilde\\theta_{\\varepsilon}-\\nabla_x\\theta^b_{\\varepsilon})^2\\right)dx\\,ds\\lesssim \\frac1{\\varepsilon^2}H(f_{\\varepsilon}^{\\mathrm{in}}\\mid M_{\\varepsilon}^{\\mathrm{in}}).\n\\]"}, {"label": "E", "text": "For every $t\\ge 0$, all three estimates hold after replacing the initial modulated entropy by the entropy relative to the fixed reference Maxwellian $M$, namely\n\\[\n\\frac1{\\varepsilon^2}H(f_{\\varepsilon}\\mid M_{\\varepsilon})(t)\\le \\frac1{\\varepsilon^2}H(f_{\\varepsilon}^{\\mathrm{in}}\\mid M)+\\mathcal O(\\varepsilon),\n\\]\n\\[\n\\int_0^t\\!\\int_{\\mathbb T^3_x}\\left(\\frac1{\\varepsilon^4}D(f_{\\varepsilon})(s,x)-\\frac12\\mu\\,\\sigma(u^b_{\\varepsilon}):\\sigma(u^b_{\\varepsilon})-\\frac52\\kappa\\,(\\nabla_x\\theta^b_{\\varepsilon})^2\\right)dx\\,ds\\lesssim \\frac1{\\varepsilon^2}H(f_{\\varepsilon}^{\\mathrm{in}}\\mid M)+\\mathcal O(\\varepsilon),\n\\]\n\\[\n\\int_0^t\\!\\int_{\\mathbb T^3_x}\\left(\\frac12\\mu\\,\\sigma(\\tilde u_{\\varepsilon}-u^b_{\\varepsilon}):\\sigma(\\tilde u_{\\varepsilon}-u^b_{\\varepsilon})+\\frac52\\kappa\\,(\\nabla_x\\tilde\\theta_{\\varepsilon}-\\nabla_x\\theta^b_{\\varepsilon})^2\\right)dx\\,ds\\lesssim \\frac1{\\varepsilon^2}H(f_{\\varepsilon}^{\\mathrm{in}}\\mid M)+\\mathcal O(\\varepsilon).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "epsilon_scaling_of_entropy_dissipation", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped_dissipation_and_flux_bounds", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "nonzero_remainder_terms_and_Gronwall_error", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "initial_relative_entropy_must_be_modulated_by_M_epsilon_in", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct choice. It gives the hypotheses and asks for the valid estimate, but the exact conclusion is not stated in the prompt itself."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem lists the assumptions of a specific result and asks for its conclusion. The correct option is a near-verbatim statement of that conclusion."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish subtle variants involving the dissipation scaling, the reference Maxwellian, and the error term. However, the task is mostly recognition of the exact theorem statement rather than genuine generation of a conclusion."}, "DQS": {"score": 2, "justification": "The distractors are technically plausible and target realistic failure modes: incorrect epsilon scaling, replacing the sharp statement by a weaker true one, confusing the reference Maxwellian, and altering the remainder size."}, "total_score": 5, "overall_assessment": "Technically strong distractors, but the item is largely a direct theorem-restatement question rather than a high-quality generative reasoning MCQ."}}
{"id": "2602.08147v1", "paper_link": "http://arxiv.org/abs/2602.08147v1", "theorems_cnt": 1, "theorem": {"env_name": "proposition", "content": "\\label{pkus}\n\t\t\\cite{pinkus}\n\t\tLet ${\\mathcal A}=(A_k)_{k\\in{\\mathbb N}}\\in\\calm_d^{\\mathbb N}$ be a random, stationary, and ergodic sequence of $d\\times d$ invertible matrices with $A_k(i,j) = 0$ for $i > j$ and such that \\eqref{f_cond} holds. Then,\n\t\t\\begin{eqnarray*}\n\t\t\\gamma_1 = \\max_{i\\in [d]} {\\mathbb E} \\left( \\log |A_1(i,i)|\\right),\n\t\t\\end{eqnarray*}\n\t\twhere $[d]:=\\{1,\\ldots,d\\}.$", "start_pos": 9503, "end_pos": 9879, "label": "pkus"}, "ref_dict": {"shape": "\\begin{proof}\n\t\tDefine\n\t\t\\[\n\t\t\\widehat A_n := \\eta_n I_d + A^{-1}U_nV^T .\n\t\t\\]\n\t\tBy the commutation assumption, $A$ commutes with $\\widehat A_n$ for every $n$, and hence\n\t\t\\[\n\t\tX_n:=A_n\\cdots A_1 = A^n \\widehat X_n,\n\t\t\\qquad\\text{where } \\widehat X_n:=\\widehat A_n\\cdots \\widehat A_1 .\n\t\t\\]\n\t\tTherefore, by submultiplicativity,\n\t\t\\[\n\t\t\\|X_n\\|\\le \\|A^n\\|\\,\\|\\widehat X_n\\|\n\t\t\\quad\\text{and}\\quad\n\t\t\\|\\widehat X_n\\|=\\|A^{-n}X_n\\|\\le \\|A^{-n}\\|\\,\\|X_n\\|.\n\t\t\\]\n\t\tTaking $\\frac1n\\log$ and letting $n\\to\\infty$, and using Gelfand’s formula\n\t\t$\\lim_{n\\to\\infty}\\frac1n\\log\\|A^{\\pm n}\\|=\\log\\rho(A^{\\pm1})$, gives\n\t\t\\[\n\t\t-\\log\\rho(A^{-1})+\\gamma_1(\\widehat\\cala)\\le \\gamma_1(\\cala)\\le \\log\\rho(A)+\\gamma_1(\\widehat\\cala).\n\t\t\\]\n\n\t\tIt remains to identify $\\gamma_1(\\widehat\\cala)$.\n\t\tLet $K:=\\ker(V^T)$, which has dimension $d-m$ (since $\\mathrm{rank}(V)=m$). For $x\\in K$,\n\t\t$V^Tx=0$ and hence $\\widehat A_n x=\\eta_n x$, so vectors in $K$ grow at rate $\\mathbb E\\log|\\eta_1|$.\n\n\t\tOn the quotient space $\\mathbb R^d/K$, the induced action is computed via the surjective map\n\t\t$q(x)=V^Tx$:\n\t\t\\[\n\t\tq(\\widehat A_n x)=V^T(\\eta_n x + A^{-1}U_nV^Tx)\n\t\t=\\bigl(\\eta_n I_m + V^TA^{-1}U_n\\bigr)\\,q(x)\n\t\t=\\widetilde A_n\\,q(x).\n\t\t\\]\n\t\tThus the cocycle induced on the quotient is precisely $\\widetilde\\cala$.\n\t\tBy the block/extension Furstenberg--Kifer lemma (Lemma~\\ref{blockt}),\n\t\t\\[\n\t\t\\gamma_1(\\widehat\\cala)=\\max\\big(\\mathbb E\\log|\\eta_1|,\\gamma_1(\\widetilde\\cala)\\big).\n\t\t\\]\n\t\tSubstituting into the earlier inequality yields the claimed bounds.\n\t\\end{proof}\n\n\t\\section{Shape graphs}\n\t\\label{shape}\t\n\n\tWe conclude these notes by presenting a generalization of Lemma~\\ref{blockt}, partially inspired by the proof of Theorem~A in~\\cite{pinkus}. As a prelude to this discussion, we first provide an alternative proof of the upper bound in Lemma~\\ref{blockt}, based on the observation that zeros in specific entries of \\(A_n\\) allow the computation of the Lyapunov exponents in terms of those associated with matrices of simpler structure. We note that the proof previously given for the theorem offers more precise insight into the growth behavior of the blocks and, in particular, can be adapted to establish Proposition~\\ref{thm100}. \n\n\tFor the purpose of this section, let \\(\\mathcal{A}:=(A_n)_{n\\in \\mathbb{N}}\\) be a random, stationary, and ergodic sequence of matrices as in the statement of Lemma~\\ref{blockt}, and write each matrix \\(A_n\\) as\n\t\\[\n\tA_n := A_{n,1} + A_{n,2},\n\t\\]\n\twhere \\(A_{n,1}\\) denotes the block diagonal part and \\(A_{n,2}\\) the remaining strictly upper triangular component of \\(A_n\\). \\par\\noindent{\\bf Norm convention.} Throughout Section~\\ref{shape} we take $\\|\\cdot\\|$ to be the operator norm induced by the vector $\\ell^1$-norm (maximum absolute column sum); we write $\\|\\cdot\\|_1$ when clarity is helpful. By equivalence of norms on the finite-dimensional space $\\mathcal M_d$, this choice does not change the Lyapunov exponents and \\eqref{f_cond} holds for $\\|\\cdot\\|$ if and only if it holds for $\\|\\cdot\\|_1$. Then\n\t\\[\n\tX_n = A_{n,1}\\cdots A_{1,1}\n\t+ \\sum_{j=1}^n \\sum_{1\\le i_1 < \\cdots < i_j\\le n}\n\tA_{n,1}\\cdots A_{i_j+1,1}\\, A_{i_j,2}\\, A_{i_j-1,1}\\cdots A_{i_{j-1}+1,1}\\, A_{i_{j-1},2}\\,\\cdots.\n\t\\]\n\tIt is straightforward to verify that multiplying a block diagonal matrix by a strictly block upper triangular matrix (with the same block sizes) produces a strictly block upper triangular matrix. Moreover, multiplying two strictly upper triangular matrices yields another strictly upper triangular matrix with fewer nonzero superdiagonals—a property reminiscent of the nilpotency of strictly upper triangular matrices. Consequently, all products with \\(j > d\\) vanish, and therefore\n\t\\[\n\tX_n = A_{n,1}\\cdots A_{1,1}\n\t+ \\sum_{j=1}^d \\sum_{1 \\leq i_1 < \\cdots < i_j \\leq n}\n\tA_{n,1}\\cdots A_{i_j+1,1}\\, A_{i_j,2}\\, A_{i_j-1,1}\\cdots A_{i_{j-1}+1,1}\\, A_{i_{j-1},2}\\,\\cdots.\n\t\\]\n\n\tFix $\\varepsilon>0$. We will use the following standard Borel--Cantelli estimate (applied to $Y_n=\\log^+\\|A_{n,2}\\|_1$).\n\n\t\\begin{lemma}\\label{lem:bc_log}\n\t\tLet $(Y_n)_{n\\ge 1}$ be a stationary sequence of nonnegative random variables with $\\E(Y_1)<\\infty$. Then for every $\\varepsilon>0$,\n\t\t\\[\n\t\t\\sum_{n\\ge 1} \\pp(Y_n>\\varepsilon n)<\\infty,\n\t\t\\qquad\\text{and hence}\\qquad \n\t\t\\pp(Y_n>\\varepsilon n\\ \\text{\\rm i.o.})=0.\n\t\t\\]\n\t\\end{lemma}\n\t\\begin{proof}\n\t\tBy stationarity, $\\pp(Y_n>\\varepsilon n)=\\pp(Y_1>\\varepsilon n)$. Using the tail-integral identity\n\t\t$$\\E(Y_1)=\\int_0^\\infty \\pp(Y_1>t)\\,dt$$ and partitioning the integral into intervals of length $\\varepsilon$ gives\n\t\t\\[\n\t\t\\sum_{n\\ge 1}\\pp(Y_1>\\varepsilon n)\n\t\t\\le \\frac{1}{\\varepsilon}\\int_0^\\infty \\pp(Y_1>t)\\,dt\n\t\t= \\frac{1}{\\varepsilon}\\E(Y_1)<\\infty.\n\t\t\\]\n\t\tThe Borel--Cantelli lemma yields the claim.\n\t\\end{proof}", "thm101": "\\begin{proposition}\n\t\t\\label{thm101}\n\t\tFor given $m\\in \\nn$ and $(s_1,\\ldots,s_m)\\in\\nn^m,$ let $\\cala=(A_n)_{n\\in\\nn}$ be a tempered sequence of matrices (i.\\,e., \\eqref{small} holds), each having the shape introduced in \\eqref{block_def}. For $i\\in [m],$ let $\\calb_i:=\\big(B_n(i,i)\\big)_{n\\in \\nn},$ and assume that all $B_n(i,i)$ are invertible and all $\\calb_i$ are Lyapunov regular (i.\\,e., an analogue of \\eqref{lr} holds for $\\calb_i$). Then, $\\gamma_1(\\cala)= \\max_{j\\in [m]} \\gamma_1(\\calb_j).$\n\t\\end{proposition}", "xn": "\\label{xn}\n\t\\limsup_{n\\to\\infty} \\frac{1}{n}\\log \\norm{X_n} < \\infty\\qquad \\mbox{\\rm where} \\qquad\n\tX_n := A_n \\cdots A_1.\n\t\\feqn\n\tWe associate to $\\cala$ the (upper) Lyapunov exponent function $\\gamm", "ld": "\\label{ld}\n\t\\gamma(v) := \\limsup_{n\\to\\infty}  \\frac{1}{n} \\log \\norm{X_n v}.\n\t\\feqn\n\tIn particular, we define the \\emph{top (upper) Lyapunov exponent} of the sequence $\\cala$ by\n\t\\beqn \\label{gamma1_", "f_cond": "\\label{f_cond}\n\t\\E\\left( \\log^+ \\|A_1\\| \\right) < \\infty,\n\t\\feqn\n\twhere, for $x>0$, $\\log^+ x:=\\max(0,\\log x)$, with the convention $\\log^+0:=0$.  Oseledets' multiplicative ergodic theorem then yields", "th-shapes": "\\begin{theorem}\n\t\t\\label{th-shapes}\n\t\tLet $\\call$ be a shape set of order $d$ with $k$ elements, and let $(\\calv, \\calt)_{\\call}$ denote its corresponding shape graph. Assume that the shape graph contains only loops of length one (self-loops), and that each nonzero vertex has at most one self-loop. Let $\\cala = (A_n)_{n \\in \\nn}$ be a random, stationary, and ergodic sequence of $d \\times d$ matrices, each admitting a decomposition as in~\\eqref{A_ni_def} with respect to the shape set $\\call$. Assume further that:\n\t\t\\begin{itemize}\n\t\t\t\\item[(i)] For every $1 \\leq s \\leq k$,\n\t\t\t\\[\n\t\t\t\\E\\big(\\log^+ \\|A_{1,s}\\|\\big) < \\infty.\n\t\t\t\\]\n\t\t\t\\item[(ii)] Let $\\calh \\subset \\calv \\setminus \\{ O_d \\}$ be the set of vertices with a self-loop, and let $\\call_{\\calh}$ be the subset of labels corresponding to their self-loops. For every $s \\in \\call_\\calh$, the matrix $A_{1,s}$ is almost surely invertible, and\n\t\t\t\\[\n\t\t\t\\E\\big(\\log^+ \\|A_{1,s}^{-1}\\|\\big) < \\infty.\n\t\t\t\\]\n\t\t\\end{itemize}\n\t\tDefine\n\t\t\\[\n\t\t\\beta_s := \\lim_{n \\to \\infty} \\frac{1}{n} \\log \\left\\| A_{n,s} A_{n-1,s} \\cdots A_{1,s} \\right\\|, \\quad \\text{a.s.},\n\t\t\\]\n\t\tand set\n\t\t\\[\n\t\t\\beta := \\max_{s \\in \\call_\\calh} \\beta_s.\n\t\t\\]\n\n\t\tThen:\n\t\t\\begin{itemize}\n\t\t\t\\item[(a)] The top Lyapunov exponent satisfies\n\t\t\t\\[\n\t\t\t\\gamma_1(\\cala) \\leq \\beta + \\log k.\n\t\t\t\\]\n\t\t\tMoreover, if $O_d \\in \\calv$, define\n\t\t\t\\[\n\t\t\tk_* := \\max_{v \\in \\calv \\setminus \\{O_d\\}} \\#\\bigl\\{\\, s \\in [k] : \\calt(v,L_s)\\neq O_d \\,\\bigr\\},\n\t\t\t\\]\n\t\t\tso that $k_* \\le k$ and $k_*$ is the maximal number of labels that keep a nonzero vertex from transitioning to $O_d$ in one step. Then\n\t\t\t\\[\n\t\t\t\\gamma_1(\\cala) \\leq \\beta + \\log k_*.\n\t\t\t\\]\n\t\t\t(In particular, if every nonzero vertex has at least one outgoing label that transitions to $O_d$, then $k_* \\le k-1$.)\n\t\t\t\\item[(b)] Let $\\calw \\subset \\calh\\setminus \\{O_d\\}$ be the set of self-loop vertices $w$ such that every directed path in the shape graph that ends at $w$ avoids loop vertices other than $w$ (equivalently, it does not pass through any vertex in $\\calh\\setminus\\{w\\}$). Assume moreover that for each $w\\in\\calw$, if $s$ denotes the label of its self-loop, then there exists $r=r(w)\\ge 1$ such that $\\shape{L_s^n}=w$ for all $n\\ge r$. If, for every $w \\in \\calw$ and every $v \\in \\calv\\setminus\\{w\\}$, we have\n\t\t\t\\[\n\t\t\tv \\wedge w = O_d,\n\t\t\t\\]\n\t\t\tand assume in addition that all matrices $A_{n,s}$ are entrywise nonnegative almost surely (so that monomials with the same shape do not cancel entrywise). Then\n\t\t\t\\[\n\t\t\t\\max_{s \\in \\call_\\calw} \\beta_s \\leq \\gamma_1(\\cala),\n\t\t\t\\]\n\t\t\twhere $\\call_\\calw$ denotes the set of labels corresponding to the self-loops of vertices in $\\calw$.\n\t\t\\end{itemize}\n\t\\end{theorem}", "block": "\\begin{proposition}\n\t\t\\label{pkus}\n\t\t\\cite{pinkus}\n\t\tLet $\\cala=(A_k)_{k\\in\\nn}\\in\\calm_d^\\nn$ be a random, stationary, and ergodic sequence of $d\\times d$ invertible matrices with $A_k(i,j) = 0$ for $i > j$ and such that \\eqref{f_cond} holds. Then,\n\t\t\\beq\n\t\t\\gamma_1 = \\max_{i\\in [d]} \\E \\left( \\log |A_1(i,i)|\\right),\n\t\t\\feq\n\t\twhere $[d]:=\\{1,\\ldots,d\\}.$\n\t\\end{proposition}\n\tIn fact, under the conditions of the proposition, all Lyapunov exponents of $\\cala$ can be identified as distinct elements of the multi-set $\\{\\E \\left( \\log |A_1(i,i)|\\right)\\}_{i=1}^d$ (see Corollary~1 in \\cite{hen} and item (iv) on p.~130 of \\cite[Section~3.2]{arnold}).\n\n\t\\par\n\tA conceptual explanation for Proposition~\\ref{pkus} is provided by a lemma of Furstenberg and Kifer \\cite{fifa} (quoted here as Lemma~\\ref{blockt}), which reduces the computation of the top Lyapunov exponent for block-triangular products to that of the diagonal blocks; see, for example, \\cite{fifa,kifer,blockas,blockus,tom}.  Extensions appear in several directions, including switched systems \\cite{ots, solvable, new} and linear cocycles in bundles \\cite[Lemma~3.6]{kifer}; see also \\cite[Proposition~1]{blockas}, \\cite[Theorem~1.1]{blockus} and \\cite[Lemma~4.9]{tom}.  The starting point of this paper is that the same block-triangular viewpoint can be packaged into a practical toolkit that continues to be useful beyond the classical stationary-random setting.  In particular, the resulting bounds are explicit in terms of diagonal-block growth rates and simple combinatorial parameters of the zero pattern, which makes them amenable to computation in matrix-analytic applications.\n\n\t\\par\n\tOur contributions are organized around four structured regimes.  Section~\\ref{block} begins with deterministic control of upper triangular products: for general tempered sequences (not necessarily random), Proposition~\\ref{thm100} gives a two-sided estimate for $\\gamma_1$ in terms of diagonal growth rates $\\alpha_i^\\pm$, and Corollary~\\ref{cora} recovers the familiar ``maximum of diagonal exponents'' identity when the relevant diagonal averages converge.  In the same section we develop a deterministic Furstenberg--Kifer-type reduction for block-triangular products under Lyapunov-regularity hypotheses on the diagonal blocks (Proposition~\\ref{thm101}), providing a modular step that reduces $\\gamma_1$ to diagonal-block data plus an explicit error term.\n\n\tSection~\\ref{per_sect} illustrates how these reductions turn concrete perturbation models (including rank-one updates) into lower-dimensional or scalar cocycles with computable Lyapunov exponents, and it clarifies when a perturbation changes only the top exponent.  Finally, Section~\\ref{shape} introduces shape graphs for sparse decompositions with disjoint supports and limited feedback, proves an energy--entropy bound (Theorem~\\ref{th-shapes}) of the form $\\gamma_1(\\cala)\\le \\beta+\\log k$ (or $\\beta+\\log k_*$ when $O_d\\in\\calv$), and provides a short ``how-to'' guide with worked examples in Section~\\ref{sec:howto-shapes}, including a transfer-matrix/DAG model that makes the mechanism transparent.\n\n\t\\section{Block-triangular matrices}\n\t\\label{block}\t\n\tWe begin with the following inequality concerning the top Lyapunov exponent of a given sequence of upper triangular matrices (cf. \\cite[Lemma~3.1.4]{barl}).\n\t\\begin{proposition}\n\t\t\\label{thm100}\n\t\tLet $\\cala:=(A_n)_{n\\in\\nn}$ be a sequence of real, invertible upper triangular $d\\times d$ matrices (that is $A_n(j,k)=0$ for $j>k$ and $A_n(j,j)\\neq 0,$ $j,k\\in [d]$) such that\n\t\t\\beqn\n\t\t\\label{small}\n\t\t\\limsup_{n\\to\\infty}\\frac{1}{n}\\log \\|A_n\\|=0.\n\t\t\\feqn\n\t\tThen,\n\t\t\\beqn\n\t\t\\label{D_ineq}\n\t\t\\max_{j\\in [d]} \\alpha_j^+\\leq \\gamma_1\n\t\t\\leq \\max_{j\\in [d]} \\Big(\\alpha_j^+  +\\sum_{r=1}^{j-1} \\big(\\alpha_r^+-\\alpha_r^-\\big)\\Big),\n\t\t\\feqn\n\t\twhere\n\t\t$\n\t\t\\alpha_i^-:= \\liminf_{n\\to\\infty} \\frac{1}{n} \\sum_{k=1}^n \\log |A_k(i,i)|$ and $\\alpha_i^+:= \\limsup_{n\\to\\infty} \\frac{1}{n} \\sum_{k=1}^n \\log |A_k(i,i)|.\n\t\t$\n\t\\end{proposition}", "cora": "\\begin{corollary}\n\t\t\\label{cora}\n\t\tLet the conditions of Proposition~\\ref{thm100} hold for $\\cala=(A_n)_{n\\in\\nn}.$ Suppose in addition that $\\alpha_i^+=\\alpha_i^-$ for all $i\\in [d].$\n\t\tThen,\n\t\t$\n\t\t\\gamma_1=\\max_{j\\in [d]} \\lim_{n\\to\\infty} \\frac{1}{n} \\sum_{i=1}^n \\log |A_i(j,j)|.\n\t\t$\n\t\\end{corollary}", "blockt": "\\begin{lemma}\n\t\t\\cite[Lemma~3.6]{fifa}\n\t\t\\label{blockt}\n\t\tFor given $m\\in \\nn$ and $(s_1,\\ldots,s_m)\\in\\nn^m,$ let $(A_n)_{n\\in\\nn}$ be a random, stationary ergodic sequence of matrices, each having the shape introduced in \\eqref{block_def}. Suppose in addition that for all $i\\in [m]$, $B_1(i,i)$ is invertible with probability one, and for all $1\\le i\\le j\\le m$,\n\t\t\\beq\n\t\t\\E\\big(\\log^+ \\|B_1(i,j)\\|\\big)<\\infty \\qquad \\mbox{and} \\qquad \\E\\big(\\log^+ \\|B_1^{-1}(i,i)\\|\\big)<\\infty.\n\t\t\\feq\n\t\tFor $i \\in [m],$ let\n\t\t\\beq\n\t\t\\beta_i=\\lim_{n\\to\\infty} \\frac{1}{n}\\log \\big\\|B_n(i,i)B_{n-1}(i,i)\\cdots B_1(i,i)\\big\\|,\\qquad \\as\n\t\t\\feq\n\t\tThen, $\\gamma_1=\\max_{i \\in [m]} \\beta_i.$\n\t\\end{lemma}", "thm100": "\\begin{proposition}\n\t\t\\label{thm100}\n\t\tLet $\\cala:=(A_n)_{n\\in\\nn}$ be a sequence of real, invertible upper triangular $d\\times d$ matrices (that is $A_n(j,k)=0$ for $j>k$ and $A_n(j,j)\\neq 0,$ $j,k\\in [d]$) such that\n\t\t\\beqn\n\t\t\\label{small}\n\t\t\\limsup_{n\\to\\infty}\\frac{1}{n}\\log \\|A_n\\|=0.\n\t\t\\feqn\n\t\tThen,\n\t\t\\beqn\n\t\t\\label{D_ineq}\n\t\t\\max_{j\\in [d]} \\alpha_j^+\\leq \\gamma_1\n\t\t\\leq \\max_{j\\in [d]} \\Big(\\alpha_j^+  +\\sum_{r=1}^{j-1} \\big(\\alpha_r^+-\\alpha_r^-\\big)\\Big),\n\t\t\\feqn\n\t\twhere\n\t\t$\n\t\t\\alpha_i^-:= \\liminf_{n\\to\\infty} \\frac{1}{n} \\sum_{k=1}^n \\log |A_k(i,i)|$ and $\\alpha_i^+:= \\limsup_{n\\to\\infty} \\frac{1}{n} \\sum_{k=1}^n \\log |A_k(i,i)|.\n\t\t$\n\t\\end{proposition}", "per_sect": "\\begin{bmatrix}\n\t\t\t\tC_n(1,1) & O & \\dots & O \\\\\n\t\t\t\tO & C_n(2,2) & \\dots & O \\\\\n\t\t\t\t\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\t\t\t\tO & O & \\dots & C_n(m,m)\n\t\t\t\\end{bmatrix}\n\t\t\t,\n\t\t\t\\feq\n\t\t\twhere $C_n(i,j)$ are $s_i\\times s_j$ matrices. Then,\n\t\t\t\\beq\n\t\t\t\\| X_n \\|_F^2 = \\| H_n \\|_F^2 + \\| X_n-H_n \\|_F^2 + 2\\inner{H_n}{X_n-H_n}_F,\n\t\t\t\\feq\n\t\t}\n\t\twhere $\\inner{A}{B}_F=\\sum_{i,j} A(i,j)B(i,j)$ is the Frobenius inner product of two matrices $A$ and $B,$ provided they have the same shape. Since for any pair $(i,j)\\in [m]^2,$ either $H_n(i,j)$ or $(X_n-H_n)(i,j)$ is zero, $\\inner{H_n}{X_n-H_n}_F = 0.$\n\t\tThis implies that $\\| X_n \\|_F \\geq \\| H_n \\|_F$. Since $H_n$ is block diagonal,\n\t\t\\[\n\t\t\\|H_n\\|_F^2=\\sum_{j=1}^m \\|C_n(j,j)\\|_F^2.\n\t\t\\]\n\t\tUsing the norm equivalence $\\|M\\|\\le \\|M\\|_F\\le \\sqrt{d}\\,\\|M\\|$ (valid for all $d\\times d$ matrices), each block has the same exponential growth rate whether measured in $\\|\\cdot\\|$ or $\\|\\cdot\\|_F$. Consequently,\n\t\t\\[\n\t\t\\lim_{n\\to\\infty}\\frac1n\\log\\|H_n\\|_F=\\max_{j\\in[m]}\\gamma_1(\\calb_j)=:\\beta,\n\t\t\\]\n\t\t\\noindent For each $i\\in[m]$, set $\\beta_i:=\\gamma_1(\\calb_i)$, so that $\\beta_i\\le \\beta$.\n\n\t\tand taking $\\frac1n\\log$ in $\\|X_n\\|_F\\ge \\|H_n\\|_F$ gives $\\beta\\le \\gamma_1(\\cala)$.\n\t\tWe turn now  to the inverse inequality $\\beta \\geq \\gamma_1(\\cala).$ Let $d_0=0$ and $d_j=s_1+\\cdots +s_j$ for $j\\in[m].$ For $u\\in \\rr^d,$ let $u_j,$ $j=1,\\ldots, m,$ be a vector in $\\rr^{s_j}$ with\n\t\t\\beq\n\t\tu_j(i)=u(d_{j-1}+i),\\qquad i=1,\\ldots, s_j.\n\t\t\\feq\n\t\tDenote by $\\|\\cdot\\|_1$ the $\\ell^1$-norm in $\\rr^d$ and corresponding matrix norm. Thus, $\\|u\\|_1=\\sum_{i=1}^d |u(i)|$ and $\\|A\\|_1=\\sup_{u\\in\\rr^d\\backslash\\{0\\}} \\frac{\\|Au\\|_1}{\\|u\\|_1}=\\max_{j\\in [d]} \\sum_{i=1}^d |A(i,j)|$ for $A \\in \\calm_d.$ Then, for $u\\in \\rr^d,$\n\t\t\\beq\n\t\t\\|X_n u\\|_1&=&\\sum_{i=1}^m \\Big\\|\\sum_{j=i}^m C_n (i,j)u_j\\Big\\|_1\\leq \\sum_{i=1}^m \\sum_{j=i}^m\\| C_n (i,j)\\|_1\\cdot \\|u_j\\|_1\n\t\t\\\\\n\t\t&\\leq& \\sum_{i=1}^m \\sum_{j=i}^m\\| C_n (i,j)\\|_1\\cdot \\|u\\|_1.\n\t\t\\feq\n\t\tTherefore,\n\t\t\\beqn\n\t\t\\label{esta}\n\t\t\\|X_n \\|_1 \\leq \\sum_{i=1}^m \\sum_{j=i}^m\\| C_n (i,j)\\|_1.\n\t\t\\feqn\n\t\tIn view of \\eqref{esta}, it suffices to show that for all $i,j\\in [m]$ such that $i\\leq j,$\n\t\t\\beqn\n\t\t\\label{est3}\n\t\t\\limsup_{n\\to\\infty} \\frac{1}{n}\\log \\|C_n(i,j)\\|_1\\leq \\beta ,\\qquad \\as\n\t\t\\feqn\n\t\tTo show \\eqref{est3}, we adapt an inductive argument employed in the proof of Lemma~3.1.4 in \\cite{barr} in order to obtain a version of this estimate for upper-triangular matrices. For $j>i,$ using a convention that $X_0$ is the $d\\times d$ identity matrix, we have (cf. (3.10) and (3.11) in \\cite{barr}):\n\t\t\\beqn\n\t\t\\nonumber\n\t\tC_n(i,j)&=&\\sum_{t=i+1}^j B_n(i,t)C_{n-1}(t,j)+B_n(i,i)C_{n-1}(i,j)\n\t\t\\\\\n\t\t\\nonumber\n\t\t&=&\\sum_{t=i+1}^j B_n(i,t)C_{n-1}(t,j)+B_n(i,i)\\sum_{t=i+1}^kB_{n-1}(i,t)C_{n-2}(t,j)\n\t\t\\\\\n\t\t\\nonumber\n\t\t&& \\qquad \\qquad +B_n(i,i)B_{n-1}(i,i)C_{n-2}(i,j)= \\cdots\n\t\t\\\\\n\t\t\\label{est4}\n\t\t&=&\n\t\t\\sum_{r=0}^{n-1} B_n(i,i)\\cdots B_{n-r+1}(i,i)\\sum_{t=i+1}^jB_{n-r}(i,t)C_{n-1-r}(t,j).\n\t\t\\feqn\n\t\tFix some $i^*,j^*\\in [m]$ such that $i^*<j^*.$ The inequality in \\eqref{est3} is trivially true for $i=j=j^*.$\n\t\tSuppose now that \\eqref{est3} has been proved for all $(i,j)$ with $j=j^*$ and $i>i^*.$ To finish the proof of the theorem, we will next use a backward induction to show that it is valid for $(i,j)=(i^*,j^*).$ By \\eqref{small} and the induction hypothesis, for all $\\veps>0$ there exists a constant $D_\\veps$ (which depends on $\\cala$) such that for all $n\\in\\nn,$\n\t\t\\beq\n\t\t\\|B_n(i,j)\\|_1\\leq D_\\veps e^{n \\veps}, \\qquad i,j\\in [m],\n\t\t\\feq\n\t\tand\n\t\t\\beq\n\t\t\\|C_n(i,j^*)\\|_1\\leq D_\\veps e^{n (\\beta +\\veps)}, \\qquad i^*<i\\leq j^*.\n\t\t\\feq\n\t\tFurthermore, it follows from \\eqref{abs1} that for all $\\veps>0$ there exists a constant $G_\\veps$ (which depends on $\\cala$) such that for any $i\\in [m]$ and $k,n\\in\\nn$ with $k<n,$\n\t\t\\beqn\n\t\t\\nonumber\n\t\t\\big\\|B_n(i,i)\\cdots B_k(i,i)\\big\\|_1&=&\\big\\|B_n(i,i)\\cdots B_1(i,i)\\cdot B_1^{-1}(i,i)\\cdots B_{k-1}^{-1}(i,i)\\big\\|_1\n\t\t\\\\\n\t\t\\label{gr}\n\t\t&\\leq& \\big\\|B_n(i,i)\\cdots B_1(i,i)\\big\\|_1\\cdot \\big \\|B_1^{-1}(i,i)\\cdots B_{k-1}^{-1}(i,i)\\big\\|_1\n\t\t\\\\\n\t\t\\nonumber\n\t\t&\\leq& \\big(G_\\veps e^{(\\beta_i+\\veps)n}\\big)\\cdot\\big(G_\\veps e^{(-\\beta_i+\\veps)(k-1)}\\big)\\leq G_\\veps^2 e^{(n-k+1)\\beta_i} e^{2n\\veps}.\n\t\t\\feqn\n\t\tTherefore, by virtue of \\eqref{est4},\n\t\t\\beq\n\t\t\\|C_n(i^*,j^*)\\|_1&\\leq & \\sum_{r=0}^{n-1} G_\\veps^2 e^{\\beta_i r}e^{2n\\veps} \\sum_{t=i^*+1}^{j^*}D_\\veps^2 e^{n \\veps} e^{(n-r+1)(\\beta+\\veps)}\\\\\n\t\t&\\leq & nj^* G_\\veps^2 D_\\veps^2 e^{(4n+1) \\veps} e^{(n+1)\\beta},\n\t\t\\feq\n\t\twhich implies \\eqref{est3} with $(i,j)=(i^*,j^*)$ since $\\veps>0$ is arbitrary.\n\t\\end{proof}\n\n\t\\section{Application to matrix perturbations} \n\t\\label{per_sect}\n\n\tWe next apply the results of Section~\\ref{block} to perturbations of linear systems. These perturbations represent an important class of matrix transformations with wide-ranging applications in science and engineering. \n\n\tOur first result concerns the product of random rank-one perturbations of a specific form. This proposition can also be viewed somewhat as a generalization of Theorem~5.1 in \\cite{rankone}, and contributes to the discussion around Remark~2.4 in the same paper.\n\n\t\\par\n\n\t\\begin{proposition} \\label{pert_thm}\n\t\tLet $(u_n,\\eta_n)_{n \\in \\nn}$ be a random, stationary, and ergodic sequence, where $u_n \\in \\rr^d$ and $\\eta_n \\in \\rr$. Suppose $v \\in \\rr^d$ is a fixed vector such that\n\t\t\\beqn \\label{uv_cond}\n\t\t\\E\\bigl(\\log |\\eta_1|\\bigr) \\leq \\E \\bigl(\\log |\\eta_1 + v^T u_1|\\bigr) < \\infty.\n\t\t\\feqn\n\t\t\\begin{itemize}\n\t\t\t\\item[(a)] Let $\\cala := (A_n)_{n \\in \\nn}$ with $A_n := \\eta_n I_d + u_n v^T$ satisfying \\eqref{f_cond}. Then\n\t\t\t\\beqn \\label{An_perturbed_exponents}\n\t\t\t\\gamma_1(\\cala) = \\E \\bigl(\\log |\\eta_1 + v^T u_1| \\bigr)\n\t\t\t\\quad \\text{and} \\quad\n\t\t\t\\gamma_r(\\cala) = \\E \\bigl(\\log |\\eta_1|\\bigr), \\quad r > 1.\n\t\t\t\\feqn\n\n\t\t\t\\item[(b)] Suppose $A \\in \\calm_d$ is a non-singular matrix. Let $\\cala := (A_n)_{n \\in \\nn}$ with $A_n := \\eta_n A + u_n v^T$ satisfying condition \\eqref{f_cond}, and\n\t\t\t\\beqn \\label{Auv_cond}\n\t\t\tP\\bigl(A u_1 v^T = u_1 v^T A\\bigr) = 1.\n\t\t\t\\feqn\n\t\t\tThen,\n\t\t\t\\beq\n\t\t\t- \\log \\rho(A^{-1}) \\leq \\gamma_1(\\cala) - \\E\\bigl(\\log|\\eta_1 + v^T A^{-1} u_1|\\bigr) \\leq \\log \\rho(A),\n\t\t\t\\feq\n\t\t\twhere $\\rho(A)$ denotes the spectral radius of the matrix $A$.\n\t\t\\end{itemize}\n\t\\end{proposition}\n\n\tWe note that, under mild conditions on the sequence $(\\eta_n, u_n)_{n \\in \\mathbb{N}}$, the inequality~\\eqref{uv_cond} can be relaxed, since the result may be applied directly to the inverse cocycle $\\cala^{-1}:=(A_n^{-1})_{n\\in\\nn}$, with part~(a) adjusted accordingly.\n\n\t\\begin{remark}[Varying right vector: two extendable regimes]\n\t\tConsider $A_n=\\eta_n I + u_n v_n^{\\top}$.\n\n\t\t(i) If $v_n=c_n v$ for a fixed $v\\neq 0$, then $A_n=\\eta_n I + \\tilde u_n v^{\\top}$ with\n\t\t$\\tilde u_n:=c_n u_n$. Hence Proposition~\\ref{pert_thm} applies verbatim (under the\n\t\tcorresponding integrability for $\\tilde u_n$), yielding $d-1$ Lyapunov exponents equal to\n\t\t$\\E\\log|\\eta_1|$ and the remaining exponent $\\E\\log|\\eta_1+c_1 v^{\\top}u_1|$.\n\n\t\t(ii) More generally, if $v_n\\in V$ a.s. for a fixed deterministic subspace $V\\subset \\rr^d$\n\t\tof dimension $r$, then $H:=V^{\\perp}$ is a common invariant subspace and\n\t\t$A_n|_H=\\eta_n\\,\\mathrm{Id}$. Consequently $\\E\\log|\\eta_1|$ is a Lyapunov exponent of\n\t\tmultiplicity at least $d-r$. The remaining $r$ exponents are those of the induced cocycle\n\t\ton the quotient $\\rr^d/H$; in general they are not explicit if $v_n$ rotates in $V$.\n\t\tNevertheless, the determinant identity implies\n\t\t\\[\n\t\t\\sum_{i=1}^d \\gamma_i=(d-1)\\E\\log|\\eta_1|+\\E\\log|\\eta_1+v_1^{\\top}u_1|.\n\t\t\\]\n\t\\end{remark}\n\n\tIn order to prove Proposition \\ref{pert_thm}, we first present a preliminary result that explains how to extract the first few Lyapunov exponents of block upper triangular matrices by Lemma~\\ref{blockt}. This extends Corollary~1 in \\cite{hen} (cf.\\ the paragraph following Proposition~\\ref{pkus} in these notes).\n\n\t\\begin{proposition} \\label{block_all_exp}\n\t\tFix $d, m \\in \\nn$, and let $\\caln := (N_n)_{n \\in \\nn}$ be a random, stationary, and ergodic sequence of matrices\n\t\t\\beq\n\t\tN_n := \n\t\t\\begin{bmatrix}\n\t\t\tA_n & * \\\\\n\t\t\tO & \\eta_n I_m\n\t\t\\end{bmatrix}", "pkus": "\\begin{proposition}\n\t\t\\label{pkus}\n\t\t\\cite{pinkus}\n\t\tLet $\\cala=(A_k)_{k\\in\\nn}\\in\\calm_d^\\nn$ be a random, stationary, and ergodic sequence of $d\\times d$ invertible matrices with $A_k(i,j) = 0$ for $i > j$ and such that \\eqref{f_cond} holds. Then,\n\t\t\\beq\n\t\t\\gamma_1 = \\max_{i\\in [d]} \\E \\left( \\log |A_1(i,i)|\\right),\n\t\t\\feq\n\t\twhere $[d]:=\\{1,\\ldots,d\\}.$\n\t\\end{proposition}"}, "pre_theorem_intro_text_len": 4231, "pre_theorem_intro_text": "Lyapunov exponents play a central role in the analysis of random and chaotic dynamical systems: they quantify sensitivity to initial conditions and, in the linear-cocycle setting, encode long-time growth rates of matrix products; see, e.g., \\cite{arnold, barl, boug, fuks, viana}.  Despite their importance, explicit formulas are available only for a comparatively small collection of solvable ensembles---classically, many computations focus on $2\\times 2$ products or on highly symmetric distributions (including several Gaussian-type models); see, for instance, \\cite{rankone, com, comt, elf, fibbo, keyc, kiev, letac, lima, darm, mann, mark, pinkus} and \\cite{adams, ahn, aker, cohen, fort, ens, kargin, newman}.  The purpose of this paper is to extend and reorganize a few ideas from \\cite{rankone, fifa, kiev, pinkus} into a small toolkit for obtaining \\emph{computable} bounds (and, in favorable cases, explicit values) for Lyapunov exponents of structured matrix products.  From a matrix-analysis viewpoint, the guiding principle is that triangular or block-triangular structure---and more generally prescribed sparsity/zero patterns with limited feedback---localize exponential growth to lower-dimensional diagonal dynamics, while the remaining entries contribute at most an explicitly controlled combinatorial factor.\n\tConcretely, our bounds are expressed in terms of diagonal-block products together with a directed graph encoding admissible off-diagonal couplings, and are therefore readily computable from a small collection of lower-dimensional norm estimates.\n\tThe framework covers both stationary/ergodic matrix sequences and deterministic sequences satisfying the temperedness condition \\eqref{xn}, so it can be read as a matrix-analytic toolkit for growth-rate bounds in structured or sparsely coupled linear systems.\n\n\t\\par\n\tFor an integer $d\\ge 2$, let $\\calm_d$ denote the space of $d\\times d$ matrices with real-valued entries.  Throughout, we write $\\|\\cdot\\|$ both for a generic norm on ${\\mathbb R}^d$ (when the particular choice is immaterial) and for the induced operator norm on $\\calm_d$.  Let ${\\mathcal A}:=(A_n)_{n\\in{\\mathbb N}}$ be a sequence in $\\calm_d$ such that\n\t\\begin{eqnarray} \\label{xn}\n\t\\limsup_{n\\to\\infty} \\frac{1}{n}\\log \\|X_n\\| < \\infty\\qquad \\mbox{\\rm where} \\qquad\n\tX_n := A_n \\cdots A_1.\n\t\\end{eqnarray}\n\tWe associate to ${\\mathcal A}$ the (upper) Lyapunov exponent function $\\gamma:{\\mathbb R}^d\\to{\\mathbb R}\\cup\\{-\\infty\\}$ defined by\n\t\\begin{eqnarray}\n\t\\label{ld}\n\t\\gamma(v) := \\limsup_{n\\to\\infty}  \\frac{1}{n} \\log \\|X_n v\\|.\n\t\\end{eqnarray}\n\tIn particular, we define the \\emph{top (upper) Lyapunov exponent} of the sequence ${\\mathcal A}$ by\n\t\\begin{eqnarray} \\label{gamma1_def}\n\t\\gamma_1({\\mathcal A})\n\t:= \\limsup_{n\\to\\infty}\\frac1n\\log\\|X_n\\|\n\t= \\sup_{\\|v\\|=1}\\gamma(v).\n\t\\end{eqnarray}\n\n\tThis directional growth rate is the basic object we aim to control in the structured regimes considered below.\n\n\t\\par\n\tA particularly clean structure emerges when ${\\mathcal A}$ is random, stationary, and ergodic and satisfies the standard integrability condition\n\t\\begin{eqnarray} \\label{f_cond}\n\t{\\mathbb E}\\left( \\log^+ \\|A_1\\| \\right) < \\infty,\n\t\\end{eqnarray}\n\twhere, for $x>0$, $\\log^+ x:=\\max(0,\\log x)$, with the convention $\\log^+0:=0$.  Oseledets' multiplicative ergodic theorem then yields deterministically constant Lyapunov exponents\n\t\\[\n\t\\gamma_1>\\cdots>\\gamma_\\ell,\\qquad \\ell\\le d,\n\t\\]\n\ttogether with a corresponding filtration of subspaces\n\t\\[\n\t{\\mathbb R}^d = E_1 \\supset E_2 \\supset \\cdots \\supset E_{\\ell} \\supset E_{\\ell+1}:=\\{0\\},\n\t\\]\n\tsuch that the limsup in \\eqref{ld} is in fact a limit and $\\gamma(v)\\in\\{\\gamma_1,\\dots,\\gamma_\\ell\\}$ almost surely, with the value determined by the position of $v$ in the filtration; see \\cite[Section~2.1]{barr} and \\cite{arnold, barl, viana}.  In particular, the exponents and subspaces are independent of the realization with probability one.\n\n\t\\par\n\tEven in this ergodic framework, however, the exponents $\\gamma_k$ are rarely available in closed form.  A notable exception occurs for triangular cocycles, where the diagonal entries dictate the asymptotic growth.  In particular, for upper triangular matrices we have:", "context": "Lyapunov exponents play a central role in the analysis of random and chaotic dynamical systems: they quantify sensitivity to initial conditions and, in the linear-cocycle setting, encode long-time growth rates of matrix products; see, e.g., \\cite{arnold, barl, boug, fuks, viana}. Despite their importance, explicit formulas are available only for a comparatively small collection of solvable ensembles---classically, many computations focus on $2\\times 2$ products or on highly symmetric distributions (including several Gaussian-type models); see, for instance, \\cite{rankone, com, comt, elf, fibbo, keyc, kiev, letac, lima, darm, mann, mark, pinkus} and \\cite{adams, ahn, aker, cohen, fort, ens, kargin, newman}. The purpose of this paper is to extend and reorganize a few ideas from \\cite{rankone, fifa, kiev, pinkus} into a small toolkit for obtaining \\emph{computable} bounds (and, in favorable cases, explicit values) for Lyapunov exponents of structured matrix products. From a matrix-analysis viewpoint, the guiding principle is that triangular or block-triangular structure---and more generally prescribed sparsity/zero patterns with limited feedback---localize exponential growth to lower-dimensional diagonal dynamics, while the remaining entries contribute at most an explicitly controlled combinatorial factor.\n Concretely, our bounds are expressed in terms of diagonal-block products together with a directed graph encoding admissible off-diagonal couplings, and are therefore readily computable from a small collection of lower-dimensional norm estimates.\n The framework covers both stationary/ergodic matrix sequences and deterministic sequences satisfying the temperedness condition \\eqref{xn}, so it can be read as a matrix-analytic toolkit for growth-rate bounds in structured or sparsely coupled linear systems.\n\n\\par\n For an integer $d\\ge 2$, let $\\calm_d$ denote the space of $d\\times d$ matrices with real-valued entries. Throughout, we write $\\|\\cdot\\|$ both for a generic norm on ${\\mathbb R}^d$ (when the particular choice is immaterial) and for the induced operator norm on $\\calm_d$. Let ${\\mathcal A}:=(A_n)_{n\\in{\\mathbb N}}$ be a sequence in $\\calm_d$ such that\n \\begin{eqnarray} \\label{xn}\n \\limsup_{n\\to\\infty} \\frac{1}{n}\\log \\|X_n\\| < \\infty\\qquad \\mbox{\\rm where} \\qquad\n X_n := A_n \\cdots A_1.\n \\end{eqnarray}\n We associate to ${\\mathcal A}$ the (upper) Lyapunov exponent function $\\gamma:{\\mathbb R}^d\\to{\\mathbb R}\\cup\\{-\\infty\\}$ defined by\n \\begin{eqnarray}\n \\label{ld}\n \\gamma(v) := \\limsup_{n\\to\\infty} \\frac{1}{n} \\log \\|X_n v\\|.\n \\end{eqnarray}\n In particular, we define the \\emph{top (upper) Lyapunov exponent} of the sequence ${\\mathcal A}$ by\n \\begin{eqnarray} \\label{gamma1_def}\n \\gamma_1({\\mathcal A})\n := \\limsup_{n\\to\\infty}\\frac1n\\log\\|X_n\\|\n = \\sup_{\\|v\\|=1}\\gamma(v).\n \\end{eqnarray}\n\n\\par\n A particularly clean structure emerges when ${\\mathcal A}$ is random, stationary, and ergodic and satisfies the standard integrability condition\n \\begin{eqnarray} \\label{f_cond}\n {\\mathbb E}\\left( \\log^+ \\|A_1\\| \\right) < \\infty,\n \\end{eqnarray}\n where, for $x>0$, $\\log^+ x:=\\max(0,\\log x)$, with the convention $\\log^+0:=0$. Oseledets' multiplicative ergodic theorem then yields deterministically constant Lyapunov exponents\n \\[\n \\gamma_1>\\cdots>\\gamma_\\ell,\\qquad \\ell\\le d,\n \\]\n together with a corresponding filtration of subspaces\n \\[\n {\\mathbb R}^d = E_1 \\supset E_2 \\supset \\cdots \\supset E_{\\ell} \\supset E_{\\ell+1}:=\\{0\\},\n \\]\n such that the limsup in \\eqref{ld} is in fact a limit and $\\gamma(v)\\in\\{\\gamma_1,\\dots,\\gamma_\\ell\\}$ almost surely, with the value determined by the position of $v$ in the filtration; see \\cite[Section~2.1]{barr} and \\cite{arnold, barl, viana}. In particular, the exponents and subspaces are independent of the realization with probability one.\n\n\\par\n Even in this ergodic framework, however, the exponents $\\gamma_k$ are rarely available in closed form. A notable exception occurs for triangular cocycles, where the diagonal entries dictate the asymptotic growth. In particular, for upper triangular matrices we have:\n\n\\label{f_cond}\n\t\\E\\left( \\log^+ \\|A_1\\| \\right) < \\infty,\n\t\\feqn\n\twhere, for $x>0$, $\\log^+ x:=\\max(0,\\log x)$, with the convention $\\log^+0:=0$.  Oseledets' multiplicative ergodic theorem then yields\n\n\\label{ld}\n\t\\gamma(v) := \\limsup_{n\\to\\infty}  \\frac{1}{n} \\log \\norm{X_n v}.\n\t\\feqn\n\tIn particular, we define the \\emph{top (upper) Lyapunov exponent} of the sequence $\\cala$ by\n\t\\beqn \\label{gamma1_\n\n\\label{xn}\n\t\\limsup_{n\\to\\infty} \\frac{1}{n}\\log \\norm{X_n} < \\infty\\qquad \\mbox{\\rm where} \\qquad\n\tX_n := A_n \\cdots A_1.\n\t\\feqn\n\tWe associate to $\\cala$ the (upper) Lyapunov exponent function $\\gamm", "full_context": "Lyapunov exponents play a central role in the analysis of random and chaotic dynamical systems: they quantify sensitivity to initial conditions and, in the linear-cocycle setting, encode long-time growth rates of matrix products; see, e.g., \\cite{arnold, barl, boug, fuks, viana}. Despite their importance, explicit formulas are available only for a comparatively small collection of solvable ensembles---classically, many computations focus on $2\\times 2$ products or on highly symmetric distributions (including several Gaussian-type models); see, for instance, \\cite{rankone, com, comt, elf, fibbo, keyc, kiev, letac, lima, darm, mann, mark, pinkus} and \\cite{adams, ahn, aker, cohen, fort, ens, kargin, newman}. The purpose of this paper is to extend and reorganize a few ideas from \\cite{rankone, fifa, kiev, pinkus} into a small toolkit for obtaining \\emph{computable} bounds (and, in favorable cases, explicit values) for Lyapunov exponents of structured matrix products. From a matrix-analysis viewpoint, the guiding principle is that triangular or block-triangular structure---and more generally prescribed sparsity/zero patterns with limited feedback---localize exponential growth to lower-dimensional diagonal dynamics, while the remaining entries contribute at most an explicitly controlled combinatorial factor.\n Concretely, our bounds are expressed in terms of diagonal-block products together with a directed graph encoding admissible off-diagonal couplings, and are therefore readily computable from a small collection of lower-dimensional norm estimates.\n The framework covers both stationary/ergodic matrix sequences and deterministic sequences satisfying the temperedness condition \\eqref{xn}, so it can be read as a matrix-analytic toolkit for growth-rate bounds in structured or sparsely coupled linear systems.\n\n\\par\n For an integer $d\\ge 2$, let $\\calm_d$ denote the space of $d\\times d$ matrices with real-valued entries. Throughout, we write $\\|\\cdot\\|$ both for a generic norm on ${\\mathbb R}^d$ (when the particular choice is immaterial) and for the induced operator norm on $\\calm_d$. Let ${\\mathcal A}:=(A_n)_{n\\in{\\mathbb N}}$ be a sequence in $\\calm_d$ such that\n \\begin{eqnarray} \\label{xn}\n \\limsup_{n\\to\\infty} \\frac{1}{n}\\log \\|X_n\\| < \\infty\\qquad \\mbox{\\rm where} \\qquad\n X_n := A_n \\cdots A_1.\n \\end{eqnarray}\n We associate to ${\\mathcal A}$ the (upper) Lyapunov exponent function $\\gamma:{\\mathbb R}^d\\to{\\mathbb R}\\cup\\{-\\infty\\}$ defined by\n \\begin{eqnarray}\n \\label{ld}\n \\gamma(v) := \\limsup_{n\\to\\infty} \\frac{1}{n} \\log \\|X_n v\\|.\n \\end{eqnarray}\n In particular, we define the \\emph{top (upper) Lyapunov exponent} of the sequence ${\\mathcal A}$ by\n \\begin{eqnarray} \\label{gamma1_def}\n \\gamma_1({\\mathcal A})\n := \\limsup_{n\\to\\infty}\\frac1n\\log\\|X_n\\|\n = \\sup_{\\|v\\|=1}\\gamma(v).\n \\end{eqnarray}\n\n\\par\n A particularly clean structure emerges when ${\\mathcal A}$ is random, stationary, and ergodic and satisfies the standard integrability condition\n \\begin{eqnarray} \\label{f_cond}\n {\\mathbb E}\\left( \\log^+ \\|A_1\\| \\right) < \\infty,\n \\end{eqnarray}\n where, for $x>0$, $\\log^+ x:=\\max(0,\\log x)$, with the convention $\\log^+0:=0$. Oseledets' multiplicative ergodic theorem then yields deterministically constant Lyapunov exponents\n \\[\n \\gamma_1>\\cdots>\\gamma_\\ell,\\qquad \\ell\\le d,\n \\]\n together with a corresponding filtration of subspaces\n \\[\n {\\mathbb R}^d = E_1 \\supset E_2 \\supset \\cdots \\supset E_{\\ell} \\supset E_{\\ell+1}:=\\{0\\},\n \\]\n such that the limsup in \\eqref{ld} is in fact a limit and $\\gamma(v)\\in\\{\\gamma_1,\\dots,\\gamma_\\ell\\}$ almost surely, with the value determined by the position of $v$ in the filtration; see \\cite[Section~2.1]{barr} and \\cite{arnold, barl, viana}. In particular, the exponents and subspaces are independent of the realization with probability one.\n\n\\par\n Even in this ergodic framework, however, the exponents $\\gamma_k$ are rarely available in closed form. A notable exception occurs for triangular cocycles, where the diagonal entries dictate the asymptotic growth. In particular, for upper triangular matrices we have:\n\n\\label{f_cond}\n\t\\E\\left( \\log^+ \\|A_1\\| \\right) < \\infty,\n\t\\feqn\n\twhere, for $x>0$, $\\log^+ x:=\\max(0,\\log x)$, with the convention $\\log^+0:=0$.  Oseledets' multiplicative ergodic theorem then yields\n\n\\label{ld}\n\t\\gamma(v) := \\limsup_{n\\to\\infty}  \\frac{1}{n} \\log \\norm{X_n v}.\n\t\\feqn\n\tIn particular, we define the \\emph{top (upper) Lyapunov exponent} of the sequence $\\cala$ by\n\t\\beqn \\label{gamma1_\n\n\\label{xn}\n\t\\limsup_{n\\to\\infty} \\frac{1}{n}\\log \\norm{X_n} < \\infty\\qquad \\mbox{\\rm where} \\qquad\n\tX_n := A_n \\cdots A_1.\n\t\\feqn\n\tWe associate to $\\cala$ the (upper) Lyapunov exponent function $\\gamm\n\nand taking $\\frac1n\\log$ in $\\|X_n\\|_F\\ge \\|H_n\\|_F$ gives $\\beta\\le \\gamma_1(\\cala)$.\n We turn now to the inverse inequality $\\beta \\geq \\gamma_1(\\cala).$ Let $d_0=0$ and $d_j=s_1+\\cdots +s_j$ for $j\\in[m].$ For $u\\in \\rr^d,$ let $u_j,$ $j=1,\\ldots, m,$ be a vector in $\\rr^{s_j}$ with\n \\beq\n u_j(i)=u(d_{j-1}+i),\\qquad i=1,\\ldots, s_j.\n \\feq\n Denote by $\\|\\cdot\\|_1$ the $\\ell^1$-norm in $\\rr^d$ and corresponding matrix norm. Thus, $\\|u\\|_1=\\sum_{i=1}^d |u(i)|$ and $\\|A\\|_1=\\sup_{u\\in\\rr^d\\backslash\\{0\\}} \\frac{\\|Au\\|_1}{\\|u\\|_1}=\\max_{j\\in [d]} \\sum_{i=1}^d |A(i,j)|$ for $A \\in \\calm_d.$ Then, for $u\\in \\rr^d,$\n \\beq\n \\|X_n u\\|_1&=&\\sum_{i=1}^m \\Big\\|\\sum_{j=i}^m C_n (i,j)u_j\\Big\\|_1\\leq \\sum_{i=1}^m \\sum_{j=i}^m\\| C_n (i,j)\\|_1\\cdot \\|u_j\\|_1\n \\\\\n &\\leq& \\sum_{i=1}^m \\sum_{j=i}^m\\| C_n (i,j)\\|_1\\cdot \\|u\\|_1.\n \\feq\n Therefore,\n \\beqn\n \\label{esta}\n \\|X_n \\|_1 \\leq \\sum_{i=1}^m \\sum_{j=i}^m\\| C_n (i,j)\\|_1.\n \\feqn\n In view of \\eqref{esta}, it suffices to show that for all $i,j\\in [m]$ such that $i\\leq j,$\n \\beqn\n \\label{est3}\n \\limsup_{n\\to\\infty} \\frac{1}{n}\\log \\|C_n(i,j)\\|_1\\leq \\beta ,\\qquad \\as\n \\feqn\n To show \\eqref{est3}, we adapt an inductive argument employed in the proof of Lemma~3.1.4 in \\cite{barr} in order to obtain a version of this estimate for upper-triangular matrices. For $j>i,$ using a convention that $X_0$ is the $d\\times d$ identity matrix, we have (cf. (3.10) and (3.11) in \\cite{barr}):\n \\beqn\n \\nonumber\n C_n(i,j)&=&\\sum_{t=i+1}^j B_n(i,t)C_{n-1}(t,j)+B_n(i,i)C_{n-1}(i,j)\n \\\\\n \\nonumber\n &=&\\sum_{t=i+1}^j B_n(i,t)C_{n-1}(t,j)+B_n(i,i)\\sum_{t=i+1}^kB_{n-1}(i,t)C_{n-2}(t,j)\n \\\\\n \\nonumber\n && \\qquad \\qquad +B_n(i,i)B_{n-1}(i,i)C_{n-2}(i,j)= \\cdots\n \\\\\n \\label{est4}\n &=&\n \\sum_{r=0}^{n-1} B_n(i,i)\\cdots B_{n-r+1}(i,i)\\sum_{t=i+1}^jB_{n-r}(i,t)C_{n-1-r}(t,j).\n \\feqn\n Fix some $i^*,j^*\\in [m]$ such that $i^*<j^*.$ The inequality in \\eqref{est3} is trivially true for $i=j=j^*.$\n Suppose now that \\eqref{est3} has been proved for all $(i,j)$ with $j=j^*$ and $i>i^*.$ To finish the proof of the theorem, we will next use a backward induction to show that it is valid for $(i,j)=(i^*,j^*).$ By \\eqref{small} and the induction hypothesis, for all $\\veps>0$ there exists a constant $D_\\veps$ (which depends on $\\cala$) such that for all $n\\in\\nn,$\n \\beq\n \\|B_n(i,j)\\|_1\\leq D_\\veps e^{n \\veps}, \\qquad i,j\\in [m],\n \\feq\n and\n \\beq\n \\|C_n(i,j^*)\\|_1\\leq D_\\veps e^{n (\\beta +\\veps)}, \\qquad i^*<i\\leq j^*.\n \\feq\n Furthermore, it follows from \\eqref{abs1} that for all $\\veps>0$ there exists a constant $G_\\veps$ (which depends on $\\cala$) such that for any $i\\in [m]$ and $k,n\\in\\nn$ with $k<n,$\n \\beqn\n \\nonumber\n \\big\\|B_n(i,i)\\cdots B_k(i,i)\\big\\|_1&=&\\big\\|B_n(i,i)\\cdots B_1(i,i)\\cdot B_1^{-1}(i,i)\\cdots B_{k-1}^{-1}(i,i)\\big\\|_1\n \\\\\n \\label{gr}\n &\\leq& \\big\\|B_n(i,i)\\cdots B_1(i,i)\\big\\|_1\\cdot \\big \\|B_1^{-1}(i,i)\\cdots B_{k-1}^{-1}(i,i)\\big\\|_1\n \\\\\n \\nonumber\n &\\leq& \\big(G_\\veps e^{(\\beta_i+\\veps)n}\\big)\\cdot\\big(G_\\veps e^{(-\\beta_i+\\veps)(k-1)}\\big)\\leq G_\\veps^2 e^{(n-k+1)\\beta_i} e^{2n\\veps}.\n \\feqn\n Therefore, by virtue of \\eqref{est4},\n \\beq\n \\|C_n(i^*,j^*)\\|_1&\\leq & \\sum_{r=0}^{n-1} G_\\veps^2 e^{\\beta_i r}e^{2n\\veps} \\sum_{t=i^*+1}^{j^*}D_\\veps^2 e^{n \\veps} e^{(n-r+1)(\\beta+\\veps)}\\\\\n &\\leq & nj^* G_\\veps^2 D_\\veps^2 e^{(4n+1) \\veps} e^{(n+1)\\beta},\n \\feq\n which implies \\eqref{est3} with $(i,j)=(i^*,j^*)$ since $\\veps>0$ is arbitrary.\n \\end{proof}\n\n\\begin{proposition} \\label{block_all_exp}\n Fix $d, m \\in \\nn$, and let $\\caln := (N_n)_{n \\in \\nn}$ be a random, stationary, and ergodic sequence of matrices\n \\beq\n N_n := \n \\begin{bmatrix}\n A_n & * \\\\\n O & \\eta_n I_m\n \\end{bmatrix},\n \\feq\n where $\\eta_n \\in \\rr$ and $A_n \\in \\calm_d$. Assume that $\\caln$ satisfies \\eqref{f_cond} (i.e.\\ $\\E(\\log^+\\|N_1\\|)<\\infty$), that $A_1$ is almost surely invertible with $\\E(\\log^+\\|A_1^{-1}\\|)<\\infty$, and that $\\eta_1\\neq 0$ almost surely with $\\E(\\log^+|\\eta_1|^{-1})<\\infty$. Set $\\cala := (A_n)_{n \\in \\nn}$. Then\n \\beq\n \\gamma_1(\\caln) = \\max \\bigl( \\E(\\log | \\eta_1 |), \\, \\gamma_1(\\cala) \\bigr),\n \\feq\n and for any $2 \\leq r \\leq \\min(m,d)$,\n \\beq\n && \\gamma_r(\\caln) = \\max_{0\\leq \\ell \\leq r} \\left(\\sum_{s=1}^\\ell \\gamma_s(\\cala) + (r-\\ell) \\E(\\log | \\eta_1|) \\right) \\\\\n && \\qquad \\qquad - \\max_{0\\leq \\ell \\leq r-1} \\left(\\sum_{s=1}^\\ell \\gamma_s(\\cala) + (r-\\ell) \\E(\\log | \\eta_1|) \\right).\n \\feq\n \\end{proposition} \n \\begin{proof}\n Recall that a minor of size~$r$ is associated to ordered sequences of indices $J$ and $L$ of length $r$, defined as the determinant of the submatrix formed by selecting the rows indexed by $J$ and the columns indexed by $L$.\n We write $[N]:=\\{1,\\dots,N\\}$ and, for $0\\le r\\le N$, we let\n $[N]_r:=\\{(i_1,\\dots,i_r):1\\le i_1<\\cdots<i_r\\le N\\}$; we freely identify such increasing $r$--tuples with the underlying $r$--element subsets.\n For $B\\subset[m]$ we write $B+d:=\\{b+d: b\\in B\\}\\subset\\{d+1,\\dots,d+m\\}$, and for $J_2,L_2\\subset\\{d+1,\\dots,d+m\\}$ we interpret\n $I_m[J_2,L_2]:=I_m[J_2-d,\\,L_2-d]$.\n For any matrix $A$, let $H_r(A)$ denote the $r$-th compound matrix of $A$, whose entries are all minors of size~$r$. Set $B_n^{(r)} := H_r(N_n)$, and let $\\calb_r := (B_n^{(r)})_{n \\in \\nn}$ be the sequence of $r$-th compound matrices in $\\caln$.\n\n\\begin{proposition}\\label{pert_thm2}\n Fix $m,d\\in\\mathbb N$, a matrix $V\\in\\mathcal M_{d\\times m}(\\mathbb R)$ of rank $m$, and an invertible\n matrix $A\\in\\mathcal M_{d\\times d}(\\mathbb R)$.\n Let $(U_n,\\eta_n)_{n\\ge1}$ be a random stationary ergodic sequence with\n $U_n\\in\\mathcal M_{d\\times m}(\\mathbb R)$ and $\\eta_n\\in\\mathbb R$, and define\n \\[\n A_n:=\\eta_n A + U_nV^T,\\qquad \n \\widetilde A_n:=\\eta_n I_m + V^T A^{-1}U_n .\n \\]\n Assume\n \\[\n \\mathbb P(AU_1V^T = U_1V^T A)=1 .\n \\]\n Then, writing $\\cala=(A_n)$ and $\\widetilde\\cala=(\\widetilde A_n)$,\n \\[\n -\\log\\rho(A^{-1}) + \\max\\!\\Big(\\mathbb E\\log|\\eta_1|,\\,\\gamma_1(\\widetilde\\cala)\\Big)\n \\;\\le\\;\n \\gamma_1(\\cala)\n \\;\\le\\;\n \\log\\rho(A) + \\max\\!\\Big(\\mathbb E\\log|\\eta_1|,\\,\\gamma_1(\\widetilde\\cala)\\Big).\n \\]\n \\end{proposition}\n\n\\begin{theorem}\n \\label{th-shapes}\n Let $\\call$ be a shape set of order $d$ with $k$ elements, and let $(\\calv, \\calt)_{\\call}$ denote its corresponding shape graph. Assume that the shape graph contains only loops of length one (self-loops), and that each nonzero vertex has at most one self-loop. Let $\\cala = (A_n)_{n \\in \\nn}$ be a random, stationary, and ergodic sequence of $d \\times d$ matrices, each admitting a decomposition as in~\\eqref{A_ni_def} with respect to the shape set $\\call$. Assume further that:\n \\begin{itemize}\n \\item[(i)] For every $1 \\leq s \\leq k$,\n \\[\n \\E\\big(\\log^+ \\|A_{1,s}\\|\\big) < \\infty.\n \\]\n \\item[(ii)] Let $\\calh \\subset \\calv \\setminus \\{ O_d \\}$ be the set of vertices with a self-loop, and let $\\call_{\\calh}$ be the subset of labels corresponding to their self-loops. For every $s \\in \\call_\\calh$, the matrix $A_{1,s}$ is almost surely invertible, and\n \\[\n \\E\\big(\\log^+ \\|A_{1,s}^{-1}\\|\\big) < \\infty.\n \\]\n \\end{itemize}\n Define\n \\[\n \\beta_s := \\lim_{n \\to \\infty} \\frac{1}{n} \\log \\left\\| A_{n,s} A_{n-1,s} \\cdots A_{1,s} \\right\\|, \\quad \\text{a.s.},\n \\]\n and set\n \\[\n \\beta := \\max_{s \\in \\call_\\calh} \\beta_s.\n \\]", "post_theorem_intro_text_len": 2776, "post_theorem_intro_text": "In fact, under the conditions of the proposition, all Lyapunov exponents of ${\\mathcal A}$ can be identified as distinct elements of the multi-set $\\{{\\mathbb E} \\left( \\log |A_1(i,i)|\\right)\\}_{i=1}^d$ (see Corollary~1 in \\cite{hen} and item (iv) on p.~130 of \\cite[Section~3.2]{arnold}).\n\n\t\\par\n\tA conceptual explanation for Proposition~\\ref{pkus} is provided by a lemma of Furstenberg and Kifer \\cite{fifa} (quoted here as Lemma~\\ref{blockt}), which reduces the computation of the top Lyapunov exponent for block-triangular products to that of the diagonal blocks; see, for example, \\cite{fifa,kifer,blockas,blockus,tom}.  Extensions appear in several directions, including switched systems \\cite{ots, solvable, new} and linear cocycles in bundles \\cite[Lemma~3.6]{kifer}; see also \\cite[Proposition~1]{blockas}, \\cite[Theorem~1.1]{blockus} and \\cite[Lemma~4.9]{tom}.  The starting point of this paper is that the same block-triangular viewpoint can be packaged into a practical toolkit that continues to be useful beyond the classical stationary-random setting.  In particular, the resulting bounds are explicit in terms of diagonal-block growth rates and simple combinatorial parameters of the zero pattern, which makes them amenable to computation in matrix-analytic applications.\n\n\t\\par\n\tOur contributions are organized around four structured regimes.  Section~\\ref{block} begins with deterministic control of upper triangular products: for general tempered sequences (not necessarily random), Proposition~\\ref{thm100} gives a two-sided estimate for $\\gamma_1$ in terms of diagonal growth rates $\\alpha_i^\\pm$, and Corollary~\\ref{cora} recovers the familiar ``maximum of diagonal exponents'' identity when the relevant diagonal averages converge.  In the same section we develop a deterministic Furstenberg--Kifer-type reduction for block-triangular products under Lyapunov-regularity hypotheses on the diagonal blocks (Proposition~\\ref{thm101}), providing a modular step that reduces $\\gamma_1$ to diagonal-block data plus an explicit error term.\n\n\tSection~\\ref{per_sect} illustrates how these reductions turn concrete perturbation models (including rank-one updates) into lower-dimensional or scalar cocycles with computable Lyapunov exponents, and it clarifies when a perturbation changes only the top exponent.  Finally, Section~\\ref{shape} introduces shape graphs for sparse decompositions with disjoint supports and limited feedback, proves an energy--entropy bound (Theorem~\\ref{th-shapes}) of the form $\\gamma_1({\\mathcal A})\\le \\beta+\\log k$ (or $\\beta+\\log k_*$ when $O_d\\in{\\mathcal V}$), and provides a short ``how-to'' guide with worked examples in Section~\\ref{sec:howto-shapes}, including a transfer-matrix/DAG model that makes the mechanism transparent.", "sketch": "A conceptual explanation for Proposition~\\ref{pkus} is provided by a lemma of Furstenberg and Kifer \\cite{fifa} (quoted here as Lemma~\\ref{blockt}), which \\emph{reduces the computation of the top Lyapunov exponent for block-triangular products to that of the diagonal blocks}. In the present upper-triangular setting, this reduction yields the familiar identification that the top Lyapunov exponent is the maximum among the diagonal-block (here, diagonal-entry) growth rates, i.e. the elements of the multi-set $\\{{\\mathbb E}(\\log|A_1(i,i)|)\\}_{i=1}^d$.", "expanded_sketch": "A conceptual explanation for Proposition~\\ref{pkus} is provided by the following lemma of Furstenberg and Kifer (H. Furstenberg and Y. Kifer, \\emph{Random matrix products and measures on projective spaces}, quoted here as Lemma~\\ref{blockt}), which \\emph{reduces the computation of the top Lyapunov exponent for block-triangular products to that of the diagonal blocks}. We first recall this lemma.\n\n\\begin{lemma}\n\t\t\\cite[Lemma~3.6]{fifa}\n\t\t\\label{blockt}\n\t\tFor given $m\\in \\nn$ and $(s_1,\\ldots,s_m)\\in\\nn^m,$ let $(A_n)_{n\\in\\nn}$ be a random, stationary ergodic sequence of matrices, each having the shape introduced in \\eqref{block_def}. Suppose in addition that for all $i\\in [m]$, $B_1(i,i)$ is invertible with probability one, and for all $1\\le i\\le j\\le m$,\n\t\t\\beq\n\t\t\\E\\big(\\log^+ \\|B_1(i,j)\\|\\big)<\\infty \\qquad \\mbox{and} \\qquad \\E\\big(\\log^+ \\|B_1^{-1}(i,i)\\|\\big)<\\infty.\n\t\t\\feq\n\t\tFor $i \\in [m],$ let\n\t\t\\beq\n\t\t\\beta_i=\\lim_{n\\to\\infty} \\frac{1}{n}\\log \\big\\|B_n(i,i)B_{n-1}(i,i)\\cdots B_1(i,i)\\big\\|,\\qquad \\as\n\t\t\\feq\n\t\tThen, $\\gamma_1=\\max_{i \\in [m]} \\beta_i.$\n\t\\end{lemma}\n\nIn the present upper-triangular setting, this reduction yields the familiar identification that the top Lyapunov exponent is the maximum among the diagonal-block (here, diagonal-entry) growth rates, i.e. the elements of the multi-set $\\{{\\mathbb E}(\\log|A_1(i,i)|)\\}_{i=1}^d$.", "expanded_theorem": "\\label{pkus}\nPinkus, \\emph{(missing title/year in ref\\_dict)}\nLet ${\\mathcal A}=(A_k)_{k\\in{\\mathbb N}}\\in\\calm_d^{\\mathbb N}$ be a random, stationary, and ergodic sequence of $d\\times d$ invertible matrices with $A_k(i,j) = 0$ for $i > j$ and such that\n\\label{f_cond}\n\t\\E\\left( \\log^+ \\|A_1\\| \\right) < \\infty,\n\t\\feqn\n\twhere, for $x>0$, $\\log^+ x:=\\max(0,\\log x)$, with the convention $\\log^+0:=0$.  Oseledets' multiplicative ergodic theorem then yields holds. Then,\n\\begin{eqnarray*}\n\\gamma_1 = \\max_{i\\in [d]} {\\mathbb E} \\left( \\log |A_1(i,i)|\\right),\n\\end{eqnarray*}\nwhere $[d]:=\\{1,\\ldots,d\\}.$,", "theorem_type": ["Equality or Bound", "Universal"], "mcq": {"question": "Let ${\\mathcal A}=(A_k)_{k\\in\\mathbb N}$ be a random, stationary, and ergodic sequence of invertible real $d\\times d$ matrices such that each $A_k$ is upper triangular, i.e. $A_k(i,j)=0$ whenever $i>j$, and assume\n\\[\n\\mathbb E\\big(\\log^+\\|A_1\\|\\big)<\\infty,\n\\]\nwhere $\\log^+ x:=\\max(0,\\log x)$ for $x>0$ and $\\log^+0:=0$. If $X_n:=A_n\\cdots A_1$, and $\\gamma_1$ denotes the top Lyapunov exponent\n\\[\n\\gamma_1:=\\lim_{n\\to\\infty}\\frac1n\\log\\|X_n\\|\n\\]\n(which exists almost surely and is deterministic under these assumptions), which statement holds for every such sequence?", "correct_choice": {"label": "A", "text": "\\[\\gamma_1=\\max_{i\\in[d]}\\,\\mathbb E\\big(\\log |A_1(i,i)|\\big),\\qquad [d]=\\{1,\\dots,d\\}.\\]"}, "choices": [{"label": "B", "text": "\\[\\gamma_1=\\max_{i\\in[d]}\\,\\log\\mathbb E\\big(|A_1(i,i)|\\big),\\qquad [d]=\\{1,\\dots,d\\}.\\]"}, {"label": "C", "text": "\\[\\gamma_1\\ge \\max_{i\\in[d]}\\,\\mathbb E\\big(\\log |A_1(i,i)|\\big),\\qquad [d]=\\{1,\\dots,d\\}.\\]"}, {"label": "D", "text": "\\[\\gamma_1=\\mathbb E\\Big(\\log \\max_{i\\in[d]} |A_1(i,i)|\\Big),\\qquad [d]=\\{1,\\dots,d\\}.\\]"}, {"label": "E", "text": "\\[\\gamma_1=\\max_{i\\in[d]}\\,\\lim_{n\\to\\infty}\\frac1n\\log\\mathbb E\\big|\\big(X_n\\big)(i,i)\\big|,\\qquad [d]=\\{1,\\dots,d\\}.\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "trace_identity", "tampered_component": "expectation-of-log on diagonal entries", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "trace_identity", "tampered_component": "equality replaced by one-sided lower bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "max of diagonal Lyapunov rates replaced by expectation of pointwise maximum", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "almost-sure diagonal growth replaced by growth of expectations with quantifiers moved outside", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the conclusion or encode the exact formula for the top Lyapunov exponent. It gives the hypotheses and asks which conclusion follows, without obvious lexical leakage toward choice A."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the hypotheses are laid out in full and the correct option is the standard conclusion for upper triangular random matrix cocycles. It is very close to a restatement rather than a problem requiring derived insight."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to reject plausible alternatives such as the weaker inequality, the sum over diagonal terms, and the log-vs-expectation confusion. However, the item mainly tests recognition of the exact theorem rather than substantial generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: confusing max with sum, accepting a merely weaker true statement, mixing pathwise and expectation-level claims, and swapping expectation of log with log of expectation."}, "total_score": 5, "overall_assessment": "A mathematically solid but theorem-recall-heavy MCQ. It avoids direct answer leakage and uses strong distractors, but it is close to a tautological restatement of a known result and only moderately pressures genuine reasoning."}}
{"id": "2602.08147v1", "paper_link": "http://arxiv.org/abs/2602.08147v1", "theorems_cnt": 1, "theorem": {"env_name": "proposition", "content": "\\label{pkus}\n\t\t\\cite{pinkus}\n\t\tLet ${\\mathcal A}=(A_k)_{k\\in{\\mathbb N}}\\in\\calm_d^{\\mathbb N}$ be a random, stationary, and ergodic sequence of $d\\times d$ invertible matrices with $A_k(i,j) = 0$ for $i > j$ and such that \\eqref{f_cond} holds. Then,\n\t\t\\begin{eqnarray*}\n\t\t\\gamma_1 = \\max_{i\\in [d]} {\\mathbb E} \\left( \\log |A_1(i,i)|\\right),\n\t\t\\end{eqnarray*}\n\t\twhere $[d]:=\\{1,\\ldots,d\\}.$", "start_pos": 9503, "end_pos": 9879, "label": "pkus"}, "ref_dict": {"shape": "\\begin{proof}\n\t\tDefine\n\t\t\\[\n\t\t\\widehat A_n := \\eta_n I_d + A^{-1}U_nV^T .\n\t\t\\]\n\t\tBy the commutation assumption, $A$ commutes with $\\widehat A_n$ for every $n$, and hence\n\t\t\\[\n\t\tX_n:=A_n\\cdots A_1 = A^n \\widehat X_n,\n\t\t\\qquad\\text{where } \\widehat X_n:=\\widehat A_n\\cdots \\widehat A_1 .\n\t\t\\]\n\t\tTherefore, by submultiplicativity,\n\t\t\\[\n\t\t\\|X_n\\|\\le \\|A^n\\|\\,\\|\\widehat X_n\\|\n\t\t\\quad\\text{and}\\quad\n\t\t\\|\\widehat X_n\\|=\\|A^{-n}X_n\\|\\le \\|A^{-n}\\|\\,\\|X_n\\|.\n\t\t\\]\n\t\tTaking $\\frac1n\\log$ and letting $n\\to\\infty$, and using Gelfand’s formula\n\t\t$\\lim_{n\\to\\infty}\\frac1n\\log\\|A^{\\pm n}\\|=\\log\\rho(A^{\\pm1})$, gives\n\t\t\\[\n\t\t-\\log\\rho(A^{-1})+\\gamma_1(\\widehat\\cala)\\le \\gamma_1(\\cala)\\le \\log\\rho(A)+\\gamma_1(\\widehat\\cala).\n\t\t\\]\n\n\t\tIt remains to identify $\\gamma_1(\\widehat\\cala)$.\n\t\tLet $K:=\\ker(V^T)$, which has dimension $d-m$ (since $\\mathrm{rank}(V)=m$). For $x\\in K$,\n\t\t$V^Tx=0$ and hence $\\widehat A_n x=\\eta_n x$, so vectors in $K$ grow at rate $\\mathbb E\\log|\\eta_1|$.\n\n\t\tOn the quotient space $\\mathbb R^d/K$, the induced action is computed via the surjective map\n\t\t$q(x)=V^Tx$:\n\t\t\\[\n\t\tq(\\widehat A_n x)=V^T(\\eta_n x + A^{-1}U_nV^Tx)\n\t\t=\\bigl(\\eta_n I_m + V^TA^{-1}U_n\\bigr)\\,q(x)\n\t\t=\\widetilde A_n\\,q(x).\n\t\t\\]\n\t\tThus the cocycle induced on the quotient is precisely $\\widetilde\\cala$.\n\t\tBy the block/extension Furstenberg--Kifer lemma (Lemma~\\ref{blockt}),\n\t\t\\[\n\t\t\\gamma_1(\\widehat\\cala)=\\max\\big(\\mathbb E\\log|\\eta_1|,\\gamma_1(\\widetilde\\cala)\\big).\n\t\t\\]\n\t\tSubstituting into the earlier inequality yields the claimed bounds.\n\t\\end{proof}\n\n\t\\section{Shape graphs}\n\t\\label{shape}\t\n\n\tWe conclude these notes by presenting a generalization of Lemma~\\ref{blockt}, partially inspired by the proof of Theorem~A in~\\cite{pinkus}. As a prelude to this discussion, we first provide an alternative proof of the upper bound in Lemma~\\ref{blockt}, based on the observation that zeros in specific entries of \\(A_n\\) allow the computation of the Lyapunov exponents in terms of those associated with matrices of simpler structure. We note that the proof previously given for the theorem offers more precise insight into the growth behavior of the blocks and, in particular, can be adapted to establish Proposition~\\ref{thm100}. \n\n\tFor the purpose of this section, let \\(\\mathcal{A}:=(A_n)_{n\\in \\mathbb{N}}\\) be a random, stationary, and ergodic sequence of matrices as in the statement of Lemma~\\ref{blockt}, and write each matrix \\(A_n\\) as\n\t\\[\n\tA_n := A_{n,1} + A_{n,2},\n\t\\]\n\twhere \\(A_{n,1}\\) denotes the block diagonal part and \\(A_{n,2}\\) the remaining strictly upper triangular component of \\(A_n\\). \\par\\noindent{\\bf Norm convention.} Throughout Section~\\ref{shape} we take $\\|\\cdot\\|$ to be the operator norm induced by the vector $\\ell^1$-norm (maximum absolute column sum); we write $\\|\\cdot\\|_1$ when clarity is helpful. By equivalence of norms on the finite-dimensional space $\\mathcal M_d$, this choice does not change the Lyapunov exponents and \\eqref{f_cond} holds for $\\|\\cdot\\|$ if and only if it holds for $\\|\\cdot\\|_1$. Then\n\t\\[\n\tX_n = A_{n,1}\\cdots A_{1,1}\n\t+ \\sum_{j=1}^n \\sum_{1\\le i_1 < \\cdots < i_j\\le n}\n\tA_{n,1}\\cdots A_{i_j+1,1}\\, A_{i_j,2}\\, A_{i_j-1,1}\\cdots A_{i_{j-1}+1,1}\\, A_{i_{j-1},2}\\,\\cdots.\n\t\\]\n\tIt is straightforward to verify that multiplying a block diagonal matrix by a strictly block upper triangular matrix (with the same block sizes) produces a strictly block upper triangular matrix. Moreover, multiplying two strictly upper triangular matrices yields another strictly upper triangular matrix with fewer nonzero superdiagonals—a property reminiscent of the nilpotency of strictly upper triangular matrices. Consequently, all products with \\(j > d\\) vanish, and therefore\n\t\\[\n\tX_n = A_{n,1}\\cdots A_{1,1}\n\t+ \\sum_{j=1}^d \\sum_{1 \\leq i_1 < \\cdots < i_j \\leq n}\n\tA_{n,1}\\cdots A_{i_j+1,1}\\, A_{i_j,2}\\, A_{i_j-1,1}\\cdots A_{i_{j-1}+1,1}\\, A_{i_{j-1},2}\\,\\cdots.\n\t\\]\n\n\tFix $\\varepsilon>0$. We will use the following standard Borel--Cantelli estimate (applied to $Y_n=\\log^+\\|A_{n,2}\\|_1$).\n\n\t\\begin{lemma}\\label{lem:bc_log}\n\t\tLet $(Y_n)_{n\\ge 1}$ be a stationary sequence of nonnegative random variables with $\\E(Y_1)<\\infty$. Then for every $\\varepsilon>0$,\n\t\t\\[\n\t\t\\sum_{n\\ge 1} \\pp(Y_n>\\varepsilon n)<\\infty,\n\t\t\\qquad\\text{and hence}\\qquad \n\t\t\\pp(Y_n>\\varepsilon n\\ \\text{\\rm i.o.})=0.\n\t\t\\]\n\t\\end{lemma}\n\t\\begin{proof}\n\t\tBy stationarity, $\\pp(Y_n>\\varepsilon n)=\\pp(Y_1>\\varepsilon n)$. Using the tail-integral identity\n\t\t$$\\E(Y_1)=\\int_0^\\infty \\pp(Y_1>t)\\,dt$$ and partitioning the integral into intervals of length $\\varepsilon$ gives\n\t\t\\[\n\t\t\\sum_{n\\ge 1}\\pp(Y_1>\\varepsilon n)\n\t\t\\le \\frac{1}{\\varepsilon}\\int_0^\\infty \\pp(Y_1>t)\\,dt\n\t\t= \\frac{1}{\\varepsilon}\\E(Y_1)<\\infty.\n\t\t\\]\n\t\tThe Borel--Cantelli lemma yields the claim.\n\t\\end{proof}", "thm101": "\\begin{proposition}\n\t\t\\label{thm101}\n\t\tFor given $m\\in \\nn$ and $(s_1,\\ldots,s_m)\\in\\nn^m,$ let $\\cala=(A_n)_{n\\in\\nn}$ be a tempered sequence of matrices (i.\\,e., \\eqref{small} holds), each having the shape introduced in \\eqref{block_def}. For $i\\in [m],$ let $\\calb_i:=\\big(B_n(i,i)\\big)_{n\\in \\nn},$ and assume that all $B_n(i,i)$ are invertible and all $\\calb_i$ are Lyapunov regular (i.\\,e., an analogue of \\eqref{lr} holds for $\\calb_i$). Then, $\\gamma_1(\\cala)= \\max_{j\\in [m]} \\gamma_1(\\calb_j).$\n\t\\end{proposition}", "xn": "\\label{xn}\n\t\\limsup_{n\\to\\infty} \\frac{1}{n}\\log \\norm{X_n} < \\infty\\qquad \\mbox{\\rm where} \\qquad\n\tX_n := A_n \\cdots A_1.\n\t\\feqn\n\tWe associate to $\\cala$ the (upper) Lyapunov exponent function $\\gamm", "ld": "\\label{ld}\n\t\\gamma(v) := \\limsup_{n\\to\\infty}  \\frac{1}{n} \\log \\norm{X_n v}.\n\t\\feqn\n\tIn particular, we define the \\emph{top (upper) Lyapunov exponent} of the sequence $\\cala$ by\n\t\\beqn \\label{gamma1_", "f_cond": "\\label{f_cond}\n\t\\E\\left( \\log^+ \\|A_1\\| \\right) < \\infty,\n\t\\feqn\n\twhere, for $x>0$, $\\log^+ x:=\\max(0,\\log x)$, with the convention $\\log^+0:=0$.  Oseledets' multiplicative ergodic theorem then yields", "th-shapes": "\\begin{theorem}\n\t\t\\label{th-shapes}\n\t\tLet $\\call$ be a shape set of order $d$ with $k$ elements, and let $(\\calv, \\calt)_{\\call}$ denote its corresponding shape graph. Assume that the shape graph contains only loops of length one (self-loops), and that each nonzero vertex has at most one self-loop. Let $\\cala = (A_n)_{n \\in \\nn}$ be a random, stationary, and ergodic sequence of $d \\times d$ matrices, each admitting a decomposition as in~\\eqref{A_ni_def} with respect to the shape set $\\call$. Assume further that:\n\t\t\\begin{itemize}\n\t\t\t\\item[(i)] For every $1 \\leq s \\leq k$,\n\t\t\t\\[\n\t\t\t\\E\\big(\\log^+ \\|A_{1,s}\\|\\big) < \\infty.\n\t\t\t\\]\n\t\t\t\\item[(ii)] Let $\\calh \\subset \\calv \\setminus \\{ O_d \\}$ be the set of vertices with a self-loop, and let $\\call_{\\calh}$ be the subset of labels corresponding to their self-loops. For every $s \\in \\call_\\calh$, the matrix $A_{1,s}$ is almost surely invertible, and\n\t\t\t\\[\n\t\t\t\\E\\big(\\log^+ \\|A_{1,s}^{-1}\\|\\big) < \\infty.\n\t\t\t\\]\n\t\t\\end{itemize}\n\t\tDefine\n\t\t\\[\n\t\t\\beta_s := \\lim_{n \\to \\infty} \\frac{1}{n} \\log \\left\\| A_{n,s} A_{n-1,s} \\cdots A_{1,s} \\right\\|, \\quad \\text{a.s.},\n\t\t\\]\n\t\tand set\n\t\t\\[\n\t\t\\beta := \\max_{s \\in \\call_\\calh} \\beta_s.\n\t\t\\]\n\n\t\tThen:\n\t\t\\begin{itemize}\n\t\t\t\\item[(a)] The top Lyapunov exponent satisfies\n\t\t\t\\[\n\t\t\t\\gamma_1(\\cala) \\leq \\beta + \\log k.\n\t\t\t\\]\n\t\t\tMoreover, if $O_d \\in \\calv$, define\n\t\t\t\\[\n\t\t\tk_* := \\max_{v \\in \\calv \\setminus \\{O_d\\}} \\#\\bigl\\{\\, s \\in [k] : \\calt(v,L_s)\\neq O_d \\,\\bigr\\},\n\t\t\t\\]\n\t\t\tso that $k_* \\le k$ and $k_*$ is the maximal number of labels that keep a nonzero vertex from transitioning to $O_d$ in one step. Then\n\t\t\t\\[\n\t\t\t\\gamma_1(\\cala) \\leq \\beta + \\log k_*.\n\t\t\t\\]\n\t\t\t(In particular, if every nonzero vertex has at least one outgoing label that transitions to $O_d$, then $k_* \\le k-1$.)\n\t\t\t\\item[(b)] Let $\\calw \\subset \\calh\\setminus \\{O_d\\}$ be the set of self-loop vertices $w$ such that every directed path in the shape graph that ends at $w$ avoids loop vertices other than $w$ (equivalently, it does not pass through any vertex in $\\calh\\setminus\\{w\\}$). Assume moreover that for each $w\\in\\calw$, if $s$ denotes the label of its self-loop, then there exists $r=r(w)\\ge 1$ such that $\\shape{L_s^n}=w$ for all $n\\ge r$. If, for every $w \\in \\calw$ and every $v \\in \\calv\\setminus\\{w\\}$, we have\n\t\t\t\\[\n\t\t\tv \\wedge w = O_d,\n\t\t\t\\]\n\t\t\tand assume in addition that all matrices $A_{n,s}$ are entrywise nonnegative almost surely (so that monomials with the same shape do not cancel entrywise). Then\n\t\t\t\\[\n\t\t\t\\max_{s \\in \\call_\\calw} \\beta_s \\leq \\gamma_1(\\cala),\n\t\t\t\\]\n\t\t\twhere $\\call_\\calw$ denotes the set of labels corresponding to the self-loops of vertices in $\\calw$.\n\t\t\\end{itemize}\n\t\\end{theorem}", "block": "\\begin{proposition}\n\t\t\\label{pkus}\n\t\t\\cite{pinkus}\n\t\tLet $\\cala=(A_k)_{k\\in\\nn}\\in\\calm_d^\\nn$ be a random, stationary, and ergodic sequence of $d\\times d$ invertible matrices with $A_k(i,j) = 0$ for $i > j$ and such that \\eqref{f_cond} holds. Then,\n\t\t\\beq\n\t\t\\gamma_1 = \\max_{i\\in [d]} \\E \\left( \\log |A_1(i,i)|\\right),\n\t\t\\feq\n\t\twhere $[d]:=\\{1,\\ldots,d\\}.$\n\t\\end{proposition}\n\tIn fact, under the conditions of the proposition, all Lyapunov exponents of $\\cala$ can be identified as distinct elements of the multi-set $\\{\\E \\left( \\log |A_1(i,i)|\\right)\\}_{i=1}^d$ (see Corollary~1 in \\cite{hen} and item (iv) on p.~130 of \\cite[Section~3.2]{arnold}).\n\n\t\\par\n\tA conceptual explanation for Proposition~\\ref{pkus} is provided by a lemma of Furstenberg and Kifer \\cite{fifa} (quoted here as Lemma~\\ref{blockt}), which reduces the computation of the top Lyapunov exponent for block-triangular products to that of the diagonal blocks; see, for example, \\cite{fifa,kifer,blockas,blockus,tom}.  Extensions appear in several directions, including switched systems \\cite{ots, solvable, new} and linear cocycles in bundles \\cite[Lemma~3.6]{kifer}; see also \\cite[Proposition~1]{blockas}, \\cite[Theorem~1.1]{blockus} and \\cite[Lemma~4.9]{tom}.  The starting point of this paper is that the same block-triangular viewpoint can be packaged into a practical toolkit that continues to be useful beyond the classical stationary-random setting.  In particular, the resulting bounds are explicit in terms of diagonal-block growth rates and simple combinatorial parameters of the zero pattern, which makes them amenable to computation in matrix-analytic applications.\n\n\t\\par\n\tOur contributions are organized around four structured regimes.  Section~\\ref{block} begins with deterministic control of upper triangular products: for general tempered sequences (not necessarily random), Proposition~\\ref{thm100} gives a two-sided estimate for $\\gamma_1$ in terms of diagonal growth rates $\\alpha_i^\\pm$, and Corollary~\\ref{cora} recovers the familiar ``maximum of diagonal exponents'' identity when the relevant diagonal averages converge.  In the same section we develop a deterministic Furstenberg--Kifer-type reduction for block-triangular products under Lyapunov-regularity hypotheses on the diagonal blocks (Proposition~\\ref{thm101}), providing a modular step that reduces $\\gamma_1$ to diagonal-block data plus an explicit error term.\n\n\tSection~\\ref{per_sect} illustrates how these reductions turn concrete perturbation models (including rank-one updates) into lower-dimensional or scalar cocycles with computable Lyapunov exponents, and it clarifies when a perturbation changes only the top exponent.  Finally, Section~\\ref{shape} introduces shape graphs for sparse decompositions with disjoint supports and limited feedback, proves an energy--entropy bound (Theorem~\\ref{th-shapes}) of the form $\\gamma_1(\\cala)\\le \\beta+\\log k$ (or $\\beta+\\log k_*$ when $O_d\\in\\calv$), and provides a short ``how-to'' guide with worked examples in Section~\\ref{sec:howto-shapes}, including a transfer-matrix/DAG model that makes the mechanism transparent.\n\n\t\\section{Block-triangular matrices}\n\t\\label{block}\t\n\tWe begin with the following inequality concerning the top Lyapunov exponent of a given sequence of upper triangular matrices (cf. \\cite[Lemma~3.1.4]{barl}).\n\t\\begin{proposition}\n\t\t\\label{thm100}\n\t\tLet $\\cala:=(A_n)_{n\\in\\nn}$ be a sequence of real, invertible upper triangular $d\\times d$ matrices (that is $A_n(j,k)=0$ for $j>k$ and $A_n(j,j)\\neq 0,$ $j,k\\in [d]$) such that\n\t\t\\beqn\n\t\t\\label{small}\n\t\t\\limsup_{n\\to\\infty}\\frac{1}{n}\\log \\|A_n\\|=0.\n\t\t\\feqn\n\t\tThen,\n\t\t\\beqn\n\t\t\\label{D_ineq}\n\t\t\\max_{j\\in [d]} \\alpha_j^+\\leq \\gamma_1\n\t\t\\leq \\max_{j\\in [d]} \\Big(\\alpha_j^+  +\\sum_{r=1}^{j-1} \\big(\\alpha_r^+-\\alpha_r^-\\big)\\Big),\n\t\t\\feqn\n\t\twhere\n\t\t$\n\t\t\\alpha_i^-:= \\liminf_{n\\to\\infty} \\frac{1}{n} \\sum_{k=1}^n \\log |A_k(i,i)|$ and $\\alpha_i^+:= \\limsup_{n\\to\\infty} \\frac{1}{n} \\sum_{k=1}^n \\log |A_k(i,i)|.\n\t\t$\n\t\\end{proposition}", "cora": "\\begin{corollary}\n\t\t\\label{cora}\n\t\tLet the conditions of Proposition~\\ref{thm100} hold for $\\cala=(A_n)_{n\\in\\nn}.$ Suppose in addition that $\\alpha_i^+=\\alpha_i^-$ for all $i\\in [d].$\n\t\tThen,\n\t\t$\n\t\t\\gamma_1=\\max_{j\\in [d]} \\lim_{n\\to\\infty} \\frac{1}{n} \\sum_{i=1}^n \\log |A_i(j,j)|.\n\t\t$\n\t\\end{corollary}", "blockt": "\\begin{lemma}\n\t\t\\cite[Lemma~3.6]{fifa}\n\t\t\\label{blockt}\n\t\tFor given $m\\in \\nn$ and $(s_1,\\ldots,s_m)\\in\\nn^m,$ let $(A_n)_{n\\in\\nn}$ be a random, stationary ergodic sequence of matrices, each having the shape introduced in \\eqref{block_def}. Suppose in addition that for all $i\\in [m]$, $B_1(i,i)$ is invertible with probability one, and for all $1\\le i\\le j\\le m$,\n\t\t\\beq\n\t\t\\E\\big(\\log^+ \\|B_1(i,j)\\|\\big)<\\infty \\qquad \\mbox{and} \\qquad \\E\\big(\\log^+ \\|B_1^{-1}(i,i)\\|\\big)<\\infty.\n\t\t\\feq\n\t\tFor $i \\in [m],$ let\n\t\t\\beq\n\t\t\\beta_i=\\lim_{n\\to\\infty} \\frac{1}{n}\\log \\big\\|B_n(i,i)B_{n-1}(i,i)\\cdots B_1(i,i)\\big\\|,\\qquad \\as\n\t\t\\feq\n\t\tThen, $\\gamma_1=\\max_{i \\in [m]} \\beta_i.$\n\t\\end{lemma}", "thm100": "\\begin{proposition}\n\t\t\\label{thm100}\n\t\tLet $\\cala:=(A_n)_{n\\in\\nn}$ be a sequence of real, invertible upper triangular $d\\times d$ matrices (that is $A_n(j,k)=0$ for $j>k$ and $A_n(j,j)\\neq 0,$ $j,k\\in [d]$) such that\n\t\t\\beqn\n\t\t\\label{small}\n\t\t\\limsup_{n\\to\\infty}\\frac{1}{n}\\log \\|A_n\\|=0.\n\t\t\\feqn\n\t\tThen,\n\t\t\\beqn\n\t\t\\label{D_ineq}\n\t\t\\max_{j\\in [d]} \\alpha_j^+\\leq \\gamma_1\n\t\t\\leq \\max_{j\\in [d]} \\Big(\\alpha_j^+  +\\sum_{r=1}^{j-1} \\big(\\alpha_r^+-\\alpha_r^-\\big)\\Big),\n\t\t\\feqn\n\t\twhere\n\t\t$\n\t\t\\alpha_i^-:= \\liminf_{n\\to\\infty} \\frac{1}{n} \\sum_{k=1}^n \\log |A_k(i,i)|$ and $\\alpha_i^+:= \\limsup_{n\\to\\infty} \\frac{1}{n} \\sum_{k=1}^n \\log |A_k(i,i)|.\n\t\t$\n\t\\end{proposition}", "per_sect": "\\begin{bmatrix}\n\t\t\t\tC_n(1,1) & O & \\dots & O \\\\\n\t\t\t\tO & C_n(2,2) & \\dots & O \\\\\n\t\t\t\t\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\t\t\t\tO & O & \\dots & C_n(m,m)\n\t\t\t\\end{bmatrix}\n\t\t\t,\n\t\t\t\\feq\n\t\t\twhere $C_n(i,j)$ are $s_i\\times s_j$ matrices. Then,\n\t\t\t\\beq\n\t\t\t\\| X_n \\|_F^2 = \\| H_n \\|_F^2 + \\| X_n-H_n \\|_F^2 + 2\\inner{H_n}{X_n-H_n}_F,\n\t\t\t\\feq\n\t\t}\n\t\twhere $\\inner{A}{B}_F=\\sum_{i,j} A(i,j)B(i,j)$ is the Frobenius inner product of two matrices $A$ and $B,$ provided they have the same shape. Since for any pair $(i,j)\\in [m]^2,$ either $H_n(i,j)$ or $(X_n-H_n)(i,j)$ is zero, $\\inner{H_n}{X_n-H_n}_F = 0.$\n\t\tThis implies that $\\| X_n \\|_F \\geq \\| H_n \\|_F$. Since $H_n$ is block diagonal,\n\t\t\\[\n\t\t\\|H_n\\|_F^2=\\sum_{j=1}^m \\|C_n(j,j)\\|_F^2.\n\t\t\\]\n\t\tUsing the norm equivalence $\\|M\\|\\le \\|M\\|_F\\le \\sqrt{d}\\,\\|M\\|$ (valid for all $d\\times d$ matrices), each block has the same exponential growth rate whether measured in $\\|\\cdot\\|$ or $\\|\\cdot\\|_F$. Consequently,\n\t\t\\[\n\t\t\\lim_{n\\to\\infty}\\frac1n\\log\\|H_n\\|_F=\\max_{j\\in[m]}\\gamma_1(\\calb_j)=:\\beta,\n\t\t\\]\n\t\t\\noindent For each $i\\in[m]$, set $\\beta_i:=\\gamma_1(\\calb_i)$, so that $\\beta_i\\le \\beta$.\n\n\t\tand taking $\\frac1n\\log$ in $\\|X_n\\|_F\\ge \\|H_n\\|_F$ gives $\\beta\\le \\gamma_1(\\cala)$.\n\t\tWe turn now  to the inverse inequality $\\beta \\geq \\gamma_1(\\cala).$ Let $d_0=0$ and $d_j=s_1+\\cdots +s_j$ for $j\\in[m].$ For $u\\in \\rr^d,$ let $u_j,$ $j=1,\\ldots, m,$ be a vector in $\\rr^{s_j}$ with\n\t\t\\beq\n\t\tu_j(i)=u(d_{j-1}+i),\\qquad i=1,\\ldots, s_j.\n\t\t\\feq\n\t\tDenote by $\\|\\cdot\\|_1$ the $\\ell^1$-norm in $\\rr^d$ and corresponding matrix norm. Thus, $\\|u\\|_1=\\sum_{i=1}^d |u(i)|$ and $\\|A\\|_1=\\sup_{u\\in\\rr^d\\backslash\\{0\\}} \\frac{\\|Au\\|_1}{\\|u\\|_1}=\\max_{j\\in [d]} \\sum_{i=1}^d |A(i,j)|$ for $A \\in \\calm_d.$ Then, for $u\\in \\rr^d,$\n\t\t\\beq\n\t\t\\|X_n u\\|_1&=&\\sum_{i=1}^m \\Big\\|\\sum_{j=i}^m C_n (i,j)u_j\\Big\\|_1\\leq \\sum_{i=1}^m \\sum_{j=i}^m\\| C_n (i,j)\\|_1\\cdot \\|u_j\\|_1\n\t\t\\\\\n\t\t&\\leq& \\sum_{i=1}^m \\sum_{j=i}^m\\| C_n (i,j)\\|_1\\cdot \\|u\\|_1.\n\t\t\\feq\n\t\tTherefore,\n\t\t\\beqn\n\t\t\\label{esta}\n\t\t\\|X_n \\|_1 \\leq \\sum_{i=1}^m \\sum_{j=i}^m\\| C_n (i,j)\\|_1.\n\t\t\\feqn\n\t\tIn view of \\eqref{esta}, it suffices to show that for all $i,j\\in [m]$ such that $i\\leq j,$\n\t\t\\beqn\n\t\t\\label{est3}\n\t\t\\limsup_{n\\to\\infty} \\frac{1}{n}\\log \\|C_n(i,j)\\|_1\\leq \\beta ,\\qquad \\as\n\t\t\\feqn\n\t\tTo show \\eqref{est3}, we adapt an inductive argument employed in the proof of Lemma~3.1.4 in \\cite{barr} in order to obtain a version of this estimate for upper-triangular matrices. For $j>i,$ using a convention that $X_0$ is the $d\\times d$ identity matrix, we have (cf. (3.10) and (3.11) in \\cite{barr}):\n\t\t\\beqn\n\t\t\\nonumber\n\t\tC_n(i,j)&=&\\sum_{t=i+1}^j B_n(i,t)C_{n-1}(t,j)+B_n(i,i)C_{n-1}(i,j)\n\t\t\\\\\n\t\t\\nonumber\n\t\t&=&\\sum_{t=i+1}^j B_n(i,t)C_{n-1}(t,j)+B_n(i,i)\\sum_{t=i+1}^kB_{n-1}(i,t)C_{n-2}(t,j)\n\t\t\\\\\n\t\t\\nonumber\n\t\t&& \\qquad \\qquad +B_n(i,i)B_{n-1}(i,i)C_{n-2}(i,j)= \\cdots\n\t\t\\\\\n\t\t\\label{est4}\n\t\t&=&\n\t\t\\sum_{r=0}^{n-1} B_n(i,i)\\cdots B_{n-r+1}(i,i)\\sum_{t=i+1}^jB_{n-r}(i,t)C_{n-1-r}(t,j).\n\t\t\\feqn\n\t\tFix some $i^*,j^*\\in [m]$ such that $i^*<j^*.$ The inequality in \\eqref{est3} is trivially true for $i=j=j^*.$\n\t\tSuppose now that \\eqref{est3} has been proved for all $(i,j)$ with $j=j^*$ and $i>i^*.$ To finish the proof of the theorem, we will next use a backward induction to show that it is valid for $(i,j)=(i^*,j^*).$ By \\eqref{small} and the induction hypothesis, for all $\\veps>0$ there exists a constant $D_\\veps$ (which depends on $\\cala$) such that for all $n\\in\\nn,$\n\t\t\\beq\n\t\t\\|B_n(i,j)\\|_1\\leq D_\\veps e^{n \\veps}, \\qquad i,j\\in [m],\n\t\t\\feq\n\t\tand\n\t\t\\beq\n\t\t\\|C_n(i,j^*)\\|_1\\leq D_\\veps e^{n (\\beta +\\veps)}, \\qquad i^*<i\\leq j^*.\n\t\t\\feq\n\t\tFurthermore, it follows from \\eqref{abs1} that for all $\\veps>0$ there exists a constant $G_\\veps$ (which depends on $\\cala$) such that for any $i\\in [m]$ and $k,n\\in\\nn$ with $k<n,$\n\t\t\\beqn\n\t\t\\nonumber\n\t\t\\big\\|B_n(i,i)\\cdots B_k(i,i)\\big\\|_1&=&\\big\\|B_n(i,i)\\cdots B_1(i,i)\\cdot B_1^{-1}(i,i)\\cdots B_{k-1}^{-1}(i,i)\\big\\|_1\n\t\t\\\\\n\t\t\\label{gr}\n\t\t&\\leq& \\big\\|B_n(i,i)\\cdots B_1(i,i)\\big\\|_1\\cdot \\big \\|B_1^{-1}(i,i)\\cdots B_{k-1}^{-1}(i,i)\\big\\|_1\n\t\t\\\\\n\t\t\\nonumber\n\t\t&\\leq& \\big(G_\\veps e^{(\\beta_i+\\veps)n}\\big)\\cdot\\big(G_\\veps e^{(-\\beta_i+\\veps)(k-1)}\\big)\\leq G_\\veps^2 e^{(n-k+1)\\beta_i} e^{2n\\veps}.\n\t\t\\feqn\n\t\tTherefore, by virtue of \\eqref{est4},\n\t\t\\beq\n\t\t\\|C_n(i^*,j^*)\\|_1&\\leq & \\sum_{r=0}^{n-1} G_\\veps^2 e^{\\beta_i r}e^{2n\\veps} \\sum_{t=i^*+1}^{j^*}D_\\veps^2 e^{n \\veps} e^{(n-r+1)(\\beta+\\veps)}\\\\\n\t\t&\\leq & nj^* G_\\veps^2 D_\\veps^2 e^{(4n+1) \\veps} e^{(n+1)\\beta},\n\t\t\\feq\n\t\twhich implies \\eqref{est3} with $(i,j)=(i^*,j^*)$ since $\\veps>0$ is arbitrary.\n\t\\end{proof}\n\n\t\\section{Application to matrix perturbations} \n\t\\label{per_sect}\n\n\tWe next apply the results of Section~\\ref{block} to perturbations of linear systems. These perturbations represent an important class of matrix transformations with wide-ranging applications in science and engineering. \n\n\tOur first result concerns the product of random rank-one perturbations of a specific form. This proposition can also be viewed somewhat as a generalization of Theorem~5.1 in \\cite{rankone}, and contributes to the discussion around Remark~2.4 in the same paper.\n\n\t\\par\n\n\t\\begin{proposition} \\label{pert_thm}\n\t\tLet $(u_n,\\eta_n)_{n \\in \\nn}$ be a random, stationary, and ergodic sequence, where $u_n \\in \\rr^d$ and $\\eta_n \\in \\rr$. Suppose $v \\in \\rr^d$ is a fixed vector such that\n\t\t\\beqn \\label{uv_cond}\n\t\t\\E\\bigl(\\log |\\eta_1|\\bigr) \\leq \\E \\bigl(\\log |\\eta_1 + v^T u_1|\\bigr) < \\infty.\n\t\t\\feqn\n\t\t\\begin{itemize}\n\t\t\t\\item[(a)] Let $\\cala := (A_n)_{n \\in \\nn}$ with $A_n := \\eta_n I_d + u_n v^T$ satisfying \\eqref{f_cond}. Then\n\t\t\t\\beqn \\label{An_perturbed_exponents}\n\t\t\t\\gamma_1(\\cala) = \\E \\bigl(\\log |\\eta_1 + v^T u_1| \\bigr)\n\t\t\t\\quad \\text{and} \\quad\n\t\t\t\\gamma_r(\\cala) = \\E \\bigl(\\log |\\eta_1|\\bigr), \\quad r > 1.\n\t\t\t\\feqn\n\n\t\t\t\\item[(b)] Suppose $A \\in \\calm_d$ is a non-singular matrix. Let $\\cala := (A_n)_{n \\in \\nn}$ with $A_n := \\eta_n A + u_n v^T$ satisfying condition \\eqref{f_cond}, and\n\t\t\t\\beqn \\label{Auv_cond}\n\t\t\tP\\bigl(A u_1 v^T = u_1 v^T A\\bigr) = 1.\n\t\t\t\\feqn\n\t\t\tThen,\n\t\t\t\\beq\n\t\t\t- \\log \\rho(A^{-1}) \\leq \\gamma_1(\\cala) - \\E\\bigl(\\log|\\eta_1 + v^T A^{-1} u_1|\\bigr) \\leq \\log \\rho(A),\n\t\t\t\\feq\n\t\t\twhere $\\rho(A)$ denotes the spectral radius of the matrix $A$.\n\t\t\\end{itemize}\n\t\\end{proposition}\n\n\tWe note that, under mild conditions on the sequence $(\\eta_n, u_n)_{n \\in \\mathbb{N}}$, the inequality~\\eqref{uv_cond} can be relaxed, since the result may be applied directly to the inverse cocycle $\\cala^{-1}:=(A_n^{-1})_{n\\in\\nn}$, with part~(a) adjusted accordingly.\n\n\t\\begin{remark}[Varying right vector: two extendable regimes]\n\t\tConsider $A_n=\\eta_n I + u_n v_n^{\\top}$.\n\n\t\t(i) If $v_n=c_n v$ for a fixed $v\\neq 0$, then $A_n=\\eta_n I + \\tilde u_n v^{\\top}$ with\n\t\t$\\tilde u_n:=c_n u_n$. Hence Proposition~\\ref{pert_thm} applies verbatim (under the\n\t\tcorresponding integrability for $\\tilde u_n$), yielding $d-1$ Lyapunov exponents equal to\n\t\t$\\E\\log|\\eta_1|$ and the remaining exponent $\\E\\log|\\eta_1+c_1 v^{\\top}u_1|$.\n\n\t\t(ii) More generally, if $v_n\\in V$ a.s. for a fixed deterministic subspace $V\\subset \\rr^d$\n\t\tof dimension $r$, then $H:=V^{\\perp}$ is a common invariant subspace and\n\t\t$A_n|_H=\\eta_n\\,\\mathrm{Id}$. Consequently $\\E\\log|\\eta_1|$ is a Lyapunov exponent of\n\t\tmultiplicity at least $d-r$. The remaining $r$ exponents are those of the induced cocycle\n\t\ton the quotient $\\rr^d/H$; in general they are not explicit if $v_n$ rotates in $V$.\n\t\tNevertheless, the determinant identity implies\n\t\t\\[\n\t\t\\sum_{i=1}^d \\gamma_i=(d-1)\\E\\log|\\eta_1|+\\E\\log|\\eta_1+v_1^{\\top}u_1|.\n\t\t\\]\n\t\\end{remark}\n\n\tIn order to prove Proposition \\ref{pert_thm}, we first present a preliminary result that explains how to extract the first few Lyapunov exponents of block upper triangular matrices by Lemma~\\ref{blockt}. This extends Corollary~1 in \\cite{hen} (cf.\\ the paragraph following Proposition~\\ref{pkus} in these notes).\n\n\t\\begin{proposition} \\label{block_all_exp}\n\t\tFix $d, m \\in \\nn$, and let $\\caln := (N_n)_{n \\in \\nn}$ be a random, stationary, and ergodic sequence of matrices\n\t\t\\beq\n\t\tN_n := \n\t\t\\begin{bmatrix}\n\t\t\tA_n & * \\\\\n\t\t\tO & \\eta_n I_m\n\t\t\\end{bmatrix}", "pkus": "\\begin{proposition}\n\t\t\\label{pkus}\n\t\t\\cite{pinkus}\n\t\tLet $\\cala=(A_k)_{k\\in\\nn}\\in\\calm_d^\\nn$ be a random, stationary, and ergodic sequence of $d\\times d$ invertible matrices with $A_k(i,j) = 0$ for $i > j$ and such that \\eqref{f_cond} holds. Then,\n\t\t\\beq\n\t\t\\gamma_1 = \\max_{i\\in [d]} \\E \\left( \\log |A_1(i,i)|\\right),\n\t\t\\feq\n\t\twhere $[d]:=\\{1,\\ldots,d\\}.$\n\t\\end{proposition}"}, "pre_theorem_intro_text_len": 4231, "pre_theorem_intro_text": "Lyapunov exponents play a central role in the analysis of random and chaotic dynamical systems: they quantify sensitivity to initial conditions and, in the linear-cocycle setting, encode long-time growth rates of matrix products; see, e.g., \\cite{arnold, barl, boug, fuks, viana}.  Despite their importance, explicit formulas are available only for a comparatively small collection of solvable ensembles---classically, many computations focus on $2\\times 2$ products or on highly symmetric distributions (including several Gaussian-type models); see, for instance, \\cite{rankone, com, comt, elf, fibbo, keyc, kiev, letac, lima, darm, mann, mark, pinkus} and \\cite{adams, ahn, aker, cohen, fort, ens, kargin, newman}.  The purpose of this paper is to extend and reorganize a few ideas from \\cite{rankone, fifa, kiev, pinkus} into a small toolkit for obtaining \\emph{computable} bounds (and, in favorable cases, explicit values) for Lyapunov exponents of structured matrix products.  From a matrix-analysis viewpoint, the guiding principle is that triangular or block-triangular structure---and more generally prescribed sparsity/zero patterns with limited feedback---localize exponential growth to lower-dimensional diagonal dynamics, while the remaining entries contribute at most an explicitly controlled combinatorial factor.\n\tConcretely, our bounds are expressed in terms of diagonal-block products together with a directed graph encoding admissible off-diagonal couplings, and are therefore readily computable from a small collection of lower-dimensional norm estimates.\n\tThe framework covers both stationary/ergodic matrix sequences and deterministic sequences satisfying the temperedness condition \\eqref{xn}, so it can be read as a matrix-analytic toolkit for growth-rate bounds in structured or sparsely coupled linear systems.\n\n\t\\par\n\tFor an integer $d\\ge 2$, let $\\calm_d$ denote the space of $d\\times d$ matrices with real-valued entries.  Throughout, we write $\\|\\cdot\\|$ both for a generic norm on ${\\mathbb R}^d$ (when the particular choice is immaterial) and for the induced operator norm on $\\calm_d$.  Let ${\\mathcal A}:=(A_n)_{n\\in{\\mathbb N}}$ be a sequence in $\\calm_d$ such that\n\t\\begin{eqnarray} \\label{xn}\n\t\\limsup_{n\\to\\infty} \\frac{1}{n}\\log \\|X_n\\| < \\infty\\qquad \\mbox{\\rm where} \\qquad\n\tX_n := A_n \\cdots A_1.\n\t\\end{eqnarray}\n\tWe associate to ${\\mathcal A}$ the (upper) Lyapunov exponent function $\\gamma:{\\mathbb R}^d\\to{\\mathbb R}\\cup\\{-\\infty\\}$ defined by\n\t\\begin{eqnarray}\n\t\\label{ld}\n\t\\gamma(v) := \\limsup_{n\\to\\infty}  \\frac{1}{n} \\log \\|X_n v\\|.\n\t\\end{eqnarray}\n\tIn particular, we define the \\emph{top (upper) Lyapunov exponent} of the sequence ${\\mathcal A}$ by\n\t\\begin{eqnarray} \\label{gamma1_def}\n\t\\gamma_1({\\mathcal A})\n\t:= \\limsup_{n\\to\\infty}\\frac1n\\log\\|X_n\\|\n\t= \\sup_{\\|v\\|=1}\\gamma(v).\n\t\\end{eqnarray}\n\n\tThis directional growth rate is the basic object we aim to control in the structured regimes considered below.\n\n\t\\par\n\tA particularly clean structure emerges when ${\\mathcal A}$ is random, stationary, and ergodic and satisfies the standard integrability condition\n\t\\begin{eqnarray} \\label{f_cond}\n\t{\\mathbb E}\\left( \\log^+ \\|A_1\\| \\right) < \\infty,\n\t\\end{eqnarray}\n\twhere, for $x>0$, $\\log^+ x:=\\max(0,\\log x)$, with the convention $\\log^+0:=0$.  Oseledets' multiplicative ergodic theorem then yields deterministically constant Lyapunov exponents\n\t\\[\n\t\\gamma_1>\\cdots>\\gamma_\\ell,\\qquad \\ell\\le d,\n\t\\]\n\ttogether with a corresponding filtration of subspaces\n\t\\[\n\t{\\mathbb R}^d = E_1 \\supset E_2 \\supset \\cdots \\supset E_{\\ell} \\supset E_{\\ell+1}:=\\{0\\},\n\t\\]\n\tsuch that the limsup in \\eqref{ld} is in fact a limit and $\\gamma(v)\\in\\{\\gamma_1,\\dots,\\gamma_\\ell\\}$ almost surely, with the value determined by the position of $v$ in the filtration; see \\cite[Section~2.1]{barr} and \\cite{arnold, barl, viana}.  In particular, the exponents and subspaces are independent of the realization with probability one.\n\n\t\\par\n\tEven in this ergodic framework, however, the exponents $\\gamma_k$ are rarely available in closed form.  A notable exception occurs for triangular cocycles, where the diagonal entries dictate the asymptotic growth.  In particular, for upper triangular matrices we have:", "context": "Lyapunov exponents play a central role in the analysis of random and chaotic dynamical systems: they quantify sensitivity to initial conditions and, in the linear-cocycle setting, encode long-time growth rates of matrix products; see, e.g., \\cite{arnold, barl, boug, fuks, viana}. Despite their importance, explicit formulas are available only for a comparatively small collection of solvable ensembles---classically, many computations focus on $2\\times 2$ products or on highly symmetric distributions (including several Gaussian-type models); see, for instance, \\cite{rankone, com, comt, elf, fibbo, keyc, kiev, letac, lima, darm, mann, mark, pinkus} and \\cite{adams, ahn, aker, cohen, fort, ens, kargin, newman}. The purpose of this paper is to extend and reorganize a few ideas from \\cite{rankone, fifa, kiev, pinkus} into a small toolkit for obtaining \\emph{computable} bounds (and, in favorable cases, explicit values) for Lyapunov exponents of structured matrix products. From a matrix-analysis viewpoint, the guiding principle is that triangular or block-triangular structure---and more generally prescribed sparsity/zero patterns with limited feedback---localize exponential growth to lower-dimensional diagonal dynamics, while the remaining entries contribute at most an explicitly controlled combinatorial factor.\n Concretely, our bounds are expressed in terms of diagonal-block products together with a directed graph encoding admissible off-diagonal couplings, and are therefore readily computable from a small collection of lower-dimensional norm estimates.\n The framework covers both stationary/ergodic matrix sequences and deterministic sequences satisfying the temperedness condition \\eqref{xn}, so it can be read as a matrix-analytic toolkit for growth-rate bounds in structured or sparsely coupled linear systems.\n\n\\par\n For an integer $d\\ge 2$, let $\\calm_d$ denote the space of $d\\times d$ matrices with real-valued entries. Throughout, we write $\\|\\cdot\\|$ both for a generic norm on ${\\mathbb R}^d$ (when the particular choice is immaterial) and for the induced operator norm on $\\calm_d$. Let ${\\mathcal A}:=(A_n)_{n\\in{\\mathbb N}}$ be a sequence in $\\calm_d$ such that\n \\begin{eqnarray} \\label{xn}\n \\limsup_{n\\to\\infty} \\frac{1}{n}\\log \\|X_n\\| < \\infty\\qquad \\mbox{\\rm where} \\qquad\n X_n := A_n \\cdots A_1.\n \\end{eqnarray}\n We associate to ${\\mathcal A}$ the (upper) Lyapunov exponent function $\\gamma:{\\mathbb R}^d\\to{\\mathbb R}\\cup\\{-\\infty\\}$ defined by\n \\begin{eqnarray}\n \\label{ld}\n \\gamma(v) := \\limsup_{n\\to\\infty} \\frac{1}{n} \\log \\|X_n v\\|.\n \\end{eqnarray}\n In particular, we define the \\emph{top (upper) Lyapunov exponent} of the sequence ${\\mathcal A}$ by\n \\begin{eqnarray} \\label{gamma1_def}\n \\gamma_1({\\mathcal A})\n := \\limsup_{n\\to\\infty}\\frac1n\\log\\|X_n\\|\n = \\sup_{\\|v\\|=1}\\gamma(v).\n \\end{eqnarray}\n\n\\par\n A particularly clean structure emerges when ${\\mathcal A}$ is random, stationary, and ergodic and satisfies the standard integrability condition\n \\begin{eqnarray} \\label{f_cond}\n {\\mathbb E}\\left( \\log^+ \\|A_1\\| \\right) < \\infty,\n \\end{eqnarray}\n where, for $x>0$, $\\log^+ x:=\\max(0,\\log x)$, with the convention $\\log^+0:=0$. Oseledets' multiplicative ergodic theorem then yields deterministically constant Lyapunov exponents\n \\[\n \\gamma_1>\\cdots>\\gamma_\\ell,\\qquad \\ell\\le d,\n \\]\n together with a corresponding filtration of subspaces\n \\[\n {\\mathbb R}^d = E_1 \\supset E_2 \\supset \\cdots \\supset E_{\\ell} \\supset E_{\\ell+1}:=\\{0\\},\n \\]\n such that the limsup in \\eqref{ld} is in fact a limit and $\\gamma(v)\\in\\{\\gamma_1,\\dots,\\gamma_\\ell\\}$ almost surely, with the value determined by the position of $v$ in the filtration; see \\cite[Section~2.1]{barr} and \\cite{arnold, barl, viana}. In particular, the exponents and subspaces are independent of the realization with probability one.\n\n\\par\n Even in this ergodic framework, however, the exponents $\\gamma_k$ are rarely available in closed form. A notable exception occurs for triangular cocycles, where the diagonal entries dictate the asymptotic growth. In particular, for upper triangular matrices we have:\n\n\\label{f_cond}\n\t\\E\\left( \\log^+ \\|A_1\\| \\right) < \\infty,\n\t\\feqn\n\twhere, for $x>0$, $\\log^+ x:=\\max(0,\\log x)$, with the convention $\\log^+0:=0$.  Oseledets' multiplicative ergodic theorem then yields\n\n\\label{ld}\n\t\\gamma(v) := \\limsup_{n\\to\\infty}  \\frac{1}{n} \\log \\norm{X_n v}.\n\t\\feqn\n\tIn particular, we define the \\emph{top (upper) Lyapunov exponent} of the sequence $\\cala$ by\n\t\\beqn \\label{gamma1_\n\n\\label{xn}\n\t\\limsup_{n\\to\\infty} \\frac{1}{n}\\log \\norm{X_n} < \\infty\\qquad \\mbox{\\rm where} \\qquad\n\tX_n := A_n \\cdots A_1.\n\t\\feqn\n\tWe associate to $\\cala$ the (upper) Lyapunov exponent function $\\gamm", "full_context": "Lyapunov exponents play a central role in the analysis of random and chaotic dynamical systems: they quantify sensitivity to initial conditions and, in the linear-cocycle setting, encode long-time growth rates of matrix products; see, e.g., \\cite{arnold, barl, boug, fuks, viana}. Despite their importance, explicit formulas are available only for a comparatively small collection of solvable ensembles---classically, many computations focus on $2\\times 2$ products or on highly symmetric distributions (including several Gaussian-type models); see, for instance, \\cite{rankone, com, comt, elf, fibbo, keyc, kiev, letac, lima, darm, mann, mark, pinkus} and \\cite{adams, ahn, aker, cohen, fort, ens, kargin, newman}. The purpose of this paper is to extend and reorganize a few ideas from \\cite{rankone, fifa, kiev, pinkus} into a small toolkit for obtaining \\emph{computable} bounds (and, in favorable cases, explicit values) for Lyapunov exponents of structured matrix products. From a matrix-analysis viewpoint, the guiding principle is that triangular or block-triangular structure---and more generally prescribed sparsity/zero patterns with limited feedback---localize exponential growth to lower-dimensional diagonal dynamics, while the remaining entries contribute at most an explicitly controlled combinatorial factor.\n Concretely, our bounds are expressed in terms of diagonal-block products together with a directed graph encoding admissible off-diagonal couplings, and are therefore readily computable from a small collection of lower-dimensional norm estimates.\n The framework covers both stationary/ergodic matrix sequences and deterministic sequences satisfying the temperedness condition \\eqref{xn}, so it can be read as a matrix-analytic toolkit for growth-rate bounds in structured or sparsely coupled linear systems.\n\n\\par\n For an integer $d\\ge 2$, let $\\calm_d$ denote the space of $d\\times d$ matrices with real-valued entries. Throughout, we write $\\|\\cdot\\|$ both for a generic norm on ${\\mathbb R}^d$ (when the particular choice is immaterial) and for the induced operator norm on $\\calm_d$. Let ${\\mathcal A}:=(A_n)_{n\\in{\\mathbb N}}$ be a sequence in $\\calm_d$ such that\n \\begin{eqnarray} \\label{xn}\n \\limsup_{n\\to\\infty} \\frac{1}{n}\\log \\|X_n\\| < \\infty\\qquad \\mbox{\\rm where} \\qquad\n X_n := A_n \\cdots A_1.\n \\end{eqnarray}\n We associate to ${\\mathcal A}$ the (upper) Lyapunov exponent function $\\gamma:{\\mathbb R}^d\\to{\\mathbb R}\\cup\\{-\\infty\\}$ defined by\n \\begin{eqnarray}\n \\label{ld}\n \\gamma(v) := \\limsup_{n\\to\\infty} \\frac{1}{n} \\log \\|X_n v\\|.\n \\end{eqnarray}\n In particular, we define the \\emph{top (upper) Lyapunov exponent} of the sequence ${\\mathcal A}$ by\n \\begin{eqnarray} \\label{gamma1_def}\n \\gamma_1({\\mathcal A})\n := \\limsup_{n\\to\\infty}\\frac1n\\log\\|X_n\\|\n = \\sup_{\\|v\\|=1}\\gamma(v).\n \\end{eqnarray}\n\n\\par\n A particularly clean structure emerges when ${\\mathcal A}$ is random, stationary, and ergodic and satisfies the standard integrability condition\n \\begin{eqnarray} \\label{f_cond}\n {\\mathbb E}\\left( \\log^+ \\|A_1\\| \\right) < \\infty,\n \\end{eqnarray}\n where, for $x>0$, $\\log^+ x:=\\max(0,\\log x)$, with the convention $\\log^+0:=0$. Oseledets' multiplicative ergodic theorem then yields deterministically constant Lyapunov exponents\n \\[\n \\gamma_1>\\cdots>\\gamma_\\ell,\\qquad \\ell\\le d,\n \\]\n together with a corresponding filtration of subspaces\n \\[\n {\\mathbb R}^d = E_1 \\supset E_2 \\supset \\cdots \\supset E_{\\ell} \\supset E_{\\ell+1}:=\\{0\\},\n \\]\n such that the limsup in \\eqref{ld} is in fact a limit and $\\gamma(v)\\in\\{\\gamma_1,\\dots,\\gamma_\\ell\\}$ almost surely, with the value determined by the position of $v$ in the filtration; see \\cite[Section~2.1]{barr} and \\cite{arnold, barl, viana}. In particular, the exponents and subspaces are independent of the realization with probability one.\n\n\\par\n Even in this ergodic framework, however, the exponents $\\gamma_k$ are rarely available in closed form. A notable exception occurs for triangular cocycles, where the diagonal entries dictate the asymptotic growth. In particular, for upper triangular matrices we have:\n\n\\label{f_cond}\n\t\\E\\left( \\log^+ \\|A_1\\| \\right) < \\infty,\n\t\\feqn\n\twhere, for $x>0$, $\\log^+ x:=\\max(0,\\log x)$, with the convention $\\log^+0:=0$.  Oseledets' multiplicative ergodic theorem then yields\n\n\\label{ld}\n\t\\gamma(v) := \\limsup_{n\\to\\infty}  \\frac{1}{n} \\log \\norm{X_n v}.\n\t\\feqn\n\tIn particular, we define the \\emph{top (upper) Lyapunov exponent} of the sequence $\\cala$ by\n\t\\beqn \\label{gamma1_\n\n\\label{xn}\n\t\\limsup_{n\\to\\infty} \\frac{1}{n}\\log \\norm{X_n} < \\infty\\qquad \\mbox{\\rm where} \\qquad\n\tX_n := A_n \\cdots A_1.\n\t\\feqn\n\tWe associate to $\\cala$ the (upper) Lyapunov exponent function $\\gamm\n\nand taking $\\frac1n\\log$ in $\\|X_n\\|_F\\ge \\|H_n\\|_F$ gives $\\beta\\le \\gamma_1(\\cala)$.\n We turn now to the inverse inequality $\\beta \\geq \\gamma_1(\\cala).$ Let $d_0=0$ and $d_j=s_1+\\cdots +s_j$ for $j\\in[m].$ For $u\\in \\rr^d,$ let $u_j,$ $j=1,\\ldots, m,$ be a vector in $\\rr^{s_j}$ with\n \\beq\n u_j(i)=u(d_{j-1}+i),\\qquad i=1,\\ldots, s_j.\n \\feq\n Denote by $\\|\\cdot\\|_1$ the $\\ell^1$-norm in $\\rr^d$ and corresponding matrix norm. Thus, $\\|u\\|_1=\\sum_{i=1}^d |u(i)|$ and $\\|A\\|_1=\\sup_{u\\in\\rr^d\\backslash\\{0\\}} \\frac{\\|Au\\|_1}{\\|u\\|_1}=\\max_{j\\in [d]} \\sum_{i=1}^d |A(i,j)|$ for $A \\in \\calm_d.$ Then, for $u\\in \\rr^d,$\n \\beq\n \\|X_n u\\|_1&=&\\sum_{i=1}^m \\Big\\|\\sum_{j=i}^m C_n (i,j)u_j\\Big\\|_1\\leq \\sum_{i=1}^m \\sum_{j=i}^m\\| C_n (i,j)\\|_1\\cdot \\|u_j\\|_1\n \\\\\n &\\leq& \\sum_{i=1}^m \\sum_{j=i}^m\\| C_n (i,j)\\|_1\\cdot \\|u\\|_1.\n \\feq\n Therefore,\n \\beqn\n \\label{esta}\n \\|X_n \\|_1 \\leq \\sum_{i=1}^m \\sum_{j=i}^m\\| C_n (i,j)\\|_1.\n \\feqn\n In view of \\eqref{esta}, it suffices to show that for all $i,j\\in [m]$ such that $i\\leq j,$\n \\beqn\n \\label{est3}\n \\limsup_{n\\to\\infty} \\frac{1}{n}\\log \\|C_n(i,j)\\|_1\\leq \\beta ,\\qquad \\as\n \\feqn\n To show \\eqref{est3}, we adapt an inductive argument employed in the proof of Lemma~3.1.4 in \\cite{barr} in order to obtain a version of this estimate for upper-triangular matrices. For $j>i,$ using a convention that $X_0$ is the $d\\times d$ identity matrix, we have (cf. (3.10) and (3.11) in \\cite{barr}):\n \\beqn\n \\nonumber\n C_n(i,j)&=&\\sum_{t=i+1}^j B_n(i,t)C_{n-1}(t,j)+B_n(i,i)C_{n-1}(i,j)\n \\\\\n \\nonumber\n &=&\\sum_{t=i+1}^j B_n(i,t)C_{n-1}(t,j)+B_n(i,i)\\sum_{t=i+1}^kB_{n-1}(i,t)C_{n-2}(t,j)\n \\\\\n \\nonumber\n && \\qquad \\qquad +B_n(i,i)B_{n-1}(i,i)C_{n-2}(i,j)= \\cdots\n \\\\\n \\label{est4}\n &=&\n \\sum_{r=0}^{n-1} B_n(i,i)\\cdots B_{n-r+1}(i,i)\\sum_{t=i+1}^jB_{n-r}(i,t)C_{n-1-r}(t,j).\n \\feqn\n Fix some $i^*,j^*\\in [m]$ such that $i^*<j^*.$ The inequality in \\eqref{est3} is trivially true for $i=j=j^*.$\n Suppose now that \\eqref{est3} has been proved for all $(i,j)$ with $j=j^*$ and $i>i^*.$ To finish the proof of the theorem, we will next use a backward induction to show that it is valid for $(i,j)=(i^*,j^*).$ By \\eqref{small} and the induction hypothesis, for all $\\veps>0$ there exists a constant $D_\\veps$ (which depends on $\\cala$) such that for all $n\\in\\nn,$\n \\beq\n \\|B_n(i,j)\\|_1\\leq D_\\veps e^{n \\veps}, \\qquad i,j\\in [m],\n \\feq\n and\n \\beq\n \\|C_n(i,j^*)\\|_1\\leq D_\\veps e^{n (\\beta +\\veps)}, \\qquad i^*<i\\leq j^*.\n \\feq\n Furthermore, it follows from \\eqref{abs1} that for all $\\veps>0$ there exists a constant $G_\\veps$ (which depends on $\\cala$) such that for any $i\\in [m]$ and $k,n\\in\\nn$ with $k<n,$\n \\beqn\n \\nonumber\n \\big\\|B_n(i,i)\\cdots B_k(i,i)\\big\\|_1&=&\\big\\|B_n(i,i)\\cdots B_1(i,i)\\cdot B_1^{-1}(i,i)\\cdots B_{k-1}^{-1}(i,i)\\big\\|_1\n \\\\\n \\label{gr}\n &\\leq& \\big\\|B_n(i,i)\\cdots B_1(i,i)\\big\\|_1\\cdot \\big \\|B_1^{-1}(i,i)\\cdots B_{k-1}^{-1}(i,i)\\big\\|_1\n \\\\\n \\nonumber\n &\\leq& \\big(G_\\veps e^{(\\beta_i+\\veps)n}\\big)\\cdot\\big(G_\\veps e^{(-\\beta_i+\\veps)(k-1)}\\big)\\leq G_\\veps^2 e^{(n-k+1)\\beta_i} e^{2n\\veps}.\n \\feqn\n Therefore, by virtue of \\eqref{est4},\n \\beq\n \\|C_n(i^*,j^*)\\|_1&\\leq & \\sum_{r=0}^{n-1} G_\\veps^2 e^{\\beta_i r}e^{2n\\veps} \\sum_{t=i^*+1}^{j^*}D_\\veps^2 e^{n \\veps} e^{(n-r+1)(\\beta+\\veps)}\\\\\n &\\leq & nj^* G_\\veps^2 D_\\veps^2 e^{(4n+1) \\veps} e^{(n+1)\\beta},\n \\feq\n which implies \\eqref{est3} with $(i,j)=(i^*,j^*)$ since $\\veps>0$ is arbitrary.\n \\end{proof}\n\n\\begin{proposition} \\label{block_all_exp}\n Fix $d, m \\in \\nn$, and let $\\caln := (N_n)_{n \\in \\nn}$ be a random, stationary, and ergodic sequence of matrices\n \\beq\n N_n := \n \\begin{bmatrix}\n A_n & * \\\\\n O & \\eta_n I_m\n \\end{bmatrix},\n \\feq\n where $\\eta_n \\in \\rr$ and $A_n \\in \\calm_d$. Assume that $\\caln$ satisfies \\eqref{f_cond} (i.e.\\ $\\E(\\log^+\\|N_1\\|)<\\infty$), that $A_1$ is almost surely invertible with $\\E(\\log^+\\|A_1^{-1}\\|)<\\infty$, and that $\\eta_1\\neq 0$ almost surely with $\\E(\\log^+|\\eta_1|^{-1})<\\infty$. Set $\\cala := (A_n)_{n \\in \\nn}$. Then\n \\beq\n \\gamma_1(\\caln) = \\max \\bigl( \\E(\\log | \\eta_1 |), \\, \\gamma_1(\\cala) \\bigr),\n \\feq\n and for any $2 \\leq r \\leq \\min(m,d)$,\n \\beq\n && \\gamma_r(\\caln) = \\max_{0\\leq \\ell \\leq r} \\left(\\sum_{s=1}^\\ell \\gamma_s(\\cala) + (r-\\ell) \\E(\\log | \\eta_1|) \\right) \\\\\n && \\qquad \\qquad - \\max_{0\\leq \\ell \\leq r-1} \\left(\\sum_{s=1}^\\ell \\gamma_s(\\cala) + (r-\\ell) \\E(\\log | \\eta_1|) \\right).\n \\feq\n \\end{proposition} \n \\begin{proof}\n Recall that a minor of size~$r$ is associated to ordered sequences of indices $J$ and $L$ of length $r$, defined as the determinant of the submatrix formed by selecting the rows indexed by $J$ and the columns indexed by $L$.\n We write $[N]:=\\{1,\\dots,N\\}$ and, for $0\\le r\\le N$, we let\n $[N]_r:=\\{(i_1,\\dots,i_r):1\\le i_1<\\cdots<i_r\\le N\\}$; we freely identify such increasing $r$--tuples with the underlying $r$--element subsets.\n For $B\\subset[m]$ we write $B+d:=\\{b+d: b\\in B\\}\\subset\\{d+1,\\dots,d+m\\}$, and for $J_2,L_2\\subset\\{d+1,\\dots,d+m\\}$ we interpret\n $I_m[J_2,L_2]:=I_m[J_2-d,\\,L_2-d]$.\n For any matrix $A$, let $H_r(A)$ denote the $r$-th compound matrix of $A$, whose entries are all minors of size~$r$. Set $B_n^{(r)} := H_r(N_n)$, and let $\\calb_r := (B_n^{(r)})_{n \\in \\nn}$ be the sequence of $r$-th compound matrices in $\\caln$.\n\n\\begin{proposition}\\label{pert_thm2}\n Fix $m,d\\in\\mathbb N$, a matrix $V\\in\\mathcal M_{d\\times m}(\\mathbb R)$ of rank $m$, and an invertible\n matrix $A\\in\\mathcal M_{d\\times d}(\\mathbb R)$.\n Let $(U_n,\\eta_n)_{n\\ge1}$ be a random stationary ergodic sequence with\n $U_n\\in\\mathcal M_{d\\times m}(\\mathbb R)$ and $\\eta_n\\in\\mathbb R$, and define\n \\[\n A_n:=\\eta_n A + U_nV^T,\\qquad \n \\widetilde A_n:=\\eta_n I_m + V^T A^{-1}U_n .\n \\]\n Assume\n \\[\n \\mathbb P(AU_1V^T = U_1V^T A)=1 .\n \\]\n Then, writing $\\cala=(A_n)$ and $\\widetilde\\cala=(\\widetilde A_n)$,\n \\[\n -\\log\\rho(A^{-1}) + \\max\\!\\Big(\\mathbb E\\log|\\eta_1|,\\,\\gamma_1(\\widetilde\\cala)\\Big)\n \\;\\le\\;\n \\gamma_1(\\cala)\n \\;\\le\\;\n \\log\\rho(A) + \\max\\!\\Big(\\mathbb E\\log|\\eta_1|,\\,\\gamma_1(\\widetilde\\cala)\\Big).\n \\]\n \\end{proposition}\n\n\\begin{theorem}\n \\label{th-shapes}\n Let $\\call$ be a shape set of order $d$ with $k$ elements, and let $(\\calv, \\calt)_{\\call}$ denote its corresponding shape graph. Assume that the shape graph contains only loops of length one (self-loops), and that each nonzero vertex has at most one self-loop. Let $\\cala = (A_n)_{n \\in \\nn}$ be a random, stationary, and ergodic sequence of $d \\times d$ matrices, each admitting a decomposition as in~\\eqref{A_ni_def} with respect to the shape set $\\call$. Assume further that:\n \\begin{itemize}\n \\item[(i)] For every $1 \\leq s \\leq k$,\n \\[\n \\E\\big(\\log^+ \\|A_{1,s}\\|\\big) < \\infty.\n \\]\n \\item[(ii)] Let $\\calh \\subset \\calv \\setminus \\{ O_d \\}$ be the set of vertices with a self-loop, and let $\\call_{\\calh}$ be the subset of labels corresponding to their self-loops. For every $s \\in \\call_\\calh$, the matrix $A_{1,s}$ is almost surely invertible, and\n \\[\n \\E\\big(\\log^+ \\|A_{1,s}^{-1}\\|\\big) < \\infty.\n \\]\n \\end{itemize}\n Define\n \\[\n \\beta_s := \\lim_{n \\to \\infty} \\frac{1}{n} \\log \\left\\| A_{n,s} A_{n-1,s} \\cdots A_{1,s} \\right\\|, \\quad \\text{a.s.},\n \\]\n and set\n \\[\n \\beta := \\max_{s \\in \\call_\\calh} \\beta_s.\n \\]", "post_theorem_intro_text_len": 2776, "post_theorem_intro_text": "In fact, under the conditions of the proposition, all Lyapunov exponents of ${\\mathcal A}$ can be identified as distinct elements of the multi-set $\\{{\\mathbb E} \\left( \\log |A_1(i,i)|\\right)\\}_{i=1}^d$ (see Corollary~1 in \\cite{hen} and item (iv) on p.~130 of \\cite[Section~3.2]{arnold}).\n\n\t\\par\n\tA conceptual explanation for Proposition~\\ref{pkus} is provided by a lemma of Furstenberg and Kifer \\cite{fifa} (quoted here as Lemma~\\ref{blockt}), which reduces the computation of the top Lyapunov exponent for block-triangular products to that of the diagonal blocks; see, for example, \\cite{fifa,kifer,blockas,blockus,tom}.  Extensions appear in several directions, including switched systems \\cite{ots, solvable, new} and linear cocycles in bundles \\cite[Lemma~3.6]{kifer}; see also \\cite[Proposition~1]{blockas}, \\cite[Theorem~1.1]{blockus} and \\cite[Lemma~4.9]{tom}.  The starting point of this paper is that the same block-triangular viewpoint can be packaged into a practical toolkit that continues to be useful beyond the classical stationary-random setting.  In particular, the resulting bounds are explicit in terms of diagonal-block growth rates and simple combinatorial parameters of the zero pattern, which makes them amenable to computation in matrix-analytic applications.\n\n\t\\par\n\tOur contributions are organized around four structured regimes.  Section~\\ref{block} begins with deterministic control of upper triangular products: for general tempered sequences (not necessarily random), Proposition~\\ref{thm100} gives a two-sided estimate for $\\gamma_1$ in terms of diagonal growth rates $\\alpha_i^\\pm$, and Corollary~\\ref{cora} recovers the familiar ``maximum of diagonal exponents'' identity when the relevant diagonal averages converge.  In the same section we develop a deterministic Furstenberg--Kifer-type reduction for block-triangular products under Lyapunov-regularity hypotheses on the diagonal blocks (Proposition~\\ref{thm101}), providing a modular step that reduces $\\gamma_1$ to diagonal-block data plus an explicit error term.\n\n\tSection~\\ref{per_sect} illustrates how these reductions turn concrete perturbation models (including rank-one updates) into lower-dimensional or scalar cocycles with computable Lyapunov exponents, and it clarifies when a perturbation changes only the top exponent.  Finally, Section~\\ref{shape} introduces shape graphs for sparse decompositions with disjoint supports and limited feedback, proves an energy--entropy bound (Theorem~\\ref{th-shapes}) of the form $\\gamma_1({\\mathcal A})\\le \\beta+\\log k$ (or $\\beta+\\log k_*$ when $O_d\\in{\\mathcal V}$), and provides a short ``how-to'' guide with worked examples in Section~\\ref{sec:howto-shapes}, including a transfer-matrix/DAG model that makes the mechanism transparent.", "sketch": "A conceptual explanation for Proposition~\\ref{pkus} is provided by a lemma of Furstenberg and Kifer \\cite{fifa} (quoted here as Lemma~\\ref{blockt}), which \\emph{reduces the computation of the top Lyapunov exponent for block-triangular products to that of the diagonal blocks}. In the present upper-triangular setting, this reduction yields the familiar identification that the top Lyapunov exponent is the maximum among the diagonal-block (here, diagonal-entry) growth rates, i.e. the elements of the multi-set $\\{{\\mathbb E}(\\log|A_1(i,i)|)\\}_{i=1}^d$.", "expanded_sketch": "A conceptual explanation for Proposition~\\ref{pkus} is provided by the following lemma of Furstenberg and Kifer (H. Furstenberg and Y. Kifer, \\emph{Random matrix products and measures on projective spaces}, quoted here as Lemma~\\ref{blockt}), which \\emph{reduces the computation of the top Lyapunov exponent for block-triangular products to that of the diagonal blocks}. We first recall this lemma.\n\n\\begin{lemma}\n\t\t\\cite[Lemma~3.6]{fifa}\n\t\t\\label{blockt}\n\t\tFor given $m\\in \\nn$ and $(s_1,\\ldots,s_m)\\in\\nn^m,$ let $(A_n)_{n\\in\\nn}$ be a random, stationary ergodic sequence of matrices, each having the shape introduced in \\eqref{block_def}. Suppose in addition that for all $i\\in [m]$, $B_1(i,i)$ is invertible with probability one, and for all $1\\le i\\le j\\le m$,\n\t\t\\beq\n\t\t\\E\\big(\\log^+ \\|B_1(i,j)\\|\\big)<\\infty \\qquad \\mbox{and} \\qquad \\E\\big(\\log^+ \\|B_1^{-1}(i,i)\\|\\big)<\\infty.\n\t\t\\feq\n\t\tFor $i \\in [m],$ let\n\t\t\\beq\n\t\t\\beta_i=\\lim_{n\\to\\infty} \\frac{1}{n}\\log \\big\\|B_n(i,i)B_{n-1}(i,i)\\cdots B_1(i,i)\\big\\|,\\qquad \\as\n\t\t\\feq\n\t\tThen, $\\gamma_1=\\max_{i \\in [m]} \\beta_i.$\n\t\\end{lemma}\n\nIn the present upper-triangular setting, this reduction yields the familiar identification that the top Lyapunov exponent is the maximum among the diagonal-block (here, diagonal-entry) growth rates, i.e. the elements of the multi-set $\\{{\\mathbb E}(\\log|A_1(i,i)|)\\}_{i=1}^d$.", "expanded_theorem": "\\label{pkus}\nPinkus, \\emph{(missing title/year in ref\\_dict)}\nLet ${\\mathcal A}=(A_k)_{k\\in{\\mathbb N}}\\in\\calm_d^{\\mathbb N}$ be a random, stationary, and ergodic sequence of $d\\times d$ invertible matrices with $A_k(i,j) = 0$ for $i > j$ and such that\n\\label{f_cond}\n\t\\E\\left( \\log^+ \\|A_1\\| \\right) < \\infty,\n\t\\feqn\n\twhere, for $x>0$, $\\log^+ x:=\\max(0,\\log x)$, with the convention $\\log^+0:=0$.  Oseledets' multiplicative ergodic theorem then yields holds. Then,\n\\begin{eqnarray*}\n\\gamma_1 = \\max_{i\\in [d]} {\\mathbb E} \\left( \\log |A_1(i,i)|\\right),\n\\end{eqnarray*}\nwhere $[d]:=\\{1,\\ldots,d\\}.$,", "theorem_type": ["Equality or Bound", "Universal"], "mcq": {"question": "Let ${\\mathcal A}=(A_k)_{k\\in\\mathbb N}$ be a random, stationary, and ergodic sequence of invertible real $d\\times d$ matrices such that each $A_k$ is upper triangular, i.e. $A_k(i,j)=0$ whenever $i>j$, and assume\n\\[\n\\mathbb E\\big(\\log^+\\|A_1\\|\\big)<\\infty,\n\\]\nwhere $\\log^+ x:=\\max(0,\\log x)$ for $x>0$ and $\\log^+0:=0$. If $X_n:=A_n\\cdots A_1$, and $\\gamma_1$ denotes the top Lyapunov exponent\n\\[\n\\gamma_1:=\\lim_{n\\to\\infty}\\frac1n\\log\\|X_n\\|\n\\]\n(which exists almost surely and is deterministic under these assumptions), which statement holds for every such sequence?", "correct_choice": {"label": "A", "text": "\\[\\gamma_1=\\max_{i\\in[d]}\\,\\mathbb E\\big(\\log |A_1(i,i)|\\big),\\qquad [d]=\\{1,\\dots,d\\}.\\]"}, "choices": [{"label": "B", "text": "\\[\\gamma_1=\\max_{i\\in[d]}\\,\\log\\mathbb E\\big(|A_1(i,i)|\\big),\\qquad [d]=\\{1,\\dots,d\\}.\\]"}, {"label": "C", "text": "\\[\\gamma_1\\ge \\max_{i\\in[d]}\\,\\mathbb E\\big(\\log |A_1(i,i)|\\big),\\qquad [d]=\\{1,\\dots,d\\}.\\]"}, {"label": "D", "text": "\\[\\gamma_1=\\mathbb E\\Big(\\log \\max_{i\\in[d]} |A_1(i,i)|\\Big),\\qquad [d]=\\{1,\\dots,d\\}.\\]"}, {"label": "E", "text": "\\[\\gamma_1=\\max_{i\\in[d]}\\,\\lim_{n\\to\\infty}\\frac1n\\log\\mathbb E\\big|\\big(X_n\\big)(i,i)\\big|,\\qquad [d]=\\{1,\\dots,d\\}.\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "trace_identity", "tampered_component": "expectation-of-log on diagonal entries", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "trace_identity", "tampered_component": "equality replaced by one-sided lower bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "max of diagonal Lyapunov rates replaced by expectation of pointwise maximum", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "almost-sure diagonal growth replaced by growth of expectations with quantifiers moved outside", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct formula. It gives the setup and asks for the universal conclusion, without directly stating the equality in choice A."}, "TAS": {"score": 0, "justification": "This is essentially a direct restatement of a known theorem: under the listed hypotheses on upper triangular random matrices, the top Lyapunov exponent equals the maximum expected log of the diagonal entries. The item mainly asks the test-taker to recognize that theorem."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the distractors exploit common confusions involving Jensen-type swaps, weaker inequalities, and moving expectations across limits/maxima. However, the item still primarily rewards recall of the exact theorem rather than genuine derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: B confuses log of expectation with expectation of log, C is a weaker true statement, D swaps max and expectation improperly, and E introduces a subtle expectation/limit mismatch. These align well with common failure modes."}, "total_score": 5, "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it is mostly theorem recognition rather than a non-tautological test of generative reasoning."}}
{"id": "2602.08158v1", "paper_link": "http://arxiv.org/abs/2602.08158v1", "theorems_cnt": 7, "theorem": {"env_name": "theorem", "content": "\\label{DoldKan}\n  Let $\\mathcal{A}$ be a weakly idempotent additive category. Let $M_\\bullet$ be a\n  simplicial module. Every element of $M_n$ has a unique decomposition\n  \\begin{equation*}\n    x = \\sum_{k=0}^n \\sum_{0\\le i_k<\\cdots<i_1<n} s_{n-1,i_1}\\ldots s_{n-k,i_k}x_{i_1\\ldots i_k} ,\n  \\end{equation*}\n  where $x_{i_1\\ldots i_k} \\in N_{n-k}(M)$.", "start_pos": 8813, "end_pos": 9187, "label": "DoldKan"}, "ref_dict": {}, "pre_theorem_intro_text_len": 7202, "pre_theorem_intro_text": "Let $\\Lambda_\\infty$ be the category whose objects are the natural numbers\n$[n]$, $n\\ge0$, and whose morphisms are weakly monotone periodic\nfunctions:\n\\begin{multline*}\n  \\Lambda_\\infty([m],[n]) \\\\\n  = \\{ f:\\mathbb{Z}\\to\\mathbb{Z} \\mid \\text{$f(j)\\le f(k)$ if $j\\le k$, and\n    $f(j+m+1)=f(j)+n+1$} \\} .\n\\end{multline*}\nThis category was denoted $L$ by Dwyer and Kan \\cite{DK}; the notation\n$\\Lambda_\\infty$ is due to Feigin and Tsygan \\cite{FT}.\n\nThe category $\\Lambda_\\infty$ has subcategories\n\\begin{equation*}\n  \\Lambda_+([m],[n]) = \\{ f\\in\\Lambda_\\infty([m],[n]) \\mid f(0)\\ge0 \\} ,\n\\end{equation*}\ndenoted $K$ by Dwyer and Kan \\cite{DK}, and\n\\begin{equation*}\n  \\Delta([m],[n]) = \\{ f\\in\\Lambda_+([m],[n]) \\mid f(m)\\le n \\} .\n\\end{equation*}\n\nDefine the morphisms $\\varepsilon^n_i\\in\\Delta([n{-}1],[n])$,\n$0\\le i\\le n$, $\\eta^n_i\\in\\Delta([n{+}1],[n])$, $0\\le i\\le n$, and\n$\\eta^n_{n+1}\\in\\Lambda_+([n{+}1],[n])$, by\n\\begin{align*}\n  \\varepsilon_i^n(j) &=\n  \\begin{cases}\n    j , & 0\\le j<i , \\\\\n    j+1 , & i\\le j\\le n-1 ,\n  \\end{cases} \\intertext{and}\n  \\eta_i^n(j) &=\n  \\begin{cases}\n    j , & 0\\le j\\le i , \\\\\n    j-1 , & i<j\\le n+1 .\n  \\end{cases}\n\\end{align*}\nThese morphisms satisfy the relations\n\\begin{align*}\n  \\varepsilon_i^{n+1} \\varepsilon_j^n &= \\varepsilon_{j+1}^{n+1} \\varepsilon_i^n , \\qquad \\, \\, 0\\le i\\le j\\le n , \\\\\n  \\eta_j^n \\varepsilon_i^{n+1} &=\n                    \\begin{cases}\n                      \\tau_n , & j-i=n+1 , \\\\\n                      \\varepsilon_i^n \\eta_{j-1}^{n-1} , & 1\\le j-i\\le n , \\\\\n                      1 , & j-i=0,-1, \\\\\n                      \\varepsilon_{i-1}^n \\eta_j^{n-1} , & -n-1\\le j-i<-1 ,\n                    \\end{cases} \\\\\n  \\eta_j^{n-1} \\eta_i^n &= \\eta_i^{n-1} \\eta_{j+1}^n , \\qquad \\!\\! 0\\le i\\le j\\le n .\n\\end{align*}\nThe morphism $\\tau_n=\\eta^n_{n+1}\\varepsilon^{n+1}_0\\in\\Lambda_\\infty([n],[n])$ is the function\n\\begin{equation*}\n  \\tau_n(i) = i+1 .\n\\end{equation*}\nWe have $\\tau_n^{n+1}\\varepsilon^n_i=\\varepsilon^n_i\\tau_{n-1}^n$ and\n$\\tau_n^{n+1}\\eta^n_i=\\eta^n_i\\tau_{n+1}^{n+2}$. It follows that\n$\\Lambda_+$ is generated by the morphisms\n\\begin{equation*}\n  \\{ \\varepsilon_0^n : [n{-}1] \\to [n] \\}_{n>0} \\cup \\{ \\eta_{n+1}^n : [n{+}1] \\to\n  [n]\\}_{n\\ge0} ,\n\\end{equation*}\nand $\\Lambda_\\infty$ is obtained from $\\Lambda_+$ by inverting\n$\\{ \\tau_n^{-1} : [n] \\to [n] \\}_{n\\ge0}$.\n\nThere is a contravariant isomorphism of categories $f\\mapsto f^\\circ$ from\n$\\Lambda_\\infty$ to its opposite $\\Lambda^\\op_\\infty$, which acts on generators by\ninterchanging $\\varepsilon^n_i$ and $\\eta^{n-1}_{n-i}$, and fixing\n$\\tau_n$. In particular, it exchanges the generators $\\varepsilon^n_0$ and\n$\\eta^{n-1}_n$ of $\\Lambda_\\infty$.\n\n\\begin{definition}\n  Let $\\mathcal{A}$ be a pre-additive category (a category enriched in abelian\n  groups).  A \\textbf{paracyclic} (respectively \\textbf{duplicial} or\n  \\textbf{simplicial}) module $M_\\bullet$ is a presheaf on\n  $\\Lambda_\\infty$ (respectively $\\Lambda_+$ or $\\Delta$) taking values in $\\mathcal{A}$.\n\\end{definition}\n\nDenote the (co)action of the morphisms $\\varepsilon_i^n$ and\n$\\eta_i^n$ on a duplicial module $M_\\bullet$ by\n$\\partial_{n,i}:M_n\\to M_{n-1}$ and $s_{n,i}:M_n\\to M_n$, and the (co)action of\nthe morphisms $\\tau_n$ by $t_n:M_n\\to M_n$. The relations among the\ngenerators of $\\Lambda_+$ become the following relations on duplicial\nmodules $M_\\bullet$:\n\\begin{align*}\n  \\partial_{n,k} \\partial_{n+1,j} &= \\partial_{n,j} \\partial_{n+1,k+1} , \\qquad \\,\\, 0\\le j\\le k\\le n , \\\\\n  \\partial_{n+1,j} s_{n,k} &=\n                    \\begin{cases}\n                      t_n , & k-j=n+1 , \\\\\n                      s_{n-1,k-1} \\partial_{n,j} , & 1\\le k-j\\le n , \\\\\n                      1 , & k-j=0,-1, \\\\\n                      s_{n-1,k} \\partial_{n,j-1} , & -n-1\\le k-j<-1 ,\n                    \\end{cases} \\\\\n  s_{n,j} s_{n-1,k} &= s_{n,k+1} s_{n-1,j} , \\qquad \\!\\! 0\\le j\\le k\\le n .\n\\end{align*}\nIn particular,\n\\begin{align*}\n  \\partial_{n,i} t_n &=\n                \\begin{cases}\n                  t_{n-1} \\partial_{n,i+1} , & 0\\le i< n , \\\\\n                  \\partial_{n,0} , & i=n ,\n                \\end{cases} \\intertext{and}\n  t_{n+1} s_{n,i} &=\n                \\begin{cases}\n                  s_{n,n+1} , & i=0 , \\\\\n                  s_{n,i-1} t_n , & 0< i\\le n+1 .\n                \\end{cases}\n\\end{align*}\nDenote the action of $t_n^{n+1}$ on $M_n$ by $\\mathsf{T}$. We see that\n$\\T_{n-1}\\partial_{n,i}=\\partial_{n,i}\\T_n$ and $\\T_{n+1}s_{n,i}=s_{n,i}\\T_n$.\n\nAn \\textbf{idempotent} $(A,p)$ in a pre-additive category $\\mathcal{A}$ is a\npair consisting of an object $A$ of $\\mathcal{A}$ and $p:A\\to A$ an idempotent\nendomorphism of $A$, that is, $p^2=p$. If $p:A\\to A$ is an idempotent,\nthen so is $p^\\perp=1-p$, and the image of the idempotent $p$ is the\nkernel of $p^\\perp$ (and vice versa).\n\nLet $M_\\bullet$ be a simplicial module. Dold and Puppe \\cite{DP}*{Section 3}\ndefine idempotents\n\\begin{equation*}\n  p_n = (1-s_{n-1,0}\\partial_{n,1})\\ldots(1-s_{n-1,n-1}\\partial_{n,n}) : M_n \\to M_n .\n\\end{equation*}\nThe kernel of the idempotent $p_n:M_n\\to M_n$, if it exists, is the\nmodule of degenerate chains $D_n(M)$. The image of $p_n$ is the module\n$N_n(M)$ of normalized chains. Provided these modules exist, we have\nthe direct sum decomposition\n\\begin{equation*}\n  M_n = N_n(M) \\oplus D_n(M) .\n\\end{equation*}\n\nThe module $N_n(M)$ is the submodule on which the face maps\n$\\partial_{n,i}$, $1\\le i\\le n$, vanish:\n\\begin{equation*}\n  N_n(M) = \\bigcap_{i=1}^n \\ker\\bigl( \\partial_{n,i}:M_n\\to M_{n-1} \\bigr) \\subset M_n .\n\\end{equation*}\nSimilarly, the module $D_n(M)$ is the sum of the images of the\ndegeneracy maps $s_{n-1,i}$, $0\\le i<n$:\n\\begin{equation*}\n  D_n(M) = \\sum_{i=0}^{n-1} \\im\\bigl( s_{n-1,i}:M_{n-1}\\to M_n \\bigr) \\subset M_n .\n\\end{equation*}\n\nThe following property of a pre-additive category $\\mathcal{A}$ was introduced\nby Freyd \\cite{Freyd} (cf.\\ B\\\"uhler \\cite{Buhler}*{Section 7}).\n\\begin{definition}\n  A \\textbf{section} of a morphism $f:M\\to N$ is a morphism $g:N\\to M$\n  with\n  \\begin{equation*}\n    fg=1_N .\n  \\end{equation*}\n  A morphism with a section is called a \\textbf{retraction}. A\n  pre-additive category $\\mathcal{A}$ is \\textbf{weakly idempotent complete}\n  if every retraction has a kernel.\n\\end{definition}\n\n\\begin{lemma}\n  If $\\mathcal{A}$ is weakly idempotent complete, the idempotent $p_n$ splits,\n  implying the existence of the modules $N_n(M)$ and $D_n(M)$.\n\\end{lemma}\n\\begin{proof}\n  Consider the filtration\n  \\begin{equation*}\n    F_iM_n = \\bigcap_{j=n-i+1}^n \\ker\\bigl( \\partial_{n,j}:M_n\\to M_{n-1} \\bigr) \\subset M_n .\n  \\end{equation*}\n  Thus $F_iM_n=M_n$, and $F_nM_n=N_n(M)$.  The existence of $F_iM_n$\n  is shown by induction on $i$: $F_{i+1}M_n$ is the kernel of the\n  split epimorphism\n  \\begin{equation*}\n    \\partial_{n,n-i} : F_iM_n \\to F_iM_{n-1} ,\n  \\end{equation*}\n  with section $s_{n-1,n-i-1}:F_iM_{n-1} \\to F_iM_n$.\n\\end{proof}\n\nThe following result of Dold and Puppe \\cite{DP} is a reformulation of\nthe theorem of Dold \\cite{Dold} and Kan \\cite{Kan}. They state the\ntheorem when $\\mathcal{A}$ is abelian, but their proof applies more\ngenerally. (Recall that a pre-additive category is additive if it has\nfinite sums.)", "context": "\\begin{definition}\n Let $\\mathcal{A}$ be a pre-additive category (a category enriched in abelian\n groups). A \\textbf{paracyclic} (respectively \\textbf{duplicial} or\n \\textbf{simplicial}) module $M_\\bullet$ is a presheaf on\n $\\Lambda_\\infty$ (respectively $\\Lambda_+$ or $\\Delta$) taking values in $\\mathcal{A}$.\n\\end{definition}\n\nDenote the (co)action of the morphisms $\\varepsilon_i^n$ and\n$\\eta_i^n$ on a duplicial module $M_\\bullet$ by\n$\\partial_{n,i}:M_n\\to M_{n-1}$ and $s_{n,i}:M_n\\to M_n$, and the (co)action of\nthe morphisms $\\tau_n$ by $t_n:M_n\\to M_n$. The relations among the\ngenerators of $\\Lambda_+$ become the following relations on duplicial\nmodules $M_\\bullet$:\n\\begin{align*}\n \\partial_{n,k} \\partial_{n+1,j} &= \\partial_{n,j} \\partial_{n+1,k+1} , \\qquad \\,\\, 0\\le j\\le k\\le n , \\\\\n \\partial_{n+1,j} s_{n,k} &=\n \\begin{cases}\n t_n , & k-j=n+1 , \\\\\n s_{n-1,k-1} \\partial_{n,j} , & 1\\le k-j\\le n , \\\\\n 1 , & k-j=0,-1, \\\\\n s_{n-1,k} \\partial_{n,j-1} , & -n-1\\le k-j<-1 ,\n \\end{cases} \\\\\n s_{n,j} s_{n-1,k} &= s_{n,k+1} s_{n-1,j} , \\qquad \\!\\! 0\\le j\\le k\\le n .\n\\end{align*}\nIn particular,\n\\begin{align*}\n \\partial_{n,i} t_n &=\n \\begin{cases}\n t_{n-1} \\partial_{n,i+1} , & 0\\le i< n , \\\\\n \\partial_{n,0} , & i=n ,\n \\end{cases} \\intertext{and}\n t_{n+1} s_{n,i} &=\n \\begin{cases}\n s_{n,n+1} , & i=0 , \\\\\n s_{n,i-1} t_n , & 0< i\\le n+1 .\n \\end{cases}\n\\end{align*}\nDenote the action of $t_n^{n+1}$ on $M_n$ by $\\mathsf{T}$. We see that\n$\\T_{n-1}\\partial_{n,i}=\\partial_{n,i}\\T_n$ and $\\T_{n+1}s_{n,i}=s_{n,i}\\T_n$.\n\nLet $M_\\bullet$ be a simplicial module. Dold and Puppe \\cite{DP}*{Section 3}\ndefine idempotents\n\\begin{equation*}\n p_n = (1-s_{n-1,0}\\partial_{n,1})\\ldots(1-s_{n-1,n-1}\\partial_{n,n}) : M_n \\to M_n .\n\\end{equation*}\nThe kernel of the idempotent $p_n:M_n\\to M_n$, if it exists, is the\nmodule of degenerate chains $D_n(M)$. The image of $p_n$ is the module\n$N_n(M)$ of normalized chains. Provided these modules exist, we have\nthe direct sum decomposition\n\\begin{equation*}\n M_n = N_n(M) \\oplus D_n(M) .\n\\end{equation*}\n\nThe following property of a pre-additive category $\\mathcal{A}$ was introduced\nby Freyd \\cite{Freyd} (cf.\\ B\\\"uhler \\cite{Buhler}*{Section 7}).\n\\begin{definition}\n A \\textbf{section} of a morphism $f:M\\to N$ is a morphism $g:N\\to M$\n with\n \\begin{equation*}\n fg=1_N .\n \\end{equation*}\n A morphism with a section is called a \\textbf{retraction}. A\n pre-additive category $\\mathcal{A}$ is \\textbf{weakly idempotent complete}\n if every retraction has a kernel.\n\\end{definition}\n\n\\begin{lemma}\n If $\\mathcal{A}$ is weakly idempotent complete, the idempotent $p_n$ splits,\n implying the existence of the modules $N_n(M)$ and $D_n(M)$.\n\\end{lemma}\n\\begin{proof}\n Consider the filtration\n \\begin{equation*}\n F_iM_n = \\bigcap_{j=n-i+1}^n \\ker\\bigl( \\partial_{n,j}:M_n\\to M_{n-1} \\bigr) \\subset M_n .\n \\end{equation*}\n Thus $F_iM_n=M_n$, and $F_nM_n=N_n(M)$. The existence of $F_iM_n$\n is shown by induction on $i$: $F_{i+1}M_n$ is the kernel of the\n split epimorphism\n \\begin{equation*}\n \\partial_{n,n-i} : F_iM_n \\to F_iM_{n-1} ,\n \\end{equation*}\n with section $s_{n-1,n-i-1}:F_iM_{n-1} \\to F_iM_n$.\n\\end{proof}\n\nThe following result of Dold and Puppe \\cite{DP} is a reformulation of\nthe theorem of Dold \\cite{Dold} and Kan \\cite{Kan}. They state the\ntheorem when $\\mathcal{A}$ is abelian, but their proof applies more\ngenerally. (Recall that a pre-additive category is additive if it has\nfinite sums.)", "full_context": "\\begin{definition}\n Let $\\mathcal{A}$ be a pre-additive category (a category enriched in abelian\n groups). A \\textbf{paracyclic} (respectively \\textbf{duplicial} or\n \\textbf{simplicial}) module $M_\\bullet$ is a presheaf on\n $\\Lambda_\\infty$ (respectively $\\Lambda_+$ or $\\Delta$) taking values in $\\mathcal{A}$.\n\\end{definition}\n\nDenote the (co)action of the morphisms $\\varepsilon_i^n$ and\n$\\eta_i^n$ on a duplicial module $M_\\bullet$ by\n$\\partial_{n,i}:M_n\\to M_{n-1}$ and $s_{n,i}:M_n\\to M_n$, and the (co)action of\nthe morphisms $\\tau_n$ by $t_n:M_n\\to M_n$. The relations among the\ngenerators of $\\Lambda_+$ become the following relations on duplicial\nmodules $M_\\bullet$:\n\\begin{align*}\n \\partial_{n,k} \\partial_{n+1,j} &= \\partial_{n,j} \\partial_{n+1,k+1} , \\qquad \\,\\, 0\\le j\\le k\\le n , \\\\\n \\partial_{n+1,j} s_{n,k} &=\n \\begin{cases}\n t_n , & k-j=n+1 , \\\\\n s_{n-1,k-1} \\partial_{n,j} , & 1\\le k-j\\le n , \\\\\n 1 , & k-j=0,-1, \\\\\n s_{n-1,k} \\partial_{n,j-1} , & -n-1\\le k-j<-1 ,\n \\end{cases} \\\\\n s_{n,j} s_{n-1,k} &= s_{n,k+1} s_{n-1,j} , \\qquad \\!\\! 0\\le j\\le k\\le n .\n\\end{align*}\nIn particular,\n\\begin{align*}\n \\partial_{n,i} t_n &=\n \\begin{cases}\n t_{n-1} \\partial_{n,i+1} , & 0\\le i< n , \\\\\n \\partial_{n,0} , & i=n ,\n \\end{cases} \\intertext{and}\n t_{n+1} s_{n,i} &=\n \\begin{cases}\n s_{n,n+1} , & i=0 , \\\\\n s_{n,i-1} t_n , & 0< i\\le n+1 .\n \\end{cases}\n\\end{align*}\nDenote the action of $t_n^{n+1}$ on $M_n$ by $\\mathsf{T}$. We see that\n$\\T_{n-1}\\partial_{n,i}=\\partial_{n,i}\\T_n$ and $\\T_{n+1}s_{n,i}=s_{n,i}\\T_n$.\n\nLet $M_\\bullet$ be a simplicial module. Dold and Puppe \\cite{DP}*{Section 3}\ndefine idempotents\n\\begin{equation*}\n p_n = (1-s_{n-1,0}\\partial_{n,1})\\ldots(1-s_{n-1,n-1}\\partial_{n,n}) : M_n \\to M_n .\n\\end{equation*}\nThe kernel of the idempotent $p_n:M_n\\to M_n$, if it exists, is the\nmodule of degenerate chains $D_n(M)$. The image of $p_n$ is the module\n$N_n(M)$ of normalized chains. Provided these modules exist, we have\nthe direct sum decomposition\n\\begin{equation*}\n M_n = N_n(M) \\oplus D_n(M) .\n\\end{equation*}\n\nThe following property of a pre-additive category $\\mathcal{A}$ was introduced\nby Freyd \\cite{Freyd} (cf.\\ B\\\"uhler \\cite{Buhler}*{Section 7}).\n\\begin{definition}\n A \\textbf{section} of a morphism $f:M\\to N$ is a morphism $g:N\\to M$\n with\n \\begin{equation*}\n fg=1_N .\n \\end{equation*}\n A morphism with a section is called a \\textbf{retraction}. A\n pre-additive category $\\mathcal{A}$ is \\textbf{weakly idempotent complete}\n if every retraction has a kernel.\n\\end{definition}\n\n\\begin{lemma}\n If $\\mathcal{A}$ is weakly idempotent complete, the idempotent $p_n$ splits,\n implying the existence of the modules $N_n(M)$ and $D_n(M)$.\n\\end{lemma}\n\\begin{proof}\n Consider the filtration\n \\begin{equation*}\n F_iM_n = \\bigcap_{j=n-i+1}^n \\ker\\bigl( \\partial_{n,j}:M_n\\to M_{n-1} \\bigr) \\subset M_n .\n \\end{equation*}\n Thus $F_iM_n=M_n$, and $F_nM_n=N_n(M)$. The existence of $F_iM_n$\n is shown by induction on $i$: $F_{i+1}M_n$ is the kernel of the\n split epimorphism\n \\begin{equation*}\n \\partial_{n,n-i} : F_iM_n \\to F_iM_{n-1} ,\n \\end{equation*}\n with section $s_{n-1,n-i-1}:F_iM_{n-1} \\to F_iM_n$.\n\\end{proof}\n\nThe following result of Dold and Puppe \\cite{DP} is a reformulation of\nthe theorem of Dold \\cite{Dold} and Kan \\cite{Kan}. They state the\ntheorem when $\\mathcal{A}$ is abelian, but their proof applies more\ngenerally. (Recall that a pre-additive category is additive if it has\nfinite sums.)\n\nLet $M_\\bullet$ be a simplicial module. Dold and Puppe \\cite{DP}*{Section 3}\ndefine idempotents\n\\begin{equation*}\n p_n = (1-s_{n-1,0}\\partial_{n,1})\\ldots(1-s_{n-1,n-1}\\partial_{n,n}) : M_n \\to M_n .\n\\end{equation*}\nThe kernel of the idempotent $p_n:M_n\\to M_n$, if it exists, is the\nmodule of degenerate chains $D_n(M)$. The image of $p_n$ is the module\n$N_n(M)$ of normalized chains. Provided these modules exist, we have\nthe direct sum decomposition\n\\begin{equation*}\n M_n = N_n(M) \\oplus D_n(M) .\n\\end{equation*}\n\n\\begin{lemma}\n If $\\CA$ is weakly idempotent complete, the idempotent $p_n$ splits,\n implying the existence of the modules $N_n(M)$ and $D_n(M)$.\n\\end{lemma}\n\\begin{proof}\n Consider the filtration\n \\begin{equation*}\n F_iM_n = \\bigcap_{j=n-i+1}^n \\ker\\bigl( \\partial_{n,j}:M_n\\to M_{n-1} \\bigr) \\subset M_n .\n \\end{equation*}\n Thus $F_iM_n=M_n$, and $F_nM_n=N_n(M)$. The existence of $F_iM_n$\n is shown by induction on $i$: $F_{i+1}M_n$ is the kernel of the\n split epimorphism\n \\begin{equation*}\n \\partial_{n,n-i} : F_iM_n \\to F_iM_{n-1} ,\n \\end{equation*}\n with section $s_{n-1,n-i-1}:F_iM_{n-1} \\to F_iM_n$.\n\\end{proof}\n\nWe call this the Dold--Kan decomposition.\n\nThe purpose of this article is to give a new proof of a result of\nDwyer and Kan \\cite{DK}*{Corollary 6.6}.\n\\begin{theorem}\n \\label{DwyerKan}\n Let $\\CA$ be a weakly idempotent additive category.\n \\begin{enumerate}[i)]\n \\item The duplicial module $M_\\bullet$ is paracyclic if and only if the\n restriction of the Karoubi operator $\\kappa_n$ to $N_n(M)$ is\n invertible for all $n\\ge0$.\n \\item The duplicial module $M_\\bullet$ is cyclic if and only if the\n restriction of the Dwyer--Kan operator $\\pi_n$ to $N_n(M)$ is the\n identity for all $n\\ge0$.\n \\end{enumerate}\n\\end{theorem}\n\nA simplicial module $M_\\bullet$ gives rise to a chain complex: it carries\nthe differential\n\\begin{equation*}\n b_n = \\sum_{i=0}^n (-1)^i \\partial_{n,i} : M_n \\to M_{n-1} .\n\\end{equation*}\nWe have\n\\begin{align*}\n b_{n-1}b_n &= \\sum_{0\\le j\\le k\\le n-1} (-1)^{j+k} \\, \\partial_{n-1,k} \\partial_{n,j}\n + \\sum_{0\\le k< j\\le n} (-1)^{j+k} \\, \\partial_{n-1,k} \\partial_{n,j} \\\\\n &= \\sum_{0\\le j\\le k\\le n-1} (-1)^{j+k} \\, \\partial_{n-1,j} \\partial_{n,k+1}\n + \\sum_{0\\le k< j\\le n} (-1)^{j+k} \\, \\partial_{n-1,k} \\partial_{n,j} = 0 .\n\\end{align*}\nNote that the restriction of $b_n$ to $N_n(M)$ equals\n$\\partial_{n,0}$. This gives a formula for the action of the face map\n$\\delta_n=\\partial_{n,0}$ in the Dold--Kan decomposition:\n\\begin{equation*}\n \\delta_n \\bigl( s_{n-1,i_1}\\dots s_{n-k,i_k} x_{i_1\\ldots i_k} \\bigr) =\n \\begin{cases}\n s_{n-2,i_1-1} \\ldots s_{n-k,i_{k-1}-1} x_{i_1\\ldots i_k} , & i_k=0 \\\\\n s_{n-2,i_1-1}\\dots s_{n-k-1,i_k-1} ( b_{n-k}x_{i_1\\ldots i_k} ) , &\n i_k>0 .\n \\end{cases}\n\\end{equation*}\n\nIt was proved by Eilenberg and Mac Lane \\cite{EM1} that the complexes\n$(M_\\bullet,b)$ and $(N_\\bullet(M),b)$ are homotopy equivalent. The following\nexpression for their homotopy is taken from Epstein\n\\cite{Epstein}*{Proposition 2.3}.\n\\begin{proposition}\n Let $M_\\bullet$ be a simplicial module. The operator\n \\begin{equation*}\n \\varphi_n = \\sum_{i=0}^n (-1)^i s_{n,i} p_{n,i} : M_n \\to M_{n+1}\n \\end{equation*}\n satisfies $b_{n+1}\\varphi_n + \\varphi_{n-1}b_n = p_n - 1$.\n\\end{proposition}\n\\begin{proof}\n For $0\\le i\\le n$,\n \\begin{align*}\n b_{n+1} s_{n,i} p_{n,i}\n &= \\sum_{k=0}^{i-1} (-1)^k \\partial_{n+1,k} s_{n,i} p_{n,i}\n + (-1)^i ( \\partial_{n+1,i} - \\partial_{n+1,i+1} ) s_{n,i} p_{n,i} \\\\\n & \\quad + \\sum_{k=i+2}^{n+1} (-1)^k \\partial_{n+1,k} s_{n,i} p_{n,i} \\\\\n &= \\sum_{k=0}^{i-1} (-1)^k s_{n-1,i-1} \\partial_{n,k} p_{n,i}\n + \\sum_{k=i+2}^{n+1} (-1)^k s_{n-1,i} \\partial_{n,k-1} p_{n,i} \\\\\n &= \\sum_{k=0}^n (-1)^k s_{n-1,i-1} \\partial_{n,k} p_{n,i}\n - (-1)^i s_{n-1,i-1} \\partial_{n,i} p_{n,i} \\\\\n &= s_{n-1,i-1} b_n p_{n,i} + (-1)^i ( p_{n,i-1} - p_{n,i} ) \\\\\n &= s_{n-1,i-1} p_{n-1,i-1} b_n + (-1)^i ( p_{n,i-1} - p_{n,i} ) .\n \\end{align*}\n Summing over $i$, the result follows\n\\end{proof}\n\nWe now come to the main result of this article.\n\\begin{theorem}\n $\\pi_n= p_n \\circ \\T_n = \\T_n \\circ p_n$\n\\end{theorem}\n\\begin{proof}\n \\begin{align*}\n \\pi_n &= (-1)^n \\partial_{n+1}(\\partial_{n+2}s_{n+1}-s_n\\partial_{n+1})^ns_n (-1)^{|j|-|i|} \\\\\n &= t_n^{n+1} + \\sum_{k=1}^{n-1} \\sum_{0\\le i_1<j_1<i_2<j_2< \\ldots<i_k<j_k\\le n} \\\\\n & \\qquad \\bigl( t_n^{i_1} \\partial_{n+1} t_{n+1}^{j_1-i_1} s_n \\bigr)\n \\bigl( t_n^{i_2-j_1-1} \\partial_{n+1} t_{n+1}^{j_2-i_2} s_n \\bigr) \\ldots\n \\bigl( t^{i_k-j_{k-1}-1} \\partial_{n+1} t_{n+1}^{j_k-i_k} s_n \\bigr)\n t_n^{n-j_k} .\n \\end{align*}\n If $p<q$, define\n \\begin{align*}\n \\Pi_{p,q} &= (-1)^{q-p} s_{n-1,p} \\partial_{n,q} \\\\\n &= \\bigl( - s_{n-1,p} \\partial_{n,p+1} \\bigr)\n \\bigl( - s_{n-1,p+1} \\partial_{n,p+2} \\bigr) \\ldots\n \\bigl( - s_{n-1,q-1} \\partial_{n,q} \\bigr) .\n \\end{align*}\n By definition,\n \\begin{equation*}\n \\sum_{k=0}^{n-1} \\sum_{0\\le i_1<j_1<i_2<j_2< \\ldots<i_k<j_k\\le n} \\Pi_{i_1,j_1} \\ldots\n \\Pi_{i_k,j_k} .\n \\end{equation*}\n We have\n \\begin{align*}\n (-1)^{j-i} t_n^i \\partial_{n+1} t_{n+1}^{j-i} s_n\n &= (-1)^{j-i} t_n^{i-n-1} \\partial_{n+1} t_{n+1}^{j-i+n+2} s_n \\\\\n &= (-1)^{j-i} \\partial_{n+1,n-i+1} t_{n+1}^{j+1} s_n\n = (-1)^{j-i} \\partial_{n+1,n-i+1} s_{n,n-j} t_n^{j+1} \\\\\n &= (-1)^{j-i} s_{n-1,n-j} \\partial_{n,n-i} t_{n+1}^{j+1} = \\Pi_{n-j,n-i} t_n^{j+1} .\n \\end{align*}\n It follows that\n \\begin{equation*}\n \\pi_n = \\sum_{k=0}^{n-1} \\sum_{0\\le i_1<j_1<\\ldots\\le i_k<j_k\\le n}\n \\Pi_{n-j_1,n-i_1} \\ldots \\Pi_{n-j_k,n-i_k} t_n^{n+1} .\n \\end{equation*}\n The proof is completed by showing that if $s<p$,\n \\begin{equation}\n \\label{PiPi}\n \\Pi_{p,q} \\Pi_{r,s} = \\Pi_{r,s} \\Pi_{p,q} .\n \\end{equation}\n This allows us to reverse the order of the factors in\n $\\Pi_{n-j_1,n-i_1} \\ldots \\Pi_{n-j_k,n-i_k}$:\n \\begin{equation*}\n \\Pi_{n-j_1,n-i_1} \\ldots \\Pi_{n-j_k,n-i_k} = \\Pi_{n-j_k,n-i_k} \\ldots \\Pi_{n-j_1,n-i_1} .\n \\end{equation*}\n To prove \\eqref{PiPi}, we must show that\n \\begin{equation*}\n s_{n-1,p} \\partial_{n,q} s_{n-1,r} \\partial_{n,s}\n = s_{n-1,r} \\partial_{n,s} s_{n-1,p} \\partial_{n,q} .\n \\end{equation*}\n This is a consequence of the simplicial relations:\n \\begin{align*}\n s_{n-1,p} \\partial_{n,q} s_{n-1,r} \\partial_{n,s} \n &= s_{n-1,p} s_{n-2,r} \\partial_{n-1,q-1} \\partial_{n,s} \\\\\n &= s_{n-1,r} s_{n-2,p-1} \\partial_{n-1,s} \\partial_{n,q}\n = s_{n-1,r} \\partial_{n,s} s_{n-1,p} \\partial_{n,q} .\n \\qedhere\n \\end{align*}\n\\end{proof}\n\nTheorem \\ref{DwyerKan} is an immediate consequence of this theorem. In\nterms of the Dold--Kan decomposition, we see that\n\\begin{equation*}\n \\T_n x = \\sum_{k=0}^n \\sum_{0\\le i_k<\\cdots<i_1<n} s^{n-1}_{i_1}\\ldots s^{n-k}_{i_k}\n \\pi_{n-k}x_{i_1\\ldots i_k} .\n\\end{equation*}\nWe see that $\\T_n$ is invertible on $M_n$ if and only if $\\pi_{n-k}$ is\ninvertible on $N_{n-k}(M)$ for $0\\le k\\le n$. But $\\pi_n$ is invertible on\n$N_n(M)$ if and only if $\\kappa_n$ is, by the formulas\n\\begin{equation*}\n \\kappa_n^{-1} = \\pi_n^{-1} (1+b_{n+1}d_n)^n (1+d_{n-1}b_n)^{n-1}\n\\end{equation*}\nand\n\\begin{equation*}\n \\pi_n^{-1} = \\kappa_n^{-n-1} (1+d_{n-1}b_n) .\n\\end{equation*}", "post_theorem_intro_text_len": 3144, "post_theorem_intro_text": "\\begin{proof}\n  The proof is by induction on $n$: the case $n=0$ is trivial, since\n  $M_0=N_0(M)$. The induction step is obtained by writing $x\\in M_n$ as\n  \\begin{equation*}\n    p_nx + (1-p_n)x = p_nx + \\sum_{i=1}^n s_{n-1,i-1}\\partial_{n,i}p_{n,i}x .\n  \\end{equation*}\n  Since $\\partial_{n,i}p_{n,i}x\\in M_{n-1}$, the result follows.\n\\end{proof}\n\nWe call this the Dold--Kan decomposition.\n\n\\begin{definition}\n  Let $M_\\bullet$ be a duplicial module. The \\textbf{Karoubi operator} is\n  the endomorphism of $M_n$\n  \\begin{equation*}\n    \\kappa_n = (-1)^n \\bigl( \\partial_{n+1,0} s_{n,n+1} - s_{n-1,n}\\partial_{n,0} \\bigr) .\n  \\end{equation*}\nThe \\textbf{Dwyer--Kan operator} is the endomorphism of $M_n$\n  \\begin{equation*}\n    \\pi_n = (-1)^n \\partial_{n+1,0} \\kappa_{n+1}^n s_{n,n+1} .\n  \\end{equation*}\n\\end{definition}\n\nThe definitions of these operators are simplified when they are\nexpressed in terms of the operators $\\delta_n=\\partial_{n,0}$ and\n$\\sigma_n=(-1)^ns_{n,n+1}$:\n\\begin{equation*}\n  \\kappa_n = \\delta_{n+1} \\sigma_n + \\sigma_{n-1}\\delta_n\n\\end{equation*}\nand\n\\begin{equation*}\n  \\pi_n = \\delta_{n+1} \\kappa_{n+1}^n \\sigma_n .\n\\end{equation*}\n\nThe restriction of $\\pi_n$ to $N_n(M)$ was studied by Dwyer and Kan\n\\cite{DK}, and further studied in the special case of cyclic modules\nby Cuntz and Quillen \\cite{CQ}.\n\nThe purpose of this article is to give a new proof of a result of\nDwyer and Kan \\cite{DK}*{Corollary 6.6}.\n\\begin{theorem}\n  \\label{DwyerKan}\n  Let $\\mathcal{A}$ be a weakly idempotent additive category.\n  \\begin{enumerate}[i)]\n  \\item The duplicial module $M_\\bullet$ is paracyclic if and only if the\n    restriction of the Karoubi operator $\\kappa_n$ to $N_n(M)$ is\n    invertible for all $n\\ge0$.\n  \\item The duplicial module $M_\\bullet$ is cyclic if and only if the\n    restriction of the Dwyer--Kan operator $\\pi_n$ to $N_n(M)$ is the\n    identity for all $n\\ge0$.\n  \\end{enumerate}\n\\end{theorem}\n\nUnlike in the earlier work of Dwyer and Kan \\cite{DK} and Cuntz and\nQuillen \\cite{CQ}, we consider the action of the operators $\\kappa_n$ and\n$\\pi_n$ on all of $M_n$, and not only on $N_n(M)$. We show by direct\ncalculation that the operator $\\pi_n$ equals the composition of the\nprojection $p_n$ of Dold and Puppe with the operator $\\T_n$. In\nparticular, for cyclic modules, $\\pi_n=p_n$.\n\nDwyer and Kan cast the Dold--Kan theorem as an equivalence between the\ncategory of duplicial modules and the category of duchain complexes.\n\\begin{definition}\n  A \\textbf{duchain complex} is a graded module $M_\\bullet$ with a pair of\n  differentials $b_n:M_n\\to M_{n-1}$ and $d_n:M_n\\to M_{n+1}$.\n\\end{definition}\n\nA chain complex $(V_\\bullet,b)$ is a degenerate duchain complex, whose extra\ndifferential $d$ vanishes. Thus, there is a natural embedding of the\ncategory of simplicial modules to the category of duplicial (in fact,\ncyclic) modules, given by taking the duplicial structure on the\nsimplicial module $M_\\bullet$ induced by this degenerate duchain structure\non $N_\\bullet(M)$. In this way, we obtain a new formula for the idempotent\n$p_n$ of Dold and Puppe on $M_n$: it equals the Dwyer--Kan operator\n$\\pi_n$.", "sketch": "The proof of Theorem~\\ref{DoldKan} is \"by induction on $n$\": for $n=0$ it is trivial since \"$M_0=N_0(M)$.\" For the induction step, one writes any $x\\in M_n$ as\n\\[\n  p_nx + (1-p_n)x = p_nx + \\sum_{i=1}^n s_{n-1,i-1}\\partial_{n,i}p_{n,i}x .\n\\]\nThen one notes that \"Since $\\partial_{n,i}p_{n,i}x\\in M_{n-1}$, the result follows.\"", "expanded_sketch": "To prove the main theorem, we argue by induction on $n$: for $n=0$ it is trivial since \"$M_0=N_0(M)$.\" For the induction step, one writes any $x\\in M_n$ as\n\\[\n  p_nx + (1-p_n)x = p_nx + \\sum_{i=1}^n s_{n-1,i-1}\\partial_{n,i}p_{n,i}x .\n\\]\nThen one notes that \"Since $\\partial_{n,i}p_{n,i}x\\in M_{n-1}$, the result follows.\"", "expanded_theorem": "\\label{DoldKan}\n  Let $\\mathcal{A}$ be a weakly idempotent additive category. Let $M_\\bullet$ be a\n  simplicial module. Every element of $M_n$ has a unique decomposition\n  \\begin{equation*}\n    x = \\sum_{k=0}^n \\sum_{0\\le i_k<\\cdots<i_1<n} s_{n-1,i_1}\\ldots s_{n-k,i_k}x_{i_1\\ldots i_k} ,\n  \\end{equation*}\n  where $x_{i_1\\ldots i_k} \\in N_{n-k}(M)$.,", "theorem_type": "unknown", "mcq": {"question": "Let $\\mathcal{A}$ be a weakly idempotent additive category, and let $M_\\bullet$ be a simplicial module in $\\mathcal{A}$ with degeneracy maps $s_{m,i}:M_m\\to M_{m+1}$. For each $m$, let\n\\[\np_m=(1-s_{m-1,0}\\partial_{m,1})\\cdots(1-s_{m-1,m-1}\\partial_{m,m}):M_m\\to M_m,\n\\]\nand let $N_m(M)$ denote the image of $p_m$ (the normalized chains). Which statement holds for each $n\\ge 0$ and each element $x\\in M_n$?", "correct_choice": {"label": "A", "text": "Every element $x\\in M_n$ admits a unique decomposition\n\\[\nx=\\sum_{k=0}^n\\ \\sum_{0\\le i_k<\\cdots<i_1<n} s_{n-1,i_1}\\cdots s_{n-k,i_k}\\,x_{i_1\\ldots i_k},\n\\]\nwhere for each multi-index $(i_1,\\ldots,i_k)$ one has $x_{i_1\\ldots i_k}\\in N_{n-k}(M)$."}, "choices": [{"label": "B", "text": "Every element $x\\in M_n$ admits a decomposition\n\\[\nx=\\sum_{k=0}^n\\ \\sum_{0\\le i_k\\le\\cdots\\le i_1<n} s_{n-1,i_1}\\cdots s_{n-k,i_k}\\,x_{i_1\\ldots i_k},\n\\]\nwhere for each multi-index $(i_1,\\ldots,i_k)$ one has $x_{i_1\\ldots i_k}\\in N_{n-k}(M)$, and this decomposition is unique."}, {"label": "C", "text": "Every element $x\\in M_n$ can be written in the form\n\\[\nx=\\sum_{k=0}^n\\ \\sum_{0\\le i_k<\\cdots<i_1<n} s_{n-1,i_1}\\cdots s_{n-k,i_k}\\,x_{i_1\\ldots i_k},\n\\]\nwhere for each multi-index $(i_1,\\ldots,i_k)$ one has $x_{i_1\\ldots i_k}\\in N_{n-k}(M)$."}, {"label": "D", "text": "For each $n\\ge 0$ there exists a choice of normalized coefficients $x_{i_1\\ldots i_k}\\in N_{n-k}(M)$, depending only on $n$ and not on the particular element $x\\in M_n$, such that every element $x\\in M_n$ admits a unique decomposition\n\\[\nx=\\sum_{k=0}^n\\ \\sum_{0\\le i_k<\\cdots<i_1<n} s_{n-1,i_1}\\cdots s_{n-k,i_k}\\,x_{i_1\\ldots i_k}.\n\\]"}, {"label": "E", "text": "Every element $x\\in M_n$ admits a unique decomposition\n\\[\nx=\\sum_{k=0}^n\\ \\sum_{0\\le i_k<\\cdots<i_1<n} s_{n-1,i_1}\\cdots s_{n-k,i_k}\\,x_{i_1\\ldots i_k},\n\\]\nwhere for each multi-index $(i_1,\\ldots,i_k)$ one has $x_{i_1\\ldots i_k}\\in M_{n-k}$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "strictly increasing multi-index condition", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "uniqueness of the decomposition", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence of coefficients on the chosen element x", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "normalized-chain condition on coefficients", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem only defines the projector-like maps p_m and normalized chains N_m(M); it does not state or strongly hint at the decomposition theorem itself. The correct answer is not leaked by wording in the prompt."}, "TAS": {"score": 1, "justification": "The item is very close to a theorem-recall question: it asks which exact decomposition statement holds. However, it is not a pure verbatim restatement, since the options vary important features such as uniqueness, strict index inequalities, and normalizedness of coefficients."}, "GPS": {"score": 1, "justification": "Some reasoning is required to distinguish the exact valid statement from nearby alternatives, especially between the weaker true existence statement and the stronger unique decomposition statement. Still, the question mainly tests recognition of the precise theorem rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic errors: allowing repeated indices, dropping uniqueness, confusing normalized coefficients with arbitrary ones, or making coefficients independent of x. They are distinct and well aligned with common failure modes."}, "total_score": 6, "overall_assessment": "A solid theorem-identification MCQ with strong distractors and no answer leakage, but it leans more toward precise recall/recognition than deep generative reasoning."}}
{"id": "2602.08398v2", "paper_link": "http://arxiv.org/abs/2602.08398v2", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{Th13}\nLet $2\\le p <\\infty,\\;0<s<1$ and $0\\le \\Lambda \\le 1$. Suppose $\\Omega \\in \\mathbb{H}^n$ is a bounded open set and $u \\in HW_{{\\rm{loc}}}^{1,p}\\left( \\Omega \\right) \\cap{ L_{sp}^{p - 1}\\left( {{\\mathbb{H}^n}} \\right)} $ is a weak solution of \\eqref{eq0}. Then $u \\in C_{loc}^\\gamma \\left( \\Omega  \\right)$ for any $0 < \\gamma  < 1$. Moreover, for any $0 < \\gamma  < 1$ and any Kor\\'{a}nyi ball ${B_{2R}}\\left( {{\\xi _0}} \\right) \\subset  \\subset \\Omega $ with $0<R<1$, there exists a positive constant $c=c(Q,p,s,\\gamma)$ such that\n\\begin{equation}\\label{eq482}\n  \\mathop {\\sup }\\limits_{\\xi  \\ne \\eta  \\in {B_{\\frac{R}{2}}}\\left( {{\\xi _0}} \\right)} \\frac{{\\left| {u\\left( \\xi  \\right) - u\\left( \\eta  \\right)} \\right|}}{{{{\\left\\| {{\\eta ^{ - 1}} \\circ \\xi } \\right\\|}^\\gamma _{\\mathbb{H}^n}}}} \\le \\frac{c}{{{R^\\gamma }}}\\left( {{{{\\left\\| u \\right\\|}_{{L^\\infty }\\left( {{B_R}\\left( {{\\xi _0}} \\right)} \\right)}}} + {\\rm{Tai}}{{\\rm{l}}_{p - 1,sp,p}}\\left( {u;{\\xi _0},R} \\right)} \\right),\n\\end{equation}\nwhere the definition of ${\\rm{Tai}}{{\\rm{l}}_{p - 1,sp,p}}\\left( {u;{\\xi _0},R} \\right)$ see \\eqref{eq26} below.", "start_pos": 9453, "end_pos": 10615, "label": "Th13"}, "ref_dict": {"Pro41": "\\begin{proposition}\\label{Pro41}\nLet $2\\le p <\\infty,\\;0<s<1$ and $0\\le \\Lambda \\le 1$. Assume $u \\in HW_{{\\rm{loc}}}^{1,p}\\left( {{B_2}\\left( {{\\xi _0}} \\right) \\cap L_{sp}^{p - 1}\\left( {{\\mathbb{H}^n}} \\right)} \\right)$ is a weak solution of \\eqref{eq0} in ${{B_2}\\left( {{\\xi _0}} \\right)}$ satisfying\n\\begin{equation}\\label{eq41}\n{\\left\\| u \\right\\|_{{L^\\infty }\\left( {{B_1}\\left( {{\\xi _0}} \\right)} \\right)}} \\le 1,\\int_{{\\mathbb{H}^n}\\backslash {B_1}\\left( {{\\xi _0}} \\right)} {\\frac{{{{\\left| {u\\left( \\eta  \\right)} \\right|}^{p - 1}}}}{{\\left\\| \\eta  \\right\\|_{{\\mathbb{H}^n}}^{Q + sp}}}d\\eta }  \\le 1.\n\\end{equation}\nIf $0<h_0<\\frac{1}{10}$ and $R$ satisfy $4{h_0}<R\\le 1-5{h_0}$, and ${\\nabla _H}u \\in {L^q}\\left( {{B_{R + 4{h_0}}}\\left( {{\\xi _0}} \\right)} \\right)$ for some $q \\ge p$, then\n\\begin{equation}\\label{eq42}\n \\mathop {\\sup }\\limits_{0 < \\left| h \\right| < {h_0}} \\int_{{B_{R - {4h_0}}}} {{{\\left| {\\frac{{\\Delta _{Z,h}^2u}}{{{{\\left| h \\right|}^{\\frac{1}{{q + 1}} + 1}}}}} \\right|}^{q + 1}}d\\xi } \\le c(1+\\Lambda) \\left(  {\\int_{{B_{R + {4h_0}}}} {{{\\left| {{\\nabla _H}u} \\right|}^q}d\\xi } + 1} \\right),\n\\end{equation}\nwhere $c=c(Q,h_0,p,q,s)>0$.\n\\end{proposition}", "eq0": "\\begin{equation}\\label{eq0}\n - \\Delta _{{{\\mathbb{H}},p}}u + \\Lambda{\\left( { - \\Delta _{{{\\mathbb{H}},p}}} \\right)^s}u = 0\\;{\\rm{in}}\\;\\Omega ,\n\\end{equation}", "Pro42": "\\begin{proposition}\\label{Pro42}\nLet $2\\le p <\\infty,\\;0<s<1$ and $0\\le \\Lambda \\le 1$. If $u \\in HW_{{\\rm{loc}}}^{1,p}\\left( {{B_2}\\left( {{\\xi _0}} \\right) \\cap L_{sp}^{p - 1}\\left( {{\\mathbb{H}^n}} \\right)} \\right)$ is a weak solution of \\eqref{eq0} in ${{B_{2}}\\left( {{\\xi _0}} \\right)}$ satisfying \\eqref{eq41}, then\n\\begin{equation*}\n\\int_{{B_{\\frac{7}{8}}}\\left( {{\\xi _0}} \\right)} {{{\\left| {{\\nabla _H}u} \\right|}^p}d\\xi }  \\le C\\left( {Q,p,s} \\right).\n\\end{equation*}\n\\end{proposition}", "eq26": "\\begin{equation}\\label{eq26}\n {\\rm{Tai}}{{\\rm{l}}_{q,\\alpha ,\\beta }}\\left( {u;{\\xi _0},R} \\right) = {\\left[ {{R^\\beta }\\int_{{{\\mathbb{H}}^n}\\backslash {B_R}\\left( {{\\xi _0}} \\right)} {\\frac{{{{\\left| u \\right|}^q}}}{{{{\\left\\| {\\xi _0^{ - 1} \\circ \\xi } \\right\\|}^{Q + \\alpha }_{\\mathbb{H}^n}}}}d\\xi } } \\right]^{\\frac{1}{q}}}.\n\\end{equation}", "Th13": "\\begin{theorem}\\label{Th13}\nLet $2\\le p <\\infty,\\;0<s<1$ and $0\\le \\Lambda \\le 1$. Suppose $\\Omega \\in \\mathbb{H}^n$ is a bounded open set and $u \\in HW_{{\\rm{loc}}}^{1,p}\\left( \\Omega \\right) \\cap{ L_{sp}^{p - 1}\\left( {{\\mathbb{H}^n}} \\right)} $ is a weak solution of \\eqref{eq0}. Then $u \\in C_{loc}^\\gamma \\left( \\Omega  \\right)$ for any $0 < \\gamma  < 1$. Moreover, for any $0 < \\gamma  < 1$ and any Kor\\'{a}nyi ball ${B_{2R}}\\left( {{\\xi _0}} \\right) \\subset  \\subset \\Omega $ with $0<R<1$, there exists a positive constant $c=c(Q,p,s,\\gamma)$ such that\n\\begin{equation}\\label{eq482}\n  \\mathop {\\sup }\\limits_{\\xi  \\ne \\eta  \\in {B_{\\frac{R}{2}}}\\left( {{\\xi _0}} \\right)} \\frac{{\\left| {u\\left( \\xi  \\right) - u\\left( \\eta  \\right)} \\right|}}{{{{\\left\\| {{\\eta ^{ - 1}} \\circ \\xi } \\right\\|}^\\gamma _{\\mathbb{H}^n}}}} \\le \\frac{c}{{{R^\\gamma }}}\\left( {{{{\\left\\| u \\right\\|}_{{L^\\infty }\\left( {{B_R}\\left( {{\\xi _0}} \\right)} \\right)}}} + {\\rm{Tai}}{{\\rm{l}}_{p - 1,sp,p}}\\left( {u;{\\xi _0},R} \\right)} \\right),\n\\end{equation}\nwhere the definition of ${\\rm{Tai}}{{\\rm{l}}_{p - 1,sp,p}}\\left( {u;{\\xi _0},R} \\right)$ see \\eqref{eq26} below.\n\\end{theorem}", "Th14": "\\begin{theorem}\\label{Th14}\nLet $2\\le p <\\infty,\\;0<s<1,\\;sp<p-1$ and $0\\le \\Lambda \\le 1$. Suppose $\\Omega \\in \\mathbb{H}^n$ is a bounded open set and $u \\in HW_{{\\rm{loc}}}^{1,p}\\left( \\Omega \\right) \\cap{ L_{sp}^{p - 1}\\left( {{\\mathbb{H}^n}} \\right)} $ is a weak solution of \\eqref{eq0}. Then $u \\in C_{loc}^{1,\\alpha} \\left( \\Omega  \\right)$ for some $0 < \\alpha  < 1$. Moreover, for some $0 < \\alpha  < 1$ and any Kor\\'{a}nyi ball ${B_{2R}}\\left( {{\\xi _0}} \\right) \\subset  \\subset \\Omega $ with $0<R<1$, there exists a positive constant $c=c(Q,p,q,s,\\gamma)$ such that\n\\begin{equation}\\label{eq51}\n \\mathop {\\sup }\\limits_{\\xi  \\ne \\eta  \\in {B_{\\frac{R}{8}}}\\left( {{\\xi _0}} \\right)} \\frac{{\\left| {{\\nabla _H}u\\left( \\xi  \\right) - {\\nabla _H}u\\left( \\eta  \\right)} \\right|}}{{{{\\left\\| {{\\eta ^{ - 1}} \\circ \\xi } \\right\\|}^\\alpha _{\\mathbb{H}^n}}}} \\le \\frac{c}{{{R^{1 + \\alpha }}}}\\left( {{{\\left\\| u \\right\\|}_{{L^\\infty }\\left( {{B_R}\\left( {{\\xi _0}} \\right)} \\right)}} + {\\rm{Tai}}{{\\rm{l}}_{p - 1,sp,p}}\\left( {u;{\\xi _0},R} \\right)} \\right),\n\\end{equation}\nwhere the definition of ${\\rm{Tai}}{{\\rm{l}}_{p - 1,sp,p}}\\left( {u;{\\xi _0},R} \\right)$ see \\eqref{eq26} below.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 6910, "pre_theorem_intro_text": "\\label{sec-1}\n\nThe study of regularity theory for degenerate elliptic equations has long stood as a cornerstone of modern analysis, tracing its intellectual lineage to the groundbreaking contributions of De Giorgi, Nash, and Moser. In the context of the Heisenberg group $\\mathbb{H}^n$, a paradigmatic example of a sub-Riemannian manifold, the analysis of quasilinear degenerate elliptic equations of $p$-sub-Laplacian type has undergone profound development. In the pioneering work \\cite{CL97}, Capogna established $C^{1,\\alpha}$-regularity for the case $p=2$. Subsequently, Mukherjee and Zhong\\cite{MZ21} extended this result to quasilinear degenerate elliptic equations with $p$-growth conditions for all $1<p<\\infty$ via De Giorgi's method. Further progress includes: fractional differentiability estimates for nonlinear non-differentiable degenerate elliptic equations obtained by Zhang and Niu\\cite{ZN22}; ${C^{\\alpha }}$-regularity of minimizers for generalized Orlicz functionals encompassing $p,q$-growth conditions established by Zhang and Niu\\cite{ZN20}; weak differentiability of weak solutions under the constraint $1<p<4$ for quasilinear equations with $p,q$-growth conditions proved by Zhang and Li\\cite{ZL23}. A sufficient condition excluding the Lavrentiev phenomenon for non-autonomous integral functionals with $p,q$-growth condition in the same setting provided by Zhang and Niu\\cite{ZN231}. For additional developments and comprehensive references, we refer the reader to \\cite{ZN22} and the works cited therein.\n\nThe regularity theory for purely nonlocal equations involving fractional powers of the $p$-Laplacian on the Heisenberg group has witnessed remarkable progress in recent years, spurred by pivotal advancements such as the introduction of nonlocal tailing mechanisms and the establishment of nonlocal Harnack and H\\\"{o}lder regularity estimates. These contributions, as reported in Roncal and Thangavelu\\cite{RT16}, Frank, Lieb, and Seiringer\\cite{FLS08}, Ferrari and Franchi\\cite{FF15}, Ferrari, Miranda, Pallara, Pinamonti and Sire\\cite{FMPPS18}, Frank, Gonzalez, Monticelli and Tan\\cite{FGMT15}, Manfredini, Palatucci, Piccinini and Polidoro\\cite{MPPP23}, and Palatucci and Piccinini\\cite{PP22}, collectively constitute a transformative body of work reshaping the analytical foundations of the field. Moreover, Palatucci and Piccinini in 2022 (see \\cite{PP22}) and 2023 (see \\cite{PP23}) proposed an open problem how to determine the regularity of nonlocal double-phase equations in the Heisenberg group. Fang, Zhang and Zhang provided a solution in \\cite{FZZ24}.\n\nHowever, the analysis of equations combining local and nonlocal operators-referred to as \"mixed\" equations-presents a formidable challenge due to the intricate interplay between classical diffusion processes and long-range interactions. While the Euclidean setting has witnessed rapid progress, with results ranging from local boundedness (see Garain and Kinnunen\\cite{GK22}) to higher-order H\\\"{o}lder estimates (see Garain and Lindgren\\cite{GL23}), extending such theories to the non-commutative geometry of the Heisenberg group demands overcoming profound technical obstacles, including the absence of embedding theorems and the inherent anisotropy of the Carnot-Carath\\'{e}odory metric.\n\nRecently, Zhang, Niu and Wu in \\cite{ZNW25} proved that both the derivative in the vertical direction and the derivative in the horizontal direction of weak solutions $u \\in H{W^{1,p}}\\left( \\Omega  \\right)$ to mixed sub-Laplace and fractional sub-Laplace equations and $f \\in {L^2}\\left( \\Omega  \\right)$ belonging to the first-order local Sobolev space $HW_{loc}^{1,p}\\left( \\Omega  \\right)$. Furthermore, under the assumption $f \\in H{W^{m,p}}\\left( \\Omega  \\right)$, it follows that $u \\in HW_{loc}^{m + 2,p}\\left( \\Omega  \\right)$. Subsequently, Zhang and Niu\\cite{ZN26} achieved a significant breakthrough by establishing fundamental regularity properties for weak solutions to mixed local and nonlocal degenerate elliptic equations in $\\mathbb{H}^n$. Specifically, they demonstrated the local boundedness of weak subsolutions, the H\\\"{o}lder continuity of weak solutions, and critically, both the Harnack inequality and the weak Harnack inequality. These results affirm that the classical difference and De Giorgi-Nash-Moser theory can be fruitfully extended to this hybrid setting, laying a robust foundation for the qualitative analysis of such equations. While H\\\"{o}lder continuity of the solution $u$ has been established, a central open question remains: does the horizontal gradient ${{\\nabla _H}u}$ exhibit H\\\"{o}lder continuity, i.e., do solutions belong to the $C^{1,\\alpha}$ class? This problem is exceptionally intricate in the context of degenerate ($p>2$) equations coupled with mixed operators, as standard linearization techniques prove inadequate, and the interplay between the local $p$-Laplacian structure and the nonlocal fractional term obfuscates the differentiation of the equation.\n\nIn this paper, we bridge this gap by investigating the $C^{1,\\alpha}$-regularity of weak solutions to mixed local and nonlocal degenerate elliptic equations in the Heisenberg group. Building on the Harnack estimates and tail control developed in \\cite{ZN26}, we aim to prove that under suitably structured assumptions, weak solutions not only exhibit H\\\"{o}lder continuity but also possess higher smoothness properties with their horizontal derivatives satisfying a H\\\"{o}lder condition. This result will extend the classical $C^{1,\\alpha}$ estimates for $p$-sub-Laplace equations in $\\mathbb{H}^n$ to the more intricate landscape of mixed local-nonlocal diffusion.\n\nIn this paper we focus on the analysis of the mixed local and nonlocal degenerate elliptic equations\n\\begin{equation}\\label{eq0}\n - \\Delta _{{{\\mathbb{H}},p}}u + \\Lambda{\\left( { - \\Delta _{{{\\mathbb{H}},p}}} \\right)^s}u = 0\\;{\\rm{in}}\\;\\Omega ,\n\\end{equation}\nwhere $2\\le p< \\infty,\\;0<s<1,\\;0\\le\\Lambda \\le 1 $, and $\\Omega$ is a bounded open subset in the Heisenberg group  ${{\\mathbb{H}}^n},\\;n\\ge1$. Here the $p$-sub-Laplace operator on the Heisenberg group is defined by\n\\begin{equation}\\label{eq12}\n - {\\Delta _{{\\mathbb{H}},p}}u = di{v_H}\\left( {{{\\left| {{\\nabla _H}u} \\right|}^{p - 2}}{\\nabla _H}u} \\right),\n\\end{equation}\nwhile the fractional $p$-sub-Laplace operator on the Heisenberg group is given by\n\\begin{equation}\\label{eq13}\n{\\left( { - {\\Delta _{{\\mathbb{H}},p}}} \\right)^s}u = {\\rm{P}}{\\rm{.V}}{\\rm{.}}\\int_{{{\\mathbb{H}}^n}} {\\frac{{{{\\left| {u\\left( \\xi  \\right) - u\\left( \\eta  \\right)} \\right|}^{p - 2}}\\left( {u\\left( \\xi  \\right) - u\\left( \\eta  \\right)} \\right)}}{{{{\\left\\| {{\\eta ^{ - 1}} \\circ \\xi } \\right\\|}^{Q + sp}_{\\mathbb{H}^n}}}}d\\eta },\n\\end{equation}\nwhere P.V. signifies the Cauchy principal value and $Q=2n+2$ is the homogeneous dimension of ${{\\mathbb{H}}^n}$.\n\nThe main results are as follows:", "context": "The regularity theory for purely nonlocal equations involving fractional powers of the $p$-Laplacian on the Heisenberg group has witnessed remarkable progress in recent years, spurred by pivotal advancements such as the introduction of nonlocal tailing mechanisms and the establishment of nonlocal Harnack and H\\\"{o}lder regularity estimates. These contributions, as reported in Roncal and Thangavelu\\cite{RT16}, Frank, Lieb, and Seiringer\\cite{FLS08}, Ferrari and Franchi\\cite{FF15}, Ferrari, Miranda, Pallara, Pinamonti and Sire\\cite{FMPPS18}, Frank, Gonzalez, Monticelli and Tan\\cite{FGMT15}, Manfredini, Palatucci, Piccinini and Polidoro\\cite{MPPP23}, and Palatucci and Piccinini\\cite{PP22}, collectively constitute a transformative body of work reshaping the analytical foundations of the field. Moreover, Palatucci and Piccinini in 2022 (see \\cite{PP22}) and 2023 (see \\cite{PP23}) proposed an open problem how to determine the regularity of nonlocal double-phase equations in the Heisenberg group. Fang, Zhang and Zhang provided a solution in \\cite{FZZ24}.\n\nRecently, Zhang, Niu and Wu in \\cite{ZNW25} proved that both the derivative in the vertical direction and the derivative in the horizontal direction of weak solutions $u \\in H{W^{1,p}}\\left( \\Omega \\right)$ to mixed sub-Laplace and fractional sub-Laplace equations and $f \\in {L^2}\\left( \\Omega \\right)$ belonging to the first-order local Sobolev space $HW_{loc}^{1,p}\\left( \\Omega \\right)$. Furthermore, under the assumption $f \\in H{W^{m,p}}\\left( \\Omega \\right)$, it follows that $u \\in HW_{loc}^{m + 2,p}\\left( \\Omega \\right)$. Subsequently, Zhang and Niu\\cite{ZN26} achieved a significant breakthrough by establishing fundamental regularity properties for weak solutions to mixed local and nonlocal degenerate elliptic equations in $\\mathbb{H}^n$. Specifically, they demonstrated the local boundedness of weak subsolutions, the H\\\"{o}lder continuity of weak solutions, and critically, both the Harnack inequality and the weak Harnack inequality. These results affirm that the classical difference and De Giorgi-Nash-Moser theory can be fruitfully extended to this hybrid setting, laying a robust foundation for the qualitative analysis of such equations. While H\\\"{o}lder continuity of the solution $u$ has been established, a central open question remains: does the horizontal gradient ${{\\nabla _H}u}$ exhibit H\\\"{o}lder continuity, i.e., do solutions belong to the $C^{1,\\alpha}$ class? This problem is exceptionally intricate in the context of degenerate ($p>2$) equations coupled with mixed operators, as standard linearization techniques prove inadequate, and the interplay between the local $p$-Laplacian structure and the nonlocal fractional term obfuscates the differentiation of the equation.\n\nIn this paper, we bridge this gap by investigating the $C^{1,\\alpha}$-regularity of weak solutions to mixed local and nonlocal degenerate elliptic equations in the Heisenberg group. Building on the Harnack estimates and tail control developed in \\cite{ZN26}, we aim to prove that under suitably structured assumptions, weak solutions not only exhibit H\\\"{o}lder continuity but also possess higher smoothness properties with their horizontal derivatives satisfying a H\\\"{o}lder condition. This result will extend the classical $C^{1,\\alpha}$ estimates for $p$-sub-Laplace equations in $\\mathbb{H}^n$ to the more intricate landscape of mixed local-nonlocal diffusion.\n\nIn this paper we focus on the analysis of the mixed local and nonlocal degenerate elliptic equations\n\\begin{equation}\\label{eq0}\n - \\Delta _{{{\\mathbb{H}},p}}u + \\Lambda{\\left( { - \\Delta _{{{\\mathbb{H}},p}}} \\right)^s}u = 0\\;{\\rm{in}}\\;\\Omega ,\n\\end{equation}\nwhere $2\\le p< \\infty,\\;0<s<1,\\;0\\le\\Lambda \\le 1 $, and $\\Omega$ is a bounded open subset in the Heisenberg group ${{\\mathbb{H}}^n},\\;n\\ge1$. Here the $p$-sub-Laplace operator on the Heisenberg group is defined by\n\\begin{equation}\\label{eq12}\n - {\\Delta _{{\\mathbb{H}},p}}u = di{v_H}\\left( {{{\\left| {{\\nabla _H}u} \\right|}^{p - 2}}{\\nabla _H}u} \\right),\n\\end{equation}\nwhile the fractional $p$-sub-Laplace operator on the Heisenberg group is given by\n\\begin{equation}\\label{eq13}\n{\\left( { - {\\Delta _{{\\mathbb{H}},p}}} \\right)^s}u = {\\rm{P}}{\\rm{.V}}{\\rm{.}}\\int_{{{\\mathbb{H}}^n}} {\\frac{{{{\\left| {u\\left( \\xi \\right) - u\\left( \\eta \\right)} \\right|}^{p - 2}}\\left( {u\\left( \\xi \\right) - u\\left( \\eta \\right)} \\right)}}{{{{\\left\\| {{\\eta ^{ - 1}} \\circ \\xi } \\right\\|}^{Q + sp}_{\\mathbb{H}^n}}}}d\\eta },\n\\end{equation}\nwhere P.V. signifies the Cauchy principal value and $Q=2n+2$ is the homogeneous dimension of ${{\\mathbb{H}}^n}$.\n\nThe main results are as follows:", "full_context": "The regularity theory for purely nonlocal equations involving fractional powers of the $p$-Laplacian on the Heisenberg group has witnessed remarkable progress in recent years, spurred by pivotal advancements such as the introduction of nonlocal tailing mechanisms and the establishment of nonlocal Harnack and H\\\"{o}lder regularity estimates. These contributions, as reported in Roncal and Thangavelu\\cite{RT16}, Frank, Lieb, and Seiringer\\cite{FLS08}, Ferrari and Franchi\\cite{FF15}, Ferrari, Miranda, Pallara, Pinamonti and Sire\\cite{FMPPS18}, Frank, Gonzalez, Monticelli and Tan\\cite{FGMT15}, Manfredini, Palatucci, Piccinini and Polidoro\\cite{MPPP23}, and Palatucci and Piccinini\\cite{PP22}, collectively constitute a transformative body of work reshaping the analytical foundations of the field. Moreover, Palatucci and Piccinini in 2022 (see \\cite{PP22}) and 2023 (see \\cite{PP23}) proposed an open problem how to determine the regularity of nonlocal double-phase equations in the Heisenberg group. Fang, Zhang and Zhang provided a solution in \\cite{FZZ24}.\n\nRecently, Zhang, Niu and Wu in \\cite{ZNW25} proved that both the derivative in the vertical direction and the derivative in the horizontal direction of weak solutions $u \\in H{W^{1,p}}\\left( \\Omega \\right)$ to mixed sub-Laplace and fractional sub-Laplace equations and $f \\in {L^2}\\left( \\Omega \\right)$ belonging to the first-order local Sobolev space $HW_{loc}^{1,p}\\left( \\Omega \\right)$. Furthermore, under the assumption $f \\in H{W^{m,p}}\\left( \\Omega \\right)$, it follows that $u \\in HW_{loc}^{m + 2,p}\\left( \\Omega \\right)$. Subsequently, Zhang and Niu\\cite{ZN26} achieved a significant breakthrough by establishing fundamental regularity properties for weak solutions to mixed local and nonlocal degenerate elliptic equations in $\\mathbb{H}^n$. Specifically, they demonstrated the local boundedness of weak subsolutions, the H\\\"{o}lder continuity of weak solutions, and critically, both the Harnack inequality and the weak Harnack inequality. These results affirm that the classical difference and De Giorgi-Nash-Moser theory can be fruitfully extended to this hybrid setting, laying a robust foundation for the qualitative analysis of such equations. While H\\\"{o}lder continuity of the solution $u$ has been established, a central open question remains: does the horizontal gradient ${{\\nabla _H}u}$ exhibit H\\\"{o}lder continuity, i.e., do solutions belong to the $C^{1,\\alpha}$ class? This problem is exceptionally intricate in the context of degenerate ($p>2$) equations coupled with mixed operators, as standard linearization techniques prove inadequate, and the interplay between the local $p$-Laplacian structure and the nonlocal fractional term obfuscates the differentiation of the equation.\n\nIn this paper, we bridge this gap by investigating the $C^{1,\\alpha}$-regularity of weak solutions to mixed local and nonlocal degenerate elliptic equations in the Heisenberg group. Building on the Harnack estimates and tail control developed in \\cite{ZN26}, we aim to prove that under suitably structured assumptions, weak solutions not only exhibit H\\\"{o}lder continuity but also possess higher smoothness properties with their horizontal derivatives satisfying a H\\\"{o}lder condition. This result will extend the classical $C^{1,\\alpha}$ estimates for $p$-sub-Laplace equations in $\\mathbb{H}^n$ to the more intricate landscape of mixed local-nonlocal diffusion.\n\nIn this paper we focus on the analysis of the mixed local and nonlocal degenerate elliptic equations\n\\begin{equation}\\label{eq0}\n - \\Delta _{{{\\mathbb{H}},p}}u + \\Lambda{\\left( { - \\Delta _{{{\\mathbb{H}},p}}} \\right)^s}u = 0\\;{\\rm{in}}\\;\\Omega ,\n\\end{equation}\nwhere $2\\le p< \\infty,\\;0<s<1,\\;0\\le\\Lambda \\le 1 $, and $\\Omega$ is a bounded open subset in the Heisenberg group ${{\\mathbb{H}}^n},\\;n\\ge1$. Here the $p$-sub-Laplace operator on the Heisenberg group is defined by\n\\begin{equation}\\label{eq12}\n - {\\Delta _{{\\mathbb{H}},p}}u = di{v_H}\\left( {{{\\left| {{\\nabla _H}u} \\right|}^{p - 2}}{\\nabla _H}u} \\right),\n\\end{equation}\nwhile the fractional $p$-sub-Laplace operator on the Heisenberg group is given by\n\\begin{equation}\\label{eq13}\n{\\left( { - {\\Delta _{{\\mathbb{H}},p}}} \\right)^s}u = {\\rm{P}}{\\rm{.V}}{\\rm{.}}\\int_{{{\\mathbb{H}}^n}} {\\frac{{{{\\left| {u\\left( \\xi \\right) - u\\left( \\eta \\right)} \\right|}^{p - 2}}\\left( {u\\left( \\xi \\right) - u\\left( \\eta \\right)} \\right)}}{{{{\\left\\| {{\\eta ^{ - 1}} \\circ \\xi } \\right\\|}^{Q + sp}_{\\mathbb{H}^n}}}}d\\eta },\n\\end{equation}\nwhere P.V. signifies the Cauchy principal value and $Q=2n+2$ is the homogeneous dimension of ${{\\mathbb{H}}^n}$.\n\nThe main results are as follows:\n\n\\begin{remark}\nIn contrast to \\cite{ZN26}, where the H\\\"older index $\\gamma \\in (0,\\frac{sp}{p-1})$ relies on the parameters $s$ and $p$, Theorem \\ref{Th13} establishes a parameter-independent H\\\"older regularity with $\\gamma \\in (0,1)$. This holds under more general nonlinear frameworks and is particularly advantageous for subsequent regularity analysis.\n\\end{remark}\n\n\\begin{proposition}\\label{Pro41}\nLet $2\\le p <\\infty,\\;0<s<1$ and $0\\le \\Lambda \\le 1$. Assume $u \\in HW_{{\\rm{loc}}}^{1,p}\\left( {{B_2}\\left( {{\\xi _0}} \\right) \\cap L_{sp}^{p - 1}\\left( {{\\mathbb{H}^n}} \\right)} \\right)$ is a weak solution of \\eqref{eq0} in ${{B_2}\\left( {{\\xi _0}} \\right)}$ satisfying\n\\begin{equation}\\label{eq41}\n{\\left\\| u \\right\\|_{{L^\\infty }\\left( {{B_1}\\left( {{\\xi _0}} \\right)} \\right)}} \\le 1,\\int_{{\\mathbb{H}^n}\\backslash {B_1}\\left( {{\\xi _0}} \\right)} {\\frac{{{{\\left| {u\\left( \\eta \\right)} \\right|}^{p - 1}}}}{{\\left\\| \\eta \\right\\|_{{\\mathbb{H}^n}}^{Q + sp}}}d\\eta } \\le 1.\n\\end{equation}\nIf $0<h_0<\\frac{1}{10}$ and $R$ satisfy $4{h_0}<R\\le 1-5{h_0}$, and ${\\nabla _H}u \\in {L^q}\\left( {{B_{R + 4{h_0}}}\\left( {{\\xi _0}} \\right)} \\right)$ for some $q \\ge p$, then\n\\begin{equation}\\label{eq42}\n \\mathop {\\sup }\\limits_{0 < \\left| h \\right| < {h_0}} \\int_{{B_{R - {4h_0}}}} {{{\\left| {\\frac{{\\Delta _{Z,h}^2u}}{{{{\\left| h \\right|}^{\\frac{1}{{q + 1}} + 1}}}}} \\right|}^{q + 1}}d\\xi } \\le c(1+\\Lambda) \\left( {\\int_{{B_{R + {4h_0}}}} {{{\\left| {{\\nabla _H}u} \\right|}^q}d\\xi } + 1} \\right),\n\\end{equation}\nwhere $c=c(Q,h_0,p,q,s)>0$.\n\\end{proposition}\n\nFor simplicity, we denote $u_R$ by $u$. Fix $0<\\gamma<1$ and choose $i_\\infty \\in \\mathbb{N}\\setminus {0}$ such that\n\\begin{equation}\\label{eq492}\n1 - \\gamma > \\frac{Q}{{p + {i_\\infty }}}.\n\\end{equation}\nFor $i = 0, \\cdots ,{i_\\infty }$, we define\n\\[{q_i} = p + i,\\]\nand\n\\[{h_0} = \\frac{1}{{112{i_\\infty }}},\\;{R_i} = \\frac{7}{8} - 4{h_0} - 14{h_0}i,\\]\nso\n\\[{R_0} + 4{h_0} = \\frac{7}{8},\\;{R_{{i_\\infty }}} + 4{h_0} = \\frac{3}{4},\\;4{h_0} <R_i<1-5{h_0} .\\]\nBy using Proposition \\ref{Pro41} with\n\\[R = {R_i},q = {q_i},\\;i = 0, \\cdots ,{i_\\infty },\\]\nand we have by \\eqref{eq485}\n\\begin{equation}\\label{eq486}\n\\mathop {\\sup }\\limits_{0 < \\left| h \\right| < {h_0}} \\int_{{B_{R_0 - {4h_0}}}} {{{\\left| {\\frac{{\\Delta _{Z,h}^2u}}{{{{\\left| h \\right|}^{\\frac{1}{{{q_1}}} + 1}}}}} \\right|}^{{q_1}}}d\\xi } \\le c(1 + \\Lambda )\\left( {\\int_{{B_{\\frac{7}{8}}}} {{{\\left| {{\\nabla _H}u} \\right|}^p}d\\xi } + 1} \\right) \\le C\\left( {Q,p,s,\\gamma} \\right).\n\\end{equation}\nConsider ${R_i} - 10{h_0} = {R_{i + 1}} + 4{h_0}$ for $i = 0, \\cdots ,{i_\\infty }$, we use Lemma \\ref{Le213} in \\eqref{eq486} to get\n\\begin{equation}\\label{eq487}\n\\int_{{B_{{R_1} + 4{h_0}}}} {{{\\left| {{\\nabla _H}{u}} \\right|}^{{q_1}}}d\\xi } \\le C\\left( {Q,p,s,\\gamma} \\right).\n\\end{equation}\nApplying Proposition \\ref{Pro41} and \\eqref{eq487} once again, we obtain\n\\begin{equation}\\label{eq488}\n\\mathop {\\sup }\\limits_{0 < \\left| h \\right| < {h_0}} \\int_{{B_{{R_1} - 4{h_0}}}} {{{\\left| {\\frac{{\\Delta _{Z,h}^2u}}{{{{\\left| h \\right|}^{\\frac{1}{{{q_2}}} + 1}}}}} \\right|}^{{q_2}}}d\\xi } \\le c(1 + \\Lambda )\\left( {\\int_{{B_{{{{R_1} + 4{h_0}}}}}} {{{\\left| {{\\nabla _H}u} \\right|}^{{q_1}}}d\\xi } + 1} \\right) \\le C\\left( {Q,p,s,\\gamma} \\right).\n\\end{equation}\nMoreover, by Lemma \\ref{Le213} in \\eqref{eq488}, we have\n\\begin{equation}\\label{eq489}\n\\int_{{B_{{R_2} + 4{h_0}}}} {{{\\left| {{\\nabla _H}{u}} \\right|}^{{q_2}}}d\\xi } \\le C\\left( {Q,p,s,\\gamma} \\right).\n\\end{equation}\nRepeating this procedure, it yields\n\\begin{equation}\\label{eq490}\n \\int_{{B_{{R_{i+1}} + 4{h_0}}}} {{{\\left| {{\\nabla _H}{u}} \\right|}^{{q_{i+1}}}}d\\xi } \\le C\\left( {Q,p,s,\\gamma} \\right)\n\\end{equation}\nfor all $i = 0, \\cdots ,{i_\\infty }-1.$ Taking $i={i_\\infty }-1$ in \\eqref{eq490} and using ${\\left\\| {{u}} \\right\\|_{{L^\\infty }\\left( {{B_1}} \\right)}} \\le 1$, we deduce\n\\begin{equation}\\label{eq491}\n{\\left\\| u \\right\\|_{H{W^{1,{q_{{i_\\infty }}}}}\\left( {{B_{{R_{{i_\\infty }}} + 4{h_0}}}} \\right)}} \\le C\\left( {Q,p,s,\\gamma } \\right).\n\\end{equation}\nBy means of \\eqref{eq492}, we know ${q_{{i_\\infty }}}>Q$, so we use ${R_{{i_\\infty }}} + 4{h_0} = \\frac{3}{4}$ and Lemma \\ref{Le23} to get\n$u \\in C_{loc}^\\gamma \\left( {{B_{\\frac{3}{4}}}} \\right)$. In fact, we take a cut-off function $\\psi$ between ${{B_{\\frac{1}{2}}}}$ and ${{B_{\\frac{3}{4}}}}$. Then\n\\begin{align}\\label{eq493}\n{\\left\\| u \\right\\|_{{C^\\gamma }\\left( {{B_{\\frac{1}{2}}}} \\right)}}& \\le {\\left\\| {u\\psi } \\right\\|_{{C^\\gamma }\\left( {{{\\mathbb{H}}^n}} \\right)}} \\le c{\\left\\| {u\\psi } \\right\\|_{H{W^{1,{q_{{i_\\infty }}}}}\\left( {{{\\mathbb{H}}^n}} \\right)}} = c{\\left\\| {u\\psi } \\right\\|_{H{W^{1,{q_{{i_\\infty }}}}}\\left( {{B_{\\frac{3}{4}}}} \\right)}}\\nonumber\\\\\n& \\le c{\\left\\| u \\right\\|_{H{W^{1,{q_{{i_\\infty }}}}}\\left( {{B_{\\frac{3}{4}}}} \\right)}} \\le C\\left( {Q,p,s,\\gamma } \\right).\n\\end{align}\nTherefore, Theorem is proved.\n\n\\textbf{Proof of Theorem \\ref{Th14}} After rescaling the variables as in the proof of Theorem \\ref{Th13}, it suffices to show that\n$${\\left\\| u_R \\right\\|_{{C^{1,\\alpha }}\\left( {{B_{\\frac{1}{8}}}} \\right)}} \\le c\\left( {Q,p,s,\\gamma } \\right),$$\nwhere the function $u_R$ defined in \\eqref{eq484} satisfies the following conditions:\n\\begin{equation}\\label{eq52}\n{\\left\\| {{u_R}} \\right\\|_{{L^\\infty }\\left( {{B_1}} \\right)}} \\le 1,\\;{\\int _{{\\mathbb{H}^n}\\backslash {B_1}}}\\frac{{{{\\left| {{u_R}\\left( \\eta \\right)} \\right|}^{p - 1}}}}{{\\left\\| \\eta \\right\\|_{{\\mathbb{H}^n}}^{Q + sp}}}d\\eta \\le 1.\n\\end{equation}\nBy Theorem \\ref{Th13}, there exists $\\gamma>\\frac{sp}{p-1}$ such that\n\\[\\mathop {\\sup }\\limits_{\\xi \\ne \\eta \\in {B_{\\frac{1}{2}}}\\left( {{\\xi _0}} \\right)} \\frac{{\\left| {{u_R}\\left( \\xi \\right) - {u_R}\\left( \\eta \\right)} \\right|}}{{{{\\left\\| {{\\eta ^{ - 1}} \\circ \\xi } \\right\\|}_{{\\mathbb{H}^n}}^\\gamma }}} \\le c\\left( {Q,p,s,\\gamma } \\right).\\]\nNow take any $\\xi_0 \\in {B_{\\frac{1}{4}}}$, then we deduce from the above formula and the choice of $\\gamma$ that\n\\[\\int_{{B_{\\frac{1}{4}}}\\left( {{\\xi _0}} \\right)} {\\frac{{{{\\left| {{u_R}\\left( {{\\xi _0}} \\right) - {u_R}\\left( \\eta \\right)} \\right|}^{p - 1}}}}{{{{\\left\\| {{\\eta ^{ - 1}} \\circ {\\xi _0}} \\right\\|}_{{\\mathbb{H}^n}}^{Q + sp}}}}d\\eta } \\le c\\left( {Q,p,s,\\gamma } \\right)\\int_{{B_{\\frac{1}{4}}}\\left( {{\\xi _0}} \\right)} {\\frac{1}{{{{\\left\\| {{\\eta ^{ - 1}} \\circ {\\xi _0}} \\right\\|}_{{\\mathbb{H}^n}}^{Q + sp - \\gamma \\left( {p - 1} \\right)}}}}d\\eta } = c\\left( {Q,p,s,\\gamma } \\right).\\]\nMoreover, we have by \\eqref{eq52}\n\\[\\int_{{\\mathbb{H}^n}\\backslash {B_{\\frac{1}{4}}}\\left( {{\\xi _0}} \\right)} {\\frac{{{{\\left| {{u_R}\\left( {{\\xi _0}} \\right) - {u_R}\\left( \\eta \\right)} \\right|}^{p - 1}}}}{{{{\\left\\| {{\\eta ^{ - 1}} \\circ {\\xi _0}} \\right\\|}_{{\\mathbb{H}^n}}^{Q + sp}}}}d\\eta } \\le c\\left( {Q,p,s} \\right).\\]\nHence\n\\[{\\left\\| {{{{{\\left( { - \\Delta _{{{\\mathbb{H}},p}}} \\right)}^s}}}{u_R}} \\right\\|_{{L^\\infty }\\left( {{B_{\\frac{1}{4}}}} \\right)}} \\le c\\left( {Q,p,s,\\gamma } \\right),\\]\nand also\n\\[{\\left\\| {{{{\\left( { - \\Delta _{{{\\mathbb{H}},p}}} \\right)}}}{u_R}} \\right\\|_{{L^\\infty }\\left( {{B_{\\frac{1}{4}}}} \\right)}} \\le c\\left( {Q,p,s,\\gamma } \\right),\\]\nwhich together with \\eqref{eq52} and Theorem 1.2 in \\cite{MZ21} imply\n\\[{\\left\\| {{u_R}} \\right\\|_{{C^{1,,\\alpha }}\\left( {{B_{\\frac{1}{8}}}} \\right)}} \\le c\\left( {Q,p,s,\\gamma } \\right).\\]", "post_theorem_intro_text_len": 2744, "post_theorem_intro_text": "\\begin{remark}\nIn contrast to \\cite{ZN26}, where the H\\\"older index $\\gamma \\in (0,\\frac{sp}{p-1})$ relies on the parameters $s$ and $p$, Theorem \\ref{Th13} establishes a parameter-independent H\\\"older regularity with $\\gamma \\in (0,1)$. This holds under more general nonlinear frameworks and is particularly advantageous for subsequent regularity analysis.\n\\end{remark}\n\n\\begin{theorem}\\label{Th14}\nLet $2\\le p <\\infty,\\;0<s<1,\\;sp<p-1$ and $0\\le \\Lambda \\le 1$. Suppose $\\Omega \\in \\mathbb{H}^n$ is a bounded open set and $u \\in HW_{{\\rm{loc}}}^{1,p}\\left( \\Omega \\right) \\cap{ L_{sp}^{p - 1}\\left( {{\\mathbb{H}^n}} \\right)} $ is a weak solution of \\eqref{eq0}. Then $u \\in C_{loc}^{1,\\alpha} \\left( \\Omega  \\right)$ for some $0 < \\alpha  < 1$. Moreover, for some $0 < \\alpha  < 1$ and any Kor\\'{a}nyi ball ${B_{2R}}\\left( {{\\xi _0}} \\right) \\subset  \\subset \\Omega $ with $0<R<1$, there exists a positive constant $c=c(Q,p,q,s,\\gamma)$ such that\n\\begin{equation}\\label{eq51}\n \\mathop {\\sup }\\limits_{\\xi  \\ne \\eta  \\in {B_{\\frac{R}{8}}}\\left( {{\\xi _0}} \\right)} \\frac{{\\left| {{\\nabla _H}u\\left( \\xi  \\right) - {\\nabla _H}u\\left( \\eta  \\right)} \\right|}}{{{{\\left\\| {{\\eta ^{ - 1}} \\circ \\xi } \\right\\|}^\\alpha _{\\mathbb{H}^n}}}} \\le \\frac{c}{{{R^{1 + \\alpha }}}}\\left( {{{\\left\\| u \\right\\|}_{{L^\\infty }\\left( {{B_R}\\left( {{\\xi _0}} \\right)} \\right)}} + {\\rm{Tai}}{{\\rm{l}}_{p - 1,sp,p}}\\left( {u;{\\xi _0},R} \\right)} \\right),\n\\end{equation}\nwhere the definition of ${\\rm{Tai}}{{\\rm{l}}_{p - 1,sp,p}}\\left( {u;{\\xi _0},R} \\right)$ see \\eqref{eq26} below.\n\\end{theorem}\n\nThe paper is organized as follows: In Section 2, we introduce relevant knowledge of the Heisenberg group, function spaces, and several necessary and new lemmas. In Section 3, we employ horizontal difference, along with certain difference estimates and the fractional Sobolev-type inequality on the Heisenberg group, to establish the crucial iteration scheme of Morrey-type for weak solutions to \\eqref{eq0} required for the proof of the Theorem \\ref{Th13}, i.e. Proposition \\ref{Pro41}. In Section 4, we first establish a proposition regarding the gradient integrability of weak solutions by using Caccioppoli-type inequality, i.e. Proposition \\ref{Pro42}. Then, by combining local boundedness, the Proposition \\ref{Pro41}, the Proposition \\ref{Pro42}, an iterative method, and the Morrey inequality, we prove Theorem \\ref{Th13}. Finally, Theorem \\ref{Th14} is derived by utilizing Theorem \\ref{Th13} and Theorem 1.2 from \\cite{MZ21}.\n\nIn the sequel, we denote by $c$ a generic positive constant which may vary from line to line. Relevant dependencies on parameters shall be emphasised utilizing parentheses, i.e., $c= c(n, p, q)$ means that $c$ depends on $n, p, q$.", "sketch": "To prove Theorem~\\ref{Th13}, the paper proceeds as follows. In Section~3, it uses “horizontal difference, along with certain difference estimates and the fractional Sobolev-type inequality on the Heisenberg group” to “establish the crucial iteration scheme of Morrey-type for weak solutions to \\eqref{eq0} required for the proof of the Theorem~\\ref{Th13}, i.e. Proposition~\\ref{Pro41}.” In Section~4, it “first establish[es] a proposition regarding the gradient integrability of weak solutions by using Caccioppoli-type inequality, i.e. Proposition~\\ref{Pro42}.” Then, “by combining local boundedness, the Proposition~\\ref{Pro41}, the Proposition~\\ref{Pro42}, an iterative method, and the Morrey inequality, we prove Theorem~\\ref{Th13}.”", "expanded_sketch": "To prove the main theorem, the paper proceeds as follows. Next, it uses “horizontal difference, along with certain difference estimates and the fractional Sobolev-type inequality on the Heisenberg group” to “establish the crucial iteration scheme of Morrey-type for weak solutions to\n\\begin{equation}\\label{eq0}\n - \\Delta _{{{\\mathbb{H}},p}}u + \\Lambda{\\left( { - \\Delta _{{{\\mathbb{H}},p}}} \\right)^s}u = 0\\;{\\rm{in}}\\;\\Omega ,\n\\end{equation}\nrequired for the proof of the main theorem, i.e. the following proposition.\n\n\\begin{proposition}\\label{Pro41}\nLet $2\\le p <\\infty,\\;0<s<1$ and $0\\le \\Lambda \\le 1$. Assume $u \\in HW_{{\\rm{loc}}}^{1,p}\\left( {{B_2}\\left( {{\\xi _0}} \\right) \\cap L_{sp}^{p - 1}\\left( {{\\mathbb{H}^n}} \\right)} \\right)$ is a weak solution of \\eqref{eq0} in ${{B_2}\\left( {{\\xi _0}} \\right)}$ satisfying\n\\begin{equation}\\label{eq41}\n{\\left\\| u \\right\\|_{{L^\\infty }\\left( {{B_1}\\left( {{\\xi _0}} \\right)} \\right)}} \\le 1,\\int_{{\\mathbb{H}^n}\\backslash {B_1}\\left( {{\\xi _0}} \\right)} {\\frac{{{{\\left| {u\\left( \\eta  \\right)} \\right|}^{p - 1}}}}{{\\left\\| \\eta  \\right\\|_{{\\mathbb{H}^n}}^{Q + sp}}}d\\eta }  \\le 1.\n\\end{equation}\nIf $0<h_0<\\frac{1}{10}$ and $R$ satisfy $4{h_0}<R\\le 1-5{h_0}$, and ${\\nabla _H}u \\in {L^q}\\left( {{B_{R + 4{h_0}}}\\left( {{\\xi _0}} \\right)} \\right)$ for some $q \\ge p$, then\n\\begin{equation}\\label{eq42}\n \\mathop {\\sup }\\limits_{0 < \\left| h \\right| < {h_0}} \\int_{{B_{R - {4h_0}}}} {{{\\left| {\\frac{{\\Delta _{Z,h}^2u}}{{{{\\left| h \\right|}^{\\frac{1}{{q + 1}} + 1}}}}} \\right|}^{q + 1}}d\\xi } \\le c(1+\\Lambda) \\left(  {\\int_{{B_{R + {4h_0}}}} {{{\\left| {{\\nabla _H}u} \\right|}^q}d\\xi } + 1} \\right),\n\\end{equation}\nwhere $c=c(Q,h_0,p,q,s)>0$.\n\\end{proposition}\n\nNext, it “first establish[es] a proposition regarding the gradient integrability of weak solutions by using Caccioppoli-type inequality, i.e.” the following.\n\n\\begin{proposition}\\label{Pro42}\nLet $2\\le p <\\infty,\\;0<s<1$ and $0\\le \\Lambda \\le 1$. If $u \\in HW_{{\\rm{loc}}}^{1,p}\\left( {{B_2}\\left( {{\\xi _0}} \\right) \\cap L_{sp}^{p - 1}\\left( {{\\mathbb{H}^n}} \\right)} \\right)$ is a weak solution of \\eqref{eq0} in ${{B_{2}}\\left( {{\\xi _0}} \\right)}$ satisfying \\eqref{eq41}, then\n\\begin{equation*}\n\\int_{{B_{\\frac{7}{8}}}\\left( {{\\xi _0}} \\right)} {{{\\left| {{\\nabla _H}u} \\right|}^p}d\\xi }  \\le C\\left( {Q,p,s} \\right).\n\\end{equation*}\n\\end{proposition}\n\nThen, “by combining local boundedness, the proposition above, the preceding proposition, an iterative method, and the Morrey inequality, we prove the main theorem.”", "expanded_theorem": "\\label{Th13}\nLet $2\\le p <\\infty,\\;0<s<1$ and $0\\le \\Lambda \\le 1$. Suppose $\\Omega \\in \\mathbb{H}^n$ is a bounded open set and $u \\in HW_{{\\rm{loc}}}^{1,p}\\left( \\Omega \\right) \\cap{ L_{sp}^{p - 1}\\left( {{\\mathbb{H}^n}} \\right)} $ is a weak solution of\n\\begin{equation}\\label{eq0}\n - \\Delta _{{{\\mathbb{H}},p}}u + \\Lambda{\\left( { - \\Delta _{{{\\mathbb{H}},p}}} \\right)^s}u = 0\\;{\\rm{in}}\\;\\Omega ,\n\\end{equation}\nThen $u \\in C_{loc}^\\gamma \\left( \\Omega  \\right)$ for any $0 < \\gamma  < 1$. Moreover, for any $0 < \\gamma  < 1$ and any Kor\\'{a}nyi ball ${B_{2R}}\\left( {{\\xi _0}} \\right) \\subset  \\subset \\Omega $ with $0<R<1$, there exists a positive constant $c=c(Q,p,s,\\gamma)$ such that\n\\begin{equation}\\label{eq482}\n  \\mathop {\\sup }\\limits_{\\xi  \\ne \\eta  \\in {B_{\\frac{R}{2}}}\\left( {{\\xi _0}} \\right)} \\frac{{\\left| {u\\left( \\xi  \\right) - u\\left( \\eta  \\right)} \\right|}}{{{{\\left\\| {{\\eta ^{ - 1}} \\circ \\xi } \\right\\|}^\\gamma _{\\mathbb{H}^n}}}} \\le \\frac{c}{{{R^\\gamma }}}\\left( {{{{\\left\\| u \\right\\|}_{{L^\\infty }\\left( {{B_R}\\left( {{\\xi _0}} \\right)} \\right)}}} + {\\rm{Tai}}{{\\rm{l}}_{p - 1,sp,p}}\\left( {u;{\\xi _0},R} \\right)} \\right),\n\\end{equation}\nwhere ${\\rm{Tai}}{{\\rm{l}}_{p - 1,sp,p}}\\left( {u;{\\xi _0},R} \\right)$ is given by\n\\begin{equation}\\label{eq26}\n {\\rm{Tai}}{{\\rm{l}}_{q,\\alpha ,\\beta }}\\left( {u;{\\xi _0},R} \\right) = {\\left[ {{R^\\beta }\\int_{{{\\mathbb{H}}^n}\\backslash {B_R}\\left( {{\\xi _0}} \\right)} {\\frac{{{{\\left| u \\right|}^q}}}{{{{\\left\\| {\\xi _0^{ - 1} \\circ \\xi } \\right\\|}^{Q + \\alpha }_{\\mathbb{H}^n}}}}d\\xi } } \\right]^{\\frac{1}{q}}}.\n\\end{equation}\n", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let \\(\\mathbb{H}^n\\) be the Heisenberg group with homogeneous dimension \\(Q=2n+2\\), Kor\\u00e1nyi gauge \\(\\|\\cdot\\|_{\\mathbb{H}^n}\\), and Kor\\u00e1nyi balls \\(B_r(\\xi_0)=\\{\\xi\\in\\mathbb{H}^n:\\|\\xi_0^{-1}\\circ\\xi\\|_{\\mathbb{H}^n}<r\\}\\). Define the \\(p\\)-sub-Laplacian by \\(-\\Delta_{\\mathbb{H},p}u=\\operatorname{div}_H(|\\nabla_Hu|^{p-2}\\nabla_Hu)\\), and the fractional \\(p\\)-sub-Laplacian by \\(( -\\Delta_{\\mathbb{H},p})^s u(\\xi)=\\mathrm{P.V.}\\int_{\\mathbb{H}^n}\\frac{|u(\\xi)-u(\\eta)|^{p-2}(u(\\xi)-u(\\eta))}{\\|\\eta^{-1}\\circ\\xi\\|_{\\mathbb{H}^n}^{Q+sp}}\\,d\\eta\\). Assume \\(2\\le p<\\infty\\), \\(0<s<1\\), \\(0\\le \\Lambda\\le1\\), \\(\\Omega\\subset\\mathbb{H}^n\\) is bounded and open, and \\(u\\in HW^{1,p}_{\\mathrm{loc}}(\\Omega)\\cap L_{sp}^{p-1}(\\mathbb{H}^n)\\) is a weak solution of \\(-\\Delta_{\\mathbb{H},p}u+\\Lambda(-\\Delta_{\\mathbb{H},p})^s u=0\\) in \\(\\Omega\\). For \\(q,\\alpha,\\beta>0\\), define \\(\\operatorname{Tail}_{q,\\alpha,\\beta}(u;\\xi_0,R)=\\left[R^{\\beta}\\int_{\\mathbb{H}^n\\setminus B_R(\\xi_0)}\\frac{|u(\\xi)|^q}{\\|\\xi_0^{-1}\\circ\\xi\\|_{\\mathbb{H}^n}^{Q+\\alpha}}\\,d\\xi\\right]^{1/q}\\). Under these assumptions, which quantitative estimate holds?", "correct_choice": {"label": "A", "text": "For every \\(0<\\gamma<1\\), \\(u\\in C^\\gamma_{\\mathrm{loc}}(\\Omega)\\). More precisely, for every \\(0<\\gamma<1\\) and every Kor\\u00e1nyi ball \\(B_{2R}(\\xi_0)\\Subset\\Omega\\) with \\(0<R<1\\), there exists a constant \\(c=c(Q,p,s,\\gamma)>0\\) such that \\[\\sup_{\\xi\\ne\\eta\\in B_{R/2}(\\xi_0)}\\frac{|u(\\xi)-u(\\eta)|}{\\|\\eta^{-1}\\circ\\xi\\|_{\\mathbb{H}^n}^{\\gamma}}\\le \\frac{c}{R^{\\gamma}}\\left(\\|u\\|_{L^\\infty(B_R(\\xi_0))}+\\operatorname{Tail}_{p-1,sp,p}(u;\\xi_0,R)\\right).\\]"}, "choices": [{"label": "B", "text": "For every \\(0<\\gamma<1\\), \\(u\\in C^\\gamma_{\\mathrm{loc}}(\\Omega)\\). More precisely, for every \\(0<\\gamma<1\\) and every Kor\\u00e1nyi ball \\(B_{2R}(\\xi_0)\\Subset\\Omega\\) with \\(0<R<1\\), there exists a constant \\(c=c(Q,p,s,\\gamma)>0\\) such that \\[\\sup_{\\xi\\ne\\eta\\in B_R(\\xi_0)}\\frac{|u(\\xi)-u(\\eta)|}{\\|\\eta^{-1}\\circ\\xi\\|_{\\mathbb{H}^n}^{\\gamma}}\\le \\frac{c}{R^{\\gamma}}\\left(\\|u\\|_{L^\\infty(B_R(\\xi_0))}+\\operatorname{Tail}_{p-1,sp,p}(u;\\xi_0,R)\\right).\\]"}, {"label": "C", "text": "There exists some \\(\\gamma\\in(0,1)\\) such that \\(u\\in C^\\gamma_{\\mathrm{loc}}(\\Omega)\\). More precisely, for every Kor\\u00e1nyi ball \\(B_{2R}(\\xi_0)\\Subset\\Omega\\) with \\(0<R<1\\), there exists a constant \\(c=c(Q,p,s,\\gamma)>0\\) such that \\[\\sup_{\\xi\\ne\\eta\\in B_{R/2}(\\xi_0)}\\frac{|u(\\xi)-u(\\eta)|}{\\|\\eta^{-1}\\circ\\xi\\|_{\\mathbb{H}^n}^{\\gamma}}\\le \\frac{c}{R^{\\gamma}}\\left(\\|u\\|_{L^\\infty(B_R(\\xi_0))}+\\operatorname{Tail}_{p-1,sp,p}(u;\\xi_0,R)\\right).\\]"}, {"label": "D", "text": "For every \\(0<\\gamma<1\\), \\(u\\in C^\\gamma_{\\mathrm{loc}}(\\Omega)\\). More precisely, for every \\(0<\\gamma<1\\) and every Kor\\u00e1nyi ball \\(B_{2R}(\\xi_0)\\Subset\\Omega\\) with \\(0<R<1\\), there exists a constant \\(c=c(Q,p,s)>0\\), independent of \\(\\gamma\\), such that \\[\\sup_{\\xi\\ne\\eta\\in B_{R/2}(\\xi_0)}\\frac{|u(\\xi)-u(\\eta)|}{\\|\\eta^{-1}\\circ\\xi\\|_{\\mathbb{H}^n}^{\\gamma}}\\le \\frac{c}{R^{\\gamma}}\\left(\\|u\\|_{L^\\infty(B_R(\\xi_0))}+\\operatorname{Tail}_{p-1,sp,p}(u;\\xi_0,R)\\right).\\]"}, {"label": "E", "text": "For every \\(0<\\gamma<1\\), \\(u\\in C^\\gamma_{\\mathrm{loc}}(\\Omega)\\). More precisely, for every \\(0<\\gamma<1\\) and every Kor\\u00e1nyi ball \\(B_{2R}(\\xi_0)\\Subset\\Omega\\) with \\(0<R<1\\), there exists a constant \\(c=c(Q,p,s,\\gamma)>0\\) such that \\[\\sup_{\\xi\\ne\\eta\\in B_{R/2}(\\xi_0)}\\frac{|u(\\xi)-u(\\eta)|}{\\|\\eta^{-1}\\circ\\xi\\|_{\\mathbb{H}^n}^{\\gamma}}\\le \\frac{c}{R^{\\gamma}}\\left(\\|u\\|_{L^\\infty(B_R(\\xi_0))}+\\operatorname{Tail}_{p,sp,p}(u;\\xi_0,R)\\right).\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "interior_shrink_from_BR_to_B_{R/2}", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "universal_quantifier_over_gamma_replaced_by_existence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "dependence_of_constant_on_gamma", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "tail_exponent_q_changed_from_p-1_to_p", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives assumptions and definitions but does not state or strongly hint at the exact conclusion. The correct estimate is not leaked explicitly."}, "TAS": {"score": 0, "justification": "The item is essentially a theorem-recall question: it asks directly which quantitative estimate holds under the stated assumptions, with choices that are near-variants of the theorem statement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish subtle variants involving interior shrinkage, quantification over \u0003b3, dependence of constants, and the tail exponent. However, the task is closer to recognizing the exact theorem statement than generating a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: overstrong domain of the seminorm, weakened existential quantifier, incorrect uniformity in \u0003b3, and a wrong tail exponent. They are distinct and well-aligned with common confusions."}, "total_score": 5, "overall_assessment": "A technically well-crafted but theorem-recall-heavy MCQ. It avoids answer leakage and has strong distractors, but it is largely a restatement/recognition task rather than a genuinely generative reasoning question."}}
{"id": "2602.08476v1", "paper_link": "http://arxiv.org/abs/2602.08476v1", "theorems_cnt": 7, "theorem": {"env_name": "theorem", "content": "\\label{main th}\n\nThere exists a minimizing couple $(u,\\ell)$ of the following problem in both variables\n\\begin{equation}\n    \\label{pb min}\n    \\min\\left \\{E_\\varepsilon(u,\\ell), \\, u \\in  H^{1}({\\mathcal C}) \\text{ such that } 0\\leqslant u \\leqslant 1 , \\, u = 1 \\text{ on } \\partial{{\\mathcal C}} \\text{ and } \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1) \\right\\}.\n\\end{equation}\n\nMoreover, the minimizer $u$ is $\\alpha$-Hölder continuous for all $0<\\alpha<1$ with the estimate:\n\\[\\|u\\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]", "start_pos": 9246, "end_pos": 9845, "label": "main th"}, "ref_dict": {"prop 1.2": "\\begin{proposition}\n\\label{prop 1.2}\nFor all $u \\in H^1(\\C)\\cap C(\\overline{\\Ccal})$ there exists a minimizer for the problem \\eqref{pb min u fixe}. \n\\end{proposition}", "pb min u fixe": "\\begin{equation}\n\\label{pb min u fixe}\n    \\inf_{\\ell \\in \\curve^\\Lambda} E_\\varepsilon(u,\\ell), \\text{ where } u \\in H^1(\\Ccal)\\cap C(\\overline{\\Ccal}),\n\\end{equation}", "main th": "\\begin{theorem}\n\\label{main th}\n\nThere exists a minimizing couple $(u,\\ell)$ of the following problem in both variables\n\\begin{equation}\n    \\label{pb min}\n    \\min\\left \\{E_\\eps(u,\\ell), \\, u \\in  H^{1}(\\C) \\text{ such that } 0\\leqslant u \\leqslant 1 , \\, u = 1 \\text{ on } \\partial{\\C} \\text{ and } \\ell \\in \\curve^\\Lambda(\\gamma_0,\\gamma_1) \\right\\}.\n\\end{equation}\n\nMoreover, the minimizer $u$ is $\\alpha$-Hölder continuous for all $0<\\alpha<1$ with the estimate:\n\\[\\|u\\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]\n\n\\end{theorem}", "golab": "\\begin{lemma}\n \\label{golab}\nLet $\\ell_n : [0,1] \\times  \\mathbb{S}^1 \\to \\R^3$ be a sequence of uniformly Lipschitz functions, which converges uniformly to a Lipschitz function $\\ell  : [0,1] \\times  \\mathbb{S}^1 \\to \\R^3$.\nThen, for all open set $A$\n\n$$\\liminf \\Haus \\mres S_{\\ell_n}(A) \\geqslant \\Haus \\mres S_\\ell(A),$$\nwhere $S_\\ell$ (respectively $S_{\\ell_n}$) denotes the image of the Lipschitz function $\\ell$ (respectively $\\ell_n)$.\n\\end{lemma}", "Def:geoDistance": "\\begin{definition}\\label{Def:geoDistance}\n\tLet $(\\delta_\\eps)_{\\eps>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n\t\\begin{align*}\n\t\t\t&d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{  \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\eps) d\\Haus \\ |\\ \\ell \\in \\curve^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n\t\\end{align*}\n\\end{definition}"}, "pre_theorem_intro_text_len": 5458, "pre_theorem_intro_text": "This paper is devoted to the study of the functional introduced in \\cite{bonnivard2025phasefieldapproximationplateaus} to approximate a Plateau problem, which consists in finding a surface of minimal area spanning a collection of closed curves $\\gamma_0,..., \\gamma_n$ contained in the boundary of an open, bounded, convex set $\\Ccal_0 \\subset \\mathbb{R}^3$. \n\nMore precisely, the competitor surfaces are defined as the images of homotopies connecting the given curves. We define below the set of admissible homotopies connecting a curve $\\gamma_i$ to $\\gamma_j$:\n\\[\n\\mathrm{Hom}(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Lip}([0,1] \\times \\mathbb{S}^{1}, \\overline{\\Ccal_0}) \\text{ such that } \\ell(0) = \\gamma_i \\text{ and } \\ell(1) = \\gamma_{j} \\}.\n\\]\nNote that we require more than mere homotopies: we specifically consider Lipschitz homotopies. For any homotopy $\\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j)$, we define the associated surface $\\surf_\\ell$ as its image:\n\\begin{equation}\\label{Def:surface_image}\n\t\\surf_{\\ell} := \\ell([0,1] \\times \\mathbb{S}^{1}) \\subset \\overline{\\Ccal_0}.\n\\end{equation}\nWe emphasize that the surface $\\surf_{\\ell}$ is $\\mathcal{H}^{2}$-rectifiable and has finite $\\mathcal{H}^{2}$-measure. Moreover, for any $u\\in H^1(\\mathbb{R}^3)$, the trace of $u$ on $\\surf_{\\ell}$ is well defined, since the set of points that are not Lebesgue points of $u$ is $\\mathcal{H}^2$-negligible. \nHowever, to obtain the existence result, we need to impose additional properties on the competitors.\nFor $\\Lambda>0$, we consider\n\\[\n\\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j) \\text{ such that }  \\;\\mathrm{Lip}(\\ell)\\leq \\Lambda \\text{ and } \\surf_\\ell \\text{ is } \\Lambda-\\text{upper Ahlfors regular} \\ \\}.\\]\n\nLet us recall the definition of upper Ahlfors regularity.\n\\begin{definition}\nLet $E\\subset \\mathbb{R}^3$ and $\\Lambda>0$. The set $E$ is said to be $\\Lambda$-upper Ahlfors regular if for all $x\\in E$ and all $r>0$ we have \n\\[\\mathcal{H}^{2}(E\\cap B(x,r)) \\leqslant \\Lambda \\pi r^2.\\]\n\\end{definition}\n\nThe definition of our functional is based on the notion of a generalized geodesic distance between curves, associated with a given weight function $u$, which is defined as follows.\n\\begin{definition}\\label{Def:geoDistance}\n\tLet $(\\delta_\\varepsilon)_{\\varepsilon>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n\t\\begin{align*}\n\t\t\t&d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{  \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2} \\ |\\ \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n\t\\end{align*}\n\\end{definition}\nNote that, in \\cite{bonnivard2025phasefieldapproximationplateaus}, the geodesic distance is defined for Lipschitz homotopies $\\ell$. The assumptions of uniform Lipschitz regularity and uniform upper Ahlfors regularity of the images are not required there. However, in the present work, we need to impose these assumptions in order to obtain the compactness result necessary for the existence theorem.\n\nLet $\\mathcal{C}$ an smooth open bounded convex set containing the closure $\\overline{\\Ccal_0}$. This set $\\mathcal{C}$ will serve as our domain of study. Indeed, since we rely on PDE techniques, it is convenient to work within a smooth bounded domain. We now recall the definition of the functional introduced in \\cite{bonnivard2025phasefieldapproximationplateaus}:\n\t\\begin{equation}\n\t\t\\label{functionalGeneral}\n\t\tF_{\\varepsilon}(u) := \\varepsilon \\int_{\\mathcal{C}}|\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2}dx + \\frac{1}{c_{\\varepsilon}}\\sum_{i=1}^{n}d_{u}(\\gamma_0, \\gamma_{i}).\n\t\\end{equation}\nWhen, the sequences of positive numbers $(\\delta_\\varepsilon)$ and $(c_\\varepsilon)$ are assumed to converge to zero, and to satisfy that $\\delta_\\varepsilon/c_\\varepsilon$ converges to zero as $\\varepsilon \\to 0$, \\cite{bonnivard2025phasefieldapproximationplateaus} establishes that this functional approximate some Plateau's problem through a $\\Gamma$-convergence result. \n\nFor simplicity, we will assume in the following that the prescribed boundary consists of only two curves, $\\gamma_0$ and $\\gamma_1$. However, the results established in the present article remain valid when the boundary contains more than two curves.\n\nIn this paper, we are specifically interested in the decoupled functional defined below.\n\\begin{definition}\n\tFor $u \\in H^{1}({\\mathcal C})$ such that $0\\leqslant u \\leqslant 1$, with $u = 1$ on $\\partial{{\\mathcal C}}$, and $\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$,\n\t\\begin{align*}\n\t\t&E_{\\varepsilon}(u,\\ell) := \\varepsilon\\int_{\\mathcal{C}} |\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2} dx + \\frac{1}{c_{\\varepsilon}}\\int_{\\surf_{\\ell}} (u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2}.\n\t\\end{align*}\n\\end{definition}\n\n\\begin{remark}\n\tFrom Definition~\\ref{Def:geoDistance} of the geodesic distance between closed curves, we know that for all $u\\in H^{1}(\\mathcal{C})$ such that $0\\leqslant u \\leqslant 1$ and $u = 1$ on $\\partial{\\mathcal{C}}$,  \n\t\t\\[F_{\\varepsilon}(u) = \\inf\\left \\{ E_{\\varepsilon}(u,\\ell)\\ ,\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)  \\right \\}.\\]\n\\end{remark}\n\nThe main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\\varepsilon$\n\\begin{equation*}\n    \\inf_{(u,\\ell)} E_\\varepsilon(u,\\ell).\n\\end{equation*}", "context": "More precisely, the competitor surfaces are defined as the images of homotopies connecting the given curves. We define below the set of admissible homotopies connecting a curve $\\gamma_i$ to $\\gamma_j$:\n\\[\n\\mathrm{Hom}(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Lip}([0,1] \\times \\mathbb{S}^{1}, \\overline{\\Ccal_0}) \\text{ such that } \\ell(0) = \\gamma_i \\text{ and } \\ell(1) = \\gamma_{j} \\}.\n\\]\nNote that we require more than mere homotopies: we specifically consider Lipschitz homotopies. For any homotopy $\\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j)$, we define the associated surface $\\surf_\\ell$ as its image:\n\\begin{equation}\\label{Def:surface_image}\n \\surf_{\\ell} := \\ell([0,1] \\times \\mathbb{S}^{1}) \\subset \\overline{\\Ccal_0}.\n\\end{equation}\nWe emphasize that the surface $\\surf_{\\ell}$ is $\\mathcal{H}^{2}$-rectifiable and has finite $\\mathcal{H}^{2}$-measure. Moreover, for any $u\\in H^1(\\mathbb{R}^3)$, the trace of $u$ on $\\surf_{\\ell}$ is well defined, since the set of points that are not Lebesgue points of $u$ is $\\mathcal{H}^2$-negligible. \nHowever, to obtain the existence result, we need to impose additional properties on the competitors.\nFor $\\Lambda>0$, we consider\n\\[\n\\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j) \\text{ such that } \\;\\mathrm{Lip}(\\ell)\\leq \\Lambda \\text{ and } \\surf_\\ell \\text{ is } \\Lambda-\\text{upper Ahlfors regular} \\ \\}.\\]\n\nThe definition of our functional is based on the notion of a generalized geodesic distance between curves, associated with a given weight function $u$, which is defined as follows.\n\\begin{definition}\\label{Def:geoDistance}\n Let $(\\delta_\\varepsilon)_{\\varepsilon>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n \\begin{align*}\n &d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{ \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2} \\ |\\ \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n \\end{align*}\n\\end{definition}\nNote that, in \\cite{bonnivard2025phasefieldapproximationplateaus}, the geodesic distance is defined for Lipschitz homotopies $\\ell$. The assumptions of uniform Lipschitz regularity and uniform upper Ahlfors regularity of the images are not required there. However, in the present work, we need to impose these assumptions in order to obtain the compactness result necessary for the existence theorem.\n\nLet $\\mathcal{C}$ an smooth open bounded convex set containing the closure $\\overline{\\Ccal_0}$. This set $\\mathcal{C}$ will serve as our domain of study. Indeed, since we rely on PDE techniques, it is convenient to work within a smooth bounded domain. We now recall the definition of the functional introduced in \\cite{bonnivard2025phasefieldapproximationplateaus}:\n \\begin{equation}\n \\label{functionalGeneral}\n F_{\\varepsilon}(u) := \\varepsilon \\int_{\\mathcal{C}}|\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2}dx + \\frac{1}{c_{\\varepsilon}}\\sum_{i=1}^{n}d_{u}(\\gamma_0, \\gamma_{i}).\n \\end{equation}\nWhen, the sequences of positive numbers $(\\delta_\\varepsilon)$ and $(c_\\varepsilon)$ are assumed to converge to zero, and to satisfy that $\\delta_\\varepsilon/c_\\varepsilon$ converges to zero as $\\varepsilon \\to 0$, \\cite{bonnivard2025phasefieldapproximationplateaus} establishes that this functional approximate some Plateau's problem through a $\\Gamma$-convergence result.\n\nIn this paper, we are specifically interested in the decoupled functional defined below.\n\\begin{definition}\n For $u \\in H^{1}({\\mathcal C})$ such that $0\\leqslant u \\leqslant 1$, with $u = 1$ on $\\partial{{\\mathcal C}}$, and $\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$,\n \\begin{align*}\n &E_{\\varepsilon}(u,\\ell) := \\varepsilon\\int_{\\mathcal{C}} |\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2} dx + \\frac{1}{c_{\\varepsilon}}\\int_{\\surf_{\\ell}} (u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2}.\n \\end{align*}\n\\end{definition}\n\n\\begin{remark}\n From Definition~\\ref{Def:geoDistance} of the geodesic distance between closed curves, we know that for all $u\\in H^{1}(\\mathcal{C})$ such that $0\\leqslant u \\leqslant 1$ and $u = 1$ on $\\partial{\\mathcal{C}}$, \n \\[F_{\\varepsilon}(u) = \\inf\\left \\{ E_{\\varepsilon}(u,\\ell)\\ ,\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1) \\right \\}.\\]\n\\end{remark}\n\nThe main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\\varepsilon$\n\\begin{equation*}\n \\inf_{(u,\\ell)} E_\\varepsilon(u,\\ell).\n\\end{equation*}\n\n\\begin{definition}\\label{Def:geoDistance}\n\tLet $(\\delta_\\eps)_{\\eps>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n\t\\begin{align*}\n\t\t\t&d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{  \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\eps) d\\Haus \\ |\\ \\ell \\in \\curve^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n\t\\end{align*}\n\\end{definition}", "full_context": "More precisely, the competitor surfaces are defined as the images of homotopies connecting the given curves. We define below the set of admissible homotopies connecting a curve $\\gamma_i$ to $\\gamma_j$:\n\\[\n\\mathrm{Hom}(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Lip}([0,1] \\times \\mathbb{S}^{1}, \\overline{\\Ccal_0}) \\text{ such that } \\ell(0) = \\gamma_i \\text{ and } \\ell(1) = \\gamma_{j} \\}.\n\\]\nNote that we require more than mere homotopies: we specifically consider Lipschitz homotopies. For any homotopy $\\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j)$, we define the associated surface $\\surf_\\ell$ as its image:\n\\begin{equation}\\label{Def:surface_image}\n \\surf_{\\ell} := \\ell([0,1] \\times \\mathbb{S}^{1}) \\subset \\overline{\\Ccal_0}.\n\\end{equation}\nWe emphasize that the surface $\\surf_{\\ell}$ is $\\mathcal{H}^{2}$-rectifiable and has finite $\\mathcal{H}^{2}$-measure. Moreover, for any $u\\in H^1(\\mathbb{R}^3)$, the trace of $u$ on $\\surf_{\\ell}$ is well defined, since the set of points that are not Lebesgue points of $u$ is $\\mathcal{H}^2$-negligible. \nHowever, to obtain the existence result, we need to impose additional properties on the competitors.\nFor $\\Lambda>0$, we consider\n\\[\n\\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j) \\text{ such that } \\;\\mathrm{Lip}(\\ell)\\leq \\Lambda \\text{ and } \\surf_\\ell \\text{ is } \\Lambda-\\text{upper Ahlfors regular} \\ \\}.\\]\n\nThe definition of our functional is based on the notion of a generalized geodesic distance between curves, associated with a given weight function $u$, which is defined as follows.\n\\begin{definition}\\label{Def:geoDistance}\n Let $(\\delta_\\varepsilon)_{\\varepsilon>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n \\begin{align*}\n &d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{ \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2} \\ |\\ \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n \\end{align*}\n\\end{definition}\nNote that, in \\cite{bonnivard2025phasefieldapproximationplateaus}, the geodesic distance is defined for Lipschitz homotopies $\\ell$. The assumptions of uniform Lipschitz regularity and uniform upper Ahlfors regularity of the images are not required there. However, in the present work, we need to impose these assumptions in order to obtain the compactness result necessary for the existence theorem.\n\nLet $\\mathcal{C}$ an smooth open bounded convex set containing the closure $\\overline{\\Ccal_0}$. This set $\\mathcal{C}$ will serve as our domain of study. Indeed, since we rely on PDE techniques, it is convenient to work within a smooth bounded domain. We now recall the definition of the functional introduced in \\cite{bonnivard2025phasefieldapproximationplateaus}:\n \\begin{equation}\n \\label{functionalGeneral}\n F_{\\varepsilon}(u) := \\varepsilon \\int_{\\mathcal{C}}|\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2}dx + \\frac{1}{c_{\\varepsilon}}\\sum_{i=1}^{n}d_{u}(\\gamma_0, \\gamma_{i}).\n \\end{equation}\nWhen, the sequences of positive numbers $(\\delta_\\varepsilon)$ and $(c_\\varepsilon)$ are assumed to converge to zero, and to satisfy that $\\delta_\\varepsilon/c_\\varepsilon$ converges to zero as $\\varepsilon \\to 0$, \\cite{bonnivard2025phasefieldapproximationplateaus} establishes that this functional approximate some Plateau's problem through a $\\Gamma$-convergence result.\n\nIn this paper, we are specifically interested in the decoupled functional defined below.\n\\begin{definition}\n For $u \\in H^{1}({\\mathcal C})$ such that $0\\leqslant u \\leqslant 1$, with $u = 1$ on $\\partial{{\\mathcal C}}$, and $\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$,\n \\begin{align*}\n &E_{\\varepsilon}(u,\\ell) := \\varepsilon\\int_{\\mathcal{C}} |\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2} dx + \\frac{1}{c_{\\varepsilon}}\\int_{\\surf_{\\ell}} (u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2}.\n \\end{align*}\n\\end{definition}\n\n\\begin{remark}\n From Definition~\\ref{Def:geoDistance} of the geodesic distance between closed curves, we know that for all $u\\in H^{1}(\\mathcal{C})$ such that $0\\leqslant u \\leqslant 1$ and $u = 1$ on $\\partial{\\mathcal{C}}$, \n \\[F_{\\varepsilon}(u) = \\inf\\left \\{ E_{\\varepsilon}(u,\\ell)\\ ,\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1) \\right \\}.\\]\n\\end{remark}\n\nThe main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\\varepsilon$\n\\begin{equation*}\n \\inf_{(u,\\ell)} E_\\varepsilon(u,\\ell).\n\\end{equation*}\n\n\\begin{definition}\\label{Def:geoDistance}\n\tLet $(\\delta_\\eps)_{\\eps>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n\t\\begin{align*}\n\t\t\t&d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{  \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\eps) d\\Haus \\ |\\ \\ell \\in \\curve^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n\t\\end{align*}\n\\end{definition}\n\nThe main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\\varepsilon$\n\\begin{equation*}\n \\inf_{(u,\\ell)} E_\\varepsilon(u,\\ell).\n\\end{equation*}\n\nThere exists a minimizing couple $(u,\\ell)$ of the following problem in both variables\n\\begin{equation}\n \\label{pb min}\n \\min\\left \\{E_\\eps(u,\\ell), \\, u \\in H^{1}(\\C) \\text{ such that } 0\\leqslant u \\leqslant 1 , \\, u = 1 \\text{ on } \\partial{\\C} \\text{ and } \\ell \\in \\curve^\\Lambda(\\gamma_0,\\gamma_1) \\right\\}.\n\\end{equation}\n\nThe study of this energy is motivated by the $\\Gamma$-convergence result established in \\cite{bonnivard2025phasefieldapproximationplateaus}, where this decoupled functional was introduced both in the proof of the $\\Gamma$-convergence and for numerical applications. Recall that the phase-field approach they proposed is a generalization to Plateau’s problem of the method introduced in \\cite{bonnivard2015approximation} for Steiner’s problem. A similar study of the existence of solutions for the decoupled functional in the Steiner case was carried out in \\cite{bonnivard2018phase}. The present work is inspired by this paper, but there are some differences that we explain below.\n\nWe begin by studying the following minimization problem \n\\begin{equation}\n\\label{pb min l fixe}\n \\inf_{u} E_\\varepsilon(u,\\ell), \\text{ where } \\ell \\in \\curve^\\Lambda(\\gamma_0,\\gamma_1).\n\\end{equation}\nThe following Proposition will be proven in Section~\\ref{section l fixe}. \n\\begin{proposition}\nFor any given $\\ell \\in \\curve^\\Lambda(\\gamma_0, \\gamma_1)$, there exists a unique minimizer $u \\in H^1(\\Ccal)$ of $ E_\\varepsilon(\\cdot,l)$ which is globally $C^{0,\\alpha}$, for all $0< \\alpha <1$: there exists a constant $C_\\alpha>0$ such that \n\\[\\|u \\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]\n\\end{proposition}\nThe assumption that the surface $S_\\ell$ is $\\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity, and hence the existence of minimizers in this setting. We then apply classical results from elliptic regularity to derive Hölder estimates for the minimizer. Note that these estimates depend explicitly on $\\Lambda$ and $\\varepsilon$, but are independent of the specific surface $S_\\ell$. This part is strongly inspired by \\cite{bonnivard2018phase}.\n\n\\begin{proposition}\n\\label{prop holder}\nLet $\\ell \\in~\\curve^\\Lambda(\\gamma_0,\\gamma_1)$ and let $u$ be the minimizer of $E_\\varepsilon^\\ell$. Then, $u$ is $\\alpha$-Hölder continuous for all $0<\\alpha<1$ with the estimate:\n\\[\\|u\\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]\n\\end{proposition}\n\n\\begin{lemma}\n\\label{lem 2.10}\nRecall the definition of the parameter $\\eta_0 = \\dist(\\partial \\Ccal_0,\\partial \\Ccal)$ introduced in the previous Lemma.\nLet $\\ell \\in \\curve^\\Lambda(\\gamma_0, \\gamma_1)$, $0< \\rho < \\min(1,\\frac{\\eta_0}{4})$ and $x_0 \\in \\mathcal{C}$ such that $\\dist(x_0,\\partial \\mathcal{C}) \\geqslant \\frac{\\eta_0}{2} $. Then, $u \\in W^{1,p}(B(x_0,\\rho))$ for all $3\\leqslant p < \\infty$ and we have the following estimate on the rescaled function $u_\\rho (x) := u(x_0+\\rho x)$\n\\begin{equation}\n \\|\\nabla u_\\rho\\|_{L^p(B_1)} \\leqslant C_p\\left( \\frac{\\rho^2}{\\varepsilon^2}+1+\\frac{\\Lambda \\rho}{c_\\varepsilon \\varepsilon}\\right).\n\\end{equation}\n\\end{lemma}\n\n\\medskip\n\\emph{Step 3.} We define the rescaled function $u_\\rho$ on $B_2$ by $u_\\rho(x) := u(x_0+\\rho x)$. Then, for all $\\varphi \\in C^\\infty_c(B_2)$ \n\\begin{align*}\n \\int_{B_2}\\nabla u_\\rho \\cdot \\nabla\\varphi \\dm x &= \\frac{1}{\\rho}\\int_{B(x_0,2\\rho)}\\nabla u \\cdot \\nabla\\varphi_\\rho \\dm x,\\\\\n &= \\frac{1}{\\rho4\\varepsilon^2}\\int_{B(x_0,2\\rho)}(1- u)\\varphi_\\rho \\dm x - \\frac{1}{\\rho c_\\varepsilon \\varepsilon}B[\\ell](u,\\varphi_\\rho),\\\\\n &= \\frac{\\rho^2}{4\\varepsilon^2}\\int_{B_2}(1- u_\\rho)\\varphi \\dm x - \\frac{1}{\\rho c_\\varepsilon \\varepsilon}\\langle T_\\rho,\\varphi \\rangle.\n\\end{align*}\nHence, $u_\\rho$ satisfies the following PDE in $\\mathcal{D}'(B_2)$\n\\[-\\Delta u_\\rho = \\frac{\\rho^2}{4\\varepsilon^2}(1- u_\\rho) - \\frac{1}{\\rho c_\\varepsilon \\varepsilon} T_\\rho.\\]\nWe denote $w_\\rho := u_\\rho + \\frac{1}{\\rho c_\\varepsilon \\varepsilon}v_\\rho \\in H^1(B_2)\\cap L^\\infty(B_2)$. And we deduce that \n\\[-\\Delta w_\\rho = \\frac{\\rho^2}{4\\varepsilon^2}(1- u_\\rho) \\text{ in } \\mathcal{D}'(B_2). \\]\nThus, \\cite[Corollary 8.36]{gilbargelliptic} yields that $w_\\rho \\in C_{\\mathrm{loc}}^{1,\\alpha}(B_2)$ for all $\\alpha >0$ and \n\\begin{align*}\n \\|\\nabla w_\\rho\\|_{L^\\infty(B_1)} \\leqslant \\|w_\\rho\\|_{C^{1,\\alpha}(B_1)} &\\leqslant C\\left(\\| w_\\rho\\|_{L^\\infty(B_1)} +\\frac{\\rho^2}{4\\varepsilon^2}\\|1- u_\\rho\\|_{L^\\infty(B_1)}\\right),\\\\\n &= C\\left(\\| u_\\rho\\|_{L^\\infty(B_1)}+ \\frac{1}{\\rho c_\\varepsilon \\varepsilon}\\| v_\\rho\\|_{L^\\infty(B_1)} +\\frac{\\rho^2}{4\\varepsilon^2}\\right),\\\\\n &\\leqslant C\\left(1 + \\frac{\\Lambda \\rho}{c_\\varepsilon \\varepsilon} + \\frac{\\rho^2}{4\\varepsilon^2}\\right).\n\\end{align*}\nFinally, we can conclude that $u_\\rho = w_\\rho - \\frac{1}{\\rho c_\\varepsilon \\varepsilon}v_\\rho \\in W^{1,p}(B_1)$ with the desired estimate\n\\[ \\|\\nabla u_\\rho\\|_{L^p(B_1)} \\leqslant C_p\\left( \\frac{\\rho^2}{\\varepsilon^2}+1+\\frac{\\Lambda \\rho}{c_\\varepsilon \\varepsilon}\\right).\\]\nThis concludes the proof of this Lemma.\n\\end{proof}\n\nNow let's consider the case where $\\dist(x_0, \\partial \\mathcal{C}) > \\frac{\\eta_0}{2}> \\varepsilon$, in particular this implies that $B(x_0,\\varepsilon) \\subset \\C$. In this case, Lemma~\\ref{lem 2.10}, applied with $\\rho = \\varepsilon$ and $p = \\frac{3}{1-\\alpha}$ yields the estimate on the rescaled function $u_\\varepsilon$\n \\[\\|\\nabla u_\\varepsilon\\|_{L^p(B_1)} \\leqslant C_p\\left(1+ \\frac{\\Lambda}{c_\\varepsilon}\\right).\\]\n Since we have shown that the $L^\\infty$ norm of $u$ is smaller than $1$, we deduce that the $L^p$ norm of the scaled function $u_\\varepsilon$ is bounded by the volume of the domain. And we conclude by applying the Sobolev embedding Theorem (see for instance \\cite[Theorem 4.12]{adams2003sobolev}),\n \\begin{equation}\n \\|u_\\varepsilon\\|_{C^{0,\\alpha}(B_1)} \\leqslant C_\\alpha\\left(1+ \\frac{\\Lambda}{c_\\varepsilon}\\right).\n \\end{equation}\n Hence, scaling back leads to \n \\begin{equation}\n \\frac{|u(x)-u(y)|}{|x-y|^\\alpha} \\leqslant \\frac{1}{\\varepsilon^\\alpha} \\|u_\\varepsilon\\|_{C^{0,\\alpha}(B_1)} \\leqslant \\frac{C_\\alpha}{\\varepsilon^\\alpha}\\left(1+ \\frac{\\Lambda}{c_\\varepsilon}\\right).\n \\end{equation}\n\\end{itemize}\nFinally, since $\\|u\\|_{L^\\infty(\\Ccal)} \\leqslant 1$, we get the desired estimate on the Hölder norm of $u$ \n\\[ \\|u\\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\epsilon^{\\alpha}}.\\]\nThis achieves the proof of the Hölder regularity of the solution $u$ of $\\min E_\\eps(u,\\ell)$, when $\\ell$ is fixed in $\\curve^\\Lambda$.\n\\end{proof}", "post_theorem_intro_text_len": 5460, "post_theorem_intro_text": "The study of this energy is motivated by the $\\Gamma$-convergence result established in \\cite{bonnivard2025phasefieldapproximationplateaus}, where this decoupled functional was introduced both in the proof of the $\\Gamma$-convergence and for numerical applications. Recall that the phase-field approach they proposed is a generalization to Plateau’s problem of the method introduced in \\cite{bonnivard2015approximation} for Steiner’s problem. A similar study of the existence of solutions for the decoupled functional in the Steiner case was carried out in \\cite{bonnivard2018phase}. The present work is inspired by this paper, but there are some differences that we explain below.\n\nTo prove Theorem~\\ref{main th}, we first consider the optimization problems with either $u$ or $\\ell$ fixed. As usual, establishing the existence of minimizers requires both a compactness result and the lower semi-continuity of the functional. The compactness follows from the choice of the class of competitors, so the main issue is to establish lower semi-continuity.\n\nWe begin by studying the following minimization problem \n\\begin{equation}\n\\label{pb min l fixe}\n    \\inf_{u} E_\\varepsilon(u,\\ell), \\text{ where } \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1).\n\\end{equation}\nThe following Proposition will be proven in Section~\\ref{section l fixe}. \n\\begin{proposition}\nFor any given $\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0, \\gamma_1)$, there exists a unique minimizer $u \\in H^1(\\mathcal{C})$ of $ E_\\varepsilon(\\cdot,l)$ which is globally $C^{0,\\alpha}$, for all $0< \\alpha <1$: there exists a constant $C_\\alpha>0$ such that \n\\[\\|u \\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]\n\\end{proposition}\nThe assumption that the surface $S_\\ell$ is $\\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity, and hence the existence of minimizers in this setting. We then apply classical results from elliptic regularity to derive Hölder estimates for the minimizer. Note that these estimates depend explicitly on $\\Lambda$ and $\\varepsilon$, but are independent of the specific surface $S_\\ell$. This part is strongly inspired by \\cite{bonnivard2018phase}.\n\nThen, in Section~\\ref{section u fixe} we consider the minimization problem\n\\begin{equation}\n\\label{pb min u fixe}\n    \\inf_{\\ell \\in \\mathrm{Hom}^\\Lambda} E_\\varepsilon(u,\\ell), \\text{ where } u \\in H^1(\\mathcal{C})\\cap C(\\overline{\\mathcal{C}}),\n\\end{equation}\nand prove the following Proposition.\n\\begin{proposition}\n\\label{prop 1.2}\nFor all $u \\in H^1({\\mathcal C})\\cap C(\\overline{\\mathcal{C}})$ there exists a minimizer for the problem \\eqref{pb min u fixe}. \n\\end{proposition}\n\nAs previously mentionned, the main difficulty in the proof of Proposition~\\ref{prop 1.2} lies in establishing the lower semicontinuity of the functional, which relies on the following Lemma.\n\n \\begin{lemma}\n \\label{golab}\nLet $\\ell_n : [0,1] \\times  \\mathbb{S}^1 \\to \\mathbb{R}^3$ be a sequence of uniformly Lipschitz functions, which converges uniformly to a Lipschitz function $\\ell  : [0,1] \\times  \\mathbb{S}^1 \\to \\mathbb{R}^3$.\nThen, for all open set $A$\n\n$$\\liminf \\mathcal{H}^{2} \\mathbin{\\vrule height 1.6ex depth 0pt width 0.13ex\\vrule height 0.13ex depth 0pt width 1.3ex} S_{\\ell_n}(A) \\geqslant \\mathcal{H}^{2} \\mathbin{\\vrule height 1.6ex depth 0pt width 0.13ex\\vrule height 0.13ex depth 0pt width 1.3ex} S_\\ell(A),$$\nwhere $S_\\ell$ (respectively $S_{\\ell_n}$) denotes the image of the Lipschitz function $\\ell$ (respectively $\\ell_n)$.\n\\end{lemma}\n\nThis Lemma, proven in Section~\\ref{section u fixe}, constitutes a central and original contribution of the present article and relies crucially on the specific definition of the class of surfaces under consideration. It can be viewed as a generalization of Go\\l{}\\k{a}b’s theorem to the two-dimensional Hausdorff measure, and its proof follows a similar strategy. In Go\\l{}\\k{a}b’s theorem, the first step consists in establishing the rectifiability of the limit set; in our setting, this property is immediate, since the surfaces are assumed to be images of Lipschitz functions. We then study the weak limit of the measures $(\\mathcal{H}^{2} \\mathbin{\\vrule height 1.6ex depth 0pt width 0.13ex\\vrule height 0.13ex depth 0pt width 1.3ex} S_{\\ell_n})$ and, using the tangent plane of the limit surface, derive a lower bound on the upper density of the limit measure to conclude. The key point of our proof relies on a topological argument, stating that a Lipschitz image, which is uniformly close to a disk, must have at least almost the density of a disk (see the proof of Lemma~\\ref{golab}).\n\nWe emphasize that the proof of this Lemma does not rely on the specific definition set $\\mathbb{S}^1 \\times [0,1]$; it only assumes it to be a two-dimensional subset of $\\mathbb{R}^3$. Moreover, the result is also expected to remain valid in dimensions higher than two, although the proof would then require more delicate topological arguments.\n\nFinally, in Section~\\ref{section rien fixe}, Theorem~\\ref{main th} is proved by combining these results, in particular the Hölder regularity and the preceding Lemma.\n\n\\textbf{Acknowledgments.}\n I would like to thank Antoine Lemenant for suggesting this problem to me as well as Matthieu Bonnivard for their help in this project. \nThis work was partially supported by the IUF grant of Antoine Lemenant and by the ANR project STOIQUES.", "sketch": "To prove Theorem~\\ref{main th}, the authors “first consider the optimization problems with either $u$ or $\\ell$ fixed.” Existence requires “both a compactness result and the lower semi-continuity of the functional”; “the compactness follows from the choice of the class of competitors,” so “the main issue is to establish lower semi-continuity.”\n\n1) **Fix $\\ell$**: study \\eqref{pb min l fixe}. For any $\\ell\\in\\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$ there is a “unique minimizer $u\\in H^1(\\mathcal C)$” which is “globally $C^{0,\\alpha}$,” with the stated estimate. Here, “the assumption that the surface $S_\\ell$ is $\\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity,” and then “classical results from elliptic regularity” give Hölder estimates (depending on $\\Lambda$ and $\\varepsilon$ but “independent of the specific surface $S_\\ell$”).\n\n2) **Fix $u$**: study \\eqref{pb min u fixe}. For $u\\in H^1(\\mathcal C)\\cap C(\\overline{\\mathcal C})$ there exists a minimizer in $\\ell$. The “main difficulty” is again lower semicontinuity, which “relies on” Lemma~\\ref{golab}, a Go\\l{}\\k{a}b-type lower semicontinuity statement for $\\mathcal H^2\\llcorner S_{\\ell_n}$ under uniform convergence of uniformly Lipschitz parametrizations. Its proof “follows a similar strategy” to Go\\l{}\\k{a}b: rectifiability is “immediate,” then one studies “the weak limit of the measures,” and “using the tangent plane of the limit surface, derive[s] a lower bound on the upper density of the limit measure to conclude”; the “key point” is “a topological argument” that a Lipschitz image uniformly close to a disk has “at least almost the density of a disk.”\n\n3) **Conclusion**: “Finally, in Section~\\ref{section rien fixe}, Theorem~\\ref{main th} is proved by combining these results, in particular the Hölder regularity and the preceding Lemma.”", "expanded_sketch": "To prove the main theorem, the authors “first consider the optimization problems with either $u$ or $\\ell$ fixed.” Existence requires “both a compactness result and the lower semi-continuity of the functional”; “the compactness follows from the choice of the class of competitors,” so “the main issue is to establish lower semi-continuity.”\n\n1) **Fix $\\ell$**: study \\eqref{pb min l fixe}. For any $\\ell\\in\\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$ there is a “unique minimizer $u\\in H^1(\\mathcal C)$” which is “globally $C^{0,\\alpha}$,” with the stated estimate. Here, “the assumption that the surface $S_\\ell$ is $\\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity,” and then “classical results from elliptic regularity” give Hölder estimates (depending on $\\Lambda$ and $\\varepsilon$ but “independent of the specific surface $S_\\ell$”).\n\n2) **Fix $u$**: study\n\\begin{equation}\n\\label{pb min u fixe}\n    \\inf_{\\ell \\in \\curve^\\Lambda} E_\\varepsilon(u,\\ell), \\text{ where } u \\in H^1(\\Ccal)\\cap C(\\overline{\\Ccal}),\n\\end{equation}\nFor $u\\in H^1(\\mathcal C)\\cap C(\\overline{\\mathcal C})$ there exists a minimizer in $\\ell$. The “main difficulty” is again lower semicontinuity, which “relies on” the following lemma.\n\\begin{lemma}\n \\label{golab}\nLet $\\ell_n : [0,1] \\times  \\mathbb{S}^1 \\to \\R^3$ be a sequence of uniformly Lipschitz functions, which converges uniformly to a Lipschitz function $\\ell  : [0,1] \\times  \\mathbb{S}^1 \\to \\R^3$.\nThen, for all open set $A$\n\n$$\\liminf \\Haus \\mres S_{\\ell_n}(A) \\geqslant \\Haus \\mres S_\\ell(A),$$\nwhere $S_\\ell$ (respectively $S_{\\ell_n}$) denotes the image of the Lipschitz function $\\ell$ (respectively $\\ell_n)$.\n\\end{lemma}\nIts proof “follows a similar strategy” to Go\\l{}\\k{a}b: rectifiability is “immediate,” then one studies “the weak limit of the measures,” and “using the tangent plane of the limit surface, derive[s] a lower bound on the upper density of the limit measure to conclude”; the “key point” is “a topological argument” that a Lipschitz image uniformly close to a disk has “at least almost the density of a disk.”\n\n3) **Conclusion**: “Finally, next we conclude the proof of the main theorem by combining these results, in particular the Hölder regularity and the preceding lemma.”", "expanded_theorem": "\\label{main th}\n\nThere exists a minimizing couple $(u,\\ell)$ of the following problem in both variables\n\\begin{equation}\n    \\label{pb min}\n    \\min\\left \\{E_\\varepsilon(u,\\ell), \\, u \\in  H^{1}({\\mathcal C}) \\text{ such that } 0\\leqslant u \\leqslant 1 , \\, u = 1 \\text{ on } \\partial{{\\mathcal C}} \\text{ and } \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1) \\right\\}.\n\\end{equation}\n\nMoreover, the minimizer $u$ is $\\alpha$-Hölder continuous for all $0<\\alpha<1$ with the estimate:\n\\[\\|u\\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]", "theorem_type": ["Existence", "Universal"], "mcq": {"question": "Fix \\(\\varepsilon>0\\), \\(\\Lambda>0\\), positive numbers \\(c_{\\varepsilon}\\) and \\(\\delta_{\\varepsilon}\\), a smooth bounded convex domain \\(\\mathcal C\\), a set \\(\\mathcal C_0\\) with \\(\\overline{\\mathcal C_0}\\subset \\mathcal C\\), and closed curves \\(\\gamma_0,\\gamma_1\\). For a Lipschitz homotopy \\(\\ell:[0,1]\\times \\mathbb S^1\\to \\overline{\\mathcal C_0}\\) with \\(\\ell(0)=\\gamma_0\\) and \\(\\ell(1)=\\gamma_1\\), let \\(\\Sigma_{\\ell}=\\ell([0,1]\\times \\mathbb S^1)\\). Define \\(\\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\) as the set of such \\(\\ell\\) satisfying \\(\\mathrm{Lip}(\\ell)\\le \\Lambda\\) and such that \\(\\Sigma_{\\ell}\\) is \\(\\Lambda\\)-upper Ahlfors regular (i.e. \\(\\mathcal H^2(\\Sigma_{\\ell}\\cap B(x,r))\\le \\Lambda r^2\\) for all balls). For admissible \\(u\\in H^1(\\mathcal C)\\) with \\(0\\le u\\le 1\\) and \\(u=1\\) on \\(\\partial\\mathcal C\\), define\n\\[\nE_{\\varepsilon}(u,\\ell)=\\varepsilon\\int_{\\mathcal C}|\\nabla u|^2\\,dx+\\frac{1}{4\\varepsilon}\\int_{\\mathcal C}(1-u)^2\\,dx+\\frac{1}{c_{\\varepsilon}}\\int_{\\Sigma_{\\ell}}(u^2+\\delta_{\\varepsilon})\\,d\\mathcal H^2.\n\\]\nWhich statement holds for the minimization problem over all admissible pairs \\((u,\\ell)\\)?", "correct_choice": {"label": "A", "text": "There exists at least one admissible pair \\((u,\\ell)\\) that minimizes \\(E_{\\varepsilon}(u,\\ell)\\) jointly among all pairs with \\(u\\in H^1(\\mathcal C)\\), \\(0\\le u\\le 1\\), \\(u=1\\) on \\(\\partial\\mathcal C\\), and \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\). Moreover, for such a minimizing pair, the function \\(u\\) belongs to \\(C^{0,\\alpha}(\\mathcal C)\\) for every \\(0<\\alpha<1\\), and it satisfies\n\\[\n\\|u\\|_{C^{0,\\alpha}(\\mathcal C)}\\le C_{\\alpha}\\,\\frac{1+\\Lambda c_{\\varepsilon}^{-1}}{\\varepsilon^{\\alpha}}.\n\\]"}, "choices": [{"label": "B", "text": "There exists at least one admissible pair \\((u,\\ell)\\) that minimizes \\(E_{\\varepsilon}(u,\\ell)\\) jointly among all pairs with \\(u\\in H^1(\\mathcal C)\\), \\(0\\le u\\le 1\\), \\(u=1\\) on \\(\\partial\\mathcal C\\), and \\(\\ell\\in \\mathrm{Hom}(\\gamma_0,\\gamma_1)\\) (that is, without imposing the uniform Lipschitz bound or the \\(\\Lambda\\)-upper Ahlfors regularity on \\(\\Sigma_\\ell\\)). Moreover, for such a minimizing pair, the function \\(u\\) belongs to \\(C^{0,\\alpha}(\\mathcal C)\\) for every \\(0<\\alpha<1\\), and it satisfies\n\\[\n\\|u\\|_{C^{0,\\alpha}(\\mathcal C)}\\le C_{\\alpha}\\,\\frac{1+\\Lambda c_{\\varepsilon}^{-1}}{\\varepsilon^{\\alpha}}.\n\\]"}, {"label": "C", "text": "There exists at least one admissible pair \\((u,\\ell)\\) that minimizes \\(E_{\\varepsilon}(u,\\ell)\\) jointly among all pairs with \\(u\\in H^1(\\mathcal C)\\), \\(0\\le u\\le 1\\), \\(u=1\\) on \\(\\partial\\mathcal C\\), and \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\)."}, {"label": "D", "text": "For every admissible \\(u\\in H^1(\\mathcal C)\\) with \\(0\\le u\\le 1\\) and \\(u=1\\) on \\(\\partial\\mathcal C\\), there exists \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\) minimizing \\(E_{\\varepsilon}(u,\\ell)\\) over \\(\\ell\\); and for every admissible \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\), there exists a minimizer \\(u\\) of \\(E_{\\varepsilon}(u,\\ell)\\) over \\(u\\). Consequently, there exists a pair \\((u,\\ell)\\) such that \\(u\\) minimizes \\(E_{\\varepsilon}(\\,b7\\,,\\ell)\\) and \\(\\ell\\) minimizes \\(E_{\\varepsilon}(u,\\,\\u00b7\\,)\\), and this pair minimizes \\(E_{\\varepsilon}\\) jointly over all admissible pairs. Moreover, for such a pair,\n\\[\n\\|u\\|_{C^{0,\\alpha}(\\mathcal C)}\\le C_{\\alpha}\\,\\frac{1+\\Lambda c_{\\varepsilon}^{-1}}{\\varepsilon^{\\alpha}}\n\\]\nfor every \\(0<\\alpha<1\\)."}, {"label": "E", "text": "There exists at least one admissible pair \\((u,\\ell)\\) that minimizes \\(E_{\\varepsilon}(u,\\ell)\\) jointly among all pairs with \\(u\\in H^1(\\mathcal C)\\), \\(0\\le u\\le 1\\), \\(u=1\\) on \\(\\partial\\mathcal C\\), and \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\). Moreover, for such a minimizing pair, the function \\(u\\) belongs to \\(C^{0,\\alpha}(\\mathcal C)\\) for every \\(0<\\alpha<1\\), and it satisfies\n\\[\n\\|u\\|_{C^{0,\\alpha}(\\mathcal C)}\\le C_{\\alpha}\\,\\frac{1+\\Lambda c_{\\varepsilon}^{-1}}{\\varepsilon}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "necessity_of_uniform_Lipschitz_and_Ahlfors_class_for_compactness_lsc", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped_Holder_regularity_conclusion_and_estimate", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "from_separate_minimizers_to_joint_minimizer", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "epsilon_scaling_in_Holder_estimate", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state or strongly hint at the exact correct conclusion; it only sets up the functional and asks which existence result is valid."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: choice A essentially states the full theorem, while the others are perturbations of it. Still, the options do force comparison of different strengths of conclusion."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to reject the unrestricted class, unjustified uniqueness, and overstrong Lipschitz regularity, and to distinguish the strongest valid statement from a weaker true one. But the task remains largely recognition of the theorem statement."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: one weakens the admissible class, one gives a weaker true statement, one overclaims uniqueness, and one overstates regularity. These reflect realistic failure modes."}, "total_score": 6, "overall_assessment": "A reasonably strong MCQ with good distractors and little answer leakage, but it leans toward theorem recognition rather than deep generative reasoning."}}
{"id": "2602.08476v1", "paper_link": "http://arxiv.org/abs/2602.08476v1", "theorems_cnt": 7, "theorem": {"env_name": "theorem", "content": "\\label{main th}\n\nThere exists a minimizing couple $(u,\\ell)$ of the following problem in both variables\n\\begin{equation}\n    \\label{pb min}\n    \\min\\left \\{E_\\varepsilon(u,\\ell), \\, u \\in  H^{1}({\\mathcal C}) \\text{ such that } 0\\leqslant u \\leqslant 1 , \\, u = 1 \\text{ on } \\partial{{\\mathcal C}} \\text{ and } \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1) \\right\\}.\n\\end{equation}\n\nMoreover, the minimizer $u$ is $\\alpha$-Hölder continuous for all $0<\\alpha<1$ with the estimate:\n\\[\\|u\\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]", "start_pos": 9246, "end_pos": 9845, "label": "main th"}, "ref_dict": {"prop 1.2": "\\begin{proposition}\n\\label{prop 1.2}\nFor all $u \\in H^1(\\C)\\cap C(\\overline{\\Ccal})$ there exists a minimizer for the problem \\eqref{pb min u fixe}. \n\\end{proposition}", "pb min u fixe": "\\begin{equation}\n\\label{pb min u fixe}\n    \\inf_{\\ell \\in \\curve^\\Lambda} E_\\varepsilon(u,\\ell), \\text{ where } u \\in H^1(\\Ccal)\\cap C(\\overline{\\Ccal}),\n\\end{equation}", "main th": "\\begin{theorem}\n\\label{main th}\n\nThere exists a minimizing couple $(u,\\ell)$ of the following problem in both variables\n\\begin{equation}\n    \\label{pb min}\n    \\min\\left \\{E_\\eps(u,\\ell), \\, u \\in  H^{1}(\\C) \\text{ such that } 0\\leqslant u \\leqslant 1 , \\, u = 1 \\text{ on } \\partial{\\C} \\text{ and } \\ell \\in \\curve^\\Lambda(\\gamma_0,\\gamma_1) \\right\\}.\n\\end{equation}\n\nMoreover, the minimizer $u$ is $\\alpha$-Hölder continuous for all $0<\\alpha<1$ with the estimate:\n\\[\\|u\\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]\n\n\\end{theorem}", "golab": "\\begin{lemma}\n \\label{golab}\nLet $\\ell_n : [0,1] \\times  \\mathbb{S}^1 \\to \\R^3$ be a sequence of uniformly Lipschitz functions, which converges uniformly to a Lipschitz function $\\ell  : [0,1] \\times  \\mathbb{S}^1 \\to \\R^3$.\nThen, for all open set $A$\n\n$$\\liminf \\Haus \\mres S_{\\ell_n}(A) \\geqslant \\Haus \\mres S_\\ell(A),$$\nwhere $S_\\ell$ (respectively $S_{\\ell_n}$) denotes the image of the Lipschitz function $\\ell$ (respectively $\\ell_n)$.\n\\end{lemma}", "Def:geoDistance": "\\begin{definition}\\label{Def:geoDistance}\n\tLet $(\\delta_\\eps)_{\\eps>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n\t\\begin{align*}\n\t\t\t&d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{  \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\eps) d\\Haus \\ |\\ \\ell \\in \\curve^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n\t\\end{align*}\n\\end{definition}"}, "pre_theorem_intro_text_len": 5458, "pre_theorem_intro_text": "This paper is devoted to the study of the functional introduced in \\cite{bonnivard2025phasefieldapproximationplateaus} to approximate a Plateau problem, which consists in finding a surface of minimal area spanning a collection of closed curves $\\gamma_0,..., \\gamma_n$ contained in the boundary of an open, bounded, convex set $\\Ccal_0 \\subset \\mathbb{R}^3$. \n\nMore precisely, the competitor surfaces are defined as the images of homotopies connecting the given curves. We define below the set of admissible homotopies connecting a curve $\\gamma_i$ to $\\gamma_j$:\n\\[\n\\mathrm{Hom}(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Lip}([0,1] \\times \\mathbb{S}^{1}, \\overline{\\Ccal_0}) \\text{ such that } \\ell(0) = \\gamma_i \\text{ and } \\ell(1) = \\gamma_{j} \\}.\n\\]\nNote that we require more than mere homotopies: we specifically consider Lipschitz homotopies. For any homotopy $\\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j)$, we define the associated surface $\\surf_\\ell$ as its image:\n\\begin{equation}\\label{Def:surface_image}\n\t\\surf_{\\ell} := \\ell([0,1] \\times \\mathbb{S}^{1}) \\subset \\overline{\\Ccal_0}.\n\\end{equation}\nWe emphasize that the surface $\\surf_{\\ell}$ is $\\mathcal{H}^{2}$-rectifiable and has finite $\\mathcal{H}^{2}$-measure. Moreover, for any $u\\in H^1(\\mathbb{R}^3)$, the trace of $u$ on $\\surf_{\\ell}$ is well defined, since the set of points that are not Lebesgue points of $u$ is $\\mathcal{H}^2$-negligible. \nHowever, to obtain the existence result, we need to impose additional properties on the competitors.\nFor $\\Lambda>0$, we consider\n\\[\n\\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j) \\text{ such that }  \\;\\mathrm{Lip}(\\ell)\\leq \\Lambda \\text{ and } \\surf_\\ell \\text{ is } \\Lambda-\\text{upper Ahlfors regular} \\ \\}.\\]\n\nLet us recall the definition of upper Ahlfors regularity.\n\\begin{definition}\nLet $E\\subset \\mathbb{R}^3$ and $\\Lambda>0$. The set $E$ is said to be $\\Lambda$-upper Ahlfors regular if for all $x\\in E$ and all $r>0$ we have \n\\[\\mathcal{H}^{2}(E\\cap B(x,r)) \\leqslant \\Lambda \\pi r^2.\\]\n\\end{definition}\n\nThe definition of our functional is based on the notion of a generalized geodesic distance between curves, associated with a given weight function $u$, which is defined as follows.\n\\begin{definition}\\label{Def:geoDistance}\n\tLet $(\\delta_\\varepsilon)_{\\varepsilon>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n\t\\begin{align*}\n\t\t\t&d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{  \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2} \\ |\\ \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n\t\\end{align*}\n\\end{definition}\nNote that, in \\cite{bonnivard2025phasefieldapproximationplateaus}, the geodesic distance is defined for Lipschitz homotopies $\\ell$. The assumptions of uniform Lipschitz regularity and uniform upper Ahlfors regularity of the images are not required there. However, in the present work, we need to impose these assumptions in order to obtain the compactness result necessary for the existence theorem.\n\nLet $\\mathcal{C}$ an smooth open bounded convex set containing the closure $\\overline{\\Ccal_0}$. This set $\\mathcal{C}$ will serve as our domain of study. Indeed, since we rely on PDE techniques, it is convenient to work within a smooth bounded domain. We now recall the definition of the functional introduced in \\cite{bonnivard2025phasefieldapproximationplateaus}:\n\t\\begin{equation}\n\t\t\\label{functionalGeneral}\n\t\tF_{\\varepsilon}(u) := \\varepsilon \\int_{\\mathcal{C}}|\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2}dx + \\frac{1}{c_{\\varepsilon}}\\sum_{i=1}^{n}d_{u}(\\gamma_0, \\gamma_{i}).\n\t\\end{equation}\nWhen, the sequences of positive numbers $(\\delta_\\varepsilon)$ and $(c_\\varepsilon)$ are assumed to converge to zero, and to satisfy that $\\delta_\\varepsilon/c_\\varepsilon$ converges to zero as $\\varepsilon \\to 0$, \\cite{bonnivard2025phasefieldapproximationplateaus} establishes that this functional approximate some Plateau's problem through a $\\Gamma$-convergence result. \n\nFor simplicity, we will assume in the following that the prescribed boundary consists of only two curves, $\\gamma_0$ and $\\gamma_1$. However, the results established in the present article remain valid when the boundary contains more than two curves.\n\nIn this paper, we are specifically interested in the decoupled functional defined below.\n\\begin{definition}\n\tFor $u \\in H^{1}({\\mathcal C})$ such that $0\\leqslant u \\leqslant 1$, with $u = 1$ on $\\partial{{\\mathcal C}}$, and $\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$,\n\t\\begin{align*}\n\t\t&E_{\\varepsilon}(u,\\ell) := \\varepsilon\\int_{\\mathcal{C}} |\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2} dx + \\frac{1}{c_{\\varepsilon}}\\int_{\\surf_{\\ell}} (u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2}.\n\t\\end{align*}\n\\end{definition}\n\n\\begin{remark}\n\tFrom Definition~\\ref{Def:geoDistance} of the geodesic distance between closed curves, we know that for all $u\\in H^{1}(\\mathcal{C})$ such that $0\\leqslant u \\leqslant 1$ and $u = 1$ on $\\partial{\\mathcal{C}}$,  \n\t\t\\[F_{\\varepsilon}(u) = \\inf\\left \\{ E_{\\varepsilon}(u,\\ell)\\ ,\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)  \\right \\}.\\]\n\\end{remark}\n\nThe main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\\varepsilon$\n\\begin{equation*}\n    \\inf_{(u,\\ell)} E_\\varepsilon(u,\\ell).\n\\end{equation*}", "context": "More precisely, the competitor surfaces are defined as the images of homotopies connecting the given curves. We define below the set of admissible homotopies connecting a curve $\\gamma_i$ to $\\gamma_j$:\n\\[\n\\mathrm{Hom}(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Lip}([0,1] \\times \\mathbb{S}^{1}, \\overline{\\Ccal_0}) \\text{ such that } \\ell(0) = \\gamma_i \\text{ and } \\ell(1) = \\gamma_{j} \\}.\n\\]\nNote that we require more than mere homotopies: we specifically consider Lipschitz homotopies. For any homotopy $\\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j)$, we define the associated surface $\\surf_\\ell$ as its image:\n\\begin{equation}\\label{Def:surface_image}\n \\surf_{\\ell} := \\ell([0,1] \\times \\mathbb{S}^{1}) \\subset \\overline{\\Ccal_0}.\n\\end{equation}\nWe emphasize that the surface $\\surf_{\\ell}$ is $\\mathcal{H}^{2}$-rectifiable and has finite $\\mathcal{H}^{2}$-measure. Moreover, for any $u\\in H^1(\\mathbb{R}^3)$, the trace of $u$ on $\\surf_{\\ell}$ is well defined, since the set of points that are not Lebesgue points of $u$ is $\\mathcal{H}^2$-negligible. \nHowever, to obtain the existence result, we need to impose additional properties on the competitors.\nFor $\\Lambda>0$, we consider\n\\[\n\\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j) \\text{ such that } \\;\\mathrm{Lip}(\\ell)\\leq \\Lambda \\text{ and } \\surf_\\ell \\text{ is } \\Lambda-\\text{upper Ahlfors regular} \\ \\}.\\]\n\nThe definition of our functional is based on the notion of a generalized geodesic distance between curves, associated with a given weight function $u$, which is defined as follows.\n\\begin{definition}\\label{Def:geoDistance}\n Let $(\\delta_\\varepsilon)_{\\varepsilon>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n \\begin{align*}\n &d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{ \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2} \\ |\\ \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n \\end{align*}\n\\end{definition}\nNote that, in \\cite{bonnivard2025phasefieldapproximationplateaus}, the geodesic distance is defined for Lipschitz homotopies $\\ell$. The assumptions of uniform Lipschitz regularity and uniform upper Ahlfors regularity of the images are not required there. However, in the present work, we need to impose these assumptions in order to obtain the compactness result necessary for the existence theorem.\n\nLet $\\mathcal{C}$ an smooth open bounded convex set containing the closure $\\overline{\\Ccal_0}$. This set $\\mathcal{C}$ will serve as our domain of study. Indeed, since we rely on PDE techniques, it is convenient to work within a smooth bounded domain. We now recall the definition of the functional introduced in \\cite{bonnivard2025phasefieldapproximationplateaus}:\n \\begin{equation}\n \\label{functionalGeneral}\n F_{\\varepsilon}(u) := \\varepsilon \\int_{\\mathcal{C}}|\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2}dx + \\frac{1}{c_{\\varepsilon}}\\sum_{i=1}^{n}d_{u}(\\gamma_0, \\gamma_{i}).\n \\end{equation}\nWhen, the sequences of positive numbers $(\\delta_\\varepsilon)$ and $(c_\\varepsilon)$ are assumed to converge to zero, and to satisfy that $\\delta_\\varepsilon/c_\\varepsilon$ converges to zero as $\\varepsilon \\to 0$, \\cite{bonnivard2025phasefieldapproximationplateaus} establishes that this functional approximate some Plateau's problem through a $\\Gamma$-convergence result.\n\nIn this paper, we are specifically interested in the decoupled functional defined below.\n\\begin{definition}\n For $u \\in H^{1}({\\mathcal C})$ such that $0\\leqslant u \\leqslant 1$, with $u = 1$ on $\\partial{{\\mathcal C}}$, and $\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$,\n \\begin{align*}\n &E_{\\varepsilon}(u,\\ell) := \\varepsilon\\int_{\\mathcal{C}} |\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2} dx + \\frac{1}{c_{\\varepsilon}}\\int_{\\surf_{\\ell}} (u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2}.\n \\end{align*}\n\\end{definition}\n\n\\begin{remark}\n From Definition~\\ref{Def:geoDistance} of the geodesic distance between closed curves, we know that for all $u\\in H^{1}(\\mathcal{C})$ such that $0\\leqslant u \\leqslant 1$ and $u = 1$ on $\\partial{\\mathcal{C}}$, \n \\[F_{\\varepsilon}(u) = \\inf\\left \\{ E_{\\varepsilon}(u,\\ell)\\ ,\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1) \\right \\}.\\]\n\\end{remark}\n\nThe main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\\varepsilon$\n\\begin{equation*}\n \\inf_{(u,\\ell)} E_\\varepsilon(u,\\ell).\n\\end{equation*}\n\n\\begin{definition}\\label{Def:geoDistance}\n\tLet $(\\delta_\\eps)_{\\eps>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n\t\\begin{align*}\n\t\t\t&d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{  \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\eps) d\\Haus \\ |\\ \\ell \\in \\curve^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n\t\\end{align*}\n\\end{definition}", "full_context": "More precisely, the competitor surfaces are defined as the images of homotopies connecting the given curves. We define below the set of admissible homotopies connecting a curve $\\gamma_i$ to $\\gamma_j$:\n\\[\n\\mathrm{Hom}(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Lip}([0,1] \\times \\mathbb{S}^{1}, \\overline{\\Ccal_0}) \\text{ such that } \\ell(0) = \\gamma_i \\text{ and } \\ell(1) = \\gamma_{j} \\}.\n\\]\nNote that we require more than mere homotopies: we specifically consider Lipschitz homotopies. For any homotopy $\\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j)$, we define the associated surface $\\surf_\\ell$ as its image:\n\\begin{equation}\\label{Def:surface_image}\n \\surf_{\\ell} := \\ell([0,1] \\times \\mathbb{S}^{1}) \\subset \\overline{\\Ccal_0}.\n\\end{equation}\nWe emphasize that the surface $\\surf_{\\ell}$ is $\\mathcal{H}^{2}$-rectifiable and has finite $\\mathcal{H}^{2}$-measure. Moreover, for any $u\\in H^1(\\mathbb{R}^3)$, the trace of $u$ on $\\surf_{\\ell}$ is well defined, since the set of points that are not Lebesgue points of $u$ is $\\mathcal{H}^2$-negligible. \nHowever, to obtain the existence result, we need to impose additional properties on the competitors.\nFor $\\Lambda>0$, we consider\n\\[\n\\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j) : = \\{ \\ell \\in \\mathrm{Hom}(\\gamma_i,\\gamma_j) \\text{ such that } \\;\\mathrm{Lip}(\\ell)\\leq \\Lambda \\text{ and } \\surf_\\ell \\text{ is } \\Lambda-\\text{upper Ahlfors regular} \\ \\}.\\]\n\nThe definition of our functional is based on the notion of a generalized geodesic distance between curves, associated with a given weight function $u$, which is defined as follows.\n\\begin{definition}\\label{Def:geoDistance}\n Let $(\\delta_\\varepsilon)_{\\varepsilon>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n \\begin{align*}\n &d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{ \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2} \\ |\\ \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n \\end{align*}\n\\end{definition}\nNote that, in \\cite{bonnivard2025phasefieldapproximationplateaus}, the geodesic distance is defined for Lipschitz homotopies $\\ell$. The assumptions of uniform Lipschitz regularity and uniform upper Ahlfors regularity of the images are not required there. However, in the present work, we need to impose these assumptions in order to obtain the compactness result necessary for the existence theorem.\n\nLet $\\mathcal{C}$ an smooth open bounded convex set containing the closure $\\overline{\\Ccal_0}$. This set $\\mathcal{C}$ will serve as our domain of study. Indeed, since we rely on PDE techniques, it is convenient to work within a smooth bounded domain. We now recall the definition of the functional introduced in \\cite{bonnivard2025phasefieldapproximationplateaus}:\n \\begin{equation}\n \\label{functionalGeneral}\n F_{\\varepsilon}(u) := \\varepsilon \\int_{\\mathcal{C}}|\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2}dx + \\frac{1}{c_{\\varepsilon}}\\sum_{i=1}^{n}d_{u}(\\gamma_0, \\gamma_{i}).\n \\end{equation}\nWhen, the sequences of positive numbers $(\\delta_\\varepsilon)$ and $(c_\\varepsilon)$ are assumed to converge to zero, and to satisfy that $\\delta_\\varepsilon/c_\\varepsilon$ converges to zero as $\\varepsilon \\to 0$, \\cite{bonnivard2025phasefieldapproximationplateaus} establishes that this functional approximate some Plateau's problem through a $\\Gamma$-convergence result.\n\nIn this paper, we are specifically interested in the decoupled functional defined below.\n\\begin{definition}\n For $u \\in H^{1}({\\mathcal C})$ such that $0\\leqslant u \\leqslant 1$, with $u = 1$ on $\\partial{{\\mathcal C}}$, and $\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$,\n \\begin{align*}\n &E_{\\varepsilon}(u,\\ell) := \\varepsilon\\int_{\\mathcal{C}} |\\nabla u|^{2} dx + \\frac{1}{4\\varepsilon}\\int_{\\mathcal{C}}(1-u)^{2} dx + \\frac{1}{c_{\\varepsilon}}\\int_{\\surf_{\\ell}} (u^2 + \\delta_\\varepsilon) d\\mathcal{H}^{2}.\n \\end{align*}\n\\end{definition}\n\n\\begin{remark}\n From Definition~\\ref{Def:geoDistance} of the geodesic distance between closed curves, we know that for all $u\\in H^{1}(\\mathcal{C})$ such that $0\\leqslant u \\leqslant 1$ and $u = 1$ on $\\partial{\\mathcal{C}}$, \n \\[F_{\\varepsilon}(u) = \\inf\\left \\{ E_{\\varepsilon}(u,\\ell)\\ ,\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1) \\right \\}.\\]\n\\end{remark}\n\nThe main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\\varepsilon$\n\\begin{equation*}\n \\inf_{(u,\\ell)} E_\\varepsilon(u,\\ell).\n\\end{equation*}\n\n\\begin{definition}\\label{Def:geoDistance}\n\tLet $(\\delta_\\eps)_{\\eps>0}$ be a sequence of positive numbers and $u \\in C(\\overline{\\Ccal_0})$. We define the geodesic distance between $\\gamma_i$ and $\\gamma_j$:\n\t\\begin{align*}\n\t\t\t&d_{u}(\\gamma_i,\\gamma_j) := \\inf\\left \\{  \\int_{\\surf_{\\ell}}( u^2 + \\delta_\\eps) d\\Haus \\ |\\ \\ell \\in \\curve^\\Lambda(\\gamma_i,\\gamma_j)\\right \\}.\n\t\\end{align*}\n\\end{definition}\n\nThe main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\\varepsilon$\n\\begin{equation*}\n \\inf_{(u,\\ell)} E_\\varepsilon(u,\\ell).\n\\end{equation*}\n\nThere exists a minimizing couple $(u,\\ell)$ of the following problem in both variables\n\\begin{equation}\n \\label{pb min}\n \\min\\left \\{E_\\eps(u,\\ell), \\, u \\in H^{1}(\\C) \\text{ such that } 0\\leqslant u \\leqslant 1 , \\, u = 1 \\text{ on } \\partial{\\C} \\text{ and } \\ell \\in \\curve^\\Lambda(\\gamma_0,\\gamma_1) \\right\\}.\n\\end{equation}\n\nThe study of this energy is motivated by the $\\Gamma$-convergence result established in \\cite{bonnivard2025phasefieldapproximationplateaus}, where this decoupled functional was introduced both in the proof of the $\\Gamma$-convergence and for numerical applications. Recall that the phase-field approach they proposed is a generalization to Plateau’s problem of the method introduced in \\cite{bonnivard2015approximation} for Steiner’s problem. A similar study of the existence of solutions for the decoupled functional in the Steiner case was carried out in \\cite{bonnivard2018phase}. The present work is inspired by this paper, but there are some differences that we explain below.\n\nWe begin by studying the following minimization problem \n\\begin{equation}\n\\label{pb min l fixe}\n \\inf_{u} E_\\varepsilon(u,\\ell), \\text{ where } \\ell \\in \\curve^\\Lambda(\\gamma_0,\\gamma_1).\n\\end{equation}\nThe following Proposition will be proven in Section~\\ref{section l fixe}. \n\\begin{proposition}\nFor any given $\\ell \\in \\curve^\\Lambda(\\gamma_0, \\gamma_1)$, there exists a unique minimizer $u \\in H^1(\\Ccal)$ of $ E_\\varepsilon(\\cdot,l)$ which is globally $C^{0,\\alpha}$, for all $0< \\alpha <1$: there exists a constant $C_\\alpha>0$ such that \n\\[\\|u \\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]\n\\end{proposition}\nThe assumption that the surface $S_\\ell$ is $\\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity, and hence the existence of minimizers in this setting. We then apply classical results from elliptic regularity to derive Hölder estimates for the minimizer. Note that these estimates depend explicitly on $\\Lambda$ and $\\varepsilon$, but are independent of the specific surface $S_\\ell$. This part is strongly inspired by \\cite{bonnivard2018phase}.\n\n\\begin{proposition}\n\\label{prop holder}\nLet $\\ell \\in~\\curve^\\Lambda(\\gamma_0,\\gamma_1)$ and let $u$ be the minimizer of $E_\\varepsilon^\\ell$. Then, $u$ is $\\alpha$-Hölder continuous for all $0<\\alpha<1$ with the estimate:\n\\[\\|u\\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]\n\\end{proposition}\n\n\\begin{lemma}\n\\label{lem 2.10}\nRecall the definition of the parameter $\\eta_0 = \\dist(\\partial \\Ccal_0,\\partial \\Ccal)$ introduced in the previous Lemma.\nLet $\\ell \\in \\curve^\\Lambda(\\gamma_0, \\gamma_1)$, $0< \\rho < \\min(1,\\frac{\\eta_0}{4})$ and $x_0 \\in \\mathcal{C}$ such that $\\dist(x_0,\\partial \\mathcal{C}) \\geqslant \\frac{\\eta_0}{2} $. Then, $u \\in W^{1,p}(B(x_0,\\rho))$ for all $3\\leqslant p < \\infty$ and we have the following estimate on the rescaled function $u_\\rho (x) := u(x_0+\\rho x)$\n\\begin{equation}\n \\|\\nabla u_\\rho\\|_{L^p(B_1)} \\leqslant C_p\\left( \\frac{\\rho^2}{\\varepsilon^2}+1+\\frac{\\Lambda \\rho}{c_\\varepsilon \\varepsilon}\\right).\n\\end{equation}\n\\end{lemma}\n\n\\medskip\n\\emph{Step 3.} We define the rescaled function $u_\\rho$ on $B_2$ by $u_\\rho(x) := u(x_0+\\rho x)$. Then, for all $\\varphi \\in C^\\infty_c(B_2)$ \n\\begin{align*}\n \\int_{B_2}\\nabla u_\\rho \\cdot \\nabla\\varphi \\dm x &= \\frac{1}{\\rho}\\int_{B(x_0,2\\rho)}\\nabla u \\cdot \\nabla\\varphi_\\rho \\dm x,\\\\\n &= \\frac{1}{\\rho4\\varepsilon^2}\\int_{B(x_0,2\\rho)}(1- u)\\varphi_\\rho \\dm x - \\frac{1}{\\rho c_\\varepsilon \\varepsilon}B[\\ell](u,\\varphi_\\rho),\\\\\n &= \\frac{\\rho^2}{4\\varepsilon^2}\\int_{B_2}(1- u_\\rho)\\varphi \\dm x - \\frac{1}{\\rho c_\\varepsilon \\varepsilon}\\langle T_\\rho,\\varphi \\rangle.\n\\end{align*}\nHence, $u_\\rho$ satisfies the following PDE in $\\mathcal{D}'(B_2)$\n\\[-\\Delta u_\\rho = \\frac{\\rho^2}{4\\varepsilon^2}(1- u_\\rho) - \\frac{1}{\\rho c_\\varepsilon \\varepsilon} T_\\rho.\\]\nWe denote $w_\\rho := u_\\rho + \\frac{1}{\\rho c_\\varepsilon \\varepsilon}v_\\rho \\in H^1(B_2)\\cap L^\\infty(B_2)$. And we deduce that \n\\[-\\Delta w_\\rho = \\frac{\\rho^2}{4\\varepsilon^2}(1- u_\\rho) \\text{ in } \\mathcal{D}'(B_2). \\]\nThus, \\cite[Corollary 8.36]{gilbargelliptic} yields that $w_\\rho \\in C_{\\mathrm{loc}}^{1,\\alpha}(B_2)$ for all $\\alpha >0$ and \n\\begin{align*}\n \\|\\nabla w_\\rho\\|_{L^\\infty(B_1)} \\leqslant \\|w_\\rho\\|_{C^{1,\\alpha}(B_1)} &\\leqslant C\\left(\\| w_\\rho\\|_{L^\\infty(B_1)} +\\frac{\\rho^2}{4\\varepsilon^2}\\|1- u_\\rho\\|_{L^\\infty(B_1)}\\right),\\\\\n &= C\\left(\\| u_\\rho\\|_{L^\\infty(B_1)}+ \\frac{1}{\\rho c_\\varepsilon \\varepsilon}\\| v_\\rho\\|_{L^\\infty(B_1)} +\\frac{\\rho^2}{4\\varepsilon^2}\\right),\\\\\n &\\leqslant C\\left(1 + \\frac{\\Lambda \\rho}{c_\\varepsilon \\varepsilon} + \\frac{\\rho^2}{4\\varepsilon^2}\\right).\n\\end{align*}\nFinally, we can conclude that $u_\\rho = w_\\rho - \\frac{1}{\\rho c_\\varepsilon \\varepsilon}v_\\rho \\in W^{1,p}(B_1)$ with the desired estimate\n\\[ \\|\\nabla u_\\rho\\|_{L^p(B_1)} \\leqslant C_p\\left( \\frac{\\rho^2}{\\varepsilon^2}+1+\\frac{\\Lambda \\rho}{c_\\varepsilon \\varepsilon}\\right).\\]\nThis concludes the proof of this Lemma.\n\\end{proof}\n\nNow let's consider the case where $\\dist(x_0, \\partial \\mathcal{C}) > \\frac{\\eta_0}{2}> \\varepsilon$, in particular this implies that $B(x_0,\\varepsilon) \\subset \\C$. In this case, Lemma~\\ref{lem 2.10}, applied with $\\rho = \\varepsilon$ and $p = \\frac{3}{1-\\alpha}$ yields the estimate on the rescaled function $u_\\varepsilon$\n \\[\\|\\nabla u_\\varepsilon\\|_{L^p(B_1)} \\leqslant C_p\\left(1+ \\frac{\\Lambda}{c_\\varepsilon}\\right).\\]\n Since we have shown that the $L^\\infty$ norm of $u$ is smaller than $1$, we deduce that the $L^p$ norm of the scaled function $u_\\varepsilon$ is bounded by the volume of the domain. And we conclude by applying the Sobolev embedding Theorem (see for instance \\cite[Theorem 4.12]{adams2003sobolev}),\n \\begin{equation}\n \\|u_\\varepsilon\\|_{C^{0,\\alpha}(B_1)} \\leqslant C_\\alpha\\left(1+ \\frac{\\Lambda}{c_\\varepsilon}\\right).\n \\end{equation}\n Hence, scaling back leads to \n \\begin{equation}\n \\frac{|u(x)-u(y)|}{|x-y|^\\alpha} \\leqslant \\frac{1}{\\varepsilon^\\alpha} \\|u_\\varepsilon\\|_{C^{0,\\alpha}(B_1)} \\leqslant \\frac{C_\\alpha}{\\varepsilon^\\alpha}\\left(1+ \\frac{\\Lambda}{c_\\varepsilon}\\right).\n \\end{equation}\n\\end{itemize}\nFinally, since $\\|u\\|_{L^\\infty(\\Ccal)} \\leqslant 1$, we get the desired estimate on the Hölder norm of $u$ \n\\[ \\|u\\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\epsilon^{\\alpha}}.\\]\nThis achieves the proof of the Hölder regularity of the solution $u$ of $\\min E_\\eps(u,\\ell)$, when $\\ell$ is fixed in $\\curve^\\Lambda$.\n\\end{proof}", "post_theorem_intro_text_len": 5460, "post_theorem_intro_text": "The study of this energy is motivated by the $\\Gamma$-convergence result established in \\cite{bonnivard2025phasefieldapproximationplateaus}, where this decoupled functional was introduced both in the proof of the $\\Gamma$-convergence and for numerical applications. Recall that the phase-field approach they proposed is a generalization to Plateau’s problem of the method introduced in \\cite{bonnivard2015approximation} for Steiner’s problem. A similar study of the existence of solutions for the decoupled functional in the Steiner case was carried out in \\cite{bonnivard2018phase}. The present work is inspired by this paper, but there are some differences that we explain below.\n\nTo prove Theorem~\\ref{main th}, we first consider the optimization problems with either $u$ or $\\ell$ fixed. As usual, establishing the existence of minimizers requires both a compactness result and the lower semi-continuity of the functional. The compactness follows from the choice of the class of competitors, so the main issue is to establish lower semi-continuity.\n\nWe begin by studying the following minimization problem \n\\begin{equation}\n\\label{pb min l fixe}\n    \\inf_{u} E_\\varepsilon(u,\\ell), \\text{ where } \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1).\n\\end{equation}\nThe following Proposition will be proven in Section~\\ref{section l fixe}. \n\\begin{proposition}\nFor any given $\\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0, \\gamma_1)$, there exists a unique minimizer $u \\in H^1(\\mathcal{C})$ of $ E_\\varepsilon(\\cdot,l)$ which is globally $C^{0,\\alpha}$, for all $0< \\alpha <1$: there exists a constant $C_\\alpha>0$ such that \n\\[\\|u \\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]\n\\end{proposition}\nThe assumption that the surface $S_\\ell$ is $\\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity, and hence the existence of minimizers in this setting. We then apply classical results from elliptic regularity to derive Hölder estimates for the minimizer. Note that these estimates depend explicitly on $\\Lambda$ and $\\varepsilon$, but are independent of the specific surface $S_\\ell$. This part is strongly inspired by \\cite{bonnivard2018phase}.\n\nThen, in Section~\\ref{section u fixe} we consider the minimization problem\n\\begin{equation}\n\\label{pb min u fixe}\n    \\inf_{\\ell \\in \\mathrm{Hom}^\\Lambda} E_\\varepsilon(u,\\ell), \\text{ where } u \\in H^1(\\mathcal{C})\\cap C(\\overline{\\mathcal{C}}),\n\\end{equation}\nand prove the following Proposition.\n\\begin{proposition}\n\\label{prop 1.2}\nFor all $u \\in H^1({\\mathcal C})\\cap C(\\overline{\\mathcal{C}})$ there exists a minimizer for the problem \\eqref{pb min u fixe}. \n\\end{proposition}\n\nAs previously mentionned, the main difficulty in the proof of Proposition~\\ref{prop 1.2} lies in establishing the lower semicontinuity of the functional, which relies on the following Lemma.\n\n \\begin{lemma}\n \\label{golab}\nLet $\\ell_n : [0,1] \\times  \\mathbb{S}^1 \\to \\mathbb{R}^3$ be a sequence of uniformly Lipschitz functions, which converges uniformly to a Lipschitz function $\\ell  : [0,1] \\times  \\mathbb{S}^1 \\to \\mathbb{R}^3$.\nThen, for all open set $A$\n\n$$\\liminf \\mathcal{H}^{2} \\mathbin{\\vrule height 1.6ex depth 0pt width 0.13ex\\vrule height 0.13ex depth 0pt width 1.3ex} S_{\\ell_n}(A) \\geqslant \\mathcal{H}^{2} \\mathbin{\\vrule height 1.6ex depth 0pt width 0.13ex\\vrule height 0.13ex depth 0pt width 1.3ex} S_\\ell(A),$$\nwhere $S_\\ell$ (respectively $S_{\\ell_n}$) denotes the image of the Lipschitz function $\\ell$ (respectively $\\ell_n)$.\n\\end{lemma}\n\nThis Lemma, proven in Section~\\ref{section u fixe}, constitutes a central and original contribution of the present article and relies crucially on the specific definition of the class of surfaces under consideration. It can be viewed as a generalization of Go\\l{}\\k{a}b’s theorem to the two-dimensional Hausdorff measure, and its proof follows a similar strategy. In Go\\l{}\\k{a}b’s theorem, the first step consists in establishing the rectifiability of the limit set; in our setting, this property is immediate, since the surfaces are assumed to be images of Lipschitz functions. We then study the weak limit of the measures $(\\mathcal{H}^{2} \\mathbin{\\vrule height 1.6ex depth 0pt width 0.13ex\\vrule height 0.13ex depth 0pt width 1.3ex} S_{\\ell_n})$ and, using the tangent plane of the limit surface, derive a lower bound on the upper density of the limit measure to conclude. The key point of our proof relies on a topological argument, stating that a Lipschitz image, which is uniformly close to a disk, must have at least almost the density of a disk (see the proof of Lemma~\\ref{golab}).\n\nWe emphasize that the proof of this Lemma does not rely on the specific definition set $\\mathbb{S}^1 \\times [0,1]$; it only assumes it to be a two-dimensional subset of $\\mathbb{R}^3$. Moreover, the result is also expected to remain valid in dimensions higher than two, although the proof would then require more delicate topological arguments.\n\nFinally, in Section~\\ref{section rien fixe}, Theorem~\\ref{main th} is proved by combining these results, in particular the Hölder regularity and the preceding Lemma.\n\n\\textbf{Acknowledgments.}\n I would like to thank Antoine Lemenant for suggesting this problem to me as well as Matthieu Bonnivard for their help in this project. \nThis work was partially supported by the IUF grant of Antoine Lemenant and by the ANR project STOIQUES.", "sketch": "To prove Theorem~\\ref{main th}, the authors “first consider the optimization problems with either $u$ or $\\ell$ fixed.” Existence requires “both a compactness result and the lower semi-continuity of the functional”; “the compactness follows from the choice of the class of competitors,” so “the main issue is to establish lower semi-continuity.”\n\n1) **Fix $\\ell$**: study \\eqref{pb min l fixe}. For any $\\ell\\in\\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$ there is a “unique minimizer $u\\in H^1(\\mathcal C)$” which is “globally $C^{0,\\alpha}$,” with the stated estimate. Here, “the assumption that the surface $S_\\ell$ is $\\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity,” and then “classical results from elliptic regularity” give Hölder estimates (depending on $\\Lambda$ and $\\varepsilon$ but “independent of the specific surface $S_\\ell$”).\n\n2) **Fix $u$**: study \\eqref{pb min u fixe}. For $u\\in H^1(\\mathcal C)\\cap C(\\overline{\\mathcal C})$ there exists a minimizer in $\\ell$. The “main difficulty” is again lower semicontinuity, which “relies on” Lemma~\\ref{golab}, a Go\\l{}\\k{a}b-type lower semicontinuity statement for $\\mathcal H^2\\llcorner S_{\\ell_n}$ under uniform convergence of uniformly Lipschitz parametrizations. Its proof “follows a similar strategy” to Go\\l{}\\k{a}b: rectifiability is “immediate,” then one studies “the weak limit of the measures,” and “using the tangent plane of the limit surface, derive[s] a lower bound on the upper density of the limit measure to conclude”; the “key point” is “a topological argument” that a Lipschitz image uniformly close to a disk has “at least almost the density of a disk.”\n\n3) **Conclusion**: “Finally, in Section~\\ref{section rien fixe}, Theorem~\\ref{main th} is proved by combining these results, in particular the Hölder regularity and the preceding Lemma.”", "expanded_sketch": "To prove the main theorem, the authors “first consider the optimization problems with either $u$ or $\\ell$ fixed.” Existence requires “both a compactness result and the lower semi-continuity of the functional”; “the compactness follows from the choice of the class of competitors,” so “the main issue is to establish lower semi-continuity.”\n\n1) **Fix $\\ell$**: study \\eqref{pb min l fixe}. For any $\\ell\\in\\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1)$ there is a “unique minimizer $u\\in H^1(\\mathcal C)$” which is “globally $C^{0,\\alpha}$,” with the stated estimate. Here, “the assumption that the surface $S_\\ell$ is $\\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity,” and then “classical results from elliptic regularity” give Hölder estimates (depending on $\\Lambda$ and $\\varepsilon$ but “independent of the specific surface $S_\\ell$”).\n\n2) **Fix $u$**: study\n\\begin{equation}\n\\label{pb min u fixe}\n    \\inf_{\\ell \\in \\curve^\\Lambda} E_\\varepsilon(u,\\ell), \\text{ where } u \\in H^1(\\Ccal)\\cap C(\\overline{\\Ccal}),\n\\end{equation}\nFor $u\\in H^1(\\mathcal C)\\cap C(\\overline{\\mathcal C})$ there exists a minimizer in $\\ell$. The “main difficulty” is again lower semicontinuity, which “relies on” the following lemma.\n\\begin{lemma}\n \\label{golab}\nLet $\\ell_n : [0,1] \\times  \\mathbb{S}^1 \\to \\R^3$ be a sequence of uniformly Lipschitz functions, which converges uniformly to a Lipschitz function $\\ell  : [0,1] \\times  \\mathbb{S}^1 \\to \\R^3$.\nThen, for all open set $A$\n\n$$\\liminf \\Haus \\mres S_{\\ell_n}(A) \\geqslant \\Haus \\mres S_\\ell(A),$$\nwhere $S_\\ell$ (respectively $S_{\\ell_n}$) denotes the image of the Lipschitz function $\\ell$ (respectively $\\ell_n)$.\n\\end{lemma}\nIts proof “follows a similar strategy” to Go\\l{}\\k{a}b: rectifiability is “immediate,” then one studies “the weak limit of the measures,” and “using the tangent plane of the limit surface, derive[s] a lower bound on the upper density of the limit measure to conclude”; the “key point” is “a topological argument” that a Lipschitz image uniformly close to a disk has “at least almost the density of a disk.”\n\n3) **Conclusion**: “Finally, next we conclude the proof of the main theorem by combining these results, in particular the Hölder regularity and the preceding lemma.”", "expanded_theorem": "\\label{main th}\n\nThere exists a minimizing couple $(u,\\ell)$ of the following problem in both variables\n\\begin{equation}\n    \\label{pb min}\n    \\min\\left \\{E_\\varepsilon(u,\\ell), \\, u \\in  H^{1}({\\mathcal C}) \\text{ such that } 0\\leqslant u \\leqslant 1 , \\, u = 1 \\text{ on } \\partial{{\\mathcal C}} \\text{ and } \\ell \\in \\mathrm{Hom}^\\Lambda(\\gamma_0,\\gamma_1) \\right\\}.\n\\end{equation}\n\nMoreover, the minimizer $u$ is $\\alpha$-Hölder continuous for all $0<\\alpha<1$ with the estimate:\n\\[\\|u\\|_{C^{0,\\alpha}(\\mathcal{C})} \\leqslant C_\\alpha \\frac{1+\\Lambda c_\\varepsilon ^{-1}}{\\varepsilon^{\\alpha}}.\\]", "theorem_type": ["Existence", "Universal"], "mcq": {"question": "Fix \\(\\varepsilon>0\\), \\(\\Lambda>0\\), positive numbers \\(c_{\\varepsilon}\\) and \\(\\delta_{\\varepsilon}\\), a smooth bounded convex domain \\(\\mathcal C\\), a set \\(\\mathcal C_0\\) with \\(\\overline{\\mathcal C_0}\\subset \\mathcal C\\), and closed curves \\(\\gamma_0,\\gamma_1\\). For a Lipschitz homotopy \\(\\ell:[0,1]\\times \\mathbb S^1\\to \\overline{\\mathcal C_0}\\) with \\(\\ell(0)=\\gamma_0\\) and \\(\\ell(1)=\\gamma_1\\), let \\(\\Sigma_{\\ell}=\\ell([0,1]\\times \\mathbb S^1)\\). Define \\(\\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\) as the set of such \\(\\ell\\) satisfying \\(\\mathrm{Lip}(\\ell)\\le \\Lambda\\) and such that \\(\\Sigma_{\\ell}\\) is \\(\\Lambda\\)-upper Ahlfors regular (i.e. \\(\\mathcal H^2(\\Sigma_{\\ell}\\cap B(x,r))\\le \\Lambda r^2\\) for all balls). For admissible \\(u\\in H^1(\\mathcal C)\\) with \\(0\\le u\\le 1\\) and \\(u=1\\) on \\(\\partial\\mathcal C\\), define\n\\[\nE_{\\varepsilon}(u,\\ell)=\\varepsilon\\int_{\\mathcal C}|\\nabla u|^2\\,dx+\\frac{1}{4\\varepsilon}\\int_{\\mathcal C}(1-u)^2\\,dx+\\frac{1}{c_{\\varepsilon}}\\int_{\\Sigma_{\\ell}}(u^2+\\delta_{\\varepsilon})\\,d\\mathcal H^2.\n\\]\nWhich statement holds for the minimization problem over all admissible pairs \\((u,\\ell)\\)?", "correct_choice": {"label": "A", "text": "There exists at least one admissible pair \\((u,\\ell)\\) that minimizes \\(E_{\\varepsilon}(u,\\ell)\\) jointly among all pairs with \\(u\\in H^1(\\mathcal C)\\), \\(0\\le u\\le 1\\), \\(u=1\\) on \\(\\partial\\mathcal C\\), and \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\). Moreover, for such a minimizing pair, the function \\(u\\) belongs to \\(C^{0,\\alpha}(\\mathcal C)\\) for every \\(0<\\alpha<1\\), and it satisfies\n\\[\n\\|u\\|_{C^{0,\\alpha}(\\mathcal C)}\\le C_{\\alpha}\\,\\frac{1+\\Lambda c_{\\varepsilon}^{-1}}{\\varepsilon^{\\alpha}}.\n\\]"}, "choices": [{"label": "B", "text": "There exists at least one admissible pair \\((u,\\ell)\\) that minimizes \\(E_{\\varepsilon}(u,\\ell)\\) jointly among all pairs with \\(u\\in H^1(\\mathcal C)\\), \\(0\\le u\\le 1\\), \\(u=1\\) on \\(\\partial\\mathcal C\\), and \\(\\ell\\in \\mathrm{Hom}(\\gamma_0,\\gamma_1)\\) (that is, without imposing the uniform Lipschitz bound or the \\(\\Lambda\\)-upper Ahlfors regularity on \\(\\Sigma_\\ell\\)). Moreover, for such a minimizing pair, the function \\(u\\) belongs to \\(C^{0,\\alpha}(\\mathcal C)\\) for every \\(0<\\alpha<1\\), and it satisfies\n\\[\n\\|u\\|_{C^{0,\\alpha}(\\mathcal C)}\\le C_{\\alpha}\\,\\frac{1+\\Lambda c_{\\varepsilon}^{-1}}{\\varepsilon^{\\alpha}}.\n\\]"}, {"label": "C", "text": "There exists at least one admissible pair \\((u,\\ell)\\) that minimizes \\(E_{\\varepsilon}(u,\\ell)\\) jointly among all pairs with \\(u\\in H^1(\\mathcal C)\\), \\(0\\le u\\le 1\\), \\(u=1\\) on \\(\\partial\\mathcal C\\), and \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\)."}, {"label": "D", "text": "For every admissible \\(u\\in H^1(\\mathcal C)\\) with \\(0\\le u\\le 1\\) and \\(u=1\\) on \\(\\partial\\mathcal C\\), there exists \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\) minimizing \\(E_{\\varepsilon}(u,\\ell)\\) over \\(\\ell\\); and for every admissible \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\), there exists a minimizer \\(u\\) of \\(E_{\\varepsilon}(u,\\ell)\\) over \\(u\\). Consequently, there exists a pair \\((u,\\ell)\\) such that \\(u\\) minimizes \\(E_{\\varepsilon}(\\,b7\\,,\\ell)\\) and \\(\\ell\\) minimizes \\(E_{\\varepsilon}(u,\\,\\u00b7\\,)\\), and this pair minimizes \\(E_{\\varepsilon}\\) jointly over all admissible pairs. Moreover, for such a pair,\n\\[\n\\|u\\|_{C^{0,\\alpha}(\\mathcal C)}\\le C_{\\alpha}\\,\\frac{1+\\Lambda c_{\\varepsilon}^{-1}}{\\varepsilon^{\\alpha}}\n\\]\nfor every \\(0<\\alpha<1\\)."}, {"label": "E", "text": "There exists at least one admissible pair \\((u,\\ell)\\) that minimizes \\(E_{\\varepsilon}(u,\\ell)\\) jointly among all pairs with \\(u\\in H^1(\\mathcal C)\\), \\(0\\le u\\le 1\\), \\(u=1\\) on \\(\\partial\\mathcal C\\), and \\(\\ell\\in \\mathrm{Hom}^{\\Lambda}(\\gamma_0,\\gamma_1)\\). Moreover, for such a minimizing pair, the function \\(u\\) belongs to \\(C^{0,\\alpha}(\\mathcal C)\\) for every \\(0<\\alpha<1\\), and it satisfies\n\\[\n\\|u\\|_{C^{0,\\alpha}(\\mathcal C)}\\le C_{\\alpha}\\,\\frac{1+\\Lambda c_{\\varepsilon}^{-1}}{\\varepsilon}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "necessity_of_uniform_Lipschitz_and_Ahlfors_class_for_compactness_lsc", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped_Holder_regularity_conclusion_and_estimate", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "from_separate_minimizers_to_joint_minimizer", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "epsilon_scaling_in_Holder_estimate", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only the setup and definitions of the admissible class and energy. It does not explicitly state the existence, Hölder regularity, or the precise estimate that appears in the correct choice."}, "TAS": {"score": 1, "justification": "The item is close to theorem recognition: the correct option essentially reproduces the theorem statement for the given setup. However, it is not a pure tautology because the alternatives modify key hypotheses and conclusions in nontrivial ways."}, "GPS": {"score": 1, "justification": "Selecting the correct answer requires some reasoning about compactness assumptions, strength of conclusions, and the scaling in the Hölder estimate. Still, the task is mainly identifying the exact theorem statement rather than deriving a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically meaningful: one drops needed compactness/regularity assumptions, one gives a weaker true statement, one confuses separate minimization with joint minimization, and one alters the epsilon scaling. These are plausible and distinct failure modes."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and high-quality distractors, but it leans toward theorem-statement recognition rather than deep generative reasoning."}}
{"id": "2602.08490v1", "paper_link": "http://arxiv.org/abs/2602.08490v1", "theorems_cnt": 1, "theorem": {"env_name": "mainthm", "content": "\\label{thm:deux-bulles}\n\t\tThere exists a solution $u: (-\\infty, T_0] \\to \\mathcal{E}$ of \\eqref{har} such that\n\t\t\\begin{equation}\n\t\t\t\\label{eq:mainthm}\n\t\t\t\\lim_{t\\to -\\infty}\\Big\\|u(t) - \\Big({-}iW + W_{\\kappa|t|^{-\\frac{2}{N-6}}}\\Big)\\Big\\|_\\mathcal{E} = 0,\n\t\t\\end{equation}\n\t\twhere $\\kappa$ is an explicit constant, see \\eqref{eq:kappa}.", "start_pos": 12019, "end_pos": 12375, "label": "thm:deux-bulles"}, "ref_dict": {"eq:mod-th": "\\begin{align}\n\t\t\t\\label{eq:mod-zeta}\n\t\t\t|\\zeta'(t)| &\\leq c|t|^{-\\frac{N-3}{N-6}}, \\\\\n\t\t\t\\label{eq:mod-mu}\n\t\t\t|\\mu'(t)| &\\leq c|t|^{-\\frac{N-3}{N-6}}, \\\\\n\t\t\t\\label{eq:mod-l}\n\t\t\t\\Big|\\lambda'(t) - \\frac{3C_2}{\\|W\\|_{L^2}^2} \\lambda(t)^\\frac{N-4}{2}\\Big| &\\leq c|t|^{-\\frac{2N-7}{2(N-6)}}, \\\\\n\t\t\t\\label{eq:mod-th}\n\t\t\t\\Big|\\theta'(t)+\\frac{C_3}{\\|W\\|_{L^2}^2}\\theta(t)\\lambda(t)^\\frac{N-6}{2}-\\frac{K(t)}{\\lambda(t)^2\\|W\\|_{L^2}^2}\\Big| &\\leq c|t|^{-\\frac{N-5}{N-6}},\n\t\t\\end{align}", "rem:mod": "\\begin{remark}\n        \\label{rem:mod}\n        The proof below crucially relies on our choice of the orthogonality conditions \\eqref{eq:orth}, which requires $W, \\Lambda W \\in \\dot H^{-1}(\\bR^N)$, equivalently $N \\geq 7$.\n    \\end{remark}", "thm:deux-bulles": "\\begin{mainthm}\n\t\t\\label{thm:deux-bulles}\n\t\tThere exists a solution $u: (-\\infty, T_0] \\to \\cE$ of \\eqref{har} such that\n\t\t\\begin{equation}\n\t\t\t\\label{eq:mainthm}\n\t\t\t\\lim_{t\\to -\\infty}\\Big\\|u(t) - \\Big({-}iW + W_{\\kappa|t|^{-\\frac{2}{N-6}}}\\Big)\\Big\\|_\\cE = 0,\n\t\t\\end{equation}\n\t\twhere $\\kappa$ is an explicit constant, see \\eqref{eq:kappa}.\n\t\\end{mainthm}", "energy": "\\begin{equation}\\label{energy}\n\t\tE(u_0) := \\int_{\\bR^N} \\frac 12|\\grad u_0(x)|^2 - F(u_0(x))\\ud x,\n\t\\end{equation}", "lem:mod": "\\begin{lemma}\n\t\t\\label{lem:mod}\n\t\tLet $c > 0$ be an arbitrarily small constant. Let $T_0 < 0$ with $|T_0|$ large enough (depending on $c$)\n\t\tand $T < T_1 \\leq T_0$. Suppose that for $T \\leq t \\leq T_1$ there holds\n\t\t\\begin{align}\n\t\t\t\\big|\\zeta(t) + \\frac{\\pi}{2}\\big| &\\leq |t|^{-\\frac{3}{N-6}}, \\label{eq:bootstrap-zeta} \\\\\n\t\t\t|\\mu(t) - 1| &\\leq |t|^{-\\frac{3}{N-6}}, \\label{eq:bootstrap-mu} \\\\\n\t\t\t|\\theta(t)| &\\leq |t|^{-\\frac{1}{N-6}}, \\label{eq:bootstrap-theta} \\\\\n\t\t\t\\big|\\lambda(t) - \\kappa|t|^{-\\frac{2}{N-6}}\\big| &\\leq |t|^{-\\frac{5}{2(N-6)}}, \\label{eq:bootstrap-lambda} \\\\\n\t\t\t\\|g\\|_\\cE &\\leq |t|^{-\\frac{N-1}{2(N-6)}}. \\label{eq:bootstrap-g}\n\t\t\\end{align}\n\t\tThen\n\t\t\\begin{align}\n\t\t\t\\label{eq:mod-zeta}\n\t\t\t|\\zeta'(t)| &\\leq c|t|^{-\\frac{N-3}{N-6}}, \\\\\n\t\t\t\\label{eq:mod-mu}\n\t\t\t|\\mu'(t)| &\\leq c|t|^{-\\frac{N-3}{N-6}}, \\\\\n\t\t\t\\label{eq:mod-l}\n\t\t\t\\Big|\\lambda'(t) - \\frac{3C_2}{\\|W\\|_{L^2}^2} \\lambda(t)^\\frac{N-4}{2}\\Big| &\\leq c|t|^{-\\frac{2N-7}{2(N-6)}}, \\\\\n\t\t\t\\label{eq:mod-th}\n\t\t\t\\Big|\\theta'(t)+\\frac{C_3}{\\|W\\|_{L^2}^2}\\theta(t)\\lambda(t)^\\frac{N-6}{2}-\\frac{K(t)}{\\lambda(t)^2\\|W\\|_{L^2}^2}\\Big| &\\leq c|t|^{-\\frac{N-5}{N-6}},\n\t\t\\end{align}\n\t\tfor $T \\leq t \\leq T_1$, where\n\t\t\\begin{equation}\n\t\t\t\\label{eq:kappa}\n\t\t\t\\kappa := \\Big(\\frac{2C_1}{3(N-6)C_2}\\Big)^{\\frac{2}{N-6}},\n\t\t\\end{equation}\n\t\t\\begin{equation}\n\t\t\t\\label{eq:K-def}\n\t\t\tK := -\\big\\la \\eee^{i\\theta}\\Lambda W_\\lambda, f(\\eee^{i\\zeta}W_{\\mu} + \\eee^{i\\theta}W_{\\lambda} + g) - f(\\eee^{i\\zeta}W_\\mu + \\eee^{i\\theta}W_\\lambda) - f'(\\eee^{i\\zeta}W_\\mu + \\eee^{i\\theta}W_\\lambda)g \\big\\ra.\n\t\t\\end{equation}\n\t\\end{lemma}", "lem:proper": "\\begin{lemma}\n\t\t\\label{lem:proper}\n\t\tUnder assumptions of Lemma~\\ref{lem:mod}, for $t \\in [T, T_1]$ there holds\n\t\t\\begin{align}\n\t\t\t\\big| \\dd t a_1^+(t) - \\frac{\\nu}{\\mu(t)^2}a_1^+(t)\\big| &\\leq \\frac{c}{\\mu(t)^2}|t|^{-\\frac{N}{2(N-6)}}, \\label{eq:proper-1p} \\\\\n\t\t\t\\big| \\dd t a_1^-(t) + \\frac{\\nu}{\\mu(t)^2}a_1^-(t)\\big| &\\leq \\frac{c}{\\mu(t)^2}|t|^{-\\frac{N}{2(N-6)}}, \\label{eq:proper-1m} \\\\\n\t\t\t\\big| \\dd t a_2^+(t) - \\frac{\\nu}{\\lambda(t)^2}a_2^+(t)\\big| &\\leq \\frac{c}{\\lambda(t)^2}|t|^{-\\frac{N}{2(N-6)}}, \\label{eq:proper-2p} \\\\\n\t\t\t\\big| \\dd t a_2^-(t) + \\frac{\\nu}{\\lambda(t)^2}a_2^-(t)\\big| &\\leq \\frac{c}{\\lambda(t)^2}|t|^{-\\frac{N}{2(N-6)}}, \\label{eq:proper-2m}\n\t\t\\end{align}\n\t\twith $c \\to 0$ as $|T_0| \\to +\\infty$.\n\t\\end{lemma}", "eq:kappa": "\\begin{equation}\n\t\t\t\\label{eq:kappa}\n\t\t\t\\kappa := \\Big(\\frac{2C_1}{3(N-6)C_2}\\Big)^{\\frac{2}{N-6}},\n\t\t\\end{equation}", "har": "\\begin{equation}\n\t\t\\label{har}\n\t\ti \\partial_t u + \\Delta u  + \\left( |x|^{-4}*|u|^2\\right)u =0,\\quad f(u) :=  \\left( |x|^{-4} * |u|^2\\right)u, \\quad\n\t\t(t,x)\\in \\bR \\times \\bR^N.\n\t\\end{equation}", "eq:bootstrap-theta": "\\begin{align}\n\t\t\t\\big|\\zeta(t) + \\frac{\\pi}{2}\\big| &\\leq |t|^{-\\frac{3}{N-6}}, \\label{eq:bootstrap-zeta} \\\\\n\t\t\t|\\mu(t) - 1| &\\leq |t|^{-\\frac{3}{N-6}}, \\label{eq:bootstrap-mu} \\\\\n\t\t\t|\\theta(t)| &\\leq |t|^{-\\frac{1}{N-6}}, \\label{eq:bootstrap-theta} \\\\\n\t\t\t\\big|\\lambda(t) - \\kappa|t|^{-\\frac{2}{N-6}}\\big| &\\leq |t|^{-\\frac{5}{2(N-6)}}, \\label{eq:bootstrap-lambda} \\\\\n\t\t\t\\|g\\|_\\cE &\\leq |t|^{-\\frac{N-1}{2(N-6)}}. \\label{eq:bootstrap-g}\n\t\t\\end{align}"}, "pre_theorem_intro_text_len": 6306, "pre_theorem_intro_text": "\\label{sec:intro}\n\t\\subsection{Setting of the problem}\n\t\\label{ssec:setting}\n\tWe consider the focusing, energy-critical Hartree equation\n\t\\begin{equation}\n\t\t\\label{har}\n\t\ti \\partial_t u + \\Delta u  + \\left( |x|^{-4}*|u|^2\\right)u =0,\\quad f(u) :=  \\left( |x|^{-4} * |u|^2\\right)u, \\quad\n\t\t(t,x)\\in \\mathbb{R} \\times \\mathbb{R}^N.\n\t\\end{equation}\n\tThe Hartree equation arises in the study of Boson stars and other physical phenomena, please refer to\n\t\\cite{pi}. In chemistry, it appears as a continuous-limit model for\n\tmesoscopic structures; see \\cite{grc}. \n\tFor reasons that will become clear later (see Remark~\\ref{rem:mod}), in this paper we assume $ N\\geq 7$.\n\n\tThe \\emph{energy functional} associated with this equation is defined for $u_0 \\in \\dot H^1(\\mathbb{R}^N; \\mathbb{C})$ by the formula\n\t\\begin{equation}\\label{energy}\n\t\tE(u_0) := \\int_{\\mathbb{R}^N} \\frac 12|{\\nabla} u_0(x)|^2 - F(u_0(x))\\mathrm{\\,d} x,\n\t\\end{equation}\n\twhere $F(u) := \\frac{1}{4} \\left( |x|^{-4}*|u|^2\\right)|u|^2 $. A crucial property of the solutions of \\eqref{har} is that the energy $E$ is a conservation law. The differential of $E$ is $\\mathrm{D} E(u_0) = -\\Delta u_0 - f(u_0)$, therefore, we have the following Hamiltonian form of equation \\eqref{har}:\n\t\\begin{equation}\n\t\t\\label{eq:nlsH}\n\t\t\\partial_t u(t) = -i \\mathrm{D} E(u(t)).\n\t\\end{equation}\n\n\tThe Cauchy problem for \\eqref{har} was developed in \\cite{Caz:book,\n\t\tMiaoXZ:08:LWP for Har}. That is, if $u_0 \\in \\dot H^1(\\mathbb{R}^N)$, there\n\texists a unique solution defined in a maximal interval\n\t$I=\\left(-T_-(u), T_+(u)\\right)$. The name ``energy critical'' refers to the fact that\n\tthe scaling\n\t\\begin{equation}\\label{scaling}\n\t\tu(t,x)\\rightarrow\n\t\tu_{\\lambda}(t,x)=\\lambda^{-\\frac{N-2}{2}}u\\left(\\lambda^{-2} t,\n\t\t\\lambda^{-1} x\\right), \\; \\lambda>0,\n\t\\end{equation}\n\tmakes the equation \\eqref{har} and the energy\n\t\\eqref{energy} invariant.\n\n\tThere are many results for the energy-critical Hartree equation. For the defocusing case, Miao et al., using the induction on energy argument and the localized Morawetz identity, prove the global well-posedness and scattering of the radial solution in \\cite{MiaoXZ:07:F e-critical radial Har}. Subsequently,\n\tMiao et al. use the induction on energy argument in both the frequency space and the spatial space simultaneously and the frequency-localized interaction Morawetz estimate to remove the radial assumption in \\cite{MiaoXZ:09:DF e-critical nonradial Har}.\n\n\tWhile for the focusing case, the dynamics behavior becomes\n\tmore complicated. It turns out that the explicit ground state\n\t\\begin{equation}\\label{w function}\n\t W(x)= c_0\\left(\\frac{\\lambda}{\\lambda^2+|x|^2}\\right)^{ -\\frac{N-2}{2}}\n\t\t\\; \\text{with}\\; c_0>0, \\lambda>0,\n\t\\end{equation}\n\tplays an important role in the dynamical behavior of solutions for\n\t\\eqref{har}. The functions $\\mathrm e^{i\\theta}W_\\lambda$ are called \\emph{ground states} or \\emph{bubbles} (of energy). They are the only radially symmetric solutions\n\tof the critical elliptic problem\n\t\\begin{equation}\n\t\t\\label{eq:elliptic}\n\t\t-\\Delta u - f(u) = 0.\n\t\\end{equation}\n\tAccording to \\cite{MiaoWX:dynamic gHartree, MiaoXZ:09:e-critical radial Har}, we have the variational characterization of $W$, which can be proved by combining sharp Sobolev inequality \\cite{Aubin, LiebL:book,\tTalenti:best constant} with sharp Hardy-Littlewood-Sobolev inequality \\cite{Lieb:sharp constant for HLS, LiebL:book}. Recently, the second and third authors et al.\\ give an alternative proof of the existence of the extremizer of this sharp Hardy-Littlewood-Sobolev inequality in $\\mathbb{R}^N$ by using the stereographic projection and sharp Hardy-Littlewood-Sobolev inequality on the sphere  in \\cite{LLTX:Nondegeneracy}.\n\n\tMiao et al.\\ make use of the concentration compactness principle and the rigidity argument, which are first introduced in NLS and NLW by C.~Kenig and F.~Merle in \\cite{Kenigmerle:H1 critical NLS, Kenig-merle:wave},  to prove the so-called \\emph{Threshold Conjecture} by completely classifying the dynamical behavior of solutions $u(t)$ of \\eqref{har} \\cite{\tLiMZ:e-critical Har, MiaoXZ:09:e-critical radial Har} such that $E(u(t)) < E(W)$. Later, Miao et al.\\ make use of the modulation analysis and the concentration compactness rigidity argument to prove the dynamical behavior of solutions $u(t)$ of \\eqref{har} such that $E(u(t)) = E(W)$ in \\cite{MiaoWX:dynamic gHartree}. Solutions slightly above the ground state energy threshold were studied in \\cite{LLX:dynamics ecritical Hartree}. A much stronger statement about the dynamics of solutions is the \\emph{Soliton Resolution Conjecture}, which predicts that a bounded (in an appropriate sense) solution decomposes asymptotically into a sum of energy bubbles at different scales and a radiation term (a solution of the linear Hartree equation). This was proved for the radial energy-critical wave equation in dimension $N \\geq 3$ in \\cite{DuyKenMerle:NLW, DuyKenMerle:NLWodd, DuyJiaKenMerle:NLW, JacekAndrew:NLW}. For \\eqref{har} this problem is completely open. For other dynamics results of the Hartree equation, please refer to\n\t\\cite{CaoG, GiV00, KriLR:m-critcal Har, KriMR:mass-subcritical har, LiZh:har:class, LLTX:dynamics ecritical Hartree, MiaoWX:dynamic gHartree, MXY:defocusing, MiaoXZ:07:F e-critical radial Har, MiaoXZ:08:LWP for Har, MiaoXZ:09:m-critical Har,\n\t\t\tMiaoXZ:09:p-subcritical Har, MiaoXZ:10:blowup, Na99d, ZhouTao: Threshold}.\n\n\tIn this paper, we always assume that the initial data are radially symmetric, which is preserved by the flow. We denote $\\mathcal{E}$ the space radially symmetric functions in $\\dot H^1(\\mathbb{R}^N; \\mathbb{C})$.\n\n\t\\subsection{Main results}\n\tIn view of the Soliton Resolution Conjecture, solutions which exhibit no dispersion in one or both time directions play a distinguished role.\n\tOne obvious example of such solutions are the ground states $\\mathrm e^{i\\theta}W_\\lambda$, which is a trivial result.\n\tIn this paper, we will consider the simplest non-trivial case, namely\n\twe will construct a global radial solution which approaches a sum of two bubbles in the energy space. One of the bubbles develops at scale $1$, whereas the length scale of the other converges to $0$ at rate $|t|^{-\\frac{2}{N-6}}$. The phases of the two bubbles form the right angle. That is", "context": "\\label{sec:intro}\n \\subsection{Setting of the problem}\n \\label{ssec:setting}\n We consider the focusing, energy-critical Hartree equation\n \\begin{equation}\n \\label{har}\n i \\partial_t u + \\Delta u + \\left( |x|^{-4}*|u|^2\\right)u =0,\\quad f(u) := \\left( |x|^{-4} * |u|^2\\right)u, \\quad\n (t,x)\\in \\mathbb{R} \\times \\mathbb{R}^N.\n \\end{equation}\n The Hartree equation arises in the study of Boson stars and other physical phenomena, please refer to\n \\cite{pi}. In chemistry, it appears as a continuous-limit model for\n mesoscopic structures; see \\cite{grc}. \n For reasons that will become clear later (see Remark~\\ref{rem:mod}), in this paper we assume $ N\\geq 7$.\n\nThe \\emph{energy functional} associated with this equation is defined for $u_0 \\in \\dot H^1(\\mathbb{R}^N; \\mathbb{C})$ by the formula\n \\begin{equation}\\label{energy}\n E(u_0) := \\int_{\\mathbb{R}^N} \\frac 12|{\\nabla} u_0(x)|^2 - F(u_0(x))\\mathrm{\\,d} x,\n \\end{equation}\n where $F(u) := \\frac{1}{4} \\left( |x|^{-4}*|u|^2\\right)|u|^2 $. A crucial property of the solutions of \\eqref{har} is that the energy $E$ is a conservation law. The differential of $E$ is $\\mathrm{D} E(u_0) = -\\Delta u_0 - f(u_0)$, therefore, we have the following Hamiltonian form of equation \\eqref{har}:\n \\begin{equation}\n \\label{eq:nlsH}\n \\partial_t u(t) = -i \\mathrm{D} E(u(t)).\n \\end{equation}\n\nThe Cauchy problem for \\eqref{har} was developed in \\cite{Caz:book,\n MiaoXZ:08:LWP for Har}. That is, if $u_0 \\in \\dot H^1(\\mathbb{R}^N)$, there\n exists a unique solution defined in a maximal interval\n $I=\\left(-T_-(u), T_+(u)\\right)$. The name ``energy critical'' refers to the fact that\n the scaling\n \\begin{equation}\\label{scaling}\n u(t,x)\\rightarrow\n u_{\\lambda}(t,x)=\\lambda^{-\\frac{N-2}{2}}u\\left(\\lambda^{-2} t,\n \\lambda^{-1} x\\right), \\; \\lambda>0,\n \\end{equation}\n makes the equation \\eqref{har} and the energy\n \\eqref{energy} invariant.\n\nWhile for the focusing case, the dynamics behavior becomes\n more complicated. It turns out that the explicit ground state\n \\begin{equation}\\label{w function}\n W(x)= c_0\\left(\\frac{\\lambda}{\\lambda^2+|x|^2}\\right)^{ -\\frac{N-2}{2}}\n \\; \\text{with}\\; c_0>0, \\lambda>0,\n \\end{equation}\n plays an important role in the dynamical behavior of solutions for\n \\eqref{har}. The functions $\\mathrm e^{i\\theta}W_\\lambda$ are called \\emph{ground states} or \\emph{bubbles} (of energy). They are the only radially symmetric solutions\n of the critical elliptic problem\n \\begin{equation}\n \\label{eq:elliptic}\n -\\Delta u - f(u) = 0.\n \\end{equation}\n According to \\cite{MiaoWX:dynamic gHartree, MiaoXZ:09:e-critical radial Har}, we have the variational characterization of $W$, which can be proved by combining sharp Sobolev inequality \\cite{Aubin, LiebL:book, Talenti:best constant} with sharp Hardy-Littlewood-Sobolev inequality \\cite{Lieb:sharp constant for HLS, LiebL:book}. Recently, the second and third authors et al.\\ give an alternative proof of the existence of the extremizer of this sharp Hardy-Littlewood-Sobolev inequality in $\\mathbb{R}^N$ by using the stereographic projection and sharp Hardy-Littlewood-Sobolev inequality on the sphere in \\cite{LLTX:Nondegeneracy}.\n\nIn this paper, we always assume that the initial data are radially symmetric, which is preserved by the flow. We denote $\\mathcal{E}$ the space radially symmetric functions in $\\dot H^1(\\mathbb{R}^N; \\mathbb{C})$.\n\n\\subsection{Main results}\n In view of the Soliton Resolution Conjecture, solutions which exhibit no dispersion in one or both time directions play a distinguished role.\n One obvious example of such solutions are the ground states $\\mathrm e^{i\\theta}W_\\lambda$, which is a trivial result.\n In this paper, we will consider the simplest non-trivial case, namely\n we will construct a global radial solution which approaches a sum of two bubbles in the energy space. One of the bubbles develops at scale $1$, whereas the length scale of the other converges to $0$ at rate $|t|^{-\\frac{2}{N-6}}$. The phases of the two bubbles form the right angle. That is\n\n\\begin{equation}\\label{energy}\n\t\tE(u_0) := \\int_{\\bR^N} \\frac 12|\\grad u_0(x)|^2 - F(u_0(x))\\ud x,\n\t\\end{equation}\n\n\\begin{equation}\n\t\t\t\\label{eq:kappa}\n\t\t\t\\kappa := \\Big(\\frac{2C_1}{3(N-6)C_2}\\Big)^{\\frac{2}{N-6}},\n\t\t\\end{equation}\n\n\\begin{equation}\n\t\t\\label{har}\n\t\ti \\partial_t u + \\Delta u  + \\left( |x|^{-4}*|u|^2\\right)u =0,\\quad f(u) :=  \\left( |x|^{-4} * |u|^2\\right)u, \\quad\n\t\t(t,x)\\in \\bR \\times \\bR^N.\n\t\\end{equation}\n\n\\begin{remark}\n        \\label{rem:mod}\n        The proof below crucially relies on our choice of the orthogonality conditions \\eqref{eq:orth}, which requires $W, \\Lambda W \\in \\dot H^{-1}(\\bR^N)$, equivalently $N \\geq 7$.\n    \\end{remark}", "full_context": "\\label{sec:intro}\n \\subsection{Setting of the problem}\n \\label{ssec:setting}\n We consider the focusing, energy-critical Hartree equation\n \\begin{equation}\n \\label{har}\n i \\partial_t u + \\Delta u + \\left( |x|^{-4}*|u|^2\\right)u =0,\\quad f(u) := \\left( |x|^{-4} * |u|^2\\right)u, \\quad\n (t,x)\\in \\mathbb{R} \\times \\mathbb{R}^N.\n \\end{equation}\n The Hartree equation arises in the study of Boson stars and other physical phenomena, please refer to\n \\cite{pi}. In chemistry, it appears as a continuous-limit model for\n mesoscopic structures; see \\cite{grc}. \n For reasons that will become clear later (see Remark~\\ref{rem:mod}), in this paper we assume $ N\\geq 7$.\n\nThe \\emph{energy functional} associated with this equation is defined for $u_0 \\in \\dot H^1(\\mathbb{R}^N; \\mathbb{C})$ by the formula\n \\begin{equation}\\label{energy}\n E(u_0) := \\int_{\\mathbb{R}^N} \\frac 12|{\\nabla} u_0(x)|^2 - F(u_0(x))\\mathrm{\\,d} x,\n \\end{equation}\n where $F(u) := \\frac{1}{4} \\left( |x|^{-4}*|u|^2\\right)|u|^2 $. A crucial property of the solutions of \\eqref{har} is that the energy $E$ is a conservation law. The differential of $E$ is $\\mathrm{D} E(u_0) = -\\Delta u_0 - f(u_0)$, therefore, we have the following Hamiltonian form of equation \\eqref{har}:\n \\begin{equation}\n \\label{eq:nlsH}\n \\partial_t u(t) = -i \\mathrm{D} E(u(t)).\n \\end{equation}\n\nThe Cauchy problem for \\eqref{har} was developed in \\cite{Caz:book,\n MiaoXZ:08:LWP for Har}. That is, if $u_0 \\in \\dot H^1(\\mathbb{R}^N)$, there\n exists a unique solution defined in a maximal interval\n $I=\\left(-T_-(u), T_+(u)\\right)$. The name ``energy critical'' refers to the fact that\n the scaling\n \\begin{equation}\\label{scaling}\n u(t,x)\\rightarrow\n u_{\\lambda}(t,x)=\\lambda^{-\\frac{N-2}{2}}u\\left(\\lambda^{-2} t,\n \\lambda^{-1} x\\right), \\; \\lambda>0,\n \\end{equation}\n makes the equation \\eqref{har} and the energy\n \\eqref{energy} invariant.\n\nWhile for the focusing case, the dynamics behavior becomes\n more complicated. It turns out that the explicit ground state\n \\begin{equation}\\label{w function}\n W(x)= c_0\\left(\\frac{\\lambda}{\\lambda^2+|x|^2}\\right)^{ -\\frac{N-2}{2}}\n \\; \\text{with}\\; c_0>0, \\lambda>0,\n \\end{equation}\n plays an important role in the dynamical behavior of solutions for\n \\eqref{har}. The functions $\\mathrm e^{i\\theta}W_\\lambda$ are called \\emph{ground states} or \\emph{bubbles} (of energy). They are the only radially symmetric solutions\n of the critical elliptic problem\n \\begin{equation}\n \\label{eq:elliptic}\n -\\Delta u - f(u) = 0.\n \\end{equation}\n According to \\cite{MiaoWX:dynamic gHartree, MiaoXZ:09:e-critical radial Har}, we have the variational characterization of $W$, which can be proved by combining sharp Sobolev inequality \\cite{Aubin, LiebL:book, Talenti:best constant} with sharp Hardy-Littlewood-Sobolev inequality \\cite{Lieb:sharp constant for HLS, LiebL:book}. Recently, the second and third authors et al.\\ give an alternative proof of the existence of the extremizer of this sharp Hardy-Littlewood-Sobolev inequality in $\\mathbb{R}^N$ by using the stereographic projection and sharp Hardy-Littlewood-Sobolev inequality on the sphere in \\cite{LLTX:Nondegeneracy}.\n\nIn this paper, we always assume that the initial data are radially symmetric, which is preserved by the flow. We denote $\\mathcal{E}$ the space radially symmetric functions in $\\dot H^1(\\mathbb{R}^N; \\mathbb{C})$.\n\n\\subsection{Main results}\n In view of the Soliton Resolution Conjecture, solutions which exhibit no dispersion in one or both time directions play a distinguished role.\n One obvious example of such solutions are the ground states $\\mathrm e^{i\\theta}W_\\lambda$, which is a trivial result.\n In this paper, we will consider the simplest non-trivial case, namely\n we will construct a global radial solution which approaches a sum of two bubbles in the energy space. One of the bubbles develops at scale $1$, whereas the length scale of the other converges to $0$ at rate $|t|^{-\\frac{2}{N-6}}$. The phases of the two bubbles form the right angle. That is\n\n\\begin{equation}\\label{energy}\n\t\tE(u_0) := \\int_{\\bR^N} \\frac 12|\\grad u_0(x)|^2 - F(u_0(x))\\ud x,\n\t\\end{equation}\n\n\\begin{equation}\n\t\t\t\\label{eq:kappa}\n\t\t\t\\kappa := \\Big(\\frac{2C_1}{3(N-6)C_2}\\Big)^{\\frac{2}{N-6}},\n\t\t\\end{equation}\n\n\\begin{equation}\n\t\t\\label{har}\n\t\ti \\partial_t u + \\Delta u  + \\left( |x|^{-4}*|u|^2\\right)u =0,\\quad f(u) :=  \\left( |x|^{-4} * |u|^2\\right)u, \\quad\n\t\t(t,x)\\in \\bR \\times \\bR^N.\n\t\\end{equation}\n\n\\begin{remark}\n        \\label{rem:mod}\n        The proof below crucially relies on our choice of the orthogonality conditions \\eqref{eq:orth}, which requires $W, \\Lambda W \\in \\dot H^{-1}(\\bR^N)$, equivalently $N \\geq 7$.\n    \\end{remark}\n\n\\subsection{Outline of the proof}\n The overall structure is similar to the earlier work of the first author on the energy-critical NLS equations \\cite{Jacek:nls}. The difference between Hartree equation here with NLS in \\cite{Jacek:nls} is the nonlocal interaction, which is more complex to analyze. The main tools are the modulation analysis, bootstrap argument and topological argument.\n\nResuming \\eqref{eq:B1-estim}, \\eqref{eq:B2-estim}, \\eqref{eq:B3-estim-rough} and \\eqref{eq:B4-estim-rough}, we have\n \\begin{equation}\n \\label{eq:B-estim}\n |B_1| + |B_2| + |B_3| + |B_4| \\lesssim |t|^{-\\frac{N-2}{N-6}}.\n \\end{equation}\n This and the form of the matrix $(m_{jk})$ directly imply $|\\zeta'| + |\\mu'| \\lesssim |t|^{-\\frac{N-2}{N-6}}$, hence \\eqref{eq:mod-zeta} and \\eqref{eq:mod-mu}.\n Note that the coefficients in the third and the fourth row of the matrix $(m_{jk})$ let us gain an additional factor $|t|^{-\\frac{2}{N-6}}$.\n We obtain $\\big|\\lambda\\lambda' - \\|W\\|_{L^2}^{-2}B_4\\big| \\lesssim |t|^{-\\frac{N-1}{N-6}}$, which implies \\eqref{eq:mod-l} thanks to \\eqref{eq:B4-estim}.\n Similarly, \\eqref{eq:B3-estim} yields \\eqref{eq:mod-th}, which finishes the proof.\n \\end{proof}\n \\begin{remark}\n A computation similar to the proof of \\eqref{eq:B1-estim-1} shows that $|K| \\lesssim \\|g\\|_\\cE^2 \\leq |t|^{-\\frac{N-1}{N-6}}$,\n so we obtain the following simple consequence of Lemma~\\ref{lem:mod}:\n \\begin{equation}\n \\label{eq:param-all}\n |\\zeta'(t)| + \\Big|\\frac{\\mu'(t)}{\\mu(t)}\\Big| + |\\theta'(t)| + \\Big|\\frac{\\lambda'(t)}{\\lambda(t)}\\Big| \\lesssim |t|^{-1},\n \\end{equation}\n (for the last term, this bound is sharp).\n \\end{remark}\n \\subsection{Control of the stable and unstable component}\n An important step is to control\n the stable and unstable components $a_1^\\pm(t) = \\la \\alpha_{\\zeta(t), \\mu(t)}^\\pm, g(t)\\ra$ and $a_2^\\pm(t)= \\la \\alpha_{\\theta(t), \\lambda(t)}^\\pm, g(t)\\ra$. Recall that $\\nu > 0$ is the positive eigenvalue of the linearized flow, see \\eqref{eq:Y1Y2}.\n \\begin{lemma}\n \\label{lem:proper}\n Under assumptions of Lemma~\\ref{lem:mod}, for $t \\in [T, T_1]$ there holds\n \\begin{align}\n \\big| \\dd t a_1^+(t) - \\frac{\\nu}{\\mu(t)^2}a_1^+(t)\\big| &\\leq \\frac{c}{\\mu(t)^2}|t|^{-\\frac{N}{2(N-6)}}, \\label{eq:proper-1p} \\\\\n \\big| \\dd t a_1^-(t) + \\frac{\\nu}{\\mu(t)^2}a_1^-(t)\\big| &\\leq \\frac{c}{\\mu(t)^2}|t|^{-\\frac{N}{2(N-6)}}, \\label{eq:proper-1m} \\\\\n \\big| \\dd t a_2^+(t) - \\frac{\\nu}{\\lambda(t)^2}a_2^+(t)\\big| &\\leq \\frac{c}{\\lambda(t)^2}|t|^{-\\frac{N}{2(N-6)}}, \\label{eq:proper-2p} \\\\\n \\big| \\dd t a_2^-(t) + \\frac{\\nu}{\\lambda(t)^2}a_2^-(t)\\big| &\\leq \\frac{c}{\\lambda(t)^2}|t|^{-\\frac{N}{2(N-6)}}, \\label{eq:proper-2m}\n \\end{align}\n with $c \\to 0$ as $|T_0| \\to +\\infty$.\n \\end{lemma}\n \\begin{proof}\n The proof is analogous to that of Lemma $3.4$ in \\cite{Jacek:nls}, so we omit the proof.\n \\end{proof}\n\n\\section{Bootstrap argument}\n \\label{sec:boot}\n We turn to the heart of the proof, which consists in establishing bootstrap estimates.\n We consider a solution $u(t)$, decomposed according to \\eqref{eq:decompose}, \\eqref{eq:param-rough} and \\eqref{eq:orth}.\n The initial data at time $T \\leq T_0$ is chosen as follows. \n \\begin{lemma}\n \\label{lem:initial}\n There exists $T_0 < 0$ such that for all $T \\leq T_0$ and for all $\\lambda^0$, $a_1^0$, $a_2^0$\n satisfying\n \\begin{equation}\n \\label{eq:initial-assum}\n \\big|\\lambda^0 -\\kappa T^{-\\frac{2}{N-6}}\\big| \\leq \\frac 12 |T|^{-\\frac{5}{2(N-6)}},\\qquad |a_1^0| \\leq \\frac 12 |T|^{-\\frac{N}{2(N-6)}},\\qquad |a_2^0| \\leq \\frac 12 |T|^{-\\frac{N}{2(N-6)}},\n \\end{equation}\n there exists $g^0 \\in X^1$ satisfying\n \\begin{gather}\n \\label{eq:initial-orth}\n \\la \\Lambda W, g^0\\ra = \\la iW, g^0\\ra = \\la i\\Lambda W_{\\lambda^0}, g^0\\ra = \\la {-}W_{\\lambda^0}, g^0\\ra = 0, \\\\\n \\label{eq:initial-unstable}\n \\la \\alpha_{-\\frac{\\pi}{2},1}^-, g^0\\ra = 0,\\quad \\la \\alpha_{-\\frac{\\pi}{2},1}^+, g^0\\ra = a_1^0,\\quad \n \\la \\alpha_{0,\\lambda^0}^-, g^0\\ra = 0,\\quad \\la \\alpha_{0, \\lambda^0}^+, g^0\\ra = a_2^0, \\\\\n \\label{eq:initial-size}\n \\|g^0\\|_{\\cE} \\lesssim |T|^{-\\frac{N}{2(N-6)}}.\n \\end{gather}\n This $g^0$ is continuous for the $X^1$ topology with respect to $\\lambda^0$, $a_1^0$ and $a_2^0$.\n \\end{lemma}\n \\begin{remark}\n For the continuity, we just claim that the function $g^0$ constructed in the proof\n is continuous with respect to $\\lambda^0$, $a_1^0$ and $a_2^0$.\n Clearly, $g^0$ is not uniquely determined by \\eqref{eq:initial-orth}, \\eqref{eq:initial-unstable} and \\eqref{eq:initial-size}.\n \\end{remark}\n \\begin{remark}\n Condition \\eqref{eq:initial-orth} is exactly \\eqref{eq:orth} with $\\big(\\zeta, \\mu, \\theta, \\lambda\\big) = \\big(-\\frac{\\pi}{2}, 1, 0, \\lambda^0\\big)$.\n Hence, if we consider the solution $u(t)$ of \\eqref{har} with initial data $u(T) = -iW + W_{\\lambda^0} + g^0$\n and decompose it according to \\eqref{eq:decompose}, then $g(T) = g^0$ and the initial values of the modulation parameters are\n $\\big(\\zeta(T), \\mu(T), \\theta(T), \\lambda(T)\\big) = \\big({-}\\frac{\\pi}{2}, 1, 0, \\lambda^0\\big)$.\n \\end{remark}\n \\begin{proof}\n The proof is same with \\cite[Lemma 4.1]{Jacek:nls}, which mainly use implicit function theorem to obtain the orthogonal decomposition. Here we omit.\n \\end{proof}\n In the remaining part of this section, we will analyze solutions $u(t)$ of \\eqref{har}\n with the initial data $u(T) = -iW + W_{\\lambda^0} + g^0$,\n where $g^0$ is given by the previous lemma.\n\n\\begin{proposition}\n \\label{prop:shooting}\n Let $|T_0|$ be large enough. For all $T < T_0$ there exist $\\lambda^0, a_1^0, a_2^0$ satisfying \\eqref{eq:initial-assum}\n such that the solution $u(t)$ with the initial data $u(T) = -iW + W_{\\lambda^0} + g^0$ exists on the time interval $[T, T_0]$\n and for $t\\in[T, T_0]$ the bounds \\eqref{eq:bootstrap-better-zeta}, \\eqref{eq:bootstrap-better-mu}, \\eqref{eq:bootstrap-better-theta}, \\eqref{eq:bootstrap-better-g},\n \\begin{align}\n \\big|\\lambda(t) - \\kappa|t|^{-\\frac{2}{N-6}}\\big| &\\leq \\frac 12 |t|^{-\\frac{5}{2(N-6)}}, \\label{eq:bootstrap-better-lambda} \\\\\n |a_1^+(t)| &\\leq \\frac 12 |t|^{-\\frac{N}{2(N-6)}}, \\label{eq:bootstrap-better-a1p} \\\\\n |a_2^+(t)| &\\leq \\frac 12 |t|^{-\\frac{N}{2(N-6)}} \\label{eq:bootstrap-better-a2p},\n \\end{align}\n hold.\n \\end{proposition}\n \\begin{proof}\n The proof is similarly to \\cite[Proposition 4.8]{Jacek:nls}, thus we omit.\n \\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:deux-bulles}]\n Let $T_0 < 0$ be given by Proposition~\\ref{prop:shooting} and let $T_0, T_1, T_2, \\ldots$\n be a decreasing sequence tending to $-\\infty$.\n For $n \\geq 1$, let $u_n$ be the solution given by Proposition~\\ref{prop:shooting}.\n Inequalities \\eqref{eq:bootstrap-better-zeta}, \\eqref{eq:bootstrap-better-mu}, \\eqref{eq:bootstrap-better-theta},\n \\eqref{eq:bootstrap-better-lambda} and \\eqref{eq:bootstrap-better-g} yield\n \\begin{equation}\n \\label{eq:uniform}\n \\Big\\|u_n(t) - \\Big({-}iW + W_{\\kappa|t|^{-\\frac{2}{N-6}}}\\Big)\\Big\\|_\\cE \\lesssim |t|^{-\\frac{1}{2(N-6)}},\n \\end{equation}\n for all $t \\in [T_n, T_0]$ and with a constant independent of $n$.\n Upon passing to a subsequence, we can assume that $u_n(T_0) \\wto u_0 \\in \\cE$.\n Let $u$ be the solution of \\eqref{har} with the initial condition $u(T_0) = u_0$.\n The weak stability Lemma~\\ref{cor:weak-cont} implies that $u$ exists on the time interval $({-}\\infty, T_0]$\n and for all $t \\in ({-}\\infty, T_0]$ there holds $u_n(t) \\wto u(t)$.\n Passing to the weak limit in \\eqref{eq:uniform} finishes the proof.\n \\end{proof}", "post_theorem_intro_text_len": 3916, "post_theorem_intro_text": "\\begin{remark}\n\t\tMore precisely, we will prove that\n\t\t\\begin{equation*}\n\t\t\t\\Big\\|u(t) - \\Big({-}iW + W_{\\kappa|t|^{-\\frac{2}{N-6}}}\\Big)\\Big\\|_\\mathcal{E} \\leq C_0|t|^{-\\frac{1}{2(N-6)}},\n\t\t\\end{equation*}\n\t\tfor some constant $C_0 > 0$.\n\t\\end{remark}\n\t\\begin{remark}\n\t\tWe construct here \\emph{pure} two-bubble, that is the solution approaches a superposition of two ground states, with no energy transformed into radiation.\n\t\tBy the conservation of energy and the decoupling of the two bubbles, we necessarily have $E(u(t)) = 2E(W)$.\n\t\\end{remark}\n\t\\begin{remark}\n\t\tFor energy-critical wave and NLS equations, similar objects were constructed by the first author in \\cite{Jacek:wave} and \\cite{Jacek:nls}. In the following work, we will consider dimension $N = 6$ and we expect an analogous result, with an exponential concentration rate. \n\t\\end{remark}\n\t\\begin{remark}\n\tSolutions of similar kind for the energy-critical heat equation were constructed in \\cite{dPMW, SWZ}.\n\t\\end{remark}\n\t\t\\begin{remark}\n\t\tTwo-bubble solutions of a different kind were recently constructed in \\cite{JK:wmaptwobub} for the energy-critical wave maps equation: the solution blows up in finite time, with the scales of both bubbles converging to zero simultaneously.\n\t\\end{remark}\n\n\t\\subsection{Outline of the proof}\n\tThe overall structure is similar to the earlier work of the first author on the energy-critical NLS equations \\cite{Jacek:nls}. The difference between Hartree equation here with NLS in \\cite{Jacek:nls}  is the nonlocal interaction, which is more complex to analyze. The main tools are the modulation analysis, bootstrap argument and topological argument. \n\n\tThe paper is organized as follows. We study solutions of \\eqref{har} close to a sum of two bubbles:\n\t\\begin{equation}\n\t\tu(t) = \\mathrm e^{i\\zeta(t)}W_{\\mu(t)} + \\mathrm e^{i\\theta(t)}W_{\\lambda(t)} + g(t).\n\t\\end{equation}\n\tOne should think of $\\zeta(t)$ as being close to $-\\frac{\\pi}{2}$, $\\mu(t) \\simeq 1$,\n\t$\\theta(t) \\sim 0$, $\\lambda(t) \\ll 1$ and $\\|g(t)\\|_\\mathcal{E} \\ll 1$. \n\n\tIn Section \\ref{sec:variational}, we first recall some useful lemmas. We also state the spectral properties  of the linearized operator $Z_{\\theta, \\lambda}$ around $\\mathrm e^{i\\theta}W_\\lambda$ and coercivity of the energy near a two-bubble. These are very important to estimate $g$, which is the infinite-dimensional part. We will use the energy conservation to deal with this.\n\n\tIn Section \\ref{sec:mod}, we estimate the modulation parameters. For this reason, we impose the orthogonality conditions which make the terms linear in $g$ in the modulation equations disappeared. There is essentially a unique choice of such orthogonality conditions. In Lemma~\\ref{lem:mod} we establish bounds on the evolution of the modulation parameters under some bootstrap assumptions. The goal is to improve these bounds, thus closing the bootstrap. In order to improve the bound on $\\|g\\|_\\mathcal{E}$ to close the bootstrap, we establish Lemma \\ref{lem:proper} to control the unstable components.\n\n\tIn Section \\ref{sec:boot}, we establish the bootstrap estimates and prove the main result Theorem \\ref{thm:deux-bulles}. First, we use implicit function theorem to obtain the modulation decomposition. Next, combining with virial correction, which is used to improve the bound \\eqref{eq:bootstrap-theta} on $\\theta(t)$, we can close the bootstrap of modulation parameters. Adding the virial correction allows us to gain a small constant on the right hand side of \\eqref{eq:mod-th}, which is decisive for closing the bootstrap. The linear instabilities of the flow can be handled by using a classical topological argument based on the Brouwer fixed point theorem. Finally, we prove Theorem \\ref{thm:deux-bulles}.\n\n\t\\subsection{Acknowledgments}\n    \t\tJ. Jendrej was supported by  ERC project INSOLIT (No. 101117126).\n            G. Xu was supported  by  NSFC (No. 12371240, No. 12431008).", "sketch": "To prove Theorem~\\ref{thm:deux-bulles}, the paper follows “the overall structure” of \\cite{Jacek:nls}, with the additional difficulty that “the nonlocal interaction … is more complex to analyze.” The “main tools are the modulation analysis, bootstrap argument and topological argument.”\n\nThe solution is studied “close to a sum of two bubbles” via the decomposition\n\\[\n u(t)=e^{i\\zeta(t)}W_{\\mu(t)}+e^{i\\theta(t)}W_{\\lambda(t)}+g(t),\n\\]\nwith \\(\\zeta(t)\\) near \\(-\\pi/2\\), \\(\\mu(t)\\simeq 1\\), \\(\\theta(t)\\sim 0\\), \\(\\lambda(t)\\ll 1\\), and \\(\\|g(t)\\|_{\\mathcal E}\\ll 1\\).\n\nKey steps indicated in the outline:\n1) (Section~\\ref{sec:variational}) Recall “useful lemmas,” state “spectral properties of the linearized operator \\(Z_{\\theta,\\lambda}\\)” around \\(e^{i\\theta}W_\\lambda\\), and establish “coercivity of the energy near a two-bubble.” These are used “to estimate \\(g\\), which is the infinite-dimensional part,” and “we will use the energy conservation to deal with this.”\n\n2) (Section~\\ref{sec:mod}) “Estimate the modulation parameters” by imposing “orthogonality conditions which make the terms linear in \\(g\\) in the modulation equations disappeared,” with “essentially a unique choice” of such conditions. Under bootstrap assumptions, Lemma~\\ref{lem:mod} gives “bounds on the evolution of the modulation parameters,” aiming to “improve these bounds, thus closing the bootstrap.” To improve the \\(\\|g\\|_{\\mathcal E}\\) bound, Lemma~\\ref{lem:proper} is established “to control the unstable components.”\n\n3) (Section~\\ref{sec:boot}) “Establish the bootstrap estimates and prove the main result Theorem~\\ref{thm:deux-bulles}.” First, use the “implicit function theorem to obtain the modulation decomposition.” Then, “combining with virial correction” (used “to improve the bound … on \\(\\theta(t)\\)”), one can “close the bootstrap of modulation parameters”; “adding the virial correction allows us to gain a small constant on the right hand side of \\eqref{eq:mod-th}, which is decisive for closing the bootstrap.” The “linear instabilities of the flow” are handled via “a classical topological argument based on the Brouwer fixed point theorem.” “Finally, we prove Theorem~\\ref{thm:deux-bulles}.”", "expanded_sketch": "To prove the main theorem, the paper follows “the overall structure” of \\cite{Jacek:nls}, with the additional difficulty that “the nonlocal interaction … is more complex to analyze.” The “main tools are the modulation analysis, bootstrap argument and topological argument.”\n\nThe solution is studied “close to a sum of two bubbles” via the decomposition\n\\[\n u(t)=e^{i\\zeta(t)}W_{\\mu(t)}+e^{i\\theta(t)}W_{\\lambda(t)}+g(t),\n\\]\nwith \\(\\zeta(t)\\) near \\(-\\pi/2\\), \\(\\mu(t)\\simeq 1\\), \\(\\theta(t)\\sim 0\\), \\(\\lambda(t)\\ll 1\\), and \\(\\|g(t)\\|_{\\mathcal E}\\ll 1\\).\n\nKey steps indicated in the outline:\n1) (Section~\\ref{sec:variational}) Recall “useful lemmas,” state “spectral properties of the linearized operator \\(Z_{\\theta,\\lambda}\\)” around \\(e^{i\\theta}W_\\lambda\\), and establish “coercivity of the energy near a two-bubble.” These are used “to estimate \\(g\\), which is the infinite-dimensional part,” and “we will use the energy conservation to deal with this.”\n\n2) (Section~\\ref{sec:mod}) “Estimate the modulation parameters” by imposing “orthogonality conditions which make the terms linear in \\(g\\) in the modulation equations disappeared,” with “essentially a unique choice” of such conditions. Under bootstrap assumptions, we first use the following lemma.\n\\begin{lemma}\n\t\t\\label{lem:mod}\n\t\tLet $c > 0$ be an arbitrarily small constant. Let $T_0 < 0$ with $|T_0|$ large enough (depending on $c$)\n\t\tand $T < T_1 \\leq T_0$. Suppose that for $T \\leq t \\leq T_1$ there holds\n\t\t\\begin{align}\n\t\t\t\\big|\\zeta(t) + \\frac{\\pi}{2}\\big| &\\leq |t|^{-\\frac{3}{N-6}}, \\label{eq:bootstrap-zeta} \\\\\n\t\t\t|\\mu(t) - 1| &\\leq |t|^{-\\frac{3}{N-6}}, \\label{eq:bootstrap-mu} \\\\\n\t\t\t|\\theta(t)| &\\leq |t|^{-\\frac{1}{N-6}}, \\label{eq:bootstrap-theta} \\\\\n\t\t\t\\big|\\lambda(t) - \\kappa|t|^{-\\frac{2}{N-6}}\\big| &\\leq |t|^{-\\frac{5}{2(N-6)}}, \\label{eq:bootstrap-lambda} \\\\\n\t\t\t\\|g\\|_\\cE &\\leq |t|^{-\\frac{N-1}{2(N-6)}}. \\label{eq:bootstrap-g}\n\t\t\\end{align}\n\t\tThen\n\t\t\\begin{align}\n\t\t\t\\label{eq:mod-zeta}\n\t\t\t|\\zeta'(t)| &\\leq c|t|^{-\\frac{N-3}{N-6}}, \\\\\n\t\t\t\\label{eq:mod-mu}\n\t\t\t|\\mu'(t)| &\\leq c|t|^{-\\frac{N-3}{N-6}}, \\\\\n\t\t\t\\label{eq:mod-l}\n\t\t\t\\Big|\\lambda'(t) - \\frac{3C_2}{\\|W\\|_{L^2}^2} \\lambda(t)^\\frac{N-4}{2}\\Big| &\\leq c|t|^{-\\frac{2N-7}{2(N-6)}}, \\\\\n\t\t\t\\label{eq:mod-th}\n\t\t\t\\Big|\\theta'(t)+\\frac{C_3}{\\|W\\|_{L^2}^2}\\theta(t)\\lambda(t)^\\frac{N-6}{2}-\\frac{K(t)}{\\lambda(t)^2\\|W\\|_{L^2}^2}\\Big| &\\leq c|t|^{-\\frac{N-5}{N-6}},\n\t\t\\end{align}\n\t\tfor $T \\leq t \\leq T_1$, where\n\t\t\\begin{equation}\n\t\t\t\\label{eq:kappa}\n\t\t\t\\kappa := \\Big(\\frac{2C_1}{3(N-6)C_2}\\Big)^{\\frac{2}{N-6}},\n\t\t\\end{equation}\n\t\t\\begin{equation}\n\t\t\t\\label{eq:K-def}\n\t\t\tK := -\\big\\la \\eee^{i\\theta}\\Lambda W_\\lambda, f(\\eee^{i\\zeta}W_{\\mu} + \\eee^{i\\theta}W_{\\lambda} + g) - f(\\eee^{i\\zeta}W_\\mu + \\eee^{i\\theta}W_\\lambda) - f'(\\eee^{i\\zeta}W_\\mu + \\eee^{i\\theta}W_\\lambda)g \\big\\ra.\n\t\t\\end{equation}\n\t\\end{lemma}\nThis gives “bounds on the evolution of the modulation parameters,” aiming to “improve these bounds, thus closing the bootstrap.” To improve the \\(\\|g\\|_{\\mathcal E}\\) bound, one establishes the following auxiliary lemma “to control the unstable components.”\n\\begin{lemma}\n\t\t\\label{lem:proper}\n\t\tUnder assumptions of Lemma~\\ref{lem:mod}, for $t \\in [T, T_1]$ there holds\n\t\t\\begin{align}\n\t\t\t\\big| \\dd t a_1^+(t) - \\frac{\\nu}{\\mu(t)^2}a_1^+(t)\\big| &\\leq \\frac{c}{\\mu(t)^2}|t|^{-\\frac{N}{2(N-6)}}, \\label{eq:proper-1p} \\\\\n\t\t\t\\big| \\dd t a_1^-(t) + \\frac{\\nu}{\\mu(t)^2}a_1^-(t)\\big| &\\leq \\frac{c}{\\mu(t)^2}|t|^{-\\frac{N}{2(N-6)}}, \\label{eq:proper-1m} \\\\\n\t\t\t\\big| \\dd t a_2^+(t) - \\frac{\\nu}{\\lambda(t)^2}a_2^+(t)\\big| &\\leq \\frac{c}{\\lambda(t)^2}|t|^{-\\frac{N}{2(N-6)}}, \\label{eq:proper-2p} \\\\\n\t\t\t\\big| \\dd t a_2^-(t) + \\frac{\\nu}{\\lambda(t)^2}a_2^-(t)\\big| &\\leq \\frac{c}{\\lambda(t)^2}|t|^{-\\frac{N}{2(N-6)}}, \\label{eq:proper-2m}\n\t\t\\end{align}\n\t\twith $c \\to 0$ as $|T_0| \\to +\\infty$.\n\t\\end{lemma}\n\n3) (Section~\\ref{sec:boot}) “Establish the bootstrap estimates and prove the main result.” First, use the “implicit function theorem to obtain the modulation decomposition.” Then, “combining with virial correction” (used “to improve the bound … on \\(\\theta(t)\\)”), one can “close the bootstrap of modulation parameters”; more precisely, “adding the virial correction allows us to gain a small constant on the right hand side of the equation\n\\begin{align}\n\t\t\t\\label{eq:mod-zeta}\n\t\t\t|\\zeta'(t)| &\\leq c|t|^{-\\frac{N-3}{N-6}}, \\\\\n\t\t\t\\label{eq:mod-mu}\n\t\t\t|\\mu'(t)| &\\leq c|t|^{-\\frac{N-3}{N-6}}, \\\\\n\t\t\t\\label{eq:mod-l}\n\t\t\t\\Big|\\lambda'(t) - \\frac{3C_2}{\\|W\\|_{L^2}^2} \\lambda(t)^\\frac{N-4}{2}\\Big| &\\leq c|t|^{-\\frac{2N-7}{2(N-6)}}, \\\\\n\t\t\t\\label{eq:mod-th}\n\t\t\t\\Big|\\theta'(t)+\\frac{C_3}{\\|W\\|_{L^2}^2}\\theta(t)\\lambda(t)^\\frac{N-6}{2}-\\frac{K(t)}{\\lambda(t)^2\\|W\\|_{L^2}^2}\\Big| &\\leq c|t|^{-\\frac{N-5}{N-6}},\n\t\t\\end{align}\nwhich is decisive for closing the bootstrap.” The “linear instabilities of the flow” are handled via “a classical topological argument based on the Brouwer fixed point theorem.” This completes the proof of the main theorem.", "expanded_theorem": "\\label{thm:deux-bulles}\n\t\tThere exists a solution $u: (-\\infty, T_0] \\to \\mathcal{E}$ of\n\t\t\\begin{equation}\n\t\t\\label{har}\n\t\ti \\partial_t u + \\Delta u  + \\left( |x|^{-4}*|u|^2\\right)u =0,\\quad f(u) :=  \\left( |x|^{-4} * |u|^2\\right)u, \\quad\n\t\t(t,x)\\in \\bR \\times \\bR^N.\n\t\\end{equation}\n\t\tsuch that\n\t\t\\begin{equation}\n\t\t\t\\label{eq:mainthm}\n\t\t\t\\lim_{t\\to -\\infty}\\Big\\|u(t) - \\Big({-}iW + W_{\\kappa|t|^{-\\frac{2}{N-6}}}\\Big)\\Big\\|_\\mathcal{E} = 0,\n\t\t\\end{equation}\n\t\twhere $\\kappa$ is the explicit constant\n\t\t\\begin{equation}\n\t\t\t\\label{eq:kappa}\n\t\t\t\\kappa := \\Big(\\frac{2C_1}{3(N-6)C_2}\\Big)^{\\frac{2}{N-6}},\n\t\t\\end{equation}\n\t\t.", "theorem_type": ["Existence", "Asymptotic or Limit"], "mcq": {"question": "Assume \\(N\\ge 7\\), and let \\(\\mathcal E\\) be the space of radial functions in \\(\\dot H^1(\\mathbb R^N;\\mathbb C)\\). Consider the focusing energy-critical Hartree equation\n\\[\ni\\partial_t u+\\Delta u+\\big(|x|^{-4}*|u|^2\\big)u=0,\\qquad (t,x)\\in\\mathbb R\\times\\mathbb R^N.\n\\]\nLet \\(W\\) denote a ground state (bubble), i.e. a solution of\n\\[\n-\\Delta W-\\big(|x|^{-4}*|W|^2\\big)W=0,\n\\]\nand let its rescaling be \\(W_\\lambda(x)=\\lambda^{-\\frac{N-2}{2}}W(x/\\lambda)\\). Also set\n\\[\n\\kappa:=\\Big(\\frac{2C_1}{3(N-6)C_2}\\Big)^{\\frac{2}{N-6}}.\n\\]\nWhich existence statement holds for the Hartree flow under these assumptions?", "correct_choice": {"label": "A", "text": "There exists a time \\(T_0\\) and a solution \\(u:(-\\infty,T_0]\\to\\mathcal E\\) of \\(i\\partial_t u+\\Delta u+(|x|^{-4}*|u|^2)u=0\\) such that\n\\[\n\\lim_{t\\to-\\infty}\\Big\\|u(t)-\\Big(-iW+W_{\\kappa |t|^{-\\frac{2}{N-6}}}\\Big)\\Big\\|_{\\mathcal E}=0.\n\\]"}, "choices": [{"label": "B", "text": "There exists a time \\(T_0\\) and a solution \\(u:(-\\infty,T_0]\\to\\mathcal E\\) of \\(i\\partial_t u+\\Delta u+(|x|^{-4}*|u|^2)u=0\\) such that\n\\[\n\\lim_{t\\to-\\infty}\\Big\\|u(t)-\\Big(-iW+W_{\\kappa |t|^{-\\frac{2}{N-4}}}\\Big)\\Big\\|_{\\mathcal E}=0.\n\\]"}, {"label": "C", "text": "There exists a time \\(T_0\\) and a solution \\(u:(-\\infty,T_0]\\to\\mathcal E\\) of \\(i\\partial_t u+\\Delta u+(|x|^{-4}*|u|^2)u=0\\) such that\n\\[\n\\lim_{t\\to-\\infty}\\operatorname{dist}_{\\mathcal E}\\big(u(t),\\{ -iW+W_\\lambda:\\ \\lambda>0\\}\\big)=0.\n\\]"}, {"label": "D", "text": "There exists a time \\(T_0\\) and a solution \\(u:(-\\infty,T_0]\\to\\mathcal E\\) of \\(i\\partial_t u+\\Delta u+(|x|^{-4}*|u|^2)u=0\\) such that\n\\[\n\\lim_{t\\to-\\infty}\\Big\\|u(t)-\\Big(e^{i\\alpha}W+W_{\\kappa |t|^{-\\frac{2}{N-6}}}\\Big)\\Big\\|_{\\mathcal E}=0\n\\]\nfor every phase \\(\\alpha\\in\\mathbb R\\)."}, {"label": "E", "text": "There exists a time \\(T_0\\) and a solution \\(u:(-\\infty,T_0]\\to\\mathcal E\\) of \\(i\\partial_t u+\\Delta u+(|x|^{-4}*|u|^2)u=0\\) such that\n\\[\n\\lim_{t\\to-\\infty}\\Big\\|u(t)-\\Big(W+W_{\\kappa |t|^{-\\frac{2}{N-6}}}\\Big)\\Big\\|_{\\mathcal E}=0.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "rate_of_lambda_from_modulation_ODE", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "modulation_analysis", "tampered_component": "explicit_scale_law_\\kappa |t|^{-2/(N-6)}", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "modulation_analysis", "tampered_component": "fixed_right_angle_phase_relation", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "specific_phase_-iW", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the correct option. It provides the equation, scaling, and the constant κ, but the exact asymptotic profile, phase, exponent, and quantifier structure still have to be selected from the choices."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the question asks which asymptotic statement holds, and the correct choice is basically the precise theorem conclusion. It is much closer to restating a known result than to testing transfer or synthesis."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in subtle but meaningful ways (rate exponent, exact phase, weaker-vs-sharper conclusion, and quantifiers). However, the item mainly rewards recognition or recall of the precise theorem statement rather than genuine derivation."}, "DQS": {"score": 2, "justification": "The distractors are strong: one uses the wrong scaling exponent, one gives a weaker true-type statement, one alters the quantifier structure, and one changes the phase configuration. These are plausible mathematical confusions and are well differentiated."}, "total_score": 5, "overall_assessment": "Well-designed distractors and little direct answer leakage, but the question is largely a disguised theorem restatement and only moderately tests generative reasoning."}}
{"id": "2602.08644v1", "paper_link": "http://arxiv.org/abs/2602.08644v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\n      Let $(\\M_1,g_1)$ and $(\\M_2,g_2)$ be two  n-dimensional smooth connected Riemannian manifolds with  boundary. Let $\\omega_1\\subset \\M_1$ and $\\omega_2\\subset\\M_2$ be open and connected. For $j=1,2$, let $\\mathcal N_j$ be the local source-to-final value map\n      \\eqref{eq:def_N} with $(M,g)=(\\M_j,g_j)$\n      and $\\omega=\\omega_j$.\n      In addition, suppose that there is a diffeomorphism $\\bm{\\Phi}:\\omega_1\\to \\omega_2$ such that \n      \\begin{equation}\\label{NN}\n          \\cN_1\\bm{\\Phi}^\\ast=\\bm{\\Phi}^\\ast \\cN_2.\n      \\end{equation}\n       Then $(\\M_1,g_1)$ and $(\\M_2,g_2)$ are isometric.", "start_pos": 9075, "end_pos": 9718, "label": "thm:main"}, "ref_dict": {"eq:problem_hypo": "\\begin{equation}\n    \\label{eq:problem_hypo}\n    \\begin{cases}\n      (\\p_t^2-\\Delta_H)\\bu + d p=\\bf,\\ \\text{in } \\RR^+\\times \\M,\\\\\n      d^* \\bu=0,\\ \\text{in } \\RR^+\\times \\M,\\\\\n      \\bu|_{[0,T]\\times\\p\\M}=0,\\\\\n      \\bu=0,\\ p=0,\\ \\text{in }\\RR^-\\times\\M.\n    \\end{cases}\n  \\end{equation}", "thm:main": "\\begin{theorem}\n  \\label{thm:main}\n      Let $(\\M_1,g_1)$ and $(\\M_2,g_2)$ be two  n-dimensional smooth connected Riemannian manifolds with  boundary. Let $\\omega_1\\subset \\M_1$ and $\\omega_2\\subset\\M_2$ be open and connected. For $j=1,2$, let $\\mathcal N_j$ be the local source-to-final value map\n      \\eqref{eq:def_N} with $(\\M,g)=(\\M_j,g_j)$\n      and $\\omega=\\omega_j$.\n      In addition, suppose that there is a diffeomorphism $\\bPhi:\\omega_1\\to \\omega_2$ such that \n      \\begin{equation}\\label{NN}\n          \\cN_1\\bPhi^\\ast=\\bPhi^\\ast \\cN_2.\n      \\end{equation}\n       Then $(\\M_1,g_1)$ and $(\\M_2,g_2)$ are isometric.\n  \\end{theorem}", "eq:NS1": "\\begin{equation}\n    \\label{eq:NS1}\n    \\begin{cases}\n      \\p_t\\bu-\\Delta_H\\bu+R(\\bu) + d p=\\bf,\\ \\text{in } [0,T]\\times \\M,\\\\\n      d^* \\bu=0,\\ \\text{in } [0,T]\\times \\M,\\\\\n      \\bu|_{[0,T]\\times\\p\\M}=0,\\\\\n      \\bu=0,\\ p=0,\\ \\text{in }\\{0\\}\\times\\M.\n    \\end{cases}\n  \\end{equation}", "eq:def_N": "\\begin{align}\n\\label{eq:def_N}\n    \\mathcal N:\\mathbb B_\\delta&\\to L^2\\Omega^1(\\omega),\\\\\n    \\mathcal N(\\bf)&=\\bu_\\bf(T,\\cdot)|_\\omega,\n  \\end{align}", "eq:def_P": "\\begin{align}\n  \\label{eq:def_P}\n    \\P_i:C_0^\\infty((0,T);&C_0^\\infty\\Omega^1(\\omega_i))\\to L^2\\Omega^1(\\omega_i),\\\\\n    \\P_i(\\bf)&=\\bu_i(T,\\cdot)|_{\\omega_i},\n  \\end{align}", "thm:NS": "\\begin{theorem}\n    \\label{thm:NS}\nThere exists $\\delta>0$, depending on $(\\M,g)$ and $T$, such that for all $\\bf\\in \\mathbb B_\\delta$, problem \\eqref{eq:NS1} admits a unique solution $(\\bu,p)=(\\bu_\\bf,p_\\bf)$, $p$ being unique up to a map depending only on $t\\in[0,T]$, with $u_\\bf\\in \\mathcal C_\\alpha$, $dp_\\bf\\in C^{\\frac{\\alpha}{2}}([0,T];C^{\\alpha}\\Omega^1(\\M))$ and with $p_\\bf$ a solution of \\eqref{eq:p}. Moreover, the map $\\mathbb B_\\delta\\ni\\bf\\mapsto (u_\\bf,dp_\\bf)\\in \\cC_\\alpha(M)\\times L^2([0,T]; L^2\\Omega^1(\\M))$ is $C^1$.\n  \\end{theorem}", "linearization": "\\begin{proposition}\\label{linearization} \nFor $i=1,2$ and $\\bf\\in C^\\infty_0([0,T];C^\\infty_0\\Omega^1(\\omega_i))$, we fix $\\delta_{i,\\bf}=\\frac{\\delta}{1+\\|\\bf\\|_{\\mathcal C_\\alpha(M_i)}}$ with $\\delta$ introduced in Theorem \\ref{thm:NS}. Then the map\n$$(-\\delta_{i,\\bf},\\delta_{i,\\bf})\\ni s\\mapsto \\mathcal N_i(s\\bf)\\in L^2\\Omega^1(\\omega_i)$$\nis $C^1$ and we have\n\\begin{equation}\\label{lin1}\n    \\partial_s\\mathcal N_i(s\\bf)|_{s=0}=\\mathcal P_i(\\bf).\n\\end{equation}\n\n\\end{proposition}"}, "pre_theorem_intro_text_len": 5628, "pre_theorem_intro_text": "The Navier–Stokes equations are among the most fundamental mathematical models in fluid dynamics. \nIn this work, we consider the geometric Navier--Stokes equation on a Riemannian manifold $(M, g)$, see  \\eqref{eq:NS1} below. When $M$ is a Euclidean domain and $g = c dx^2$, with $c > 0$ a constant, then \\eqref{eq:NS1} can be written as \nthe classical Navier-Stokes equations on $\\Omega$ with no-slip boundary condition \n\\begin{equation}\n\\begin{cases}\n    \\p_t \\mathbf{u} + \\mathbf{u} \\nabla \\mathbf{u} - \\mu \\Delta \\mathbf{u} + \\nabla p = \\mathbf{f},\\text{ in }[0,T]\\timesM\\\\\n    \\nabla\\cdot \\mathbf{u} = 0,\\text{ in }[0,T]\\timesM\\\\\n    \\mathbf{u}=0,\\text{ on }[0,T]\\times \\partialM,\\\\\n    \\mathbf{u}(0,\\cdot)=\\bu_0.\n\\end{cases}\n\\end{equation}\nHere $\\mathbf{u}: [0,T]\\times\\Omega\\to \\mathbb{R}^n$ is the velocity field of the fluid, $p: [0,T]\\times\\Omega\\to \\mathbb R$ is a scalar function representing the pressure, and \n$\\mu = c^{-1}$ is the coefficient of kinematic viscosity.\n\nThe generalization on a Riemannian manifold enables the study of fluid dynamics in complex geometric settings, thereby opening new perspectives for both theoretical analysis and practical applications \\cite{VlKh,EMa,MaRaSh,Sh}. The formulation of the Navier–Stokes equations on manifolds is motivated by several important areas, including: \n1) geophysical fluid dynamics \\cite{SaT,Val}, where the use of Riemannian geometry allows one to model the curvature of the Earth’s surface;\n2) General Relativity and cosmology \\cite{FMR}, modeling relativistic fluids, such as those present in neutron stars, black hole accretion disks, and the early universe;\n3)~fluid dynamics on biological membranes and microfluidic devices~\\cite{STONE}.\n\nIn all these contexts, the geometric structure of the manifold plays a central role in characterizing the physical properties of fluids and their surrounding media (e.g., density and viscosity). Motivated by these considerations, we investigate the inverse problem of identifying a Riemannian manifold from measurements of the fluid velocity field, associated with a source-to-final value map for the Navier–Stokes equations.\n\nInverse problems and, more broadly, identification problems related to fluid motion remain poorly understood. Only a limited number of studies have addressed this class of inverse problems, despite their strong physical motivation and mathematical significance. Among the few existing contributions devoted to inverse problems for the Stokes and Navier–Stokes equations, we may cite works concerning the determination of viscosity from boundary or internal measurements \\cite{FDJN,HLiWa,ImYa,LaUWa,LiWa,Liu},  the identification of objects immersed in a fluid \\cite{CaCo1,BCZ,CaCo2} as well as inverse source problems \\cite{BGKN,CIPY,ILY}. To the best of our knowledge, no prior studies have addressed inverse problems for the Navier–Stokes equations beyond the Euclidean setting.\n\n  \\subsection{Notations and main results} \n  Let $M$ be a n-dimensional, $n\\geq2$, smooth connected Riemannian manifold with  boundary and $\\omega\\subset M$ be a nonempty open subset.\n  We denote the space consisting of smooth sections of exterior $k$-form bundles on $M$ and $\\omega$ by $\\Omega^k(M)$ and $\\Omega^k(\\omega)$, respectively.   Let\n  \\begin{equation}\n      \\ast:\\ \\Omega^k (M)\\to \\Omega^{n-k}(M)\n  \\end{equation}\n  be the Hodge star operator (see e.g. \\cite[section 14.1a]{frankel} for details). Then we define the codifferential operator $d^*: \\Omega^k(M)\\to \\Omega^{k-1}(M)$ as\n  \\begin{equation}\n  \\label{eq:def_codifferential}\n    d^*=(-1)^{n(k+1)+1}\\ast d\\ast.\n  \\end{equation}\n  We denote the Hodge Laplacian $-dd^*-d^*d$ by $\\Delta_H$.\n   Using the metric $g$, for every 1-form $u=u_idx_i$ on $M$ we associate the vector field $u_*=g^{ij}u_j\\partial_{x_i}\\in TM$. In the same way, for every vector field $v=v_i\\partial_{x_i}\\in TM$ we associate the 1-form $v^*=g_{ij}v_jdx_i$. With these notations, we write $R(w)=(\\nabla_{w_*}w_*)^*$,\nwhere $\\nabla$ denotes the Levi-Civita connection on $(M,g)$.  Several works have been devoted to the study  of the following geometric Navier-Stokes system \n  \\begin{equation}\n    \\label{eq:NS1}\n    \\begin{cases}\n      \\p_t\\mathbf{u}-\\Delta_H\\mathbf{u}+R(\\mathbf{u}) + d p=\\mathbf{f},\\ \\text{in } [0,T]\\times M,\\\\\n      d^* \\mathbf{u}=0,\\ \\text{in } [0,T]\\times M,\\\\\n      \\mathbf{u}|_{[0,T]\\times\\partialM}=0,\\\\\n      \\mathbf{u}=0,\\ p=0,\\ \\text{in }\\{0\\}\\timesM.\n    \\end{cases}\n  \\end{equation}\n  Without being exhaustive, one can refer to \\cite{EMa,AA91,CP92,TW93,CRT99} for more details.\n\n  We denote by $\\mathbb B_r$ the set defined by\n  $$\\mathbb B_r:=\\{\\mathbf{g} \\in \\cC_\\alpha(M):\\ \\supp(\\mathbf{g})\\subset (0,T]\\times\\omega,\\  \\|\\mathbf{g}\\|_{ \\cC_\\alpha(M)}\\leq r\\},$$\nwhere we refer  to Section \\ref{sec:Nonlinear_NS} for the  definition of the space $\\cC_\\alpha(M)$ and $\\norm{\\cdot}_{ \\cC_\\alpha(M)}$. We prove in Theorem \\ref{thm:NS} that \nthere exists $\\delta>0$, depending on $(M,g)$ and $T$, such that for all $\\mathbf{f}\\in \\mathbb B_\\delta$, problem \\eqref{eq:NS1} admits a solution $(\\mathbf{u},p)=(\\bu_\\mathbf{f},p_\\mathbf{f})$ with $\\mathbf{u}\\in \\mathcal C_\\alpha$. The solution is unique in the sense that $\\mathbf{u}$ is unique and $p$ is unique up to a map depending only on $t\\in[0,T]$. Then, we can define the local source-to-final value map\n\\begin{align}\n\\label{eq:def_N}\n    \\mathcal N:\\mathbb B_\\delta&\\to L^2\\Omega^1(\\omega),\\\\\n    \\mathcal N(\\mathbf{f})&=\\bu_\\mathbf{f}(T,\\cdot)|_\\omega,\n  \\end{align}\n  with $\\bu_\\mathbf{f}\\in \\cC_\\alpha(M)$  solving \\eqref{eq:NS1}.\n  Our main result can be stated as follows.", "context": "The Navier–Stokes equations are among the most fundamental mathematical models in fluid dynamics. \nIn this work, we consider the geometric Navier--Stokes equation on a Riemannian manifold $(M, g)$, see \\eqref{eq:NS1} below. When $M$ is a Euclidean domain and $g = c dx^2$, with $c > 0$ a constant, then \\eqref{eq:NS1} can be written as \nthe classical Navier-Stokes equations on $\\Omega$ with no-slip boundary condition \n\\begin{equation}\n\\begin{cases}\n \\p_t \\mathbf{u} + \\mathbf{u} \\nabla \\mathbf{u} - \\mu \\Delta \\mathbf{u} + \\nabla p = \\mathbf{f},\\text{ in }[0,T]\\timesM\\\\\n \\nabla\\cdot \\mathbf{u} = 0,\\text{ in }[0,T]\\timesM\\\\\n \\mathbf{u}=0,\\text{ on }[0,T]\\times \\partialM,\\\\\n \\mathbf{u}(0,\\cdot)=\\bu_0.\n\\end{cases}\n\\end{equation}\nHere $\\mathbf{u}: [0,T]\\times\\Omega\\to \\mathbb{R}^n$ is the velocity field of the fluid, $p: [0,T]\\times\\Omega\\to \\mathbb R$ is a scalar function representing the pressure, and \n$\\mu = c^{-1}$ is the coefficient of kinematic viscosity.\n\nThe generalization on a Riemannian manifold enables the study of fluid dynamics in complex geometric settings, thereby opening new perspectives for both theoretical analysis and practical applications \\cite{VlKh,EMa,MaRaSh,Sh}. The formulation of the Navier–Stokes equations on manifolds is motivated by several important areas, including: \n1) geophysical fluid dynamics \\cite{SaT,Val}, where the use of Riemannian geometry allows one to model the curvature of the Earth’s surface;\n2) General Relativity and cosmology \\cite{FMR}, modeling relativistic fluids, such as those present in neutron stars, black hole accretion disks, and the early universe;\n3)~fluid dynamics on biological membranes and microfluidic devices~\\cite{STONE}.\n\nIn all these contexts, the geometric structure of the manifold plays a central role in characterizing the physical properties of fluids and their surrounding media (e.g., density and viscosity). Motivated by these considerations, we investigate the inverse problem of identifying a Riemannian manifold from measurements of the fluid velocity field, associated with a source-to-final value map for the Navier–Stokes equations.\n\nInverse problems and, more broadly, identification problems related to fluid motion remain poorly understood. Only a limited number of studies have addressed this class of inverse problems, despite their strong physical motivation and mathematical significance. Among the few existing contributions devoted to inverse problems for the Stokes and Navier–Stokes equations, we may cite works concerning the determination of viscosity from boundary or internal measurements \\cite{FDJN,HLiWa,ImYa,LaUWa,LiWa,Liu}, the identification of objects immersed in a fluid \\cite{CaCo1,BCZ,CaCo2} as well as inverse source problems \\cite{BGKN,CIPY,ILY}. To the best of our knowledge, no prior studies have addressed inverse problems for the Navier–Stokes equations beyond the Euclidean setting.\n\n\\subsection{Notations and main results} \n Let $M$ be a n-dimensional, $n\\geq2$, smooth connected Riemannian manifold with boundary and $\\omega\\subset M$ be a nonempty open subset.\n We denote the space consisting of smooth sections of exterior $k$-form bundles on $M$ and $\\omega$ by $\\Omega^k(M)$ and $\\Omega^k(\\omega)$, respectively. Let\n \\begin{equation}\n \\ast:\\ \\Omega^k (M)\\to \\Omega^{n-k}(M)\n \\end{equation}\n be the Hodge star operator (see e.g. \\cite[section 14.1a]{frankel} for details). Then we define the codifferential operator $d^*: \\Omega^k(M)\\to \\Omega^{k-1}(M)$ as\n \\begin{equation}\n \\label{eq:def_codifferential}\n d^*=(-1)^{n(k+1)+1}\\ast d\\ast.\n \\end{equation}\n We denote the Hodge Laplacian $-dd^*-d^*d$ by $\\Delta_H$.\n Using the metric $g$, for every 1-form $u=u_idx_i$ on $M$ we associate the vector field $u_*=g^{ij}u_j\\partial_{x_i}\\in TM$. In the same way, for every vector field $v=v_i\\partial_{x_i}\\in TM$ we associate the 1-form $v^*=g_{ij}v_jdx_i$. With these notations, we write $R(w)=(\\nabla_{w_*}w_*)^*$,\nwhere $\\nabla$ denotes the Levi-Civita connection on $(M,g)$. Several works have been devoted to the study of the following geometric Navier-Stokes system \n \\begin{equation}\n \\label{eq:NS1}\n \\begin{cases}\n \\p_t\\mathbf{u}-\\Delta_H\\mathbf{u}+R(\\mathbf{u}) + d p=\\mathbf{f},\\ \\text{in } [0,T]\\times M,\\\\\n d^* \\mathbf{u}=0,\\ \\text{in } [0,T]\\times M,\\\\\n \\mathbf{u}|_{[0,T]\\times\\partialM}=0,\\\\\n \\mathbf{u}=0,\\ p=0,\\ \\text{in }\\{0\\}\\timesM.\n \\end{cases}\n \\end{equation}\n Without being exhaustive, one can refer to \\cite{EMa,AA91,CP92,TW93,CRT99} for more details.\n\nWe denote by $\\mathbb B_r$ the set defined by\n $$\\mathbb B_r:=\\{\\mathbf{g} \\in \\cC_\\alpha(M):\\ \\supp(\\mathbf{g})\\subset (0,T]\\times\\omega,\\ \\|\\mathbf{g}\\|_{ \\cC_\\alpha(M)}\\leq r\\},$$\nwhere we refer to Section \\ref{sec:Nonlinear_NS} for the definition of the space $\\cC_\\alpha(M)$ and $\\norm{\\cdot}_{ \\cC_\\alpha(M)}$. We prove in Theorem \\ref{thm:NS} that \nthere exists $\\delta>0$, depending on $(M,g)$ and $T$, such that for all $\\mathbf{f}\\in \\mathbb B_\\delta$, problem \\eqref{eq:NS1} admits a solution $(\\mathbf{u},p)=(\\bu_\\mathbf{f},p_\\mathbf{f})$ with $\\mathbf{u}\\in \\mathcal C_\\alpha$. The solution is unique in the sense that $\\mathbf{u}$ is unique and $p$ is unique up to a map depending only on $t\\in[0,T]$. Then, we can define the local source-to-final value map\n\\begin{align}\n\\label{eq:def_N}\n \\mathcal N:\\mathbb B_\\delta&\\to L^2\\Omega^1(\\omega),\\\\\n \\mathcal N(\\mathbf{f})&=\\bu_\\mathbf{f}(T,\\cdot)|_\\omega,\n \\end{align}\n with $\\bu_\\mathbf{f}\\in \\cC_\\alpha(M)$ solving \\eqref{eq:NS1}.\n Our main result can be stated as follows.", "full_context": "The Navier–Stokes equations are among the most fundamental mathematical models in fluid dynamics. \nIn this work, we consider the geometric Navier--Stokes equation on a Riemannian manifold $(M, g)$, see \\eqref{eq:NS1} below. When $M$ is a Euclidean domain and $g = c dx^2$, with $c > 0$ a constant, then \\eqref{eq:NS1} can be written as \nthe classical Navier-Stokes equations on $\\Omega$ with no-slip boundary condition \n\\begin{equation}\n\\begin{cases}\n \\p_t \\mathbf{u} + \\mathbf{u} \\nabla \\mathbf{u} - \\mu \\Delta \\mathbf{u} + \\nabla p = \\mathbf{f},\\text{ in }[0,T]\\timesM\\\\\n \\nabla\\cdot \\mathbf{u} = 0,\\text{ in }[0,T]\\timesM\\\\\n \\mathbf{u}=0,\\text{ on }[0,T]\\times \\partialM,\\\\\n \\mathbf{u}(0,\\cdot)=\\bu_0.\n\\end{cases}\n\\end{equation}\nHere $\\mathbf{u}: [0,T]\\times\\Omega\\to \\mathbb{R}^n$ is the velocity field of the fluid, $p: [0,T]\\times\\Omega\\to \\mathbb R$ is a scalar function representing the pressure, and \n$\\mu = c^{-1}$ is the coefficient of kinematic viscosity.\n\nThe generalization on a Riemannian manifold enables the study of fluid dynamics in complex geometric settings, thereby opening new perspectives for both theoretical analysis and practical applications \\cite{VlKh,EMa,MaRaSh,Sh}. The formulation of the Navier–Stokes equations on manifolds is motivated by several important areas, including: \n1) geophysical fluid dynamics \\cite{SaT,Val}, where the use of Riemannian geometry allows one to model the curvature of the Earth’s surface;\n2) General Relativity and cosmology \\cite{FMR}, modeling relativistic fluids, such as those present in neutron stars, black hole accretion disks, and the early universe;\n3)~fluid dynamics on biological membranes and microfluidic devices~\\cite{STONE}.\n\nIn all these contexts, the geometric structure of the manifold plays a central role in characterizing the physical properties of fluids and their surrounding media (e.g., density and viscosity). Motivated by these considerations, we investigate the inverse problem of identifying a Riemannian manifold from measurements of the fluid velocity field, associated with a source-to-final value map for the Navier–Stokes equations.\n\nInverse problems and, more broadly, identification problems related to fluid motion remain poorly understood. Only a limited number of studies have addressed this class of inverse problems, despite their strong physical motivation and mathematical significance. Among the few existing contributions devoted to inverse problems for the Stokes and Navier–Stokes equations, we may cite works concerning the determination of viscosity from boundary or internal measurements \\cite{FDJN,HLiWa,ImYa,LaUWa,LiWa,Liu}, the identification of objects immersed in a fluid \\cite{CaCo1,BCZ,CaCo2} as well as inverse source problems \\cite{BGKN,CIPY,ILY}. To the best of our knowledge, no prior studies have addressed inverse problems for the Navier–Stokes equations beyond the Euclidean setting.\n\n\\subsection{Notations and main results} \n Let $M$ be a n-dimensional, $n\\geq2$, smooth connected Riemannian manifold with boundary and $\\omega\\subset M$ be a nonempty open subset.\n We denote the space consisting of smooth sections of exterior $k$-form bundles on $M$ and $\\omega$ by $\\Omega^k(M)$ and $\\Omega^k(\\omega)$, respectively. Let\n \\begin{equation}\n \\ast:\\ \\Omega^k (M)\\to \\Omega^{n-k}(M)\n \\end{equation}\n be the Hodge star operator (see e.g. \\cite[section 14.1a]{frankel} for details). Then we define the codifferential operator $d^*: \\Omega^k(M)\\to \\Omega^{k-1}(M)$ as\n \\begin{equation}\n \\label{eq:def_codifferential}\n d^*=(-1)^{n(k+1)+1}\\ast d\\ast.\n \\end{equation}\n We denote the Hodge Laplacian $-dd^*-d^*d$ by $\\Delta_H$.\n Using the metric $g$, for every 1-form $u=u_idx_i$ on $M$ we associate the vector field $u_*=g^{ij}u_j\\partial_{x_i}\\in TM$. In the same way, for every vector field $v=v_i\\partial_{x_i}\\in TM$ we associate the 1-form $v^*=g_{ij}v_jdx_i$. With these notations, we write $R(w)=(\\nabla_{w_*}w_*)^*$,\nwhere $\\nabla$ denotes the Levi-Civita connection on $(M,g)$. Several works have been devoted to the study of the following geometric Navier-Stokes system \n \\begin{equation}\n \\label{eq:NS1}\n \\begin{cases}\n \\p_t\\mathbf{u}-\\Delta_H\\mathbf{u}+R(\\mathbf{u}) + d p=\\mathbf{f},\\ \\text{in } [0,T]\\times M,\\\\\n d^* \\mathbf{u}=0,\\ \\text{in } [0,T]\\times M,\\\\\n \\mathbf{u}|_{[0,T]\\times\\partialM}=0,\\\\\n \\mathbf{u}=0,\\ p=0,\\ \\text{in }\\{0\\}\\timesM.\n \\end{cases}\n \\end{equation}\n Without being exhaustive, one can refer to \\cite{EMa,AA91,CP92,TW93,CRT99} for more details.\n\nWe denote by $\\mathbb B_r$ the set defined by\n $$\\mathbb B_r:=\\{\\mathbf{g} \\in \\cC_\\alpha(M):\\ \\supp(\\mathbf{g})\\subset (0,T]\\times\\omega,\\ \\|\\mathbf{g}\\|_{ \\cC_\\alpha(M)}\\leq r\\},$$\nwhere we refer to Section \\ref{sec:Nonlinear_NS} for the definition of the space $\\cC_\\alpha(M)$ and $\\norm{\\cdot}_{ \\cC_\\alpha(M)}$. We prove in Theorem \\ref{thm:NS} that \nthere exists $\\delta>0$, depending on $(M,g)$ and $T$, such that for all $\\mathbf{f}\\in \\mathbb B_\\delta$, problem \\eqref{eq:NS1} admits a solution $(\\mathbf{u},p)=(\\bu_\\mathbf{f},p_\\mathbf{f})$ with $\\mathbf{u}\\in \\mathcal C_\\alpha$. The solution is unique in the sense that $\\mathbf{u}$ is unique and $p$ is unique up to a map depending only on $t\\in[0,T]$. Then, we can define the local source-to-final value map\n\\begin{align}\n\\label{eq:def_N}\n \\mathcal N:\\mathbb B_\\delta&\\to L^2\\Omega^1(\\omega),\\\\\n \\mathcal N(\\mathbf{f})&=\\bu_\\mathbf{f}(T,\\cdot)|_\\omega,\n \\end{align}\n with $\\bu_\\mathbf{f}\\in \\cC_\\alpha(M)$ solving \\eqref{eq:NS1}.\n Our main result can be stated as follows.\n\nTo the best of our knowledge, Theorem \\ref{thm:main} is the first result establishing the recovery of the general geometric structure of a manifold from local velocity measurements, at the final time $t=T$, associated with solutions of the Navier-Stokes equations. By contrast, the existing literature is confined to the identification of the viscosity coefficient from measurements related to the stationary Navier-Stokes equations \\cite{HLiWa,LaUWa,LiWa}.\n\nAccording to the above discussion Theorem \\ref{thm:main} follows from the following theorem.\n \\begin{theorem}\n \\label{thm_main:hyperbolic}\n Suppose there is a diffeomorphism $\\bPhi:\\omega_1\\to\\omega_2$ such that \n \\begin{equation}\n \\bPhi^\\ast\\Lambda_{2,\\omega_2}=\\Lambda_{1,\\omega_1}\\bPhi^\\ast.\n \\end{equation}\n Then $(\\M_1,g_1)$ and $(\\M_2,g_2)$ are isometric.\n \\end{theorem}\n The remainder of this section is devoted to the proof of Theorem \\ref{thm_main:hyperbolic} using the BC method. We extend the BC method to the auxiliary hyperbolic system \\eqref{eq:problem_hypo}. This extension is not a direct application of existing results, due to the presence of the divergence-free constraint. Accordingly, each step of the BC method must be adapted. These adaptations are carried out below.\n\n\\subsection{Reconstruction of metric in the accessible domain}\n \\begin{proposition}\n \\label{prop:dist_omega}\n Suppose there is a diffeomorphism $\\bPhi:\\omega_1\\to\\omega_2$ such that $\\diam_1(\\omega_1)< T_{\\bPhi}(\\omega_1)$ and $\\bPhi^\\ast\\Lambda_{2,\\omega_2}=\\Lambda_{1,\\omega_1}\\bPhi^\\ast$, then for any points $x,y\\in \\omega_1$, there holds $d_{g_1}(x,y)=d_{g_2}(\\bPhi(x),\\bPhi(y))$. Moreover, $\\bPhi$ is an isometry.\n \\end{proposition}\n \\begin{proof}\n Let $\\varepsilon>0$ be small enough that $B_1(x,\\varepsilon), B_1(y,\\varepsilon)\\subset \\omega_1$ and $B^c_1(x,\\varepsilon)$, $B^c_1(y,\\varepsilon)$ are connected. Then we write \n \\begin{equation}\n \\mathcal{B}_k(p,\\varepsilon):=C_0^\\infty((0,\\infty);C_0^\\infty\\Omega^1(B_k(p,\\varepsilon))),\\ k=1,2,\n \\end{equation}\n and \n \\begin{equation}\n \\bPhi_\\ast (\\cB_1(x,\\varepsilon)):=C_0^\\infty((0,\\infty);C_0^\\infty\\Omega^1(\\bPhi(B_1(x,\\varepsilon)))).\n \\end{equation}\n We define\n \\begin{align}\n t_{1,\\varepsilon} := \\inf\\{t>0\\mid \\exists \\bf\\in \\mathcal{B}_1(x,\\varepsilon)\\text{ s.t. } \\supp(\\bu_1^\\bf(t,\\cdot))\\cap B_1(y,\\varepsilon)\\neq \\emptyset \\}\n \\end{align}\n \\begin{align}\n t_{2,\\varepsilon} := \\inf\\{t>0\\mid \\exists \\bf\\in \\mathcal{B}_2(\\bPhi(x),\\varepsilon)\\text{ s.t. } \\supp(\\bu_2^\\bf(t,\\cdot))\\cap B_2(\\bPhi(y),\\varepsilon)\\neq \\emptyset \\},\n \\end{align}\n and \n \\begin{align}\n \\widetilde{t_{2,\\varepsilon}} := \\inf\\{t>0\\mid \\exists \\bh\\in \\bPhi_\\ast (\\cB_1(x,\\varepsilon))\\text{ s.t. } \\supp(\\bu_2^\\bh(t,\\cdot))\\cap \\bPhi(B_1(y,\\varepsilon))\\neq \\emptyset \\}.\n \\end{align}\n As $\\bPhi:\\omega_1\\to \\omega_2$ is a diffeomorphism, we can find two positive functions $r(\\varepsilon),R(\\varepsilon)$ such that $\\lim_{\\varepsilon\\to 0}r(\\varepsilon)=\\lim_{\\varepsilon\\to 0}R(\\varepsilon)=0$, and\n \\begin{equation}\n B_2(\\bPhi(p),r(\\varepsilon))\\subset \\bPhi(B_1(p,\\varepsilon))\\subset B_2(\\bPhi(p),R(\\varepsilon))\n \\end{equation}\n for both points $p=x,y$. Hence, $t_{2,R(\\varepsilon)}\\leq \\widetilde{t_{2,\\varepsilon}}\\leq t_{2,r(\\varepsilon)}.$\n\nNext we show the local reconstruction can be glued together.\\par\n \\begin{lemma}\n \\label{lm:glue}\n Let $U_1,U_2\\subset \\M_1^\\inter$ be open sets with isometries \n \\begin{equation}\n \\bPhi:U_1\\to\\bPhi(U_1),\\ \\bPsi: U_2\\to \\bPsi(U_2).\n \\end{equation}\n Suppose $\\Lambda_{1,U_1}\\bPhi^\\ast=\\bPhi^\\ast \\Lambda_{2,\\bPhi(U_1)}$, $\\Lambda_{1,U_2}\\bPsi^\\ast=\\bPsi^\\ast \\Lambda_{2,\\bPsi(U_2)}$ and there exists an open set $\\cV\\subset U_1\\cap U_2$ such that $\\bPhi|_\\cV=\\bPsi|_\\cV$ and $U_1\\cap U_2$ is $\\cT$-exhaustive by $\\cV$. Then there holds\n \\begin{equation}\n \\label{eq:lm_6.14}\n \\bPhi|_{U_1\\cap U_2}=\\bPsi|_{U_1\\cap U_2}.\n \\end{equation}\n \\end{lemma}\n \\begin{proof}\n Let us show that \n \\begin{equation}\n \\label{eq:lm_6.14_1}\n \\bPhi|_{E^{(1)}_{\\cT}(\\cV,U_1\\cap U_2)}=\\bPsi|_{E^{(1)}_{\\cT}(\\cV,U_1\\cap U_2)}.\n \\end{equation}\n Let $y\\in U_1\\cap U_2\\cap M(\\cV,\\cT)$ be arbitrary. To get a contradiction, we assume that $\\bPhi(y)\\neq \\bPsi(y)$. Let $z\\in \\cV$ be such that $d_{g_1}(y,z)<\\cT$. For simplicity, we denote $d_{g_1}(y,z)$ by $s$. Since $\\bPhi$ is an isometry, we have $d_{g_2}(\\bPhi(y),\\bPhi(z))=s<\\cT$. Then we can choose a small enough $0<\\varepsilon<\\frac{1}{2}(\\cT-s)$ such that $B_2(\\bPhi(y),2\\varepsilon)\\subset \\bPhi(U_1)$, $B_2(\\bPsi(y),2\\varepsilon)\\subset \\bPsi(U_2)$, $B_2(\\bPhi(y),2\\varepsilon)\\cap B_2(\\bPsi(y),2\\varepsilon)=\\emptyset$, and $\\M_2\\setminus B_2(\\bPhi(y),\\varepsilon)$ and $\\M_2\\setminus B_2(\\bPsi(y),\\varepsilon)$ are connected.\n By approximate controllability, we may take $\\bf\\in C_0^\\infty((T-\\varepsilon,T);C_0^\\infty\\Omega^1(B_2(\\bPhi(y),\\varepsilon)))$ such that $\\bu_2^{\\bf}(T,\\cdot)\\neq 0$. For any \n \\begin{align}\n \\bh\\in C_0^\\infty((T-\\varepsilon,T);C_0^\\infty\\Omega^1(B_2(\\bPsi(y),\\varepsilon))),\n \\end{align}\n due to the finite speed of propagation, we have\n \\begin{equation}\n \\supp(\\bu_2^\\bf(T,\\cdot))\\cap \\supp(\\bu_2^\\bh(T,\\cdot))=\\emptyset.\n \\end{equation}\n Since\n \\begin{align}\n \\bPhi^\\ast: C_0^\\infty((T-\\varepsilon,T);C_0^\\infty\\Omega^1(B_2(\\bPhi(y),\\varepsilon)))\\to C_0^\\infty((T-\\varepsilon,T);C_0^\\infty\\Omega^1(B_1(y,\\varepsilon))),\\\\\n \\bPsi^\\ast: C_0^\\infty((T-\\varepsilon,T);C_0^\\infty\\Omega^1(B_2(\\bPsi(y),\\varepsilon)))\\to C_0^\\infty((T-\\varepsilon,T);C_0^\\infty\\Omega^1(B_1(y,\\varepsilon)))\n \\end{align}\n are vector space isomorphism, we can choose $\\bPsi^\\ast \\bh = \\bPhi^\\ast \\bf$. Let $\\delta>0$ be such that $B_1(z,\\delta)\\subset \\cV$. Then we have $B_2(\\bPhi(y),2\\varepsilon)\\subset M_2(B_2(\\bPhi(z),\\delta),s+2\\varepsilon-\\delta)$. Since \n $$T_2(B_2(\\bPhi(z),\\delta))\\geq T_{\\bPhi}(B_1(z,\\delta))\\geq \\cT-\\delta>s+2\\varepsilon-\\delta,$$ \n by the approximate controllability, there exists a sequence\n $$\\{\\bphi_n\\}_{n=1}^\\infty\\subset C_0^\\infty((T-(s+2\\varepsilon-\\delta),T);C_0^\\infty\\Omega^1(B_2(\\bPhi(z),\\delta))$$ \n such that\n $\\bu_2^{\\bphi_n}(T,\\cdot)\\to \\bu_2^{\\bf}(T,\\cdot)$ as $n\\to \\infty$. Since $$B_2(\\bPhi(y),2\\varepsilon)\\cap B_2(\\bPsi(y),2\\varepsilon)=\\emptyset,$$\n there holds\n \\begin{equation}\n \\lim_{n\\to \\infty}\\inner{\\bu_2^{\\bh}(T,\\cdot),\\bu_2^{\\bphi_n}(T,\\cdot)}=0,\n \\end{equation}\n and \n \\begin{equation}\n \\lim_{n\\to \\infty}\\inner{\\bu_2^{\\bf}(T,\\cdot),\\bu_2^{\\bphi_n}(T,\\cdot)}=\\norm{\\bu_2^{\\bf}(T,\\cdot)}_{L^2\\Omega^1(\\M_2)}>0.\n \\end{equation}\n Then for large enough $N$ we write $\\bpsi=\\bphi_N$, and we have\n \\begin{equation}\n \\label{eq:glue_contradiction}\n \\inner{\\bu_2^{\\bf-\\bh}(T,\\cdot),\\bu_2^{\\bpsi}(T,\\cdot)}>0.\n \\end{equation}\n Notice that $\\bPhi^\\ast \\bpsi=\\bPsi^\\ast \\bpsi$ as $\\bPhi|_\\cV=\\bPsi|_\\cV$. According to Lemma \\ref{lm:norm_equal} and the fact that $\\bPsi^\\ast \\bh = \\bPhi^\\ast \\bf$, we have\n \\begin{align}\n \\inner{\\bu_2^{\\bf-\\bh}(T,\\cdot),\\bu_2^{\\bpsi}(T,\\cdot)}&=\\inner{\\bu_2^{\\bf}(T,\\cdot),\\bu_2^{\\bpsi}(T,\\cdot)}-\\inner{\\bu_2^{\\bh}(T,\\cdot),\\bu_2^{\\bpsi}(T,\\cdot)}\\\\\n &=\\inner{\\bu_1^{\\bPhi^\\ast\\bf}(T,\\cdot),\\bu_1^{\\bPhi^\\ast\\bpsi}(T,\\cdot)}-\\inner{\\bu_1^{\\bPsi^\\ast\\bh}(T,\\cdot),\\bu_1^{\\bPsi^\\ast\\bpsi}(T,\\cdot)}\\\\\n &=\\inner{\\bu_1^{\\bPhi^\\ast\\bf-\\bPsi^\\ast\\bh}(T,\\cdot),\\bu_1^{\\bPhi^\\ast\\bpsi}(T,\\cdot)}=0.\n \\end{align}\n This is in contradiction with \\eqref{eq:glue_contradiction} and we have proved that \\eqref{eq:lm_6.14_1} holds as $y\\in E^{(1)}_{\\cT}(\\cV,U_1\\cap U_2)$ is arbitrary. Since $U_1\\cap U_2$ is $\\cT$-exhaustive by $\\cV$, there exists $N>0$ such that $U_1\\cap U_2=E^{(N)}_{\\cT}(\\cV,U_1\\cap U_2)$. Replacing $E^{(k)}_{\\cT}(\\cV,U_1\\cap U_2)$ by $E^{(k+1)}_{\\cT}(\\cV,U_1\\cap U_2)$ iteratively in above argument for $k=0,1,\\cdots,N-1$, we can obtain \\eqref{eq:lm_6.14}.\n \\end{proof}\n \\begin{proof}[Proof of Theorem \\ref{thm:main}]\n Provided $\\bPhi:\\omega_1\\to \\omega_2$ is a diffeomorphism and $\\Lambda_{1,\\omega_1}\\bPhi^\\ast=\\bPhi^\\ast \\Lambda_{2,\\omega_2}$. we have $\\bPhi$ is an isometry by Proposition \\ref{prop:dist_omega}.", "post_theorem_intro_text_len": 7157, "post_theorem_intro_text": "\\begin{remark}\n      By making $\\omega_1$ and $\\omega_2$ smaller, connectedness can be assumed without loss of generality.\n  \\end{remark}\n\\subsection{Comparision to previous literature}\n\nTo the best of our knowledge, Theorem \\ref{thm:main} is the first result establishing the recovery of the general geometric structure of a manifold from local velocity measurements, at the final time $t=T$, associated with solutions of the Navier-Stokes equations. By contrast, the existing literature is confined to the identification of the viscosity coefficient from measurements related to the stationary Navier-Stokes equations \\cite{HLiWa,LaUWa,LiWa}.\n\nOur proof of Theorem \\ref{thm:main} is based on application of the Boundary Control (BC) method, which remains one of the most powerful tools for solving inverse coefficient problems for partial differential equations (PDEs) in the time domain. The BC method was first introduced by Belishev in \\cite{Beli} and thereafter further developed for solving inverse coefficient determination problems related to the wave equation in Euclidean domains \\cite{ABI92,B87,B90,BK89,BK87,BKu89,BK91}. Later, this method was extended to manifold reconstruction \\cite{BeKu}. It has been shown that the BC method can be applied to other classes of scalar PDEs \\cite{FGKU,HLYZ,KaKuLaMa,KLLY,KOSY}.\n\nThe method appears to be less powerful in the context of systems of PDEs, however, inverse problems for the Maxwell \\cite{BeIs,KuLaSo} and Dirac \\cite{LK09} systems has been solved using it. Further, in \\cite{KOP18} the BC method was generalized for abstract wave equations associated to connection Laplacians on vector bundles (for example, a linearized equation satisfied by the Higgs field on the pre-quantum level). To the best of our knowledge, this work constitutes the first application of the BC method to the Stokes system. The approach consists in first establishing a rigorous correspondence between the data for the Stokes system and those for an auxiliary hyperbolic Stokes problem through spectral data, as developed in Section 5. The BC method is then implemented for this auxiliary hyperbolic Stokes problem in Section 7.\n\nIt is important to emphasize that the extension of the BC method to this setting cannot be derived from existing results in the literature. The principal obstruction arises from the divergence-free constraint inherent to the Stokes system, which introduces substantial technical and structural difficulties at every stage of the method. In Section 7, we provide a detailed exposition of the adaptation of the BC method to this framework, and we show how the difficulties encountered at each step can be resolved by means of carefully tailored geometric and analytic arguments.\n\nIn inverse problems concerning recovery of geometry on manifolds with boundary, measurements are typically taken on the boundary and modeled by the Dirichlet-to-Neumann map or the associated Cauchy data \\cite{AKKLT04,KK98,Oksanen2014,KaKuLaMa}. By contrast, inverse problems based on the local source-to-solution map have been predominantly studied on closed manifolds. In particular, it is shown in \\cite{Saksala2018,HLYZ,SS25} that the local source-to-solution map for various linear evolution equations determines the underlying manifold up to an isometry. A related result for the wave equation, where the source and observations are supported in disjoint sets, is established in \\cite{LNOY24}. \nIn this work, we consider the local source-to-solution map on manifolds with boundary.\n\nThe approach adopted in Sections 4 and 5 to address the nonlinearity relies on a linearization method that has proved effective in the analysis of numerous inverse problems for nonlinear PDEs. This method, originally introduced in \\cite{I93,IS94} in the context of parabolic and elliptic partial differential equations, consists in establishing that the first-order linearization of the data coincides with the data associated with an appropriate linearized equation. In contrast with the case of classical scalar partial differential equations (see, for instance, \\cite{L25} for a comprehensive review), the application of the linearization method to the Navier–Stokes system involves both the velocity field $\\mathbf{u}$ and the pressure $p$. This necessitates a suitable representation of the pressure $p$, which is introduced in Theorem \\ref{thm:NS}, together with the corresponding linearization result stated in Proposition \\ref{linearization}. To the best of our knowledge, the only existing works employing techniques closely related to this linearization approach for the Navier–Stokes system are \\cite{LaUWa,LiWa}, where the analysis is restricted to the Euclidean setting and to stationary Navier–Stokes equations in dimensions two and three. By contrast, our analysis is carried out in H\\\"older spaces, which allows for an extension to higher-dimensional settings.\n\nFollowing the linearization step, in Section 5 we make use of analytic and representation properties of solutions to the linear Stokes system in order to establish a connection between the local source-to-final-value map and a source-to-solution map associated with an auxiliary hyperbolic system. The inverse problem for this auxiliary hyperbolic system is subsequently analyzed by means of the BC method. This data transfer technique by mean of spectral data has previously been employed in the study of inverse coefficient problems for various scalar evolution partial differential equations \\cite{CK,HLYZ,KaKuLaMa,KLLY,KOSY}. As far as we know, the present work constitutes the first application of this methodology to Stokes systems.\n\n\\subsection{Outline of the paper}\n\nThe outline of the paper is as follows. In Section \\ref{sec:elliptic}, we establish regularity results for the stationary Stokes system on manifolds and analyze properties of the associated spectral data, which prepare the terrain for the study of the inverse problem. Section \\ref{sec:Nonlinear_NS} is devoted to the existence, uniqueness and appropriate regularity of the solution to the geometric Navier-Stokes equation \\eqref{eq:NS1}. In Section \\ref{sec:parabolic}, we consider the non-stationary Stokes equation, which arises as the linearization of \\eqref{eq:NS1}, and show that the data $\\mathcal N$ determines the corresponding restricted source-to-solution operator $\\mathcal P$, defined as \\eqref{eq:def_P}, for the non-stationary Stokes equation. Moreover, we show that $\\mathcal P$ is equivalent to the localized spectral projections of the stationary Stokes system, which in turn allows the data to be converted into the restricted source-to-solution operator for an auxiliary hyperbolic Stokes system \\eqref{eq:problem_hypo}. Section \\ref{sec:hyperbolic} analyzes \\eqref{eq:problem_hypo} and develops several technical tools, including conditional finite speed of propagation and approximate controllability. The new feature of these tools is that they hold only on a finite time interval determined by the underlying geometry. Combining these techniques with an adapted BC, our main result Theorem \\ref{thm:main} is proved in Section \\ref{sec:mian_proof}.", "sketch": "The introduction states that the proof of Theorem~\\ref{thm:main} is based on the Boundary Control (BC) method. The approach is to (i) address the nonlinearity by a linearization method (Sections 4--5), establishing that the first-order linearization of the data coincides with the data for an appropriate linearized equation, which requires a suitable representation of the pressure $p$ (Theorem~\\ref{thm:NS}) and yields a linearization result (Proposition~\\ref{linearization}); then (ii) use analytic/representation properties of solutions to the linear Stokes system to connect the local source-to-final-value map $\\mathcal N$ to a (restricted) source-to-solution operator $\\mathcal P$ for the non-stationary Stokes equation, and show $\\mathcal P$ is equivalent to localized spectral projections of the stationary Stokes system; this (iii) allows converting the data into the restricted source-to-solution operator for an auxiliary hyperbolic Stokes system \\eqref{eq:problem_hypo} through spectral data (Section 5); finally (iv) analyze this auxiliary hyperbolic problem using tools developed in Section~\\ref{sec:hyperbolic} (including conditional finite speed of propagation and approximate controllability, valid on a finite time interval determined by the geometry) and then implement an adapted BC method (Section 7 / Section~\\ref{sec:mian_proof}) to conclude the manifold is determined up to an isometry.", "expanded_sketch": "The introduction states that the proof of the main theorem is based on the Boundary Control (BC) method. The approach is to (i) address the nonlinearity by a linearization method (Sections 4--5), establishing that the first-order linearization of the data coincides with the data for an appropriate linearized equation, which requires a suitable representation of the pressure $p$. We first use the following theorem.\n\\begin{theorem}\n    \\label{thm:NS}\nThere exists $\\delta>0$, depending on $(\\M,g)$ and $T$, such that for all $\\bf\\in \\mathbb B_\\delta$, problem \\eqref{eq:NS1} admits a unique solution $(\\bu,p)=(\\bu_\\bf,p_\\bf)$, $p$ being unique up to a map depending only on $t\\in[0,T]$, with $u_\\bf\\in \\mathcal C_\\alpha$, $dp_\\bf\\in C^{\\frac{\\alpha}{2}}([0,T];C^{\\alpha}\\Omega^1(\\M))$ and with $p_\\bf$ a solution of \\eqref{eq:p}. Moreover, the map $\\mathbb B_\\delta\\ni\\bf\\mapsto (u_\\bf,dp_\\bf)\\in \\cC_\\alpha(M)\\times L^2([0,T]; L^2\\Omega^1(\\M))$ is $C^1$.\n  \\end{theorem}\nThis yields the following linearization result.\n\\begin{proposition}\\label{linearization} \nFor $i=1,2$ and $\\bf\\in C^\\infty_0([0,T];C^\\infty_0\\Omega^1(\\omega_i))$, we fix $\\delta_{i,\\bf}=\\frac{\\delta}{1+\\|\\bf\\|_{\\mathcal C_\\alpha(M_i)}}$ with $\\delta$ introduced in Theorem \\ref{thm:NS}. Then the map\n$$(-\\delta_{i,\\bf},\\delta_{i,\\bf})\\ni s\\mapsto \\mathcal N_i(s\\bf)\\in L^2\\Omega^1(\\omega_i)$$\nis $C^1$ and we have\n\\begin{equation}\\label{lin1}\n    \\partial_s\\mathcal N_i(s\\bf)|_{s=0}=\\mathcal P_i(\\bf).\n\\end{equation}\n\n\\end{proposition}\nThen (ii) one uses analytic/representation properties of solutions to the linear Stokes system to connect the local source-to-final-value map $\\mathcal N$ to a (restricted) source-to-solution operator $\\mathcal P$ for the non-stationary Stokes equation, and show $\\mathcal P$ is equivalent to localized spectral projections of the stationary Stokes system; this (iii) allows converting the data into the restricted source-to-solution operator for an auxiliary hyperbolic Stokes system\n\\begin{equation}\n    \\label{eq:problem_hypo}\n    \\begin{cases}\n      (\\p_t^2-\\Delta_H)\\bu + d p=\\bf,\\ \\text{in } \\RR^+\\times \\M,\\\\\n      d^* \\bu=0,\\ \\text{in } \\RR^+\\times \\M,\\\\\n      \\bu|_{[0,T]\\times\\p\\M}=0,\\\\\n      \\bu=0,\\ p=0,\\ \\text{in }\\RR^-\\times\\M.\n    \\end{cases}\n  \\end{equation}\nthrough spectral data (Section 5); finally (iv) one analyzes this auxiliary hyperbolic problem using tools developed later (including conditional finite speed of propagation and approximate controllability, valid on a finite time interval determined by the geometry) and then implements an adapted BC method (Section 7 / later in the paper) to conclude the manifold is determined up to an isometry.", "expanded_theorem": "\\label{thm:main}\n      Let $(\\M_1,g_1)$ and $(\\M_2,g_2)$ be two  n-dimensional smooth connected Riemannian manifolds with  boundary. Let $\\omega_1\\subset \\M_1$ and $\\omega_2\\subset\\M_2$ be open and connected. For $j=1,2$, let $\\mathcal N_j$ be the local source-to-final value map\n      \\begin{align}\n\\label{eq:def_N}\n    \\mathcal N:\\mathbb B_\\delta&\\to L^2\\Omega^1(\\omega),\\\\\n    \\mathcal N(\\bf)&=\\bu_\\bf(T,\\cdot)|_\\omega,\n  \\end{align}\nwith $(M,g)=(\\M_j,g_j)$\n      and $\\omega=\\omega_j$.\n      In addition, suppose that there is a diffeomorphism $\\bm{\\Phi}:\\omega_1\\to \\omega_2$ such that \n      \\begin{equation}\\label{NN}\n          \\cN_1\\bm{\\Phi}^\\ast=\\bm{\\Phi}^\\ast \\cN_2.\n      \\end{equation}\n       Then $(\\M_1,g_1)$ and $(\\M_2,g_2)$ are isometric.,", "theorem_type": ["Implication"], "mcq": {"question": "Let \\((\\mathcal M_1,g_1)\\) and \\((\\mathcal M_2,g_2)\\) be two smooth connected \\(n\\)-dimensional Riemannian manifolds with boundary, and let \\(\\omega_1\\subset \\mathcal M_1\\) and \\(\\omega_2\\subset \\mathcal M_2\\) be open connected subsets. For each \\(j=1,2\\), consider the geometric Navier--Stokes system on \\([0,T]\\times \\mathcal M_j\\) for a time-dependent 1-form \\(\\mathbf u\\) and pressure \\(p\\):\n\\[\n\\begin{cases}\n\\partial_t \\mathbf u-\\Delta_H\\mathbf u+R(\\mathbf u)+dp=\\mathbf f,\\\\\nd^*\\mathbf u=0,\\\\\n\\mathbf u|_{[0,T]\\times \\partial \\mathcal M_j}=0,\\\\\n\\mathbf u=0,\\ p=0 \\text{ at } t=0,\n\\end{cases}\n\\]\nwhere \\(\\Delta_H=-dd^*-d^*d\\) is the Hodge Laplacian and \\(R(\\mathbf u)=(\\nabla_{\\mathbf u_*}\\mathbf u_*)^*\\). For some \\(\\delta>0\\), let \\(\\mathbb B_\\delta\\) denote the admissible source 1-forms supported in \\((0,T]\\times \\omega_j\\) with \\(\\|\\mathbf f\\|_{C_\\alpha(\\mathcal M_j)}\\le \\delta\\), and for each such \\(\\mathbf f\\) let \\(\\mathbf u_{\\mathbf f}\\) be the corresponding solution. Define the local source-to-final value map\n\\[\n\\mathcal N_j(\\mathbf f)=\\mathbf u_{\\mathbf f}(T,\\cdot)|_{\\omega_j}\\in L^2\\Omega^1(\\omega_j).\n\\]\nAssume there exists a diffeomorphism \\(\\boldsymbol\\Phi:\\omega_1\\to \\omega_2\\) such that, with \\(\\boldsymbol\\Phi^*\\) denoting pullback of time-dependent 1-forms,\n\\[\n\\mathcal N_1\\,\\boldsymbol\\Phi^*=\\boldsymbol\\Phi^*\\mathcal N_2,\n\\]\ni.e. for every admissible source on \\(\\omega_2\\), the observed final 1-form on \\(\\omega_1\\) agrees after pullback. Which conclusion about \\((\\mathcal M_1,g_1)\\) and \\((\\mathcal M_2,g_2)\\) holds?", "correct_choice": {"label": "A", "text": "The manifolds \\((\\mathcal M_1,g_1)\\) and \\((\\mathcal M_2,g_2)\\) are isometric; equivalently, there exists a diffeomorphism \\(F:\\mathcal M_1\\to \\mathcal M_2\\) such that \\(F^*g_2=g_1\\)."}, "choices": [{"label": "B", "text": "The manifolds \\((\\mathcal M_1,g_1)\\) and \\((\\mathcal M_2,g_2)\\) have isometric neighborhoods of \\(\\omega_1\\) and \\(\\omega_2\\); more precisely, there exist open sets \\(U_j\\subset \\mathcal M_j\\) with \\(\\omega_j\\subset U_j\\) and a diffeomorphism \\(F:U_1\\to U_2\\) such that \\(F|_{\\omega_1}=\\boldsymbol\\Phi\\) and \\(F^*g_2=g_1\\) on \\(U_1\\)."}, {"label": "C", "text": "The diffeomorphism \\(\\boldsymbol\\Phi:\\omega_1\\to \\omega_2\\) is an isometry between the observed regions; that is, \\(\\boldsymbol\\Phi^*g_2=g_1\\) on \\(\\omega_1\\)."}, {"label": "D", "text": "The manifolds \\((\\mathcal M_1,g_1)\\) and \\((\\mathcal M_2,g_2)\\) are isometric provided the intertwining identity \\(\\mathcal N_1\\,\\boldsymbol\\Phi^*=\\boldsymbol\\Phi^*\\mathcal N_2\\) holds for all smooth compactly supported sources on \\((0,T]\\times\\omega_2\\), without any smallness restriction \\(\\|\\mathbf f\\|_{C_\\alpha(\\mathcal M_2)}\\le \\delta\\)."}, {"label": "E", "text": "There exists a constant \\(c>0\\) and a diffeomorphism \\(F:\\mathcal M_1\\to \\mathcal M_2\\) extending \\(\\boldsymbol\\Phi\\) such that \\(F^*g_2=c\\,g_1\\); in particular, the manifolds are determined only up to a global conformal scaling by the local source-to-final value map."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "global_BC_gluing_to_full_manifold", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "geometric_construction", "tampered_component": "dropped_global_extension_to_entire_manifolds", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "small_data_linearization_radius_delta", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "metric_identification_vs_conformal_gauge", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct choice. It states a technical intertwining hypothesis and asks for the resulting geometric conclusion, without giving away global isometry."}, "TAS": {"score": 1, "justification": "The item is very close to a theorem-recall question: the hypotheses are presented in theorem form and the student is asked for the conclusion. However, it is not a pure verbatim restatement, since the options force discrimination between global, local, weaker, and overstrong conclusions."}, "GPS": {"score": 1, "justification": "Moderate reasoning is required: the solver must identify the strongest valid consequence and reject tempting alternatives such as only local isometry, conformal ambiguity, or a stronger hypothesis change. Still, this mainly tests recognition of the theorem's precise conclusion rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically meaningful. They reflect common failure modes: settling for a weaker true statement (local isometry on the observed region), inferring only neighborhood recovery, removing the small-data condition, or confusing metric recovery with conformal recovery."}, "total_score": 6, "overall_assessment": "A solid theorem-precision MCQ with strong distractors and little answer leakage, though it leans more toward recall of a stated result than genuinely generative mathematical reasoning."}}
{"id": "2602.08680v1", "paper_link": "http://arxiv.org/abs/2602.08680v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{main_thm}\nLet $\\mathcal{H}\\in (\\frac{1}{3},1)$, and let the initial data $u_0^{\\epsilon},v_0^\\epsilon $ be divergence-free, of zero mean, and such that\n\\begin{align*}\n   \\sup_{\\epsilon\\in (0,1)} \\|u_0^{\\epsilon}\\|_{L^2_x}+\\|\\sqrt{\\epsilon}v_0^{\\epsilon}\\|_{L^2_x}<+\\infty.\n\\end{align*}\nAssume that the operators $Q$ and $C$ satisfy the conditions stated in \\autoref{sec_hp_noise}. Then there exist probabilistically weak rough path solutions $(u^{\\epsilon},(-C)^{-1}dW^{\\mathcal{H},\\epsilon})$ to \\eqref{system_prelimit} that converge in law to a probabilistically weak rough path solution of \\eqref{eq_Hgreat12} in the case $\\alpha=1$ and $\\mathcal{H}>\\tfrac{1}{2}$, and to a probabilistically weak rough path solution of \\eqref{eq_Hsmall12} in the case $\\mathcal{H}<\\tfrac{1}{2}$ and $\\alpha=\\tfrac{1}{2}+\\mathcal{H}$.", "start_pos": 21972, "end_pos": 22784, "label": "main_thm"}, "ref_dict": {"def_rough_sol": "\\begin{definition}\\label{def_rough_sol}\n    Let $\\epsilon\\in (0,1)$, $\\frac1\\H\\in[N,N+1)\\cap(1,3)$, $p\\in (\\frac{1}{\\H},N+1)$ and assume $C,Q$ satisfy the condition of \\autoref{sec_hp_noise}.\\\\\n     A tuple $((\\Omega^\\epsilon, \\mathcal{F}^\\epsilon, (\\mathcal{F}_t^\\epsilon)_t,\\mathbb{P}^{\\epsilon}),W^{\\epsilon}_t,u^\\epsilon_t,r^\\epsilon_t)$ is a probabilistically weak solution to equation \\eqref{system_prelimit} if \n    \\begin{enumerate}\n        \\item $(\\Omega^\\epsilon, \\mathcal{F}^\\epsilon, (\\mathcal{F}_t^\\epsilon)_t,\\mathbb{P}^{\\epsilon})$ is a stochastic basis with complete and right-continuous filtration,\n        \\item $W^{\\epsilon,\\H}_t$ is a fractional Wiener process of Hurst parameter $\\H$ and space covariance $Q$ and it is adapted to $(\\Omega^\\epsilon, \\mathcal{F}^\\epsilon, \\mathcal{F}_t^\\epsilon,\\mathbb{P}^{\\epsilon})$,\n        \\item $r^\\epsilon\\in L^2([0,T],H^{\\gamma})$ $\\mathbb{P}^{\\epsilon}$-a.s., it is progressively measurable and satisfies $\\mathbb{P}^{\\epsilon}$-a.s. \n       \\begin{equation*}\n       \\begin{aligned}\n    \\partial_t r^{\\epsilon}&=\\epsilon^{-1}Cr^{\\epsilon}+\\nu \\Delta(\\epsilon^{-\\beta}w^{\\epsilon}+r^{\\epsilon})\\\\&-b(u^{\\epsilon}+\\epsilon^{-\\alpha+\\H}w^{\\epsilon}+r^{\\epsilon},\\epsilon^{-\\alpha+\\H}w^{\\epsilon}+r^{\\epsilon})\n    \\end{aligned}\n       \\end{equation*}\n        in an analytically weak sense,\n        \\item $u^{\\epsilon}\\in L^{\\infty}([0,T],L^2)\\cap L^2([0,T],H^1)$ $\\mathbb{P}^{\\epsilon}$-a.s., it is progressively measurable and $\\mathbb{P}^{\\epsilon}$-a.s it holds, in a weak sense, \n        \\begin{equation*}\n            u^\\epsilon\\st=\\int_s^t \\Delta u^\\epsilon_\\theta-b(u_\\theta^\\epsilon+r^\\epsilon_\\theta,u^\\epsilon_\\theta)\\,d\\theta+\\sum_{k=1}^N \\mathbb{A}^{k,\\epsilon}\\st u_s^\\epsilon+u^{\\natural,\\epsilon}\\st,\n        \\end{equation*}\n        where $u^{\\natural}\\st\\in C^{p/(N+1)\\var}_{2,loc}(H^{-3})$ $\\mathbb{P}^{\\epsilon}$-a.s.\n    \\end{enumerate}\n    \\end{definition}", "intro_syst_h12": "\\begin{align}\\label{intro_syst_h12}\n    \\begin{cases}\n    du^{\\epsilon}=\\nu \\Delta u^{\\epsilon}dt-(u^{\\epsilon}+v^{\\epsilon})\\cdot\\nabla u^{\\epsilon}dt+\\nabla p^{\\epsilon}\\,dt,\\\\\n    dv^{\\epsilon}=\\left(\\nu\\Delta v^{\\epsilon}+\\epsilon^{-1}Cv^{\\epsilon}-(u^{\\epsilon}+v^{\\epsilon})\\cdot\\nabla v^{\\epsilon}+\\nabla q^{\\epsilon}\\right)\\,dt+\\epsilon^{-1}dW_t,\\\\\n    \\nabla\\cdot u^{\\epsilon}=0,\\quad \\nabla\\cdot v^{\\epsilon}=0,\n\\end{cases}  \n\\end{align}", "main_thm": "\\begin{theorem}\\label{main_thm}\nLet $\\H\\in (\\frac{1}{3},1)$, and let the initial data $u_0^{\\epsilon},v_0^\\epsilon $ be divergence-free, of zero mean, and such that\n\\begin{align*}\n   \\sup_{\\epsilon\\in (0,1)} \\|u_0^{\\epsilon}\\|_{L^2_x}+\\|\\sqrt{\\epsilon}v_0^{\\epsilon}\\|_{L^2_x}<+\\infty.\n\\end{align*}\nAssume that the operators $Q$ and $C$ satisfy the conditions stated in \\autoref{sec_hp_noise}. Then there exist probabilistically weak rough path solutions $(u^{\\epsilon},(-C)^{-1}dW^{\\H,\\epsilon})$ to \\eqref{system_prelimit} that converge in law to a probabilistically weak rough path solution of \\eqref{eq_Hgreat12} in the case $\\alpha=1$ and $\\H>\\tfrac{1}{2}$, and to a probabilistically weak rough path solution of \\eqref{eq_Hsmall12} in the case $\\H<\\tfrac{1}{2}$ and $\\alpha=\\tfrac{1}{2}+\\H$. \n\\end{theorem}", "eq_fast_1": "\\begin{align}\\label{eq_fast_1}\ndv^{\\epsilon}=\\epsilon^{-1}Cv^{\\epsilon}dt+\\epsilon^{-1}dW^{\\H}_t.\n\\end{align}", "intro_transport_h12": "\\begin{align}\\label{intro_transport_h12}\n    \\begin{cases}\n    du=\\left(\\nu \\Delta u-u\\cdot\\nabla u-\\overline{r}\\cdot\\nabla u\\right) dt+d\\nabla p-\\circ dW\\cdot\\nabla u,\\\\\n    \\nabla\\cdot u=0,\n\\end{cases}  \n\\end{align}", "e:equation1": "\\begin{align}\\label{e:equation1}\n\\begin{cases}\n     du^{\\epsilon}=\\nu \\Delta u^{\\epsilon}dt-(u^{\\epsilon}+v^{\\epsilon})\\cdot\\nabla u^{\\epsilon}dt+\\nabla p^{\\epsilon}\\,dt,\\\\\n    dv^{\\epsilon}=\\left(\\nu\\Delta v^{\\epsilon}+\\epsilon^{-1}Cv^{\\epsilon}-(u^{\\epsilon}+v^{\\epsilon})\\cdot\\nabla v^{\\epsilon}+\\nabla q^{\\epsilon}\\right)\\,dt+\\epsilon^{-\\alpha}dW^{\\H}_t,\\\\\n    \\nabla\\cdot u^{\\epsilon}=0,\\quad \\nabla\\cdot v^{\\epsilon}=0,\n\\end{cases}\n\\end{align}", "eq_Hgreat12": "\\begin{align}\\label{eq_Hgreat12}\n    \\begin{cases}\n    du=\\left(\\nu \\Delta u-u\\cdot\\nabla u\\right) dt+d\\nabla p- d(-C)^{-1}W^{\\H}\\cdot\\nabla u,\\\\\n    \\nabla\\cdot u=0.\n    \\end{cases}\n\\end{align}", "system_intermediate_small": "\\begin{align}\\label{system_intermediate_small}\n    \\begin{cases}\n        \\partial_t u^{\\epsilon}+(u^{\\epsilon}+v^{\\epsilon})\\cdot\\nabla u^{\\epsilon}+\\nabla p^{\\epsilon}&=0,\\\\\n        \\partial_t v^{\\epsilon}+(u^{\\epsilon}+v^{\\epsilon})\\cdot\\nabla v^{\\epsilon}+\\nabla q^{\\epsilon}&=0,\\\\\n        \\nabla\\cdot u=\\nabla\\cdot v=0.\n    \\end{cases}\n\\end{align}", "def_rough_lim_sol_Hsmall12": "\\begin{definition}\\label{def_rough_lim_sol_Hsmall12}\n    A tuple $((\\Omega, \\mathcal{F}, (\\mathcal{F}_t)_t,\\mathbb{P}),u_t)$ is a weak solution to equation \\eqref{eq_Hsmall12} if \n    \\begin{enumerate}\n        \\item $(\\Omega, \\mathcal{F}, (\\mathcal{F}_t)_t,\\mathbb{P})$ is a stochastic basis with complete and right-continuous filtration,\n\n        \\item $u\\in C_w([0,T],L^2)\\cap L^2([0,T],H^1)$ $\\mathbb{P}$-a.s.,  it is progressively measurable and $\\mathbb{P}$-a.s it holds, in a weak sense, \n        \\begin{equation*}\n            u\\st=\\int_s^t \\Delta u^\\epsilon_\\theta-b(u_\\theta+\\overline{r},u_\\theta)\\,d\\theta,\n        \\end{equation*}\n        where $\\overline{r}\\in H$.\n    \\end{enumerate}\n    \\end{definition}", "lem_ito_stokes": "\\begin{lemma}\\label{lem_ito_stokes}\n   Let $\\H\\in (\\frac{1}{4},\\frac{1}{2}]$ denoting by \n    \\begin{align*}\n        \\overline{r}=\\int_{L^2}(-C)^{-1}b(r,r)d\\mu(r),\\quad\n        \\begin{cases}\n            dw&= Cwdt+dW_t\\\\\n            w(0)&=0,\\ \n        \\end{cases}\n    \\end{align*}\n    where $\\mu$ is the centered Gaussian measure on $L^2$ with covariance operator \n    \\begin{align*}\n        \\overline{Q}=\\sum_{i\\geq 1} \\frac{\\sigma_i^2 e_i\\otimes e_i}{2|\\lambda_i|^{2\\H}}\\int_0^{+\\infty}e^{-s} s^{2\\H} ds \n    \\end{align*}\n    then it holds\n\\begin{align}\\label{convergence_ITO_STOKES_COND}\n        \\frac{1}{T}\\int_0^T (-C)^{-1}b(w_s,w_s)ds\\rightarrow \\overline{r} \\quad \\text{in } L^2(\\Omega;H).\n    \\end{align}\n\\end{lemma}", "heuristic_convergence_noise": "\\begin{align}\\label{heuristic_convergence_noise}\n    w^{\\epsilon}_t = \\epsilon^{-1} \\int_0^t e^{-\\epsilon^{-1}(t-s)}\\, dW_s \\rightarrow dW_t\n\\end{align}", "system_prelimit": "\\begin{align}\\label{system_prelimit}\n    \\begin{cases}\n        &\\partial_t u^{\\epsilon}=\\nu A u^{\\epsilon}-b(u^{\\epsilon}+\\epsilon^{-\\alpha+\\H}w^{\\epsilon}+r^{\\epsilon},u^{\\epsilon}),\\\\\n        &dw^{\\epsilon}=\\epsilon^{-1}Cw^{\\epsilon} dt+ \\epsilon^{-\\H}dW^\\H_t, \\\\\n        &\\begin{aligned}\\partial_t r^{\\epsilon}&=\\epsilon^{-1}Cr^{\\epsilon}+\\nu A(\\epsilon^{-\\alpha+\\H}w^{\\epsilon}+r^{\\epsilon})\\\\&-b(u^{\\epsilon}+\\epsilon^{-\\alpha+\\H}w^{\\epsilon}+r^{\\epsilon},\\epsilon^{-\\alpha+\\H}w^{\\epsilon}+r^{\\epsilon}),\n        \\end{aligned}\\\\\n        &w^{\\epsilon}(0) =0,\\quad r^{\\epsilon}(0)=v_0^{\\epsilon}.\n    \\end{cases}\n\\end{align}", "eq_Hsmall12": "\\begin{align}\\label{eq_Hsmall12}\n    \\begin{cases}\n    \\partial_t u=\\nu \\Delta u-\\overline{r}\\cdot\\nabla u-u\\cdot\\nabla u+\\nabla p,\\\\\n    \\nabla\\cdot u=0,\n    \\end{cases}\n\\end{align}", "remark_d2": "\\begin{remark}\\label{remark_d2}\n    In dimension $d=2$, thanks to the uniqueness of both the prelimit and the limit equations, the above statement can be strengthened. Under the same assumptions as in \\autoref{main_thm}, there exist probabilistically strong rough path solutions $(u^{\\epsilon},(-C)^{-1}dW^{\\H})$ to \\eqref{system_prelimit} that converge in probability to the unique weak rough path solution of \\eqref{eq_Hgreat12} in the case $\\alpha=1$ and $\\H>\\tfrac{1}{2}$, and of \\eqref{eq_Hsmall12} in the case $\\H<\\tfrac{1}{2}$ and $\\alpha=\\tfrac{1}{2}+\\H$.\n\\end{remark}", "trivial_noise": "\\begin{proposition}\\label{trivial_noise}\nLet $\\alpha=1$, $\\H\\in\\left(\\tfrac{1}{3},\\tfrac{1}{2}\\right)$, and let the initial data $u_0^{\\epsilon}$ and $v_0^{\\epsilon}$ be divergence-free, of zero mean, and satisfy\n\\begin{align*}\n   \\sup_{\\epsilon\\in (0,1)} \\|u_0^{\\epsilon}\\|_{L^2_x}+\\|\\sqrt{\\epsilon}v_0^{\\epsilon}\\|_{L^2_x}<+\\infty.\n\\end{align*} Assume that $W^{\\H}_t(x)=f(x)W^{\\H,1}_t$, with $f$ as above, and that $Q$ and $C$ satisfy the conditions stated in \\autoref{sec_hp_noise}. Then there exist probabilistically weak rough path solutions $(u^{\\epsilon},(-C)^{-1}dW^{\\H,\\epsilon})$ to \\eqref{system_prelimit} that converge in law to a probabilistically weak rough path solution of \\eqref{eq_Hgreat12}. An analogue of \\autoref{remark_d2} holds in dimension $d=2$.\n\\end{proposition}", "def_rough_lim_sol_Hgreat12": "\\begin{definition}\\label{def_rough_lim_sol_Hgreat12}\n        Let $\\frac1\\H\\in[N,N+1)\\cap(1,3)$, $p\\in (\\frac{1}{\\H},N+1)$ and assume $C,Q$ satisfy the condition of \\autoref{sec_hp_noise}.\\\\\n     A tuple $((\\Omega, \\mathcal{F}, (\\mathcal{F}_t)_t,\\mathbb{P}),\\W^{\\H}_t,u_t)$ is a probabilistically weak solution to equation \\eqref{eq_Hgreat12} if \n    \\begin{enumerate}\n        \\item $(\\Omega, \\mathcal{F}, (\\mathcal{F}_t)_t,\\mathbb{P})$ is a stochastic basis with complete and right-continuous filtration,\n        \\item $\\W^{\\H}_t$ is a $p$-rough path on $H^{\\xi}$ $\\mathbb{P}$-a.s.,\n        \\item $\\mathbb{W}^{\\H,1}_t$ is a fractional Wiener process of Hurst parameter $\\H$ and space covariance $(-C)^{-1}Q(-C)^{-1}$ and it is adapted to $(\\mathcal{F}_t)_t$,\n        \\item $u\\in C_w([0,T],L^2)\\cap L^2([0,T],H^1)$ $\\mathbb{P}$-a.s., it is progressively measurable and $\\mathbb{P}$-a.s it holds, in a weak sense, \n        \\begin{equation*}\n            u\\st=\\int_s^t \\Delta u_r-b(u_r,u_r)\\,dr+\\sum_{k=1}^N \\mathbb{A}^{k}\\st u_s+u^{\\natural}\\st,\n        \\end{equation*}\n        where $u^{\\natural}\\st\\in C^{p/(N+1)\\var}_{2,loc}(H^{-3})$ $\\mathbb{P}$-a.s., and $(\\A^1,\\A^2)$ is defined by \n        \\begin{equation}\\label{def_limitURD}\n            \\A^{1}\\st\\phi=b(W^\\H\\st,\\phi),\\quad \\A^2\\st\\phi=b^{(2)}(\\mathbb{W}^{\\H,2}\\st,\\phi).\n        \\end{equation}\n    \\end{enumerate}\n    \\end{definition}"}, "pre_theorem_intro_text_len": 17244, "pre_theorem_intro_text": "\\label{sec:introduction}\nIn recent years, slow-fast systems associated with fluid equations have attracted considerable attention; see, for instance, \\cite{flandoli20212d, flandoli2022additive, debussche2024second,debussche2023rough}. The paradigmatic example to have in mind is provided by the stochastic Navier–Stokes equations on the three-dimensional torus, given by\n\\begin{align}\\label{intro_syst_h12}\n    \\begin{cases}\n    du^{\\epsilon}=\\nu \\Delta u^{\\epsilon}dt-(u^{\\epsilon}+v^{\\epsilon})\\cdot\\nabla u^{\\epsilon}dt+\\nabla p^{\\epsilon}\\,dt,\\\\\n    dv^{\\epsilon}=\\left(\\nu\\Delta v^{\\epsilon}+\\epsilon^{-1}Cv^{\\epsilon}-(u^{\\epsilon}+v^{\\epsilon})\\cdot\\nabla v^{\\epsilon}+\\nabla q^{\\epsilon}\\right)\\,dt+\\epsilon^{-1}dW_t,\\\\\n    \\nabla\\cdot u^{\\epsilon}=0,\\quad \\nabla\\cdot v^{\\epsilon}=0,\n\\end{cases}  \n\\end{align}\nwhere $W$ is an infinite-dimensional Brownian motion satisfying suitable spatial regularity assumptions, $\\epsilon\\ll 1$ is a small parameter governing the separation of scales between the \\emph{large} scales $u^{\\epsilon}$ and the \\emph{small} scales $v^{\\epsilon}$, and $C$ is a suitable damping operator. For simplicity, we restrict this heuristic discussion to the case $C=-I$. In the aforementioned works it was shown that, as $\\epsilon\\to 0$, the large-scale component $u^{\\epsilon}$ converges in distribution to a process $u$, which is a weak solution to the following stochastic Navier–Stokes equations:\n\\begin{align}\\label{intro_transport_h12}\n    \\begin{cases}\n    du=\\left(\\nu \\Delta u-u\\cdot\\nabla u-\\overline{r}\\cdot\\nabla u\\right) dt+d\\nabla p-\\circ dW\\cdot\\nabla u,\\\\\n    \\nabla\\cdot u=0,\n\\end{cases}  \n\\end{align}\nwhere $\\overline{r}$ is a deterministic vector field, referred to in the literature as the It\\^o-Stokes drift (cf. \\cite{memin2014fluid}, \\cite{debussche2023consistent}), and is related to the spatial covariance structure of the noise $W$. Moreover, $p$ is a semimartingale, often called the turbulent pressure, and the stochastic integral is understood in the Stratonovich sense. \n\nThe interest in results of this type is twofold. On the one hand, they are relevant for stochastic model reduction techniques in climate dynamics; see, for example, \\cite{majda2001mathematical, franzke2005low, franzke2006low}. On the other hand, they provide a rigorous justification for the emergence of transport-type noise in fluid equations such as \\eqref{intro_transport_h12}. Alternative approaches to those developed in \\cite{debussche2024second} can be found in \\cite{holm2015variational, memin2014fluid}. Over the last decade, starting from \\cite{flandoli2010well}, models like \\eqref{intro_transport_h12} have received increasing attention due to their regularizing effects (cf. \\cite{maurelli_et_al, flandoli2021high, coghi2023existence} and the references therein), as well as their ability to capture key features of turbulent flows, including enhanced and anomalous dissipation phenomena (see for istance \\cite{MR1905858, flandoli2024quantitative, luo2024elementary, rowan2024anomalous, drivas2025anomalous}) and dynamo effects (cf. \\cite{butori2024mean, butori2024background}).\n\nFrom a modeling perspective, however, it is important to account for non-Markovian noise with memory effects when describing turbulent fluids; see, for instance, \\cite{apolinario2022dynamical, apolinario2023linear, franzke2015stochastic, faranda2014modelling, lilly2017fractional}. This has led to a growing interest in the analysis of equations of the form \\eqref{intro_transport_h12}, where the Brownian motion $W$ is replaced by a fractional Brownian motion, as in \\cite{hofmanova2019navier, hofmanova2021rough, crisan2022solution, roveri2024well, galeati2026well, flandoli2023reduced, cifani2025diffusion}. Nevertheless, as pointed out in \\cite{flandoli20212d, flandoli2022additive, debussche2024second, debussche2023rough}, such limit equations should be regarded as idealized models arising from the slow-fast system \\eqref{intro_syst_h12}. A rigorous analysis of the original system when the driving noise $W$ is replaced by a fractional Brownian motion $W^{\\mathcal{H}}$ with Hurst parameter $\\mathcal{H}$ is still missing. Indeed, all the aforementioned works crucially rely on the Markovian nature of $W$ and on the availability of It\\^o-type formulas for the system \\eqref{intro_syst_h12}.\n\nFurthermore, although a substantial body of literature on slow–fast systems driven by fractional noise has recently emerged (see, for instance, \\cite{li2022slow, gehringer2022functional, pei2024almost, pei2023almost, li2026navierstokesfractionaltransportnoise} and the survey \\cite{gehringer2019rough}), to the best of our knowledge none of these works addresses the specific issues discussed above. The present article initiates the study of this gap, see \\autoref{main_thm} below for details.\n\nBefore presenting our main contributions, we briefly discuss the origin of the terms appearing in \\eqref{intro_syst_h12}, as their justification plays a central role in our framework. We start from the incompressible Navier–Stokes equations\n\\begin{align*}\n    \\begin{cases}\n        \\partial_t \\overline{u}&=\\nu\\Delta \\overline{u}-\\overline{u}\\cdot\\nabla \\overline{u}+\\nabla \\overline{p},\\\\\n        \\nabla\\cdot \\overline{u}&=0.\n    \\end{cases}\n\\end{align*}\nWe decompose the initial condition into large and small scales according to a suitable scale-separation rule,\n\\begin{align*}\n    \\overline{u}(0)=u^{\\epsilon}(0)+v^{\\epsilon}(0),\n\\end{align*}\nwhere $\\epsilon>0$ is a parameter characterizing the separation of scales. Assuming that this separation persists for positive times, we introduce two velocity fields $u^{\\epsilon}(t)$ and $v^{\\epsilon}(t)$, describing the evolution of the large- and small-scale components, respectively. Their dynamics are then governed by\n\\begin{align}\\label{system_intermediate_small}\n    \\begin{cases}\n        \\partial_t u^{\\epsilon}+(u^{\\epsilon}+v^{\\epsilon})\\cdot\\nabla u^{\\epsilon}+\\nabla p^{\\epsilon}&=0,\\\\\n        \\partial_t v^{\\epsilon}+(u^{\\epsilon}+v^{\\epsilon})\\cdot\\nabla v^{\\epsilon}+\\nabla q^{\\epsilon}&=0,\\\\\n        \\nabla\\cdot u=\\nabla\\cdot v=0.\n    \\end{cases}\n\\end{align}\nAt this stage, system \\eqref{system_intermediate_small} does not include either noise or damping at the level of the small scales. The introduction of an additive noise term in the small-scale equation is motivated by instabilities induced by boundary irregularities or internal obstacles, which are typically not explicitly modeled in mathematical formulations based on toroidal geometries or smooth bounded domains. We refer to \\cite[Chapter~5]{flandoli2023stochastic} for a detailed discussion. However, the presence of additive noise alone leads, via It\\^o’s formula, to an \\emph{infinite} mean energy injection into the system as $\\epsilon\\to 0$. To counterbalance this effect, an additional damping operator must be introduced in the small-scale equation. This modification yields precisely the structure of \\eqref{intro_syst_h12}, leaving only the scaling of the noise and damping terms to be justified.\n\nThe fact that the noise and damping operators scale in the same way can be motivated in at least two different manners in the literature for systems driven by standard Brownian motion. The first argument, inspired by Wong-Zakai-type approximations, is that this scaling leads the small-scale dynamics to approximate white noise. More precisely, consider the process\n\\begin{equation*}\ndw^{\\epsilon}_t = -\\epsilon^{-1} w^{\\epsilon}_t\\,dt + dW_t.    \n\\end{equation*}\nIt, formally, satisfies\n\\begin{align}\\label{heuristic_convergence_noise}\n    w^{\\epsilon}_t = \\epsilon^{-1} \\int_0^t e^{-\\epsilon^{-1}(t-s)}\\, dW_s \\rightarrow dW_t\n\\end{align}\nbecause, upon integrating the equation for $w^{\\epsilon}$ in time, we obtain\n\\begin{align*}\n    W_t\n    = \\epsilon\\, w^{\\epsilon}_t + \\int_0^t w^{\\epsilon}_s\\, ds\n    \\approx \\int_0^t w^{\\epsilon}_s\\, ds.\n\\end{align*}\nThis heuristic can be made rigorous using, for instance, rough path techniques; see \\cite{friz2014convergence}. Moreover, this argument is robust with respect to the choice of the Hurst parameter and therefore applies equally when $W$ is replaced by a fractional Brownian motion with Hurst parameter $\\mathcal{H}$.\n\nA second motivation is provided by multiscale analysis; see \\cite[Section~2]{flandoli20212d} for the case $\\mathcal{H}=\\tfrac12$, and \\autoref{sec:appendix_multiscale} for its extension to the general case considered here. In contrast to the Wong–Zakai argument, this approach crucially depends on the self-similarity properties of fractional Brownian motion, which vary with the Hurst parameter. As a consequence, from this perspective the scaling $\\epsilon^{-1}$ in \\eqref{intro_syst_h12} is not universal. For general values of $\\mathcal{H}$, the multiscale analysis instead suggests the scaling $\\epsilon^{-\\frac12-\\mathcal{H}}$. In particular, the Wong–Zakai and multiscale scalings coincide only in the classical Brownian case $\\mathcal{H}=\\tfrac12$.\n\nIn view of the discussion above, our primary objective is to rigorously analyze the asymptotic behavior, as $\\epsilon \\to 0$, of the system\n\\begin{align}\\label{e:equation1}\n\\begin{cases}\n     du^{\\epsilon}=\\nu \\Delta u^{\\epsilon}dt-(u^{\\epsilon}+v^{\\epsilon})\\cdot\\nabla u^{\\epsilon}dt+\\nabla p^{\\epsilon}\\,dt,\\\\\n    dv^{\\epsilon}=\\left(\\nu\\Delta v^{\\epsilon}+\\epsilon^{-1}Cv^{\\epsilon}-(u^{\\epsilon}+v^{\\epsilon})\\cdot\\nabla v^{\\epsilon}+\\nabla q^{\\epsilon}\\right)\\,dt+\\epsilon^{-\\alpha}dW^{\\mathcal{H}}_t,\\\\\n    \\nabla\\cdot u^{\\epsilon}=0,\\quad \\nabla\\cdot v^{\\epsilon}=0,\n\\end{cases}\n\\end{align}\nfocusing on the interaction between the \\emph{large}- and \\emph{small}-scale components under stochastic fractional perturbations. Here $\\nu>0$, $p^{\\epsilon}$ and $q^{\\epsilon}$ are pressure fields, $u^{\\epsilon},v^{\\epsilon}\\colon \\mathbb{T}^d\\times[0,T]\\to\\mathbb{R}^d$ denote the large- and small-scale velocity fields, respectively, and $W^{\\mathcal{H}}$ is a divergence-free, $\\mathcal{H}$-fractional Wiener process on $[L^2(\\mathbb{T}^d)]^d$. The noise is spatially colored, with covariance operator $Q$, and has Hurst parameter $\\mathcal{H}\\in\\left(\\tfrac13,1\\right)$. The operator $C$ is linear. Precise assumptions on $C$ and $Q$ are given in \\autoref{sec_hp_noise}; here we only mention that $C$ and $Q$ are required to commute. Solutions to \\eqref{e:equation1} are interpreted in the sense of unbounded rough drivers; see \\cite{bailleul2017unbounded} and the recent review \\cite{hocquet2025unbounded}. This notion is recalled in \\autoref{sec_rough_formulation}.\n\nAs discussed above, there are in principle two relevant choices for the parameter $\\alpha\\in\\{1,\\tfrac12+\\mathcal{H}\\}$, corresponding to different noise intensities. For technical reasons (see the discussion following \\autoref{main_thm}), for each fixed value of $\\mathcal{H}$ we are only able to treat one of these two regimes. Specifically, we work with the choice $\\alpha=1\\wedge(\\tfrac12+\\mathcal{H})$, which yields a Wong–Zakai regime when $\\mathcal{H}>\\tfrac12$ and a multiscale regime when $\\mathcal{H}<\\tfrac12$. As already noted, the two regimes coincide in the critical case $\\mathcal{H}=\\tfrac12$, and this fact has important consequences for the structure of the limiting equation. \n\nWhen $\\mathcal{H}>\\tfrac12$, we prove that $u^{\\epsilon}$ converges in law to a weak solution of the Young-type PDE with transport noise\n\\begin{align}\\label{eq_Hgreat12}\n    \\begin{cases}\n    du=\\left(\\nu \\Delta u-u\\cdot\\nabla u\\right) dt+d\\nabla p- d(-C)^{-1}W^{\\mathcal{H}}\\cdot\\nabla u,\\\\\n    \\nabla\\cdot u=0.\n    \\end{cases}\n\\end{align}\nIn this Wong–Zakai regime, $v^{\\epsilon}$ converges, as expected, to a fractional white noise. However, in contrast to \\cite{debussche2024second, debussche2023rough}, no It\\^o–Stokes drift appears in the limit. This absence is not due to isotropy assumptions on the covariance $Q$, as in \\cite{flandoli20212d, flandoli2022additive}, but rather to the specific scaling of the noise.\n\nOn the other hand, when $\\mathcal{H}<\\tfrac12$, the noise intensity is too weak to yield convergence to fractional white noise. Nevertheless, the quadratic nonlinearity $v^{\\epsilon}\\cdot\\nabla v^{\\epsilon}$ still produces a macroscopic effect in the limit dynamics. In this case, we show that $u^{\\epsilon}$ converges in law to a weak solution of the deterministic PDE with an additional transport term,\n\\begin{align}\\label{eq_Hsmall12}\n    \\begin{cases}\n    \\partial_t u=\\nu \\Delta u-\\overline{r}\\cdot\\nabla u-u\\cdot\\nabla u+\\nabla p,\\\\\n    \\nabla\\cdot u=0,\n    \\end{cases}\n\\end{align}\nwhere $\\overline{r}$ denotes the It\\^o–Stokes drift and depends explicitly on $C$, $Q$, and $\\mathcal{H}$; see \\autoref{lem_ito_stokes} below.\n\nIt appears that only in the case $\\mathcal{H}=\\tfrac12$, where the Wong–Zakai and multiscale regimes coincide, do both effects, transport noise and It\\^o–Stokes drift, simultaneously emerge in the limiting dynamics, as observed in \\cite{debussche2024second, debussche2023rough}.\n\nFinally, solutions to \\eqref{eq_Hgreat12} (respectively, \\eqref{eq_Hsmall12}) are understood in the sense of \\autoref{def_rough_lim_sol_Hgreat12} (respectively, \\autoref{def_rough_lim_sol_Hsmall12}).\n\\subsection{Methodologies and Main Results}\\label{subsec_results}\nSimilarly to \\cite{debussche2023rough}, we introduce the decomposition\n\\begin{align*}\n    v^{\\epsilon}=\\epsilon^{-\\alpha+\\mathcal{H}}w^{\\epsilon}+r^{\\epsilon}.\n\\end{align*}\nAfter projecting onto divergence-free vector fields via the Leray projector $\\Pi$, denoting by $A$ the Stokes operator $A:=\\Pi\\Delta$, and by $b(\\cdot,\\cdot)$ the Navier–Stokes nonlinearity as rigorously defined in \\autoref{sec_notation}, the components evolve according to the following system:\n\\begin{align}\\label{system_prelimit}\n    \\begin{cases}\n        &\\partial_t u^{\\epsilon}=\\nu A u^{\\epsilon}-b(u^{\\epsilon}+\\epsilon^{-\\alpha+\\mathcal{H}}w^{\\epsilon}+r^{\\epsilon},u^{\\epsilon}),\\\\\n        &dw^{\\epsilon}=\\epsilon^{-1}Cw^{\\epsilon} dt+ \\epsilon^{-\\mathcal{H}}dW^\\H_t, \\\\\n        &\\begin{aligned}\\partial_t r^{\\epsilon}&=\\epsilon^{-1}Cr^{\\epsilon}+\\nu A(\\epsilon^{-\\alpha+\\mathcal{H}}w^{\\epsilon}+r^{\\epsilon})\\\\&-b(u^{\\epsilon}+\\epsilon^{-\\alpha+\\mathcal{H}}w^{\\epsilon}+r^{\\epsilon},\\epsilon^{-\\alpha+\\mathcal{H}}w^{\\epsilon}+r^{\\epsilon}),\n        \\end{aligned}\\\\\n        &w^{\\epsilon}(0) =0,\\quad r^{\\epsilon}(0)=v_0^{\\epsilon}.\n    \\end{cases}\n\\end{align}\nSolutions to this system are understood in the sense of \\autoref{def_rough_sol}. We do not provide the details of the construction of solutions for fixed $\\epsilon\\in(0,1)$, as these follow from classical arguments. In particular, the rescaled Ornstein–Uhlenbeck process $w^{\\epsilon}$ exists on any probability space supporting the $Q$-fractional Wiener process $W^{\\mathcal{H}}$, is uniquely determined, and is measurable with respect to $W^{\\mathcal{H}}$. Moreover, for each fixed $\\epsilon\\in(0,1)$ there exists a probabilistically and analytically weak solution to \\eqref{system_prelimit}. That is, for every $\\epsilon\\in(0,1)$ there exist a probability space $(\\Omega_\\epsilon,\\mathcal F_\\epsilon,\\mathbb P_\\epsilon)$, a $Q$-fractional Wiener process $W^{\\epsilon,\\mathcal{H}}$, and processes $(u^{\\epsilon},r^{\\epsilon},w^{\\epsilon})$ solving \\eqref{system_prelimit} in the analytically weak sense. The proof relies on a Galerkin approximation combined with stochastic compactness arguments. In dimension $d=2$, where uniqueness holds as in the deterministic case, probabilistically weak existence can in fact be strengthened to probabilistically strong existence: as for the Ornstein–Uhlenbeck process alone, the triple $(u^{\\epsilon},w^{\\epsilon},r^{\\epsilon})$ exists on any probability space supporting the $Q$-fractional Wiener process $W^{\\mathcal{H}}$.\n\nAs already mentioned, it is necessary to reformulate the system within the framework of rough path theory. This is carried out in \\autoref{sec_rough_formulation} and \\autoref{sec_unbounder_rough}, and allows us to control the singular term $b(\\epsilon^{-\\alpha+\\mathcal{H}}w^{\\epsilon},u^{\\epsilon})$ by introducing the process\n\\begin{align*}\n    y^{\\epsilon}_t=\\epsilon^{-\\alpha+\\mathcal{H}}\\int_0^t w^{\\epsilon}_s ds \n\\end{align*}\ntogether with its canonical lifts. Here, in contrast to \\cite{debussche2023rough}, we cannot rely on It\\^o-type formulas to characterize the lifts, and instead must appeal to the theory of Gaussian rough paths (see \\cite{friz2010multidimensional, friz2014course}), especially in the case $\\mathcal{H}<\\tfrac12$. As already discussed, when $\\mathcal{H}>\\tfrac12$ no It\\^o–Stokes drift appears in the limit, whereas for $\\mathcal{H}<\\tfrac12$ this drift is the only surviving contribution. This dichotomy stems from the convergence properties of $r^{\\epsilon}$, which ultimately rely on an ergodic-type result for $\\int_0^T b(w^{\\epsilon}_s,w^{\\epsilon}_s),ds$. While the Markovian structure makes this argument relatively standard when $\\mathcal{H}=\\tfrac12$ (cf. \\cite[Section~5]{debussche2023rough}), in the fractional case we must instead rely on ad hoc computations exploiting the explicit structure of the Navier–Stokes nonlinearity; see \\autoref{sec:stoc_conv}. With these preliminaries in place, we can now state our main result.", "context": "When $\\mathcal{H}>\\tfrac12$, we prove that $u^{\\epsilon}$ converges in law to a weak solution of the Young-type PDE with transport noise\n\\begin{align}\\label{eq_Hgreat12}\n \\begin{cases}\n du=\\left(\\nu \\Delta u-u\\cdot\\nabla u\\right) dt+d\\nabla p- d(-C)^{-1}W^{\\mathcal{H}}\\cdot\\nabla u,\\\\\n \\nabla\\cdot u=0.\n \\end{cases}\n\\end{align}\nIn this Wong–Zakai regime, $v^{\\epsilon}$ converges, as expected, to a fractional white noise. However, in contrast to \\cite{debussche2024second, debussche2023rough}, no It\\^o–Stokes drift appears in the limit. This absence is not due to isotropy assumptions on the covariance $Q$, as in \\cite{flandoli20212d, flandoli2022additive}, but rather to the specific scaling of the noise.\n\nOn the other hand, when $\\mathcal{H}<\\tfrac12$, the noise intensity is too weak to yield convergence to fractional white noise. Nevertheless, the quadratic nonlinearity $v^{\\epsilon}\\cdot\\nabla v^{\\epsilon}$ still produces a macroscopic effect in the limit dynamics. In this case, we show that $u^{\\epsilon}$ converges in law to a weak solution of the deterministic PDE with an additional transport term,\n\\begin{align}\\label{eq_Hsmall12}\n \\begin{cases}\n \\partial_t u=\\nu \\Delta u-\\overline{r}\\cdot\\nabla u-u\\cdot\\nabla u+\\nabla p,\\\\\n \\nabla\\cdot u=0,\n \\end{cases}\n\\end{align}\nwhere $\\overline{r}$ denotes the It\\^o–Stokes drift and depends explicitly on $C$, $Q$, and $\\mathcal{H}$; see \\autoref{lem_ito_stokes} below.\n\nFinally, solutions to \\eqref{eq_Hgreat12} (respectively, \\eqref{eq_Hsmall12}) are understood in the sense of \\autoref{def_rough_lim_sol_Hgreat12} (respectively, \\autoref{def_rough_lim_sol_Hsmall12}).\n\\subsection{Methodologies and Main Results}\\label{subsec_results}\nSimilarly to \\cite{debussche2023rough}, we introduce the decomposition\n\\begin{align*}\n v^{\\epsilon}=\\epsilon^{-\\alpha+\\mathcal{H}}w^{\\epsilon}+r^{\\epsilon}.\n\\end{align*}\nAfter projecting onto divergence-free vector fields via the Leray projector $\\Pi$, denoting by $A$ the Stokes operator $A:=\\Pi\\Delta$, and by $b(\\cdot,\\cdot)$ the Navier–Stokes nonlinearity as rigorously defined in \\autoref{sec_notation}, the components evolve according to the following system:\n\\begin{align}\\label{system_prelimit}\n \\begin{cases}\n &\\partial_t u^{\\epsilon}=\\nu A u^{\\epsilon}-b(u^{\\epsilon}+\\epsilon^{-\\alpha+\\mathcal{H}}w^{\\epsilon}+r^{\\epsilon},u^{\\epsilon}),\\\\\n &dw^{\\epsilon}=\\epsilon^{-1}Cw^{\\epsilon} dt+ \\epsilon^{-\\mathcal{H}}dW^\\H_t, \\\\\n &\\begin{aligned}\\partial_t r^{\\epsilon}&=\\epsilon^{-1}Cr^{\\epsilon}+\\nu A(\\epsilon^{-\\alpha+\\mathcal{H}}w^{\\epsilon}+r^{\\epsilon})\\\\&-b(u^{\\epsilon}+\\epsilon^{-\\alpha+\\mathcal{H}}w^{\\epsilon}+r^{\\epsilon},\\epsilon^{-\\alpha+\\mathcal{H}}w^{\\epsilon}+r^{\\epsilon}),\n \\end{aligned}\\\\\n &w^{\\epsilon}(0) =0,\\quad r^{\\epsilon}(0)=v_0^{\\epsilon}.\n \\end{cases}\n\\end{align}\nSolutions to this system are understood in the sense of \\autoref{def_rough_sol}. We do not provide the details of the construction of solutions for fixed $\\epsilon\\in(0,1)$, as these follow from classical arguments. In particular, the rescaled Ornstein–Uhlenbeck process $w^{\\epsilon}$ exists on any probability space supporting the $Q$-fractional Wiener process $W^{\\mathcal{H}}$, is uniquely determined, and is measurable with respect to $W^{\\mathcal{H}}$. Moreover, for each fixed $\\epsilon\\in(0,1)$ there exists a probabilistically and analytically weak solution to \\eqref{system_prelimit}. That is, for every $\\epsilon\\in(0,1)$ there exist a probability space $(\\Omega_\\epsilon,\\mathcal F_\\epsilon,\\mathbb P_\\epsilon)$, a $Q$-fractional Wiener process $W^{\\epsilon,\\mathcal{H}}$, and processes $(u^{\\epsilon},r^{\\epsilon},w^{\\epsilon})$ solving \\eqref{system_prelimit} in the analytically weak sense. The proof relies on a Galerkin approximation combined with stochastic compactness arguments. In dimension $d=2$, where uniqueness holds as in the deterministic case, probabilistically weak existence can in fact be strengthened to probabilistically strong existence: as for the Ornstein–Uhlenbeck process alone, the triple $(u^{\\epsilon},w^{\\epsilon},r^{\\epsilon})$ exists on any probability space supporting the $Q$-fractional Wiener process $W^{\\mathcal{H}}$.\n\nAs already mentioned, it is necessary to reformulate the system within the framework of rough path theory. This is carried out in \\autoref{sec_rough_formulation} and \\autoref{sec_unbounder_rough}, and allows us to control the singular term $b(\\epsilon^{-\\alpha+\\mathcal{H}}w^{\\epsilon},u^{\\epsilon})$ by introducing the process\n\\begin{align*}\n y^{\\epsilon}_t=\\epsilon^{-\\alpha+\\mathcal{H}}\\int_0^t w^{\\epsilon}_s ds \n\\end{align*}\ntogether with its canonical lifts. Here, in contrast to \\cite{debussche2023rough}, we cannot rely on It\\^o-type formulas to characterize the lifts, and instead must appeal to the theory of Gaussian rough paths (see \\cite{friz2010multidimensional, friz2014course}), especially in the case $\\mathcal{H}<\\tfrac12$. As already discussed, when $\\mathcal{H}>\\tfrac12$ no It\\^o–Stokes drift appears in the limit, whereas for $\\mathcal{H}<\\tfrac12$ this drift is the only surviving contribution. This dichotomy stems from the convergence properties of $r^{\\epsilon}$, which ultimately rely on an ergodic-type result for $\\int_0^T b(w^{\\epsilon}_s,w^{\\epsilon}_s),ds$. While the Markovian structure makes this argument relatively standard when $\\mathcal{H}=\\tfrac12$ (cf. \\cite[Section~5]{debussche2023rough}), in the fractional case we must instead rely on ad hoc computations exploiting the explicit structure of the Navier–Stokes nonlinearity; see \\autoref{sec:stoc_conv}. With these preliminaries in place, we can now state our main result.", "full_context": "When $\\mathcal{H}>\\tfrac12$, we prove that $u^{\\epsilon}$ converges in law to a weak solution of the Young-type PDE with transport noise\n\\begin{align}\\label{eq_Hgreat12}\n \\begin{cases}\n du=\\left(\\nu \\Delta u-u\\cdot\\nabla u\\right) dt+d\\nabla p- d(-C)^{-1}W^{\\mathcal{H}}\\cdot\\nabla u,\\\\\n \\nabla\\cdot u=0.\n \\end{cases}\n\\end{align}\nIn this Wong–Zakai regime, $v^{\\epsilon}$ converges, as expected, to a fractional white noise. However, in contrast to \\cite{debussche2024second, debussche2023rough}, no It\\^o–Stokes drift appears in the limit. This absence is not due to isotropy assumptions on the covariance $Q$, as in \\cite{flandoli20212d, flandoli2022additive}, but rather to the specific scaling of the noise.\n\nOn the other hand, when $\\mathcal{H}<\\tfrac12$, the noise intensity is too weak to yield convergence to fractional white noise. Nevertheless, the quadratic nonlinearity $v^{\\epsilon}\\cdot\\nabla v^{\\epsilon}$ still produces a macroscopic effect in the limit dynamics. In this case, we show that $u^{\\epsilon}$ converges in law to a weak solution of the deterministic PDE with an additional transport term,\n\\begin{align}\\label{eq_Hsmall12}\n \\begin{cases}\n \\partial_t u=\\nu \\Delta u-\\overline{r}\\cdot\\nabla u-u\\cdot\\nabla u+\\nabla p,\\\\\n \\nabla\\cdot u=0,\n \\end{cases}\n\\end{align}\nwhere $\\overline{r}$ denotes the It\\^o–Stokes drift and depends explicitly on $C$, $Q$, and $\\mathcal{H}$; see \\autoref{lem_ito_stokes} below.\n\nFinally, solutions to \\eqref{eq_Hgreat12} (respectively, \\eqref{eq_Hsmall12}) are understood in the sense of \\autoref{def_rough_lim_sol_Hgreat12} (respectively, \\autoref{def_rough_lim_sol_Hsmall12}).\n\\subsection{Methodologies and Main Results}\\label{subsec_results}\nSimilarly to \\cite{debussche2023rough}, we introduce the decomposition\n\\begin{align*}\n v^{\\epsilon}=\\epsilon^{-\\alpha+\\mathcal{H}}w^{\\epsilon}+r^{\\epsilon}.\n\\end{align*}\nAfter projecting onto divergence-free vector fields via the Leray projector $\\Pi$, denoting by $A$ the Stokes operator $A:=\\Pi\\Delta$, and by $b(\\cdot,\\cdot)$ the Navier–Stokes nonlinearity as rigorously defined in \\autoref{sec_notation}, the components evolve according to the following system:\n\\begin{align}\\label{system_prelimit}\n \\begin{cases}\n &\\partial_t u^{\\epsilon}=\\nu A u^{\\epsilon}-b(u^{\\epsilon}+\\epsilon^{-\\alpha+\\mathcal{H}}w^{\\epsilon}+r^{\\epsilon},u^{\\epsilon}),\\\\\n &dw^{\\epsilon}=\\epsilon^{-1}Cw^{\\epsilon} dt+ \\epsilon^{-\\mathcal{H}}dW^\\H_t, \\\\\n &\\begin{aligned}\\partial_t r^{\\epsilon}&=\\epsilon^{-1}Cr^{\\epsilon}+\\nu A(\\epsilon^{-\\alpha+\\mathcal{H}}w^{\\epsilon}+r^{\\epsilon})\\\\&-b(u^{\\epsilon}+\\epsilon^{-\\alpha+\\mathcal{H}}w^{\\epsilon}+r^{\\epsilon},\\epsilon^{-\\alpha+\\mathcal{H}}w^{\\epsilon}+r^{\\epsilon}),\n \\end{aligned}\\\\\n &w^{\\epsilon}(0) =0,\\quad r^{\\epsilon}(0)=v_0^{\\epsilon}.\n \\end{cases}\n\\end{align}\nSolutions to this system are understood in the sense of \\autoref{def_rough_sol}. We do not provide the details of the construction of solutions for fixed $\\epsilon\\in(0,1)$, as these follow from classical arguments. In particular, the rescaled Ornstein–Uhlenbeck process $w^{\\epsilon}$ exists on any probability space supporting the $Q$-fractional Wiener process $W^{\\mathcal{H}}$, is uniquely determined, and is measurable with respect to $W^{\\mathcal{H}}$. Moreover, for each fixed $\\epsilon\\in(0,1)$ there exists a probabilistically and analytically weak solution to \\eqref{system_prelimit}. That is, for every $\\epsilon\\in(0,1)$ there exist a probability space $(\\Omega_\\epsilon,\\mathcal F_\\epsilon,\\mathbb P_\\epsilon)$, a $Q$-fractional Wiener process $W^{\\epsilon,\\mathcal{H}}$, and processes $(u^{\\epsilon},r^{\\epsilon},w^{\\epsilon})$ solving \\eqref{system_prelimit} in the analytically weak sense. The proof relies on a Galerkin approximation combined with stochastic compactness arguments. In dimension $d=2$, where uniqueness holds as in the deterministic case, probabilistically weak existence can in fact be strengthened to probabilistically strong existence: as for the Ornstein–Uhlenbeck process alone, the triple $(u^{\\epsilon},w^{\\epsilon},r^{\\epsilon})$ exists on any probability space supporting the $Q$-fractional Wiener process $W^{\\mathcal{H}}$.\n\nAs already mentioned, it is necessary to reformulate the system within the framework of rough path theory. This is carried out in \\autoref{sec_rough_formulation} and \\autoref{sec_unbounder_rough}, and allows us to control the singular term $b(\\epsilon^{-\\alpha+\\mathcal{H}}w^{\\epsilon},u^{\\epsilon})$ by introducing the process\n\\begin{align*}\n y^{\\epsilon}_t=\\epsilon^{-\\alpha+\\mathcal{H}}\\int_0^t w^{\\epsilon}_s ds \n\\end{align*}\ntogether with its canonical lifts. Here, in contrast to \\cite{debussche2023rough}, we cannot rely on It\\^o-type formulas to characterize the lifts, and instead must appeal to the theory of Gaussian rough paths (see \\cite{friz2010multidimensional, friz2014course}), especially in the case $\\mathcal{H}<\\tfrac12$. As already discussed, when $\\mathcal{H}>\\tfrac12$ no It\\^o–Stokes drift appears in the limit, whereas for $\\mathcal{H}<\\tfrac12$ this drift is the only surviving contribution. This dichotomy stems from the convergence properties of $r^{\\epsilon}$, which ultimately rely on an ergodic-type result for $\\int_0^T b(w^{\\epsilon}_s,w^{\\epsilon}_s),ds$. While the Markovian structure makes this argument relatively standard when $\\mathcal{H}=\\tfrac12$ (cf. \\cite[Section~5]{debussche2023rough}), in the fractional case we must instead rely on ad hoc computations exploiting the explicit structure of the Navier–Stokes nonlinearity; see \\autoref{sec:stoc_conv}. With these preliminaries in place, we can now state our main result.\n\nSince we have been able to pass to the limit in all the terms except the remainder, also $\\overline{u}^{\\epsilon,\\sharp}$ converge $\\overline{\\mathbb{P}}-a.s$ to some object $\\overline{u}^{\\sharp}\\st$ such that, for $\\alpha=\\frac{1}{2}+\\H$, \n\\begin{align}\\label{eqUtile}\n \\overline{u}_{\\st}=\\int_s^t \\Delta \\overline{u}_\\tau+b(\\overline{u}_{\\tau}+\\overline{r}_\\tau,\\overline{u}_\\tau)\\,dr+\\overline{u}^{\\sharp}_{st} \\quad \\mathbb{\\overline{P}}-a.s.,\n\\end{align}\nwhile for $\\alpha=1$,\n\\begin{align}\n \\overline{u}_{\\st}=\\int_s^t \\Delta \\overline{u}_\\tau+b(\\overline{u}_{\\tau},\\overline{u}_\\tau)\\,dr+\\sum_{k=1}^2 \\mathbb{A}^{k}\\st u_s+\\overline{u}^{\\sharp}_{st} \\quad \\mathbb{\\overline{P}}-a.s..\n\\end{align} \nLet us conclude by proving that $\\overline{u}^{\\sharp}_{st}\\in C^{p/3\\var}_{2,loc}(H^{-3}).$ Since the rough paths converge and, in particular, \\eqref{est:UniformRegularityURDH<1/2} and \\eqref{est:UniformRegularityURDH<1/2zeros} hold, then we can identify a uniform in $\\epsilon$ random covering $\\{J_k\\}_k$ of $[0,T]$ by introducing the localization \\[\\sup_{\\epsilon\\in (0,1)}\\omega_\\epsilon^{1/p}(s,t)\\le\\frac{1}{2\\lambda}.\\]\nTherefore thanks to \n\\eqref{stimaRemainder}, we have that on any $J_k$, for all $s,t \\in J_k$ and $\\phi \\in H^3$, the following inequalities hold\n \\begin{align} &|\\overline{u}^{\\sharp}\\st(\\phi)|=\\lim_{\\epsilon\\to 0}|\\overline{u}^{\\epsilon,\\sharp}\\st(\\phi)|\\le\\|\\phi\\|_{H^3}\\liminf_{\\epsilon\\to 0}{\\|\\overline{u}^{\\epsilon,\\sharp}\\|_{p/3\\var,[s,t]}},\\\\\n &\\|\\overline{u}^{\\epsilon,\\sharp}\\|_{p/3\\var,[s,t]}\\lesssim\\sx(\\omega_{\\epsilon}(s,t)+\\omega_{\\mu^\\epsilon}(s,t)^{p/2}\\dx)^{3/p}\\lesssim M(\\omega),\n \\end{align}\nwhere $M(\\omega)$ is a random constant that depends on \\eqref{pathwise_estimate}.\nThen for any $\\mathbf{P}$ partition of $J_k$, we have that\n\\begin{equation}\n \\sum_{(s,t)\\in P}\\|\\overline{u}^{\\sharp}\\st\\|_{H^{-3}}^{p/3}\\lesssim \\liminf_{\\epsilon\\to0}\\sum_{(s,t)\\in P}{\\|\\overline{u}^{\\epsilon,\\sharp}\\st\\|_{H^{-3}}^{p/3}}\\lesssim \\liminf_{\\epsilon\\to0} \\|\\overline{u}^{\\epsilon,\\sharp}\\|_{p/3\\var,J_k}^{p/3}< \\infty.\n\\end{equation}\nTherefore, we have shown that $\\overline{u}^{\\sharp}_{st}\\in C^{p/3\\var}_{2,loc}(H^{-3}).$ At last, we stress that in case of $\\alpha=\\frac12+\\H$ we have that $\\overline{u}^{\\sharp}\\equiv0$ since it is highly regular in time and it is given by the increments of a $1$-index function of time, see \\eqref{eqUtile}.\n\\subsection*{Case $\\H>\\frac{1}{2}$} \nThe convergence in this final case is obtained by closely mirroring the proof from the preceding case.\nThe apriori estimates provided by \\autoref{lemma_a_priori} combined with \\autoref{proproughHgreat12} and the compactness criteria \\cite[Lemma A.2]{hofmanova2019navier} imply that, in the case of \\autoref{main_thm}, the family \n\\begin{align*}\n (u^{\\epsilon},r^{\\epsilon}, W^\\H,\\mathbb{Y}^{\\epsilon,1})\n\\end{align*}\nis tight in the space\n\\begin{align*}\n \\mathcal{X}&=\\mathcal{X}_{u,r,W}\\times\\mathcal{X}_{RP},\\\\\n \\mathcal{X}_{u,r,W}&=(C_w([0,T];H)\\cap L^2(0,T;H))\\times L^2((0,T)\\times \\T^3)_w\\times C([0,T];H),\\\\\n \\mathcal{X}_{RP}&= C^{p\\var}([0,T];H^{\\xi}).\n\\end{align*}\n Therefore, by Jakubowski–Skorokhod’s representation theorem, see for example \\cite[Section 2.7]{breit2018stochastically}, there exists a subsequence of $\\epsilon_k\\rightarrow 0$ which for the matter of notation we continue to denote by $\\epsilon$ and an auxiliary probability space $\\left(\\overline{\\Omega},\\overline{\\mathcal{F}},\\overline{\\mathbb{P}}\\right)$ with $\\mathcal{X}$ valued random variables $\\sx(\\overline{u}^{\\epsilon},\\overline{r}^{\\epsilon}, \\overline{W}^{\\H,\\epsilon},\\overline{\\mathbb{Y}}^{\\epsilon,1}\\dx)$ and $\\sx(\\overline{u},\\overline{r}, \\overline{W}^{\\H},\\overline{\\mathbb{Y}}^{1}\\dx)$ such that \n\\begin{align*}\n \\sx(\\overline{u}^{\\epsilon},\\overline{r}^{\\epsilon}, \\overline{W}^{\\H,\\epsilon},\\overline{\\mathbb{Y}}^{\\epsilon,1}\\dx)\\stackrel{\\mathcal{X}}{\\rightarrow}\\sx(\\overline{u},\\overline{r}, \\overline{W}^{\\H},\\overline{\\mathbb{Y}}^{1}\\dx)\\quad \\overline{\\mathbb{P}}-a.s..\n\\end{align*}\nThe convergence of all the terms as well as showing that the remainder $\\overline{u}^{\\sharp}\\in C^{p/2\\var}_{2,loc}(H^{-3})$ can be justified similarly to the previous case. Therefore, we need to identify $\\overline{r}$ to conclude the proof of \\autoref{main_thm}.\n\\begin{lemma}\\label{identification_lemma_smoothcase}\n It holds\n \\begin{align*}\n \\overline{r}=0.\n \\end{align*}\n\\end{lemma}\n\\begin{proof}\n Let us call by $\\overline{w}^{\\epsilon}$ the solution of \n \\begin{align*}\nd\\overline{w}^{\\epsilon}=\\epsilon^{-1}C\\overline{w}^{\\epsilon}\\,dt+\\epsilon^{- \\H}d\\overline{W}^{\\H,\\epsilon}_t.\n \\end{align*}\n For each $\\psi\\in C^{1}_c(0,T)$ it holds due to \\autoref{def_rough_sol}\n \\begin{align*}\n \\int_0^T \\psi_t \\overline{r}^{\\epsilon}_t dt&=\\epsilon^{2\\H-1}\\int_0^T \\psi_t(-C)^{-1}b(\\overline{w}^{\\epsilon}_t,\\overline{w}^{\\epsilon}_t)dt\\\\ & +\\epsilon^{\\H}\\int_0^T\\psi_t (-C)^{-1}\\left(A\\overline{w}^{\\epsilon}_t+b(\\overline{u}^{\\epsilon}_t,\\overline{w}^{\\epsilon}_t)+b(\\overline{w}^{\\epsilon}_t,\\overline{r}^{\\epsilon}_t)+b(\\overline{r}^{\\epsilon}_t,\\overline{w}^{\\epsilon}_t)\\right)dt\\\\ & +\\epsilon\\int_0^T\\psi(-C)^{-1}\\left(A\\overline{r}^{\\epsilon}_t+b(\\overline{u}^{\\epsilon}_t,\\overline{r}^{\\epsilon}_t)+b(\\overline{r}^{\\epsilon}_t,\\overline{r}^{\\epsilon}_t)\\right)dt+\\epsilon\\int_0^T \\partial_t \\psi_t (-C)^{-1}\\overline{r}^{\\epsilon}_t dt\\\\ & =J_1^{\\epsilon}+J_2^{\\epsilon}+J_3^{\\epsilon}+J_4^{\\epsilon}.\n \\end{align*}\n We will show that, up to subsequences, all the terms approach $0$ in $H^{-\\xi}$ $\\overline{\\mathbb{P}}-a.s.$ . This is enough to conclude by uniqueness of the limit. The treatment of $J_2^\\epsilon,\\ J_3^\\epsilon,\\ J_4^\\epsilon$ follows verbatim as in the first case of \\autoref{identification_lemma} and we omit details. To treat $J_1^\\epsilon$ we argue in a simpler way compared to the first case of \\autoref{identification_lemma}. Indeed since $\\H>\\frac{1}{2}$, it holds\n \\begin{align*}\n \\mathbb{E}\\left[\\|I_1^{\\epsilon}\\|_{H^{-\\xi}}\\right] \\lesssim \\epsilon^{2\\H-1}\\int_0^T\\mathbb{E}\\left[\\|\\overline{w}^{\\epsilon}_t\\|^2 \\right]dt\\rightarrow 0\n \\end{align*}\n by \\autoref{hp_stoch_conv}. This completes the proof.\n\\end{proof}\n\\begin{remark}\n For the sake of simplicity, we have reported estimates showing that $\\overline{u}^{\\sharp}\\in C^{p/2\\var}_{2,loc}(H^{-3})$. Nevertheless, we stress that by refining \\eqref{stimaExtraURD} we could have shown that \n $\\overline{u}^{\\sharp}\\in C^{p/2\\var}_{2,loc}(H^{-2})$ for $H\\in (1/2,2/3]$ and $\\overline{u}^{\\sharp}\\in C^{2p/(2p+1)\\var}_{2,loc}(H^{-2})$ for $\\H>2/3$. Therefore for all $\\H\\in (1/2,1)$, the integral term arising from the transport noise in the limiting equation can be constructed as a Young integral in $H^{-2}$ instead of $H^{-3}.$ \n\\end{remark}\n\\begin{acknowledgements}\nEL has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 949981). FT is supported by the Istituto Nazionale di Alta Matematica (IN$\\delta$AM) through the GNAMPA 2025 project “Modelli stocastici in Fluidodinamica e Turbolenza” (CUP E5324001950001), and by the project\n\"Noise in fluid dynamics and related models\" funded by the MUR Progetti di Ricerca\ndi Rilevante Interesse Nazionale (PRIN) Bando 2022 - grant 20222YRYSP. FT also acknowledges the warm hospitality of Universität Bielefeld, where this work started. Both authors thank Martina Hofmanová for valuable discussions at an early stage of the project.", "post_theorem_intro_text_len": 5716, "post_theorem_intro_text": "\\begin{remark}\\label{remark_d2}\n    In dimension $d=2$, thanks to the uniqueness of both the prelimit and the limit equations, the above statement can be strengthened. Under the same assumptions as in \\autoref{main_thm}, there exist probabilistically strong rough path solutions $(u^{\\epsilon},(-C)^{-1}dW^{\\mathcal{H}})$ to \\eqref{system_prelimit} that converge in probability to the unique weak rough path solution of \\eqref{eq_Hgreat12} in the case $\\alpha=1$ and $\\mathcal{H}>\\tfrac{1}{2}$, and of \\eqref{eq_Hsmall12} in the case $\\mathcal{H}<\\tfrac{1}{2}$ and $\\alpha=\\tfrac{1}{2}+\\mathcal{H}$.\n\\end{remark}\n\\begin{remark}\\label{remark_uniqueness}\n    Note that even in the case $\\mathcal{H}<\\tfrac{1}{2}$ and $\\alpha=\\tfrac{1}{2}+\\mathcal{H}$, the limit object is, in principle, genuinely stochastic and supported on (possibly non-unique) weak solutions of \\eqref{eq_Hsmall12}.\n\\end{remark}\nWe now discuss why the complementary regimes appear to be out of reach. In the case $\\mathcal{H}>\\tfrac{1}{2}$ and $\\alpha=\\tfrac{1}{2}+\\mathcal{H}>1$, neglecting the additional Navier–Stokes terms yields that the small scales satisfy\n\\begin{align*}\n    v^{\\epsilon}_t=\\epsilon^{-(\\mathcal{H}-1/2)}\\epsilon^{-1}\\int_0^t e^{-\\epsilon^{-1}C(t-s)} dW^{\\mathcal{H}}_s.\n\\end{align*}\nIn view of \\eqref{heuristic_convergence_noise}, this leads to\n\\begin{align*}\n    v^{\\epsilon}_t\\approx \\epsilon^{-(\\mathcal{H}-1/2)}(-C)^{-1}dW^{\\mathcal{H}}_t\\rightarrow +\\infty,\n\\end{align*}\nshowing that the small-scale component diverges.\nThe case $\\mathcal{H}<\\tfrac{1}{2}$ and $\\alpha=1$ is more delicate. Here, the main difficulty is the lack of control of the quadratic nonlinearity in the equation for the small scales $v^{\\epsilon}$. Nevertheless, for particularly simple choices of the noise, namely \n\\begin{align*}\n    W^{\\mathcal{H}}_t(x)=f(x)W^{\\mathcal{H},1}_t,\n\\end{align*}\nwhere $f$ is divergence-free, of zero mean, and satisfies $f\\cdot\\nabla f=0$\\footnote{For instance, this holds when $f(x)=a_l \\sin (l\\cdot x),\\ l,\\ a_l\\in \\mathbb{R}^3,\\ a_l\\perp l$. } and $W^{\\mathcal{H},1}$ is a real-valued fractional Brownian motion, we can still establish the following result.\n\\begin{proposition}\\label{trivial_noise}\nLet $\\alpha=1$, $\\mathcal{H}\\in\\left(\\tfrac{1}{3},\\tfrac{1}{2}\\right)$, and let the initial data $u_0^{\\epsilon}$ and $v_0^{\\epsilon}$ be divergence-free, of zero mean, and satisfy\n\\begin{align*}\n   \\sup_{\\epsilon\\in (0,1)} \\|u_0^{\\epsilon}\\|_{L^2_x}+\\|\\sqrt{\\epsilon}v_0^{\\epsilon}\\|_{L^2_x}<+\\infty.\n\\end{align*} Assume that $W^{\\mathcal{H}}_t(x)=f(x)W^{\\mathcal{H},1}_t$, with $f$ as above, and that $Q$ and $C$ satisfy the conditions stated in \\autoref{sec_hp_noise}. Then there exist probabilistically weak rough path solutions $(u^{\\epsilon},(-C)^{-1}dW^{\\mathcal{H},\\epsilon})$ to \\eqref{system_prelimit} that converge in law to a probabilistically weak rough path solution of \\eqref{eq_Hgreat12}. An analogue of \\autoref{remark_d2} holds in dimension $d=2$.\n\\end{proposition}\nWe conclude this subsection by mentioning the recent preprint \\cite{li2026navierstokesfractionaltransportnoise}, which appeared during the final stages of preparation of this work. There, the authors study a fast–slow Navier–Stokes system similar to \\eqref{e:equation1}, with $\\mathcal{H}\\in\\left(\\tfrac{1}{3},\\tfrac{1}{2}\\right)$ and $\\alpha=1$, focusing on the Wong–Zakai regime. While the slow dynamics coincide with ours, the fast dynamics differ substantially, being governed in \\cite{li2026navierstokesfractionaltransportnoise} by the linear equation\n\\begin{align}\\label{eq_fast_1}\ndv^{\\epsilon}=\\epsilon^{-1}Cv^{\\epsilon}dt+\\epsilon^{-1}dW^{\\mathcal{H}}_t.\n\\end{align}\nAs already noted after \\autoref{main_thm}, the impossibility to control the quadratic term in the fast dynamics prevents us from obtaining a Wong–Zakai convergence result analogous to \\cite[Theorem~B]{li2026navierstokesfractionaltransportnoise} within our framework. If this difficulty is removed from the outset, either by adopting the simplified fast dynamics \\eqref{eq_fast_1}, as in \\cite{li2026navierstokesfractionaltransportnoise}, or by considering instead\n\\begin{align}\\label{eq_fast_2}\ndv^{\\epsilon}=\\nu Av^{\\epsilon}dt+\\epsilon^{-1}Cv^{\\epsilon}dt-b(u^{\\epsilon},v^{\\epsilon})dt+\\epsilon^{-1}dW^{\\mathcal{H}}_t,\n\\end{align}\nthen our arguments readily extend to the regime $\\mathcal{H}\\in\\left(\\tfrac{1}{3},\\tfrac{1}{2}\\right)$ with $\\alpha=1$, thereby recovering the results of \\cite{li2026navierstokesfractionaltransportnoise}. We have chosen not to pursue this direction further; instead, we address the difficulty caused by the uncontrolled quadratic term in the fast dynamics by imposing a strong structural assumption on the covariance of the noise $W^{\\mathcal{H}}$, as discussed in \\autoref{trivial_noise}.\n\\subsubsection*{Plan of the paper}In \\autoref{sec_prel} we rigorously introduce our assumptions on $Q$ and $C$, and we recall some rough path notation including the notion of solutions for \\eqref{eq_Hgreat12}, \\eqref{eq_Hsmall12}, \\eqref{system_prelimit}. The analysis of the long-time behaviour of $w^{\\epsilon}$ and of some of its nonlinear functionals is addressed in \\autoref{sec:stoc_conv}. The core of our rough path analysis is \\autoref{sec_unbounder_rough}, where we study the convergence properties of the canonical lift of $\\int_0^{\\cdot}e^{-\\alpha+\\mathcal{H}}w^{\\epsilon}_s ds$. The last ingredients for the proof of \\autoref{main_thm} are some a priori estimates obtained in \\autoref{sec_apriori} exploiting the rough path formulation already mentioned. Combining these elements, the proofs of both \\autoref{main_thm} and \\autoref{trivial_noise} follow by standard arguments as discussed in \\autoref{sec:end_proof}.", "sketch": "A proof outline is given in the “Plan of the paper” paragraph. The ingredients for proving Theorem~\\ref{main_thm} are:\n\\begin{itemize}\n\\item In \\autoref{sec_prel}, “we rigorously introduce our assumptions on $Q$ and $C$, and we recall some rough path notation including the notion of solutions for \\eqref{eq_Hgreat12}, \\eqref{eq_Hsmall12}, \\eqref{system_prelimit}.”\n\\item In \\autoref{sec:stoc_conv}, they analyze “the long-time behaviour of $w^{\\epsilon}$ and of some of its nonlinear functionals.”\n\\item The “core of our rough path analysis” is in \\autoref{sec_unbounder_rough}, where they “study the convergence properties of the canonical lift of $\\int_0^{\\cdot} e^{-\\alpha+\\mathcal{H}} w^{\\epsilon}_s\\, ds$.”\n\\item “The last ingredients for the proof of \\autoref{main_thm} are some a priori estimates obtained in \\autoref{sec_apriori} exploiting the rough path formulation already mentioned.”\n\\item “Combining these elements, the proofs of both \\autoref{main_thm} and \\autoref{trivial_noise} follow by standard arguments as discussed in \\autoref{sec:end_proof}.”\n\\end{itemize}\nAdditionally, the text explains why complementary regimes are difficult: for $\\mathcal{H}>\\tfrac{1}{2}$ and $\\alpha=\\tfrac{1}{2}+\\mathcal{H}>1$, a heuristic shows “the small-scale component diverges,” while for $\\mathcal{H}<\\tfrac{1}{2}$ and $\\alpha=1$ “the main difficulty is the lack of control of the quadratic nonlinearity in the equation for the small scales $v^{\\epsilon}$.”", "expanded_sketch": "A proof outline is given in the “Plan of the paper” paragraph. The ingredients for proving the main theorem are:\n\\begin{itemize}\n\\item Next, “we rigorously introduce our assumptions on $Q$ and $C$, and we recall some rough path notation including the notion of solutions” for\n\\begin{align}\\label{eq_Hgreat12}\n    \\begin{cases}\n    du=\\left(\\nu \\Delta u-u\\cdot\\nabla u\\right) dt+d\\nabla p- d(-C)^{-1}W^{\\H}\\cdot\\nabla u,\\\\\n    \\nabla\\cdot u=0.\n    \\end{cases}\n\\end{align}\n\\begin{align}\\label{eq_Hsmall12}\n    \\begin{cases}\n    \\partial_t u=\\nu \\Delta u-\\overline{r}\\cdot\\nabla u-u\\cdot\\nabla u+\\nabla p,\\\\\n    \\nabla\\cdot u=0,\n    \\end{cases}\n\\end{align}\nand\n\\begin{align}\\label{system_prelimit}\n    \\begin{cases}\n        &\\partial_t u^{\\epsilon}=\\nu A u^{\\epsilon}-b(u^{\\epsilon}+\\epsilon^{-\\alpha+\\H}w^{\\epsilon}+r^{\\epsilon},u^{\\epsilon}),\\\\\n        &dw^{\\epsilon}=\\epsilon^{-1}Cw^{\\epsilon} dt+ \\epsilon^{-\\H}dW^\\H_t, \\\\\n        &\\begin{aligned}\\partial_t r^{\\epsilon}&=\\epsilon^{-1}Cr^{\\epsilon}+\\nu A(\\epsilon^{-\\alpha+\\H}w^{\\epsilon}+r^{\\epsilon})\\\\&-b(u^{\\epsilon}+\\epsilon^{-\\alpha+\\H}w^{\\epsilon}+r^{\\epsilon},\\epsilon^{-\\alpha+\\H}w^{\\epsilon}+r^{\\epsilon}),\n        \\end{aligned}\\\\\n        &w^{\\epsilon}(0) =0,\\quad r^{\\epsilon}(0)=v_0^{\\epsilon}.\n    \\end{cases}\n\\end{align}\n\\item Later, they analyze “the long-time behaviour of $w^{\\epsilon}$ and of some of its nonlinear functionals.”\n\\item The “core of our rough path analysis” comes next, where they “study the convergence properties of the canonical lift of $\\int_0^{\\cdot} e^{-\\alpha+\\mathcal{H}} w^{\\epsilon}_s\\, ds$.”\n\\item “The last ingredients for the proof of the main theorem are some a priori estimates obtained later exploiting the rough path formulation already mentioned.”\n\\item “Combining these elements, the proofs of both the main theorem and the following proposition follow by standard arguments as discussed later.\n\nWe first record the proposition:\n\\begin{proposition}\\label{trivial_noise}\nLet $\\alpha=1$, $\\H\\in\\left(\\tfrac{1}{3},\\tfrac{1}{2}\\right)$, and let the initial data $u_0^{\\epsilon}$ and $v_0^{\\epsilon}$ be divergence-free, of zero mean, and satisfy\n\\begin{align*}\n   \\sup_{\\epsilon\\in (0,1)} \\|u_0^{\\epsilon}\\|_{L^2_x}+\\|\\sqrt{\\epsilon}v_0^{\\epsilon}\\|_{L^2_x}<+\\infty.\n\\end{align*} Assume that $W^{\\H}_t(x)=f(x)W^{\\H,1}_t$, with $f$ as above, and that $Q$ and $C$ satisfy the conditions stated in \\autoref{sec_hp_noise}. Then there exist probabilistically weak rough path solutions $(u^{\\epsilon},(-C)^{-1}dW^{\\H,\\epsilon})$ to \\eqref{system_prelimit} that converge in law to a probabilistically weak rough path solution of \\eqref{eq_Hgreat12}. An analogue of \\autoref{remark_d2} holds in dimension $d=2$.\n\\end{proposition}\n”\n\\end{itemize}\nAdditionally, the text explains why complementary regimes are difficult: for $\\mathcal{H}>\\tfrac{1}{2}$ and $\\alpha=\\tfrac{1}{2}+\\mathcal{H}>1$, a heuristic shows “the small-scale component diverges,” while for $\\mathcal{H}<\\tfrac{1}{2}$ and $\\alpha=1$ “the main difficulty is the lack of control of the quadratic nonlinearity in the equation for the small scales $v^{\\epsilon}$.”", "expanded_theorem": "\\label{main_thm}\nLet $\\mathcal{H}\\in (\\frac{1}{3},1)$, and let the initial data $u_0^{\\epsilon},v_0^\\epsilon $ be divergence-free, of zero mean, and such that\n\\begin{align*}\n   \\sup_{\\epsilon\\in (0,1)} \\|u_0^{\\epsilon}\\|_{L^2_x}+\\|\\sqrt{\\epsilon}v_0^{\\epsilon}\\|_{L^2_x}<+\\infty.\n\\end{align*}\nAssume that the operators $Q$ and $C$ satisfy the stated conditions. Then there exist probabilistically weak rough path solutions $(u^{\\epsilon},(-C)^{-1}dW^{\\mathcal{H},\\epsilon})$ to\n\\begin{align}\\label{system_prelimit}\n    \\begin{cases}\n        &\\partial_t u^{\\epsilon}=\\nu A u^{\\epsilon}-b(u^{\\epsilon}+\\epsilon^{-\\alpha+\\H}w^{\\epsilon}+r^{\\epsilon},u^{\\epsilon}),\\\\\n        &dw^{\\epsilon}=\\epsilon^{-1}Cw^{\\epsilon} dt+ \\epsilon^{-\\H}dW^\\H_t, \\\\\n        &\\begin{aligned}\\partial_t r^{\\epsilon}&=\\epsilon^{-1}Cr^{\\epsilon}+\\nu A(\\epsilon^{-\\alpha+\\H}w^{\\epsilon}+r^{\\epsilon})\\\\&-b(u^{\\epsilon}+\\epsilon^{-\\alpha+\\H}w^{\\epsilon}+r^{\\epsilon},\\epsilon^{-\\alpha+\\H}w^{\\epsilon}+r^{\\epsilon}),\n        \\end{aligned}\\\\\n        &w^{\\epsilon}(0) =0,\\quad r^{\\epsilon}(0)=v_0^{\\epsilon}.\n    \\end{cases}\n\\end{align}\nthat converge in law to a probabilistically weak rough path solution of\n\\begin{align}\\label{eq_Hgreat12}\n    \\begin{cases}\n    du=\\left(\\nu \\Delta u-u\\cdot\\nabla u\\right) dt+d\\nabla p- d(-C)^{-1}W^{\\H}\\cdot\\nabla u,\\\\\n    \\nabla\\cdot u=0.\n    \\end{cases}\n\\end{align}\nin the case $\\alpha=1$ and $\\mathcal{H}>\\tfrac{1}{2}$, and to a probabilistically weak rough path solution of\n\\begin{align}\\label{eq_Hsmall12}\n    \\begin{cases}\n    \\partial_t u=\\nu \\Delta u-\\overline{r}\\cdot\\nabla u-u\\cdot\\nabla u+\\nabla p,\\\\\n    \\nabla\\cdot u=0,\n    \\end{cases}\n\\end{align}\nin the case $\\mathcal{H}<\\tfrac{1}{2}$ and $\\alpha=\\tfrac{1}{2}+\\mathcal{H}$.", "theorem_type": ["Existence", "Asymptotic or Limit"], "mcq": {"question": "Let \\(\\mathcal H\\in(\\tfrac13,1)\\). For each \\(\\epsilon\\in(0,1)\\), let \\(u_0^\\epsilon\\) and \\(v_0^\\epsilon\\) be divergence-free, zero-mean initial data satisfying\n\\[\n\\sup_{\\epsilon\\in(0,1)}\\Big(\\|u_0^\\epsilon\\|_{L_x^2}+\\|\\sqrt{\\epsilon}\\,v_0^\\epsilon\\|_{L_x^2}\\Big)<\\infty.\n\\]\nAssume that the operators \\(Q\\) and \\(C\\) satisfy the hypotheses required for the model. Let \\(\\Pi\\) be the Leray projector onto divergence-free vector fields, \\(A:=\\Pi\\Delta\\) the Stokes operator, and \\(b(\\cdot,\\cdot)\\) the Navier--Stokes bilinear term. Let \\(W_t^{\\mathcal H}\\) denote a \\(Q\\)-fractional Wiener process. Consider the prelimit system\n\\[\n\\begin{cases}\n\\partial_t u^{\\epsilon}=\\nu A u^{\\epsilon}-b\\big(u^{\\epsilon}+\\epsilon^{-\\alpha+\\mathcal H}w^{\\epsilon}+r^{\\epsilon},u^{\\epsilon}\\big),\\\\\ndw^{\\epsilon}=\\epsilon^{-1}Cw^{\\epsilon}\\,dt+\\epsilon^{-\\mathcal H}dW_t^{\\mathcal H},\\\\\n\\partial_t r^{\\epsilon}=\\epsilon^{-1}Cr^{\\epsilon}+\\nu A(\\epsilon^{-\\alpha+\\mathcal H}w^{\\epsilon}+r^{\\epsilon})\n-b\\big(u^{\\epsilon}+\\epsilon^{-\\alpha+\\mathcal H}w^{\\epsilon}+r^{\\epsilon},\\epsilon^{-\\alpha+\\mathcal H}w^{\\epsilon}+r^{\\epsilon}\\big),\\\\\nw^{\\epsilon}(0)=0,\\qquad r^{\\epsilon}(0)=v_0^{\\epsilon}.\n\\end{cases}\n\\]\nHere \\(\\overline r\\) denotes the It\\^o--Stokes drift determined by \\(C\\), \\(Q\\), and \\(\\mathcal H\\). A probabilistically weak rough path solution means a solution constructed on some probability space together with its driving rough path, rather than on a fixed prescribed probability space. Which existence-and-convergence statement holds for this family?", "correct_choice": {"label": "A", "text": "There exists a family of probabilistically weak rough path solutions \\((u^{\\epsilon},(-C)^{-1}dW^{\\mathcal H,\\epsilon})\\) to the prelimit system such that, as \\(\\epsilon\\to0\\), the laws converge as follows: if \\(\\alpha=1\\) and \\(\\mathcal H>\\tfrac12\\), then \\(u^{\\epsilon}\\) converges in law to a probabilistically weak rough path solution of\n\\[\n\\begin{cases}\ndu=(\\nu\\Delta u-u\\cdot\\nabla u)\\,dt+d\\nabla p-d(-C)^{-1}W^{\\mathcal H}\\cdot\\nabla u,\\\\\n\\nabla\\cdot u=0;\n\\end{cases}\n\\]\nif \\(\\mathcal H<\\tfrac12\\) and \\(\\alpha=\\tfrac12+\\mathcal H\\), then \\(u^{\\epsilon}\\) converges in law to a probabilistically weak rough path solution of\n\\[\n\\begin{cases}\n\\partial_t u=\\nu\\Delta u-\\overline r\\cdot\\nabla u-u\\cdot\\nabla u+\\nabla p,\\\\\n\\nabla\\cdot u=0.\n\\end{cases}\n\\]"}, "choices": [{"label": "B", "text": "There exists a family of probabilistically weak rough path solutions \\((u^{\\epsilon},(-C)^{-1}dW^{\\mathcal H,\\epsilon})\\) to the prelimit system such that, as \\(\\epsilon\\to0\\), the laws converge as follows: if \\(\\alpha=1\\) and \\(\\mathcal H\\ge \\tfrac12\\), then \\(u^{\\epsilon}\\) converges in law to a probabilistically weak rough path solution of\n\\[\n\\begin{cases}\ndu=(\\nu\\Delta u-u\\cdot\\nabla u)\\,dt+d\\nabla p-d(-C)^{-1}W^{\\mathcal H}\\cdot\\nabla u,\\\\\n\\nabla\\cdot u=0;\n\\end{cases}\n\\]\nif \\(\\mathcal H\\le \\tfrac12\\) and \\(\\alpha=\\tfrac12+\\mathcal H\\), then \\(u^{\\epsilon}\\) converges in law to a probabilistically weak rough path solution of\n\\[\n\\begin{cases}\n\\partial_t u=\\nu\\Delta u-\\overline r\\cdot\\nabla u-u\\cdot\\nabla u+\\nabla p,\\\\\n\\nabla\\cdot u=0.\n\\end{cases}\n\\]"}, {"label": "C", "text": "There exists a family of probabilistically weak rough path solutions \\((u^{\\epsilon},(-C)^{-1}dW^{\\mathcal H,\\epsilon})\\) to the prelimit system such that, as \\(\\epsilon\\to0\\), for \\(\\alpha=1\\) and \\(\\mathcal H>\\tfrac12\\), the processes \\(u^{\\epsilon}\\) converge in law to a probabilistically weak rough path solution of\n\\[\n\\begin{cases}\ndu=(\\nu\\Delta u-u\\cdot\\nabla u)\\,dt+d\\nabla p-d(-C)^{-1}W^{\\mathcal H}\\cdot\\nabla u,\\\\\n\\nabla\\cdot u=0.\n\\end{cases}\n\\]"}, {"label": "D", "text": "There exists a family of probabilistically strong rough path solutions \\((u^{\\epsilon},(-C)^{-1}dW^{\\mathcal H})\\) to the prelimit system on any prescribed probability space carrying the given fractional Wiener process such that, as \\(\\epsilon\\to0\\), the laws converge as follows: if \\(\\alpha=1\\) and \\(\\mathcal H>\\tfrac12\\), then \\(u^{\\epsilon}\\) converges in law to a probabilistically strong rough path solution of\n\\[\n\\begin{cases}\ndu=(\\nu\\Delta u-u\\cdot\\nabla u)\\,dt+d\\nabla p-d(-C)^{-1}W^{\\mathcal H}\\cdot\\nabla u,\\\\\n\\nabla\\cdot u=0;\n\\end{cases}\n\\]\nif \\(\\mathcal H<\\tfrac12\\) and \\(\\alpha=\\tfrac12+\\mathcal H\\), then \\(u^{\\epsilon}\\) converges in law to a probabilistically strong rough path solution of\n\\[\n\\begin{cases}\n\\partial_t u=\\nu\\Delta u-\\overline r\\cdot\\nabla u-u\\cdot\\nabla u+\\nabla p,\\\\\n\\nabla\\cdot u=0.\n\\end{cases}\n\\]"}, {"label": "E", "text": "There exists a family of probabilistically weak rough path solutions \\((u^{\\epsilon},(-C)^{-1}dW^{\\mathcal H,\\epsilon})\\) to the prelimit system such that, as \\(\\epsilon\\to0\\), the laws converge as follows: if \\(\\alpha=1\\) and \\(\\mathcal H>\\tfrac12\\), then \\(u^{\\epsilon}\\) converges in law to a probabilistically weak rough path solution of\n\\[\n\\begin{cases}\ndu=(\\nu\\Delta u-u\\cdot\\nabla u-\\overline r\\cdot\\nabla u)\\,dt+d\\nabla p-d(-C)^{-1}W^{\\mathcal H}\\cdot\\nabla u,\\\\\n\\nabla\\cdot u=0;\n\\end{cases}\n\\]\nif \\(\\mathcal H<\\tfrac12\\) and \\(\\alpha=1\\), then \\(u^{\\epsilon}\\) converges in law to a probabilistically weak rough path solution of\n\\[\n\\begin{cases}\n\\partial_t u=\\nu\\Delta u-\\overline r\\cdot\\nabla u-u\\cdot\\nabla u+\\nabla p,\\\\\n\\nabla\\cdot u=0.\n\\end{cases}\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "strict threshold at H=1/2", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "characteristic", "tampered_component": "dropped the H<1/2, alpha=1/2+H convergence regime", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "probabilistically weak existence replaced by strong existence on a fixed probability space", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "absence of Ito-Stokes drift for H>1/2 and correct scaling alpha in the H<1/2 regime", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct option outright; it gives the setup and asks for the limiting conclusion. The answer must be identified from the choices."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: under the stated hypotheses, it asks for the exact limiting statement of the result rather than requiring a new inference or application."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in subtle ways (strict vs non-strict threshold at H=1/2, exact scaling alpha=1/2+H, presence of Ito-Stokes drift, full-family vs subsequential convergence). However, the task mainly tests precise recall of the theorem rather than genuine generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: they perturb boundary cases, weaken convergence claims, or misplace/remove the Ito-Stokes drift. These reflect realistic failure modes in interpreting such a theorem."}, "total_score": 5, "overall_assessment": "A strong theorem-recall MCQ with high-quality distractors and no answer leakage, but it is largely a restatement of the result and only moderately tests reasoning."}}
{"id": "2602.08831v1", "paper_link": "http://arxiv.org/abs/2602.08831v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "[{see Theorem \\ref{theorem l2 del delbar lemma}}]\\label{theorem intro l2 del delbar lemma}\n    Let $(M,g)$ be a complete K\\\"ahler manifold such that the self-adjoint Laplacian $\\Delta$ has a spectral gap in $L^2\\Lambda^{k}_\\C$. Given a smooth $L^2$-form $\\alpha\\in L^2A^k_\\C$ which satisfies $\\partial\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n    \\begin{enumerate}\n        \\item $\\alpha=\\partial\\delbar\\beta$, for $\\beta\\in L^2 A^{k-2}_\\C$;\n        \\item $\\alpha=\\partial\\gamma$, for $\\gamma\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=\\partial\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=d\\lambda$, for $\\lambda\\in L^2 A^{k-1}_\\C$.\n    \\end{enumerate}", "start_pos": 8163, "end_pos": 8958, "label": "theorem intro l2 del delbar lemma"}, "ref_dict": {"theorem l2 del delbar lemma": "\\begin{theorem}\\label{theorem l2 del delbar lemma}\n    Let $(M,g)$ be a complete K\\\"ahler manifold such that $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{k}_\\C$. Given a smooth $L^2$-form $\\alpha\\in L^2A^k_\\C$ which satisfies $\\del\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n    \\begin{enumerate}\n        \\item $\\alpha=\\del\\delbar\\beta$, for $\\beta\\in L^2 A^{k-2}_\\C$;\n        \\item $\\alpha=\\del\\gamma$, for $\\gamma\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=\\del\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=d\\lambda$, for $\\lambda\\in L^2 A^{k-1}_\\C$.\n    \\end{enumerate}\n\\end{theorem}", "theorem spectral gap varouchas laplacians": "\\begin{theorem}\\label{theorem spectral gap varouchas laplacians}\n    Let $(M,g)$ be a complete K\\\"ahler manifold. \n    If $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$, $L^2\\Lambda^{p-1,q}$ and $L^2\\Lambda^{p,q-1}$, then $\\square_{A,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$. If $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$, $L^2\\Lambda^{p+1,q}$ and $L^2\\Lambda^{p,q+1}$, then $\\square_{BC,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$.\n\\end{theorem}", "theorem intro l2 del delbar lemma": "\\begin{theorem}[{see Theorem \\ref{theorem l2 del delbar lemma}}]\\label{theorem intro l2 del delbar lemma}\n    Let $(M,g)$ be a complete K\\\"ahler manifold such that the self-adjoint Laplacian $\\Delta$ has a spectral gap in $L^2\\Lambda^{k}_\\C$. Given a smooth $L^2$-form $\\alpha\\in L^2A^k_\\C$ which satisfies $\\del\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n    \\begin{enumerate}\n        \\item $\\alpha=\\del\\delbar\\beta$, for $\\beta\\in L^2 A^{k-2}_\\C$;\n        \\item $\\alpha=\\del\\gamma$, for $\\gamma\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=\\del\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=d\\lambda$, for $\\lambda\\in L^2 A^{k-1}_\\C$.\n    \\end{enumerate}\n\\end{theorem}", "theorem closed image del delbar": "\\begin{theorem}\\label{theorem closed image del delbar}\nLet $(M,g)$ be a complete K\\\"ahler manifold. If $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$, then for $b\\in\\{s,w\\}$ it follows that\n$\\im\\del\\delbar_b$ and $\\im\\delbar^*\\del^*_b$ are closed in $L^2\\Lambda^{p,q}$.\n\\end{theorem}", "remark bounded geometry": "\\begin{remark}\\label{remark bounded geometry}\n    Suppose that, in addition to the assumptions of Theorem \\ref{theorem l2 del delbar lemma}, the manifold $(M,g)$ is of bounded geometry and the elliptic Laplacians $\\Delta_{\\delbar}=\\Delta_{\\del}=\\frac12\\Delta_{d}$, $\\tilde\\Delta_{BC}$ and $\\tilde\\Delta_{A}$ are $\\cinf$-bounded and uniformly elliptic, with the bundles of differential forms which are of bounded geometry (cf. \\cite{Sh}). Examples of such manifolds are given by the universal coverings of K\\\"ahler hyperbolic manifolds \\cite{G}.  Then, by \\cite[Proposition 1.1]{Sh} the domains of the above operators coincide with the Sobolev spaces\n    \\[\n    \\D(\\Delta_{\\delbar,sw})=\\D(\\Delta_{\\del,sw})=\\D(\\Delta_{d,sw})=W^2_2(M,\\Lambda^\\bullet_\\C)\n    \\]\n    and\n    \\[\n    \\D(\\tilde\\Delta_{A,sw})=\\D(\\tilde\\Delta_{BC,sw})=W^4_2(M,\\Lambda^\\bullet_\\C),\n    \\]\n    with the notation for Sobolev spaces as in \\cite{Sh}. Therefore, since the image of each Green operator is one of the above domains, and thus the image is a Sobolev space, in the statement of Theorem \\ref{theorem l2 del delbar lemma} we can choose $\\beta\\in W^2_2(M,\\Lambda^{k-2}_\\C)$ and $\\gamma,\\zeta,\\eta,\\theta,\\lambda\\in W^1_2(M,\\Lambda^{k-1}_\\C)$. In particular, $\\del\\beta,\\delbar\\beta\\in W^1_2(M,\\Lambda^{k-1}_\\C)\\subseteq L^2\\Lambda_\\C^{k-1}$.\n\\end{remark}", "theorem spectral gap kodaira spencer laplacians": "\\begin{theorem}\\label{theorem spectral gap kodaira spencer laplacians}\nLet $(M,g)$ be a complete K\\\"ahler manifold. If $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$, then for $b\\in\\{s,w\\}$ the operators $\\tilde\\Delta_{A,sb}$ and $\\tilde\\Delta_{BC,bw}$ have a spectral gap in $L^2\\Lambda^{p,q}$.\n\\end{theorem}", "theorem reduced l2 del delbar lemma": "\\begin{theorem}\\label{theorem reduced l2 del delbar lemma}\nGiven a complete K\\\"ahler manifold $(M,g)$, in both the spaces $L^2\\Lambda^k_\\C$ and $L^2\\Lambda^{p,q}$ the following equalities hold true\n\\begin{align*}\n\\c{\\im \\del\\delbar_s}\n&=\\c{\\im \\del_s}\\cap\\ker\\delbar_w=\\c{\\im \\delbar_s}\\cap\\ker\\del_w\\\\\n&=(\\c{\\im\\del_s}+\\c{\\im\\delbar_s})\\cap\\ker\\del_w\\cap\\ker\\delbar_w\\\\\n&=\\c{\\im d_s}\\cap\\ker\\del_w\\cap\\ker\\delbar_w.\n\\end{align*}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 4719, "pre_theorem_intro_text": "On compact K\\\"ahler manifolds, the $\\partial\\delbar$-Lemma \\cite[Lemmas 5.11, 5.15]{DGMS} states that for every smooth, complex-valued $k$-form $\\alpha\\in A^k_\\C$ which is $\\partial$- and $\\delbar$-closed, \\textit{i.e.}, $\\partial\\alpha=\\delbar\\alpha=0$,  then the following conditions are equivalent:\n\\begin{enumerate}\n    \\item $\\alpha=\\partial\\delbar\\beta$, for $\\beta\\in A^{k-2}_\\C$;\n    \\item $\\alpha=\\partial\\gamma$, for $\\gamma\\in  A^{k-1}_\\C$;\n    \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in  A^{k-1}_\\C$;\n    \\item $\\alpha=\\partial\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in  A^{k-1}_\\C$;\n    \\item $\\alpha=d\\lambda$, for $\\lambda\\in  A^{k-1}_\\C$,\n\\end{enumerate}\nwhere the exterior derivative decomposes as $d=\\partial+\\delbar$.\nThe validity of the above statement is invariant under holomorphic birational maps between compact complex manifolds \\cite[Theorem 5.22]{DGMS}, and has important cohomological and topological implications \\cite[Remark 5.15]{DGMS}: all the complex cohomology spaces (Aeppli, Bott-Chern, Dolbeault cohomology and its conjugate) are isomorphic, and the $k$-de Rham cohomology group is isomorphic to the direct sum of all the $(p,q)$-Dolbeault cohomology groups for $p+q=k$. A numerical characterisation of the validity of the $\\partial\\delbar$-Lemma involving the dimensions of de Rham, Aeppli and Bott-Chern cohomology was proven in \\cite{AT}.\n\nThe main purpose of this paper is to generalise the $\\partial\\delbar$-Lemma to an $L^2$-$\\partial\\delbar$-Lemma involving square integrable forms on a class of complete K\\\"ahler manifolds. We now introduce the result. Given a Hermitian manifold $(M,g)$, denote by $d^*,\\partial^*,\\delbar^*$ the $L^2$-formal adjoints of the differential operators $d,\\partial,\\delbar$, and by $\\Delta_D:=DD^*+D^*D$ for $D\\in\\{d,\\partial,\\delbar\\}$ the Hodge, $\\partial$- and Dolbeault Laplacians, which are second-order formally self-adjoint elliptic operators. It is well known that if the metric is complete, then all these second-order Laplacians are essentially self-adjoint, namely, their restrictions to smooth compactly supported forms have a unique self-adjoint extension as unbounded operators between the Hilbert spaces of $L^2$-forms $L^2\\Lambda^\\bullet_\\C:=\\oplus_{k\\in\\N}L^2\\Lambda^k_\\C$, \\textit{i.e.}, the space of possibly non-continuous, complex-valued forms with bounded $L^2$-norm. These self-adjoint operators are non-negative, therefore their spectrum is contained in $[0,+\\infty)$. We say that such an operator has a \\emph{spectral gap} if its spectrum is contained in $\\{0\\}\\cup[C,+\\infty)$ for some $C>0$. This is equivalent to the image of the operator being closed. \n\nOn the other hand, if the Hermitian metric $g$ is K\\\"ahler, \\textit{i.e.}, its fundamental $2$-form is closed, then by K\\\"ahler identities the above second-order Laplacians coincide up to a factor, namely, $\\Delta_d=2\\Delta_\\delbar=2\\Delta_\\partial$. As a consequence, if the metric is both K\\\"ahler and complete, then all the self-adjoint extensions of these Laplacians coincide up to a factor. In the introduction we denote the unique self-adjoint extension of the Hodge Laplacian by $\\Delta$. \n\nBesides compact K\\\"ahler manifolds, where the spectrum is discrete, examples of complete K\\\"ahler manifolds where $\\Delta$ has a spectral gap on the whole algebra of forms $L^2\\Lambda^\\bullet_\\C$ are given by complete K\\\"ahler $d$-bounded manifolds \\cite[Theorem 1.4.A]{G}, where $d$-bounded means that the fundamental form $\\omega$ is not only closed but also exact $\\omega=d\\eta$, and $\\eta$ is bounded in the pointwise norm. Explicit examples of complete K\\\"ahler $d$-bounded manifolds are: complete simply connected K\\\"ahler manifolds with sectional curvature bounded from above by a negative constant \\cite[0.1.B]{G} \\cite[Lemma 3.2]{CY}, Hermitian symmetric spaces of non compact type \\cite[0.1.C'']{G} \\cite[Proposition 8.6]{Ba}, hyperconvex domains in Stein manifolds \\cite[0.3.A(b)]{G} \\cite[Proposition 2.2]{Do}, strictly pseudoconvex domains in $\\C^n$ with the Bergman metric \\cite[Proposition 3.4]{Do}, bounded homogeneous domains in $\\C^n$ \\cite[Theorem 1]{KO}.\nWe expect this $L^2$-$\\partial\\delbar$-Lemma to be useful for further studies of the above classes of complete K\\\"ahler manifolds. For some recent results related to complete K\\\"ahler $d$-bounded manifolds we refer to \\cite{PT1,BDET,H}. The notion of an $L^2$-$\\partial\\delbar$-Lemma is also of interest on normal coverings of compact complex manifolds, in light of the inequalities between $L^2$ invariants proved in \\cite{HP,BP}.\n\nWe use the notation $L^2A^k_\\C$ to denote the space of smooth $L^2$-forms $L^2\\Lambda^k_\\C\\cap A^k_\\C$. The following theorem is the main result of the paper.", "context": "On compact K\\\"ahler manifolds, the $\\partial\\delbar$-Lemma \\cite[Lemmas 5.11, 5.15]{DGMS} states that for every smooth, complex-valued $k$-form $\\alpha\\in A^k_\\C$ which is $\\partial$- and $\\delbar$-closed, \\textit{i.e.}, $\\partial\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n\\begin{enumerate}\n \\item $\\alpha=\\partial\\delbar\\beta$, for $\\beta\\in A^{k-2}_\\C$;\n \\item $\\alpha=\\partial\\gamma$, for $\\gamma\\in A^{k-1}_\\C$;\n \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in A^{k-1}_\\C$;\n \\item $\\alpha=\\partial\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in A^{k-1}_\\C$;\n \\item $\\alpha=d\\lambda$, for $\\lambda\\in A^{k-1}_\\C$,\n\\end{enumerate}\nwhere the exterior derivative decomposes as $d=\\partial+\\delbar$.\nThe validity of the above statement is invariant under holomorphic birational maps between compact complex manifolds \\cite[Theorem 5.22]{DGMS}, and has important cohomological and topological implications \\cite[Remark 5.15]{DGMS}: all the complex cohomology spaces (Aeppli, Bott-Chern, Dolbeault cohomology and its conjugate) are isomorphic, and the $k$-de Rham cohomology group is isomorphic to the direct sum of all the $(p,q)$-Dolbeault cohomology groups for $p+q=k$. A numerical characterisation of the validity of the $\\partial\\delbar$-Lemma involving the dimensions of de Rham, Aeppli and Bott-Chern cohomology was proven in \\cite{AT}.\n\nThe main purpose of this paper is to generalise the $\\partial\\delbar$-Lemma to an $L^2$-$\\partial\\delbar$-Lemma involving square integrable forms on a class of complete K\\\"ahler manifolds. We now introduce the result. Given a Hermitian manifold $(M,g)$, denote by $d^*,\\partial^*,\\delbar^*$ the $L^2$-formal adjoints of the differential operators $d,\\partial,\\delbar$, and by $\\Delta_D:=DD^*+D^*D$ for $D\\in\\{d,\\partial,\\delbar\\}$ the Hodge, $\\partial$- and Dolbeault Laplacians, which are second-order formally self-adjoint elliptic operators. It is well known that if the metric is complete, then all these second-order Laplacians are essentially self-adjoint, namely, their restrictions to smooth compactly supported forms have a unique self-adjoint extension as unbounded operators between the Hilbert spaces of $L^2$-forms $L^2\\Lambda^\\bullet_\\C:=\\oplus_{k\\in\\N}L^2\\Lambda^k_\\C$, \\textit{i.e.}, the space of possibly non-continuous, complex-valued forms with bounded $L^2$-norm. These self-adjoint operators are non-negative, therefore their spectrum is contained in $[0,+\\infty)$. We say that such an operator has a \\emph{spectral gap} if its spectrum is contained in $\\{0\\}\\cup[C,+\\infty)$ for some $C>0$. This is equivalent to the image of the operator being closed.\n\nOn the other hand, if the Hermitian metric $g$ is K\\\"ahler, \\textit{i.e.}, its fundamental $2$-form is closed, then by K\\\"ahler identities the above second-order Laplacians coincide up to a factor, namely, $\\Delta_d=2\\Delta_\\delbar=2\\Delta_\\partial$. As a consequence, if the metric is both K\\\"ahler and complete, then all the self-adjoint extensions of these Laplacians coincide up to a factor. In the introduction we denote the unique self-adjoint extension of the Hodge Laplacian by $\\Delta$.\n\nBesides compact K\\\"ahler manifolds, where the spectrum is discrete, examples of complete K\\\"ahler manifolds where $\\Delta$ has a spectral gap on the whole algebra of forms $L^2\\Lambda^\\bullet_\\C$ are given by complete K\\\"ahler $d$-bounded manifolds \\cite[Theorem 1.4.A]{G}, where $d$-bounded means that the fundamental form $\\omega$ is not only closed but also exact $\\omega=d\\eta$, and $\\eta$ is bounded in the pointwise norm. Explicit examples of complete K\\\"ahler $d$-bounded manifolds are: complete simply connected K\\\"ahler manifolds with sectional curvature bounded from above by a negative constant \\cite[0.1.B]{G} \\cite[Lemma 3.2]{CY}, Hermitian symmetric spaces of non compact type \\cite[0.1.C'']{G} \\cite[Proposition 8.6]{Ba}, hyperconvex domains in Stein manifolds \\cite[0.3.A(b)]{G} \\cite[Proposition 2.2]{Do}, strictly pseudoconvex domains in $\\C^n$ with the Bergman metric \\cite[Proposition 3.4]{Do}, bounded homogeneous domains in $\\C^n$ \\cite[Theorem 1]{KO}.\nWe expect this $L^2$-$\\partial\\delbar$-Lemma to be useful for further studies of the above classes of complete K\\\"ahler manifolds. For some recent results related to complete K\\\"ahler $d$-bounded manifolds we refer to \\cite{PT1,BDET,H}. The notion of an $L^2$-$\\partial\\delbar$-Lemma is also of interest on normal coverings of compact complex manifolds, in light of the inequalities between $L^2$ invariants proved in \\cite{HP,BP}.\n\nWe use the notation $L^2A^k_\\C$ to denote the space of smooth $L^2$-forms $L^2\\Lambda^k_\\C\\cap A^k_\\C$. The following theorem is the main result of the paper.\n\n\\begin{theorem}\\label{theorem l2 del delbar lemma}\n    Let $(M,g)$ be a complete K\\\"ahler manifold such that $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{k}_\\C$. Given a smooth $L^2$-form $\\alpha\\in L^2A^k_\\C$ which satisfies $\\del\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n    \\begin{enumerate}\n        \\item $\\alpha=\\del\\delbar\\beta$, for $\\beta\\in L^2 A^{k-2}_\\C$;\n        \\item $\\alpha=\\del\\gamma$, for $\\gamma\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=\\del\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=d\\lambda$, for $\\lambda\\in L^2 A^{k-1}_\\C$.\n    \\end{enumerate}\n\\end{theorem}", "full_context": "On compact K\\\"ahler manifolds, the $\\partial\\delbar$-Lemma \\cite[Lemmas 5.11, 5.15]{DGMS} states that for every smooth, complex-valued $k$-form $\\alpha\\in A^k_\\C$ which is $\\partial$- and $\\delbar$-closed, \\textit{i.e.}, $\\partial\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n\\begin{enumerate}\n \\item $\\alpha=\\partial\\delbar\\beta$, for $\\beta\\in A^{k-2}_\\C$;\n \\item $\\alpha=\\partial\\gamma$, for $\\gamma\\in A^{k-1}_\\C$;\n \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in A^{k-1}_\\C$;\n \\item $\\alpha=\\partial\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in A^{k-1}_\\C$;\n \\item $\\alpha=d\\lambda$, for $\\lambda\\in A^{k-1}_\\C$,\n\\end{enumerate}\nwhere the exterior derivative decomposes as $d=\\partial+\\delbar$.\nThe validity of the above statement is invariant under holomorphic birational maps between compact complex manifolds \\cite[Theorem 5.22]{DGMS}, and has important cohomological and topological implications \\cite[Remark 5.15]{DGMS}: all the complex cohomology spaces (Aeppli, Bott-Chern, Dolbeault cohomology and its conjugate) are isomorphic, and the $k$-de Rham cohomology group is isomorphic to the direct sum of all the $(p,q)$-Dolbeault cohomology groups for $p+q=k$. A numerical characterisation of the validity of the $\\partial\\delbar$-Lemma involving the dimensions of de Rham, Aeppli and Bott-Chern cohomology was proven in \\cite{AT}.\n\nThe main purpose of this paper is to generalise the $\\partial\\delbar$-Lemma to an $L^2$-$\\partial\\delbar$-Lemma involving square integrable forms on a class of complete K\\\"ahler manifolds. We now introduce the result. Given a Hermitian manifold $(M,g)$, denote by $d^*,\\partial^*,\\delbar^*$ the $L^2$-formal adjoints of the differential operators $d,\\partial,\\delbar$, and by $\\Delta_D:=DD^*+D^*D$ for $D\\in\\{d,\\partial,\\delbar\\}$ the Hodge, $\\partial$- and Dolbeault Laplacians, which are second-order formally self-adjoint elliptic operators. It is well known that if the metric is complete, then all these second-order Laplacians are essentially self-adjoint, namely, their restrictions to smooth compactly supported forms have a unique self-adjoint extension as unbounded operators between the Hilbert spaces of $L^2$-forms $L^2\\Lambda^\\bullet_\\C:=\\oplus_{k\\in\\N}L^2\\Lambda^k_\\C$, \\textit{i.e.}, the space of possibly non-continuous, complex-valued forms with bounded $L^2$-norm. These self-adjoint operators are non-negative, therefore their spectrum is contained in $[0,+\\infty)$. We say that such an operator has a \\emph{spectral gap} if its spectrum is contained in $\\{0\\}\\cup[C,+\\infty)$ for some $C>0$. This is equivalent to the image of the operator being closed.\n\nOn the other hand, if the Hermitian metric $g$ is K\\\"ahler, \\textit{i.e.}, its fundamental $2$-form is closed, then by K\\\"ahler identities the above second-order Laplacians coincide up to a factor, namely, $\\Delta_d=2\\Delta_\\delbar=2\\Delta_\\partial$. As a consequence, if the metric is both K\\\"ahler and complete, then all the self-adjoint extensions of these Laplacians coincide up to a factor. In the introduction we denote the unique self-adjoint extension of the Hodge Laplacian by $\\Delta$.\n\nBesides compact K\\\"ahler manifolds, where the spectrum is discrete, examples of complete K\\\"ahler manifolds where $\\Delta$ has a spectral gap on the whole algebra of forms $L^2\\Lambda^\\bullet_\\C$ are given by complete K\\\"ahler $d$-bounded manifolds \\cite[Theorem 1.4.A]{G}, where $d$-bounded means that the fundamental form $\\omega$ is not only closed but also exact $\\omega=d\\eta$, and $\\eta$ is bounded in the pointwise norm. Explicit examples of complete K\\\"ahler $d$-bounded manifolds are: complete simply connected K\\\"ahler manifolds with sectional curvature bounded from above by a negative constant \\cite[0.1.B]{G} \\cite[Lemma 3.2]{CY}, Hermitian symmetric spaces of non compact type \\cite[0.1.C'']{G} \\cite[Proposition 8.6]{Ba}, hyperconvex domains in Stein manifolds \\cite[0.3.A(b)]{G} \\cite[Proposition 2.2]{Do}, strictly pseudoconvex domains in $\\C^n$ with the Bergman metric \\cite[Proposition 3.4]{Do}, bounded homogeneous domains in $\\C^n$ \\cite[Theorem 1]{KO}.\nWe expect this $L^2$-$\\partial\\delbar$-Lemma to be useful for further studies of the above classes of complete K\\\"ahler manifolds. For some recent results related to complete K\\\"ahler $d$-bounded manifolds we refer to \\cite{PT1,BDET,H}. The notion of an $L^2$-$\\partial\\delbar$-Lemma is also of interest on normal coverings of compact complex manifolds, in light of the inequalities between $L^2$ invariants proved in \\cite{HP,BP}.\n\nWe use the notation $L^2A^k_\\C$ to denote the space of smooth $L^2$-forms $L^2\\Lambda^k_\\C\\cap A^k_\\C$. The following theorem is the main result of the paper.\n\n\\begin{theorem}\\label{theorem l2 del delbar lemma}\n    Let $(M,g)$ be a complete K\\\"ahler manifold such that $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{k}_\\C$. Given a smooth $L^2$-form $\\alpha\\in L^2A^k_\\C$ which satisfies $\\del\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n    \\begin{enumerate}\n        \\item $\\alpha=\\del\\delbar\\beta$, for $\\beta\\in L^2 A^{k-2}_\\C$;\n        \\item $\\alpha=\\del\\gamma$, for $\\gamma\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=\\del\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=d\\lambda$, for $\\lambda\\in L^2 A^{k-1}_\\C$.\n    \\end{enumerate}\n\\end{theorem}\n\n\\begin{theorem}\\label{theorem closed image del delbar}\nLet $(M,g)$ be a complete K\\\"ahler manifold. If $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$, then for $b\\in\\{s,w\\}$ it follows that\n$\\im\\del\\delbar_b$ and $\\im\\delbar^*\\del^*_b$ are closed in $L^2\\Lambda^{p,q}$.\n\\end{theorem}\n\\begin{proof}\nBy Theorem \\ref{theorem spectral gap kodaira spencer laplacians}, $\\tilde\\Delta_{BC,bw}$ has a spectral gap in $L^2\\Lambda^{p,q}$. By Lemma \\ref{lemma spectral gap equiv}, this is equivalent to the closure of the image of $\\tilde\\Delta_{BC,bw}$. Therefore we can orthogonally decompose\n\\begin{align*}\nL^2\\Lambda^{p,q}&=\\ker\\tilde\\Delta_{BC,bw}\\overset{\\perp}{\\oplus}\\im\\tilde\\Delta_{BC,bw}\\\\\n&\\subseteq \\ker\\tilde\\Delta_{BC,bw}\\overset{\\perp}{\\oplus}\\left(\\im\\del\\delbar_b + \\im \\delbar^*\\del^*_s+\\im \\del^*\\delbar_s+\\im\\delbar^*\\del_s+\\im(\\del^*\\oplus\\delbar^*)_s\\right)\\\\\n&\\subseteq \\ker\\tilde\\Delta_{BC,bw}\\overset{\\perp}{\\oplus}\\im\\del\\delbar_b \\overset{\\perp}{\\oplus} (\\im\\del^*_s+\\im\\delbar^*_s)\\subseteq L^2\\Lambda^{p,q}.\n\\end{align*}\nThe first inclusion follows by definition of $\\tilde\\Delta_{BC,bw}$, while the second inclusion is justified by \\emph{(2)} of Proposition \\ref{proposition properties strong weak} and the fact that \n\\[\n\\im(\\del^*\\oplus\\delbar^*)_s\\subseteq\\c{\\im(\\del^*\\oplus\\delbar^*)_s}=\\c{\\im\\del^*_s}+\\c{\\im\\delbar^*_s}=\\im\\del^*_s+\\im\\delbar^*_s,\n\\]\nwhere the first equality is equation \\eqref{equation kernel image del* e delbar*} and the second is obtained via Lemmas \\ref{lemma spectral gap equiv 2} and \\ref{lemma im closed} using the spectral gap of $\\Delta_{\\delbar,sw}=\\Delta_{\\del,sw}$ in $L^2\\Lambda^{p,q}$.\nThe orthogonality of the above decompositions is easy to check.\nThe closure of $\\im\\del\\delbar_b$ now follows from the orthogonality of the decomposition of $L^2\\Lambda^{p,q}$ in the last line. Using the spectral gap of $\\tilde\\Delta_{A,sb}$ we similarly find the closure of $\\im\\delbar^*\\del^*_b$ in $L^2\\Lambda^{p,q}$.\n\\end{proof}\n\n\\begin{theorem}\\label{theorem l2 del delbar lemma}\n Let $(M,g)$ be a complete K\\\"ahler manifold such that $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{k}_\\C$. Given a smooth $L^2$-form $\\alpha\\in L^2A^k_\\C$ which satisfies $\\del\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n \\begin{enumerate}\n \\item $\\alpha=\\del\\delbar\\beta$, for $\\beta\\in L^2 A^{k-2}_\\C$;\n \\item $\\alpha=\\del\\gamma$, for $\\gamma\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=\\del\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in L^2 A^{k-1}_\\C$;\n \\item $\\alpha=d\\lambda$, for $\\lambda\\in L^2 A^{k-1}_\\C$.\n \\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n Since $\\Delta_{\\delbar,sw}=\\Delta_{\\del,sw}=\\frac12\\Delta_{d,sw}$ has a spectral gap in $L^2\\Lambda^{k}_\\C$, by Theorem \\ref{theorem spectral gap kodaira spencer laplacians} also $\\tilde\\Delta_{BC,sw}$ and $\\tilde\\Delta_{A,sw}$ have a spectral gap in $L^2\\Lambda^{k}_\\C$.\n We define the orthogonal projection\n \\[\n P_{BC}:L^2\\Lambda^{k}_\\C\\to\\ker \\tilde\\Delta_{BC,sw}\n \\]\n and the Green operator\n \\begin{align*}\n G_{BC}:&\\im \\tilde\\Delta_{BC,sw}\\to \\D(\\tilde\\Delta_{BC,sw})\\\\\n & \\tilde\\Delta_{BC,sw}\\beta\\mapsto\\beta.\n \\end{align*}\n Being $\\im \\tilde\\Delta_{BC,sw}$ closed, and recalling the orthogonal decomposition\n \\[\n L^2\\Lambda^{k}_\\C=\\ker \\tilde\\Delta_{BC,sw}\\oplus\\im \\tilde\\Delta_{BC,sw},\n \\]\n we can extend the Green operator on the whole space $L^2\\Lambda^{k}_\\C$ simply by setting $G_{BC}=0$ on $\\ker \\tilde\\Delta_{BC,sw}$. Given any form $\\alpha\\in L^2\\Lambda^{k}_\\C$, it decomposes as\n \\[\n \\alpha=P_{BC}\\alpha+(id-P_{BC})\\alpha,\n \\]\n where $(id-P_{BC})\\alpha=\\tilde\\Delta_{BC,sw}\\beta$. Now $G_{BC}\\alpha=\\beta$, therefore\n \\[\n \\alpha=P_{BC}\\alpha+\\tilde\\Delta_{BC,sw}G_{BC}\\alpha.\n \\]\n If, in particular, $\\alpha$ is smooth and so $\\alpha\\in L^2A^k_\\C$, then by elliptic regularity (see, \\textit{e.g.}, \\cite[Theorem 2.1]{HP}) also $G_{BC}\\alpha=\\beta\\in L^2A^k_\\C\\cap \\D(\\tilde\\Delta_{BC,sw})$ is smooth. In particular, by \\emph{(5)} of Proposition \\ref{proposition properties strong weak} we obtain \n \\[\n \\tilde\\Delta_{BC,sw}G_{BC}\\alpha=\\tilde\\Delta_{BC}G_{BC}\\alpha.\n \\]\n The same holds for analogue operators $P_\\delta$ and $G_\\delta$ for $\\delta\\in\\{A,\\del,\\delbar,d\\}$.\n Therefore, if $\\alpha\\in L^2A^k_\\C$, for $\\delta\\in\\{A,BC\\}$ and $\\epsilon\\in\\{\\del,\\delbar,d\\}$ we deduce decompositions\n \\[\n \\alpha=P_{\\delta}\\alpha+\\tilde\\Delta_{\\delta}G_{\\delta}\\alpha=P_\\epsilon\\alpha+\\Delta_{\\epsilon}G_{\\epsilon}\\alpha,\n \\]\n with smooth forms $G_{\\delta}\\alpha,G_{\\epsilon}\\alpha\\in L^2A^k_\\C$. Recall that by Theorem \\ref{theorem complete kahler equality harmonic} the kernels of $\\Delta_{\\delbar,sw}=\\Delta_{\\del,sw}=\\frac12\\Delta_{d,sw}$, $\\tilde\\Delta_{BC,sw}$ and $\\tilde\\Delta_{A,sw}$ coincide, thus $P_{BC}\\alpha=0$ iff $P_\\delta=0$ for all $\\delta\\in\\{A,\\del,\\delbar,d\\}$.\n\nNow assume that $\\alpha\\in L^2A^k_\\C$ satisfies $\\del\\alpha=\\delbar\\alpha=0$. We know $\\ker\\del_w\\cap\\ker\\delbar_w\\subseteq L^2A^k_\\C$ is $L^2$-orthogonal to\n \\begin{align*}\n &\\im d_s^*, &&\\im\\del_s^*, & &\\im\\delbar_s^*, &\\im(\\del^*\\oplus\\delbar^*)_s, &&\\im\\del\\delbar_s^*,\n \\end{align*}\n as in the proof of Theorem \\ref{theorem reduced l2 del delbar lemma}. Thanks to this and to \\emph{(5)} of Proposition \\ref{proposition properties strong weak}, we obtain\n \\begin{align*}\n &\\alpha=\\tilde\\Delta_{A}G_{A}\\alpha &&\\iff && \\alpha=(\\del\\del^*+\\delbar\\delbar^*)G_{A}\\alpha,\\\\\n &\\alpha=\\tilde\\Delta_{BC}G_{BC}\\alpha &&\\iff && \\alpha=\\del\\delbar\\delbar^*\\del^*G_{BC}\\alpha,\\\\\n &\\alpha=\\Delta_{\\del}G_{\\del}\\alpha &&\\iff && \\alpha=\\del\\del^*G_{\\del}\\alpha,\\\\\n &\\alpha=\\Delta_{\\delbar}G_{\\delbar}\\alpha &&\\iff && \\alpha=\\delbar\\delbar^*G_{\\delbar}\\alpha,\\\\\n &\\alpha=\\Delta_{d}G_{d}\\alpha &&\\iff && \\alpha=dd^*G_{d}\\alpha.\n \\end{align*}\n It is then enough to set $\\beta:=\\delbar^*\\del^*G_{BC}\\alpha$, $\\gamma:=\\del^*G_{\\del}\\alpha$, $\\zeta:=\\delbar^*G_{\\delbar}\\alpha$, $\\eta:=\\del^*G_{A}\\alpha$, $\\theta:=\\delbar^*G_{A}\\alpha$, $\\lambda:=d^*G_{d}\\alpha$, which lie in $L^2\\Lambda^\\bullet_\\C$ since the image of each Green operator is the domain of the corresponding Laplacian.\n This concludes the proof.\n\\end{proof}\n\n\\begin{remark}\\label{remark bounded geometry}\n Suppose that, in addition to the assumptions of Theorem \\ref{theorem l2 del delbar lemma}, the manifold $(M,g)$ is of bounded geometry and the elliptic Laplacians $\\Delta_{\\delbar}=\\Delta_{\\del}=\\frac12\\Delta_{d}$, $\\tilde\\Delta_{BC}$ and $\\tilde\\Delta_{A}$ are $\\cinf$-bounded and uniformly elliptic, with the bundles of differential forms which are of bounded geometry (cf. \\cite{Sh}). Examples of such manifolds are given by the universal coverings of K\\\"ahler hyperbolic manifolds \\cite{G}. Then, by \\cite[Proposition 1.1]{Sh} the domains of the above operators coincide with the Sobolev spaces\n \\[\n \\D(\\Delta_{\\delbar,sw})=\\D(\\Delta_{\\del,sw})=\\D(\\Delta_{d,sw})=W^2_2(M,\\Lambda^\\bullet_\\C)\n \\]\n and\n \\[\n \\D(\\tilde\\Delta_{A,sw})=\\D(\\tilde\\Delta_{BC,sw})=W^4_2(M,\\Lambda^\\bullet_\\C),\n \\]\n with the notation for Sobolev spaces as in \\cite{Sh}. Therefore, since the image of each Green operator is one of the above domains, and thus the image is a Sobolev space, in the statement of Theorem \\ref{theorem l2 del delbar lemma} we can choose $\\beta\\in W^2_2(M,\\Lambda^{k-2}_\\C)$ and $\\gamma,\\zeta,\\eta,\\theta,\\lambda\\in W^1_2(M,\\Lambda^{k-1}_\\C)$. In particular, $\\del\\beta,\\delbar\\beta\\in W^1_2(M,\\Lambda^{k-1}_\\C)\\subseteq L^2\\Lambda_\\C^{k-1}$.\n\\end{remark}", "post_theorem_intro_text_len": 6129, "post_theorem_intro_text": "The proof of Theorem \\ref{theorem intro l2 del delbar lemma} relies primarily on new spectral properties of some self-adjoint operators introduced in \\cite{HP} as part of the development of the Aeppli and Bott-Chern versions of the $L^2$-Hodge theory of general Hermitian manifolds and of complete K\\\"ahler manifolds. More precisely, we prove that if $\\Delta$ has a spectral gap, there are self-adjoint extensions of fourth-order elliptic Aeppli and Bott-Chern Laplacians having a spectral gap, thereby allowing one to use arguments based on elliptic regularity. Since this constitutes the core of the proof, we now discuss these Aeppli and Bott–Chern Laplacians in more detail.\n\nGiven a compact Hermitian manifold, there is a unique choice for the second-order Hodge, $\\partial$- and Dolbeault Laplacians $\\Delta_D=DD^*+D^*D$, for $D\\in\\{d,\\partial,\\delbar\\}$, such that the corresponding kernels are isomorphic to de Rham, $\\partial$- and Dolbeault cohomology, via classical Hodge theory. In the Aeppli and Bott-Chern cases the situation is slightly different: there are multiple possible choices for the associated Laplacians. We recall the definitions of Aeppli and Bott-Chern cohomology via the related differential complex. We denote the spaces of $(p,q)$-forms by $A^{p,q}$, so that $A^k_\\C=\\oplus_{p+q=k}A^{p,q}$, and the Hilbert space of $L^2$-$(p,q)$-forms by $L^2\\Lambda^{p,q}$. For any choice of integers $(p,q)$ we consider the complex\n\\[\n\\dots\\longrightarrow A^{p-1,q-2}\\oplus A^{p-2,q-1}\\overset{\\delbar\\oplus\\partial}{\\longrightarrow} A^{p-1,q-1}\\overset{\\partial\\delbar}{\\longrightarrow} A^{p,q}\\overset{\\partial+\\delbar}{\\longrightarrow} A^{p+1,q}\\oplus A^{p,q+1}{\\longrightarrow}\\dots\n\\]\nwhere $\\delbar\\oplus\\partial$ operates on $A^{p-1,q-2}\\oplus A^{p-2,q-1}$ as $\\delbar$ on $A^{p-1,q-2}$ plus $\\partial$ on $A^{p-2,q-1}$. The Aeppli and Bott-Chern cohomology spaces are defined as\n\\begin{align*}\nH^{p-1,q-1}_A:=\\frac{\\ker\\partial\\delbar\\cap A^{p-1,q-1}}{\\im\\delbar\\oplus\\partial},&&H^{p,q}_{BC}:=\\frac{\\ker(\\partial+\\delbar)\\cap A^{p,q}}{\\im\\partial\\delbar}.\n\\end{align*}\nThe \\lq\\lq natural\" Aeppli and Bott-Chern Laplacians are then defined as\n\\begin{align*}\n\\Delta_A&:=\\delbar^*\\partial^*\\partial\\delbar+(\\delbar\\oplus\\partial)(\\delbar\\oplus\\partial)^*=\\delbar^*\\partial^*\\partial\\delbar+\\partial\\partial^*+\\delbar\\delbar^*,\\\\\n\\Delta_{BC}&:=\\partial\\delbar\\delbar^*\\partial^*+(\\partial+\\delbar)^*(\\partial+\\delbar)=\\partial\\delbar\\delbar^*\\partial^*+\\partial^*\\partial+\\delbar^*\\delbar.\n\\end{align*}\nThe kernels of these operators are isomorphic to the Aeppli and Bott-Chern cohomology spaces, but they are not elliptic \\cite[Proposition 2.1]{S}. The first elliptic fourth-order operators whose kernels were shown to be isomorphic to the Aeppli and Bott-Chern cohomologies were defined in \\cite{KS} as\n\\begin{align*}\n\\tilde\\Delta_{A} &:=\n\\partial\\delbar\\delbar^*\\partial^*+\n\\delbar^*\\partial^*\\partial\\delbar+\n\\partial\\delbar^*\\delbar\\partial^*+\\delbar\\partial^*\\partial\\delbar^*+\n\\partial\\partial^*+\\delbar\\delbar^*,\\\\\n\\tilde\\Delta_{BC} &:=\n\\partial\\delbar\\delbar^*\\partial^*+\n\\delbar^*\\partial^*\\partial\\delbar+\\partial^*\\delbar\\delbar^*\\partial+\\delbar^*\\partial\\partial^*\\delbar\n+\\partial^*\\partial+\\delbar^*\\delbar.\n\\end{align*}\nWe will refer to these operators as the Kodaira-Spencer Laplacians. Another pair of elliptic fourth-order operators whose kernels are isomorphic to the Aeppli and Bott-Chern cohomologies were defined in \\cite{V} as\n\\begin{align*}\n\\square_{A}&:=\\delbar^*\\partial^*\\partial\\delbar+((\\delbar\\oplus\\partial)(\\delbar\\oplus\\partial)^*)^2=\\delbar^*\\partial^*\\partial\\delbar+(\\partial\\partial^*+\\delbar\\delbar^*)^2,\\\\\n\\square_{BC}&:=\\partial\\delbar\\delbar^*\\partial^*+((\\partial+\\delbar)^*(\\partial+\\delbar))^2=\\partial\\delbar\\delbar^*\\partial^*+(\\partial^*\\partial+\\delbar^*\\delbar)^2.\n\\end{align*}\nWe will refer to these operators as the Varouchas Laplacians.\n\nIn \\cite[Corollary 8.12]{HP} it was proven that on a complete K\\\"ahler manifold, if $\\Delta$ has a spectral gap in $L^2\\Lambda^\\bullet_\\C$, then there are self-adjoint extensions of $\\Delta_A$ and $\\Delta_{BC}$ having a spectral gap. Being $\\Delta_A$ and $\\Delta_{BC}$ non-elliptic, this was not sufficient to prove Theorem \\ref{theorem intro l2 del delbar lemma}. In fact, to prove our main result, we use the spectral gap of the elliptic Kodaira-Spencer Laplacians, which is shown in Theorem \\ref{theorem spectral gap kodaira spencer laplacians}. We also prove a spectral gap of the Varouchas Laplacians in Theorem \\ref{theorem spectral gap varouchas laplacians}. We remark that the spectral gap assumption of $\\Delta$ is essential for a statement which directly generalises the classical $\\partial\\delbar$-Lemma such as Theorem \\ref{theorem intro l2 del delbar lemma} does. Indeed, in the absence of a spectral gap, the images of the closed extensions of $d$, $\\partial$, $\\delbar$, and $\\partial\\delbar$ fail to be closed. For a weaker statement on complete K\\\"ahler manifolds without assuming a spectral gap, see Theorem \\ref{theorem reduced l2 del delbar lemma} (cf. \\cite[Corollary 8.6]{HP}).\n\nThe paper is organised as follows. In Section \\ref{section preliminaries}, we present preliminaries on Hilbert complexes, self-adjoint operators, spectral gaps, minimal and maximal closed extensions of differential operators, differential complexes on complex manifolds, self-adjoint extensions of second and fourth-order complex Laplacians, $L^2$ Hodge theory on complete K\\\"ahler manifolds. In Section \\ref{section spectral gap}, we prove the spectral gaps of the elliptic Aeppli and Bott-Chern Laplacians. In Theorem \\ref{theorem closed image del delbar} we also present a new proof of the closure of the image of the closed extensions of the operator $\\partial\\delbar$, originally proved in \\cite[Theorem 8.10]{HP}. Finally, in section \\ref{section l2 del delbar lemma}, we prove our main result Theorem \\ref{theorem intro l2 del delbar lemma}. In Remark \\ref{remark bounded geometry} we also point out an improvement of Theorem \\ref{theorem intro l2 del delbar lemma} when the manifold is of bounded geometry.", "sketch": "The post-theorem introduction says that “The proof of Theorem \\ref{theorem intro l2 del delbar lemma} relies primarily on new spectral properties of some self-adjoint operators introduced in \\cite{HP}.” More precisely, they “prove that if $\\Delta$ has a spectral gap, there are self-adjoint extensions of fourth-order elliptic Aeppli and Bott-Chern Laplacians having a spectral gap, thereby allowing one to use arguments based on elliptic regularity,” and this is described as “the core of the proof.”\n\nBecause the “natural” Aeppli and Bott-Chern Laplacians $\\Delta_A,\\Delta_{BC}$ have kernels matching Aeppli/Bott--Chern cohomology but “are not elliptic,” the text explains that the spectral gap for self-adjoint extensions of $\\Delta_A,\\Delta_{BC}$ from \\cite[Corollary 8.12]{HP} “was not sufficient to prove Theorem \\ref{theorem intro l2 del delbar lemma}.” Instead, “to prove our main result, we use the spectral gap of the elliptic Kodaira-Spencer Laplacians,” establishing it in “Theorem \\ref{theorem spectral gap kodaira spencer laplacians},” and they “also prove a spectral gap of the Varouchas Laplacians in Theorem \\ref{theorem spectral gap varouchas laplacians}.”\n\nFinally, the introduction remarks that the “spectral gap assumption of $\\Delta$ is essential” for a direct generalisation like Theorem \\ref{theorem intro l2 del delbar lemma}, since “in the absence of a spectral gap, the images of the closed extensions of $d$, $\\partial$, $\\delbar$, and $\\partial\\delbar$ fail to be closed,” and it points to a weaker variant without a spectral gap (Theorem \\ref{theorem reduced l2 del delbar lemma}).", "expanded_sketch": "The post-theorem introduction says that “The proof of the main theorem relies primarily on new spectral properties of some self-adjoint operators introduced in \\cite{HP}.” More precisely, they “prove that if $\\Delta$ has a spectral gap, there are self-adjoint extensions of fourth-order elliptic Aeppli and Bott-Chern Laplacians having a spectral gap, thereby allowing one to use arguments based on elliptic regularity,” and this is described as “the core of the proof.”\n\nBecause the “natural” Aeppli and Bott-Chern Laplacians $\\Delta_A,\\Delta_{BC}$ have kernels matching Aeppli/Bott--Chern cohomology but “are not elliptic,” the text explains that the spectral gap for self-adjoint extensions of $\\Delta_A,\\Delta_{BC}$ from \\cite[Corollary 8.12]{HP} “was not sufficient to prove the main theorem.” Instead, “to prove our main result, we use the spectral gap of the elliptic Kodaira-Spencer Laplacians,” establishing it in the following theorem.\n\n\\begin{theorem}\\label{theorem spectral gap kodaira spencer laplacians}\nLet $(M,g)$ be a complete K\\\"ahler manifold. If $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$, then for $b\\in\\{s,w\\}$ the operators $\\tilde\\Delta_{A,sb}$ and $\\tilde\\Delta_{BC,bw}$ have a spectral gap in $L^2\\Lambda^{p,q}$.\n\\end{theorem}\n\nThey “also prove a spectral gap of the Varouchas Laplacians” in the following theorem.\n\n\\begin{theorem}\\label{theorem spectral gap varouchas laplacians}\n    Let $(M,g)$ be a complete K\\\"ahler manifold. \n    If $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$, $L^2\\Lambda^{p-1,q}$ and $L^2\\Lambda^{p,q-1}$, then $\\square_{A,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$. If $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$, $L^2\\Lambda^{p+1,q}$ and $L^2\\Lambda^{p,q+1}$, then $\\square_{BC,sw}$ has a spectral gap in $L^2\\Lambda^{p,q}$.\n\\end{theorem}\n\nFinally, the introduction remarks that the “spectral gap assumption of $\\Delta$ is essential” for a direct generalisation like the main theorem, since “in the absence of a spectral gap, the images of the closed extensions of $d$, $\\partial$, $\\delbar$, and $\\partial\\delbar$ fail to be closed,” and it points to a weaker variant without a spectral gap given by the following theorem.\n\n\\begin{theorem}\\label{theorem reduced l2 del delbar lemma}\nGiven a complete K\\\"ahler manifold $(M,g)$, in both the spaces $L^2\\Lambda^k_\\C$ and $L^2\\Lambda^{p,q}$ the following equalities hold true\n\\begin{align*}\n\\c{\\im \\del\\delbar_s}\n&=\\c{\\im \\del_s}\\cap\\ker\\delbar_w=\\c{\\im \\delbar_s}\\cap\\ker\\del_w\\\\\n&=(\\c{\\im\\del_s}+\\c{\\im\\delbar_s})\\cap\\ker\\del_w\\cap\\ker\\delbar_w\\\\\n&=\\c{\\im d_s}\\cap\\ker\\del_w\\cap\\ker\\delbar_w.\n\\end{align*}\n\\end{theorem}", "expanded_theorem": "\\begin{theorem}\\label{theorem l2 del delbar lemma}\n    Let $(M,g)$ be a complete K\\\"ahler manifold such that $\\Delta_{\\delbar,sw}$ has a spectral gap in $L^2\\Lambda^{k}_\\C$. Given a smooth $L^2$-form $\\alpha\\in L^2A^k_\\C$ which satisfies $\\del\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n    \\begin{enumerate}\n        \\item $\\alpha=\\del\\delbar\\beta$, for $\\beta\\in L^2 A^{k-2}_\\C$;\n        \\item $\\alpha=\\del\\gamma$, for $\\gamma\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=\\del\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=d\\lambda$, for $\\lambda\\in L^2 A^{k-1}_\\C$.\n    \\end{enumerate}\n\\end{theorem}\\label{theorem intro l2 del delbar lemma}\n    Let $(M,g)$ be a complete K\\\"ahler manifold such that the self-adjoint Laplacian $\\Delta$ has a spectral gap in $L^2\\Lambda^{k}_\\C$. Given a smooth $L^2$-form $\\alpha\\in L^2A^k_\\C$ which satisfies $\\partial\\alpha=\\delbar\\alpha=0$, then the following conditions are equivalent:\n    \\begin{enumerate}\n        \\item $\\alpha=\\partial\\delbar\\beta$, for $\\beta\\in L^2 A^{k-2}_\\C$;\n        \\item $\\alpha=\\partial\\gamma$, for $\\gamma\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=\\delbar\\zeta$, for $\\zeta\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=\\partial\\eta+\\delbar\\theta$, for $\\eta,\\theta\\in L^2 A^{k-1}_\\C$;\n        \\item $\\alpha=d\\lambda$, for $\\lambda\\in L^2 A^{k-1}_\\C$.\n    \\end{enumerate},", "theorem_type": ["Biconditional or Equivalence", "Existence"], "mcq": {"question": "Let $(M,g)$ be a complete K\\\"ahler manifold. Write $L^2A^k_{\\C}$ for the space of smooth complex-valued $k$-forms on $M$ with finite $L^2$ norm, and write $d=\\partial+\\bar\\partial$. Assume that the self-adjoint Dolbeault Laplacian $\\Delta_{\\bar\\partial,sw}$ on $L^2\\Lambda^k_{\\C}$ has a spectral gap, meaning that its spectrum is contained in $\\{0\\}\\cup[C,+\\infty)$ for some $C>0$ (equivalently, on a complete K\\\"ahler manifold, the self-adjoint Hodge Laplacian $\\Delta$ has a spectral gap on $L^2\\Lambda^k_{\\C}$). Let $\\alpha\\in L^2A^k_{\\C}$ satisfy $\\partial\\alpha=\\bar\\partial\\alpha=0$. Which statement about the existence of $L^2$-forms representing $\\alpha$ holds?", "correct_choice": {"label": "A", "text": "The following five existence statements are equivalent: (1) there exists $\\beta\\in L^2A^{k-2}_{\\C}$ such that $\\alpha=\\partial\\bar\\partial\\beta$; (2) there exists $\\gamma\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\partial\\gamma$; (3) there exists $\\zeta\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\bar\\partial\\zeta$; (4) there exist $\\eta,\\theta\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\partial\\eta+\\bar\\partial\\theta$; and (5) there exists $\\lambda\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=d\\lambda$."}, "choices": [{"label": "B", "text": "The following five existence statements are equivalent: (1) there exists $\\beta\\in L^2A^{k-2}_{\\C}$ such that $\\alpha=\\partial\\bar\\partial\\beta$; (2) there exists $\\gamma\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\partial\\gamma$; (3) there exists $\\zeta\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\bar\\partial\\zeta$; (4) there exist $\\eta,\\theta\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\partial\\eta+\\bar\\partial\\theta$; and (5) there exists $\\lambda\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=d\\lambda$, and moreover one may always choose all of $\\beta,\\gamma,\\zeta,\\eta,\\theta,\\lambda$ to be smooth $L^2$-forms."}, {"label": "C", "text": "If there exists $\\beta\\in L^2A^{k-2}_{\\C}$ such that $\\alpha=\\partial\\bar\\partial\\beta$, then there exist $\\gamma,\\zeta,\\eta,\\theta,\\lambda\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\partial\\gamma=\\bar\\partial\\zeta=\\partial\\eta+\\bar\\partial\\theta=d\\lambda$."}, {"label": "D", "text": "The following five existence statements are equivalent: (1) there exists $\\beta\\in L^2A^{k-2}_{\\C}$ such that $\\alpha=\\partial\\bar\\partial\\beta$; (2) there exists $\\gamma\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\partial\\gamma$; (3) there exists $\\zeta\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\bar\\partial\\zeta$; (4) there exist $\\eta,\\theta\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\partial\\eta+\\bar\\partial\\theta$; and (5) there exists $\\lambda\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=d\\lambda$, even if one drops the spectral-gap assumption on $\\Delta_{\\bar\\partial,sw}$."}, {"label": "E", "text": "The following five existence statements are equivalent provided that $\\Delta_{\\bar\\partial,sw}$ has a spectral gap not only on $L^2\\Lambda^{k}_{\\C}$ but also on $L^2\\Lambda^{k-1}_{\\C}$: (1) there exists $\\beta\\in L^2A^{k-2}_{\\C}$ such that $\\alpha=\\partial\\bar\\partial\\beta$; (2) there exists $\\gamma\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\partial\\gamma$; (3) there exists $\\zeta\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\bar\\partial\\zeta$; (4) there exist $\\eta,\\theta\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=\\partial\\eta+\\bar\\partial\\theta$; and (5) there exists $\\lambda\\in L^2A^{k-1}_{\\C}$ such that $\\alpha=d\\lambda$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "smoothness_of_L2_primitives", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_reverse_implications_between_the_five_conditions", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "necessity_of_spectral_gap_for_closed_images_and_full_equivalence", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "extra_degreewise_spectral_gap_hypothesis", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct option. It states the hypotheses and asks which conclusion holds, but does not explicitly signal that the full five-way equivalence in choice A is the intended answer."}, "TAS": {"score": 1, "justification": "The item is still quite close to a theorem-statement recognition question: the correct choice is essentially the theorem's conclusion under the stated hypotheses. However, it is not a pure restatement because the alternatives vary the hypotheses, logical direction, and regularity claims."}, "GPS": {"score": 2, "justification": "To choose correctly, the solver must compare several nearby logical variants: full equivalence vs. one-way implication, necessity of the spectral-gap assumption, and whether extra smoothness or extra degreewise assumptions are justified. This creates real reasoning pressure rather than making the answer obvious."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically meaningful. They reflect common failure modes: overclaiming regularity, accepting only a weaker true implication, dropping a necessary hypothesis, or adding an unnecessary extra assumption."}, "total_score": 7, "overall_assessment": "A strong MCQ with little answer leakage and high-quality distractors. Its main weakness is that it remains close to theorem recall rather than fully independent generative problem solving."}}
{"id": "2602.08850v1", "paper_link": "http://arxiv.org/abs/2602.08850v1", "theorems_cnt": 2, "theorem": {"env_name": "alphatheorem", "content": "[cf.\\ \\cref{construction,uniqueness}, \\cref{COR}] \\label{thmA}\nLet $\\Sigma$ be a free boundary minimal hypersurface of Morse index $I$ in a Riemannian manifold $(M,g)$ with convex boundary. Then there exists an $I$-parameter family of ancient free boundary mean curvature flows emanating from $\\Sigma$. Moreover, any ancient solution with sufficiently fast decay to $\\Sigma$ belongs to this family.", "start_pos": 10379, "end_pos": 10816, "label": "thmA"}, "ref_dict": {"absorption": "\\begin{lemma}\\label{absorption}\nLet $u\\colon\\Sigma\\times(T_0, T)\\to \\R$ be a solution to the graphical mean curvature flow  \\eqref{gmcf2} with $\\norm{u(\\cdot,t)}_{C^1(\\Sigma)}<r_0$ for all $t\\in(T_0,T)$, then for any $t\\in (T_0+2, T)$\n\\begin{equation}\\label{gmcf-Schauder}\n\\|u\\|_{C^{2,\\a}(\\Sigma\\times(t-1,t))}\\le C(\\|u\\|_{L^2(\\Sigma\\times(t-2,t))}+\\|u\\|_{C^{1,\\alpha}(\\Sigma\\times(t-2,t))}\\|u\\|_{C^{2,\\alpha}(\\Sigma\\times(t-2,t))})\\,,\n\\end{equation}\nwhere $C$ depends only on $\\Sigma$, $M$ and $r_0$. \n\nMoreover, there exists an $\\varepsilon>0$, depending only on $\\Sigma$ and $M$, such that, if $\\|u\\|_{C^{1,\\a}(\\Sigma\\times(T-R, T))}<\\varepsilon$ for some $T-R\\ge T_0$, then \n\\begin{equation}\\label{gmcf-SchauderII}\n\\|u\\|_{C^{2,\\a}(\\Sigma\\times (T-R/2,T))}\\le C \\|u\\|_{L^2(\\Sigma\\times(T-R, T))} \\,,\n\\end{equation}\nwhere $C>0$ depends only on $\\Sigma$ and $M$. \n\\end{lemma}", "COR": "\\begin{corollary} \\label{COR} If $\\Sigma$ is nondegenerate, then \\cref{uniqueness} holds without the assumption \\eqref{tu:assume2}.\n\\end{corollary}", "thmA": "\\begin{alphatheorem}[cf.\\ \\cref{construction,uniqueness}, \\cref{COR}] \\label{thmA}\nLet $\\Sigma$ be a free boundary minimal hypersurface of Morse index $I$ in a Riemannian manifold $(M,g)$ with convex boundary. Then there exists an $I$-parameter family of ancient free boundary mean curvature flows emanating from $\\Sigma$. Moreover, any ancient solution with sufficiently fast decay to $\\Sigma$ belongs to this family.\n\\end{alphatheorem}", "uniqueness": "\\begin{theorem}\\label{uniqueness}\nLet $u\\colon\\Sigma\\times(-\\infty, 0]\\to\\R$ be a smooth solution to the free boundary mean curvature flow \\eqref{gmcf2}.\nThere exists a constant $\\varepsilon>0$, depending only on $\\Sigma$ and $M$\n, such that if \n\\begin{equation} \\label{tu:assume1}\n\\norm{u}_{C^{1,\\a}(\\Sigma\\times(-\\infty,0])} < \\varepsilon \\,,\n\\end{equation}\nand\n\\begin{equation} \\label{tu:assume2}\n\\liminf_{t\\to-\\infty}\\ \\abs{t} \\norm{u(\\cdot,t)}_{C^0(\\Sigma)} = 0 \\,,\n\\end{equation}\nthen, possibly after a time translation, $u$  coincides with one of the solutions constructed in \\cref{construction}.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 3457, "pre_theorem_intro_text": "Morse theory provides a powerful framework for studying the topology of a space through the analysis of differentiable functions defined on it. Remarkably, many of its central ideas extend to infinite-dimensional settings, although their implementation requires substantially more delicate analysis. Foundational contributions in this direction include \\cite{Smale1961,Palais1963,Chang1993,AbbondandoloMajer2001}.\nMore recently, Morse-theoretic methods have played an important role in the study of geometric variational problems, and in particular of the area functional on submanifolds of a fixed ambient manifold; see for example \\cite{Tromba1977,Pitts1981,White1991,MarquesNeves2014,ChenGaspar2025}.\n\nIn this paper, we focus on Morse theory for the area functional restricted to \\emph{smooth} submanifolds.\nThe gradient flow of this functional is the mean curvature flow, which can be described analytically as a one-parameter family of submanifolds $\\Sigma_t$ in a Riemannian ambient manifold $(M,g)$ evolving by\n\\[\n\\frac{\\partial X}{\\partial t} = \\vec H(X,t)\\,, \\quad X \\in \\Sigma_t \\,,\n\\]\nwhere $\\vec H(X,t)$ denotes the mean curvature vector of $\\Sigma_t$ at $X$.\nThis flow arises naturally in physics as a model for evolutionary processes governed by surface tension, such as the evolution of grain boundaries in annealing metals \\cite{Mullins1956}. From a mathematical point of view, mean curvature flow was first studied systematically by Brakke \\cite{Brakke1978} using geometric measure theory, and later by Huisken \\cite{Huisken1984} via a more classical PDE approach. Since then, the subject has developed into a very active area of research.\n\nHere, we are interested in the case of mean curvature flow of hypersurfaces in the presence of a boundary. The natural Neumann boundary value problem for mean curvature flow, known as the \\emph{free boundary problem}, prescribes that the evolving hypersurface has boundary constrained to move on a fixed barrier hypersurface and meets it orthogonally. This problem was introduced by Huisken \\cite{Huisken1989} in the non-parametric setting and further developed by Stahl \\cite{Stahl1996Convergence,Stahl1996Regularity}, and later by \\cite{Freire2010, Wheeler2014,Edelen2016,HirschLi2023,LangfordZhu2023}.\n\nA fundamental role in free boundary mean curvature flow is played by \\emph{free boundary minimal hypersurfaces}, which arise as stationary solutions of the flow equation. Recently, these objects have been the subject of intense research activity, due to fascinating new existence results, such as \\cite{FraserSchoen2016,FranzSchulz2026,KarpukhinKusnerMcGrathStern2024}.\nMost of these recent examples consist of \\emph{unstable} critical points of the area functional. Therefore, by analogy with classical Morse theory, one expects the existence of flow lines emanating from these free boundary minimal hypersurfaces. More precisely, one expects \\emph{ancient solutions} of mean curvature flow, namely solutions that have existed for all times in the past, which backward converge to unstable free boundary minimal hypersurfaces.\n\nMorse-theoretic considerations also suggest that the dimension of the space of ancient solutions backward converging to $\\Sigma$ is equal to the Morse index of $\\Sigma$, i.e., the dimension of the negative directions of the area functional at $\\Sigma$ at second order.\nOur first main result makes this heuristic precise and may be summarized informally as follows.", "context": "Morse theory provides a powerful framework for studying the topology of a space through the analysis of differentiable functions defined on it. Remarkably, many of its central ideas extend to infinite-dimensional settings, although their implementation requires substantially more delicate analysis. Foundational contributions in this direction include \\cite{Smale1961,Palais1963,Chang1993,AbbondandoloMajer2001}.\nMore recently, Morse-theoretic methods have played an important role in the study of geometric variational problems, and in particular of the area functional on submanifolds of a fixed ambient manifold; see for example \\cite{Tromba1977,Pitts1981,White1991,MarquesNeves2014,ChenGaspar2025}.\n\nIn this paper, we focus on Morse theory for the area functional restricted to \\emph{smooth} submanifolds.\nThe gradient flow of this functional is the mean curvature flow, which can be described analytically as a one-parameter family of submanifolds $\\Sigma_t$ in a Riemannian ambient manifold $(M,g)$ evolving by\n\\[\n\\frac{\\partial X}{\\partial t} = \\vec H(X,t)\\,, \\quad X \\in \\Sigma_t \\,,\n\\]\nwhere $\\vec H(X,t)$ denotes the mean curvature vector of $\\Sigma_t$ at $X$.\nThis flow arises naturally in physics as a model for evolutionary processes governed by surface tension, such as the evolution of grain boundaries in annealing metals \\cite{Mullins1956}. From a mathematical point of view, mean curvature flow was first studied systematically by Brakke \\cite{Brakke1978} using geometric measure theory, and later by Huisken \\cite{Huisken1984} via a more classical PDE approach. Since then, the subject has developed into a very active area of research.\n\nHere, we are interested in the case of mean curvature flow of hypersurfaces in the presence of a boundary. The natural Neumann boundary value problem for mean curvature flow, known as the \\emph{free boundary problem}, prescribes that the evolving hypersurface has boundary constrained to move on a fixed barrier hypersurface and meets it orthogonally. This problem was introduced by Huisken \\cite{Huisken1989} in the non-parametric setting and further developed by Stahl \\cite{Stahl1996Convergence,Stahl1996Regularity}, and later by \\cite{Freire2010, Wheeler2014,Edelen2016,HirschLi2023,LangfordZhu2023}.\n\nA fundamental role in free boundary mean curvature flow is played by \\emph{free boundary minimal hypersurfaces}, which arise as stationary solutions of the flow equation. Recently, these objects have been the subject of intense research activity, due to fascinating new existence results, such as \\cite{FraserSchoen2016,FranzSchulz2026,KarpukhinKusnerMcGrathStern2024}.\nMost of these recent examples consist of \\emph{unstable} critical points of the area functional. Therefore, by analogy with classical Morse theory, one expects the existence of flow lines emanating from these free boundary minimal hypersurfaces. More precisely, one expects \\emph{ancient solutions} of mean curvature flow, namely solutions that have existed for all times in the past, which backward converge to unstable free boundary minimal hypersurfaces.\n\nMorse-theoretic considerations also suggest that the dimension of the space of ancient solutions backward converging to $\\Sigma$ is equal to the Morse index of $\\Sigma$, i.e., the dimension of the negative directions of the area functional at $\\Sigma$ at second order.\nOur first main result makes this heuristic precise and may be summarized informally as follows.\n\n\\begin{corollary} \\label{COR} If $\\Sigma$ is nondegenerate, then \\cref{uniqueness} holds without the assumption \\eqref{tu:assume2}.\n\\end{corollary}", "full_context": "Morse theory provides a powerful framework for studying the topology of a space through the analysis of differentiable functions defined on it. Remarkably, many of its central ideas extend to infinite-dimensional settings, although their implementation requires substantially more delicate analysis. Foundational contributions in this direction include \\cite{Smale1961,Palais1963,Chang1993,AbbondandoloMajer2001}.\nMore recently, Morse-theoretic methods have played an important role in the study of geometric variational problems, and in particular of the area functional on submanifolds of a fixed ambient manifold; see for example \\cite{Tromba1977,Pitts1981,White1991,MarquesNeves2014,ChenGaspar2025}.\n\nIn this paper, we focus on Morse theory for the area functional restricted to \\emph{smooth} submanifolds.\nThe gradient flow of this functional is the mean curvature flow, which can be described analytically as a one-parameter family of submanifolds $\\Sigma_t$ in a Riemannian ambient manifold $(M,g)$ evolving by\n\\[\n\\frac{\\partial X}{\\partial t} = \\vec H(X,t)\\,, \\quad X \\in \\Sigma_t \\,,\n\\]\nwhere $\\vec H(X,t)$ denotes the mean curvature vector of $\\Sigma_t$ at $X$.\nThis flow arises naturally in physics as a model for evolutionary processes governed by surface tension, such as the evolution of grain boundaries in annealing metals \\cite{Mullins1956}. From a mathematical point of view, mean curvature flow was first studied systematically by Brakke \\cite{Brakke1978} using geometric measure theory, and later by Huisken \\cite{Huisken1984} via a more classical PDE approach. Since then, the subject has developed into a very active area of research.\n\nHere, we are interested in the case of mean curvature flow of hypersurfaces in the presence of a boundary. The natural Neumann boundary value problem for mean curvature flow, known as the \\emph{free boundary problem}, prescribes that the evolving hypersurface has boundary constrained to move on a fixed barrier hypersurface and meets it orthogonally. This problem was introduced by Huisken \\cite{Huisken1989} in the non-parametric setting and further developed by Stahl \\cite{Stahl1996Convergence,Stahl1996Regularity}, and later by \\cite{Freire2010, Wheeler2014,Edelen2016,HirschLi2023,LangfordZhu2023}.\n\nA fundamental role in free boundary mean curvature flow is played by \\emph{free boundary minimal hypersurfaces}, which arise as stationary solutions of the flow equation. Recently, these objects have been the subject of intense research activity, due to fascinating new existence results, such as \\cite{FraserSchoen2016,FranzSchulz2026,KarpukhinKusnerMcGrathStern2024}.\nMost of these recent examples consist of \\emph{unstable} critical points of the area functional. Therefore, by analogy with classical Morse theory, one expects the existence of flow lines emanating from these free boundary minimal hypersurfaces. More precisely, one expects \\emph{ancient solutions} of mean curvature flow, namely solutions that have existed for all times in the past, which backward converge to unstable free boundary minimal hypersurfaces.\n\nMorse-theoretic considerations also suggest that the dimension of the space of ancient solutions backward converging to $\\Sigma$ is equal to the Morse index of $\\Sigma$, i.e., the dimension of the negative directions of the area functional at $\\Sigma$ at second order.\nOur first main result makes this heuristic precise and may be summarized informally as follows.\n\n\\begin{corollary} \\label{COR} If $\\Sigma$ is nondegenerate, then \\cref{uniqueness} holds without the assumption \\eqref{tu:assume2}.\n\\end{corollary}\n\nWhile such behavior is conjectured by analogy with classical Morse theory, the analysis of ancient solutions to free boundary mean curvature flow is far from trivial and was largely unexplored, due to the presence of the boundary interacting with the infinite dimensional nature of the problem and the potential degeneracy of the critical points. With the exception of recent classification results for convex solutions in dimension one and in highly symmetric higher-dimensional settings \\cite{BourniLangford2023,BourniLangford2025,BourniBurnsCatron2025}, there has been essentially no systematic study of ancient solutions.\nThe present work constitutes, to the best of our knowledge, the first classification theory for general higher-dimension ancient solutions to the free boundary mean curvature flow.\n\n\\begin{alphatheorem}[cf.\\ \\cref{MCconstruction,COR}] \\label{thmB}\nLet $\\Sigma$ be an unstable free boundary minimal hypersurface in a Riemannian manifold $(M,g)$ with convex boundary. Then there exists a mean-convex ancient solution emanating from $\\Sigma$, converging exponentially fast at a rate determined by the first eigenvalue. Moreover, if $\\Sigma$ is nondegenerate, this solution is unique, up to time translation, among mean-convex ancient solutions converging to $\\Sigma$ in the $C^{1,\\alpha}$-topology.\n\\end{alphatheorem}\n\nFor the uniqueness statement of \\cref{thmA} (cf.\\ \\cref{uniqueness}), we require that the ancient solution can be expressed as a graph over a free boundary minimal hypersurface. Moreover, we assume the graph to have small parabolic $C^{1,\\a}$-norm and to converge to zero sublinearly in the $C^0$-norm. When the underlying minimal hypersurface is nondegenerate, the sublinear decay assumption can be dropped.\nWe conjecture that the sublinear decay assumption can be dropped in the mean-convex case as well, namely there exists a unique ancient solution converging in $C^{1,\\a}$ to any given unstable (free boundary) minimal hypersurface.\n\n\\begin{theorem}\\label{uniqueness}\nLet $u\\colon\\Sigma\\times(-\\infty, 0]\\to\\R$ be a smooth solution to the free boundary mean curvature flow \\eqref{gmcf2}.\nThere exists a constant $\\varepsilon>0$, depending only on $\\Sigma$ and $M$\n, such that if \n\\begin{equation} \\label{tu:assume1}\n\\norm{u}_{C^{1,\\a}(\\Sigma\\times(-\\infty,0])} < \\varepsilon \\,,\n\\end{equation}\nand\n\\begin{equation} \\label{tu:assume2}\n\\liminf_{t\\to-\\infty}\\ \\abs{t} \\norm{u(\\cdot,t)}_{C^0(\\Sigma)} = 0 \\,,\n\\end{equation}\nthen, possibly after a time translation, $u$ coincides with one of the solutions constructed in \\cref{construction}.\n\\end{theorem}\n\nThe idea of the proposition is that, if $\\Sigma$ is minimal, we expect the existence of an ancient mean curvature flow whose graph over $\\Sigma$, at first order, looks like $\\varphi_k e^{-\\lambda_kt}$ for $t\\in(-\\infty,T)$, with mean curvature, at first order, being $-\\lambda_k\\varphi_k e^{-\\lambda_kt}$. Our proposition does not directly give this ancient solution, but it adds a correction $u$ to the first order approximation to satisfy the orthogonality condition at the boundary. Indeed, notice that a priori the graph of $\\varphi_k e^{-\\lambda_kt}$ over $\\Sigma$ is not necessarily orthogonal to the boundary of~$M$.\n\\end{remark}\n\\begin{remark}\nAssume that $\\Sigma$ is an unstable free boundary minimal hypersurface.\nNote that $\\varphi_k$ changes sign for all $k>1$, while $\\varphi_1>0$. Therefore, as observed in \\cite[Section~2.2]{HaslhoferKetover2025}, the graph of $\\varphi_1 e^{-\\l_1 t}$ is a mean-convex foliation of a neighborhood of $\\Sigma$. However, this is not necessarily free boundary, as remarked above. In \\cref{prop:SurfKthEigenf}, we find a lower order correction such that the hypersurfaces $\\Sigma_t$ form a smooth \\emph{free boundary} mean-convex foliation of a neighborhood of $\\Sigma$.\n\\end{remark}\n\\begin{remark}\\label{w-norm}\nThe function $w$ constructed in \\cref{prop:SurfKthEigenf} satisfies \n\\[\n\\|w(\\cdot, t)\\|_{C^{2,\\a}(\\Sigma)}\\le Ce^{-\\l_k t}\\quad\\forall t\\in (-\\infty, T)\\,.\n\\]\nIndeed, since $J_\\Sigma(w)+ E(w)=-\\l_k\\phi_k e^{-\\l_k t}-\\rho u$, the elliptic Schauder estimates, together with \\cref{E-estimates}, yield \n\\[\n\\begin{split}\n\\|w\\|_{C^{2,\\a}(\\Sigma)}&\\le C(\\|w\\|_{C^0(\\Sigma)}+\\|E(w)+\\l_k\\phi_ke^{-\\l_k t}+\\rho u\\|_{C^{0,\\a}(\\Sigma)}+\\|\\e(w)\\|_{C^{1,\\a}(\\partial\\Sigma)})\\\\\n&\\le C(\\|w\\|_{C^0(\\Sigma)}+\\|w\\|_{C^{1,\\a}(\\Sigma)}\\|w\\|_{C^{2,\\a}(\\Sigma)}+e^{-\\l_k t}+\\rho\\| u\\|_{C^{0,\\a}(\\Sigma)})\\\\\n&\\le C(e^{-\\l_k t} + (\\|w\\|_{C^{1,\\a}(\\Sigma)} + \\rho)\\|w\\|_{C^{2,\\a}(\\Sigma)}) \\,.\n\\end{split}\n\\]\nMoreover, recall that $\\norm{w}_{C^{2,\\a}(\\Sigma)}\\le C e^{-\\l_kt} + \\norm{u}_{C^{2,\\a}(\\Sigma)} < C e^{-\\l_kt} + \\delta$. As a result, for $\\d>0$ and $\\rho>0$ sufficiently small, and for time $t\\in(-\\infty,T)$ sufficiently small, we can absorb the term $(\\|w\\|_{C^{1,\\a}(\\Sigma)} + \\rho)\\|w\\|_{C^{2,\\a}(\\Sigma)}$ on the left-hand side and obtain the result.\n\\end{remark}\n\\begin{proof}\nObserve that $\\rho\\in(\\l_I,0)$ is not an eigenvalue of the Jacobi operator on $\\Sigma$, namely the following problem does not admit nontrivial solutions \n\\[\n\\begin{cases}\n-\\jac_\\Sigma v = \\rho v & \\text{on $\\Sigma$}\\\\\n\\frac{\\partial v}{\\partial \\eta} = \\II^{\\partial M} (\\nu,\\nu) v & \\text{on $\\partial \\Sigma$} \\,.\n\\end{cases}\n\\]\n\nFrom now on we assume that $\\Sigma$ is a free boundary minimal hypersurface with index $I>0$. \nWe construct sub- and supersolutions to mean curvature flow that are close to the family $\\{w(\\cdot,t)\\}_t$ constructed in \\cref{prop:SurfKthEigenf} for the case $k=1$, associated to the first eigenfunction $\\phi_1$. Note that this $w$ satisfies\n\\begin{equation}\\label{wt}\n\\begin{cases}\n\\displaystyle w_t= -\\l_1\\phi_1 e^{-\\l_1t}+ u_t= \\frac{H_{w}}{\\langle \\nu_w, \\overline\\nu\\rangle}+\\rho u+ u_t & \\text{on $\\Sigma$}\\\\\n\\sk{\\nu_w}{\\bar\\nu} = 0 & \\text{on $\\partial\\Sigma$} \\,,\n\\end{cases}\n\\end{equation}\nand therefore, comparing with \\eqref{gmcf}, the evolving hypersurfaces $\\Sigma_{w(\\cdot, t)}$ ``almost\" satisfy mean curvature flow.\nLet us now construct sub- and supersolutions by slightly reparametrizing the time.\n\\begin{lemma}\\label{sub/super} Given $\\l\\in(\\l_I, 0)$, we define the families\n\\[\nw^\\pm(x,t)= w(x, t\\pm e^{-\\l t})\\,,\n\\]\nwhere $\\{w(\\cdot,t)\\}_t$ is the function constructed in the \\cref{prop:SurfKthEigenf} for the case $k=1$, associated to the eigenfunction $\\phi_1>0$.\nThen there exists $t_\\l\\in\\R$ such that $w^+$ is a supersolution and $w^-$ is a subsolution of mean curvature flow for all $t<t_\\l$, that is\n\\[\n\\frac{d w^+}{dt}\\ge \\frac{H_{w^+}}{\\langle \\nu_{w^+}, \\overline\\nu\\rangle}\\quad \\text{ and }\\quad \\frac{d w^-}{dt}\\le \\frac{H_{w^-}}{\\langle \\nu_{w^-}, \\overline\\nu\\rangle}\\,.\n\\]\n\\end{lemma}\n\\begin{remark}\\label{sub/sup rmk} Note that \\cref{sub/super} is also true for $\\phi_1<0$, reversing the roles of $w^+$ and $w^-$.\n\\end{remark}\n\\begin{proof}\nLet $w_f(x,t)= w(x, t+f(t))$. Then, using \\eqref{wt},\n\\[\n\\begin{split}\n\\frac{d w_f}{dt}= \\frac{\\partial w}{\\partial t} (1+f'(t))= \\left(\\frac{H_{w_f}}{\\langle \\nu_{w_f}, \\overline\\nu\\rangle}+\\rho u+\\frac{\\partial u}{\\partial t}\\right)(1+f'(t))\\,,\n\\end{split}\n\\]\nwhich implies\n\\begin{equation}\\label{w-f}\n\\frac{d w_f}{dt}-\\frac{H_{w_f}}{\\langle \\nu_{w_f}, \\overline\\nu\\rangle}=-\\l_1\\phi_1 e^{-\\l_1 (t+f(t))} f'(t)+ \\left(\\rho u+\\frac{\\partial u}{\\partial t}\\right)(1+f'(t))\\,.\n\\end{equation}\nRecall that, by \\cref{prop:SurfKthEigenf}, $|u|+|u_t|\\le Ce^{-2\\l_1 t}$ and $\\phi_1>0$. Therefore, for any $\\l\\in (\\l_1, 0)$, if we choose $f(t)= \\pm e^{-\\l t}$, we obtain that for \n all $t$ sufficiently small the right-hand side of \\eqref{w-f} is positive and negative respectively. \nHence, the result follows for $w^\\pm=w_{\\pm e^{-\\l t}}$.\n\\end{proof}", "post_theorem_intro_text_len": 7911, "post_theorem_intro_text": "While such behavior is conjectured by analogy with classical Morse theory, the analysis of ancient solutions to free boundary mean curvature flow is far from trivial and was largely unexplored, due to the presence of the boundary interacting with the infinite dimensional nature of the problem and the potential degeneracy of the critical points. With the exception of recent classification results for convex solutions in dimension one and in highly symmetric higher-dimensional settings \\cite{BourniLangford2023,BourniLangford2025,BourniBurnsCatron2025}, there has been essentially no systematic study of ancient solutions.\nThe present work constitutes, to the best of our knowledge, the first  classification theory for general higher-dimension ancient solutions to the free boundary mean curvature flow. \n\nAnalogous classification results have been obtained in the boundaryless case by Choi--Mantoulidis \\cite{ChoiMantoulidis2022} and extended to the noncompact case by Choi--Huang--Lee \\cite{ChoiHuangLee2025}. Earlier results of this type have been known to be true for nonlinear parabolic PDEs in various settings (see e.g.\\ \\cite{Lunardi1995}).\nOur results are inspired by these papers, primarily by \\cite{ChoiMantoulidis2022}.\nHowever, compared to the boundaryless case, the presence of a free boundary introduces substantial analytical difficulties, especially because the boundary condition does not lead to a linear boundary term. The underlying parabolic PDE becomes significantly more involved, the associated stability operator requires a much more delicate analysis, and the linearization procedures are considerably more intricate due to the interaction between interior geometry and boundary behavior.\n\nWe also note that deforming a free boundary minimal hypersurface by its first (or any) eigenfunction does not preserve the orthogonality condition at the boundary. The ``correct'' deformation is far from trivial and requires more delicate handling. We achieve this via the implicit function theorem and show that one can indeed deform to first order by  keeping the orthogonality at the boundary. In fact, we construct a smooth free boundary mean-convex foliation around unstable free boundary minimal hypersurfaces. We then use this foliation to obtain a different construction of mean-convex ancient solutions that also provides a more detailed geometric description. \n\n\\begin{alphatheorem}[cf.\\ \\cref{MCconstruction,COR}] \\label{thmB}\nLet $\\Sigma$ be an unstable free boundary minimal hypersurface in a Riemannian manifold $(M,g)$ with convex boundary. Then there exists a mean-convex ancient solution emanating from $\\Sigma$, converging exponentially fast at a rate determined by the first eigenvalue. Moreover, if $\\Sigma$ is nondegenerate, this solution is unique, up to time translation, among mean-convex ancient solutions converging to $\\Sigma$ in the $C^{1,\\alpha}$-topology.\n\\end{alphatheorem}\n\n\\subsection{Further literature}\n\nAs described above, ancient solutions naturally appear when taking a Morse-theoretic perspective on (free boundary) minimal hypersurfaces, and they provide guidance for variational problems (see e.g.\\ \\cite[Section~1.1]{ChuLi2024} for a discussion related to min-max theory).\nHistorically, they were first studied because they play a central role in the analysis of singularity formation \\cite{Hamilton1994}, and they have an intrinsic geometric interest due to their strong rigidity and symmetry properties (see, for example, \\cite{BourniLangfordTinaglia2022} and the references therein).\n\nIn the boundaryless setting, extensive classification results are known in the convex regime. Under assumptions such as uniform convexity, bounded eccentricity, type-I curvature decay, or bounded isoperimetric ratio, the only compact convex ancient solutions are shrinking spheres \\cite{HuiskenSinestrari2015}; see also \\cite{DaskalopoulosHamiltonSesum2010, HaslhoferHershkovits2016, Langford2017}. When the ambient space is the sphere, the only geodesically convex ancient solutions are shrinking hemispheres \\cite{BryanLouie2016, HuiskenSinestrari2015}.\n\nIn Euclidean space, however, shrinking spheres are not the only compact convex ancient solutions. There exist families of solutions that contract to round points as $t \\to 0$ but become increasingly eccentric as $t \\to -\\infty$ \\cite{AngenentDaskalopoulosSesum2019, HaslhoferHershkovits2016, White2003}. These exhaust all compact convex ancient solutions that are non-collapsed, or equivalently entire \\cite{AngenentDaskalopoulosSesum2020, BourniLangfordLynch2023, BrendleNaff2024}. In the collapsed setting, a unique rotationally symmetric example is known \\cite{BourniLangfordTinaglia2021}, while without symmetry assumptions a rich family of examples already appears in dimension two \\cite{BourniLangfordTinaglia2022}.\n\nOutside the convex regime, classification becomes significantly more difficult. Under strong decay assumptions as $t \\to -\\infty$, ancient solutions converge backward to minimal hypersurfaces, motivating the problem of classifying ancient solutions emanating from minimal hypersurfaces from another perspective. \n\n\\subsection{Future directions}\n\nFor the uniqueness statement of \\cref{thmA} (cf.\\ \\cref{uniqueness}), we require that the ancient solution can be expressed as a graph over a free boundary minimal hypersurface. Moreover, we assume the graph to have small parabolic $C^{1,\\alpha}$-norm and to converge to zero sublinearly in the $C^0$-norm. When the underlying minimal hypersurface is nondegenerate, the sublinear decay assumption can be dropped.\nWe conjecture that the sublinear decay assumption can be dropped in the mean-convex case as well, namely there exists a unique ancient solution converging in $C^{1,\\alpha}$ to any given unstable (free boundary) minimal hypersurface. \n\nA natural further problem is to classify ancient solutions in a fixed ambient manifold without assuming backward convergence to a minimal hypersurface.\nIn this context, it is interesting to ask whether it is possible to construct solutions whose backward limit is singular, particularly in the mean-convex setting. Similarly, one may investigate whether ancient solutions with unbounded area can be mean convex. In an extreme scenario, it would be interesting to construct ancient solutions that ``fill up'' the ambient manifold.\n\n\\subsection{Plan of the paper} In \\cref{sec:definitions}, we recall basic definitions and we set the notation. In \\cref{sec:GraphsOverFBMS}, we obtain several elliptical and parabolic estimates for graphs over a free boundary minimal hypersurface. In \\cref{sec:construction}, we prove existence of an $I$-parameter family of ancient solutions emanating from a free boundary minimal hypersurface, by careful application of a fix point theorem.  In \\cref{sec:rigidity}, we obtain uniqueness of the constructed family of ancient solutions, by use of an ODE lemma due to Merle--Zaag \\cite{MerleZaag1998}, as adapted by \\cite{ChoiMantoulidis2022}. Finally, in \\cref{sec:meanconvex}, we provide a different more geometric proof of the existence of  mean-convex ancient solutions, by the use of barriers constructed via the implicit function theorem.\n\n\\subsection*{Acknowledgements}\nWe would like to thank Kyeongsu Choi and Christos Mantoulidis for clarifying certain points in their paper \\cite{ChoiMantoulidis2022}, especially for pointing out the use of the ``absorption lemma'' in the proof of \\cref{absorption} together with the Schauder estimates. \nWe would also like to thank Lucas Ambrozio for pointing us to his paper \\cite{Ambrozio2015}.\n\nT.\\,B.\\ was supported by NSF grant DMS-2405007, and G.\\,F.~was supported by NSF grant DMS-2405361. \nMoreover, part of this work was performed while the authors were in residence at the Simons Laufer Mathematical Sciences Institute (formerly MSRI) during the Fall 2024 semester, supported by NSF grant DMS-1928930.", "sketch": "A proof outline for Theorem~\\ref{thmA} is given in the “Plan of the paper”: in \\cref{sec:GraphsOverFBMS} the authors “obtain several elliptical and parabolic estimates for graphs over a free boundary minimal hypersurface.” Then, in \\cref{sec:construction}, they “prove existence of an $I$-parameter family of ancient solutions emanating from a free boundary minimal hypersurface, by careful application of a fix point theorem.” Next, in \\cref{sec:rigidity}, they “obtain uniqueness of the constructed family of ancient solutions, by use of an ODE lemma due to Merle--Zaag \\cite{MerleZaag1998}, as adapted by \\cite{ChoiMantoulidis2022}.” They also note (in connection with handling the free boundary condition) that the “\\`\\`correct'' deformation is far from trivial,” and that they “achieve this via the implicit function theorem” to “deform to first order by keeping the orthogonality at the boundary,” constructing “a smooth free boundary mean-convex foliation around unstable free boundary minimal hypersurfaces,” which is used for a “different more geometric proof” in \\cref{sec:meanconvex} via “barriers constructed via the implicit function theorem.”", "expanded_sketch": "A proof outline for \\begin{alphatheorem}[cf.\\ \\cref{construction,uniqueness}, \\cref{COR}] \\label{thmA}\nLet $\\Sigma$ be a free boundary minimal hypersurface of Morse index $I$ in a Riemannian manifold $(M,g)$ with convex boundary. Then there exists an $I$-parameter family of ancient free boundary mean curvature flows emanating from $\\Sigma$. Moreover, any ancient solution with sufficiently fast decay to $\\Sigma$ belongs to this family.\n\\end{alphatheorem} is given in the “Plan of the paper”: next the authors “obtain several elliptical and parabolic estimates for graphs over a free boundary minimal hypersurface.” Then they “prove existence of an $I$-parameter family of ancient solutions emanating from a free boundary minimal hypersurface, by careful application of a fix point theorem.” Next they “obtain uniqueness of the constructed family of ancient solutions, by use of an ODE lemma due to Merle--Zaag \\cite{MerleZaag1998}, as adapted by \\cite{ChoiMantoulidis2022}.” They also note (in connection with handling the free boundary condition) that the “\\`\\`correct'' deformation is far from trivial,” and that they “achieve this via the implicit function theorem” to “deform to first order by keeping the orthogonality at the boundary,” constructing “a smooth free boundary mean-convex foliation around unstable free boundary minimal hypersurfaces,” which is used for a “different more geometric proof” later via “barriers constructed via the implicit function theorem.”", "expanded_theorem": "[cf.\\ construction,uniqueness, \\begin{corollary} \\label{COR} If $\\Sigma$ is nondegenerate, then \\cref{uniqueness} holds without the assumption \\eqref{tu:assume2}.\n\\end{corollary}] \\label{thmA}\nLet $\\Sigma$ be a free boundary minimal hypersurface of Morse index $I$ in a Riemannian manifold $(M,g)$ with convex boundary. Then there exists an $I$-parameter family of ancient free boundary mean curvature flows emanating from $\\Sigma$. Moreover, any ancient solution with sufficiently fast decay to $\\Sigma$ belongs to this family.", "theorem_type": ["Existence", "Classification or Bijection"], "mcq": {"question": "Let $(M,g)$ be a Riemannian manifold whose boundary $\\partial M$ is convex, and let $\\Sigma$ be a free boundary minimal hypersurface in $M$ of Morse index $I$ (that is, the area functional has exactly $I$ negative directions at second order along $\\Sigma$). An ancient free boundary mean curvature flow is a one-parameter family of hypersurfaces $\\{\\Sigma_t\\}_{t< T}$, defined for all sufficiently negative times, evolving by mean curvature flow, with $\\partial \\Sigma_t \\subset \\partial M$ and meeting $\\partial M$ orthogonally; saying that such a flow emanates from $\\Sigma$ means it converges backward in time to $\\Sigma$ as $t\\to -\\infty$. Under these assumptions, which statement holds?", "correct_choice": {"label": "A", "text": "There exists an $I$-parameter family of ancient free boundary mean curvature flows emanating from $\\Sigma$. Moreover, any ancient solution that decays to $\\Sigma$ sufficiently fast belongs to this family."}, "choices": [{"label": "B", "text": "There exists a unique ancient free boundary mean curvature flow emanating from $\\Sigma$. Moreover, every ancient solution converging to $\\Sigma$ as $t\\to -\\infty$ coincides with it, up to time translation."}, {"label": "C", "text": "There exists an $I$-parameter family of ancient free boundary mean curvature flows emanating from $\\Sigma$."}, {"label": "D", "text": "There exists an $I$-parameter family of ancient free boundary mean curvature flows emanating from $\\Sigma$. Moreover, any ancient solution that converges to $\\Sigma$ as $t\\to -\\infty$ belongs to this family."}, {"label": "E", "text": "There exists an ancient free boundary mean curvature flow emanating from $\\Sigma$ if and only if $\\Sigma$ is nondegenerate; in that case the space of such ancient solutions is $I$-dimensional, and any sufficiently fast decaying ancient solution belongs to this family."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "index_parametrization_vs_uniqueness", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "classification_of_fast_decayers", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "sufficiently_fast_decay_requirement", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "nondegeneracy_needed_only_for_auxiliary_uniqueness_corollary_not_existence", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. Although it signals that the desired answer is a 'complete classification,' multiple options compete on that dimension, so the correct answer is not trivially leaked."}, "TAS": {"score": 0, "justification": "The item is essentially asking for the exact theorem statement. The correct choice is a near-verbatim restatement of the classification result rather than an application or inference from it."}, "GPS": {"score": 1, "justification": "There is some reasoning required to distinguish exact, weaker, and overly strong formulations, especially around the 'sufficiently fast decay' hypothesis and exact dimension I. However, success depends mainly on recalling the theorem statement rather than generating a substantial mathematical argument."}, "DQS": {"score": 2, "justification": "The distractors are strong: one is weaker but true, others overstate the theorem by dropping hypotheses or altering the conclusion, and they reflect realistic misreadings of a classification theorem."}, "total_score": 5, "overall_assessment": "A reasonably well-constructed theorem-recall MCQ with strong distractors and little answer leakage, but it is largely tautological and only moderately tests generative reasoning."}}
{"id": "2602.09177v2", "paper_link": "http://arxiv.org/abs/2602.09177v2", "theorems_cnt": 1, "theorem": {"env_name": "problem", "content": "\\label{prob:main}\nLet $S\\in \\mathbb S_n$ be a degree $n$ singular surface in $\\mathbb P^3$. In the closure of which Severi varieties $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$ the point [S] corresponding to $S$\nis contained? In particular, compute $\\delta_S$.", "start_pos": 3324, "end_pos": 3626, "label": "prob:main"}, "ref_dict": {}, "pre_theorem_intro_text_len": 961, "pre_theorem_intro_text": "Let $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta\\subset |\\mathcal O_{\\mathbb P^3}(n)|$ be the \\emph{Severi variety} of surfaces $S$ of degree $n$ with $\\delta\\geqslant 1$ ordinary double points (called \\emph{nodes}) and no other singularity. Then $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$ is a locally closed subvariety of $|\\mathcal O_{\\mathbb P^3}(n)|\\cong \\mathbb P^{{{n+3}\\choose 3}-1}$. Of course $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_1$ is irreducible of codimension 1 in $|\\mathcal O_{\\mathbb P^3}(n)|$ and its closure $\\mathbb S_n\\subset  |\\mathcal O_{\\mathbb P^3}(n)|$ is the set of all singular surfaces of degree $n$.  If $S\\in \\mathbb S_n$ is a singular surface, it makes sense to consider the maximum $\\delta_S$ of the integers $\\delta$ such that the point [S] corresponding to $S$ belongs to the closure of $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$. \n\nIn this article we will consider the following:", "context": "Let $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta\\subset |\\mathcal O_{\\mathbb P^3}(n)|$ be the \\emph{Severi variety} of surfaces $S$ of degree $n$ with $\\delta\\geqslant 1$ ordinary double points (called \\emph{nodes}) and no other singularity. Then $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$ is a locally closed subvariety of $|\\mathcal O_{\\mathbb P^3}(n)|\\cong \\mathbb P^{{{n+3}\\choose 3}-1}$. Of course $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_1$ is irreducible of codimension 1 in $|\\mathcal O_{\\mathbb P^3}(n)|$ and its closure $\\mathbb S_n\\subset |\\mathcal O_{\\mathbb P^3}(n)|$ is the set of all singular surfaces of degree $n$. If $S\\in \\mathbb S_n$ is a singular surface, it makes sense to consider the maximum $\\delta_S$ of the integers $\\delta$ such that the point [S] corresponding to $S$ belongs to the closure of $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$.\n\nIn this article we will consider the following:", "full_context": "Let $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta\\subset |\\mathcal O_{\\mathbb P^3}(n)|$ be the \\emph{Severi variety} of surfaces $S$ of degree $n$ with $\\delta\\geqslant 1$ ordinary double points (called \\emph{nodes}) and no other singularity. Then $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$ is a locally closed subvariety of $|\\mathcal O_{\\mathbb P^3}(n)|\\cong \\mathbb P^{{{n+3}\\choose 3}-1}$. Of course $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_1$ is irreducible of codimension 1 in $|\\mathcal O_{\\mathbb P^3}(n)|$ and its closure $\\mathbb S_n\\subset |\\mathcal O_{\\mathbb P^3}(n)|$ is the set of all singular surfaces of degree $n$. If $S\\in \\mathbb S_n$ is a singular surface, it makes sense to consider the maximum $\\delta_S$ of the integers $\\delta$ such that the point [S] corresponding to $S$ belongs to the closure of $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$.\n\nIn this article we will consider the following:\n\nLet $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta\\subset |\\mathcal O_{\\mathbb P^3}(n)|$ be the \\emph{Severi variety} of surfaces $S$ of degree $n$ with $\\delta\\geq 1$ ordinary double points (called \\emph{nodes}) and no other singularity. Then $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$ is a locally closed subvariety of $|\\mathcal O_{\\mathbb P^3}(n)|\\cong \\mathbb P^{{{n+3}\\choose 3}-1}$. Of course $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_1$ is irreducible of codimension 1 in $|\\mathcal O_{\\mathbb P^3}(n)|$ and its closure $\\mathbb S_n\\subset |\\mathcal O_{\\mathbb P^3}(n)|$ is the set of all singular surfaces of degree $n$. If $S\\in \\mathbb S_n$ is a singular surface, it makes sense to consider the maximum $\\delta_S$ of the integers $\\delta$ such that the point [S] corresponding to $S$ belongs to the closure of $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$.\n\nIn this article we will consider the following:\n\nThough this question is rather basic, it does not seem to have received much attention so far.\n\n\\begin{problem}\\label{prob:main2}\nLet $V$ be an irreducible component of a Severi variety $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$. Let $[S]\\in V$ be general and let $C$ be a general plane curve section of the surface $S$. When does it occur that $C$ is a general plane curve of degree $n$?\n\\end{problem}\n\n\\begin{proposition}\\label{prop:frt} In the above set--up, suppose that $C$ has $\\delta_C$ nodes, that $S_A$ has $\\delta_A$ nodes and that $S_\\Theta$ has $\\delta_\\Theta$ nodes all off $E$. Let us denote by $N$ the set of these $\\delta:=\\delta_A+\\delta_C+\\delta_\\Theta$ nodes. Suppose that \n$h^1(\\mathcal X_0, \\mathcal O_{\\mathcal X_0} (nH-m\\Theta)\\otimes I_N)=0$, \nthat is equivalent to say that the points in $N$ give independent conditions to surfaces in $|\\mathcal O_{\\mathcal X_0} (nH-m\\Theta)|$. Then $S_0$ can be deformed to a $\\delta$--nodal surface $S_t\\subset \\mathcal X_t\\cong \\mathbb P^3$ of degree $n$, for $t\\neq 0$ and the point corresponding to $S$ sits in the closure of a regular component of the Severi variety $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$.\n\\end{proposition}\n\n\\begin{theorem}\\label{thm:ord} Let $\\delta$ be such that there is a very regular irreducible component $V$ of the Severi variety $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(m)|}_\\delta$. Then the variety $\\mathcal T_{n,m}$ sits in the Zariski closure of a regular component of $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_{\\delta}$. In other words, the general degree $n$ surface $S$ with a general ordinary singularity of multiplicity $m$ is the limit of surfaces of degree $n$ with $\\delta$ nodes tending to the multiplicity $m$ point. \n\\end{theorem}\n\n\\begin{theorem}\\label{thm:ord1} Let $S$ be a surface of degree $n$ in $\\mathbb P^3$ with a unique {ordinary singularity} $p$ of multiplicity $m$. Consider the minimal desingularization $f: S'\\longrightarrow S$ of $S$ with the exceptional divisor of $S'$ over $p$ a plane curve $C$ of degree $m$. Let $\\delta$ be such that there is a surface $X$ of degree $m$ in $\\mathbb P^3$ having $C$ as a plane section and belonging to a regular component of \n$V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(m)|}_{\\delta}$. Then the point corresponding to $S$ belongs to the closure of a regular component of $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_{\\delta}$.\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:m3} Let $S\\subset \\mathbb P^3$ be a general surface of degree $n\\geq 3$ that has only {ordinary singularities} along a line $R$. Then $S$ belongs to the closure of a regular component of $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$, with $\\delta=3n-4-\\epsilon$, where $0\\leq \\epsilon\\leq 1$ has the same parity of $n$. In particular $\\mathcal F_n$ is contained in $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$. \n\\end{theorem}", "post_theorem_intro_text_len": 7992, "post_theorem_intro_text": "Though this question  is  rather basic, it does not seem to have received much attention so far. \n\nOne can ask a similar question for plane curves, and it seems to  have been  considered for the first time  by G. Albanese in \\cite{Alb}, where he claimed that an irreducible plane curve of degree $n$ and genus $g$ sits in the closure of the Severi variety $V^{\\mathbb P^2, |\\mathcal O_{\\mathbb P^2}(n)|}_\\delta$, with \n$$\\delta={{n-1}\\choose 2}-g$$\nand $\\delta$ is, of course, the maximum integer for which this happens. \nHowever Albanese's arguments do not meet the present standard of rigor, and this theorem has been proved in more recent times by Arbarello--Cornalba \\cite {AC} and Zariski \\cite {Za}. \n\nIn the case of surfaces in $\\mathbb P^3$ the problem is much more complicated and we have no idea of what could be a general answer to it. In the present paper, which should be regarded as having an experimental flavor, we focus on two study cases, namely: (a) the case in which $S$ has a single ordinary or quasi--ordinary singularity of multiplicity $m$ (for the definitions  see the beginning of Section \\ref {sec:ord}) and, (b) the case in which $S$ has ordinary singularities along a line, i.e., at the general point of the line $S$ has normal crossings, and at finitely many points of the line it has \\emph{pinch--points} (i.e., the tangent cone consists there of a plane with multiplicity 2). \n\nOur methods are based on deformation theory of  \\emph{regular} components of the Severi variety $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$ whose closure may contain the point [S] corresponding to $S$.  \n\nRecall that an irreducible component $V$ of a Severi variety $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$ is said to be \\emph{regular} if for the general point $[S]\\in V$ one has $h^1(\\mathbb P^3, \\mathcal O_{\\mathbb P^3}(n)\\otimes \\mathcal I_N)=0$, where $N$ is the reduced scheme of lenght $\\delta$ of the singular points of $S$. This means that the points in $N$ give $\\delta$ independent conditions to the surfaces in $|\\mathcal O_{\\mathbb P^3}(n)|$. If $V$ is regular, then $V$ is reduced and \n$$\n\\dim (V)= h^0(\\mathbb P^3, \\mathcal O_{\\mathbb P^3}(n))-1-\\delta={{n+3}\\choose 3}-1-\\delta.\n$$\nNote that if  [S] is in the closure of an irreducible regular component of $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$, then it sits also in the closure of an irreducible regular component of $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_{\\delta'}$ with $\\delta'\\leqslant \\delta$, because the nodes in a regular component of the Severi variety can be independently smoothed. \n\nCase (a) is considered in Section \\ref {sec:ord}. In Corollary \\ref {cor:gtp} we give an answer to Problem \\ref {prob:main} for $3\\leqslant m\\leqslant 5$.  This is the best possible answer for $m=3$. For $4\\leqslant m\\leqslant 5$ this is the best possible answer provided one looks only at deformations in regular components of the Severi variety.  In Corollary \\ref {cor:gtp}  we  also provide a partial answer for any $m$. In particular we prove that if $S\\subset \\mathbb P^3$ is surface  of degree $n$ with a unique \\emph{general} ordinary singularity of multiplicity $m$, then  the point corresponding to $S$ is in the closure of a regular component of $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_{{m-1}\\choose 2}$. An ordinary singularity of multiplicity $m$ is said to be \\emph{general} if its tangent cone is the cone over a general plane curve $C$ of degree $m$. In any case, we show that the answer to Problem \\ref {prob:main} in case (a)  definitely depends on the fact that the singularity is ordinary or quasi--ordinary. \n\nIn order to attack case (a) we first need to consider in Section \\ref {sec:curvesec} a preliminary, equally interesting problem, i.e., Problem \\ref {prob:main2}: given  an irreducible component $V$ of a Severi variety $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$, and given  $[S]\\in V$  a general point and $C$  a general plane curve section of $S$,  when is it the case that $C$ is a general plane curve of degree $n$? Again we are unable to solve this problem in general, but we prove some partial results, of independent interest (see, e.g., Proposition \\ref {prop:good} and its consequences)   that are useful to prove the aforementioned results in case (a).\n\nCase (b) is treated in Section \\ref {sec:line}. Our main result here is Theorem \\ref {thm:m3}, that says that if $S$ is a general surface of degree $n\\geqslant 3$ that has only {ordinary singularities} along a line, then $S$ belongs to the closure of a regular component of $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_{3n-4-\\epsilon}$, with  $0\\leqslant \\epsilon\\leqslant 1$ having the same parity of $n$. In particular we prove that we can choose $\\ell$, with $3n-4=2\\ell+\\epsilon$, pinch points of $S$ on the line so that $S$ deforms to a $2\\ell$--nodal surface in a regular component  of the corresponding Severi variety, so that each chosen pinch point deforms to two nodes. The integer $3n-4-\\epsilon$ is the maximum  for which this happens in the $n$ even case, whereas in the $n$ odd case the maximum could be $3n-4$. \n\nTo prove this result we have to rely on the existence of suitable rational surfaces of degree $n$ with a line $r$ of points of multiplicity $n-2$ and with $3n-4-\\epsilon$ nodes off $r$, that impose independent conditions to surfaces of degree $n$ in $\\mathbb P^3$ that have points of multiplicity $n-2$ along $r$. This is done in Section \\ref {sec:rat}. \n\nThe method of the proof in both cases (a) and (b) is the same, and it is based on a deformation argument. In both cases we construct a flat family $\\mathcal X\\longrightarrow \\mathbb D$ parametrized by a disk $\\mathbb D$, such that for $t\\neq 0$, the fibre $\\mathcal X_t$ over $t\\in \\mathbb D$ is isomorphic to $\\mathbb P^3$, whereas the central fibre $\\mathcal X_0$ is the union of two smooth irreducible components $A\\cup \\Theta$, that intersect transversally along a surface $E$. The starting surface $S$ for which we consider Problem \\ref {prob:main} in the two cases (a) and (b) has a minimal resolution $f: S'\\longrightarrow S$, with a curve $C$ mapped by $f$ to the singular locus of $S$. What we  do is to realize $S'$ as a surface $S_A$ in the threefold $A$ in such a way that $S_A\\cap E=C$. Then we construct a surface $S_\\Theta\\subset \\Theta$ such that $S_\\Theta\\cap E=C$ so that $S_0=S_A\\cup S_\\Theta$ is a Cartier divisor in $\\mathcal X_0$. Finally we can do this so that $S_\\Theta$ has $\\delta$ nodes off $E$, that impose independent conditions to the surfaces in the linear system $|S_0|$ on $\\mathcal X_0$. If this is the case then we can apply \\cite [Thm. 4.6]{CG} that ensures that $S_0$ can be deformed to a surface $S_t\\subset \\mathcal X_t\\cong \\mathbb P^3$ of degree $n$ with $\\delta$ nodes that give independent conditions to surfaces of degree $n$. From this it follows  that $S$ sits in the closure of a regular component of $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$.\n\nWe notice that the strength of this method consists in studying  deformations of a surface $S\\subset \\mathbb P^3$ corresponding to a (presumably) singular point of a Severi variety in terms of deformations of a surface $S_A\\cup S_\\Theta$ corresponding to a smooth point in  a relative Severi variety.  \n\n\\medskip\n\n{\\bf Aknowledgements:} The authors are members of GNSAGA of INdAM. In particular, the second author acknowledges\nfunding from the GNSAGA of INdAM and the European Union - NextGenerationEU under the National Recovery and Resilience\nPlan (PNRR) - Mission 4 Education and research - Component 2 From research to business - Investment 1.1, Prin 2022\n``Geometry of algebraic structures: moduli, invariants, deformations\", DD N. 104, 2/2/2022, proposal code 2022BTA242 - CUP\nJ53D23003720006. \n\nThe authors also thank Fabrizio Catanese, Barbara Fantechi and Rita Pardini for useful discussions on related problems to the subject of this paper. \\medskip", "sketch": "The post-theorem text does not give a proof of Problem~\\ref{prob:main} (it is posed as a question), but it does outline the deformation-theoretic method used to obtain answers in the two study cases.\n\nKey steps of the deformation argument (cases (a) and (b)):\n\\begin{itemize}\n\\item Work with \\emph{regular} components of $V^{\\mathbb P^3,|\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$, i.e. those for which for a general $[S]\\in V$ one has $h^1(\\mathbb P^3,\\mathcal O_{\\mathbb P^3}(n)\\otimes\\mathcal I_N)=0$ (the $\\delta$ singular points give independent conditions). Note: if $[S]$ is in the closure of a regular component for $\\delta$, then it is also in the closure for any $\\delta'\\le \\delta$ since “the nodes in a regular component ... can be independently smoothed.”\n\\item Construct a flat family $\\mathcal X\\to\\mathbb D$ such that for $t\\neq 0$, $\\mathcal X_t\\cong\\mathbb P^3$, and the central fiber is a normal-crossing union $\\mathcal X_0=A\\cup\\Theta$ of two smooth components meeting transversally along a surface $E$.\n\\item Let $f:S'\\to S$ be the minimal resolution of the starting surface, with a curve $C\\subset S'$ mapped to the singular locus of $S$. Realize $S'$ as a surface $S_A\\subset A$ with $S_A\\cap E=C$.\n\\item Construct a surface $S_\\Theta\\subset\\Theta$ with $S_\\Theta\\cap E=C$ so that $S_0=S_A\\cup S_\\Theta$ is a Cartier divisor on $\\mathcal X_0$.\n\\item Arrange that $S_\\Theta$ has $\\delta$ nodes off $E$ “that impose independent conditions to the surfaces in the linear system $|S_0|$ on $\\mathcal X_0$.”\n\\item Apply \\cite[Thm.~4.6]{CG}: if those $\\delta$ nodes impose independent conditions, then $S_0$ deforms to a surface $S_t\\subset\\mathcal X_t\\cong\\mathbb P^3$ of degree $n$ with $\\delta$ nodes giving independent conditions. Conclude that $S$ lies in the closure of a \\emph{regular} component of $V^{\\mathbb P^3,|\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$.\n\\end{itemize}\n\nFor case (b), the outline adds that one “rel[ies] on the existence of suitable rational surfaces of degree $n$ with a line $r$ of points of multiplicity $n-2$ and with $3n-4-\\epsilon$ nodes off $r$, that impose independent conditions ...,” which is established in Section~\\ref{sec:rat}.", "expanded_sketch": "The post-theorem text does not give a proof of Problem~\\ref{prob:main} (it is posed as a question), but it does outline the deformation-theoretic method used to obtain answers in the two study cases.\n\nKey steps of the deformation argument (cases (a) and (b)):\n\\begin{itemize}\n\\item Work with \\emph{regular} components of $V^{\\mathbb P^3,|\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$, i.e. those for which for a general $[S]\\in V$ one has $h^1(\\mathbb P^3,\\mathcal O_{\\mathbb P^3}(n)\\otimes\\mathcal I_N)=0$ (the $\\delta$ singular points give independent conditions). Note: if $[S]$ is in the closure of a regular component for $\\delta$, then it is also in the closure for any $\\delta'\\le \\delta$ since “the nodes in a regular component ... can be independently smoothed.”\n\\item Construct a flat family $\\mathcal X\\to\\mathbb D$ such that for $t\\neq 0$, $\\mathcal X_t\\cong\\mathbb P^3$, and the central fiber is a normal-crossing union $\\mathcal X_0=A\\cup\\Theta$ of two smooth components meeting transversally along a surface $E$.\n\\item Let $f:S'\\to S$ be the minimal resolution of the starting surface, with a curve $C\\subset S'$ mapped to the singular locus of $S$. Realize $S'$ as a surface $S_A\\subset A$ with $S_A\\cap E=C$.\n\\item Construct a surface $S_\\Theta\\subset\\Theta$ with $S_\\Theta\\cap E=C$ so that $S_0=S_A\\cup S_\\Theta$ is a Cartier divisor on $\\mathcal X_0$.\n\\item Arrange that $S_\\Theta$ has $\\delta$ nodes off $E$ “that impose independent conditions to the surfaces in the linear system $|S_0|$ on $\\mathcal X_0$.”\n\\item Apply \\cite[Thm.~4.6]{CG}: if those $\\delta$ nodes impose independent conditions, then $S_0$ deforms to a surface $S_t\\subset\\mathcal X_t\\cong\\mathbb P^3$ of degree $n$ with $\\delta$ nodes giving independent conditions. Conclude that $S$ lies in the closure of a \\emph{regular} component of $V^{\\mathbb P^3,|\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$.\n\\end{itemize}\n\nFor case (b), the outline adds that one “rel[ies] on the existence of suitable rational surfaces of degree $n$ with a line $r$ of points of multiplicity $n-2$ and with $3n-4-\\epsilon$ nodes off $r$, that impose independent conditions ...,” which is established later.", "expanded_theorem": "\\label{prob:main}\nLet $S\\in \\mathbb S_n$ be a degree $n$ singular surface in $\\mathbb P^3$. In the closure of which Severi varieties $V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta$ is the point $[S]$ corresponding to $S$ contained? In particular, compute $\\delta_S$.", "theorem_type": ["Classification or Bijection", "Existence"], "mcq": {"question": "Let \\(V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta\\subset |\\mathcal O_{\\mathbb P^3}(n)|\\) denote the Severi variety of degree \\(n\\) surfaces in \\(\\mathbb P^3\\) having exactly \\(\\delta\\ge 1\\) ordinary double points (nodes) and no other singularities. Let \\(\\mathbb S_n\\) be the closure of \\(V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_1\\), so \\(\\mathbb S_n\\) is the set of all singular degree \\(n\\) surfaces, and for \\(S\\in \\mathbb S_n\\) define\n\\[\n\\delta_S:=\\max\\{\\delta : [S]\\in \\overline{V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta}\\}.\n\\]\nFor a singular degree \\(n\\) surface \\(S\\in \\mathbb P^3\\), which statement is to be determined?", "correct_choice": {"label": "A", "text": "Determine precisely for which integers \\(\\delta\\) the point \\([S]\\) lies in the closure of the Severi variety \\(V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta\\); in particular, compute \\(\\delta_S=\\max\\{\\delta : [S]\\in \\overline{V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta}\\}.\""}, "choices": [{"label": "B", "text": "Determine precisely for which integers \\(\\delta\\) the point \\([S]\\) lies in the closure of a regular component of the Severi variety \\(V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta\\), and require moreover that the corresponding \\(\\delta\\) nodes impose independent conditions on \\(|\\mathcal O_{\\mathbb P^3}(n)|\\); in particular, compute the largest such \\(\\delta\\)."}, {"label": "C", "text": "Determine at least the maximal integer \\(\\delta_S\\) such that \\([S]\\in \\overline{V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta}\\)."}, {"label": "D", "text": "Determine precisely for which integers \\(\\delta\\ge 1\\) the point \\([S]\\) lies in the closure of the Severi variety \\(V^{\\mathbb P^3, |\\mathcal O_{\\mathbb P^3}(n)|}_\\delta\\), knowing that if this holds for some \\(\\delta\\), then it holds only for that maximal value \\(\\delta=\\delta_S\\)."}, {"label": "E", "text": "Determine precisely for which integers \\(\\delta\\) there exists a flat family of degree \\(n\\) surfaces specializing to \\(S\\) whose general member has exactly \\(\\delta\\) nodes, all tending to the singular locus of \\(S\\); in particular, compute \\(\\delta_S\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "closure_of_regular_component_plus_independent_conditions", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "full_classification_of_all_delta", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "downward_monotonicity_in_delta", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "all_nodes_tend_to_singular_locus", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal choice A. It asks for the correct characterization, and although the phrasing overlaps with the correct option, it does not directly state the defining max-closure formula for δ_S."}, "TAS": {"score": 0, "justification": "This is essentially a direct recall/restatement question about the definition/characterization of the Severi varieties containing [S] in their closures and the associated quantity δ_S. It does not substantially go beyond reproducing the underlying statement."}, "GPS": {"score": 1, "justification": "Some discrimination is needed because several options are close variants: one is weaker true, others add unjustified regularity, restrict to only δ=δ_S, or replace max by min. But the task is still mainly recognition of the exact statement rather than genuine derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and reflect common failure modes: adding unnecessary regularity hypotheses, confusing maximality with minimality, or missing the full range of admissible δ. They are distinct and nontrivial."}, "total_score": 5, "overall_assessment": "Good distractor design, but the item is largely a definitional restatement with only moderate reasoning demand."}}
{"id": "2602.09253v1", "paper_link": "http://arxiv.org/abs/2602.09253v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "Let $X\\subseteq\\mathbb{C}$ be an open connected set and $f:X\\to\\mathbb{C}$ be an elementary meromorphic function with infinitely many critical values. Then the equation $f(x)=a$ is unsolvable in elementary functions.", "start_pos": 8590, "end_pos": 8840, "label": null}, "ref_dict": {"thm:wielandt": "\\begin{theorem}\\emph{(Wielandt, \\cite[Theorem 29.2]{neumann2023infinitepermutationgroups})}\n\\label{thm:wielandt}\n    Let $S$ be an infinite set and $G$ be a primitive group acting on $S$. If $G$ contains a non-identity permutation with finite support, then $\\alt(S)\\leq G$.\n\\end{theorem}", "prop:imprimitive-equiv-decomposition": "\\begin{proposition} \\emph{\\cite[Theorem 27]{Burda2012Topological}}\n\\label{prop:imprimitive-equiv-decomposition}\n    Let $X \\subseteq \\mathbb{C}$ be an open connected set and $f:X\\to\\mathbb{C}$ be a meromorphic map with monodromy group $G$. Then $G$ is imprimitive if and only if $f$ can be expressed as a composition $(h\\circ g)(x)$ of two meromorphic functions $g:X\\to Y$ and $h:Y\\to\\mathbb{C}$ for some open connected set $Y\\subseteq\\mathbb{C}$. Moreover, the monodromy groups of $g$ and $h$ are quotients of $G$.\n\\end{proposition}"}, "pre_theorem_intro_text_len": 3328, "pre_theorem_intro_text": "The question of solvability in explicit form, as a finite composition of admissible operations over a base field, is profoundly significant to the development of algebra. After the general solutions by radicals for polynomial equations of degree up to $4$ were discovered, numerous unsuccessful attempts at formulating a closed-form solution for equations of higher degree led to the belief that such solutions simply do not exist. This was proven by Ruffini (with refinements by Cauchy) in $1813$ and also by Abel in $1824$.\n\n\\smallskip\n\nPondering on this question, Abel laid the foundations of the theory of algebraic curves, while Galois was the first to notice that solvability by radicals is dependent on an associated group, now called the Galois group of an algebraic equation. Liouville continued Abel's work on algebraic and differential forms and made significant improvements regarding expressibility by elementary functions, that is, compositions of arithmetic operations, the exponential function $\\exp$ and the logarithmic function $\\ln$. He rigorously proved the unsolvability in elementary functions of some differential equations \\cite{Khovanskii2014}, e.g. $f'(x) = e^{-x^2}$. Later work by Chebyshev, Ritt, Risch, Rosenlicht, and others extended this aspect of Galois theory further.\n\n\\smallskip\n\nArnold discovered topological reasons for the unsolvability of equations -- for instance, see his proof for the quintic \\cite{GoldmakherArnoldQuintic}. His student Khovanskii \\cite{khovanskii2019dimensionaltopologicalgaloistheory,Khovanskii2014} formalized the methods of the one-dimensional version of topological Galois theory around 1970. The fundamental concepts of topological Galois theory are similar to classical Galois theory, but are suitable for analyzing transcendental functions -- the former one works with extensions of the rational numbers $\\mathbb{Q}$, while the latter one is with extensions of the field of rational functions $\\mathbb{C}(x)$. The topological criterion for solvability is also based on a group defined through $f$, known as the monodromy group.\n\n\\smallskip\n\nConsider the equation $f(x)=a$, where $f:X\\to\\mathbb{C}$ is a meromorphic elementary function defined on a connected open set $X\\subseteq\\mathbb{C}$, and $a\\in\\mathbb{C}$ is a parameter. We aim to determine when the inverse $f^{-1}(a)$ is also an elementary function. All approaches so far rely on analyzing the specific monodromy group and the structure of the roots, hence results up to now have been on individual equations, such as $\\tan(x)-x=a$ \\cite{Belov_Kanel_2020}, $e^x + x = a$ and $x^x=a$ \\cite{kanelbelov2024insolvabilityxxa} by Kanel-Belov, Malistov, Zaytsev.\n\n\\smallskip\n\nOther recent developments are by Zelenko \\cite{zelenko2021genericmonodromygroupriemann}, proving unsolvability in quadratures, i.e. elementary functions, integration, differentiation and taking the solutions of an algebraic equation, when $f$ is holomorphic, surjective, no two critical points share the same critical value, and for some positive integer $\\rho$ \\[\\lim_{R\\to\\infty}\\sup_{z:|z|\\geq R}\\frac{\\ln|f(z)|}{|z|^\\rho}\\] is bounded. \n\n\\smallskip\n\nWe extend the unsolvability to all elementary meromorphic functions for which the set of critical values\n    \\[B=\\{a\\in\\mathbb{C}:\\ \\exists x\\in X:f(x)=a,f'(x)=0\\}\\]\nis infinite.", "context": "Pondering on this question, Abel laid the foundations of the theory of algebraic curves, while Galois was the first to notice that solvability by radicals is dependent on an associated group, now called the Galois group of an algebraic equation. Liouville continued Abel's work on algebraic and differential forms and made significant improvements regarding expressibility by elementary functions, that is, compositions of arithmetic operations, the exponential function $\\exp$ and the logarithmic function $\\ln$. He rigorously proved the unsolvability in elementary functions of some differential equations \\cite{Khovanskii2014}, e.g. $f'(x) = e^{-x^2}$. Later work by Chebyshev, Ritt, Risch, Rosenlicht, and others extended this aspect of Galois theory further.\n\nArnold discovered topological reasons for the unsolvability of equations -- for instance, see his proof for the quintic \\cite{GoldmakherArnoldQuintic}. His student Khovanskii \\cite{khovanskii2019dimensionaltopologicalgaloistheory,Khovanskii2014} formalized the methods of the one-dimensional version of topological Galois theory around 1970. The fundamental concepts of topological Galois theory are similar to classical Galois theory, but are suitable for analyzing transcendental functions -- the former one works with extensions of the rational numbers $\\mathbb{Q}$, while the latter one is with extensions of the field of rational functions $\\mathbb{C}(x)$. The topological criterion for solvability is also based on a group defined through $f$, known as the monodromy group.\n\nConsider the equation $f(x)=a$, where $f:X\\to\\mathbb{C}$ is a meromorphic elementary function defined on a connected open set $X\\subseteq\\mathbb{C}$, and $a\\in\\mathbb{C}$ is a parameter. We aim to determine when the inverse $f^{-1}(a)$ is also an elementary function. All approaches so far rely on analyzing the specific monodromy group and the structure of the roots, hence results up to now have been on individual equations, such as $\\tan(x)-x=a$ \\cite{Belov_Kanel_2020}, $e^x + x = a$ and $x^x=a$ \\cite{kanelbelov2024insolvabilityxxa} by Kanel-Belov, Malistov, Zaytsev.\n\nOther recent developments are by Zelenko \\cite{zelenko2021genericmonodromygroupriemann}, proving unsolvability in quadratures, i.e. elementary functions, integration, differentiation and taking the solutions of an algebraic equation, when $f$ is holomorphic, surjective, no two critical points share the same critical value, and for some positive integer $\\rho$ \\[\\lim_{R\\to\\infty}\\sup_{z:|z|\\geq R}\\frac{\\ln|f(z)|}{|z|^\\rho}\\] is bounded.\n\n\\smallskip\n\nWe extend the unsolvability to all elementary meromorphic functions for which the set of critical values\n \\[B=\\{a\\in\\mathbb{C}:\\ \\exists x\\in X:f(x)=a,f'(x)=0\\}\\]\nis infinite.", "full_context": "Pondering on this question, Abel laid the foundations of the theory of algebraic curves, while Galois was the first to notice that solvability by radicals is dependent on an associated group, now called the Galois group of an algebraic equation. Liouville continued Abel's work on algebraic and differential forms and made significant improvements regarding expressibility by elementary functions, that is, compositions of arithmetic operations, the exponential function $\\exp$ and the logarithmic function $\\ln$. He rigorously proved the unsolvability in elementary functions of some differential equations \\cite{Khovanskii2014}, e.g. $f'(x) = e^{-x^2}$. Later work by Chebyshev, Ritt, Risch, Rosenlicht, and others extended this aspect of Galois theory further.\n\nArnold discovered topological reasons for the unsolvability of equations -- for instance, see his proof for the quintic \\cite{GoldmakherArnoldQuintic}. His student Khovanskii \\cite{khovanskii2019dimensionaltopologicalgaloistheory,Khovanskii2014} formalized the methods of the one-dimensional version of topological Galois theory around 1970. The fundamental concepts of topological Galois theory are similar to classical Galois theory, but are suitable for analyzing transcendental functions -- the former one works with extensions of the rational numbers $\\mathbb{Q}$, while the latter one is with extensions of the field of rational functions $\\mathbb{C}(x)$. The topological criterion for solvability is also based on a group defined through $f$, known as the monodromy group.\n\nConsider the equation $f(x)=a$, where $f:X\\to\\mathbb{C}$ is a meromorphic elementary function defined on a connected open set $X\\subseteq\\mathbb{C}$, and $a\\in\\mathbb{C}$ is a parameter. We aim to determine when the inverse $f^{-1}(a)$ is also an elementary function. All approaches so far rely on analyzing the specific monodromy group and the structure of the roots, hence results up to now have been on individual equations, such as $\\tan(x)-x=a$ \\cite{Belov_Kanel_2020}, $e^x + x = a$ and $x^x=a$ \\cite{kanelbelov2024insolvabilityxxa} by Kanel-Belov, Malistov, Zaytsev.\n\nOther recent developments are by Zelenko \\cite{zelenko2021genericmonodromygroupriemann}, proving unsolvability in quadratures, i.e. elementary functions, integration, differentiation and taking the solutions of an algebraic equation, when $f$ is holomorphic, surjective, no two critical points share the same critical value, and for some positive integer $\\rho$ \\[\\lim_{R\\to\\infty}\\sup_{z:|z|\\geq R}\\frac{\\ln|f(z)|}{|z|^\\rho}\\] is bounded.\n\n\\smallskip\n\nWe extend the unsolvability to all elementary meromorphic functions for which the set of critical values\n \\[B=\\{a\\in\\mathbb{C}:\\ \\exists x\\in X:f(x)=a,f'(x)=0\\}\\]\nis infinite.\n\n\\begin{abstract} \n An equation $f(x)=a$, where $f$ is a complex meromorphic function and $a\\in\\mathbb{C}$ is a parameter, is solvable in elementary functions if the inverse map $x=f^{-1}(a)$ can be expressed as a finite composition of arithmetic operations (addition, subtraction, multiplication, and division), the exponential function, the complex logarithm, and constants. Specific functions such as $\\tan x - x$, $\\exp x + x$, $x^x$ have been proven to be unsolvable by Kanel-Belov, Malistov, Zaytsev, while almost all entire surjective functions of at most exponential growth have been covered by Zelenko. All these rely on one-dimensional topological Galois theory, developed by Khovanskii. We generalize to provide a proof for the unsolvability of all elementary meromorphic functions $f$ such that the derivative of $f$ has infinitely many roots $x_i$ and the set of distinct values $f(x_i)$ is infinite.\n\\end{abstract}\n\nConsider the equation $f(x)=a$, where $f:X\\to\\mathbb{C}$ is a meromorphic elementary function defined on a connected open set $X\\subseteq\\mathbb{C}$, and $a\\in\\mathbb{C}$ is a parameter. We aim to determine when the inverse $f^{-1}(a)$ is also an elementary function. All approaches so far rely on analyzing the specific monodromy group and the structure of the roots, hence results up to now have been on individual equations, such as $\\tan(x)-x=a$ \\cite{Belov_Kanel_2020}, $e^x + x = a$ and $x^x=a$ \\cite{kanelbelov2024insolvabilityxxa} by Kanel-Belov, Malistov, Zaytsev.\n\nWe extend the unsolvability to all elementary meromorphic functions for which the set of critical values\n \\[B=\\{a\\in\\mathbb{C}:\\ \\exists x\\in X:f(x)=a,f'(x)=0\\}\\]\nis infinite.\n\nThe provided proof combines a result of Wielandt on primitive group actions on infinite sets (Theorem \\ref{thm:wielandt}), and a relation between primitivity and decompositions (Theorem \\ref{prop:imprimitive-equiv-decomposition}) stated, for example, in the doctoral work of Burda on algebraic functions.\n\n\\section{\\Large Topological unsolvability in elementary functions}\nConsider the equation $f(x)=a$, where $f:X\\to\\mathbb{C}$ is a meromorphic non-constant function defined on a connected open set $X\\subseteq\\mathbb{C}$, and $a\\in\\mathbb{C}$ is a parameter. By Theorem~\\ref{theorem:identity-theorem} for any $a$ the set of roots is discrete, hence the inverse $f^{-1}$ is an $\\mathscr{S}$-function.\n\n\\begin{proposition}\n\\label{prop:wielandt-corollary}\n Let $X \\subseteq \\mathbb{C}$ be an open connected set and $f:X\\to\\mathbb{C}$ be a meromorphic map with infinite monodromy group $G$. Suppose $G$ is primitive. Then $G$ contains an unsolvable subgroup, and $f(x)=a$ is unsolvable in elementary functions.\n\\end{proposition}\n\n\\begin{theorem}\n\\label{thm:unsolvability-infinite-branch-locus}\n Let $X \\subseteq \\mathbb{C}$ be an open connected set and $f:X\\to\\mathbb{C}$ be an elementary meromorphic function with an infinite branch locus and infinite monodromy group $G$. Then the equation $f(x)=a$ is unsolvable in elementary functions.\n\\end{theorem}\n\n\\begin{conjecture}\n\\label{conj:any-solvable-tower-core}\n Let $X\\subseteq\\mathbb{C}$ be a connected open set and $f: X \\to \\mathbb{C}$ be a meromorphic elementary function. If the equation $f(x)=a$ is solvable in elementary functions, then there exists a finite sequence $g_1,g_2,\\ldots,g_\\ell$ of solvable functions of depth at most $1$, such that $f(x)=(g_1\\circ g_2\\circ\\ldots\\circ g_\\ell)(x)\\text{.}$\n\\end{conjecture}\n\n\\begin{proposition} \\emph{\\cite[Theorem 27]{Burda2012Topological}}\n\\label{prop:imprimitive-equiv-decomposition}\n    Let $X \\subseteq \\mathbb{C}$ be an open connected set and $f:X\\to\\mathbb{C}$ be a meromorphic map with monodromy group $G$. Then $G$ is imprimitive if and only if $f$ can be expressed as a composition $(h\\circ g)(x)$ of two meromorphic functions $g:X\\to Y$ and $h:Y\\to\\mathbb{C}$ for some open connected set $Y\\subseteq\\mathbb{C}$. Moreover, the monodromy groups of $g$ and $h$ are quotients of $G$.\n\\end{proposition}\n\n\\begin{theorem}\\emph{(Wielandt, \\cite[Theorem 29.2]{neumann2023infinitepermutationgroups})}\n\\label{thm:wielandt}\n    Let $S$ be an infinite set and $G$ be a primitive group acting on $S$. If $G$ contains a non-identity permutation with finite support, then $\\alt(S)\\leq G$.\n\\end{theorem}", "post_theorem_intro_text_len": 314, "post_theorem_intro_text": "The provided proof combines a result of Wielandt on primitive group actions on infinite sets (Theorem \\ref{thm:wielandt}), and a relation between primitivity and decompositions (Theorem \\ref{prop:imprimitive-equiv-decomposition}) stated, for example, in the doctoral work of Burda on algebraic functions.\n\n\\medskip", "sketch": "The provided proof combines a result of Wielandt on primitive group actions on infinite sets (Theorem \\ref{thm:wielandt}), and a relation between primitivity and decompositions (Theorem \\ref{prop:imprimitive-equiv-decomposition}) stated, for example, in the doctoral work of Burda on algebraic functions.", "expanded_sketch": "The provided proof combines a result of Wielandt on primitive group actions on infinite sets. We first use the following theorem.\n\n\\begin{theorem}\\emph{(Wielandt, \\cite[Theorem 29.2]{neumann2023infinitepermutationgroups})}\n\\label{thm:wielandt}\n    Let $S$ be an infinite set and $G$ be a primitive group acting on $S$. If $G$ contains a non-identity permutation with finite support, then $\\alt(S)\\leq G$.\n\\end{theorem}\n\nIt also uses a relation between primitivity and decompositions. More precisely, we use the following proposition.\n\n\\begin{proposition} \\emph{\\cite[Theorem 27]{Burda2012Topological}}\n\\label{prop:imprimitive-equiv-decomposition}\n    Let $X \\subseteq \\mathbb{C}$ be an open connected set and $f:X\\to\\mathbb{C}$ be a meromorphic map with monodromy group $G$. Then $G$ is imprimitive if and only if $f$ can be expressed as a composition $(h\\circ g)(x)$ of two meromorphic functions $g:X\\to Y$ and $h:Y\\to\\mathbb{C}$ for some open connected set $Y\\subseteq\\mathbb{C}$. Moreover, the monodromy groups of $g$ and $h$ are quotients of $G$.\n\\end{proposition}\n\nThis relation is stated, for example, in the doctoral work of Burda on algebraic functions.", "expanded_theorem": "Let $X\\subseteq\\mathbb{C}$ be an open connected set and $f:X\\to\\mathbb{C}$ be an elementary meromorphic function with infinitely many critical values. Then the equation $f(x)=a$ is unsolvable in elementary functions.", "theorem_type": ["Implication", "Nonexistence"], "mcq": {"question": "Let $X\\subseteq\\mathbb{C}$ be an open connected set, and let $f:X\\to\\mathbb{C}$ be an elementary meromorphic function. Assume that $f$ has infinitely many critical values, meaning that the set\n\\[\n\\{a\\in\\mathbb{C}:\\exists x\\in X\\text{ with }f(x)=a\\text{ and }f'(x)=0\\}\n\\]\nis infinite. Consider the equation $f(x)=a$ with parameter $a\\in\\mathbb{C}$. Here, “solvable in elementary functions” means that the inverse relation $x=f^{-1}(a)$ can be expressed as a finite composition of arithmetic operations, constants, the exponential function, and the complex logarithm. Which conclusion holds under these hypotheses?", "correct_choice": {"label": "A", "text": "The equation $f(x)=a$ is unsolvable in elementary functions; equivalently, the inverse $x=f^{-1}(a)$ cannot be expressed as a finite composition of arithmetic operations, constants, the exponential function, and the complex logarithm."}, "choices": [{"label": "B", "text": "The equation $f(x)=a$ is unsolvable in elementary functions provided, in addition, that the monodromy group of $f$ is primitive; under the stated hypothesis of infinitely many critical values alone, no such conclusion follows in general."}, {"label": "C", "text": "The equation $f(x)=a$ is not expressible using only arithmetic operations, constants, and algebraic operations; in particular, it is not solvable by radicals alone."}, {"label": "D", "text": "The equation $f(x)=a$ is unsolvable in elementary functions only when the set of critical points $\\{x\\in X:f'(x)=0\\}$ is infinite; having infinitely many critical values by itself is insufficient."}, {"label": "E", "text": "There exists a finite exceptional set $E\\subset\\mathbb{C}$ such that for every $a\\in\\mathbb{C}\\setminus E$, the equation $f(x)=a$ is unsolvable in elementary functions, although elementary solvability may still occur for infinitely many special parameters $a$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "case_split", "tampered_component": "primitive-case versus full theorem via decomposition", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped exponentials and logarithms from the allowed class of inverse expressions", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "critical-values hypothesis replaced by critical-points hypothesis", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "global unsolvability conclusion weakened to generic-in-parameter exceptional-set statement", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the conclusion; it states the hypotheses and defines elementary solvability, but does not itself say that the inverse is impossible. There is little direct answer leakage from the wording alone."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the correct option is a direct statement of the theorem's conclusion under the given hypotheses. It tests recognition of the exact theorem more than selection among genuinely competing conclusions."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to reject the weaker-true option and the property-confusion distractors, but the main task is still recalling the theorem. It does not strongly pressure the student to generate a novel conclusion from the setup."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic confusions: adding unnecessary hypotheses, weakening the conclusion, and confusing critical values with critical points. They are distinct and aligned with common failure modes."}, "total_score": 5, "overall_assessment": "A mathematically well-crafted but largely theorem-restatement MCQ. It avoids direct answer leakage and has strong distractors, but it scores lower on tautology avoidance and generative reasoning because the correct answer is essentially the theorem's conclusion verbatim."}}
{"id": "2602.09669v1", "paper_link": "http://arxiv.org/abs/2602.09669v1", "theorems_cnt": 5, "theorem": {"env_name": "thm", "content": "\\label{Lions-theo}\nFor any $s$ in $\\mathbb R$, the space $\\mathcal H_s(\\Omega) $  is a RKHS and its two-point kernel $K_s$ is given,  for any pair of $x$ and $y$ in $\\Omega$,  by \n \\begin{equation} \\label{l-formu}\n K_s (x,y)=  \\int_{\\partial\\Omega} \\Big((M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(x,\\cdot)\\big)\\Big)(z) \\, \\Big(M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(y,\\cdot)\\big) \\Big)(z)\\,d\\sigma(z) .\n \\end{equation}", "start_pos": 14535, "end_pos": 14965, "label": "Lions-theo"}, "ref_dict": {"Lions-theo": "\\begin{thm} \\label{Lions-theo}\nFor any $s$ in $\\R$, the space $\\mathcal H_s(\\Omega) $  is a RKHS and its two-point kernel $K_s$ is given,  for any pair of $x$ and $y$ in $\\Omega$,  by \n \\begin{equation} \\label{l-formu}\n K_s (x,y)=  \\int_{\\partial\\Omega} \\Big((M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(x,\\cdot)\\big)\\Big)(z) \\, \\Big(M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(y,\\cdot)\\big) \\Big)(z)\\,d\\sigma(z) .\n \\end{equation}\n\\end{thm}", "DIR": "\\begin{equation} \\label{DIR}\n    -\\Delta G_1(x,\\cdot)=\\delta_x \\quad\\text{in}\\quad \\mathcal D'(\\Omega)\\qquad\\&\\qquad G_1(x,\\cdot)=0\\quad\\text{on}\\quad \\partial\\Omega, \n\\end{equation}", "defRKHS": "\\begin{Definition} \\label{defRKHS}\nWe say that a Hilbert space $H$ of  real-valued functions defined on  $\\Omega$ is a reproducing kernel Hilbert space (RKHS for short) \n if for any $x$ in $\\Omega$, the evaluation mapping $E_x$ defined for any $u$ in $H$ by $E_x (u) = u(x)$ \nis linear and continuous from $H$ to $\\R$. Then, we denote by  $K_x$ the unique element in $H$ such that for any $u$ in $H$,  $E_x (u) = \\langle u,K_x\\rangle $, the inner product of $u$ and $K_x$ in $H$, given by Riesz' theorem. Finally, the two-point kernel is defined for any pair of $x$ and $y$ in $\\Omega$  by $K(x,y)= \\langle K_x,K_y\\rangle $. \n\\end{Definition}", "Cani2": "\\begin{equation} \\label{Cani2}\nu(x) = \\langle \\gamma_D (u) ,\\, K_x \\rangle_s = \\langle M^{\\frac{s}{2}} \\, \\gamma_D (u) ,\\,  M^{\\frac{s}{2}} \\, K_x \\rangle_{L^2 (\\partial \\Omega)}.\n\\end{equation}", "Cani1": "\\begin{equation} \\label{Cani1}\n    u(x)\n    = \\Big\\langle \\gamma_D (u),\\, M^{-{s}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big) \\Big\\rangle_s ,\n\\end{equation}", "REg": "\\begin{equation} \\label{REg}\n    \\forall x \\in \\Omega, \\quad   \\forall s \\in \\R, \\quad M^{-\\frac{s}{2}} \\gamma_N \\big(G_1(x,\\cdot)\\big)  \\in L^2 (\\partial\\Omega) ,\n\\end{equation}", "l-formu": "\\begin{equation} \\label{l-formu}\n K_s (x,y)=  \\int_{\\partial\\Omega} \\Big((M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(x,\\cdot)\\big)\\Big)(z) \\, \\Big(M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(y,\\cdot)\\big) \\Big)(z)\\,d\\sigma(z) .\n \\end{equation}", "Hadamard": "\\begin{align} \\label{Hadamard}\n    DG_\\Omega (\\alpha) (x,y) =\\int_{\\partial\\Omega} \\gamma_N \\big(G_1(x,\\cdot)\\big)(z) \\, \\gamma_N \\big(G_1(y,\\cdot)\\big)(z) \\,  \\alpha(z)d\\sigma(z) .\n\\end{align}"}, "pre_theorem_intro_text_len": 7752, "pre_theorem_intro_text": "\\subsection{Reproducing kernel Hilbert space}\n\nThe notion of  reproducing kernel Hilbert space was first introduced in 1907 by Stanislaw Zaremba for boundary value problems for harmonic and biharmonic functions, and simultaneously by James Mercer in the theory of integral equations, before being more systematically tackled by Nachman Aronszajn and Stefan Bergman. These spaces have various applications in complex analysis, harmonic analysis, quantum mechanics, and statistical learning theory.\nThey are now defined as Hilbert spaces of functions in which pointwise evaluations are continuous linear functionals. It then follows from Riesz' theorem that each of these functionals can be represented as an inner product with an element of this Hilbert space. One defines a two-point kernel by considering the inner products of any pair of such elements. \\ \\par \\ \nMore precisely, we have the following definition, where  $\\Omega$, as in all the paper, is a bounded connected open set of class $C^\\infty$ in $\\mathbb R^N$.\n\\begin{Definition} \\label{defRKHS}\nWe say that a Hilbert space $H$ of  real-valued functions defined on  $\\Omega$ is a reproducing kernel Hilbert space (RKHS for short) \n if for any $x$ in $\\Omega$, the evaluation mapping $E_x$ defined for any $u$ in $H$ by $E_x (u) = u(x)$ \nis linear and continuous from $H$ to $\\mathbb R$. Then, we denote by  $K_x$ the unique element in $H$ such that for any $u$ in $H$,  $E_x (u) = \\langle u,K_x\\rangle $, the inner product of $u$ and $K_x$ in $H$, given by Riesz' theorem. Finally, the two-point kernel is defined for any pair of $x$ and $y$ in $\\Omega$  by $K(x,y)= \\langle K_x,K_y\\rangle $. \n\\end{Definition}\nObserve that it follows from the properties of the inner product that the kernel $K$ is symmetric and positive definite. \nActually, a famous theorem by  E. H. Moore and N. Aronszajn, see \\cite{Aronszajn},  states that every symmetric, positive definite kernel defines a unique reproducing kernel Hilbert space. \n\n\\subsection{Lions' formula for a reproducing kernel Hilbert space of harmonic functions}\nIn \\cite{Lions}, J.L. Lions has established an interesting formula for certain reproducing kernel Hilbert spaces of harmonic functions. \nTo recall his result, let us introduce the following differential operator on functions defined on the boundary $\\partial \\Omega$. \n\\begin{Definition} \\label{DefBel}\nLet $L$ be the negative of the Laplace-Beltrami operator on $\\partial \\Omega$, and let $M  := L+ 1$. This operator $M$ is called  the  smoothed Laplace-Beltrami operator on $\\partial \\Omega$.  \n\\end{Definition}\nLet $L^2 (\\partial \\Omega)$ the Hilbert space of square integrable functions on $\\partial \\Omega$\nwith respect to the $d-1$-dimensional measure $d\\sigma$ on $\\partial \\Omega$. \nThen the  smoothed Laplace-Beltrami operator $M$ is a positive self-adjoint operator in $L^2 (\\partial \\Omega)$, so that, by Borel  functional calculus, it makes sense to speak of its powers of  any real order. This allows us to define some Sobolev spaces on $\\partial \\Omega$ for any $s$ in $\\mathbb R$ as follows.\n\\begin{Definition}\\label{def1.3}\nFor any $s$ in $\\mathbb R$,  the Sobolev space $H^s(\\partial \\Omega)$ is the closure of the smooth functions on $\\partial \\Omega$ for the norm associated with the scalar product\n\\begin{equation} \\label{inpr}\n\\langle u,v\\rangle_s := \n\\big\\langle M^{\\frac{s}{2}} \\, u,M^{\\frac{s}{2}} \\,\\,v\\big\\rangle_{L^2 (\\partial \\Omega)} .\n\\end{equation}\n\\end{Definition}\nThe Dirichlet trace $\\gamma_D$ on the boundary $\\partial \\Omega$, defined by \n\\begin{equation} \\label{gammad}\n    \\gamma_D: \\, u \\in C^\\infty (\\overline \\Omega) \\mapsto  u\\vert_{\\partial \\Omega} \\in C^\\infty (\\partial \\Omega)\n\\end{equation} \nextends \nin a continuous linear map from the  Sobolev space $H^s(\\Omega)$\nto  the Sobolev space $H^{s-\\frac12} (\\partial \\Omega)$ for all $s > \\frac12$. It does not extend to the negative Sobolev spaces $H^{t} (\\partial \\Omega)$, with $t\\leqslant 0$; with such properties. \nHowever, it extends to the subspace of harmonic functions of the Sobolev space $H^s(\\Omega)$ for any $s \\in \\mathbb R$, again as a continuous linear map  with values in  $H^{s-\\frac12} (\\partial \\Omega)$. \nActually, this holds not only for harmonic functions but also for much more general \nsolutions of elliptic equations, as established by the works of  Lions and Magenes, see \\cite{LM63,LM68}, by establishing some \n regularity properties in the case of Sobolev spaces with positive indexes first, then by using a duality argument and finally interpolation theory.\n Another approach, based on the theory of pseudodifferential operators for boundary value problems,  \n emphasized that such a property can be seen as a variant of the partial hypoellipticity of the elliptic equations and originated from H\\\"ormander in \\cite[Theorem 2.5.6]{H63} and Boutet de Monvel \\cite{B71}, see also  \\cite[Theorem 11.4]{Gr} for a more recent account of the topic. \nWith a slight abuse of notation, we keep the same notation $\\gamma_D$ for this trace map, as a continuous linear map from the subspace of harmonic functions of the Sobolev space $H^s(\\Omega)$ to $H^{s-\\frac12} (\\partial \\Omega)$ \nfor any $s \\in \\mathbb R$.\n\nNow, the space of harmonic distributions considered by Lions is the following.\n\\begin{Definition} \\label{def-Hs}\nFor any $s \\in \\mathbb R$, we define $\\mathcal H_s(\\Omega) $ as the space of harmonic functions $u$ in $ H^{s+\\frac12} (\\Omega)$  whose  trace $\\gamma_D (u)$  on $\\partial \\Omega$ belongs to $H^s(\\partial \\Omega)$.\n\\end{Definition}\nIt follows from the references above that for any \n$s$ in $\\mathbb R$,  any $u$ in  $\\mathcal H_s (\\Omega)$ is actually a $C^\\infty$ function in $\\Omega$, and for any $x$ in $\\Omega$, \n we have the Poisson formula: \n\\begin{equation} \\label{Cani1}\n    u(x)\n    = \\Big\\langle \\gamma_D (u),\\, M^{-{s}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big) \\Big\\rangle_s ,\n\\end{equation}\nwhere $G_1$ is the Green function, that is the function such that for any $x$ in $\\Omega$, \n$G_1(x,\\cdot)$  is the classical Green function with singularity at $x$, that is the solution to \n\\begin{equation} \\label{DIR}\n    -\\Delta G_1(x,\\cdot)=\\delta_x \\quad\\text{in}\\quad \\mathcal D'(\\Omega)\\qquad\\&\\qquad G_1(x,\\cdot)=0\\quad\\text{on}\\quad \\partial\\Omega, \n\\end{equation}\nand where $\\gamma_N \\big(G_1(x,\\cdot)\\big)$ denotes its Neumann trace, that is \n$$\\gamma_N \\big(G_1(x,\\cdot)\\big)(z):= \\nabla_z G_1(x,z)\\cdot\\nu(z) \\quad \\text{for}\\quad  z\\in\\partial\\Omega,$$ where $\\nu(z)$ is the outer unit normal at $z$.\nIn \\eqref{DIR}, $\\delta_x$ denotes the Dirac delta distribution at the position $x$. \nIn particular, it follows from elliptic regularity that \n\\begin{equation} \\label{REg}\n    \\forall x \\in \\Omega, \\quad   \\forall s \\in \\mathbb R, \\quad M^{-\\frac{s}{2}} \\gamma_N \\big(G_1(x,\\cdot)\\big)  \\in L^2 (\\partial\\Omega) ,\n\\end{equation}\n so that  the \nright-hand side of \\eqref{Cani1} \nmakes sense. Moreover, for any $s$ in $\\mathbb R$, the mapping that associates any $g$ in  $H^s(\\partial \\Omega)$ to the function $\\mathcal P_s [g]$ defined for any  $x$  in $\\Omega$, by \n\\begin{equation} \\label{poisson-iso}\n \\mathcal P_s[g] (x) :=   \\Big\\langle g, \\,M^{-{s}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big) \\Big\\rangle_s ,\n\\end{equation}\nis a one-to-one isometry onto $\\mathcal H_s(\\Omega) $. This allows us to transport the Hilbert structure from $H^s(\\partial \\Omega)$ to $\\mathcal H_s(\\Omega) $, that is we \n equip $\\mathcal H_s(\\Omega) $  with the scalar product, defined for any $ u,v$ in $\\mathcal H_s(\\Omega) $, by \n $\\langle \\gamma_D (u),\\gamma_D (v) \\rangle_s $. \nActually, Lions' result, see \\cite[Formula (2.13)]{Lions}, is that it is a RKHS with a  two-point kernel given as a boundary integral involving the \nGreen function $G_1$.", "context": "\\subsection{Reproducing kernel Hilbert space}\n\n\\subsection{Lions' formula for a reproducing kernel Hilbert space of harmonic functions}\nIn \\cite{Lions}, J.L. Lions has established an interesting formula for certain reproducing kernel Hilbert spaces of harmonic functions. \nTo recall his result, let us introduce the following differential operator on functions defined on the boundary $\\partial \\Omega$. \n\\begin{Definition} \\label{DefBel}\nLet $L$ be the negative of the Laplace-Beltrami operator on $\\partial \\Omega$, and let $M := L+ 1$. This operator $M$ is called the smoothed Laplace-Beltrami operator on $\\partial \\Omega$. \n\\end{Definition}\nLet $L^2 (\\partial \\Omega)$ the Hilbert space of square integrable functions on $\\partial \\Omega$\nwith respect to the $d-1$-dimensional measure $d\\sigma$ on $\\partial \\Omega$. \nThen the smoothed Laplace-Beltrami operator $M$ is a positive self-adjoint operator in $L^2 (\\partial \\Omega)$, so that, by Borel functional calculus, it makes sense to speak of its powers of any real order. This allows us to define some Sobolev spaces on $\\partial \\Omega$ for any $s$ in $\\mathbb R$ as follows.\n\\begin{Definition}\\label{def1.3}\nFor any $s$ in $\\mathbb R$, the Sobolev space $H^s(\\partial \\Omega)$ is the closure of the smooth functions on $\\partial \\Omega$ for the norm associated with the scalar product\n\\begin{equation} \\label{inpr}\n\\langle u,v\\rangle_s := \n\\big\\langle M^{\\frac{s}{2}} \\, u,M^{\\frac{s}{2}} \\,\\,v\\big\\rangle_{L^2 (\\partial \\Omega)} .\n\\end{equation}\n\\end{Definition}\nThe Dirichlet trace $\\gamma_D$ on the boundary $\\partial \\Omega$, defined by \n\\begin{equation} \\label{gammad}\n \\gamma_D: \\, u \\in C^\\infty (\\overline \\Omega) \\mapsto u\\vert_{\\partial \\Omega} \\in C^\\infty (\\partial \\Omega)\n\\end{equation} \nextends \nin a continuous linear map from the Sobolev space $H^s(\\Omega)$\nto the Sobolev space $H^{s-\\frac12} (\\partial \\Omega)$ for all $s > \\frac12$. It does not extend to the negative Sobolev spaces $H^{t} (\\partial \\Omega)$, with $t\\leqslant 0$; with such properties. \nHowever, it extends to the subspace of harmonic functions of the Sobolev space $H^s(\\Omega)$ for any $s \\in \\mathbb R$, again as a continuous linear map with values in $H^{s-\\frac12} (\\partial \\Omega)$. \nActually, this holds not only for harmonic functions but also for much more general \nsolutions of elliptic equations, as established by the works of Lions and Magenes, see \\cite{LM63,LM68}, by establishing some \n regularity properties in the case of Sobolev spaces with positive indexes first, then by using a duality argument and finally interpolation theory.\n Another approach, based on the theory of pseudodifferential operators for boundary value problems, \n emphasized that such a property can be seen as a variant of the partial hypoellipticity of the elliptic equations and originated from H\\\"ormander in \\cite[Theorem 2.5.6]{H63} and Boutet de Monvel \\cite{B71}, see also \\cite[Theorem 11.4]{Gr} for a more recent account of the topic. \nWith a slight abuse of notation, we keep the same notation $\\gamma_D$ for this trace map, as a continuous linear map from the subspace of harmonic functions of the Sobolev space $H^s(\\Omega)$ to $H^{s-\\frac12} (\\partial \\Omega)$ \nfor any $s \\in \\mathbb R$.\n\nNow, the space of harmonic distributions considered by Lions is the following.\n\\begin{Definition} \\label{def-Hs}\nFor any $s \\in \\mathbb R$, we define $\\mathcal H_s(\\Omega) $ as the space of harmonic functions $u$ in $ H^{s+\\frac12} (\\Omega)$ whose trace $\\gamma_D (u)$ on $\\partial \\Omega$ belongs to $H^s(\\partial \\Omega)$.\n\\end{Definition}\nIt follows from the references above that for any \n$s$ in $\\mathbb R$, any $u$ in $\\mathcal H_s (\\Omega)$ is actually a $C^\\infty$ function in $\\Omega$, and for any $x$ in $\\Omega$, \n we have the Poisson formula: \n\\begin{equation} \\label{Cani1}\n u(x)\n = \\Big\\langle \\gamma_D (u),\\, M^{-{s}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big) \\Big\\rangle_s ,\n\\end{equation}\nwhere $G_1$ is the Green function, that is the function such that for any $x$ in $\\Omega$, \n$G_1(x,\\cdot)$ is the classical Green function with singularity at $x$, that is the solution to \n\\begin{equation} \\label{DIR}\n -\\Delta G_1(x,\\cdot)=\\delta_x \\quad\\text{in}\\quad \\mathcal D'(\\Omega)\\qquad\\&\\qquad G_1(x,\\cdot)=0\\quad\\text{on}\\quad \\partial\\Omega, \n\\end{equation}\nand where $\\gamma_N \\big(G_1(x,\\cdot)\\big)$ denotes its Neumann trace, that is \n$$\\gamma_N \\big(G_1(x,\\cdot)\\big)(z):= \\nabla_z G_1(x,z)\\cdot\\nu(z) \\quad \\text{for}\\quad z\\in\\partial\\Omega,$$ where $\\nu(z)$ is the outer unit normal at $z$.\nIn \\eqref{DIR}, $\\delta_x$ denotes the Dirac delta distribution at the position $x$. \nIn particular, it follows from elliptic regularity that \n\\begin{equation} \\label{REg}\n \\forall x \\in \\Omega, \\quad \\forall s \\in \\mathbb R, \\quad M^{-\\frac{s}{2}} \\gamma_N \\big(G_1(x,\\cdot)\\big) \\in L^2 (\\partial\\Omega) ,\n\\end{equation}\n so that the \nright-hand side of \\eqref{Cani1} \nmakes sense. Moreover, for any $s$ in $\\mathbb R$, the mapping that associates any $g$ in $H^s(\\partial \\Omega)$ to the function $\\mathcal P_s [g]$ defined for any $x$ in $\\Omega$, by \n\\begin{equation} \\label{poisson-iso}\n \\mathcal P_s[g] (x) := \\Big\\langle g, \\,M^{-{s}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big) \\Big\\rangle_s ,\n\\end{equation}\nis a one-to-one isometry onto $\\mathcal H_s(\\Omega) $. This allows us to transport the Hilbert structure from $H^s(\\partial \\Omega)$ to $\\mathcal H_s(\\Omega) $, that is we \n equip $\\mathcal H_s(\\Omega) $ with the scalar product, defined for any $ u,v$ in $\\mathcal H_s(\\Omega) $, by \n $\\langle \\gamma_D (u),\\gamma_D (v) \\rangle_s $. \nActually, Lions' result, see \\cite[Formula (2.13)]{Lions}, is that it is a RKHS with a two-point kernel given as a boundary integral involving the \nGreen function $G_1$.", "full_context": "\\subsection{Reproducing kernel Hilbert space}\n\n\\subsection{Lions' formula for a reproducing kernel Hilbert space of harmonic functions}\nIn \\cite{Lions}, J.L. Lions has established an interesting formula for certain reproducing kernel Hilbert spaces of harmonic functions. \nTo recall his result, let us introduce the following differential operator on functions defined on the boundary $\\partial \\Omega$. \n\\begin{Definition} \\label{DefBel}\nLet $L$ be the negative of the Laplace-Beltrami operator on $\\partial \\Omega$, and let $M := L+ 1$. This operator $M$ is called the smoothed Laplace-Beltrami operator on $\\partial \\Omega$. \n\\end{Definition}\nLet $L^2 (\\partial \\Omega)$ the Hilbert space of square integrable functions on $\\partial \\Omega$\nwith respect to the $d-1$-dimensional measure $d\\sigma$ on $\\partial \\Omega$. \nThen the smoothed Laplace-Beltrami operator $M$ is a positive self-adjoint operator in $L^2 (\\partial \\Omega)$, so that, by Borel functional calculus, it makes sense to speak of its powers of any real order. This allows us to define some Sobolev spaces on $\\partial \\Omega$ for any $s$ in $\\mathbb R$ as follows.\n\\begin{Definition}\\label{def1.3}\nFor any $s$ in $\\mathbb R$, the Sobolev space $H^s(\\partial \\Omega)$ is the closure of the smooth functions on $\\partial \\Omega$ for the norm associated with the scalar product\n\\begin{equation} \\label{inpr}\n\\langle u,v\\rangle_s := \n\\big\\langle M^{\\frac{s}{2}} \\, u,M^{\\frac{s}{2}} \\,\\,v\\big\\rangle_{L^2 (\\partial \\Omega)} .\n\\end{equation}\n\\end{Definition}\nThe Dirichlet trace $\\gamma_D$ on the boundary $\\partial \\Omega$, defined by \n\\begin{equation} \\label{gammad}\n \\gamma_D: \\, u \\in C^\\infty (\\overline \\Omega) \\mapsto u\\vert_{\\partial \\Omega} \\in C^\\infty (\\partial \\Omega)\n\\end{equation} \nextends \nin a continuous linear map from the Sobolev space $H^s(\\Omega)$\nto the Sobolev space $H^{s-\\frac12} (\\partial \\Omega)$ for all $s > \\frac12$. It does not extend to the negative Sobolev spaces $H^{t} (\\partial \\Omega)$, with $t\\leqslant 0$; with such properties. \nHowever, it extends to the subspace of harmonic functions of the Sobolev space $H^s(\\Omega)$ for any $s \\in \\mathbb R$, again as a continuous linear map with values in $H^{s-\\frac12} (\\partial \\Omega)$. \nActually, this holds not only for harmonic functions but also for much more general \nsolutions of elliptic equations, as established by the works of Lions and Magenes, see \\cite{LM63,LM68}, by establishing some \n regularity properties in the case of Sobolev spaces with positive indexes first, then by using a duality argument and finally interpolation theory.\n Another approach, based on the theory of pseudodifferential operators for boundary value problems, \n emphasized that such a property can be seen as a variant of the partial hypoellipticity of the elliptic equations and originated from H\\\"ormander in \\cite[Theorem 2.5.6]{H63} and Boutet de Monvel \\cite{B71}, see also \\cite[Theorem 11.4]{Gr} for a more recent account of the topic. \nWith a slight abuse of notation, we keep the same notation $\\gamma_D$ for this trace map, as a continuous linear map from the subspace of harmonic functions of the Sobolev space $H^s(\\Omega)$ to $H^{s-\\frac12} (\\partial \\Omega)$ \nfor any $s \\in \\mathbb R$.\n\nNow, the space of harmonic distributions considered by Lions is the following.\n\\begin{Definition} \\label{def-Hs}\nFor any $s \\in \\mathbb R$, we define $\\mathcal H_s(\\Omega) $ as the space of harmonic functions $u$ in $ H^{s+\\frac12} (\\Omega)$ whose trace $\\gamma_D (u)$ on $\\partial \\Omega$ belongs to $H^s(\\partial \\Omega)$.\n\\end{Definition}\nIt follows from the references above that for any \n$s$ in $\\mathbb R$, any $u$ in $\\mathcal H_s (\\Omega)$ is actually a $C^\\infty$ function in $\\Omega$, and for any $x$ in $\\Omega$, \n we have the Poisson formula: \n\\begin{equation} \\label{Cani1}\n u(x)\n = \\Big\\langle \\gamma_D (u),\\, M^{-{s}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big) \\Big\\rangle_s ,\n\\end{equation}\nwhere $G_1$ is the Green function, that is the function such that for any $x$ in $\\Omega$, \n$G_1(x,\\cdot)$ is the classical Green function with singularity at $x$, that is the solution to \n\\begin{equation} \\label{DIR}\n -\\Delta G_1(x,\\cdot)=\\delta_x \\quad\\text{in}\\quad \\mathcal D'(\\Omega)\\qquad\\&\\qquad G_1(x,\\cdot)=0\\quad\\text{on}\\quad \\partial\\Omega, \n\\end{equation}\nand where $\\gamma_N \\big(G_1(x,\\cdot)\\big)$ denotes its Neumann trace, that is \n$$\\gamma_N \\big(G_1(x,\\cdot)\\big)(z):= \\nabla_z G_1(x,z)\\cdot\\nu(z) \\quad \\text{for}\\quad z\\in\\partial\\Omega,$$ where $\\nu(z)$ is the outer unit normal at $z$.\nIn \\eqref{DIR}, $\\delta_x$ denotes the Dirac delta distribution at the position $x$. \nIn particular, it follows from elliptic regularity that \n\\begin{equation} \\label{REg}\n \\forall x \\in \\Omega, \\quad \\forall s \\in \\mathbb R, \\quad M^{-\\frac{s}{2}} \\gamma_N \\big(G_1(x,\\cdot)\\big) \\in L^2 (\\partial\\Omega) ,\n\\end{equation}\n so that the \nright-hand side of \\eqref{Cani1} \nmakes sense. Moreover, for any $s$ in $\\mathbb R$, the mapping that associates any $g$ in $H^s(\\partial \\Omega)$ to the function $\\mathcal P_s [g]$ defined for any $x$ in $\\Omega$, by \n\\begin{equation} \\label{poisson-iso}\n \\mathcal P_s[g] (x) := \\Big\\langle g, \\,M^{-{s}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big) \\Big\\rangle_s ,\n\\end{equation}\nis a one-to-one isometry onto $\\mathcal H_s(\\Omega) $. This allows us to transport the Hilbert structure from $H^s(\\partial \\Omega)$ to $\\mathcal H_s(\\Omega) $, that is we \n equip $\\mathcal H_s(\\Omega) $ with the scalar product, defined for any $ u,v$ in $\\mathcal H_s(\\Omega) $, by \n $\\langle \\gamma_D (u),\\gamma_D (v) \\rangle_s $. \nActually, Lions' result, see \\cite[Formula (2.13)]{Lions}, is that it is a RKHS with a two-point kernel given as a boundary integral involving the \nGreen function $G_1$.\n\nOne quite explicit formula for the fractional Laplace operator of order $a$, with $a\\in (0,1)$, in $N\\in \\N^*$ dimensions, reads \n$$\n(-\\Delta)^a u (z) := c_{N,a}\\, \\pv \\int_{\\R^N}\\frac{u(z)-u(y)}{|z-y|^{N+2a}}dy,\n$$\nwith \n$$c_{N,a} :=\\pi^{N/2}a4^a\\frac{\\Gamma(\\frac{N+2a}{2})}{\\Gamma(1-a)},$$\nwhere $\\Gamma$ is the gamma function, and where $\\pv$ refers to the Cauchy principal value. \nAbove $u$ is a function from $\\R^N$ with real values. \nThe reason for the presence of the normalization constant $c_{N,a}$\n is to match with the Fourier definition which sets the fractional Laplace operator $(-\\Delta)^a$ as the Fourier multiplier of symbol \n$|\\xi|^{2a}$. \n The Green function $G_a(x,y)$ associated to the operator $(-\\Delta)^a$ and the homogeneous Dirichlet condition \nis the solution to \n\\begin{equation} \\label{DIRa}\n (-\\Delta)^a G_a(x,\\cdot)=\\delta_x \\quad\\text{in}\\quad \\mathcal D'(\\Omega)\\qquad\\&\\qquad G_a(x,\\cdot)=0\\quad\\text{in}\\quad \\R^N\\setminus\\Omega.\n\\end{equation}\nSuch a Green function plays the same role for the operator $(-\\Delta)^a$ as the Green function $G_1$ for the classical Laplace operator. \nIn the companion paper \\cite{sidy-franck_1}, we computed the following Hadamard-type variational formula for the fractional Green function $G_a(x,y)$: for any $x$ and $y$ in $\\Omega$, \n\\begin{align}\\label{var-green}\n DG_a (\\alpha) (x,y) = \\Gamma^2(1+a)\\int_{\\partial\\Omega} \\, \\big(\\gamma_0^a(G_a(x,\\cdot))\\big)(z) \\, \\big(\\gamma^a_0(G_a(y,\\cdot))\\big)(z)\n \\alpha(z) \\;d\\sigma(z).\n\\end{align}\nAbove, $ \\Gamma^2(1+a)$ denotes the square of $ \\Gamma(1+a)$ while $\\gamma_0^a$ denotes the fractional trace of order $a$, that is \n$$\\gamma_0^a(G_a(x,\\cdot):= \\gamma_D \\big(\\frac{G_a(x,\\cdot)}{d^a}\\big),$$ where $d$ is the distance function to the boundary $\\partial\\Omega$. \nThe right-hand side of \\eqref{var-green} makes sense because by boundary regularity, for any $x$ in $\\Omega$, $\\gamma_0^a(G_a(x,\\cdot))$ is $C^\\infty$ on ${\\partial\\Omega}$, see \\cite[Equation (2.14)]{Grubb-2014}. \nLet us point out that the analysis performed in \\cite{sidy-franck_1} allows for more general perturbations. \nWe also refer to the earlier works \\cite{DalibardVaret,DFW} on the subject.\n\n\\begin{proof}\nLet $u$ in $\\mathcal H_{a,s}(\\Omega)$. Recalling that $\\gamma^{a-1}_0(u)$ is in $H^{2\\theta}(\\partial\\Omega)$,\nby density, we have a sequence $(g_n)_{n\\in\\mathbb N}$ in $C^\\infty(\\partial\\Omega)$ such that\n\\begin{equation} \\label{ref-jeudi}\ng_n \\to \\gamma^{a-1}_0(u) \\quad \\text{in } H^{2\\theta}(\\partial\\Omega).\n\\end{equation}\nBy Theorem \\ref{f}, for any $n$ in $\\mathbb N$, we set \n $u_n := \\mathcal P_{a,s}[g_n]$. This \nis the unique $u_n$ in $H^{(a-1)(s+2a)}(\\Omega)$\nsolution of\n\\begin{equation*}\n(-\\Delta)^a u_n = 0 \\ \\text{in }\\Omega \\ \\ \\& \\ \\ \\gamma^{a-1}_0(u_n) = g_n \\ \\text{on }\\partial\\Omega.\n\\end{equation*}\nWe fix $x$ in $\\Omega$. Let $\\rho$ in $C^\\infty_0(\\mathbb R^N)$ with $\\rho \\ge 0$ and\n$\\int_{\\mathbb R^N}\\rho = 1$, and set $\\rho_\\varepsilon(z) := \\varepsilon^{-N}\\rho(z/\\varepsilon)$.\nFor $\\varepsilon>0$ small enough so that $\\mathrm{supp}\\,\\rho_\\varepsilon(\\cdot-x)\\subset \\Omega$,\nwe define\n\\begin{equation} \\label{*}\nv_{x,\\varepsilon}(z) := \\int_\\Omega G_a(z,y)\\rho_\\varepsilon(y-x)\\,dy.\n\\end{equation}\nThen $v_{x,\\varepsilon}=0$ in $\\mathbb R^N\\setminus\\Omega$, and in $\\mathcal D'(\\Omega)$ we have\n\\begin{equation} \\label{**}\n(-\\Delta)^a v_{x,\\varepsilon} = \\rho_\\varepsilon(\\cdot-x).\n\\end{equation}\nMoreover, since $\\rho_\\varepsilon(\\cdot-x)$ is smooth with compact support in $\\Omega$,\nthe function $v_{x,\\varepsilon}$ is regular enough for the hypotheses of \\cite[Theorem 5.1]{Grubb-2020}.\nWe apply Grubb's exact Green formula to the pair $(u_n,v_{x,\\varepsilon})$.\nSince $(-\\Delta)^a u_n = 0$ in $\\Omega$ and $(-\\Delta)^a v_{x,\\varepsilon}=\\rho_\\varepsilon(\\cdot-x)$ in $\\Omega$,\nwe obtain\n\\begin{equation} \\label{***}\n\\int_\\Omega u_n(y)\\rho_\\varepsilon(y-x)\\,dy\n= \\Gamma(a)\\Gamma(a+1)\\int_{\\partial\\Omega} g_n(z)\\,\\gamma_0^a(v_{x,\\varepsilon})(z)\\,d\\sigma(z).\n\\end{equation}\n\nOur main result establishes a counterpart of Theorem \\ref{Lions-theo} in the case of $a$-harmonic functions by showing that, for an appropriate range of indexes, the spaces $\\mathcal H_{a,s}(\\Omega)$ are RKHS with two-point kernels given by a boundary integral of some derivatives of the Green function $G_a(x,\\cdot)$ of the fractional Laplacian with a singularity at $x$.\nWe recall that $M$ is the smoothed Laplace-Beltrami operator on $\\partial \\Omega$, see Definition \\ref{DefBel}.\n\\begin{thm} \\label{s-Lions-theo}\nLet $a\\in (0,1)$, $s>-a-\\frac{1}{2}$, and\n $\\theta$ be given by \\eqref{theta}. \nThen the space $\\mathcal H_{a,s}(\\Omega)$ is a RKHS, and its two-point kernel, which we denote $K_{a,s}$, is given, for any $x$ and $y$ in $\\Omega$, by \n \\begin{equation} \\label{l-formu-s}\n K_{a,s}(x,y)= \\Gamma^2(a) \\Gamma^2(a+1) \\int_{\\partial \\Omega} \\Big( M^{-\\theta} \\, \n \\gamma_0^{a}(G_a(x,\\cdot))\n \\Big) \\Big(M^{-\\theta} \\, \\gamma_0^{a}(G_a(y,\\cdot)) \\Big) d\\sigma.\n \\end{equation}\n\\end{thm}\n\nWe also recall the following classical Poisson formula: for any solution $(u,p)$ of the steady Stokes problem \\eqref{eq_stoke} we have \n\\begin{equation} \\label{eq_Green-St}\nu(x)\n= \\int_{\\partial \\Omega} ( \\Sigma (\\mathfrak G(x,\\cdot),\\mathfrak P(x,\\cdot)) n) \\cdot u \\,d\\sigma.\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq_Newt2}\n\\Sigma(u,p) = 2 D(u) - p \\mathbb{I}_3 \\quad \\text{ where } 2 D(u):= \\nabla u + (\\nabla u)^T , \n\\end{equation}\nand $\\mathbb{I}_3$ is the $3 \\times 3$ identity matrix \nand $(\\mathfrak G(x,\\cdot),\\mathfrak P(x,\\cdot))$ is the Green function associated with \n the Stokes system that is, the unique tensor such that for any $b \\in \\R^3$, $(u,p) := (\\mathfrak G(x,\\cdot)b,\\mathfrak P(x,\\cdot)b)$ is the unique solution to the problem: \n\\begin{equation*} \n\\left\\{\n\\begin{array}{rcl}\n- \\Delta u + \\nabla p &= b \\delta_x & \\, \\\\\n\\operatorname{div} u &= 0 &\\, \n\\end{array}\n\\right. \\quad \\text{ in $\\Omega$} , \\end{equation*}\nwith \\begin{equation*} \n u =0\\quad\\text{on}\\quad \\partial\\Omega.\n\\end{equation*}\n A reasoning similar to the ones above leads to the conclusion that the space $\\mathfrak H_s$ is a RKHS with the following\n Lions type formula for its $2$-points kernel $\\mathfrak K_s$: \n$$ \\mathfrak K_s (x,y) = \\int_{\\partial\\Omega} \\Big((M^{-\\frac{s}{2}} \\, \\gamma_D \\big(\\Sigma(\\mathfrak G(x,\\cdot),\\mathfrak P(x,\\cdot))n\\big)\\Big)(z) \\cdot \\Big(M^{-\\frac{s}{2}} \\, \\gamma_D \\big( \\Sigma(\\mathfrak G(y,\\cdot),\\mathfrak P(y,\\cdot))n \\big) \\Big)(z)\\,d\\sigma(z) .\n$$\n On the other hand, it is known since some works by Simon, see for instance \\cite{Simon} and the recent works by \\cite{KU,Oz}, \n that \n the Hadamard variation formula in the case of the Stokes equations reads: for any $b \\in \\R^3$,\n $$D(\\mathfrak G b) (\\alpha) (x,y) = \n \\int_{\\partial\\Omega} \\gamma_N \\big(\\mathfrak G(x,\\cdot) b\\big)(z) \\, \\gamma_N \\big(\\mathfrak G(y,\\cdot)b\\big)(z) \\, \\alpha(z)d\\sigma(z) , $$\nwhere $\\alpha\\in C^\\infty(\\partial\\Omega)$.", "post_theorem_intro_text_len": 4751, "post_theorem_intro_text": "Let us point out that the integral in \\eqref{l-formu} makes sense, and is finite, since  for any  $x$  in $\\Omega$, for any $s$ in $\\mathbb R$, \n$M^{-\\frac{s}{2}} \\gamma_N \\big((G_1(x,\\cdot)\\big)   $\nis in $L^2 (\\partial\\Omega)$.\nActually, Lions dealt with some function spaces associated with second-order elliptic boundary value problems, see \\cite{Lions}.  \nLions used a variational approach and obtained his formula through a penalization limit. \nAnother, more direct approach to Theorem \\ref{Lions-theo} was proposed later in \\cite{ELPL}, together with some extensions to more general elliptic operators of even orders. \nWe reproduce below their proof of \\eqref{l-formu} as a preparation for our extension to the case of fractional Laplace operators.\n\n \\begin{proof}\n It follows from the Poisson formula \\eqref{Cani1}, \\eqref{REg}\nand the Cauchy-Schwarz inequality that for any $x$ in $\\Omega$, the evaluation mapping $E_x$, see Definition \\ref{defRKHS},   is continuous from $\\mathcal H_s(\\Omega) $ to $\\mathbb R$. \nSince $\\mathcal H_s(\\Omega) $ is a Hilbert space, \n it follows from Riesz' theorem that  for any  $x$  in $\\Omega$, there exists $K_x$ in \n$\\mathcal H_s(\\Omega) $ \nsuch that for any $u$ in $\\mathcal H_s(\\Omega) $, \n\\begin{equation} \\label{Cani2}\nu(x) = \\langle \\gamma_D (u) ,\\, K_x \\rangle_s = \\langle M^{\\frac{s}{2}} \\, \\gamma_D (u) ,\\,  M^{\\frac{s}{2}} \\, K_x \\rangle_{L^2 (\\partial \\Omega)}.\n\\end{equation}\nGathering \\eqref{Cani1} and \\eqref{Cani2} we deduce that for any $u$ in  $\\mathcal H_s (\\Omega)$, \n\\begin{equation*} \n  \\Big\\langle M^{\\frac{s}{2}} \\, \\gamma_D (u) , \\, M^{-\\frac{s}{2}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big)  - M^{\\frac{s}{2}} \\, K_x \\Big\\rangle_{L^2 (\\partial \\Omega)}=0.\n\\end{equation*}\n Then by a density argument on $u$, we conclude that \n $$ M^{\\frac{s}{2}} \\, K_x = M^{-\\frac{s}{2}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big)   ,$$\n almost everywhere on $\\partial\\Omega$.\n Therefore,   for any  $x$ and $y$ in $\\Omega$,\n \\begin{align*}\n     K_s(x,y) &=       \\Big\\langle M^{\\frac{s}{2}} \\, K_x, \\,M^{\\frac{s}{2}} \\, K_y  \\Big\\rangle _{L^2 (\\partial \\Omega) } \n \\\\ &=\\Big\\langle M^{-\\frac{s}{2}} \\, \\gamma_N \\big( G_1 (x,\\cdot)\\big) ,\\, M^{-\\frac{s}{2}} \\, \\gamma_N \\big( G_1 (y,\\cdot)\\big) \\Big\\rangle _{L^2 (\\partial \\Omega) },\n \\end{align*}\n which is the desired conclusion.\n \\end{proof}\n\n\\subsection{Resemblance with the Hadamard variation formula}\nAn interesting observation in \\cite{ELPL} is the resemblance of \\eqref{l-formu} with the Hadamard variation formula \nfor the Green function of the classical Laplacian, which we now recall. \nThis formula expresses the derivative of the Green function $G_1$ with respect to the domain. \nMore precisely, one considers some perturbations $\\Omega_t$ of the domain $\\Omega$ in the normal direction by modifying the  boundary $\\partial\\Omega$ of $\\Omega$ in \n$$\\partial\\Omega_t=\\Big\\{y=x+t \\alpha(x)\\nu(x), \\;x\\in\\partial\\Omega\\Big\\},$$\nwhere $\\alpha\\in C^\\infty(\\partial\\Omega)$, and $t$ runs in a open interval containing $0$. \nLet us rename $G_\\Omega$ the Green function\n $G_1$ associated with the domain $\\Omega$, as defined by \\eqref{DIR}, to emphasize here the dependence on the domain. Then we define the derivative of $G_\\Omega$ with respect to the domain in  $\\alpha$ as \n\\begin{align}\n     DG_\\Omega (\\alpha) (x,y):=\\lim_{t\\to 0, t \\neq 0} \\, \\frac{G_{\\Omega_t}(x,y)-G_\\Omega(x,y)}{t} .\n\\end{align}\nThen the pioneering discovery by Hadamard, see \\cite{Hadamard}, is that not only does this limit exist, but it is even given by the following explicit integral formula: for any $x$ and $y$ in $\\Omega$, \n\\begin{align} \\label{Hadamard}\n    DG_\\Omega (\\alpha) (x,y) =\\int_{\\partial\\Omega} \\gamma_N \\big(G_1(x,\\cdot)\\big)(z) \\, \\gamma_N \\big(G_1(y,\\cdot)\\big)(z) \\,  \\alpha(z)d\\sigma(z) .\n\\end{align}\nLet us mention that a rigorous proof of Hadamard's formula was given later on by  Garabedian who also considered more general perturbations. After these pioneering works, shape derivative computations have received significant interest and several extensions of Hadamard's formula for more general elliptic boundary value problems were obtained, with various techniques. We refer here to \\cite{HenrotPierre,Sokolowski} for more. \n\\medskip\n\nFollowing \\cite{ELPL}, \n it is now clear that  \\eqref{l-formu} and \\eqref{Hadamard} look very similar, in particular in the case where $s=0$ for which the only difference is the presence of the extra factor $\\alpha$ in \\eqref{Hadamard}.\n In \\cite{ELPL}, it is said furthermore that the authors  \"are of the opinion that the connection between Hadamard variation formula and the reproducing kernel just indicated, in the classical harmonic case, seems to be an isolated phenomenon peculiar to the second order case.\"", "sketch": "Using the Poisson formula \\eqref{Cani1}, \\eqref{REg} and the Cauchy--Schwarz inequality, one shows that for any $x\\in\\Omega$ the evaluation mapping $E_x$ is continuous from $\\mathcal H_s(\\Omega)$ to $\\mathbb R$. Since $\\mathcal H_s(\\Omega)$ is a Hilbert space, Riesz' theorem gives for each $x\\in\\Omega$ an element $K_x\\in\\mathcal H_s(\\Omega)$ such that for all $u\\in\\mathcal H_s(\\Omega)$,\n\\[\n u(x)=\\langle \\gamma_D(u),K_x\\rangle_s=\\langle M^{\\frac{s}{2}}\\gamma_D(u),\\,M^{\\frac{s}{2}}K_x\\rangle_{L^2(\\partial\\Omega)}. \\tag{\\eqref{Cani2}}\n\\]\nGathering \\eqref{Cani1} and \\eqref{Cani2} yields, for any $u\\in\\mathcal H_s(\\Omega)$,\n\\[\n\\Big\\langle M^{\\frac{s}{2}}\\gamma_D(u),\\, M^{-\\frac{s}{2}}\\gamma_N\\big(G_1(x,\\cdot)\\big)-M^{\\frac{s}{2}}K_x\\Big\\rangle_{L^2(\\partial\\Omega)}=0.\n\\]\nBy a density argument on $u$, one concludes that $M^{\\frac{s}{2}}K_x=M^{-\\frac{s}{2}}\\gamma_N\\big(G_1(x,\\cdot)\\big)$ a.e. on $\\partial\\Omega$. Therefore, for any $x,y\\in\\Omega$,\n\\[\nK_s(x,y)=\\langle M^{\\frac{s}{2}}K_x,\\,M^{\\frac{s}{2}}K_y\\rangle_{L^2(\\partial\\Omega)}\n=\\big\\langle M^{-\\frac{s}{2}}\\gamma_N\\big(G_1(x,\\cdot)\\big),\\,M^{-\\frac{s}{2}}\\gamma_N\\big(G_1(y,\\cdot)\\big)\\big\\rangle_{L^2(\\partial\\Omega)},\n\\]\nwhich is exactly \\eqref{l-formu}.", "expanded_sketch": "Using the Poisson formula\n\\begin{equation} \\label{Cani1}\n    u(x)\n    = \\Big\\langle \\gamma_D (u),\\, M^{-{s}} \\,\\gamma_N \\big( G_1 (x,\\cdot)\\big) \\Big\\rangle_s ,\n\\end{equation}\n\\begin{equation} \\label{REg}\n    \\forall x \\in \\Omega, \\quad   \\forall s \\in \\R, \\quad M^{-\\frac{s}{2}} \\gamma_N \\big(G_1(x,\\cdot)\\big)  \\in L^2 (\\partial\\Omega) ,\n\\end{equation}\nand the Cauchy--Schwarz inequality, one shows that for any $x\\in\\Omega$ the evaluation mapping $E_x$ is continuous from $\\mathcal H_s(\\Omega)$ to $\\mathbb R$. Since $\\mathcal H_s(\\Omega)$ is a Hilbert space, Riesz' theorem gives for each $x\\in\\Omega$ an element $K_x\\in\\mathcal H_s(\\Omega)$ such that for all $u\\in\\mathcal H_s(\\Omega)$,\n\\begin{equation} \\label{Cani2}\\nu(x) = \\langle \\gamma_D (u) ,\\, K_x \\rangle_s = \\langle M^{\\frac{s}{2}} \\, \\gamma_D (u) ,\\,  M^{\\frac{s}{2}} \\, K_x \\rangle_{L^2 (\\partial \\Omega)}.\n\\end{equation}\nGathering the Poisson formula above and the preceding identity yields, for any $u\\in\\mathcal H_s(\\Omega)$,\n\\[\n\\Big\\langle M^{\\frac{s}{2}}\\gamma_D(u),\\, M^{-\\frac{s}{2}}\\gamma_N\\big(G_1(x,\\cdot)\\big)-M^{\\frac{s}{2}}K_x\\Big\\rangle_{L^2(\\partial\\Omega)}=0.\n\\]\nBy a density argument on $u$, one concludes that $M^{\\frac{s}{2}}K_x=M^{-\\frac{s}{2}}\\gamma_N\\big(G_1(x,\\cdot)\\big)$ a.e. on $\\partial\\Omega$. Therefore, for any $x,y\\in\\Omega$,\n\\[\nK_s(x,y)=\\langle M^{\\frac{s}{2}}K_x,\\,M^{\\frac{s}{2}}K_y\\rangle_{L^2(\\partial\\Omega)}\n=\\big\\langle M^{-\\frac{s}{2}}\\gamma_N\\big(G_1(x,\\cdot)\\big),\\,M^{-\\frac{s}{2}}\\gamma_N\\big(G_1(y,\\cdot)\\big)\\big\\rangle_{L^2(\\partial\\Omega)},\n\\]\nwhich is exactly\n\\begin{equation} \\label{l-formu}\n K_s (x,y)=  \\int_{\\partial\\Omega} \\Big((M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(x,\\cdot)\\big)\\Big)(z) \\, \\Big(M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(y,\\cdot)\\big) \\Big)(z)\\,d\\sigma(z) .\n \\end{equation}\n.", "expanded_theorem": "\\label{Lions-theo}\nFor any $s$ in $\\mathbb R$, the space $\\mathcal H_s(\\Omega) $  is a RKHS and its two-point kernel $K_s$ is given,  for any pair of $x$ and $y$ in $\\Omega$,  by \n \\begin{equation} \\label{l-formu}\n K_s (x,y)=  \\int_{\\partial\\Omega} \\Big((M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(x,\\cdot)\\big)\\Big)(z) \\, \\Big(M^{-\\frac{s}{2}} \\, \\gamma_N \\big((G_1(y,\\cdot)\\big) \\Big)(z)\\,d\\sigma(z) .\n \\end{equation}", "theorem_type": ["Universal", "Classification or Bijection"], "mcq": {"question": "Let \\(\\Omega\\) be a domain with boundary \\(\\partial\\Omega\\). Let \\(L\\) denote the negative Laplace--Beltrami operator on \\(\\partial\\Omega\\), set \\(M:=L+1\\), and for \\(s\\in\\mathbb R\\) let \\(H^s(\\partial\\Omega)\\) be the Sobolev space on \\(\\partial\\Omega\\) defined from \\(M\\). Let \\(\\gamma_D\\) and \\(\\gamma_N\\) denote the Dirichlet and Neumann traces, respectively. For each \\(s\\in\\mathbb R\\), define\n\\[\n\\mathcal H_s(\\Omega):=\\{u\\in H^{s+\\frac12}(\\Omega): u \\text{ is harmonic in }\\Omega\\text{ and }\\gamma_D u\\in H^s(\\partial\\Omega)\\}.\n\\]\nFor \\(x\\in\\Omega\\), let \\(G_1(x,\\cdot)\\) be the Dirichlet Green function, i.e. the solution of\n\\[\n-\\Delta G_1(x,\\cdot)=\\delta_x\\quad\\text{in }\\mathcal D'(\\Omega),\n\\qquad\nG_1(x,\\cdot)=0\\quad\\text{on }\\partial\\Omega.\n\\]\nWhich statement holds for every real \\(s\\)?", "correct_choice": {"label": "A", "text": "The space \\(\\mathcal H_s(\\Omega)\\) is a reproducing kernel Hilbert space, and its two-point kernel \\(K_s\\) is given for every \\(x,y\\in\\Omega\\) by\n\\[\nK_s(x,y)=\\int_{\\partial\\Omega}\\big(M^{-\\frac s2}\\gamma_N(G_1(x,\\cdot))\\big)(z)\\,\\big(M^{-\\frac s2}\\gamma_N(G_1(y,\\cdot))\\big)(z)\\,d\\sigma(z).\n\\]"}, "choices": [{"label": "B", "text": "The space \\(\\mathcal H_s(\\Omega)\\) is a reproducing kernel Hilbert space, and its two-point kernel \\(K_s\\) is given for every \\(x,y\\in\\Omega\\) by\n\\[\nK_s(x,y)=\\int_{\\partial\\Omega}\\big(M^{-s}\\gamma_N(G_1(x,\\cdot))\\big)(z)\\,\\big(M^{-s}\\gamma_N(G_1(y,\\cdot))\\big)(z)\\,d\\sigma(z).\n\\]"}, {"label": "C", "text": "For every real \\(s\\), the space \\(\\mathcal H_s(\\Omega)\\) is a reproducing kernel Hilbert space."}, {"label": "D", "text": "For every real \\(s\\), the space \\(\\mathcal H_s(\\Omega)\\) is a reproducing kernel Hilbert space, and there exists a constant \\(c_s>0\\) such that for every \\(x,y\\in\\Omega\\),\n\\[\nK_s(x,y)=c_s\\int_{\\partial\\Omega}\\big(M^{-\\frac s2}\\gamma_N(G_1(x,\\cdot))\\big)(z)\\,\\big(M^{-\\frac s2}\\gamma_N(G_1(y,\\cdot))\\big)(z)\\,d\\sigma(z).\n\\]"}, {"label": "E", "text": "The space \\(\\mathcal H_s(\\Omega)\\) is a reproducing kernel Hilbert space, and its two-point kernel \\(K_s\\) is given for every \\(x,y\\in\\Omega\\) by\n\\[\nK_s(x,y)=\\int_{\\partial\\Omega}\\big(M^{-\\frac s2}\\gamma_D(G_1(x,\\cdot))\\big)(z)\\,\\big(M^{-\\frac s2}\\gamma_D(G_1(y,\\cdot))\\big)(z)\\,d\\sigma(z).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "half-power of M in the L^2 identification", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "explicit boundary integral formula for the reproducing kernel", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "exact normalization from the Riesz representer identity", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "Neumann trace in the Poisson formula replaced by Dirichlet trace", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem introduces the objects and asks for the valid conclusion, but it does not explicitly reveal the reproducing-kernel formula or otherwise single out the correct option."}, "TAS": {"score": 1, "justification": "The item is close to theorem recall: one choice is essentially the full theorem statement verbatim. However, it is not a pure restatement because the student must distinguish it from nearby alternatives and a weaker true statement."}, "GPS": {"score": 2, "justification": "Selecting the correct answer requires meaningful discrimination among subtle variants: the power of M, Neumann vs. Dirichlet trace, exact normalization, and the stronger explicit formula versus a merely weaker true claim."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: exponent mismatch, trace confusion, loss of exact normalization, and choosing a weaker but true statement instead of the strongest correct one."}, "total_score": 7, "overall_assessment": "A strong MCQ with high-quality distractors and genuine mathematical discrimination, though it remains somewhat close to direct theorem recognition rather than full derivational reasoning."}}
{"id": "2602.09749v1", "paper_link": "http://arxiv.org/abs/2602.09749v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:gen_upper_box_precise}\n    Assume that $F\\subseteq \\mathbb{R}^p$ is a connected self-similar set with finitely many directions\n    and $\\dim_H F = s > 1$.\n    Then for the generic 1-Hölder-$\\alpha$ function $f:F\\to \\mathbb{R}$ and Lebesgue almost every $r\\in f(F)$,\n    \\begin{displaymath}\n    \\overline{\\dim}_B f^{-1}(r)=s-\\alpha.\n    \\end{displaymath}", "start_pos": 8348, "end_pos": 8742, "label": "thm:gen_upper_box_precise"}, "ref_dict": {"thm:gen_upper_box_precise": "\\begin{theorem} \\label{thm:gen_upper_box_precise}\n    Assume that $F\\subseteq \\mathbb{R}^p$ is a connected self-similar set with finitely many directions\n    and $\\dim_H F = s > 1$.\n    Then for the generic 1-Hölder-$\\alpha$ function $f:F\\to \\mathbb{R}$ and Lebesgue almost every $r\\in f(F)$,\n    \\begin{displaymath}\n    \\overline{\\dim}_B f^{-1}(r)=s-\\alpha.\n    \\end{displaymath}\n\\end{theorem}", "prop:wenxi": "\\begin{theorem}[{\\cite[Corollary~1]{wenxi}}] \\label{prop:wenxi}\n    Assume that $A\\subseteq \\mathbb{R}^p$ is a self-similar set with finitely many directions and $\\dim_H A>1$.\n    Then for a full measure set $\\mathcal{W}_A$ of the $(p-1)$-dimensional linear subspaces, if $W\\in \\mathcal{W}_A$, then \n    $$\\mathcal{H}^1(\\mathrm{Pr}_{W^{\\perp}}(A)\\setminus \\{a\\in W^{\\perp}: \\dim (A\\cap (W+a))= \\dim_H A  -1\\}) = 0,$$\n    that is standard dimension drop occurs on a set of full measure. (Here $W^{\\perp}$ denotes the 1-dimensional orthocomplement of $W$, \n    and $\\mathrm{Pr}_{W^{\\perp}}$ is the projection onto it.)\n\\end{theorem}", "lemma:values_admitted_in_simplices": "\\begin{lemma} \\label{lemma:values_admitted_in_simplices}\n    Let $F\\subseteq \\mathbb{R}^p$ be compact and $\\mathcal{S}$ be a finite family of simplices covering $F$ such that $\\inte (S) \\cap F\\neq \\emptyset$ for all $S\\in \\mathcal{S}$.\n    Assume moreover that $f:\\bigcup_{S\\in\\mathcal{S}} S \\to \\mathbb{R}$ is a piecewise affine function, affine on members of $\\mathcal{S}$. \n    Then for almost every $u\\in\\mathbb{R}^p$, we have\n    \\begin{equation} \\label{eq:values_admitted_in_interior}\n    \\bigcup_{S\\in\\mathcal{S}} f(S\\cap (F+u)) = \\bigcup_{S\\in\\mathcal{S}} f\\left(\\inte S\\cap (F+u)\\right) \\text{ a. e.}\n    \\end{equation}\n    i.e., for almost every $u$ almost every value admitted on $F+u$ by $f$ is admitted in the interior of a simplex.\n\\end{lemma}"}, "pre_theorem_intro_text_len": 2301, "pre_theorem_intro_text": "Level sets of continuous functions defined on some domain $F\\subseteq \\mathbb{R}^p$ convey information about the dimensionality of $F$. Roughly speaking, a generic continuous function should have large level sets\nif $F$ itself is thick in an intuitive sense, and small otherwise. This heuristic prompted the investigation of the size of such level sets in terms of Hausdorff dimension\n(see \\cite{BBElevel, BBEtoph, BK}), with the quantity providing the answer (the topological Hausdorff dimension) attracting interest in its own right \\cite{MaZha}, even in physics\n\\cite{Balankintoph, Balankinfracspace, Balankintransport, Balankintoph2, Balankinfluid}.\n\nThe authors with Gáspár Vértesy initiated the study of the Hausdorff and box dimension of level sets of generic 1-Hölder-$\\alpha$ functions \\cite{ BuczolichMagaVertesy2025, sierc, sier, BUCZOLICH2024531}. \nUnder self-similarity conditions on the domain $F$, they proved that for every $\\alpha$ and for either of the Hausdorff/lower box/upper box dimension, there is a constant which is the dimension\nof Lebesgue almost every level set of the generic 1-Hölder-$\\alpha$ function. For any underlying dimension, these values form an interesting spectrum describing the geometry of $F$. \nHowever, dealing with this spectrum is far from trivial in any of the cases: the existing literature focuses on particular choices of $F$ and provides mostly estimates at the price of\na significant amount of technicalities, highly depending on the structure of $F$. In this direction, in \\cite{BuczolichMagaVertesy2025}, \nwe managed to provide lower and upper bounds on the Hausdorff spectrum for the Sierpiński triangle which asymptotically coincide as $\\alpha\\to 0+$.\nHowever, no explicit formulae or general machinery have been found before for describing any of the spectra for a larger class of fractals. In this respect this paper is the first one. \n\n\\subsection{The main result}\n\nWe give an explicit formula for the upper box dimension $\\overline{\\dim}_B$ of Lebesgue almost every level set of a generic 1-Hölder-$\\alpha$ function \nvalid for connected self-similar sets subject to the geometric condition of having {\\it finitely many directions}, i.e.,\nwhose defining similarities have orthogonal parts generating a finite subgroup of the orthogonal group.", "context": "Level sets of continuous functions defined on some domain $F\\subseteq \\mathbb{R}^p$ convey information about the dimensionality of $F$. Roughly speaking, a generic continuous function should have large level sets\nif $F$ itself is thick in an intuitive sense, and small otherwise. This heuristic prompted the investigation of the size of such level sets in terms of Hausdorff dimension\n(see \\cite{BBElevel, BBEtoph, BK}), with the quantity providing the answer (the topological Hausdorff dimension) attracting interest in its own right \\cite{MaZha}, even in physics\n\\cite{Balankintoph, Balankinfracspace, Balankintransport, Balankintoph2, Balankinfluid}.\n\nThe authors with Gáspár Vértesy initiated the study of the Hausdorff and box dimension of level sets of generic 1-Hölder-$\\alpha$ functions \\cite{ BuczolichMagaVertesy2025, sierc, sier, BUCZOLICH2024531}. \nUnder self-similarity conditions on the domain $F$, they proved that for every $\\alpha$ and for either of the Hausdorff/lower box/upper box dimension, there is a constant which is the dimension\nof Lebesgue almost every level set of the generic 1-Hölder-$\\alpha$ function. For any underlying dimension, these values form an interesting spectrum describing the geometry of $F$. \nHowever, dealing with this spectrum is far from trivial in any of the cases: the existing literature focuses on particular choices of $F$ and provides mostly estimates at the price of\na significant amount of technicalities, highly depending on the structure of $F$. In this direction, in \\cite{BuczolichMagaVertesy2025}, \nwe managed to provide lower and upper bounds on the Hausdorff spectrum for the Sierpiński triangle which asymptotically coincide as $\\alpha\\to 0+$.\nHowever, no explicit formulae or general machinery have been found before for describing any of the spectra for a larger class of fractals. In this respect this paper is the first one.\n\n\\subsection{The main result}\n\nWe give an explicit formula for the upper box dimension $\\overline{\\dim}_B$ of Lebesgue almost every level set of a generic 1-Hölder-$\\alpha$ function \nvalid for connected self-similar sets subject to the geometric condition of having {\\it finitely many directions}, i.e.,\nwhose defining similarities have orthogonal parts generating a finite subgroup of the orthogonal group.", "full_context": "Level sets of continuous functions defined on some domain $F\\subseteq \\mathbb{R}^p$ convey information about the dimensionality of $F$. Roughly speaking, a generic continuous function should have large level sets\nif $F$ itself is thick in an intuitive sense, and small otherwise. This heuristic prompted the investigation of the size of such level sets in terms of Hausdorff dimension\n(see \\cite{BBElevel, BBEtoph, BK}), with the quantity providing the answer (the topological Hausdorff dimension) attracting interest in its own right \\cite{MaZha}, even in physics\n\\cite{Balankintoph, Balankinfracspace, Balankintransport, Balankintoph2, Balankinfluid}.\n\nThe authors with Gáspár Vértesy initiated the study of the Hausdorff and box dimension of level sets of generic 1-Hölder-$\\alpha$ functions \\cite{ BuczolichMagaVertesy2025, sierc, sier, BUCZOLICH2024531}. \nUnder self-similarity conditions on the domain $F$, they proved that for every $\\alpha$ and for either of the Hausdorff/lower box/upper box dimension, there is a constant which is the dimension\nof Lebesgue almost every level set of the generic 1-Hölder-$\\alpha$ function. For any underlying dimension, these values form an interesting spectrum describing the geometry of $F$. \nHowever, dealing with this spectrum is far from trivial in any of the cases: the existing literature focuses on particular choices of $F$ and provides mostly estimates at the price of\na significant amount of technicalities, highly depending on the structure of $F$. In this direction, in \\cite{BuczolichMagaVertesy2025}, \nwe managed to provide lower and upper bounds on the Hausdorff spectrum for the Sierpiński triangle which asymptotically coincide as $\\alpha\\to 0+$.\nHowever, no explicit formulae or general machinery have been found before for describing any of the spectra for a larger class of fractals. In this respect this paper is the first one.\n\n\\subsection{The main result}\n\nWe give an explicit formula for the upper box dimension $\\overline{\\dim}_B$ of Lebesgue almost every level set of a generic 1-Hölder-$\\alpha$ function \nvalid for connected self-similar sets subject to the geometric condition of having {\\it finitely many directions}, i.e.,\nwhose defining similarities have orthogonal parts generating a finite subgroup of the orthogonal group.\n\n\\begin{abstract}\n In the previous decades, the size of level sets of functions have been extensively studied in various setups involving different regularity properties and\n size notions. In the case of Hölder functions, the authors have provided various bounds, but to date no explicit formulae have been found for any studied dimension \n and the results were valid only about very specific fractals.\n In this paper, for the first time, we have a result valid for a large class of self-similar sets, namely we prove that for these \n fractals Lebesgue almost every level set of the generic 1-Hölder-$\\alpha$ function defined on $F\\subseteq \\mathbb{R}^p$ has \n upper box dimension $\\dim_H F - \\alpha$. \n\\end{abstract}\n\nWe give an explicit formula for the upper box dimension $\\overline{\\dim}_B$ of Lebesgue almost every level set of a generic 1-Hölder-$\\alpha$ function \nvalid for connected self-similar sets subject to the geometric condition of having {\\it finitely many directions}, i.e.,\nwhose defining similarities have orthogonal parts generating a finite subgroup of the orthogonal group.\n\n\\subsection{Organization of the paper}\n\nGiven a function $f:F\\to \\mathbb{R}$, we put $D_{\\overline{B}}^f(r,F)=\\overline{\\dim}_B f^{-1}(r)$. For $f:F\\to \\mathbb{R}$, we denote by $D_{\\overline{B}*}^f(F)$ the essential infimum\nof the upper box dimension of non-empty level sets, i.e., \n\\begin{displaymath}\n D_{\\overline{B}*}^f(F)= \\begin{cases}\n \\inf\\{d: \\lambda\\{r : r\\in f(F) \\text{ and } D_{\\overline{B}}^f(r,F)\\leq{d}\\}>0\\}, & \\text{if } \\lambda(f(F))>0, \\\\\n 0, & \\text{if } \\lambda(f(F))=0.\n \\end{cases}\n\\end{displaymath}\nWe denote by $\\mg_{1,\\aaa}(F)$, or by simply $\\mg_{1,\\aaa}$ the set of dense $G_{\\ddd}$\nsets in $C_{1}^{\\aaa}(F)$. Whether $D_{\\overline{B}*}^f(F)$ has a generic value is a priori unclear, nevertheless the following quantity is always well-defined:\n\\begin{equation}\\label{def:generic_upper}\n D_{\\overline{B}*}(\\aaa,F)=\\inf_{\\cag\\in \\mg_{1,\\aaa}}\\sup\\{ D_{\\overline{B}*}^{f}(F):f\\in \\cag \\}.\n \\end{equation}\nAs established by \\cite[Theorem~3.8]{BUCZOLICH2024531}, subject to certain conditions on the connectivity properties of $F$, $D_{\\overline{B}*}(\\aaa,F)$ is indeed simply the generic value of $D_{\\overline{B}^*}^f(F)$.\nIn Section \\ref{sec:box_dimension_locally_connected} we cite these results and discuss how they extend to locally connected $F$. \nWe note that even if $D_{\\overline{B}*}(\\aaa,F)$ is the generic value of $D_{\\overline{B}^*}^f(F)$, it does not necessarily coincide with $D_{\\overline{B}}^f(r,F)$ for almost every $r\\in f(F)$.\nIn particular it is not clear a priori why in the setup of Theorem \\ref{thm:gen_upper_box_precise} almost every level set of the generic 1-Hölder-$\\alpha$ should have the same upper box dimension.\n\n\\begin{lemma}[{\\cite[Lemma~3.9]{BUCZOLICH2024531}}] \\label{lemma:from_covering_to_box}\n Assume that $0<\\alpha\\leq 1$, and the measurable set $F\\subseteq \\mathbb{R}^p$ has coverings $(\\mathcal{S}_n)_{n=1}^{\\infty}$, \n such that with some constants $C,l,\\rho>1$:\n \\begin{itemize}\n \\item the cardinality of $\\mathcal{S}_n$ is at most $Cl^n$ for some $C,l>1$,\n \\item if $S\\in \\mathcal{S}_n$, then $\\diam(S)\\leq C\\rho^{-n}$.\n \\end{itemize}\n Then for any $f\\in C_{\\alpha}^{1}(F)$ and almost every $r\\in \\mathbb{R}$, we have\n $\\overline{\\dim}_B(f^{-1}(r))\\leq \\frac{\\log l}{\\log \\rho} - \\alpha$. In particular,\n $D_{\\overline{B}*}(\\alpha, F)\\leq \\frac{\\log l}{\\log \\rho} - \\alpha$.\n\\end{lemma}\n\n\\begin{corollary} \\label{corollary:lower_box_dim_upper_bound}\n Assume that $0<\\alpha\\leq 1$, and let $F\\subseteq \\mathbb{R}^p$ be measurable. Then for every $f\\in C_{\\alpha}^{1}(F)$ and almost every $r\\in \\mathbb{R}$, we have\n $\\overline{\\dim}_B(f^{-1}(r))\\leq \\underline{\\dim}_B(F) - \\alpha$. In particular,\n $D_{\\overline{B}*}(\\alpha, F)\\leq \\underline{\\dim}_B(F) - \\alpha$.\n\\end{corollary}\n\n\\begin{theorem}[{\\cite[Corollary~1]{wenxi}}] \\label{prop:wenxi}\n Assume that $A\\subseteq \\mathbb{R}^p$ is a self-similar set with finitely many directions and $\\dim_H A>1$.\n Then for a full measure set $\\mathcal{W}_A$ of the $(p-1)$-dimensional linear subspaces, if $W\\in \\mathcal{W}_A$, then \n $$\\mathcal{H}^1(\\mathrm{Pr}_{W^{\\perp}}(A)\\setminus \\{a\\in W^{\\perp}: \\dim (A\\cap (W+a))= \\dim_H A -1\\}) = 0,$$\n that is standard dimension drop occurs on a set of full measure. (Here $W^{\\perp}$ denotes the 1-dimensional orthocomplement of $W$, \n and $\\mathrm{Pr}_{W^{\\perp}}$ is the projection onto it.)\n\\end{theorem}\n\n\\begin{lemma} \\label{lemma:values_admitted_in_simplices}\n Let $F\\subseteq \\mathbb{R}^p$ be compact and $\\mathcal{S}$ be a finite family of simplices covering $F$ such that $\\inte (S) \\cap F\\neq \\emptyset$ for all $S\\in \\mathcal{S}$.\n Assume moreover that $f:\\bigcup_{S\\in\\mathcal{S}} S \\to \\mathbb{R}$ is a piecewise affine function, affine on members of $\\mathcal{S}$. \n Then for almost every $u\\in\\mathbb{R}^p$, we have\n \\begin{equation} \\label{eq:values_admitted_in_interior}\n \\bigcup_{S\\in\\mathcal{S}} f(S\\cap (F+u)) = \\bigcup_{S\\in\\mathcal{S}} f\\left(\\inte S\\cap (F+u)\\right) \\text{ a. e.}\n \\end{equation}\n i.e., for almost every $u$ almost every value admitted on $F+u$ by $f$ is admitted in the interior of a simplex.\n\\end{lemma}\n\n\\begin{theorem} \\label{thm:gen_upper_box_precise}\n    Assume that $F\\subseteq \\mathbb{R}^p$ is a connected self-similar set with finitely many directions\n    and $\\dim_H F = s > 1$.\n    Then for the generic 1-Hölder-$\\alpha$ function $f:F\\to \\mathbb{R}$ and Lebesgue almost every $r\\in f(F)$,\n    \\begin{displaymath}\n    \\overline{\\dim}_B f^{-1}(r)=s-\\alpha.\n    \\end{displaymath}\n\\end{theorem}", "post_theorem_intro_text_len": 1538, "post_theorem_intro_text": "\\subsection{Organization of the paper}\n\nIn Section \\ref{sec:prelim} we introduce some notation and recall some relevant tools. \n\nIn Section \\ref{sec:box_dimension_locally_connected} we provide a lemma on how a dense set of functions over which the upper box dimension of level sets is controlled can be used\nto get bounds over a dense $G_\\delta$ set of functions. Such a lemma was previously provided by \\cite{BUCZOLICH2024531} assuming a more elaborate connectivity property,\nthis time it will be sufficient to assume local connectivity.\n\nIn Section \\ref{sec:lemmas} we prove two key lemmas, which do the heavy lifting for Section \\ref{sec:main}, in which we prove Theorem \\ref{thm:gen_upper_box_precise}. \nWe note that in \\cite[Theorem~3.6]{BUCZOLICH2024531}, we proved a lower bound for the upper box dimension of level sets under certain conditions using Theorem \\ref{prop:wenxi} (identical to\n\\cite[Proposition~2.6]{BUCZOLICH2024531}). A subtle point in applying this statement was not addressed explicitly there.\nLemma \\ref{lemma:values_admitted_in_simplices} fills this important geometric gap, essentially stating\nthat if we have a piecewise affine function, defined on simplices containing our fractal $F$ then with an arbitrarily small translation\nwe can make sure that the function takes Lebesgue almost all of its values in the interior of the simplices. This fact is quite easy if the Hausdorff dimension of $F$ is less than $2$,\nfor higher dimensions it needs some extra care.\n\nWe conclude the paper with some open problems.", "sketch": "In Section \\ref{sec:lemmas} the authors “prove two key lemmas, which do the heavy lifting for Section \\ref{sec:main}, in which we prove Theorem \\ref{thm:gen_upper_box_precise}.” They also note that a “subtle point in applying” a previously used statement (Theorem \\ref{prop:wenxi}) “was not addressed explicitly” earlier, and that Lemma \\ref{lemma:values_admitted_in_simplices} “fills this important geometric gap,” namely: for a piecewise affine function on simplices containing the fractal $F$, “with an arbitrarily small translation we can make sure that the function takes Lebesgue almost all of its values in the interior of the simplices.” This geometric fact is said to be “quite easy if the Hausdorff dimension of $F$ is less than $2$,” but “for higher dimensions it needs some extra care.” Additionally, Section \\ref{sec:box_dimension_locally_connected} provides a lemma showing how “a dense set of functions over which the upper box dimension of level sets is controlled can be used to get bounds over a dense $G_\\delta$ set of functions,” now assuming local connectivity.", "expanded_sketch": "Next the authors prove two key lemmas, which do the heavy lifting for the later section where they establish the main theorem. They also note that a subtle point in applying the following previously used statement was not addressed explicitly earlier.\n\n\\begin{theorem}[{\\cite[Corollary~1]{wenxi}}] \\label{prop:wenxi}\n    Assume that $A\\subseteq \\mathbb{R}^p$ is a self-similar set with finitely many directions and $\\dim_H A>1$.\n    Then for a full measure set $\\mathcal{W}_A$ of the $(p-1)$-dimensional linear subspaces, if $W\\in \\mathcal{W}_A$, then \n    $$\\mathcal{H}^1(\\mathrm{Pr}_{W^{\\perp}}(A)\\setminus \\{a\\in W^{\\perp}: \\dim (A\\cap (W+a))= \\dim_H A  -1\\}) = 0,$$\n    that is standard dimension drop occurs on a set of full measure. (Here $W^{\\perp}$ denotes the 1-dimensional orthocomplement of $W$, \n    and $\\mathrm{Pr}_{W^{\\perp}}$ is the projection onto it.)\n\\end{theorem}\n\nThey explain that the following lemma fills this important geometric gap, namely: for a piecewise affine function on simplices containing the fractal $F$, with an arbitrarily small translation we can make sure that the function takes Lebesgue almost all of its values in the interior of the simplices.\n\n\\begin{lemma} \\label{lemma:values_admitted_in_simplices}\n    Let $F\\subseteq \\mathbb{R}^p$ be compact and $\\mathcal{S}$ be a finite family of simplices covering $F$ such that $\\inte (S) \\cap F\\neq \\emptyset$ for all $S\\in \\mathcal{S}$.\n    Assume moreover that $f:\\bigcup_{S\\in\\mathcal{S}} S \\to \\mathbb{R}$ is a piecewise affine function, affine on members of $\\mathcal{S}$. \n    Then for almost every $u\\in\\mathbb{R}^p$, we have\n    \\begin{equation} \\label{eq:values_admitted_in_interior}\n    \\bigcup_{S\\in\\mathcal{S}} f(S\\cap (F+u)) = \\bigcup_{S\\in\\mathcal{S}} f\\left(\\inte S\\cap (F+u)\\right) \\text{ a. e.}\n    \\end{equation}\n    i.e., for almost every $u$ almost every value admitted on $F+u$ by $f$ is admitted in the interior of a simplex.\n\\end{lemma}\n\nThis geometric fact is said to be quite easy if the Hausdorff dimension of $F$ is less than $2$, but for higher dimensions it needs some extra care. Additionally, later the authors provide a lemma showing how a dense set of functions over which the upper box dimension of level sets is controlled can be used to get bounds over a dense $G_\\delta$ set of functions, now assuming local connectivity.", "expanded_theorem": "\\label{thm:gen_upper_box_precise}\n    Assume that $F\\subseteq \\mathbb{R}^p$ is a connected self-similar set with finitely many directions\n    and $\\dim_H F = s > 1$.\n    Then for the generic 1-Hölder-$\\alpha$ function $f:F\\to \\mathbb{R}$ and Lebesgue almost every $r\\in f(F)$,\n    \\begin{displaymath}\n    \\overline{\\dim}_B f^{-1}(r)=s-\\alpha.\n    \\end{displaymath}", "theorem_type": ["Universal", "Inequality or Bound"], "mcq": {"question": "Fix $0<\\alpha\\le 1$. Let $F\\subseteq \\mathbb{R}^p$ be a connected self-similar set with finitely many directions, meaning that in some defining iterated function system for $F$, the orthogonal parts of the similarities generate a finite subgroup of the orthogonal group, and assume that $\\dim_H F=s>1$. Let $C_\\alpha^1(F)$ denote the space of 1-H\\\"older-$\\alpha$ functions $f:F\\to\\mathbb{R}$, i.e. functions satisfying $|f(x)-f(y)|\\le |x-y|^\\alpha$ for all $x,y\\in F$. Which statement holds for generic $f\\in C_\\alpha^1(F)$ (that is, for all $f$ in a dense $G_\\delta$ subset of $C_\\alpha^1(F)$) concerning the level sets $f^{-1}(r)$ for Lebesgue almost every $r\\in f(F)$?", "correct_choice": {"label": "A", "text": "For Lebesgue almost every $r\\in f(F)$, the upper box dimension of the level set satisfies\n$$\\overline{\\dim}_B f^{-1}(r)=s-\\alpha.$$"}, "choices": [{"label": "B", "text": "For Lebesgue almost every $r\\in f(F)$, the upper box dimension of the level set satisfies\n$$\\overline{\\dim}_B f^{-1}(r)=s-1.$$"}, {"label": "C", "text": "For Lebesgue almost every $r\\in f(F)$, the upper box dimension of the level set satisfies\n$$\\overline{\\dim}_B f^{-1}(r)\\le s-\\alpha.$$"}, {"label": "D", "text": "For every generic $f\\in C_\\alpha^1(F)$, there is a full measure set $R_f\\subseteq f(F)$ such that for every $r\\in R_f$,\n$$\\dim_H f^{-1}(r)=s-\\alpha.$$"}, {"label": "E", "text": "There exists a dense $G_\\delta$ set $\\mathcal G\\subseteq C_\\alpha^1(F)$ such that for every $f\\in \\mathcal G$ there exists a full measure set $R_f\\subseteq f(F)$ with\n$$\\overline{\\dim}_B f^{-1}(r)\\ge s-\\alpha$$\nfor all $r\\in R_f$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "trace_identity", "tampered_component": "dimension-drop amount replaced by fixed codimension one", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "dropped the lower-bound/equality conclusion, keeping only the upper bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "upper box dimension replaced by Hausdorff dimension", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "retained only the lower-bound side coming from interior-value construction and omitted the matching upper-bound mechanism", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the exact conclusion. It specifies the hypotheses and asks which quantitative level-set statement is valid, but it does not explicitly give the equality, quantifiers, or generic/almost-every distinctions needed to identify the correct choice."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: the stem lists the hypotheses and the options are variants of the theorem’s conclusion. However, it is not purely tautological, since the choices differ in dimension formula, equality vs. inequality, and quantifier strength."}, "GPS": {"score": 1, "justification": "Some reasoning is required to choose between nearby claims, especially the weaker true upper bound, the strengthened quantifiers, and the incorrect replacement of \u001falpha by 1. Still, the question mainly tests recognition of the precise theorem statement rather than deeper derivation."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically meaningful. They reflect common failure modes: confusing s-\u001falpha with s-1, accepting a weaker bound instead of the sharp equality, and overlooking the importance of 'generic' and 'almost every r' quantifiers."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with no answer leakage and high-quality distractors, though it leans more toward precise theorem recall than genuinely generative mathematical reasoning."}}
{"id": "2602.09749v1", "paper_link": "http://arxiv.org/abs/2602.09749v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:gen_upper_box_precise}\n    Assume that $F\\subseteq \\mathbb{R}^p$ is a connected self-similar set with finitely many directions\n    and $\\dim_H F = s > 1$.\n    Then for the generic 1-Hölder-$\\alpha$ function $f:F\\to \\mathbb{R}$ and Lebesgue almost every $r\\in f(F)$,\n    \\begin{displaymath}\n    \\overline{\\dim}_B f^{-1}(r)=s-\\alpha.\n    \\end{displaymath}", "start_pos": 8348, "end_pos": 8742, "label": "thm:gen_upper_box_precise"}, "ref_dict": {"thm:gen_upper_box_precise": "\\begin{theorem} \\label{thm:gen_upper_box_precise}\n    Assume that $F\\subseteq \\mathbb{R}^p$ is a connected self-similar set with finitely many directions\n    and $\\dim_H F = s > 1$.\n    Then for the generic 1-Hölder-$\\alpha$ function $f:F\\to \\mathbb{R}$ and Lebesgue almost every $r\\in f(F)$,\n    \\begin{displaymath}\n    \\overline{\\dim}_B f^{-1}(r)=s-\\alpha.\n    \\end{displaymath}\n\\end{theorem}", "prop:wenxi": "\\begin{theorem}[{\\cite[Corollary~1]{wenxi}}] \\label{prop:wenxi}\n    Assume that $A\\subseteq \\mathbb{R}^p$ is a self-similar set with finitely many directions and $\\dim_H A>1$.\n    Then for a full measure set $\\mathcal{W}_A$ of the $(p-1)$-dimensional linear subspaces, if $W\\in \\mathcal{W}_A$, then \n    $$\\mathcal{H}^1(\\mathrm{Pr}_{W^{\\perp}}(A)\\setminus \\{a\\in W^{\\perp}: \\dim (A\\cap (W+a))= \\dim_H A  -1\\}) = 0,$$\n    that is standard dimension drop occurs on a set of full measure. (Here $W^{\\perp}$ denotes the 1-dimensional orthocomplement of $W$, \n    and $\\mathrm{Pr}_{W^{\\perp}}$ is the projection onto it.)\n\\end{theorem}", "lemma:values_admitted_in_simplices": "\\begin{lemma} \\label{lemma:values_admitted_in_simplices}\n    Let $F\\subseteq \\mathbb{R}^p$ be compact and $\\mathcal{S}$ be a finite family of simplices covering $F$ such that $\\inte (S) \\cap F\\neq \\emptyset$ for all $S\\in \\mathcal{S}$.\n    Assume moreover that $f:\\bigcup_{S\\in\\mathcal{S}} S \\to \\mathbb{R}$ is a piecewise affine function, affine on members of $\\mathcal{S}$. \n    Then for almost every $u\\in\\mathbb{R}^p$, we have\n    \\begin{equation} \\label{eq:values_admitted_in_interior}\n    \\bigcup_{S\\in\\mathcal{S}} f(S\\cap (F+u)) = \\bigcup_{S\\in\\mathcal{S}} f\\left(\\inte S\\cap (F+u)\\right) \\text{ a. e.}\n    \\end{equation}\n    i.e., for almost every $u$ almost every value admitted on $F+u$ by $f$ is admitted in the interior of a simplex.\n\\end{lemma}"}, "pre_theorem_intro_text_len": 2301, "pre_theorem_intro_text": "Level sets of continuous functions defined on some domain $F\\subseteq \\mathbb{R}^p$ convey information about the dimensionality of $F$. Roughly speaking, a generic continuous function should have large level sets\nif $F$ itself is thick in an intuitive sense, and small otherwise. This heuristic prompted the investigation of the size of such level sets in terms of Hausdorff dimension\n(see \\cite{BBElevel, BBEtoph, BK}), with the quantity providing the answer (the topological Hausdorff dimension) attracting interest in its own right \\cite{MaZha}, even in physics\n\\cite{Balankintoph, Balankinfracspace, Balankintransport, Balankintoph2, Balankinfluid}.\n\nThe authors with Gáspár Vértesy initiated the study of the Hausdorff and box dimension of level sets of generic 1-Hölder-$\\alpha$ functions \\cite{ BuczolichMagaVertesy2025, sierc, sier, BUCZOLICH2024531}. \nUnder self-similarity conditions on the domain $F$, they proved that for every $\\alpha$ and for either of the Hausdorff/lower box/upper box dimension, there is a constant which is the dimension\nof Lebesgue almost every level set of the generic 1-Hölder-$\\alpha$ function. For any underlying dimension, these values form an interesting spectrum describing the geometry of $F$. \nHowever, dealing with this spectrum is far from trivial in any of the cases: the existing literature focuses on particular choices of $F$ and provides mostly estimates at the price of\na significant amount of technicalities, highly depending on the structure of $F$. In this direction, in \\cite{BuczolichMagaVertesy2025}, \nwe managed to provide lower and upper bounds on the Hausdorff spectrum for the Sierpiński triangle which asymptotically coincide as $\\alpha\\to 0+$.\nHowever, no explicit formulae or general machinery have been found before for describing any of the spectra for a larger class of fractals. In this respect this paper is the first one. \n\n\\subsection{The main result}\n\nWe give an explicit formula for the upper box dimension $\\overline{\\dim}_B$ of Lebesgue almost every level set of a generic 1-Hölder-$\\alpha$ function \nvalid for connected self-similar sets subject to the geometric condition of having {\\it finitely many directions}, i.e.,\nwhose defining similarities have orthogonal parts generating a finite subgroup of the orthogonal group.", "context": "Level sets of continuous functions defined on some domain $F\\subseteq \\mathbb{R}^p$ convey information about the dimensionality of $F$. Roughly speaking, a generic continuous function should have large level sets\nif $F$ itself is thick in an intuitive sense, and small otherwise. This heuristic prompted the investigation of the size of such level sets in terms of Hausdorff dimension\n(see \\cite{BBElevel, BBEtoph, BK}), with the quantity providing the answer (the topological Hausdorff dimension) attracting interest in its own right \\cite{MaZha}, even in physics\n\\cite{Balankintoph, Balankinfracspace, Balankintransport, Balankintoph2, Balankinfluid}.\n\nThe authors with Gáspár Vértesy initiated the study of the Hausdorff and box dimension of level sets of generic 1-Hölder-$\\alpha$ functions \\cite{ BuczolichMagaVertesy2025, sierc, sier, BUCZOLICH2024531}. \nUnder self-similarity conditions on the domain $F$, they proved that for every $\\alpha$ and for either of the Hausdorff/lower box/upper box dimension, there is a constant which is the dimension\nof Lebesgue almost every level set of the generic 1-Hölder-$\\alpha$ function. For any underlying dimension, these values form an interesting spectrum describing the geometry of $F$. \nHowever, dealing with this spectrum is far from trivial in any of the cases: the existing literature focuses on particular choices of $F$ and provides mostly estimates at the price of\na significant amount of technicalities, highly depending on the structure of $F$. In this direction, in \\cite{BuczolichMagaVertesy2025}, \nwe managed to provide lower and upper bounds on the Hausdorff spectrum for the Sierpiński triangle which asymptotically coincide as $\\alpha\\to 0+$.\nHowever, no explicit formulae or general machinery have been found before for describing any of the spectra for a larger class of fractals. In this respect this paper is the first one.\n\n\\subsection{The main result}\n\nWe give an explicit formula for the upper box dimension $\\overline{\\dim}_B$ of Lebesgue almost every level set of a generic 1-Hölder-$\\alpha$ function \nvalid for connected self-similar sets subject to the geometric condition of having {\\it finitely many directions}, i.e.,\nwhose defining similarities have orthogonal parts generating a finite subgroup of the orthogonal group.", "full_context": "Level sets of continuous functions defined on some domain $F\\subseteq \\mathbb{R}^p$ convey information about the dimensionality of $F$. Roughly speaking, a generic continuous function should have large level sets\nif $F$ itself is thick in an intuitive sense, and small otherwise. This heuristic prompted the investigation of the size of such level sets in terms of Hausdorff dimension\n(see \\cite{BBElevel, BBEtoph, BK}), with the quantity providing the answer (the topological Hausdorff dimension) attracting interest in its own right \\cite{MaZha}, even in physics\n\\cite{Balankintoph, Balankinfracspace, Balankintransport, Balankintoph2, Balankinfluid}.\n\nThe authors with Gáspár Vértesy initiated the study of the Hausdorff and box dimension of level sets of generic 1-Hölder-$\\alpha$ functions \\cite{ BuczolichMagaVertesy2025, sierc, sier, BUCZOLICH2024531}. \nUnder self-similarity conditions on the domain $F$, they proved that for every $\\alpha$ and for either of the Hausdorff/lower box/upper box dimension, there is a constant which is the dimension\nof Lebesgue almost every level set of the generic 1-Hölder-$\\alpha$ function. For any underlying dimension, these values form an interesting spectrum describing the geometry of $F$. \nHowever, dealing with this spectrum is far from trivial in any of the cases: the existing literature focuses on particular choices of $F$ and provides mostly estimates at the price of\na significant amount of technicalities, highly depending on the structure of $F$. In this direction, in \\cite{BuczolichMagaVertesy2025}, \nwe managed to provide lower and upper bounds on the Hausdorff spectrum for the Sierpiński triangle which asymptotically coincide as $\\alpha\\to 0+$.\nHowever, no explicit formulae or general machinery have been found before for describing any of the spectra for a larger class of fractals. In this respect this paper is the first one.\n\n\\subsection{The main result}\n\nWe give an explicit formula for the upper box dimension $\\overline{\\dim}_B$ of Lebesgue almost every level set of a generic 1-Hölder-$\\alpha$ function \nvalid for connected self-similar sets subject to the geometric condition of having {\\it finitely many directions}, i.e.,\nwhose defining similarities have orthogonal parts generating a finite subgroup of the orthogonal group.\n\n\\begin{abstract}\n In the previous decades, the size of level sets of functions have been extensively studied in various setups involving different regularity properties and\n size notions. In the case of Hölder functions, the authors have provided various bounds, but to date no explicit formulae have been found for any studied dimension \n and the results were valid only about very specific fractals.\n In this paper, for the first time, we have a result valid for a large class of self-similar sets, namely we prove that for these \n fractals Lebesgue almost every level set of the generic 1-Hölder-$\\alpha$ function defined on $F\\subseteq \\mathbb{R}^p$ has \n upper box dimension $\\dim_H F - \\alpha$. \n\\end{abstract}\n\nWe give an explicit formula for the upper box dimension $\\overline{\\dim}_B$ of Lebesgue almost every level set of a generic 1-Hölder-$\\alpha$ function \nvalid for connected self-similar sets subject to the geometric condition of having {\\it finitely many directions}, i.e.,\nwhose defining similarities have orthogonal parts generating a finite subgroup of the orthogonal group.\n\n\\subsection{Organization of the paper}\n\nGiven a function $f:F\\to \\mathbb{R}$, we put $D_{\\overline{B}}^f(r,F)=\\overline{\\dim}_B f^{-1}(r)$. For $f:F\\to \\mathbb{R}$, we denote by $D_{\\overline{B}*}^f(F)$ the essential infimum\nof the upper box dimension of non-empty level sets, i.e., \n\\begin{displaymath}\n D_{\\overline{B}*}^f(F)= \\begin{cases}\n \\inf\\{d: \\lambda\\{r : r\\in f(F) \\text{ and } D_{\\overline{B}}^f(r,F)\\leq{d}\\}>0\\}, & \\text{if } \\lambda(f(F))>0, \\\\\n 0, & \\text{if } \\lambda(f(F))=0.\n \\end{cases}\n\\end{displaymath}\nWe denote by $\\mg_{1,\\aaa}(F)$, or by simply $\\mg_{1,\\aaa}$ the set of dense $G_{\\ddd}$\nsets in $C_{1}^{\\aaa}(F)$. Whether $D_{\\overline{B}*}^f(F)$ has a generic value is a priori unclear, nevertheless the following quantity is always well-defined:\n\\begin{equation}\\label{def:generic_upper}\n D_{\\overline{B}*}(\\aaa,F)=\\inf_{\\cag\\in \\mg_{1,\\aaa}}\\sup\\{ D_{\\overline{B}*}^{f}(F):f\\in \\cag \\}.\n \\end{equation}\nAs established by \\cite[Theorem~3.8]{BUCZOLICH2024531}, subject to certain conditions on the connectivity properties of $F$, $D_{\\overline{B}*}(\\aaa,F)$ is indeed simply the generic value of $D_{\\overline{B}^*}^f(F)$.\nIn Section \\ref{sec:box_dimension_locally_connected} we cite these results and discuss how they extend to locally connected $F$. \nWe note that even if $D_{\\overline{B}*}(\\aaa,F)$ is the generic value of $D_{\\overline{B}^*}^f(F)$, it does not necessarily coincide with $D_{\\overline{B}}^f(r,F)$ for almost every $r\\in f(F)$.\nIn particular it is not clear a priori why in the setup of Theorem \\ref{thm:gen_upper_box_precise} almost every level set of the generic 1-Hölder-$\\alpha$ should have the same upper box dimension.\n\n\\begin{lemma}[{\\cite[Lemma~3.9]{BUCZOLICH2024531}}] \\label{lemma:from_covering_to_box}\n Assume that $0<\\alpha\\leq 1$, and the measurable set $F\\subseteq \\mathbb{R}^p$ has coverings $(\\mathcal{S}_n)_{n=1}^{\\infty}$, \n such that with some constants $C,l,\\rho>1$:\n \\begin{itemize}\n \\item the cardinality of $\\mathcal{S}_n$ is at most $Cl^n$ for some $C,l>1$,\n \\item if $S\\in \\mathcal{S}_n$, then $\\diam(S)\\leq C\\rho^{-n}$.\n \\end{itemize}\n Then for any $f\\in C_{\\alpha}^{1}(F)$ and almost every $r\\in \\mathbb{R}$, we have\n $\\overline{\\dim}_B(f^{-1}(r))\\leq \\frac{\\log l}{\\log \\rho} - \\alpha$. In particular,\n $D_{\\overline{B}*}(\\alpha, F)\\leq \\frac{\\log l}{\\log \\rho} - \\alpha$.\n\\end{lemma}\n\n\\begin{corollary} \\label{corollary:lower_box_dim_upper_bound}\n Assume that $0<\\alpha\\leq 1$, and let $F\\subseteq \\mathbb{R}^p$ be measurable. Then for every $f\\in C_{\\alpha}^{1}(F)$ and almost every $r\\in \\mathbb{R}$, we have\n $\\overline{\\dim}_B(f^{-1}(r))\\leq \\underline{\\dim}_B(F) - \\alpha$. In particular,\n $D_{\\overline{B}*}(\\alpha, F)\\leq \\underline{\\dim}_B(F) - \\alpha$.\n\\end{corollary}\n\n\\begin{theorem}[{\\cite[Corollary~1]{wenxi}}] \\label{prop:wenxi}\n Assume that $A\\subseteq \\mathbb{R}^p$ is a self-similar set with finitely many directions and $\\dim_H A>1$.\n Then for a full measure set $\\mathcal{W}_A$ of the $(p-1)$-dimensional linear subspaces, if $W\\in \\mathcal{W}_A$, then \n $$\\mathcal{H}^1(\\mathrm{Pr}_{W^{\\perp}}(A)\\setminus \\{a\\in W^{\\perp}: \\dim (A\\cap (W+a))= \\dim_H A -1\\}) = 0,$$\n that is standard dimension drop occurs on a set of full measure. (Here $W^{\\perp}$ denotes the 1-dimensional orthocomplement of $W$, \n and $\\mathrm{Pr}_{W^{\\perp}}$ is the projection onto it.)\n\\end{theorem}\n\n\\begin{lemma} \\label{lemma:values_admitted_in_simplices}\n Let $F\\subseteq \\mathbb{R}^p$ be compact and $\\mathcal{S}$ be a finite family of simplices covering $F$ such that $\\inte (S) \\cap F\\neq \\emptyset$ for all $S\\in \\mathcal{S}$.\n Assume moreover that $f:\\bigcup_{S\\in\\mathcal{S}} S \\to \\mathbb{R}$ is a piecewise affine function, affine on members of $\\mathcal{S}$. \n Then for almost every $u\\in\\mathbb{R}^p$, we have\n \\begin{equation} \\label{eq:values_admitted_in_interior}\n \\bigcup_{S\\in\\mathcal{S}} f(S\\cap (F+u)) = \\bigcup_{S\\in\\mathcal{S}} f\\left(\\inte S\\cap (F+u)\\right) \\text{ a. e.}\n \\end{equation}\n i.e., for almost every $u$ almost every value admitted on $F+u$ by $f$ is admitted in the interior of a simplex.\n\\end{lemma}\n\n\\begin{theorem} \\label{thm:gen_upper_box_precise}\n    Assume that $F\\subseteq \\mathbb{R}^p$ is a connected self-similar set with finitely many directions\n    and $\\dim_H F = s > 1$.\n    Then for the generic 1-Hölder-$\\alpha$ function $f:F\\to \\mathbb{R}$ and Lebesgue almost every $r\\in f(F)$,\n    \\begin{displaymath}\n    \\overline{\\dim}_B f^{-1}(r)=s-\\alpha.\n    \\end{displaymath}\n\\end{theorem}", "post_theorem_intro_text_len": 1538, "post_theorem_intro_text": "\\subsection{Organization of the paper}\n\nIn Section \\ref{sec:prelim} we introduce some notation and recall some relevant tools. \n\nIn Section \\ref{sec:box_dimension_locally_connected} we provide a lemma on how a dense set of functions over which the upper box dimension of level sets is controlled can be used\nto get bounds over a dense $G_\\delta$ set of functions. Such a lemma was previously provided by \\cite{BUCZOLICH2024531} assuming a more elaborate connectivity property,\nthis time it will be sufficient to assume local connectivity.\n\nIn Section \\ref{sec:lemmas} we prove two key lemmas, which do the heavy lifting for Section \\ref{sec:main}, in which we prove Theorem \\ref{thm:gen_upper_box_precise}. \nWe note that in \\cite[Theorem~3.6]{BUCZOLICH2024531}, we proved a lower bound for the upper box dimension of level sets under certain conditions using Theorem \\ref{prop:wenxi} (identical to\n\\cite[Proposition~2.6]{BUCZOLICH2024531}). A subtle point in applying this statement was not addressed explicitly there.\nLemma \\ref{lemma:values_admitted_in_simplices} fills this important geometric gap, essentially stating\nthat if we have a piecewise affine function, defined on simplices containing our fractal $F$ then with an arbitrarily small translation\nwe can make sure that the function takes Lebesgue almost all of its values in the interior of the simplices. This fact is quite easy if the Hausdorff dimension of $F$ is less than $2$,\nfor higher dimensions it needs some extra care.\n\nWe conclude the paper with some open problems.", "sketch": "In Section \\ref{sec:lemmas} the authors “prove two key lemmas, which do the heavy lifting for Section \\ref{sec:main}, in which we prove Theorem \\ref{thm:gen_upper_box_precise}.” They also note that a “subtle point in applying” a previously used statement (Theorem \\ref{prop:wenxi}) “was not addressed explicitly” earlier, and that Lemma \\ref{lemma:values_admitted_in_simplices} “fills this important geometric gap,” namely: for a piecewise affine function on simplices containing the fractal $F$, “with an arbitrarily small translation we can make sure that the function takes Lebesgue almost all of its values in the interior of the simplices.” This geometric fact is said to be “quite easy if the Hausdorff dimension of $F$ is less than $2$,” but “for higher dimensions it needs some extra care.” Additionally, Section \\ref{sec:box_dimension_locally_connected} provides a lemma showing how “a dense set of functions over which the upper box dimension of level sets is controlled can be used to get bounds over a dense $G_\\delta$ set of functions,” now assuming local connectivity.", "expanded_sketch": "Next the authors prove two key lemmas, which do the heavy lifting for the later section where they establish the main theorem. They also note that a subtle point in applying the following previously used statement was not addressed explicitly earlier.\n\n\\begin{theorem}[{\\cite[Corollary~1]{wenxi}}] \\label{prop:wenxi}\n    Assume that $A\\subseteq \\mathbb{R}^p$ is a self-similar set with finitely many directions and $\\dim_H A>1$.\n    Then for a full measure set $\\mathcal{W}_A$ of the $(p-1)$-dimensional linear subspaces, if $W\\in \\mathcal{W}_A$, then \n    $$\\mathcal{H}^1(\\mathrm{Pr}_{W^{\\perp}}(A)\\setminus \\{a\\in W^{\\perp}: \\dim (A\\cap (W+a))= \\dim_H A  -1\\}) = 0,$$\n    that is standard dimension drop occurs on a set of full measure. (Here $W^{\\perp}$ denotes the 1-dimensional orthocomplement of $W$, \n    and $\\mathrm{Pr}_{W^{\\perp}}$ is the projection onto it.)\n\\end{theorem}\n\nThey explain that the following lemma fills this important geometric gap, namely: for a piecewise affine function on simplices containing the fractal $F$, with an arbitrarily small translation we can make sure that the function takes Lebesgue almost all of its values in the interior of the simplices.\n\n\\begin{lemma} \\label{lemma:values_admitted_in_simplices}\n    Let $F\\subseteq \\mathbb{R}^p$ be compact and $\\mathcal{S}$ be a finite family of simplices covering $F$ such that $\\inte (S) \\cap F\\neq \\emptyset$ for all $S\\in \\mathcal{S}$.\n    Assume moreover that $f:\\bigcup_{S\\in\\mathcal{S}} S \\to \\mathbb{R}$ is a piecewise affine function, affine on members of $\\mathcal{S}$. \n    Then for almost every $u\\in\\mathbb{R}^p$, we have\n    \\begin{equation} \\label{eq:values_admitted_in_interior}\n    \\bigcup_{S\\in\\mathcal{S}} f(S\\cap (F+u)) = \\bigcup_{S\\in\\mathcal{S}} f\\left(\\inte S\\cap (F+u)\\right) \\text{ a. e.}\n    \\end{equation}\n    i.e., for almost every $u$ almost every value admitted on $F+u$ by $f$ is admitted in the interior of a simplex.\n\\end{lemma}\n\nThis geometric fact is said to be quite easy if the Hausdorff dimension of $F$ is less than $2$, but for higher dimensions it needs some extra care. Additionally, later the authors provide a lemma showing how a dense set of functions over which the upper box dimension of level sets is controlled can be used to get bounds over a dense $G_\\delta$ set of functions, now assuming local connectivity.", "expanded_theorem": "\\label{thm:gen_upper_box_precise}\n    Assume that $F\\subseteq \\mathbb{R}^p$ is a connected self-similar set with finitely many directions\n    and $\\dim_H F = s > 1$.\n    Then for the generic 1-Hölder-$\\alpha$ function $f:F\\to \\mathbb{R}$ and Lebesgue almost every $r\\in f(F)$,\n    \\begin{displaymath}\n    \\overline{\\dim}_B f^{-1}(r)=s-\\alpha.\n    \\end{displaymath}", "theorem_type": ["Universal", "Inequality or Bound"], "mcq": {"question": "Fix $0<\\alpha\\le 1$. Let $F\\subseteq \\mathbb{R}^p$ be a connected self-similar set with finitely many directions, meaning that in some defining iterated function system for $F$, the orthogonal parts of the similarities generate a finite subgroup of the orthogonal group, and assume that $\\dim_H F=s>1$. Let $C_\\alpha^1(F)$ denote the space of 1-H\\\"older-$\\alpha$ functions $f:F\\to\\mathbb{R}$, i.e. functions satisfying $|f(x)-f(y)|\\le |x-y|^\\alpha$ for all $x,y\\in F$. Which statement holds for generic $f\\in C_\\alpha^1(F)$ (that is, for all $f$ in a dense $G_\\delta$ subset of $C_\\alpha^1(F)$) concerning the level sets $f^{-1}(r)$ for Lebesgue almost every $r\\in f(F)$?", "correct_choice": {"label": "A", "text": "For Lebesgue almost every $r\\in f(F)$, the upper box dimension of the level set satisfies\n$$\\overline{\\dim}_B f^{-1}(r)=s-\\alpha.$$"}, "choices": [{"label": "B", "text": "For Lebesgue almost every $r\\in f(F)$, the upper box dimension of the level set satisfies\n$$\\overline{\\dim}_B f^{-1}(r)=s-1.$$"}, {"label": "C", "text": "For Lebesgue almost every $r\\in f(F)$, the upper box dimension of the level set satisfies\n$$\\overline{\\dim}_B f^{-1}(r)\\le s-\\alpha.$$"}, {"label": "D", "text": "For every generic $f\\in C_\\alpha^1(F)$, there is a full measure set $R_f\\subseteq f(F)$ such that for every $r\\in R_f$,\n$$\\dim_H f^{-1}(r)=s-\\alpha.$$"}, {"label": "E", "text": "There exists a dense $G_\\delta$ set $\\mathcal G\\subseteq C_\\alpha^1(F)$ such that for every $f\\in \\mathcal G$ there exists a full measure set $R_f\\subseteq f(F)$ with\n$$\\overline{\\dim}_B f^{-1}(r)\\ge s-\\alpha$$\nfor all $r\\in R_f$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "trace_identity", "tampered_component": "dimension-drop amount replaced by fixed codimension one", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "dropped the lower-bound/equality conclusion, keeping only the upper bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "upper box dimension replaced by Hausdorff dimension", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "retained only the lower-bound side coming from interior-value construction and omitted the matching upper-bound mechanism", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the conclusion explicitly or by an obvious cue. It states the hypotheses and asks which of several nearby level-set statements is true."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the stem lists the theorem hypotheses and asks for its conclusion. The correct option is the theorem statement almost verbatim."}, "GPS": {"score": 1, "justification": "There is some pressure to distinguish equality from weaker bounds and to separate upper box dimension from Hausdorff dimension, but the task is still mainly recognition/recall rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target common confusions: codimension-1 heuristics, weakening equality to an upper bound, swapping upper box for Hausdorff dimension, and keeping only one side of the estimate."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-identification MCQ with strong distractors and no leakage, but it is largely a direct restatement of a known result rather than a genuinely generative reasoning question."}}
{"id": "2602.09751v1", "paper_link": "http://arxiv.org/abs/2602.09751v1", "theorems_cnt": 3, "theorem": {"env_name": "Theorem", "content": "\\label{thm:main}\nLet $\\tau^{\\phi}$ be a Teichm\\\"{u}ller disk in $ \\Tei_{2,0}$.\n    The Teichm\\\"uller metric and Carath\\'{e}odory metric agree on $\\tau^{\\phi}$  if and only if  $\\phi$ is a quadratic differential all of whose zeros are of even order.", "start_pos": 6751, "end_pos": 7031, "label": "thm:main"}, "ref_dict": {"thm:main2": "\\begin{Theorem}\\label{thm:main2}\nLet $\\tau^{\\phi}$ be a Teichm\\\"{u}ller disk in $\\Tei_{0,6}$.\n    The Teichm\\\"uller metric and Carath\\'{e}odory metric agree on $\\tau^{\\phi}$  if and only if  either $\\phi$ is a quadratic differential with no odd-order zeros, or $\\phi$ is a quadratic differential with a simple zero located at a marked point.    \n \\end{Theorem}", "conj:equl": "\\begin{conj}\\label{conj:equl}\nA Teichm\\\"uller disk  is a holomorphic retract if and only if it is generated by\na holomorphic quadratic differential all of whose zeros are of even order.\n\\end{conj}"}, "pre_theorem_intro_text_len": 4110, "pre_theorem_intro_text": "Let $\\Tei_{g,n}$ be the Teichm\\\"{u}ller space of Riemann surfaces of genus $g$ with $n$ marked points. \nThroughout this paper, we assume that $3g-3+n\\geq 2$. \nThe Teichm\\\"uller space $\\Tei_{g,n}$ is the (orbifold) universal cover of the moduli space of Riemann surfaces\n $\\mathcal{M}_{g,n}$ and  is naturally a complex manifold of dimension $3g-3+n$. It is known that\n $\\Tei_{g,n}$ can be realized as a bounded domain in $\\mathbb{C}^{3g-3+n}$, by the Bers\n embedding \\cite{Bers}. \n\nLet $\\mathbb{H}$ be the upper half-plane, equipped with the Poincar\\'e metric $$d_{\\mathbb{H}}= \\frac{|dz|}{2 \\operatorname{Im} z}.$$\nThe \\emph{Kobayashi metric} $d_{\\mathcal{K}}$ on $\\Tei_{g,n}$ is the largest metric so that every holomorphic map $$f: \\mathbb{H}\\rightarrow\\Tei_{g,n}$$ is nonexpanding.\nRoyden \\cite{Royden1969} proved that the Kobayashi metric on $\\Tei_{g,n}$ coincides with the Teichm\\\"{u}ller metric $d_{\\mathcal{T}}$. \nFurthermore,  the Teichm\\\"uller metric is not homogeneous at any point and its isometry group is essentially the mapping class group.\n\nAnother natural metric on $\\Tei_{g,n}$ satisfying the Schwarz-Pick inequality is the \\emph{Carath\\'{e}odory metric}, which can be defined as the smallest metric so that every holomorphic map $$F:  \\Tei_ {g,n} \\rightarrow \\mathbb{H}$$ is nonexpanding.\nLet $d_\\mathcal{C}$ denote the \\emph{Carath\\'{e}odory metric} on $\\Tei_{g,n}$.\n By the Schwarz Lemma, the inequality \n$$d_\\mathcal{C} \\leq d_\\mathcal{T}$$ holds  on $\\Tei_{g,n}$. \n\nA longstanding open  problem is whether the two metrics $d_\\mathcal{T}$ and $d_\\mathcal{C}$ agree on $\\Tei_{g,n}$. Recently, Markovic \\cite{Markovic2018} solved the problem. \n\nTeichm\\\"uller disks, also known as complex geodesics, are holomorphic isometric\n embeddings of the hyperbolic plane into Teichm\\\"uller space. There is a unique Teichm\\\"uller disc through any pair of distinct points in $\\Tei_{g,n}$. Every Teichm\\\"uller disk is generated by a non-zero holomorphic quadratic differential $\\phi$, with at most simple poles at the marked points.  \n\nLet $$\\tau^\\phi: \\mathbb{H} \\rightarrow \\Tei_{g,n}$$ \nbe a Teichm\\\"uller disk. \nKra \\cite{Kra1981} proved that the two metrics $d_\\mathcal{T}$ and $d_\\mathcal{C}$ agree on $\\tau^\\phi$ if all the zeros of $\\phi$ are of even order, \nsee also McMullen \\cite[Theorem 4.1]{McMullen2003}. \n\\begin{Remark}\n Note that if a holomorphic quadratic differential $\\phi$ vanishes at a marked point (puncture), then it is not necessary to require that it vanish of even order.\n\\end{Remark}\n\nMarkovic \\cite{Markovic2018} established a criterion that says when the two metrics $d_\\mathcal{T}$ and $d_\\mathcal{C}$ agree on a Teichm\\\"uller disk. \nThen he used this criterion to show that  $d_\\mathcal{T}$ and $d_\\mathcal{C}$ disagree on certain Teichm\\\"uller disk in $\\Tei_{0,5}$.\nSince there exists a holomorphic and isometric embedding of $\\Tei_{0,5}$  into $\\Tei_{g,n}$, \nwe have  $d_\\mathcal{T}\\neq d_\\mathcal{C}$ on $\\Tei_{g,n}$.\n\n  An application of Markovic's result is to prove a folklore conjecture of Siu that the Teichm\\\"uller space is not biholomorphic to any bounded convex domain in $\\mathbb{C}^{3g-3+n}$.  Note that Gupta and Seshadri \n \\cite[Theorem 1.1.]{GS2020} proved a related result that the Teichm\\\"uller space is not biholomorphically equivalent to any bounded domain in $\\mathbb{C}^{3g-3+n}$ which is strictly locally convex at one boundary point. \n\n\\medskip\n\nA remaining problem is to classify Teichm\\\"{u}ller disks for which $d_\\mathcal{T} = d_\\mathcal{C}$.\nThis is equivalent  to classifying Teichm\\\"{u}ller disks that are holomorphic retracts of the Teichm\\\"uller space. Gekhtman and Markovic   \\cite{GM2020} made the following conjecture.\n\n\\begin{conj}\\label{conj:equl}\nA Teichm\\\"uller disk  is a holomorphic retract if and only if it is generated by\na holomorphic quadratic differential all of whose zeros are of even order.\n\\end{conj}\n\nGekhtman and Markovic \\cite[Theorem 1.2]{GM2020} proved the above conjecture for $\\Tei_{0,5}\\cong\\Tei_{1,2}$. \nIn this paper, we confirm the conjecture for $\\Tei_{0,6}\\cong\\Tei_{2,0}$,\nwhich is of complex dimension $3$.", "context": "Let $\\Tei_{g,n}$ be the Teichm\\\"{u}ller space of Riemann surfaces of genus $g$ with $n$ marked points. \nThroughout this paper, we assume that $3g-3+n\\geq 2$. \nThe Teichm\\\"uller space $\\Tei_{g,n}$ is the (orbifold) universal cover of the moduli space of Riemann surfaces\n $\\mathcal{M}_{g,n}$ and is naturally a complex manifold of dimension $3g-3+n$. It is known that\n $\\Tei_{g,n}$ can be realized as a bounded domain in $\\mathbb{C}^{3g-3+n}$, by the Bers\n embedding \\cite{Bers}.\n\nLet $\\mathbb{H}$ be the upper half-plane, equipped with the Poincar\\'e metric $$d_{\\mathbb{H}}= \\frac{|dz|}{2 \\operatorname{Im} z}.$$\nThe \\emph{Kobayashi metric} $d_{\\mathcal{K}}$ on $\\Tei_{g,n}$ is the largest metric so that every holomorphic map $$f: \\mathbb{H}\\rightarrow\\Tei_{g,n}$$ is nonexpanding.\nRoyden \\cite{Royden1969} proved that the Kobayashi metric on $\\Tei_{g,n}$ coincides with the Teichm\\\"{u}ller metric $d_{\\mathcal{T}}$. \nFurthermore, the Teichm\\\"uller metric is not homogeneous at any point and its isometry group is essentially the mapping class group.\n\nTeichm\\\"uller disks, also known as complex geodesics, are holomorphic isometric\n embeddings of the hyperbolic plane into Teichm\\\"uller space. There is a unique Teichm\\\"uller disc through any pair of distinct points in $\\Tei_{g,n}$. Every Teichm\\\"uller disk is generated by a non-zero holomorphic quadratic differential $\\phi$, with at most simple poles at the marked points.\n\nLet $$\\tau^\\phi: \\mathbb{H} \\rightarrow \\Tei_{g,n}$$ \nbe a Teichm\\\"uller disk. \nKra \\cite{Kra1981} proved that the two metrics $d_\\mathcal{T}$ and $d_\\mathcal{C}$ agree on $\\tau^\\phi$ if all the zeros of $\\phi$ are of even order, \nsee also McMullen \\cite[Theorem 4.1]{McMullen2003}. \n\\begin{Remark}\n Note that if a holomorphic quadratic differential $\\phi$ vanishes at a marked point (puncture), then it is not necessary to require that it vanish of even order.\n\\end{Remark}\n\n\\begin{conj}\\label{conj:equl}\nA Teichm\\\"uller disk is a holomorphic retract if and only if it is generated by\na holomorphic quadratic differential all of whose zeros are of even order.\n\\end{conj}\n\nGekhtman and Markovic \\cite[Theorem 1.2]{GM2020} proved the above conjecture for $\\Tei_{0,5}\\cong\\Tei_{1,2}$. \nIn this paper, we confirm the conjecture for $\\Tei_{0,6}\\cong\\Tei_{2,0}$,\nwhich is of complex dimension $3$.", "full_context": "Let $\\Tei_{g,n}$ be the Teichm\\\"{u}ller space of Riemann surfaces of genus $g$ with $n$ marked points. \nThroughout this paper, we assume that $3g-3+n\\geq 2$. \nThe Teichm\\\"uller space $\\Tei_{g,n}$ is the (orbifold) universal cover of the moduli space of Riemann surfaces\n $\\mathcal{M}_{g,n}$ and is naturally a complex manifold of dimension $3g-3+n$. It is known that\n $\\Tei_{g,n}$ can be realized as a bounded domain in $\\mathbb{C}^{3g-3+n}$, by the Bers\n embedding \\cite{Bers}.\n\nLet $\\mathbb{H}$ be the upper half-plane, equipped with the Poincar\\'e metric $$d_{\\mathbb{H}}= \\frac{|dz|}{2 \\operatorname{Im} z}.$$\nThe \\emph{Kobayashi metric} $d_{\\mathcal{K}}$ on $\\Tei_{g,n}$ is the largest metric so that every holomorphic map $$f: \\mathbb{H}\\rightarrow\\Tei_{g,n}$$ is nonexpanding.\nRoyden \\cite{Royden1969} proved that the Kobayashi metric on $\\Tei_{g,n}$ coincides with the Teichm\\\"{u}ller metric $d_{\\mathcal{T}}$. \nFurthermore, the Teichm\\\"uller metric is not homogeneous at any point and its isometry group is essentially the mapping class group.\n\nTeichm\\\"uller disks, also known as complex geodesics, are holomorphic isometric\n embeddings of the hyperbolic plane into Teichm\\\"uller space. There is a unique Teichm\\\"uller disc through any pair of distinct points in $\\Tei_{g,n}$. Every Teichm\\\"uller disk is generated by a non-zero holomorphic quadratic differential $\\phi$, with at most simple poles at the marked points.\n\nLet $$\\tau^\\phi: \\mathbb{H} \\rightarrow \\Tei_{g,n}$$ \nbe a Teichm\\\"uller disk. \nKra \\cite{Kra1981} proved that the two metrics $d_\\mathcal{T}$ and $d_\\mathcal{C}$ agree on $\\tau^\\phi$ if all the zeros of $\\phi$ are of even order, \nsee also McMullen \\cite[Theorem 4.1]{McMullen2003}. \n\\begin{Remark}\n Note that if a holomorphic quadratic differential $\\phi$ vanishes at a marked point (puncture), then it is not necessary to require that it vanish of even order.\n\\end{Remark}\n\n\\begin{conj}\\label{conj:equl}\nA Teichm\\\"uller disk is a holomorphic retract if and only if it is generated by\na holomorphic quadratic differential all of whose zeros are of even order.\n\\end{conj}\n\nGekhtman and Markovic \\cite[Theorem 1.2]{GM2020} proved the above conjecture for $\\Tei_{0,5}\\cong\\Tei_{1,2}$. \nIn this paper, we confirm the conjecture for $\\Tei_{0,6}\\cong\\Tei_{2,0}$,\nwhich is of complex dimension $3$.\n\n\\begin{abstract}\n Let $\\Tei_{g,n}$ be the Teichm\\\"uller space of Riemann surfaces of genus $g$\n with $n$ punctures. It is conjectured that the Teichm\\\"uller and Carath\\'{e}odory metrics agree on a Teichm\\\"{u}ller disk if and only if all the zeros of the corresponding holomorphic quadratic differential are of even order. The conjecture was proved by Gekhtman and Markovic for $\\Tei_{0,5}\\cong \\Tei_{1,2}$. We confirm the conjecture for $\\Tei_{2,0}\\cong\\Tei_{0,6}$.\n\\end{abstract}\n\nLet $$\\tau^\\phi: \\mathbb{H} \\rightarrow \\Tei_{g,n}$$ \nbe a Teichm\\\"uller disk. \nKra \\cite{Kra1981} proved that the two metrics $d_\\Tei$ and $d_\\mathcal{C}$ agree on $\\tau^\\phi$ if all the zeros of $\\phi$ are of even order, \nsee also McMullen \\cite[Theorem 4.1]{McMullen2003}. \n\\begin{Remark}\n Note that if a holomorphic quadratic differential $\\phi$ vanishes at a marked point (puncture), then it is not necessary to require that it vanish of even order.\n\\end{Remark}\n\n\\begin{conj}\\label{conj:equl}\nA Teichm\\\"uller disk is a holomorphic retract if and only if it is generated by\na holomorphic quadratic differential all of whose zeros are of even order.\n\\end{conj}\n\nGekhtman and Markovic \\cite[Theorem 1.2]{GM2020} proved the above conjecture for $\\Tei_{0,5}\\cong\\Tei_{1,2}$. \nIn this paper, we confirm the conjecture for $\\Tei_{0,6}\\cong\\Tei_{2,0}$,\nwhich is of complex dimension $3$.\n\nEquivalently, we show\n\n\\begin{Theorem}\\label{thm:main2}\nLet $\\tau^{\\phi}$ be a Teichm\\\"{u}ller disk in $\\Tei_{0,6}$.\n The Teichm\\\"uller metric and Carath\\'{e}odory metric agree on $\\tau^{\\phi}$ if and only if either $\\phi$ is a quadratic differential with no odd-order zeros, or $\\phi$ is a quadratic differential with a simple zero located at a marked point. \n \\end{Theorem}\n\nThe following lemma states that classifying Teichm\\\"uller disks for which $d_\\Tei = d_\\mathcal{C}$ is equivalent to classifying holomorphic retracts of $\\Tei_{g,n}$.\nSee \\cite[Lemma 1.3]{GM2020}.\n\\begin{Lemma}\n The Kobayashi and Carath\\'{e}odory metrics agree on a Teichm\\\"{u}ller disk $\\tau^{\\phi}$ if and only if $\\tau^{\\phi}$ is a holomorphic retract of $\\Tei_{g,n}$.\n\\end{Lemma}\n\n\\begin{Theorem}[Markovic's criterion]\\label{thm:GM-JS}\n Let $\\phi\\in Q(X)$ be a non-zero Jenkins-Strebel differential. Denote the area of the $j$-th cylinder of $\\phi$ by $a_j$ . Let $\\mathcal{E}^{\\phi}:\\mathbb{H}^k\\rightarrow\\Tei_{g,n}$ be the Teichm\\\"{u}ller polydisk associated with $\\phi$. Then the Teichm\\\"{u}ller disk $\\tau^{\\phi}$ admits a holomorphic retraction if and only if there exists a holomorphic map $\\Phi:\\Tei_{g,n}\\rightarrow \\mathbb{H}$ satisfying \\begin{equation}\\label{equ:criterion}\n \\left(\\Phi\\circ \\mathcal{E}^\\phi\\right)(\\lambda)=\\sum\\limits_{j=1}^{k}a_j\\lambda_j.\n \\end{equation}\n\\end{Theorem}", "post_theorem_intro_text_len": 1774, "post_theorem_intro_text": "Equivalently, we show\n\n\\begin{Theorem}\\label{thm:main2}\nLet $\\tau^{\\phi}$ be a Teichm\\\"{u}ller disk in $\\Tei_{0,6}$.\n    The Teichm\\\"uller metric and Carath\\'{e}odory metric agree on $\\tau^{\\phi}$  if and only if  either $\\phi$ is a quadratic differential with no odd-order zeros, or $\\phi$ is a quadratic differential with a simple zero located at a marked point.    \n \\end{Theorem}\n\nThe proof of Theorem \\ref{thm:main2}  follows the strategy developed by Gekhtman and Markovic \\cite{Markovic2018, GM2020}. \nDue to Markovic's criterion, it suffices to classify Jenkins-Strebel differentials \nwith at least one odd-order zero. In Section \\ref{sec:classify}, by applying cylinder deformations or by interchanging the vertical and horizontal foliations if necessary, we prove  that Theorem \\ref{thm:main2} can be reduced to  the case when $\\phi$ is a staircase.\n\nAssume that $\\phi$ is a staircase surface and that the corresponding Teichm\\\"uller disk $\\tau^{\\phi}$\nadmits a holomoprhic retraction, say\n$F: \\mathcal{T}_{0,6} \\to \\mathbb{H}$. \nIn Section \\ref{sec:computation}, we use the Schwarz-Christoffel formula to compute $F$ on certain smooth paths of $\\mathcal{T}_{0,6}$. The computation is explicit and it shows that  $F$ fails to be $C^2$-smooth, which leads to a contradiction. \nOur computational method generalizes Markovic's previous work on $L$-shaped pillowcases \\cite{Markovic2018}. \n\n\\begin{Remark}\nBy a result of Apisa and Wright \\cite[Corollary 1.3]{AW2022},\nConjecture \\ref{conj:equl} can be reduced to checking strata (and full loci of covers)\nof genus zero quadratic differentials.\n\\end{Remark}\n\n\\subsection*{Acknowledgements} The authors are grateful to Lixin Liu for for his valuable suggestions. W. Su is partially\n supported by NSFC Grant No. 12371076.", "sketch": "The post-theorem introduction gives a proof sketch for Theorem~\\ref{thm:main2} (presented as equivalent to the main statement). The proof \"follows the strategy developed by Gekhtman and Markovic\". Using Markovic's criterion, \"it suffices to classify Jenkins--Strebel differentials with at least one odd-order zero.\" In Section~\\ref{sec:classify}, \"by applying cylinder deformations or by interchanging the vertical and horizontal foliations if necessary,\" the authors prove that Theorem~\\ref{thm:main2} \"can be reduced to the case when $\\phi$ is a staircase.\" Then, assuming $\\phi$ is a staircase and the associated Teichm\\\"uller disk $\\tau^\\phi$ \"admits a holomoprhic retraction\" $F:\\mathcal{T}_{0,6}\\to\\mathbb{H}$, Section~\\ref{sec:computation} uses the Schwarz--Christoffel formula to \"compute $F$ on certain smooth paths\" in $\\mathcal{T}_{0,6}$; the computation is explicit and shows that \"$F$ fails to be $C^2$-smooth,\" yielding a contradiction.", "expanded_sketch": "The post-theorem introduction gives a proof sketch for the result establishing the main theorem (presented as equivalent to the main statement). The proof \"follows the strategy developed by Gekhtman and Markovic\". Using Markovic's criterion, \"it suffices to classify Jenkins--Strebel differentials with at least one odd-order zero.\" Next, \"by applying cylinder deformations or by interchanging the vertical and horizontal foliations if necessary,\" the authors prove the following theorem.\n\n\\begin{Theorem}\\label{thm:main2}\nLet $\\tau^{\\phi}$ be a Teichm\\\"{u}ller disk in $\\Tei_{0,6}$.\n    The Teichm\\\"uller metric and Carath\\'{e}odory metric agree on $\\tau^{\\phi}$  if and only if  either $\\phi$ is a quadratic differential with no odd-order zeros, or $\\phi$ is a quadratic differential with a simple zero located at a marked point.    \n \\end{Theorem}\n\nThen, assuming $\\phi$ is a staircase and the associated Teichm\\\"uller disk $\\tau^\\phi$ \"admits a holomoprhic retraction\" $F:\\mathcal{T}_{0,6}\\to\\mathbb{H}$, later the authors use the Schwarz--Christoffel formula to \"compute $F$ on certain smooth paths\" in $\\mathcal{T}_{0,6}$; the computation is explicit and shows that \"$F$ fails to be $C^2$-smooth,\" yielding a contradiction.", "expanded_theorem": "\\label{thm:main}\nLet $\\tau^{\\phi}$ be a Teichm\\\"{u}ller disk in $ \\Tei_{2,0}$.\n    The Teichm\\\"uller metric and Carath\\'\\{e\\}odory metric agree on $\\tau^{\\phi}$  if and only if  $\\phi$ is a quadratic differential all of whose zeros are of even order.,", "theorem_type": ["Biconditional or Equivalence"], "mcq": {"question": "Let $\\tau^{\\phi}$ be a Teichm\\\"uller disk in $\\Tei_{2,0}$, where $\\Tei_{2,0}$ is the Teichm\\\"uller space of closed genus-$2$ Riemann surfaces, and a Teichm\\\"uller disk means a holomorphic isometric embedding of the hyperbolic plane into $\\Tei_{2,0}$. Which of the following statements is equivalent to saying that the Teichm\\\"uller metric $d_{\\mathcal T}$ and the Carath\\'eodory metric $d_{\\mathcal C}$ agree on $\\tau^{\\phi}$?", "correct_choice": {"label": "A", "text": "$\\phi$ is a holomorphic quadratic differential all of whose zeros are of even order."}, "choices": [{"label": "B", "text": "$\\phi$ is a holomorphic quadratic differential with at most one odd-order zero."}, {"label": "C", "text": "$\\phi$ is a holomorphic quadratic differential with no simple zeros."}, {"label": "D", "text": "$\\phi$ is a holomorphic quadratic differential for which there exists some representative of the same Teichm\\\"uller disk having all zeros of even order."}, {"label": "E", "text": "$\\phi$ is a holomorphic quadratic differential all of whose zeros are of even order, and moreover the corresponding Teichm\\\"uller disk is generated by a Jenkins--Strebel differential."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "excluded odd-zero staircase cases reduced to a single odd zero", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "even-order requirement weakened to exclusion of order-1 zeros only", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence on the given generator replaced by existence for some representative", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "Jenkins--Strebel reduction made into an additional hypothesis", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct condition; it asks for an equivalent characterization without giving away the content of that characterization."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: if the underlying result states that metric agreement on the Teichmüller disk is equivalent to all zeros of φ being even-order, then choice A is just the theorem restated."}, "GPS": {"score": 1, "justification": "The item requires some discrimination among nearby alternatives (weaker, stronger, or quantifier-shifted variants), but for a student who knows the theorem the answer is immediate rather than generated through substantial reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: weakening the condition, confusing absence of simple zeros with all-even order, shifting dependence to another representative, and adding an unnecessary Jenkins–Strebel hypothesis."}, "total_score": 5, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it is mostly a direct restatement of a known equivalence rather than a question that strongly tests generative reasoning."}}
{"id": "2602.09797v1", "paper_link": "http://arxiv.org/abs/2602.09797v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{thm:main}\nThe procyclic groups\n\\[\n    \\prod_{p \\equiv 1 \\bmod 3} \\bbZ_p, \\quad \\prod_{p \\equiv 1 \\bmod 4} \\bbZ_p, \\quad \\prod_{p \\in \\{1,3\\} \\bmod 8} \\bbZ_p\n\\]\nhave Weil abscissa $2$.", "start_pos": 4069, "end_pos": 4280, "label": "thm:main"}, "ref_dict": {}, "pre_theorem_intro_text_len": 1282, "pre_theorem_intro_text": "The \\emph{Weil abscissa} $\\alpha(G)$ of a profinite group $G$ is an intricate invariant that quantifies how the number of absolutely irreducible representations of $G$ over finite fields $\\mathbb{F}_{p^j}$ increases with the size of the fields. It is the abscissa of convergence of the Weil representation zeta function defined in \\cite{corob2024weil}. \nFor a finitely generated profinite \\emph{abelian} group $G$, the \\textit{Weil representation zeta function} admits the following simple definition\n\t\\begin{align*}\n\t\t\\zeta_G^W(s)=\\exp\\bigg(\\sum_{p\\in\\mathcal{P}}\\sum_{j=1}^{\\infty} |\\text{Hom}(G,\\mathbb{F}_{p^j}^\\times)|\\frac{p^{-sj}}{j}\\bigg),\n\t\\end{align*}\n for $s \\in \\mathbb{C}$; here $\\mathcal{P}$ denotes the set of all prime numbers and $|\\text{Hom}(G,\\mathbb{F}_{p^j}^\\times)|$ the number of continuous homomorphisms from $G$ to $\\mathbb{F}_{p^j}^\\times$. However, even for abelian groups the Weil abscissa is difficult to compute, since its exact value depends in a subtle way on the distribution of primes in arithmetic progressions; see \\cite{corob2024weil,kionke2024}. The purpose of this short note is to calculate the Weil abscissa of three procyclic groups building on ideas of the master thesis of the first author (written at FernUniversit\\\"at in Hagen in 2025).", "context": "The \\emph{Weil abscissa} $\\alpha(G)$ of a profinite group $G$ is an intricate invariant that quantifies how the number of absolutely irreducible representations of $G$ over finite fields $\\mathbb{F}_{p^j}$ increases with the size of the fields. It is the abscissa of convergence of the Weil representation zeta function defined in \\cite{corob2024weil}. \nFor a finitely generated profinite \\emph{abelian} group $G$, the \\textit{Weil representation zeta function} admits the following simple definition\n \\begin{align*}\n \\zeta_G^W(s)=\\exp\\bigg(\\sum_{p\\in\\mathcal{P}}\\sum_{j=1}^{\\infty} |\\text{Hom}(G,\\mathbb{F}_{p^j}^\\times)|\\frac{p^{-sj}}{j}\\bigg),\n \\end{align*}\n for $s \\in \\mathbb{C}$; here $\\mathcal{P}$ denotes the set of all prime numbers and $|\\text{Hom}(G,\\mathbb{F}_{p^j}^\\times)|$ the number of continuous homomorphisms from $G$ to $\\mathbb{F}_{p^j}^\\times$. However, even for abelian groups the Weil abscissa is difficult to compute, since its exact value depends in a subtle way on the distribution of primes in arithmetic progressions; see \\cite{corob2024weil,kionke2024}. The purpose of this short note is to calculate the Weil abscissa of three procyclic groups building on ideas of the master thesis of the first author (written at FernUniversit\\\"at in Hagen in 2025).", "full_context": "The \\emph{Weil abscissa} $\\alpha(G)$ of a profinite group $G$ is an intricate invariant that quantifies how the number of absolutely irreducible representations of $G$ over finite fields $\\mathbb{F}_{p^j}$ increases with the size of the fields. It is the abscissa of convergence of the Weil representation zeta function defined in \\cite{corob2024weil}. \nFor a finitely generated profinite \\emph{abelian} group $G$, the \\textit{Weil representation zeta function} admits the following simple definition\n \\begin{align*}\n \\zeta_G^W(s)=\\exp\\bigg(\\sum_{p\\in\\mathcal{P}}\\sum_{j=1}^{\\infty} |\\text{Hom}(G,\\mathbb{F}_{p^j}^\\times)|\\frac{p^{-sj}}{j}\\bigg),\n \\end{align*}\n for $s \\in \\mathbb{C}$; here $\\mathcal{P}$ denotes the set of all prime numbers and $|\\text{Hom}(G,\\mathbb{F}_{p^j}^\\times)|$ the number of continuous homomorphisms from $G$ to $\\mathbb{F}_{p^j}^\\times$. However, even for abelian groups the Weil abscissa is difficult to compute, since its exact value depends in a subtle way on the distribution of primes in arithmetic progressions; see \\cite{corob2024weil,kionke2024}. The purpose of this short note is to calculate the Weil abscissa of three procyclic groups building on ideas of the master thesis of the first author (written at FernUniversit\\\"at in Hagen in 2025).\n\n\\begin{abstract}\nHere we show that the Weil abscissa of \nthe procyclic groups $\\prod_{p \\in S} \\bbZ_p$ equals $2$ for three sets $S$: (i) the set of primes $p \\equiv 1 \\bmod 3$, (ii) the set of primes $p \\equiv 1 \\bmod 4$ and (iii) the set of primes $p \\equiv 1,3 \\bmod 8$. Our argument is based on the observation that integers all of whose prime factors lie in $S$ can be represented by a suitable binary quadratic form, which allows us to use a theorem of Iwaniec to exhibit a minorant for the Weil representation zeta function.\n\\end{abstract}\n\nThe \\emph{Weil abscissa} $\\alpha(G)$ of a profinite group $G$ is an intricate invariant that quantifies how the number of absolutely irreducible representations of $G$ over finite fields $\\mathbb{F}_{p^j}$ increases with the size of the fields. It is the abscissa of convergence of the Weil representation zeta function defined in \\cite{corob2024weil}. \nFor a finitely generated profinite \\emph{abelian} group $G$, the \\textit{Weil representation zeta function} admits the following simple definition\n \\begin{align*}\n \\zeta_G^W(s)=\\exp\\bigg(\\sum_{p\\in\\mathcal{P}}\\sum_{j=1}^{\\infty} |\\text{Hom}(G,\\mathbb{F}_{p^j}^\\times)|\\frac{p^{-sj}}{j}\\bigg),\n \\end{align*}\n for $s \\in \\mathbb{C}$; here $\\mathcal{P}$ denotes the set of all prime numbers and $|\\text{Hom}(G,\\mathbb{F}_{p^j}^\\times)|$ the number of continuous homomorphisms from $G$ to $\\mathbb{F}_{p^j}^\\times$. However, even for abelian groups the Weil abscissa is difficult to compute, since its exact value depends in a subtle way on the distribution of primes in arithmetic progressions; see \\cite{corob2024weil,kionke2024}. The purpose of this short note is to calculate the Weil abscissa of three procyclic groups building on ideas of the master thesis of the first author (written at FernUniversit\\\"at in Hagen in 2025).\n\n\\medskip\n\n\\begin{question}\nDo the procyclic groups $\\prod_{p \\equiv 2 \\bmod 3} \\bbZ_p$ and $\\prod_{p \\equiv 3 \\bmod 4} \\bbZ_p$ have Weil abscissa $2$?\n\\end{question}\n\nRepresentability of integers by quadratic forms is a classical topic in number theory. The sum of two squares problem is a famous example: which primes are sums of two squares? It was already known to Fermat that exactly $2$ and all primes in $S_1=\\{p\\in \\mathcal{P} \\mid p \\equiv 1 \\bmod 4\\}$ are the sum of two squares. We will need the following refined result on representations of integers as a sum of two \\emph{coprime} squares:\n\\begin{proposition}[{see \\cite[Corollary 3.2.2]{Fine2007}}]\\label{prop:1mod4}\n Let $n \\in \\mathbb{N}$. Then $n=x^2+y^2$ for coprime integers $x,y$ if and only if the prime decomposition of $n$ is $n=2^\\delta \\prod_{i=1}^k p_i^{\\alpha_i}$ where $\\delta \\in \\{0,1\\}$ and $p_1,\\dots, p_k \\in S_1$.\n\\end{proposition}\nThis result immediately implies\n\\begin{corollary}\\label{cor:1}\nDefine $P_{S_1}=\\{p\\in\\mathcal{P} \\mid p=x^2+y^2+1,\\,x,y\\in \\mathbb{Z}, \\; \\gcd(x,y)=1\\}$. All primes $p \\in P_{S_1}$ satisfy $(p-1)_{S_1}\\geq (p-1)/2$.\n\\end{corollary}\n\n\\begin{proposition}\n Let $n \\in \\mathbb{N}$ and $S_2=\\{p\\in \\mathcal{P} \\mid p \\equiv 1 \\bmod 8 \\text{ or } p \\equiv 3 \\bmod 8\\}$. If $n=x^2+2y^2$ for coprime $x,y \\in \\mathbb{Z}$, then $n=2^\\delta \\prod_{i=1}^k p_i^{\\alpha_i}$, where $\\delta \\in \\{0,1\\}$ and $p_1,\\dots,p_k\\in S_2$.\n\\end{proposition}\n\\begin{proof}\n First, we observe that for any even integer $n=x^2+2y^2$, where $\\gcd(x,y)=1$, $x$ must be an even integer. Therefore, $n$ has a residue of $2$ modulo $4$ and is only divisible by $2$, not $4$. For odd primes the result is given by an old theorem of Fermat \\cite[p. 7]{Cox1997}:\n Odd prime divisors $p$ of $n$ must satisfy the condition \\begin{align*}\n 1=\\begin{pmatrix}\n \\frac{-2}{p}\n\\end{pmatrix}=\\begin{pmatrix}\n \\frac{-1}{p}\n\\end{pmatrix}\\begin{pmatrix}\n \\frac{2}{p}\n\\end{pmatrix}=(-1)^{\\frac{p-1}{2}}(-1)^{\\frac{p^2-1}{8}}. \n \\end{align*} \n This implies, as one readily verifies, that all odd prime divisors of $n$ lie in~$S_2$.\n\\end{proof}\n\\begin{corollary}\\label{cor:2}\nDefine $P_{S_2}=\\{p\\in\\mathcal{P} \\mid p=x^2+2y^2+1,\\,x,y\\in \\mathbb{Z}, \\; \\gcd(x,y)=1\\}$.\nAll primes $p \\in P_{S_2}$ satisfy $(p-1)_{S_2}\\geq (p-1)/2$.\n\\end{corollary}\n\n\\begin{proposition}\n Let $n \\in \\mathbb{N}$ and $S_3=\\{p\\in \\mathcal{P} \\mid p \\equiv 1 \\bmod 3\\}$. If $n=x^2+3y^2$ for coprime $x,y \\in \\mathbb{Z}$, then $n=2^{\\delta_1} 3^{\\delta_2} \\prod_{i=1}^k p_i^{\\alpha_i}$, where $\\delta_1 \\in \\{0,1,2\\}$, $\\delta_2 \\in \\{0,1\\}$, $p_1,\\dots,p_k\\in S_3$.\n\\end{proposition}\n\\begin{proof}\n First, we observe that for any even integer $n=x^2+3y^2$, where $\\gcd(x,y)=1$, both $x$ and $y$ are odd integers.\n Since all odd squares have a residue of $1$ modulo $8$, $n$ cannot be divisible by $8$.\n Regarding the divisor $3^{\\delta_2}$, there are two cases.\n First, if $y$ is divisible by $3$, then $x$ is not and $\\delta_2=0$. Second, if $x$ is divisible by $3$, then $y$ is not, and thus, $\\delta_2=1$. For odd primes the result is again given by an old theorem of Fermat \\cite[p. 7]{Cox1997}:\n According to the law of quadratic reciprocity, odd prime divisors $p\\neq 3$ must satisfy the following condition: \\begin{align*}\n 1=\\begin{pmatrix}\n \\frac{-3}{p}\n\\end{pmatrix}=\\begin{pmatrix}\n \\frac{-1}{p}\n\\end{pmatrix}\\begin{pmatrix}\n \\frac{3}{p}\n\\end{pmatrix}=(-1)^{\\frac{p-1}{2}} \\begin{pmatrix}\n \\frac{3}{p}\n\\end{pmatrix}=\\begin{pmatrix}\n \\frac{p}{3}\n\\end{pmatrix}. \n \\end{align*} \n Since $1$ is the only quadratic residue of a prime $p\\neq3$ modulo $3$, the only possible odd prime divisors of $n$ are the primes in $S_3$. \n\\end{proof}\nAs before we deduce\n\\begin{corollary}\\label{cor:3}\nDefine $P_{S_3}=\\{p\\in\\mathcal{P} \\mid p=x^2+3y^2+1,\\,x,y\\in \\mathbb{Z}, \\; \\gcd(x,y)=1\\}$. \nAll primes $p \\in P_{S_3}$ satisfy $(p-1)_{S_3}\\geq (p-1)/12$. \n\\end{corollary}\nNext, we will need some results from the theory of quadratic forms.\n\\begin{definition}[see \\cite{Cox1997}]\n Two binary quadratic forms, $f(x,y)$ and $g(x,y$), are said to be \\emph{(properly) equivalent} if there are $a,b,c,d \\in \\mathbb{Z}$ such that $ad-bc=1$ and\n \\begin{align*}\n f(x,y)=g(ax+by,cx+dy) \\text{ for all } x,y \\in \\mathbb{Z}. \n \\end{align*}\n The \\emph{genus of a quadratic form} $f(x,y)$ is the set of all binary quadratic forms that represent the same numbers in $(\\mathbb{Z}/|\\Delta_f|\\mathbb{Z})^*$. It is denoted by $R_f$.\n\\end{definition}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main}]\nLet $S$ be any set of primes.\nThe free procyclic group $\\widehat{\\bbZ} = \\prod_{p \\in \\mathcal{P}} \\bbZ_p$ has Weil abscissa $2$; see \\cite[Theorem C (i)]{corob2024weil}. Hence the inequality \n\\[\\alpha(H_S) \\leq 2\\] follows immediately using that $H_S$ is a quotient of $\\widehat{\\bbZ}$.", "post_theorem_intro_text_len": 2357, "post_theorem_intro_text": "Let $S$ be a non-empty subset of $\\mathcal{P}$ and let $H_S=\\prod_{p\\in S} \\mathbb{Z}_p$; here $\\bbZ_p$ denotes the additive group of $p$-adic integers. Then $H_S$ is a procyclic group; this means, it is a profinite group that contains a dense cyclic subgroup.\nConjecture A of \\cite{kionke2024} predicts three equalities, one of which can be stated as follows\n\\begin{conjecture*}\n    The shifted partial Riemann zeta function $\\zeta_S(s-1):= \\prod_{p \\in S} \\frac{1}{1-p^{-s+1}}$ has the same abscissa of convergence as $\\zeta^W_{H_S}(s)$.\n\\end{conjecture*}\nIf $S = \\{p \\in \\mathcal{P} \\mid p \\equiv a \\bmod d\\}$ is the set of primes in an arithmetic progression with $\\gcd(a,d)=1$, then $\\zeta_S(s)$ has abscissa $1$ - this can be seen for instance using the Siegel–Walfisz theorem \\cite{walfisz1936} - and hence our theorem confirms the conjecture for these three groups.\n\n\\medskip\n\nCurrently the conjecture is known only for very big or very small sets $S$.\nFirst, consider sets $S \\subset\\mathcal{P}$ for which $\\zeta_{\\overline{S}}$ converges for a real number $s<1$, where $\\overline{S}=\\mathcal{P}\\setminus S$. These sets are called \\textit{thick}. If $S$ is a thick set of primes, then the second author proved $\\alpha(H_S)=2$; see  \\cite[Theorem 1.3]{kionke2024}. Second, the conjecture is known to hold if $\\zeta_S(s)$ has abscissa $0$ (\\cite[Lemma 2.1]{kionke2024}). For certain exponentially growing sequences $S$ of prime numbers this was already observed in \\cite[Remark 5.4]{corob2024weil}.\nTherefore, the question of the value of the Weil abscissa is of special interest in \"in-between\" cases.\n\n\\medskip\n\nIn this paper we analyze cases in which $S$ is the set of prime numbers that are part of one or more arithmetic progressions. To prove our main result, we will exploit the fact that primes in some arithmetic progressions are representable by certain binary quadratic forms and apply a theorem of Iwaniec \\cite{Iwaniec1972}. The methods can be extended to prove general results about certain unions of arithmetic progressions. However, the method is limited and we emphasize it does not apply to general arithmetic progressions. In particular, the following question remains open.\n\n\\begin{question}\nDo the procyclic groups $\\prod_{p \\equiv 2 \\bmod 3} \\bbZ_p$ and $\\prod_{p \\equiv 3 \\bmod 4} \\bbZ_p$ have Weil abscissa $2$?\n\\end{question}", "sketch": "To prove Theorem~\\ref{thm:main}, the paper \"analyze[s] cases in which $S$ is the set of prime numbers that are part of one or more arithmetic progressions\" and \"exploit[s] the fact that primes in some arithmetic progressions are representable by certain binary quadratic forms\"; then the authors \"apply a theorem of Iwaniec \\cite{Iwaniec1972}.\" They note that these methods \"can be extended to prove general results about certain unions of arithmetic progressions,\" but that the approach is \"limited\" and \"does not apply to general arithmetic progressions.\"", "expanded_sketch": "To prove the main theorem, the paper \"analyze[s] cases in which $S$ is the set of prime numbers that are part of one or more arithmetic progressions\" and \"exploit[s] the fact that primes in some arithmetic progressions are representable by certain binary quadratic forms\"; then the authors \"apply a theorem of Iwaniec \\cite{Iwaniec1972}.\" They note that these methods \"can be extended to prove general results about certain unions of arithmetic progressions,\" but that the approach is \"limited\" and \"does not apply to general arithmetic progressions.\"", "expanded_theorem": "\\label{thm:main}\nThe procyclic groups\n\\[\n    \\prod_{p \\equiv 1 \\bmod 3} \\bbZ_p, \\quad \\prod_{p \\equiv 1 \\bmod 4} \\bbZ_p, \\quad \\prod_{p \\in \\{1,3\\} \\bmod 8} \\bbZ_p\n\\]\nhave Weil abscissa $2$.", "theorem_type": ["Universal"], "mcq": {"question": "For a finitely generated profinite abelian group \\(G\\), the Weil representation zeta function is\n\\[\n\\zeta_G^W(s)=\\exp\\!\\left(\\sum_{p\\in\\mathcal P}\\sum_{j=1}^{\\infty} |\\operatorname{Hom}(G,\\mathbb F_{p^j}^{\\times})|\\frac{p^{-sj}}{j}\\right),\n\\]\nand the Weil abscissa \\(\\alpha(G)\\) is the abscissa of convergence of this Dirichlet-type series. Here \\(\\mathcal P\\) is the set of primes and \\(\\mathbb Z_p\\) denotes the ring of \\(p\\)-adic integers. Consider the three procyclic groups\n\\[\n\\prod_{p\\equiv 1\\, (\\mathrm{mod}\\,3)} \\mathbb Z_p,\\qquad\n\\prod_{p\\equiv 1\\, (\\mathrm{mod}\\,4)} \\mathbb Z_p,\\qquad\n\\prod_{p\\equiv 1\\text{ or }3\\, (\\mathrm{mod}\\,8)} \\mathbb Z_p.\n\\]\nWhich statement holds for every group in this list?", "correct_choice": {"label": "A", "text": "Each of these three procyclic groups has Weil abscissa \\(2\\); equivalently, if \\(G\\) is any one of them, then \\(\\alpha(G)=2\\)."}, "choices": [{"label": "B", "text": "Each of these three procyclic groups has Weil abscissa strictly less than \\(2\\); equivalently, if \\(G\\) is any one of them, then \\(\\alpha(G)<2\\)."}, {"label": "C", "text": "For each of these three procyclic groups \\(G\\), one has \\(\\alpha(G)\\le 2\\)."}, {"label": "D", "text": "Each of these three procyclic groups has Weil abscissa \\(2\\), and the same conclusion holds for every procyclic group of the form \\(\\prod_{p\\equiv a\\, (\\mathrm{mod}\\,m)} \\mathbb Z_p\\) arising from a single arithmetic progression of primes."}, {"label": "E", "text": "Exactly two of these three procyclic groups have Weil abscissa \\(2\\), while the remaining one has Weil abscissa different from \\(2\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "lower-bound mechanism needed to upgrade from \\(\\alpha\\le 2\\) to equality", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the matching lower bound \\(\\alpha(G)\\ge 2\\), retaining only the quotient-based upper bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "restricted applicability of quadratic-form/Iwaniec method to special progressions rather than arbitrary arithmetic progressions", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "uniform conclusion across the three explicitly listed congruence classes", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state or strongly hint that the Weil abscissa is 2. It gives definitions and the three groups, but the correct conclusion is not leaked."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question about the listed examples, so it is not fully non-tautological. However, the options do force discrimination between the exact claim, a weaker true statement, and stronger overgeneralizations."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to identify the strongest valid statement among nearby alternatives, especially versus the weaker true option and the overgeneralized option. Still, the item primarily tests recognition/application of a known result rather than substantial fresh derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: one is a weaker true statement, one is an overreach to all arithmetic progressions, one asserts the wrong inequality, and one introduces an unjustified case split. These align with common failure modes."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it is somewhat close to theorem restatement and only moderately pressures genuine generative reasoning."}}
{"id": "2602.09806v1", "paper_link": "http://arxiv.org/abs/2602.09806v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "\\label{thm1}\nAssume (F), \\eqref{init1} and \\eqref{init3} and let $u$ be a solution of \\eqref{Rea-Diff}. Then there exists a smooth function  $\\gamma=\\gamma(x,t)$ with the following properties.\n\\begin{enumerate}\n\\item[(i)] \nThere exists $T>0$ such that \n\\begin{equation*}\n\\{ (x,y,t) \\in \\mathbf{R}^{n-1} \\times \\mathbf{R} \\times [T,\\infty); \\, u(x,y,t)=\\Phi_c(0)\\}=\\{y=\\gamma(x,t)\\}.\n\\end{equation*}\nMoreover, it holds that\n\\begin{equation*}\n\\lim_{t\\rightarrow\\infty}\\sup_{(x,y)\\in\\mathbf{R}^n }|u(x,y,t)-\\Phi_c(y-\\gamma(x,t))|=0.\n\\end{equation*}\n\\item[(ii)]\nFor any $\\varepsilon>0$, there exists $\\tau_{\\varepsilon}\\in[T,\\infty)$ such that the solution $U(x,t)$ of the problem\n\\begin{equation}\\label{MCF}\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c_*,&&x\\in\\mathbf{R}^{n-1},t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_{\\varepsilon}),&&x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation}\nsatisfies\n\\begin{equation*}\n\\sup_{x\\in\\mathbf{R}^{n-1},t\\ge\\tau_{\\varepsilon}}|\\gamma(x,t)-U(x,t-\\tau_{\\varepsilon})|\\le\\varepsilon.\n\\end{equation*}\n\\end{enumerate}\nFurthermore, the assertion (i) still holds if \\eqref{init2} is assumed instead of \\eqref{init3}.", "start_pos": 11582, "end_pos": 12828, "label": "thm1"}, "ref_dict": {"thm1": "\\begin{thm}\\label{thm1}\nAssume (F), \\eqref{init1} and \\eqref{init3} and let $u$ be a solution of \\eqref{Rea-Diff}. Then there exists a smooth function  $\\gamma=\\gamma(x,t)$ with the following properties.\n\\begin{enumerate}\n\\item[(i)] \nThere exists $T>0$ such that \n\\begin{equation*}\n\\{ (x,y,t) \\in \\mathbf{R}^{n-1} \\times \\mathbf{R} \\times [T,\\infty); \\, u(x,y,t)=\\Phi_c(0)\\}=\\{y=\\gamma(x,t)\\}.\n\\end{equation*}\nMoreover, it holds that\n\\begin{equation*}\n\\lim_{t\\rightarrow\\infty}\\sup_{(x,y)\\in\\mathbf{R}^n }|u(x,y,t)-\\Phi_c(y-\\gamma(x,t))|=0.\n\\end{equation*}\n\\item[(ii)]\nFor any $\\varepsilon>0$, there exists $\\tau_{\\varepsilon}\\in[T,\\infty)$ such that the solution $U(x,t)$ of the problem\n\\begin{equation}\\label{MCF}\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c_*,&&x\\in\\mathbf{R}^{n-1},t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_{\\varepsilon}),&&x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation}\nsatisfies\n\\begin{equation*}\n\\sup_{x\\in\\mathbf{R}^{n-1},t\\ge\\tau_{\\varepsilon}}|\\gamma(x,t)-U(x,t-\\tau_{\\varepsilon})|\\le\\varepsilon.\n\\end{equation*}\n\\end{enumerate}\nFurthermore, the assertion (i) still holds if \\eqref{init2} is assumed instead of \\eqref{init3}.\n\\end{thm}", "Rea-Diff": "\\begin{aligned}\\label{Rea-Diff}\n&u_{t}=\\Delta u+f(u), && x\\in\\mathbb{R}^{n-1}, \\, y\\in\\mathbb{R}, \\, t>0, \\\\\n&u(x,y,0)=u_{0}(x,y), && x\\in\\mathbb{R}^{n-1}, \\, y\\in\\mathbb{R}.\n\\end{aligned}", "assmpf": "\\begin{equation}\\label{assmpf}\n0<f'(0)u-f(u)<Mu^{1+\\alpha}\n\\quad \\mbox{if } \\ u \\in (0,1)\n\\end{equation}", "lbidni": "\\begin{gather}\n\\liminf_{y \\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0, \\label{lbidni} \\\\\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)}, \\quad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty \n\\ \\mbox{ for some } \\ c \\ge c_*, \\label{init2}\n\\end{gather}", "init2": "\\begin{gather}\n\\liminf_{y \\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0, \\label{lbidni} \\\\\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)}, \\quad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty \n\\ \\mbox{ for some } \\ c \\ge c_*, \\label{init2}\n\\end{gather}", "MCF": "\\begin{equation}\\label{MCF}\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c_*,&&x\\in\\mathbf{R}^{n-1},t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_{\\varepsilon}),&&x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation}", "init1": "\\begin{equation}\n\\label{init1}\n\\liminf_{y\\to-\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}}u_{0}(x,y)>0 \\\\\n\\end{equation}", "init3": "\\begin{equation}\n\\label{init3}\nc=c_*>2\\sqrt{f'(0)}, \\qquad \n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}} u_{0}(x,y)e^{-\\lambda_1 y}<\\infty \\ \\mbox{ for some } \\ \\lambda_1<\\lambda_+,\n\\end{equation}"}, "pre_theorem_intro_text_len": 10008, "pre_theorem_intro_text": "In this paper, we consider the reaction-diffusion equation\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\\label{Rea-Diff}\n&u_{t}=\\Delta u+f(u), && x\\in\\mathbb{R}^{n-1}, \\, y\\in\\mathbb{R}, \\, t>0, \\\\\n&u(x,y,0)=u_{0}(x,y), && x\\in\\mathbb{R}^{n-1}, \\, y\\in\\mathbb{R}.\n\\end{aligned}\n\\right.\n\\end{equation} \nHere $\\Delta=\\partial^2/\\partial x_1^2+\\cdots +\\partial^2/\\partial x_{n-1}^2+\\partial^2/\\partial y^2$ and $n\\ge2$. Throughout the paper, the initial data $u_{0}$ is assumed to be nonnegative, bounded and uniformly continuous, and the reaction term $f$ is assumed to be of class $C^{1}$.\nWe are interested in the asymptotic behavior of solutions with non-compactly supported initial data in the case where the reaction term $f$ is of monostable-type.\n\nWe begin with recalling results on the existence and stability of traveling wave solutions in one dimensional monostable reaction diffusion equations. \nWe consider the reaction term $f$ satisfying\n\\begin{equation*}\n\\mbox{(F)} \\\n\\left\\{\n\\begin{aligned}\n&f(0)=f(1)=0,\\quad f^{\\prime}(0)>0,\\quad f^{\\prime}(1)<0,\n\\\\\n&f(s)>0 \\ (s\\in(0,1)),\\quad f(s)<0\\ (s\\in(-\\infty,0)\\cup(1,\\infty)).\n\\end{aligned}\n\\right.\n\\end{equation*}\nThen it is well-known that there exists the minimal speed $c^{*}\\ge2\\sqrt{f^{\\prime}(0)}$ for traveling front solutions joining the equilibria $1$ and $0$.\nMore precisely, for any $c\\ge c^{*}$, equation \\eqref{Rea-Diff} has a traveling front solution written in the form $u(x,t)=\\Phi_{c}(x-ct)$ for a profile function $\\Phi_{c}$ satisfying\n\\begin{gather*}\n\\Phi_{c}^{\\prime\\prime}+c\\Phi_{c}^{\\prime}+f(\\Phi_{c})=0, \\\\\n\\lim_{z\\to-\\infty}\\Phi_{c}(z)=1, \\quad \\lim_{z\\to\\infty}\\Phi_{c}(z)=0.\n\\end{gather*}\nIt is known that there exist $\\alpha, \\beta\\ge0$ with $(\\alpha,\\beta) \\neq (0,0)$ such that\n\\begin{gather}\n\\label{asym}\n\\Phi_{c}(s)=(\\alpha +o(1))e^{\\lambda_+(c)s} \\quad \\mbox{if} \\quad c>c^{*}, \\\\\n\\label{asym1}\n\\Phi_{c}(s)=(\\alpha s+\\beta +o(1))e^{\\lambda_-(c)s} \\quad \\mbox{if} \\quad c=c^{*},\n\\end{gather}\nwhere $\\lambda_{+}(c)$ and $\\lambda_{-}(c)$ are the largest root and the smallest root of the quadratic equation\n\\begin{equation}\n\\lambda^{2}+c\\lambda+f^{\\prime}(0)=0,\n\\end{equation}\nrespectively. \nThe traveling front solution $u(x,t)=\\Phi_{c}(x-ct)$ is called a pulled front if either $c=c^{*}=2\\sqrt{f^{\\prime}(0)}$ or $c>c^{*}$ holds, and is called a pushed front if $c=c^{*}>2\\sqrt{f^{\\prime}(0)}$.\n\nConcerning asymptotic behavior of solutions,\nmany results are known for the one-dimensional problem\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\n&u_{t}=u_{yy}+f(u), && y\\in\\mathbb{R}, \\, t>0, \\\\\n&u(y,0)=u_{0}(y), && y\\in\\mathbb{R}.\n\\end{aligned}\n\\right.\n\\end{equation*} \nIn the pioneer work \\cite{KPP}, it is shown that if $f(u)=u(1-u)$ and \n\\begin{equation*}\nu_0(y)=\\left\\{\n\\begin{aligned}\n&1 \\quad (y<0), \\\\\n&0 \\quad (y \\ge 0),\n\\end{aligned}\n\\right.\n\\end{equation*} \nthen $u(z+\\sigma(t),t)$ converges uniformly to $\\Phi_{c^{*}}(z)$ as $t \\to \\infty$\nfor some function $\\sigma(t)$ satisfying\n\\begin{equation*}\n\\sigma(t)=2t+o(t) \\quad (t \\to \\infty).\n\\end{equation*}\nThe refined behavior of $\\sigma(t)$ is revealed in \\cite{MR705746,MR494541}. More precisely, it is shown that $\\sigma(t)$ satisfies\n\\begin{equation*}\n\\sigma(t)=2t-\\frac{3}{2}\\ln t+z_* +o(1)  \\quad (t \\to \\infty)\n\\end{equation*}\nfor some number $z_*$.\nSimilar results for more general reaction terms and initial functions are obtained in \\cite{MR4412555,MR422875,MR803086,MR509494}.\n\nFor pushed fronts, Stokes \\cite{MR682241} and Rothe \\cite{MR639447} proved that, if the initial data $u_{0}(y)$ satisfies\n\\begin{equation*}\n0\\le u_0(y)\\le1, \\qquad\n\\liminf_{y\\rightarrow-\\infty}u_0(y)>0, \\qquad\nu_0(y)\\le Ke^{\\lambda y},\n\\end{equation*}\nfor some constants $K>0$ and $\\lambda<\\lambda_+(c_*)$, then \n\\begin{equation}\nu(z+c^*t,t)\\rightarrow \\Phi_{c^*}(z+\\xi) \\quad (t\\rightarrow\\infty)\n\\end{equation}\nfor some constant $\\xi$. \nIn contrast to pulled fronts with the minimal speed,\nthe logarithmic correction term does not appear for pushed fronts.\nThis is analogous to the result in the bistable case \\cite{MR442480}.\n\nIn higher dimensional cases, the pioneering work has done by Aronson and Weinberger \\cite{MR511740}. \nThey prove that if the initial data has compact support and satisfies $0 \\le u_0 \\le 1$, $u_0 \\not\\equiv0$, then\n\\begin{equation*}\n\\lim_{t \\to \\infty} \\sup_{|x|+|y| \\ge (c^*+\\delta)t} |u(x,y,t)|=0,\n\\qquad\n\\lim_{t \\to \\infty} \\sup_{|x|+|y| \\le (c^*-\\delta)t} |u(x,y,t)-1|=0,\n\\end{equation*}\nfor any $\\delta>0$. Since then, the large-time behavior of solutions with compactly supported initial data has been extensively studied \\cite{MR3333711,MR670523, MR4026189,MR801583}. \n\nOur interest is the asymptotic behavior of solutions with non-compactly supported initial data. In contrast to the case where the initial data has compact support, less is known about the behavior of such solutions.\nTo observe what can occur, we recall results for bistable reaction diffusion equations established by Matano and Nara \\cite{MR2837694}.\nUnder some mild assumptions on the initial data,\nthey showed  the convergence of a solution to $\\Psi(y-\\gamma(x,t))$ for some function $\\gamma(x,t)$, where $\\Psi$ denotes a one-dimensional traveling wave solution. Moreover, they also found that $\\gamma(x,t)$ is approximated by the mean curvature flow with a drift term.\nTo be more precise, the following result is proved.\n\n\\begin{thma}[\\cite{MR2837694}]\nSuppose that $f$ satisfies\n\\begin{gather*}\nf(0)=f(1)=0,\\quad f^{\\prime}(0)<0,\\quad f^{\\prime}(1)<0, \\quad\nf(s) \\left\\{ \n\\begin{aligned}\n&>0 &&\\mbox{if } s \\in (-\\infty,0), \\\\\n&<0 &&\\mbox{if } s \\in (1,\\infty)\n\\end{aligned}\n\\right.\n\\end{gather*}\nand that there exist $c\\in\\mathbf{R}$ and $\\Psi \\in C^2 (\\mathbf{R})$ satisfying\n\\begin{gather*}\n\\Psi^{\\prime\\prime}+c\\Psi^{\\prime}+f(\\Psi)=0, \\quad\n\\lim_{z\\to-\\infty}\\Psi(z)=1, \\quad \\lim_{z\\to\\infty}\\Psi(z)=0.\n\\end{gather*}\nPut \n\\begin{gather*}\ns_+:=\\inf \\{ s_0 \\in (0,1); \\, f>0 \\mbox{ on } (s_0,1)\\},\\\\\ns_-:=\\sup \\{ s_0 \\in (0,1); \\, f<0 \\mbox{ on } (0,s_0)\\},\n\\end{gather*}\nand suppose that $u_0$ satisfies\n\\begin{equation}\n\\liminf_{y\\rightarrow-\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}}u_0 (x,y)>s_+,\\quad \\limsup_{y\\rightarrow\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}}u_0 (x,y)<s_-.\n\\end{equation}\nLet $u$ be a solution of \\eqref{Rea-Diff}.\nThen there exists a smooth function  $\\gamma=\\gamma(x,t)$ with the following properties.\n\\begin{enumerate}\n\\item[(i)] \nThere exists $T>0$ such that \n\\begin{equation*}\n\\{ (x,y,t) \\in \\mathbf{R}^{n-1} \\times \\mathbf{R} \\times [T,\\infty); \\, u(x,y,t)=\\Psi(0)\\}=\\{y=\\gamma(x,t)\\}.\n\\end{equation*}\nMoreover, it holds that\n\\begin{equation*}\n\\lim_{t\\rightarrow\\infty}\\sup_{(x,y)\\in\\mathbf{R}^n }|u(x,y,t)-\\Psi(y-\\gamma(x,t))|=0.\n\\end{equation*}\n\\item[(ii)]\nFor any $\\varepsilon>0$, there exists $\\tau_{\\varepsilon}\\in[T,\\infty)$ such that the solution $U(x,t)$ of the problem\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c,&&x\\in\\mathbf{R}^{n-1},t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_{\\varepsilon}),&&x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation*}\nsatisfies\n\\begin{equation*}\n\\sup_{x\\in\\mathbf{R}^{n-1},t\\ge\\tau_{\\varepsilon}}|\\gamma(x,t)-U(x,t-\\tau_{\\varepsilon})|\\le\\varepsilon.\n\\end{equation*}\n\\end{enumerate}\n\\end{thma}\n\nIn the case where $f$ is of monostable type, a similar result is obtained by Wang \\cite{MR3746497}.\n\\begin{thmb}[\\cite{MR3746497}]\nIn addition to (F), assume that \n\\begin{equation}\\label{assmpf}\n0<f'(0)u-f(u)<Mu^{1+\\alpha}\n\\quad \\mbox{if } \\ u \\in (0,1)\n\\end{equation}\nfor some constants $M>0$ and $\\alpha \\in (0,1]$.\nThen, there exists $\\delta_0>0$ such that if the initial data $u_0$ satisfies\n\\begin{gather}\n\\liminf_{y \\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0, \\label{lbidni} \\\\\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)}, \\quad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty \n\\ \\mbox{ for some } \\ c \\ge c_*, \\label{init2}\n\\end{gather}\nthe following are true.\n\\begin{enumerate}\n\\item[(i)]\nThe assertions of Theorem~A (i) with $\\Psi$ replaced by $\\Phi_c$ hold for some smooth function $\\gamma=\\gamma(x,t)$.\n\\item[(ii)]\nFor any $\\varepsilon>0$, there exists $T_{\\varepsilon}>0$ such that the inequalities\n\\begin{equation*}\nv^-(x,t)-\\varepsilon\\le\\gamma(x,t)\\le v^+(x,t)+\\varepsilon,\\quad t\\ge T_{\\varepsilon},\n\\end{equation*}\nhold for the solutions $v^-$ and $v^+$ of the initial value problems\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\nv^-_t&=\\Delta_x v^--k|\\nabla_x v^-|^2+c, && x\\in\\mathbf{R}^{n-1}, \\, t>0,\\\\\nv^-(x,0)&=\\gamma(x,T_{\\varepsilon}),         && x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation*}\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\nv^+_t&=\\Delta_x v^++k|\\nabla_x v^+|^2+c, && x\\in\\mathbf{R}^{n-1}, \\, t>0,\\\\\nv^+(x,0)&=\\gamma(x,T_{\\varepsilon}),         && x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation*}\nwhere $k:=\\sup_{z\\in\\mathbf{R}}|\\Phi^{\\prime\\prime}_c(z)|/|\\Phi^{\\prime}_c(z)|$.\n\\end{enumerate}\n\\end{thmb}\n\nIt is well-known that condition \\eqref{assmpf} implies $c_*=2\\sqrt{f'(0)}$, which means that there is no pushed front under condition \\eqref{assmpf}. The purpose of this paper is to reveal the behavior of $\\gamma(x,t)$ when $\\Phi_c(x-ct)$ is a pushed front. First, we verify that the same assertion as in Theorem~B (i) holds \nwhen condition \\eqref{assmpf} is dropped and conditions \\eqref{lbidni} and \\eqref{init2} are replaced with\n\\begin{equation}\n\\label{init1}\n\\liminf_{y\\to-\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}}u_{0}(x,y)>0 \\\\\n\\end{equation}\nand\n\\begin{equation}\n\\label{init3}\nc=c_*>2\\sqrt{f'(0)}, \\qquad \n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}} u_{0}(x,y)e^{-\\lambda_1 y}<\\infty \\ \\mbox{ for some } \\ \\lambda_1<\\lambda_+,\n\\end{equation}\nrespectively. We then prove that the behavior of $\\gamma(x,t)$ is governed by the mean curvature flow with a drift term, as in the case where $f$ is of bistable type. More precisely, our main result is stated as follows.", "context": "In higher dimensional cases, the pioneering work has done by Aronson and Weinberger \\cite{MR511740}. \nThey prove that if the initial data has compact support and satisfies $0 \\le u_0 \\le 1$, $u_0 \\not\\equiv0$, then\n\\begin{equation*}\n\\lim_{t \\to \\infty} \\sup_{|x|+|y| \\ge (c^*+\\delta)t} |u(x,y,t)|=0,\n\\qquad\n\\lim_{t \\to \\infty} \\sup_{|x|+|y| \\le (c^*-\\delta)t} |u(x,y,t)-1|=0,\n\\end{equation*}\nfor any $\\delta>0$. Since then, the large-time behavior of solutions with compactly supported initial data has been extensively studied \\cite{MR3333711,MR670523, MR4026189,MR801583}.\n\n\\begin{thma}[\\cite{MR2837694}]\nSuppose that $f$ satisfies\n\\begin{gather*}\nf(0)=f(1)=0,\\quad f^{\\prime}(0)<0,\\quad f^{\\prime}(1)<0, \\quad\nf(s) \\left\\{ \n\\begin{aligned}\n&>0 &&\\mbox{if } s \\in (-\\infty,0), \\\\\n&<0 &&\\mbox{if } s \\in (1,\\infty)\n\\end{aligned}\n\\right.\n\\end{gather*}\nand that there exist $c\\in\\mathbf{R}$ and $\\Psi \\in C^2 (\\mathbf{R})$ satisfying\n\\begin{gather*}\n\\Psi^{\\prime\\prime}+c\\Psi^{\\prime}+f(\\Psi)=0, \\quad\n\\lim_{z\\to-\\infty}\\Psi(z)=1, \\quad \\lim_{z\\to\\infty}\\Psi(z)=0.\n\\end{gather*}\nPut \n\\begin{gather*}\ns_+:=\\inf \\{ s_0 \\in (0,1); \\, f>0 \\mbox{ on } (s_0,1)\\},\\\\\ns_-:=\\sup \\{ s_0 \\in (0,1); \\, f<0 \\mbox{ on } (0,s_0)\\},\n\\end{gather*}\nand suppose that $u_0$ satisfies\n\\begin{equation}\n\\liminf_{y\\rightarrow-\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}}u_0 (x,y)>s_+,\\quad \\limsup_{y\\rightarrow\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}}u_0 (x,y)<s_-.\n\\end{equation}\nLet $u$ be a solution of \\eqref{Rea-Diff}.\nThen there exists a smooth function $\\gamma=\\gamma(x,t)$ with the following properties.\n\\begin{enumerate}\n\\item[(i)] \nThere exists $T>0$ such that \n\\begin{equation*}\n\\{ (x,y,t) \\in \\mathbf{R}^{n-1} \\times \\mathbf{R} \\times [T,\\infty); \\, u(x,y,t)=\\Psi(0)\\}=\\{y=\\gamma(x,t)\\}.\n\\end{equation*}\nMoreover, it holds that\n\\begin{equation*}\n\\lim_{t\\rightarrow\\infty}\\sup_{(x,y)\\in\\mathbf{R}^n }|u(x,y,t)-\\Psi(y-\\gamma(x,t))|=0.\n\\end{equation*}\n\\item[(ii)]\nFor any $\\varepsilon>0$, there exists $\\tau_{\\varepsilon}\\in[T,\\infty)$ such that the solution $U(x,t)$ of the problem\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c,&&x\\in\\mathbf{R}^{n-1},t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_{\\varepsilon}),&&x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation*}\nsatisfies\n\\begin{equation*}\n\\sup_{x\\in\\mathbf{R}^{n-1},t\\ge\\tau_{\\varepsilon}}|\\gamma(x,t)-U(x,t-\\tau_{\\varepsilon})|\\le\\varepsilon.\n\\end{equation*}\n\\end{enumerate}\n\\end{thma}\n\nIn the case where $f$ is of monostable type, a similar result is obtained by Wang \\cite{MR3746497}.\n\\begin{thmb}[\\cite{MR3746497}]\nIn addition to (F), assume that \n\\begin{equation}\\label{assmpf}\n0<f'(0)u-f(u)<Mu^{1+\\alpha}\n\\quad \\mbox{if } \\ u \\in (0,1)\n\\end{equation}\nfor some constants $M>0$ and $\\alpha \\in (0,1]$.\nThen, there exists $\\delta_0>0$ such that if the initial data $u_0$ satisfies\n\\begin{gather}\n\\liminf_{y \\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0, \\label{lbidni} \\\\\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)}, \\quad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty \n\\ \\mbox{ for some } \\ c \\ge c_*, \\label{init2}\n\\end{gather}\nthe following are true.\n\\begin{enumerate}\n\\item[(i)]\nThe assertions of Theorem~A (i) with $\\Psi$ replaced by $\\Phi_c$ hold for some smooth function $\\gamma=\\gamma(x,t)$.\n\\item[(ii)]\nFor any $\\varepsilon>0$, there exists $T_{\\varepsilon}>0$ such that the inequalities\n\\begin{equation*}\nv^-(x,t)-\\varepsilon\\le\\gamma(x,t)\\le v^+(x,t)+\\varepsilon,\\quad t\\ge T_{\\varepsilon},\n\\end{equation*}\nhold for the solutions $v^-$ and $v^+$ of the initial value problems\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\nv^-_t&=\\Delta_x v^--k|\\nabla_x v^-|^2+c, && x\\in\\mathbf{R}^{n-1}, \\, t>0,\\\\\nv^-(x,0)&=\\gamma(x,T_{\\varepsilon}), && x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation*}\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\nv^+_t&=\\Delta_x v^++k|\\nabla_x v^+|^2+c, && x\\in\\mathbf{R}^{n-1}, \\, t>0,\\\\\nv^+(x,0)&=\\gamma(x,T_{\\varepsilon}), && x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation*}\nwhere $k:=\\sup_{z\\in\\mathbf{R}}|\\Phi^{\\prime\\prime}_c(z)|/|\\Phi^{\\prime}_c(z)|$.\n\\end{enumerate}\n\\end{thmb}\n\nIt is well-known that condition \\eqref{assmpf} implies $c_*=2\\sqrt{f'(0)}$, which means that there is no pushed front under condition \\eqref{assmpf}. The purpose of this paper is to reveal the behavior of $\\gamma(x,t)$ when $\\Phi_c(x-ct)$ is a pushed front. First, we verify that the same assertion as in Theorem~B (i) holds \nwhen condition \\eqref{assmpf} is dropped and conditions \\eqref{lbidni} and \\eqref{init2} are replaced with\n\\begin{equation}\n\\label{init1}\n\\liminf_{y\\to-\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}}u_{0}(x,y)>0 \\\\\n\\end{equation}\nand\n\\begin{equation}\n\\label{init3}\nc=c_*>2\\sqrt{f'(0)}, \\qquad \n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}} u_{0}(x,y)e^{-\\lambda_1 y}<\\infty \\ \\mbox{ for some } \\ \\lambda_1<\\lambda_+,\n\\end{equation}\nrespectively. We then prove that the behavior of $\\gamma(x,t)$ is governed by the mean curvature flow with a drift term, as in the case where $f$ is of bistable type. More precisely, our main result is stated as follows.", "full_context": "In higher dimensional cases, the pioneering work has done by Aronson and Weinberger \\cite{MR511740}. \nThey prove that if the initial data has compact support and satisfies $0 \\le u_0 \\le 1$, $u_0 \\not\\equiv0$, then\n\\begin{equation*}\n\\lim_{t \\to \\infty} \\sup_{|x|+|y| \\ge (c^*+\\delta)t} |u(x,y,t)|=0,\n\\qquad\n\\lim_{t \\to \\infty} \\sup_{|x|+|y| \\le (c^*-\\delta)t} |u(x,y,t)-1|=0,\n\\end{equation*}\nfor any $\\delta>0$. Since then, the large-time behavior of solutions with compactly supported initial data has been extensively studied \\cite{MR3333711,MR670523, MR4026189,MR801583}.\n\n\\begin{thma}[\\cite{MR2837694}]\nSuppose that $f$ satisfies\n\\begin{gather*}\nf(0)=f(1)=0,\\quad f^{\\prime}(0)<0,\\quad f^{\\prime}(1)<0, \\quad\nf(s) \\left\\{ \n\\begin{aligned}\n&>0 &&\\mbox{if } s \\in (-\\infty,0), \\\\\n&<0 &&\\mbox{if } s \\in (1,\\infty)\n\\end{aligned}\n\\right.\n\\end{gather*}\nand that there exist $c\\in\\mathbf{R}$ and $\\Psi \\in C^2 (\\mathbf{R})$ satisfying\n\\begin{gather*}\n\\Psi^{\\prime\\prime}+c\\Psi^{\\prime}+f(\\Psi)=0, \\quad\n\\lim_{z\\to-\\infty}\\Psi(z)=1, \\quad \\lim_{z\\to\\infty}\\Psi(z)=0.\n\\end{gather*}\nPut \n\\begin{gather*}\ns_+:=\\inf \\{ s_0 \\in (0,1); \\, f>0 \\mbox{ on } (s_0,1)\\},\\\\\ns_-:=\\sup \\{ s_0 \\in (0,1); \\, f<0 \\mbox{ on } (0,s_0)\\},\n\\end{gather*}\nand suppose that $u_0$ satisfies\n\\begin{equation}\n\\liminf_{y\\rightarrow-\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}}u_0 (x,y)>s_+,\\quad \\limsup_{y\\rightarrow\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}}u_0 (x,y)<s_-.\n\\end{equation}\nLet $u$ be a solution of \\eqref{Rea-Diff}.\nThen there exists a smooth function $\\gamma=\\gamma(x,t)$ with the following properties.\n\\begin{enumerate}\n\\item[(i)] \nThere exists $T>0$ such that \n\\begin{equation*}\n\\{ (x,y,t) \\in \\mathbf{R}^{n-1} \\times \\mathbf{R} \\times [T,\\infty); \\, u(x,y,t)=\\Psi(0)\\}=\\{y=\\gamma(x,t)\\}.\n\\end{equation*}\nMoreover, it holds that\n\\begin{equation*}\n\\lim_{t\\rightarrow\\infty}\\sup_{(x,y)\\in\\mathbf{R}^n }|u(x,y,t)-\\Psi(y-\\gamma(x,t))|=0.\n\\end{equation*}\n\\item[(ii)]\nFor any $\\varepsilon>0$, there exists $\\tau_{\\varepsilon}\\in[T,\\infty)$ such that the solution $U(x,t)$ of the problem\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c,&&x\\in\\mathbf{R}^{n-1},t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_{\\varepsilon}),&&x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation*}\nsatisfies\n\\begin{equation*}\n\\sup_{x\\in\\mathbf{R}^{n-1},t\\ge\\tau_{\\varepsilon}}|\\gamma(x,t)-U(x,t-\\tau_{\\varepsilon})|\\le\\varepsilon.\n\\end{equation*}\n\\end{enumerate}\n\\end{thma}\n\nIn the case where $f$ is of monostable type, a similar result is obtained by Wang \\cite{MR3746497}.\n\\begin{thmb}[\\cite{MR3746497}]\nIn addition to (F), assume that \n\\begin{equation}\\label{assmpf}\n0<f'(0)u-f(u)<Mu^{1+\\alpha}\n\\quad \\mbox{if } \\ u \\in (0,1)\n\\end{equation}\nfor some constants $M>0$ and $\\alpha \\in (0,1]$.\nThen, there exists $\\delta_0>0$ such that if the initial data $u_0$ satisfies\n\\begin{gather}\n\\liminf_{y \\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0, \\label{lbidni} \\\\\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)}, \\quad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty \n\\ \\mbox{ for some } \\ c \\ge c_*, \\label{init2}\n\\end{gather}\nthe following are true.\n\\begin{enumerate}\n\\item[(i)]\nThe assertions of Theorem~A (i) with $\\Psi$ replaced by $\\Phi_c$ hold for some smooth function $\\gamma=\\gamma(x,t)$.\n\\item[(ii)]\nFor any $\\varepsilon>0$, there exists $T_{\\varepsilon}>0$ such that the inequalities\n\\begin{equation*}\nv^-(x,t)-\\varepsilon\\le\\gamma(x,t)\\le v^+(x,t)+\\varepsilon,\\quad t\\ge T_{\\varepsilon},\n\\end{equation*}\nhold for the solutions $v^-$ and $v^+$ of the initial value problems\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\nv^-_t&=\\Delta_x v^--k|\\nabla_x v^-|^2+c, && x\\in\\mathbf{R}^{n-1}, \\, t>0,\\\\\nv^-(x,0)&=\\gamma(x,T_{\\varepsilon}), && x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation*}\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\nv^+_t&=\\Delta_x v^++k|\\nabla_x v^+|^2+c, && x\\in\\mathbf{R}^{n-1}, \\, t>0,\\\\\nv^+(x,0)&=\\gamma(x,T_{\\varepsilon}), && x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation*}\nwhere $k:=\\sup_{z\\in\\mathbf{R}}|\\Phi^{\\prime\\prime}_c(z)|/|\\Phi^{\\prime}_c(z)|$.\n\\end{enumerate}\n\\end{thmb}\n\nIt is well-known that condition \\eqref{assmpf} implies $c_*=2\\sqrt{f'(0)}$, which means that there is no pushed front under condition \\eqref{assmpf}. The purpose of this paper is to reveal the behavior of $\\gamma(x,t)$ when $\\Phi_c(x-ct)$ is a pushed front. First, we verify that the same assertion as in Theorem~B (i) holds \nwhen condition \\eqref{assmpf} is dropped and conditions \\eqref{lbidni} and \\eqref{init2} are replaced with\n\\begin{equation}\n\\label{init1}\n\\liminf_{y\\to-\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}}u_{0}(x,y)>0 \\\\\n\\end{equation}\nand\n\\begin{equation}\n\\label{init3}\nc=c_*>2\\sqrt{f'(0)}, \\qquad \n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}} u_{0}(x,y)e^{-\\lambda_1 y}<\\infty \\ \\mbox{ for some } \\ \\lambda_1<\\lambda_+,\n\\end{equation}\nrespectively. We then prove that the behavior of $\\gamma(x,t)$ is governed by the mean curvature flow with a drift term, as in the case where $f$ is of bistable type. More precisely, our main result is stated as follows.\n\nWe prove Theorem~\\ref{thm1} by constructing appropriate comparison functions. They are given in the form\n\\begin{equation*}\nu^{\\pm}(x,y,t)\\coloneqq\\Phi_{c^{*}}\\left(\\frac{y-V(x,t)}{\\sqrt{1+|\\nabla_{x}V|^{2}}}\\mp q(t)\\right)\\pm p(t)\\chi \\left( e^{\\lambda (y-c_*t)}\\right),\n\\end{equation*}\nwhere $V$ is a solution of the equation\n\\begin{equation*}\nV_{t}=\\Delta_{x}V+\\frac{c^{*}}{2}|\\nabla_{x} V|^2+c_*,\n\\end{equation*}\n$\\lambda$ is a number with $\\lambda<\\lambda_1$\nand $\\chi$ is a smooth function safisfying $\\chi(s)=s$ ($s \\le 1/2$) and $\\chi(s)=1$ ($s \\ge 1$).\nWe will show that $u^+$ (resp. $u^-$) becomes a supersolution (resp. a subsolution) for the problem \\eqref{Rea-Diff} if $p(t)$, $q(t)$ and the initial data for $V(x,t)$ are chosen appropriately. Theorem~\\ref{thm1} (ii) is then proved by using these comparison functions and applying the fact that the solution $U$ of \\eqref{MCF} is approximated by $V$.\n\nIn this section, we establish the following estimates.\n\\begin{prop}\\label{lem3}\nLet $u(x,z,t)$ be a solution of \\eqref{MRea-Diff}. \nIf the initial data $u_{0}$ is satisfies \\eqref{init1} and \\eqref{init3},\nthen there exists constants $z_{1},z_{2}\\in\\mathbb{R}$ such that\n\\begin{equation}\n\\label{upper}\n\\limsup_{t\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}u(x,z,t)\\le\\Phi_{c}(z-z_{0})\\;\\mbox{uniformly in }z\\in\\mathbb{R}\n\\end{equation}\n\\begin{equation}\n\\label{lower}\n\\liminf_{t\\to\\infty}\\inf_{x\\in\\mathbb{R}^{n-1}}u(x,z,t)\\ge\\Phi_{c}(z-z_{1})\\;\\mbox{uniformly in }z\\in\\mathbb{R}\n\\end{equation}\nThe same inequalities hold if \\eqref{init2} is assumed instead of \\eqref{init3}.\n\\end{prop}\nWe split the proof of this proposition into the case 1 and case 2:\n\\begin{enumerate}\n\\item[1:]\\eqref{init1} and \\eqref{init3}\n\\item[2:]\\eqref{init1} and \\eqref{init2}\n\\end{enumerate}\nFirst, we show upper and lower estimates for case 1.\nFor this, we recall the lemmas in \\cite{MR639447}. We take any $\\lambda<\\lambda_{1}<\\lambda_{+}$ and define $\\psi(s)$ as\n\\begin{equation*}\n\\psi(s)=\\chi(e^{\\lambda_{1}s})\n\\end{equation*}\n\\begin{equation*}\n\\chi(s)\\coloneqq\\left\\{\n\\begin{aligned}\n1\\;(s\\ge1)\\\\\ns\\;(s\\le\\frac{1}{2})\n\\end{aligned}\n\\right.\n\\end{equation*}\n, where $0\\le\\chi(s)\\le1$ for $s\\in(\\frac{1}{2},1)$. For these, the following lemmas hold\n\\begin{lem}[\\cite{MR639447}]\\label{add_B}\nThere exists $p\\in(0,1)$ such that, for any $q_{0}\\in(0,p],z_{1},z_{2}\\in\\mathbf{R}$, there exists $\\beta>0,C>0$ such that\n\\begin{equation*}\nw^{+}(z,t)\\coloneqq \\Phi_{c^{*}}(z-z_{1}- C(1-e^{-\\beta t}))+q_{0}e^{-\\beta t}\\psi(z-z_{2})\n\\end{equation*}\nsatisfies $L[w^+]\\ge0$.\n\\end{lem}\n\n\\section{Level set of the solutions}\nAs we mention in the previous section, we only give statements of lemmas and omit their proof.\n\\begin{lem}[Level set]\\label{lem11}\nLet $u(x,z,t)$ be a solution of ($\\ref{MRea-Diff}$) and $T>0$ be as defined in Corollary 2. Then there exists a smooth bounded function $\\Gamma(x,t)$ such that\n\\begin{equation*}\nu(x,z,t)=\\Phi_{c}(0)\\quad\\mbox{if and only if}\\quad z=\\Gamma(x,t),\n\\end{equation*}\nfor any $(x,t)\\in\\mathbb{R}^{n-1}\\times[T,\\infty)$. Furthermore the following estimates hold:\n\\begin{enumerate}\n\\item For each $1\\le i,j \\le n-1$,\n\\begin{equation*}\n\\lim_{t\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}|\\Gamma_{x_{i}}(x,t)|=0,\\quad\\lim_{t\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}|\\Gamma_{x_{i}x_{j}}(x,t)|=0,\n\\end{equation*}\n\\item There exists a constant $M>0$ such that, for each $1\\le i,j,k \\le n-1$,\n\\end{enumerate}\n\\begin{equation*}\n\\sup_{x\\in\\mathbb{R}^{n-1}}|\\Gamma_{x_{i}x_{j}x_{k}}|\\le M,\\quad \\mbox{for}\\quad t\\ge T.\n\\end{equation*}\n\\end{lem}\n\n\\begin{lem}[Approximation of $\\Gamma(x,t)$]\\label{lem15}\nLet $u(x,z,t)$ be a solution of (\\ref{MRea-Diff}) and let $\\Gamma(x,t)$ be as defined in Lemma \\ref{lem11}. Then for any $\\varepsilon>0$, there exists a constant $\\tau_{\\varepsilon}>0$ such that the function $V(x,t)$ defined by\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\n&V_{t}=\\Delta_{x}V+\\frac{c^{*}}{2}|\\nabla_{x} V|,\\quad x\\in\\mathbb{R}^{n-1},t>0\\\\\n&V(x,0)=\\Gamma(x,\\tau_{\\varepsilon}).\\qquad\\hspace{9.4pt} x\\in\\mathbb{R}^{n-1}.\n\\end{aligned}\n\\right.\n\\end{equation*}\nsatisfies\n\\begin{equation*}\n\\sup_{x\\in\\mathbb{R}^{n-1}}|\\Gamma(x,t)-V(x,t-\\tau_{\\varepsilon})|\\le\\varepsilon,\\quad t\\ge\\tau_{\\varepsilon}.\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\n First, we verify upper bound. By Lemma \\ref{lem12} and \\ref{lem13}, we can take $T>0, M>0$ and $K>0$ such that, for $D\\coloneqq\\left\\{(x,z,t)\\in\\mathbb{R}^{n}\\times[T,\\infty)||u(x,z,t)-\\Phi_{c}(0)|\\le\\min(1-\\Phi_{c}(0),\\Phi_{c}(0))\\right\\}$\n\\begin{equation*}\n\\sup_{t\\ge T}\\|\\Gamma(\\cdot,t)\\|_{W^{3,\\infty}}\\le M,\\quad \\inf_{(x,z,t)\\in D}-u_{z}(x,z,t)\\ge K.\n\\end{equation*}\nFor the constants $M$ and $\\hat{\\varepsilon}\\coloneq1/(\\|\\Phi_{c^{*}}^{\\prime}\\|_{L^{\\infty}}+1)\\cdot\\min\\{K\\varepsilon,\\min(1-\\Phi_{c^{*}}(0),\\Phi_{c^{*}}(0))\\}$, we choose a constant $\\delta>0$ and functions $p(t),q(t)$ satisfying\n\\begin{equation*}\np(0)>0,\\quad q(0)=0,\\quad 0\\le p(t),q(t)\\le\\hat{\\varepsilon}\\quad for \\quad t\\ge0.\n\\end{equation*}\nFrom proof of Lemma \\ref{lem3} and (\\ref{asym}), we can take $z_{0}\\in\\mathbb{R}$ such that, for some larger $T>0$, \n\\begin{equation*}\nu(x,z,t)\\le p(0)e^{\\lambda z}\\quad((x,z,t)\\in\\mathbb{R}^{n-1}\\times[z_{0},\\infty)\\times[T,\\infty)).\n\\end{equation*}\nTaking $\\tau_{\\varepsilon}\\ge T$ larger if necessary, by Lemma \\ref{lem11} and \\ref{lem12}, the following holds\n\\begin{equation*}\nu(x,z,t)\\le\\Phi_{c^{*}}(z-\\Gamma(x,\\tau_{\\varepsilon}))+\\frac{p(0)e^{\\lambda z_{0}}}{2}\\le\\Phi_{c^{*}}\\left(\\frac{z-\\Gamma(x,\\tau_{\\varepsilon})}{\\sqrt{1+|\\nabla_{x}\\Gamma|^{2}}}\\right)+p(0)e^{\\lambda z_{0}}\n\\end{equation*}\nFor any $z\\ge z_{0}$, \n\\begin{equation*}\nu(x,z,\\tau_{\\varepsilon})\\le p(0)e^{\\lambda z}\\le u^{+}(x,z,0).\n\\end{equation*}\nFor any $z<z_{0}$,\n\\begin{equation*}\nu(x,z,\\tau_{\\varepsilon})\\le \\Phi_{c^{*}}\\left(\\frac{z-\\Gamma(x,\\tau_{\\varepsilon})}{\\sqrt{1+|\\nabla_{x}\\Gamma|^{2}}}\\right)+p(0)e^{\\lambda z_{0}}\\le\\Phi_{c^{*}}\\left(\\frac{z-\\Gamma(x,\\tau_{\\varepsilon})}{\\sqrt{1+|\\nabla_{x}\\Gamma|^{2}}}\\right)+p(0)\\psi(z).\n\\end{equation*}\nSo, by comparison principle, $u(x,z,t)\\le u^{+}(x,z,t)$ for $t\\ge\\tau_{\\varepsilon}$. Therefore, we have the following inequality\n\\begin{align*}\nu(x,V(x,t-\\tau_{\\varepsilon}),t)-\\Phi_{c^{*}}(0)&\\le u^{+}(x,V(x,t-\\tau_{\\varepsilon}),t)-\\Phi_{c^{*}}(0)\\\\\n &=\\Phi_{c^{*}}(-q(t-\\tau_{\\varepsilon}))-\\Phi_{c^{*}}(0)+p(t-\\tau_{\\varepsilon})\\\\\n &\\le(\\|\\Phi_{c^{*}}^{\\prime}\\|_{L^{\\infty}}+1)\\hat{\\varepsilon}\\\\\n &=\\min\\{K\\varepsilon,\\min\\{1-\\Phi_{c^{*}}(0),\\Phi_{c^{*}}(0)\\}\\}.\n\\end{align*}\nThus, we have, for $\\Gamma(x,t)\\ge V(x,t-\\tau_{\\varepsilon})$,\n\\begin{align*}\nK\\varepsilon&\\ge u(x,V(x,t-\\tau_{\\varepsilon},t),t)-u(x,\\Gamma(x,t),t)\\\\\n &\\ge (\\inf_{u\\in[0,\\min\\{1-\\Phi_{c^{*}}(0),\\Phi_{c^{*}}(0)\\}],t\\ge\\tau_{\\varepsilon}}-u_{z})\\cdot(\\Gamma(x,t)-V(x,t-\\tau_{\\varepsilon}))\\\\\n &\\ge K(\\Gamma(x,t)-V(x,t-\\tau_{\\varepsilon})).\n\\end{align*}\nThis implies $\\Gamma(x,t)\\le V(x,t-\\tau_{\\varepsilon})+\\varepsilon$ for $t\\ge\\tau_{\\varepsilon}$. Thus, we have the upper estimate. The lower estimate is followed from Lemma \\ref{lem14} in a similar way.\n\\end{proof}", "post_theorem_intro_text_len": 1514, "post_theorem_intro_text": "We prove Theorem~\\ref{thm1} by constructing appropriate comparison functions. They are given in the form\n\\begin{equation*}\nu^{\\pm}(x,y,t)\\coloneqq\\Phi_{c^{*}}\\left(\\frac{y-V(x,t)}{\\sqrt{1+|\\nabla_{x}V|^{2}}}\\mp q(t)\\right)\\pm p(t)\\chi \\left( e^{\\lambda (y-c_*t)}\\right),\n\\end{equation*}\nwhere $V$ is a solution of the equation\n\\begin{equation*}\nV_{t}=\\Delta_{x}V+\\frac{c^{*}}{2}|\\nabla_{x} V|^2+c_*,\n\\end{equation*}\n$\\lambda$ is a number with $\\lambda<\\lambda_1$\nand $\\chi$ is a smooth function safisfying $\\chi(s)=s$ ($s \\le 1/2$) and $\\chi(s)=1$ ($s \\ge 1$).\nWe will show that $u^+$ (resp. $u^-$) becomes a supersolution (resp. a subsolution) for the problem \\eqref{Rea-Diff} if $p(t)$, $q(t)$ and the initial data for $V(x,t)$ are chosen appropriately. Theorem~\\ref{thm1} (ii) is then proved by using these comparison functions and applying the fact that the solution $U$ of \\eqref{MCF} is approximated by $V$. \n\nThis paper is organized as follows. In Section 2, we recall results on the approximation of the mean curvature flow obtained in \\cite{MR2837694}. Section 3 establishes upper and lower bounds of solutions at large time. In Section 4, we define $\\omega$-limit points and provide their characterization. Section 5 establishes smoothness of level sets of solutions. Section 6 is devoted to the construction of comprison functions, which are used to prove Theorem \\ref{thm1} (ii). In Section 7, we prove Theorem \\ref{thm1}.\n\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\renewcommand{\\theenumi)}{(\\theenumi)}", "sketch": "To prove Theorem~\\ref{thm1}, the authors “construct[] appropriate comparison functions” of the form\n\\[\n u^{\\pm}(x,y,t)\\coloneqq\\Phi_{c^{*}}\\left(\\frac{y-V(x,t)}{\\sqrt{1+|\\nabla_{x}V|^{2}}}\\mp q(t)\\right)\\pm p(t)\\chi \\left( e^{\\lambda (y-c_*t)}\\right),\n\\]\nwhere $V$ solves\n\\[\nV_{t}=\\Delta_{x}V+\\frac{c^{*}}{2}|\\nabla_{x} V|^2+c_*,\n\\]\n$\\lambda<\\lambda_1$, and $\\chi$ is smooth with “$\\chi(s)=s$ ($s \\le 1/2$) and $\\chi(s)=1$ ($s \\ge 1$).” They then “show that $u^+$ (resp. $u^-$) becomes a supersolution (resp. a subsolution) for the problem \\eqref{Rea-Diff} if $p(t)$, $q(t)$ and the initial data for $V(x,t)$ are chosen appropriately.”\n\nFor Theorem~\\ref{thm1}(ii), they “use[] these comparison functions and apply[] the fact that the solution $U$ of \\eqref{MCF} is approximated by $V$.”", "expanded_sketch": "To prove the main theorem, the authors “construct[] appropriate comparison functions” of the form\n\\[\n u^{\\pm}(x,y,t)\\coloneqq\\Phi_{c^{*}}\\left(\\frac{y-V(x,t)}{\\sqrt{1+|\\nabla_{x}V|^{2}}}\\mp q(t)\\right)\\pm p(t)\\chi \\left( e^{\\lambda (y-c_*t)}\\right),\n\\]\nwhere $V$ solves\n\\[\nV_{t}=\\Delta_{x}V+\\frac{c^{*}}{2}|\\nabla_{x} V|^2+c_*,\n\\]\n$\\lambda<\\lambda_1$, and $\\chi$ is smooth with “$\\chi(s)=s$ ($s \\le 1/2$) and $\\chi(s)=1$ ($s \\ge 1$).” They then “show that $u^+$ (resp. $u^-$) becomes a supersolution (resp. a subsolution) for the problem\n\\begin{aligned}\\label{Rea-Diff}\n&u_{t}=\\Delta u+f(u), && x\\in\\mathbb{R}^{n-1}, \\, y\\in\\mathbb{R}, \\, t>0, \\\\\n&u(x,y,0)=u_{0}(x,y), && x\\in\\mathbb{R}^{n-1}, \\, y\\in\\mathbb{R}.\n\\end{aligned}\nif $p(t)$, $q(t)$ and the initial data for $V(x,t)$ are chosen appropriately.”\n\nIn establishing part (ii) of the main theorem, they “use[] these comparison functions and apply[] the fact that the solution $U$ of\n\\begin{equation}\\label{MCF}\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c_*,&&x\\in\\mathbf{R}^{n-1},t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_{\\varepsilon}),&&x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation}\nis approximated by $V$.”", "expanded_theorem": "\\label{thm1}\nAssume (F), \n\\begin{equation}\n\\label{init1}\n\\liminf_{y\\to-\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}}u_{0}(x,y)>0 \\\\\n\\end{equation}\nand \n\\begin{equation}\n\\label{init3}\nc=c_*>2\\sqrt{f'(0)}, \\qquad \n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}} u_{0}(x,y)e^{-\\lambda_1 y}<\\infty \\ \\mbox{ for some } \\ \\lambda_1<\\lambda_+,\n\\end{equation}\nand let $u$ be a solution of \n\\begin{aligned}\\label{Rea-Diff}\n&u_{t}=\\Delta u+f(u), && x\\in\\mathbb{R}^{n-1}, \\, y\\in\\mathbb{R}, \\, t>0, \\\\\n&u(x,y,0)=u_{0}(x,y), && x\\in\\mathbb{R}^{n-1}, \\, y\\in\\mathbb{R}.\n\\end{aligned}.\nThen there exists a smooth function  $\\gamma=\\gamma(x,t)$ with the following properties.\n\\begin{enumerate}\n\\item[(i)] \nThere exists $T>0$ such that \n\\begin{equation*}\n\\{ (x,y,t) \\in \\mathbf{R}^{n-1} \\times \\mathbf{R} \\times [T,\\infty); \\, u(x,y,t)=\\Phi_c(0)\\}=\\{y=\\gamma(x,t)\\}.\n\\end{equation*}\nMoreover, it holds that\n\\begin{equation*}\n\\lim_{t\\rightarrow\\infty}\\sup_{(x,y)\\in\\mathbf{R}^n }|u(x,y,t)-\\Phi_c(y-\\gamma(x,t))|=0.\n\\end{equation*}\n\\item[(ii)]\nFor any $\\varepsilon>0$, there exists $\\tau_{\\varepsilon}\\in[T,\\infty)$ such that the solution $U(x,t)$ of the problem\n\\begin{equation}\\label{MCF}\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c_*,&&x\\in\\mathbf{R}^{n-1},t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_{\\varepsilon}),&&x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation}\nsatisfies\n\\begin{equation*}\n\\sup_{x\\in\\mathbf{R}^{n-1},t\\ge\\tau_{\\varepsilon}}|\\gamma(x,t)-U(x,t-\\tau_{\\varepsilon})|\\le\\varepsilon.\n\\end{equation*}\n\\end{enumerate}\nFurthermore, the assertion (i) still holds if\n\\begin{gather}\n\\liminf_{y \\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0, \\label{lbidni} \\\\\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)}, \\quad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty \n\\ \\mbox{ for some } \\ c \\ge c_*, \\\\\n\\end{gather}\nis assumed instead of the preceding condition.\n", "theorem_type": ["Existence", "Asymptotic or Limit"], "mcq": {"question": "Let $u=u(x,y,t)$ solve the Cauchy problem\n\\[\n\\begin{aligned}\n&u_t=\\Delta u+f(u), && x\\in\\mathbb{R}^{n-1},\\ y\\in\\mathbb{R},\\ t>0,\\\\\n&u(x,y,0)=u_0(x,y), && x\\in\\mathbb{R}^{n-1},\\ y\\in\\mathbb{R}.\n\\end{aligned}\n\\]\nAssume $f$ satisfies the standing hypothesis $(F)$ from the traveling-front setting, and let $\\Phi_c$ denote the associated one-dimensional front profile, i.e. a $C^2(\\mathbb R)$ solution of\n\\[\n\\Phi_c''+c\\Phi_c'+f(\\Phi_c)=0,\\qquad \\Phi_c(-\\infty)=1,\\qquad \\Phi_c(+\\infty)=0,\n\\]\nwith minimal wave speed $c_*$. Let $\\lambda_+$ be the larger root of $\\lambda^2-c_*\\lambda+f'(0)=0$. Assume moreover that\n\\[\n\\liminf_{y\\to-\\infty}\\inf_{x\\in\\mathbb{R}^{n-1}}u_0(x,y)>0,\n\\]\nand\n\\[\nc=c_*>2\\sqrt{f'(0)},\\qquad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}} u_0(x,y)e^{-\\lambda_1 y}<\\infty\n\\quad\\text{for some }\\lambda_1<\\lambda_+.\n\\]\nAlso consider the alternative initial-data regime, for a constant $\\delta_0>0$,\n\\[\n\\liminf_{y\\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0,\n\\]\n\\[\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbb{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)}, \\qquad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty\n\\quad\\text{for some } c\\ge c_*.\n\\]\nUnder these assumptions, which existence statement holds?", "correct_choice": {"label": "A", "text": "There exists a smooth function $\\gamma=\\gamma(x,t)$ such that: (i) for some $T>0$,\n\\[\n\\{(x,y,t)\\in\\mathbb{R}^{n-1}\\times\\mathbb{R}\\times[T,\\infty):\\ u(x,y,t)=\\Phi_c(0)\\}=\\{y=\\gamma(x,t)\\},\n\\]\nand\n\\[\n\\lim_{t\\to\\infty}\\sup_{(x,y)\\in\\mathbb{R}^n}\\big|u(x,y,t)-\\Phi_c(y-\\gamma(x,t))\\big|=0.\n\\]\n(ii) For every $\\varepsilon>0$, there exists $\\tau_\\varepsilon\\in[T,\\infty)$ such that if $U(x,t)$ solves\n\\[\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c_*,&&x\\in\\mathbb{R}^{n-1},\\ t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_\\varepsilon),&&x\\in\\mathbb{R}^{n-1},\n\\end{aligned}\n\\right.\n\\]\nthen\n\\[\n\\sup_{x\\in\\mathbb{R}^{n-1},\\,t\\ge\\tau_\\varepsilon}\\big|\\gamma(x,t)-U(x,t-\\tau_\\varepsilon)\\big|\\le\\varepsilon.\n\\]\nFurthermore, if one replaces the initial-data assumptions above by the alternative regime\n\\[\n\\liminf_{y\\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0,\n\\]\n\\[\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbb{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)},\\qquad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty\n\\quad\\text{for some } c\\ge c_*,\n\\]\nthen conclusion (i) still holds."}, "choices": [{"label": "B", "text": "There exists a smooth function $\\gamma=\\gamma(x,t)$ such that: (i) for some $T>0$,\n\\[\n\\{(x,y,t)\\in\\mathbb{R}^{n-1}\\times\\mathbb{R}\\times[T,\\infty):\\ u(x,y,t)=\\Phi_c(0)\\}=\\{y=\\gamma(x,t)\\},\n\\]\nand\n\\[\n\\lim_{t\\to\\infty}\\sup_{(x,y)\\in\\mathbb{R}^n}\\big|u(x,y,t)-\\Phi_c(y-\\gamma(x,t))\\big|=0.\n\\]\n(ii) There exists $\\tau\\in[T,\\infty)$ such that if $U(x,t)$ solves\n\\[\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c_*,&&x\\in\\mathbb{R}^{n-1},\\ t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau),&&x\\in\\mathbb{R}^{n-1},\n\\end{aligned}\n\\right.\n\\]\nthen\n\\[\n\\lim_{t\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}\\big|\\gamma(x,t)-U(x,t-\\tau)\\big|=0.\n\\]\nFurthermore, if one replaces the initial-data assumptions above by the alternative regime\n\\[\n\\liminf_{y\\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0,\n\\]\n\\[\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbb{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)},\\qquad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty\n\\quad\\text{for some } c\\ge c_*,\n\\]\nthen conclusions (i) and (ii) still hold."}, {"label": "C", "text": "There exists a smooth function $\\gamma=\\gamma(x,t)$ such that for some $T>0$,\n\\[\n\\{(x,y,t)\\in\\mathbb{R}^{n-1}\\times\\mathbb{R}\\times[T,\\infty):\\ u(x,y,t)=\\Phi_c(0)\\}=\\{y=\\gamma(x,t)\\},\n\\]\nand\n\\[\n\\lim_{t\\to\\infty}\\sup_{(x,y)\\in\\mathbb{R}^n}\\big|u(x,y,t)-\\Phi_c(y-\\gamma(x,t))\\big|=0.\n\\]\nMoreover, if one replaces the initial-data assumptions above by the alternative regime\n\\[\n\\liminf_{y\\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0,\n\\]\n\\[\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbb{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)},\\qquad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty\n\\quad\\text{for some } c\\ge c_*,\n\\]\nthen the same conclusion still holds."}, {"label": "D", "text": "There exists a smooth function $\\gamma=\\gamma(x,t)$ such that: (i) for some $T>0$,\n\\[\n\\{(x,y,t)\\in\\mathbb{R}^{n-1}\\times\\mathbb{R}\\times[T,\\infty):\\ u(x,y,t)=\\Phi_c(0)\\}=\\{y=\\gamma(x,t)\\},\n\\]\nand\n\\[\n\\lim_{t\\to\\infty}\\sup_{(x,y)\\in\\mathbb{R}^n}\\big|u(x,y,t)-\\Phi_c(y-\\gamma(x,t))\\big|=0.\n\\]\n(ii) For every $\\varepsilon>0$, there exists $\\tau_\\varepsilon\\in[T,\\infty)$ such that if $U(x,t)$ solves\n\\[\n\\left\\{\n\\begin{aligned}\n&U_t=\\Delta_x U+\\frac{c_*}{2}|\\nabla_x U|^2+c_*,&&x\\in\\mathbb{R}^{n-1},\\ t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_\\varepsilon),&&x\\in\\mathbb{R}^{n-1},\n\\end{aligned}\n\\right.\n\\]\nthen\n\\[\n\\sup_{x\\in\\mathbb{R}^{n-1},\\,t\\ge\\tau_\\varepsilon}\\big|\\gamma(x,t)-U(x,t-\\tau_\\varepsilon)\\big|\\le\\varepsilon.\n\\]\nFurthermore, if one replaces the initial-data assumptions above by the alternative regime\n\\[\n\\liminf_{y\\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0,\n\\]\n\\[\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbb{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)},\\qquad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty\n\\quad\\text{for some } c\\ge c_*,\n\\]\nthen conclusion (i) still holds."}, {"label": "E", "text": "There exists a smooth function $\\gamma=\\gamma(x,t)$ such that: (i) for some $T>0$,\n\\[\n\\{(x,y,t)\\in\\mathbb{R}^{n-1}\\times\\mathbb{R}\\times[T,\\infty):\\ u(x,y,t)=\\Phi_c(0)\\}=\\{y=\\gamma(x,t)\\},\n\\]\nand\n\\[\n\\lim_{t\\to\\infty}\\sup_{(x,y)\\in\\mathbb{R}^n}\\big|u(x,y,t)-\\Phi_c(y-\\gamma(x,t))\\big|=0.\n\\]\n(ii) For every $\\varepsilon>0$, there exists $\\tau_\\varepsilon\\in[T,\\infty)$ such that if $U(x,t)$ solves\n\\[\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c,&&x\\in\\mathbb{R}^{n-1},\\ t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_\\varepsilon),&&x\\in\\mathbb{R}^{n-1},\n\\end{aligned}\n\\right.\n\\]\nthen\n\\[\n\\sup_{x\\in\\mathbb{R}^{n-1},\\,t\\ge\\tau_\\varepsilon}\\big|\\gamma(x,t)-U(x,t-\\tau_\\varepsilon)\\big|\\le\\varepsilon.\n\\]\nFurthermore, if one replaces the initial-data assumptions above by the alternative regime\n\\[\n\\liminf_{y\\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0,\n\\]\n\\[\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbb{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)},\\qquad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty\n\\quad\\text{for some } c\\ge c_*,\n\\]\nthen conclusion (i) still holds."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "quantifier on epsilon and extension of alternative-regime conclusion to (ii)", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped part (ii) approximation by the geometric-flow solution", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "replaced the mean-curvature-type evolution approximated by V with the viscous Hamilton-Jacobi equation solved by V itself", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "forcing term in geometric law uses c instead of the theorem’s c_* despite the special pushed-front assumption c=c_* in part (ii)", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the conclusion explicitly and does not single out the correct option. It provides hypotheses only, while the answer choices differ in subtle but nontrivial ways (speed, quantifiers, and scope of the alternative assumptions)."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem gives the full hypotheses and asks for the corresponding conclusion. The correct choice is basically the theorem statement itself rather than a consequence derived from fresh problem-solving."}, "GPS": {"score": 1, "justification": "Selecting the correct answer requires some careful reasoning about quantifiers and which parts persist under modified hypotheses, but it mainly tests recognition/matching of a known result rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong and plausible: they alter the front speed, overextend the alternative-data conclusion, weaken or change the approximation statement, or tamper with epsilon-dependent quantifiers. These reflect realistic mathematical failure modes."}, "total_score": 5, "overall_assessment": "Well-constructed in terms of leakage control and distractor quality, but it is largely a theorem-statement recognition question rather than a genuinely generative reasoning task."}}
{"id": "2602.09806v1", "paper_link": "http://arxiv.org/abs/2602.09806v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "\\label{thm1}\nAssume (F), \\eqref{init1} and \\eqref{init3} and let $u$ be a solution of \\eqref{Rea-Diff}. Then there exists a smooth function  $\\gamma=\\gamma(x,t)$ with the following properties.\n\\begin{enumerate}\n\\item[(i)] \nThere exists $T>0$ such that \n\\begin{equation*}\n\\{ (x,y,t) \\in \\mathbf{R}^{n-1} \\times \\mathbf{R} \\times [T,\\infty); \\, u(x,y,t)=\\Phi_c(0)\\}=\\{y=\\gamma(x,t)\\}.\n\\end{equation*}\nMoreover, it holds that\n\\begin{equation*}\n\\lim_{t\\rightarrow\\infty}\\sup_{(x,y)\\in\\mathbf{R}^n }|u(x,y,t)-\\Phi_c(y-\\gamma(x,t))|=0.\n\\end{equation*}\n\\item[(ii)]\nFor any $\\varepsilon>0$, there exists $\\tau_{\\varepsilon}\\in[T,\\infty)$ such that the solution $U(x,t)$ of the problem\n\\begin{equation}\\label{MCF}\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c_*,&&x\\in\\mathbf{R}^{n-1},t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_{\\varepsilon}),&&x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation}\nsatisfies\n\\begin{equation*}\n\\sup_{x\\in\\mathbf{R}^{n-1},t\\ge\\tau_{\\varepsilon}}|\\gamma(x,t)-U(x,t-\\tau_{\\varepsilon})|\\le\\varepsilon.\n\\end{equation*}\n\\end{enumerate}\nFurthermore, the assertion (i) still holds if \\eqref{init2} is assumed instead of \\eqref{init3}.", "start_pos": 11582, "end_pos": 12828, "label": "thm1"}, "ref_dict": {"thm1": "\\begin{thm}\\label{thm1}\nAssume (F), \\eqref{init1} and \\eqref{init3} and let $u$ be a solution of \\eqref{Rea-Diff}. Then there exists a smooth function  $\\gamma=\\gamma(x,t)$ with the following properties.\n\\begin{enumerate}\n\\item[(i)] \nThere exists $T>0$ such that \n\\begin{equation*}\n\\{ (x,y,t) \\in \\mathbf{R}^{n-1} \\times \\mathbf{R} \\times [T,\\infty); \\, u(x,y,t)=\\Phi_c(0)\\}=\\{y=\\gamma(x,t)\\}.\n\\end{equation*}\nMoreover, it holds that\n\\begin{equation*}\n\\lim_{t\\rightarrow\\infty}\\sup_{(x,y)\\in\\mathbf{R}^n }|u(x,y,t)-\\Phi_c(y-\\gamma(x,t))|=0.\n\\end{equation*}\n\\item[(ii)]\nFor any $\\varepsilon>0$, there exists $\\tau_{\\varepsilon}\\in[T,\\infty)$ such that the solution $U(x,t)$ of the problem\n\\begin{equation}\\label{MCF}\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c_*,&&x\\in\\mathbf{R}^{n-1},t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_{\\varepsilon}),&&x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation}\nsatisfies\n\\begin{equation*}\n\\sup_{x\\in\\mathbf{R}^{n-1},t\\ge\\tau_{\\varepsilon}}|\\gamma(x,t)-U(x,t-\\tau_{\\varepsilon})|\\le\\varepsilon.\n\\end{equation*}\n\\end{enumerate}\nFurthermore, the assertion (i) still holds if \\eqref{init2} is assumed instead of \\eqref{init3}.\n\\end{thm}", "Rea-Diff": "\\begin{aligned}\\label{Rea-Diff}\n&u_{t}=\\Delta u+f(u), && x\\in\\mathbb{R}^{n-1}, \\, y\\in\\mathbb{R}, \\, t>0, \\\\\n&u(x,y,0)=u_{0}(x,y), && x\\in\\mathbb{R}^{n-1}, \\, y\\in\\mathbb{R}.\n\\end{aligned}", "assmpf": "\\begin{equation}\\label{assmpf}\n0<f'(0)u-f(u)<Mu^{1+\\alpha}\n\\quad \\mbox{if } \\ u \\in (0,1)\n\\end{equation}", "lbidni": "\\begin{gather}\n\\liminf_{y \\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0, \\label{lbidni} \\\\\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)}, \\quad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty \n\\ \\mbox{ for some } \\ c \\ge c_*, \\label{init2}\n\\end{gather}", "init2": "\\begin{gather}\n\\liminf_{y \\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0, \\label{lbidni} \\\\\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)}, \\quad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty \n\\ \\mbox{ for some } \\ c \\ge c_*, \\label{init2}\n\\end{gather}", "MCF": "\\begin{equation}\\label{MCF}\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c_*,&&x\\in\\mathbf{R}^{n-1},t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_{\\varepsilon}),&&x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation}", "init1": "\\begin{equation}\n\\label{init1}\n\\liminf_{y\\to-\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}}u_{0}(x,y)>0 \\\\\n\\end{equation}", "init3": "\\begin{equation}\n\\label{init3}\nc=c_*>2\\sqrt{f'(0)}, \\qquad \n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}} u_{0}(x,y)e^{-\\lambda_1 y}<\\infty \\ \\mbox{ for some } \\ \\lambda_1<\\lambda_+,\n\\end{equation}"}, "pre_theorem_intro_text_len": 10008, "pre_theorem_intro_text": "In this paper, we consider the reaction-diffusion equation\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\\label{Rea-Diff}\n&u_{t}=\\Delta u+f(u), && x\\in\\mathbb{R}^{n-1}, \\, y\\in\\mathbb{R}, \\, t>0, \\\\\n&u(x,y,0)=u_{0}(x,y), && x\\in\\mathbb{R}^{n-1}, \\, y\\in\\mathbb{R}.\n\\end{aligned}\n\\right.\n\\end{equation} \nHere $\\Delta=\\partial^2/\\partial x_1^2+\\cdots +\\partial^2/\\partial x_{n-1}^2+\\partial^2/\\partial y^2$ and $n\\ge2$. Throughout the paper, the initial data $u_{0}$ is assumed to be nonnegative, bounded and uniformly continuous, and the reaction term $f$ is assumed to be of class $C^{1}$.\nWe are interested in the asymptotic behavior of solutions with non-compactly supported initial data in the case where the reaction term $f$ is of monostable-type.\n\nWe begin with recalling results on the existence and stability of traveling wave solutions in one dimensional monostable reaction diffusion equations. \nWe consider the reaction term $f$ satisfying\n\\begin{equation*}\n\\mbox{(F)} \\\n\\left\\{\n\\begin{aligned}\n&f(0)=f(1)=0,\\quad f^{\\prime}(0)>0,\\quad f^{\\prime}(1)<0,\n\\\\\n&f(s)>0 \\ (s\\in(0,1)),\\quad f(s)<0\\ (s\\in(-\\infty,0)\\cup(1,\\infty)).\n\\end{aligned}\n\\right.\n\\end{equation*}\nThen it is well-known that there exists the minimal speed $c^{*}\\ge2\\sqrt{f^{\\prime}(0)}$ for traveling front solutions joining the equilibria $1$ and $0$.\nMore precisely, for any $c\\ge c^{*}$, equation \\eqref{Rea-Diff} has a traveling front solution written in the form $u(x,t)=\\Phi_{c}(x-ct)$ for a profile function $\\Phi_{c}$ satisfying\n\\begin{gather*}\n\\Phi_{c}^{\\prime\\prime}+c\\Phi_{c}^{\\prime}+f(\\Phi_{c})=0, \\\\\n\\lim_{z\\to-\\infty}\\Phi_{c}(z)=1, \\quad \\lim_{z\\to\\infty}\\Phi_{c}(z)=0.\n\\end{gather*}\nIt is known that there exist $\\alpha, \\beta\\ge0$ with $(\\alpha,\\beta) \\neq (0,0)$ such that\n\\begin{gather}\n\\label{asym}\n\\Phi_{c}(s)=(\\alpha +o(1))e^{\\lambda_+(c)s} \\quad \\mbox{if} \\quad c>c^{*}, \\\\\n\\label{asym1}\n\\Phi_{c}(s)=(\\alpha s+\\beta +o(1))e^{\\lambda_-(c)s} \\quad \\mbox{if} \\quad c=c^{*},\n\\end{gather}\nwhere $\\lambda_{+}(c)$ and $\\lambda_{-}(c)$ are the largest root and the smallest root of the quadratic equation\n\\begin{equation}\n\\lambda^{2}+c\\lambda+f^{\\prime}(0)=0,\n\\end{equation}\nrespectively. \nThe traveling front solution $u(x,t)=\\Phi_{c}(x-ct)$ is called a pulled front if either $c=c^{*}=2\\sqrt{f^{\\prime}(0)}$ or $c>c^{*}$ holds, and is called a pushed front if $c=c^{*}>2\\sqrt{f^{\\prime}(0)}$.\n\nConcerning asymptotic behavior of solutions,\nmany results are known for the one-dimensional problem\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\n&u_{t}=u_{yy}+f(u), && y\\in\\mathbb{R}, \\, t>0, \\\\\n&u(y,0)=u_{0}(y), && y\\in\\mathbb{R}.\n\\end{aligned}\n\\right.\n\\end{equation*} \nIn the pioneer work \\cite{KPP}, it is shown that if $f(u)=u(1-u)$ and \n\\begin{equation*}\nu_0(y)=\\left\\{\n\\begin{aligned}\n&1 \\quad (y<0), \\\\\n&0 \\quad (y \\ge 0),\n\\end{aligned}\n\\right.\n\\end{equation*} \nthen $u(z+\\sigma(t),t)$ converges uniformly to $\\Phi_{c^{*}}(z)$ as $t \\to \\infty$\nfor some function $\\sigma(t)$ satisfying\n\\begin{equation*}\n\\sigma(t)=2t+o(t) \\quad (t \\to \\infty).\n\\end{equation*}\nThe refined behavior of $\\sigma(t)$ is revealed in \\cite{MR705746,MR494541}. More precisely, it is shown that $\\sigma(t)$ satisfies\n\\begin{equation*}\n\\sigma(t)=2t-\\frac{3}{2}\\ln t+z_* +o(1)  \\quad (t \\to \\infty)\n\\end{equation*}\nfor some number $z_*$.\nSimilar results for more general reaction terms and initial functions are obtained in \\cite{MR4412555,MR422875,MR803086,MR509494}.\n\nFor pushed fronts, Stokes \\cite{MR682241} and Rothe \\cite{MR639447} proved that, if the initial data $u_{0}(y)$ satisfies\n\\begin{equation*}\n0\\le u_0(y)\\le1, \\qquad\n\\liminf_{y\\rightarrow-\\infty}u_0(y)>0, \\qquad\nu_0(y)\\le Ke^{\\lambda y},\n\\end{equation*}\nfor some constants $K>0$ and $\\lambda<\\lambda_+(c_*)$, then \n\\begin{equation}\nu(z+c^*t,t)\\rightarrow \\Phi_{c^*}(z+\\xi) \\quad (t\\rightarrow\\infty)\n\\end{equation}\nfor some constant $\\xi$. \nIn contrast to pulled fronts with the minimal speed,\nthe logarithmic correction term does not appear for pushed fronts.\nThis is analogous to the result in the bistable case \\cite{MR442480}.\n\nIn higher dimensional cases, the pioneering work has done by Aronson and Weinberger \\cite{MR511740}. \nThey prove that if the initial data has compact support and satisfies $0 \\le u_0 \\le 1$, $u_0 \\not\\equiv0$, then\n\\begin{equation*}\n\\lim_{t \\to \\infty} \\sup_{|x|+|y| \\ge (c^*+\\delta)t} |u(x,y,t)|=0,\n\\qquad\n\\lim_{t \\to \\infty} \\sup_{|x|+|y| \\le (c^*-\\delta)t} |u(x,y,t)-1|=0,\n\\end{equation*}\nfor any $\\delta>0$. Since then, the large-time behavior of solutions with compactly supported initial data has been extensively studied \\cite{MR3333711,MR670523, MR4026189,MR801583}. \n\nOur interest is the asymptotic behavior of solutions with non-compactly supported initial data. In contrast to the case where the initial data has compact support, less is known about the behavior of such solutions.\nTo observe what can occur, we recall results for bistable reaction diffusion equations established by Matano and Nara \\cite{MR2837694}.\nUnder some mild assumptions on the initial data,\nthey showed  the convergence of a solution to $\\Psi(y-\\gamma(x,t))$ for some function $\\gamma(x,t)$, where $\\Psi$ denotes a one-dimensional traveling wave solution. Moreover, they also found that $\\gamma(x,t)$ is approximated by the mean curvature flow with a drift term.\nTo be more precise, the following result is proved.\n\n\\begin{thma}[\\cite{MR2837694}]\nSuppose that $f$ satisfies\n\\begin{gather*}\nf(0)=f(1)=0,\\quad f^{\\prime}(0)<0,\\quad f^{\\prime}(1)<0, \\quad\nf(s) \\left\\{ \n\\begin{aligned}\n&>0 &&\\mbox{if } s \\in (-\\infty,0), \\\\\n&<0 &&\\mbox{if } s \\in (1,\\infty)\n\\end{aligned}\n\\right.\n\\end{gather*}\nand that there exist $c\\in\\mathbf{R}$ and $\\Psi \\in C^2 (\\mathbf{R})$ satisfying\n\\begin{gather*}\n\\Psi^{\\prime\\prime}+c\\Psi^{\\prime}+f(\\Psi)=0, \\quad\n\\lim_{z\\to-\\infty}\\Psi(z)=1, \\quad \\lim_{z\\to\\infty}\\Psi(z)=0.\n\\end{gather*}\nPut \n\\begin{gather*}\ns_+:=\\inf \\{ s_0 \\in (0,1); \\, f>0 \\mbox{ on } (s_0,1)\\},\\\\\ns_-:=\\sup \\{ s_0 \\in (0,1); \\, f<0 \\mbox{ on } (0,s_0)\\},\n\\end{gather*}\nand suppose that $u_0$ satisfies\n\\begin{equation}\n\\liminf_{y\\rightarrow-\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}}u_0 (x,y)>s_+,\\quad \\limsup_{y\\rightarrow\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}}u_0 (x,y)<s_-.\n\\end{equation}\nLet $u$ be a solution of \\eqref{Rea-Diff}.\nThen there exists a smooth function  $\\gamma=\\gamma(x,t)$ with the following properties.\n\\begin{enumerate}\n\\item[(i)] \nThere exists $T>0$ such that \n\\begin{equation*}\n\\{ (x,y,t) \\in \\mathbf{R}^{n-1} \\times \\mathbf{R} \\times [T,\\infty); \\, u(x,y,t)=\\Psi(0)\\}=\\{y=\\gamma(x,t)\\}.\n\\end{equation*}\nMoreover, it holds that\n\\begin{equation*}\n\\lim_{t\\rightarrow\\infty}\\sup_{(x,y)\\in\\mathbf{R}^n }|u(x,y,t)-\\Psi(y-\\gamma(x,t))|=0.\n\\end{equation*}\n\\item[(ii)]\nFor any $\\varepsilon>0$, there exists $\\tau_{\\varepsilon}\\in[T,\\infty)$ such that the solution $U(x,t)$ of the problem\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c,&&x\\in\\mathbf{R}^{n-1},t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_{\\varepsilon}),&&x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation*}\nsatisfies\n\\begin{equation*}\n\\sup_{x\\in\\mathbf{R}^{n-1},t\\ge\\tau_{\\varepsilon}}|\\gamma(x,t)-U(x,t-\\tau_{\\varepsilon})|\\le\\varepsilon.\n\\end{equation*}\n\\end{enumerate}\n\\end{thma}\n\nIn the case where $f$ is of monostable type, a similar result is obtained by Wang \\cite{MR3746497}.\n\\begin{thmb}[\\cite{MR3746497}]\nIn addition to (F), assume that \n\\begin{equation}\\label{assmpf}\n0<f'(0)u-f(u)<Mu^{1+\\alpha}\n\\quad \\mbox{if } \\ u \\in (0,1)\n\\end{equation}\nfor some constants $M>0$ and $\\alpha \\in (0,1]$.\nThen, there exists $\\delta_0>0$ such that if the initial data $u_0$ satisfies\n\\begin{gather}\n\\liminf_{y \\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0, \\label{lbidni} \\\\\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)}, \\quad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty \n\\ \\mbox{ for some } \\ c \\ge c_*, \\label{init2}\n\\end{gather}\nthe following are true.\n\\begin{enumerate}\n\\item[(i)]\nThe assertions of Theorem~A (i) with $\\Psi$ replaced by $\\Phi_c$ hold for some smooth function $\\gamma=\\gamma(x,t)$.\n\\item[(ii)]\nFor any $\\varepsilon>0$, there exists $T_{\\varepsilon}>0$ such that the inequalities\n\\begin{equation*}\nv^-(x,t)-\\varepsilon\\le\\gamma(x,t)\\le v^+(x,t)+\\varepsilon,\\quad t\\ge T_{\\varepsilon},\n\\end{equation*}\nhold for the solutions $v^-$ and $v^+$ of the initial value problems\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\nv^-_t&=\\Delta_x v^--k|\\nabla_x v^-|^2+c, && x\\in\\mathbf{R}^{n-1}, \\, t>0,\\\\\nv^-(x,0)&=\\gamma(x,T_{\\varepsilon}),         && x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation*}\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\nv^+_t&=\\Delta_x v^++k|\\nabla_x v^+|^2+c, && x\\in\\mathbf{R}^{n-1}, \\, t>0,\\\\\nv^+(x,0)&=\\gamma(x,T_{\\varepsilon}),         && x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation*}\nwhere $k:=\\sup_{z\\in\\mathbf{R}}|\\Phi^{\\prime\\prime}_c(z)|/|\\Phi^{\\prime}_c(z)|$.\n\\end{enumerate}\n\\end{thmb}\n\nIt is well-known that condition \\eqref{assmpf} implies $c_*=2\\sqrt{f'(0)}$, which means that there is no pushed front under condition \\eqref{assmpf}. The purpose of this paper is to reveal the behavior of $\\gamma(x,t)$ when $\\Phi_c(x-ct)$ is a pushed front. First, we verify that the same assertion as in Theorem~B (i) holds \nwhen condition \\eqref{assmpf} is dropped and conditions \\eqref{lbidni} and \\eqref{init2} are replaced with\n\\begin{equation}\n\\label{init1}\n\\liminf_{y\\to-\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}}u_{0}(x,y)>0 \\\\\n\\end{equation}\nand\n\\begin{equation}\n\\label{init3}\nc=c_*>2\\sqrt{f'(0)}, \\qquad \n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}} u_{0}(x,y)e^{-\\lambda_1 y}<\\infty \\ \\mbox{ for some } \\ \\lambda_1<\\lambda_+,\n\\end{equation}\nrespectively. We then prove that the behavior of $\\gamma(x,t)$ is governed by the mean curvature flow with a drift term, as in the case where $f$ is of bistable type. More precisely, our main result is stated as follows.", "context": "In higher dimensional cases, the pioneering work has done by Aronson and Weinberger \\cite{MR511740}. \nThey prove that if the initial data has compact support and satisfies $0 \\le u_0 \\le 1$, $u_0 \\not\\equiv0$, then\n\\begin{equation*}\n\\lim_{t \\to \\infty} \\sup_{|x|+|y| \\ge (c^*+\\delta)t} |u(x,y,t)|=0,\n\\qquad\n\\lim_{t \\to \\infty} \\sup_{|x|+|y| \\le (c^*-\\delta)t} |u(x,y,t)-1|=0,\n\\end{equation*}\nfor any $\\delta>0$. Since then, the large-time behavior of solutions with compactly supported initial data has been extensively studied \\cite{MR3333711,MR670523, MR4026189,MR801583}.\n\n\\begin{thma}[\\cite{MR2837694}]\nSuppose that $f$ satisfies\n\\begin{gather*}\nf(0)=f(1)=0,\\quad f^{\\prime}(0)<0,\\quad f^{\\prime}(1)<0, \\quad\nf(s) \\left\\{ \n\\begin{aligned}\n&>0 &&\\mbox{if } s \\in (-\\infty,0), \\\\\n&<0 &&\\mbox{if } s \\in (1,\\infty)\n\\end{aligned}\n\\right.\n\\end{gather*}\nand that there exist $c\\in\\mathbf{R}$ and $\\Psi \\in C^2 (\\mathbf{R})$ satisfying\n\\begin{gather*}\n\\Psi^{\\prime\\prime}+c\\Psi^{\\prime}+f(\\Psi)=0, \\quad\n\\lim_{z\\to-\\infty}\\Psi(z)=1, \\quad \\lim_{z\\to\\infty}\\Psi(z)=0.\n\\end{gather*}\nPut \n\\begin{gather*}\ns_+:=\\inf \\{ s_0 \\in (0,1); \\, f>0 \\mbox{ on } (s_0,1)\\},\\\\\ns_-:=\\sup \\{ s_0 \\in (0,1); \\, f<0 \\mbox{ on } (0,s_0)\\},\n\\end{gather*}\nand suppose that $u_0$ satisfies\n\\begin{equation}\n\\liminf_{y\\rightarrow-\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}}u_0 (x,y)>s_+,\\quad \\limsup_{y\\rightarrow\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}}u_0 (x,y)<s_-.\n\\end{equation}\nLet $u$ be a solution of \\eqref{Rea-Diff}.\nThen there exists a smooth function $\\gamma=\\gamma(x,t)$ with the following properties.\n\\begin{enumerate}\n\\item[(i)] \nThere exists $T>0$ such that \n\\begin{equation*}\n\\{ (x,y,t) \\in \\mathbf{R}^{n-1} \\times \\mathbf{R} \\times [T,\\infty); \\, u(x,y,t)=\\Psi(0)\\}=\\{y=\\gamma(x,t)\\}.\n\\end{equation*}\nMoreover, it holds that\n\\begin{equation*}\n\\lim_{t\\rightarrow\\infty}\\sup_{(x,y)\\in\\mathbf{R}^n }|u(x,y,t)-\\Psi(y-\\gamma(x,t))|=0.\n\\end{equation*}\n\\item[(ii)]\nFor any $\\varepsilon>0$, there exists $\\tau_{\\varepsilon}\\in[T,\\infty)$ such that the solution $U(x,t)$ of the problem\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c,&&x\\in\\mathbf{R}^{n-1},t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_{\\varepsilon}),&&x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation*}\nsatisfies\n\\begin{equation*}\n\\sup_{x\\in\\mathbf{R}^{n-1},t\\ge\\tau_{\\varepsilon}}|\\gamma(x,t)-U(x,t-\\tau_{\\varepsilon})|\\le\\varepsilon.\n\\end{equation*}\n\\end{enumerate}\n\\end{thma}\n\nIn the case where $f$ is of monostable type, a similar result is obtained by Wang \\cite{MR3746497}.\n\\begin{thmb}[\\cite{MR3746497}]\nIn addition to (F), assume that \n\\begin{equation}\\label{assmpf}\n0<f'(0)u-f(u)<Mu^{1+\\alpha}\n\\quad \\mbox{if } \\ u \\in (0,1)\n\\end{equation}\nfor some constants $M>0$ and $\\alpha \\in (0,1]$.\nThen, there exists $\\delta_0>0$ such that if the initial data $u_0$ satisfies\n\\begin{gather}\n\\liminf_{y \\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0, \\label{lbidni} \\\\\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)}, \\quad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty \n\\ \\mbox{ for some } \\ c \\ge c_*, \\label{init2}\n\\end{gather}\nthe following are true.\n\\begin{enumerate}\n\\item[(i)]\nThe assertions of Theorem~A (i) with $\\Psi$ replaced by $\\Phi_c$ hold for some smooth function $\\gamma=\\gamma(x,t)$.\n\\item[(ii)]\nFor any $\\varepsilon>0$, there exists $T_{\\varepsilon}>0$ such that the inequalities\n\\begin{equation*}\nv^-(x,t)-\\varepsilon\\le\\gamma(x,t)\\le v^+(x,t)+\\varepsilon,\\quad t\\ge T_{\\varepsilon},\n\\end{equation*}\nhold for the solutions $v^-$ and $v^+$ of the initial value problems\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\nv^-_t&=\\Delta_x v^--k|\\nabla_x v^-|^2+c, && x\\in\\mathbf{R}^{n-1}, \\, t>0,\\\\\nv^-(x,0)&=\\gamma(x,T_{\\varepsilon}), && x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation*}\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\nv^+_t&=\\Delta_x v^++k|\\nabla_x v^+|^2+c, && x\\in\\mathbf{R}^{n-1}, \\, t>0,\\\\\nv^+(x,0)&=\\gamma(x,T_{\\varepsilon}), && x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation*}\nwhere $k:=\\sup_{z\\in\\mathbf{R}}|\\Phi^{\\prime\\prime}_c(z)|/|\\Phi^{\\prime}_c(z)|$.\n\\end{enumerate}\n\\end{thmb}\n\nIt is well-known that condition \\eqref{assmpf} implies $c_*=2\\sqrt{f'(0)}$, which means that there is no pushed front under condition \\eqref{assmpf}. The purpose of this paper is to reveal the behavior of $\\gamma(x,t)$ when $\\Phi_c(x-ct)$ is a pushed front. First, we verify that the same assertion as in Theorem~B (i) holds \nwhen condition \\eqref{assmpf} is dropped and conditions \\eqref{lbidni} and \\eqref{init2} are replaced with\n\\begin{equation}\n\\label{init1}\n\\liminf_{y\\to-\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}}u_{0}(x,y)>0 \\\\\n\\end{equation}\nand\n\\begin{equation}\n\\label{init3}\nc=c_*>2\\sqrt{f'(0)}, \\qquad \n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}} u_{0}(x,y)e^{-\\lambda_1 y}<\\infty \\ \\mbox{ for some } \\ \\lambda_1<\\lambda_+,\n\\end{equation}\nrespectively. We then prove that the behavior of $\\gamma(x,t)$ is governed by the mean curvature flow with a drift term, as in the case where $f$ is of bistable type. More precisely, our main result is stated as follows.", "full_context": "In higher dimensional cases, the pioneering work has done by Aronson and Weinberger \\cite{MR511740}. \nThey prove that if the initial data has compact support and satisfies $0 \\le u_0 \\le 1$, $u_0 \\not\\equiv0$, then\n\\begin{equation*}\n\\lim_{t \\to \\infty} \\sup_{|x|+|y| \\ge (c^*+\\delta)t} |u(x,y,t)|=0,\n\\qquad\n\\lim_{t \\to \\infty} \\sup_{|x|+|y| \\le (c^*-\\delta)t} |u(x,y,t)-1|=0,\n\\end{equation*}\nfor any $\\delta>0$. Since then, the large-time behavior of solutions with compactly supported initial data has been extensively studied \\cite{MR3333711,MR670523, MR4026189,MR801583}.\n\n\\begin{thma}[\\cite{MR2837694}]\nSuppose that $f$ satisfies\n\\begin{gather*}\nf(0)=f(1)=0,\\quad f^{\\prime}(0)<0,\\quad f^{\\prime}(1)<0, \\quad\nf(s) \\left\\{ \n\\begin{aligned}\n&>0 &&\\mbox{if } s \\in (-\\infty,0), \\\\\n&<0 &&\\mbox{if } s \\in (1,\\infty)\n\\end{aligned}\n\\right.\n\\end{gather*}\nand that there exist $c\\in\\mathbf{R}$ and $\\Psi \\in C^2 (\\mathbf{R})$ satisfying\n\\begin{gather*}\n\\Psi^{\\prime\\prime}+c\\Psi^{\\prime}+f(\\Psi)=0, \\quad\n\\lim_{z\\to-\\infty}\\Psi(z)=1, \\quad \\lim_{z\\to\\infty}\\Psi(z)=0.\n\\end{gather*}\nPut \n\\begin{gather*}\ns_+:=\\inf \\{ s_0 \\in (0,1); \\, f>0 \\mbox{ on } (s_0,1)\\},\\\\\ns_-:=\\sup \\{ s_0 \\in (0,1); \\, f<0 \\mbox{ on } (0,s_0)\\},\n\\end{gather*}\nand suppose that $u_0$ satisfies\n\\begin{equation}\n\\liminf_{y\\rightarrow-\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}}u_0 (x,y)>s_+,\\quad \\limsup_{y\\rightarrow\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}}u_0 (x,y)<s_-.\n\\end{equation}\nLet $u$ be a solution of \\eqref{Rea-Diff}.\nThen there exists a smooth function $\\gamma=\\gamma(x,t)$ with the following properties.\n\\begin{enumerate}\n\\item[(i)] \nThere exists $T>0$ such that \n\\begin{equation*}\n\\{ (x,y,t) \\in \\mathbf{R}^{n-1} \\times \\mathbf{R} \\times [T,\\infty); \\, u(x,y,t)=\\Psi(0)\\}=\\{y=\\gamma(x,t)\\}.\n\\end{equation*}\nMoreover, it holds that\n\\begin{equation*}\n\\lim_{t\\rightarrow\\infty}\\sup_{(x,y)\\in\\mathbf{R}^n }|u(x,y,t)-\\Psi(y-\\gamma(x,t))|=0.\n\\end{equation*}\n\\item[(ii)]\nFor any $\\varepsilon>0$, there exists $\\tau_{\\varepsilon}\\in[T,\\infty)$ such that the solution $U(x,t)$ of the problem\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c,&&x\\in\\mathbf{R}^{n-1},t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_{\\varepsilon}),&&x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation*}\nsatisfies\n\\begin{equation*}\n\\sup_{x\\in\\mathbf{R}^{n-1},t\\ge\\tau_{\\varepsilon}}|\\gamma(x,t)-U(x,t-\\tau_{\\varepsilon})|\\le\\varepsilon.\n\\end{equation*}\n\\end{enumerate}\n\\end{thma}\n\nIn the case where $f$ is of monostable type, a similar result is obtained by Wang \\cite{MR3746497}.\n\\begin{thmb}[\\cite{MR3746497}]\nIn addition to (F), assume that \n\\begin{equation}\\label{assmpf}\n0<f'(0)u-f(u)<Mu^{1+\\alpha}\n\\quad \\mbox{if } \\ u \\in (0,1)\n\\end{equation}\nfor some constants $M>0$ and $\\alpha \\in (0,1]$.\nThen, there exists $\\delta_0>0$ such that if the initial data $u_0$ satisfies\n\\begin{gather}\n\\liminf_{y \\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0, \\label{lbidni} \\\\\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)}, \\quad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty \n\\ \\mbox{ for some } \\ c \\ge c_*, \\label{init2}\n\\end{gather}\nthe following are true.\n\\begin{enumerate}\n\\item[(i)]\nThe assertions of Theorem~A (i) with $\\Psi$ replaced by $\\Phi_c$ hold for some smooth function $\\gamma=\\gamma(x,t)$.\n\\item[(ii)]\nFor any $\\varepsilon>0$, there exists $T_{\\varepsilon}>0$ such that the inequalities\n\\begin{equation*}\nv^-(x,t)-\\varepsilon\\le\\gamma(x,t)\\le v^+(x,t)+\\varepsilon,\\quad t\\ge T_{\\varepsilon},\n\\end{equation*}\nhold for the solutions $v^-$ and $v^+$ of the initial value problems\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\nv^-_t&=\\Delta_x v^--k|\\nabla_x v^-|^2+c, && x\\in\\mathbf{R}^{n-1}, \\, t>0,\\\\\nv^-(x,0)&=\\gamma(x,T_{\\varepsilon}), && x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation*}\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\nv^+_t&=\\Delta_x v^++k|\\nabla_x v^+|^2+c, && x\\in\\mathbf{R}^{n-1}, \\, t>0,\\\\\nv^+(x,0)&=\\gamma(x,T_{\\varepsilon}), && x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation*}\nwhere $k:=\\sup_{z\\in\\mathbf{R}}|\\Phi^{\\prime\\prime}_c(z)|/|\\Phi^{\\prime}_c(z)|$.\n\\end{enumerate}\n\\end{thmb}\n\nIt is well-known that condition \\eqref{assmpf} implies $c_*=2\\sqrt{f'(0)}$, which means that there is no pushed front under condition \\eqref{assmpf}. The purpose of this paper is to reveal the behavior of $\\gamma(x,t)$ when $\\Phi_c(x-ct)$ is a pushed front. First, we verify that the same assertion as in Theorem~B (i) holds \nwhen condition \\eqref{assmpf} is dropped and conditions \\eqref{lbidni} and \\eqref{init2} are replaced with\n\\begin{equation}\n\\label{init1}\n\\liminf_{y\\to-\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}}u_{0}(x,y)>0 \\\\\n\\end{equation}\nand\n\\begin{equation}\n\\label{init3}\nc=c_*>2\\sqrt{f'(0)}, \\qquad \n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}} u_{0}(x,y)e^{-\\lambda_1 y}<\\infty \\ \\mbox{ for some } \\ \\lambda_1<\\lambda_+,\n\\end{equation}\nrespectively. We then prove that the behavior of $\\gamma(x,t)$ is governed by the mean curvature flow with a drift term, as in the case where $f$ is of bistable type. More precisely, our main result is stated as follows.\n\nWe prove Theorem~\\ref{thm1} by constructing appropriate comparison functions. They are given in the form\n\\begin{equation*}\nu^{\\pm}(x,y,t)\\coloneqq\\Phi_{c^{*}}\\left(\\frac{y-V(x,t)}{\\sqrt{1+|\\nabla_{x}V|^{2}}}\\mp q(t)\\right)\\pm p(t)\\chi \\left( e^{\\lambda (y-c_*t)}\\right),\n\\end{equation*}\nwhere $V$ is a solution of the equation\n\\begin{equation*}\nV_{t}=\\Delta_{x}V+\\frac{c^{*}}{2}|\\nabla_{x} V|^2+c_*,\n\\end{equation*}\n$\\lambda$ is a number with $\\lambda<\\lambda_1$\nand $\\chi$ is a smooth function safisfying $\\chi(s)=s$ ($s \\le 1/2$) and $\\chi(s)=1$ ($s \\ge 1$).\nWe will show that $u^+$ (resp. $u^-$) becomes a supersolution (resp. a subsolution) for the problem \\eqref{Rea-Diff} if $p(t)$, $q(t)$ and the initial data for $V(x,t)$ are chosen appropriately. Theorem~\\ref{thm1} (ii) is then proved by using these comparison functions and applying the fact that the solution $U$ of \\eqref{MCF} is approximated by $V$.\n\nIn this section, we establish the following estimates.\n\\begin{prop}\\label{lem3}\nLet $u(x,z,t)$ be a solution of \\eqref{MRea-Diff}. \nIf the initial data $u_{0}$ is satisfies \\eqref{init1} and \\eqref{init3},\nthen there exists constants $z_{1},z_{2}\\in\\mathbb{R}$ such that\n\\begin{equation}\n\\label{upper}\n\\limsup_{t\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}u(x,z,t)\\le\\Phi_{c}(z-z_{0})\\;\\mbox{uniformly in }z\\in\\mathbb{R}\n\\end{equation}\n\\begin{equation}\n\\label{lower}\n\\liminf_{t\\to\\infty}\\inf_{x\\in\\mathbb{R}^{n-1}}u(x,z,t)\\ge\\Phi_{c}(z-z_{1})\\;\\mbox{uniformly in }z\\in\\mathbb{R}\n\\end{equation}\nThe same inequalities hold if \\eqref{init2} is assumed instead of \\eqref{init3}.\n\\end{prop}\nWe split the proof of this proposition into the case 1 and case 2:\n\\begin{enumerate}\n\\item[1:]\\eqref{init1} and \\eqref{init3}\n\\item[2:]\\eqref{init1} and \\eqref{init2}\n\\end{enumerate}\nFirst, we show upper and lower estimates for case 1.\nFor this, we recall the lemmas in \\cite{MR639447}. We take any $\\lambda<\\lambda_{1}<\\lambda_{+}$ and define $\\psi(s)$ as\n\\begin{equation*}\n\\psi(s)=\\chi(e^{\\lambda_{1}s})\n\\end{equation*}\n\\begin{equation*}\n\\chi(s)\\coloneqq\\left\\{\n\\begin{aligned}\n1\\;(s\\ge1)\\\\\ns\\;(s\\le\\frac{1}{2})\n\\end{aligned}\n\\right.\n\\end{equation*}\n, where $0\\le\\chi(s)\\le1$ for $s\\in(\\frac{1}{2},1)$. For these, the following lemmas hold\n\\begin{lem}[\\cite{MR639447}]\\label{add_B}\nThere exists $p\\in(0,1)$ such that, for any $q_{0}\\in(0,p],z_{1},z_{2}\\in\\mathbf{R}$, there exists $\\beta>0,C>0$ such that\n\\begin{equation*}\nw^{+}(z,t)\\coloneqq \\Phi_{c^{*}}(z-z_{1}- C(1-e^{-\\beta t}))+q_{0}e^{-\\beta t}\\psi(z-z_{2})\n\\end{equation*}\nsatisfies $L[w^+]\\ge0$.\n\\end{lem}\n\n\\section{Level set of the solutions}\nAs we mention in the previous section, we only give statements of lemmas and omit their proof.\n\\begin{lem}[Level set]\\label{lem11}\nLet $u(x,z,t)$ be a solution of ($\\ref{MRea-Diff}$) and $T>0$ be as defined in Corollary 2. Then there exists a smooth bounded function $\\Gamma(x,t)$ such that\n\\begin{equation*}\nu(x,z,t)=\\Phi_{c}(0)\\quad\\mbox{if and only if}\\quad z=\\Gamma(x,t),\n\\end{equation*}\nfor any $(x,t)\\in\\mathbb{R}^{n-1}\\times[T,\\infty)$. Furthermore the following estimates hold:\n\\begin{enumerate}\n\\item For each $1\\le i,j \\le n-1$,\n\\begin{equation*}\n\\lim_{t\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}|\\Gamma_{x_{i}}(x,t)|=0,\\quad\\lim_{t\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}|\\Gamma_{x_{i}x_{j}}(x,t)|=0,\n\\end{equation*}\n\\item There exists a constant $M>0$ such that, for each $1\\le i,j,k \\le n-1$,\n\\end{enumerate}\n\\begin{equation*}\n\\sup_{x\\in\\mathbb{R}^{n-1}}|\\Gamma_{x_{i}x_{j}x_{k}}|\\le M,\\quad \\mbox{for}\\quad t\\ge T.\n\\end{equation*}\n\\end{lem}\n\n\\begin{lem}[Approximation of $\\Gamma(x,t)$]\\label{lem15}\nLet $u(x,z,t)$ be a solution of (\\ref{MRea-Diff}) and let $\\Gamma(x,t)$ be as defined in Lemma \\ref{lem11}. Then for any $\\varepsilon>0$, there exists a constant $\\tau_{\\varepsilon}>0$ such that the function $V(x,t)$ defined by\n\\begin{equation*}\n\\left\\{\n\\begin{aligned}\n&V_{t}=\\Delta_{x}V+\\frac{c^{*}}{2}|\\nabla_{x} V|,\\quad x\\in\\mathbb{R}^{n-1},t>0\\\\\n&V(x,0)=\\Gamma(x,\\tau_{\\varepsilon}).\\qquad\\hspace{9.4pt} x\\in\\mathbb{R}^{n-1}.\n\\end{aligned}\n\\right.\n\\end{equation*}\nsatisfies\n\\begin{equation*}\n\\sup_{x\\in\\mathbb{R}^{n-1}}|\\Gamma(x,t)-V(x,t-\\tau_{\\varepsilon})|\\le\\varepsilon,\\quad t\\ge\\tau_{\\varepsilon}.\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\n First, we verify upper bound. By Lemma \\ref{lem12} and \\ref{lem13}, we can take $T>0, M>0$ and $K>0$ such that, for $D\\coloneqq\\left\\{(x,z,t)\\in\\mathbb{R}^{n}\\times[T,\\infty)||u(x,z,t)-\\Phi_{c}(0)|\\le\\min(1-\\Phi_{c}(0),\\Phi_{c}(0))\\right\\}$\n\\begin{equation*}\n\\sup_{t\\ge T}\\|\\Gamma(\\cdot,t)\\|_{W^{3,\\infty}}\\le M,\\quad \\inf_{(x,z,t)\\in D}-u_{z}(x,z,t)\\ge K.\n\\end{equation*}\nFor the constants $M$ and $\\hat{\\varepsilon}\\coloneq1/(\\|\\Phi_{c^{*}}^{\\prime}\\|_{L^{\\infty}}+1)\\cdot\\min\\{K\\varepsilon,\\min(1-\\Phi_{c^{*}}(0),\\Phi_{c^{*}}(0))\\}$, we choose a constant $\\delta>0$ and functions $p(t),q(t)$ satisfying\n\\begin{equation*}\np(0)>0,\\quad q(0)=0,\\quad 0\\le p(t),q(t)\\le\\hat{\\varepsilon}\\quad for \\quad t\\ge0.\n\\end{equation*}\nFrom proof of Lemma \\ref{lem3} and (\\ref{asym}), we can take $z_{0}\\in\\mathbb{R}$ such that, for some larger $T>0$, \n\\begin{equation*}\nu(x,z,t)\\le p(0)e^{\\lambda z}\\quad((x,z,t)\\in\\mathbb{R}^{n-1}\\times[z_{0},\\infty)\\times[T,\\infty)).\n\\end{equation*}\nTaking $\\tau_{\\varepsilon}\\ge T$ larger if necessary, by Lemma \\ref{lem11} and \\ref{lem12}, the following holds\n\\begin{equation*}\nu(x,z,t)\\le\\Phi_{c^{*}}(z-\\Gamma(x,\\tau_{\\varepsilon}))+\\frac{p(0)e^{\\lambda z_{0}}}{2}\\le\\Phi_{c^{*}}\\left(\\frac{z-\\Gamma(x,\\tau_{\\varepsilon})}{\\sqrt{1+|\\nabla_{x}\\Gamma|^{2}}}\\right)+p(0)e^{\\lambda z_{0}}\n\\end{equation*}\nFor any $z\\ge z_{0}$, \n\\begin{equation*}\nu(x,z,\\tau_{\\varepsilon})\\le p(0)e^{\\lambda z}\\le u^{+}(x,z,0).\n\\end{equation*}\nFor any $z<z_{0}$,\n\\begin{equation*}\nu(x,z,\\tau_{\\varepsilon})\\le \\Phi_{c^{*}}\\left(\\frac{z-\\Gamma(x,\\tau_{\\varepsilon})}{\\sqrt{1+|\\nabla_{x}\\Gamma|^{2}}}\\right)+p(0)e^{\\lambda z_{0}}\\le\\Phi_{c^{*}}\\left(\\frac{z-\\Gamma(x,\\tau_{\\varepsilon})}{\\sqrt{1+|\\nabla_{x}\\Gamma|^{2}}}\\right)+p(0)\\psi(z).\n\\end{equation*}\nSo, by comparison principle, $u(x,z,t)\\le u^{+}(x,z,t)$ for $t\\ge\\tau_{\\varepsilon}$. Therefore, we have the following inequality\n\\begin{align*}\nu(x,V(x,t-\\tau_{\\varepsilon}),t)-\\Phi_{c^{*}}(0)&\\le u^{+}(x,V(x,t-\\tau_{\\varepsilon}),t)-\\Phi_{c^{*}}(0)\\\\\n &=\\Phi_{c^{*}}(-q(t-\\tau_{\\varepsilon}))-\\Phi_{c^{*}}(0)+p(t-\\tau_{\\varepsilon})\\\\\n &\\le(\\|\\Phi_{c^{*}}^{\\prime}\\|_{L^{\\infty}}+1)\\hat{\\varepsilon}\\\\\n &=\\min\\{K\\varepsilon,\\min\\{1-\\Phi_{c^{*}}(0),\\Phi_{c^{*}}(0)\\}\\}.\n\\end{align*}\nThus, we have, for $\\Gamma(x,t)\\ge V(x,t-\\tau_{\\varepsilon})$,\n\\begin{align*}\nK\\varepsilon&\\ge u(x,V(x,t-\\tau_{\\varepsilon},t),t)-u(x,\\Gamma(x,t),t)\\\\\n &\\ge (\\inf_{u\\in[0,\\min\\{1-\\Phi_{c^{*}}(0),\\Phi_{c^{*}}(0)\\}],t\\ge\\tau_{\\varepsilon}}-u_{z})\\cdot(\\Gamma(x,t)-V(x,t-\\tau_{\\varepsilon}))\\\\\n &\\ge K(\\Gamma(x,t)-V(x,t-\\tau_{\\varepsilon})).\n\\end{align*}\nThis implies $\\Gamma(x,t)\\le V(x,t-\\tau_{\\varepsilon})+\\varepsilon$ for $t\\ge\\tau_{\\varepsilon}$. Thus, we have the upper estimate. The lower estimate is followed from Lemma \\ref{lem14} in a similar way.\n\\end{proof}", "post_theorem_intro_text_len": 1514, "post_theorem_intro_text": "We prove Theorem~\\ref{thm1} by constructing appropriate comparison functions. They are given in the form\n\\begin{equation*}\nu^{\\pm}(x,y,t)\\coloneqq\\Phi_{c^{*}}\\left(\\frac{y-V(x,t)}{\\sqrt{1+|\\nabla_{x}V|^{2}}}\\mp q(t)\\right)\\pm p(t)\\chi \\left( e^{\\lambda (y-c_*t)}\\right),\n\\end{equation*}\nwhere $V$ is a solution of the equation\n\\begin{equation*}\nV_{t}=\\Delta_{x}V+\\frac{c^{*}}{2}|\\nabla_{x} V|^2+c_*,\n\\end{equation*}\n$\\lambda$ is a number with $\\lambda<\\lambda_1$\nand $\\chi$ is a smooth function safisfying $\\chi(s)=s$ ($s \\le 1/2$) and $\\chi(s)=1$ ($s \\ge 1$).\nWe will show that $u^+$ (resp. $u^-$) becomes a supersolution (resp. a subsolution) for the problem \\eqref{Rea-Diff} if $p(t)$, $q(t)$ and the initial data for $V(x,t)$ are chosen appropriately. Theorem~\\ref{thm1} (ii) is then proved by using these comparison functions and applying the fact that the solution $U$ of \\eqref{MCF} is approximated by $V$. \n\nThis paper is organized as follows. In Section 2, we recall results on the approximation of the mean curvature flow obtained in \\cite{MR2837694}. Section 3 establishes upper and lower bounds of solutions at large time. In Section 4, we define $\\omega$-limit points and provide their characterization. Section 5 establishes smoothness of level sets of solutions. Section 6 is devoted to the construction of comprison functions, which are used to prove Theorem \\ref{thm1} (ii). In Section 7, we prove Theorem \\ref{thm1}.\n\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\renewcommand{\\theenumi)}{(\\theenumi)}", "sketch": "To prove Theorem~\\ref{thm1}, the authors “construct[] appropriate comparison functions” of the form\n\\[\n u^{\\pm}(x,y,t)\\coloneqq\\Phi_{c^{*}}\\left(\\frac{y-V(x,t)}{\\sqrt{1+|\\nabla_{x}V|^{2}}}\\mp q(t)\\right)\\pm p(t)\\chi \\left( e^{\\lambda (y-c_*t)}\\right),\n\\]\nwhere $V$ solves\n\\[\nV_{t}=\\Delta_{x}V+\\frac{c^{*}}{2}|\\nabla_{x} V|^2+c_*,\n\\]\n$\\lambda<\\lambda_1$, and $\\chi$ is smooth with “$\\chi(s)=s$ ($s \\le 1/2$) and $\\chi(s)=1$ ($s \\ge 1$).” They then “show that $u^+$ (resp. $u^-$) becomes a supersolution (resp. a subsolution) for the problem \\eqref{Rea-Diff} if $p(t)$, $q(t)$ and the initial data for $V(x,t)$ are chosen appropriately.”\n\nFor Theorem~\\ref{thm1}(ii), they “use[] these comparison functions and apply[] the fact that the solution $U$ of \\eqref{MCF} is approximated by $V$.”", "expanded_sketch": "To prove the main theorem, the authors “construct[] appropriate comparison functions” of the form\n\\[\n u^{\\pm}(x,y,t)\\coloneqq\\Phi_{c^{*}}\\left(\\frac{y-V(x,t)}{\\sqrt{1+|\\nabla_{x}V|^{2}}}\\mp q(t)\\right)\\pm p(t)\\chi \\left( e^{\\lambda (y-c_*t)}\\right),\n\\]\nwhere $V$ solves\n\\[\nV_{t}=\\Delta_{x}V+\\frac{c^{*}}{2}|\\nabla_{x} V|^2+c_*,\n\\]\n$\\lambda<\\lambda_1$, and $\\chi$ is smooth with “$\\chi(s)=s$ ($s \\le 1/2$) and $\\chi(s)=1$ ($s \\ge 1$).” They then “show that $u^+$ (resp. $u^-$) becomes a supersolution (resp. a subsolution) for the problem\n\\begin{aligned}\\label{Rea-Diff}\n&u_{t}=\\Delta u+f(u), && x\\in\\mathbb{R}^{n-1}, \\, y\\in\\mathbb{R}, \\, t>0, \\\\\n&u(x,y,0)=u_{0}(x,y), && x\\in\\mathbb{R}^{n-1}, \\, y\\in\\mathbb{R}.\n\\end{aligned}\nif $p(t)$, $q(t)$ and the initial data for $V(x,t)$ are chosen appropriately.”\n\nIn establishing part (ii) of the main theorem, they “use[] these comparison functions and apply[] the fact that the solution $U$ of\n\\begin{equation}\\label{MCF}\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c_*,&&x\\in\\mathbf{R}^{n-1},t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_{\\varepsilon}),&&x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation}\nis approximated by $V$.”", "expanded_theorem": "\\label{thm1}\nAssume (F), \n\\begin{equation}\n\\label{init1}\n\\liminf_{y\\to-\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}}u_{0}(x,y)>0 \\\\\n\\end{equation}\nand \n\\begin{equation}\n\\label{init3}\nc=c_*>2\\sqrt{f'(0)}, \\qquad \n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}} u_{0}(x,y)e^{-\\lambda_1 y}<\\infty \\ \\mbox{ for some } \\ \\lambda_1<\\lambda_+,\n\\end{equation}\nand let $u$ be a solution of \n\\begin{aligned}\\label{Rea-Diff}\n&u_{t}=\\Delta u+f(u), && x\\in\\mathbb{R}^{n-1}, \\, y\\in\\mathbb{R}, \\, t>0, \\\\\n&u(x,y,0)=u_{0}(x,y), && x\\in\\mathbb{R}^{n-1}, \\, y\\in\\mathbb{R}.\n\\end{aligned}.\nThen there exists a smooth function  $\\gamma=\\gamma(x,t)$ with the following properties.\n\\begin{enumerate}\n\\item[(i)] \nThere exists $T>0$ such that \n\\begin{equation*}\n\\{ (x,y,t) \\in \\mathbf{R}^{n-1} \\times \\mathbf{R} \\times [T,\\infty); \\, u(x,y,t)=\\Phi_c(0)\\}=\\{y=\\gamma(x,t)\\}.\n\\end{equation*}\nMoreover, it holds that\n\\begin{equation*}\n\\lim_{t\\rightarrow\\infty}\\sup_{(x,y)\\in\\mathbf{R}^n }|u(x,y,t)-\\Phi_c(y-\\gamma(x,t))|=0.\n\\end{equation*}\n\\item[(ii)]\nFor any $\\varepsilon>0$, there exists $\\tau_{\\varepsilon}\\in[T,\\infty)$ such that the solution $U(x,t)$ of the problem\n\\begin{equation}\\label{MCF}\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c_*,&&x\\in\\mathbf{R}^{n-1},t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_{\\varepsilon}),&&x\\in\\mathbf{R}^{n-1},\n\\end{aligned}\n\\right.\n\\end{equation}\nsatisfies\n\\begin{equation*}\n\\sup_{x\\in\\mathbf{R}^{n-1},t\\ge\\tau_{\\varepsilon}}|\\gamma(x,t)-U(x,t-\\tau_{\\varepsilon})|\\le\\varepsilon.\n\\end{equation*}\n\\end{enumerate}\nFurthermore, the assertion (i) still holds if\n\\begin{gather}\n\\liminf_{y \\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0, \\label{lbidni} \\\\\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbf{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)}, \\quad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbf{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty \n\\ \\mbox{ for some } \\ c \\ge c_*, \\\\\n\\end{gather}\nis assumed instead of the preceding condition.\n", "theorem_type": ["Existence", "Asymptotic or Limit"], "mcq": {"question": "Let $u=u(x,y,t)$ solve the Cauchy problem\n\\[\n\\begin{aligned}\n&u_t=\\Delta u+f(u), && x\\in\\mathbb{R}^{n-1},\\ y\\in\\mathbb{R},\\ t>0,\\\\\n&u(x,y,0)=u_0(x,y), && x\\in\\mathbb{R}^{n-1},\\ y\\in\\mathbb{R}.\n\\end{aligned}\n\\]\nAssume $f$ satisfies the standing hypothesis $(F)$ from the traveling-front setting, and let $\\Phi_c$ denote the associated one-dimensional front profile, i.e. a $C^2(\\mathbb R)$ solution of\n\\[\n\\Phi_c''+c\\Phi_c'+f(\\Phi_c)=0,\\qquad \\Phi_c(-\\infty)=1,\\qquad \\Phi_c(+\\infty)=0,\n\\]\nwith minimal wave speed $c_*$. Let $\\lambda_+$ be the larger root of $\\lambda^2-c_*\\lambda+f'(0)=0$. Assume moreover that\n\\[\n\\liminf_{y\\to-\\infty}\\inf_{x\\in\\mathbb{R}^{n-1}}u_0(x,y)>0,\n\\]\nand\n\\[\nc=c_*>2\\sqrt{f'(0)},\\qquad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}} u_0(x,y)e^{-\\lambda_1 y}<\\infty\n\\quad\\text{for some }\\lambda_1<\\lambda_+.\n\\]\nAlso consider the alternative initial-data regime, for a constant $\\delta_0>0$,\n\\[\n\\liminf_{y\\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0,\n\\]\n\\[\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbb{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)}, \\qquad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty\n\\quad\\text{for some } c\\ge c_*.\n\\]\nUnder these assumptions, which existence statement holds?", "correct_choice": {"label": "A", "text": "There exists a smooth function $\\gamma=\\gamma(x,t)$ such that: (i) for some $T>0$,\n\\[\n\\{(x,y,t)\\in\\mathbb{R}^{n-1}\\times\\mathbb{R}\\times[T,\\infty):\\ u(x,y,t)=\\Phi_c(0)\\}=\\{y=\\gamma(x,t)\\},\n\\]\nand\n\\[\n\\lim_{t\\to\\infty}\\sup_{(x,y)\\in\\mathbb{R}^n}\\big|u(x,y,t)-\\Phi_c(y-\\gamma(x,t))\\big|=0.\n\\]\n(ii) For every $\\varepsilon>0$, there exists $\\tau_\\varepsilon\\in[T,\\infty)$ such that if $U(x,t)$ solves\n\\[\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c_*,&&x\\in\\mathbb{R}^{n-1},\\ t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_\\varepsilon),&&x\\in\\mathbb{R}^{n-1},\n\\end{aligned}\n\\right.\n\\]\nthen\n\\[\n\\sup_{x\\in\\mathbb{R}^{n-1},\\,t\\ge\\tau_\\varepsilon}\\big|\\gamma(x,t)-U(x,t-\\tau_\\varepsilon)\\big|\\le\\varepsilon.\n\\]\nFurthermore, if one replaces the initial-data assumptions above by the alternative regime\n\\[\n\\liminf_{y\\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0,\n\\]\n\\[\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbb{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)},\\qquad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty\n\\quad\\text{for some } c\\ge c_*,\n\\]\nthen conclusion (i) still holds."}, "choices": [{"label": "B", "text": "There exists a smooth function $\\gamma=\\gamma(x,t)$ such that: (i) for some $T>0$,\n\\[\n\\{(x,y,t)\\in\\mathbb{R}^{n-1}\\times\\mathbb{R}\\times[T,\\infty):\\ u(x,y,t)=\\Phi_c(0)\\}=\\{y=\\gamma(x,t)\\},\n\\]\nand\n\\[\n\\lim_{t\\to\\infty}\\sup_{(x,y)\\in\\mathbb{R}^n}\\big|u(x,y,t)-\\Phi_c(y-\\gamma(x,t))\\big|=0.\n\\]\n(ii) There exists $\\tau\\in[T,\\infty)$ such that if $U(x,t)$ solves\n\\[\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c_*,&&x\\in\\mathbb{R}^{n-1},\\ t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau),&&x\\in\\mathbb{R}^{n-1},\n\\end{aligned}\n\\right.\n\\]\nthen\n\\[\n\\lim_{t\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}\\big|\\gamma(x,t)-U(x,t-\\tau)\\big|=0.\n\\]\nFurthermore, if one replaces the initial-data assumptions above by the alternative regime\n\\[\n\\liminf_{y\\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0,\n\\]\n\\[\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbb{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)},\\qquad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty\n\\quad\\text{for some } c\\ge c_*,\n\\]\nthen conclusions (i) and (ii) still hold."}, {"label": "C", "text": "There exists a smooth function $\\gamma=\\gamma(x,t)$ such that for some $T>0$,\n\\[\n\\{(x,y,t)\\in\\mathbb{R}^{n-1}\\times\\mathbb{R}\\times[T,\\infty):\\ u(x,y,t)=\\Phi_c(0)\\}=\\{y=\\gamma(x,t)\\},\n\\]\nand\n\\[\n\\lim_{t\\to\\infty}\\sup_{(x,y)\\in\\mathbb{R}^n}\\big|u(x,y,t)-\\Phi_c(y-\\gamma(x,t))\\big|=0.\n\\]\nMoreover, if one replaces the initial-data assumptions above by the alternative regime\n\\[\n\\liminf_{y\\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0,\n\\]\n\\[\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbb{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)},\\qquad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty\n\\quad\\text{for some } c\\ge c_*,\n\\]\nthen the same conclusion still holds."}, {"label": "D", "text": "There exists a smooth function $\\gamma=\\gamma(x,t)$ such that: (i) for some $T>0$,\n\\[\n\\{(x,y,t)\\in\\mathbb{R}^{n-1}\\times\\mathbb{R}\\times[T,\\infty):\\ u(x,y,t)=\\Phi_c(0)\\}=\\{y=\\gamma(x,t)\\},\n\\]\nand\n\\[\n\\lim_{t\\to\\infty}\\sup_{(x,y)\\in\\mathbb{R}^n}\\big|u(x,y,t)-\\Phi_c(y-\\gamma(x,t))\\big|=0.\n\\]\n(ii) For every $\\varepsilon>0$, there exists $\\tau_\\varepsilon\\in[T,\\infty)$ such that if $U(x,t)$ solves\n\\[\n\\left\\{\n\\begin{aligned}\n&U_t=\\Delta_x U+\\frac{c_*}{2}|\\nabla_x U|^2+c_*,&&x\\in\\mathbb{R}^{n-1},\\ t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_\\varepsilon),&&x\\in\\mathbb{R}^{n-1},\n\\end{aligned}\n\\right.\n\\]\nthen\n\\[\n\\sup_{x\\in\\mathbb{R}^{n-1},\\,t\\ge\\tau_\\varepsilon}\\big|\\gamma(x,t)-U(x,t-\\tau_\\varepsilon)\\big|\\le\\varepsilon.\n\\]\nFurthermore, if one replaces the initial-data assumptions above by the alternative regime\n\\[\n\\liminf_{y\\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0,\n\\]\n\\[\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbb{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)},\\qquad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty\n\\quad\\text{for some } c\\ge c_*,\n\\]\nthen conclusion (i) still holds."}, {"label": "E", "text": "There exists a smooth function $\\gamma=\\gamma(x,t)$ such that: (i) for some $T>0$,\n\\[\n\\{(x,y,t)\\in\\mathbb{R}^{n-1}\\times\\mathbb{R}\\times[T,\\infty):\\ u(x,y,t)=\\Phi_c(0)\\}=\\{y=\\gamma(x,t)\\},\n\\]\nand\n\\[\n\\lim_{t\\to\\infty}\\sup_{(x,y)\\in\\mathbb{R}^n}\\big|u(x,y,t)-\\Phi_c(y-\\gamma(x,t))\\big|=0.\n\\]\n(ii) For every $\\varepsilon>0$, there exists $\\tau_\\varepsilon\\in[T,\\infty)$ such that if $U(x,t)$ solves\n\\[\n\\left\\{\n\\begin{aligned}\n&\\frac{U_t}{\\sqrt{1+|\\nabla_x U|^2}}=\\mathrm{div}\\left(\\frac{\\nabla_x U}{\\sqrt{1+|\\nabla_x U|}}\\right)+c,&&x\\in\\mathbb{R}^{n-1},\\ t>0,\\\\\n&U(x,0)=\\gamma(x,\\tau_\\varepsilon),&&x\\in\\mathbb{R}^{n-1},\n\\end{aligned}\n\\right.\n\\]\nthen\n\\[\n\\sup_{x\\in\\mathbb{R}^{n-1},\\,t\\ge\\tau_\\varepsilon}\\big|\\gamma(x,t)-U(x,t-\\tau_\\varepsilon)\\big|\\le\\varepsilon.\n\\]\nFurthermore, if one replaces the initial-data assumptions above by the alternative regime\n\\[\n\\liminf_{y\\to -\\infty} \\inf_{x \\in \\mathbb{R}^{n-1}} u_0(x,y)>1-\\delta_0,\n\\]\n\\[\n0<\\liminf_{y\\to\\infty}\\inf_{x\\in\\mathbb{R}^{n-1}} \\frac{u_0(x,y)}{\\Phi_c(y)},\\qquad\n\\limsup_{y\\to\\infty}\\sup_{x\\in\\mathbb{R}^{n-1}}\\frac{u_0(x,y)}{\\Phi_c(y)}<\\infty\n\\quad\\text{for some } c\\ge c_*,\n\\]\nthen conclusion (i) still holds."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "quantifier on epsilon and extension of alternative-regime conclusion to (ii)", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped part (ii) approximation by the geometric-flow solution", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "replaced the mean-curvature-type evolution approximated by V with the viscous Hamilton-Jacobi equation solved by V itself", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "forcing term in geometric law uses c instead of the theorem’s c_* despite the special pushed-front assumption c=c_* in part (ii)", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option, and no single phrase directly points to choice A. Although the setup is highly specialized and theorem-specific, the answer is not leaked in a trivial way."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem lists the full hypotheses and asks which existence statement holds. The correct choice is basically the theorem conclusion, so the task is close to matching premises to a restated result rather than reasoning from competing conclusions."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the choices differ by subtle quantifiers, PDE laws, and scope of the alternative-regime conclusion. However, the item mainly tests precise recognition/recall of the theorem statement, not genuine generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and targeted: one is weaker-but-true, others alter quantifiers or replace the geometric evolution law in realistic ways. These reflect common failure modes in reading advanced theorem statements."}, "total_score": 5, "overall_assessment": "Technically well-constructed with strong distractors, but it is largely a theorem-restatement/recognition question rather than a non-tautological test of generative reasoning."}}
{"id": "2602.10302v1", "paper_link": "http://arxiv.org/abs/2602.10302v1", "theorems_cnt": 1, "theorem": {"env_name": "main", "content": "[\\Cref{main}]\nIf $(A,\\sigma)$ is a totally decomposable $K$-algebra with involution of the first kind with $\\ind A\\leqslant 2$, then ${\\bf PSim}^+(A,\\sigma)$ is $R$-trivial.", "start_pos": 7007, "end_pos": 7195, "label": null}, "ref_dict": {"main": "\\begin{cor}\\label{main}   \nLet $(A,\\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind such that $\\ind A\\leq 2$. Then ${\\bf PSim}^+(A,\\s)$ is $R$-trivial. \n\\end{cor}"}, "pre_theorem_intro_text_len": 1629, "pre_theorem_intro_text": "Let $K$ be a field of characteristic different from $2$.\nLet $(A,\\sigma)$ be a $K$-algebra with involution. Let ${\\bf PSim}(A,\\sigma)$ denote the algebraic group of its projective similitudes and ${\\bf PSim}^+(A,\\sigma)$ the connected component of the identity.\nThe latter is a classical semi-simple connected linear algebraic group of adjoint type.\n(In various cases, one has ${\\bf PSim}^+(A,\\sigma)={\\bf PSim}(A,\\sigma)$, e.g.~if $\\sigma$ is symplectic.)\nIn \\cite{Mer96}, Merkurjev studied \n the geometric properties of ${\\bf PSim}^+(A,\\sigma)$ as a variety, regarding the question whether it is rational, stably rational or $R$-trivial. \n\nIn this article, we assume that the involution $\\sigma$ is $K$-linear (i.e.~an involution of the first kind).\nWe call $(A,\\sigma)$ \\emph{totally decomposable} \nif $$(A,\\sigma)\\simeq \\Ad(\\psi)\\otimes\\left(\\bigotimes_{i=1}^r (Q_i,\\tau_i)\\right)$$\nfor some $r\\in\\mathbb{N}$, $K$-quaternion algebras with involution $(Q_i,\\tau_i)$ for $1\\leqslant i\\leqslant r$ and the adjoint algebra with involution $\\Ad(\\psi)$ of some odd-dimensional quadratic form $\\psi$ over $K$. (Taking $\\psi=\\langle 1\\rangle$ corresponds to the situation without the first factor.)\n\nIf $\\sigma$ is orthogonal and $A$ is split, then we obtain by \\cite[Theorem 1]{Bec08} that $(A,\\sigma)\\simeq \\Ad(\\psi\\otimes \\pi)$\nfor some Pfister form $\\pi$ over $K$, and in this case it follows by \\cite[Prop.~7]{Mer96} that ${\\bf PSim}^+(A,\\sigma)$ is stably rational, and in particular $R$-trivial, by \\cite[Prop.~1]{Mer96}.\nOne may ask whether these facts still hold without assuming that $A$ is split.\nIn this article, we show:", "context": "Let $K$ be a field of characteristic different from $2$.\nLet $(A,\\sigma)$ be a $K$-algebra with involution. Let ${\\bf PSim}(A,\\sigma)$ denote the algebraic group of its projective similitudes and ${\\bf PSim}^+(A,\\sigma)$ the connected component of the identity.\nThe latter is a classical semi-simple connected linear algebraic group of adjoint type.\n(In various cases, one has ${\\bf PSim}^+(A,\\sigma)={\\bf PSim}(A,\\sigma)$, e.g.~if $\\sigma$ is symplectic.)\nIn \\cite{Mer96}, Merkurjev studied \n the geometric properties of ${\\bf PSim}^+(A,\\sigma)$ as a variety, regarding the question whether it is rational, stably rational or $R$-trivial.\n\nIn this article, we assume that the involution $\\sigma$ is $K$-linear (i.e.~an involution of the first kind).\nWe call $(A,\\sigma)$ \\emph{totally decomposable} \nif $$(A,\\sigma)\\simeq \\Ad(\\psi)\\otimes\\left(\\bigotimes_{i=1}^r (Q_i,\\tau_i)\\right)$$\nfor some $r\\in\\mathbb{N}$, $K$-quaternion algebras with involution $(Q_i,\\tau_i)$ for $1\\leqslant i\\leqslant r$ and the adjoint algebra with involution $\\Ad(\\psi)$ of some odd-dimensional quadratic form $\\psi$ over $K$. (Taking $\\psi=\\langle 1\\rangle$ corresponds to the situation without the first factor.)\n\nIf $\\sigma$ is orthogonal and $A$ is split, then we obtain by \\cite[Theorem 1]{Bec08} that $(A,\\sigma)\\simeq \\Ad(\\psi\\otimes \\pi)$\nfor some Pfister form $\\pi$ over $K$, and in this case it follows by \\cite[Prop.~7]{Mer96} that ${\\bf PSim}^+(A,\\sigma)$ is stably rational, and in particular $R$-trivial, by \\cite[Prop.~1]{Mer96}.\nOne may ask whether these facts still hold without assuming that $A$ is split.\nIn this article, we show:\n\n\\begin{cor}\\label{main}   \nLet $(A,\\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind such that $\\ind A\\leq 2$. Then ${\\bf PSim}^+(A,\\s)$ is $R$-trivial. \n\\end{cor}", "full_context": "Let $K$ be a field of characteristic different from $2$.\nLet $(A,\\sigma)$ be a $K$-algebra with involution. Let ${\\bf PSim}(A,\\sigma)$ denote the algebraic group of its projective similitudes and ${\\bf PSim}^+(A,\\sigma)$ the connected component of the identity.\nThe latter is a classical semi-simple connected linear algebraic group of adjoint type.\n(In various cases, one has ${\\bf PSim}^+(A,\\sigma)={\\bf PSim}(A,\\sigma)$, e.g.~if $\\sigma$ is symplectic.)\nIn \\cite{Mer96}, Merkurjev studied \n the geometric properties of ${\\bf PSim}^+(A,\\sigma)$ as a variety, regarding the question whether it is rational, stably rational or $R$-trivial.\n\nIn this article, we assume that the involution $\\sigma$ is $K$-linear (i.e.~an involution of the first kind).\nWe call $(A,\\sigma)$ \\emph{totally decomposable} \nif $$(A,\\sigma)\\simeq \\Ad(\\psi)\\otimes\\left(\\bigotimes_{i=1}^r (Q_i,\\tau_i)\\right)$$\nfor some $r\\in\\mathbb{N}$, $K$-quaternion algebras with involution $(Q_i,\\tau_i)$ for $1\\leqslant i\\leqslant r$ and the adjoint algebra with involution $\\Ad(\\psi)$ of some odd-dimensional quadratic form $\\psi$ over $K$. (Taking $\\psi=\\langle 1\\rangle$ corresponds to the situation without the first factor.)\n\nIf $\\sigma$ is orthogonal and $A$ is split, then we obtain by \\cite[Theorem 1]{Bec08} that $(A,\\sigma)\\simeq \\Ad(\\psi\\otimes \\pi)$\nfor some Pfister form $\\pi$ over $K$, and in this case it follows by \\cite[Prop.~7]{Mer96} that ${\\bf PSim}^+(A,\\sigma)$ is stably rational, and in particular $R$-trivial, by \\cite[Prop.~1]{Mer96}.\nOne may ask whether these facts still hold without assuming that $A$ is split.\nIn this article, we show:\n\n\\begin{cor}\\label{main}   \nLet $(A,\\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind such that $\\ind A\\leq 2$. Then ${\\bf PSim}^+(A,\\s)$ is $R$-trivial. \n\\end{cor}\n\n\\begin{abstract}\nWe show that the group of proper projective similitudes of a totally decomposable algebra with involution of the first kind over a field of characteristic different from $2$ is $R$-trivial.\n\nLet $K$ be a field of characteristic different from $2$.\nLet $(A,\\s)$ be a $K$-algebra with involution. Let ${\\bf PSim}(A,\\s)$ denote the algebraic group of its projective similitudes and ${\\bf PSim}^+(A,\\s)$ the connected component of the identity.\nThe latter is a classical semi-simple connected linear algebraic group of adjoint type.\n(In various cases, one has ${\\bf PSim}^+(A,\\s)={\\bf PSim}(A,\\s)$, e.g.~if $\\s$ is symplectic.)\nIn \\cite{Mer96}, Merkurjev studied \n the geometric properties of ${\\bf PSim}^+(A,\\s)$ as a variety, regarding the question whether it is rational, stably rational or $R$-trivial.\n\nIn this article, we assume that the involution $\\s$ is $K$-linear (i.e.~an involution of the first kind).\nWe call $(A,\\s)$ \\emph{totally decomposable} \nif $$(A,\\s)\\simeq \\Ad(\\psi)\\otimes\\left(\\bigotimes_{i=1}^r (Q_i,\\tau_i)\\right)$$\nfor some $r\\in\\nat$, $K$-quaternion algebras with involution $(Q_i,\\tau_i)$ for $1\\leq i\\leq r$ and the adjoint algebra with involution $\\Ad(\\psi)$ of some odd-dimensional quadratic form $\\psi$ over $K$. (Taking $\\psi=\\la 1\\ra$ corresponds to the situation without the first factor.)\n\nIf $\\s$ is orthogonal and $A$ is split, then we obtain by \\cite[Theorem 1]{Bec08} that $(A,\\s)\\simeq \\Ad(\\psi\\otimes \\pi)$\nfor some Pfister form $\\pi$ over $K$, and in this case it follows by \\cite[Prop.~7]{Mer96} that ${\\bf PSim}^+(A,\\s)$ is stably rational, and in particular $R$-trivial, by \\cite[Prop.~1]{Mer96}.\nOne may ask whether these facts still hold without assuming that $A$ is split.\nIn this article, we show:\n\nThe most interesting case is that where $\\s$ is orthogonal and $\\ind A=2$.\nTo our knowledge, this is the first general positive result on $R$-triviality of ${\\bf PSim}^+(A,\\s)$ that applies to non-split algebras with orthogonal involution $(A,\\s)$ of degree divisible by $8$.\n\nLet $(A, \\sigma)$ be an $K$-algebra with involution of the first kind. (That is, $A$ is a central simple $K$-algebra and $\\s$ is a $K$-linear involution on $A$.)\nIn $\\mg{K}$, we consider the subgroup\n$$\\G(A,\\s)=\\{\\s(x)x\\mid x\\in\\mg{A}\\}\\cap\\mg{K}\\,.$$\nIts elements are called \\emph{similarity factors} or \\emph{multipliers of similitude of $(A,\\s)$}. We refer to \\cite[\\S12]{KMRT98} for a discussion of similitudes and their multipliers.\n\n\\begin{thm}\\label{P:main}\n Let $(A,\\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind of even degree and with $\\ind A\\leq 2$. \n For every $c\\in\\G(A,\\s)$ there exist $d_1,d_2\\in\\mg{K}$ such that $\\s_{K(\\sqrt{d_i})}$ is hyperbolic for $i\\in\\{1,2\\}$ and $c\\in\\D\\lla d_1\\rra\\cdot \\D\\lla d_2\\rra$.\n\\end{thm}\n\\begin{proof}\nIt follows from the hypothesis by \\cite[Corollary]{Bec08} if $\\s$ is symplectic and by \\cite[Theorem 2]{Bec08} if $\\s$ is orthogonal that there exists a decomposition\n$$(A,\\s)\\simeq \\Ad(\\psi\\otimes \\rho)\\otimes (Q,\\gamma)$$\nwith a $K$-quaternion algebra with involution of the first kind $(Q,\\tau)$, a Pfister form $\\rho$ over $K$ and an odd-dimensional quadratic form $\\psi$ over $K$.\n\n\\begin{cor}\\label{main} \nLet $(A,\\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind such that $\\ind A\\leq 2$. Then ${\\bf PSim}^+(A,\\s)$ is $R$-trivial. \n\\end{cor}\n\\begin{proof}\nUsing \\cite[Theorem 1]{Mer96}, this follows immediately from \\Cref{P:main}.\n\\end{proof}\n\n\\begin{cor}\\label{main}   \nLet $(A,\\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind such that $\\ind A\\leq 2$. Then ${\\bf PSim}^+(A,\\s)$ is $R$-trivial. \n\\end{cor}", "post_theorem_intro_text_len": 881, "post_theorem_intro_text": "The most interesting case is that where $\\sigma$ is orthogonal and $\\ind A=2$.\nTo our knowledge, this is the first general positive result on $R$-triviality of ${\\bf PSim}^+(A,\\sigma)$ that applies to non-split algebras with orthogonal involution $(A,\\sigma)$ of degree divisible by $8$.\n\nFor the proof, we crucially rely on the characterization of $R$-equivalence in ${\\bf PSim}^+(A,\\sigma)$ established in \\cite[Theorem 1]{Mer96}. It translates the question into that of comparing the group of similarity factors with the subgroup of the multiplicative group ${K}^{\\times}$ generated by the norms from finite field extensions where the involution $\\sigma$ becomes hyperbolic.\nThe main part of our proof consists in describing the similarity factors of $(A,\\sigma)$ in terms of quadratic forms over $K$.\nTo this end, we combine a variety of key results from quadratic form theory.", "sketch": "For the proof, we crucially rely on the characterization of $R$-equivalence in ${\\bf PSim}^+(A,\\sigma)$ established in \\cite[Theorem 1]{Mer96}. It translates the question into that of comparing the group of similarity factors with the subgroup of the multiplicative group ${K}^{\\times}$ generated by the norms from finite field extensions where the involution $\\sigma$ becomes hyperbolic. The main part of our proof consists in describing the similarity factors of $(A,\\sigma)$ in terms of quadratic forms over $K$. To this end, we combine a variety of key results from quadratic form theory.", "expanded_sketch": "For the proof, we crucially rely on the characterization of $R$-equivalence in ${\\bf PSim}^+(A,\\sigma)$ established in \\cite[Theorem 1]{Mer96}. It translates the question into that of comparing the group of similarity factors with the subgroup of the multiplicative group ${K}^{\\times}$ generated by the norms from finite field extensions where the involution $\\sigma$ becomes hyperbolic. The main part of our proof consists in describing the similarity factors of $(A,\\sigma)$ in terms of quadratic forms over $K$. To this end, we combine a variety of key results from quadratic form theory.,", "expanded_theorem": "\\begin{cor}\\label{main}   \nLet $(A,\\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind such that $\\ind A\\leq 2$. Then ${\\bf PSim}^+(A,\\s)$ is $R$-trivial. \n\\end{cor}\nIf $(A,\\sigma)$ is a totally decomposable $K$-algebra with involution of the first kind with $\\ind A\\leqslant 2$, then ${\\bf PSim}^+(A,\\sigma)$ is $R$-trivial.,", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let $K$ be a field of characteristic different from $2$. A $K$-algebra with involution of the first kind means a central simple $K$-algebra $A$ equipped with a $K$-linear involution $\\sigma$. Call $(A,\\sigma)$ totally decomposable if\n$$\n(A,\\sigma)\\simeq \\operatorname{Ad}(\\psi)\\otimes\\left(\\bigotimes_{i=1}^r (Q_i,\\tau_i)\\right)\n$$\nfor some $r\\in\\mathbb N$, some $K$-quaternion algebras with involution $(Q_i,\\tau_i)$, and the adjoint algebra with involution $\\operatorname{Ad}(\\psi)$ of an odd-dimensional quadratic form $\\psi$ over $K$. Let ${\\bf PSim}(A,\\sigma)$ denote the algebraic group of projective similitudes of $(A,\\sigma)$, and let ${\\bf PSim}^+(A,\\sigma)$ be its connected component of the identity. Which statement holds for every totally decomposable $K$-algebra with involution of the first kind satisfying $\\operatorname{ind} A\\le 2$?", "correct_choice": {"label": "A", "text": "${\\bf PSim}^+(A,\\sigma)$ is $R$-trivial."}, "choices": [{"label": "B", "text": "${\\bf PSim}(A,\\sigma)$ is $R$-trivial."}, {"label": "C", "text": "${\\bf PSim}^+(A,\\sigma)$ is unirational over $K$."}, {"label": "D", "text": "For every such $(A,\\sigma)$, the variety ${\\bf PSim}^+(A,\\sigma)$ is stably rational."}, {"label": "E", "text": "For every such $(A,\\sigma)$ and every $c\\in K^{\\times}$, there exist quadratic extensions $L_1/K$ and $L_2/K$ over which $\\sigma$ becomes hyperbolic such that $c\\in N_{L_1/K}(L_1^{\\times})\\,N_{L_2/K}(L_2^{\\times})$; hence ${\\bf PSim}^+(A,\\sigma)$ is $R$-trivial."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "connected_component_vs_full_projective_similitude_group", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_R_triviality_conclusion_to_birational_weaker_property", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "R_triviality_vs_stable_rationality", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "similarity_factors_only_replaced_by_all_of_K_times", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It states hypotheses and definitions, but it does not directly announce that ${\\bf PSim}^+(A,\\sigma)$ is $R$-trivial."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the hypotheses are stated in full, and the correct choice is just the theorem's conclusion restated almost verbatim."}, "GPS": {"score": 1, "justification": "There is some pressure to distinguish the exact conclusion from nearby variants (split-only, finite quotient, after extension, full group vs identity component), but the item mainly tests recognition/recall rather than genuine derivation or synthesis."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic confusion points: weakening the conclusion, adding unnecessary split hypotheses, shifting to a field extension, or confusing ${\\bf PSim}^+$ with the full ${\\bf PSim}$."}, "total_score": 5, "overall_assessment": "A mathematically well-constructed recall question with strong distractors, but it is largely theorem restatement rather than a generative reasoning task."}}
{"id": "2602.10302v1", "paper_link": "http://arxiv.org/abs/2602.10302v1", "theorems_cnt": 1, "theorem": {"env_name": "main", "content": "[\\Cref{main}]\nIf $(A,\\sigma)$ is a totally decomposable $K$-algebra with involution of the first kind with $\\ind A\\leqslant 2$, then ${\\bf PSim}^+(A,\\sigma)$ is $R$-trivial.", "start_pos": 7007, "end_pos": 7195, "label": null}, "ref_dict": {"main": "\\begin{cor}\\label{main}   \nLet $(A,\\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind such that $\\ind A\\leq 2$. Then ${\\bf PSim}^+(A,\\s)$ is $R$-trivial. \n\\end{cor}"}, "pre_theorem_intro_text_len": 1629, "pre_theorem_intro_text": "Let $K$ be a field of characteristic different from $2$.\nLet $(A,\\sigma)$ be a $K$-algebra with involution. Let ${\\bf PSim}(A,\\sigma)$ denote the algebraic group of its projective similitudes and ${\\bf PSim}^+(A,\\sigma)$ the connected component of the identity.\nThe latter is a classical semi-simple connected linear algebraic group of adjoint type.\n(In various cases, one has ${\\bf PSim}^+(A,\\sigma)={\\bf PSim}(A,\\sigma)$, e.g.~if $\\sigma$ is symplectic.)\nIn \\cite{Mer96}, Merkurjev studied \n the geometric properties of ${\\bf PSim}^+(A,\\sigma)$ as a variety, regarding the question whether it is rational, stably rational or $R$-trivial. \n\nIn this article, we assume that the involution $\\sigma$ is $K$-linear (i.e.~an involution of the first kind).\nWe call $(A,\\sigma)$ \\emph{totally decomposable} \nif $$(A,\\sigma)\\simeq \\Ad(\\psi)\\otimes\\left(\\bigotimes_{i=1}^r (Q_i,\\tau_i)\\right)$$\nfor some $r\\in\\mathbb{N}$, $K$-quaternion algebras with involution $(Q_i,\\tau_i)$ for $1\\leqslant i\\leqslant r$ and the adjoint algebra with involution $\\Ad(\\psi)$ of some odd-dimensional quadratic form $\\psi$ over $K$. (Taking $\\psi=\\langle 1\\rangle$ corresponds to the situation without the first factor.)\n\nIf $\\sigma$ is orthogonal and $A$ is split, then we obtain by \\cite[Theorem 1]{Bec08} that $(A,\\sigma)\\simeq \\Ad(\\psi\\otimes \\pi)$\nfor some Pfister form $\\pi$ over $K$, and in this case it follows by \\cite[Prop.~7]{Mer96} that ${\\bf PSim}^+(A,\\sigma)$ is stably rational, and in particular $R$-trivial, by \\cite[Prop.~1]{Mer96}.\nOne may ask whether these facts still hold without assuming that $A$ is split.\nIn this article, we show:", "context": "Let $K$ be a field of characteristic different from $2$.\nLet $(A,\\sigma)$ be a $K$-algebra with involution. Let ${\\bf PSim}(A,\\sigma)$ denote the algebraic group of its projective similitudes and ${\\bf PSim}^+(A,\\sigma)$ the connected component of the identity.\nThe latter is a classical semi-simple connected linear algebraic group of adjoint type.\n(In various cases, one has ${\\bf PSim}^+(A,\\sigma)={\\bf PSim}(A,\\sigma)$, e.g.~if $\\sigma$ is symplectic.)\nIn \\cite{Mer96}, Merkurjev studied \n the geometric properties of ${\\bf PSim}^+(A,\\sigma)$ as a variety, regarding the question whether it is rational, stably rational or $R$-trivial.\n\nIn this article, we assume that the involution $\\sigma$ is $K$-linear (i.e.~an involution of the first kind).\nWe call $(A,\\sigma)$ \\emph{totally decomposable} \nif $$(A,\\sigma)\\simeq \\Ad(\\psi)\\otimes\\left(\\bigotimes_{i=1}^r (Q_i,\\tau_i)\\right)$$\nfor some $r\\in\\mathbb{N}$, $K$-quaternion algebras with involution $(Q_i,\\tau_i)$ for $1\\leqslant i\\leqslant r$ and the adjoint algebra with involution $\\Ad(\\psi)$ of some odd-dimensional quadratic form $\\psi$ over $K$. (Taking $\\psi=\\langle 1\\rangle$ corresponds to the situation without the first factor.)\n\nIf $\\sigma$ is orthogonal and $A$ is split, then we obtain by \\cite[Theorem 1]{Bec08} that $(A,\\sigma)\\simeq \\Ad(\\psi\\otimes \\pi)$\nfor some Pfister form $\\pi$ over $K$, and in this case it follows by \\cite[Prop.~7]{Mer96} that ${\\bf PSim}^+(A,\\sigma)$ is stably rational, and in particular $R$-trivial, by \\cite[Prop.~1]{Mer96}.\nOne may ask whether these facts still hold without assuming that $A$ is split.\nIn this article, we show:\n\n\\begin{cor}\\label{main}   \nLet $(A,\\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind such that $\\ind A\\leq 2$. Then ${\\bf PSim}^+(A,\\s)$ is $R$-trivial. \n\\end{cor}", "full_context": "Let $K$ be a field of characteristic different from $2$.\nLet $(A,\\sigma)$ be a $K$-algebra with involution. Let ${\\bf PSim}(A,\\sigma)$ denote the algebraic group of its projective similitudes and ${\\bf PSim}^+(A,\\sigma)$ the connected component of the identity.\nThe latter is a classical semi-simple connected linear algebraic group of adjoint type.\n(In various cases, one has ${\\bf PSim}^+(A,\\sigma)={\\bf PSim}(A,\\sigma)$, e.g.~if $\\sigma$ is symplectic.)\nIn \\cite{Mer96}, Merkurjev studied \n the geometric properties of ${\\bf PSim}^+(A,\\sigma)$ as a variety, regarding the question whether it is rational, stably rational or $R$-trivial.\n\nIn this article, we assume that the involution $\\sigma$ is $K$-linear (i.e.~an involution of the first kind).\nWe call $(A,\\sigma)$ \\emph{totally decomposable} \nif $$(A,\\sigma)\\simeq \\Ad(\\psi)\\otimes\\left(\\bigotimes_{i=1}^r (Q_i,\\tau_i)\\right)$$\nfor some $r\\in\\mathbb{N}$, $K$-quaternion algebras with involution $(Q_i,\\tau_i)$ for $1\\leqslant i\\leqslant r$ and the adjoint algebra with involution $\\Ad(\\psi)$ of some odd-dimensional quadratic form $\\psi$ over $K$. (Taking $\\psi=\\langle 1\\rangle$ corresponds to the situation without the first factor.)\n\nIf $\\sigma$ is orthogonal and $A$ is split, then we obtain by \\cite[Theorem 1]{Bec08} that $(A,\\sigma)\\simeq \\Ad(\\psi\\otimes \\pi)$\nfor some Pfister form $\\pi$ over $K$, and in this case it follows by \\cite[Prop.~7]{Mer96} that ${\\bf PSim}^+(A,\\sigma)$ is stably rational, and in particular $R$-trivial, by \\cite[Prop.~1]{Mer96}.\nOne may ask whether these facts still hold without assuming that $A$ is split.\nIn this article, we show:\n\n\\begin{cor}\\label{main}   \nLet $(A,\\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind such that $\\ind A\\leq 2$. Then ${\\bf PSim}^+(A,\\s)$ is $R$-trivial. \n\\end{cor}\n\n\\begin{abstract}\nWe show that the group of proper projective similitudes of a totally decomposable algebra with involution of the first kind over a field of characteristic different from $2$ is $R$-trivial.\n\nLet $K$ be a field of characteristic different from $2$.\nLet $(A,\\s)$ be a $K$-algebra with involution. Let ${\\bf PSim}(A,\\s)$ denote the algebraic group of its projective similitudes and ${\\bf PSim}^+(A,\\s)$ the connected component of the identity.\nThe latter is a classical semi-simple connected linear algebraic group of adjoint type.\n(In various cases, one has ${\\bf PSim}^+(A,\\s)={\\bf PSim}(A,\\s)$, e.g.~if $\\s$ is symplectic.)\nIn \\cite{Mer96}, Merkurjev studied \n the geometric properties of ${\\bf PSim}^+(A,\\s)$ as a variety, regarding the question whether it is rational, stably rational or $R$-trivial.\n\nIn this article, we assume that the involution $\\s$ is $K$-linear (i.e.~an involution of the first kind).\nWe call $(A,\\s)$ \\emph{totally decomposable} \nif $$(A,\\s)\\simeq \\Ad(\\psi)\\otimes\\left(\\bigotimes_{i=1}^r (Q_i,\\tau_i)\\right)$$\nfor some $r\\in\\nat$, $K$-quaternion algebras with involution $(Q_i,\\tau_i)$ for $1\\leq i\\leq r$ and the adjoint algebra with involution $\\Ad(\\psi)$ of some odd-dimensional quadratic form $\\psi$ over $K$. (Taking $\\psi=\\la 1\\ra$ corresponds to the situation without the first factor.)\n\nIf $\\s$ is orthogonal and $A$ is split, then we obtain by \\cite[Theorem 1]{Bec08} that $(A,\\s)\\simeq \\Ad(\\psi\\otimes \\pi)$\nfor some Pfister form $\\pi$ over $K$, and in this case it follows by \\cite[Prop.~7]{Mer96} that ${\\bf PSim}^+(A,\\s)$ is stably rational, and in particular $R$-trivial, by \\cite[Prop.~1]{Mer96}.\nOne may ask whether these facts still hold without assuming that $A$ is split.\nIn this article, we show:\n\nThe most interesting case is that where $\\s$ is orthogonal and $\\ind A=2$.\nTo our knowledge, this is the first general positive result on $R$-triviality of ${\\bf PSim}^+(A,\\s)$ that applies to non-split algebras with orthogonal involution $(A,\\s)$ of degree divisible by $8$.\n\nLet $(A, \\sigma)$ be an $K$-algebra with involution of the first kind. (That is, $A$ is a central simple $K$-algebra and $\\s$ is a $K$-linear involution on $A$.)\nIn $\\mg{K}$, we consider the subgroup\n$$\\G(A,\\s)=\\{\\s(x)x\\mid x\\in\\mg{A}\\}\\cap\\mg{K}\\,.$$\nIts elements are called \\emph{similarity factors} or \\emph{multipliers of similitude of $(A,\\s)$}. We refer to \\cite[\\S12]{KMRT98} for a discussion of similitudes and their multipliers.\n\n\\begin{thm}\\label{P:main}\n Let $(A,\\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind of even degree and with $\\ind A\\leq 2$. \n For every $c\\in\\G(A,\\s)$ there exist $d_1,d_2\\in\\mg{K}$ such that $\\s_{K(\\sqrt{d_i})}$ is hyperbolic for $i\\in\\{1,2\\}$ and $c\\in\\D\\lla d_1\\rra\\cdot \\D\\lla d_2\\rra$.\n\\end{thm}\n\\begin{proof}\nIt follows from the hypothesis by \\cite[Corollary]{Bec08} if $\\s$ is symplectic and by \\cite[Theorem 2]{Bec08} if $\\s$ is orthogonal that there exists a decomposition\n$$(A,\\s)\\simeq \\Ad(\\psi\\otimes \\rho)\\otimes (Q,\\gamma)$$\nwith a $K$-quaternion algebra with involution of the first kind $(Q,\\tau)$, a Pfister form $\\rho$ over $K$ and an odd-dimensional quadratic form $\\psi$ over $K$.\n\n\\begin{cor}\\label{main} \nLet $(A,\\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind such that $\\ind A\\leq 2$. Then ${\\bf PSim}^+(A,\\s)$ is $R$-trivial. \n\\end{cor}\n\\begin{proof}\nUsing \\cite[Theorem 1]{Mer96}, this follows immediately from \\Cref{P:main}.\n\\end{proof}\n\n\\begin{cor}\\label{main}   \nLet $(A,\\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind such that $\\ind A\\leq 2$. Then ${\\bf PSim}^+(A,\\s)$ is $R$-trivial. \n\\end{cor}", "post_theorem_intro_text_len": 881, "post_theorem_intro_text": "The most interesting case is that where $\\sigma$ is orthogonal and $\\ind A=2$.\nTo our knowledge, this is the first general positive result on $R$-triviality of ${\\bf PSim}^+(A,\\sigma)$ that applies to non-split algebras with orthogonal involution $(A,\\sigma)$ of degree divisible by $8$.\n\nFor the proof, we crucially rely on the characterization of $R$-equivalence in ${\\bf PSim}^+(A,\\sigma)$ established in \\cite[Theorem 1]{Mer96}. It translates the question into that of comparing the group of similarity factors with the subgroup of the multiplicative group ${K}^{\\times}$ generated by the norms from finite field extensions where the involution $\\sigma$ becomes hyperbolic.\nThe main part of our proof consists in describing the similarity factors of $(A,\\sigma)$ in terms of quadratic forms over $K$.\nTo this end, we combine a variety of key results from quadratic form theory.", "sketch": "For the proof, we crucially rely on the characterization of $R$-equivalence in ${\\bf PSim}^+(A,\\sigma)$ established in \\cite[Theorem 1]{Mer96}. It translates the question into that of comparing the group of similarity factors with the subgroup of the multiplicative group ${K}^{\\times}$ generated by the norms from finite field extensions where the involution $\\sigma$ becomes hyperbolic. The main part of our proof consists in describing the similarity factors of $(A,\\sigma)$ in terms of quadratic forms over $K$. To this end, we combine a variety of key results from quadratic form theory.", "expanded_sketch": "For the proof, we crucially rely on the characterization of $R$-equivalence in ${\\bf PSim}^+(A,\\sigma)$ established in \\cite[Theorem 1]{Mer96}. It translates the question into that of comparing the group of similarity factors with the subgroup of the multiplicative group ${K}^{\\times}$ generated by the norms from finite field extensions where the involution $\\sigma$ becomes hyperbolic. The main part of our proof consists in describing the similarity factors of $(A,\\sigma)$ in terms of quadratic forms over $K$. To this end, we combine a variety of key results from quadratic form theory.,", "expanded_theorem": "\\begin{cor}\\label{main}   \nLet $(A,\\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind such that $\\ind A\\leq 2$. Then ${\\bf PSim}^+(A,\\s)$ is $R$-trivial. \n\\end{cor}\nIf $(A,\\sigma)$ is a totally decomposable $K$-algebra with involution of the first kind with $\\ind A\\leqslant 2$, then ${\\bf PSim}^+(A,\\sigma)$ is $R$-trivial.,", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let $K$ be a field of characteristic different from $2$. A $K$-algebra with involution of the first kind means a central simple $K$-algebra $A$ equipped with a $K$-linear involution $\\sigma$. Call $(A,\\sigma)$ totally decomposable if\n$$\n(A,\\sigma)\\simeq \\operatorname{Ad}(\\psi)\\otimes\\left(\\bigotimes_{i=1}^r (Q_i,\\tau_i)\\right)\n$$\nfor some $r\\in\\mathbb N$, some $K$-quaternion algebras with involution $(Q_i,\\tau_i)$, and the adjoint algebra with involution $\\operatorname{Ad}(\\psi)$ of an odd-dimensional quadratic form $\\psi$ over $K$. Let ${\\bf PSim}(A,\\sigma)$ denote the algebraic group of projective similitudes of $(A,\\sigma)$, and let ${\\bf PSim}^+(A,\\sigma)$ be its connected component of the identity. Which statement holds for every totally decomposable $K$-algebra with involution of the first kind satisfying $\\operatorname{ind} A\\le 2$?", "correct_choice": {"label": "A", "text": "${\\bf PSim}^+(A,\\sigma)$ is $R$-trivial."}, "choices": [{"label": "B", "text": "${\\bf PSim}(A,\\sigma)$ is $R$-trivial."}, {"label": "C", "text": "${\\bf PSim}^+(A,\\sigma)$ is unirational over $K$."}, {"label": "D", "text": "For every such $(A,\\sigma)$, the variety ${\\bf PSim}^+(A,\\sigma)$ is stably rational."}, {"label": "E", "text": "For every such $(A,\\sigma)$ and every $c\\in K^{\\times}$, there exist quadratic extensions $L_1/K$ and $L_2/K$ over which $\\sigma$ becomes hyperbolic such that $c\\in N_{L_1/K}(L_1^{\\times})\\,N_{L_2/K}(L_2^{\\times})$; hence ${\\bf PSim}^+(A,\\sigma)$ is $R$-trivial."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "connected_component_vs_full_projective_similitude_group", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_R_triviality_conclusion_to_birational_weaker_property", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "R_triviality_vs_stable_rationality", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "similarity_factors_only_replaced_by_all_of_K_times", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option, nor does it contain a near-verbatim hint of the conclusion. The answer must be identified from the options rather than extracted from the wording."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the hypotheses are stated in full and the student is asked which conclusion holds for all such objects. That makes it very close to a direct restatement of a known result."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options are nearby variants: connected component vs full group, R-triviality vs unirationality vs stable rationality, and a technical norm-condition overreach. Still, the item mainly tests recognition/recall rather than genuine derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: confusing the connected component with the full group, replacing the correct statement by a weaker true one, strengthening to stable rationality, or overgeneralizing proof ingredients into an all-purpose norm statement."}, "total_score": 5, "overall_assessment": "A mathematically well-constructed theorem-discrimination MCQ with strong distractors and little answer leakage, but it is largely a direct recall/restatement question rather than one that strongly tests generative reasoning."}}
{"id": "2602.10509v1", "paper_link": "http://arxiv.org/abs/2602.10509v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "Assume that (F1-5) hold true. \n    Then there exists a nontrivial periodic solution for~\\eqref{eq:NDE-spatial}, which is in~$C^1$.", "start_pos": 12614, "end_pos": 12770, "label": null}, "ref_dict": {"eq:NLD with external field": "\\begin{align}\\label{eq:NLD with external field}\n    -i\\gamma^0\\gamma^k\\p_k\\psi + m\\gamma^0\\psi -a\\psi - M\\psi -\\nabla F(x, \\psi)=0,\n\\end{align}", "eq:F Soler": "\\begin{align}\\label{eq:F Soler}\n    F(\\psi)\n    =\\frac{1}{2}G(\\bar{\\psi}\\psi)\n\\end{align}", "eq:nonlinearity Finkelstein": "\\begin{align}\\label{eq:nonlinearity Finkelstein}\n    F(\\psi) = \\frac{1}{2}|\\bar{\\psi}\\psi|^2 + b|\\bar{\\psi}\\gamma^5\\psi|^2\n\\end{align}", "rmk:multiplicity": "\\begin{rmk}\\label{rmk:multiplicity}\n    It is tempting to hope for multiple solutions at this stage, especially when~$F$ is even in~$\\psi$ or has more symmetries, as in~\\cite{DingRuf2008Solutions} and~\\cite{DingLiu2014periodic}. \n    This would require a restrcition of the value of~$a$, or equivalently, that~$m\\gg a$. \n    The latter condition on~$m$ is not too restrictive since, in the orginal setting, we should have~$mc^2$ where~$c$ stands for the speed of light. \n    There are plenty of works on the limiting behavior as~$c\\to+\\infty$. \n    We leave this, together with the multiplicity, to a future work. \n\\end{rmk}", "eq:NDE-spatial": "\\begin{align}\\label{eq:NDE-spatial}\\tag{*}\n    -i\\gamma^0 \\gamma^k\\p_k \\psi + m\\gamma^0\\psi -a\\psi-\\nabla F(\\psi)=0.\n\\end{align}", "eq:4D nonlinear Dirac": "\\begin{align}\\label{eq:4D nonlinear Dirac}\n    \\sum_{\\mu=0}^{3} i\\gamma^\\mu\\p_\\mu\\Psi-m \\Psi + \\gamma^0 \\nabla F(\\Psi)=0\n\\end{align}", "eq:small oscillation": "\\begin{align}\\label{eq:small oscillation}\n    0< a \\leq a + M < m, \n\\end{align}"}, "pre_theorem_intro_text_len": 8033, "pre_theorem_intro_text": "In physical models for fermionic fields, the equations of motion often appear as Dirac equations with various nonlinearities.\nIn this work we consider periodic solutions on~$\\mathbb{R}^{1+3}$, which are actually spatially periodic solutions and stationary in physical sense i.e. they propagate via a complex phase factor without changing shapes. \nSuch solutions fails to decay at infinity, and we have to employ a perturbative variational scheme to obtain a nontrivial solution.\n\n\\\n\nMore precisely, we will be concerned with the following nonlinear Dirac equation\n\\begin{align}\\label{eq:4D nonlinear Dirac}\n    \\sum_{\\mu=0}^{3} i\\gamma^\\mu\\p_\\mu\\Psi-m \\Psi + \\gamma^0 \\nabla F(\\Psi)=0\n\\end{align}\nwhere~$\\Psi\\colon \\mathbb{R}^3\\to \\mathbb{C}^4$ stands for a spinor field,~$m>0$ corresponds to the mass of the particles, and~$F\\colon \\mathbb{C}^4\\to \\mathbb{R}$ is the self-coupling of particles in the system, which are chosen according to the complexity of the physical and chemical models, see e.g.~\\cite{BartschDing2006Solutions,Ding2007variational, Esteban2002overview,Psarelli2005Maxwell,Soler1970Classical, Thaller1992Dirac} and the references therein.\nThe~$\\gamma^\\mu$'s are as given follows.\nThe matrix~$\\gamma^0$ takes the form\n\\begin{align}\n    \\gamma^0= \\begin{pmatrix} I_2 & 0 \\\\ 0 & -I_2\\end{pmatrix}\n    =\\begin{pmatrix}\n        1 & & &  \\\\ & 1 & & \\\\ & & -1 & \\\\ & &  & -1\n    \\end{pmatrix}\n\\end{align}\nand induces a decomposition of the spinors into~$\\pm1$ eigenspaces of~$\\gamma^0$.\nAs for the others, we denote by\n\\begin{align}\n    {\\sigma^1} = \\left( {\\begin{array}{*{20}{c}}\n\t\t\t{0}&1 \\\\\n\t\t\t1&{0}\n\t\\end{array}} \\right),\\;{\\sigma^2} = \\left( {\\begin{array}{*{20}{c}}\n\t\t\t{0}&{ - i} \\\\\n\t\t\ti&{0}\n\t\\end{array}} \\right),\\;{\\sigma^3} = \\left( {\\begin{array}{*{20}{c}}\n\t\t\t1&{0} \\\\\n\t\t\t{0}&{ - 1}\n\t\\end{array}} \\right)\n\\end{align}\nfor the Pauli matrices, and the gamma matrices are\n\\begin{align}\n    {\\gamma^k} = \\left( {\\begin{array}{*{20}{c}}\n\t\t\t{0}&{{\\sigma _k}} \\\\\n\t\t\t{{-\\sigma _k}}&{0}\n\t\\end{array}} \\right),\\quad\\text{for}\\quad k= 1, 2, 3.\n\\end{align}\nNote that they satisfy the following Clifford relations: for~$j,k\\in\\left\\{1,2,3\\right\\}$,\n\\begin{align}\n    \\sigma^j\\sigma^k+\\sigma^k\\sigma^j= 2\\delta^{jk}I_2, & &\n    \\gamma^j\\gamma^k + \\gamma^k\\gamma^j= -2\\delta^{jk}I_4.\n\\end{align}\nMoreover,\n\\begin{align}\n    \\gamma^0\\gamma^k + \\gamma^k \\gamma^0 =0, & &\n    (\\gamma^0)^2=I_4.\n\\end{align}\nThen~$\\sum_{\\mu} i\\gamma^\\mu \\p_\\mu$ is the 4D space-time Dirac operator.\nWith the time direction involved, it is not elliptic.\n\nAn typical ansatz for the solutions are the so-called \\emph{stationary solutions} or~\\emph{solitary solutions} (see e.g.~\\cite{EstebanSere1995stationary}) which takes the form\n\\begin{align}\n    \\Psi(t,\\vec{x})= e^{-i a t}\\psi(\\vec{x})\n\\end{align}\nSuch solutions are regarded as the “particle-like solutions” and hence attract much attention. \nFor spinors of such form and for~$F$ invariant under rotations ( i.e. ~$F(e^{i\\theta}\\Psi)=F(\\Psi)$),~\\eqref{eq:4D nonlinear Dirac} is transformed into\n\\begin{align}\n    i\\gamma^k\\p_k\\psi + a\\gamma^0\\psi-m\\psi +\\gamma^0\\nabla F(\\psi)=0.\n\\end{align}\nEquivalently, we are led to consider the equation\n\\begin{align}\\label{eq:NDE-spatial}\\tag{*}\n    -i\\gamma^0 \\gamma^k\\p_k \\psi + m\\gamma^0\\psi -a\\psi-\\nabla F(\\psi)=0.\n\\end{align}\nEquations of this form have been studied in depth in literature, see e.g.~\\cite{BalabaneCazenave1988Existence, Balabane1988Existence, Balabane1990Existence, Cazenave1986Existence,EstebanSere1995stationary,Merle1988Existence} and the references quoted therein.\nDifferent from~\\cite{EstebanSere1995stationary}, here we are concerned with periodic solutions, namely solutions satisfying\n\\begin{align}\n    \\psi(\\theta+\\tau)=\\psi(\\theta), \\quad \\forall \\tau\\in \\Gamma\n\\end{align}\nwhere~$\\Gamma\\subset \\mathbb{R}^3$ is a 3-dimensional lattice.\nThey are referred as the standing periodic waves for~\\eqref{eq:4D nonlinear Dirac}. \n\n\\ \n\nIn quantum fields theory it is often required that the Hamiltonian has some symmetry, and the periodicity is thus frequently imposed. \nIn one-dimensional dynamics this is usually the case. \nAccording to the features of the particle system, sometimes it is require that the nonlinearity has spatial periodicity, which is not the case here since our~$F$ has no explicit dependence on the space variable~$x$. \nHere we are in a similar setting as in~\\cite{DingLiu2014periodic}, namely we seek for periodic solutions for the nonlinear Dirac equation~\\eqref{eq:NDE-spatial}. \nThe difference to ~\\cite{DingLiu2014periodic} lies in the nonlinearity, which will become clear later. \n\nGeometrically, this amounts to consider the equation~\\eqref{eq:NDE-spatial} on the compact manifold~$\\mathbb{T}^3=\\faktor{\\mathbb{R}^3}{\\Gamma}$.\nThe spectral properties for the linearized operators on torus is in great contrast to the one on~$\\mathbb{R}^3$, so is the analysis needed for the variational scheme. \nFor example, we cannot appeal to the Fourier space to obtain enough linking structures, and we do not have a global Pohozaev identity to control the kinetic energy and the potential energy separately, which are crucial for the proof in~\\cite{EstebanSere1995stationary} to obtain solutions on~$\\mathbb{R}^3$. \nOn the other hand, the torus is still good enough to allow for variational analysis and we can changle local strategies to carry out the variational method in~\\cite{EstebanSere1995stationary} to obtain nontrivial solutions. \n\n\\ \n\nThe main motivation of the nonlinearities comes from the following examples. \nIn the Soler model~\\cite{Soler1970Classical}, the self-coupling has the form \n\\begin{align}\\label{eq:F Soler}\n    F(\\psi)\n    =\\frac{1}{2}G(\\bar{\\psi}\\psi)\n\\end{align}\nwith~$G\\in C^2(\\mathbb{R},\\R_+)$, and~$G(s)=0$ for~$s\\leq 0$.\nTo solve the corresponding nonlinear Dirac equation on~$\\mathbb{R}^3$, a suitable “radial” Ansatz were used to reduce the equation to ODEs and then the shooting method can provide some solutions. \nThese methods cannot be applied in the periodic setting for the lack of a good radail Ansatz. \nA more general form of~$F$ arise in~\\cite{Finkelstein1951Non-linea} where the physicists were studying the symmetric coupling between nucleons, muons, and leptons, for which \n\\begin{align}\\label{eq:nonlinearity Finkelstein}\n    F(\\psi) = \\frac{1}{2}|\\bar{\\psi}\\psi|^2 + b|\\bar{\\psi}\\gamma^5\\psi|^2\n\\end{align}\nwhere~$\\gamma^5= \\gamma^0\\gamma^1\\gamma^2\\gamma^3$.\nIn such models, the existence theory remains still largely open. \nIndeed, for~\\eqref{eq:nonlinearity Finkelstein}, it has growth order~$|\\psi|^4$ which is super-critical for the Sobolev embeddings, thus not appropriate for the variational scheme. \n\n\\\n\nUp to now we can only deal with the sub-critical case. \nTo be precise, we impose the following hypothese on the nonlinearity~$F$.\nSuppose that there exist constants~$A_j>0$,~$1\\leq j\\leq 4$, and~$\\nu>1$,~$\\alpha>2$,~$\\beta>3$,~$2<\\alpha_1\\leq\\alpha_2<3$ such that\n\\begin{itemize}\n    \\item[(F1)] \\( 0 \\leq F(\\psi) \\leq A_1 \\big( |\\psi|^{\\alpha_1} + |\\psi|^{\\alpha_2} \\big) \\quad \\forall \\psi \\in \\mathbb{C}^4 \\),\n\n    \\item[(F2)] \\( F \\in C^2 \\), \\( F(0) = F'(0) = F''(0) = 0 \\), and\n          \\( |F''(\\psi)| \\leq A_2 |\\psi|^{\\alpha_2 - 2} \\) for \\( |\\psi| \\) large,\n\n    \\item[(F3)] \\( \\alpha F(\\psi) \\leq dF(\\psi)[\\psi] \\) for some \\( \\alpha > 2 \\),\n\n    \\item[(F4)] \\( F(\\psi) \\geq A_3 |\\psi\\overline{\\psi}|^\\nu - A_4 \\),\n\n    \\item[(F5)]  $\\exists\\quad\\beta > 3$, $\\forall\\quad\\delta > 0$, $\\exists \\quad C_\\delta > 0$, $\\forall \\psi \\in \\mathbb{C}^4$, such that\n\\begin{align}\n     |\\nabla F(\\psi)| \\leq \\left( \\delta + C_\\delta F(\\psi)^{\\frac{1}{\\beta}} \\right) |\\psi|.\n\\end{align}\n\\end{itemize}\nThe practical nonlinearity that we aim to model include the Soler nonlinearity in~\\eqref{eq:F Soler} as well as \n\\begin{align}\n    F(\\psi)= |\\bar{\\psi}\\psi|^\\alpha+ b|\\bar{\\psi}\\gamma^5\\psi|^\\beta,\n\\end{align}\nwhere~$\\gamma^5=\\gamma^0\\gamma^1\\gamma^2\\gamma^3$ and~$b\\geq 0$,~$1<\\alpha,\\beta<\\frac{3}{2}$.\n\nThe main result of this work is the following.", "context": "An typical ansatz for the solutions are the so-called \\emph{stationary solutions} or~\\emph{solitary solutions} (see e.g.~\\cite{EstebanSere1995stationary}) which takes the form\n\\begin{align}\n \\Psi(t,\\vec{x})= e^{-i a t}\\psi(\\vec{x})\n\\end{align}\nSuch solutions are regarded as the “particle-like solutions” and hence attract much attention. \nFor spinors of such form and for~$F$ invariant under rotations ( i.e. ~$F(e^{i\\theta}\\Psi)=F(\\Psi)$),~\\eqref{eq:4D nonlinear Dirac} is transformed into\n\\begin{align}\n i\\gamma^k\\p_k\\psi + a\\gamma^0\\psi-m\\psi +\\gamma^0\\nabla F(\\psi)=0.\n\\end{align}\nEquivalently, we are led to consider the equation\n\\begin{align}\\label{eq:NDE-spatial}\\tag{*}\n -i\\gamma^0 \\gamma^k\\p_k \\psi + m\\gamma^0\\psi -a\\psi-\\nabla F(\\psi)=0.\n\\end{align}\nEquations of this form have been studied in depth in literature, see e.g.~\\cite{BalabaneCazenave1988Existence, Balabane1988Existence, Balabane1990Existence, Cazenave1986Existence,EstebanSere1995stationary,Merle1988Existence} and the references quoted therein.\nDifferent from~\\cite{EstebanSere1995stationary}, here we are concerned with periodic solutions, namely solutions satisfying\n\\begin{align}\n \\psi(\\theta+\\tau)=\\psi(\\theta), \\quad \\forall \\tau\\in \\Gamma\n\\end{align}\nwhere~$\\Gamma\\subset \\mathbb{R}^3$ is a 3-dimensional lattice.\nThey are referred as the standing periodic waves for~\\eqref{eq:4D nonlinear Dirac}.\n\nIn quantum fields theory it is often required that the Hamiltonian has some symmetry, and the periodicity is thus frequently imposed. \nIn one-dimensional dynamics this is usually the case. \nAccording to the features of the particle system, sometimes it is require that the nonlinearity has spatial periodicity, which is not the case here since our~$F$ has no explicit dependence on the space variable~$x$. \nHere we are in a similar setting as in~\\cite{DingLiu2014periodic}, namely we seek for periodic solutions for the nonlinear Dirac equation~\\eqref{eq:NDE-spatial}. \nThe difference to ~\\cite{DingLiu2014periodic} lies in the nonlinearity, which will become clear later.\n\nGeometrically, this amounts to consider the equation~\\eqref{eq:NDE-spatial} on the compact manifold~$\\mathbb{T}^3=\\faktor{\\mathbb{R}^3}{\\Gamma}$.\nThe spectral properties for the linearized operators on torus is in great contrast to the one on~$\\mathbb{R}^3$, so is the analysis needed for the variational scheme. \nFor example, we cannot appeal to the Fourier space to obtain enough linking structures, and we do not have a global Pohozaev identity to control the kinetic energy and the potential energy separately, which are crucial for the proof in~\\cite{EstebanSere1995stationary} to obtain solutions on~$\\mathbb{R}^3$. \nOn the other hand, the torus is still good enough to allow for variational analysis and we can changle local strategies to carry out the variational method in~\\cite{EstebanSere1995stationary} to obtain nontrivial solutions.\n\nThe main motivation of the nonlinearities comes from the following examples. \nIn the Soler model~\\cite{Soler1970Classical}, the self-coupling has the form \n\\begin{align}\\label{eq:F Soler}\n F(\\psi)\n =\\frac{1}{2}G(\\bar{\\psi}\\psi)\n\\end{align}\nwith~$G\\in C^2(\\mathbb{R},\\R_+)$, and~$G(s)=0$ for~$s\\leq 0$.\nTo solve the corresponding nonlinear Dirac equation on~$\\mathbb{R}^3$, a suitable “radial” Ansatz were used to reduce the equation to ODEs and then the shooting method can provide some solutions. \nThese methods cannot be applied in the periodic setting for the lack of a good radail Ansatz. \nA more general form of~$F$ arise in~\\cite{Finkelstein1951Non-linea} where the physicists were studying the symmetric coupling between nucleons, muons, and leptons, for which \n\\begin{align}\\label{eq:nonlinearity Finkelstein}\n F(\\psi) = \\frac{1}{2}|\\bar{\\psi}\\psi|^2 + b|\\bar{\\psi}\\gamma^5\\psi|^2\n\\end{align}\nwhere~$\\gamma^5= \\gamma^0\\gamma^1\\gamma^2\\gamma^3$.\nIn such models, the existence theory remains still largely open. \nIndeed, for~\\eqref{eq:nonlinearity Finkelstein}, it has growth order~$|\\psi|^4$ which is super-critical for the Sobolev embeddings, thus not appropriate for the variational scheme.\n\n\\item[(F5)] $\\exists\\quad\\beta > 3$, $\\forall\\quad\\delta > 0$, $\\exists \\quad C_\\delta > 0$, $\\forall \\psi \\in \\mathbb{C}^4$, such that\n\\begin{align}\n |\\nabla F(\\psi)| \\leq \\left( \\delta + C_\\delta F(\\psi)^{\\frac{1}{\\beta}} \\right) |\\psi|.\n\\end{align}\n\\end{itemize}\nThe practical nonlinearity that we aim to model include the Soler nonlinearity in~\\eqref{eq:F Soler} as well as \n\\begin{align}\n F(\\psi)= |\\bar{\\psi}\\psi|^\\alpha+ b|\\bar{\\psi}\\gamma^5\\psi|^\\beta,\n\\end{align}\nwhere~$\\gamma^5=\\gamma^0\\gamma^1\\gamma^2\\gamma^3$ and~$b\\geq 0$,~$1<\\alpha,\\beta<\\frac{3}{2}$.\n\nThe main result of this work is the following.\n\n\\begin{align}\\label{eq:4D nonlinear Dirac}\n    \\sum_{\\mu=0}^{3} i\\gamma^\\mu\\p_\\mu\\Psi-m \\Psi + \\gamma^0 \\nabla F(\\Psi)=0\n\\end{align}\n\n\\begin{align}\\label{eq:F Soler}\n    F(\\psi)\n    =\\frac{1}{2}G(\\bar{\\psi}\\psi)\n\\end{align}\n\n\\begin{align}\\label{eq:NDE-spatial}\\tag{*}\n    -i\\gamma^0 \\gamma^k\\p_k \\psi + m\\gamma^0\\psi -a\\psi-\\nabla F(\\psi)=0.\n\\end{align}\n\n\\begin{align}\\label{eq:nonlinearity Finkelstein}\n    F(\\psi) = \\frac{1}{2}|\\bar{\\psi}\\psi|^2 + b|\\bar{\\psi}\\gamma^5\\psi|^2\n\\end{align}", "full_context": "An typical ansatz for the solutions are the so-called \\emph{stationary solutions} or~\\emph{solitary solutions} (see e.g.~\\cite{EstebanSere1995stationary}) which takes the form\n\\begin{align}\n \\Psi(t,\\vec{x})= e^{-i a t}\\psi(\\vec{x})\n\\end{align}\nSuch solutions are regarded as the “particle-like solutions” and hence attract much attention. \nFor spinors of such form and for~$F$ invariant under rotations ( i.e. ~$F(e^{i\\theta}\\Psi)=F(\\Psi)$),~\\eqref{eq:4D nonlinear Dirac} is transformed into\n\\begin{align}\n i\\gamma^k\\p_k\\psi + a\\gamma^0\\psi-m\\psi +\\gamma^0\\nabla F(\\psi)=0.\n\\end{align}\nEquivalently, we are led to consider the equation\n\\begin{align}\\label{eq:NDE-spatial}\\tag{*}\n -i\\gamma^0 \\gamma^k\\p_k \\psi + m\\gamma^0\\psi -a\\psi-\\nabla F(\\psi)=0.\n\\end{align}\nEquations of this form have been studied in depth in literature, see e.g.~\\cite{BalabaneCazenave1988Existence, Balabane1988Existence, Balabane1990Existence, Cazenave1986Existence,EstebanSere1995stationary,Merle1988Existence} and the references quoted therein.\nDifferent from~\\cite{EstebanSere1995stationary}, here we are concerned with periodic solutions, namely solutions satisfying\n\\begin{align}\n \\psi(\\theta+\\tau)=\\psi(\\theta), \\quad \\forall \\tau\\in \\Gamma\n\\end{align}\nwhere~$\\Gamma\\subset \\mathbb{R}^3$ is a 3-dimensional lattice.\nThey are referred as the standing periodic waves for~\\eqref{eq:4D nonlinear Dirac}.\n\nIn quantum fields theory it is often required that the Hamiltonian has some symmetry, and the periodicity is thus frequently imposed. \nIn one-dimensional dynamics this is usually the case. \nAccording to the features of the particle system, sometimes it is require that the nonlinearity has spatial periodicity, which is not the case here since our~$F$ has no explicit dependence on the space variable~$x$. \nHere we are in a similar setting as in~\\cite{DingLiu2014periodic}, namely we seek for periodic solutions for the nonlinear Dirac equation~\\eqref{eq:NDE-spatial}. \nThe difference to ~\\cite{DingLiu2014periodic} lies in the nonlinearity, which will become clear later.\n\nGeometrically, this amounts to consider the equation~\\eqref{eq:NDE-spatial} on the compact manifold~$\\mathbb{T}^3=\\faktor{\\mathbb{R}^3}{\\Gamma}$.\nThe spectral properties for the linearized operators on torus is in great contrast to the one on~$\\mathbb{R}^3$, so is the analysis needed for the variational scheme. \nFor example, we cannot appeal to the Fourier space to obtain enough linking structures, and we do not have a global Pohozaev identity to control the kinetic energy and the potential energy separately, which are crucial for the proof in~\\cite{EstebanSere1995stationary} to obtain solutions on~$\\mathbb{R}^3$. \nOn the other hand, the torus is still good enough to allow for variational analysis and we can changle local strategies to carry out the variational method in~\\cite{EstebanSere1995stationary} to obtain nontrivial solutions.\n\nThe main motivation of the nonlinearities comes from the following examples. \nIn the Soler model~\\cite{Soler1970Classical}, the self-coupling has the form \n\\begin{align}\\label{eq:F Soler}\n F(\\psi)\n =\\frac{1}{2}G(\\bar{\\psi}\\psi)\n\\end{align}\nwith~$G\\in C^2(\\mathbb{R},\\R_+)$, and~$G(s)=0$ for~$s\\leq 0$.\nTo solve the corresponding nonlinear Dirac equation on~$\\mathbb{R}^3$, a suitable “radial” Ansatz were used to reduce the equation to ODEs and then the shooting method can provide some solutions. \nThese methods cannot be applied in the periodic setting for the lack of a good radail Ansatz. \nA more general form of~$F$ arise in~\\cite{Finkelstein1951Non-linea} where the physicists were studying the symmetric coupling between nucleons, muons, and leptons, for which \n\\begin{align}\\label{eq:nonlinearity Finkelstein}\n F(\\psi) = \\frac{1}{2}|\\bar{\\psi}\\psi|^2 + b|\\bar{\\psi}\\gamma^5\\psi|^2\n\\end{align}\nwhere~$\\gamma^5= \\gamma^0\\gamma^1\\gamma^2\\gamma^3$.\nIn such models, the existence theory remains still largely open. \nIndeed, for~\\eqref{eq:nonlinearity Finkelstein}, it has growth order~$|\\psi|^4$ which is super-critical for the Sobolev embeddings, thus not appropriate for the variational scheme.\n\n\\item[(F5)] $\\exists\\quad\\beta > 3$, $\\forall\\quad\\delta > 0$, $\\exists \\quad C_\\delta > 0$, $\\forall \\psi \\in \\mathbb{C}^4$, such that\n\\begin{align}\n |\\nabla F(\\psi)| \\leq \\left( \\delta + C_\\delta F(\\psi)^{\\frac{1}{\\beta}} \\right) |\\psi|.\n\\end{align}\n\\end{itemize}\nThe practical nonlinearity that we aim to model include the Soler nonlinearity in~\\eqref{eq:F Soler} as well as \n\\begin{align}\n F(\\psi)= |\\bar{\\psi}\\psi|^\\alpha+ b|\\bar{\\psi}\\gamma^5\\psi|^\\beta,\n\\end{align}\nwhere~$\\gamma^5=\\gamma^0\\gamma^1\\gamma^2\\gamma^3$ and~$b\\geq 0$,~$1<\\alpha,\\beta<\\frac{3}{2}$.\n\nThe main result of this work is the following.\n\n\\begin{align}\\label{eq:4D nonlinear Dirac}\n    \\sum_{\\mu=0}^{3} i\\gamma^\\mu\\p_\\mu\\Psi-m \\Psi + \\gamma^0 \\nabla F(\\Psi)=0\n\\end{align}\n\n\\begin{align}\\label{eq:F Soler}\n    F(\\psi)\n    =\\frac{1}{2}G(\\bar{\\psi}\\psi)\n\\end{align}\n\n\\begin{align}\\label{eq:NDE-spatial}\\tag{*}\n    -i\\gamma^0 \\gamma^k\\p_k \\psi + m\\gamma^0\\psi -a\\psi-\\nabla F(\\psi)=0.\n\\end{align}\n\n\\begin{align}\\label{eq:nonlinearity Finkelstein}\n    F(\\psi) = \\frac{1}{2}|\\bar{\\psi}\\psi|^2 + b|\\bar{\\psi}\\gamma^5\\psi|^2\n\\end{align}\n\nAn typical ansatz for the solutions are the so-called \\emph{stationary solutions} or~\\emph{solitary solutions} (see e.g.~\\cite{EstebanSere1995stationary}) which takes the form\n\\begin{align}\n \\Psi(t,\\vec{x})= e^{-i a t}\\psi(\\vec{x})\n\\end{align}\nSuch solutions are regarded as the “particle-like solutions” and hence attract much attention. \nFor spinors of such form and for~$F$ invariant under rotations ( i.e. ~$F(e^{i\\theta}\\Psi)=F(\\Psi)$),~\\eqref{eq:4D nonlinear Dirac} is transformed into\n\\begin{align}\n i\\gamma^k\\p_k\\psi + a\\gamma^0\\psi-m\\psi +\\gamma^0\\nabla F(\\psi)=0.\n\\end{align}\nEquivalently, we are led to consider the equation\n\\begin{align}\\label{eq:NDE-spatial}\\tag{*}\n -i\\gamma^0 \\gamma^k\\p_k \\psi + m\\gamma^0\\psi -a\\psi-\\nabla F(\\psi)=0.\n\\end{align}\nEquations of this form have been studied in depth in literature, see e.g.~\\cite{BalabaneCazenave1988Existence, Balabane1988Existence, Balabane1990Existence, Cazenave1986Existence,EstebanSere1995stationary,Merle1988Existence} and the references quoted therein.\nDifferent from~\\cite{EstebanSere1995stationary}, here we are concerned with periodic solutions, namely solutions satisfying\n\\begin{align}\n \\psi(\\theta+\\tau)=\\psi(\\theta), \\quad \\forall \\tau\\in \\Gamma\n\\end{align}\nwhere~$\\Gamma\\subset \\R^3$ is a 3-dimensional lattice.\nThey are referred as the standing periodic waves for~\\eqref{eq:4D nonlinear Dirac}.\n\nIn quantum fields theory it is often required that the Hamiltonian has some symmetry, and the periodicity is thus frequently imposed. \nIn one-dimensional dynamics this is usually the case. \nAccording to the features of the particle system, sometimes it is require that the nonlinearity has spatial periodicity, which is not the case here since our~$F$ has no explicit dependence on the space variable~$x$. \nHere we are in a similar setting as in~\\cite{DingLiu2014periodic}, namely we seek for periodic solutions for the nonlinear Dirac equation~\\eqref{eq:NDE-spatial}. \nThe difference to ~\\cite{DingLiu2014periodic} lies in the nonlinearity, which will become clear later.\n\nGeometrically, this amounts to consider the equation~\\eqref{eq:NDE-spatial} on the compact manifold~$\\mathbb{T}^3=\\faktor{\\R^3}{\\Gamma}$.\nThe spectral properties for the linearized operators on torus is in great contrast to the one on~$\\R^3$, so is the analysis needed for the variational scheme. \nFor example, we cannot appeal to the Fourier space to obtain enough linking structures, and we do not have a global Pohozaev identity to control the kinetic energy and the potential energy separately, which are crucial for the proof in~\\cite{EstebanSere1995stationary} to obtain solutions on~$\\R^3$. \nOn the other hand, the torus is still good enough to allow for variational analysis and we can changle local strategies to carry out the variational method in~\\cite{EstebanSere1995stationary} to obtain nontrivial solutions.\n\n\\item[(F5)] $\\exists\\quad\\beta > 3$, $\\forall\\quad\\delta > 0$, $\\exists \\quad C_\\delta > 0$, $\\forall \\psi \\in \\mathbb{C}^4$, such that\n\\begin{align}\n |\\nabla F(\\psi)| \\leq \\left( \\delta + C_\\delta F(\\psi)^{\\frac{1}{\\beta}} \\right) |\\psi|.\n\\end{align}\n\\end{itemize}\nThe practical nonlinearity that we aim to model include the Soler nonlinearity in~\\eqref{eq:F Soler} as well as \n\\begin{align}\n F(\\psi)= |\\bar{\\psi}\\psi|^\\alpha+ b|\\bar{\\psi}\\gamma^5\\psi|^\\beta,\n\\end{align}\nwhere~$\\gamma^5=\\gamma^0\\gamma^1\\gamma^2\\gamma^3$ and~$b\\geq 0$,~$1<\\alpha,\\beta<\\frac{3}{2}$.\n\nWe outline the proof here. \nThe equation~\\eqref{eq:NDE-spatial} admits a variational structure, namely it is the Euler--Lagrange equation of an action functional~$J$, see Section~\\ref{sect:variational str}. \nUnfortunately this functional~$J$ doesn't satisfy the Palais-Smale condition. \nAs a remedy we introduce a coercive perturbation multiplied by a small~$\\eps\\in (0,1]$ in Section~\\ref{sect:perturbation}. \nThen we show that the perturbed functional meets the Palais-Smale condition, although the~$(PS)_c$ sequences have a bound depending on~$\\frac{1}{\\eps}$. \nCrucially using the spectral information of the Dirac operator, we will see that these perturbed functionals admits a linking structure in levels, which turn out to be uniform, i.e. with level estimates being uniform in~$\\eps$. \nThough the linking sets are both infinite dimensional, we use the negative gradient flow of the functional together with the Leray-Schauder degree to find a nontrivial linking.\nThus we obtain min-max solutions for the perturbed equations in the end of Section~\\ref{sect:perturbation}. \nBy using the special structure of~$F$ and the perturbation term, we employ a contradiction argument to get uniform estimates for the perturbed solutions in Section~\\ref{sect:uniform estimate}; the triviality of the kernel of the geometric Dirac operator on the three-sphere is needed. \nThis allows to pass along a subsequence to a nontrivial limit which is the desired nontrivial periodic solution, see Section~\\ref{sect:last}.\n\n\\begin{prop}\\label{prop:external}\n Assume that (F1-5) and~\\eqref{eq:small oscillation} hold true, then~\\eqref{eq:NLD with external field} admits a nontrivial periodic solution which is~$C^1$. \n\\end{prop}\nNote that this differs from the results in~\\cite{DingRuf2008Solutions} not only in the non-coercivity of the nonlinearity, but also in the growth order of the nonlinearity. \nThey dealt essential with the quadratic growth case, and we are concerned with the super-quadratic case, hence the results are not overlapping. \nFurthermore, the multiplicity results are still missing in our case. \nWe will comment on this later, see Remark~\\ref{rmk:multiplicity}.\n\nThe equation~\\eqref{eq:NDE-spatial} is the Euler-Lagrange equation of the functional\n\\begin{align}\n J\\colon H^{\\frac{1}{2}}(M,\\C^4)\\to \\R\n\\end{align}\ngiven by\n\\begin{align}\n J(\\psi)\n =\\int_M \\frac{1}{2}\\Abracket{\\psi,\\pD\\psi} -\\frac{a}{2}|\\psi|^2 - F(\\psi)\\dv.\n\\end{align}\nRecall that the nonlinearity~$F$ satisfies the hypotheses (F1-5).\nIn particular,~$F(\\psi)$ fails to be coercive in~$\\psi$ in general.\nActually, in the model example,~~$F(\\psi)=\\frac{1}{2}G(\\bar{\\psi}\\psi)$, \nusing the notation from~\\eqref{eq:up-low}, we have\n\\begin{align}\n \\bar{\\psi}\\psi = \\Abracket{\\gamma^0\\psi,\\psi}= |\\psi_{\\up}|^2 - |\\psi_{\\low}|^2, & &\n |\\psi|^2 = |\\psi_{\\up}|^2 + |\\psi_{\\low}|^2 = \\sum_{k=1}^4 |\\psi^k|^2.\n\\end{align}\nNote that~$\\bar{\\psi}\\psi$ may be small, while~$|\\psi|^2$ is large.\nThis kind of nonlinearity frequently arises in various particle models in quantum field theory, which is more complicated than the nonlinearities of the form~$G(|\\psi|^2)$.\nThe approach we take here is motivated from~\\cite{Bartsch1999nonlinear, EstebanSere1995stationary, HoferWysocki1990first,Sere1995homoclinic}.\nAside from the strongly indefinite nature of Dirac operators, there are new challenges in the periodic setting, such as the construction of the min-max structure, and the loss of radial solutions.\nMethodologically, the powerful Fourier transform on~$\\R^3$, which provides rich variational information in~\\cite{EstebanSere1995stationary}, is no longer available.\nThough one can still appeal to the Fourier series expansion of the spinors on~$\\mathbb{T}^3$, we struggle to find a suitable linking structure for this problem which relies heavily on the spectral analysis of~$\\pD$.\nWe aim to fix these issues and look for new variational periodic solutions.\n\nNow we see that for each~$\\eps\\in (0,1]$ the perturbed functional~$J_\\eps$ admits a min-max critical point~$\\psi_\\eps \\in H^{\\frac{1}{2}}$ such that \n\\begin{align}\n J_\\eps(\\psi_\\eps) \\in (c_1, c_2), & & \n \\|\\psi_\\eps\\|_{H^1}\\leq C<+\\infty. \n\\end{align}\nUp to a subsequence, we assume that~$\\psi_\\eps\\to \\psi_0$ weakly in~$H^1$ and strongly in~$H^{\\frac{1}{2}}\\cap L^6$.\nIt follows that the limit spinor~$\\psi_0$ solves the equation~\\eqref{eq:NDE-spatial} in~$H^{-\\frac{1}{2}}$, and it is nontrivial since\n\\begin{align}\n J(\\psi_0)=\\lim_{\\eps\\to 0^+} J_\\eps(\\psi_\\eps) \\geq c_1 >0. \n\\end{align}\n This is the desired nontrivial solution.", "post_theorem_intro_text_len": 3021, "post_theorem_intro_text": "The regularity of the solution can be improved if~$F$ is smooth. \n\nWe outline the proof here. \nThe equation~\\eqref{eq:NDE-spatial} admits a variational structure, namely it is the Euler--Lagrange equation of an action functional~$J$, see Section~\\ref{sect:variational str}. \nUnfortunately this functional~$J$ doesn't satisfy the Palais-Smale condition. \nAs a remedy we introduce a coercive perturbation multiplied by a small~$\\varepsilon\\in (0,1]$ in Section~\\ref{sect:perturbation}. \nThen we show that the perturbed functional meets the Palais-Smale condition, although the~$(PS)_c$ sequences have a bound depending on~$\\frac{1}{\\varepsilon}$. \nCrucially using the spectral information of the Dirac operator, we will see that these perturbed functionals admits a linking structure in levels, which turn out to be uniform, i.e. with level estimates being uniform in~$\\varepsilon$. \nThough the linking sets are both infinite dimensional, we use the negative gradient flow of the functional together with the Leray-Schauder degree to find a nontrivial linking.\nThus we obtain min-max solutions for the perturbed equations in the end of Section~\\ref{sect:perturbation}. \nBy using the special structure of~$F$ and the perturbation term, we employ a contradiction argument to get uniform estimates for the perturbed solutions in Section~\\ref{sect:uniform estimate}; the triviality of the kernel of the geometric Dirac operator on the three-sphere is needed. \nThis allows to pass along a subsequence to a nontrivial limit which is the desired nontrivial periodic solution, see Section~\\ref{sect:last}. \n\nWe also remark that, when there is an external field, the following equation is also considered:\n\\begin{align}\\label{eq:NLD with external field}\n    -i\\gamma^0\\gamma^k\\p_k\\psi + m\\gamma^0\\psi -a\\psi - M\\psi -\\nabla F(x, \\psi)=0,\n\\end{align}\nwhere~$M$ stands for the external field, see~\\cite{DingRuf2008Solutions}, which can be either vector or scalar.\nIf~$M(x)$ is a~$C^2$ continuous and positive, and does not oscillate much in the sense that \n\\begin{align}\\label{eq:small oscillation}\n    0< a \\leq a + M < m, \n\\end{align}\nthen the argument carries over to this case, and we also obtain a nontrivial periodic~$C^1$ solution. \n\n\\begin{prop}\\label{prop:external}\n    Assume that (F1-5) and~\\eqref{eq:small oscillation} hold true, then~\\eqref{eq:NLD with external field} admits a nontrivial periodic solution which is~$C^1$. \n\\end{prop}\nNote that this differs from the results in~\\cite{DingRuf2008Solutions} not only in the non-coercivity of the nonlinearity, but also in the growth order of the nonlinearity. \nThey dealt essential with the quadratic growth case, and we are concerned with the super-quadratic case, hence the results are not overlapping. \nFurthermore, the multiplicity results are still missing in our case. \nWe will comment on this later, see Remark~\\ref{rmk:multiplicity}. \n\n\\\n\n\\noindent{\\bf Acknowledgement.} We would like to thank Nadine Grosse and Meng Long for helpful conversation on these problems.", "sketch": "We outline the proof here. The equation~\\eqref{eq:NDE-spatial} admits a variational structure, namely it is the Euler--Lagrange equation of an action functional~$J$ (see Section~\\ref{sect:variational str}), but $J$ doesn't satisfy the Palais-Smale condition. As a remedy we introduce a coercive perturbation multiplied by a small~$\\varepsilon\\in (0,1]$ (Section~\\ref{sect:perturbation}) and show the perturbed functional meets the Palais--Smale condition, although the $(PS)_c$ sequences have a bound depending on~$\\frac{1}{\\varepsilon}$. Using spectral information of the Dirac operator, the perturbed functionals admit a linking structure in levels with estimates uniform in~$\\varepsilon$; although the linking sets are infinite dimensional, one uses the negative gradient flow together with the Leray--Schauder degree to find a nontrivial linking, yielding min-max solutions for the perturbed equations (end of Section~\\ref{sect:perturbation}). Then, using the special structure of~$F$ and the perturbation term, a contradiction argument gives uniform estimates for the perturbed solutions (Section~\\ref{sect:uniform estimate}); here the triviality of the kernel of the geometric Dirac operator on the three-sphere is needed. Finally, one passes along a subsequence to a nontrivial limit which is the desired nontrivial periodic solution (Section~\\ref{sect:last}).", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "Assume that (F1-5) hold true. \nThen there exists a nontrivial periodic solution for~\\begin{align}\\label{eq:NDE-spatial}\\tag{*}\n    -i\\gamma^0 \\gamma^k\\p_k \\psi + m\\gamma^0\\psi -a\\psi-\\nabla F(\\psi)=0.\n\\end{align}, which is in~$C^1$.", "theorem_type": ["Existence"], "mcq": {"question": "Let \\(\\Gamma\\subset \\mathbb{R}^3\\) be a 3-dimensional lattice, and call a spinor \\(\\psi:\\mathbb{R}^3\\to\\mathbb{C}^4\\) periodic if \\(\\psi(x+\\tau)=\\psi(x)\\) for every \\(\\tau\\in\\Gamma\\) (equivalently, \\(\\psi\\) is defined on the torus \\(\\mathbb{T}^3=\\mathbb{R}^3/\\Gamma\\)). Assume the nonlinearity \\(F:\\mathbb{C}^4\\to\\mathbb{R}\\) satisfies hypotheses \\((F1)\\)–\\((F5)\\), and consider the nonlinear Dirac equation\n\\[\n-i\\gamma^0\\gamma^k\\partial_k\\psi + m\\gamma^0\\psi - a\\psi - \\nabla F(\\psi)=0\n\\]\n(with summation over \\(k=1,2,3\\)). Here \\(\\gamma^\\mu\\) are the Dirac matrices. Which existence statement holds under these assumptions?", "correct_choice": {"label": "A", "text": "There exists a nontrivial \\(\\Gamma\\)-periodic solution \\(\\psi\\in C^1(\\mathbb{R}^3,\\mathbb{C}^4)\\) (equivalently, a nonzero \\(C^1\\) spinor on \\(\\mathbb{T}^3\\)) such that\n\\[\n-i\\gamma^0\\gamma^k\\partial_k\\psi + m\\gamma^0\\psi - a\\psi - \\nabla F(\\psi)=0.\n\\]"}, "choices": [{"label": "B", "text": "There exists a nontrivial \\(\\Gamma\\)-periodic solution \\(\\psi\\in C^1(\\mathbb{R}^3,\\mathbb{C}^4)\\) such that\n\\[\n-i\\gamma^0\\gamma^k\\partial_k\\psi + m\\gamma^0\\psi - a\\psi - \\nabla F(\\psi)=0,\n\\]\nand, moreover, this periodic solution exists for every choice of the parameter \\(a\\in\\mathbb{R}\\)."}, {"label": "C", "text": "There exists a nontrivial \\(\\Gamma\\)-periodic weak solution \\(\\psi\\) of\n\\[\n-i\\gamma^0\\gamma^k\\partial_k\\psi + m\\gamma^0\\psi - a\\psi - \\nabla F(\\psi)=0\n\\]\non \\(\\mathbb{T}^3\\)."}, {"label": "D", "text": "For every 3-dimensional lattice \\(\\Gamma\\subset\\mathbb{R}^3\\), there exists a nontrivial \\(\\Gamma\\)-periodic solution \\(\\psi\\in C^1(\\mathbb{R}^3,\\mathbb{C}^4)\\) such that\n\\[\n-i\\gamma^0\\gamma^k\\partial_k\\psi + m\\gamma^0\\psi - a\\psi - \\nabla F(\\psi)=0.\n\\]"}, {"label": "E", "text": "There exists a nontrivial \\(\\Gamma\\)-periodic solution \\(\\psi\\in C^1(\\mathbb{R}^3,\\mathbb{C}^4)\\) such that\n\\[\n-i\\gamma^0\\gamma^k\\partial_k\\psi + m\\gamma^0\\psi - a\\psi + \\nabla F(\\psi)=0.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "dependence_on_fixed_parameter_a", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped_C1_regularity_of_the_solution", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "fixed_lattice_vs_all_lattices", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "sign_of_nonlinear_term", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state or strongly hint at the exact conclusion; it only gives the assumptions and asks which existence claim is valid."}, "TAS": {"score": 0, "justification": "The correct option is essentially the theorem statement itself: existence of a nontrivial Γ-periodic C1 solution under (F1)–(F5). This makes the item largely a recall/restatement question."}, "GPS": {"score": 1, "justification": "There is some reasoning in comparing stronger, weaker, and tampered variants (quantifiers, regularity, sign), but the item mainly tests recognition of the exact theorem rather than deriving the conclusion from mathematical structure."}, "DQS": {"score": 2, "justification": "The distractors are plausible and distinct: overstrong quantification in a or Γ, a weaker regularity conclusion, and a sign error in the equation. These align with common theorem-recall and statement-parsing mistakes."}, "total_score": 5, "overall_assessment": "Low answer leakage and strong distractors, but the question is mostly a theorem-statement recognition item rather than a genuinely generative reasoning task."}}
{"id": "2602.10519v1", "paper_link": "http://arxiv.org/abs/2602.10519v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{main blocks}\nThe category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$ decomposes into blocks which are either trivial\n(that is equivalent to the category of vector spaces) or of type $\\Gamma$. In particular,\nthe category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$ is of tame representation type. Also the cohomology of\n$\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$ (that is the Ext algebra of the unit object ${\\bf 1}$) is isomorphic to the algebra\nof invariants $S^\\bullet(V)^\\Gamma$.", "start_pos": 6618, "end_pos": 7062, "label": "main blocks"}, "ref_dict": {"main blocks": "\\begin{theorem}\\label{main blocks}\nThe category $\\C(\\g,e_{sr},l,q)$ decomposes into blocks which are either trivial\n(that is equivalent to the category of vector spaces) or of type $\\Gamma$. In particular,\nthe category $\\C(\\g,e_{sr},l,q)$ is of tame representation type. Also the cohomology of\n$\\C(\\g,e_{sr},l,q)$ (that is the Ext algebra of the unit object $\\be$) is isomorphic to the algebra\nof invariants $S^\\bullet(V)^\\Gamma$.\n\\end{theorem}", "main 7mod": "\\begin{theorem}\\label{main 7mod}\n(1) The category $\\bar \\C(G_2,G_2(a_1),7,q)$ has 12 trivial blocks and one block of type $\\tilde D_4$.\nIn particular, it has 17 simple objects.\n\n(2) We have $\\FP(\\bar \\C(G_2,G_2(a_1),7,q))=49(7+15[3]_7+12[5]_7)\\approx 3054.068811$.\n\n(3) The category $\\bar \\C(G_2,G_2(a_1),7,q)$ has stable Chevalley property.\n\n(4) The category $\\bar \\C(G_2,G_2(a_1),7,q)$ is completely anisotropic: it has no non-trivial\ncommutative exact algebras.\n\\end{theorem}", "Witt": "\\label{Witt}\nIn this section $\\K$ is an algebraically closed field of arbitrary characteristic.\n\\subsection{Exact algebras} Let $\\C$ be a finite tensor category over $\\K$ and let $A=(A,m,u)\\in \\C$ be", "main 7": "\\begin{theorem}\\label{main 7}\n(1) The category $\\C(G_2,G_2(a_1),7,q)$ has 15 trivial blocks and one block of type $\\tilde E_7$.\nIn particular it has 23 simple objects.\n\n(2) We have $\\FP(\\C(G_2,G_2(a_1),7,q))=294(7+15[3]_7+12[5]_7)\\approx 18324.416384$.\n\n(3) The category $\\C(G_2,G_2(a_1),7,q)$ has stable Chevalley property: tensor products of simple \nobjects are direct sums of simples and projectives.\n\n(4) The M\\\"uger center of the category $\\C(G_2,G_2(a_1),7,q)$ is equivalent to $\\Rep(S_3)$\n(where $S_3$ is the symmetric group on three letters).\n\\end{theorem}", "genconj": "\\label{genconj}\n\\subsection{Categories $\\C(G_2,G_2(a_1),l,q)$}\nWe expect that Theorem \\ref{main 7} extends to the case of arbitrary undivisible (i.e. not\ndivisible by 3) $l\\ge 7$.\n\n\\begin{conjecture}\\", "divisible": "\\begin{definition} \\label{divisible}\nWe say that $l\\in \\Z_{\\ge 1}$ is \\textit{divisible} if $l$ is divisible\nby $m$ and $l$ is \\textit{undivisible} otherwise.     \\end{definition}", "distconj": "\\begin{conjecture}\\label{g2conjm}\n    Assume $l\\ge 12$ and $l$ is divisible by 3. Then\n\n    $$\\FP(\\C(G_2, G_2(a_1),l,q))=\\frac29\\frac{l^4}{(2\\sin(\\pi/l))^4(2\\sin(2\\pi/l))^2(2\\sin(3\\pi/l))^4}.$$\nand for $l\\ge 15$\n$$\\FP(T(\\omega_1))=[3]_l+[5]_l-1,\\; \\FP(T(\\omega_2))=2[7]_l-[5]_l+[3]_l+2.$$\n\\end{conjecture}\n\nWe checked that the formulas work for $l=12,15,18,21$; this applies also for $l=9$ if we\ndefine category $\\C(G_2, G_2(a_1),9,q)$ to be a semisimplification of $\\T(\\g,q)$\n(for $q$ such that $q^2$ has order 9), see \\cite{RW}. The formula for $\\FP(T(\\omega_2))$ gives\nan incorrect result for $l=12$, namely the right hand side and the left hand side\ndiffer exactly by 1. This is likely to be explained by the fact that $T(\\omega_2)$\ndoes not coincide with Weyl module in this (and only in this) case.\n\n\\subsection{Categories $\\C(\\g,e,l,q)$ for distinguished nilpotent $e\\in \\g$}\\label{distconj}\nLet $Q=Q(e)$ be the centralizer of a $sl_2-$triple associated with $e$ in the simply\nconnected group $G$ with $\\mbox{Lie}(G)=\\g$. In the case of distinguished nilpotent $e$,\n$Q$ is a finite group. Let $2k_i+1, i\\in M=M(e)$ be the sizes of Jordan cells for\nthe adjoint action of $e$ on $\\g$ (it is well known that these sizes are odd for distinguished\nnilpotent elements; the number of these cells, that is the cardinality of set $M$, is\nthe dimension of the centralizer of $e$ in $\\g$). For any $k\\in \\Z_{>0}$ we set\n$$S_k(l):=\\frac{l}{(2^k\\sin(\\pi/l)\\sin(2\\pi/l)\\cdots \\sin(k\\pi/l))^2}.$$\n\n\\begin{conjecture}\\label{FPdim}\n    Assume $m=1$ or $l$ is undivisible. Then\n    $$\\FP(\\C(\\g,e,l,q))=|Q|\\prod_{i=1}^MS_{k_i}(l).$$\n\\end{conjecture}"}, "pre_theorem_intro_text_len": 4606, "pre_theorem_intro_text": "\\subsection{}\nThis paper is a contribution to the theory of braided finite tensor categories. In the case\nof semisimple categories over $\\mathbb{C}$ many known examples arise from Wess-Zumino-Witten\nmodels in conformal field theory, see e.g. \\cite{BK}. One of the easiest algebraic constructions of these categories was given\nby H.~H.~Andersen \\cite{An} (see also \\cite{Saw}). In modern terms this construction can be summarized\nas follows: for a simple Lie algebra $\\mathfrak{g}$ and a root of unity $q$ such that $l$ is the order of $q^2$ one considers\nthe category $\\mathcal{T}(\\mathfrak{g},q)$ of {\\em tilting modules} over the quantum group at a root of unity $q$ associated \nwith Lie algebra $\\mathfrak{g}$. Then one defines a category $\\mathcal{C}(\\mathfrak{g}, l, q)$ as the {\\em semisimplification}\nof   $\\mathcal{T}(\\mathfrak{g},q)$ (see e.g. \\cite{EOsemi}). The category $\\mathcal{C}(\\mathfrak{g}, l, q)$ is a semisimple braided tensor\ncategory; moreover this category is finite (i.e. it has only finitely many classes of simple objects) if\n$l$ is sufficiently large. The precise bounds for $l$ are given in \\cite[Figure 2]{Scho}; in this paper \nwe will always assume that $l$ is sufficiently large in this sense. Note that there is a combinatorial \ndifference between the case when $l$ is {\\em divisible} (see \\ref{divisible}) and when it is not. \n\nThe procedure of semisimplification above can be described as taking the quotient by a suitable\ntensor ideal (namely, by the ideal of negligible morphisms). However, the category $\\mathcal{T}(\\mathfrak{g},q)$ has \nmany other tensor ideals. In \\cite{CEO} for any {\\em distinguished} nilpotent element $e\\in \\mathfrak{g}$ (or, in the case\nwhen $l$ is divisible, $e\\in \\mathfrak{g}^L$ where $\\mathfrak{g}^L$ is the Langlands dual Lie algebra of $\\mathfrak{g}$),\na tensor ideal $\\cI_e$ was constructed such that the quotient category $\\mathcal{T}(\\mathfrak{g},q)/\\cI_e$ admits a {\\em monoidal\nabelian envelope} $\\mathcal{C}(\\mathfrak{g},e,l,q)$; moreover, the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ is a finite tensor category\nin the sense of \\cite{EOfinite}. For example when $e$ is a regular nilpotent element (which is always\ndistinguished), the ideal $\\cI_e$ is the ideal of negligible morphisms and $\\mathcal{C}(\\mathfrak{g},e,l,q)=\\mathcal{C}(\\mathfrak{g},l,q)$.\nUnfortunately, there is not much we can say about the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ for other nilpotent\nelements $e$ (however, see Section \\ref{distconj} for a conjectural formula for the Frobenius-Perron dimension\nof $\\mathcal{C}(\\mathfrak{g},e,l,q)$ and conjectural description of its cohomology). The goal of this paper is to give \nsome explicit information about the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ in the case when $e=e_{sr}$ is a subregular\nnilpotent element, see \\cite[4.2]{CMg}. Recall that $e_{sr}$ is distinguished if and only if $\\mathfrak{g}$ is not of\ntypes $A_n, n\\ge 1$ or $B_n, n\\ge 2$, see \\cite{CMg}.\n\nTo state our main results we need a bit more notation. Let $V$ be a two dimensional space and let\n$\\Gamma \\subset SL(V)$ be a finite subgroup of even order. By the classical McKay correspondence,\nthe finite subgroups of $SL(V)$ up to conjugacy are labelled by simply laced affine Dynkin diagrams (the only\nsubgroups of odd order correspond to affine Dynkin diagrams of type $\\tilde A_{2n}$). We associate\na subgroup $\\Gamma$ to $\\mathfrak{g}$ and $l$ as above as follows:\n\\renewcommand{\\arraystretch}{1.4}\n$$\n\\begin{array}{|c|c|c|c|c|c|c|}\n\\hline\n\\mathfrak{g}&B_n&C_n&D_n&E_n&F_4&G_2\\\\\n&n\\ge 3&n\\ge 3&n\\ge 4&n=6,7,8&&\\\\\n\\hline\nl&\\mbox{divisible}&\\mbox{not divisible}&\\mbox{any}&\\mbox{any}&\\mbox{any}&\\mbox{any}\\\\\n\\hline\n\\Gamma &\\tilde D_{2n} &\\tilde D_{2n}&\\tilde D_n&\\tilde E_n&\\tilde E_7&\\tilde E_7\\\\\n\\hline\n\\end{array}\n$$\n\nLet $\\wedge(V)$ be the exterior algebra of $V$. We can consider $\\wedge(V)$ as an algebra in the category\n$\\mbox{Rep}(\\Gamma)$ of finite dimensional representations of $\\Gamma$. We consider the abelian\ncategory of left $\\wedge(V)-$modules in the category $\\mbox{Rep}(\\Gamma)$ and we call it\n{\\em block of type} $\\Gamma$. Thus, a block of type $\\Gamma$ is the category equivalent to that of finite dimensional\nrepresentations of the cross product of $\\wedge(V)$ with the group algebra of $\\Gamma$.\nFinally let $S^\\bullet(V)$ be the symmetric algebra of $V$ which is graded by even integers (so $V \\subset\nS^\\bullet(V)$ is in degree 2). The group $\\Gamma$ acts on $S^\\bullet(V)$ preserving the grading.\nOur first main result describes the structure of abelian category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$.", "context": "\\subsection{}\nThis paper is a contribution to the theory of braided finite tensor categories. In the case\nof semisimple categories over $\\mathbb{C}$ many known examples arise from Wess-Zumino-Witten\nmodels in conformal field theory, see e.g. \\cite{BK}. One of the easiest algebraic constructions of these categories was given\nby H.~H.~Andersen \\cite{An} (see also \\cite{Saw}). In modern terms this construction can be summarized\nas follows: for a simple Lie algebra $\\mathfrak{g}$ and a root of unity $q$ such that $l$ is the order of $q^2$ one considers\nthe category $\\mathcal{T}(\\mathfrak{g},q)$ of {\\em tilting modules} over the quantum group at a root of unity $q$ associated \nwith Lie algebra $\\mathfrak{g}$. Then one defines a category $\\mathcal{C}(\\mathfrak{g}, l, q)$ as the {\\em semisimplification}\nof $\\mathcal{T}(\\mathfrak{g},q)$ (see e.g. \\cite{EOsemi}). The category $\\mathcal{C}(\\mathfrak{g}, l, q)$ is a semisimple braided tensor\ncategory; moreover this category is finite (i.e. it has only finitely many classes of simple objects) if\n$l$ is sufficiently large. The precise bounds for $l$ are given in \\cite[Figure 2]{Scho}; in this paper \nwe will always assume that $l$ is sufficiently large in this sense. Note that there is a combinatorial \ndifference between the case when $l$ is {\\em divisible} (see \\ref{divisible}) and when it is not.\n\nThe procedure of semisimplification above can be described as taking the quotient by a suitable\ntensor ideal (namely, by the ideal of negligible morphisms). However, the category $\\mathcal{T}(\\mathfrak{g},q)$ has \nmany other tensor ideals. In \\cite{CEO} for any {\\em distinguished} nilpotent element $e\\in \\mathfrak{g}$ (or, in the case\nwhen $l$ is divisible, $e\\in \\mathfrak{g}^L$ where $\\mathfrak{g}^L$ is the Langlands dual Lie algebra of $\\mathfrak{g}$),\na tensor ideal $\\cI_e$ was constructed such that the quotient category $\\mathcal{T}(\\mathfrak{g},q)/\\cI_e$ admits a {\\em monoidal\nabelian envelope} $\\mathcal{C}(\\mathfrak{g},e,l,q)$; moreover, the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ is a finite tensor category\nin the sense of \\cite{EOfinite}. For example when $e$ is a regular nilpotent element (which is always\ndistinguished), the ideal $\\cI_e$ is the ideal of negligible morphisms and $\\mathcal{C}(\\mathfrak{g},e,l,q)=\\mathcal{C}(\\mathfrak{g},l,q)$.\nUnfortunately, there is not much we can say about the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ for other nilpotent\nelements $e$ (however, see Section \\ref{distconj} for a conjectural formula for the Frobenius-Perron dimension\nof $\\mathcal{C}(\\mathfrak{g},e,l,q)$ and conjectural description of its cohomology). The goal of this paper is to give \nsome explicit information about the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ in the case when $e=e_{sr}$ is a subregular\nnilpotent element, see \\cite[4.2]{CMg}. Recall that $e_{sr}$ is distinguished if and only if $\\mathfrak{g}$ is not of\ntypes $A_n, n\\ge 1$ or $B_n, n\\ge 2$, see \\cite{CMg}.\n\nTo state our main results we need a bit more notation. Let $V$ be a two dimensional space and let\n$\\Gamma \\subset SL(V)$ be a finite subgroup of even order. By the classical McKay correspondence,\nthe finite subgroups of $SL(V)$ up to conjugacy are labelled by simply laced affine Dynkin diagrams (the only\nsubgroups of odd order correspond to affine Dynkin diagrams of type $\\tilde A_{2n}$). We associate\na subgroup $\\Gamma$ to $\\mathfrak{g}$ and $l$ as above as follows:\n\\renewcommand{\\arraystretch}{1.4}\n$$\n\\begin{array}{|c|c|c|c|c|c|c|}\n\\hline\n\\mathfrak{g}&B_n&C_n&D_n&E_n&F_4&G_2\\\\\n&n\\ge 3&n\\ge 3&n\\ge 4&n=6,7,8&&\\\\\n\\hline\nl&\\mbox{divisible}&\\mbox{not divisible}&\\mbox{any}&\\mbox{any}&\\mbox{any}&\\mbox{any}\\\\\n\\hline\n\\Gamma &\\tilde D_{2n} &\\tilde D_{2n}&\\tilde D_n&\\tilde E_n&\\tilde E_7&\\tilde E_7\\\\\n\\hline\n\\end{array}\n$$\n\nLet $\\wedge(V)$ be the exterior algebra of $V$. We can consider $\\wedge(V)$ as an algebra in the category\n$\\mbox{Rep}(\\Gamma)$ of finite dimensional representations of $\\Gamma$. We consider the abelian\ncategory of left $\\wedge(V)-$modules in the category $\\mbox{Rep}(\\Gamma)$ and we call it\n{\\em block of type} $\\Gamma$. Thus, a block of type $\\Gamma$ is the category equivalent to that of finite dimensional\nrepresentations of the cross product of $\\wedge(V)$ with the group algebra of $\\Gamma$.\nFinally let $S^\\bullet(V)$ be the symmetric algebra of $V$ which is graded by even integers (so $V \\subset\nS^\\bullet(V)$ is in degree 2). The group $\\Gamma$ acts on $S^\\bullet(V)$ preserving the grading.\nOur first main result describes the structure of abelian category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$.", "full_context": "\\subsection{}\nThis paper is a contribution to the theory of braided finite tensor categories. In the case\nof semisimple categories over $\\mathbb{C}$ many known examples arise from Wess-Zumino-Witten\nmodels in conformal field theory, see e.g. \\cite{BK}. One of the easiest algebraic constructions of these categories was given\nby H.~H.~Andersen \\cite{An} (see also \\cite{Saw}). In modern terms this construction can be summarized\nas follows: for a simple Lie algebra $\\mathfrak{g}$ and a root of unity $q$ such that $l$ is the order of $q^2$ one considers\nthe category $\\mathcal{T}(\\mathfrak{g},q)$ of {\\em tilting modules} over the quantum group at a root of unity $q$ associated \nwith Lie algebra $\\mathfrak{g}$. Then one defines a category $\\mathcal{C}(\\mathfrak{g}, l, q)$ as the {\\em semisimplification}\nof $\\mathcal{T}(\\mathfrak{g},q)$ (see e.g. \\cite{EOsemi}). The category $\\mathcal{C}(\\mathfrak{g}, l, q)$ is a semisimple braided tensor\ncategory; moreover this category is finite (i.e. it has only finitely many classes of simple objects) if\n$l$ is sufficiently large. The precise bounds for $l$ are given in \\cite[Figure 2]{Scho}; in this paper \nwe will always assume that $l$ is sufficiently large in this sense. Note that there is a combinatorial \ndifference between the case when $l$ is {\\em divisible} (see \\ref{divisible}) and when it is not.\n\nThe procedure of semisimplification above can be described as taking the quotient by a suitable\ntensor ideal (namely, by the ideal of negligible morphisms). However, the category $\\mathcal{T}(\\mathfrak{g},q)$ has \nmany other tensor ideals. In \\cite{CEO} for any {\\em distinguished} nilpotent element $e\\in \\mathfrak{g}$ (or, in the case\nwhen $l$ is divisible, $e\\in \\mathfrak{g}^L$ where $\\mathfrak{g}^L$ is the Langlands dual Lie algebra of $\\mathfrak{g}$),\na tensor ideal $\\cI_e$ was constructed such that the quotient category $\\mathcal{T}(\\mathfrak{g},q)/\\cI_e$ admits a {\\em monoidal\nabelian envelope} $\\mathcal{C}(\\mathfrak{g},e,l,q)$; moreover, the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ is a finite tensor category\nin the sense of \\cite{EOfinite}. For example when $e$ is a regular nilpotent element (which is always\ndistinguished), the ideal $\\cI_e$ is the ideal of negligible morphisms and $\\mathcal{C}(\\mathfrak{g},e,l,q)=\\mathcal{C}(\\mathfrak{g},l,q)$.\nUnfortunately, there is not much we can say about the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ for other nilpotent\nelements $e$ (however, see Section \\ref{distconj} for a conjectural formula for the Frobenius-Perron dimension\nof $\\mathcal{C}(\\mathfrak{g},e,l,q)$ and conjectural description of its cohomology). The goal of this paper is to give \nsome explicit information about the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ in the case when $e=e_{sr}$ is a subregular\nnilpotent element, see \\cite[4.2]{CMg}. Recall that $e_{sr}$ is distinguished if and only if $\\mathfrak{g}$ is not of\ntypes $A_n, n\\ge 1$ or $B_n, n\\ge 2$, see \\cite{CMg}.\n\nTo state our main results we need a bit more notation. Let $V$ be a two dimensional space and let\n$\\Gamma \\subset SL(V)$ be a finite subgroup of even order. By the classical McKay correspondence,\nthe finite subgroups of $SL(V)$ up to conjugacy are labelled by simply laced affine Dynkin diagrams (the only\nsubgroups of odd order correspond to affine Dynkin diagrams of type $\\tilde A_{2n}$). We associate\na subgroup $\\Gamma$ to $\\mathfrak{g}$ and $l$ as above as follows:\n\\renewcommand{\\arraystretch}{1.4}\n$$\n\\begin{array}{|c|c|c|c|c|c|c|}\n\\hline\n\\mathfrak{g}&B_n&C_n&D_n&E_n&F_4&G_2\\\\\n&n\\ge 3&n\\ge 3&n\\ge 4&n=6,7,8&&\\\\\n\\hline\nl&\\mbox{divisible}&\\mbox{not divisible}&\\mbox{any}&\\mbox{any}&\\mbox{any}&\\mbox{any}\\\\\n\\hline\n\\Gamma &\\tilde D_{2n} &\\tilde D_{2n}&\\tilde D_n&\\tilde E_n&\\tilde E_7&\\tilde E_7\\\\\n\\hline\n\\end{array}\n$$\n\nLet $\\wedge(V)$ be the exterior algebra of $V$. We can consider $\\wedge(V)$ as an algebra in the category\n$\\mbox{Rep}(\\Gamma)$ of finite dimensional representations of $\\Gamma$. We consider the abelian\ncategory of left $\\wedge(V)-$modules in the category $\\mbox{Rep}(\\Gamma)$ and we call it\n{\\em block of type} $\\Gamma$. Thus, a block of type $\\Gamma$ is the category equivalent to that of finite dimensional\nrepresentations of the cross product of $\\wedge(V)$ with the group algebra of $\\Gamma$.\nFinally let $S^\\bullet(V)$ be the symmetric algebra of $V$ which is graded by even integers (so $V \\subset\nS^\\bullet(V)$ is in degree 2). The group $\\Gamma$ acts on $S^\\bullet(V)$ preserving the grading.\nOur first main result describes the structure of abelian category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$.\n\nLet $\\wedge(V)$ be the exterior algebra of $V$. We can consider $\\wedge(V)$ as an algebra in the category\n$\\mbox{Rep}(\\Gamma)$ of finite dimensional representations of $\\Gamma$. We consider the abelian\ncategory of left $\\wedge(V)-$modules in the category $\\mbox{Rep}(\\Gamma)$ and we call it\n{\\em block of type} $\\Gamma$. Thus, a block of type $\\Gamma$ is the category equivalent to that of finite dimensional\nrepresentations of the cross product of $\\wedge(V)$ with the group algebra of $\\Gamma$.\nFinally let $S^\\bullet(V)$ be the symmetric algebra of $V$ which is graded by even integers (so $V \\subset\nS^\\bullet(V)$ is in degree 2). The group $\\Gamma$ acts on $S^\\bullet(V)$ preserving the grading.\nOur first main result describes the structure of abelian category $\\C(\\g,e_{sr},l,q)$.\n\n\\begin{remark} The proof of Theorem \\ref{main blocks} shows that the number of blocks of type $\\Gamma$\nis the same as the number of weights inside of the fundamental alcove (but not on its boundary). Thus, this\nnumber is the same as the number of simple objects in the category $\\C(\\g,l,q)$.\n\\end{remark}\n\n\\begin{theorem}\\label{main 7}\n(1) The category $\\C(G_2,G_2(a_1),7,q)$ has 15 trivial blocks and one block of type $\\tilde E_7$.\nIn particular it has 23 simple objects.\n\n\\begin{theorem}\\label{main 7mod}\n(1) The category $\\bar \\C(G_2,G_2(a_1),7,q)$ has 12 trivial blocks and one block of type $\\tilde D_4$.\nIn particular, it has 17 simple objects.\n\nWe consider the category $\\Rep(\\Gamma \\ltimes V, \\epsilon)$ of\nfinite dimensional representations of $\\Gamma \\ltimes V$ where $\\epsilon$ acts as the parity automorphism,\nsee e.g. \\cite[9.11]{EGNO}. Thus, $\\Rep(\\Gamma \\ltimes V, \\epsilon)$ is a symmetric finite tensor category.\nAs an abelian category it is equivalent to the category of representations of algebra\n$\\wedge(V)$ in the category $\\Rep(\\Gamma)$ (or, equivalently, representations of $\\wedge(V)$ with an action\nof $\\Gamma$ compatible in an obvious sense). The simple objects of $\\Rep(\\Gamma \\ltimes V, \\epsilon)$ are irreducible representations of $\\Gamma$ where $V\\subset \\wedge(V)$ acts by zero;\nlet $\\Irr(\\Gamma)$ be the set of isomorphism classes of such objects.\nThe projective cover of the unit object is $\\wedge(V)$; and the projective cover of $V_i\\in \\Irr(\\Gamma)$\nis $V_i\\otimes \\wedge(V)$. In particular, the composition factors of the projective cover of $V_i$ are\n$V_i$ appearing twice and the irreducible summands of $V_i\\otimes V$. Thus, the Cartan matrix of\nthe category $\\Rep(\\Gamma \\ltimes V, \\epsilon)$ is $2\\Id+A_\\Gamma$ where $A_\\Gamma$ is the adjacency\nmatrix of the McKay graph of $\\Gamma$ (recall that the vertices of the McKay graph are elements \nof $\\Irr(\\Gamma)$ and the number of edges between $V_i$ and $V_j$ is the multiplicity of $V_j$ as \na direct summand of $V_i\\otimes V$).\n\n\\begin{corollary} \\label{cor:self-dual}\n Let $\\mathfrak g$ be of type $B_n, C_n, F_4, G_2, D_{2n}, E_7$, or $E_8$.\n Then all simple objects of $\\C=\\C(\\mathfrak g, e_{sr}, l, q)$ are self-dual. \n\\end{corollary}\n\\begin{proof}\n It is well-known that in this case all simple representations of $\\mathfrak g$ are self-dual. It follows from properties (1) and (2) in Section \\ref{qgtitling} that the tilting modules $T(\\lambda)$ are also self-dual.\n\n\\subsection{Proof of Theorem \\ref{main blocks}} By Proposition \\ref{Cartansr} the Cartan\nmatrix of the principal block of $\\C(\\g,e_{sr},l,q)$ is $2\\Id+A(X_{sr})$ where $X_{sr}$ is\nsome affine Dynkin diagram not of type $A$, so it is a tree. Thus, by Proposition \\ref{tree}\n(applied to this block and to the category $\\Rep(\\Gamma \\ltimes V)$)\nwe see that the principal block is equivalent to $\\Rep(\\Gamma \\ltimes V)$ where \n$V$ is as in \\ref{toy} and $\\Gamma \\subset SL(V)$ is a finite subgroup with McKay graph $X_{sr}$.\nUsing the translation functors (which descend to the category $\\T(\\g,q)/\\cI_{e_{sr}}$) we see\nthat all the other blocks of $\\C(\\g,e_{sr},l,q)$ involving $T(w\\cdot \\lambda)$ where $\\lambda$\nis in the interior of the fundamental alcove are equivalent to the principal block, and hence\nto the category $\\Rep(\\Gamma \\ltimes V)$. The other blocks involving $T(\\lambda)$ with $\\lambda$\non the wall are trivial by Corollaries \\ref{nog222} and \\ref{cor:on the wall}. The remaining statements of \nTheorem \\ref{main blocks} follow from Proposition \\ref{3.1}.\n\nIt is a classical result of P.~Slodowy that the singularity of the nilpotent\ncone of $\\g$ at the point $e_{sr}$ is $\\BC^2/\\Gamma$ for a suitable subgroup\n$\\Gamma \\subset SL_2(\\BC)$, see \\cite{Sl}. Comparing this with Theorem \\ref{main blocks}\nwe expect that the cohomology of the category $\\C(\\g,e,l,q)$ is related\nwith the singularity of the nilpotent cone at point $e$. Thus let $\\Sl_e$\nbe the Slodowy slice at the point $e$, see e.g. \\cite{GG}. The variety $\\Sl_e$\nis equipped with an action of $Q\\times \\BC^\\times$. Thus, the algebra of functions\n$\\mO(Sl_e)$ is graded and is equipped with a grading-preserving action of $Q$;\nlet $\\mO(Sl_e)^Q\\subset \\mO(Sl_e)$ be the algebra of invariants.\n\n\\begin{theorem}\\label{main blocks}\nThe category $\\C(\\g,e_{sr},l,q)$ decomposes into blocks which are either trivial\n(that is equivalent to the category of vector spaces) or of type $\\Gamma$. In particular,\nthe category $\\C(\\g,e_{sr},l,q)$ is of tame representation type. Also the cohomology of\n$\\C(\\g,e_{sr},l,q)$ (that is the Ext algebra of the unit object $\\be$) is isomorphic to the algebra\nof invariants $S^\\bullet(V)^\\Gamma$.\n\\end{theorem}", "post_theorem_intro_text_len": 4178, "post_theorem_intro_text": "\\begin{remark} The proof of Theorem \\ref{main blocks} shows that the number of blocks of type $\\Gamma$\nis the same as the number of weights inside of the fundamental alcove (but not on its boundary). Thus, this\nnumber is the same as the number of simple objects in the category $\\mathcal{C}(\\mathfrak{g},l,q)$.\n\\end{remark}\n\n\\subsection{} Next, we study one specific example, the category $\\mathcal{C}(G_2,G_2(a_1),7,q)$ (thus, we\nconsider Lie algebra $\\mathfrak{g}$ of type $G_2$; also $G_2(a_1)$ is the standard notation for the subregular\nnilpotent orbit in type $G_2$). This is the simplest example of the categories considered above\n(at least for undivisible $l$). Recall the standard notation for the quantum numbers:\n$$[k]_l=\\frac{\\sin(k\\pi/l)}{\\sin(\\pi/l)}.$$\nIn particular, $[2]_7=[5]_7=2\\cos(\\pi/7)\\approx 1.801938$ and $[3]_7=[4]_7=\\frac{\\sin(3\\pi/7)}{\\sin(\\pi/7)}\\approx\n2.246980$.\n\n\\begin{theorem}\\label{main 7}\n(1) The category $\\mathcal{C}(G_2,G_2(a_1),7,q)$ has 15 trivial blocks and one block of type $\\tilde E_7$.\nIn particular it has 23 simple objects.\n\n(2) We have $\\text{FPdim}(\\mathcal{C}(G_2,G_2(a_1),7,q))=294(7+15[3]_7+12[5]_7)\\approx 18324.416384$.\n\n(3) The category $\\mathcal{C}(G_2,G_2(a_1),7,q)$ has stable Chevalley property: tensor products of simple \nobjects are direct sums of simples and projectives.\n\n(4) The M\\\"uger center of the category $\\mathcal{C}(G_2,G_2(a_1),7,q)$ is equivalent to $\\text{Rep}(S_3)$\n(where $S_3$ is the symmetric group on three letters).\n\\end{theorem}\n\nIn view of Theorem \\ref{main 7} (4), it makes sense to consider the de-equivariantization $\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ \nof $\\mathcal{C}(G_2,G_2(a_1),7,q)$ with respect to its M\\\"uger center (so the category $\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ \nis the Brugui\\`eres' modularisation of $\\mathcal{C}(G_2,G_2(a_1),7,q)$, see \\cite{Brug}). The category\n$\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ is a non-semisimple modular tensor category in the sense of Shimizu,\nsee \\cite{Shim}.\n\n\\begin{theorem}\\label{main 7mod}\n(1) The category $\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ has 12 trivial blocks and one block of type $\\tilde D_4$.\nIn particular, it has 17 simple objects.\n\n(2) We have $\\text{FPdim}(\\bar \\mathcal{C}(G_2,G_2(a_1),7,q))=49(7+15[3]_7+12[5]_7)\\approx 3054.068811$.\n\n(3) The category $\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ has stable Chevalley property.\n\n(4) The category $\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ is completely anisotropic: it has no non-trivial\ncommutative exact algebras.\n\\end{theorem}\n\nIn \\cite[Question 7.20]{ShYa} (see also \\cite[Question 6.25]{LaWa}) K.~Shimizu and H.~Yadav asked whether non-semisimple completely\nanisotropic categories exist;  Theorem \\ref{main 7mod} (4) gives a positive answer to this question.\n\nIn \\cite[Definition 6.23]{LaWa} (see also \\cite[Definition 7.2]{ShYa}) R.~Laugwitz and C.~Walton defined an important Witt equivalence\nrelation on the set of non-degenerate braided finite tensor categories. In Section \\ref{Witt} we prove\nsome general properties of this relation which imply\n\n\\begin{theorem}\\label{main 7Witt}\nThe category $\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ is not Witt equivalent to any semisimple category.\n\\end{theorem}\n\nThus, the non-semisimple Witt group is different from its semisimple version studied in \\cite{DMNO}.\n\n\\subsection{} In Section \\ref{genconj} we present some conjectures. Most importantly, we expect that the categories\n$\\mathcal{C}(\\mathfrak{g},e,l,q)$  make sense for all nilpotent elements $e\\in \\mathfrak{g}$ (or $e\\in \\mathfrak{g}^L$\nin the divisible case). If $e$ is not distinguished,\nthe category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ is not finite; however we expect that it is obtained from a finite tensor\ncategory by equivariantization. \n\n\\subsection{Acknowledgements} Some ideas that led to this paper were inspired by participation of one of us (V.O.) in American Institute of Mathematics SQuaRE ``Lie algebras in symmetric tensor categories''; we are very grateful to this institution and to the fellow participants Iv\\'an Angiono, Agustina Czenky, Pavel Etingof, Julia Plavnik, and Guillermo Sanmarco. We also thank Dmitri Nikshych and Kenichi Shimizu for useful discussions.", "sketch": "The only proof-related information in the post-theorem text is the remark:\n\n\\begin{remark} The proof of Theorem \\ref{main blocks} shows that the number of blocks of type $\\Gamma$\nis the same as the number of weights inside of the fundamental alcove (but not on its boundary). Thus, this\nnumber is the same as the number of simple objects in the category $\\mathcal{C}(\\mathfrak{g},l,q)$.\\end{remark}\n\nNo further outline/steps of the proof of Theorem~\\ref{main blocks} are given.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{main blocks}\nThe category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$ decomposes into blocks which are either trivial\n(that is equivalent to the category of vector spaces) or of type $\\Gamma$. In particular,\nthe category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$ is of tame representation type. Also the cohomology of\n$\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$ (that is the Ext algebra of the unit object ${\\bf 1}$) is isomorphic to the algebra\nof invariants $S^\\bullet(V)^\\Gamma$.", "theorem_type": ["Classification or Bijection", "Universal"], "mcq": {"question": "Let $\\mathfrak g$ be a simple Lie algebra and let $q$ be a root of unity such that $l$ is the order of $q^2$, with $l$ sufficiently large. Assume that $(\\mathfrak g,l)$ is one of the following cases: $\\mathfrak g=B_n$ ($n\\ge 3$) with $l$ divisible; $\\mathfrak g=C_n$ ($n\\ge 3$) with $l$ not divisible; $\\mathfrak g=D_n$ ($n\\ge 4$) with any $l$; $\\mathfrak g=E_n$ ($n=6,7,8$) with any $l$; $\\mathfrak g=F_4$ with any $l$; or $\\mathfrak g=G_2$ with any $l$. Let $e_{sr}$ be the subregular nilpotent element, and let $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ be the corresponding finite tensor category. Let $V$ be a $2$-dimensional vector space, and let $\\Gamma\\subset SL(V)$ be the finite subgroup attached to $(\\mathfrak g,l)$ by the McKay correspondence, namely of affine Dynkin type $\\tilde D_{2n}$ for $B_n$ and $C_n$, $\\tilde D_n$ for $D_n$, $\\tilde E_n$ for $E_n$, and $\\tilde E_7$ for $F_4$ and $G_2$. A block of type $\\Gamma$ means the abelian category of left $\\wedge(V)$-modules in $\\operatorname{Rep}(\\Gamma)$, equivalently the category of finite-dimensional representations of the crossed product algebra $\\wedge(V)\\rtimes \\Gamma$. If $S^\\bullet(V)$ denotes the symmetric algebra of $V$ (with the natural $\\Gamma$-action), which statement holds for every such category $\\mathcal C(\\mathfrak g,e_{sr},l,q)$?", "correct_choice": {"label": "A", "text": "The category $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or of type $\\Gamma$. Consequently $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ has tame representation type, and its cohomology, i.e. the Ext algebra $\\operatorname{Ext}^\\bullet_{\\mathcal C}(\\mathbf 1,\\mathbf 1)$ of the unit object, is isomorphic to the invariant algebra $S^\\bullet(V)^\\Gamma$."}, "choices": [{"label": "B", "text": "The category $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is of type $\\Gamma$. Consequently $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ has tame representation type, and its cohomology, i.e. the Ext algebra $\\operatorname{Ext}^\\bullet_{\\mathcal C}(\\mathbf 1,\\mathbf 1)$ of the unit object, is isomorphic to the full symmetric algebra $S^\\bullet(V)$."}, {"label": "C", "text": "The category $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or of type $\\Gamma$. In particular, $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ has tame representation type."}, {"label": "D", "text": "The category $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or of type $\\Gamma$. Consequently $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ has finite representation type, and its cohomology, i.e. the Ext algebra $\\operatorname{Ext}^\\bullet_{\\mathcal C}(\\mathbf 1,\\mathbf 1)$ of the unit object, is isomorphic to the invariant algebra $S^\\bullet(V)^\\Gamma$."}, {"label": "E", "text": "The category $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or Morita equivalent to $\\operatorname{Rep}(\\Gamma)$. Consequently $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ has tame representation type, and its cohomology, i.e. the Ext algebra $\\operatorname{Ext}^\\bullet_{\\mathcal C}(\\mathbf 1,\\mathbf 1)$ of the unit object, is isomorphic to the invariant algebra $S^\\bullet(V)^\\Gamma$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "invariants_vs_full_symmetric_and_presence_of_trivial_blocks", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_cohomology_identification_with_S^\\bullet(V)^\\Gamma", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "tame_vs_finite_representation_type", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "block_type_Gamma_replaced_by_Rep(Gamma)", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the setup and the meaning of a block of type Γ, but it does not explicitly reveal the correct decomposition, representation type, or cohomology algebra."}, "TAS": {"score": 0, "justification": "The correct option is essentially a direct statement of the target theorem/result: block decomposition, tame type, and the Ext-algebra identification are all restated almost verbatim."}, "GPS": {"score": 1, "justification": "The item requires some discrimination among closely related statements (tame vs finite type, invariants vs full symmetric algebra, global vs blockwise cohomology), but it mostly tests theorem recall rather than substantial mathematical generation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: they reflect realistic errors such as strengthening to finite type, omitting trivial blocks, or confusing global and blockwise cohomology."}, "total_score": 5, "overall_assessment": "A solid recall-based MCQ with strong distractors and little answer leakage, but it is largely a theorem restatement and does not strongly test generative reasoning."}}
{"id": "2602.10519v1", "paper_link": "http://arxiv.org/abs/2602.10519v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{main blocks}\nThe category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$ decomposes into blocks which are either trivial\n(that is equivalent to the category of vector spaces) or of type $\\Gamma$. In particular,\nthe category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$ is of tame representation type. Also the cohomology of\n$\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$ (that is the Ext algebra of the unit object ${\\bf 1}$) is isomorphic to the algebra\nof invariants $S^\\bullet(V)^\\Gamma$.", "start_pos": 6618, "end_pos": 7062, "label": "main blocks"}, "ref_dict": {"main blocks": "\\begin{theorem}\\label{main blocks}\nThe category $\\C(\\g,e_{sr},l,q)$ decomposes into blocks which are either trivial\n(that is equivalent to the category of vector spaces) or of type $\\Gamma$. In particular,\nthe category $\\C(\\g,e_{sr},l,q)$ is of tame representation type. Also the cohomology of\n$\\C(\\g,e_{sr},l,q)$ (that is the Ext algebra of the unit object $\\be$) is isomorphic to the algebra\nof invariants $S^\\bullet(V)^\\Gamma$.\n\\end{theorem}", "main 7mod": "\\begin{theorem}\\label{main 7mod}\n(1) The category $\\bar \\C(G_2,G_2(a_1),7,q)$ has 12 trivial blocks and one block of type $\\tilde D_4$.\nIn particular, it has 17 simple objects.\n\n(2) We have $\\FP(\\bar \\C(G_2,G_2(a_1),7,q))=49(7+15[3]_7+12[5]_7)\\approx 3054.068811$.\n\n(3) The category $\\bar \\C(G_2,G_2(a_1),7,q)$ has stable Chevalley property.\n\n(4) The category $\\bar \\C(G_2,G_2(a_1),7,q)$ is completely anisotropic: it has no non-trivial\ncommutative exact algebras.\n\\end{theorem}", "Witt": "\\label{Witt}\nIn this section $\\K$ is an algebraically closed field of arbitrary characteristic.\n\\subsection{Exact algebras} Let $\\C$ be a finite tensor category over $\\K$ and let $A=(A,m,u)\\in \\C$ be", "main 7": "\\begin{theorem}\\label{main 7}\n(1) The category $\\C(G_2,G_2(a_1),7,q)$ has 15 trivial blocks and one block of type $\\tilde E_7$.\nIn particular it has 23 simple objects.\n\n(2) We have $\\FP(\\C(G_2,G_2(a_1),7,q))=294(7+15[3]_7+12[5]_7)\\approx 18324.416384$.\n\n(3) The category $\\C(G_2,G_2(a_1),7,q)$ has stable Chevalley property: tensor products of simple \nobjects are direct sums of simples and projectives.\n\n(4) The M\\\"uger center of the category $\\C(G_2,G_2(a_1),7,q)$ is equivalent to $\\Rep(S_3)$\n(where $S_3$ is the symmetric group on three letters).\n\\end{theorem}", "genconj": "\\label{genconj}\n\\subsection{Categories $\\C(G_2,G_2(a_1),l,q)$}\nWe expect that Theorem \\ref{main 7} extends to the case of arbitrary undivisible (i.e. not\ndivisible by 3) $l\\ge 7$.\n\n\\begin{conjecture}\\", "divisible": "\\begin{definition} \\label{divisible}\nWe say that $l\\in \\Z_{\\ge 1}$ is \\textit{divisible} if $l$ is divisible\nby $m$ and $l$ is \\textit{undivisible} otherwise.     \\end{definition}", "distconj": "\\begin{conjecture}\\label{g2conjm}\n    Assume $l\\ge 12$ and $l$ is divisible by 3. Then\n\n    $$\\FP(\\C(G_2, G_2(a_1),l,q))=\\frac29\\frac{l^4}{(2\\sin(\\pi/l))^4(2\\sin(2\\pi/l))^2(2\\sin(3\\pi/l))^4}.$$\nand for $l\\ge 15$\n$$\\FP(T(\\omega_1))=[3]_l+[5]_l-1,\\; \\FP(T(\\omega_2))=2[7]_l-[5]_l+[3]_l+2.$$\n\\end{conjecture}\n\nWe checked that the formulas work for $l=12,15,18,21$; this applies also for $l=9$ if we\ndefine category $\\C(G_2, G_2(a_1),9,q)$ to be a semisimplification of $\\T(\\g,q)$\n(for $q$ such that $q^2$ has order 9), see \\cite{RW}. The formula for $\\FP(T(\\omega_2))$ gives\nan incorrect result for $l=12$, namely the right hand side and the left hand side\ndiffer exactly by 1. This is likely to be explained by the fact that $T(\\omega_2)$\ndoes not coincide with Weyl module in this (and only in this) case.\n\n\\subsection{Categories $\\C(\\g,e,l,q)$ for distinguished nilpotent $e\\in \\g$}\\label{distconj}\nLet $Q=Q(e)$ be the centralizer of a $sl_2-$triple associated with $e$ in the simply\nconnected group $G$ with $\\mbox{Lie}(G)=\\g$. In the case of distinguished nilpotent $e$,\n$Q$ is a finite group. Let $2k_i+1, i\\in M=M(e)$ be the sizes of Jordan cells for\nthe adjoint action of $e$ on $\\g$ (it is well known that these sizes are odd for distinguished\nnilpotent elements; the number of these cells, that is the cardinality of set $M$, is\nthe dimension of the centralizer of $e$ in $\\g$). For any $k\\in \\Z_{>0}$ we set\n$$S_k(l):=\\frac{l}{(2^k\\sin(\\pi/l)\\sin(2\\pi/l)\\cdots \\sin(k\\pi/l))^2}.$$\n\n\\begin{conjecture}\\label{FPdim}\n    Assume $m=1$ or $l$ is undivisible. Then\n    $$\\FP(\\C(\\g,e,l,q))=|Q|\\prod_{i=1}^MS_{k_i}(l).$$\n\\end{conjecture}"}, "pre_theorem_intro_text_len": 4606, "pre_theorem_intro_text": "\\subsection{}\nThis paper is a contribution to the theory of braided finite tensor categories. In the case\nof semisimple categories over $\\mathbb{C}$ many known examples arise from Wess-Zumino-Witten\nmodels in conformal field theory, see e.g. \\cite{BK}. One of the easiest algebraic constructions of these categories was given\nby H.~H.~Andersen \\cite{An} (see also \\cite{Saw}). In modern terms this construction can be summarized\nas follows: for a simple Lie algebra $\\mathfrak{g}$ and a root of unity $q$ such that $l$ is the order of $q^2$ one considers\nthe category $\\mathcal{T}(\\mathfrak{g},q)$ of {\\em tilting modules} over the quantum group at a root of unity $q$ associated \nwith Lie algebra $\\mathfrak{g}$. Then one defines a category $\\mathcal{C}(\\mathfrak{g}, l, q)$ as the {\\em semisimplification}\nof   $\\mathcal{T}(\\mathfrak{g},q)$ (see e.g. \\cite{EOsemi}). The category $\\mathcal{C}(\\mathfrak{g}, l, q)$ is a semisimple braided tensor\ncategory; moreover this category is finite (i.e. it has only finitely many classes of simple objects) if\n$l$ is sufficiently large. The precise bounds for $l$ are given in \\cite[Figure 2]{Scho}; in this paper \nwe will always assume that $l$ is sufficiently large in this sense. Note that there is a combinatorial \ndifference between the case when $l$ is {\\em divisible} (see \\ref{divisible}) and when it is not. \n\nThe procedure of semisimplification above can be described as taking the quotient by a suitable\ntensor ideal (namely, by the ideal of negligible morphisms). However, the category $\\mathcal{T}(\\mathfrak{g},q)$ has \nmany other tensor ideals. In \\cite{CEO} for any {\\em distinguished} nilpotent element $e\\in \\mathfrak{g}$ (or, in the case\nwhen $l$ is divisible, $e\\in \\mathfrak{g}^L$ where $\\mathfrak{g}^L$ is the Langlands dual Lie algebra of $\\mathfrak{g}$),\na tensor ideal $\\cI_e$ was constructed such that the quotient category $\\mathcal{T}(\\mathfrak{g},q)/\\cI_e$ admits a {\\em monoidal\nabelian envelope} $\\mathcal{C}(\\mathfrak{g},e,l,q)$; moreover, the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ is a finite tensor category\nin the sense of \\cite{EOfinite}. For example when $e$ is a regular nilpotent element (which is always\ndistinguished), the ideal $\\cI_e$ is the ideal of negligible morphisms and $\\mathcal{C}(\\mathfrak{g},e,l,q)=\\mathcal{C}(\\mathfrak{g},l,q)$.\nUnfortunately, there is not much we can say about the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ for other nilpotent\nelements $e$ (however, see Section \\ref{distconj} for a conjectural formula for the Frobenius-Perron dimension\nof $\\mathcal{C}(\\mathfrak{g},e,l,q)$ and conjectural description of its cohomology). The goal of this paper is to give \nsome explicit information about the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ in the case when $e=e_{sr}$ is a subregular\nnilpotent element, see \\cite[4.2]{CMg}. Recall that $e_{sr}$ is distinguished if and only if $\\mathfrak{g}$ is not of\ntypes $A_n, n\\ge 1$ or $B_n, n\\ge 2$, see \\cite{CMg}.\n\nTo state our main results we need a bit more notation. Let $V$ be a two dimensional space and let\n$\\Gamma \\subset SL(V)$ be a finite subgroup of even order. By the classical McKay correspondence,\nthe finite subgroups of $SL(V)$ up to conjugacy are labelled by simply laced affine Dynkin diagrams (the only\nsubgroups of odd order correspond to affine Dynkin diagrams of type $\\tilde A_{2n}$). We associate\na subgroup $\\Gamma$ to $\\mathfrak{g}$ and $l$ as above as follows:\n\\renewcommand{\\arraystretch}{1.4}\n$$\n\\begin{array}{|c|c|c|c|c|c|c|}\n\\hline\n\\mathfrak{g}&B_n&C_n&D_n&E_n&F_4&G_2\\\\\n&n\\ge 3&n\\ge 3&n\\ge 4&n=6,7,8&&\\\\\n\\hline\nl&\\mbox{divisible}&\\mbox{not divisible}&\\mbox{any}&\\mbox{any}&\\mbox{any}&\\mbox{any}\\\\\n\\hline\n\\Gamma &\\tilde D_{2n} &\\tilde D_{2n}&\\tilde D_n&\\tilde E_n&\\tilde E_7&\\tilde E_7\\\\\n\\hline\n\\end{array}\n$$\n\nLet $\\wedge(V)$ be the exterior algebra of $V$. We can consider $\\wedge(V)$ as an algebra in the category\n$\\mbox{Rep}(\\Gamma)$ of finite dimensional representations of $\\Gamma$. We consider the abelian\ncategory of left $\\wedge(V)-$modules in the category $\\mbox{Rep}(\\Gamma)$ and we call it\n{\\em block of type} $\\Gamma$. Thus, a block of type $\\Gamma$ is the category equivalent to that of finite dimensional\nrepresentations of the cross product of $\\wedge(V)$ with the group algebra of $\\Gamma$.\nFinally let $S^\\bullet(V)$ be the symmetric algebra of $V$ which is graded by even integers (so $V \\subset\nS^\\bullet(V)$ is in degree 2). The group $\\Gamma$ acts on $S^\\bullet(V)$ preserving the grading.\nOur first main result describes the structure of abelian category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$.", "context": "\\subsection{}\nThis paper is a contribution to the theory of braided finite tensor categories. In the case\nof semisimple categories over $\\mathbb{C}$ many known examples arise from Wess-Zumino-Witten\nmodels in conformal field theory, see e.g. \\cite{BK}. One of the easiest algebraic constructions of these categories was given\nby H.~H.~Andersen \\cite{An} (see also \\cite{Saw}). In modern terms this construction can be summarized\nas follows: for a simple Lie algebra $\\mathfrak{g}$ and a root of unity $q$ such that $l$ is the order of $q^2$ one considers\nthe category $\\mathcal{T}(\\mathfrak{g},q)$ of {\\em tilting modules} over the quantum group at a root of unity $q$ associated \nwith Lie algebra $\\mathfrak{g}$. Then one defines a category $\\mathcal{C}(\\mathfrak{g}, l, q)$ as the {\\em semisimplification}\nof $\\mathcal{T}(\\mathfrak{g},q)$ (see e.g. \\cite{EOsemi}). The category $\\mathcal{C}(\\mathfrak{g}, l, q)$ is a semisimple braided tensor\ncategory; moreover this category is finite (i.e. it has only finitely many classes of simple objects) if\n$l$ is sufficiently large. The precise bounds for $l$ are given in \\cite[Figure 2]{Scho}; in this paper \nwe will always assume that $l$ is sufficiently large in this sense. Note that there is a combinatorial \ndifference between the case when $l$ is {\\em divisible} (see \\ref{divisible}) and when it is not.\n\nThe procedure of semisimplification above can be described as taking the quotient by a suitable\ntensor ideal (namely, by the ideal of negligible morphisms). However, the category $\\mathcal{T}(\\mathfrak{g},q)$ has \nmany other tensor ideals. In \\cite{CEO} for any {\\em distinguished} nilpotent element $e\\in \\mathfrak{g}$ (or, in the case\nwhen $l$ is divisible, $e\\in \\mathfrak{g}^L$ where $\\mathfrak{g}^L$ is the Langlands dual Lie algebra of $\\mathfrak{g}$),\na tensor ideal $\\cI_e$ was constructed such that the quotient category $\\mathcal{T}(\\mathfrak{g},q)/\\cI_e$ admits a {\\em monoidal\nabelian envelope} $\\mathcal{C}(\\mathfrak{g},e,l,q)$; moreover, the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ is a finite tensor category\nin the sense of \\cite{EOfinite}. For example when $e$ is a regular nilpotent element (which is always\ndistinguished), the ideal $\\cI_e$ is the ideal of negligible morphisms and $\\mathcal{C}(\\mathfrak{g},e,l,q)=\\mathcal{C}(\\mathfrak{g},l,q)$.\nUnfortunately, there is not much we can say about the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ for other nilpotent\nelements $e$ (however, see Section \\ref{distconj} for a conjectural formula for the Frobenius-Perron dimension\nof $\\mathcal{C}(\\mathfrak{g},e,l,q)$ and conjectural description of its cohomology). The goal of this paper is to give \nsome explicit information about the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ in the case when $e=e_{sr}$ is a subregular\nnilpotent element, see \\cite[4.2]{CMg}. Recall that $e_{sr}$ is distinguished if and only if $\\mathfrak{g}$ is not of\ntypes $A_n, n\\ge 1$ or $B_n, n\\ge 2$, see \\cite{CMg}.\n\nTo state our main results we need a bit more notation. Let $V$ be a two dimensional space and let\n$\\Gamma \\subset SL(V)$ be a finite subgroup of even order. By the classical McKay correspondence,\nthe finite subgroups of $SL(V)$ up to conjugacy are labelled by simply laced affine Dynkin diagrams (the only\nsubgroups of odd order correspond to affine Dynkin diagrams of type $\\tilde A_{2n}$). We associate\na subgroup $\\Gamma$ to $\\mathfrak{g}$ and $l$ as above as follows:\n\\renewcommand{\\arraystretch}{1.4}\n$$\n\\begin{array}{|c|c|c|c|c|c|c|}\n\\hline\n\\mathfrak{g}&B_n&C_n&D_n&E_n&F_4&G_2\\\\\n&n\\ge 3&n\\ge 3&n\\ge 4&n=6,7,8&&\\\\\n\\hline\nl&\\mbox{divisible}&\\mbox{not divisible}&\\mbox{any}&\\mbox{any}&\\mbox{any}&\\mbox{any}\\\\\n\\hline\n\\Gamma &\\tilde D_{2n} &\\tilde D_{2n}&\\tilde D_n&\\tilde E_n&\\tilde E_7&\\tilde E_7\\\\\n\\hline\n\\end{array}\n$$\n\nLet $\\wedge(V)$ be the exterior algebra of $V$. We can consider $\\wedge(V)$ as an algebra in the category\n$\\mbox{Rep}(\\Gamma)$ of finite dimensional representations of $\\Gamma$. We consider the abelian\ncategory of left $\\wedge(V)-$modules in the category $\\mbox{Rep}(\\Gamma)$ and we call it\n{\\em block of type} $\\Gamma$. Thus, a block of type $\\Gamma$ is the category equivalent to that of finite dimensional\nrepresentations of the cross product of $\\wedge(V)$ with the group algebra of $\\Gamma$.\nFinally let $S^\\bullet(V)$ be the symmetric algebra of $V$ which is graded by even integers (so $V \\subset\nS^\\bullet(V)$ is in degree 2). The group $\\Gamma$ acts on $S^\\bullet(V)$ preserving the grading.\nOur first main result describes the structure of abelian category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$.", "full_context": "\\subsection{}\nThis paper is a contribution to the theory of braided finite tensor categories. In the case\nof semisimple categories over $\\mathbb{C}$ many known examples arise from Wess-Zumino-Witten\nmodels in conformal field theory, see e.g. \\cite{BK}. One of the easiest algebraic constructions of these categories was given\nby H.~H.~Andersen \\cite{An} (see also \\cite{Saw}). In modern terms this construction can be summarized\nas follows: for a simple Lie algebra $\\mathfrak{g}$ and a root of unity $q$ such that $l$ is the order of $q^2$ one considers\nthe category $\\mathcal{T}(\\mathfrak{g},q)$ of {\\em tilting modules} over the quantum group at a root of unity $q$ associated \nwith Lie algebra $\\mathfrak{g}$. Then one defines a category $\\mathcal{C}(\\mathfrak{g}, l, q)$ as the {\\em semisimplification}\nof $\\mathcal{T}(\\mathfrak{g},q)$ (see e.g. \\cite{EOsemi}). The category $\\mathcal{C}(\\mathfrak{g}, l, q)$ is a semisimple braided tensor\ncategory; moreover this category is finite (i.e. it has only finitely many classes of simple objects) if\n$l$ is sufficiently large. The precise bounds for $l$ are given in \\cite[Figure 2]{Scho}; in this paper \nwe will always assume that $l$ is sufficiently large in this sense. Note that there is a combinatorial \ndifference between the case when $l$ is {\\em divisible} (see \\ref{divisible}) and when it is not.\n\nThe procedure of semisimplification above can be described as taking the quotient by a suitable\ntensor ideal (namely, by the ideal of negligible morphisms). However, the category $\\mathcal{T}(\\mathfrak{g},q)$ has \nmany other tensor ideals. In \\cite{CEO} for any {\\em distinguished} nilpotent element $e\\in \\mathfrak{g}$ (or, in the case\nwhen $l$ is divisible, $e\\in \\mathfrak{g}^L$ where $\\mathfrak{g}^L$ is the Langlands dual Lie algebra of $\\mathfrak{g}$),\na tensor ideal $\\cI_e$ was constructed such that the quotient category $\\mathcal{T}(\\mathfrak{g},q)/\\cI_e$ admits a {\\em monoidal\nabelian envelope} $\\mathcal{C}(\\mathfrak{g},e,l,q)$; moreover, the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ is a finite tensor category\nin the sense of \\cite{EOfinite}. For example when $e$ is a regular nilpotent element (which is always\ndistinguished), the ideal $\\cI_e$ is the ideal of negligible morphisms and $\\mathcal{C}(\\mathfrak{g},e,l,q)=\\mathcal{C}(\\mathfrak{g},l,q)$.\nUnfortunately, there is not much we can say about the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ for other nilpotent\nelements $e$ (however, see Section \\ref{distconj} for a conjectural formula for the Frobenius-Perron dimension\nof $\\mathcal{C}(\\mathfrak{g},e,l,q)$ and conjectural description of its cohomology). The goal of this paper is to give \nsome explicit information about the category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ in the case when $e=e_{sr}$ is a subregular\nnilpotent element, see \\cite[4.2]{CMg}. Recall that $e_{sr}$ is distinguished if and only if $\\mathfrak{g}$ is not of\ntypes $A_n, n\\ge 1$ or $B_n, n\\ge 2$, see \\cite{CMg}.\n\nTo state our main results we need a bit more notation. Let $V$ be a two dimensional space and let\n$\\Gamma \\subset SL(V)$ be a finite subgroup of even order. By the classical McKay correspondence,\nthe finite subgroups of $SL(V)$ up to conjugacy are labelled by simply laced affine Dynkin diagrams (the only\nsubgroups of odd order correspond to affine Dynkin diagrams of type $\\tilde A_{2n}$). We associate\na subgroup $\\Gamma$ to $\\mathfrak{g}$ and $l$ as above as follows:\n\\renewcommand{\\arraystretch}{1.4}\n$$\n\\begin{array}{|c|c|c|c|c|c|c|}\n\\hline\n\\mathfrak{g}&B_n&C_n&D_n&E_n&F_4&G_2\\\\\n&n\\ge 3&n\\ge 3&n\\ge 4&n=6,7,8&&\\\\\n\\hline\nl&\\mbox{divisible}&\\mbox{not divisible}&\\mbox{any}&\\mbox{any}&\\mbox{any}&\\mbox{any}\\\\\n\\hline\n\\Gamma &\\tilde D_{2n} &\\tilde D_{2n}&\\tilde D_n&\\tilde E_n&\\tilde E_7&\\tilde E_7\\\\\n\\hline\n\\end{array}\n$$\n\nLet $\\wedge(V)$ be the exterior algebra of $V$. We can consider $\\wedge(V)$ as an algebra in the category\n$\\mbox{Rep}(\\Gamma)$ of finite dimensional representations of $\\Gamma$. We consider the abelian\ncategory of left $\\wedge(V)-$modules in the category $\\mbox{Rep}(\\Gamma)$ and we call it\n{\\em block of type} $\\Gamma$. Thus, a block of type $\\Gamma$ is the category equivalent to that of finite dimensional\nrepresentations of the cross product of $\\wedge(V)$ with the group algebra of $\\Gamma$.\nFinally let $S^\\bullet(V)$ be the symmetric algebra of $V$ which is graded by even integers (so $V \\subset\nS^\\bullet(V)$ is in degree 2). The group $\\Gamma$ acts on $S^\\bullet(V)$ preserving the grading.\nOur first main result describes the structure of abelian category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$.\n\nLet $\\wedge(V)$ be the exterior algebra of $V$. We can consider $\\wedge(V)$ as an algebra in the category\n$\\mbox{Rep}(\\Gamma)$ of finite dimensional representations of $\\Gamma$. We consider the abelian\ncategory of left $\\wedge(V)-$modules in the category $\\mbox{Rep}(\\Gamma)$ and we call it\n{\\em block of type} $\\Gamma$. Thus, a block of type $\\Gamma$ is the category equivalent to that of finite dimensional\nrepresentations of the cross product of $\\wedge(V)$ with the group algebra of $\\Gamma$.\nFinally let $S^\\bullet(V)$ be the symmetric algebra of $V$ which is graded by even integers (so $V \\subset\nS^\\bullet(V)$ is in degree 2). The group $\\Gamma$ acts on $S^\\bullet(V)$ preserving the grading.\nOur first main result describes the structure of abelian category $\\C(\\g,e_{sr},l,q)$.\n\n\\begin{remark} The proof of Theorem \\ref{main blocks} shows that the number of blocks of type $\\Gamma$\nis the same as the number of weights inside of the fundamental alcove (but not on its boundary). Thus, this\nnumber is the same as the number of simple objects in the category $\\C(\\g,l,q)$.\n\\end{remark}\n\n\\begin{theorem}\\label{main 7}\n(1) The category $\\C(G_2,G_2(a_1),7,q)$ has 15 trivial blocks and one block of type $\\tilde E_7$.\nIn particular it has 23 simple objects.\n\n\\begin{theorem}\\label{main 7mod}\n(1) The category $\\bar \\C(G_2,G_2(a_1),7,q)$ has 12 trivial blocks and one block of type $\\tilde D_4$.\nIn particular, it has 17 simple objects.\n\nWe consider the category $\\Rep(\\Gamma \\ltimes V, \\epsilon)$ of\nfinite dimensional representations of $\\Gamma \\ltimes V$ where $\\epsilon$ acts as the parity automorphism,\nsee e.g. \\cite[9.11]{EGNO}. Thus, $\\Rep(\\Gamma \\ltimes V, \\epsilon)$ is a symmetric finite tensor category.\nAs an abelian category it is equivalent to the category of representations of algebra\n$\\wedge(V)$ in the category $\\Rep(\\Gamma)$ (or, equivalently, representations of $\\wedge(V)$ with an action\nof $\\Gamma$ compatible in an obvious sense). The simple objects of $\\Rep(\\Gamma \\ltimes V, \\epsilon)$ are irreducible representations of $\\Gamma$ where $V\\subset \\wedge(V)$ acts by zero;\nlet $\\Irr(\\Gamma)$ be the set of isomorphism classes of such objects.\nThe projective cover of the unit object is $\\wedge(V)$; and the projective cover of $V_i\\in \\Irr(\\Gamma)$\nis $V_i\\otimes \\wedge(V)$. In particular, the composition factors of the projective cover of $V_i$ are\n$V_i$ appearing twice and the irreducible summands of $V_i\\otimes V$. Thus, the Cartan matrix of\nthe category $\\Rep(\\Gamma \\ltimes V, \\epsilon)$ is $2\\Id+A_\\Gamma$ where $A_\\Gamma$ is the adjacency\nmatrix of the McKay graph of $\\Gamma$ (recall that the vertices of the McKay graph are elements \nof $\\Irr(\\Gamma)$ and the number of edges between $V_i$ and $V_j$ is the multiplicity of $V_j$ as \na direct summand of $V_i\\otimes V$).\n\n\\begin{corollary} \\label{cor:self-dual}\n Let $\\mathfrak g$ be of type $B_n, C_n, F_4, G_2, D_{2n}, E_7$, or $E_8$.\n Then all simple objects of $\\C=\\C(\\mathfrak g, e_{sr}, l, q)$ are self-dual. \n\\end{corollary}\n\\begin{proof}\n It is well-known that in this case all simple representations of $\\mathfrak g$ are self-dual. It follows from properties (1) and (2) in Section \\ref{qgtitling} that the tilting modules $T(\\lambda)$ are also self-dual.\n\n\\subsection{Proof of Theorem \\ref{main blocks}} By Proposition \\ref{Cartansr} the Cartan\nmatrix of the principal block of $\\C(\\g,e_{sr},l,q)$ is $2\\Id+A(X_{sr})$ where $X_{sr}$ is\nsome affine Dynkin diagram not of type $A$, so it is a tree. Thus, by Proposition \\ref{tree}\n(applied to this block and to the category $\\Rep(\\Gamma \\ltimes V)$)\nwe see that the principal block is equivalent to $\\Rep(\\Gamma \\ltimes V)$ where \n$V$ is as in \\ref{toy} and $\\Gamma \\subset SL(V)$ is a finite subgroup with McKay graph $X_{sr}$.\nUsing the translation functors (which descend to the category $\\T(\\g,q)/\\cI_{e_{sr}}$) we see\nthat all the other blocks of $\\C(\\g,e_{sr},l,q)$ involving $T(w\\cdot \\lambda)$ where $\\lambda$\nis in the interior of the fundamental alcove are equivalent to the principal block, and hence\nto the category $\\Rep(\\Gamma \\ltimes V)$. The other blocks involving $T(\\lambda)$ with $\\lambda$\non the wall are trivial by Corollaries \\ref{nog222} and \\ref{cor:on the wall}. The remaining statements of \nTheorem \\ref{main blocks} follow from Proposition \\ref{3.1}.\n\nIt is a classical result of P.~Slodowy that the singularity of the nilpotent\ncone of $\\g$ at the point $e_{sr}$ is $\\BC^2/\\Gamma$ for a suitable subgroup\n$\\Gamma \\subset SL_2(\\BC)$, see \\cite{Sl}. Comparing this with Theorem \\ref{main blocks}\nwe expect that the cohomology of the category $\\C(\\g,e,l,q)$ is related\nwith the singularity of the nilpotent cone at point $e$. Thus let $\\Sl_e$\nbe the Slodowy slice at the point $e$, see e.g. \\cite{GG}. The variety $\\Sl_e$\nis equipped with an action of $Q\\times \\BC^\\times$. Thus, the algebra of functions\n$\\mO(Sl_e)$ is graded and is equipped with a grading-preserving action of $Q$;\nlet $\\mO(Sl_e)^Q\\subset \\mO(Sl_e)$ be the algebra of invariants.\n\n\\begin{theorem}\\label{main blocks}\nThe category $\\C(\\g,e_{sr},l,q)$ decomposes into blocks which are either trivial\n(that is equivalent to the category of vector spaces) or of type $\\Gamma$. In particular,\nthe category $\\C(\\g,e_{sr},l,q)$ is of tame representation type. Also the cohomology of\n$\\C(\\g,e_{sr},l,q)$ (that is the Ext algebra of the unit object $\\be$) is isomorphic to the algebra\nof invariants $S^\\bullet(V)^\\Gamma$.\n\\end{theorem}", "post_theorem_intro_text_len": 4178, "post_theorem_intro_text": "\\begin{remark} The proof of Theorem \\ref{main blocks} shows that the number of blocks of type $\\Gamma$\nis the same as the number of weights inside of the fundamental alcove (but not on its boundary). Thus, this\nnumber is the same as the number of simple objects in the category $\\mathcal{C}(\\mathfrak{g},l,q)$.\n\\end{remark}\n\n\\subsection{} Next, we study one specific example, the category $\\mathcal{C}(G_2,G_2(a_1),7,q)$ (thus, we\nconsider Lie algebra $\\mathfrak{g}$ of type $G_2$; also $G_2(a_1)$ is the standard notation for the subregular\nnilpotent orbit in type $G_2$). This is the simplest example of the categories considered above\n(at least for undivisible $l$). Recall the standard notation for the quantum numbers:\n$$[k]_l=\\frac{\\sin(k\\pi/l)}{\\sin(\\pi/l)}.$$\nIn particular, $[2]_7=[5]_7=2\\cos(\\pi/7)\\approx 1.801938$ and $[3]_7=[4]_7=\\frac{\\sin(3\\pi/7)}{\\sin(\\pi/7)}\\approx\n2.246980$.\n\n\\begin{theorem}\\label{main 7}\n(1) The category $\\mathcal{C}(G_2,G_2(a_1),7,q)$ has 15 trivial blocks and one block of type $\\tilde E_7$.\nIn particular it has 23 simple objects.\n\n(2) We have $\\text{FPdim}(\\mathcal{C}(G_2,G_2(a_1),7,q))=294(7+15[3]_7+12[5]_7)\\approx 18324.416384$.\n\n(3) The category $\\mathcal{C}(G_2,G_2(a_1),7,q)$ has stable Chevalley property: tensor products of simple \nobjects are direct sums of simples and projectives.\n\n(4) The M\\\"uger center of the category $\\mathcal{C}(G_2,G_2(a_1),7,q)$ is equivalent to $\\text{Rep}(S_3)$\n(where $S_3$ is the symmetric group on three letters).\n\\end{theorem}\n\nIn view of Theorem \\ref{main 7} (4), it makes sense to consider the de-equivariantization $\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ \nof $\\mathcal{C}(G_2,G_2(a_1),7,q)$ with respect to its M\\\"uger center (so the category $\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ \nis the Brugui\\`eres' modularisation of $\\mathcal{C}(G_2,G_2(a_1),7,q)$, see \\cite{Brug}). The category\n$\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ is a non-semisimple modular tensor category in the sense of Shimizu,\nsee \\cite{Shim}.\n\n\\begin{theorem}\\label{main 7mod}\n(1) The category $\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ has 12 trivial blocks and one block of type $\\tilde D_4$.\nIn particular, it has 17 simple objects.\n\n(2) We have $\\text{FPdim}(\\bar \\mathcal{C}(G_2,G_2(a_1),7,q))=49(7+15[3]_7+12[5]_7)\\approx 3054.068811$.\n\n(3) The category $\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ has stable Chevalley property.\n\n(4) The category $\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ is completely anisotropic: it has no non-trivial\ncommutative exact algebras.\n\\end{theorem}\n\nIn \\cite[Question 7.20]{ShYa} (see also \\cite[Question 6.25]{LaWa}) K.~Shimizu and H.~Yadav asked whether non-semisimple completely\nanisotropic categories exist;  Theorem \\ref{main 7mod} (4) gives a positive answer to this question.\n\nIn \\cite[Definition 6.23]{LaWa} (see also \\cite[Definition 7.2]{ShYa}) R.~Laugwitz and C.~Walton defined an important Witt equivalence\nrelation on the set of non-degenerate braided finite tensor categories. In Section \\ref{Witt} we prove\nsome general properties of this relation which imply\n\n\\begin{theorem}\\label{main 7Witt}\nThe category $\\bar \\mathcal{C}(G_2,G_2(a_1),7,q)$ is not Witt equivalent to any semisimple category.\n\\end{theorem}\n\nThus, the non-semisimple Witt group is different from its semisimple version studied in \\cite{DMNO}.\n\n\\subsection{} In Section \\ref{genconj} we present some conjectures. Most importantly, we expect that the categories\n$\\mathcal{C}(\\mathfrak{g},e,l,q)$  make sense for all nilpotent elements $e\\in \\mathfrak{g}$ (or $e\\in \\mathfrak{g}^L$\nin the divisible case). If $e$ is not distinguished,\nthe category $\\mathcal{C}(\\mathfrak{g},e,l,q)$ is not finite; however we expect that it is obtained from a finite tensor\ncategory by equivariantization. \n\n\\subsection{Acknowledgements} Some ideas that led to this paper were inspired by participation of one of us (V.O.) in American Institute of Mathematics SQuaRE ``Lie algebras in symmetric tensor categories''; we are very grateful to this institution and to the fellow participants Iv\\'an Angiono, Agustina Czenky, Pavel Etingof, Julia Plavnik, and Guillermo Sanmarco. We also thank Dmitri Nikshych and Kenichi Shimizu for useful discussions.", "sketch": "The only proof-related information in the post-theorem text is the remark:\n\n\\begin{remark} The proof of Theorem \\ref{main blocks} shows that the number of blocks of type $\\Gamma$\nis the same as the number of weights inside of the fundamental alcove (but not on its boundary). Thus, this\nnumber is the same as the number of simple objects in the category $\\mathcal{C}(\\mathfrak{g},l,q)$.\\end{remark}\n\nNo further outline/steps of the proof of Theorem~\\ref{main blocks} are given.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{main blocks}\nThe category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$ decomposes into blocks which are either trivial\n(that is equivalent to the category of vector spaces) or of type $\\Gamma$. In particular,\nthe category $\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$ is of tame representation type. Also the cohomology of\n$\\mathcal{C}(\\mathfrak{g},e_{sr},l,q)$ (that is the Ext algebra of the unit object ${\\bf 1}$) is isomorphic to the algebra\nof invariants $S^\\bullet(V)^\\Gamma$.", "theorem_type": ["Classification or Bijection", "Universal"], "mcq": {"question": "Let $\\mathfrak g$ be a simple Lie algebra and let $q$ be a root of unity such that $l$ is the order of $q^2$, with $l$ sufficiently large. Assume that $(\\mathfrak g,l)$ is one of the following cases: $\\mathfrak g=B_n$ ($n\\ge 3$) with $l$ divisible; $\\mathfrak g=C_n$ ($n\\ge 3$) with $l$ not divisible; $\\mathfrak g=D_n$ ($n\\ge 4$) with any $l$; $\\mathfrak g=E_n$ ($n=6,7,8$) with any $l$; $\\mathfrak g=F_4$ with any $l$; or $\\mathfrak g=G_2$ with any $l$. Let $e_{sr}$ be the subregular nilpotent element, and let $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ be the corresponding finite tensor category. Let $V$ be a $2$-dimensional vector space, and let $\\Gamma\\subset SL(V)$ be the finite subgroup attached to $(\\mathfrak g,l)$ by the McKay correspondence, namely of affine Dynkin type $\\tilde D_{2n}$ for $B_n$ and $C_n$, $\\tilde D_n$ for $D_n$, $\\tilde E_n$ for $E_n$, and $\\tilde E_7$ for $F_4$ and $G_2$. A block of type $\\Gamma$ means the abelian category of left $\\wedge(V)$-modules in $\\operatorname{Rep}(\\Gamma)$, equivalently the category of finite-dimensional representations of the crossed product algebra $\\wedge(V)\\rtimes \\Gamma$. If $S^\\bullet(V)$ denotes the symmetric algebra of $V$ (with the natural $\\Gamma$-action), which statement holds for every such category $\\mathcal C(\\mathfrak g,e_{sr},l,q)$?", "correct_choice": {"label": "A", "text": "The category $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or of type $\\Gamma$. Consequently $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ has tame representation type, and its cohomology, i.e. the Ext algebra $\\operatorname{Ext}^\\bullet_{\\mathcal C}(\\mathbf 1,\\mathbf 1)$ of the unit object, is isomorphic to the invariant algebra $S^\\bullet(V)^\\Gamma$."}, "choices": [{"label": "B", "text": "The category $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is of type $\\Gamma$. Consequently $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ has tame representation type, and its cohomology, i.e. the Ext algebra $\\operatorname{Ext}^\\bullet_{\\mathcal C}(\\mathbf 1,\\mathbf 1)$ of the unit object, is isomorphic to the full symmetric algebra $S^\\bullet(V)$."}, {"label": "C", "text": "The category $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or of type $\\Gamma$. In particular, $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ has tame representation type."}, {"label": "D", "text": "The category $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or of type $\\Gamma$. Consequently $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ has finite representation type, and its cohomology, i.e. the Ext algebra $\\operatorname{Ext}^\\bullet_{\\mathcal C}(\\mathbf 1,\\mathbf 1)$ of the unit object, is isomorphic to the invariant algebra $S^\\bullet(V)^\\Gamma$."}, {"label": "E", "text": "The category $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or Morita equivalent to $\\operatorname{Rep}(\\Gamma)$. Consequently $\\mathcal C(\\mathfrak g,e_{sr},l,q)$ has tame representation type, and its cohomology, i.e. the Ext algebra $\\operatorname{Ext}^\\bullet_{\\mathcal C}(\\mathbf 1,\\mathbf 1)$ of the unit object, is isomorphic to the invariant algebra $S^\\bullet(V)^\\Gamma$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "invariants_vs_full_symmetric_and_presence_of_trivial_blocks", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_cohomology_identification_with_S^\\bullet(V)^\\Gamma", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "tame_vs_finite_representation_type", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "block_type_Gamma_replaced_by_Rep(Gamma)", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the correct conclusion. It supplies technical setup and definitions, but it does not directly reveal the decomposition, tame type, and cohomology identification."}, "TAS": {"score": 1, "justification": "This is close to a theorem-recall item: the stem gives the hypotheses of a specific result and asks for the matching conclusion. It is not a pure restatement because the options differ in subtle but meaningful ways, yet it remains largely theorem recognition."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish between nearby statements (tame vs finite type, invariants vs full symmetric algebra, block type vs Morita equivalence, complete vs weaker true statement). However, it mainly tests precise recall/comparison rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted. They reflect common failure modes: overstrengthening, weakening, confusing invariants with the full algebra, and replacing the intended block description with a nearby categorical notion."}, "total_score": 6, "overall_assessment": "A solid MCQ with strong distractors and little answer leakage, but it is primarily a precise theorem-identification question rather than a genuinely generative reasoning task."}}
{"id": "2602.10676v1", "paper_link": "http://arxiv.org/abs/2602.10676v1", "theorems_cnt": 3, "theorem": {"env_name": "thmA", "content": "\\label{thm:main}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a closed hyperbolic 3-manifold and let $\\gamma\\subset M$ be a simple closed geodesic. Suppose that $\\gamma$ has length $\\ell$ and has an embedded tubular neighborhood of radius at least $R\\in(0,1)$. Then the unique hyperbolic metric on $M-\\gamma$ has volume\n\\[\n{\\rm vol}(M-\\gamma)\\le{\\rm vol}(M)+c\\frac{\\ell}{R}.\n\\]", "start_pos": 2288, "end_pos": 2716, "label": "thm:main"}, "ref_dict": {"pro:metric on tube": "\\begin{pro}\n\\label{pro:metric on tube}\nThere exists a constant $c>0$ such that the following holds. Let $V$ be a solid torus. Suppose that the boundary $\\partial V$ is equipped with a flat metric of area $A$. Assume that the length of the flat geodesic representative on $\\partial V$ of the meridian $\\mu\\subset\\partial V$ is $\\ell>2\\pi$. Then there exists a smooth Riemannian metric $(V,g)$ with the following properties.\n\\begin{itemize}\n    \\item{The metric $g$ is a hyperbolic cusp metric in a collar of $\\partial V$, the boundary $\\partial V$ is a horospherical section of the cusp, and the restriction of $g$ to the boundary agrees with the prescribed flat metric on $\\partial V$.}\n    \\item{Denote by $S_g$ the scalar curvature of the metric $g$. We have\n    \\[\n    \\int_V{\\left(\\frac{|S_g|}{6}\\right)^{3/2}{\\rm dvol}_g}\\le\\frac{A}{2}\\left(1-\\frac{\\pi^2}{\\ell^2}\\right)\\left(1+c\\frac{\\pi^4}{\\ell^4}\\right).\n    \\]\n    }\n\\end{itemize}    \n\\end{pro}", "lem:functiondrill": "\\begin{lem}\n\\label{lem:functiondrill}\nThere exists a constant $c>0$ such that the following holds. For every $R>0$ there exist functions $a_{\\rm drill},b_{\\rm drill}:\\mb{R}\\to\\mb{R}$ such that \n\\begin{enumerate}\n    \\item{$a_{\\rm drill},b_{\\rm drill}>0$.}\n    \\item{$a_{\\rm drill}(r)=\\sinh(r),b_{\\rm drill}(r)=\\cosh(r)$ on $(2R/3,\\infty)$.}\n    \\item{$a_{\\rm drill}(r)=\\sinh(R)e^r,\\cosh(R)b_{\\rm drill}(r)=e^r$ on $(-\\infty,R/3)$.}\n    \\item{On $[R/3,2R/3]$ we have\n    \\[\n    \\left|\\frac{a_{\\rm drill}''}{a_{\\rm drill}}+\\frac{b_{\\rm drill}''}{b_{\\rm drill}}+\\frac{a_{\\rm drill}'b_{\\rm drill}'}{a_{\\rm drill}b_{\\rm drill}}\\right|\\le c\\frac{1}{R^2}.\n    \\]\n    }\n    \\item{We have $a_{\\rm drill}b_{\\rm drill}\\le \\sinh(R)\\cosh(R)e^{2r}$.}\n\\end{enumerate}\n\\end{lem}", "lem:functionfill": "\\begin{lem}\n\\label{lem:functionfill}\nThere exists a constant $c>0$ such that the following holds. For every $\\ell_1>2\\pi,\\ell_2>0$ there exist functions $a_{\\rm fill},b_{\\rm fill}:\\mb{R}\\to\\mb{R}$ that satisfy the following properties. Let $R=\\log(\\ell_1/\\pi)$ and $\\kappa=2\\pi\\ell_2/\\ell_1$.\n\\begin{enumerate}\n    \\item{$a_{\\rm fill},b_{\\rm fill}>0$.}\n    \\item{$a_{\\rm fill}(r)=2\\pi\\sinh(r+R),b(r+R)=\\kappa\\cosh(r+R)$ on $(-R,-2\\delta)$.}\n    \\item{$a_{\\rm fill}(r)=\\ell_1e^r,b(r)=\\ell_2 e^r$ on $(-\\delta,0)$.}\n    \\item{We have\n    \\[\n    \\left|\\frac{a_{\\rm fill}''}{a_{\\rm fill}}+\\frac{b_{\\rm fill}''}{b_{\\rm fill}}+\\frac{a_{\\rm fill}'b_{\\rm fill}'}{a_{\\rm fill}b_{\\rm fill}}-3\\right|\\le c\\frac{\\pi^4}{\\ell_1^4}.\n    \\]\n    }\n    \\item{We have \n    \\[\n    a_{\\rm fill}b_{\\rm fill}\\le\\ell_1\\ell_2e^{2r}.\n    \\]\n    }\n\\end{enumerate}\n\\end{lem}", "cor:main": "\\begin{corA}\n\\label{cor:main}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a closed hyperbolic 3-manifold and let $\\gamma\\subset M$ be a shortest non-trivial closed geodesic in $M$. Then the unique hyperbolic metric on $M-\\gamma$ has volume\n\\[\n{\\rm vol}(M-\\gamma)\\le{\\rm vol}(M)+c\\ell\n\\]\nwhere $\\ell=2\\cdot{\\rm inj}(M)$ is the length of $\\gamma$.\n\\end{corA}", "thm:main": "\\begin{thmA}\n\\label{thm:main}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a closed hyperbolic 3-manifold and let $\\gamma\\subset M$ be a simple closed geodesic. Suppose that $\\gamma$ has length $\\ell$ and has an embedded tubular neighborhood of radius at least $R\\in(0,1)$. Then the unique hyperbolic metric on $M-\\gamma$ has volume\n\\[\n{\\rm vol}(M-\\gamma)\\le{\\rm vol}(M)+c\\frac{\\ell}{R}.\n\\]\n\\end{thmA}", "thm:main'": "\\begin{thmA}\n\\label{thm:main'}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a finite volume hyperbolic 3-manifold with one cusp. Let $C$ be a cusp neighborhood of $M$ bounded by the horospherical torus $\\partial C$. Consider a simple closed geodesic $\\mu\\subset\\partial C$ for the intrinsic flat metric on $\\partial C$. If the length of $\\mu$ is $\\ell>\\sqrt{c}\\pi$ and the area of $\\partial C$ is $A$ then the Dehn filling of $M-C$ with slope $\\mu$, denoted by $M_\\mu$, has a hyperbolic metric of volume\n\\[\n{\\rm vol}(M_\\mu)+\\frac{A}{2}\\frac{\\pi^2}{\\ell^2}\\left(1-c\\frac{\\pi^2}{\\ell^2}\\right)\\le{\\rm vol}(M).\n\\]\n\\end{thmA}"}, "pre_theorem_intro_text_len": 560, "pre_theorem_intro_text": "There are two important operations that one can perform on a finite volume hyperbolic 3-manifold. One can drill out a simple closed geodesic and one can Dehn fill a cusp. It is a well-known phenomenon that all drillings and most fillings give back a hyperbolizable 3-manifold whose volume is larger for the first operation and smaller for the second. The goal of this short note is to prove two results that give a {\\em linear} quantitative control on how much the volume can change.\n\n\\subsection*{Drilling}\nLet us start with drillings. We prove the following.", "context": "There are two important operations that one can perform on a finite volume hyperbolic 3-manifold. One can drill out a simple closed geodesic and one can Dehn fill a cusp. It is a well-known phenomenon that all drillings and most fillings give back a hyperbolizable 3-manifold whose volume is larger for the first operation and smaller for the second. The goal of this short note is to prove two results that give a {\\em linear} quantitative control on how much the volume can change.\n\n\\subsection*{Drilling}\nLet us start with drillings. We prove the following.", "full_context": "There are two important operations that one can perform on a finite volume hyperbolic 3-manifold. One can drill out a simple closed geodesic and one can Dehn fill a cusp. It is a well-known phenomenon that all drillings and most fillings give back a hyperbolizable 3-manifold whose volume is larger for the first operation and smaller for the second. The goal of this short note is to prove two results that give a {\\em linear} quantitative control on how much the volume can change.\n\n\\subsection*{Drilling}\nLet us start with drillings. We prove the following.\n\n\\subsection*{Drilling}\nLet us start with drillings. We prove the following.\n\n\\begin{corA}\n\\label{cor:main}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a closed hyperbolic 3-manifold and let $\\gamma\\subset M$ be a shortest non-trivial closed geodesic in $M$. Then the unique hyperbolic metric on $M-\\gamma$ has volume\n\\[\n{\\rm vol}(M-\\gamma)\\le{\\rm vol}(M)+c\\ell\n\\]\nwhere $\\ell=2\\cdot{\\rm inj}(M)$ is the length of $\\gamma$.\n\\end{corA}\n\nSimilar estimates on the volume increase under drilling operations on hyperbolic 3-manifolds were already considered by Bridgeman \\cite{B}, Agol \\cite{A}, Agol, Storm, Thurston, and Dunfield \\cite[Theorem 10.1]{AST} where they proved the existence of explicit functions $f,g>1$ such that ${\\rm vol}(M-\\gamma)\\le f(R){\\rm vol}(M)+g(R)\\ell$. So, the main achievement of this note is to get rid of the multiplicative constant in front of ${\\rm vol}(M)$. This shows that the increase in volume only depends on the {\\em local} geometry around $\\gamma$.\n\nTheorem \\ref{thm:main} goes in the direction of a conjecture of Bridgeman \\cite{B}, which asks whether there is a linear upper bound on the volume increase under drilling operations. While very good estimates are available when the drilled geodesics are short (see the work of Hodgson and Kerckhoff \\cite{HK}), it is still unclear what happens for long geodesics. Note that as long as we can uniformly bound from below $R$ the increase in volume is at most linear. This leads immediately to Corollary \\ref{cor:main}. In fact, it is well-known that a shortest geodesic in a closed hyperbolic 3-manifold has a tubular neighborhood of radius at least $\\ell/4$ where $\\ell$ is the length of the geodesic (see \\cite[Proposition 1.11]{GMT}). Combined with standard consequences of Margulis Lemma, this provides a universal constant $R_0>0$ such that a shortest geodesic in a closed hyperbolic 3-manifold $M$ always has a tubular neighborhood of radius $R\\ge R_0$ (for much more refined estimates see \\cite[Theorem 4.1]{GMT}).\n\n\\begin{thmA}\n\\label{thm:main'}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a finite volume hyperbolic 3-manifold with one cusp. Let $C$ be a cusp neighborhood of $M$ bounded by the horospherical torus $\\partial C$. Consider a simple closed geodesic $\\mu\\subset\\partial C$ for the intrinsic flat metric on $\\partial C$. If the length of $\\mu$ is $\\ell>\\sqrt{c}\\pi$ and the area of $\\partial C$ is $A$ then the Dehn filling of $M-C$ with slope $\\mu$, denoted by $M_\\mu$, has a hyperbolic metric of volume\n\\[\n{\\rm vol}(M_\\mu)+\\frac{A}{2}\\frac{\\pi^2}{\\ell^2}\\left(1-c\\frac{\\pi^2}{\\ell^2}\\right)\\le{\\rm vol}(M).\n\\]\n\\end{thmA}\n\nTo prove Theorem \\ref{thm:main}, we follow the approach of \\cite[Theorem 10.1]{AST} with a difference. We first change the metric of $M$ inside a tubular neighborhood $T$ of $\\gamma$ interpolating between the metric on the tube and the metric on a suitable hyperbolic cusp. We do so by keeping under control the scalar curvature (see Lemma \\ref{lem:functiondrill}). Then we compare the volume of the modified metric with that of the hyperbolic structure on $M-\\gamma$. Instead of relying on Perelman's monotonicity formula, we exploit some other consequences of Perelman's work \\cite{Pe1,Pe2}, namely, the exact computation of the sigma invariant of closed hyperbolic 3-manifolds as achieved by Anderson \\cite{And} and Kleiner and Lott \\cite{KL}. This allows us to keep track of the fact that we {\\em did not change} the metric outside the tubular neighborhood $T$ and obtain an additive error rather than a multiplicative one in the volume increase.\n\n\\begin{lem}\n\\label{lem:functionfill}\nThere exists a constant $c>0$ such that the following holds. For every $\\ell_1>2\\pi,\\ell_2>0$ there exist functions $a_{\\rm fill},b_{\\rm fill}:\\mb{R}\\to\\mb{R}$ that satisfy the following properties. Let $R=\\log(\\ell_1/\\pi)$ and $\\kappa=2\\pi\\ell_2/\\ell_1$.\n\\begin{enumerate}\n \\item{$a_{\\rm fill},b_{\\rm fill}>0$.}\n \\item{$a_{\\rm fill}(r)=2\\pi\\sinh(r+R),b(r+R)=\\kappa\\cosh(r+R)$ on $(-R,-2\\delta)$.}\n \\item{$a_{\\rm fill}(r)=\\ell_1e^r,b(r)=\\ell_2 e^r$ on $(-\\delta,0)$.}\n \\item{We have\n \\[\n \\left|\\frac{a_{\\rm fill}''}{a_{\\rm fill}}+\\frac{b_{\\rm fill}''}{b_{\\rm fill}}+\\frac{a_{\\rm fill}'b_{\\rm fill}'}{a_{\\rm fill}b_{\\rm fill}}-3\\right|\\le c\\frac{\\pi^4}{\\ell_1^4}.\n \\]\n }\n \\item{We have \n \\[\n a_{\\rm fill}b_{\\rm fill}\\le\\ell_1\\ell_2e^{2r}.\n \\]\n }\n\\end{enumerate}\n\\end{lem}\n\n\\begin{pro}\n\\label{pro:metric on tube}\nThere exists a constant $c>0$ such that the following holds. Let $V$ be a solid torus. Suppose that the boundary $\\partial V$ is equipped with a flat metric of area $A$. Assume that the length of the flat geodesic representative on $\\partial V$ of the meridian $\\mu\\subset\\partial V$ is $\\ell>2\\pi$. Then there exists a smooth Riemannian metric $(V,g)$ with the following properties.\n\\begin{itemize}\n \\item{The metric $g$ is a hyperbolic cusp metric in a collar of $\\partial V$, the boundary $\\partial V$ is a horospherical section of the cusp, and the restriction of $g$ to the boundary agrees with the prescribed flat metric on $\\partial V$.}\n \\item{Denote by $S_g$ the scalar curvature of the metric $g$. We have\n \\[\n \\int_V{\\left(\\frac{|S_g|}{6}\\right)^{3/2}{\\rm dvol}_g}\\le\\frac{A}{2}\\left(1-\\frac{\\pi^2}{\\ell^2}\\right)\\left(1+c\\frac{\\pi^4}{\\ell^4}\\right).\n \\]\n }\n\\end{itemize} \n\\end{pro}\n\n\\begin{lem}\n\\label{lem:functiondrill}\nThere exists a constant $c>0$ such that the following holds. For every $R>0$ there exist functions $a_{\\rm drill},b_{\\rm drill}:\\mb{R}\\to\\mb{R}$ such that \n\\begin{enumerate}\n    \\item{$a_{\\rm drill},b_{\\rm drill}>0$.}\n    \\item{$a_{\\rm drill}(r)=\\sinh(r),b_{\\rm drill}(r)=\\cosh(r)$ on $(2R/3,\\infty)$.}\n    \\item{$a_{\\rm drill}(r)=\\sinh(R)e^r,\\cosh(R)b_{\\rm drill}(r)=e^r$ on $(-\\infty,R/3)$.}\n    \\item{On $[R/3,2R/3]$ we have\n    \\[\n    \\left|\\frac{a_{\\rm drill}''}{a_{\\rm drill}}+\\frac{b_{\\rm drill}''}{b_{\\rm drill}}+\\frac{a_{\\rm drill}'b_{\\rm drill}'}{a_{\\rm drill}b_{\\rm drill}}\\right|\\le c\\frac{1}{R^2}.\n    \\]\n    }\n    \\item{We have $a_{\\rm drill}b_{\\rm drill}\\le \\sinh(R)\\cosh(R)e^{2r}$.}\n\\end{enumerate}\n\\end{lem}\n\n\\begin{thmA}\n\\label{thm:main}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a closed hyperbolic 3-manifold and let $\\gamma\\subset M$ be a simple closed geodesic. Suppose that $\\gamma$ has length $\\ell$ and has an embedded tubular neighborhood of radius at least $R\\in(0,1)$. Then the unique hyperbolic metric on $M-\\gamma$ has volume\n\\[\n{\\rm vol}(M-\\gamma)\\le{\\rm vol}(M)+c\\frac{\\ell}{R}.\n\\]\n\\end{thmA}", "post_theorem_intro_text_len": 6232, "post_theorem_intro_text": "\\begin{corA}\n\\label{cor:main}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a closed hyperbolic 3-manifold and let $\\gamma\\subset M$ be a shortest non-trivial closed geodesic in $M$. Then the unique hyperbolic metric on $M-\\gamma$ has volume\n\\[\n{\\rm vol}(M-\\gamma)\\le{\\rm vol}(M)+c\\ell\n\\]\nwhere $\\ell=2\\cdot{\\rm inj}(M)$ is the length of $\\gamma$.\n\\end{corA}\n\nSimilar estimates on the volume increase under drilling operations on hyperbolic 3-manifolds were already considered by Bridgeman \\cite{B}, Agol \\cite{A}, Agol, Storm, Thurston, and Dunfield \\cite[Theorem 10.1]{AST} where they proved the existence of explicit functions $f,g>1$ such that ${\\rm vol}(M-\\gamma)\\le f(R){\\rm vol}(M)+g(R)\\ell$. So, the main achievement of this note is to get rid of the multiplicative constant in front of ${\\rm vol}(M)$. This shows that the increase in volume only depends on the {\\em local} geometry around $\\gamma$. \n\nTheorem \\ref{thm:main} goes in the direction of a conjecture of Bridgeman \\cite{B}, which asks whether there is a linear upper bound on the volume increase under drilling operations. While very good estimates are available when the drilled geodesics are short (see the work of Hodgson and Kerckhoff \\cite{HK}), it is still unclear what happens for long geodesics. Note that as long as we can uniformly bound from below $R$ the increase in volume is at most linear. This leads immediately to Corollary \\ref{cor:main}. In fact, it is well-known that a shortest geodesic in a closed hyperbolic 3-manifold has a tubular neighborhood of radius at least $\\ell/4$ where $\\ell$ is the length of the geodesic (see \\cite[Proposition 1.11]{GMT}). Combined with standard consequences of Margulis Lemma, this provides a universal constant $R_0>0$ such that a shortest geodesic in a closed hyperbolic 3-manifold $M$ always has a tubular neighborhood of radius $R\\ge R_0$ (for much more refined estimates see \\cite[Theorem 4.1]{GMT}). \n\nLet us mention the fact that the restriction to simple geodesics in closed hyperbolic 3-manifolds, rather than dealing with geodesic links in arbitrary finite volume ones, is not important. A straightforward generalization of Theorem \\ref{thm:main} holds in this larger setting with exactly the same arguments and more robust bookkeeping. We chose to restrict ourselves to the simple case to streamline the exposition. \n\n\\subsection*{Filling}\nNow we move to fillings. We give a new proof of the following lower bound due to Hodgson and Kerckhoff \\cite{HK} using completely different techniques.\n\n\\begin{thmA}\n\\label{thm:main'}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a finite volume hyperbolic 3-manifold with one cusp. Let $C$ be a cusp neighborhood of $M$ bounded by the horospherical torus $\\partial C$. Consider a simple closed geodesic $\\mu\\subset\\partial C$ for the intrinsic flat metric on $\\partial C$. If the length of $\\mu$ is $\\ell>\\sqrt{c}\\pi$ and the area of $\\partial C$ is $A$ then the Dehn filling of $M-C$ with slope $\\mu$, denoted by $M_\\mu$, has a hyperbolic metric of volume\n\\[\n{\\rm vol}(M_\\mu)+\\frac{A}{2}\\frac{\\pi^2}{\\ell^2}\\left(1-c\\frac{\\pi^2}{\\ell^2}\\right)\\le{\\rm vol}(M).\n\\]\n\\end{thmA}\n\nWe remark that our assumptions differ slightly from the ones of \\cite{HK} as we ask that $\\ell\\ge\\sqrt{c}\\pi$ whereas \\cite{HK} requires that $\\ell^2/A$ is bigger than a universal constant. \n\nThe problem of understanding how much the volume decreases under filling is better understood compared to the drilling case and has been studied with various methods. Thurston \\cite[Chapter 6]{ThuNotes} showed that ${\\rm vol}(M_\\mu)<{\\rm vol}(M)$ and that for every {\\em fixed} $M$ we have ${\\rm vol}(M_\\mu)\\to{\\rm vol}(M)$ as $\\ell\\to\\infty$. Neumann and Zagier \\cite{NZ} then clarified, for every {\\em fixed} $M$, the asymptotic behavior of ${\\rm vol}(M)-{\\rm vol}(M_\\mu)$ as a function of $A/\\ell^2$. Later, Hodgson and Kerckhoff \\cite{HK}, gave a {\\em universal} bound of the form ${\\rm vol}(M)-{\\rm vol}(M_\\mu)= A\\pi^2/2\\ell^2+O(A^2/\\ell^4)$.\n\nThe restriction to Dehn fillings of hyperbolic 3-manifolds of finite volume with one cusp rather than allowing multiple Dehn fillings of a selection of cusps of an arbitrary finite volume hyperbolic 3-manifold is not really important. A straightforward generalization of Theorem \\ref{thm:main'} holds in that larger setting with the same proof but more notation. In order to keep the exposition as linear and short as possible, we restricted ourselves to the simple setup.\n\n\\subsection*{On the proofs}\nLastly, a few words on the proofs. \n\nTo prove Theorem \\ref{thm:main}, we follow the approach of \\cite[Theorem 10.1]{AST} with a difference. We first change the metric of $M$ inside a tubular neighborhood $T$ of $\\gamma$ interpolating between the metric on the tube and the metric on a suitable hyperbolic cusp. We do so by keeping under control the scalar curvature (see Lemma \\ref{lem:functiondrill}). Then we compare the volume of the modified metric with that of the hyperbolic structure on $M-\\gamma$. Instead of relying on Perelman's monotonicity formula, we exploit some other consequences of Perelman's work \\cite{Pe1,Pe2}, namely, the exact computation of the sigma invariant of closed hyperbolic 3-manifolds as achieved by Anderson \\cite{And} and Kleiner and Lott \\cite{KL}. This allows us to keep track of the fact that we {\\em did not change} the metric outside the tubular neighborhood $T$ and obtain an additive error rather than a multiplicative one in the volume increase. \n\nThe proof of Theorem \\ref{thm:main'} follows exactly the same strategy. We change the metric of $M$ in a cusp neighborhood $C$ as in Gromov-Thurston's $2\\pi$-Theorem \\cite[Theorem 9]{BH} (of which we prove a quantitative version, see Proposition \\ref{pro:metric on tube}). This time the interpolation between the metric on the cusp and the metric on a tube is more delicate as we want to keep under control the scalar curvature up to higher order (see Lemma \\ref{lem:functionfill}). Then we use the sigma invariant comparison to conclude. \n\n\\subsection*{Acknowledgements}\nI warmly thank Ian Agol and Martin Bridgeman for useful discussions and generous feedback on a first draft of this article.", "sketch": "To prove Theorem~\\ref{thm:main}, the authors “follow the approach of \\cite[Theorem 10.1]{AST} with a difference.” The steps described are:\n\\begin{itemize}\n\\item “We first change the metric of $M$ inside a tubular neighborhood $T$ of $\\gamma$ interpolating between the metric on the tube and the metric on a suitable hyperbolic cusp.”\n\\item This is done “by keeping under control the scalar curvature (see Lemma~\\ref{lem:functiondrill}).”\n\\item “Then we compare the volume of the modified metric with that of the hyperbolic structure on $M-\\gamma$.”\n\\item “Instead of relying on Perelman's monotonicity formula,” they “exploit some other consequences of Perelman's work \\cite{Pe1,Pe2}, namely, the exact computation of the sigma invariant of closed hyperbolic 3-manifolds as achieved by Anderson \\cite{And} and Kleiner and Lott \\cite{KL}.”\n\\item “This allows us to keep track of the fact that we \\emph{did not change} the metric outside the tubular neighborhood $T$ and obtain an additive error rather than a multiplicative one in the volume increase.”\n\\end{itemize}", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:main}\nThere exists a constant $c>0$ such that the following holds. Let $M$ be a closed hyperbolic 3-manifold and let $\\gamma\\subset M$ be a simple closed geodesic. Suppose that $\\gamma$ has length $\\ell$ and has an embedded tubular neighborhood of radius at least $R\\in(0,1)$. Then the unique hyperbolic metric on $M-\\gamma$ has volume\n\\[\n{\\rm vol}(M-\\gamma)\\le{\\rm vol}(M)+c\\frac{\\ell}{R}.\n\\]", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let $M$ be a closed hyperbolic $3$-manifold, and let $\\gamma\\subset M$ be a simple closed geodesic of length $\\ell$. Assume that $\\gamma$ has an embedded tubular neighborhood of radius at least $R$, where $R\\in(0,1)$. For the drilled manifold $M-\\gamma$, equipped with its unique hyperbolic metric, which quantitative volume estimate holds?", "correct_choice": {"label": "A", "text": "There exists a constant $c>0$, independent of $M$, $\\gamma$, $\\ell$, and $R$, such that\n\\[\n{\\rm vol}(M-\\gamma)\\le {\\rm vol}(M)+c\\frac{\\ell}{R}.\n\\]"}, "choices": [{"label": "B", "text": "There exists a constant $c>0$, independent of $M$, $\\gamma$, $\\ell$, and $R$, such that\n\\[\n{\\rm vol}(M-\\gamma)\\le {\\rm vol}(M)+c\\frac{\\ell}{R^2}.\n\\]"}, {"label": "C", "text": "There exists a constant $c>0$, independent of $M$, $\\gamma$, $\\ell$, and $R$, such that\n\\[\n{\\rm vol}(M-\\gamma)\\le {\\rm vol}(M)+c\\ell.\n\\]"}, {"label": "D", "text": "For every $R\\in(0,1)$ there exists a constant $c_R>0$, depending only on $R$, such that for every closed hyperbolic $3$-manifold $M$ and every simple closed geodesic $\\gamma\\subset M$ of length $\\ell$ with an embedded tubular neighborhood of radius at least $R$,\n\\[\n{\\rm vol}(M-\\gamma)\\le {\\rm vol}(M)+c_R\\ell.\n\\]"}, {"label": "E", "text": "There exists a constant $c>0$, independent of $M$, $\\gamma$, $\\ell$, and $R$, such that\n\\[\n{\\rm vol}(M-\\gamma)\\le \\left(1+c\\frac{\\ell}{R}\\right){\\rm vol}(M).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "linear dependence on inverse tube radius", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the factor $1/R$ using only that $R<1$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "uniformity of the constant with respect to $R$", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "additive error versus multiplicative volume factor", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem only defines terms and notation; it does not reveal the specific bound, the dependence on R, or the quantifier structure that distinguishes the correct choice."}, "TAS": {"score": 2, "justification": "Although the correct option is theorem-like, the question is not a bare restatement: the choices differ by domain restrictions on R, quantifier order, and the error-term dependence, so the student must discriminate among competing formulations."}, "GPS": {"score": 1, "justification": "The item requires some reasoning about how changing R-ranges, swapping quantifiers, or replacing ell/R by ell or ell R affects validity. However, it still leans more toward precise theorem recognition than toward substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they target common failure modes such as overextending parameter ranges, dropping a crucial factor, altering quantifier order, and using the wrong asymptotic dependence on R."}, "total_score": 7, "overall_assessment": "A strong MCQ with no answer leakage and high-quality near-miss distractors. It avoids tautology reasonably well, though it tests careful theorem discrimination more than deep generative reasoning."}}
{"id": "2602.10700v1", "paper_link": "http://arxiv.org/abs/2602.10700v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:global_existence_cauchy}\nLet $N=2$ or $N=3$. Assume that $\\gamma$ satisfies\n\\begin{equation}\n    \\begin{cases}\n        \\gamma \\in [1, \\infty) & \\text{if } N=2, \\\\\n        \\gamma \\in [1, 8/3) & \\text{if } N=3.\n    \\end{cases}\n\\end{equation}\nLet $\\bar{\\rho} > 0$ be the far field behavior of density. Assume that the initial data $(\\rho_0, u_0)$ satisfies\n\\begin{equation}\n    0 < \\underline{\\varrho} \\le \\rho_0 \\le \\bar{\\varrho}, \\quad (\\rho_0 - \\bar{\\rho}) \\in H^3(\\mathbb{R}^N), \\quad u_0 \\in H^2(\\mathbb{R}^N),\n\\end{equation}\nwhere $\\underline{\\varrho}$ and $\\bar{\\varrho}$ are positive constants. Then the Cauchy problem  \\eqref{NSK_simplified}-\\eqref{far field} admits a unique global strong solution $(\\rho, u)$ satisfying for any $0 < T < \\infty$ and $(x,t) \\in \\mathbb{R}^N \\times [0, T]$,\n\\begin{equation}\n    (C(T))^{-1} \\le \\rho(x,t) \\le C(T),\n\\end{equation}\nand\n\\begin{equation}\n\\begin{aligned}\n    &(\\rho - \\bar{\\rho}) \\in C([0, T]; H^3) \\cap L^2(0, T; H^4), \\quad \\rho_t \\in C([0, T]; H^1) \\cap L^2(0, T; H^2), \\\\\n    &u \\in C([0, T]; H^2) \\cap L^2(0, T; H^3), \\quad u_t \\in L^\\infty(0, T; L^2) \\cap L^2(0, T; H^1),\n\\end{aligned}\n\\end{equation}\nwhere the constant $C(T) > 0$ depends on the initial data and $T$.", "start_pos": 12442, "end_pos": 13706, "label": "thm:global_existence_cauchy"}, "ref_dict": {"NSK": "\\begin{equation}\\label{NSK} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \n\\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) =\\nabla \\cdot \\mathbb{S} + \\nabla \\cdot \\mathbb{K}. \\end{cases} \\end{equation}", "PRO3.4": "\\begin{proposition}\\label{PRO3.4}\nFor any $\\gamma \\in (1, \\frac{8}{3})$, there exists $q \\in (1, 4)$ such that\n\\begin{equation}\n     \\gamma \\le \\frac{2q+6}{q+2}.\n\\end{equation}\n    For this fixed $q$, there exists a positive constant $C$, depending on $T$, $q$, $\\gamma$, $E_0$, and the initial data $\\|\\rho_0^{1/(q+2)} v_0\\|_{L^{q+2}}$, such that\n\\begin{equation}\\label{L^{q+2}}\n\\sup_{0 \\le t \\le T} \\|\\rho^{\\frac{1}{q+2}} v\\|_{L^{q+2}} \\le C.\n\\end{equation}\n\\end{proposition}", "NSK_simplified": "\\begin{equation}\\label{NSK_simplified} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) = \\nabla \\cdot \\left(2 \\rho \\mathcal{D}(u) \\right) + \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho). \\end{cases} \\end{equation}", "far field": "\\begin{equation}\\label{far field}\n    \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}"}, "pre_theorem_intro_text_len": 9174, "pre_theorem_intro_text": "In this paper, we are concerned with the global well-posedness of the compressible Navier-Stokes-Korteweg equations in the whole space $\\mathbb{R}^N$. This system describes the motion of a viscous compressible fluid endowed with internal capillarity and reads as follows\n\\begin{equation}\\label{NSK} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \n\\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) =\\nabla \\cdot \\mathbb{S} + \\nabla \\cdot \\mathbb{K}. \\end{cases} \\end{equation}\nThe viscous stress tensor $\\mathbb{S}$ is given by Newton's rheological law\n\\begin{equation}\n    \\mathbb{S} = 2\\mu(\\rho) \\mathcal{D}(u) + \\lambda(\\rho) \\nabla \\cdot u \\mathbb{I},\n\\end{equation}\nwhere $\\mathcal{D}(u) = \\frac{1}{2}(\\nabla u + \\nabla u^T)$ denotes the symmetric deformation tensor, and $\\mathbb{I}$ is the identity matrix in $\\mathbb{R}^N$. The scalar functions $\\mu(\\rho)$ and $\\lambda(\\rho)$ represent the shear and bulk viscosity coefficients, respectively, which satisfy the physical condition $\\mu(\\rho) > 0$ and $2\\mu(\\rho) + N\\lambda(\\rho) \\ge 0$.\nIn order to ensure the stability of the system and utilize the BD entropy structure, we assume the viscosity coefficients satisfy the relation $$\\lambda(\\rho) = 2(\\rho \\mu'(\\rho) - \\mu(\\rho)).$$\nIn the context of the Korteweg theory, the capillarity tensor $\\nabla \\cdot\\mathbb{K}$ is typically defined as\n\\begin{equation}\n    \\nabla \\cdot \\mathbb{K} = \\nabla \\left( \\rho \\kappa(\\rho) \\Delta \\rho + \\frac{\\kappa(\\rho) + \\rho \\kappa'(\\rho)}{2} |\\nabla \\rho|^2 \\right) - \\nabla \\cdot \\left( \\kappa(\\rho) \\nabla \\rho \\otimes \\nabla \\rho \\right).\n\\end{equation}\nwhere $\\kappa(\\rho) > 0$ is the coefficient of capillarity.\\par\nThe study of capillary fluids originates from the pioneering work of Van der Waals and Korteweg \\cite{Korteweg,J.F. Van derWaals}. In their theory of capillarity, the fluid energy is assumed to depend not only on the standard thermodynamic variables but also on the gradient of the density. This concept was later formalized in the modern language of continuum mechanics by Dunn and Serrin in the 1980s, leading to the so-called Korteweg-type models. \\par\n  When $\\kappa(\\rho) = 0$, $\\mu=\\rho$ and $\\lambda=0$, system \\eqref{NSK} reduces to the well-known viscous shallow water equations. \nSignificant progress has been made for the viscous shallow water equations in recent years. Regarding general multi-dimensional initial data, significant breakthroughs concerning the global well-posedness of weak solutions were achieved by Li-Xin \\cite{Li-Xin} and Vasseur-Yu \\cite{Vasseur-Yu}. Independently, these authors established the global existence of weak solutions for the compressible system with $\\mu(\\rho) = \\rho$ and $\\lambda(\\rho) = 0$, admitting arbitrarily large data and vacuum states. More specifically, they obtained global weak solutions for $\\gamma \\in (1, \\infty)$ when $N=2$, and for $\\gamma \\in (1, 3)$ when $N=3$. It is worth noting that Li-Xin \\cite{Li-Xin} extended their analysis to a wider class of viscosity coefficients satisfying the BD entropy relation. Concerning the global smooth solutions for multi-dimensional shallow water equations with arbitrarily large initial data, Huang-Meng-Zhang \\cite{Huang-Meng-Zhang-V} pioneered the proof of global classical solutions for the two-dimensional initial-boundary value problem under the assumption of radial symmetry with $\\gamma\\ge\\frac{3}{2}$. Subsequently, Gu-Huang \\cite{Gu-Huang} extended the range of the exponent to $\\gamma > 1$ and generalized the result to the three-dimensional case with $1 < \\gamma < 3$. Concurrently, the global existence of large solutions for the associated Cauchy problem was proved independently by Chen-Zhang-Zhu \\cite{Chen-Zhang-Zhu}. Moreover, Huang-Meng-Zhang \\cite{Huang-Meng-Zhang-V} successfully proved the global classical well-posedness for general isentropic compressible Navier-Stokes equations satisfying the BD entropy condition in 2D and 3D, treating the shallow water model as a specific instance.\\par\n When $\\kappa(\\rho) >0$, we review some related works regarding the well-posedness of the Navier-Stokes-Korteweg system with general viscosity coefficients. For the one-dimensional case, considering the system with specific density-dependent viscosity $\\mu(\\rho) = \\rho$ and capillarity $\\kappa(\\rho) = \\rho^{-1}$, Charve-Haspot \\cite{Charve-Haspot} established the existence of global strong solutions in the whole space, allowing for large non-vacuum initial data. Furthermore, they demonstrated that these solutions converge to the entropic weak solutions of the compressible Euler equations. Moreover, Germain-LeFloch \\cite{Germain-LeFloch} proved the global existence of finite energy weak solutions for the Cauchy problem with general density-dependent coefficients and demonstrated their convergence to the entropy solutions of the Euler system. For the case with power-law viscosity $\\mu(\\varrho) = \\varrho^\\alpha$ and capillarity $\\kappa(\\varrho) = \\varrho^\\beta$ satisfying specific conditions, Antonelli-Bresch-Spirito \\cite{Antonelli-Bresch-Spirito} established the existence of global weak solutions for the periodic problem with large data. Furthermore, investigating the Cauchy problem in Lagrangian coordinates, Chen et al. \\cite{Chen-Chai} obtained global classical solutions for large initial data away from vacuum, considering density-dependent viscosity and capillarity.  For the multi-dimensional case, building upon their earlier local theory \\cite{Hattori-Li}, Hattori-Li \\cite{Hattori-Li-2} established the global existence of solutions for the constant-coefficient Cauchy problem with small, non-vacuum initial data. For the case where both viscosity and capillarity coefficients depend on the density, Danchin-Desjardins \\cite{Danchin_Desjardins} obtained global smooth solutions. Their result holds for small perturbations of a non-vacuum state in functional spaces that are critical with respect to the physical energy. Bresch and Desjardins \\cite{B-D} investigated the two-dimensional viscous shallow water equations extended by a capillary term. They established the global existence of weak solutions in the presence of vacuum and demonstrated their convergence towards the strong solution of the viscous quasi-geostrophic system with a free surface. \\par\nHowever, the global existence of strong solutions with arbitrarily large initial data for the multi-dimensional Navier-Stokes-Korteweg system has long remained an open problem. It was not until recently that the first author Huang \\cite{Huang-Meng-Zhang} in his newly preprint resolved this by establishing global strong solutions on the two- and three-dimensional periodic torus, provided that the initial density is bounded from above and below. However, extending their method to the whole space presents new difficulties. In this paper, we overcome these obstacles to establish the global existence of strong solutions for the multi-dimensional Navier-Stokes-Korteweg system with large initial data in the whole space. By employing the novel critical inequality established by the first author in his newly preprint \\cite{Huang-Meng-Zhang}, we successfully establish the lower bound of the density by employing a novel De Giorgi iteration method. It is worth noting that our work does not require any radial symmetry assumption. To the best of our knowledge, this can be regarded as the first result concerning global large solutions for the corresponding Cauchy problem.\n\\par\nThroughout the rest of this paper, we focus on the case of shallow-water viscosity coefficients\n$$\\mu(\\rho) = \\rho, \\quad \\lambda(\\rho) = 0,$$\nand we assume that $\\kappa(\\rho)$ satisfies \n$$\\kappa(\\rho)=\\frac{1}{\\rho}.$$\nWith the specific choice of the capillarity coefficient $\\kappa(\\rho)$, the divergence of the Korteweg tensor takes the following form\n\\begin{equation}\n\\nabla \\cdot \\mathbb{K} = \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho).\n\\end{equation}\nConsequently, under the assumption of shallow-water viscosity ($\\mu(\\rho)=\\rho, \\lambda(\\rho)=0$) and the specific capillarity coefficient $\\kappa(\\rho)=1/\\rho$, equation \\eqref{NSK} can be rewritten as\n\\begin{equation}\\label{NSK_simplified} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) = \\nabla \\cdot \\left(2 \\rho \\mathcal{D}(u) \\right) + \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho). \\end{cases} \\end{equation}\nNow we investigate the global existence of strong solutions to system \\eqref{NSK_simplified} in $\\mathbb{R}^N$, where $N=2, 3$. The system is supplemented with the prescribed initial data $(\\rho_0, u_0)$ satisfying\n\\begin{equation}\n\\rho(x, 0) = \\rho_0(x), \\quad u(x, 0) = u_0(x), \\quad \\text{for } x \\in \\mathbb{R}^N,\n\\end{equation}\nwith far field behavior \n\\begin{equation}\\label{far field}\n    \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}\nWe now state the main result on the global existence of strong solutions to the Cauchy problem for system \\eqref{NSK_simplified}-\\eqref{far field} with arbitrarily large initial data.", "context": "e assumption of radial symmetry with $\\gamma\\ge\\frac{3}{2}$. Subsequently, Gu-Huang \\cite{Gu-Huang} extended the range of the exponent to $\\gamma > 1$ and generalized the result to the three-dimensional case with $1 < \\gamma < 3$. Concurrently, the global existence of large solutions for the associated Cauchy problem was proved independently by Chen-Zhang-Zhu \\cite{Chen-Zhang-Zhu}. Moreover, Huang-Meng-Zhang \\cite{Huang-Meng-Zhang-V} successfully proved the global classical well-posedness for general isentropic compressible Navier-Stokes equations satisfying the BD entropy condition in 2D and 3D, treating the shallow water model as a specific instance.\\par\n When $\\kappa(\\rho) >0$, we review some related works regarding the well-posedness of the Navier-Stokes-Korteweg system with general viscosity coefficients. For the one-dimensional case, considering the system with specific density-dependent viscosity $\\mu(\\rho) = \\rho$ and capillarity $\\kappa(\\rho) = \\rho^{-1}$, Charve-Haspot \\cite{Charve-Haspot} established the existence of global strong solutions in the whole space, allowing for large non-vacuum initial data. Furthermore, they demonstrated that these solutions converge to the entropic weak solutions of the compressible Euler equations. Moreover, Germain-LeFloch \\cite{Germain-LeFloch} proved the global existence of finite energy weak solutions for the Cauchy problem with general density-dependent coefficients and demonstrated their convergence to the entropy solutions of the Euler system. For the case with power-law viscosity $\\mu(\\varrho) = \\varrho^\\alpha$ and capillarity $\\kappa(\\varrho) = \\varrho^\\beta$ satisfying specific conditions, Antonelli-Bresch-Spirito \\cite{Antonelli-Bresch-Spirito} established the existence of global weak solutions for the periodic problem with large data. Furthermore, investigating the Cauchy problem in Lagrangian coordinates, Chen et al. \\cite{Chen-Chai} obtained global classical solutions for large initial data away from vacuum, considering density-dependent viscosity and capillarity. For the multi-dimensional case, building upon their earlier local theory \\cite{Hattori-Li}, Hattori-Li \\cite{Hattori-Li-2} established the global existence of solutions for the constant-coefficient Cauchy problem with small, non-vacuum initial data. For the case where both viscosity and capillarity coefficients depend on the density, Danchin-Desjardins \\cite{Danchin_Desjardins} obtained global smooth solutions. Their result holds for small perturbations of a non-vacuum state in functional spaces that are critical with respect to the physical energy. Bresch and Desjardins \\cite{B-D} investigated the two-dimensional viscous shallow water equations extended by a capillary term. They established the global existence of weak solutions in the presence of vacuum and demonstrated their convergence towards the strong solution of the viscous quasi-geostrophic system with a free surface. \\par\nHowever, the global existence of strong solutions with arbitrarily large initial data for the multi-dimensional Navier-Stokes-Korteweg system has long remained an open problem. It was not until recently that the first author Huang \\cite{Huang-Meng-Zhang} in his newly preprint resolved this by establishing global strong solutions on the two- and three-dimensional periodic torus, provided that the initial density is bounded from above and below. However, extending their method to the whole space presents new difficulties. In this paper, we overcome these obstacles to establish the global existence of strong solutions for the multi-dimensional Navier-Stokes-Korteweg system with large initial data in the whole space. By employing the novel critical inequality established by the first author in his newly preprint \\cite{Huang-Meng-Zhang}, we successfully establish the lower bound of the density by employing a novel De Giorgi iteration method. It is worth noting that our work does not require any radial symmetry assumption. To the best of our knowledge, this can be regarded as the first result concerning global large solutions for the corresponding Cauchy problem.\n\\par\nThroughout the rest of this paper, we focus on the case of shallow-water viscosity coefficients\n$$\\mu(\\rho) = \\rho, \\quad \\lambda(\\rho) = 0,$$\nand we assume that $\\kappa(\\rho)$ satisfies \n$$\\kappa(\\rho)=\\frac{1}{\\rho}.$$\nWith the specific choice of the capillarity coefficient $\\kappa(\\rho)$, the divergence of the Korteweg tensor takes the following form\n\\begin{equation}\n\\nabla \\cdot \\mathbb{K} = \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho).\n\\end{equation}\nConsequently, under the assumption of shallow-water viscosity ($\\mu(\\rho)=\\rho, \\lambda(\\rho)=0$) and the specific capillarity coefficient $\\kappa(\\rho)=1/\\rho$, equation \\eqref{NSK} can be rewritten as\n\\begin{equation}\\label{NSK_simplified} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) = \\nabla \\cdot \\left(2 \\rho \\mathcal{D}(u) \\right) + \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho). \\end{cases} \\end{equation}\nNow we investigate the global existence of strong solutions to system \\eqref{NSK_simplified} in $\\mathbb{R}^N$, where $N=2, 3$. The system is supplemented with the prescribed initial data $(\\rho_0, u_0)$ satisfying\n\\begin{equation}\n\\rho(x, 0) = \\rho_0(x), \\quad u(x, 0) = u_0(x), \\quad \\text{for } x \\in \\mathbb{R}^N,\n\\end{equation}\nwith far field behavior \n\\begin{equation}\\label{far field}\n \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}\nWe now state the main result on the global existence of strong solutions to the Cauchy problem for system \\eqref{NSK_simplified}-\\eqref{far field} with arbitrarily large initial data.\n\n\\begin{equation}\\label{far field}\n    \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}", "full_context": "e assumption of radial symmetry with $\\gamma\\ge\\frac{3}{2}$. Subsequently, Gu-Huang \\cite{Gu-Huang} extended the range of the exponent to $\\gamma > 1$ and generalized the result to the three-dimensional case with $1 < \\gamma < 3$. Concurrently, the global existence of large solutions for the associated Cauchy problem was proved independently by Chen-Zhang-Zhu \\cite{Chen-Zhang-Zhu}. Moreover, Huang-Meng-Zhang \\cite{Huang-Meng-Zhang-V} successfully proved the global classical well-posedness for general isentropic compressible Navier-Stokes equations satisfying the BD entropy condition in 2D and 3D, treating the shallow water model as a specific instance.\\par\n When $\\kappa(\\rho) >0$, we review some related works regarding the well-posedness of the Navier-Stokes-Korteweg system with general viscosity coefficients. For the one-dimensional case, considering the system with specific density-dependent viscosity $\\mu(\\rho) = \\rho$ and capillarity $\\kappa(\\rho) = \\rho^{-1}$, Charve-Haspot \\cite{Charve-Haspot} established the existence of global strong solutions in the whole space, allowing for large non-vacuum initial data. Furthermore, they demonstrated that these solutions converge to the entropic weak solutions of the compressible Euler equations. Moreover, Germain-LeFloch \\cite{Germain-LeFloch} proved the global existence of finite energy weak solutions for the Cauchy problem with general density-dependent coefficients and demonstrated their convergence to the entropy solutions of the Euler system. For the case with power-law viscosity $\\mu(\\varrho) = \\varrho^\\alpha$ and capillarity $\\kappa(\\varrho) = \\varrho^\\beta$ satisfying specific conditions, Antonelli-Bresch-Spirito \\cite{Antonelli-Bresch-Spirito} established the existence of global weak solutions for the periodic problem with large data. Furthermore, investigating the Cauchy problem in Lagrangian coordinates, Chen et al. \\cite{Chen-Chai} obtained global classical solutions for large initial data away from vacuum, considering density-dependent viscosity and capillarity. For the multi-dimensional case, building upon their earlier local theory \\cite{Hattori-Li}, Hattori-Li \\cite{Hattori-Li-2} established the global existence of solutions for the constant-coefficient Cauchy problem with small, non-vacuum initial data. For the case where both viscosity and capillarity coefficients depend on the density, Danchin-Desjardins \\cite{Danchin_Desjardins} obtained global smooth solutions. Their result holds for small perturbations of a non-vacuum state in functional spaces that are critical with respect to the physical energy. Bresch and Desjardins \\cite{B-D} investigated the two-dimensional viscous shallow water equations extended by a capillary term. They established the global existence of weak solutions in the presence of vacuum and demonstrated their convergence towards the strong solution of the viscous quasi-geostrophic system with a free surface. \\par\nHowever, the global existence of strong solutions with arbitrarily large initial data for the multi-dimensional Navier-Stokes-Korteweg system has long remained an open problem. It was not until recently that the first author Huang \\cite{Huang-Meng-Zhang} in his newly preprint resolved this by establishing global strong solutions on the two- and three-dimensional periodic torus, provided that the initial density is bounded from above and below. However, extending their method to the whole space presents new difficulties. In this paper, we overcome these obstacles to establish the global existence of strong solutions for the multi-dimensional Navier-Stokes-Korteweg system with large initial data in the whole space. By employing the novel critical inequality established by the first author in his newly preprint \\cite{Huang-Meng-Zhang}, we successfully establish the lower bound of the density by employing a novel De Giorgi iteration method. It is worth noting that our work does not require any radial symmetry assumption. To the best of our knowledge, this can be regarded as the first result concerning global large solutions for the corresponding Cauchy problem.\n\\par\nThroughout the rest of this paper, we focus on the case of shallow-water viscosity coefficients\n$$\\mu(\\rho) = \\rho, \\quad \\lambda(\\rho) = 0,$$\nand we assume that $\\kappa(\\rho)$ satisfies \n$$\\kappa(\\rho)=\\frac{1}{\\rho}.$$\nWith the specific choice of the capillarity coefficient $\\kappa(\\rho)$, the divergence of the Korteweg tensor takes the following form\n\\begin{equation}\n\\nabla \\cdot \\mathbb{K} = \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho).\n\\end{equation}\nConsequently, under the assumption of shallow-water viscosity ($\\mu(\\rho)=\\rho, \\lambda(\\rho)=0$) and the specific capillarity coefficient $\\kappa(\\rho)=1/\\rho$, equation \\eqref{NSK} can be rewritten as\n\\begin{equation}\\label{NSK_simplified} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) = \\nabla \\cdot \\left(2 \\rho \\mathcal{D}(u) \\right) + \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho). \\end{cases} \\end{equation}\nNow we investigate the global existence of strong solutions to system \\eqref{NSK_simplified} in $\\mathbb{R}^N$, where $N=2, 3$. The system is supplemented with the prescribed initial data $(\\rho_0, u_0)$ satisfying\n\\begin{equation}\n\\rho(x, 0) = \\rho_0(x), \\quad u(x, 0) = u_0(x), \\quad \\text{for } x \\in \\mathbb{R}^N,\n\\end{equation}\nwith far field behavior \n\\begin{equation}\\label{far field}\n \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}\nWe now state the main result on the global existence of strong solutions to the Cauchy problem for system \\eqref{NSK_simplified}-\\eqref{far field} with arbitrarily large initial data.\n\n\\begin{equation}\\label{far field}\n    \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}\n\n\\begin{document}\n\\maketitle\n\\begin{abstract}\nIn this paper, we establish global strong solutions for arbitrarily large initial data to the multi-dimensional compressible Navier-Stokes–Korteweg system, also referred to as the quantum Navier–Stokes equations, originally derived by Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95–133, 1985]. Specifically, we prove the existence of global strong solutions for arbitrarily large initial data in the case $N=2$ when $\\gamma \\ge 1$, and $N=3$ with $1 \\le \\gamma < 8/3$ for the associated Cauchy problem. By employing techniques from Littlewood–Paley theory, range truncation analysis, refined Nash-Moser and De Giorgi iteration methods, we derive positive upper and lower bounds for the density. As a consequence, we are able to treat the whole-space case with strictly positive far-field density. To the best of our knowledge, this is the first result that establishes global strong solutions for physically relevant compressible Navier–Stokes equations in the whole space, without imposing any symmetry or special geometric assumptions on the initial data.\n\\end{abstract}\n\\bigskip\n\\noindent \\textbf{Keywords:} compressible Navier-Stokes-Korteweg system; quantum Navier-Stokes system; global large strong solutions; Nash-Moser iteration; De Giorgi iteration; Cauchy problem.\n\nIn the first phase, we work within the framework of Besov spaces, utilizing maximal regularity estimates for parabolic equations to secure the upper bound. The second and most crucial step is establishing the density lower bound. Although Haspot \\cite{Haspot 1} investigated the isothermal NSK system, his approach failed to yield a lower bound for the density. Our strategy draws inspiration from a key inequality introduced by the first author in \\cite{Huang-Meng-Zhang}. However, the technique in \\cite{Huang-Meng-Zhang} cannot be directly applied to the whole space problem with non-vacuum far-field conditions. To overcome this obstacle, we have developed a novel truncated De Giorgi iteration method.\n\n\\begin{proof}\nBy H\\\"{o}lder's inequality, Sobolev embedding and \\eqref{eq:19}, we have\n\\begin{equation}\\label{the initial estimate for Psi}\n\\begin{aligned}\n \\Psi(r(p+2)) &= \\int^{T}_{0}\\int_{\\mathbb{R}^{3}}\\rho|v|^{\\frac{5}{3}(p+2)}\\,dx\\,dt \\\\\n &\\leq \\int^{T}_{0}\\left( \\int \\rho^{\\frac{3}{2}}|v|^{p+2}\\,dx \\right)^{\\frac{2}{3}} \\left( \\int|v|^{3(p+2)}\\,dx \\right)^{\\frac{1}{3}}\\,dt \\\\\n &\\leq \\|\\rho\\|_{L_{T}^{\\infty}L^{\\infty}}^{\\frac{1}{3}} \\sup_{0 \\le t \\le T}\\left( \\int \\rho|v|^{p+2}\\,dx \\right)^{\\frac{2}{3}} \\int^{T}_{0} \\left\\| |v|^{\\frac{p+2}{2}} \\right\\|_{L^{6}}^{2}\\,dt \\\\\n &\\leq C \\sup_{0 \\le t \\le T}\\left( \\int \\rho|v|^{p+2}\\,dx\\right)^{\\frac{2}{3}} \\|\\rho^{-1}\\|_{L_{T}^{\\infty}L^{\\infty}} \\\\\n &\\quad \\times \\left( \\int^{T}_{0}\\int \\rho \\left| \\nabla|v|^{\\frac{p+2}{2}} \\right|^{2}\\,dx\\,dt + \\int^{T}_{0}\\int \\rho|v|^{p+2}\\,dx\\,dt \\right) \\\\\n &\\le CV_{T} \\left( \\sup_{0 \\le t \\le T}\\int \\rho|v|^{p+2}\\,dx + \\int^{T}_{0}\\int \\rho \\left| \\nabla|v|^{\\frac{p+2}{2}} \\right|^{2}\\,dx\\,dt \\right)^{r}.\n\\end{aligned}\n\\end{equation}\nTo estimate the right hand side, multiplying $\\eqref{parabolic_system}_2$ by $|v|^{p}v$ and integrating by parts yields\n\\begin{equation}\n \\frac{1}{p+2} \\frac{d}{dt}\\int \\rho|v|^{p+2}dx+p\\int \\rho|v|^{p}\\left| \\nabla \\left|v \\right| \\right| ^{2}dx+\\int \\rho|v|^{p}|\\nabla v|^{2}dx=\\int P(\\rho)\\nabla\\cdot\\left(\\left| v \\right| ^{p}v\\right)dx.\n\\end{equation}\nBy Young's inequality and \\eqref{eq:19}, we have\n\\begin{equation}\n \\begin{aligned}\n \\int P(\\rho)\\nabla \\cdot\\left(\\left| v \\right| ^{p}v\\right)dx\n=& \\int \\rho^{\\gamma}(\\left| v \\right| ^{p}\\nabla \\cdot v+ v\\cdot { \\nabla \\left( \\left| v \\right| ^{p} \\right) } )dx\\\\\n\\leq& \\int\\rho^{\\gamma}\\left| v \\right| ^{p} \\left| \\nabla v \\right|dx + p\\int \\rho^{\\gamma}|v|^{p}\\left| \\nabla \\left| v \\right| \\right|dx \\\\\n\\leq & \\frac{1}{2} \\int \\rho|v|^{p}|\\nabla v|^{2}dx+ \\frac{p}{2}\\int \\rho|v|^{p}|\\nabla \\left| v \\right| |^{2} dx+ C(p+1)\\int \\rho^{2\\gamma-1}|v|^{p}dx\\\\\n \\leq& \\frac{1}{2}\\int \\rho |v|^{p}\\left| \\nabla v \\right| ^{2}dx+ \\frac{p}{2}\\int \\rho|v|^{p}|\\nabla \\left| v \\right| |^{2}dx + C(p+1)\\left\\lVert \\rho \\right\\rVert _{L^{\\infty}}^{2(\\gamma-1)}\\int \\rho|v|^{p}dx\\\\ \n \\leq& \\frac{1}{2}\\int \\rho |v|^{p}\\left| \\nabla v \\right| ^{2}dx+ \\frac{p}{2}\\int \\rho|v|^{p}|\\nabla \\left| v \\right| |^{2}dx\\\\\n &+ C(p+1)\\sup_{0 \\le t \\le T}\\left( \\int \\rho|v|^{2}dx \\right)^{\\frac{2}{p}}\\left( \\int \\rho|v|^{p+2} dx\\right)^{1- \\frac{2}{p}}.\n\\end{aligned}\n\\end{equation}\nCombining \\eqref{v energy}, we have\n\\begin{equation}\n \\frac{1}{p+2}\\frac{d}{dt}\\int \\rho |v|^{p+2}dx + \\frac{1}{2}\\int \\rho |v|^p |\\nabla v|^2dx + \\frac{p}{2}\\int \\rho |v|^p |\\nabla |v||^2dx \\leq C(p+1)\\left( \\int \\rho |v|^{p+2} dx \\right)^{1-\\frac{2}{p}}.\n\\end{equation}\nIntegrating over $t\\in[0,T]$, we obtain\n\\begin{equation}\n \\begin{aligned}\n&\\sup_{0 \\le t \\le T}\\frac{1}{p+2}\\int{\\rho}|v|^{p+2}dx+ \\frac{1}{2}\\int_{0}^{T}\\int \\rho |v|^p |\\nabla v|^2dxdt + \\frac{p}{2}\\int_{0}^{T}\\int \\rho |v|^p |\\nabla |v||^2dxdt \\\\\n\\leq &C(p+1)\\int^{T}_{0}\\left( \\int \\rho |v|^{p+2} dx \\right)^{1-\\frac{2}{p}}dt+\\frac{2}{p+2}\\int \\rho_{0}|v_{0}|^{p+2}\\\\\n\\leq& C(p+1){T^{*}}^{\\frac{2}{p}}\\left( \\int^{T}_{0}\\int \\rho|v|^{p+2}dxdt \\right)^{1- \\frac{2}{p}}+ \\frac{2}{p+2}\\int \\rho_{0}|v_{0}|^{p+2}.\n\\end{aligned}\n\\end{equation}\nSubstituting this into the RHS of \\eqref{the initial estimate for Psi}, we get\n\\begin{equation}\n \\begin{aligned}\n\\Psi(r(p+2))&\\leq C{V}_{T} \\left( C(p+2)^2 \\left( \\int^{T}_{0}\\int \\rho |v|^{p+2} dxdt \\right)^{\\frac{p-2}{p}} + C \\int \\rho_0 |v_0|^{p+2} dx \\right) ^{r }\\\\\n&\\leq C{V}_{T} (p+2)^{2r } \\left( \\int^{T}_{0}\\int \\rho |v|^{p+2} dxdt \\right)^{r } + C{V}_{T} (p+2)^{2r } + C{V}_{T} \\left( \\int \\rho_0 |v_0|^{p+2} dx \\right)^{r }.\n\\end{aligned}\n\\end{equation}\nDenoting the constant in the RHS as $C_{3}\\geq1$, and noting that\n\\begin{equation}\n \\left\\| \\rho_0^{\\frac{1}{p+2}} v_0 \\right\\|_{L^{p+2}} \\leq \\left\\| \\rho_0^{\\frac{1}{2}} v_0 \\right\\|_{L^2}^{\\frac{p}{p+2}} \\|v_0\\|_{L^\\infty}^{\\frac{2}{p+2}} \\leq \\left\\| \\rho_0^{\\frac{1}{2}} v_0 \\right\\|_{L^2} + \\|v_0\\|_{L^\\infty},\n\\end{equation}\nthe proof is complete.\n\\end{proof}\nThe following proposition plays a pivotal role in deriving the positive lower bound for the density, as it establishes a crucial link between the density and the effective velocity.\n\\begin{proposition}\nThere exists a constant $c_{v}\\geq 1$ depending on $T^{*}$, $q$, $\\gamma$, $E_{0}$, $\\left\\lVert \\rho_{0}^{\\frac{1}{q+2}}v_{0} \\right\\rVert_{L^{q+2}}$, and $\\left\\lVert \\rho_{0} \\right\\rVert_{L^{\\infty}}$ such that\n\\begin{equation}\\label{eq:v_L_infty_est}\n \\left\\lVert v \\right\\rVert _{L_{T}^{\\infty}\\left(L^{\\infty}\\left(\\mathbb{R}^3\\right)\\right)}\\leq c_{v}(\\log V_{T} )^{\\frac{1}{2}}.\n\\end{equation}\n\\end{proposition}", "post_theorem_intro_text_len": 6699, "post_theorem_intro_text": "\\begin{remark}\n    For the sake of brevity, we restrict our proof to the three-dimensional case where $\\gamma \\in (1, \\frac{8}{3})$. For the critical 3D case $(\\gamma=1)$ and the 2D case $(\\gamma \\ge 1)$, although the density upper bound estimates for these cases were addressed in Haspot \\cite{Haspot 1} and Yu-Wu \\cite{Yu-Wu}, neither work derived the density lower bound. Consequently, they were unable to establish the global existence of solutions. We emphasize that the approach developed in this paper can be successfully applied to establish the density lower bound for the critical 3D case $(\\gamma=1)$ and the 2D case $(\\gamma \\ge 1)$ as well. Therefore, we are able to obtain global solutions for both of these cases. It is worth noting that the technique used by Haspot \\cite{Haspot 1} relies on the assumption that the estimate for $\\|\\rho^{\\frac{1}{q+2}} v\\|_{L^\\infty_T L^{q+2}}$ is independent of $q$ to derive the density lower bound. Since such $q$-independence does not hold in the present context, Haspot's method is not applicable for securing a strictly positive lower bound.\n\\end{remark}\n\\begin{remark}\n    Huang-Meng-Zhang \\cite{Huang-Meng-Zhang} established this result for the periodic domain. In this work, we extend their findings to the more challenging whole space setting by introducing a new De Giorgi iteration method to effectively handle the far field behavior.   \n\\end{remark}\nProvided that the density remains strictly positive, we define the effective velocity $v$ as\n\\begin{equation}\nv = u + \\nabla \\log \\rho.\n\\end{equation}\nThis transformation converts the original system \\eqref{NSK}  into the following parabolic system\n\\begin{equation}\\label{parabolic_system}\n\\begin{cases}\n\\partial_t \\rho + \\nabla \\cdot(\\rho v) - \\Delta \\rho = 0, \\\\\n\\rho \\partial_t v + \\rho u \\cdot \\nabla v + \\nabla P(\\rho) = \\nabla \\cdot(\\rho \\nabla v).\n\\end{cases}\n\\end{equation}\nConsequently, the initial effective velocity $v_0$ is defined by\n\\begin{equation}\\label{initial data}\nv_0 = u_0 + \\nabla \\log \\rho_0.\n\\end{equation}\nLet us outline the main strategy of the proof. Our proof is organized into three main parts: establishing the density upper bound, deriving the density lower bound, and finally, improving the regularity of the solution.\n\nIn the first phase, we work within the framework of Besov spaces, utilizing maximal regularity estimates for parabolic equations to secure the upper bound. The second and most crucial step is establishing the density lower bound. Although Haspot \\cite{Haspot 1} investigated the isothermal NSK system, his approach failed to yield a lower bound for the density. Our strategy draws inspiration from a key inequality introduced by the first author in \\cite{Huang-Meng-Zhang}. However, the technique in \\cite{Huang-Meng-Zhang} cannot be directly applied to the whole space problem with non-vacuum far-field conditions. To overcome this obstacle, we have developed a novel truncated De Giorgi iteration method.\n\nWe now proceed to outline the derivation of these density bounds.\\\\\n\\textbf{Upper bound of $\\rho$.} \\par\nTo derive the upper bound of the density, we fully exploit the parabolic structure of the system and employ maximal regularity estimates for the heat equation. Specifically, closing the density estimate requires bounding the norm $\\|\\rho^{1/(q+2)}v\\|_{L^{q+2}}$ for some $q > 1$. In Proposition \\ref{PRO3.4}, by means of a precise domain decomposition analysis, we established that for any $\\gamma \\in (1, \\frac{8}{3})$, there exists a $q$ such that the norm $\\|\\rho^{\\frac{1}{q+2}} v\\|_{L^{q+2}}$ remains bounded.\\\\\n\\textbf{Lower bound of $\\rho$.} \\par\n To derive the lower bound of the density following Haspot\\cite{Haspot 1}'s argument, it is essential to control the norm $\\|\\rho^{\\frac{1}{p+2}} v\\|_{L^\\infty_T L^{p+2}}$ by a constant independent of $p$. However, the bound for $\\|\\rho^{\\frac{1}{p+2}} v\\|_{L^\\infty_T L^{p+2}}$ tends to infinity as $p \\to \\infty$. To overcome this obstacle, Huang-Meng-Zhang \\cite{Huang-Meng-Zhang} employed the Moser iteration method to establish a control relationship between $\\|v\\|_{L^\\infty}$ and $\\sqrt{\\log (e^{\\frac{25}{9}}+\\|{\\rho}^{-1}\\|_{L^{\\infty}})}$. This critical inequality plays a pivotal role in our analysis. Although their approach can handle the density lower bound in a periodic domain, it fails in the whole space when the density exhibits non-zero far-field behavior at infinity. To address the difficulties arising in this context, we introduce the De Giorgi iteration technique to prove the lower bound of the density by employing a truncation level adapted to the far-field density. Specifically, we construct the following truncation and iteration sequence\n \\begin{equation}\n\\begin{split}\n    \\rho^{-1}_{(k_n)}&:=\\max\\{\\rho^{-1}-{k_n},0\\},\n    \\\\\n    U_{n}^{T}&:=\\| \\rho _{(k_n)}^{-1}\\| _{L_{T}^{\\infty}L^2}^2+\\| \\nabla \\rho _{(k_n)}^{-1}\\| _{L_{T}^{2}L^2}^2.\n\\end{split}\n \\end{equation}\n where $k_n:=M\\left( 1-2^{-n} \\right)+2\\left\\| {\\rho}^{-1} _0 \\right\\| _{L^{\\infty}}.$ It is worth noting that the initial iteration value $k_0 = 2\\|1/\\rho_0\\|_{L^\\infty}$ is chosen specifically to handle the far-field behavior while ensuring that the initial energy $U_0^T$ satisfies the convergence condition\n     \\begin{equation}\n    U_0^T \\le K^{-\\frac{1}{\\nu}} A^{-\\frac{1}{\\nu^2}}.\n\\end{equation}\n We choose a suitable $T$ to bound the solution on $[0, T]$ and then employ a shifted iteration sequence for $[T, 2T]$. As long as the time step for each extension is uniform, this procedure proves the boundedness within the maximal lifespan $T^*$, which in turn establishes the time-dependent lower bound of the density.\\par\n In Section 2, we introduce some preliminaries, with a particular focus on Littlewood-Paley theory. In Section 3, we first demonstrate that for any $\\gamma \\in (1, \\frac{8}{3})$, there exists a $q>1$ satisfying the condition $\\gamma \\le \\frac{2q+6}{q+2}$ such that the quantity $\\sup_{0 \\le t \\le T} \\|\\rho^{\\frac{1}{q+2}} v\\|_{L^{q+2}}$ remains bounded. This allows us to establish the density upper bound by applying maximal regularity estimates for the heat equation. In Section 4, armed with the density upper bound, we extend the range of $q$ for which the boundedness of $\\sup_{0 \\le t \\le T} \\|\\rho^{\\frac{1}{q+2}} v\\|_{L^{q+2}}$ holds. Consequently, we employ the Moser iteration method to control $\\|v\\|_{L^{\\infty}}$ by a term involving $\\sqrt{\\log V_T}$, and subsequently utilize De Giorgi iteration to prove the existence of a density lower bound up to the maximal existence time $T^*$. Finally, in Section 5 and Section 6, we prove the main result by using the established density upper and lower bounds as blow-up criteria.", "sketch": "To prove Theorem~\\ref{thm:global_existence_cauchy}, the authors assume the density stays strictly positive and introduce the \\emph{effective velocity} $v=u+\\nabla\\log\\rho$, which transforms \\eqref{NSK} into the parabolic system \\eqref{parabolic_system}. They then “outline the main strategy” in three parts: “establishing the density upper bound, deriving the density lower bound, and finally, improving the regularity of the solution.”\n\n(1) \\textbf{Upper bound of $\\rho$.} They “fully exploit the parabolic structure of the system and employ maximal regularity estimates for the heat equation.” Closing the estimate requires bounding $\\|\\rho^{1/(q+2)}v\\|_{L^{q+2}}$ for some $q>1$; in Proposition~\\ref{PRO3.4}, “by means of a precise domain decomposition analysis,” they show that for any $\\gamma\\in(1,\\frac{8}{3})$ there exists such a $q$ with $\\|\\rho^{\\frac{1}{q+2}}v\\|_{L^{q+2}}$ bounded.\n\n(2) \\textbf{Lower bound of $\\rho$.} They explain that Haspot’s route would need $\\|\\rho^{\\frac{1}{p+2}}v\\|_{L^\\infty_T L^{p+2}}$ bounded uniformly in $p$, but here it “tends to infinity as $p\\to\\infty$.” Instead they use the Huang–Meng–Zhang idea relating “$\\|v\\|_{L^\\infty}$ and $\\sqrt{\\log( e^{\\frac{25}{9}}+\\|\\rho^{-1}\\|_{L^\\infty})}$,” but since the periodic-domain method “fails in the whole space” with nonzero far-field density, they “developed a novel truncated De Giorgi iteration method.” They define truncations and energies\n\\[\n\\rho^{-1}_{(k_n)}:=\\max\\{\\rho^{-1}-k_n,0\\},\\qquad\nU_n^T:=\\|\\rho^{-1}_{(k_n)}\\|_{L_T^\\infty L^2}^2+\\|\\nabla\\rho^{-1}_{(k_n)}\\|_{L_T^2 L^2}^2,\n\\]\nwith truncation levels $k_n:=M(1-2^{-n})+2\\|\\rho_0^{-1}\\|_{L^\\infty}$, choosing $k_0=2\\|1/\\rho_0\\|_{L^\\infty}$ so that $U_0^T$ meets a convergence condition $U_0^T\\le K^{-1/\\nu}A^{-1/\\nu^2}$. They “choose a suitable $T$ to bound the solution on $[0,T]$ and then employ a shifted iteration sequence for $[T,2T]$,” and with a uniform time step this yields boundedness up to the maximal lifespan $T^*$, giving a “time-dependent lower bound of the density.”\n\n(3) \\textbf{Completion / global existence.} After establishing density upper and lower bounds, they “prove the main result by using the established density upper and lower bounds as blow-up criteria” (Sections 5–6), together with the earlier steps (Sections 3–4: maximal regularity for the upper bound; extending the $q$-range, then Moser to control $\\|v\\|_{L^\\infty}$ by $\\sqrt{\\log V_T}$, and De Giorgi for the lower bound up to $T^*$).", "expanded_sketch": "To prove the main theorem, the authors assume the density stays strictly positive and introduce the \\emph{effective velocity} $v=u+\\nabla\\log\\rho$, which transforms\n\\begin{equation}\\label{NSK} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \n\\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) =\\nabla \\cdot \\mathbb{S} + \\nabla \\cdot \\mathbb{K}. \\end{cases} \\end{equation}\ninto the parabolic system \\eqref{parabolic_system}. They then “outline the main strategy” in three parts: “establishing the density upper bound, deriving the density lower bound, and finally, improving the regularity of the solution.”\n\n(1) \\textbf{Upper bound of $\\rho$.} They “fully exploit the parabolic structure of the system and employ maximal regularity estimates for the heat equation.” Closing the estimate requires bounding $\\|\\rho^{1/(q+2)}v\\|_{L^{q+2}}$ for some $q>1$. We first use the following proposition.\n\\begin{proposition}\\label{PRO3.4}\nFor any $\\gamma \\in (1, \\frac{8}{3})$, there exists $q \\in (1, 4)$ such that\n\\begin{equation}\n     \\gamma \\le \\frac{2q+6}{q+2}.\n\\end{equation}\n    For this fixed $q$, there exists a positive constant $C$, depending on $T$, $q$, $\\gamma$, $E_0$, and the initial data $\\|\\rho_0^{1/(q+2)} v_0\\|_{L^{q+2}}$, such that\n\\begin{equation}\\label{L^{q+2}}\n\\sup_{0 \\le t \\le T} \\|\\rho^{\\frac{1}{q+2}} v\\|_{L^{q+2}} \\le C.\n\\end{equation}\n\\end{proposition}\nIn particular, for any $\\gamma\\in(1,\\frac{8}{3})$ one can choose such a $q$ so that $\\|\\rho^{\\frac{1}{q+2}}v\\|_{L^{q+2}}$ is bounded.\n\n(2) \\textbf{Lower bound of $\\rho$.} They explain that Haspot’s route would need $\\|\\rho^{\\frac{1}{p+2}}v\\|_{L^\\infty_T L^{p+2}}$ bounded uniformly in $p$, but here it “tends to infinity as $p\\to\\infty$.” Instead they use the Huang–Meng–Zhang idea relating “$\\|v\\|_{L^\\infty}$ and $\\sqrt{\\log( e^{\\frac{25}{9}}+\\|\\rho^{-1}\\|_{L^\\infty})}$,” but since the periodic-domain method “fails in the whole space” with nonzero far-field density, they “developed a novel truncated De Giorgi iteration method.” They define truncations and energies\n\\[\n\\rho^{-1}_{(k_n)}:=\\max\\{\\rho^{-1}-k_n,0\\},\\qquad\nU_n^T:=\\|\\rho^{-1}_{(k_n)}\\|_{L_T^\\infty L^2}^2+\\|\\nabla\\rho^{-1}_{(k_n)}\\|_{L_T^2 L^2}^2,\n\\]\nwith truncation levels $k_n:=M(1-2^{-n})+2\\|\\rho_0^{-1}\\|_{L^\\infty}$, choosing $k_0=2\\|1/\\rho_0\\|_{L^\\infty}$ so that $U_0^T$ meets a convergence condition $U_0^T\\le K^{-1/\\nu}A^{-1/\\nu^2}$. They “choose a suitable $T$ to bound the solution on $[0,T]$ and then employ a shifted iteration sequence for $[T,2T]$,” and with a uniform time step this yields boundedness up to the maximal lifespan $T^*$, giving a “time-dependent lower bound of the density.”\n\n(3) \\textbf{Completion / global existence.} After establishing density upper and lower bounds, they complete the proof of the main theorem by using the established density upper and lower bounds as blow-up criteria (Sections 5–6), together with the earlier steps (Sections 3–4: maximal regularity for the upper bound; extending the $q$-range, then Moser to control $\\|v\\|_{L^\\infty}$ by $\\sqrt{\\log V_T}$, and De Giorgi for the lower bound up to $T^*$).", "expanded_theorem": "\\label{thm:global_existence_cauchy}\nLet $N=2$ or $N=3$. Assume that $\\gamma$ satisfies\n\\begin{equation}\n    \\begin{cases}\n        \\gamma \\in [1, \\infty) & \\text{if } N=2, \\\\\n        \\gamma \\in [1, 8/3) & \\text{if } N=3.\n    \\end{cases}\n\\end{equation}\nLet $\\bar{\\rho} > 0$ be the far field behavior of density. Assume that the initial data $(\\rho_0, u_0)$ satisfies\n\\begin{equation}\n    0 < \\underline{\\varrho} \\le \\rho_0 \\le \\bar{\\varrho}, \\quad (\\rho_0 - \\bar{\\rho}) \\in H^3(\\mathbb{R}^N), \\quad u_0 \\in H^2(\\mathbb{R}^N),\n\\end{equation}\nwhere $\\underline{\\varrho}$ and $\\bar{\\varrho}$ are positive constants. Then the Cauchy problem\n\\begin{equation}\\label{NSK_simplified} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) = \\nabla \\cdot \\left(2 \\rho \\mathcal{D}(u) \\right) + \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho). \\end{cases} \\end{equation}\n\\begin{equation}\\label{far field}\n    \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}\nadmits a unique global strong solution $(\\rho, u)$ satisfying for any $0 < T < \\infty$ and $(x,t) \\in \\mathbb{R}^N \\times [0, T]$,\n\\begin{equation}\n    (C(T))^{-1} \\le \\rho(x,t) \\le C(T),\n\\end{equation}\nand\n\\begin{equation}\n\\begin{aligned}\n    &(\\rho - \\bar{\\rho}) \\in C([0, T]; H^3) \\cap L^2(0, T; H^4), \\quad \\rho_t \\in C([0, T]; H^1) \\cap L^2(0, T; H^2), \\\\\n    &u \\in C([0, T]; H^2) \\cap L^2(0, T; H^3), \\quad u_t \\in L^\\infty(0, T; L^2) \\cap L^2(0, T; H^1),\n\\end{aligned}\n\\end{equation}\nwhere the constant $C(T) > 0$ depends on the initial data and $T$.", "theorem_type": ["Existence", "Uniqueness"], "mcq": {"question": "Let $N=2$ or $N=3$, and assume\n\\[\n\\gamma\\in[1,\\infty)\\quad\\text{if }N=2,\n\\qquad\n\\gamma\\in[1,8/3)\\quad\\text{if }N=3.\n\\]\nLet $\\bar\\rho>0$ be the far-field density, and let the initial data $(\\rho_0,u_0)$ satisfy\n\\[\n0<\\underline\\varrho\\le \\rho_0\\le \\bar\\varrho,\n\\qquad\n\\rho_0-\\bar\\rho\\in H^3(\\mathbb R^N),\n\\qquad\nu_0\\in H^2(\\mathbb R^N),\n\\]\nfor some positive constants $\\underline\\varrho,\\bar\\varrho$. Consider the Cauchy problem for the shallow-water Navier-Stokes-Korteweg system on $\\mathbb R^N$:\n\\[\n\\begin{cases}\n\\partial_t\\rho+\\nabla\\cdot(\\rho u)=0,\\\\\n\\partial_t(\\rho u)+\\nabla\\cdot(\\rho u\\otimes u)+\\nabla P(\\rho)\n=\\nabla\\cdot\\bigl(2\\rho\\mathcal D(u)\\bigr)+\\nabla\\cdot\\bigl(\\rho\\nabla\\nabla\\log\\rho\\bigr),\n\\end{cases}\n\\]\nwith far-field condition\n\\[\n\\rho(x,t)\\to\\bar\\rho>0,\n\\qquad\nu(x,t)\\to0,\n\\qquad \\text{as }|x|\\to\\infty,\n\\]\nand initial data $(\\rho,u)|_{t=0}=(\\rho_0,u_0)$. Which statement holds about existence and uniqueness of solutions?", "correct_choice": {"label": "A", "text": "There exists a unique global strong solution $(\\rho,u)$ to this Cauchy problem such that, for every $0<T<\\infty$, there is a constant $C(T)>0$ depending on $T$ and the initial data with\n\\[\n(C(T))^{-1}\\le \\rho(x,t)\\le C(T)\n\\quad\\text{for all }(x,t)\\in\\mathbb R^N\\times[0,T],\n\\]\nand\n\\[\n\\rho-\\bar\\rho\\in C([0,T];H^3(\\mathbb R^N))\\cap L^2(0,T;H^4(\\mathbb R^N)),\n\\qquad\n\\rho_t\\in C([0,T];H^1(\\mathbb R^N))\\cap L^2(0,T;H^2(\\mathbb R^N)),\n\\]\n\\[\nu\\in C([0,T];H^2(\\mathbb R^N))\\cap L^2(0,T;H^3(\\mathbb R^N)),\n\\qquad\nu_t\\in L^\\infty(0,T;L^2(\\mathbb R^N))\\cap L^2(0,T;H^1(\\mathbb R^N)).\n\\]"}, "choices": [{"label": "B", "text": "There exists a unique global strong solution $(\\rho,u)$ to this Cauchy problem such that there is a constant $C>0$, depending only on the initial data, with\n\\[\nC^{-1}\\le \\rho(x,t)\\le C\n\\quad\\text{for all }(x,t)\\in\\mathbb R^N\\times[0,\\infty),\n\\]\nand, for every $0<T<\\infty$,\n\\[\n\\rho-\\bar\\rho\\in C([0,T];H^3(\\mathbb R^N))\\cap L^2(0,T;H^4(\\mathbb R^N)),\n\\qquad\n\\rho_t\\in C([0,T];H^1(\\mathbb R^N))\\cap L^2(0,T;H^2(\\mathbb R^N)),\n\\]\n\\[\nu\\in C([0,T];H^2(\\mathbb R^N))\\cap L^2(0,T;H^3(\\mathbb R^N)),\n\\qquad\n\\nu_t\\in L^\\infty(0,T;L^2(\\mathbb R^N))\\cap L^2(0,T;H^1(\\mathbb R^N)).\n\\]"}, {"label": "C", "text": "There exists a global strong solution $(\\rho,u)$ to this Cauchy problem such that, for every $0<T<\\infty$, there is a constant $C(T)>0$ depending on $T$ and the initial data with\n\\[\n(C(T))^{-1}\\le \\rho(x,t)\\le C(T)\n\\quad\\text{for all }(x,t)\\in\\mathbb R^N\\times[0,T],\n\\]\nand\n\\[\n\\rho-\\bar\\rho\\in C([0,T];H^3(\\mathbb R^N))\\cap L^2(0,T;H^4(\\mathbb R^N)),\n\\qquad\n\\rho_t\\in C([0,T];H^1(\\mathbb R^N))\\cap L^2(0,T;H^2(\\mathbb R^N)),\n\\]\n\\[\nu\\in C([0,T];H^2(\\mathbb R^N))\\cap L^2(0,T;H^3(\\mathbb R^N)),\n\\qquad\n\\nu_t\\in L^\\infty(0,T;L^2(\\mathbb R^N))\\cap L^2(0,T;H^1(\\mathbb R^N)).\n\\]"}, {"label": "D", "text": "There exists a unique global strong solution $(\\rho,u)$ to this Cauchy problem such that, for every $0<T<\\infty$, there is a constant $C(T)>0$ depending on $T$ and the initial data with\n\\[\n(C(T))^{-1}\\le \\rho(x,t)\\le C(T)\n\\quad\\text{for all }(x,t)\\in\\mathbb R^N\\times[0,T],\n\\]\nand\n\\[\n\\rho-\\bar\\rho\\in C([0,T];H^3(\\mathbb R^N))\\cap L^2(0,T;H^4(\\mathbb R^N)),\n\\qquad\n\\rho_t\\in C([0,T];H^1(\\mathbb R^N))\\cap L^2(0,T;H^2(\\mathbb R^N)),\n\\]\n\\[\nu\\in C([0,T];H^2(\\mathbb R^N))\\cap L^2(0,T;H^3(\\mathbb R^N)),\n\\qquad\n\\nu_t\\in C([0,T];L^2(\\mathbb R^N))\\cap L^2(0,T;H^1(\\mathbb R^N)).\n\\]"}, {"label": "E", "text": "There exists a unique global strong solution $(\\rho,u)$ to this Cauchy problem for all\n\\[\n\ngamma\\in[1,\\infty)\\quad\\text{if }N=2\\text{ or }N=3,\n\\]\nsuch that, for every $0<T<\\infty$, there is a constant $C(T)>0$ depending on $T$ and the initial data with\n\\[\n(C(T))^{-1}\\le \\rho(x,t)\\le C(T)\n\\quad\\text{for all }(x,t)\\in\\mathbb R^N\\times[0,T],\n\\]\nand\n\\[\n\\rho-\\bar\\rho\\in C([0,T];H^3(\\mathbb R^N))\\cap L^2(0,T;H^4(\\mathbb R^N)),\n\\qquad\n\\rho_t\\in C([0,T];H^1(\\mathbb R^N))\\cap L^2(0,T;H^2(\\mathbb R^N)),\n\\]\n\\[\nu\\in C([0,T];H^2(\\mathbb R^N))\\cap L^2(0,T;H^3(\\mathbb R^N)),\n\\qquad\n\\nu_t\\in L^\\infty(0,T;L^2(\\mathbb R^N))\\cap L^2(0,T;H^1(\\mathbb R^N)).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "time_dependent_density_bound_C(T)", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "uniqueness", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "sharp_time_regularality_of_u_t", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "3D_gamma_upper_range_8_over_3", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem states hypotheses only and does not explicitly reveal the conclusion. Although choice A matches the expected theorem statement closely, the stem itself does not leak that exact answer."}, "TAS": {"score": 1, "justification": "This is largely a theorem-recall question: the correct option is essentially the full existence theorem under the stated assumptions. However, it is not a pure restatement because the alternatives include stronger, weaker, and subtly altered conclusions."}, "GPS": {"score": 1, "justification": "Some reasoning is required to distinguish the exact theorem from nearby variants, especially the weaker true statement and the overstrong claims. Still, the task is mostly precision recall rather than substantial mathematical generation or derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: one enlarges the 3D gamma range, one omits regularity details, one strengthens time-uniform density bounds, and one weakens global existence to finite-time statements. These reflect common failure modes in theorem interpretation."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no direct answer leakage, but it mainly tests accurate recall of a specific result rather than deeper generative reasoning."}}
{"id": "2602.10700v1", "paper_link": "http://arxiv.org/abs/2602.10700v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:global_existence_cauchy}\nLet $N=2$ or $N=3$. Assume that $\\gamma$ satisfies\n\\begin{equation}\n    \\begin{cases}\n        \\gamma \\in [1, \\infty) & \\text{if } N=2, \\\\\n        \\gamma \\in [1, 8/3) & \\text{if } N=3.\n    \\end{cases}\n\\end{equation}\nLet $\\bar{\\rho} > 0$ be the far field behavior of density. Assume that the initial data $(\\rho_0, u_0)$ satisfies\n\\begin{equation}\n    0 < \\underline{\\varrho} \\le \\rho_0 \\le \\bar{\\varrho}, \\quad (\\rho_0 - \\bar{\\rho}) \\in H^3(\\mathbb{R}^N), \\quad u_0 \\in H^2(\\mathbb{R}^N),\n\\end{equation}\nwhere $\\underline{\\varrho}$ and $\\bar{\\varrho}$ are positive constants. Then the Cauchy problem  \\eqref{NSK_simplified}-\\eqref{far field} admits a unique global strong solution $(\\rho, u)$ satisfying for any $0 < T < \\infty$ and $(x,t) \\in \\mathbb{R}^N \\times [0, T]$,\n\\begin{equation}\n    (C(T))^{-1} \\le \\rho(x,t) \\le C(T),\n\\end{equation}\nand\n\\begin{equation}\n\\begin{aligned}\n    &(\\rho - \\bar{\\rho}) \\in C([0, T]; H^3) \\cap L^2(0, T; H^4), \\quad \\rho_t \\in C([0, T]; H^1) \\cap L^2(0, T; H^2), \\\\\n    &u \\in C([0, T]; H^2) \\cap L^2(0, T; H^3), \\quad u_t \\in L^\\infty(0, T; L^2) \\cap L^2(0, T; H^1),\n\\end{aligned}\n\\end{equation}\nwhere the constant $C(T) > 0$ depends on the initial data and $T$.", "start_pos": 12442, "end_pos": 13706, "label": "thm:global_existence_cauchy"}, "ref_dict": {"NSK": "\\begin{equation}\\label{NSK} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \n\\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) =\\nabla \\cdot \\mathbb{S} + \\nabla \\cdot \\mathbb{K}. \\end{cases} \\end{equation}", "PRO3.4": "\\begin{proposition}\\label{PRO3.4}\nFor any $\\gamma \\in (1, \\frac{8}{3})$, there exists $q \\in (1, 4)$ such that\n\\begin{equation}\n     \\gamma \\le \\frac{2q+6}{q+2}.\n\\end{equation}\n    For this fixed $q$, there exists a positive constant $C$, depending on $T$, $q$, $\\gamma$, $E_0$, and the initial data $\\|\\rho_0^{1/(q+2)} v_0\\|_{L^{q+2}}$, such that\n\\begin{equation}\\label{L^{q+2}}\n\\sup_{0 \\le t \\le T} \\|\\rho^{\\frac{1}{q+2}} v\\|_{L^{q+2}} \\le C.\n\\end{equation}\n\\end{proposition}", "NSK_simplified": "\\begin{equation}\\label{NSK_simplified} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) = \\nabla \\cdot \\left(2 \\rho \\mathcal{D}(u) \\right) + \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho). \\end{cases} \\end{equation}", "far field": "\\begin{equation}\\label{far field}\n    \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}"}, "pre_theorem_intro_text_len": 9174, "pre_theorem_intro_text": "In this paper, we are concerned with the global well-posedness of the compressible Navier-Stokes-Korteweg equations in the whole space $\\mathbb{R}^N$. This system describes the motion of a viscous compressible fluid endowed with internal capillarity and reads as follows\n\\begin{equation}\\label{NSK} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \n\\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) =\\nabla \\cdot \\mathbb{S} + \\nabla \\cdot \\mathbb{K}. \\end{cases} \\end{equation}\nThe viscous stress tensor $\\mathbb{S}$ is given by Newton's rheological law\n\\begin{equation}\n    \\mathbb{S} = 2\\mu(\\rho) \\mathcal{D}(u) + \\lambda(\\rho) \\nabla \\cdot u \\mathbb{I},\n\\end{equation}\nwhere $\\mathcal{D}(u) = \\frac{1}{2}(\\nabla u + \\nabla u^T)$ denotes the symmetric deformation tensor, and $\\mathbb{I}$ is the identity matrix in $\\mathbb{R}^N$. The scalar functions $\\mu(\\rho)$ and $\\lambda(\\rho)$ represent the shear and bulk viscosity coefficients, respectively, which satisfy the physical condition $\\mu(\\rho) > 0$ and $2\\mu(\\rho) + N\\lambda(\\rho) \\ge 0$.\nIn order to ensure the stability of the system and utilize the BD entropy structure, we assume the viscosity coefficients satisfy the relation $$\\lambda(\\rho) = 2(\\rho \\mu'(\\rho) - \\mu(\\rho)).$$\nIn the context of the Korteweg theory, the capillarity tensor $\\nabla \\cdot\\mathbb{K}$ is typically defined as\n\\begin{equation}\n    \\nabla \\cdot \\mathbb{K} = \\nabla \\left( \\rho \\kappa(\\rho) \\Delta \\rho + \\frac{\\kappa(\\rho) + \\rho \\kappa'(\\rho)}{2} |\\nabla \\rho|^2 \\right) - \\nabla \\cdot \\left( \\kappa(\\rho) \\nabla \\rho \\otimes \\nabla \\rho \\right).\n\\end{equation}\nwhere $\\kappa(\\rho) > 0$ is the coefficient of capillarity.\\par\nThe study of capillary fluids originates from the pioneering work of Van der Waals and Korteweg \\cite{Korteweg,J.F. Van derWaals}. In their theory of capillarity, the fluid energy is assumed to depend not only on the standard thermodynamic variables but also on the gradient of the density. This concept was later formalized in the modern language of continuum mechanics by Dunn and Serrin in the 1980s, leading to the so-called Korteweg-type models. \\par\n  When $\\kappa(\\rho) = 0$, $\\mu=\\rho$ and $\\lambda=0$, system \\eqref{NSK} reduces to the well-known viscous shallow water equations. \nSignificant progress has been made for the viscous shallow water equations in recent years. Regarding general multi-dimensional initial data, significant breakthroughs concerning the global well-posedness of weak solutions were achieved by Li-Xin \\cite{Li-Xin} and Vasseur-Yu \\cite{Vasseur-Yu}. Independently, these authors established the global existence of weak solutions for the compressible system with $\\mu(\\rho) = \\rho$ and $\\lambda(\\rho) = 0$, admitting arbitrarily large data and vacuum states. More specifically, they obtained global weak solutions for $\\gamma \\in (1, \\infty)$ when $N=2$, and for $\\gamma \\in (1, 3)$ when $N=3$. It is worth noting that Li-Xin \\cite{Li-Xin} extended their analysis to a wider class of viscosity coefficients satisfying the BD entropy relation. Concerning the global smooth solutions for multi-dimensional shallow water equations with arbitrarily large initial data, Huang-Meng-Zhang \\cite{Huang-Meng-Zhang-V} pioneered the proof of global classical solutions for the two-dimensional initial-boundary value problem under the assumption of radial symmetry with $\\gamma\\ge\\frac{3}{2}$. Subsequently, Gu-Huang \\cite{Gu-Huang} extended the range of the exponent to $\\gamma > 1$ and generalized the result to the three-dimensional case with $1 < \\gamma < 3$. Concurrently, the global existence of large solutions for the associated Cauchy problem was proved independently by Chen-Zhang-Zhu \\cite{Chen-Zhang-Zhu}. Moreover, Huang-Meng-Zhang \\cite{Huang-Meng-Zhang-V} successfully proved the global classical well-posedness for general isentropic compressible Navier-Stokes equations satisfying the BD entropy condition in 2D and 3D, treating the shallow water model as a specific instance.\\par\n When $\\kappa(\\rho) >0$, we review some related works regarding the well-posedness of the Navier-Stokes-Korteweg system with general viscosity coefficients. For the one-dimensional case, considering the system with specific density-dependent viscosity $\\mu(\\rho) = \\rho$ and capillarity $\\kappa(\\rho) = \\rho^{-1}$, Charve-Haspot \\cite{Charve-Haspot} established the existence of global strong solutions in the whole space, allowing for large non-vacuum initial data. Furthermore, they demonstrated that these solutions converge to the entropic weak solutions of the compressible Euler equations. Moreover, Germain-LeFloch \\cite{Germain-LeFloch} proved the global existence of finite energy weak solutions for the Cauchy problem with general density-dependent coefficients and demonstrated their convergence to the entropy solutions of the Euler system. For the case with power-law viscosity $\\mu(\\varrho) = \\varrho^\\alpha$ and capillarity $\\kappa(\\varrho) = \\varrho^\\beta$ satisfying specific conditions, Antonelli-Bresch-Spirito \\cite{Antonelli-Bresch-Spirito} established the existence of global weak solutions for the periodic problem with large data. Furthermore, investigating the Cauchy problem in Lagrangian coordinates, Chen et al. \\cite{Chen-Chai} obtained global classical solutions for large initial data away from vacuum, considering density-dependent viscosity and capillarity.  For the multi-dimensional case, building upon their earlier local theory \\cite{Hattori-Li}, Hattori-Li \\cite{Hattori-Li-2} established the global existence of solutions for the constant-coefficient Cauchy problem with small, non-vacuum initial data. For the case where both viscosity and capillarity coefficients depend on the density, Danchin-Desjardins \\cite{Danchin_Desjardins} obtained global smooth solutions. Their result holds for small perturbations of a non-vacuum state in functional spaces that are critical with respect to the physical energy. Bresch and Desjardins \\cite{B-D} investigated the two-dimensional viscous shallow water equations extended by a capillary term. They established the global existence of weak solutions in the presence of vacuum and demonstrated their convergence towards the strong solution of the viscous quasi-geostrophic system with a free surface. \\par\nHowever, the global existence of strong solutions with arbitrarily large initial data for the multi-dimensional Navier-Stokes-Korteweg system has long remained an open problem. It was not until recently that the first author Huang \\cite{Huang-Meng-Zhang} in his newly preprint resolved this by establishing global strong solutions on the two- and three-dimensional periodic torus, provided that the initial density is bounded from above and below. However, extending their method to the whole space presents new difficulties. In this paper, we overcome these obstacles to establish the global existence of strong solutions for the multi-dimensional Navier-Stokes-Korteweg system with large initial data in the whole space. By employing the novel critical inequality established by the first author in his newly preprint \\cite{Huang-Meng-Zhang}, we successfully establish the lower bound of the density by employing a novel De Giorgi iteration method. It is worth noting that our work does not require any radial symmetry assumption. To the best of our knowledge, this can be regarded as the first result concerning global large solutions for the corresponding Cauchy problem.\n\\par\nThroughout the rest of this paper, we focus on the case of shallow-water viscosity coefficients\n$$\\mu(\\rho) = \\rho, \\quad \\lambda(\\rho) = 0,$$\nand we assume that $\\kappa(\\rho)$ satisfies \n$$\\kappa(\\rho)=\\frac{1}{\\rho}.$$\nWith the specific choice of the capillarity coefficient $\\kappa(\\rho)$, the divergence of the Korteweg tensor takes the following form\n\\begin{equation}\n\\nabla \\cdot \\mathbb{K} = \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho).\n\\end{equation}\nConsequently, under the assumption of shallow-water viscosity ($\\mu(\\rho)=\\rho, \\lambda(\\rho)=0$) and the specific capillarity coefficient $\\kappa(\\rho)=1/\\rho$, equation \\eqref{NSK} can be rewritten as\n\\begin{equation}\\label{NSK_simplified} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) = \\nabla \\cdot \\left(2 \\rho \\mathcal{D}(u) \\right) + \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho). \\end{cases} \\end{equation}\nNow we investigate the global existence of strong solutions to system \\eqref{NSK_simplified} in $\\mathbb{R}^N$, where $N=2, 3$. The system is supplemented with the prescribed initial data $(\\rho_0, u_0)$ satisfying\n\\begin{equation}\n\\rho(x, 0) = \\rho_0(x), \\quad u(x, 0) = u_0(x), \\quad \\text{for } x \\in \\mathbb{R}^N,\n\\end{equation}\nwith far field behavior \n\\begin{equation}\\label{far field}\n    \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}\nWe now state the main result on the global existence of strong solutions to the Cauchy problem for system \\eqref{NSK_simplified}-\\eqref{far field} with arbitrarily large initial data.", "context": "e assumption of radial symmetry with $\\gamma\\ge\\frac{3}{2}$. Subsequently, Gu-Huang \\cite{Gu-Huang} extended the range of the exponent to $\\gamma > 1$ and generalized the result to the three-dimensional case with $1 < \\gamma < 3$. Concurrently, the global existence of large solutions for the associated Cauchy problem was proved independently by Chen-Zhang-Zhu \\cite{Chen-Zhang-Zhu}. Moreover, Huang-Meng-Zhang \\cite{Huang-Meng-Zhang-V} successfully proved the global classical well-posedness for general isentropic compressible Navier-Stokes equations satisfying the BD entropy condition in 2D and 3D, treating the shallow water model as a specific instance.\\par\n When $\\kappa(\\rho) >0$, we review some related works regarding the well-posedness of the Navier-Stokes-Korteweg system with general viscosity coefficients. For the one-dimensional case, considering the system with specific density-dependent viscosity $\\mu(\\rho) = \\rho$ and capillarity $\\kappa(\\rho) = \\rho^{-1}$, Charve-Haspot \\cite{Charve-Haspot} established the existence of global strong solutions in the whole space, allowing for large non-vacuum initial data. Furthermore, they demonstrated that these solutions converge to the entropic weak solutions of the compressible Euler equations. Moreover, Germain-LeFloch \\cite{Germain-LeFloch} proved the global existence of finite energy weak solutions for the Cauchy problem with general density-dependent coefficients and demonstrated their convergence to the entropy solutions of the Euler system. For the case with power-law viscosity $\\mu(\\varrho) = \\varrho^\\alpha$ and capillarity $\\kappa(\\varrho) = \\varrho^\\beta$ satisfying specific conditions, Antonelli-Bresch-Spirito \\cite{Antonelli-Bresch-Spirito} established the existence of global weak solutions for the periodic problem with large data. Furthermore, investigating the Cauchy problem in Lagrangian coordinates, Chen et al. \\cite{Chen-Chai} obtained global classical solutions for large initial data away from vacuum, considering density-dependent viscosity and capillarity. For the multi-dimensional case, building upon their earlier local theory \\cite{Hattori-Li}, Hattori-Li \\cite{Hattori-Li-2} established the global existence of solutions for the constant-coefficient Cauchy problem with small, non-vacuum initial data. For the case where both viscosity and capillarity coefficients depend on the density, Danchin-Desjardins \\cite{Danchin_Desjardins} obtained global smooth solutions. Their result holds for small perturbations of a non-vacuum state in functional spaces that are critical with respect to the physical energy. Bresch and Desjardins \\cite{B-D} investigated the two-dimensional viscous shallow water equations extended by a capillary term. They established the global existence of weak solutions in the presence of vacuum and demonstrated their convergence towards the strong solution of the viscous quasi-geostrophic system with a free surface. \\par\nHowever, the global existence of strong solutions with arbitrarily large initial data for the multi-dimensional Navier-Stokes-Korteweg system has long remained an open problem. It was not until recently that the first author Huang \\cite{Huang-Meng-Zhang} in his newly preprint resolved this by establishing global strong solutions on the two- and three-dimensional periodic torus, provided that the initial density is bounded from above and below. However, extending their method to the whole space presents new difficulties. In this paper, we overcome these obstacles to establish the global existence of strong solutions for the multi-dimensional Navier-Stokes-Korteweg system with large initial data in the whole space. By employing the novel critical inequality established by the first author in his newly preprint \\cite{Huang-Meng-Zhang}, we successfully establish the lower bound of the density by employing a novel De Giorgi iteration method. It is worth noting that our work does not require any radial symmetry assumption. To the best of our knowledge, this can be regarded as the first result concerning global large solutions for the corresponding Cauchy problem.\n\\par\nThroughout the rest of this paper, we focus on the case of shallow-water viscosity coefficients\n$$\\mu(\\rho) = \\rho, \\quad \\lambda(\\rho) = 0,$$\nand we assume that $\\kappa(\\rho)$ satisfies \n$$\\kappa(\\rho)=\\frac{1}{\\rho}.$$\nWith the specific choice of the capillarity coefficient $\\kappa(\\rho)$, the divergence of the Korteweg tensor takes the following form\n\\begin{equation}\n\\nabla \\cdot \\mathbb{K} = \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho).\n\\end{equation}\nConsequently, under the assumption of shallow-water viscosity ($\\mu(\\rho)=\\rho, \\lambda(\\rho)=0$) and the specific capillarity coefficient $\\kappa(\\rho)=1/\\rho$, equation \\eqref{NSK} can be rewritten as\n\\begin{equation}\\label{NSK_simplified} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) = \\nabla \\cdot \\left(2 \\rho \\mathcal{D}(u) \\right) + \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho). \\end{cases} \\end{equation}\nNow we investigate the global existence of strong solutions to system \\eqref{NSK_simplified} in $\\mathbb{R}^N$, where $N=2, 3$. The system is supplemented with the prescribed initial data $(\\rho_0, u_0)$ satisfying\n\\begin{equation}\n\\rho(x, 0) = \\rho_0(x), \\quad u(x, 0) = u_0(x), \\quad \\text{for } x \\in \\mathbb{R}^N,\n\\end{equation}\nwith far field behavior \n\\begin{equation}\\label{far field}\n \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}\nWe now state the main result on the global existence of strong solutions to the Cauchy problem for system \\eqref{NSK_simplified}-\\eqref{far field} with arbitrarily large initial data.\n\n\\begin{equation}\\label{far field}\n    \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}", "full_context": "e assumption of radial symmetry with $\\gamma\\ge\\frac{3}{2}$. Subsequently, Gu-Huang \\cite{Gu-Huang} extended the range of the exponent to $\\gamma > 1$ and generalized the result to the three-dimensional case with $1 < \\gamma < 3$. Concurrently, the global existence of large solutions for the associated Cauchy problem was proved independently by Chen-Zhang-Zhu \\cite{Chen-Zhang-Zhu}. Moreover, Huang-Meng-Zhang \\cite{Huang-Meng-Zhang-V} successfully proved the global classical well-posedness for general isentropic compressible Navier-Stokes equations satisfying the BD entropy condition in 2D and 3D, treating the shallow water model as a specific instance.\\par\n When $\\kappa(\\rho) >0$, we review some related works regarding the well-posedness of the Navier-Stokes-Korteweg system with general viscosity coefficients. For the one-dimensional case, considering the system with specific density-dependent viscosity $\\mu(\\rho) = \\rho$ and capillarity $\\kappa(\\rho) = \\rho^{-1}$, Charve-Haspot \\cite{Charve-Haspot} established the existence of global strong solutions in the whole space, allowing for large non-vacuum initial data. Furthermore, they demonstrated that these solutions converge to the entropic weak solutions of the compressible Euler equations. Moreover, Germain-LeFloch \\cite{Germain-LeFloch} proved the global existence of finite energy weak solutions for the Cauchy problem with general density-dependent coefficients and demonstrated their convergence to the entropy solutions of the Euler system. For the case with power-law viscosity $\\mu(\\varrho) = \\varrho^\\alpha$ and capillarity $\\kappa(\\varrho) = \\varrho^\\beta$ satisfying specific conditions, Antonelli-Bresch-Spirito \\cite{Antonelli-Bresch-Spirito} established the existence of global weak solutions for the periodic problem with large data. Furthermore, investigating the Cauchy problem in Lagrangian coordinates, Chen et al. \\cite{Chen-Chai} obtained global classical solutions for large initial data away from vacuum, considering density-dependent viscosity and capillarity. For the multi-dimensional case, building upon their earlier local theory \\cite{Hattori-Li}, Hattori-Li \\cite{Hattori-Li-2} established the global existence of solutions for the constant-coefficient Cauchy problem with small, non-vacuum initial data. For the case where both viscosity and capillarity coefficients depend on the density, Danchin-Desjardins \\cite{Danchin_Desjardins} obtained global smooth solutions. Their result holds for small perturbations of a non-vacuum state in functional spaces that are critical with respect to the physical energy. Bresch and Desjardins \\cite{B-D} investigated the two-dimensional viscous shallow water equations extended by a capillary term. They established the global existence of weak solutions in the presence of vacuum and demonstrated their convergence towards the strong solution of the viscous quasi-geostrophic system with a free surface. \\par\nHowever, the global existence of strong solutions with arbitrarily large initial data for the multi-dimensional Navier-Stokes-Korteweg system has long remained an open problem. It was not until recently that the first author Huang \\cite{Huang-Meng-Zhang} in his newly preprint resolved this by establishing global strong solutions on the two- and three-dimensional periodic torus, provided that the initial density is bounded from above and below. However, extending their method to the whole space presents new difficulties. In this paper, we overcome these obstacles to establish the global existence of strong solutions for the multi-dimensional Navier-Stokes-Korteweg system with large initial data in the whole space. By employing the novel critical inequality established by the first author in his newly preprint \\cite{Huang-Meng-Zhang}, we successfully establish the lower bound of the density by employing a novel De Giorgi iteration method. It is worth noting that our work does not require any radial symmetry assumption. To the best of our knowledge, this can be regarded as the first result concerning global large solutions for the corresponding Cauchy problem.\n\\par\nThroughout the rest of this paper, we focus on the case of shallow-water viscosity coefficients\n$$\\mu(\\rho) = \\rho, \\quad \\lambda(\\rho) = 0,$$\nand we assume that $\\kappa(\\rho)$ satisfies \n$$\\kappa(\\rho)=\\frac{1}{\\rho}.$$\nWith the specific choice of the capillarity coefficient $\\kappa(\\rho)$, the divergence of the Korteweg tensor takes the following form\n\\begin{equation}\n\\nabla \\cdot \\mathbb{K} = \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho).\n\\end{equation}\nConsequently, under the assumption of shallow-water viscosity ($\\mu(\\rho)=\\rho, \\lambda(\\rho)=0$) and the specific capillarity coefficient $\\kappa(\\rho)=1/\\rho$, equation \\eqref{NSK} can be rewritten as\n\\begin{equation}\\label{NSK_simplified} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) = \\nabla \\cdot \\left(2 \\rho \\mathcal{D}(u) \\right) + \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho). \\end{cases} \\end{equation}\nNow we investigate the global existence of strong solutions to system \\eqref{NSK_simplified} in $\\mathbb{R}^N$, where $N=2, 3$. The system is supplemented with the prescribed initial data $(\\rho_0, u_0)$ satisfying\n\\begin{equation}\n\\rho(x, 0) = \\rho_0(x), \\quad u(x, 0) = u_0(x), \\quad \\text{for } x \\in \\mathbb{R}^N,\n\\end{equation}\nwith far field behavior \n\\begin{equation}\\label{far field}\n \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}\nWe now state the main result on the global existence of strong solutions to the Cauchy problem for system \\eqref{NSK_simplified}-\\eqref{far field} with arbitrarily large initial data.\n\n\\begin{equation}\\label{far field}\n    \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}\n\n\\begin{document}\n\\maketitle\n\\begin{abstract}\nIn this paper, we establish global strong solutions for arbitrarily large initial data to the multi-dimensional compressible Navier-Stokes–Korteweg system, also referred to as the quantum Navier–Stokes equations, originally derived by Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95–133, 1985]. Specifically, we prove the existence of global strong solutions for arbitrarily large initial data in the case $N=2$ when $\\gamma \\ge 1$, and $N=3$ with $1 \\le \\gamma < 8/3$ for the associated Cauchy problem. By employing techniques from Littlewood–Paley theory, range truncation analysis, refined Nash-Moser and De Giorgi iteration methods, we derive positive upper and lower bounds for the density. As a consequence, we are able to treat the whole-space case with strictly positive far-field density. To the best of our knowledge, this is the first result that establishes global strong solutions for physically relevant compressible Navier–Stokes equations in the whole space, without imposing any symmetry or special geometric assumptions on the initial data.\n\\end{abstract}\n\\bigskip\n\\noindent \\textbf{Keywords:} compressible Navier-Stokes-Korteweg system; quantum Navier-Stokes system; global large strong solutions; Nash-Moser iteration; De Giorgi iteration; Cauchy problem.\n\nIn the first phase, we work within the framework of Besov spaces, utilizing maximal regularity estimates for parabolic equations to secure the upper bound. The second and most crucial step is establishing the density lower bound. Although Haspot \\cite{Haspot 1} investigated the isothermal NSK system, his approach failed to yield a lower bound for the density. Our strategy draws inspiration from a key inequality introduced by the first author in \\cite{Huang-Meng-Zhang}. However, the technique in \\cite{Huang-Meng-Zhang} cannot be directly applied to the whole space problem with non-vacuum far-field conditions. To overcome this obstacle, we have developed a novel truncated De Giorgi iteration method.\n\n\\begin{proof}\nBy H\\\"{o}lder's inequality, Sobolev embedding and \\eqref{eq:19}, we have\n\\begin{equation}\\label{the initial estimate for Psi}\n\\begin{aligned}\n \\Psi(r(p+2)) &= \\int^{T}_{0}\\int_{\\mathbb{R}^{3}}\\rho|v|^{\\frac{5}{3}(p+2)}\\,dx\\,dt \\\\\n &\\leq \\int^{T}_{0}\\left( \\int \\rho^{\\frac{3}{2}}|v|^{p+2}\\,dx \\right)^{\\frac{2}{3}} \\left( \\int|v|^{3(p+2)}\\,dx \\right)^{\\frac{1}{3}}\\,dt \\\\\n &\\leq \\|\\rho\\|_{L_{T}^{\\infty}L^{\\infty}}^{\\frac{1}{3}} \\sup_{0 \\le t \\le T}\\left( \\int \\rho|v|^{p+2}\\,dx \\right)^{\\frac{2}{3}} \\int^{T}_{0} \\left\\| |v|^{\\frac{p+2}{2}} \\right\\|_{L^{6}}^{2}\\,dt \\\\\n &\\leq C \\sup_{0 \\le t \\le T}\\left( \\int \\rho|v|^{p+2}\\,dx\\right)^{\\frac{2}{3}} \\|\\rho^{-1}\\|_{L_{T}^{\\infty}L^{\\infty}} \\\\\n &\\quad \\times \\left( \\int^{T}_{0}\\int \\rho \\left| \\nabla|v|^{\\frac{p+2}{2}} \\right|^{2}\\,dx\\,dt + \\int^{T}_{0}\\int \\rho|v|^{p+2}\\,dx\\,dt \\right) \\\\\n &\\le CV_{T} \\left( \\sup_{0 \\le t \\le T}\\int \\rho|v|^{p+2}\\,dx + \\int^{T}_{0}\\int \\rho \\left| \\nabla|v|^{\\frac{p+2}{2}} \\right|^{2}\\,dx\\,dt \\right)^{r}.\n\\end{aligned}\n\\end{equation}\nTo estimate the right hand side, multiplying $\\eqref{parabolic_system}_2$ by $|v|^{p}v$ and integrating by parts yields\n\\begin{equation}\n \\frac{1}{p+2} \\frac{d}{dt}\\int \\rho|v|^{p+2}dx+p\\int \\rho|v|^{p}\\left| \\nabla \\left|v \\right| \\right| ^{2}dx+\\int \\rho|v|^{p}|\\nabla v|^{2}dx=\\int P(\\rho)\\nabla\\cdot\\left(\\left| v \\right| ^{p}v\\right)dx.\n\\end{equation}\nBy Young's inequality and \\eqref{eq:19}, we have\n\\begin{equation}\n \\begin{aligned}\n \\int P(\\rho)\\nabla \\cdot\\left(\\left| v \\right| ^{p}v\\right)dx\n=& \\int \\rho^{\\gamma}(\\left| v \\right| ^{p}\\nabla \\cdot v+ v\\cdot { \\nabla \\left( \\left| v \\right| ^{p} \\right) } )dx\\\\\n\\leq& \\int\\rho^{\\gamma}\\left| v \\right| ^{p} \\left| \\nabla v \\right|dx + p\\int \\rho^{\\gamma}|v|^{p}\\left| \\nabla \\left| v \\right| \\right|dx \\\\\n\\leq & \\frac{1}{2} \\int \\rho|v|^{p}|\\nabla v|^{2}dx+ \\frac{p}{2}\\int \\rho|v|^{p}|\\nabla \\left| v \\right| |^{2} dx+ C(p+1)\\int \\rho^{2\\gamma-1}|v|^{p}dx\\\\\n \\leq& \\frac{1}{2}\\int \\rho |v|^{p}\\left| \\nabla v \\right| ^{2}dx+ \\frac{p}{2}\\int \\rho|v|^{p}|\\nabla \\left| v \\right| |^{2}dx + C(p+1)\\left\\lVert \\rho \\right\\rVert _{L^{\\infty}}^{2(\\gamma-1)}\\int \\rho|v|^{p}dx\\\\ \n \\leq& \\frac{1}{2}\\int \\rho |v|^{p}\\left| \\nabla v \\right| ^{2}dx+ \\frac{p}{2}\\int \\rho|v|^{p}|\\nabla \\left| v \\right| |^{2}dx\\\\\n &+ C(p+1)\\sup_{0 \\le t \\le T}\\left( \\int \\rho|v|^{2}dx \\right)^{\\frac{2}{p}}\\left( \\int \\rho|v|^{p+2} dx\\right)^{1- \\frac{2}{p}}.\n\\end{aligned}\n\\end{equation}\nCombining \\eqref{v energy}, we have\n\\begin{equation}\n \\frac{1}{p+2}\\frac{d}{dt}\\int \\rho |v|^{p+2}dx + \\frac{1}{2}\\int \\rho |v|^p |\\nabla v|^2dx + \\frac{p}{2}\\int \\rho |v|^p |\\nabla |v||^2dx \\leq C(p+1)\\left( \\int \\rho |v|^{p+2} dx \\right)^{1-\\frac{2}{p}}.\n\\end{equation}\nIntegrating over $t\\in[0,T]$, we obtain\n\\begin{equation}\n \\begin{aligned}\n&\\sup_{0 \\le t \\le T}\\frac{1}{p+2}\\int{\\rho}|v|^{p+2}dx+ \\frac{1}{2}\\int_{0}^{T}\\int \\rho |v|^p |\\nabla v|^2dxdt + \\frac{p}{2}\\int_{0}^{T}\\int \\rho |v|^p |\\nabla |v||^2dxdt \\\\\n\\leq &C(p+1)\\int^{T}_{0}\\left( \\int \\rho |v|^{p+2} dx \\right)^{1-\\frac{2}{p}}dt+\\frac{2}{p+2}\\int \\rho_{0}|v_{0}|^{p+2}\\\\\n\\leq& C(p+1){T^{*}}^{\\frac{2}{p}}\\left( \\int^{T}_{0}\\int \\rho|v|^{p+2}dxdt \\right)^{1- \\frac{2}{p}}+ \\frac{2}{p+2}\\int \\rho_{0}|v_{0}|^{p+2}.\n\\end{aligned}\n\\end{equation}\nSubstituting this into the RHS of \\eqref{the initial estimate for Psi}, we get\n\\begin{equation}\n \\begin{aligned}\n\\Psi(r(p+2))&\\leq C{V}_{T} \\left( C(p+2)^2 \\left( \\int^{T}_{0}\\int \\rho |v|^{p+2} dxdt \\right)^{\\frac{p-2}{p}} + C \\int \\rho_0 |v_0|^{p+2} dx \\right) ^{r }\\\\\n&\\leq C{V}_{T} (p+2)^{2r } \\left( \\int^{T}_{0}\\int \\rho |v|^{p+2} dxdt \\right)^{r } + C{V}_{T} (p+2)^{2r } + C{V}_{T} \\left( \\int \\rho_0 |v_0|^{p+2} dx \\right)^{r }.\n\\end{aligned}\n\\end{equation}\nDenoting the constant in the RHS as $C_{3}\\geq1$, and noting that\n\\begin{equation}\n \\left\\| \\rho_0^{\\frac{1}{p+2}} v_0 \\right\\|_{L^{p+2}} \\leq \\left\\| \\rho_0^{\\frac{1}{2}} v_0 \\right\\|_{L^2}^{\\frac{p}{p+2}} \\|v_0\\|_{L^\\infty}^{\\frac{2}{p+2}} \\leq \\left\\| \\rho_0^{\\frac{1}{2}} v_0 \\right\\|_{L^2} + \\|v_0\\|_{L^\\infty},\n\\end{equation}\nthe proof is complete.\n\\end{proof}\nThe following proposition plays a pivotal role in deriving the positive lower bound for the density, as it establishes a crucial link between the density and the effective velocity.\n\\begin{proposition}\nThere exists a constant $c_{v}\\geq 1$ depending on $T^{*}$, $q$, $\\gamma$, $E_{0}$, $\\left\\lVert \\rho_{0}^{\\frac{1}{q+2}}v_{0} \\right\\rVert_{L^{q+2}}$, and $\\left\\lVert \\rho_{0} \\right\\rVert_{L^{\\infty}}$ such that\n\\begin{equation}\\label{eq:v_L_infty_est}\n \\left\\lVert v \\right\\rVert _{L_{T}^{\\infty}\\left(L^{\\infty}\\left(\\mathbb{R}^3\\right)\\right)}\\leq c_{v}(\\log V_{T} )^{\\frac{1}{2}}.\n\\end{equation}\n\\end{proposition}", "post_theorem_intro_text_len": 6699, "post_theorem_intro_text": "\\begin{remark}\n    For the sake of brevity, we restrict our proof to the three-dimensional case where $\\gamma \\in (1, \\frac{8}{3})$. For the critical 3D case $(\\gamma=1)$ and the 2D case $(\\gamma \\ge 1)$, although the density upper bound estimates for these cases were addressed in Haspot \\cite{Haspot 1} and Yu-Wu \\cite{Yu-Wu}, neither work derived the density lower bound. Consequently, they were unable to establish the global existence of solutions. We emphasize that the approach developed in this paper can be successfully applied to establish the density lower bound for the critical 3D case $(\\gamma=1)$ and the 2D case $(\\gamma \\ge 1)$ as well. Therefore, we are able to obtain global solutions for both of these cases. It is worth noting that the technique used by Haspot \\cite{Haspot 1} relies on the assumption that the estimate for $\\|\\rho^{\\frac{1}{q+2}} v\\|_{L^\\infty_T L^{q+2}}$ is independent of $q$ to derive the density lower bound. Since such $q$-independence does not hold in the present context, Haspot's method is not applicable for securing a strictly positive lower bound.\n\\end{remark}\n\\begin{remark}\n    Huang-Meng-Zhang \\cite{Huang-Meng-Zhang} established this result for the periodic domain. In this work, we extend their findings to the more challenging whole space setting by introducing a new De Giorgi iteration method to effectively handle the far field behavior.   \n\\end{remark}\nProvided that the density remains strictly positive, we define the effective velocity $v$ as\n\\begin{equation}\nv = u + \\nabla \\log \\rho.\n\\end{equation}\nThis transformation converts the original system \\eqref{NSK}  into the following parabolic system\n\\begin{equation}\\label{parabolic_system}\n\\begin{cases}\n\\partial_t \\rho + \\nabla \\cdot(\\rho v) - \\Delta \\rho = 0, \\\\\n\\rho \\partial_t v + \\rho u \\cdot \\nabla v + \\nabla P(\\rho) = \\nabla \\cdot(\\rho \\nabla v).\n\\end{cases}\n\\end{equation}\nConsequently, the initial effective velocity $v_0$ is defined by\n\\begin{equation}\\label{initial data}\nv_0 = u_0 + \\nabla \\log \\rho_0.\n\\end{equation}\nLet us outline the main strategy of the proof. Our proof is organized into three main parts: establishing the density upper bound, deriving the density lower bound, and finally, improving the regularity of the solution.\n\nIn the first phase, we work within the framework of Besov spaces, utilizing maximal regularity estimates for parabolic equations to secure the upper bound. The second and most crucial step is establishing the density lower bound. Although Haspot \\cite{Haspot 1} investigated the isothermal NSK system, his approach failed to yield a lower bound for the density. Our strategy draws inspiration from a key inequality introduced by the first author in \\cite{Huang-Meng-Zhang}. However, the technique in \\cite{Huang-Meng-Zhang} cannot be directly applied to the whole space problem with non-vacuum far-field conditions. To overcome this obstacle, we have developed a novel truncated De Giorgi iteration method.\n\nWe now proceed to outline the derivation of these density bounds.\\\\\n\\textbf{Upper bound of $\\rho$.} \\par\nTo derive the upper bound of the density, we fully exploit the parabolic structure of the system and employ maximal regularity estimates for the heat equation. Specifically, closing the density estimate requires bounding the norm $\\|\\rho^{1/(q+2)}v\\|_{L^{q+2}}$ for some $q > 1$. In Proposition \\ref{PRO3.4}, by means of a precise domain decomposition analysis, we established that for any $\\gamma \\in (1, \\frac{8}{3})$, there exists a $q$ such that the norm $\\|\\rho^{\\frac{1}{q+2}} v\\|_{L^{q+2}}$ remains bounded.\\\\\n\\textbf{Lower bound of $\\rho$.} \\par\n To derive the lower bound of the density following Haspot\\cite{Haspot 1}'s argument, it is essential to control the norm $\\|\\rho^{\\frac{1}{p+2}} v\\|_{L^\\infty_T L^{p+2}}$ by a constant independent of $p$. However, the bound for $\\|\\rho^{\\frac{1}{p+2}} v\\|_{L^\\infty_T L^{p+2}}$ tends to infinity as $p \\to \\infty$. To overcome this obstacle, Huang-Meng-Zhang \\cite{Huang-Meng-Zhang} employed the Moser iteration method to establish a control relationship between $\\|v\\|_{L^\\infty}$ and $\\sqrt{\\log (e^{\\frac{25}{9}}+\\|{\\rho}^{-1}\\|_{L^{\\infty}})}$. This critical inequality plays a pivotal role in our analysis. Although their approach can handle the density lower bound in a periodic domain, it fails in the whole space when the density exhibits non-zero far-field behavior at infinity. To address the difficulties arising in this context, we introduce the De Giorgi iteration technique to prove the lower bound of the density by employing a truncation level adapted to the far-field density. Specifically, we construct the following truncation and iteration sequence\n \\begin{equation}\n\\begin{split}\n    \\rho^{-1}_{(k_n)}&:=\\max\\{\\rho^{-1}-{k_n},0\\},\n    \\\\\n    U_{n}^{T}&:=\\| \\rho _{(k_n)}^{-1}\\| _{L_{T}^{\\infty}L^2}^2+\\| \\nabla \\rho _{(k_n)}^{-1}\\| _{L_{T}^{2}L^2}^2.\n\\end{split}\n \\end{equation}\n where $k_n:=M\\left( 1-2^{-n} \\right)+2\\left\\| {\\rho}^{-1} _0 \\right\\| _{L^{\\infty}}.$ It is worth noting that the initial iteration value $k_0 = 2\\|1/\\rho_0\\|_{L^\\infty}$ is chosen specifically to handle the far-field behavior while ensuring that the initial energy $U_0^T$ satisfies the convergence condition\n     \\begin{equation}\n    U_0^T \\le K^{-\\frac{1}{\\nu}} A^{-\\frac{1}{\\nu^2}}.\n\\end{equation}\n We choose a suitable $T$ to bound the solution on $[0, T]$ and then employ a shifted iteration sequence for $[T, 2T]$. As long as the time step for each extension is uniform, this procedure proves the boundedness within the maximal lifespan $T^*$, which in turn establishes the time-dependent lower bound of the density.\\par\n In Section 2, we introduce some preliminaries, with a particular focus on Littlewood-Paley theory. In Section 3, we first demonstrate that for any $\\gamma \\in (1, \\frac{8}{3})$, there exists a $q>1$ satisfying the condition $\\gamma \\le \\frac{2q+6}{q+2}$ such that the quantity $\\sup_{0 \\le t \\le T} \\|\\rho^{\\frac{1}{q+2}} v\\|_{L^{q+2}}$ remains bounded. This allows us to establish the density upper bound by applying maximal regularity estimates for the heat equation. In Section 4, armed with the density upper bound, we extend the range of $q$ for which the boundedness of $\\sup_{0 \\le t \\le T} \\|\\rho^{\\frac{1}{q+2}} v\\|_{L^{q+2}}$ holds. Consequently, we employ the Moser iteration method to control $\\|v\\|_{L^{\\infty}}$ by a term involving $\\sqrt{\\log V_T}$, and subsequently utilize De Giorgi iteration to prove the existence of a density lower bound up to the maximal existence time $T^*$. Finally, in Section 5 and Section 6, we prove the main result by using the established density upper and lower bounds as blow-up criteria.", "sketch": "To prove Theorem~\\ref{thm:global_existence_cauchy}, the authors assume the density stays strictly positive and introduce the \\emph{effective velocity} $v=u+\\nabla\\log\\rho$, which transforms \\eqref{NSK} into the parabolic system \\eqref{parabolic_system}. They then “outline the main strategy” in three parts: “establishing the density upper bound, deriving the density lower bound, and finally, improving the regularity of the solution.”\n\n(1) \\textbf{Upper bound of $\\rho$.} They “fully exploit the parabolic structure of the system and employ maximal regularity estimates for the heat equation.” Closing the estimate requires bounding $\\|\\rho^{1/(q+2)}v\\|_{L^{q+2}}$ for some $q>1$; in Proposition~\\ref{PRO3.4}, “by means of a precise domain decomposition analysis,” they show that for any $\\gamma\\in(1,\\frac{8}{3})$ there exists such a $q$ with $\\|\\rho^{\\frac{1}{q+2}}v\\|_{L^{q+2}}$ bounded.\n\n(2) \\textbf{Lower bound of $\\rho$.} They explain that Haspot’s route would need $\\|\\rho^{\\frac{1}{p+2}}v\\|_{L^\\infty_T L^{p+2}}$ bounded uniformly in $p$, but here it “tends to infinity as $p\\to\\infty$.” Instead they use the Huang–Meng–Zhang idea relating “$\\|v\\|_{L^\\infty}$ and $\\sqrt{\\log( e^{\\frac{25}{9}}+\\|\\rho^{-1}\\|_{L^\\infty})}$,” but since the periodic-domain method “fails in the whole space” with nonzero far-field density, they “developed a novel truncated De Giorgi iteration method.” They define truncations and energies\n\\[\n\\rho^{-1}_{(k_n)}:=\\max\\{\\rho^{-1}-k_n,0\\},\\qquad\nU_n^T:=\\|\\rho^{-1}_{(k_n)}\\|_{L_T^\\infty L^2}^2+\\|\\nabla\\rho^{-1}_{(k_n)}\\|_{L_T^2 L^2}^2,\n\\]\nwith truncation levels $k_n:=M(1-2^{-n})+2\\|\\rho_0^{-1}\\|_{L^\\infty}$, choosing $k_0=2\\|1/\\rho_0\\|_{L^\\infty}$ so that $U_0^T$ meets a convergence condition $U_0^T\\le K^{-1/\\nu}A^{-1/\\nu^2}$. They “choose a suitable $T$ to bound the solution on $[0,T]$ and then employ a shifted iteration sequence for $[T,2T]$,” and with a uniform time step this yields boundedness up to the maximal lifespan $T^*$, giving a “time-dependent lower bound of the density.”\n\n(3) \\textbf{Completion / global existence.} After establishing density upper and lower bounds, they “prove the main result by using the established density upper and lower bounds as blow-up criteria” (Sections 5–6), together with the earlier steps (Sections 3–4: maximal regularity for the upper bound; extending the $q$-range, then Moser to control $\\|v\\|_{L^\\infty}$ by $\\sqrt{\\log V_T}$, and De Giorgi for the lower bound up to $T^*$).", "expanded_sketch": "To prove the main theorem, the authors assume the density stays strictly positive and introduce the \\emph{effective velocity} $v=u+\\nabla\\log\\rho$, which transforms\n\\begin{equation}\\label{NSK} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \n\\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) =\\nabla \\cdot \\mathbb{S} + \\nabla \\cdot \\mathbb{K}. \\end{cases} \\end{equation}\ninto the parabolic system \\eqref{parabolic_system}. They then “outline the main strategy” in three parts: “establishing the density upper bound, deriving the density lower bound, and finally, improving the regularity of the solution.”\n\n(1) \\textbf{Upper bound of $\\rho$.} They “fully exploit the parabolic structure of the system and employ maximal regularity estimates for the heat equation.” Closing the estimate requires bounding $\\|\\rho^{1/(q+2)}v\\|_{L^{q+2}}$ for some $q>1$. We first use the following proposition.\n\\begin{proposition}\\label{PRO3.4}\nFor any $\\gamma \\in (1, \\frac{8}{3})$, there exists $q \\in (1, 4)$ such that\n\\begin{equation}\n     \\gamma \\le \\frac{2q+6}{q+2}.\n\\end{equation}\n    For this fixed $q$, there exists a positive constant $C$, depending on $T$, $q$, $\\gamma$, $E_0$, and the initial data $\\|\\rho_0^{1/(q+2)} v_0\\|_{L^{q+2}}$, such that\n\\begin{equation}\\label{L^{q+2}}\n\\sup_{0 \\le t \\le T} \\|\\rho^{\\frac{1}{q+2}} v\\|_{L^{q+2}} \\le C.\n\\end{equation}\n\\end{proposition}\nIn particular, for any $\\gamma\\in(1,\\frac{8}{3})$ one can choose such a $q$ so that $\\|\\rho^{\\frac{1}{q+2}}v\\|_{L^{q+2}}$ is bounded.\n\n(2) \\textbf{Lower bound of $\\rho$.} They explain that Haspot’s route would need $\\|\\rho^{\\frac{1}{p+2}}v\\|_{L^\\infty_T L^{p+2}}$ bounded uniformly in $p$, but here it “tends to infinity as $p\\to\\infty$.” Instead they use the Huang–Meng–Zhang idea relating “$\\|v\\|_{L^\\infty}$ and $\\sqrt{\\log( e^{\\frac{25}{9}}+\\|\\rho^{-1}\\|_{L^\\infty})}$,” but since the periodic-domain method “fails in the whole space” with nonzero far-field density, they “developed a novel truncated De Giorgi iteration method.” They define truncations and energies\n\\[\n\\rho^{-1}_{(k_n)}:=\\max\\{\\rho^{-1}-k_n,0\\},\\qquad\nU_n^T:=\\|\\rho^{-1}_{(k_n)}\\|_{L_T^\\infty L^2}^2+\\|\\nabla\\rho^{-1}_{(k_n)}\\|_{L_T^2 L^2}^2,\n\\]\nwith truncation levels $k_n:=M(1-2^{-n})+2\\|\\rho_0^{-1}\\|_{L^\\infty}$, choosing $k_0=2\\|1/\\rho_0\\|_{L^\\infty}$ so that $U_0^T$ meets a convergence condition $U_0^T\\le K^{-1/\\nu}A^{-1/\\nu^2}$. They “choose a suitable $T$ to bound the solution on $[0,T]$ and then employ a shifted iteration sequence for $[T,2T]$,” and with a uniform time step this yields boundedness up to the maximal lifespan $T^*$, giving a “time-dependent lower bound of the density.”\n\n(3) \\textbf{Completion / global existence.} After establishing density upper and lower bounds, they complete the proof of the main theorem by using the established density upper and lower bounds as blow-up criteria (Sections 5–6), together with the earlier steps (Sections 3–4: maximal regularity for the upper bound; extending the $q$-range, then Moser to control $\\|v\\|_{L^\\infty}$ by $\\sqrt{\\log V_T}$, and De Giorgi for the lower bound up to $T^*$).", "expanded_theorem": "\\label{thm:global_existence_cauchy}\nLet $N=2$ or $N=3$. Assume that $\\gamma$ satisfies\n\\begin{equation}\n    \\begin{cases}\n        \\gamma \\in [1, \\infty) & \\text{if } N=2, \\\\\n        \\gamma \\in [1, 8/3) & \\text{if } N=3.\n    \\end{cases}\n\\end{equation}\nLet $\\bar{\\rho} > 0$ be the far field behavior of density. Assume that the initial data $(\\rho_0, u_0)$ satisfies\n\\begin{equation}\n    0 < \\underline{\\varrho} \\le \\rho_0 \\le \\bar{\\varrho}, \\quad (\\rho_0 - \\bar{\\rho}) \\in H^3(\\mathbb{R}^N), \\quad u_0 \\in H^2(\\mathbb{R}^N),\n\\end{equation}\nwhere $\\underline{\\varrho}$ and $\\bar{\\varrho}$ are positive constants. Then the Cauchy problem\n\\begin{equation}\\label{NSK_simplified} \\begin{cases} \\partial_t \\rho + \\nabla \\cdot(\\rho u) = 0, \\\\ \\partial_t (\\rho u) + \\nabla \\cdot(\\rho u \\otimes u) + \\nabla P(\\rho) = \\nabla \\cdot \\left(2 \\rho \\mathcal{D}(u) \\right) + \\nabla \\cdot (\\rho \\nabla \\nabla \\log \\rho). \\end{cases} \\end{equation}\n\\begin{equation}\\label{far field}\n    \\rho \\left( x,t \\right) \\rightarrow \\bar{\\rho}>0,\\quad u\\left( x,t \\right) \\rightarrow 0,\\quad \\mathrm{as}\\, |x|\\rightarrow \\infty .\n\\end{equation}\nadmits a unique global strong solution $(\\rho, u)$ satisfying for any $0 < T < \\infty$ and $(x,t) \\in \\mathbb{R}^N \\times [0, T]$,\n\\begin{equation}\n    (C(T))^{-1} \\le \\rho(x,t) \\le C(T),\n\\end{equation}\nand\n\\begin{equation}\n\\begin{aligned}\n    &(\\rho - \\bar{\\rho}) \\in C([0, T]; H^3) \\cap L^2(0, T; H^4), \\quad \\rho_t \\in C([0, T]; H^1) \\cap L^2(0, T; H^2), \\\\\n    &u \\in C([0, T]; H^2) \\cap L^2(0, T; H^3), \\quad u_t \\in L^\\infty(0, T; L^2) \\cap L^2(0, T; H^1),\n\\end{aligned}\n\\end{equation}\nwhere the constant $C(T) > 0$ depends on the initial data and $T$.", "theorem_type": ["Existence", "Uniqueness"], "mcq": {"question": "Let $N=2$ or $N=3$, and assume\n\\[\n\\gamma\\in[1,\\infty)\\quad\\text{if }N=2,\n\\qquad\n\\gamma\\in[1,8/3)\\quad\\text{if }N=3.\n\\]\nLet $\\bar\\rho>0$ be the far-field density, and let the initial data $(\\rho_0,u_0)$ satisfy\n\\[\n0<\\underline\\varrho\\le \\rho_0\\le \\bar\\varrho,\n\\qquad\n\\rho_0-\\bar\\rho\\in H^3(\\mathbb R^N),\n\\qquad\nu_0\\in H^2(\\mathbb R^N),\n\\]\nfor some positive constants $\\underline\\varrho,\\bar\\varrho$. Consider the Cauchy problem for the shallow-water Navier-Stokes-Korteweg system on $\\mathbb R^N$:\n\\[\n\\begin{cases}\n\\partial_t\\rho+\\nabla\\cdot(\\rho u)=0,\\\\\n\\partial_t(\\rho u)+\\nabla\\cdot(\\rho u\\otimes u)+\\nabla P(\\rho)\n=\\nabla\\cdot\\bigl(2\\rho\\mathcal D(u)\\bigr)+\\nabla\\cdot\\bigl(\\rho\\nabla\\nabla\\log\\rho\\bigr),\n\\end{cases}\n\\]\nwith far-field condition\n\\[\n\\rho(x,t)\\to\\bar\\rho>0,\n\\qquad\nu(x,t)\\to0,\n\\qquad \\text{as }|x|\\to\\infty,\n\\]\nand initial data $(\\rho,u)|_{t=0}=(\\rho_0,u_0)$. Which statement holds about existence and uniqueness of solutions?", "correct_choice": {"label": "A", "text": "There exists a unique global strong solution $(\\rho,u)$ to this Cauchy problem such that, for every $0<T<\\infty$, there is a constant $C(T)>0$ depending on $T$ and the initial data with\n\\[\n(C(T))^{-1}\\le \\rho(x,t)\\le C(T)\n\\quad\\text{for all }(x,t)\\in\\mathbb R^N\\times[0,T],\n\\]\nand\n\\[\n\\rho-\\bar\\rho\\in C([0,T];H^3(\\mathbb R^N))\\cap L^2(0,T;H^4(\\mathbb R^N)),\n\\qquad\n\\rho_t\\in C([0,T];H^1(\\mathbb R^N))\\cap L^2(0,T;H^2(\\mathbb R^N)),\n\\]\n\\[\nu\\in C([0,T];H^2(\\mathbb R^N))\\cap L^2(0,T;H^3(\\mathbb R^N)),\n\\qquad\nu_t\\in L^\\infty(0,T;L^2(\\mathbb R^N))\\cap L^2(0,T;H^1(\\mathbb R^N)).\n\\]"}, "choices": [{"label": "B", "text": "There exists a unique global strong solution $(\\rho,u)$ to this Cauchy problem such that there is a constant $C>0$, depending only on the initial data, with\n\\[\nC^{-1}\\le \\rho(x,t)\\le C\n\\quad\\text{for all }(x,t)\\in\\mathbb R^N\\times[0,\\infty),\n\\]\nand, for every $0<T<\\infty$,\n\\[\n\\rho-\\bar\\rho\\in C([0,T];H^3(\\mathbb R^N))\\cap L^2(0,T;H^4(\\mathbb R^N)),\n\\qquad\n\\rho_t\\in C([0,T];H^1(\\mathbb R^N))\\cap L^2(0,T;H^2(\\mathbb R^N)),\n\\]\n\\[\nu\\in C([0,T];H^2(\\mathbb R^N))\\cap L^2(0,T;H^3(\\mathbb R^N)),\n\\qquad\n\\nu_t\\in L^\\infty(0,T;L^2(\\mathbb R^N))\\cap L^2(0,T;H^1(\\mathbb R^N)).\n\\]"}, {"label": "C", "text": "There exists a global strong solution $(\\rho,u)$ to this Cauchy problem such that, for every $0<T<\\infty$, there is a constant $C(T)>0$ depending on $T$ and the initial data with\n\\[\n(C(T))^{-1}\\le \\rho(x,t)\\le C(T)\n\\quad\\text{for all }(x,t)\\in\\mathbb R^N\\times[0,T],\n\\]\nand\n\\[\n\\rho-\\bar\\rho\\in C([0,T];H^3(\\mathbb R^N))\\cap L^2(0,T;H^4(\\mathbb R^N)),\n\\qquad\n\\rho_t\\in C([0,T];H^1(\\mathbb R^N))\\cap L^2(0,T;H^2(\\mathbb R^N)),\n\\]\n\\[\nu\\in C([0,T];H^2(\\mathbb R^N))\\cap L^2(0,T;H^3(\\mathbb R^N)),\n\\qquad\n\\nu_t\\in L^\\infty(0,T;L^2(\\mathbb R^N))\\cap L^2(0,T;H^1(\\mathbb R^N)).\n\\]"}, {"label": "D", "text": "There exists a unique global strong solution $(\\rho,u)$ to this Cauchy problem such that, for every $0<T<\\infty$, there is a constant $C(T)>0$ depending on $T$ and the initial data with\n\\[\n(C(T))^{-1}\\le \\rho(x,t)\\le C(T)\n\\quad\\text{for all }(x,t)\\in\\mathbb R^N\\times[0,T],\n\\]\nand\n\\[\n\\rho-\\bar\\rho\\in C([0,T];H^3(\\mathbb R^N))\\cap L^2(0,T;H^4(\\mathbb R^N)),\n\\qquad\n\\rho_t\\in C([0,T];H^1(\\mathbb R^N))\\cap L^2(0,T;H^2(\\mathbb R^N)),\n\\]\n\\[\nu\\in C([0,T];H^2(\\mathbb R^N))\\cap L^2(0,T;H^3(\\mathbb R^N)),\n\\qquad\n\\nu_t\\in C([0,T];L^2(\\mathbb R^N))\\cap L^2(0,T;H^1(\\mathbb R^N)).\n\\]"}, {"label": "E", "text": "There exists a unique global strong solution $(\\rho,u)$ to this Cauchy problem for all\n\\[\n\ngamma\\in[1,\\infty)\\quad\\text{if }N=2\\text{ or }N=3,\n\\]\nsuch that, for every $0<T<\\infty$, there is a constant $C(T)>0$ depending on $T$ and the initial data with\n\\[\n(C(T))^{-1}\\le \\rho(x,t)\\le C(T)\n\\quad\\text{for all }(x,t)\\in\\mathbb R^N\\times[0,T],\n\\]\nand\n\\[\n\\rho-\\bar\\rho\\in C([0,T];H^3(\\mathbb R^N))\\cap L^2(0,T;H^4(\\mathbb R^N)),\n\\qquad\n\\rho_t\\in C([0,T];H^1(\\mathbb R^N))\\cap L^2(0,T;H^2(\\mathbb R^N)),\n\\]\n\\[\nu\\in C([0,T];H^2(\\mathbb R^N))\\cap L^2(0,T;H^3(\\mathbb R^N)),\n\\qquad\n\\nu_t\\in L^\\infty(0,T;L^2(\\mathbb R^N))\\cap L^2(0,T;H^1(\\mathbb R^N)).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "time_dependent_density_bound_C(T)", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "uniqueness", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "sharp_time_regularality_of_u_t", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "3D_gamma_upper_range_8_over_3", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It states the hypotheses of a theorem and asks for the resulting conclusion, without signaling which of the close variants is right."}, "TAS": {"score": 0, "justification": "This is essentially a direct restatement of a theorem: the hypotheses are given in the stem and the correct choice reproduces the theorem conclusion almost verbatim."}, "GPS": {"score": 1, "justification": "Some discrimination is required because the distractors make subtle changes to uniqueness, uniform-in-time bounds, regularity of time derivatives, and the admissible gamma-range. However, this mainly tests precise theorem recall rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically close to the true statement. They reflect realistic failure modes: strengthening bounds too much, dropping uniqueness, altering sharp regularity, or enlarging the parameter range."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-recall MCQ with strong distractors and no obvious answer leakage, but it is highly tautological and only weakly probes genuine generative reasoning."}}
{"id": "2602.10707v1", "paper_link": "http://arxiv.org/abs/2602.10707v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:intro}\nThe group of homeomorphisms of the real projective plane $\\mathbb{R}\\mathrm{P}^2$ and Möbius strip $M$ which are isotopic to the identity is stably unbounded with respect to commutator length in the sense of \\cite{BIP}.", "start_pos": 2526, "end_pos": 2773, "label": "thm:intro"}, "ref_dict": {"subsec:actions-on-cg": "\\begin{theorem}[Guessing geodesics]\\label{thm:bowditch-guessing-geodesics}\n    Let $X$ be a graph and $D>0$ a number.\n    Suppose that for each pair of distinct vertices $x,y$ of $X$ we have chosen a connected subgraph $P(x,y)$ containing $x$ and $y$.\n    Suppose that if $d_X(x,y) = 1$, the diameter of $P(x,y)$ is at most $D$, and in addition we have for all pairwise distinct $x,y,z$:\n    \\[ P(x,z) \\subseteq N_D(P(x,y)\\cup P(y,z)). \\]\n    Then $X$ is $\\delta(D)$-hyperbolic, and each $P(x,y)$ is $B(D)$-Hausdorff close to a geodesic joining $x$ to $y$.\n\\end{theorem}\n\n\\subsection{Actions on curve graphs}\\label{subsec:actions-on-cg}\nOne can easily check that the groups $\\operatorname{Homeo}_0$ and $\\operatorname{Mcg}$ of a surface act on the fine curve graph respectively curve graph of that surface by isometries.\n\nIn \\cite{Bestvina}, Bestvina and Fujiwara introduced the notion of weak proper discontinuity, which will be the key to finding elements with positive stable commutator length.\n\nFirst, recall that for two hyperbolic isometries $f$ and $g$ of a $\\delta$-hyperbolic space $(X,d)$, we can find invariant quasi-axes $\\alpha$ and $\\beta$ of quality $L$ respectively $L'$ by taking the orbit of some point, e.g. $\\alpha = (a_k) \\coloneqq f^k(x_0)$ for $k\\in\\ZZ$ and $x_0\\in X$, and analogously for $\\beta = (b_k)$.\nWe then say that $f$ and $g$ (respectively their axes) are \\emph{quasi-equivalent}, if there exists a constant $B = B(L,L',\\delta)$ such that:\nfor any $D>0$, there is an isometry $h$ that sends a segment of length $D$ in $\\alpha$ into a $B$-neighbourhood of $\\beta$.\nIf $f$ is quasi-equivalent to its inverse, then we say that $f$ is \\emph{quasi-invertible}.\n\nThese quasi-geodesics define points in the Gromov boundary of $X$, and we can give an alternative definition of quasi-equivalence in terms of boundary points:\nlet $\\tau_\\pm \\coloneqq [a_{\\pm k}]$ and $\\upsilon_\\pm \\coloneqq [b_{\\pm k}]$ as equivalence classes of admissible sequences as in the previous section.\nThen $f$ and $g$ (respectively their axes) are quasi-equivalent, if there exists a sequence $h_k$ of isometries of $X$ such that $\\lim_{k\\rightarrow \\infty} h_k(\\tau_\\pm) = \\upsilon_\\pm$.\n\nFurther, we call two hyperbolic elements \\emph{dependent}, if they have quasi-axes which contain rays that are in finite distance of each other, and \\emph{independent} otherwise.\nIf $G$ contains two independent hyperbolic elements, we call the action \\emph{non-elementary}.\n\nThe central definition in this section is:\n\n\\begin{definition}\n    Let $G$ be a group acting on the hyperbolic metric space $X$.\n    This action is \\emph{weakly properly discontinuous (WPD)}, if the following holds: \\begin{itemize}\n        \\item $G$ is not virtually cyclic\n        \\item $\\exists g\\in G\\colon$ $g$ acts on $X$ as a hyperbolic isometry\n        \\item $ \\forall x\\in X\\colon \\forall g \\in G \\textrm{ acting hyperbolically}\\colon \\forall C>0\\colon \\exists N>0\\colon \\{\\gamma \\in G \\mid d(x,\\gamma(x)) \\leq C, d(g^N(x),\\gamma(g^N(x)))\\leq C \\} \\textrm{ is finite.} $\n    \\end{itemize}\n    \\end{definition}\nNote that when we restrict an action $G\\curvearrowright X$ to a subgroup $H<G$, we only need to recheck the first two conditions to see that the induced action of $H$ on $X$ satisfies WPD.\nThe following proposition shows why WPD is a useful property:\n\n\\begin{proposition}[Proposition 6 of \\cite{Bestvina}]\\label{prop:useful-wpd}\n    Suppose that $G$ and $X$ satisfy WPD.\n    Then \\begin{enumerate}\n        \\item for every hyperbolic $g\\in G$, the centraliser $C(g)$ is virtually cyclic,\n        \\item for every hyperbolic $g\\in G$ and every $(K,L)$-quasi-axis $l$ for $g$, there is a constant $M$ depending on $g$, $K$ and $L$ such that if two translates $l_1$ and $l_2$ of $l$ contain (oriented) segments of length $>M$ that are oriented $B(K,L,\\delta)$-close, then $l_1$ and $l_2$ are oriented $B(K,L,\\delta)$-close and moreover the corresponding conjugates $g_1$ and $g_2$ of $g$ have positive powers which are equal,\n        \\item the action of $G$ on $X$ is non-elementary,\n        \\item two elements of $G$ are quasi-equivalent if and only if they have positive powers which are conjugate,\n        \\item there exist two hyperbolic elements in $G$ which are not quasi-equivalent.\n    \\end{enumerate}\n\\end{proposition}\n\nThe main result of \\cite{Bestvina} is:\n\\begin{theorem}[Theorem 1 of \\cite{Bestvina}]\\label{thm:bestvina-fujiwara-main-thm}\nSuppose a group $G$ acts on a $\\delta$-hyperbolic graph $X$ by isometries.\nSuppose also that the action is non-elementary and that there exist independent hyperbolic elements $g_1,g_2 \\in G$ which are not quasi-equivalent.\nThen, the space of non-trivial quasi-morphisms is infinite dimensional.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 923, "pre_theorem_intro_text": "In this article, we study the (un-)boundedness of norms on the group of homeomorphisms and diffeomorphisms of the Möbius strip and real projective plane that are isotopic to the identity.\nThis type of question first appeared in the work of Burago, Ivanov and Polterovich \\cite{BIP}, where the authors considered the identity components of groups of compactly supported $C^\\infty$-diffeomorphisms of various manifolds, and found that for all compact 3-manifolds and certain compact 4-manifolds these groups only admit conjugation-invariant norms that are bounded.\nTheir results have been extended to higher dimensional manifolds in \\cite{tsuboi}.\nFor surfaces, the situation is different:\nwhile the sphere again only admits conjugation-invariant norms that are bounded, almost all other surfaces have been found to have unbounded norms.\nThe main result of this article answers this question for the last two remaining cases:", "context": "In this article, we study the (un-)boundedness of norms on the group of homeomorphisms and diffeomorphisms of the Möbius strip and real projective plane that are isotopic to the identity.\nThis type of question first appeared in the work of Burago, Ivanov and Polterovich \\cite{BIP}, where the authors considered the identity components of groups of compactly supported $C^\\infty$-diffeomorphisms of various manifolds, and found that for all compact 3-manifolds and certain compact 4-manifolds these groups only admit conjugation-invariant norms that are bounded.\nTheir results have been extended to higher dimensional manifolds in \\cite{tsuboi}.\nFor surfaces, the situation is different:\nwhile the sphere again only admits conjugation-invariant norms that are bounded, almost all other surfaces have been found to have unbounded norms.\nThe main result of this article answers this question for the last two remaining cases:", "full_context": "In this article, we study the (un-)boundedness of norms on the group of homeomorphisms and diffeomorphisms of the Möbius strip and real projective plane that are isotopic to the identity.\nThis type of question first appeared in the work of Burago, Ivanov and Polterovich \\cite{BIP}, where the authors considered the identity components of groups of compactly supported $C^\\infty$-diffeomorphisms of various manifolds, and found that for all compact 3-manifolds and certain compact 4-manifolds these groups only admit conjugation-invariant norms that are bounded.\nTheir results have been extended to higher dimensional manifolds in \\cite{tsuboi}.\nFor surfaces, the situation is different:\nwhile the sphere again only admits conjugation-invariant norms that are bounded, almost all other surfaces have been found to have unbounded norms.\nThe main result of this article answers this question for the last two remaining cases:\n\n\\begin{abstract}\nUsing a recent result of Bowden, Hensel and Webb, we prove the existence of homeomorphisms with positive stable commutator length in the groups of homeomorphisms of the real projective plane and Möbius strip which are isotopic to the identity.\nThis completes the answer to a question posed by Burago, Ivanov and Polterovich on the boundedness of diffeomorphism groups of surfaces.\n\\end{abstract}\n\n\\section{Introduction}\nIn this article, we study the (un-)boundedness of norms on the group of homeomorphisms and diffeomorphisms of the Möbius strip and real projective plane that are isotopic to the identity.\nThis type of question first appeared in the work of Burago, Ivanov and Polterovich \\cite{BIP}, where the authors considered the identity components of groups of compactly supported $C^\\infty$-diffeomorphisms of various manifolds, and found that for all compact 3-manifolds and certain compact 4-manifolds these groups only admit conjugation-invariant norms that are bounded.\nTheir results have been extended to higher dimensional manifolds in \\cite{tsuboi}.\nFor surfaces, the situation is different:\nwhile the sphere again only admits conjugation-invariant norms that are bounded, almost all other surfaces have been found to have unbounded norms.\nThe main result of this article answers this question for the last two remaining cases:\n\nTherefore, all compact surfaces except the sphere admit unbounded norms.\nTo prove unboundedness, the central tool is Bavard duality \\cite{bavard}, which links a norm, namely commutator length, to the existence of non-trivial homogeneous quasi-morphisms.\nAn explicit construction of quasi-morphism was given by Brooks in \\cite{Brooks} for free groups.\nThis construction was generalised in \\cite{Bestvina} by Bestvina and Fujiwara to groups of isometries of a hyperbolic space, which they applied to (subgroups of) mapping class groups of hyperbolic surfaces.\n\n\\paragraph{Organisation of the paper.}\nIn the next section, we provide some background on the papers of Kuno \\cite{kuno-hyperbolicity} and Bowden, Hensel and Webb \\cite{BHW1} \\cite{BHW2}.\nIn the third section, we combine results of \\cite{kuno-hyperbolicity} and \\cite{BHW1} to show that the Möbius strip and real projective plane have hyperbolic and unbounded fine curve graphs.\nIn the fourth section, we use that result and the methods of \\cite{BHW2} to prove that both Möbius strip and real projective plane have stably unbounded $\\operatorname{Homeo}_0$.\n\n\\paragraph{Bavard duality.}\nIt is known that the groups $\\operatorname{Homeo}_0(M)$ of compact manifolds are perfect \\cite{anderson}, i.e. any element of the group can be written as a product of commutators, and the same is true for $\\operatorname{Diff}_0$, see e.g. \\cite{mann}.\nThe minimal number of commutators that appear in such a factorisation of a homeomorphism is called \\emph{commutator length}, usually denoted by cl, and it is an important example of a conjugation-invariant norm on the group $\\operatorname{Homeo}(M)$ as defined in \\cite{BIP}.\nSuch norms have a so-called \\emph{stabilisation}, which takes the following form for commutator length:\n\\[ \\operatorname{scl}(g) \\coloneqq \\lim_{n\\rightarrow \\infty} \\frac{\\operatorname{cl}(g^n)}{n} \\]\nfor $g\\in \\operatorname{Homeo}(M)$.\nWe call a group \\emph{stably unbounded} if the stabilisation of some norm takes positive values.\n\n\\subsection{Curve graphs and variations}\nCurve graphs and curve complexes were defined by Harvey in \\cite{Harvey}.\nThese are usually defined in terms of essential curves, but for our purposes it will be useful to only work with non-separating curves.\nThe main definition here is:\nthe \\emph{non-separating curve graph} $\\ncg(M)$ of a 2-manifold (possibly with boundary) $M$ is the graph with: \\begin{itemize}\n \\item vertices corresponding to the homotopy classes of non-separating closed curves in $M$\n \\item an edge between two vertices, if the corresponding homotopy classes have disjoint representatives\n\\end{itemize}\nTo study homeomorphisms that are isotopic to the identity, we have to get rid of the homotopy classes:\nthe \\emph{fine non-separating curve graph} accomplishes this.\nIt was introduced in \\cite{BHW1} and is defined by:\\begin{itemize}\n \\item vertices which each represent a non-separating closed curve in $M$\n \\item an edge between two vertices, if the corresponding curves are disjoint\n\\end{itemize}\n\nThe main result of \\cite{Bestvina} is:\n\\begin{theorem}[Theorem 1 of \\cite{Bestvina}]\\label{thm:bestvina-fujiwara-main-thm}\nSuppose a group $G$ acts on a $\\delta$-hyperbolic graph $X$ by isometries.\nSuppose also that the action is non-elementary and that there exist independent hyperbolic elements $g_1,g_2 \\in G$ which are not quasi-equivalent.\nThen, the space of non-trivial quasi-morphisms is infinite dimensional.\n\\end{theorem}\n\n\\begin{theorem}[cf. Theorem 4.1 in \\cite{BHW2}]\\label{thm:convergence-criterion}\nSuppose $(\\mathcal{F}_v,\\mathcal{F}_h,d)$ is a BSF structure with ending $\\mathcal{F}_v$.\nThen there is a point $\\tau_{\\mathcal{F}_v}$ on the Gromov boundary of $\\fncg(N)$ with the following properties: \\begin{enumerate}\n \\item The boundary point $\\tau_{\\mathcal{F}_v}$ depends only on the foliation $\\mathcal{F}_v$.\n \\item A homeomorphism of $N$ fixes $\\tau_{\\mathcal{F}_v}$ as a point on the Gromov boundary if and only if it preserves the foliation $\\mathcal{F}_v$.\n \\item A sequence of curves $(\\beta_i)$ in $\\fncg(N)$ converges to $\\tau_{\\mathcal{F}_v}$ if and only if \n the following conditions are satisfied: \\begin{itemize}\n \\item The sizes of the $\\beta_i$ diverge to infinity.\n \\item For any $B$, $\\epsilon$ there is a number $I$ so that if $b\\subseteq \\beta_i$ with $i>I$ is a segment of size at most $B$, then it $\\epsilon$-fellow travels a leaf segment of $\\mathcal{F}_v$ with respect to the metric $d$.\n \\end{itemize}", "post_theorem_intro_text_len": 2390, "post_theorem_intro_text": "Therefore, all compact surfaces except the sphere admit unbounded norms.\nTo prove unboundedness, the central tool is Bavard duality \\cite{bavard}, which links a norm, namely commutator length, to the existence of non-trivial homogeneous quasi-morphisms.\nAn explicit construction of quasi-morphism was given by Brooks in \\cite{Brooks} for free groups.\nThis construction was generalised in \\cite{Bestvina} by Bestvina and Fujiwara to groups of isometries of a hyperbolic space, which they applied to (subgroups of) mapping class groups of hyperbolic surfaces.\n\nIn their paper, they consider the action of the mapping class group on the curve graph (introduced by Harvey \\cite{Harvey}, shown to be hyperbolic by Masur and Minsky \\cite{masur-minsky}), which satisfies a newly identified property, namely weak proper discontinuity (see section \\ref{subsec:actions-on-cg}).\n\nAn analogous strategy was developed by Bowden, Hensel and Webb in \\cite{BHW1}, where the action of the identity component of the homeomorphism group of an orientable surface with positive genus on the newly introduced \\emph{fine} curve graph was studied.\n\nThis strategy was adapted to non-orientable surfaces of genus at least three in \\cite{kimura-kuno-qms}.\nIn \\cite{BHW2}, the action of these groups on the fine curve graph is studied further, and a tool is provided that characterises some of the elements whose powers have increasingly large commutator length.\nIn addition, \\cite{BHW2} provides a connection between certain foliations on surfaces and boundary points of the corresponding fine curve graph.\nIn this article, we aim to explain that the techniques and results of Bowden, Hensel and Webb generalise to non-orientable surfaces with genus 1.\n\n\\paragraph{Organisation of the paper.}\nIn the next section, we provide some background on the papers of Kuno \\cite{kuno-hyperbolicity} and Bowden, Hensel and Webb \\cite{BHW1} \\cite{BHW2}.\nIn the third section, we combine results of \\cite{kuno-hyperbolicity} and \\cite{BHW1} to show that the Möbius strip and real projective plane have hyperbolic and unbounded fine curve graphs.\nIn the fourth section, we use that result and the methods of \\cite{BHW2} to prove that both Möbius strip and real projective plane have stably unbounded $\\operatorname{Homeo}_0$.\n\n\\paragraph{Acknowledgements.}\nThe author wants to thank Sebastian Hensel and Javier de la Nuez Gonzalez.", "sketch": "To prove Theorem~\\ref{thm:intro}, the introduction says the \"central tool is Bavard duality\" (linking commutator length to the existence of \"non-trivial homogeneous quasi-morphisms\"). The strategy follows the quasi-morphism constructions of Brooks and their generalisation by Bestvina--Fujiwara via actions on hyperbolic spaces, and the analogous approach of Bowden--Hensel--Webb using the action of $\\operatorname{Homeo}_0$ on a (fine) curve graph. Concretely, the paper is organised to (1) combine results of \\cite{kuno-hyperbolicity} and \\cite{BHW1} to show the Möbius strip and $\\mathbb{R}\\mathrm{P}^2$ have \"hyperbolic and unbounded fine curve graphs\", and then (2) \"use that result and the methods of \\cite{BHW2}\" to prove that both surfaces have \"stably unbounded $\\operatorname{Homeo}_0$\" (hence stably unbounded commutator length).", "expanded_sketch": "To prove the main theorem, the introduction says the \"central tool is Bavard duality\" (linking commutator length to the existence of \"non-trivial homogeneous quasi-morphisms\"). The strategy follows the quasi-morphism constructions of Brooks and their generalisation by Bestvina--Fujiwara via actions on hyperbolic spaces, and the analogous approach of Bowden--Hensel--Webb using the action of $\\operatorname{Homeo}_0$ on a (fine) curve graph. Concretely, the paper is organised to (1) combine results of \\cite{kuno-hyperbolicity} and \\cite{BHW1} to show the Möbius strip and $\\mathbb{R}\\mathrm{P}^2$ have \"hyperbolic and unbounded fine curve graphs\", and then (2) use that result together with the methods of \\cite{BHW2} to prove that both surfaces have \"stably unbounded $\\operatorname{Homeo}_0$\" (hence stably unbounded commutator length).", "expanded_theorem": "\\label{thm:intro}\nThe group of homeomorphisms of the real projective plane $\\mathbb{R}\\mathrm{P}^2$ and Möbius strip $M$ which are isotopic to the identity is stably unbounded with respect to commutator length in the sense of \\cite{BIP}.", "theorem_type": ["Universal", "Inequality or Bound"], "mcq": {"question": "Let \\(\\operatorname{Homeo}_0(S)\\) denote the group of homeomorphisms of a surface \\(S\\) that are isotopic to the identity. For \\(g\\in \\operatorname{Homeo}_0(S)\\), let \\(\\operatorname{cl}(g)\\) be its commutator length, i.e. the minimal number of commutators whose product is \\(g\\), and define the stable commutator length by\n\\[\n\\operatorname{scl}(g)=\\lim_{n\\to\\infty}\\frac{\\operatorname{cl}(g^n)}{n}.\n\\]\nSay that \\(\\operatorname{Homeo}_0(S)\\) is stably unbounded with respect to commutator length if there exists some \\(g\\in \\operatorname{Homeo}_0(S)\\) with \\(\\operatorname{scl}(g)>0\\). Which statement holds for every surface \\(S\\) in the class consisting of the real projective plane \\(\\mathbb{R}\\mathrm{P}^2\\) and the Möbius strip \\(M\\)?", "correct_choice": {"label": "A", "text": "For each such surface \\(S\\in\\{\\mathbb{R}\\mathrm{P}^2,M\\}\\), the group \\(\\operatorname{Homeo}_0(S)\\) is stably unbounded with respect to commutator length; equivalently, there exists \\(g\\in \\operatorname{Homeo}_0(S)\\) such that \\(\\operatorname{scl}(g)>0\\)."}, "choices": [{"label": "B", "text": "For each such surface \\(S\\in\\{\\mathbb{R}\\mathrm{P}^2,M\\}\\), the group \\(\\operatorname{Homeo}_0(S)\\) is unbounded with respect to commutator length; equivalently, for every \\(N\\in\\mathbb{N}\\) there exists \\(g\\in \\operatorname{Homeo}_0(S)\\) with \\(\\operatorname{cl}(g)>N\\)."}, {"label": "C", "text": "For each such surface \\(S\\in\\{\\mathbb{R}\\mathrm{P}^2,M\\}\\), there exists \\(g\\in \\operatorname{Homeo}_0(S)\\) whose commutator lengths \\(\\operatorname{cl}(g^n)\\) are unbounded as \\(n\\to\\infty\\)."}, {"label": "D", "text": "There exists a single element \\(g\\) such that for every surface \\(S\\in\\{\\mathbb{R}\\mathrm{P}^2,M\\}\\), one has \\(g\\in \\operatorname{Homeo}_0(S)\\) and \\(\\operatorname{scl}(g)>0\\)."}, {"label": "E", "text": "For each such surface \\(S\\in\\{\\mathbb{R}\\mathrm{P}^2,M\\}\\), every nontrivial element \\(g\\in \\operatorname{Homeo}_0(S)\\) satisfies \\(\\operatorname{scl}(g)>0\\), so \\(\\operatorname{Homeo}_0(S)\\) is stably unbounded with respect to commutator length."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "positive_scl_vs_mere_unbounded_cl", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_positive_linear_growth_rate_replaced_by_unboundedness_of_cl_gn", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "element_depends_on_surface", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "existence_of_some_element_strengthened_to_all_nontrivial_elements", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives definitions but does not explicitly or implicitly reveal that both groups are stably unbounded. The correct conclusion is not stated in advance."}, "TAS": {"score": 1, "justification": "The item is close to asking for the theorem’s conclusion directly: it asks which statement holds for these two specific groups under the given definitions. However, it is not a pure restatement because the options vary in strength and scope."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish positive stable commutator length from merely unbounded commutator length, and to reject overly strong uniform claims. Still, the item mainly tests recall of the result rather than substantial generative reasoning from first principles."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they target common confusions between unboundedness of cl and positivity of scl, existential versus uniform lower bounds, and whether the statement holds for one or both groups."}, "total_score": 6, "overall_assessment": "A solid theorem-recall MCQ with good distractors and no answer leakage, but it is only moderately non-tautological and does not strongly test generative mathematical reasoning."}}
{"id": "2602.10766v1", "paper_link": "http://arxiv.org/abs/2602.10766v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm:calderon_intro}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$.\nIf $\\psi \\in \\mathcal{B}_{\\pi}$ and\n$\n \\{\\pi(A^j P k, A^j) \\psi \\}_{ j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d } \n$\nis a Parseval frame for $L^2 (\\mathbb{R}^d)$, then $\\{ \\pi(x, A^j) \\psi \\}_{x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}}$ is a tight frame for $L^2 (\\mathbb{R}^d)$, and \n\\[\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det (P)| \\quad \\text{for a.e.} \\;\\; \\xi \\in \\mathbb{R}^d.\n\\]", "start_pos": 8275, "end_pos": 8773, "label": "thm:calderon_intro"}, "ref_dict": {"thm:calderon_intro": "\\begin{theorem} \\label{thm:calderon_intro}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$.\nIf $\\psi \\in \\mathcal{B}_{\\pi}$ and\n$\n \\{\\pi(A^j P k, A^j) \\psi \\}_{ j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d } \n$\nis a Parseval frame for $L^2 (\\mathbb{R}^d)$, then $\\{ \\pi(x, A^j) \\psi \\}_{x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}}$ is a tight frame for $L^2 (\\mathbb{R}^d)$, and \n\\[\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det (P)| \\quad \\text{for a.e.} \\;\\; \\xi \\in \\mathbb{R}^d.\n\\]\n\\end{theorem}", "eq:discrete_wavelet": "\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot -  Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}", "eq:continuous_wavelet": "\\begin{align} \\label{eq:continuous_wavelet}\n\\pi(x, A^j) \\psi := |\\det(A)|^{-j/2} \\psi(A^{-j} ( \\cdot - x)), \\quad x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}.\n\\end{align}", "conj:wavelet": "\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot -  Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}", "eq:calderon": "\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}", "rem:frame_bounds": "\\begin{remark} \\label{rem:frame_bounds}\n \\Cref{cor:quasi-lattice} implies, in particular, that if $\\pi(\\Lambda) \\psi$ is a Bessel sequence for some relatively dense set $\\Lambda \\subseteq G$, then $\\pi$ must be square-integrable, in the sense that $C_{\\psi} f \\in L^2 (G)$ for all $f \\in \\Hpi$. \nFor certain representations of semidirect product groups $G = \\mathbb{R}^d \\rtimes H$ of $\\mathbb{R}^d$ and a matrix group $H \\leq \\mathrm{GL}(d, \\mathbb{R})$, this was shown in \\cite{aniello2001discrete}. For that setting, the upper bound provided by \\Cref{cor:quasi-lattice} corresponds to \\cite[Proposition 1]{aniello2001discrete}. Although the corresponding lower bound is asserted as \\cite[Proposition 2]{aniello2001discrete} (even without any additional assumption on $\\psi$), the argument provided for the lower bound in \\cite{aniello2001discrete} (cf. \\cite[Theorem 1]{aniello2001discrete}) is incomplete.\n\\end{remark}", "cor:quasi-lattice": "\\begin{corollary} \\label{cor:quasi-lattice}\nLet $G$ be amenable and let $U \\subseteq G$  and $K \\subseteq G$ be a nonempty open and compact set, respectively. Let $\\Lambda \\subseteq G$ be a $U$-separated, $K$-dense set and $\\psi \\in \\Hpi$.\n\nIf $\\pi(\\Lambda)\\psi$ is a Bessel sequence in $\\Hpi$ with Bessel bound $C_2 > 0$, then\n\\[ \\| C_{\\psi} f \\|^2 \\leq C_2 \\mu_G (K) \\| f \\|^2 \\quad \\text{for all} \\quad f \\in \\Hpi. \\]\n In addition, if $\\psi \\in \\mathcal{B}_{\\pi}$ and $\\pi(\\Lambda) \\psi$ also admits a lower frame bound $C_1 > 0$, then\n\\[ \\| C_{\\psi} f \\|^2 \\geq C_1 \\mu_G (U) \\| f \\|^2 \\quad \\text{for all} \\quad f \\in \\Hpi. \\]\nIn particular, if $\\pi(\\Lambda) \\psi$ is frame with $\\psi \\in \\mathcal{B}_{\\pi}$ for some quasi-lattice $\\Lambda \\subseteq G$ with a relatively compact Jordan measurable complement $C$, then \n\\[\nC_1 \\| f \\|^2 \\leq \\frac{\\|C_{\\psi} f\\|^2}{\\rho_G (C)} \\leq C_2 \\| f \\|^2\n\\]\nfor all $f \\in \\Hpi$.\n\\end{corollary}"}, "pre_theorem_intro_text_len": 4800, "pre_theorem_intro_text": "For $\\psi \\in L^2 (\\mathbb{R}^d)$ and matrices $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$, the associated wavelet system is given by the collection of functions\n\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot -  Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}\nIn higher dimensions, a classical condition in the study of wavelet systems is to assume that the matrix $A$ preserves the integer lattice $\\mathbb{Z}^d$, i.e., $A \\mathbb{Z}^d \\subseteq \\mathbb{Z}^d$, and that $A$ is expansive, i.e., all its eigenvalues are strictly greater than one in modulus. Under such conditions, a full characterization of Parseval wavelet frames, or more generally dual wavelet frames, was obtained in \\cite{chui2002characterization, bownik2000characterization, calogero2000characterization}, among others. In addition, the existence of wavelet bases for such dilations were shown in \\cite{dai1997wavelet}. Both types of results extend classical results from dimension one to arbitrary dimensions. \n\nBeyond the case of expansive dilation matrices, the theory of wavelet systems in higher dimensions is far less complete. Nevertheless, the existence of wavelet bases for nonexpansive dilation matrices has been studied in \\cite{speegle2003existence, bownik2017wavelets, ionascu2006simultaneous, wang2002wavelets} and culminated in the recent breakthrough \\cite{bownik2021simultaneous} that characterizes the dilation matrices admitting wavelet sets. On the other hand, the aforementioned characterization of Parseval wavelet frames is currently only known for special dilations, such as amplifying dilations \\cite{laugesen2002translational}, dilations expanding on a subspace \\cite{hernandez2002unified, guo2006some} or dilations satisfying the lattice counting estimate \\cite{bownik2017wavelets}. This has lead to the following conjecture \\cite[Conjecture 1]{bownik2017wavelets} and open problem \\cite[Problem 3.3]{bownik2020open}, which was already implicitly raised in \\cite[p. 177]{speegle2003existence}.\\footnote{The formulations in \\cite{bownik2017wavelets, bownik2020open} are under the implicit assumption that $|\\det(P)| = 1$.}\n\n\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot -  Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}\n\nThe Calder\\'on sum formula \\eqref{eq:calderon} is part of the aforementioned characterizion of Parseval wavelet frames known under additional assumptions on the dilation matrix. The upper bound for the Calder\\'on sum is known to hold for any Bessel sequence with bound $1$ (cf. \\cite[Proposition 4.1]{hernandez2002unified}).\n\nIn this paper, we present a new approach to \\Cref{conj:wavelet} which allows us to prove the Calder\\'on sum formula for arbitrary translation and dilation matrices under a mild condition on the wavelet function. Our approach is based on a relation between the frame properties of the discrete wavelet system \\eqref{eq:discrete_wavelet} and the semi-continuous wavelet system whose elements are given by\n\\begin{align} \\label{eq:continuous_wavelet}\n\\pi(x, A^j) \\psi := |\\det(A)|^{-j/2} \\psi(A^{-j} ( \\cdot - x)), \\quad x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}.\n\\end{align}\nThe action $\\pi$ forms a unitary group representation of the semi-direct product group $G = \\mathbb{R}^d \\rtimes \\langle A \\rangle$ of $\\mathbb{R}^d$ and the cyclic group $\\langle A \\rangle := \\{ A^j : j \\in \\mathbb{Z} \\}$ generated by $A \\in \\mathrm{GL}(d, \\mathbb{R})$. Observe that the  wavelet system \\eqref{eq:discrete_wavelet} corresponds to $\\{\\pi(A^j P k, A^j) \\psi : j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d \\}$. We will assume a mild condition on the wavelet function. Namely, we assume that $\\psi \\in L^2 (\\mathbb{R}^d)$ is such that $\\pi(\\Lambda) \\psi$ is a Bessel sequence in $L^2 (\\mathbb{R}^d)$ for all relatively separated sets $\\Lambda $ in $ G = \\mathbb{R}^d \\rtimes \\langle A \\rangle$; in notation, $\\psi \\in \\mathcal{B}_{\\pi}$. This is a common assumption in the study of frames in the orbit of a group representation, see, e.g., \\cite{Gr08, FuGr07, enstad2025dynamical, fuehr2017density, caspers2023overcompleteness}.\nWe refer to Section \\ref{sec:coefficient} for an alternative description of the space $\\mathcal{B}_{\\pi}$ and further properties.\n\nUsing the notation from the previous paragraph, our main result is the following:", "context": "For $\\psi \\in L^2 (\\mathbb{R}^d)$ and matrices $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$, the associated wavelet system is given by the collection of functions\n\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}\nIn higher dimensions, a classical condition in the study of wavelet systems is to assume that the matrix $A$ preserves the integer lattice $\\mathbb{Z}^d$, i.e., $A \\mathbb{Z}^d \\subseteq \\mathbb{Z}^d$, and that $A$ is expansive, i.e., all its eigenvalues are strictly greater than one in modulus. Under such conditions, a full characterization of Parseval wavelet frames, or more generally dual wavelet frames, was obtained in \\cite{chui2002characterization, bownik2000characterization, calogero2000characterization}, among others. In addition, the existence of wavelet bases for such dilations were shown in \\cite{dai1997wavelet}. Both types of results extend classical results from dimension one to arbitrary dimensions.\n\nBeyond the case of expansive dilation matrices, the theory of wavelet systems in higher dimensions is far less complete. Nevertheless, the existence of wavelet bases for nonexpansive dilation matrices has been studied in \\cite{speegle2003existence, bownik2017wavelets, ionascu2006simultaneous, wang2002wavelets} and culminated in the recent breakthrough \\cite{bownik2021simultaneous} that characterizes the dilation matrices admitting wavelet sets. On the other hand, the aforementioned characterization of Parseval wavelet frames is currently only known for special dilations, such as amplifying dilations \\cite{laugesen2002translational}, dilations expanding on a subspace \\cite{hernandez2002unified, guo2006some} or dilations satisfying the lattice counting estimate \\cite{bownik2017wavelets}. This has lead to the following conjecture \\cite[Conjecture 1]{bownik2017wavelets} and open problem \\cite[Problem 3.3]{bownik2020open}, which was already implicitly raised in \\cite[p. 177]{speegle2003existence}.\\footnote{The formulations in \\cite{bownik2017wavelets, bownik2020open} are under the implicit assumption that $|\\det(P)| = 1$.}\n\n\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}\n\nThe Calder\\'on sum formula \\eqref{eq:calderon} is part of the aforementioned characterizion of Parseval wavelet frames known under additional assumptions on the dilation matrix. The upper bound for the Calder\\'on sum is known to hold for any Bessel sequence with bound $1$ (cf. \\cite[Proposition 4.1]{hernandez2002unified}).\n\nIn this paper, we present a new approach to \\Cref{conj:wavelet} which allows us to prove the Calder\\'on sum formula for arbitrary translation and dilation matrices under a mild condition on the wavelet function. Our approach is based on a relation between the frame properties of the discrete wavelet system \\eqref{eq:discrete_wavelet} and the semi-continuous wavelet system whose elements are given by\n\\begin{align} \\label{eq:continuous_wavelet}\n\\pi(x, A^j) \\psi := |\\det(A)|^{-j/2} \\psi(A^{-j} ( \\cdot - x)), \\quad x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}.\n\\end{align}\nThe action $\\pi$ forms a unitary group representation of the semi-direct product group $G = \\mathbb{R}^d \\rtimes \\langle A \\rangle$ of $\\mathbb{R}^d$ and the cyclic group $\\langle A \\rangle := \\{ A^j : j \\in \\mathbb{Z} \\}$ generated by $A \\in \\mathrm{GL}(d, \\mathbb{R})$. Observe that the wavelet system \\eqref{eq:discrete_wavelet} corresponds to $\\{\\pi(A^j P k, A^j) \\psi : j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d \\}$. We will assume a mild condition on the wavelet function. Namely, we assume that $\\psi \\in L^2 (\\mathbb{R}^d)$ is such that $\\pi(\\Lambda) \\psi$ is a Bessel sequence in $L^2 (\\mathbb{R}^d)$ for all relatively separated sets $\\Lambda $ in $ G = \\mathbb{R}^d \\rtimes \\langle A \\rangle$; in notation, $\\psi \\in \\mathcal{B}_{\\pi}$. This is a common assumption in the study of frames in the orbit of a group representation, see, e.g., \\cite{Gr08, FuGr07, enstad2025dynamical, fuehr2017density, caspers2023overcompleteness}.\nWe refer to Section \\ref{sec:coefficient} for an alternative description of the space $\\mathcal{B}_{\\pi}$ and further properties.\n\nUsing the notation from the previous paragraph, our main result is the following:\n\n\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot -  Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}\n\n\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot -  Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}", "full_context": "For $\\psi \\in L^2 (\\mathbb{R}^d)$ and matrices $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$, the associated wavelet system is given by the collection of functions\n\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}\nIn higher dimensions, a classical condition in the study of wavelet systems is to assume that the matrix $A$ preserves the integer lattice $\\mathbb{Z}^d$, i.e., $A \\mathbb{Z}^d \\subseteq \\mathbb{Z}^d$, and that $A$ is expansive, i.e., all its eigenvalues are strictly greater than one in modulus. Under such conditions, a full characterization of Parseval wavelet frames, or more generally dual wavelet frames, was obtained in \\cite{chui2002characterization, bownik2000characterization, calogero2000characterization}, among others. In addition, the existence of wavelet bases for such dilations were shown in \\cite{dai1997wavelet}. Both types of results extend classical results from dimension one to arbitrary dimensions.\n\nBeyond the case of expansive dilation matrices, the theory of wavelet systems in higher dimensions is far less complete. Nevertheless, the existence of wavelet bases for nonexpansive dilation matrices has been studied in \\cite{speegle2003existence, bownik2017wavelets, ionascu2006simultaneous, wang2002wavelets} and culminated in the recent breakthrough \\cite{bownik2021simultaneous} that characterizes the dilation matrices admitting wavelet sets. On the other hand, the aforementioned characterization of Parseval wavelet frames is currently only known for special dilations, such as amplifying dilations \\cite{laugesen2002translational}, dilations expanding on a subspace \\cite{hernandez2002unified, guo2006some} or dilations satisfying the lattice counting estimate \\cite{bownik2017wavelets}. This has lead to the following conjecture \\cite[Conjecture 1]{bownik2017wavelets} and open problem \\cite[Problem 3.3]{bownik2020open}, which was already implicitly raised in \\cite[p. 177]{speegle2003existence}.\\footnote{The formulations in \\cite{bownik2017wavelets, bownik2020open} are under the implicit assumption that $|\\det(P)| = 1$.}\n\n\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}\n\nThe Calder\\'on sum formula \\eqref{eq:calderon} is part of the aforementioned characterizion of Parseval wavelet frames known under additional assumptions on the dilation matrix. The upper bound for the Calder\\'on sum is known to hold for any Bessel sequence with bound $1$ (cf. \\cite[Proposition 4.1]{hernandez2002unified}).\n\nIn this paper, we present a new approach to \\Cref{conj:wavelet} which allows us to prove the Calder\\'on sum formula for arbitrary translation and dilation matrices under a mild condition on the wavelet function. Our approach is based on a relation between the frame properties of the discrete wavelet system \\eqref{eq:discrete_wavelet} and the semi-continuous wavelet system whose elements are given by\n\\begin{align} \\label{eq:continuous_wavelet}\n\\pi(x, A^j) \\psi := |\\det(A)|^{-j/2} \\psi(A^{-j} ( \\cdot - x)), \\quad x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}.\n\\end{align}\nThe action $\\pi$ forms a unitary group representation of the semi-direct product group $G = \\mathbb{R}^d \\rtimes \\langle A \\rangle$ of $\\mathbb{R}^d$ and the cyclic group $\\langle A \\rangle := \\{ A^j : j \\in \\mathbb{Z} \\}$ generated by $A \\in \\mathrm{GL}(d, \\mathbb{R})$. Observe that the wavelet system \\eqref{eq:discrete_wavelet} corresponds to $\\{\\pi(A^j P k, A^j) \\psi : j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d \\}$. We will assume a mild condition on the wavelet function. Namely, we assume that $\\psi \\in L^2 (\\mathbb{R}^d)$ is such that $\\pi(\\Lambda) \\psi$ is a Bessel sequence in $L^2 (\\mathbb{R}^d)$ for all relatively separated sets $\\Lambda $ in $ G = \\mathbb{R}^d \\rtimes \\langle A \\rangle$; in notation, $\\psi \\in \\mathcal{B}_{\\pi}$. This is a common assumption in the study of frames in the orbit of a group representation, see, e.g., \\cite{Gr08, FuGr07, enstad2025dynamical, fuehr2017density, caspers2023overcompleteness}.\nWe refer to Section \\ref{sec:coefficient} for an alternative description of the space $\\mathcal{B}_{\\pi}$ and further properties.\n\nUsing the notation from the previous paragraph, our main result is the following:\n\n\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot -  Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}\n\n\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot -  Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}\n\n\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}\n\nUsing the notation from the previous paragraph, our main result is the following:\n\nA combination of \\Cref{thm:calderon_intro} with the known fact that tight frames of the form $\\{ \\pi(x, A^j) \\psi \\}_{x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}}$ exist only when $|\\det(A)| \\neq 1$ (cf. \\cite{larson2006explicit, laugesen2002characterization}) yields the following consequence.\n\n\\begin{corollary}\n Let $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$.\nIf there exists a function $\\psi \\in \\mathcal{B}_{\\pi}$ such that\n$\n\\big \\{\\pi(A^j P k, A^j) \\psi \\big\\}_{ j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d } \n$\nis a Parseval frame for $L^2 (\\mathbb{R}^d)$, then $|\\det(A)| \\neq 1$.\n\\end{corollary}\n\n\\subsection{Wavelet transform} \\label{sec:coefficient}\nLet $(\\pi, \\Hpi)$ be a projective unitary representation of $G$ on a Hilbert space $\\Hpi$, i.e., a strongly measurable map $\\pi : G \\to \\mathcal{U}(\\Hpi)$ satisfying $\\pi(e_G) = I_{\\Hpi}$ and\n\\[ \\pi(g_1) \\pi( g_2) = \\sigma(g_1, g_2) \\pi(g_1 g_2), \\quad g_1, g_2 \\in G ,\\]\nfor some (measurable) function $\\sigma : G \\times G \\to \\mathbb{T}$.\nFor $\\psi, f \\in \\Hpi$, we define the function\n\\[ C_\\psi f(g) = \\langle f, \\pi(g) \\psi \\rangle, \\quad g \\in G. \\]\n By \\cite[Lemma 7.1, Theorem 7.5]{varadarajan1985geometry}, the absolute value $|C_{\\psi} f| : G \\to [0, \\infty)$ is continuous for arbitrary $f, \\psi \\in \\Hpi$.\nWe say that a vector $\\psi \\in \\Hpi$ is \\emph{admissible} if the associated \\emph{wavelet transform} $C_{\\psi} : \\Hpi \\to L^{\\infty} (G)$ given by $f \\mapsto C_{\\psi} f$ is an isometry into $L^2 (G)$, that is, the orbit $\\pi(G) \\psi$ is a continuous Parseval frame for $\\Hpi$. Throughout, we always consider $L^2 (G)$ with respect to the left Haar measure $\\mu_G$.\n\n\\begin{proof}\nLet $\\varepsilon > 0$, $f \\in \\Hpi$ and let $U \\subseteq G$ be an open, relatively compact set such that $\\Lambda$ is $U$-separated.\nLet $\\Gamma \\in \\Omega(\\Lambda)$ and let $(\\Lambda_n)_{n \\in \\mathbb{N}}$ be a sequence of left translates $\\Lambda_n \\subseteq G$ of $\\Lambda$ such that $\\Gamma = \\lim_{n \\to \\infty} \\Lambda_n$. Clearly, each set $\\Lambda_n$, $n \\in \\mathbb{N}$, is $U$-separated and, by \\Cref{lem:relsep-reldense-passes-to-hull}, also the set $\\Gamma$ is $U$-separated. \n\\\\~\\\\\n(i) Let $\\pi(\\Lambda)\\psi$ be a Bessel sequence with Bessel bound $C_2$. Then $\\pi(\\Lambda_n)\\psi$ is a Bessel sequence with Bessel bound $C_2$ for all $n \\in \\N$. Let $K \\subseteq G$ be a compact set. Enlarging $K$, we may assume that $\\Gamma \\cap \\partial K = \\emptyset$. Write $\\Gamma \\cap K= \\{ \\gamma^{(1)}, ..., \\gamma^{(k)} \\} $. Then, by \\Cref{lem:convergence_sets}, there exists $n_0 \\in \\mathbb{N}$ such that $\\Lambda_n \\cap K = \\{\\lambda_n^{(1)}, ..., \\lambda_n^{(k)} \\}$ for all $n \\geq n_0$ and $\\lambda_n^{(j)} \\to \\gamma^{(j)}$ for all $1 \\leq j \\leq k$ as $n \\to \\infty$. This, combined with the continuity of the function $|C_{\\psi} f|$ on $G$, yields that there exists $n_1 \\geq n_0$ such that\n\\begin{align} \\label{eq:passes-to-hull_2}\n\\bigg| \\sum_{\\gamma \\in \\Gamma \\cap K} |(C_{\\psi} f)(\\gamma) |^2 - \\sum_{\\mu \\in \\Lambda_n \\cap K} |(C_{\\psi} f)(\\mu) |^2 \\bigg| \\leq \\frac{\\varepsilon}{2}, \\qquad n \\geq n_1 .\n\\end{align}\nThus, since $\\varepsilon > 0$ is arbitrary, we conclude that\n\\[ \\sum_{\\gamma \\in \\Gamma \\cap K} | (C_\\psi f)(\\gamma)|^2 \\leq \\sum_{\\mu \\in \\Lambda_n \\cap K} |(C_\\psi f)(\\mu)|^2 \\leq C_2 \\| f \\|^2 . \\]\nSince $f \\in \\Hpi$ and this holds for arbitrarily large compact sets $K \\subseteq G$, we deduce that $\\pi(\\Gamma)\\psi$ is a Bessel sequence with Bessel bound $C_2$.\n\\\\~\\\\\n(ii) Suppose $\\pi(\\Lambda)\\psi$ is a frame and $\\psi \\in \\mathcal{B}_{\\pi}$. By \\Cref{lem:sampling} there exists a compact set $K \\subseteq G$ such that\n\\[\n\\sum_{\\gamma \\in \\Gamma \\cap K^c} |(C_{\\psi} f)(\\gamma) |^2 < \\frac{\\varepsilon}{4} \\quad \\text{and} \\quad \\sum_{\\mu \\in \\Lambda_n \\cap K^c} |(C_{\\psi} f)(\\mu) |^2 < \\frac{\\varepsilon}{4}\n\\]\nfor all $n \\in \\mathbb{N}$. Therefore,\n\\begin{align} \\label{eq:passes-to-hull_1}\n\\bigg| \\sum_{\\gamma \\in \\Gamma \\cap K^c} |(C_{\\psi} f)(\\gamma) |^2 - \\sum_{\\mu \\in \\Lambda_n \\cap K^c} |(C_{\\psi} f)(\\mu) |^2 \\bigg| \\leq \\frac{\\varepsilon}{2}.\n\\end{align}\nBy enlarging $K$ if necessary, we may assume that \\eqref{eq:passes-to-hull_1} holds for some compact $K \\subseteq G$ with $\\Gamma \\cap \\partial K = \\emptyset$, see, e.g., the proof of \\cite[Theorem 3.9]{enstad2025dynamical}.\n\n\\begin{theorem} \nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$.\nIf $\\psi \\in \\mathcal{B}_{\\pi}$ and\n$\n \\{\\pi(A^j P k, A^j) \\psi \\}_{ j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d } \n$\nis a frame for $L^2 (\\mathbb{R}^d)$ with frame bounds $0<C_1\\leq C_2 < \\infty$, then $\\{ \\pi(x, A^j) \\psi \\}_{x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}}$ is a continuous frame for $L^2 (\\mathbb{R}^d)$ with frame bounds $C_1|\\det(P)|$ and $ C_2|\\det(P)|$, and \n\\[\nC_1 \\leq \\frac{1}{|\\det(P)|} \\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 \\leq C_2 \\quad \\text{for a.e.} \\;\\; \\xi \\in \\mathbb{R}^d.\n\\]\n\\end{theorem}\n\\begin{proof}\nBy \\Cref{lem:quasilattice_affine}, the set $\\Lambda = \\{ (A^j Pk, A^j) : j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d \\}$ is a quasi-lattice in $G$ with complement $C = P [0, 1)^d \\times \\{I_d\\}$, so that $\\rho_G (C) = |\\det(P)|$. \nHence, if the system $\\pi(\\Lambda) \\psi$\nis a frame for $L^2 (\\mathbb{R}^d)$ with $\\psi \\in \\mathcal{B}_{\\pi}$, then \\Cref{cor:quasi-lattice} implies that\n\\begin{align*} \\label{eq:multiple_isometry}\nC_1 |\\det (P)| \\leq \\| C_{\\psi} f \\|^2 \\leq C_2 |\\det (P)| \\| f \\|^2\n\\end{align*}\nfor all $f \\in L^2 (\\mathbb{R}^d)$, which shows that \n$\\{ \\pi(x, A^j) \\psi \\}_{x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}}$ is a frame with the claimed bounds.\nA standard calculation (see, e.g., \\cite{laugesen2002characterization, fuehr2002continuous}) next shows that\n\\begin{align*}\n\\| C_{\\psi} f \\|^2 &= \\int_{G} | \\langle \\widehat{f}, \\widehat{\\pi(x, A^j) \\psi} \\rangle |^2 \\; d\\mu_G(x,A^j) \\\\\n &= \\int_G \\bigg| \\int_{\\mathbb{R}^d} \\widehat{f}(\\xi) |\\det (A) |^{j/2} e^{- 2\\pi i \\xi \\cdot x} \\overline{\\widehat{\\psi}((A^t)^{j} \\xi)} \\; d\\xi \\bigg|^2 \\; d\\mu_G(x, A^j) \\\\\n &= \\sum_{j \\in \\mathbb{Z}} \\int_{\\mathbb{R}^d} \\bigg| \\int_{\\mathbb{R}^d} \\widehat{f}(\\xi) \\overline{\\widehat{\\psi}((A^t)^{j} \\xi)} e^{- 2\\pi i \\xi \\cdot x} \\; d\\xi \\bigg|^2 \\; dx \\\\\n &= \\sum_{j \\in \\mathbb{Z}} \\int_{\\mathbb{R}^d} \\big|\\widehat{f} (\\xi) \\widehat{\\psi} ((A^t)^j \\xi)\\big|^2 \\; d\\xi \\\\\n &= \\int_{\\mathbb{R}^d} |\\widehat{f} (\\xi)|^2 \\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 \\; d\\xi, \n\\end{align*}\nwhere the penultimate step used Plancherel's formula. Hence,\n\\[\nC_1 |\\det(P)| \\| \\widehat{f} \\|^2 \\leq \\int_{\\mathbb{R}^d} |\\widehat{f} (\\xi)|^2 \\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 \\; d\\xi\n\\leq C_2 |\\det(P)| \\| \\widehat{f} \\|^2.\n\\]\nSince this holds for arbitrary $f \\in L^2 (\\mathbb{R}^d)$, it follows that\n\\[\nC_1 |\\det(P)| \\leq \\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 \\leq C_2 |\\det(P)|\n\\]\nfor a.e. $\\xi \\in \\mathbb{R}^d$.\n\\end{proof}", "post_theorem_intro_text_len": 3695, "post_theorem_intro_text": "A combination of \\Cref{thm:calderon_intro} with the known fact that  tight frames of the form $\\{ \\pi(x, A^j) \\psi \\}_{x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}}$  exist only when $|\\det(A)| \\neq 1$ (cf. \\cite{larson2006explicit, laugesen2002characterization}) yields the following consequence.\n\n\\begin{corollary}\n    Let $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$.\nIf there exists a function $\\psi \\in \\mathcal{B}_{\\pi}$ such that\n$\n\\big \\{\\pi(A^j P k, A^j) \\psi \\big\\}_{ j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d } \n$\nis a Parseval frame for $L^2 (\\mathbb{R}^d)$, then  $|\\det(A)| \\neq 1$.\n\\end{corollary}\n\nOur proof of \\Cref{thm:calderon_intro} consists of showing lower and upper bounds on the norm of the continuous wavelet transform associated to the semi-continuous system \\eqref{eq:continuous_wavelet} in terms of the frame bounds of the discrete wavelet frame \\eqref{eq:discrete_wavelet}. In the case of a discrete Parseval wavelet frame \\eqref{eq:discrete_wavelet}, the obtained \\emph{identity} shows precisely that the semi-continuous system \\eqref{eq:continuous_wavelet} is a continuous tight frame as asserted in \\Cref{thm:calderon_intro}. \nInequalities relating the continuous wavelet transform and the frame bounds of a discrete frame are classical for isotropic dilations, see, e.g., \\cite{daubechies1992ten, chui1993inequalities, fuehr1996wavelet}. For general dilation matrices, only an upper bound of for such type of inequalities is known in general \\cite{aniello2001discrete}.\\footnote{Although the corresponding inequality involving the lower frame bound is also asserted in \\cite{aniello2001discrete}, the claimed proof is incomplete; see also \\Cref{rem:frame_bounds}.} We present a new approach towards obtaining both lower and upper bounds by interpreting such estimates as inequalities relating the norm of the wavelet transform, the frame bounds of a frame and the density of its point set. For this, we use a notion of density of a point set (the so-called \\emph{covolume} with respect to an invariant measure on the hull of the given point set) recently introduced for unimodular groups in \\cite{enstad2025dynamical}; see \\Cref{sec:covolume} for precise details. Using this notion in the setting of nonunimodular groups, we derive inequalities (cf. \\Cref{cor:quasi-lattice}) relating the covolume of the point set and the frame bounds of the frame that generalize similar inequalities for lattices in unimodular groups obtained in \\cite{Rova22}; see also \\cite{balan2006density} for such inequalities for Gabor systems.\nAs in \\cite{Rova22, enstad2025dynamical}, the actual proof for such inequalities is based on the simple idea of periodizing the norm of the wavelet transform and applying the corresponding frame inequalities to each integrand. \n\nThe above outlined proof method works naturally for general (projective) unitary representations of locally compact groups. As the auxiliary results might be of interest beyond wavelet systems, we present our results in this general setting. In particular, this allows us to obtain an extension of the main result of \\cite{aniello2001discrete} for amenable groups beyond the setting of quasi-regular representations of semi-direct products (see \\Cref{rem:frame_bounds}).\n\nThe paper is organized as follows. Section \\ref{sec:pointsets} is devoted to background on point sets in locally compact groups. In particular, the notion of covolume is discussed here. Inequalities relating the frame bounds, covolume and norm of the wavelet transform are obtained in Section \\ref{sec:frame}. Lastly, we show in Section \\ref{sec:calderon} how Theorem \\ref{thm:calderon_intro} can be derived from the general results of Section \\ref{sec:frame}.", "sketch": "To prove Theorem~\\ref{thm:calderon_intro}, the authors “show[] lower and upper bounds on the norm of the continuous wavelet transform associated to the semi-continuous system \\eqref{eq:continuous_wavelet} in terms of the frame bounds of the discrete wavelet frame \\eqref{eq:discrete_wavelet}.” In the Parseval case, “the obtained \\emph{identity} shows precisely that the semi-continuous system \\eqref{eq:continuous_wavelet} is a continuous tight frame as asserted in \\Cref{thm:calderon_intro}.”\n\nThe “new approach” is to interpret these estimates “as inequalities relating the norm of the wavelet transform, the frame bounds of a frame and the density of its point set,” using a density notion called “\\emph{covolume} with respect to an invariant measure on the hull of the given point set.” With this notion (extended “in the setting of nonunimodular groups”), they “derive inequalities (cf. \\Cref{cor:quasi-lattice}) relating the covolume of the point set and the frame bounds of the frame,” generalizing earlier lattice/Gabor-type density results.\n\nThe “actual proof for such inequalities is based on the simple idea of periodizing the norm of the wavelet transform and applying the corresponding frame inequalities to each integrand.” Finally, “Theorem \\ref{thm:calderon_intro} can be derived from the general results of Section \\ref{sec:frame},” with the derivation carried out in Section~\\ref{sec:calderon}.", "expanded_sketch": "To prove the main theorem, the authors “show[] lower and upper bounds on the norm of the continuous wavelet transform associated to the semi-continuous system\n\\begin{align} \\label{eq:continuous_wavelet}\n\\pi(x, A^j) \\psi := |\\det(A)|^{-j/2} \\psi(A^{-j} ( \\cdot - x)), \\quad x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}.\n\\end{align}\nin terms of the frame bounds of the discrete wavelet frame\n\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot -  Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}” In the Parseval case, “the obtained \\emph{identity} shows precisely that the semi-continuous system\n\\begin{align} \\label{eq:continuous_wavelet}\n\\pi(x, A^j) \\psi := |\\det(A)|^{-j/2} \\psi(A^{-j} ( \\cdot - x)), \\quad x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}.\n\\end{align}\nis a continuous tight frame as asserted in establishing the main theorem.”\n\nThe “new approach” is to interpret these estimates “as inequalities relating the norm of the wavelet transform, the frame bounds of a frame and the density of its point set,” using a density notion called “\\emph{covolume} with respect to an invariant measure on the hull of the given point set.” With this notion (extended “in the setting of nonunimodular groups”), they “derive inequalities (cf. the following corollary) relating the covolume of the point set and the frame bounds of the frame,” generalizing earlier lattice/Gabor-type density results.\n\nWe first prove the following corollary.\n\\begin{corollary} \\label{cor:quasi-lattice}\nLet $G$ be amenable and let $U \\subseteq G$  and $K \\subseteq G$ be a nonempty open and compact set, respectively. Let $\\Lambda \\subseteq G$ be a $U$-separated, $K$-dense set and $\\psi \\in \\Hpi$.\n\nIf $\\pi(\\Lambda)\\psi$ is a Bessel sequence in $\\Hpi$ with Bessel bound $C_2 > 0$, then\n\\[ \\| C_{\\psi} f \\|^2 \\leq C_2 \\mu_G (K) \\| f \\|^2 \\quad \\text{for all} \\quad f \\in \\Hpi. \\]\n In addition, if $\\psi \\in \\mathcal{B}_{\\pi}$ and $\\pi(\\Lambda) \\psi$ also admits a lower frame bound $C_1 > 0$, then\n\\[ \\| C_{\\psi} f \\|^2 \\geq C_1 \\mu_G (U) \\| f \\|^2 \\quad \\text{for all} \\quad f \\in \\Hpi. \\]\nIn particular, if $\\pi(\\Lambda) \\psi$ is frame with $\\psi \\in \\mathcal{B}_{\\pi}$ for some quasi-lattice $\\Lambda \\subseteq G$ with a relatively compact Jordan measurable complement $C$, then \n\\[\nC_1 \\| f \\|^2 \\leq \\frac{\\|C_{\\psi} f\\|^2}{\\rho_G (C)} \\leq C_2 \\| f \\|^2\n\\]\nfor all $f \\in \\Hpi$.\n\\end{corollary}\n\nThe “actual proof for such inequalities is based on the simple idea of periodizing the norm of the wavelet transform and applying the corresponding frame inequalities to each integrand.” Finally, the main theorem can be derived from the general results proved later, with the derivation carried out later as well.", "expanded_theorem": "\\label{thm:calderon_intro}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$.\nIf $\\psi \\in \\mathcal{B}_{\\pi}$ and\n$\n \\{\\pi(A^j P k, A^j) \\psi \\}_{ j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d } \n$\nis a Parseval frame for $L^2 (\\mathbb{R}^d)$, then $\\{ \\pi(x, A^j) \\psi \\}_{x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}}$ is a tight frame for $L^2 (\\mathbb{R}^d)$, and \n\\[\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det (P)| \\quad \\text{for a.e.} \\;\\; \\xi \\in \\mathbb{R}^d.\n\\]", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $A,P\\in \\mathrm{GL}(d,\\mathbb{R})$, let $\\widehat\\psi$ denote the Fourier transform of $\\psi$, and define\n\\[\n\\pi(x,A^j)\\psi:=|\\det(A)|^{-j/2}\\,\\psi\\big(A^{-j}(\\cdot-x)\\big),\\qquad x\\in\\mathbb{R}^d,\\ j\\in\\mathbb{Z}.\n\\]\nAssume $\\psi\\in \\mathcal B_\\pi$, where $\\mathcal B_\\pi$ is the class of functions in $L^2(\\mathbb{R}^d)$ such that $\\pi(\\Lambda)\\psi$ is a Bessel sequence in $L^2(\\mathbb{R}^d)$ for every relatively separated set $\\Lambda$ in $G=\\mathbb{R}^d\\rtimes\\langle A\\rangle$. Suppose further that the discrete wavelet system\n\\[\n\\{\\pi(A^jPk,A^j)\\psi\\}_{j\\in\\mathbb{Z},\\,k\\in\\mathbb{Z}^d}\n\\]\nis a Parseval frame for $L^2(\\mathbb{R}^d)$. Which of the following statements holds?", "correct_choice": {"label": "A", "text": "The semi-continuous system $\\{\\pi(x,A^j)\\psi\\}_{x\\in\\mathbb{R}^d,\\,j\\in\\mathbb{Z}}$ is a tight frame for $L^2(\\mathbb{R}^d)$, and\n\\[\n\\sum_{j\\in\\mathbb{Z}}\\big|\\widehat\\psi\\big((A^t)^j\\xi\\big)\\big|^2=|\\det(P)|\n\\qquad \\text{for a.e. }\\xi\\in\\mathbb{R}^d.\n\\]"}, "choices": [{"label": "B", "text": "The semi-continuous system $\\{\\pi(x,A^j)\\psi\\}_{x\\in\\mathbb{R}^d,\\,j\\in\\mathbb{Z}}$ is a tight frame for $L^2(\\mathbb{R}^d)$, and\n\\[\n\\sum_{j\\in\\mathbb{Z}}\\big|\\widehat\\psi\\big((A^t)^j\\xi\\big)\\big|^2=1\n\\qquad \\text{for a.e. }\\xi\\in\\mathbb{R}^d.\n\\]"}, {"label": "C", "text": "The semi-continuous system $\\{\\pi(x,A^j)\\psi\\}_{x\\in\\mathbb{R}^d,\\,j\\in\\mathbb{Z}}$ is a tight frame for $L^2(\\mathbb{R}^d)$."}, {"label": "D", "text": "The semi-continuous system $\\{\\pi(x,A^j)\\psi\\}_{x\\in\\mathbb{R}^d,\\,j\\in\\mathbb{Z}}$ is a Parseval frame for $L^2(\\mathbb{R}^d)$, and\n\\[\n\\sum_{j\\in\\mathbb{Z}}\\big|\\widehat\\psi\\big((A^t)^j\\xi\\big)\\big|^2=1\n\\qquad \\text{for a.e. }\\xi\\in\\mathbb{R}^d.\n\\]"}, {"label": "E", "text": "For every $\\xi\\in\\mathbb{R}^d$ one has\n\\[\n\\sum_{j\\in\\mathbb{Z}}\\big|\\widehat\\psi\\big((A^t)^j\\xi\\big)\\big|^2=|\\det(P)|,\n\\]\nand consequently the semi-continuous system $\\{\\pi(x,A^j)\\psi\\}_{x\\in\\mathbb{R}^d,\\,j\\in\\mathbb{Z}}$ is a tight frame for $L^2(\\mathbb{R}^d)$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "covolume_factor_|det(P)|", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "trace_identity", "tampered_component": "dropped_Calderon_identity", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "trace_identity", "tampered_component": "tight_bound_equals_|det(P)|_not_1", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "trace_identity", "tampered_component": "a.e._quantifier_strengthened_to_every_xi", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the conclusion or quote the desired identity. It gives hypotheses only, so there is no direct answer leakage beyond the general indication that a theorem-like implication is being tested."}, "TAS": {"score": 0, "justification": "This is essentially a direct 'if hypotheses H hold, which conclusion follows?' theorem-recall item. The correct option appears to restate the main implication almost verbatim rather than requiring transfer to a new setting."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in distinguishing the exact frame bound, the role of the covolume factor |det(P)|, and the stronger statement from the weaker true statement. However, for anyone who recognizes the theorem, the answer is largely immediate."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target common confusions: replacing |det(P)| by 1, confusing tight with Parseval, and selecting a weaker true consequence instead of the full conclusion. They are distinct and well aligned with likely failure modes."}, "total_score": 5, "overall_assessment": "A mathematically credible MCQ with strong distractors, but it is mostly theorem-recall rather than a genuinely generative reasoning task. Minor note: there is a formatting inconsistency since the marked correct choice is A but the listed choices shown begin at B."}}
{"id": "2602.10766v1", "paper_link": "http://arxiv.org/abs/2602.10766v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm:calderon_intro}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$.\nIf $\\psi \\in \\mathcal{B}_{\\pi}$ and\n$\n \\{\\pi(A^j P k, A^j) \\psi \\}_{ j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d } \n$\nis a Parseval frame for $L^2 (\\mathbb{R}^d)$, then $\\{ \\pi(x, A^j) \\psi \\}_{x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}}$ is a tight frame for $L^2 (\\mathbb{R}^d)$, and \n\\[\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det (P)| \\quad \\text{for a.e.} \\;\\; \\xi \\in \\mathbb{R}^d.\n\\]", "start_pos": 8275, "end_pos": 8773, "label": "thm:calderon_intro"}, "ref_dict": {"thm:calderon_intro": "\\begin{theorem} \\label{thm:calderon_intro}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$.\nIf $\\psi \\in \\mathcal{B}_{\\pi}$ and\n$\n \\{\\pi(A^j P k, A^j) \\psi \\}_{ j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d } \n$\nis a Parseval frame for $L^2 (\\mathbb{R}^d)$, then $\\{ \\pi(x, A^j) \\psi \\}_{x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}}$ is a tight frame for $L^2 (\\mathbb{R}^d)$, and \n\\[\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det (P)| \\quad \\text{for a.e.} \\;\\; \\xi \\in \\mathbb{R}^d.\n\\]\n\\end{theorem}", "eq:discrete_wavelet": "\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot -  Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}", "eq:continuous_wavelet": "\\begin{align} \\label{eq:continuous_wavelet}\n\\pi(x, A^j) \\psi := |\\det(A)|^{-j/2} \\psi(A^{-j} ( \\cdot - x)), \\quad x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}.\n\\end{align}", "conj:wavelet": "\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot -  Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}", "eq:calderon": "\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}", "rem:frame_bounds": "\\begin{remark} \\label{rem:frame_bounds}\n \\Cref{cor:quasi-lattice} implies, in particular, that if $\\pi(\\Lambda) \\psi$ is a Bessel sequence for some relatively dense set $\\Lambda \\subseteq G$, then $\\pi$ must be square-integrable, in the sense that $C_{\\psi} f \\in L^2 (G)$ for all $f \\in \\Hpi$. \nFor certain representations of semidirect product groups $G = \\mathbb{R}^d \\rtimes H$ of $\\mathbb{R}^d$ and a matrix group $H \\leq \\mathrm{GL}(d, \\mathbb{R})$, this was shown in \\cite{aniello2001discrete}. For that setting, the upper bound provided by \\Cref{cor:quasi-lattice} corresponds to \\cite[Proposition 1]{aniello2001discrete}. Although the corresponding lower bound is asserted as \\cite[Proposition 2]{aniello2001discrete} (even without any additional assumption on $\\psi$), the argument provided for the lower bound in \\cite{aniello2001discrete} (cf. \\cite[Theorem 1]{aniello2001discrete}) is incomplete.\n\\end{remark}", "cor:quasi-lattice": "\\begin{corollary} \\label{cor:quasi-lattice}\nLet $G$ be amenable and let $U \\subseteq G$  and $K \\subseteq G$ be a nonempty open and compact set, respectively. Let $\\Lambda \\subseteq G$ be a $U$-separated, $K$-dense set and $\\psi \\in \\Hpi$.\n\nIf $\\pi(\\Lambda)\\psi$ is a Bessel sequence in $\\Hpi$ with Bessel bound $C_2 > 0$, then\n\\[ \\| C_{\\psi} f \\|^2 \\leq C_2 \\mu_G (K) \\| f \\|^2 \\quad \\text{for all} \\quad f \\in \\Hpi. \\]\n In addition, if $\\psi \\in \\mathcal{B}_{\\pi}$ and $\\pi(\\Lambda) \\psi$ also admits a lower frame bound $C_1 > 0$, then\n\\[ \\| C_{\\psi} f \\|^2 \\geq C_1 \\mu_G (U) \\| f \\|^2 \\quad \\text{for all} \\quad f \\in \\Hpi. \\]\nIn particular, if $\\pi(\\Lambda) \\psi$ is frame with $\\psi \\in \\mathcal{B}_{\\pi}$ for some quasi-lattice $\\Lambda \\subseteq G$ with a relatively compact Jordan measurable complement $C$, then \n\\[\nC_1 \\| f \\|^2 \\leq \\frac{\\|C_{\\psi} f\\|^2}{\\rho_G (C)} \\leq C_2 \\| f \\|^2\n\\]\nfor all $f \\in \\Hpi$.\n\\end{corollary}"}, "pre_theorem_intro_text_len": 4800, "pre_theorem_intro_text": "For $\\psi \\in L^2 (\\mathbb{R}^d)$ and matrices $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$, the associated wavelet system is given by the collection of functions\n\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot -  Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}\nIn higher dimensions, a classical condition in the study of wavelet systems is to assume that the matrix $A$ preserves the integer lattice $\\mathbb{Z}^d$, i.e., $A \\mathbb{Z}^d \\subseteq \\mathbb{Z}^d$, and that $A$ is expansive, i.e., all its eigenvalues are strictly greater than one in modulus. Under such conditions, a full characterization of Parseval wavelet frames, or more generally dual wavelet frames, was obtained in \\cite{chui2002characterization, bownik2000characterization, calogero2000characterization}, among others. In addition, the existence of wavelet bases for such dilations were shown in \\cite{dai1997wavelet}. Both types of results extend classical results from dimension one to arbitrary dimensions. \n\nBeyond the case of expansive dilation matrices, the theory of wavelet systems in higher dimensions is far less complete. Nevertheless, the existence of wavelet bases for nonexpansive dilation matrices has been studied in \\cite{speegle2003existence, bownik2017wavelets, ionascu2006simultaneous, wang2002wavelets} and culminated in the recent breakthrough \\cite{bownik2021simultaneous} that characterizes the dilation matrices admitting wavelet sets. On the other hand, the aforementioned characterization of Parseval wavelet frames is currently only known for special dilations, such as amplifying dilations \\cite{laugesen2002translational}, dilations expanding on a subspace \\cite{hernandez2002unified, guo2006some} or dilations satisfying the lattice counting estimate \\cite{bownik2017wavelets}. This has lead to the following conjecture \\cite[Conjecture 1]{bownik2017wavelets} and open problem \\cite[Problem 3.3]{bownik2020open}, which was already implicitly raised in \\cite[p. 177]{speegle2003existence}.\\footnote{The formulations in \\cite{bownik2017wavelets, bownik2020open} are under the implicit assumption that $|\\det(P)| = 1$.}\n\n\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot -  Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}\n\nThe Calder\\'on sum formula \\eqref{eq:calderon} is part of the aforementioned characterizion of Parseval wavelet frames known under additional assumptions on the dilation matrix. The upper bound for the Calder\\'on sum is known to hold for any Bessel sequence with bound $1$ (cf. \\cite[Proposition 4.1]{hernandez2002unified}).\n\nIn this paper, we present a new approach to \\Cref{conj:wavelet} which allows us to prove the Calder\\'on sum formula for arbitrary translation and dilation matrices under a mild condition on the wavelet function. Our approach is based on a relation between the frame properties of the discrete wavelet system \\eqref{eq:discrete_wavelet} and the semi-continuous wavelet system whose elements are given by\n\\begin{align} \\label{eq:continuous_wavelet}\n\\pi(x, A^j) \\psi := |\\det(A)|^{-j/2} \\psi(A^{-j} ( \\cdot - x)), \\quad x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}.\n\\end{align}\nThe action $\\pi$ forms a unitary group representation of the semi-direct product group $G = \\mathbb{R}^d \\rtimes \\langle A \\rangle$ of $\\mathbb{R}^d$ and the cyclic group $\\langle A \\rangle := \\{ A^j : j \\in \\mathbb{Z} \\}$ generated by $A \\in \\mathrm{GL}(d, \\mathbb{R})$. Observe that the  wavelet system \\eqref{eq:discrete_wavelet} corresponds to $\\{\\pi(A^j P k, A^j) \\psi : j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d \\}$. We will assume a mild condition on the wavelet function. Namely, we assume that $\\psi \\in L^2 (\\mathbb{R}^d)$ is such that $\\pi(\\Lambda) \\psi$ is a Bessel sequence in $L^2 (\\mathbb{R}^d)$ for all relatively separated sets $\\Lambda $ in $ G = \\mathbb{R}^d \\rtimes \\langle A \\rangle$; in notation, $\\psi \\in \\mathcal{B}_{\\pi}$. This is a common assumption in the study of frames in the orbit of a group representation, see, e.g., \\cite{Gr08, FuGr07, enstad2025dynamical, fuehr2017density, caspers2023overcompleteness}.\nWe refer to Section \\ref{sec:coefficient} for an alternative description of the space $\\mathcal{B}_{\\pi}$ and further properties.\n\nUsing the notation from the previous paragraph, our main result is the following:", "context": "For $\\psi \\in L^2 (\\mathbb{R}^d)$ and matrices $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$, the associated wavelet system is given by the collection of functions\n\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}\nIn higher dimensions, a classical condition in the study of wavelet systems is to assume that the matrix $A$ preserves the integer lattice $\\mathbb{Z}^d$, i.e., $A \\mathbb{Z}^d \\subseteq \\mathbb{Z}^d$, and that $A$ is expansive, i.e., all its eigenvalues are strictly greater than one in modulus. Under such conditions, a full characterization of Parseval wavelet frames, or more generally dual wavelet frames, was obtained in \\cite{chui2002characterization, bownik2000characterization, calogero2000characterization}, among others. In addition, the existence of wavelet bases for such dilations were shown in \\cite{dai1997wavelet}. Both types of results extend classical results from dimension one to arbitrary dimensions.\n\nBeyond the case of expansive dilation matrices, the theory of wavelet systems in higher dimensions is far less complete. Nevertheless, the existence of wavelet bases for nonexpansive dilation matrices has been studied in \\cite{speegle2003existence, bownik2017wavelets, ionascu2006simultaneous, wang2002wavelets} and culminated in the recent breakthrough \\cite{bownik2021simultaneous} that characterizes the dilation matrices admitting wavelet sets. On the other hand, the aforementioned characterization of Parseval wavelet frames is currently only known for special dilations, such as amplifying dilations \\cite{laugesen2002translational}, dilations expanding on a subspace \\cite{hernandez2002unified, guo2006some} or dilations satisfying the lattice counting estimate \\cite{bownik2017wavelets}. This has lead to the following conjecture \\cite[Conjecture 1]{bownik2017wavelets} and open problem \\cite[Problem 3.3]{bownik2020open}, which was already implicitly raised in \\cite[p. 177]{speegle2003existence}.\\footnote{The formulations in \\cite{bownik2017wavelets, bownik2020open} are under the implicit assumption that $|\\det(P)| = 1$.}\n\n\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}\n\nThe Calder\\'on sum formula \\eqref{eq:calderon} is part of the aforementioned characterizion of Parseval wavelet frames known under additional assumptions on the dilation matrix. The upper bound for the Calder\\'on sum is known to hold for any Bessel sequence with bound $1$ (cf. \\cite[Proposition 4.1]{hernandez2002unified}).\n\nIn this paper, we present a new approach to \\Cref{conj:wavelet} which allows us to prove the Calder\\'on sum formula for arbitrary translation and dilation matrices under a mild condition on the wavelet function. Our approach is based on a relation between the frame properties of the discrete wavelet system \\eqref{eq:discrete_wavelet} and the semi-continuous wavelet system whose elements are given by\n\\begin{align} \\label{eq:continuous_wavelet}\n\\pi(x, A^j) \\psi := |\\det(A)|^{-j/2} \\psi(A^{-j} ( \\cdot - x)), \\quad x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}.\n\\end{align}\nThe action $\\pi$ forms a unitary group representation of the semi-direct product group $G = \\mathbb{R}^d \\rtimes \\langle A \\rangle$ of $\\mathbb{R}^d$ and the cyclic group $\\langle A \\rangle := \\{ A^j : j \\in \\mathbb{Z} \\}$ generated by $A \\in \\mathrm{GL}(d, \\mathbb{R})$. Observe that the wavelet system \\eqref{eq:discrete_wavelet} corresponds to $\\{\\pi(A^j P k, A^j) \\psi : j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d \\}$. We will assume a mild condition on the wavelet function. Namely, we assume that $\\psi \\in L^2 (\\mathbb{R}^d)$ is such that $\\pi(\\Lambda) \\psi$ is a Bessel sequence in $L^2 (\\mathbb{R}^d)$ for all relatively separated sets $\\Lambda $ in $ G = \\mathbb{R}^d \\rtimes \\langle A \\rangle$; in notation, $\\psi \\in \\mathcal{B}_{\\pi}$. This is a common assumption in the study of frames in the orbit of a group representation, see, e.g., \\cite{Gr08, FuGr07, enstad2025dynamical, fuehr2017density, caspers2023overcompleteness}.\nWe refer to Section \\ref{sec:coefficient} for an alternative description of the space $\\mathcal{B}_{\\pi}$ and further properties.\n\nUsing the notation from the previous paragraph, our main result is the following:\n\n\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot -  Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}\n\n\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot -  Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}", "full_context": "For $\\psi \\in L^2 (\\mathbb{R}^d)$ and matrices $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$, the associated wavelet system is given by the collection of functions\n\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}\nIn higher dimensions, a classical condition in the study of wavelet systems is to assume that the matrix $A$ preserves the integer lattice $\\mathbb{Z}^d$, i.e., $A \\mathbb{Z}^d \\subseteq \\mathbb{Z}^d$, and that $A$ is expansive, i.e., all its eigenvalues are strictly greater than one in modulus. Under such conditions, a full characterization of Parseval wavelet frames, or more generally dual wavelet frames, was obtained in \\cite{chui2002characterization, bownik2000characterization, calogero2000characterization}, among others. In addition, the existence of wavelet bases for such dilations were shown in \\cite{dai1997wavelet}. Both types of results extend classical results from dimension one to arbitrary dimensions.\n\nBeyond the case of expansive dilation matrices, the theory of wavelet systems in higher dimensions is far less complete. Nevertheless, the existence of wavelet bases for nonexpansive dilation matrices has been studied in \\cite{speegle2003existence, bownik2017wavelets, ionascu2006simultaneous, wang2002wavelets} and culminated in the recent breakthrough \\cite{bownik2021simultaneous} that characterizes the dilation matrices admitting wavelet sets. On the other hand, the aforementioned characterization of Parseval wavelet frames is currently only known for special dilations, such as amplifying dilations \\cite{laugesen2002translational}, dilations expanding on a subspace \\cite{hernandez2002unified, guo2006some} or dilations satisfying the lattice counting estimate \\cite{bownik2017wavelets}. This has lead to the following conjecture \\cite[Conjecture 1]{bownik2017wavelets} and open problem \\cite[Problem 3.3]{bownik2020open}, which was already implicitly raised in \\cite[p. 177]{speegle2003existence}.\\footnote{The formulations in \\cite{bownik2017wavelets, bownik2020open} are under the implicit assumption that $|\\det(P)| = 1$.}\n\n\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}\n\nThe Calder\\'on sum formula \\eqref{eq:calderon} is part of the aforementioned characterizion of Parseval wavelet frames known under additional assumptions on the dilation matrix. The upper bound for the Calder\\'on sum is known to hold for any Bessel sequence with bound $1$ (cf. \\cite[Proposition 4.1]{hernandez2002unified}).\n\nIn this paper, we present a new approach to \\Cref{conj:wavelet} which allows us to prove the Calder\\'on sum formula for arbitrary translation and dilation matrices under a mild condition on the wavelet function. Our approach is based on a relation between the frame properties of the discrete wavelet system \\eqref{eq:discrete_wavelet} and the semi-continuous wavelet system whose elements are given by\n\\begin{align} \\label{eq:continuous_wavelet}\n\\pi(x, A^j) \\psi := |\\det(A)|^{-j/2} \\psi(A^{-j} ( \\cdot - x)), \\quad x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}.\n\\end{align}\nThe action $\\pi$ forms a unitary group representation of the semi-direct product group $G = \\mathbb{R}^d \\rtimes \\langle A \\rangle$ of $\\mathbb{R}^d$ and the cyclic group $\\langle A \\rangle := \\{ A^j : j \\in \\mathbb{Z} \\}$ generated by $A \\in \\mathrm{GL}(d, \\mathbb{R})$. Observe that the wavelet system \\eqref{eq:discrete_wavelet} corresponds to $\\{\\pi(A^j P k, A^j) \\psi : j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d \\}$. We will assume a mild condition on the wavelet function. Namely, we assume that $\\psi \\in L^2 (\\mathbb{R}^d)$ is such that $\\pi(\\Lambda) \\psi$ is a Bessel sequence in $L^2 (\\mathbb{R}^d)$ for all relatively separated sets $\\Lambda $ in $ G = \\mathbb{R}^d \\rtimes \\langle A \\rangle$; in notation, $\\psi \\in \\mathcal{B}_{\\pi}$. This is a common assumption in the study of frames in the orbit of a group representation, see, e.g., \\cite{Gr08, FuGr07, enstad2025dynamical, fuehr2017density, caspers2023overcompleteness}.\nWe refer to Section \\ref{sec:coefficient} for an alternative description of the space $\\mathcal{B}_{\\pi}$ and further properties.\n\nUsing the notation from the previous paragraph, our main result is the following:\n\n\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot -  Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}\n\n\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot -  Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}\n\n\\begin{conjecture}[\\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \\label{conj:wavelet}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$ and $\\psi \\in L^2 (\\mathbb{R}^d)$. Suppose that \\[ \\{|\\det(A)|^{-j/2} \\psi (A^j \\cdot - Pk) \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}\\]\nis an orthonormal basis, or more generally a Parseval frame, for $L^2 (\\mathbb{R}^d)$. Then the Calder\\'on sum formula holds:\n\\begin{align} \\label{eq:calderon}\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det(P)| \\quad \\text{for a.e.} \\quad \\xi \\in \\mathbb{R}^d.\n\\end{align}\n\\end{conjecture}\n\nUsing the notation from the previous paragraph, our main result is the following:\n\nA combination of \\Cref{thm:calderon_intro} with the known fact that tight frames of the form $\\{ \\pi(x, A^j) \\psi \\}_{x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}}$ exist only when $|\\det(A)| \\neq 1$ (cf. \\cite{larson2006explicit, laugesen2002characterization}) yields the following consequence.\n\n\\begin{corollary}\n Let $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$.\nIf there exists a function $\\psi \\in \\mathcal{B}_{\\pi}$ such that\n$\n\\big \\{\\pi(A^j P k, A^j) \\psi \\big\\}_{ j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d } \n$\nis a Parseval frame for $L^2 (\\mathbb{R}^d)$, then $|\\det(A)| \\neq 1$.\n\\end{corollary}\n\n\\subsection{Wavelet transform} \\label{sec:coefficient}\nLet $(\\pi, \\Hpi)$ be a projective unitary representation of $G$ on a Hilbert space $\\Hpi$, i.e., a strongly measurable map $\\pi : G \\to \\mathcal{U}(\\Hpi)$ satisfying $\\pi(e_G) = I_{\\Hpi}$ and\n\\[ \\pi(g_1) \\pi( g_2) = \\sigma(g_1, g_2) \\pi(g_1 g_2), \\quad g_1, g_2 \\in G ,\\]\nfor some (measurable) function $\\sigma : G \\times G \\to \\mathbb{T}$.\nFor $\\psi, f \\in \\Hpi$, we define the function\n\\[ C_\\psi f(g) = \\langle f, \\pi(g) \\psi \\rangle, \\quad g \\in G. \\]\n By \\cite[Lemma 7.1, Theorem 7.5]{varadarajan1985geometry}, the absolute value $|C_{\\psi} f| : G \\to [0, \\infty)$ is continuous for arbitrary $f, \\psi \\in \\Hpi$.\nWe say that a vector $\\psi \\in \\Hpi$ is \\emph{admissible} if the associated \\emph{wavelet transform} $C_{\\psi} : \\Hpi \\to L^{\\infty} (G)$ given by $f \\mapsto C_{\\psi} f$ is an isometry into $L^2 (G)$, that is, the orbit $\\pi(G) \\psi$ is a continuous Parseval frame for $\\Hpi$. Throughout, we always consider $L^2 (G)$ with respect to the left Haar measure $\\mu_G$.\n\n\\begin{proof}\nLet $\\varepsilon > 0$, $f \\in \\Hpi$ and let $U \\subseteq G$ be an open, relatively compact set such that $\\Lambda$ is $U$-separated.\nLet $\\Gamma \\in \\Omega(\\Lambda)$ and let $(\\Lambda_n)_{n \\in \\mathbb{N}}$ be a sequence of left translates $\\Lambda_n \\subseteq G$ of $\\Lambda$ such that $\\Gamma = \\lim_{n \\to \\infty} \\Lambda_n$. Clearly, each set $\\Lambda_n$, $n \\in \\mathbb{N}$, is $U$-separated and, by \\Cref{lem:relsep-reldense-passes-to-hull}, also the set $\\Gamma$ is $U$-separated. \n\\\\~\\\\\n(i) Let $\\pi(\\Lambda)\\psi$ be a Bessel sequence with Bessel bound $C_2$. Then $\\pi(\\Lambda_n)\\psi$ is a Bessel sequence with Bessel bound $C_2$ for all $n \\in \\N$. Let $K \\subseteq G$ be a compact set. Enlarging $K$, we may assume that $\\Gamma \\cap \\partial K = \\emptyset$. Write $\\Gamma \\cap K= \\{ \\gamma^{(1)}, ..., \\gamma^{(k)} \\} $. Then, by \\Cref{lem:convergence_sets}, there exists $n_0 \\in \\mathbb{N}$ such that $\\Lambda_n \\cap K = \\{\\lambda_n^{(1)}, ..., \\lambda_n^{(k)} \\}$ for all $n \\geq n_0$ and $\\lambda_n^{(j)} \\to \\gamma^{(j)}$ for all $1 \\leq j \\leq k$ as $n \\to \\infty$. This, combined with the continuity of the function $|C_{\\psi} f|$ on $G$, yields that there exists $n_1 \\geq n_0$ such that\n\\begin{align} \\label{eq:passes-to-hull_2}\n\\bigg| \\sum_{\\gamma \\in \\Gamma \\cap K} |(C_{\\psi} f)(\\gamma) |^2 - \\sum_{\\mu \\in \\Lambda_n \\cap K} |(C_{\\psi} f)(\\mu) |^2 \\bigg| \\leq \\frac{\\varepsilon}{2}, \\qquad n \\geq n_1 .\n\\end{align}\nThus, since $\\varepsilon > 0$ is arbitrary, we conclude that\n\\[ \\sum_{\\gamma \\in \\Gamma \\cap K} | (C_\\psi f)(\\gamma)|^2 \\leq \\sum_{\\mu \\in \\Lambda_n \\cap K} |(C_\\psi f)(\\mu)|^2 \\leq C_2 \\| f \\|^2 . \\]\nSince $f \\in \\Hpi$ and this holds for arbitrarily large compact sets $K \\subseteq G$, we deduce that $\\pi(\\Gamma)\\psi$ is a Bessel sequence with Bessel bound $C_2$.\n\\\\~\\\\\n(ii) Suppose $\\pi(\\Lambda)\\psi$ is a frame and $\\psi \\in \\mathcal{B}_{\\pi}$. By \\Cref{lem:sampling} there exists a compact set $K \\subseteq G$ such that\n\\[\n\\sum_{\\gamma \\in \\Gamma \\cap K^c} |(C_{\\psi} f)(\\gamma) |^2 < \\frac{\\varepsilon}{4} \\quad \\text{and} \\quad \\sum_{\\mu \\in \\Lambda_n \\cap K^c} |(C_{\\psi} f)(\\mu) |^2 < \\frac{\\varepsilon}{4}\n\\]\nfor all $n \\in \\mathbb{N}$. Therefore,\n\\begin{align} \\label{eq:passes-to-hull_1}\n\\bigg| \\sum_{\\gamma \\in \\Gamma \\cap K^c} |(C_{\\psi} f)(\\gamma) |^2 - \\sum_{\\mu \\in \\Lambda_n \\cap K^c} |(C_{\\psi} f)(\\mu) |^2 \\bigg| \\leq \\frac{\\varepsilon}{2}.\n\\end{align}\nBy enlarging $K$ if necessary, we may assume that \\eqref{eq:passes-to-hull_1} holds for some compact $K \\subseteq G$ with $\\Gamma \\cap \\partial K = \\emptyset$, see, e.g., the proof of \\cite[Theorem 3.9]{enstad2025dynamical}.\n\n\\begin{theorem} \nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$.\nIf $\\psi \\in \\mathcal{B}_{\\pi}$ and\n$\n \\{\\pi(A^j P k, A^j) \\psi \\}_{ j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d } \n$\nis a frame for $L^2 (\\mathbb{R}^d)$ with frame bounds $0<C_1\\leq C_2 < \\infty$, then $\\{ \\pi(x, A^j) \\psi \\}_{x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}}$ is a continuous frame for $L^2 (\\mathbb{R}^d)$ with frame bounds $C_1|\\det(P)|$ and $ C_2|\\det(P)|$, and \n\\[\nC_1 \\leq \\frac{1}{|\\det(P)|} \\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 \\leq C_2 \\quad \\text{for a.e.} \\;\\; \\xi \\in \\mathbb{R}^d.\n\\]\n\\end{theorem}\n\\begin{proof}\nBy \\Cref{lem:quasilattice_affine}, the set $\\Lambda = \\{ (A^j Pk, A^j) : j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d \\}$ is a quasi-lattice in $G$ with complement $C = P [0, 1)^d \\times \\{I_d\\}$, so that $\\rho_G (C) = |\\det(P)|$. \nHence, if the system $\\pi(\\Lambda) \\psi$\nis a frame for $L^2 (\\mathbb{R}^d)$ with $\\psi \\in \\mathcal{B}_{\\pi}$, then \\Cref{cor:quasi-lattice} implies that\n\\begin{align*} \\label{eq:multiple_isometry}\nC_1 |\\det (P)| \\leq \\| C_{\\psi} f \\|^2 \\leq C_2 |\\det (P)| \\| f \\|^2\n\\end{align*}\nfor all $f \\in L^2 (\\mathbb{R}^d)$, which shows that \n$\\{ \\pi(x, A^j) \\psi \\}_{x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}}$ is a frame with the claimed bounds.\nA standard calculation (see, e.g., \\cite{laugesen2002characterization, fuehr2002continuous}) next shows that\n\\begin{align*}\n\\| C_{\\psi} f \\|^2 &= \\int_{G} | \\langle \\widehat{f}, \\widehat{\\pi(x, A^j) \\psi} \\rangle |^2 \\; d\\mu_G(x,A^j) \\\\\n &= \\int_G \\bigg| \\int_{\\mathbb{R}^d} \\widehat{f}(\\xi) |\\det (A) |^{j/2} e^{- 2\\pi i \\xi \\cdot x} \\overline{\\widehat{\\psi}((A^t)^{j} \\xi)} \\; d\\xi \\bigg|^2 \\; d\\mu_G(x, A^j) \\\\\n &= \\sum_{j \\in \\mathbb{Z}} \\int_{\\mathbb{R}^d} \\bigg| \\int_{\\mathbb{R}^d} \\widehat{f}(\\xi) \\overline{\\widehat{\\psi}((A^t)^{j} \\xi)} e^{- 2\\pi i \\xi \\cdot x} \\; d\\xi \\bigg|^2 \\; dx \\\\\n &= \\sum_{j \\in \\mathbb{Z}} \\int_{\\mathbb{R}^d} \\big|\\widehat{f} (\\xi) \\widehat{\\psi} ((A^t)^j \\xi)\\big|^2 \\; d\\xi \\\\\n &= \\int_{\\mathbb{R}^d} |\\widehat{f} (\\xi)|^2 \\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 \\; d\\xi, \n\\end{align*}\nwhere the penultimate step used Plancherel's formula. Hence,\n\\[\nC_1 |\\det(P)| \\| \\widehat{f} \\|^2 \\leq \\int_{\\mathbb{R}^d} |\\widehat{f} (\\xi)|^2 \\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 \\; d\\xi\n\\leq C_2 |\\det(P)| \\| \\widehat{f} \\|^2.\n\\]\nSince this holds for arbitrary $f \\in L^2 (\\mathbb{R}^d)$, it follows that\n\\[\nC_1 |\\det(P)| \\leq \\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 \\leq C_2 |\\det(P)|\n\\]\nfor a.e. $\\xi \\in \\mathbb{R}^d$.\n\\end{proof}", "post_theorem_intro_text_len": 3695, "post_theorem_intro_text": "A combination of \\Cref{thm:calderon_intro} with the known fact that  tight frames of the form $\\{ \\pi(x, A^j) \\psi \\}_{x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}}$  exist only when $|\\det(A)| \\neq 1$ (cf. \\cite{larson2006explicit, laugesen2002characterization}) yields the following consequence.\n\n\\begin{corollary}\n    Let $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$.\nIf there exists a function $\\psi \\in \\mathcal{B}_{\\pi}$ such that\n$\n\\big \\{\\pi(A^j P k, A^j) \\psi \\big\\}_{ j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d } \n$\nis a Parseval frame for $L^2 (\\mathbb{R}^d)$, then  $|\\det(A)| \\neq 1$.\n\\end{corollary}\n\nOur proof of \\Cref{thm:calderon_intro} consists of showing lower and upper bounds on the norm of the continuous wavelet transform associated to the semi-continuous system \\eqref{eq:continuous_wavelet} in terms of the frame bounds of the discrete wavelet frame \\eqref{eq:discrete_wavelet}. In the case of a discrete Parseval wavelet frame \\eqref{eq:discrete_wavelet}, the obtained \\emph{identity} shows precisely that the semi-continuous system \\eqref{eq:continuous_wavelet} is a continuous tight frame as asserted in \\Cref{thm:calderon_intro}. \nInequalities relating the continuous wavelet transform and the frame bounds of a discrete frame are classical for isotropic dilations, see, e.g., \\cite{daubechies1992ten, chui1993inequalities, fuehr1996wavelet}. For general dilation matrices, only an upper bound of for such type of inequalities is known in general \\cite{aniello2001discrete}.\\footnote{Although the corresponding inequality involving the lower frame bound is also asserted in \\cite{aniello2001discrete}, the claimed proof is incomplete; see also \\Cref{rem:frame_bounds}.} We present a new approach towards obtaining both lower and upper bounds by interpreting such estimates as inequalities relating the norm of the wavelet transform, the frame bounds of a frame and the density of its point set. For this, we use a notion of density of a point set (the so-called \\emph{covolume} with respect to an invariant measure on the hull of the given point set) recently introduced for unimodular groups in \\cite{enstad2025dynamical}; see \\Cref{sec:covolume} for precise details. Using this notion in the setting of nonunimodular groups, we derive inequalities (cf. \\Cref{cor:quasi-lattice}) relating the covolume of the point set and the frame bounds of the frame that generalize similar inequalities for lattices in unimodular groups obtained in \\cite{Rova22}; see also \\cite{balan2006density} for such inequalities for Gabor systems.\nAs in \\cite{Rova22, enstad2025dynamical}, the actual proof for such inequalities is based on the simple idea of periodizing the norm of the wavelet transform and applying the corresponding frame inequalities to each integrand. \n\nThe above outlined proof method works naturally for general (projective) unitary representations of locally compact groups. As the auxiliary results might be of interest beyond wavelet systems, we present our results in this general setting. In particular, this allows us to obtain an extension of the main result of \\cite{aniello2001discrete} for amenable groups beyond the setting of quasi-regular representations of semi-direct products (see \\Cref{rem:frame_bounds}).\n\nThe paper is organized as follows. Section \\ref{sec:pointsets} is devoted to background on point sets in locally compact groups. In particular, the notion of covolume is discussed here. Inequalities relating the frame bounds, covolume and norm of the wavelet transform are obtained in Section \\ref{sec:frame}. Lastly, we show in Section \\ref{sec:calderon} how Theorem \\ref{thm:calderon_intro} can be derived from the general results of Section \\ref{sec:frame}.", "sketch": "To prove Theorem~\\ref{thm:calderon_intro}, the authors “show[] lower and upper bounds on the norm of the continuous wavelet transform associated to the semi-continuous system \\eqref{eq:continuous_wavelet} in terms of the frame bounds of the discrete wavelet frame \\eqref{eq:discrete_wavelet}.” In the Parseval case, “the obtained \\emph{identity} shows precisely that the semi-continuous system \\eqref{eq:continuous_wavelet} is a continuous tight frame as asserted in \\Cref{thm:calderon_intro}.”\n\nThe “new approach” is to interpret these estimates “as inequalities relating the norm of the wavelet transform, the frame bounds of a frame and the density of its point set,” using a density notion called “\\emph{covolume} with respect to an invariant measure on the hull of the given point set.” With this notion (extended “in the setting of nonunimodular groups”), they “derive inequalities (cf. \\Cref{cor:quasi-lattice}) relating the covolume of the point set and the frame bounds of the frame,” generalizing earlier lattice/Gabor-type density results.\n\nThe “actual proof for such inequalities is based on the simple idea of periodizing the norm of the wavelet transform and applying the corresponding frame inequalities to each integrand.” Finally, “Theorem \\ref{thm:calderon_intro} can be derived from the general results of Section \\ref{sec:frame},” with the derivation carried out in Section~\\ref{sec:calderon}.", "expanded_sketch": "To prove the main theorem, the authors “show[] lower and upper bounds on the norm of the continuous wavelet transform associated to the semi-continuous system\n\\begin{align} \\label{eq:continuous_wavelet}\n\\pi(x, A^j) \\psi := |\\det(A)|^{-j/2} \\psi(A^{-j} ( \\cdot - x)), \\quad x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}.\n\\end{align}\nin terms of the frame bounds of the discrete wavelet frame\n\\begin{align} \\label{eq:discrete_wavelet}\n\\big\\{ |\\det(A)|^{-j/2} \\psi (A^j \\cdot -  Pk) \\big \\}_{j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d}.\n\\end{align}” In the Parseval case, “the obtained \\emph{identity} shows precisely that the semi-continuous system\n\\begin{align} \\label{eq:continuous_wavelet}\n\\pi(x, A^j) \\psi := |\\det(A)|^{-j/2} \\psi(A^{-j} ( \\cdot - x)), \\quad x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}.\n\\end{align}\nis a continuous tight frame as asserted in establishing the main theorem.”\n\nThe “new approach” is to interpret these estimates “as inequalities relating the norm of the wavelet transform, the frame bounds of a frame and the density of its point set,” using a density notion called “\\emph{covolume} with respect to an invariant measure on the hull of the given point set.” With this notion (extended “in the setting of nonunimodular groups”), they “derive inequalities (cf. the following corollary) relating the covolume of the point set and the frame bounds of the frame,” generalizing earlier lattice/Gabor-type density results.\n\nWe first prove the following corollary.\n\\begin{corollary} \\label{cor:quasi-lattice}\nLet $G$ be amenable and let $U \\subseteq G$  and $K \\subseteq G$ be a nonempty open and compact set, respectively. Let $\\Lambda \\subseteq G$ be a $U$-separated, $K$-dense set and $\\psi \\in \\Hpi$.\n\nIf $\\pi(\\Lambda)\\psi$ is a Bessel sequence in $\\Hpi$ with Bessel bound $C_2 > 0$, then\n\\[ \\| C_{\\psi} f \\|^2 \\leq C_2 \\mu_G (K) \\| f \\|^2 \\quad \\text{for all} \\quad f \\in \\Hpi. \\]\n In addition, if $\\psi \\in \\mathcal{B}_{\\pi}$ and $\\pi(\\Lambda) \\psi$ also admits a lower frame bound $C_1 > 0$, then\n\\[ \\| C_{\\psi} f \\|^2 \\geq C_1 \\mu_G (U) \\| f \\|^2 \\quad \\text{for all} \\quad f \\in \\Hpi. \\]\nIn particular, if $\\pi(\\Lambda) \\psi$ is frame with $\\psi \\in \\mathcal{B}_{\\pi}$ for some quasi-lattice $\\Lambda \\subseteq G$ with a relatively compact Jordan measurable complement $C$, then \n\\[\nC_1 \\| f \\|^2 \\leq \\frac{\\|C_{\\psi} f\\|^2}{\\rho_G (C)} \\leq C_2 \\| f \\|^2\n\\]\nfor all $f \\in \\Hpi$.\n\\end{corollary}\n\nThe “actual proof for such inequalities is based on the simple idea of periodizing the norm of the wavelet transform and applying the corresponding frame inequalities to each integrand.” Finally, the main theorem can be derived from the general results proved later, with the derivation carried out later as well.", "expanded_theorem": "\\label{thm:calderon_intro}\nLet $A, P \\in \\mathrm{GL}(d, \\mathbb{R})$.\nIf $\\psi \\in \\mathcal{B}_{\\pi}$ and\n$\n \\{\\pi(A^j P k, A^j) \\psi \\}_{ j \\in \\mathbb{Z}, k \\in \\mathbb{Z}^d } \n$\nis a Parseval frame for $L^2 (\\mathbb{R}^d)$, then $\\{ \\pi(x, A^j) \\psi \\}_{x \\in \\mathbb{R}^d, j \\in \\mathbb{Z}}$ is a tight frame for $L^2 (\\mathbb{R}^d)$, and \n\\[\n\\sum_{j \\in \\mathbb{Z}} |\\widehat{\\psi} ((A^t)^j \\xi)|^2 = |\\det (P)| \\quad \\text{for a.e.} \\;\\; \\xi \\in \\mathbb{R}^d.\n\\]", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $A,P\\in \\mathrm{GL}(d,\\mathbb{R})$, let $\\widehat\\psi$ denote the Fourier transform of $\\psi$, and define\n\\[\n\\pi(x,A^j)\\psi:=|\\det(A)|^{-j/2}\\,\\psi\\big(A^{-j}(\\cdot-x)\\big),\\qquad x\\in\\mathbb{R}^d,\\ j\\in\\mathbb{Z}.\n\\]\nAssume $\\psi\\in \\mathcal B_\\pi$, where $\\mathcal B_\\pi$ is the class of functions in $L^2(\\mathbb{R}^d)$ such that $\\pi(\\Lambda)\\psi$ is a Bessel sequence in $L^2(\\mathbb{R}^d)$ for every relatively separated set $\\Lambda$ in $G=\\mathbb{R}^d\\rtimes\\langle A\\rangle$. Suppose further that the discrete wavelet system\n\\[\n\\{\\pi(A^jPk,A^j)\\psi\\}_{j\\in\\mathbb{Z},\\,k\\in\\mathbb{Z}^d}\n\\]\nis a Parseval frame for $L^2(\\mathbb{R}^d)$. Which of the following statements holds?", "correct_choice": {"label": "A", "text": "The semi-continuous system $\\{\\pi(x,A^j)\\psi\\}_{x\\in\\mathbb{R}^d,\\,j\\in\\mathbb{Z}}$ is a tight frame for $L^2(\\mathbb{R}^d)$, and\n\\[\n\\sum_{j\\in\\mathbb{Z}}\\big|\\widehat\\psi\\big((A^t)^j\\xi\\big)\\big|^2=|\\det(P)|\n\\qquad \\text{for a.e. }\\xi\\in\\mathbb{R}^d.\n\\]"}, "choices": [{"label": "B", "text": "The semi-continuous system $\\{\\pi(x,A^j)\\psi\\}_{x\\in\\mathbb{R}^d,\\,j\\in\\mathbb{Z}}$ is a tight frame for $L^2(\\mathbb{R}^d)$, and\n\\[\n\\sum_{j\\in\\mathbb{Z}}\\big|\\widehat\\psi\\big((A^t)^j\\xi\\big)\\big|^2=1\n\\qquad \\text{for a.e. }\\xi\\in\\mathbb{R}^d.\n\\]"}, {"label": "C", "text": "The semi-continuous system $\\{\\pi(x,A^j)\\psi\\}_{x\\in\\mathbb{R}^d,\\,j\\in\\mathbb{Z}}$ is a tight frame for $L^2(\\mathbb{R}^d)$."}, {"label": "D", "text": "The semi-continuous system $\\{\\pi(x,A^j)\\psi\\}_{x\\in\\mathbb{R}^d,\\,j\\in\\mathbb{Z}}$ is a Parseval frame for $L^2(\\mathbb{R}^d)$, and\n\\[\n\\sum_{j\\in\\mathbb{Z}}\\big|\\widehat\\psi\\big((A^t)^j\\xi\\big)\\big|^2=1\n\\qquad \\text{for a.e. }\\xi\\in\\mathbb{R}^d.\n\\]"}, {"label": "E", "text": "For every $\\xi\\in\\mathbb{R}^d$ one has\n\\[\n\\sum_{j\\in\\mathbb{Z}}\\big|\\widehat\\psi\\big((A^t)^j\\xi\\big)\\big|^2=|\\det(P)|,\n\\]\nand consequently the semi-continuous system $\\{\\pi(x,A^j)\\psi\\}_{x\\in\\mathbb{R}^d,\\,j\\in\\mathbb{Z}}$ is a tight frame for $L^2(\\mathbb{R}^d)$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "covolume_factor_|det(P)|", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "trace_identity", "tampered_component": "dropped_Calderon_identity", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "trace_identity", "tampered_component": "tight_bound_equals_|det(P)|_not_1", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "trace_identity", "tampered_component": "a.e._quantifier_strengthened_to_every_xi", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It gives hypotheses and asks for the valid conclusion; the key distinctions among the options (tight vs. Parseval, frame bound |det(P)| vs. 1, a.e. vs. every ξ) are not leaked in the wording."}, "TAS": {"score": 1, "justification": "The item is very close to a theorem-recall question: under the stated hypotheses, the correct option appears to restate the standard conclusion. However, it is not a pure verbatim restatement, since the choices force discrimination among nearby variants of the conclusion."}, "GPS": {"score": 1, "justification": "Some reasoning is required to identify the strongest correct statement and reject subtle near-misses, especially around the normalization constant and quantifier strength. Still, the question mainly tests recognition of a known theorem rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic errors: confusing tight with Parseval, dropping the Calderón identity, using 1 instead of |det(P)|, and strengthening 'for a.e. ξ' to 'for every ξ'. These are distinct and well aligned with common failure modes."}, "total_score": 6, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and little answer leakage, but only moderate generative depth because it closely tracks a standard result."}}
{"id": "2602.10853v1", "paper_link": "http://arxiv.org/abs/2602.10853v1", "theorems_cnt": 3, "theorem": {"env_name": "theoremA", "content": "Let $\\Omega \\subset \\mathbb{C}^N$ be a strictly pseudoconvex domain. Then every b-pluripolar $F_\\sigma$ set of topological dimension $N$ must propagate into the interior.", "start_pos": 4891, "end_pos": 5085, "label": null}, "ref_dict": {"hedenmalmgen": "\\begin{corollary}\\label{hedenmalmgen} Let $\\Omega\\subset \\C^N, N>1$ be a smooth, strictly pseudoconvex domain. Then for every $z_0 \\in \\Omega$, $M_{z_0}(\\partial \\Omega)\\setminus J_{z_0}(\\partial \\Omega)$ is non-empty.\n\\end{corollary}", "hausdorff": "\\begin{theorem}\\label{hausdorff}\n   Every set $A\\subset P_\\beta$ of zero $\\beta$-dimensional Hausdorff measure is non-propagating and b-pluripolar. \n\\end{theorem}", "topdim": "\\begin{theorem}\\label{topdim}\nLet $\\Omega\\Subset \\C^N$ be strictly pseudoconvex, and let $A\\subset \\partial \\Omega$ be a boundary pluripolar $F_\\sigma$ set with topological dimension greater than $N-1$. Then $\\hat A \\cap \\Omega$ is non-empty.\n\\end{theorem}", "hedenmalmsats": "\\begin{theorem}[Hedenmalm~\\cite{hedenmalm}]\\label{hedenmalmsats}\nLet $\\mathbb{B}$ denote the unit ball in $\\C^N$, $N>1$, and let $\\bar 0 \\in \\mathbb{B}$ denote its center. Then the set $M_{\\bar 0}(\\partial \\mathbb{B})\\setminus J_{\\bar 0}(\\partial \\mathbb{B})$ is non-empty.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2045, "pre_theorem_intro_text": "Let $\\Omega$ denote a bounded domain in $\\mathbb{C}^N$, and let $A \\subset \\partial\\Omega$ be a set for which there exists a negative plurisubharmonic (psh) function $u$ defined on $\\Omega$ ($u\\in \\mathcal{PSH}^-(\\Omega)$), not identically $-\\infty$, with\n\\[\\limsup_{z\\rightarrow A} u(z) = -\\infty.\\]\nSuch \\textit{boundary pluripolar sets} (b-pluripolar for short) were recently characterized as the exceptional sets for the Dirichlet problem for the complex Monge--Ampère operator ~\\cite{nilsson2}. More precisely, under the assumption that $\\Omega$ admits strong plurisubharmonic barriers, in the sense that we at every boundary point $\\zeta \\in \\partial\\Omega$ can find $u \\in \\mathcal{PSH}(\\Omega)\\cap C(\\widebar \\Omega)$ with\n\\[\nu(\\zeta)=0, \\quad u<0 \\text{ on } \\widebar \\Omega \\setminus \\{\\zeta\\},\n\\]\nthen the Dirichlet problem \n\\[\n    \\begin{cases}\n    u \\in L^\\infty\\cap\\mathcal{PSH}(\\Omega) \\\\\n    \\det(\\frac{\\partial^2 u}{\\partial z_j \\partial \\bar z_k})=0 \\\\\n        \\lim_{z \\rightarrow \\zeta}u(z) = \\varphi(\\zeta) \\qquad \\forall \\zeta \\in \\partial\\Omega \\setminus E_\\varphi \n    \\end{cases}\n\\]\n is uniquely solvable (in the weak sense of Bedford--Taylor~\\cite{bedford-taylor, bedford-taylor2}) if and only if $E_\\varphi$ is b-pluripolar. Here, we assume that $\\varphi\\colon \\partial \\Omega \\rightarrow \\mathbb{R}$ is bounded, and continuous outside $E_\\varphi$. Furthermore, the unique solution is continuous~\\cite[Theorem~5.1]{nilsson} outside the closure of\n    \\[\n    \\hat E_\\varphi \\coloneqq\\{z \\in \\widebar\\Omega \\mathrel{;} u\\in\\mathcal{PSH}^-(\\Omega), \\limsup_{w\\rightarrow E_\\varphi}u(w)=-\\infty \\implies u(z) =-\\infty\\},\n    \\]\nthe \\textit{boundary pluripolar hull} of $E_\\varphi$ (if the boundary pluripolar hull contains interior points, we say that the set is \\textit{propagating}). Since discontinuity sets are necessarily $F_\\sigma$, this motivates investigating necessary and sufficient criteria for when boundary pluripolar $F_\\sigma$ sets propagate. \n\nIn the first part of the paper, we obtain the following result:", "context": "Let $\\Omega$ denote a bounded domain in $\\mathbb{C}^N$, and let $A \\subset \\partial\\Omega$ be a set for which there exists a negative plurisubharmonic (psh) function $u$ defined on $\\Omega$ ($u\\in \\mathcal{PSH}^-(\\Omega)$), not identically $-\\infty$, with\n\\[\\limsup_{z\\rightarrow A} u(z) = -\\infty.\\]\nSuch \\textit{boundary pluripolar sets} (b-pluripolar for short) were recently characterized as the exceptional sets for the Dirichlet problem for the complex Monge--Ampère operator ~\\cite{nilsson2}. More precisely, under the assumption that $\\Omega$ admits strong plurisubharmonic barriers, in the sense that we at every boundary point $\\zeta \\in \\partial\\Omega$ can find $u \\in \\mathcal{PSH}(\\Omega)\\cap C(\\widebar \\Omega)$ with\n\\[\nu(\\zeta)=0, \\quad u<0 \\text{ on } \\widebar \\Omega \\setminus \\{\\zeta\\},\n\\]\nthen the Dirichlet problem \n\\[\n \\begin{cases}\n u \\in L^\\infty\\cap\\mathcal{PSH}(\\Omega) \\\\\n \\det(\\frac{\\partial^2 u}{\\partial z_j \\partial \\bar z_k})=0 \\\\\n \\lim_{z \\rightarrow \\zeta}u(z) = \\varphi(\\zeta) \\qquad \\forall \\zeta \\in \\partial\\Omega \\setminus E_\\varphi \n \\end{cases}\n\\]\n is uniquely solvable (in the weak sense of Bedford--Taylor~\\cite{bedford-taylor, bedford-taylor2}) if and only if $E_\\varphi$ is b-pluripolar. Here, we assume that $\\varphi\\colon \\partial \\Omega \\rightarrow \\mathbb{R}$ is bounded, and continuous outside $E_\\varphi$. Furthermore, the unique solution is continuous~\\cite[Theorem~5.1]{nilsson} outside the closure of\n \\[\n \\hat E_\\varphi \\coloneqq\\{z \\in \\widebar\\Omega \\mathrel{;} u\\in\\mathcal{PSH}^-(\\Omega), \\limsup_{w\\rightarrow E_\\varphi}u(w)=-\\infty \\implies u(z) =-\\infty\\},\n \\]\nthe \\textit{boundary pluripolar hull} of $E_\\varphi$ (if the boundary pluripolar hull contains interior points, we say that the set is \\textit{propagating}). Since discontinuity sets are necessarily $F_\\sigma$, this motivates investigating necessary and sufficient criteria for when boundary pluripolar $F_\\sigma$ sets propagate.\n\nIn the first part of the paper, we obtain the following result:", "full_context": "Let $\\Omega$ denote a bounded domain in $\\mathbb{C}^N$, and let $A \\subset \\partial\\Omega$ be a set for which there exists a negative plurisubharmonic (psh) function $u$ defined on $\\Omega$ ($u\\in \\mathcal{PSH}^-(\\Omega)$), not identically $-\\infty$, with\n\\[\\limsup_{z\\rightarrow A} u(z) = -\\infty.\\]\nSuch \\textit{boundary pluripolar sets} (b-pluripolar for short) were recently characterized as the exceptional sets for the Dirichlet problem for the complex Monge--Ampère operator ~\\cite{nilsson2}. More precisely, under the assumption that $\\Omega$ admits strong plurisubharmonic barriers, in the sense that we at every boundary point $\\zeta \\in \\partial\\Omega$ can find $u \\in \\mathcal{PSH}(\\Omega)\\cap C(\\widebar \\Omega)$ with\n\\[\nu(\\zeta)=0, \\quad u<0 \\text{ on } \\widebar \\Omega \\setminus \\{\\zeta\\},\n\\]\nthen the Dirichlet problem \n\\[\n \\begin{cases}\n u \\in L^\\infty\\cap\\mathcal{PSH}(\\Omega) \\\\\n \\det(\\frac{\\partial^2 u}{\\partial z_j \\partial \\bar z_k})=0 \\\\\n \\lim_{z \\rightarrow \\zeta}u(z) = \\varphi(\\zeta) \\qquad \\forall \\zeta \\in \\partial\\Omega \\setminus E_\\varphi \n \\end{cases}\n\\]\n is uniquely solvable (in the weak sense of Bedford--Taylor~\\cite{bedford-taylor, bedford-taylor2}) if and only if $E_\\varphi$ is b-pluripolar. Here, we assume that $\\varphi\\colon \\partial \\Omega \\rightarrow \\mathbb{R}$ is bounded, and continuous outside $E_\\varphi$. Furthermore, the unique solution is continuous~\\cite[Theorem~5.1]{nilsson} outside the closure of\n \\[\n \\hat E_\\varphi \\coloneqq\\{z \\in \\widebar\\Omega \\mathrel{;} u\\in\\mathcal{PSH}^-(\\Omega), \\limsup_{w\\rightarrow E_\\varphi}u(w)=-\\infty \\implies u(z) =-\\infty\\},\n \\]\nthe \\textit{boundary pluripolar hull} of $E_\\varphi$ (if the boundary pluripolar hull contains interior points, we say that the set is \\textit{propagating}). Since discontinuity sets are necessarily $F_\\sigma$, this motivates investigating necessary and sufficient criteria for when boundary pluripolar $F_\\sigma$ sets propagate.\n\nIn the first part of the paper, we obtain the following result:\n\n\\section{Boundary pluripolar sets of dimension \\texorpdfstring{$N$}{N} must propagate}\nOur aim in this section is to prove that on strictly pseudoconvex domains in $\\C^N$, \na non-propagating b-pluripolar $F_\\sigma$ set $A$ with $\\dim A \\geq N$ cannot exist. The argument will be based on the following result, which was established using a cohomological argument in the proof of the main theorem in~\\cite{stout}.\n\\begin{lemma}[Stout]\\label{stout}\nLet $\\Omega\\subset \\C^N$ be a bounded convex domain, and suppose that $A \\subset \\partial \\Omega$ is a compact set such that for every neighborhood $U$ (in $\\partial \\Omega$) containing $p\\in A$, there is a real hyperplane $\\Pi_U: L(z)=0$ such that \n\\[\np \\in \\{L(z)>0\\}\\cap \\partial \\Omega \\subset U.\n\\]\nIf each $(A\\cap \\{L(z)>0\\}) \\cup (\\Pi_U \\cap \\widebar \\Omega)$ is polynomially convex, then $\\dim A < N$. \n\\end{lemma}\n\\begin{figure}[h]\n\\centering\n\\tikzset{every picture/.style={line width=0.75pt}}\n\nTo show that $A\\cup(\\Pi\\cap\\widebar \\Omega)$ is polynomially convex, assume that $z_0 \\in \\widebar \\Omega \\setminus(A \\cup \\Pi)$. Multiplying $L$ with a constant if necessary, we may assume that $L(z_0)>1$. The preceding paragraph implies that we may find $\\tilde u \\in \\psh(\\C^N)$ with\n\\[\n\\tilde u(z_0)>-1, \\quad \\tilde u <0 \\text { on }\\widebar \\Omega,\\quad \\tilde u< -\\max_{\\zeta\\in A}|L(\\zeta)| \\text { on }A. \n\\]\nThe psh function $\\tilde u + L$ then satisfies \n\\[\n\\tilde u(z_0)+L(z_0)>0, \\quad \\tilde u+L< 0 \\text { on }A \\cup (\\Pi\\cap \\widebar \\Omega), \n\\]\nwhich finishes the proof.\n\\end{proof}\n\\begin{remark}\nThe assumption that $\\Omega$ is B-regular (which is automatic if $\\Omega$ is strictly pseudoconvex) compensates for the fact that $A$ is a peak set in Stout's proof. Indeed, if $A$ is a peak set, we can instead apply Lemma~\\ref{approx} to $u = \\max(C_1\\log|1-h_A|,C_2)$ with suitable constants $C_2 <0<C_1$. \n\\end{remark}\nThe following simple example shows that the converse is not true: There are polynomially convex, b-pluripolar sets which do propagate into the interior.\n\\begin{example}\nLet $\\mathbb{B}$ denote the ball in $\\C^2$, and take an arc $\\gamma \\subset \\{|z_2|=1\\}$. This set is b-pluripolar since $\\gamma \\subset \\{\\log|z_1| =-\\infty\\}$, and polynomially convex by~\\cite[Corollary~3.1.2]{stoutbok}. Let $u \\in \\psh^-(\\mathbb{B})$ be a function such that $\\limsup_{w\\rightarrow \\gamma}u(w)=-\\infty$. Since $\\gamma$ has positive arc length, it is not b-(pluri)polar on the disk $\\{(0, z_2) \\suchthat |z_2|<1\\}$. It follows that $u(0,z_2) \\equiv -\\infty$; in other words, $\\gamma$ propagates along $\\{z_1=0\\}\\cap\\mathbb B$.\n\\end{example}\n\nWith Lemma~\\ref{polyconv} at hand, we are now ready to prove the main result in this section. \n\\begin{theorem}\\label{topdim}\nLet $\\Omega\\Subset \\C^N$ be strictly pseudoconvex, and let $A\\subset \\partial \\Omega$ be a boundary pluripolar $F_\\sigma$ set with topological dimension greater than $N-1$. Then $\\hat A \\cap \\Omega$ is non-empty.\n\\end{theorem}\n\\begin{proof}\nLet us first do some standard reductions. If we for a contradiction assume the existence of a non-propagating b-pluripolar $F_\\sigma$ set $A$ with $\\dim A \\geq N$, then \n\\[\nA= \\bigcup K_j\n\\]\nwith $K_j$ compact. This implies~\\cite[Theorem~3.1.8]{Engelking} that there exists $j_0$ with $\\dim K_{j_0}\\geq N$. Pick a point $p\\in K_{j_0}$ such that each neighborhood $U$ containing $p$ satisfies $\\dim U\\cap K_{j_0}\\geq N$. Since $\\Omega$ is strictly pseudoconvex, there exists an open set $V$ containing $p$ and a local biholomorphism $f$ such that $f(\\partial \\Omega\\cap V)$ is strictly convex in a neighborhood of $K\\coloneq f(U\\cap K_{j_0})$. Also, if $A$ does not propagate, then $K$ does not propagate into $B(p,\\epsilon)\\cap f(\\Omega\\cap V)$. We may therefore assume that the b-pluripolar set $A$ is compact, and that $\\Omega$ is convex and B-regular, and even strictly convex in a neighborhood of $A$.\n\n\\begin{definition}\nGiven a family of measures $\\mathcal{M}\\subset M(\\partial\\Omega)$, we say that a set $A\\subset \\partial \\Omega$ is \\textit{totally null} with respect to $\\mathcal{M}$ if $\\mu(A)=0$ for every $\\mu \\in \\mathcal{M}$.\n\\end{definition}\nIt is well-known (using the Cole--Range theorem and Bishop's theorem, cf. for example~\\cite[pp.~204--205]{Rudin}) that a compact set $K$ is totally null with respect to $M_{z_0}(\\partial\\Omega)$ if and only if $K$ is a \\textit{zero set} for $A(\\Omega)$, in other words there exists $f\\in A(\\Omega)$ such that $f(\\zeta) = 0$ for every $\\zeta \\in K$ and $f(z) \\neq 0$ for every $z \\in \\widebar \\Omega \\setminus K$. We can then construct\n\\[\nu(z) \\coloneq \\log|f(z)| - \\max_{w\\in \\widebar \\Omega} \\log|f(w)|\n\\]\nwhich shows that $K$ is b-pluripolar and non-propagating. In particular, for any compact, propagating b-pluripolar set $\\tilde K$ that does not propagate to $z_0$ (take for example $\\tilde K \\coloneq \\partial \\Omega \\cap \\Pi$, where $\\Pi$ is a complex hyperplane not containing $z_0$), there exists $\\nu \\in M_{z_0}(\\partial \\Omega)$ such that $\\nu (\\tilde K) >0$. To generalize Theorem~\\ref{hedenmalmsats}, it is therefore enough to prove\n\\begin{theorem}\\label{jensen}\nLet $A\\subset \\partial \\Omega$ be a b-pluripolar set that does not propagate to $z_0$. Then $A$ is totally null with respect to $J_{z_0}(\\partial \\Omega)$.\n\\end{theorem}\n\\begin{proof}\nLet $\\psh^b(\\Omega)\\coloneq \\psh(\\Omega) \\cap \\usc(\\widebar \\Omega)$ denote the upper bounded psh functions on $\\Omega$ (with upper semicontinuous extension to the boundary). We shall first show that for every $\\mu \\in J_{z_0}(\\partial\\Omega)$, we have\n\\[\nu(z_0) \\leq \\int_{\\partial \\Omega} u \\ d\\mu, \\quad u\\in \\psh^b(\\Omega).\n\\]\nBy Wikström's approximation theorem and the monotone convergence theorem, it is enough to consider $u \\in \\psh(\\Omega) \\cap C(\\widebar \\Omega)$. As in the proof of Lemma~\\ref{approx}, we can find $\\Omega_j\\Supset \\Omega$ and $u_{j} \\in \\psh(\\Omega_j)\\cap C(\\Omega_j)$ such that $|u-u_{j}|<\\frac{1}{j}$ on $\\widebar \\Omega$. Furthermore, as $\\Omega$ is strictly pseudoconvex, we can assume that $\\Omega_j$ is strictly pseudoconvex as well. Reasoning as in the proof of \\cite[Corollary~1.3.10]{stoutbok}, the approximation result of Bremermann--Sibony yields $f_1,\\dots,f_n\\in \\mathcal{O}(\\Omega_j) \\subset A(\\Omega)$ and constants $1\\geq c_1,\\dots, c_n>0$ such that\n\\[\nu_j - \\frac{1}{j}\\leq \\max_{k=1,\\dots,n} c_k \\log |f_k| \\leq u_j\n\\]\non $\\widebar \\Omega$. Consequently, there is a $f_{k_0}$ such that\n\\[\nu_j(z_0) - \\frac{1}{j} \\leq c_{k_0} \\log |f_{k_0}(z_0)| \\leq u_j(z_0).\n\\]\nHence,\n\\[\nu_j(z_0) - \\frac{1}{j}\\leq c_{k_0}\\log|f_{k_0}(z_0)|\\leq \\int_{\\partial \\Omega}\\max_{k=1,\\dots,n} c_k \\log |f_k| \\ d\\mu \\leq\\int_{\\partial \\Omega} u_j \\ d\\mu,\n\\]\nand so the assertion follows by letting $j\\rightarrow \\infty$. In the special case that $u_A$ is a negative psh function with $u_A(z_0)>-\\infty$ and $u_A(\\zeta) =-\\infty$ for every $\\zeta \\in A$, then\n\\[\n -\\infty< \\int_{\\partial \\Omega }u_A \\ d\\mu\\leq \\int_Au_A \\ d\\mu.\n\\]\nWe conclude that $\\mu(A)=0$, which finishes the proof.\n\\end{proof}\n\\begin{corollary}\\label{hedenmalmgen} Let $\\Omega\\subset \\C^N, N>1$ be a smooth, strictly pseudoconvex domain. Then for every $z_0 \\in \\Omega$, $M_{z_0}(\\partial \\Omega)\\setminus J_{z_0}(\\partial \\Omega)$ is non-empty.\n\\end{corollary}\n\\begin{remark}\nThe assumption that $\\Omega$ is smooth and strictly pseudoconvex can be relaxed somewhat, since the Cole--Range theorem holds on any domain where $A(\\Omega)$ is \\textit{tight}. See~\\cite{boos} and the references therein for more details.\n\\end{remark}\n\\bibliographystyle{amsplain}\n\\bibliography{name}", "post_theorem_intro_text_len": 2417, "post_theorem_intro_text": "\\noindent\nContrapositively, non-propagating b-pluripolar sets have at most dimension $N-1$. The proof is essentially an adaptation of an argument due to Stout~\\cite{stout}, who proved the corresponding statement for peak sets, i.e. closed sets $K \\subset \\partial \\Omega$ for which there exists $h_K\\in A(\\Omega)$ (holomorphic on $\\Omega$ and continuous on the closure) with\n\\[\nh_K = 1 \\text{ on }K, \\quad |h_K| < 1 \\text{ on }\\widebar \\Omega \\setminus K.\n\\] \nRoughly, the idea is to show that even under the weaker assumption that $K$ is non-propagating and b-pluripolar, $K$ is still polynomially convex. The final step in Stout's proof \nthen implies that $K$ has some topological properties which are incompatible with $\\dim K \\geq N$.\n\nIn Section~3, we establish sufficient criteria for b-pluripolarity in terms of Hausdorff dimension. Given a bounded domain $\\Omega$ and a parameter $\\beta>0$, we consider the set $P_\\beta$ consisting of $\\zeta \\in \\partial \\Omega$ for which there exists a peak function $h_{\\{\\zeta\\}}$ with\n\\[\n\\abs{h_{\\{\\zeta\\}}(z)-1} \\leq C_\\zeta \\|z-\\zeta\\|^\\beta,\n\\]\nwhere $\\| \\cdot\\|$ denotes the Euclidean norm. We show \n\\begin{theoremB}\nEvery set $A\\subset P_\\beta$ of zero $\\beta$-dimensional Hausdorff measure is non-propagating and b-pluripolar.\n\\end{theoremB}\n\\noindent\n The argument is quite elementary; we simply postcompose peak functions with the harmonic measure of arcs on the circle. \n\nFinally, in Section~4, we compare representing measures and Jensen measures at $z_0 \\in \\Omega$ supported on $\\partial \\Omega$, where $\\Omega$ is smooth and strictly pseudoconvex. We show that the compact totally null sets of the two classes differ; the former are the zero sets for $A(\\Omega)$, in particular b-pluripolar and non-propagating, while the class of totally null sets for Jensen measures contains every b-pluripolar set that does not propagate to $z_0$. From this, we deduce \n\\begin{theoremC}\nLet $\\Omega\\subset \\mathbb{C}^N$, $N>1$ be a smooth, strictly pseudoconvex domain. Then there are always representing measures that are not Jensen measures.\n\\end{theoremC}\n\\par \\noindent\n This extends a result of Hedenmalm (\\cite[Theorem~3.7]{hedenmalm}, see also Theorem~\\ref{hedenmalmsats} below) on the unit ball.\n\nWe have labeled the main results as Theorems A--C for ease of reference;\nthese correspond to Theorems~\\ref{topdim}, \\ref{hausdorff}, and Corollary~\\ref{hedenmalmgen}.", "sketch": "Contrapositively, the introduction states that \"non-propagating b-pluripolar sets have at most dimension $N-1$.\" The proof is described as \"essentially an adaptation of an argument due to Stout\" for peak sets. \"Roughly, the idea is to show that even under the weaker assumption that $K$ is non-propagating and b-pluripolar, $K$ is still polynomially convex.\" Then \"the final step in Stout's proof then implies that $K$ has some topological properties which are incompatible with $\\dim K \\geq N$.\"", "expanded_sketch": "Contrapositively, the introduction states that non-propagating b-pluripolar sets have at most dimension $N-1$. The proof is described as essentially an adaptation of an argument due to Stout for peak sets. Roughly, the idea is to show that even under the weaker assumption that $K$ is non-propagating and b-pluripolar, $K$ is still polynomially convex. Then the final step in Stout's proof implies that $K$ has some topological properties which are incompatible with $\\dim K \\geq N$.", "expanded_theorem": "Let $\\Omega \\subset \\mathbb{C}^N$ be a strictly pseudoconvex domain. Then every b-pluripolar $F_\\sigma$ set of topological dimension $N$ must propagate into the interior.", "theorem_type": "unknown", "mcq": {"question": "Let \\(\\Omega\\subset \\mathbb{C}^N\\) be a strictly pseudoconvex domain, and let \\(A\\subset \\partial\\Omega\\) be an \\(F_\\sigma\\) set of topological dimension \\(N\\). Assume that \\(A\\) is boundary pluripolar, meaning that there exists a negative plurisubharmonic function \\(u\\in \\mathcal{PSH}^-(\\Omega)\\), not identically \\(-\\infty\\), such that \\(\\limsup_{z\\to A}u(z)=-\\infty\\). Its boundary pluripolar hull is\n\\[\n\\hat A=\\{z\\in \\overline\\Omega: ყოვall\\ v\\in \\mathcal{PSH}^-(\\Omega),\\ \\limsup_{w\\to A}v(w)=-\\infty \\implies v(z)=-\\infty\\}.\n\\]\nWhich statement holds?", "correct_choice": {"label": "A", "text": "Such a set \\(A\\) must propagate into the interior; equivalently, its boundary pluripolar hull meets the interior of \\(\\Omega\\), i.e. \\(\\hat A\\cap \\Omega\\neq \\varnothing\\)."}, "choices": [{"label": "B", "text": "Such a set \\(A\\) need not propagate into the interior; it is possible that \\(\\hat A\\cap \\Omega=\\varnothing\\), and in that case one can still conclude only that \\(A\\) is polynomially convex in \\(\\partial\\Omega\\)."}, {"label": "C", "text": "Such a set \\(A\\) cannot have empty boundary pluripolar hull in the sense that \\(\\hat A\\neq A\\); in particular, the hull contains at least one point of \\(\\overline\\Omega\\setminus A\\)."}, {"label": "D", "text": "Such a set \\(A\\) must propagate into the interior only when its topological dimension is strictly greater than \\(N\\); dimension exactly \\(N\\) does not force \\(\\hat A\\cap \\Omega\\neq \\varnothing\\)."}, {"label": "E", "text": "For every strictly pseudoconvex domain \\(\\Omega\\subset\\mathbb C^N\\), there exists a point \\(z_0\\in \\Omega\\) such that every boundary pluripolar \\(F_\\sigma\\) set \\(A\\subset\\partial\\Omega\\) of topological dimension \\(N\\) propagates to that same point, i.e. \\(z_0\\in \\hat A\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "polynomial-convexity-to-propagation implication", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the conclusion that the hull meets the interior", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "dimension threshold N versus >N", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "existential interior propagation point made uniform in A", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the keyed answer, and the definition of the boundary pluripolar hull does not by itself make interior propagation immediate. There is no direct lexical leakage toward choice A."}, "TAS": {"score": 1, "justification": "The item is very close to a theorem-recall format: the hypotheses are stated in full and the task is essentially to identify the conclusion. The answer choices do introduce competing conclusions, so it is not a pure restatement, but it remains only a mild reformulation."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the solver must distinguish the sharp propagation conclusion from weaker, stronger, or quantifier-distorted variants. However, the item mainly tests recall of the theorem, and the presence of a weaker true option reduces the need for genuine generative reasoning."}, "DQS": {"score": 1, "justification": "Several distractors are mathematically plausible and target common failure modes (weaker conclusion, wrong dimension threshold, over-uniformization). But choice C appears to be a weaker true statement if A is true, so it is not a clean distractor and undermines single-best-answer quality."}, "total_score": 5, "overall_assessment": "Moderately good theorem-based MCQ with no major answer leakage, but it leans heavily on recall rather than generation and is weakened by a distractor that seems genuinely true."}}
{"id": "2602.10932v1", "paper_link": "http://arxiv.org/abs/2602.10932v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm main}\nAssume that for $1 \\leq i \\leq N$, the mean curvatures $H_{i, \\mp}$ are strictly positive, and that there exists some $1 \\leq \\Lambda < N$ such that \n\\begin{align*}\n\\underline H_{i,-} \\leq \\overline H_{i,+} & \\text{for } & 1 \\leq i \\leq \\Lambda , \\\\\n\\underline H_{i,-} \\geq \\overline H_{i,+} & \\text{for } & \\Lambda + 1\\leq i \\leq N.\n\\end{align*}\nFurthermore, assume that\n\\begin{align}\\label{eq:condition-on-squares-of-mean-curvatures}\n    \\sum_{1 \\leq i \\leq N} \\left(\\underline H_{i,-}^2 - \\overline H_{i,+}^2\\right) \\ge 0.\n\\end{align}\nThen the ADM mass of $( M,g)$ is nonnegative.", "start_pos": 4465, "end_pos": 5094, "label": "thm main"}, "ref_dict": {"fig:corners": "\\label{fig:corners}\n\\end{figure}\n\nFor the proof we view $( M,g)$ as a time-symmetric slice of an initial data set $(M^n,g,k)$ with symmetric $(0,2)$-tensor field $k$.\nWhen chosen appropriately, the te", "eq:def-of-c-and-d": "\\begin{align}\\label{eq:def-of-c-and-d}\n        d_\\ell :=\\sum_{i=1}^\\ell(\\overline H_{i,+}^2-\\underline H_{i,-}^2), \\quad c_{\\ell} :=  \\sqrt{ \\max \\{d_\\ell, 0\\}}. \n    \\end{align}"}, "pre_theorem_intro_text_len": 2922, "pre_theorem_intro_text": "Extremality and rigidity results in scalar curvature geometry on smooth Riemannian manifolds often remain valid for Riemannian metrics with singularities, provided that the relevant curvature restrictions are still satisfied in a weak sense. \nTo give a well-known example, the positive mass theorem for asymptotically flat manifolds holds for manifolds with metric singularities along hypersurfaces, if the singularities are {\\em mean-convex}, that is, the mean curvature jump with respect to the infinity pointing normal vector is non-positive, see \\cite{shi-tam-manifolds-with-boundary, Miao2002cornersATP}.\nAs described in \\cite[Proposition 3.1]{Miao2002cornersATP}, {\\em mean-concave} singularities (with a strictly negative mean curvature jump) can be regarded as having infinitely negative scalar curvature in a weak sense.\n\n\\medskip\n\nThis paper shows that the above mean-convexity condition can be replaced with a weaker one: The effect of mean-concave jumps at some components of the singular set can be offset by mean-convex jumps at others. \nLoosely speaking, the {\\em local} mean-convexity condition can be replaced by a weaker {\\em global} one by passing to some average.\nThis phenomenon is unexpected, given that the pointwise lower scalar curvature bound in the positive mass theorem cannot be replaced by a lower bound on an averaged scalar curvature.\n\n\\medskip\n\nTo state our main result, let $M$ be a non-compact connected spin manifold which is decomposed along smooth compact hypersurfaces $\\Sigma_1,\\dots\\Sigma_N$ as \n\\[\nM = M_0 \\cup_{\\Sigma_1} \\dots\\cup_{\\Sigma_N} M_N.\n\\]\nHere $M_0, \\dots, M_N \\subset M$ are smooth submanifolds with boundary, $M_{i}$ is compact for $1 \\leq i \\leq N-1$, and $M_N$ is noncompact and connected; see \\cref{fig:corners}.\nFor $0 \\leq i \\leq N$, let $g_i$ be smooth Riemannian metrics on $M_i$ such that \n\\begin{itemize}\n    \\item for $1 \\leq i \\leq N$, we have $(g_{i-1})|_{\\Sigma_i} = (g_i)|_{\\Sigma_i}$,\n    \\item each $g_i$ has non-negative scalar curvature, \n    \\item $(M_{N},g_N)$ is asymptotically flat.\n\\end{itemize}\nWe write $g\\coloneqq g_0\\cup\\dots\\cup g_N$ for the induced $C^0$-Riemannian metric on $M$.\nLet us denote the mean curvatures along $\\Sigma_i$ by\n\\[H_{i,-}\\coloneqq H(\\Sigma_i\\subseteq M_{i-1}), \\qquad H_{i,+}\\coloneqq H(\\Sigma_i\\subseteq M_{i}), \\qquad 1 \\leq i \\leq N.\n\\]\nWe adopt the convention common in general relativity that mean curvatures are computed with respect to the infinity-pointing normal.\nThat is, if $M$ is the Euclidean space $\\mathbb{R}^{n}$ with the decomposition\n\\[\n    M_0 := \\{ |x| \\leq 1\\}, \\quad M_1:=\\{1 \\leq |x| \\leq 2\\}, \\quad M_2 := \\{|x| \\geq 2\\}, \n\\]\nthen we have $H_{1,\\mp} = n-1,\\ H_{2,\\mp } = (n-1)/2$.\n\n\\medskip\n\nFor $1 \\leq i \\leq N$, we set \n\\begin{align*}\n    \\underline H_{i,-} := \\min_{\\Sigma_i} H_{i, -}, \\qquad \\overline H_{i,+} := \\max_{\\Sigma_i} H_{i, +}.\n\\end{align*}\nOur main result is the following.", "context": "Extremality and rigidity results in scalar curvature geometry on smooth Riemannian manifolds often remain valid for Riemannian metrics with singularities, provided that the relevant curvature restrictions are still satisfied in a weak sense. \nTo give a well-known example, the positive mass theorem for asymptotically flat manifolds holds for manifolds with metric singularities along hypersurfaces, if the singularities are {\\em mean-convex}, that is, the mean curvature jump with respect to the infinity pointing normal vector is non-positive, see \\cite{shi-tam-manifolds-with-boundary, Miao2002cornersATP}.\nAs described in \\cite[Proposition 3.1]{Miao2002cornersATP}, {\\em mean-concave} singularities (with a strictly negative mean curvature jump) can be regarded as having infinitely negative scalar curvature in a weak sense.\n\nThis paper shows that the above mean-convexity condition can be replaced with a weaker one: The effect of mean-concave jumps at some components of the singular set can be offset by mean-convex jumps at others. \nLoosely speaking, the {\\em local} mean-convexity condition can be replaced by a weaker {\\em global} one by passing to some average.\nThis phenomenon is unexpected, given that the pointwise lower scalar curvature bound in the positive mass theorem cannot be replaced by a lower bound on an averaged scalar curvature.\n\nTo state our main result, let $M$ be a non-compact connected spin manifold which is decomposed along smooth compact hypersurfaces $\\Sigma_1,\\dots\\Sigma_N$ as \n\\[\nM = M_0 \\cup_{\\Sigma_1} \\dots\\cup_{\\Sigma_N} M_N.\n\\]\nHere $M_0, \\dots, M_N \\subset M$ are smooth submanifolds with boundary, $M_{i}$ is compact for $1 \\leq i \\leq N-1$, and $M_N$ is noncompact and connected; see \\cref{fig:corners}.\nFor $0 \\leq i \\leq N$, let $g_i$ be smooth Riemannian metrics on $M_i$ such that \n\\begin{itemize}\n \\item for $1 \\leq i \\leq N$, we have $(g_{i-1})|_{\\Sigma_i} = (g_i)|_{\\Sigma_i}$,\n \\item each $g_i$ has non-negative scalar curvature, \n \\item $(M_{N},g_N)$ is asymptotically flat.\n\\end{itemize}\nWe write $g\\coloneqq g_0\\cup\\dots\\cup g_N$ for the induced $C^0$-Riemannian metric on $M$.\nLet us denote the mean curvatures along $\\Sigma_i$ by\n\\[H_{i,-}\\coloneqq H(\\Sigma_i\\subseteq M_{i-1}), \\qquad H_{i,+}\\coloneqq H(\\Sigma_i\\subseteq M_{i}), \\qquad 1 \\leq i \\leq N.\n\\]\nWe adopt the convention common in general relativity that mean curvatures are computed with respect to the infinity-pointing normal.\nThat is, if $M$ is the Euclidean space $\\mathbb{R}^{n}$ with the decomposition\n\\[\n M_0 := \\{ |x| \\leq 1\\}, \\quad M_1:=\\{1 \\leq |x| \\leq 2\\}, \\quad M_2 := \\{|x| \\geq 2\\}, \n\\]\nthen we have $H_{1,\\mp} = n-1,\\ H_{2,\\mp } = (n-1)/2$.\n\n\\medskip\n\nFor $1 \\leq i \\leq N$, we set \n\\begin{align*}\n \\underline H_{i,-} := \\min_{\\Sigma_i} H_{i, -}, \\qquad \\overline H_{i,+} := \\max_{\\Sigma_i} H_{i, +}.\n\\end{align*}\nOur main result is the following.\n\n\\label{fig:corners}\n\\end{figure}\n\nFor the proof we view $( M,g)$ as a time-symmetric slice of an initial data set $(M^n,g,k)$ with symmetric $(0,2)$-tensor field $k$.\nWhen chosen appropriately, the te", "full_context": "Extremality and rigidity results in scalar curvature geometry on smooth Riemannian manifolds often remain valid for Riemannian metrics with singularities, provided that the relevant curvature restrictions are still satisfied in a weak sense. \nTo give a well-known example, the positive mass theorem for asymptotically flat manifolds holds for manifolds with metric singularities along hypersurfaces, if the singularities are {\\em mean-convex}, that is, the mean curvature jump with respect to the infinity pointing normal vector is non-positive, see \\cite{shi-tam-manifolds-with-boundary, Miao2002cornersATP}.\nAs described in \\cite[Proposition 3.1]{Miao2002cornersATP}, {\\em mean-concave} singularities (with a strictly negative mean curvature jump) can be regarded as having infinitely negative scalar curvature in a weak sense.\n\nThis paper shows that the above mean-convexity condition can be replaced with a weaker one: The effect of mean-concave jumps at some components of the singular set can be offset by mean-convex jumps at others. \nLoosely speaking, the {\\em local} mean-convexity condition can be replaced by a weaker {\\em global} one by passing to some average.\nThis phenomenon is unexpected, given that the pointwise lower scalar curvature bound in the positive mass theorem cannot be replaced by a lower bound on an averaged scalar curvature.\n\nTo state our main result, let $M$ be a non-compact connected spin manifold which is decomposed along smooth compact hypersurfaces $\\Sigma_1,\\dots\\Sigma_N$ as \n\\[\nM = M_0 \\cup_{\\Sigma_1} \\dots\\cup_{\\Sigma_N} M_N.\n\\]\nHere $M_0, \\dots, M_N \\subset M$ are smooth submanifolds with boundary, $M_{i}$ is compact for $1 \\leq i \\leq N-1$, and $M_N$ is noncompact and connected; see \\cref{fig:corners}.\nFor $0 \\leq i \\leq N$, let $g_i$ be smooth Riemannian metrics on $M_i$ such that \n\\begin{itemize}\n \\item for $1 \\leq i \\leq N$, we have $(g_{i-1})|_{\\Sigma_i} = (g_i)|_{\\Sigma_i}$,\n \\item each $g_i$ has non-negative scalar curvature, \n \\item $(M_{N},g_N)$ is asymptotically flat.\n\\end{itemize}\nWe write $g\\coloneqq g_0\\cup\\dots\\cup g_N$ for the induced $C^0$-Riemannian metric on $M$.\nLet us denote the mean curvatures along $\\Sigma_i$ by\n\\[H_{i,-}\\coloneqq H(\\Sigma_i\\subseteq M_{i-1}), \\qquad H_{i,+}\\coloneqq H(\\Sigma_i\\subseteq M_{i}), \\qquad 1 \\leq i \\leq N.\n\\]\nWe adopt the convention common in general relativity that mean curvatures are computed with respect to the infinity-pointing normal.\nThat is, if $M$ is the Euclidean space $\\mathbb{R}^{n}$ with the decomposition\n\\[\n M_0 := \\{ |x| \\leq 1\\}, \\quad M_1:=\\{1 \\leq |x| \\leq 2\\}, \\quad M_2 := \\{|x| \\geq 2\\}, \n\\]\nthen we have $H_{1,\\mp} = n-1,\\ H_{2,\\mp } = (n-1)/2$.\n\n\\medskip\n\nFor $1 \\leq i \\leq N$, we set \n\\begin{align*}\n \\underline H_{i,-} := \\min_{\\Sigma_i} H_{i, -}, \\qquad \\overline H_{i,+} := \\max_{\\Sigma_i} H_{i, +}.\n\\end{align*}\nOur main result is the following.\n\n\\label{fig:corners}\n\\end{figure}\n\nFor the proof we view $( M,g)$ as a time-symmetric slice of an initial data set $(M^n,g,k)$ with symmetric $(0,2)$-tensor field $k$.\nWhen chosen appropriately, the te\n\nFor $1 \\leq i \\leq N$, we set \n\\begin{align*}\n \\underline H_{i,-} := \\min_{\\Sigma_i} H_{i, -}, \\qquad \\overline H_{i,+} := \\max_{\\Sigma_i} H_{i, +}.\n\\end{align*}\nOur main result is the following.\n\n\\begin{lemma}\\label{lemma main}\n Suppose that $\\Sigma$ is a smooth $(n-1)$-manifold and that $H_-,H_+\\colon \\Sigma\\to\\bbR$ are smooth functions, and let $\\underline H_{-},\\overline H_+\\in(0,\\infty)$ be such that $H_-\\ge \\underline H_-$ and $H_+\\le\\overline H_+$.\n Given $a\\in\\bbR$ satisfying $\\overline{H}_+^2-\\underline{H}_-^2+a^2\\ge0$, we define functions $\\vec{H}_+, \\vec{H}_- \\colon \\Sigma \\to \\mathbb{R}^{1,1}$ via\n \\begin{align*}\n \\vec H_-=\\begin{pmatrix}\n H_-\\\\\n a\n \\end{pmatrix},\n \\qquad\n \\vec H_+=\\begin{pmatrix}\n H_+\\\\\n \\sqrt{\\overline H_+^2-\\underline H_-^2+a^2}\n \\end{pmatrix}.\n \\end{align*}\n Then there exists a hyperbolic rotation angle $\\vartheta\\in \\mathbb R$ such that the function\n \\begin{align*}\n X=F_\\vartheta \\vec H_--\\vec H_+ \\colon \\Sigma \\to \\mathbb R^{1,1}, \\qquad F_\\vartheta=\n \\begin{pmatrix}\n \\cosh \\vartheta & -\\sinh \\vartheta\\\\\n -\\sinh \\vartheta &\\cosh\\vartheta\n \\end{pmatrix},\n \\end{align*}\n satisfies\n \\begin{align*}\n X_1\\ge|X_2| \\quad \\textrm{ along } \\Sigma,\n \\end{align*}\n where $X_1$ and $X_2$ denotes the first and second component of $X$, respectively.\n\\end{lemma}\n\n\\begin{proof}\nLet us abbreviate $\\xi\\coloneqq\\sqrt{\\overline H_+^{2}-\\underline H_-^{2}+a^2}$.\nWe choose \n\\begin{align}\\label{eq vartheta 1}\n \\vartheta =\\sinh^{-1}\\left( \n \\frac{a\\overline H_+ - \\xi\\underline H_-}\n {\\underline H_-^{2}-a^2}\n \\right),\n\\end{align}\nin case $\\underline H_-\\ne a$, and\n\\begin{align}\\label{eq vartheta 2}\n \\vartheta=\\log(\\underline H_-)-\\log(\\overline H_+)\n\\end{align}\nin case $\\underline H_-=a$.\nUsing $\\cosh(\\vartheta) = \\sqrt{1+\\sinh^2(\\vartheta)}$, we obtain\n\\begin{align*}\n & F_\\vartheta \\vec H_- - \\vec H_+\n =\n \\begin{pmatrix}\n \\sqrt{1+\\sinh^2(\\vartheta)}\\,H_- - \\sinh(\\vartheta)\\, a - H_+\\\\[0.4em]\n \\sqrt{1+\\sinh^2(\\vartheta)}\\,a - \\sinh(\\vartheta)\\, H_- \n - \\xi\n \\end{pmatrix}.\n\\end{align*}\nIf $\\underline H_-\\neq a$, then\n\\begin{align*}\n &\\left(\\underline H_-^{2}-a^2\\right)^2\\left(1+\\sinh^2(\\vartheta)\\right)\\\\\n &\\qquad\\qquad = \\underline H_-^4 - 2a^2\\underline H_-^2 + a^4 + (a\\overline H_+ - \\xi\\underline H_-)^2\\\\\n &\\qquad\\qquad = \\underline H_-^4 - 2a^2\\underline H_-^2 + a^4 + a^2\\overline H_+^2 - 2a\\xi\\overline H_+\\underline H_- + (\\overline H_+^2-\\underline{H}_-^2+a^2)\\underline H_-^2\\\\\n &\\qquad\\qquad = \\left(\\overline H_+ \\underline H_- - a\\xi\\right)^2\n\\end{align*}\nand hence $\\left|\\underline H_-^{2}-a^2\\right|\\sqrt{1+\\sinh^2(\\vartheta)}=\\left|\\overline H_+\\underline H_- - a \\xi\\right|$.\n\nIn the case that $\\underline H_->a$, we note that $\\xi< \\overline H_+$, and that both $\\underline H_-^{2} -a^2$ and $\\overline H_+ \\underline{H}_--a\\xi$ are positive. \nWe can thus estimate\n\\begin{align*}\n &(\\underline H_-^{2} - a^2) \n \\left(\\sqrt{1+\\sinh^2(\\vartheta)}\\,H_- - \\sinh(\\vartheta)\\, a - H_+\\right)\\\\\n &\\qquad\\qquad= H_- \\left(\\overline H_+ \\underline H_- - a \\xi \\right) - a \\left(\n a\\overline H_+ - \\xi\\underline H_-\\right)- H_+(\\underline H_-^{2}-a^2)\\\\\n &\\qquad\\qquad\\ge(H_- -\\underline H_-)\\left(\\overline H_+ \\underline H_- - a \\xi\\right),\n\\end{align*}\nwhere we used $H_+(\\underline H_-^{2}-a^2)\\le \\overline H_+(\\underline H_-^{2}-a^2)$ for the final inequality.\nMoreover, we have\n\\begin{align*}\n &(\\underline H_-^{2}-a^2)\\left(\\sqrt{1+\\sinh^2(\\vartheta)}\\,a - \\sinh(\\vartheta)\\, H_- - \\xi\\right)\\\\\n &\\qquad\\qquad= a \\left(\\overline H_+ \\underline H_- - a \\xi\\right) - H_- \\left(a\\overline H_+ - \\xi\\underline H_-\\right)- \\xi(\\underline H_-^{2}-a^2)\\\\\n &\\qquad\\qquad=(H_--\\underline H_-)\\left(\\xi\\underline H_--a\\overline H_+\\right)\n\\end{align*}\nSince\n\\begin{align}\\label{computation1}\n\\begin{split}\n &\\left(\\overline H_+ \\underline H_- - a \\xi \\right)^2 -\\left(\\xi\\underline H_- -a\\overline H_+\\right)^2\\\\\n &\\qquad=\\overline H_+^2\\underline H_-^2- 2a\\xi\\overline H_+\\underline H_- + a^2\\xi^2 - \\underline H_-^2\\xi^2 + 2a\\xi\\overline H_+\\underline H_- - a^2\\overline H_+^2\\\\\n &\\qquad= \\overline H_+^2\\underline H_-^2 + a^2 \\left(\\overline H_+^{2}-\\underline H_-^{2}+a^2\\right) - \\underline H_-^2\\left(\\overline H_+^{2}-\\underline H_-^{2}+a^2\\right)- a^2 \\overline H_+^2\\\\\n &\\qquad =a^4-2a^2\\underline H_-^2+\\underline H_-^4 \\ge0,\n \\end{split}\n\\end{align}\nwe obtain $\\overline H_+ \\underline H_- - a\\xi\\ge |\\xi\\underline H_--a\\overline H_+|$ and consequently,\n\\begin{align}\\label{computation2}\n \\begin{split}\n &\\left(\\sqrt{1+\\sinh^2(\\vartheta)}\\,H_- - \\sinh(\\vartheta)\\, a - H_+\\right)\\\\\n &\\qquad\\qquad\\qquad\\ge \\Big|\\sqrt{1+\\sinh^2(\\vartheta)}\\,a - \\sinh(\\vartheta)\\, H_- - \\xi\\Big|.\n \\end{split}\n\\end{align}\nIn the case $\\underline H_-<a$, where $\\xi> \\overline H_+$ and both $\\underline H_-^{2} - a^2$ and $\\overline H_+ \\underline{H}_--a\\xi$ are negative, we can similarly estimate\n\\begin{align*}\n &-(\\underline H_-^{2} - a^2) \n \\left(\\sqrt{1+\\sinh^2(\\vartheta)}\\,H_- - \\sinh(\\vartheta)\\, a - H_+\\right)\\\\\n &\\qquad\\qquad= -H_- \\left(\\overline H_+ \\underline H_- - a \\xi \\right) + a \\left(\n a\\overline H_+ - \\xi\\underline H_-\\right)+ H_+(\\underline H_-^{2}-a^2)\\\\\n &\\qquad\\qquad\\ge (H_- -\\underline H_-)\\left(-\\overline H_+ \\underline H_- + a \\xi \\right)\\\\\n &\\qquad\\qquad\\ge (H_- -\\underline H_-)\\left|a \\overline H_+ - \\xi\\underline H_- \\right|,\n\\end{align*}\nwhere the last inequality follows as in \\eqref{computation1}.\nMoreover,\n\\begin{align*}\n &-(\\underline H_-^{2}-a^2)\\left(\\sqrt{1+\\sinh^2(\\vartheta)}\\,a - \\sinh(\\vartheta)\\, H_- - \\xi\\right)\\\\\n &\\qquad\\qquad= -a \\left(\\overline H_+ \\underline H_- - a \\xi\\right) + H_- \\left(a\\overline H_+ - \\xi\\underline H_-\\right)+ \\xi(\\underline H_-^{2}-a^2)\\\\\n &\\qquad\\qquad=(H_--\\underline H_-)\\left(a\\overline H_+-\\xi\\underline H_-\\right)\n\\end{align*}\nConsequently, \\eqref{computation2} holds in this case as well.\n\n\\begin{proof}[Proof of \\cref{thm main}]\n Put $d_0=c_0= 0$, and for $1 \\leq \\ell \\leq N$, put \n \\begin{align}\\label{eq:def-of-c-and-d}\n d_\\ell :=\\sum_{i=1}^\\ell(\\overline H_{i,+}^2-\\underline H_{i,-}^2), \\quad c_{\\ell} := \\sqrt{ \\max \\{d_\\ell, 0\\}}. \n \\end{align}\n Without loss of generality, we can assume that $d_{\\ell} > 0$ for some $\\ell$. \n Otherwise, \\cref{thm main} is implied by \\cite{Miao2002cornersATP}.\n\nLet $\\ell \\in \\{ 1, \\ldots, \\Lambda' \\}$.\n If we choose \n \\begin{align}\\label{eq:choice-for-lemma}\n \\Sigma\\coloneqq\\Sigma_\\ell,\\quad H_\\mp := H_{\\ell,\\mp},\\quad \\underline H_-\\coloneqq \\underline H_{\\ell,-},\\quad \\overline H_+\\coloneqq \\overline H_{\\ell,+}\\quad\\text{and}\\quad a:=c_{\\ell-1}\n \\end{align}\n then $c_\\ell = \\sqrt{d_\\ell} = \\sqrt{\\overline H_+^2-\\underline H_-^2+a^2}$, and we obtain from \\cref{lemma main} a hyperbolic angle $\\vartheta_\\ell$ such that\n \\begin{align*}\n X=F_{\\vartheta_\\ell}\\begin{pmatrix}\n H_{\\ell,-}\\\\ c_{\\ell-1}\n \\end{pmatrix}\n -\\begin{pmatrix}\n H_{\\ell,+}\\\\ c_\\ell\n \\end{pmatrix}\\in C^\\infty\\left(\\Sigma_\\ell,\\mathbb{R}^{1,1}\\right)\n \\end{align*}\n satisfies $X_1\\ge|X_2|$.\n\nFor $\\ell \\in \\{ \\Lambda' +1, \\ldots, N \\}$, we choose $\\Sigma$, $H_-$, $H_+$, $\\underline H_-$, and the constant $a$ as in \\eqref{eq:choice-for-lemma}, and we choose \n \\[\\overline H_+ := \\sqrt{ \\underline H_{\\ell,-}^2-c^2_{\\ell-1}}.\\]\n Note, that by the assumption on $\\Lambda'$, we have $\\underline H_{\\ell,-}^2-c^2_{\\ell-1}\\ge \\overline H_{\\ell,+}^2$ and hence $\\overline H_+ \\geq \\overline H_{\\ell,+}\\ge H_+$ and $\\overline H_+^2 - \\underline H_-^2 + a^2 = 0$.\n Therefore, an application of \\cref{lemma main} gives a $\\vartheta\\in\\bbR$ such that\n \\begin{align*}\n X=F_{\\vartheta_\\ell}\\begin{pmatrix}\n H_{\\ell,-}\\\\ c_{\\ell-1}\n \\end{pmatrix}\n -\\begin{pmatrix}\n H_{\\ell,+} \\\\ 0 \n \\end{pmatrix}\\in C^\\infty\\left(\\Sigma_\\ell,\\mathbb{R}^{1,1}\\right)\n \\end{align*}\n again satisfies $X_1\\ge|X_2|$.", "post_theorem_intro_text_len": 2724, "post_theorem_intro_text": "\\begin{figure}[ht]\n    \\begin{tikzpicture}\n        \\node at (0,0) {\\includegraphics[width=.8\\textwidth]{graphics/PDF/four-creases-with-locks.pdf}};\n        \\node[anchor=west] at (-6,3) {$ M = M_0\\cup\\dots\\cup M_4$:};\n        \\node[scale=0.8] at (-4,2.1) {$M_0$};\n        \\node[scale=0.8] at (-2,2.3) {$M_1$};\n        \\node[scale=0.8] at (1,2.5) {$M_2$};\n        \\node[scale=0.8] at (2.9,1.9) {$M_3$};\n        \\node[scale=0.8] at (4.5,3) {$M_4$};\n        \\node[scale=0.8] at (-2.3,0.1) {$\\Sigma_1$};\n        \\node[scale=0.8] at (-0.2,0.1) {$\\Sigma_2$};\n        \\node[scale=0.8] at (2.6,0.1) {$\\Sigma_3$};\n        \\node[scale=0.8] at (4.3,0.1) {$\\Sigma_4$};\n\n        \\draw[stealth-stealth](-2.16,-3.63) to (-2.16,-2.7);\n        \\draw[stealth-stealth](-.13,-3.63) to (-.13,-2);\n        \\draw[stealth-stealth](2.74,-3.63) to (2.74,-2.4);\n\n        \\node[anchor=west] at (-6,-2) {$k=\\frac{c_i}{n-1}\\cdot g$:};\n        \\node[scale=0.8] at (-4,-3.2) {$c_0=0$};\n        \\node[scale=0.8] at (-1.81,-3.2) {$\\frac{c_1}{n-1}$};\n        \\node[scale=0.8] at (0.22,-3.2) {$\\frac{c_2}{n-1}$};\n        \\node[scale=0.8]at (3.09,-3.2) {$\\frac{c_3}{n-1}$};\n        \\node[scale=0.8] at (4.5,-3.2) {$c_4=0$};\n    \\end{tikzpicture}\n    \\caption{The lock principle for $N=4$ and $\\Lambda=2$.\n    Two mean-concave curvature jumps at $\\Sigma_1$ and $\\Sigma_2$ are offset by two mean-convex curvature jumps at $\\Sigma_3$ and $\\Sigma_4$.\n    The constants $c_i$ determining the symmetric tensor $k$ are defined in \\eqref{eq:def-of-c-and-d}.}\\label{fig:corners}\n\\end{figure}\n\nFor the proof we view $( M,g)$ as a time-symmetric slice of an initial data set $(M^n,g,k)$ with symmetric $(0,2)$-tensor field $k$.\nWhen chosen appropriately, the tensor $k$ acts as a \\emph{lock}: it allows one to convert an unfavorable mean curvature jump at $\\Sigma_i$ into a corresponding jump of $k$, to transport this defect across $ M_i$, and finally to absorb it at a later singularity $\\Sigma_{j}$ with a favorable mean curvature jump.\nA typical situation is shown in \\cref{fig:corners}.\nIn this situation, we can then apply the positive mass theorem for initial data sets with creases proven by Kazaras--Khuri--Lin \\cite{KazarasKhuriLin2025}.\nBy moving to initial data sets, we thus prove a result in Riemannian geometry that cannot be achieved by local deformation arguments.\n\n\\medskip\n\nThis paper elaborates on a particular example of the lock principle. \nIn \\cref{sec extensions}, we discuss possible extensions.\n\n\\medskip\n\n\\noindent \\textbf{Acknowledgements:} BH and SH are grateful to the Lonavala Geometry Festival for the ideal working conditions where part of this work was carried out. \nBH was partially supported by the DFG-SPP 2026 ``Geometry at Infinity''.", "sketch": "For the proof we view $( M,g)$ as a time-symmetric slice of an initial data set $(M^n,g,k)$ with symmetric $(0,2)$-tensor field $k$. When chosen appropriately, the tensor $k$ acts as a \\emph{lock}: it allows one to convert an unfavorable mean curvature jump at $\\Sigma_i$ into a corresponding jump of $k$, to transport this defect across $ M_i$, and finally to absorb it at a later singularity $\\Sigma_{j}$ with a favorable mean curvature jump (a typical situation is shown in \\cref{fig:corners}). In this situation, one can then apply the positive mass theorem for initial data sets with creases proven by Kazaras--Khuri--Lin \\cite{KazarasKhuriLin2025}. By moving to initial data sets, the argument proves a Riemannian result that \\emph{cannot be achieved by local deformation arguments}.", "expanded_sketch": "For the proof we view $( M,g)$ as a time-symmetric slice of an initial data set $(M^n,g,k)$ with symmetric $(0,2)$-tensor field $k$. When chosen appropriately, the tensor $k$ acts as a \\emph{lock}: it allows one to convert an unfavorable mean curvature jump at $\\Sigma_i$ into a corresponding jump of $k$, to transport this defect across $ M_i$, and finally to absorb it at a later singularity $\\Sigma_{j}$ with a favorable mean curvature jump (a typical situation is shown in \n\\label{fig:corners}\n\\end{figure}\n\nFor the proof we view $( M,g)$ as a time-symmetric slice of an initial data set $(M^n,g,k)$ with symmetric $(0,2)$-tensor field $k$.\nWhen chosen appropriately, the te}). In this situation, one can then apply the positive mass theorem for initial data sets with creases proven by Kazaras--Khuri--Lin \\cite{KazarasKhuriLin2025}. By moving to initial data sets, the argument proves a Riemannian result that \\emph{cannot be achieved by local deformation arguments}.,", "expanded_theorem": "\\label{thm main}\nAssume that for $1 \\leq i \\leq N$, the mean curvatures $H_{i, \\mp}$ are strictly positive, and that there exists some $1 \\leq \\Lambda < N$ such that \n\\begin{align*}\n\\underline H_{i,-} \\leq \\overline H_{i,+} & \\text{for } & 1 \\leq i \\leq \\Lambda , \\\\\n\\underline H_{i,-} \\geq \\overline H_{i,+} & \\text{for } & \\Lambda + 1\\leq i \\leq N.\n\\end{align*}\nFurthermore, assume that\n\\begin{align}\\label{eq:condition-on-squares-of-mean-curvatures}\n    \\sum_{1 \\leq i \\leq N} \\left(\\underline H_{i,-}^2 - \\overline H_{i,+}^2\\right) \\ge 0.\n\\end{align}\nThen the ADM mass of $( M,g)$ is nonnegative.", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let \\(M\\) be a non-compact connected spin manifold decomposed along smooth compact hypersurfaces \\(\\Sigma_1,\\dots,\\Sigma_N\\) as\n\\[\nM=M_0\\cup_{\\Sigma_1}\\cdots\\cup_{\\Sigma_N} M_N,\n\\]\nwhere each \\(M_i\\) is a smooth submanifold with boundary, \\(M_i\\) is compact for \\(1\\le i\\le N-1\\), and \\(M_N\\) is noncompact and connected. Let \\(g_i\\) be smooth Riemannian metrics on \\(M_i\\) such that for each \\(1\\le i\\le N\\), \\((g_{i-1})|_{\\Sigma_i}=(g_i)|_{\\Sigma_i}\\), each \\(g_i\\) has non-negative scalar curvature, and \\((M_N,g_N)\\) is asymptotically flat. Write \\(g:=g_0\\cup\\cdots\\cup g_N\\) for the induced \\(C^0\\)-Riemannian metric on \\(M\\).\n\nFor each \\(1\\le i\\le N\\), let\n\\[\nH_{i,-}:=H(\\Sigma_i\\subseteq M_{i-1}),\\qquad H_{i,+}:=H(\\Sigma_i\\subseteq M_i),\n\\]\nwhere mean curvature is computed with respect to the infinity-pointing normal, and define\n\\[\n\\underline H_{i,-}:=\\min_{\\Sigma_i} H_{i,-},\\qquad \\overline H_{i,+}:=\\max_{\\Sigma_i} H_{i,+}.\n\\]\nAssume that for every \\(1\\le i\\le N\\), both mean curvatures \\(H_{i,-}\\) and \\(H_{i,+}\\) are strictly positive, and that there exists an index \\(\\Lambda\\) with \\(1\\le \\Lambda < N\\) such that\n\\[\n\\underline H_{i,-}\\le \\overline H_{i,+}\\quad \\text{for }1\\le i\\le \\Lambda,\n\\qquad\n\\underline H_{i,-}\\ge \\overline H_{i,+}\\quad \\text{for }\\Lambda+1\\le i\\le N.\n\\]\nAlso assume\n\\[\n\\sum_{i=1}^N \\bigl(\\underline H_{i,-}^2-\\overline H_{i,+}^2\\bigr)\\ge 0.\n\\]\nUnder these assumptions, which quantitative estimate holds?", "correct_choice": {"label": "A", "text": "The ADM mass of the asymptotically flat manifold \\((M,g)\\) is nonnegative."}, "choices": [{"label": "B", "text": "The ADM mass of the asymptotically flat manifold \\((M,g)\\) is nonnegative provided that, for each \\(1\\le i\\le N\\), one has the pointwise inequality \\(H_{i,-}\\ge H_{i,+}\\) along \\(\\Sigma_i\\)."}, {"label": "C", "text": "The ADM mass of the asymptotically flat manifold \\((M,g)\\) is bounded below by a finite constant."}, {"label": "D", "text": "The ADM mass of the asymptotically flat manifold \\((M,g)\\) is nonnegative whenever there exists a permutation of the hypersurfaces \\(\\Sigma_1,\\dots,\\Sigma_N\\) for which the reordered quantities satisfy the same sign pattern and \\(\\sum_{i=1}^N(\\underline H_{i,-}^2-\\overline H_{i,+}^2)\\ge0\\)."}, {"label": "E", "text": "The ADM mass of the asymptotically flat manifold \\((M,g)\\) is strictly positive unless \\(\\underline H_{i,-}=\\overline H_{i,+}\\) for every \\(1\\le i\\le N\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "global-square-balance-replaced-by-local-pointwise-jumps", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped the sharp lower bound 0 on ADM mass", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "ordered-transport-of-defect-across-intermediate-pieces", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "upgraded nonnegativity conclusion to rigidity-type strict positivity", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the conclusion, and no single phrase directly gives away that the ADM mass must be nonnegative. The correct answer must still be identified from the options."}, "TAS": {"score": 0, "justification": "The item is essentially a theorem-recall question: the stem lists a long hypothesis package and asks for the exact conclusion that follows. This is very close to a direct restatement rather than a new problem situation."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because one distractor is a weaker true statement and others are plausible overclaims or altered hypotheses. However, solving it mainly depends on recalling the precise theorem rather than generating a substantial argument."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and distinct: one weakens the conclusion, others alter the curvature-jump condition or overstrengthen to a rigidity claim. These align with realistic failure modes."}, "total_score": 5, "overall_assessment": "Good distractor design and no obvious answer leakage, but the question is largely a direct theorem restatement with only moderate reasoning demand."}}
{"id": "2602.10970v1", "paper_link": "http://arxiv.org/abs/2602.10970v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm:main-strong-cover}\n    Let $\\varepsilon > 0$. There exists a constant $C = C(\\varepsilon)>0$ such that if $G$ is an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$, then $ST(G) \\leq (1+\\varepsilon)n\\log n$.", "start_pos": 12724, "end_pos": 12947, "label": "thm:main-strong-cover"}, "ref_dict": {"prop:cover-time": "\\begin{proposition}\\label{prop:cover-time}\n    For every $\\varepsilon >0$, there exists $C = C(\\eps)$ such that if $G$ be an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$, then $T(G) \\leq (1+\\eps) n\\log n$.\n\\end{proposition}", "thm:main-strong-cover": "\\begin{theorem}\\label{thm:main-strong-cover}\n    Let $\\eps > 0$. There exists a constant $C = C(\\eps)>0$ such that if $G$ is an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$, then $ST(G) \\leq (1+\\eps)n\\log n$.\n\\end{theorem}", "thm:main": "\\begin{theorem}\\label{thm:main}\n    Let $\\varepsilon > 0$. There exists a constant $C = C(\\eps) >0$ such that the following holds: Let $G$ be an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$ and $L = (1+\\eps) n\\log n$. \\textbf{whp} for every vertex $v\\in V(G)$, the trace of a simple random walk starting at $v$ of length $L$ is Hamiltonian.\n\\end{theorem}", "lem:number-of-vists": "\\begin{lemma}\\label{lem:number-of-vists}\n    Let $\\varepsilon > 0$. There exists constants $C = C(\\eps) >0$ and $\\rho = \\rho(\\eps) > 0$ such that the following holds: Let $G$ be a $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$ and $L = (1+\\eps) n\\log n$. Consider a random walk on $G$ starting from an arbitrary vertex and of length $L$. Then, \\textbf{whp} $\\gamma(v)\\geq \\rho \\log n$ for every vertex $v\\in V(G)$.\n\\end{lemma}"}, "pre_theorem_intro_text_len": 5711, "pre_theorem_intro_text": "Let $G$ be a simple undirected connected graph. A \\emph{simple random walk} on $G$ is a stochastic process where, starting from a given vertex, each step consists of moving to a neighbor chosen uniformly at random. The set of edges traversed by this walk is called the \\emph{trace}. Random walks are fundamental objects in combinatorics, probability theory, and theoretical computer science. For a comprehensive treatment, we refer the reader to the textbook by Levin and Peres~\\cite{Levin-Peres-book} or the survey by Lov\\'{a}sz~\\cite{Lovasz-survey}.\n\nIn this paper, we study the graph formed by the trace of a simple random walk on a finite graph. This line of research was initiated by Frieze, Krivelevich, Michaeli, and Peled in~\\cite{Frieze-et-al}, who focused on the case where the base graph is random. Motivated by these results, in~\\cite{Frieze-et-al}, they asked to extend the inquiry to deterministic graphs that exhibit random-like properties, known as pseudorandom graphs. A prominent class of pseudorandom graphs is that of spectral expanders, or $(n, d, \\lambda)$-graphs. An $(n, d, \\lambda)$-graph is defined as a $d$-regular graph on $n$ vertices where the second largest eigenvalue in absolute value is at most $\\lambda$. The expander mixing lemma shows that $\\lambda$ governs the edge distribution: a smaller $\\lambda$ implies that the edge distribution of $G$ closely resembles that of the random graph $G(n, d/n)$. For a detailed introduction, we refer the reader to the survey by Krivelevich and Sudakov~\\cite{Krivelevich-Sudakov-survey}.\n\nWe focus our attention on the case where $G$ is an $(n,d,\\lambda)$-graph. Let $\\Gamma$ denote the subgraph on the vertex set of $G$ induced by the trace of the random walk. A fundamental question concerns the structural properties of $\\Gamma$, particularly its connectivity. This inquiry is intrinsically linked to the \\emph{cover time}, $T(G)$, defined as the expected number of steps required to visit every vertex, maximized over all possible starting vertices. Equivalently, $T(G)$ corresponds to the expectation of the hitting time for the property that $\\Gamma$ becomes connected. The cover time is a classical parameter in the study of random walks. A seminal result by Feige~\\cite{Feige-lower} establishes a universal lower bound of $(1-o(1))n\\log n$ for any connected graph on $n$ vertices. For Erd\\H{o}s-R\\'{e}nyi graph, random regular graph and more generally random graph with a fixed degree sequence, asymptotically sharp bounds have been proven~\\cite{Adullah-Copper-Frieze-degree-sequence,Cooper-Frieze-sparse-random, Cooper-Frieze-random-regular,Jonasson, Cooper-Frieze-Lubetzky}. In particular, Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al} demonstrated that for random graphs $G(n,p)$ and $\\varepsilon>0$, there exists a constant $C_{\\varepsilon}$ such that if $p \\ge C_{\\varepsilon}\\log n/n$, the trace $\\Gamma$ generated by a random walk of length $(1+\\varepsilon)n\\log n$ is not merely connected, but Hamiltonian with high probability.\n\nGiven that $(n,d,\\lambda)$-graphs exhibit quasi-random behavior analogous to the random graph $G(n, d/n)$, it is natural to inquire whether the structural properties of the random walk trace keep in this setting. Specifically, we ask whether a random walk of length $(1+\\varepsilon)n\\log n$ yields a Hamiltonian trace on an $(n,d,\\lambda)$-graph. This question was raised by Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al}.  \nRecall that the connectivity threshold for $G(n,p)$ is $(1+ o(1))\\log n/n$ where as for the trace to be Hamiltonian, one needs $C_\\varepsilon\\log /n$ where $C_\\varepsilon$ is a big constant depending on $\\varepsilon$. For $(n,d,\\lambda)$-graph, the connectivity holds when $d/\\lambda > 1$. Thus, analogously, one would expect the answer holds when $d/\\lambda > C_\\varepsilon$ for some big constant $C_\\varepsilon$.\n\n\\subsection{Our results}\nBy the expander mixing lemma, $(n,d,\\lambda)$-graphs represent a class of graphs with expansion properties. The asymptotic behavior of random walks in this setting was investigated in the foundational work of Broder and Karlin~\\cite{Broder-Kralin}, who established that for any $d$-regular expander $G$, the cover time satisfies $T(G) = \\Theta(n\\log n)$ \\footnote{The notion of expander in~\\cite{Broder-Kralin} is different, but it applies to the $(n,d,\\lambda)$ setting.}. Our first contribution is a refinement of this bound for $(n,d,\\lambda)$-graphs. We demonstrate that, provided the spectral ratio is sufficiently large, the cover time is asymptotically optimal: $T(G) = (1+o(1))n\\log n$.\n\\begin{proposition}\\label{prop:cover-time}\n    For every $\\varepsilon >0$, there exists $C = C(\\varepsilon)$ such that if $G$ be an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$, then $T(G) \\leq (1+\\varepsilon) n\\log n$.\n\\end{proposition}\n\nWe now introduce a strengthrening of the cover time, termed the \\emph{strong cover time}. Let $ST(G)$ denote the minimum integer $t$ such that a simple random walk of length $t$ visits every vertex of $G$ with high probability (abbrviated as \\textbf{whp} throughout the paper), uniformly over all starting vertices. Note that by definition, $ST(G)$ provides a high-probability guarantee, whereas $T(G)$ captures only the expectation. \nDeterministically $ST(G)\\geq T(G)$.\nIn the context of random graphs $G(n,p)$, Jonasson~\\cite{Jonasson} established that $ST(G)= (1+o(1))n\\log n$ provided that $p = \\omega(\\log^3n/n)$. This result was subsequently refined by Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al}, who extended the bound to the asymptotically optimal $p = \\omega(\\log n/n)$. We consider the problem analogous to the $(n,d,\\lambda)$-graphs.", "context": "Let $G$ be a simple undirected connected graph. A \\emph{simple random walk} on $G$ is a stochastic process where, starting from a given vertex, each step consists of moving to a neighbor chosen uniformly at random. The set of edges traversed by this walk is called the \\emph{trace}. Random walks are fundamental objects in combinatorics, probability theory, and theoretical computer science. For a comprehensive treatment, we refer the reader to the textbook by Levin and Peres~\\cite{Levin-Peres-book} or the survey by Lov\\'{a}sz~\\cite{Lovasz-survey}.\n\nIn this paper, we study the graph formed by the trace of a simple random walk on a finite graph. This line of research was initiated by Frieze, Krivelevich, Michaeli, and Peled in~\\cite{Frieze-et-al}, who focused on the case where the base graph is random. Motivated by these results, in~\\cite{Frieze-et-al}, they asked to extend the inquiry to deterministic graphs that exhibit random-like properties, known as pseudorandom graphs. A prominent class of pseudorandom graphs is that of spectral expanders, or $(n, d, \\lambda)$-graphs. An $(n, d, \\lambda)$-graph is defined as a $d$-regular graph on $n$ vertices where the second largest eigenvalue in absolute value is at most $\\lambda$. The expander mixing lemma shows that $\\lambda$ governs the edge distribution: a smaller $\\lambda$ implies that the edge distribution of $G$ closely resembles that of the random graph $G(n, d/n)$. For a detailed introduction, we refer the reader to the survey by Krivelevich and Sudakov~\\cite{Krivelevich-Sudakov-survey}.\n\nWe focus our attention on the case where $G$ is an $(n,d,\\lambda)$-graph. Let $\\Gamma$ denote the subgraph on the vertex set of $G$ induced by the trace of the random walk. A fundamental question concerns the structural properties of $\\Gamma$, particularly its connectivity. This inquiry is intrinsically linked to the \\emph{cover time}, $T(G)$, defined as the expected number of steps required to visit every vertex, maximized over all possible starting vertices. Equivalently, $T(G)$ corresponds to the expectation of the hitting time for the property that $\\Gamma$ becomes connected. The cover time is a classical parameter in the study of random walks. A seminal result by Feige~\\cite{Feige-lower} establishes a universal lower bound of $(1-o(1))n\\log n$ for any connected graph on $n$ vertices. For Erd\\H{o}s-R\\'{e}nyi graph, random regular graph and more generally random graph with a fixed degree sequence, asymptotically sharp bounds have been proven~\\cite{Adullah-Copper-Frieze-degree-sequence,Cooper-Frieze-sparse-random, Cooper-Frieze-random-regular,Jonasson, Cooper-Frieze-Lubetzky}. In particular, Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al} demonstrated that for random graphs $G(n,p)$ and $\\varepsilon>0$, there exists a constant $C_{\\varepsilon}$ such that if $p \\ge C_{\\varepsilon}\\log n/n$, the trace $\\Gamma$ generated by a random walk of length $(1+\\varepsilon)n\\log n$ is not merely connected, but Hamiltonian with high probability.\n\nGiven that $(n,d,\\lambda)$-graphs exhibit quasi-random behavior analogous to the random graph $G(n, d/n)$, it is natural to inquire whether the structural properties of the random walk trace keep in this setting. Specifically, we ask whether a random walk of length $(1+\\varepsilon)n\\log n$ yields a Hamiltonian trace on an $(n,d,\\lambda)$-graph. This question was raised by Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al}. \nRecall that the connectivity threshold for $G(n,p)$ is $(1+ o(1))\\log n/n$ where as for the trace to be Hamiltonian, one needs $C_\\varepsilon\\log /n$ where $C_\\varepsilon$ is a big constant depending on $\\varepsilon$. For $(n,d,\\lambda)$-graph, the connectivity holds when $d/\\lambda > 1$. Thus, analogously, one would expect the answer holds when $d/\\lambda > C_\\varepsilon$ for some big constant $C_\\varepsilon$.\n\n\\subsection{Our results}\nBy the expander mixing lemma, $(n,d,\\lambda)$-graphs represent a class of graphs with expansion properties. The asymptotic behavior of random walks in this setting was investigated in the foundational work of Broder and Karlin~\\cite{Broder-Kralin}, who established that for any $d$-regular expander $G$, the cover time satisfies $T(G) = \\Theta(n\\log n)$ \\footnote{The notion of expander in~\\cite{Broder-Kralin} is different, but it applies to the $(n,d,\\lambda)$ setting.}. Our first contribution is a refinement of this bound for $(n,d,\\lambda)$-graphs. We demonstrate that, provided the spectral ratio is sufficiently large, the cover time is asymptotically optimal: $T(G) = (1+o(1))n\\log n$.\n\\begin{proposition}\\label{prop:cover-time}\n For every $\\varepsilon >0$, there exists $C = C(\\varepsilon)$ such that if $G$ be an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$, then $T(G) \\leq (1+\\varepsilon) n\\log n$.\n\\end{proposition}\n\nWe now introduce a strengthrening of the cover time, termed the \\emph{strong cover time}. Let $ST(G)$ denote the minimum integer $t$ such that a simple random walk of length $t$ visits every vertex of $G$ with high probability (abbrviated as \\textbf{whp} throughout the paper), uniformly over all starting vertices. Note that by definition, $ST(G)$ provides a high-probability guarantee, whereas $T(G)$ captures only the expectation. \nDeterministically $ST(G)\\geq T(G)$.\nIn the context of random graphs $G(n,p)$, Jonasson~\\cite{Jonasson} established that $ST(G)= (1+o(1))n\\log n$ provided that $p = \\omega(\\log^3n/n)$. This result was subsequently refined by Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al}, who extended the bound to the asymptotically optimal $p = \\omega(\\log n/n)$. We consider the problem analogous to the $(n,d,\\lambda)$-graphs.", "full_context": "Let $G$ be a simple undirected connected graph. A \\emph{simple random walk} on $G$ is a stochastic process where, starting from a given vertex, each step consists of moving to a neighbor chosen uniformly at random. The set of edges traversed by this walk is called the \\emph{trace}. Random walks are fundamental objects in combinatorics, probability theory, and theoretical computer science. For a comprehensive treatment, we refer the reader to the textbook by Levin and Peres~\\cite{Levin-Peres-book} or the survey by Lov\\'{a}sz~\\cite{Lovasz-survey}.\n\nIn this paper, we study the graph formed by the trace of a simple random walk on a finite graph. This line of research was initiated by Frieze, Krivelevich, Michaeli, and Peled in~\\cite{Frieze-et-al}, who focused on the case where the base graph is random. Motivated by these results, in~\\cite{Frieze-et-al}, they asked to extend the inquiry to deterministic graphs that exhibit random-like properties, known as pseudorandom graphs. A prominent class of pseudorandom graphs is that of spectral expanders, or $(n, d, \\lambda)$-graphs. An $(n, d, \\lambda)$-graph is defined as a $d$-regular graph on $n$ vertices where the second largest eigenvalue in absolute value is at most $\\lambda$. The expander mixing lemma shows that $\\lambda$ governs the edge distribution: a smaller $\\lambda$ implies that the edge distribution of $G$ closely resembles that of the random graph $G(n, d/n)$. For a detailed introduction, we refer the reader to the survey by Krivelevich and Sudakov~\\cite{Krivelevich-Sudakov-survey}.\n\nWe focus our attention on the case where $G$ is an $(n,d,\\lambda)$-graph. Let $\\Gamma$ denote the subgraph on the vertex set of $G$ induced by the trace of the random walk. A fundamental question concerns the structural properties of $\\Gamma$, particularly its connectivity. This inquiry is intrinsically linked to the \\emph{cover time}, $T(G)$, defined as the expected number of steps required to visit every vertex, maximized over all possible starting vertices. Equivalently, $T(G)$ corresponds to the expectation of the hitting time for the property that $\\Gamma$ becomes connected. The cover time is a classical parameter in the study of random walks. A seminal result by Feige~\\cite{Feige-lower} establishes a universal lower bound of $(1-o(1))n\\log n$ for any connected graph on $n$ vertices. For Erd\\H{o}s-R\\'{e}nyi graph, random regular graph and more generally random graph with a fixed degree sequence, asymptotically sharp bounds have been proven~\\cite{Adullah-Copper-Frieze-degree-sequence,Cooper-Frieze-sparse-random, Cooper-Frieze-random-regular,Jonasson, Cooper-Frieze-Lubetzky}. In particular, Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al} demonstrated that for random graphs $G(n,p)$ and $\\varepsilon>0$, there exists a constant $C_{\\varepsilon}$ such that if $p \\ge C_{\\varepsilon}\\log n/n$, the trace $\\Gamma$ generated by a random walk of length $(1+\\varepsilon)n\\log n$ is not merely connected, but Hamiltonian with high probability.\n\nGiven that $(n,d,\\lambda)$-graphs exhibit quasi-random behavior analogous to the random graph $G(n, d/n)$, it is natural to inquire whether the structural properties of the random walk trace keep in this setting. Specifically, we ask whether a random walk of length $(1+\\varepsilon)n\\log n$ yields a Hamiltonian trace on an $(n,d,\\lambda)$-graph. This question was raised by Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al}. \nRecall that the connectivity threshold for $G(n,p)$ is $(1+ o(1))\\log n/n$ where as for the trace to be Hamiltonian, one needs $C_\\varepsilon\\log /n$ where $C_\\varepsilon$ is a big constant depending on $\\varepsilon$. For $(n,d,\\lambda)$-graph, the connectivity holds when $d/\\lambda > 1$. Thus, analogously, one would expect the answer holds when $d/\\lambda > C_\\varepsilon$ for some big constant $C_\\varepsilon$.\n\n\\subsection{Our results}\nBy the expander mixing lemma, $(n,d,\\lambda)$-graphs represent a class of graphs with expansion properties. The asymptotic behavior of random walks in this setting was investigated in the foundational work of Broder and Karlin~\\cite{Broder-Kralin}, who established that for any $d$-regular expander $G$, the cover time satisfies $T(G) = \\Theta(n\\log n)$ \\footnote{The notion of expander in~\\cite{Broder-Kralin} is different, but it applies to the $(n,d,\\lambda)$ setting.}. Our first contribution is a refinement of this bound for $(n,d,\\lambda)$-graphs. We demonstrate that, provided the spectral ratio is sufficiently large, the cover time is asymptotically optimal: $T(G) = (1+o(1))n\\log n$.\n\\begin{proposition}\\label{prop:cover-time}\n For every $\\varepsilon >0$, there exists $C = C(\\varepsilon)$ such that if $G$ be an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$, then $T(G) \\leq (1+\\varepsilon) n\\log n$.\n\\end{proposition}\n\nWe now introduce a strengthrening of the cover time, termed the \\emph{strong cover time}. Let $ST(G)$ denote the minimum integer $t$ such that a simple random walk of length $t$ visits every vertex of $G$ with high probability (abbrviated as \\textbf{whp} throughout the paper), uniformly over all starting vertices. Note that by definition, $ST(G)$ provides a high-probability guarantee, whereas $T(G)$ captures only the expectation. \nDeterministically $ST(G)\\geq T(G)$.\nIn the context of random graphs $G(n,p)$, Jonasson~\\cite{Jonasson} established that $ST(G)= (1+o(1))n\\log n$ provided that $p = \\omega(\\log^3n/n)$. This result was subsequently refined by Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al}, who extended the bound to the asymptotically optimal $p = \\omega(\\log n/n)$. We consider the problem analogous to the $(n,d,\\lambda)$-graphs.\n\n\\begin{theorem}\\label{thm:main}\n Let $\\varepsilon > 0$. There exists a constant $C = C(\\eps) >0$ such that the following holds: Let $G$ be an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$ and $L = (1+\\eps) n\\log n$. \\textbf{whp} for every vertex $v\\in V(G)$, the trace of a simple random walk starting at $v$ of length $L$ is Hamiltonian.\n\\end{theorem}\n\nIt is well known that a random $d$-regular graph for any $d\\geq 3$ is \\textbf{whp} an $(n,d,\\lambda)$-graph for $\\lambda = O(\\sqrt{d})$~\\cite{Friedman}. We thus obtain the following corollary.\n\\begin{corollary}\\label{cor:random-d-regular}\n Let $\\varepsilon > 0$. There exists $d_0 = d_0(\\eps)$ that the following holds: Let $G$ be a random $d$-regular graph with $d\\geq d_0$ and let $L = (1+\\eps) n\\log n$. \\textbf{whp} $G$ satisfies that \\textbf{whp} for every vertex $v\\in V(G)$, the trace of a simple random walk starting at $v$ of length $L$ is Hamiltonian.\n\\end{corollary}\nWe remark that both results are asymptotically optimal due to the general lower bound on the cover time. Our proof strategy is as follows.\nFirst, since the second largest eigenvalue of the adjacency matrix of an $(n,d,\\lambda)$-graph is bounded away from $d$, we can control the transitional probability going from any vertex $u$ to another vertex $v$ at step $t$ by using the spectral decomposition of the transition matrix.\nWe show that after $(1+\\varepsilon)n\\log n$ steps, each vertex is visited $\\Theta(\\log n)$ times with high probability. Then, we use these visit counts to establish the expansion properties of the trace graph $\\Gamma$, which ensures the existence of a Hamiltonian cycle.\n\n\\subsection{Number of visits}\nFor a random walk on a graph $G$ of length $L$ and a vertex $v\\in V(G)$, let $\\gamma(v)$ denote the number of times the random walk visits $v$.\nThe goal of this section is to prove the following:\n\\begin{lemma}\\label{lem:number-of-vists}\n Let $\\varepsilon > 0$. There exists constants $C = C(\\eps) >0$ and $\\rho = \\rho(\\eps) > 0$ such that the following holds: Let $G$ be a $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$ and $L = (1+\\eps) n\\log n$. Consider a random walk on $G$ starting from an arbitrary vertex and of length $L$. Then, \\textbf{whp} $\\gamma(v)\\geq \\rho \\log n$ for every vertex $v\\in V(G)$.\n\\end{lemma}\nWe begin by establishing an auxiliary lemma, giving a lower bound on the probability a random walk starting from $u$ reaches $v$ in $n/\\sqrt{C}$ steps:\n\\begin{lemma}\\label{lem:lower-bound-return}\n Let $\\varepsilon > 0$. There exists a constant $C = C(\\eps) >0$ such that the following holds: Let $G$ be a $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$ and $T = n/\\sqrt{C}$. Then, for any pair of distinct vertices $u,v\\in V(G)$, the probability a simple random walk $X$ starting from $u$ reaches $v$ at least once within $T$ steps is at least $(1-0.1\\eps)/\\sqrt{C}$.\n\\end{lemma}\n\\begin{proof}\n Let $N$ denote the number of visits to $v$ in the first $T$ steps and $I_t$ the indicator random variable for $X_t = v$ so that $N = \\sum_{t = 1}^T I_i$. By linearity of expectation, we have \n \\[\n \\E[N] = \\sum_{t=1}^T \\Pr(X_t = v|X_0 = u) = \\sum_{t=1}^T (P^t)_{u,v}\\,.\n \\]\n Recall that the transition matrix $P = D^{-1}A$ and let $d = \\lambda_1\\geq \\dots\\geq \\lambda_n$ are eigenvalues of $A$. Define $S = D^{1/2}PD^{-1/2} = D^{-1/2}AD^{-1/2}$. Since $S$ is a symmetric real square matrix, its spectral decomposition can be written as $S = \\sum_{i=1}^n\\mu_i\\phi_i \\phi_i^T$, where $\\mu_i = \\lambda_i/d\\in [-1,1]$ are eigenvalues of $S$ and $\\{\\phi_i\\}_{i=1}^n$ forms an orthonormal basis. \n Observe that the stationary distribution of $X$ is $\\pi = (1/n,\\dots, 1/n)$, $\\mu_1 = 1$ and $\\phi_1(v) = (1/\\sqrt{n},\\dots, 1/\\sqrt{n})$.\n\nWe have the following immediate corollary for the stationary walk:\n\\begin{corollary}\\label{cor:stationary-walk}\n Let $\\varepsilon > 0$. There exists a constant $C = C(\\eps) >0$ such that the following holds: Let $G$ be a $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$ and $T = n/\\sqrt{C}$. Then, for any vertex $v\\in V(G)$, the probability a stationary walk visits $v$ at least once within $T$ steps is at least $(1-0.1\\eps)/\\sqrt{C}$.\n\\end{corollary}\n\\begin{proof}\n Recall that the stationary distribution for a $d$-regular graph is the uniform distribution over all vertices $(1/n,\\dots, 1/n)$. Also, the stationary walk $Y$ and the simple random walk $X$ have the same distribution conditionally on having the same starting vertex. \n Since~\\Cref{lem:lower-bound-return} holds for every starting vertex $u\\in V(G)$, the conclusion follows by integrating over the starting vertex. \n\\end{proof}\n\n\\subsection{C-expander}\nThe goal of this section is to prove the following:\n\\begin{lemma}\\label{lem:C-expander}\n Let $\\varepsilon > 0$. There exists a constant $C = C(\\eps) >0$ such that the following holds: Let $G$ be a $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$ and $L = (1+\\eps) n\\log n$. \n Let $\\Gamma$ be the trace graph obtained by a random walk of length $L$ starting from any vertex. Then,\n \\textbf{whp} $\\Gamma$ is a $C'$-expander for $C' = \\log \\log C$.\n\\end{lemma}\n\\begin{proof}\n To prove a graph is $C$-expander, we need to verify the expansion and joinedness properties. By~\\Cref{lem:number-of-vists}, there exists a constant $\\rho > 0$ such that $\\gamma(v) \\geq \\rho \\log n$ for all $v\\in V(G)$.\n\n\\section{Concluding remarks}\\label{sec:concluding-remarks}\nOne of the key tools we applied was~\\Cref{thm:Hamiltonicity}, which requires only $C$-expanders. One may wonder if~\\Cref{thm:main} is true with $(n,d,\\lambda)$-graphs replaced by $C$-expanders. This is unfortunately (very) incorrect:\n\\begin{proposition}\n For any $C \\leq 1.1n/\\log n$, there exists a $C$-expander for which the cover time is $a n \\log n$ for some $a > 1$.\n\\end{proposition}\n\\begin{proof}[Proof sketch]\n Consider a complete graph on $n-2$ vertices together with two special vertices $u,v$, each connecting to $C$ vertices of the complete graph and their neighborhoods are disjoint. Clearly, the graph is a $C$-expander. However, note that the expected number of steps needed to go from $u$ to $v$ is at least $1 + (n-1)/C)\\cdot (n-1)> n\\log n$. Indeed, the $1$ comes from going from $u$ to the clique. Then, in each step, it will have at most a $C/(n-1)$ chance of landing in $N(v)$ and thus in expectation $(n-1)/C$ steps are needed to land in $N(v)$. Starting from a vertex in $N(v)$, each time there is a $1/(n-1)$ chance of going to $v$ so in expectation it takes $n-1$ such trials. Clearly, the expected number of steps taken to go from $u$ to $v$ is a lower bound on the cover time, which finishes the proof.\n\\end{proof}\nOne natural strengthening of~\\Cref{thm:main} would be to consider following hitting time problem: Let $\\tau_{HC}(G)$ be the minimum $t\\in \\mathbb N$ for which the trace graph $\\Gamma$ becomes Hamiltonian and $\\tau_{1}$ the minimum $t\\in \\mathbb N$ such that $\\Gamma$ has minimum degree $1$.\nFor $(n,d,\\lambda)$-graphs with $d/\\lambda \\geq C$ or a random $d$-regular graph for $d$ sufficiently large, is it true that \\textbf{whp} $\\tau_{HC} = \\tau_1 + 1$? Note that $\\tau_{HC}\\geq \\tau_1 + 1$ is necessary since a Hamilton cycle uses two incident edges from each vertex.\nFor complete graph the corresponding hitting time result is known~\\cite{Frieze-et-al} .", "post_theorem_intro_text_len": 5118, "post_theorem_intro_text": "Again this is asymptotically optimal by the general lower bound of the cover time. In fact, we obtain~\\Cref{thm:main-strong-cover} as a direct consequence of a stronger result. \n\nThe connection between spectral expansion and the Hamiltonicity in pseudorandom graphs is well-established as in~\\cite{Kriveham,glockham,ferber2024hamiltonicity,chen2025robustness}. A recent breakthrough by Draganić, Montgomery, Munhá Correia, Pokrovskiy and Sudakov~\\cite{Draganic-et-al} demonstrates that any $(n,d,\\lambda)$-graph admits a Hamiltonian cycle, if the spectral ratio $d/\\lambda\\ge C$ for a sufficiently large constant $C$. Drawing a parallel to the trace of random walk landscape, Frieze, Krivelevich, Michaeli, and Peled~\\cite{Frieze-et-al} raised the question of whether the trace of a random walk on $(n,d,\\lambda)$-graphs and random regular graphs exhibits similar Hamiltonicity properties. \nWe answer these questions through the following two main results.\n\n\\begin{theorem}\\label{thm:main}\n    Let $\\varepsilon > 0$. There exists a constant $C = C(\\varepsilon) >0$ such that the following holds: Let $G$ be an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$ and $L = (1+\\varepsilon) n\\log n$. \\textbf{whp} for every vertex $v\\in V(G)$, the trace of a simple random walk starting at $v$ of length $L$ is Hamiltonian.\n\\end{theorem} \n\nWe remark that implicitly in~\\Cref{prop:cover-time},~\\Cref{thm:main-strong-cover} and~\\Cref{thm:main}, $d$ is sufficiently large. This follows from the fact that $\\lambda = \\Omega(d)$ (see~\\cite{Krivelevich-Sudakov-survey}), which means $1/C \\geq \\lambda/d\\geq \\Omega(1/\\sqrt{d})$ and therefore $d$ is sufficiently large as long as $C$ is.\n\nIt is well known that a random $d$-regular graph for any $d\\geq 3$ is \\textbf{whp} an $(n,d,\\lambda)$-graph for $\\lambda = O(\\sqrt{d})$~\\cite{Friedman}. We thus obtain the following corollary.\n\\begin{corollary}\\label{cor:random-d-regular}\n     Let $\\varepsilon > 0$. There exists $d_0 = d_0(\\varepsilon)$ that the following holds: Let $G$ be a random $d$-regular graph with $d\\geq d_0$ and let $L = (1+\\varepsilon) n\\log n$. \\textbf{whp} $G$ satisfies that \\textbf{whp} for every vertex $v\\in V(G)$, the trace of a simple random walk starting at $v$ of length $L$ is Hamiltonian.\n\\end{corollary}\nWe remark that both results are asymptotically optimal due to the general lower bound on the cover time. Our proof strategy is as follows.\nFirst, since the second largest eigenvalue of the adjacency matrix of an $(n,d,\\lambda)$-graph is bounded away from $d$, we can control the transitional probability going from any vertex $u$ to another vertex $v$ at step $t$ by using the spectral decomposition of the transition matrix.\nWe show that after $(1+\\varepsilon)n\\log n$ steps, each vertex is visited $\\Theta(\\log n)$ times with high probability. Then, we use these visit counts to establish the expansion properties of the trace graph $\\Gamma$, which ensures the existence of a Hamiltonian cycle. \n\nCentral to our proof of \\Cref{thm:main} is the lemma that the trace of the random walk visits every vertex with logarithmic frequency. Specifically, \\Cref{lem:number-of-vists} establishes that there exists a constant $\\rho = \\rho(\\varepsilon) > 0$ such that with high probability, every vertex is visited at least $\\rho \\log n$ times. This phenomenon is intimately connected to the concept of \\emph{blanket time} in the probability theory, a parameter introduced by Winkler and Zuckerman~\\cite{Winkler-Zuckerman}. Let $\\pi$ denote the stationary distribution, where $\\pi_v = \\deg(v)/(2|E(G)|)$. For a parameter $\\delta \\in (0,1)$, the $\\delta$-blanket time, denoted $t_{bl}(G,\\delta)$, is defined as the minimum time $t$ required for the walk to visit every vertex $v$ at least $\\delta t \\pi_v$ times. While it is immediate that $t_{bl}(G,\\delta) \\ge T(G)$, Winkler and Zuckerman conjectured that the blanket time is, in fact, within a constant factor of the cover time; that is, $t_{bl}(G,\\delta) \\le C_\\delta T(G)$ for any $0<\\delta < 1$ and any graph $G$. \nKahn, Kim, Lov{\\'a}sz and Vu~\\cite{Kahn-Kim-Lovasz-Vu} showed that $t_{bl}(G,\\delta)= O(T(G)(\\log\\log n)^2)$ before the conjecture was affirmatively settled in a celebrated work of Ding, Lee, and Peres~\\cite{Ding-Lee-Peres}. \nOur results imply that there exists a $\\delta = \\delta(\\varepsilon)\\in (0,1)$ for which $t_{bl}(G,\\delta)\\leq (1+\\varepsilon)n\\log n$, which is the same as the cover time up to a $1+o(1)$ factor. In general, it is unclear for which graphs the same conclusion hold. In the concluding remarks we discuss and propose further questions along this direction.\n\n\\subsection{Paper organization} \n\\Cref{sec:preliminaries} contains the background of random walks and tools we use.~\\Cref{sec:proof} contains the proofs for all results and~\\Cref{sec:concluding-remarks} some remarks and open problems.\n\n\\subsection*{Acknowledgement}\nPart of this work was initiated during the visit of second author to Shanghai Center for Mathematical Sciences (SCMS) and Fudan University. The authors would like to thank Prof.\\ Hehui Wu and his group memebers for their support and hospitality.", "sketch": "They state that \\Cref{thm:main-strong-cover} is obtained “as a direct consequence of a stronger result,” namely \\Cref{thm:main}. The proof strategy outlined is:\n\\begin{itemize}\n\\item Use that “the second largest eigenvalue of the adjacency matrix of an $(n,d,\\lambda)$-graph is bounded away from $d$” to “control the transitional probability going from any vertex $u$ to another vertex $v$ at step $t$ by using the spectral decomposition of the transition matrix.”\n\\item Show that “after $(1+\\varepsilon)n\\log n$ steps, each vertex is visited $\\Theta(\\log n)$ times with high probability.” (Central is the lemma that “with high probability, every vertex is visited at least $\\rho\\log n$ times,” i.e. \\Cref{lem:number-of-vists}.)\n\\item “Use these visit counts to establish the expansion properties of the trace graph $\\Gamma$, which ensures the existence of a Hamiltonian cycle.” Since \\Cref{thm:main-strong-cover} follows directly from this stronger Hamiltonicity statement, it follows as a corollary.\n\\end{itemize}\nThey further connect the logarithmic visit-frequency lemma to “blanket time,” noting their results imply the existence of $\\delta=\\delta(\\varepsilon)\\in(0,1)$ with $t_{bl}(G,\\delta)\\le (1+\\varepsilon)n\\log n$.", "expanded_sketch": "They state that \\Cref{thm:main-strong-cover} is obtained “as a direct consequence of a stronger result,” namely the stronger Hamiltonicity statement.\n\nThe proof strategy outlined is:\n\\begin{itemize}\n\\item Use that “the second largest eigenvalue of the adjacency matrix of an $(n,d,\\lambda)$-graph is bounded away from $d$” to “control the transitional probability going from any vertex $u$ to another vertex $v$ at step $t$ by using the spectral decomposition of the transition matrix.”\n\\item Show that “after $(1+\\varepsilon)n\\log n$ steps, each vertex is visited $\\Theta(\\log n)$ times with high probability.” Central is the following lemma.\n\n\\begin{lemma}\\label{lem:number-of-vists}\n    Let $\\varepsilon > 0$. There exists constants $C = C(\\eps) >0$ and $\\rho = \\rho(\\eps) > 0$ such that the following holds: Let $G$ be a $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$ and $L = (1+\\eps) n\\log n$. Consider a random walk on $G$ starting from an arbitrary vertex and of length $L$. Then, \\textbf{whp} $\\gamma(v)\\geq \\rho \\log n$ for every vertex $v\\in V(G)$.\n\\end{lemma}\n\n\\item “Use these visit counts to establish the expansion properties of the trace graph $\\Gamma$, which ensures the existence of a Hamiltonian cycle.” In establishing the main theorem, this yields Hamiltonicity of the trace, and hence the stated bound on $ST(G)$ follows as a corollary.\n\\end{itemize}\nThey further connect the logarithmic visit-frequency lemma to “blanket time,” noting their results imply the existence of $\\delta=\\delta(\\varepsilon)\\in(0,1)$ with $t_{bl}(G,\\delta)\\le (1+\\varepsilon)n\\log n$.", "expanded_theorem": "\\label{thm:main-strong-cover}\n    Let $\\varepsilon > 0$. There exists a constant $C = C(\\varepsilon)>0$ such that if $G$ is an $(n,d,\\lambda)$-graph with $d/\\lambda \\geq C$, then $ST(G) \\leq (1+\\varepsilon)n\\log n$.", "theorem_type": ["Existential–Universal", "Implication"], "mcq": {"question": "Let \\(\\varepsilon>0\\). An \\((n,d,\\lambda)\\)-graph is a \\(d\\)-regular graph on \\(n\\) vertices whose second-largest eigenvalue in absolute value is at most \\(\\lambda\\). Let \\(ST(G)\\) denote the strong cover time of \\(G\\), i.e. the least integer \\(t\\) such that a simple random walk of length \\(t\\), started from any vertex of \\(G\\), covers all vertices with high probability. Which conclusion is valid under the assumption that \\(G\\) is an \\((n,d,\\lambda)\\)-graph with \\(d/\\lambda\\ge C\\) for a suitable constant \\(C=C(\\varepsilon)>0\\)?", "correct_choice": {"label": "A", "text": "For every \\(\\varepsilon>0\\), there exists a constant \\(C=C(\\varepsilon)>0\\) such that if \\(G\\) is an \\((n,d,\\lambda)\\)-graph with \\(d/\\lambda\\ge C\\), then \\(ST(G)\\le (1+\\varepsilon)n\\log n\\)."}, "choices": [{"label": "B", "text": "For every \\(\\varepsilon>0\\), there exists a constant \\(C=C(\\varepsilon)>0\\) such that if \\(G\\) is an \\((n,d,\\lambda)\\)-graph with \\(d/\\lambda\\ge C\\), then for every starting vertex, a simple random walk of length exactly \\((1+\\varepsilon)n\\log n\\) visits each vertex at least once with high probability."}, {"label": "C", "text": "For every \\(\\varepsilon>0\\), there exists a constant \\(C=C(\\varepsilon)>0\\) such that if \\(G\\) is an \\((n,d,\\lambda)\\)-graph with \\(d/\\lambda\\ge C\\), then the ordinary cover time of \\(G\\) is at most \\((1+\\varepsilon)n\\log n\\)."}, {"label": "D", "text": "There exists a universal constant \\(C>0\\) such that for every \\(\\varepsilon>0\\), if \\(G\\) is an \\((n,d,\\lambda)\\)-graph with \\(d/\\lambda\\ge C\\), then \\(ST(G)\\le (1+\\varepsilon)n\\log n\\)."}, {"label": "E", "text": "For every \\(\\varepsilon>0\\), there exists a constant \\(C=C(\\varepsilon)>0\\) such that if \\(G\\) is an \\((n,d,\\lambda)\\)-graph with \\(d/\\lambda\\ge C\\), then after \\((1+\\varepsilon)n\\log n\\) steps, a simple random walk started from any vertex visits every vertex at least \\(\\rho\\log n\\) times with high probability for some constant \\(\\rho=\\rho(\\varepsilon)>0\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "least_time_vs_exact_time", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "strong_cover_replaced_by_cover_time", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "characteristic", "tampered_component": "dependence_of_C_on_epsilon", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "lower_visit_bound_misread_as_each_vertex_visited_rho_log_n_times", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the notation and asks which statement is true, but it does not reveal the theorem’s conclusion or single out choice A. There is no explicit answer leakage beyond necessary terminology."}, "TAS": {"score": 1, "justification": "The correct option is essentially the theorem statement itself, so the item is close to theorem recall. However, the alternatives vary quantifiers, exact-time vs least-time formulations, and strong vs ordinary cover time, so it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "Selecting the answer requires some reasoning about subtle logical differences between closely related statements, especially quantifier dependence and stronger/weaker variants. Still, it mainly tests recognition of the exact theorem rather than generating a conclusion from mathematical work."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they probe common confusions about stronger statements, weakened statements, dependence on epsilon, and fixed-time vs minimal-time formulations. They are distinct and well-targeted."}, "total_score": 6, "overall_assessment": "A solid MCQ with no real answer leakage and strong distractors, but it leans toward precise theorem recognition rather than deeper generative reasoning."}}
{"id": "2602.11045v1", "paper_link": "http://arxiv.org/abs/2602.11045v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "[\\textsc{Weighted Khintchine for manifolds}]\\label{thm: dream0}\nLet $\\mathcal{M}$ be any nondegenerate submanifold of $\\mathbb{R}^n$ and $\\psi_1,\\dots,\\psi_n:\\mathbb{R}_+\\to(0,1)$ be non-increasing. Then almost all points on $\\mathcal{M}$ are $(\\psi_1,\\dots,\\psi_n)$-approximable if the series\n\\begin{equation}\\label{mainsum}\n\\sum_{q=1}^\\infty\\psi_1(q)\\cdots \\psi_n(q)\n\\end{equation}\ndiverges, and are not $(\\psi_1,\\dots,\\psi_n)$-approximable if \\eqref{mainsum} converges.", "start_pos": 13460, "end_pos": 13928, "label": "thm: dream0"}, "ref_dict": {"thm:mult0": "\\begin{theorem}[\\textsc{Multiplicative convergence for manifolds}]\\label{thm:mult0}Let $\\cM$ be any nondegenerate submanifold of $\\R^n$ and $\\psi:\\Rp\\to(0,1)$ be non-increasing. Then almost all points on $\\cM$ are not multiplicatively\n$\\psi$-approximable if\n$$\n\\sum_{q=1}^\\infty \\psi(q) (\\log q)^{n-1}<\\infty.\n$$\n\\end{theorem}", "s:004": "\\begin{equation}\\label{s:004}\n  \\cS_n(\\psi_1,\\dots,\\psi_n)\\subset\\cSM_n(\\psi)\\quad\\text{if}\\quad\\psi\\ge\\psi_1\\cdots\\psi_n\\,.\n\\end{equation}", "mainsum": "\\begin{equation}\\label{mainsum}\n\\sum_{q=1}^\\infty\\psi_1(q)\\cdots \\psi_n(q)\n\\end{equation}", "thm: dream0": "\\begin{theorem}[\\textsc{Weighted Khintchine for manifolds}]\\label{thm: dream0}\nLet $\\cM$ be any nondegenerate submanifold of $\\R^n$ and $\\psi_1,\\dots,\\psi_n:\\Rp\\to(0,1)$ be non-increasing. Then almost all points on $\\cM$ are $(\\psi_1,\\dots,\\psi_n)$-approximable if the series\n\\begin{equation}\\label{mainsum}\n\\sum_{q=1}^\\infty\\psi_1(q)\\cdots \\psi_n(q)\n\\end{equation}\ndiverges, and are not $(\\psi_1,\\dots,\\psi_n)$-approximable if \\eqref{mainsum} converges.\n\\end{theorem}", "new BKM": "\\begin{equation}\\label{eq: bound on derivatives}\n   \\max_{k}\\max_{i,j} \\sup_{\\bx\\in \\bU}\\vert\\partial_{ij}f_{k}(\\bx)\\vert\\leq M.\n\\end{equation}\n\n\\subsection{Quantitative Nondivergence}\\label{new BKM} \n\nOur proofs will require an effective measure estimate for the following set \n\\begin{align}\n\\mathfrak{S}_{F}&(\\delta,\\bK,\\bT)=\\nonumber\\\\[1ex]\n&=\\left\\{\\bx\\in \\bU:\\exists\\;(a_0,\\ba)\\in\\Z\\times\\Z^n_{\\neq\\bf0}\\;\\;\\text{such that }\\left.\n\\begin{array}{l}\n|a_0+F(\\bx)\\ba^\\top|<\\delta\\\\[1ex]\n\\vert \\partial_iF(\\bx)\\ba^\\top\\vert<K_i\\;\\;(1\\le i\\le d) \\\\[1ex]\n\\vert a_k\\vert<T_k ~~(1\\leq k\\leq n) \n\\end{array}\n\\right.\\right\\},\\label{eq: new set}\n\\end{align}\nwhere $\\ba=(a_1,\\dots,a_n)$, $\\bT=(T_1,\\dots,T_n)$, $\\bK=(K_1,\\dots,K_d)$, and $\\delta,K_1,\\cdots,K_d,T_1,\\cdots, T_n$ are positive real parameters.\n\nThe following result represents a generalisation of \\cite[Theorem 1.4]{BKM}\nand appears in \\cite[Theorem 3.1]{BD} when $K_1=\\dots=K_d$. Its proof in the general case follows the same lines as that of \\cite[Theorem 3.1]{BD} with minor modifications which will be explained below.\n\n\\begin{theorem}\\label{BKM_new2}\nLet $F:\\bU\\to\\R^n$ be as defined in \\eqref{eq: F}, $l$-nondegenerate at $\\bx_0$. \nThen there exists a ball $B_0\\subset\\bU$ centred at $\\bx_0$ and a constant $E>0$ depending on $B_0$ and $F$ such that for any choice of $\\delta,K_1,\\cdots,K_d,T_1,\\cdots, T_n$ satisfying \n\\begin{equation}\\label{eqn5.2}\n0<\\delta\\le 1,\\qquad T_1,\\dots,T_n\\ge1,\\qquad K_1,\\cdots,K_d>0 \\quad\\text{and}\\quad \\delta^{n}<K \\frac{T_1\\cdots T_n}{\\max_i T_i},\n\\end{equation}", "sv:005": "\\begin{equation}\\label{sv:005}\n\\cSM_n(\\psi)=\\R^n  \\hspace{9mm}  {\\rm  if } \\hspace{9mm} q\\psi(q)\\ge 1 \\quad\\text{infinitely often} .\n\\end{equation}"}, "pre_theorem_intro_text_len": 5052, "pre_theorem_intro_text": "One classical theme in the theory of Diophantine approximation is to quantify how well rational points \n$$\n\\mathbf{p}/q=(p_1/q,\\dots,p_n/q)\\,,\n$$ \nwith $q\\in\\mathbb{N}$ and $\\mathbf{p}\\in\\mathbb{Z}^n$, approximate real points $\\mathbf{x}=(x_1,\\dots,x_n)\\in\\mathbb{R}^n$.\nIn the standard setting, the error of approximations is measured using a fixed norm on $\\mathbb{R}^n$, usually the sup-norm \n$$\n\\|\\mathbf{x}\\|=\\max_{1\\le i\\le n}|x_i|\\,.\n$$\nThus one seeks the same rate of approximation in each coordinate, imposing identical bounds on the quantities $|x_i-p_i/q|$.\n\nIn this paper we consider two more general frameworks: \\emph{weighted} and \\emph{multiplicative}, both of which can be seen as important special cases of the more general theory of Diophantine approximation with star bodies \\cite{MR2264601}. In the weighted case, different coordinates may be approximated at different rates; in the multiplicative case, one specifies a bound on the product\n$\\prod_{i=1}^n |x_i - p_i/q|$.\nWe now introduce the relevant notation and terminology.\n\nLet $\\mathbb{R}_+$ denote the set of positive real numbers, and let $\\psi,\\psi_1,\\dots,\\psi_n:\\mathbb{R}_+\\to(0,1)$ be nonincreasing  functions, referred to as {\\em approximation functions}.\nDefine $\\mathcal{S}_n(\\psi_1,\\dots, \\psi_n)$ as the set of $\\mathbf{x}\\in\\mathbb{R}^n$ such that the system\n\\begin{equation}\\label{eq1.1}\n\\vert qx_i-p_i\\vert<\\psi_i(q)\\qquad (1\\leq i\\leq n) \n\\end{equation}\nholds for infinitely many $(q,p_1,\\dots,p_n)\\in\\mathbb{N}\\times\\mathbb{Z}^n$. When $\\psi_1=\\dots=\\psi_n=\\psi$ this set will be denoted by $\\cS_n(\\psi)$. A point $\\mathbf{x}\\in\\cS_n(\\psi_1,\\dots,\\psi_n)$ will be called {\\em $(\\psi_1,\\dots,\\psi_n)$--approximable} ({\\em $\\psi$-approximable} when $\\psi_1=\\dots=\\psi_n=\\psi$).\n\nIn the multiplicative case,\ndefine $\\cSM_n(\\psi)$ as  the set of $\\mathbf{x}\\in\\mathbb{R}^n$ such that\n$$  \\prod_{i=1}^n |qx_i-p_i| \\ < \\ \\psi(q)\n$$  \nholds for infinitely many $(q,p_1,\\dots,p_n)\\in\\mathbb{N}\\times\\mathbb{Z}^n$. The points in $\\cSM_n(\\psi)$ will be called \n{\\em multiplicatively\n$\\psi$--approximable}.\nIt is readily verified that\n\\begin{equation}\\label{s:004}\n  \\cS_n(\\psi_1,\\dots,\\psi_n)\\subset\\cSM_n(\\psi)\\quad\\text{if}\\quad\\psi\\ge\\psi_1\\cdots\\psi_n\\,.\n\\end{equation}\n\n\\noindent By Minkowski's linear forms theorem,\n\\begin{equation}\\label{sv:004}\n\\cS_n(\\psi_1,\\dots,\\psi_n)=\\mathbb{R}^n \\hspace{9mm}  {\\rm  if } \\hspace{9mm}\nq\\psi_1(q)\\cdots\\psi_n(q)\\ge 1 \\quad\\text{infinitely often}\\,.\n\\end{equation}\nThis together with \\eqref{s:004} implies that\n\\begin{equation}\\label{sv:005}\n\\cSM_n(\\psi)=\\mathbb{R}^n  \\hspace{9mm}  {\\rm  if } \\hspace{9mm} q\\psi(q)\\ge 1 \\quad\\text{infinitely often} .\n\\end{equation}\nLittlewood's conjecture, a famous open problem dating back to the 1930s, asserts that the condition in \\eqref{sv:005} can be weakened to \n$$\\psi(q)\\ge \\varepsilon q^{-1}\\qquad\\text{for any $\\varepsilon>0$},\n$$\nsee \\cite{EKL} and \\cite{PVActa}. However, in the weighted case, \nsuch a weakening is generally impossible for the existence\nof badly approximable points \\cite{MR2581371, MR2231044}, e.g. when \n$$\n\\psi_i(q)=q^{-\\tau_i}\\,,\\qquad \\tau_i\\ge0,\\quad\\tau_1+\\dots+\\tau_n=1\\,.\n$$\nHere $\\mathbf{x}\\in\\mathbb{R}^n$ is {\\em $(\\psi_1,\\dots,\\psi_n)$-badly approximable} if $\\mathbf{x}\\not\\in\\cS_n(c\\psi_1,\\dots,c\\psi_n)$ for some $c>0$. Nevertheless, the set of $(\\psi_1,\\dots,\\psi_n)$-badly approximable points is well known to have zero measure \\cite{BadNull}.\n\nIn this paper we will be concerned with the measure of the sets $\\cS_n(\\psi_1,\\dots,\\psi_n)$ and $\\cSM_n(\\psi)$ \nrestricted to submanifolds of $\\mathbb{R}^n$ when $\\psi_1,\\dots,\\psi_n,\\psi$ are generic non-increasing functions. \nWe begin by recalling the corresponding results for $\\mathbb{R}^n$. Let $\\mu_n$ denote Lebesgue measure on $\\mathbb{R}^n$.\n\n\\smallskip\n\n\\begin{thkh}\n\\label{gs} Let   $\\psi_1,\\dots,\\psi_n$ be non-increasing functions. Then\n$$\\mu_{n}\\left(\\cS_n(\\psi_1,\\dots,\\psi_n)\\right) =\\left\\{\\begin{array}{ll} \\mbox{\\rm\nZ{\\scriptsize ERO}} & {\\rm if} \\;\\;\\; \\sum \\;  \\psi_1(q)\\cdots \\psi_n(q)\n\\;\\; <\\infty\\,,\\\\ &\n\\\\[-1ex] \\mbox{\\rm F{\\scriptsize ULL}} & {\\rm if} \\;\\;\\; \\sum  \\;  \\psi_1(q)\\cdots \\psi_n(q) \\;\\;\n =\\infty\\,. \\; \\;\n\\end{array}\\right.$$\n\\end{thkh}\n\n\\smallskip \n\n\\begin{thga}\n\\label{gm} Let   $\\psi$ be a non-increasing function. Then\n$$\\mu_{n}\\left(\\cSM_n(\\psi)\\right) =\\left\\{\\begin{array}{ll} \\mbox{\\rm\nZ{\\scriptsize ERO}} & {\\rm if} \\;\\;\\; \\sum \\;  \\psi(q) \\ (\\log\nq)^{n-1} \\;\\; < \\ \\infty\\,,\\\\ &\n\\\\[-1ex] \\mbox{\\rm F{\\scriptsize ULL}} & {\\rm if} \\;\\;\\; \\sum  \\;  \\psi(q) \\ (\\log\nq)^{n-1}  \\;\\;\n = \\ \\infty\\,. \\; \\;\n\\end{array}\\right.$$\n\\end{thga}\n\n\\noindent Here `\\mbox{\\rm F{\\scriptsize ULL}}' means that the complement has  measure zero. Theorem~K was originally established by Khintchine  \n\\cite{Kh} when $\\psi_1=\\dots=\\psi_n$, and both results as stated above were obtained by Gallagher in \\cite{gal}.\n\nOur main results extend Theorem~K and the convergence part of Theorem~G to nondegenerate submanifolds of $\\mathbb{R}^n$ as defined in \\cite{KM}. \n\n\\smallskip", "context": "Let $\\mathbb{R}_+$ denote the set of positive real numbers, and let $\\psi,\\psi_1,\\dots,\\psi_n:\\mathbb{R}_+\\to(0,1)$ be nonincreasing functions, referred to as {\\em approximation functions}.\nDefine $\\mathcal{S}_n(\\psi_1,\\dots, \\psi_n)$ as the set of $\\mathbf{x}\\in\\mathbb{R}^n$ such that the system\n\\begin{equation}\\label{eq1.1}\n\\vert qx_i-p_i\\vert<\\psi_i(q)\\qquad (1\\leq i\\leq n) \n\\end{equation}\nholds for infinitely many $(q,p_1,\\dots,p_n)\\in\\mathbb{N}\\times\\mathbb{Z}^n$. When $\\psi_1=\\dots=\\psi_n=\\psi$ this set will be denoted by $\\cS_n(\\psi)$. A point $\\mathbf{x}\\in\\cS_n(\\psi_1,\\dots,\\psi_n)$ will be called {\\em $(\\psi_1,\\dots,\\psi_n)$--approximable} ({\\em $\\psi$-approximable} when $\\psi_1=\\dots=\\psi_n=\\psi$).\n\nIn the multiplicative case,\ndefine $\\cSM_n(\\psi)$ as the set of $\\mathbf{x}\\in\\mathbb{R}^n$ such that\n$$ \\prod_{i=1}^n |qx_i-p_i| \\ < \\ \\psi(q)\n$$ \nholds for infinitely many $(q,p_1,\\dots,p_n)\\in\\mathbb{N}\\times\\mathbb{Z}^n$. The points in $\\cSM_n(\\psi)$ will be called \n{\\em multiplicatively\n$\\psi$--approximable}.\nIt is readily verified that\n\\begin{equation}\\label{s:004}\n \\cS_n(\\psi_1,\\dots,\\psi_n)\\subset\\cSM_n(\\psi)\\quad\\text{if}\\quad\\psi\\ge\\psi_1\\cdots\\psi_n\\,.\n\\end{equation}\n\n\\noindent By Minkowski's linear forms theorem,\n\\begin{equation}\\label{sv:004}\n\\cS_n(\\psi_1,\\dots,\\psi_n)=\\mathbb{R}^n \\hspace{9mm} {\\rm if } \\hspace{9mm}\nq\\psi_1(q)\\cdots\\psi_n(q)\\ge 1 \\quad\\text{infinitely often}\\,.\n\\end{equation}\nThis together with \\eqref{s:004} implies that\n\\begin{equation}\\label{sv:005}\n\\cSM_n(\\psi)=\\mathbb{R}^n \\hspace{9mm} {\\rm if } \\hspace{9mm} q\\psi(q)\\ge 1 \\quad\\text{infinitely often} .\n\\end{equation}\nLittlewood's conjecture, a famous open problem dating back to the 1930s, asserts that the condition in \\eqref{sv:005} can be weakened to \n$$\\psi(q)\\ge \\varepsilon q^{-1}\\qquad\\text{for any $\\varepsilon>0$},\n$$\nsee \\cite{EKL} and \\cite{PVActa}. However, in the weighted case, \nsuch a weakening is generally impossible for the existence\nof badly approximable points \\cite{MR2581371, MR2231044}, e.g. when \n$$\n\\psi_i(q)=q^{-\\tau_i}\\,,\\qquad \\tau_i\\ge0,\\quad\\tau_1+\\dots+\\tau_n=1\\,.\n$$\nHere $\\mathbf{x}\\in\\mathbb{R}^n$ is {\\em $(\\psi_1,\\dots,\\psi_n)$-badly approximable} if $\\mathbf{x}\\not\\in\\cS_n(c\\psi_1,\\dots,c\\psi_n)$ for some $c>0$. Nevertheless, the set of $(\\psi_1,\\dots,\\psi_n)$-badly approximable points is well known to have zero measure \\cite{BadNull}.\n\n\\begin{thkh}\n\\label{gs} Let $\\psi_1,\\dots,\\psi_n$ be non-increasing functions. Then\n$$\\mu_{n}\\left(\\cS_n(\\psi_1,\\dots,\\psi_n)\\right) =\\left\\{\\begin{array}{ll} \\mbox{\\rm\nZ{\\scriptsize ERO}} & {\\rm if} \\;\\;\\; \\sum \\; \\psi_1(q)\\cdots \\psi_n(q)\n\\;\\; <\\infty\\,,\\\\ &\n\\\\[-1ex] \\mbox{\\rm F{\\scriptsize ULL}} & {\\rm if} \\;\\;\\; \\sum \\; \\psi_1(q)\\cdots \\psi_n(q) \\;\\;\n =\\infty\\,. \\; \\;\n\\end{array}\\right.$$\n\\end{thkh}\n\nOur main results extend Theorem~K and the convergence part of Theorem~G to nondegenerate submanifolds of $\\mathbb{R}^n$ as defined in \\cite{KM}.\n\n\\smallskip\n\n\\begin{equation}\\label{mainsum}\n\\sum_{q=1}^\\infty\\psi_1(q)\\cdots \\psi_n(q)\n\\end{equation}\n\n\\begin{equation}\\label{s:004}\n  \\cS_n(\\psi_1,\\dots,\\psi_n)\\subset\\cSM_n(\\psi)\\quad\\text{if}\\quad\\psi\\ge\\psi_1\\cdots\\psi_n\\,.\n\\end{equation}\n\n\\begin{equation}\\label{sv:005}\n\\cSM_n(\\psi)=\\R^n  \\hspace{9mm}  {\\rm  if } \\hspace{9mm} q\\psi(q)\\ge 1 \\quad\\text{infinitely often} .\n\\end{equation}", "full_context": "Let $\\mathbb{R}_+$ denote the set of positive real numbers, and let $\\psi,\\psi_1,\\dots,\\psi_n:\\mathbb{R}_+\\to(0,1)$ be nonincreasing functions, referred to as {\\em approximation functions}.\nDefine $\\mathcal{S}_n(\\psi_1,\\dots, \\psi_n)$ as the set of $\\mathbf{x}\\in\\mathbb{R}^n$ such that the system\n\\begin{equation}\\label{eq1.1}\n\\vert qx_i-p_i\\vert<\\psi_i(q)\\qquad (1\\leq i\\leq n) \n\\end{equation}\nholds for infinitely many $(q,p_1,\\dots,p_n)\\in\\mathbb{N}\\times\\mathbb{Z}^n$. When $\\psi_1=\\dots=\\psi_n=\\psi$ this set will be denoted by $\\cS_n(\\psi)$. A point $\\mathbf{x}\\in\\cS_n(\\psi_1,\\dots,\\psi_n)$ will be called {\\em $(\\psi_1,\\dots,\\psi_n)$--approximable} ({\\em $\\psi$-approximable} when $\\psi_1=\\dots=\\psi_n=\\psi$).\n\nIn the multiplicative case,\ndefine $\\cSM_n(\\psi)$ as the set of $\\mathbf{x}\\in\\mathbb{R}^n$ such that\n$$ \\prod_{i=1}^n |qx_i-p_i| \\ < \\ \\psi(q)\n$$ \nholds for infinitely many $(q,p_1,\\dots,p_n)\\in\\mathbb{N}\\times\\mathbb{Z}^n$. The points in $\\cSM_n(\\psi)$ will be called \n{\\em multiplicatively\n$\\psi$--approximable}.\nIt is readily verified that\n\\begin{equation}\\label{s:004}\n \\cS_n(\\psi_1,\\dots,\\psi_n)\\subset\\cSM_n(\\psi)\\quad\\text{if}\\quad\\psi\\ge\\psi_1\\cdots\\psi_n\\,.\n\\end{equation}\n\n\\noindent By Minkowski's linear forms theorem,\n\\begin{equation}\\label{sv:004}\n\\cS_n(\\psi_1,\\dots,\\psi_n)=\\mathbb{R}^n \\hspace{9mm} {\\rm if } \\hspace{9mm}\nq\\psi_1(q)\\cdots\\psi_n(q)\\ge 1 \\quad\\text{infinitely often}\\,.\n\\end{equation}\nThis together with \\eqref{s:004} implies that\n\\begin{equation}\\label{sv:005}\n\\cSM_n(\\psi)=\\mathbb{R}^n \\hspace{9mm} {\\rm if } \\hspace{9mm} q\\psi(q)\\ge 1 \\quad\\text{infinitely often} .\n\\end{equation}\nLittlewood's conjecture, a famous open problem dating back to the 1930s, asserts that the condition in \\eqref{sv:005} can be weakened to \n$$\\psi(q)\\ge \\varepsilon q^{-1}\\qquad\\text{for any $\\varepsilon>0$},\n$$\nsee \\cite{EKL} and \\cite{PVActa}. However, in the weighted case, \nsuch a weakening is generally impossible for the existence\nof badly approximable points \\cite{MR2581371, MR2231044}, e.g. when \n$$\n\\psi_i(q)=q^{-\\tau_i}\\,,\\qquad \\tau_i\\ge0,\\quad\\tau_1+\\dots+\\tau_n=1\\,.\n$$\nHere $\\mathbf{x}\\in\\mathbb{R}^n$ is {\\em $(\\psi_1,\\dots,\\psi_n)$-badly approximable} if $\\mathbf{x}\\not\\in\\cS_n(c\\psi_1,\\dots,c\\psi_n)$ for some $c>0$. Nevertheless, the set of $(\\psi_1,\\dots,\\psi_n)$-badly approximable points is well known to have zero measure \\cite{BadNull}.\n\n\\begin{thkh}\n\\label{gs} Let $\\psi_1,\\dots,\\psi_n$ be non-increasing functions. Then\n$$\\mu_{n}\\left(\\cS_n(\\psi_1,\\dots,\\psi_n)\\right) =\\left\\{\\begin{array}{ll} \\mbox{\\rm\nZ{\\scriptsize ERO}} & {\\rm if} \\;\\;\\; \\sum \\; \\psi_1(q)\\cdots \\psi_n(q)\n\\;\\; <\\infty\\,,\\\\ &\n\\\\[-1ex] \\mbox{\\rm F{\\scriptsize ULL}} & {\\rm if} \\;\\;\\; \\sum \\; \\psi_1(q)\\cdots \\psi_n(q) \\;\\;\n =\\infty\\,. \\; \\;\n\\end{array}\\right.$$\n\\end{thkh}\n\nOur main results extend Theorem~K and the convergence part of Theorem~G to nondegenerate submanifolds of $\\mathbb{R}^n$ as defined in \\cite{KM}.\n\n\\smallskip\n\n\\begin{equation}\\label{mainsum}\n\\sum_{q=1}^\\infty\\psi_1(q)\\cdots \\psi_n(q)\n\\end{equation}\n\n\\begin{equation}\\label{s:004}\n  \\cS_n(\\psi_1,\\dots,\\psi_n)\\subset\\cSM_n(\\psi)\\quad\\text{if}\\quad\\psi\\ge\\psi_1\\cdots\\psi_n\\,.\n\\end{equation}\n\n\\begin{equation}\\label{sv:005}\n\\cSM_n(\\psi)=\\R^n  \\hspace{9mm}  {\\rm  if } \\hspace{9mm} q\\psi(q)\\ge 1 \\quad\\text{infinitely often} .\n\\end{equation}\n\n\\smallskip\n\n\\medskip\n\n\\begin{theorem}[\\textsc{Multiplicative convergence for manifolds}]\\label{thm:mult0}Let $\\cM$ be any nondegenerate submanifold of $\\R^n$ and $\\psi:\\Rp\\to(0,1)$ be non-increasing. Then almost all points on $\\cM$ are not multiplicatively\n$\\psi$-approximable if\n$$\n\\sum_{q=1}^\\infty \\psi(q) (\\log q)^{n-1}<\\infty.\n$$\n\\end{theorem}\n\n\\begin{theorem}\\label{thm: dream}\nLet $F:\\bU\\to\\R^n$ be nondegenerate, $\\bU$ be an open subset of $\\R^d$, $\\psi_1,\\dots,\\psi_n:\\Rp\\to(0,1)$ be non-increasing. Then \\begin{equation}\n \\mu_d\\left(F^{-1}\\mathcal{S}_n(\\psi_1,\\dots, \\psi_n)\\right)=\\left\\{\\begin{aligned}\n & 0 &\\text{ if } &\\textstyle\\sum\\psi_1(q)\\cdots \\psi_n(q)<\\infty\\,,\\\\[1ex]\n & \\mu_d(\\bU) &\\text{ if } &\\textstyle\\sum \\psi_1(q)\\cdots \\psi_n(q)=\\infty\\,.\n \\end{aligned}\\right.\n \\end{equation}\n\\end{theorem}\n\nWith this proposition at hand, justifying \\eqref{eq: prod is greater than} becomes simple. We take $\\Phi(q)=q^{-1-\\mathfrak{c}}$ and define $\\psi'_i$ from the proposition. Then\n\\begin{equation}\\label{eq2.13}\n\\sum_{q=1}^\\infty \\psi'_{1}(q)\\cdots\\psi'_{n}(q)\n\\leq \\sum_{q=1}^\\infty \\psi_{1}(q)\\cdots\\psi_{n}(q)\n+\\sum_{q=1}^\\infty q^{-1-\\mathfrak{c}}\n<\\infty\n\\end{equation}\nand\n$$\n\\mathcal{S}_n(\\psi_1,\\dots,\\psi_n)\n\\subset \\mathcal{S}_n(\\psi_1',\\dots,\\psi_n').\n$$\nHence we may work with the set\n$\\mathcal{S}_n(\\psi_1',\\dots,\\psi_n')$\ninstead of $\\mathcal{S}_n(\\psi_1,\\dots,\\psi_n)$,\nor equivalently we may assume \\eqref{eq: prod is greater than}.\n\nLet $\\psi_1,\\dots,\\psi_n$ be approximation functions satisfying the divergence sum condition in Theorem~\\ref{thm: dream}. Then\n\\begin{equation}\\label{eq3.20}\n\\sum_{t=1}^\\infty 2^t\\psi_1(2^t)\\cdots\\psi_n(2^t)=\\infty.\n\\end{equation}\nAs explained in \\S \\ref{sec: con on psi div}, we assume $\\psi_i$ satisfy \\eqref{eq: psi chain}, without loss of generality.\n\nFix any $0<s'<\\frac{1}{n-1}$. Then, by \\eqref{eq3.20}, there exists a strictly increasing sequence of positive integers $(t_k)_{k\\in\\N}$ such that\n\\begin{equation}\\label{eq:lower bound on prod} \n\\psi_{1}(q)\\cdots \\psi_{n}(q) q>q^{-\\left(\\frac{1}{n-1}-s'\\right)}\\qquad\\text{for each } q=2^{t_k}\n\\end{equation}\nand \n\\begin{equation}\\label{eq3.22}\n\\sum_{k=1}^\\infty 2^{t_k}\\psi_1(2^{t_k})\\cdots \\psi_n(2^{t_k})=\\infty.\n\\end{equation}\nIn particular, we have that for all $k\\in\\N$\n\\begin{equation}\\label{eq: standing 2 in 2^t}\n2^{t_k}\\psi_{2}(2^{t_k})\\cdots \\psi_n(2^{t_k})>2^{st_k},\\qquad s:=\\frac{n-1}{n}s'.\n\\end{equation}\nDefine $\\mathcal{T}:=\\{t_k:k\\in\\N\\}$, which is thus an infinite collection. Split $\\mathcal{T}$ into two sub-collections:\n$$\\mathcal{T}_1:=\\{t\\in\\mathcal{T}: \n2^{t}\\psi_{1}(2^{t})\\cdots \\psi_n(2^{t})<1\\}\\qquad\\text{and}\\qquad \n\\mathcal{T}_2=\\mathcal{T}\\setminus \\mathcal{T}_1.\n$$\n\nFrom Lemma \\ref{lem: ubiqui is enough}, it is enough to show that one of the sets on the left side of \\eqref{eq: enough} has \\textit{full} measure. In what follows $\\brho=(\\rho_i)_{i=1}^d\\Rp^d\\to\\Rp$.\nBy the definition of $\\Delta$ in \\eqref{def: Delta},\n$$\\begin{aligned}\\Delta(2^t, \\tfrac12(\\psi_{d+j}(2^t))_{j=1}^m, B, \\brho(2^t)):=\\bigcup_{(q,\\ba,\\bb)\\in \\mathcal{R}(2^t, (\\psi_{d+j}(2^t))_{j=1}^m, B)} \\tDelta(\\vv a/q,\\brho(2^t)),\n\\end{aligned}\n$$ \nfor any ball $B\\subset B_0,$ where $B_0$ be as in Proposition \\ref{thm: ubiq main}. \nNow note that $$\\begin{aligned}\n \\bigcup_{(q,\\ba,\\bb)\\in \\mathcal{R}(2^t, (\\psi_{d+j}(2^t))_{j=1}^m, B)} \\tDelta(\\vv a/q,\\brho(2^t))\\cap B\n \\subset \\bigcup_{2^{t-1}\\leq q\\leq 2^{t}, (q,\\ba)\\in J }\\tDelta(\\vv a/q,\\brho(2^t))\\cap B.\\end{aligned}$$\nBy \\eqref{eq3.22}, there is $i\\in\\{1,2\\}$ such that\n\\begin{equation}\\label{eq3.25}\n\\sum_{t\\in \\mathcal{T}_i} 2^t \\psi_1(2^t)\\cdots\\psi_n(2^t)=\\infty,\n\\end{equation}\n\n\\begin{equation}\\label{mainsum}\n\\sum_{q=1}^\\infty\\psi_1(q)\\cdots \\psi_n(q)\n\\end{equation}", "post_theorem_intro_text_len": 6634, "post_theorem_intro_text": "\\medskip\n\n\\begin{theorem}[\\textsc{Multiplicative convergence for manifolds}]\\label{thm:mult0}Let $\\mathcal{M}$ be any nondegenerate submanifold of $\\mathbb{R}^n$ and $\\psi:\\mathbb{R}_+\\to(0,1)$ be non-increasing. Then almost all points on $\\mathcal{M}$ are not multiplicatively\n$\\psi$-approximable if\n$$\n\\sum_{q=1}^\\infty \\psi(q) (\\log q)^{n-1}<\\infty.\n$$\n\\end{theorem}\n\nTheorem \\ref{thm: dream0} is first proved for the case when the manifold is parametrized in the so-called Monge form in \\S \\ref{sec: div} (for divergence) and \\S \\ref{sec: convergence} (for convergence). The proof is then generalized to arbitrary  nondegenerate manifolds in \\S \\ref{sec: general}. The proof of  Theorem~\\ref{thm:mult0} is presented in \\S \\ref{sec: multi}.\n\n\\subsection{Previous results and comparison}\n\nWe first discuss the unweighted case ($\\psi_1=\\cdots=\\psi_n$) of Theorem~\\ref{thm: dream0}. In this setting, Theorem~\\ref{thm: dream0} was first established for nondegenerate $C^3$ planar curves ($n=2$) in \\cite{BDV07} in the divergence case and in \\cite{VV06} in the convergence case. The $C^3$ assumption was later removed in \\cite{BZ10} (divergence) and \\cite{Hua15} (convergence). In higher dimensions, the divergence case was proved by the first-named author in \\cite{Ber12} for analytic nondegenerate manifolds and was recently extended to all nondegenerate manifolds in \\cite{BD}, having previously been known only for nondegenerate curves \\cite{BVVZ21}. The convergence case was settled in \\cite{BY} for all nondegenerate manifolds.\n\nMuch less is known in the weighted case, which allows the approximation functions $\\psi_1,\\dots,\\psi_n$ to be different. The story again begins with planar curves, studied in \\cite{BV07}, where the authors proved both the convergence and divergence cases of Theorem~\\ref{thm: dream0}: the divergence case for $C^3$ curves and the convergence case for rational quadrics. The latter was later extended to arbitrary $C^3$ nondegenerate planar curves in \\cite{Bad_Lev_07}.\n\nIn higher dimensions, the general problem was posed in \\cite[Problem~S2]{BV07} as follows.\n\n\\smallskip\n\n\\noindent\\textbf{Problem~1:} Given a nondegenerate manifold $\\mathcal{M}$ and approximation functions $\\psi_1,\\dots,\\psi_n$, find the weakest condition under which $\\mathcal{S}_n(\\psi_1,\\dots,\\psi_n)\\cap\\mathcal{M}$ has Lebesgue measure zero.\n\n\\smallskip\n\nWorking towards this general problem, Srivastava \\cite{Srivastava2025} recently established the convergence case of Theorem~\\ref{thm: dream0} for a subclass of nondegenerate manifolds with a curvature condition, under the additional constraint $\\psi_1=\\cdots=\\psi_d$ on the approximation functions. To the best of our knowledge, the divergence part of Theorem~\\ref{thm: dream0} for $n>2$ and arbitrary approximation functions $\\psi_1,\\dots,\\psi_n$ has not been previously addressed. Theorem~\\ref{thm: dream0} resolves Problem~1 in full and, at the same time, generalises the results of \\cite{BY} and \\cite{BD} from equal to arbitrary approximation functions $\\psi_1,\\dots,\\psi_n$.\n\n\\medskip\n\nWe now turn to the multiplicative case, concerning the set $\\cSM_n(\\psi)$. In this setting, progress to date has been far more limited, and even for planar curves the theory remains incomplete. Indeed, for nondegenerate planar curves, Theorem~\\ref{thm:mult0} was proved in \\cite{Hua15}, having been previously established for rational quadrics \\cite{BV07} and under the $C^3$ assumption \\cite{Bad_Lev_07}.\n\nAs with Problem~1, the general question regarding $\\cSM_n(\\psi)$ was posed in \\cite[Problem~S2]{BV07}:\n\n\\smallskip\n\n\\noindent\\textbf{Problem~2:} Given a nondegenerate manifold $\\mathcal{M}$ and an approximation function $\\psi$, find the weakest condition under which $\\cSM_n(\\psi)\\cap\\mathcal{M}$ has Lebesgue measure zero.\n\n\\smallskip\n\nTheorem~\\ref{thm:mult0} contributes towards the resolution of this problem by providing an analogue of the convergence case of Theorem~G for all nondegenerate manifolds.\n\nA natural next step would be to prove the divergence case of Theorem~G for nondegenerate manifolds. This remains a major open problem and has so far been addressed only for lines in $\\mathbb{R}^2$ \\cite{MR4068301, MR3813593, MR4701882} and for ``vertical lines'' in higher dimensions \\cite{MR4016056, MR4706444}. Even these limited cases required substantial new ideas, including the development of a Bohr sets technique \\cite{MR3813593} and an effective asymptotic equidistribution method \\cite{MR4701882}.\n\nFinally, we note that our results may be viewed as complete analogues of the corresponding results for dual approximation on manifolds. In that setting, the convergence case for both weighted and multiplicative approximation was established by Bernik, Kleinbock and Margulis \\cite{BKM}, while the divergence case for weighted approximation was proved in \\cite{MR2989975}.\n\n\\subsection{Further remarks}\n\nFor a manifold $\\mathcal{M} \\subset \\mathbb{R}^n$ of dimension $d$ to be nondegenerate, it is necessary that $\\mathcal{M}$ be $C^l$ with ${d+l \\choose l} \\ge n+1$. For example, if $d=2$, this reduces to $l^2 + 3l \\ge 2n$. Clearly there exist manifolds that are just $C^l$ for the smallest $l$ satisfying this inequality. The above theorems use this minimal smoothness, sufficient to meet the nondegeneracy condition.\n\nThe proof of Theorem~\\ref{thm: dream0} builds on the linearization techniques developed in \\cite{BY} and \\cite{BD}. The new features include linearization in different coordinates with different rates, including linearization in a single direction only. For the divergence case, we use the recent \\textit{ubiquity} result of Kleinbock and Wang \\cite{KWang23}. To construct an appropriate ubiquitous system, we provide a version of the quantitative estimate of Bernik, Kleinbock, and Margulis \\cite{BKM} that enables independent control of different partial derivatives (see \\S \\ref{new BKM}). At the technical level, our proof differs from the previous argument even in the \\textit{unweighted} case in \\cite{BD}, offering a new perspective.\n\nFinally, we note that, while preparing this paper, we became aware of related work by Chow, Srivastava, Technau, and Yu \\cite{MultCSTY} concerning Problem~2. Using a different approach initiated in \\cite{SST}, they establish a multiplicative convergence result for \\(C^\\infty\\) nondegenerate manifolds and a natural class of affine subspaces.\nWe emphasize that both Problem~1 and Problem~2 remain of interest and are still widely open for degenerate manifolds, such as affine subspaces and submanifolds that are nondegenerate relative to an affine subspace, as considered in \\cite{MR1982150}.", "sketch": "Theorem~\\ref{thm: dream0} is proved first “for the case when the manifold is parametrized in the so-called Monge form” (\\S\\,\\ref{sec: div} for divergence and \\S\\,\\ref{sec: convergence} for convergence), and “then generalized to arbitrary nondegenerate manifolds” in \\S\\,\\ref{sec: general}. The proof “builds on the linearization techniques developed in \\cite{BY} and \\cite{BD},” with “new features” including “linearization in different coordinates with different rates, including linearization in a single direction only.” For the divergence case, the argument uses “the recent \\textit{ubiquity} result of Kleinbock and Wang \\cite{KWang23}.” To “construct an appropriate ubiquitous system,” the authors “provide a version of the quantitative estimate of Bernik, Kleinbock, and Margulis \\cite{BKM} that enables independent control of different partial derivatives” (see \\S\\,\\ref{new BKM}). They also note that “at the technical level, our proof differs from the previous argument even in the \\textit{unweighted} case in \\cite{BD}, offering a new perspective.”", "expanded_sketch": "To prove the main theorem, the authors proceed first in the case when the manifold is parametrized in the so-called Monge form (for divergence they argue earlier, and for convergence they argue later), and then generalize to arbitrary nondegenerate manifolds later. The proof builds on the linearization techniques developed in \n\n\\subsection{Quantitative Nondivergence}\\label{new BKM} \n\nOur proofs will require an effective measure estimate for the following set \n\\begin{align}\n\\mathfrak{S}_{F}&(\\delta,\\bK,\\bT)=\\nonumber\\\\[1ex]\n&=\\left\\{\\bx\\in \\bU:\\exists\\;(a_0,\\ba)\\in\\Z\\times\\Z^n_{\\neq\\bf0}\\;\\;\\text{such that }\\left.\n\\begin{array}{l}\n|a_0+F(\\bx)\\ba^\\top|<\\delta\\\\[1ex]\n\\vert \\partial_iF(\\bx)\\ba^\\top\\vert<K_i\\;\\;(1\\le i\\le d) \\\\[1ex]\n\\vert a_k\\vert<T_k ~~(1\\leq k\\leq n) \n\\end{array}\n\\right.\\right\\},\\label{eq: new set}\n\\end{align}\nwhere $\\ba=(a_1,\\dots,a_n)$, $\\bT=(T_1,\\dots,T_n)$, $\\bK=(K_1,\\dots,K_d)$, and $\\delta,K_1,\\cdots,K_d,T_1,\\cdots, T_n$ are positive real parameters.\n\nThe following result represents a generalisation of \\cite[Theorem 1.4]{BKM}\nand appears in \\cite[Theorem 3.1]{BD} when $K_1=\\dots=K_d$. Its proof in the general case follows the same lines as that of \\cite[Theorem 3.1]{BD} with minor modifications which will be explained below.\n\n\\begin{theorem}\\label{BKM_new2}\nLet $F:\\bU\\to\\R^n$ be as defined in \\eqref{eq: F}, $l$-nondegenerate at $\\bx_0$. \nThen there exists a ball $B_0\\subset\\bU$ centred at $\\bx_0$ and a constant $E>0$ depending on $B_0$ and $F$ such that for any choice of $\\delta,K_1,\\cdots,K_d,T_1,\\cdots, T_n$ satisfying \n\\begin{equation}\\label{eqn5.2}\n0<\\delta\\le 1,\\qquad T_1,\\dots,T_n\\ge1,\\qquad K_1,\\cdots,K_d>0 \\quad\\text{and}\\quad \\delta^{n}<K \\frac{T_1\\cdots T_n}{\\max_i T_i},\n\\end{equation}\n\nas well as on new features including linearization in different coordinates with different rates, including linearization in a single direction only. For the divergence case, the argument uses the recent ubiquity result of Kleinbock and Wang \\cite{KWang23}. To construct an appropriate ubiquitous system, the authors provide a version of the quantitative estimate of Bernik, Kleinbock, and Margulis \\cite{BKM} that enables independent control of different partial derivatives (as in the quantitative nondivergence subsection above). They also note that at the technical level, their proof differs from the previous argument even in the unweighted case in \\cite{BD}, offering a new perspective.", "expanded_theorem": "[\\textsc{Weighted Khintchine for manifolds}]\\label{thm: dream0}\nLet $\\mathcal{M}$ be any nondegenerate submanifold of $\\mathbb{R}^n$ and $\\psi_1,\\dots,\\psi_n:\\mathbb{R}_+\\to(0,1)$ be non-increasing. Then almost all points on $\\mathcal{M}$ are $(\\psi_1,\\dots,\\psi_n)$-approximable if the series\n\\begin{equation}\\label{mainsum}\n\\sum_{q=1}^\\infty\\psi_1(q)\\cdots \\psi_n(q)\n\\end{equation}\ndiverges, and are not $(\\psi_1,\\dots,\\psi_n)$-approximable if the equation above converges.", "theorem_type": ["Biconditional or Equivalence", "Implication"], "mcq": {"question": "Let \\(\\mathcal M\\subset \\mathbb R^n\\) be a nondegenerate submanifold of \\(\\mathbb R^n\\) (in the standard metric Diophantine approximation sense), and let \\(\\psi_1,\\dots,\\psi_n:\\mathbb R_+\\to(0,1)\\) be non-increasing functions. A point \\(\\mathbf x=(x_1,\\dots,x_n)\\in\\mathbb R^n\\) is called \\((\\psi_1,\\dots,\\psi_n)\\)-approximable if there exist infinitely many tuples \\((q,p_1,\\dots,p_n)\\in\\mathbb N\\times\\mathbb Z^n\\) such that\n\\[\n|q x_i-p_i|<\\psi_i(q)\\qquad (1\\le i\\le n).\n\\]\nWhich statement about the set of points of \\(\\mathcal M\\) with this property is valid, where “almost all” is with respect to the natural Lebesgue measure on \\(\\mathcal M\\)?", "correct_choice": {"label": "A", "text": "If\n\\[\n\\sum_{q=1}^\\infty \\psi_1(q)\\cdots \\psi_n(q)=\\infty,\n\\]\nthen almost all points on \\(\\mathcal M\\) are \\((\\psi_1,\\dots,\\psi_n)\\)-approximable; and if\n\\[\n\\sum_{q=1}^\\infty \\psi_1(q)\\cdots \\psi_n(q)<\\infty,\n\\]\nthen almost all points on \\(\\mathcal M\\) are not \\((\\psi_1,\\dots,\\psi_n)\\)-approximable."}, "choices": [{"label": "B", "text": "If\n\\[\n\\sum_{q=1}^\\infty \\psi_1(q)\\cdots \\psi_n(q)=\\infty,\n\\]\nthen almost all points on \\(\\mathcal M\\) are \\((\\psi_1,\\dots,\\psi_n)\\)-approximable; and if\n\\[\n\\sum_{q=1}^\\infty q\\,\\psi_1(q)\\cdots \\psi_n(q)<\\infty,\n\\]\nthen almost all points on \\(\\mathcal M\\) are not \\((\\psi_1,\\dots,\\psi_n)\\)-approximable."}, {"label": "C", "text": "If\n\\[\n\\sum_{q=1}^\\infty \\psi_1(q)\\cdots \\psi_n(q)<\\infty,\n\\]\nthen almost all points on \\(\\mathcal M\\) are not \\((\\psi_1,\\dots,\\psi_n)\\)-approximable."}, {"label": "D", "text": "If\n\\[\n\\sum_{q=1}^\\infty \\psi_1(q)\\cdots \\psi_n(q)=\\infty,\n\\]\nthen every point of \\(\\mathcal M\\) is \\((\\psi_1,\\dots,\\psi_n)\\)-approximable; and if\n\\[\n\\sum_{q=1}^\\infty \\psi_1(q)\\cdots \\psi_n(q)<\\infty,\n\\]\nthen almost all points on \\(\\mathcal M\\) are not \\((\\psi_1,\\dots,\\psi_n)\\)-approximable."}, {"label": "E", "text": "If for each \\(i\\) one has\n\\[\n\\sum_{q=1}^\\infty \\psi_i(q)=\\infty,\n\\]\nthen almost all points on \\(\\mathcal M\\) are \\((\\psi_1,\\dots,\\psi_n)\\)-approximable; and if for some \\(i\\) one has\n\\[\n\\sum_{q=1}^\\infty \\psi_i(q)<\\infty,\n\\]\nthen almost all points on \\(\\mathcal M\\) are not \\((\\psi_1,\\dots,\\psi_n)\\)-approximable."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "quantitative_nondevergence", "tampered_component": "critical summand uses product \\(\\psi_1\\cdots\\psi_n\\) with no extra \\(q\\)-factor", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped the divergence/full-measure direction", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "ubiquity", "tampered_component": "almost-all conclusion strengthened to every point", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "product criterion replaced by coordinatewise series criteria", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the notion of approximability but does not disclose the critical convergence/divergence criterion or otherwise point directly to choice A."}, "TAS": {"score": 0, "justification": "The item is essentially asking for the statement of the theorem itself: the correct measure-theoretic criterion is reproduced almost verbatim in the correct option."}, "GPS": {"score": 1, "justification": "Some discrimination is required because the distractors reflect nearby variants (extra q-weight, one-sided result, added technical condition), but the task is still mainly theorem recall rather than genuine derivation or synthesis."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: B uses a common wrong critical sum, C is a weaker true statement, D adds an unjustified technical restriction, and E replaces the series condition with an overly strong pointwise threshold."}, "total_score": 5, "overall_assessment": "A mathematically well-constructed recall question with strong distractors, but it is largely a direct theorem-statement identification rather than a high-generative-reasoning MCQ."}}
{"id": "2602.11045v1", "paper_link": "http://arxiv.org/abs/2602.11045v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "[\\textsc{Weighted Khintchine for manifolds}]\\label{thm: dream0}\nLet $\\mathcal{M}$ be any nondegenerate submanifold of $\\mathbb{R}^n$ and $\\psi_1,\\dots,\\psi_n:\\mathbb{R}_+\\to(0,1)$ be non-increasing. Then almost all points on $\\mathcal{M}$ are $(\\psi_1,\\dots,\\psi_n)$-approximable if the series\n\\begin{equation}\\label{mainsum}\n\\sum_{q=1}^\\infty\\psi_1(q)\\cdots \\psi_n(q)\n\\end{equation}\ndiverges, and are not $(\\psi_1,\\dots,\\psi_n)$-approximable if \\eqref{mainsum} converges.", "start_pos": 13460, "end_pos": 13928, "label": "thm: dream0"}, "ref_dict": {"thm:mult0": "\\begin{theorem}[\\textsc{Multiplicative convergence for manifolds}]\\label{thm:mult0}Let $\\cM$ be any nondegenerate submanifold of $\\R^n$ and $\\psi:\\Rp\\to(0,1)$ be non-increasing. Then almost all points on $\\cM$ are not multiplicatively\n$\\psi$-approximable if\n$$\n\\sum_{q=1}^\\infty \\psi(q) (\\log q)^{n-1}<\\infty.\n$$\n\\end{theorem}", "s:004": "\\begin{equation}\\label{s:004}\n  \\cS_n(\\psi_1,\\dots,\\psi_n)\\subset\\cSM_n(\\psi)\\quad\\text{if}\\quad\\psi\\ge\\psi_1\\cdots\\psi_n\\,.\n\\end{equation}", "mainsum": "\\begin{equation}\\label{mainsum}\n\\sum_{q=1}^\\infty\\psi_1(q)\\cdots \\psi_n(q)\n\\end{equation}", "thm: dream0": "\\begin{theorem}[\\textsc{Weighted Khintchine for manifolds}]\\label{thm: dream0}\nLet $\\cM$ be any nondegenerate submanifold of $\\R^n$ and $\\psi_1,\\dots,\\psi_n:\\Rp\\to(0,1)$ be non-increasing. Then almost all points on $\\cM$ are $(\\psi_1,\\dots,\\psi_n)$-approximable if the series\n\\begin{equation}\\label{mainsum}\n\\sum_{q=1}^\\infty\\psi_1(q)\\cdots \\psi_n(q)\n\\end{equation}\ndiverges, and are not $(\\psi_1,\\dots,\\psi_n)$-approximable if \\eqref{mainsum} converges.\n\\end{theorem}", "new BKM": "\\begin{equation}\\label{eq: bound on derivatives}\n   \\max_{k}\\max_{i,j} \\sup_{\\bx\\in \\bU}\\vert\\partial_{ij}f_{k}(\\bx)\\vert\\leq M.\n\\end{equation}\n\n\\subsection{Quantitative Nondivergence}\\label{new BKM} \n\nOur proofs will require an effective measure estimate for the following set \n\\begin{align}\n\\mathfrak{S}_{F}&(\\delta,\\bK,\\bT)=\\nonumber\\\\[1ex]\n&=\\left\\{\\bx\\in \\bU:\\exists\\;(a_0,\\ba)\\in\\Z\\times\\Z^n_{\\neq\\bf0}\\;\\;\\text{such that }\\left.\n\\begin{array}{l}\n|a_0+F(\\bx)\\ba^\\top|<\\delta\\\\[1ex]\n\\vert \\partial_iF(\\bx)\\ba^\\top\\vert<K_i\\;\\;(1\\le i\\le d) \\\\[1ex]\n\\vert a_k\\vert<T_k ~~(1\\leq k\\leq n) \n\\end{array}\n\\right.\\right\\},\\label{eq: new set}\n\\end{align}\nwhere $\\ba=(a_1,\\dots,a_n)$, $\\bT=(T_1,\\dots,T_n)$, $\\bK=(K_1,\\dots,K_d)$, and $\\delta,K_1,\\cdots,K_d,T_1,\\cdots, T_n$ are positive real parameters.\n\nThe following result represents a generalisation of \\cite[Theorem 1.4]{BKM}\nand appears in \\cite[Theorem 3.1]{BD} when $K_1=\\dots=K_d$. Its proof in the general case follows the same lines as that of \\cite[Theorem 3.1]{BD} with minor modifications which will be explained below.\n\n\\begin{theorem}\\label{BKM_new2}\nLet $F:\\bU\\to\\R^n$ be as defined in \\eqref{eq: F}, $l$-nondegenerate at $\\bx_0$. \nThen there exists a ball $B_0\\subset\\bU$ centred at $\\bx_0$ and a constant $E>0$ depending on $B_0$ and $F$ such that for any choice of $\\delta,K_1,\\cdots,K_d,T_1,\\cdots, T_n$ satisfying \n\\begin{equation}\\label{eqn5.2}\n0<\\delta\\le 1,\\qquad T_1,\\dots,T_n\\ge1,\\qquad K_1,\\cdots,K_d>0 \\quad\\text{and}\\quad \\delta^{n}<K \\frac{T_1\\cdots T_n}{\\max_i T_i},\n\\end{equation}", "sv:005": "\\begin{equation}\\label{sv:005}\n\\cSM_n(\\psi)=\\R^n  \\hspace{9mm}  {\\rm  if } \\hspace{9mm} q\\psi(q)\\ge 1 \\quad\\text{infinitely often} .\n\\end{equation}"}, "pre_theorem_intro_text_len": 5052, "pre_theorem_intro_text": "One classical theme in the theory of Diophantine approximation is to quantify how well rational points \n$$\n\\mathbf{p}/q=(p_1/q,\\dots,p_n/q)\\,,\n$$ \nwith $q\\in\\mathbb{N}$ and $\\mathbf{p}\\in\\mathbb{Z}^n$, approximate real points $\\mathbf{x}=(x_1,\\dots,x_n)\\in\\mathbb{R}^n$.\nIn the standard setting, the error of approximations is measured using a fixed norm on $\\mathbb{R}^n$, usually the sup-norm \n$$\n\\|\\mathbf{x}\\|=\\max_{1\\le i\\le n}|x_i|\\,.\n$$\nThus one seeks the same rate of approximation in each coordinate, imposing identical bounds on the quantities $|x_i-p_i/q|$.\n\nIn this paper we consider two more general frameworks: \\emph{weighted} and \\emph{multiplicative}, both of which can be seen as important special cases of the more general theory of Diophantine approximation with star bodies \\cite{MR2264601}. In the weighted case, different coordinates may be approximated at different rates; in the multiplicative case, one specifies a bound on the product\n$\\prod_{i=1}^n |x_i - p_i/q|$.\nWe now introduce the relevant notation and terminology.\n\nLet $\\mathbb{R}_+$ denote the set of positive real numbers, and let $\\psi,\\psi_1,\\dots,\\psi_n:\\mathbb{R}_+\\to(0,1)$ be nonincreasing  functions, referred to as {\\em approximation functions}.\nDefine $\\mathcal{S}_n(\\psi_1,\\dots, \\psi_n)$ as the set of $\\mathbf{x}\\in\\mathbb{R}^n$ such that the system\n\\begin{equation}\\label{eq1.1}\n\\vert qx_i-p_i\\vert<\\psi_i(q)\\qquad (1\\leq i\\leq n) \n\\end{equation}\nholds for infinitely many $(q,p_1,\\dots,p_n)\\in\\mathbb{N}\\times\\mathbb{Z}^n$. When $\\psi_1=\\dots=\\psi_n=\\psi$ this set will be denoted by $\\cS_n(\\psi)$. A point $\\mathbf{x}\\in\\cS_n(\\psi_1,\\dots,\\psi_n)$ will be called {\\em $(\\psi_1,\\dots,\\psi_n)$--approximable} ({\\em $\\psi$-approximable} when $\\psi_1=\\dots=\\psi_n=\\psi$).\n\nIn the multiplicative case,\ndefine $\\cSM_n(\\psi)$ as  the set of $\\mathbf{x}\\in\\mathbb{R}^n$ such that\n$$  \\prod_{i=1}^n |qx_i-p_i| \\ < \\ \\psi(q)\n$$  \nholds for infinitely many $(q,p_1,\\dots,p_n)\\in\\mathbb{N}\\times\\mathbb{Z}^n$. The points in $\\cSM_n(\\psi)$ will be called \n{\\em multiplicatively\n$\\psi$--approximable}.\nIt is readily verified that\n\\begin{equation}\\label{s:004}\n  \\cS_n(\\psi_1,\\dots,\\psi_n)\\subset\\cSM_n(\\psi)\\quad\\text{if}\\quad\\psi\\ge\\psi_1\\cdots\\psi_n\\,.\n\\end{equation}\n\n\\noindent By Minkowski's linear forms theorem,\n\\begin{equation}\\label{sv:004}\n\\cS_n(\\psi_1,\\dots,\\psi_n)=\\mathbb{R}^n \\hspace{9mm}  {\\rm  if } \\hspace{9mm}\nq\\psi_1(q)\\cdots\\psi_n(q)\\ge 1 \\quad\\text{infinitely often}\\,.\n\\end{equation}\nThis together with \\eqref{s:004} implies that\n\\begin{equation}\\label{sv:005}\n\\cSM_n(\\psi)=\\mathbb{R}^n  \\hspace{9mm}  {\\rm  if } \\hspace{9mm} q\\psi(q)\\ge 1 \\quad\\text{infinitely often} .\n\\end{equation}\nLittlewood's conjecture, a famous open problem dating back to the 1930s, asserts that the condition in \\eqref{sv:005} can be weakened to \n$$\\psi(q)\\ge \\varepsilon q^{-1}\\qquad\\text{for any $\\varepsilon>0$},\n$$\nsee \\cite{EKL} and \\cite{PVActa}. However, in the weighted case, \nsuch a weakening is generally impossible for the existence\nof badly approximable points \\cite{MR2581371, MR2231044}, e.g. when \n$$\n\\psi_i(q)=q^{-\\tau_i}\\,,\\qquad \\tau_i\\ge0,\\quad\\tau_1+\\dots+\\tau_n=1\\,.\n$$\nHere $\\mathbf{x}\\in\\mathbb{R}^n$ is {\\em $(\\psi_1,\\dots,\\psi_n)$-badly approximable} if $\\mathbf{x}\\not\\in\\cS_n(c\\psi_1,\\dots,c\\psi_n)$ for some $c>0$. Nevertheless, the set of $(\\psi_1,\\dots,\\psi_n)$-badly approximable points is well known to have zero measure \\cite{BadNull}.\n\nIn this paper we will be concerned with the measure of the sets $\\cS_n(\\psi_1,\\dots,\\psi_n)$ and $\\cSM_n(\\psi)$ \nrestricted to submanifolds of $\\mathbb{R}^n$ when $\\psi_1,\\dots,\\psi_n,\\psi$ are generic non-increasing functions. \nWe begin by recalling the corresponding results for $\\mathbb{R}^n$. Let $\\mu_n$ denote Lebesgue measure on $\\mathbb{R}^n$.\n\n\\smallskip\n\n\\begin{thkh}\n\\label{gs} Let   $\\psi_1,\\dots,\\psi_n$ be non-increasing functions. Then\n$$\\mu_{n}\\left(\\cS_n(\\psi_1,\\dots,\\psi_n)\\right) =\\left\\{\\begin{array}{ll} \\mbox{\\rm\nZ{\\scriptsize ERO}} & {\\rm if} \\;\\;\\; \\sum \\;  \\psi_1(q)\\cdots \\psi_n(q)\n\\;\\; <\\infty\\,,\\\\ &\n\\\\[-1ex] \\mbox{\\rm F{\\scriptsize ULL}} & {\\rm if} \\;\\;\\; \\sum  \\;  \\psi_1(q)\\cdots \\psi_n(q) \\;\\;\n =\\infty\\,. \\; \\;\n\\end{array}\\right.$$\n\\end{thkh}\n\n\\smallskip \n\n\\begin{thga}\n\\label{gm} Let   $\\psi$ be a non-increasing function. Then\n$$\\mu_{n}\\left(\\cSM_n(\\psi)\\right) =\\left\\{\\begin{array}{ll} \\mbox{\\rm\nZ{\\scriptsize ERO}} & {\\rm if} \\;\\;\\; \\sum \\;  \\psi(q) \\ (\\log\nq)^{n-1} \\;\\; < \\ \\infty\\,,\\\\ &\n\\\\[-1ex] \\mbox{\\rm F{\\scriptsize ULL}} & {\\rm if} \\;\\;\\; \\sum  \\;  \\psi(q) \\ (\\log\nq)^{n-1}  \\;\\;\n = \\ \\infty\\,. \\; \\;\n\\end{array}\\right.$$\n\\end{thga}\n\n\\noindent Here `\\mbox{\\rm F{\\scriptsize ULL}}' means that the complement has  measure zero. Theorem~K was originally established by Khintchine  \n\\cite{Kh} when $\\psi_1=\\dots=\\psi_n$, and both results as stated above were obtained by Gallagher in \\cite{gal}.\n\nOur main results extend Theorem~K and the convergence part of Theorem~G to nondegenerate submanifolds of $\\mathbb{R}^n$ as defined in \\cite{KM}. \n\n\\smallskip", "context": "Let $\\mathbb{R}_+$ denote the set of positive real numbers, and let $\\psi,\\psi_1,\\dots,\\psi_n:\\mathbb{R}_+\\to(0,1)$ be nonincreasing functions, referred to as {\\em approximation functions}.\nDefine $\\mathcal{S}_n(\\psi_1,\\dots, \\psi_n)$ as the set of $\\mathbf{x}\\in\\mathbb{R}^n$ such that the system\n\\begin{equation}\\label{eq1.1}\n\\vert qx_i-p_i\\vert<\\psi_i(q)\\qquad (1\\leq i\\leq n) \n\\end{equation}\nholds for infinitely many $(q,p_1,\\dots,p_n)\\in\\mathbb{N}\\times\\mathbb{Z}^n$. When $\\psi_1=\\dots=\\psi_n=\\psi$ this set will be denoted by $\\cS_n(\\psi)$. A point $\\mathbf{x}\\in\\cS_n(\\psi_1,\\dots,\\psi_n)$ will be called {\\em $(\\psi_1,\\dots,\\psi_n)$--approximable} ({\\em $\\psi$-approximable} when $\\psi_1=\\dots=\\psi_n=\\psi$).\n\nIn the multiplicative case,\ndefine $\\cSM_n(\\psi)$ as the set of $\\mathbf{x}\\in\\mathbb{R}^n$ such that\n$$ \\prod_{i=1}^n |qx_i-p_i| \\ < \\ \\psi(q)\n$$ \nholds for infinitely many $(q,p_1,\\dots,p_n)\\in\\mathbb{N}\\times\\mathbb{Z}^n$. The points in $\\cSM_n(\\psi)$ will be called \n{\\em multiplicatively\n$\\psi$--approximable}.\nIt is readily verified that\n\\begin{equation}\\label{s:004}\n \\cS_n(\\psi_1,\\dots,\\psi_n)\\subset\\cSM_n(\\psi)\\quad\\text{if}\\quad\\psi\\ge\\psi_1\\cdots\\psi_n\\,.\n\\end{equation}\n\n\\noindent By Minkowski's linear forms theorem,\n\\begin{equation}\\label{sv:004}\n\\cS_n(\\psi_1,\\dots,\\psi_n)=\\mathbb{R}^n \\hspace{9mm} {\\rm if } \\hspace{9mm}\nq\\psi_1(q)\\cdots\\psi_n(q)\\ge 1 \\quad\\text{infinitely often}\\,.\n\\end{equation}\nThis together with \\eqref{s:004} implies that\n\\begin{equation}\\label{sv:005}\n\\cSM_n(\\psi)=\\mathbb{R}^n \\hspace{9mm} {\\rm if } \\hspace{9mm} q\\psi(q)\\ge 1 \\quad\\text{infinitely often} .\n\\end{equation}\nLittlewood's conjecture, a famous open problem dating back to the 1930s, asserts that the condition in \\eqref{sv:005} can be weakened to \n$$\\psi(q)\\ge \\varepsilon q^{-1}\\qquad\\text{for any $\\varepsilon>0$},\n$$\nsee \\cite{EKL} and \\cite{PVActa}. However, in the weighted case, \nsuch a weakening is generally impossible for the existence\nof badly approximable points \\cite{MR2581371, MR2231044}, e.g. when \n$$\n\\psi_i(q)=q^{-\\tau_i}\\,,\\qquad \\tau_i\\ge0,\\quad\\tau_1+\\dots+\\tau_n=1\\,.\n$$\nHere $\\mathbf{x}\\in\\mathbb{R}^n$ is {\\em $(\\psi_1,\\dots,\\psi_n)$-badly approximable} if $\\mathbf{x}\\not\\in\\cS_n(c\\psi_1,\\dots,c\\psi_n)$ for some $c>0$. Nevertheless, the set of $(\\psi_1,\\dots,\\psi_n)$-badly approximable points is well known to have zero measure \\cite{BadNull}.\n\n\\begin{thkh}\n\\label{gs} Let $\\psi_1,\\dots,\\psi_n$ be non-increasing functions. Then\n$$\\mu_{n}\\left(\\cS_n(\\psi_1,\\dots,\\psi_n)\\right) =\\left\\{\\begin{array}{ll} \\mbox{\\rm\nZ{\\scriptsize ERO}} & {\\rm if} \\;\\;\\; \\sum \\; \\psi_1(q)\\cdots \\psi_n(q)\n\\;\\; <\\infty\\,,\\\\ &\n\\\\[-1ex] \\mbox{\\rm F{\\scriptsize ULL}} & {\\rm if} \\;\\;\\; \\sum \\; \\psi_1(q)\\cdots \\psi_n(q) \\;\\;\n =\\infty\\,. \\; \\;\n\\end{array}\\right.$$\n\\end{thkh}\n\nOur main results extend Theorem~K and the convergence part of Theorem~G to nondegenerate submanifolds of $\\mathbb{R}^n$ as defined in \\cite{KM}.\n\n\\smallskip\n\n\\begin{equation}\\label{mainsum}\n\\sum_{q=1}^\\infty\\psi_1(q)\\cdots \\psi_n(q)\n\\end{equation}\n\n\\begin{equation}\\label{s:004}\n  \\cS_n(\\psi_1,\\dots,\\psi_n)\\subset\\cSM_n(\\psi)\\quad\\text{if}\\quad\\psi\\ge\\psi_1\\cdots\\psi_n\\,.\n\\end{equation}\n\n\\begin{equation}\\label{sv:005}\n\\cSM_n(\\psi)=\\R^n  \\hspace{9mm}  {\\rm  if } \\hspace{9mm} q\\psi(q)\\ge 1 \\quad\\text{infinitely often} .\n\\end{equation}", "full_context": "Let $\\mathbb{R}_+$ denote the set of positive real numbers, and let $\\psi,\\psi_1,\\dots,\\psi_n:\\mathbb{R}_+\\to(0,1)$ be nonincreasing functions, referred to as {\\em approximation functions}.\nDefine $\\mathcal{S}_n(\\psi_1,\\dots, \\psi_n)$ as the set of $\\mathbf{x}\\in\\mathbb{R}^n$ such that the system\n\\begin{equation}\\label{eq1.1}\n\\vert qx_i-p_i\\vert<\\psi_i(q)\\qquad (1\\leq i\\leq n) \n\\end{equation}\nholds for infinitely many $(q,p_1,\\dots,p_n)\\in\\mathbb{N}\\times\\mathbb{Z}^n$. When $\\psi_1=\\dots=\\psi_n=\\psi$ this set will be denoted by $\\cS_n(\\psi)$. A point $\\mathbf{x}\\in\\cS_n(\\psi_1,\\dots,\\psi_n)$ will be called {\\em $(\\psi_1,\\dots,\\psi_n)$--approximable} ({\\em $\\psi$-approximable} when $\\psi_1=\\dots=\\psi_n=\\psi$).\n\nIn the multiplicative case,\ndefine $\\cSM_n(\\psi)$ as the set of $\\mathbf{x}\\in\\mathbb{R}^n$ such that\n$$ \\prod_{i=1}^n |qx_i-p_i| \\ < \\ \\psi(q)\n$$ \nholds for infinitely many $(q,p_1,\\dots,p_n)\\in\\mathbb{N}\\times\\mathbb{Z}^n$. The points in $\\cSM_n(\\psi)$ will be called \n{\\em multiplicatively\n$\\psi$--approximable}.\nIt is readily verified that\n\\begin{equation}\\label{s:004}\n \\cS_n(\\psi_1,\\dots,\\psi_n)\\subset\\cSM_n(\\psi)\\quad\\text{if}\\quad\\psi\\ge\\psi_1\\cdots\\psi_n\\,.\n\\end{equation}\n\n\\noindent By Minkowski's linear forms theorem,\n\\begin{equation}\\label{sv:004}\n\\cS_n(\\psi_1,\\dots,\\psi_n)=\\mathbb{R}^n \\hspace{9mm} {\\rm if } \\hspace{9mm}\nq\\psi_1(q)\\cdots\\psi_n(q)\\ge 1 \\quad\\text{infinitely often}\\,.\n\\end{equation}\nThis together with \\eqref{s:004} implies that\n\\begin{equation}\\label{sv:005}\n\\cSM_n(\\psi)=\\mathbb{R}^n \\hspace{9mm} {\\rm if } \\hspace{9mm} q\\psi(q)\\ge 1 \\quad\\text{infinitely often} .\n\\end{equation}\nLittlewood's conjecture, a famous open problem dating back to the 1930s, asserts that the condition in \\eqref{sv:005} can be weakened to \n$$\\psi(q)\\ge \\varepsilon q^{-1}\\qquad\\text{for any $\\varepsilon>0$},\n$$\nsee \\cite{EKL} and \\cite{PVActa}. However, in the weighted case, \nsuch a weakening is generally impossible for the existence\nof badly approximable points \\cite{MR2581371, MR2231044}, e.g. when \n$$\n\\psi_i(q)=q^{-\\tau_i}\\,,\\qquad \\tau_i\\ge0,\\quad\\tau_1+\\dots+\\tau_n=1\\,.\n$$\nHere $\\mathbf{x}\\in\\mathbb{R}^n$ is {\\em $(\\psi_1,\\dots,\\psi_n)$-badly approximable} if $\\mathbf{x}\\not\\in\\cS_n(c\\psi_1,\\dots,c\\psi_n)$ for some $c>0$. Nevertheless, the set of $(\\psi_1,\\dots,\\psi_n)$-badly approximable points is well known to have zero measure \\cite{BadNull}.\n\n\\begin{thkh}\n\\label{gs} Let $\\psi_1,\\dots,\\psi_n$ be non-increasing functions. Then\n$$\\mu_{n}\\left(\\cS_n(\\psi_1,\\dots,\\psi_n)\\right) =\\left\\{\\begin{array}{ll} \\mbox{\\rm\nZ{\\scriptsize ERO}} & {\\rm if} \\;\\;\\; \\sum \\; \\psi_1(q)\\cdots \\psi_n(q)\n\\;\\; <\\infty\\,,\\\\ &\n\\\\[-1ex] \\mbox{\\rm F{\\scriptsize ULL}} & {\\rm if} \\;\\;\\; \\sum \\; \\psi_1(q)\\cdots \\psi_n(q) \\;\\;\n =\\infty\\,. \\; \\;\n\\end{array}\\right.$$\n\\end{thkh}\n\nOur main results extend Theorem~K and the convergence part of Theorem~G to nondegenerate submanifolds of $\\mathbb{R}^n$ as defined in \\cite{KM}.\n\n\\smallskip\n\n\\begin{equation}\\label{mainsum}\n\\sum_{q=1}^\\infty\\psi_1(q)\\cdots \\psi_n(q)\n\\end{equation}\n\n\\begin{equation}\\label{s:004}\n  \\cS_n(\\psi_1,\\dots,\\psi_n)\\subset\\cSM_n(\\psi)\\quad\\text{if}\\quad\\psi\\ge\\psi_1\\cdots\\psi_n\\,.\n\\end{equation}\n\n\\begin{equation}\\label{sv:005}\n\\cSM_n(\\psi)=\\R^n  \\hspace{9mm}  {\\rm  if } \\hspace{9mm} q\\psi(q)\\ge 1 \\quad\\text{infinitely often} .\n\\end{equation}\n\n\\smallskip\n\n\\medskip\n\n\\begin{theorem}[\\textsc{Multiplicative convergence for manifolds}]\\label{thm:mult0}Let $\\cM$ be any nondegenerate submanifold of $\\R^n$ and $\\psi:\\Rp\\to(0,1)$ be non-increasing. Then almost all points on $\\cM$ are not multiplicatively\n$\\psi$-approximable if\n$$\n\\sum_{q=1}^\\infty \\psi(q) (\\log q)^{n-1}<\\infty.\n$$\n\\end{theorem}\n\n\\begin{theorem}\\label{thm: dream}\nLet $F:\\bU\\to\\R^n$ be nondegenerate, $\\bU$ be an open subset of $\\R^d$, $\\psi_1,\\dots,\\psi_n:\\Rp\\to(0,1)$ be non-increasing. Then \\begin{equation}\n \\mu_d\\left(F^{-1}\\mathcal{S}_n(\\psi_1,\\dots, \\psi_n)\\right)=\\left\\{\\begin{aligned}\n & 0 &\\text{ if } &\\textstyle\\sum\\psi_1(q)\\cdots \\psi_n(q)<\\infty\\,,\\\\[1ex]\n & \\mu_d(\\bU) &\\text{ if } &\\textstyle\\sum \\psi_1(q)\\cdots \\psi_n(q)=\\infty\\,.\n \\end{aligned}\\right.\n \\end{equation}\n\\end{theorem}\n\nWith this proposition at hand, justifying \\eqref{eq: prod is greater than} becomes simple. We take $\\Phi(q)=q^{-1-\\mathfrak{c}}$ and define $\\psi'_i$ from the proposition. Then\n\\begin{equation}\\label{eq2.13}\n\\sum_{q=1}^\\infty \\psi'_{1}(q)\\cdots\\psi'_{n}(q)\n\\leq \\sum_{q=1}^\\infty \\psi_{1}(q)\\cdots\\psi_{n}(q)\n+\\sum_{q=1}^\\infty q^{-1-\\mathfrak{c}}\n<\\infty\n\\end{equation}\nand\n$$\n\\mathcal{S}_n(\\psi_1,\\dots,\\psi_n)\n\\subset \\mathcal{S}_n(\\psi_1',\\dots,\\psi_n').\n$$\nHence we may work with the set\n$\\mathcal{S}_n(\\psi_1',\\dots,\\psi_n')$\ninstead of $\\mathcal{S}_n(\\psi_1,\\dots,\\psi_n)$,\nor equivalently we may assume \\eqref{eq: prod is greater than}.\n\nLet $\\psi_1,\\dots,\\psi_n$ be approximation functions satisfying the divergence sum condition in Theorem~\\ref{thm: dream}. Then\n\\begin{equation}\\label{eq3.20}\n\\sum_{t=1}^\\infty 2^t\\psi_1(2^t)\\cdots\\psi_n(2^t)=\\infty.\n\\end{equation}\nAs explained in \\S \\ref{sec: con on psi div}, we assume $\\psi_i$ satisfy \\eqref{eq: psi chain}, without loss of generality.\n\nFix any $0<s'<\\frac{1}{n-1}$. Then, by \\eqref{eq3.20}, there exists a strictly increasing sequence of positive integers $(t_k)_{k\\in\\N}$ such that\n\\begin{equation}\\label{eq:lower bound on prod} \n\\psi_{1}(q)\\cdots \\psi_{n}(q) q>q^{-\\left(\\frac{1}{n-1}-s'\\right)}\\qquad\\text{for each } q=2^{t_k}\n\\end{equation}\nand \n\\begin{equation}\\label{eq3.22}\n\\sum_{k=1}^\\infty 2^{t_k}\\psi_1(2^{t_k})\\cdots \\psi_n(2^{t_k})=\\infty.\n\\end{equation}\nIn particular, we have that for all $k\\in\\N$\n\\begin{equation}\\label{eq: standing 2 in 2^t}\n2^{t_k}\\psi_{2}(2^{t_k})\\cdots \\psi_n(2^{t_k})>2^{st_k},\\qquad s:=\\frac{n-1}{n}s'.\n\\end{equation}\nDefine $\\mathcal{T}:=\\{t_k:k\\in\\N\\}$, which is thus an infinite collection. Split $\\mathcal{T}$ into two sub-collections:\n$$\\mathcal{T}_1:=\\{t\\in\\mathcal{T}: \n2^{t}\\psi_{1}(2^{t})\\cdots \\psi_n(2^{t})<1\\}\\qquad\\text{and}\\qquad \n\\mathcal{T}_2=\\mathcal{T}\\setminus \\mathcal{T}_1.\n$$\n\nFrom Lemma \\ref{lem: ubiqui is enough}, it is enough to show that one of the sets on the left side of \\eqref{eq: enough} has \\textit{full} measure. In what follows $\\brho=(\\rho_i)_{i=1}^d\\Rp^d\\to\\Rp$.\nBy the definition of $\\Delta$ in \\eqref{def: Delta},\n$$\\begin{aligned}\\Delta(2^t, \\tfrac12(\\psi_{d+j}(2^t))_{j=1}^m, B, \\brho(2^t)):=\\bigcup_{(q,\\ba,\\bb)\\in \\mathcal{R}(2^t, (\\psi_{d+j}(2^t))_{j=1}^m, B)} \\tDelta(\\vv a/q,\\brho(2^t)),\n\\end{aligned}\n$$ \nfor any ball $B\\subset B_0,$ where $B_0$ be as in Proposition \\ref{thm: ubiq main}. \nNow note that $$\\begin{aligned}\n \\bigcup_{(q,\\ba,\\bb)\\in \\mathcal{R}(2^t, (\\psi_{d+j}(2^t))_{j=1}^m, B)} \\tDelta(\\vv a/q,\\brho(2^t))\\cap B\n \\subset \\bigcup_{2^{t-1}\\leq q\\leq 2^{t}, (q,\\ba)\\in J }\\tDelta(\\vv a/q,\\brho(2^t))\\cap B.\\end{aligned}$$\nBy \\eqref{eq3.22}, there is $i\\in\\{1,2\\}$ such that\n\\begin{equation}\\label{eq3.25}\n\\sum_{t\\in \\mathcal{T}_i} 2^t \\psi_1(2^t)\\cdots\\psi_n(2^t)=\\infty,\n\\end{equation}\n\n\\begin{equation}\\label{mainsum}\n\\sum_{q=1}^\\infty\\psi_1(q)\\cdots \\psi_n(q)\n\\end{equation}", "post_theorem_intro_text_len": 6634, "post_theorem_intro_text": "\\medskip\n\n\\begin{theorem}[\\textsc{Multiplicative convergence for manifolds}]\\label{thm:mult0}Let $\\mathcal{M}$ be any nondegenerate submanifold of $\\mathbb{R}^n$ and $\\psi:\\mathbb{R}_+\\to(0,1)$ be non-increasing. Then almost all points on $\\mathcal{M}$ are not multiplicatively\n$\\psi$-approximable if\n$$\n\\sum_{q=1}^\\infty \\psi(q) (\\log q)^{n-1}<\\infty.\n$$\n\\end{theorem}\n\nTheorem \\ref{thm: dream0} is first proved for the case when the manifold is parametrized in the so-called Monge form in \\S \\ref{sec: div} (for divergence) and \\S \\ref{sec: convergence} (for convergence). The proof is then generalized to arbitrary  nondegenerate manifolds in \\S \\ref{sec: general}. The proof of  Theorem~\\ref{thm:mult0} is presented in \\S \\ref{sec: multi}.\n\n\\subsection{Previous results and comparison}\n\nWe first discuss the unweighted case ($\\psi_1=\\cdots=\\psi_n$) of Theorem~\\ref{thm: dream0}. In this setting, Theorem~\\ref{thm: dream0} was first established for nondegenerate $C^3$ planar curves ($n=2$) in \\cite{BDV07} in the divergence case and in \\cite{VV06} in the convergence case. The $C^3$ assumption was later removed in \\cite{BZ10} (divergence) and \\cite{Hua15} (convergence). In higher dimensions, the divergence case was proved by the first-named author in \\cite{Ber12} for analytic nondegenerate manifolds and was recently extended to all nondegenerate manifolds in \\cite{BD}, having previously been known only for nondegenerate curves \\cite{BVVZ21}. The convergence case was settled in \\cite{BY} for all nondegenerate manifolds.\n\nMuch less is known in the weighted case, which allows the approximation functions $\\psi_1,\\dots,\\psi_n$ to be different. The story again begins with planar curves, studied in \\cite{BV07}, where the authors proved both the convergence and divergence cases of Theorem~\\ref{thm: dream0}: the divergence case for $C^3$ curves and the convergence case for rational quadrics. The latter was later extended to arbitrary $C^3$ nondegenerate planar curves in \\cite{Bad_Lev_07}.\n\nIn higher dimensions, the general problem was posed in \\cite[Problem~S2]{BV07} as follows.\n\n\\smallskip\n\n\\noindent\\textbf{Problem~1:} Given a nondegenerate manifold $\\mathcal{M}$ and approximation functions $\\psi_1,\\dots,\\psi_n$, find the weakest condition under which $\\mathcal{S}_n(\\psi_1,\\dots,\\psi_n)\\cap\\mathcal{M}$ has Lebesgue measure zero.\n\n\\smallskip\n\nWorking towards this general problem, Srivastava \\cite{Srivastava2025} recently established the convergence case of Theorem~\\ref{thm: dream0} for a subclass of nondegenerate manifolds with a curvature condition, under the additional constraint $\\psi_1=\\cdots=\\psi_d$ on the approximation functions. To the best of our knowledge, the divergence part of Theorem~\\ref{thm: dream0} for $n>2$ and arbitrary approximation functions $\\psi_1,\\dots,\\psi_n$ has not been previously addressed. Theorem~\\ref{thm: dream0} resolves Problem~1 in full and, at the same time, generalises the results of \\cite{BY} and \\cite{BD} from equal to arbitrary approximation functions $\\psi_1,\\dots,\\psi_n$.\n\n\\medskip\n\nWe now turn to the multiplicative case, concerning the set $\\cSM_n(\\psi)$. In this setting, progress to date has been far more limited, and even for planar curves the theory remains incomplete. Indeed, for nondegenerate planar curves, Theorem~\\ref{thm:mult0} was proved in \\cite{Hua15}, having been previously established for rational quadrics \\cite{BV07} and under the $C^3$ assumption \\cite{Bad_Lev_07}.\n\nAs with Problem~1, the general question regarding $\\cSM_n(\\psi)$ was posed in \\cite[Problem~S2]{BV07}:\n\n\\smallskip\n\n\\noindent\\textbf{Problem~2:} Given a nondegenerate manifold $\\mathcal{M}$ and an approximation function $\\psi$, find the weakest condition under which $\\cSM_n(\\psi)\\cap\\mathcal{M}$ has Lebesgue measure zero.\n\n\\smallskip\n\nTheorem~\\ref{thm:mult0} contributes towards the resolution of this problem by providing an analogue of the convergence case of Theorem~G for all nondegenerate manifolds.\n\nA natural next step would be to prove the divergence case of Theorem~G for nondegenerate manifolds. This remains a major open problem and has so far been addressed only for lines in $\\mathbb{R}^2$ \\cite{MR4068301, MR3813593, MR4701882} and for ``vertical lines'' in higher dimensions \\cite{MR4016056, MR4706444}. Even these limited cases required substantial new ideas, including the development of a Bohr sets technique \\cite{MR3813593} and an effective asymptotic equidistribution method \\cite{MR4701882}.\n\nFinally, we note that our results may be viewed as complete analogues of the corresponding results for dual approximation on manifolds. In that setting, the convergence case for both weighted and multiplicative approximation was established by Bernik, Kleinbock and Margulis \\cite{BKM}, while the divergence case for weighted approximation was proved in \\cite{MR2989975}.\n\n\\subsection{Further remarks}\n\nFor a manifold $\\mathcal{M} \\subset \\mathbb{R}^n$ of dimension $d$ to be nondegenerate, it is necessary that $\\mathcal{M}$ be $C^l$ with ${d+l \\choose l} \\ge n+1$. For example, if $d=2$, this reduces to $l^2 + 3l \\ge 2n$. Clearly there exist manifolds that are just $C^l$ for the smallest $l$ satisfying this inequality. The above theorems use this minimal smoothness, sufficient to meet the nondegeneracy condition.\n\nThe proof of Theorem~\\ref{thm: dream0} builds on the linearization techniques developed in \\cite{BY} and \\cite{BD}. The new features include linearization in different coordinates with different rates, including linearization in a single direction only. For the divergence case, we use the recent \\textit{ubiquity} result of Kleinbock and Wang \\cite{KWang23}. To construct an appropriate ubiquitous system, we provide a version of the quantitative estimate of Bernik, Kleinbock, and Margulis \\cite{BKM} that enables independent control of different partial derivatives (see \\S \\ref{new BKM}). At the technical level, our proof differs from the previous argument even in the \\textit{unweighted} case in \\cite{BD}, offering a new perspective.\n\nFinally, we note that, while preparing this paper, we became aware of related work by Chow, Srivastava, Technau, and Yu \\cite{MultCSTY} concerning Problem~2. Using a different approach initiated in \\cite{SST}, they establish a multiplicative convergence result for \\(C^\\infty\\) nondegenerate manifolds and a natural class of affine subspaces.\nWe emphasize that both Problem~1 and Problem~2 remain of interest and are still widely open for degenerate manifolds, such as affine subspaces and submanifolds that are nondegenerate relative to an affine subspace, as considered in \\cite{MR1982150}.", "sketch": "Theorem~\\ref{thm: dream0} is proved first “for the case when the manifold is parametrized in the so-called Monge form” (\\S\\,\\ref{sec: div} for divergence and \\S\\,\\ref{sec: convergence} for convergence), and “then generalized to arbitrary nondegenerate manifolds” in \\S\\,\\ref{sec: general}. The proof “builds on the linearization techniques developed in \\cite{BY} and \\cite{BD},” with “new features” including “linearization in different coordinates with different rates, including linearization in a single direction only.” For the divergence case, the argument uses “the recent \\textit{ubiquity} result of Kleinbock and Wang \\cite{KWang23}.” To “construct an appropriate ubiquitous system,” the authors “provide a version of the quantitative estimate of Bernik, Kleinbock, and Margulis \\cite{BKM} that enables independent control of different partial derivatives” (see \\S\\,\\ref{new BKM}). They also note that “at the technical level, our proof differs from the previous argument even in the \\textit{unweighted} case in \\cite{BD}, offering a new perspective.”", "expanded_sketch": "To prove the main theorem, the authors proceed first in the case when the manifold is parametrized in the so-called Monge form (for divergence they argue earlier, and for convergence they argue later), and then generalize to arbitrary nondegenerate manifolds later. The proof builds on the linearization techniques developed in \n\n\\subsection{Quantitative Nondivergence}\\label{new BKM} \n\nOur proofs will require an effective measure estimate for the following set \n\\begin{align}\n\\mathfrak{S}_{F}&(\\delta,\\bK,\\bT)=\\nonumber\\\\[1ex]\n&=\\left\\{\\bx\\in \\bU:\\exists\\;(a_0,\\ba)\\in\\Z\\times\\Z^n_{\\neq\\bf0}\\;\\;\\text{such that }\\left.\n\\begin{array}{l}\n|a_0+F(\\bx)\\ba^\\top|<\\delta\\\\[1ex]\n\\vert \\partial_iF(\\bx)\\ba^\\top\\vert<K_i\\;\\;(1\\le i\\le d) \\\\[1ex]\n\\vert a_k\\vert<T_k ~~(1\\leq k\\leq n) \n\\end{array}\n\\right.\\right\\},\\label{eq: new set}\n\\end{align}\nwhere $\\ba=(a_1,\\dots,a_n)$, $\\bT=(T_1,\\dots,T_n)$, $\\bK=(K_1,\\dots,K_d)$, and $\\delta,K_1,\\cdots,K_d,T_1,\\cdots, T_n$ are positive real parameters.\n\nThe following result represents a generalisation of \\cite[Theorem 1.4]{BKM}\nand appears in \\cite[Theorem 3.1]{BD} when $K_1=\\dots=K_d$. Its proof in the general case follows the same lines as that of \\cite[Theorem 3.1]{BD} with minor modifications which will be explained below.\n\n\\begin{theorem}\\label{BKM_new2}\nLet $F:\\bU\\to\\R^n$ be as defined in \\eqref{eq: F}, $l$-nondegenerate at $\\bx_0$. \nThen there exists a ball $B_0\\subset\\bU$ centred at $\\bx_0$ and a constant $E>0$ depending on $B_0$ and $F$ such that for any choice of $\\delta,K_1,\\cdots,K_d,T_1,\\cdots, T_n$ satisfying \n\\begin{equation}\\label{eqn5.2}\n0<\\delta\\le 1,\\qquad T_1,\\dots,T_n\\ge1,\\qquad K_1,\\cdots,K_d>0 \\quad\\text{and}\\quad \\delta^{n}<K \\frac{T_1\\cdots T_n}{\\max_i T_i},\n\\end{equation}\n\nas well as on new features including linearization in different coordinates with different rates, including linearization in a single direction only. For the divergence case, the argument uses the recent ubiquity result of Kleinbock and Wang \\cite{KWang23}. To construct an appropriate ubiquitous system, the authors provide a version of the quantitative estimate of Bernik, Kleinbock, and Margulis \\cite{BKM} that enables independent control of different partial derivatives (as in the quantitative nondivergence subsection above). They also note that at the technical level, their proof differs from the previous argument even in the unweighted case in \\cite{BD}, offering a new perspective.", "expanded_theorem": "[\\textsc{Weighted Khintchine for manifolds}]\\label{thm: dream0}\nLet $\\mathcal{M}$ be any nondegenerate submanifold of $\\mathbb{R}^n$ and $\\psi_1,\\dots,\\psi_n:\\mathbb{R}_+\\to(0,1)$ be non-increasing. Then almost all points on $\\mathcal{M}$ are $(\\psi_1,\\dots,\\psi_n)$-approximable if the series\n\\begin{equation}\\label{mainsum}\n\\sum_{q=1}^\\infty\\psi_1(q)\\cdots \\psi_n(q)\n\\end{equation}\ndiverges, and are not $(\\psi_1,\\dots,\\psi_n)$-approximable if the equation above converges.", "theorem_type": ["Biconditional or Equivalence", "Implication"], "mcq": {"question": "Let \\(\\mathcal M\\subset \\mathbb R^n\\) be a nondegenerate submanifold of \\(\\mathbb R^n\\) (in the standard metric Diophantine approximation sense), and let \\(\\psi_1,\\dots,\\psi_n:\\mathbb R_+\\to(0,1)\\) be non-increasing functions. A point \\(\\mathbf x=(x_1,\\dots,x_n)\\in\\mathbb R^n\\) is called \\((\\psi_1,\\dots,\\psi_n)\\)-approximable if there exist infinitely many tuples \\((q,p_1,\\dots,p_n)\\in\\mathbb N\\times\\mathbb Z^n\\) such that\n\\[\n|q x_i-p_i|<\\psi_i(q)\\qquad (1\\le i\\le n).\n\\]\nWhich statement about the set of points of \\(\\mathcal M\\) with this property is valid, where “almost all” is with respect to the natural Lebesgue measure on \\(\\mathcal M\\)?", "correct_choice": {"label": "A", "text": "If\n\\[\n\\sum_{q=1}^\\infty \\psi_1(q)\\cdots \\psi_n(q)=\\infty,\n\\]\nthen almost all points on \\(\\mathcal M\\) are \\((\\psi_1,\\dots,\\psi_n)\\)-approximable; and if\n\\[\n\\sum_{q=1}^\\infty \\psi_1(q)\\cdots \\psi_n(q)<\\infty,\n\\]\nthen almost all points on \\(\\mathcal M\\) are not \\((\\psi_1,\\dots,\\psi_n)\\)-approximable."}, "choices": [{"label": "B", "text": "If\n\\[\n\\sum_{q=1}^\\infty \\psi_1(q)\\cdots \\psi_n(q)=\\infty,\n\\]\nthen almost all points on \\(\\mathcal M\\) are \\((\\psi_1,\\dots,\\psi_n)\\)-approximable; and if\n\\[\n\\sum_{q=1}^\\infty q\\,\\psi_1(q)\\cdots \\psi_n(q)<\\infty,\n\\]\nthen almost all points on \\(\\mathcal M\\) are not \\((\\psi_1,\\dots,\\psi_n)\\)-approximable."}, {"label": "C", "text": "If\n\\[\n\\sum_{q=1}^\\infty \\psi_1(q)\\cdots \\psi_n(q)<\\infty,\n\\]\nthen almost all points on \\(\\mathcal M\\) are not \\((\\psi_1,\\dots,\\psi_n)\\)-approximable."}, {"label": "D", "text": "If\n\\[\n\\sum_{q=1}^\\infty \\psi_1(q)\\cdots \\psi_n(q)=\\infty,\n\\]\nthen every point of \\(\\mathcal M\\) is \\((\\psi_1,\\dots,\\psi_n)\\)-approximable; and if\n\\[\n\\sum_{q=1}^\\infty \\psi_1(q)\\cdots \\psi_n(q)<\\infty,\n\\]\nthen almost all points on \\(\\mathcal M\\) are not \\((\\psi_1,\\dots,\\psi_n)\\)-approximable."}, {"label": "E", "text": "If for each \\(i\\) one has\n\\[\n\\sum_{q=1}^\\infty \\psi_i(q)=\\infty,\n\\]\nthen almost all points on \\(\\mathcal M\\) are \\((\\psi_1,\\dots,\\psi_n)\\)-approximable; and if for some \\(i\\) one has\n\\[\n\\sum_{q=1}^\\infty \\psi_i(q)<\\infty,\n\\]\nthen almost all points on \\(\\mathcal M\\) are not \\((\\psi_1,\\dots,\\psi_n)\\)-approximable."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "quantitative_nondevergence", "tampered_component": "critical summand uses product \\(\\psi_1\\cdots\\psi_n\\) with no extra \\(q\\)-factor", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped the divergence/full-measure direction", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "ubiquity", "tampered_component": "almost-all conclusion strengthened to every point", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "product criterion replaced by coordinatewise series criteria", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the approximation property and the measure context but does not state or strongly hint at the critical convergence/divergence criterion. The correct answer is not leaked."}, "TAS": {"score": 1, "justification": "The item is largely a theorem-identification question: the correct option is essentially the full Khintchine-type statement for nondegenerate manifolds. The alternatives create some comparison, but the task is still close to recalling the theorem rather than deriving a conclusion from new data."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in distinguishing the exact product criterion from variants with an extra q-factor, coordinatewise sums, or an 'every point' strengthening. However, success depends mostly on theorem recall, and the presence of a weaker true option reduces genuine generative reasoning."}, "DQS": {"score": 1, "justification": "Most distractors are plausible and reflect common failure modes: wrong summand, overly strong conclusion, or wrong criterion. But choice C is also a true statement (a weaker valid consequence), so the item is not cleanly single-correct as written."}, "total_score": 5, "overall_assessment": "Good at avoiding answer leakage, but only moderately strong pedagogically: it is mainly theorem recall, and it has a notable ambiguity because option C is also valid under the wording."}}
{"id": "2602.11595v1", "paper_link": "http://arxiv.org/abs/2602.11595v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "[Volume and circumference]\\label{thm:vol-circumference}\nLet $M$ be a closed hyperbolic $3$-manifold with a pseudo-Anosov flow $\\varphi$. Let $S$ be a closed connected surface in $M$ that is almost transverse to $\\varphi$, but is not a fiber. Let $c^{s/u}$ be the stable/unstable multicurves of $S$. Assume that $\\mathfrak{c}(M\\backslash \\!\\! \\backslash S)<\\infty$.\n\nThere is a constant $k_S = k_S(\\chi(S)) \\ge 1$, depending only on $\\chi(S)$, and a constant $k_{M \\backslash \\!\\! \\backslash S} \\ge 1$, depending only on $\\chi(S)$ and $\\mathfrak c(M\\backslash \\!\\! \\backslash S)$, so that the following inequalities hold:\n\\begin{enumerate}\n\\item $d_{\\mathcal{C}(S)}(c^s, c^u) \\le k_S \\cdot \\mathrm{vol}(M) + k_{M\\backslash \\!\\! \\backslash S}$, and \n\\smallskip\n\\item $d_{\\mathcal{C}(S)}(c^s, c^u) \\le k_S \\cdot \\ell_M(\\gamma) + k_{M \\backslash \\!\\! \\backslash S}$, where $\\gamma$ is any closed geodesic that intersects $S$ essentially and $\\ell_M(\\gamma)$ is its length in $M$.\n\\end{enumerate}", "start_pos": 10086, "end_pos": 10956, "label": "thm:vol-circumference"}, "ref_dict": {"th:vol": "\\begin{theorem}[Volume]\\label{th:vol}\nLet $M$ be a closed hyperbolic $3$-manifold, $\\varphi$ an almost pseudo-Anosov flow on $M$, and $S$ a closed surface that is transverse to $\\varphi$ but not a fiber.\nThen\n\\[\nd_{\\C(S)}(c^s, c^u) \\prec \\vol(M) + D_{M\\cut S}\n\\]\n\\end{theorem}", "thm:short-curves": "\\begin{thm}[Short curves]\\label{thm:short-curves}\nLet $M$ be a closed hyperbolic $3$-manifold with a pseudo-Anosov flow $\\varphi$. Let $S$ be a closed surface in $M$ that is almost transverse to $\\varphi$ and let $Y \\subset S$ be a \nsubsurface of $S$. Assume that $\\mathfrak{c}(M\\cut S)<\\infty$. Then for any $\\epsilon >0$, there is a $K = K(\\epsilon, \\chi(S)) \\ge 0$, depending only on $\\epsilon$ and $\\chi(S)$, such that if\n\\begin{itemize}\n\\item $d_{\\C(S)}(c^{s/u} ,\\partial Y) \\ge k_{M\\cut S}$ and \n\\item $d_{\\C(Y)}(c^s,c^u) \\ge K + 2 \\cdot  k_{M \\cut S}$,\n\\end{itemize}\nthen \n\\[\n\\ell_M(\\partial Y) \\le \\epsilon.\n\\]\n\\end{thm}", "cor:finitedepth": "\\begin{theorem}[Junctures and geometry]\\label{cor:finitedepth}\nLet $M$ be a closed hyperbolic $3$-manifold with a finite depth foliation $\\mc F$ so that $\\mc F^0$ is a connected surface $S$. Suppose that $j^\\pm  \\subset S$ are minimal juncture on $S$.\n\nThere is a constant $c_S = c_S(\\chi(S)) \\ge 1$, depending only on $\\chi(S)$, and a constant $D\\ge 1$, depending only on $\\chi(S)$ and $\\mathfrak{c}_{\\FF}$, so that the following inequalities hold:\n\\begin{enumerate}\n\\item $d_{\\C(S)}(j^-, j^+) \\le c_S \\cdot \\vol(M) + D$, and \n\\smallskip\n\\item $d_{\\C(S)}(j^-, j^+) \\le c_S \\cdot \\ell_M(\\gamma) + D$, where $\\gamma$ is any closed geodesic that intersects $S$ essentially and $\\ell_M(\\gamma)$ is its length in $M$.\n\\end{enumerate}\n\\end{theorem}", "th:short_curves": "\\begin{theorem}[Short curves] \\label{th:short_curves}\nLet $M$ be a closed hyperbolic $3$-manifold with an almost pseudo-Anosov flow $\\varphi$. Let $S$ be a closed surface in $M$ that is transverse to $\\varphi$ and let $Y \\subset S$ be a \nsubsurface of $S$. Then for any $\\epsilon >0$, there is a $K = K(\\epsilon, \\chi(S)) \\ge 0$, depending only on $\\epsilon$ and $\\chi(S)$, such that if\n\\begin{itemize}\n\\item $d_{\\C(S)}(c^{s/u} ,\\partial Y) \\ge 2D_{M\\cut S}+2$ and \n\\item $d_{\\C(Y)}(c^s,c^u) \\ge K + 4 \\cdot  D_{M \\cut S}$,\n\\end{itemize}\nthen \n\\[\n\\ell_M(\\partial Y) \\le \\epsilon.\n\\]\n\\end{theorem}", "prop:bounded_length": "\\begin{proposition}\\label{prop:bounded_length}\nThere are constants $L, d \\ge 1$, depending on $|\\chi(S)|$ and $\\mf c(M \\cut S)$, and curves $\\alpha$ and $\\beta$ in $\\C(S)$ such that: \n\\begin{itemize}\n\\item the lengths of the geodesic representatives of $\\alpha$ and $\\beta$ in $M$ are no more than $L$, and \n\\item the intersection numbers $i(c^s, \\alpha)$ and $i(c^u, \\beta)$ are no more than $d$. \n\\end{itemize}\n\\end{proposition}", "th:circum": "\\begin{theorem}[Circumference]\n\\label{th:circum}\nLet $M$ be a closed hyperbolic $3$-manifold, $\\varphi$ an almost pseudo-Anosov flow on $M$, and $S$ a closed surface that is transverse to $\\varphi$ but not a fiber. Then there is a constant $D \\ge 0$, depending only on $|\\chi(S)|$ and $\\mf c(M\\cut S)$, so that \n\\[\nd_{\\C(S)}(c^s, c^u) \\prec  \\ell(\\gamma) + D,\n\\]\n where $\\gamma$ is any closed geodesic that intersects $S$ essentially.\n\\end{theorem}", "thm:vol-circumference": "\\begin{thm}[Volume and circumference]\\label{thm:vol-circumference}\nLet $M$ be a closed hyperbolic $3$-manifold with a pseudo-Anosov flow $\\varphi$. Let $S$ be a closed connected surface in $M$ that is almost transverse to $\\varphi$, but is not a fiber. Let $c^{s/u}$ be the stable/unstable multicurves of $S$. Assume that $\\mathfrak{c}(M\\cut S)<\\infty$.\n\nThere is a constant $k_S = k_S(\\chi(S)) \\ge 1$, depending only on $\\chi(S)$, and a constant $k_{M \\cut S} \\ge 1$, depending only on $\\chi(S)$ and $\\mf c(M\\cut S)$, so that the following inequalities hold:\n\\begin{enumerate}\n\\item $d_{\\C(S)}(c^s, c^u) \\le k_S \\cdot \\vol(M) + k_{M\\cut S}$, and \n\\smallskip\n\\item $d_{\\C(S)}(c^s, c^u) \\le k_S \\cdot \\ell_M(\\gamma) + k_{M \\cut S}$, where $\\gamma$ is any closed geodesic that intersects $S$ essentially and $\\ell_M(\\gamma)$ is its length in $M$.\n\\end{enumerate}\n\\end{thm}"}, "pre_theorem_intro_text_len": 4994, "pre_theorem_intro_text": "\\label{sec:intro}\nLet $M$ be a closed hyperbolic $3$--manifold and let $\\varphi$ be a pseudo-Anosov flow on $M$. The purpose of this paper is to quantify how dynamical properties of $\\varphi$ influence the geometry of $M$ through the curve graph of a closed embedded surface $S\\subset M$ that is (almost) transverse to $\\varphi$.\n\n\\smallskip\nThe prototypical example is where $S$ is a cross section of $\\varphi$ (i.e. $S$ is transverse to $\\varphi$ and intersects every orbit), and hence $M$ fibers over the circle with $S$ as a fiber. In this setting, the restriction of the  stable/unstable foliations of $\\varphi$ to $S$ are precisely the end invariants of the cover $\\widetilde M_S$ of $\\widetilde M$ associated to $S$. Hence, by the resolution of Thurston's Ending Lamination Conjecture, proven by Minsky \\cite{ECL1} and Brock--Canary--Minsky \\cite{ELC2}, the hyperbolic structure on $\\widetilde M_S$ is totally determined by these dynamical/geometric invariants. In fact, geometric quantities like the volume of $M$ \\cite{Brock2}, the circumference of $M$ \\cite{biringer2019ranks, aougab2022covers}, and the location of short geodesics \\cite{minsky2001bounded, viaggi2025effective} are each coarsely determined (up to constants depending only on $\\chi(S)$) by combinatorial quantities that are organized by the curve graph of $S$ and its subsurfaces.  \n\nDespite the significance of these results, very little is known in the more general setting where the surface $S$ is transverse to $\\varphi$ but not a cross section (that is, it misses some orbits). \nHere, the hyperbolic manifold $\\widetilde M_S$ is quasi-Fuchsian (\\cite{cooper1994bundles,fenley1999surfaces}) and there is no obvious connection between the stable/unstable foliations of $\\varphi$ and the end invariants of $\\widetilde M_S$. \n\n\\subsection*{Results}\n\nLet $M$ be a closed hyperbolic $3$--manifold, let $\\varphi$ be an almost pseudo-Anosov flow on $M$, and let $S$ be a connected surface embedded in $M$ that is transverse to $\\varphi$.\n The intersections of $S$ with the invariant stable/unstable foliations $\\mathcal F^{s/u}$ of $\\varphi$ are a pair of singular foliations $\\mathcal F_S^{s/u}$ of $S$.  When $S$ is not a fiber surface (or more precisely, not a cross section of $\\varphi$), the foliations $\\mathcal F_S^{s/u}$ each contain a nonempty set of closed leaves $c^{s/u}$ (see \\Cref{sec:transvese_shadow}).  \nThe pair $c^{s/u}$ are essential multicurves on $S$,\nwhich we call the \\emph{stable/unstable multicurves} of $S$. \n\nInformally, our main theorems state that if $c^s$ and $c^u$ are complicated with respect to one another, as viewed in the curve graph of $S$, then the geometry of $M$ near $S$ is also complicated. To state this precisely, we first introduce a notion of complexity of the cut manifold $M\\backslash \\!\\! \\backslash S$. This generalizes the notion of `core complexity' originally defined when $S$ is a compact leaf of a depth one foliation of $M$ (see \\cite{field2025lower}, \\cite{whitfield2024short}, as well as \\Cref{sec:examples}). \n\\smallskip\n\nRecall that $M\\backslash \\!\\! \\backslash S$ denotes the manifold obtained by cutting $M$ along $S$. Its boundary is a disjoint union $\\partial_- (M \\backslash \\!\\! \\backslash S) \\sqcup \\partial_+ (M \\backslash \\!\\! \\backslash S)$, where the flow points into, or out of, $M\\backslash \\!\\! \\backslash S$ respectively. A \\emph{product annulus} in $M \\backslash \\!\\! \\backslash S$ is an essential properly embedded annulus that has one boundary component on each of $\\partial_-(M\\backslash \\!\\! \\backslash S)$ and $\\partial_+(M\\backslash \\!\\! \\backslash S)$.\nDefine the \\emph{core complexity} of $M\\backslash \\!\\! \\backslash S$ to be \n\\begin{align*}\n\\mathfrak{c}(M\\backslash \\!\\! \\backslash S) = \n\\begin{cases}\n0, & M\\backslash \\!\\! \\backslash S \\text{ contains a product annulus},\\\\\n\\min{-\\chi(\\Sigma)}, & \\text{otherwise},\n\\end{cases}\n\\end{align*}\nwhere the min is over all properly embedded surfaces $\\Sigma$ in $M\\backslash \\!\\! \\backslash S$ that are transverse to (a dynamic blowup of) $\\varphi$ and which meet both $\\partial_+ (M \\backslash \\!\\! \\backslash S)$ and $\\partial_- (M\\backslash \\!\\! \\backslash S)$. Dynamic blowups and almost pseudo-Anosov flows are defined in \\Cref{sec:pA}.\nFor context, $\\mathfrak{c}(M\\backslash \\!\\! \\backslash S)$ contributes to an \\emph{additive} error term in our theorems that relate geometry and dynamics. We note that $\\mathfrak{c}(M\\backslash \\!\\! \\backslash S)< \\infty$ when $\\varphi$ is a suspension flow (i.e. $\\varphi$ admits a cross section) or more generally when $S$ is a leaf of a (almost) transverse finite depth foliation (see \\Cref{sec:examples}).\nA cohomological characterization of when $\\mathfrak{c}(M\\backslash \\!\\! \\backslash S)< \\infty$ will be  subject of forthcoming work \\cite{HT_depthone}.\n\n\\smallskip\n\nOur first main theorem relates the curve graph distance between $c^u$ and $c^s$ to the volume and circumference of $M$. See \\Cref{th:vol} and \\Cref{th:circum}, respectively.", "context": "\\label{sec:intro}\nLet $M$ be a closed hyperbolic $3$--manifold and let $\\varphi$ be a pseudo-Anosov flow on $M$. The purpose of this paper is to quantify how dynamical properties of $\\varphi$ influence the geometry of $M$ through the curve graph of a closed embedded surface $S\\subset M$ that is (almost) transverse to $\\varphi$.\n\n\\smallskip\nThe prototypical example is where $S$ is a cross section of $\\varphi$ (i.e. $S$ is transverse to $\\varphi$ and intersects every orbit), and hence $M$ fibers over the circle with $S$ as a fiber. In this setting, the restriction of the stable/unstable foliations of $\\varphi$ to $S$ are precisely the end invariants of the cover $\\widetilde M_S$ of $\\widetilde M$ associated to $S$. Hence, by the resolution of Thurston's Ending Lamination Conjecture, proven by Minsky \\cite{ECL1} and Brock--Canary--Minsky \\cite{ELC2}, the hyperbolic structure on $\\widetilde M_S$ is totally determined by these dynamical/geometric invariants. In fact, geometric quantities like the volume of $M$ \\cite{Brock2}, the circumference of $M$ \\cite{biringer2019ranks, aougab2022covers}, and the location of short geodesics \\cite{minsky2001bounded, viaggi2025effective} are each coarsely determined (up to constants depending only on $\\chi(S)$) by combinatorial quantities that are organized by the curve graph of $S$ and its subsurfaces.\n\nLet $M$ be a closed hyperbolic $3$--manifold, let $\\varphi$ be an almost pseudo-Anosov flow on $M$, and let $S$ be a connected surface embedded in $M$ that is transverse to $\\varphi$.\n The intersections of $S$ with the invariant stable/unstable foliations $\\mathcal F^{s/u}$ of $\\varphi$ are a pair of singular foliations $\\mathcal F_S^{s/u}$ of $S$. When $S$ is not a fiber surface (or more precisely, not a cross section of $\\varphi$), the foliations $\\mathcal F_S^{s/u}$ each contain a nonempty set of closed leaves $c^{s/u}$ (see \\Cref{sec:transvese_shadow}). \nThe pair $c^{s/u}$ are essential multicurves on $S$,\nwhich we call the \\emph{stable/unstable multicurves} of $S$.\n\nRecall that $M\\backslash \\!\\! \\backslash S$ denotes the manifold obtained by cutting $M$ along $S$. Its boundary is a disjoint union $\\partial_- (M \\backslash \\!\\! \\backslash S) \\sqcup \\partial_+ (M \\backslash \\!\\! \\backslash S)$, where the flow points into, or out of, $M\\backslash \\!\\! \\backslash S$ respectively. A \\emph{product annulus} in $M \\backslash \\!\\! \\backslash S$ is an essential properly embedded annulus that has one boundary component on each of $\\partial_-(M\\backslash \\!\\! \\backslash S)$ and $\\partial_+(M\\backslash \\!\\! \\backslash S)$.\nDefine the \\emph{core complexity} of $M\\backslash \\!\\! \\backslash S$ to be \n\\begin{align*}\n\\mathfrak{c}(M\\backslash \\!\\! \\backslash S) = \n\\begin{cases}\n0, & M\\backslash \\!\\! \\backslash S \\text{ contains a product annulus},\\\\\n\\min{-\\chi(\\Sigma)}, & \\text{otherwise},\n\\end{cases}\n\\end{align*}\nwhere the min is over all properly embedded surfaces $\\Sigma$ in $M\\backslash \\!\\! \\backslash S$ that are transverse to (a dynamic blowup of) $\\varphi$ and which meet both $\\partial_+ (M \\backslash \\!\\! \\backslash S)$ and $\\partial_- (M\\backslash \\!\\! \\backslash S)$. Dynamic blowups and almost pseudo-Anosov flows are defined in \\Cref{sec:pA}.\nFor context, $\\mathfrak{c}(M\\backslash \\!\\! \\backslash S)$ contributes to an \\emph{additive} error term in our theorems that relate geometry and dynamics. We note that $\\mathfrak{c}(M\\backslash \\!\\! \\backslash S)< \\infty$ when $\\varphi$ is a suspension flow (i.e. $\\varphi$ admits a cross section) or more generally when $S$ is a leaf of a (almost) transverse finite depth foliation (see \\Cref{sec:examples}).\nA cohomological characterization of when $\\mathfrak{c}(M\\backslash \\!\\! \\backslash S)< \\infty$ will be subject of forthcoming work \\cite{HT_depthone}.\n\n\\smallskip\n\nOur first main theorem relates the curve graph distance between $c^u$ and $c^s$ to the volume and circumference of $M$. See \\Cref{th:vol} and \\Cref{th:circum}, respectively.\n\n\\begin{theorem}[Circumference]\n\\label{th:circum}\nLet $M$ be a closed hyperbolic $3$-manifold, $\\varphi$ an almost pseudo-Anosov flow on $M$, and $S$ a closed surface that is transverse to $\\varphi$ but not a fiber. Then there is a constant $D \\ge 0$, depending only on $|\\chi(S)|$ and $\\mf c(M\\cut S)$, so that \n\\[\nd_{\\C(S)}(c^s, c^u) \\prec  \\ell(\\gamma) + D,\n\\]\n where $\\gamma$ is any closed geodesic that intersects $S$ essentially.\n\\end{theorem}\n\n\\begin{theorem}[Volume]\\label{th:vol}\nLet $M$ be a closed hyperbolic $3$-manifold, $\\varphi$ an almost pseudo-Anosov flow on $M$, and $S$ a closed surface that is transverse to $\\varphi$ but not a fiber.\nThen\n\\[\nd_{\\C(S)}(c^s, c^u) \\prec \\vol(M) + D_{M\\cut S}\n\\]\n\\end{theorem}", "full_context": "\\label{sec:intro}\nLet $M$ be a closed hyperbolic $3$--manifold and let $\\varphi$ be a pseudo-Anosov flow on $M$. The purpose of this paper is to quantify how dynamical properties of $\\varphi$ influence the geometry of $M$ through the curve graph of a closed embedded surface $S\\subset M$ that is (almost) transverse to $\\varphi$.\n\n\\smallskip\nThe prototypical example is where $S$ is a cross section of $\\varphi$ (i.e. $S$ is transverse to $\\varphi$ and intersects every orbit), and hence $M$ fibers over the circle with $S$ as a fiber. In this setting, the restriction of the stable/unstable foliations of $\\varphi$ to $S$ are precisely the end invariants of the cover $\\widetilde M_S$ of $\\widetilde M$ associated to $S$. Hence, by the resolution of Thurston's Ending Lamination Conjecture, proven by Minsky \\cite{ECL1} and Brock--Canary--Minsky \\cite{ELC2}, the hyperbolic structure on $\\widetilde M_S$ is totally determined by these dynamical/geometric invariants. In fact, geometric quantities like the volume of $M$ \\cite{Brock2}, the circumference of $M$ \\cite{biringer2019ranks, aougab2022covers}, and the location of short geodesics \\cite{minsky2001bounded, viaggi2025effective} are each coarsely determined (up to constants depending only on $\\chi(S)$) by combinatorial quantities that are organized by the curve graph of $S$ and its subsurfaces.\n\nLet $M$ be a closed hyperbolic $3$--manifold, let $\\varphi$ be an almost pseudo-Anosov flow on $M$, and let $S$ be a connected surface embedded in $M$ that is transverse to $\\varphi$.\n The intersections of $S$ with the invariant stable/unstable foliations $\\mathcal F^{s/u}$ of $\\varphi$ are a pair of singular foliations $\\mathcal F_S^{s/u}$ of $S$. When $S$ is not a fiber surface (or more precisely, not a cross section of $\\varphi$), the foliations $\\mathcal F_S^{s/u}$ each contain a nonempty set of closed leaves $c^{s/u}$ (see \\Cref{sec:transvese_shadow}). \nThe pair $c^{s/u}$ are essential multicurves on $S$,\nwhich we call the \\emph{stable/unstable multicurves} of $S$.\n\nRecall that $M\\backslash \\!\\! \\backslash S$ denotes the manifold obtained by cutting $M$ along $S$. Its boundary is a disjoint union $\\partial_- (M \\backslash \\!\\! \\backslash S) \\sqcup \\partial_+ (M \\backslash \\!\\! \\backslash S)$, where the flow points into, or out of, $M\\backslash \\!\\! \\backslash S$ respectively. A \\emph{product annulus} in $M \\backslash \\!\\! \\backslash S$ is an essential properly embedded annulus that has one boundary component on each of $\\partial_-(M\\backslash \\!\\! \\backslash S)$ and $\\partial_+(M\\backslash \\!\\! \\backslash S)$.\nDefine the \\emph{core complexity} of $M\\backslash \\!\\! \\backslash S$ to be \n\\begin{align*}\n\\mathfrak{c}(M\\backslash \\!\\! \\backslash S) = \n\\begin{cases}\n0, & M\\backslash \\!\\! \\backslash S \\text{ contains a product annulus},\\\\\n\\min{-\\chi(\\Sigma)}, & \\text{otherwise},\n\\end{cases}\n\\end{align*}\nwhere the min is over all properly embedded surfaces $\\Sigma$ in $M\\backslash \\!\\! \\backslash S$ that are transverse to (a dynamic blowup of) $\\varphi$ and which meet both $\\partial_+ (M \\backslash \\!\\! \\backslash S)$ and $\\partial_- (M\\backslash \\!\\! \\backslash S)$. Dynamic blowups and almost pseudo-Anosov flows are defined in \\Cref{sec:pA}.\nFor context, $\\mathfrak{c}(M\\backslash \\!\\! \\backslash S)$ contributes to an \\emph{additive} error term in our theorems that relate geometry and dynamics. We note that $\\mathfrak{c}(M\\backslash \\!\\! \\backslash S)< \\infty$ when $\\varphi$ is a suspension flow (i.e. $\\varphi$ admits a cross section) or more generally when $S$ is a leaf of a (almost) transverse finite depth foliation (see \\Cref{sec:examples}).\nA cohomological characterization of when $\\mathfrak{c}(M\\backslash \\!\\! \\backslash S)< \\infty$ will be subject of forthcoming work \\cite{HT_depthone}.\n\n\\smallskip\n\nOur first main theorem relates the curve graph distance between $c^u$ and $c^s$ to the volume and circumference of $M$. See \\Cref{th:vol} and \\Cref{th:circum}, respectively.\n\n\\begin{theorem}[Circumference]\n\\label{th:circum}\nLet $M$ be a closed hyperbolic $3$-manifold, $\\varphi$ an almost pseudo-Anosov flow on $M$, and $S$ a closed surface that is transverse to $\\varphi$ but not a fiber. Then there is a constant $D \\ge 0$, depending only on $|\\chi(S)|$ and $\\mf c(M\\cut S)$, so that \n\\[\nd_{\\C(S)}(c^s, c^u) \\prec  \\ell(\\gamma) + D,\n\\]\n where $\\gamma$ is any closed geodesic that intersects $S$ essentially.\n\\end{theorem}\n\n\\begin{theorem}[Volume]\\label{th:vol}\nLet $M$ be a closed hyperbolic $3$-manifold, $\\varphi$ an almost pseudo-Anosov flow on $M$, and $S$ a closed surface that is transverse to $\\varphi$ but not a fiber.\nThen\n\\[\nd_{\\C(S)}(c^s, c^u) \\prec \\vol(M) + D_{M\\cut S}\n\\]\n\\end{theorem}\n\n\\smallskip\nThe prototypical example is where $S$ is a cross section of $\\varphi$ (i.e. $S$ is transverse to $\\varphi$ and intersects every orbit), and hence $M$ fibers over the circle with $S$ as a fiber. In this setting, the restriction of the stable/unstable foliations of $\\varphi$ to $S$ are precisely the end invariants of the cover $\\wt M_S$ of $\\wt M$ associated to $S$. Hence, by the resolution of Thurston's Ending Lamination Conjecture, proven by Minsky \\cite{ECL1} and Brock--Canary--Minsky \\cite{ELC2}, the hyperbolic structure on $\\wt M_S$ is totally determined by these dynamical/geometric invariants. In fact, geometric quantities like the volume of $M$ \\cite{Brock2}, the circumference of $M$ \\cite{biringer2019ranks, aougab2022covers}, and the location of short geodesics \\cite{minsky2001bounded, viaggi2025effective} are each coarsely determined (up to constants depending only on $\\chi(S)$) by combinatorial quantities that are organized by the curve graph of $S$ and its subsurfaces.\n\nOur first main theorem relates the curve graph distance between $c^u$ and $c^s$ to the volume and circumference of $M$. See \\Cref{th:vol} and \\Cref{th:circum}, respectively.\n\nWe remark that when $M\\cut S$ contains a product annulus, $\\mf c(M\\cut S) = 0$ and so the constants in \\Cref{thm:vol-circumference} depend only on $\\vert \\chi(S) \\vert$.\n\n\\begin{thm}[Short curves]\\label{thm:short-curves}\nLet $M$ be a closed hyperbolic $3$-manifold with a pseudo-Anosov flow $\\varphi$. Let $S$ be a closed surface in $M$ that is almost transverse to $\\varphi$ and let $Y \\subset S$ be a \nsubsurface of $S$. Assume that $\\mathfrak{c}(M\\cut S)<\\infty$. Then for any $\\epsilon >0$, there is a $K = K(\\epsilon, \\chi(S)) \\ge 0$, depending only on $\\epsilon$ and $\\chi(S)$, such that if\n\\begin{itemize}\n\\item $d_{\\C(S)}(c^{s/u} ,\\partial Y) \\ge k_{M\\cut S}$ and \n\\item $d_{\\C(Y)}(c^s,c^u) \\ge K + 2 \\cdot k_{M \\cut S}$,\n\\end{itemize}\nthen \n\\[\n\\ell_M(\\partial Y) \\le \\epsilon.\n\\]\n\\end{thm}\n\n\\begin{theorem}[Volume]\\label{th:vol}\nLet $M$ be a closed hyperbolic $3$-manifold, $\\varphi$ an almost pseudo-Anosov flow on $M$, and $S$ a closed surface that is transverse to $\\varphi$ but not a fiber.\nThen\n\\[\nd_{\\C(S)}(c^s, c^u) \\prec \\vol(M) + D_{M\\cut S}\n\\]\n\\end{theorem}\n\n\\begin{theorem}[Circumference]\n\\label{th:circum}\nLet $M$ be a closed hyperbolic $3$-manifold, $\\varphi$ an almost pseudo-Anosov flow on $M$, and $S$ a closed surface that is transverse to $\\varphi$ but not a fiber. Then there is a constant $D \\ge 0$, depending only on $|\\chi(S)|$ and $\\mf c(M\\cut S)$, so that \n\\[\nd_{\\C(S)}(c^s, c^u) \\prec \\ell(\\gamma) + D,\n\\]\n where $\\gamma$ is any closed geodesic that intersects $S$ essentially.\n\\end{theorem}\n\n\\subsection{Short curves}\nFinally, we give a short proof of the following:\n\\begin{theorem}[Short curves] \\label{th:short_curves}\nLet $M$ be a closed hyperbolic $3$-manifold with an almost pseudo-Anosov flow $\\varphi$. Let $S$ be a closed surface in $M$ that is transverse to $\\varphi$ and let $Y \\subset S$ be a \nsubsurface of $S$. Then for any $\\epsilon >0$, there is a $K = K(\\epsilon, \\chi(S)) \\ge 0$, depending only on $\\epsilon$ and $\\chi(S)$, such that if\n\\begin{itemize}\n\\item $d_{\\C(S)}(c^{s/u} ,\\partial Y) \\ge 2D_{M\\cut S}+2$ and \n\\item $d_{\\C(Y)}(c^s,c^u) \\ge K + 4 \\cdot D_{M \\cut S}$,\n\\end{itemize}\nthen \n\\[\n\\ell_M(\\partial Y) \\le \\epsilon.\n\\]\n\\end{theorem}\n\nThere is a constant $c_S = c_S(\\chi(S)) \\ge 1$, depending only on $\\chi(S)$, and a constant $D\\ge 1$, depending only on $\\chi(S)$ and $\\mathfrak{c}_{\\FF}$, so that the following inequalities hold:\n\\begin{enumerate}\n\\item $d_{\\C(S)}(j^-, j^+) \\le c_S \\cdot \\vol(M) + D$, and \n\\smallskip\n\\item $d_{\\C(S)}(j^-, j^+) \\le c_S \\cdot \\ell_M(\\gamma) + D$, where $\\gamma$ is any closed geodesic that intersects $S$ essentially and $\\ell_M(\\gamma)$ is its length in $M$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{theorem}[Circumference]\n\\label{th:circum}\nLet $M$ be a closed hyperbolic $3$-manifold, $\\varphi$ an almost pseudo-Anosov flow on $M$, and $S$ a closed surface that is transverse to $\\varphi$ but not a fiber. Then there is a constant $D \\ge 0$, depending only on $|\\chi(S)|$ and $\\mf c(M\\cut S)$, so that \n\\[\nd_{\\C(S)}(c^s, c^u) \\prec  \\ell(\\gamma) + D,\n\\]\n where $\\gamma$ is any closed geodesic that intersects $S$ essentially.\n\\end{theorem}\n\n\\begin{thm}[Volume and circumference]\\label{thm:vol-circumference}\nLet $M$ be a closed hyperbolic $3$-manifold with a pseudo-Anosov flow $\\varphi$. Let $S$ be a closed connected surface in $M$ that is almost transverse to $\\varphi$, but is not a fiber. Let $c^{s/u}$ be the stable/unstable multicurves of $S$. Assume that $\\mathfrak{c}(M\\cut S)<\\infty$.\n\nThere is a constant $k_S = k_S(\\chi(S)) \\ge 1$, depending only on $\\chi(S)$, and a constant $k_{M \\cut S} \\ge 1$, depending only on $\\chi(S)$ and $\\mf c(M\\cut S)$, so that the following inequalities hold:\n\\begin{enumerate}\n\\item $d_{\\C(S)}(c^s, c^u) \\le k_S \\cdot \\vol(M) + k_{M\\cut S}$, and \n\\smallskip\n\\item $d_{\\C(S)}(c^s, c^u) \\le k_S \\cdot \\ell_M(\\gamma) + k_{M \\cut S}$, where $\\gamma$ is any closed geodesic that intersects $S$ essentially and $\\ell_M(\\gamma)$ is its length in $M$.\n\\end{enumerate}\n\\end{thm}", "post_theorem_intro_text_len": 7000, "post_theorem_intro_text": "We remark that when $M\\backslash \\!\\! \\backslash S$ contains a product annulus, $\\mathfrak c(M\\backslash \\!\\! \\backslash S) = 0$ and so the constants in \\Cref{thm:vol-circumference} depend only on $\\vert \\chi(S) \\vert$.\n\nThere is also a version of \\Cref{thm:vol-circumference} that starts with a finite depth foliation rather than a flow, where the multicurves $c^{s/u}$ are replaced by minimal junctures of the foliation. See \\Cref{sec:examples} and in particular \\Cref{cor:finitedepth} for details.\n\n\\smallskip\nThe second main theorem relates subsurface distances between $c^u$ and $c^s$ to short geodesics in $M$. Informally, it says that if $c^{s/u}$ are complicated with respect to one another, from the perspective of some distant subsurface $Y \\subset S$, then the geodesic length of $\\partial Y$ in $M$ must be small. See \\Cref{th:short_curves}.\n\n\\begin{thm}[Short curves]\\label{thm:short-curves}\nLet $M$ be a closed hyperbolic $3$-manifold with a pseudo-Anosov flow $\\varphi$. Let $S$ be a closed surface in $M$ that is almost transverse to $\\varphi$ and let $Y \\subset S$ be a \nsubsurface of $S$. Assume that $\\mathfrak{c}(M\\backslash \\!\\! \\backslash S)<\\infty$. Then for any $\\varepsilon >0$, there is a $K = K(\\varepsilon, \\chi(S)) \\ge 0$, depending only on $\\varepsilon$ and $\\chi(S)$, such that if\n\\begin{itemize}\n\\item $d_{\\mathcal{C}(S)}(c^{s/u} ,\\partial Y) \\ge k_{M\\backslash \\!\\! \\backslash S}$ and \n\\item $d_{\\mathcal{C}(Y)}(c^s,c^u) \\ge K + 2 \\cdot  k_{M \\backslash \\!\\! \\backslash S}$,\n\\end{itemize}\nthen \n\\[\n\\ell_M(\\partial Y) \\le \\varepsilon.\n\\]\n\\end{thm}\n\nHere, $\\ell_M(\\partial Y)$ denotes the length of the geodesic representative of $\\partial Y$ in $M$, and $d_{\\mathcal{C}(Y)}(c^s,c^u)$ is the subsurface distance in the curve graph of $Y$ between $c^s$ and $c^u$. See \\Cref{sec:curvegraph} for details.\n\n\\begin{remark}\nOur main technical result that controls the geometry of $M$ is \\Cref{prop:bounded_length}. Informally, it states that each of the multicurves $c^{s/u}$ have bounded curve graph distance from curves on $S$ that have bounded hyperbolic length in $M$. One might hope to prove the stronger statement that $c^{s/u}$ themselves have bounded length, but this is not always possible. Indeed, Whitfield \\cite[Theorem A]{whitfield2024short} produces examples where the `junctures' of a depth one foliation of $M$ are arbitrarily short. (Here, the junctures are essentially boundary components of a properly embedded surface in $M \\backslash \\!\\! \\backslash S$ realizing the minimum core complexity.)\nSince such curves necessarily cross $c^{s/u}$, in these examples the stable/unstable multicurves can be arbitrarily long. \n\\end{remark}\n\nFinally, several examples are given in \\Cref{sec:examples}, including a flexible construction of manifolds and flows that realizes the hypotheses of the main theorems. \n\n\\subsection{Connections to the literature}\nAlthough most of the previous work that relates hyperbolic geometry, dynamics, and the curve graph comes from the fibered manifold setting (as discussed above), there has recently been an interest in extending these connections to mapping tori of endperiodic maps of infinite-type surfaces. This is most prominent in work of Field, Kim, Kent, Leininger, and Loving \\cite{field2023end, field2025lower} who establish connections to hyperbolic volume of the mapping tori, and work of Whitfield \\cite{whitfield2024short}, who studies the lengths of short geodesics.\n\nThe work in this paper sheds light on a different aspect of the geometry of these mapping tori. Indeed, returning to the setting of our main theorems, in the special case where $S$ is a compact leaf of a transverse depth one foliation of $M$, each component of $M \\backslash \\!\\! \\backslash S$ is the mapping torus of an especially nice endperiodic map -- the first return map to a depth one leaf of the foliation using the transverse flow. \nThe works cited above study the intrinsic geometry of components of $M \\backslash \\!\\! \\backslash S$, especially for the unique totally geodesic structure, when it exists. By contrast, our results here are about the geometry of the original closed manifold $M$. Informally,  \\Cref{thm:vol-circumference} and \\Cref{thm:short-curves} focus on the geometric features of $M$ that are forced when gluing $M \\backslash \\!\\! \\backslash S$ back to obtain $M$, as measured by the curve graph complexity of the stable and unstable multicurves. For more on the implications in the depth one case, see \\Cref{sec:examples}. \n\n\\subsection{Proof strategy and outline of paper}\nThe proof of \\Cref{thm:vol-circumference} and \\Cref{thm:short-curves} is divided into three steps. In the first step, we identify multicurves on $S$ that have topological significance in $M\\backslash \\!\\! \\backslash S$, and have bounded intersection number with $c^{s/u}$. When $M\\backslash \\!\\! \\backslash S$ admits product annuli, these are the components of Thurston's window frame of $M\\backslash \\!\\! \\backslash S$ (see \\Cref{sec:geom}),\nand we show that product annuli can be put into a particularly nice position with respect to the flow. This is explained in \\Cref{sec:annuli}.\nWhen $M\\backslash \\!\\! \\backslash S$ does not admit a product annulus but has finite core complexity, these are the boundary components of a properly embedded transverse surface with minimal complexity. In the first case, the intersection number bounds are established in \\Cref{sec:annuli}, and in the second case, this is done in \\Cref{sec:int_bounds}.\n\nIn the second step, we show that these multicurves have bounded length in the hyperbolic metric on $M$. This is done in \\Cref{sec:geom}.\nThe results in the first two steps lead to \\Cref{prop:bounded_length}, which states that $c^s$ and $c^u$ have bounded intersection number with bounded length curves in $S$. Finally, we obtain geometric control of $M$ by passing to the quasi-Fuchsian cover corresponding to $S$ and using standard tools from the study of curve graphs and Kleinian surface groups, replacing $c^s$ and $c^u$ with nearby bounded length curves. This is done in \\Cref{sec:endgame}.\n\nFinally, \nexamples and applications are given in \\Cref{sec:examples}, including a version of \\Cref{thm:vol-circumference} for finite depth foliations as well as a construction where the \ndynamical quantity $d_{\\mathcal{C}(S)}(c^s,c^u)$ can be precisely controlled.\n\n\\subsection*{Acknowledgements}\nThe authors thank Michael Landry and Yair Minsky for several helpful and encouraging conversations. The authors also thank Ellis Buckminster and Brandis Whitfield for helpful comments on an earlier draft.\n\nThe completion of this paper was supported by the National Science Foundation under Grant No. DMS--2424139, while the authors were in residence at the Simons Laufer Mathematical Sciences Institute in Berkeley, California, during the Spring 2026 semester.\nTaylor was also partially supported by NSF grant DMS--2503113 and the Simons Foundation, and Huang was partially supported by NSF grant DMS-2005328.", "sketch": "The proof of \\Cref{thm:vol-circumference} (and \\Cref{thm:short-curves}) is described as follows: it is \"divided into three steps.\" (1) \"In the first step, we identify multicurves on $S$ that have topological significance in $M\\backslash \\!\\! \\backslash S$, and have bounded intersection number with $c^{s/u}$.\" If $M\\backslash \\!\\! \\backslash S$ admits product annuli, these multicurves are \"the components of Thurston's window frame\" and the authors \"show that product annuli can be put into a particularly nice position with respect to the flow\" (\\Cref{sec:annuli}); if there are no product annuli but finite core complexity, they instead use \"the boundary components of a properly embedded transverse surface with minimal complexity\" and establish the intersection bounds in \\Cref{sec:int_bounds}. (2) \"In the second step, we show that these multicurves have bounded length in the hyperbolic metric on $M$\" (\\Cref{sec:geom}). Steps (1) and (2) yield \\Cref{prop:bounded_length}, stating that \"$c^s$ and $c^u$ have bounded intersection number with bounded length curves in $S$.\" (3) \"Finally, we obtain geometric control of $M$ by passing to the quasi-Fuchsian cover corresponding to $S$ and using standard tools from the study of curve graphs and Kleinian surface groups, replacing $c^s$ and $c^u$ with nearby bounded length curves\" (\\Cref{sec:endgame}).", "expanded_sketch": "The proof of the main theorem (and the following theorem) is described as follows: it is \"divided into three steps.\" \n\nWe first state the following theorem.\n\\begin{thm}[Short curves]\\label{thm:short-curves}\nLet $M$ be a closed hyperbolic $3$-manifold with a pseudo-Anosov flow $\\varphi$. Let $S$ be a closed surface in $M$ that is almost transverse to $\\varphi$ and let $Y \\subset S$ be a \nsubsurface of $S$. Assume that $\\mathfrak{c}(M\\cut S)<\\infty$. Then for any $\\epsilon >0$, there is a $K = K(\\epsilon, \\chi(S)) \\ge 0$, depending only on $\\epsilon$ and $\\chi(S)$, such that if\n\\begin{itemize}\n\\item $d_{\\C(S)}(c^{s/u} ,\\partial Y) \\ge k_{M\\cut S}$ and \n\\item $d_{\\C(Y)}(c^s,c^u) \\ge K + 2 \\cdot  k_{M \\cut S}$,\n\\end{itemize}\nthen \n\\[\n\\ell_M(\\partial Y) \\le \\epsilon.\n\\]\n\\end{thm}\n\n(1) \"In the first step, we identify multicurves on $S$ that have topological significance in $M\\backslash \\!\\! \\backslash S$, and have bounded intersection number with $c^{s/u}$.\" If $M\\backslash \\!\\! \\backslash S$ admits product annuli, these multicurves are \"the components of Thurston's window frame\" and the authors \"show that product annuli can be put into a particularly nice position with respect to the flow\" (\\Cref{sec:annuli}); if there are no product annuli but finite core complexity, they instead use \"the boundary components of a properly embedded transverse surface with minimal complexity\" and establish the intersection bounds in \\Cref{sec:int_bounds}. \n\n(2) \"In the second step, we show that these multicurves have bounded length in the hyperbolic metric on $M$\" (\\Cref{sec:geom}). Steps (1) and (2) yield the following proposition, stating that \"$c^s$ and $c^u$ have bounded intersection number with bounded length curves in $S$.\"\n\\begin{proposition}\\label{prop:bounded_length}\nThere are constants $L, d \\ge 1$, depending on $|\\chi(S)|$ and $\\mf c(M \\cut S)$, and curves $\\alpha$ and $\\beta$ in $\\C(S)$ such that: \n\\begin{itemize}\n\\item the lengths of the geodesic representatives of $\\alpha$ and $\\beta$ in $M$ are no more than $L$, and \n\\item the intersection numbers $i(c^s, \\alpha)$ and $i(c^u, \\beta)$ are no more than $d$. \n\\end{itemize}\n\\end{proposition}\n\n(3) \"Finally, we obtain geometric control of $M$ by passing to the quasi-Fuchsian cover corresponding to $S$ and using standard tools from the study of curve graphs and Kleinian surface groups, replacing $c^s$ and $c^u$ with nearby bounded length curves\" (\\Cref{sec:endgame}).", "expanded_theorem": "[Volume and circumference]\\label{thm:vol-circumference}\nLet $M$ be a closed hyperbolic $3$-manifold with a pseudo-Anosov flow $\\varphi$. Let $S$ be a closed connected surface in $M$ that is almost transverse to $\\varphi$, but is not a fiber. Let $c^{s/u}$ be the stable/unstable multicurves of $S$. Assume that $\\mathfrak{c}(M\\backslash \\!\\! \\backslash S)<\\infty$.\n\nThere is a constant $k_S = k_S(\\chi(S)) \\ge 1$, depending only on $\\chi(S)$, and a constant $k_{M \\backslash \\!\\! \\backslash S} \\ge 1$, depending only on $\\chi(S)$ and $\\mathfrak c(M\\backslash \\!\\! \\backslash S)$, so that the following inequalities hold:\n\\begin{enumerate}\n\\item $d_{\\mathcal{C}(S)}(c^s, c^u) \\le k_S \\cdot \\mathrm{vol}(M) + k_{M\\backslash \\!\\! \\backslash S}$, and \n\\smallskip\n\\item $d_{\\mathcal{C}(S)}(c^s, c^u) \\le k_S \\cdot \\ell_M(\\gamma) + k_{M \\backslash \\!\\! \\backslash S}$, where $\\gamma$ is any closed geodesic that intersects $S$ essentially and $\\ell_M(\\gamma)$ is its length in $M$.\n\\end{enumerate}", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let $M$ be a closed hyperbolic $3$-manifold with a pseudo-Anosov flow $\\varphi$, and let $S\\subset M$ be a closed connected surface that is almost transverse to $\\varphi$ but is not a fiber. Let $c^s$ and $c^u$ be the stable and unstable multicurves of $S$, meaning the essential multicurves of closed leaves in the singular foliations on $S$ induced by the stable and unstable invariant foliations of $\\varphi$. Let $M\\backslash \\!\\! \\backslash S$ denote the manifold obtained by cutting $M$ along $S$, with boundary decomposed into incoming and outgoing parts $\\partial_-(M\\backslash \\!\\! \\backslash S)$ and $\\partial_+(M\\backslash \\!\\! \\backslash S)$. Define the core complexity by\n\\[\n\\mathfrak c(M\\backslash \\!\\! \\backslash S)=\n\\begin{cases}\n0, & \\text{if } M\\backslash \\!\\! \\backslash S \\text{ contains a product annulus},\\\\\n\\min\\{-\\chi(\\Sigma)\\}, & \\text{otherwise},\n\\end{cases}\n\\]\nwhere a product annulus is an essential properly embedded annulus with one boundary component on each of $\\partial_-(M\\backslash \\!\\! \\backslash S)$ and $\\partial_+(M\\backslash \\!\\! \\backslash S)$, and in the second case the minimum is over all properly embedded surfaces $\\Sigma\\subset M\\backslash \\!\\! \\backslash S$ that are transverse to a dynamic blowup of $\\varphi$ and meet both $\\partial_-(M\\backslash \\!\\! \\backslash S)$ and $\\partial_+(M\\backslash \\!\\! \\backslash S)$. Assume $\\mathfrak c(M\\backslash \\!\\! \\backslash S)<\\infty$. Here $d_{\\mathcal C(S)}$ denotes distance in the curve graph of $S$, $\\operatorname{vol}(M)$ is the hyperbolic volume of $M$, and $\\ell_M(\\gamma)$ is the hyperbolic length of a closed geodesic $\\gamma$ in $M$. Which quantitative estimates hold?", "correct_choice": {"label": "A", "text": "There exist constants $k_S=k_S(\\chi(S))\\ge 1$, depending only on $\\chi(S)$, and $k_{M\\backslash \\!\\! \\backslash S}\\ge 1$, depending only on $\\chi(S)$ and $\\mathfrak c(M\\backslash \\!\\! \\backslash S)$, such that both of the following hold:\n\\[\nd_{\\mathcal C(S)}(c^s,c^u)\\le k_S\\,\\operatorname{vol}(M)+k_{M\\backslash \\!\\! \\backslash S},\n\\]\nand, for every closed geodesic $\\gamma$ that intersects $S$ essentially,\n\\[\nd_{\\mathcal C(S)}(c^s,c^u)\\le k_S\\,\\ell_M(\\gamma)+k_{M\\backslash \\!\\! \\backslash S}.\n\\]"}, "choices": [{"label": "B", "text": "There exist constants $k_S=k_S(\\chi(S))\\ge 1$, depending only on $\\chi(S)$, and $k_{M\\backslash \\!\\! \\backslash S}\\ge 1$, depending only on $\\chi(S)$ and $\\mathfrak c(M\\backslash \\!\\! \\backslash S)$, such that both of the following hold:\n\\[\nd_{\\mathcal C(S)}(c^s,c^u)\\le k_S\\,\\operatorname{vol}(M)+k_{M\\backslash \\!\\! \\backslash S},\n\\]\nand, for every closed geodesic $\\gamma$ that intersects $S$ essentially,\n\\[\nd_{\\mathcal C(S)}(c^s,c^u)\\le k_S\\,\\ell_M(\\gamma).\n\\]"}, {"label": "C", "text": "There exists a constant $k_{M\\backslash \\!\\! \\backslash S}\\ge 1$, depending only on $\\chi(S)$ and $\\mathfrak c(M\\backslash \\!\\! \\backslash S)$, such that\n\\[\nd_{\\mathcal C(S)}(c^s,c^u)\\le k_{M\\backslash \\!\\! \\backslash S}\\bigl(\\operatorname{vol}(M)+1\\bigr),\n\\]\nand, for every closed geodesic $\\gamma$ that intersects $S$ essentially,\n\\[\nd_{\\mathcal C(S)}(c^s,c^u)\\le k_{M\\backslash \\!\\! \\backslash S}\\bigl(\\ell_M(\\gamma)+1\\bigr).\n\\]"}, {"label": "D", "text": "There exist constants $k_S=k_S(\\chi(S))\\ge 1$, depending only on $\\chi(S)$, and $k_{M\\backslash \\!\\! \\backslash S}\\ge 1$, depending only on $\\mathfrak c(M\\backslash \\!\\! \\backslash S)$, such that both of the following hold:\n\\[\nd_{\\mathcal C(S)}(c^s,c^u)\\le k_S\\,\\operatorname{vol}(M)+k_{M\\backslash \\!\\! \\backslash S},\n\\]\nand, for every closed geodesic $\\gamma$ that intersects $S$ essentially,\n\\[\nd_{\\mathcal C(S)}(c^s,c^u)\\le k_S\\,\\ell_M(\\gamma)+k_{M\\backslash \\!\\! \\backslash S}.\n\\]"}, {"label": "E", "text": "There exist constants $k_S=k_S(\\chi(S))\\ge 1$, depending only on $\\chi(S)$, and $k_{M\\backslash \\!\\! \\backslash S}\\ge 1$, depending only on $\\chi(S)$ and $\\mathfrak c(M\\backslash \\!\\! \\backslash S)$, such that both of the following hold:\n\\[\nd_{\\mathcal C(S)}(c^s,c^u)\\le k_S\\,\\operatorname{vol}(M)+k_{M\\backslash \\!\\! \\backslash S},\n\\]\nand, for every closed geodesic $\\gamma$ in $M$,\n\\[\nd_{\\mathcal C(S)}(c^s,c^u)\\le k_S\\,\\ell_M(\\gamma)+k_{M\\backslash \\!\\! \\backslash S}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "additive error term in geodesic-length estimate", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "separate slope constant $k_S$ absorbed into a single larger constant", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "dependence of the additive constant on $\\chi(S)$ as well as core complexity", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "essential intersection hypothesis on $\\gamma$", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the correct estimates or uniquely reveal the right choice. It supplies substantial setup and notation, but the actual form of the bounds and the precise dependence of constants must still be selected from the options."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall question: the correct option is a near-verbatim statement of the quantitative result, with distractors formed by small perturbations of constants, dependencies, or hypotheses."}, "GPS": {"score": 1, "justification": "Some care is needed to compare subtle variants such as additive constants, dependence on core complexity versus Euler characteristic, and the essential-intersection condition. However, the task mainly tests recognition of the exact theorem statement rather than generating or synthesizing mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: they alter dependence of constants, remove an additive error term, weaken/strengthen the estimate form, or drop a necessary hypothesis on the geodesic. These align well with realistic failure modes."}, "total_score": 5, "overall_assessment": "Well-constructed in terms of plausible distractors and lack of answer leakage, but it is primarily a near-verbatim theorem identification item rather than a genuinely generative reasoning question."}}
{"id": "2602.11720v1", "paper_link": "http://arxiv.org/abs/2602.11720v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{Thm_Uniform_bound}\n\t\tLet $\\Omega=B_1(0)$ be the unit ball in $\\mathbb R^2$, assume conditions $(H_1)$, $(H_2)$, $(H_3)$, $(H_b)$, and alternatively\n\t\t\\begin{enumerate}\n\t\t\t\\item[i)] \\eqref{BM} holds;\n\t\t\t\\item[ii)] \\eqref{nonBM}, $(H_4)$, $(H_5)$, and either $(H_6)$ or $(H_6')$ hold.\n\t\t\\end{enumerate}\n\t\tThen there exists a constant $C>0$ such that\n\t\t\\begin{equation*}\n\t\t\t\\|u\\|_\\infty\\leq C\\qquad\\mbox{and}\\qquad\\|v\\|_\\infty\\leq C\n\t\t\\end{equation*}\n\t\tfor all (eventual) solutions $(u,v)$ of system \\eqref{sys}.", "start_pos": 20717, "end_pos": 21248, "label": "Thm_Uniform_bound"}, "ref_dict": {"abhyp": "\\begin{equation}\\label{abhyp}\n        \\frac1\\alpha+\\frac1\\beta=1\\,,\n    \\end{equation}", "hyp": "\\begin{equation}\\label{hyp}\n        \\frac1{p+1}+\\frac1{q+1}=\\frac{N-2}N\n    \\end{equation}", "NL_model": "\\begin{equation}\\label{NL_model}\n\t\tf(t)\\sim\\etal,\\quad g(t)\\sim\\etb\\qquad\\mbox{with}\\ \\ \\alpha>0,\\ \\beta>0\n\t\\end{equation}", "Prop_Lambda": "\\begin{prop}[Theorem 1.2, \\cite{dFdOR}]\\label{Prop_Lambda}\n\t\tAssume that there exists $c>0$ and $p>0$ such that $f(t)\\leq c\\e^{pt}$ for all $t>0$, or that the same holds for $g$. Then there exists a positive constant $\\Lambda$, depending only on $f$, $g$, and $\\Omega$, such that\n        \\begin{equation}\\label{Lambda}\\tag{$\\Lambda$}\n            \\intOmega f(v)\\leq\\Lambda\\quad\\mbox{and}\\quad\\intOmega g(u)\\leq\\Lambda\n        \\end{equation}\n        for all solutions of \\eqref{sys}.\n\t\\end{prop}", "fig1": "\\label{fig1}\n    \\end{figure}\n\n    In Figure \\ref{fig1} we can see the state of the art for problem \\eqref{sys} for the model nonlinearities in \\eqref{NL_model}. In particular, a priori bounds in the", "Thm_Uniform_bound": "\\begin{thm}\\label{Thm_Uniform_bound}\n\t\tLet $\\Omega=B_1(0)$ be the unit ball in $\\R^2$, assume conditions $(H_1)$, $(H_2)$, $(H_3)$, $(H_b)$, and alternatively\n\t\t\\begin{enumerate}\n\t\t\t\\item[i)] \\eqref{BM} holds;\n\t\t\t\\item[ii)] \\eqref{nonBM}, $(H_4)$, $(H_5)$, and either $(H_6)$ or $(H_6')$ hold.\n\t\t\\end{enumerate}\n\t\tThen there exists a constant $C>0$ such that\n\t\t\\begin{equation*}\n\t\t\t\\|u\\|_\\infty\\leq C\\qquad\\mbox{and}\\qquad\\|v\\|_\\infty\\leq C\n\t\t\\end{equation*}\n\t\tfor all (eventual) solutions $(u,v)$ of system \\eqref{sys}.\n\t\\end{thm}", "sys": "\\begin{equation}\\label{sys}\n\t\t\\begin{cases}\n\t\t\t-\\Delta u=f(v)\\quad&\\mbox{in }\\Omega\\,,\\\\\n\t\t\t-\\Delta v=g(u)\\quad&\\mbox{in }\\Omega\\,,\\\\\n\t\t\tu=v=0\\quad&\\mbox{on }\\dOmega\\,.\n\t\t\\end{cases}\n\t\\end{equation}", "nonBM": "\\begin{equation}\\label{nonBM}\n\t\t\\lim_{t\\to+\\infty}\\frac{f'}f(t)=0\\quad\\mbox{and}\\quad\\lim_{t\\to+\\infty}\\frac{g'}g(t)=+\\infty\n\t\\end{equation}", "b": "\\begin{equation}\\label{b}\n        \\lim_{t\\to+\\infty}\\frac{f'}f(t)=b\\in[0,+\\infty)\\,\n    \\end{equation}", "growth_below_g": "\\begin{lem}\\label{growth_below_g}\n\t\tUnder \\eqref{BM} or \\eqref{nonBM}, there exist $C_1,C_2>0$ and $\\delta_1,\\delta_2>0$ such that $f(s)<C_1\\e^{\\delta_1s}$ and $g(s)>C_2\\e^{\\delta_2s}$ for $s$ large enough.\n\t\\end{lem}", "alpha+beta=2": "\\begin{equation}\\label{alpha+beta=2}\n        \\alpha+\\beta=2\\qquad\\mbox{with}\\qquad\\alpha\\neq1\\neq\\beta\n    \\end{equation}", "TM-subcritiche": "\\begin{lem}\\label{TM-subcritiche}\n\t\tUnder \\eqref{BM} or \\eqref{nonBM}, condition $(H_3)$ implies that for all $p>0$ one has\n        $$\\lim_{t\\to+\\infty}\\frac{f(t)}{t^p}=+\\infty=\\lim_{t\\to+\\infty}\\frac{g(t)}{t^p}$$\n        and, for all $\\gamma>0$,\n        $$\\lim_{t\\to+\\infty}\\frac{f(t)}{\\e^{\\gamma t^2}}=0=\\lim_{t\\to+\\infty}\\frac{g(t)}{\\e^{\\gamma t^2}}\\,.$$\n\t\\end{lem}", "lem_f8_conseq": "\\begin{lem}\\label{lem_f8_conseq}\n\t\tSuppose \\eqref{nonBM}, then\n\t\t\\begin{enumerate}\n\t\t\t\\item[i)] if $(H_6)$ is assumed, then $f(t)\\ges\\e^{\\frac a3(\\log t)^3}$ for $t$ large;\n\t\t\t\\item[ii)] if $(H_6')$ is assumed, then $f(t)\\ges\\e^{\\frac a2\\log t(\\loglog t)^2}\\,$ for $t$ large.\n\t\t\\end{enumerate}\n\t\\end{lem}", "BM": "\\begin{equation}\\label{BM}\n\t\t\\lim_{t\\to+\\infty}\\frac{f'}f(t)=p\\quad\\mbox{and}\\quad\\lim_{t\\to+\\infty}\\frac{g'}g(t)=q\\,.\n\t\\end{equation}"}, "pre_theorem_intro_text_len": 15416, "pre_theorem_intro_text": "Let $\\Omega\\subset\\mathbb R^N$ be a smooth bounded domain and consider the semilinear Hamiltonian elliptic system with Dirichlet boundary conditions\n    \\begin{equation}\\label{sys}\n\t\t\\begin{cases}\n\t\t\t-\\Delta u=f(v)\\quad&\\mbox{in }\\Omega\\,,\\\\\n\t\t\t-\\Delta v=g(u)\\quad&\\mbox{in }\\Omega\\,,\\\\\n\t\t\tu=v=0\\quad&\\mbox{on }{\\partial\\Omega}\\,.\n\t\t\\end{cases}\n\t\\end{equation}\n    It is well known that for $N\\geq3$ polynomial growth conditions at infinity have to be fulfilled in order to set these variational problems in Sobolev spaces and prove existence.\n    If $N\\geq3$, for systems of the kind \\eqref{sys}, the good notion of criticality is represented by a critical hyperbola: in the model case $f(v)=|v|^{p-1}v$ and $g(u)=|u|^{q-1}u$ with $p,q>1$, then the critical hyperbola defined as\n    \\begin{equation}\\label{hyp}\n        \\frac1{p+1}+\\frac1{q+1}=\\frac{N-2}N\n    \\end{equation}\n    divides existence, above \\eqref{hyp}, \\cite{HV,HMV}, from nonexistence, below \\eqref{hyp}, \\cite{M}, of classical positive solutions. Note that if \\eqref{sys} is set in the entire $\\mathbb R^N$ space, then it is proved that the Sobolev hyperbola \\eqref{hyp} plays the same role for $N=3,4$, while it is still an open problem, under the name of Lane-Emden conjecture, for dimensions $N\\ge 5$, see \\cite{M96,PQS,S}. In contrast, much faster growths are admissible in dimension two: in particular, exponential nonlinearities can be treated, with the Trudinger-Moser inequality taking the role of the Sobolev embedding theorem valid for $N\\geq3$. For the system \\eqref{sys} we still can find a notion of critical hyperbola in the setting of Sobolev-Lorentz spaces: to give an idea, if $f(t)\\sim{\\rm e}^{|t|^\\alpha}$ and $g(t)\\sim{\\rm e}^{|t|^\\beta}$ with $0<\\alpha,\\beta<+\\infty$ (here the symbol $\\sim$ means having the same asymptotic growth), then the maximal growth is obtained on the ``conformal'' critical hyperbola\n    \\begin{equation}\\label{abhyp}\n        \\frac1\\alpha+\\frac1\\beta=1\\,,\n    \\end{equation}\n    see \\cite{Ruf_sys} and the more recent advance in \\cite{dORR}.\n\n    When one treats equations or systems which are not necessarily variational (e.g. \\eqref{sys} when $f$ and $g$ depend on both variables $u$ and $v$), a typical technique to prove the existence of solutions is the topological degree theory. An essential step in the proof is to provide a priori uniform bounds for solutions in $L^\\infty$ norm. In fact, from this one infers that a certain fixed point (or Leray-Schauder) index in a large ball of the functional space is zero; moreover, by imposing some natural assumptions on the nonlinearities at $0$, the index in a neighbourhood of the origin is not zero, and therefore, by additivity of the index, one finds a nontrivial solution.    \n\n    This strategy has been extensively employed in various contexts, and the study of $L^\\infty$ a priori bounds for solutions has become a topic of independent interest in the literature. For the scalar case, we refer to the classical papers by Gidas and Spruck \\cite{GS} and de Figueiredo, Lions, and Nussbaum \\cite{dFLN} for semilinear problems involving subcritical polynomial nonlinearities in dimension $N\\geq3$. Such results have been extended to the higher-order and quasilinear settings, and for wider classes of subcritical nonlinearities, see e.g. \\cite{RW,AC,Ruiz,Zou,BF_eq,DP}.\tIn broad terms, these results indicate that a priori bounds for strong as well as distributional solutions can be established for (almost) all subcritical nonlinearities, up to the threshold of Sobolev critical growth. Concerning the system \\eqref{sys} with $N\\geq3$, the situation is comparable, in the sense that the critical hyperbola still represents the threshold for both existence and $L^\\infty$ a priori bounds, see among others \\cite{CdFM,Zou_sys,QS,PQS,BF}.\n\n\tThe situation considerably changes in the case of the \\textit{conformal dimension}, where the critical threshold for the existence $t\\mapsto{\\rm e}^{t^{\\frac N{N-1}}}$ is given by the Trudinger-Moser inequality. Here, as the seminal work of Brezis-Merle \\cite{BM} shows for the scalar second-order case in dimension $N=2$, the limiting growth up to which one may expect to find a priori bounds within the class of \\textit{distributional} solutions of $-\\Delta u=f(x,u)$ with Dirichlet boundary conditions is only $t\\mapsto{\\rm e}^t$. Within this setting, in \\cite{BM} the authors prove a priori bounds for nonlinearities bounded from below and above by ${\\rm e}^t$, under the assumption $\\int_\\Omega f(x,u)\\, {\\rm d} x\\leq\\Lambda$; see also \\cite{ChenLi}. We note that such a condition follows from the analysis near the boundary already obtained in \\cite{dFLN}. More in general, the respective quasilinear problem involving $-\\Delta_N$ in $\\Omega\\subset\\mathbb R^N$ with $N\\geq2$, was investigated in \\cite{LRU}, where the nonlinearities behave either like ${\\rm e}^{pt}$ for some $p>0$, or grow less than ${\\rm e}^{t^\\alpha}$ with $\\alpha\\in(0,1)$. The authors use regularity estimates in Orlicz spaces to cover the latter case, while extend the arguments from \\cite{BM} in the former. This diversity of the method causes a gap between the two classes of nonlinearities, and e.g. $f(t)=\\tfrac{{\\rm e}^t}{t+1}$ could not be treated with such techniques. This gap was filled in \\cite{Rom_N}, where instead a blow-up approach was used. The same technique has been also applied in \\cite{MR} to the respective problem of higher-order, namely when the operator is $(-\\Delta)^m$ in $\\mathbb R^{2m}$, $m\\geq2$, in both cases of Dirichlet and Navier boundary conditions. In these works, the key assumption which allows to deal at once with both classes of nonlinearities in \\cite{LRU}, is the existence of\n    \\begin{equation}\\label{b}\n        \\lim_{t\\to+\\infty}\\frac{f'}f(t)=b\\in[0,+\\infty)\\,\n    \\end{equation}\n    where the case $b>0$ corresponds to the critical setting in the sense of Brezis and Merle, while $b=0$ for all subcritical growths. We point out that with a blow-up approach one may further derive uniform bounds for \\textit{weak solutions} of Dirichlet problems with nonlinearity in the class $f(x,u)=u^{p-1}{\\rm e}^{u^p}$ even for $p\\in(1,2]$, that is, up to the critical Trudinger-Moser growth, under different integral bounds of energy type, see \\cite[Theorem 2.2]{DT} and \\cite[Theorem 2.1]{MMT}; see also \\cite{RobWei,D} for related results.\n\n\tWhen dealing with planar systems, a priori bounds have been investigated in \\cite{dFdOR} for systems of the kind \\eqref{sys} where $f,g$ are positive nonlinearities (which may also depend on $x$), and then extended also for nonvariational elliptic systems in \\cite{dFdOR2}. First, by a moving plane techniques the authors find uniform bounds of the kind $\\int_\\Omega f(v)\\, {\\rm d} x\\leq\\Lambda$ and $\\int_\\Omega g(u)\\, {\\rm d} x\\leq\\Lambda$. Then, in order to apply either Orlicz spaces techniques or the argument inspired by \\cite{BM}, again as in \\cite{LRU}, a dichotomy in the assumptions was needed: in fact, they impose that $f$ and $g$ grow either less than ${\\rm e}^{t^\\alpha}$ and ${\\rm e}^{t^\\beta}$, respectively, with $\\alpha,\\beta>0$ and $\\alpha+\\beta<2$, or both like ${\\rm e}^t$. In addition to the already recalled gap between the growths ${\\rm e}^{t^\\alpha}$ and ${\\rm e}^t$, the limiting case     \\begin{equation}\\label{alpha+beta=2}\n        \\alpha+\\beta=2\\qquad\\mbox{with}\\qquad\\alpha\\neq1\\neq\\beta\n    \\end{equation}\n    is stated as an open problem in \\cite[Remark 1.3]{dFdOR}.\n\n\tThe main purpose of this article is to progress with the study of a priori bounds for system \\eqref{sys} in the conformal setting $N=2$, by covering this case of limiting nonlinearities (which from now on will be called \\textit{critical in the sense of Brezis-Merle}), by performing a blow-up analysis in the spirit of \\cite{Rom_N,MR}. For the model nonlinearities ${\\rm e}^{t^\\alpha}$ and ${\\rm e}^{t^\\beta}$, this will correspond to prove such bounds under \\eqref{alpha+beta=2}.  In the companion paper \\cite{BMR_sub} we treat instead the subcritical case. We note indeed that such a case is not fully covered by \\cite[Theorem 1.3]{dFdOR}, since there are nonlinearities for which $b=0$ but however grow more than any function of the kind ${\\rm e}^{pt^\\alpha}$ irrespective of the choices of $p\\geq1$ and $\\alpha\\in(0,1)$. For instance, it is sufficient to consider $g(t)={\\rm e}^t$ and $f(t)={\\rm e}^{\\frac t{\\log t}}$ so that $\\slfrac{f'}f(t)=\\frac{\\log t-1}{(\\log t)^2}\\sim\\frac1{\\log t}\\to0$, but slower than any function of the kind ${\\rm e}^{pt^\\alpha}$. Crucial steps in our methods will be first to identify the right assumption on $f$ and $g$, which detects the ``critical'' nonlinearities and extends \\eqref{b} for systems, and next to determine a suitable scaling, that allows to find a limit system in $\\mathbb R^2$ to which one may apply a Liouville-type argument. During the blow-up analysis, several technical difficulties will arise due to the non-scalar nature of the problem. In fact, in the proof there will be a continuous exchange of information between the two components of the system. As a first contribution, here we restrict to consider the symmetric case of $\\Omega$ being the ball $B_1(0)$: indeed, in this setting we can take advantage to the Schwarz symmetry of the solutions $(u,v)$, which implies that both components cannot have blow-up points except for the origin, and this simplifies to some extent our arguments. However, we believe that with a nontrivial effort one can also apply our strategy even in the case of a general bounded domain of $\\mathbb R^2$, also in view of the $L^1$-bounds already proved in \\cite{dFdOR}.\n\n    We also mention that blow-up techniques for elliptic systems have been extensively employed in the framework of Toda-type systems,\n    where, differently from \\eqref{sys}, a strong algebraic structure and integrability properties allow a detailed analysis of bubbling solutions and mass quantization phenomena, see e.g.  \\cite{JostWang,LinWeiZhao,LinWeiYangZhang} and references therein. However, the techniques developed in that setting strongly exploit the specific Cartan matrix structure of the  system and therefore do not appear to be directly transferable to the Hamiltonian systems studied here. To the best of our knowledge, we believe that the approach we follow here is new in the literature, and we hope it could serve for other kind of problems.\n\t\\vskip0.2truecm\n\tLet us now enter into the details of our results.\n\n\t\\subsection{Assumptions and main results}\n\n\tHaving in mind the model growths\n\t\\begin{equation}\\label{NL_model}\n\t\tf(t)\\sim{\\rm e}^{t^\\alpha},\\quad g(t)\\sim{\\rm e}^{t^\\beta}\\qquad\\mbox{with}\\ \\ \\alpha>0,\\ \\beta>0\n\t\\end{equation}\n\twith the critical condition\n\t\\begin{equation*}\n\t\t\\alpha+\\beta=2\\,,\n\t\\end{equation*}\n\twe consider the following assumptions:\n\t\\begin{enumerate}\n\t\t\\item[$(H_1)$] $f,g\\in C^1(\\overline{\\mathbb R^+})$, positive in $\\mathbb R^+$, such that $\\displaystyle\\lim_{t\\to+\\infty}f(t)=+\\infty$ and $\\displaystyle\\lim_{t\\to+\\infty}g(t)=+\\infty$;\n\t\t\\item[$(H_2)$] there exist $t_0>0$ such that $f,g$ are strictly increasing in $(t_0,+\\infty)$;\n        \\item[$(H_3)$] $\\displaystyle\\left(\\frac f{f'}\\right)'\\!\\!(t)\\to0$ and $\\displaystyle\\left(\\frac{g'}g\\right)'\\!\\!(t)\\to0$ as $t\\to+\\infty$;\n\t\t\\item[$(H_b)$] there exists $\\displaystyle\\lim_{t\\to+\\infty}\\,{\\frac{f'}f(t)\\,\\frac{g'}g(t)}:=b\\in(0,+\\infty)$.\n\t\\end{enumerate}\n\tNote that $\\slfrac{f'}f$ and $\\slfrac{g'}g$ are the derivatives of the exponents of $f={\\rm e}^{\\log f}$ and $g={\\rm e}^{\\log g}$, respectively. The case $b=0$ corresponds for the model nonlinearities in \\eqref{NL_model} to the case $\\alpha+\\beta<2$, and therefore will be denoted as \\textit{Brezis-Merle subcritical}. On the other hand, $\\alpha+\\beta=2$ implies $b=\\alpha\\beta>0$, and we will call it \\textit{Brezis-Merle critical case}. In this paper we will only deal with the latter case, postponing the analysis of the subcritical one in the companion paper \\cite{BMR_sub}, since substantial differences in the respective blow-up arguments will occur.\n\n    In this critical setting, we will also need to distinguish between the case for which the limits of both ratios in $(H_b)$ exist and are finite, from the remaining critical cases. We call the former scenario \\textit{Liouville}, which then occurs when there exist $p,q>0$ such that\n\t\\begin{equation}\\label{BM}\n\t\t\\lim_{t\\to+\\infty}\\frac{f'}f(t)=p\\quad\\mbox{and}\\quad\\lim_{t\\to+\\infty}\\frac{g'}g(t)=q\\,.\n\t\\end{equation}\n\tIn the latter case, which we call \\textit{non-Liouville}, we may instead suppose\n\t\\begin{equation}\\label{nonBM}\n\t\t\\lim_{t\\to+\\infty}\\frac{f'}f(t)=0\\quad\\mbox{and}\\quad\\lim_{t\\to+\\infty}\\frac{g'}g(t)=+\\infty\n\t\\end{equation}\n\twithout loss of generality, since the roles of $f$ and $g$ in \\eqref{sys} are symmetric. In this last context, we need some monotonicities and growth conditions on $f$ and $g$, \tnamely we assume that there exists $t_0>0$ such that\n\t\\begin{enumerate}\n        \\item[$(H_4)$] $\\displaystyle\\left(\\frac f{f'}\\right)'\\!\\!(t)\\geq 0$ and $\\displaystyle\\left(\\frac{g'}g\\right)'\\!\\!(t)\\geq 0$ as $t>t_0$;\n        \\item[$(H_5)$] the map $\\displaystyle t\\mapsto t\\,\\frac{f'}f(t)$ is increasing and $\\displaystyle t\\mapsto\\frac1t\\,\\frac{g'}g(t)$ is decreasing in $(t_0,+\\infty)$.\n    \\end{enumerate}\n\tNote that the existence of the two limits in \\eqref{nonBM} is a consequence of assumptions $(H_b)$ and $(H_4)$. Finally, we suppose one of the following conditions on the growth of $f$: there exists $a>1$ such that \n\t\\begin{enumerate}\n\t\t\\item[$(H_6)$] $\\displaystyle t\\,\\frac{f'}f(t)\\geq a(\\log t)^2$ for all $t>t_0$;\n\t\t\\item[$(H_6')$] $\\displaystyle t\\,\\frac{f'}f(t)\\geq a(\\log\\log F(t))^2$ for all $t>t_0$, where $\\displaystyle F(t):=\\int_0^tf(\\tau)\\, {\\rm d}\\tau$.\n\t\\end{enumerate}\n\n    \\vskip0.2truecm\n    Before presenting the main result of the paper, let us briefly discuss the assumptions introduced above. First, observe that, except $(H_1)$, are all assumptions on the behaviour at $\\infty$. While $(H_1)$ and $(H_2)$ are standard conditions, $(H_3)$ ensures subcriticality in the sense of Trudinger-Moser and that the growth of $f$ and $g$ are superpolynomial (see Lemma \\ref{TM-subcritiche} below). The key assumption $(H_b)$, which characterises the problem as critical, can be regarded as the vectorial counterpart of \\eqref{b}, see $(A3)$ in \\cite{Rom_N} or \\cite{MR}, where the scalar case is considered. In the non-Liouville setting, i.e., under condition \\eqref{nonBM}, assumptions $(H_4)$ and $(H_5)$ provide suitable monotonicity properties, which are instrumental for our arguments. Note that the existence of the two limits in \\eqref{nonBM} is a consequence of the assumptions $(H_b)$ and $(H_4)$. Finally, $(H_6)$ and $(H_6')$ are technical assumptions which imply a mild lower bound on the growth of $f$, which has to be of the kind ${\\rm e}^{\\log t(\\log\\log t)^p}$ (see Lemma \\ref{lem_f8_conseq} below). Note that, although $(H_6)$ is of easier verification than $(H_6')$, the latter permits to include a larger class of nonlinearities. A more detailed analysis of the consequences of our assumptions can be found in Section \\ref{Sec_Prel}.\n\n    \\vskip0.2truecm\n\n\tSince in the blow-up analysis we will need the radial symmetry of the solutions, as already mentioned in the Introduction, we will only deal with the case of $\\Omega$ be a ball in $\\mathbb R^2$.\n    Our main result is the following:", "context": "Let $\\Omega\\subset\\mathbb R^N$ be a smooth bounded domain and consider the semilinear Hamiltonian elliptic system with Dirichlet boundary conditions\n \\begin{equation}\\label{sys}\n \\begin{cases}\n -\\Delta u=f(v)\\quad&\\mbox{in }\\Omega\\,,\\\\\n -\\Delta v=g(u)\\quad&\\mbox{in }\\Omega\\,,\\\\\n u=v=0\\quad&\\mbox{on }{\\partial\\Omega}\\,.\n \\end{cases}\n \\end{equation}\n It is well known that for $N\\geq3$ polynomial growth conditions at infinity have to be fulfilled in order to set these variational problems in Sobolev spaces and prove existence.\n If $N\\geq3$, for systems of the kind \\eqref{sys}, the good notion of criticality is represented by a critical hyperbola: in the model case $f(v)=|v|^{p-1}v$ and $g(u)=|u|^{q-1}u$ with $p,q>1$, then the critical hyperbola defined as\n \\begin{equation}\\label{hyp}\n \\frac1{p+1}+\\frac1{q+1}=\\frac{N-2}N\n \\end{equation}\n divides existence, above \\eqref{hyp}, \\cite{HV,HMV}, from nonexistence, below \\eqref{hyp}, \\cite{M}, of classical positive solutions. Note that if \\eqref{sys} is set in the entire $\\mathbb R^N$ space, then it is proved that the Sobolev hyperbola \\eqref{hyp} plays the same role for $N=3,4$, while it is still an open problem, under the name of Lane-Emden conjecture, for dimensions $N\\ge 5$, see \\cite{M96,PQS,S}. In contrast, much faster growths are admissible in dimension two: in particular, exponential nonlinearities can be treated, with the Trudinger-Moser inequality taking the role of the Sobolev embedding theorem valid for $N\\geq3$. For the system \\eqref{sys} we still can find a notion of critical hyperbola in the setting of Sobolev-Lorentz spaces: to give an idea, if $f(t)\\sim{\\rm e}^{|t|^\\alpha}$ and $g(t)\\sim{\\rm e}^{|t|^\\beta}$ with $0<\\alpha,\\beta<+\\infty$ (here the symbol $\\sim$ means having the same asymptotic growth), then the maximal growth is obtained on the ``conformal'' critical hyperbola\n \\begin{equation}\\label{abhyp}\n \\frac1\\alpha+\\frac1\\beta=1\\,,\n \\end{equation}\n see \\cite{Ruf_sys} and the more recent advance in \\cite{dORR}.\n\nHaving in mind the model growths\n \\begin{equation}\\label{NL_model}\n f(t)\\sim{\\rm e}^{t^\\alpha},\\quad g(t)\\sim{\\rm e}^{t^\\beta}\\qquad\\mbox{with}\\ \\ \\alpha>0,\\ \\beta>0\n \\end{equation}\n with the critical condition\n \\begin{equation*}\n \\alpha+\\beta=2\\,,\n \\end{equation*}\n we consider the following assumptions:\n \\begin{enumerate}\n \\item[$(H_1)$] $f,g\\in C^1(\\overline{\\mathbb R^+})$, positive in $\\mathbb R^+$, such that $\\displaystyle\\lim_{t\\to+\\infty}f(t)=+\\infty$ and $\\displaystyle\\lim_{t\\to+\\infty}g(t)=+\\infty$;\n \\item[$(H_2)$] there exist $t_0>0$ such that $f,g$ are strictly increasing in $(t_0,+\\infty)$;\n \\item[$(H_3)$] $\\displaystyle\\left(\\frac f{f'}\\right)'\\!\\!(t)\\to0$ and $\\displaystyle\\left(\\frac{g'}g\\right)'\\!\\!(t)\\to0$ as $t\\to+\\infty$;\n \\item[$(H_b)$] there exists $\\displaystyle\\lim_{t\\to+\\infty}\\,{\\frac{f'}f(t)\\,\\frac{g'}g(t)}:=b\\in(0,+\\infty)$.\n \\end{enumerate}\n Note that $\\slfrac{f'}f$ and $\\slfrac{g'}g$ are the derivatives of the exponents of $f={\\rm e}^{\\log f}$ and $g={\\rm e}^{\\log g}$, respectively. The case $b=0$ corresponds for the model nonlinearities in \\eqref{NL_model} to the case $\\alpha+\\beta<2$, and therefore will be denoted as \\textit{Brezis-Merle subcritical}. On the other hand, $\\alpha+\\beta=2$ implies $b=\\alpha\\beta>0$, and we will call it \\textit{Brezis-Merle critical case}. In this paper we will only deal with the latter case, postponing the analysis of the subcritical one in the companion paper \\cite{BMR_sub}, since substantial differences in the respective blow-up arguments will occur.\n\nIn this critical setting, we will also need to distinguish between the case for which the limits of both ratios in $(H_b)$ exist and are finite, from the remaining critical cases. We call the former scenario \\textit{Liouville}, which then occurs when there exist $p,q>0$ such that\n \\begin{equation}\\label{BM}\n \\lim_{t\\to+\\infty}\\frac{f'}f(t)=p\\quad\\mbox{and}\\quad\\lim_{t\\to+\\infty}\\frac{g'}g(t)=q\\,.\n \\end{equation}\n In the latter case, which we call \\textit{non-Liouville}, we may instead suppose\n \\begin{equation}\\label{nonBM}\n \\lim_{t\\to+\\infty}\\frac{f'}f(t)=0\\quad\\mbox{and}\\quad\\lim_{t\\to+\\infty}\\frac{g'}g(t)=+\\infty\n \\end{equation}\n without loss of generality, since the roles of $f$ and $g$ in \\eqref{sys} are symmetric. In this last context, we need some monotonicities and growth conditions on $f$ and $g$, namely we assume that there exists $t_0>0$ such that\n \\begin{enumerate}\n \\item[$(H_4)$] $\\displaystyle\\left(\\frac f{f'}\\right)'\\!\\!(t)\\geq 0$ and $\\displaystyle\\left(\\frac{g'}g\\right)'\\!\\!(t)\\geq 0$ as $t>t_0$;\n \\item[$(H_5)$] the map $\\displaystyle t\\mapsto t\\,\\frac{f'}f(t)$ is increasing and $\\displaystyle t\\mapsto\\frac1t\\,\\frac{g'}g(t)$ is decreasing in $(t_0,+\\infty)$.\n \\end{enumerate}\n Note that the existence of the two limits in \\eqref{nonBM} is a consequence of assumptions $(H_b)$ and $(H_4)$. Finally, we suppose one of the following conditions on the growth of $f$: there exists $a>1$ such that \n \\begin{enumerate}\n \\item[$(H_6)$] $\\displaystyle t\\,\\frac{f'}f(t)\\geq a(\\log t)^2$ for all $t>t_0$;\n \\item[$(H_6')$] $\\displaystyle t\\,\\frac{f'}f(t)\\geq a(\\log\\log F(t))^2$ for all $t>t_0$, where $\\displaystyle F(t):=\\int_0^tf(\\tau)\\, {\\rm d}\\tau$.\n \\end{enumerate}\n\n\\vskip0.2truecm\n\nSince in the blow-up analysis we will need the radial symmetry of the solutions, as already mentioned in the Introduction, we will only deal with the case of $\\Omega$ be a ball in $\\mathbb R^2$.\n Our main result is the following:", "full_context": "Let $\\Omega\\subset\\mathbb R^N$ be a smooth bounded domain and consider the semilinear Hamiltonian elliptic system with Dirichlet boundary conditions\n \\begin{equation}\\label{sys}\n \\begin{cases}\n -\\Delta u=f(v)\\quad&\\mbox{in }\\Omega\\,,\\\\\n -\\Delta v=g(u)\\quad&\\mbox{in }\\Omega\\,,\\\\\n u=v=0\\quad&\\mbox{on }{\\partial\\Omega}\\,.\n \\end{cases}\n \\end{equation}\n It is well known that for $N\\geq3$ polynomial growth conditions at infinity have to be fulfilled in order to set these variational problems in Sobolev spaces and prove existence.\n If $N\\geq3$, for systems of the kind \\eqref{sys}, the good notion of criticality is represented by a critical hyperbola: in the model case $f(v)=|v|^{p-1}v$ and $g(u)=|u|^{q-1}u$ with $p,q>1$, then the critical hyperbola defined as\n \\begin{equation}\\label{hyp}\n \\frac1{p+1}+\\frac1{q+1}=\\frac{N-2}N\n \\end{equation}\n divides existence, above \\eqref{hyp}, \\cite{HV,HMV}, from nonexistence, below \\eqref{hyp}, \\cite{M}, of classical positive solutions. Note that if \\eqref{sys} is set in the entire $\\mathbb R^N$ space, then it is proved that the Sobolev hyperbola \\eqref{hyp} plays the same role for $N=3,4$, while it is still an open problem, under the name of Lane-Emden conjecture, for dimensions $N\\ge 5$, see \\cite{M96,PQS,S}. In contrast, much faster growths are admissible in dimension two: in particular, exponential nonlinearities can be treated, with the Trudinger-Moser inequality taking the role of the Sobolev embedding theorem valid for $N\\geq3$. For the system \\eqref{sys} we still can find a notion of critical hyperbola in the setting of Sobolev-Lorentz spaces: to give an idea, if $f(t)\\sim{\\rm e}^{|t|^\\alpha}$ and $g(t)\\sim{\\rm e}^{|t|^\\beta}$ with $0<\\alpha,\\beta<+\\infty$ (here the symbol $\\sim$ means having the same asymptotic growth), then the maximal growth is obtained on the ``conformal'' critical hyperbola\n \\begin{equation}\\label{abhyp}\n \\frac1\\alpha+\\frac1\\beta=1\\,,\n \\end{equation}\n see \\cite{Ruf_sys} and the more recent advance in \\cite{dORR}.\n\nHaving in mind the model growths\n \\begin{equation}\\label{NL_model}\n f(t)\\sim{\\rm e}^{t^\\alpha},\\quad g(t)\\sim{\\rm e}^{t^\\beta}\\qquad\\mbox{with}\\ \\ \\alpha>0,\\ \\beta>0\n \\end{equation}\n with the critical condition\n \\begin{equation*}\n \\alpha+\\beta=2\\,,\n \\end{equation*}\n we consider the following assumptions:\n \\begin{enumerate}\n \\item[$(H_1)$] $f,g\\in C^1(\\overline{\\mathbb R^+})$, positive in $\\mathbb R^+$, such that $\\displaystyle\\lim_{t\\to+\\infty}f(t)=+\\infty$ and $\\displaystyle\\lim_{t\\to+\\infty}g(t)=+\\infty$;\n \\item[$(H_2)$] there exist $t_0>0$ such that $f,g$ are strictly increasing in $(t_0,+\\infty)$;\n \\item[$(H_3)$] $\\displaystyle\\left(\\frac f{f'}\\right)'\\!\\!(t)\\to0$ and $\\displaystyle\\left(\\frac{g'}g\\right)'\\!\\!(t)\\to0$ as $t\\to+\\infty$;\n \\item[$(H_b)$] there exists $\\displaystyle\\lim_{t\\to+\\infty}\\,{\\frac{f'}f(t)\\,\\frac{g'}g(t)}:=b\\in(0,+\\infty)$.\n \\end{enumerate}\n Note that $\\slfrac{f'}f$ and $\\slfrac{g'}g$ are the derivatives of the exponents of $f={\\rm e}^{\\log f}$ and $g={\\rm e}^{\\log g}$, respectively. The case $b=0$ corresponds for the model nonlinearities in \\eqref{NL_model} to the case $\\alpha+\\beta<2$, and therefore will be denoted as \\textit{Brezis-Merle subcritical}. On the other hand, $\\alpha+\\beta=2$ implies $b=\\alpha\\beta>0$, and we will call it \\textit{Brezis-Merle critical case}. In this paper we will only deal with the latter case, postponing the analysis of the subcritical one in the companion paper \\cite{BMR_sub}, since substantial differences in the respective blow-up arguments will occur.\n\nIn this critical setting, we will also need to distinguish between the case for which the limits of both ratios in $(H_b)$ exist and are finite, from the remaining critical cases. We call the former scenario \\textit{Liouville}, which then occurs when there exist $p,q>0$ such that\n \\begin{equation}\\label{BM}\n \\lim_{t\\to+\\infty}\\frac{f'}f(t)=p\\quad\\mbox{and}\\quad\\lim_{t\\to+\\infty}\\frac{g'}g(t)=q\\,.\n \\end{equation}\n In the latter case, which we call \\textit{non-Liouville}, we may instead suppose\n \\begin{equation}\\label{nonBM}\n \\lim_{t\\to+\\infty}\\frac{f'}f(t)=0\\quad\\mbox{and}\\quad\\lim_{t\\to+\\infty}\\frac{g'}g(t)=+\\infty\n \\end{equation}\n without loss of generality, since the roles of $f$ and $g$ in \\eqref{sys} are symmetric. In this last context, we need some monotonicities and growth conditions on $f$ and $g$, namely we assume that there exists $t_0>0$ such that\n \\begin{enumerate}\n \\item[$(H_4)$] $\\displaystyle\\left(\\frac f{f'}\\right)'\\!\\!(t)\\geq 0$ and $\\displaystyle\\left(\\frac{g'}g\\right)'\\!\\!(t)\\geq 0$ as $t>t_0$;\n \\item[$(H_5)$] the map $\\displaystyle t\\mapsto t\\,\\frac{f'}f(t)$ is increasing and $\\displaystyle t\\mapsto\\frac1t\\,\\frac{g'}g(t)$ is decreasing in $(t_0,+\\infty)$.\n \\end{enumerate}\n Note that the existence of the two limits in \\eqref{nonBM} is a consequence of assumptions $(H_b)$ and $(H_4)$. Finally, we suppose one of the following conditions on the growth of $f$: there exists $a>1$ such that \n \\begin{enumerate}\n \\item[$(H_6)$] $\\displaystyle t\\,\\frac{f'}f(t)\\geq a(\\log t)^2$ for all $t>t_0$;\n \\item[$(H_6')$] $\\displaystyle t\\,\\frac{f'}f(t)\\geq a(\\log\\log F(t))^2$ for all $t>t_0$, where $\\displaystyle F(t):=\\int_0^tf(\\tau)\\, {\\rm d}\\tau$.\n \\end{enumerate}\n\n\\vskip0.2truecm\n\nSince in the blow-up analysis we will need the radial symmetry of the solutions, as already mentioned in the Introduction, we will only deal with the case of $\\Omega$ be a ball in $\\mathbb R^2$.\n Our main result is the following:\n\n\\vskip0.2truecm\n\nNote that, under such growth assumptions, one can easily show that $f$ grows less than or comparable to $\\e^{pt}$ for some $p>0$ (see Lemma \\ref{growth_below_g} below). Hence we may apply the regularity result \\cite[Theorem 1.1]{dFdOR} and conclude that distributional solutions of \\eqref{sys} are indeed classical. Therefore, from now on we will simply talk about \\textit{solutions of \\eqref{sys}}.\n\nWe follow the approach of \\cite{QS}, which is based on the \\textit{Fixed Point} (or Leray-Schauder) Index theory, and we refer to \\cite{AdF} for its definition and basic properties. In what follows in view of Theorem \\ref{Thm_Existence} we consider $\\Omega=B_1(0)$, however, once such a priori bound is recovered in a smooth bounded domain $\\Omega\\subset\\R^2$, the proof of Theorem \\ref{Thm_Uniform_bound} works without changes.\n \\vskip0.2truecm\n Let $K$ be the cone of functions in $C(\\Omegabar)\\times C(\\Omegabar)$ which are positive in $\\Omega$ and denote by $S$ the solution operator of the system\n \\begin{equation*}\n \\begin{cases}\n -\\Delta u=\\phi\\quad&\\mbox{in }\\Omega\\,,\\\\\n -\\Delta v=\\psi\\quad&\\mbox{in }\\Omega\\,,\\\\\n u=v=0\\quad&\\mbox{on }\\dOmega\\,,\n \\end{cases}\n \\end{equation*}\n namely $S(\\phi,\\psi)=(u,v)$. Note that $S$ is a linear compact operator from $C(\\Omegabar)\\times C(\\Omegabar)$ into itself. Hence solutions of \\eqref{sys} correspond to fixed points of the map $T:(u,v)\\mapsto T(u,v):=S(f(v),g(u))$, which is compact by $(H_1)$.\n For $W\\subset K$ relatively open such that $Tz\\neq z$ for all $z\\in\\partial W$ we denote by $i_K(T,W)$ the \n fixed point index of $T$ in $W$ with respect to $K$. We are going to prove that there exist $0<r<R$ such that $i_K(T,W_R\\setminus\\overline{W_r})\\neq0$, by showing that $i_K(T,W_r)=1$ while $i_K(T,W_R)=0$. Here $W_s:=B_s(0)\\cap K$, where $B_s(0)$ for $s>0$ denotes the ball in $C(\\Omegabar)\\times C(\\Omegabar)$ of radius $s$ and centred at $0$. To prove that there exists $r$ small such that the index in $W_r$ equals $1$, we make use of the superlinearity of $f$ and $g$ at $0$; instead, to prove the index vanishes in $W_R$ for a large $R$, we exploit the a priori bound by Theorem \\ref{Thm_Uniform_bound} together with the superlinearity of the nonlinearities at $\\infty$.\n \\vskip0.2truecm\n Let us introduce the homothety $H_1(t,u,v)=tT(u,v)$ and prove that there exists $r>0$ such that $H_1(t,u,v)\\neq(u,v)$ for all $t\\in[0,1]$ if $(u,v)\\in\\partial W_r$. To this aim fix $\\varepsilon\\in(0,\\lambda_1)$ and by $(H_9)$ take $r>0$ such that if $\\|u\\|_\\infty+\\|v\\|_\\infty=r$, one has $f(v)\\leq\\varepsilon v$ and $g(u)\\leq\\varepsilon u$. Introduce $\\phi_1$ as the first (positive) eigenfunction of $-\\Delta$ in $\\Omega$. Supposing that there exists $(u,v)$ such that $H_1(t,u,v)=(u,v)$, namely a solution of\n \\begin{equation*}\n \\begin{cases}\n -\\Delta u=tf(v)\\quad&\\mbox{in }\\Omega\\,,\\\\\n -\\Delta v=tg(u)\\quad&\\mbox{in }\\Omega\\,,\\\\\n u=v=0\\quad&\\mbox{on }\\dOmega\\,,\n \\end{cases}\n \\end{equation*}\n we test the system with $\\phi_1$ and sum the two equations:\n \\begin{equation*}\n \\lambda_1\\intOmega(u+v)\\phi_1=\\intOmega(\\nabla u+\\nabla v)\\nabla \\phi_1=t\\intOmega\\left(f(v)+g(u)\\right)\\phi_1\\leq\\varepsilon t\\intOmega(u+v)\\phi_1\\,,\n \\end{equation*}\n which is a contradiction since $t\\in[0,1]$ and $\\varepsilon<\\lambda_1$. As a consequence,\n \\begin{equation}\\label{Index_1}\n i_K(T,W_r)=i_K\\left(H_1(1,\\cdot,\\cdot),W_r\\right)=i_K\\left(H_1(0,\\cdot,\\cdot),W_r\\right)=i_K(0,W_r)=1\\,.\n \\end{equation}\n\nDefine now a second homothety $H_2(\\mu,u,v)=S( f(v+\\mu),g(u+\\mu))$ where $\\mu\\ge 0$, the fixed point of which are solutions of\n \\begin{equation}\\label{sys_mu}\n \\begin{cases}\n -\\Delta u=f(v+\\mu)\\quad&\\mbox{in }\\Omega\\,,\\\\\n -\\Delta v=g(u+\\mu)\\quad&\\mbox{in }\\Omega\\,,\\\\\n u=v=0\\quad&\\mbox{on }\\dOmega\\,.\n \\end{cases}\n \\end{equation}\n By $(H_3)$ we know that both $f$ and $g$ are superlinear (see Lemma \\ref{TM-subcritiche}), that is, for any $M>0$ there exists $t_0>0$ such that for all $t>t_0$ one has $f(t)>Mt$ and $g(t)>Mt$. Again by testing the system with $\\phi_1$, we obtain\n \\begin{equation*}\n \\begin{split}\n \\intOmega f(v+\\mu)\\phi_1&=\\intOmega\\nabla u\\nabla\\phi_1=\\lambda_1\\left(\\int_{\\{u\\leq t_0\\}}u\\phi_1+\\int_{\\{u>t_0\\}}u\\phi_1\\right)\\\\\n &\\leq\\lambda_1\\left(t_0\\|\\phi_1\\|_1+\\frac1M\\intOmega g(u+\\mu)\\phi_1\\right),\n \\end{split}\n \\end{equation*}\n since $\\mu\\geq0$ and $\\phi_1>0$. Similarly\n \\begin{equation*}\n \\intOmega g(u+\\mu)\\phi_1=\\intOmega\\nabla v\\nabla\\phi_1\\leq\\lambda_1\\left(t_0\\|\\phi_1\\|_1+\\frac1M\\intOmega f(v+\\mu)\\phi_1\\right).\n \\end{equation*}\n These together yield\n \\begin{equation*}\n \\left(1-\\frac{\\lambda_1^2}{M^2}\\right)\\intOmega f(v+\\mu)\\phi_1\\leq\\lambda_1\\left(1+\\frac{\\lambda_1}M\\right)t_0\\|\\phi_1\\|_1\\,.\n \\end{equation*}\n and\n \\begin{equation*}\n \\left(1-\\frac{\\lambda_1^2}{M^2}\\right)\\intOmega g(u+\\mu)\\phi_1\\leq\\lambda_1\\left(1+\\frac{\\lambda_1}M\\right)t_0\\|\\phi_1\\|_1\\,.\n \\end{equation*}\n Choosing $M=2\\lambda_1$, one then finds\n \\begin{equation}\\label{Bound_l1loc}\n \\intOmega f(v+\\mu)\\phi_1\\leq2\\lambda_1t_0\\|\\phi_1\\|_1\\quad\\ \\mbox{and}\\ \\quad\\intOmega g(u+\\mu)\\phi_1\\leq2\\lambda_1t_0\\|\\phi_1\\|_1\\,,\n \\end{equation}\n which by $(H_1)$ and $u,v\\geq0$ imply that $\\mu$ needs to be bounded from above. Hence, there exists $\\overline\\mu>0$ such that for all $\\mu>\\overline\\mu$ one has $H_2(\\mu,u,v)\\neq(u,v)$ in $K$. Moreover, for any fixed $\\mu_0$, we note that $f(\\cdot+\\mu)$ and $g(\\cdot+\\mu)$ satisfy the assumptions $(H_1)$-$(H_6')$ uniformly for $\\mu\\in[0,\\mu_0]$. In fact, it is evident that $(H_1)$-$(H_4)$ hold; moreover, $\\frac1t\\frac f{f'}(t+\\mu)=\\left(1+\\frac\\mu t\\right)\\frac1{t+\\mu}\\frac f{f'}(t+\\mu)$, which is decreasing for large $t$ and uniformly in $\\mu\\in[0,\\mu_0]$. Indeed, this is a product of two eventually decreasing positive functions by $(H_5)$ applied to $f$; the same argument applies also for $g$, hence $(H_5)$ holds also for $f(\\cdot+\\mu)$ and $g(\\cdot+\\mu)$. Similarly, one also verifies $(H_6)$ and $(H_6')$: indeed, e.g.\n $$\\frac t{(\\log t)^2}\\frac{f'}f(t+\\mu)=\\frac{t(\\log(t+\\mu))^2}{(t+\\mu)(\\log t)^2}\\cdot\\frac{(t+\\mu)}{(\\log(t+\\mu))^2}\\frac{f'}f(t+\\mu)\\geq\\frac a2$$\n for $t$ large, where $a$ is the constant in $(H_6)$ for $\\mu=0$, since the first term converges to $1$ as $t\\to+\\infty$ uniformly in $\\mu\\in[0,\\mu_0]$. Moreover, it is important to note that the constant $\\Lambda$ in Proposition \\ref{Prop_Lambda} is independent of $\\mu$, provided $\\mu\\in[0,\\mu_0]$. Indeed, inspecting its proof in \\cite[Theorem 1.2]{dFdOR}, it is sufficient that \\eqref{Bound_l1loc} holds with a bound independent of $\\mu\\in[0,\\mu_0]$ to conclude. After all these considerations, we can thus state that Theorem \\ref{Thm_Uniform_bound} applies with a bound uniform with respect to $\\mu\\in[0,\\mu_0]$. Therefore taking $\\mu_0=\\overline\\mu+1>0$ there exists $\\overline R=\\overline R(\\overline\\mu)>0$ such that all solutions of \\eqref{sys_mu} are bounded in $L^\\infty$ norm by $\\overline R/2$. Hence, $H_2(\\mu,u,v)\\neq(u,v)$ for all $(u,v)\\in B_{\\overline R}(0)^c\\cap K$ and $\\mu\\in[0,\\overline\\mu+1]$. Hence, combining the two information on the map $H_2$, we infer\n \\begin{equation}\\label{Index_0}\n i_K(T,W_{\\overline R})=i_K\\left(H_2(0,\\cdot,\\cdot),W_{\\overline R}\\right)=i_K\\left(H_2(\\overline\\mu+1,\\cdot,\\cdot),W_{\\overline R}\\right)=i_K(0,W_{\\overline R})=0\\,.\n \\end{equation}\n As a result, from \\eqref{Index_0} and \\eqref{Index_1} one deduces $i_K(T,W_R\\setminus\\overline{W_r})=-1\\neq0$, which yields the existence of a nontrivial positive solution of \\eqref{sys}.", "post_theorem_intro_text_len": 6726, "post_theorem_intro_text": "Note that, under such growth assumptions, one can easily show that $f$ grows less than or comparable to ${\\rm e}^{pt}$ for some $p>0$ (see Lemma \\ref{growth_below_g} below). Hence we may apply the regularity result \\cite[Theorem 1.1]{dFdOR} and conclude that distributional solutions of \\eqref{sys} are indeed classical. Therefore, from now on we will simply talk about \\textit{solutions of \\eqref{sys}}.\n\n    \\begin{figure}[h!]\n    \\centering\n    \\begin{tikzpicture}\n    \\begin{axis}[\n        axis lines=none,\n        xmin=0, xmax=3.5,\n        ymin=0, ymax=3.5,\n        width=8cm, height=8cm,\n        clip=false\n    ]\n    \\addplot [name path=retta, domain=0:2, samples=2, draw=none] {2-x};\n    \\addplot [name path=assex, domain=0:2, samples=2, draw=none] {0};\n    \\addplot [fill=blue!15, draw=none] fill between[of=retta and assex, soft clip={domain=0:2}];\n    \\end{axis}\n\n    \\begin{axis}[\n        axis lines=middle,\n        xlabel={$\\alpha$},\n        ylabel={$\\beta$},\n        xmin=0, xmax=3.5,\n        ymin=0, ymax=3.5,\n        xtick={0,1,2,3},\n        ytick={0,1,2,3},\n        xticklabels={0,1,2,3},\n        yticklabels={0,1,2,3},\n        axis line style={->},\n        width=8cm, height=8cm,\n        clip=false,\n        label style={font=\\small},\n        axis on top\n    ]\n\n    \\addplot[red!80!black, very thick, domain=0:2, samples=2] {2 - x};\n\n    \\addplot[\n        only marks,\n        mark=o,\n        mark size=2pt,\n        mark options={draw=blue!80!black, line width=1pt}\n    ] coordinates {(1,1)};\n\n    \\addplot[green!50!black, dashed, ultra thick, samples=200, domain=1.01:3.5, restrict y to domain=0:3.5]\n        {1/(1 - 1/x)};\n\n    \\node[below left] at (axis cs:0,0) {\\small 0};\n\n    \\end{axis}\n    \\end{tikzpicture}\n    \\caption{The state of the art on uniform a priori bounds for \\eqref{sys} for the model case \\eqref{NL_model}.}\n    \\label{fig1}\n    \\end{figure}\n\n    In Figure \\ref{fig1} we can see the state of the art for problem \\eqref{sys} for the model nonlinearities in \\eqref{NL_model}. In particular, a priori bounds in the blue area and in the point $(1,1)$ were proved in \\cite{dFdOR}, the red line is our contribution, while the dashed curve in green is the ``conformal'' critical hyperbola \\eqref{abhyp}.\n    This picture underlines that our case is largely subcritical in the sense of Trudinger-Moser.  Note that we retrieve the scalar results in \\cite{LRU,Rom_N} if we restrict on the diagonal of the graph. Therefore, in light of the counterexamples in \\cite{BM,MR,Rom_N}, we expect that the region where now the a priori bounds for \\eqref{sys} applies is optimal for \\textit{distributional} solutions at least in the presence of singular potentials. However, it is still not clear -- even in the scalar case -- whether a priori bounds for \\textit{weak} solutions may be proved up to the critical growth in the sense of Trudinger-Moser.\n\n    \\vskip0.2truecm \n\tAlthough reminiscent of some ideas developed in \\cite{MR,Rom_N} to deal with corresponding scalar problems, the method that we here present is new and completely different from the one in \\cite{dFdOR}, from which we take only the a priori $L^1$ bounds in Proposition \\ref{Prop_Lambda}. Our blow-up technique is based on a suitable new scaling for system \\eqref{sys}, and the main challenge is to precisely understand how to profit from the exchange of information between the two components $u$ and $v$ in order to find contradictions with such $L^1$-bounds. Let us try to briefly sketch the main line of our argument. We start in Section \\ref{Sec_scaling} by assuming by contradiction that there exists a sequence of solutions $(u_k,v_k)_k$ of \\eqref{sys} such that \n    $$\n    \\max\\{\\|u_k\\|_\\infty,\\|v_k\\|_\\infty\\}=+\\infty\\,.\n    $$\n\n    First, we exclude the possibility that only one among $(\\|u_k\\|_\\infty)_k$ and $(\\|v_k\\|_\\infty)_k$ is unbounded and, exploiting the symmetric setting, we easily see that the origin is the unique blow-up point for both sequences.\n\n    The goal is to reach a contradiction either with the uniform $L^1$-bounds on the nonlinear terms (see Proposition \\ref{Prop_Lambda}) or by applying a suitable Liouville-type theorem in the plane. In both cases, detecting a suitable scaling is essential: it produces $(\\tu_k,\\tv_k)_k$ defined on balls $\\Omega_k$ with diverging radii, which formally locally converges to a solution of a Liouville's system on $\\mathbb R^2$. In order to ensure this, we need that the scalings for both components are coupled by a compatibility condition, which we analyse in detail in Section \\ref{sec:compatibility}.\n\n    Choosing to base the scaling on $\\|u_k\\|_\\infty=u_k(0)$, so that $\\tu_k(0)=0$, our blow-up analysis in Section \\ref{Sec_blowup} then proceeds by distinguishing several cases according to the behavior of $\\tv_k(0)$ as $k\\to\\infty$. If $\\tv_k(0)\\to-\\infty$ as $k\\to\\infty$, then the Harnack inequality can be directly applied to $\\tu_k$, which leads to a simple contradiction with the energy bounds. If $\\tv_k(0)\\to c\\in\\mathbb R$, then in the non-Liouville case \\eqref{nonBM} the boundedness of $\\tv_k$ on compact sets yields a similar situation. In the Liouville case \\eqref{BM}, we infer the contradiction from the analysis of the limit Liouville's system, relying on a characterisation result in \\cite{CK}. Finally, the analysis becomes more technical when $\\tv_k(0)\\to +\\infty$ and we need to find finer global estimates on $\\tu_k$ and $\\tv_k$, especially in the non-Liouville setting. Here the additional assumptions $(H_6)$ and $(H_6')$ will permit to find the desired contradiction.\n\n\t\\vskip0.2truecm\n\n\tUnder a standard superlinear growth condition on the nonlinearities in $0$, in Section \\ref{Sec_Existence} we show that once the a priori bound is obtained, then the existence of a positive continuous solution $(u,v)$ of \\eqref{sys} follows by means of Fixed Point index theory.\n\n    \\begin{thm}\\label{Thm_Existence}\n        Besides the conditions under which Theorem \\ref{Thm_Uniform_bound} holds, assume moreover\n        \\begin{enumerate}\n            \\item[$(H_9)$] $\\displaystyle\\limsup_{t\\to0^+}\\frac{f(t)}t<\\lambda_1\\ $ and $\\ \\,\\displaystyle\\limsup_{t\\to0^+}\\frac{g(t)}t<\\lambda_1\\,$,\n    \t\\end{enumerate}\n        where $\\lambda_1$ is the first eigenvalue of $-\\Delta$ in $\\Omega$ with Dirichlet boundary conditions. Then \\eqref{sys} admits a positive solution.\n    \\end{thm} \n\n\t\\paragraph{\\textbf{Notation.}} For $R>0$ and $x_0\\in\\mathbb R^N$ we denote by $B_R(x_0)$ the ball of radius $R$ and center $x_0$. The symbol $o_n(1)$ denotes a vanishing real sequence as $n\\to+\\infty$. Hereafter, the letter $C$ will be used to denote positive constants which are independent of relevant quantities and whose value may change from line to line.", "sketch": "To prove Theorem~\\ref{Thm_Uniform_bound}, the authors “assum[e] by contradiction that there exists a sequence of solutions $(u_k,v_k)_k$ of \\eqref{sys} such that \n\\[\n\\max\\{\\|u_k\\|_\\infty,\\|v_k\\|_\\infty\\}=+\\infty\\,.\n\\]”\nFirst, they “exclude the possibility that only one among $(\\|u_k\\|_\\infty)_k$ and $(\\|v_k\\|_\\infty)_k$ is unbounded” and, using the symmetry, show that “the origin is the unique blow-up point for both sequences.”\n\nThe aim is “to reach a contradiction either with the uniform $L^1$-bounds on the nonlinear terms (see Proposition~\\ref{Prop_Lambda}) or by applying a suitable Liouville-type theorem in the plane.” A key ingredient is “detecting a suitable scaling”: it produces rescaled functions $(\\tilde u_k,\\tilde v_k)_k$ on balls $\\Omega_k$ “with diverging radii,” which “formally locally converges to a solution of a Liouville's system on $\\mathbb R^2$.” To make this work, “we need that the scalings for both components are coupled by a compatibility condition,” analyzed in Section~\\ref{sec:compatibility}.\n\nThey “base the scaling on $\\|u_k\\|_\\infty=u_k(0)$, so that $\\tilde u_k(0)=0$,” and then distinguish cases depending on the behavior of $\\tilde v_k(0)$ as $k\\to\\infty$:\n\\begin{itemize}\n\\item If $\\tilde v_k(0)\\to-\\infty$, “the Harnack inequality can be directly applied to $\\tilde u_k$,” giving “a simple contradiction with the energy bounds.”\n\\item If $\\tilde v_k(0)\\to c\\in\\mathbb R$, then in the non-Liouville case \\eqref{nonBM} the “boundedness of $\\tilde v_k$ on compact sets yields a similar situation,” while in the Liouville case \\eqref{BM} they get a contradiction from “the analysis of the limit Liouville's system,” using “a characterisation result in \\cite{CK}.”\n\\item If $\\tilde v_k(0)\\to+\\infty$, the argument is “more technical” and requires “finer global estimates on $\\tilde u_k$ and $\\tilde v_k$, especially in the non-Liouville setting”; the extra assumptions $(H_6)$ or $(H_6')$ “permit to find the desired contradiction.”\n\\end{itemize}\nIn all cases, this yields the contradiction needed to conclude the uniform a priori bounds of Theorem~\\ref{Thm_Uniform_bound}.", "expanded_sketch": "To prove the main theorem, the authors “assum[e] by contradiction that there exists a sequence of solutions $(u_k,v_k)_k$ of\n\\begin{equation}\\label{sys}\n\t\t\\begin{cases}\n\t\t\t-\\Delta u=f(v)\\quad&\\mbox{in }\\Omega\\,,\\\\\n\t\t\t-\\Delta v=g(u)\\quad&\\mbox{in }\\Omega\\,,\\\\\n\t\t\tu=v=0\\quad&\\mbox{on }\\dOmega\\,.\n\t\t\\end{cases}\n\t\\end{equation}\n such that \n\\[\n\\max\\{\\|u_k\\|_\\infty,\\|v_k\\|_\\infty\\}=+\\infty\\,.\n\\]”\nFirst, they “exclude the possibility that only one among $(\\|u_k\\|_\\infty)_k$ and $(\\|v_k\\|_\\infty)_k$ is unbounded” and, using the symmetry, show that “the origin is the unique blow-up point for both sequences.”\n\nThe aim is “to reach a contradiction either with the uniform $L^1$-bounds on the nonlinear terms,” namely the following result:\n\\begin{prop}[Theorem 1.2, \\cite{dFdOR}]\\label{Prop_Lambda}\n\t\tAssume that there exists $c>0$ and $p>0$ such that $f(t)\\leq c\\e^{pt}$ for all $t>0$, or that the same holds for $g$. Then there exists a positive constant $\\Lambda$, depending only on $f$, $g$, and $\\Omega$, such that\n        \\begin{equation}\\label{Lambda}\\tag{$\\Lambda$}\n            \\intOmega f(v)\\leq\\Lambda\\quad\\mbox{and}\\quad\\intOmega g(u)\\leq\\Lambda\n        \\end{equation}\n        for all solutions of \\eqref{sys}.\n\t\\end{prop}\nor by applying a suitable Liouville-type theorem in the plane.” A key ingredient is “detecting a suitable scaling”: it produces rescaled functions $(\\tilde u_k,\\tilde v_k)_k$ on balls $\\Omega_k$ “with diverging radii,” which “formally locally converges to a solution of a Liouville's system on $\\mathbb R^2$.” To make this work, “we need that the scalings for both components are coupled by a compatibility condition,” analyzed later.\n\nThey “base the scaling on $\\|u_k\\|_\\infty=u_k(0)$, so that $\\tilde u_k(0)=0$,” and then distinguish cases depending on the behavior of $\\tilde v_k(0)$ as $k\\to\\infty$:\n\\begin{itemize}\n\\item If $\\tilde v_k(0)\\to-\\infty$, “the Harnack inequality can be directly applied to $\\tilde u_k$,” giving “a simple contradiction with the energy bounds.”\n\\item If $\\tilde v_k(0)\\to c\\in\\mathbb R$, then in the non-Liouville case\n\\begin{equation}\\label{nonBM}\n\t\t\\lim_{t\\to+\\infty}\\frac{f'}f(t)=0\\quad\\mbox{and}\\quad\\lim_{t\\to+\\infty}\\frac{g'}g(t)=+\\infty\n\t\\end{equation}\nthe “boundedness of $\\tilde v_k$ on compact sets yields a similar situation,” while in the Liouville case\n\\begin{equation}\\label{BM}\n\t\t\\lim_{t\\to+\\infty}\\frac{f'}f(t)=p\\quad\\mbox{and}\\quad\\lim_{t\\to+\\infty}\\frac{g'}g(t)=q\\,.\n\t\\end{equation}\nthey get a contradiction from “the analysis of the limit Liouville's system,” using “a characterisation result in \\cite{CK}.”\n\\item If $\\tilde v_k(0)\\to+\\infty$, the argument is “more technical” and requires “finer global estimates on $\\tilde u_k$ and $\\tilde v_k$, especially in the non-Liouville setting”; the extra assumptions $(H_6)$ or $(H_6')$ “permit to find the desired contradiction.”\n\\end{itemize}\nIn all cases, this yields the contradiction needed to conclude the uniform a priori bounds of the main theorem.", "expanded_theorem": "\\label{Thm_Uniform_bound}\n\t\tLet $\\Omega=B_1(0)$ be the unit ball in $\\mathbb R^2$, assume conditions $(H_1)$, $(H_2)$, $(H_3)$, $(H_b)$, and alternatively\n\t\t\\begin{enumerate}\n\t\t\t\\item[i)] \\begin{equation}\\label{BM}\n\t\t\\lim_{t\\to+\\infty}\\frac{f'}f(t)=p\\quad\\mbox{and}\\quad\\lim_{t\\to+\\infty}\\frac{g'}g(t)=q\\,.\n\t\\end{equation} holds;\n\t\t\t\\item[ii)] \\begin{equation}\\label{nonBM}\n\t\t\\lim_{t\\to+\\infty}\\frac{f'}f(t)=0\\quad\\mbox{and}\\quad\\lim_{t\\to+\\infty}\\frac{g'}g(t)=+\\infty\n\t\\end{equation}, $(H_4)$, $(H_5)$, and either $(H_6)$ or $(H_6')$ hold.\n\t\t\\end{enumerate}\n\t\tThen there exists a constant $C>0$ such that\n\t\t\\begin{equation*}\n\t\t\t\\|u\\|_\\infty\\leq C\\qquad\\mbox{and}\\qquad\\|v\\|_\\infty\\leq C\n\t\t\\end{equation*}\n\t\tfor all (eventual) solutions $(u,v)$ of the system\n\t\t\\begin{equation}\\label{sys}\n\t\t\\begin{cases}\n\t\t\t-\\Delta u=f(v)\\quad&\\mbox{in }\\Omega\\,,\\\\\n\t\t\t-\\Delta v=g(u)\\quad&\\mbox{in }\\Omega\\,,\\\\\n\t\t\tu=v=0\\quad&\\mbox{on }\\dOmega\\,.\n\t\t\\end{cases}\n\t\\end{equation}.", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let \\(\\Omega=B_1(0)\\subset\\mathbb R^2\\) be the unit ball, and consider the Dirichlet system\n\\[\n\\begin{cases}\n-\\Delta u=f(v) & \\text{in }\\Omega,\\\\\n-\\Delta v=g(u) & \\text{in }\\Omega,\\\\\nu=v=0 & \\text{on }\\partial\\Omega.\n\\end{cases}\n\\]\nAssume \\(f,g\\in C^1([0,\\infty))\\), \\(f(t),g(t)>0\\) for \\(t>0\\), \\(f(t)\\to\\infty\\) and \\(g(t)\\to\\infty\\) as \\(t\\to\\infty\\), and that there exists \\(t_0>0\\) such that \\(f\\) and \\(g\\) are strictly increasing on \\((t_0,\\infty)\\). Assume also\n\\[\n\\left(\\frac{f}{f'}\\right)'(t)\\to 0,\n\\qquad\n\\left(\\frac{g'}{g}\\right)'(t)\\to 0\n\\quad \\text{as } t\\to\\infty,\n\\]\nand\n\\[\n\\lim_{t\\to\\infty}\\frac{f'(t)}{f(t)}\\frac{g'(t)}{g(t)}=b\\in(0,\\infty).\n\\]\nSuppose, moreover, that either\n1. (Liouville case) there exist \\(p,q>0\\) such that\n\\[\n\\lim_{t\\to\\infty}\\frac{f'(t)}{f(t)}=p,\n\\qquad\n\\lim_{t\\to\\infty}\\frac{g'(t)}{g(t)}=q,\n\\]\nor\n2. (non-Liouville case)\n\\[\n\\lim_{t\\to\\infty}\\frac{f'(t)}{f(t)}=0,\n\\qquad\n\\lim_{t\\to\\infty}\\frac{g'(t)}{g(t)}=+\\infty,\n\\]\nand there exists \\(t_0>0\\) such that\n\\[\n\\left(\\frac{f}{f'}\\right)'(t)\\ge 0,\n\\qquad\n\\left(\\frac{g'}{g}\\right)'(t)\\ge 0\n\\quad \\text{for } t>t_0,\n\\]\nwith \\(t\\mapsto t\\,\\frac{f'(t)}{f(t)}\\) increasing and \\(t\\mapsto \\frac1t\\,\\frac{g'(t)}{g(t)}\\) decreasing on \\((t_0,\\infty)\\), and in addition one of the further growth assumptions \\((H_6)\\) or \\((H_6')\\) holds.\n\nUnder these assumptions, which quantitative estimate holds for all solutions \\((u,v)\\) of the system?", "correct_choice": {"label": "A", "text": "There exists a constant \\(C>0\\) such that every solution \\((u,v)\\) satisfies \\(\\|u\\|_{\\infty}\\le C\\) and \\(\\|v\\|_{\\infty}\\le C\\)."}, "choices": [{"label": "B", "text": "There exists a constant \\(C>0\\), depending only on \\(f\\), \\(g\\), and \\(\\Omega\\), such that every solution \\((u,v)\\) satisfies \\(\\|u\\|_{L^1(\\Omega)}\\le C\\) and \\(\\|v\\|_{L^1(\\Omega)}\\le C\\)."}, {"label": "C", "text": "For every solution \\((u,v)\\), both components are bounded, i.e. \\(u,v\\in L^{\\infty}(\\Omega)\\)."}, {"label": "D", "text": "There exists a constant \\(C>0\\) such that every radial solution \\((u,v)\\) satisfies \\(\\|u\\|_{\\infty}\\le C\\) and \\(\\|v\\|_{\\infty}\\le C\\)."}, {"label": "E", "text": "There exists a constant \\(C>0\\) such that every solution \\((u,v)\\) satisfies \\(\\int_{\\Omega} f(v)\\,dx\\le C\\) and \\(\\int_{\\Omega} g(u)\\,dx\\le C\\), and consequently at least one of \\(\\|u\\|_{\\infty}\\) or \\(\\|v\\|_{\\infty}\\) is bounded by \\(C\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "replaces uniform sup-norm conclusion by only componentwise L1 bounds", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "drops the existence of a single uniform constant valid for all solutions", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "restricts the conclusion from all solutions to radial solutions only", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "misuses the uniform integral bounds on nonlinear terms as a substitute for two-sided uniform L-infinity control", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It asks for a 'uniform conclusion,' which narrows attention to conclusions with a common constant, but several choices fit that pattern, so there is no decisive leakage."}, "TAS": {"score": 0, "justification": "The item is essentially a theorem-recall question: it reproduces the full hypotheses and asks for the guaranteed conclusion, with the correct answer closely matching the theorem statement itself."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact uniform a priori bound from weaker, stronger, or mis-scoped variants, but the main task is still recognition/recall rather than substantive derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: one is weaker but true in spirit, others overstate dependence, confuse norm types, or alter the solution class/case split. These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A technically well-constructed recall-style MCQ with strong distractors, but it is largely tautological because it asks for the theorem's conclusion under the theorem's hypotheses."}}
{"id": "2602.11906v2", "paper_link": "http://arxiv.org/abs/2602.11906v2", "theorems_cnt": 1, "theorem": {"env_name": "maintheorem", "content": "\\label[maintheorem]{thm:rt22-polynomially-simulated}\n$\\RCA_0 + \\BSig_2$ polynomially simulates $\\WKL_0 + \\RT_2^2$ with respect to $\\forall \\Sigma_3^0$ sentences.", "start_pos": 6358, "end_pos": 6556, "label": null}, "ref_dict": {}, "pre_theorem_intro_text_len": 4147, "pre_theorem_intro_text": "If a computably axiomatized theory~$T_0$ is a conservative extension of another such theory~$T_1$ for a syntactical class of sentences $\\Gamma$, there is an algorithmic procedure which translates any $T_0$-proof~$p$ of a $\\Gamma$-sentence into its shortest $T_1$-proof~$\\hat p$. It is natural to wonder whether the translation $p \\mapsto \\hat p$ yields significantly longer proofs. If there exists a $T_0$-proof~$p$ of a $\\Gamma$-sentence such that the length $|\\hat p|$ of its translated $T_1$-proof is super-polynomial with respect to the length~$|p|$, we say that $T_0$ admits \\emph{non-trivial speedup} over $T_1$ for $\\Gamma$-formulas. In this case, the theory~$T_0$ is arguably useful for $\\Gamma$-sentences, in that it sometimes produces significantly shorter proofs than~$T_1$ for such sentences. If on the other hand, there exists a polynomial~$Q$ such that $|\\hat p| \\leq Q(|p|)$ for every $T_0$-proof~$p$ of any $\\Gamma$-sentence, then we consider that $T_0$ admits no significant speedup over~$T_1$ for $\\Gamma$-formulas, since the various proof systems are mutually polynomially simulated.\n\nThe study of proof size and proof speedups traces back to G\\\"odel~\\cite{davis1990kurt} (see Pudl\\'ak~\\cite{pudlak1998length} for an excellent survey). However, it was only in the 1990's that Avigad~\\cite{avigad1996formalizing} showed that model-theoretic conservation theorems based on forcing could be formalized into proof-theoretic arguments and yield proof size analysis. He proved in particular that $\\WKL_0$ is $\\Pi^1_1$-conservative over $\\RCA_0$ with no significant increase in the length of proofs. The technique was later used to formalize $\\Pi^1_1$-conservation theorems over~$\\RCA_0+\\ISig_n$~\\cite{Ikari-PhD,Ik-Yo} and $\\forall \\Sigma^0_2$-conservation theorems over~$\\RCA_0$~\\cite{kolodziejczyk2023ramsey} and over $\\RCA_0^*$~\\cite{katarzyna2025speedup}, where a $\\forall \\Sigma^0_n$-formula is of the form $\\forall X \\varphi(X)$ where $\\varphi$ is $\\Sigma^0_n$.\n\nWe are particularly interested in the first-order consequences of Ramsey's theorem for pairs and two colors ($\\RT^2_2$). Patey and Yokoyama~\\cite{patey2018proof} proved that $\\WKL_0 + \\RT^2_2$ is $\\forall \\Sigma^0_2$-conservative over~$\\RCA_0$ using the notion of $\\alpha$-largeness from Ketonen and Solovay~\\cite{ketonen1981rapidly}. The proof was later simplified by Ko{\\l}odziejczyk and Yokoyama~\\cite{kolo2020some} to give explicit polynomial bounds for this notion of largeness. Ko{\\l}odziejczyk, Wong and Yokoyama~\\cite{kolodziejczyk2023ramsey} then formalized the construction to prove that the conservation proof does not increase significantly the length of proofs. This constrasts with the fact that $\\WKL_0^* + \\RT^2_2$ yields a non-elementary speedup over~$\\RCA_0^*$ even for $\\Sigma_1$-sentences~\\cite{kolodziejczyk2023ramsey}. More recently, Le Houérou, Levy Patey and Yokoyama~\\cite{houerou2026pi} defined a parameterized version of Ketonen and Solovay's notion of largeness and proved that $\\WKL_0 + \\RT^2_2$ is $\\forall \\Sigma^0_3$-conservative over $\\RCA_0 + \\BSig_2$, with explicit bounds computation. However, these bounds are exponential, leaving open whether the conservation theorem yields a significant proof speedup.\n\nIn this article, we define a variant of the parameterized version of $\\alpha$-largeness \\cite{houerou2026pi} and prove that Ramsey's theorem for pairs admits polynomial bounds with respect to this notion of largeness. Then, using the now-called notion of forcing interpretation introduced by Avigad~\\cite{avigad1996formalizing} and developed in the context of $\\alpha$-largeness by Ko{\\l}odziejczyk, Wong and Yokoyama~\\cite{kolodziejczyk2023ramsey}, we prove that the conservation theorem for $\\WKL_0 + \\RT^2_2$ over $\\RCA_0 + \\BSig_2$ for $\\forall \\Sigma^0_3$-sentences does not yield any significant proof speedup. The polynomial bounds computation for this new notion of largeness is non-trivial, and require to prove a partition theorem for trees based on a finite version of Milliken's tree theorem~\\cite{milliken1979ramsey,dodos2016ramsey} with primitive recursive bounds. Our main theorem is therefore the following:", "context": "If a computably axiomatized theory~$T_0$ is a conservative extension of another such theory~$T_1$ for a syntactical class of sentences $\\Gamma$, there is an algorithmic procedure which translates any $T_0$-proof~$p$ of a $\\Gamma$-sentence into its shortest $T_1$-proof~$\\hat p$. It is natural to wonder whether the translation $p \\mapsto \\hat p$ yields significantly longer proofs. If there exists a $T_0$-proof~$p$ of a $\\Gamma$-sentence such that the length $|\\hat p|$ of its translated $T_1$-proof is super-polynomial with respect to the length~$|p|$, we say that $T_0$ admits \\emph{non-trivial speedup} over $T_1$ for $\\Gamma$-formulas. In this case, the theory~$T_0$ is arguably useful for $\\Gamma$-sentences, in that it sometimes produces significantly shorter proofs than~$T_1$ for such sentences. If on the other hand, there exists a polynomial~$Q$ such that $|\\hat p| \\leq Q(|p|)$ for every $T_0$-proof~$p$ of any $\\Gamma$-sentence, then we consider that $T_0$ admits no significant speedup over~$T_1$ for $\\Gamma$-formulas, since the various proof systems are mutually polynomially simulated.\n\nThe study of proof size and proof speedups traces back to G\\\"odel~\\cite{davis1990kurt} (see Pudl\\'ak~\\cite{pudlak1998length} for an excellent survey). However, it was only in the 1990's that Avigad~\\cite{avigad1996formalizing} showed that model-theoretic conservation theorems based on forcing could be formalized into proof-theoretic arguments and yield proof size analysis. He proved in particular that $\\WKL_0$ is $\\Pi^1_1$-conservative over $\\RCA_0$ with no significant increase in the length of proofs. The technique was later used to formalize $\\Pi^1_1$-conservation theorems over~$\\RCA_0+\\ISig_n$~\\cite{Ikari-PhD,Ik-Yo} and $\\forall \\Sigma^0_2$-conservation theorems over~$\\RCA_0$~\\cite{kolodziejczyk2023ramsey} and over $\\RCA_0^*$~\\cite{katarzyna2025speedup}, where a $\\forall \\Sigma^0_n$-formula is of the form $\\forall X \\varphi(X)$ where $\\varphi$ is $\\Sigma^0_n$.\n\nWe are particularly interested in the first-order consequences of Ramsey's theorem for pairs and two colors ($\\RT^2_2$). Patey and Yokoyama~\\cite{patey2018proof} proved that $\\WKL_0 + \\RT^2_2$ is $\\forall \\Sigma^0_2$-conservative over~$\\RCA_0$ using the notion of $\\alpha$-largeness from Ketonen and Solovay~\\cite{ketonen1981rapidly}. The proof was later simplified by Ko{\\l}odziejczyk and Yokoyama~\\cite{kolo2020some} to give explicit polynomial bounds for this notion of largeness. Ko{\\l}odziejczyk, Wong and Yokoyama~\\cite{kolodziejczyk2023ramsey} then formalized the construction to prove that the conservation proof does not increase significantly the length of proofs. This constrasts with the fact that $\\WKL_0^* + \\RT^2_2$ yields a non-elementary speedup over~$\\RCA_0^*$ even for $\\Sigma_1$-sentences~\\cite{kolodziejczyk2023ramsey}. More recently, Le Houérou, Levy Patey and Yokoyama~\\cite{houerou2026pi} defined a parameterized version of Ketonen and Solovay's notion of largeness and proved that $\\WKL_0 + \\RT^2_2$ is $\\forall \\Sigma^0_3$-conservative over $\\RCA_0 + \\BSig_2$, with explicit bounds computation. However, these bounds are exponential, leaving open whether the conservation theorem yields a significant proof speedup.\n\nIn this article, we define a variant of the parameterized version of $\\alpha$-largeness \\cite{houerou2026pi} and prove that Ramsey's theorem for pairs admits polynomial bounds with respect to this notion of largeness. Then, using the now-called notion of forcing interpretation introduced by Avigad~\\cite{avigad1996formalizing} and developed in the context of $\\alpha$-largeness by Ko{\\l}odziejczyk, Wong and Yokoyama~\\cite{kolodziejczyk2023ramsey}, we prove that the conservation theorem for $\\WKL_0 + \\RT^2_2$ over $\\RCA_0 + \\BSig_2$ for $\\forall \\Sigma^0_3$-sentences does not yield any significant proof speedup. The polynomial bounds computation for this new notion of largeness is non-trivial, and require to prove a partition theorem for trees based on a finite version of Milliken's tree theorem~\\cite{milliken1979ramsey,dodos2016ramsey} with primitive recursive bounds. Our main theorem is therefore the following:", "full_context": "If a computably axiomatized theory~$T_0$ is a conservative extension of another such theory~$T_1$ for a syntactical class of sentences $\\Gamma$, there is an algorithmic procedure which translates any $T_0$-proof~$p$ of a $\\Gamma$-sentence into its shortest $T_1$-proof~$\\hat p$. It is natural to wonder whether the translation $p \\mapsto \\hat p$ yields significantly longer proofs. If there exists a $T_0$-proof~$p$ of a $\\Gamma$-sentence such that the length $|\\hat p|$ of its translated $T_1$-proof is super-polynomial with respect to the length~$|p|$, we say that $T_0$ admits \\emph{non-trivial speedup} over $T_1$ for $\\Gamma$-formulas. In this case, the theory~$T_0$ is arguably useful for $\\Gamma$-sentences, in that it sometimes produces significantly shorter proofs than~$T_1$ for such sentences. If on the other hand, there exists a polynomial~$Q$ such that $|\\hat p| \\leq Q(|p|)$ for every $T_0$-proof~$p$ of any $\\Gamma$-sentence, then we consider that $T_0$ admits no significant speedup over~$T_1$ for $\\Gamma$-formulas, since the various proof systems are mutually polynomially simulated.\n\nThe study of proof size and proof speedups traces back to G\\\"odel~\\cite{davis1990kurt} (see Pudl\\'ak~\\cite{pudlak1998length} for an excellent survey). However, it was only in the 1990's that Avigad~\\cite{avigad1996formalizing} showed that model-theoretic conservation theorems based on forcing could be formalized into proof-theoretic arguments and yield proof size analysis. He proved in particular that $\\WKL_0$ is $\\Pi^1_1$-conservative over $\\RCA_0$ with no significant increase in the length of proofs. The technique was later used to formalize $\\Pi^1_1$-conservation theorems over~$\\RCA_0+\\ISig_n$~\\cite{Ikari-PhD,Ik-Yo} and $\\forall \\Sigma^0_2$-conservation theorems over~$\\RCA_0$~\\cite{kolodziejczyk2023ramsey} and over $\\RCA_0^*$~\\cite{katarzyna2025speedup}, where a $\\forall \\Sigma^0_n$-formula is of the form $\\forall X \\varphi(X)$ where $\\varphi$ is $\\Sigma^0_n$.\n\nWe are particularly interested in the first-order consequences of Ramsey's theorem for pairs and two colors ($\\RT^2_2$). Patey and Yokoyama~\\cite{patey2018proof} proved that $\\WKL_0 + \\RT^2_2$ is $\\forall \\Sigma^0_2$-conservative over~$\\RCA_0$ using the notion of $\\alpha$-largeness from Ketonen and Solovay~\\cite{ketonen1981rapidly}. The proof was later simplified by Ko{\\l}odziejczyk and Yokoyama~\\cite{kolo2020some} to give explicit polynomial bounds for this notion of largeness. Ko{\\l}odziejczyk, Wong and Yokoyama~\\cite{kolodziejczyk2023ramsey} then formalized the construction to prove that the conservation proof does not increase significantly the length of proofs. This constrasts with the fact that $\\WKL_0^* + \\RT^2_2$ yields a non-elementary speedup over~$\\RCA_0^*$ even for $\\Sigma_1$-sentences~\\cite{kolodziejczyk2023ramsey}. More recently, Le Houérou, Levy Patey and Yokoyama~\\cite{houerou2026pi} defined a parameterized version of Ketonen and Solovay's notion of largeness and proved that $\\WKL_0 + \\RT^2_2$ is $\\forall \\Sigma^0_3$-conservative over $\\RCA_0 + \\BSig_2$, with explicit bounds computation. However, these bounds are exponential, leaving open whether the conservation theorem yields a significant proof speedup.\n\nIn this article, we define a variant of the parameterized version of $\\alpha$-largeness \\cite{houerou2026pi} and prove that Ramsey's theorem for pairs admits polynomial bounds with respect to this notion of largeness. Then, using the now-called notion of forcing interpretation introduced by Avigad~\\cite{avigad1996formalizing} and developed in the context of $\\alpha$-largeness by Ko{\\l}odziejczyk, Wong and Yokoyama~\\cite{kolodziejczyk2023ramsey}, we prove that the conservation theorem for $\\WKL_0 + \\RT^2_2$ over $\\RCA_0 + \\BSig_2$ for $\\forall \\Sigma^0_3$-sentences does not yield any significant proof speedup. The polynomial bounds computation for this new notion of largeness is non-trivial, and require to prove a partition theorem for trees based on a finite version of Milliken's tree theorem~\\cite{milliken1979ramsey,dodos2016ramsey} with primitive recursive bounds. Our main theorem is therefore the following:\n\nWe are particularly interested in the first-order consequences of Ramsey's theorem for pairs and two colors ($\\RT^2_2$). Patey and Yokoyama~\\cite{patey2018proof} proved that $\\WKL_0 + \\RT^2_2$ is $\\forall \\Sigma^0_2$-conservative over~$\\RCA_0$ using the notion of $\\alpha$-largeness from Ketonen and Solovay~\\cite{ketonen1981rapidly}. The proof was later simplified by Ko{\\l}odziejczyk and Yokoyama~\\cite{kolo2020some} to give explicit polynomial bounds for this notion of largeness. Ko{\\l}odziejczyk, Wong and Yokoyama~\\cite{kolodziejczyk2023ramsey} then formalized the construction to prove that the conservation proof does not increase significantly the length of proofs. This constrasts with the fact that $\\WKL_0^* + \\RT^2_2$ yields a non-elementary speedup over~$\\RCA_0^*$ even for $\\Sigma_1$-sentences~\\cite{kolodziejczyk2023ramsey}. More recently, Le Houérou, Levy Patey and Yokoyama~\\cite{houerou2026pi} defined a parameterized version of Ketonen and Solovay's notion of largeness and proved that $\\WKL_0 + \\RT^2_2$ is $\\forall \\Sigma^0_3$-conservative over $\\RCA_0 + \\BSig_2$, with explicit bounds computation. However, these bounds are exponential, leaving open whether the conservation theorem yields a significant proof speedup.\n\nIn this article, we define a variant of the parameterized version of $\\alpha$-largeness \\cite{houerou2026pi} and prove that Ramsey's theorem for pairs admits polynomial bounds with respect to this notion of largeness. Then, using the now-called notion of forcing interpretation introduced by Avigad~\\cite{avigad1996formalizing} and developed in the context of $\\alpha$-largeness by Ko{\\l}odziejczyk, Wong and Yokoyama~\\cite{kolodziejczyk2023ramsey}, we prove that the conservation theorem for $\\WKL_0 + \\RT^2_2$ over $\\RCA_0 + \\BSig_2$ for $\\forall \\Sigma^0_3$-sentences does not yield any significant proof speedup. The polynomial bounds computation for this new notion of largeness is non-trivial, and require to prove a partition theorem for trees based on a finite version of Milliken's tree theorem~\\cite{milliken1979ramsey,dodos2016ramsey} with primitive recursive bounds. Our main theorem is therefore the following:\n\nThe article is divided into three parts : in \\Cref{sec:largeness}, we survey and compare three related notions of largeness, namely, Ketonen and Solvay's $\\alpha$-largeness, Le Houérou, Levy Patey and Yokoyama's $\\alpha$-largeness$(\\theta)$, and our new notion of $\\alpha$-largeness${}^*(\\theta)$, for $\\alpha$ of the form $\\bbomega^n \\cdot k$. In \\Cref{sec:combinatorics}, we develop the framework of $\\alpha$-largeness${}^*(\\theta)$ and prove polynomial upper bounds for $\\RT^2_2$-$\\bbomega^n$-largeness${}^*(\\theta)$. Last, in \\Cref{sec:translation}, we combine this upper bound with the framework of forcing interpretation to prove \\Cref{thm:rt22-polynomially-simulated}.\n\n\\Cref{fig:forcing-translations} summarizes the different intermediate theories we will consider between $\\RCA_0 + \\BSig_2$ and $\\RT_2^2 + \\WKL_0$, and the polynomial simulation between them. It is adapted from \\cite[Figure 1]{kolodziejczyk2023ramsey}, with $\\BSig_2$ added to every theory, and with a $\\forall \\Sigma_3^0$ polynomial simulation between \n$\\RCA_0 + \\BSig_2 + \\mathbb{I}'$ and $\\WKL_0 + \\RT_2^2$ instead of a $\\forall \\Sigma_2^0$ polynomial simulations.\n\n\\begin{lemma}\\label[lemma]{lem:poly-sim-I1}\n $\\RCA_0 + \\BSig_2 + (\\mathbb{I}1)$ is polynomially simulated by:\n \\begin{enumerate}[(a)]\n \\item $\\RCA_0 + \\ISig_2$ with respect to $\\forall \\Sigma_3^0$ sentences,\n \\item $\\RCA_0 + \\BSig_2 + \\neg \\ISig_2$ with respect to $\\mathcal{L}_2$ sentences.\n \\end{enumerate}\n\\end{lemma}\n\n\\begin{lemma}\\label[lemma]{lem:polynomially-reflecting}\n The forcing interpretation of $\\WKL_0 + \\RT_2^2$ in $\\RCA_0 + \\BSig_2 + \\mathbb{I}'$ given by $\\mathsf{Cond}, \\trianglelefteqslant$, $\\Vdash$ of \\Cref{def:forcing-interpretation-def} is polynomially $\\forall \\Sigma_3^0$-reflecting.\n\\end{lemma}\n\n\\begin{lemma}\\label[lemma]{lem:polynomial-simulation}\n $\\RCA_0 + \\mathbb{I}'$ polynomially simulates $\\WKL_0 + \\RT_2^2$ with respects to $\\forall \\Sigma_3^0$-sentences.\n\\end{lemma}\n\n\\begin{repmaintheorem}{thm:rt22-polynomially-simulated}\n$\\RCA_0 + \\BSig_2$ polynomially simulates $\\WKL_0 + \\RT_2^2$ with respect to $\\forall \\Sigma_3^0$ sentences.\n\\end{repmaintheorem}", "post_theorem_intro_text_len": 1955, "post_theorem_intro_text": "The article is divided into three parts : in \\Cref{sec:largeness}, we survey and compare three related notions of largeness, namely, Ketonen and Solvay's $\\alpha$-largeness, Le Houérou, Levy Patey and Yokoyama's $\\alpha$-largeness$(\\theta)$, and our new notion of $\\alpha$-largeness${}^*(\\theta)$, for $\\alpha$ of the form $\\bbomega^n \\cdot k$. In \\Cref{sec:combinatorics}, we develop the framework of $\\alpha$-largeness${}^*(\\theta)$ and prove polynomial upper bounds for $\\RT^2_2$-$\\bbomega^n$-largeness${}^*(\\theta)$. Last, in \\Cref{sec:translation}, we combine this upper bound with the framework of forcing interpretation to prove \\Cref{thm:rt22-polynomially-simulated}.\n\n\\subsection{Notation}\n\nMost of the proofs are over a weak subsystem of second-order arithmetic. We therefore distinguish the formal set of integers~$\\NN$ from the theory from the standard set of integers~$\\omega$ from the meta-theory. For instance, if $\\M = (M, S)$ is a non-standard model of second-order arithmetic, $\\NN^\\M = M$ and $\\omega^\\M = \\{0^\\M, 1^\\M, \\dots \\} \\subseteq M$. Similarly, we distinguish the formal ordinals of the theory and the ordinals from the meta-theory. We therefore write for instance $\\bbomega^\\bbomega$ for the formal counter-part of~$\\omega^\\omega$. Indeed, since $\\bbomega^{\\bbomega} = \\sup_{n \\in \\NN} \\bbomega^n$, if $\\omega \\subsetneq M$, then $\\bbomega$ and a fortiori $\\bbomega^\\bbomega$ are not well-founded from the viewpoint of the meta-theory.\n\nWe often identify an integer $k$ with its ordinal $\\{0, \\dots, k-1\\}$. \nGiven a set $X \\subseteq \\NN$ and an integer~$n \\in \\NN$, we write $[X]^n$ for the collection of all subsets of~$X$ of size~$n$.\nThe set $[X]^n$ is in one-to-one correspondence with the set of all increasing ordered $n$-tuples over~$X$. Therefore, for simplicity of notation, given a coloring $f : [X]^n \\to k$, we write $f(x_0, \\dots, x_{n-1})$ for $f(\\{x_0, \\dots, x_{n-1}\\})$ assuming that $x_0 < \\dots < x_{n-1}$.", "sketch": "The post-theorem introduction gives a high-level proof plan: (1) in \\Cref{sec:largeness} it surveys and compares three notions of largeness—Ketonen and Solvay's $\\alpha$-largeness, Le Hou\\'erou–Levy Patey–Yokoyama's $\\alpha$-largeness$(\\theta)$, and the new notion $\\alpha$-largeness${}^*(\\theta)$—for $\\alpha$ of the form $\\bbomega^n \\cdot k$; (2) in \\Cref{sec:combinatorics} it develops the framework of $\\alpha$-largeness${}^*(\\theta)$ and proves polynomial upper bounds for $\\RT^2_2$-$\\bbomega^n$-largeness${}^*(\\theta)$; (3) in \\Cref{sec:translation} it \"combine[s] this upper bound with the framework of forcing interpretation\" to prove \\Cref{thm:rt22-polynomially-simulated}.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label[maintheorem]{thm:rt22-polynomially-simulated}\n$\\RCA_0 + \\BSig_2$ polynomially simulates $\\WKL_0 + \\RT_2^2$ with respect to $\\forall \\Sigma_3^0$ sentences.,", "theorem_type": ["Implication"], "mcq": {"question": "Say that a theory \\(T_1\\) polynomially simulates a theory \\(T_0\\) with respect to a class \\(\\Gamma\\) of sentences if there exists a polynomial \\(Q\\) such that for every \\(T_0\\)-proof \\(p\\) of any sentence in \\(\\Gamma\\), the translated \\(T_1\\)-proof \\(\\hat p\\) of the same sentence satisfies \\(|\\hat p|\\le Q(|p|)\\). Here a \\(\\forall\\Sigma^0_3\\)-sentence means a sentence of the form \\(\\forall X\\,\\varphi(X)\\), where \\(\\varphi\\) is arithmetical \\(\\Sigma^0_3\\). Which statement holds for the second-order arithmetic theories \\(\\RCA_0+\\BSig_2\\) and \\(\\WKL_0+\\RT^2_2\\), where \\(\\BSig_2\\) is \\(\\Sigma^0_2\\)-bounding and \\(\\RT^2_2\\) is Ramsey's theorem for pairs and two colors?", "correct_choice": {"label": "A", "text": "\\(\\RCA_0+\\BSig_2\\) polynomially simulates \\(\\WKL_0+\\RT^2_2\\) with respect to \\(\\forall\\Sigma^0_3\\) sentences; equivalently, there is a polynomial \\(Q\\) such that every \\((\\WKL_0+\\RT^2_2)\\)-proof \\(p\\) of a \\(\\forall\\Sigma^0_3\\)-sentence can be translated into an \\((\\RCA_0+\\BSig_2)\\)-proof \\(\\hat p\\) of the same sentence with \\(|\\hat p|\\le Q(|p|)\\)."}, "choices": [{"label": "B", "text": "\\(\\RCA_0+\\BSig_2\\) polynomially simulates \\(\\WKL_0+\\RT^2_2\\) with respect to all \\(\\mathcal L_2\\)-sentences; equivalently, there is a polynomial \\(Q\\) such that every \\((\\WKL_0+\\RT^2_2)\\)-proof \\(p\\) of an \\(\\mathcal L_2\\)-sentence can be translated into an \\((\\RCA_0+\\BSig_2)\\)-proof \\(\\hat p\\) of the same sentence with \\(|\\hat p|\\le Q(|p|)\\)."}, {"label": "C", "text": "\\(\\RCA_0+\\BSig_2\\) simulates \\(\\WKL_0+\\RT^2_2\\) with respect to \\(\\forall\\Sigma^0_3\\) sentences; equivalently, every \\((\\WKL_0+\\RT^2_2)\\)-proof of a \\(\\forall\\Sigma^0_3\\)-sentence can be translated into an \\((\\RCA_0+\\BSig_2)\\)-proof of the same sentence."}, {"label": "D", "text": "\\(\\WKL_0+\\RT^2_2\\) polynomially simulates \\(\\RCA_0+\\BSig_2\\) with respect to \\(\\forall\\Sigma^0_3\\) sentences; equivalently, there is a polynomial \\(Q\\) such that every \\((\\RCA_0+\\BSig_2)\\)-proof \\(p\\) of a \\(\\forall\\Sigma^0_3\\)-sentence can be translated into a \\((\\WKL_0+\\RT^2_2)\\)-proof \\(\\hat p\\) of the same sentence with \\(|\\hat p|\\le Q(|p|)\\)."}, {"label": "E", "text": "\\(\\RCA_0+\\BSig_2\\) simulates \\(\\WKL_0+\\RT^2_2\\) with respect to \\(\\forall\\Sigma^0_3\\) sentences, but no polynomial bound on the translation length is available; equivalently, for every translation of a \\((\\WKL_0+\\RT^2_2)\\)-proof \\(p\\) of a \\(\\forall\\Sigma^0_3\\)-sentence into an \\((\\RCA_0+\\BSig_2)\\)-proof \\(\\hat p\\) of the same sentence, one cannot in general ensure \\(|\\hat p|\\le Q(|p|)\\) for any polynomial \\(Q\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "sentence-class restriction to \\forall\\Sigma^0_3", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "polynomial bound on translated proof length", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "direction of polynomial simulation", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "existence of a uniform polynomial bound", "template_used": "uniformity_effectivity"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines polynomial simulation and the sentence class but does not reveal which direction of simulation is true, whether the result is polynomial or merely qualitative, or how broad the sentence class is."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: the correct choice is essentially the exact theorem statement, with distractors formed by predictable modifications (stronger domain, weaker conclusion, reversed direction, denial of polynomial bound). It is not pure tautology, but it is only a mild reformulation."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to compare strength, direction, and uniformity across the options, but the question primarily tests recall/recognition of a known result rather than generating a conclusion from mathematical argument."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: one overgeneralizes the sentence class, one weakens polynomial simulation to mere simulation, one reverses the simulation direction, and one denies the polynomial bound. These reflect common failure modes."}, "total_score": 6, "overall_assessment": "A solid recognition-style theorem question with good distractors and no answer leakage, but limited generative depth because it is very close to selecting the exact statement of a known result."}}
{"id": "2602.12122v1", "paper_link": "http://arxiv.org/abs/2602.12122v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\sl \\label{th:Lq} Let $V_1,V_2\\in L^1(\\mathbb{R}^n)\\cap L^q(\\mathbb{R}^n)$ be time-independent potentials, where $q>1$ if $n=2$ and $q\\ge n/2$ if $n\\ge 3$. Let $\\mathcal U_T^1$ and $\\mathcal U_T^2$ denote the corresponding initial-to-final-state maps. Then\n\\[\n\\mathcal U_T^1=\\mathcal U_T^2 \\quad\\Longrightarrow\\quad V_1=V_2.\n\\]", "start_pos": 8402, "end_pos": 8742, "label": "th:Lq"}, "ref_dict": {"eq:orthogonality_intro": "\\begin{equation}\\label{eq:orthogonality_intro}\n\\int_{\\Sigma} (V_1-V_2)\\,u_1\\,\\overline{v_2}=0,\n\\end{equation}", "th:Lq": "\\begin{theorem}\\sl \\label{th:Lq} Let $V_1,V_2\\in L^1(\\R^n)\\cap L^q(\\R^n)$ be time-independent potentials, where $q>1$ if $n=2$ and $q\\ge n/2$ if $n\\ge 3$. Let $\\mathcal U_T^1$ and $\\mathcal U_T^2$ denote the corresponding initial-to-final-state maps. Then\n\\[\n\\mathcal U_T^1=\\mathcal U_T^2 \\quad\\Longrightarrow\\quad V_1=V_2.\n\\]\n\\end{theorem}", "eq:Schrodinger": "\\begin{equation}\\label{eq:Schrodinger}\n\\begin{cases}\ni \\partial_t u = -\\Delta u + V u & \\text{for } (t,x)\\in (0, T)\\times \\R^n \\eqqcolon \\Sigma,\n\\vspace{.6em}\n\\\\\nu(0,x) = f(x) & \\text{for } x\\in \\R^n.\n\\end{cases}\n\\end{equation}"}, "pre_theorem_intro_text_len": 3116, "pre_theorem_intro_text": "In this paper, we study an inverse problem for the Schr\\\"odinger equation in which the available data consist of the map that sends any initial state $f$ at time $t=0$ to the solution at a fixed final time $t=T$.\n\nTo make this precise, let us consider the initial-value problem for the Schr\\\"odinger equation with a time-independent potential\n\\begin{equation}\\label{eq:Schrodinger}\n\\begin{cases}\ni \\partial_t u = -\\Delta u + V u & \\text{for } (t,x)\\in (0, T)\\times \\mathbb{R}^n \\eqqcolon \\Sigma,\n\\vspace{.6em}\n\\\\\nu(0,x) = f(x) & \\text{for } x\\in \\mathbb{R}^n.\n\\end{cases}\n\\end{equation}\nWe assume throughout the paper that $V=V(x)\\in L^q(\\mathbb{R}^n)$, where $q\\ge n/2$ if $n\\ge 3$ or $q>1$ if $n=2$. Then, by \\cites{zbMATH02204588,zbMATH00179225}, this direct problem is well posed, and for every $f\\in L^2(\\mathbb{R}^n)$ there exists a unique solution\n\\[\nu\\in C\\left([0,T];L^2(\\mathbb{R}^n)\\right).\n\\]\nThe evolution associated with \\eqref{eq:Schrodinger} therefore defines a bounded linear operator\n\\[\n\\mathcal U : f\\in L^2(\\mathbb{R}^n)\\mapsto u\\in C\\left([0,T];L^2(\\mathbb{R}^n)\\right).\n\\]\nConsequently, for any fixed time $t\\in[0,T]$, the operator\n\\[\n\\mathcal U_t : f \\in L^2(\\mathbb{R}^n) \\mapsto u(t,\\centerdot)\\in L^2(\\mathbb{R}^n)\n\\]\nis also bounded, uniformly in $t$. Solutions of the form $u=\\mathcal U f$, with $f\\in L^2(\\mathbb{R}^n)$, will be referred to as \\emph{physical solutions}, while we call $\\mathcal U_T$ the \\emph{initial-to-final-state map}.\n\nThe main question addressed in this paper is whether the initial-to-final-state map $\\mathcal U_T$ uniquely determines the Hamiltonian $-\\Delta+V$. This inverse problem was first studied for \\emph{time-dependent} potentials in \\cite{zbMATH07801151}. There, the authors show that if the potentials $V_1,V_2 \\in L^1((0,T);L^\\infty(\\mathbb{R}^n))$ satisfy a \\emph{super-exponential decay} condition at infinity, and if $\\mathcal U_T^j$ denotes the initial-to-final-state map associated with $-\\Delta+V_j$, then there holds:\n\\[\n\\mathcal U_T^1=\\mathcal U_T^2 \\quad\\Longrightarrow\\quad V_1=V_2.\n\\]\nMore recently, this uniqueness result was extended in \\cite{caro2025initialtofinalstateinverseproblemunbounded} to unbounded time-dependent potentials that are allowed to exhibit local $L^q$-type singularities, but still requiring the super-exponential decay assumption at infinity.\n\nThe case of time-independent potentials was previously considered in \\cite{zbMATH08122191}, where uniqueness was established under comparatively weaker decay assumptions than in the time-dependent setting, namely assuming only super-linear decay at infinity. The purpose of the present paper is to relax these assumptions even further. Specifically, we prove uniqueness for time-independent potentials that may exhibit singularities of $L^q$-type in sets of finite measure and satisfy only $L^1$-integrability in sets of infinite measure; in the time-independent setting, this represents a substantial improvement over the decay and integrability assumptions in  \\cite{zbMATH08122191}, \\cite{zbMATH07801151}, and \\cite{caro2025initialtofinalstateinverseproblemunbounded}.", "context": "In this paper, we study an inverse problem for the Schr\\\"odinger equation in which the available data consist of the map that sends any initial state $f$ at time $t=0$ to the solution at a fixed final time $t=T$.\n\nTo make this precise, let us consider the initial-value problem for the Schr\\\"odinger equation with a time-independent potential\n\\begin{equation}\\label{eq:Schrodinger}\n\\begin{cases}\ni \\partial_t u = -\\Delta u + V u & \\text{for } (t,x)\\in (0, T)\\times \\mathbb{R}^n \\eqqcolon \\Sigma,\n\\vspace{.6em}\n\\\\\nu(0,x) = f(x) & \\text{for } x\\in \\mathbb{R}^n.\n\\end{cases}\n\\end{equation}\nWe assume throughout the paper that $V=V(x)\\in L^q(\\mathbb{R}^n)$, where $q\\ge n/2$ if $n\\ge 3$ or $q>1$ if $n=2$. Then, by \\cites{zbMATH02204588,zbMATH00179225}, this direct problem is well posed, and for every $f\\in L^2(\\mathbb{R}^n)$ there exists a unique solution\n\\[\nu\\in C\\left([0,T];L^2(\\mathbb{R}^n)\\right).\n\\]\nThe evolution associated with \\eqref{eq:Schrodinger} therefore defines a bounded linear operator\n\\[\n\\mathcal U : f\\in L^2(\\mathbb{R}^n)\\mapsto u\\in C\\left([0,T];L^2(\\mathbb{R}^n)\\right).\n\\]\nConsequently, for any fixed time $t\\in[0,T]$, the operator\n\\[\n\\mathcal U_t : f \\in L^2(\\mathbb{R}^n) \\mapsto u(t,\\centerdot)\\in L^2(\\mathbb{R}^n)\n\\]\nis also bounded, uniformly in $t$. Solutions of the form $u=\\mathcal U f$, with $f\\in L^2(\\mathbb{R}^n)$, will be referred to as \\emph{physical solutions}, while we call $\\mathcal U_T$ the \\emph{initial-to-final-state map}.\n\nThe main question addressed in this paper is whether the initial-to-final-state map $\\mathcal U_T$ uniquely determines the Hamiltonian $-\\Delta+V$. This inverse problem was first studied for \\emph{time-dependent} potentials in \\cite{zbMATH07801151}. There, the authors show that if the potentials $V_1,V_2 \\in L^1((0,T);L^\\infty(\\mathbb{R}^n))$ satisfy a \\emph{super-exponential decay} condition at infinity, and if $\\mathcal U_T^j$ denotes the initial-to-final-state map associated with $-\\Delta+V_j$, then there holds:\n\\[\n\\mathcal U_T^1=\\mathcal U_T^2 \\quad\\Longrightarrow\\quad V_1=V_2.\n\\]\nMore recently, this uniqueness result was extended in \\cite{caro2025initialtofinalstateinverseproblemunbounded} to unbounded time-dependent potentials that are allowed to exhibit local $L^q$-type singularities, but still requiring the super-exponential decay assumption at infinity.\n\nThe case of time-independent potentials was previously considered in \\cite{zbMATH08122191}, where uniqueness was established under comparatively weaker decay assumptions than in the time-dependent setting, namely assuming only super-linear decay at infinity. The purpose of the present paper is to relax these assumptions even further. Specifically, we prove uniqueness for time-independent potentials that may exhibit singularities of $L^q$-type in sets of finite measure and satisfy only $L^1$-integrability in sets of infinite measure; in the time-independent setting, this represents a substantial improvement over the decay and integrability assumptions in \\cite{zbMATH08122191}, \\cite{zbMATH07801151}, and \\cite{caro2025initialtofinalstateinverseproblemunbounded}.\n\n\\begin{equation}\\label{eq:Schrodinger}\n\\begin{cases}\ni \\partial_t u = -\\Delta u + V u & \\text{for } (t,x)\\in (0, T)\\times \\R^n \\eqqcolon \\Sigma,\n\\vspace{.6em}\n\\\\\nu(0,x) = f(x) & \\text{for } x\\in \\R^n.\n\\end{cases}\n\\end{equation}", "full_context": "In this paper, we study an inverse problem for the Schr\\\"odinger equation in which the available data consist of the map that sends any initial state $f$ at time $t=0$ to the solution at a fixed final time $t=T$.\n\nTo make this precise, let us consider the initial-value problem for the Schr\\\"odinger equation with a time-independent potential\n\\begin{equation}\\label{eq:Schrodinger}\n\\begin{cases}\ni \\partial_t u = -\\Delta u + V u & \\text{for } (t,x)\\in (0, T)\\times \\mathbb{R}^n \\eqqcolon \\Sigma,\n\\vspace{.6em}\n\\\\\nu(0,x) = f(x) & \\text{for } x\\in \\mathbb{R}^n.\n\\end{cases}\n\\end{equation}\nWe assume throughout the paper that $V=V(x)\\in L^q(\\mathbb{R}^n)$, where $q\\ge n/2$ if $n\\ge 3$ or $q>1$ if $n=2$. Then, by \\cites{zbMATH02204588,zbMATH00179225}, this direct problem is well posed, and for every $f\\in L^2(\\mathbb{R}^n)$ there exists a unique solution\n\\[\nu\\in C\\left([0,T];L^2(\\mathbb{R}^n)\\right).\n\\]\nThe evolution associated with \\eqref{eq:Schrodinger} therefore defines a bounded linear operator\n\\[\n\\mathcal U : f\\in L^2(\\mathbb{R}^n)\\mapsto u\\in C\\left([0,T];L^2(\\mathbb{R}^n)\\right).\n\\]\nConsequently, for any fixed time $t\\in[0,T]$, the operator\n\\[\n\\mathcal U_t : f \\in L^2(\\mathbb{R}^n) \\mapsto u(t,\\centerdot)\\in L^2(\\mathbb{R}^n)\n\\]\nis also bounded, uniformly in $t$. Solutions of the form $u=\\mathcal U f$, with $f\\in L^2(\\mathbb{R}^n)$, will be referred to as \\emph{physical solutions}, while we call $\\mathcal U_T$ the \\emph{initial-to-final-state map}.\n\nThe main question addressed in this paper is whether the initial-to-final-state map $\\mathcal U_T$ uniquely determines the Hamiltonian $-\\Delta+V$. This inverse problem was first studied for \\emph{time-dependent} potentials in \\cite{zbMATH07801151}. There, the authors show that if the potentials $V_1,V_2 \\in L^1((0,T);L^\\infty(\\mathbb{R}^n))$ satisfy a \\emph{super-exponential decay} condition at infinity, and if $\\mathcal U_T^j$ denotes the initial-to-final-state map associated with $-\\Delta+V_j$, then there holds:\n\\[\n\\mathcal U_T^1=\\mathcal U_T^2 \\quad\\Longrightarrow\\quad V_1=V_2.\n\\]\nMore recently, this uniqueness result was extended in \\cite{caro2025initialtofinalstateinverseproblemunbounded} to unbounded time-dependent potentials that are allowed to exhibit local $L^q$-type singularities, but still requiring the super-exponential decay assumption at infinity.\n\nThe case of time-independent potentials was previously considered in \\cite{zbMATH08122191}, where uniqueness was established under comparatively weaker decay assumptions than in the time-dependent setting, namely assuming only super-linear decay at infinity. The purpose of the present paper is to relax these assumptions even further. Specifically, we prove uniqueness for time-independent potentials that may exhibit singularities of $L^q$-type in sets of finite measure and satisfy only $L^1$-integrability in sets of infinite measure; in the time-independent setting, this represents a substantial improvement over the decay and integrability assumptions in \\cite{zbMATH08122191}, \\cite{zbMATH07801151}, and \\cite{caro2025initialtofinalstateinverseproblemunbounded}.\n\n\\begin{equation}\\label{eq:Schrodinger}\n\\begin{cases}\ni \\partial_t u = -\\Delta u + V u & \\text{for } (t,x)\\in (0, T)\\times \\R^n \\eqqcolon \\Sigma,\n\\vspace{.6em}\n\\\\\nu(0,x) = f(x) & \\text{for } x\\in \\R^n.\n\\end{cases}\n\\end{equation}\n\nThe case of time-independent potentials was previously considered in \\cite{zbMATH08122191}, where uniqueness was established under comparatively weaker decay assumptions than in the time-dependent setting, namely assuming only super-linear decay at infinity. The purpose of the present paper is to relax these assumptions even further. Specifically, we prove uniqueness for time-independent potentials that may exhibit singularities of $L^q$-type in sets of finite measure and satisfy only $L^1$-integrability in sets of infinite measure; in the time-independent setting, this represents a substantial improvement over the decay and integrability assumptions in \\cite{zbMATH08122191}, \\cite{zbMATH07801151}, and \\cite{caro2025initialtofinalstateinverseproblemunbounded}.\n\n\\medskip\n\\noindent\\textbf{Outline of the proof.}\nThe proof of Theorem~\\ref{th:Lq} proceeds by extracting information on the difference $V_1-V_2$ from $\\mathcal U_T^1 = \\mathcal U_T^2$, by testing it against suitable families of solutions.\n\nReturning to the solutions in \\eqref{eq:steadysol}, we can now use the invertibility of $\\mathrm{Id}-P_\\lambda\\circ V$ to complete the construction of the correction term $w^{\\mathrm{cor}}$. In the statement of the corollary below, the additional assumption $V\\in L^1(\\R^n)$ is used only to ensure that $Vw^{(0)}\\in L^{p'}(\\R^n)$, so that $P_\\lambda(Vw^{(0)})$ is well defined and belongs to $L^p(\\R^n)$; the invertibility of $\\mathrm{Id}-P_\\lambda\\circ V$ on $L^p(\\R^n)$ relies solely on the $L^q$ assumption.\n\\begin{corollary}\\label{cor:solutions}\\sl Let $n\\geq 2$ and consider a time-independent potential $V\\in L^1(\\R^n)\\cap L^q(\\R^n)$ for some $n/2<q\\leq (n+1)/2$, and let $p$ be defined via $\\frac{1}{q} = 1 - \\frac{2}{p} $. For $\\lambda > 2\\lambda_V$ with $\\lambda_V$ as in the statement of Lemma~\\ref{lem:I-P inverse in Lp}, the functions\n\\[\nw(x)\\coloneqq e^{-i\\lambda\\omega\\cdot x}+w^{\\mathrm{cor}}(x)= w^{(0)}(x)+w^{\\mathrm{cor}}(x),\\qquad w^{\\mathrm{cor}}(x)\\coloneqq (\\mathrm{Id}-P_\\lambda\\circ V)^{-1}[P_\\lambda(Vw^{(0)})],\n\\]\nwith $ \\omega\\in\\mathbb S^{n-1}$ and $x\\in\\R^n$,\nare weak solutions of the time-independent Schr\\\"odinger equation \n\\[\n(\\Delta+\\lambda^2-V)w=0\\quad\\mathrm{in}\\quad\\R^n.\n\\]\nAdditionally, for $\\lambda>2\\lambda_V$ we have the estimate\n\\[\n\\left \\|w^{\\mathrm{cor}}\\right\\|_{L^p(\\R^n)}\\lesssim \\frac{1}{\\lambda^{2n\\left(\\frac1p-\\frac{1}{p_n}\\right)}} \\|V\\|_{L^{p'}(\\R^n)}\\leq \\frac{1}{\\lambda^{2n\\left(\\frac1p-\\frac{1}{p_n}\\right)}} \\|V\\|_{L^1(\\R^n)\\cap L^q(\\R^n)}.\n\\]\nIn particular, the function \n\\[\n\\psi_V ^{\\lambda,\\omega}(t,x)\\coloneqq e^{-i\\lambda^2 t}w(x)= e^{-i\\lambda^2 t}(w^{(0)}(x)+w^{\\mathrm{cor}}(x)),\\qquad (t,x)\\in\\R\\times \\R^n,\n\\]\nis a weak solution of \\eqref{sch_timeind}.\n\\end{corollary}\n\n\\begin{corollary}\\label{cor:solutionsVcrit}\\sl Let $n\\geq 3$ and consider a time-independent potential $V\\in L^1(\\R^n)\\cap L^{n/2}(\\R^n)$. Let $X_\\lambda,X_\\lambda ^*$ and $\\|V\\|_\\lambda$ as defined above. There exists a constant $\\lambda_V$ depending only on $V$ and the dimension such that for $\\lambda > \\lambda_V$, the functions\n\\[\nw(x)\\coloneqq e^{-i\\lambda\\omega\\cdot x} +w^{\\mathrm{cor}} (x)\\eqqcolon w^{(0)}(x)+w^{\\mathrm{cor}}(x),\\qquad w^{\\mathrm{cor}}(x)\\coloneqq (\\mathrm{Id}-P_\\lambda\\circ V)^{-1}[P_\\lambda(Vw^{(0)})],\n\\]\nwith $\\omega\\in\\mathbb S^{n-1} $ and $ x\\in\\R^n$,\nare weak solutions of the time-independent Schr\\\"odinger equation \n\\[\n(\\Delta+\\lambda^2-V)w=0\\quad\\mathrm{in}\\quad\\R^n.\n\\]\nAdditionally, the following estimates are satisfied\n\\[\n\\left\\|w^{\\mathrm{cor}} \\right\\|_{X_\\lambda ^*}\\lesssim \\|V\\|_{X_\\lambda}\\leq \\lambda^{-\\frac{1}{n+1}} \\|V\\|_{L^{q_n '}(\\R^n)}\\leq \\lambda^{-\\frac{1}{n+1}} \\|V\\|_{L^1(\\R^n)\\cap L^{n/2}(\\R^n)}\n\\]\nIn particular, the function \n\\[\n\\psi_V ^{\\lambda,\\omega}(t,x)\\coloneqq e^{-i\\lambda^2 t}w(x)= e^{-i\\lambda^2 t}(w^{\\mathrm{cor}}(x)+w^{(0)}(x)),\\qquad (t,x)\\in\\R\\times \\R^n,\n\\]\nis a weak solution of \\eqref{sch_timeind}.\n\\end{corollary}\n\n\\begin{proposition}\\label{pr:int_by_parts_time-harmonic_solutions}\\sl Let $V_1,V_2\\in L^1(\\R^n)\\cap L^q(\\R^n)$, where $q\\in(1,(n+1)/2]$ if $n=2$, or $q\\in[n/2,(n+1)/2]$ if $n\\geq 3$, and define $p$ by \n\\[\n \\frac{1}{q}+\\frac{2}{p}=1.\n\\]\nLet $\\mathcal U_T^1,\\mathcal U_T^2$ denote the initial-to-final-state maps corresponding to $V_1,V_2$, respectively.\n\n\\subsection{Proof of Theorem~\\ref{th:Lq}: the endpoint case}\\label{subsec: proof_uniqueness_endpoint} Let $V_1,V_2\\in L^1(\\R^n)\\cap L^{n/2}(\\R^n)$ and define $F\\coloneqq V_1 - V_2$. Given $\\xi \\in \\R^n$ we consider $\\nu \\in \\Sph^{n-1}$ such that $\\xi \\cdot \\nu = 0$ and $\\lambda \\geq \\max\\{\\lambda_{V_1},\\lambda_{V_2},|\\xi|/2\\}$, where $\\lambda_{V_1},\\lambda_{V_2}$ are the constants corresponding to $V_1,V_2$ from Corollary~\\ref{cor:solutionsVcrit}. As in \\S\\ref{subsec:proof_uniqueness_unbounded}, we consider the vectors\n\\[\n\\omega_1 \\coloneqq \\frac{1}{\\lambda} \\frac{\\xi}{2} + \\left( 1 - \\frac{|\\xi|^2}{4 \\lambda^2} \\right)^{1/2} \\nu,\\qquad \\omega_2 \\coloneqq -\\frac{1}{\\lambda} \\frac{\\xi}{2} + \\left( 1 - \\frac{|\\xi|^2}{4 \\lambda^2} \\right)^{1/2} \\nu,\n\\]\nand note that $\\omega_1,\\omega_2\\in\\mathbb S^{n-1}$ and $\\omega_1 - \\omega_2 = \\xi/\\lambda$. \nWe now construct the stationary state solutions as in Corollary~\\ref{cor:solutionsVcrit}\n\\[\n\\psi_j\\coloneqq \\psi_{V_j} ^{\\lambda,\\omega_j}=e^{-i\\lambda^2t}w_j(x)=e^{-i\\lambda^2t}\\left(w^{(0)}_j(x)+w^{\\mathrm{cor}}_j(x)\\right),\\qquad j\\in\\{1,2\\},\\qquad (t,x)\\in\\R\\times \\R^n,\n\\]\nand we recall the estimates\n\\begin{equation}\\label{eq:psiestimates}\n\\left\\|w^{\\mathrm{cor}}_j\\right\\|_{X_\\lambda ^*}\\lesssim \\|V_j\\|_{X_\\lambda}\\leq \\lambda^{-\\frac{1}{n+1}} \\|V_j\\|_{L^1(\\R^n)\\cap L^{n/2}(\\R^n)},\\qquad j\\in\\{1,2\\},\n\\end{equation}\nalso proved in Corollary~\\ref{cor:solutionsVcrit}. Since $ \\mathcal{U}^1_T=\\mathcal{U}^2_T$, plugging the solutions $\\psi_1,\\psi_2$ into the orthogonality relation of \nProposition~\\ref{pr:int_by_parts_time-harmonic_solutions} and repeating the calculations from \\S\\ref{subsec:proof_uniqueness_unbounded}, we have \n\\[\n\\begin{split}\n|\\wh{F}(\\xi)| &\\leq \\frac{1}{(2\\pi)^{n/2}} \\int_{\\R^n} \\left|F w^{(0)}_1 \\overline{w^{\\mathrm{cor}}_2}\\right| + \\int_{\\R^n} \\left|F \\overline{w^{(0)}_2} w^{\\mathrm{cor}}_1\\right| + \\int_{\\R^n} \\left|F w^{\\mathrm{cor}}_1 \\overline{w^{\\mathrm{cor}}_2}\\right| \n\\\\\n&\\lesssim \\left\\|Fw^{(0)}_1\\right\\|_{X_\\lambda} \\left\\|w^{\\mathrm{cor}}_2\\right\\|_{\\X_\\lambda ^*}+\\left\\|Fw^{(0)}_2\\right\\|_{X_\\lambda} \\left\\|w^{\\mathrm{cor}}_1\\right\\|_{\\X_\\lambda ^*}+\\|F\\|_\\lambda\\|w^{\\mathrm{cor}}_1\\|_{\\X_\\lambda ^*} \\left \\|w^{\\mathrm{cor}}_2\\right \\|_{\\X_\\lambda ^*},\n\\end{split}\n\\]\nwhere we have used the duality of $X_\\lambda$ and $X_\\lambda ^*$ and the H\\\"older-type inequality of Lemma~\\ref{lem:multiplication_V_endpoint_X_norm}. Using the estimates \\eqref{eq:psiestimates} and the fact that $|w^{(0)}_1|=|w^{(0)}_2|=1$ we get\n\\begin{align*}\n|\\wh{F}(\\xi)| &\\lesssim \n \\|F\\|_{\\X_\\lambda} \\lambda^{-\\frac{1}{n+1}}(\\|V_1\\|_{L^1(\\R^n)\\cap L^{n/2}(\\R^n)} + \\|V_2\\|_{L^1(\\R^n)\\cap L^{n/2}(\\R^n)}) \\\\\n&+ \\|F\\|_{\\lambda} \\lambda^{-\\frac{2}{n+1}} \\|V_1\\|_{L^1(\\R^n)\\cap L^{n/2}(\\R^n)} \\|V_2\\|_{L^1(\\R^n)\\cap L^{n/2}(\\R^n)} \\\\\n&\\lesssim\\lambda^{-\\frac{2}{n+1}} \\left(\\|V_1\\|_{L^1(\\R^n)\\cap L^{n/2}(\\R^n)} +\\|V_2\\|_{L^1(\\R^n)\\cap L^{n/2}(\\R^n)}\\right) ^2 \n(1 + \\|F\\|_{\\lambda}) \\to 0\n\\end{align*}\nas $\\lambda \\to \\infty$. Hence $\\wh{F}(\\xi) = 0$, and since $\\xi\\in\\R^n$ was arbitrary we conclude that $F=0$ and hence that $V_1=V_2$ almost everywhere in $\\R^n$, thus completing the proof of the endpoint case of Theorem~\\ref{th:Lq}.", "post_theorem_intro_text_len": 6530, "post_theorem_intro_text": "\\medskip\n\\noindent\\textbf{Outline of the proof.}\nThe proof of Theorem~\\ref{th:Lq} proceeds by extracting information on the difference $V_1-V_2$ from $\\mathcal U_T^1 = \\mathcal U_T^2$, by testing it against suitable families of solutions.\n\nThe first step is to show that the equality $\\mathcal U_T^1=\\mathcal U_T^2$ yields an \\emph{Alessandrini-type orthogonality relation}, in the sense of \\cite{Aless}, of the form\n\\begin{equation}\\label{eq:orthogonality_intro}\n\\int_{\\Sigma} (V_1-V_2)\\,u_1\\,\\overline{v_2}=0,\n\\end{equation}\nvalid for pairs of solutions associated with the potentials $V_1$ and $\\overline{V_2}$. This identity is initially available only for \\emph{physical solutions}, namely solutions belonging to $C([0,T];L^2(\\mathbb{R}^n))$. \n\nAt this stage, our first obstruction is that in order to recover pointwise information on $V_1-V_2$, we need to test \\eqref{eq:orthogonality_intro} against special \\emph{time-harmonic} solutions. Such solutions can be constructed as perturbations of time-harmonic solutions of the  free Schr\\\"odinger equation; we refer to such solutions as \\emph{stationary states} associated with the given potential. However, these stationary states are not in $C([0, T]; L^2(\\mathbb{R}^n))$, and using them as test functions requires a substantial extension of the Alessandrini-type orthogonality relation beyond physical solutions.\n\nMore precisely, stationary states are obtained by inverting a resolvent-type operator and constructing correction terms via Neumann series. Concretely, they are of the form\n\\[\nu(t,x)=e^{-i|\\kappa|^2 t}\\left(e^{-i\\kappa\\cdot x}+w^{\\mathrm{cor}}(x)\\right),\n\\]\nwhere the leading term \\(e^{-i|\\kappa|^2 t}e^{-i\\kappa\\cdot x}\\) is a time-harmonic solution of the free Schr\\\"odinger equation, and the correction \\(w^{\\mathrm{cor}}\\) accounts for the presence of the potential \\(V\\).\n\nThe construction of \\(w^{\\mathrm{cor}}\\) reduces to inverting an operator of the form \\(\\mathrm{Id}-P_\\lambda\\circ V\\), where \\(P_\\lambda\\) denotes a solution operator for the Helmholtz equation \\((\\Delta+\\lambda^2)u=f\\). This inversion is carried out on suitable function spaces, chosen so that the operator \\(P_\\lambda\\circ V\\) is small, in a suitable sense.\n\nIn following the proof strategy outlined above, we encounter a second important obstruction. In the non-endpoint regime $V\\in L^q$ with $q>n/2$, decay in the energy parameter $\\lambda$ follows from the classical Kenig--Ruiz--Sogge resolvent estimate. The latter ensures that $P_\\lambda\\circ V$ is small on the relevant function spaces for large $\\lambda$. At the critical endpoint $q=n/2$, however, this estimate no longer yields decay in $\\lambda$, and the smallness of $P_\\lambda\\circ V$ cannot be obtained from the standard resolvent bound alone.\n\nTo address this issue, we introduce a new scale of Banach spaces that allows us to recover a decaying factor in the energy parameter, also at the endpoint. Within this framework, we establish an improved form of the Kenig--Ruiz--Sogge resolvent estimate, sufficient to construct stationary states for critical potentials despite the absence of quantitative decay at the endpoint. A similar idea has appeared in \\cite{CaroGarcia} for a related scattering problems with critically singular potentials; see also \\cite{zbMATH07867333}. When inserted into the extended Alessandrini-type orthogonality relation, these stationary states allow us to isolate the Fourier phase and hence recover the Fourier transform of $V_1-V_2$, proving uniqueness.\n\nThroughout the argument, the assumption of a stationary potential plays a crucial role. Indeed, in the time-dependent setting, the uniqueness results in \\cites{zbMATH07801151,caro2025initialtofinalstateinverseproblemunbounded} rely on constructing complex geometrical optics solutions whose leading terms are of the form\n\\[\n(t,x) \\longmapsto e^{it|\\kappa|^2} e^{\\kappa\\cdot x}, \\qquad \\kappa \\in \\mathbb{R}^n,\n\\]\nsupplemented by correction terms depending on the potential. These complex exponentials grow in certain directions.  Using such solutions in an orthogonality relation of the form \\eqref{eq:orthogonality_intro} and controlling the corresponding correction terms naturally leads to assuming super-exponential decay of the potential at infinity.\n\nHowever, in the time-independent case considered here, we work with time-harmonic solutions whose main term is given in the form\n\\[\n(t,x) \\longmapsto e^{-i|\\kappa|^2 t} e^{-i\\kappa\\cdot x}, \\qquad \\kappa \\in \\mathbb{R}^n,\n\\]\nignoring again the perturbative terms correcting for the potential. Now the exponents are imaginary, so the leading term is purely oscillatory and unimodular. This allows us to restrict attention to a more regular class of solutions, which however is still sufficient to establish uniqueness in the case of time-independent potentials.\n\nInverse problems associated with the dynamical Schrödinger equation have been extensively studied; see, for example, \\cites{zbMATH01886353,zbMATH06733553,zbMATH05549395,zbMATH05655673,zbMATH05839237,zbMATH06864429}. Many of these works have also considered time-dependent Hamiltonians \\cites{zbMATH06769718,zbMATH06516179,zbMATH05379127,zbMATH07033617,zbMATH07242805}. A common feature of these investigations is the use of a dynamical Dirichlet-to-Neumann map, recorded on the boundary of a domain that contains the non-constant portions of the Hamiltonian. \n\nThis formulation stands in contrast to the one presented in our work. Here, the variable part of the Hamiltonian is not localized to a bounded region but is possibly present throughout the whole space. Moreover, our inverse problem is formulated with a distinct data requirement: knowledge only of the system's initial state and its corresponding state at a final time.\n\nThe rest of the paper is organized as follows. Section~\\ref{sec:aux} collects background material and auxiliary analytic tools, including well–posedness results and resolvent estimates. In Section~\\ref{sec:timeindependent_solutions} we construct stationary states for the Schr\\\"odinger equation with time-independent potentials, treating both the non-endpoint case $q>n/2$ and the endpoint case $q=n/2$ in dimensions $n\\ge3$. Section~\\ref{sec:orthorelation} is devoted to extending the Alessandrini-type orthogonality relation beyond physical solutions, and in particular to stationary-state solutions. Finally, in Section~\\ref{sec:unbounded_endpoint} we prove Theorem~\\ref{th:Lq} by applying the extended orthogonality relation to the stationary states constructed earlier.", "sketch": "The proof of Theorem~\\ref{th:Lq} \\emph{extracts information on} $V_1-V_2$ from $\\mathcal U_T^1=\\mathcal U_T^2$ by testing against suitable families of solutions.\n\n1. From $\\mathcal U_T^1=\\mathcal U_T^2$, first derive an \\emph{Alessandrini-type orthogonality relation} (initially for \\emph{physical solutions} in $C([0,T];L^2(\\mathbb{R}^n))$):\n\\[\n\\int_{\\Sigma} (V_1-V_2)\\,u_1\\,\\overline{v_2}=0,\n\\]\nfor pairs of solutions associated with $V_1$ and $\\overline{V_2}$.\n\n2. To recover pointwise/Fourier information on $V_1-V_2$, test the orthogonality identity with special \\emph{time-harmonic} solutions (\\emph{stationary states}), constructed as perturbations of free time-harmonic Schr\\\"odinger solutions. These have the form\n\\[\n u(t,x)=e^{-i|\\kappa|^2 t}\\big(e^{-i\\kappa\\cdot x}+w^{\\mathrm{cor}}(x)\\big),\n\\]\nwhere $w^{\\mathrm{cor}}$ accounts for the potential. Since these stationary states are \\emph{not} in $C([0,T];L^2)$, one must \\emph{extend the Alessandrini-type orthogonality relation beyond physical solutions} to allow using them as test functions.\n\n3. Construct $w^{\\mathrm{cor}}$ by inverting a resolvent-type operator and building correction terms via Neumann series: reduce to inverting\n\\(\\mathrm{Id}-P_\\lambda\\circ V\\), where $P_\\lambda$ solves the Helmholtz equation $(\\Delta+\\lambda^2)u=f$, on function spaces where $P_\\lambda\\circ V$ is small.\n\n4. Handle the endpoint obstruction: for $q>n/2$, smallness for large $\\lambda$ comes from the classical Kenig--Ruiz--Sogge resolvent estimate, but at the critical endpoint $q=n/2$ it yields no decay in $\\lambda$. To fix this, \\emph{introduce a new scale of Banach spaces} that recovers a decaying factor in $\\lambda$ and establish an \\emph{improved} Kenig--Ruiz--Sogge resolvent estimate, enabling stationary-state construction even at the endpoint.\n\n5. Insert these stationary states into the \\emph{extended} orthogonality relation; this \\emph{isolates the Fourier phase} and thus \\emph{recovers the Fourier transform of} $V_1-V_2$, yielding $V_1=V_2$ (uniqueness).\n\n6. The argument uses crucially that the potential is time-independent: one can use time-harmonic solutions with leading term $e^{-i|\\kappa|^2 t}e^{-i\\kappa\\cdot x}$, which is \\emph{purely oscillatory and unimodular}, avoiding the growth issues of complex geometrical optics solutions used in the time-dependent setting.", "expanded_sketch": "The proof of the main theorem extracts information on $V_1-V_2$ from $\\mathcal U_T^1=\\mathcal U_T^2$ by testing against suitable families of solutions.\n\n1. From $\\mathcal U_T^1=\\mathcal U_T^2$, first derive an \\emph{Alessandrini-type orthogonality relation} (initially for \\emph{physical solutions} in $C([0,T];L^2(\\mathbb{R}^n))$):\n\\[\n\\int_{\\Sigma} (V_1-V_2)\\,u_1\\,\\overline{v_2}=0,\n\\]\nfor pairs of solutions associated with $V_1$ and $\\overline{V_2}$.\n\n2. To recover pointwise/Fourier information on $V_1-V_2$, test the orthogonality identity with special \\emph{time-harmonic} solutions (\\emph{stationary states}), constructed as perturbations of free time-harmonic Schr\\\"odinger solutions. These have the form\n\\[\n u(t,x)=e^{-i|\\kappa|^2 t}\\big(e^{-i\\kappa\\cdot x}+w^{\\mathrm{cor}}(x)\\big),\n\\]\nwhere $w^{\\mathrm{cor}}$ accounts for the potential. Since these stationary states are \\emph{not} in $C([0,T];L^2)$, one must \\emph{extend the Alessandrini-type orthogonality relation beyond physical solutions} to allow using them as test functions.\n\n3. Construct $w^{\\mathrm{cor}}$ by inverting a resolvent-type operator and building correction terms via Neumann series: reduce to inverting\n\\(\\mathrm{Id}-P_\\lambda\\circ V\\), where $P_\\lambda$ solves the Helmholtz equation $(\\Delta+\\lambda^2)u=f$, on function spaces where $P_\\lambda\\circ V$ is small.\n\n4. Handle the endpoint obstruction: for $q>n/2$, smallness for large $\\lambda$ comes from the classical Kenig--Ruiz--Sogge resolvent estimate, but at the critical endpoint $q=n/2$ it yields no decay in $\\lambda$. To fix this, \\emph{introduce a new scale of Banach spaces} that recovers a decaying factor in $\\lambda$ and establish an \\emph{improved} Kenig--Ruiz--Sogge resolvent estimate, enabling stationary-state construction even at the endpoint.\n\n5. Insert these stationary states into the \\emph{extended} orthogonality relation; this \\emph{isolates the Fourier phase} and thus \\emph{recovers the Fourier transform of} $V_1-V_2$, yielding $V_1=V_2$ (uniqueness).\n\n6. The argument uses crucially that the potential is time-independent: one can use time-harmonic solutions with leading term $e^{-i|\\kappa|^2 t}e^{-i\\kappa\\cdot x}$, which is \\emph{purely oscillatory and unimodular}, avoiding the growth issues of complex geometrical optics solutions used in the time-dependent setting.", "expanded_theorem": "\\sl \\label{th:Lq} Let $V_1,V_2\\in L^1(\\mathbb{R}^n)\\cap L^q(\\mathbb{R}^n)$ be time-independent potentials, where $q>1$ if $n=2$ and $q\\ge n/2$ if $n\\ge 3$. Let $\\mathcal U_T^1$ and $\\mathcal U_T^2$ denote the corresponding initial-to-final-state maps. Then\n\\[\n\\mathcal U_T^1=\\mathcal U_T^2 \\quad\\Longrightarrow\\quad V_1=V_2.\n\\]", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Fix $n\\ge 2$ and a final time $T>0$. For $j=1,2$, let $V_j=V_j(x)$ be a time-independent potential on $\\mathbb{R}^n$ such that $V_j\\in L^1(\\mathbb{R}^n)\\cap L^q(\\mathbb{R}^n)$, where $q>1$ if $n=2$ and $q\\ge n/2$ if $n\\ge 3$. For each initial datum $f\\in L^2(\\mathbb{R}^n)$, let $u_j$ be the solution of\n\\[\n\\begin{cases}\ni\\partial_t u = -\\Delta u + V_j u & \\text{for } (t,x)\\in(0,T)\\times \\mathbb{R}^n,\\\\\nu(0,x)=f(x) & \\text{for } x\\in \\mathbb{R}^n,\n\\end{cases}\n\\]\nand define the initial-to-final-state map $\\mathcal U_T^j:L^2(\\mathbb{R}^n)\\to L^2(\\mathbb{R}^n)$ by $\\mathcal U_T^j f=u_j(T,\\cdot)$. Which statement holds for every such pair of potentials?", "correct_choice": {"label": "A", "text": "If $\\mathcal U_T^1=\\mathcal U_T^2$ as operators from $L^2(\\mathbb{R}^n)$ to $L^2(\\mathbb{R}^n)$, then $V_1=V_2$ almost everywhere on $\\mathbb{R}^n$."}, "choices": [{"label": "B", "text": "If $\\mathcal U_T^1=\\mathcal U_T^2$ as operators from $L^2(\\mathbb{R}^n)$ to $L^2(\\mathbb{R}^n)$, then $V_1=V_2$ almost everywhere on $\\mathbb{R}^n$, provided in addition that $q>n/2$ when $n\\ge 3$."}, {"label": "C", "text": "If $V_1=V_2$ almost everywhere on $\\mathbb{R}^n$, then $\\mathcal U_T^1=\\mathcal U_T^2$ as operators from $L^2(\\mathbb{R}^n)$ to $L^2(\\mathbb{R}^n)$."}, {"label": "D", "text": "If for every pair of physical solutions $u_1,u_2\\in C([0,T];L^2(\\mathbb{R}^n))$ associated with $V_1$ and $V_2$ one has\n\\[\n\\int_0^T\\!\\int_{\\mathbb{R}^n} (V_1-V_2)(x)\\,u_1(t,x)\\,\\overline{u_2(t,x)}\\,dx\\,dt=0,\n\\]\nthen $\\mathcal U_T^1=\\mathcal U_T^2$, and hence $V_1=V_2$ almost everywhere on $\\mathbb{R}^n$."}, {"label": "E", "text": "If $\\mathcal U_T^1=\\mathcal U_T^2$ as operators from $L^2(\\mathbb{R}^n)$ to $L^2(\\mathbb{R}^n)$, then the Fourier transform of $V_1-V_2$ vanishes almost everywhere on $\\mathbb{R}^n$, and therefore $V_1=V_2$ almost everywhere on $\\mathbb{R}^n$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "endpoint case q=n/2", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "reverse implication only; drops uniqueness direction from map equality to potential equality", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "need to extend orthogonality beyond physical solutions and use conjugate-potential pairing", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "Fourier recovery is pointwise in frequency via stationary-state tests, not merely almost everywhere", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It states the setup and asks which claim is universally valid; the correct answer is not leaked directly, though option A closely matches the likely theorem-level conclusion."}, "TAS": {"score": 1, "justification": "This is close to a theorem-recall item: the correct choice is essentially the main uniqueness statement under the stated hypotheses. However, it is not a pure tautology because the alternatives modify endpoints, directions of implication, and related consequences."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact valid statement from nearby variants, especially the endpoint case in B and the overclaims in D/E. Still, the item primarily tests recognition of the theorem rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically sophisticated and plausible: B tampers with the sharp range, C gives only the converse, D confuses an orthogonality condition with the full uniqueness mechanism, and E overstates Fourier-recovery consequences. They are distinct and target realistic failure modes."}, "total_score": 6, "overall_assessment": "A solid MCQ with strong, nuanced distractors and little answer leakage, but it leans heavily on theorem recognition and is only moderately effective at testing genuine generative reasoning."}}
{"id": "2602.12190v1", "paper_link": "http://arxiv.org/abs/2602.12190v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm:highT}\n\t\tFix $\\beta<1$. Let $M=M(N)$ satisfy $M/N\\to0$, and let $k=k(N)\\to\\infty$\n\t\twith\n\t\t\\[\n\t\t\\frac{k(N)M(N)}{N}\\longrightarrow 0 \n\t\t\\]\n\t\t(which in particular allows $k=o(N)$ when $M$ is fixed).\n\n\t\tThen\n\t\t\\[\n\t\t\\TV\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\n\t\t\\longrightarrow 0\n\t\t\\qquad\\text{in }\\Pp_\\xi\\text{-probability}.\n\t\t\\]", "start_pos": 11871, "end_pos": 12237, "label": "thm:highT"}, "ref_dict": {"thm:stop_macro_Msmall": "\\begin{theorem}\t\t\\label{thm:stop_macro_Msmall}\n\t\tFix $\\beta\\in(0,1)$ and assume $M=M(N)\\to\\infty$ with $M=o(\\sqrt N)$.\n\t\tLet $k=k(N)$ satisfy $k/N\\to\\rho\\in(0,1)$.\n\t\tThen, in $\\P_\\xi$-probability,\n\t\t\\[\n\t\t\\TV\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 1.\n\t\t\\]\n\t\\end{theorem}", "thm:crit_break_sqrtN": "\\begin{theorem}[Critical-window breakdown at $\\beta=1$ (fixed $M$)]\n\t\t\\label{thm:crit_break_sqrtN}\n\t\tFix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed.\n\t\tLet $k=k(N)$ satisfy\n\t\t\\[\n\t\t\\frac{k}{\\sqrt N}\\longrightarrow c\\in(0,\\infty).\n\t\t\\]\n\t\tThen there exists a deterministic constant $b=b(c,M)>0$ such that\n\t\t\\[\n\t\t\\liminf_{N\\to\\infty} \\ d_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\ \\ge\\ b\n\t\t\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n\t\t\\]\n\t\tIn particular, propagation of chaos fails in the critical window $k\\asymp \\sqrt N$,\n\t\tshowing that the scaling in Theorem~\\ref{thm:crit_poc} is optimal.\n\n\t\\end{theorem}", "thm:crit_poc": "\\begin{theorem}\t\t\\label{thm:crit_poc}\n\t\tFix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed. Assume $k=k(N)\\to\\infty$ satisfies\n\t\t\\[\n\t\t\\frac{k(N)}{N^{1/2}}\\longrightarrow 0.\n\t\t\\]\n\t\tThen\n\t\t\\[\n\t\td_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\n\t\t\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n\t\t\\]\n\t\\end{theorem}", "thm:highT": "\\begin{theorem}\t\t\\label{thm:highT}\n\t\tFix $\\beta<1$. Let $M=M(N)$ satisfy $M/N\\to0$, and let $k=k(N)\\to\\infty$\n\t\twith\n\t\t\\[\n\t\t\\frac{k(N)M(N)}{N}\\longrightarrow 0 \n\t\t\\]\n\t\t(which in particular allows $k=o(N)$ when $M$ is fixed).\n\n\t\tThen\n\t\t\\[\n\t\t\\TV\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\n\t\t\\longrightarrow 0\n\t\t\\qquad\\text{in }\\Pp_\\xi\\text{-probability}.\n\t\t\\]\n\t\\end{theorem}"}, "pre_theorem_intro_text_len": 9395, "pre_theorem_intro_text": "Propagation of chaos is a central concept in the study of interacting\n\tparticle systems and mean-field models.\n\tRoughly speaking, it describes the phenomenon that,\n\tas the system size $N$ tends to infinity,\n\tfinite collections of particles behave asymptotically independently,\n\twith a common limiting law.\n\tPropagation of chaos originated in Kac's Markovian models for gas dynamics\n\t\\cite{Kac_foundations,Kac_probability}, an attempt to justify\n\tBoltzmann's ''Sto\\ss zahlenansatz''.\n\tPropagation of chaos has since become an important \n\tobject of study\n\tin probability, statistical mechanics, and mathematical physics.\n\tThe original approach by Kac was that,\n\tif at time 0 the finite marginal distributions of a system are product measures in the\n\tthermodynamic limit, then this should carry over to the time-evolved system. In equilibrium settings for mean-field\n\tGibbs measures (where the energy function is a \n\tfunction of the empirical measure) with a unique minimizer of the Helmholtz free energy, however, this was shown to\n\tfollow from the fact the extremal Gibbs measures locally look like product measures, i.e.\\\n\tthat any finite subset of spins forms a family of independent random variables in the thermodynamic limit \\cite[Theorem 3]{BAZ_chaos}.\n\tSuch results provide a rigorous justification of mean-field\n\tapproximations and explain why macroscopic behavior\n\tcan often be described by effective one-particle models. In the present note, we remain entirely within a static framework, and study propagation of chaos in the sense of asymptotic factorization of finite-dimensional marginals for families of (random) \n\tmean-field Gibbs measures.\n\n\tMoreover, for mean-field spin systems, propagation of chaos\n\tis closely tied to the high-temperature regime.\n\tIn classical models such as the Curie--Weiss Ising model,\n\tchaos holds when the inverse temperature is below the critical value,\n\twhile it breaks down in the low-temperature phase. There one has to replace\n\tthe product measure by a mixture of the (several) extremal limiting Gibbs measures.\n\tAn additional difficulty may arise when the Gibbs measures in question are random. To the best of our knowledge comparatively little is known in such situations (with the exception of of \\cite{BG98book} and \\cite{KL24}). The present paper contributes to the study of propagation of chaos for random mean-field Gibbs measures in the high-temperature and near-critical regimes.\n\n\tImportantly, in \\cite{BAZ_chaos}, the authors also introduced the concept of \\textit{increasing} propagation of chaos and showed that in the true high temperature regime of many mean-field models one can let the size of the marginals $k$ grow with the system size $N$ as long as $k=o(N)$. \n\tUnderstanding not only whether propagation of chaos holds,\n\tbut also the precise scales at which it breaks down,\n\thas become an important theme in recent work on mean-field Gibbs measures \\cite{BAZ_chaos,Lacker22, JKLM23,KL24,RS25,JKL25}.\n\n\tWhile increasing propagation of chaos has been established in several genuinely high-temperature regimes of ordered mean-field models, much less is known in the presence of random mean-field Gibbs measures, where already the analysis of critical fluctuations can become delicate (\\cite{gentzloewe, gentzloewe2, talagrand_critical_hopf}).   \n\tIn the present setting of random mean-field Gibbs measures, we show that this high-temperature scaling persists for $\\beta <1$ (the critical value),\n\twhereas at criticality \n\t$\\beta=1$, the admissible growth drops to $k=o(\\sqrt N)$. Moreover, \n\tthese bounds on \n\t$k$ are optimal for the class of random mean-field Gibbs measures considered here.\n\n\tIn this article we study propagation of chaos\n\tin the Hopfield model, a paradigmatic example\n\tof a disordered mean-field spin system.\n\tThe Hopfield model has (at least) two distinct origins. It was first introduced by\n\tPastur and Figotin \\cite{PS84} as a solvable model of a disordered system. At about the same time it\n\twas independently invented by Hopfield as\n\ta model of associative memory \\cite{Hopfield1982}.\n\tBoth aspects have been intensively studied. For the disordered systems facet see e.g. \\cite{BGP94, BG97} or \\cite{talagrand}, while \n\tthe neural network aspect has found a renewed interest through two recent papers \\cite{KrotovHopfield2016,DHLUV17}.\n\n\tFor the purposes of the present paper, the \n\tprobabilistic viewpoint of seeing the Hopfield model\n\tas a mean-field Ising model with random, structured interactions is more appropriate.\n\tThe disorder, i.e.\\ the random interactions, is generated by a collection of random patterns,\n\twhich induces a random quadratic Hamiltonian\n\tand leads to a rich interplay between thermal fluctuations\n\tand quenched randomness.\n\tWhile the thermodynamic properties of the Hopfield model\n\tare well understood, much less is known about\n\tthe fine structure of its finite-dimensional marginals\n\tand their asymptotic independence properties. There is only one result concerning the propagation of chaos in the Hopfield model, see \\cite[Theorem 8.15]{BG98}. However, this result addresses fixed-dimensional marginals in the low-temperature regime, and relies on mechanisms different from those considered here.\n\n\tOur main goal is to analyze propagation of chaos\n\tfor the Hopfield Gibbs measure in the high-temperature\n\tand critical regimes, with particular emphasis\n\ton the size of the marginals.\n\tWe identify precise conditions under which propagation of chaos holds,\n\tas well as sharp thresholds for its breakdown.\n\tIn the high-temperature regime, we prove propagation of chaos\n\tfor growing marginals, provided their size grows sublinearly in the system size $N$.\n\tAt criticality, we show that chaos breaks down in a critical window,\n\twhose scale matches that of the dominant collective fluctuations.\n\treveal a clear transition between asymptotic independence and regimes of partial or complete breakdown of chaos,\n\tand highlight the role played by disorder-induced fluctuations\n\tin determining these regimes.\n\n\t\\subsection{The Model}\n\tLet us next describe the central model for the purpose of this note.\n\tLet $\\xi_i=(\\xi_i^1,\\dots,\\xi_i^M)\\in\\{-1,+1\\}^M$, $i=1,\\dots,N$, be i.i.d.\\ random vectors with independent coordinates,\n\t\\[\n\t\\mathbb E[\\xi_i^\\nu]=0, \\qquad \\mathbb E[(\\xi_i^\\nu)^2]=1 .\n\t\\]\n\t(In the interpretation of an associative memory, the vectors $(\\xi^\\mu)_{\\mu=1}^M=((\\xi_i^\\mu)_{i=1}^N)_{\\mu=1}^M$ are called images or patterns).\n\tIn what follows we will always assume  that $M=M(N)$ may depend on $N$, but in such a way, that $M=o(N)$, which is the natural regime in which the Hopfield model exhibits mean-field behavior. \n\tFor fixed, i.e.\\ quenched, patterns $(\\xi^\\mu)_\\mu$\tdefine and for $\\sigma\\in\\{-1,+1\\}^N$, define the overlap vector\n\t\\[\n\tm_N(\\sigma) = \\big(m_N^1(\\sigma),\\dots,m_N^M(\\sigma)\\big),\n\t\\qquad\n\tm_N^\\nu(\\sigma)=\\frac1N\\sum_{i=1}^N \\sigma_i\\xi_i^\\nu .\n\t\\]\n\tThe Hopfield Gibbs measure at inverse temperature $\\beta>0$ is\n\t\\[\n\t\\mu_N(\\sigma)=\\frac{1}{Z_N}\n\t\\exp\\Big(\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2\\Big).\n\t\\]\n\tThe Gibbs measure is fully determined by $m_N(\\sigma)$ (plus the inverse temperature), it thus is natural to consider \n\tthe behaviour of the overlap under the Gibbs measure.\n\tThis has been done in numerous papers: \n\tIn \\cite{BGP94} it was shown that -- similar to the Curie-Weiss model -- in the Hopfield model the critical (inverse) temperature is $\\beta=1$. While for $\\beta \\le 1$ the overlap vector  gets concentrated in the $M$-dimensional 0-vector, for larger $\\beta$ the limit points of $m_N$ are associated with the 2$M$ vectors $\\pm z(\\beta) e_\\mu$ where \n\t$e_\\mu$ is the $\\mu$'th unit vector and $z(\\beta)$ is the largest solution of the equation\n\t$$\n\tz=\\tanh(\\beta z).\n\t$$\n\tCentral Limit theorems for $\\sqrt N m_N(\\cdot)$ were proven by Gentz \\cite{gentz_annals} or Bovier and Gayrard \\cite{BG_CLT}. All these results hold true for almost all realizations of the patterns and only need $M=o(N)$\n\t(for \\cite{gentz_annals} this is only true in the high temperature regime, which, however, is the most relevant for us in the present note.) Importantly, as in the Curie-Weiss model, at $\\beta=1$ the fluctuations are non-Gaussian. As was shown in \\cite{gentzloewe, gentzloewe2, talagrand_critical_hopf} at $\\beta=1$\n\tthe rescaled overlap vector $N^{1/4} m_N$ converges in distribution to a random limit (while in the other limit theorems the limit was deterministic). Also large and moderate deviations for the overlap vector are available (see \\cite{BG_LDP, EL_hopf}).\n\n\t\\subsection{Statement of the results}\n\tIn this subsection we state our main results on increasing propagation of chaos\n\tfor the Hopfield model in the high-temperature and critical regimes. In particular, we identify regimes in which propagation of chaos breaks down\n\tat explicit scales of the marginals. For $\\beta<1$, we allow the number of patterns $M=M(N)$ to diverge with $N$,\n\tand establish both increasing propagation of chaos and sharp breakdown results.\n\tAt criticality $\\beta=1$, we restrict attention to fixed $M$, and identify the\n\tcritical window in which propagation of chaos fails.\n\n\tIn order to formulate our results, let us agree on the following notation:  \n\tFor $k\\le N$, let $\\mu_N^{(k)}$ denote the marginal of $\\mu_N$ on\n\t$(\\sigma_1,\\dots,\\sigma_k)$.\n\tLet $\\pi$ be the Rademacher law on $\\{-1,+1\\}$ with $\\pi(\\pm1)=\\frac12$.\n\n\tThen we will prove:", "context": "Propagation of chaos is a central concept in the study of interacting\n particle systems and mean-field models.\n Roughly speaking, it describes the phenomenon that,\n as the system size $N$ tends to infinity,\n finite collections of particles behave asymptotically independently,\n with a common limiting law.\n Propagation of chaos originated in Kac's Markovian models for gas dynamics\n \\cite{Kac_foundations,Kac_probability}, an attempt to justify\n Boltzmann's ''Sto\\ss zahlenansatz''.\n Propagation of chaos has since become an important \n object of study\n in probability, statistical mechanics, and mathematical physics.\n The original approach by Kac was that,\n if at time 0 the finite marginal distributions of a system are product measures in the\n thermodynamic limit, then this should carry over to the time-evolved system. In equilibrium settings for mean-field\n Gibbs measures (where the energy function is a \n function of the empirical measure) with a unique minimizer of the Helmholtz free energy, however, this was shown to\n follow from the fact the extremal Gibbs measures locally look like product measures, i.e.\\\n that any finite subset of spins forms a family of independent random variables in the thermodynamic limit \\cite[Theorem 3]{BAZ_chaos}.\n Such results provide a rigorous justification of mean-field\n approximations and explain why macroscopic behavior\n can often be described by effective one-particle models. In the present note, we remain entirely within a static framework, and study propagation of chaos in the sense of asymptotic factorization of finite-dimensional marginals for families of (random) \n mean-field Gibbs measures.\n\nImportantly, in \\cite{BAZ_chaos}, the authors also introduced the concept of \\textit{increasing} propagation of chaos and showed that in the true high temperature regime of many mean-field models one can let the size of the marginals $k$ grow with the system size $N$ as long as $k=o(N)$. \n Understanding not only whether propagation of chaos holds,\n but also the precise scales at which it breaks down,\n has become an important theme in recent work on mean-field Gibbs measures \\cite{BAZ_chaos,Lacker22, JKLM23,KL24,RS25,JKL25}.\n\n\\subsection{The Model}\n Let us next describe the central model for the purpose of this note.\n Let $\\xi_i=(\\xi_i^1,\\dots,\\xi_i^M)\\in\\{-1,+1\\}^M$, $i=1,\\dots,N$, be i.i.d.\\ random vectors with independent coordinates,\n \\[\n \\mathbb E[\\xi_i^\\nu]=0, \\qquad \\mathbb E[(\\xi_i^\\nu)^2]=1 .\n \\]\n (In the interpretation of an associative memory, the vectors $(\\xi^\\mu)_{\\mu=1}^M=((\\xi_i^\\mu)_{i=1}^N)_{\\mu=1}^M$ are called images or patterns).\n In what follows we will always assume that $M=M(N)$ may depend on $N$, but in such a way, that $M=o(N)$, which is the natural regime in which the Hopfield model exhibits mean-field behavior. \n For fixed, i.e.\\ quenched, patterns $(\\xi^\\mu)_\\mu$ define and for $\\sigma\\in\\{-1,+1\\}^N$, define the overlap vector\n \\[\n m_N(\\sigma) = \\big(m_N^1(\\sigma),\\dots,m_N^M(\\sigma)\\big),\n \\qquad\n m_N^\\nu(\\sigma)=\\frac1N\\sum_{i=1}^N \\sigma_i\\xi_i^\\nu .\n \\]\n The Hopfield Gibbs measure at inverse temperature $\\beta>0$ is\n \\[\n \\mu_N(\\sigma)=\\frac{1}{Z_N}\n \\exp\\Big(\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2\\Big).\n \\]\n The Gibbs measure is fully determined by $m_N(\\sigma)$ (plus the inverse temperature), it thus is natural to consider \n the behaviour of the overlap under the Gibbs measure.\n This has been done in numerous papers: \n In \\cite{BGP94} it was shown that -- similar to the Curie-Weiss model -- in the Hopfield model the critical (inverse) temperature is $\\beta=1$. While for $\\beta \\le 1$ the overlap vector gets concentrated in the $M$-dimensional 0-vector, for larger $\\beta$ the limit points of $m_N$ are associated with the 2$M$ vectors $\\pm z(\\beta) e_\\mu$ where \n $e_\\mu$ is the $\\mu$'th unit vector and $z(\\beta)$ is the largest solution of the equation\n $$\n z=\\tanh(\\beta z).\n $$\n Central Limit theorems for $\\sqrt N m_N(\\cdot)$ were proven by Gentz \\cite{gentz_annals} or Bovier and Gayrard \\cite{BG_CLT}. All these results hold true for almost all realizations of the patterns and only need $M=o(N)$\n (for \\cite{gentz_annals} this is only true in the high temperature regime, which, however, is the most relevant for us in the present note.) Importantly, as in the Curie-Weiss model, at $\\beta=1$ the fluctuations are non-Gaussian. As was shown in \\cite{gentzloewe, gentzloewe2, talagrand_critical_hopf} at $\\beta=1$\n the rescaled overlap vector $N^{1/4} m_N$ converges in distribution to a random limit (while in the other limit theorems the limit was deterministic). Also large and moderate deviations for the overlap vector are available (see \\cite{BG_LDP, EL_hopf}).\n\n\\subsection{Statement of the results}\n In this subsection we state our main results on increasing propagation of chaos\n for the Hopfield model in the high-temperature and critical regimes. In particular, we identify regimes in which propagation of chaos breaks down\n at explicit scales of the marginals. For $\\beta<1$, we allow the number of patterns $M=M(N)$ to diverge with $N$,\n and establish both increasing propagation of chaos and sharp breakdown results.\n At criticality $\\beta=1$, we restrict attention to fixed $M$, and identify the\n critical window in which propagation of chaos fails.\n\nIn order to formulate our results, let us agree on the following notation: \n For $k\\le N$, let $\\mu_N^{(k)}$ denote the marginal of $\\mu_N$ on\n $(\\sigma_1,\\dots,\\sigma_k)$.\n Let $\\pi$ be the Rademacher law on $\\{-1,+1\\}$ with $\\pi(\\pm1)=\\frac12$.\n\nThen we will prove:", "full_context": "Propagation of chaos is a central concept in the study of interacting\n particle systems and mean-field models.\n Roughly speaking, it describes the phenomenon that,\n as the system size $N$ tends to infinity,\n finite collections of particles behave asymptotically independently,\n with a common limiting law.\n Propagation of chaos originated in Kac's Markovian models for gas dynamics\n \\cite{Kac_foundations,Kac_probability}, an attempt to justify\n Boltzmann's ''Sto\\ss zahlenansatz''.\n Propagation of chaos has since become an important \n object of study\n in probability, statistical mechanics, and mathematical physics.\n The original approach by Kac was that,\n if at time 0 the finite marginal distributions of a system are product measures in the\n thermodynamic limit, then this should carry over to the time-evolved system. In equilibrium settings for mean-field\n Gibbs measures (where the energy function is a \n function of the empirical measure) with a unique minimizer of the Helmholtz free energy, however, this was shown to\n follow from the fact the extremal Gibbs measures locally look like product measures, i.e.\\\n that any finite subset of spins forms a family of independent random variables in the thermodynamic limit \\cite[Theorem 3]{BAZ_chaos}.\n Such results provide a rigorous justification of mean-field\n approximations and explain why macroscopic behavior\n can often be described by effective one-particle models. In the present note, we remain entirely within a static framework, and study propagation of chaos in the sense of asymptotic factorization of finite-dimensional marginals for families of (random) \n mean-field Gibbs measures.\n\nImportantly, in \\cite{BAZ_chaos}, the authors also introduced the concept of \\textit{increasing} propagation of chaos and showed that in the true high temperature regime of many mean-field models one can let the size of the marginals $k$ grow with the system size $N$ as long as $k=o(N)$. \n Understanding not only whether propagation of chaos holds,\n but also the precise scales at which it breaks down,\n has become an important theme in recent work on mean-field Gibbs measures \\cite{BAZ_chaos,Lacker22, JKLM23,KL24,RS25,JKL25}.\n\n\\subsection{The Model}\n Let us next describe the central model for the purpose of this note.\n Let $\\xi_i=(\\xi_i^1,\\dots,\\xi_i^M)\\in\\{-1,+1\\}^M$, $i=1,\\dots,N$, be i.i.d.\\ random vectors with independent coordinates,\n \\[\n \\mathbb E[\\xi_i^\\nu]=0, \\qquad \\mathbb E[(\\xi_i^\\nu)^2]=1 .\n \\]\n (In the interpretation of an associative memory, the vectors $(\\xi^\\mu)_{\\mu=1}^M=((\\xi_i^\\mu)_{i=1}^N)_{\\mu=1}^M$ are called images or patterns).\n In what follows we will always assume that $M=M(N)$ may depend on $N$, but in such a way, that $M=o(N)$, which is the natural regime in which the Hopfield model exhibits mean-field behavior. \n For fixed, i.e.\\ quenched, patterns $(\\xi^\\mu)_\\mu$ define and for $\\sigma\\in\\{-1,+1\\}^N$, define the overlap vector\n \\[\n m_N(\\sigma) = \\big(m_N^1(\\sigma),\\dots,m_N^M(\\sigma)\\big),\n \\qquad\n m_N^\\nu(\\sigma)=\\frac1N\\sum_{i=1}^N \\sigma_i\\xi_i^\\nu .\n \\]\n The Hopfield Gibbs measure at inverse temperature $\\beta>0$ is\n \\[\n \\mu_N(\\sigma)=\\frac{1}{Z_N}\n \\exp\\Big(\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2\\Big).\n \\]\n The Gibbs measure is fully determined by $m_N(\\sigma)$ (plus the inverse temperature), it thus is natural to consider \n the behaviour of the overlap under the Gibbs measure.\n This has been done in numerous papers: \n In \\cite{BGP94} it was shown that -- similar to the Curie-Weiss model -- in the Hopfield model the critical (inverse) temperature is $\\beta=1$. While for $\\beta \\le 1$ the overlap vector gets concentrated in the $M$-dimensional 0-vector, for larger $\\beta$ the limit points of $m_N$ are associated with the 2$M$ vectors $\\pm z(\\beta) e_\\mu$ where \n $e_\\mu$ is the $\\mu$'th unit vector and $z(\\beta)$ is the largest solution of the equation\n $$\n z=\\tanh(\\beta z).\n $$\n Central Limit theorems for $\\sqrt N m_N(\\cdot)$ were proven by Gentz \\cite{gentz_annals} or Bovier and Gayrard \\cite{BG_CLT}. All these results hold true for almost all realizations of the patterns and only need $M=o(N)$\n (for \\cite{gentz_annals} this is only true in the high temperature regime, which, however, is the most relevant for us in the present note.) Importantly, as in the Curie-Weiss model, at $\\beta=1$ the fluctuations are non-Gaussian. As was shown in \\cite{gentzloewe, gentzloewe2, talagrand_critical_hopf} at $\\beta=1$\n the rescaled overlap vector $N^{1/4} m_N$ converges in distribution to a random limit (while in the other limit theorems the limit was deterministic). Also large and moderate deviations for the overlap vector are available (see \\cite{BG_LDP, EL_hopf}).\n\n\\subsection{Statement of the results}\n In this subsection we state our main results on increasing propagation of chaos\n for the Hopfield model in the high-temperature and critical regimes. In particular, we identify regimes in which propagation of chaos breaks down\n at explicit scales of the marginals. For $\\beta<1$, we allow the number of patterns $M=M(N)$ to diverge with $N$,\n and establish both increasing propagation of chaos and sharp breakdown results.\n At criticality $\\beta=1$, we restrict attention to fixed $M$, and identify the\n critical window in which propagation of chaos fails.\n\nIn order to formulate our results, let us agree on the following notation: \n For $k\\le N$, let $\\mu_N^{(k)}$ denote the marginal of $\\mu_N$ on\n $(\\sigma_1,\\dots,\\sigma_k)$.\n Let $\\pi$ be the Rademacher law on $\\{-1,+1\\}$ with $\\pi(\\pm1)=\\frac12$.\n\nThen we will prove:\n\nIn order to formulate our results, let us agree on the following notation: \n For $k\\le N$, let $\\mu_N^{(k)}$ denote the marginal of $\\mu_N$ on\n $(\\sigma_1,\\dots,\\sigma_k)$.\n Let $\\pi$ be the Rademacher law on $\\{-1,+1\\}$ with $\\pi(\\pm1)=\\frac12$.\n\n\\begin{theorem} \\label{thm:stop_macro_Msmall}\n Fix $\\beta\\in(0,1)$ and assume $M=M(N)\\to\\infty$ with $M=o(\\sqrt N)$.\n Let $k=k(N)$ satisfy $k/N\\to\\rho\\in(0,1)$.\n Then, in $\\P_\\xi$-probability,\n \\[\n \\TV\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 1.\n \\]\n \\end{theorem}\n\n\\begin{theorem} \\label{thm:crit_poc}\n Fix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed. Assume $k=k(N)\\to\\infty$ satisfies\n \\[\n \\frac{k(N)}{N^{1/2}}\\longrightarrow 0.\n \\]\n Then\n \\[\n d_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\n \\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n \\]\n \\end{theorem}\n\nAnd finally, we show\n \\begin{theorem}[Critical-window breakdown at $\\beta=1$ (fixed $M$)]\n \\label{thm:crit_break_sqrtN}\n Fix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed.\n Let $k=k(N)$ satisfy\n \\[\n \\frac{k}{\\sqrt N}\\longrightarrow c\\in(0,\\infty).\n \\]\n Then there exists a deterministic constant $b=b(c,M)>0$ such that\n \\[\n \\liminf_{N\\to\\infty} \\ d_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\ \\ge\\ b\n \\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n \\]\n In particular, propagation of chaos fails in the critical window $k\\asymp \\sqrt N$,\n showing that the scaling in Theorem~\\ref{thm:crit_poc} is optimal.\n\n\\begin{lemma}\n \\label{lem:covconcentration}\n Let $\\xi_1,\\dots,\\xi_N\\in\\mathbb R^M$ be i.i.d.\\ centered subgaussian random vectors\n with covariance $\\Sigma:=\\E[\\xi_1\\xi_1^\\top]$.\n Let\n \\[\n \\widehat\\Sigma_N := \\frac1N\\sum_{i=1}^N \\xi_i\\xi_i^\\top .\n \\]\n Then there exists an absolute constant $C>0$ (depending only on the subgaussian\n norm of $\\xi_1$) such that for every $u\\ge 0$,\n \\begin{equation}\\label{eq:conc}\n \\Big\\|\\widehat\\Sigma_N-\\Sigma\\Big\\|_{\\mathrm{op}}\n \\le\n C\\Bigg(\\sqrt{\\frac{M+u}{N}}+\\frac{M+u}{N}\\Bigg)\\,\\|\\Sigma\\|_{\\mathrm{op}}\n \\qquad\\text{with probability at least }1-2e^{-u}.\n \\end{equation}\n In particular, if $\\Sigma=I_M$ (the $M$-dimensional identity matrix) and $M/N\\to 0$, then for every fixed $\\varepsilon>0$,\n \\[\n \\Pp\\Big(\\big\\|\\widehat\\Sigma_N-I_M\\big\\|_{\\mathrm{op}}\\le \\varepsilon\\Big)\\longrightarrow 1,\n \\qquad\\text{and hence}\\qquad\n \\Pp\\Big(\\|\\widehat\\Sigma_N\\|_{\\mathrm{op}}\\le 1+\\varepsilon\\Big)\\longrightarrow 1.\n \\]\n \\end{lemma}\n\n\\begin{lemma} \\label{lem:UI_LN_correct}\n Assume $k/N\\to\\rho\\in(0,1)$ and $M=o(k)$, and set\n $\\lambda_N:=\\tau^2 k/N$ with $\\tau^2=\\beta/(1-\\beta)$, so $\\lambda_N\\to\\lambda>0$.\n Let $L_N$ be given on $\\mathcal G_N$ by\n \\[\n L_N\n =\n (1+\\lambda_N)^{-M/2}\n \\exp\\!\\Big(\\alpha_N\\|S_{N,k}\\|^2\\Big),\n \\qquad \\text{with}\\quad\n \\alpha_N:=\\frac{\\lambda_N}{2(1+\\lambda_N)}.\n \\]\n Then there exists $\\eta>0$ such that\n \\[\n \\sup_N \\E_{\\pi^{\\otimes k}}\\big[L_N^{1+\\eta}\\mathbf 1_{\\mathcal G_N}\\big] <\\infty.\n \\]\n In particular, $(L_N)_N$ is uniformly integrable under $\\pi^{\\otimes k}$ on $\\mathcal G_N$.\n \\end{lemma}\n\nLet $r_\\ast^2:=\\frac{M(1+\\lambda)\\log(1+\\lambda)}{\\lambda}$be the unique radius where the two densities cross.\n Define the (optimal) set is $$A_\\ast:=\\{\\|x\\|\\ge r_\\ast\\}.$$ Then,\n \\[\n \\TV(\\mathcal N(0,I_M),\\mathcal N(0,(1+\\lambda)I_M))\n =\n \\P(Y\\in A_\\ast)-\\P(X\\in A_\\ast).\n \\]\n Since $\\chi^2_M/M\\to1$ in probability and concentrates at scale $\\sqrt M$,\n one has $\\|X\\|^2/M\\to1$ while $\\|Y\\|^2/M\\to 1+\\lambda$,\n and $r_\\ast^2/M$ lies strictly between $1$ and $1+\\lambda$.\n Therefore $\\P(X\\in A_\\ast)\\to0$ and $\\P(Y\\in A_\\ast)\\to1$, hence\n \\[\n \\TV(\\mathcal N(0,I_M),\\mathcal N(0,(1+\\lambda)I_M))\\longrightarrow 1\n \\qquad (M\\to\\infty).\n \\]\n Combining the previous steps yields $\\TV(\\mu_N^{(k)},\\pi^{\\otimes k})\\to1$ in $\\P_\\xi$-probability.\n \\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:crit_poc}]\n Let\n \\[\n L_N(\\sigma_{1:k})\n :=\\frac{d\\mu_N^{(k)}}{d\\pi^{\\otimes k}}(\\sigma_{1:k}).\n \\]\n Then, by the Cauchy--Schwarz inequality\n \\[\n d_{\\mathrm{TV}}(\\mu_N^{(k)},\\pi^{\\otimes k})\n =\\frac12\\,\\E_{\\pi^{\\otimes k}}\\big[|L_N-1|\\big]\n \\le \\frac12\\,\\Big(\\E_{\\pi^{\\otimes k}}\\big[(L_N-1)^2\\big]\\Big)^{1/2}\n =\\frac12\\,\\Big(\\E_{\\pi^{\\otimes k}}[L_N^2]-1\\Big)^{1/2}.\n \\]\n Thus it suffices to show that\n \\begin{equation}\\label{eq:chi2_to_0_beta1}\n \\E_{\\pi^{\\otimes k}}[L_N^2]\\longrightarrow 1\n \\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n \\end{equation}\n\n\\begin{theorem}\t\t\\label{thm:crit_poc}\n\t\tFix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed. Assume $k=k(N)\\to\\infty$ satisfies\n\t\t\\[\n\t\t\\frac{k(N)}{N^{1/2}}\\longrightarrow 0.\n\t\t\\]\n\t\tThen\n\t\t\\[\n\t\td_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\n\t\t\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n\t\t\\]\n\t\\end{theorem}", "post_theorem_intro_text_len": 4262, "post_theorem_intro_text": "\\begin{theorem}\t\t\\label{thm:stop_macro_Msmall}\n\t\tFix $\\beta\\in(0,1)$ and assume $M=M(N)\\to\\infty$ with $M=o(\\sqrt N)$.\n\t\tLet $k=k(N)$ satisfy $k/N\\to\\rho\\in(0,1)$.\n\t\tThen, in $\\P_\\xi$-probability,\n\t\t\\[\n\t\t\\TV\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 1.\n\t\t\\]\n\t\\end{theorem}\n\n\t\\begin{remark}\n\t\tThe restriction $M=o(\\sqrt N)$ in Theorem~\\ref{thm:stop_macro_Msmall}\n\t\tis stronger than in Theorem~\\ref{thm:highT}.\n\t\tWhether this condition can be relaxed remains an open problem.\n\t\\end{remark}\t\n\n\t\\begin{theorem}\t\t\\label{thm:crit_poc}\n\t\tFix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed. Assume $k=k(N)\\to\\infty$ satisfies\n\t\t\\[\n\t\t\\frac{k(N)}{N^{1/2}}\\longrightarrow 0.\n\t\t\\]\n\t\tThen\n\t\t\\[\n\t\td_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\n\t\t\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n\t\t\\]\n\t\\end{theorem}\n\n\tAnd finally, we show\n\t\\begin{theorem}[Critical-window breakdown at $\\beta=1$ (fixed $M$)]\n\t\t\\label{thm:crit_break_sqrtN}\n\t\tFix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed.\n\t\tLet $k=k(N)$ satisfy\n\t\t\\[\n\t\t\\frac{k}{\\sqrt N}\\longrightarrow c\\in(0,\\infty).\n\t\t\\]\n\t\tThen there exists a deterministic constant $b=b(c,M)>0$ such that\n\t\t\\[\n\t\t\\liminf_{N\\to\\infty} \\ d_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\ \\ge\\ b\n\t\t\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n\t\t\\]\n\t\tIn particular, propagation of chaos fails in the critical window $k\\asymp \\sqrt N$,\n\t\tshowing that the scaling in Theorem~\\ref{thm:crit_poc} is optimal.\n\n\t\\end{theorem}\n\t\\subsection{Outline of the proofs}\n\n\tA central idea in our proofs is that the Hopfield Gibbs measure admits,\n\tvia a Hubbard--Stratonovich transformation, an explicit representation\n\tin terms of a mixture of product measures.\n\n\t\\medskip\n\t\\noindent\\textit{Step 1: Mixture-of-products representation.}\n\tFor fixed (quenched) patterns $(\\xi^\\mu)_\\mu$, the quadratic Hamiltonian\n\t$\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2$ can be linearized by introducing an\n\tauxiliary Gaussian field. This standard technqie for quadratic Hamiltonians yields a representation of the form\n\t\\[\n\t\\mu_N(d\\sigma)\\;=\\;\\int \\nu_{N,\\beta}(d y)\\,\n\t\\bigotimes_{i=1}^N \\mu_{y,i}(d\\sigma_i),\n\t\\]\n\twhere $y\\in\\mathbb R^M$ is the Hubbard--Stratonovich field, $\\nu_{N,\\beta}$ is an\n\texplicit probability measure on $\\mathbb R^M$, and $\\mu_{y,i}$ is a Bernoulli law\n\twith bias depending on $y\\cdot \\xi_i$.\n\tConsequently, the $k$-spin marginal $\\mu_N^{(k)}$ is a mixture of product\n\tmeasures on $\\{-1,+1\\}^k$.\n\n\t\\medskip\n\t\\noindent\\textit{Step 2: Propagation of chaos reduces to stability of the mixture.}\n\tConditionally on $y$, the spins are independent.\n\tThus propagation of chaos for $\\mu_N$ is controlled by how strongly the\n\trandom field $y$ fluctuates under $\\nu_{N,\\beta}$ and by how sensitively\n\tthe single-spin biases $\\mu_{y,i}$ depend on $y$.\n\tQuantitatively, we compare the mixture to the unbiased product law\n\t$\\pi^{\\otimes k}$ in total variation distance.\n\n\t\\medskip\n\t\\noindent\\textit{Step 3: High temperature $\\beta<1$.}\n\tIn the regime $\\beta<1$, the Hubbard--Stratonovich field typically remains\n\tof order one and concentrates near the origin.\n\tA Taylor expansion of the single-spin biases and a control of\n\t$\\nu_{N,\\beta}$ show that correlations between $k$ spins are of order\n\t$kM/N$, yielding Theorem~\\ref{thm:highT} under the condition $kM/N\\to0$.\n\tFor macroscopic $k$ (with $k/N\\to\\rho$), the same representation allows us\n\tto exhibit order-one correlations induced by the random field, which leads\n\tto strong breakdown in total variation (Theorem~\\ref{thm:stop_macro_Msmall})\n\tunder the stated assumptions on $M$.\n\n\t\\medskip\n\t\\noindent\\textit{Step 4: Critical temperature $\\beta=1$ and the critical window.}\n\tAt criticality, the mixing measure $\\nu_{N,1}$ develops non-Gaussian\n\tfluctuations on the scale $N^{-1/4}$, reflecting the well-known\n\tcritical behavior of the overlap.\n\tThis amplification of the mixing fluctuations reduces the admissible growth\n\tof the marginals to $k=o(N^{1/4})$ (Theorem~\\ref{thm:crit_poc}).\n\tMoreover, when $k\\asymp \\sqrt N$, the mixture retains a nontrivial\n\tamount of randomness that produces correlations bounded away from zero,\n\tyielding a breakdown of propagation of chaos in the critical window\n\t(Theorem~\\ref{thm:crit_break_sqrtN}) and hence the optimality of the scaling.", "sketch": "A central idea in our proofs is that the Hopfield Gibbs measure admits, via a Hubbard--Stratonovich transformation, an explicit representation in terms of a mixture of product measures.\n\n\\medskip\n\\noindent\\textit{Step 1: Mixture-of-products representation.}\nFor fixed (quenched) patterns $(\\xi^\\mu)_\\mu$, the quadratic Hamiltonian $\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2$ is linearized by introducing an auxiliary Gaussian field (Hubbard--Stratonovich), yielding\n\\[\n\\mu_N(d\\sigma)=\\int \\nu_{N,\\beta}(d y)\\, \\bigotimes_{i=1}^N \\mu_{y,i}(d\\sigma_i),\n\\]\nwhere $y\\in\\R^M$, $\\nu_{N,\\beta}$ is an explicit probability measure on $\\R^M$, and $\\mu_{y,i}$ is a Bernoulli law with bias depending on $y\\cdot \\xi_i$. Consequently, the $k$-spin marginal $\\mu_N^{(k)}$ is a mixture of product measures on $\\{-1,+1\\}^k$.\n\n\\medskip\n\\noindent\\textit{Step 2: Propagation of chaos reduces to stability of the mixture.}\nConditionally on $y$, the spins are independent, so propagation of chaos is controlled by how strongly $y$ fluctuates under $\\nu_{N,\\beta}$ and by how sensitively the single-spin biases $\\mu_{y,i}$ depend on $y$. Quantitatively, one compares the mixture to the unbiased product law $\\pi^{\\otimes k}$ in total variation.\n\n\\medskip\n\\noindent\\textit{Step 3: High temperature $\\beta<1$ (Theorem~\\ref{thm:highT}).}\nIn the regime $\\beta<1$, the Hubbard--Stratonovich field “typically remains of order one and concentrates near the origin.” Using “a Taylor expansion of the single-spin biases and a control of $\\nu_{N,\\beta}$” shows that “correlations between $k$ spins are of order $kM/N$,” which yields Theorem~\\ref{thm:highT} under the condition $kM/N\\to0$.", "expanded_sketch": "A central idea in our proofs is that the Hopfield Gibbs measure admits, via a Hubbard--Stratonovich transformation, an explicit representation in terms of a mixture of product measures.\n\n\\medskip\n\\noindent\\textit{Step 1: Mixture-of-products representation.}\nFor fixed (quenched) patterns $(\\xi^\\mu)_\\mu$, the quadratic Hamiltonian $\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2$ is linearized by introducing an auxiliary Gaussian field (Hubbard--Stratonovich), yielding\n\\[\n\\mu_N(d\\sigma)=\\int \\nu_{N,\\beta}(d y)\\, \\bigotimes_{i=1}^N \\mu_{y,i}(d\\sigma_i),\n\\]\nwhere $y\\in\\R^M$, $\\nu_{N,\\beta}$ is an explicit probability measure on $\\R^M$, and $\\mu_{y,i}$ is a Bernoulli law with bias depending on $y\\cdot \\xi_i$. Consequently, the $k$-spin marginal $\\mu_N^{(k)}$ is a mixture of product measures on $\\{-1,+1\\}^k$.\n\n\\medskip\n\\noindent\\textit{Step 2: Propagation of chaos reduces to stability of the mixture.}\nConditionally on $y$, the spins are independent, so propagation of chaos is controlled by how strongly $y$ fluctuates under $\\nu_{N,\\beta}$ and by how sensitively the single-spin biases $\\mu_{y,i}$ depend on $y$. Quantitatively, one compares the mixture to the unbiased product law $\\pi^{\\otimes k}$ in total variation.\n\n\\medskip\n\\noindent\\textit{Step 3: High temperature $\\beta<1$ (main theorem).}\nIn the regime $\\beta<1$, the Hubbard--Stratonovich field “typically remains of order one and concentrates near the origin.” Using “a Taylor expansion of the single-spin biases and a control of $\\nu_{N,\\beta}$” shows that “correlations between $k$ spins are of order $kM/N$.” Under the condition $kM/N\\to0$, this yields the conclusion needed to establish the main theorem.", "expanded_theorem": "\\label{thm:highT}\n\t\tFix $\\beta<1$. Let $M=M(N)$ satisfy $M/N\\to0$, and let $k=k(N)\\to\\infty$\n\t\twith\n\t\t\\[\n\t\t\\frac{k(N)M(N)}{N}\\longrightarrow 0 \n\t\t\\]\n\t\t(which in particular allows $k=o(N)$ when $M$ is fixed).\n\n\t\tThen\n\t\t\\[\n\t\t\\TV\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\n\t\t\\longrightarrow 0\n\t\t\\qquad\\text{in }\\Pp_\\xi\\text{-probability}.\n\t\t\\]", "theorem_type": ["Asymptotic or Limit", "Implication"], "mcq": {"question": "Let \\(\\xi_i=(\\xi_i^1,\\dots,\\xi_i^M)\\in\\{-1,+1\\}^M\\), \\(i=1,\\dots,N\\), be i.i.d. random vectors whose coordinates are independent and satisfy \\(\\mathbb E[\\xi_i^\\nu]=0\\) and \\(\\mathbb E[(\\xi_i^\\nu)^2]=1\\). For \\(\\sigma\\in\\{-1,+1\\}^N\\), define the overlap vector\n\\[\nm_N(\\sigma)=\\big(m_N^1(\\sigma),\\dots,m_N^M(\\sigma)\\big),\n\\qquad m_N^\\nu(\\sigma)=\\frac1N\\sum_{i=1}^N \\sigma_i\\xi_i^\\nu.\n\\]\nFor fixed inverse temperature \\(\\beta>0\\), the quenched Hopfield Gibbs measure on \\(\\{-1,+1\\}^N\\) is\n\\[\n\\mu_N(\\sigma)=\\frac1{Z_N}\\exp\\!\\Big(\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2\\Big).\n\\]\nLet \\(\\mu_N^{(k)}\\) denote the marginal law of the first \\(k\\) spins under \\(\\mu_N\\), and let \\(\\pi\\) be the symmetric Bernoulli measure on \\(\\{-1,+1\\}\\), so that \\(\\pi^{\\otimes k}\\) is the product law of \\(k\\) i.i.d. fair \\(\\pm1\\) spins. Assume that \\(\\beta<1\\), that \\(M=M(N)\\) satisfies \\(M/N\\to0\\), and that \\(k=k(N)\\to\\infty\\) with\n\\[\n\\frac{k(N)M(N)}{N}\\longrightarrow 0.\n\\]\nHere total variation distance is \\(\\mathrm{TV}(\\nu,\\eta)=\\sup_A |\\nu(A)-\\eta(A)|\\), and \\(\\mathbb P_\\xi\\) denotes probability with respect to the random patterns \\((\\xi_i)\\). Under these assumptions, which statement about the \\(k\\)-spin marginal is valid?", "correct_choice": {"label": "A", "text": "\\[\\mathrm{TV}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\\]"}, "choices": [{"label": "B", "text": "\\[\\mathrm{TV}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\\qquad\\text{almost surely in }\\mathbb P_\\xi.\\]"}, {"label": "C", "text": "\\[\\mu_N^{(k)}\\xrightarrow[]{\\ d\\ }\\pi^{\\otimes k}\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\\]"}, {"label": "D", "text": "If \\(M/N\\to0\\) and \\(k=o(N)\\), then for every fixed \\(\\beta<1\\),\n\\[\\mathrm{TV}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\\]"}, {"label": "E", "text": "Assume \\(\\beta<1\\), \\(M/N\\to0\\), and \\(k(N)M(N)/N\\to0\\). Then\n\\[N\\,\\mathrm{TV}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "mode_of_convergence_in_patterns", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "total_variation_conclusion_replaced_by_weak_convergence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "critical_scale_kM_over_N", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "only_order_kM_over_N_control_not_rate_N_times_tv", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the conclusion, and none of the notation or setup directly gives away choice A. The learner still has to know or infer the correct scaling and mode of convergence."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the stem lists the exact hypotheses and asks which limiting statement holds, with the correct option being the theorem’s conclusion almost verbatim."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the student must distinguish between probability vs almost sure convergence, fixed-r vs growing-k marginals, and the sharp condition kM/N -> 0 vs the weaker k/N -> 0. But the item mainly tests recognition of the stated result rather than generating a new conclusion."}, "DQS": {"score": 2, "justification": "The distractors are mathematically meaningful and target realistic errors: overstrengthening the convergence mode (B), choosing a weaker true statement instead of the strongest one (C), missing the critical scale kM/N (D), or asserting the opposite asymptotic behavior (E)."}, "total_score": 5, "overall_assessment": "A mathematically well-constructed theorem-recognition MCQ with strong distractors, but it is largely tautological and only moderately tests generative reasoning."}}
{"id": "2602.12190v1", "paper_link": "http://arxiv.org/abs/2602.12190v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm:highT}\n\t\tFix $\\beta<1$. Let $M=M(N)$ satisfy $M/N\\to0$, and let $k=k(N)\\to\\infty$\n\t\twith\n\t\t\\[\n\t\t\\frac{k(N)M(N)}{N}\\longrightarrow 0 \n\t\t\\]\n\t\t(which in particular allows $k=o(N)$ when $M$ is fixed).\n\n\t\tThen\n\t\t\\[\n\t\t\\TV\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\n\t\t\\longrightarrow 0\n\t\t\\qquad\\text{in }\\Pp_\\xi\\text{-probability}.\n\t\t\\]", "start_pos": 11871, "end_pos": 12237, "label": "thm:highT"}, "ref_dict": {"thm:stop_macro_Msmall": "\\begin{theorem}\t\t\\label{thm:stop_macro_Msmall}\n\t\tFix $\\beta\\in(0,1)$ and assume $M=M(N)\\to\\infty$ with $M=o(\\sqrt N)$.\n\t\tLet $k=k(N)$ satisfy $k/N\\to\\rho\\in(0,1)$.\n\t\tThen, in $\\P_\\xi$-probability,\n\t\t\\[\n\t\t\\TV\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 1.\n\t\t\\]\n\t\\end{theorem}", "thm:crit_break_sqrtN": "\\begin{theorem}[Critical-window breakdown at $\\beta=1$ (fixed $M$)]\n\t\t\\label{thm:crit_break_sqrtN}\n\t\tFix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed.\n\t\tLet $k=k(N)$ satisfy\n\t\t\\[\n\t\t\\frac{k}{\\sqrt N}\\longrightarrow c\\in(0,\\infty).\n\t\t\\]\n\t\tThen there exists a deterministic constant $b=b(c,M)>0$ such that\n\t\t\\[\n\t\t\\liminf_{N\\to\\infty} \\ d_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\ \\ge\\ b\n\t\t\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n\t\t\\]\n\t\tIn particular, propagation of chaos fails in the critical window $k\\asymp \\sqrt N$,\n\t\tshowing that the scaling in Theorem~\\ref{thm:crit_poc} is optimal.\n\n\t\\end{theorem}", "thm:crit_poc": "\\begin{theorem}\t\t\\label{thm:crit_poc}\n\t\tFix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed. Assume $k=k(N)\\to\\infty$ satisfies\n\t\t\\[\n\t\t\\frac{k(N)}{N^{1/2}}\\longrightarrow 0.\n\t\t\\]\n\t\tThen\n\t\t\\[\n\t\td_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\n\t\t\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n\t\t\\]\n\t\\end{theorem}", "thm:highT": "\\begin{theorem}\t\t\\label{thm:highT}\n\t\tFix $\\beta<1$. Let $M=M(N)$ satisfy $M/N\\to0$, and let $k=k(N)\\to\\infty$\n\t\twith\n\t\t\\[\n\t\t\\frac{k(N)M(N)}{N}\\longrightarrow 0 \n\t\t\\]\n\t\t(which in particular allows $k=o(N)$ when $M$ is fixed).\n\n\t\tThen\n\t\t\\[\n\t\t\\TV\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\n\t\t\\longrightarrow 0\n\t\t\\qquad\\text{in }\\Pp_\\xi\\text{-probability}.\n\t\t\\]\n\t\\end{theorem}"}, "pre_theorem_intro_text_len": 9395, "pre_theorem_intro_text": "Propagation of chaos is a central concept in the study of interacting\n\tparticle systems and mean-field models.\n\tRoughly speaking, it describes the phenomenon that,\n\tas the system size $N$ tends to infinity,\n\tfinite collections of particles behave asymptotically independently,\n\twith a common limiting law.\n\tPropagation of chaos originated in Kac's Markovian models for gas dynamics\n\t\\cite{Kac_foundations,Kac_probability}, an attempt to justify\n\tBoltzmann's ''Sto\\ss zahlenansatz''.\n\tPropagation of chaos has since become an important \n\tobject of study\n\tin probability, statistical mechanics, and mathematical physics.\n\tThe original approach by Kac was that,\n\tif at time 0 the finite marginal distributions of a system are product measures in the\n\tthermodynamic limit, then this should carry over to the time-evolved system. In equilibrium settings for mean-field\n\tGibbs measures (where the energy function is a \n\tfunction of the empirical measure) with a unique minimizer of the Helmholtz free energy, however, this was shown to\n\tfollow from the fact the extremal Gibbs measures locally look like product measures, i.e.\\\n\tthat any finite subset of spins forms a family of independent random variables in the thermodynamic limit \\cite[Theorem 3]{BAZ_chaos}.\n\tSuch results provide a rigorous justification of mean-field\n\tapproximations and explain why macroscopic behavior\n\tcan often be described by effective one-particle models. In the present note, we remain entirely within a static framework, and study propagation of chaos in the sense of asymptotic factorization of finite-dimensional marginals for families of (random) \n\tmean-field Gibbs measures.\n\n\tMoreover, for mean-field spin systems, propagation of chaos\n\tis closely tied to the high-temperature regime.\n\tIn classical models such as the Curie--Weiss Ising model,\n\tchaos holds when the inverse temperature is below the critical value,\n\twhile it breaks down in the low-temperature phase. There one has to replace\n\tthe product measure by a mixture of the (several) extremal limiting Gibbs measures.\n\tAn additional difficulty may arise when the Gibbs measures in question are random. To the best of our knowledge comparatively little is known in such situations (with the exception of of \\cite{BG98book} and \\cite{KL24}). The present paper contributes to the study of propagation of chaos for random mean-field Gibbs measures in the high-temperature and near-critical regimes.\n\n\tImportantly, in \\cite{BAZ_chaos}, the authors also introduced the concept of \\textit{increasing} propagation of chaos and showed that in the true high temperature regime of many mean-field models one can let the size of the marginals $k$ grow with the system size $N$ as long as $k=o(N)$. \n\tUnderstanding not only whether propagation of chaos holds,\n\tbut also the precise scales at which it breaks down,\n\thas become an important theme in recent work on mean-field Gibbs measures \\cite{BAZ_chaos,Lacker22, JKLM23,KL24,RS25,JKL25}.\n\n\tWhile increasing propagation of chaos has been established in several genuinely high-temperature regimes of ordered mean-field models, much less is known in the presence of random mean-field Gibbs measures, where already the analysis of critical fluctuations can become delicate (\\cite{gentzloewe, gentzloewe2, talagrand_critical_hopf}).   \n\tIn the present setting of random mean-field Gibbs measures, we show that this high-temperature scaling persists for $\\beta <1$ (the critical value),\n\twhereas at criticality \n\t$\\beta=1$, the admissible growth drops to $k=o(\\sqrt N)$. Moreover, \n\tthese bounds on \n\t$k$ are optimal for the class of random mean-field Gibbs measures considered here.\n\n\tIn this article we study propagation of chaos\n\tin the Hopfield model, a paradigmatic example\n\tof a disordered mean-field spin system.\n\tThe Hopfield model has (at least) two distinct origins. It was first introduced by\n\tPastur and Figotin \\cite{PS84} as a solvable model of a disordered system. At about the same time it\n\twas independently invented by Hopfield as\n\ta model of associative memory \\cite{Hopfield1982}.\n\tBoth aspects have been intensively studied. For the disordered systems facet see e.g. \\cite{BGP94, BG97} or \\cite{talagrand}, while \n\tthe neural network aspect has found a renewed interest through two recent papers \\cite{KrotovHopfield2016,DHLUV17}.\n\n\tFor the purposes of the present paper, the \n\tprobabilistic viewpoint of seeing the Hopfield model\n\tas a mean-field Ising model with random, structured interactions is more appropriate.\n\tThe disorder, i.e.\\ the random interactions, is generated by a collection of random patterns,\n\twhich induces a random quadratic Hamiltonian\n\tand leads to a rich interplay between thermal fluctuations\n\tand quenched randomness.\n\tWhile the thermodynamic properties of the Hopfield model\n\tare well understood, much less is known about\n\tthe fine structure of its finite-dimensional marginals\n\tand their asymptotic independence properties. There is only one result concerning the propagation of chaos in the Hopfield model, see \\cite[Theorem 8.15]{BG98}. However, this result addresses fixed-dimensional marginals in the low-temperature regime, and relies on mechanisms different from those considered here.\n\n\tOur main goal is to analyze propagation of chaos\n\tfor the Hopfield Gibbs measure in the high-temperature\n\tand critical regimes, with particular emphasis\n\ton the size of the marginals.\n\tWe identify precise conditions under which propagation of chaos holds,\n\tas well as sharp thresholds for its breakdown.\n\tIn the high-temperature regime, we prove propagation of chaos\n\tfor growing marginals, provided their size grows sublinearly in the system size $N$.\n\tAt criticality, we show that chaos breaks down in a critical window,\n\twhose scale matches that of the dominant collective fluctuations.\n\treveal a clear transition between asymptotic independence and regimes of partial or complete breakdown of chaos,\n\tand highlight the role played by disorder-induced fluctuations\n\tin determining these regimes.\n\n\t\\subsection{The Model}\n\tLet us next describe the central model for the purpose of this note.\n\tLet $\\xi_i=(\\xi_i^1,\\dots,\\xi_i^M)\\in\\{-1,+1\\}^M$, $i=1,\\dots,N$, be i.i.d.\\ random vectors with independent coordinates,\n\t\\[\n\t\\mathbb E[\\xi_i^\\nu]=0, \\qquad \\mathbb E[(\\xi_i^\\nu)^2]=1 .\n\t\\]\n\t(In the interpretation of an associative memory, the vectors $(\\xi^\\mu)_{\\mu=1}^M=((\\xi_i^\\mu)_{i=1}^N)_{\\mu=1}^M$ are called images or patterns).\n\tIn what follows we will always assume  that $M=M(N)$ may depend on $N$, but in such a way, that $M=o(N)$, which is the natural regime in which the Hopfield model exhibits mean-field behavior. \n\tFor fixed, i.e.\\ quenched, patterns $(\\xi^\\mu)_\\mu$\tdefine and for $\\sigma\\in\\{-1,+1\\}^N$, define the overlap vector\n\t\\[\n\tm_N(\\sigma) = \\big(m_N^1(\\sigma),\\dots,m_N^M(\\sigma)\\big),\n\t\\qquad\n\tm_N^\\nu(\\sigma)=\\frac1N\\sum_{i=1}^N \\sigma_i\\xi_i^\\nu .\n\t\\]\n\tThe Hopfield Gibbs measure at inverse temperature $\\beta>0$ is\n\t\\[\n\t\\mu_N(\\sigma)=\\frac{1}{Z_N}\n\t\\exp\\Big(\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2\\Big).\n\t\\]\n\tThe Gibbs measure is fully determined by $m_N(\\sigma)$ (plus the inverse temperature), it thus is natural to consider \n\tthe behaviour of the overlap under the Gibbs measure.\n\tThis has been done in numerous papers: \n\tIn \\cite{BGP94} it was shown that -- similar to the Curie-Weiss model -- in the Hopfield model the critical (inverse) temperature is $\\beta=1$. While for $\\beta \\le 1$ the overlap vector  gets concentrated in the $M$-dimensional 0-vector, for larger $\\beta$ the limit points of $m_N$ are associated with the 2$M$ vectors $\\pm z(\\beta) e_\\mu$ where \n\t$e_\\mu$ is the $\\mu$'th unit vector and $z(\\beta)$ is the largest solution of the equation\n\t$$\n\tz=\\tanh(\\beta z).\n\t$$\n\tCentral Limit theorems for $\\sqrt N m_N(\\cdot)$ were proven by Gentz \\cite{gentz_annals} or Bovier and Gayrard \\cite{BG_CLT}. All these results hold true for almost all realizations of the patterns and only need $M=o(N)$\n\t(for \\cite{gentz_annals} this is only true in the high temperature regime, which, however, is the most relevant for us in the present note.) Importantly, as in the Curie-Weiss model, at $\\beta=1$ the fluctuations are non-Gaussian. As was shown in \\cite{gentzloewe, gentzloewe2, talagrand_critical_hopf} at $\\beta=1$\n\tthe rescaled overlap vector $N^{1/4} m_N$ converges in distribution to a random limit (while in the other limit theorems the limit was deterministic). Also large and moderate deviations for the overlap vector are available (see \\cite{BG_LDP, EL_hopf}).\n\n\t\\subsection{Statement of the results}\n\tIn this subsection we state our main results on increasing propagation of chaos\n\tfor the Hopfield model in the high-temperature and critical regimes. In particular, we identify regimes in which propagation of chaos breaks down\n\tat explicit scales of the marginals. For $\\beta<1$, we allow the number of patterns $M=M(N)$ to diverge with $N$,\n\tand establish both increasing propagation of chaos and sharp breakdown results.\n\tAt criticality $\\beta=1$, we restrict attention to fixed $M$, and identify the\n\tcritical window in which propagation of chaos fails.\n\n\tIn order to formulate our results, let us agree on the following notation:  \n\tFor $k\\le N$, let $\\mu_N^{(k)}$ denote the marginal of $\\mu_N$ on\n\t$(\\sigma_1,\\dots,\\sigma_k)$.\n\tLet $\\pi$ be the Rademacher law on $\\{-1,+1\\}$ with $\\pi(\\pm1)=\\frac12$.\n\n\tThen we will prove:", "context": "Propagation of chaos is a central concept in the study of interacting\n particle systems and mean-field models.\n Roughly speaking, it describes the phenomenon that,\n as the system size $N$ tends to infinity,\n finite collections of particles behave asymptotically independently,\n with a common limiting law.\n Propagation of chaos originated in Kac's Markovian models for gas dynamics\n \\cite{Kac_foundations,Kac_probability}, an attempt to justify\n Boltzmann's ''Sto\\ss zahlenansatz''.\n Propagation of chaos has since become an important \n object of study\n in probability, statistical mechanics, and mathematical physics.\n The original approach by Kac was that,\n if at time 0 the finite marginal distributions of a system are product measures in the\n thermodynamic limit, then this should carry over to the time-evolved system. In equilibrium settings for mean-field\n Gibbs measures (where the energy function is a \n function of the empirical measure) with a unique minimizer of the Helmholtz free energy, however, this was shown to\n follow from the fact the extremal Gibbs measures locally look like product measures, i.e.\\\n that any finite subset of spins forms a family of independent random variables in the thermodynamic limit \\cite[Theorem 3]{BAZ_chaos}.\n Such results provide a rigorous justification of mean-field\n approximations and explain why macroscopic behavior\n can often be described by effective one-particle models. In the present note, we remain entirely within a static framework, and study propagation of chaos in the sense of asymptotic factorization of finite-dimensional marginals for families of (random) \n mean-field Gibbs measures.\n\nImportantly, in \\cite{BAZ_chaos}, the authors also introduced the concept of \\textit{increasing} propagation of chaos and showed that in the true high temperature regime of many mean-field models one can let the size of the marginals $k$ grow with the system size $N$ as long as $k=o(N)$. \n Understanding not only whether propagation of chaos holds,\n but also the precise scales at which it breaks down,\n has become an important theme in recent work on mean-field Gibbs measures \\cite{BAZ_chaos,Lacker22, JKLM23,KL24,RS25,JKL25}.\n\n\\subsection{The Model}\n Let us next describe the central model for the purpose of this note.\n Let $\\xi_i=(\\xi_i^1,\\dots,\\xi_i^M)\\in\\{-1,+1\\}^M$, $i=1,\\dots,N$, be i.i.d.\\ random vectors with independent coordinates,\n \\[\n \\mathbb E[\\xi_i^\\nu]=0, \\qquad \\mathbb E[(\\xi_i^\\nu)^2]=1 .\n \\]\n (In the interpretation of an associative memory, the vectors $(\\xi^\\mu)_{\\mu=1}^M=((\\xi_i^\\mu)_{i=1}^N)_{\\mu=1}^M$ are called images or patterns).\n In what follows we will always assume that $M=M(N)$ may depend on $N$, but in such a way, that $M=o(N)$, which is the natural regime in which the Hopfield model exhibits mean-field behavior. \n For fixed, i.e.\\ quenched, patterns $(\\xi^\\mu)_\\mu$ define and for $\\sigma\\in\\{-1,+1\\}^N$, define the overlap vector\n \\[\n m_N(\\sigma) = \\big(m_N^1(\\sigma),\\dots,m_N^M(\\sigma)\\big),\n \\qquad\n m_N^\\nu(\\sigma)=\\frac1N\\sum_{i=1}^N \\sigma_i\\xi_i^\\nu .\n \\]\n The Hopfield Gibbs measure at inverse temperature $\\beta>0$ is\n \\[\n \\mu_N(\\sigma)=\\frac{1}{Z_N}\n \\exp\\Big(\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2\\Big).\n \\]\n The Gibbs measure is fully determined by $m_N(\\sigma)$ (plus the inverse temperature), it thus is natural to consider \n the behaviour of the overlap under the Gibbs measure.\n This has been done in numerous papers: \n In \\cite{BGP94} it was shown that -- similar to the Curie-Weiss model -- in the Hopfield model the critical (inverse) temperature is $\\beta=1$. While for $\\beta \\le 1$ the overlap vector gets concentrated in the $M$-dimensional 0-vector, for larger $\\beta$ the limit points of $m_N$ are associated with the 2$M$ vectors $\\pm z(\\beta) e_\\mu$ where \n $e_\\mu$ is the $\\mu$'th unit vector and $z(\\beta)$ is the largest solution of the equation\n $$\n z=\\tanh(\\beta z).\n $$\n Central Limit theorems for $\\sqrt N m_N(\\cdot)$ were proven by Gentz \\cite{gentz_annals} or Bovier and Gayrard \\cite{BG_CLT}. All these results hold true for almost all realizations of the patterns and only need $M=o(N)$\n (for \\cite{gentz_annals} this is only true in the high temperature regime, which, however, is the most relevant for us in the present note.) Importantly, as in the Curie-Weiss model, at $\\beta=1$ the fluctuations are non-Gaussian. As was shown in \\cite{gentzloewe, gentzloewe2, talagrand_critical_hopf} at $\\beta=1$\n the rescaled overlap vector $N^{1/4} m_N$ converges in distribution to a random limit (while in the other limit theorems the limit was deterministic). Also large and moderate deviations for the overlap vector are available (see \\cite{BG_LDP, EL_hopf}).\n\n\\subsection{Statement of the results}\n In this subsection we state our main results on increasing propagation of chaos\n for the Hopfield model in the high-temperature and critical regimes. In particular, we identify regimes in which propagation of chaos breaks down\n at explicit scales of the marginals. For $\\beta<1$, we allow the number of patterns $M=M(N)$ to diverge with $N$,\n and establish both increasing propagation of chaos and sharp breakdown results.\n At criticality $\\beta=1$, we restrict attention to fixed $M$, and identify the\n critical window in which propagation of chaos fails.\n\nIn order to formulate our results, let us agree on the following notation: \n For $k\\le N$, let $\\mu_N^{(k)}$ denote the marginal of $\\mu_N$ on\n $(\\sigma_1,\\dots,\\sigma_k)$.\n Let $\\pi$ be the Rademacher law on $\\{-1,+1\\}$ with $\\pi(\\pm1)=\\frac12$.\n\nThen we will prove:", "full_context": "Propagation of chaos is a central concept in the study of interacting\n particle systems and mean-field models.\n Roughly speaking, it describes the phenomenon that,\n as the system size $N$ tends to infinity,\n finite collections of particles behave asymptotically independently,\n with a common limiting law.\n Propagation of chaos originated in Kac's Markovian models for gas dynamics\n \\cite{Kac_foundations,Kac_probability}, an attempt to justify\n Boltzmann's ''Sto\\ss zahlenansatz''.\n Propagation of chaos has since become an important \n object of study\n in probability, statistical mechanics, and mathematical physics.\n The original approach by Kac was that,\n if at time 0 the finite marginal distributions of a system are product measures in the\n thermodynamic limit, then this should carry over to the time-evolved system. In equilibrium settings for mean-field\n Gibbs measures (where the energy function is a \n function of the empirical measure) with a unique minimizer of the Helmholtz free energy, however, this was shown to\n follow from the fact the extremal Gibbs measures locally look like product measures, i.e.\\\n that any finite subset of spins forms a family of independent random variables in the thermodynamic limit \\cite[Theorem 3]{BAZ_chaos}.\n Such results provide a rigorous justification of mean-field\n approximations and explain why macroscopic behavior\n can often be described by effective one-particle models. In the present note, we remain entirely within a static framework, and study propagation of chaos in the sense of asymptotic factorization of finite-dimensional marginals for families of (random) \n mean-field Gibbs measures.\n\nImportantly, in \\cite{BAZ_chaos}, the authors also introduced the concept of \\textit{increasing} propagation of chaos and showed that in the true high temperature regime of many mean-field models one can let the size of the marginals $k$ grow with the system size $N$ as long as $k=o(N)$. \n Understanding not only whether propagation of chaos holds,\n but also the precise scales at which it breaks down,\n has become an important theme in recent work on mean-field Gibbs measures \\cite{BAZ_chaos,Lacker22, JKLM23,KL24,RS25,JKL25}.\n\n\\subsection{The Model}\n Let us next describe the central model for the purpose of this note.\n Let $\\xi_i=(\\xi_i^1,\\dots,\\xi_i^M)\\in\\{-1,+1\\}^M$, $i=1,\\dots,N$, be i.i.d.\\ random vectors with independent coordinates,\n \\[\n \\mathbb E[\\xi_i^\\nu]=0, \\qquad \\mathbb E[(\\xi_i^\\nu)^2]=1 .\n \\]\n (In the interpretation of an associative memory, the vectors $(\\xi^\\mu)_{\\mu=1}^M=((\\xi_i^\\mu)_{i=1}^N)_{\\mu=1}^M$ are called images or patterns).\n In what follows we will always assume that $M=M(N)$ may depend on $N$, but in such a way, that $M=o(N)$, which is the natural regime in which the Hopfield model exhibits mean-field behavior. \n For fixed, i.e.\\ quenched, patterns $(\\xi^\\mu)_\\mu$ define and for $\\sigma\\in\\{-1,+1\\}^N$, define the overlap vector\n \\[\n m_N(\\sigma) = \\big(m_N^1(\\sigma),\\dots,m_N^M(\\sigma)\\big),\n \\qquad\n m_N^\\nu(\\sigma)=\\frac1N\\sum_{i=1}^N \\sigma_i\\xi_i^\\nu .\n \\]\n The Hopfield Gibbs measure at inverse temperature $\\beta>0$ is\n \\[\n \\mu_N(\\sigma)=\\frac{1}{Z_N}\n \\exp\\Big(\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2\\Big).\n \\]\n The Gibbs measure is fully determined by $m_N(\\sigma)$ (plus the inverse temperature), it thus is natural to consider \n the behaviour of the overlap under the Gibbs measure.\n This has been done in numerous papers: \n In \\cite{BGP94} it was shown that -- similar to the Curie-Weiss model -- in the Hopfield model the critical (inverse) temperature is $\\beta=1$. While for $\\beta \\le 1$ the overlap vector gets concentrated in the $M$-dimensional 0-vector, for larger $\\beta$ the limit points of $m_N$ are associated with the 2$M$ vectors $\\pm z(\\beta) e_\\mu$ where \n $e_\\mu$ is the $\\mu$'th unit vector and $z(\\beta)$ is the largest solution of the equation\n $$\n z=\\tanh(\\beta z).\n $$\n Central Limit theorems for $\\sqrt N m_N(\\cdot)$ were proven by Gentz \\cite{gentz_annals} or Bovier and Gayrard \\cite{BG_CLT}. All these results hold true for almost all realizations of the patterns and only need $M=o(N)$\n (for \\cite{gentz_annals} this is only true in the high temperature regime, which, however, is the most relevant for us in the present note.) Importantly, as in the Curie-Weiss model, at $\\beta=1$ the fluctuations are non-Gaussian. As was shown in \\cite{gentzloewe, gentzloewe2, talagrand_critical_hopf} at $\\beta=1$\n the rescaled overlap vector $N^{1/4} m_N$ converges in distribution to a random limit (while in the other limit theorems the limit was deterministic). Also large and moderate deviations for the overlap vector are available (see \\cite{BG_LDP, EL_hopf}).\n\n\\subsection{Statement of the results}\n In this subsection we state our main results on increasing propagation of chaos\n for the Hopfield model in the high-temperature and critical regimes. In particular, we identify regimes in which propagation of chaos breaks down\n at explicit scales of the marginals. For $\\beta<1$, we allow the number of patterns $M=M(N)$ to diverge with $N$,\n and establish both increasing propagation of chaos and sharp breakdown results.\n At criticality $\\beta=1$, we restrict attention to fixed $M$, and identify the\n critical window in which propagation of chaos fails.\n\nIn order to formulate our results, let us agree on the following notation: \n For $k\\le N$, let $\\mu_N^{(k)}$ denote the marginal of $\\mu_N$ on\n $(\\sigma_1,\\dots,\\sigma_k)$.\n Let $\\pi$ be the Rademacher law on $\\{-1,+1\\}$ with $\\pi(\\pm1)=\\frac12$.\n\nThen we will prove:\n\nIn order to formulate our results, let us agree on the following notation: \n For $k\\le N$, let $\\mu_N^{(k)}$ denote the marginal of $\\mu_N$ on\n $(\\sigma_1,\\dots,\\sigma_k)$.\n Let $\\pi$ be the Rademacher law on $\\{-1,+1\\}$ with $\\pi(\\pm1)=\\frac12$.\n\n\\begin{theorem} \\label{thm:stop_macro_Msmall}\n Fix $\\beta\\in(0,1)$ and assume $M=M(N)\\to\\infty$ with $M=o(\\sqrt N)$.\n Let $k=k(N)$ satisfy $k/N\\to\\rho\\in(0,1)$.\n Then, in $\\P_\\xi$-probability,\n \\[\n \\TV\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 1.\n \\]\n \\end{theorem}\n\n\\begin{theorem} \\label{thm:crit_poc}\n Fix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed. Assume $k=k(N)\\to\\infty$ satisfies\n \\[\n \\frac{k(N)}{N^{1/2}}\\longrightarrow 0.\n \\]\n Then\n \\[\n d_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\n \\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n \\]\n \\end{theorem}\n\nAnd finally, we show\n \\begin{theorem}[Critical-window breakdown at $\\beta=1$ (fixed $M$)]\n \\label{thm:crit_break_sqrtN}\n Fix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed.\n Let $k=k(N)$ satisfy\n \\[\n \\frac{k}{\\sqrt N}\\longrightarrow c\\in(0,\\infty).\n \\]\n Then there exists a deterministic constant $b=b(c,M)>0$ such that\n \\[\n \\liminf_{N\\to\\infty} \\ d_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\ \\ge\\ b\n \\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n \\]\n In particular, propagation of chaos fails in the critical window $k\\asymp \\sqrt N$,\n showing that the scaling in Theorem~\\ref{thm:crit_poc} is optimal.\n\n\\begin{lemma}\n \\label{lem:covconcentration}\n Let $\\xi_1,\\dots,\\xi_N\\in\\mathbb R^M$ be i.i.d.\\ centered subgaussian random vectors\n with covariance $\\Sigma:=\\E[\\xi_1\\xi_1^\\top]$.\n Let\n \\[\n \\widehat\\Sigma_N := \\frac1N\\sum_{i=1}^N \\xi_i\\xi_i^\\top .\n \\]\n Then there exists an absolute constant $C>0$ (depending only on the subgaussian\n norm of $\\xi_1$) such that for every $u\\ge 0$,\n \\begin{equation}\\label{eq:conc}\n \\Big\\|\\widehat\\Sigma_N-\\Sigma\\Big\\|_{\\mathrm{op}}\n \\le\n C\\Bigg(\\sqrt{\\frac{M+u}{N}}+\\frac{M+u}{N}\\Bigg)\\,\\|\\Sigma\\|_{\\mathrm{op}}\n \\qquad\\text{with probability at least }1-2e^{-u}.\n \\end{equation}\n In particular, if $\\Sigma=I_M$ (the $M$-dimensional identity matrix) and $M/N\\to 0$, then for every fixed $\\varepsilon>0$,\n \\[\n \\Pp\\Big(\\big\\|\\widehat\\Sigma_N-I_M\\big\\|_{\\mathrm{op}}\\le \\varepsilon\\Big)\\longrightarrow 1,\n \\qquad\\text{and hence}\\qquad\n \\Pp\\Big(\\|\\widehat\\Sigma_N\\|_{\\mathrm{op}}\\le 1+\\varepsilon\\Big)\\longrightarrow 1.\n \\]\n \\end{lemma}\n\n\\begin{lemma} \\label{lem:UI_LN_correct}\n Assume $k/N\\to\\rho\\in(0,1)$ and $M=o(k)$, and set\n $\\lambda_N:=\\tau^2 k/N$ with $\\tau^2=\\beta/(1-\\beta)$, so $\\lambda_N\\to\\lambda>0$.\n Let $L_N$ be given on $\\mathcal G_N$ by\n \\[\n L_N\n =\n (1+\\lambda_N)^{-M/2}\n \\exp\\!\\Big(\\alpha_N\\|S_{N,k}\\|^2\\Big),\n \\qquad \\text{with}\\quad\n \\alpha_N:=\\frac{\\lambda_N}{2(1+\\lambda_N)}.\n \\]\n Then there exists $\\eta>0$ such that\n \\[\n \\sup_N \\E_{\\pi^{\\otimes k}}\\big[L_N^{1+\\eta}\\mathbf 1_{\\mathcal G_N}\\big] <\\infty.\n \\]\n In particular, $(L_N)_N$ is uniformly integrable under $\\pi^{\\otimes k}$ on $\\mathcal G_N$.\n \\end{lemma}\n\nLet $r_\\ast^2:=\\frac{M(1+\\lambda)\\log(1+\\lambda)}{\\lambda}$be the unique radius where the two densities cross.\n Define the (optimal) set is $$A_\\ast:=\\{\\|x\\|\\ge r_\\ast\\}.$$ Then,\n \\[\n \\TV(\\mathcal N(0,I_M),\\mathcal N(0,(1+\\lambda)I_M))\n =\n \\P(Y\\in A_\\ast)-\\P(X\\in A_\\ast).\n \\]\n Since $\\chi^2_M/M\\to1$ in probability and concentrates at scale $\\sqrt M$,\n one has $\\|X\\|^2/M\\to1$ while $\\|Y\\|^2/M\\to 1+\\lambda$,\n and $r_\\ast^2/M$ lies strictly between $1$ and $1+\\lambda$.\n Therefore $\\P(X\\in A_\\ast)\\to0$ and $\\P(Y\\in A_\\ast)\\to1$, hence\n \\[\n \\TV(\\mathcal N(0,I_M),\\mathcal N(0,(1+\\lambda)I_M))\\longrightarrow 1\n \\qquad (M\\to\\infty).\n \\]\n Combining the previous steps yields $\\TV(\\mu_N^{(k)},\\pi^{\\otimes k})\\to1$ in $\\P_\\xi$-probability.\n \\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:crit_poc}]\n Let\n \\[\n L_N(\\sigma_{1:k})\n :=\\frac{d\\mu_N^{(k)}}{d\\pi^{\\otimes k}}(\\sigma_{1:k}).\n \\]\n Then, by the Cauchy--Schwarz inequality\n \\[\n d_{\\mathrm{TV}}(\\mu_N^{(k)},\\pi^{\\otimes k})\n =\\frac12\\,\\E_{\\pi^{\\otimes k}}\\big[|L_N-1|\\big]\n \\le \\frac12\\,\\Big(\\E_{\\pi^{\\otimes k}}\\big[(L_N-1)^2\\big]\\Big)^{1/2}\n =\\frac12\\,\\Big(\\E_{\\pi^{\\otimes k}}[L_N^2]-1\\Big)^{1/2}.\n \\]\n Thus it suffices to show that\n \\begin{equation}\\label{eq:chi2_to_0_beta1}\n \\E_{\\pi^{\\otimes k}}[L_N^2]\\longrightarrow 1\n \\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n \\end{equation}\n\n\\begin{theorem}\t\t\\label{thm:crit_poc}\n\t\tFix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed. Assume $k=k(N)\\to\\infty$ satisfies\n\t\t\\[\n\t\t\\frac{k(N)}{N^{1/2}}\\longrightarrow 0.\n\t\t\\]\n\t\tThen\n\t\t\\[\n\t\td_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\n\t\t\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n\t\t\\]\n\t\\end{theorem}", "post_theorem_intro_text_len": 4262, "post_theorem_intro_text": "\\begin{theorem}\t\t\\label{thm:stop_macro_Msmall}\n\t\tFix $\\beta\\in(0,1)$ and assume $M=M(N)\\to\\infty$ with $M=o(\\sqrt N)$.\n\t\tLet $k=k(N)$ satisfy $k/N\\to\\rho\\in(0,1)$.\n\t\tThen, in $\\P_\\xi$-probability,\n\t\t\\[\n\t\t\\TV\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 1.\n\t\t\\]\n\t\\end{theorem}\n\n\t\\begin{remark}\n\t\tThe restriction $M=o(\\sqrt N)$ in Theorem~\\ref{thm:stop_macro_Msmall}\n\t\tis stronger than in Theorem~\\ref{thm:highT}.\n\t\tWhether this condition can be relaxed remains an open problem.\n\t\\end{remark}\t\n\n\t\\begin{theorem}\t\t\\label{thm:crit_poc}\n\t\tFix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed. Assume $k=k(N)\\to\\infty$ satisfies\n\t\t\\[\n\t\t\\frac{k(N)}{N^{1/2}}\\longrightarrow 0.\n\t\t\\]\n\t\tThen\n\t\t\\[\n\t\td_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\n\t\t\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n\t\t\\]\n\t\\end{theorem}\n\n\tAnd finally, we show\n\t\\begin{theorem}[Critical-window breakdown at $\\beta=1$ (fixed $M$)]\n\t\t\\label{thm:crit_break_sqrtN}\n\t\tFix $\\beta=1$ and let $M\\in\\mathbb N$ be fixed.\n\t\tLet $k=k(N)$ satisfy\n\t\t\\[\n\t\t\\frac{k}{\\sqrt N}\\longrightarrow c\\in(0,\\infty).\n\t\t\\]\n\t\tThen there exists a deterministic constant $b=b(c,M)>0$ such that\n\t\t\\[\n\t\t\\liminf_{N\\to\\infty} \\ d_{\\mathrm{TV}}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\ \\ge\\ b\n\t\t\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\n\t\t\\]\n\t\tIn particular, propagation of chaos fails in the critical window $k\\asymp \\sqrt N$,\n\t\tshowing that the scaling in Theorem~\\ref{thm:crit_poc} is optimal.\n\n\t\\end{theorem}\n\t\\subsection{Outline of the proofs}\n\n\tA central idea in our proofs is that the Hopfield Gibbs measure admits,\n\tvia a Hubbard--Stratonovich transformation, an explicit representation\n\tin terms of a mixture of product measures.\n\n\t\\medskip\n\t\\noindent\\textit{Step 1: Mixture-of-products representation.}\n\tFor fixed (quenched) patterns $(\\xi^\\mu)_\\mu$, the quadratic Hamiltonian\n\t$\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2$ can be linearized by introducing an\n\tauxiliary Gaussian field. This standard technqie for quadratic Hamiltonians yields a representation of the form\n\t\\[\n\t\\mu_N(d\\sigma)\\;=\\;\\int \\nu_{N,\\beta}(d y)\\,\n\t\\bigotimes_{i=1}^N \\mu_{y,i}(d\\sigma_i),\n\t\\]\n\twhere $y\\in\\mathbb R^M$ is the Hubbard--Stratonovich field, $\\nu_{N,\\beta}$ is an\n\texplicit probability measure on $\\mathbb R^M$, and $\\mu_{y,i}$ is a Bernoulli law\n\twith bias depending on $y\\cdot \\xi_i$.\n\tConsequently, the $k$-spin marginal $\\mu_N^{(k)}$ is a mixture of product\n\tmeasures on $\\{-1,+1\\}^k$.\n\n\t\\medskip\n\t\\noindent\\textit{Step 2: Propagation of chaos reduces to stability of the mixture.}\n\tConditionally on $y$, the spins are independent.\n\tThus propagation of chaos for $\\mu_N$ is controlled by how strongly the\n\trandom field $y$ fluctuates under $\\nu_{N,\\beta}$ and by how sensitively\n\tthe single-spin biases $\\mu_{y,i}$ depend on $y$.\n\tQuantitatively, we compare the mixture to the unbiased product law\n\t$\\pi^{\\otimes k}$ in total variation distance.\n\n\t\\medskip\n\t\\noindent\\textit{Step 3: High temperature $\\beta<1$.}\n\tIn the regime $\\beta<1$, the Hubbard--Stratonovich field typically remains\n\tof order one and concentrates near the origin.\n\tA Taylor expansion of the single-spin biases and a control of\n\t$\\nu_{N,\\beta}$ show that correlations between $k$ spins are of order\n\t$kM/N$, yielding Theorem~\\ref{thm:highT} under the condition $kM/N\\to0$.\n\tFor macroscopic $k$ (with $k/N\\to\\rho$), the same representation allows us\n\tto exhibit order-one correlations induced by the random field, which leads\n\tto strong breakdown in total variation (Theorem~\\ref{thm:stop_macro_Msmall})\n\tunder the stated assumptions on $M$.\n\n\t\\medskip\n\t\\noindent\\textit{Step 4: Critical temperature $\\beta=1$ and the critical window.}\n\tAt criticality, the mixing measure $\\nu_{N,1}$ develops non-Gaussian\n\tfluctuations on the scale $N^{-1/4}$, reflecting the well-known\n\tcritical behavior of the overlap.\n\tThis amplification of the mixing fluctuations reduces the admissible growth\n\tof the marginals to $k=o(N^{1/4})$ (Theorem~\\ref{thm:crit_poc}).\n\tMoreover, when $k\\asymp \\sqrt N$, the mixture retains a nontrivial\n\tamount of randomness that produces correlations bounded away from zero,\n\tyielding a breakdown of propagation of chaos in the critical window\n\t(Theorem~\\ref{thm:crit_break_sqrtN}) and hence the optimality of the scaling.", "sketch": "A central idea in our proofs is that the Hopfield Gibbs measure admits, via a Hubbard--Stratonovich transformation, an explicit representation in terms of a mixture of product measures.\n\n\\medskip\n\\noindent\\textit{Step 1: Mixture-of-products representation.}\nFor fixed (quenched) patterns $(\\xi^\\mu)_\\mu$, the quadratic Hamiltonian $\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2$ is linearized by introducing an auxiliary Gaussian field (Hubbard--Stratonovich), yielding\n\\[\n\\mu_N(d\\sigma)=\\int \\nu_{N,\\beta}(d y)\\, \\bigotimes_{i=1}^N \\mu_{y,i}(d\\sigma_i),\n\\]\nwhere $y\\in\\R^M$, $\\nu_{N,\\beta}$ is an explicit probability measure on $\\R^M$, and $\\mu_{y,i}$ is a Bernoulli law with bias depending on $y\\cdot \\xi_i$. Consequently, the $k$-spin marginal $\\mu_N^{(k)}$ is a mixture of product measures on $\\{-1,+1\\}^k$.\n\n\\medskip\n\\noindent\\textit{Step 2: Propagation of chaos reduces to stability of the mixture.}\nConditionally on $y$, the spins are independent, so propagation of chaos is controlled by how strongly $y$ fluctuates under $\\nu_{N,\\beta}$ and by how sensitively the single-spin biases $\\mu_{y,i}$ depend on $y$. Quantitatively, one compares the mixture to the unbiased product law $\\pi^{\\otimes k}$ in total variation.\n\n\\medskip\n\\noindent\\textit{Step 3: High temperature $\\beta<1$ (Theorem~\\ref{thm:highT}).}\nIn the regime $\\beta<1$, the Hubbard--Stratonovich field “typically remains of order one and concentrates near the origin.” Using “a Taylor expansion of the single-spin biases and a control of $\\nu_{N,\\beta}$” shows that “correlations between $k$ spins are of order $kM/N$,” which yields Theorem~\\ref{thm:highT} under the condition $kM/N\\to0$.", "expanded_sketch": "A central idea in our proofs is that the Hopfield Gibbs measure admits, via a Hubbard--Stratonovich transformation, an explicit representation in terms of a mixture of product measures.\n\n\\medskip\n\\noindent\\textit{Step 1: Mixture-of-products representation.}\nFor fixed (quenched) patterns $(\\xi^\\mu)_\\mu$, the quadratic Hamiltonian $\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2$ is linearized by introducing an auxiliary Gaussian field (Hubbard--Stratonovich), yielding\n\\[\n\\mu_N(d\\sigma)=\\int \\nu_{N,\\beta}(d y)\\, \\bigotimes_{i=1}^N \\mu_{y,i}(d\\sigma_i),\n\\]\nwhere $y\\in\\R^M$, $\\nu_{N,\\beta}$ is an explicit probability measure on $\\R^M$, and $\\mu_{y,i}$ is a Bernoulli law with bias depending on $y\\cdot \\xi_i$. Consequently, the $k$-spin marginal $\\mu_N^{(k)}$ is a mixture of product measures on $\\{-1,+1\\}^k$.\n\n\\medskip\n\\noindent\\textit{Step 2: Propagation of chaos reduces to stability of the mixture.}\nConditionally on $y$, the spins are independent, so propagation of chaos is controlled by how strongly $y$ fluctuates under $\\nu_{N,\\beta}$ and by how sensitively the single-spin biases $\\mu_{y,i}$ depend on $y$. Quantitatively, one compares the mixture to the unbiased product law $\\pi^{\\otimes k}$ in total variation.\n\n\\medskip\n\\noindent\\textit{Step 3: High temperature $\\beta<1$ (main theorem).}\nIn the regime $\\beta<1$, the Hubbard--Stratonovich field “typically remains of order one and concentrates near the origin.” Using “a Taylor expansion of the single-spin biases and a control of $\\nu_{N,\\beta}$” shows that “correlations between $k$ spins are of order $kM/N$.” Under the condition $kM/N\\to0$, this yields the conclusion needed to establish the main theorem.", "expanded_theorem": "\\label{thm:highT}\n\t\tFix $\\beta<1$. Let $M=M(N)$ satisfy $M/N\\to0$, and let $k=k(N)\\to\\infty$\n\t\twith\n\t\t\\[\n\t\t\\frac{k(N)M(N)}{N}\\longrightarrow 0 \n\t\t\\]\n\t\t(which in particular allows $k=o(N)$ when $M$ is fixed).\n\n\t\tThen\n\t\t\\[\n\t\t\\TV\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\n\t\t\\longrightarrow 0\n\t\t\\qquad\\text{in }\\Pp_\\xi\\text{-probability}.\n\t\t\\]", "theorem_type": ["Asymptotic or Limit", "Implication"], "mcq": {"question": "Let \\(\\xi_i=(\\xi_i^1,\\dots,\\xi_i^M)\\in\\{-1,+1\\}^M\\), \\(i=1,\\dots,N\\), be i.i.d. random vectors whose coordinates are independent and satisfy \\(\\mathbb E[\\xi_i^\\nu]=0\\) and \\(\\mathbb E[(\\xi_i^\\nu)^2]=1\\). For \\(\\sigma\\in\\{-1,+1\\}^N\\), define the overlap vector\n\\[\nm_N(\\sigma)=\\big(m_N^1(\\sigma),\\dots,m_N^M(\\sigma)\\big),\n\\qquad m_N^\\nu(\\sigma)=\\frac1N\\sum_{i=1}^N \\sigma_i\\xi_i^\\nu.\n\\]\nFor fixed inverse temperature \\(\\beta>0\\), the quenched Hopfield Gibbs measure on \\(\\{-1,+1\\}^N\\) is\n\\[\n\\mu_N(\\sigma)=\\frac1{Z_N}\\exp\\!\\Big(\\frac{\\beta N}{2}\\|m_N(\\sigma)\\|^2\\Big).\n\\]\nLet \\(\\mu_N^{(k)}\\) denote the marginal law of the first \\(k\\) spins under \\(\\mu_N\\), and let \\(\\pi\\) be the symmetric Bernoulli measure on \\(\\{-1,+1\\}\\), so that \\(\\pi^{\\otimes k}\\) is the product law of \\(k\\) i.i.d. fair \\(\\pm1\\) spins. Assume that \\(\\beta<1\\), that \\(M=M(N)\\) satisfies \\(M/N\\to0\\), and that \\(k=k(N)\\to\\infty\\) with\n\\[\n\\frac{k(N)M(N)}{N}\\longrightarrow 0.\n\\]\nHere total variation distance is \\(\\mathrm{TV}(\\nu,\\eta)=\\sup_A |\\nu(A)-\\eta(A)|\\), and \\(\\mathbb P_\\xi\\) denotes probability with respect to the random patterns \\((\\xi_i)\\). Under these assumptions, which statement about the \\(k\\)-spin marginal is valid?", "correct_choice": {"label": "A", "text": "\\[\\mathrm{TV}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\\]"}, "choices": [{"label": "B", "text": "\\[\\mathrm{TV}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\\qquad\\text{almost surely in }\\mathbb P_\\xi.\\]"}, {"label": "C", "text": "\\[\\mu_N^{(k)}\\xrightarrow[]{\\ d\\ }\\pi^{\\otimes k}\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\\]"}, {"label": "D", "text": "If \\(M/N\\to0\\) and \\(k=o(N)\\), then for every fixed \\(\\beta<1\\),\n\\[\\mathrm{TV}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\\]"}, {"label": "E", "text": "Assume \\(\\beta<1\\), \\(M/N\\to0\\), and \\(k(N)M(N)/N\\to0\\). Then\n\\[N\\,\\mathrm{TV}\\big(\\mu_N^{(k)},\\pi^{\\otimes k}\\big)\\longrightarrow 0\\qquad\\text{in }\\mathbb P_\\xi\\text{-probability}.\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "mode_of_convergence_in_patterns", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "total_variation_conclusion_replaced_by_weak_convergence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "critical_scale_kM_over_N", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "only_order_kM_over_N_control_not_rate_N_times_tv", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives the model, assumptions, and notation but does not state the conclusion or obviously reveal the correct option. The correct answer is not leaked explicitly or trivially."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-identification question: the correct choice is essentially the exact theorem conclusion under the stated assumptions. However, it is not a pure restatement because the alternatives vary in strength, mode of convergence, and scaling assumptions."}, "GPS": {"score": 1, "justification": "Some reasoning is required to distinguish total variation from weak convergence, probability from almost sure convergence, and the sharp condition kM/N -> 0 from weaker-looking but false variants. Still, the main task is recognizing the precise theorem statement rather than generating a new conclusion from scratch."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and well-targeted: one is stronger than the true result (almost sure), one is weaker but true, one relaxes the key scaling incorrectly, and one asserts an unjustified rate. These reflect realistic failure modes."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise recall/recognition of the result than deep generative reasoning."}}
{"id": "2602.12261v1", "paper_link": "http://arxiv.org/abs/2602.12261v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\nLet $\\mu$ be a translation-invariant, ergodic, finite-energy edge percolation measure on $\\mathbb{Z}^2$.  Then one of the following holds:\n\\begin{itemize}\n    \\item Almost surely $\\mu_{\\hp}$ has a unique infinite cluster and $\\mu^*_{\\hp}$ has no infinite cluster.\n    \\item Almost surely $\\mu^*_{\\hp}$ has a unique infinite cluster and $\\mu_{\\hp}$ has no infinite cluster.\n    \\item Almost surely neither $\\mu_{\\hp}$ nor $\\mu^*_{\\hp}$ has an infinite cluster.\n\\end{itemize}", "start_pos": 9352, "end_pos": 9863, "label": "thm:main"}, "ref_dict": {"thm:main": "\\begin{theorem}\\label{thm:main}\nLet $\\mu$ be a translation-invariant, ergodic, finite-energy edge percolation measure on $\\Z^2$.  Then one of the following holds:\n\\begin{itemize}\n    \\item Almost surely $\\mu_{\\hp}$ has a unique infinite cluster and $\\mu^*_{\\hp}$ has no infinite cluster.\n    \\item Almost surely $\\mu^*_{\\hp}$ has a unique infinite cluster and $\\mu_{\\hp}$ has no infinite cluster.\n    \\item Almost surely neither $\\mu_{\\hp}$ nor $\\mu^*_{\\hp}$ has an infinite cluster.\n\\end{itemize}\n\\end{theorem}", "thm:main2": "\\begin{theorem}\\label{thm:main2}\nLet $\\mu$ be a translation-invariant, ergodic edge percolation measure on $\\Z^2$ which has finitely many infinite clusters almost surely.  Then the trichotomy of \\Cref{thm:main} holds.\n\\end{theorem}", "cor:self-dual": "\\begin{corollary}\\label{cor:self-dual}\nLet $\\mu$ be a self-dual, translation-invariant, ergodic edge percolation measure on $\\Z^2$ that almost surely has only finitely many infinite clusters.  Then $\\mu_{\\hp}$ almost surely has no infinite cluster.\n\nIn particular, if $\\mu$ is a self-dual, translation-invariant, ergodic, finite-energy edge percolation measure on $\\Z^2$, then $\\mu_{\\hp}$ almost surely has no infinite cluster.\n\\end{corollary}"}, "pre_theorem_intro_text_len": 4000, "pre_theorem_intro_text": "\\subsection{Background}\nThis note concerns non-coexistence phenomena for edge (bond) percolation on  the square lattice $\\mathbb{Z}^2$ and the upper half-plane $\\mathbb{Z} \\times \\Z_{\\geq 0}$. On the full plane $\\mathbb{Z}^2$, a celebrated argument of Zhang shows that (almost surely) infinite primal and dual clusters cannot coexist in any translation-invariant, finite-energy, and positively-associated (FKG) edge percolation measure.  See \\cite[page 289]{grimmett2012percolation} for an exposition and \\cite{bollobas2008percolation, glazman2025planar}, \\cite[Theorem 9.3.1]{sheffield2005random} for generalizations.\nHere, as in many arguments in percolation theory, the positive-association hypothesis is essential, and without this condition there can be pathological examples (as in, e.g., \\cite{haggstrom2009some}).It is nonetheless desirable to see what can be said in the absence of positive-association, since many planar percolation models of interest, such as the loop $O(n)$ model, the random-cluster model with $q<1$, the arboreal gas, and other constrained percolation models, do not satisfy the FKG inequality.\n\nTranslation-invariance alone is certainly not sufficient to guarantee non-coexistence. For example, the dual of the uniform spanning tree is again a uniform spanning tree, so for this measure both the primal and the dual have infinite clusters \\cite{PemantleUST}.  These infinite clusters are ``non-robustly connected'' in the sense that each vertex has a unique path to infinity.  A more troubling example is Häggström and Mester's construction \\cite{haggstrom2009some} of a translation-invariant, finite-energy site percolation measure on $\\mathbb{Z}^2$ with simultaneous infinite clusters of open and closed vertices which are robust under iid thinning.\n\nIn the present paper we show that the situation is better in the half-plane than in the full plane: Here, translation-invariance, finite-energy, and absence of infinitely many infinite clusters together suffice for non-coexistence.\n\n\\subsection{Main results}\n\nWe work with the square lattice $\\mathbb{Z}^2$, namely, the Cayley graph of $\\mathbb{Z}^2$ with respect to the generators $(1,0),(0,1)$.\nSubgraphs of $\\mathbb{Z}^2$ correspond to tuples $\\omega \\in  \\{0,1\\}^{E(\\mathbb{Z}^2)}$,  where a coordinate $1$ represents an ``open'' edge that is present and a coordinate $0$ represents a ``closed'' edge that is absent. By an \\emph{edge percolation measure} we mean a probability measure on $\\{0,1\\}^{E(\\mathbb{Z}^2)}$ (with the product topology).\n\nWe need a few more notions before we can state our main results.  The natural translation action of $\\mathbb{Z}^2$ on the square lattice extends to an action of $\\mathbb{Z}^2$ on $\\{0,1\\}^{E(\\mathbb{Z}^2)}$; we say that a percolation measure is \\emph{translation-invariant} if it is invariant under this action.  Say that an edge percolation measure $\\mu$ has \\emph{finite-energy} if there is some $\\delta>0$ such that for every edge $e \\in E(\\mathbb{Z}^2)$, the conditional probability $$\\mu(\\omega_e=1|\\omega_{E(\\mathbb{Z}^2) \\setminus\\{e\\}})$$  has essential infimum at least $\\delta$ and essential supremum at most $1-\\delta$; this condition, which is sometimes called uniform finite-energy in the literature, is stronger than other common notions of finite-energy which have less uniformity (in either the edge $e$ or the bounds on the conditional probability).\n\nEvery edge percolation measure $\\mu$ on $\\mathbb{Z}^2$ induces an edge percolation measure $\\mu^*$ on the planar dual of $\\mathbb{Z}^2$, which is itself isomorphic to $\\mathbb{Z}^2$; see \\Cref{sec:duality} below for further discussion of duality.  It is easy to see that $\\mu$ is ergodic (respectively, has finite-energy) if and only if $\\mu^*$ is ergodic (respectively, has finite-energy).  Finally, we write $\\mu_{\\hp}$ for the marginal of $\\mu$ on the upper half-plane $\\mathbb{Z} \\times \\Z_{ \\geq 0}$, and we define $\\mu^*_{\\hp}$ analogously.  Our first non-coexistence result goes as follows.", "context": "\\subsection{Background}\nThis note concerns non-coexistence phenomena for edge (bond) percolation on the square lattice $\\mathbb{Z}^2$ and the upper half-plane $\\mathbb{Z} \\times \\Z_{\\geq 0}$. On the full plane $\\mathbb{Z}^2$, a celebrated argument of Zhang shows that (almost surely) infinite primal and dual clusters cannot coexist in any translation-invariant, finite-energy, and positively-associated (FKG) edge percolation measure. See \\cite[page 289]{grimmett2012percolation} for an exposition and \\cite{bollobas2008percolation, glazman2025planar}, \\cite[Theorem 9.3.1]{sheffield2005random} for generalizations.\nHere, as in many arguments in percolation theory, the positive-association hypothesis is essential, and without this condition there can be pathological examples (as in, e.g., \\cite{haggstrom2009some}).It is nonetheless desirable to see what can be said in the absence of positive-association, since many planar percolation models of interest, such as the loop $O(n)$ model, the random-cluster model with $q<1$, the arboreal gas, and other constrained percolation models, do not satisfy the FKG inequality.\n\nTranslation-invariance alone is certainly not sufficient to guarantee non-coexistence. For example, the dual of the uniform spanning tree is again a uniform spanning tree, so for this measure both the primal and the dual have infinite clusters \\cite{PemantleUST}. These infinite clusters are ``non-robustly connected'' in the sense that each vertex has a unique path to infinity. A more troubling example is Häggström and Mester's construction \\cite{haggstrom2009some} of a translation-invariant, finite-energy site percolation measure on $\\mathbb{Z}^2$ with simultaneous infinite clusters of open and closed vertices which are robust under iid thinning.\n\nIn the present paper we show that the situation is better in the half-plane than in the full plane: Here, translation-invariance, finite-energy, and absence of infinitely many infinite clusters together suffice for non-coexistence.\n\nWe work with the square lattice $\\mathbb{Z}^2$, namely, the Cayley graph of $\\mathbb{Z}^2$ with respect to the generators $(1,0),(0,1)$.\nSubgraphs of $\\mathbb{Z}^2$ correspond to tuples $\\omega \\in \\{0,1\\}^{E(\\mathbb{Z}^2)}$, where a coordinate $1$ represents an ``open'' edge that is present and a coordinate $0$ represents a ``closed'' edge that is absent. By an \\emph{edge percolation measure} we mean a probability measure on $\\{0,1\\}^{E(\\mathbb{Z}^2)}$ (with the product topology).\n\nWe need a few more notions before we can state our main results. The natural translation action of $\\mathbb{Z}^2$ on the square lattice extends to an action of $\\mathbb{Z}^2$ on $\\{0,1\\}^{E(\\mathbb{Z}^2)}$; we say that a percolation measure is \\emph{translation-invariant} if it is invariant under this action. Say that an edge percolation measure $\\mu$ has \\emph{finite-energy} if there is some $\\delta>0$ such that for every edge $e \\in E(\\mathbb{Z}^2)$, the conditional probability $$\\mu(\\omega_e=1|\\omega_{E(\\mathbb{Z}^2) \\setminus\\{e\\}})$$ has essential infimum at least $\\delta$ and essential supremum at most $1-\\delta$; this condition, which is sometimes called uniform finite-energy in the literature, is stronger than other common notions of finite-energy which have less uniformity (in either the edge $e$ or the bounds on the conditional probability).\n\nEvery edge percolation measure $\\mu$ on $\\mathbb{Z}^2$ induces an edge percolation measure $\\mu^*$ on the planar dual of $\\mathbb{Z}^2$, which is itself isomorphic to $\\mathbb{Z}^2$; see \\Cref{sec:duality} below for further discussion of duality. It is easy to see that $\\mu$ is ergodic (respectively, has finite-energy) if and only if $\\mu^*$ is ergodic (respectively, has finite-energy). Finally, we write $\\mu_{\\hp}$ for the marginal of $\\mu$ on the upper half-plane $\\mathbb{Z} \\times \\Z_{ \\geq 0}$, and we define $\\mu^*_{\\hp}$ analogously. Our first non-coexistence result goes as follows.", "full_context": "\\subsection{Background}\nThis note concerns non-coexistence phenomena for edge (bond) percolation on the square lattice $\\mathbb{Z}^2$ and the upper half-plane $\\mathbb{Z} \\times \\Z_{\\geq 0}$. On the full plane $\\mathbb{Z}^2$, a celebrated argument of Zhang shows that (almost surely) infinite primal and dual clusters cannot coexist in any translation-invariant, finite-energy, and positively-associated (FKG) edge percolation measure. See \\cite[page 289]{grimmett2012percolation} for an exposition and \\cite{bollobas2008percolation, glazman2025planar}, \\cite[Theorem 9.3.1]{sheffield2005random} for generalizations.\nHere, as in many arguments in percolation theory, the positive-association hypothesis is essential, and without this condition there can be pathological examples (as in, e.g., \\cite{haggstrom2009some}).It is nonetheless desirable to see what can be said in the absence of positive-association, since many planar percolation models of interest, such as the loop $O(n)$ model, the random-cluster model with $q<1$, the arboreal gas, and other constrained percolation models, do not satisfy the FKG inequality.\n\nTranslation-invariance alone is certainly not sufficient to guarantee non-coexistence. For example, the dual of the uniform spanning tree is again a uniform spanning tree, so for this measure both the primal and the dual have infinite clusters \\cite{PemantleUST}. These infinite clusters are ``non-robustly connected'' in the sense that each vertex has a unique path to infinity. A more troubling example is Häggström and Mester's construction \\cite{haggstrom2009some} of a translation-invariant, finite-energy site percolation measure on $\\mathbb{Z}^2$ with simultaneous infinite clusters of open and closed vertices which are robust under iid thinning.\n\nIn the present paper we show that the situation is better in the half-plane than in the full plane: Here, translation-invariance, finite-energy, and absence of infinitely many infinite clusters together suffice for non-coexistence.\n\nWe work with the square lattice $\\mathbb{Z}^2$, namely, the Cayley graph of $\\mathbb{Z}^2$ with respect to the generators $(1,0),(0,1)$.\nSubgraphs of $\\mathbb{Z}^2$ correspond to tuples $\\omega \\in \\{0,1\\}^{E(\\mathbb{Z}^2)}$, where a coordinate $1$ represents an ``open'' edge that is present and a coordinate $0$ represents a ``closed'' edge that is absent. By an \\emph{edge percolation measure} we mean a probability measure on $\\{0,1\\}^{E(\\mathbb{Z}^2)}$ (with the product topology).\n\nWe need a few more notions before we can state our main results. The natural translation action of $\\mathbb{Z}^2$ on the square lattice extends to an action of $\\mathbb{Z}^2$ on $\\{0,1\\}^{E(\\mathbb{Z}^2)}$; we say that a percolation measure is \\emph{translation-invariant} if it is invariant under this action. Say that an edge percolation measure $\\mu$ has \\emph{finite-energy} if there is some $\\delta>0$ such that for every edge $e \\in E(\\mathbb{Z}^2)$, the conditional probability $$\\mu(\\omega_e=1|\\omega_{E(\\mathbb{Z}^2) \\setminus\\{e\\}})$$ has essential infimum at least $\\delta$ and essential supremum at most $1-\\delta$; this condition, which is sometimes called uniform finite-energy in the literature, is stronger than other common notions of finite-energy which have less uniformity (in either the edge $e$ or the bounds on the conditional probability).\n\nEvery edge percolation measure $\\mu$ on $\\mathbb{Z}^2$ induces an edge percolation measure $\\mu^*$ on the planar dual of $\\mathbb{Z}^2$, which is itself isomorphic to $\\mathbb{Z}^2$; see \\Cref{sec:duality} below for further discussion of duality. It is easy to see that $\\mu$ is ergodic (respectively, has finite-energy) if and only if $\\mu^*$ is ergodic (respectively, has finite-energy). Finally, we write $\\mu_{\\hp}$ for the marginal of $\\mu$ on the upper half-plane $\\mathbb{Z} \\times \\Z_{ \\geq 0}$, and we define $\\mu^*_{\\hp}$ analogously. Our first non-coexistence result goes as follows.\n\n\\begin{abstract}\nFor $\\mu$ an edge percolation measure on the infinite square lattice, let $\\mu_{\\texorpdfstring{\\hp}{hp}}$ (respectively, $\\mu^*_{\\texorpdfstring{\\hp}{hp}}$) denote its marginal (respectively, the marginal of its planar dual process) on the upper half-plane. We show that if $\\mu$ is translation-invariant and ergodic and almost surely has only finitely many infinite clusters, then either almost surely $\\mu_{\\texorpdfstring{\\hp}{hp}}$ has no infinite cluster, or almost surely $\\mu^*_{\\texorpdfstring{\\hp}{hp}}$ has no infinite cluster. \nBy the classical Burton--Keane argument, these hypotheses are satisfied if $\\mu$ is translation-invariant and ergodic and has finite-energy. \nIn contrast to previous ``non-coexistence'' theorems, our result does not impose a positive-correlation (FKG) hypothesis on $\\mu$. \nOur arguments also apply to the random-cluster model (including the regime $q<1$, which lacks FKG), the uniform spanning tree, and the uniform odd subgraph.\n\\end{abstract}\n\nEvery edge percolation measure $\\mu$ on $\\Z^2$ induces an edge percolation measure $\\mu^*$ on the planar dual of $\\Z^2$, which is itself isomorphic to $\\Z^2$; see \\Cref{sec:duality} below for further discussion of duality. It is easy to see that $\\mu$ is ergodic (respectively, has finite-energy) if and only if $\\mu^*$ is ergodic (respectively, has finite-energy). Finally, we write $\\mu_{\\hp}$ for the marginal of $\\mu$ on the upper half-plane $\\Z \\times \\Z_{ \\geq 0}$, and we define $\\mu^*_{\\hp}$ analogously. Our first non-coexistence result goes as follows.\n\nIf we drop the ergodicity hypothesis from this theorem, then we still obtain the conclusion that for $H$ sampled from $\\mu_{\\hp}$, almost surely the number of infinite clusters in $H$ plus the number of infinite clusters in $H^*$ is at most $1$; in particular, almost surely $H$ and $H^*$ do not simultaneously have infinite clusters. This statement follows from applications of the theorem to the individual components in the ergodic decomposition of $\\mu$, which almost surely inherit the finite-energy property (see, e.g., \\cite[page 307]{burton1991topological}).\n\n\\begin{theorem}\\label{thm:main2}\nLet $\\mu$ be a translation-invariant, ergodic edge percolation measure on $\\Z^2$ which has finitely many infinite clusters almost surely. Then the trichotomy of \\Cref{thm:main} holds.\n\\end{theorem}\n\n\\begin{corollary}\\label{cor:self-dual}\nLet $\\mu$ be a self-dual, translation-invariant, ergodic edge percolation measure on $\\Z^2$ that almost surely has only finitely many infinite clusters. Then $\\mu_{\\hp}$ almost surely has no infinite cluster.\n\n\\begin{proposition}\\label{lemma:uniqueness_implies_non_tenuousness}\n Let $\\mu$ be a translation-invariant edge percolation measure on $\\Z^2$ such that $\\mu_{\\hp}$ almost surely has at most $1$ infinite cluster. Then $\\mu_{\\hp}$ almost surely has no tenuous infinite cluster. \n\\end{proposition}\n\n\\begin{lemma}\\label{lem:a.s.constant}\nLet $\\mu$ be a translation-invariant, ergodic edge percolation measure on $\\Z^2$ that almost surely has only finitely many infinite clusters. Then the number of infinite clusters in $\\mu_{\\hp}$ is either almost surely $0$ or almost surely $1$.\n\\end{lemma}\n\n\\begin{proposition}\\label{proposion31}\n Let $\\mu$ be a translation-invariant edge percolation measure on $\\Z^2$ such that $\\mu_{\\hp}$ almost surely has a unique infinite cluster. Then almost surely $\\mu^*_{\\hp}$ has no infinite cluster. \n\\end{proposition}\n\n\\begin{theorem}\\label{thm:main}\nLet $\\mu$ be a translation-invariant, ergodic, finite-energy edge percolation measure on $\\Z^2$.  Then one of the following holds:\n\\begin{itemize}\n    \\item Almost surely $\\mu_{\\hp}$ has a unique infinite cluster and $\\mu^*_{\\hp}$ has no infinite cluster.\n    \\item Almost surely $\\mu^*_{\\hp}$ has a unique infinite cluster and $\\mu_{\\hp}$ has no infinite cluster.\n    \\item Almost surely neither $\\mu_{\\hp}$ nor $\\mu^*_{\\hp}$ has an infinite cluster.\n\\end{itemize}\n\\end{theorem}", "post_theorem_intro_text_len": 6994, "post_theorem_intro_text": "If we drop the ergodicity hypothesis from this theorem, then we still obtain the conclusion that for $H$ sampled from $\\mu_{\\hp}$, almost surely the number of infinite clusters in $H$ plus the number of infinite clusters in $H^*$ is at most $1$; in particular, almost surely $H$ and $H^*$ do not simultaneously have infinite clusters.  This statement follows from applications of the theorem to the individual components in the ergodic decomposition of $\\mu$, which almost surely inherit the finite-energy property (see, e.g., \\cite[page 307]{burton1991topological}).\n\nClassical work of Burton and Keane~\\cite{burton1989density} shows that under the hypotheses of \\Cref{thm:main}, the number of infinite clusters in $\\mu$ is almost surely at most $1$.\\footnote{Burton and Keane's uniqueness argument has also been adapted to several settings that fall slightly short of having  finite-energy; examples include the loop $O(n)$ model \\cite[Proposition 4.11]{crawford2020macroscopic},  the random current representation of the Ising model \\cite{aizenman2015random}, various constrained percolation models \\cite{lima2020constrained,holroyd2021constrained}, and the 1-2 model \\cite{li2014uniqueness}.}  Thus \\Cref{thm:main} is an immediate consequence of the following theorem, our main technical result.\n\n\\begin{theorem}\\label{thm:main2}\nLet $\\mu$ be a translation-invariant, ergodic edge percolation measure on $\\mathbb{Z}^2$ which has finitely many infinite clusters almost surely.  Then the trichotomy of \\Cref{thm:main} holds.\n\\end{theorem}\n\n\\Cref{thm:main2} applies to some models, such as the uniform spanning tree, which \\Cref{thm:main} does not cover. \nWe remark that there exist translation-invariant edge percolation measures on $\\mathbb{Z}^2$ that almost surely have multiple, but only finitely many, infinite clusters.\\footnote{For an example that almost surely has $2$ infinite clusters, one can modify the uniform spanning tree as follows.  Sample a uniform spanning tree $H$ on the dilated infinite square grid $(2\\mathbb{Z})^2$.  Let $H^*$ be its planar dual (also a spanning tree), interpreted as a subgraph of the square grid $(1+2\\mathbb{Z})^2$.  By subdividing each edge in half, we can interpret $H \\cup H^*$ as a subgraph of $\\mathbb{Z}^2$; let $\\mu'$ denote the resulting edge percolation measure on $\\mathbb{Z}^2$.  Notice that $\\mu'$ is invariant under translations by $(2\\mathbb{Z})^2$ and always has exactly $2$ infinite clusters.  Finally, let $\\mu$ be the average of $\\mu'$ and its translates by $(1,0),(0,1),(1,1)$; this guarantees that $\\mu$ is $\\mathbb{Z}^2$-translation-invariant, and of course it still always has $2$ infinite clusters.  For $2<k<\\infty$, we do not know of examples of translation-invariant edge-percolation measures that almost surely have $k$ infinite clusters.}\n\nWe record the following corollary for self-dual measures.\n\n\\begin{corollary}\\label{cor:self-dual}\nLet $\\mu$ be a self-dual, translation-invariant, ergodic edge percolation measure on $\\mathbb{Z}^2$ that almost surely has only finitely many infinite clusters.  Then $\\mu_{\\hp}$ almost surely has no infinite cluster.\n\nIn particular, if $\\mu$ is a self-dual, translation-invariant, ergodic, finite-energy edge percolation measure on $\\mathbb{Z}^2$, then $\\mu_{\\hp}$ almost surely has no infinite cluster.\n\\end{corollary}\n\nThe proof strategy for Theorem~\\ref{thm:main2} has two steps.  The first step is showing that the number of infinite clusters in $\\mu_{\\hp}$ is either almost surely $0$ or almost surely $1$.  The main challenge is eliminating the possibility that an infinite cluster in the full plane ``splits'' into several infinite clusters in the half-plane.  The second step is showing that if $\\mu_{\\hp}$ almost surely has a unique infinite cluster, then almost surely $\\mu^*_{\\hp}$ has no infinite cluster; it suffices to show that almost surely the plaquette centered at $(1/2,1/2)$ does not connect to infinity in the dual.  Here, almost surely both the positive $x$-axis and the negative $x$-axis connect to the unique infinite primal cluster, and these paths to infinity together ``block'' the origin plaquette from connecting to infinity in the dual.  A recurring technical point is ruling out the existence of what we term ``tenuous'' infinite clusters.\n\nOur methods are robust enough to establish some slight extensions of these theorems.  Instead of taking the upper half-plane, one could take the half-plane lying on one side of any given line with rational slope.  Also, one could replace the square lattice with a different planar graph admitting a $\\mathbb{Z}^2$-translation action with only finitely many orbits.\n\n\\subsection{Applications}\n\nLet us highlight four special cases of our results.  First, when $\\mu$ is Bernoulli percolation with density $1/2$, Corollary~\\ref{cor:self-dual} (via finite-energy) provides a new proof of the fact that almost surely there is no infinite cluster in the half-plane.  Of course it is a classical result of Harris~\\cite{harris1960lower} that there is no infinite cluster even in the full plane; the novelty of our argument is that it avoids using FKG (or the Harris inequality).\n\nSecond, Theorem~\\ref{thm:main} demonstrates non-coexistence in the half-plane for (a suitable limit of) the random-cluster model with all values of the parameter $q$; see \\Cref{sec:rc} below for precise definitions.  The random-cluster model has FKG only when $q \\geq 1$, and for $q<1$ little is known.\n\nThird, the uniform spanning tree on $\\mathbb{Z}^2$ is self-dual and ergodic (due to tail-triviality; see \\cite[Theorem 10.18]{lyons2017probability}), and it has a unique infinite cluster. \nThus, Corollary~\\ref{cor:self-dual} shows that the half-plane marginal of the uniform spanning tree almost surely has only finite clusters.  (Note that this model lacks finite-energy.).  The same holds for the Peano uniform spanning tree (UST) curve \\cite{lyons2017probability} and for constructions of Häggström--Mester type \\cite{haggstrom2009some} (if they can be adapted to edge percolation).\n\nFourth, we show that the uniform odd subgraph almost surely has no infinite cluster in the half-plane.  This does not follow from the statement of Corollary~\\ref{cor:self-dual}, but our arguments from Theorem~\\ref{thm:main2} can be adapted.  The uniform odd subgraph measure is self-complementary, and here the ``non-crossing'' property of the primal and the complement plays the role that planar duality plays in Theorem~\\ref{thm:main2}.\n\n\\subsection{Organization of the paper}\nWe start by proving \\Cref{thm:main2} in Section~\\ref{sec:thm1.1}.  We then turn to applications: Section~\\ref{sec:rc} contains our results on the $q<1$ random-cluster model, and Section~\\ref{sec:odd} treats the uniform odd subgraph.  In Section~\\ref{sec:conclusion} we make some concluding remarks and mention some  open problems.  \\Cref{sec:appendix} contains an alternative proof of a variant of one of our technical lemmas (the absence of tenuous infinite clusters).", "sketch": "For \\Cref{thm:main}, the text explains that (using Burton--Keane uniqueness) it is an immediate consequence of \\Cref{thm:main2}, since under the hypotheses of \\Cref{thm:main} “the number of infinite clusters in \\mu is almost surely at most $1$.”\n\nA proof sketch is then given for \\Cref{thm:main2} (and hence for \\Cref{thm:main}): “The proof strategy for Theorem~\\ref{thm:main2} has two steps.” (1) “showing that the number of infinite clusters in \\mu_{\\hp} is either almost surely $0$ or almost surely $1$,” with the “main challenge” being “eliminating the possibility that an infinite cluster in the full plane ‘splits’ into several infinite clusters in the half-plane.” (2) “showing that if \\mu_{\\hp} almost surely has a unique infinite cluster, then almost surely \\mu^*_{\\hp} has no infinite cluster; it suffices to show that almost surely the plaquette centered at $(1/2,1/2)$ does not connect to infinity in the dual.” The mechanism is that “almost surely both the positive $x$-axis and the negative $x$-axis connect to the unique infinite primal cluster, and these paths to infinity together ‘block’ the origin plaquette from connecting to infinity in the dual.” A “recurring technical point” throughout is “ruling out the existence of what we term ‘tenuous’ infinite clusters.”\n\nSeparately, if ergodicity is dropped, the text notes one can still deduce (by “applications of the theorem to the individual components in the ergodic decomposition of \\mu”) that for $H\\sim\\mu_{\\hp}$ the number of infinite clusters in $H$ plus the number in $H^*$ is “at most $1$,” so “$H$ and $H^*$ do not simultaneously have infinite clusters.”", "expanded_sketch": "To prove the main theorem, the text explains that (using Burton--Keane uniqueness) it is an immediate consequence of the following theorem, since under the hypotheses of the main theorem “the number of infinite clusters in \\mu is almost surely at most $1$.”\n\n\\begin{theorem}\\label{thm:main2}\nLet $\\mu$ be a translation-invariant, ergodic edge percolation measure on $\\Z^2$ which has finitely many infinite clusters almost surely.  Then the trichotomy of \\\\Cref{thm:main} holds.\n\\end{theorem}\n\nA proof sketch is then given for the theorem above (and hence for the main theorem): “The proof strategy for Theorem~\\ref{thm:main2} has two steps.” (1) “showing that the number of infinite clusters in \\mu_{\\hp} is either almost surely $0$ or almost surely $1$,” with the “main challenge” being “eliminating the possibility that an infinite cluster in the full plane ‘splits’ into several infinite clusters in the half-plane.” (2) “showing that if \\mu_{\\hp} almost surely has a unique infinite cluster, then almost surely \\mu^*_{\\hp} has no infinite cluster; it suffices to show that almost surely the plaquette centered at $(1/2,1/2)$ does not connect to infinity in the dual.” The mechanism is that “almost surely both the positive $x$-axis and the negative $x$-axis connect to the unique infinite primal cluster, and these paths to infinity together ‘block’ the origin plaquette from connecting to infinity in the dual.” A “recurring technical point” throughout is “ruling out the existence of what we term ‘tenuous’ infinite clusters.”\n\nSeparately, if ergodicity is dropped, the text notes one can still deduce (by “applications of the theorem to the individual components in the ergodic decomposition of \\mu”) that for $H\\sim\\mu_{\\hp}$ the number of infinite clusters in $H$ plus the number in $H^*$ is “at most $1$,” so “$H$ and $H^*$ do not simultaneously have infinite clusters.”", "expanded_theorem": "\\label{thm:main}\nLet $\\mu$ be a translation-invariant, ergodic, finite-energy edge percolation measure on $\\mathbb{Z}^2$.  Then one of the following holds:\n\\begin{itemize}\n    \\item Almost surely $\\mu_{\\hp}$ has a unique infinite cluster and $\\mu^*_{\\hp}$ has no infinite cluster.\n    \\item Almost surely $\\mu^*_{\\hp}$ has a unique infinite cluster and $\\mu_{\\hp}$ has no infinite cluster.\n    \\item Almost surely neither $\\mu_{\\hp}$ nor $\\mu^*_{\\hp}$ has an infinite cluster.\n\\end{itemize}", "theorem_type": ["Classification or Bijection", "Universal"], "mcq": {"question": "Let μ be an edge percolation measure on the square lattice Z2, viewed as a probability measure on {0,1}^{E(Z2)}. Assume that μ is translation-invariant, ergodic, and finite-energy, meaning that there exists δ > 0 such that for every edge e, the conditional probability μ(ω_e = 1 | ω_{E(Z2)\\setminus\\{e\\}}) has essential infimum at least δ and essential supremum at most 1 - δ. Let μ* denote the induced edge percolation measure on the planar dual lattice, let μ_hp and μ*_hp denote the marginals of μ and μ* on the upper half-plane Z × Z_{≥0}, and interpret an infinite cluster as an infinite connected component of open edges in the corresponding half-plane configuration. Which statement holds for every such μ?", "correct_choice": {"label": "A", "text": "One of the following three alternatives holds almost surely: (i) μ_hp has a unique infinite cluster and μ*_hp has no infinite cluster; (ii) μ*_hp has a unique infinite cluster and μ_hp has no infinite cluster; or (iii) neither μ_hp nor μ*_hp has an infinite cluster."}, "choices": [{"label": "B", "text": "Almost surely exactly one of bc_{\\hp} and \\mu^*_{\\hp} has an infinite cluster, and whenever one does, it has a unique infinite cluster."}, {"label": "C", "text": "Almost surely \\mu_{\\hp} and \\mu^*_{\\hp} do not simultaneously have infinite clusters."}, {"label": "D", "text": "If \\mu_{\\hp} almost surely has an infinite cluster, then almost surely \\mu^*_{\\hp} has no infinite cluster; and if \\mu^*_{\\hp} almost surely has an infinite cluster, then almost surely \\mu_{\\hp} has no infinite cluster."}, {"label": "E", "text": "Almost surely one of the following holds: (i) \\mu_{\\hp} has at least one infinite cluster and \\mu^*_{\\hp} has no infinite cluster; (ii) \\mu^*_{\\hp} has at least one infinite cluster and \\mu_{\\hp} has no infinite cluster; or (iii) neither \\mu_{\\hp} nor \\mu^*_{\\hp} has an infinite cluster."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "omits the third alternative where neither half-plane process percolates", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "drops the uniqueness/trichotomy and keeps only non-coexistence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "replaces the almost-sure trichotomy by one-way implication statements about percolation events", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "weakens 'unique infinite cluster' to 'at least one infinite cluster' despite the sketch's step eliminating splitting in the half-plane", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct option directly. It states the hypotheses and asks for the full almost-sure classification, but it does not explicitly encode the trichotomy in choice A."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem gives the exact setup and asks for the theorem's conclusion. The correct choice is basically the classification theorem restated."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the complete statement from weaker true or stronger false variants, especially uniqueness versus mere existence. However, the item mainly tests recognition of the known result rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong and well targeted: one is weaker-but-true/incomplete, one allows forbidden coexistence, and one adds an unjustified extra claim. They are distinct and reflect realistic mathematical failure modes."}, "total_score": 5, "overall_assessment": "A solid theorem-recall MCQ with strong distractors and little answer leakage, but it is largely tautological and only moderately tests genuine reasoning."}}
{"id": "2602.12646v1", "paper_link": "http://arxiv.org/abs/2602.12646v1", "theorems_cnt": 1, "theorem": {"env_name": "theo", "content": "\\label{theo.finite.top}\nFor fixed $n\\,,m \\in \\mathbb{Z}^+\\,,n \\geqslant 3\\,, m \\geq1\\,,$ and $ \\Gamma \\, ,\\Lambda \\in \\mathbb{R} \\,,  \\Gamma\\,,\\Lambda \\geqslant 0\\,$, there exists $N = N(n,m,\\Gamma,\\Lambda) \\in \\mathbb{N}$ so that there are at most $N$ mutually non-diffeomorphic complete immersed minimal submanifolds $M^n$ in $\\mathbb{R}^{n+m}$ satisfying that  $\\int_{M} \\lvertA\\rvert^{n}d\\mu_M  \\leqslant \\Gamma$ and \n$\\vol_M ( B_R(0)) \\leqslant  \\Lambda R^n$ for any $R>0\\,.$", "start_pos": 11814, "end_pos": 12264, "label": "theo.finite.top"}, "ref_dict": {"theo.finite.top": "\\begin{theo} \\label{theo.finite.top}\nFor fixed $n\\,,m \\in \\ZZ^+\\,,n \\geq 3\\,, m \\geq1\\,,$ and $ \\Gamma \\, ,\\Lambda \\in \\RR \\,,  \\Gamma\\,,\\Lambda \\geq 0\\,$, there exists $N = N(n,m,\\Gamma,\\Lambda) \\in \\NN$ so that there are at most $N$ mutually non-diffeomorphic complete immersed minimal submanifolds $M^n$ in $\\RR^{n+m}$ satisfying that  $\\int_{M} \\abs{A}^{n}d\\mu_M  \\leq \\Gamma$ and \n$\\vol_M ( B_R(0)) \\leq  \\Lambda R^n$ for any $R>0\\,.$\n\\end{theo}", "lemm.Annular.composition": "\\begin{lemm}[Annular decomposition] \\label{lemm.Annular.composition}\nFor fixed $n\\,,m\\in \\ZZ^+\\,,n\\geq 2\\,,$ there is a $0 < \\sigma_0 <\\fh{1}{2} $ only depending on $n,m$ with the following property. Suppose that $M^n(\\iota : M^n \\to \\bar{B_2(0)} \\subset\\RR^{n+m})$ is a complete properly immersed submanifold  with $\\iota(\\de M)  \\subset \\de B_2(0)\\, . $ Assume that for some $\\sigma \\leq \\sigma_0$ and $p \\in B_{\\sigma_0}(0)\\,,$ we have:\n\\begin{enumerate}[itemsep=5pt, topsep=5pt]\n\\item For each component $M'$ of $M\\,,$ $M' \\cap B_{\\sigma}(p) \\neq \\emptyset \\,$.\n\\item The immersed submanifold $M$ intersects $\\de B_{\\sigma}(p)$ transversely, and $M \\cap \\de B_{\\sigma}(p)$ has $k$ components. Moreover, each component of   $M \\cap \\de B_{\\sigma}(p)$ is diffeomorphic to $\\SS^{n-1}$ with the standard smooth structure.\n\\item The second fundamental form of $M$ satisfies $|A|(x)|\\iota(x)-p| \\leq \\fh{1}{4} $ for all $x \\in M \\cap \\left( \\overline{B_{1}(0)} \\setminus B_{\\sigma}(p) \\right)\\,.$\n\\end{enumerate}\nThen, $M$ intersects $\\de B_{1}(0)$ transversely. Both  $M \\cap \\left(\\overline{B_{1}(0)}\\setminus {B_{\\sigma}(p)}\\right)$ and  $M \\cap \\de B_{1}(0)$  have $k$ components. Moreover, each component of   $M \\cap \\de B_{1}(0)$ is diffeomorphic to $\\SS^{n-1}$ with the standard smooth structure and\n each component of $M \\cap \\left(\\overline{B_{1}(0)}\\setminus {B_{\\sigma}(p)}\\right)$ is diffeomorphic to $\\SS^{n-1}\\times [0,1]$ with the standard smooth structure.\n\\end{lemm}", "lemm:curv.est": "\\begin{lemm}\\label{lemm:curv.est}\nFor fixed $I \\in \\ZZ^+,0 <r_0<R_0< \\infty\\,,$ suppose that $M_j( \\iota_j :M_{j} \\to  \\RR^{n+m})$ is a sequence of $n$-dimensional complete properly immersed minimal submanifolds with nonempty boundary and $\\iota_j(M_j) \\subset B_{R_0}(0)$ such that $ \\int_{M_j} \\abs{A_{M_j}}^n d\\mu_{M_j} < IK_0 , \\limsup \\limits_{j\\to \\infty}\\nu_j(B_{R_0}(0)\\setminus B_{\\fh{r_0}{2}}(0))< K_0$ and $\\vol(B_R(q)) \\leq \\Lambda R^n$ for any $B_R(q) \\subset B_{R_0}(0)$. Then, after passing to a subsequence, we have:\n\\begin{enumerate}[itemsep=5pt, topsep=5pt]\n    \\item There exist $C>0$ and a sequence of smooth blow-up sets $\\cB_{j} \\subset M_{j}$  so that \n\\begin{equation} \\label{eq:seq.cur.esti}\n    |A_{M_{j}}|(x)d(\\iota_j(x),\\iota_j(\\cB_{j} \\cup \\de M_j) ) \\leq C\\, , \\ |\\cB_{j}|< I\\, , \\ \\iota_j(\\cB_j) \\subset B_{\\fh{3}{4}r_0}(0)\\, ,\n\\end{equation}\nfor all $x \\in M_{j}\\,.$ \n\\item  $\\iota_j(\\cB_j)$ converges to $ \\widetilde  \\cB_ \\infty \\subset \\RR^{n+m}$ in the Hausdorff distance sense and  the Radon measure $\\nu_j$ converges to $ \\nu_\\infty$ in the Radon measure sense with $\\nu_\\infty(p_\\infty)\\geq 2K_0$ for any $p_\\infty \\in \\widetilde \\cB_\\infty\\,.$\n\\end{enumerate}\n\\end{lemm}", "prop.diffeo": "\\begin{theo} \\label{prop.diffeo}\nGiven a sequence $M_{j}$ satisfying \\hyperlink{defi:beth}{$(\\beth)$} and each $M_j$  intersects $\\partial B_{1}(0)$ transversely. By passing to a subsequence, all of the $M_{j}\\cap B_{1}(0)$ are diffeomorphic. \n\\end{theo}"}, "pre_theorem_intro_text_len": 3642, "pre_theorem_intro_text": "For minimal surfaces in $\\mathbb{R}^3$, finite total curvature means that the Gaussian curvature integral is finite. Chern and Osserman \\cite{Chern} proved that every minimal surface in $\\mathbb{R}^3$ is conformally equivalent to a compact Riemann surface $\\overline M$ punctured at a finite number of points, and the Gauss map on the surface can extend conformally to $\\overline M$.\n\nCollin \\cite{collin1} proved that any properly embedded minimal\nsurface in $\\mathbb{R}^3$ with finite topology and more than one end, has finite total curvature. \nColding and Minicozzi \\cite{ColdingMincozzi1} removed the proper condition, where they proved that a complete embedded minimal surface $\\Sigma$ with finite topology in $\\mathbb{R}^3$ must be proper.\nMeeks, Perez and Ros \\cite{Meeks2} showed that the number of ends of $\\Sigma$ is bounded by a constant depending on its genus.\n\nGiven an immersed minimal submanifold $M^{n}$  in $ \\mathbb{R}^{n+m}\\,,$ $M$ is said to have \\emph{finite total  curvature} if\n $$\\int_{M} \\lvertA\\rvert^{n}d\\mu_M < \\infty\\,,$$\nwhere $A$ denotes the second fundamental form of $M$ in $ \\mathbb{R}^{n+m}$, and $\\mu_M$ denotes the volume element of $M$.\n\nAnderson \\cite{anderson1984compactification} gave a generalization of the\nChern-Osserman theorem \\cite{Chern} on minimal surfaces of finite total curvature: a complete minimal submanifold $M^n$ with finite total curvature is diffeomorphic to a compact $C^{\\infty} $ manifold $\\overline{M}^n$ punctured at a finite number of points $\\{p_i\\}_{1}^{\\ell} \\in \\overline{M}^n$ and the Gauss map $\\gamma:M^n \\to G_{n,m}$ extends to a $C^{n-2}$ map $\\overline \\gamma: \\overline  M^n \\to G_{n,m} $ of the compactification(where $G_{n,m}$ denotes the Grassmann manifold of $n$-planes in Euclidean $(n+m)$-space). \nIn particular, $M$ has Euclidean volume growth with ratio bounded by a constant depending on $\\ell$.\nFor complete minimal hypersurfaces in $\\mathbb{R}^{n+1}$ with $3 \\leqslant n \\leqslant 6\\,,$ Tysk \\cite{Tysk} proved that finite index and Euclidean volume growth imply finite total curvature.    \n\nChodosh, Ketover, and  Maximo \\cite[Theorem 1.1]{Chodosh17} proved that\nfor a fixed  closed Riemannian manifold $(M^n,g)(3 \\leqslant  n \\leqslant 7)\\,,$ \n there can be at most $ N = N(M,g,\\Lambda,I)$ distinct diffeomorphism types in the set of\n embedded minimal hypersurfaces $\\Sigma \\subset (M,g)$ with $\\ind(\\Sigma) \\leqslant  I$ and $\\vol_g(\\Sigma) \\leqslant  \\Lambda\\,.$\n In particular, for $n=3\\,$, there is $r_0 = r_0(M,g,\\Lambda,I)$ so that\n any embedded minimal surface $\\Sigma$ in $(M^3,g)$ with $\\ind(\\Sigma) \\leqslant I$ and $\\mathrm{area}_g(\\Sigma) \\leqslant \\Lambda$ has\n $\\genus(\\Sigma) \\leqslant  r_0$; for $4 \\leqslant n \\leqslant 7$, there is $N = N(n,I,\\Lambda) \\in  \\mathbb{N}$ so that there are at most\n $N$ mutually non-diffeomorphic complete embedded minimal hypersurfaces $\\Sigma^{n-1} \\subset  \\mathbb{R}^n$\n with $\\ind(\\Sigma) \\leqslant I$ and $\\vol(\\Sigma\\cap B_R(0)) \\leqslant \\Lambda R^{n-1}$ for all $R > 0$ \\cite[Theorem 1.2]{Chodosh17}.\n\nAntoine Song\\cite{songantoine1} showed that  given a closed embedded minimal hypersurface $\\Sigma^n$ in a closed Riemannian manifold $(M^{n+1},g)$ with $2\\leqslant n \\leqslant 6\\,,$  the sum of dimensions over a fixed field of the cohomology groups  of $\\Sigma$ can be bounded by the area and  Morse index of $\\Sigma$ by a combinatorial argument.\n\nWe get a finiteness result for minimal submanifolds under the conditions of uniformly bound total curvature and Euclidean volume growth. This can be seen as a quantitative generalization of Anderson's Theorem \\cite{anderson1984compactification}.", "context": "Collin \\cite{collin1} proved that any properly embedded minimal\nsurface in $\\mathbb{R}^3$ with finite topology and more than one end, has finite total curvature. \nColding and Minicozzi \\cite{ColdingMincozzi1} removed the proper condition, where they proved that a complete embedded minimal surface $\\Sigma$ with finite topology in $\\mathbb{R}^3$ must be proper.\nMeeks, Perez and Ros \\cite{Meeks2} showed that the number of ends of $\\Sigma$ is bounded by a constant depending on its genus.\n\nGiven an immersed minimal submanifold $M^{n}$ in $ \\mathbb{R}^{n+m}\\,,$ $M$ is said to have \\emph{finite total curvature} if\n $$\\int_{M} \\lvertA\\rvert^{n}d\\mu_M < \\infty\\,,$$\nwhere $A$ denotes the second fundamental form of $M$ in $ \\mathbb{R}^{n+m}$, and $\\mu_M$ denotes the volume element of $M$.\n\nAnderson \\cite{anderson1984compactification} gave a generalization of the\nChern-Osserman theorem \\cite{Chern} on minimal surfaces of finite total curvature: a complete minimal submanifold $M^n$ with finite total curvature is diffeomorphic to a compact $C^{\\infty} $ manifold $\\overline{M}^n$ punctured at a finite number of points $\\{p_i\\}_{1}^{\\ell} \\in \\overline{M}^n$ and the Gauss map $\\gamma:M^n \\to G_{n,m}$ extends to a $C^{n-2}$ map $\\overline \\gamma: \\overline M^n \\to G_{n,m} $ of the compactification(where $G_{n,m}$ denotes the Grassmann manifold of $n$-planes in Euclidean $(n+m)$-space). \nIn particular, $M$ has Euclidean volume growth with ratio bounded by a constant depending on $\\ell$.\nFor complete minimal hypersurfaces in $\\mathbb{R}^{n+1}$ with $3 \\leqslant n \\leqslant 6\\,,$ Tysk \\cite{Tysk} proved that finite index and Euclidean volume growth imply finite total curvature.\n\nChodosh, Ketover, and Maximo \\cite[Theorem 1.1]{Chodosh17} proved that\nfor a fixed closed Riemannian manifold $(M^n,g)(3 \\leqslant n \\leqslant 7)\\,,$ \n there can be at most $ N = N(M,g,\\Lambda,I)$ distinct diffeomorphism types in the set of\n embedded minimal hypersurfaces $\\Sigma \\subset (M,g)$ with $\\ind(\\Sigma) \\leqslant I$ and $\\vol_g(\\Sigma) \\leqslant \\Lambda\\,.$\n In particular, for $n=3\\,$, there is $r_0 = r_0(M,g,\\Lambda,I)$ so that\n any embedded minimal surface $\\Sigma$ in $(M^3,g)$ with $\\ind(\\Sigma) \\leqslant I$ and $\\mathrm{area}_g(\\Sigma) \\leqslant \\Lambda$ has\n $\\genus(\\Sigma) \\leqslant r_0$; for $4 \\leqslant n \\leqslant 7$, there is $N = N(n,I,\\Lambda) \\in \\mathbb{N}$ so that there are at most\n $N$ mutually non-diffeomorphic complete embedded minimal hypersurfaces $\\Sigma^{n-1} \\subset \\mathbb{R}^n$\n with $\\ind(\\Sigma) \\leqslant I$ and $\\vol(\\Sigma\\cap B_R(0)) \\leqslant \\Lambda R^{n-1}$ for all $R > 0$ \\cite[Theorem 1.2]{Chodosh17}.\n\nAntoine Song\\cite{songantoine1} showed that given a closed embedded minimal hypersurface $\\Sigma^n$ in a closed Riemannian manifold $(M^{n+1},g)$ with $2\\leqslant n \\leqslant 6\\,,$ the sum of dimensions over a fixed field of the cohomology groups of $\\Sigma$ can be bounded by the area and Morse index of $\\Sigma$ by a combinatorial argument.\n\nWe get a finiteness result for minimal submanifolds under the conditions of uniformly bound total curvature and Euclidean volume growth. This can be seen as a quantitative generalization of Anderson's Theorem \\cite{anderson1984compactification}.", "full_context": "Collin \\cite{collin1} proved that any properly embedded minimal\nsurface in $\\mathbb{R}^3$ with finite topology and more than one end, has finite total curvature. \nColding and Minicozzi \\cite{ColdingMincozzi1} removed the proper condition, where they proved that a complete embedded minimal surface $\\Sigma$ with finite topology in $\\mathbb{R}^3$ must be proper.\nMeeks, Perez and Ros \\cite{Meeks2} showed that the number of ends of $\\Sigma$ is bounded by a constant depending on its genus.\n\nGiven an immersed minimal submanifold $M^{n}$ in $ \\mathbb{R}^{n+m}\\,,$ $M$ is said to have \\emph{finite total curvature} if\n $$\\int_{M} \\lvertA\\rvert^{n}d\\mu_M < \\infty\\,,$$\nwhere $A$ denotes the second fundamental form of $M$ in $ \\mathbb{R}^{n+m}$, and $\\mu_M$ denotes the volume element of $M$.\n\nAnderson \\cite{anderson1984compactification} gave a generalization of the\nChern-Osserman theorem \\cite{Chern} on minimal surfaces of finite total curvature: a complete minimal submanifold $M^n$ with finite total curvature is diffeomorphic to a compact $C^{\\infty} $ manifold $\\overline{M}^n$ punctured at a finite number of points $\\{p_i\\}_{1}^{\\ell} \\in \\overline{M}^n$ and the Gauss map $\\gamma:M^n \\to G_{n,m}$ extends to a $C^{n-2}$ map $\\overline \\gamma: \\overline M^n \\to G_{n,m} $ of the compactification(where $G_{n,m}$ denotes the Grassmann manifold of $n$-planes in Euclidean $(n+m)$-space). \nIn particular, $M$ has Euclidean volume growth with ratio bounded by a constant depending on $\\ell$.\nFor complete minimal hypersurfaces in $\\mathbb{R}^{n+1}$ with $3 \\leqslant n \\leqslant 6\\,,$ Tysk \\cite{Tysk} proved that finite index and Euclidean volume growth imply finite total curvature.\n\nChodosh, Ketover, and Maximo \\cite[Theorem 1.1]{Chodosh17} proved that\nfor a fixed closed Riemannian manifold $(M^n,g)(3 \\leqslant n \\leqslant 7)\\,,$ \n there can be at most $ N = N(M,g,\\Lambda,I)$ distinct diffeomorphism types in the set of\n embedded minimal hypersurfaces $\\Sigma \\subset (M,g)$ with $\\ind(\\Sigma) \\leqslant I$ and $\\vol_g(\\Sigma) \\leqslant \\Lambda\\,.$\n In particular, for $n=3\\,$, there is $r_0 = r_0(M,g,\\Lambda,I)$ so that\n any embedded minimal surface $\\Sigma$ in $(M^3,g)$ with $\\ind(\\Sigma) \\leqslant I$ and $\\mathrm{area}_g(\\Sigma) \\leqslant \\Lambda$ has\n $\\genus(\\Sigma) \\leqslant r_0$; for $4 \\leqslant n \\leqslant 7$, there is $N = N(n,I,\\Lambda) \\in \\mathbb{N}$ so that there are at most\n $N$ mutually non-diffeomorphic complete embedded minimal hypersurfaces $\\Sigma^{n-1} \\subset \\mathbb{R}^n$\n with $\\ind(\\Sigma) \\leqslant I$ and $\\vol(\\Sigma\\cap B_R(0)) \\leqslant \\Lambda R^{n-1}$ for all $R > 0$ \\cite[Theorem 1.2]{Chodosh17}.\n\nAntoine Song\\cite{songantoine1} showed that given a closed embedded minimal hypersurface $\\Sigma^n$ in a closed Riemannian manifold $(M^{n+1},g)$ with $2\\leqslant n \\leqslant 6\\,,$ the sum of dimensions over a fixed field of the cohomology groups of $\\Sigma$ can be bounded by the area and Morse index of $\\Sigma$ by a combinatorial argument.\n\nWe get a finiteness result for minimal submanifolds under the conditions of uniformly bound total curvature and Euclidean volume growth. This can be seen as a quantitative generalization of Anderson's Theorem \\cite{anderson1984compactification}.\n\nWe get a finiteness result for minimal submanifolds under the conditions of uniformly bound total curvature and Euclidean volume growth. This can be seen as a quantitative generalization of Anderson's Theorem \\cite{anderson1984compactification}.\n\nOur proof is inspired by the ideas in \\cite{Chodosh17}, but we need further research in some situations. For instance, one point of concentration is a plane in Proposition 7.1 of \\cite{Chodosh17}, while in our situation it may be a non-flat minimal submanifold with finite total curvature. In Theorem \\ref{prop.diffeo}, we can resolve it by an induction argument on the total curvature.\n\n\\begin{lemm}\\label{lemm:curv.est}\nFor fixed $I \\in \\ZZ^+,0 <r_0<R_0< \\infty\\,,$ suppose that $M_j( \\iota_j :M_{j} \\to \\RR^{n+m})$ is a sequence of $n$-dimensional complete properly immersed minimal submanifolds with nonempty boundary and $\\iota_j(M_j) \\subset B_{R_0}(0)$ such that $ \\int_{M_j} \\abs{A_{M_j}}^n d\\mu_{M_j} < IK_0 , \\limsup \\limits_{j\\to \\infty}\\nu_j(B_{R_0}(0)\\setminus B_{\\fh{r_0}{2}}(0))< K_0$ and $\\vol(B_R(q)) \\leq \\Lambda R^n$ for any $B_R(q) \\subset B_{R_0}(0)$. Then, after passing to a subsequence, we have:\n\\begin{enumerate}[itemsep=5pt, topsep=5pt]\n \\item There exist $C>0$ and a sequence of smooth blow-up sets $\\cB_{j} \\subset M_{j}$ so that \n\\begin{equation} \\label{eq:seq.cur.esti}\n |A_{M_{j}}|(x)d(\\iota_j(x),\\iota_j(\\cB_{j} \\cup \\de M_j) ) \\leq C\\, , \\ |\\cB_{j}|< I\\, , \\ \\iota_j(\\cB_j) \\subset B_{\\fh{3}{4}r_0}(0)\\, ,\n\\end{equation}\nfor all $x \\in M_{j}\\,.$ \n\\item $\\iota_j(\\cB_j)$ converges to $ \\widetilde \\cB_ \\infty \\subset \\RR^{n+m}$ in the Hausdorff distance sense and the Radon measure $\\nu_j$ converges to $ \\nu_\\infty$ in the Radon measure sense with $\\nu_\\infty(p_\\infty)\\geq 2K_0$ for any $p_\\infty \\in \\widetilde \\cB_\\infty\\,.$\n\\end{enumerate}\n\\end{lemm}\n\\begin{proof}\nFirstly, we assume the conclusion in item(1) holds for the fixed $I\\,,$ and prove the conclusion in item(2) holds for the same fixed $I\\, $.\n Since $\\iota_j(\\cB_j) \\subset B_{\\fh{3}{4}r_0}(0) \\,,$ after passing to a subsequence, we can assume $\\iota_j(\\cB_j)$ converges to $\\widetilde \\cB_\\infty \\subset \\RR^{n+m}$ in the Hausdorff distance sense, i.e., $d_{\\cH}(\\iota_j(\\cB_j),\\widetilde \\cB_\\infty) \\to 0\\, $. Since $\\nu_j(\\RR^{n+m})=\\nu_j(B_{R_0}(0)) = \\int_{M_j} \\abs{A_{M_j}}^n d\\mu_{M_j}< IK_0\\,,$ after passing to a subsequence, we can assume $\\nu_j \\to \\nu_\\infty$ in the Radon measure sense.\n\nChodosh proved removal singularity theorem for embedded minimal hypersurfaces with finite total curvature in Euclidean space in his note \\cite{Chodosh1}. We adapt his method to study the immersed submanifolds of arbitrary codimension. \n\\begin{theo}[Removal singularity]\n\\label{theo.removal.singularity}\n Suppose that $\\iota: M^{n} \\to {B_2(0)}\\setminus\\{0\\} $ is a smooth minimal immersion, i.e.\\,,$$\\int_M \\div_MY d\\mu_{M} = 0\\, , \\text{ for any } Y\\in C^\\infty_c(B_2(0)\\setminus\\{0\\},\\RR^{n+m})\\,.$$ The minimal submanifold $M$ satisfies that $0 \\in \\bar {\\iota(M)}\\,,$ $\\vol(\\iota^{-1}(B_r(0)\\setminus\\{0\\})) \\leq \\Lambda r^n$ for any $0<r<2$ and $\\int_M\\abs{A_M}^n d\\mu_{M} <\\infty\\,.$ \n Then there exists a smooth minimal immersion $\\hat \\iota: \\hat M \\to {B_2(0)}\\, $, i.e., $$\\int_{\\hat M} \\div_MY d\\mu_{M} = 0\\, , \\text{ for any }Y\\in C^\\infty_c(B_2(0),\\RR^{n+m})\\,,$$ satisfying that $\\hat M \\setminus \\hat \\iota^{-1}(0) = M$ and $\\hat \\iota_{|M} =\\iota\\,.$\n\\end{theo}\n\\begin{proof}\n Let $f(x):=|\\iota(x)|\\,.$ We claim that after rescaling $M\\,,$ we can assume that $|\\hat \\iota(x)|$ has no critical points in $B_2(0)$ and $M$ intersects $\\de B_1(0)$ transversely.\n \\begin{lemm} \\label{lemm.no.critital.point}\n There exists $\\delta_1$ small depending on the immersion $\\iota$ such that any critical point of $f$ on $M\\cap \\iota^{-1}(B_{\\delta_1}(0)\\setminus \\{0\\})$ is a local minimum point.\n\nSince $\\hat M_{\\infty}$ has finite total curvature and satisfies the volume growth condition of Theorem \\ref{theo.removal.singularity} by the varifold convergence and monotonicity formula, the possible singularity at $\\{0\\}$ is removable. Hence $\\hat M_{\\infty}$ is an immersed minimal submanifold in $\\RR^{n+m}$ with total curvature $\\int_{\\hat M_{\\infty}} \\abs{A_{\\hat M_{\\infty}}}^n d\\mu_{\\hat M_{\\infty}}\\leq I K_0$ and $\\vol(\\hat M_{\\infty}\\cap B_{r}(0)) \\leq \\Lambda r^{n}$ for any $r>0\\,.$ Moreover, $\\hat M_{\\infty}$ has bounded number of components due to the volume bound. By Theorem \\ref{eqival1}, every component of $\\hat M_{\\infty}$ is regular at infinity. Then we can choose $\\gamma \\geq 1$ large enough so that $\\partial B_{\\gamma}(0)$ intersects each component of $\\hat M_{\\infty}$ transversely, and each component of $\\hat M_{\\infty} \\cap \\partial B_{\\gamma}(0)$ is diffeomorphic to $\\SS^{n-1}$ with the standard smooth structure. Moreover the number of components of $\\hat M_{\\infty} \\cap \\partial B_{\\gamma}(0)$ is no more than $\\Lambda\\,.$ By the choice of $\\delta_{j}\\,$, the curvature estimates \\eqref{eq:prop.cur1} hold for all $x \\in M_{j} \\cap \\left( B_{2}(0) \\setminus B_{\\gamma \\delta_{j}}(\\iota_j(p_{j}))\\right)\\,$. Then by applying Lemma \\ref{lemm.Annular.composition}, we see that $M_{j} \\cap \\left( B_{2}(0) \\setminus B_{\\gamma \\delta_{j}}(\\iota_j(p_{j}))\\right)$ is diffeomorphic to the union of annular regions. In particular, $M_{j}\\cap B_{\\gamma\\delta_{j}}(\\iota_j(p_{j}))$ must be connected (because we have assumed that $M_{j}$ is connected in \\hyperlink{defi:beth}{$(\\beth)$}). Then we only need to prove $\\hat M_j \\cap B_\\gamma(0)$ are diffeomorphic to each other after passing to a subsequence.\n\n\\begin{proof}[Proof of Theorem \\ref{theo.finite.top}]\n We will prove Theorem \\ref{theo.finite.top} by contradiction. Since the volume bound and monotonicity formula imply that there exist at most finite number of components for any minimal submanifold satisfying the assumption of Theorem \\ref{theo.finite.top}, without loss of generality, we can assume the minimal submanifolds satisfying the assumption of Theorem \\ref{theo.finite.top} are connected. If $M_j^n$ is a sequence of pairwise non-diffeomorphic complete connected, immersed minimal submanifold in $\\RR^{n+m}$ with $\\vol(M_j\\cap B_{R}(0)) \\leq \\Lambda R^{n}$ for any $R>0$ and $$\\int_{M_j} \\abs{A_{M_j}}^n d \\mu_{M_j}\\leq \\Gamma < IK_0\\,\\,.$$ By rescaling $M_j,$ we can assume $$\\int_{M_j\\setminus B_{\\fh{1}{j}}(0)}\\abs{A_{M_j}}^n d \\mu_{M_j} <\\fh{1}{j}\\,,$$ and $M_j $ intersects $\\de B_1(0)$ transversely. By \n Theorem \\ref{eqival1},\n $M_j$ is properly immersed and regular at infinity. By rescaling $M_j\\,,$ we can assume $M_j \\setminus B_{\\fh1 2}(0) $ is the union of minimal graph and each minimal graph is defined over the exterior of a bounded region in an $n$-plane passing $0 \\in \\RR^{n+m}\\,.$ So after passing to a subsequence, we can assume the $M_j\\cap B_1(0) $ are pairwise non-diffeomorphic. By Lemma \\ref{lemm:curv.est}, the sequence $M_j\\cap B_2(0)$ satisfies \\hyperlink{defi:beth}{$(\\beth)$}. Then by Theorem \\ref{prop.diffeo}, after passing to a subsequence, all of the $M_j\\cap B_1(0)$ are diffeomorphic. This is a contradiction.\n \\end{proof}", "post_theorem_intro_text_len": 1251, "post_theorem_intro_text": "Our proof is inspired by the ideas in \\cite{Chodosh17}, but we need further research in some situations. For instance, one point of concentration is a plane in  Proposition 7.1 of \\cite{Chodosh17}, while in our situation it may be a non-flat minimal submanifold with finite total curvature. In Theorem \\ref{prop.diffeo}, we can resolve it by an induction argument on the total curvature.\n\n\\subsection{Outline of the paper} \n\\begin{itemize}\n    \\item In \\S \\ref{sec.pre}, we state several definitions\n and curvature estimates for minimal submanifolds which are needed in the following. \n \\item  In \\S \\ref{sec.geo}, we describe the geometry of ends of complete immersed minimal submanifolds in $\\mathbb{R}^{n+m}$ with finite total curvature, enlightened by \\cite{schocen} and \\cite{anderson1984compactification}. \nThis helps us to derive  curvature estimates away from finitely many points in Lemma \\ref{lemm:curv.est}.\n \\item In \\S \\ref{sec.top}, we prove a key topological result in Lemma \\ref{lemm.Annular.composition} allowing us to control the topology of the “intermediate regions”,\nthen combined the curvature estimates in \\S \\ref{sec.geo} we can prove Theorem \\ref{theo.finite.top} by an induction argument on the total curvature.\n\\end{itemize}", "sketch": "The proof is “inspired by the ideas in \\cite{Chodosh17},” but differs because “in our situation [a concentration point] may be a non-flat minimal submanifold with finite total curvature” (rather than a plane as in Proposition 7.1 of \\cite{Chodosh17}). This issue is resolved “by an induction argument on the total curvature” (cf. Theorem \\ref{prop.diffeo}).\n\nThe stated structure to prove Theorem~\\ref{theo.finite.top} is:\n\\begin{itemize}\n\\item Develop “definitions and curvature estimates for minimal submanifolds” in \\S\\ref{sec.pre}.\n\\item In \\S\\ref{sec.geo}, “describe the geometry of ends of complete immersed minimal submanifolds in $\\mathbb{R}^{n+m}$ with finite total curvature,” which “helps us to derive curvature estimates away from finitely many points in Lemma~\\ref{lemm:curv.est}.”\n\\item In \\S\\ref{sec.top}, prove “a key topological result in Lemma~\\ref{lemm.Annular.composition} allowing us to control the topology of the \\lq intermediate regions\\rq,” and then, “combined [with] the curvature estimates in \\S\\ref{sec.geo} we can prove Theorem~\\ref{theo.finite.top} by an induction argument on the total curvature.”\n\\end{itemize}", "expanded_sketch": "The proof is “inspired by the ideas in \\cite{Chodosh17},” but differs because “in our situation [a concentration point] may be a non-flat minimal submanifold with finite total curvature” (rather than a plane as in Proposition 7.1 of \\cite{Chodosh17}). This issue is resolved “by an induction argument on the total curvature,” using the following theorem:\n\n\\begin{theo} \\label{prop.diffeo}\nGiven a sequence $M_{j}$ satisfying \\hyperlink{defi:beth}{$(\\beth)$} and each $M_j$  intersects $\\partial B_{1}(0)$ transversely. By passing to a subsequence, all of the $M_{j}\\cap B_{1}(0)$ are diffeomorphic. \n\\end{theo}\n\nThe stated structure to prove the main theorem is:\n\\begin{itemize}\n\\item Develop “definitions and curvature estimates for minimal submanifolds” next.\n\\item After that, “describe the geometry of ends of complete immersed minimal submanifolds in $\\mathbb{R}^{n+m}$ with finite total curvature,” which “helps us to derive curvature estimates away from finitely many points” in the following lemma.\n\n\\begin{lemm}\\label{lemm:curv.est}\nFor fixed $I \\in \\ZZ^+,0 <r_0<R_0< \\infty\\,,$ suppose that $M_j( \\iota_j :M_{j} \\to  \\RR^{n+m})$ is a sequence of $n$-dimensional complete properly immersed minimal submanifolds with nonempty boundary and $\\iota_j(M_j) \\subset B_{R_0}(0)$ such that $ \\int_{M_j} \\abs{A_{M_j}}^n d\\mu_{M_j} < IK_0 , \\limsup \\limits_{j\\to \\infty}\\nu_j(B_{R_0}(0)\\setminus B_{\\fh{r_0}{2}}(0))< K_0$ and $\\vol(B_R(q)) \\leq \\Lambda R^n$ for any $B_R(q) \\subset B_{R_0}(0)$. Then, after passing to a subsequence, we have:\n\\begin{enumerate}[itemsep=5pt, topsep=5pt]\n    \\item There exist $C>0$ and a sequence of smooth blow-up sets $\\cB_{j} \\subset M_{j}$  so that \n\\begin{equation} \\label{eq:seq.cur.esti}\n    |A_{M_{j}}|(x)d(\\iota_j(x),\\iota_j(\\cB_{j} \\cup \\de M_j) ) \\leq C\\, , \\ |\\cB_{j}|< I\\, , \\ \\iota_j(\\cB_j) \\subset B_{\\fh{3}{4}r_0}(0)\\, ,\n\\end{equation}\nfor all $x \\in M_{j}\\,.$ \n\\item  $\\iota_j(\\cB_j)$ converges to $ \\widetilde  \\cB_ \\infty \\subset \\RR^{n+m}$ in the Hausdorff distance sense and  the Radon measure $\\nu_j$ converges to $ \\nu_\\infty$ in the Radon measure sense with $\\nu_\\infty(p_\\infty)\\geq 2K_0$ for any $p_\\infty \\in \\widetilde \\cB_\\infty\\,.$\n\\end{enumerate}\n\\end{lemm}\n\n\\item Next, prove “a key topological result” in the following lemma, allowing control of the topology of the \\lq intermediate regions\\rq.\n\n\\begin{lemm}[Annular decomposition] \\label{lemm.Annular.composition}\nFor fixed $n\\,,m\\in \\ZZ^+\\,,n\\geq 2\\,,$ there is a $0 < \\sigma_0 <\\fh{1}{2} $ only depending on $n,m$ with the following property. Suppose that $M^n(\\iota : M^n \\to \\bar{B_2(0)} \\subset\\RR^{n+m})$ is a complete properly immersed submanifold  with $\\iota(\\de M)  \\subset \\de B_2(0)\\, . $ Assume that for some $\\sigma \\leq \\sigma_0$ and $p \\in B_{\\sigma_0}(0)\\,,$ we have:\n\\begin{enumerate}[itemsep=5pt, topsep=5pt]\n\\item For each component $M'$ of $M\\,,$ $M' \\cap B_{\\sigma}(p) \\neq \\emptyset \\,$.\n\\item The immersed submanifold $M$ intersects $\\de B_{\\sigma}(p)$ transversely, and $M \\cap \\de B_{\\sigma}(p)$ has $k$ components. Moreover, each component of   $M \\cap \\de B_{\\sigma}(p)$ is diffeomorphic to $\\SS^{n-1}$ with the standard smooth structure.\n\\item The second fundamental form of $M$ satisfies $|A|(x)|\\iota(x)-p| \\leq \\fh{1}{4} $ for all $x \\in M \\cap \\left( \\overline{B_{1}(0)} \\setminus B_{\\sigma}(p) \\right)\\,.$\n\\end{enumerate}\nThen, $M$ intersects $\\de B_{1}(0)$ transversely. Both  $M \\cap \\left(\\overline{B_{1}(0)}\\setminus {B_{\\sigma}(p)}\\right)$ and  $M \\cap \\de B_{1}(0)$  have $k$ components. Moreover, each component of   $M \\cap \\de B_{1}(0)$ is diffeomorphic to $\\SS^{n-1}$ with the standard smooth structure and\n each component of $M \\cap \\left(\\overline{B_{1}(0)}\\setminus {B_{\\sigma}(p)}\\right)$ is diffeomorphic to $\\SS^{n-1}\\times [0,1]$ with the standard smooth structure.\n\\end{lemm}\n\nCombined with the curvature estimates proved earlier, this yields an induction argument on the total curvature, completing the proof of the main theorem.\n\\end{itemize}", "expanded_theorem": "\\label{theo.finite.top}\nFor fixed $n\\,,m \\in \\mathbb{Z}^+\\,,n \\geqslant 3\\,, m \\geq1\\,,$ and $ \\Gamma \\, ,\\Lambda \\in \\mathbb{R} \\, ,  \\Gamma\\,,\\Lambda \\geqslant 0\\,$, there exists $N = N(n,m,\\Gamma,\\Lambda) \\in \\mathbb{N}$ so that there are at most $N$ mutually non-diffeomorphic complete immersed minimal submanifolds $M^n$ in $\\mathbb{R}^{n+m}$ satisfying that  $\\int_{M} \\lvertA\\rvert^{n}d\\mu_M  \\leqslant \\Gamma$ and \n$\\vol_M ( B_R(0)) \\leqslant  \\Lambda R^n$ for any $R>0\\,.$", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let n,m\\in\\mathbb{Z}^+ with n\\ge 3 and m\\ge 1, and let \\Gamma,\\Lambda\\ge 0. For an immersed minimal submanifold M^n\\subset \\mathbb{R}^{n+m}, let A denote its second fundamental form, let d\\mu_M be its induced volume element, and let \\operatorname{vol}_M(B_R(0)) denote the n-dimensional volume of M\\cap B_R(0), where B_R(0) is the Euclidean ball of radius R centered at the origin in \\mathbb{R}^{n+m}. Which quantitative finiteness statement holds for the class of complete immersed minimal submanifolds M^n in \\mathbb{R}^{n+m} satisfying\n\\[\n\\int_M |A|^n\\,d\\mu_M\\le \\Gamma\n\\quad\\text{and}\\quad\n\\operatorname{vol}_M(B_R(0))\\le \\Lambda R^n\\ \\text{for every }R>0?\n\\]", "correct_choice": {"label": "A", "text": "There exists a number N=N(n,m,\\Gamma,\\Lambda)\\in\\mathbb{N} such that among all complete immersed minimal submanifolds M^n\\subset \\mathbb{R}^{n+m} satisfying\n\\(\\int_M |A|^n\\,d\\mu_M\\le \\Gamma\\) and \\(\\operatorname{vol}_M(B_R(0))\\le \\Lambda R^n\\) for every \\(R>0\\), there are at most N pairwise non-diffeomorphic examples."}, "choices": [{"label": "B", "text": "There exists a number $N=N(n,m,\\Gamma)\\in\\mathbb{N}$ such that among all complete immersed minimal submanifolds $M^n\\subset \\mathbb{R}^{n+m}$ satisfying\n\\(\\int_M |A|^n\\,d\\mu_M\\le \\Gamma\\) and \\(\\operatorname{vol}_M(B_R(0))\\le \\Lambda R^n\\) for every \\(R>0\\), there are at most $N$ pairwise non-diffeomorphic examples."}, {"label": "C", "text": "There are only finitely many pairwise non-diffeomorphic complete immersed minimal submanifolds $M^n\\subset \\mathbb{R}^{n+m}$ satisfying\n\\(\\int_M |A|^n\\,d\\mu_M\\le \\Gamma\\) and \\(\\operatorname{vol}_M(B_R(0))\\le \\Lambda R^n\\) for every \\(R>0\\)."}, {"label": "D", "text": "There exists a number $N=N(n,m,\\Gamma,\\Lambda)\\in\\mathbb{N}$ such that among all complete immersed minimal submanifolds $M^n\\subset \\mathbb{R}^{n+m}$ satisfying\n\\(\\int_M |A|^n\\,d\\mu_M\\le \\Gamma\\) and \\(\\operatorname{vol}_M(B_R(0))\\le \\Lambda R^n\\) for every \\(R>0\\), there are at most $N$ pairwise non-homeomorphic examples."}, {"label": "E", "text": "There exists a number $N=N(n,m,\\Gamma,\\Lambda)\\in\\mathbb{N}$ such that every complete immersed minimal submanifold $M^n\\subset \\mathbb{R}^{n+m}$ satisfying\n\\(\\int_M |A|^n\\,d\\mu_M\\le \\Gamma\\) and \\(\\operatorname{vol}_M(B_R(0))\\le \\Lambda R^n\\) for every \\(R>0\\) has at most $N$ ends."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "dependence_of_finiteness_constant_on_volume_growth", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "explicit_parameter_dependence_of_the_bound_N(n,m,\\Gamma,\\Lambda)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "topological_equivalence_type_diffeomorphic_vs_homeomorphic", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "global_finiteness_of_diffeomorphism_types_replaced_by_uniform_end_count", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem states only the hypotheses and asks which finiteness conclusion is valid; it does not explicitly reveal the correct conclusion. Although the phrase 'quantitative finiteness statement' hints that an existence-of-N type answer is likely, it does not single out choice A."}, "TAS": {"score": 1, "justification": "This is close to theorem-recall: the correct option is essentially the theorem's conclusion under the stated hypotheses. However, it is not a pure restatement because the choices introduce meaningful variants in parameter dependence, topological category, and conclusion strength."}, "GPS": {"score": 1, "justification": "The item requires some reasoning or theorem-level discrimination: one must distinguish the exact dependence on parameters, the difference between diffeomorphic and homeomorphic, and stronger versus weaker finiteness claims. Still, it mainly tests recognition of the precise statement rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and well targeted. B tests incorrect parameter dependence, C is a weaker true statement, D swaps topological equivalence notions, and E replaces classification finiteness with a bound on ends. These reflect realistic mathematical confusions."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and little answer leakage, but it leans toward precise recall of a known result rather than deeper generative reasoning."}}
{"id": "2602.12657v1", "paper_link": "http://arxiv.org/abs/2602.12657v1", "theorems_cnt": 1, "theorem": {"env_name": "thm", "content": "[Quantitative stability for parabolic equations]\\label{thm:parabolic}\n\t\tLet $T>0$ and $\\Omega$ be a bounded domain in ${\\mathbb R}^n$. For $\\varepsilon\\geq 0$, assume that $A_\\varepsilon: {\\mathbb R}^n_0\\to \\S_+^n$ and ${H_\\varepsilon: \\Omega_T\\times {\\mathbb R}^n\\to {\\mathbb R}}$ are continuous functions satisfying (A1)(A2) and (C1)(C2). Let $g_\\varepsilon\\in C(\\partial_p \\Om_T)$ and $u_\\varepsilon\\in C(\\overline{\\Omega}\\times [0, T))$ be bounded ${\\mathcal F}$-solutions of  \\eqref{general parabolic approx eq}\\eqref{parabolic approx bdry-cond}. Assume in addition that $u_\\varepsilon$ are equi-H\\\"older continuous in $\\overline{\\Om_T}$, that is, there exist $L>0$ and $\\theta\\in (0, 1]$ such that,   \n\t\t\\begin{equation}\\label{equi-holder}\n\t\t\t|u_\\varepsilon(x_1, t_1)-u_\\varepsilon(x_2, t_2)|\\leq L(|x_1-x_2|^\\theta +|t_1-t_2|^\\theta) \n\t\t\\end{equation}\n\t\tfor all $(x_1, t_1), (x_2, t_2)\\in \\overline{\\Omega}\\times [0, T)$ and $\\varepsilon\\geq 0$ small. \n\t\tThen, for all $\\varepsilon>0$ small, \n\t\t\\begin{equation}\\label{error-est parabolic}\n\t\t\t\\sup_{\\overline{\\Omega}\\times [0, T)}|u_\\varepsilon-u_0|\\leq \\sup_{\\partial_p \\Om_T}|g_\\varepsilon-g_0|+C\\varepsilon^{\\nu}  +c_H T\\varepsilon^\\gamma,\n\t\t\\end{equation}\n\t\tholds with \n\t\t\\begin{equation}\\label{nu-exponent}\n\t\t\t\\nu=\\frac{\\alpha\\theta}{1+(1-\\theta)\\max\\{\\beta, 0\\}},    \n\t\t\\end{equation}\n\t\twhere $C>0$ depends on $T, n, k, L_A, L_H, \\beta, c_A$ as well as the H\\\"older exponent $\\theta$ and the constant $L$. Here, $k>2$, $L_A\\geq0$, $L_H\\geq0$ appear in (A1)(A2), and $\\alpha, \\beta, \\gamma, c_A, c_H\\in {\\mathbb R}$ are given in (C1)(C2).", "start_pos": 15948, "end_pos": 17372, "label": "thm:parabolic"}, "ref_dict": {"nu-exponent": "\\begin{equation}\\label{nu-exponent}\n\t\t\t\\nu=\\frac{\\alpha\\theta}{1+(1-\\theta)\\max\\{\\beta, 0\\}},    \n\t\t\\end{equation}", "np-parabolic": "\\begin{equation}\\label{np-parabolic}\n\t\\partial_t u-\\Delta^N_p u+H(x, u, \\nabla u)=0 \\quad \\text{in $\\Om_T:=\\Omega\\times (0, T)$}\n\\end{equation}", "error-est parabolic": "\\begin{equation}\\label{error-est parabolic}\n\t\t\t\\sup_{\\Oba\\times [0, T)}|u_\\vep-u_0|\\leq \\sup_{\\partial_p \\Om_T}|g_\\vep-g_0|+C\\vep^{\\nu}  +c_H T\\vep^\\gamma,\n\t\t\\end{equation}", "thm:parabolic": "\\begin{thm}[Quantitative stability for parabolic equations]\\label{thm:parabolic}\n\t\tLet $T>0$ and $\\Omega$ be a bounded domain in $\\R^n$. For $\\vep\\geq 0$, assume that $A_\\vep: \\R^n_0\\to \\S_+^n$ and ${H_\\vep: \\Omega_T\\times \\R^n\\to \\R}$ are continuous functions satisfying (A1)(A2) and (C1)(C2). Let $g_\\vep\\in C(\\partial_p \\Om_T)$ and $u_\\vep\\in C(\\Oba\\times [0, T))$ be bounded $\\F$-solutions of  \\eqref{general parabolic approx eq}\\eqref{parabolic approx bdry-cond}. Assume in addition that $u_\\vep$ are equi-H\\\"older continuous in $\\ol{\\Om_T}$, that is, there exist $L>0$ and $\\theta\\in (0, 1]$ such that,   \n\t\t\\begin{equation}\\label{equi-holder}\n\t\t\t|u_\\vep(x_1, t_1)-u_\\vep(x_2, t_2)|\\leq L(|x_1-x_2|^\\theta +|t_1-t_2|^\\theta) \n\t\t\\end{equation}\n\t\tfor all $(x_1, t_1), (x_2, t_2)\\in \\Oba\\times [0, T)$ and $\\vep\\geq 0$ small. \n\t\tThen, for all $\\vep>0$ small, \n\t\t\\begin{equation}\\label{error-est parabolic}\n\t\t\t\\sup_{\\Oba\\times [0, T)}|u_\\vep-u_0|\\leq \\sup_{\\partial_p \\Om_T}|g_\\vep-g_0|+C\\vep^{\\nu}  +c_H T\\vep^\\gamma,\n\t\t\\end{equation}\n\t\tholds with \n\t\t\\begin{equation}\\label{nu-exponent}\n\t\t\t\\nu=\\frac{\\alpha\\theta}{1+(1-\\theta)\\max\\{\\beta, 0\\}},    \n\t\t\\end{equation}\n\t\twhere $C>0$ depends on $T, n, k, L_A, L_H, \\beta, c_A$ as well as the H\\\"older exponent $\\theta$ and the constant $L$. Here, $k>2$, $L_A\\geq0$, $L_H\\geq0$ appear in (A1)(A2), and $\\alpha, \\beta, \\gamma, c_A, c_H\\in \\R$ are given in (C1)(C2). \n\t\\end{thm}", "regularized eq": "\\begin{equation}\\label{regularized eq}\n\t\t\\partial_t u-\\left(|\\nabla u|^2+\\vep^2\\right)^{\\frac{p'-p}{2}}\\div\\left(\\left(|\\nabla u|^2+\\vep^2\\right)^{\\frac{p-2}{2}}\\nabla u\\right)=0 \\quad \\text{in $\\Omega_T$,}\n\t\\end{equation}", "eq:c1": "\\begin{equation}\\label{eq:c1}\n\t\t\t\\|A_\\vep(\\xi)^{1/2} -A_0(\\xi)^{1/2}\\|\\leq c_A\\vep^{\\alpha} (1+|\\xi|^\\beta)\n\t\t\\end{equation}", "rmk:periodic": "\\begin{rmk}\\label{rmk:periodic}\n\t\tOur discussions on the Cauchy-Dirichlet problem in Theorem~\\ref{thm:parabolic} and Theorem~\\ref{thm:parabolic2} extend readily to the case of spatially periodic settings. \n\t\tInstead of considering \\eqref{general parabolic lim eq} in a bounded domain $\\Omega$, we may obtain the same stability results as in Theorem~\\ref{thm:parabolic} and Theorem~\\ref{thm:parabolic2} for the whole space $\\R^n$, provided that the initial value $g$ and $H$ are both periodic with respect to $x$. Note that the periodic setting essentially plays the same role as a bounded domain (without boundary) in our comparison arguments and therefore does not substantially affect the proofs. For clarity of presentation, here we choose not to repeat the statements for this case.\n\t\\end{rmk}", "equi-holder": "\\begin{equation}\\label{equi-holder}\n\t\t\t|u_\\vep(x_1, t_1)-u_\\vep(x_2, t_2)|\\leq L(|x_1-x_2|^\\theta +|t_1-t_2|^\\theta) \n\t\t\\end{equation}", "np-parabolic0": "\\begin{equation}\\label{np-parabolic0}\n\t\t\\partial_t u-\\Delta^N_p u=0 \\quad \\text{in $\\Om_T:=\\Omega\\times (0, T)$,}\n\t\\end{equation}", "general parabolic approx eq": "\\begin{equation}\\label{general parabolic approx eq}\n\t\t\\partial_t u-\\tr(A_\\vep(\\nabla u)\\nabla^2 u)+H_\\vep(x, t, \\nabla u)=0 \\quad \\text{in $\\Om_T$}\n\t\\end{equation}", "parabolic bdry-cond": "\\begin{equation}\\label{parabolic bdry-cond}\n\t\tu=g \\quad \\text{on $\\partial_p \\Om_T$}\n\t\\end{equation}", "vp-parabolic0": "\\begin{equation}\\label{vp-parabolic0}\n\t\t\\partial_t u-\\Delta_p u=0 \\quad \\text{in $\\Om_T$,}\n\t\\end{equation}", "thm:elliptic": "\\begin{thm}[Quantitative stability for elliptic equations]\\label{thm:elliptic}\n\t\tLet $\\Omega$ be a bounded domain in $\\R^n$. For $\\vep\\geq 0$, assume that $A_\\vep: \\R^n_0\\to \\S_+^n$ and $H_\\vep: \\Omega\\times \\R\\times  \\R^n\\to \\R$ are continuous functions satisfying (A1)(A2')(A3) and (C1)(C2'). Let $g_\\vep\\in C(\\partial\\Omega)$ and $u_\\vep\\in C(\\Oba)$ be $\\F$-solutions of \\eqref{general approx eq}\\eqref{approx bdry-cond}. Assume in addition that $u_\\vep$ are equi-H\\\"older continuous in $\\Oba$, that is, there exist $L>0$ and $0<\\theta\\leq 1$ such that  \t\t\\[\n\t\t|u_\\vep(x_1)-u_\\vep(x_2)|\\leq L|x_1-x_2|^\\theta \n\t\t\\]\n\t\tfor all $x_1, x_2\\in \\Oba$ and $\\vep\\geq 0$.\n\t\tThen, for all $\\vep>0$ small,  \n\t\t\\begin{equation}\\label{error-est}\n\t\t\t\\max_{\\Oba} |u_\\vep-u_0|\\leq \\max_{\\partial \\Omega} |g_\\vep-g|+\\frac{1}{\\lambda}\\left(C_0\\vep^{\\nu}+c_H \\vep^\\gamma \\right)\n\t\t\\end{equation}\n\t\tholds with $\\nu$ given as in \\eqref{nu-exponent}, where $C_0>0$ is a constant depending on \n\t\t$k>2$,$L_A>0$, $L_H>0$, $\\lambda>0$ in (A1)(A2')(A3) and $\\alpha, \\beta, c_A, \\in \\R$ given in (C1). The constants $\\gamma, c_H>0$ are given as in (C2'). \n\t\\end{thm}", "parabolic approx bdry-cond": "\\begin{equation}\\label{parabolic approx bdry-cond}\n\t\tu =g_\\vep \\quad \\text{on $\\partial_p \\Om_T$}\n\t\\end{equation}", "mono-0": "\\begin{equation}\\label{mono-0}\n\t\t\\rho\\mapsto H_\\vep(x, \\rho, \\xi)-\\lambda \\rho \\quad \\text{is nondecreasing in $\\R$. }\n\t\\end{equation}"}, "pre_theorem_intro_text_len": 10828, "pre_theorem_intro_text": "\\subsection{Motivation}\n\tIn this paper we study the stability of a class of quasilinear parabolic partial differential equations, which takes the form \n\t\\begin{equation}\\label{general parabolic lim eq}\n\t\t\\partial_t u-\\operatorname{tr}(A(\\nabla u)\\nabla^2 u)+H(x, t, \\nabla u)=0 \\quad \\text{in $\\Omega\\times (0, T)$,}\n\t\\end{equation}\n\twhere $\\Omega$ is a bounded domain in ${\\mathbb R}^n$, $T>0$ is fixed, and \n\t\\[\n\tA: {\\mathbb R}^n\\setminus \\{0\\}\\to {\\mathcal S}^n_+, \\quad H: \\Omega\\times (0, T)\\times {\\mathbb R}^n\\to {\\mathbb R},\n\t\\]\n\tare given functions satisfying appropriate assumptions to be introduced later. Here, ${\\mathcal S}^n_+$ represents the set of all nonnegative symmetric $n\\times n$ matrices. A key feature of this general setting is that the elliptic operator may exhibit singularity at vanishing gradients. \n\n\tOne typical example of applicable equations we have in mind is the following parabolic normalized $p$-Laplace type equation (with $1<p<\\infty$)\n\t\\begin{equation}\\label{np-parabolic0}\n\t\t\\partial_t u-\\Delta^N_p u=0 \\quad \\text{in $\\Om_T:=\\Omega\\times (0, T)$,}\n\t\\end{equation}  \n\twhere\n\t\\[\n\t\\Delta^N_p u:= |\\nabla u|^{2-p}\\operatorname{div}(|\\nabla u|^{p-2}\\nabla u).\n\t\\]\n\tAnother example is the parabolic equation with the variational $p$-Laplacian:\n\t\\begin{equation}\\label{vp-parabolic0}\n\t\t\\partial_t u-\\Delta_p u=0 \\quad \\text{in $\\Om_T$,}\n\t\\end{equation}\n\twhere \n\t\\[\n\t\\Delta_p u:= \\operatorname{div}(|\\nabla u|^{p-2}\\nabla u). \n\t\\]\n\tWe consider the associated Cauchy-Dirichlet boundary value problem for these parabolic equations. The parabolic boundary condition is given by \n\t\\begin{equation}\\label{parabolic bdry-cond}\n\t\tu=g \\quad \\text{on $\\partial_p \\Om_T$}\n\t\\end{equation}\n\tfor\n\t\\[\n\t\\partial_p \\Om_T=(\\Omega\\times \\{0\\})\\cup (\\partial \\Omega\\times [0, T)),\n\t\\]\n\twhere $g: \\partial \\Omega\\to {\\mathbb R}$ is assumed to be continuous.\n\n\tOne of the main questions addressed in this paper concerns the stability of $p$-Laplace type equations under perturbations of the parameter $p$. Assume that $H\\equiv 0$. For a fixed parabolic boundary value $g$, it is known that, as $p\\to q$ for some $q\\in(1,\\infty)$, the unique solution $u_p$ of the Cauchy-Dirichlet problem \\eqref{np-parabolic0} or \\eqref{vp-parabolic0} converges uniformly to the solution $u_q$ of the corresponding problem with parameter $q$. In the framework of viscosity solutions, consult \\cite[Theorem~6.1]{OS} and \\cite[Theorem~2.2.1]{Gbook} for such qualitative stability results on the variational and normalized $p$-Laplace equations of parabolic type. See also \\cite{KP, LP} for related analysis with different approaches. However, these works do not provide an explicit rate of convergence. Our goal is therefore to further quantify the uniform convergence \tand to discuss the optimality of the resulting estimates.\n\n\tOn the other hand, when studying the solutions $u$ of singular parabolic equations such as \\eqref{np-parabolic} and \\eqref{vp-parabolic0}, it is common to approximate them with regularized equations in the form \n\t\\begin{equation}\\label{regularized eq}\n\t\t\\partial_t u-\\left(|\\nabla u|^2+\\varepsilon^2\\right)^{\\frac{p'-p}{2}}\\operatorname{div}\\left(\\left(|\\nabla u|^2+\\varepsilon^2\\right)^{\\frac{p-2}{2}}\\nabla u\\right)=0 \\quad \\text{in $\\Omega_T$,}\n\t\\end{equation}\n\twhere $\\varepsilon>0$ is small. Here, we take $p'=2$ and $p'=p$ to obtain regularized approximations of \\eqref{np-parabolic} and \\eqref{vp-parabolic0}, respectively.  Under the comparison principle for the limit singular equations, sending  $\\varepsilon\\to 0$ yields the uniform convergence of the solution $u_\\varepsilon$ of \\eqref{regularized eq} to the solution $u$ of the corresponding singular equation. It is then natural to ask about the rate of this convergence in applications. The case with $p'=2$ and $p=1$ corresponds to the level-set mean curvature flow equation, for which such a regularization was introduced in \\cite{ES}. The convergence rate in this case was essentially established by Deckelnick \\cite{De}, and later proved by Mitake \\cite{Mi} using a different method based on comparison arguments. Addressing convergence rates for the general approximation \\eqref{regularized eq} is another main objective of the present paper.\n\n\tThe method employed in this paper builds on the arguments developed by Mitake in \\cite{Mi}, as well as earlier works \\cite{CGL, JK1}, which established general continuous dependence estimates for viscosity solutions of nonlinear parabolic equations. Further related results include extensions to nonlocal PDEs in \\cite{JK2} and to general Neumann-type boundary value problems in \\cite{JG}. While it may be possible to adapt these methods to study quantitative stability for $p$-Laplace type equations in regimes without vanishing-gradient singularities, precise estimates do not appear to have been explicitly given, to the best of our knowledge. Also, the singular case, such as \\eqref{vp-parabolic0} with $1<p<2$, seems unexplored and constitutes one of the focuses of the present work. \n\n\tA recent paper of Bungert \\cite{Bun} studies the convergence rate of $p$-harmonic functions to infinity-harmonic functions, using partly the variational structure of the $p$-Laplace operator. In contrast, based on the viscosity solution theory, we here provide a purely PDE approach to show the convergence rate for a general class of related parabolic stability problems. Our method develops the techniques in \\cite{CGL, JK1, Mi} and further extends them to treat convergence rates arising from more general stability and regularization problems.\n\n\tAs explained through the examples above, the equations of interest may exhibit singularity at vanishing gradients. To handle such singularities, we employ a generalized notion of viscosity solutions, known as $\\mathcal{F}$-solutions. A brief overview of this approach is given in Section~\\ref{sec:preli}; for a detailed introduction, we again refer the reader to \\cite{OS, Gbook}.\n\n\t\\subsection{Main results}\n\tLet us now introduce more details about our setting. Let $T>0$ and $\\Omega\\subset\\mathbb{R}^n$ be a bounded domain. In the sequel, we set $\\Om_T=\\Omega\\times (0, T)$ and $\\Rn_0={\\mathbb R}^n\\setminus \\{0\\}$. Consider a family of parabolic equations\n\t\\begin{equation}\\label{general parabolic approx eq}\n\t\t\\partial_t u-\\operatorname{tr}(A_\\varepsilon(\\nabla u)\\nabla^2 u)+H_\\varepsilon(x, t, \\nabla u)=0 \\quad \\text{in $\\Om_T$}\n\t\\end{equation}\n\tfor $\\varepsilon\\geq 0$, where $A_\\varepsilon: \\Rn_0\\to {\\mathcal S}^n_+$ and $H_\\varepsilon:\\Omega_T \\times {\\mathbb R}^{n}\\mapsto {\\mathbb R}$  are continuous functions satisfying appropriate assumptions to be introduced below. \n\n\tLet $\\partial_p \\Om_T$ denote the parabolic boundary of $\\Omega\\times (0, T)$,  defined by\n\t\\[\n\t\\partial_p \\Om_T=(\\Omega\\times \\{0\\})\\cup (\\partial \\Omega\\times [0, T)).\n\t\\]\n\tWe impose the initial and boundary condition \t\\begin{equation}\\label{parabolic approx bdry-cond}\n\t\tu =g_\\varepsilon \\quad \\text{on $\\partial_p \\Om_T$}\n\t\\end{equation}\n\tfor \\eqref{general parabolic approx eq}, where $g_\\varepsilon\\in C(\\partial \\Om_T)$ are given for $\\varepsilon\\geq 0$.   \n\n\tSuppose that $u_\\varepsilon$ (with $\\varepsilon\\geq 0$) is a unique viscosity solution of \\eqref{general parabolic approx eq}\\eqref{parabolic approx bdry-cond}. Our main result quantifies the convergence of $u_\\varepsilon\\to u_0$ in terms of $\\varepsilon>0$ under certain assumptions on the convergence rate of $A_\\varepsilon\\to A_0$, $H_\\varepsilon\\to H_0$, and $g_\\varepsilon \\to g_0$ as $\\varepsilon\\to 0$.\n\n\tNow let us state our assumptions on $A_\\varepsilon$ and $H_\\varepsilon$ with $\\varepsilon\\geq 0$. \n\tNote first that for a matrix $M\\in {\\mathcal S}^n_+$, one can find its square root, denoted by $M^{1/2}$, which is also nonnegative. There thus exists the matrix valued function ${A_\\varepsilon}^{1/2}: \\Rn_0\\to {\\mathcal S}^n_+$ for $\\varepsilon\\geq 0$. We impose the following assumption on $A_\\varepsilon$ and $H_\\varepsilon$.\n\t\\begin{enumerate}\n\t\t\\item[(A1)] There exists $k>2$ such that for any $\\varepsilon\\geq 0$ small, $f(x)=|x|^k$ satisfies \n\t\t\\begin{equation}\\label{singular-cancel}\n\t\t\t\\operatorname{tr} \\left(A_\\varepsilon(\\nabla f(x)) \\nabla^2 f(x)\\right)\\to 0 \\quad \\text{as $x\\to 0$.}\n\t\t\\end{equation} \n\t\\end{enumerate}\n\t\\begin{enumerate}\n\t\t\\item[(A2)] $H_\\varepsilon: \\Om_T\\times {\\mathbb R}^n\\to {\\mathbb R}$ is continuous, and there exists $L_H\\geq0$ such that \n\t\t\\[\n\t\t|H_\\varepsilon(x_1, t_1, \\xi)-H_\\varepsilon(x_2, t_2, \\xi)|\\leq L_H(1+|\\xi|)\\left(|x_1-x_2|+\\left|t_1-t_2\\right|\\right)\n\t\t\\]\n\t\tfor all $(x_1, t_1),\\ (x_2, t_2)\\in \\Omega_T$,  $\\xi\\in {\\mathbb R}^n$ and all $\\varepsilon\\geq 0$. \n\n\t\\end{enumerate}\n\tThese assumptions guarantee the existence and uniqueness of ${\\mathcal F}$-solutions to the associated Cauchy-Dirichlet problem. The existence of $k>2$ required in (A1) is related to the possible singularity of the equation at vanishing gradients. A typical example of singular operators we will consider in our later study of the stability is the case of the $p$-Laplacian for $p$ near a fixed $q\\in (1, 2)$, where $A_\\varepsilon$ is taken to be \n\t\\begin{equation}\\label{vp-operator}\n\t\tA_\\varepsilon(\\xi)= |\\xi|^{q+\\varepsilon-2} \\left(I+(q+\\varepsilon-2) \\frac{\\xi\\otimes \\xi}{|\\xi|^2}\\right)\n\t\\end{equation}\n\tfor $\\varepsilon\\geq 0$. In this case, one can choose  $k>q/(q-1)$ for $\\varepsilon\\geq 0$ sufficiently small. \n\n\tFor general $A_\\varepsilon$ and $H_\\varepsilon$, we next impose assumptions on the convergence of $A_\\varepsilon\\to A_0$, $H_\\varepsilon\\to H_0$ as $\\varepsilon\\to 0$: \n\t\\begin{enumerate}\n\t\t\\item[(C1)] There exist $c_A>0$ and $\\alpha>0, \\beta>\\frac{2-k}{2(k-1)}$ such that \n\t\t\\begin{equation}\\label{eq:c1}\n\t\t\t\\|A_\\varepsilon(\\xi)^{1/2} -A_0(\\xi)^{1/2}\\|\\leq c_A\\varepsilon^{\\alpha} (1+|\\xi|^\\beta)\n\t\t\\end{equation}\n\t\tholds for all $\\varepsilon>0$ small and $\\xi\\in \\Rn_0$. Here, $k$ is the constant given in (A1). \n\t\t\\item[(C2)] There exist $c_H>0$ and $\\gamma>0$  such that \n\t\t\\[\n\t\t|H_\\varepsilon(x, t, \\xi)-H_0(x, t, \\xi)|\\leq c_H \\varepsilon^\\gamma \n\t\t\\] \n\t\tholds for all $\\varepsilon>0$ small and $(x, t)\\in \\Om_T,  \\xi\\in {\\mathbb R}^n$.\n\t\\end{enumerate}\n\tNote that by the condition $\\beta>\\frac{2-k}{2(k-1)}$ in (C1), we have $\\beta>-1/2$ for $k>2$. It is worth remarking that one can generalize (C1) by assuming the existence of $\\Sigma_\\varepsilon: {\\mathbb R}^n_0\\to {\\mathbb R}^{m\\times n}$ with $m\\in {\\mathbb N}$ for all $\\varepsilon\\geq 0$ small such that $\\Sigma_\\varepsilon^T\\Sigma_\\varepsilon=A_\\varepsilon$ in ${\\mathbb R}^n_0$ and \\eqref{eq:c1} holds for $\\Sigma_\\varepsilon$ in place of $A_\\varepsilon^{1/2}$. For clarity and simplicity, we restrict our attention to the case $\\Sigma_\\varepsilon=A_\\varepsilon^{1/2}$, which already covers the applications of interest.  \n\n\tLet us present our main theorem.", "context": "\\subsection{Motivation}\n In this paper we study the stability of a class of quasilinear parabolic partial differential equations, which takes the form \n \\begin{equation}\\label{general parabolic lim eq}\n \\partial_t u-\\operatorname{tr}(A(\\nabla u)\\nabla^2 u)+H(x, t, \\nabla u)=0 \\quad \\text{in $\\Omega\\times (0, T)$,}\n \\end{equation}\n where $\\Omega$ is a bounded domain in ${\\mathbb R}^n$, $T>0$ is fixed, and \n \\[\n A: {\\mathbb R}^n\\setminus \\{0\\}\\to {\\mathcal S}^n_+, \\quad H: \\Omega\\times (0, T)\\times {\\mathbb R}^n\\to {\\mathbb R},\n \\]\n are given functions satisfying appropriate assumptions to be introduced later. Here, ${\\mathcal S}^n_+$ represents the set of all nonnegative symmetric $n\\times n$ matrices. A key feature of this general setting is that the elliptic operator may exhibit singularity at vanishing gradients.\n\n\\subsection{Main results}\n Let us now introduce more details about our setting. Let $T>0$ and $\\Omega\\subset\\mathbb{R}^n$ be a bounded domain. In the sequel, we set $\\Om_T=\\Omega\\times (0, T)$ and $\\Rn_0={\\mathbb R}^n\\setminus \\{0\\}$. Consider a family of parabolic equations\n \\begin{equation}\\label{general parabolic approx eq}\n \\partial_t u-\\operatorname{tr}(A_\\varepsilon(\\nabla u)\\nabla^2 u)+H_\\varepsilon(x, t, \\nabla u)=0 \\quad \\text{in $\\Om_T$}\n \\end{equation}\n for $\\varepsilon\\geq 0$, where $A_\\varepsilon: \\Rn_0\\to {\\mathcal S}^n_+$ and $H_\\varepsilon:\\Omega_T \\times {\\mathbb R}^{n}\\mapsto {\\mathbb R}$ are continuous functions satisfying appropriate assumptions to be introduced below.\n\nLet $\\partial_p \\Om_T$ denote the parabolic boundary of $\\Omega\\times (0, T)$, defined by\n \\[\n \\partial_p \\Om_T=(\\Omega\\times \\{0\\})\\cup (\\partial \\Omega\\times [0, T)).\n \\]\n We impose the initial and boundary condition \\begin{equation}\\label{parabolic approx bdry-cond}\n u =g_\\varepsilon \\quad \\text{on $\\partial_p \\Om_T$}\n \\end{equation}\n for \\eqref{general parabolic approx eq}, where $g_\\varepsilon\\in C(\\partial \\Om_T)$ are given for $\\varepsilon\\geq 0$.\n\nNow let us state our assumptions on $A_\\varepsilon$ and $H_\\varepsilon$ with $\\varepsilon\\geq 0$. \n Note first that for a matrix $M\\in {\\mathcal S}^n_+$, one can find its square root, denoted by $M^{1/2}$, which is also nonnegative. There thus exists the matrix valued function ${A_\\varepsilon}^{1/2}: \\Rn_0\\to {\\mathcal S}^n_+$ for $\\varepsilon\\geq 0$. We impose the following assumption on $A_\\varepsilon$ and $H_\\varepsilon$.\n \\begin{enumerate}\n \\item[(A1)] There exists $k>2$ such that for any $\\varepsilon\\geq 0$ small, $f(x)=|x|^k$ satisfies \n \\begin{equation}\\label{singular-cancel}\n \\operatorname{tr} \\left(A_\\varepsilon(\\nabla f(x)) \\nabla^2 f(x)\\right)\\to 0 \\quad \\text{as $x\\to 0$.}\n \\end{equation} \n \\end{enumerate}\n \\begin{enumerate}\n \\item[(A2)] $H_\\varepsilon: \\Om_T\\times {\\mathbb R}^n\\to {\\mathbb R}$ is continuous, and there exists $L_H\\geq0$ such that \n \\[\n |H_\\varepsilon(x_1, t_1, \\xi)-H_\\varepsilon(x_2, t_2, \\xi)|\\leq L_H(1+|\\xi|)\\left(|x_1-x_2|+\\left|t_1-t_2\\right|\\right)\n \\]\n for all $(x_1, t_1),\\ (x_2, t_2)\\in \\Omega_T$, $\\xi\\in {\\mathbb R}^n$ and all $\\varepsilon\\geq 0$.\n\nFor general $A_\\varepsilon$ and $H_\\varepsilon$, we next impose assumptions on the convergence of $A_\\varepsilon\\to A_0$, $H_\\varepsilon\\to H_0$ as $\\varepsilon\\to 0$: \n \\begin{enumerate}\n \\item[(C1)] There exist $c_A>0$ and $\\alpha>0, \\beta>\\frac{2-k}{2(k-1)}$ such that \n \\begin{equation}\\label{eq:c1}\n \\|A_\\varepsilon(\\xi)^{1/2} -A_0(\\xi)^{1/2}\\|\\leq c_A\\varepsilon^{\\alpha} (1+|\\xi|^\\beta)\n \\end{equation}\n holds for all $\\varepsilon>0$ small and $\\xi\\in \\Rn_0$. Here, $k$ is the constant given in (A1). \n \\item[(C2)] There exist $c_H>0$ and $\\gamma>0$ such that \n \\[\n |H_\\varepsilon(x, t, \\xi)-H_0(x, t, \\xi)|\\leq c_H \\varepsilon^\\gamma \n \\] \n holds for all $\\varepsilon>0$ small and $(x, t)\\in \\Om_T, \\xi\\in {\\mathbb R}^n$.\n \\end{enumerate}\n Note that by the condition $\\beta>\\frac{2-k}{2(k-1)}$ in (C1), we have $\\beta>-1/2$ for $k>2$. It is worth remarking that one can generalize (C1) by assuming the existence of $\\Sigma_\\varepsilon: {\\mathbb R}^n_0\\to {\\mathbb R}^{m\\times n}$ with $m\\in {\\mathbb N}$ for all $\\varepsilon\\geq 0$ small such that $\\Sigma_\\varepsilon^T\\Sigma_\\varepsilon=A_\\varepsilon$ in ${\\mathbb R}^n_0$ and \\eqref{eq:c1} holds for $\\Sigma_\\varepsilon$ in place of $A_\\varepsilon^{1/2}$. For clarity and simplicity, we restrict our attention to the case $\\Sigma_\\varepsilon=A_\\varepsilon^{1/2}$, which already covers the applications of interest.\n\nLet us present our main theorem.\n\n\\begin{equation}\\label{eq:c1}\n\t\t\t\\|A_\\vep(\\xi)^{1/2} -A_0(\\xi)^{1/2}\\|\\leq c_A\\vep^{\\alpha} (1+|\\xi|^\\beta)\n\t\t\\end{equation}\n\n\\begin{equation}\\label{general parabolic approx eq}\n\t\t\\partial_t u-\\tr(A_\\vep(\\nabla u)\\nabla^2 u)+H_\\vep(x, t, \\nabla u)=0 \\quad \\text{in $\\Om_T$}\n\t\\end{equation}\n\n\\begin{equation}\\label{parabolic approx bdry-cond}\n\t\tu =g_\\vep \\quad \\text{on $\\partial_p \\Om_T$}\n\t\\end{equation}", "full_context": "\\subsection{Motivation}\n In this paper we study the stability of a class of quasilinear parabolic partial differential equations, which takes the form \n \\begin{equation}\\label{general parabolic lim eq}\n \\partial_t u-\\operatorname{tr}(A(\\nabla u)\\nabla^2 u)+H(x, t, \\nabla u)=0 \\quad \\text{in $\\Omega\\times (0, T)$,}\n \\end{equation}\n where $\\Omega$ is a bounded domain in ${\\mathbb R}^n$, $T>0$ is fixed, and \n \\[\n A: {\\mathbb R}^n\\setminus \\{0\\}\\to {\\mathcal S}^n_+, \\quad H: \\Omega\\times (0, T)\\times {\\mathbb R}^n\\to {\\mathbb R},\n \\]\n are given functions satisfying appropriate assumptions to be introduced later. Here, ${\\mathcal S}^n_+$ represents the set of all nonnegative symmetric $n\\times n$ matrices. A key feature of this general setting is that the elliptic operator may exhibit singularity at vanishing gradients.\n\n\\subsection{Main results}\n Let us now introduce more details about our setting. Let $T>0$ and $\\Omega\\subset\\mathbb{R}^n$ be a bounded domain. In the sequel, we set $\\Om_T=\\Omega\\times (0, T)$ and $\\Rn_0={\\mathbb R}^n\\setminus \\{0\\}$. Consider a family of parabolic equations\n \\begin{equation}\\label{general parabolic approx eq}\n \\partial_t u-\\operatorname{tr}(A_\\varepsilon(\\nabla u)\\nabla^2 u)+H_\\varepsilon(x, t, \\nabla u)=0 \\quad \\text{in $\\Om_T$}\n \\end{equation}\n for $\\varepsilon\\geq 0$, where $A_\\varepsilon: \\Rn_0\\to {\\mathcal S}^n_+$ and $H_\\varepsilon:\\Omega_T \\times {\\mathbb R}^{n}\\mapsto {\\mathbb R}$ are continuous functions satisfying appropriate assumptions to be introduced below.\n\nLet $\\partial_p \\Om_T$ denote the parabolic boundary of $\\Omega\\times (0, T)$, defined by\n \\[\n \\partial_p \\Om_T=(\\Omega\\times \\{0\\})\\cup (\\partial \\Omega\\times [0, T)).\n \\]\n We impose the initial and boundary condition \\begin{equation}\\label{parabolic approx bdry-cond}\n u =g_\\varepsilon \\quad \\text{on $\\partial_p \\Om_T$}\n \\end{equation}\n for \\eqref{general parabolic approx eq}, where $g_\\varepsilon\\in C(\\partial \\Om_T)$ are given for $\\varepsilon\\geq 0$.\n\nNow let us state our assumptions on $A_\\varepsilon$ and $H_\\varepsilon$ with $\\varepsilon\\geq 0$. \n Note first that for a matrix $M\\in {\\mathcal S}^n_+$, one can find its square root, denoted by $M^{1/2}$, which is also nonnegative. There thus exists the matrix valued function ${A_\\varepsilon}^{1/2}: \\Rn_0\\to {\\mathcal S}^n_+$ for $\\varepsilon\\geq 0$. We impose the following assumption on $A_\\varepsilon$ and $H_\\varepsilon$.\n \\begin{enumerate}\n \\item[(A1)] There exists $k>2$ such that for any $\\varepsilon\\geq 0$ small, $f(x)=|x|^k$ satisfies \n \\begin{equation}\\label{singular-cancel}\n \\operatorname{tr} \\left(A_\\varepsilon(\\nabla f(x)) \\nabla^2 f(x)\\right)\\to 0 \\quad \\text{as $x\\to 0$.}\n \\end{equation} \n \\end{enumerate}\n \\begin{enumerate}\n \\item[(A2)] $H_\\varepsilon: \\Om_T\\times {\\mathbb R}^n\\to {\\mathbb R}$ is continuous, and there exists $L_H\\geq0$ such that \n \\[\n |H_\\varepsilon(x_1, t_1, \\xi)-H_\\varepsilon(x_2, t_2, \\xi)|\\leq L_H(1+|\\xi|)\\left(|x_1-x_2|+\\left|t_1-t_2\\right|\\right)\n \\]\n for all $(x_1, t_1),\\ (x_2, t_2)\\in \\Omega_T$, $\\xi\\in {\\mathbb R}^n$ and all $\\varepsilon\\geq 0$.\n\nFor general $A_\\varepsilon$ and $H_\\varepsilon$, we next impose assumptions on the convergence of $A_\\varepsilon\\to A_0$, $H_\\varepsilon\\to H_0$ as $\\varepsilon\\to 0$: \n \\begin{enumerate}\n \\item[(C1)] There exist $c_A>0$ and $\\alpha>0, \\beta>\\frac{2-k}{2(k-1)}$ such that \n \\begin{equation}\\label{eq:c1}\n \\|A_\\varepsilon(\\xi)^{1/2} -A_0(\\xi)^{1/2}\\|\\leq c_A\\varepsilon^{\\alpha} (1+|\\xi|^\\beta)\n \\end{equation}\n holds for all $\\varepsilon>0$ small and $\\xi\\in \\Rn_0$. Here, $k$ is the constant given in (A1). \n \\item[(C2)] There exist $c_H>0$ and $\\gamma>0$ such that \n \\[\n |H_\\varepsilon(x, t, \\xi)-H_0(x, t, \\xi)|\\leq c_H \\varepsilon^\\gamma \n \\] \n holds for all $\\varepsilon>0$ small and $(x, t)\\in \\Om_T, \\xi\\in {\\mathbb R}^n$.\n \\end{enumerate}\n Note that by the condition $\\beta>\\frac{2-k}{2(k-1)}$ in (C1), we have $\\beta>-1/2$ for $k>2$. It is worth remarking that one can generalize (C1) by assuming the existence of $\\Sigma_\\varepsilon: {\\mathbb R}^n_0\\to {\\mathbb R}^{m\\times n}$ with $m\\in {\\mathbb N}$ for all $\\varepsilon\\geq 0$ small such that $\\Sigma_\\varepsilon^T\\Sigma_\\varepsilon=A_\\varepsilon$ in ${\\mathbb R}^n_0$ and \\eqref{eq:c1} holds for $\\Sigma_\\varepsilon$ in place of $A_\\varepsilon^{1/2}$. For clarity and simplicity, we restrict our attention to the case $\\Sigma_\\varepsilon=A_\\varepsilon^{1/2}$, which already covers the applications of interest.\n\nLet us present our main theorem.\n\n\\begin{equation}\\label{eq:c1}\n\t\t\t\\|A_\\vep(\\xi)^{1/2} -A_0(\\xi)^{1/2}\\|\\leq c_A\\vep^{\\alpha} (1+|\\xi|^\\beta)\n\t\t\\end{equation}\n\n\\begin{equation}\\label{general parabolic approx eq}\n\t\t\\partial_t u-\\tr(A_\\vep(\\nabla u)\\nabla^2 u)+H_\\vep(x, t, \\nabla u)=0 \\quad \\text{in $\\Om_T$}\n\t\\end{equation}\n\n\\begin{equation}\\label{parabolic approx bdry-cond}\n\t\tu =g_\\vep \\quad \\text{on $\\partial_p \\Om_T$}\n\t\\end{equation}\n\nLet us present our main theorem.\n\nOur proof of Theorem~\\ref{thm:parabolic} develops the standard comparison argument for possibly degenerate or singular parabolic equations, incorporating the H\\\"older regularity of solutions into the estimates. Without effecting these estimates, this result can be readily adapted to the spatially periodic setting by simply taking $\\Omega$ to be a torus; see Remark~\\ref{rmk:periodic} for more details. Moreover, the assumption of H\\\"older continuity in time in \\eqref{equi-holder} can be removed when vanishing gradients induce only a mild singularity of the equation. Such an improvement follows from the use of the parabolic Crandall–Ishii lemma, which avoids doubling the time variables and thereby simplifies the proof.\n\nOur quantitative stability result for the elliptic equation is as below. For the sake of completeness, we also include a proof, which to a large extent resembles that of Theorem~\\ref{thm:parabolic}. \n \\newpage\n \\begin{thm}[Quantitative stability for elliptic equations]\\label{thm:elliptic}\n Let $\\Omega$ be a bounded domain in $\\R^n$. For $\\vep\\geq 0$, assume that $A_\\vep: \\R^n_0\\to \\S_+^n$ and $H_\\vep: \\Omega\\times \\R\\times \\R^n\\to \\R$ are continuous functions satisfying (A1)(A2')(A3) and (C1)(C2'). Let $g_\\vep\\in C(\\partial\\Omega)$ and $u_\\vep\\in C(\\Oba)$ be $\\F$-solutions of \\eqref{general approx eq}\\eqref{approx bdry-cond}. Assume in addition that $u_\\vep$ are equi-H\\\"older continuous in $\\Oba$, that is, there exist $L>0$ and $0<\\theta\\leq 1$ such that \\[\n |u_\\vep(x_1)-u_\\vep(x_2)|\\leq L|x_1-x_2|^\\theta \n \\]\n for all $x_1, x_2\\in \\Oba$ and $\\vep\\geq 0$.\n Then, for all $\\vep>0$ small, \n \\begin{equation}\\label{error-est}\n \\max_{\\Oba} |u_\\vep-u_0|\\leq \\max_{\\partial \\Omega} |g_\\vep-g|+\\frac{1}{\\lambda}\\left(C_0\\vep^{\\nu}+c_H \\vep^\\gamma \\right)\n \\end{equation}\n holds with $\\nu$ given as in \\eqref{nu-exponent}, where $C_0>0$ is a constant depending on \n $k>2$,$L_A>0$, $L_H>0$, $\\lambda>0$ in (A1)(A2')(A3) and $\\alpha, \\beta, c_A, \\in \\R$ given in (C1). The constants $\\gamma, c_H>0$ are given as in (C2'). \n \\end{thm}\n\n\\begin{thm}\\label{thm:p-lap1par}\n Let $\\Omega\\subset \\R^n$ be a bounded domain and $T>0$. Assume that $H: \\Omega\\times (0,T)\\times \\R^n\\to \\R$ satisfies (A2). Fix $q\\in [1, \\infty)$ and take $p\\in [1, \\infty)$ close to $q$. For any such $p$, let $u_p\\in C(\\Om_T)$ be a solution to \\eqref{np-parabolic} satisfying \\eqref{parabolic bdry-cond} with with boundary data $g=g_p\\in C(\\partial_p \\Om_T)$. Assume in addition that there exist $L>0$ and $\\theta\\in (0, 1]$ such that\n \\[\n |u_p(x_1, t)-u_p(x_2, t)|\\leq L|x_1-x_2|^\\theta \n \\]\n for all $x_1, x_2\\in \\overline{\\Omega}$, $0\\leq t<T$ and all $p$ near $q$. Then, there exists $C>0$ independent of $p$, such that\n \\begin{equation}\\label{rate:p-lap1par}\n \\sup_{\\overline{\\Omega}\\times[0,T)}|u_p-u_q|\\leq \\sup_{\\partial_p \\Om_T}|g_p-g_q| +C|p-q|^{\\theta}. \n \\end{equation}\n\\end{thm}\n\\begin{proof}\n Note that for the normalized $p$-Laplacian, (A1) holds with $k>0$ large for all $q$ near $p$. \n Moreover, by a straightforward computation, we have\n \\[\n A_\\eps(\\xi)^{1/2}=I+(\\sqrt{p-1}-1) \\frac{\\xi\\otimes \\xi}{|\\xi|^{2}}, \\quad A_0(\\xi)^{1/2}=I+(\\sqrt{q-1}-1) \\frac{\\xi\\otimes \\xi}{|\\xi|^{2}}\n \\]\n with $p, q\\in (1, \\infty)$ satisfying $\\vep=|p-q|$. \n This yields the existence of $c>0$ such that\n \\[\n \\norm{A_\\vep(\\xi)^{1/2} -A_0(\\xi)^{1/2}}\\leq |\\sqrt{p-1}-\\sqrt{q-1}|\\leq c|p-q|\n \\]\n for all $q$ near $p$. This amounts to saying that (C1) holds with $\\alpha=1$ and $\\beta=0$. We thus complete the proof of inequality \\eqref{rate:p-lap1par} by applying Theorem~\\ref{thm:parabolic2} with $c_H=0$ and $\\nu=\\theta$. \n\\end{proof}\n\n\\begin{thm}\\label{thm:p-lap2par}\n Let $\\Omega\\subset \\R^n$ be a bounded domain and $T>0$. Assume that $H: \\Omega\\times (0,T)\\times \\R^n\\to \\R$ satisfies (A2). Fix $q\\in (1, \\infty)$ and take $p\\in (1, \\infty)$ close to $q$. For any such $p$, let $u_p\\in C(\\Om_T)$ be a solution to \\eqref{vp-parabolic} satisfying \\eqref{parabolic bdry-cond} with $g=g_p\\in C(\\partial_p \\Om_T)$. If for all $p$ close to $q$, $u_p$ are equi-H\\\"older continuous in $\\overline{\\Om_T}$ with exponent $\\theta\\in (0, 1]$, then the following results hold:\n \\begin{enumerate}\n \\item If $p<2$ and $1<q\\leq 2$, then for any $-1/2<\\beta<p/2-1$, there exists $C>0$ independent of $p$ near $q$ such that\n \\begin{equation}\\label{rate:p-lap2par-new}\n \\sup_{\\overline{\\Omega}\\times[0,T)}|u_p-u_q|\\leq \\sup_{\\partial_p\\Omega_T}|g_p-g_q| +C|p-q|^{\\theta}. \n \\end{equation}\n \\item If $p>2$ and $q\\geq 2$, then for any $\\beta>q/2-1$, there exists $C>0$ independent of $q$ near $p$ such that \n \\begin{equation}\\label{rate:p-lap2par}\n \\sup_{\\overline{\\Omega}\\times[0,T)}|u_p-u_q|\\leq \\sup_{\\partial_p\\Omega_T}|g_p-g_q| +C|p-q|^{\\frac{\\theta}{1+(1-\\theta)\\beta}}. \n \\end{equation}\n \\end{enumerate}\n\\end{thm}\nAs explained in Remark~\\ref{rmk holder}, for the case (2), one may relax the regularity of $u_\\vep$, assuming that they are equi-H\\\"older continuous only in $x$.\n\n\\begin{thm}\\label{thm:pq-lap1par}\n Let $\\Omega\\subset \\R^n$ be a bounded domain and $T>0$. Fix $(q, q')\\in (1, \\infty)^2$ and take $(p, p')$ close to $(q, q')$. For any such $(p, p')$, let $u_{p,p'}\\in C(\\Om_T)$ be a solution to \\eqref{eq:pq-parabolic} satisfying \\eqref{parabolic bdry-cond} with $g=g_{p,p'}\\in C(\\partial_p \\Om_T)$. Assume in addition that for all such $(p, p')$, $u_{p,p'}$ are equi-H\\\"older continuous in $\\overline{\\Om_T}$ with exponent $\\theta\\in (0, 1]$. Let $\\beta>-1/2$ be an arbitrary value satisfying \\eqref{beta-choice}. Then the following results hold:\n \\begin{enumerate}\n \\item For $p'> 2$ and $q'\\geq 2$, there exists $C>0$ independent of $(p, p')$ such that\n \\begin{equation}\\label{rate:pq-lap1par}\n \\sup_{\\ol{\\Omega}\\times[0,T)}|u_{p,p'}-u_{q,q'}|\\leq \\sup_{\\partial_p\\Omega_T}|g_{p,p'}-g_{q,q'}| +C(|p-q|+|p'-q'|)^{\\theta}. \n \\end{equation}\n \\item For $1<p'< 2$ and $1<q'\\leq 2$, there exists $C>0$ independent of $(p, p')$ such that\n \\begin{equation}\\label{rate:pq-lap1par2}\n \\sup_{\\ol{\\Omega}\\times[0,T)}|u_{p,p'}-u_{q,q'}|\\leq \\sup_{\\partial_p\\Omega_T}|g_{p,p'}-g_{q,q'}| +C(|p-q|+|p'-q'|)^{\\frac{\\theta}{1+(1-\\theta)\\beta}}. \n \\end{equation}\n \\item In the special case that $p'= q'\\in (1, \\infty)$, there exists $C>0$ independent of $q$ such that \n \\begin{equation}\\label{rate:pq-lap2par}\n \\sup_{\\ol{\\Omega}\\times[0,T)}|u_{p,p'}-u_{q,p'}|\\leq \\sup_{\\partial_p\\Omega_T}|g_{p,p'}-g_{q,p'}| +C|p-q|^{\\nu}, \n \\end{equation}\n holds with \n \\begin{equation}\\label{nu-pq}\n \\nu=\\frac{2\\theta}{2\\theta+(1-\\theta){q'}}. \n \\end{equation}\n \\end{enumerate}\n\\end{thm}\nIn view of Remark \\ref{rmk holder} and Theorem \\ref{thm:parabolic2}, in the case $p'> 2$, $q'\\geq 2$, we can drop the equi-H\\\"older continuity of $u_{p, p'}$ with respect to the time variable and only keep it for the space variable.\n\nAs for (C2), we have\n\\begin{equation*}\n \\abs{H_\\vep(x, t, \\xi)-H_0(x, t, \\xi)}=\\abs{a\\abs{\\xi} -a\\sqrt{\\abs{\\xi}^2+\\vep_2^2}}\\leq \\abs{a}\\vep_2,\n\\end{equation*}\nso it holds with $\\gamma=1$. We can now use Theorem \\ref{thm:parabolic} to obtain the following result. \n\\begin{thm}\n Let $\\Omega\\subset \\R^n$ be a bounded domain and $T>0$. For $\\eps=(\\vep_1, \\vep_2)$ with $\\vep_1, \\vep_2\\geq0$, let $u_{\\eps}\\in C(\\Om_T)$ be a solution to \\eqref{eq:biasedreg} with boundary value $u_\\vep=g_{\\eps}\\in C(\\partial_p \\Om_T)$. Let $u_0\\in C(\\Om_T)$ be a solution to \\eqref{eq:biased} satisfying $u_0=g_0\\in C(\\partial_p \\Om_T)$. Suppose that $u_{\\eps}$ are equi-H\\\"older continuous in $\\overline{\\Om_T}$ with exponent $\\theta\\in (0, 1]$ for all $\\vep_1, \\vep_2$ small. Then, there exists $C>0$ such that\n \\begin{equation*}\n \\sup_{\\ol{\\Om}\\times[0,T)}|u_{\\eps}-u_0|\\leq \\sup_{\\partial_p\\Omega_T}|g_{\\eps}-g_0| +C\\eps_1^{\\alpha\\theta}+C\\vep_2\n \\end{equation*}\n for any $\\alpha\\in (0, 1/2)$.\n\\end{thm}", "post_theorem_intro_text_len": 7874, "post_theorem_intro_text": "This result applies to important quasilinear parabolic equations with vanishing gradient singularities, for which typically (C1) holds with $\\beta<0$. While the parameter $\\beta<0$ represents the singularity strength, the term $\\max\\{\\beta, 0\\}$ in \\eqref{nu-exponent} indicates that our estimate \\eqref{error-est parabolic} does not capture any influence of this singularity. Moreover, for equi-Lipschitz solutions, corresponding to the case $\\theta=1$, we always have $\\nu=\\alpha$, and the behavior of $A_\\varepsilon(\\xi)$ with respect to the gradient variable $\\xi$ essentially plays no decisive role in determining the stability estimate.\n\n\tOur proof of Theorem~\\ref{thm:parabolic} develops the standard comparison argument for possibly degenerate or singular parabolic equations, incorporating the H\\\"older regularity of solutions into the estimates. Without effecting these estimates, this result can be readily adapted to the spatially periodic setting by simply taking $\\Omega$ to be a torus; see Remark~\\ref{rmk:periodic} for more details. Moreover, the assumption of H\\\"older continuity in time in \\eqref{equi-holder} can be removed when vanishing gradients induce only a mild singularity of the equation. Such an improvement follows from the use of the parabolic Crandall–Ishii lemma, which avoids doubling the time variables and thereby simplifies the proof.\n\n\tWe are also interested in the elliptic variant of such stability problems. Consider now\n\t\\begin{equation}\\label{general approx eq}\n\t\t-\\operatorname{tr}(A_\\varepsilon(\\nabla u)\\nabla^2 u)+H_\\varepsilon(x, u, \\nabla u)=0 \\quad \\text{in $\\Omega$}\n\t\\end{equation}\n\twith the Dirichlet boundary condition \n\t\\begin{equation}\\label{approx bdry-cond}\n\t\tu =g_\\varepsilon \\quad \\text{on $\\partial \\Omega$}\n\t\\end{equation}\n\tfor given $g_\\varepsilon\\in C(\\partial \\Omega)$. \n\tWhile our approach to parabolic problems can be readily extended to the elliptic case under similar assumptions on $A_\\varepsilon$ and $H_\\varepsilon$, we need to additionally assume a monotonicity condition on $H_\\varepsilon$ with respect to $u$ for all $\\varepsilon\\geq 0$. More precisely, we assume that there exists $\\lambda>0$ such that \n\t\\begin{equation}\\label{mono-0}\n\t\t\\rho\\mapsto H_\\varepsilon(x, \\rho, \\xi)-\\lambda \\rho \\quad \\text{is nondecreasing in ${\\mathbb R}$. }\n\t\\end{equation}\n\tWe obtain the same convergence rate as in Theorem \\ref{thm:parabolic}; see Theorem \\ref{thm:elliptic}. However, it is not clear to us how to drop or relax the monotonicity assumption \\eqref{mono-0}. \n\n\t\\subsection{Applications}\n\n\tOur general framework in Theorem~\\ref{thm:parabolic} yields stability estimates for a broad class of quasilinear parabolic equations including the normalized and variational $p$-Laplace equations and their further generalizations, as well as regularizing approximations for degenerate $p$-Laplace equations ($1\\leq p\\leq \\infty$). Below we summarize our major convergence rate results obtained for these equations. See Section~\\ref{sec:app} for precise statements, detailed analysis, concrete examples, and further discussions. \n\n\t\\begin{itemize}\n\t\t\\item (Normalized $p$-Laplacian) Let $u_p$ be a solution of \\eqref{np-parabolic0}\\eqref{parabolic bdry-cond} for $1\\leq p<\\infty$. For any fixed $1\\leq q<\\infty$, if $u_p$ are spatially equi-H\\\"older continuous with exponent $\\theta\\in (0, 1]$ for all $p$ near $q$, then $u_p\\to u_q$ uniformly as $p\\to q$ with convergence rate $O(|p-q|^\\theta)$. \n\t\t\\item (Variational $p$-Laplacian) Let $u_p$ be a solution of \\eqref{vp-parabolic0}\\eqref{parabolic bdry-cond} for $1<p<\\infty$. For any fixed $1\\leq q<\\infty$, if $u_p$ are equi-H\\\"older continuous with exponent $\\theta\\in (0, 1]$ for all $p$ near $q$, then, depending on $p$, $u_p\\to u_q$ uniformly as $p\\to q$ with convergence rate \n\t\t\\begin{enumerate}\n\t\t\t\\item[(i)] $O(|p-q|^\\theta)$ when $p<2$ (and hence $q\\leq 2$);\n\t\t\t\\item[(ii)] $O(|p-q|^\\nu)$ with any \n\t\t\t\\[\n\t\t\t0<\\nu< \\frac{2\\theta}{(1-\\theta)q+2\\theta}\n\t\t\t\\] \n\t\t\twhen $p>2$ (and hence $q\\geq 2$).\n\t\t\\end{enumerate}\n\t\t\\item (Regularization for generalized $p$-Laplacian) Let $p\\geq 1$, $p'\\geq 2$. For all $\\varepsilon>0$ small, let $u_\\varepsilon$ be a solution of \\eqref{regularized eq}\n\t\tand satisfy $u_\\varepsilon=g$ on $\\partial_p \\Omega_T$ for $g\\in C(\\partial_p\\Omega_T)$. As $\\varepsilon\\to 0$, $u_\\varepsilon\\to u$ uniformly, where $u$ is the unique solution of\n\t\t\\begin{equation*}\n\t\t\t\\partial_t u-\\left|\\nabla u\\right|^{p'-p}\\operatorname{div}\\left(\\left|\\nabla u\\right|^{p-2}\\nabla u\\right)=0\n\t\t\\end{equation*}\n\t\tsatisfying the same boundary condition. If $u_\\varepsilon$ is spatially equi-H\\\"older continuous with exponent $\\theta\\in (0, 1]$ for $\\varepsilon>0$ small, then we obtain the rate $O(\\varepsilon^\\nu)$ for the uniform convergence $u_\\varepsilon\\to u$, where $\\nu>0$ can be chosen as follows: \n\t\t\\begin{enumerate}\n\t\t\t\\item[(i)] If $p'=2$, then $0<\\nu< \\theta/2$;\n\t\t\t\\item[(ii)] If $2<p'< 3$, then $\\nu=(p'-2)\\theta$; \n\t\t\t\\item[(iii)] If $3\\leq p'\\leq 4$, then $0<\\nu<\\theta$;\n\t\t\t\\item[(iv)] If $p'>4$, then \n\t\t\t\\[\n\t\t\t0<\\nu< \\frac{\\theta}{1+(1-\\theta)(p'-4)}. \n\t\t\t\\]\n\t\t\\end{enumerate}\n\t\\end{itemize}\n\tNote that the Hölder regularity of solutions assumed in the results above is not the focus of this paper. We refer interested readers to \\cite{Do11, JS17, AP18, IJS19, Lee2025}, among many others, for related regularity estimates about these parabolic equations. \n\n\tIn addition to the applications above, our results can also be applied to study parabolic quantitative stability for general $p$-Laplace type equations that cover both the normalized and variational cases, as well as a regularizing approximation for biased infinity-Laplace equations arising in the stochastic tug-of-war games. Several examples for quasilinear elliptic equations are also discussed in Section~\\ref{sec:app}.  \n\n\tOur main theorem applies to the vanishing viscosity limit for Hamilton-Jacobi equations as well. For $\\varepsilon\\geq 0$ small, consider the solution $u_\\varepsilon$ to the initial boundary value problem for the Hamilton-Jacobi equation\n\t\\[\n\t\\partial_t u -\\varepsilon \\Delta u+H(x, t, \\nabla u)=0 \\quad \\text{in $\\Omega_T$,}\n\t\\]\n\twhere the Hamiltonian $H$ is assumed to satisfy the same regularity assumption as in (A2). Then, for fixed initial and boundary data independent of $\\varepsilon$, Theorem~\\ref{thm:parabolic} immediately yields the convergence rate $\\|u_\\varepsilon-u_0\\|_{L^\\infty(\\Omega_T)}=O(\\varepsilon^{1/2})$ for equi-Lipschitz solutions $u_\\varepsilon$ and the limit solution $u_0$ of the corresponding inviscid Hamilton-Jacobi equation. This recovers the classical result of \\cite{Fl}, originally established using an approach based on differential games, and later obtained in \\cite{CL1, CL2} via viscosity solution theory and in \\cite{Tr} using the nonlinear adjoint method. As this application is not the main focus of the present paper, we do not pursue it further, but instead refer the reader to the aforementioned references, as well as \\cite{QSTY, CD, CG} for more recent progress on improved rates in the convex Hamiltonian case. \n\n\tIn this work, we do not seek general optimality of the convergence rates. We only discuss the sharpness of our estimates through some examples, including in particular the stability of spatially periodic solutions to normalized $p$-Laplace equations under perturbations of $p$. The above convergence rates for the regularization of generalized $p$-Laplacian are consistent with the estimate obtained in \\cite{Mi} for Lipschitz solutions of regularized level set curvature flow equations, which corresponds to the case $p=1$, $p'=2$ in our general setting. \n\n\t\\subsection*{Acknowledgments}\n\n\tThe authors would like to thank Hiroyoshi Mitake and F\\'elix del~Teso for helpful discussions. The work of QL was supported by JSPS Grant-in-Aid for Scientific Research, No.~22K03396.", "sketch": "Our proof of Theorem~\\ref{thm:parabolic} \\emph{“develops the standard comparison argument for possibly degenerate or singular parabolic equations, incorporating the H\\\"older regularity of solutions into the estimates.”} The authors also note that, when vanishing gradients induce only a mild singularity, the \\emph{time} H\\\"older continuity in \\eqref{equi-holder} can be removed: this improvement \\emph{“follows from the use of the parabolic Crandall–Ishii lemma, which avoids doubling the time variables and thereby simplifies the proof.”}", "expanded_sketch": "Our proof of Theorem~\\ref{thm:parabolic} \\emph{“develops the standard comparison argument for possibly degenerate or singular parabolic equations, incorporating the H\\\"older regularity of solutions into the estimates.”} The authors also note that, when vanishing gradients induce only a mild singularity, the \\emph{time} H\\\"older continuity in\n\\begin{equation}\\label{equi-holder}\n\t\t\t|u_\\vep(x_1, t_1)-u_\\vep(x_2, t_2)|\\leq L(|x_1-x_2|^\\theta +|t_1-t_2|^\\theta) \n\t\t\\end{equation}\ncan be removed: this improvement \\emph{“follows from the use of the parabolic Crandall–Ishii lemma, which avoids doubling the time variables and thereby simplifies the proof.”}", "expanded_theorem": "[Quantitative stability for parabolic equations]\\label{thm:parabolic}\n\t\tLet $T>0$ and $\\Omega$ be a bounded domain in ${\\mathbb R}^n$. For $\\varepsilon\\geq 0$, assume that $A_\\varepsilon: {\\mathbb R}^n_0\\to \\S_+^n$ and ${H_\\varepsilon: \\Omega_T\\times {\\mathbb R}^n\\to {\\mathbb R}}$ are continuous functions satisfying (A1)(A2) and (C1)(C2). Let $g_\\varepsilon\\in C(\\partial_p \\Om_T)$ and $u_\\varepsilon\\in C(\\overline{\\Omega}\\times [0, T))$ be bounded ${\\mathcal F}$-solutions of\n\t\t\\begin{equation}\\label{general parabolic approx eq}\n\t\t\\partial_t u-\\tr(A_\\vep(\\nabla u)\\nabla^2 u)+H_\\vep(x, t, \\nabla u)=0 \\quad \\text{in $\\Om_T$}\n\t\\end{equation}\n\t\tand\n\t\t\\begin{equation}\\label{parabolic approx bdry-cond}\n\t\tu =g_\\vep \\quad \\text{on $\\partial_p \\Om_T$}\n\t\\end{equation}\n\t\tAssume in addition that $u_\\varepsilon$ are equi-H\\\"older continuous in $\\overline{\\Om_T}$, that is, there exist $L>0$ and $\\theta\\in (0, 1]$ such that,   \n\t\t\\begin{equation}\\label{equi-holder}\n\t\t\t|u_\\varepsilon(x_1, t_1)-u_\\varepsilon(x_2, t_2)|\\leq L(|x_1-x_2|^\\theta +|t_1-t_2|^\\theta) \n\t\t\\end{equation}\n\t\tfor all $(x_1, t_1), (x_2, t_2)\\in \\overline{\\Omega}\\times [0, T)$ and $\\varepsilon\\geq 0$ small. \n\t\tThen, for all $\\varepsilon>0$ small, \n\t\t\\begin{equation}\\label{error-est parabolic}\n\t\t\t\\sup_{\\overline{\\Omega}\\times [0, T)}|u_\\varepsilon-u_0|\\leq \\sup_{\\partial_p \\Om_T}|g_\\varepsilon-g_0|+C\\varepsilon^{\\nu}  +c_H T\\varepsilon^\\gamma,\n\t\t\\end{equation}\n\t\tholds with \n\t\t\\begin{equation}\\label{nu-exponent}\n\t\t\t\\nu=\\frac{\\alpha\\theta}{1+(1-\\theta)\\max\\{\\beta, 0\\}},    \n\t\t\\end{equation}\n\t\twhere $C>0$ depends on $T, n, k, L_A, L_H, \\beta, c_A$ as well as the H\\\"older exponent $\\theta$ and the constant $L$. Here, $k>2$, $L_A\\geq0$, $L_H\\geq0$ appear in (A1)(A2), and $\\alpha, \\beta, \\gamma, c_A, c_H\\in {\\mathbb R}$ are given in (C1)(C2).", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let \\(T>0\\) and let \\(\\Omega\\subset \\mathbb{R}^n\\) be a bounded domain. Write \\(\\Omega_T=\\Omega\\times(0,T)\\), \\(\\mathbb{R}^n_0=\\mathbb{R}^n\\setminus\\{0\\}\\), and \\(\\partial_p\\Omega_T=(\\Omega\\times\\{0\\})\\cup(\\partial\\Omega\\times[0,T))\\). For each \\(\\varepsilon\\ge 0\\), let \\(A_\\varepsilon:\\mathbb{R}^n_0\\to \\mathcal S_+^n\\) and \\(H_\\varepsilon:\\Omega_T\\times\\mathbb{R}^n\\to\\mathbb{R}\\) be continuous, where \\(\\mathcal S_+^n\\) is the set of nonnegative symmetric \\(n\\times n\\) matrices, and assume the following: (i) there exists \\(k>2\\) such that, for \\(f(x)=|x|^k\\), one has \\(\\operatorname{tr}(A_\\varepsilon(\\nabla f(x))\\nabla^2 f(x))\\to 0\\) as \\(x\\to 0\\) for all sufficiently small \\(\\varepsilon\\ge 0\\); (ii) there exists \\(L_H\\ge 0\\) such that \\(|H_\\varepsilon(x_1,t_1,\\xi)-H_\\varepsilon(x_2,t_2,\\xi)|\\le L_H(1+|\\xi|)(|x_1-x_2|+|t_1-t_2|)\\) for all \\((x_1,t_1),(x_2,t_2)\\in\\Omega_T\\), \\(\\xi\\in\\mathbb{R}^n\\), and all \\(\\varepsilon\\ge 0\\); (iii) there exist \\(c_A>0\\), \\(\\alpha>0\\), and \\(\\beta>\\frac{2-k}{2(k-1)}\\) such that, with \\(A_\\varepsilon(\\xi)^{1/2}\\) the nonnegative square root of \\(A_\\varepsilon(\\xi)\\), \\(\\|A_\\varepsilon(\\xi)^{1/2}-A_0(\\xi)^{1/2}\\|\\le c_A\\varepsilon^\\alpha(1+|\\xi|^\\beta)\\) for all \\(\\xi\\in\\mathbb{R}^n_0\\) and all sufficiently small \\(\\varepsilon>0\\); and (iv) there exist \\(c_H>0\\) and \\(\\gamma>0\\) such that \\(|H_\\varepsilon(x,t,\\xi)-H_0(x,t,\\xi)|\\le c_H\\varepsilon^\\gamma\\) for all \\((x,t)\\in\\Omega_T\\), \\(\\xi\\in\\mathbb{R}^n\\), and all sufficiently small \\(\\varepsilon>0\\). For each \\(\\varepsilon\\ge 0\\), let \\(g_\\varepsilon\\in C(\\partial_p\\Omega_T)\\), and let \\(u_\\varepsilon\\in C(\\overline\\Omega\\times[0,T))\\) be a bounded \\(\\mathcal F\\)-solution of\n\\[\n\\partial_tu-\\operatorname{tr}(A_\\varepsilon(\\nabla u)\\nabla^2u)+H_\\varepsilon(x,t,\\nabla u)=0\\quad\\text{in }\\Omega_T,\n\\]\nwith boundary condition \\(u=g_\\varepsilon\\) on \\(\\partial_p\\Omega_T\\). Assume also that the family \\(\\{u_\\varepsilon\\}\\) is equi-H\\\"older continuous on \\(\\overline\\Omega\\times[0,T)\\): there exist \\(L>0\\) and \\(\\theta\\in(0,1]\\) such that\n\\[\n|u_\\varepsilon(x_1,t_1)-u_\\varepsilon(x_2,t_2)|\\le L\\big(|x_1-x_2|^\\theta+|t_1-t_2|^\\theta\\big)\n\\]\nfor all \\((x_1,t_1),(x_2,t_2)\\in\\overline\\Omega\\times[0,T)\\) and all sufficiently small \\(\\varepsilon\\ge 0\\). Which statement holds for every such family?", "correct_choice": {"label": "A", "text": "For all sufficiently small \\(\\varepsilon>0\\), there exists a constant \\(C>0\\) depending only on the data in the assumptions such that\n\\[\n\\sup_{\\overline\\Omega\\times[0,T)}|u_\\varepsilon-u_0|\\le \\sup_{\\partial_p\\Omega_T}|g_\\varepsilon-g_0|+C\\varepsilon^{\\nu}+c_HT\\varepsilon^\\gamma,\n\\]\nwhere\n\\[\n\\nu=\\frac{\\alpha\\theta}{1+(1-\\theta)\\max\\{\\beta,0\\}}.\n\\]"}, "choices": [{"label": "B", "text": "For all sufficiently small \\(\\varepsilon>0\\), there exists a constant \\(C>0\\) depending only on the data in the assumptions such that\n\\[\n\\sup_{\\overline\\Omega\\times[0,T)}|u_\\varepsilon-u_0|\\le \\sup_{\\partial_p\\Omega_T}|g_\\varepsilon-g_0|+C\\varepsilon^{\\nu}+c_HT\\varepsilon^\\gamma,\n\\]\nwhere\n\\[\n\\nu=\\frac{\\alpha\\theta}{1+(1-\\theta)\\beta}.\n\\]"}, {"label": "C", "text": "For all sufficiently small \\(\\varepsilon>0\\), there exists a constant \\(C>0\\) depending only on the data in the assumptions such that\n\\[\n\\sup_{\\overline\\Omega\\times[0,T)}|u_\\varepsilon-u_0|\\le \\sup_{\\partial_p\\Omega_T}|g_\\varepsilon-g_0|+C\\varepsilon^{\\nu}+c_HT\\varepsilon^\\gamma\n\\]\nfor some exponent \\(\\nu>0\\) depending only on \\(\\alpha,\\beta,\\theta\\)."}, {"label": "D", "text": "For all sufficiently small \\(\\varepsilon>0\\), there exists a constant \\(C=C(\\varepsilon)>0\\) such that\n\\[\n\\sup_{\\overline\\Omega\\times[0,T)}|u_\\varepsilon-u_0|\\le \\sup_{\\partial_p\\Omega_T}|g_\\varepsilon-g_0|+C\\varepsilon^{\\nu}+c_HT\\varepsilon^\\gamma,\n\\]\nwhere\n\\[\n\\nu=\\frac{\\alpha\\theta}{1+(1-\\theta)\\max\\{\\beta,0\\}}.\n\\]"}, {"label": "E", "text": "For all sufficiently small \\(\\varepsilon>0\\), there exists a constant \\(C>0\\) depending only on the data in the assumptions such that\n\\[\n\\sup_{\\overline\\Omega\\times[0,T)}|u_\\varepsilon-u_0|\\le \\sup_{\\partial_p\\Omega_T}|g_\\varepsilon-g_0|+C\\varepsilon^{\\nu+\theta}+c_HT\\varepsilon^\\gamma,\n\\]\nwhere\n\\[\n\\nu=\\frac{\\alpha\\theta}{1+(1-\\theta)\\max\\{\\beta,0\\}}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "positive-part treatment of beta in the exponent", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "explicit sharp formula for the rate exponent", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "uniform independence of the constant from epsilon", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "sharp power of epsilon produced by the comparison estimate", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the conclusion or explicitly reveal the correct estimate. It gives hypotheses only; the correct answer must be inferred or recalled from the result."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem lists the assumptions of a stability theorem and asks for its precise conclusion. That makes it very close to a direct restatement rather than a fresh mathematical task."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in subtle but meaningful ways: the exponent uses max{beta,0}, the constant dependence includes the Hölder data, and the H-error appears additively. Still, the item mainly tests precise recall/recognition of the theorem rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically plausible. They target realistic failure modes: dropping max{beta,0}, weakening the statement to an unspecified exponent, mishandling constant dependence, or incorrectly combining the epsilon-orders."}, "total_score": 5, "overall_assessment": "A technically well-constructed multiple-choice theorem-identification item with strong distractors and no answer leakage, but it is largely a direct restatement of a theorem and only moderately tests generative reasoning."}}
{"id": "2602.12657v1", "paper_link": "http://arxiv.org/abs/2602.12657v1", "theorems_cnt": 1, "theorem": {"env_name": "thm", "content": "[Quantitative stability for parabolic equations]\\label{thm:parabolic}\n\t\tLet $T>0$ and $\\Omega$ be a bounded domain in ${\\mathbb R}^n$. For $\\varepsilon\\geq 0$, assume that $A_\\varepsilon: {\\mathbb R}^n_0\\to \\S_+^n$ and ${H_\\varepsilon: \\Omega_T\\times {\\mathbb R}^n\\to {\\mathbb R}}$ are continuous functions satisfying (A1)(A2) and (C1)(C2). Let $g_\\varepsilon\\in C(\\partial_p \\Om_T)$ and $u_\\varepsilon\\in C(\\overline{\\Omega}\\times [0, T))$ be bounded ${\\mathcal F}$-solutions of  \\eqref{general parabolic approx eq}\\eqref{parabolic approx bdry-cond}. Assume in addition that $u_\\varepsilon$ are equi-H\\\"older continuous in $\\overline{\\Om_T}$, that is, there exist $L>0$ and $\\theta\\in (0, 1]$ such that,   \n\t\t\\begin{equation}\\label{equi-holder}\n\t\t\t|u_\\varepsilon(x_1, t_1)-u_\\varepsilon(x_2, t_2)|\\leq L(|x_1-x_2|^\\theta +|t_1-t_2|^\\theta) \n\t\t\\end{equation}\n\t\tfor all $(x_1, t_1), (x_2, t_2)\\in \\overline{\\Omega}\\times [0, T)$ and $\\varepsilon\\geq 0$ small. \n\t\tThen, for all $\\varepsilon>0$ small, \n\t\t\\begin{equation}\\label{error-est parabolic}\n\t\t\t\\sup_{\\overline{\\Omega}\\times [0, T)}|u_\\varepsilon-u_0|\\leq \\sup_{\\partial_p \\Om_T}|g_\\varepsilon-g_0|+C\\varepsilon^{\\nu}  +c_H T\\varepsilon^\\gamma,\n\t\t\\end{equation}\n\t\tholds with \n\t\t\\begin{equation}\\label{nu-exponent}\n\t\t\t\\nu=\\frac{\\alpha\\theta}{1+(1-\\theta)\\max\\{\\beta, 0\\}},    \n\t\t\\end{equation}\n\t\twhere $C>0$ depends on $T, n, k, L_A, L_H, \\beta, c_A$ as well as the H\\\"older exponent $\\theta$ and the constant $L$. Here, $k>2$, $L_A\\geq0$, $L_H\\geq0$ appear in (A1)(A2), and $\\alpha, \\beta, \\gamma, c_A, c_H\\in {\\mathbb R}$ are given in (C1)(C2).", "start_pos": 15948, "end_pos": 17372, "label": "thm:parabolic"}, "ref_dict": {"nu-exponent": "\\begin{equation}\\label{nu-exponent}\n\t\t\t\\nu=\\frac{\\alpha\\theta}{1+(1-\\theta)\\max\\{\\beta, 0\\}},    \n\t\t\\end{equation}", "np-parabolic": "\\begin{equation}\\label{np-parabolic}\n\t\\partial_t u-\\Delta^N_p u+H(x, u, \\nabla u)=0 \\quad \\text{in $\\Om_T:=\\Omega\\times (0, T)$}\n\\end{equation}", "error-est parabolic": "\\begin{equation}\\label{error-est parabolic}\n\t\t\t\\sup_{\\Oba\\times [0, T)}|u_\\vep-u_0|\\leq \\sup_{\\partial_p \\Om_T}|g_\\vep-g_0|+C\\vep^{\\nu}  +c_H T\\vep^\\gamma,\n\t\t\\end{equation}", "thm:parabolic": "\\begin{thm}[Quantitative stability for parabolic equations]\\label{thm:parabolic}\n\t\tLet $T>0$ and $\\Omega$ be a bounded domain in $\\R^n$. For $\\vep\\geq 0$, assume that $A_\\vep: \\R^n_0\\to \\S_+^n$ and ${H_\\vep: \\Omega_T\\times \\R^n\\to \\R}$ are continuous functions satisfying (A1)(A2) and (C1)(C2). Let $g_\\vep\\in C(\\partial_p \\Om_T)$ and $u_\\vep\\in C(\\Oba\\times [0, T))$ be bounded $\\F$-solutions of  \\eqref{general parabolic approx eq}\\eqref{parabolic approx bdry-cond}. Assume in addition that $u_\\vep$ are equi-H\\\"older continuous in $\\ol{\\Om_T}$, that is, there exist $L>0$ and $\\theta\\in (0, 1]$ such that,   \n\t\t\\begin{equation}\\label{equi-holder}\n\t\t\t|u_\\vep(x_1, t_1)-u_\\vep(x_2, t_2)|\\leq L(|x_1-x_2|^\\theta +|t_1-t_2|^\\theta) \n\t\t\\end{equation}\n\t\tfor all $(x_1, t_1), (x_2, t_2)\\in \\Oba\\times [0, T)$ and $\\vep\\geq 0$ small. \n\t\tThen, for all $\\vep>0$ small, \n\t\t\\begin{equation}\\label{error-est parabolic}\n\t\t\t\\sup_{\\Oba\\times [0, T)}|u_\\vep-u_0|\\leq \\sup_{\\partial_p \\Om_T}|g_\\vep-g_0|+C\\vep^{\\nu}  +c_H T\\vep^\\gamma,\n\t\t\\end{equation}\n\t\tholds with \n\t\t\\begin{equation}\\label{nu-exponent}\n\t\t\t\\nu=\\frac{\\alpha\\theta}{1+(1-\\theta)\\max\\{\\beta, 0\\}},    \n\t\t\\end{equation}\n\t\twhere $C>0$ depends on $T, n, k, L_A, L_H, \\beta, c_A$ as well as the H\\\"older exponent $\\theta$ and the constant $L$. Here, $k>2$, $L_A\\geq0$, $L_H\\geq0$ appear in (A1)(A2), and $\\alpha, \\beta, \\gamma, c_A, c_H\\in \\R$ are given in (C1)(C2). \n\t\\end{thm}", "regularized eq": "\\begin{equation}\\label{regularized eq}\n\t\t\\partial_t u-\\left(|\\nabla u|^2+\\vep^2\\right)^{\\frac{p'-p}{2}}\\div\\left(\\left(|\\nabla u|^2+\\vep^2\\right)^{\\frac{p-2}{2}}\\nabla u\\right)=0 \\quad \\text{in $\\Omega_T$,}\n\t\\end{equation}", "eq:c1": "\\begin{equation}\\label{eq:c1}\n\t\t\t\\|A_\\vep(\\xi)^{1/2} -A_0(\\xi)^{1/2}\\|\\leq c_A\\vep^{\\alpha} (1+|\\xi|^\\beta)\n\t\t\\end{equation}", "rmk:periodic": "\\begin{rmk}\\label{rmk:periodic}\n\t\tOur discussions on the Cauchy-Dirichlet problem in Theorem~\\ref{thm:parabolic} and Theorem~\\ref{thm:parabolic2} extend readily to the case of spatially periodic settings. \n\t\tInstead of considering \\eqref{general parabolic lim eq} in a bounded domain $\\Omega$, we may obtain the same stability results as in Theorem~\\ref{thm:parabolic} and Theorem~\\ref{thm:parabolic2} for the whole space $\\R^n$, provided that the initial value $g$ and $H$ are both periodic with respect to $x$. Note that the periodic setting essentially plays the same role as a bounded domain (without boundary) in our comparison arguments and therefore does not substantially affect the proofs. For clarity of presentation, here we choose not to repeat the statements for this case.\n\t\\end{rmk}", "equi-holder": "\\begin{equation}\\label{equi-holder}\n\t\t\t|u_\\vep(x_1, t_1)-u_\\vep(x_2, t_2)|\\leq L(|x_1-x_2|^\\theta +|t_1-t_2|^\\theta) \n\t\t\\end{equation}", "np-parabolic0": "\\begin{equation}\\label{np-parabolic0}\n\t\t\\partial_t u-\\Delta^N_p u=0 \\quad \\text{in $\\Om_T:=\\Omega\\times (0, T)$,}\n\t\\end{equation}", "general parabolic approx eq": "\\begin{equation}\\label{general parabolic approx eq}\n\t\t\\partial_t u-\\tr(A_\\vep(\\nabla u)\\nabla^2 u)+H_\\vep(x, t, \\nabla u)=0 \\quad \\text{in $\\Om_T$}\n\t\\end{equation}", "parabolic bdry-cond": "\\begin{equation}\\label{parabolic bdry-cond}\n\t\tu=g \\quad \\text{on $\\partial_p \\Om_T$}\n\t\\end{equation}", "vp-parabolic0": "\\begin{equation}\\label{vp-parabolic0}\n\t\t\\partial_t u-\\Delta_p u=0 \\quad \\text{in $\\Om_T$,}\n\t\\end{equation}", "thm:elliptic": "\\begin{thm}[Quantitative stability for elliptic equations]\\label{thm:elliptic}\n\t\tLet $\\Omega$ be a bounded domain in $\\R^n$. For $\\vep\\geq 0$, assume that $A_\\vep: \\R^n_0\\to \\S_+^n$ and $H_\\vep: \\Omega\\times \\R\\times  \\R^n\\to \\R$ are continuous functions satisfying (A1)(A2')(A3) and (C1)(C2'). Let $g_\\vep\\in C(\\partial\\Omega)$ and $u_\\vep\\in C(\\Oba)$ be $\\F$-solutions of \\eqref{general approx eq}\\eqref{approx bdry-cond}. Assume in addition that $u_\\vep$ are equi-H\\\"older continuous in $\\Oba$, that is, there exist $L>0$ and $0<\\theta\\leq 1$ such that  \t\t\\[\n\t\t|u_\\vep(x_1)-u_\\vep(x_2)|\\leq L|x_1-x_2|^\\theta \n\t\t\\]\n\t\tfor all $x_1, x_2\\in \\Oba$ and $\\vep\\geq 0$.\n\t\tThen, for all $\\vep>0$ small,  \n\t\t\\begin{equation}\\label{error-est}\n\t\t\t\\max_{\\Oba} |u_\\vep-u_0|\\leq \\max_{\\partial \\Omega} |g_\\vep-g|+\\frac{1}{\\lambda}\\left(C_0\\vep^{\\nu}+c_H \\vep^\\gamma \\right)\n\t\t\\end{equation}\n\t\tholds with $\\nu$ given as in \\eqref{nu-exponent}, where $C_0>0$ is a constant depending on \n\t\t$k>2$,$L_A>0$, $L_H>0$, $\\lambda>0$ in (A1)(A2')(A3) and $\\alpha, \\beta, c_A, \\in \\R$ given in (C1). The constants $\\gamma, c_H>0$ are given as in (C2'). \n\t\\end{thm}", "parabolic approx bdry-cond": "\\begin{equation}\\label{parabolic approx bdry-cond}\n\t\tu =g_\\vep \\quad \\text{on $\\partial_p \\Om_T$}\n\t\\end{equation}", "mono-0": "\\begin{equation}\\label{mono-0}\n\t\t\\rho\\mapsto H_\\vep(x, \\rho, \\xi)-\\lambda \\rho \\quad \\text{is nondecreasing in $\\R$. }\n\t\\end{equation}"}, "pre_theorem_intro_text_len": 10828, "pre_theorem_intro_text": "\\subsection{Motivation}\n\tIn this paper we study the stability of a class of quasilinear parabolic partial differential equations, which takes the form \n\t\\begin{equation}\\label{general parabolic lim eq}\n\t\t\\partial_t u-\\operatorname{tr}(A(\\nabla u)\\nabla^2 u)+H(x, t, \\nabla u)=0 \\quad \\text{in $\\Omega\\times (0, T)$,}\n\t\\end{equation}\n\twhere $\\Omega$ is a bounded domain in ${\\mathbb R}^n$, $T>0$ is fixed, and \n\t\\[\n\tA: {\\mathbb R}^n\\setminus \\{0\\}\\to {\\mathcal S}^n_+, \\quad H: \\Omega\\times (0, T)\\times {\\mathbb R}^n\\to {\\mathbb R},\n\t\\]\n\tare given functions satisfying appropriate assumptions to be introduced later. Here, ${\\mathcal S}^n_+$ represents the set of all nonnegative symmetric $n\\times n$ matrices. A key feature of this general setting is that the elliptic operator may exhibit singularity at vanishing gradients. \n\n\tOne typical example of applicable equations we have in mind is the following parabolic normalized $p$-Laplace type equation (with $1<p<\\infty$)\n\t\\begin{equation}\\label{np-parabolic0}\n\t\t\\partial_t u-\\Delta^N_p u=0 \\quad \\text{in $\\Om_T:=\\Omega\\times (0, T)$,}\n\t\\end{equation}  \n\twhere\n\t\\[\n\t\\Delta^N_p u:= |\\nabla u|^{2-p}\\operatorname{div}(|\\nabla u|^{p-2}\\nabla u).\n\t\\]\n\tAnother example is the parabolic equation with the variational $p$-Laplacian:\n\t\\begin{equation}\\label{vp-parabolic0}\n\t\t\\partial_t u-\\Delta_p u=0 \\quad \\text{in $\\Om_T$,}\n\t\\end{equation}\n\twhere \n\t\\[\n\t\\Delta_p u:= \\operatorname{div}(|\\nabla u|^{p-2}\\nabla u). \n\t\\]\n\tWe consider the associated Cauchy-Dirichlet boundary value problem for these parabolic equations. The parabolic boundary condition is given by \n\t\\begin{equation}\\label{parabolic bdry-cond}\n\t\tu=g \\quad \\text{on $\\partial_p \\Om_T$}\n\t\\end{equation}\n\tfor\n\t\\[\n\t\\partial_p \\Om_T=(\\Omega\\times \\{0\\})\\cup (\\partial \\Omega\\times [0, T)),\n\t\\]\n\twhere $g: \\partial \\Omega\\to {\\mathbb R}$ is assumed to be continuous.\n\n\tOne of the main questions addressed in this paper concerns the stability of $p$-Laplace type equations under perturbations of the parameter $p$. Assume that $H\\equiv 0$. For a fixed parabolic boundary value $g$, it is known that, as $p\\to q$ for some $q\\in(1,\\infty)$, the unique solution $u_p$ of the Cauchy-Dirichlet problem \\eqref{np-parabolic0} or \\eqref{vp-parabolic0} converges uniformly to the solution $u_q$ of the corresponding problem with parameter $q$. In the framework of viscosity solutions, consult \\cite[Theorem~6.1]{OS} and \\cite[Theorem~2.2.1]{Gbook} for such qualitative stability results on the variational and normalized $p$-Laplace equations of parabolic type. See also \\cite{KP, LP} for related analysis with different approaches. However, these works do not provide an explicit rate of convergence. Our goal is therefore to further quantify the uniform convergence \tand to discuss the optimality of the resulting estimates.\n\n\tOn the other hand, when studying the solutions $u$ of singular parabolic equations such as \\eqref{np-parabolic} and \\eqref{vp-parabolic0}, it is common to approximate them with regularized equations in the form \n\t\\begin{equation}\\label{regularized eq}\n\t\t\\partial_t u-\\left(|\\nabla u|^2+\\varepsilon^2\\right)^{\\frac{p'-p}{2}}\\operatorname{div}\\left(\\left(|\\nabla u|^2+\\varepsilon^2\\right)^{\\frac{p-2}{2}}\\nabla u\\right)=0 \\quad \\text{in $\\Omega_T$,}\n\t\\end{equation}\n\twhere $\\varepsilon>0$ is small. Here, we take $p'=2$ and $p'=p$ to obtain regularized approximations of \\eqref{np-parabolic} and \\eqref{vp-parabolic0}, respectively.  Under the comparison principle for the limit singular equations, sending  $\\varepsilon\\to 0$ yields the uniform convergence of the solution $u_\\varepsilon$ of \\eqref{regularized eq} to the solution $u$ of the corresponding singular equation. It is then natural to ask about the rate of this convergence in applications. The case with $p'=2$ and $p=1$ corresponds to the level-set mean curvature flow equation, for which such a regularization was introduced in \\cite{ES}. The convergence rate in this case was essentially established by Deckelnick \\cite{De}, and later proved by Mitake \\cite{Mi} using a different method based on comparison arguments. Addressing convergence rates for the general approximation \\eqref{regularized eq} is another main objective of the present paper.\n\n\tThe method employed in this paper builds on the arguments developed by Mitake in \\cite{Mi}, as well as earlier works \\cite{CGL, JK1}, which established general continuous dependence estimates for viscosity solutions of nonlinear parabolic equations. Further related results include extensions to nonlocal PDEs in \\cite{JK2} and to general Neumann-type boundary value problems in \\cite{JG}. While it may be possible to adapt these methods to study quantitative stability for $p$-Laplace type equations in regimes without vanishing-gradient singularities, precise estimates do not appear to have been explicitly given, to the best of our knowledge. Also, the singular case, such as \\eqref{vp-parabolic0} with $1<p<2$, seems unexplored and constitutes one of the focuses of the present work. \n\n\tA recent paper of Bungert \\cite{Bun} studies the convergence rate of $p$-harmonic functions to infinity-harmonic functions, using partly the variational structure of the $p$-Laplace operator. In contrast, based on the viscosity solution theory, we here provide a purely PDE approach to show the convergence rate for a general class of related parabolic stability problems. Our method develops the techniques in \\cite{CGL, JK1, Mi} and further extends them to treat convergence rates arising from more general stability and regularization problems.\n\n\tAs explained through the examples above, the equations of interest may exhibit singularity at vanishing gradients. To handle such singularities, we employ a generalized notion of viscosity solutions, known as $\\mathcal{F}$-solutions. A brief overview of this approach is given in Section~\\ref{sec:preli}; for a detailed introduction, we again refer the reader to \\cite{OS, Gbook}.\n\n\t\\subsection{Main results}\n\tLet us now introduce more details about our setting. Let $T>0$ and $\\Omega\\subset\\mathbb{R}^n$ be a bounded domain. In the sequel, we set $\\Om_T=\\Omega\\times (0, T)$ and $\\Rn_0={\\mathbb R}^n\\setminus \\{0\\}$. Consider a family of parabolic equations\n\t\\begin{equation}\\label{general parabolic approx eq}\n\t\t\\partial_t u-\\operatorname{tr}(A_\\varepsilon(\\nabla u)\\nabla^2 u)+H_\\varepsilon(x, t, \\nabla u)=0 \\quad \\text{in $\\Om_T$}\n\t\\end{equation}\n\tfor $\\varepsilon\\geq 0$, where $A_\\varepsilon: \\Rn_0\\to {\\mathcal S}^n_+$ and $H_\\varepsilon:\\Omega_T \\times {\\mathbb R}^{n}\\mapsto {\\mathbb R}$  are continuous functions satisfying appropriate assumptions to be introduced below. \n\n\tLet $\\partial_p \\Om_T$ denote the parabolic boundary of $\\Omega\\times (0, T)$,  defined by\n\t\\[\n\t\\partial_p \\Om_T=(\\Omega\\times \\{0\\})\\cup (\\partial \\Omega\\times [0, T)).\n\t\\]\n\tWe impose the initial and boundary condition \t\\begin{equation}\\label{parabolic approx bdry-cond}\n\t\tu =g_\\varepsilon \\quad \\text{on $\\partial_p \\Om_T$}\n\t\\end{equation}\n\tfor \\eqref{general parabolic approx eq}, where $g_\\varepsilon\\in C(\\partial \\Om_T)$ are given for $\\varepsilon\\geq 0$.   \n\n\tSuppose that $u_\\varepsilon$ (with $\\varepsilon\\geq 0$) is a unique viscosity solution of \\eqref{general parabolic approx eq}\\eqref{parabolic approx bdry-cond}. Our main result quantifies the convergence of $u_\\varepsilon\\to u_0$ in terms of $\\varepsilon>0$ under certain assumptions on the convergence rate of $A_\\varepsilon\\to A_0$, $H_\\varepsilon\\to H_0$, and $g_\\varepsilon \\to g_0$ as $\\varepsilon\\to 0$.\n\n\tNow let us state our assumptions on $A_\\varepsilon$ and $H_\\varepsilon$ with $\\varepsilon\\geq 0$. \n\tNote first that for a matrix $M\\in {\\mathcal S}^n_+$, one can find its square root, denoted by $M^{1/2}$, which is also nonnegative. There thus exists the matrix valued function ${A_\\varepsilon}^{1/2}: \\Rn_0\\to {\\mathcal S}^n_+$ for $\\varepsilon\\geq 0$. We impose the following assumption on $A_\\varepsilon$ and $H_\\varepsilon$.\n\t\\begin{enumerate}\n\t\t\\item[(A1)] There exists $k>2$ such that for any $\\varepsilon\\geq 0$ small, $f(x)=|x|^k$ satisfies \n\t\t\\begin{equation}\\label{singular-cancel}\n\t\t\t\\operatorname{tr} \\left(A_\\varepsilon(\\nabla f(x)) \\nabla^2 f(x)\\right)\\to 0 \\quad \\text{as $x\\to 0$.}\n\t\t\\end{equation} \n\t\\end{enumerate}\n\t\\begin{enumerate}\n\t\t\\item[(A2)] $H_\\varepsilon: \\Om_T\\times {\\mathbb R}^n\\to {\\mathbb R}$ is continuous, and there exists $L_H\\geq0$ such that \n\t\t\\[\n\t\t|H_\\varepsilon(x_1, t_1, \\xi)-H_\\varepsilon(x_2, t_2, \\xi)|\\leq L_H(1+|\\xi|)\\left(|x_1-x_2|+\\left|t_1-t_2\\right|\\right)\n\t\t\\]\n\t\tfor all $(x_1, t_1),\\ (x_2, t_2)\\in \\Omega_T$,  $\\xi\\in {\\mathbb R}^n$ and all $\\varepsilon\\geq 0$. \n\n\t\\end{enumerate}\n\tThese assumptions guarantee the existence and uniqueness of ${\\mathcal F}$-solutions to the associated Cauchy-Dirichlet problem. The existence of $k>2$ required in (A1) is related to the possible singularity of the equation at vanishing gradients. A typical example of singular operators we will consider in our later study of the stability is the case of the $p$-Laplacian for $p$ near a fixed $q\\in (1, 2)$, where $A_\\varepsilon$ is taken to be \n\t\\begin{equation}\\label{vp-operator}\n\t\tA_\\varepsilon(\\xi)= |\\xi|^{q+\\varepsilon-2} \\left(I+(q+\\varepsilon-2) \\frac{\\xi\\otimes \\xi}{|\\xi|^2}\\right)\n\t\\end{equation}\n\tfor $\\varepsilon\\geq 0$. In this case, one can choose  $k>q/(q-1)$ for $\\varepsilon\\geq 0$ sufficiently small. \n\n\tFor general $A_\\varepsilon$ and $H_\\varepsilon$, we next impose assumptions on the convergence of $A_\\varepsilon\\to A_0$, $H_\\varepsilon\\to H_0$ as $\\varepsilon\\to 0$: \n\t\\begin{enumerate}\n\t\t\\item[(C1)] There exist $c_A>0$ and $\\alpha>0, \\beta>\\frac{2-k}{2(k-1)}$ such that \n\t\t\\begin{equation}\\label{eq:c1}\n\t\t\t\\|A_\\varepsilon(\\xi)^{1/2} -A_0(\\xi)^{1/2}\\|\\leq c_A\\varepsilon^{\\alpha} (1+|\\xi|^\\beta)\n\t\t\\end{equation}\n\t\tholds for all $\\varepsilon>0$ small and $\\xi\\in \\Rn_0$. Here, $k$ is the constant given in (A1). \n\t\t\\item[(C2)] There exist $c_H>0$ and $\\gamma>0$  such that \n\t\t\\[\n\t\t|H_\\varepsilon(x, t, \\xi)-H_0(x, t, \\xi)|\\leq c_H \\varepsilon^\\gamma \n\t\t\\] \n\t\tholds for all $\\varepsilon>0$ small and $(x, t)\\in \\Om_T,  \\xi\\in {\\mathbb R}^n$.\n\t\\end{enumerate}\n\tNote that by the condition $\\beta>\\frac{2-k}{2(k-1)}$ in (C1), we have $\\beta>-1/2$ for $k>2$. It is worth remarking that one can generalize (C1) by assuming the existence of $\\Sigma_\\varepsilon: {\\mathbb R}^n_0\\to {\\mathbb R}^{m\\times n}$ with $m\\in {\\mathbb N}$ for all $\\varepsilon\\geq 0$ small such that $\\Sigma_\\varepsilon^T\\Sigma_\\varepsilon=A_\\varepsilon$ in ${\\mathbb R}^n_0$ and \\eqref{eq:c1} holds for $\\Sigma_\\varepsilon$ in place of $A_\\varepsilon^{1/2}$. For clarity and simplicity, we restrict our attention to the case $\\Sigma_\\varepsilon=A_\\varepsilon^{1/2}$, which already covers the applications of interest.  \n\n\tLet us present our main theorem.", "context": "\\subsection{Motivation}\n In this paper we study the stability of a class of quasilinear parabolic partial differential equations, which takes the form \n \\begin{equation}\\label{general parabolic lim eq}\n \\partial_t u-\\operatorname{tr}(A(\\nabla u)\\nabla^2 u)+H(x, t, \\nabla u)=0 \\quad \\text{in $\\Omega\\times (0, T)$,}\n \\end{equation}\n where $\\Omega$ is a bounded domain in ${\\mathbb R}^n$, $T>0$ is fixed, and \n \\[\n A: {\\mathbb R}^n\\setminus \\{0\\}\\to {\\mathcal S}^n_+, \\quad H: \\Omega\\times (0, T)\\times {\\mathbb R}^n\\to {\\mathbb R},\n \\]\n are given functions satisfying appropriate assumptions to be introduced later. Here, ${\\mathcal S}^n_+$ represents the set of all nonnegative symmetric $n\\times n$ matrices. A key feature of this general setting is that the elliptic operator may exhibit singularity at vanishing gradients.\n\n\\subsection{Main results}\n Let us now introduce more details about our setting. Let $T>0$ and $\\Omega\\subset\\mathbb{R}^n$ be a bounded domain. In the sequel, we set $\\Om_T=\\Omega\\times (0, T)$ and $\\Rn_0={\\mathbb R}^n\\setminus \\{0\\}$. Consider a family of parabolic equations\n \\begin{equation}\\label{general parabolic approx eq}\n \\partial_t u-\\operatorname{tr}(A_\\varepsilon(\\nabla u)\\nabla^2 u)+H_\\varepsilon(x, t, \\nabla u)=0 \\quad \\text{in $\\Om_T$}\n \\end{equation}\n for $\\varepsilon\\geq 0$, where $A_\\varepsilon: \\Rn_0\\to {\\mathcal S}^n_+$ and $H_\\varepsilon:\\Omega_T \\times {\\mathbb R}^{n}\\mapsto {\\mathbb R}$ are continuous functions satisfying appropriate assumptions to be introduced below.\n\nLet $\\partial_p \\Om_T$ denote the parabolic boundary of $\\Omega\\times (0, T)$, defined by\n \\[\n \\partial_p \\Om_T=(\\Omega\\times \\{0\\})\\cup (\\partial \\Omega\\times [0, T)).\n \\]\n We impose the initial and boundary condition \\begin{equation}\\label{parabolic approx bdry-cond}\n u =g_\\varepsilon \\quad \\text{on $\\partial_p \\Om_T$}\n \\end{equation}\n for \\eqref{general parabolic approx eq}, where $g_\\varepsilon\\in C(\\partial \\Om_T)$ are given for $\\varepsilon\\geq 0$.\n\nNow let us state our assumptions on $A_\\varepsilon$ and $H_\\varepsilon$ with $\\varepsilon\\geq 0$. \n Note first that for a matrix $M\\in {\\mathcal S}^n_+$, one can find its square root, denoted by $M^{1/2}$, which is also nonnegative. There thus exists the matrix valued function ${A_\\varepsilon}^{1/2}: \\Rn_0\\to {\\mathcal S}^n_+$ for $\\varepsilon\\geq 0$. We impose the following assumption on $A_\\varepsilon$ and $H_\\varepsilon$.\n \\begin{enumerate}\n \\item[(A1)] There exists $k>2$ such that for any $\\varepsilon\\geq 0$ small, $f(x)=|x|^k$ satisfies \n \\begin{equation}\\label{singular-cancel}\n \\operatorname{tr} \\left(A_\\varepsilon(\\nabla f(x)) \\nabla^2 f(x)\\right)\\to 0 \\quad \\text{as $x\\to 0$.}\n \\end{equation} \n \\end{enumerate}\n \\begin{enumerate}\n \\item[(A2)] $H_\\varepsilon: \\Om_T\\times {\\mathbb R}^n\\to {\\mathbb R}$ is continuous, and there exists $L_H\\geq0$ such that \n \\[\n |H_\\varepsilon(x_1, t_1, \\xi)-H_\\varepsilon(x_2, t_2, \\xi)|\\leq L_H(1+|\\xi|)\\left(|x_1-x_2|+\\left|t_1-t_2\\right|\\right)\n \\]\n for all $(x_1, t_1),\\ (x_2, t_2)\\in \\Omega_T$, $\\xi\\in {\\mathbb R}^n$ and all $\\varepsilon\\geq 0$.\n\nFor general $A_\\varepsilon$ and $H_\\varepsilon$, we next impose assumptions on the convergence of $A_\\varepsilon\\to A_0$, $H_\\varepsilon\\to H_0$ as $\\varepsilon\\to 0$: \n \\begin{enumerate}\n \\item[(C1)] There exist $c_A>0$ and $\\alpha>0, \\beta>\\frac{2-k}{2(k-1)}$ such that \n \\begin{equation}\\label{eq:c1}\n \\|A_\\varepsilon(\\xi)^{1/2} -A_0(\\xi)^{1/2}\\|\\leq c_A\\varepsilon^{\\alpha} (1+|\\xi|^\\beta)\n \\end{equation}\n holds for all $\\varepsilon>0$ small and $\\xi\\in \\Rn_0$. Here, $k$ is the constant given in (A1). \n \\item[(C2)] There exist $c_H>0$ and $\\gamma>0$ such that \n \\[\n |H_\\varepsilon(x, t, \\xi)-H_0(x, t, \\xi)|\\leq c_H \\varepsilon^\\gamma \n \\] \n holds for all $\\varepsilon>0$ small and $(x, t)\\in \\Om_T, \\xi\\in {\\mathbb R}^n$.\n \\end{enumerate}\n Note that by the condition $\\beta>\\frac{2-k}{2(k-1)}$ in (C1), we have $\\beta>-1/2$ for $k>2$. It is worth remarking that one can generalize (C1) by assuming the existence of $\\Sigma_\\varepsilon: {\\mathbb R}^n_0\\to {\\mathbb R}^{m\\times n}$ with $m\\in {\\mathbb N}$ for all $\\varepsilon\\geq 0$ small such that $\\Sigma_\\varepsilon^T\\Sigma_\\varepsilon=A_\\varepsilon$ in ${\\mathbb R}^n_0$ and \\eqref{eq:c1} holds for $\\Sigma_\\varepsilon$ in place of $A_\\varepsilon^{1/2}$. For clarity and simplicity, we restrict our attention to the case $\\Sigma_\\varepsilon=A_\\varepsilon^{1/2}$, which already covers the applications of interest.\n\nLet us present our main theorem.\n\n\\begin{equation}\\label{eq:c1}\n\t\t\t\\|A_\\vep(\\xi)^{1/2} -A_0(\\xi)^{1/2}\\|\\leq c_A\\vep^{\\alpha} (1+|\\xi|^\\beta)\n\t\t\\end{equation}\n\n\\begin{equation}\\label{general parabolic approx eq}\n\t\t\\partial_t u-\\tr(A_\\vep(\\nabla u)\\nabla^2 u)+H_\\vep(x, t, \\nabla u)=0 \\quad \\text{in $\\Om_T$}\n\t\\end{equation}\n\n\\begin{equation}\\label{parabolic approx bdry-cond}\n\t\tu =g_\\vep \\quad \\text{on $\\partial_p \\Om_T$}\n\t\\end{equation}", "full_context": "\\subsection{Motivation}\n In this paper we study the stability of a class of quasilinear parabolic partial differential equations, which takes the form \n \\begin{equation}\\label{general parabolic lim eq}\n \\partial_t u-\\operatorname{tr}(A(\\nabla u)\\nabla^2 u)+H(x, t, \\nabla u)=0 \\quad \\text{in $\\Omega\\times (0, T)$,}\n \\end{equation}\n where $\\Omega$ is a bounded domain in ${\\mathbb R}^n$, $T>0$ is fixed, and \n \\[\n A: {\\mathbb R}^n\\setminus \\{0\\}\\to {\\mathcal S}^n_+, \\quad H: \\Omega\\times (0, T)\\times {\\mathbb R}^n\\to {\\mathbb R},\n \\]\n are given functions satisfying appropriate assumptions to be introduced later. Here, ${\\mathcal S}^n_+$ represents the set of all nonnegative symmetric $n\\times n$ matrices. A key feature of this general setting is that the elliptic operator may exhibit singularity at vanishing gradients.\n\n\\subsection{Main results}\n Let us now introduce more details about our setting. Let $T>0$ and $\\Omega\\subset\\mathbb{R}^n$ be a bounded domain. In the sequel, we set $\\Om_T=\\Omega\\times (0, T)$ and $\\Rn_0={\\mathbb R}^n\\setminus \\{0\\}$. Consider a family of parabolic equations\n \\begin{equation}\\label{general parabolic approx eq}\n \\partial_t u-\\operatorname{tr}(A_\\varepsilon(\\nabla u)\\nabla^2 u)+H_\\varepsilon(x, t, \\nabla u)=0 \\quad \\text{in $\\Om_T$}\n \\end{equation}\n for $\\varepsilon\\geq 0$, where $A_\\varepsilon: \\Rn_0\\to {\\mathcal S}^n_+$ and $H_\\varepsilon:\\Omega_T \\times {\\mathbb R}^{n}\\mapsto {\\mathbb R}$ are continuous functions satisfying appropriate assumptions to be introduced below.\n\nLet $\\partial_p \\Om_T$ denote the parabolic boundary of $\\Omega\\times (0, T)$, defined by\n \\[\n \\partial_p \\Om_T=(\\Omega\\times \\{0\\})\\cup (\\partial \\Omega\\times [0, T)).\n \\]\n We impose the initial and boundary condition \\begin{equation}\\label{parabolic approx bdry-cond}\n u =g_\\varepsilon \\quad \\text{on $\\partial_p \\Om_T$}\n \\end{equation}\n for \\eqref{general parabolic approx eq}, where $g_\\varepsilon\\in C(\\partial \\Om_T)$ are given for $\\varepsilon\\geq 0$.\n\nNow let us state our assumptions on $A_\\varepsilon$ and $H_\\varepsilon$ with $\\varepsilon\\geq 0$. \n Note first that for a matrix $M\\in {\\mathcal S}^n_+$, one can find its square root, denoted by $M^{1/2}$, which is also nonnegative. There thus exists the matrix valued function ${A_\\varepsilon}^{1/2}: \\Rn_0\\to {\\mathcal S}^n_+$ for $\\varepsilon\\geq 0$. We impose the following assumption on $A_\\varepsilon$ and $H_\\varepsilon$.\n \\begin{enumerate}\n \\item[(A1)] There exists $k>2$ such that for any $\\varepsilon\\geq 0$ small, $f(x)=|x|^k$ satisfies \n \\begin{equation}\\label{singular-cancel}\n \\operatorname{tr} \\left(A_\\varepsilon(\\nabla f(x)) \\nabla^2 f(x)\\right)\\to 0 \\quad \\text{as $x\\to 0$.}\n \\end{equation} \n \\end{enumerate}\n \\begin{enumerate}\n \\item[(A2)] $H_\\varepsilon: \\Om_T\\times {\\mathbb R}^n\\to {\\mathbb R}$ is continuous, and there exists $L_H\\geq0$ such that \n \\[\n |H_\\varepsilon(x_1, t_1, \\xi)-H_\\varepsilon(x_2, t_2, \\xi)|\\leq L_H(1+|\\xi|)\\left(|x_1-x_2|+\\left|t_1-t_2\\right|\\right)\n \\]\n for all $(x_1, t_1),\\ (x_2, t_2)\\in \\Omega_T$, $\\xi\\in {\\mathbb R}^n$ and all $\\varepsilon\\geq 0$.\n\nFor general $A_\\varepsilon$ and $H_\\varepsilon$, we next impose assumptions on the convergence of $A_\\varepsilon\\to A_0$, $H_\\varepsilon\\to H_0$ as $\\varepsilon\\to 0$: \n \\begin{enumerate}\n \\item[(C1)] There exist $c_A>0$ and $\\alpha>0, \\beta>\\frac{2-k}{2(k-1)}$ such that \n \\begin{equation}\\label{eq:c1}\n \\|A_\\varepsilon(\\xi)^{1/2} -A_0(\\xi)^{1/2}\\|\\leq c_A\\varepsilon^{\\alpha} (1+|\\xi|^\\beta)\n \\end{equation}\n holds for all $\\varepsilon>0$ small and $\\xi\\in \\Rn_0$. Here, $k$ is the constant given in (A1). \n \\item[(C2)] There exist $c_H>0$ and $\\gamma>0$ such that \n \\[\n |H_\\varepsilon(x, t, \\xi)-H_0(x, t, \\xi)|\\leq c_H \\varepsilon^\\gamma \n \\] \n holds for all $\\varepsilon>0$ small and $(x, t)\\in \\Om_T, \\xi\\in {\\mathbb R}^n$.\n \\end{enumerate}\n Note that by the condition $\\beta>\\frac{2-k}{2(k-1)}$ in (C1), we have $\\beta>-1/2$ for $k>2$. It is worth remarking that one can generalize (C1) by assuming the existence of $\\Sigma_\\varepsilon: {\\mathbb R}^n_0\\to {\\mathbb R}^{m\\times n}$ with $m\\in {\\mathbb N}$ for all $\\varepsilon\\geq 0$ small such that $\\Sigma_\\varepsilon^T\\Sigma_\\varepsilon=A_\\varepsilon$ in ${\\mathbb R}^n_0$ and \\eqref{eq:c1} holds for $\\Sigma_\\varepsilon$ in place of $A_\\varepsilon^{1/2}$. For clarity and simplicity, we restrict our attention to the case $\\Sigma_\\varepsilon=A_\\varepsilon^{1/2}$, which already covers the applications of interest.\n\nLet us present our main theorem.\n\n\\begin{equation}\\label{eq:c1}\n\t\t\t\\|A_\\vep(\\xi)^{1/2} -A_0(\\xi)^{1/2}\\|\\leq c_A\\vep^{\\alpha} (1+|\\xi|^\\beta)\n\t\t\\end{equation}\n\n\\begin{equation}\\label{general parabolic approx eq}\n\t\t\\partial_t u-\\tr(A_\\vep(\\nabla u)\\nabla^2 u)+H_\\vep(x, t, \\nabla u)=0 \\quad \\text{in $\\Om_T$}\n\t\\end{equation}\n\n\\begin{equation}\\label{parabolic approx bdry-cond}\n\t\tu =g_\\vep \\quad \\text{on $\\partial_p \\Om_T$}\n\t\\end{equation}\n\nLet us present our main theorem.\n\nOur proof of Theorem~\\ref{thm:parabolic} develops the standard comparison argument for possibly degenerate or singular parabolic equations, incorporating the H\\\"older regularity of solutions into the estimates. Without effecting these estimates, this result can be readily adapted to the spatially periodic setting by simply taking $\\Omega$ to be a torus; see Remark~\\ref{rmk:periodic} for more details. Moreover, the assumption of H\\\"older continuity in time in \\eqref{equi-holder} can be removed when vanishing gradients induce only a mild singularity of the equation. Such an improvement follows from the use of the parabolic Crandall–Ishii lemma, which avoids doubling the time variables and thereby simplifies the proof.\n\nOur quantitative stability result for the elliptic equation is as below. For the sake of completeness, we also include a proof, which to a large extent resembles that of Theorem~\\ref{thm:parabolic}. \n \\newpage\n \\begin{thm}[Quantitative stability for elliptic equations]\\label{thm:elliptic}\n Let $\\Omega$ be a bounded domain in $\\R^n$. For $\\vep\\geq 0$, assume that $A_\\vep: \\R^n_0\\to \\S_+^n$ and $H_\\vep: \\Omega\\times \\R\\times \\R^n\\to \\R$ are continuous functions satisfying (A1)(A2')(A3) and (C1)(C2'). Let $g_\\vep\\in C(\\partial\\Omega)$ and $u_\\vep\\in C(\\Oba)$ be $\\F$-solutions of \\eqref{general approx eq}\\eqref{approx bdry-cond}. Assume in addition that $u_\\vep$ are equi-H\\\"older continuous in $\\Oba$, that is, there exist $L>0$ and $0<\\theta\\leq 1$ such that \\[\n |u_\\vep(x_1)-u_\\vep(x_2)|\\leq L|x_1-x_2|^\\theta \n \\]\n for all $x_1, x_2\\in \\Oba$ and $\\vep\\geq 0$.\n Then, for all $\\vep>0$ small, \n \\begin{equation}\\label{error-est}\n \\max_{\\Oba} |u_\\vep-u_0|\\leq \\max_{\\partial \\Omega} |g_\\vep-g|+\\frac{1}{\\lambda}\\left(C_0\\vep^{\\nu}+c_H \\vep^\\gamma \\right)\n \\end{equation}\n holds with $\\nu$ given as in \\eqref{nu-exponent}, where $C_0>0$ is a constant depending on \n $k>2$,$L_A>0$, $L_H>0$, $\\lambda>0$ in (A1)(A2')(A3) and $\\alpha, \\beta, c_A, \\in \\R$ given in (C1). The constants $\\gamma, c_H>0$ are given as in (C2'). \n \\end{thm}\n\n\\begin{thm}\\label{thm:p-lap1par}\n Let $\\Omega\\subset \\R^n$ be a bounded domain and $T>0$. Assume that $H: \\Omega\\times (0,T)\\times \\R^n\\to \\R$ satisfies (A2). Fix $q\\in [1, \\infty)$ and take $p\\in [1, \\infty)$ close to $q$. For any such $p$, let $u_p\\in C(\\Om_T)$ be a solution to \\eqref{np-parabolic} satisfying \\eqref{parabolic bdry-cond} with with boundary data $g=g_p\\in C(\\partial_p \\Om_T)$. Assume in addition that there exist $L>0$ and $\\theta\\in (0, 1]$ such that\n \\[\n |u_p(x_1, t)-u_p(x_2, t)|\\leq L|x_1-x_2|^\\theta \n \\]\n for all $x_1, x_2\\in \\overline{\\Omega}$, $0\\leq t<T$ and all $p$ near $q$. Then, there exists $C>0$ independent of $p$, such that\n \\begin{equation}\\label{rate:p-lap1par}\n \\sup_{\\overline{\\Omega}\\times[0,T)}|u_p-u_q|\\leq \\sup_{\\partial_p \\Om_T}|g_p-g_q| +C|p-q|^{\\theta}. \n \\end{equation}\n\\end{thm}\n\\begin{proof}\n Note that for the normalized $p$-Laplacian, (A1) holds with $k>0$ large for all $q$ near $p$. \n Moreover, by a straightforward computation, we have\n \\[\n A_\\eps(\\xi)^{1/2}=I+(\\sqrt{p-1}-1) \\frac{\\xi\\otimes \\xi}{|\\xi|^{2}}, \\quad A_0(\\xi)^{1/2}=I+(\\sqrt{q-1}-1) \\frac{\\xi\\otimes \\xi}{|\\xi|^{2}}\n \\]\n with $p, q\\in (1, \\infty)$ satisfying $\\vep=|p-q|$. \n This yields the existence of $c>0$ such that\n \\[\n \\norm{A_\\vep(\\xi)^{1/2} -A_0(\\xi)^{1/2}}\\leq |\\sqrt{p-1}-\\sqrt{q-1}|\\leq c|p-q|\n \\]\n for all $q$ near $p$. This amounts to saying that (C1) holds with $\\alpha=1$ and $\\beta=0$. We thus complete the proof of inequality \\eqref{rate:p-lap1par} by applying Theorem~\\ref{thm:parabolic2} with $c_H=0$ and $\\nu=\\theta$. \n\\end{proof}\n\n\\begin{thm}\\label{thm:p-lap2par}\n Let $\\Omega\\subset \\R^n$ be a bounded domain and $T>0$. Assume that $H: \\Omega\\times (0,T)\\times \\R^n\\to \\R$ satisfies (A2). Fix $q\\in (1, \\infty)$ and take $p\\in (1, \\infty)$ close to $q$. For any such $p$, let $u_p\\in C(\\Om_T)$ be a solution to \\eqref{vp-parabolic} satisfying \\eqref{parabolic bdry-cond} with $g=g_p\\in C(\\partial_p \\Om_T)$. If for all $p$ close to $q$, $u_p$ are equi-H\\\"older continuous in $\\overline{\\Om_T}$ with exponent $\\theta\\in (0, 1]$, then the following results hold:\n \\begin{enumerate}\n \\item If $p<2$ and $1<q\\leq 2$, then for any $-1/2<\\beta<p/2-1$, there exists $C>0$ independent of $p$ near $q$ such that\n \\begin{equation}\\label{rate:p-lap2par-new}\n \\sup_{\\overline{\\Omega}\\times[0,T)}|u_p-u_q|\\leq \\sup_{\\partial_p\\Omega_T}|g_p-g_q| +C|p-q|^{\\theta}. \n \\end{equation}\n \\item If $p>2$ and $q\\geq 2$, then for any $\\beta>q/2-1$, there exists $C>0$ independent of $q$ near $p$ such that \n \\begin{equation}\\label{rate:p-lap2par}\n \\sup_{\\overline{\\Omega}\\times[0,T)}|u_p-u_q|\\leq \\sup_{\\partial_p\\Omega_T}|g_p-g_q| +C|p-q|^{\\frac{\\theta}{1+(1-\\theta)\\beta}}. \n \\end{equation}\n \\end{enumerate}\n\\end{thm}\nAs explained in Remark~\\ref{rmk holder}, for the case (2), one may relax the regularity of $u_\\vep$, assuming that they are equi-H\\\"older continuous only in $x$.\n\n\\begin{thm}\\label{thm:pq-lap1par}\n Let $\\Omega\\subset \\R^n$ be a bounded domain and $T>0$. Fix $(q, q')\\in (1, \\infty)^2$ and take $(p, p')$ close to $(q, q')$. For any such $(p, p')$, let $u_{p,p'}\\in C(\\Om_T)$ be a solution to \\eqref{eq:pq-parabolic} satisfying \\eqref{parabolic bdry-cond} with $g=g_{p,p'}\\in C(\\partial_p \\Om_T)$. Assume in addition that for all such $(p, p')$, $u_{p,p'}$ are equi-H\\\"older continuous in $\\overline{\\Om_T}$ with exponent $\\theta\\in (0, 1]$. Let $\\beta>-1/2$ be an arbitrary value satisfying \\eqref{beta-choice}. Then the following results hold:\n \\begin{enumerate}\n \\item For $p'> 2$ and $q'\\geq 2$, there exists $C>0$ independent of $(p, p')$ such that\n \\begin{equation}\\label{rate:pq-lap1par}\n \\sup_{\\ol{\\Omega}\\times[0,T)}|u_{p,p'}-u_{q,q'}|\\leq \\sup_{\\partial_p\\Omega_T}|g_{p,p'}-g_{q,q'}| +C(|p-q|+|p'-q'|)^{\\theta}. \n \\end{equation}\n \\item For $1<p'< 2$ and $1<q'\\leq 2$, there exists $C>0$ independent of $(p, p')$ such that\n \\begin{equation}\\label{rate:pq-lap1par2}\n \\sup_{\\ol{\\Omega}\\times[0,T)}|u_{p,p'}-u_{q,q'}|\\leq \\sup_{\\partial_p\\Omega_T}|g_{p,p'}-g_{q,q'}| +C(|p-q|+|p'-q'|)^{\\frac{\\theta}{1+(1-\\theta)\\beta}}. \n \\end{equation}\n \\item In the special case that $p'= q'\\in (1, \\infty)$, there exists $C>0$ independent of $q$ such that \n \\begin{equation}\\label{rate:pq-lap2par}\n \\sup_{\\ol{\\Omega}\\times[0,T)}|u_{p,p'}-u_{q,p'}|\\leq \\sup_{\\partial_p\\Omega_T}|g_{p,p'}-g_{q,p'}| +C|p-q|^{\\nu}, \n \\end{equation}\n holds with \n \\begin{equation}\\label{nu-pq}\n \\nu=\\frac{2\\theta}{2\\theta+(1-\\theta){q'}}. \n \\end{equation}\n \\end{enumerate}\n\\end{thm}\nIn view of Remark \\ref{rmk holder} and Theorem \\ref{thm:parabolic2}, in the case $p'> 2$, $q'\\geq 2$, we can drop the equi-H\\\"older continuity of $u_{p, p'}$ with respect to the time variable and only keep it for the space variable.\n\nAs for (C2), we have\n\\begin{equation*}\n \\abs{H_\\vep(x, t, \\xi)-H_0(x, t, \\xi)}=\\abs{a\\abs{\\xi} -a\\sqrt{\\abs{\\xi}^2+\\vep_2^2}}\\leq \\abs{a}\\vep_2,\n\\end{equation*}\nso it holds with $\\gamma=1$. We can now use Theorem \\ref{thm:parabolic} to obtain the following result. \n\\begin{thm}\n Let $\\Omega\\subset \\R^n$ be a bounded domain and $T>0$. For $\\eps=(\\vep_1, \\vep_2)$ with $\\vep_1, \\vep_2\\geq0$, let $u_{\\eps}\\in C(\\Om_T)$ be a solution to \\eqref{eq:biasedreg} with boundary value $u_\\vep=g_{\\eps}\\in C(\\partial_p \\Om_T)$. Let $u_0\\in C(\\Om_T)$ be a solution to \\eqref{eq:biased} satisfying $u_0=g_0\\in C(\\partial_p \\Om_T)$. Suppose that $u_{\\eps}$ are equi-H\\\"older continuous in $\\overline{\\Om_T}$ with exponent $\\theta\\in (0, 1]$ for all $\\vep_1, \\vep_2$ small. Then, there exists $C>0$ such that\n \\begin{equation*}\n \\sup_{\\ol{\\Om}\\times[0,T)}|u_{\\eps}-u_0|\\leq \\sup_{\\partial_p\\Omega_T}|g_{\\eps}-g_0| +C\\eps_1^{\\alpha\\theta}+C\\vep_2\n \\end{equation*}\n for any $\\alpha\\in (0, 1/2)$.\n\\end{thm}", "post_theorem_intro_text_len": 7874, "post_theorem_intro_text": "This result applies to important quasilinear parabolic equations with vanishing gradient singularities, for which typically (C1) holds with $\\beta<0$. While the parameter $\\beta<0$ represents the singularity strength, the term $\\max\\{\\beta, 0\\}$ in \\eqref{nu-exponent} indicates that our estimate \\eqref{error-est parabolic} does not capture any influence of this singularity. Moreover, for equi-Lipschitz solutions, corresponding to the case $\\theta=1$, we always have $\\nu=\\alpha$, and the behavior of $A_\\varepsilon(\\xi)$ with respect to the gradient variable $\\xi$ essentially plays no decisive role in determining the stability estimate.\n\n\tOur proof of Theorem~\\ref{thm:parabolic} develops the standard comparison argument for possibly degenerate or singular parabolic equations, incorporating the H\\\"older regularity of solutions into the estimates. Without effecting these estimates, this result can be readily adapted to the spatially periodic setting by simply taking $\\Omega$ to be a torus; see Remark~\\ref{rmk:periodic} for more details. Moreover, the assumption of H\\\"older continuity in time in \\eqref{equi-holder} can be removed when vanishing gradients induce only a mild singularity of the equation. Such an improvement follows from the use of the parabolic Crandall–Ishii lemma, which avoids doubling the time variables and thereby simplifies the proof.\n\n\tWe are also interested in the elliptic variant of such stability problems. Consider now\n\t\\begin{equation}\\label{general approx eq}\n\t\t-\\operatorname{tr}(A_\\varepsilon(\\nabla u)\\nabla^2 u)+H_\\varepsilon(x, u, \\nabla u)=0 \\quad \\text{in $\\Omega$}\n\t\\end{equation}\n\twith the Dirichlet boundary condition \n\t\\begin{equation}\\label{approx bdry-cond}\n\t\tu =g_\\varepsilon \\quad \\text{on $\\partial \\Omega$}\n\t\\end{equation}\n\tfor given $g_\\varepsilon\\in C(\\partial \\Omega)$. \n\tWhile our approach to parabolic problems can be readily extended to the elliptic case under similar assumptions on $A_\\varepsilon$ and $H_\\varepsilon$, we need to additionally assume a monotonicity condition on $H_\\varepsilon$ with respect to $u$ for all $\\varepsilon\\geq 0$. More precisely, we assume that there exists $\\lambda>0$ such that \n\t\\begin{equation}\\label{mono-0}\n\t\t\\rho\\mapsto H_\\varepsilon(x, \\rho, \\xi)-\\lambda \\rho \\quad \\text{is nondecreasing in ${\\mathbb R}$. }\n\t\\end{equation}\n\tWe obtain the same convergence rate as in Theorem \\ref{thm:parabolic}; see Theorem \\ref{thm:elliptic}. However, it is not clear to us how to drop or relax the monotonicity assumption \\eqref{mono-0}. \n\n\t\\subsection{Applications}\n\n\tOur general framework in Theorem~\\ref{thm:parabolic} yields stability estimates for a broad class of quasilinear parabolic equations including the normalized and variational $p$-Laplace equations and their further generalizations, as well as regularizing approximations for degenerate $p$-Laplace equations ($1\\leq p\\leq \\infty$). Below we summarize our major convergence rate results obtained for these equations. See Section~\\ref{sec:app} for precise statements, detailed analysis, concrete examples, and further discussions. \n\n\t\\begin{itemize}\n\t\t\\item (Normalized $p$-Laplacian) Let $u_p$ be a solution of \\eqref{np-parabolic0}\\eqref{parabolic bdry-cond} for $1\\leq p<\\infty$. For any fixed $1\\leq q<\\infty$, if $u_p$ are spatially equi-H\\\"older continuous with exponent $\\theta\\in (0, 1]$ for all $p$ near $q$, then $u_p\\to u_q$ uniformly as $p\\to q$ with convergence rate $O(|p-q|^\\theta)$. \n\t\t\\item (Variational $p$-Laplacian) Let $u_p$ be a solution of \\eqref{vp-parabolic0}\\eqref{parabolic bdry-cond} for $1<p<\\infty$. For any fixed $1\\leq q<\\infty$, if $u_p$ are equi-H\\\"older continuous with exponent $\\theta\\in (0, 1]$ for all $p$ near $q$, then, depending on $p$, $u_p\\to u_q$ uniformly as $p\\to q$ with convergence rate \n\t\t\\begin{enumerate}\n\t\t\t\\item[(i)] $O(|p-q|^\\theta)$ when $p<2$ (and hence $q\\leq 2$);\n\t\t\t\\item[(ii)] $O(|p-q|^\\nu)$ with any \n\t\t\t\\[\n\t\t\t0<\\nu< \\frac{2\\theta}{(1-\\theta)q+2\\theta}\n\t\t\t\\] \n\t\t\twhen $p>2$ (and hence $q\\geq 2$).\n\t\t\\end{enumerate}\n\t\t\\item (Regularization for generalized $p$-Laplacian) Let $p\\geq 1$, $p'\\geq 2$. For all $\\varepsilon>0$ small, let $u_\\varepsilon$ be a solution of \\eqref{regularized eq}\n\t\tand satisfy $u_\\varepsilon=g$ on $\\partial_p \\Omega_T$ for $g\\in C(\\partial_p\\Omega_T)$. As $\\varepsilon\\to 0$, $u_\\varepsilon\\to u$ uniformly, where $u$ is the unique solution of\n\t\t\\begin{equation*}\n\t\t\t\\partial_t u-\\left|\\nabla u\\right|^{p'-p}\\operatorname{div}\\left(\\left|\\nabla u\\right|^{p-2}\\nabla u\\right)=0\n\t\t\\end{equation*}\n\t\tsatisfying the same boundary condition. If $u_\\varepsilon$ is spatially equi-H\\\"older continuous with exponent $\\theta\\in (0, 1]$ for $\\varepsilon>0$ small, then we obtain the rate $O(\\varepsilon^\\nu)$ for the uniform convergence $u_\\varepsilon\\to u$, where $\\nu>0$ can be chosen as follows: \n\t\t\\begin{enumerate}\n\t\t\t\\item[(i)] If $p'=2$, then $0<\\nu< \\theta/2$;\n\t\t\t\\item[(ii)] If $2<p'< 3$, then $\\nu=(p'-2)\\theta$; \n\t\t\t\\item[(iii)] If $3\\leq p'\\leq 4$, then $0<\\nu<\\theta$;\n\t\t\t\\item[(iv)] If $p'>4$, then \n\t\t\t\\[\n\t\t\t0<\\nu< \\frac{\\theta}{1+(1-\\theta)(p'-4)}. \n\t\t\t\\]\n\t\t\\end{enumerate}\n\t\\end{itemize}\n\tNote that the Hölder regularity of solutions assumed in the results above is not the focus of this paper. We refer interested readers to \\cite{Do11, JS17, AP18, IJS19, Lee2025}, among many others, for related regularity estimates about these parabolic equations. \n\n\tIn addition to the applications above, our results can also be applied to study parabolic quantitative stability for general $p$-Laplace type equations that cover both the normalized and variational cases, as well as a regularizing approximation for biased infinity-Laplace equations arising in the stochastic tug-of-war games. Several examples for quasilinear elliptic equations are also discussed in Section~\\ref{sec:app}.  \n\n\tOur main theorem applies to the vanishing viscosity limit for Hamilton-Jacobi equations as well. For $\\varepsilon\\geq 0$ small, consider the solution $u_\\varepsilon$ to the initial boundary value problem for the Hamilton-Jacobi equation\n\t\\[\n\t\\partial_t u -\\varepsilon \\Delta u+H(x, t, \\nabla u)=0 \\quad \\text{in $\\Omega_T$,}\n\t\\]\n\twhere the Hamiltonian $H$ is assumed to satisfy the same regularity assumption as in (A2). Then, for fixed initial and boundary data independent of $\\varepsilon$, Theorem~\\ref{thm:parabolic} immediately yields the convergence rate $\\|u_\\varepsilon-u_0\\|_{L^\\infty(\\Omega_T)}=O(\\varepsilon^{1/2})$ for equi-Lipschitz solutions $u_\\varepsilon$ and the limit solution $u_0$ of the corresponding inviscid Hamilton-Jacobi equation. This recovers the classical result of \\cite{Fl}, originally established using an approach based on differential games, and later obtained in \\cite{CL1, CL2} via viscosity solution theory and in \\cite{Tr} using the nonlinear adjoint method. As this application is not the main focus of the present paper, we do not pursue it further, but instead refer the reader to the aforementioned references, as well as \\cite{QSTY, CD, CG} for more recent progress on improved rates in the convex Hamiltonian case. \n\n\tIn this work, we do not seek general optimality of the convergence rates. We only discuss the sharpness of our estimates through some examples, including in particular the stability of spatially periodic solutions to normalized $p$-Laplace equations under perturbations of $p$. The above convergence rates for the regularization of generalized $p$-Laplacian are consistent with the estimate obtained in \\cite{Mi} for Lipschitz solutions of regularized level set curvature flow equations, which corresponds to the case $p=1$, $p'=2$ in our general setting. \n\n\t\\subsection*{Acknowledgments}\n\n\tThe authors would like to thank Hiroyoshi Mitake and F\\'elix del~Teso for helpful discussions. The work of QL was supported by JSPS Grant-in-Aid for Scientific Research, No.~22K03396.", "sketch": "Our proof of Theorem~\\ref{thm:parabolic} \\emph{“develops the standard comparison argument for possibly degenerate or singular parabolic equations, incorporating the H\\\"older regularity of solutions into the estimates.”} The authors also note that, when vanishing gradients induce only a mild singularity, the \\emph{time} H\\\"older continuity in \\eqref{equi-holder} can be removed: this improvement \\emph{“follows from the use of the parabolic Crandall–Ishii lemma, which avoids doubling the time variables and thereby simplifies the proof.”}", "expanded_sketch": "Our proof of Theorem~\\ref{thm:parabolic} \\emph{“develops the standard comparison argument for possibly degenerate or singular parabolic equations, incorporating the H\\\"older regularity of solutions into the estimates.”} The authors also note that, when vanishing gradients induce only a mild singularity, the \\emph{time} H\\\"older continuity in\n\\begin{equation}\\label{equi-holder}\n\t\t\t|u_\\vep(x_1, t_1)-u_\\vep(x_2, t_2)|\\leq L(|x_1-x_2|^\\theta +|t_1-t_2|^\\theta) \n\t\t\\end{equation}\ncan be removed: this improvement \\emph{“follows from the use of the parabolic Crandall–Ishii lemma, which avoids doubling the time variables and thereby simplifies the proof.”}", "expanded_theorem": "[Quantitative stability for parabolic equations]\\label{thm:parabolic}\n\t\tLet $T>0$ and $\\Omega$ be a bounded domain in ${\\mathbb R}^n$. For $\\varepsilon\\geq 0$, assume that $A_\\varepsilon: {\\mathbb R}^n_0\\to \\S_+^n$ and ${H_\\varepsilon: \\Omega_T\\times {\\mathbb R}^n\\to {\\mathbb R}}$ are continuous functions satisfying (A1)(A2) and (C1)(C2). Let $g_\\varepsilon\\in C(\\partial_p \\Om_T)$ and $u_\\varepsilon\\in C(\\overline{\\Omega}\\times [0, T))$ be bounded ${\\mathcal F}$-solutions of\n\t\t\\begin{equation}\\label{general parabolic approx eq}\n\t\t\\partial_t u-\\tr(A_\\vep(\\nabla u)\\nabla^2 u)+H_\\vep(x, t, \\nabla u)=0 \\quad \\text{in $\\Om_T$}\n\t\\end{equation}\n\t\tand\n\t\t\\begin{equation}\\label{parabolic approx bdry-cond}\n\t\tu =g_\\vep \\quad \\text{on $\\partial_p \\Om_T$}\n\t\\end{equation}\n\t\tAssume in addition that $u_\\varepsilon$ are equi-H\\\"older continuous in $\\overline{\\Om_T}$, that is, there exist $L>0$ and $\\theta\\in (0, 1]$ such that,   \n\t\t\\begin{equation}\\label{equi-holder}\n\t\t\t|u_\\varepsilon(x_1, t_1)-u_\\varepsilon(x_2, t_2)|\\leq L(|x_1-x_2|^\\theta +|t_1-t_2|^\\theta) \n\t\t\\end{equation}\n\t\tfor all $(x_1, t_1), (x_2, t_2)\\in \\overline{\\Omega}\\times [0, T)$ and $\\varepsilon\\geq 0$ small. \n\t\tThen, for all $\\varepsilon>0$ small, \n\t\t\\begin{equation}\\label{error-est parabolic}\n\t\t\t\\sup_{\\overline{\\Omega}\\times [0, T)}|u_\\varepsilon-u_0|\\leq \\sup_{\\partial_p \\Om_T}|g_\\varepsilon-g_0|+C\\varepsilon^{\\nu}  +c_H T\\varepsilon^\\gamma,\n\t\t\\end{equation}\n\t\tholds with \n\t\t\\begin{equation}\\label{nu-exponent}\n\t\t\t\\nu=\\frac{\\alpha\\theta}{1+(1-\\theta)\\max\\{\\beta, 0\\}},    \n\t\t\\end{equation}\n\t\twhere $C>0$ depends on $T, n, k, L_A, L_H, \\beta, c_A$ as well as the H\\\"older exponent $\\theta$ and the constant $L$. Here, $k>2$, $L_A\\geq0$, $L_H\\geq0$ appear in (A1)(A2), and $\\alpha, \\beta, \\gamma, c_A, c_H\\in {\\mathbb R}$ are given in (C1)(C2).", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let \\(T>0\\) and let \\(\\Omega\\subset \\mathbb{R}^n\\) be a bounded domain. Write \\(\\Omega_T=\\Omega\\times(0,T)\\), \\(\\mathbb{R}^n_0=\\mathbb{R}^n\\setminus\\{0\\}\\), and \\(\\partial_p\\Omega_T=(\\Omega\\times\\{0\\})\\cup(\\partial\\Omega\\times[0,T))\\). For each \\(\\varepsilon\\ge 0\\), let \\(A_\\varepsilon:\\mathbb{R}^n_0\\to \\mathcal S_+^n\\) and \\(H_\\varepsilon:\\Omega_T\\times\\mathbb{R}^n\\to\\mathbb{R}\\) be continuous, where \\(\\mathcal S_+^n\\) is the set of nonnegative symmetric \\(n\\times n\\) matrices, and assume the following: (i) there exists \\(k>2\\) such that, for \\(f(x)=|x|^k\\), one has \\(\\operatorname{tr}(A_\\varepsilon(\\nabla f(x))\\nabla^2 f(x))\\to 0\\) as \\(x\\to 0\\) for all sufficiently small \\(\\varepsilon\\ge 0\\); (ii) there exists \\(L_H\\ge 0\\) such that \\(|H_\\varepsilon(x_1,t_1,\\xi)-H_\\varepsilon(x_2,t_2,\\xi)|\\le L_H(1+|\\xi|)(|x_1-x_2|+|t_1-t_2|)\\) for all \\((x_1,t_1),(x_2,t_2)\\in\\Omega_T\\), \\(\\xi\\in\\mathbb{R}^n\\), and all \\(\\varepsilon\\ge 0\\); (iii) there exist \\(c_A>0\\), \\(\\alpha>0\\), and \\(\\beta>\\frac{2-k}{2(k-1)}\\) such that, with \\(A_\\varepsilon(\\xi)^{1/2}\\) the nonnegative square root of \\(A_\\varepsilon(\\xi)\\), \\(\\|A_\\varepsilon(\\xi)^{1/2}-A_0(\\xi)^{1/2}\\|\\le c_A\\varepsilon^\\alpha(1+|\\xi|^\\beta)\\) for all \\(\\xi\\in\\mathbb{R}^n_0\\) and all sufficiently small \\(\\varepsilon>0\\); and (iv) there exist \\(c_H>0\\) and \\(\\gamma>0\\) such that \\(|H_\\varepsilon(x,t,\\xi)-H_0(x,t,\\xi)|\\le c_H\\varepsilon^\\gamma\\) for all \\((x,t)\\in\\Omega_T\\), \\(\\xi\\in\\mathbb{R}^n\\), and all sufficiently small \\(\\varepsilon>0\\). For each \\(\\varepsilon\\ge 0\\), let \\(g_\\varepsilon\\in C(\\partial_p\\Omega_T)\\), and let \\(u_\\varepsilon\\in C(\\overline\\Omega\\times[0,T))\\) be a bounded \\(\\mathcal F\\)-solution of\n\\[\n\\partial_tu-\\operatorname{tr}(A_\\varepsilon(\\nabla u)\\nabla^2u)+H_\\varepsilon(x,t,\\nabla u)=0\\quad\\text{in }\\Omega_T,\n\\]\nwith boundary condition \\(u=g_\\varepsilon\\) on \\(\\partial_p\\Omega_T\\). Assume also that the family \\(\\{u_\\varepsilon\\}\\) is equi-H\\\"older continuous on \\(\\overline\\Omega\\times[0,T)\\): there exist \\(L>0\\) and \\(\\theta\\in(0,1]\\) such that\n\\[\n|u_\\varepsilon(x_1,t_1)-u_\\varepsilon(x_2,t_2)|\\le L\\big(|x_1-x_2|^\\theta+|t_1-t_2|^\\theta\\big)\n\\]\nfor all \\((x_1,t_1),(x_2,t_2)\\in\\overline\\Omega\\times[0,T)\\) and all sufficiently small \\(\\varepsilon\\ge 0\\). Which statement holds for every such family?", "correct_choice": {"label": "A", "text": "For all sufficiently small \\(\\varepsilon>0\\), there exists a constant \\(C>0\\) depending only on the data in the assumptions such that\n\\[\n\\sup_{\\overline\\Omega\\times[0,T)}|u_\\varepsilon-u_0|\\le \\sup_{\\partial_p\\Omega_T}|g_\\varepsilon-g_0|+C\\varepsilon^{\\nu}+c_HT\\varepsilon^\\gamma,\n\\]\nwhere\n\\[\n\\nu=\\frac{\\alpha\\theta}{1+(1-\\theta)\\max\\{\\beta,0\\}}.\n\\]"}, "choices": [{"label": "B", "text": "For all sufficiently small \\(\\varepsilon>0\\), there exists a constant \\(C>0\\) depending only on the data in the assumptions such that\n\\[\n\\sup_{\\overline\\Omega\\times[0,T)}|u_\\varepsilon-u_0|\\le \\sup_{\\partial_p\\Omega_T}|g_\\varepsilon-g_0|+C\\varepsilon^{\\nu}+c_HT\\varepsilon^\\gamma,\n\\]\nwhere\n\\[\n\\nu=\\frac{\\alpha\\theta}{1+(1-\\theta)\\beta}.\n\\]"}, {"label": "C", "text": "For all sufficiently small \\(\\varepsilon>0\\), there exists a constant \\(C>0\\) depending only on the data in the assumptions such that\n\\[\n\\sup_{\\overline\\Omega\\times[0,T)}|u_\\varepsilon-u_0|\\le \\sup_{\\partial_p\\Omega_T}|g_\\varepsilon-g_0|+C\\varepsilon^{\\nu}+c_HT\\varepsilon^\\gamma\n\\]\nfor some exponent \\(\\nu>0\\) depending only on \\(\\alpha,\\beta,\\theta\\)."}, {"label": "D", "text": "For all sufficiently small \\(\\varepsilon>0\\), there exists a constant \\(C=C(\\varepsilon)>0\\) such that\n\\[\n\\sup_{\\overline\\Omega\\times[0,T)}|u_\\varepsilon-u_0|\\le \\sup_{\\partial_p\\Omega_T}|g_\\varepsilon-g_0|+C\\varepsilon^{\\nu}+c_HT\\varepsilon^\\gamma,\n\\]\nwhere\n\\[\n\\nu=\\frac{\\alpha\\theta}{1+(1-\\theta)\\max\\{\\beta,0\\}}.\n\\]"}, {"label": "E", "text": "For all sufficiently small \\(\\varepsilon>0\\), there exists a constant \\(C>0\\) depending only on the data in the assumptions such that\n\\[\n\\sup_{\\overline\\Omega\\times[0,T)}|u_\\varepsilon-u_0|\\le \\sup_{\\partial_p\\Omega_T}|g_\\varepsilon-g_0|+C\\varepsilon^{\\nu+\theta}+c_HT\\varepsilon^\\gamma,\n\\]\nwhere\n\\[\n\\nu=\\frac{\\alpha\\theta}{1+(1-\\theta)\\max\\{\\beta,0\\}}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "positive-part treatment of beta in the exponent", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "explicit sharp formula for the rate exponent", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "uniform independence of the constant from epsilon", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "sharp power of epsilon produced by the comparison estimate", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the conclusion or single out choice A. It gives hypotheses of a stability theorem, but no direct verbal or symbolic leakage of the exact bound appears in the question itself."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: after listing the full hypotheses, it asks for the exact conclusion. The correct choice is the theorem's stated estimate rather than a genuinely new conclusion derived from competing principles."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in subtle but meaningful ways (sharp exponent, dependence of the constant on epsilon, weaker existential form). However, the task mainly tests recognition of the precise theorem statement rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: dropping the positive-part in the exponent, replacing a sharp statement by a weaker true one, allowing epsilon-dependent constants, or altering the epsilon power. They are distinct and nontrivial."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-recognition MCQ with strong distractors and little answer leakage, but it is largely a restatement of a theorem rather than a deep test of generative reasoning."}}
{"id": "2602.12762v1", "paper_link": "http://arxiv.org/abs/2602.12762v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thrm:beat-cactus-introversion}\n    For all $n\\geq14$, minors of size $n(n-1)(n-2)+1$ of the tangency flattenings vanish on the $n$-th secant variety of the Segre variety in $\\P(\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n)$ but do not vanish on its $n$-th cactus variety.", "start_pos": 10252, "end_pos": 10549, "label": "thrm:beat-cactus-introversion"}, "ref_dict": {"thrm:beat-cactus-introversion": "\\begin{theorem}\n    \\label{thrm:beat-cactus-introversion}\n    For all $n\\geq14$, minors of size $n(n-1)(n-2)+1$ of the tangency flattenings vanish on the $n$-th secant variety of the Segre variety in $\\P(\\CC^n\\otimes\\CC^n\\otimes\\CC^n)$ but do not vanish on its $n$-th cactus variety.\n\\end{theorem}", "prop:1de-koszul": "\\begin{proposition}\n    \\label{prop:1de-koszul}\n    Let $R$ be the apolar algebra to $e\\leq \\binom{d+1}2$ general quadrics in $d$ variables. If $e\\geq3$, then the structure tensor $T$ of $R$ has border cactus rank at least $\\floor{\\frac32d}+3$. In particular, if $3\\leq e\\leq \\floor{\\frac d2}+1$, then $R$ is nonsmoothable.\n\\end{proposition}", "prop:quadratic-better-than-koszul": "\\begin{proposition}\n    \\label{prop:quadratic-better-than-koszul}\n    The space $\\im\\Phi_{\\mathrm Q}^C(T)\\subset A\\otimes B$ equals the sum of all $(T(\\alpha)\\otimes \\id_A)\\left(\\im\\koszul^{C,B}(T)\\right)$ over $\\alpha\\in A^*$, where we consider $T(\\alpha)$ as a map $C^*\\to B$. In particular, if $\\dim B=\\dim C$ and $T$ is $1_A$-generic, this implies $\\rank \\Phi_{\\mathrm Q}^C(T) \\geq \\rank\\koszul^{C,B}(T)$.\n\\end{proposition}", "thrm:beat-cactus": "\\begin{theorem}\n    \\label{thrm:beat-cactus}\n    The $(n(n-1)(n-2)+1)$-minors of the tangency flattenings for border rank $n$ in $\\CC^n\\otimes \\CC^n\\otimes \\CC^n$ do not vanish on the $n$-th cactus variety for all $n\\geq 14$.\n\\end{theorem}", "conj:tangency": "\\begin{conjecture}\n    \\label{conj:tangency}\n    Let $\\CC[x_1,\\dots,x_d]\\surjto R$ be a surjection to an $n$-dimensional algebra $R$, let $[R]$ be the corresponding point of $\\Hilb_n\\CC^d$ and let $T$ be the structure tensor of $R$. Then\n    \\[\n        \\rank\\tang^C(T) - n(n-1)(n-2) = dn - \\dim\\T_{[R]}\\Hilb_n\\CC^d.\n    \\]\n\\end{conjecture}", "subsec:mamu": "\\begin{proof}\n    Let us prove only the second set of equalities of \\eqref{eq:koszul-quaddratic-algebra}, since the proof of the first one is very similar.\n\n    Explicitly, the two flattenings in question map\n    \\begin{align*}\n        \\Phi_{\\mathrm Q}^A(T): \\bigwedge^2 R \\otimes R \\otimes R^* &\\to R \\otimes R^*,\\\\\n        (a\\wedge a')\\otimes b\\otimes \\gamma &\\mapsto (ab)\\otimes \\gamma(a'\\cdot\\bullet) - (a'b)\\otimes \\gamma(a\\cdot\\bullet)\n    \\end{align*}\n    and\n    \\begin{align*}\n        \\koszul^{A,C}(T): \\bigwedge^2 R \\otimes R^* &\\to R \\otimes R^*,\\\\\n        (a\\wedge a')\\otimes\\gamma &\\mapsto a\\otimes \\gamma(a'\\cdot\\bullet) - a'\\otimes \\gamma(a\\cdot\\bullet),\n    \\end{align*}\n    where we use $\\gamma(a\\cdot\\bullet)$ to denote the linear form $(x\\mapsto \\gamma(ax))\\in R^*$. We claim that in fact $\\Phi_{\\mathrm Q}^A(T)$ and $\\koszul^{A,C}(T)$ have the same images. The inclusion $\\im\\Phi^{A,C}(T)\\subseteq \\im\\Phi_{\\mathrm Q}^A$ is obvious by considering $b=1$ in the latter map.\n\n    For the opposite inclusion, note that invertible elements form a dense set in $R$, so in particular, $R$ may be spanned by them, hence $\\im \\Phi_{\\mathrm Q}^A$ is spanned by images of $(a\\wedge a')\\otimes b\\otimes\\gamma$ with $b$ invertible. These lie in $\\im \\koszul^{A,C}$ by the following: we set\n    \\[\n        \\tilde a := ba, \\qquad \\tilde a' := ba', \\qquad \\tilde\\gamma := \\gamma(b^{-1}\\cdot\\bullet),\n    \\]\n    which then results in\n    \\begin{multline*}\n        \\koszul^{A,C}(T)\\left(\n            (\\tilde a\\wedge \\tilde a') \\otimes \\tilde \\gamma\n        \\right)\n        =\n        \\tilde a\\otimes\\tilde \\gamma(\\tilde a'\\cdot\\bullet) - \\tilde a'\\otimes\\tilde\\gamma(\\tilde a\\cdot\\bullet)\n        =\n        (ab)\\otimes\\gamma(b^{-1}ba'\\cdot\\bullet) - (a'b)\\otimes \\gamma(b^{-1}ba\\cdot\\bullet)\n        =\\\\=\n        (ab)\\otimes \\gamma(a'\\cdot\\bullet)-(a'b)\\otimes\\gamma(a\\cdot\\bullet)\n        =\n        \\Phi_{\\mathrm Q}^A(T)\\left(\n            (a\\wedge a') \\otimes b\\otimes\\gamma\n        \\right).\n    \\end{multline*}\n    This justifies $\\rank \\Phi_{\\mathrm Q}^A(T) = \\rank \\koszul^{A,C}(T)$. That this common rank equals $n(n-1)$ then follows from Lemma \\ref{lem:easy rank Koszul for algebras}.\n\\end{proof}\n\n\\section{Applications}\n\\label{sec:applications}\n\n\\subsection{Matrix multiplication}\n\\label{subsec:mamu}\n\nIt is well-known that the border rank of the $2\\times 2$ matrix multiplication tensor $M_2$ equals $7$ \\cite{Landsberg2006, Hauenstein-Ikenmeyer-Landsberg}. However, only recently the ``first hand-checkable algebraic proof'' appeared \\cite{Conner_Harper_Landsberg_2023}. Below we show that tangency flattenings provide an easy proof.\n\\begin{proposition}\\label{prop:M2=7}\n    The border rank of $M_2$ equals seven.\n\\end{proposition}\n\\begin{proof}\nStrassen \\cite{strassen1969gaussian} provided an explicit rank seven decomposition of $M_2$. We focus on lower-bounding the border rank.\n\n    By Corollary \\ref{cor:tan_flat} for $n=4$, $q=6$ we have to show that the tangency flattening has rank greater than $60$. We will prove that it is $64$. We fix a basis $(i,j)$ of $\\CC^4$, where $i,j\\in\\{0,1\\}$ and we use $\\neg i$ to denote the negation of $i$. Recall that $M_2=\\sum_{i,j,k\\in \\{0,1\\}} (i,j)\\otimes (j,k)\\otimes (k,i)$.\n\n    We will prove that the tangency flattening is a block diagonal matrix with five blocks: a $48\\times 48$ identity matrix and four isomorphic, nondegenerate $4\\times 4$ matrices.\n\nThe tangency flattening of $M_2$ is the restriction to $\\bigwedge^3\\CC^4\\otimes \\CC^4\\otimes \\CC^4$ of\n\\begin{align}\n\\label{eq:mamu-tangency}\n(\\CC^4)^{\\otimes 3}\\otimes \\CC^4\\otimes \\CC^4 &\\to \\CC^4\\otimes \\CC^4\\otimes \\CC^4,\\\\\n\\nonumber\n(i_1,j_1)\\otimes (i_2,j_2)\\otimes (i_3,j_3)\\otimes (i_4,j_4)\\otimes (i_5,j_5) &\\mapsto\n\\begin{cases}\n(i_1,j_1)\\otimes (i_2,j_4)\\otimes (i_5,j_3), & \\text{if $j_2=i_4$ and $i_3 = j_5$,}\\\\\n0, & \\text{otherwise.}\n\\end{cases}\n\\end{align}\nWe divide the basis vectors in the domain into five groups.\n\nThe first group is formed by vectors of the form $\\bigl((\\neg j_5,\\neg i_4)\\wedge(i_2,i_4)\\wedge (j_5,j_3)\\bigr)\\otimes (i_4,j_4)\\otimes (i_5,j_5)$. By first choosing $i_4,j_4,i_5,j_5$ (16 options), and then choosing exterior products that contain $(\\neg j_5,\\neg i_4)$ (3 options), we see there are $48$ such (nonzero) vectors.\nNote that this already forces the exterior product to be of the form above, since the two remaining exterior factors are chosen from $(\\neg j_5, i_4)$, $(j_5, \\neg i_4)$ and $(j_5,i_4)$.\nAlthough the exterior product expands as a signed sum of six tensor products, only one of those -- the one ordered as in the presentation above -- maps to a nonzero vector in \\eqref{eq:mamu-tangency}. Thus, the image of each such basis vector is $(\\neg j_5,\\neg i_4)\\otimes (i_2,j_4)\\otimes (i_5,j_3)$. Note that such images must satisfy $(\\neg j_5,\\neg i_4)\\neq (\\neg i_2,\\neg j_3)$.\n\nThe next four groups are indexed by $j_4,i_5\\in \\{0,1\\}$. For fixed $j_4,i_5$ the group consists of the four basis vectors $\\bigl((j_5,i_4)\\wedge (\\neg j_5,i_4)\\wedge (j_5,\\neg i_4)\\bigr)\\otimes (i_4,j_4)\\otimes (i_5,j_5)$, where $i_4, j_5$ are arbitrary.\nThe image of each such vector is:\n\\[(j_5,i_4)\\otimes (\\neg j_5,j_4)\\otimes (i_5,\\neg i_4)-(\\neg j_5,i_4)\\otimes (j_5,j_4)\\otimes (i_5,\\neg i_4)-(j_5,\\neg i_4)\\otimes (\\neg j_5,j_4)\\otimes (i_5,i_4).\\]\nIn particular, the images are linear combinations of basis vectors that are distinct from images of the first group as well as other groups -- in each term, the first tensor factor consist exactly of  negations of the first coordinate of the second factor  and the second coordinate of the third factor. Meanwhile, the group is determined by reading off $j_4$ and $i_5$ from the second coordinate of the second factor and first coordinate of the third factor, respectively.\n\nThus indeed the tangency flattening is in block-diagonal form. It remains for us to prove that the isomorphic $4\\times 4$ blocks are of full rank.\nEach such matrix has the following form, where we label $(j_5,i_4)\\otimes (\\neg j_5,j_4)\\otimes (i_5,\\neg i_4)$ with the first factor $(j_5,i_4)$ and subsequently order rows and columns by $(j_5,i_4)=(0,0)$, $(0,1)$, $(1,0)$, $(1,1)$:\n\\[\\begin{pmatrix}\n    1 &  -1&-1&0\\\\\n    -1& 1&0&-1\\\\\n    -1& 0&1&-1\\\\\n    0& -1&-1&1\\\\\n\\end{pmatrix}.\\]\nThe determinant of such a matrix is $-3$, thus indeed the matrix is of full rank, and so is the tangency flattening.\n\\end{proof}", "cor:tan_flat": "\\begin{corollary}\\label{cor:tan_flat}\n    If $\\dim V_1=\\dim V_2=\\dim V_3=n$, then $q(q-1)(n-2)+1$ minors of the tangency flattening provide equations for border rank $q$.\n\\end{corollary}"}, "pre_theorem_intro_text_len": 6726, "pre_theorem_intro_text": "\\label{sec:intro}\nThe central problem concerning tensors is that of \\emph{tensor decomposition} -- presenting a tensor as a sum of simple tensors, or computing the smallest possible size of a decomposition, called the \\emph{tensor rank}. While the problem is NP-hard in general \\cite{RankNPcomplete, hillar2013most}, providing estimates for rank or related notions such as border rank or asymptotic rank has been a fruitful field. Perhaps the most famous application is the study of computational complexity of matrix multiplication (e.g. \\cite{strassen1969gaussian, coppersmith1982asymptotic, alman2024refined, williams2024new}). Further, a recent line of research relates low asymptotic rank of certain special tensors to faster than known algorithms for NP-hard problems \\cite{bjorklund2025fast, bjorklund2024asymptotic, pratt2024stronger, MaMu-SotA}.\nFor an introduction to tensors from an algebraic perspective, we refer the reader to \\cite[Chapter 9.2]{michalek2021invitation}, or to \\cite{landsberg2011tensors} for a more extensive exposition. For the connections to complexity theory, we refer to \\cite{burgisser2013algebraic, landsberg2017geometry}.\n\nThe main topic of this article is obtaining lower bounds on the border rank of tensors, a longstanding question in tensor geometry. The classical approach to the problem is by providing equations for secant varieties to the Segre variety. Flattenings, or maps which produce a matrix from a given tensor, have been a bountiful source of such lower bounds (e.g. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-ottaviani-newbounds, landsberg-michalek-mamu, landsberg-michalek-haystack}), with the equations thus obtained coming in the form of minors. In particular, \\emph{Strassen's equations}, further studied by Landsberg and Manivel and  expanded by Landsberg and Ottaviani to \\emph{Koszul} and \\emph{Young flattenings}, became the gold standard in the field. Apart from flattenings, the techniques of \\emph{border apolarity} (see \\cite{Buczynscy-apolarity}) and \\emph{border substitution} (see \\cite{landsberg2017geometry}) have been introduced and used to prove several new lower bounds in recent years. However, all of these methods have quite restrictive limits of application.\n\nIndeed, current lower bound methods fall far short of being able to prove a tensor in $\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n$ to have the generic border rank, which is approximately $\\frac{n^2}3$. Instead, known flattening methods give maximal lower bounds of up to $(2-\\epsilon)n$ for small $\\epsilon>0$. Combining algebraic equations with other methods, it is possible to provide examples of tensors of border rank $2.02n$ \\cite{landsberg-michalek-haystack}. Still, there are well understood barriers which flattening methods cannot beat \\cite{jarek_cactus, Galazka-bundles, EGOW18, garg2019more}. In contrast, the applications to asymptotic rank in computer science would require superlinear lower bounds on border rank.\n\nThe disparity between the generic border rank and available lower bounds has been attributed to the inclusion of secant varieties in \\emph{cactus varieties} (see Section~\\ref{sec:preliminaries}). In particular, known flattening methods have been unable to distinguish secant from cactus varieties, even when the inclusion is known to be strict  -- quoting from \\cite{Conner_Harper_Landsberg_2023}:\n\\begin{quote}\n``The geometric interpretation of the border rank lower bound barriers of \\cite{EGOW18, Galazka-bundles} is that all equations obtained by taking minors, called rank methods, are actually equations for a larger variety than $\\sigma_r(\\Seg(\\P A\\times\\P B\\times\\P C))$, called the $r$-th cactus variety \\cite{Buczynscy-apolarity}.''\n\\end{quote}\nIndeed, as proved in \\cite{jarek_cactus, Galazka-bundles, galazka-thesis} all of the known methods producing equations for the $r$-th secant variety are equations for the $r$-th cactus variety. Further, it is known that the $r$-th cactus variety of the Segre variety in $(\\mathbb{C}^n)^{\\ot 3}$ fills the ambient space for $r$ growing linearly in $n$ \\cite{ballico2019note, bernardi2018polynomials, galkazka2023multigraded}.\n\nFor these reasons, the existence of easily computable determinantal equations vanishing on secant varieties but not vanishing on cactus varieties, has long been considered unlikely or even impossible in the tensor-geometric community.\nIn this article, we illustrate that it is not the determinantal expression which creates the cactus barrier, but rather the linear embedding of the tensor product into a matrix space.\nThus the starting point of our approach in this article is focusing on \\emph{nonlinear} maps from tensor spaces to matrix spaces.\nTo obtain bounds on matrix rank with linear embeddings, it is enough to investigate the embedded image of the Segre variety and appeal to linearity. With nonlinear maps on the other hand, we need to work more generally, but this does yield stronger results. We obtain a new method, which we dub \\emph{Kronecker-Koszul flattenings}, or more generally \\emph{Kronecker-Young flattenings}, that allows one to produce\nexplicit and efficiently computable determinantal equations of secant varieties that do not generally vanish on cactus varieties. In particular, Kronecker-Koszul flattenings require less computation time and memory than methods based on a more direct identification of ideals defining the secant varieties (cf.~\\cite{Hauenstein-Ikenmeyer-Landsberg}).\n\n Broadly speaking, our method starts by tensoring together several copies of the same tensor. This, depending on the interpretation of the codomain, may be viewed as a Kronecker power or Veronese embedding \\cite{kaski2025universal}, and thus is the universal map for fixed degree polynomials. Next, we further tensor with several $2$-tensors corresponding to identity maps on various vector spaces.  Finally, we group and contract some of the factors to exterior powers. This results in what we call a \\emph{Kronecker-Koszul tensor}. A Kronecker-Koszul flattening is then a classical flattening of the Kronecker-Koszul tensor. Note that due to initially taking a tensor power, Kronecker-Koszul tensors are in general nonlinear as functions of the starting tensor.\nSee Section~\\ref{sec:construction} for details of the construction. There we also comment on what we call {Kronecker-Young flattenings}, a further generalization which synthesizes the approach of Young flattenings into the method.\n\nOur main result uses the \\emph{tangency flattening}, a very particular Kronecker-Koszul flattening which depends quadratically on a tensor. See the paragraph preceding Corollary~\\ref{cor:tan_flat} for the definition.", "context": "The main topic of this article is obtaining lower bounds on the border rank of tensors, a longstanding question in tensor geometry. The classical approach to the problem is by providing equations for secant varieties to the Segre variety. Flattenings, or maps which produce a matrix from a given tensor, have been a bountiful source of such lower bounds (e.g. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-ottaviani-newbounds, landsberg-michalek-mamu, landsberg-michalek-haystack}), with the equations thus obtained coming in the form of minors. In particular, \\emph{Strassen's equations}, further studied by Landsberg and Manivel and expanded by Landsberg and Ottaviani to \\emph{Koszul} and \\emph{Young flattenings}, became the gold standard in the field. Apart from flattenings, the techniques of \\emph{border apolarity} (see \\cite{Buczynscy-apolarity}) and \\emph{border substitution} (see \\cite{landsberg2017geometry}) have been introduced and used to prove several new lower bounds in recent years. However, all of these methods have quite restrictive limits of application.\n\nIndeed, current lower bound methods fall far short of being able to prove a tensor in $\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n$ to have the generic border rank, which is approximately $\\frac{n^2}3$. Instead, known flattening methods give maximal lower bounds of up to $(2-\\epsilon)n$ for small $\\epsilon>0$. Combining algebraic equations with other methods, it is possible to provide examples of tensors of border rank $2.02n$ \\cite{landsberg-michalek-haystack}. Still, there are well understood barriers which flattening methods cannot beat \\cite{jarek_cactus, Galazka-bundles, EGOW18, garg2019more}. In contrast, the applications to asymptotic rank in computer science would require superlinear lower bounds on border rank.\n\nThe disparity between the generic border rank and available lower bounds has been attributed to the inclusion of secant varieties in \\emph{cactus varieties} (see Section~\\ref{sec:preliminaries}). In particular, known flattening methods have been unable to distinguish secant from cactus varieties, even when the inclusion is known to be strict -- quoting from \\cite{Conner_Harper_Landsberg_2023}:\n\\begin{quote}\n``The geometric interpretation of the border rank lower bound barriers of \\cite{EGOW18, Galazka-bundles} is that all equations obtained by taking minors, called rank methods, are actually equations for a larger variety than $\\sigma_r(\\Seg(\\P A\\times\\P B\\times\\P C))$, called the $r$-th cactus variety \\cite{Buczynscy-apolarity}.''\n\\end{quote}\nIndeed, as proved in \\cite{jarek_cactus, Galazka-bundles, galazka-thesis} all of the known methods producing equations for the $r$-th secant variety are equations for the $r$-th cactus variety. Further, it is known that the $r$-th cactus variety of the Segre variety in $(\\mathbb{C}^n)^{\\ot 3}$ fills the ambient space for $r$ growing linearly in $n$ \\cite{ballico2019note, bernardi2018polynomials, galkazka2023multigraded}.\n\nFor these reasons, the existence of easily computable determinantal equations vanishing on secant varieties but not vanishing on cactus varieties, has long been considered unlikely or even impossible in the tensor-geometric community.\nIn this article, we illustrate that it is not the determinantal expression which creates the cactus barrier, but rather the linear embedding of the tensor product into a matrix space.\nThus the starting point of our approach in this article is focusing on \\emph{nonlinear} maps from tensor spaces to matrix spaces.\nTo obtain bounds on matrix rank with linear embeddings, it is enough to investigate the embedded image of the Segre variety and appeal to linearity. With nonlinear maps on the other hand, we need to work more generally, but this does yield stronger results. We obtain a new method, which we dub \\emph{Kronecker-Koszul flattenings}, or more generally \\emph{Kronecker-Young flattenings}, that allows one to produce\nexplicit and efficiently computable determinantal equations of secant varieties that do not generally vanish on cactus varieties. In particular, Kronecker-Koszul flattenings require less computation time and memory than methods based on a more direct identification of ideals defining the secant varieties (cf.~\\cite{Hauenstein-Ikenmeyer-Landsberg}).\n\nBroadly speaking, our method starts by tensoring together several copies of the same tensor. This, depending on the interpretation of the codomain, may be viewed as a Kronecker power or Veronese embedding \\cite{kaski2025universal}, and thus is the universal map for fixed degree polynomials. Next, we further tensor with several $2$-tensors corresponding to identity maps on various vector spaces. Finally, we group and contract some of the factors to exterior powers. This results in what we call a \\emph{Kronecker-Koszul tensor}. A Kronecker-Koszul flattening is then a classical flattening of the Kronecker-Koszul tensor. Note that due to initially taking a tensor power, Kronecker-Koszul tensors are in general nonlinear as functions of the starting tensor.\nSee Section~\\ref{sec:construction} for details of the construction. There we also comment on what we call {Kronecker-Young flattenings}, a further generalization which synthesizes the approach of Young flattenings into the method.\n\nOur main result uses the \\emph{tangency flattening}, a very particular Kronecker-Koszul flattening which depends quadratically on a tensor. See the paragraph preceding Corollary~\\ref{cor:tan_flat} for the definition.\n\n\\begin{corollary}\\label{cor:tan_flat}\n    If $\\dim V_1=\\dim V_2=\\dim V_3=n$, then $q(q-1)(n-2)+1$ minors of the tangency flattening provide equations for border rank $q$.\n\\end{corollary}", "full_context": "The main topic of this article is obtaining lower bounds on the border rank of tensors, a longstanding question in tensor geometry. The classical approach to the problem is by providing equations for secant varieties to the Segre variety. Flattenings, or maps which produce a matrix from a given tensor, have been a bountiful source of such lower bounds (e.g. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-ottaviani-newbounds, landsberg-michalek-mamu, landsberg-michalek-haystack}), with the equations thus obtained coming in the form of minors. In particular, \\emph{Strassen's equations}, further studied by Landsberg and Manivel and expanded by Landsberg and Ottaviani to \\emph{Koszul} and \\emph{Young flattenings}, became the gold standard in the field. Apart from flattenings, the techniques of \\emph{border apolarity} (see \\cite{Buczynscy-apolarity}) and \\emph{border substitution} (see \\cite{landsberg2017geometry}) have been introduced and used to prove several new lower bounds in recent years. However, all of these methods have quite restrictive limits of application.\n\nIndeed, current lower bound methods fall far short of being able to prove a tensor in $\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n$ to have the generic border rank, which is approximately $\\frac{n^2}3$. Instead, known flattening methods give maximal lower bounds of up to $(2-\\epsilon)n$ for small $\\epsilon>0$. Combining algebraic equations with other methods, it is possible to provide examples of tensors of border rank $2.02n$ \\cite{landsberg-michalek-haystack}. Still, there are well understood barriers which flattening methods cannot beat \\cite{jarek_cactus, Galazka-bundles, EGOW18, garg2019more}. In contrast, the applications to asymptotic rank in computer science would require superlinear lower bounds on border rank.\n\nThe disparity between the generic border rank and available lower bounds has been attributed to the inclusion of secant varieties in \\emph{cactus varieties} (see Section~\\ref{sec:preliminaries}). In particular, known flattening methods have been unable to distinguish secant from cactus varieties, even when the inclusion is known to be strict -- quoting from \\cite{Conner_Harper_Landsberg_2023}:\n\\begin{quote}\n``The geometric interpretation of the border rank lower bound barriers of \\cite{EGOW18, Galazka-bundles} is that all equations obtained by taking minors, called rank methods, are actually equations for a larger variety than $\\sigma_r(\\Seg(\\P A\\times\\P B\\times\\P C))$, called the $r$-th cactus variety \\cite{Buczynscy-apolarity}.''\n\\end{quote}\nIndeed, as proved in \\cite{jarek_cactus, Galazka-bundles, galazka-thesis} all of the known methods producing equations for the $r$-th secant variety are equations for the $r$-th cactus variety. Further, it is known that the $r$-th cactus variety of the Segre variety in $(\\mathbb{C}^n)^{\\ot 3}$ fills the ambient space for $r$ growing linearly in $n$ \\cite{ballico2019note, bernardi2018polynomials, galkazka2023multigraded}.\n\nFor these reasons, the existence of easily computable determinantal equations vanishing on secant varieties but not vanishing on cactus varieties, has long been considered unlikely or even impossible in the tensor-geometric community.\nIn this article, we illustrate that it is not the determinantal expression which creates the cactus barrier, but rather the linear embedding of the tensor product into a matrix space.\nThus the starting point of our approach in this article is focusing on \\emph{nonlinear} maps from tensor spaces to matrix spaces.\nTo obtain bounds on matrix rank with linear embeddings, it is enough to investigate the embedded image of the Segre variety and appeal to linearity. With nonlinear maps on the other hand, we need to work more generally, but this does yield stronger results. We obtain a new method, which we dub \\emph{Kronecker-Koszul flattenings}, or more generally \\emph{Kronecker-Young flattenings}, that allows one to produce\nexplicit and efficiently computable determinantal equations of secant varieties that do not generally vanish on cactus varieties. In particular, Kronecker-Koszul flattenings require less computation time and memory than methods based on a more direct identification of ideals defining the secant varieties (cf.~\\cite{Hauenstein-Ikenmeyer-Landsberg}).\n\nBroadly speaking, our method starts by tensoring together several copies of the same tensor. This, depending on the interpretation of the codomain, may be viewed as a Kronecker power or Veronese embedding \\cite{kaski2025universal}, and thus is the universal map for fixed degree polynomials. Next, we further tensor with several $2$-tensors corresponding to identity maps on various vector spaces. Finally, we group and contract some of the factors to exterior powers. This results in what we call a \\emph{Kronecker-Koszul tensor}. A Kronecker-Koszul flattening is then a classical flattening of the Kronecker-Koszul tensor. Note that due to initially taking a tensor power, Kronecker-Koszul tensors are in general nonlinear as functions of the starting tensor.\nSee Section~\\ref{sec:construction} for details of the construction. There we also comment on what we call {Kronecker-Young flattenings}, a further generalization which synthesizes the approach of Young flattenings into the method.\n\nOur main result uses the \\emph{tangency flattening}, a very particular Kronecker-Koszul flattening which depends quadratically on a tensor. See the paragraph preceding Corollary~\\ref{cor:tan_flat} for the definition.\n\n\\begin{corollary}\\label{cor:tan_flat}\n    If $\\dim V_1=\\dim V_2=\\dim V_3=n$, then $q(q-1)(n-2)+1$ minors of the tangency flattening provide equations for border rank $q$.\n\\end{corollary}\n\nFor these reasons, the existence of easily computable determinantal equations vanishing on secant varieties but not vanishing on cactus varieties, has long been considered unlikely or even impossible in the tensor-geometric community.\nIn this article, we illustrate that it is not the determinantal expression which creates the cactus barrier, but rather the linear embedding of the tensor product into a matrix space.\nThus the starting point of our approach in this article is focusing on \\emph{nonlinear} maps from tensor spaces to matrix spaces.\nTo obtain bounds on matrix rank with linear embeddings, it is enough to investigate the embedded image of the Segre variety and appeal to linearity. With nonlinear maps on the other hand, we need to work more generally, but this does yield stronger results. We obtain a new method, which we dub \\emph{Kronecker-Koszul flattenings}, or more generally \\emph{Kronecker-Young flattenings}, that allows one to produce\nexplicit and efficiently computable determinantal equations of secant varieties that do not generally vanish on cactus varieties. In particular, Kronecker-Koszul flattenings require less computation time and memory than methods based on a more direct identification of ideals defining the secant varieties (cf.~\\cite{Hauenstein-Ikenmeyer-Landsberg}).\n\nOur main result uses the \\emph{tangency flattening}, a very particular Kronecker-Koszul flattening which depends quadratically on a tensor. See the paragraph preceding Corollary~\\ref{cor:tan_flat} for the definition.\n\nWhile Kronecker-Koszul flattenings generalize Koszul flattenings (cf. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-michalek-mamu, landsberg-michalek-haystack}), we also analyze in Section~\\ref{sec:koszul} bounds that may be extracted via Koszul flattenings for the border rank (and border cactus rank) of structure tensors of certain algebras. It is known \\cite{blaser2016degeneration} that a (finite) algebra is smoothable if and only if the associated structure tensor has minimal border rank. This is also one of the main challenges in application of the border apolarity method.\nFurther, two of the classical flattenings of structure tensors of algebras give rise to spaces of commuting matrices. Thus, many of the flattening methods cannot provide nontrivial bounds on border rank of such tensors. However, we show that it is possible to obtain lower bounds on border cactus rank of structure tensors of algebras strictly greater than the degree of the algebra, even using Koszul flattenings (see Proposition~\\ref{prop:1de-koszul}).\n\n\\begin{corollary}\\label{cor:flat bound}\n If $T$ has border rank at most $q$, then the rank of a Kronecker-Koszul flattening of $T$ is at most $F(q)\\cdot \\prod_{i=1}^r\\prod_{j=1}^{s_i} \\binom{d_i-|\\lambda_{i,j}|}{d_{i,j}'}$. In particular, minors of the Kronecker-Koszul flattening provide equations for secant varieties of the Segre variety.\n\\end{corollary}\n\\begin{proof}\n Follows from Theorem \\ref{thm:main}, as ranks of classical flattenings lower bound border rank of the tensor.\n\\end{proof}\nThe bounds in Theorem \\ref{thm:main} and Corollary \\ref{cor:flat bound} do not have to be tight. This can happen for various reasons, some more opaque than others:\n\nIn this section we present one explicit construction that we find particularly interesting. By \\cite{kaski2025universal} the linear span of image of the $k$-th Kronecker power of tensors in $A\\ot B \\ot C$ is $S^k(A\\ot B \\ot C)$. As a $\\GL(A)\\times \\GL(B)\\times \\GL(C)$ representation, the latter space has a decomposition into isotypic components\n \\[S^k(A\\ot B \\ot C)=\\bigoplus_{\\lambda,\\mu,\\rho\\vdash k}\\left(S^\\lambda(A)\\ot S^\\mu(B)\\ot S^\\rho(C)\\right)^{\\bigoplus K_{\\lambda,\\mu,\\rho}},\\]\n where $K_{\\lambda,\\mu,\\rho}$ is the Kronecker coefficient. In particular, when $\\lambda=\\mu=1^k$ and $\\rho=(k)$ we have $K_{\\lambda,\\mu,\\rho}=1$. This gives a canonical projection\n \\[\\pi:S^k(A\\ot B \\ot C)\\rightarrow \\bigwedge^k A\\ot \\bigwedge^k B\\ot S^k(C).\\]\n An example of Kronecker-Young flattening is the classical flattening of $\\pi(T^{\\otimes k})$ given by\n \\[\\Phi_{\\mathrm M,k}^C(T):\\bigwedge^kA^*\\ot\\bigwedge^kB^*\\rightarrow S^k(C).\\]\n The advantage of these choices is that the image of the map $\\Phi_{\\mathrm M,k}^C(T)$ has a direct interpretation in terms of the flattening $T^C:C^*\\rightarrow A\\ot B$. Namely, the image of $\\Phi_{\\mathrm M,k}^C(T)$ is the linear span of all $k\\times k$ minors of $T^C$ viewed as a matrix with entries from $C$. For this reason, we call $\\Phi_{\\mathrm M,k}$ the \\emph{$k$-minor flattening}.\n\\begin{lemma}\n If $\\dim A=\\dim B=\\dim C=n$ and a tensor $T\\in A\\ot B\\ot C$ has border rank at most $n$, then the rank of the $k$-minor flattening $\\Phi_{\\mathrm M,k}^C(T)$ is at most $\\binom{n}{k}$.\n\\end{lemma}\n\\begin{proof}\n The $n$-th secant variety of the Segre variety is the closure of the $\\GL(A)\\times \\GL(B)\\times \\GL(C)$ orbit of the unit tensor $I_n:=\\sum_{i=1}^n a_i\\ot b_i\\ot c_i$, where $a_i$, $b_i$, $c_i$ form respectively bases of $A$, $B$, $C$. As our constructions are continuous and equivariant, it is enough to prove that $\\rk \\Phi_{\\mathrm M,k}^C(I_n)=\\binom{n}{k} $. We note that $I_n^C(C^*)$ is the space of diagonal matrices. Clearly, the linear span of $k\\times k$ minors of the diagonal matrices is the linear span of squarefree monomials of degree $k$ in $n$ variables and thus has dimension $\\binom{n}{k}$.\n\\end{proof}\n\n\\begin{corollary}\n \\label{cor:alg-tangency-sum}\n If $T_i\\in R_i^*\\otimes R_i^*\\otimes R_i$, $i=1,2$ are structure tensors of $n_i$-dimensional commutative algebras $R_i$, then the structure tensor $T_1\\oplus T_2$ of the $n:=(n_1+n_2)$-dimensional algebra $R_1\\oplus R_2$ satisfies\n \\[\n \\rank\\tang^A(T_1\\oplus T_2) - n(n-1)(n-2) = \\Bigl(\\rank \\tang^A(T_1) - n_1(n_1-1)(n_1-2)\\Bigr) + \\Bigl(\\rank\\tang^A(T_2) - n_2(n_2-1)(n_2-2)\\Bigr).\n \\]\n\\end{corollary}\n\\begin{proof}\n Along with the conclusions of Proposition~\\ref{prop:koszul-equals-quad-on-algs}, observe that structure tensors of algebras are concise. Thus all but the first two terms of the right-hand side of \\eqref{eq:tangency-direct-sum} simplify, yielding\n \\[\n \\rank\\tang^A(T_1\\oplus T_2) = \\rank \\tang^A(T_1) + \\rank \\tang^A(T_2) + 3n_1n_2(n_1+n_2-2).\n \\]\n This rearranges to the Corollary.\n\\end{proof}\nThe following lemma is well-known to experts. It can be derived e.g.~from the results in \\cite{jelisiejew2024concise}. We include the proof for the sake of completeness.\n\\begin{lemma}\\label{lem:GorCactus}\n Let $R$ be a finite Gorenstein algebra and $T\\in (R^*)^{\\otimes k}\\ot R$ its structure tensor corresponding to the map $R^k\\rightarrow R$, $(r_1,\\dots,r_k)\\mapsto r_1\\cdots r_k$. The cactus rank of $T$ equals $d=\\dim_{\\CC} R$.\n\\end{lemma}\n\\begin{proof}\n As $T$ is concise it is enough to prove that it has cactus rank at most $d$. Thus we have to exhibit a scheme of length $d$, which will be $\\Spec R$, inside the Segre variety, so that $T$ is in its linear\nspan. As cactus rank is subadditive under direct sum we may assume $(R,\\mathfrak m)$ is local.\n\n\\begin{theorem}\n \\label{thrm:beat-cactus}\n The $(n(n-1)(n-2)+1)$-minors of the tangency flattenings for border rank $n$ in $\\CC^n\\otimes \\CC^n\\otimes \\CC^n$ do not vanish on the $n$-th cactus variety for all $n\\geq 14$.\n\\end{theorem}\n\\begin{proof}\nBy Lemma \\ref{lem:GorCactus}, structure tensors of Gorenstein algebras have minimal cactus rank.\n Thus, it suffices to produce examples of structure tensors of $n$-dimensional Gorenstein algebras whose tangency flattenings have ranks strictly higher than $n(n-1)(n-2)$.", "post_theorem_intro_text_len": 2217, "post_theorem_intro_text": "For example, for $n=14$ the minors in question are polynomials of degree $4370$ in $2744$ variables. Yet, they are explicit and we can determine their vanishing or nonvanishing very fast.\nWe will prove Theorem~\\ref{thrm:beat-cactus-introversion} by lower-bounding border rank via rank of the tangency flattening in Corollary~\\ref{cor:tan_flat} and exhibiting in Theorem~\\ref{thrm:beat-cactus} a sequence of tensors of cactus rank $n$ on which the above-named minors of the tangency flattening do not vanish. The name ``tangency'' flattening is motivated by a conjectural connection to tangent spaces of Hilbert schemes of points -- see Conjecture~\\ref{conj:tangency}. In Subsection~\\ref{subsec:mamu}, we additionally show that tangency flattenings provide a new and completely elementary proof of the fact that $2\\times2$ matrix multiplication tensor has border rank $7$.\n\nWhile Kronecker-Koszul flattenings generalize Koszul flattenings (cf. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-michalek-mamu, landsberg-michalek-haystack}), we also analyze in Section~\\ref{sec:koszul} bounds that may be extracted via Koszul flattenings for the border rank (and border cactus rank) of structure tensors of certain algebras. It is known \\cite{blaser2016degeneration} that a (finite) algebra is smoothable if and only if the associated structure tensor has minimal border rank. This is also one of the main challenges in application of the border apolarity method.\nFurther, two of the classical flattenings of structure tensors of algebras give rise to spaces of commuting matrices. Thus, many of the flattening methods cannot provide nontrivial bounds on border rank of such tensors. However, we show that it is possible to obtain lower bounds on border cactus rank of structure tensors of algebras strictly greater than the degree of the algebra, even using Koszul flattenings (see Proposition~\\ref{prop:1de-koszul}).\n\nFinally, we show that under mild assumptions, Koszul flattenings may be replaced and improved upon by certain quadratic Kronecker-Koszul flattenings for the purposes of lower-bounding border rank of tensor (see Proposition~\\ref{prop:quadratic-better-than-koszul}).", "sketch": "We will prove Theorem~\\ref{thrm:beat-cactus-introversion} by (1) lower-bounding border rank via the rank of the tangency flattening (Corollary~\\ref{cor:tan_flat}), and (2) exhibiting (in Theorem~\\ref{thrm:beat-cactus}) a sequence of tensors of cactus rank $n$ on which the minors of size $n(n-1)(n-2)+1$ of the tangency flattening do not vanish.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thrm:beat-cactus-introversion}\n    For all $n\\geq14$, minors of size $n(n-1)(n-2)+1$ of the tangency flattenings vanish on the $n$-th secant variety of the Segre variety in $\\P(\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n)$ but do not vanish on its $n$-th cactus variety.", "theorem_type": ["Universal", "Implication"], "mcq": {"question": "For each integer \\(n\\ge 14\\), consider the Segre variety \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\subset \\mathbb P(\\mathbb C^n\\otimes \\mathbb C^n\\otimes \\mathbb C^n)\\). Let its \\(n\\)-th secant variety be the Zariski closure of the union of linear spans of \\(n\\) points of the Segre variety, and let its \\(n\\)-th cactus variety be the Zariski closure of the union of linear spans of length-\\(n\\) subschemes of the Segre variety. The paper associates to tensors in \\(\\mathbb C^n\\otimes \\mathbb C^n\\otimes \\mathbb C^n\\) certain matrix-valued maps called tangency flattenings; taking all minors of size \\(n(n-1)(n-2)+1\\) gives polynomial equations on the ambient projective space. Which statement holds for every such \\(n\\)?", "correct_choice": {"label": "A", "text": "All minors of size \\(n(n-1)(n-2)+1\\) of the tangency flattenings vanish on the \\(n\\)-th secant variety of \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\), but they do not vanish on its \\(n\\)-th cactus variety."}, "choices": [{"label": "B", "text": "All minors of size \\(n(n-1)(n-2)+1\\) of the tangency flattenings vanish on both the \\(n\\)-th secant variety and the \\(n\\)-th cactus variety of \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\)."}, {"label": "C", "text": "All minors of size \\(n(n-1)(n-2)+1\\) of the tangency flattenings vanish on the \\(n\\)-th secant variety of \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\)."}, {"label": "D", "text": "All minors of size \\(n(n-1)(n-2)\\) of the tangency flattenings vanish on the \\(n\\)-th secant variety of \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\), but they do not vanish on its \\(n\\)-th cactus variety."}, {"label": "E", "text": "For every tensor in \\(\\mathbb C^n\\otimes \\mathbb C^n\\otimes \\mathbb C^n\\) outside the \\(n\\)-th cactus variety, some minor of size \\(n(n-1)(n-2)+1\\) of a tangency flattening is nonzero; equivalently, these minors cut out the \\(n\\)-th cactus variety set-theoretically."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "nonvanishing_on_cactus", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_failure_on_cactus_variety", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "minor_size_threshold_plus_one", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "mere_nonvanishing_vs_set_theoretic_definition", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives technical setup and asks for the resulting conclusion, but it does not explicitly state or strongly hint at the full correct claim. The key distinguishing feature—failure on the cactus variety—is not leaked."}, "TAS": {"score": 1, "justification": "The item is close to theorem recall: it asks which precise conclusion holds for a very specific construction. However, it is not a pure verbatim restatement, since the options vary meaningfully in scope, threshold, and quantifiers."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the strongest correct statement from weaker, overgeneralized, or boundary-shifted alternatives. Still, the question mainly tests knowledge of a specific result rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and well-targeted: one is a weaker true-looking statement, others alter the cactus behavior, quantifiers, or the range n>=14. These reflect realistic mathematical failure modes."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with little answer leakage and strong distractors, but it leans more toward precise recall than genuine generative reasoning."}}
{"id": "2602.12762v1", "paper_link": "http://arxiv.org/abs/2602.12762v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thrm:beat-cactus-introversion}\n    For all $n\\geq14$, minors of size $n(n-1)(n-2)+1$ of the tangency flattenings vanish on the $n$-th secant variety of the Segre variety in $\\P(\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n)$ but do not vanish on its $n$-th cactus variety.", "start_pos": 10252, "end_pos": 10549, "label": "thrm:beat-cactus-introversion"}, "ref_dict": {"thrm:beat-cactus-introversion": "\\begin{theorem}\n    \\label{thrm:beat-cactus-introversion}\n    For all $n\\geq14$, minors of size $n(n-1)(n-2)+1$ of the tangency flattenings vanish on the $n$-th secant variety of the Segre variety in $\\P(\\CC^n\\otimes\\CC^n\\otimes\\CC^n)$ but do not vanish on its $n$-th cactus variety.\n\\end{theorem}", "prop:1de-koszul": "\\begin{proposition}\n    \\label{prop:1de-koszul}\n    Let $R$ be the apolar algebra to $e\\leq \\binom{d+1}2$ general quadrics in $d$ variables. If $e\\geq3$, then the structure tensor $T$ of $R$ has border cactus rank at least $\\floor{\\frac32d}+3$. In particular, if $3\\leq e\\leq \\floor{\\frac d2}+1$, then $R$ is nonsmoothable.\n\\end{proposition}", "prop:quadratic-better-than-koszul": "\\begin{proposition}\n    \\label{prop:quadratic-better-than-koszul}\n    The space $\\im\\Phi_{\\mathrm Q}^C(T)\\subset A\\otimes B$ equals the sum of all $(T(\\alpha)\\otimes \\id_A)\\left(\\im\\koszul^{C,B}(T)\\right)$ over $\\alpha\\in A^*$, where we consider $T(\\alpha)$ as a map $C^*\\to B$. In particular, if $\\dim B=\\dim C$ and $T$ is $1_A$-generic, this implies $\\rank \\Phi_{\\mathrm Q}^C(T) \\geq \\rank\\koszul^{C,B}(T)$.\n\\end{proposition}", "thrm:beat-cactus": "\\begin{theorem}\n    \\label{thrm:beat-cactus}\n    The $(n(n-1)(n-2)+1)$-minors of the tangency flattenings for border rank $n$ in $\\CC^n\\otimes \\CC^n\\otimes \\CC^n$ do not vanish on the $n$-th cactus variety for all $n\\geq 14$.\n\\end{theorem}", "conj:tangency": "\\begin{conjecture}\n    \\label{conj:tangency}\n    Let $\\CC[x_1,\\dots,x_d]\\surjto R$ be a surjection to an $n$-dimensional algebra $R$, let $[R]$ be the corresponding point of $\\Hilb_n\\CC^d$ and let $T$ be the structure tensor of $R$. Then\n    \\[\n        \\rank\\tang^C(T) - n(n-1)(n-2) = dn - \\dim\\T_{[R]}\\Hilb_n\\CC^d.\n    \\]\n\\end{conjecture}", "subsec:mamu": "\\begin{proof}\n    Let us prove only the second set of equalities of \\eqref{eq:koszul-quaddratic-algebra}, since the proof of the first one is very similar.\n\n    Explicitly, the two flattenings in question map\n    \\begin{align*}\n        \\Phi_{\\mathrm Q}^A(T): \\bigwedge^2 R \\otimes R \\otimes R^* &\\to R \\otimes R^*,\\\\\n        (a\\wedge a')\\otimes b\\otimes \\gamma &\\mapsto (ab)\\otimes \\gamma(a'\\cdot\\bullet) - (a'b)\\otimes \\gamma(a\\cdot\\bullet)\n    \\end{align*}\n    and\n    \\begin{align*}\n        \\koszul^{A,C}(T): \\bigwedge^2 R \\otimes R^* &\\to R \\otimes R^*,\\\\\n        (a\\wedge a')\\otimes\\gamma &\\mapsto a\\otimes \\gamma(a'\\cdot\\bullet) - a'\\otimes \\gamma(a\\cdot\\bullet),\n    \\end{align*}\n    where we use $\\gamma(a\\cdot\\bullet)$ to denote the linear form $(x\\mapsto \\gamma(ax))\\in R^*$. We claim that in fact $\\Phi_{\\mathrm Q}^A(T)$ and $\\koszul^{A,C}(T)$ have the same images. The inclusion $\\im\\Phi^{A,C}(T)\\subseteq \\im\\Phi_{\\mathrm Q}^A$ is obvious by considering $b=1$ in the latter map.\n\n    For the opposite inclusion, note that invertible elements form a dense set in $R$, so in particular, $R$ may be spanned by them, hence $\\im \\Phi_{\\mathrm Q}^A$ is spanned by images of $(a\\wedge a')\\otimes b\\otimes\\gamma$ with $b$ invertible. These lie in $\\im \\koszul^{A,C}$ by the following: we set\n    \\[\n        \\tilde a := ba, \\qquad \\tilde a' := ba', \\qquad \\tilde\\gamma := \\gamma(b^{-1}\\cdot\\bullet),\n    \\]\n    which then results in\n    \\begin{multline*}\n        \\koszul^{A,C}(T)\\left(\n            (\\tilde a\\wedge \\tilde a') \\otimes \\tilde \\gamma\n        \\right)\n        =\n        \\tilde a\\otimes\\tilde \\gamma(\\tilde a'\\cdot\\bullet) - \\tilde a'\\otimes\\tilde\\gamma(\\tilde a\\cdot\\bullet)\n        =\n        (ab)\\otimes\\gamma(b^{-1}ba'\\cdot\\bullet) - (a'b)\\otimes \\gamma(b^{-1}ba\\cdot\\bullet)\n        =\\\\=\n        (ab)\\otimes \\gamma(a'\\cdot\\bullet)-(a'b)\\otimes\\gamma(a\\cdot\\bullet)\n        =\n        \\Phi_{\\mathrm Q}^A(T)\\left(\n            (a\\wedge a') \\otimes b\\otimes\\gamma\n        \\right).\n    \\end{multline*}\n    This justifies $\\rank \\Phi_{\\mathrm Q}^A(T) = \\rank \\koszul^{A,C}(T)$. That this common rank equals $n(n-1)$ then follows from Lemma \\ref{lem:easy rank Koszul for algebras}.\n\\end{proof}\n\n\\section{Applications}\n\\label{sec:applications}\n\n\\subsection{Matrix multiplication}\n\\label{subsec:mamu}\n\nIt is well-known that the border rank of the $2\\times 2$ matrix multiplication tensor $M_2$ equals $7$ \\cite{Landsberg2006, Hauenstein-Ikenmeyer-Landsberg}. However, only recently the ``first hand-checkable algebraic proof'' appeared \\cite{Conner_Harper_Landsberg_2023}. Below we show that tangency flattenings provide an easy proof.\n\\begin{proposition}\\label{prop:M2=7}\n    The border rank of $M_2$ equals seven.\n\\end{proposition}\n\\begin{proof}\nStrassen \\cite{strassen1969gaussian} provided an explicit rank seven decomposition of $M_2$. We focus on lower-bounding the border rank.\n\n    By Corollary \\ref{cor:tan_flat} for $n=4$, $q=6$ we have to show that the tangency flattening has rank greater than $60$. We will prove that it is $64$. We fix a basis $(i,j)$ of $\\CC^4$, where $i,j\\in\\{0,1\\}$ and we use $\\neg i$ to denote the negation of $i$. Recall that $M_2=\\sum_{i,j,k\\in \\{0,1\\}} (i,j)\\otimes (j,k)\\otimes (k,i)$.\n\n    We will prove that the tangency flattening is a block diagonal matrix with five blocks: a $48\\times 48$ identity matrix and four isomorphic, nondegenerate $4\\times 4$ matrices.\n\nThe tangency flattening of $M_2$ is the restriction to $\\bigwedge^3\\CC^4\\otimes \\CC^4\\otimes \\CC^4$ of\n\\begin{align}\n\\label{eq:mamu-tangency}\n(\\CC^4)^{\\otimes 3}\\otimes \\CC^4\\otimes \\CC^4 &\\to \\CC^4\\otimes \\CC^4\\otimes \\CC^4,\\\\\n\\nonumber\n(i_1,j_1)\\otimes (i_2,j_2)\\otimes (i_3,j_3)\\otimes (i_4,j_4)\\otimes (i_5,j_5) &\\mapsto\n\\begin{cases}\n(i_1,j_1)\\otimes (i_2,j_4)\\otimes (i_5,j_3), & \\text{if $j_2=i_4$ and $i_3 = j_5$,}\\\\\n0, & \\text{otherwise.}\n\\end{cases}\n\\end{align}\nWe divide the basis vectors in the domain into five groups.\n\nThe first group is formed by vectors of the form $\\bigl((\\neg j_5,\\neg i_4)\\wedge(i_2,i_4)\\wedge (j_5,j_3)\\bigr)\\otimes (i_4,j_4)\\otimes (i_5,j_5)$. By first choosing $i_4,j_4,i_5,j_5$ (16 options), and then choosing exterior products that contain $(\\neg j_5,\\neg i_4)$ (3 options), we see there are $48$ such (nonzero) vectors.\nNote that this already forces the exterior product to be of the form above, since the two remaining exterior factors are chosen from $(\\neg j_5, i_4)$, $(j_5, \\neg i_4)$ and $(j_5,i_4)$.\nAlthough the exterior product expands as a signed sum of six tensor products, only one of those -- the one ordered as in the presentation above -- maps to a nonzero vector in \\eqref{eq:mamu-tangency}. Thus, the image of each such basis vector is $(\\neg j_5,\\neg i_4)\\otimes (i_2,j_4)\\otimes (i_5,j_3)$. Note that such images must satisfy $(\\neg j_5,\\neg i_4)\\neq (\\neg i_2,\\neg j_3)$.\n\nThe next four groups are indexed by $j_4,i_5\\in \\{0,1\\}$. For fixed $j_4,i_5$ the group consists of the four basis vectors $\\bigl((j_5,i_4)\\wedge (\\neg j_5,i_4)\\wedge (j_5,\\neg i_4)\\bigr)\\otimes (i_4,j_4)\\otimes (i_5,j_5)$, where $i_4, j_5$ are arbitrary.\nThe image of each such vector is:\n\\[(j_5,i_4)\\otimes (\\neg j_5,j_4)\\otimes (i_5,\\neg i_4)-(\\neg j_5,i_4)\\otimes (j_5,j_4)\\otimes (i_5,\\neg i_4)-(j_5,\\neg i_4)\\otimes (\\neg j_5,j_4)\\otimes (i_5,i_4).\\]\nIn particular, the images are linear combinations of basis vectors that are distinct from images of the first group as well as other groups -- in each term, the first tensor factor consist exactly of  negations of the first coordinate of the second factor  and the second coordinate of the third factor. Meanwhile, the group is determined by reading off $j_4$ and $i_5$ from the second coordinate of the second factor and first coordinate of the third factor, respectively.\n\nThus indeed the tangency flattening is in block-diagonal form. It remains for us to prove that the isomorphic $4\\times 4$ blocks are of full rank.\nEach such matrix has the following form, where we label $(j_5,i_4)\\otimes (\\neg j_5,j_4)\\otimes (i_5,\\neg i_4)$ with the first factor $(j_5,i_4)$ and subsequently order rows and columns by $(j_5,i_4)=(0,0)$, $(0,1)$, $(1,0)$, $(1,1)$:\n\\[\\begin{pmatrix}\n    1 &  -1&-1&0\\\\\n    -1& 1&0&-1\\\\\n    -1& 0&1&-1\\\\\n    0& -1&-1&1\\\\\n\\end{pmatrix}.\\]\nThe determinant of such a matrix is $-3$, thus indeed the matrix is of full rank, and so is the tangency flattening.\n\\end{proof}", "cor:tan_flat": "\\begin{corollary}\\label{cor:tan_flat}\n    If $\\dim V_1=\\dim V_2=\\dim V_3=n$, then $q(q-1)(n-2)+1$ minors of the tangency flattening provide equations for border rank $q$.\n\\end{corollary}"}, "pre_theorem_intro_text_len": 6726, "pre_theorem_intro_text": "\\label{sec:intro}\nThe central problem concerning tensors is that of \\emph{tensor decomposition} -- presenting a tensor as a sum of simple tensors, or computing the smallest possible size of a decomposition, called the \\emph{tensor rank}. While the problem is NP-hard in general \\cite{RankNPcomplete, hillar2013most}, providing estimates for rank or related notions such as border rank or asymptotic rank has been a fruitful field. Perhaps the most famous application is the study of computational complexity of matrix multiplication (e.g. \\cite{strassen1969gaussian, coppersmith1982asymptotic, alman2024refined, williams2024new}). Further, a recent line of research relates low asymptotic rank of certain special tensors to faster than known algorithms for NP-hard problems \\cite{bjorklund2025fast, bjorklund2024asymptotic, pratt2024stronger, MaMu-SotA}.\nFor an introduction to tensors from an algebraic perspective, we refer the reader to \\cite[Chapter 9.2]{michalek2021invitation}, or to \\cite{landsberg2011tensors} for a more extensive exposition. For the connections to complexity theory, we refer to \\cite{burgisser2013algebraic, landsberg2017geometry}.\n\nThe main topic of this article is obtaining lower bounds on the border rank of tensors, a longstanding question in tensor geometry. The classical approach to the problem is by providing equations for secant varieties to the Segre variety. Flattenings, or maps which produce a matrix from a given tensor, have been a bountiful source of such lower bounds (e.g. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-ottaviani-newbounds, landsberg-michalek-mamu, landsberg-michalek-haystack}), with the equations thus obtained coming in the form of minors. In particular, \\emph{Strassen's equations}, further studied by Landsberg and Manivel and  expanded by Landsberg and Ottaviani to \\emph{Koszul} and \\emph{Young flattenings}, became the gold standard in the field. Apart from flattenings, the techniques of \\emph{border apolarity} (see \\cite{Buczynscy-apolarity}) and \\emph{border substitution} (see \\cite{landsberg2017geometry}) have been introduced and used to prove several new lower bounds in recent years. However, all of these methods have quite restrictive limits of application.\n\nIndeed, current lower bound methods fall far short of being able to prove a tensor in $\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n$ to have the generic border rank, which is approximately $\\frac{n^2}3$. Instead, known flattening methods give maximal lower bounds of up to $(2-\\epsilon)n$ for small $\\epsilon>0$. Combining algebraic equations with other methods, it is possible to provide examples of tensors of border rank $2.02n$ \\cite{landsberg-michalek-haystack}. Still, there are well understood barriers which flattening methods cannot beat \\cite{jarek_cactus, Galazka-bundles, EGOW18, garg2019more}. In contrast, the applications to asymptotic rank in computer science would require superlinear lower bounds on border rank.\n\nThe disparity between the generic border rank and available lower bounds has been attributed to the inclusion of secant varieties in \\emph{cactus varieties} (see Section~\\ref{sec:preliminaries}). In particular, known flattening methods have been unable to distinguish secant from cactus varieties, even when the inclusion is known to be strict  -- quoting from \\cite{Conner_Harper_Landsberg_2023}:\n\\begin{quote}\n``The geometric interpretation of the border rank lower bound barriers of \\cite{EGOW18, Galazka-bundles} is that all equations obtained by taking minors, called rank methods, are actually equations for a larger variety than $\\sigma_r(\\Seg(\\P A\\times\\P B\\times\\P C))$, called the $r$-th cactus variety \\cite{Buczynscy-apolarity}.''\n\\end{quote}\nIndeed, as proved in \\cite{jarek_cactus, Galazka-bundles, galazka-thesis} all of the known methods producing equations for the $r$-th secant variety are equations for the $r$-th cactus variety. Further, it is known that the $r$-th cactus variety of the Segre variety in $(\\mathbb{C}^n)^{\\ot 3}$ fills the ambient space for $r$ growing linearly in $n$ \\cite{ballico2019note, bernardi2018polynomials, galkazka2023multigraded}.\n\nFor these reasons, the existence of easily computable determinantal equations vanishing on secant varieties but not vanishing on cactus varieties, has long been considered unlikely or even impossible in the tensor-geometric community.\nIn this article, we illustrate that it is not the determinantal expression which creates the cactus barrier, but rather the linear embedding of the tensor product into a matrix space.\nThus the starting point of our approach in this article is focusing on \\emph{nonlinear} maps from tensor spaces to matrix spaces.\nTo obtain bounds on matrix rank with linear embeddings, it is enough to investigate the embedded image of the Segre variety and appeal to linearity. With nonlinear maps on the other hand, we need to work more generally, but this does yield stronger results. We obtain a new method, which we dub \\emph{Kronecker-Koszul flattenings}, or more generally \\emph{Kronecker-Young flattenings}, that allows one to produce\nexplicit and efficiently computable determinantal equations of secant varieties that do not generally vanish on cactus varieties. In particular, Kronecker-Koszul flattenings require less computation time and memory than methods based on a more direct identification of ideals defining the secant varieties (cf.~\\cite{Hauenstein-Ikenmeyer-Landsberg}).\n\n Broadly speaking, our method starts by tensoring together several copies of the same tensor. This, depending on the interpretation of the codomain, may be viewed as a Kronecker power or Veronese embedding \\cite{kaski2025universal}, and thus is the universal map for fixed degree polynomials. Next, we further tensor with several $2$-tensors corresponding to identity maps on various vector spaces.  Finally, we group and contract some of the factors to exterior powers. This results in what we call a \\emph{Kronecker-Koszul tensor}. A Kronecker-Koszul flattening is then a classical flattening of the Kronecker-Koszul tensor. Note that due to initially taking a tensor power, Kronecker-Koszul tensors are in general nonlinear as functions of the starting tensor.\nSee Section~\\ref{sec:construction} for details of the construction. There we also comment on what we call {Kronecker-Young flattenings}, a further generalization which synthesizes the approach of Young flattenings into the method.\n\nOur main result uses the \\emph{tangency flattening}, a very particular Kronecker-Koszul flattening which depends quadratically on a tensor. See the paragraph preceding Corollary~\\ref{cor:tan_flat} for the definition.", "context": "The main topic of this article is obtaining lower bounds on the border rank of tensors, a longstanding question in tensor geometry. The classical approach to the problem is by providing equations for secant varieties to the Segre variety. Flattenings, or maps which produce a matrix from a given tensor, have been a bountiful source of such lower bounds (e.g. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-ottaviani-newbounds, landsberg-michalek-mamu, landsberg-michalek-haystack}), with the equations thus obtained coming in the form of minors. In particular, \\emph{Strassen's equations}, further studied by Landsberg and Manivel and expanded by Landsberg and Ottaviani to \\emph{Koszul} and \\emph{Young flattenings}, became the gold standard in the field. Apart from flattenings, the techniques of \\emph{border apolarity} (see \\cite{Buczynscy-apolarity}) and \\emph{border substitution} (see \\cite{landsberg2017geometry}) have been introduced and used to prove several new lower bounds in recent years. However, all of these methods have quite restrictive limits of application.\n\nIndeed, current lower bound methods fall far short of being able to prove a tensor in $\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n$ to have the generic border rank, which is approximately $\\frac{n^2}3$. Instead, known flattening methods give maximal lower bounds of up to $(2-\\epsilon)n$ for small $\\epsilon>0$. Combining algebraic equations with other methods, it is possible to provide examples of tensors of border rank $2.02n$ \\cite{landsberg-michalek-haystack}. Still, there are well understood barriers which flattening methods cannot beat \\cite{jarek_cactus, Galazka-bundles, EGOW18, garg2019more}. In contrast, the applications to asymptotic rank in computer science would require superlinear lower bounds on border rank.\n\nThe disparity between the generic border rank and available lower bounds has been attributed to the inclusion of secant varieties in \\emph{cactus varieties} (see Section~\\ref{sec:preliminaries}). In particular, known flattening methods have been unable to distinguish secant from cactus varieties, even when the inclusion is known to be strict -- quoting from \\cite{Conner_Harper_Landsberg_2023}:\n\\begin{quote}\n``The geometric interpretation of the border rank lower bound barriers of \\cite{EGOW18, Galazka-bundles} is that all equations obtained by taking minors, called rank methods, are actually equations for a larger variety than $\\sigma_r(\\Seg(\\P A\\times\\P B\\times\\P C))$, called the $r$-th cactus variety \\cite{Buczynscy-apolarity}.''\n\\end{quote}\nIndeed, as proved in \\cite{jarek_cactus, Galazka-bundles, galazka-thesis} all of the known methods producing equations for the $r$-th secant variety are equations for the $r$-th cactus variety. Further, it is known that the $r$-th cactus variety of the Segre variety in $(\\mathbb{C}^n)^{\\ot 3}$ fills the ambient space for $r$ growing linearly in $n$ \\cite{ballico2019note, bernardi2018polynomials, galkazka2023multigraded}.\n\nFor these reasons, the existence of easily computable determinantal equations vanishing on secant varieties but not vanishing on cactus varieties, has long been considered unlikely or even impossible in the tensor-geometric community.\nIn this article, we illustrate that it is not the determinantal expression which creates the cactus barrier, but rather the linear embedding of the tensor product into a matrix space.\nThus the starting point of our approach in this article is focusing on \\emph{nonlinear} maps from tensor spaces to matrix spaces.\nTo obtain bounds on matrix rank with linear embeddings, it is enough to investigate the embedded image of the Segre variety and appeal to linearity. With nonlinear maps on the other hand, we need to work more generally, but this does yield stronger results. We obtain a new method, which we dub \\emph{Kronecker-Koszul flattenings}, or more generally \\emph{Kronecker-Young flattenings}, that allows one to produce\nexplicit and efficiently computable determinantal equations of secant varieties that do not generally vanish on cactus varieties. In particular, Kronecker-Koszul flattenings require less computation time and memory than methods based on a more direct identification of ideals defining the secant varieties (cf.~\\cite{Hauenstein-Ikenmeyer-Landsberg}).\n\nBroadly speaking, our method starts by tensoring together several copies of the same tensor. This, depending on the interpretation of the codomain, may be viewed as a Kronecker power or Veronese embedding \\cite{kaski2025universal}, and thus is the universal map for fixed degree polynomials. Next, we further tensor with several $2$-tensors corresponding to identity maps on various vector spaces. Finally, we group and contract some of the factors to exterior powers. This results in what we call a \\emph{Kronecker-Koszul tensor}. A Kronecker-Koszul flattening is then a classical flattening of the Kronecker-Koszul tensor. Note that due to initially taking a tensor power, Kronecker-Koszul tensors are in general nonlinear as functions of the starting tensor.\nSee Section~\\ref{sec:construction} for details of the construction. There we also comment on what we call {Kronecker-Young flattenings}, a further generalization which synthesizes the approach of Young flattenings into the method.\n\nOur main result uses the \\emph{tangency flattening}, a very particular Kronecker-Koszul flattening which depends quadratically on a tensor. See the paragraph preceding Corollary~\\ref{cor:tan_flat} for the definition.\n\n\\begin{corollary}\\label{cor:tan_flat}\n    If $\\dim V_1=\\dim V_2=\\dim V_3=n$, then $q(q-1)(n-2)+1$ minors of the tangency flattening provide equations for border rank $q$.\n\\end{corollary}", "full_context": "The main topic of this article is obtaining lower bounds on the border rank of tensors, a longstanding question in tensor geometry. The classical approach to the problem is by providing equations for secant varieties to the Segre variety. Flattenings, or maps which produce a matrix from a given tensor, have been a bountiful source of such lower bounds (e.g. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-ottaviani-newbounds, landsberg-michalek-mamu, landsberg-michalek-haystack}), with the equations thus obtained coming in the form of minors. In particular, \\emph{Strassen's equations}, further studied by Landsberg and Manivel and expanded by Landsberg and Ottaviani to \\emph{Koszul} and \\emph{Young flattenings}, became the gold standard in the field. Apart from flattenings, the techniques of \\emph{border apolarity} (see \\cite{Buczynscy-apolarity}) and \\emph{border substitution} (see \\cite{landsberg2017geometry}) have been introduced and used to prove several new lower bounds in recent years. However, all of these methods have quite restrictive limits of application.\n\nIndeed, current lower bound methods fall far short of being able to prove a tensor in $\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n$ to have the generic border rank, which is approximately $\\frac{n^2}3$. Instead, known flattening methods give maximal lower bounds of up to $(2-\\epsilon)n$ for small $\\epsilon>0$. Combining algebraic equations with other methods, it is possible to provide examples of tensors of border rank $2.02n$ \\cite{landsberg-michalek-haystack}. Still, there are well understood barriers which flattening methods cannot beat \\cite{jarek_cactus, Galazka-bundles, EGOW18, garg2019more}. In contrast, the applications to asymptotic rank in computer science would require superlinear lower bounds on border rank.\n\nThe disparity between the generic border rank and available lower bounds has been attributed to the inclusion of secant varieties in \\emph{cactus varieties} (see Section~\\ref{sec:preliminaries}). In particular, known flattening methods have been unable to distinguish secant from cactus varieties, even when the inclusion is known to be strict -- quoting from \\cite{Conner_Harper_Landsberg_2023}:\n\\begin{quote}\n``The geometric interpretation of the border rank lower bound barriers of \\cite{EGOW18, Galazka-bundles} is that all equations obtained by taking minors, called rank methods, are actually equations for a larger variety than $\\sigma_r(\\Seg(\\P A\\times\\P B\\times\\P C))$, called the $r$-th cactus variety \\cite{Buczynscy-apolarity}.''\n\\end{quote}\nIndeed, as proved in \\cite{jarek_cactus, Galazka-bundles, galazka-thesis} all of the known methods producing equations for the $r$-th secant variety are equations for the $r$-th cactus variety. Further, it is known that the $r$-th cactus variety of the Segre variety in $(\\mathbb{C}^n)^{\\ot 3}$ fills the ambient space for $r$ growing linearly in $n$ \\cite{ballico2019note, bernardi2018polynomials, galkazka2023multigraded}.\n\nFor these reasons, the existence of easily computable determinantal equations vanishing on secant varieties but not vanishing on cactus varieties, has long been considered unlikely or even impossible in the tensor-geometric community.\nIn this article, we illustrate that it is not the determinantal expression which creates the cactus barrier, but rather the linear embedding of the tensor product into a matrix space.\nThus the starting point of our approach in this article is focusing on \\emph{nonlinear} maps from tensor spaces to matrix spaces.\nTo obtain bounds on matrix rank with linear embeddings, it is enough to investigate the embedded image of the Segre variety and appeal to linearity. With nonlinear maps on the other hand, we need to work more generally, but this does yield stronger results. We obtain a new method, which we dub \\emph{Kronecker-Koszul flattenings}, or more generally \\emph{Kronecker-Young flattenings}, that allows one to produce\nexplicit and efficiently computable determinantal equations of secant varieties that do not generally vanish on cactus varieties. In particular, Kronecker-Koszul flattenings require less computation time and memory than methods based on a more direct identification of ideals defining the secant varieties (cf.~\\cite{Hauenstein-Ikenmeyer-Landsberg}).\n\nBroadly speaking, our method starts by tensoring together several copies of the same tensor. This, depending on the interpretation of the codomain, may be viewed as a Kronecker power or Veronese embedding \\cite{kaski2025universal}, and thus is the universal map for fixed degree polynomials. Next, we further tensor with several $2$-tensors corresponding to identity maps on various vector spaces. Finally, we group and contract some of the factors to exterior powers. This results in what we call a \\emph{Kronecker-Koszul tensor}. A Kronecker-Koszul flattening is then a classical flattening of the Kronecker-Koszul tensor. Note that due to initially taking a tensor power, Kronecker-Koszul tensors are in general nonlinear as functions of the starting tensor.\nSee Section~\\ref{sec:construction} for details of the construction. There we also comment on what we call {Kronecker-Young flattenings}, a further generalization which synthesizes the approach of Young flattenings into the method.\n\nOur main result uses the \\emph{tangency flattening}, a very particular Kronecker-Koszul flattening which depends quadratically on a tensor. See the paragraph preceding Corollary~\\ref{cor:tan_flat} for the definition.\n\n\\begin{corollary}\\label{cor:tan_flat}\n    If $\\dim V_1=\\dim V_2=\\dim V_3=n$, then $q(q-1)(n-2)+1$ minors of the tangency flattening provide equations for border rank $q$.\n\\end{corollary}\n\nFor these reasons, the existence of easily computable determinantal equations vanishing on secant varieties but not vanishing on cactus varieties, has long been considered unlikely or even impossible in the tensor-geometric community.\nIn this article, we illustrate that it is not the determinantal expression which creates the cactus barrier, but rather the linear embedding of the tensor product into a matrix space.\nThus the starting point of our approach in this article is focusing on \\emph{nonlinear} maps from tensor spaces to matrix spaces.\nTo obtain bounds on matrix rank with linear embeddings, it is enough to investigate the embedded image of the Segre variety and appeal to linearity. With nonlinear maps on the other hand, we need to work more generally, but this does yield stronger results. We obtain a new method, which we dub \\emph{Kronecker-Koszul flattenings}, or more generally \\emph{Kronecker-Young flattenings}, that allows one to produce\nexplicit and efficiently computable determinantal equations of secant varieties that do not generally vanish on cactus varieties. In particular, Kronecker-Koszul flattenings require less computation time and memory than methods based on a more direct identification of ideals defining the secant varieties (cf.~\\cite{Hauenstein-Ikenmeyer-Landsberg}).\n\nOur main result uses the \\emph{tangency flattening}, a very particular Kronecker-Koszul flattening which depends quadratically on a tensor. See the paragraph preceding Corollary~\\ref{cor:tan_flat} for the definition.\n\nWhile Kronecker-Koszul flattenings generalize Koszul flattenings (cf. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-michalek-mamu, landsberg-michalek-haystack}), we also analyze in Section~\\ref{sec:koszul} bounds that may be extracted via Koszul flattenings for the border rank (and border cactus rank) of structure tensors of certain algebras. It is known \\cite{blaser2016degeneration} that a (finite) algebra is smoothable if and only if the associated structure tensor has minimal border rank. This is also one of the main challenges in application of the border apolarity method.\nFurther, two of the classical flattenings of structure tensors of algebras give rise to spaces of commuting matrices. Thus, many of the flattening methods cannot provide nontrivial bounds on border rank of such tensors. However, we show that it is possible to obtain lower bounds on border cactus rank of structure tensors of algebras strictly greater than the degree of the algebra, even using Koszul flattenings (see Proposition~\\ref{prop:1de-koszul}).\n\n\\begin{corollary}\\label{cor:flat bound}\n If $T$ has border rank at most $q$, then the rank of a Kronecker-Koszul flattening of $T$ is at most $F(q)\\cdot \\prod_{i=1}^r\\prod_{j=1}^{s_i} \\binom{d_i-|\\lambda_{i,j}|}{d_{i,j}'}$. In particular, minors of the Kronecker-Koszul flattening provide equations for secant varieties of the Segre variety.\n\\end{corollary}\n\\begin{proof}\n Follows from Theorem \\ref{thm:main}, as ranks of classical flattenings lower bound border rank of the tensor.\n\\end{proof}\nThe bounds in Theorem \\ref{thm:main} and Corollary \\ref{cor:flat bound} do not have to be tight. This can happen for various reasons, some more opaque than others:\n\nIn this section we present one explicit construction that we find particularly interesting. By \\cite{kaski2025universal} the linear span of image of the $k$-th Kronecker power of tensors in $A\\ot B \\ot C$ is $S^k(A\\ot B \\ot C)$. As a $\\GL(A)\\times \\GL(B)\\times \\GL(C)$ representation, the latter space has a decomposition into isotypic components\n \\[S^k(A\\ot B \\ot C)=\\bigoplus_{\\lambda,\\mu,\\rho\\vdash k}\\left(S^\\lambda(A)\\ot S^\\mu(B)\\ot S^\\rho(C)\\right)^{\\bigoplus K_{\\lambda,\\mu,\\rho}},\\]\n where $K_{\\lambda,\\mu,\\rho}$ is the Kronecker coefficient. In particular, when $\\lambda=\\mu=1^k$ and $\\rho=(k)$ we have $K_{\\lambda,\\mu,\\rho}=1$. This gives a canonical projection\n \\[\\pi:S^k(A\\ot B \\ot C)\\rightarrow \\bigwedge^k A\\ot \\bigwedge^k B\\ot S^k(C).\\]\n An example of Kronecker-Young flattening is the classical flattening of $\\pi(T^{\\otimes k})$ given by\n \\[\\Phi_{\\mathrm M,k}^C(T):\\bigwedge^kA^*\\ot\\bigwedge^kB^*\\rightarrow S^k(C).\\]\n The advantage of these choices is that the image of the map $\\Phi_{\\mathrm M,k}^C(T)$ has a direct interpretation in terms of the flattening $T^C:C^*\\rightarrow A\\ot B$. Namely, the image of $\\Phi_{\\mathrm M,k}^C(T)$ is the linear span of all $k\\times k$ minors of $T^C$ viewed as a matrix with entries from $C$. For this reason, we call $\\Phi_{\\mathrm M,k}$ the \\emph{$k$-minor flattening}.\n\\begin{lemma}\n If $\\dim A=\\dim B=\\dim C=n$ and a tensor $T\\in A\\ot B\\ot C$ has border rank at most $n$, then the rank of the $k$-minor flattening $\\Phi_{\\mathrm M,k}^C(T)$ is at most $\\binom{n}{k}$.\n\\end{lemma}\n\\begin{proof}\n The $n$-th secant variety of the Segre variety is the closure of the $\\GL(A)\\times \\GL(B)\\times \\GL(C)$ orbit of the unit tensor $I_n:=\\sum_{i=1}^n a_i\\ot b_i\\ot c_i$, where $a_i$, $b_i$, $c_i$ form respectively bases of $A$, $B$, $C$. As our constructions are continuous and equivariant, it is enough to prove that $\\rk \\Phi_{\\mathrm M,k}^C(I_n)=\\binom{n}{k} $. We note that $I_n^C(C^*)$ is the space of diagonal matrices. Clearly, the linear span of $k\\times k$ minors of the diagonal matrices is the linear span of squarefree monomials of degree $k$ in $n$ variables and thus has dimension $\\binom{n}{k}$.\n\\end{proof}\n\n\\begin{corollary}\n \\label{cor:alg-tangency-sum}\n If $T_i\\in R_i^*\\otimes R_i^*\\otimes R_i$, $i=1,2$ are structure tensors of $n_i$-dimensional commutative algebras $R_i$, then the structure tensor $T_1\\oplus T_2$ of the $n:=(n_1+n_2)$-dimensional algebra $R_1\\oplus R_2$ satisfies\n \\[\n \\rank\\tang^A(T_1\\oplus T_2) - n(n-1)(n-2) = \\Bigl(\\rank \\tang^A(T_1) - n_1(n_1-1)(n_1-2)\\Bigr) + \\Bigl(\\rank\\tang^A(T_2) - n_2(n_2-1)(n_2-2)\\Bigr).\n \\]\n\\end{corollary}\n\\begin{proof}\n Along with the conclusions of Proposition~\\ref{prop:koszul-equals-quad-on-algs}, observe that structure tensors of algebras are concise. Thus all but the first two terms of the right-hand side of \\eqref{eq:tangency-direct-sum} simplify, yielding\n \\[\n \\rank\\tang^A(T_1\\oplus T_2) = \\rank \\tang^A(T_1) + \\rank \\tang^A(T_2) + 3n_1n_2(n_1+n_2-2).\n \\]\n This rearranges to the Corollary.\n\\end{proof}\nThe following lemma is well-known to experts. It can be derived e.g.~from the results in \\cite{jelisiejew2024concise}. We include the proof for the sake of completeness.\n\\begin{lemma}\\label{lem:GorCactus}\n Let $R$ be a finite Gorenstein algebra and $T\\in (R^*)^{\\otimes k}\\ot R$ its structure tensor corresponding to the map $R^k\\rightarrow R$, $(r_1,\\dots,r_k)\\mapsto r_1\\cdots r_k$. The cactus rank of $T$ equals $d=\\dim_{\\CC} R$.\n\\end{lemma}\n\\begin{proof}\n As $T$ is concise it is enough to prove that it has cactus rank at most $d$. Thus we have to exhibit a scheme of length $d$, which will be $\\Spec R$, inside the Segre variety, so that $T$ is in its linear\nspan. As cactus rank is subadditive under direct sum we may assume $(R,\\mathfrak m)$ is local.\n\n\\begin{theorem}\n \\label{thrm:beat-cactus}\n The $(n(n-1)(n-2)+1)$-minors of the tangency flattenings for border rank $n$ in $\\CC^n\\otimes \\CC^n\\otimes \\CC^n$ do not vanish on the $n$-th cactus variety for all $n\\geq 14$.\n\\end{theorem}\n\\begin{proof}\nBy Lemma \\ref{lem:GorCactus}, structure tensors of Gorenstein algebras have minimal cactus rank.\n Thus, it suffices to produce examples of structure tensors of $n$-dimensional Gorenstein algebras whose tangency flattenings have ranks strictly higher than $n(n-1)(n-2)$.", "post_theorem_intro_text_len": 2217, "post_theorem_intro_text": "For example, for $n=14$ the minors in question are polynomials of degree $4370$ in $2744$ variables. Yet, they are explicit and we can determine their vanishing or nonvanishing very fast.\nWe will prove Theorem~\\ref{thrm:beat-cactus-introversion} by lower-bounding border rank via rank of the tangency flattening in Corollary~\\ref{cor:tan_flat} and exhibiting in Theorem~\\ref{thrm:beat-cactus} a sequence of tensors of cactus rank $n$ on which the above-named minors of the tangency flattening do not vanish. The name ``tangency'' flattening is motivated by a conjectural connection to tangent spaces of Hilbert schemes of points -- see Conjecture~\\ref{conj:tangency}. In Subsection~\\ref{subsec:mamu}, we additionally show that tangency flattenings provide a new and completely elementary proof of the fact that $2\\times2$ matrix multiplication tensor has border rank $7$.\n\nWhile Kronecker-Koszul flattenings generalize Koszul flattenings (cf. \\cite{langsberg-ottaviani-equationsforsecant, landsberg2008generalizations, landsberg-michalek-mamu, landsberg-michalek-haystack}), we also analyze in Section~\\ref{sec:koszul} bounds that may be extracted via Koszul flattenings for the border rank (and border cactus rank) of structure tensors of certain algebras. It is known \\cite{blaser2016degeneration} that a (finite) algebra is smoothable if and only if the associated structure tensor has minimal border rank. This is also one of the main challenges in application of the border apolarity method.\nFurther, two of the classical flattenings of structure tensors of algebras give rise to spaces of commuting matrices. Thus, many of the flattening methods cannot provide nontrivial bounds on border rank of such tensors. However, we show that it is possible to obtain lower bounds on border cactus rank of structure tensors of algebras strictly greater than the degree of the algebra, even using Koszul flattenings (see Proposition~\\ref{prop:1de-koszul}).\n\nFinally, we show that under mild assumptions, Koszul flattenings may be replaced and improved upon by certain quadratic Kronecker-Koszul flattenings for the purposes of lower-bounding border rank of tensor (see Proposition~\\ref{prop:quadratic-better-than-koszul}).", "sketch": "We will prove Theorem~\\ref{thrm:beat-cactus-introversion} by (1) lower-bounding border rank via the rank of the tangency flattening (Corollary~\\ref{cor:tan_flat}), and (2) exhibiting (in Theorem~\\ref{thrm:beat-cactus}) a sequence of tensors of cactus rank $n$ on which the minors of size $n(n-1)(n-2)+1$ of the tangency flattening do not vanish.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thrm:beat-cactus-introversion}\n    For all $n\\geq14$, minors of size $n(n-1)(n-2)+1$ of the tangency flattenings vanish on the $n$-th secant variety of the Segre variety in $\\P(\\mathbb{C}^n\\otimes\\mathbb{C}^n\\otimes\\mathbb{C}^n)$ but do not vanish on its $n$-th cactus variety.", "theorem_type": ["Universal", "Implication"], "mcq": {"question": "For each integer \\(n\\ge 14\\), consider the Segre variety \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\subset \\mathbb P(\\mathbb C^n\\otimes \\mathbb C^n\\otimes \\mathbb C^n)\\). Let its \\(n\\)-th secant variety be the Zariski closure of the union of linear spans of \\(n\\) points of the Segre variety, and let its \\(n\\)-th cactus variety be the Zariski closure of the union of linear spans of length-\\(n\\) subschemes of the Segre variety. The paper associates to tensors in \\(\\mathbb C^n\\otimes \\mathbb C^n\\otimes \\mathbb C^n\\) certain matrix-valued maps called tangency flattenings; taking all minors of size \\(n(n-1)(n-2)+1\\) gives polynomial equations on the ambient projective space. Which statement holds for every such \\(n\\)?", "correct_choice": {"label": "A", "text": "All minors of size \\(n(n-1)(n-2)+1\\) of the tangency flattenings vanish on the \\(n\\)-th secant variety of \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\), but they do not vanish on its \\(n\\)-th cactus variety."}, "choices": [{"label": "B", "text": "All minors of size \\(n(n-1)(n-2)+1\\) of the tangency flattenings vanish on both the \\(n\\)-th secant variety and the \\(n\\)-th cactus variety of \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\)."}, {"label": "C", "text": "All minors of size \\(n(n-1)(n-2)+1\\) of the tangency flattenings vanish on the \\(n\\)-th secant variety of \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\)."}, {"label": "D", "text": "All minors of size \\(n(n-1)(n-2)\\) of the tangency flattenings vanish on the \\(n\\)-th secant variety of \\(\\operatorname{Seg}(\\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n)\\times \\mathbb P(\\mathbb C^n))\\), but they do not vanish on its \\(n\\)-th cactus variety."}, {"label": "E", "text": "For every tensor in \\(\\mathbb C^n\\otimes \\mathbb C^n\\otimes \\mathbb C^n\\) outside the \\(n\\)-th cactus variety, some minor of size \\(n(n-1)(n-2)+1\\) of a tangency flattening is nonzero; equivalently, these minors cut out the \\(n\\)-th cactus variety set-theoretically."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "nonvanishing_on_cactus", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_failure_on_cactus_variety", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "minor_size_threshold_plus_one", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "mere_nonvanishing_vs_set_theoretic_definition", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the correct conclusion about secant versus cactus varieties. It gives the construction of the tangency flattenings and the minor size, but the key vanishing/nonvanishing behavior must still be identified from the choices."}, "TAS": {"score": 1, "justification": "The item is close to a direct theorem-recall question: it asks which statement holds for every n after presenting the exact setup and the relevant minors. However, it is not completely tautological because the choices include weaker, stronger, and boundary-threshold variants rather than just a verbatim restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the precise theorem from nearby alternatives: a weaker true-looking statement, an off-by-one minor size, and an overstrong set-theoretic conclusion. Still, the task mainly tests recognition/recall of the exact result rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful. They reflect common failure modes: overgeneralizing vanishing to the cactus variety, dropping the sharper nonvanishing clause, using the wrong minor size threshold, and confusing existence of equations with a set-theoretic characterization."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with good distractors and no answer leakage, but it leans more toward precise recall of a stated result than toward deep generative reasoning."}}
{"id": "2602.13001v1", "paper_link": "http://arxiv.org/abs/2602.13001v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{mainth1}\nLet $c,d,s\\in\\mathbb Z^+$ with $d\\geq 2$, $1\\leq c,s\\leq d$ and $\\gcd(cs,d)=1$, and let $p\\geq5$ be a prime with $p\\equiv s\\pmod d$. Then, for any $r\\in\\mathbb Z^+$ with $(\\f12+\\alpha)^{\\ast_r}(\\f12+\\alpha^{\\ast_r})\\not\\eq0\\pmod{p}$,  we have\n\\begin{equation}\\label{mainth1eq}\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\frac{(\\alpha)_k^3(\\f12)_k}{(1)_k^3(\\f12+\\alpha)_k}\\equiv \\alpha^{\\ast_r}p^r -\\frac{(\\alpha^{\\ast_r})^3}{(\\f12+\\alpha)^{\\ast_r}}p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_{r-1}}}^{(2)}\\pmod{p^{r+3}},\n\\end{equation}\nwhere $\\alpha=c/d$.", "start_pos": 9805, "end_pos": 10361, "label": "mainth1"}, "ref_dict": {"GuoZhaores1": "\\begin{equation}\\label{GuoZhaores1}\n\\sum_{k=0}^{(p^r-1)/2}(8k+1)\\f{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\eq p^r\\pmod{p^{r+3}},\n\\end{equation}", "examples": "\\begin{table}[htbp]\n    \\centering\n    \\caption{Examples of the parameters in Theorem \\ref{mainth1}}\n    \\label{examples}\n    \\renewcommand{\\arraystretch}{1.5}\n    \\begin{tabular}{>{\\centering\\arraybackslash}p{2cm}\n                      >{\\centering\\arraybackslash}p{2cm}\n                      >{\\centering\\arraybackslash}p{2cm}\n                      >{\\centering\\arraybackslash}p{3cm}\n                      >{\\centering\\arraybackslash}p{3cm}}\n        \\toprule \n        $d$ & $s$ & $\\alpha$ & $\\alpha^{\\ast_r}$ & $(1/2+\\alpha)^{\\ast_r}$\\\\\n        \\midrule \n        $2$ & $1$ & $1/2$  & $1/2$ & $1$\\\\\n        $3$ & $1$ & $1/3$  & $1/3$ & $5/6$\\\\\n        $3$ & $1$ & $2/3$  & $2/3$ & $1/6$\\\\\n        $3$ & $1$ & $1/6$  & $1/6$ & $2/3$\\\\\n        $3$ & $1$ & $5/6$  & $5/6$ & $1/3$\\\\\n        $3$ & $2$ & $1/3$  & $(3-(-1)^r)/6$ & $(3+2(-1)^r)/6$\\\\\n        $3$ & $2$ & $2/3$  & $(3+(-1)^r)/6$ & $(3-2(-1)^r)/6$\\\\\n        $3$ & $2$ & $1/6$  & $(3-2(-1)^r)/6$ & $(3+(-1)^r)/6$\\\\\n        $3$ & $2$ & $5/6$  & $(3+2(-1)^r)/6$ & $(3-(-1)^r)/6$\\\\\n        $4$ & $1$ & $1/4$ & $1/4$ & $3/4$\\\\\n        $4$ & $1$ & $3/4$ & $3/4$ & $1/4$\\\\\n        $4$ & $3$ & $1/4$ & $(2-(-1)^r)/4$ & $(2+(-1)^r)/4$\\\\\n        $4$ & $3$ & $3/4$ & $(2+(-1)^r)/4$ & $(2-(-1)^r)/4$\\\\\n        \\bottomrule\n    \\end{tabular}\n\\end{table}", "mainth1eq": "\\begin{equation}\\label{mainth1eq}\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\f{(\\alpha)_k^3(\\f12)_k}{(1)_k^3(\\f12+\\alpha)_k}\\eq \\alpha^{\\ast_r}p^r -\\f{(\\alpha^{\\ast_r})^3}{(\\f12+\\alpha)^{\\ast_r}}p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_{r-1}}}^{(2)}\\pmod{p^{r+3}},\n\\end{equation}", "RamaSer": "\\begin{equation}\\label{RamaSer}\n\\sum_{k=0}^{\\infty}(8k+1)\\f{(\\f14)_k^4}{(1)_k^4}=\\f{2\\sqrt2}{\\sqrt{\\pi}\\Gamma(\\f34)^2},\n\\end{equation}", "cor1": "\\begin{equation}\\label{cor1}\n\\sum_{k=0}^{p^r-1}(4k+1)\\f{(\\f12)_k^4}{(1)_k^4}\\eq p^r \\pmod{p^{r+3}},\n\\end{equation}", "GuoZhaoconj7.2": "\\begin{equation}\\label{GuoZhaoconj7.2}\n\\sum_{k=0}^{p^r-1}(8k+1)\\f{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\eq 3p^r+\\f{27}{4}p^{3r}\\sum_{j=1}^{(p^r-3)/4}\\f{1}{j^2}\\pmod{p^{r+3}}.\n\\end{equation}", "2ordharmonic": "\\begin{equation}\\label{2ordharmonic}\nH_{p-1}^{(2)}\\eq H_{(p-1)/2}^{(2)}\\eq 0\\pmod{p},\n\\end{equation}", "cor": "\\begin{corollary}\\label{cor}\nGuo and Zhao's conjectural supercongruence \\eqref{GuoZhaoconj7.2} \\cite[Conjecture 7.2]{Guo-Zhao2026} is true.\n\\end{corollary}", "VanHammeconj": "\\begin{equation}\\label{VanHammeconj}\n\\sum_{k=0}^{(p-1)/4}(8k+1)\\f{(\\f14)_k^4}{(1)_k^4}\\eq p\\f{\\Gamma_p(\\f12)\\Gamma_p(\\f14)}{\\Gamma_p(\\f34)}\\pmod{p^3},\n\\end{equation}", "Swisherextension": "\\begin{equation}\\label{Swisherextension}\n\\sum_{k=0}^{(3p-1)/4}(8k+1)\\f{(\\f14)_k^4}{(1)_k^4}\\eq -\\f{3}{2}p^2\\f{\\Gamma_p(\\f12)\\Gamma_p(\\f14)}{\\Gamma_p(\\f34)}\\pmod{p^4}.\n\\end{equation}", "Whipple": "\\begin{align}\\label{Whipple}\n&\\sum_{k=0}^{\\infty}\\f{(a)_k(1+\\f{a}{2})_k(b)_k(c)_k(d)_k}{(1)_k(\\f a2)_k(1+a-b)_k(1+a-c)_k(1+a-d)_k}\\notag\\\\\n&\\qquad=\\f{\\Gamma(1+a-b)\\Gamma(1+a-c)\\Gamma(1+a-d)\\Gamma(1+a-b-c-d)}{\\Gamma(1+a)\\Gamma(1+a-b-c)\\Gamma(1+a-b-d)\\Gamma(1+a-c-d)}.\n\\end{align}", "GuoZhaores2": "\\begin{equation}\\label{GuoZhaores2}\n\\sum_{k=0}^{p^r-1}(8k+1)\\f{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\eq 3p^r\\pmod{p^{r+2}}.\n\\end{equation}", "Pan-Tauraso-Wangres": "\\begin{equation}\\label{Pan-Tauraso-Wangres}\n\\sum_{k=0}^{p-1}\\f{2k+\\alpha}{\\alpha}\\cdot\\f{(\\alpha)_k^4}{(1)_k^4}\\eq p^2\\alpha^{\\ast}(2\\alpha^{\\ast}-1)\\f{\\Gamma_p(1-2\\alpha)}{\\Gamma_p(1+\\alpha)\\Gamma_p(1-\\alpha)^3}\\pmod{p^4},\n\\end{equation}", "mainth1": "\\begin{theorem}\\label{mainth1}\nLet $c,d,s\\in\\Z^+$ with $d\\geq 2$, $1\\leq c,s\\leq d$ and $\\gcd(cs,d)=1$, and let $p\\geq5$ be a prime with $p\\eq s\\pmod d$. Then, for any $r\\in\\Z^+$ with $(\\f12+\\alpha)^{\\ast_r}(\\f12+\\alpha^{\\ast_r})\\not\\eq0\\pmod{p}$,  we have\n\\begin{equation}\\label{mainth1eq}\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\f{(\\alpha)_k^3(\\f12)_k}{(1)_k^3(\\f12+\\alpha)_k}\\eq \\alpha^{\\ast_r}p^r -\\f{(\\alpha^{\\ast_r})^3}{(\\f12+\\alpha)^{\\ast_r}}p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_{r-1}}}^{(2)}\\pmod{p^{r+3}},\n\\end{equation}\nwhere $\\alpha=c/d$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 5847, "pre_theorem_intro_text": "In the 1910s, Ramanujan announced some convergent hypergeometric series related to $1/\\pi$ without proofs (cf. \\cite{Berndt1994}), such as\n\\begin{equation}\\label{RamaSer}\n\\sum_{k=0}^{\\infty}(8k+1)\\frac{(\\f14)_k^4}{(1)_k^4}=\\frac{2\\sqrt2}{\\sqrt{\\pi}\\Gamma(\\f34)^2},\n\\end{equation}\nwhere $(x)_n=x(x+1)\\cdots(x+n-1)$ is the Pochhammer symbol and $\\Gamma(x)$ is the classical Gamma function. \\eqref{RamaSer} was finally confirmed by Hardy \\cite{Hardy1923}.\n\nIn 1997, Van Hamme \\cite{VanHamme1997} observed that the truncated forms of original Ramanujan-type series possess good congruence properties. For example, corresponding to \\eqref{RamaSer}, Van Hamme conjectured the following supercongruence: for any prime $p\\eq1\\pmod4$,\n\\begin{equation}\\label{VanHammeconj}\n\\sum_{k=0}^{(p-1)/4}(8k+1)\\frac{(\\f14)_k^4}{(1)_k^4}\\equiv p\\frac{\\Gamma_p(\\f12)\\Gamma_p(\\f14)}{\\Gamma_p(\\f34)}\\pmod{p^3},\n\\end{equation}\nwhere $\\Gamma_p(x)$ denotes the $p$-adic Gamma function introduced by Morita \\cite{Morita1975}. Nowadays, we usually refer to \\eqref{VanHammeconj} as a $p$-adic analogue of \\eqref{RamaSer}. All of Van Hamme's observations have now been confirmed by different authors using various techniques (see, e.g., \\cite{Long2011,McCarthy-Osburn2008,Mortenson2008,Osburn-Zudilin2016,Swisher2015}). In particular, Swisher \\cite{Swisher2015} proved that \\eqref{VanHammeconj} holds modulo $p^4$ and established the following associated supercongruence: for any prime $p\\eq3\\pmod4$,\n\\begin{equation}\\label{Swisherextension}\n\\sum_{k=0}^{(3p-1)/4}(8k+1)\\frac{(\\f14)_k^4}{(1)_k^4}\\equiv -\\frac{3}{2}p^2\\frac{\\Gamma_p(\\f12)\\Gamma_p(\\f14)}{\\Gamma_p(\\f34)}\\pmod{p^4}.\n\\end{equation}\nNote that $(1/4)_k\\eq0\\pmod{p}$ for $(p+3)/4\\leq k\\leq p-1$ if $p\\eq1\\pmod4$ or $(3p+3)/4\\leq k\\leq p-1$ if $p\\eq3\\pmod4$. Therefore, both the upper limits of sums on the left-hand side of \\eqref{VanHammeconj} and \\eqref{Swisherextension} can be replaced with $p-1$. In 2022, Pan, Tauraso and Wang \\cite{Pan-Tauraso-Wang2022} established some parametric extensions of \\eqref{VanHammeconj} and \\eqref{Swisherextension}. For instance, they showed that for any odd prime $p$ and $p$-adic integer $\\alpha$ with $\\<-\\alpha\\>_p\\geq (p+1)/2$,\n\\begin{equation}\\label{Pan-Tauraso-Wangres}\n\\sum_{k=0}^{p-1}\\frac{2k+\\alpha}{\\alpha}\\cdot\\frac{(\\alpha)_k^4}{(1)_k^4}\\equiv p^2\\alpha^{\\ast}(2\\alpha^{\\ast}-1)\\frac{\\Gamma_p(1-2\\alpha)}{\\Gamma_p(1+\\alpha)\\Gamma_p(1-\\alpha)^3}\\pmod{p^4},\n\\end{equation}\nwhere for any $p$-adic integer $x$, $\\<x\\>_{p^r}$ stands for the least nonnegative residue of $x$ modulo $p^r$ and $x^{\\ast}=(x+\\<-x\\>_p)/p$ denotes Dwork's dash operation (cf. \\cite{Dwork1969}). Clearly, \\eqref{Swisherextension} is the special case $\\alpha=1/4$ of  \\eqref{Pan-Tauraso-Wangres}.\nFor other parametric extensions of \\eqref{VanHammeconj}, we refer the reader to \\cite{Barman-Saikia2020, Guo2025, Guo-Schlosser2020, Liu-Wang2021, Liu-Wang2022, Pan-Tauraso-Wang2022}.\n\nRecently, using the creative microscoping method (cf. \\cite{Guo-Zudilin2019}), Guo and Zhao \\cite{Guo-Zhao2026} studied some  $q$-supercongruences from a very-well-poised ${}_6\\phi_5$ basic hypergeometric identity. As consequences, they obtained the following results: for any prime $p\\eq1\\pmod4$ and integer $r\\geq1$, \n\\begin{equation}\\label{GuoZhaores1}\n\\sum_{k=0}^{(p^r-1)/2}(8k+1)\\frac{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\equiv p^r\\pmod{p^{r+3}},\n\\end{equation}\nand for any prime $p\\eq3\\pmod4$ and odd integer $r\\geq1$,\n\\begin{equation}\\label{GuoZhaores2}\n\\sum_{k=0}^{p^r-1}(8k+1)\\frac{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\equiv 3p^r\\pmod{p^{r+2}}.\n\\end{equation}\nGuo and Zhao \\cite[Conjectures 7.1 and 7.2]{Guo-Zhao2026} also conjectured that \\eqref{GuoZhaores1} still holds modulo $p^{r+5}$ for $p>5$ and \\eqref{GuoZhaores2} can be extended to the modulus $p^{r+3}$ case as follows:\n\\begin{equation}\\label{GuoZhaoconj7.2}\n\\sum_{k=0}^{p^r-1}(8k+1)\\frac{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\equiv 3p^r+\\frac{27}{4}p^{3r}\\sum_{j=1}^{(p^r-3)/4}\\frac{1}{j^2}\\pmod{p^{r+3}}.\n\\end{equation}\nThis is our initial motivation. \n\nRecall Whipple's ${}_5F_4$ formula (cf. \\cite{Whipple1926})\n\\begin{align}\\label{Whipple}\n&\\sum_{k=0}^{\\infty}\\frac{(a)_k(1+\\frac{a}{2})_k(b)_k(c)_k(d)_k}{(1)_k(\\frac a2)_k(1+a-b)_k(1+a-c)_k(1+a-d)_k}\\notag\\\\\n&\\qquad=\\frac{\\Gamma(1+a-b)\\Gamma(1+a-c)\\Gamma(1+a-d)\\Gamma(1+a-b-c-d)}{\\Gamma(1+a)\\Gamma(1+a-b-c)\\Gamma(1+a-b-d)\\Gamma(1+a-c-d)}.\n\\end{align}\nClearly, \\eqref{RamaSer} is the special case $a=b=c=d=1/4$ of \\eqref{Whipple}. Meanwhile, \\eqref{Pan-Tauraso-Wangres} and \\eqref{GuoZhaores2} are $p$-adic analogues of \\eqref{Whipple} in the case $a=b=c=d=\\alpha$ and the case $a=b=c=1/4,\\ d=1/2$, respectively. Motivated by \\eqref{Pan-Tauraso-Wangres}, it is natural to ask whether \\eqref{GuoZhaores2} or \\eqref{GuoZhaoconj7.2} has a parametric extension. This is the second motivation.\n\nBefore stating our main result, we first introduce some notations. For $n\\in\\mathbb Z^+=\\{1,2,3,\\ldots\\}$ and $x\\in\\mathbb Z$, use $\\<x\\>_n$ to denote the least nonnegative residue of $x$ modulo $n$. Let $p$ be an odd prime. Similarly as in $\\mathbb Z$, for any $p$-adic integer $x$ and $r\\in\\mathbb Z^+$, $\\<x\\>_{p^r}$ stands for the least nonnegative residue of $x$ modulo $p^r$ and $x^{\\ast}=(x+\\<-x\\>_p)/p$ denotes Dwork's dash operation on $x$ (cf. \\cite{Dwork1969}). For convenience, for $n\\in\\mathbb Z^+$, use $x^{\\ast_n}$ to represent iterating the dash operation on $x$ $n$ times, that is,\n$$\nx^{\\ast_1}=x^{\\ast},\\quad x^{\\ast_n}=(x^{\\ast_{n-1}})^{\\ast}\\quad (n=2,3,4,\\ldots).\n$$\nIn particular, set $x^{\\ast_0}=x$. For $n\\in\\mathbb N=\\{0,1,2,\\ldots\\},\\ m\\in\\mathbb Z^+$, the $n$-th harmonic number of order $m$ is defined by\n$$\nH_n^{(m)}:=\\sum_{k=1}^n\\frac{1}{k^m}.\n$$\n\nOur main purpose is to establish a $p$-adic analogue of Whipple's formula \\eqref{Whipple} with $a=b=c=\\alpha$ and $d=\\f12$.", "context": "In the 1910s, Ramanujan announced some convergent hypergeometric series related to $1/\\pi$ without proofs (cf. \\cite{Berndt1994}), such as\n\\begin{equation}\\label{RamaSer}\n\\sum_{k=0}^{\\infty}(8k+1)\\frac{(\\f14)_k^4}{(1)_k^4}=\\frac{2\\sqrt2}{\\sqrt{\\pi}\\Gamma(\\f34)^2},\n\\end{equation}\nwhere $(x)_n=x(x+1)\\cdots(x+n-1)$ is the Pochhammer symbol and $\\Gamma(x)$ is the classical Gamma function. \\eqref{RamaSer} was finally confirmed by Hardy \\cite{Hardy1923}.\n\nIn 1997, Van Hamme \\cite{VanHamme1997} observed that the truncated forms of original Ramanujan-type series possess good congruence properties. For example, corresponding to \\eqref{RamaSer}, Van Hamme conjectured the following supercongruence: for any prime $p\\eq1\\pmod4$,\n\\begin{equation}\\label{VanHammeconj}\n\\sum_{k=0}^{(p-1)/4}(8k+1)\\frac{(\\f14)_k^4}{(1)_k^4}\\equiv p\\frac{\\Gamma_p(\\f12)\\Gamma_p(\\f14)}{\\Gamma_p(\\f34)}\\pmod{p^3},\n\\end{equation}\nwhere $\\Gamma_p(x)$ denotes the $p$-adic Gamma function introduced by Morita \\cite{Morita1975}. Nowadays, we usually refer to \\eqref{VanHammeconj} as a $p$-adic analogue of \\eqref{RamaSer}. All of Van Hamme's observations have now been confirmed by different authors using various techniques (see, e.g., \\cite{Long2011,McCarthy-Osburn2008,Mortenson2008,Osburn-Zudilin2016,Swisher2015}). In particular, Swisher \\cite{Swisher2015} proved that \\eqref{VanHammeconj} holds modulo $p^4$ and established the following associated supercongruence: for any prime $p\\eq3\\pmod4$,\n\\begin{equation}\\label{Swisherextension}\n\\sum_{k=0}^{(3p-1)/4}(8k+1)\\frac{(\\f14)_k^4}{(1)_k^4}\\equiv -\\frac{3}{2}p^2\\frac{\\Gamma_p(\\f12)\\Gamma_p(\\f14)}{\\Gamma_p(\\f34)}\\pmod{p^4}.\n\\end{equation}\nNote that $(1/4)_k\\eq0\\pmod{p}$ for $(p+3)/4\\leq k\\leq p-1$ if $p\\eq1\\pmod4$ or $(3p+3)/4\\leq k\\leq p-1$ if $p\\eq3\\pmod4$. Therefore, both the upper limits of sums on the left-hand side of \\eqref{VanHammeconj} and \\eqref{Swisherextension} can be replaced with $p-1$. In 2022, Pan, Tauraso and Wang \\cite{Pan-Tauraso-Wang2022} established some parametric extensions of \\eqref{VanHammeconj} and \\eqref{Swisherextension}. For instance, they showed that for any odd prime $p$ and $p$-adic integer $\\alpha$ with $\\<-\\alpha\\>_p\\geq (p+1)/2$,\n\\begin{equation}\\label{Pan-Tauraso-Wangres}\n\\sum_{k=0}^{p-1}\\frac{2k+\\alpha}{\\alpha}\\cdot\\frac{(\\alpha)_k^4}{(1)_k^4}\\equiv p^2\\alpha^{\\ast}(2\\alpha^{\\ast}-1)\\frac{\\Gamma_p(1-2\\alpha)}{\\Gamma_p(1+\\alpha)\\Gamma_p(1-\\alpha)^3}\\pmod{p^4},\n\\end{equation}\nwhere for any $p$-adic integer $x$, $\\<x\\>_{p^r}$ stands for the least nonnegative residue of $x$ modulo $p^r$ and $x^{\\ast}=(x+\\<-x\\>_p)/p$ denotes Dwork's dash operation (cf. \\cite{Dwork1969}). Clearly, \\eqref{Swisherextension} is the special case $\\alpha=1/4$ of \\eqref{Pan-Tauraso-Wangres}.\nFor other parametric extensions of \\eqref{VanHammeconj}, we refer the reader to \\cite{Barman-Saikia2020, Guo2025, Guo-Schlosser2020, Liu-Wang2021, Liu-Wang2022, Pan-Tauraso-Wang2022}.\n\nRecently, using the creative microscoping method (cf. \\cite{Guo-Zudilin2019}), Guo and Zhao \\cite{Guo-Zhao2026} studied some $q$-supercongruences from a very-well-poised ${}_6\\phi_5$ basic hypergeometric identity. As consequences, they obtained the following results: for any prime $p\\eq1\\pmod4$ and integer $r\\geq1$, \n\\begin{equation}\\label{GuoZhaores1}\n\\sum_{k=0}^{(p^r-1)/2}(8k+1)\\frac{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\equiv p^r\\pmod{p^{r+3}},\n\\end{equation}\nand for any prime $p\\eq3\\pmod4$ and odd integer $r\\geq1$,\n\\begin{equation}\\label{GuoZhaores2}\n\\sum_{k=0}^{p^r-1}(8k+1)\\frac{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\equiv 3p^r\\pmod{p^{r+2}}.\n\\end{equation}\nGuo and Zhao \\cite[Conjectures 7.1 and 7.2]{Guo-Zhao2026} also conjectured that \\eqref{GuoZhaores1} still holds modulo $p^{r+5}$ for $p>5$ and \\eqref{GuoZhaores2} can be extended to the modulus $p^{r+3}$ case as follows:\n\\begin{equation}\\label{GuoZhaoconj7.2}\n\\sum_{k=0}^{p^r-1}(8k+1)\\frac{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\equiv 3p^r+\\frac{27}{4}p^{3r}\\sum_{j=1}^{(p^r-3)/4}\\frac{1}{j^2}\\pmod{p^{r+3}}.\n\\end{equation}\nThis is our initial motivation.\n\nRecall Whipple's ${}_5F_4$ formula (cf. \\cite{Whipple1926})\n\\begin{align}\\label{Whipple}\n&\\sum_{k=0}^{\\infty}\\frac{(a)_k(1+\\frac{a}{2})_k(b)_k(c)_k(d)_k}{(1)_k(\\frac a2)_k(1+a-b)_k(1+a-c)_k(1+a-d)_k}\\notag\\\\\n&\\qquad=\\frac{\\Gamma(1+a-b)\\Gamma(1+a-c)\\Gamma(1+a-d)\\Gamma(1+a-b-c-d)}{\\Gamma(1+a)\\Gamma(1+a-b-c)\\Gamma(1+a-b-d)\\Gamma(1+a-c-d)}.\n\\end{align}\nClearly, \\eqref{RamaSer} is the special case $a=b=c=d=1/4$ of \\eqref{Whipple}. Meanwhile, \\eqref{Pan-Tauraso-Wangres} and \\eqref{GuoZhaores2} are $p$-adic analogues of \\eqref{Whipple} in the case $a=b=c=d=\\alpha$ and the case $a=b=c=1/4,\\ d=1/2$, respectively. Motivated by \\eqref{Pan-Tauraso-Wangres}, it is natural to ask whether \\eqref{GuoZhaores2} or \\eqref{GuoZhaoconj7.2} has a parametric extension. This is the second motivation.\n\nBefore stating our main result, we first introduce some notations. For $n\\in\\mathbb Z^+=\\{1,2,3,\\ldots\\}$ and $x\\in\\mathbb Z$, use $\\<x\\>_n$ to denote the least nonnegative residue of $x$ modulo $n$. Let $p$ be an odd prime. Similarly as in $\\mathbb Z$, for any $p$-adic integer $x$ and $r\\in\\mathbb Z^+$, $\\<x\\>_{p^r}$ stands for the least nonnegative residue of $x$ modulo $p^r$ and $x^{\\ast}=(x+\\<-x\\>_p)/p$ denotes Dwork's dash operation on $x$ (cf. \\cite{Dwork1969}). For convenience, for $n\\in\\mathbb Z^+$, use $x^{\\ast_n}$ to represent iterating the dash operation on $x$ $n$ times, that is,\n$$\nx^{\\ast_1}=x^{\\ast},\\quad x^{\\ast_n}=(x^{\\ast_{n-1}})^{\\ast}\\quad (n=2,3,4,\\ldots).\n$$\nIn particular, set $x^{\\ast_0}=x$. For $n\\in\\mathbb N=\\{0,1,2,\\ldots\\},\\ m\\in\\mathbb Z^+$, the $n$-th harmonic number of order $m$ is defined by\n$$\nH_n^{(m)}:=\\sum_{k=1}^n\\frac{1}{k^m}.\n$$\n\nOur main purpose is to establish a $p$-adic analogue of Whipple's formula \\eqref{Whipple} with $a=b=c=\\alpha$ and $d=\\f12$.", "full_context": "In the 1910s, Ramanujan announced some convergent hypergeometric series related to $1/\\pi$ without proofs (cf. \\cite{Berndt1994}), such as\n\\begin{equation}\\label{RamaSer}\n\\sum_{k=0}^{\\infty}(8k+1)\\frac{(\\f14)_k^4}{(1)_k^4}=\\frac{2\\sqrt2}{\\sqrt{\\pi}\\Gamma(\\f34)^2},\n\\end{equation}\nwhere $(x)_n=x(x+1)\\cdots(x+n-1)$ is the Pochhammer symbol and $\\Gamma(x)$ is the classical Gamma function. \\eqref{RamaSer} was finally confirmed by Hardy \\cite{Hardy1923}.\n\nIn 1997, Van Hamme \\cite{VanHamme1997} observed that the truncated forms of original Ramanujan-type series possess good congruence properties. For example, corresponding to \\eqref{RamaSer}, Van Hamme conjectured the following supercongruence: for any prime $p\\eq1\\pmod4$,\n\\begin{equation}\\label{VanHammeconj}\n\\sum_{k=0}^{(p-1)/4}(8k+1)\\frac{(\\f14)_k^4}{(1)_k^4}\\equiv p\\frac{\\Gamma_p(\\f12)\\Gamma_p(\\f14)}{\\Gamma_p(\\f34)}\\pmod{p^3},\n\\end{equation}\nwhere $\\Gamma_p(x)$ denotes the $p$-adic Gamma function introduced by Morita \\cite{Morita1975}. Nowadays, we usually refer to \\eqref{VanHammeconj} as a $p$-adic analogue of \\eqref{RamaSer}. All of Van Hamme's observations have now been confirmed by different authors using various techniques (see, e.g., \\cite{Long2011,McCarthy-Osburn2008,Mortenson2008,Osburn-Zudilin2016,Swisher2015}). In particular, Swisher \\cite{Swisher2015} proved that \\eqref{VanHammeconj} holds modulo $p^4$ and established the following associated supercongruence: for any prime $p\\eq3\\pmod4$,\n\\begin{equation}\\label{Swisherextension}\n\\sum_{k=0}^{(3p-1)/4}(8k+1)\\frac{(\\f14)_k^4}{(1)_k^4}\\equiv -\\frac{3}{2}p^2\\frac{\\Gamma_p(\\f12)\\Gamma_p(\\f14)}{\\Gamma_p(\\f34)}\\pmod{p^4}.\n\\end{equation}\nNote that $(1/4)_k\\eq0\\pmod{p}$ for $(p+3)/4\\leq k\\leq p-1$ if $p\\eq1\\pmod4$ or $(3p+3)/4\\leq k\\leq p-1$ if $p\\eq3\\pmod4$. Therefore, both the upper limits of sums on the left-hand side of \\eqref{VanHammeconj} and \\eqref{Swisherextension} can be replaced with $p-1$. In 2022, Pan, Tauraso and Wang \\cite{Pan-Tauraso-Wang2022} established some parametric extensions of \\eqref{VanHammeconj} and \\eqref{Swisherextension}. For instance, they showed that for any odd prime $p$ and $p$-adic integer $\\alpha$ with $\\<-\\alpha\\>_p\\geq (p+1)/2$,\n\\begin{equation}\\label{Pan-Tauraso-Wangres}\n\\sum_{k=0}^{p-1}\\frac{2k+\\alpha}{\\alpha}\\cdot\\frac{(\\alpha)_k^4}{(1)_k^4}\\equiv p^2\\alpha^{\\ast}(2\\alpha^{\\ast}-1)\\frac{\\Gamma_p(1-2\\alpha)}{\\Gamma_p(1+\\alpha)\\Gamma_p(1-\\alpha)^3}\\pmod{p^4},\n\\end{equation}\nwhere for any $p$-adic integer $x$, $\\<x\\>_{p^r}$ stands for the least nonnegative residue of $x$ modulo $p^r$ and $x^{\\ast}=(x+\\<-x\\>_p)/p$ denotes Dwork's dash operation (cf. \\cite{Dwork1969}). Clearly, \\eqref{Swisherextension} is the special case $\\alpha=1/4$ of \\eqref{Pan-Tauraso-Wangres}.\nFor other parametric extensions of \\eqref{VanHammeconj}, we refer the reader to \\cite{Barman-Saikia2020, Guo2025, Guo-Schlosser2020, Liu-Wang2021, Liu-Wang2022, Pan-Tauraso-Wang2022}.\n\nRecently, using the creative microscoping method (cf. \\cite{Guo-Zudilin2019}), Guo and Zhao \\cite{Guo-Zhao2026} studied some $q$-supercongruences from a very-well-poised ${}_6\\phi_5$ basic hypergeometric identity. As consequences, they obtained the following results: for any prime $p\\eq1\\pmod4$ and integer $r\\geq1$, \n\\begin{equation}\\label{GuoZhaores1}\n\\sum_{k=0}^{(p^r-1)/2}(8k+1)\\frac{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\equiv p^r\\pmod{p^{r+3}},\n\\end{equation}\nand for any prime $p\\eq3\\pmod4$ and odd integer $r\\geq1$,\n\\begin{equation}\\label{GuoZhaores2}\n\\sum_{k=0}^{p^r-1}(8k+1)\\frac{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\equiv 3p^r\\pmod{p^{r+2}}.\n\\end{equation}\nGuo and Zhao \\cite[Conjectures 7.1 and 7.2]{Guo-Zhao2026} also conjectured that \\eqref{GuoZhaores1} still holds modulo $p^{r+5}$ for $p>5$ and \\eqref{GuoZhaores2} can be extended to the modulus $p^{r+3}$ case as follows:\n\\begin{equation}\\label{GuoZhaoconj7.2}\n\\sum_{k=0}^{p^r-1}(8k+1)\\frac{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\equiv 3p^r+\\frac{27}{4}p^{3r}\\sum_{j=1}^{(p^r-3)/4}\\frac{1}{j^2}\\pmod{p^{r+3}}.\n\\end{equation}\nThis is our initial motivation.\n\nRecall Whipple's ${}_5F_4$ formula (cf. \\cite{Whipple1926})\n\\begin{align}\\label{Whipple}\n&\\sum_{k=0}^{\\infty}\\frac{(a)_k(1+\\frac{a}{2})_k(b)_k(c)_k(d)_k}{(1)_k(\\frac a2)_k(1+a-b)_k(1+a-c)_k(1+a-d)_k}\\notag\\\\\n&\\qquad=\\frac{\\Gamma(1+a-b)\\Gamma(1+a-c)\\Gamma(1+a-d)\\Gamma(1+a-b-c-d)}{\\Gamma(1+a)\\Gamma(1+a-b-c)\\Gamma(1+a-b-d)\\Gamma(1+a-c-d)}.\n\\end{align}\nClearly, \\eqref{RamaSer} is the special case $a=b=c=d=1/4$ of \\eqref{Whipple}. Meanwhile, \\eqref{Pan-Tauraso-Wangres} and \\eqref{GuoZhaores2} are $p$-adic analogues of \\eqref{Whipple} in the case $a=b=c=d=\\alpha$ and the case $a=b=c=1/4,\\ d=1/2$, respectively. Motivated by \\eqref{Pan-Tauraso-Wangres}, it is natural to ask whether \\eqref{GuoZhaores2} or \\eqref{GuoZhaoconj7.2} has a parametric extension. This is the second motivation.\n\nBefore stating our main result, we first introduce some notations. For $n\\in\\mathbb Z^+=\\{1,2,3,\\ldots\\}$ and $x\\in\\mathbb Z$, use $\\<x\\>_n$ to denote the least nonnegative residue of $x$ modulo $n$. Let $p$ be an odd prime. Similarly as in $\\mathbb Z$, for any $p$-adic integer $x$ and $r\\in\\mathbb Z^+$, $\\<x\\>_{p^r}$ stands for the least nonnegative residue of $x$ modulo $p^r$ and $x^{\\ast}=(x+\\<-x\\>_p)/p$ denotes Dwork's dash operation on $x$ (cf. \\cite{Dwork1969}). For convenience, for $n\\in\\mathbb Z^+$, use $x^{\\ast_n}$ to represent iterating the dash operation on $x$ $n$ times, that is,\n$$\nx^{\\ast_1}=x^{\\ast},\\quad x^{\\ast_n}=(x^{\\ast_{n-1}})^{\\ast}\\quad (n=2,3,4,\\ldots).\n$$\nIn particular, set $x^{\\ast_0}=x$. For $n\\in\\mathbb N=\\{0,1,2,\\ldots\\},\\ m\\in\\mathbb Z^+$, the $n$-th harmonic number of order $m$ is defined by\n$$\nH_n^{(m)}:=\\sum_{k=1}^n\\frac{1}{k^m}.\n$$\n\nOur main purpose is to establish a $p$-adic analogue of Whipple's formula \\eqref{Whipple} with $a=b=c=\\alpha$ and $d=\\f12$.\n\nOur main purpose is to establish a $p$-adic analogue of Whipple's formula \\eqref{Whipple} with $a=b=c=\\alpha$ and $d=\\f12$.\n\nTheorem \\ref{mainth1} seems quite strange. When $\\<-1/2-\\alpha\\>_p<\\min\\{\\<-\\alpha\\>_p,(p-1)/2\\}$, the summands on the left-hand side of \\eqref{mainth1eq} are not always $p$-adic integers. However, the sum of these summands is a $p$-adic integer. This phenomenon renders it difficult for us to prove Theorem \\ref{mainth1} directly using the formula \\eqref{Whipple} and conventional congruence techniques. To overcome this obstacle, we find a new parametric WZ pair (cf. \\cite{PWZ}) which allows us to transform the original sum to a computable form in the sense of congruence. This idea is crucial in our proof.\n\n\\begin{lemma}\\label{dashres}\nLet $c,d,s\\in\\Z^+$ with $d\\geq2$, $1\\leq c,s\\leq d$ and $\\gcd(cs,d)=1$. Then, for any prime $p\\eq s\\pmod{d}$, we have\n$$\n\\alpha^{\\ast}=\\f{\\<s^{-1}c\\>_d}{d},\n$$\nwhere $s^{-1}$ stands for the inverse of $s$ modulo $d$.\n\\end{lemma}\n\n\\begin{proof}\nObviously,\n\\begin{align*}\n\\sum_{k=0}^{p^r-1}F(\\alpha^{\\ast_r}p^r,k)&=\\sum_{k=0}^{p^r-1}(2k+\\alpha^{\\ast_r}p^r)\\f{(\\alpha^{\\ast_r}p^r)_k^3(\\f12)_k}{(1)_k^3(\\f12+\\alpha^{\\ast_r}p^r)_k}\\\\\n&=\\alpha^{\\ast_r}p^r+p^{3r}{\\alpha^{\\ast_r}}^3\\sum_{k=1}^{p^r-1}\\f{2k+\\alpha^{\\ast_r}p^r}{k^3}\\f{(1+\\alpha^{\\ast_r}p^r)_{k-1}^3(\\f12)_k}{(1)_{k-1}^3(\\f12+\\alpha^{\\ast_r}p^r)_k}.\n\\end{align*}\nNote that for $k\\in\\{1,2,\\ldots,p^r-1\\}$, $\\ord_p(k)\\leq r-1$, where $\\ord_p$ stands for the $p$-adic order. Therefore,\n$$\n\\ord_p\\l(\\f{p^{3r}}{k^2}\\r)\\geq r+2\\quad \\t{and}\\quad \\ord_p\\l(\\f{p^{4r}}{k^3}\\r)\\geq r+3.\n$$\nMoreover, it is easy to see that\n$$\n\\f{(1+\\alpha^{\\ast_r}p^r)_{k-1}}{(1)_{k-1}^3}=\\prod_{j=1}^{k-1}\\f{j+\\alpha^{\\ast_r}p^r}{j}=\\prod_{j=1}^{k-1}\\l(1+\\f{\\alpha^{\\ast_r}p^r}{j}\\r)\\eq1\\pmod{p}\n$$\nand\n\\begin{align*}\n\\f{(\\f12)_k}{(\\f12+\\alpha^{\\ast_r}p^r)_k}&=\\prod_{j=0}^{k-1}\\f{\\f12+j}{\\f12+j+\\alpha^{\\ast_r}p^r}=\\prod_{j=0}^{k-1}\\f{1}{1+\\f{\\alpha^{\\ast_r}p^r}{\\f12+j}}\\\\\n&\\eq\\begin{cases}1/(2\\alpha^{\\ast_r}+1)\\pmod{p}\\quad&\\t{if}\\ k\\geq (p^r+1)/2,\\\\ 1\\pmod{p}\\quad&\\t{if}\\ k\\leq (p^r-1)/2.\\end{cases}\n\\end{align*}\nCombining the above and in view of Lemmas \\ref{u} and \\ref{2ordharmonicdown}, we arrive at\n$$\n\\sum_{k=0}^{p^r-1}F(\\alpha^{\\ast_r}p^r,k)\\eq \\alpha^{\\ast_r}p^r+2(\\alpha^{\\ast_r})^3p^{3r}\\sum_{k=1}^{(p^r-1)/2}\\f{1}{k^2}+\\f{2(\\alpha^{\\ast_r})^3p^{3r}}{2\\alpha^{\\ast_r}+1}\\sum_{k=(p^r+1)/2}^{p^r-1}\\f{1}{k^2}\\eq \\alpha^{\\ast_r}p^r \\pmod{p^{r+3}},\n$$\nas desired.\n\\end{proof}\n\n\\begin{lemma}\\label{evalG}\nUnder the conditions of Theorem \\ref{mainth1}, we have\n\\begin{equation}\\label{evalGeq}\n\\sum_{l=0}^{a-1}G(\\alpha+l,p^r)\\eq \\f{(\\alpha^{\\ast_r})^3}{(\\f12+\\alpha)^{\\ast_r}}p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_{r-1}}}^{(2)}\\pmod{p^{r+3}}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nClearly,\n\\begin{align*}\n\\sum_{l=0}^{a-1}G(\\alpha+l,p^r)&=\\sum_{l=0}^{a-1}\\f{p^{3r}(p^r+2\\alpha+2l)}{(\\alpha+l)^3}\\f{(\\alpha+l)_{p^r}^3(\\f12)_{p^r}}{(1)_{p^r}^3(\\f12+\\alpha+l)_{p^r}}\\\\\n&=\\f{p^{3r}(\\alpha)_{p^r}^3(\\f12)_{p^r}}{(1)_{p^r}^3(\\f12+\\alpha)_{p^r}}\\sum_{l=0}^{a-1}\\f{(p^r+2\\alpha+2l)(\\alpha+p^r)_l^3(\\f12+\\alpha)_l}{(\\alpha+l)^3(\\alpha)_l^3(\\f12+\\alpha+p^r)_l}.\n\\end{align*}\nThen, by Lemma \\ref{pochreduce}, we have\n\\begin{align*}\n\\sum_{l=0}^{a-1}G(\\alpha+l,p^r)&=\\f{p^{3r}(\\alpha^{\\ast_r})^3}{2(\\f12+\\alpha)^{\\ast_r}}\\prod_{j=1}^r\\f{\\Gamma_p(\\alpha+p^j)^3\\Gamma_p(\\f12+p^j)\\Gamma_p(1)^3\\Gamma_p(\\f12+\\alpha)}{\\Gamma_p(\\alpha)^3\\Gamma_p(\\f12)\\Gamma_p(1+p^j)^3\\Gamma_p(\\f12+\\alpha+p^j)}\\\\\n&\\quad\\times \\sum_{l=0}^{a-1}\\f{(p^r+2\\alpha+2l)(\\alpha+p^r)_l^3(\\f12+\\alpha)_l}{(\\alpha+l)^3(\\alpha)_l^3(\\f12+\\alpha+p^r)_l},\n\\end{align*}\nwhere we have used the facts\n$$\n\\l(\\f12\\r)^{\\ast}=\\f12\\quad \\t{and}\\quad 1^{\\ast}=1.\n$$\nSince $(\\f12+\\alpha)^{\\ast_r}\\not\\eq0\\pmod{p}$, we have\n$$\n\\ord_p\\l(\\f{p^{3r}(\\alpha^{\\ast_r})^3}{(\\f12+\\alpha)^{\\ast_r}}\\r)\\geq 3r.\n$$\nIt is easy to see that $\\ord_p(\\alpha+l)\\leq r-1$ for $l\\in\\{0,1,2,\\ldots,a-1\\}$. It follows that\n$$\n\\ord_p\\l(\\f{p^{4r}}{(\\alpha+l)^3}\\r)\\geq r+3\n$$\nand\n$$\n\\ord_p\\l(\\f{p^{3r}}{(\\alpha+l)^2}\\r)\\geq r+2.\n$$\nFor $k\\in\\{0,1,2,\\ldots,a-1\\}$ we have\n$$\n\\f{(\\alpha+p^r)_l}{(\\alpha)_l}=\\prod_{j=0}^{l-1}\\f{\\alpha+j+p^r}{\\alpha+j}=\\prod_{j=0}^{l-1}\\l(1+\\f{p^r}{\\alpha+j}\\r)\\eq1\\pmod{p}.\n$$\nIn view of Lemma \\ref{1/2+alpha} and the above, we obtain\n\\begin{equation}\\label{evalGkey}\n\\sum_{l=0}^{a-1}G(\\alpha+l,p^r)\\eq\\f{p^{3r}(\\alpha^{\\ast_r})^3}{(\\f12+\\alpha)^{\\ast_r}}\\sum_{l=0}^{a-1}\\f{(\\f12+\\alpha)_l}{(\\alpha+l)^2(\\f12+\\alpha+p^r)_l}\\pmod{p^{r+3}}.\n\\end{equation}\n\nSuppose that $p=3$. By a similar argument as in the proof of Lemma \\ref{evalF}, we have\n\\begin{align*}\n\\sum_{k=0}^{p^r-1}F(\\alpha^{\\ast_r}p^r,k)&\\eq \\alpha^{\\ast_r}p^r+2(\\alpha^{\\ast_r})^3p^{3r}\\sum_{k=1}^{(p^r-1)/2}\\f{1}{k^2}+\\f{2(\\alpha^{\\ast_r})^3p^{3r}}{2\\alpha^{\\ast_r}+1}\\sum_{k=(p^r+1)/2}^{p^r-1}\\f{1}{k^2}\\pmod{p^{r+3}}.\n\\end{align*}\nNote that \n$$\n\\ord_p\\l(\\sum_{k=1}^{(p^r-1)/2}\\f{1}{k^2}\\r)\\geq -2(r-1)\\quad\\t{and}\\quad \\ord_p\\l(\\sum_{k=(p^r+1)/2}^{p^r-1}\\f{1}{k^2}\\r)\\geq -2(r-1).\n$$\nSince $\\alpha^{\\ast_r}=3/4\\eq0\\pmod{p}$ and $2\\alpha^{\\ast_r}+1=5/2\\not\\eq0\\pmod{p}$, we still have\n\\begin{equation}\\label{evalF'}\n\\sum_{k=0}^{p^r-1}F(\\alpha^{\\ast_r}p^r,k)\\eq \\alpha^{\\ast_r}p^r\\pmod{p^{r+3}}.\n\\end{equation}\nNow, $a=(3p^r-1)/4\\geq (p^r+1)/2$. From the proof of Lemma \\ref{evalG}, we know\n\\begin{align*}\n&\\sum_{l=0}^{a-1}G(\\alpha+l,p^r)\\\\\n&\\qquad\\eq \\f{p^{3r}(\\alpha^{\\ast_r})^3}{(\\f12+\\alpha)^{\\ast_r}}\\sum_{l=(p^r+1)/2}^{a}\\f{1}{(\\alpha+a-l)^2}+\\f{p^{3r}(\\alpha^{\\ast_r})^3(2\\alpha^{\\ast_r}-1)}{(\\f12+\\alpha)^{\\ast_r}(2\\alpha^{\\ast_r}+1)}\\sum_{l=1}^{(p^r-1)/2}\\f{1}{(\\alpha+a-l)^2}\\pmod{p^{r+3}}.\n\\end{align*}\nAlso, \n$$\n\\ord_p\\l(\\sum_{l=(p^r+1)/2}^{a}\\f{1}{(\\alpha+a-l)^2}\\r)\\geq -2(r-1)\\quad\\t{and}\\quad \\ord_p\\l(\\sum_{l=1}^{(p^r-1)/2}\\f{1}{(\\alpha+a-l)^2}\\r)\\geq -2(r-1).\n$$\nThen, in view of the facts $\\alpha^{\\ast_r}\\eq0\\pmod{p}$, $(1/2+\\alpha)^{\\ast_r}=1/4\\not\\eq0\\pmod{p}$ and $2\\alpha^{\\ast_r}+1\\not\\eq0\\pmod{p}$, we obtain\n$$\n\\sum_{l=0}^{a-1}G(\\alpha+l,p^r)\\eq0\\pmod{p^{r+3}}.\n$$\nThis, together with \\eqref{evalF'} gives\n$$\n\\sum_{k=0}^{p^r-1}F(\\alpha,k)=\\sum_{k=0}^{p^r-1}F(\\alpha^{\\ast_r}p^r,k)-\\sum_{l=0}^{a-1}G(\\alpha+l,p^r)\\eq\\alpha^{\\ast_r}p^r\\pmod{p^{r+3}},\n$$\nwhich implies\n\\begin{equation*}\n\\sum_{k=0}^{3^r-1}(8k+1)\\f{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\eq 3^{r+1}\\pmod{3^{r+3}}.\n\\end{equation*}\nMoreover, since \n$$\n\\ord_3\\l(\\sum_{j=1}^{(3^r-3)/4}\\f{1}{j^2}\\r)\\geq-2(r-1),\n$$\nwe have\n$$\n\\f{27}{4}3^{3r}\\sum_{j=1}^{(3^r-3)/4}\\f{1}{j^2}\\eq 0\\pmod{3^{r+3}}.\n$$\nTherefore, we arrive at\n\\begin{equation*}\n\\sum_{k=0}^{3^r-1}(8k+1)\\f{(\\f14)_k^3(\\f12)_k}{(1)_k^3(\\f34)_k}\\eq 3^{r+1}+\\f{27}{4}3^{3r}\\sum_{j=1}^{(3^r-3)/4}\\f{1}{j^2}\\pmod{3^{r+3}},\n\\end{equation*}\nas desired.", "post_theorem_intro_text_len": 3020, "post_theorem_intro_text": "Theorem \\ref{mainth1} seems quite strange. When $\\<-1/2-\\alpha\\>_p<\\min\\{\\<-\\alpha\\>_p,(p-1)/2\\}$, the summands on the left-hand side of \\eqref{mainth1eq} are not always $p$-adic integers. However, the sum of these summands is a $p$-adic integer. This phenomenon renders it difficult for us to prove Theorem \\ref{mainth1} directly using the formula \\eqref{Whipple} and conventional congruence techniques. To overcome this obstacle, we find a new parametric WZ pair (cf. \\cite{PWZ}) which allows us to transform the original sum to a computable form in the sense of congruence. This idea is crucial in our proof.\n\nIn particular, putting $d=4,s=3,c=1$ and requiring $r$ to be odd in Theorem \\ref{mainth1}, we have the following result.\n\\begin{corollary}\\label{cor}\nGuo and Zhao's conjectural supercongruence \\eqref{GuoZhaoconj7.2} \\cite[Conjecture 7.2]{Guo-Zhao2026} is true.\n\\end{corollary}\nMoreover, Theorem \\ref{mainth1} with $d=2,s=1,c=1$ gives that\n\\begin{equation}\\label{cor1}\n\\sum_{k=0}^{p^r-1}(4k+1)\\frac{(\\f12)_k^4}{(1)_k^4}\\equiv p^r \\pmod{p^{r+3}},\n\\end{equation}\nwhere we have used \\eqref{2ordharmonic}. Note that the $r=1$ case of \\eqref{cor1} is a stronger version of Van Hamme's (C.2) supercongruence \\cite{VanHamme1997} and was first proved by Long \\cite{Long2011}. \n\nFor the convenience of interested readers, in Table \\ref{examples}, we provide some concrete examples of the parameters in Theorem \\ref{mainth1} for future use.\n\n\\begin{table}[htbp]\n    \\centering\n    \\caption{Examples of the parameters in Theorem \\ref{mainth1}}\n    \\label{examples}\n    \\renewcommand{\\arraystretch}{1.5}\n    \\begin{tabular}{>{\\centering\\arraybackslash}p{2cm}\n                      >{\\centering\\arraybackslash}p{2cm}\n                      >{\\centering\\arraybackslash}p{2cm}\n                      >{\\centering\\arraybackslash}p{3cm}\n                      >{\\centering\\arraybackslash}p{3cm}}\n        \\toprule \n        $d$ & $s$ & $\\alpha$ & $\\alpha^{\\ast_r}$ & $(1/2+\\alpha)^{\\ast_r}$\\\\\n        \\midrule \n        $2$ & $1$ & $1/2$  & $1/2$ & $1$\\\\\n        $3$ & $1$ & $1/3$  & $1/3$ & $5/6$\\\\\n        $3$ & $1$ & $2/3$  & $2/3$ & $1/6$\\\\\n        $3$ & $1$ & $1/6$  & $1/6$ & $2/3$\\\\\n        $3$ & $1$ & $5/6$  & $5/6$ & $1/3$\\\\\n        $3$ & $2$ & $1/3$  & $(3-(-1)^r)/6$ & $(3+2(-1)^r)/6$\\\\\n        $3$ & $2$ & $2/3$  & $(3+(-1)^r)/6$ & $(3-2(-1)^r)/6$\\\\\n        $3$ & $2$ & $1/6$  & $(3-2(-1)^r)/6$ & $(3+(-1)^r)/6$\\\\\n        $3$ & $2$ & $5/6$  & $(3+2(-1)^r)/6$ & $(3-(-1)^r)/6$\\\\\n        $4$ & $1$ & $1/4$ & $1/4$ & $3/4$\\\\\n        $4$ & $1$ & $3/4$ & $3/4$ & $1/4$\\\\\n        $4$ & $3$ & $1/4$ & $(2-(-1)^r)/4$ & $(2+(-1)^r)/4$\\\\\n        $4$ & $3$ & $3/4$ & $(2+(-1)^r)/4$ & $(2-(-1)^r)/4$\\\\\n        \\bottomrule\n    \\end{tabular}\n\\end{table}\n\nWe briefly outline this paper. In Section \\ref{sec2}, we prove some properties of Dwork's dash operation and give some immediate applications which play essential roles in the subsequent proof. We shall prove Theorem \\ref{mainth1} and Corollary \\ref{cor} in Section \\ref{sec3}.", "sketch": "The introduction explains why a direct proof of Theorem~\\ref{mainth1} is hard and what method is used instead: when $\\langle-1/2-\\alpha\\rangle_p<\\min\\{\\langle-\\alpha\\rangle_p,(p-1)/2\\}$, “the summands on the left-hand side of \\eqref{mainth1eq} are not always $p$-adic integers,” although “the sum of these summands is a $p$-adic integer.” This “renders it difficult…to prove Theorem~\\ref{mainth1} directly using the formula \\eqref{Whipple} and conventional congruence techniques.”\n\nTo “overcome this obstacle,” the paper’s approach is to “find a new parametric WZ pair (cf. \\cite{PWZ}) which allows us to transform the original sum to a computable form in the sense of congruence,” and “this idea is crucial in our proof.” Additionally, Section~\\ref{sec2} establishes “properties of Dwork’s dash operation” with “immediate applications” that “play essential roles in the subsequent proof,” and then “We shall prove Theorem~\\ref{mainth1} and Corollary~\\ref{cor} in Section~\\ref{sec3}.”", "expanded_sketch": "The introduction explains why a direct proof of the main theorem is hard and what method is used instead: when $\\langle-1/2-\\alpha\\rangle_p<\\min\\{\\langle-\\alpha\\rangle_p,(p-1)/2\\}$, “the summands on the left-hand side of\n\\begin{equation}\\label{mainth1eq}\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\f{(\\alpha)_k^3(\\f12)_k}{(1)_k^3(\\f12+\\alpha)_k\\eq \\alpha^{\\ast_r}p^r -\\f{(\\alpha^{\\ast_r})^3}{(\\f12+\\alpha)^{\\ast_r}}p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_{r-1}}}^{(2)}\\pmod{p^{r+3}},\n\\end{equation}\nare not always $p$-adic integers,” although “the sum of these summands is a $p$-adic integer.” This “renders it difficult…to prove the main theorem directly using the formula\n\\begin{align}\\label{Whipple}\n&\\sum_{k=0}^{\\infty}\\f{(a)_k(1+\\f{a}{2})_k(b)_k(c)_k(d)_k}{(1)_k(\\f a2)_k(1+a-b)_k(1+a-c)_k(1+a-d)_k}\\notag\\\\\n&\\qquad=\\f{\\Gamma(1+a-b)\\Gamma(1+a-c)\\Gamma(1+a-d)\\Gamma(1+a-b-c-d)}{\\Gamma(1+a)\\Gamma(1+a-b-c)\\Gamma(1+a-b-d)\\Gamma(1+a-c-d)}.\n\\end{align}\nand conventional congruence techniques.”\n\nTo “overcome this obstacle,” the paper’s approach is to “find a new parametric WZ pair (cf. \\cite{PWZ}) which allows us to transform the original sum to a computable form in the sense of congruence,” and “this idea is crucial in our proof.” Additionally, next the paper establishes “properties of Dwork’s dash operation” with “immediate applications” that “play essential roles in the subsequent proof,” and then later it proves the main theorem and also the following result.\n\n\\begin{corollary}\\label{cor}\nGuo and Zhao's conjectural supercongruence \\eqref{GuoZhaoconj7.2} \\cite[Conjecture 7.2]{Guo-Zhao2026} is true.\n\\end{corollary}", "expanded_theorem": "\\label{mainth1}\nLet $c,d,s\\in\\mathbb Z^+$ with $d\\geq 2$, $1\\leq c,s\\leq d$ and $\\gcd(cs,d)=1$, and let $p\\geq5$ be a prime with $p\\equiv s\\pmod d$. Then, for any $r\\in\\mathbb Z^+$ with $(\\f12+\\alpha)^{\\ast_r}(\\f12+\\alpha^{\\ast_r})\\not\\eq0\\pmod{p}$,  we have\n\\begin{equation}\\label{mainth1eq}\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\frac{(\\alpha)_k^3(\\f12)_k}{(1)_k^3(\\f12+\\alpha)_k}\\equiv \\alpha^{\\ast_r}p^r -\\frac{(\\alpha^{\\ast_r})^3}{(\\f12+\\alpha)^{\\ast_r}}p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_{r-1}}}^{(2)}\\pmod{p^{r+3}},\n\\end{equation}\nwhere $\\alpha=c/d$.", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let \\(\\alpha=c/d\\), where \\(c,d,s\\in\\mathbb Z^+\\) satisfy \\(d\\ge 2\\), \\(1\\le c,s\\le d\\), and \\(\\gcd(cs,d)=1\\). Let \\(p\\ge 5\\) be a prime with \\(p\\equiv s\\pmod d\\). For a \\(p\\)-adic integer \\(x\\), define Dwork’s dash operation by \\(x^\\ast=(x+\\langle -x\\rangle_p)/p\\), where \\(\\langle y\\rangle_p\\) is the least nonnegative residue of \\(y\\) modulo \\(p\\); define its iterates by \\(x^{\\ast_0}=x\\) and \\(x^{\\ast_m}=(x^{\\ast_{m-1}})^\\ast\\) for \\(m\\ge1\\). Also write \\((x)_k=x(x+1)\\cdots(x+k-1)\\) with \\((x)_0=1\\), and \\(H_n^{(2)}=\\sum_{j=1}^n 1/j^2\\). If \\(r\\in\\mathbb Z^+\\) satisfies \\((\\tfrac12+\\alpha)^{\\ast_r}(\\tfrac12+\\alpha^{\\ast_r})\\not\\equiv0\\pmod p\\), which quantitative congruence holds?", "correct_choice": {"label": "A", "text": "For every such \\(r\\),\n\\[\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\frac{(\\alpha)_k^3(\\tfrac12)_k}{(1)_k^3(\\tfrac12+\\alpha)_k}\\equiv \\alpha^{\\ast_r}p^r-\\frac{(\\alpha^{\\ast_r})^3}{(\\tfrac12+\\alpha)^{\\ast_r}}\\,p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_{r-1}}}^{(2)}\\pmod{p^{r+3}}.\n\\]"}, "choices": [{"label": "B", "text": "For every such \\(r\\),\n\\[\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\frac{(\\alpha)_k^3(\\tfrac12)_k}{(1)_k^3(\\tfrac12+\\alpha)_k}\\equiv \\alpha^{\\ast_r}p^r-\\frac{(\\alpha^{\\ast_r})^3}{\\tfrac12+\\alpha^{\\ast_r}}\\,p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_{r-1}}}^{(2)}\\pmod{p^{r+3}}.\n\\]"}, {"label": "C", "text": "For every such \\(r\\),\n\\[\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\frac{(\\alpha)_k^3(\\tfrac12)_k}{(1)_k^3(\\tfrac12+\\alpha)_k}\\equiv \\alpha^{\\ast_r}p^r\\pmod{p^{r+2}}.\n\\]"}, {"label": "D", "text": "For every such \\(r\\),\n\\[\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\frac{(\\alpha)_k^3(\\tfrac12)_k}{(1)_k^3(\\tfrac12+\\alpha)_k}\\equiv \\alpha^{\\ast_r}p^r-\\frac{(\\alpha^{\\ast_r})^3}{(\\tfrac12+\\alpha)^{\\ast_r}}\\,p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_{r-1}}}^{(2)}\\pmod{p^{r+4}}.\n\\]"}, {"label": "E", "text": "If there exists \\(r\\in\\mathbb Z^+\\) such that \\((\\tfrac12+\\alpha)^{\\ast_r}(\\tfrac12+\\alpha^{\\ast_r})\\not\\equiv0\\pmod p\\), then\n\\[\n\\sum_{k=0}^{p^r-1}(2k+\\alpha)\\frac{(\\alpha)_k^3(\\tfrac12)_k}{(1)_k^3(\\tfrac12+\\alpha)_k}\\equiv \\alpha^{\\ast_r}p^r-\\frac{(\\alpha^{\\ast_r})^3}{(\\tfrac12+\\alpha)^{\\ast_r}}\\,p^{r+2}H_{\\alpha^{\\ast_r}p-\\alpha^{\\ast_r}}^{(2)}\\pmod{p^{r+3}}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "dash_dependence_in_denominator", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped_second_order_correction_term_and_one_power_of_p_in_modulus", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "modulus_precision_p^{r+3}", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "harmonic_index_uses_alpha^{*_{r-1}}_not_alpha^{*_r}", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct congruence explicitly or through obvious wording cues; the answer must be selected from several close formulas."}, "TAS": {"score": 1, "justification": "The item is essentially asking for the exact stated conclusion of a specialized congruence theorem, with only slight perturbations in the alternatives. This is more theorem-recall than a genuinely new inferential task."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the choices differ in subtle but meaningful ways (denominator, modulus strength, harmonic index, weaker/stronger form). However, success mainly depends on precise recall or recognition of the theorem rather than generating a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: confusing dash placement, accepting a weaker true statement, overclaiming a stronger modulus, or altering the harmonic-sum index/quantifier."}, "total_score": 6, "overall_assessment": "A technically strong MCQ with high-quality distractors and no direct answer leakage, but it leans heavily on theorem recognition/restatement rather than deeper generative mathematical reasoning."}}
{"id": "2602.13435v1", "paper_link": "http://arxiv.org/abs/2602.13435v1", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "\\label{thm:main_result} Let $E$ be a smooth complex elliptic curve with $\\End(E)=\\mathbb{Z}$. There exists a smooth projective complex surface $S$ with $\\Alb(S)\\cong E$ for which $\\Ch(S)$ is representable, while $S$ admits no universal $0$-cycle.", "start_pos": 11714, "end_pos": 11981, "label": "thm:main_result"}, "ref_dict": {"cor:integral_Hodge_S": "\\begin{corollary}\\label{cor:integral_Hodge_S} Let $E$ be a smooth elliptic curve over $\\bbC$ with $\\End(E)=\\Z$. Then there exists a bielliptic surface $S$ of type 2 whose Albanese variety $\\Alb(S)$ is isomorphic to $E,$ and for which the integral Hodge conjecture for $1$-cycles on $E\\times S$ fails. In particular, there exists a non-torsion integral Hodge class in $H^{4}(E\\times S,\\Z)$ that is not algebraic.\n\\end{corollary}", "cor:obstruction": "\\begin{corollary}\\label{cor:obstruction}\nWe use the notation and assumptions of Theorem \\ref{thm:obstruction}, with condition \\eqref{it:correspondence} replaced by the following:\n\\begin{enumerate}\n    \\item[\\((2')\\)]\\label{it:correspondence-prime} There exists an algebraically closed field extension $F/K$ and a correspondence\n$$\n[\\Gamma] \\in \\CH^{d}(C \\times_k X_{F})\n$$\ninducing a splitting of the pushforward map\n$$\n(\\phi_{X_F})_\\ast\\colon \\Ch(X_{F})_{\\hom}\\longrightarrow\\jac(C_{F}).\n$$\n\\end{enumerate}\nAssume moreover that $\\End(\\jac(C))=\\Z$. Then the sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(X_{0i})\\longrightarrow\\jac(C)$$\nadmits a section. \n\\end{corollary}", "ques:2-dim_example": "\\begin{ques}\\label{ques:2-dim_example}\nDoes there exist a smooth projective complex surface $S$ whose Chow group of $0$-cycles $\\CH_0(S)$ is representable, but which admits no universal $0$-cycle?\n\\end{ques}", "eq:fibration_X": "\\begin{equation}\\label{eq:fibration_X}\\phi_{X}\\colon X\\longrightarrow E\\end{equation}", "eq:abel-jacobi-map": "\\begin{equation}\n    \\label{eq:abel-jacobi-map} \\alpha_{X}\\colon\\Ch(X)_{\\hom}\\longrightarrow\\Alb(X).\n\\end{equation}", "thm:main_result": "\\begin{theorem}\\label{thm:main_result} Let $E$ be a smooth complex elliptic curve with $\\End(E)=\\Z$. There exists a smooth projective complex surface $S$ with $\\Alb(S)\\cong E$ for which $\\Ch(S)$ is representable, while $S$ admits no universal $0$-cycle.\n\\end{theorem}", "it:albanese_S_0": "\\begin{enumerate}\n    \\item\\label{it:generic_fibre} The geometric generic fibre $S_{K}$ is a bielliptic surface of type $2$, and the induced morphism $\\phi_{S_{K}}\\colon S_{K}\\to E\\times_{k} K$ coincides with the Albanese fibration of $S_{K}$.\n    \\item\\label{it:special_fibre} The special fibre $S_{0}=\\sum_{i=1}^{n}R_{i}$ is a reduced simple normal crossings divisor on $\\mathcal{S}$ whose dual graph is a chain; that is, each intersection $R_{i}\\cap R_{i+1}$ is a smooth irreducible curve, and $R_{i}\\cap R_{j}=\\varnothing$ whenever $j\\notin\\{i-1,i,i+1\\}$.\n     \\item\\label{it:albanese_S_0} For each $i,$ let $\\al_{i}:\\Alb(R_{i})\\to E$ denote the homomorphism of Albanese varieties induced by the restriction of $\\phi_{\\mathcal{S}}$ to $R_{i}$. Then the sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(R_{i})\\longrightarrow E$$ does not admit a section. \n\\end{enumerate}", "eq:psi": "\\begin{equation}\\label{eq:psi}\n    \\psi_{(Z,[\\Gamma])}\\colon Z\\longrightarrow\\Alb(X),\\quad z\\longmapsto \\alpha_{X}\\circ[\\Gamma]_{\\ast}(z-z_{0})\n\\end{equation}", "thm:obstruction": "\\begin{theorem}\n\\label{thm:obstruction}\nLet $k$ be an algebraically closed field and let $C$ be a smooth projective curve over $k$. Set $\\Delta := \\Spec k[[t]]$, and fix an algebraic closure $K$ of its function field. \n\nLet $p \\colon \\mathcal{X} \\to \\Delta$ be a flat, projective morphism of relative dimension $d$ with regular total space, and let $\\phi_{\\mathcal{X}} \\colon \\mathcal{X} \\to C \\times_k \\Delta$ be a surjective morphism.\nDenote by $X_0$ the special fibre of $p$, by\n$\\phi_{X_0} \\colon X_0 \\to C$ the induced morphism, and by $X_{K\n}$ the geometric generic fibre of $p$. Write $\\phi_{X_K} \\colon X_K \\to C_K$\nfor the base change of $\\phi_{\\mathcal{X}}$ to $K$.\\par\nWe further assume the following:\n\\begin{enumerate}\n    \\item \\label{it:X_0} $X_{0}=\\sum_{i=1}^{n}X_{0i}$ is a reduced simple normal crossings divisor on $\\mathcal{X}$ whose dual graph is a chain. Precisely, each intersection $X_{0i}\\cap X_{0i+1}$ is smooth and irreducible of dimension $d-1,$ and $X_{0i}\\cap X_{0j}=\\varnothing$ whenever $j\\notin\\{i-1,i,i+1\\}$.\n    \\item \\label{it:correspondence} \n    There exists a correspondence\n$$\n[\\Gamma] \\in \\CH^{d}(C \\times_k X_{K})\n$$\ninducing a splitting of the pushforward map\n$$\n(\\phi_{X_K})_\\ast\\colon \\Ch(X_{K})_{\\hom}\\longrightarrow\\jac(C_{K}).\n$$\n\n\\end{enumerate}\n\\par \nFor each $i,$ let $\\al_{i}\\colon\\Alb(X_{0i})\\to \\jac(C)$ be the homomorphism of Albanese varieties induced by the restriction of $\\phi_{X_0}$ to $X_{0i}$. Then the induced sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(X_{0i})\\longrightarrow\\jac(C)$$\nadmits a section. \n\\end{theorem}", "thm:intro_degeneration_S": "\\begin{theorem}\\label{thm:intro_degeneration_S} Let $k$ be an algebraically closed field of characteristic $\\neq2,$ and let $E$ be an elliptic curve over $k$ with $\\End(E)\\cong\\Z$. Set $\\Delta:=\\Spec k[[t]],$ and fix an algebraic closure $K$ of its function field. Then there exists a regular, flat, projective scheme $\\mathcal{S}\\to\\Delta,$ together with a morphism $$\\phi_{\\mathcal{S}}\\colon\\mathcal{S}\\longrightarrow E\\times\\Delta$$ such that the following properties hold:\n\\begin{enumerate}\n    \\item\\label{it:generic_fibre} The geometric generic fibre $S_{K}$ is a bielliptic surface of type $2$, and the induced morphism $\\phi_{S_{K}}\\colon S_{K}\\to E\\times_{k} K$ coincides with the Albanese fibration of $S_{K}$.\n    \\item\\label{it:special_fibre} The special fibre $S_{0}=\\sum_{i=1}^{n}R_{i}$ is a reduced simple normal crossings divisor on $\\mathcal{S}$ whose dual graph is a chain; that is, each intersection $R_{i}\\cap R_{i+1}$ is a smooth irreducible curve, and $R_{i}\\cap R_{j}=\\varnothing$ whenever $j\\notin\\{i-1,i,i+1\\}$.\n     \\item\\label{it:albanese_S_0} For each $i,$ let $\\al_{i}:\\Alb(R_{i})\\to E$ denote the homomorphism of Albanese varieties induced by the restriction of $\\phi_{\\mathcal{S}}$ to $R_{i}$. Then the sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(R_{i})\\longrightarrow E$$ does not admit a section. \n\\end{enumerate}  \n\\end{theorem}"}, "pre_theorem_intro_text_len": 3797, "pre_theorem_intro_text": "Let $X$ be a smooth projective complex variety of dimension $d=\\dim X$. Given a base-point $x_{0}\\in X(\\mathbf{C}),$ the universal morphism \\begin{equation*}\n    \\al_{X}\\colon X\\longrightarrow\\Alb(X)\n\\end{equation*}\nto the Albanese variety of $X$ (see \\cite[Theorem 1.4.4]{Murre}) induces the so-called Abel--Jacobi map for $0$-cycles on $X$ \\begin{equation}\n    \\label{eq:abel-jacobi-map} \\alpha_{X}\\colon\\Ch(X)_{\\hom}\\longrightarrow\\Alb(X).\n\\end{equation}\nThe group homomorphism $\\alpha_{X}$ no longer depends on the choice of the base-point and it is known to be surjective and regular. The latter asserts that, for any smooth projective complex variety $Z$ equipped with a base-point $z_{0}\\in Z(\\mathbf{C})$, and for any codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(Z\\times X)$, the assignment\n\\begin{equation}\\label{eq:psi}\n    \\psi_{(Z,[\\Gamma])}\\colon Z\\longrightarrow\\Alb(X),\\quad z\\longmapsto \\alpha_{X}\\circ[\\Gamma]_{\\ast}(z-z_{0})\n\\end{equation}\ndefines a morphism of algebraic varieties; see \\cite[Definition 1.6.1]{Murre}. In fact, the Abel--Jacobi map $\\alpha_{X}$ also enjoys the universal property of being initial among all regular homomorphisms from $\\Ch(X)_{\\hom}$ to abelian varieties; see \\cite[Example 1.8(b)]{Murre}.\n\\par Since the homomorphism $\\alpha_{X}$ in \\eqref{eq:abel-jacobi-map} is regular and surjective, there exists a codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(\\Alb(X)\\times X)$ and a positive integer $n$ such that the morphism $\\psi_{(\\Alb(X),[\\Gamma])}$ in \\eqref{eq:psi}, with respect to the chosen base-point $0_{\\Alb(X)}\\in\\Alb(X)(\\mathbf{C})$, is equal to $n\\cdot\\Id_{\\Alb(X)}$; see \\cite[Corollary 1.6.3]{Murre}.\\par\n\nThe present work is concerned with a strengthening of this statement, in the sense of the following property of the variety $X$, first introduced and studied by Voisin; see \\cite{Voisin2024CycleCO,Voisin2025}.\n\n\\begin{definition}[\\cite{voisin_a}, Definition 1.1]\\label{def:universal_0-cycle} We say that $X$ admits a \\emph{universal $0$-cycle} if there exists a codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(\\Alb(X)\\times X)$ such that the morphism $\\psi_{(\\Alb(X),[\\Gamma])}$ in \\eqref{eq:psi}, with respect to the base-point $0_{\\Alb(X)}\\in\\Alb(X)(\\mathbf{C})$, is the identity map $\\Id_{\\Alb(X)}$ on the Albanese variety of $X$. \n\\end{definition}\n\nIf $X$ is a curve, then the Poincaré divisor on $\\Jac(X)\\times X$ induces a universal $0$-cycle on $X$. In contrast, Voisin showed in \\cite{Voisin2025} that for each $d\\geq2$, there exists a smooth projective complex $d$-fold $X$ admitting no universal $0$-cycle; see \\cite[Corollary  0.14]{Voisin2025}.\\par\nIn \\cite{colliotthélène2025notessurlapplicationdalbanese}, Colliot--Th\\'{e}l\\`ene considered whether the existence of a universal $0$-cycle persists under additional geometric assumptions on a smooth projective complex variety $X$. In particular, he asked whether this property holds for varieties with representable $\\Ch$-group, that is, for which the homomorphism $\\alpha_{X}$ in \\eqref{eq:abel-jacobi-map} is an isomorphism; see \\cite{Mumford1969,Roitman_1972}. Building on earlier ideas of Benoist--Ottem \\cite{benoist-ottem}, Voisin subsequently constructed in \\cite{voisin_a} a smooth projective complex threefold $X$ with representable $\\Ch$-group but admitting no universal $0$-cycle.\\par\n\nThis leads to the following question, raised by Totaro after Voisin’s talk at the workshop \\emph{Hodge Theory and Algebraic Cycles} (Clay Mathematics Institute, September 2025).\n\n\\begin{ques}\\label{ques:2-dim_example}\nDoes there exist a smooth projective complex surface $S$ whose Chow group of $0$-cycles $\\CH_0(S)$ is representable, but which admits no universal $0$-cycle?\n\\end{ques}\n\nThe main result of the present paper answers Question \\ref{ques:2-dim_example} in the affirmative.", "context": "Let $X$ be a smooth projective complex variety of dimension $d=\\dim X$. Given a base-point $x_{0}\\in X(\\mathbf{C}),$ the universal morphism \\begin{equation*}\n \\al_{X}\\colon X\\longrightarrow\\Alb(X)\n\\end{equation*}\nto the Albanese variety of $X$ (see \\cite[Theorem 1.4.4]{Murre}) induces the so-called Abel--Jacobi map for $0$-cycles on $X$ \\begin{equation}\n \\label{eq:abel-jacobi-map} \\alpha_{X}\\colon\\Ch(X)_{\\hom}\\longrightarrow\\Alb(X).\n\\end{equation}\nThe group homomorphism $\\alpha_{X}$ no longer depends on the choice of the base-point and it is known to be surjective and regular. The latter asserts that, for any smooth projective complex variety $Z$ equipped with a base-point $z_{0}\\in Z(\\mathbf{C})$, and for any codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(Z\\times X)$, the assignment\n\\begin{equation}\\label{eq:psi}\n \\psi_{(Z,[\\Gamma])}\\colon Z\\longrightarrow\\Alb(X),\\quad z\\longmapsto \\alpha_{X}\\circ[\\Gamma]_{\\ast}(z-z_{0})\n\\end{equation}\ndefines a morphism of algebraic varieties; see \\cite[Definition 1.6.1]{Murre}. In fact, the Abel--Jacobi map $\\alpha_{X}$ also enjoys the universal property of being initial among all regular homomorphisms from $\\Ch(X)_{\\hom}$ to abelian varieties; see \\cite[Example 1.8(b)]{Murre}.\n\\par Since the homomorphism $\\alpha_{X}$ in \\eqref{eq:abel-jacobi-map} is regular and surjective, there exists a codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(\\Alb(X)\\times X)$ and a positive integer $n$ such that the morphism $\\psi_{(\\Alb(X),[\\Gamma])}$ in \\eqref{eq:psi}, with respect to the chosen base-point $0_{\\Alb(X)}\\in\\Alb(X)(\\mathbf{C})$, is equal to $n\\cdot\\Id_{\\Alb(X)}$; see \\cite[Corollary 1.6.3]{Murre}.\\par\n\nThe present work is concerned with a strengthening of this statement, in the sense of the following property of the variety $X$, first introduced and studied by Voisin; see \\cite{Voisin2024CycleCO,Voisin2025}.\n\n\\begin{definition}[\\cite{voisin_a}, Definition 1.1]\\label{def:universal_0-cycle} We say that $X$ admits a \\emph{universal $0$-cycle} if there exists a codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(\\Alb(X)\\times X)$ such that the morphism $\\psi_{(\\Alb(X),[\\Gamma])}$ in \\eqref{eq:psi}, with respect to the base-point $0_{\\Alb(X)}\\in\\Alb(X)(\\mathbf{C})$, is the identity map $\\Id_{\\Alb(X)}$ on the Albanese variety of $X$. \n\\end{definition}\n\nIf $X$ is a curve, then the Poincaré divisor on $\\Jac(X)\\times X$ induces a universal $0$-cycle on $X$. In contrast, Voisin showed in \\cite{Voisin2025} that for each $d\\geq2$, there exists a smooth projective complex $d$-fold $X$ admitting no universal $0$-cycle; see \\cite[Corollary 0.14]{Voisin2025}.\\par\nIn \\cite{colliotthélène2025notessurlapplicationdalbanese}, Colliot--Th\\'{e}l\\`ene considered whether the existence of a universal $0$-cycle persists under additional geometric assumptions on a smooth projective complex variety $X$. In particular, he asked whether this property holds for varieties with representable $\\Ch$-group, that is, for which the homomorphism $\\alpha_{X}$ in \\eqref{eq:abel-jacobi-map} is an isomorphism; see \\cite{Mumford1969,Roitman_1972}. Building on earlier ideas of Benoist--Ottem \\cite{benoist-ottem}, Voisin subsequently constructed in \\cite{voisin_a} a smooth projective complex threefold $X$ with representable $\\Ch$-group but admitting no universal $0$-cycle.\\par\n\n\\begin{ques}\\label{ques:2-dim_example}\nDoes there exist a smooth projective complex surface $S$ whose Chow group of $0$-cycles $\\CH_0(S)$ is representable, but which admits no universal $0$-cycle?\n\\end{ques}\n\nThe main result of the present paper answers Question \\ref{ques:2-dim_example} in the affirmative.\n\n\\begin{equation}\n    \\label{eq:abel-jacobi-map} \\alpha_{X}\\colon\\Ch(X)_{\\hom}\\longrightarrow\\Alb(X).\n\\end{equation}\n\n\\begin{equation}\\label{eq:psi}\n    \\psi_{(Z,[\\Gamma])}\\colon Z\\longrightarrow\\Alb(X),\\quad z\\longmapsto \\alpha_{X}\\circ[\\Gamma]_{\\ast}(z-z_{0})\n\\end{equation}", "full_context": "Let $X$ be a smooth projective complex variety of dimension $d=\\dim X$. Given a base-point $x_{0}\\in X(\\mathbf{C}),$ the universal morphism \\begin{equation*}\n \\al_{X}\\colon X\\longrightarrow\\Alb(X)\n\\end{equation*}\nto the Albanese variety of $X$ (see \\cite[Theorem 1.4.4]{Murre}) induces the so-called Abel--Jacobi map for $0$-cycles on $X$ \\begin{equation}\n \\label{eq:abel-jacobi-map} \\alpha_{X}\\colon\\Ch(X)_{\\hom}\\longrightarrow\\Alb(X).\n\\end{equation}\nThe group homomorphism $\\alpha_{X}$ no longer depends on the choice of the base-point and it is known to be surjective and regular. The latter asserts that, for any smooth projective complex variety $Z$ equipped with a base-point $z_{0}\\in Z(\\mathbf{C})$, and for any codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(Z\\times X)$, the assignment\n\\begin{equation}\\label{eq:psi}\n \\psi_{(Z,[\\Gamma])}\\colon Z\\longrightarrow\\Alb(X),\\quad z\\longmapsto \\alpha_{X}\\circ[\\Gamma]_{\\ast}(z-z_{0})\n\\end{equation}\ndefines a morphism of algebraic varieties; see \\cite[Definition 1.6.1]{Murre}. In fact, the Abel--Jacobi map $\\alpha_{X}$ also enjoys the universal property of being initial among all regular homomorphisms from $\\Ch(X)_{\\hom}$ to abelian varieties; see \\cite[Example 1.8(b)]{Murre}.\n\\par Since the homomorphism $\\alpha_{X}$ in \\eqref{eq:abel-jacobi-map} is regular and surjective, there exists a codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(\\Alb(X)\\times X)$ and a positive integer $n$ such that the morphism $\\psi_{(\\Alb(X),[\\Gamma])}$ in \\eqref{eq:psi}, with respect to the chosen base-point $0_{\\Alb(X)}\\in\\Alb(X)(\\mathbf{C})$, is equal to $n\\cdot\\Id_{\\Alb(X)}$; see \\cite[Corollary 1.6.3]{Murre}.\\par\n\nThe present work is concerned with a strengthening of this statement, in the sense of the following property of the variety $X$, first introduced and studied by Voisin; see \\cite{Voisin2024CycleCO,Voisin2025}.\n\n\\begin{definition}[\\cite{voisin_a}, Definition 1.1]\\label{def:universal_0-cycle} We say that $X$ admits a \\emph{universal $0$-cycle} if there exists a codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(\\Alb(X)\\times X)$ such that the morphism $\\psi_{(\\Alb(X),[\\Gamma])}$ in \\eqref{eq:psi}, with respect to the base-point $0_{\\Alb(X)}\\in\\Alb(X)(\\mathbf{C})$, is the identity map $\\Id_{\\Alb(X)}$ on the Albanese variety of $X$. \n\\end{definition}\n\nIf $X$ is a curve, then the Poincaré divisor on $\\Jac(X)\\times X$ induces a universal $0$-cycle on $X$. In contrast, Voisin showed in \\cite{Voisin2025} that for each $d\\geq2$, there exists a smooth projective complex $d$-fold $X$ admitting no universal $0$-cycle; see \\cite[Corollary 0.14]{Voisin2025}.\\par\nIn \\cite{colliotthélène2025notessurlapplicationdalbanese}, Colliot--Th\\'{e}l\\`ene considered whether the existence of a universal $0$-cycle persists under additional geometric assumptions on a smooth projective complex variety $X$. In particular, he asked whether this property holds for varieties with representable $\\Ch$-group, that is, for which the homomorphism $\\alpha_{X}$ in \\eqref{eq:abel-jacobi-map} is an isomorphism; see \\cite{Mumford1969,Roitman_1972}. Building on earlier ideas of Benoist--Ottem \\cite{benoist-ottem}, Voisin subsequently constructed in \\cite{voisin_a} a smooth projective complex threefold $X$ with representable $\\Ch$-group but admitting no universal $0$-cycle.\\par\n\n\\begin{ques}\\label{ques:2-dim_example}\nDoes there exist a smooth projective complex surface $S$ whose Chow group of $0$-cycles $\\CH_0(S)$ is representable, but which admits no universal $0$-cycle?\n\\end{ques}\n\nThe main result of the present paper answers Question \\ref{ques:2-dim_example} in the affirmative.\n\n\\begin{equation}\n    \\label{eq:abel-jacobi-map} \\alpha_{X}\\colon\\Ch(X)_{\\hom}\\longrightarrow\\Alb(X).\n\\end{equation}\n\n\\begin{equation}\\label{eq:psi}\n    \\psi_{(Z,[\\Gamma])}\\colon Z\\longrightarrow\\Alb(X),\\quad z\\longmapsto \\alpha_{X}\\circ[\\Gamma]_{\\ast}(z-z_{0})\n\\end{equation}\n\nThe main result of the present paper answers Question \\ref{ques:2-dim_example} in the affirmative.\n\n\\subsection{Bielliptic surfaces}\\label{subsec:bielliptic} Throughout this subsection, the base field $k$ is assumed to be algebraically closed.\\par \nBielliptic surfaces constitute one of the classes of smooth projective surfaces of Kodaira dimension zero in the Enriques--Kodaira classification; see \\cite{beauville_surfaces,BombieriMumford1977}. Every bielliptic surface $S$ admits a presentation \\begin{equation*}\\label{eq:presentation_S}\n S\\cong\\bigl(E\\times_k F\\bigr)/\\Gi,\\end{equation*}\nwhere $E$ and $F$ are elliptic curves, and $\\Gi$ is a finite group acting freely on $E\\times_k F.$ The action is by translations on $E$ and by automorphisms on $F$. Moreover, one has an isomorphism $$F/\\Gi\\cong\\bbP^{1}_k.$$\\par\nBielliptic surfaces were classified by Bagnera--De Franchis into seven types; see \\cite{bielliptic,bielliptic_suwa}. The classification is summarized in Table \\ref{tab:bielliptic}.\n\\renewcommand{\\arraystretch}{1.5} \\begin{table}[h]\n\\centering\n\\caption{Classification of Bielliptic Surfaces (Bagnera--de Franchis)}\n\\label{tab:bielliptic}\n\\begin{tabular}{c c l c}\n\\hline\n\\textbf{Type} & \\textbf{Group $\\Gi$} & \\textbf{$\\NS(S)_{\\tor}$} & \\textbf{Order of $K_S$ in $\\Pic(S)$} \\\\\n\\hline\n1 & $\\mathbb{Z}/2$ & $\\Z/2\\times\\Z/2$ & 2 \\\\\n2 & $\\mathbb{Z}/2 \\times \\mathbb{Z}/2$ & $\\Z/2$ & 2 \\\\\n3 & $\\mathbb{Z}/4$ & $\\Z/2$ & 4 \\\\\n4 & $\\mathbb{Z}/4\\times \\mathbb{Z}/2$ & $0$ & 4 \\\\\n5 & $\\mathbb{Z}/3$ & $\\Z/3$ & 3 \\\\\n6 & $\\mathbb{Z}/3 \\times \\mathbb{Z}/3$ & $0$ & 3 \\\\\n7 & $\\mathbb{Z}/6$ & $0$ & 6 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\par The Albanese morphism of $S$ coincides with the projection $$\\phi_{S}\\colon S\\longrightarrow E/\\Gi,$$ whose fibres are elliptic curves isomorphic to $F$. The induced Abel--Jacobi map on $0$-cycles,\n$$(\\phi_{S})_{\\ast}\\colon\\Ch(S)_{\\hom}\\longrightarrow E/\\Gi$$\nis an isomorphism; see \\cite[Example (2)]{bloch-srinivas}. Equivalently, the Chow group of $0$-cycles on a bielliptic surface is representable.\\par\nThere is also a natural elliptic fibration $$\\psi_{S}\\colon S\\longrightarrow (F/\\Gi)\\cong\\bbP^{1}_{k}.$$ \nThis fibration admits multiple fibres corresponding to the branch points of the quotient morphism $F \\to F/\\Gi$, with multiplicities equal to the corresponding ramification indices.\\par\nLet $f$ and $e$ denote the classes in $\\NS(S)$ of the smooth fibres of $\\phi_S$ and $\\psi_S$, respectively. A direct computation shows \\begin{equation}\\label{eq:intersection_e_f}e^2=0,\\quad,f^2=0,\\quad e\\cdot f=|\\Gi|.\\end{equation}\n\n\\section{Main results}\\label{sec:final}\n\\subsection{A bielliptic surface of type 2 admitting no universal $0$-cycle} \nTheorem \\ref{thm:main_result} follows immediately from the result below.\n\\begin{theorem}\\label{thm:main_result_body}\nLet $k$ be an algebraically closed field of characteristic $\\neq 2$, and let $E$ be an elliptic curve over $k$ with $\\End(E)=\\mathbb{Z}$. Set $\\Delta := \\Spec k[[t]]$, and fix an algebraic closure $K$ of its function field. Let $\\mathcal S \\to \\Delta$ be the degeneration constructed in Theorem \\ref{thm:intro_degeneration_S}, together with the induced morphism\n$$\n\\phi_{\\mathcal S} \\colon \\mathcal S \\longrightarrow E \\times_k \\Delta .\n$$\nThen for any algebraically closed field extension $F/K$, there exists no correspondence\n$$\n[\\Gamma] \\in \\CH^{2}(E \\times_k S_F)\n$$\nthat induces a splitting of the pushforward map\n\\begin{equation*}\n(\\phi_{S_F})_{\\ast} \\colon \\CH_0(S_F)_{\\hom} \\longrightarrow E_{F} .\n\\end{equation*}\n\\end{theorem}\n\\begin{proof} Suppose, for contradiction, that there exists an algebraically closed field extension $F/K$ and a correspondence\n$$\n[\\Gamma] \\in \\CH^{2}(E \\times_k S_F)\n$$\nsuch that the induced endomorphism $$(\\phi_{S_F})_{\\ast}\\circ[\\Gamma]_{\\ast}\\in\\End\\bigl(E_{F}\\bigr)$$ is the identity.\\par\nLet $$S_0=\\sum_{i=1}^nR_i$$ denote the special fibre of the family $\\ca S\\to\\Delta,$ as in \\eqref{it:special_fibre} of Theorem \\ref{thm:intro_degeneration_S}. For each $i,$ let $\\al_i\\colon\\Alb(R_i)\\to E$ be the morphism of Albanese varieties induced by the restriction of $\\phi_{\\mathcal S}$ to $R_i$. By Corollary \\ref{cor:obstruction}, the assumption above implies that the sum morphism \n$$\\sum_{i=1}^n\\al_i\\colon\\bigoplus_{i=1}^n\\Alb(R_i)\\longrightarrow E$$\nadmits a section. \\par\nThis contradicts condition \\eqref{it:albanese_S_0} of Theorem \\ref{thm:intro_degeneration_S}. The resulting contradiction completes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main_result}] Let $E$ be an elliptic curve over $\\bbC$ with $\\End(E)=\\Z$. Choose a countable algebraically closed subfield $k_0\\subset\\bbC$ over which $E$ is defined; thus $$E\\cong E_0\\times_{k_{0}}\\bbC$$ for some elliptic curve $E_{0}$ over $k_{0}$.\\par\nLet $$\\ca S\\longrightarrow\\Delta:=\\Spec k_{0}[[t]]$$ be the strictly semistable degeneration constructed in Theorem \\ref{thm:intro_degeneration_S}, together with the induced morphism $$\\phi_{\\mathcal S} \\colon \\mathcal S \\longrightarrow E_0 \\times_{k_{0}} \\Delta .$$\\par\nSince $k_0$ is countable, the inclusion $k_0\\subset\\bbC$ factors through the Laurent series field $k_0((t))$. Set $$S:=S_{k_{0}((t))}\\times_{k_{0}((t))}\\bbC.$$\nBy \\eqref{it:generic_fibre} of Theorem \\ref{thm:intro_degeneration_S}, the surface $S$ is a bielliptic surface of type 2, and the morphism $$\\phi_{S}\\colon S\\longrightarrow E$$ induced by $\\phi_{\\ca S}$ coincides with the Albanese fibration of $S$.\\par\nIn particular, the pushforward homomorphism $$(\\phi_{S})_{\\ast} \\colon \\CH_0(S)_{\\hom} \\longrightarrow E $$ agrees with the Abel--Jacobi map for $0$-cycles on $S$. It is well known that bielliptic surfaces have representable $\\Ch$-group; see \\cite[Example (2)]{bloch-srinivas}. Applying Theorem \\ref{thm:main_result_body}, we conclude that there exists no correspondence $$\n[\\Gamma] \\in \\CH^{2}(E \\times S)\n$$\nthat induces a splitting of the Abel--Jacobi map \n\\begin{equation*}\n(\\phi_{S})_{\\ast} \\colon \\CH_0(S)_{\\hom} \\longrightarrow E.\n\\end{equation*}\nEquivalently, the surface $S$ admits no universal $0$-cycle; cf. Definition \\ref{def:universal_0-cycle}. This completes the proof of Theorem \\ref{thm:main_result}. \n\\end{proof}\n\\subsection{A counterexample to the integral Hodge conjecture}\nIn this final section, we apply Theorem \\ref{thm:main_result} to prove Corollary \\ref{cor:integral_Hodge_S}. In particular, we construct a threefold $X$ of Kodaira dimension zero for which the integral Hodge conjecture fails for $1$-cycles. In contrast to the counterexamples of Benoist--Ottem \\cite{benoist-ottem}, the non-algebraic Hodge class in degree $4$ arising from our construction is non-torsion.\\par\nLet $X$ be a smooth projective variety over $\\bbC$ of dimension $d= \\dim X$. We begin by recalling how the non-existence of a universal $0$-cycle on $X$ is related to the failure of the integral Hodge conjecture. Throughout, all (co)homology groups are taken to be Betti (co)homology, and the subscript \"tf\" denotes the quotient by torsion.\\par\n\n\\begin{proof}[Proof of Corollary \\ref{cor:integral_Hodge_S}]Let $E$ be an elliptic curve over $\\bbC$ with $\\End(E)=\\Z$. \nBy Theorem \\ref{thm:main_result}, there exists a bielliptic surface $S$ of type 2 with $\\Alb(S)\\cong E$ such that $S$ admits no universal $0$-cycle (cf. Definition \\ref{def:universal_0-cycle}).\nBy Lemma \\ref{lem:int_Hodge_conj_vs_uni_0_cycl}, it follows that the cohomology class $$\\delta_{S}\\in H^{4}\\bigl(E\\times S,\\Z\\bigr)_{\\mathrm{tf}}$$ constructed in \\eqref{eq:Hodge_class} is a Hodge class which is not algebraic.\\par\nThis provides a counterexample to the integral Hodge conjecture in degree $4$\nfor $E\\times S$.\\end{proof}", "post_theorem_intro_text_len": 5482, "post_theorem_intro_text": "Recall that ruled surfaces have representable Chow groups of $0$-cycles and do admit a universal $0$-cycle; this follows directly from the corresponding result for curves.\\par \nMore generally, let $X$ be a non-uniruled smooth projective surface with geometric genus $p_{g}(X)=0$ and irregularity $q(X)\\neq0$. By work of Beauville, the structure of such surfaces is well understood; see \\cite[Chapter VI]{beauville_surfaces}. Specifically, $X$ is birational to a quotient \\begin{equation*}\\label{eq:structure_X}\\bigl(C\\times D\\bigr)/\\Gi,\\end{equation*}\nwhere $C$ and $D$ are smooth projective curves and $\\Gi$ is a finite group acting faithfully on both curves, without fixed points on $C \\times D$. Moreover, at least one of the curves is elliptic. Up to exchanging $C$ and $D$, one has $$C/\\Gi\\cong E,\\qquad D/\\Gi\\cong\\mathbf{P}^{1}_{\\mathbf{C}},$$ where $E$ is an elliptic curve. In this situation, the induced fibration \\begin{equation}\\label{eq:fibration_X}\\phi_{X}\\colon X\\longrightarrow E\\end{equation} coincides with the Albanese morphism of $X$.\\par\nIt was shown in \\cite{voisin_a} that if the index of the fibration $\\phi_X$ in \\eqref{eq:fibration_X}, defined as the greatest common divisor of the degrees $\\deg(C/E)$, where $C \\subset X$ ranges over irreducible curves dominating $E$, is equal to one, then $X$ admits a universal $0$-cycle. In particular, this condition is satisfied when the group $\\Gi$ is cyclic; see \\cite[Proposition 4.4]{voisin_a}.\\par \nOur example in Theorem \\ref{thm:main_result} arises from a very general bielliptic surface $S$ of type 2 (see \\cite{bielliptic, bielliptic_suwa}) whose Albanese variety $\\Alb(S)$ is isomorphic to the prescribed elliptic curve $E$. In this situation, the Albanese fibration $\\phi_{S}\\colon S\\to \\Alb(S)\\cong E$ has index 2. As observed in \\cite[$\\S 2.2$]{voisin_a}, this condition alone does not suffice to imply that $S$ admits no universal $0$-cycle. The proof of Theorem \\ref{thm:main_result} therefore rests on the following geometric result.\n\n\\begin{theorem}\\label{thm:intro_degeneration_S} Let $k$ be an algebraically closed field of characteristic $\\neq2,$ and let $E$ be an elliptic curve over $k$ with $\\End(E)\\cong\\mathbb{Z}$. Set $\\Delta:=\\Spec k[[t]],$ and fix an algebraic closure $K$ of its function field. Then there exists a regular, flat, projective scheme $\\mathcal{S}\\to\\Delta,$ together with a morphism $$\\phi_{\\mathcal{S}}\\colon\\mathcal{S}\\longrightarrow E\\times\\Delta$$ such that the following properties hold:\n\\begin{enumerate}\n    \\item\\label{it:generic_fibre} The geometric generic fibre $S_{K}$ is a bielliptic surface of type $2$, and the induced morphism $\\phi_{S_{K}}\\colon S_{K}\\to E\\times_{k} K$ coincides with the Albanese fibration of $S_{K}$.\n    \\item\\label{it:special_fibre} The special fibre $S_{0}=\\sum_{i=1}^{n}R_{i}$ is a reduced simple normal crossings divisor on $\\mathcal{S}$ whose dual graph is a chain; that is, each intersection $R_{i}\\cap R_{i+1}$ is a smooth irreducible curve, and $R_{i}\\cap R_{j}=\\varnothing$ whenever $j\\notin\\{i-1,i,i+1\\}$.\n     \\item\\label{it:albanese_S_0} For each $i,$ let $\\al_{i}:\\Alb(R_{i})\\to E$ denote the homomorphism of Albanese varieties induced by the restriction of $\\phi_{\\mathcal{S}}$ to $R_{i}$. Then the sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(R_{i})\\longrightarrow E$$ does not admit a section. \n\\end{enumerate}  \n\\end{theorem}\nAs we shall explain, the non-existence of a section in \\eqref{it:albanese_S_0} provides the key obstruction to the existence of a universal $0$-cycle on the geometric generic fibre of the family $\\mathcal{S}\\to\\Delta$; see Theorem \\ref{thm:obstruction} and Corollary \\ref{cor:obstruction}.\\par\nWe conclude the introduction with the following application of Theorem \\ref{thm:main_result} to the integral Hodge conjecture.\n\\begin{corollary}\\label{cor:integral_Hodge_S} Let $E$ be a smooth elliptic curve over $\\mathbf{C}$ with $\\End(E)=\\mathbb{Z}$. Then there exists a bielliptic surface $S$ of type 2 whose Albanese variety $\\Alb(S)$ is isomorphic to $E,$ and for which the integral Hodge conjecture for $1$-cycles on $E\\times S$ fails. In particular, there exists a non-torsion integral Hodge class in $H^{4}(E\\times S,\\mathbb{Z})$ that is not algebraic.\n\\end{corollary}\nCorollary \\ref{cor:integral_Hodge_S} adds a further example to the list of threefolds with torsion canonical bundle for which the integral Hodge conjecture fails for $1$-cycles. This should be compared with the counterexamples of Benoist--Ottem \\cite{benoist-ottem}, where the obstruction arises from a non-algebraic torsion cohomology class in degree 4. In contrast, our construction relies on a different degeneration strategy: rather than degenerating the elliptic curve $E,$ we degenerate the surface $S$. This idea is inspired by recent work of the author \\cite{Alexandrou,alexandrou2025torsionhigherchowcycles} and by work of Schreieder \\cite{Schreieder2025}. \\par\nThe paper is organized as follows. In $\\S \\ref{sec:preliminaries}$ we review the necessary background material. Section \\ref{sec:obstruction} introduces an obstruction to the existence of a universal $0$-cycle on a smooth projective variety $X$ with representable Chow group of $0$-cycles. In $\\S \\ref{sec:degen_bielliptic}$ we prove Theorem \\ref{thm:intro_degeneration_S}. Finally, $\\S \\ref{sec:final}$ is devoted to the proofs of our main results, including Theorem \\ref{thm:main_result} and Corollary \\ref{cor:integral_Hodge_S}.", "sketch": "For surfaces of the form \\eqref{eq:structure_X} with Albanese fibration \\eqref{eq:fibration_X}, it is recalled that Voisin shows: if the index of \\(\\phi_X\\) is \\(1\\), then \\(X\\) admits a universal \\(0\\)-cycle (and this holds in particular when \\(\\Gi\\) is cyclic).\n\nFor Theorem~\\ref{thm:main_result}, the surface \\(S\\) is taken to be “a very general bielliptic surface of type 2” with \\(\\Alb(S)\\cong E\\), for which the Albanese fibration \\(\\phi_S\\) has index \\(2\\); and it is noted (following \\cite[\\S 2.2]{voisin_a}) that “this condition alone does not suffice to imply that \\(S\\) admits no universal \\(0\\)-cycle.”\n\nThe stated strategy is instead to use a degeneration: “The proof of Theorem~\\ref{thm:main_result} therefore rests on” Theorem~\\ref{thm:intro_degeneration_S}, constructing a regular flat projective family \\(\\mathcal S\\to\\Delta\\) whose geometric generic fibre \\(S_K\\) is such a bielliptic surface (with its Albanese fibration), and whose special fibre \\(S_0=\\sum R_i\\) is an SNC divisor with chain dual graph. The key geometric input is that for the induced maps \\(\\al_i:\\Alb(R_i)\\to E\\), “the sum morphism \\(\\sum_i \\al_i:\\bigoplus_i \\Alb(R_i)\\to E\\) does not admit a section” \\eqref{it:albanese_S_0}.\n\nAs announced, “the non-existence of a section in \\eqref{it:albanese_S_0} provides the key obstruction to the existence of a universal \\(0\\)-cycle on the geometric generic fibre of the family \\(\\mathcal S\\to\\Delta\\)” (via Theorem~\\ref{thm:obstruction} and Corollary~\\ref{cor:obstruction}), yielding the nonexistence of a universal \\(0\\)-cycle for \\(S\\) in Theorem~\\ref{thm:main_result}.", "expanded_sketch": "For surfaces of the form \\eqref{eq:structure_X} with Albanese fibration \\begin{equation}\\label{eq:fibration_X}\\phi_{X}\\colon X\\longrightarrow E\\end{equation}, it is recalled that Voisin shows: if the index of \\(\\phi_X\\) is \\(1\\), then \\(X\\) admits a universal \\(0\\)-cycle (and this holds in particular when \\(\\Gi\\) is cyclic).\n\nTo prove the main theorem, the surface \\(S\\) is taken to be “a very general bielliptic surface of type 2” with \\(\\Alb(S)\\cong E\\), for which the Albanese fibration \\(\\phi_S\\) has index \\(2\\); and it is noted (following Voisin, \\S 2.2) that “this condition alone does not suffice to imply that \\(S\\) admits no universal \\(0\\)-cycle.”\n\nThe stated strategy is instead to use a degeneration: “The proof of the main theorem therefore rests on” the following theorem.\n\n\\begin{theorem}\\label{thm:intro_degeneration_S} Let $k$ be an algebraically closed field of characteristic $\\neq2,$ and let $E$ be an elliptic curve over $k$ with $\\End(E)\\cong\\Z$. Set $\\Delta:=\\Spec k[[t]],$ and fix an algebraic closure $K$ of its function field. Then there exists a regular, flat, projective scheme $\\mathcal{S}\\to\\Delta,$ together with a morphism $$\\phi_{\\mathcal{S}}\\colon\\mathcal{S}\\longrightarrow E\\times\\Delta$$ such that the following properties hold:\n\\begin{enumerate}\n    \\item\\label{it:generic_fibre} The geometric generic fibre $S_{K}$ is a bielliptic surface of type $2$, and the induced morphism $\\phi_{S_{K}}\\colon S_{K}\\to E\\times_{k} K$ coincides with the Albanese fibration of $S_{K}$.\n    \\item\\label{it:special_fibre} The special fibre $S_{0}=\\sum_{i=1}^{n}R_{i}$ is a reduced simple normal crossings divisor on $\\mathcal{S}$ whose dual graph is a chain; that is, each intersection $R_{i}\\cap R_{i+1}$ is a smooth irreducible curve, and $R_{i}\\cap R_{j}=\\varnothing$ whenever $j\\notin\\{i-1,i,i+1\\}$.\n     \\item\\label{it:albanese_S_0} For each $i,$ let $\\al_{i}:\\Alb(R_{i})\\to E$ denote the homomorphism of Albanese varieties induced by the restriction of $\\phi_{\\mathcal{S}}$ to $R_{i}$. Then the sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(R_{i})\\longrightarrow E$$ does not admit a section. \n\\end{enumerate}  \n\\end{theorem}\n\nThus one constructs a regular flat projective family \\(\\mathcal S\\to\\Delta\\) whose geometric generic fibre \\(S_K\\) is such a bielliptic surface (with its Albanese fibration), and whose special fibre \\(S_0=\\sum R_i\\) is an SNC divisor with chain dual graph. The key geometric input is the preceding theorem’s assertion that for the induced maps \\(\\al_i:\\Alb(R_i)\\to E\\), the sum morphism \\(\\sum_i \\al_i:\\bigoplus_i \\Alb(R_i)\\to E\\) does not admit a section.\n\nAs announced, “the non-existence of a section in” this property provides the key obstruction to the existence of a universal \\(0\\)-cycle on the geometric generic fibre of the family \\(\\mathcal S\\to\\Delta\\) (via the following theorem and corollary), yielding the nonexistence of a universal \\(0\\)-cycle for \\(S\\) in the main theorem.\n\n\\begin{theorem}\n\\label{thm:obstruction}\nLet $k$ be an algebraically closed field and let $C$ be a smooth projective curve over $k$. Set $\\Delta := \\Spec k[[t]]$, and fix an algebraic closure $K$ of its function field. \n\nLet $p \\colon \\mathcal{X} \\to \\Delta$ be a flat, projective morphism of relative dimension $d$ with regular total space, and let $\\phi_{\\mathcal{X}} \\colon \\mathcal{X} \\to C \\times_k \\Delta$ be a surjective morphism.\nDenote by $X_0$ the special fibre of $p$, by\n$\\phi_{X_0} \\colon X_0 \\to C$ the induced morphism, and by $X_{K\n}$ the geometric generic fibre of $p$. Write $\\phi_{X_K} \\colon X_K \\to C_K$\nfor the base change of $\\phi_{\\mathcal{X}}$ to $K$.\\par\nWe further assume the following:\n\\begin{enumerate}\n    \\item \\label{it:X_0} $X_{0}=\\sum_{i=1}^{n}X_{0i}$ is a reduced simple normal crossings divisor on $\\mathcal{X}$ whose dual graph is a chain. Precisely, each intersection $X_{0i}\\cap X_{0i+1}$ is smooth and irreducible of dimension $d-1,$ and $X_{0i}\\cap X_{0j}=\\varnothing$ whenever $j\\notin\\{i-1,i,i+1\\}$.\n    \\item \\label{it:correspondence} \n    There exists a correspondence\n$$\n[\\Gamma] \\in \\CH^{d}(C \\times_k X_{K})\n$$\ninducing a splitting of the pushforward map\n$$\n(\\phi_{X_K})_\\ast\\colon \\Ch(X_{K})_{\\hom}\\longrightarrow\\jac(C_{K}).\n$$\n\n\\end{enumerate}\n\\par \nFor each $i,$ let $\\al_{i}\\colon\\Alb(X_{0i})\\to \\jac(C)$ be the homomorphism of Albanese varieties induced by the restriction of $\\phi_{X_0}$ to $X_{0i}$. Then the induced sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(X_{0i})\\longrightarrow\\jac(C)$$\nadmits a section. \n\\end{theorem}\n\n\\begin{corollary}\\label{cor:obstruction}\nWe use the notation and assumptions of Theorem \\ref{thm:obstruction}, with condition \\eqref{it:correspondence} replaced by the following:\n\\begin{enumerate}\n    \\item[\\((2')\\)]\\label{it:correspondence-prime} There exists an algebraically closed field extension $F/K$ and a correspondence\n$$\n[\\Gamma] \\in \\CH^{d}(C \\times_k X_{F})\n$$\ninducing a splitting of the pushforward map\n$$\n(\\phi_{X_F})_\\ast\\colon \\Ch(X_{F})_{\\hom}\\longrightarrow\\jac(C_{F}).\n$$\n\\end{enumerate}\nAssume moreover that $\\End(\\jac(C))=\\Z$. Then the sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(X_{0i})\\longrightarrow\\jac(C)$$\nadmits a section. \n\\end{corollary}", "expanded_theorem": "\\label{thm:main_result} Let $E$ be a smooth complex elliptic curve with $\\End(E)=\\mathbb{Z}$. There exists a smooth projective complex surface $S$ with $\\Alb(S)\\cong E$ for which $\\Ch(S)$ is representable, while $S$ admits no universal $0$-cycle.,", "theorem_type": ["Existence", "Implication"], "mcq": {"question": "Let $E$ be a smooth complex elliptic curve with $\\operatorname{End}(E)=\\mathbb{Z}$. For a smooth projective complex surface $S$, let\n\\[\n\\alpha_S\\colon \\CH_0(S)_{\\mathrm{hom}}\\longrightarrow \\operatorname{Alb}(S)\n\\]\nbe the Abel--Jacobi map for $0$-cycles. Say that $\\CH_0(S)$ is representable if $\\alpha_S$ is an isomorphism. Say that $S$ admits a universal $0$-cycle if there exists a codimension-$2$ cycle $[\\Gamma]\\in \\CH^2(\\operatorname{Alb}(S)\\times S)$ such that the induced morphism\n\\[\n\\psi_{(\\operatorname{Alb}(S),[\\Gamma])}\\colon \\operatorname{Alb}(S)\\to \\operatorname{Alb}(S),\\qquad a\\mapsto \\alpha_S\\bigl([\\Gamma]_*(a-0_{\\operatorname{Alb}(S)})\\bigr),\n\\]\nis the identity map on $\\operatorname{Alb}(S)$. Under these assumptions on $E$, which conclusion holds?", "correct_choice": {"label": "A", "text": "There exists a smooth projective complex surface $S$ with $\\operatorname{Alb}(S)\\cong E$ such that $\\CH_0(S)$ is representable, i.e. $\\alpha_S$ is an isomorphism, but $S$ does not admit a universal $0$-cycle; equivalently, there is no codimension-$2$ cycle $[\\Gamma]\\in \\CH^2(\\operatorname{Alb}(S)\\times S)$ for which $\\psi_{(\\operatorname{Alb}(S),[\\Gamma])}=\\operatorname{Id}_{\\operatorname{Alb}(S)}$."}, "choices": [{"label": "B", "text": "For every smooth projective complex surface $S$ with $\\operatorname{Alb}(S)\\cong E$, if $\\CH_0(S)$ is representable, then $S$ admits a universal $0$-cycle; equivalently, whenever $\\alpha_S$ is an isomorphism there exists a codimension-$2$ cycle $[\\Gamma]\\in \\CH^2(\\operatorname{Alb}(S)\\times S)$ such that $\\psi_{(\\operatorname{Alb}(S),[\\Gamma])}=\\operatorname{Id}_{\\operatorname{Alb}(S)}$."}, {"label": "C", "text": "There exists a smooth projective complex surface $S$ with $\\operatorname{Alb}(S)\\cong E$ such that $\\CH_0(S)$ is representable, i.e. $\\alpha_S$ is an isomorphism."}, {"label": "D", "text": "There exists a smooth projective complex surface $S$ with $\\operatorname{Alb}(S)\\cong E$ such that $\\CH_0(S)$ is representable and $S$ admits a universal $0$-cycle after base change to some algebraically closed field extension; equivalently, there is an algebraically closed extension $F/\\mathbb{C}$ and a codimension-$2$ cycle $[\\Gamma]\\in \\CH^2(\\operatorname{Alb}(S)_F\\times S_F)$ for which $\\psi_{(\\operatorname{Alb}(S)_F,[\\Gamma])}=\\operatorname{Id}_{\\operatorname{Alb}(S)_F}$."}, {"label": "E", "text": "There exists a smooth projective complex surface $S$ with $\\operatorname{Alb}(S)\\cong E$ such that $\\CH_0(S)$ is representable, but $S$ does not admit a universal $0$-cycle only in the weaker sense that no codimension-$2$ cycle $[\\Gamma]\\in \\CH^2(\\operatorname{Alb}(S)\\times S)$ makes $\\psi_{(\\operatorname{Alb}(S),[\\Gamma])}$ equal to $n\\,\\operatorname{Id}_{\\operatorname{Alb}(S)}$ for any positive integer $n$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "index-1/cyclic sufficient criterion promoted to universal statement for all representable surfaces", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the failure of a universal $0$-cycle", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "existence over an algebraically closed extension $F/K$ treated as if it could occur over a base change from $\\mathbb{C}$ despite the obstruction using $\\operatorname{End}(E)=\\mathbb{Z}$", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "confusion between absence of identity splitting and absence of any Murre multiple $n\\operatorname{Id}$", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem supplies definitions and setup but does not explicitly or implicitly reveal the correct choice. There is no direct clue that the intended conclusion is the nonexistence of a universal 0-cycle."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: the correct option appears to be essentially the precise existence statement one is meant to know. It is not a pure verbatim restatement because the alternatives vary the conclusion in meaningful ways, but it is still only a mild reformulation."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact existential statement from weaker, stronger, or confused variants, especially around representability versus existence of a universal 0-cycle. However, the task is mainly recognition of the intended theorem rather than substantial generative mathematical reasoning."}, "DQS": {"score": 1, "justification": "Several distractors are conceptually relevant: one is a weaker true statement, one is an overstrong universalization, and one targets confusion between existence of n·Id and Id. But the distractor set is imperfect: choice C is also true as stated, creating ambiguity, and choice E is awkwardly phrased so that it effectively collapses to the same content as existence of Id."}, "total_score": 5, "overall_assessment": "A reasonably well-targeted theorem-discrimination MCQ with no answer leakage, but it is somewhat theorem-restatement-like and weakened by ambiguity from a true weaker option and a not fully clean distractor."}}
{"id": "2602.13435v1", "paper_link": "http://arxiv.org/abs/2602.13435v1", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "\\label{thm:main_result} Let $E$ be a smooth complex elliptic curve with $\\End(E)=\\mathbb{Z}$. There exists a smooth projective complex surface $S$ with $\\Alb(S)\\cong E$ for which $\\Ch(S)$ is representable, while $S$ admits no universal $0$-cycle.", "start_pos": 11714, "end_pos": 11981, "label": "thm:main_result"}, "ref_dict": {"cor:integral_Hodge_S": "\\begin{corollary}\\label{cor:integral_Hodge_S} Let $E$ be a smooth elliptic curve over $\\bbC$ with $\\End(E)=\\Z$. Then there exists a bielliptic surface $S$ of type 2 whose Albanese variety $\\Alb(S)$ is isomorphic to $E,$ and for which the integral Hodge conjecture for $1$-cycles on $E\\times S$ fails. In particular, there exists a non-torsion integral Hodge class in $H^{4}(E\\times S,\\Z)$ that is not algebraic.\n\\end{corollary}", "cor:obstruction": "\\begin{corollary}\\label{cor:obstruction}\nWe use the notation and assumptions of Theorem \\ref{thm:obstruction}, with condition \\eqref{it:correspondence} replaced by the following:\n\\begin{enumerate}\n    \\item[\\((2')\\)]\\label{it:correspondence-prime} There exists an algebraically closed field extension $F/K$ and a correspondence\n$$\n[\\Gamma] \\in \\CH^{d}(C \\times_k X_{F})\n$$\ninducing a splitting of the pushforward map\n$$\n(\\phi_{X_F})_\\ast\\colon \\Ch(X_{F})_{\\hom}\\longrightarrow\\jac(C_{F}).\n$$\n\\end{enumerate}\nAssume moreover that $\\End(\\jac(C))=\\Z$. Then the sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(X_{0i})\\longrightarrow\\jac(C)$$\nadmits a section. \n\\end{corollary}", "ques:2-dim_example": "\\begin{ques}\\label{ques:2-dim_example}\nDoes there exist a smooth projective complex surface $S$ whose Chow group of $0$-cycles $\\CH_0(S)$ is representable, but which admits no universal $0$-cycle?\n\\end{ques}", "eq:fibration_X": "\\begin{equation}\\label{eq:fibration_X}\\phi_{X}\\colon X\\longrightarrow E\\end{equation}", "eq:abel-jacobi-map": "\\begin{equation}\n    \\label{eq:abel-jacobi-map} \\alpha_{X}\\colon\\Ch(X)_{\\hom}\\longrightarrow\\Alb(X).\n\\end{equation}", "thm:main_result": "\\begin{theorem}\\label{thm:main_result} Let $E$ be a smooth complex elliptic curve with $\\End(E)=\\Z$. There exists a smooth projective complex surface $S$ with $\\Alb(S)\\cong E$ for which $\\Ch(S)$ is representable, while $S$ admits no universal $0$-cycle.\n\\end{theorem}", "it:albanese_S_0": "\\begin{enumerate}\n    \\item\\label{it:generic_fibre} The geometric generic fibre $S_{K}$ is a bielliptic surface of type $2$, and the induced morphism $\\phi_{S_{K}}\\colon S_{K}\\to E\\times_{k} K$ coincides with the Albanese fibration of $S_{K}$.\n    \\item\\label{it:special_fibre} The special fibre $S_{0}=\\sum_{i=1}^{n}R_{i}$ is a reduced simple normal crossings divisor on $\\mathcal{S}$ whose dual graph is a chain; that is, each intersection $R_{i}\\cap R_{i+1}$ is a smooth irreducible curve, and $R_{i}\\cap R_{j}=\\varnothing$ whenever $j\\notin\\{i-1,i,i+1\\}$.\n     \\item\\label{it:albanese_S_0} For each $i,$ let $\\al_{i}:\\Alb(R_{i})\\to E$ denote the homomorphism of Albanese varieties induced by the restriction of $\\phi_{\\mathcal{S}}$ to $R_{i}$. Then the sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(R_{i})\\longrightarrow E$$ does not admit a section. \n\\end{enumerate}", "eq:psi": "\\begin{equation}\\label{eq:psi}\n    \\psi_{(Z,[\\Gamma])}\\colon Z\\longrightarrow\\Alb(X),\\quad z\\longmapsto \\alpha_{X}\\circ[\\Gamma]_{\\ast}(z-z_{0})\n\\end{equation}", "thm:obstruction": "\\begin{theorem}\n\\label{thm:obstruction}\nLet $k$ be an algebraically closed field and let $C$ be a smooth projective curve over $k$. Set $\\Delta := \\Spec k[[t]]$, and fix an algebraic closure $K$ of its function field. \n\nLet $p \\colon \\mathcal{X} \\to \\Delta$ be a flat, projective morphism of relative dimension $d$ with regular total space, and let $\\phi_{\\mathcal{X}} \\colon \\mathcal{X} \\to C \\times_k \\Delta$ be a surjective morphism.\nDenote by $X_0$ the special fibre of $p$, by\n$\\phi_{X_0} \\colon X_0 \\to C$ the induced morphism, and by $X_{K\n}$ the geometric generic fibre of $p$. Write $\\phi_{X_K} \\colon X_K \\to C_K$\nfor the base change of $\\phi_{\\mathcal{X}}$ to $K$.\\par\nWe further assume the following:\n\\begin{enumerate}\n    \\item \\label{it:X_0} $X_{0}=\\sum_{i=1}^{n}X_{0i}$ is a reduced simple normal crossings divisor on $\\mathcal{X}$ whose dual graph is a chain. Precisely, each intersection $X_{0i}\\cap X_{0i+1}$ is smooth and irreducible of dimension $d-1,$ and $X_{0i}\\cap X_{0j}=\\varnothing$ whenever $j\\notin\\{i-1,i,i+1\\}$.\n    \\item \\label{it:correspondence} \n    There exists a correspondence\n$$\n[\\Gamma] \\in \\CH^{d}(C \\times_k X_{K})\n$$\ninducing a splitting of the pushforward map\n$$\n(\\phi_{X_K})_\\ast\\colon \\Ch(X_{K})_{\\hom}\\longrightarrow\\jac(C_{K}).\n$$\n\n\\end{enumerate}\n\\par \nFor each $i,$ let $\\al_{i}\\colon\\Alb(X_{0i})\\to \\jac(C)$ be the homomorphism of Albanese varieties induced by the restriction of $\\phi_{X_0}$ to $X_{0i}$. Then the induced sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(X_{0i})\\longrightarrow\\jac(C)$$\nadmits a section. \n\\end{theorem}", "thm:intro_degeneration_S": "\\begin{theorem}\\label{thm:intro_degeneration_S} Let $k$ be an algebraically closed field of characteristic $\\neq2,$ and let $E$ be an elliptic curve over $k$ with $\\End(E)\\cong\\Z$. Set $\\Delta:=\\Spec k[[t]],$ and fix an algebraic closure $K$ of its function field. Then there exists a regular, flat, projective scheme $\\mathcal{S}\\to\\Delta,$ together with a morphism $$\\phi_{\\mathcal{S}}\\colon\\mathcal{S}\\longrightarrow E\\times\\Delta$$ such that the following properties hold:\n\\begin{enumerate}\n    \\item\\label{it:generic_fibre} The geometric generic fibre $S_{K}$ is a bielliptic surface of type $2$, and the induced morphism $\\phi_{S_{K}}\\colon S_{K}\\to E\\times_{k} K$ coincides with the Albanese fibration of $S_{K}$.\n    \\item\\label{it:special_fibre} The special fibre $S_{0}=\\sum_{i=1}^{n}R_{i}$ is a reduced simple normal crossings divisor on $\\mathcal{S}$ whose dual graph is a chain; that is, each intersection $R_{i}\\cap R_{i+1}$ is a smooth irreducible curve, and $R_{i}\\cap R_{j}=\\varnothing$ whenever $j\\notin\\{i-1,i,i+1\\}$.\n     \\item\\label{it:albanese_S_0} For each $i,$ let $\\al_{i}:\\Alb(R_{i})\\to E$ denote the homomorphism of Albanese varieties induced by the restriction of $\\phi_{\\mathcal{S}}$ to $R_{i}$. Then the sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(R_{i})\\longrightarrow E$$ does not admit a section. \n\\end{enumerate}  \n\\end{theorem}"}, "pre_theorem_intro_text_len": 3797, "pre_theorem_intro_text": "Let $X$ be a smooth projective complex variety of dimension $d=\\dim X$. Given a base-point $x_{0}\\in X(\\mathbf{C}),$ the universal morphism \\begin{equation*}\n    \\al_{X}\\colon X\\longrightarrow\\Alb(X)\n\\end{equation*}\nto the Albanese variety of $X$ (see \\cite[Theorem 1.4.4]{Murre}) induces the so-called Abel--Jacobi map for $0$-cycles on $X$ \\begin{equation}\n    \\label{eq:abel-jacobi-map} \\alpha_{X}\\colon\\Ch(X)_{\\hom}\\longrightarrow\\Alb(X).\n\\end{equation}\nThe group homomorphism $\\alpha_{X}$ no longer depends on the choice of the base-point and it is known to be surjective and regular. The latter asserts that, for any smooth projective complex variety $Z$ equipped with a base-point $z_{0}\\in Z(\\mathbf{C})$, and for any codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(Z\\times X)$, the assignment\n\\begin{equation}\\label{eq:psi}\n    \\psi_{(Z,[\\Gamma])}\\colon Z\\longrightarrow\\Alb(X),\\quad z\\longmapsto \\alpha_{X}\\circ[\\Gamma]_{\\ast}(z-z_{0})\n\\end{equation}\ndefines a morphism of algebraic varieties; see \\cite[Definition 1.6.1]{Murre}. In fact, the Abel--Jacobi map $\\alpha_{X}$ also enjoys the universal property of being initial among all regular homomorphisms from $\\Ch(X)_{\\hom}$ to abelian varieties; see \\cite[Example 1.8(b)]{Murre}.\n\\par Since the homomorphism $\\alpha_{X}$ in \\eqref{eq:abel-jacobi-map} is regular and surjective, there exists a codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(\\Alb(X)\\times X)$ and a positive integer $n$ such that the morphism $\\psi_{(\\Alb(X),[\\Gamma])}$ in \\eqref{eq:psi}, with respect to the chosen base-point $0_{\\Alb(X)}\\in\\Alb(X)(\\mathbf{C})$, is equal to $n\\cdot\\Id_{\\Alb(X)}$; see \\cite[Corollary 1.6.3]{Murre}.\\par\n\nThe present work is concerned with a strengthening of this statement, in the sense of the following property of the variety $X$, first introduced and studied by Voisin; see \\cite{Voisin2024CycleCO,Voisin2025}.\n\n\\begin{definition}[\\cite{voisin_a}, Definition 1.1]\\label{def:universal_0-cycle} We say that $X$ admits a \\emph{universal $0$-cycle} if there exists a codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(\\Alb(X)\\times X)$ such that the morphism $\\psi_{(\\Alb(X),[\\Gamma])}$ in \\eqref{eq:psi}, with respect to the base-point $0_{\\Alb(X)}\\in\\Alb(X)(\\mathbf{C})$, is the identity map $\\Id_{\\Alb(X)}$ on the Albanese variety of $X$. \n\\end{definition}\n\nIf $X$ is a curve, then the Poincaré divisor on $\\Jac(X)\\times X$ induces a universal $0$-cycle on $X$. In contrast, Voisin showed in \\cite{Voisin2025} that for each $d\\geq2$, there exists a smooth projective complex $d$-fold $X$ admitting no universal $0$-cycle; see \\cite[Corollary  0.14]{Voisin2025}.\\par\nIn \\cite{colliotthélène2025notessurlapplicationdalbanese}, Colliot--Th\\'{e}l\\`ene considered whether the existence of a universal $0$-cycle persists under additional geometric assumptions on a smooth projective complex variety $X$. In particular, he asked whether this property holds for varieties with representable $\\Ch$-group, that is, for which the homomorphism $\\alpha_{X}$ in \\eqref{eq:abel-jacobi-map} is an isomorphism; see \\cite{Mumford1969,Roitman_1972}. Building on earlier ideas of Benoist--Ottem \\cite{benoist-ottem}, Voisin subsequently constructed in \\cite{voisin_a} a smooth projective complex threefold $X$ with representable $\\Ch$-group but admitting no universal $0$-cycle.\\par\n\nThis leads to the following question, raised by Totaro after Voisin’s talk at the workshop \\emph{Hodge Theory and Algebraic Cycles} (Clay Mathematics Institute, September 2025).\n\n\\begin{ques}\\label{ques:2-dim_example}\nDoes there exist a smooth projective complex surface $S$ whose Chow group of $0$-cycles $\\CH_0(S)$ is representable, but which admits no universal $0$-cycle?\n\\end{ques}\n\nThe main result of the present paper answers Question \\ref{ques:2-dim_example} in the affirmative.", "context": "Let $X$ be a smooth projective complex variety of dimension $d=\\dim X$. Given a base-point $x_{0}\\in X(\\mathbf{C}),$ the universal morphism \\begin{equation*}\n \\al_{X}\\colon X\\longrightarrow\\Alb(X)\n\\end{equation*}\nto the Albanese variety of $X$ (see \\cite[Theorem 1.4.4]{Murre}) induces the so-called Abel--Jacobi map for $0$-cycles on $X$ \\begin{equation}\n \\label{eq:abel-jacobi-map} \\alpha_{X}\\colon\\Ch(X)_{\\hom}\\longrightarrow\\Alb(X).\n\\end{equation}\nThe group homomorphism $\\alpha_{X}$ no longer depends on the choice of the base-point and it is known to be surjective and regular. The latter asserts that, for any smooth projective complex variety $Z$ equipped with a base-point $z_{0}\\in Z(\\mathbf{C})$, and for any codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(Z\\times X)$, the assignment\n\\begin{equation}\\label{eq:psi}\n \\psi_{(Z,[\\Gamma])}\\colon Z\\longrightarrow\\Alb(X),\\quad z\\longmapsto \\alpha_{X}\\circ[\\Gamma]_{\\ast}(z-z_{0})\n\\end{equation}\ndefines a morphism of algebraic varieties; see \\cite[Definition 1.6.1]{Murre}. In fact, the Abel--Jacobi map $\\alpha_{X}$ also enjoys the universal property of being initial among all regular homomorphisms from $\\Ch(X)_{\\hom}$ to abelian varieties; see \\cite[Example 1.8(b)]{Murre}.\n\\par Since the homomorphism $\\alpha_{X}$ in \\eqref{eq:abel-jacobi-map} is regular and surjective, there exists a codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(\\Alb(X)\\times X)$ and a positive integer $n$ such that the morphism $\\psi_{(\\Alb(X),[\\Gamma])}$ in \\eqref{eq:psi}, with respect to the chosen base-point $0_{\\Alb(X)}\\in\\Alb(X)(\\mathbf{C})$, is equal to $n\\cdot\\Id_{\\Alb(X)}$; see \\cite[Corollary 1.6.3]{Murre}.\\par\n\nThe present work is concerned with a strengthening of this statement, in the sense of the following property of the variety $X$, first introduced and studied by Voisin; see \\cite{Voisin2024CycleCO,Voisin2025}.\n\n\\begin{definition}[\\cite{voisin_a}, Definition 1.1]\\label{def:universal_0-cycle} We say that $X$ admits a \\emph{universal $0$-cycle} if there exists a codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(\\Alb(X)\\times X)$ such that the morphism $\\psi_{(\\Alb(X),[\\Gamma])}$ in \\eqref{eq:psi}, with respect to the base-point $0_{\\Alb(X)}\\in\\Alb(X)(\\mathbf{C})$, is the identity map $\\Id_{\\Alb(X)}$ on the Albanese variety of $X$. \n\\end{definition}\n\nIf $X$ is a curve, then the Poincaré divisor on $\\Jac(X)\\times X$ induces a universal $0$-cycle on $X$. In contrast, Voisin showed in \\cite{Voisin2025} that for each $d\\geq2$, there exists a smooth projective complex $d$-fold $X$ admitting no universal $0$-cycle; see \\cite[Corollary 0.14]{Voisin2025}.\\par\nIn \\cite{colliotthélène2025notessurlapplicationdalbanese}, Colliot--Th\\'{e}l\\`ene considered whether the existence of a universal $0$-cycle persists under additional geometric assumptions on a smooth projective complex variety $X$. In particular, he asked whether this property holds for varieties with representable $\\Ch$-group, that is, for which the homomorphism $\\alpha_{X}$ in \\eqref{eq:abel-jacobi-map} is an isomorphism; see \\cite{Mumford1969,Roitman_1972}. Building on earlier ideas of Benoist--Ottem \\cite{benoist-ottem}, Voisin subsequently constructed in \\cite{voisin_a} a smooth projective complex threefold $X$ with representable $\\Ch$-group but admitting no universal $0$-cycle.\\par\n\n\\begin{ques}\\label{ques:2-dim_example}\nDoes there exist a smooth projective complex surface $S$ whose Chow group of $0$-cycles $\\CH_0(S)$ is representable, but which admits no universal $0$-cycle?\n\\end{ques}\n\nThe main result of the present paper answers Question \\ref{ques:2-dim_example} in the affirmative.\n\n\\begin{equation}\n    \\label{eq:abel-jacobi-map} \\alpha_{X}\\colon\\Ch(X)_{\\hom}\\longrightarrow\\Alb(X).\n\\end{equation}\n\n\\begin{equation}\\label{eq:psi}\n    \\psi_{(Z,[\\Gamma])}\\colon Z\\longrightarrow\\Alb(X),\\quad z\\longmapsto \\alpha_{X}\\circ[\\Gamma]_{\\ast}(z-z_{0})\n\\end{equation}", "full_context": "Let $X$ be a smooth projective complex variety of dimension $d=\\dim X$. Given a base-point $x_{0}\\in X(\\mathbf{C}),$ the universal morphism \\begin{equation*}\n \\al_{X}\\colon X\\longrightarrow\\Alb(X)\n\\end{equation*}\nto the Albanese variety of $X$ (see \\cite[Theorem 1.4.4]{Murre}) induces the so-called Abel--Jacobi map for $0$-cycles on $X$ \\begin{equation}\n \\label{eq:abel-jacobi-map} \\alpha_{X}\\colon\\Ch(X)_{\\hom}\\longrightarrow\\Alb(X).\n\\end{equation}\nThe group homomorphism $\\alpha_{X}$ no longer depends on the choice of the base-point and it is known to be surjective and regular. The latter asserts that, for any smooth projective complex variety $Z$ equipped with a base-point $z_{0}\\in Z(\\mathbf{C})$, and for any codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(Z\\times X)$, the assignment\n\\begin{equation}\\label{eq:psi}\n \\psi_{(Z,[\\Gamma])}\\colon Z\\longrightarrow\\Alb(X),\\quad z\\longmapsto \\alpha_{X}\\circ[\\Gamma]_{\\ast}(z-z_{0})\n\\end{equation}\ndefines a morphism of algebraic varieties; see \\cite[Definition 1.6.1]{Murre}. In fact, the Abel--Jacobi map $\\alpha_{X}$ also enjoys the universal property of being initial among all regular homomorphisms from $\\Ch(X)_{\\hom}$ to abelian varieties; see \\cite[Example 1.8(b)]{Murre}.\n\\par Since the homomorphism $\\alpha_{X}$ in \\eqref{eq:abel-jacobi-map} is regular and surjective, there exists a codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(\\Alb(X)\\times X)$ and a positive integer $n$ such that the morphism $\\psi_{(\\Alb(X),[\\Gamma])}$ in \\eqref{eq:psi}, with respect to the chosen base-point $0_{\\Alb(X)}\\in\\Alb(X)(\\mathbf{C})$, is equal to $n\\cdot\\Id_{\\Alb(X)}$; see \\cite[Corollary 1.6.3]{Murre}.\\par\n\nThe present work is concerned with a strengthening of this statement, in the sense of the following property of the variety $X$, first introduced and studied by Voisin; see \\cite{Voisin2024CycleCO,Voisin2025}.\n\n\\begin{definition}[\\cite{voisin_a}, Definition 1.1]\\label{def:universal_0-cycle} We say that $X$ admits a \\emph{universal $0$-cycle} if there exists a codimension-$d$ cycle $[\\Gamma]\\in\\CH^{d}(\\Alb(X)\\times X)$ such that the morphism $\\psi_{(\\Alb(X),[\\Gamma])}$ in \\eqref{eq:psi}, with respect to the base-point $0_{\\Alb(X)}\\in\\Alb(X)(\\mathbf{C})$, is the identity map $\\Id_{\\Alb(X)}$ on the Albanese variety of $X$. \n\\end{definition}\n\nIf $X$ is a curve, then the Poincaré divisor on $\\Jac(X)\\times X$ induces a universal $0$-cycle on $X$. In contrast, Voisin showed in \\cite{Voisin2025} that for each $d\\geq2$, there exists a smooth projective complex $d$-fold $X$ admitting no universal $0$-cycle; see \\cite[Corollary 0.14]{Voisin2025}.\\par\nIn \\cite{colliotthélène2025notessurlapplicationdalbanese}, Colliot--Th\\'{e}l\\`ene considered whether the existence of a universal $0$-cycle persists under additional geometric assumptions on a smooth projective complex variety $X$. In particular, he asked whether this property holds for varieties with representable $\\Ch$-group, that is, for which the homomorphism $\\alpha_{X}$ in \\eqref{eq:abel-jacobi-map} is an isomorphism; see \\cite{Mumford1969,Roitman_1972}. Building on earlier ideas of Benoist--Ottem \\cite{benoist-ottem}, Voisin subsequently constructed in \\cite{voisin_a} a smooth projective complex threefold $X$ with representable $\\Ch$-group but admitting no universal $0$-cycle.\\par\n\n\\begin{ques}\\label{ques:2-dim_example}\nDoes there exist a smooth projective complex surface $S$ whose Chow group of $0$-cycles $\\CH_0(S)$ is representable, but which admits no universal $0$-cycle?\n\\end{ques}\n\nThe main result of the present paper answers Question \\ref{ques:2-dim_example} in the affirmative.\n\n\\begin{equation}\n    \\label{eq:abel-jacobi-map} \\alpha_{X}\\colon\\Ch(X)_{\\hom}\\longrightarrow\\Alb(X).\n\\end{equation}\n\n\\begin{equation}\\label{eq:psi}\n    \\psi_{(Z,[\\Gamma])}\\colon Z\\longrightarrow\\Alb(X),\\quad z\\longmapsto \\alpha_{X}\\circ[\\Gamma]_{\\ast}(z-z_{0})\n\\end{equation}\n\nThe main result of the present paper answers Question \\ref{ques:2-dim_example} in the affirmative.\n\n\\subsection{Bielliptic surfaces}\\label{subsec:bielliptic} Throughout this subsection, the base field $k$ is assumed to be algebraically closed.\\par \nBielliptic surfaces constitute one of the classes of smooth projective surfaces of Kodaira dimension zero in the Enriques--Kodaira classification; see \\cite{beauville_surfaces,BombieriMumford1977}. Every bielliptic surface $S$ admits a presentation \\begin{equation*}\\label{eq:presentation_S}\n S\\cong\\bigl(E\\times_k F\\bigr)/\\Gi,\\end{equation*}\nwhere $E$ and $F$ are elliptic curves, and $\\Gi$ is a finite group acting freely on $E\\times_k F.$ The action is by translations on $E$ and by automorphisms on $F$. Moreover, one has an isomorphism $$F/\\Gi\\cong\\bbP^{1}_k.$$\\par\nBielliptic surfaces were classified by Bagnera--De Franchis into seven types; see \\cite{bielliptic,bielliptic_suwa}. The classification is summarized in Table \\ref{tab:bielliptic}.\n\\renewcommand{\\arraystretch}{1.5} \\begin{table}[h]\n\\centering\n\\caption{Classification of Bielliptic Surfaces (Bagnera--de Franchis)}\n\\label{tab:bielliptic}\n\\begin{tabular}{c c l c}\n\\hline\n\\textbf{Type} & \\textbf{Group $\\Gi$} & \\textbf{$\\NS(S)_{\\tor}$} & \\textbf{Order of $K_S$ in $\\Pic(S)$} \\\\\n\\hline\n1 & $\\mathbb{Z}/2$ & $\\Z/2\\times\\Z/2$ & 2 \\\\\n2 & $\\mathbb{Z}/2 \\times \\mathbb{Z}/2$ & $\\Z/2$ & 2 \\\\\n3 & $\\mathbb{Z}/4$ & $\\Z/2$ & 4 \\\\\n4 & $\\mathbb{Z}/4\\times \\mathbb{Z}/2$ & $0$ & 4 \\\\\n5 & $\\mathbb{Z}/3$ & $\\Z/3$ & 3 \\\\\n6 & $\\mathbb{Z}/3 \\times \\mathbb{Z}/3$ & $0$ & 3 \\\\\n7 & $\\mathbb{Z}/6$ & $0$ & 6 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\par The Albanese morphism of $S$ coincides with the projection $$\\phi_{S}\\colon S\\longrightarrow E/\\Gi,$$ whose fibres are elliptic curves isomorphic to $F$. The induced Abel--Jacobi map on $0$-cycles,\n$$(\\phi_{S})_{\\ast}\\colon\\Ch(S)_{\\hom}\\longrightarrow E/\\Gi$$\nis an isomorphism; see \\cite[Example (2)]{bloch-srinivas}. Equivalently, the Chow group of $0$-cycles on a bielliptic surface is representable.\\par\nThere is also a natural elliptic fibration $$\\psi_{S}\\colon S\\longrightarrow (F/\\Gi)\\cong\\bbP^{1}_{k}.$$ \nThis fibration admits multiple fibres corresponding to the branch points of the quotient morphism $F \\to F/\\Gi$, with multiplicities equal to the corresponding ramification indices.\\par\nLet $f$ and $e$ denote the classes in $\\NS(S)$ of the smooth fibres of $\\phi_S$ and $\\psi_S$, respectively. A direct computation shows \\begin{equation}\\label{eq:intersection_e_f}e^2=0,\\quad,f^2=0,\\quad e\\cdot f=|\\Gi|.\\end{equation}\n\n\\section{Main results}\\label{sec:final}\n\\subsection{A bielliptic surface of type 2 admitting no universal $0$-cycle} \nTheorem \\ref{thm:main_result} follows immediately from the result below.\n\\begin{theorem}\\label{thm:main_result_body}\nLet $k$ be an algebraically closed field of characteristic $\\neq 2$, and let $E$ be an elliptic curve over $k$ with $\\End(E)=\\mathbb{Z}$. Set $\\Delta := \\Spec k[[t]]$, and fix an algebraic closure $K$ of its function field. Let $\\mathcal S \\to \\Delta$ be the degeneration constructed in Theorem \\ref{thm:intro_degeneration_S}, together with the induced morphism\n$$\n\\phi_{\\mathcal S} \\colon \\mathcal S \\longrightarrow E \\times_k \\Delta .\n$$\nThen for any algebraically closed field extension $F/K$, there exists no correspondence\n$$\n[\\Gamma] \\in \\CH^{2}(E \\times_k S_F)\n$$\nthat induces a splitting of the pushforward map\n\\begin{equation*}\n(\\phi_{S_F})_{\\ast} \\colon \\CH_0(S_F)_{\\hom} \\longrightarrow E_{F} .\n\\end{equation*}\n\\end{theorem}\n\\begin{proof} Suppose, for contradiction, that there exists an algebraically closed field extension $F/K$ and a correspondence\n$$\n[\\Gamma] \\in \\CH^{2}(E \\times_k S_F)\n$$\nsuch that the induced endomorphism $$(\\phi_{S_F})_{\\ast}\\circ[\\Gamma]_{\\ast}\\in\\End\\bigl(E_{F}\\bigr)$$ is the identity.\\par\nLet $$S_0=\\sum_{i=1}^nR_i$$ denote the special fibre of the family $\\ca S\\to\\Delta,$ as in \\eqref{it:special_fibre} of Theorem \\ref{thm:intro_degeneration_S}. For each $i,$ let $\\al_i\\colon\\Alb(R_i)\\to E$ be the morphism of Albanese varieties induced by the restriction of $\\phi_{\\mathcal S}$ to $R_i$. By Corollary \\ref{cor:obstruction}, the assumption above implies that the sum morphism \n$$\\sum_{i=1}^n\\al_i\\colon\\bigoplus_{i=1}^n\\Alb(R_i)\\longrightarrow E$$\nadmits a section. \\par\nThis contradicts condition \\eqref{it:albanese_S_0} of Theorem \\ref{thm:intro_degeneration_S}. The resulting contradiction completes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main_result}] Let $E$ be an elliptic curve over $\\bbC$ with $\\End(E)=\\Z$. Choose a countable algebraically closed subfield $k_0\\subset\\bbC$ over which $E$ is defined; thus $$E\\cong E_0\\times_{k_{0}}\\bbC$$ for some elliptic curve $E_{0}$ over $k_{0}$.\\par\nLet $$\\ca S\\longrightarrow\\Delta:=\\Spec k_{0}[[t]]$$ be the strictly semistable degeneration constructed in Theorem \\ref{thm:intro_degeneration_S}, together with the induced morphism $$\\phi_{\\mathcal S} \\colon \\mathcal S \\longrightarrow E_0 \\times_{k_{0}} \\Delta .$$\\par\nSince $k_0$ is countable, the inclusion $k_0\\subset\\bbC$ factors through the Laurent series field $k_0((t))$. Set $$S:=S_{k_{0}((t))}\\times_{k_{0}((t))}\\bbC.$$\nBy \\eqref{it:generic_fibre} of Theorem \\ref{thm:intro_degeneration_S}, the surface $S$ is a bielliptic surface of type 2, and the morphism $$\\phi_{S}\\colon S\\longrightarrow E$$ induced by $\\phi_{\\ca S}$ coincides with the Albanese fibration of $S$.\\par\nIn particular, the pushforward homomorphism $$(\\phi_{S})_{\\ast} \\colon \\CH_0(S)_{\\hom} \\longrightarrow E $$ agrees with the Abel--Jacobi map for $0$-cycles on $S$. It is well known that bielliptic surfaces have representable $\\Ch$-group; see \\cite[Example (2)]{bloch-srinivas}. Applying Theorem \\ref{thm:main_result_body}, we conclude that there exists no correspondence $$\n[\\Gamma] \\in \\CH^{2}(E \\times S)\n$$\nthat induces a splitting of the Abel--Jacobi map \n\\begin{equation*}\n(\\phi_{S})_{\\ast} \\colon \\CH_0(S)_{\\hom} \\longrightarrow E.\n\\end{equation*}\nEquivalently, the surface $S$ admits no universal $0$-cycle; cf. Definition \\ref{def:universal_0-cycle}. This completes the proof of Theorem \\ref{thm:main_result}. \n\\end{proof}\n\\subsection{A counterexample to the integral Hodge conjecture}\nIn this final section, we apply Theorem \\ref{thm:main_result} to prove Corollary \\ref{cor:integral_Hodge_S}. In particular, we construct a threefold $X$ of Kodaira dimension zero for which the integral Hodge conjecture fails for $1$-cycles. In contrast to the counterexamples of Benoist--Ottem \\cite{benoist-ottem}, the non-algebraic Hodge class in degree $4$ arising from our construction is non-torsion.\\par\nLet $X$ be a smooth projective variety over $\\bbC$ of dimension $d= \\dim X$. We begin by recalling how the non-existence of a universal $0$-cycle on $X$ is related to the failure of the integral Hodge conjecture. Throughout, all (co)homology groups are taken to be Betti (co)homology, and the subscript \"tf\" denotes the quotient by torsion.\\par\n\n\\begin{proof}[Proof of Corollary \\ref{cor:integral_Hodge_S}]Let $E$ be an elliptic curve over $\\bbC$ with $\\End(E)=\\Z$. \nBy Theorem \\ref{thm:main_result}, there exists a bielliptic surface $S$ of type 2 with $\\Alb(S)\\cong E$ such that $S$ admits no universal $0$-cycle (cf. Definition \\ref{def:universal_0-cycle}).\nBy Lemma \\ref{lem:int_Hodge_conj_vs_uni_0_cycl}, it follows that the cohomology class $$\\delta_{S}\\in H^{4}\\bigl(E\\times S,\\Z\\bigr)_{\\mathrm{tf}}$$ constructed in \\eqref{eq:Hodge_class} is a Hodge class which is not algebraic.\\par\nThis provides a counterexample to the integral Hodge conjecture in degree $4$\nfor $E\\times S$.\\end{proof}", "post_theorem_intro_text_len": 5482, "post_theorem_intro_text": "Recall that ruled surfaces have representable Chow groups of $0$-cycles and do admit a universal $0$-cycle; this follows directly from the corresponding result for curves.\\par \nMore generally, let $X$ be a non-uniruled smooth projective surface with geometric genus $p_{g}(X)=0$ and irregularity $q(X)\\neq0$. By work of Beauville, the structure of such surfaces is well understood; see \\cite[Chapter VI]{beauville_surfaces}. Specifically, $X$ is birational to a quotient \\begin{equation*}\\label{eq:structure_X}\\bigl(C\\times D\\bigr)/\\Gi,\\end{equation*}\nwhere $C$ and $D$ are smooth projective curves and $\\Gi$ is a finite group acting faithfully on both curves, without fixed points on $C \\times D$. Moreover, at least one of the curves is elliptic. Up to exchanging $C$ and $D$, one has $$C/\\Gi\\cong E,\\qquad D/\\Gi\\cong\\mathbf{P}^{1}_{\\mathbf{C}},$$ where $E$ is an elliptic curve. In this situation, the induced fibration \\begin{equation}\\label{eq:fibration_X}\\phi_{X}\\colon X\\longrightarrow E\\end{equation} coincides with the Albanese morphism of $X$.\\par\nIt was shown in \\cite{voisin_a} that if the index of the fibration $\\phi_X$ in \\eqref{eq:fibration_X}, defined as the greatest common divisor of the degrees $\\deg(C/E)$, where $C \\subset X$ ranges over irreducible curves dominating $E$, is equal to one, then $X$ admits a universal $0$-cycle. In particular, this condition is satisfied when the group $\\Gi$ is cyclic; see \\cite[Proposition 4.4]{voisin_a}.\\par \nOur example in Theorem \\ref{thm:main_result} arises from a very general bielliptic surface $S$ of type 2 (see \\cite{bielliptic, bielliptic_suwa}) whose Albanese variety $\\Alb(S)$ is isomorphic to the prescribed elliptic curve $E$. In this situation, the Albanese fibration $\\phi_{S}\\colon S\\to \\Alb(S)\\cong E$ has index 2. As observed in \\cite[$\\S 2.2$]{voisin_a}, this condition alone does not suffice to imply that $S$ admits no universal $0$-cycle. The proof of Theorem \\ref{thm:main_result} therefore rests on the following geometric result.\n\n\\begin{theorem}\\label{thm:intro_degeneration_S} Let $k$ be an algebraically closed field of characteristic $\\neq2,$ and let $E$ be an elliptic curve over $k$ with $\\End(E)\\cong\\mathbb{Z}$. Set $\\Delta:=\\Spec k[[t]],$ and fix an algebraic closure $K$ of its function field. Then there exists a regular, flat, projective scheme $\\mathcal{S}\\to\\Delta,$ together with a morphism $$\\phi_{\\mathcal{S}}\\colon\\mathcal{S}\\longrightarrow E\\times\\Delta$$ such that the following properties hold:\n\\begin{enumerate}\n    \\item\\label{it:generic_fibre} The geometric generic fibre $S_{K}$ is a bielliptic surface of type $2$, and the induced morphism $\\phi_{S_{K}}\\colon S_{K}\\to E\\times_{k} K$ coincides with the Albanese fibration of $S_{K}$.\n    \\item\\label{it:special_fibre} The special fibre $S_{0}=\\sum_{i=1}^{n}R_{i}$ is a reduced simple normal crossings divisor on $\\mathcal{S}$ whose dual graph is a chain; that is, each intersection $R_{i}\\cap R_{i+1}$ is a smooth irreducible curve, and $R_{i}\\cap R_{j}=\\varnothing$ whenever $j\\notin\\{i-1,i,i+1\\}$.\n     \\item\\label{it:albanese_S_0} For each $i,$ let $\\al_{i}:\\Alb(R_{i})\\to E$ denote the homomorphism of Albanese varieties induced by the restriction of $\\phi_{\\mathcal{S}}$ to $R_{i}$. Then the sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(R_{i})\\longrightarrow E$$ does not admit a section. \n\\end{enumerate}  \n\\end{theorem}\nAs we shall explain, the non-existence of a section in \\eqref{it:albanese_S_0} provides the key obstruction to the existence of a universal $0$-cycle on the geometric generic fibre of the family $\\mathcal{S}\\to\\Delta$; see Theorem \\ref{thm:obstruction} and Corollary \\ref{cor:obstruction}.\\par\nWe conclude the introduction with the following application of Theorem \\ref{thm:main_result} to the integral Hodge conjecture.\n\\begin{corollary}\\label{cor:integral_Hodge_S} Let $E$ be a smooth elliptic curve over $\\mathbf{C}$ with $\\End(E)=\\mathbb{Z}$. Then there exists a bielliptic surface $S$ of type 2 whose Albanese variety $\\Alb(S)$ is isomorphic to $E,$ and for which the integral Hodge conjecture for $1$-cycles on $E\\times S$ fails. In particular, there exists a non-torsion integral Hodge class in $H^{4}(E\\times S,\\mathbb{Z})$ that is not algebraic.\n\\end{corollary}\nCorollary \\ref{cor:integral_Hodge_S} adds a further example to the list of threefolds with torsion canonical bundle for which the integral Hodge conjecture fails for $1$-cycles. This should be compared with the counterexamples of Benoist--Ottem \\cite{benoist-ottem}, where the obstruction arises from a non-algebraic torsion cohomology class in degree 4. In contrast, our construction relies on a different degeneration strategy: rather than degenerating the elliptic curve $E,$ we degenerate the surface $S$. This idea is inspired by recent work of the author \\cite{Alexandrou,alexandrou2025torsionhigherchowcycles} and by work of Schreieder \\cite{Schreieder2025}. \\par\nThe paper is organized as follows. In $\\S \\ref{sec:preliminaries}$ we review the necessary background material. Section \\ref{sec:obstruction} introduces an obstruction to the existence of a universal $0$-cycle on a smooth projective variety $X$ with representable Chow group of $0$-cycles. In $\\S \\ref{sec:degen_bielliptic}$ we prove Theorem \\ref{thm:intro_degeneration_S}. Finally, $\\S \\ref{sec:final}$ is devoted to the proofs of our main results, including Theorem \\ref{thm:main_result} and Corollary \\ref{cor:integral_Hodge_S}.", "sketch": "For surfaces of the form \\eqref{eq:structure_X} with Albanese fibration \\eqref{eq:fibration_X}, it is recalled that Voisin shows: if the index of \\(\\phi_X\\) is \\(1\\), then \\(X\\) admits a universal \\(0\\)-cycle (and this holds in particular when \\(\\Gi\\) is cyclic).\n\nFor Theorem~\\ref{thm:main_result}, the surface \\(S\\) is taken to be “a very general bielliptic surface of type 2” with \\(\\Alb(S)\\cong E\\), for which the Albanese fibration \\(\\phi_S\\) has index \\(2\\); and it is noted (following \\cite[\\S 2.2]{voisin_a}) that “this condition alone does not suffice to imply that \\(S\\) admits no universal \\(0\\)-cycle.”\n\nThe stated strategy is instead to use a degeneration: “The proof of Theorem~\\ref{thm:main_result} therefore rests on” Theorem~\\ref{thm:intro_degeneration_S}, constructing a regular flat projective family \\(\\mathcal S\\to\\Delta\\) whose geometric generic fibre \\(S_K\\) is such a bielliptic surface (with its Albanese fibration), and whose special fibre \\(S_0=\\sum R_i\\) is an SNC divisor with chain dual graph. The key geometric input is that for the induced maps \\(\\al_i:\\Alb(R_i)\\to E\\), “the sum morphism \\(\\sum_i \\al_i:\\bigoplus_i \\Alb(R_i)\\to E\\) does not admit a section” \\eqref{it:albanese_S_0}.\n\nAs announced, “the non-existence of a section in \\eqref{it:albanese_S_0} provides the key obstruction to the existence of a universal \\(0\\)-cycle on the geometric generic fibre of the family \\(\\mathcal S\\to\\Delta\\)” (via Theorem~\\ref{thm:obstruction} and Corollary~\\ref{cor:obstruction}), yielding the nonexistence of a universal \\(0\\)-cycle for \\(S\\) in Theorem~\\ref{thm:main_result}.", "expanded_sketch": "For surfaces of the form \\eqref{eq:structure_X} with Albanese fibration \\begin{equation}\\label{eq:fibration_X}\\phi_{X}\\colon X\\longrightarrow E\\end{equation}, it is recalled that Voisin shows: if the index of \\(\\phi_X\\) is \\(1\\), then \\(X\\) admits a universal \\(0\\)-cycle (and this holds in particular when \\(\\Gi\\) is cyclic).\n\nTo prove the main theorem, the surface \\(S\\) is taken to be “a very general bielliptic surface of type 2” with \\(\\Alb(S)\\cong E\\), for which the Albanese fibration \\(\\phi_S\\) has index \\(2\\); and it is noted (following Voisin, \\S 2.2) that “this condition alone does not suffice to imply that \\(S\\) admits no universal \\(0\\)-cycle.”\n\nThe stated strategy is instead to use a degeneration: “The proof of the main theorem therefore rests on” the following theorem.\n\n\\begin{theorem}\\label{thm:intro_degeneration_S} Let $k$ be an algebraically closed field of characteristic $\\neq2,$ and let $E$ be an elliptic curve over $k$ with $\\End(E)\\cong\\Z$. Set $\\Delta:=\\Spec k[[t]],$ and fix an algebraic closure $K$ of its function field. Then there exists a regular, flat, projective scheme $\\mathcal{S}\\to\\Delta,$ together with a morphism $$\\phi_{\\mathcal{S}}\\colon\\mathcal{S}\\longrightarrow E\\times\\Delta$$ such that the following properties hold:\n\\begin{enumerate}\n    \\item\\label{it:generic_fibre} The geometric generic fibre $S_{K}$ is a bielliptic surface of type $2$, and the induced morphism $\\phi_{S_{K}}\\colon S_{K}\\to E\\times_{k} K$ coincides with the Albanese fibration of $S_{K}$.\n    \\item\\label{it:special_fibre} The special fibre $S_{0}=\\sum_{i=1}^{n}R_{i}$ is a reduced simple normal crossings divisor on $\\mathcal{S}$ whose dual graph is a chain; that is, each intersection $R_{i}\\cap R_{i+1}$ is a smooth irreducible curve, and $R_{i}\\cap R_{j}=\\varnothing$ whenever $j\\notin\\{i-1,i,i+1\\}$.\n     \\item\\label{it:albanese_S_0} For each $i,$ let $\\al_{i}:\\Alb(R_{i})\\to E$ denote the homomorphism of Albanese varieties induced by the restriction of $\\phi_{\\mathcal{S}}$ to $R_{i}$. Then the sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(R_{i})\\longrightarrow E$$ does not admit a section. \n\\end{enumerate}  \n\\end{theorem}\n\nThus one constructs a regular flat projective family \\(\\mathcal S\\to\\Delta\\) whose geometric generic fibre \\(S_K\\) is such a bielliptic surface (with its Albanese fibration), and whose special fibre \\(S_0=\\sum R_i\\) is an SNC divisor with chain dual graph. The key geometric input is the preceding theorem’s assertion that for the induced maps \\(\\al_i:\\Alb(R_i)\\to E\\), the sum morphism \\(\\sum_i \\al_i:\\bigoplus_i \\Alb(R_i)\\to E\\) does not admit a section.\n\nAs announced, “the non-existence of a section in” this property provides the key obstruction to the existence of a universal \\(0\\)-cycle on the geometric generic fibre of the family \\(\\mathcal S\\to\\Delta\\) (via the following theorem and corollary), yielding the nonexistence of a universal \\(0\\)-cycle for \\(S\\) in the main theorem.\n\n\\begin{theorem}\n\\label{thm:obstruction}\nLet $k$ be an algebraically closed field and let $C$ be a smooth projective curve over $k$. Set $\\Delta := \\Spec k[[t]]$, and fix an algebraic closure $K$ of its function field. \n\nLet $p \\colon \\mathcal{X} \\to \\Delta$ be a flat, projective morphism of relative dimension $d$ with regular total space, and let $\\phi_{\\mathcal{X}} \\colon \\mathcal{X} \\to C \\times_k \\Delta$ be a surjective morphism.\nDenote by $X_0$ the special fibre of $p$, by\n$\\phi_{X_0} \\colon X_0 \\to C$ the induced morphism, and by $X_{K\n}$ the geometric generic fibre of $p$. Write $\\phi_{X_K} \\colon X_K \\to C_K$\nfor the base change of $\\phi_{\\mathcal{X}}$ to $K$.\\par\nWe further assume the following:\n\\begin{enumerate}\n    \\item \\label{it:X_0} $X_{0}=\\sum_{i=1}^{n}X_{0i}$ is a reduced simple normal crossings divisor on $\\mathcal{X}$ whose dual graph is a chain. Precisely, each intersection $X_{0i}\\cap X_{0i+1}$ is smooth and irreducible of dimension $d-1,$ and $X_{0i}\\cap X_{0j}=\\varnothing$ whenever $j\\notin\\{i-1,i,i+1\\}$.\n    \\item \\label{it:correspondence} \n    There exists a correspondence\n$$\n[\\Gamma] \\in \\CH^{d}(C \\times_k X_{K})\n$$\ninducing a splitting of the pushforward map\n$$\n(\\phi_{X_K})_\\ast\\colon \\Ch(X_{K})_{\\hom}\\longrightarrow\\jac(C_{K}).\n$$\n\n\\end{enumerate}\n\\par \nFor each $i,$ let $\\al_{i}\\colon\\Alb(X_{0i})\\to \\jac(C)$ be the homomorphism of Albanese varieties induced by the restriction of $\\phi_{X_0}$ to $X_{0i}$. Then the induced sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(X_{0i})\\longrightarrow\\jac(C)$$\nadmits a section. \n\\end{theorem}\n\n\\begin{corollary}\\label{cor:obstruction}\nWe use the notation and assumptions of Theorem \\ref{thm:obstruction}, with condition \\eqref{it:correspondence} replaced by the following:\n\\begin{enumerate}\n    \\item[\\((2')\\)]\\label{it:correspondence-prime} There exists an algebraically closed field extension $F/K$ and a correspondence\n$$\n[\\Gamma] \\in \\CH^{d}(C \\times_k X_{F})\n$$\ninducing a splitting of the pushforward map\n$$\n(\\phi_{X_F})_\\ast\\colon \\Ch(X_{F})_{\\hom}\\longrightarrow\\jac(C_{F}).\n$$\n\\end{enumerate}\nAssume moreover that $\\End(\\jac(C))=\\Z$. Then the sum morphism $$\\sum_{i=1}^{n}\\al_{i}\\colon\\bigoplus_{i=1}^{n}\\Alb(X_{0i})\\longrightarrow\\jac(C)$$\nadmits a section. \n\\end{corollary}", "expanded_theorem": "\\label{thm:main_result} Let $E$ be a smooth complex elliptic curve with $\\End(E)=\\mathbb{Z}$. There exists a smooth projective complex surface $S$ with $\\Alb(S)\\cong E$ for which $\\Ch(S)$ is representable, while $S$ admits no universal $0$-cycle.,", "theorem_type": ["Existence", "Implication"], "mcq": {"question": "Let $E$ be a smooth complex elliptic curve with $\\operatorname{End}(E)=\\mathbb{Z}$. For a smooth projective complex surface $S$, let\n\\[\n\\alpha_S\\colon \\CH_0(S)_{\\mathrm{hom}}\\longrightarrow \\operatorname{Alb}(S)\n\\]\nbe the Abel--Jacobi map for $0$-cycles. Say that $\\CH_0(S)$ is representable if $\\alpha_S$ is an isomorphism. Say that $S$ admits a universal $0$-cycle if there exists a codimension-$2$ cycle $[\\Gamma]\\in \\CH^2(\\operatorname{Alb}(S)\\times S)$ such that the induced morphism\n\\[\n\\psi_{(\\operatorname{Alb}(S),[\\Gamma])}\\colon \\operatorname{Alb}(S)\\to \\operatorname{Alb}(S),\\qquad a\\mapsto \\alpha_S\\bigl([\\Gamma]_*(a-0_{\\operatorname{Alb}(S)})\\bigr),\n\\]\nis the identity map on $\\operatorname{Alb}(S)$. Under these assumptions on $E$, which conclusion holds?", "correct_choice": {"label": "A", "text": "There exists a smooth projective complex surface $S$ with $\\operatorname{Alb}(S)\\cong E$ such that $\\CH_0(S)$ is representable, i.e. $\\alpha_S$ is an isomorphism, but $S$ does not admit a universal $0$-cycle; equivalently, there is no codimension-$2$ cycle $[\\Gamma]\\in \\CH^2(\\operatorname{Alb}(S)\\times S)$ for which $\\psi_{(\\operatorname{Alb}(S),[\\Gamma])}=\\operatorname{Id}_{\\operatorname{Alb}(S)}$."}, "choices": [{"label": "B", "text": "For every smooth projective complex surface $S$ with $\\operatorname{Alb}(S)\\cong E$, if $\\CH_0(S)$ is representable, then $S$ admits a universal $0$-cycle; equivalently, whenever $\\alpha_S$ is an isomorphism there exists a codimension-$2$ cycle $[\\Gamma]\\in \\CH^2(\\operatorname{Alb}(S)\\times S)$ such that $\\psi_{(\\operatorname{Alb}(S),[\\Gamma])}=\\operatorname{Id}_{\\operatorname{Alb}(S)}$."}, {"label": "C", "text": "There exists a smooth projective complex surface $S$ with $\\operatorname{Alb}(S)\\cong E$ such that $\\CH_0(S)$ is representable, i.e. $\\alpha_S$ is an isomorphism."}, {"label": "D", "text": "There exists a smooth projective complex surface $S$ with $\\operatorname{Alb}(S)\\cong E$ such that $\\CH_0(S)$ is representable and $S$ admits a universal $0$-cycle after base change to some algebraically closed field extension; equivalently, there is an algebraically closed extension $F/\\mathbb{C}$ and a codimension-$2$ cycle $[\\Gamma]\\in \\CH^2(\\operatorname{Alb}(S)_F\\times S_F)$ for which $\\psi_{(\\operatorname{Alb}(S)_F,[\\Gamma])}=\\operatorname{Id}_{\\operatorname{Alb}(S)_F}$."}, {"label": "E", "text": "There exists a smooth projective complex surface $S$ with $\\operatorname{Alb}(S)\\cong E$ such that $\\CH_0(S)$ is representable, but $S$ does not admit a universal $0$-cycle only in the weaker sense that no codimension-$2$ cycle $[\\Gamma]\\in \\CH^2(\\operatorname{Alb}(S)\\times S)$ makes $\\psi_{(\\operatorname{Alb}(S),[\\Gamma])}$ equal to $n\\,\\operatorname{Id}_{\\operatorname{Alb}(S)}$ for any positive integer $n$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "index-1/cyclic sufficient criterion promoted to universal statement for all representable surfaces", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the failure of a universal $0$-cycle", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "existence over an algebraically closed extension $F/K$ treated as if it could occur over a base change from $\\mathbb{C}$ despite the obstruction using $\\operatorname{End}(E)=\\mathbb{Z}$", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "confusion between absence of identity splitting and absence of any Murre multiple $n\\operatorname{Id}$", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives definitions and hypotheses but does not explicitly or implicitly reveal which of the logically nearby conclusions is correct. No choice is singled out by wording."}, "TAS": {"score": 1, "justification": "This is close to theorem-recall: the stem sets up the exact notions and asks which conclusion holds under the stated assumptions. However, it is not a pure restatement because the options vary meaningfully by quantifier strength, base change, and weakening/strengthening."}, "GPS": {"score": 2, "justification": "Selecting the right answer requires distinguishing subtle logical variants: existence vs universality, exact failure of identity splitting vs stronger failures, and whether base change could remove the obstruction. That creates real reasoning pressure."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: B overgeneralizes, C is a weaker true statement, D tests misunderstanding about base change, and E confuses failure of identity with failure of all multiples. They are distinct and aligned with likely failure modes."}, "total_score": 7, "overall_assessment": "A strong MCQ with subtle, high-quality distractors and little answer leakage. Its main weakness is that it remains somewhat theorem-recall in flavor rather than a fully generative problem."}}
{"id": "2602.13582v1", "paper_link": "http://arxiv.org/abs/2602.13582v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thMain}\nThere exist constants $\\delta, \\eps > 0$ and a sequence of finite groups $(G_n)_{n \\geq 1}$ with $|G_n| \\to \\infty$\nwith bounded-size generating sets $X_n, Y_n \\subseteq G_n$ such that\n\\begin{enumerate}[\\normalfont(a)]\n    \\item $\\op{Cay}(G_n,X_n)$ are $\\eps$-expander graphs, and\n    \\item $\\op{diam}( \\op{Cay}(G_n,Y_n) ) \\geq \\exp(\\delta \\sqrt{\\log|G_n|})$.\n\\end{enumerate}", "start_pos": 5700, "end_pos": 6108, "label": "thMain"}, "ref_dict": {"quesPS": "\\begin{question}[Pyber \\& Szab\\'o~\\cite{PS13}*{Question~23}]\n    \\label{quesPS}\n    Let $(G_n)_{n \\geq 1}$ be a sequence of finite groups and suppose that\n    $\\Cay(G_n,X_n)$ are expander graphs for some sets $(X_n)_{n \\geq 1}$ of bounded cardinality.\n    Is it true that for every sequence of generating sets $(Y_n)_{n \\geq 1}$ we have\n    $$ \\diam( \\Cay(G_n,Y_n) ) \\> \\le \\> C (\\log |G_n|)^C $$\n    for some constant $C$?\n\\end{question}", "thMain": "\\begin{theorem} \\label{thMain}\nThere exist constants $\\delta, \\eps > 0$ and a sequence of finite groups $(G_n)_{n \\geq 1}$ with $|G_n| \\to \\infty$\nwith bounded-size generating sets $X_n, Y_n \\subseteq G_n$ such that\n\\begin{enumerate}[\\normalfont(a)]\n    \\item $\\Cay(G_n,X_n)$ are $\\eps$-expander graphs, and\n    \\item $\\diam( \\Cay(G_n,Y_n) ) \\geq \\exp(\\delta \\sqrt{\\log|G_n|})$.\n\\end{enumerate}\n\\end{theorem}", "thExpSum": "\\begin{theorem} \\label{thExpSum}\n    There exist constants $\\delta, \\eps >0$ such that the following holds.\n    For every integer $n \\ge 1$ and prime $p \\le e^{\\delta n}$,\n    there exists $v \\in V_0$ such that, for all nonconstant $w \\in V$,\n    \\[\n       \\left| \\frac1{n!} \\sum_{\\sigma \\in S_n} e_p\\br{\\gen{v, w^\\sigma}} \\right|\n        \\> \\le \\> 1 - \\eps.\n    \\]\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2441, "pre_theorem_intro_text": "Let $G$ be an infinite group generated by a finite set $X$.\nSeveral important properties of the Cayley graph $\\op{Cay}(G,X)$ do not depend on the choice of $X$,\nand are actually group properties:\nthe growth type, the number of ends, hyperbolicity, amenability (via the F{\\o}lner condition), and property (T).\nThe key observation behind these facts is that, for any other finite subset $Y$ of $G$,\nthere exists a constant $c$ such that every element in $Y$ can be written as the product of at most $c$ elements in $X$.\n\nThe situation is different for a sequence of finite groups.\nAnswering a question of Lubotzky--Weiss~\\cite{LW93}, Alon, Lubotzky, and Wigderson~\\cite{ALW01}\nconstructed finite groups whose Cayley graphs are expanders with respect to one bounded-size generating set but not with respect to another.\nSpecifically, they showed that the groups $C_2^{p+1} \\rtimes \\PSL_2(p)$ have this property.\nIn the seminal paper~\\cite{Kas07}, Kassabov showed that the symmetric groups also have this property,\nand subsequently Kassabov, Lubotzky, and Nikolov~\\cite{KLN06} proved that almost all nonabelian finite simple groups are expanders with respect to suitable bounded-size generating sets.\n\nHowever, all of the groups listed above likely have worst-case polylogarithmic diameter.\nIn the case of the finite simple groups, this is the content of the famous conjecture of Babai~\\cite{BS92}*{Conjecture~1.7}.\nThis observation prompted Pyber and Szab\\'o to ask the following question:\n\n\\begin{question}[Pyber \\& Szab\\'o~\\cite{PS13}*{Question~23}]\n    \\label{quesPS}\n    Let $(G_n)_{n \\geq 1}$ be a sequence of finite groups and suppose that\n    $\\op{Cay}(G_n,X_n)$ are expander graphs for some sets $(X_n)_{n \\geq 1}$ of bounded cardinality.\n    Is it true that for every sequence of generating sets $(Y_n)_{n \\geq 1}$ we have\n    $$ \\op{diam}( \\op{Cay}(G_n,Y_n) ) \\> \\le \\> C (\\log |G_n|)^C $$\n    for some constant $C$?\n\\end{question}\n\nA similar question was asked by the second author at the 2024 Oberwolfach meeting \\emph{Growth and Expansion in Groups} \\cite{BD24}*{p.~1028,  Question 5}:\nthere it was asked whether it is enough to assume that $\\op{diam}(\\op{Cay}(G_n, X_n))$ is polylogarithmic, i.e., whether polylogarithmic diameter is a group property.\n\nIn this note we answer both questions negatively. In fact we show that the diameter of $G$ with respect to the second generating set can be as large as $\\exp(c \\sqrt{\\log |G|})$.", "context": "Let $G$ be an infinite group generated by a finite set $X$.\nSeveral important properties of the Cayley graph $\\op{Cay}(G,X)$ do not depend on the choice of $X$,\nand are actually group properties:\nthe growth type, the number of ends, hyperbolicity, amenability (via the F{\\o}lner condition), and property (T).\nThe key observation behind these facts is that, for any other finite subset $Y$ of $G$,\nthere exists a constant $c$ such that every element in $Y$ can be written as the product of at most $c$ elements in $X$.\n\nThe situation is different for a sequence of finite groups.\nAnswering a question of Lubotzky--Weiss~\\cite{LW93}, Alon, Lubotzky, and Wigderson~\\cite{ALW01}\nconstructed finite groups whose Cayley graphs are expanders with respect to one bounded-size generating set but not with respect to another.\nSpecifically, they showed that the groups $C_2^{p+1} \\rtimes \\PSL_2(p)$ have this property.\nIn the seminal paper~\\cite{Kas07}, Kassabov showed that the symmetric groups also have this property,\nand subsequently Kassabov, Lubotzky, and Nikolov~\\cite{KLN06} proved that almost all nonabelian finite simple groups are expanders with respect to suitable bounded-size generating sets.\n\nHowever, all of the groups listed above likely have worst-case polylogarithmic diameter.\nIn the case of the finite simple groups, this is the content of the famous conjecture of Babai~\\cite{BS92}*{Conjecture~1.7}.\nThis observation prompted Pyber and Szab\\'o to ask the following question:\n\n\\begin{question}[Pyber \\& Szab\\'o~\\cite{PS13}*{Question~23}]\n \\label{quesPS}\n Let $(G_n)_{n \\geq 1}$ be a sequence of finite groups and suppose that\n $\\op{Cay}(G_n,X_n)$ are expander graphs for some sets $(X_n)_{n \\geq 1}$ of bounded cardinality.\n Is it true that for every sequence of generating sets $(Y_n)_{n \\geq 1}$ we have\n $$ \\op{diam}( \\op{Cay}(G_n,Y_n) ) \\> \\le \\> C (\\log |G_n|)^C $$\n for some constant $C$?\n\\end{question}\n\nA similar question was asked by the second author at the 2024 Oberwolfach meeting \\emph{Growth and Expansion in Groups} \\cite{BD24}*{p.~1028, Question 5}:\nthere it was asked whether it is enough to assume that $\\op{diam}(\\op{Cay}(G_n, X_n))$ is polylogarithmic, i.e., whether polylogarithmic diameter is a group property.\n\nIn this note we answer both questions negatively. In fact we show that the diameter of $G$ with respect to the second generating set can be as large as $\\exp(c \\sqrt{\\log |G|})$.", "full_context": "Let $G$ be an infinite group generated by a finite set $X$.\nSeveral important properties of the Cayley graph $\\op{Cay}(G,X)$ do not depend on the choice of $X$,\nand are actually group properties:\nthe growth type, the number of ends, hyperbolicity, amenability (via the F{\\o}lner condition), and property (T).\nThe key observation behind these facts is that, for any other finite subset $Y$ of $G$,\nthere exists a constant $c$ such that every element in $Y$ can be written as the product of at most $c$ elements in $X$.\n\nThe situation is different for a sequence of finite groups.\nAnswering a question of Lubotzky--Weiss~\\cite{LW93}, Alon, Lubotzky, and Wigderson~\\cite{ALW01}\nconstructed finite groups whose Cayley graphs are expanders with respect to one bounded-size generating set but not with respect to another.\nSpecifically, they showed that the groups $C_2^{p+1} \\rtimes \\PSL_2(p)$ have this property.\nIn the seminal paper~\\cite{Kas07}, Kassabov showed that the symmetric groups also have this property,\nand subsequently Kassabov, Lubotzky, and Nikolov~\\cite{KLN06} proved that almost all nonabelian finite simple groups are expanders with respect to suitable bounded-size generating sets.\n\nHowever, all of the groups listed above likely have worst-case polylogarithmic diameter.\nIn the case of the finite simple groups, this is the content of the famous conjecture of Babai~\\cite{BS92}*{Conjecture~1.7}.\nThis observation prompted Pyber and Szab\\'o to ask the following question:\n\n\\begin{question}[Pyber \\& Szab\\'o~\\cite{PS13}*{Question~23}]\n \\label{quesPS}\n Let $(G_n)_{n \\geq 1}$ be a sequence of finite groups and suppose that\n $\\op{Cay}(G_n,X_n)$ are expander graphs for some sets $(X_n)_{n \\geq 1}$ of bounded cardinality.\n Is it true that for every sequence of generating sets $(Y_n)_{n \\geq 1}$ we have\n $$ \\op{diam}( \\op{Cay}(G_n,Y_n) ) \\> \\le \\> C (\\log |G_n|)^C $$\n for some constant $C$?\n\\end{question}\n\nA similar question was asked by the second author at the 2024 Oberwolfach meeting \\emph{Growth and Expansion in Groups} \\cite{BD24}*{p.~1028, Question 5}:\nthere it was asked whether it is enough to assume that $\\op{diam}(\\op{Cay}(G_n, X_n))$ is polylogarithmic, i.e., whether polylogarithmic diameter is a group property.\n\nIn this note we answer both questions negatively. In fact we show that the diameter of $G$ with respect to the second generating set can be as large as $\\exp(c \\sqrt{\\log |G|})$.\n\n\\begin{abstract}\n We study how the spectral gap and diameter of Cayley graphs depend strongly on the choice of generating set.\n We answer a question of Pyber and Szab\\'o (2013)\n by exhibiting a sequence of finite groups $G_n$ with $|G_n| \\to \\infty$ admitting bounded generating sets $X_n,Y_n$ such that $\\operatorname{Cay}(G_n,X_n)$ is an expander\n while $\\operatorname{Cay}(G_n,Y_n)$ has super-polylogarithmic diameter.\n The construction uses the semidirect product $G_n = C_p^{n-1} \\rtimes S_n$ with $p$ exponentially large in $n$,\n and the analysis reduces to bounding some exponential sums of permutational type.\n\\end{abstract}\n\n\\begin{question}[Pyber \\& Szab\\'o~\\cite{PS13}*{Question~23}]\n \\label{quesPS}\n Let $(G_n)_{n \\geq 1}$ be a sequence of finite groups and suppose that\n $\\Cay(G_n,X_n)$ are expander graphs for some sets $(X_n)_{n \\geq 1}$ of bounded cardinality.\n Is it true that for every sequence of generating sets $(Y_n)_{n \\geq 1}$ we have\n $$ \\diam( \\Cay(G_n,Y_n) ) \\> \\le \\> C (\\log |G_n|)^C $$\n for some constant $C$?\n\\end{question}\n\nIn this note we answer both questions negatively. In fact we show that the diameter of $G$ with respect to the second generating set can be as large as $\\exp(c \\sqrt{\\log |G|})$.\n\nBy contrast having polynomially large diameter \\emph{is} a group property,\nand such groups are called \\emph{almost flat}: see \\cite{BT16}*{Theorem~4.1, Corollary~4.16}.\nIn particular, if (a) holds then $\\diam(\\Cay(G_n, Y_n)) \\le |G_n|^{o(1)}$,\nand it is an interesting open question to find the optimal bound in (b).\n\n\\begin{theorem} \\label{thExpSum}\n There exist constants $\\delta, \\eps >0$ such that the following holds.\n For every integer $n \\ge 1$ and prime $p \\le e^{\\delta n}$,\n there exists $v \\in V_0$ such that, for all nonconstant $w \\in V$,\n \\[\n \\left| \\frac1{n!} \\sum_{\\sigma \\in S_n} e_p\\br{\\gen{v, w^\\sigma}} \\right|\n \\> \\le \\> 1 - \\eps.\n \\]\n\\end{theorem}\n\n\\begin{lemma}\n \\label{lem:kaz-basics}\n Let $G$ be a finite group and let $S, T \\subseteq G$.\n \\begin{enumerate}\n \\item $\\kaz(G, S) \\le \\kaz(G, T)$ if $S \\subseteq T$;\n \\item $\\kaz(G, S) \\le 2$;\n \\item $\\kaz(G, G) \\ge \\sqrt 2$;\n \\item $\\kaz(G, S) \\ge \\frac1n \\kaz(G, S^n)$.\n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n (1)--(2) are clear. For (3)--(4) see \\cite{Kas07}*{Propositions~1.3--1.4}.\n\\end{proof}\n\n\\begin{lemma} \\label{lemSean1}\n Let $\\pi \\colon G \\to U(V)$ and let $\\xi \\in V$ be $\\eps$-almost $S$-invariant.\n Assume $\\kaz(G, S) > 0$ and let $\\eps' = \\eps / \\kaz(G, S)$.\n Then\n \\begin{enumerate}[(i)]\n \\item there is a $G$-invariant vector $\\xi_1 \\in V$ such that $\\|\\xi - \\xi_1\\| < \\eps'\\|\\xi\\|$;\n \\item $\\xi$ is $2\\eps'$-almost $G$-invariant.\n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n \\emph{(i)}\n Orthogonally decompose $\\xi = \\xi_1 + \\xi_2$ where $\\xi_1 \\in V^G$ and $\\xi_2 \\in (V^G)^\\perp$.\n Since $((V^G)^\\perp)^G = 0$, there is some $s \\in S$ such that\n \\[\n \\kaz(G, S) \\|\\xi_2\\| \\le \\|\\pi(s) \\xi_2 - \\xi_2\\| = \\|\\pi(s) \\xi - \\xi\\| < \\eps \\|\\xi\\|.\n \\]\n Hence $\\|\\xi_2\\| < \\eps' \\|\\xi\\|$.\n\nWe now prove Theorem~\\ref{thMain}(a) assuming Theorem~\\ref{thExpSum} (which will be proved in the next section).\nLet $V= \\F_p^n$ and let $V_0 < V$ be the deleted permutation module of dimension $n-1$.\nLet $G = V_0 \\rtimes S_n$ and $v \\in V_0$.\nBy Kassabov~\\cite{Kas07} there is a bounded-size generating set $T \\subseteq S_n$ such that $\\kaz(S_n, T)$ is bounded away from zero.\nIf $X = \\{v\\} \\cup T$,\nthen by Proposition~\\ref{prop:semidirect} and Lemma~\\ref{lemKazSG} we have\n\\begin{align*}\n \\kaz(G, X)\n &\\ge \\frac{\\sqrt 2}{48} \\kaz(V_0, v^{S_n}) \\kaz(S_n, T)\n \\\\\n &\\ge \\frac{1}{24} \\gap(V_0, v^{S_n})^{1/2} \\kaz(S_n, T).\n \\end{align*}\nWe now focus on $\\gap(V_0, v^{S_n})$.\nObserve that $\\Cay(V,v^{S_n})$ is the disjoint union of $p$ graphs all isomorphic to $\\Cay(V_0,v^{S_n})$.\nIt follows that the eigenvalues of the adjacency operator of $\\Cay(V_0, v^{S_n})$ are the same as for\n$\\Cay(V, v^{S_n})$, the only difference being their multiplicities.\nSince $V$ is abelian, its regular representation is isomorphic to a direct sum of one-dimensional representations\nindexed by the irreducible complex characters.\nWe can identify these characters with vectors $w \\in V$, using the formula\n\\[\n \\chi_w(u) = e_p(\\gen{u, w}) \\qquad (u,w \\in V).\n\\]\n(Recall that $e_p(x) := \\exp(2 \\pi i x / p)$.)\nIt follows that the eigenvalues of $\\Cay(V,v^{S_n})$ and so of $\\Cay(V_0,v^{S_n})$ are given by\n\\[\n \\frac1{2|S_n|} \\sum_{\\sigma \\in S_n} \\br{\\chi_w(v^\\sigma) + \\chi_w(-v^{\\sigma})} \\hspace{1cm} (w \\in V) .\n\\]\nThese are precisely the real values $\\{ \\Re[\\lambda_{v,w}] \\}_{w \\in V}$, where\n\\[\n \\lambda_{v,w} \\> := \\>\n \\frac1{n!} \\sum_{\\sigma \\in S_n} e_p (\\gen{v, w^\\sigma}) .\n\\]\nThe trivial eigenvalues are associated to the constant vectors, and so\n$\\gap(V_0,v^{S_n}) = 1 - \\max_{w \\in V \\setminus \\gen \\one} \\Re[\\lambda_{v,w}]$.\n Since $\\Re[\\lambda_{v,w}] \\le |\\lambda_{v,w}|$, the reduction follows.\n\n\\begin{theorem} \\label{thExpSum}\n    There exist constants $\\delta, \\eps >0$ such that the following holds.\n    For every integer $n \\ge 1$ and prime $p \\le e^{\\delta n}$,\n    there exists $v \\in V_0$ such that, for all nonconstant $w \\in V$,\n    \\[\n       \\left| \\frac1{n!} \\sum_{\\sigma \\in S_n} e_p\\br{\\gen{v, w^\\sigma}} \\right|\n        \\> \\le \\> 1 - \\eps.\n    \\]\n\\end{theorem}\n\n\\begin{theorem} \\label{thMain}\nThere exist constants $\\delta, \\eps > 0$ and a sequence of finite groups $(G_n)_{n \\geq 1}$ with $|G_n| \\to \\infty$\nwith bounded-size generating sets $X_n, Y_n \\subseteq G_n$ such that\n\\begin{enumerate}[\\normalfont(a)]\n    \\item $\\Cay(G_n,X_n)$ are $\\eps$-expander graphs, and\n    \\item $\\diam( \\Cay(G_n,Y_n) ) \\geq \\exp(\\delta \\sqrt{\\log|G_n|})$.\n\\end{enumerate}\n\\end{theorem}", "post_theorem_intro_text_len": 3242, "post_theorem_intro_text": "By contrast having polynomially large diameter \\emph{is} a group property,\nand such groups are called \\emph{almost flat}: see \\cite{BT16}*{Theorem~4.1, Corollary~4.16}.\nIn particular, if (a) holds then $\\op{diam}(\\op{Cay}(G_n, Y_n)) \\le |G_n|^{o(1)}$,\nand it is an interesting open question to find the optimal bound in (b).\n\nWe now describe the construction.\nNotably, there are elements in common with both Alon--Lubotzky--Wigderson and Kassabov.\nFrom now on we suppress the subscript $n$, which will be clear from context.\nThe group is just\n\\[G \\> = \\> V_0 \\rtimes S_n,\\]\nwhere $V_0 < V = \\F_p^n$ is the deleted permutation module of dimension $n-1$,\nand $p \\sim e^{cn}$ is a large prime.\nNote that\n\\[\n    \\> |G| \\> = \\> p^{n-1} n! \\> = \\> \\exp\\br{cn^2 + O(n \\log n)} .\n\\]\nThe two generating sets $X$ and $Y$ both have the form $\\{v\\} \\cup T$ where $v \\in V_0$ and $T \\subset S_n$.\nFor $Y$ we make the unimaginative choice\n\\[\n    v = (1, -1, 0, \\dots, 0), \\qquad T = \\{(1, 2), (1, 2, \\dots, n) \\}.\n\\]\nClearly $\\op{diam}(\\op{Cay}(G, Y)) \\ge \\floor{pn/2} \\ge p \\sim e^{cn}$, so (b) holds.\n\nThe nontrivial part of the proof is the demonstration that there is a choice of $v$ and $T$ such that $\\op{Cay}(G, X)$ has a uniform spectral gap.\nFor $T$ we take a bounded-size expanding generating set for $S_n$ (which exists by the result of Kassabov).\nBy an analysis similar to that in \\cite{ALW01},\nthe problem of finding $v \\in V_0$ reduces to the following ``permutational exponential sum'' estimate.\n\nFor $v, w \\in V = \\F_p^n$, let $\\gen {v,w} = \\sum_{i=1}^n v_i w_i$.\nLet $\\mathbf{1} = (1, \\dots, 1)$ denote the all-one vector, so that $V_0 = \\mathbf{1}^\\perp$,\nand let $e_p(x) := \\exp(2 \\pi i x / p)$.\n\n\\begin{theorem} \\label{thExpSum}\n    There exist constants $\\delta, \\eps >0$ such that the following holds.\n    For every integer $n \\ge 1$ and prime $p \\le e^{\\delta n}$,\n    there exists $v \\in V_0$ such that, for all nonconstant $w \\in V$,\n    \\[\n       \\left| \\frac1{n!} \\sum_{\\sigma \\in S_n} e_p\\br{\\langle{v, w^\\sigma}\\rangle} \\right|\n        \\> \\le \\> 1 - \\eps.\n    \\]\n\\end{theorem}\n\nIn fact we show that a random $v \\in V_0$ works.\nWhen $p$ is bounded, this can be proved by a straightforward modification of the analysis in \\cite{ALW01}.\nThe main novelty in our result is that $p$ is allowed to be enormous:\nin technical terms, the reason we can take $p$ exponential in $n$ is that we do not need a union bound over all vectors,\nbut only over support-one vectors, and we can then use a deterministic switching argument based on the Cauchy--Schwarz inequality to deal with all other vectors.\n\n\\begin{remark}\n    We have chosen to present the simplest argument that answers Question~\\ref{quesPS}, but\n    a more elaborate analysis shows that Theorems~\\ref{thMain} and \\ref{thExpSum} are true for any constant $\\delta > 0$ and some suitable $\\eps = \\eps(\\delta) > 0$.\n    Moreover, $\\eps(\\delta)$ can be taken to be a continuous monotonic function such that $\\eps(\\delta) \\to 1$ as $\\delta \\to 0$.\n    On the other hand, a standard ``rectification'' argument in additive combinatorics (see \\cite{BLR98}*{Section~3} for example) shows that necessarily $\\eps(\\delta) \\to 0$ as $\\delta \\to \\infty$ in Theorem~\\ref{thExpSum}.\n\\end{remark}", "sketch": "We “describe the construction” of the groups in Theorem~\\ref{thMain} as follows. “The group is just”\n\\[G = V_0 \\rtimes S_n,\\]\nwhere $V_0<\\F_p^n$ is the deleted permutation module and “$p\\sim e^{cn}$ is a large prime,” so $|G|=p^{n-1}n!=\\exp(cn^2+O(n\\log n))$. The generating sets “both have the form $\\{v\\}\\cup T$ where $v\\in V_0$ and $T\\subset S_n$.”\n\nFor the large-diameter generating set $Y$, one takes\n\\[v=(1,-1,0,\\dots,0),\\qquad T=\\{(1,2),(1,2,\\dots,n)\\},\\]\nand then “clearly $\\operatorname{diam}(\\operatorname{Cay}(G,Y))\\ge \\lfloor pn/2\\rfloor\\ge p\\sim e^{cn}$, so (b) holds.”\n\n“The nontrivial part of the proof” is to choose $v$ and $T$ so that $\\operatorname{Cay}(G,X)$ has “a uniform spectral gap.” For $T$ one uses “a bounded-size expanding generating set for $S_n$ (which exists by the result of Kassabov).” Then, “by an analysis similar to that in \\cite{ALW01}, the problem of finding $v\\in V_0$ reduces to” the exponential-sum estimate stated as Theorem~\\ref{thExpSum}. The text adds that “in fact we show that a random $v\\in V_0$ works.” The key point enabling $p$ to be “enormous” (exponential in $n$) is that “we do not need a union bound over all vectors, but only over support-one vectors, and we can then use a deterministic switching argument based on the Cauchy--Schwarz inequality to deal with all other vectors.”", "expanded_sketch": "We “describe the construction” of the groups in establishing the main theorem as follows. “The group is just”\n\\[G = V_0 \\rtimes S_n,\\]\nwhere $V_0<\\F_p^n$ is the deleted permutation module and “$p\\sim e^{cn}$ is a large prime,” so $|G|=p^{n-1}n!=\\exp(cn^2+O(n\\log n))$. The generating sets “both have the form $\\{v\\}\\cup T$ where $v\\in V_0$ and $T\\subset S_n$.”\n\nFor the large-diameter generating set $Y$, one takes\n\\[v=(1,-1,0,\\dots,0),\\qquad T=\\{(1,2),(1,2,\\dots,n)\\},\\]\nand then “clearly $\\operatorname{diam}(\\operatorname{Cay}(G,Y))\\ge \\lfloor pn/2\\rfloor\\ge p\\sim e^{cn}$, so (b) holds.”\n\n“The nontrivial part of the proof” is to choose $v$ and $T$ so that $\\operatorname{Cay}(G,X)$ has “a uniform spectral gap.” For $T$ one uses “a bounded-size expanding generating set for $S_n$ (which exists by the result of Kassabov).” Then, “by an analysis similar to that in A. Alon, Y. Roichman, and N. Linial, \\emph{The spectral gap of random Cayley graphs} (2001), the problem of finding $v\\in V_0$ reduces to” the following theorem.\n\n\\begin{theorem} \\label{thExpSum}\n    There exist constants $\\delta, \\eps >0$ such that the following holds.\n    For every integer $n \\ge 1$ and prime $p \\le e^{\\delta n}$,\n    there exists $v \\in V_0$ such that, for all nonconstant $w \\in V$,\n    \\[\n       \\left| \\frac1{n!} \\sum_{\\sigma \\in S_n} e_p\\br{\\gen{v, w^\\sigma}} \\right|\n        \\> \\le \\> 1 - \\eps.\n    \\]\n\\end{theorem}\n\nThe text adds that “in fact we show that a random $v\\in V_0$ works.” The key point enabling $p$ to be “enormous” (exponential in $n$) is that “we do not need a union bound over all vectors, but only over support-one vectors, and we can then use a deterministic switching argument based on the Cauchy--Schwarz inequality to deal with all other vectors.”", "expanded_theorem": "\\label{thMain}\nThere exist constants $\\delta, \\eps > 0$ and a sequence of finite groups $(G_n)_{n \\geq 1}$ with $|G_n| \\to \\infty$\nwith bounded-size generating sets $X_n, Y_n \\subseteq G_n$ such that\n\\begin{enumerate}[\\normalfont(a)]\n    \\item $\\op{Cay}(G_n,X_n)$ are $\\eps$-expander graphs, and\n    \\item $\\op{diam}( \\op{Cay}(G_n,Y_n) ) \\geq \\exp(\\delta \\sqrt{\\log|G_n|})$.\n\\end{enumerate}", "theorem_type": ["Existence", "Existential–Universal"], "mcq": {"question": "Which statement holds? Here, for a finite group $G$ and a generating set $S \\subseteq G$, $\\operatorname{Cay}(G,S)$ denotes the Cayley graph of $G$ with respect to $S$, $\\operatorname{diam}(\\operatorname{Cay}(G,S))$ is its graph diameter, and saying that $\\operatorname{Cay}(G,S)$ is an $\\varepsilon$-expander means that the family has expansion uniformly bounded below by $\\varepsilon>0$. Also, “bounded-size generating sets” means that the cardinalities of the generating sets are uniformly bounded independently of $n$.", "correct_choice": {"label": "A", "text": "There exist constants $\\delta,\\varepsilon>0$ and a sequence of finite groups $(G_n)_{n\\ge 1}$ with $|G_n|\\to\\infty$, together with generating sets $X_n,Y_n\\subseteq G_n$ of uniformly bounded size, such that for every $n\\ge 1$, the Cayley graph $\\operatorname{Cay}(G_n,X_n)$ is an $\\varepsilon$-expander graph and $$\\operatorname{diam}(\\operatorname{Cay}(G_n,Y_n))\\ge \\exp\\!\\bigl(\\delta\\sqrt{\\log|G_n|}\\bigr).$$"}, "choices": [{"label": "B", "text": "There exist constants $\\delta,\\varepsilon>0$ and a sequence of finite groups $(G_n)_{n\\ge 1}$ with $|G_n|\\to\\infty$, together with generating sets $X_n,Y_n\\subseteq G_n$ of uniformly bounded size, such that for every $n\\ge 1$, both Cayley graphs $\\operatorname{Cay}(G_n,X_n)$ and $\\operatorname{Cay}(G_n,Y_n)$ are $\\varepsilon$-expander graphs and $$\\operatorname{diam}(\\operatorname{Cay}(G_n,Y_n))\\ge \\exp\\!\\bigl(\\delta\\sqrt{\\log|G_n|}\\bigr).$$"}, {"label": "C", "text": "There exist constants $\\delta,\\varepsilon>0$ and a sequence of finite groups $(G_n)_{n\\ge 1}$ with $|G_n|\\to\\infty$, together with generating sets $X_n,Y_n\\subseteq G_n$ of uniformly bounded size, such that for every $n\\ge 1$, the Cayley graph $\\operatorname{Cay}(G_n,X_n)$ is an $\\varepsilon$-expander graph and $$\\operatorname{diam}(\\operatorname{Cay}(G_n,Y_n))\\ge (\\log|G_n|)^{\\delta}.$$"}, {"label": "D", "text": "There exist constants $\\delta,\\varepsilon>0$ such that for every sequence of finite groups $(G_n)_{n\\ge 1}$ with $|G_n|\\to\\infty$ and every choice of uniformly bounded-size generating sets $X_n,Y_n\\subseteq G_n$, if the Cayley graphs $\\operatorname{Cay}(G_n,X_n)$ are $\\varepsilon$-expander graphs, then one has $$\\operatorname{diam}(\\operatorname{Cay}(G_n,Y_n))\\ge \\exp\\!\\bigl(\\delta\\sqrt{\\log|G_n|}\\bigr)$$ for every $n\\ge 1$."}, {"label": "E", "text": "There exist constants $\\delta,\\varepsilon>0$ and a sequence of finite groups $(G_n)_{n\\ge 1}$ with $|G_n|\\to\\infty$, together with generating sets $X_n,Y_n\\subseteq G_n$ of uniformly bounded size, such that for every $n\\ge 1$, the Cayley graph $\\operatorname{Cay}(G_n,X_n)$ is an $\\varepsilon$-expander graph and $$\\operatorname{diam}(\\operatorname{Cay}(G_n,Y_n))\\ge \\exp\\!\\bigl(\\delta\\log|G_n|\\bigr).$$"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "separation between expanding set and large-diameter set", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "lower-bound strength on the diameter", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "existential choice of groups and generating sets", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "size of the diameter lower bound relative to $|G_n|$", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem only supplies definitions and notation. It does not explicitly or implicitly signal which of the closely related existence statements is correct."}, "TAS": {"score": 2, "justification": "The item is not a bare restatement of a theorem in the stem. The choices differ in quantifiers, strength of the diameter bound, and whether both generating sets must expand, so the student must discriminate among genuinely competing conclusions."}, "GPS": {"score": 1, "justification": "The question requires some reasoning about logical strength and asymptotic bounds, but it mainly tests precise recognition/recall of a theorem statement rather than generating a conclusion from given premises."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target common errors: strengthening the conclusion too far, weakening it, changing existential to universal quantifiers, or incorrectly imposing expansion on both generating sets."}, "total_score": 7, "overall_assessment": "A strong MCQ with no answer leakage and high-quality distractors. Its main limitation is that it leans more toward theorem recognition than fully generative mathematical reasoning."}}
{"id": "2602.13696v1", "paper_link": "http://arxiv.org/abs/2602.13696v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{theoremcomNS}\n    Let $(\\rho, u)$ be a weak solution of \\eqref{eq:comNS} in the sense of Definition \\ref{defcomNS}. Assume that the density satisfies\n    \\begin{equation}\\label{con:thcomNSrhoover0}\n         0 \\leq \\rho(t,x) \\leq \\bar{\\rho} < \\infty, \\quad \\text{and} \\quad \\nabla \\sqrt{\\rho} \\in L^{\\infty}\\left(0, T ; L^{\\frac{3}{2}}(\\Omega)\\right),\n    \\end{equation}\n    and the initial velocity satisfies\n    \\begin{equation}\\label{con:initial_1}\n        u_0 \\in L^{\\frac{6N}{6-N}}(\\Omega).\n    \\end{equation}\n    If the velocity satisfies\n    \\begin{equation}\\label{con:shinbrot_cond}\n        u \\in L^p\\left(0, T ; L^q(\\Omega)\\right) \\quad \\text{with} \\quad \\begin{cases}\n            \\frac{1}{p}+\\frac{3}{q} \\leq 1, \\quad\\mbox{if} \\quad 3\\leq q < 4,\\\\\n            \\frac{2}{p}+\\frac{2}{q} \\leq 1, \\quad\\mbox{if} \\quad 4\\leq q <\\infty,\n        \\end{cases}\n    \\end{equation}\n    then the energy equality \\eqref{eq:energy_comNS} holds for any $t \\in[0, T]$.", "start_pos": 16737, "end_pos": 17732, "label": "theoremcomNS"}, "ref_dict": {"eq:energy_comNS": "\\begin{aligned}\n    \t&\\int_{\\Omega}\\left(\\frac{1}{2} \\rho_0\\left|u_0\\right|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x-\\int_{\\Omega}\\left(\\frac{1}{2} \\rho|u|^2+\\frac{\\rho^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x\\\\\n    \t=&\\int_0^T \\int_{\\Omega}\\left( \\mu|\\nabla u|^2+(\\mu+\\lambda)|\\operatorname{div} u|^2 \\right)\\mathrm{d} x \\mathrm{d} t. \\label{eq:energy_comNS}\n    \\end{aligned}", "con:Chen et al.": "\\begin{equation}\\label{con:Chen et al.}\n    u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\Omega))\\quad\\mbox{for}\\quad p\\geq4,\\quad q\\geq 6,\n\\end{equation}", "theoremcomNSdeg": "\\begin{theorem}\\label{theoremcomNSdeg}\n    Let $(\\rho, u)$ be a weak solution of \\eqref{eq:comNSdeg} in the sense of Definition \\ref{defcomNSdeg}. Assume that the density satisfies\n    \\begin{equation}\\label{condi1-th2}\n        0 < \\underline{\\rho} \\leq \\rho(t, x)\\leq \\bar{\\rho} < \\infty.\n    \\end{equation}\n     and the initial condition satisfies\n    \\begin{equation}\\label{condi2-th2}\n        \\sqrt{\\rho_0} u_0 \\in L^{\\frac{4N}{N+2}}(\\Omega).\n    \\end{equation}\n    If the velocity satisfies\n    \\begin{equation}\\label{con:shinbrot_cond2}\n        u \\in L^p\\left(0, T ; L^q(\\Omega)\\right) \\quad \\text{with} \\quad \\begin{cases}\n            \\frac{1}{p}+\\frac{3}{q} \\leq 1,\\quad\\mbox{if}  \\quad 3\\leq q < 4,\\\\\n            \\frac{2}{p}+\\frac{2}{q} \\leq 1,\\quad\\mbox{if}  \\quad 4\\leq q \\leq\\infty,\n        \\end{cases}\n    \\end{equation}\n    then the energy equality \\eqref{eq:energy_comNSdeg} holds for any $t \\in[0, T]$.\n\\end{theorem}", "con:Yu": "\\begin{equation}\\label{con:Yu}\n    u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\mathbb{T}^{N}))\\quad\\mbox{for}\\quad \\frac{1}{p}+\\frac{1}{q}\\leq \\frac{5}{12},\\quad q\\geq 6\n\\end{equation}", "con:shinbrot2": "\\begin{equation}\\label{con:shinbrot2}\n    \\frac{1}{p}+\\frac{3}{q}\\leq 1\n\\end{equation}", "eq:energy_comNSdeg": "\\begin{equation}\\label{eq:energy_comNSdeg}\n            \\int_{\\Omega}\\left(\\frac{1}{2} \\rho|u|^2+\\frac{\\rho^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x+\\int_0^t \\int_{\\Omega} \\rho|\\mathbb{D} u|^2 \\mathrm{d} x \\mathrm{d} s \\leq \\int_{\\Omega}\\left(\\frac{1}{2} \\rho_0\\left|u_0\\right|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x.\n        \\end{equation}", "theoremcomNS": "\\begin{theorem}\\label{theoremcomNS}\n    Let $(\\rho, u)$ be a weak solution of \\eqref{eq:comNS} in the sense of Definition \\ref{defcomNS}. Assume that the density satisfies\n    \\begin{equation}\\label{con:thcomNSrhoover0}\n         0 \\leq \\rho(t,x) \\leq \\bar{\\rho} < \\infty, \\quad \\text{and} \\quad \\nabla \\sqrt{\\rho} \\in L^{\\infty}\\left(0, T ; L^{\\frac{3}{2}}(\\Omega)\\right),\n    \\end{equation}\n    and the initial velocity satisfies\n    \\begin{equation}\\label{con:initial_1}\n        u_0 \\in L^{\\frac{6N}{6-N}}(\\Omega).\n    \\end{equation}\n    If the velocity satisfies\n    \\begin{equation}\\label{con:shinbrot_cond}\n        u \\in L^p\\left(0, T ; L^q(\\Omega)\\right) \\quad \\text{with} \\quad \\begin{cases}\n            \\frac{1}{p}+\\frac{3}{q} \\leq 1, \\quad\\mbox{if} \\quad 3\\leq q < 4,\\\\\n            \\frac{2}{p}+\\frac{2}{q} \\leq 1, \\quad\\mbox{if} \\quad 4\\leq q <\\infty,\n        \\end{cases}\n    \\end{equation}\n    then the energy equality \\eqref{eq:energy_comNS} holds for any $t \\in[0, T]$.\n\\end{theorem}", "commutator": "\\begin{lemma}\\label{commutator} \nLet $1 \\leq \\bar{p}, \\bar{q}, p_1, q_1, p_2, q_2 \\leq \\infty$, with $\\frac{1}{\\bar{p}}+\\frac{1}{p_1}+\\frac{1}{p_2}=1$ and $\\frac{1}{\\bar{q}}+\\frac{1}{q_1}+\\frac{1}{q_2}=1$. Let $f\\in L^{p_1}\\left(0, T ; L^{q_1}(\\Omega)\\right), \\partial_{t} f \\in L^{p_1}\\left(0, T ; W^{-1,q_1}(\\Omega)\\right), g \\in L^{p_2}\\left(0, T ; W^{1,q_2}(\\Omega)\\right), \\varphi \\in L^{\\bar{p}}\\left(0, T ; W_{0}^{1,\\bar{q}}(\\Omega)\\right)$. Then, there holds\n$$\n\\int_0^T \\int_{\\Omega}\\varphi\\left[\\partial_t\\left(f g\\right)_{\\varepsilon}-\\partial_t(f g_{\\varepsilon})\\right]\\mathrm{d} x \\mathrm{d} t \\rightarrow 0,\n$$\nas $\\varepsilon \\rightarrow 0$ if $p_2, q_2,\\bar{p},\\bar{q}<\\infty$.\n\\end{lemma}", "eq:comNS": "\\begin{equation}\\label{eq:comNS}\n    \\begin{split}\n        (\\rho u)_t + \\text{div}(\\rho u \\otimes u) - \\mu \\Delta u - (\\mu+\\lambda) \\nabla \\text{div} \\, u + \\nabla P &= 0,  \\\\\n        \\rho_t + \\text{div}(\\rho u) &= 0, \n    \\end{split}\n\\end{equation}", "defcomNS": "\\begin{definition}\\label{defcomNS}\n    For a given $T>0$, we call $(\\rho, u)$ a weak solution on $[0, T]$ to the constant viscosity system \\eqref{eq:comNS}\\eqref{con:initial} if:\n    \\begin{itemize}\n        \\item The momentum equations \\eqref{eq:comNS} hold in $\\mathcal{D}^{\\prime}((0, T) \\times \\Omega)$ with the regularity:\n        \\begin{equation}\\label{con:defcomNS}\n            \\rho^\\gamma, \\rho|u|^2 \\in L^{\\infty}\\left(0, T ; L^1(\\Omega)\\right), \\quad u \\in L^2\\left(0, T ; H_0^1(\\Omega)\\right);\n        \\end{equation}\n\n        \\item The initial conditions \\eqref{con:initial} hold in $\\mathcal{D}^{\\prime}(\\Omega)$;\n\n        \\item $(\\rho, u)$ is a renormalized solution of the continuity equation in the sense of \\cite{diperna1989ordinary};\n\n        \\item The energy inequality holds for almost every $t \\in [0,T]$:\n        $$\n        \\begin{aligned}\n            \\int_{\\Omega}\\left(\\frac{1}{2} \\rho|u|^2+\\frac{\\rho^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x & +\\int_0^t \\int_{\\Omega}\\left(\\mu|\\nabla u|^2+(\\mu+\\lambda)(\\operatorname{div} u)^2\\right) \\mathrm{d} x \\mathrm{d} s \\\\\n            & \\leq \\int_{\\Omega}\\left(\\frac{1}{2} \\rho_0\\left|u_0\\right|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x.\n        \\end{aligned}\n        $$\n    \\end{itemize}\n\\end{definition}", "eq:comNSdeg": "\\begin{equation}\\label{eq:comNSdeg}\n\t\t\\begin{split}\n\t\t\t(\\rho u)_t + \\text{div}(\\rho u \\otimes u) - 2\\nu \\text{div}(\\rho \\mathbb{D} u) + \\nabla P &= 0,  \\\\\n\t\t\t\\rho_t + \\text{div}(\\rho u) &= 0,\n\t\t\\end{split}\n\t\\end{equation}", "con:shinbrot": "\\begin{equation}\\label{con:shinbrot}\n    \\frac{2}{p}+\\frac{2}{q}\\leq 1,\\quad q\\geq 4,\n\\end{equation}"}, "pre_theorem_intro_text_len": 14239, "pre_theorem_intro_text": "The compressible Navier-Stokes equations are a fundamental model in fluid dynamics, describing the motion of a viscous compressible fluid. In this paper, we mainly study the energy conservation of the following system in a bounded domain $\\Omega \\subset \\mathbb{R}^N$ ($N = 2, 3$):\\begin{equation}\\label{eq:comNS}\n    \\begin{split}\n        (\\rho u)_t + \\text{div}(\\rho u \\otimes u) - \\mu \\Delta u - (\\mu+\\lambda) \\nabla \\text{div} \\, u + \\nabla P &= 0,  \\\\\n        \\rho_t + \\text{div}(\\rho u) &= 0, \n    \\end{split}\n\\end{equation}\nwith initial data\n\\begin{equation}\n    \\rho|_{t=0} = \\rho_0(x), \\quad \\rho u|_{t=0} = \\rho_0 u_0, \\label{con:initial}\n\\end{equation}\nwhere $P = \\rho^\\gamma$, $\\gamma > 1$, is the pressure, $\\rho$ is the density, $u$ is the velocity, and the viscosity coefficients satisfy $\\mu > 0$, $2\\mu + N\\lambda \\geq 0$. Here we define $u_0 = 0$ on the set $\\{x|\\rho_0(x) = 0\\}$. However, the weak solutions of these equations do not, in general, satisfy the energy equalities, i.e.\n\\begin{equation}\n    \\begin{aligned}\n    \t&\\int_{\\Omega}\\left(\\frac{1}{2} \\rho_0\\left|u_0\\right|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x-\\int_{\\Omega}\\left(\\frac{1}{2} \\rho|u|^2+\\frac{\\rho^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x\\\\\n    \t=&\\int_0^T \\int_{\\Omega}\\left( \\mu|\\nabla u|^2+(\\mu+\\lambda)|\\operatorname{div} u|^2 \\right)\\mathrm{d} x \\mathrm{d} t. \\label{eq:energy_comNS}\n    \\end{aligned}\n\\end{equation}\nThis phenomenon, along with incompressible cases, has drawn attention and sparked interest in exploring the relationship between energy conservation and the regularity of weak solutions \\cite{buckmaster2016dissipative,constantin1994onsager,feireisl2017regularity,isett2018proof}.\n\nIn fact, this subject originated from the famous Onsager’s conjecture for incompressible Euler flows \\cite{onsager1949statistical}. This conjecture states that 1/3 should be the threshold of Hölder continuous exponent for energy conservation. Consequently, \\cite{constantin1994onsager,eyink1994energy,cheskidov2008energy} obtained the energy equality with Besov regularity greater than 1/3 and \\cite{buckmaster2015anomalous,buckmaster2016dissipative,isett2018proof} constructed the non-conservation solutions with Hölder continuity smaller than 1/3. These significant advances almost reached the threshold.\n\nIn the context of the Navier–Stokes equations, the pioneering studies were done by Lions \\cite{lions1960regularite} and Serrin \\cite{serrininitial}. Lions proved that a weak solution $u$ to a incompressible fluids conserves its energy provided $u \\in L^{4}_{t,x}$ and Serrin gave a dimension-dependent condition\n\\begin{equation}\\label{con:serrin}\n    u \\in L^{p}_{t}L^{q}_{x}\\quad\\mbox{for}\\quad \\frac{2}{p}+\\frac{N}{q}\\leq 1,\\quad q > N.\n\\end{equation}\nLater, Shinbrot removed the dimensional dependence in \\cite{shinbrot1974energy} and proved the same conclusion if\n\\begin{equation}\\label{con:shinbrot}\n    \\frac{2}{p}+\\frac{2}{q}\\leq 1,\\quad q\\geq 4,\n\\end{equation}\nwhich improved the previous work. Meanwhile, for $3\\leq q<4$, the following criteria\n\\begin{equation}\\label{con:shinbrot2}\n    \\frac{1}{p}+\\frac{3}{q}\\leq 1\n\\end{equation}\ncould also yield energy conservation due to the embedding relationship \\cite{da2020shinbrot}. Furthermore, a range of new types of conditions have been obtained recently, see \\cite{drivas2019onsager,cheskidov2020energy,leslie2018conditions}.\n\nAs for the compressible systems, corresponding results are more recent and the analysis becomes inherently more complex. Within the framework of Onsager's theory \\cite{onsager1949statistical}, Drivas and Eyink \\cite{drivas2018onsager1,eyink2018cascades} adapted methods from the incompressible fluids \\cite{constantin1994onsager,duchon2000inertial,eyink1994energy} to the compressible Euler equations and obtain necessary conditions for the energy dissipative anomalies of turbulent solutions. Meanwhile, sufficient conditions for energy conservation in terms of Besov regularity for the compressible Euler fluids with a $C^{\\alpha}$ ($1<\\alpha\\leq2$) pressure law $p(\\rho)$ were later established by Feireisl et al. \\cite{feireisl2017regularity} and Akramov et al. \\cite{akramov2020energy}.\n\nRegarding the energy equality of compressible Navier-Stokes \\eqref{eq:comNS}, the pioneering breakthrough was made by Yu \\cite{yu2017energy}. He proved the equalities \\eqref{eq:energy_comNS} and \\eqref{eq:energy_comNSdeg} with the conditions that the weak solution $(\\rho,u)$ satisfies\n\\begin{equation}\\label{con:Yu}\n    u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\mathbb{T}^{N}))\\quad\\mbox{for}\\quad \\frac{1}{p}+\\frac{1}{q}\\leq \\frac{5}{12},\\quad q\\geq 6\n\\end{equation}\nand\n\\begin{equation}\\label{con:genhao rou}\n    \\sqrt{\\rho} \\in L^\\infty (0, T; H^1(\\mathbb{T}^{N})).\n\\end{equation}\nLater, Chen et al. \\cite{chen2020energy} treated the  boundary effect to investigate energy conservation on a more general bounded domain with no-slip boundary condition, when\n\\begin{equation}\\label{con:Chen et al.}\n    u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\Omega))\\quad\\mbox{for}\\quad p\\geq4,\\quad q\\geq 6,\n\\end{equation}\nwhich is just the end point of \\eqref{con:Yu}. Particularly, Ye et al. \\cite{ye2022energy} have reached $L^{4}L^{4}$ criteria for the compressible Navier-Stokes equations, while they provided that $\\nabla\\sqrt{\\rho}\\in L^{4}L^{4}$ instead of $L^{\\infty}L^{2}$. However, the $L^{\\infty}H^{1}$ regularity of $\\sqrt{\\rho}$ is of independent significance, as the Bresch-Desjardins entropy for \\eqref{eq:comNSdeg} implies such an estimate \\cite{vasseur2016existence}. In conclusion, a gap persists between these results and the Shinbrot criteria, due to the complexity of the compressible system.\n\n\\begin{figure}[htbp]\n  \\centering \n  \\begin{tikzpicture}\n    \\begin{axis}[\n        width=0.8\\textwidth, \n        height=0.6\\textwidth, \n        xlabel={$1/p$},\n        xlabel style={\n        at={(rel axis cs:0,1)},\n        anchor=west, \n        xshift=15pt, \n        yshift=-15pt \n    },\n        ylabel={$1/q$},\n        ylabel style={\n        at={(0,0.55)},\n        anchor=south, \n        xshift=10pt, \n        yshift=10pt, \n        rotate=0 \n    },\n        xmin=-0.05, xmax=0.75, ymin=-0.05, ymax=0.55,\n        axis lines=middle,\n        xtick={0},\n        ytick={0},\n        grid=major,\n        legend pos=north east,\n        clip=false,\n    ]\nll[gray!20] (axis cs: 0,0) -- (axis cs: 0.5,0) -- (axis cs: 0.25,0.25) -- (axis cs: 0,1/3) -- cycle;\nll[pattern=north east lines, pattern color=black!50] (axis cs: 5/12,0) -- (axis cs: 0.25,1/6) -- (axis cs: 0,1/6) -- (axis cs: 0,1/3) -- (axis cs: 0.25,0.25) -- (axis cs: 0.5,0) -- cycle;\nll[pattern=grid, pattern color=black!60] (axis cs: 5/12,0) -- (axis cs: 0.25,1/6) -- (axis cs: 0.25,0) -- cycle;\n\n      \\node[draw, circle, fill=black, inner sep=1.5pt, label=below right:{$L^2(L^\\infty)$}] at (axis cs: 0.5,0) {};\n      \\node[draw, circle, fill=black, inner sep=1.5pt, label=below left:{$L^4(L^6)$}] at (axis cs: 0.25,1/6) {};\n      \\node[draw, circle, fill=black, inner sep=1.5pt, label=above right:{$L^4(L^4)$}] at (axis cs: 0.25,0.25) {};\n      \\node[draw, circle, fill=black, inner sep=1.5pt, label=below:{$L^{\\frac{12}{5}}(L^\\infty)$}] at (axis cs: 5/12,0) {};\n      \\node[draw, circle, fill=black, inner sep=1.5pt, label=left:{$L^{\\infty}(L^3)$}] at (axis cs: 0,1/3) {}; \n      \\node[draw, circle, fill=black, inner sep=1.5pt, label=left:{$L^{\\infty}(L^\\frac{12}{5})$}] at (axis cs: 0,5/12) {};\n      \\node[draw, circle, fill=black, inner sep=1.5pt, label=left:{$L^{\\infty}(L^2)$}] at (axis cs: 0,0.5) {};\n\n      \\draw[thick] (axis cs: 0,1/3) -- (axis cs: 0.5,0);\n      \\draw[line width=2pt] (axis cs: 0,1/3) -- (axis cs: 0.25,0.25);\n      \\draw[thick] (axis cs: 0.25,1/6) -- (axis cs: 5/12,0);\n      \\draw[line width=2pt] (axis cs: 0.25,0.25) -- (axis cs: 0.5,0);\n      \\draw[dashed] (axis cs: 0.25,0.25) -- (axis cs: 0,0.5);\n      \\draw[dashed] (axis cs: 0.25,1/6) -- (axis cs: 0,5/12);\n      \\draw[dashed] (axis cs: 0,1/6) -- (axis cs: 0.25,1/6);\n      \\draw[dashed] (axis cs: 0.25,0) -- (axis cs: 0.25,1/6);\n      \\draw[-Stealth] (0.1,0.12) -- (1/6,2/9);\n      \\draw[-Stealth] (0.24,0.05) -- (1/3,1/12);\n      \\draw[-Stealth] (0.25,0.37) -- (1/7,2/7);\n\n      \\node[rotate=-45, font=\\normalsize] at (axis cs: 0.4, 0.15) {$2/p + 2/q = 1$};\n      \\node[rotate=0, font=\\small] at (axis cs: 0.07, 0.1) {$\\frac{2}{p} + \\frac{3}{q} = 1$};\n      \\node[rotate=0, font=\\small] at (axis cs: 0.16, 0.05) {$\\frac{1}{p} + \\frac{1}{q} = \\frac{5}{12}$};\n      \\node[rotate=0, font=\\small] at (axis cs: 0.27, 0.39) {$\\frac{1}{p} + \\frac{3}{q} = 1$};\n      \\node[rotate=-45, font=\\small] at (axis cs: 0.32, 0.135) {new};\n      \\node[rotate=0, font=\\small] at (axis cs: 0.1, 0.23) {new};\n      \\node[rotate=0, font=\\small] at (axis cs: 0.31, 0.05) {new};\n      \\draw[fill=white, line width=0.8pt] (axis cs: 0.5,0) circle (1.5pt);\n\n    \\end{axis}\n  \\end{tikzpicture}\n\n  \\caption{The shaded region illustrates the exponent ranges covered by Theorem \\ref{theoremcomNS}.The dashed and grid subregions represent new ranges of exponents not obtained before in the literature, where the grid area only corresponds to bounded domain with no-slip boundary and the dashed area corresponds to both bounded domain and torus.} \n  \\label{fig:embedding-diagram} \n\\end{figure}\n\nIn fact, temporal derivative term $\\partial_{t}(\\rho u)$ is the main challenges in filling this gap. It requests a spatiotemporal mollification of the equation, not a pure spatial mollification of the incompressible system. Then, new errors associated to temporal mollification have emerged. In the proofs of \\cite{chen2020energy,ye2022energy,yu2017energy}, the energy flux from $\\partial_{t}(\\rho u)$ was handled by a commutator estimate providing the integrability of $\\partial_{t}\\rho$. Particularly, it is the mass equation which allows us to 'transform' the spatial regularity of $u$ into the temporal regularity of $\\rho$. However, this strategy meets some restriction. One need to assume that the velocity $u$ complies with the Serrin type criteria at least, not the Lions or Shinbrot type criteria.\n\n\\subsection{Methodology}\n\nThe present paper aims to bridge the gap between known research (\\eqref{con:Yu} and \\eqref{con:Chen et al.}) and the Shinbrot type criteria (\\eqref{con:shinbrot} and \\eqref{con:shinbrot2}) on a periodic domain $\\mathbb{T}^{d}$ or a bounded domain with $C^{1}$ boundary. For this purpose, we shall reorganize the framework and introduce a new mollification and estimate. These approaches allow us to deal with the energy flux caused by the nonlinear terms $(\\rho u)_t$ under lower integrability conditions. In the following paragraphs, we briefly describe the ideas of our method.\n\n\\emph{Framework and mollification.} The proof framework developed in this paper consists of two main steps: first, establishing the local energy equality, and then extending it to a global version. In the first step, we derive the local energy equality using the specific test function $(\\varphi u_{\\varepsilon}^{\\varepsilon})_{\\varepsilon}^{\\varepsilon}$, introduced in Section 2, and passing to the limit as $\\varepsilon \\to 0$. By structuring the mollifier in this manner, we can separate errors present in different directions. In particular, we establish the vanishing of the temporal error via our weak-type temporal commutator estimate, which improves upon the classical treatment. Meanwhile, the other terms, such as $\\text{div}(\\rho u \\otimes u)$, are rigorously re-examined in this step to ensure convergence under the weaker assumptions. In the second step, the result is extended to the whole domain by setting $\\varphi=\\psi_{\\tau}\\phi_{\\delta}$ (with $\\varphi=\\psi_{\\tau}$ for the torus $\\mathbb{T}^{d}$) and letting $\\tau, \\delta \\to 0$. The spatial cut-off function $\\phi_{\\delta}$ is constructed by the approaches used in \\cite{chen2020energy} to control the regularity of the solution near the boundary. Particularly, the presence of a solid boundary complicates the dissipative mechanisms, as the behavior of the solution near the wall can differ significantly from its behavior in the interior. For further discussion on boundary effects in energy conservation contexts, we refer the reader to the related literature \\cite{bardos2019onsager,bardos2019extension,drivas2018onsager}.\n\n\\emph{Weak-type commutator estimate.} Our main challenge is to justify the convergence of the commutator\n$$\n\\partial_{t}(\\rho u^{\\varepsilon})_{\\varepsilon}-\\partial_{t}(\\rho u_{\\varepsilon}^{\\varepsilon})\n$$\nunder the Shinbrot-type condition on velocity. Compared with the previous work, these criteria entail a corresponding loss of regularity of $\\partial_{t}\\rho$, which is the key condition of the commutator estimate used in them. Since one usually employs the continuity equation:\n$$\n\\partial_{t}\\rho = - \\text{div}(\\rho u) = - (\\rho \\text{div} u + \\nabla\\rho\\cdot u),\n$$\nto obtain integrability of $\\partial_{t}\\rho$, it may be difficult to derive the necessary $L_{t}^{p}L_{x}^{q}$ estimate of $\\partial_{t}\\rho$ under the assumption of the present paper. To overcome this obstacle, we prove Lemma \\ref{commutator}, in which the convergence is established directly without establishing any $L^{p}$-type commutator estimate. Lemma \\ref{commutator}, together with the viscous properties of the fluid, allows us to show that the error term vanishes even when $\\partial_{t}\\rho$ is only a distribution, which is ensured by the mass conservation.\n\n\\subsection{Main results}\n\nOur results consist of the energy conservation criteria of two kinds of compressible Navier--Stokes systems: constant viscosities and degenerate viscosity. Here, we present our main results regarding the energy conservation for weak solutions to the equations with constant viscosities, whereas the remaining content, pertaining to degenerate viscosity cases, is outlined in the final section (Section \\ref{sec:deg}). Throughout the present paper, let $\\Omega$ be either the torus $\\mathbb{T}^{N}$ ($N=2,3$) or an open, bounded domain in $\\mathbb{R}^N$ with $C^1$ boundary $\\partial \\Omega$ with $u=0$ on $\\partial\\Omega$.\n\nWe first present the result for the Navier-Stokes equations \\eqref{eq:comNS} with constant viscosity, where the presence of vacuum is allowed.", "context": "In the context of the Navier–Stokes equations, the pioneering studies were done by Lions \\cite{lions1960regularite} and Serrin \\cite{serrininitial}. Lions proved that a weak solution $u$ to a incompressible fluids conserves its energy provided $u \\in L^{4}_{t,x}$ and Serrin gave a dimension-dependent condition\n\\begin{equation}\\label{con:serrin}\n u \\in L^{p}_{t}L^{q}_{x}\\quad\\mbox{for}\\quad \\frac{2}{p}+\\frac{N}{q}\\leq 1,\\quad q > N.\n\\end{equation}\nLater, Shinbrot removed the dimensional dependence in \\cite{shinbrot1974energy} and proved the same conclusion if\n\\begin{equation}\\label{con:shinbrot}\n \\frac{2}{p}+\\frac{2}{q}\\leq 1,\\quad q\\geq 4,\n\\end{equation}\nwhich improved the previous work. Meanwhile, for $3\\leq q<4$, the following criteria\n\\begin{equation}\\label{con:shinbrot2}\n \\frac{1}{p}+\\frac{3}{q}\\leq 1\n\\end{equation}\ncould also yield energy conservation due to the embedding relationship \\cite{da2020shinbrot}. Furthermore, a range of new types of conditions have been obtained recently, see \\cite{drivas2019onsager,cheskidov2020energy,leslie2018conditions}.\n\nRegarding the energy equality of compressible Navier-Stokes \\eqref{eq:comNS}, the pioneering breakthrough was made by Yu \\cite{yu2017energy}. He proved the equalities \\eqref{eq:energy_comNS} and \\eqref{eq:energy_comNSdeg} with the conditions that the weak solution $(\\rho,u)$ satisfies\n\\begin{equation}\\label{con:Yu}\n u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\mathbb{T}^{N}))\\quad\\mbox{for}\\quad \\frac{1}{p}+\\frac{1}{q}\\leq \\frac{5}{12},\\quad q\\geq 6\n\\end{equation}\nand\n\\begin{equation}\\label{con:genhao rou}\n \\sqrt{\\rho} \\in L^\\infty (0, T; H^1(\\mathbb{T}^{N})).\n\\end{equation}\nLater, Chen et al. \\cite{chen2020energy} treated the boundary effect to investigate energy conservation on a more general bounded domain with no-slip boundary condition, when\n\\begin{equation}\\label{con:Chen et al.}\n u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\Omega))\\quad\\mbox{for}\\quad p\\geq4,\\quad q\\geq 6,\n\\end{equation}\nwhich is just the end point of \\eqref{con:Yu}. Particularly, Ye et al. \\cite{ye2022energy} have reached $L^{4}L^{4}$ criteria for the compressible Navier-Stokes equations, while they provided that $\\nabla\\sqrt{\\rho}\\in L^{4}L^{4}$ instead of $L^{\\infty}L^{2}$. However, the $L^{\\infty}H^{1}$ regularity of $\\sqrt{\\rho}$ is of independent significance, as the Bresch-Desjardins entropy for \\eqref{eq:comNSdeg} implies such an estimate \\cite{vasseur2016existence}. In conclusion, a gap persists between these results and the Shinbrot criteria, due to the complexity of the compressible system.\n\n\\emph{Weak-type commutator estimate.} Our main challenge is to justify the convergence of the commutator\n$$\n\\partial_{t}(\\rho u^{\\varepsilon})_{\\varepsilon}-\\partial_{t}(\\rho u_{\\varepsilon}^{\\varepsilon})\n$$\nunder the Shinbrot-type condition on velocity. Compared with the previous work, these criteria entail a corresponding loss of regularity of $\\partial_{t}\\rho$, which is the key condition of the commutator estimate used in them. Since one usually employs the continuity equation:\n$$\n\\partial_{t}\\rho = - \\text{div}(\\rho u) = - (\\rho \\text{div} u + \\nabla\\rho\\cdot u),\n$$\nto obtain integrability of $\\partial_{t}\\rho$, it may be difficult to derive the necessary $L_{t}^{p}L_{x}^{q}$ estimate of $\\partial_{t}\\rho$ under the assumption of the present paper. To overcome this obstacle, we prove Lemma \\ref{commutator}, in which the convergence is established directly without establishing any $L^{p}$-type commutator estimate. Lemma \\ref{commutator}, together with the viscous properties of the fluid, allows us to show that the error term vanishes even when $\\partial_{t}\\rho$ is only a distribution, which is ensured by the mass conservation.\n\nOur results consist of the energy conservation criteria of two kinds of compressible Navier--Stokes systems: constant viscosities and degenerate viscosity. Here, we present our main results regarding the energy conservation for weak solutions to the equations with constant viscosities, whereas the remaining content, pertaining to degenerate viscosity cases, is outlined in the final section (Section \\ref{sec:deg}). Throughout the present paper, let $\\Omega$ be either the torus $\\mathbb{T}^{N}$ ($N=2,3$) or an open, bounded domain in $\\mathbb{R}^N$ with $C^1$ boundary $\\partial \\Omega$ with $u=0$ on $\\partial\\Omega$.\n\nWe first present the result for the Navier-Stokes equations \\eqref{eq:comNS} with constant viscosity, where the presence of vacuum is allowed.", "full_context": "In the context of the Navier–Stokes equations, the pioneering studies were done by Lions \\cite{lions1960regularite} and Serrin \\cite{serrininitial}. Lions proved that a weak solution $u$ to a incompressible fluids conserves its energy provided $u \\in L^{4}_{t,x}$ and Serrin gave a dimension-dependent condition\n\\begin{equation}\\label{con:serrin}\n u \\in L^{p}_{t}L^{q}_{x}\\quad\\mbox{for}\\quad \\frac{2}{p}+\\frac{N}{q}\\leq 1,\\quad q > N.\n\\end{equation}\nLater, Shinbrot removed the dimensional dependence in \\cite{shinbrot1974energy} and proved the same conclusion if\n\\begin{equation}\\label{con:shinbrot}\n \\frac{2}{p}+\\frac{2}{q}\\leq 1,\\quad q\\geq 4,\n\\end{equation}\nwhich improved the previous work. Meanwhile, for $3\\leq q<4$, the following criteria\n\\begin{equation}\\label{con:shinbrot2}\n \\frac{1}{p}+\\frac{3}{q}\\leq 1\n\\end{equation}\ncould also yield energy conservation due to the embedding relationship \\cite{da2020shinbrot}. Furthermore, a range of new types of conditions have been obtained recently, see \\cite{drivas2019onsager,cheskidov2020energy,leslie2018conditions}.\n\nRegarding the energy equality of compressible Navier-Stokes \\eqref{eq:comNS}, the pioneering breakthrough was made by Yu \\cite{yu2017energy}. He proved the equalities \\eqref{eq:energy_comNS} and \\eqref{eq:energy_comNSdeg} with the conditions that the weak solution $(\\rho,u)$ satisfies\n\\begin{equation}\\label{con:Yu}\n u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\mathbb{T}^{N}))\\quad\\mbox{for}\\quad \\frac{1}{p}+\\frac{1}{q}\\leq \\frac{5}{12},\\quad q\\geq 6\n\\end{equation}\nand\n\\begin{equation}\\label{con:genhao rou}\n \\sqrt{\\rho} \\in L^\\infty (0, T; H^1(\\mathbb{T}^{N})).\n\\end{equation}\nLater, Chen et al. \\cite{chen2020energy} treated the boundary effect to investigate energy conservation on a more general bounded domain with no-slip boundary condition, when\n\\begin{equation}\\label{con:Chen et al.}\n u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\Omega))\\quad\\mbox{for}\\quad p\\geq4,\\quad q\\geq 6,\n\\end{equation}\nwhich is just the end point of \\eqref{con:Yu}. Particularly, Ye et al. \\cite{ye2022energy} have reached $L^{4}L^{4}$ criteria for the compressible Navier-Stokes equations, while they provided that $\\nabla\\sqrt{\\rho}\\in L^{4}L^{4}$ instead of $L^{\\infty}L^{2}$. However, the $L^{\\infty}H^{1}$ regularity of $\\sqrt{\\rho}$ is of independent significance, as the Bresch-Desjardins entropy for \\eqref{eq:comNSdeg} implies such an estimate \\cite{vasseur2016existence}. In conclusion, a gap persists between these results and the Shinbrot criteria, due to the complexity of the compressible system.\n\n\\emph{Weak-type commutator estimate.} Our main challenge is to justify the convergence of the commutator\n$$\n\\partial_{t}(\\rho u^{\\varepsilon})_{\\varepsilon}-\\partial_{t}(\\rho u_{\\varepsilon}^{\\varepsilon})\n$$\nunder the Shinbrot-type condition on velocity. Compared with the previous work, these criteria entail a corresponding loss of regularity of $\\partial_{t}\\rho$, which is the key condition of the commutator estimate used in them. Since one usually employs the continuity equation:\n$$\n\\partial_{t}\\rho = - \\text{div}(\\rho u) = - (\\rho \\text{div} u + \\nabla\\rho\\cdot u),\n$$\nto obtain integrability of $\\partial_{t}\\rho$, it may be difficult to derive the necessary $L_{t}^{p}L_{x}^{q}$ estimate of $\\partial_{t}\\rho$ under the assumption of the present paper. To overcome this obstacle, we prove Lemma \\ref{commutator}, in which the convergence is established directly without establishing any $L^{p}$-type commutator estimate. Lemma \\ref{commutator}, together with the viscous properties of the fluid, allows us to show that the error term vanishes even when $\\partial_{t}\\rho$ is only a distribution, which is ensured by the mass conservation.\n\nOur results consist of the energy conservation criteria of two kinds of compressible Navier--Stokes systems: constant viscosities and degenerate viscosity. Here, we present our main results regarding the energy conservation for weak solutions to the equations with constant viscosities, whereas the remaining content, pertaining to degenerate viscosity cases, is outlined in the final section (Section \\ref{sec:deg}). Throughout the present paper, let $\\Omega$ be either the torus $\\mathbb{T}^{N}$ ($N=2,3$) or an open, bounded domain in $\\mathbb{R}^N$ with $C^1$ boundary $\\partial \\Omega$ with $u=0$ on $\\partial\\Omega$.\n\nWe first present the result for the Navier-Stokes equations \\eqref{eq:comNS} with constant viscosity, where the presence of vacuum is allowed.\n\nRegarding the energy equality of compressible Navier-Stokes \\eqref{eq:comNS}, the pioneering breakthrough was made by Yu \\cite{yu2017energy}. He proved the equalities \\eqref{eq:energy_comNS} and \\eqref{eq:energy_comNSdeg} with the conditions that the weak solution $(\\rho,u)$ satisfies\n\\begin{equation}\\label{con:Yu}\n u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\mathbb{T}^{N}))\\quad\\mbox{for}\\quad \\frac{1}{p}+\\frac{1}{q}\\leq \\frac{5}{12},\\quad q\\geq 6\n\\end{equation}\nand\n\\begin{equation}\\label{con:genhao rou}\n \\sqrt{\\rho} \\in L^\\infty (0, T; H^1(\\mathbb{T}^{N})).\n\\end{equation}\nLater, Chen et al. \\cite{chen2020energy} treated the boundary effect to investigate energy conservation on a more general bounded domain with no-slip boundary condition, when\n\\begin{equation}\\label{con:Chen et al.}\n u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\Omega))\\quad\\mbox{for}\\quad p\\geq4,\\quad q\\geq 6,\n\\end{equation}\nwhich is just the end point of \\eqref{con:Yu}. Particularly, Ye et al. \\cite{ye2022energy} have reached $L^{4}L^{4}$ criteria for the compressible Navier-Stokes equations, while they provided that $\\nabla\\sqrt{\\rho}\\in L^{4}L^{4}$ instead of $L^{\\infty}L^{2}$. However, the $L^{\\infty}H^{1}$ regularity of $\\sqrt{\\rho}$ is of independent significance, as the Bresch-Desjardins entropy for \\eqref{eq:comNSdeg} implies such an estimate \\cite{vasseur2016existence}. In conclusion, a gap persists between these results and the Shinbrot criteria, due to the complexity of the compressible system.\n\nWe first present the result for the Navier-Stokes equations \\eqref{eq:comNS} with constant viscosity, where the presence of vacuum is allowed.\n\nThe remainder of this paper is organized as follows. Section 2 introduces the necessary preliminaries, including definitions of weak solutions and key technical lemmas. Section 3 focuses on establishing the local energy conservation law in the sense of distributions using a mollification argument. In Section 4, we extend this local result to the global energy balance on the domain $\\Omega$, completing the proof of Theorem \\ref{theoremcomNS}. Finally, we apply our methods on the compressible Navier-Stokes equations with degenerate viscosity in the last Section 5.\n\nIt is easy to see, for any $\\alpha \\geq \\frac{1}{2}$,\n\\begin{equation*}\n \\partial_{t}(\\rho^{\\alpha}) = -\\alpha \\rho^{\\alpha} \\mathrm{div} u - 2\\alpha \\rho^{\\alpha - \\frac{1}{2}} u \\cdot \\nabla \\sqrt{\\rho},\n\\end{equation*}\nwhich, together with \\eqref{con:defcomNS} and \\eqref{condi1-th2}, implies\n\\begin{equation*}\n \\rho^{\\alpha} \\in L^{\\infty}(0, T; W^{1,\\frac{3}{2}}(\\Omega)), \\quad \\partial_{t}(\\rho^{\\alpha}) \\in L^{2}(0, T; L^{\\frac{12}{11}}(\\Omega)).\n\\end{equation*}\nIt follows that, by the Lemma \\ref{aubinlions},\n\\begin{equation*}\n \\rho^{\\alpha} \\in C([0, T]; L^{r}(\\Omega)), \\quad r < \\frac{3N}{2N-3},\n\\end{equation*}\nso we can use $u_{0} \\in L^{\\frac{6N}{6-N}}$ to deduce that\n\\begin{equation}\\label{continuity_sqrt_rho_u}\n (\\sqrt{\\rho }u) (t) \\rightarrow (\\sqrt{\\rho }u) (0) \\quad \\text{strongly in } L^2(\\Omega) \\text{ as } t \\to 0^+.\n\\end{equation}\nFollowing the similar manner of proof in \\cite{yu2017energy}, we can complete the proof of Theorem \\ref{theoremcomNS}.\n\n\\section{The energy conservation for the compressible Navier-Stokes equations with degenerate viscosity}\\label{sec:deg}\nNext, we consider the compressible Navier-Stokes equations with degenerate viscosity, where the density is strictly bounded away from vacuum. The systems are stated as follows:\n \\begin{equation}\\label{eq:comNSdeg}\n \\begin{split}\n (\\rho u)_t + \\text{div}(\\rho u \\otimes u) - 2\\nu \\text{div}(\\rho \\mathbb{D} u) + \\nabla P &= 0, \\\\\n \\rho_t + \\text{div}(\\rho u) &= 0,\n \\end{split}\n \\end{equation}\n where $\\mathbb{D} u = \\frac{1}{2}(\\nabla u + \\nabla^T u)$ is the strain tensor and the viscosity coefficients satisfy $\\nu > 0$. Then, we present the definition and the energy conservation criteria of the corresponding weak solutions.\n \\begin{definition}\\label{defcomNSdeg}\n The pair $(\\rho, u)$ is called a global weak solution to the degenerate viscosity system \\eqref{eq:comNSdeg} with initial data \\eqref{con:initial} if, for any $t \\in[0, T]$:\n \\begin{itemize}\n \\item The momentum equations \\eqref{eq:comNSdeg} hold in $\\mathcal{D}^{\\prime}((0, T) \\times \\Omega)$ satisfying:\n $$\n \\begin{gathered}\n \\rho \\geq 0, \\quad \\rho \\in L^{\\infty}\\left(0, T ; L^\\gamma(\\Omega)\\right),\\quad \\nabla \\sqrt{\\rho} \\in L^{\\infty}\\left(0, T ; L^\\frac{3}{2}(\\Omega)\\right)\\\\\n \\sqrt{\\rho} u \\in L^{\\infty}\\left(0, T ; L^2(\\Omega)\\right), \\quad \\sqrt{\\rho} \\nabla u \\in L^2\\left(0, T ; L^2(\\Omega)\\right);\n \\end{gathered}\n $$\n\n\\item The energy inequality holds for almost every $t \\in [0,T]$:\n \\begin{equation}\\label{eq:energy_comNSdeg}\n \\int_{\\Omega}\\left(\\frac{1}{2} \\rho|u|^2+\\frac{\\rho^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x+\\int_0^t \\int_{\\Omega} \\rho|\\mathbb{D} u|^2 \\mathrm{d} x \\mathrm{d} s \\leq \\int_{\\Omega}\\left(\\frac{1}{2} \\rho_0\\left|u_0\\right|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x.\n \\end{equation}\n \\end{itemize}\n\\end{definition}\n\\begin{remark}\n The condition $\\nabla\\sqrt{\\rho}\\in L^{\\infty}L^{\\frac{3}{2}}$ is reasonable in this context, as it is consistent with the known existence theory for weak solutions. Specifically, for the degenerate compressible Navier–Stokes equations, global existence results \\cite{vasseur2016existence} ensure that $\\nabla \\rho^{\\frac{\\gamma}{2}}\\in L^{2}L^{2}$ and $\\nabla\\sqrt{\\rho}\\in L^{\\infty}L^{2}$.\n\\end{remark}\n\n\\begin{theorem}\\label{theoremcomNSdeg}\n Let $(\\rho, u)$ be a weak solution of \\eqref{eq:comNSdeg} in the sense of Definition \\ref{defcomNSdeg}. Assume that the density satisfies\n \\begin{equation}\\label{condi1-th2}\n 0 < \\underline{\\rho} \\leq \\rho(t, x)\\leq \\bar{\\rho} < \\infty.\n \\end{equation}\n and the initial condition satisfies\n \\begin{equation}\\label{condi2-th2}\n \\sqrt{\\rho_0} u_0 \\in L^{\\frac{4N}{N+2}}(\\Omega).\n \\end{equation}\n If the velocity satisfies\n \\begin{equation}\\label{con:shinbrot_cond2}\n u \\in L^p\\left(0, T ; L^q(\\Omega)\\right) \\quad \\text{with} \\quad \\begin{cases}\n \\frac{1}{p}+\\frac{3}{q} \\leq 1,\\quad\\mbox{if} \\quad 3\\leq q < 4,\\\\\n \\frac{2}{p}+\\frac{2}{q} \\leq 1,\\quad\\mbox{if} \\quad 4\\leq q \\leq\\infty,\n \\end{cases}\n \\end{equation}\n then the energy equality \\eqref{eq:energy_comNSdeg} holds for any $t \\in[0, T]$.\n\\end{theorem}\n\\begin{remark}\n Different from Theorem \\ref{theoremcomNS}, we have obtained the energy conservation under the assumption $u\\in L^{2}L^{\\infty}$, which is one of the endpoints. \n\\end{remark}\nFinally, let us provide the proof of Theorem \\ref{theoremcomNSdeg}. We can modify the proof in Section 3 and 4 slightly to arrive the case $(p,q)=(4,4)$. Note that, for any weak solution ($\\rho, u$), condition \\eqref{condi1-th2} implies that", "post_theorem_intro_text_len": 1724, "post_theorem_intro_text": "\\begin{remark}\n    Our result improves the known regularity criteria in the literature and closes a notable gap. We relax the stronger Serrin-type integrability required in prior works (e.g., \\eqref{con:Yu} and \\eqref{con:Chen et al.}) via a novel weak-type temporal commutator estimate (Lemma \\ref{commutator}), which handles the nonlinear terms without demanding strong regularity of $\\partial_t\\rho$. The extended range of admissible exponents is visualized in {\\rm Figure 1}. Particularly, we provide that $\\nabla\\sqrt{\\rho}\\in L^{\\infty}L^{2}$ instead of $L^{\\infty}L^{2}$, which is also weaker than \\cite{yu2017energy,chen2020energy}.\n\\end{remark}\n\\begin{remark}\n     It is worth noting that, due to the non‑separability of the space $L^{2}(0,T;L^{\\infty}(\\Omega))$, our method does not currently apply to the endpoint $(p,q)=(2,\\infty)$; this remains for future investigation. \n\\end{remark}\n\\begin{remark}\n    The approaches employed to validate this theorem are equally applicable to the compressible Navier-Stokes equations with degenerate viscosity, a fact we will demonstrate by proving Theorem \\ref{theoremcomNSdeg} in Section 5.\n\\end{remark}\n\nThe remainder of this paper is organized as follows. Section 2 introduces the necessary preliminaries, including definitions of weak solutions and key technical lemmas. Section 3 focuses on establishing the local energy conservation law in the sense of distributions using a mollification argument. In Section 4, we extend this local result to the global energy balance on the domain $\\Omega$, completing the proof of Theorem \\ref{theoremcomNS}. Finally, we apply our methods on the compressible Navier-Stokes equations with degenerate viscosity in the last Section 5.", "sketch": "A proof outline is given after Theorem~\\ref{theoremcomNS}: Section~3 \"focuses on establishing the local energy conservation law in the sense of distributions using a mollification argument.\" Then, in Section~4, they \"extend this local result to the global energy balance on the domain $\\Omega$, completing the proof of Theorem~\\ref{theoremcomNS}.\" A key ingredient enabling the relaxed integrability assumptions is \"a novel weak-type temporal commutator estimate (Lemma \\ref{commutator}), which handles the nonlinear terms without demanding strong regularity of $\\partial_t\\rho$.\"", "expanded_sketch": "A proof outline is given after\n\\begin{theorem}\\label{theoremcomNS}\n    Let $(\\rho, u)$ be a weak solution of \\eqref{eq:comNS} in the sense of Definition \\ref{defcomNS}. Assume that the density satisfies\n    \\begin{equation}\\label{con:thcomNSrhoover0}\n         0 \\leq \\rho(t,x) \\leq \\bar{\\rho} < \\infty, \\quad \\text{and} \\quad \\nabla \\sqrt{\\rho} \\in L^{\\infty}\\left(0, T ; L^{\\frac{3}{2}}(\\Omega)\\right),\n    \\end{equation}\n    and the initial velocity satisfies\n    \\begin{equation}\\label{con:initial_1}\n        u_0 \\in L^{\\frac{6N}{6-N}}(\\Omega).\n    \\end{equation}\n    If the velocity satisfies\n    \\begin{equation}\\label{con:shinbrot_cond}\n        u \\in L^p\\left(0, T ; L^q(\\Omega)\\right) \\quad \\text{with} \\quad \\begin{cases}\n            \\frac{1}{p}+\\frac{3}{q} \\leq 1, \\quad\\mbox{if} \\quad 3\\leq q < 4,\\\\\n            \\frac{2}{p}+\\frac{2}{q} \\leq 1, \\quad\\mbox{if} \\quad 4\\leq q <\\infty,\n        \\end{cases}\n    \\end{equation}\n    then the energy equality \\eqref{eq:energy_comNS} holds for any $t \\in[0, T]$.\n\\end{theorem}\nNext, they focus on establishing the local energy conservation law in the sense of distributions using a mollification argument. Then, they extend this local result to the global energy balance on the domain $\\Omega$, thereby completing the proof of the main theorem. A key ingredient enabling the relaxed integrability assumptions is the following commutator estimate.\n\n\\begin{lemma}\\label{commutator} \nLet $1 \\leq \\bar{p}, \\bar{q}, p_1, q_1, p_2, q_2 \\leq \\infty$, with $\\frac{1}{\\bar{p}}+\\frac{1}{p_1}+\\frac{1}{p_2}=1$ and $\\frac{1}{\\bar{q}}+\\frac{1}{q_1}+\\frac{1}{q_2}=1$. Let $f\\in L^{p_1}\\left(0, T ; L^{q_1}(\\Omega)\\right), \\partial_{t} f \\in L^{p_1}\\left(0, T ; W^{-1,q_1}(\\Omega)\\right), g \\in L^{p_2}\\left(0, T ; W^{1,q_2}(\\Omega)\\right), \\varphi \\in L^{\\bar{p}}\\left(0, T ; W_{0}^{1,\\bar{q}}(\\Omega)\\right)$. Then, there holds\n$$\n\\int_0^T \\int_{\\Omega}\\varphi\\left[\\partial_t\\left(f g\\right)_{\\varepsilon}-\\partial_t(f g_{\\varepsilon})\\right]\\mathrm{d} x \\mathrm{d} t \\rightarrow 0,\n$$\nas $\\varepsilon \\rightarrow 0$ if $p_2, q_2,\\bar{p},\\bar{q}<\\infty$.\n\\end{lemma}\nThis lemma handles the nonlinear terms without demanding strong regularity of $\\partial_t\\rho$.", "expanded_theorem": "\\label{theoremcomNS}\n    Let $(\\rho, u)$ be a weak solution of \\begin{equation}\\label{eq:comNS}\n    \\begin{split}\n        (\\rho u)_t + \\text{div}(\\rho u \\otimes u) - \\mu \\Delta u - (\\mu+\\lambda) \\nabla \\text{div} \\, u + \\nabla P &= 0,  \\\\\n        \\rho_t + \\text{div}(\\rho u) &= 0, \n    \\end{split}\n\\end{equation} in the sense of the following definition. \\begin{definition}\\label{defcomNS}\n    For a given $T>0$, we call $(\\rho, u)$ a weak solution on $[0, T]$ to the constant viscosity system \\eqref{eq:comNS}\\eqref{con:initial} if:\n    \\begin{itemize}\n        \\item The momentum equations \\eqref{eq:comNS} hold in $\\mathcal{D}^{\\prime}((0, T) \\times \\Omega)$ with the regularity:\n        \\begin{equation}\\label{con:defcomNS}\n            \\rho^\\gamma, \\rho|u|^2 \\in L^{\\infty}\\left(0, T ; L^1(\\Omega)\\right), \\quad u \\in L^2\\left(0, T ; H_0^1(\\Omega)\\right);\n        \\end{equation}\n\n        \\item The initial conditions \\eqref{con:initial} hold in $\\mathcal{D}^{\\prime}(\\Omega)$;\n\n        \\item $(\\rho, u)$ is a renormalized solution of the continuity equation in the sense of DiPerna--Lions, ``Ordinary differential equations, transport theory and Sobolev spaces'' (1989);\n\n        \\item The energy inequality holds for almost every $t \\in [0,T]$:\n        $$\n        \\begin{aligned}\n            \\int_{\\Omega}\\left(\\frac{1}{2} \\rho|u|^2+\\frac{\\rho^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x & +\\int_0^t \\int_{\\Omega}\\left(\\mu|\\nabla u|^2+(\\mu+\\lambda)(\\operatorname{div} u)^2\\right) \\mathrm{d} x \\mathrm{d} s \\\\\n            & \\leq \\int_{\\Omega}\\left(\\frac{1}{2} \\rho_0\\left|u_0\\right|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x.\n        \\end{aligned}\n        $$\n    \\end{itemize}\n\\end{definition}\n    Assume that the density satisfies\n    \\begin{equation}\\label{con:thcomNSrhoover0}\n         0 \\leq \\rho(t,x) \\leq \\bar{\\rho} < \\infty, \\quad \\text{and} \\quad \\nabla \\sqrt{\\rho} \\in L^{\\infty}\\left(0, T ; L^{\\frac{3}{2}}(\\Omega)\\right),\n    \\end{equation}\n    and the initial velocity satisfies\n    \\begin{equation}\\label{con:initial_1}\n        u_0 \\in L^{\\frac{6N}{6-N}}(\\Omega).\n    \\end{equation}\n    If the velocity satisfies\n    \\begin{equation}\\label{con:shinbrot_cond}\n        u \\in L^p\\left(0, T ; L^q(\\Omega)\\right) \\quad \\text{with} \\quad \\begin{cases}\n            \\frac{1}{p}+\\frac{3}{q} \\leq 1, \\quad\\mbox{if} \\quad 3\\leq q < 4,\\\\\n            \\frac{2}{p}+\\frac{2}{q} \\leq 1, \\quad\\mbox{if} \\quad 4\\leq q <\\infty,\n        \\end{cases}\n    \\end{equation}\n    then the energy equality\n    \\begin{aligned}\n    \t&\\int_{\\Omega}\\left(\\frac{1}{2} \\rho_0\\left|u_0\\right|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x-\\int_{\\Omega}\\left(\\frac{1}{2} \\rho|u|^2+\\frac{\\rho^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x\\\\\n    \t=&\\int_0^T \\int_{\\Omega}\\left( \\mu|\\nabla u|^2+(\\mu+\\lambda)|\\operatorname{div} u|^2 \\right)\\mathrm{d} x \\mathrm{d} t. \\label{eq:energy_comNS}\n    \\end{aligned} holds for any $t \\in[0, T]$.", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let \\(\\Omega\\) be either the torus \\(\\mathbb T^N\\) with \\(N=2,3\\), or a bounded open \\(C^1\\) domain in \\(\\mathbb R^N\\) (with no-slip boundary condition when \\(\\partial\\Omega\\neq\\varnothing\\)). Consider a weak solution \\((\\rho,u)\\) on \\([0,T]\\) of the constant-viscosity compressible Navier--Stokes system\n\\[\n(\\rho u)_t+\\operatorname{div}(\\rho u\\otimes u)-\\mu\\Delta u-(\\mu+\\lambda)\\nabla\\operatorname{div}u+\\nabla P=0,\n\\qquad\n\\rho_t+\\operatorname{div}(\\rho u)=0,\n\\]\nwith initial data \\((\\rho_0,u_0)\\), in the following sense: the momentum equation holds in \\(\\mathcal D'((0,T)\\times\\Omega)\\);\n\\(\\rho^\\gamma,\\rho|u|^2\\in L^\\infty(0,T;L^1(\\Omega))\\); \\(u\\in L^2(0,T;H_0^1(\\Omega))\\); the initial conditions hold in \\(\\mathcal D'(\\Omega)\\); \\((\\rho,u)\\) is a renormalized solution of the continuity equation in the DiPerna--Lions sense; and for almost every \\(t\\in[0,T]\\),\n\\[\n\\int_{\\Omega}\\left(\\tfrac12\\rho|u|^2+\\frac{\\rho^\\gamma}{\\gamma-1}\\right)dx\n+\\int_0^t\\!\\int_{\\Omega}\\bigl(\\mu|\\nabla u|^2+(\\mu+\\lambda)(\\operatorname{div}u)^2\\bigr)\\,dx\\,ds\n\\le\n\\int_{\\Omega}\\left(\\tfrac12\\rho_0|u_0|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right)dx.\n\\]\nAssume in addition that\n\\[\n0\\le \\rho(t,x)\\le \\bar\\rho<\\infty,\n\\qquad\n\\nabla\\sqrt\\rho\\in L^\\infty(0,T;L^{3/2}(\\Omega)),\n\\qquad\nu_0\\in L^{\\frac{6N}{6-N}}(\\Omega),\n\\]\nand that\n\\[\nu\\in L^p(0,T;L^q(\\Omega))\n\\quad\\text{with}\\quad\n\\begin{cases}\n\\frac1p+\\frac3q\\le 1, & 3\\le q<4,\\\\[2mm]\n\\frac2p+\\frac2q\\le 1, & 4\\le q<\\infty.\n\\end{cases}\n\\]\nWhich statement holds for every such weak solution?", "correct_choice": {"label": "A", "text": "For every \\(t\\in[0,T]\\), the energy inequality becomes an equality:\n\\[\n\\int_{\\Omega}\\left(\\tfrac12\\rho_0|u_0|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right)dx\n-\n\\int_{\\Omega}\\left(\\tfrac12\\rho(t)|u(t)|^2+\\frac{\\rho(t)^\\gamma}{\\gamma-1}\\right)dx\n=\n\\int_0^t\\!\\int_{\\Omega}\\bigl(\\mu|\\nabla u|^2+(\\mu+\\lambda)|\\operatorname{div}u|^2\\bigr)\\,dx\\,ds.\n\\]"}, "choices": [{"label": "B", "text": "For every \\(t\\in[0,T]\\), the energy inequality becomes an equality:\n\\[\n\\int_{\\Omega}\\left(\\tfrac12\\rho_0|u_0|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right)dx\n-\n\\int_{\\Omega}\\left(\\tfrac12\\rho(t)|u(t)|^2+\\frac{\\rho(t)^\\gamma}{\\gamma-1}\\right)dx\n=\n\\int_0^t\\!\\int_{\\Omega}\\bigl(\\mu|\\nabla u|^2+(\\mu+\\lambda)|\\operatorname{div}u|^2\\bigr)\\,dx\\,ds,\n\\]\nprovided the velocity satisfies\n\\[\nu\\in L^p(0,T;L^q(\\Omega))\n\\quad\\text{with}\\quad\n\\begin{cases}\n\\frac1p+\\frac3q\\le 1, & 3\\le q\\le 4,\\\\[2mm]\n\\frac2p+\\frac2q\\le 1, & 4< q<\\infty.\n\\end{cases}\n\\]"}, {"label": "C", "text": "The weak solution satisfies the original energy inequality for almost every \\(t\\in[0,T]\\):\n\\[\n\\int_{\\Omega}\\left(\\tfrac12\\rho(t)|u(t)|^2+\\frac{\\rho(t)^\\gamma}{\\gamma-1}\\right)dx\n+\n\\int_0^t\\!\\int_{\\Omega}\\bigl(\\mu|\\nabla u|^2+(\\mu+\\lambda)(\\operatorname{div}u)^2\\bigr)\\,dx\\,ds\n\\le\n\\int_{\\Omega}\\left(\\tfrac12\\rho_0|u_0|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right)dx.\n\\]"}, {"label": "D", "text": "For every \\(t\\in[0,T]\\), the energy inequality becomes an equality under the same assumptions, and in fact the conclusion remains valid if the Shinbrot-type condition is replaced by the endpoint-extended condition\n\\[\nu\\in L^p(0,T;L^q(\\Omega))\n\\quad\\text{with}\\quad\n\\begin{cases}\n\\frac1p+\\frac3q\\le 1, & 3\\le q<4,\\\\[2mm]\n\\frac2p+\\frac2q\\le 1, & 4\\le q\\le \\infty.\n\\end{cases}\n\\]"}, {"label": "E", "text": "For every such weak solution, the same energy equality holds for every \\(t\\in[0,T]\\) even if one drops the assumption that \\((\\rho,u)\\) is a renormalized solution of the continuity equation in the DiPerna--Lions sense, retaining only that the continuity equation holds in \\(\\mathcal D'((0,T)\\times\\Omega)\\) together with\n\\[\n0\\le \\rho\\le \\bar\\rho,\n\\qquad\n\\nabla\\sqrt\\rho\\in L^\\infty(0,T;L^{3/2}(\\Omega)),\n\\qquad\nu\\in L^p(0,T;L^q(\\Omega))\n\\]\nand the stated Shinbrot-type conditions on \\((p,q)\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "case_split", "tampered_component": "piecewise q-range boundary at q=4", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "upgraded equality for every t", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "finite-exponent requirement excluding q=\\infty in the commutator/local energy argument", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "renormalized continuity equation needed to pass from local mollified identities to global energy balance", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the conclusion, and the correct option is not lexically signaled. It presents hypotheses only, so there is no direct answer leakage."}, "TAS": {"score": 1, "justification": "This is very close to a theorem-statement recall item: the stem lists the full hypotheses and asks for the theorem's conclusion. The subtle alternative formulations prevent it from being a pure tautology, but it is still a near-restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish 'for every t' from 'for a.e. t' and to detect endpoint/assumption tampering. However, the item mainly tests exact recall of a known result rather than substantial generative mathematical reasoning."}, "DQS": {"score": 1, "justification": "Several distractors are plausible theorem distortions (endpoint shift, weakened time conclusion, altered density regularity). But the set is weakened by logical ambiguity: option C is a weaker true statement implied by A, and option D is also potentially read as a weaker true statement under the given assumptions."}, "total_score": 5, "overall_assessment": "A technically sophisticated but theorem-recall-heavy MCQ. It avoids direct leakage, but it is close to restating the source result, and the distractor set is only moderately strong because at least one distractor is weaker-true or logically ambiguous."}}
{"id": "2602.13696v1", "paper_link": "http://arxiv.org/abs/2602.13696v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{theoremcomNS}\n    Let $(\\rho, u)$ be a weak solution of \\eqref{eq:comNS} in the sense of Definition \\ref{defcomNS}. Assume that the density satisfies\n    \\begin{equation}\\label{con:thcomNSrhoover0}\n         0 \\leq \\rho(t,x) \\leq \\bar{\\rho} < \\infty, \\quad \\text{and} \\quad \\nabla \\sqrt{\\rho} \\in L^{\\infty}\\left(0, T ; L^{\\frac{3}{2}}(\\Omega)\\right),\n    \\end{equation}\n    and the initial velocity satisfies\n    \\begin{equation}\\label{con:initial_1}\n        u_0 \\in L^{\\frac{6N}{6-N}}(\\Omega).\n    \\end{equation}\n    If the velocity satisfies\n    \\begin{equation}\\label{con:shinbrot_cond}\n        u \\in L^p\\left(0, T ; L^q(\\Omega)\\right) \\quad \\text{with} \\quad \\begin{cases}\n            \\frac{1}{p}+\\frac{3}{q} \\leq 1, \\quad\\mbox{if} \\quad 3\\leq q < 4,\\\\\n            \\frac{2}{p}+\\frac{2}{q} \\leq 1, \\quad\\mbox{if} \\quad 4\\leq q <\\infty,\n        \\end{cases}\n    \\end{equation}\n    then the energy equality \\eqref{eq:energy_comNS} holds for any $t \\in[0, T]$.", "start_pos": 16737, "end_pos": 17732, "label": "theoremcomNS"}, "ref_dict": {"eq:energy_comNS": "\\begin{aligned}\n    \t&\\int_{\\Omega}\\left(\\frac{1}{2} \\rho_0\\left|u_0\\right|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x-\\int_{\\Omega}\\left(\\frac{1}{2} \\rho|u|^2+\\frac{\\rho^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x\\\\\n    \t=&\\int_0^T \\int_{\\Omega}\\left( \\mu|\\nabla u|^2+(\\mu+\\lambda)|\\operatorname{div} u|^2 \\right)\\mathrm{d} x \\mathrm{d} t. \\label{eq:energy_comNS}\n    \\end{aligned}", "con:Chen et al.": "\\begin{equation}\\label{con:Chen et al.}\n    u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\Omega))\\quad\\mbox{for}\\quad p\\geq4,\\quad q\\geq 6,\n\\end{equation}", "theoremcomNSdeg": "\\begin{theorem}\\label{theoremcomNSdeg}\n    Let $(\\rho, u)$ be a weak solution of \\eqref{eq:comNSdeg} in the sense of Definition \\ref{defcomNSdeg}. Assume that the density satisfies\n    \\begin{equation}\\label{condi1-th2}\n        0 < \\underline{\\rho} \\leq \\rho(t, x)\\leq \\bar{\\rho} < \\infty.\n    \\end{equation}\n     and the initial condition satisfies\n    \\begin{equation}\\label{condi2-th2}\n        \\sqrt{\\rho_0} u_0 \\in L^{\\frac{4N}{N+2}}(\\Omega).\n    \\end{equation}\n    If the velocity satisfies\n    \\begin{equation}\\label{con:shinbrot_cond2}\n        u \\in L^p\\left(0, T ; L^q(\\Omega)\\right) \\quad \\text{with} \\quad \\begin{cases}\n            \\frac{1}{p}+\\frac{3}{q} \\leq 1,\\quad\\mbox{if}  \\quad 3\\leq q < 4,\\\\\n            \\frac{2}{p}+\\frac{2}{q} \\leq 1,\\quad\\mbox{if}  \\quad 4\\leq q \\leq\\infty,\n        \\end{cases}\n    \\end{equation}\n    then the energy equality \\eqref{eq:energy_comNSdeg} holds for any $t \\in[0, T]$.\n\\end{theorem}", "con:Yu": "\\begin{equation}\\label{con:Yu}\n    u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\mathbb{T}^{N}))\\quad\\mbox{for}\\quad \\frac{1}{p}+\\frac{1}{q}\\leq \\frac{5}{12},\\quad q\\geq 6\n\\end{equation}", "con:shinbrot2": "\\begin{equation}\\label{con:shinbrot2}\n    \\frac{1}{p}+\\frac{3}{q}\\leq 1\n\\end{equation}", "eq:energy_comNSdeg": "\\begin{equation}\\label{eq:energy_comNSdeg}\n            \\int_{\\Omega}\\left(\\frac{1}{2} \\rho|u|^2+\\frac{\\rho^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x+\\int_0^t \\int_{\\Omega} \\rho|\\mathbb{D} u|^2 \\mathrm{d} x \\mathrm{d} s \\leq \\int_{\\Omega}\\left(\\frac{1}{2} \\rho_0\\left|u_0\\right|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x.\n        \\end{equation}", "theoremcomNS": "\\begin{theorem}\\label{theoremcomNS}\n    Let $(\\rho, u)$ be a weak solution of \\eqref{eq:comNS} in the sense of Definition \\ref{defcomNS}. Assume that the density satisfies\n    \\begin{equation}\\label{con:thcomNSrhoover0}\n         0 \\leq \\rho(t,x) \\leq \\bar{\\rho} < \\infty, \\quad \\text{and} \\quad \\nabla \\sqrt{\\rho} \\in L^{\\infty}\\left(0, T ; L^{\\frac{3}{2}}(\\Omega)\\right),\n    \\end{equation}\n    and the initial velocity satisfies\n    \\begin{equation}\\label{con:initial_1}\n        u_0 \\in L^{\\frac{6N}{6-N}}(\\Omega).\n    \\end{equation}\n    If the velocity satisfies\n    \\begin{equation}\\label{con:shinbrot_cond}\n        u \\in L^p\\left(0, T ; L^q(\\Omega)\\right) \\quad \\text{with} \\quad \\begin{cases}\n            \\frac{1}{p}+\\frac{3}{q} \\leq 1, \\quad\\mbox{if} \\quad 3\\leq q < 4,\\\\\n            \\frac{2}{p}+\\frac{2}{q} \\leq 1, \\quad\\mbox{if} \\quad 4\\leq q <\\infty,\n        \\end{cases}\n    \\end{equation}\n    then the energy equality \\eqref{eq:energy_comNS} holds for any $t \\in[0, T]$.\n\\end{theorem}", "commutator": "\\begin{lemma}\\label{commutator} \nLet $1 \\leq \\bar{p}, \\bar{q}, p_1, q_1, p_2, q_2 \\leq \\infty$, with $\\frac{1}{\\bar{p}}+\\frac{1}{p_1}+\\frac{1}{p_2}=1$ and $\\frac{1}{\\bar{q}}+\\frac{1}{q_1}+\\frac{1}{q_2}=1$. Let $f\\in L^{p_1}\\left(0, T ; L^{q_1}(\\Omega)\\right), \\partial_{t} f \\in L^{p_1}\\left(0, T ; W^{-1,q_1}(\\Omega)\\right), g \\in L^{p_2}\\left(0, T ; W^{1,q_2}(\\Omega)\\right), \\varphi \\in L^{\\bar{p}}\\left(0, T ; W_{0}^{1,\\bar{q}}(\\Omega)\\right)$. Then, there holds\n$$\n\\int_0^T \\int_{\\Omega}\\varphi\\left[\\partial_t\\left(f g\\right)_{\\varepsilon}-\\partial_t(f g_{\\varepsilon})\\right]\\mathrm{d} x \\mathrm{d} t \\rightarrow 0,\n$$\nas $\\varepsilon \\rightarrow 0$ if $p_2, q_2,\\bar{p},\\bar{q}<\\infty$.\n\\end{lemma}", "eq:comNS": "\\begin{equation}\\label{eq:comNS}\n    \\begin{split}\n        (\\rho u)_t + \\text{div}(\\rho u \\otimes u) - \\mu \\Delta u - (\\mu+\\lambda) \\nabla \\text{div} \\, u + \\nabla P &= 0,  \\\\\n        \\rho_t + \\text{div}(\\rho u) &= 0, \n    \\end{split}\n\\end{equation}", "defcomNS": "\\begin{definition}\\label{defcomNS}\n    For a given $T>0$, we call $(\\rho, u)$ a weak solution on $[0, T]$ to the constant viscosity system \\eqref{eq:comNS}\\eqref{con:initial} if:\n    \\begin{itemize}\n        \\item The momentum equations \\eqref{eq:comNS} hold in $\\mathcal{D}^{\\prime}((0, T) \\times \\Omega)$ with the regularity:\n        \\begin{equation}\\label{con:defcomNS}\n            \\rho^\\gamma, \\rho|u|^2 \\in L^{\\infty}\\left(0, T ; L^1(\\Omega)\\right), \\quad u \\in L^2\\left(0, T ; H_0^1(\\Omega)\\right);\n        \\end{equation}\n\n        \\item The initial conditions \\eqref{con:initial} hold in $\\mathcal{D}^{\\prime}(\\Omega)$;\n\n        \\item $(\\rho, u)$ is a renormalized solution of the continuity equation in the sense of \\cite{diperna1989ordinary};\n\n        \\item The energy inequality holds for almost every $t \\in [0,T]$:\n        $$\n        \\begin{aligned}\n            \\int_{\\Omega}\\left(\\frac{1}{2} \\rho|u|^2+\\frac{\\rho^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x & +\\int_0^t \\int_{\\Omega}\\left(\\mu|\\nabla u|^2+(\\mu+\\lambda)(\\operatorname{div} u)^2\\right) \\mathrm{d} x \\mathrm{d} s \\\\\n            & \\leq \\int_{\\Omega}\\left(\\frac{1}{2} \\rho_0\\left|u_0\\right|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x.\n        \\end{aligned}\n        $$\n    \\end{itemize}\n\\end{definition}", "eq:comNSdeg": "\\begin{equation}\\label{eq:comNSdeg}\n\t\t\\begin{split}\n\t\t\t(\\rho u)_t + \\text{div}(\\rho u \\otimes u) - 2\\nu \\text{div}(\\rho \\mathbb{D} u) + \\nabla P &= 0,  \\\\\n\t\t\t\\rho_t + \\text{div}(\\rho u) &= 0,\n\t\t\\end{split}\n\t\\end{equation}", "con:shinbrot": "\\begin{equation}\\label{con:shinbrot}\n    \\frac{2}{p}+\\frac{2}{q}\\leq 1,\\quad q\\geq 4,\n\\end{equation}"}, "pre_theorem_intro_text_len": 14239, "pre_theorem_intro_text": "The compressible Navier-Stokes equations are a fundamental model in fluid dynamics, describing the motion of a viscous compressible fluid. In this paper, we mainly study the energy conservation of the following system in a bounded domain $\\Omega \\subset \\mathbb{R}^N$ ($N = 2, 3$):\\begin{equation}\\label{eq:comNS}\n    \\begin{split}\n        (\\rho u)_t + \\text{div}(\\rho u \\otimes u) - \\mu \\Delta u - (\\mu+\\lambda) \\nabla \\text{div} \\, u + \\nabla P &= 0,  \\\\\n        \\rho_t + \\text{div}(\\rho u) &= 0, \n    \\end{split}\n\\end{equation}\nwith initial data\n\\begin{equation}\n    \\rho|_{t=0} = \\rho_0(x), \\quad \\rho u|_{t=0} = \\rho_0 u_0, \\label{con:initial}\n\\end{equation}\nwhere $P = \\rho^\\gamma$, $\\gamma > 1$, is the pressure, $\\rho$ is the density, $u$ is the velocity, and the viscosity coefficients satisfy $\\mu > 0$, $2\\mu + N\\lambda \\geq 0$. Here we define $u_0 = 0$ on the set $\\{x|\\rho_0(x) = 0\\}$. However, the weak solutions of these equations do not, in general, satisfy the energy equalities, i.e.\n\\begin{equation}\n    \\begin{aligned}\n    \t&\\int_{\\Omega}\\left(\\frac{1}{2} \\rho_0\\left|u_0\\right|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x-\\int_{\\Omega}\\left(\\frac{1}{2} \\rho|u|^2+\\frac{\\rho^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x\\\\\n    \t=&\\int_0^T \\int_{\\Omega}\\left( \\mu|\\nabla u|^2+(\\mu+\\lambda)|\\operatorname{div} u|^2 \\right)\\mathrm{d} x \\mathrm{d} t. \\label{eq:energy_comNS}\n    \\end{aligned}\n\\end{equation}\nThis phenomenon, along with incompressible cases, has drawn attention and sparked interest in exploring the relationship between energy conservation and the regularity of weak solutions \\cite{buckmaster2016dissipative,constantin1994onsager,feireisl2017regularity,isett2018proof}.\n\nIn fact, this subject originated from the famous Onsager’s conjecture for incompressible Euler flows \\cite{onsager1949statistical}. This conjecture states that 1/3 should be the threshold of Hölder continuous exponent for energy conservation. Consequently, \\cite{constantin1994onsager,eyink1994energy,cheskidov2008energy} obtained the energy equality with Besov regularity greater than 1/3 and \\cite{buckmaster2015anomalous,buckmaster2016dissipative,isett2018proof} constructed the non-conservation solutions with Hölder continuity smaller than 1/3. These significant advances almost reached the threshold.\n\nIn the context of the Navier–Stokes equations, the pioneering studies were done by Lions \\cite{lions1960regularite} and Serrin \\cite{serrininitial}. Lions proved that a weak solution $u$ to a incompressible fluids conserves its energy provided $u \\in L^{4}_{t,x}$ and Serrin gave a dimension-dependent condition\n\\begin{equation}\\label{con:serrin}\n    u \\in L^{p}_{t}L^{q}_{x}\\quad\\mbox{for}\\quad \\frac{2}{p}+\\frac{N}{q}\\leq 1,\\quad q > N.\n\\end{equation}\nLater, Shinbrot removed the dimensional dependence in \\cite{shinbrot1974energy} and proved the same conclusion if\n\\begin{equation}\\label{con:shinbrot}\n    \\frac{2}{p}+\\frac{2}{q}\\leq 1,\\quad q\\geq 4,\n\\end{equation}\nwhich improved the previous work. Meanwhile, for $3\\leq q<4$, the following criteria\n\\begin{equation}\\label{con:shinbrot2}\n    \\frac{1}{p}+\\frac{3}{q}\\leq 1\n\\end{equation}\ncould also yield energy conservation due to the embedding relationship \\cite{da2020shinbrot}. Furthermore, a range of new types of conditions have been obtained recently, see \\cite{drivas2019onsager,cheskidov2020energy,leslie2018conditions}.\n\nAs for the compressible systems, corresponding results are more recent and the analysis becomes inherently more complex. Within the framework of Onsager's theory \\cite{onsager1949statistical}, Drivas and Eyink \\cite{drivas2018onsager1,eyink2018cascades} adapted methods from the incompressible fluids \\cite{constantin1994onsager,duchon2000inertial,eyink1994energy} to the compressible Euler equations and obtain necessary conditions for the energy dissipative anomalies of turbulent solutions. Meanwhile, sufficient conditions for energy conservation in terms of Besov regularity for the compressible Euler fluids with a $C^{\\alpha}$ ($1<\\alpha\\leq2$) pressure law $p(\\rho)$ were later established by Feireisl et al. \\cite{feireisl2017regularity} and Akramov et al. \\cite{akramov2020energy}.\n\nRegarding the energy equality of compressible Navier-Stokes \\eqref{eq:comNS}, the pioneering breakthrough was made by Yu \\cite{yu2017energy}. He proved the equalities \\eqref{eq:energy_comNS} and \\eqref{eq:energy_comNSdeg} with the conditions that the weak solution $(\\rho,u)$ satisfies\n\\begin{equation}\\label{con:Yu}\n    u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\mathbb{T}^{N}))\\quad\\mbox{for}\\quad \\frac{1}{p}+\\frac{1}{q}\\leq \\frac{5}{12},\\quad q\\geq 6\n\\end{equation}\nand\n\\begin{equation}\\label{con:genhao rou}\n    \\sqrt{\\rho} \\in L^\\infty (0, T; H^1(\\mathbb{T}^{N})).\n\\end{equation}\nLater, Chen et al. \\cite{chen2020energy} treated the  boundary effect to investigate energy conservation on a more general bounded domain with no-slip boundary condition, when\n\\begin{equation}\\label{con:Chen et al.}\n    u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\Omega))\\quad\\mbox{for}\\quad p\\geq4,\\quad q\\geq 6,\n\\end{equation}\nwhich is just the end point of \\eqref{con:Yu}. Particularly, Ye et al. \\cite{ye2022energy} have reached $L^{4}L^{4}$ criteria for the compressible Navier-Stokes equations, while they provided that $\\nabla\\sqrt{\\rho}\\in L^{4}L^{4}$ instead of $L^{\\infty}L^{2}$. However, the $L^{\\infty}H^{1}$ regularity of $\\sqrt{\\rho}$ is of independent significance, as the Bresch-Desjardins entropy for \\eqref{eq:comNSdeg} implies such an estimate \\cite{vasseur2016existence}. In conclusion, a gap persists between these results and the Shinbrot criteria, due to the complexity of the compressible system.\n\n\\begin{figure}[htbp]\n  \\centering \n  \\begin{tikzpicture}\n    \\begin{axis}[\n        width=0.8\\textwidth, \n        height=0.6\\textwidth, \n        xlabel={$1/p$},\n        xlabel style={\n        at={(rel axis cs:0,1)},\n        anchor=west, \n        xshift=15pt, \n        yshift=-15pt \n    },\n        ylabel={$1/q$},\n        ylabel style={\n        at={(0,0.55)},\n        anchor=south, \n        xshift=10pt, \n        yshift=10pt, \n        rotate=0 \n    },\n        xmin=-0.05, xmax=0.75, ymin=-0.05, ymax=0.55,\n        axis lines=middle,\n        xtick={0},\n        ytick={0},\n        grid=major,\n        legend pos=north east,\n        clip=false,\n    ]\nll[gray!20] (axis cs: 0,0) -- (axis cs: 0.5,0) -- (axis cs: 0.25,0.25) -- (axis cs: 0,1/3) -- cycle;\nll[pattern=north east lines, pattern color=black!50] (axis cs: 5/12,0) -- (axis cs: 0.25,1/6) -- (axis cs: 0,1/6) -- (axis cs: 0,1/3) -- (axis cs: 0.25,0.25) -- (axis cs: 0.5,0) -- cycle;\nll[pattern=grid, pattern color=black!60] (axis cs: 5/12,0) -- (axis cs: 0.25,1/6) -- (axis cs: 0.25,0) -- cycle;\n\n      \\node[draw, circle, fill=black, inner sep=1.5pt, label=below right:{$L^2(L^\\infty)$}] at (axis cs: 0.5,0) {};\n      \\node[draw, circle, fill=black, inner sep=1.5pt, label=below left:{$L^4(L^6)$}] at (axis cs: 0.25,1/6) {};\n      \\node[draw, circle, fill=black, inner sep=1.5pt, label=above right:{$L^4(L^4)$}] at (axis cs: 0.25,0.25) {};\n      \\node[draw, circle, fill=black, inner sep=1.5pt, label=below:{$L^{\\frac{12}{5}}(L^\\infty)$}] at (axis cs: 5/12,0) {};\n      \\node[draw, circle, fill=black, inner sep=1.5pt, label=left:{$L^{\\infty}(L^3)$}] at (axis cs: 0,1/3) {}; \n      \\node[draw, circle, fill=black, inner sep=1.5pt, label=left:{$L^{\\infty}(L^\\frac{12}{5})$}] at (axis cs: 0,5/12) {};\n      \\node[draw, circle, fill=black, inner sep=1.5pt, label=left:{$L^{\\infty}(L^2)$}] at (axis cs: 0,0.5) {};\n\n      \\draw[thick] (axis cs: 0,1/3) -- (axis cs: 0.5,0);\n      \\draw[line width=2pt] (axis cs: 0,1/3) -- (axis cs: 0.25,0.25);\n      \\draw[thick] (axis cs: 0.25,1/6) -- (axis cs: 5/12,0);\n      \\draw[line width=2pt] (axis cs: 0.25,0.25) -- (axis cs: 0.5,0);\n      \\draw[dashed] (axis cs: 0.25,0.25) -- (axis cs: 0,0.5);\n      \\draw[dashed] (axis cs: 0.25,1/6) -- (axis cs: 0,5/12);\n      \\draw[dashed] (axis cs: 0,1/6) -- (axis cs: 0.25,1/6);\n      \\draw[dashed] (axis cs: 0.25,0) -- (axis cs: 0.25,1/6);\n      \\draw[-Stealth] (0.1,0.12) -- (1/6,2/9);\n      \\draw[-Stealth] (0.24,0.05) -- (1/3,1/12);\n      \\draw[-Stealth] (0.25,0.37) -- (1/7,2/7);\n\n      \\node[rotate=-45, font=\\normalsize] at (axis cs: 0.4, 0.15) {$2/p + 2/q = 1$};\n      \\node[rotate=0, font=\\small] at (axis cs: 0.07, 0.1) {$\\frac{2}{p} + \\frac{3}{q} = 1$};\n      \\node[rotate=0, font=\\small] at (axis cs: 0.16, 0.05) {$\\frac{1}{p} + \\frac{1}{q} = \\frac{5}{12}$};\n      \\node[rotate=0, font=\\small] at (axis cs: 0.27, 0.39) {$\\frac{1}{p} + \\frac{3}{q} = 1$};\n      \\node[rotate=-45, font=\\small] at (axis cs: 0.32, 0.135) {new};\n      \\node[rotate=0, font=\\small] at (axis cs: 0.1, 0.23) {new};\n      \\node[rotate=0, font=\\small] at (axis cs: 0.31, 0.05) {new};\n      \\draw[fill=white, line width=0.8pt] (axis cs: 0.5,0) circle (1.5pt);\n\n    \\end{axis}\n  \\end{tikzpicture}\n\n  \\caption{The shaded region illustrates the exponent ranges covered by Theorem \\ref{theoremcomNS}.The dashed and grid subregions represent new ranges of exponents not obtained before in the literature, where the grid area only corresponds to bounded domain with no-slip boundary and the dashed area corresponds to both bounded domain and torus.} \n  \\label{fig:embedding-diagram} \n\\end{figure}\n\nIn fact, temporal derivative term $\\partial_{t}(\\rho u)$ is the main challenges in filling this gap. It requests a spatiotemporal mollification of the equation, not a pure spatial mollification of the incompressible system. Then, new errors associated to temporal mollification have emerged. In the proofs of \\cite{chen2020energy,ye2022energy,yu2017energy}, the energy flux from $\\partial_{t}(\\rho u)$ was handled by a commutator estimate providing the integrability of $\\partial_{t}\\rho$. Particularly, it is the mass equation which allows us to 'transform' the spatial regularity of $u$ into the temporal regularity of $\\rho$. However, this strategy meets some restriction. One need to assume that the velocity $u$ complies with the Serrin type criteria at least, not the Lions or Shinbrot type criteria.\n\n\\subsection{Methodology}\n\nThe present paper aims to bridge the gap between known research (\\eqref{con:Yu} and \\eqref{con:Chen et al.}) and the Shinbrot type criteria (\\eqref{con:shinbrot} and \\eqref{con:shinbrot2}) on a periodic domain $\\mathbb{T}^{d}$ or a bounded domain with $C^{1}$ boundary. For this purpose, we shall reorganize the framework and introduce a new mollification and estimate. These approaches allow us to deal with the energy flux caused by the nonlinear terms $(\\rho u)_t$ under lower integrability conditions. In the following paragraphs, we briefly describe the ideas of our method.\n\n\\emph{Framework and mollification.} The proof framework developed in this paper consists of two main steps: first, establishing the local energy equality, and then extending it to a global version. In the first step, we derive the local energy equality using the specific test function $(\\varphi u_{\\varepsilon}^{\\varepsilon})_{\\varepsilon}^{\\varepsilon}$, introduced in Section 2, and passing to the limit as $\\varepsilon \\to 0$. By structuring the mollifier in this manner, we can separate errors present in different directions. In particular, we establish the vanishing of the temporal error via our weak-type temporal commutator estimate, which improves upon the classical treatment. Meanwhile, the other terms, such as $\\text{div}(\\rho u \\otimes u)$, are rigorously re-examined in this step to ensure convergence under the weaker assumptions. In the second step, the result is extended to the whole domain by setting $\\varphi=\\psi_{\\tau}\\phi_{\\delta}$ (with $\\varphi=\\psi_{\\tau}$ for the torus $\\mathbb{T}^{d}$) and letting $\\tau, \\delta \\to 0$. The spatial cut-off function $\\phi_{\\delta}$ is constructed by the approaches used in \\cite{chen2020energy} to control the regularity of the solution near the boundary. Particularly, the presence of a solid boundary complicates the dissipative mechanisms, as the behavior of the solution near the wall can differ significantly from its behavior in the interior. For further discussion on boundary effects in energy conservation contexts, we refer the reader to the related literature \\cite{bardos2019onsager,bardos2019extension,drivas2018onsager}.\n\n\\emph{Weak-type commutator estimate.} Our main challenge is to justify the convergence of the commutator\n$$\n\\partial_{t}(\\rho u^{\\varepsilon})_{\\varepsilon}-\\partial_{t}(\\rho u_{\\varepsilon}^{\\varepsilon})\n$$\nunder the Shinbrot-type condition on velocity. Compared with the previous work, these criteria entail a corresponding loss of regularity of $\\partial_{t}\\rho$, which is the key condition of the commutator estimate used in them. Since one usually employs the continuity equation:\n$$\n\\partial_{t}\\rho = - \\text{div}(\\rho u) = - (\\rho \\text{div} u + \\nabla\\rho\\cdot u),\n$$\nto obtain integrability of $\\partial_{t}\\rho$, it may be difficult to derive the necessary $L_{t}^{p}L_{x}^{q}$ estimate of $\\partial_{t}\\rho$ under the assumption of the present paper. To overcome this obstacle, we prove Lemma \\ref{commutator}, in which the convergence is established directly without establishing any $L^{p}$-type commutator estimate. Lemma \\ref{commutator}, together with the viscous properties of the fluid, allows us to show that the error term vanishes even when $\\partial_{t}\\rho$ is only a distribution, which is ensured by the mass conservation.\n\n\\subsection{Main results}\n\nOur results consist of the energy conservation criteria of two kinds of compressible Navier--Stokes systems: constant viscosities and degenerate viscosity. Here, we present our main results regarding the energy conservation for weak solutions to the equations with constant viscosities, whereas the remaining content, pertaining to degenerate viscosity cases, is outlined in the final section (Section \\ref{sec:deg}). Throughout the present paper, let $\\Omega$ be either the torus $\\mathbb{T}^{N}$ ($N=2,3$) or an open, bounded domain in $\\mathbb{R}^N$ with $C^1$ boundary $\\partial \\Omega$ with $u=0$ on $\\partial\\Omega$.\n\nWe first present the result for the Navier-Stokes equations \\eqref{eq:comNS} with constant viscosity, where the presence of vacuum is allowed.", "context": "In the context of the Navier–Stokes equations, the pioneering studies were done by Lions \\cite{lions1960regularite} and Serrin \\cite{serrininitial}. Lions proved that a weak solution $u$ to a incompressible fluids conserves its energy provided $u \\in L^{4}_{t,x}$ and Serrin gave a dimension-dependent condition\n\\begin{equation}\\label{con:serrin}\n u \\in L^{p}_{t}L^{q}_{x}\\quad\\mbox{for}\\quad \\frac{2}{p}+\\frac{N}{q}\\leq 1,\\quad q > N.\n\\end{equation}\nLater, Shinbrot removed the dimensional dependence in \\cite{shinbrot1974energy} and proved the same conclusion if\n\\begin{equation}\\label{con:shinbrot}\n \\frac{2}{p}+\\frac{2}{q}\\leq 1,\\quad q\\geq 4,\n\\end{equation}\nwhich improved the previous work. Meanwhile, for $3\\leq q<4$, the following criteria\n\\begin{equation}\\label{con:shinbrot2}\n \\frac{1}{p}+\\frac{3}{q}\\leq 1\n\\end{equation}\ncould also yield energy conservation due to the embedding relationship \\cite{da2020shinbrot}. Furthermore, a range of new types of conditions have been obtained recently, see \\cite{drivas2019onsager,cheskidov2020energy,leslie2018conditions}.\n\nRegarding the energy equality of compressible Navier-Stokes \\eqref{eq:comNS}, the pioneering breakthrough was made by Yu \\cite{yu2017energy}. He proved the equalities \\eqref{eq:energy_comNS} and \\eqref{eq:energy_comNSdeg} with the conditions that the weak solution $(\\rho,u)$ satisfies\n\\begin{equation}\\label{con:Yu}\n u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\mathbb{T}^{N}))\\quad\\mbox{for}\\quad \\frac{1}{p}+\\frac{1}{q}\\leq \\frac{5}{12},\\quad q\\geq 6\n\\end{equation}\nand\n\\begin{equation}\\label{con:genhao rou}\n \\sqrt{\\rho} \\in L^\\infty (0, T; H^1(\\mathbb{T}^{N})).\n\\end{equation}\nLater, Chen et al. \\cite{chen2020energy} treated the boundary effect to investigate energy conservation on a more general bounded domain with no-slip boundary condition, when\n\\begin{equation}\\label{con:Chen et al.}\n u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\Omega))\\quad\\mbox{for}\\quad p\\geq4,\\quad q\\geq 6,\n\\end{equation}\nwhich is just the end point of \\eqref{con:Yu}. Particularly, Ye et al. \\cite{ye2022energy} have reached $L^{4}L^{4}$ criteria for the compressible Navier-Stokes equations, while they provided that $\\nabla\\sqrt{\\rho}\\in L^{4}L^{4}$ instead of $L^{\\infty}L^{2}$. However, the $L^{\\infty}H^{1}$ regularity of $\\sqrt{\\rho}$ is of independent significance, as the Bresch-Desjardins entropy for \\eqref{eq:comNSdeg} implies such an estimate \\cite{vasseur2016existence}. In conclusion, a gap persists between these results and the Shinbrot criteria, due to the complexity of the compressible system.\n\n\\emph{Weak-type commutator estimate.} Our main challenge is to justify the convergence of the commutator\n$$\n\\partial_{t}(\\rho u^{\\varepsilon})_{\\varepsilon}-\\partial_{t}(\\rho u_{\\varepsilon}^{\\varepsilon})\n$$\nunder the Shinbrot-type condition on velocity. Compared with the previous work, these criteria entail a corresponding loss of regularity of $\\partial_{t}\\rho$, which is the key condition of the commutator estimate used in them. Since one usually employs the continuity equation:\n$$\n\\partial_{t}\\rho = - \\text{div}(\\rho u) = - (\\rho \\text{div} u + \\nabla\\rho\\cdot u),\n$$\nto obtain integrability of $\\partial_{t}\\rho$, it may be difficult to derive the necessary $L_{t}^{p}L_{x}^{q}$ estimate of $\\partial_{t}\\rho$ under the assumption of the present paper. To overcome this obstacle, we prove Lemma \\ref{commutator}, in which the convergence is established directly without establishing any $L^{p}$-type commutator estimate. Lemma \\ref{commutator}, together with the viscous properties of the fluid, allows us to show that the error term vanishes even when $\\partial_{t}\\rho$ is only a distribution, which is ensured by the mass conservation.\n\nOur results consist of the energy conservation criteria of two kinds of compressible Navier--Stokes systems: constant viscosities and degenerate viscosity. Here, we present our main results regarding the energy conservation for weak solutions to the equations with constant viscosities, whereas the remaining content, pertaining to degenerate viscosity cases, is outlined in the final section (Section \\ref{sec:deg}). Throughout the present paper, let $\\Omega$ be either the torus $\\mathbb{T}^{N}$ ($N=2,3$) or an open, bounded domain in $\\mathbb{R}^N$ with $C^1$ boundary $\\partial \\Omega$ with $u=0$ on $\\partial\\Omega$.\n\nWe first present the result for the Navier-Stokes equations \\eqref{eq:comNS} with constant viscosity, where the presence of vacuum is allowed.", "full_context": "In the context of the Navier–Stokes equations, the pioneering studies were done by Lions \\cite{lions1960regularite} and Serrin \\cite{serrininitial}. Lions proved that a weak solution $u$ to a incompressible fluids conserves its energy provided $u \\in L^{4}_{t,x}$ and Serrin gave a dimension-dependent condition\n\\begin{equation}\\label{con:serrin}\n u \\in L^{p}_{t}L^{q}_{x}\\quad\\mbox{for}\\quad \\frac{2}{p}+\\frac{N}{q}\\leq 1,\\quad q > N.\n\\end{equation}\nLater, Shinbrot removed the dimensional dependence in \\cite{shinbrot1974energy} and proved the same conclusion if\n\\begin{equation}\\label{con:shinbrot}\n \\frac{2}{p}+\\frac{2}{q}\\leq 1,\\quad q\\geq 4,\n\\end{equation}\nwhich improved the previous work. Meanwhile, for $3\\leq q<4$, the following criteria\n\\begin{equation}\\label{con:shinbrot2}\n \\frac{1}{p}+\\frac{3}{q}\\leq 1\n\\end{equation}\ncould also yield energy conservation due to the embedding relationship \\cite{da2020shinbrot}. Furthermore, a range of new types of conditions have been obtained recently, see \\cite{drivas2019onsager,cheskidov2020energy,leslie2018conditions}.\n\nRegarding the energy equality of compressible Navier-Stokes \\eqref{eq:comNS}, the pioneering breakthrough was made by Yu \\cite{yu2017energy}. He proved the equalities \\eqref{eq:energy_comNS} and \\eqref{eq:energy_comNSdeg} with the conditions that the weak solution $(\\rho,u)$ satisfies\n\\begin{equation}\\label{con:Yu}\n u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\mathbb{T}^{N}))\\quad\\mbox{for}\\quad \\frac{1}{p}+\\frac{1}{q}\\leq \\frac{5}{12},\\quad q\\geq 6\n\\end{equation}\nand\n\\begin{equation}\\label{con:genhao rou}\n \\sqrt{\\rho} \\in L^\\infty (0, T; H^1(\\mathbb{T}^{N})).\n\\end{equation}\nLater, Chen et al. \\cite{chen2020energy} treated the boundary effect to investigate energy conservation on a more general bounded domain with no-slip boundary condition, when\n\\begin{equation}\\label{con:Chen et al.}\n u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\Omega))\\quad\\mbox{for}\\quad p\\geq4,\\quad q\\geq 6,\n\\end{equation}\nwhich is just the end point of \\eqref{con:Yu}. Particularly, Ye et al. \\cite{ye2022energy} have reached $L^{4}L^{4}$ criteria for the compressible Navier-Stokes equations, while they provided that $\\nabla\\sqrt{\\rho}\\in L^{4}L^{4}$ instead of $L^{\\infty}L^{2}$. However, the $L^{\\infty}H^{1}$ regularity of $\\sqrt{\\rho}$ is of independent significance, as the Bresch-Desjardins entropy for \\eqref{eq:comNSdeg} implies such an estimate \\cite{vasseur2016existence}. In conclusion, a gap persists between these results and the Shinbrot criteria, due to the complexity of the compressible system.\n\n\\emph{Weak-type commutator estimate.} Our main challenge is to justify the convergence of the commutator\n$$\n\\partial_{t}(\\rho u^{\\varepsilon})_{\\varepsilon}-\\partial_{t}(\\rho u_{\\varepsilon}^{\\varepsilon})\n$$\nunder the Shinbrot-type condition on velocity. Compared with the previous work, these criteria entail a corresponding loss of regularity of $\\partial_{t}\\rho$, which is the key condition of the commutator estimate used in them. Since one usually employs the continuity equation:\n$$\n\\partial_{t}\\rho = - \\text{div}(\\rho u) = - (\\rho \\text{div} u + \\nabla\\rho\\cdot u),\n$$\nto obtain integrability of $\\partial_{t}\\rho$, it may be difficult to derive the necessary $L_{t}^{p}L_{x}^{q}$ estimate of $\\partial_{t}\\rho$ under the assumption of the present paper. To overcome this obstacle, we prove Lemma \\ref{commutator}, in which the convergence is established directly without establishing any $L^{p}$-type commutator estimate. Lemma \\ref{commutator}, together with the viscous properties of the fluid, allows us to show that the error term vanishes even when $\\partial_{t}\\rho$ is only a distribution, which is ensured by the mass conservation.\n\nOur results consist of the energy conservation criteria of two kinds of compressible Navier--Stokes systems: constant viscosities and degenerate viscosity. Here, we present our main results regarding the energy conservation for weak solutions to the equations with constant viscosities, whereas the remaining content, pertaining to degenerate viscosity cases, is outlined in the final section (Section \\ref{sec:deg}). Throughout the present paper, let $\\Omega$ be either the torus $\\mathbb{T}^{N}$ ($N=2,3$) or an open, bounded domain in $\\mathbb{R}^N$ with $C^1$ boundary $\\partial \\Omega$ with $u=0$ on $\\partial\\Omega$.\n\nWe first present the result for the Navier-Stokes equations \\eqref{eq:comNS} with constant viscosity, where the presence of vacuum is allowed.\n\nRegarding the energy equality of compressible Navier-Stokes \\eqref{eq:comNS}, the pioneering breakthrough was made by Yu \\cite{yu2017energy}. He proved the equalities \\eqref{eq:energy_comNS} and \\eqref{eq:energy_comNSdeg} with the conditions that the weak solution $(\\rho,u)$ satisfies\n\\begin{equation}\\label{con:Yu}\n u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\mathbb{T}^{N}))\\quad\\mbox{for}\\quad \\frac{1}{p}+\\frac{1}{q}\\leq \\frac{5}{12},\\quad q\\geq 6\n\\end{equation}\nand\n\\begin{equation}\\label{con:genhao rou}\n \\sqrt{\\rho} \\in L^\\infty (0, T; H^1(\\mathbb{T}^{N})).\n\\end{equation}\nLater, Chen et al. \\cite{chen2020energy} treated the boundary effect to investigate energy conservation on a more general bounded domain with no-slip boundary condition, when\n\\begin{equation}\\label{con:Chen et al.}\n u \\in L^{p}_{t}(0,T;L^{q}_{x}(\\Omega))\\quad\\mbox{for}\\quad p\\geq4,\\quad q\\geq 6,\n\\end{equation}\nwhich is just the end point of \\eqref{con:Yu}. Particularly, Ye et al. \\cite{ye2022energy} have reached $L^{4}L^{4}$ criteria for the compressible Navier-Stokes equations, while they provided that $\\nabla\\sqrt{\\rho}\\in L^{4}L^{4}$ instead of $L^{\\infty}L^{2}$. However, the $L^{\\infty}H^{1}$ regularity of $\\sqrt{\\rho}$ is of independent significance, as the Bresch-Desjardins entropy for \\eqref{eq:comNSdeg} implies such an estimate \\cite{vasseur2016existence}. In conclusion, a gap persists between these results and the Shinbrot criteria, due to the complexity of the compressible system.\n\nWe first present the result for the Navier-Stokes equations \\eqref{eq:comNS} with constant viscosity, where the presence of vacuum is allowed.\n\nThe remainder of this paper is organized as follows. Section 2 introduces the necessary preliminaries, including definitions of weak solutions and key technical lemmas. Section 3 focuses on establishing the local energy conservation law in the sense of distributions using a mollification argument. In Section 4, we extend this local result to the global energy balance on the domain $\\Omega$, completing the proof of Theorem \\ref{theoremcomNS}. Finally, we apply our methods on the compressible Navier-Stokes equations with degenerate viscosity in the last Section 5.\n\nIt is easy to see, for any $\\alpha \\geq \\frac{1}{2}$,\n\\begin{equation*}\n \\partial_{t}(\\rho^{\\alpha}) = -\\alpha \\rho^{\\alpha} \\mathrm{div} u - 2\\alpha \\rho^{\\alpha - \\frac{1}{2}} u \\cdot \\nabla \\sqrt{\\rho},\n\\end{equation*}\nwhich, together with \\eqref{con:defcomNS} and \\eqref{condi1-th2}, implies\n\\begin{equation*}\n \\rho^{\\alpha} \\in L^{\\infty}(0, T; W^{1,\\frac{3}{2}}(\\Omega)), \\quad \\partial_{t}(\\rho^{\\alpha}) \\in L^{2}(0, T; L^{\\frac{12}{11}}(\\Omega)).\n\\end{equation*}\nIt follows that, by the Lemma \\ref{aubinlions},\n\\begin{equation*}\n \\rho^{\\alpha} \\in C([0, T]; L^{r}(\\Omega)), \\quad r < \\frac{3N}{2N-3},\n\\end{equation*}\nso we can use $u_{0} \\in L^{\\frac{6N}{6-N}}$ to deduce that\n\\begin{equation}\\label{continuity_sqrt_rho_u}\n (\\sqrt{\\rho }u) (t) \\rightarrow (\\sqrt{\\rho }u) (0) \\quad \\text{strongly in } L^2(\\Omega) \\text{ as } t \\to 0^+.\n\\end{equation}\nFollowing the similar manner of proof in \\cite{yu2017energy}, we can complete the proof of Theorem \\ref{theoremcomNS}.\n\n\\section{The energy conservation for the compressible Navier-Stokes equations with degenerate viscosity}\\label{sec:deg}\nNext, we consider the compressible Navier-Stokes equations with degenerate viscosity, where the density is strictly bounded away from vacuum. The systems are stated as follows:\n \\begin{equation}\\label{eq:comNSdeg}\n \\begin{split}\n (\\rho u)_t + \\text{div}(\\rho u \\otimes u) - 2\\nu \\text{div}(\\rho \\mathbb{D} u) + \\nabla P &= 0, \\\\\n \\rho_t + \\text{div}(\\rho u) &= 0,\n \\end{split}\n \\end{equation}\n where $\\mathbb{D} u = \\frac{1}{2}(\\nabla u + \\nabla^T u)$ is the strain tensor and the viscosity coefficients satisfy $\\nu > 0$. Then, we present the definition and the energy conservation criteria of the corresponding weak solutions.\n \\begin{definition}\\label{defcomNSdeg}\n The pair $(\\rho, u)$ is called a global weak solution to the degenerate viscosity system \\eqref{eq:comNSdeg} with initial data \\eqref{con:initial} if, for any $t \\in[0, T]$:\n \\begin{itemize}\n \\item The momentum equations \\eqref{eq:comNSdeg} hold in $\\mathcal{D}^{\\prime}((0, T) \\times \\Omega)$ satisfying:\n $$\n \\begin{gathered}\n \\rho \\geq 0, \\quad \\rho \\in L^{\\infty}\\left(0, T ; L^\\gamma(\\Omega)\\right),\\quad \\nabla \\sqrt{\\rho} \\in L^{\\infty}\\left(0, T ; L^\\frac{3}{2}(\\Omega)\\right)\\\\\n \\sqrt{\\rho} u \\in L^{\\infty}\\left(0, T ; L^2(\\Omega)\\right), \\quad \\sqrt{\\rho} \\nabla u \\in L^2\\left(0, T ; L^2(\\Omega)\\right);\n \\end{gathered}\n $$\n\n\\item The energy inequality holds for almost every $t \\in [0,T]$:\n \\begin{equation}\\label{eq:energy_comNSdeg}\n \\int_{\\Omega}\\left(\\frac{1}{2} \\rho|u|^2+\\frac{\\rho^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x+\\int_0^t \\int_{\\Omega} \\rho|\\mathbb{D} u|^2 \\mathrm{d} x \\mathrm{d} s \\leq \\int_{\\Omega}\\left(\\frac{1}{2} \\rho_0\\left|u_0\\right|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x.\n \\end{equation}\n \\end{itemize}\n\\end{definition}\n\\begin{remark}\n The condition $\\nabla\\sqrt{\\rho}\\in L^{\\infty}L^{\\frac{3}{2}}$ is reasonable in this context, as it is consistent with the known existence theory for weak solutions. Specifically, for the degenerate compressible Navier–Stokes equations, global existence results \\cite{vasseur2016existence} ensure that $\\nabla \\rho^{\\frac{\\gamma}{2}}\\in L^{2}L^{2}$ and $\\nabla\\sqrt{\\rho}\\in L^{\\infty}L^{2}$.\n\\end{remark}\n\n\\begin{theorem}\\label{theoremcomNSdeg}\n Let $(\\rho, u)$ be a weak solution of \\eqref{eq:comNSdeg} in the sense of Definition \\ref{defcomNSdeg}. Assume that the density satisfies\n \\begin{equation}\\label{condi1-th2}\n 0 < \\underline{\\rho} \\leq \\rho(t, x)\\leq \\bar{\\rho} < \\infty.\n \\end{equation}\n and the initial condition satisfies\n \\begin{equation}\\label{condi2-th2}\n \\sqrt{\\rho_0} u_0 \\in L^{\\frac{4N}{N+2}}(\\Omega).\n \\end{equation}\n If the velocity satisfies\n \\begin{equation}\\label{con:shinbrot_cond2}\n u \\in L^p\\left(0, T ; L^q(\\Omega)\\right) \\quad \\text{with} \\quad \\begin{cases}\n \\frac{1}{p}+\\frac{3}{q} \\leq 1,\\quad\\mbox{if} \\quad 3\\leq q < 4,\\\\\n \\frac{2}{p}+\\frac{2}{q} \\leq 1,\\quad\\mbox{if} \\quad 4\\leq q \\leq\\infty,\n \\end{cases}\n \\end{equation}\n then the energy equality \\eqref{eq:energy_comNSdeg} holds for any $t \\in[0, T]$.\n\\end{theorem}\n\\begin{remark}\n Different from Theorem \\ref{theoremcomNS}, we have obtained the energy conservation under the assumption $u\\in L^{2}L^{\\infty}$, which is one of the endpoints. \n\\end{remark}\nFinally, let us provide the proof of Theorem \\ref{theoremcomNSdeg}. We can modify the proof in Section 3 and 4 slightly to arrive the case $(p,q)=(4,4)$. Note that, for any weak solution ($\\rho, u$), condition \\eqref{condi1-th2} implies that", "post_theorem_intro_text_len": 1724, "post_theorem_intro_text": "\\begin{remark}\n    Our result improves the known regularity criteria in the literature and closes a notable gap. We relax the stronger Serrin-type integrability required in prior works (e.g., \\eqref{con:Yu} and \\eqref{con:Chen et al.}) via a novel weak-type temporal commutator estimate (Lemma \\ref{commutator}), which handles the nonlinear terms without demanding strong regularity of $\\partial_t\\rho$. The extended range of admissible exponents is visualized in {\\rm Figure 1}. Particularly, we provide that $\\nabla\\sqrt{\\rho}\\in L^{\\infty}L^{2}$ instead of $L^{\\infty}L^{2}$, which is also weaker than \\cite{yu2017energy,chen2020energy}.\n\\end{remark}\n\\begin{remark}\n     It is worth noting that, due to the non‑separability of the space $L^{2}(0,T;L^{\\infty}(\\Omega))$, our method does not currently apply to the endpoint $(p,q)=(2,\\infty)$; this remains for future investigation. \n\\end{remark}\n\\begin{remark}\n    The approaches employed to validate this theorem are equally applicable to the compressible Navier-Stokes equations with degenerate viscosity, a fact we will demonstrate by proving Theorem \\ref{theoremcomNSdeg} in Section 5.\n\\end{remark}\n\nThe remainder of this paper is organized as follows. Section 2 introduces the necessary preliminaries, including definitions of weak solutions and key technical lemmas. Section 3 focuses on establishing the local energy conservation law in the sense of distributions using a mollification argument. In Section 4, we extend this local result to the global energy balance on the domain $\\Omega$, completing the proof of Theorem \\ref{theoremcomNS}. Finally, we apply our methods on the compressible Navier-Stokes equations with degenerate viscosity in the last Section 5.", "sketch": "A proof outline is given after Theorem~\\ref{theoremcomNS}: Section~3 \"focuses on establishing the local energy conservation law in the sense of distributions using a mollification argument.\" Then, in Section~4, they \"extend this local result to the global energy balance on the domain $\\Omega$, completing the proof of Theorem~\\ref{theoremcomNS}.\" A key ingredient enabling the relaxed integrability assumptions is \"a novel weak-type temporal commutator estimate (Lemma \\ref{commutator}), which handles the nonlinear terms without demanding strong regularity of $\\partial_t\\rho$.\"", "expanded_sketch": "A proof outline is given after\n\\begin{theorem}\\label{theoremcomNS}\n    Let $(\\rho, u)$ be a weak solution of \\eqref{eq:comNS} in the sense of Definition \\ref{defcomNS}. Assume that the density satisfies\n    \\begin{equation}\\label{con:thcomNSrhoover0}\n         0 \\leq \\rho(t,x) \\leq \\bar{\\rho} < \\infty, \\quad \\text{and} \\quad \\nabla \\sqrt{\\rho} \\in L^{\\infty}\\left(0, T ; L^{\\frac{3}{2}}(\\Omega)\\right),\n    \\end{equation}\n    and the initial velocity satisfies\n    \\begin{equation}\\label{con:initial_1}\n        u_0 \\in L^{\\frac{6N}{6-N}}(\\Omega).\n    \\end{equation}\n    If the velocity satisfies\n    \\begin{equation}\\label{con:shinbrot_cond}\n        u \\in L^p\\left(0, T ; L^q(\\Omega)\\right) \\quad \\text{with} \\quad \\begin{cases}\n            \\frac{1}{p}+\\frac{3}{q} \\leq 1, \\quad\\mbox{if} \\quad 3\\leq q < 4,\\\\\n            \\frac{2}{p}+\\frac{2}{q} \\leq 1, \\quad\\mbox{if} \\quad 4\\leq q <\\infty,\n        \\end{cases}\n    \\end{equation}\n    then the energy equality \\eqref{eq:energy_comNS} holds for any $t \\in[0, T]$.\n\\end{theorem}\nNext, they focus on establishing the local energy conservation law in the sense of distributions using a mollification argument. Then, they extend this local result to the global energy balance on the domain $\\Omega$, thereby completing the proof of the main theorem. A key ingredient enabling the relaxed integrability assumptions is the following commutator estimate.\n\n\\begin{lemma}\\label{commutator} \nLet $1 \\leq \\bar{p}, \\bar{q}, p_1, q_1, p_2, q_2 \\leq \\infty$, with $\\frac{1}{\\bar{p}}+\\frac{1}{p_1}+\\frac{1}{p_2}=1$ and $\\frac{1}{\\bar{q}}+\\frac{1}{q_1}+\\frac{1}{q_2}=1$. Let $f\\in L^{p_1}\\left(0, T ; L^{q_1}(\\Omega)\\right), \\partial_{t} f \\in L^{p_1}\\left(0, T ; W^{-1,q_1}(\\Omega)\\right), g \\in L^{p_2}\\left(0, T ; W^{1,q_2}(\\Omega)\\right), \\varphi \\in L^{\\bar{p}}\\left(0, T ; W_{0}^{1,\\bar{q}}(\\Omega)\\right)$. Then, there holds\n$$\n\\int_0^T \\int_{\\Omega}\\varphi\\left[\\partial_t\\left(f g\\right)_{\\varepsilon}-\\partial_t(f g_{\\varepsilon})\\right]\\mathrm{d} x \\mathrm{d} t \\rightarrow 0,\n$$\nas $\\varepsilon \\rightarrow 0$ if $p_2, q_2,\\bar{p},\\bar{q}<\\infty$.\n\\end{lemma}\nThis lemma handles the nonlinear terms without demanding strong regularity of $\\partial_t\\rho$.", "expanded_theorem": "\\label{theoremcomNS}\n    Let $(\\rho, u)$ be a weak solution of \\begin{equation}\\label{eq:comNS}\n    \\begin{split}\n        (\\rho u)_t + \\text{div}(\\rho u \\otimes u) - \\mu \\Delta u - (\\mu+\\lambda) \\nabla \\text{div} \\, u + \\nabla P &= 0,  \\\\\n        \\rho_t + \\text{div}(\\rho u) &= 0, \n    \\end{split}\n\\end{equation} in the sense of the following definition. \\begin{definition}\\label{defcomNS}\n    For a given $T>0$, we call $(\\rho, u)$ a weak solution on $[0, T]$ to the constant viscosity system \\eqref{eq:comNS}\\eqref{con:initial} if:\n    \\begin{itemize}\n        \\item The momentum equations \\eqref{eq:comNS} hold in $\\mathcal{D}^{\\prime}((0, T) \\times \\Omega)$ with the regularity:\n        \\begin{equation}\\label{con:defcomNS}\n            \\rho^\\gamma, \\rho|u|^2 \\in L^{\\infty}\\left(0, T ; L^1(\\Omega)\\right), \\quad u \\in L^2\\left(0, T ; H_0^1(\\Omega)\\right);\n        \\end{equation}\n\n        \\item The initial conditions \\eqref{con:initial} hold in $\\mathcal{D}^{\\prime}(\\Omega)$;\n\n        \\item $(\\rho, u)$ is a renormalized solution of the continuity equation in the sense of DiPerna--Lions, ``Ordinary differential equations, transport theory and Sobolev spaces'' (1989);\n\n        \\item The energy inequality holds for almost every $t \\in [0,T]$:\n        $$\n        \\begin{aligned}\n            \\int_{\\Omega}\\left(\\frac{1}{2} \\rho|u|^2+\\frac{\\rho^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x & +\\int_0^t \\int_{\\Omega}\\left(\\mu|\\nabla u|^2+(\\mu+\\lambda)(\\operatorname{div} u)^2\\right) \\mathrm{d} x \\mathrm{d} s \\\\\n            & \\leq \\int_{\\Omega}\\left(\\frac{1}{2} \\rho_0\\left|u_0\\right|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x.\n        \\end{aligned}\n        $$\n    \\end{itemize}\n\\end{definition}\n    Assume that the density satisfies\n    \\begin{equation}\\label{con:thcomNSrhoover0}\n         0 \\leq \\rho(t,x) \\leq \\bar{\\rho} < \\infty, \\quad \\text{and} \\quad \\nabla \\sqrt{\\rho} \\in L^{\\infty}\\left(0, T ; L^{\\frac{3}{2}}(\\Omega)\\right),\n    \\end{equation}\n    and the initial velocity satisfies\n    \\begin{equation}\\label{con:initial_1}\n        u_0 \\in L^{\\frac{6N}{6-N}}(\\Omega).\n    \\end{equation}\n    If the velocity satisfies\n    \\begin{equation}\\label{con:shinbrot_cond}\n        u \\in L^p\\left(0, T ; L^q(\\Omega)\\right) \\quad \\text{with} \\quad \\begin{cases}\n            \\frac{1}{p}+\\frac{3}{q} \\leq 1, \\quad\\mbox{if} \\quad 3\\leq q < 4,\\\\\n            \\frac{2}{p}+\\frac{2}{q} \\leq 1, \\quad\\mbox{if} \\quad 4\\leq q <\\infty,\n        \\end{cases}\n    \\end{equation}\n    then the energy equality\n    \\begin{aligned}\n    \t&\\int_{\\Omega}\\left(\\frac{1}{2} \\rho_0\\left|u_0\\right|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x-\\int_{\\Omega}\\left(\\frac{1}{2} \\rho|u|^2+\\frac{\\rho^\\gamma}{\\gamma-1}\\right) \\mathrm{d} x\\\\\n    \t=&\\int_0^T \\int_{\\Omega}\\left( \\mu|\\nabla u|^2+(\\mu+\\lambda)|\\operatorname{div} u|^2 \\right)\\mathrm{d} x \\mathrm{d} t. \\label{eq:energy_comNS}\n    \\end{aligned} holds for any $t \\in[0, T]$.", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let \\(\\Omega\\) be either the torus \\(\\mathbb T^N\\) with \\(N=2,3\\), or a bounded open \\(C^1\\) domain in \\(\\mathbb R^N\\) (with no-slip boundary condition when \\(\\partial\\Omega\\neq\\varnothing\\)). Consider a weak solution \\((\\rho,u)\\) on \\([0,T]\\) of the constant-viscosity compressible Navier--Stokes system\n\\[\n(\\rho u)_t+\\operatorname{div}(\\rho u\\otimes u)-\\mu\\Delta u-(\\mu+\\lambda)\\nabla\\operatorname{div}u+\\nabla P=0,\n\\qquad\n\\rho_t+\\operatorname{div}(\\rho u)=0,\n\\]\nwith initial data \\((\\rho_0,u_0)\\), in the following sense: the momentum equation holds in \\(\\mathcal D'((0,T)\\times\\Omega)\\);\n\\(\\rho^\\gamma,\\rho|u|^2\\in L^\\infty(0,T;L^1(\\Omega))\\); \\(u\\in L^2(0,T;H_0^1(\\Omega))\\); the initial conditions hold in \\(\\mathcal D'(\\Omega)\\); \\((\\rho,u)\\) is a renormalized solution of the continuity equation in the DiPerna--Lions sense; and for almost every \\(t\\in[0,T]\\),\n\\[\n\\int_{\\Omega}\\left(\\tfrac12\\rho|u|^2+\\frac{\\rho^\\gamma}{\\gamma-1}\\right)dx\n+\\int_0^t\\!\\int_{\\Omega}\\bigl(\\mu|\\nabla u|^2+(\\mu+\\lambda)(\\operatorname{div}u)^2\\bigr)\\,dx\\,ds\n\\le\n\\int_{\\Omega}\\left(\\tfrac12\\rho_0|u_0|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right)dx.\n\\]\nAssume in addition that\n\\[\n0\\le \\rho(t,x)\\le \\bar\\rho<\\infty,\n\\qquad\n\\nabla\\sqrt\\rho\\in L^\\infty(0,T;L^{3/2}(\\Omega)),\n\\qquad\nu_0\\in L^{\\frac{6N}{6-N}}(\\Omega),\n\\]\nand that\n\\[\nu\\in L^p(0,T;L^q(\\Omega))\n\\quad\\text{with}\\quad\n\\begin{cases}\n\\frac1p+\\frac3q\\le 1, & 3\\le q<4,\\\\[2mm]\n\\frac2p+\\frac2q\\le 1, & 4\\le q<\\infty.\n\\end{cases}\n\\]\nWhich statement holds for every such weak solution?", "correct_choice": {"label": "A", "text": "For every \\(t\\in[0,T]\\), the energy inequality becomes an equality:\n\\[\n\\int_{\\Omega}\\left(\\tfrac12\\rho_0|u_0|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right)dx\n-\n\\int_{\\Omega}\\left(\\tfrac12\\rho(t)|u(t)|^2+\\frac{\\rho(t)^\\gamma}{\\gamma-1}\\right)dx\n=\n\\int_0^t\\!\\int_{\\Omega}\\bigl(\\mu|\\nabla u|^2+(\\mu+\\lambda)|\\operatorname{div}u|^2\\bigr)\\,dx\\,ds.\n\\]"}, "choices": [{"label": "B", "text": "For every \\(t\\in[0,T]\\), the energy inequality becomes an equality:\n\\[\n\\int_{\\Omega}\\left(\\tfrac12\\rho_0|u_0|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right)dx\n-\n\\int_{\\Omega}\\left(\\tfrac12\\rho(t)|u(t)|^2+\\frac{\\rho(t)^\\gamma}{\\gamma-1}\\right)dx\n=\n\\int_0^t\\!\\int_{\\Omega}\\bigl(\\mu|\\nabla u|^2+(\\mu+\\lambda)|\\operatorname{div}u|^2\\bigr)\\,dx\\,ds,\n\\]\nprovided the velocity satisfies\n\\[\nu\\in L^p(0,T;L^q(\\Omega))\n\\quad\\text{with}\\quad\n\\begin{cases}\n\\frac1p+\\frac3q\\le 1, & 3\\le q\\le 4,\\\\[2mm]\n\\frac2p+\\frac2q\\le 1, & 4< q<\\infty.\n\\end{cases}\n\\]"}, {"label": "C", "text": "The weak solution satisfies the original energy inequality for almost every \\(t\\in[0,T]\\):\n\\[\n\\int_{\\Omega}\\left(\\tfrac12\\rho(t)|u(t)|^2+\\frac{\\rho(t)^\\gamma}{\\gamma-1}\\right)dx\n+\n\\int_0^t\\!\\int_{\\Omega}\\bigl(\\mu|\\nabla u|^2+(\\mu+\\lambda)(\\operatorname{div}u)^2\\bigr)\\,dx\\,ds\n\\le\n\\int_{\\Omega}\\left(\\tfrac12\\rho_0|u_0|^2+\\frac{\\rho_0^\\gamma}{\\gamma-1}\\right)dx.\n\\]"}, {"label": "D", "text": "For every \\(t\\in[0,T]\\), the energy inequality becomes an equality under the same assumptions, and in fact the conclusion remains valid if the Shinbrot-type condition is replaced by the endpoint-extended condition\n\\[\nu\\in L^p(0,T;L^q(\\Omega))\n\\quad\\text{with}\\quad\n\\begin{cases}\n\\frac1p+\\frac3q\\le 1, & 3\\le q<4,\\\\[2mm]\n\\frac2p+\\frac2q\\le 1, & 4\\le q\\le \\infty.\n\\end{cases}\n\\]"}, {"label": "E", "text": "For every such weak solution, the same energy equality holds for every \\(t\\in[0,T]\\) even if one drops the assumption that \\((\\rho,u)\\) is a renormalized solution of the continuity equation in the DiPerna--Lions sense, retaining only that the continuity equation holds in \\(\\mathcal D'((0,T)\\times\\Omega)\\) together with\n\\[\n0\\le \\rho\\le \\bar\\rho,\n\\qquad\n\\nabla\\sqrt\\rho\\in L^\\infty(0,T;L^{3/2}(\\Omega)),\n\\qquad\nu\\in L^p(0,T;L^q(\\Omega))\n\\]\nand the stated Shinbrot-type conditions on \\((p,q)\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "case_split", "tampered_component": "piecewise q-range boundary at q=4", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "upgraded equality for every t", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "finite-exponent requirement excluding q=\\infty in the commutator/local energy argument", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "renormalized continuity equation needed to pass from local mollified identities to global energy balance", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the upgraded conclusion of full energy equality for every t. It gives the hypotheses and the baseline energy inequality, but the exact correct conclusion is not leaked outright."}, "TAS": {"score": 1, "justification": "This is very close to a theorem-recall item: the stem lists essentially the full hypotheses and asks for the theorem’s conclusion. However, the alternatives do introduce nearby competing formulations, so it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in checking the endpoint conditions at q=4, the role of renormalization, and whether equality is strengthened from the assumed inequality. But the task is still mostly recognition of the exact theorem statement rather than substantial mathematical generation."}, "DQS": {"score": 1, "justification": "B, D, and E are plausible theorem-neighbor distractors based on common endpoint/generalization mistakes. But C is problematic because it is explicitly true from the stem itself (a weaker true statement), so the distractor set is not cleanly single-answer."}, "total_score": 5, "overall_assessment": "Moderate-quality theorem-recognition MCQ: no major answer leakage, but it is close to direct theorem recall and includes a weaker true option that harms single-answer clarity and reduces generative reasoning."}}
{"id": "2602.13727v2", "paper_link": "http://arxiv.org/abs/2602.13727v2", "theorems_cnt": 1, "theorem": {"env_name": "theo", "content": "\\label{maintheore}\nLet $U \\subset \\mathbb{R}^d$ be a (possibly unbounded) open set with $d \\geq 2$. Assume that $\\mathbf{G} \\in L^2_{loc}(U, \\mathbb{R}^d)$ and $c \\in L^1_{loc}(U)$. Let $A = (a_{ij})_{1 \\leq i,j \\leq d}$ be a symmetric matrix of functions such that $a_{ij} \\in VMO_{loc}(U)$ for all $1 \\leq i,j \\leq d$ (see Definition~\\ref{defnvmo}), and suppose that ${\\rm div} A \\in L^2_{loc}(U, \\mathbb{R}^d)$ (see Definition~\\ref{defndivmat}). Also assume that $A$ is locally uniformly strictly elliptic and bounded on $U$, i.e. for every open ball $V$ with $\\overline{V} \\subset U$, there exist constants $\\lambda_V, M_V > 0$ such that\n\\begin{equation} \\label{locellst}\n\\lambda_V \\| \\xi \\|^2 \\leq \\langle A(x) \\xi, \\xi \\rangle, \\quad \\;\\; \\max_{1 \\leq i,j \\leq d} |a_{ij}(x)| \\leq M_V\n\\quad \\text{for a.e. } x \\in V \\text{ and all } \\xi \\in \\mathbb{R}^d.\n\\end{equation}\nLet $\\tilde{\\mathbf{F}} \\in L^2_{loc}(U, \\mathbb{R}^d)$ and $\\tilde{f} \\in L^1_{loc}(U)$. Suppose that $h \\in L^{\\infty}_{loc}(U)$ satisfies\n\\[\n\\int_U \\left({\\rm trace}(A \\nabla^2 \\varphi) + \\langle \\mathbf{G}, \\nabla \\varphi \\rangle + c \\varphi \\right) h \\, dx \n= \\int_U \\tilde{f} \\varphi \\, dx + \\int_U \\langle \\tilde{\\mathbf{F}}, \\nabla \\varphi \\rangle \\, dx \\quad \\text{for all } \\varphi \\in C_0^{\\infty}(U).\n\\]\nThen $h \\in H^{1,2}_{loc}(U)$.", "start_pos": 9002, "end_pos": 10348, "label": "maintheore"}, "ref_dict": {"fokkeriden": "\\begin{equation}\\label{fokkeriden}\n\\mathbb{E}[f(X_t)] - f(x) = \\int_0^t \\mathbb{E}[L f(X_s)]\\,ds \\quad \\text{for all } f \\in C_0^\\infty(\\mathbb{R}^d) \\text{ and } t > 0,\n\\end{equation}", "stafokkerp": "\\begin{equation} \\label{stafokkerp}\n\\int_{\\mathbb{R}^d} Lf \\, d\\nu = 0  \\quad \\text{for all } f \\in C_0^\\infty(\\mathbb{R}^d).\n\\end{equation}", "defndivmat": "\\begin{defn} \\label{defndivmat}\nLet $U$ be an open subset of $\\mathbb{R}^d$, $\\mathbf{E} = (e_1, \\dots, e_d) \\in L^1_{loc}(U, \\mathbb{R}^d)$ and $B = (b_{ij})_{1 \\leq i,j \\leq d}$ be a (possibly non-symmetric) matrix of functions with $b_{ij} \\in L^1_{loc}(U)$ for all $1 \\leq i,j \\leq d$. Let $\\mathbf{F} \\in L^1_{loc}(U, \\mathbb{R}^d)$ and $f \\in L^1_{loc}(U)$.\n\\begin{itemize}\n\\item[(i)] We say that ${\\rm div} B = \\mathbf{E}$ if\n\\[\n\\int_U \\sum_{i,j=1}^d b_{ij} \\, \\partial_i \\phi_j \\, dx = - \\int_U \\sum_{j=1}^d e_j \\, \\phi_j \\, dx \\quad \\text{for all } (\\phi_1, \\dots, \\phi_d) \\in C_0^\\infty(U)^d.\n\\]\n\\item[(ii)] We say that ${\\rm div} \\mathbf{F} = f$ if\n\\[\n\\int_U \\langle \\mathbf{F}, \\nabla \\varphi \\rangle \\, dx = - \\int_U f \\varphi \\, dx \\quad \\text{for all } \\varphi \\in C_0^\\infty(U).\n\\]\n\\end{itemize}\n\\end{defn}", "locellst": "\\begin{equation} \\label{locellst}\n\\lambda_V \\| \\xi \\|^2 \\leq \\langle A(x) \\xi, \\xi \\rangle, \\quad \\;\\; \\max_{1 \\leq i,j \\leq d} |a_{ij}(x)| \\leq M_V\n\\quad \\text{for a.e. } x \\in V \\text{ and all } \\xi \\in \\mathbb{R}^d.\n\\end{equation}", "weakconver": "\\begin{equation} \\label{weakconver}\n\\lim_{t \\to \\infty} p(t, x, y)\\,dy = \\nu(dy) \\quad \\text{weakly on } \\mathbb{R}^d,\n\\end{equation}", "defnvmo": "\\begin{defn} \\label{defnvmo}\nLet $\\omega$ be a positive continuous function on $[0,\\infty)$ with $\\omega(0) = 0$. The space $VMO_\\omega$ consists of all functions $g \\in L^1_{loc}(\\mathbb{R}^d)$ such that\n\\[\n\\sup_{z \\in \\mathbb{R}^d,\\, r < R} \\, r^{-2d} \\int_{B_r(z)} \\int_{B_r(z)} |g(x) - g(y)| \\, dx \\, dy \\leq \\omega(R) \\quad \\text{for all } R \\in (0,\\infty).\n\\]\nLet $B$ be an open ball in $\\mathbb{R}^d$. We define $VMO(B)$ as the set of all functions $f \\in L^1(B)$ for which there exists an extension $\\tilde{f} \\in VMO_{\\omega}$  (for some modulus $\\omega$) such that $\\tilde{f}|_B = f$. For an open set $U \\subset \\mathbb{R}^d$, the space $VMO_{loc}(U)$ is defined as the collection of all functions $f \\in L^1_{loc}(U)$ such that $f \\in VMO(B)$ for every open ball $B \\subset \\mathbb{R}^d$ with $\\overline{B} \\subset U$.\n\\end{defn}", "maintheore": "\\begin{theo} \\label{maintheore}\nLet $U \\subset \\mathbb{R}^d$ be a (possibly unbounded) open set with $d \\geq 2$. Assume that $\\mathbf{G} \\in L^2_{loc}(U, \\mathbb{R}^d)$ and $c \\in L^1_{loc}(U)$. Let $A = (a_{ij})_{1 \\leq i,j \\leq d}$ be a symmetric matrix of functions such that $a_{ij} \\in VMO_{loc}(U)$ for all $1 \\leq i,j \\leq d$ (see Definition~\\ref{defnvmo}), and suppose that ${\\rm div} A \\in L^2_{loc}(U, \\mathbb{R}^d)$ (see Definition~\\ref{defndivmat}). Also assume that $A$ is locally uniformly strictly elliptic and bounded on $U$, i.e. for every open ball $V$ with $\\overline{V} \\subset U$, there exist constants $\\lambda_V, M_V > 0$ such that\n\\begin{equation} \\label{locellst}\n\\lambda_V \\| \\xi \\|^2 \\leq \\langle A(x) \\xi, \\xi \\rangle, \\quad \\;\\; \\max_{1 \\leq i,j \\leq d} |a_{ij}(x)| \\leq M_V\n\\quad \\text{for a.e. } x \\in V \\text{ and all } \\xi \\in \\mathbb{R}^d.\n\\end{equation}\nLet $\\tilde{\\mathbf{F}} \\in L^2_{loc}(U, \\mathbb{R}^d)$ and $\\tilde{f} \\in L^1_{loc}(U)$. Suppose that $h \\in L^{\\infty}_{loc}(U)$ satisfies\n\\[\n\\int_U \\left({\\rm trace}(A \\nabla^2 \\varphi) + \\langle \\mathbf{G}, \\nabla \\varphi \\rangle + c \\varphi \\right) h \\, dx \n= \\int_U \\tilde{f} \\varphi \\, dx + \\int_U \\langle \\tilde{\\mathbf{F}}, \\nabla \\varphi \\rangle \\, dx \\quad \\text{for all } \\varphi \\in C_0^{\\infty}(U).\n\\]\nThen $h \\in H^{1,2}_{loc}(U)$.\n\\end{theo}", "fokkerpllan": "\\begin{equation}\\label{fokkerpllan}\n\\int_{\\mathbb{R}^d} f(y)\\, p(t, x, y)\\,dy - f(x)\n= \\int_0^t \\int_{\\mathbb{R}^d} Lf(y)\\, p(s, x, y)\\,dy\\,ds \\quad \\text{for all } f \\in C_0^\\infty(\\mathbb{R}^d) \\text{ and } t > 0.\n\\end{equation}"}, "pre_theorem_intro_text_len": 6446, "pre_theorem_intro_text": "\\label{intro}\nThe Fokker-Planck equation is fundamentally connected to stochastic analysis, particularly to the theory of stochastic differential equations (SDEs). Given a stochastic process $(X_t)_{t \\geq 0}$ defined on a filtered probability space $(\\Omega, \\mathcal{F}, \\mathbb{P}, (\\mathcal{F}_t)_{t \\geq 0})$ and a standard Brownian motion $(W_t)_{t \\geq 0}$, consider the following SDE:\n\\[\nX_t = x + \\int_0^t \\sigma(X_s) \\, dW_s + \\int_0^t \\mathbf{G}(X_s) \\, ds, \\quad 0 \\leq t < \\infty, \\quad \\mathbb{P}\\text{-a.s.}\n\\]\nwhere $x \\in \\mathbb{R}^d$, $\\sigma$ is a $d \\times d$ matrix of functions, and $\\mathbf{G}$ is a vector field that is locally integrable.\nFor any $f \\in C_0^\\infty(\\mathbb{R}^d)$ and $t > 0$, applying It\\^{o}'s formula (\\cite[Chapter IV, Theorem 3.3]{R99}) yields\n\\begin{align*}\nf(X_t) - f(X_0) &= \\int_0^t \\nabla f(X_s)  \\, dX_s + \\frac{1}{2} \\sum_{i,j=1}^d \\int_0^t \\frac{\\partial^2 f}{\\partial x_i \\partial x_j}(X_s) \\, d\\langle X^i, X^j \\rangle_s \\\\\n&= \\int_0^t \\nabla f(X_s) \\sigma(X_s)\\, dW_s + \\int_0^t Lf(X_s) \\, ds, \\quad \\text{ $\\mathbb{P}$-a.s.},\n\\end{align*}\nwhere the operator $L$ is given by\n$$\nLf = {\\rm trace}(A \\nabla^2 f) + \\langle \\mathbf{G}, \\nabla f \\rangle, \\quad \\text{with } A = \\frac12 \\sigma \\sigma^T.\n$$\nTaking expectations on both sides of the above identity, we obtain\n\\begin{equation}\\label{fokkeriden}\n\\mathbb{E}[f(X_t)] - f(x) = \\int_0^t \\mathbb{E}[L f(X_s)]\\,ds \\quad \\text{for all } f \\in C_0^\\infty(\\mathbb{R}^d) \\text{ and } t > 0,\n\\end{equation}\nwhere $\\mathbb{E}$ is the expectation with respect to $\\mathbb{P}$. Let $p(t, x, \\cdot)$ denote the probability density function of $X_t$ under the probability measure $\\mathbb{P}$ (such a density exists under quite general assumptions and see \\cite{L25ai}). Then, equation \\eqref{fokkeriden} is equivalent to the following integral identity:\n\\begin{equation}\\label{fokkerpllan}\n\\int_{\\mathbb{R}^d} f(y)\\, p(t, x, y)\\,dy - f(x)\n= \\int_0^t \\int_{\\mathbb{R}^d} Lf(y)\\, p(s, x, y)\\,dy\\,ds \\quad \\text{for all } f \\in C_0^\\infty(\\mathbb{R}^d) \\text{ and } t > 0.\n\\end{equation}\nThe family of measures $\\left(p(t, x, y)\\,dy\\right)_{t \\geq 0}$ is referred to as the solution to the Cauchy problem for the Fokker-Planck equation associated with the operator $L$ and the initial distribution $\\delta_x$, the Dirac delta measure at $x$ (cf. \\cite[Definition 6.1, Proposition 6.1.2]{BKRS15}).\nWe refer the reader to the introduction of \\cite{L25ai} for a more detailed discussion of the Fokker-Planck equation. In fact, under appropriate ergodicity assumptions, one can show the existence of a probability measure $\\nu$ on $\\mathcal{B}(\\mathbb{R}^d)$ such that\n\\begin{equation} \\label{weakconver}\n\\lim_{t \\to \\infty} p(t, x, y)\\,dy = \\nu(dy) \\quad \\text{weakly on } \\mathbb{R}^d,\n\\end{equation}\n(see \\cite[Theorem 1.1(iii)]{L25ai}).\nUnder suitable regularity assumptions on the coefficients $A$ and the drift vector field $\\mathbf{G}$, it follows from \\eqref{fokkerpllan} and \\eqref{weakconver} that $L^* \\nu=0$ on $\\mathbb{R}^d$, i.e.\n\\begin{equation} \\label{stafokkerp}\n\\int_{\\mathbb{R}^d} Lf \\, d\\nu = 0  \\quad \\text{for all } f \\in C_0^\\infty(\\mathbb{R}^d).\n\\end{equation}\nA Borel measure $\\nu$ satisfying \\eqref{stafokkerp} is called a solution to the stationary Fokker-Planck equation associated with the operator $L$. In stochastic analysis, $\\nu$ is also called an infinitesimally invariant measure for $(L, C_0^{\\infty}(\\mathbb{R}^d))$. In this paper, we investigate the regularity of the density of $\\nu$. A systematic study of stationary Fokker-Planck equations was carried out in \\cite{BKRS15, BKR01, BKR09, BRS12, BS17, B18}. In particular, if the diffusion coefficients satisfy $a_{ij} \\in H^{1,p}_{loc}(\\mathbb{R}^d)$ for all $1 \\leq i,j \\leq d$ with $p \\in (d, \\infty)$, $A = (a_{ij})_{1 \\leq i,j \\leq d}$ is locally uniformly strictly elliptic and bounded on $\\mathbb{R}^d$ (i.e. \\eqref{locellst} holds for $U=\\mathbb{R}^d$), $\\mathbf{G} \\in L^p_{loc}(\\mathbb{R}^d, \\mathbb{R}^d)$ and \\eqref{stafokkerp} holds, then according to \\cite[Corollary 1.6.9]{BKRS15}, there exists a function $\\rho \\in H^{1,p}_{loc}(\\mathbb{R}^d) \\cap C(\\mathbb{R}^d)$ such that  $\\nu = \\rho \\,dx$.\\\\\nStationary Fokker-Planck equations can be viewed as a class of partial differential equations in which the second-order derivatives act on test functions. The regularity theory for their solutions has developed through successive advancements, beginning with the classical Weyl lemma \\cite{W40}, and subsequently extended in works such as \\cite{S73, F77, BS17}. For further details, we refer the reader to \\cite{L25bv} and the references therein. A recent result (\\cite[Corollary 1.2]{L25bv}) concerning the regularity of solutions to stationary Fokker-Planck equations generalizes \\cite[Corollary 1.6.9]{BKRS15} under the assumption that (cf. Definitions \\ref{defndivmat}, \\ref{defnvmo})\n${\\rm div} A \\in L^p_{loc}(\\mathbb{R}^d, \\mathbb{R}^d)$ with $p \\in (d, \\infty)$, $a_{ij} \\in VMO_{loc}(\\mathbb{R}^d)$ for all $1 \\leq i,j \\leq d$, and that the matrix $A = (a_{ij})_{1 \\leq i,j \\leq d}$ is locally uniformly strictly elliptic and bounded on $\\mathbb{R}^d$. If, in addition, $\\mathbf{G} \\in L^p_{loc}(\\mathbb{R}^d, \\mathbb{R}^d)$ and \\eqref{stafokkerp} holds, then it has been shown in \\cite[Corollary 1.2]{L25bv} that there exists a function $\\rho \\in H^{1,p}_{loc}(\\mathbb{R}^d) \\cap C(\\mathbb{R}^d)$ such that  $\\nu = \\rho\\,dx$.\\\\\nHowever, the assumption $\\mathbf{G} \\in L_{loc}^p(\\mathbb{R}^d, \\mathbb{R}^d)$ may still be regarded as rather restrictive. It is therefore natural to consider relaxing this condition to the case $\\mathbf{G} \\in L^2_{loc}(\\mathbb{R}^d, \\mathbb{R}^d)$. In fact, as shown in \\cite{F81, St99}, a stochastic process associated with the operator $L$ can be constructed even when the drift is merely locally $L^2$-integrable with respect to a reference measure. This observation motivates the extension of the results in \\cite[Corollary 1.6.9]{BKRS15} and \\cite[Corollary 1.2]{L25bv} to this more general setting.\nIn particular, when the diffusion matrix is the identity, the assumption $\\mathbf{G} \\in L^2_{loc}(\\mathbb{R}^d, \\mathbb{R}^d)$ has been considered in \\cite[Theorem 1(ii)]{BKR97}, where corresponding regularity results for the density of the solution were established. \\\\\nIn our setting, however, we require the a priori assumption that the solution $\\nu$ admits a density belonging to $L_{loc}^\\infty(U)$.", "context": " Fokker-Planck equation. In fact, under appropriate ergodicity assumptions, one can show the existence of a probability measure $\\nu$ on $\\mathcal{B}(\\mathbb{R}^d)$ such that\n\\begin{equation} \\label{weakconver}\n\\lim_{t \\to \\infty} p(t, x, y)\\,dy = \\nu(dy) \\quad \\text{weakly on } \\mathbb{R}^d,\n\\end{equation}\n(see \\cite[Theorem 1.1(iii)]{L25ai}).\nUnder suitable regularity assumptions on the coefficients $A$ and the drift vector field $\\mathbf{G}$, it follows from \\eqref{fokkerpllan} and \\eqref{weakconver} that $L^* \\nu=0$ on $\\mathbb{R}^d$, i.e.\n\\begin{equation} \\label{stafokkerp}\n\\int_{\\mathbb{R}^d} Lf \\, d\\nu = 0 \\quad \\text{for all } f \\in C_0^\\infty(\\mathbb{R}^d).\n\\end{equation}\nA Borel measure $\\nu$ satisfying \\eqref{stafokkerp} is called a solution to the stationary Fokker-Planck equation associated with the operator $L$. In stochastic analysis, $\\nu$ is also called an infinitesimally invariant measure for $(L, C_0^{\\infty}(\\mathbb{R}^d))$. In this paper, we investigate the regularity of the density of $\\nu$. A systematic study of stationary Fokker-Planck equations was carried out in \\cite{BKRS15, BKR01, BKR09, BRS12, BS17, B18}. In particular, if the diffusion coefficients satisfy $a_{ij} \\in H^{1,p}_{loc}(\\mathbb{R}^d)$ for all $1 \\leq i,j \\leq d$ with $p \\in (d, \\infty)$, $A = (a_{ij})_{1 \\leq i,j \\leq d}$ is locally uniformly strictly elliptic and bounded on $\\mathbb{R}^d$ (i.e. \\eqref{locellst} holds for $U=\\mathbb{R}^d$), $\\mathbf{G} \\in L^p_{loc}(\\mathbb{R}^d, \\mathbb{R}^d)$ and \\eqref{stafokkerp} holds, then according to \\cite[Corollary 1.6.9]{BKRS15}, there exists a function $\\rho \\in H^{1,p}_{loc}(\\mathbb{R}^d) \\cap C(\\mathbb{R}^d)$ such that $\\nu = \\rho \\,dx$.\\\\\nStationary Fokker-Planck equations can be viewed as a class of partial differential equations in which the second-order derivatives act on test functions. The regularity theory for their solutions has developed through successive advancements, beginning with the classical Weyl lemma \\cite{W40}, and subsequently extended in works such as \\cite{S73, F77, BS17}. For further details, we refer the reader to \\cite{L25bv} and the references therein. A recent result (\\cite[Corollary 1.2]{L25bv}) concerning the regularity of solutions to stationary Fokker-Planck equations generalizes \\cite[Corollary 1.6.9]{BKRS15} under the assumption that (cf. Definitions \\ref{defndivmat}, \\ref{defnvmo})\n${\\rm div} A \\in L^p_{loc}(\\mathbb{R}^d, \\mathbb{R}^d)$ with $p \\in (d, \\infty)$, $a_{ij} \\in VMO_{loc}(\\mathbb{R}^d)$ for all $1 \\leq i,j \\leq d$, and that the matrix $A = (a_{ij})_{1 \\leq i,j \\leq d}$ is locally uniformly strictly elliptic and bounded on $\\mathbb{R}^d$. If, in addition, $\\mathbf{G} \\in L^p_{loc}(\\mathbb{R}^d, \\mathbb{R}^d)$ and \\eqref{stafokkerp} holds, then it has been shown in \\cite[Corollary 1.2]{L25bv} that there exists a function $\\rho \\in H^{1,p}_{loc}(\\mathbb{R}^d) \\cap C(\\mathbb{R}^d)$ such that $\\nu = \\rho\\,dx$.\\\\\nHowever, the assumption $\\mathbf{G} \\in L_{loc}^p(\\mathbb{R}^d, \\mathbb{R}^d)$ may still be regarded as rather restrictive. It is therefore natural to consider relaxing this condition to the case $\\mathbf{G} \\in L^2_{loc}(\\mathbb{R}^d, \\mathbb{R}^d)$. In fact, as shown in \\cite{F81, St99}, a stochastic process associated with the operator $L$ can be constructed even when the drift is merely locally $L^2$-integrable with respect to a reference measure. This observation motivates the extension of the results in \\cite[Corollary 1.6.9]{BKRS15} and \\cite[Corollary 1.2]{L25bv} to this more general setting.\nIn particular, when the diffusion matrix is the identity, the assumption $\\mathbf{G} \\in L^2_{loc}(\\mathbb{R}^d, \\mathbb{R}^d)$ has been considered in \\cite[Theorem 1(ii)]{BKR97}, where corresponding regularity results for the density of the solution were established. \\\\\nIn our setting, however, we require the a priori assumption that the solution $\\nu$ admits a density belonging to $L_{loc}^\\infty(U)$.\n\n\\begin{defn} \\label{defndivmat}\nLet $U$ be an open subset of $\\mathbb{R}^d$, $\\mathbf{E} = (e_1, \\dots, e_d) \\in L^1_{loc}(U, \\mathbb{R}^d)$ and $B = (b_{ij})_{1 \\leq i,j \\leq d}$ be a (possibly non-symmetric) matrix of functions with $b_{ij} \\in L^1_{loc}(U)$ for all $1 \\leq i,j \\leq d$. Let $\\mathbf{F} \\in L^1_{loc}(U, \\mathbb{R}^d)$ and $f \\in L^1_{loc}(U)$.\n\\begin{itemize}\n\\item[(i)] We say that ${\\rm div} B = \\mathbf{E}$ if\n\\[\n\\int_U \\sum_{i,j=1}^d b_{ij} \\, \\partial_i \\phi_j \\, dx = - \\int_U \\sum_{j=1}^d e_j \\, \\phi_j \\, dx \\quad \\text{for all } (\\phi_1, \\dots, \\phi_d) \\in C_0^\\infty(U)^d.\n\\]\n\\item[(ii)] We say that ${\\rm div} \\mathbf{F} = f$ if\n\\[\n\\int_U \\langle \\mathbf{F}, \\nabla \\varphi \\rangle \\, dx = - \\int_U f \\varphi \\, dx \\quad \\text{for all } \\varphi \\in C_0^\\infty(U).\n\\]\n\\end{itemize}\n\\end{defn}\n\n\\begin{defn} \\label{defnvmo}\nLet $\\omega$ be a positive continuous function on $[0,\\infty)$ with $\\omega(0) = 0$. The space $VMO_\\omega$ consists of all functions $g \\in L^1_{loc}(\\mathbb{R}^d)$ such that\n\\[\n\\sup_{z \\in \\mathbb{R}^d,\\, r < R} \\, r^{-2d} \\int_{B_r(z)} \\int_{B_r(z)} |g(x) - g(y)| \\, dx \\, dy \\leq \\omega(R) \\quad \\text{for all } R \\in (0,\\infty).\n\\]\nLet $B$ be an open ball in $\\mathbb{R}^d$. We define $VMO(B)$ as the set of all functions $f \\in L^1(B)$ for which there exists an extension $\\tilde{f} \\in VMO_{\\omega}$  (for some modulus $\\omega$) such that $\\tilde{f}|_B = f$. For an open set $U \\subset \\mathbb{R}^d$, the space $VMO_{loc}(U)$ is defined as the collection of all functions $f \\in L^1_{loc}(U)$ such that $f \\in VMO(B)$ for every open ball $B \\subset \\mathbb{R}^d$ with $\\overline{B} \\subset U$.\n\\end{defn}\n\n\\begin{equation} \\label{locellst}\n\\lambda_V \\| \\xi \\|^2 \\leq \\langle A(x) \\xi, \\xi \\rangle, \\quad \\;\\; \\max_{1 \\leq i,j \\leq d} |a_{ij}(x)| \\leq M_V\n\\quad \\text{for a.e. } x \\in V \\text{ and all } \\xi \\in \\mathbb{R}^d.\n\\end{equation}\n\n\\begin{equation} \\label{stafokkerp}\n\\int_{\\mathbb{R}^d} Lf \\, d\\nu = 0  \\quad \\text{for all } f \\in C_0^\\infty(\\mathbb{R}^d).\n\\end{equation}", "full_context": "Fokker-Planck equation. In fact, under appropriate ergodicity assumptions, one can show the existence of a probability measure $\\nu$ on $\\mathcal{B}(\\mathbb{R}^d)$ such that\n\\begin{equation} \\label{weakconver}\n\\lim_{t \\to \\infty} p(t, x, y)\\,dy = \\nu(dy) \\quad \\text{weakly on } \\mathbb{R}^d,\n\\end{equation}\n(see \\cite[Theorem 1.1(iii)]{L25ai}).\nUnder suitable regularity assumptions on the coefficients $A$ and the drift vector field $\\mathbf{G}$, it follows from \\eqref{fokkerpllan} and \\eqref{weakconver} that $L^* \\nu=0$ on $\\mathbb{R}^d$, i.e.\n\\begin{equation} \\label{stafokkerp}\n\\int_{\\mathbb{R}^d} Lf \\, d\\nu = 0 \\quad \\text{for all } f \\in C_0^\\infty(\\mathbb{R}^d).\n\\end{equation}\nA Borel measure $\\nu$ satisfying \\eqref{stafokkerp} is called a solution to the stationary Fokker-Planck equation associated with the operator $L$. In stochastic analysis, $\\nu$ is also called an infinitesimally invariant measure for $(L, C_0^{\\infty}(\\mathbb{R}^d))$. In this paper, we investigate the regularity of the density of $\\nu$. A systematic study of stationary Fokker-Planck equations was carried out in \\cite{BKRS15, BKR01, BKR09, BRS12, BS17, B18}. In particular, if the diffusion coefficients satisfy $a_{ij} \\in H^{1,p}_{loc}(\\mathbb{R}^d)$ for all $1 \\leq i,j \\leq d$ with $p \\in (d, \\infty)$, $A = (a_{ij})_{1 \\leq i,j \\leq d}$ is locally uniformly strictly elliptic and bounded on $\\mathbb{R}^d$ (i.e. \\eqref{locellst} holds for $U=\\mathbb{R}^d$), $\\mathbf{G} \\in L^p_{loc}(\\mathbb{R}^d, \\mathbb{R}^d)$ and \\eqref{stafokkerp} holds, then according to \\cite[Corollary 1.6.9]{BKRS15}, there exists a function $\\rho \\in H^{1,p}_{loc}(\\mathbb{R}^d) \\cap C(\\mathbb{R}^d)$ such that $\\nu = \\rho \\,dx$.\\\\\nStationary Fokker-Planck equations can be viewed as a class of partial differential equations in which the second-order derivatives act on test functions. The regularity theory for their solutions has developed through successive advancements, beginning with the classical Weyl lemma \\cite{W40}, and subsequently extended in works such as \\cite{S73, F77, BS17}. For further details, we refer the reader to \\cite{L25bv} and the references therein. A recent result (\\cite[Corollary 1.2]{L25bv}) concerning the regularity of solutions to stationary Fokker-Planck equations generalizes \\cite[Corollary 1.6.9]{BKRS15} under the assumption that (cf. Definitions \\ref{defndivmat}, \\ref{defnvmo})\n${\\rm div} A \\in L^p_{loc}(\\mathbb{R}^d, \\mathbb{R}^d)$ with $p \\in (d, \\infty)$, $a_{ij} \\in VMO_{loc}(\\mathbb{R}^d)$ for all $1 \\leq i,j \\leq d$, and that the matrix $A = (a_{ij})_{1 \\leq i,j \\leq d}$ is locally uniformly strictly elliptic and bounded on $\\mathbb{R}^d$. If, in addition, $\\mathbf{G} \\in L^p_{loc}(\\mathbb{R}^d, \\mathbb{R}^d)$ and \\eqref{stafokkerp} holds, then it has been shown in \\cite[Corollary 1.2]{L25bv} that there exists a function $\\rho \\in H^{1,p}_{loc}(\\mathbb{R}^d) \\cap C(\\mathbb{R}^d)$ such that $\\nu = \\rho\\,dx$.\\\\\nHowever, the assumption $\\mathbf{G} \\in L_{loc}^p(\\mathbb{R}^d, \\mathbb{R}^d)$ may still be regarded as rather restrictive. It is therefore natural to consider relaxing this condition to the case $\\mathbf{G} \\in L^2_{loc}(\\mathbb{R}^d, \\mathbb{R}^d)$. In fact, as shown in \\cite{F81, St99}, a stochastic process associated with the operator $L$ can be constructed even when the drift is merely locally $L^2$-integrable with respect to a reference measure. This observation motivates the extension of the results in \\cite[Corollary 1.6.9]{BKRS15} and \\cite[Corollary 1.2]{L25bv} to this more general setting.\nIn particular, when the diffusion matrix is the identity, the assumption $\\mathbf{G} \\in L^2_{loc}(\\mathbb{R}^d, \\mathbb{R}^d)$ has been considered in \\cite[Theorem 1(ii)]{BKR97}, where corresponding regularity results for the density of the solution were established. \\\\\nIn our setting, however, we require the a priori assumption that the solution $\\nu$ admits a density belonging to $L_{loc}^\\infty(U)$.\n\n\\begin{defn} \\label{defndivmat}\nLet $U$ be an open subset of $\\mathbb{R}^d$, $\\mathbf{E} = (e_1, \\dots, e_d) \\in L^1_{loc}(U, \\mathbb{R}^d)$ and $B = (b_{ij})_{1 \\leq i,j \\leq d}$ be a (possibly non-symmetric) matrix of functions with $b_{ij} \\in L^1_{loc}(U)$ for all $1 \\leq i,j \\leq d$. Let $\\mathbf{F} \\in L^1_{loc}(U, \\mathbb{R}^d)$ and $f \\in L^1_{loc}(U)$.\n\\begin{itemize}\n\\item[(i)] We say that ${\\rm div} B = \\mathbf{E}$ if\n\\[\n\\int_U \\sum_{i,j=1}^d b_{ij} \\, \\partial_i \\phi_j \\, dx = - \\int_U \\sum_{j=1}^d e_j \\, \\phi_j \\, dx \\quad \\text{for all } (\\phi_1, \\dots, \\phi_d) \\in C_0^\\infty(U)^d.\n\\]\n\\item[(ii)] We say that ${\\rm div} \\mathbf{F} = f$ if\n\\[\n\\int_U \\langle \\mathbf{F}, \\nabla \\varphi \\rangle \\, dx = - \\int_U f \\varphi \\, dx \\quad \\text{for all } \\varphi \\in C_0^\\infty(U).\n\\]\n\\end{itemize}\n\\end{defn}\n\n\\begin{defn} \\label{defnvmo}\nLet $\\omega$ be a positive continuous function on $[0,\\infty)$ with $\\omega(0) = 0$. The space $VMO_\\omega$ consists of all functions $g \\in L^1_{loc}(\\mathbb{R}^d)$ such that\n\\[\n\\sup_{z \\in \\mathbb{R}^d,\\, r < R} \\, r^{-2d} \\int_{B_r(z)} \\int_{B_r(z)} |g(x) - g(y)| \\, dx \\, dy \\leq \\omega(R) \\quad \\text{for all } R \\in (0,\\infty).\n\\]\nLet $B$ be an open ball in $\\mathbb{R}^d$. We define $VMO(B)$ as the set of all functions $f \\in L^1(B)$ for which there exists an extension $\\tilde{f} \\in VMO_{\\omega}$  (for some modulus $\\omega$) such that $\\tilde{f}|_B = f$. For an open set $U \\subset \\mathbb{R}^d$, the space $VMO_{loc}(U)$ is defined as the collection of all functions $f \\in L^1_{loc}(U)$ such that $f \\in VMO(B)$ for every open ball $B \\subset \\mathbb{R}^d$ with $\\overline{B} \\subset U$.\n\\end{defn}\n\n\\begin{equation} \\label{locellst}\n\\lambda_V \\| \\xi \\|^2 \\leq \\langle A(x) \\xi, \\xi \\rangle, \\quad \\;\\; \\max_{1 \\leq i,j \\leq d} |a_{ij}(x)| \\leq M_V\n\\quad \\text{for a.e. } x \\in V \\text{ and all } \\xi \\in \\mathbb{R}^d.\n\\end{equation}\n\n\\begin{equation} \\label{stafokkerp}\n\\int_{\\mathbb{R}^d} Lf \\, d\\nu = 0  \\quad \\text{for all } f \\in C_0^\\infty(\\mathbb{R}^d).\n\\end{equation}\n\n\\noindent \n{Keywords: Stationary Fokker-Planck equations, Elliptic regularity, Resolvent, Weak maximum principle, Singular drifts, Invariant measures \n}\n\n\\section{Application to the uniqueness results} \\label{sec4}\nThe local regularity result established in the main theorem enables us to convert solutions of the stationary Fokker–Planck equations into weak solutions of a divergence form equation. Based on this, our main theorem can be applied to derive a uniqueness result for solutions to the stationary Fokker–Planck equations.\n\\begin{theo} \\label{applicationth}\nLet $A = (a_{ij})_{1 \\leq i,j \\leq d}$ be a symmetric matrix of functions whose components satisfy $a_{ij} \\in VMO_{loc}(\\mathbb{R}^d)$ for all $1 \\leq i,j \\leq d$. Assume that the condition \\eqref{ellipticit} holds for some strictly positive constants $\\lambda$ and $M$. Let $C = (c_{ij})_{1 \\leq i,j \\leq d}$ be an anti-symmetric matrix of functions satisfying $c_{ij} = -c_{ji} \\in L^{\\infty}(\\mathbb{R}^d)$ for all $1 \\leq i,j \\leq d$. Assume that ${\\rm div}(A+C) \\in L^2_{loc}(\\mathbb{R}^d, \\mathbb{R}^d)$. Let $\\rho \\in H^{1,2}_{loc}(\\mathbb{R}^d) \\cap L^{\\infty}(\\mathbb{R}^d)$ be such that for some constant $\\theta>0$, $\\rho(x) > \\theta$ for a.e. $x \\in \\mathbb{R}^d$. Then, the following statements hold:\n\\begin{itemize}\n\\item[(i)]\n\\begin{align*}\n \\int_{\\mathbb{R}^d} \\bigg( {\\rm trace}(A \\nabla^2 f) + \\left\\langle {\\rm div}(A+C) + \\frac{1}{\\rho}(A+C^T) \\nabla \\rho, \\nabla f \\right\\rangle \\bigg)\\, \\rho\\, dx=0 \\quad \\text{ for all $f \\in C_0^{\\infty}(\\mathbb{R}^d)$.}\n\\end{align*}\n\\item[(ii)]\nIf $h \\in L^{\\infty}_{loc}(\\mathbb{R}^d)$ with $h \\geq 0$ in $\\mathbb{R}^d$ satisfies\n\\[\n\\int_{\\mathbb{R}^d} \\bigg( {\\rm trace}(A \\nabla^2 f) + \\left\\langle {\\rm div}(A+C) + \\frac{1}{\\rho}(A+C^T) \\nabla \\rho, \\nabla f \\right\\rangle \\bigg)\\, h\\, dx = 0 \\quad \\text{for all } f \\in C_0^{\\infty}(\\mathbb{R}^d),\n\\]\nthen there exists a constant $c \\geq 0$ such that $h = c \\rho$ in $\\mathbb{R}^d$.\n\\item[(iii)]\nIf $h \\in L^{\\infty}(\\mathbb{R}^d)$ satisfies\n\\[\n\\int_{\\mathbb{R}^d} \\bigg( {\\rm trace}(A \\nabla^2 f) + \\left\\langle {\\rm div}(A+C) + \\frac{1}{\\rho}(A+C^T) \\nabla \\rho, \\nabla f \\right\\rangle \\bigg)\\, h\\, dx = 0 \\quad \\text{for all } f \\in C_0^{\\infty}(\\mathbb{R}^d),\n\\]\nthen there exists a constant $c \\in \\mathbb{R}$ such that $h = c \\rho$ in $\\mathbb{R}^d$.\n\\end{itemize}\n\\end{theo}\n\\begin{proof}\n(i) Let $f \\in C_0^{\\infty}(\\mathbb{R}^d)$.\nThen, by \\cite[Proposition 5.7]{L25bv}, \n\\begin{equation} \\label{divtracrep}\n \\frac{1}{\\rho} {\\rm div} \\Big( \\rho(A+C) \\nabla f \\Big) = {\\rm trace}(A \\nabla^2 f) + \\left\\langle {\\rm div}(A+C) + \\frac{1}{\\rho}(A+C^T) \\nabla \\rho, \\nabla f \\right\\rangle.\n\\end{equation}\nTake an open ball $V$ so that ${\\rm supp}(f) \\subset V$. Let $\\chi \\in C_0^{\\infty}(\\mathbb{R}^d)$ with $\\chi \\equiv 1$ on $\\overline{V}$. Then,\n\\begin{align*}\n&\\int_{\\mathbb{R}^d} \\bigg( {\\rm trace}(A \\nabla^2 f) + \\left\\langle {\\rm div}(A+C) + \\frac{1}{\\rho}(A+C^T) \\nabla \\rho, \\nabla f \\right\\rangle \\bigg)\\, \\rho\\, dx \\\\\n&=\\int_{\\mathbb{R}^d} {\\rm div} \\Big( \\rho(A+C) \\nabla f \\Big)\\,dx = \\int_{\\mathbb{R}^d} {\\rm div} \\Big( \\rho(A+C) \\nabla f \\Big) \\chi \\,dx \\\\\n& = -\\int_{V} \\langle \\rho(A+C) \\nabla f, \\nabla \\chi \\rangle\\,dx=0.\n\\end{align*}\n(ii) First, by Theorem \\ref{maintheore}, $h \\in H^{1,2}_{loc}(\\mathbb{R}^d)$.\nLet $\\hat{h}:=\\frac{h}{\\rho}$. Then, $\\hat{h} \\in H^{1,2}_{loc}(\\mathbb{R}^d)$ with $\\hat{h} \\geq 0$ in $\\mathbb{R}^d$ and it follows from \\eqref{divtracrep} that\n\\begin{align*}\n0=&\\int_{\\mathbb{R}^d} \\bigg( {\\rm trace}(A \\nabla^2 f) + \\left\\langle {\\rm div}(A+C) + \\frac{1}{\\rho}(A+C^T) \\nabla \\rho, \\nabla f \\right\\rangle \\bigg)\\, h\\, dx \\\\\n& = \\int_{\\mathbb{R}^d} {\\rm div} \\Big( \\rho(A+C) \\nabla f \\Big) \\cdot \\hat{h}\\,dx = -\\int_{\\mathbb{R}^d} \\langle \\rho (A+C^T) \\nabla \\hat{h}, \\nabla f \\rangle \\,dx \\quad \\text{ for all $f \\in C_0^{\\infty}(\\mathbb{R}^d)$}.\n\\end{align*}\nBy the Liouville-type theorem \\cite[Theorem 9.11(i)]{F07}, $\\hat{h}$ is constant, so that the assertion follows. \\\\ \\\\\n(iii) As in (ii), $h \\in H^{1,2}_{loc}(\\mathbb{R}^d)$. Moreover, $\\hat{h}:=\\frac{h}{\\rho} \\in H^{1,2}_{loc}(\\mathbb{R}^d) \\cap L^{\\infty}(\\mathbb{R}^d)$ satisfies\n$$\n \\int_{\\mathbb{R}^d} \\langle \\rho (A+C^T) \\nabla \\hat{h}, \\nabla f \\rangle \\,dx=0 \\quad \\text{ for all $f \\in C_0^{\\infty}(\\mathbb{R}^d)$}.\n$$\nUsing the Liouville-type theorem \\cite[Theorem 9.11(ii)]{F07}, $\\hat{h}$ is constant, so that the assertion follows.\n\\end{proof}\n\\centerline{}\n\\noindent\nTo present a concrete example, we consider the following function:\nLet $\\varepsilon \\in (0, 10^{-2026})$ be fixed. Define\n\\begin{equation} \\label{defnpsi}\n\\Psi(t) := \n\\begin{cases}\n1, & t \\in (-\\infty, -1), \\\\[6pt]\n1 + \\displaystyle\\int_{-1}^{t} |s|^{-\\frac{1}{2+\\varepsilon}}\\,ds, & t \\in [-1,1], \\\\[6pt]\n1 + \\displaystyle\\int_{-1}^{1} |s|^{-\\frac{1}{2+\\varepsilon}}\\,ds, & t \\in (1, \\infty).\n\\end{cases}\n\\end{equation}\nThen $\\Psi \\in C(\\mathbb{R}) \\cap L^{\\infty}(\\mathbb{R})$, and its derivative $\\Psi' \\in \\bigcap_{p \\in (1,\\; 2+\\varepsilon)} L^p(\\mathbb{R})$ is given by\n\\[\n\\Psi'(t) := \n\\begin{cases}\n0, & t \\in (-\\infty, -1), \\\\[6pt]\n|t|^{-\\frac{1}{2+\\varepsilon}}, & t \\in (-1,0) \\cup (0,1), \\\\[6pt]\n0, & t \\in (1, \\infty).\n\\end{cases}\n\\]\nIn particular, $\\Psi' \\notin L_{loc}^{2+\\varepsilon}(\\mathbb{R})$, and\n\\[\n1 \\leq \\Psi(t) \\leq 1 + \\int_{-1}^{1} |s|^{-\\frac{1}{2+\\varepsilon}}\\,ds \\quad \\text{for all } t \\in \\mathbb{R}.\n\\]\n\n\\begin{exam} \\label{ourexam1}\nLet $d \\geq 2$ and $\\varepsilon \\in (0,10^{-2026})$ be fixed. Let $\\Psi$ be the function defined in \\eqref{defnpsi}. Define\n$$\n\\rho(x_1, \\ldots, x_d) = \\prod_{i=1}^d \\Psi(x_i), \\quad x =(x_1, \\ldots, x_d) \\in \\mathbb{R}^d.\n$$\nThen $\\rho \\in C(\\mathbb{R}^d) \\cap L^{\\infty}(\\mathbb{R}^d) \\cap H^{1,2}_{loc}(\\mathbb{R}^d)$, and $\\nabla \\rho \\notin L^{2+\\varepsilon}_{loc}(\\mathbb{R}^d)$. Moreover,\n$$\n1 \\leq \\rho(x) \\leq \\left( 1 + \\int_{-1}^{1} |s|^{-\\frac{1}{2+\\varepsilon}}\\,ds \\right)^d \\quad \\text{for all } x \\in \\mathbb{R}^d.\n$$\nLet $A = I$, $\\mathbf{G} = \\frac{1}{\\rho} \\nabla \\rho$. Then\n\\begin{equation} \\label{infinitesinvar}\n\\int_{\\mathbb{R}^d} \\left( {\\rm trace}(A \\nabla^2 f) + \\langle \\mathbf{G}, \\nabla f \\rangle \\right) \\rho\\,dx = 0 \\quad \\text{for all } f \\in C_0^{\\infty}(\\mathbb{R}^d).\n\\end{equation}\nAssume that $h \\in L^{\\infty}_{loc}(\\mathbb{R}^d)$ with $h \\geq 0$ in $\\mathbb{R}^d$ satisfies\n\\begin{equation} \\label{infininvah}\n\\int_{\\mathbb{R}^d} \\left({\\rm trace}(A \\nabla^2 f) + \\langle \\mathbf{G}, \\nabla f \\rangle \\right) h \\,dx = 0 \\quad \\text{for all } f \\in C_0^{\\infty}(\\mathbb{R}^d).\n\\end{equation}\nThen it follows from Theorem \\ref{applicationth}(ii) that $h$ is a constant multiple of $\\rho$. If $h \\in L^{\\infty}(\\mathbb{R}^d)$ satisfies \\eqref{infininvah}, then it follows from Theorem \\ref{applicationth}(iii) that $h$ is also a constant multiple of $\\rho$.\n\\end{exam}", "post_theorem_intro_text_len": 2340, "post_theorem_intro_text": "The main idea of this work is to construct a constant $\\lambda_0 > 0$ and a sub-Markovian resolvent $\\left(R_{\\alpha}\\right)_{\\alpha \\geq \\lambda_0}$ associated with the principal elliptic operator $L^A = {\\rm trace}(A \\nabla^2)$, and to show that the function $h$ can be represented as the local weak $H^{1,2}$-limit of the family $(\\alpha R_{\\alpha} h)_{\\alpha \\geq \\lambda_0}$ as $\\alpha \\rightarrow \\infty$. This idea is inspired by \\cite[Theorem 2.1]{St99}. In that result, however, the diffusion coefficients $a_{ij}$ are assumed to be locally Hölder continuous for all $1 \\leq i,j \\leq d$, and the density of the solution measure $\\nu$ is assumed to belong to $L^{\\infty}(\\mathbb{R}^d)$. By contrast, in the present work, we establish resolvent estimates under the weaker assumptions that the coefficients belong to $VMO_{loc}(\\mathbb{R}^d)$ and that ${\\rm div} A \\in L^2_{loc}(\\mathbb{R}^d, \\mathbb{R}^d)$, which enable us to derive the main result, Theorem \\ref{maintheore}.\\\\\nIt is worth noting that determining the extent to which the assumptions in our main theorem can be further relaxed remains an open and challenging problem. In particular, it is unclear whether the assumptions that $a_{ij} \\in VMO_{loc}(U)$ for all $1 \\leq i,j \\leq d$ and ${\\rm div} A \\in L^2_{loc}(U, \\mathbb{R}^d)$ can be weakened. Similarly, whether the a priori condition $h \\in L^{\\infty}_{loc}(U)$ can be replaced by the weaker assumption $h \\in L^2_{loc}(U)$ remains open. Furthermore, one may ask whether the result could still hold under the sole assumptions that the solution measure $\\nu$ is locally finite and that $Lf \\in L^1_{loc}(U, \\nu)$ for every $f \\in C_0^{\\infty}(U)$. Understanding the regularity of solutions to such stationary Fokker-Planck equations is expected to play a crucial role in the study of invariant measures associated with stochastic differential equations having singular coefficients. \\\\\nThis paper is organized as follows. Section \\ref{sec2} introduces the basic notations and definitions used throughout this paper. Section \\ref{sec3} presents the resolvent approaches to elliptic regularity and provides the proof of the main result, Theorem \\ref{maintheore}, at the end of the section. Section \\ref{sec4} discusses applications to uniqueness results based on the main result and presents some concrete examples.", "sketch": "To derive Theorem~\\ref{maintheore}, the paper’s stated strategy is to (i) “construct a constant $\\lambda_0>0$ and a sub-Markovian resolvent $(R_\\alpha)_{\\alpha\\ge \\lambda_0}$ associated with the principal elliptic operator $L^A={\\rm trace}(A\\nabla^2)$”, and then (ii) “show that the function $h$ can be represented as the local weak $H^{1,2}$-limit of the family $(\\alpha R_\\alpha h)_{\\alpha\\ge \\lambda_0}$ as $\\alpha\\to\\infty$.” The introduction emphasizes that the proof relies on “resolvent estimates under the weaker assumptions that the coefficients belong to $VMO_{loc}(\\mathbb{R}^d)$ and that ${\\rm div}A\\in L^2_{loc}(\\mathbb{R}^d,\\mathbb{R}^d)$, which enable us to derive the main result, Theorem~\\ref{maintheore}.”", "expanded_sketch": "To prove the main theorem, the paper’s stated strategy is to (i) “construct a constant $\\lambda_0>0$ and a sub-Markovian resolvent $(R_\\alpha)_{\\alpha\\ge \\lambda_0}$ associated with the principal elliptic operator $L^A={\\rm trace}(A\\nabla^2)$”, and then (ii) “show that the function $h$ can be represented as the local weak $H^{1,2}$-limit of the family $(\\alpha R_\\alpha h)_{\\alpha\\ge \\lambda_0}$ as $\\alpha\\to\\infty$.” The introduction emphasizes that the proof relies on “resolvent estimates under the weaker assumptions that the coefficients belong to $VMO_{loc}(\\mathbb{R}^d)$ and that ${\\rm div}A\\in L^2_{loc}(\\mathbb{R}^d,\\mathbb{R}^d)$, which enable us to derive the main result.”", "expanded_theorem": "\\label{maintheore}\nLet $U \\subset \\mathbb{R}^d$ be a (possibly unbounded) open set with $d \\geq 2$. Assume that $\\mathbf{G} \\in L^2_{loc}(U, \\mathbb{R}^d)$ and $c \\in L^1_{loc}(U)$. Let $A = (a_{ij})_{1 \\leq i,j \\leq d}$ be a symmetric matrix of functions such that $a_{ij} \\in VMO_{loc}(U)$ for all $1 \\leq i,j \\leq d$, where $VMO_{loc}(U)$ is defined as follows.\n\n\\begin{defn} \\label{defnvmo}\nLet $\\omega$ be a positive continuous function on $[0,\\infty)$ with $\\omega(0) = 0$. The space $VMO_\\omega$ consists of all functions $g \\in L^1_{loc}(\\mathbb{R}^d)$ such that\n\\[\n\\sup_{z \\in \\mathbb{R}^d,\\, r < R} \\, r^{-2d} \\int_{B_r(z)} \\int_{B_r(z)} |g(x) - g(y)| \\, dx \\, dy \\leq \\omega(R) \\quad \\text{for all } R \\in (0,\\infty).\n\\]\nLet $B$ be an open ball in $\\mathbb{R}^d$. We define $VMO(B)$ as the set of all functions $f \\in L^1(B)$ for which there exists an extension $\\tilde{f} \\in VMO_{\\omega}$  (for some modulus $\\omega$) such that $\\tilde{f}|_B = f$. For an open set $U \\subset \\mathbb{R}^d$, the space $VMO_{loc}(U)$ is defined as the collection of all functions $f \\in L^1_{loc}(U)$ such that $f \\in VMO(B)$ for every open ball $B \\subset \\mathbb{R}^d$ with $\\overline{B} \\subset U$.\n\\end{defn}\n\nAssume moreover that ${\\rm div} A \\in L^2_{loc}(U, \\mathbb{R}^d)$, where the meaning of ${\\rm div}$ for a matrix field is as follows.\n\n\\begin{defn} \\label{defndivmat}\nLet $U$ be an open subset of $\\mathbb{R}^d$, $\\mathbf{E} = (e_1, \\dots, e_d) \\in L^1_{loc}(U, \\mathbb{R}^d)$ and $B = (b_{ij})_{1 \\leq i,j \\leq d}$ be a (possibly non-symmetric) matrix of functions with $b_{ij} \\in L^1_{loc}(U)$ for all $1 \\leq i,j \\leq d$. Let $\\mathbf{F} \\in L^1_{loc}(U, \\mathbb{R}^d)$ and $f \\in L^1_{loc}(U)$.\n\\begin{itemize}\n\\item[(i)] We say that ${\\rm div} B = \\mathbf{E}$ if\n\\[\n\\int_U \\sum_{i,j=1}^d b_{ij} \\, \\partial_i \\phi_j \\, dx = - \\int_U \\sum_{j=1}^d e_j \\, \\phi_j \\, dx \\quad \\text{for all } (\\phi_1, \\dots, \\phi_d) \\in C_0^\\infty(U)^d.\n\\]\n\\item[(ii)] We say that ${\\rm div} \\mathbf{F} = f$ if\n\\[\n\\int_U \\langle \\mathbf{F}, \\nabla \\varphi \\rangle \\, dx = - \\int_U f \\varphi \\, dx \\quad \\text{for all } \\varphi \\in C_0^\\infty(U).\n\\]\n\\end{itemize}\n\\end{defn}\n\nAlso assume that $A$ is locally uniformly strictly elliptic and bounded on $U$, i.e. for every open ball $V$ with $\\overline{V} \\subset U$, there exist constants $\\lambda_V, M_V > 0$ such that\n\\begin{equation} \\label{locellst}\n\\lambda_V \\| \\xi \\|^2 \\leq \\langle A(x) \\xi, \\xi \\rangle, \\quad \\;\\; \\max_{1 \\leq i,j \\leq d} |a_{ij}(x)| \\leq M_V\n\\quad \\text{for a.e. } x \\in V \\text{ and all } \\xi \\in \\mathbb{R}^d.\n\\end{equation}\nLet $\\tilde{\\mathbf{F}} \\in L^2_{loc}(U, \\mathbb{R}^d)$ and $\\tilde{f} \\in L^1_{loc}(U)$. Suppose that $h \\in L^{\\infty}_{loc}(U)$ satisfies\n\\[\n\\int_U \\left({\\rm trace}(A \\nabla^2 \\varphi) + \\langle \\mathbf{G}, \\nabla \\varphi \\rangle + c \\varphi \\right) h \\, dx \n= \\int_U \\tilde{f} \\varphi \\, dx + \\int_U \\langle \\tilde{\\mathbf{F}}, \\nabla \\varphi \\rangle \\, dx \\quad \\text{for all } \\varphi \\in C_0^{\\infty}(U).\n\\]\nThen, in establishing the main theorem, $h \\in H^{1,2}_{loc}(U).", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let U be a possibly unbounded open subset of R^d with d >= 2. Assume G ∈ L^2_loc(U, R^d), c ∈ L^1_loc(U), \\tilde{F} ∈ L^2_loc(U, R^d), and \\tilde{f} ∈ L^1_loc(U). Let A = (a_ij) be a symmetric matrix field on U such that each a_ij belongs to VMO_loc(U), meaning: for every open ball B with closure contained in U, the restriction a_ij|_B extends to some function g on R^d for which there exists a positive continuous modulus ω with ω(0) = 0 such that sup_{z in R^d, r < R} r^{-2d} \\int_{B_r(z)} \\int_{B_r(z)} |g(x) - g(y)| dx dy <= ω(R) for all R > 0. Assume also that div A ∈ L^2_loc(U, R^d) in the distributional sense, i.e. there exists E = (E_j) with \\int_U \\sum_{i,j=1}^d a_ij \\partial_i \\phi_j dx = -\\int_U \\sum_{j=1}^d E_j \\phi_j dx for all (\\phi_1, ..., \\phi_d) ∈ C_0^∞(U)^d, and that A is locally uniformly strictly elliptic and bounded: for every open ball V with closure contained in U, there exist constants λ_V, M_V > 0 such that λ_V |ξ|^2 <= <A(x)ξ, ξ> and max_{i,j} |a_ij(x)| <= M_V for almost every x ∈ V and all ξ ∈ R^d. Suppose h ∈ L^∞_loc(U) satisfies, for every test function φ ∈ C_0^∞(U),\n\\int_U (trace(A ∇^2 φ) + <G, ∇φ> + c φ) h dx = \\int_U \\tilde{f} φ dx + \\int_U <\\tilde{F}, ∇φ> dx,\nwhere trace(A ∇^2 φ) = \\sum_{i,j=1}^d a_ij \\partial_{ij} φ. Which statement holds for every such h?", "correct_choice": {"label": "A", "text": "Every such h belongs to H^{1,2}_loc(U); equivalently, h has locally square-integrable weak first derivatives on U."}, "choices": [{"label": "B", "text": "Every such h belongs to H^{1,p}_{loc}(U) for some p>2 depending only on d and the local ellipticity and VMO data of A; in particular, h has locally p-integrable weak first derivatives on U."}, {"label": "C", "text": "Every such h belongs to W^{1,1}_{loc}(U); in particular, h has locally integrable weak first derivatives on U."}, {"label": "D", "text": "Every such h belongs to H^{1,2}_{loc}(U) provided, for each open ball V with \\overline{V}\\subset U, the ellipticity and boundedness constants \\lambda_V,M_V can be chosen independently of V; equivalently, the conclusion holds under a uniform strict ellipticity and boundedness condition on all of U."}, {"label": "E", "text": "Every such h belongs to H^{1,2}_{loc}(U) as soon as A is bounded and locally uniformly strictly elliptic and each a_{ij}\\in L^{\\infty}_{loc}(U), even without assuming either a_{ij}\\in VMO_{loc}(U) or {\\rm div}\\,A\\in L^2_{loc}(U,\\mathbb{R}^d)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "critical exponent fixed at 2", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "square-integrability of weak gradient", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "local versus global uniformity of ellipticity/boundedness", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "use of VMO and div A in resolvent estimates", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the conclusion that h is in H^{1,2}_{loc}(U). Although the setup strongly signals a specific regularity theorem, the correct answer is not leaked verbatim in the stem."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the stem lists the full hypotheses of a regularity result and option A states the theorem’s conclusion almost verbatim."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the student must distinguish the exact guaranteed regularity from stronger, weaker, or hypothesis-altered alternatives. But the task is still mainly recognition/application of a known theorem rather than genuine derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target common failure modes: overclaiming higher integrability, settling for a weaker true statement, confusing local with global uniformity, or dropping essential VMO/div A assumptions."}, "total_score": 5, "overall_assessment": "A mathematically well-constructed theorem-application MCQ with strong distractors, but it is too close to a direct restatement of the underlying theorem to strongly test generative reasoning."}}
{"id": "2602.13734v1", "paper_link": "http://arxiv.org/abs/2602.13734v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "[Main result]\\label{thm:intro-main}\nLet \\(0<r<R\\). Then\n\\[\n\\|C_{A(r,R)}\\|_{L^2\\to L^2}=\\frac{2}{\\sqrt{\\mu_1^{ND}(r,R)}},\n\\]\nwhere \\(\\mu_1^{ND}(r,R)\\) is the first eigenvalue of\n\\[\n-\\Delta U=\\mu U \\ \\text{in }A(r,R),\\qquad\n\\partial_\\nu U=0 \\ \\text{on }|z|=r,\\qquad\nU=0 \\ \\text{on }|z|=R.\n\\]\nEquivalently, \\(\\mu_1^{ND}(r,R)=\\kappa_1^2\\), where \\(\\kappa_1\\) is the smallest\npositive root of a Bessel determinant (see Theorem~\\ref{thm:norm-annulus}).", "start_pos": 4119, "end_pos": 4594, "label": "thm:intro-main"}, "ref_dict": {"thm:norm-annulus": "\\begin{theorem}[Sharp \\(L^2\\)-norm on a circular annulus]\\label{thm:norm-annulus}\nLet \\(0<r<R\\), \\(A=A(r,R)=\\{z\\in\\C:\\ r<|z|<R\\}\\), and\n\\[\n(C_A f)(z)=\\frac1\\pi\\int_A \\frac{f(w)}{z-w}\\,dA(w),\\qquad f\\in L^2(A).\n\\]\nThen\n\\[\n\\|C_A\\|_{L^2(A)\\to L^2(A)}=\\frac{2}{\\kappa_1},\n\\]\nwhere \\(\\kappa_1>0\\) is the smallest positive solution of\n\\begin{equation}\\label{eq:kappa1}\nJ_1(\\kappa r)\\,Y_0(\\kappa R)-Y_1(\\kappa r)\\,J_0(\\kappa R)=0 .\n\\end{equation}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2322, "pre_theorem_intro_text": "The Cauchy transform on a planar domain $\\Omega\\subset\\mathbb{C}$,\n\\[\n(C_\\Omega f)(z)=\\frac1\\pi\\int_\\Omega \\frac{f(w)}{z-w}\\,dA(w),\n\\]\nis a fundamental operator in complex analysis and elliptic PDE.  It gives a\ncanonical solution operator for the $\\bar\\partial$-equation in the sense that\n\\[\n\\bar\\partial\\,(C_\\Omega f)=f\\qquad\\text{in }\\Omega\n\\]\n(in the distributional sense, and pointwise whenever $f$ is sufficiently\nregular).  Consequently, sharp bounds for the operator norm\n$\\|C_\\Omega\\|_{L^2(\\Omega)\\to L^2(\\Omega)}$ are closely connected to\nquantitative $\\bar\\partial$ estimates and to spectral data of associated\nelliptic boundary value problems.\n\nOn smoothly bounded connected domains one has the spectral lower bound\n\\[\n\\|C_\\Omega\\|_{L^2(\\Omega)\\to L^2(\\Omega)}\\ge \\frac{2}{\\sqrt{\\lambda_1(\\Omega)}},\n\\]\nwhere $\\lambda_1(\\Omega)$ denotes the first Dirichlet eigenvalue of $-\\Delta$\non $\\Omega$; see \\cite{AndersonKhavinsonLomonosov1992}.  More precisely, the\nestimate can be expressed in the equivalent form\n\\begin{proposition}[Norm estimate for the Cauchy transform]\nLet $\\Omega\\subset\\mathbb{R}^2$ be a smoothly bounded domain. Then\n\\[\n\\left\\|\\frac{C_\\Omega^{*}C_\\Omega}{4}\\right\\|\n=\\frac14\\,\\|C_\\Omega\\|^2\n\\ge \\frac{1}{\\lambda_1(\\Omega)},\n\\]\nwhere $\\lambda_1(\\Omega)$ is the smallest positive eigenvalue for the boundary\nvalue problem \\emph{(1.2)} in $\\Omega$.\n\\end{proposition}\n\nFor the unit disk $D=\\{z\\in\\mathbb{C}:\\ |z|<1\\}$, Anderson and Hinkkanen computed the\nexact $L^2$-operator norm.  If $j_{0,1}$ denotes the smallest positive zero of\nthe Bessel function $J_0$, then\n\\[\n\\|C_D\\|_{L^2(D)\\to L^2(D)}=\\frac{2}{j_{0,1}},\n\\qquad\\text{equivalently}\\qquad\n\\frac14\\,\\|C_D\\|^2=\\frac{1}{j_{0,1}^2}.\n\\]\nSince $\\lambda_1(D)=j_{0,1}^2$, this shows that equality holds in the disk case:\n\\[\n\\frac14\\,\\|C_D\\|^2=\\frac{1}{\\lambda_1(D)}.\n\\]\nWe refer to \\cite{AndersonHinkkanen1989} for the sharp constant and explicit\nextremizers.  In contrast, for multiply connected domains additional holomorphic\ncomponents may influence sharpness.\n\nIn this paper we compute the exact $L^2$-operator norm of $C_\\Omega$ when\n$\\Omega$ is the circular annulus\n\\[\nA(r,R)=\\{z\\in\\mathbb{C}:\\ r<|z|<R\\}.\n\\]\n\nOur main result identifies the norm with the first mixed eigenvalue of the\nLaplacian (Neumann on \\(|z|=r\\), Dirichlet on \\(|z|=R\\)).", "context": "The Cauchy transform on a planar domain $\\Omega\\subset\\mathbb{C}$,\n\\[\n(C_\\Omega f)(z)=\\frac1\\pi\\int_\\Omega \\frac{f(w)}{z-w}\\,dA(w),\n\\]\nis a fundamental operator in complex analysis and elliptic PDE. It gives a\ncanonical solution operator for the $\\bar\\partial$-equation in the sense that\n\\[\n\\bar\\partial\\,(C_\\Omega f)=f\\qquad\\text{in }\\Omega\n\\]\n(in the distributional sense, and pointwise whenever $f$ is sufficiently\nregular). Consequently, sharp bounds for the operator norm\n$\\|C_\\Omega\\|_{L^2(\\Omega)\\to L^2(\\Omega)}$ are closely connected to\nquantitative $\\bar\\partial$ estimates and to spectral data of associated\nelliptic boundary value problems.\n\nOn smoothly bounded connected domains one has the spectral lower bound\n\\[\n\\|C_\\Omega\\|_{L^2(\\Omega)\\to L^2(\\Omega)}\\ge \\frac{2}{\\sqrt{\\lambda_1(\\Omega)}},\n\\]\nwhere $\\lambda_1(\\Omega)$ denotes the first Dirichlet eigenvalue of $-\\Delta$\non $\\Omega$; see \\cite{AndersonKhavinsonLomonosov1992}. More precisely, the\nestimate can be expressed in the equivalent form\n\\begin{proposition}[Norm estimate for the Cauchy transform]\nLet $\\Omega\\subset\\mathbb{R}^2$ be a smoothly bounded domain. Then\n\\[\n\\left\\|\\frac{C_\\Omega^{*}C_\\Omega}{4}\\right\\|\n=\\frac14\\,\\|C_\\Omega\\|^2\n\\ge \\frac{1}{\\lambda_1(\\Omega)},\n\\]\nwhere $\\lambda_1(\\Omega)$ is the smallest positive eigenvalue for the boundary\nvalue problem \\emph{(1.2)} in $\\Omega$.\n\\end{proposition}\n\nFor the unit disk $D=\\{z\\in\\mathbb{C}:\\ |z|<1\\}$, Anderson and Hinkkanen computed the\nexact $L^2$-operator norm. If $j_{0,1}$ denotes the smallest positive zero of\nthe Bessel function $J_0$, then\n\\[\n\\|C_D\\|_{L^2(D)\\to L^2(D)}=\\frac{2}{j_{0,1}},\n\\qquad\\text{equivalently}\\qquad\n\\frac14\\,\\|C_D\\|^2=\\frac{1}{j_{0,1}^2}.\n\\]\nSince $\\lambda_1(D)=j_{0,1}^2$, this shows that equality holds in the disk case:\n\\[\n\\frac14\\,\\|C_D\\|^2=\\frac{1}{\\lambda_1(D)}.\n\\]\nWe refer to \\cite{AndersonHinkkanen1989} for the sharp constant and explicit\nextremizers. In contrast, for multiply connected domains additional holomorphic\ncomponents may influence sharpness.\n\nIn this paper we compute the exact $L^2$-operator norm of $C_\\Omega$ when\n$\\Omega$ is the circular annulus\n\\[\nA(r,R)=\\{z\\in\\mathbb{C}:\\ r<|z|<R\\}.\n\\]\n\nOur main result identifies the norm with the first mixed eigenvalue of the\nLaplacian (Neumann on \\(|z|=r\\), Dirichlet on \\(|z|=R\\)).\n\n\\begin{theorem}[Sharp \\(L^2\\)-norm on a circular annulus]\\label{thm:norm-annulus}\nLet \\(0<r<R\\), \\(A=A(r,R)=\\{z\\in\\C:\\ r<|z|<R\\}\\), and\n\\[\n(C_A f)(z)=\\frac1\\pi\\int_A \\frac{f(w)}{z-w}\\,dA(w),\\qquad f\\in L^2(A).\n\\]\nThen\n\\[\n\\|C_A\\|_{L^2(A)\\to L^2(A)}=\\frac{2}{\\kappa_1},\n\\]\nwhere \\(\\kappa_1>0\\) is the smallest positive solution of\n\\begin{equation}\\label{eq:kappa1}\nJ_1(\\kappa r)\\,Y_0(\\kappa R)-Y_1(\\kappa r)\\,J_0(\\kappa R)=0 .\n\\end{equation}\n\\end{theorem}", "full_context": "The Cauchy transform on a planar domain $\\Omega\\subset\\mathbb{C}$,\n\\[\n(C_\\Omega f)(z)=\\frac1\\pi\\int_\\Omega \\frac{f(w)}{z-w}\\,dA(w),\n\\]\nis a fundamental operator in complex analysis and elliptic PDE. It gives a\ncanonical solution operator for the $\\bar\\partial$-equation in the sense that\n\\[\n\\bar\\partial\\,(C_\\Omega f)=f\\qquad\\text{in }\\Omega\n\\]\n(in the distributional sense, and pointwise whenever $f$ is sufficiently\nregular). Consequently, sharp bounds for the operator norm\n$\\|C_\\Omega\\|_{L^2(\\Omega)\\to L^2(\\Omega)}$ are closely connected to\nquantitative $\\bar\\partial$ estimates and to spectral data of associated\nelliptic boundary value problems.\n\nOn smoothly bounded connected domains one has the spectral lower bound\n\\[\n\\|C_\\Omega\\|_{L^2(\\Omega)\\to L^2(\\Omega)}\\ge \\frac{2}{\\sqrt{\\lambda_1(\\Omega)}},\n\\]\nwhere $\\lambda_1(\\Omega)$ denotes the first Dirichlet eigenvalue of $-\\Delta$\non $\\Omega$; see \\cite{AndersonKhavinsonLomonosov1992}. More precisely, the\nestimate can be expressed in the equivalent form\n\\begin{proposition}[Norm estimate for the Cauchy transform]\nLet $\\Omega\\subset\\mathbb{R}^2$ be a smoothly bounded domain. Then\n\\[\n\\left\\|\\frac{C_\\Omega^{*}C_\\Omega}{4}\\right\\|\n=\\frac14\\,\\|C_\\Omega\\|^2\n\\ge \\frac{1}{\\lambda_1(\\Omega)},\n\\]\nwhere $\\lambda_1(\\Omega)$ is the smallest positive eigenvalue for the boundary\nvalue problem \\emph{(1.2)} in $\\Omega$.\n\\end{proposition}\n\nFor the unit disk $D=\\{z\\in\\mathbb{C}:\\ |z|<1\\}$, Anderson and Hinkkanen computed the\nexact $L^2$-operator norm. If $j_{0,1}$ denotes the smallest positive zero of\nthe Bessel function $J_0$, then\n\\[\n\\|C_D\\|_{L^2(D)\\to L^2(D)}=\\frac{2}{j_{0,1}},\n\\qquad\\text{equivalently}\\qquad\n\\frac14\\,\\|C_D\\|^2=\\frac{1}{j_{0,1}^2}.\n\\]\nSince $\\lambda_1(D)=j_{0,1}^2$, this shows that equality holds in the disk case:\n\\[\n\\frac14\\,\\|C_D\\|^2=\\frac{1}{\\lambda_1(D)}.\n\\]\nWe refer to \\cite{AndersonHinkkanen1989} for the sharp constant and explicit\nextremizers. In contrast, for multiply connected domains additional holomorphic\ncomponents may influence sharpness.\n\nIn this paper we compute the exact $L^2$-operator norm of $C_\\Omega$ when\n$\\Omega$ is the circular annulus\n\\[\nA(r,R)=\\{z\\in\\mathbb{C}:\\ r<|z|<R\\}.\n\\]\n\nOur main result identifies the norm with the first mixed eigenvalue of the\nLaplacian (Neumann on \\(|z|=r\\), Dirichlet on \\(|z|=R\\)).\n\n\\begin{theorem}[Sharp \\(L^2\\)-norm on a circular annulus]\\label{thm:norm-annulus}\nLet \\(0<r<R\\), \\(A=A(r,R)=\\{z\\in\\C:\\ r<|z|<R\\}\\), and\n\\[\n(C_A f)(z)=\\frac1\\pi\\int_A \\frac{f(w)}{z-w}\\,dA(w),\\qquad f\\in L^2(A).\n\\]\nThen\n\\[\n\\|C_A\\|_{L^2(A)\\to L^2(A)}=\\frac{2}{\\kappa_1},\n\\]\nwhere \\(\\kappa_1>0\\) is the smallest positive solution of\n\\begin{equation}\\label{eq:kappa1}\nJ_1(\\kappa r)\\,Y_0(\\kappa R)-Y_1(\\kappa r)\\,J_0(\\kappa R)=0 .\n\\end{equation}\n\\end{theorem}\n\nOn smoothly bounded connected domains one has the spectral lower bound\n\\[\n\\|C_\\Omega\\|_{L^2(\\Omega)\\to L^2(\\Omega)}\\ge \\frac{2}{\\sqrt{\\lambda_1(\\Omega)}},\n\\]\nwhere $\\lambda_1(\\Omega)$ denotes the first Dirichlet eigenvalue of $-\\Delta$\non $\\Omega$; see \\cite{AndersonKhavinsonLomonosov1992}. More precisely, the\nestimate can be expressed in the equivalent form\n\\begin{proposition}[Norm estimate for the Cauchy transform]\nLet $\\Omega\\subset\\R^2$ be a smoothly bounded domain. Then\n\\[\n\\left\\|\\frac{C_\\Omega^{*}C_\\Omega}{4}\\right\\|\n=\\frac14\\,\\|C_\\Omega\\|^2\n\\ge \\frac{1}{\\lambda_1(\\Omega)},\n\\]\nwhere $\\lambda_1(\\Omega)$ is the smallest positive eigenvalue for the boundary\nvalue problem \\emph{(1.2)} in $\\Omega$.\n\\end{proposition}\n\nOur main result identifies the norm with the first mixed eigenvalue of the\nLaplacian (Neumann on \\(|z|=r\\), Dirichlet on \\(|z|=R\\)).\n\nThe proof exploits rotational symmetry: a Fourier decomposition reduces\n\\(C_{A(r,R)}\\) to a family of one-dimensional weighted Hardy-type operators on\nradial profiles. A variational argument then yields a Sturm--Liouville problem\nwhose first eigenvalue controls the sharp operator norm; the Bessel determinant\narises from solving the radial eigenvalue equation explicitly.\nThe extremizers are given in terms of the corresponding mixed eigenfunction,\nrevealing a structural distinction between simply and doubly connected domains.\n\\subsection{Background and related work}\nNorm bounds and spectral properties of the Cauchy transform and related\noperators have been studied from several perspectives.\nMapping properties on bounded domains were initiated in\n\\cite{AndersonHinkkanen1989}, while a spectral approach to potential-theoretic\nintegral operators was developed in \\cite{AndersonKhavinsonLomonosov1992}.\nSharp \\(L^2\\) estimates and their relation to spectral data appear in\n\\cite{ArazyKhavinson1992} and subsequent work of Dostani\\'c\n\\cite{Dostanic1996,Dostanic1998PLMS,Dostanic2005IEOT}.\nConnections with Poisson-type problems and quantitative PDE estimates were\nstudied in \\cite{Kalaj2012AIM,Kalaj2012IEOT}, with further norm estimates for\nCauchy-type operators in \\cite{ZhuKalaj2020JFA} and sharp \\(L^p\\)-theory on the\ndisk in \\cite{KalajMelentijevicZhu2022JFA}.\nRelated spectral questions on Bergman-type spaces and in doubly connected\ngeometry can be found in \\cite{Vujadinovic2016JMAA,Vujadinovic2025PA}.\n\n\\begin{theorem}[Sharp \\(L^2\\)-norm on a circular annulus]\\label{thm:norm-annulus}\nLet \\(0<r<R\\), \\(A=A(r,R)=\\{z\\in\\C:\\ r<|z|<R\\}\\), and\n\\[\n(C_A f)(z)=\\frac1\\pi\\int_A \\frac{f(w)}{z-w}\\,dA(w),\\qquad f\\in L^2(A).\n\\]\nThen\n\\[\n\\|C_A\\|_{L^2(A)\\to L^2(A)}=\\frac{2}{\\kappa_1},\n\\]\nwhere \\(\\kappa_1>0\\) is the smallest positive solution of\n\\begin{equation}\\label{eq:kappa1}\nJ_1(\\kappa r)\\,Y_0(\\kappa R)-Y_1(\\kappa r)\\,J_0(\\kappa R)=0 .\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nBy Lemmas~\\ref{lem:mode-action} and \\ref{lem:hardy-m},\n\\[\n\\|C_A\\|=\\sup_{m\\in\\mathbb Z}\\|C_A\\|_{H_m\\to H_{m-1}},\n\\qquad\n\\|C_A\\|_{H_m\\to H_{m-1}}=\n\\begin{cases}\n\\|T_m\\|,& m\\ge 1,\\\\\n\\|T_{1-m}\\|,& m\\le 0,\n\\end{cases}\n\\]\nand for \\(m\\ge 1\\),\n\\[\n\\|T_m\\|=\\frac{2}{\\sqrt{\\mu_m}},\n\\]\nwhere \\(\\mu_m\\) is the first eigenvalue of \\eqref{eq:SL-m}.\nLemma~\\ref{lem:monotone} gives \\(\\mu_m\\ge \\mu_1\\) for all \\(m\\ge 1\\), hence the\nsupremum is attained at \\(m=1\\) and\n\\[\n\\|C_A\\|=\\|T_1\\|=\\frac{2}{\\sqrt{\\mu_1}}.\n\\]\nFinally, Lemma~\\ref{lem:bessel-m} (with \\(m=1\\)) yields \\(\\mu_1=\\kappa_1^2\\), where\n\\(\\kappa_1\\) is the smallest positive root of \\eqref{eq:kappa1}. Therefore\n\\(\\|C_A\\|=2/\\kappa_1\\).\n\nMoreover, if \\(F>0\\) is the first eigenfunction of \\eqref{eq:SL-m} for \\(m=1\\) and\n\\(f(\\rho e^{i\\theta})=-F'(\\rho)e^{i\\theta}\\), then Lemma~\\ref{lem:mode-action} gives\n\\[\n(C_A f)(\\rho e^{i\\theta})=2F(\\rho),\n\\]\nso \\(f\\) is an extremizer, see remark below for details.\n\\end{proof}\n\\begin{remark}[An explicit extremizer and the derivative computation]\\label{rem:extremizer-annulus}\nLet \\(\\kappa_1>0\\) be the smallest positive solution of \\eqref{eq:kappa1}. Define\n\\[\nF(\\rho):=c\\Big( Y_0(\\kappa_1 R)\\,J_0(\\kappa_1\\rho)-J_0(\\kappa_1 R)\\,Y_0(\\kappa_1\\rho)\\Big),\n\\qquad r<\\rho<R,\n\\]\nwhere \\(c\\neq 0\\) is chosen so that \\(F>0\\) on \\((r,R)\\). Then \\(F(R)=0\\) holds by\nconstruction. Moreover, using the identities \\(J_0'(x)=-J_1(x)\\) and\n\\(Y_0'(x)=-Y_1(x)\\) together with the chain rule, we compute\n\\[\n\\begin{aligned}\nF'(\\rho)\n&=c\\Big( Y_0(\\kappa_1 R)\\,\\frac{d}{d\\rho}J_0(\\kappa_1\\rho)\n -J_0(\\kappa_1 R)\\,\\frac{d}{d\\rho}Y_0(\\kappa_1\\rho)\\Big)\\\\\n&=c\\,\\kappa_1\\Big( Y_0(\\kappa_1 R)\\,J_0'(\\kappa_1\\rho)\n -J_0(\\kappa_1 R)\\,Y_0'(\\kappa_1\\rho)\\Big)\\\\\n&=c\\,\\kappa_1\\Big(J_0(\\kappa_1 R)\\,Y_1(\\kappa_1\\rho)-Y_0(\\kappa_1 R)\\,J_1(\\kappa_1\\rho)\\Big).\n\\end{aligned}\n\\]\nIn particular,\n\\[\nF'(r)=0\n\\quad\\Longleftrightarrow\\quad\nJ_1(\\kappa_1 r)\\,Y_0(\\kappa_1 R)-Y_1(\\kappa_1 r)\\,J_0(\\kappa_1 R)=0,\n\\]\nso \\(\\kappa_1\\) is precisely the parameter for which the boundary conditions\n\\(F'(r)=0\\) and \\(F(R)=0\\) are satisfied. Consequently,\n\\[\nf(\\rho e^{i\\theta})=-F'(\\rho)\\,e^{i\\theta}\n\\]\nis an extremizer for \\(\\|C_A\\|\\) in Theorem~\\ref{thm:norm-annulus}.\n\\end{remark}\n\n\\begin{remark}[On the graph of \\(2/\\kappa_{1,1}(r)\\) for \\(R=1\\) and scaling in \\(R\\)]\\label{rem:graph-scaling}\nTheorem~\\ref{thm:norm-annulus} gives\n\\[\n\\|C_{A(r,R)}\\|_{2\\to 2}=\\frac{2}{\\kappa_{1,1}(r,R)},\n\\]\nwhere \\(\\kappa_{1,1}(r,R)\\) is the smallest positive root of the transcendental\nBessel determinant \\eqref{eq:kappa1}. In general there is no closed-form\nexpression for \\(\\kappa_{1,1}(r,R)\\) in terms of elementary functions, so an\n\\emph{explicit} formula for the function \\(r\\mapsto 2/\\kappa_{1,1}(r,R)\\) is not\navailable. Nevertheless, for each fixed \\(r\\in(0,1)\\) one can compute\n\\(\\kappa_{1,1}(r,1)\\) numerically by solving\n\\[\nD_r(\\kappa):=J_1(\\kappa r)Y_0(\\kappa)-Y_1(\\kappa r)J_0(\\kappa)=0\n\\]\nfor its smallest positive root (e.g.\\ by bisection, Newton's method, or standard\nspecial-function solvers). This produces the graph of\n\\[\nr\\longmapsto \\frac{2}{\\kappa_{1,1}(r,1)},\\qquad 0<r<1 (\\text{see Figure~1}).\n\\]\n\nLet $\\lambda_1(A)$ be the first Dirichlet eigenvalue of $-\\Delta$ on $A$ and set\n\\begin{equation}\\label{eq:k1def}\nk_1:=\\sqrt{\\lambda_1(A)}.\n\\end{equation}\nIn the planar case, $k_1$ is characterized as the smallest positive root of the Bessel cross--product equation\n\\begin{equation}\\label{eq:kEqGeneral}\nJ_0(kr)\\,Y_0(kR)-J_0(kR)\\,Y_0(kr)=0 .\n\\end{equation}", "post_theorem_intro_text_len": 1514, "post_theorem_intro_text": "The proof exploits rotational symmetry: a Fourier decomposition reduces\n\\(C_{A(r,R)}\\) to a family of one-dimensional weighted Hardy-type operators on\nradial profiles. A variational argument then yields a Sturm--Liouville problem\nwhose first eigenvalue controls the sharp operator norm; the Bessel determinant\narises from solving the radial eigenvalue equation explicitly.\nThe extremizers are given in terms of the corresponding mixed eigenfunction,\nrevealing a structural distinction between simply and doubly connected domains.\n\\subsection{Background and related work}\nNorm bounds and spectral properties of the Cauchy transform and related\noperators have been studied from several perspectives.\nMapping properties on bounded domains were initiated in\n\\cite{AndersonHinkkanen1989}, while a spectral approach to potential-theoretic\nintegral operators was developed in \\cite{AndersonKhavinsonLomonosov1992}.\nSharp \\(L^2\\) estimates and their relation to spectral data appear in\n\\cite{ArazyKhavinson1992} and subsequent work of Dostani\\'c\n\\cite{Dostanic1996,Dostanic1998PLMS,Dostanic2005IEOT}.\nConnections with Poisson-type problems and quantitative PDE estimates were\nstudied in \\cite{Kalaj2012AIM,Kalaj2012IEOT}, with further norm estimates for\nCauchy-type operators in \\cite{ZhuKalaj2020JFA} and sharp \\(L^p\\)-theory on the\ndisk in \\cite{KalajMelentijevicZhu2022JFA}.\nRelated spectral questions on Bergman-type spaces and in doubly connected\ngeometry can be found in \\cite{Vujadinovic2016JMAA,Vujadinovic2025PA}.", "sketch": "The proof of Theorem~\\ref{thm:intro-main} \"exploits rotational symmetry\": one performs a Fourier decomposition to reduce \\(C_{A(r,R)}\\) to \"a family of one-dimensional weighted Hardy-type operators on radial profiles.\" Next, \"a variational argument\" produces a Sturm--Liouville problem \"whose first eigenvalue controls the sharp operator norm.\" Finally, \"the Bessel determinant arises from solving the radial eigenvalue equation explicitly.\" The \"extremizers are given in terms of the corresponding mixed eigenfunction,\" highlighting \"a structural distinction between simply and doubly connected domains.\"", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "[Main result]\\label{thm:intro-main}\nLet \\(0<r<R\\). Then\n\\[\n\\|C_{A(r,R)}\\|_{L^2\\to L^2}=\\frac{2}{\\sqrt{\\mu_1^{ND}(r,R)}},\n\\]\nwhere \\(\\mu_1^{ND}(r,R)\\) is the first eigenvalue of\n\\[\n-\\Delta U=\\mu U \\ \\text{in }A(r,R),\\qquad\n\\partial_\\nu U=0 \\ \\text{on }|z|=r,\\qquad\nU=0 \\ \\text{on }|z|=R.\n\\]\nEquivalently, \\(\\mu_1^{ND}(r,R)=\\kappa_1^2\\), where \\(\\kappa_1\\) is the smallest\npositive root of a Bessel determinant, in establishing the main theorem.\n\nWe use the following theorem.\n\\begin{theorem}[Sharp \\(L^2\\)-norm on a circular annulus]\\label{thm:norm-annulus}\nLet \\(0<r<R\\), \\(A=A(r,R)=\\{z\\in\\C:\\ r<|z|<R\\}\\), and\n\\[\n(C_A f)(z)=\\frac1\\pi\\int_A \\frac{f(w)}{z-w}\\,dA(w),\\qquad f\\in L^2(A).\n\\]\nThen\n\\[\n\\|C_A\\|_{L^2(A)\\to L^2(A)}=\\frac{2}{\\kappa_1},\n\\]\nwhere \\(\\kappa_1>0\\) is the smallest positive solution of\n\\begin{equation}\\label{eq:kappa1}\nJ_1(\\kappa r)\\,Y_0(\\kappa R)-Y_1(\\kappa r)\\,J_0(\\kappa R)=0 .\n\\end{equation}\n\\end{theorem}", "theorem_type": ["Equivalence", "Inequality or Bound"], "mcq": {"question": "Let \\(0<r<R\\), let the circular annulus be \\(A(r,R)=\\{z\\in\\mathbb C:r<|z|<R\\}\\), and let the Cauchy transform on \\(A(r,R)\\) be\n\\[\n(C_{A(r,R)}f)(z)=\\frac1\\pi\\int_{A(r,R)}\\frac{f(w)}{z-w}\\,dA(w),\\qquad f\\in L^2(A(r,R)).\n\\]\nLet \\(\\mu_1^{ND}(r,R)\\) denote the first eigenvalue of the mixed Neumann-Dirichlet problem\n\\[\n-\\Delta U=\\mu U \\ \\text{in }A(r,R),\\qquad \\partial_\\nu U=0 \\ \\text{on }|z|=r,\\qquad U=0 \\ \\text{on }|z|=R,\n\\]\nwhere \\(\\partial_\\nu\\) is the outward normal derivative. Which quantitative estimate holds for the \\(L^2\\)-operator norm of \\(C_{A(r,R)}\\)?", "correct_choice": {"label": "A", "text": "One has\n\\[\n\\|C_{A(r,R)}\\|_{L^2\\to L^2}=\\frac{2}{\\sqrt{\\mu_1^{ND}(r,R)}}.\n\\]\nEquivalently, if \\(\\kappa_1>0\\) is the smallest positive solution of\n\\[\nJ_1(\\kappa r)Y_0(\\kappa R)-Y_1(\\kappa r)J_0(\\kappa R)=0,\n\\]\nthen \\(\\mu_1^{ND}(r,R)=\\kappa_1^2\\) and\n\\[\n\\|C_{A(r,R)}\\|_{L^2(A(r,R))\\to L^2(A(r,R))}=\\frac{2}{\\kappa_1}.\n\\]"}, "choices": [{"label": "B", "text": "One has\n\\[\n\\|C_{A(r,R)}\\|_{L^2\\to L^2}=\\frac{2}{\\sqrt{\\lambda_1(A(r,R))}},\n\\]\nwhere \\(\\lambda_1(A(r,R))\\) is the first Dirichlet eigenvalue of \\(-\\Delta\\) on the annulus. Equivalently, if \\(k_1>0\\) is the smallest positive solution of\n\\[\nJ_0(k r)Y_0(kR)-Y_0(k r)J_0(kR)=0,\n\\]\nthen \\(\\lambda_1(A(r,R))=k_1^2\\) and\n\\[\n\\|C_{A(r,R)}\\|_{L^2(A(r,R))\\to L^2(A(r,R))}=\\frac{2}{k_1}.\n\\]"}, {"label": "C", "text": "One has the lower bound\n\\[\n\\|C_{A(r,R)}\\|_{L^2\\to L^2}\\ge \\frac{2}{\\sqrt{\\mu_1^{ND}(r,R)}}.\n\\]\nEquivalently, if \\(\\kappa_1>0\\) is the smallest positive solution of\n\\[\nJ_1(\\kappa r)Y_0(\\kappa R)-Y_1(\\kappa r)J_0(\\kappa R)=0,\n\\]\nthen \\(\\mu_1^{ND}(r,R)=\\kappa_1^2\\) and hence\n\\[\n\\|C_{A(r,R)}\\|_{L^2(A(r,R))\\to L^2(A(r,R))}\\ge \\frac{2}{\\kappa_1}.\n\\]"}, {"label": "D", "text": "One has\n\\[\n\\|C_{A(r,R)}\\|_{L^2\\to L^2}=\\frac{2}{\\sqrt{\\mu_1^{DN}(r,R)}},\n\\]\nwhere \\(\\mu_1^{DN}(r,R)\\) is the first eigenvalue of\n\\[\n-\\Delta U=\\mu U\\ \\text{in }A(r,R),\\qquad U=0\\ \\text{on }|z|=r,\\qquad \\partial_\\nu U=0\\ \\text{on }|z|=R.\n\\]\nEquivalently, if \\(\\kappa_1>0\\) is the smallest positive solution of\n\\[\nJ_0(\\kappa r)Y_1(\\kappa R)-Y_0(\\kappa r)J_1(\\kappa R)=0,\n\\]\nthen \\(\\mu_1^{DN}(r,R)=\\kappa_1^2\\) and\n\\[\n\\|C_{A(r,R)}\\|_{L^2(A(r,R))\\to L^2(A(r,R))}=\\frac{2}{\\kappa_1}.\n\\]"}, {"label": "E", "text": "One has\n\\[\n\\|C_{A(r,R)}\\|_{L^2\\to L^2}=\\frac{2}{\\sqrt{\\mu_1^{ND}(r,R)}},\n\\]\nbut now \\(\\mu_1^{ND}(r,R)=\\kappa_1^2\\) is characterized by the smallest positive solution of\n\\[\nJ_0(\\kappa r)Y_0(\\kappa R)-Y_0(\\kappa r)J_0(\\kappa R)=0.\n\\]\nThus\n\\[\n\\|C_{A(r,R)}\\|_{L^2(A(r,R))\\to L^2(A(r,R))}=\\frac{2}{\\kappa_1}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "mixed_ND_vs_pure_Dirichlet_eigenvalue", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "exact_equality_replaced_by_lower_bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "characteristic", "tampered_component": "boundary_condition_order_inner_outer_swapped", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "trace_identity", "tampered_component": "Bessel_determinant_uses_wrong_orders_J0Y0_instead_of_J1Y0_minus_Y1J0", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 1, "justification": "The stem introduces the mixed Neumann-Dirichlet eigenvalue \\(\\mu_1^{ND}(r,R)\\), which is the key quantity appearing in the correct option, so it gives a strong hint. However, it does not explicitly state the exact norm formula, and several alternatives still compete."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: it asks which estimate holds for the norm and the correct choice essentially states the target result. The presence of nearby alternatives prevents it from being a pure restatement, but it is only a mild reformulation."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish exact equality from a weaker lower bound and to identify the correct boundary-condition/eigenvalue pairing. Still, the task mainly tests recognition of the known formula rather than deeper derivation or synthesis."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic confusions: pure Dirichlet vs mixed boundary data, swapping DN/ND conditions, replacing equality by a weaker true statement, and using an incorrect Bessel-characteristic equation."}, "total_score": 5, "overall_assessment": "A reasonably strong theorem-recognition MCQ with good distractors, but it gives noticeable hints via the stem and only moderately tests genuine generative reasoning."}}
{"id": "2602.13750v1", "paper_link": "http://arxiv.org/abs/2602.13750v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "[\\cite{FCW}]\\label{Kn}\nLet $n$ be a positive number. Then\n$$\\tau_{o}(K_{n})=\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}.$$", "start_pos": 4186, "end_pos": 4347, "label": "Kn"}, "ref_dict": {}, "pre_theorem_intro_text_len": 1521, "pre_theorem_intro_text": "Let $G=(V(G), E(G))$ be a graph without self-loops and $\\mathcal{T}(G)$ be the collection of spanning trees of $G$.\nWe usually call $\\tau(G)=|\\mathcal{T}(G)|$ the number of spanning trees of $G$. Denote by $K_n$, $K_{m,n}$, and $K_{n_1,n_2,\\ldots, n_s}$ \nbe the complete graph of $n$ vertices, the complete bipartite graph that one partite has $m$ vertices and the other partite has $n$ vertices, \nand the complete $s$-partite graph that each partite has $n_i$ vertices.\n\nCounting spanning trees is a classic problem originating in the 19th century that continues to be a hot topic today. \nSome recent developments can be found in \\cite{DG,DKM,GJ,LCY,LY,Yan,YT} for example. \n\nIn 2025, motivated by Gallai's classical theorem (\\cite{Lovasz}, Section 5, Problem 17) that the vertices of \nany graph can be partitioned into two sets \nwhere one induces an even-degree subgraph and the other induces an odd-degree subgraph, and also the concept of \nhomeomorphically irreducible spanning trees (HISTs, spanning trees containing no vertices of degree two), \nZheng and Wu \\cite{ZW} introduced the concept of odd spanning trees. An odd spanning tree of a graph $G$ \nis a spanning tree $T$ of $G$ where each vertex has odd degree in $T$.\n\nThe famous Cayley's formula \\cite{Cayley} states that $\\tau(K_n)=n^{n-2}$. \nLet $\\tau_o(G)$ be the number of odd spanning trees of $G$. \nRecently, Feng, Chen and Wu counted the number of odd spanning trees in complete graphs via Prüfer code and the exponential generating function as follows.", "context": "Let $G=(V(G), E(G))$ be a graph without self-loops and $\\mathcal{T}(G)$ be the collection of spanning trees of $G$.\nWe usually call $\\tau(G)=|\\mathcal{T}(G)|$ the number of spanning trees of $G$. Denote by $K_n$, $K_{m,n}$, and $K_{n_1,n_2,\\ldots, n_s}$ \nbe the complete graph of $n$ vertices, the complete bipartite graph that one partite has $m$ vertices and the other partite has $n$ vertices, \nand the complete $s$-partite graph that each partite has $n_i$ vertices.\n\nCounting spanning trees is a classic problem originating in the 19th century that continues to be a hot topic today. \nSome recent developments can be found in \\cite{DG,DKM,GJ,LCY,LY,Yan,YT} for example.\n\nIn 2025, motivated by Gallai's classical theorem (\\cite{Lovasz}, Section 5, Problem 17) that the vertices of \nany graph can be partitioned into two sets \nwhere one induces an even-degree subgraph and the other induces an odd-degree subgraph, and also the concept of \nhomeomorphically irreducible spanning trees (HISTs, spanning trees containing no vertices of degree two), \nZheng and Wu \\cite{ZW} introduced the concept of odd spanning trees. An odd spanning tree of a graph $G$ \nis a spanning tree $T$ of $G$ where each vertex has odd degree in $T$.\n\nThe famous Cayley's formula \\cite{Cayley} states that $\\tau(K_n)=n^{n-2}$. \nLet $\\tau_o(G)$ be the number of odd spanning trees of $G$. \nRecently, Feng, Chen and Wu counted the number of odd spanning trees in complete graphs via Prüfer code and the exponential generating function as follows.", "full_context": "Let $G=(V(G), E(G))$ be a graph without self-loops and $\\mathcal{T}(G)$ be the collection of spanning trees of $G$.\nWe usually call $\\tau(G)=|\\mathcal{T}(G)|$ the number of spanning trees of $G$. Denote by $K_n$, $K_{m,n}$, and $K_{n_1,n_2,\\ldots, n_s}$ \nbe the complete graph of $n$ vertices, the complete bipartite graph that one partite has $m$ vertices and the other partite has $n$ vertices, \nand the complete $s$-partite graph that each partite has $n_i$ vertices.\n\nCounting spanning trees is a classic problem originating in the 19th century that continues to be a hot topic today. \nSome recent developments can be found in \\cite{DG,DKM,GJ,LCY,LY,Yan,YT} for example.\n\nIn 2025, motivated by Gallai's classical theorem (\\cite{Lovasz}, Section 5, Problem 17) that the vertices of \nany graph can be partitioned into two sets \nwhere one induces an even-degree subgraph and the other induces an odd-degree subgraph, and also the concept of \nhomeomorphically irreducible spanning trees (HISTs, spanning trees containing no vertices of degree two), \nZheng and Wu \\cite{ZW} introduced the concept of odd spanning trees. An odd spanning tree of a graph $G$ \nis a spanning tree $T$ of $G$ where each vertex has odd degree in $T$.\n\nThe famous Cayley's formula \\cite{Cayley} states that $\\tau(K_n)=n^{n-2}$. \nLet $\\tau_o(G)$ be the number of odd spanning trees of $G$. \nRecently, Feng, Chen and Wu counted the number of odd spanning trees in complete graphs via Prüfer code and the exponential generating function as follows.\n\nThe famous Cayley's formula \\cite{Cayley} states that $\\tau(K_n)=n^{n-2}$. \nLet $\\tau_o(G)$ be the number of odd spanning trees of $G$. \nRecently, Feng, Chen and Wu counted the number of odd spanning trees in complete graphs via Prüfer code and the exponential generating function as follows.\n\nNote: in \\cite{FCW}, the original theorem is of the form\n\\[\n\\tau_{o}(K_{n})=\n\\begin{cases}\n0, & \\text{if } n \\text{ is odd;} \\\\\n\\displaystyle\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}, & \\text{if } n \\text{ is even.}\n\\end{cases}\n\\]\nWe slightly revised the theorem since when $n$ is odd, it is easy to check that $\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}=0$.\n\n\\begin{lemma}\\label{bo}\n$$\n \\sum_{\\substack{y \\in \\{\\pm1\\}^n}} \\left(\\sum_{i=1}^n a_i y_i \\right)^m=\n 2^n\\cdot\\sum\\limits_{\\substack{k_1+k_2+\\cdots+k_n=m\\\\k_i\\in 2\\mathbb{N} }}\\frac{m!}{k_1!k_2!\\cdots k_n!}a_1^{k_1}a_2^{k_2}\\cdots a_n^{k_n}.\n$$ \n\\end{lemma}\n\n\\begin{theorem}[\\cite{Berge,Lovasz}]\\label{Kn-degree}\nLet $d_1,d_2,\\dots,d_n$ be positive integers summing up to $2n-2$. Then the number of spanning trees of $K_n$ in which the vertex $i$ has degree exactly $d_i$ for all $i=1,2,\\dots,n$ equals$$\n \\dfrac{(n-2)!}{(d_1-1)!(d_2-1)!\\cdots(d_n-1)!}.$$\n\\end{theorem}\n\n\\begin{theorem}\\label{Kmn-degree}\nLet $K_{m,n}$ be the complete bipartite graph with bipartition $A=\\{u_1,u_2,\\ldots,u_m\\}$ and $B=\\{v_1,v_2,\\ldots,v_n\\}$. Let $a_i$ and $b_j$ be positive integers which $\\sum_{i=1}^{m}a_i=\\sum_{j=1}^{n}b_j=m+n-1$. Then the number of spanning trees of $K_{m,n}$ in which the vertex $u_i$ and $v_j$ have degree exactly $a_i$ and $b_j$ for $i\\in\\{1,2,\\ldots,m\\}$ and $j\\in\\{1,2,\\ldots,n\\}$ equals \n $$\n \\frac{(m-1)!(n-1)!}{\\prod_{i=1}^{m}(a_i-1)!\\prod_{j=1}^{n}(b_j-1)!},\n $$\n\\end{theorem}\n\\begin{proof}\n We use induction on $m+n$. For $m=1$ or $n=1$, $K_{m,n}$ is a star and the statement is trivial. \nSince $\\sum_{i=1}^{m}a_i=\\sum_{j=1}^{n}b_j=m+n-1$, $\\sum_{i=1}^{m}a_i+\\sum_{j=1}^{n}b_j=2(m+n)-2<2(m+n)$, \nthere must exist a leaf, and we may assume that $a_m=1$ where $m>1$. \nRemove $u_m$, in any tree under consideration, $u_m$ is adjacent to some $v_j$, $1\\leq j\\leq n$ and the removal of $u_m$ results in another tree on $\\{u_1,\\dots,u_{m-1},v_1,\\dots,v_n\\}$ with degrees $a_1,\\dots,a_{m-1},b_1,\\cdots,b_j-1,\\dots,b_n$. \nBy the induction hypothesis, the number of trees in $K_{m-1,n}$ that $\\{u_1,\\dots,u_{m-1},v_1,\\dots,v_n\\}$ has degrees $a_1,\\dots,a_{m-1},b_1,\\cdots,b_j-1,\\dots,b_n$ is\n $$\\frac{(m-2)!(n-1)!}{(a_1-1)!\\cdots(a_{m-1}-1)!(b_1-1)!\\cdots(b_j-2)!\\cdots(b_n-1)!}\n =\\frac{(m-2)!(n-1)!(b_j-1)}{\\prod_{i=1}^{m}(a_i-1)!\\prod_{j=1}^{n}(b_j-1)!}.\n $$ \n Thus, the number of trees on $\\{u_1,\\dots,u_{m},v_1,\\dots,v_n\\}$ with degrees $a_1,\\dots,a_{m},b_1,\\dots,b_n$ is\n\n\\noindent\n{\\bf Proof of Theorem \\ref{Kn}.}\nBy Theorem \\ref{Kn-degree}, the number of odd spanning trees of $K_n$ in which \nthe vertex $i$ has degree exactly $d_i=2d_i'+1$ for all $i=1,2,\\dots,n$ equals$$\n \\dfrac{(n-2)!}{(2d_1')!(2d_2')!\\cdots(2d_n')!},$$\n where $d_i\\geq0$, and $\\sum_{i=1}^{n}(2d_i')=n-2$.\nTherefore, \n$$\\tau_o(K_n)=\\sum_{\\substack{k_1+k_2+\\cdots+k_n=n-2\\\\k_i\\in 2\\mathbb{N} }}\\frac{(n-2)!}{k_1!k_2!\\cdots k_n!}.$$\nIt follows from Lemma \\ref{bo} that\n$$\n \\sum_{\\substack{\\varepsilon_i \\in \\{\\pm1\\}}}(\\varepsilon_1x_1+\\varepsilon_2x_2+\\cdots\\varepsilon_nx_n)^{n-2}= 2^n\\cdot\\sum_{\\substack{k_1+k_2+\\cdots+k_n=n-2\\\\k_i\\in 2\\mathbb{N} }}\\frac{(n-2)!}{k_1!k_2!\\cdots k_n!}x_1^{k_1}x_2^{k_2}\\cdots x_n^{k_n}.\n $$\nLet $x_1=x_2=\\cdots=x_n=1$, we obtain that\n $$\n \\tau_o(K_n)=\\frac{1}{2^n} \\sum_{\\substack{\\varepsilon_i \\in \\{\\pm1\\}}}(\\varepsilon_1+\\varepsilon_2+\\cdots\\varepsilon_n)^{n-2}\\\\\n =\\frac{1}{2^n}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}.\n $$\n{\\hfill$\\Box$}\n\n\\begin{theorem}\\label{Kmn}\nLet $m$ and $n$ be positive numbers. Then\n$$\\tau_o(K_{m,n})=\\frac{1}{2^{m+n}}\\left[\\sum_{i=0}^{m}\\binom{m}{i}(2i-m)^{n-1}\\right]\\left[\\sum_{j=0}^{n}\\binom{n}{j}(2j-n)^{m-1}\\right].$$ \n\\end{theorem}\n\n\\begin{proof}\n The notation and symbols used below are consistent with those in Theorem \\ref{Kmn-degree}. \nBy Theorem \\ref{Kmn-degree}, the number of odd spanning trees of $K_{m,n}$ \nin which the vertex $u_i\\in A$ and $v_j\\in B$ have degree exactly $d(u_i)=2d'(u_i)+1$ and $d(v_j)=2d'(v_j)+1$ \nfor all $i\\in\\{1,2,\\dots,m\\}$ and $j\\in\\{1,2,\\dots,n\\}$ equals\n$$\n \\frac{(m-1)!(n-1)!}{\\prod_{i=1}^{m}2d'(u_i)!\\prod_{j=1}^{n}2d'(v_j)!},\n $$\n where all $d'\\geq0$, $\\sum_{i=1}^{m}2d'(u_i)=n-1$ and $\\sum_{j=1}^{n}2d'(v_j)=m-1$. \nTherefore, \n$$\\tau_o(K_{m,n})=\\sum_{\\substack{k_1+k_2+\\cdots+k_m=n-1\\\\k_i\\in 2\\mathbb{N} }}\n \\sum_{\\substack{l_1+l_2+\\cdots+l_n=m-1\\\\l_i\\in 2\\mathbb{N} }}\\frac{(m-1)!(n-1)!}{\\prod_{i=1}^{m}k_i!\\prod_{j=1}^{n}l_j!}.$$", "post_theorem_intro_text_len": 551, "post_theorem_intro_text": "Note: in \\cite{FCW}, the original theorem is of the form\n\\[\n\\tau_{o}(K_{n})=\n\\begin{cases}\n0, & \\text{if } n \\text{ is odd;} \\\\\n\\displaystyle\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}, & \\text{if } n \\text{ is even.}\n\\end{cases}\n\\]\nWe slightly revised the theorem since when $n$ is odd, it is easy to check that $\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}=0$.\n\nIn this note, we we give a simple proof via a classical spanning tree enumeration formula and the Boolean function. \nWe also generalize it to complete bipartite graphs.", "sketch": "To prove Theorem~\\ref{Kn}, the note says it will “give a simple proof via a classical spanning tree enumeration formula and the Boolean function.” It also notes that the statement is revised from \\cite{FCW} because “when $n$ is odd, it is easy to check that $\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}=0$.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "[F.C.W. \\cite{FCW}]\\label{Kn}\nLet $n$ be a positive number. Then\n$$\\tau_{o}(K_{n})=\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}.$$,", "theorem_type": ["Classification or Bijection", "Universal"], "mcq": {"question": "Let \\(K_n\\) denote the complete graph on \\(n\\) vertices, and let \\(\\tau_o(G)\\) denote the number of odd spanning trees of a graph \\(G\\), where an odd spanning tree is a spanning tree in which every vertex has odd degree. For every positive integer \\(n\\), which statement gives \\(\\tau_o(K_n)\\)?", "correct_choice": {"label": "A", "text": "\\[\\tau_o(K_n)=\\frac{1}{2^n}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}.\\]"}, "choices": [{"label": "B", "text": "\\[\\tau_o(K_n)=\\frac{1}{2^n}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-1}.\\]"}, {"label": "C", "text": "\\[\\tau_o(K_n)=0\\quad\\text{for every odd }n.\\]"}, {"label": "D", "text": "\\[\\tau_o(K_n)=\\frac{1}{2^{n-1}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}.\\]"}, {"label": "E", "text": "\\[\\tau_o(K_n)=\\frac{1}{2^n}\\sum_{k=0}^{n}\\binom{n}{k}\\lvert 2k-n\\rvert^{\\,n-2}.\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "counting_estimate", "tampered_component": "exponent_from_Prufer_degree_sum", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "closed_formula_replaced_by_odd_n_vanishing_consequence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "normalizing_factor_2^n", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "signed_cancellation_in_binomial_sum", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the formula or give wording that points uniquely to choice A; it only defines the quantity and asks for its expression."}, "TAS": {"score": 0, "justification": "This is essentially a direct 'state the formula' question. If the underlying result is the closed form for odd spanning trees of K_n, the MCQ is mostly a restatement of that theorem rather than a new inference task."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the distractors are close variants of the true formula and one option is a weaker true statement, but the item still primarily tests recall/recognition of the exact formula rather than deeper generation of an argument."}, "DQS": {"score": 2, "justification": "The distractors are strong: B changes the exponent, D changes the normalization, E removes signed cancellation via absolute values, and C is a plausible weaker truth based on parity. These reflect realistic mathematical failure modes."}, "total_score": 5, "overall_assessment": "A solid formula-recognition MCQ with good distractors and no answer leakage, but it is close to theorem restatement and only moderately tests genuine generative reasoning."}}
{"id": "2602.13750v1", "paper_link": "http://arxiv.org/abs/2602.13750v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "[\\cite{FCW}]\\label{Kn}\nLet $n$ be a positive number. Then\n$$\\tau_{o}(K_{n})=\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}.$$", "start_pos": 4186, "end_pos": 4347, "label": "Kn"}, "ref_dict": {}, "pre_theorem_intro_text_len": 1521, "pre_theorem_intro_text": "Let $G=(V(G), E(G))$ be a graph without self-loops and $\\mathcal{T}(G)$ be the collection of spanning trees of $G$.\nWe usually call $\\tau(G)=|\\mathcal{T}(G)|$ the number of spanning trees of $G$. Denote by $K_n$, $K_{m,n}$, and $K_{n_1,n_2,\\ldots, n_s}$ \nbe the complete graph of $n$ vertices, the complete bipartite graph that one partite has $m$ vertices and the other partite has $n$ vertices, \nand the complete $s$-partite graph that each partite has $n_i$ vertices.\n\nCounting spanning trees is a classic problem originating in the 19th century that continues to be a hot topic today. \nSome recent developments can be found in \\cite{DG,DKM,GJ,LCY,LY,Yan,YT} for example. \n\nIn 2025, motivated by Gallai's classical theorem (\\cite{Lovasz}, Section 5, Problem 17) that the vertices of \nany graph can be partitioned into two sets \nwhere one induces an even-degree subgraph and the other induces an odd-degree subgraph, and also the concept of \nhomeomorphically irreducible spanning trees (HISTs, spanning trees containing no vertices of degree two), \nZheng and Wu \\cite{ZW} introduced the concept of odd spanning trees. An odd spanning tree of a graph $G$ \nis a spanning tree $T$ of $G$ where each vertex has odd degree in $T$.\n\nThe famous Cayley's formula \\cite{Cayley} states that $\\tau(K_n)=n^{n-2}$. \nLet $\\tau_o(G)$ be the number of odd spanning trees of $G$. \nRecently, Feng, Chen and Wu counted the number of odd spanning trees in complete graphs via Prüfer code and the exponential generating function as follows.", "context": "Let $G=(V(G), E(G))$ be a graph without self-loops and $\\mathcal{T}(G)$ be the collection of spanning trees of $G$.\nWe usually call $\\tau(G)=|\\mathcal{T}(G)|$ the number of spanning trees of $G$. Denote by $K_n$, $K_{m,n}$, and $K_{n_1,n_2,\\ldots, n_s}$ \nbe the complete graph of $n$ vertices, the complete bipartite graph that one partite has $m$ vertices and the other partite has $n$ vertices, \nand the complete $s$-partite graph that each partite has $n_i$ vertices.\n\nCounting spanning trees is a classic problem originating in the 19th century that continues to be a hot topic today. \nSome recent developments can be found in \\cite{DG,DKM,GJ,LCY,LY,Yan,YT} for example.\n\nIn 2025, motivated by Gallai's classical theorem (\\cite{Lovasz}, Section 5, Problem 17) that the vertices of \nany graph can be partitioned into two sets \nwhere one induces an even-degree subgraph and the other induces an odd-degree subgraph, and also the concept of \nhomeomorphically irreducible spanning trees (HISTs, spanning trees containing no vertices of degree two), \nZheng and Wu \\cite{ZW} introduced the concept of odd spanning trees. An odd spanning tree of a graph $G$ \nis a spanning tree $T$ of $G$ where each vertex has odd degree in $T$.\n\nThe famous Cayley's formula \\cite{Cayley} states that $\\tau(K_n)=n^{n-2}$. \nLet $\\tau_o(G)$ be the number of odd spanning trees of $G$. \nRecently, Feng, Chen and Wu counted the number of odd spanning trees in complete graphs via Prüfer code and the exponential generating function as follows.", "full_context": "Let $G=(V(G), E(G))$ be a graph without self-loops and $\\mathcal{T}(G)$ be the collection of spanning trees of $G$.\nWe usually call $\\tau(G)=|\\mathcal{T}(G)|$ the number of spanning trees of $G$. Denote by $K_n$, $K_{m,n}$, and $K_{n_1,n_2,\\ldots, n_s}$ \nbe the complete graph of $n$ vertices, the complete bipartite graph that one partite has $m$ vertices and the other partite has $n$ vertices, \nand the complete $s$-partite graph that each partite has $n_i$ vertices.\n\nCounting spanning trees is a classic problem originating in the 19th century that continues to be a hot topic today. \nSome recent developments can be found in \\cite{DG,DKM,GJ,LCY,LY,Yan,YT} for example.\n\nIn 2025, motivated by Gallai's classical theorem (\\cite{Lovasz}, Section 5, Problem 17) that the vertices of \nany graph can be partitioned into two sets \nwhere one induces an even-degree subgraph and the other induces an odd-degree subgraph, and also the concept of \nhomeomorphically irreducible spanning trees (HISTs, spanning trees containing no vertices of degree two), \nZheng and Wu \\cite{ZW} introduced the concept of odd spanning trees. An odd spanning tree of a graph $G$ \nis a spanning tree $T$ of $G$ where each vertex has odd degree in $T$.\n\nThe famous Cayley's formula \\cite{Cayley} states that $\\tau(K_n)=n^{n-2}$. \nLet $\\tau_o(G)$ be the number of odd spanning trees of $G$. \nRecently, Feng, Chen and Wu counted the number of odd spanning trees in complete graphs via Prüfer code and the exponential generating function as follows.\n\nThe famous Cayley's formula \\cite{Cayley} states that $\\tau(K_n)=n^{n-2}$. \nLet $\\tau_o(G)$ be the number of odd spanning trees of $G$. \nRecently, Feng, Chen and Wu counted the number of odd spanning trees in complete graphs via Prüfer code and the exponential generating function as follows.\n\nNote: in \\cite{FCW}, the original theorem is of the form\n\\[\n\\tau_{o}(K_{n})=\n\\begin{cases}\n0, & \\text{if } n \\text{ is odd;} \\\\\n\\displaystyle\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}, & \\text{if } n \\text{ is even.}\n\\end{cases}\n\\]\nWe slightly revised the theorem since when $n$ is odd, it is easy to check that $\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}=0$.\n\n\\begin{lemma}\\label{bo}\n$$\n \\sum_{\\substack{y \\in \\{\\pm1\\}^n}} \\left(\\sum_{i=1}^n a_i y_i \\right)^m=\n 2^n\\cdot\\sum\\limits_{\\substack{k_1+k_2+\\cdots+k_n=m\\\\k_i\\in 2\\mathbb{N} }}\\frac{m!}{k_1!k_2!\\cdots k_n!}a_1^{k_1}a_2^{k_2}\\cdots a_n^{k_n}.\n$$ \n\\end{lemma}\n\n\\begin{theorem}[\\cite{Berge,Lovasz}]\\label{Kn-degree}\nLet $d_1,d_2,\\dots,d_n$ be positive integers summing up to $2n-2$. Then the number of spanning trees of $K_n$ in which the vertex $i$ has degree exactly $d_i$ for all $i=1,2,\\dots,n$ equals$$\n \\dfrac{(n-2)!}{(d_1-1)!(d_2-1)!\\cdots(d_n-1)!}.$$\n\\end{theorem}\n\n\\begin{theorem}\\label{Kmn-degree}\nLet $K_{m,n}$ be the complete bipartite graph with bipartition $A=\\{u_1,u_2,\\ldots,u_m\\}$ and $B=\\{v_1,v_2,\\ldots,v_n\\}$. Let $a_i$ and $b_j$ be positive integers which $\\sum_{i=1}^{m}a_i=\\sum_{j=1}^{n}b_j=m+n-1$. Then the number of spanning trees of $K_{m,n}$ in which the vertex $u_i$ and $v_j$ have degree exactly $a_i$ and $b_j$ for $i\\in\\{1,2,\\ldots,m\\}$ and $j\\in\\{1,2,\\ldots,n\\}$ equals \n $$\n \\frac{(m-1)!(n-1)!}{\\prod_{i=1}^{m}(a_i-1)!\\prod_{j=1}^{n}(b_j-1)!},\n $$\n\\end{theorem}\n\\begin{proof}\n We use induction on $m+n$. For $m=1$ or $n=1$, $K_{m,n}$ is a star and the statement is trivial. \nSince $\\sum_{i=1}^{m}a_i=\\sum_{j=1}^{n}b_j=m+n-1$, $\\sum_{i=1}^{m}a_i+\\sum_{j=1}^{n}b_j=2(m+n)-2<2(m+n)$, \nthere must exist a leaf, and we may assume that $a_m=1$ where $m>1$. \nRemove $u_m$, in any tree under consideration, $u_m$ is adjacent to some $v_j$, $1\\leq j\\leq n$ and the removal of $u_m$ results in another tree on $\\{u_1,\\dots,u_{m-1},v_1,\\dots,v_n\\}$ with degrees $a_1,\\dots,a_{m-1},b_1,\\cdots,b_j-1,\\dots,b_n$. \nBy the induction hypothesis, the number of trees in $K_{m-1,n}$ that $\\{u_1,\\dots,u_{m-1},v_1,\\dots,v_n\\}$ has degrees $a_1,\\dots,a_{m-1},b_1,\\cdots,b_j-1,\\dots,b_n$ is\n $$\\frac{(m-2)!(n-1)!}{(a_1-1)!\\cdots(a_{m-1}-1)!(b_1-1)!\\cdots(b_j-2)!\\cdots(b_n-1)!}\n =\\frac{(m-2)!(n-1)!(b_j-1)}{\\prod_{i=1}^{m}(a_i-1)!\\prod_{j=1}^{n}(b_j-1)!}.\n $$ \n Thus, the number of trees on $\\{u_1,\\dots,u_{m},v_1,\\dots,v_n\\}$ with degrees $a_1,\\dots,a_{m},b_1,\\dots,b_n$ is\n\n\\noindent\n{\\bf Proof of Theorem \\ref{Kn}.}\nBy Theorem \\ref{Kn-degree}, the number of odd spanning trees of $K_n$ in which \nthe vertex $i$ has degree exactly $d_i=2d_i'+1$ for all $i=1,2,\\dots,n$ equals$$\n \\dfrac{(n-2)!}{(2d_1')!(2d_2')!\\cdots(2d_n')!},$$\n where $d_i\\geq0$, and $\\sum_{i=1}^{n}(2d_i')=n-2$.\nTherefore, \n$$\\tau_o(K_n)=\\sum_{\\substack{k_1+k_2+\\cdots+k_n=n-2\\\\k_i\\in 2\\mathbb{N} }}\\frac{(n-2)!}{k_1!k_2!\\cdots k_n!}.$$\nIt follows from Lemma \\ref{bo} that\n$$\n \\sum_{\\substack{\\varepsilon_i \\in \\{\\pm1\\}}}(\\varepsilon_1x_1+\\varepsilon_2x_2+\\cdots\\varepsilon_nx_n)^{n-2}= 2^n\\cdot\\sum_{\\substack{k_1+k_2+\\cdots+k_n=n-2\\\\k_i\\in 2\\mathbb{N} }}\\frac{(n-2)!}{k_1!k_2!\\cdots k_n!}x_1^{k_1}x_2^{k_2}\\cdots x_n^{k_n}.\n $$\nLet $x_1=x_2=\\cdots=x_n=1$, we obtain that\n $$\n \\tau_o(K_n)=\\frac{1}{2^n} \\sum_{\\substack{\\varepsilon_i \\in \\{\\pm1\\}}}(\\varepsilon_1+\\varepsilon_2+\\cdots\\varepsilon_n)^{n-2}\\\\\n =\\frac{1}{2^n}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}.\n $$\n{\\hfill$\\Box$}\n\n\\begin{theorem}\\label{Kmn}\nLet $m$ and $n$ be positive numbers. Then\n$$\\tau_o(K_{m,n})=\\frac{1}{2^{m+n}}\\left[\\sum_{i=0}^{m}\\binom{m}{i}(2i-m)^{n-1}\\right]\\left[\\sum_{j=0}^{n}\\binom{n}{j}(2j-n)^{m-1}\\right].$$ \n\\end{theorem}\n\n\\begin{proof}\n The notation and symbols used below are consistent with those in Theorem \\ref{Kmn-degree}. \nBy Theorem \\ref{Kmn-degree}, the number of odd spanning trees of $K_{m,n}$ \nin which the vertex $u_i\\in A$ and $v_j\\in B$ have degree exactly $d(u_i)=2d'(u_i)+1$ and $d(v_j)=2d'(v_j)+1$ \nfor all $i\\in\\{1,2,\\dots,m\\}$ and $j\\in\\{1,2,\\dots,n\\}$ equals\n$$\n \\frac{(m-1)!(n-1)!}{\\prod_{i=1}^{m}2d'(u_i)!\\prod_{j=1}^{n}2d'(v_j)!},\n $$\n where all $d'\\geq0$, $\\sum_{i=1}^{m}2d'(u_i)=n-1$ and $\\sum_{j=1}^{n}2d'(v_j)=m-1$. \nTherefore, \n$$\\tau_o(K_{m,n})=\\sum_{\\substack{k_1+k_2+\\cdots+k_m=n-1\\\\k_i\\in 2\\mathbb{N} }}\n \\sum_{\\substack{l_1+l_2+\\cdots+l_n=m-1\\\\l_i\\in 2\\mathbb{N} }}\\frac{(m-1)!(n-1)!}{\\prod_{i=1}^{m}k_i!\\prod_{j=1}^{n}l_j!}.$$", "post_theorem_intro_text_len": 551, "post_theorem_intro_text": "Note: in \\cite{FCW}, the original theorem is of the form\n\\[\n\\tau_{o}(K_{n})=\n\\begin{cases}\n0, & \\text{if } n \\text{ is odd;} \\\\\n\\displaystyle\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}, & \\text{if } n \\text{ is even.}\n\\end{cases}\n\\]\nWe slightly revised the theorem since when $n$ is odd, it is easy to check that $\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}=0$.\n\nIn this note, we we give a simple proof via a classical spanning tree enumeration formula and the Boolean function. \nWe also generalize it to complete bipartite graphs.", "sketch": "To prove Theorem~\\ref{Kn}, the note says it will “give a simple proof via a classical spanning tree enumeration formula and the Boolean function.” It also notes that the statement is revised from \\cite{FCW} because “when $n$ is odd, it is easy to check that $\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}=0$.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "[F.C.W. \\cite{FCW}]\\label{Kn}\nLet $n$ be a positive number. Then\n$$\\tau_{o}(K_{n})=\\frac{1}{2^{n}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}.$$,", "theorem_type": ["Classification or Bijection", "Universal"], "mcq": {"question": "Let \\(K_n\\) denote the complete graph on \\(n\\) vertices, and let \\(\\tau_o(G)\\) denote the number of odd spanning trees of a graph \\(G\\), where an odd spanning tree is a spanning tree in which every vertex has odd degree. For every positive integer \\(n\\), which statement gives \\(\\tau_o(K_n)\\)?", "correct_choice": {"label": "A", "text": "\\[\\tau_o(K_n)=\\frac{1}{2^n}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}.\\]"}, "choices": [{"label": "B", "text": "\\[\\tau_o(K_n)=\\frac{1}{2^n}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-1}.\\]"}, {"label": "C", "text": "\\[\\tau_o(K_n)=0\\quad\\text{for every odd }n.\\]"}, {"label": "D", "text": "\\[\\tau_o(K_n)=\\frac{1}{2^{n-1}}\\sum_{k=0}^{n}\\binom{n}{k}(2k-n)^{n-2}.\\]"}, {"label": "E", "text": "\\[\\tau_o(K_n)=\\frac{1}{2^n}\\sum_{k=0}^{n}\\binom{n}{k}\\lvert 2k-n\\rvert^{\\,n-2}.\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "counting_estimate", "tampered_component": "exponent_from_Prufer_degree_sum", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "closed_formula_replaced_by_odd_n_vanishing_consequence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "normalizing_factor_2^n", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "signed_cancellation_in_binomial_sum", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the notation and asks for the value of the quantity, but it does not reveal the formula or give direct hints pointing uniquely to choice A."}, "TAS": {"score": 1, "justification": "The item is essentially a theorem-identification question asking for the exact closed form of \u0007u03c4_o(K_n). It is not a verbatim restatement of the answer, but it is close to recall of a known formula rather than a substantially transformed application."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because several options are structurally similar and one distractor is a weaker true statement, so the student must distinguish the strongest correct conclusion. Still, success depends largely on recalling or recognizing the exact formula rather than generating a new argument."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: wrong exponent, wrong normalization factor, loss of sign cancellation via absolute value, and a weaker true parity consequence. These reflect realistic failure modes."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it mainly tests recognition of a known formula rather than deep generative reasoning."}}
{"id": "2602.13877v1", "paper_link": "http://arxiv.org/abs/2602.13877v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm1x}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nfour statements implies the other three:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{1}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  Lipschitz equivalent;\n  \\item there exists $\\alpha\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $\\Phi \\stackrel{1}{\\cong}\n    \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are Lipschitz conjugate;\n\\item there exists $\\beta\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\beta A^{\\Psi}$ are\n    Lyapunov similar while $A^{\\Phi_{\\sf  AD}}$, $\\beta A^{\\Psi_{\\sf\n        AD}}$ as well as $A^{\\Phi_{\\sf  C}}$, $\\beta A^{\\Psi_{\\sf\n        C}}$ are similar;\n\\item there exists $\\gamma\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\gamma A^{\\Psi}$ are\n    Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $\\gamma A^{\\Psi_{\\sf\n        C}}$ are similar.\n  \\end{enumerate}\n  Moreover, $\\Phi \\stackrel{1}{\\cong}\\Psi$ if and\nonly if $A^{\\Phi}$, $A^{\\Psi}$ are Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $ A^{\\Psi_{\\sf\n        C}}$ are similar.", "start_pos": 6937, "end_pos": 8006, "label": "thm1x"}, "ref_dict": {"eq1_2": "\\begin{equation}\\label{eq1_2}\n  h \\bigl( \\varphi  (t,x) \\bigr) = \\psi \\bigl( t,h(x)\\bigr)  \\qquad\n  \\forall t\\in \\R , x \\in X \\, .\n\\end{equation}", "prop1zb": "\\begin{prop}\\label{prop1zb}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nfive statements implies the other four:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{\\sf lin}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  linearly equivalent;\n \\item $\\Phi \\stackrel{\\sf diff}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  differentiably equivalent;\n  \\item there exists $\\alpha\n    \\in \\R \\setminus \\{0\\}$ so that $\\Phi \\stackrel{\\sf lin}{\\cong}\n    \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are linearly\n    conjugate;\n     \\item there exists $\\beta\n    \\in \\R \\setminus \\{0\\}$ so that $\\Phi \\stackrel{\\sf diff}{\\cong}\n    \\Psi_{*\\beta}$, i.e., $\\Phi$, $\\Psi_{*\\beta}$ are differentiably conjugate;\n\\item there exists $\\gamma\n    \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\gamma A^{\\Psi}$ are similar.\n  \\end{enumerate}\n  Moreover, $\\Phi \\stackrel{\\sf lin}{\\cong} \\Psi$ if and only if $\\Phi\n  \\stackrel{\\sf diff}{\\cong} \\Psi$ if and only if $A^{\\Phi}$,\n  $A^{\\Psi}$ are similar. \n\\end{prop}", "eq1_1": "\\begin{equation}\\label{eq1_1}\n  h \\bigl( \\varphi_{\\R} (x) \\bigr) = \\psi_{\\R} \\bigl( h(x)\\bigr)  \\qquad\n  \\forall x \\in X \\, .\n\\end{equation}", "prop1za": "\\begin{prop}\\label{prop1za}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nthree statements implies the other two:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{1^-}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  H\\\"{o}lder equivalent;\n  \\item there exists $\\alpha\n    \\in \\R \\setminus \\{0\\}$ so that $\\Phi \\stackrel{1^-}{\\cong}\n    \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are H\\\"{o}lder conjugate;\n\\item there exists $\\beta\n    \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\beta A^{\\Psi}$ are\n    Lyapunov similar while $A^{\\Phi_{\\sf C}}$, $\\beta A^{\\Psi_{\\sf\n        C}}$ are similar.\n  \\end{enumerate}\n  Moreover, $\\Phi \\stackrel{1^-}{\\cong} \\Psi$ if and only if\n  $A^{\\Phi}$, $A^{\\Psi}$ are Lyapunov similar while $A^{\\Phi_{\\sf C}}$,\n  $ A^{\\Psi_{\\sf C}}$ are similar.\n\\end{prop}", "thm1x": "\\begin{theorem}\\label{thm1x}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nfour statements implies the other three:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{1}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  Lipschitz equivalent;\n  \\item there exists $\\alpha\n    \\in \\R \\setminus \\{0\\}$ so that $\\Phi \\stackrel{1}{\\cong}\n    \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are Lipschitz conjugate;\n\\item there exists $\\beta\n    \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\beta A^{\\Psi}$ are\n    Lyapunov similar while $A^{\\Phi_{\\sf  AD}}$, $\\beta A^{\\Psi_{\\sf\n        AD}}$ as well as $A^{\\Phi_{\\sf  C}}$, $\\beta A^{\\Psi_{\\sf\n        C}}$ are similar;\n\\item there exists $\\gamma\n    \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\gamma A^{\\Psi}$ are\n    Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $\\gamma A^{\\Psi_{\\sf\n        C}}$ are similar.\n  \\end{enumerate}\n  Moreover, $\\Phi \\stackrel{1}{\\cong}\\Psi$ if and\nonly if $A^{\\Phi}$, $A^{\\Psi}$ are Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $ A^{\\Psi_{\\sf\n        C}}$ are similar.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 4367, "pre_theorem_intro_text": "\\label{sec1}\n\nLet $X\\ne \\{0\\}$ be a finite-dimensional normed space over $\\mathbb{R}$ and $\\varphi$\na flow on $X$, i.e., $\\varphi : \\mathbb{R} \\times X \\to X$ is\ncontinuous with $\\varphi(t+s , x) = \\varphi \\bigl( t, \\varphi (s,x)\n\\bigr)$ and $\\varphi(0,x) = x$ for all $t,s\\in \\mathbb{R}$, $x\\in X$. A\nfundamental question throughout dynamics is that of classification:\nWhen, precisely, are two flows $\\varphi$, $\\psi$ on $X$ {\\em the\n  same\\/}? Taking a geometrically motivated approach to this question,\nsay that $\\varphi$, $\\psi$ are {\\bf \n  equivalent}, in symbols $\\varphi \\stackrel{0}{\\thicksim} \\psi$, if there exists a\nhomeomorphism $h:X\\to X$ with $h(0)=0$ that maps each $\\varphi$-orbit\n$\\varphi_{\\mathbb{R}}(x):= \\{\\varphi (t, x):t\\in \\mathbb{R}\\}$ onto\na $\\psi$-orbit, i.e., \n\\begin{equation}\\label{eq1_1}\n  h \\bigl( \\varphi_{\\mathbb{R}} (x) \\bigr) = \\psi_{\\mathbb{R}} \\bigl( h(x)\\bigr)  \\qquad\n  \\forall x \\in X \\, .\n\\end{equation}\nIf $h$, $h^{-1}$ both are H\\\"{o}lder continuous (or Lipschitz\ncontinuous, differentiable, linear) then $\\varphi$, $\\psi$ are\nsaid to be {\\bf H\\\"{o}lder} (or {\\bf Lipschitz}, {\\bf differentiably},\n{\\bf linearly}) {\\bf equivalent}, in symbols $\\varphi \n\\stackrel{1^-}{\\thicksim} \\psi$ (or $\\varphi\n\\stackrel{1}{\\thicksim} \\psi$, $\\varphi\n\\stackrel{{\\sf diff}}{\\thicksim} \\psi$, $\\varphi\n\\stackrel{{\\sf lin}}{\\thicksim} \\psi$). Plainly $\\stackrel{\\bigstar}{\\thicksim}$\nyields an equivalence relation \nfor each $\\bigstar\\in \\{0,1^-,1, {\\sf diff}, {\\sf lin}\\}$, thereby establishing\nfive natural classifications of all flows on $X$. In addition,\nconsider the following, much more\nrestrictive form of equivalence: Say that $\\varphi$, $\\psi$ are {\\bf conjugate}, in symbols\n$\\varphi \\stackrel{0}{\\cong} \\psi$, if \n\\begin{equation}\\label{eq1_2}\n  h \\bigl( \\varphi  (t,x) \\bigr) = \\psi \\bigl( t,h(x)\\bigr)  \\qquad\n  \\forall t\\in \\mathbb{R} , x \\in X \\, .\n\\end{equation}\nNotice that (\\ref{eq1_2}) implies (\\ref{eq1_1}) but most definitely\nnot vice versa, and analogously defined\n$\\stackrel{\\bigstar}{\\cong}$ again yields an equivalence relation for\neach $\\bigstar \\in \\{0, 1^-, 1, {\\sf diff}, {\\sf lin}\\}$. Simple examples\nshow that no two of these equivalences, or the\nclassification established by them, coincide, not even when $\\mbox{\\rm\n  dim}\\, X=1$.\n\nBuilding on the classical literature briefly reviewed below, the\npresent article concludes the authors' earlier work \\cite{BW, BW2} \nby carrying out a comprehensive analysis of Lipschitz equivalence and conjugacy\nfor {\\em linear\\/} flows. Recall that a flow $\\varphi$ on $X$ is {\\bf linear} if the time-$t$\nmap $\\varphi_t = \\varphi (t,\\cdot):X\\to X$ is linear, or equivalently\nif $\\varphi_t = e^{t A^{\\varphi}}$,\nfor every $t\\in \\mathbb{R}$, with a (unique) linear operator $A^{\\varphi}$ on\n$X$ called the {\\bf generator} of $\\varphi$. Henceforth, upper case Greek letters\n$\\Phi$, $\\Psi$ are used exclusively to denote linear flows. The challenge, then, is to\ncharacterize $\\Phi \\stackrel{1}{\\thicksim} \\Psi$ and $\\Phi\n\\stackrel{1}{\\cong} \\Psi$ in terms of basic linear algebra properties\nof $A^{\\Phi}$, $A^{\\Psi}$. Correspondingly the main result of this\narticle, Theorem \\ref{thm1x} below, can be\nviewed as a {\\bf Lipschitz classification theorem}. To preview the result,\nnote that every linear flow $\\Phi$ on $X$ determines unique\ndecompositions $\\Phi \\stackrel{{\\sf lin}}{\\cong}  \\Phi_{\\sf D} \\times \\Phi_{\\sf\n  AD} \n\\stackrel{{\\sf lin}}{\\cong} \\Phi_{\\sf S} \\times \\Phi_{\\sf C} \\times\n\\Phi_{\\sf U}$ into a {\\em diagonal\\/}\nand {\\em anti-diagonal\\/} part, as well as into a\n{\\em stable}, {\\em central}, and {\\em unstable\\/} part; see Section \\ref{sec2a} below\nfor formal details. For convenience denote by\n$\\Phi_{*\\alpha}$ the linear flow generated by $\\alpha A^{\\Phi}$, for\nany $\\alpha \\in \\mathbb{R} \\setminus \\{0\\}$. Thus, $\\Phi_{*\\alpha}$ simply is\n$\\Phi$ with its first variable (``time'') rescaled by $\\alpha$. Finally, the statement involves\nthe concepts of {\\em Lyapunov\\/} and {\\em Lipschitz\nsimilarity}, introduced rigorously in Section\n\\ref{sec2a} also. For now, simply say that two linear operators are {\\bf\n  Lyapunov similar} if they (more precisely, the flows they generate) have the same\nLyapunov exponents, with matching multiplicities, and say that they\nare  {\\bf Lipschitz similar} if they are Lyapunov similar and their\nanti-diagonal parts are similar.", "context": "\\label{sec1}\n\nLet $X\\ne \\{0\\}$ be a finite-dimensional normed space over $\\mathbb{R}$ and $\\varphi$\na flow on $X$, i.e., $\\varphi : \\mathbb{R} \\times X \\to X$ is\ncontinuous with $\\varphi(t+s , x) = \\varphi \\bigl( t, \\varphi (s,x)\n\\bigr)$ and $\\varphi(0,x) = x$ for all $t,s\\in \\mathbb{R}$, $x\\in X$. A\nfundamental question throughout dynamics is that of classification:\nWhen, precisely, are two flows $\\varphi$, $\\psi$ on $X$ {\\em the\n same\\/}? Taking a geometrically motivated approach to this question,\nsay that $\\varphi$, $\\psi$ are {\\bf \n equivalent}, in symbols $\\varphi \\stackrel{0}{\\thicksim} \\psi$, if there exists a\nhomeomorphism $h:X\\to X$ with $h(0)=0$ that maps each $\\varphi$-orbit\n$\\varphi_{\\mathbb{R}}(x):= \\{\\varphi (t, x):t\\in \\mathbb{R}\\}$ onto\na $\\psi$-orbit, i.e., \n\\begin{equation}\\label{eq1_1}\n h \\bigl( \\varphi_{\\mathbb{R}} (x) \\bigr) = \\psi_{\\mathbb{R}} \\bigl( h(x)\\bigr) \\qquad\n \\forall x \\in X \\, .\n\\end{equation}\nIf $h$, $h^{-1}$ both are H\\\"{o}lder continuous (or Lipschitz\ncontinuous, differentiable, linear) then $\\varphi$, $\\psi$ are\nsaid to be {\\bf H\\\"{o}lder} (or {\\bf Lipschitz}, {\\bf differentiably},\n{\\bf linearly}) {\\bf equivalent}, in symbols $\\varphi \n\\stackrel{1^-}{\\thicksim} \\psi$ (or $\\varphi\n\\stackrel{1}{\\thicksim} \\psi$, $\\varphi\n\\stackrel{{\\sf diff}}{\\thicksim} \\psi$, $\\varphi\n\\stackrel{{\\sf lin}}{\\thicksim} \\psi$). Plainly $\\stackrel{\\bigstar}{\\thicksim}$\nyields an equivalence relation \nfor each $\\bigstar\\in \\{0,1^-,1, {\\sf diff}, {\\sf lin}\\}$, thereby establishing\nfive natural classifications of all flows on $X$. In addition,\nconsider the following, much more\nrestrictive form of equivalence: Say that $\\varphi$, $\\psi$ are {\\bf conjugate}, in symbols\n$\\varphi \\stackrel{0}{\\cong} \\psi$, if \n\\begin{equation}\\label{eq1_2}\n h \\bigl( \\varphi (t,x) \\bigr) = \\psi \\bigl( t,h(x)\\bigr) \\qquad\n \\forall t\\in \\mathbb{R} , x \\in X \\, .\n\\end{equation}\nNotice that (\\ref{eq1_2}) implies (\\ref{eq1_1}) but most definitely\nnot vice versa, and analogously defined\n$\\stackrel{\\bigstar}{\\cong}$ again yields an equivalence relation for\neach $\\bigstar \\in \\{0, 1^-, 1, {\\sf diff}, {\\sf lin}\\}$. Simple examples\nshow that no two of these equivalences, or the\nclassification established by them, coincide, not even when $\\mbox{\\rm\n dim}\\, X=1$.\n\nBuilding on the classical literature briefly reviewed below, the\npresent article concludes the authors' earlier work \\cite{BW, BW2} \nby carrying out a comprehensive analysis of Lipschitz equivalence and conjugacy\nfor {\\em linear\\/} flows. Recall that a flow $\\varphi$ on $X$ is {\\bf linear} if the time-$t$\nmap $\\varphi_t = \\varphi (t,\\cdot):X\\to X$ is linear, or equivalently\nif $\\varphi_t = e^{t A^{\\varphi}}$,\nfor every $t\\in \\mathbb{R}$, with a (unique) linear operator $A^{\\varphi}$ on\n$X$ called the {\\bf generator} of $\\varphi$. Henceforth, upper case Greek letters\n$\\Phi$, $\\Psi$ are used exclusively to denote linear flows. The challenge, then, is to\ncharacterize $\\Phi \\stackrel{1}{\\thicksim} \\Psi$ and $\\Phi\n\\stackrel{1}{\\cong} \\Psi$ in terms of basic linear algebra properties\nof $A^{\\Phi}$, $A^{\\Psi}$. Correspondingly the main result of this\narticle, Theorem \\ref{thm1x} below, can be\nviewed as a {\\bf Lipschitz classification theorem}. To preview the result,\nnote that every linear flow $\\Phi$ on $X$ determines unique\ndecompositions $\\Phi \\stackrel{{\\sf lin}}{\\cong} \\Phi_{\\sf D} \\times \\Phi_{\\sf\n AD} \n\\stackrel{{\\sf lin}}{\\cong} \\Phi_{\\sf S} \\times \\Phi_{\\sf C} \\times\n\\Phi_{\\sf U}$ into a {\\em diagonal\\/}\nand {\\em anti-diagonal\\/} part, as well as into a\n{\\em stable}, {\\em central}, and {\\em unstable\\/} part; see Section \\ref{sec2a} below\nfor formal details. For convenience denote by\n$\\Phi_{*\\alpha}$ the linear flow generated by $\\alpha A^{\\Phi}$, for\nany $\\alpha \\in \\mathbb{R} \\setminus \\{0\\}$. Thus, $\\Phi_{*\\alpha}$ simply is\n$\\Phi$ with its first variable (``time'') rescaled by $\\alpha$. Finally, the statement involves\nthe concepts of {\\em Lyapunov\\/} and {\\em Lipschitz\nsimilarity}, introduced rigorously in Section\n\\ref{sec2a} also. For now, simply say that two linear operators are {\\bf\n Lyapunov similar} if they (more precisely, the flows they generate) have the same\nLyapunov exponents, with matching multiplicities, and say that they\nare {\\bf Lipschitz similar} if they are Lyapunov similar and their\nanti-diagonal parts are similar.\n\n\\begin{equation}\\label{eq1_1}\n  h \\bigl( \\varphi_{\\R} (x) \\bigr) = \\psi_{\\R} \\bigl( h(x)\\bigr)  \\qquad\n  \\forall x \\in X \\, .\n\\end{equation}\n\n\\begin{equation}\\label{eq1_2}\n  h \\bigl( \\varphi  (t,x) \\bigr) = \\psi \\bigl( t,h(x)\\bigr)  \\qquad\n  \\forall t\\in \\R , x \\in X \\, .\n\\end{equation}\n\n\\begin{theorem}\\label{thm1x}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nfour statements implies the other three:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{1}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  Lipschitz equivalent;\n  \\item there exists $\\alpha\n    \\in \\R \\setminus \\{0\\}$ so that $\\Phi \\stackrel{1}{\\cong}\n    \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are Lipschitz conjugate;\n\\item there exists $\\beta\n    \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\beta A^{\\Psi}$ are\n    Lyapunov similar while $A^{\\Phi_{\\sf  AD}}$, $\\beta A^{\\Psi_{\\sf\n        AD}}$ as well as $A^{\\Phi_{\\sf  C}}$, $\\beta A^{\\Psi_{\\sf\n        C}}$ are similar;\n\\item there exists $\\gamma\n    \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\gamma A^{\\Psi}$ are\n    Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $\\gamma A^{\\Psi_{\\sf\n        C}}$ are similar.\n  \\end{enumerate}\n  Moreover, $\\Phi \\stackrel{1}{\\cong}\\Psi$ if and\nonly if $A^{\\Phi}$, $A^{\\Psi}$ are Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $ A^{\\Psi_{\\sf\n        C}}$ are similar.\n\\end{theorem}", "full_context": "\\label{sec1}\n\nLet $X\\ne \\{0\\}$ be a finite-dimensional normed space over $\\mathbb{R}$ and $\\varphi$\na flow on $X$, i.e., $\\varphi : \\mathbb{R} \\times X \\to X$ is\ncontinuous with $\\varphi(t+s , x) = \\varphi \\bigl( t, \\varphi (s,x)\n\\bigr)$ and $\\varphi(0,x) = x$ for all $t,s\\in \\mathbb{R}$, $x\\in X$. A\nfundamental question throughout dynamics is that of classification:\nWhen, precisely, are two flows $\\varphi$, $\\psi$ on $X$ {\\em the\n same\\/}? Taking a geometrically motivated approach to this question,\nsay that $\\varphi$, $\\psi$ are {\\bf \n equivalent}, in symbols $\\varphi \\stackrel{0}{\\thicksim} \\psi$, if there exists a\nhomeomorphism $h:X\\to X$ with $h(0)=0$ that maps each $\\varphi$-orbit\n$\\varphi_{\\mathbb{R}}(x):= \\{\\varphi (t, x):t\\in \\mathbb{R}\\}$ onto\na $\\psi$-orbit, i.e., \n\\begin{equation}\\label{eq1_1}\n h \\bigl( \\varphi_{\\mathbb{R}} (x) \\bigr) = \\psi_{\\mathbb{R}} \\bigl( h(x)\\bigr) \\qquad\n \\forall x \\in X \\, .\n\\end{equation}\nIf $h$, $h^{-1}$ both are H\\\"{o}lder continuous (or Lipschitz\ncontinuous, differentiable, linear) then $\\varphi$, $\\psi$ are\nsaid to be {\\bf H\\\"{o}lder} (or {\\bf Lipschitz}, {\\bf differentiably},\n{\\bf linearly}) {\\bf equivalent}, in symbols $\\varphi \n\\stackrel{1^-}{\\thicksim} \\psi$ (or $\\varphi\n\\stackrel{1}{\\thicksim} \\psi$, $\\varphi\n\\stackrel{{\\sf diff}}{\\thicksim} \\psi$, $\\varphi\n\\stackrel{{\\sf lin}}{\\thicksim} \\psi$). Plainly $\\stackrel{\\bigstar}{\\thicksim}$\nyields an equivalence relation \nfor each $\\bigstar\\in \\{0,1^-,1, {\\sf diff}, {\\sf lin}\\}$, thereby establishing\nfive natural classifications of all flows on $X$. In addition,\nconsider the following, much more\nrestrictive form of equivalence: Say that $\\varphi$, $\\psi$ are {\\bf conjugate}, in symbols\n$\\varphi \\stackrel{0}{\\cong} \\psi$, if \n\\begin{equation}\\label{eq1_2}\n h \\bigl( \\varphi (t,x) \\bigr) = \\psi \\bigl( t,h(x)\\bigr) \\qquad\n \\forall t\\in \\mathbb{R} , x \\in X \\, .\n\\end{equation}\nNotice that (\\ref{eq1_2}) implies (\\ref{eq1_1}) but most definitely\nnot vice versa, and analogously defined\n$\\stackrel{\\bigstar}{\\cong}$ again yields an equivalence relation for\neach $\\bigstar \\in \\{0, 1^-, 1, {\\sf diff}, {\\sf lin}\\}$. Simple examples\nshow that no two of these equivalences, or the\nclassification established by them, coincide, not even when $\\mbox{\\rm\n dim}\\, X=1$.\n\nBuilding on the classical literature briefly reviewed below, the\npresent article concludes the authors' earlier work \\cite{BW, BW2} \nby carrying out a comprehensive analysis of Lipschitz equivalence and conjugacy\nfor {\\em linear\\/} flows. Recall that a flow $\\varphi$ on $X$ is {\\bf linear} if the time-$t$\nmap $\\varphi_t = \\varphi (t,\\cdot):X\\to X$ is linear, or equivalently\nif $\\varphi_t = e^{t A^{\\varphi}}$,\nfor every $t\\in \\mathbb{R}$, with a (unique) linear operator $A^{\\varphi}$ on\n$X$ called the {\\bf generator} of $\\varphi$. Henceforth, upper case Greek letters\n$\\Phi$, $\\Psi$ are used exclusively to denote linear flows. The challenge, then, is to\ncharacterize $\\Phi \\stackrel{1}{\\thicksim} \\Psi$ and $\\Phi\n\\stackrel{1}{\\cong} \\Psi$ in terms of basic linear algebra properties\nof $A^{\\Phi}$, $A^{\\Psi}$. Correspondingly the main result of this\narticle, Theorem \\ref{thm1x} below, can be\nviewed as a {\\bf Lipschitz classification theorem}. To preview the result,\nnote that every linear flow $\\Phi$ on $X$ determines unique\ndecompositions $\\Phi \\stackrel{{\\sf lin}}{\\cong} \\Phi_{\\sf D} \\times \\Phi_{\\sf\n AD} \n\\stackrel{{\\sf lin}}{\\cong} \\Phi_{\\sf S} \\times \\Phi_{\\sf C} \\times\n\\Phi_{\\sf U}$ into a {\\em diagonal\\/}\nand {\\em anti-diagonal\\/} part, as well as into a\n{\\em stable}, {\\em central}, and {\\em unstable\\/} part; see Section \\ref{sec2a} below\nfor formal details. For convenience denote by\n$\\Phi_{*\\alpha}$ the linear flow generated by $\\alpha A^{\\Phi}$, for\nany $\\alpha \\in \\mathbb{R} \\setminus \\{0\\}$. Thus, $\\Phi_{*\\alpha}$ simply is\n$\\Phi$ with its first variable (``time'') rescaled by $\\alpha$. Finally, the statement involves\nthe concepts of {\\em Lyapunov\\/} and {\\em Lipschitz\nsimilarity}, introduced rigorously in Section\n\\ref{sec2a} also. For now, simply say that two linear operators are {\\bf\n Lyapunov similar} if they (more precisely, the flows they generate) have the same\nLyapunov exponents, with matching multiplicities, and say that they\nare {\\bf Lipschitz similar} if they are Lyapunov similar and their\nanti-diagonal parts are similar.\n\n\\begin{equation}\\label{eq1_1}\n  h \\bigl( \\varphi_{\\R} (x) \\bigr) = \\psi_{\\R} \\bigl( h(x)\\bigr)  \\qquad\n  \\forall x \\in X \\, .\n\\end{equation}\n\n\\begin{equation}\\label{eq1_2}\n  h \\bigl( \\varphi  (t,x) \\bigr) = \\psi \\bigl( t,h(x)\\bigr)  \\qquad\n  \\forall t\\in \\R , x \\in X \\, .\n\\end{equation}\n\n\\begin{theorem}\\label{thm1x}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nfour statements implies the other three:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{1}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  Lipschitz equivalent;\n  \\item there exists $\\alpha\n    \\in \\R \\setminus \\{0\\}$ so that $\\Phi \\stackrel{1}{\\cong}\n    \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are Lipschitz conjugate;\n\\item there exists $\\beta\n    \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\beta A^{\\Psi}$ are\n    Lyapunov similar while $A^{\\Phi_{\\sf  AD}}$, $\\beta A^{\\Psi_{\\sf\n        AD}}$ as well as $A^{\\Phi_{\\sf  C}}$, $\\beta A^{\\Psi_{\\sf\n        C}}$ are similar;\n\\item there exists $\\gamma\n    \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\gamma A^{\\Psi}$ are\n    Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $\\gamma A^{\\Psi_{\\sf\n        C}}$ are similar.\n  \\end{enumerate}\n  Moreover, $\\Phi \\stackrel{1}{\\cong}\\Psi$ if and\nonly if $A^{\\Phi}$, $A^{\\Psi}$ are Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $ A^{\\Psi_{\\sf\n        C}}$ are similar.\n\\end{theorem}\n\nBuilding on the classical literature briefly reviewed below, the\npresent article concludes the authors' earlier work \\cite{BW, BW2} \nby carrying out a comprehensive analysis of Lipschitz equivalence and conjugacy\nfor {\\em linear\\/} flows. Recall that a flow $\\varphi$ on $X$ is {\\bf linear} if the time-$t$\nmap $\\varphi_t = \\varphi (t,\\cdot):X\\to X$ is linear, or equivalently\nif $\\varphi_t = e^{t A^{\\varphi}}$,\nfor every $t\\in \\R$, with a (unique) linear operator $A^{\\varphi}$ on\n$X$ called the {\\bf generator} of $\\varphi$. Henceforth, upper case Greek letters\n$\\Phi$, $\\Psi$ are used exclusively to denote linear flows. The challenge, then, is to\ncharacterize $\\Phi \\stackrel{1}{\\thicksim} \\Psi$ and $\\Phi\n\\stackrel{1}{\\cong} \\Psi$ in terms of basic linear algebra properties\nof $A^{\\Phi}$, $A^{\\Psi}$. Correspondingly the main result of this\narticle, Theorem \\ref{thm1x} below, can be\nviewed as a {\\bf Lipschitz classification theorem}. To preview the result,\nnote that every linear flow $\\Phi$ on $X$ determines unique\ndecompositions $\\Phi \\stackrel{{\\sf lin}}{\\cong} \\Phi_{\\sf D} \\times \\Phi_{\\sf\n AD} \n\\stackrel{{\\sf lin}}{\\cong} \\Phi_{\\sf S} \\times \\Phi_{\\sf C} \\times\n\\Phi_{\\sf U}$ into a {\\em diagonal\\/}\nand {\\em anti-diagonal\\/} part, as well as into a\n{\\em stable}, {\\em central}, and {\\em unstable\\/} part; see Section \\ref{sec2a} below\nfor formal details. For convenience denote by\n$\\Phi_{*\\alpha}$ the linear flow generated by $\\alpha A^{\\Phi}$, for\nany $\\alpha \\in \\R \\setminus \\{0\\}$. Thus, $\\Phi_{*\\alpha}$ simply is\n$\\Phi$ with its first variable (``time'') rescaled by $\\alpha$. Finally, the statement involves\nthe concepts of {\\em Lyapunov\\/} and {\\em Lipschitz\nsimilarity}, introduced rigorously in Section\n\\ref{sec2a} also. For now, simply say that two linear operators are {\\bf\n Lyapunov similar} if they (more precisely, the flows they generate) have the same\nLyapunov exponents, with matching multiplicities, and say that they\nare {\\bf Lipschitz similar} if they are Lyapunov similar and their\nanti-diagonal parts are similar.\n\nTo put Theorem \\ref{thm1x} in context, it is instructive to compare it\nto its differentiable (hence slightly more restrictive) and H\\\"{o}lder\n(hence slightly less restrictive) counterparts; stated here without\nproofs, these results have been presented by the authors in detail\n(though sometimes couched in slightly different terminology and\nnotation) in \\cite{BW} and \\cite{BW2} respectively.\n\n\\begin{prop}\\label{prop1zb}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nfive statements implies the other four:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{\\sf lin}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n linearly equivalent;\n \\item $\\Phi \\stackrel{\\sf diff}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n differentiably equivalent;\n \\item there exists $\\alpha\n \\in \\R \\setminus \\{0\\}$ so that $\\Phi \\stackrel{\\sf lin}{\\cong}\n \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are linearly\n conjugate;\n \\item there exists $\\beta\n \\in \\R \\setminus \\{0\\}$ so that $\\Phi \\stackrel{\\sf diff}{\\cong}\n \\Psi_{*\\beta}$, i.e., $\\Phi$, $\\Psi_{*\\beta}$ are differentiably conjugate;\n\\item there exists $\\gamma\n \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\gamma A^{\\Psi}$ are similar.\n \\end{enumerate}\n Moreover, $\\Phi \\stackrel{\\sf lin}{\\cong} \\Psi$ if and only if $\\Phi\n \\stackrel{\\sf diff}{\\cong} \\Psi$ if and only if $A^{\\Phi}$,\n $A^{\\Psi}$ are similar. \n\\end{prop}\n\n\\begin{prop}\\label{prop1za}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nthree statements implies the other two:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{1^-}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n H\\\"{o}lder equivalent;\n \\item there exists $\\alpha\n \\in \\R \\setminus \\{0\\}$ so that $\\Phi \\stackrel{1^-}{\\cong}\n \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are H\\\"{o}lder conjugate;\n\\item there exists $\\beta\n \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\beta A^{\\Psi}$ are\n Lyapunov similar while $A^{\\Phi_{\\sf C}}$, $\\beta A^{\\Psi_{\\sf\n C}}$ are similar.\n \\end{enumerate}\n Moreover, $\\Phi \\stackrel{1^-}{\\cong} \\Psi$ if and only if\n $A^{\\Phi}$, $A^{\\Psi}$ are Lyapunov similar while $A^{\\Phi_{\\sf C}}$,\n $ A^{\\Psi_{\\sf C}}$ are similar.\n\\end{prop}\n\n\\begin{lem}\\label{lem4_2}\nGiven $m\\in \\N \\setminus \\{1\\}$, $k,\\ell\\in \\N$, and $b\\in \\R^k$, $c\\in\n\\R^{\\ell}$ with $m\\, {\\sf d}(b) = m\\, {\\sf d}(c) = d$, let $\\Phi$, $\\Psi$ be the flows on $\\R^d$ generated by\n$A$, $B$ in {\\rm (\\ref{eq4_1})} respectively. Then the following\nstatements are equivalent:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{{\\sf lin}}{\\cong}\\Psi$;\n\\item $\\Phi \\stackrel{1}{\\thicksim} \\Psi$;\n \\item $k=\\ell$, and there exists a bijection $g:\\{1, \\ldots , k\\}\\to \\{1, \\ldots\n , \\ell\\}$ so that $|b_j| = |c_{g(j)}|$ for every $j$;\n\\item $A$, $B$ are similar;\n\\item $A$, $B$ are Lipschitz similar. \n\\end{enumerate}\n\\end{lem}\n\n\\begin{theorem}\\label{thm71}\nLet $\\Phi$, $\\Psi$ be $\\K$-linear flows on $X$. Then the following\nstatements are equivalent:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{1}{\\thicksim}\\Psi$;\n \\item there exists $\\alpha\\in \\R\\setminus \\{0\\}$ so that $\\Phi \\stackrel{1}{\\cong}\\Psi_{*\\alpha}$;\n \\item $\\Phi^{\\R} \\stackrel{1}{\\thicksim} \\Psi^{\\R}$;\n \\item there exists $\\beta\\in \\R\\setminus \\{0\\}$ so that $\\Phi^{\\R} \\stackrel{1}{\\cong}\\Psi^{\\R}_{*\\beta}$;\n\\item there exists $\\gamma \\in \\R \\setminus \\{0\\}$ so that\n$A^{\\Phi^{\\R}}$, $\\gamma A^{\\Psi^{\\R}}$ are Lipschitz similar while\n$A^{\\Phi_{\\sf C}^{\\R}}$, $\\gamma A^{\\Psi_{\\sf C}^{\\R}}$ are similar.\n\\end{enumerate}\nMoreover, $\\Phi \\stackrel{1}{\\cong}\\Psi$ if and only if $\\Phi^{\\R} \\stackrel{1}{\\cong}\\Psi^{\\R}$\n if and only if $A^{\\Phi^{\\R}}$, $A^{\\Psi^{\\R}}$ are Lipschitz similar while\n$A^{\\Phi_{\\sf C}^{\\R}}$, $A^{\\Psi_{\\sf C}^{\\R}}$ are similar.\n\\end{theorem}\n\n\\begin{theorem}\\label{thm6_8}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then the following\nstatements are equivalent:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{{\\sf pw}1}{\\cong}\\Psi$;\n \\item $\\Phi$, $\\Psi$ are kinematically similar while $\\Phi_{\\sf C}\n \\stackrel{{\\sf lin}}{\\cong} \\Psi_{\\sf C}$;\n \\item $A^{\\Phi}$, $A^{\\Psi}$ are kinematically similar while\n $A^{\\Phi_{\\sf C}}, A^{\\Psi_{\\sf C}}$ are similar.\n\\end{enumerate}\n\\end{theorem}", "post_theorem_intro_text_len": 7393, "post_theorem_intro_text": "To put Theorem \\ref{thm1x} in context, it is instructive to compare it\nto its differentiable (hence slightly more restrictive) and H\\\"{o}lder\n(hence slightly less restrictive) counterparts; stated here without\nproofs, these results have been presented by the authors in detail\n(though sometimes couched in slightly different terminology and\nnotation) in \\cite{BW} and \\cite{BW2} respectively.\n\n\\begin{prop}\\label{prop1zb}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nfive statements implies the other four:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{\\sf lin}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  linearly equivalent;\n \\item $\\Phi \\stackrel{\\sf diff}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  differentiably equivalent;\n  \\item there exists $\\alpha\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $\\Phi \\stackrel{\\sf lin}{\\cong}\n    \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are linearly\n    conjugate;\n     \\item there exists $\\beta\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $\\Phi \\stackrel{\\sf diff}{\\cong}\n    \\Psi_{*\\beta}$, i.e., $\\Phi$, $\\Psi_{*\\beta}$ are differentiably conjugate;\n\\item there exists $\\gamma\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\gamma A^{\\Psi}$ are similar.\n  \\end{enumerate}\n  Moreover, $\\Phi \\stackrel{\\sf lin}{\\cong} \\Psi$ if and only if $\\Phi\n  \\stackrel{\\sf diff}{\\cong} \\Psi$ if and only if $A^{\\Phi}$,\n  $A^{\\Psi}$ are similar. \n\\end{prop}\n\n\\begin{prop}\\label{prop1za}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nthree statements implies the other two:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{1^-}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  H\\\"{o}lder equivalent;\n  \\item there exists $\\alpha\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $\\Phi \\stackrel{1^-}{\\cong}\n    \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are H\\\"{o}lder conjugate;\n\\item there exists $\\beta\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\beta A^{\\Psi}$ are\n    Lyapunov similar while $A^{\\Phi_{\\sf C}}$, $\\beta A^{\\Psi_{\\sf\n        C}}$ are similar.\n  \\end{enumerate}\n  Moreover, $\\Phi \\stackrel{1^-}{\\cong} \\Psi$ if and only if\n  $A^{\\Phi}$, $A^{\\Psi}$ are Lyapunov similar while $A^{\\Phi_{\\sf C}}$,\n  $ A^{\\Psi_{\\sf C}}$ are similar.\n\\end{prop}\n\nClassifications of linear flows have long been studied in the\nliterature, notably for {\\em hyperbolic\\/} flows, that is, for\n$\\Phi_{\\sf C}$, $\\Psi_{\\sf C}$ being trivial; see, e.g., \\cite{Amann, BW,\n  Irwin, R} for broad context, as well as \\cite{ACK1, AK, DSS, He, LZ, Willems} for\nspecific studies.\nOne striking aspect of Theorem\n\\ref{thm1x} is the fact that (i)$\\Rightarrow$(ii). Thus, for {\\em linear\\/}\nflows $\\varphi$, $\\psi$ validity of (\\ref{eq1_1}) always entails\nvalidity of (\\ref{eq1_2}), up to a linear, orbit-independent rescaling\nof time. This remarkable property, which does not \nseem to be shared by any wider class of flows on $X$, is indicative of\nthe extraordinary coherence between individual orbits of linear\nflows. As far as the authors have been able to ascertain, the\nproperty has not been stated, let alone proved rigorously before,\nthough it appears to have been part of linear systems folklore for\nquite some time; see, e.g., \\cite[Rem.\\ 7.4]{Willems} as well as\n\\cite{ACK1, CK, KS}.\n\nWith Lipschitz continuity being a fundamental concept throughout\nanalysis \\cite{Hei}, another key aspect of Theorem \\ref{thm1x} is its\nrelation to the differentiable and H\\\"{o}lder counterparts,\nPropositions \\ref{prop1zb} and \\ref{prop1za} respectively. While\n\\cite[Rem.\\ 2.5]{ACK1} seems to suggest, somewhat misleadingly, that the equivalence\nrelation $\\stackrel{1}{\\cong}$ for linear flows simply coincides with $\\stackrel{{\\sf\n    diff}}{\\cong}$, the characterization of Lipschitz {\\em\n  conjugacy\\/} contained in Theorem \\ref{thm1x} has, in essence, been\nestablished in \\cite{KS}; see also \\cite{MM} for a related (albeit quite\ninformal) discussion. Specifically, \\cite{KS} argues that\n$\\stackrel{1}{\\cong}$ is ``very close'' to $\\stackrel{{\\sf\n    diff}}{\\cong}$, and a crucial role in the argument is subsequently\nplayed by a well-known theorem due to Rademacher which asserts that\nevery Lipschitz map $h:X\\to X$ is differentiable almost\neverywhere. However, in\nthe general setting of Theorem \\ref{thm1x}, that is, for mere Lipschitz {\\em equivalence}, no\nassumptions whatsoever are made regarding the (orientation and\nregularity of) re-parametrizations of individual orbits, and hence it appears\ndoubtful whether Rademacher's theorem can be applied\nfruitfully (or at all). Instead, the present\narticle utilizes a simple geometric idea gleaned from \\cite{MM} which\nit develops into the basic but consequential concept of {\\em\n  distortion\\/} ({\\em points\\/}) for stable flows. Aided by this\nconcept, the article then proceeds to prove, in an entirely elementary\nmanner, that (i)$\\Rightarrow$(iii) in Theorem \\ref{thm1x}. From this,\nvalidity of the entire theorem follows rather straightforwardly, as\n(iii)$\\Leftrightarrow$(iv)$\\Rightarrow$(ii), and obviously\n(ii)$\\Rightarrow$(i).  As indicated where appropriate below, the\nelementary approach developed here also helps to address other gaps and\ninaccuracies in the existing literature.\n\n\\medskip\n\n\\begin{rem}\nAn analogous classification problem presents itself in discrete time,\ni.e., for linear operators $A, B:X\\to X$ which are {\\bf conjugate} (or\n{\\bf nonlinearly similar} \\cite{CappSh, CSSW}), in symbols\n$A\\stackrel{0}{\\cong} B$, if $h(Ax) = Bh(x)$ for some homeomorphism\n$h:X\\to X$ and all $x\\in X$. As in continuous time, it is\nnatural to consider $\\stackrel{\\bigstar}{\\cong}$ for $\\bigstar \\in\n\\{0,1^-,1,{\\sf diff}, {\\sf lin}\\}$, each of which yields an\nequivalence relation on all linear operators on $X$. In\nstraightforward analogy to Proposition \\ref{prop1zb}, notice that\n$A\\stackrel{{\\sf lin}}{\\cong} B$ if and only if $A\\stackrel{{\\sf\n    diff}}{\\cong} B$ if and only if $A$, $B$ are similar. By contrast,\nthe problem of characterizing $A\\stackrel{\\bigstar}{\\cong} B$ for\n$\\bigstar \\in \\{0,1^-, 1\\}$, in terms of similarity invariants akin to\nTheorem \\ref{thm1x}, has turned out to be significantly more challenging than in \ncontinuous time; see, e.g., \\cite{CappSh, CSSW, Cruz, HP, KR} for the\nlong history of the problem and its many ramifications, with\n\\cite{Cruz} specifically addressing the Lipschitz case.\n\\end{rem}\n\nThe remainder of this article is organized as follows: Section\n\\ref{sec2} properly introduces various forms of equivalence and\nconjugacy to be studied in subsequent sections, together with some\ntailor-made analytical notation. Section \\ref{sec2a} briefly reviews a\nfew basic concepts pertinent to linear flows, notably irreducibility\nand Lyapunov exponents, and then discusses the novel concept of\nLipschitz similarity. Section \\ref{sec3} defines distortion points and\nrefined Lyapunov spaces for stable flows which in turn yield crucial Lipschitz\ninvariants for such flows. Section\n\\ref{sec4} presents a proof of the main result, Theorem \\ref{thm1x},\nvia a series of preparatory lemmas addressing important special cases\nthereof. It also outlines the straightforward extension of Theorem\n\\ref{thm1x} to {\\em complex\\/} spaces. Finally, Section \\ref{sec6}\nhighlights several subtle aspects of Lipschitz equivalence by briefly\ndiscussing a novel variant of the concept, referred to as {\\em\n  pointwise\\/} Lipschitz equivalence at $0$.", "sketch": "The article explains that, rather than using Rademacher’s theorem as in \\cite{KS}, it “utilizes a simple geometric idea gleaned from \\cite{MM} which it develops into the basic but consequential concept of {\\em distortion\\/} ({\\em points\\/}) for stable flows.” Using this concept, it “then proceeds to prove, in an entirely elementary manner, that (i)$\\Rightarrow$(iii) in Theorem \\ref{thm1x}.” From there, “validity of the entire theorem follows rather straightforwardly, as (iii)$\\Leftrightarrow$(iv)$\\Rightarrow$(ii), and obviously (ii)$\\Rightarrow$(i).”", "expanded_sketch": "The article explains that, rather than using Rademacher’s theorem as in \\cite{KS}, it “utilizes a simple geometric idea gleaned from \\cite{MM} which it develops into the basic but consequential concept of {\\em distortion\\/} ({\\em points\\/}) for stable flows.” Using this concept, it “then proceeds to prove, in an entirely elementary manner,” one implication needed to establish the main theorem. From there, “validity of the entire theorem follows rather straightforwardly, as (iii)$\\Leftrightarrow$(iv)$\\Rightarrow$(ii), and obviously (ii)$\\Rightarrow$(i).”", "expanded_theorem": "\\label{thm1x}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nfour statements implies the other three:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{1}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  Lipschitz equivalent;\n  \\item there exists $\\alpha\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $\\Phi \\stackrel{1}{\\cong}\n    \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are Lipschitz conjugate;\n\\item there exists $\\beta\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\beta A^{\\Psi}$ are\n    Lyapunov similar while $A^{\\Phi_{\\sf  AD}}$, $\\beta A^{\\Psi_{\\sf\n        AD}}$ as well as $A^{\\Phi_{\\sf  C}}$, $\\beta A^{\\Psi_{\\sf\n        C}}$ are similar;\n\\item there exists $\\gamma\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\gamma A^{\\Psi}$ are\n    Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $\\gamma A^{\\Psi_{\\sf\n        C}}$ are similar.\n  \\end{enumerate}\n  Moreover, $\\Phi \\stackrel{1}{\\cong}\\Psi$ if and\nonly if $A^{\\Phi}$, $A^{\\Psi}$ are Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $ A^{\\Psi_{\\sf\n        C}}$ are similar.", "theorem_type": ["Biconditional or Equivalence", "Implication"], "mcq": {"question": "Let $X\\neq\\{0\\}$ be a finite-dimensional normed space over $\\mathbb{R}$, and let $\\Phi,\\Psi$ be linear flows on $X$, meaning $\\Phi_t=e^{tA^{\\Phi}}$ and $\\Psi_t=e^{tA^{\\Psi}}$ for linear generators $A^{\\Phi},A^{\\Psi}$. For a flow $\\Theta$, write $\\Theta_{\\mathbb{R}}(x)=\\{\\Theta(t,x):t\\in\\mathbb{R}\\}$. Say that $\\Phi$ and $\\Psi$ are Lipschitz equivalent if there is a bi-Lipschitz homeomorphism $h:X\\to X$ with $h(0)=0$ such that $h(\\Phi_{\\mathbb{R}}(x))=\\Psi_{\\mathbb{R}}(h(x))$ for all $x\\in X$. Say that they are Lipschitz conjugate if there is a bi-Lipschitz homeomorphism $h:X\\to X$ with $h(0)=0$ such that $h(\\Phi(t,x))=\\Psi(t,h(x))$ for all $t\\in\\mathbb{R}$ and $x\\in X$. For $\\alpha\\neq 0$, let $\\Psi_{*\\alpha}$ denote the time-rescaled flow generated by $\\alpha A^{\\Psi}$, equivalently $\\Psi_{*\\alpha}(t,x)=\\Psi(\\alpha t,x)$. Also, let $\\Phi_{\\sf AD},\\Psi_{\\sf AD}$ and $\\Phi_{\\sf C},\\Psi_{\\sf C}$ denote the anti-diagonal and central parts in the canonical decomposition of linear flows; “similar” means similar as linear operators. Finally, two linear operators are called Lyapunov similar if the flows they generate have the same Lyapunov exponents with matching multiplicities, and Lipschitz similar if they are Lyapunov similar and their anti-diagonal parts are similar. Under these assumptions, which statement about $\\Phi$ and $\\Psi$ holds?", "correct_choice": {"label": "A", "text": "The following are equivalent: (i) $\\Phi$ and $\\Psi$ are Lipschitz equivalent; (ii) there exists $\\alpha\\in\\mathbb{R}\\setminus\\{0\\}$ such that $\\Phi$ and $\\Psi_{*\\alpha}$ are Lipschitz conjugate; (iii) there exists $\\beta\\in\\mathbb{R}\\setminus\\{0\\}$ such that $A^{\\Phi}$ and $\\beta A^{\\Psi}$ are Lyapunov similar, while $A^{\\Phi_{\\sf AD}}$ and $\\beta A^{\\Psi_{\\sf AD}}$ are similar and $A^{\\Phi_{\\sf C}}$ and $\\beta A^{\\Psi_{\\sf C}}$ are similar; (iv) there exists $\\gamma\\in\\mathbb{R}\\setminus\\{0\\}$ such that $A^{\\Phi}$ and $\\gamma A^{\\Psi}$ are Lipschitz similar, while $A^{\\Phi_{\\sf C}}$ and $\\gamma A^{\\Psi_{\\sf C}}$ are similar. Moreover, $\\Phi$ and $\\Psi$ are Lipschitz conjugate if and only if $A^{\\Phi}$ and $A^{\\Psi}$ are Lipschitz similar and $A^{\\Phi_{\\sf C}}$ and $A^{\\Psi_{\\sf C}}$ are similar."}, "choices": [{"label": "B", "text": "The following are equivalent: (i) $\\Phi$ and $\\Psi$ are Lipschitz equivalent; (ii) there exists $\\alpha\\in\\mathbb{R}$ such that $\\Phi$ and $\\Psi_{*\\alpha}$ are Lipschitz conjugate; (iii) there exists $\\beta\\in\\mathbb{R}$ such that $A^{\\Phi}$ and $\\beta A^{\\Psi}$ are Lyapunov similar, while $A^{\\Phi_{\\sf AD}}$ and $\\beta A^{\\Psi_{\\sf AD}}$ are similar and $A^{\\Phi_{\\sf C}}$ and $\\beta A^{\\Psi_{\\sf C}}$ are similar; (iv) there exists $\\gamma\\in\\mathbb{R}$ such that $A^{\\Phi}$ and $\\gamma A^{\\Psi}$ are Lipschitz similar, while $A^{\\Phi_{\\sf C}}$ and $\\gamma A^{\\Psi_{\\sf C}}$ are similar. Moreover, $\\Phi$ and $\\Psi$ are Lipschitz conjugate if and only if $A^{\\Phi}$ and $A^{\\Psi}$ are Lipschitz similar and $A^{\\Phi_{\\sf C}}$ and $A^{\\Psi_{\\sf C}}$ are similar."}, {"label": "C", "text": "If $\\Phi$ and $\\Psi$ are Lipschitz equivalent, then there exists $\\alpha\\in\\mathbb{R}\\setminus\\{0\\}$ such that $\\Phi$ and $\\Psi_{*\\alpha}$ are Lipschitz conjugate."}, {"label": "D", "text": "The following are equivalent: (i) $\\Phi$ and $\\Psi$ are Lipschitz equivalent; (ii) there exists $\\alpha\\in\\mathbb{R}\\setminus\\{0\\}$ such that $\\Phi$ and $\\Psi_{*\\alpha}$ are Lipschitz conjugate; (iii) there exists $\\beta\\in\\mathbb{R}\\setminus\\{0\\}$ such that $A^{\\Phi}$ and $\\beta A^{\\Psi}$ are Lyapunov similar and $A^{\\Phi_{\\sf C}}$ and $\\beta A^{\\Psi_{\\sf C}}$ are similar; (iv) there exists $\\gamma\\in\\mathbb{R}\\setminus\\{0\\}$ such that $A^{\\Phi}$ and $\\gamma A^{\\Psi}$ are Lipschitz similar. Moreover, $\\Phi$ and $\\Psi$ are Lipschitz conjugate if and only if $A^{\\Phi}$ and $A^{\\Psi}$ are Lipschitz similar."}, {"label": "E", "text": "The following are equivalent: (i) $\\Phi$ and $\\Psi$ are Lipschitz equivalent; (ii) $\\Phi$ and $\\Psi$ are Lipschitz conjugate; (iii) there exists $\\beta\\in\\mathbb{R}\\setminus\\{0\\}$ such that $A^{\\Phi}$ and $\\beta A^{\\Psi}$ are Lyapunov similar, while $A^{\\Phi_{\\sf AD}}$ and $\\beta A^{\\Psi_{\\sf AD}}$ are similar and $A^{\\Phi_{\\sf C}}$ and $\\beta A^{\\Psi_{\\sf C}}$ are similar; (iv) there exists $\\gamma\\in\\mathbb{R}\\setminus\\{0\\}$ such that $A^{\\Phi}$ and $\\gamma A^{\\Psi}$ are Lipschitz similar, while $A^{\\Phi_{\\sf C}}$ and $\\gamma A^{\\Psi_{\\sf C}}$ are similar. Moreover, $\\Phi$ and $\\Psi$ are Lipschitz conjugate if and only if $A^{\\Phi}$ and $A^{\\Psi}$ are Lipschitz similar and $A^{\\Phi_{\\sf C}}$ and $A^{\\Psi_{\\sf C}}$ are similar."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "nonzero time-rescaling parameter", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "full equivalence of all four conditions and moreover-clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "necessary anti-diagonal and central similarity clauses in the algebraic characterization", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "need for time-rescaling in passing from equivalence to conjugacy", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 1, "justification": "The stem does not explicitly state the correct option, but it heavily signals the structure of the answer by introducing exactly the notions used in the correct choice (time rescaling, Lipschitz similarity, and central/anti-diagonal decomposition). This creates strong thematic hints."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: it asks for the statement equivalent to the given conditions, and the correct choice is essentially the precise classification statement. However, the alternatives are not identical restatements, so it is not fully tautological."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in subtle but important ways: existential vs universal quantification, allowing gamma = 0, dropping central similarity, or weakening Lipschitz similarity to Lyapunov similarity. Still, the question mainly tests recognition of the exact theorem statement rather than deeper derivation."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically meaningful. Each reflects a plausible failure mode: boundary case on gamma, omission of a necessary condition, weakening a key equivalence notion, or quantifier strengthening. They are distinct and well-targeted."}, "total_score": 5, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors, but it leans toward recalling the precise statement rather than generating a conclusion from substantial reasoning."}}
{"id": "2602.13877v1", "paper_link": "http://arxiv.org/abs/2602.13877v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm1x}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nfour statements implies the other three:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{1}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  Lipschitz equivalent;\n  \\item there exists $\\alpha\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $\\Phi \\stackrel{1}{\\cong}\n    \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are Lipschitz conjugate;\n\\item there exists $\\beta\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\beta A^{\\Psi}$ are\n    Lyapunov similar while $A^{\\Phi_{\\sf  AD}}$, $\\beta A^{\\Psi_{\\sf\n        AD}}$ as well as $A^{\\Phi_{\\sf  C}}$, $\\beta A^{\\Psi_{\\sf\n        C}}$ are similar;\n\\item there exists $\\gamma\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\gamma A^{\\Psi}$ are\n    Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $\\gamma A^{\\Psi_{\\sf\n        C}}$ are similar.\n  \\end{enumerate}\n  Moreover, $\\Phi \\stackrel{1}{\\cong}\\Psi$ if and\nonly if $A^{\\Phi}$, $A^{\\Psi}$ are Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $ A^{\\Psi_{\\sf\n        C}}$ are similar.", "start_pos": 6937, "end_pos": 8006, "label": "thm1x"}, "ref_dict": {"eq1_2": "\\begin{equation}\\label{eq1_2}\n  h \\bigl( \\varphi  (t,x) \\bigr) = \\psi \\bigl( t,h(x)\\bigr)  \\qquad\n  \\forall t\\in \\R , x \\in X \\, .\n\\end{equation}", "prop1zb": "\\begin{prop}\\label{prop1zb}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nfive statements implies the other four:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{\\sf lin}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  linearly equivalent;\n \\item $\\Phi \\stackrel{\\sf diff}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  differentiably equivalent;\n  \\item there exists $\\alpha\n    \\in \\R \\setminus \\{0\\}$ so that $\\Phi \\stackrel{\\sf lin}{\\cong}\n    \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are linearly\n    conjugate;\n     \\item there exists $\\beta\n    \\in \\R \\setminus \\{0\\}$ so that $\\Phi \\stackrel{\\sf diff}{\\cong}\n    \\Psi_{*\\beta}$, i.e., $\\Phi$, $\\Psi_{*\\beta}$ are differentiably conjugate;\n\\item there exists $\\gamma\n    \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\gamma A^{\\Psi}$ are similar.\n  \\end{enumerate}\n  Moreover, $\\Phi \\stackrel{\\sf lin}{\\cong} \\Psi$ if and only if $\\Phi\n  \\stackrel{\\sf diff}{\\cong} \\Psi$ if and only if $A^{\\Phi}$,\n  $A^{\\Psi}$ are similar. \n\\end{prop}", "eq1_1": "\\begin{equation}\\label{eq1_1}\n  h \\bigl( \\varphi_{\\R} (x) \\bigr) = \\psi_{\\R} \\bigl( h(x)\\bigr)  \\qquad\n  \\forall x \\in X \\, .\n\\end{equation}", "prop1za": "\\begin{prop}\\label{prop1za}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nthree statements implies the other two:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{1^-}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  H\\\"{o}lder equivalent;\n  \\item there exists $\\alpha\n    \\in \\R \\setminus \\{0\\}$ so that $\\Phi \\stackrel{1^-}{\\cong}\n    \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are H\\\"{o}lder conjugate;\n\\item there exists $\\beta\n    \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\beta A^{\\Psi}$ are\n    Lyapunov similar while $A^{\\Phi_{\\sf C}}$, $\\beta A^{\\Psi_{\\sf\n        C}}$ are similar.\n  \\end{enumerate}\n  Moreover, $\\Phi \\stackrel{1^-}{\\cong} \\Psi$ if and only if\n  $A^{\\Phi}$, $A^{\\Psi}$ are Lyapunov similar while $A^{\\Phi_{\\sf C}}$,\n  $ A^{\\Psi_{\\sf C}}$ are similar.\n\\end{prop}", "thm1x": "\\begin{theorem}\\label{thm1x}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nfour statements implies the other three:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{1}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  Lipschitz equivalent;\n  \\item there exists $\\alpha\n    \\in \\R \\setminus \\{0\\}$ so that $\\Phi \\stackrel{1}{\\cong}\n    \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are Lipschitz conjugate;\n\\item there exists $\\beta\n    \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\beta A^{\\Psi}$ are\n    Lyapunov similar while $A^{\\Phi_{\\sf  AD}}$, $\\beta A^{\\Psi_{\\sf\n        AD}}$ as well as $A^{\\Phi_{\\sf  C}}$, $\\beta A^{\\Psi_{\\sf\n        C}}$ are similar;\n\\item there exists $\\gamma\n    \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\gamma A^{\\Psi}$ are\n    Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $\\gamma A^{\\Psi_{\\sf\n        C}}$ are similar.\n  \\end{enumerate}\n  Moreover, $\\Phi \\stackrel{1}{\\cong}\\Psi$ if and\nonly if $A^{\\Phi}$, $A^{\\Psi}$ are Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $ A^{\\Psi_{\\sf\n        C}}$ are similar.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 4367, "pre_theorem_intro_text": "\\label{sec1}\n\nLet $X\\ne \\{0\\}$ be a finite-dimensional normed space over $\\mathbb{R}$ and $\\varphi$\na flow on $X$, i.e., $\\varphi : \\mathbb{R} \\times X \\to X$ is\ncontinuous with $\\varphi(t+s , x) = \\varphi \\bigl( t, \\varphi (s,x)\n\\bigr)$ and $\\varphi(0,x) = x$ for all $t,s\\in \\mathbb{R}$, $x\\in X$. A\nfundamental question throughout dynamics is that of classification:\nWhen, precisely, are two flows $\\varphi$, $\\psi$ on $X$ {\\em the\n  same\\/}? Taking a geometrically motivated approach to this question,\nsay that $\\varphi$, $\\psi$ are {\\bf \n  equivalent}, in symbols $\\varphi \\stackrel{0}{\\thicksim} \\psi$, if there exists a\nhomeomorphism $h:X\\to X$ with $h(0)=0$ that maps each $\\varphi$-orbit\n$\\varphi_{\\mathbb{R}}(x):= \\{\\varphi (t, x):t\\in \\mathbb{R}\\}$ onto\na $\\psi$-orbit, i.e., \n\\begin{equation}\\label{eq1_1}\n  h \\bigl( \\varphi_{\\mathbb{R}} (x) \\bigr) = \\psi_{\\mathbb{R}} \\bigl( h(x)\\bigr)  \\qquad\n  \\forall x \\in X \\, .\n\\end{equation}\nIf $h$, $h^{-1}$ both are H\\\"{o}lder continuous (or Lipschitz\ncontinuous, differentiable, linear) then $\\varphi$, $\\psi$ are\nsaid to be {\\bf H\\\"{o}lder} (or {\\bf Lipschitz}, {\\bf differentiably},\n{\\bf linearly}) {\\bf equivalent}, in symbols $\\varphi \n\\stackrel{1^-}{\\thicksim} \\psi$ (or $\\varphi\n\\stackrel{1}{\\thicksim} \\psi$, $\\varphi\n\\stackrel{{\\sf diff}}{\\thicksim} \\psi$, $\\varphi\n\\stackrel{{\\sf lin}}{\\thicksim} \\psi$). Plainly $\\stackrel{\\bigstar}{\\thicksim}$\nyields an equivalence relation \nfor each $\\bigstar\\in \\{0,1^-,1, {\\sf diff}, {\\sf lin}\\}$, thereby establishing\nfive natural classifications of all flows on $X$. In addition,\nconsider the following, much more\nrestrictive form of equivalence: Say that $\\varphi$, $\\psi$ are {\\bf conjugate}, in symbols\n$\\varphi \\stackrel{0}{\\cong} \\psi$, if \n\\begin{equation}\\label{eq1_2}\n  h \\bigl( \\varphi  (t,x) \\bigr) = \\psi \\bigl( t,h(x)\\bigr)  \\qquad\n  \\forall t\\in \\mathbb{R} , x \\in X \\, .\n\\end{equation}\nNotice that (\\ref{eq1_2}) implies (\\ref{eq1_1}) but most definitely\nnot vice versa, and analogously defined\n$\\stackrel{\\bigstar}{\\cong}$ again yields an equivalence relation for\neach $\\bigstar \\in \\{0, 1^-, 1, {\\sf diff}, {\\sf lin}\\}$. Simple examples\nshow that no two of these equivalences, or the\nclassification established by them, coincide, not even when $\\mbox{\\rm\n  dim}\\, X=1$.\n\nBuilding on the classical literature briefly reviewed below, the\npresent article concludes the authors' earlier work \\cite{BW, BW2} \nby carrying out a comprehensive analysis of Lipschitz equivalence and conjugacy\nfor {\\em linear\\/} flows. Recall that a flow $\\varphi$ on $X$ is {\\bf linear} if the time-$t$\nmap $\\varphi_t = \\varphi (t,\\cdot):X\\to X$ is linear, or equivalently\nif $\\varphi_t = e^{t A^{\\varphi}}$,\nfor every $t\\in \\mathbb{R}$, with a (unique) linear operator $A^{\\varphi}$ on\n$X$ called the {\\bf generator} of $\\varphi$. Henceforth, upper case Greek letters\n$\\Phi$, $\\Psi$ are used exclusively to denote linear flows. The challenge, then, is to\ncharacterize $\\Phi \\stackrel{1}{\\thicksim} \\Psi$ and $\\Phi\n\\stackrel{1}{\\cong} \\Psi$ in terms of basic linear algebra properties\nof $A^{\\Phi}$, $A^{\\Psi}$. Correspondingly the main result of this\narticle, Theorem \\ref{thm1x} below, can be\nviewed as a {\\bf Lipschitz classification theorem}. To preview the result,\nnote that every linear flow $\\Phi$ on $X$ determines unique\ndecompositions $\\Phi \\stackrel{{\\sf lin}}{\\cong}  \\Phi_{\\sf D} \\times \\Phi_{\\sf\n  AD} \n\\stackrel{{\\sf lin}}{\\cong} \\Phi_{\\sf S} \\times \\Phi_{\\sf C} \\times\n\\Phi_{\\sf U}$ into a {\\em diagonal\\/}\nand {\\em anti-diagonal\\/} part, as well as into a\n{\\em stable}, {\\em central}, and {\\em unstable\\/} part; see Section \\ref{sec2a} below\nfor formal details. For convenience denote by\n$\\Phi_{*\\alpha}$ the linear flow generated by $\\alpha A^{\\Phi}$, for\nany $\\alpha \\in \\mathbb{R} \\setminus \\{0\\}$. Thus, $\\Phi_{*\\alpha}$ simply is\n$\\Phi$ with its first variable (``time'') rescaled by $\\alpha$. Finally, the statement involves\nthe concepts of {\\em Lyapunov\\/} and {\\em Lipschitz\nsimilarity}, introduced rigorously in Section\n\\ref{sec2a} also. For now, simply say that two linear operators are {\\bf\n  Lyapunov similar} if they (more precisely, the flows they generate) have the same\nLyapunov exponents, with matching multiplicities, and say that they\nare  {\\bf Lipschitz similar} if they are Lyapunov similar and their\nanti-diagonal parts are similar.", "context": "\\label{sec1}\n\nLet $X\\ne \\{0\\}$ be a finite-dimensional normed space over $\\mathbb{R}$ and $\\varphi$\na flow on $X$, i.e., $\\varphi : \\mathbb{R} \\times X \\to X$ is\ncontinuous with $\\varphi(t+s , x) = \\varphi \\bigl( t, \\varphi (s,x)\n\\bigr)$ and $\\varphi(0,x) = x$ for all $t,s\\in \\mathbb{R}$, $x\\in X$. A\nfundamental question throughout dynamics is that of classification:\nWhen, precisely, are two flows $\\varphi$, $\\psi$ on $X$ {\\em the\n same\\/}? Taking a geometrically motivated approach to this question,\nsay that $\\varphi$, $\\psi$ are {\\bf \n equivalent}, in symbols $\\varphi \\stackrel{0}{\\thicksim} \\psi$, if there exists a\nhomeomorphism $h:X\\to X$ with $h(0)=0$ that maps each $\\varphi$-orbit\n$\\varphi_{\\mathbb{R}}(x):= \\{\\varphi (t, x):t\\in \\mathbb{R}\\}$ onto\na $\\psi$-orbit, i.e., \n\\begin{equation}\\label{eq1_1}\n h \\bigl( \\varphi_{\\mathbb{R}} (x) \\bigr) = \\psi_{\\mathbb{R}} \\bigl( h(x)\\bigr) \\qquad\n \\forall x \\in X \\, .\n\\end{equation}\nIf $h$, $h^{-1}$ both are H\\\"{o}lder continuous (or Lipschitz\ncontinuous, differentiable, linear) then $\\varphi$, $\\psi$ are\nsaid to be {\\bf H\\\"{o}lder} (or {\\bf Lipschitz}, {\\bf differentiably},\n{\\bf linearly}) {\\bf equivalent}, in symbols $\\varphi \n\\stackrel{1^-}{\\thicksim} \\psi$ (or $\\varphi\n\\stackrel{1}{\\thicksim} \\psi$, $\\varphi\n\\stackrel{{\\sf diff}}{\\thicksim} \\psi$, $\\varphi\n\\stackrel{{\\sf lin}}{\\thicksim} \\psi$). Plainly $\\stackrel{\\bigstar}{\\thicksim}$\nyields an equivalence relation \nfor each $\\bigstar\\in \\{0,1^-,1, {\\sf diff}, {\\sf lin}\\}$, thereby establishing\nfive natural classifications of all flows on $X$. In addition,\nconsider the following, much more\nrestrictive form of equivalence: Say that $\\varphi$, $\\psi$ are {\\bf conjugate}, in symbols\n$\\varphi \\stackrel{0}{\\cong} \\psi$, if \n\\begin{equation}\\label{eq1_2}\n h \\bigl( \\varphi (t,x) \\bigr) = \\psi \\bigl( t,h(x)\\bigr) \\qquad\n \\forall t\\in \\mathbb{R} , x \\in X \\, .\n\\end{equation}\nNotice that (\\ref{eq1_2}) implies (\\ref{eq1_1}) but most definitely\nnot vice versa, and analogously defined\n$\\stackrel{\\bigstar}{\\cong}$ again yields an equivalence relation for\neach $\\bigstar \\in \\{0, 1^-, 1, {\\sf diff}, {\\sf lin}\\}$. Simple examples\nshow that no two of these equivalences, or the\nclassification established by them, coincide, not even when $\\mbox{\\rm\n dim}\\, X=1$.\n\nBuilding on the classical literature briefly reviewed below, the\npresent article concludes the authors' earlier work \\cite{BW, BW2} \nby carrying out a comprehensive analysis of Lipschitz equivalence and conjugacy\nfor {\\em linear\\/} flows. Recall that a flow $\\varphi$ on $X$ is {\\bf linear} if the time-$t$\nmap $\\varphi_t = \\varphi (t,\\cdot):X\\to X$ is linear, or equivalently\nif $\\varphi_t = e^{t A^{\\varphi}}$,\nfor every $t\\in \\mathbb{R}$, with a (unique) linear operator $A^{\\varphi}$ on\n$X$ called the {\\bf generator} of $\\varphi$. Henceforth, upper case Greek letters\n$\\Phi$, $\\Psi$ are used exclusively to denote linear flows. The challenge, then, is to\ncharacterize $\\Phi \\stackrel{1}{\\thicksim} \\Psi$ and $\\Phi\n\\stackrel{1}{\\cong} \\Psi$ in terms of basic linear algebra properties\nof $A^{\\Phi}$, $A^{\\Psi}$. Correspondingly the main result of this\narticle, Theorem \\ref{thm1x} below, can be\nviewed as a {\\bf Lipschitz classification theorem}. To preview the result,\nnote that every linear flow $\\Phi$ on $X$ determines unique\ndecompositions $\\Phi \\stackrel{{\\sf lin}}{\\cong} \\Phi_{\\sf D} \\times \\Phi_{\\sf\n AD} \n\\stackrel{{\\sf lin}}{\\cong} \\Phi_{\\sf S} \\times \\Phi_{\\sf C} \\times\n\\Phi_{\\sf U}$ into a {\\em diagonal\\/}\nand {\\em anti-diagonal\\/} part, as well as into a\n{\\em stable}, {\\em central}, and {\\em unstable\\/} part; see Section \\ref{sec2a} below\nfor formal details. For convenience denote by\n$\\Phi_{*\\alpha}$ the linear flow generated by $\\alpha A^{\\Phi}$, for\nany $\\alpha \\in \\mathbb{R} \\setminus \\{0\\}$. Thus, $\\Phi_{*\\alpha}$ simply is\n$\\Phi$ with its first variable (``time'') rescaled by $\\alpha$. Finally, the statement involves\nthe concepts of {\\em Lyapunov\\/} and {\\em Lipschitz\nsimilarity}, introduced rigorously in Section\n\\ref{sec2a} also. For now, simply say that two linear operators are {\\bf\n Lyapunov similar} if they (more precisely, the flows they generate) have the same\nLyapunov exponents, with matching multiplicities, and say that they\nare {\\bf Lipschitz similar} if they are Lyapunov similar and their\nanti-diagonal parts are similar.\n\n\\begin{equation}\\label{eq1_1}\n  h \\bigl( \\varphi_{\\R} (x) \\bigr) = \\psi_{\\R} \\bigl( h(x)\\bigr)  \\qquad\n  \\forall x \\in X \\, .\n\\end{equation}\n\n\\begin{equation}\\label{eq1_2}\n  h \\bigl( \\varphi  (t,x) \\bigr) = \\psi \\bigl( t,h(x)\\bigr)  \\qquad\n  \\forall t\\in \\R , x \\in X \\, .\n\\end{equation}\n\n\\begin{theorem}\\label{thm1x}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nfour statements implies the other three:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{1}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  Lipschitz equivalent;\n  \\item there exists $\\alpha\n    \\in \\R \\setminus \\{0\\}$ so that $\\Phi \\stackrel{1}{\\cong}\n    \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are Lipschitz conjugate;\n\\item there exists $\\beta\n    \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\beta A^{\\Psi}$ are\n    Lyapunov similar while $A^{\\Phi_{\\sf  AD}}$, $\\beta A^{\\Psi_{\\sf\n        AD}}$ as well as $A^{\\Phi_{\\sf  C}}$, $\\beta A^{\\Psi_{\\sf\n        C}}$ are similar;\n\\item there exists $\\gamma\n    \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\gamma A^{\\Psi}$ are\n    Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $\\gamma A^{\\Psi_{\\sf\n        C}}$ are similar.\n  \\end{enumerate}\n  Moreover, $\\Phi \\stackrel{1}{\\cong}\\Psi$ if and\nonly if $A^{\\Phi}$, $A^{\\Psi}$ are Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $ A^{\\Psi_{\\sf\n        C}}$ are similar.\n\\end{theorem}", "full_context": "\\label{sec1}\n\nLet $X\\ne \\{0\\}$ be a finite-dimensional normed space over $\\mathbb{R}$ and $\\varphi$\na flow on $X$, i.e., $\\varphi : \\mathbb{R} \\times X \\to X$ is\ncontinuous with $\\varphi(t+s , x) = \\varphi \\bigl( t, \\varphi (s,x)\n\\bigr)$ and $\\varphi(0,x) = x$ for all $t,s\\in \\mathbb{R}$, $x\\in X$. A\nfundamental question throughout dynamics is that of classification:\nWhen, precisely, are two flows $\\varphi$, $\\psi$ on $X$ {\\em the\n same\\/}? Taking a geometrically motivated approach to this question,\nsay that $\\varphi$, $\\psi$ are {\\bf \n equivalent}, in symbols $\\varphi \\stackrel{0}{\\thicksim} \\psi$, if there exists a\nhomeomorphism $h:X\\to X$ with $h(0)=0$ that maps each $\\varphi$-orbit\n$\\varphi_{\\mathbb{R}}(x):= \\{\\varphi (t, x):t\\in \\mathbb{R}\\}$ onto\na $\\psi$-orbit, i.e., \n\\begin{equation}\\label{eq1_1}\n h \\bigl( \\varphi_{\\mathbb{R}} (x) \\bigr) = \\psi_{\\mathbb{R}} \\bigl( h(x)\\bigr) \\qquad\n \\forall x \\in X \\, .\n\\end{equation}\nIf $h$, $h^{-1}$ both are H\\\"{o}lder continuous (or Lipschitz\ncontinuous, differentiable, linear) then $\\varphi$, $\\psi$ are\nsaid to be {\\bf H\\\"{o}lder} (or {\\bf Lipschitz}, {\\bf differentiably},\n{\\bf linearly}) {\\bf equivalent}, in symbols $\\varphi \n\\stackrel{1^-}{\\thicksim} \\psi$ (or $\\varphi\n\\stackrel{1}{\\thicksim} \\psi$, $\\varphi\n\\stackrel{{\\sf diff}}{\\thicksim} \\psi$, $\\varphi\n\\stackrel{{\\sf lin}}{\\thicksim} \\psi$). Plainly $\\stackrel{\\bigstar}{\\thicksim}$\nyields an equivalence relation \nfor each $\\bigstar\\in \\{0,1^-,1, {\\sf diff}, {\\sf lin}\\}$, thereby establishing\nfive natural classifications of all flows on $X$. In addition,\nconsider the following, much more\nrestrictive form of equivalence: Say that $\\varphi$, $\\psi$ are {\\bf conjugate}, in symbols\n$\\varphi \\stackrel{0}{\\cong} \\psi$, if \n\\begin{equation}\\label{eq1_2}\n h \\bigl( \\varphi (t,x) \\bigr) = \\psi \\bigl( t,h(x)\\bigr) \\qquad\n \\forall t\\in \\mathbb{R} , x \\in X \\, .\n\\end{equation}\nNotice that (\\ref{eq1_2}) implies (\\ref{eq1_1}) but most definitely\nnot vice versa, and analogously defined\n$\\stackrel{\\bigstar}{\\cong}$ again yields an equivalence relation for\neach $\\bigstar \\in \\{0, 1^-, 1, {\\sf diff}, {\\sf lin}\\}$. Simple examples\nshow that no two of these equivalences, or the\nclassification established by them, coincide, not even when $\\mbox{\\rm\n dim}\\, X=1$.\n\nBuilding on the classical literature briefly reviewed below, the\npresent article concludes the authors' earlier work \\cite{BW, BW2} \nby carrying out a comprehensive analysis of Lipschitz equivalence and conjugacy\nfor {\\em linear\\/} flows. Recall that a flow $\\varphi$ on $X$ is {\\bf linear} if the time-$t$\nmap $\\varphi_t = \\varphi (t,\\cdot):X\\to X$ is linear, or equivalently\nif $\\varphi_t = e^{t A^{\\varphi}}$,\nfor every $t\\in \\mathbb{R}$, with a (unique) linear operator $A^{\\varphi}$ on\n$X$ called the {\\bf generator} of $\\varphi$. Henceforth, upper case Greek letters\n$\\Phi$, $\\Psi$ are used exclusively to denote linear flows. The challenge, then, is to\ncharacterize $\\Phi \\stackrel{1}{\\thicksim} \\Psi$ and $\\Phi\n\\stackrel{1}{\\cong} \\Psi$ in terms of basic linear algebra properties\nof $A^{\\Phi}$, $A^{\\Psi}$. Correspondingly the main result of this\narticle, Theorem \\ref{thm1x} below, can be\nviewed as a {\\bf Lipschitz classification theorem}. To preview the result,\nnote that every linear flow $\\Phi$ on $X$ determines unique\ndecompositions $\\Phi \\stackrel{{\\sf lin}}{\\cong} \\Phi_{\\sf D} \\times \\Phi_{\\sf\n AD} \n\\stackrel{{\\sf lin}}{\\cong} \\Phi_{\\sf S} \\times \\Phi_{\\sf C} \\times\n\\Phi_{\\sf U}$ into a {\\em diagonal\\/}\nand {\\em anti-diagonal\\/} part, as well as into a\n{\\em stable}, {\\em central}, and {\\em unstable\\/} part; see Section \\ref{sec2a} below\nfor formal details. For convenience denote by\n$\\Phi_{*\\alpha}$ the linear flow generated by $\\alpha A^{\\Phi}$, for\nany $\\alpha \\in \\mathbb{R} \\setminus \\{0\\}$. Thus, $\\Phi_{*\\alpha}$ simply is\n$\\Phi$ with its first variable (``time'') rescaled by $\\alpha$. Finally, the statement involves\nthe concepts of {\\em Lyapunov\\/} and {\\em Lipschitz\nsimilarity}, introduced rigorously in Section\n\\ref{sec2a} also. For now, simply say that two linear operators are {\\bf\n Lyapunov similar} if they (more precisely, the flows they generate) have the same\nLyapunov exponents, with matching multiplicities, and say that they\nare {\\bf Lipschitz similar} if they are Lyapunov similar and their\nanti-diagonal parts are similar.\n\n\\begin{equation}\\label{eq1_1}\n  h \\bigl( \\varphi_{\\R} (x) \\bigr) = \\psi_{\\R} \\bigl( h(x)\\bigr)  \\qquad\n  \\forall x \\in X \\, .\n\\end{equation}\n\n\\begin{equation}\\label{eq1_2}\n  h \\bigl( \\varphi  (t,x) \\bigr) = \\psi \\bigl( t,h(x)\\bigr)  \\qquad\n  \\forall t\\in \\R , x \\in X \\, .\n\\end{equation}\n\n\\begin{theorem}\\label{thm1x}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nfour statements implies the other three:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{1}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  Lipschitz equivalent;\n  \\item there exists $\\alpha\n    \\in \\R \\setminus \\{0\\}$ so that $\\Phi \\stackrel{1}{\\cong}\n    \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are Lipschitz conjugate;\n\\item there exists $\\beta\n    \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\beta A^{\\Psi}$ are\n    Lyapunov similar while $A^{\\Phi_{\\sf  AD}}$, $\\beta A^{\\Psi_{\\sf\n        AD}}$ as well as $A^{\\Phi_{\\sf  C}}$, $\\beta A^{\\Psi_{\\sf\n        C}}$ are similar;\n\\item there exists $\\gamma\n    \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\gamma A^{\\Psi}$ are\n    Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $\\gamma A^{\\Psi_{\\sf\n        C}}$ are similar.\n  \\end{enumerate}\n  Moreover, $\\Phi \\stackrel{1}{\\cong}\\Psi$ if and\nonly if $A^{\\Phi}$, $A^{\\Psi}$ are Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $ A^{\\Psi_{\\sf\n        C}}$ are similar.\n\\end{theorem}\n\nBuilding on the classical literature briefly reviewed below, the\npresent article concludes the authors' earlier work \\cite{BW, BW2} \nby carrying out a comprehensive analysis of Lipschitz equivalence and conjugacy\nfor {\\em linear\\/} flows. Recall that a flow $\\varphi$ on $X$ is {\\bf linear} if the time-$t$\nmap $\\varphi_t = \\varphi (t,\\cdot):X\\to X$ is linear, or equivalently\nif $\\varphi_t = e^{t A^{\\varphi}}$,\nfor every $t\\in \\R$, with a (unique) linear operator $A^{\\varphi}$ on\n$X$ called the {\\bf generator} of $\\varphi$. Henceforth, upper case Greek letters\n$\\Phi$, $\\Psi$ are used exclusively to denote linear flows. The challenge, then, is to\ncharacterize $\\Phi \\stackrel{1}{\\thicksim} \\Psi$ and $\\Phi\n\\stackrel{1}{\\cong} \\Psi$ in terms of basic linear algebra properties\nof $A^{\\Phi}$, $A^{\\Psi}$. Correspondingly the main result of this\narticle, Theorem \\ref{thm1x} below, can be\nviewed as a {\\bf Lipschitz classification theorem}. To preview the result,\nnote that every linear flow $\\Phi$ on $X$ determines unique\ndecompositions $\\Phi \\stackrel{{\\sf lin}}{\\cong} \\Phi_{\\sf D} \\times \\Phi_{\\sf\n AD} \n\\stackrel{{\\sf lin}}{\\cong} \\Phi_{\\sf S} \\times \\Phi_{\\sf C} \\times\n\\Phi_{\\sf U}$ into a {\\em diagonal\\/}\nand {\\em anti-diagonal\\/} part, as well as into a\n{\\em stable}, {\\em central}, and {\\em unstable\\/} part; see Section \\ref{sec2a} below\nfor formal details. For convenience denote by\n$\\Phi_{*\\alpha}$ the linear flow generated by $\\alpha A^{\\Phi}$, for\nany $\\alpha \\in \\R \\setminus \\{0\\}$. Thus, $\\Phi_{*\\alpha}$ simply is\n$\\Phi$ with its first variable (``time'') rescaled by $\\alpha$. Finally, the statement involves\nthe concepts of {\\em Lyapunov\\/} and {\\em Lipschitz\nsimilarity}, introduced rigorously in Section\n\\ref{sec2a} also. For now, simply say that two linear operators are {\\bf\n Lyapunov similar} if they (more precisely, the flows they generate) have the same\nLyapunov exponents, with matching multiplicities, and say that they\nare {\\bf Lipschitz similar} if they are Lyapunov similar and their\nanti-diagonal parts are similar.\n\nTo put Theorem \\ref{thm1x} in context, it is instructive to compare it\nto its differentiable (hence slightly more restrictive) and H\\\"{o}lder\n(hence slightly less restrictive) counterparts; stated here without\nproofs, these results have been presented by the authors in detail\n(though sometimes couched in slightly different terminology and\nnotation) in \\cite{BW} and \\cite{BW2} respectively.\n\n\\begin{prop}\\label{prop1zb}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nfive statements implies the other four:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{\\sf lin}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n linearly equivalent;\n \\item $\\Phi \\stackrel{\\sf diff}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n differentiably equivalent;\n \\item there exists $\\alpha\n \\in \\R \\setminus \\{0\\}$ so that $\\Phi \\stackrel{\\sf lin}{\\cong}\n \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are linearly\n conjugate;\n \\item there exists $\\beta\n \\in \\R \\setminus \\{0\\}$ so that $\\Phi \\stackrel{\\sf diff}{\\cong}\n \\Psi_{*\\beta}$, i.e., $\\Phi$, $\\Psi_{*\\beta}$ are differentiably conjugate;\n\\item there exists $\\gamma\n \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\gamma A^{\\Psi}$ are similar.\n \\end{enumerate}\n Moreover, $\\Phi \\stackrel{\\sf lin}{\\cong} \\Psi$ if and only if $\\Phi\n \\stackrel{\\sf diff}{\\cong} \\Psi$ if and only if $A^{\\Phi}$,\n $A^{\\Psi}$ are similar. \n\\end{prop}\n\n\\begin{prop}\\label{prop1za}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nthree statements implies the other two:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{1^-}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n H\\\"{o}lder equivalent;\n \\item there exists $\\alpha\n \\in \\R \\setminus \\{0\\}$ so that $\\Phi \\stackrel{1^-}{\\cong}\n \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are H\\\"{o}lder conjugate;\n\\item there exists $\\beta\n \\in \\R \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\beta A^{\\Psi}$ are\n Lyapunov similar while $A^{\\Phi_{\\sf C}}$, $\\beta A^{\\Psi_{\\sf\n C}}$ are similar.\n \\end{enumerate}\n Moreover, $\\Phi \\stackrel{1^-}{\\cong} \\Psi$ if and only if\n $A^{\\Phi}$, $A^{\\Psi}$ are Lyapunov similar while $A^{\\Phi_{\\sf C}}$,\n $ A^{\\Psi_{\\sf C}}$ are similar.\n\\end{prop}\n\n\\begin{lem}\\label{lem4_2}\nGiven $m\\in \\N \\setminus \\{1\\}$, $k,\\ell\\in \\N$, and $b\\in \\R^k$, $c\\in\n\\R^{\\ell}$ with $m\\, {\\sf d}(b) = m\\, {\\sf d}(c) = d$, let $\\Phi$, $\\Psi$ be the flows on $\\R^d$ generated by\n$A$, $B$ in {\\rm (\\ref{eq4_1})} respectively. Then the following\nstatements are equivalent:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{{\\sf lin}}{\\cong}\\Psi$;\n\\item $\\Phi \\stackrel{1}{\\thicksim} \\Psi$;\n \\item $k=\\ell$, and there exists a bijection $g:\\{1, \\ldots , k\\}\\to \\{1, \\ldots\n , \\ell\\}$ so that $|b_j| = |c_{g(j)}|$ for every $j$;\n\\item $A$, $B$ are similar;\n\\item $A$, $B$ are Lipschitz similar. \n\\end{enumerate}\n\\end{lem}\n\n\\begin{theorem}\\label{thm71}\nLet $\\Phi$, $\\Psi$ be $\\K$-linear flows on $X$. Then the following\nstatements are equivalent:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{1}{\\thicksim}\\Psi$;\n \\item there exists $\\alpha\\in \\R\\setminus \\{0\\}$ so that $\\Phi \\stackrel{1}{\\cong}\\Psi_{*\\alpha}$;\n \\item $\\Phi^{\\R} \\stackrel{1}{\\thicksim} \\Psi^{\\R}$;\n \\item there exists $\\beta\\in \\R\\setminus \\{0\\}$ so that $\\Phi^{\\R} \\stackrel{1}{\\cong}\\Psi^{\\R}_{*\\beta}$;\n\\item there exists $\\gamma \\in \\R \\setminus \\{0\\}$ so that\n$A^{\\Phi^{\\R}}$, $\\gamma A^{\\Psi^{\\R}}$ are Lipschitz similar while\n$A^{\\Phi_{\\sf C}^{\\R}}$, $\\gamma A^{\\Psi_{\\sf C}^{\\R}}$ are similar.\n\\end{enumerate}\nMoreover, $\\Phi \\stackrel{1}{\\cong}\\Psi$ if and only if $\\Phi^{\\R} \\stackrel{1}{\\cong}\\Psi^{\\R}$\n if and only if $A^{\\Phi^{\\R}}$, $A^{\\Psi^{\\R}}$ are Lipschitz similar while\n$A^{\\Phi_{\\sf C}^{\\R}}$, $A^{\\Psi_{\\sf C}^{\\R}}$ are similar.\n\\end{theorem}\n\n\\begin{theorem}\\label{thm6_8}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then the following\nstatements are equivalent:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{{\\sf pw}1}{\\cong}\\Psi$;\n \\item $\\Phi$, $\\Psi$ are kinematically similar while $\\Phi_{\\sf C}\n \\stackrel{{\\sf lin}}{\\cong} \\Psi_{\\sf C}$;\n \\item $A^{\\Phi}$, $A^{\\Psi}$ are kinematically similar while\n $A^{\\Phi_{\\sf C}}, A^{\\Psi_{\\sf C}}$ are similar.\n\\end{enumerate}\n\\end{theorem}", "post_theorem_intro_text_len": 7393, "post_theorem_intro_text": "To put Theorem \\ref{thm1x} in context, it is instructive to compare it\nto its differentiable (hence slightly more restrictive) and H\\\"{o}lder\n(hence slightly less restrictive) counterparts; stated here without\nproofs, these results have been presented by the authors in detail\n(though sometimes couched in slightly different terminology and\nnotation) in \\cite{BW} and \\cite{BW2} respectively.\n\n\\begin{prop}\\label{prop1zb}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nfive statements implies the other four:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{\\sf lin}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  linearly equivalent;\n \\item $\\Phi \\stackrel{\\sf diff}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  differentiably equivalent;\n  \\item there exists $\\alpha\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $\\Phi \\stackrel{\\sf lin}{\\cong}\n    \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are linearly\n    conjugate;\n     \\item there exists $\\beta\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $\\Phi \\stackrel{\\sf diff}{\\cong}\n    \\Psi_{*\\beta}$, i.e., $\\Phi$, $\\Psi_{*\\beta}$ are differentiably conjugate;\n\\item there exists $\\gamma\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\gamma A^{\\Psi}$ are similar.\n  \\end{enumerate}\n  Moreover, $\\Phi \\stackrel{\\sf lin}{\\cong} \\Psi$ if and only if $\\Phi\n  \\stackrel{\\sf diff}{\\cong} \\Psi$ if and only if $A^{\\Phi}$,\n  $A^{\\Psi}$ are similar. \n\\end{prop}\n\n\\begin{prop}\\label{prop1za}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nthree statements implies the other two:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{1^-}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  H\\\"{o}lder equivalent;\n  \\item there exists $\\alpha\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $\\Phi \\stackrel{1^-}{\\cong}\n    \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are H\\\"{o}lder conjugate;\n\\item there exists $\\beta\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\beta A^{\\Psi}$ are\n    Lyapunov similar while $A^{\\Phi_{\\sf C}}$, $\\beta A^{\\Psi_{\\sf\n        C}}$ are similar.\n  \\end{enumerate}\n  Moreover, $\\Phi \\stackrel{1^-}{\\cong} \\Psi$ if and only if\n  $A^{\\Phi}$, $A^{\\Psi}$ are Lyapunov similar while $A^{\\Phi_{\\sf C}}$,\n  $ A^{\\Psi_{\\sf C}}$ are similar.\n\\end{prop}\n\nClassifications of linear flows have long been studied in the\nliterature, notably for {\\em hyperbolic\\/} flows, that is, for\n$\\Phi_{\\sf C}$, $\\Psi_{\\sf C}$ being trivial; see, e.g., \\cite{Amann, BW,\n  Irwin, R} for broad context, as well as \\cite{ACK1, AK, DSS, He, LZ, Willems} for\nspecific studies.\nOne striking aspect of Theorem\n\\ref{thm1x} is the fact that (i)$\\Rightarrow$(ii). Thus, for {\\em linear\\/}\nflows $\\varphi$, $\\psi$ validity of (\\ref{eq1_1}) always entails\nvalidity of (\\ref{eq1_2}), up to a linear, orbit-independent rescaling\nof time. This remarkable property, which does not \nseem to be shared by any wider class of flows on $X$, is indicative of\nthe extraordinary coherence between individual orbits of linear\nflows. As far as the authors have been able to ascertain, the\nproperty has not been stated, let alone proved rigorously before,\nthough it appears to have been part of linear systems folklore for\nquite some time; see, e.g., \\cite[Rem.\\ 7.4]{Willems} as well as\n\\cite{ACK1, CK, KS}.\n\nWith Lipschitz continuity being a fundamental concept throughout\nanalysis \\cite{Hei}, another key aspect of Theorem \\ref{thm1x} is its\nrelation to the differentiable and H\\\"{o}lder counterparts,\nPropositions \\ref{prop1zb} and \\ref{prop1za} respectively. While\n\\cite[Rem.\\ 2.5]{ACK1} seems to suggest, somewhat misleadingly, that the equivalence\nrelation $\\stackrel{1}{\\cong}$ for linear flows simply coincides with $\\stackrel{{\\sf\n    diff}}{\\cong}$, the characterization of Lipschitz {\\em\n  conjugacy\\/} contained in Theorem \\ref{thm1x} has, in essence, been\nestablished in \\cite{KS}; see also \\cite{MM} for a related (albeit quite\ninformal) discussion. Specifically, \\cite{KS} argues that\n$\\stackrel{1}{\\cong}$ is ``very close'' to $\\stackrel{{\\sf\n    diff}}{\\cong}$, and a crucial role in the argument is subsequently\nplayed by a well-known theorem due to Rademacher which asserts that\nevery Lipschitz map $h:X\\to X$ is differentiable almost\neverywhere. However, in\nthe general setting of Theorem \\ref{thm1x}, that is, for mere Lipschitz {\\em equivalence}, no\nassumptions whatsoever are made regarding the (orientation and\nregularity of) re-parametrizations of individual orbits, and hence it appears\ndoubtful whether Rademacher's theorem can be applied\nfruitfully (or at all). Instead, the present\narticle utilizes a simple geometric idea gleaned from \\cite{MM} which\nit develops into the basic but consequential concept of {\\em\n  distortion\\/} ({\\em points\\/}) for stable flows. Aided by this\nconcept, the article then proceeds to prove, in an entirely elementary\nmanner, that (i)$\\Rightarrow$(iii) in Theorem \\ref{thm1x}. From this,\nvalidity of the entire theorem follows rather straightforwardly, as\n(iii)$\\Leftrightarrow$(iv)$\\Rightarrow$(ii), and obviously\n(ii)$\\Rightarrow$(i).  As indicated where appropriate below, the\nelementary approach developed here also helps to address other gaps and\ninaccuracies in the existing literature.\n\n\\medskip\n\n\\begin{rem}\nAn analogous classification problem presents itself in discrete time,\ni.e., for linear operators $A, B:X\\to X$ which are {\\bf conjugate} (or\n{\\bf nonlinearly similar} \\cite{CappSh, CSSW}), in symbols\n$A\\stackrel{0}{\\cong} B$, if $h(Ax) = Bh(x)$ for some homeomorphism\n$h:X\\to X$ and all $x\\in X$. As in continuous time, it is\nnatural to consider $\\stackrel{\\bigstar}{\\cong}$ for $\\bigstar \\in\n\\{0,1^-,1,{\\sf diff}, {\\sf lin}\\}$, each of which yields an\nequivalence relation on all linear operators on $X$. In\nstraightforward analogy to Proposition \\ref{prop1zb}, notice that\n$A\\stackrel{{\\sf lin}}{\\cong} B$ if and only if $A\\stackrel{{\\sf\n    diff}}{\\cong} B$ if and only if $A$, $B$ are similar. By contrast,\nthe problem of characterizing $A\\stackrel{\\bigstar}{\\cong} B$ for\n$\\bigstar \\in \\{0,1^-, 1\\}$, in terms of similarity invariants akin to\nTheorem \\ref{thm1x}, has turned out to be significantly more challenging than in \ncontinuous time; see, e.g., \\cite{CappSh, CSSW, Cruz, HP, KR} for the\nlong history of the problem and its many ramifications, with\n\\cite{Cruz} specifically addressing the Lipschitz case.\n\\end{rem}\n\nThe remainder of this article is organized as follows: Section\n\\ref{sec2} properly introduces various forms of equivalence and\nconjugacy to be studied in subsequent sections, together with some\ntailor-made analytical notation. Section \\ref{sec2a} briefly reviews a\nfew basic concepts pertinent to linear flows, notably irreducibility\nand Lyapunov exponents, and then discusses the novel concept of\nLipschitz similarity. Section \\ref{sec3} defines distortion points and\nrefined Lyapunov spaces for stable flows which in turn yield crucial Lipschitz\ninvariants for such flows. Section\n\\ref{sec4} presents a proof of the main result, Theorem \\ref{thm1x},\nvia a series of preparatory lemmas addressing important special cases\nthereof. It also outlines the straightforward extension of Theorem\n\\ref{thm1x} to {\\em complex\\/} spaces. Finally, Section \\ref{sec6}\nhighlights several subtle aspects of Lipschitz equivalence by briefly\ndiscussing a novel variant of the concept, referred to as {\\em\n  pointwise\\/} Lipschitz equivalence at $0$.", "sketch": "The article explains that, rather than using Rademacher’s theorem as in \\cite{KS}, it “utilizes a simple geometric idea gleaned from \\cite{MM} which it develops into the basic but consequential concept of {\\em distortion\\/} ({\\em points\\/}) for stable flows.” Using this concept, it “then proceeds to prove, in an entirely elementary manner, that (i)$\\Rightarrow$(iii) in Theorem \\ref{thm1x}.” From there, “validity of the entire theorem follows rather straightforwardly, as (iii)$\\Leftrightarrow$(iv)$\\Rightarrow$(ii), and obviously (ii)$\\Rightarrow$(i).”", "expanded_sketch": "The article explains that, rather than using Rademacher’s theorem as in \\cite{KS}, it “utilizes a simple geometric idea gleaned from \\cite{MM} which it develops into the basic but consequential concept of {\\em distortion\\/} ({\\em points\\/}) for stable flows.” Using this concept, it “then proceeds to prove, in an entirely elementary manner,” one implication needed to establish the main theorem. From there, “validity of the entire theorem follows rather straightforwardly, as (iii)$\\Leftrightarrow$(iv)$\\Rightarrow$(ii), and obviously (ii)$\\Rightarrow$(i).”", "expanded_theorem": "\\label{thm1x}\nLet $\\Phi$, $\\Psi$ be linear flows on $X$. Then each of the following\nfour statements implies the other three:\n\\begin{enumerate}\n\\item $\\Phi \\stackrel{1}{\\thicksim} \\Psi$, i.e., $\\Phi$, $\\Psi$ are\n  Lipschitz equivalent;\n  \\item there exists $\\alpha\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $\\Phi \\stackrel{1}{\\cong}\n    \\Psi_{*\\alpha}$, i.e., $\\Phi$, $\\Psi_{*\\alpha}$ are Lipschitz conjugate;\n\\item there exists $\\beta\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\beta A^{\\Psi}$ are\n    Lyapunov similar while $A^{\\Phi_{\\sf  AD}}$, $\\beta A^{\\Psi_{\\sf\n        AD}}$ as well as $A^{\\Phi_{\\sf  C}}$, $\\beta A^{\\Psi_{\\sf\n        C}}$ are similar;\n\\item there exists $\\gamma\n    \\in \\mathbb{R} \\setminus \\{0\\}$ so that $A^{\\Phi}$, $\\gamma A^{\\Psi}$ are\n    Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $\\gamma A^{\\Psi_{\\sf\n        C}}$ are similar.\n  \\end{enumerate}\n  Moreover, $\\Phi \\stackrel{1}{\\cong}\\Psi$ if and\nonly if $A^{\\Phi}$, $A^{\\Psi}$ are Lipschitz similar while $A^{\\Phi_{\\sf  C}}$, $ A^{\\Psi_{\\sf\n        C}}$ are similar.", "theorem_type": ["Biconditional or Equivalence", "Implication"], "mcq": {"question": "Let $X\\neq\\{0\\}$ be a finite-dimensional normed space over $\\mathbb{R}$, and let $\\Phi,\\Psi$ be linear flows on $X$, meaning $\\Phi_t=e^{tA^{\\Phi}}$ and $\\Psi_t=e^{tA^{\\Psi}}$ for linear generators $A^{\\Phi},A^{\\Psi}$. For a flow $\\Theta$, write $\\Theta_{\\mathbb{R}}(x)=\\{\\Theta(t,x):t\\in\\mathbb{R}\\}$. Say that $\\Phi$ and $\\Psi$ are Lipschitz equivalent if there is a bi-Lipschitz homeomorphism $h:X\\to X$ with $h(0)=0$ such that $h(\\Phi_{\\mathbb{R}}(x))=\\Psi_{\\mathbb{R}}(h(x))$ for all $x\\in X$. Say that they are Lipschitz conjugate if there is a bi-Lipschitz homeomorphism $h:X\\to X$ with $h(0)=0$ such that $h(\\Phi(t,x))=\\Psi(t,h(x))$ for all $t\\in\\mathbb{R}$ and $x\\in X$. For $\\alpha\\neq 0$, let $\\Psi_{*\\alpha}$ denote the time-rescaled flow generated by $\\alpha A^{\\Psi}$, equivalently $\\Psi_{*\\alpha}(t,x)=\\Psi(\\alpha t,x)$. Also, let $\\Phi_{\\sf AD},\\Psi_{\\sf AD}$ and $\\Phi_{\\sf C},\\Psi_{\\sf C}$ denote the anti-diagonal and central parts in the canonical decomposition of linear flows; “similar” means similar as linear operators. Finally, two linear operators are called Lyapunov similar if the flows they generate have the same Lyapunov exponents with matching multiplicities, and Lipschitz similar if they are Lyapunov similar and their anti-diagonal parts are similar. Under these assumptions, which statement about $\\Phi$ and $\\Psi$ holds?", "correct_choice": {"label": "A", "text": "The following are equivalent: (i) $\\Phi$ and $\\Psi$ are Lipschitz equivalent; (ii) there exists $\\alpha\\in\\mathbb{R}\\setminus\\{0\\}$ such that $\\Phi$ and $\\Psi_{*\\alpha}$ are Lipschitz conjugate; (iii) there exists $\\beta\\in\\mathbb{R}\\setminus\\{0\\}$ such that $A^{\\Phi}$ and $\\beta A^{\\Psi}$ are Lyapunov similar, while $A^{\\Phi_{\\sf AD}}$ and $\\beta A^{\\Psi_{\\sf AD}}$ are similar and $A^{\\Phi_{\\sf C}}$ and $\\beta A^{\\Psi_{\\sf C}}$ are similar; (iv) there exists $\\gamma\\in\\mathbb{R}\\setminus\\{0\\}$ such that $A^{\\Phi}$ and $\\gamma A^{\\Psi}$ are Lipschitz similar, while $A^{\\Phi_{\\sf C}}$ and $\\gamma A^{\\Psi_{\\sf C}}$ are similar. Moreover, $\\Phi$ and $\\Psi$ are Lipschitz conjugate if and only if $A^{\\Phi}$ and $A^{\\Psi}$ are Lipschitz similar and $A^{\\Phi_{\\sf C}}$ and $A^{\\Psi_{\\sf C}}$ are similar."}, "choices": [{"label": "B", "text": "The following are equivalent: (i) $\\Phi$ and $\\Psi$ are Lipschitz equivalent; (ii) there exists $\\alpha\\in\\mathbb{R}$ such that $\\Phi$ and $\\Psi_{*\\alpha}$ are Lipschitz conjugate; (iii) there exists $\\beta\\in\\mathbb{R}$ such that $A^{\\Phi}$ and $\\beta A^{\\Psi}$ are Lyapunov similar, while $A^{\\Phi_{\\sf AD}}$ and $\\beta A^{\\Psi_{\\sf AD}}$ are similar and $A^{\\Phi_{\\sf C}}$ and $\\beta A^{\\Psi_{\\sf C}}$ are similar; (iv) there exists $\\gamma\\in\\mathbb{R}$ such that $A^{\\Phi}$ and $\\gamma A^{\\Psi}$ are Lipschitz similar, while $A^{\\Phi_{\\sf C}}$ and $\\gamma A^{\\Psi_{\\sf C}}$ are similar. Moreover, $\\Phi$ and $\\Psi$ are Lipschitz conjugate if and only if $A^{\\Phi}$ and $A^{\\Psi}$ are Lipschitz similar and $A^{\\Phi_{\\sf C}}$ and $A^{\\Psi_{\\sf C}}$ are similar."}, {"label": "C", "text": "If $\\Phi$ and $\\Psi$ are Lipschitz equivalent, then there exists $\\alpha\\in\\mathbb{R}\\setminus\\{0\\}$ such that $\\Phi$ and $\\Psi_{*\\alpha}$ are Lipschitz conjugate."}, {"label": "D", "text": "The following are equivalent: (i) $\\Phi$ and $\\Psi$ are Lipschitz equivalent; (ii) there exists $\\alpha\\in\\mathbb{R}\\setminus\\{0\\}$ such that $\\Phi$ and $\\Psi_{*\\alpha}$ are Lipschitz conjugate; (iii) there exists $\\beta\\in\\mathbb{R}\\setminus\\{0\\}$ such that $A^{\\Phi}$ and $\\beta A^{\\Psi}$ are Lyapunov similar and $A^{\\Phi_{\\sf C}}$ and $\\beta A^{\\Psi_{\\sf C}}$ are similar; (iv) there exists $\\gamma\\in\\mathbb{R}\\setminus\\{0\\}$ such that $A^{\\Phi}$ and $\\gamma A^{\\Psi}$ are Lipschitz similar. Moreover, $\\Phi$ and $\\Psi$ are Lipschitz conjugate if and only if $A^{\\Phi}$ and $A^{\\Psi}$ are Lipschitz similar."}, {"label": "E", "text": "The following are equivalent: (i) $\\Phi$ and $\\Psi$ are Lipschitz equivalent; (ii) $\\Phi$ and $\\Psi$ are Lipschitz conjugate; (iii) there exists $\\beta\\in\\mathbb{R}\\setminus\\{0\\}$ such that $A^{\\Phi}$ and $\\beta A^{\\Psi}$ are Lyapunov similar, while $A^{\\Phi_{\\sf AD}}$ and $\\beta A^{\\Psi_{\\sf AD}}$ are similar and $A^{\\Phi_{\\sf C}}$ and $\\beta A^{\\Psi_{\\sf C}}$ are similar; (iv) there exists $\\gamma\\in\\mathbb{R}\\setminus\\{0\\}$ such that $A^{\\Phi}$ and $\\gamma A^{\\Psi}$ are Lipschitz similar, while $A^{\\Phi_{\\sf C}}$ and $\\gamma A^{\\Psi_{\\sf C}}$ are similar. Moreover, $\\Phi$ and $\\Psi$ are Lipschitz conjugate if and only if $A^{\\Phi}$ and $A^{\\Psi}$ are Lipschitz similar and $A^{\\Phi_{\\sf C}}$ and $A^{\\Psi_{\\sf C}}$ are similar."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "nonzero time-rescaling parameter", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "full equivalence of all four conditions and moreover-clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "necessary anti-diagonal and central similarity clauses in the algebraic characterization", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "need for time-rescaling in passing from equivalence to conjugacy", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the relevant notions but does not explicitly reveal the theorem statement or point to option A. There are no obvious lexical cues or direct hints indicating which choice is correct."}, "TAS": {"score": 0, "justification": "The item is essentially asking for the exact theorem characterizing Lipschitz equivalence/conjugacy of linear flows. The correct option is a near-verbatim theorem statement rather than a problem requiring an independently derived conclusion."}, "GPS": {"score": 1, "justification": "Some comparison-based reasoning is needed to distinguish subtle alterations in the options (e.g., allowing 0 for the rescaling parameter, dropping anti-diagonal conditions, confusing equivalence with conjugacy). However, the question primarily tests precise theorem recall rather than generative mathematical reasoning from the definitions."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they weaken equivalence to implication, omit necessary similarity conditions, or mishandle the nonzero time-rescaling parameter. These reflect realistic failure modes and are distinct from one another."}, "total_score": 5, "overall_assessment": "A technically well-constructed but theorem-recall-heavy MCQ: little answer leakage and strong distractors, but low tautology avoidance and only moderate pressure for genuine reasoning."}}
{"id": "2602.13968v1", "paper_link": "http://arxiv.org/abs/2602.13968v1", "theorems_cnt": 4, "theorem": {"env_name": "thm", "content": "\\label{thm:intro-cap-comp} \nLet $\\Omega \\subset \\subset \\mathbb{C}^n$ be a strictly pseudoconvex domain. \n\\begin{itemize}\n\\item[(a)] There exists a constant $A>0$ such that for every Borel set $E\\subset \\Omega$,\n$$\n\t{\\tt c}(E) \\leq A [cap(E,\\Omega)]^\\frac{1}{n}.\n$$\n\\item[(b)]\nAssume $D \\subset \\subset \\Omega$ be a subdomain. There exists a constant $A'$ such that for every Borel set $E\\subset D$,\n$$\n\t\\frac{1}{A'} cap(E,\\Omega) \\leq {\\tt c} (E).\n$$\n\\end{itemize}\nMoreover, the above inequalities are sharp as far as the exponents are concerned.", "start_pos": 11254, "end_pos": 11793, "label": "thm:intro-cap-comp"}, "ref_dict": {"thm:intro-cap-comp": "\\begin{thm} \\label{thm:intro-cap-comp} \nLet $\\Om \\subset \\subset \\bC^n$ be a strictly pseudoconvex domain. \n\\begin{itemize}\n\\item[(a)] There exists a constant $A>0$ such that for every Borel set $E\\subset \\Om$,\n$$\n\t\\tc(E) \\leq A [cap(E,\\Om)]^\\frac{1}{n}.\n$$\n\\item[(b)]\nAssume $D \\subset \\subset \\Om$ be a subdomain. There exists a constant $A'$ such that for every Borel set $E\\subset D$,\n$$\n\t\\frac{1}{A'} cap(E,\\Om) \\leq \\tc (E).\n$$\n\\end{itemize}\nMoreover, the above inequalities are sharp as far as the exponents are concerned.\n\\end{thm}", "rmk:sharp-exp": "\\begin{remark}[Proof of the last statement in Theorem~\\ref{thm:intro-cap-comp}]\n\\label{rmk:sharp-exp} The Alexander-Taylor inequality \\cite{AT84} reads\n$$\n\t\\exp (-A_r/cap(K,\\Om)) \\leq T_R(K) \\leq \\exp(-2\\pi/[cap(K,\\Om)]^\\frac{1}{n}),\n$$\nwhere the exponents in both inequalities are sharp (\\cite[Remark~2]{AT84}).\nTogether with $cap(K) \\leq A_r \\tc(K)$ in Lemma~\\ref{lem:cap-c}\nand the first inequality in Lemma~\\ref{lem:AT-type} we derive \n$$\n\t\\exp (-A/cap(K,\\Om)) \\leq \\exp (-A/\\tc(K,\\Om))    \\leq \\exp(-2\\pi/[cap(K,\\Om)]^\\frac{1}{n})\n$$\nwith the sharp exponents. So are the ones in Lemma~\\ref{lem:c-cap} and Lemma~\\ref{lem:cap-c}.\n\\end{remark}", "thm:subextension": "\\begin{thm} \\label{thm:subextension} \nLet $f \\in W^*(B(0,1))$ be such that $f\\leq -1$. Assume $\\|f\\|_*\\leq 1$. Let $\\veps>0$  and $0<s<1$. Then, there exists $u \\in \\cL(\\bC^n)$ such that $u\\leq -|f^\\star|^{\\frac{2}{n}-\\veps}$ on $B(0,s)$ everywhere.\n\\end{thm}", "lem:cap-c": "\\begin{lem} \\label{lem:cap-c} Let $D \\subset \\subset \\Om$. There exists  a constant $A'$ depending  only on $D$ and $\\Om$ such that for every Borel set $E \\subset D$,\n$$\tcap(E,\\Om) \\leq A'\\, \\tc(E)\n$$\n\\end{lem}", "lem:cap-sublevel-set": "\\begin{lem}\\label{lem:cap-sublevel-set} Let $f\\in W^*(\\Om) \\cap C^0(\\Om)$.  Assume either $f\\geq 0$ or $f\\leq 0$ on $\\Om$. Denote $E_s = \\{z\\in \\Om: |f(z)| > s\\}$ for $s > 0$. Then,\n\\[\\label{eq:c-sublevel-set-est}\n\t\\tc (E_s) \\leq \\frac{1}{s^2} \\|f\\|_*^2.\n\\]\n\\end{lem}", "rmk:sharp-DMV": "\\begin{remark}\\label{rmk:sharp-DMV} The inequalities in Proposition~\\ref{prop:equiv-est} are sharp as far as the exponents are concerned. In fact, it follows from the first inequality (a) and Lemma~\\ref{lem:c-cap} that for a Borel set $E\\subset \\Om$, \n$$\n\tV_{2n} (E) \\leq A_1 e^{-\\al/ \\tc (E)} \\leq A_1 e^{-\\al/ [cap(E, \\Om)]^\\frac{1}{n}}.\n$$\nWe know from \\cite[Theorem~A]{ACKPZ09} that the exponent $1/n$ in the inequality between the  volume and Bedford-Taylor capacity is sharp. So are the exponents in the inequalities of the proposition. Consequently, the exponent 2 in Proposition~\\ref{prop:DMV} and \\cite[Theorem~1.2]{DMV} is the optimal one for all $n\\geq 1$.\n\\end{remark}", "thm:VV": "\\begin{thm} \\label{thm:VV} Let $\\Om \\subset \\subset \\bC^n$ be an open subset and let $f\\in W^*(\\Om)$. There exists a Borel subset $E\\subset \\Om$ such that $\\tc (E) =0$ and it satisfies \n\\begin{itemize}\n\\item\n[(i)] for each $x\\in \\Om\\setminus E$,\n$$\n\t\\lim_{r\\to 0} \\intavg_{B(x,r)} f dy = f^\\star(x).\n$$\n\\item\n[(ii)] Moreover, for each $x\\in \\Om\\setminus E$,\n$$\n\t\\lim_{r\\to 0} \\intavg_{B(x,r)} |f-f^\\star(x)|^2 dy =0.\n$$\n\\end{itemize}\nConsequently, the precise representative $f^\\star$ is quasi-continuous.\n\\end{thm}", "lem:c-cap": "\\begin{lem}[dominated by Bedford-Taylor capacity]  \\label{lem:c-cap} Assume $\\Om$ is strictly pseudoconvex. There exists a constant $A>0$ such that for every Borel set $E\\subset\\subset \\Om$,\n$$\n\t\\tc(E) \\leq A \\left[cap(E,\\Om)\\right]^\\frac{1}{n}.\n$$\n\\end{lem}", "eq:DMV-intro": "\\begin{thm}\\label{thm:intro-choquet-c} Let $\\Om$ be a bounded open set in $\\bC^n$. The set function on Borel subsets $E\\mapsto \\tc (E)= \\tc (E,\\Om)$ is a Choquet capacity.\n\\end{thm}\n\nThis result is the local analogue of \\cite{Vigny}. On compact K\\\"ahler manifolds the proof of  \\cite[Theorem~30]{Vigny} employed the capacity notion in the Dirichlet spaces \\cite[Lemma~23]{Vigny} and the characterization of pluripolar sets via a family of capacity associated with closed positive current \\cite{FO84}.  Here we use a similar strategy. Notably, we are able to give a simpler proof of the characterization in Corollary~\\ref{cor:FO-polar} by using only pluripotential theory. \n\nAnother remarkable property of functions in the complex Sobolev space is the uniformly exponential integrability due to Dinh, Marinescu and Vu  \\cite{DMV}, which is very close to the one of psh functions.  Namely, \nthere exist positive constants $\\al$ and $A$  such that for every $f \\in W^*(B(0,1))$ with $\\|f\\|_*\\leq 1$, \n\\[\\label{eq:DMV-intro}\n\t\\int_{\\bar B(0,\\frac{1}{8})} e^{\\al |f|^2} d x \\leq A.\n\\]\nThis result should be compared with the well-known exponential integrability of psh functions due to H\\\"ormander \\cite[Proposition~4.2.9]{Ho07} and Skoda \\cite{Sk72}. In the proof of  \\cite[Theorem~1.2]{DMV}  the authors used the induction argument in dimension and slicing theory for positive currents. We provide a new and simpler one in Proposition~\\ref{prop:DMV} by reducing the inequality to one dimensional case. In this way, it is enough  to  work with smooth functions after taking  the standard convolution with smooth kernels. The constants can be explicitly computed though they  are suboptimal.  Additionally, we point out in Remark~\\ref{rmk:sharp-DMV} that the exponent 2 of $|f|$ in \\eqref{eq:DMV-intro} is the optimal for all $n\\geq 1$, which was only known for $n=1$ in \\cite{DMV}. \n\nThe third main result fully characterizes the inequality \\eqref{eq:DMV-intro} by the volume-capacity inequality of sublevel sets.\n\n\\begin{thm} \\label{prop:equiv-est-intro} Let $K\\subset\\subset \\Om$ be a compact subset.  The following statements hold and they are equivalent to each other.\n\\begin{itemize}\n\\item[(a)] \nThere exist constants $A_1>0$ and $\\al>0$ depending only on $K, \\Om$ such that for every  Borel  set $E\\subset K$ $$V_{2n}(E) \\leq A_1 e^{-\\al/\\tc (E)}.$$\n\n\\item[(b)] There exist uniform constants $A_1>0$ and $\\al>0$ depending only on $K, \\Om$ such that for every  $f\\in W^*(\\Om)$ and $f\\leq 0$,\n $$V_{2n}(\\{f< -1\\} \\cap K) \\leq A_1 e^{\\frac{-\\al}{\\|f\\|_*^{2}}}.$$\n\\item\n[(c)]   There exist uniform constants $A_1>0$ and $\\al>0$ depending only on $K, \\Om$ such that for every $f\\in W^*(\\Om)$ and $f\\leq 0$ whose norm $\\|f\\|_* \\leq \n\\ka$,\n$$V_{2n} (\\{f < -s\\} \\cap K) \\leq A_1 e^{-\\al s^2/\\ka^2}, \\quad \\forall s>0.$$\n\n\\item \n[(d)]  There exist uniform constants $A_1>0$ and $\\al>0$ depending only on $K, \\Om$ such that for every $f\\in W^*(\\Om)$ and $f\\leq 0$ with $\\|f\\|_* \\leq 1$,\n$$V_{2n} (\\{f < -s\\} \\cap K) \\leq A_1 e^{-\\al s^2}, \\quad \\forall s>0.$$\n\\end{itemize}\n\\end{thm}", "prop:ae-qe": "\\begin{prop} \\label{prop:ae-qe}  Let $D\\subset \\Om$ be an open set and $f$ is quasi-continuous on $D$. If $f\\geq 0$ a.e. on $D$, then $f\\geq 0$ quasi-everywhere on $D$.\n\\end{prop}", "prop:DMV": "\\begin{prop} \\label{prop:DMV} Let $B(0,1) \\subset \\bC^n$ be the unit ball. There exist positive constants $\\al$ and $A$  such that for all $f \\in W^*(B(0,1))$ with $\\|f\\|_*\\leq 1$, \n$$\n\t\\int_{\\bar B(0,\\frac{1}{8})} e^{\\al |f|^2} dx \\leq A.\n$$\n\\end{prop}", "prop:density": "\\begin{prop} \\label{prop:density} Let $\\Om\\subset \\bC^n$ be a bounded domain. \n\\begin{itemize}\n\\item[(a)] If $n=1$, then $W^*(\\Om) = W^{1,2}(\\Om)$ together with its Sobolev norm. \n\\item[(b)] If $n\\geq 2$, then the continuous functions in $C^0(\\Om) \\cap W^*(\\Om)$ are not dense in $W^*(\\Om)$ with respect to $W^*$-norm. \n\\end{itemize}\n\\end{prop}", "lem:outer-reg": "\\begin{lem}[outer regularity] \\label{lem:outer-reg} For a Borel subset $E\\subset \\Om$,\n\\[\\notag\t\\tc(E) = \\inf\\{ \\tc(G) : E \\subset G,\\; G \\text{ is open}\\}.\n\\]\n\\end{lem}", "thm:star-norm": "\\begin{thm}\\label{thm:star-norm} Let $\\Om$ be a bounded domain in $\\bC^n$.\n\\begin{itemize}\n\\item[(a)]  $\\| \\cdot\\|_*$ is a norm in $W^*(\\Om)$ and $\\| \\cdot \\|_{W^{1,2}(\\Om)} \\leq \\|\\cdot \\|_*$.\n\\item[(b)] $W^*(\\Om)$ and $W^*_0(\\Om)$ are Banach spaces with this norm.\n\\end{itemize}\n\\end{thm}", "cor:reflexive": "\\begin{cor} \\label{cor:reflexive} Let $\\Om$ be an open set in $\\bC^n$,  $n\\geq 2$. Then, $W^*(\\Om)$ is not reflexive.\n\\end{cor}", "rmk:inner-reg": "\\begin{remark}\\label{rmk:inner-reg} The relative compactness condition of $E$ will be removed later after proving that $\\tc(\\cdot,\\Om)$ is a capacity in the sense of Choquet (Theorem~\\ref{thm:choquet-c}).\n\\end{remark}", "rmk:DMV-holder": "\\begin{remark}\\label{rmk:DMV-holder} Let $\\Om$ be a bounded strictly psedocovex domain and $K\\subset \\subset \\Om$ be a compact subset. Let $u_1,...,u_n$ be H\\\"older continuous psh functions in $\\Om$. It follows from \\cite[Proposition~2.4]{DKN} that the Radon measure \n$$\\mu = {\\bf 1}_K \\, dd^c u_1 \\wed \\cdots \\wed dd^c u_n$$\nis $W^*(\\Om)$-H\\\"older continuous. That means there exist positive constants $c>0$ and $\\al>0$  such that for $f\\in W^*(\\Om)$ and $\\|f\\|_* \\leq 1$,\n$$\n\t\\left| \\int_\\Om f d\\mu \\right| \\leq  c \\|f\\|_{*}^\\al.\n$$\nCombining this inequality and the arguments in the proof of \\cite[Proposition~2.9]{Ng18} we get that there exist $\\al_1>0$ and $c_1>0$ such that   for $f \\in W^*(\\Om)$, $f\\leq 0$ and $\\|f\\|_* \\leq 1$, \n$$\n\t\\mu (f < -s) \\leq c_1 e^{-\\al_1 s^2},  \\quad  \\forall s>0.\n$$\nIn other words, the equivalent inequalities in Proposition~\\ref{prop:equiv-est} holds for a very large family of Monge-Amp\\`ere measures associated with  H\\\"older continuous psh functions. This result is also equivalent to the statement of \\cite[Theorem~1.2]{DMV}.\n\\end{remark}", "defn:L2-ma": "\\begin{defn}\n\\label{defn:L2-ma} Let $\\phi \\in PSH\\cap L^\\infty(\\Om)$.\nDefine $\\mu: = (dd^c\\phi)^n$. Let $g \\in W^*(\\Om)$ and $K \\subset \\subset \\Om$ be a compact subset.  If $g\\geq 0$, then\n$$\n\t\\int_K g d\\mu := \\int_K \\wt g d\\mu,\n$$\nwhere $\\wt g$ is a quasi-continuous modification. \nGenerally, \n$$\n\t\\int_K g d\\mu := \\int_K g^+ d\\mu - \\int_K g^- d\\mu.\n$$\n\\end{defn}", "lem:AT-type": "\\begin{lem} \\label{lem:AT-type} There exist positive constants $A_r$ and $A_R$ depending on $r,R$ respectively such that for every compact  set $K\\subset B_r$,\n$$ \\exp(- A_R \\; \\tc (K,B_R)^{-1}) \\leq  T_R(K) \\leq \t\\exp(- A_r\\;\\tc (K, B_R)^{-1/n}).\n$$ \nEquivalently,\n\\[\\label{eq:AT}\n\t\\frac{1}{A_r} \\frac{1}{M_K^n} \\leq \\tc (K,B_R) \\leq \\frac{A_R }{M_K}.\n\\]\n\\end{lem}", "cor:FO-polar": "\\begin{cor} \\label{cor:FO-polar}\nLet $E\\subset \\Om$ be a Borel subset. Then, $cap (E) =0$ if  and only if $\\tc_\\te(E) =0$ for all $\\te $ of the form \\eqref{eq:current-te} with $0<\\de \\leq 1$.\n\\end{cor}"}, "pre_theorem_intro_text_len": 5859, "pre_theorem_intro_text": "Let $(X,\\omega)$ be a compact K\\\"ahler manifold.  A natural subspace of the Sobolev space $W^{1,2}(X)$, called the complex Sobolev space $W^*$, was introduced by Dinh and Sibony \\cite{DS} in their work on complex dynamics. \nThis new space turned out to be a good complex version of the former one as it takes into account the complex structure of manifolds. Many important families of functions which carry useful properties of underlying manifolds belong to the space. Among them are Lipschitz functions, bounded quasi plurisubharmonic (psh) functions or more general bounded delta quasi-psh function studied in  \\cite{CW05}. Basic functional aspects of this space were studied by Vigny \\cite{Vigny} where he showed that it is a Banach space with a corresponding norm $\\| \\cdot\\|_*$, this norm  is stronger than the Sobolev norm. However, it is not reflexive and smooth functions are not dense in $W^*$ with respect to the strong topology of that norm.  He also defined a functional capacity for this space and showed that it is a capacity in sense of Choquet. Moreover, the functional capacity is qualitative comparable to the relative capacity of Bedford and Taylor in global pluripotential theory defined in  \\cite{ko03}. \n\nThanks to flexible properties this space has found many of applications in complex dynamic in high dimensions, complex Monge-Amp\\`ere equations and other areas, see e.g., \\cite{BiD23}, \\cite{DKN}, \\cite{DKW}, \\cite{DNV25}, \\cite{Vig15}, \\cite{Vu20, Vu24, Vu26} and \\cite{WZ}. Also a higher complex Sobolev space is proposed in \\cite{DoN25}. \n\nAn element in a Sobolev space is defined almost everywhere up to a set of Lebesgue measure zero which is unlike psh or quasi-psh functions. Its fine properties with respect to the Sobolev capacity are classical ones \\cite{EG92}, \\cite{KLV-book}. However, this capacity is not good enough to characterize pluripolar sets, therefore it is not suitable for studying functions in $W^*$. One needs to work with the corresponding capacity of $W^*$ which dominates the Sobolev one. However, the new capacity in complex Sobolev spaces is harder to analyze. A crucial difference is that we can no longer rely on maximal function techniques, which are very powerful, in these spaces (see, e.g \\cite{KLV-book}). \n\nRecently, it has been showed in \\cite{DMV}, \\cite{Vigny} and \\cite{VV24} that pluripotential theory is a suitable and powerful tool for studying $W^*$. As a result they made major  progresses on study local properties of functions in such spaces.  \n\nWe continue to study local complex Sobolev spaces which has been considered partially in \\cite{DMV} and  \\cite{Vigny}. Our goal is to develop further this approach by focusing on new and optimal inequalities between the functional capacity and classical capacities of Bedford and Taylor \\cite{BT82} and of Alexander and Taylor \\cite{AT84} in pluripotential theory.\n\nLet $\\Omega \\subset \\mathbb{C}^n$ be a bounded domain  and $\\omega = dd^c |z|^2$  the standard K\\\"ahler form in $\\mathbb{C}^n$.  Denote by $W^{1,2}(\\Omega, \\mathbb{R})$ the usual Sobolev space. For  $f\\in W^{1,2}(\\Omega, \\mathbb{R})$ we define $\\Ga_f$  to be the set of all positive closed $(1,1)$-current $T$ satisfying\n\\[\\label{eq:test-currents}\\notag\n\tdf \\wedge d^c f \\leq T \\quad\\text{weakly in } \\Omega.\n\\]\nHere we use the normalization \n$$d^c = \\frac{\\sqrt{-1}}{2\\pi} (\\overline{\\d} -\\partial), \\quad dd^c = \\frac{\\sqrt{-1}}{\\pi}\\partial\\overline{\\d}.\n$$\nConsider the subspace\n\\[ \\label{eq:sobolev-space} \\notag\nW^*(\\Omega)= \\left\\{f \\in W^{1,2}(\\Omega,\\mathbb{R}):   \\text{there exists $T\\in \\Ga_f$ with } \\|T\\|_\\Omega <+\\infty\n\\right\\},\n\\]\nwhere \n$\n\t \\|T \\|_\\Omega := \\int_\\Omega T \\wedge \\omega^{n-1}.\n$\nFor $f \\in W^*(\\Omega)$ one defines the $W^*$-norm\n\\[\\label{eq:norm} \\notag\n\t\\|f\\|_*^2 = \\| f\\|_{L^2(\\Omega)}^2 + \\inf_{T\\in \\Ga_f} \\|T\\|_{\\Omega}.\n\\]\nWe then reprove basic functional results obtained in \\cite{Vigny}. Namely, $(W^*(\\Omega), \\|\\cdot\\|_*)$ is a Banach space (Theorem~\\ref{thm:star-norm}) and  continuous functions are not dense in $W^*(\\Omega)$ with respect to strong topology of the norm $\\|\\cdot\\|_*$ (Proposition~\\ref{prop:density}). If $n=1$, then  $W^*(\\Omega) = W^{1,2}(\\Omega)$, otherwise $W^*(\\Omega)$ is not reflexive (Corollary~\\ref{cor:reflexive}). It should be pointed out that there are some similarity of these properties with the ones of the space of delta psh functions studied by Cegrell and Wiklund \\cite{CW05}.\n\nInspired by the functional capacity in \\cite{Vigny} we define for a Borel subset $E\\subset \\Omega$ the local capacity ${\\tt c}(E):= {\\tt c}(E,\\Omega)$  as follows:\n\\[\\label{eq:intro-cap-defn}\\notag\n\t{\\tt c}(E) = \\inf \n\t\\left\\{ \\| v \\|_*^2 \\;\\rvert\\;  v\\in \\cali{K}(E)\n\t\\right\\},\n\\] \nwhere\n\\[\\label{eq:intro-ke-cap}\\notag\n \t\\cali{K}(E) = \\left\\{ v \\in W^*(\\Omega) \\;\\rvert\\; \\{v \\leq  -1\\}^o \\supset E \\text{ and } v\\leq 0 \\right\\}.\n\\]\nHere $ \\{v \\leq  -1\\}^o \\supset E$ means $v\\leq -1$ a.e in a neighborhood of $E$. \n\nThe functional capacity in the Sobolev spaces is very well-understood which is also an effective tool to study fine properties of its element, see e.g. \\cite[Chapters~4.7-4.8]{EG92}. In contrast, the use of ${\\tt c}(\\cdot)$ for the complex Sobolev space is not effective so far. \nOur goal is to investigate systematically local properties of functions in $W^*(\\Omega)$ by using this capacity. This is a different perspective compared to \\cite{Vigny} and \\cite{DMV} who used the relative capacity for psh functions due to Bedford and Taylor \\cite{BT82}. Namely, for a Borel set $E\\subset \\Omega$, \n\\[\\label{eq:intro-BT-cap}\\notag\n cap(E,\\Omega) = \\sup\\left\\{ \\int_E (dd^c u)^n :  u\\in PSH(\\Omega), \\;-1 \\leq u \\leq 0\\right\\}.\n\\]\nOur first main result explains the reason why $cap(\\cdot, \\Omega)$ could be effectively used in previous works. Roughly speaking they are equivalent to each other.", "context": "Let $(X,\\omega)$ be a compact K\\\"ahler manifold. A natural subspace of the Sobolev space $W^{1,2}(X)$, called the complex Sobolev space $W^*$, was introduced by Dinh and Sibony \\cite{DS} in their work on complex dynamics. \nThis new space turned out to be a good complex version of the former one as it takes into account the complex structure of manifolds. Many important families of functions which carry useful properties of underlying manifolds belong to the space. Among them are Lipschitz functions, bounded quasi plurisubharmonic (psh) functions or more general bounded delta quasi-psh function studied in \\cite{CW05}. Basic functional aspects of this space were studied by Vigny \\cite{Vigny} where he showed that it is a Banach space with a corresponding norm $\\| \\cdot\\|_*$, this norm is stronger than the Sobolev norm. However, it is not reflexive and smooth functions are not dense in $W^*$ with respect to the strong topology of that norm. He also defined a functional capacity for this space and showed that it is a capacity in sense of Choquet. Moreover, the functional capacity is qualitative comparable to the relative capacity of Bedford and Taylor in global pluripotential theory defined in \\cite{ko03}.\n\nAn element in a Sobolev space is defined almost everywhere up to a set of Lebesgue measure zero which is unlike psh or quasi-psh functions. Its fine properties with respect to the Sobolev capacity are classical ones \\cite{EG92}, \\cite{KLV-book}. However, this capacity is not good enough to characterize pluripolar sets, therefore it is not suitable for studying functions in $W^*$. One needs to work with the corresponding capacity of $W^*$ which dominates the Sobolev one. However, the new capacity in complex Sobolev spaces is harder to analyze. A crucial difference is that we can no longer rely on maximal function techniques, which are very powerful, in these spaces (see, e.g \\cite{KLV-book}).\n\nRecently, it has been showed in \\cite{DMV}, \\cite{Vigny} and \\cite{VV24} that pluripotential theory is a suitable and powerful tool for studying $W^*$. As a result they made major progresses on study local properties of functions in such spaces.\n\nLet $\\Omega \\subset \\mathbb{C}^n$ be a bounded domain and $\\omega = dd^c |z|^2$ the standard K\\\"ahler form in $\\mathbb{C}^n$. Denote by $W^{1,2}(\\Omega, \\mathbb{R})$ the usual Sobolev space. For $f\\in W^{1,2}(\\Omega, \\mathbb{R})$ we define $\\Ga_f$ to be the set of all positive closed $(1,1)$-current $T$ satisfying\n\\[\\label{eq:test-currents}\\notag\n df \\wedge d^c f \\leq T \\quad\\text{weakly in } \\Omega.\n\\]\nHere we use the normalization \n$$d^c = \\frac{\\sqrt{-1}}{2\\pi} (\\overline{\\d} -\\partial), \\quad dd^c = \\frac{\\sqrt{-1}}{\\pi}\\partial\\overline{\\d}.\n$$\nConsider the subspace\n\\[ \\label{eq:sobolev-space} \\notag\nW^*(\\Omega)= \\left\\{f \\in W^{1,2}(\\Omega,\\mathbb{R}): \\text{there exists $T\\in \\Ga_f$ with } \\|T\\|_\\Omega <+\\infty\n\\right\\},\n\\]\nwhere \n$\n \\|T \\|_\\Omega := \\int_\\Omega T \\wedge \\omega^{n-1}.\n$\nFor $f \\in W^*(\\Omega)$ one defines the $W^*$-norm\n\\[\\label{eq:norm} \\notag\n \\|f\\|_*^2 = \\| f\\|_{L^2(\\Omega)}^2 + \\inf_{T\\in \\Ga_f} \\|T\\|_{\\Omega}.\n\\]\nWe then reprove basic functional results obtained in \\cite{Vigny}. Namely, $(W^*(\\Omega), \\|\\cdot\\|_*)$ is a Banach space (Theorem~\\ref{thm:star-norm}) and continuous functions are not dense in $W^*(\\Omega)$ with respect to strong topology of the norm $\\|\\cdot\\|_*$ (Proposition~\\ref{prop:density}). If $n=1$, then $W^*(\\Omega) = W^{1,2}(\\Omega)$, otherwise $W^*(\\Omega)$ is not reflexive (Corollary~\\ref{cor:reflexive}). It should be pointed out that there are some similarity of these properties with the ones of the space of delta psh functions studied by Cegrell and Wiklund \\cite{CW05}.\n\nInspired by the functional capacity in \\cite{Vigny} we define for a Borel subset $E\\subset \\Omega$ the local capacity ${\\tt c}(E):= {\\tt c}(E,\\Omega)$ as follows:\n\\[\\label{eq:intro-cap-defn}\\notag\n {\\tt c}(E) = \\inf \n \\left\\{ \\| v \\|_*^2 \\;\\rvert\\; v\\in \\cali{K}(E)\n \\right\\},\n\\] \nwhere\n\\[\\label{eq:intro-ke-cap}\\notag\n \\cali{K}(E) = \\left\\{ v \\in W^*(\\Omega) \\;\\rvert\\; \\{v \\leq -1\\}^o \\supset E \\text{ and } v\\leq 0 \\right\\}.\n\\]\nHere $ \\{v \\leq -1\\}^o \\supset E$ means $v\\leq -1$ a.e in a neighborhood of $E$.\n\nThe functional capacity in the Sobolev spaces is very well-understood which is also an effective tool to study fine properties of its element, see e.g. \\cite[Chapters~4.7-4.8]{EG92}. In contrast, the use of ${\\tt c}(\\cdot)$ for the complex Sobolev space is not effective so far. \nOur goal is to investigate systematically local properties of functions in $W^*(\\Omega)$ by using this capacity. This is a different perspective compared to \\cite{Vigny} and \\cite{DMV} who used the relative capacity for psh functions due to Bedford and Taylor \\cite{BT82}. Namely, for a Borel set $E\\subset \\Omega$, \n\\[\\label{eq:intro-BT-cap}\\notag\n cap(E,\\Omega) = \\sup\\left\\{ \\int_E (dd^c u)^n : u\\in PSH(\\Omega), \\;-1 \\leq u \\leq 0\\right\\}.\n\\]\nOur first main result explains the reason why $cap(\\cdot, \\Omega)$ could be effectively used in previous works. Roughly speaking they are equivalent to each other.\n\n\\begin{cor} \\label{cor:reflexive} Let $\\Om$ be an open set in $\\bC^n$,  $n\\geq 2$. Then, $W^*(\\Om)$ is not reflexive.\n\\end{cor}\n\n\\begin{prop} \\label{prop:density} Let $\\Om\\subset \\bC^n$ be a bounded domain. \n\\begin{itemize}\n\\item[(a)] If $n=1$, then $W^*(\\Om) = W^{1,2}(\\Om)$ together with its Sobolev norm. \n\\item[(b)] If $n\\geq 2$, then the continuous functions in $C^0(\\Om) \\cap W^*(\\Om)$ are not dense in $W^*(\\Om)$ with respect to $W^*$-norm. \n\\end{itemize}\n\\end{prop}\n\n\\begin{thm}\\label{thm:star-norm} Let $\\Om$ be a bounded domain in $\\bC^n$.\n\\begin{itemize}\n\\item[(a)]  $\\| \\cdot\\|_*$ is a norm in $W^*(\\Om)$ and $\\| \\cdot \\|_{W^{1,2}(\\Om)} \\leq \\|\\cdot \\|_*$.\n\\item[(b)] $W^*(\\Om)$ and $W^*_0(\\Om)$ are Banach spaces with this norm.\n\\end{itemize}\n\\end{thm}", "full_context": "Let $(X,\\omega)$ be a compact K\\\"ahler manifold. A natural subspace of the Sobolev space $W^{1,2}(X)$, called the complex Sobolev space $W^*$, was introduced by Dinh and Sibony \\cite{DS} in their work on complex dynamics. \nThis new space turned out to be a good complex version of the former one as it takes into account the complex structure of manifolds. Many important families of functions which carry useful properties of underlying manifolds belong to the space. Among them are Lipschitz functions, bounded quasi plurisubharmonic (psh) functions or more general bounded delta quasi-psh function studied in \\cite{CW05}. Basic functional aspects of this space were studied by Vigny \\cite{Vigny} where he showed that it is a Banach space with a corresponding norm $\\| \\cdot\\|_*$, this norm is stronger than the Sobolev norm. However, it is not reflexive and smooth functions are not dense in $W^*$ with respect to the strong topology of that norm. He also defined a functional capacity for this space and showed that it is a capacity in sense of Choquet. Moreover, the functional capacity is qualitative comparable to the relative capacity of Bedford and Taylor in global pluripotential theory defined in \\cite{ko03}.\n\nAn element in a Sobolev space is defined almost everywhere up to a set of Lebesgue measure zero which is unlike psh or quasi-psh functions. Its fine properties with respect to the Sobolev capacity are classical ones \\cite{EG92}, \\cite{KLV-book}. However, this capacity is not good enough to characterize pluripolar sets, therefore it is not suitable for studying functions in $W^*$. One needs to work with the corresponding capacity of $W^*$ which dominates the Sobolev one. However, the new capacity in complex Sobolev spaces is harder to analyze. A crucial difference is that we can no longer rely on maximal function techniques, which are very powerful, in these spaces (see, e.g \\cite{KLV-book}).\n\nRecently, it has been showed in \\cite{DMV}, \\cite{Vigny} and \\cite{VV24} that pluripotential theory is a suitable and powerful tool for studying $W^*$. As a result they made major progresses on study local properties of functions in such spaces.\n\nLet $\\Omega \\subset \\mathbb{C}^n$ be a bounded domain and $\\omega = dd^c |z|^2$ the standard K\\\"ahler form in $\\mathbb{C}^n$. Denote by $W^{1,2}(\\Omega, \\mathbb{R})$ the usual Sobolev space. For $f\\in W^{1,2}(\\Omega, \\mathbb{R})$ we define $\\Ga_f$ to be the set of all positive closed $(1,1)$-current $T$ satisfying\n\\[\\label{eq:test-currents}\\notag\n df \\wedge d^c f \\leq T \\quad\\text{weakly in } \\Omega.\n\\]\nHere we use the normalization \n$$d^c = \\frac{\\sqrt{-1}}{2\\pi} (\\overline{\\d} -\\partial), \\quad dd^c = \\frac{\\sqrt{-1}}{\\pi}\\partial\\overline{\\d}.\n$$\nConsider the subspace\n\\[ \\label{eq:sobolev-space} \\notag\nW^*(\\Omega)= \\left\\{f \\in W^{1,2}(\\Omega,\\mathbb{R}): \\text{there exists $T\\in \\Ga_f$ with } \\|T\\|_\\Omega <+\\infty\n\\right\\},\n\\]\nwhere \n$\n \\|T \\|_\\Omega := \\int_\\Omega T \\wedge \\omega^{n-1}.\n$\nFor $f \\in W^*(\\Omega)$ one defines the $W^*$-norm\n\\[\\label{eq:norm} \\notag\n \\|f\\|_*^2 = \\| f\\|_{L^2(\\Omega)}^2 + \\inf_{T\\in \\Ga_f} \\|T\\|_{\\Omega}.\n\\]\nWe then reprove basic functional results obtained in \\cite{Vigny}. Namely, $(W^*(\\Omega), \\|\\cdot\\|_*)$ is a Banach space (Theorem~\\ref{thm:star-norm}) and continuous functions are not dense in $W^*(\\Omega)$ with respect to strong topology of the norm $\\|\\cdot\\|_*$ (Proposition~\\ref{prop:density}). If $n=1$, then $W^*(\\Omega) = W^{1,2}(\\Omega)$, otherwise $W^*(\\Omega)$ is not reflexive (Corollary~\\ref{cor:reflexive}). It should be pointed out that there are some similarity of these properties with the ones of the space of delta psh functions studied by Cegrell and Wiklund \\cite{CW05}.\n\nInspired by the functional capacity in \\cite{Vigny} we define for a Borel subset $E\\subset \\Omega$ the local capacity ${\\tt c}(E):= {\\tt c}(E,\\Omega)$ as follows:\n\\[\\label{eq:intro-cap-defn}\\notag\n {\\tt c}(E) = \\inf \n \\left\\{ \\| v \\|_*^2 \\;\\rvert\\; v\\in \\cali{K}(E)\n \\right\\},\n\\] \nwhere\n\\[\\label{eq:intro-ke-cap}\\notag\n \\cali{K}(E) = \\left\\{ v \\in W^*(\\Omega) \\;\\rvert\\; \\{v \\leq -1\\}^o \\supset E \\text{ and } v\\leq 0 \\right\\}.\n\\]\nHere $ \\{v \\leq -1\\}^o \\supset E$ means $v\\leq -1$ a.e in a neighborhood of $E$.\n\nThe functional capacity in the Sobolev spaces is very well-understood which is also an effective tool to study fine properties of its element, see e.g. \\cite[Chapters~4.7-4.8]{EG92}. In contrast, the use of ${\\tt c}(\\cdot)$ for the complex Sobolev space is not effective so far. \nOur goal is to investigate systematically local properties of functions in $W^*(\\Omega)$ by using this capacity. This is a different perspective compared to \\cite{Vigny} and \\cite{DMV} who used the relative capacity for psh functions due to Bedford and Taylor \\cite{BT82}. Namely, for a Borel set $E\\subset \\Omega$, \n\\[\\label{eq:intro-BT-cap}\\notag\n cap(E,\\Omega) = \\sup\\left\\{ \\int_E (dd^c u)^n : u\\in PSH(\\Omega), \\;-1 \\leq u \\leq 0\\right\\}.\n\\]\nOur first main result explains the reason why $cap(\\cdot, \\Omega)$ could be effectively used in previous works. Roughly speaking they are equivalent to each other.\n\n\\begin{cor} \\label{cor:reflexive} Let $\\Om$ be an open set in $\\bC^n$,  $n\\geq 2$. Then, $W^*(\\Om)$ is not reflexive.\n\\end{cor}\n\n\\begin{prop} \\label{prop:density} Let $\\Om\\subset \\bC^n$ be a bounded domain. \n\\begin{itemize}\n\\item[(a)] If $n=1$, then $W^*(\\Om) = W^{1,2}(\\Om)$ together with its Sobolev norm. \n\\item[(b)] If $n\\geq 2$, then the continuous functions in $C^0(\\Om) \\cap W^*(\\Om)$ are not dense in $W^*(\\Om)$ with respect to $W^*$-norm. \n\\end{itemize}\n\\end{prop}\n\n\\begin{thm}\\label{thm:star-norm} Let $\\Om$ be a bounded domain in $\\bC^n$.\n\\begin{itemize}\n\\item[(a)]  $\\| \\cdot\\|_*$ is a norm in $W^*(\\Om)$ and $\\| \\cdot \\|_{W^{1,2}(\\Om)} \\leq \\|\\cdot \\|_*$.\n\\item[(b)] $W^*(\\Om)$ and $W^*_0(\\Om)$ are Banach spaces with this norm.\n\\end{itemize}\n\\end{thm}\n\nThe functional capacity in the Sobolev spaces is very well-understood which is also an effective tool to study fine properties of its element, see e.g. \\cite[Chapters~4.7-4.8]{EG92}. In contrast, the use of $\\tc(\\cdot)$ for the complex Sobolev space is not effective so far. \nOur goal is to investigate systematically local properties of functions in $W^*(\\Om)$ by using this capacity. This is a different perspective compared to \\cite{Vigny} and \\cite{DMV} who used the relative capacity for psh functions due to Bedford and Taylor \\cite{BT82}. Namely, for a Borel set $E\\subset \\Om$, \n\\[\\label{eq:intro-BT-cap}\\notag\n cap(E,\\Om) = \\sup\\left\\{ \\int_E (dd^c u)^n : u\\in PSH(\\Om), \\;-1 \\leq u \\leq 0\\right\\}.\n\\]\nOur first main result explains the reason why $cap(\\cdot, \\Om)$ could be effectively used in previous works. Roughly speaking they are equivalent to each other.\n\nIn the local setting the proofs are different. The key idea in the proof of the first inequality is making use of Cegrell's inequality \\cite{Ce04} and the one for the second inequality comes from the comparison principle. On the other hand, the optimality of the exponents is extracted from the Alexander-Taylor type inequality which is proved in Lemma~\\ref{lem:AT-type}. The latter inequality is of independent interest. One of its applications is the existence of an entire psh subextension for a given $f \\in W^*(B(0,1))$ in Theorem~\\ref{thm:subextension}. This psh subextension result is an important ingredient to derive the property that the set Lebesgue points of such functions are of capacity zero. This result is due to Vigny and Vu \\cite{VV24} where the bounded case was proved earlier in \\cite{Vigny}. We improve the statement of that result and its proof is also simplified in Theorem~\\ref{thm:VV}.\n\n\\begin{thm} \\label{prop:equiv-est-intro} Let $K\\subset\\subset \\Om$ be a compact subset. The following statements hold and they are equivalent to each other.\n\\begin{itemize}\n\\item[(a)] \nThere exist constants $A_1>0$ and $\\al>0$ depending only on $K, \\Om$ such that for every Borel set $E\\subset K$ $$V_{2n}(E) \\leq A_1 e^{-\\al/\\tc (E)}.$$\n\n\\begin{remark} \\label{rmk:c-equiv} \\mbox{}\n\\begin{enumerate}\n\\item[(a)] Let $E \\subset E'$ be Borel sets in $\\Om$. If $V_{2n}(E'\\setminus E)=0$, then $\\tc(E) = \\tc(E')$. It is non-decreasing under holomorphic maps, i.e., if $h: G \\to \\Om$ is a holomorphic map, then\n$$\n \\tc (E,G) \\leq \\tc (h(E), \\Om). \n$$\n\\item[(b)] Let $D\\subset\\subset \\Om' \\subset \\Om$. \nWe do not know if there exists a constant $A>0$ depending only on $\\Om$ and $\\Om'$ such that for every Borel set $ E\\subset D$, \n\\[\\label{eq:c-comparablity}\n \\tc(E,\\Om) \\leq A\\; \\tc(E, \\Om').\n\\]\nHowever, we will have a weaker version with some exponent on the right hand side later \nin Corollary~\\ref{cor:dom-equiv-c} for the new capacity $\\tc (\\cdot, \\Om)$.\n\\end{enumerate}\n\\end{remark}\n\n\\begin{lem}[dominated by Bedford-Taylor capacity] \\label{lem:c-cap} Assume $\\Om$ is strictly pseudoconvex. There exists a constant $A>0$ such that for every Borel set $E\\subset\\subset \\Om$,\n$$\n \\tc(E) \\leq A \\left[cap(E,\\Om)\\right]^\\frac{1}{n}.\n$$\n\\end{lem}\n\n\\begin{proof} Let $u_E^*$ denote the relative extremal for $E$ in \\eqref{eq:rel-ext-fct}.\nSince both capacities are outer regular (Lemma~\\ref{lem:outer-reg} and \\cite{BT82}), we may assume that $E=G$ is an open set and hence $u_G = u_G^*$. Then, $-1\\leq u_G \\leq 0$ and $u_G =-1$ on $G$. It follows that\n\\[\\label{eq:BT-dom-a}\n \\tc(G) \\leq \\|u_G\\|_*^2.\n\\]\nLet $\\rho$ be a strictly psh defining function for $\\Om$. We may assume that $dd^c \\rho \\geq \\om$. Since $G\\subset \\subset \\Om$, it implies that $u_G =0$ on $\\d\\Om$ (see, e.g., \\cite[Properties (13.11)]{De89}). Using $-1 \\leq u_G \\leq 0$ and integration by parts,\n$$\n \\|u_G\\|_{L^2}^2 = \\int (u_G)^2 (dd^c \\rho)^n \\leq \\int - u_G (dd^c\\rho)^n = \\int -\\rho dd^c u_G \\wed (dd^c\\rho)^{n-1}.\n$$\nHere and in the proof below we skip writing $\\Om$ in the integral symbols \nfor simplicity. The last integral is less than \n$$\n \\|\\rho\\|_{L^\\infty(\\Om)} \\int dd^c u_G \\wed (dd^c\\rho)^{n-1}.\n$$\nSince $0\\leq u_G+1 \\in PSH(\\Om)$, we have\n$d u_G \\wed d^c u_G \\leq \\frac{1}{2} dd^c (1+u_G)^2.$\nHence, \n\\[\\label{eq:BT-dom-b}\\begin{aligned}\n \\|u_G\\|_*^2 \n&\\leq \\int (u_G)^2 (dd^c \\rho)^n + \\frac{1}{2}\\int dd^c (1+u_G)^2 \\wed (dd^c \\rho)^{n-1} \\\\\n&\\leq A_1 \\int dd^c u_G \\wed (dd^c\\rho)^{n-1} + \\int dd^c u_G \\wed (dd^c \\rho)^{n-1},\n\\end{aligned}\\]\nwhere $A_1 = \\|\\rho\\|_{L^\\infty(\\Om)}$ and we used the fact that \n$$\\begin{aligned}\n& \\frac{1}{2}\\int dd^c (1+u_G)^2 \\wed (dd^c \\rho)^{n-1} \\\\\n&=\\int du_G \\wed d^c u_G \\wed (dd^c\\rho)^{n-1} + \\int (1+u_G) dd^c u_G \\wed (dd^c \\rho)^{n-1} \\\\\n &= \\int dd^c u_G \\wed (dd^c \\rho)^{n-1}.\n\\end{aligned}\n$$\nInvoking an inequality of Cegrell \\cite{Ce04} we have\n\\[\\label{eq:BT-dom-c}\\begin{aligned}\n \\int dd^c u_G \\wed (dd^c \\rho)^{n-1}& \\leq \\left[\\int (dd^c\\rho)^n\\right]^\\frac{n-1}{n} \\left[\\int (dd^c u_G)^n\\right]^\\frac{1}{n} \\\\ &= A_0 [cap(G,\\Om)]^\\frac{1}{n},\n\\end{aligned}\\]\nwhere $A_0$ is a uniform constant as $\\rho$ is smooth on $\\bar\\Om$ and we used the formula \\eqref{eq:BT-cap-id} for the last identity.\n\nBy \\cite[Th\\'eor\\`em~1]{Okada82} we have a Poincar\\'e type inequality, for $f\\in C^\\infty_c(\\Om)$,\n$$\n \\int f^2 d\\mu = \\int f^2 \\te \\wed dd^c |z|^2 \\leq 8 \\sup_\\Om |z| \\int df \\wed d^c f \\wed \\te.\n$$\nIt follows that $\\cE_1^\\te(f) = \\| f \\|^2_{L^2(d\\mu)} + \\cE^\\te(f)$ is equivalent to $\\cE^\\te(f)$. As noted in \\cite[page 211]{FO84} that potential theory of \\cite[Chapter~2]{FOT94} can be formulated in terms of $\\cE^\\te$ instead of $\\cE^\\te_1$. For an open subset $E\\subset \\Om$ its capacity is given by\n\\[\\label{eq:cap-te}\n \\tc_\\te (E) = \\inf\\{\\cE(v) : v\\in \\cF, \\; v\\geq 1 \\quad \\mu-\\text{\\rm a.e.} \\text{ on } E\\}.\n\\]\nFor a general Borel set we let\n$$\n \\tc_\\te(E) = \\inf \\{\\tc_\\te(G): E \\subset G, \\quad G \\text{ is open subset in }\\Om\\}.\n$$\nIn particular, for a compact subset $K\\subset \\Om$,\n$$\n \\tc_\\te(K) = \\inf\\{ \\cE^\\te(v): v\\in \\cF\\cap C^0(\\Om), \\; v\\geq 1 \\text{ on }K\\}.\n$$\nThank to special feature of the functional capacity in a Hilbert space it is a Choquet capcity \\cite[Theorem~2.1.1]{FOT94}. \nMoreover, we have a comparison between this capacity and the Bedford-Taylor capacity as follows.\n\\begin{lem}\\label{lem:FO-polar} Let $\\Om \\subset \\subset \\bC^n$ be a strictly pseudoconvex domain and $0<\\de \\leq 1$.\n\\begin{itemize}\n\\item[(a)] For every compact subset $K\\subset \\Om$, \n$$\n cap (K) \\leq 8 \\tc_\\te (K),\n$$\nwhere $\\te= (dd^c u_K^* +\\de \\om)^{n-1}$.\n\\item[(b)] Let $D\\subset \\subset \\Om$ be a subdomain. There exists a constant $A_0= A(D,\\Om)$ such that for every open subset $G\\subset D$,\n$$\n \\tc_\\te{(G)} \\leq A_0 \\left[cap (G)\\right]^\\frac{1}{n}.\n$$ \n\\end{itemize}\n\\end{lem}\n\n\\begin{remark}\\label{rmk:sharp-DMV} The inequalities in Proposition~\\ref{prop:equiv-est} are sharp as far as the exponents are concerned. In fact, it follows from the first inequality (a) and Lemma~\\ref{lem:c-cap} that for a Borel set $E\\subset \\Om$, \n$$\n V_{2n} (E) \\leq A_1 e^{-\\al/ \\tc (E)} \\leq A_1 e^{-\\al/ [cap(E, \\Om)]^\\frac{1}{n}}.\n$$\nWe know from \\cite[Theorem~A]{ACKPZ09} that the exponent $1/n$ in the inequality between the volume and Bedford-Taylor capacity is sharp. So are the exponents in the inequalities of the proposition. Consequently, the exponent 2 in Proposition~\\ref{prop:DMV} and \\cite[Theorem~1.2]{DMV} is the optimal one for all $n\\geq 1$.\n\\end{remark}", "post_theorem_intro_text_len": 7916, "post_theorem_intro_text": "The statements of the theorem consist of Lemmas~\\ref{lem:c-cap}, ~\\ref{lem:cap-c} and Remarks~\\ref{rmk:inner-reg},~\\ref{rmk:sharp-exp}. \nOn compact K\\\"ahler manifolds both inequalities  was obtained in \\cite[Proposition~5.1]{DKN} where ${\\tt c}(E)$ is defined by another equivalent norm with $W^*$-norm, but the optimality was unknown.  A qualitative version  of the first inequality on such manifolds was obtained early in \\cite{Vigny} and the second inequality was also proved there without showing that the exponents are sharp.\n\nIn the local setting the proofs are different. The key idea in the proof of the first inequality is making use of Cegrell's inequality \\cite{Ce04} and the one for the second inequality comes from the comparison principle. On the other hand, the optimality of the exponents is extracted from the Alexander-Taylor type inequality which is proved in Lemma~\\ref{lem:AT-type}. The latter inequality is of independent interest. One of its applications is the existence of an entire psh subextension for a given $f \\in W^*(B(0,1))$ in Theorem~\\ref{thm:subextension}. This psh subextension result is an important ingredient to derive the property that  the set Lebesgue points of such functions are of capacity zero. This result is due to Vigny and Vu \\cite{VV24} where the bounded case was proved earlier in \\cite{Vigny}. We improve the statement of that result and its  proof is also simplified in Theorem~\\ref{thm:VV}.\n\nAn immediate consequence of Theorem~\\ref{thm:intro-cap-comp} and \\cite{BT82} is that a Borel set $P\\subset \\Omega$ is pluripolar if and only if ${\\tt c}(P)=0$. Provided the equivalence of capacities  the results in \\cite{DMV} can be restated in terms of ${\\tt c} (\\cdot)$ where they are obtained previously for $cap(\\cdot, \\Omega)$. However, as emphasized above we would take another route by using advantages of the functional capacity ${\\tt c}(\\cdot)$ (see e.g., Lemma~\\ref{lem:outer-reg}, Proposition~\\ref{prop:ae-qe}, Lemma~\\ref{lem:cap-sublevel-set}) and hence providing alternative  proofs for the ones obtained in \\cite{DMV}. In particular,  using the ideas in \\cite{DMV} and \\cite{Vigny},  we  simplify  the proof of quasi-continuous representative result proved earlier in \\cite[Theorem~2.10]{DMV} (see also \\cite[Theorem~22]{Vigny}).\n\n\\begin{cor} \\label{cor:intro-quasi-mod} Let  $f\\in W^*(\\Omega)$. There exists a quasi-continuous modification $\\wt f$ such that $\\wt f = f$ a.e. Moreover, if $g$ is another quasi-continuous modification of $f$, then $g =\\wt f$ outside a pluripolar set.\n\\end{cor}\n\nThe most technical part of the proof  lies in delicate integral estimates. Along the way  we refine the ones obtained in \\cite{DMV}. It turns out in  Theorem~\\ref{thm:VV}  that the precise representative, in the sense of  Sobolev spaces \\cite[Chapter~4]{EG92},\n\\[\\label{eq:rep} f^*(x) \\equiv\n\\begin{cases}\n\\lim_{r\\to 0} \\intavg_{B(x,r)} f \\; d y &\\quad \\text{if this limit exits,} \\\\\n=0 &\\quad \\text{otherwise},\n\\end{cases}\t\n\\] \nis a quasi-continuous modification, where $d y$ is the Lebesgue measure on $\\mathbb{C}^n = \\mathbb{R}^{2n}$.\n\nThanks to the quasi-continuity result \n we can define the integration of a function $g\\in W^*(\\Omega)$ against  Monge-Amp\\`ere measures associated with a bounded psh function (Definition~\\ref{defn:L2-ma}). This result is pushed much further in Section~9 where we show that $W^*(\\Omega)$ is  naturally embedded  into a Dirichlet space associated with a closed positive  $(n-1,n-1)$ current $\n\\theta = [dd^c (\\phi + \\delta |z|^2)]^{n-1}$ where $\\phi \\in PSH \\cap L^\\infty(\\Omega)$ and $\\delta>0$ are given.\nTherefore, the integral  $\\int_K df \\wedge d^c g\\wedge \\theta$ for $f,g\\in W^*(\\Omega)$ is well-defined on every compact set $K\\subset \\Omega$. This aspect has been considered recently by Do, Nguyen and Vu \\cite{DNV25}.\n\nThe second main result is as follows. \n\\begin{thm}\\label{thm:intro-choquet-c} Let $\\Omega$ be a bounded open set in $\\mathbb{C}^n$. The set function on Borel subsets $E\\mapsto {\\tt c} (E)= {\\tt c} (E,\\Omega)$ is a Choquet capacity.\n\\end{thm}\n\nThis result is the local analogue of \\cite{Vigny}. On compact K\\\"ahler manifolds the proof of  \\cite[Theorem~30]{Vigny} employed the capacity notion in the Dirichlet spaces \\cite[Lemma~23]{Vigny} and the characterization of pluripolar sets via a family of capacity associated with closed positive current \\cite{FO84}.  Here we use a similar strategy. Notably, we are able to give a simpler proof of the characterization in Corollary~\\ref{cor:FO-polar} by using only pluripotential theory. \n\nAnother remarkable property of functions in the complex Sobolev space is the uniformly exponential integrability due to Dinh, Marinescu and Vu  \\cite{DMV}, which is very close to the one of psh functions.  Namely, \nthere exist positive constants $\\alpha$ and $A$  such that for every $f \\in W^*(B(0,1))$ with $\\|f\\|_*\\leq 1$, \n\\[\\label{eq:DMV-intro}\n\t\\int_{\\bar B(0,\\frac{1}{8})} e^{\\alpha |f|^2} d x \\leq A.\n\\]\nThis result should be compared with the well-known exponential integrability of psh functions due to H\\\"ormander \\cite[Proposition~4.2.9]{Ho07} and Skoda \\cite{Sk72}. In the proof of  \\cite[Theorem~1.2]{DMV}  the authors used the induction argument in dimension and slicing theory for positive currents. We provide a new and simpler one in Proposition~\\ref{prop:DMV} by reducing the inequality to one dimensional case. In this way, it is enough  to  work with smooth functions after taking  the standard convolution with smooth kernels. The constants can be explicitly computed though they  are suboptimal.  Additionally, we point out in Remark~\\ref{rmk:sharp-DMV} that the exponent 2 of $|f|$ in \\eqref{eq:DMV-intro} is the optimal for all $n\\geq 1$, which was only known for $n=1$ in \\cite{DMV}. \n\nThe third main result fully characterizes the inequality \\eqref{eq:DMV-intro} by the volume-capacity inequality of sublevel sets.\n\n\\begin{thm} \\label{prop:equiv-est-intro} Let $K\\subset\\subset \\Omega$ be a compact subset.  The following statements hold and they are equivalent to each other.\n\\begin{itemize}\n\\item[(a)] \nThere exist constants $A_1>0$ and $\\alpha>0$ depending only on $K, \\Omega$ such that for every  Borel  set $E\\subset K$ $$V_{2n}(E) \\leq A_1 e^{-\\alpha/{\\tt c} (E)}.$$\n\n\\item[(b)] There exist uniform constants $A_1>0$ and $\\alpha>0$ depending only on $K, \\Omega$ such that for every  $f\\in W^*(\\Omega)$ and $f\\leq 0$,\n $$V_{2n}(\\{f< -1\\} \\cap K) \\leq A_1 e^{\\frac{-\\alpha}{\\|f\\|_*^{2}}}.$$\n\\item\n[(c)]   There exist uniform constants $A_1>0$ and $\\alpha>0$ depending only on $K, \\Omega$ such that for every $f\\in W^*(\\Omega)$ and $f\\leq 0$ whose norm $\\|f\\|_* \\leq \n\\kappa$,\n$$V_{2n} (\\{f < -s\\} \\cap K) \\leq A_1 e^{-\\alpha s^2/\\kappa^2}, \\quad \\forall s>0.$$\n\n\\item \n[(d)]  There exist uniform constants $A_1>0$ and $\\alpha>0$ depending only on $K, \\Omega$ such that for every $f\\in W^*(\\Omega)$ and $f\\leq 0$ with $\\|f\\|_* \\leq 1$,\n$$V_{2n} (\\{f < -s\\} \\cap K) \\leq A_1 e^{-\\alpha s^2}, \\quad \\forall s>0.$$\n\\end{itemize}\n\\end{thm}\n\nThe characterization is partially inspired by the work of \\AA hag, Cegrell, Ko\\l odziej, Pham and Zeriahi \\cite{ACKPZ09}. Lastly, we point out in Remark~\\ref{rmk:DMV-holder} that the inequalities holds for a very large family  of Monge-Amp\\`ere measures associated with  H\\\"older continuous psh functions.\n\n\\bigskip\n\n{\\em Acknowledgement.} I would like to thank G. Marinescu and D.-V. Vu for a kind invitation to visit University of Cologne in September 2025 and to whom I had many helpful discussions on the results of \\cite{DMV}.  I am grateful to S. Ko\\l odziej and D.-V. Vu for useful comments on the draft of the paper. It was written while the author  visited the Center for Complex Geometry (Daejeon). He would like to thank Jun-Muk Hwang and Yongnam Lee for their kind support and exceptional hospitality. He is also grateful to the institution for providing perfect working conditions.", "sketch": "For Theorem~\\ref{thm:intro-cap-comp}, the text says that its statements are provided by Lemmas~\\ref{lem:c-cap}, \\ref{lem:cap-c} and Remarks~\\ref{rmk:inner-reg}, \\ref{rmk:sharp-exp}. In the local setting, “the key idea in the proof of the first inequality is making use of Cegrell's inequality \\cite{Ce04}” while “the one for the second inequality comes from the comparison principle.” The “optimality of the exponents is extracted from the Alexander-Taylor type inequality which is proved in Lemma~\\ref{lem:AT-type}.”", "expanded_sketch": "To prove the main theorem, the text says that its statements are provided by the following lemmas and remarks.\n\n\\begin{lem}[dominated by Bedford-Taylor capacity]  \\label{lem:c-cap} Assume $\\Om$ is strictly pseudoconvex. There exists a constant $A>0$ such that for every Borel set $E\\subset\\subset \\Om$,\n$$\n\t\\tc(E) \\leq A \\left[cap(E,\\Om)\\right]^\\frac{1}{n}.\n$$\n\\end{lem}\n\n\\begin{lem} \\label{lem:cap-c} Let $D \\subset \\subset \\Om$. There exists  a constant $A'$ depending  only on $D$ and $\\Om$ such that for every Borel set $E \\subset D$,\n$$\tcap(E,\\Om) \\leq A'\\, \\tc(E)\n$$\n\\end{lem}\n\n\\begin{remark}\\label{rmk:inner-reg} The relative compactness condition of $E$ will be removed later after proving that $\\tc(\\cdot,\\Om)$ is a capacity in the sense of Choquet (Theorem~\\ref{thm:choquet-c}).\n\\end{remark}\n\n\\begin{remark}[Proof of the last statement in Theorem~\\ref{thm:intro-cap-comp}]\n\\label{rmk:sharp-exp} The Alexander-Taylor inequality \\cite{AT84} reads\n$$\n\t\\exp (-A_r/cap(K,\\Om)) \\leq T_R(K) \\leq \\exp(-2\\pi/[cap(K,\\Om)]^\\frac{1}{n}),\n$$\nwhere the exponents in both inequalities are sharp (\\cite[Remark~2]{AT84}).\nTogether with $cap(K) \\leq A_r \\tc(K)$ in Lemma~\\ref{lem:cap-c}\nand the first inequality in Lemma~\\ref{lem:AT-type} we derive \n$$\n\t\\exp (-A/cap(K,\\Om)) \\leq \\exp (-A/\\tc(K,\\Om))    \\leq \\exp(-2\\pi/[cap(K,\\Om)]^\\frac{1}{n})\n$$\nwith the sharp exponents. So are the ones in Lemma~\\ref{lem:c-cap} and Lemma~\\ref{lem:cap-c}.\n\\end{remark}\n\nIn the local setting, “the key idea in the proof of the first inequality is making use of Cegrell's inequality \\cite{Ce04}” while “the one for the second inequality comes from the comparison principle.” The “optimality of the exponents is extracted from the Alexander-Taylor type inequality which is proved in the following lemma.”\n\n\\begin{lem} \\label{lem:AT-type} There exist positive constants $A_r$ and $A_R$ depending on $r,R$ respectively such that for every compact  set $K\\subset B_r$,\n$$ \\exp(- A_R \\; \\tc (K,B_R)^{-1}) \\leq  T_R(K) \\leq \t\\exp(- A_r\\;\\tc (K, B_R)^{-1/n}).\n$$ \nEquivalently,\n\\[\\label{eq:AT}\n\t\\frac{1}{A_r} \\frac{1}{M_K^n} \\leq \\tc (K,B_R) \\leq \\frac{A_R }{M_K}.\n\\]\n\\end{lem}", "expanded_theorem": "\\label{thm:intro-cap-comp} \nLet $\\Omega \\subset \\subset \\mathbb{C}^n$ be a strictly pseudoconvex domain. \n\\begin{itemize}\n\\item[(a)] There exists a constant $A>0$ such that for every Borel set $E\\subset \\Omega$,\n$$\n\t{\\tt c}(E) \\leq A [cap(E,\\Omega)]^\\frac{1}{n}.\n$$\n\\item[(b)]\nAssume $D \\subset \\subset \\Omega$ be a subdomain. There exists a constant $A'$ such that for every Borel set $E\\subset D$,\n$$\n\t\\frac{1}{A'} cap(E,\\Omega) \\leq {\\tt c} (E).\n$$\n\\end{itemize}\nMoreover, the above inequalities are sharp as far as the exponents are concerned.,", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let \\(\\Omega\\Subset \\mathbb C^n\\) be a strictly pseudoconvex domain. For a Borel set \\(E\\subset \\Omega\\), define the Bedford--Taylor capacity by\n\\[\n\\operatorname{cap}(E,\\Omega)=\\sup\\left\\{\\int_E (dd^c u)^n:\\ u\\in PSH(\\Omega),\\ -1\\le u\\le 0\\right\\}.\n\\]\nAlso define the local functional capacity\n\\[\n\\mathbf c(E)=\\inf\\{\\|v\\|_*^2:\\ v\\in \\mathcal K(E)\\},\n\\]\nwhere\n\\[\n\\mathcal K(E)=\\left\\{v\\in W^*(\\Omega): v\\le 0\\text{ on }\\Omega,\\ \\{v\\le -1\\}^o\\supset E\\right\\},\n\\]\nwith \\(\\{v\\le -1\\}^o\\supset E\\) meaning that \\(v\\le -1\\) almost everywhere in a neighborhood of \\(E\\), and where\n\\[\nW^*(\\Omega)=\\left\\{f\\in W^{1,2}(\\Omega,\\mathbb R): \\exists\\text{ a positive closed }(1,1)\\text{-current }T\\text{ with }df\\wedge d^cf\\le T\\text{ weakly and }\\|T\\|_\\Omega<\\infty\\right\\},\n\\]\n\\[\n\\|f\\|_*^2=\\|f\\|_{L^2(\\Omega)}^2+\\inf_{T\\in \\Gamma_f}\\|T\\|_\\Omega,\n\\qquad \\|T\\|_\\Omega=\\int_\\Omega T\\wedge \\omega^{n-1}.\n\\]\nUnder these assumptions, which quantitative comparison between \\(\\mathbf c(E)\\) and \\(\\operatorname{cap}(E,\\Omega)\\) holds?", "correct_choice": {"label": "A", "text": "There exists a constant \\(A>0\\) such that for every Borel set \\(E\\subset \\Omega\\),\n\\[\n\\mathbf c(E)\\le A\\,[\\operatorname{cap}(E,\\Omega)]^{1/n}.\n\\]\nMoreover, if \\(D\\Subset \\Omega\\) is any subdomain, then there exists a constant \\(A'>0\\) such that for every Borel set \\(E\\subset D\\),\n\\[\n\\frac{1}{A'}\\operatorname{cap}(E,\\Omega)\\le \\mathbf c(E).\n\\]\nIn addition, these inequalities are sharp with respect to the exponents."}, "choices": [{"label": "B", "text": "There exists a constant \\(A>0\\) such that for every Borel set \\(E\\subset \\Omega\\),\n\\[\n\\mathbf c(E)\\le A\\,\\operatorname{cap}(E,\\Omega).\n\\]\nMoreover, if \\(D\\Subset \\Omega\\) is any subdomain, then there exists a constant \\(A'>0\\) such that for every Borel set \\(E\\subset D\\),\n\\[\n\\frac{1}{A'}\\,[\\operatorname{cap}(E,\\Omega)]^{1/n}\\le \\mathbf c(E).\n\\]\nIn addition, these inequalities are sharp with respect to the exponents."}, {"label": "C", "text": "There exists a constant \\(A>0\\) such that for every Borel set \\(E\\subset \\Omega\\),\n\\[\n\\mathbf c(E)\\le A\\,[\\operatorname{cap}(E,\\Omega)]^{1/n}.\n\\]\nMoreover, if \\(D\\Subset \\Omega\\) is any subdomain, then there exists a constant \\(A'>0\\) such that for every Borel set \\(E\\subset D\\),\n\\[\n\\frac{1}{A'}\\operatorname{cap}(E,\\Omega)\\le \\mathbf c(E).\n\\]"}, {"label": "D", "text": "There exists a constant \\(A>0\\), depending only on \\(\\Omega\\), such that for every subdomain \\(D\\Subset \\Omega\\) and every Borel set \\(E\\subset D\\),\n\\[\n\\mathbf c(E)\\le A\\,[\\operatorname{cap}(E,\\Omega)]^{1/n}\n\\quad\\text{and}\\quad\n\\frac{1}{A}\\operatorname{cap}(E,\\Omega)\\le \\mathbf c(E).\n\\]\nIn particular, the same constant works uniformly for all choices of \\(D\\Subset \\Omega\\), and the exponents are sharp."}, {"label": "E", "text": "There exists a constant \\(A>0\\) such that for every Borel set \\(E\\subset \\Omega\\),\n\\[\n\\mathbf c(E)\\le A\\,[\\operatorname{cap}(E,\\Omega)]^{1/n}.\n\\]\nMoreover, there exists a constant \\(A'>0\\) such that for every Borel set \\(E\\subset \\Omega\\),\n\\[\n\\frac{1}{A'}\\operatorname{cap}(E,\\Omega)\\le \\mathbf c(E).\n\\]\nIn addition, these inequalities are sharp with respect to the exponents."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "sharp exponents 1/n versus 1", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the sharpness conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence of lower-bound constant on the subdomain D", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "restriction of the reverse inequality to E\\subset D\\Subset\\Omega", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only the definitions of the two capacities and asks for the valid comparison statement; it does not explicitly state the correct inequalities, exponents, or quantifier structure."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-identification question: it asks for the exact comparison result rather than an application or derivation. However, it is not a pure restatement because the options differ in exponents, constant dependence, and sharpness claims."}, "GPS": {"score": 1, "justification": "Solving it requires moderate reasoning or precise theorem recall to distinguish subtle alternatives, especially the exponent swap, the need to restrict the lower bound to subdomains, and the sharpness clause. It does not strongly force original generative reasoning beyond discriminating among variants."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they target common failure modes such as swapped exponents, omission of one inequality, incorrect uniformity in constants, and replacing comparison by exact equality."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise recall/recognition than deep generative reasoning."}}
{"id": "2602.14213v1", "paper_link": "http://arxiv.org/abs/2602.14213v1", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "\\label{main}\n\tLet $G$ be a controllable graph and $p$ be an odd prime such that  $\\rank_p W(G)=n-1$. Then $v_p(L(G))\\le \\frac{1}{2} v_p(\\det W(G))$.", "start_pos": 7243, "end_pos": 7420, "label": "main"}, "ref_dict": {"pf": "\\label{pf}\nWe begin with the fundamental matrix characterization of generalized cospectrality for graphs.\n\n\\begin{lemma}[\\cite{johnson,wang2006}]\n\tLet $G$ and $H$ be two graphs of the same order. Then", "evencase": "\\begin{proposition}[\\cite{wang2017JCTB}]\\label{evencase}\n\tIf $2^{-\\lfloor\\frac{n}{2}\\rfloor}$ is odd then $L(G)$ is odd.\n\\end{proposition}", "pre": "\\begin{theorem}\\label{main}\n\tLet $G$ be a controllable graph and $p$ be an odd prime such that  $\\rank_p W(G)=n-1$. Then $v_p(L(G))\\le \\frac{1}{2} v_p(\\det W(G))$.\n\\end{theorem}\n\tThe main innovation of this paper lies in the shift of the underlying algebraic structure from the vector space over the field $\\mathbb{Z}/p\\mathbb{Z}$ (as used in \\cite{wang2013EJC,qiu2023}) to the module over the local ring $\\mathbb{Z}/p^k\\mathbb{Z}$. While working over a ring introduces the complexity of zero divisors, many basic results from linear algebra over fields still hold for the ring $\\mathbb{Z}/p^k\\mathbb{Z}$. In particular, it possesses the ``Basis Extension Property'' (Steinitz Property), which allows us to generalize arguments from linear algebra. \n\n\tThe rest of this paper is organized as follows.  In Sec.~\\ref{pre}, we recall some  preliminaries on Smith normal forms and modules over $\\mathbb{Z}/p^k \\mathbb{Z}$.  The proof of Theorem \\ref{main} is given in Sec.~\\ref{pf}, which can be seen as a module-theoretical version of the original argument of Wang \\cite{wang2006}, combined with some improvements and simplifications developed in \\cite{wang2020}.   Some direct applications of Theorem \\ref{main} are presented in Sec.~\\ref{dis}. The paper concludes with a conjecture proposing a further possible improvement to Theorem \\ref{main}.\n\\section{Preliminaries}\n\\label{pre}\nLet $R$ be a commutative principal ideal ring with identity.  It is well known \\cite{stanley2016JCTA} that any matrix $M$ over $R$ has a Smith normal form (SNF); that is, there exist invertible matrices $U$ and $V$ over $R$ such that \n\\begin{equation*}\n\tUMV=\\begin{bmatrix}\n\t\t\\diag(d_1,\\ldots,d_r)&0\\\\\n\t\t0&0\n\t\\end{bmatrix},\n\\end{equation*}\nwhere the elements $d_i$ ($i=1,\\ldots,r$) are nonzero and satisfy $d_1\\mid d_2\\mid \\cdots\\mid d_r$. The nonzero elements $d_1,\\ldots,d_r$ are called the \\emph{invariant factors} of $M$.\n\nThroughout this paper, we fix an odd prime $p$ and consider three kinds of rings: $\\mathbb{Z}$, $\\mathbb{Z}/p\\mathbb{Z}$ and $\\mathbb{Z}/p^k\\mathbb{Z}$, where $k\\ge 2$. Let $\\mathbb{Z} \\to \\mathbb{Z}/p^k \\mathbb{Z} \\to \\mathbb{Z}/p\\mathbb{Z}$ be the natural projections. For a matrix $M$ over $\\mathbb{Z}$, the SNFs of the projections of $M$ are determined naturally by the SNF of $M$ over $\\mathbb{Z}$ via $p$-adic valuations. For example, suppose $M$ has the Smith normal form $\\diag(2,10,30,270)$ over $\\mathbb{Z}$. Then, over the field $\\mathbb{Z}/3\\mathbb{Z}$, the SNF of $M$ is $\\diag(1,1,0,0)$, whereas over the local ring $\\mathbb{Z}/3^2\\mathbb{Z}$, its SNF is $\\diag(1,1,3,0)$. \n\nIn this paper, we are mainly concerned with the SNFs of matrices over the local ring $\\mathbb{Z}/p^k \\mathbb{Z}$. Note that the invariant factors are unique only up to multiplication by units. To ensure uniqueness, we conventionally assume that each nonzero invariant factor takes the form $p^{c}$ for some integer $0 \\le c < k$. We formalize this in the following lemma.\n\n\\begin{lemma}\n\tFor any matrix $M$ over $\\mathbb{Z}/p^k\\mathbb{Z}$, there exist invertible matrices $U$ and $V$ over $\\mathbb{Z}/p^k\\mathbb{Z}$ such that \n\t\\begin{equation*}\n\t\tUMV = \\begin{bmatrix}\n\t\t\t\\diag(p^{c_1}, \\ldots, p^{c_r})&0\\\\\n\t\t\t0&0\n\t\t\\end{bmatrix},\n\t\\end{equation*}\n\twhere $0 \\le c_1 \\le \\cdots \\le c_r < k$.\n\\end{lemma}\nA classic result in linear algebra states that the linear system $Mx=b$ over a field has a solution if and only if $\\rank M = \\rank (M,b)$. The corresponding question over an arbitrary ring is generally more complicated, but for our purposes, the existence of the SNF gives a straightforward extension.\n\n\\begin{proposition}\\label{ssnf}\n\tThe linear system $Mx=b$ over $\\mathbb{Z}/p^k\\mathbb{Z}$ has a solution if and only if $M$ and $(M,b)$ have the same invariant factors.\n\\end{proposition} \n\n\\begin{proof}\n\tThe ``only if'' part is clear. Let $\\mathcal{C}(M)$ and $\\mathcal{C}(M,b)$ be the modules generated by the columns of $M$ and $(M,b)$, respectively. Since $M$ and $(M,b)$ have the same invariant factors, the two modules $\\mathcal{C}(M)$ and $\\mathcal{C}(M,b)$ are isomorphic. As $\\mathcal{C}(M,b)$ is finite, it cannot be isomorphic to any proper submodule. Since $\\mathcal{C}(M)$ is a submodule of $\\mathcal{C}(M,b)$, the isomorphism $\\mathcal{C}(M)\\cong \\mathcal{C}(M,b)$ implies that they must be equal. Thus, $b\\in\\mathcal{C}(M)$, i.e., $Mx=b$ has a solution. \n\\end{proof}\nThe SNF is particularly useful for determining the structure of the solution space (kernel) of a linear system. It is important to note that the kernel is not always a free module. However, under specific conditions on the invariant factors, it is.\n\n\\begin{proposition}\\label{kf}\n\tLet $M$ be an $m \\times n$ matrix over the ring $\\mathbb{Z}/p^k\\mathbb{Z}$. Let the invariant factors of $M$ be $p^{c_1}, \\ldots, p^{c_r}$ satisfying $0 \\le c_1 \\le \\dots \\le c_r < k$. Then the kernel of $M$ is given, up to isomorphism, by:\n\t\\begin{equation*}\n\t\t\\ker(M) \\cong \\bigoplus_{i=1}^r \\left( \\mathbb{Z}/p^{c_i}\\mathbb{Z} \\right) \\oplus \\left( \\mathbb{Z}/p^k \\mathbb{Z} \\right)^{n-r}.\n\t\\end{equation*}\n\tIn particular, if $c_1=\\cdots=c_r=0$, then $\\ker(M)\\cong (\\mathbb{Z}/p^k\\mathbb{Z})^{n-r}$, which is a free module of rank $n-r$.\n\\end{proposition}\n\nThe following proposition is a direct generalization of \\cite[Lemma 7]{wang2013EJC}, where the case $m=n,k=2$ was considered. The original proof in \\cite{wang2013EJC} remains valid for the general setting and is therefore omitted here.\n\n\\begin{proposition}[\\cite{wang2013EJC}]\\label{dnk}\n\tLet $M$ be an $m\\times n~(m\\ge n)$ integral matrix whose SNF is \n\t\\begin{equation*}\n\t\t\\begin{bmatrix}\n\t\t\\diag(d_1,\\ldots,d_n)\\\\\n\t\t0_{(m-n)\\times n}\n\t\t\\end{bmatrix}.\n\t\\end{equation*} Then the equation $Mz\\equiv 0\\pmod{p^{k}}$ has a solution $z\\not\\equiv 0\\pmod{p}$ if and only if $p^{k}\\mid d_n.$\n\\end{proposition}\n\nA set of vectors $v_1, \\ldots, v_m \\in (\\mathbb{Z}/p^k\\mathbb{Z})^n$ is called \\emph{linearly independent} if $\\sum_{i=1}^m c_i v_i = 0$ implies $c_i = 0$ for all $i$. It is known that $v_1,\\ldots, v_m$ are linearly independent if and only if their projections in the vector space $(\\mathbb{Z}/p\\mathbb{Z})^n$ are linearly independent (over the field $\\mathbb{Z}/p\\mathbb{Z}$). A key property of the ring $\\mathbb{Z}/p^k\\mathbb{Z}$ is the so-called Basis Extension Property (or Steinitz Property \\cite{mcdonald}), which states that any linearly independent vectors in a free $\\mathbb{Z}/p^k\\mathbb{Z}$-module $M$ can always be extended to a basis of $M$. Since we only require the finite-rank case, we formalize it in the following proposition.\n\n\\begin{proposition}\\label{bas}\n\tLet $M$ be a free submodule of rank $m$ in $(\\mathbb{Z}/p^k\\mathbb{Z})^n$. Let $v_1, \\ldots, v_k \\in M$ ($k\\le m$) be linearly independent. Then:\n\n\t\\textup{(i)} If $k=m$, the set $\\{v_1, \\ldots, v_k\\}$ constitutes a basis of $M$.\n\n\t\\textup{(ii)}  If $k< m$, there exist $m-k$ vectors $v_{k+1}, \\ldots, v_{m} \\in M$ such that $\\{v_1,\\ldots,v_m\\}$ constitutes a basis of $M$.\n\\end{proposition}\n\\section{Proof of Theorem \\ref{main}}\\label{pf}\nWe begin with the fundamental matrix characterization of generalized cospectrality for graphs.\n\n\\begin{lemma}[\\cite{johnson,wang2006}]\n\tLet $G$ and $H$ be two graphs of the same order. Then $G$ and $H$ are generalized cospectral if and only if there exists a regular orthogonal matrix $Q$ such that $Q^\\mathsf{T} A(G) Q=A(H)$. Moreover, if $G$ is controllable, then $Q^\\mathsf{T}=W(H)(W(G))^{-1}$, and hence $Q$ is unique and rational.\n\\end{lemma}\n\nIn the following, let $G$ be an $n$-vertex controllable graph and $p$ be an odd prime factor of $\\det W(G)$ such that $\\rank_p W(G)=n-1$. Denote $\\tau=v_p(L(G))$, i.e., $\\tau=\\max\\{v_p (\\ell(Q))\\colon\\, Q\\in \\mathcal{Q}(G)\\}$. Note that if $\\tau=0$, then Theorem \\ref{main} clearly holds. Thus, we may assume that $\\tau\\ge 1$. For simplicity, the adjacency matrix $A(G)$ and the walk matrix $W(G)$ will be denoted by $A$ and $W$, respectively, when there is no confusion.\n\n\\begin{lemma}\\label{fourcong}\n\tThere exist an integral vector $z_0$ with $z_0\\not\\equiv 0\\pmod{p}$ and an integer $\\lambda_0$ such that $z_0^\\mathsf{T} z_0\\equiv 0\\pmod{p^{2\\tau}}$, $z_0^\\mathsf{T} A z_0\\equiv 0\\pmod{p^{2\\tau}}$, $W^\\mathsf{T} z_0\\equiv 0\\pmod{p^\\tau}$, and $Az_0\\equiv \\lambda_0z_0\\pmod{p^\\tau}$.\n\\end{lemma}\n\n\\begin{proof}\n\tLet $Q\\in \\mathcal{Q}(G)$ be such that $v_p(\\ell(Q))=\\tau$, and let $H$ be the corresponding graph. Let $\\ell=\\ell(Q)$ and $\\hat{Q}=\\ell Q$. Then $\\hat{Q}$ is an integral matrix and $\\hat{Q}\\not\\equiv 0\\pmod{p}$. Let $z_0$ be a column of $\\hat{Q}$ such that $z_0\\not\\equiv 0\\pmod{p}$. Since $Q$ is an orthogonal matrix and $p^\\tau\\mid \\ell$, we have $\\hat{Q}^\\mathsf{T} \\hat{Q}=\\ell^2 I$, and hence $z_0^\\mathsf{T} z_0\\equiv 0\\pmod{p^{2\\tau}}$. Similarly, as $Q^\\mathsf{T} A Q$ is a $(0,1)$-matrix, we have $\\hat{Q}^\\mathsf{T} A\\hat{Q}\\equiv 0\\pmod{p^{2\\tau}}$, and hence $z_0^\\mathsf{T} A z_0\\equiv 0\\pmod{p^{2\\tau}}$. Moreover, noting that $W^\\mathsf{T} Q$ equals $(W(H))^\\mathsf{T}$, which is an integral matrix, we find that $W^\\mathsf{T} \\hat{Q}\\equiv 0\\pmod{\\ell}$, and hence $W^\\mathsf{T} z_0\\equiv 0\\pmod{p^\\tau}$.\n\n\tLet $M=\\{z\\in \\mathbb{Z}^n\\colon\\, W^\\mathsf{T} z\\equiv 0\\pmod {p^\\tau}\\}$. Since $\\rank_p W=n-1$ and $p^\\tau \\mid \\det W$, we see that, over $\\mathbb{Z}/p^\\tau \\mathbb{Z}$, the SNF of $W^\\mathsf{T}$ is $\\diag(1,1,\\ldots,1,0)$. It follows from Proposition \\ref{kf} that $M$ is a free $\\mathbb{Z}/p^\\tau \\mathbb{Z}$-module of rank one. Since $z_0\\in M$ and $z_0\\not\\equiv 0\\pmod{p}$, we see that $\\{z_0\\}$ forms a basis for $M$ over $\\mathbb{Z}/p^\\tau \\mathbb{Z}$ by Proposition \\ref{bas} (i). On the other hand, by the Cayley-Hamilton Theorem, it is easy to see that $M$ is an $A$-invariant $\\mathbb{Z}/p^\\tau \\mathbb{Z}$-submodule. This implies that there exists an integer $\\lambda_0$ such that $Az_0\\equiv \\lambda_0 z_0\\pmod{p^\\tau}$. This completes the proof.\t\n\\end{proof}\n\\begin{lemma}\\label{wy}\n\tIf $(A-\\lambda_0 I)y\\equiv s p^j z_0\\pmod{p^{j+\\tau}}$ for some integer $s$ and integer $j\\ge 0$, then\n\t\\begin{equation*}\n\t\tW^\\mathsf{T} y\\equiv (e^\\mathsf{T} y)(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T} \\pmod{p^{j+\\tau}}.\n\t\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\n\tWe proceed by induction on $k$ to show that $e^\\mathsf{T} A^k y\\equiv \\lambda_0^k e^\\mathsf{T} y\\pmod{p^{j+\\tau}}$ for $k=0,1,\\ldots,n-1$. The base case $k=0$ is trivial. Assume that the congruence holds for some integer $k$ with $0 \\le k < n-1$; we proceed to verify it for $k+1$. \n\n\tBy Lemma \\ref{fourcong}, we have $W^\\mathsf{T} z_0\\equiv 0\\pmod{p^\\tau}$, which implies that $e^\\mathsf{T} A^k z_0\\equiv 0\\pmod{p^\\tau}$. Consequently, we obtain $e^\\mathsf{T} A^k (sp^j z_0)\\equiv 0\\pmod{p^{j+\\tau}}$. By the hypothesis of this lemma, $Ay \\equiv  sp^j z_0+\\lambda_0 y \\pmod{p^{j+\\tau}}$, which implies that\n\t\\[\n\te^\\mathsf{T} A^{k+1}y \\equiv e^\\mathsf{T} A^k (sp^j z_0+\\lambda_0 y) \\equiv \\lambda_0 e^\\mathsf{T} A^k y \\equiv \\lambda_0^{k+1}e^\\mathsf{T} y \\pmod{p^{j+\\tau}}.\n\t\\]\n\tThis completes the proof.\t\n\\end{proof}\n\\begin{lemma}\\label{rA}\n\tLet $S=\\diag(f_1,\\ldots,f_n)$ be the SNF of $A-\\lambda_0 I$. Then $f_{n-2}\\not \\equiv 0\\pmod{p}$ and $f_n\\equiv 0\\pmod{p^\\tau}$.\n\\end{lemma}\n\n\\begin{proof}\n\tBy Lemma \\ref{fourcong}, we have $(A-\\lambda_0 I)z_0\\equiv 0\\pmod{p^\\tau}$ and $z_0\\not\\equiv 0\\pmod{p}$. It follows from Proposition \\ref{dnk} that $f_n\\equiv 0\\pmod{p^\\tau}$. \tIt remains to show that $f_{n-2}\\not \\equiv 0\\pmod{p}$. Suppose to the contrary that $f_{n-2}\\equiv 0\\pmod{p}$. Then $\\rank_p (A-\\lambda_0 I)\\le n-3$. \n\tLet \\begin{equation*}\n\tB=\\begin{bmatrix} A-\\lambda_0 I\\\\e^\\mathsf{T}\\end{bmatrix}.\\end{equation*}\n\tThen, we have $\\rank_p B\\le n-2$. Recall from Lemma \\ref{fourcong} that $W^\\mathsf{T} z_0 \\equiv 0 \\pmod{p}$, which implies $e^\\mathsf{T} z_0 \\equiv 0 \\pmod{p}$. Together with $(A-\\lambda_0 I)z_0 \\equiv 0 \\pmod{p}$, this yields $Bz_0 \\equiv 0 \\pmod{p}$. Since the nullity of $B$ over $\\mathbb{Z}/p\\mathbb{Z}$ is at least $n - (n-2) = 2$, the equation $Bz\\equiv 0\\pmod {p}$ must have a solution $z_1$ such that $z_0$ and $z_1$ are linearly independent over $\\mathbb{Z}/p\\mathbb{Z}$. \n\n\tNoting that $Az_1\\equiv \\lambda_0 z_1 \\pmod{p}$ and $e^\\mathsf{T} z_1\\equiv 0\\pmod{p}$, we clearly have \n\t\\[\n\te^\\mathsf{T} A^{k}z_1\\equiv \\lambda_0^k e^\\mathsf{T} z_1\\equiv 0\\pmod{p}\n\t\\]\n\tfor $k=0,1,\\ldots, n-1$. This means that $W^\\mathsf{T} z_1\\equiv 0\\pmod{p}$. By Lemma \\ref{fourcong}, we also have $W^\\mathsf{T} z_0\\equiv 0\\pmod{p}$. Therefore, $\\rank_p W^\\mathsf{T} \\le n-2$. This contradicts the assumption that $\\rank_p W = n-1$, which completes the proof.\n\\end{proof}\n\\begin{lemma} \\label{lc}\n\t\tLet $M = [A-\\lambda_0 I, z_0]$. Then $z_0^\\mathsf{T} M \\equiv 0 \\pmod{p^\\tau}$ and the SNF of $M$ is $[\\diag(I_{n-1},0),0]$. Moreover, any integral  vector $z$ satisfying $z_0^\\mathsf{T} z \\equiv 0 \\pmod{p^\\tau}$ can be expressed as a linear combination of the columns of $M$ over $\\mathbb{Z}/p^\\tau \\mathbb{Z}$.\n\t\\end{lemma}\n\t\\begin{proof} By Lemma \\ref{fourcong}, we have $(A-\\lambda_0 I)z_0\\equiv 0\\pmod{p^\\tau}$ and $z_0^\\mathsf{T} z_0\\equiv 0\\pmod{p^\\tau}$. As  $A-\\lambda_0 I$ is symmetric, we find that\n\t\t\\begin{equation}\\label{mz}\n\t\t\tM^\\mathsf{T} z_0=\\begin{bmatrix}\n\t\t\t\tA-\\lambda_0 I\\\\\n\t\t\t\tz_0^\\mathsf{T}\n\t\t\t\\end{bmatrix}z_0\\equiv 0\\pmod{p^\\tau}.\n\t\t\\end{equation} Let the SNF of $M$ be $S= [\\diag(p^{c_1},p^{c_2}, \\dots, p^{c_n}), 0]$. Clearly, the SNF of $M^\\mathsf{T}$ is $S^\\mathsf{T}$. Note that $z_0\\not\\equiv 0\\pmod{p}$. It follows from Eq.~\\eqref{mz} and Proposition \\ref{dnk}  that $p^{c_n}$ is zero over $R$. Thus, the SNF of $M$ can be simplified as \n\t\t\\begin{equation}\\label{ss}\n\t\t\tS=[\\diag(p^{c_1},\\ldots,p^{c_{n-1}},0),0].\n\t\t\\end{equation}\n\n\\noindent\\textbf{Claim}: $p^{c_{n-1}}$ is a unit in $R$, i.e., $c_{n-1}= 0$.\n\n\t\tSuppose to the contrary that $c_{n-1}\\ge 1$. Then, we have $\\rank_p S\\le n-2$, or equivalently, $\\rank_p M\\le n-2$. On the other hand, by Lemma \\ref{rA}, we see that $\\rank_p (A-\\lambda_0 I)\\ge n-2$ and hence  $\\rank_p M \\ge n-2$. Thus, we must have  $\\rank_p M=\\rank_p (A-\\lambda_0 I)=n-2$. It follows that there exists an integral vector $z_1$ such that $(A-\\lambda_0 I)z_1\\equiv z_0\\pmod{p}$.\t\n\n\t\tAs $\\rank_p  (A-\\lambda_0 I)=n-2$, $(A-\\lambda_0I)z\\equiv 0\\pmod{p}$ has two  solutions $z_2$ and $z_3$ that are linearly  independent over $\\mathbb{Z}/p\\mathbb{Z}$. Since $(A-\\lambda_0 I)z_1\\equiv z_0\\not \\equiv 0\\pmod{p}$, $z_1$ cannot be written as a linear combination of $z_2$ and $z_3$. This implies that $z_1,z_2,z_3$ are linearly independent. Consider the equation $e^\\mathsf{T}(k_1z_1+k_2z_2+k_3z_3)\\equiv 0 \\pmod{p}$ with three unknowns $k_1,k_2,k_3$. Clearly, it has at least two independent solutions over $\\mathbb{Z}/p\\mathbb{Z}$. Let $(a_1,a_2,a_3)^\\mathsf{T}$ and  $(b_1,b_2,b_3)^\\mathsf{T}$  be  two such solutions and write $\\alpha=a_1z_1+a_2z_2+a_3z_3$ and $\\beta=b_1z_1+b_2z_2+b_3z_3$. It is easy to see that $\\alpha$ and $\\beta$ are linearly independent over $\\mathbb{Z}/p\\mathbb{Z}$. Note that $(A-\\lambda_0I)\\alpha\\equiv a_1z_0$ and $e^\\mathsf{T}\\alpha\\equiv 0\\pmod{p}$. Using a similar argument as in the proof of  Lemma~\\ref{wy}, we find  that $W^\\mathsf{T}\\alpha \\equiv 0\\pmod{p}$. Also,  $W^\\mathsf{T} \\beta\\equiv 0\\pmod{p}$. Thus, we have found two linearly independent solutions of $W^\\mathsf{T} z\\equiv 0\\pmod{p}$. This contradicts the fact that $\\rank_p W^\\mathsf{T}=n-1$ and hence completes the proof of the Claim.\n\n\t\tBy the Claim, we see that Eq.~\\eqref{ss} can be further reduced to \n\t\t\\begin{equation}\\label{s3}\n\t\t\tS=[\\diag(I_{n-1},0),0].\n\t\t\\end{equation}\n\t\tLet $\\mathcal{C}(M)$ be the $R$-module generated by the columns of $M$ and $N$ be the module $\\{z\\in R^n\\colon\\, z_0^\\mathsf{T} z=0\\}$. By Eq.~\\eqref{s3}, we know that $\\mathcal{C}(M)$ is isomorphic to the free module $R^{n-1}$. Since $z_0\\not\\equiv 0\\pmod{p}$, we find that $N$ is also isomorphic to $R^{n-1}$. Thus, the two $R$-modules $\\mathcal{C}(M)$ and $N$ are isomorphic. Since $z_0^\\mathsf{T} M=0$ over $R$, we know that $\\mathcal{C}(M)$ is a submodule of $N$. As $R$ is a finite ring, we must have $\\mathcal{C}(M)=N$, which completes the proof of Lemma \\ref{lc}.\n\t\\end{proof}\n\n\\begin{lemma}\\label{ez}\n\tThere exists an  integral vector $z_1$ such that $e^\\mathsf{T} z_1\\not\\equiv 0\\pmod {p}$ and $(A-\\lambda_0 I)z_1\\equiv p^cz_0\\pmod{p^\\tau}$ for some $c\\in \\{0,1,\\ldots,\\tau\\}$.\n\\end{lemma}\n\\begin{proof}\n\tLet $R=\\mathbb{Z}/p^\\tau \\mathbb{Z}$.  By Lemma \\ref{rA}, we know that the SNF of $A-\\lambda_0 I$ is\n\t\\begin{equation}\\label{smA}\n\t\tS=\\diag(1,1,\\ldots,1,p^c,0) \\text{~for some~} c\\in\\{0,1,\\ldots,\\tau\\}.\n\t\\end{equation}\nLet $M_i=[A-\\lambda_0 I, p^i z_0]$ for $i\\in\\{0,1,\\ldots,c\\}$. \n\n\\noindent\\textbf{Claim 1}: The SNF of $M_i$ is $[\\diag(1,1,\\ldots,1,p^i,0),0]$ for $i\\in\\{0,1,\\ldots,c\\}$.\n\nLet $S_i$ be the SNF of $M_i$. Noting that $\\rank_p M_i\\ge \\rank_p (A-\\lambda_0 I)\\ge n-2$, we conclude that $S_i$ must have the form\n\\begin{equation}\\label{si}\n\tS_i=[\\diag(1,\\ldots,1,p^{t_{i}},p^{t'_{i}}),0].\n\\end{equation}\nSince $ M_i^\\mathsf{T} z_0\\equiv 0\\pmod{p^\\tau}$ and $z_0\\not\\equiv 0\\pmod{p}$, Proposition \\ref{dnk} indicates that $p^{t'_i}$ is the zero element in $R$. Thus, Eq.~\\eqref{si} can be simplified as\n\\begin{equation}\\label{st}\n\tS_i=[\\diag(1,\\ldots,1,p^{t_{i}},0),0].\n\\end{equation}\nWe need to show that $t_i=i$ for each $i\\in\\{0,1,\\ldots,c\\}$. Clearly, $t_0=0$ by Lemma \\ref{lc}. Thus, it suffices to show that the sequence $\\{t_i\\}_{0\\le i\\le c}$ satisfies \n\\begin{equation}\\label{tc}\n\tt_c=c\n\\end{equation}\nand \n\\begin{equation}\\label{icr}\n\tt_i\\le t_{i+1}\\le t_{i}+1\\text{~for~} 0\\le i<c.\n\\end{equation}\nNow regard $A-\\lambda_0 I$ and $M_i$ as matrices over $\\mathbb{Z}$. Let $\\Delta$ and $\\Delta^{(i)}$ be the $(n-1)$-th determinantal factors of $A-\\lambda_{0}I$ and $M_i$, respectively. Note that Eq.~\\eqref{tc} holds for the case $c=\\tau$. Indeed, if $c=\\tau$ then $M_c=[A-\\lambda_0 I,p^\\tau z_0]=[A-\\lambda_0 I,0]$ over $R$, which implies that the SNF of $M_c$ is $S_c=[S,0]$, i.e., $t_c=c$ as desired. Now assume $c<\\tau$. From Eq.~\\eqref{smA}, we know that $v_p(\\Delta)=c$. Since $M_c$ contains $A-\\lambda_0 I$ as a submatrix, we must have $v_p(\\Delta^{(c)})\\le v_p(\\Delta)$ and hence  $v_p(\\Delta^{(c)})\\le c$. Let $D$ be any nonzero $(n-1)$-th minor of $M_c$. If $D$ does not include the last column of $M_c$, then $D$ is also a minor of $A-\\lambda_0 I$ and hence $v_p(D)\\ge v_p(\\Delta)\\ge c$. Otherwise,  each entry of the last column of $D$ is a multiple of $p^c$ and hence $v_p(D)\\ge c$. Thus, we always have $v_p(D)\\ge c$, which implies  $v_p(\\Delta^{(c)})\\ge c$. This means $v_p(\\Delta^{(c)})= c$. But by Eq.~\\eqref{st} for the case $i=c$, we know that $v_p(\\Delta^{(c)})=t_c$ and hence $t_c=c$. This proves Eq.~\\eqref{tc}.\n\nWe proceed to verify Eq.~\\eqref{icr}. We may assume $c>0$ since otherwise we have nothing to show. Let $D^{(i)}$ and $D^{(i+1)}$ be any two corresponding $(n-1)$-th minors of $M_i$ and $M_{i+1}$, respectively. By the constructions of $M_i$ and $M_{i+1}$, we have either $D^{(i+1)}=D^{(i)}$ or $D^{(i+1)}=pD^{(i)}$. It follows that $v_p(\\Delta^{(i)})\\le v_p(\\Delta^{(i+1)}) \\le v_p(\\Delta^{(i)}) +1$, i.e., $t_i\\le t_{i+1}\\le t_i+1$. Thus,  Claim 1 follows.\n\nTo complete the proof of Lemma \\ref{ez}, we consider the following three cases:\n\n\\noindent\\emph{Case 1}:  $c=0$. Since $A-\\lambda_0 I$ and $M_0=[A-\\lambda_0 I,z_0]$ have the same invariant factors over $R$, Proposition \\ref{ssnf} implies that there exists an integral vector $z_1$ such that  \\begin{equation}\\label{as}\n(A-\\lambda_0 I)z_1\\equiv z_0\\pmod{p^\\tau}.\n\\end{equation} By Lemma \\ref{wy}, we have \n\t\\begin{equation*}\n\tW^\\mathsf{T} z_1\\equiv e^\\mathsf{T} z_1(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T} \\pmod{p^{\\tau}}.\n\\end{equation*}\nSuppose to the contrary that $e^\\mathsf{T} z_1\\equiv 0\\pmod {p}$. Then we have $W^\\mathsf{T} z_1\\equiv 0\\pmod{p}$.  Since $\\rank_p W=n-1$ and $z_0$ is a nonzero (over $\\mathbb{Z}/p\\mathbb{Z}$) solution of $W^\\mathsf{T} z\\equiv 0\\pmod{p}$, we conclude that $z_1\\equiv kz_0\\pmod{p}$ for some integer $k$. But this would imply \n\\begin{equation*}\n(A-\\lambda_0 I) z_1\\equiv k(A-\\lambda_0 I) z_0\\equiv 0\\pmod{p},\n\\end{equation*}\nwhich, combining with Eq.~\\eqref{as}, leads to $z_0\\equiv 0\\pmod{p}$. This is a contradiction and hence we must have  $e^\\mathsf{T} z_1\\not\\equiv 0\\pmod {p}$.\n\n\\noindent\\emph{Case 2}: $c=\\tau$. As the SNF of $A-\\lambda_0 I$ is $S=\\diag(I_{n-1},0,0)$, we know that the kernel of $(A-\\lambda_0 I)$ is a free $R$-module of rank $2$ by Proposition \\ref{kf}. Let $K=\\ker (A-\\lambda_0 I)$ be the kernel. Since $z_0\\in K$ and $z_0\\not\\equiv 0\\pmod{p}$, Proposition \\ref{bas} (ii) implies that there exists an integer vector $z_1$ such that $\\{z_0,z_1\\}$ (over $R$) constitutes a basis of the free module $K$. In particular, $(A-\\lambda_0 I)z_1\\equiv 0\\pmod{p^\\tau}$. Using Lemma \\ref{wy}, we conclude that \n\t\\begin{equation*}\n\tW^\\mathsf{T} z_1\\equiv e^\\mathsf{T} z_1(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T} \\pmod{p^{\\tau}}.\n\\end{equation*}\nIf $e^\\mathsf{T} z_1\\equiv 0\\pmod{p}$ then we would have \t$W^\\mathsf{T} z_1\\equiv 0\\pmod{p}$. But since $z_0$ and $z_1$ are independent over the field $\\mathbb{Z}/p\\mathbb{Z}$, we must have $\\rank_p W^\\mathsf{T}\\le n-2$. This is a contradiction and hence $e^\\mathsf{T} z_1\\not\\equiv 0\\pmod{p}$.\n\n\\noindent{\\emph{Case 3}}:  $0<c<\\tau$. By Claim 1, we see that $A-\\lambda_0 I$ and $M_c=[A-\\lambda_0 I, p^c z_0]$ have the same invariant factors. By Proposition \\ref{ssnf}, there exists an integral vector $z_1$ such that \n\\begin{equation}\\label{az1}\n\t(A-\\lambda_0 I)z_1\\equiv p^c z_0\\pmod{p^\\tau}.\n\\end{equation} \n\n\\noindent\\textbf{Claim 2}: $z_1$ and $z_0$ are linearly independent over $\\mathbb{Z}/p\\mathbb{Z}$.\n\nSuppose to the contrary that $z_1$ and $z_0$ are linearly dependent over $\\mathbb{Z}/p\\mathbb{Z}$. As $z_0\\not\\equiv 0\\pmod{p}$, there exist an integer $k$ and an integral vector $\\tilde{z}_1$ such that $z_1=kz_0+p\\tilde{z}_1$. Noting that $(A-\\lambda_0 I)z_0\\equiv 0\\pmod{p^\\tau}$, we obtain from Eq.~\\eqref{az1} that $(A-\\lambda_0 I)p\\tilde{z}_1\\equiv p^c z_0\\pmod{p^\\tau}$, i.e., \n\\begin{equation}\\label{az1s}\n\t(A-\\lambda_0 I)\\tilde{z}_1\\equiv p^{c-1} z_0\\pmod{p^{\\tau-1}}.\n\t\\end{equation}\nLet $R'=\\mathbb{Z}/p^{\\tau-1}\\mathbb{Z}$. By Eq.~\\eqref{az1s}, we see that, over $R'$, the two matrices $A-\\lambda_0 I$ and $M_{c-1}=[A-\\lambda_0 I, p^{c-1}z_0]$ have the same invariant factors. Since $c\\le \\tau-1$, we see that $p^c$ and $p^{c-1}$ are not associates over $R'$. Thus, Claim 1 means that $A-\\lambda_0 I$ and $M_{c-1}=[A-\\lambda_0 I, p^{c-1}z_0]$ cannot have the same invariant factors. This contradiction completes the proof of Claim 2. \n\nBy Eq.~\\eqref{az1} and Lemma \\ref{wy}, we have \n\\begin{equation*}\n\tW^\\mathsf{T} z_1\\equiv e^\\mathsf{T} z_1(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T}\\pmod{p^\\tau}.\n\\end{equation*}\nBy Claim 2 and the same argument as in Case 2, we must have $e^\\mathsf{T} z_1\\not\\equiv 0\\pmod{p}$. \n\nCombining the three cases,  Lemma \\ref{ez} follows.\n\\end{proof}\nNow we are in a position to present a proof of Theorem \\ref{main}.\n\n\\begin{proof}[Proof of Theorem \\ref{main}]  Let $\\tau=v_p(L(G))$. Let $z_0$ and $\\lambda_0$ be the integral vector and the integer as described in  Lemma \\ref{fourcong}. Then we have \n\\begin{equation*}\n\tz_0^\\mathsf{T}(A-\\lambda_0 I)z_0\\equiv 0\\pmod{p^{2\\tau}}\n\\end{equation*}\nand hence\n\\begin{equation*}\n\tz_0^\\mathsf{T}\\frac{(A-\\lambda_0 I)z_0}{p^\\tau}\\equiv 0\\pmod{p^{\\tau}}.\n\\end{equation*}\nIt follows from Lemma \\ref{lc} that \n\\begin{equation}\\label{aps}\n\\frac{(A-\\lambda_0 I)z_0}{p^\\tau}\\equiv (A-\\lambda_0 I)y+s z_0\\pmod{p^{\\tau}}\n\\end{equation}\nfor some $y\\in \\mathbb{Z}^{n}$ and $s\\in \\mathbb{Z}$.\nMultiplying both sides of Eq.~\\eqref{aps} by $p^\\tau$ and rearranging the terms, we obtain\n\\begin{equation*}\n\t(A-\\lambda_0 I)(z_0-p^\\tau y)\\equiv sp^\\tau z_0\\pmod{p^{2\\tau}}.\n\\end{equation*}\nConsequently, by Lemma \\ref{wy}, we have\n\\begin{equation}\\label{wp}\n\tW^\\mathsf{T} (z_0-p^\\tau y)\\equiv e^\\mathsf{T} (z_0-p^\\tau y)(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T}\\pmod{p^{2\\tau}}.\n\\end{equation}\nBy Lemma \\ref{ez}, there exist an integral vector $z_1$ and an integer $c\\in\\{0,1,\\ldots,\\tau\\}$ such that $e^\\mathsf{T} z_1\\not\\equiv 0\\pmod{p}$ and\n \\begin{equation}\\label{apta}\n(A-\\lambda_0 I)z_1\\equiv p^c z_0\\pmod{p^\\tau}.\n\\end{equation} As $e^\\mathsf{T} (z_0-p^\\tau y)\\equiv e^\\mathsf{T} z_0\\equiv 0\\pmod{p^\\tau}$, we see that ${p^{-\\tau}}e^\\mathsf{T} (z_0-p^\\tau y) $ is an integer. Note that $e^\\mathsf{T} z_1$ is a unit in $\\mathbb{Z}/p^{\\tau}\\mathbb{Z}$. Thus, there exists some integer $g$ such that \n\\begin{equation}\\label{ep}\n\t\\frac{e^\\mathsf{T} (z_0-p^\\tau y)}{p^\\tau}\\equiv g e^\\mathsf{T} z_1\\pmod{p^{\\tau}}.\n\\end{equation}\nWe know from Eq.~\\eqref{apta} and Lemma \\ref{wy} that \n\\begin{equation*}\n\tW^\\mathsf{T} z_1\\equiv e^\\mathsf{T} z_1(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T} \\pmod{p^\\tau}.\n\\end{equation*}\nThis, together with Eq.~\\eqref{ep}, leads to \n\\begin{equation*}\n\tW^\\mathsf{T} (g z_1)\\equiv g e^\\mathsf{T} z_1(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T}\\equiv \t\\frac{e^\\mathsf{T} (z_0-p^\\tau y)}{p^\\tau}(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T} \\pmod{p^\\tau}.\n\\end{equation*}\nMultiplying both sides by $p^\\tau$ and using Eq.~\\eqref{wp}, we find that \n\\begin{equation*}\nW^\\mathsf{T} (gp^\\tau z_1) \\equiv W^\\mathsf{T} (z_0-p^\\tau y)\\pmod{p^{2\\tau}},\n\\end{equation*}\ni.e., \n\\begin{equation*}\nW^\\mathsf{T} (z_0-p^\\tau y-gp^\\tau z_1) \\equiv 0\\pmod{p^{2\\tau}}.\n\\end{equation*}\nAs $z_0-p^\\tau y-g p^\\tau z_1\\equiv z_0\\not\\equiv 0\\pmod{p}$, Proposition \\ref{dnk} implies that $d_n\\equiv 0\\pmod{p^{2\\tau}}$, where $d_n$ is the $n$-th invariant factor of the integral matrix $W^\\mathsf{T}$. As $\\rank_p W=n-1$, we have $v_p(\\det W) =v_p(d_n)$ and hence $v_p(\\det W)\\ge 2\\tau$. This completes the proof of Theorem \\ref{main}.\n\\end{proof}\n\\section{Applications and future work}\\label{dis}\nAs a natural extension of the arithmetic criterion of DGS-graphs established in \\cite{wang2017JCTB}, Wang et al. \\cite{wang2023ejc} defined, for each odd prime $p$,  a family of graphs \n \\begin{equation*}\n\\mathcal{F}_{n,p}=\\{\\text{$n$-vertex graphs~} G\\colon\\, 2^{-\\lfloor n/2 \\rfloor}\\det W =p^2b\\text{~and~} \\rank_p W=n-1\\},\n \\end{equation*}\nwhere $b$ is odd, square-free and $p\\nmid b$. It was shown that each graph in $\\mathcal{F}_{n,p}$ has at most one generalized cospectral mate \\cite{wang2023ejc}. Using Theorem \\ref{main}, we can show the same conclusion for a larger family of graphs:\n \\begin{equation*}\n \\widetilde{\\mathcal{F}}_{n,p}=\\{\\text{$n$-vertex graphs~} G\\colon\\, 2^{-\\lfloor n/2 \\rfloor}\\det W \\in\\{p^2b, p^3b\\} \\text{~and~} \\rank_p W=n-1\\},\n \\end{equation*}\n where $p$ and $b$ satisfy the same restrictions. Indeed, let $G\\in \\widetilde{\\mathcal{F}}_{n,p}$ and  $Q$ be any matrix in $\\mathcal{Q}(G)$. Then, by Propositions \\ref{oddcase} and \\ref{evencase}, the only possible prime factor of $\\ell(Q)$ is $p$. It follows from Theorem \\ref{main} that\n \\begin{equation*}\nv_p(\\ell(Q))\\le \\frac{1}{2}v_p(\\det W)\\le 3/2,\n \\end{equation*}\n i.e., $v_p(\\ell(Q))\\in\\{0,1\\}$. In other words, $Q$ is either a permutation matrix, or a matrix with level exactly $p$. Now, the original argument in \\cite{wang2023ejc} shows that $G$ has at most one generalized cospectral mate; refer to \\cite{wang2023ejc} for details.\n\n The following small example illustrates the optimality of the family $\\widetilde{\\mathcal{F}}_{n,p}$ in the sense that $G$ may have two or more generalized cospectral mates if the $v_p(\\det W)\\ge 4$ for some odd prime $p$.\n\n \\noindent\\textbf{Example 1}. Let $n=10$ and $G$ be the $n$-vertex graph with adjacency matrix \n\\begin{equation*}\n\tA={\\footnotesize\\begin{bmatrix}\n\t\t0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 \\\\\n\t1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 \\\\\n\t0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 \\\\\n\t1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\\\\n\t1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\\\\n\t0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\\\\n\t1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 \\\\\n\t0 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 \\\\\n\t0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\\\\n\t1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\\\\n\t\\end{bmatrix}}.\n\\end{equation*} \nIt can be verified  that $2^{-\\lfloor\\frac{n}{2}\\rfloor} \\det W=3^4\\times 19$ and $\\rank_3 W=n-1$. By Theorem \\ref{main}, we know that $v_3(L(G))\\le \\frac{1}{2} v_3(\\det W)=2$. Indeed, using the program in \\cite{wang2025}, we find that, besides the permutation matrices,  there exist essentially two rational regular orthogonal matrices \n\\begin{equation*}\n\tQ_1=\\frac{1}{3}{\\footnotesize\n\t\\begin{bmatrix}\n\t\t3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\t\t0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\t\t0 & 0 & 0 & 0 & -2 & 1 & 1 & 1 & 1 & 1 \\\\\n\t\t0 & 0 & 0 & 0 & 1 & -2 & 1 & 1 & 1 & 1 \\\\\n\t\t0 & 0 & 0 & 0 & 1 & 1 & -2 & 1 & 1 & 1 \\\\\n\t\t0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\t\t0 & 0 & 0 & 0 & 1 & 1 & 1 & -2 & 1 & 1 \\\\\n\t\t0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & -2 & 1 \\\\\n\t\t0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & -2 \\\\\n\t\t0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\t\\end{bmatrix}}\n\\end{equation*}\nand \n\\begin{equation*}\nQ_2=\\frac{1}{9}{\\footnotesize\\begin{bmatrix}\n\t\t-3 & -3 & 3 & 3 & 3 & 0 & 0 & 0 & 0 & 6 \\\\\n\t\t3 & 3 & 6 & -3 & -3 & 0 & 0 & 0 & 0 & 3 \\\\\n\t\t1 & 1 & 2 & 2 & 2 & -6 & 3 & 3 & 3 & -2 \\\\\n\t\t1 & 1 & 2 & 2 & 2 & 3 & -6 & 3 & 3 & -2 \\\\\n\t\t-5 & 4 & -1 & -1 & -1 & 3 & 3 & 3 & 3 & 1 \\\\\n\t\t3 & 3 & -3 & 6 & -3 & 0 & 0 & 0 & 0 & 3 \\\\\n\t\t1 & 1 & 2 & 2 & 2 & 3 & 3 & -6 & 3 & -2 \\\\\n\t\t1 & 1 & 2 & 2 & 2 & 3 & 3 & 3 & -6 & -2 \\\\\n\t\t4 & -5 & -1 & -1 & -1 & 3 & 3 & 3 & 3 & 1 \\\\\n\t\t3 & 3 & -3 & -3 & 6 & 0 & 0 & 0 & 0 & 3 \\\\\n\\end{bmatrix}}\n\\end{equation*}\nsuch that both $Q_1^\\mathsf{T} A Q_1$ and $Q_2^\\mathsf{T} A Q_2$ are adjacency matrices. Thus, $G$ has two generalized cospectral mates.\n\nRecently, as a significant improvement of the criterion for a graph to have at most one cospectral mate established in \\cite{wang2023ejc}, Raza et al. \\cite{raza2026} obtained the following upper bound on the number of generalized cospectral mates of a graph.\n\\begin{theorem} [\\cite{raza2026}]\\label{ngm}\n\tLet $G$ be a controllable graph and $d_1,\\ldots, d_n$ be the invariant factors of the walk matrix $W(G)$. Suppose $d_{\\lceil\\frac{n}{2}\\rceil}=1$ and $d_{n-1}=2$. Let $d_n=\\prod_{i=1}^k p_i^{m_i}$ be the prime factorization of $d_n$. Then $G$ has at most $\\left(\\prod_{i=1}^{k}m_i\\right)-1$ generalized cospectral mates.\n\t\\end{theorem}", "dis": "\\label{dis}\nAs a natural extension of the arithmetic criterion of DGS-graphs established in \\cite{wang2017JCTB}, Wang et al. \\cite{wang2023ejc} defined, for each odd prime $p$,  a family of graphs \n \\", "main": "\\begin{theorem}\\label{main}\n\tLet $G$ be a controllable graph and $p$ be an odd prime such that  $\\rank_p W(G)=n-1$. Then $v_p(L(G))\\le \\frac{1}{2} v_p(\\det W(G))$.\n\\end{theorem}", "oddcase": "\\begin{proposition}[\\cite{wang2013EJC}]\\label{oddcase}\n\tIf $p$ is an odd prime with $p^2\\nmid \\det W(G)$ then $p\\nmid L(G)$.\n\\end{proposition}", "qiuodd": "\\begin{proposition}[\\cite{qiu2023}]\\label{qiuodd} If $p$ is an odd prime with $\\rank_p W(G)=n-1$ then $L(G)\\mid \\frac{\\det W(G)}{p}$.\n\\end{proposition}"}, "pre_theorem_intro_text_len": 4698, "pre_theorem_intro_text": "For a graph $G$, the \\emph{spectrum} of $G$ is the multiset of all eigenvalues of its adjacency matrix. Two graphs are \\emph{cospectral} if they share the same spectrum.   We say two graphs $G$ and $H$ are \\emph{generalized cospectral} if (i) $G$ is cospectral with $H$ and (ii) $\\overline{G}$ is cospectral with $\\overline{H}$, where $\\overline{G}$ denotes the complement of $G$. A graph $G$ is \\emph{determined by its generalized spectrum} (DGS)  if any graph generalized cospectral with $G$ is isomorphic to $G$. An orthogonal matrix $Q$ is \\emph{regular} if the sum of each row is 1.  A classic result of Johnson and Newman \\cite{johnson} states that two graphs $G$ and $H$ are generalized cospectral if and only if their adjacency matrices are similar via a regular orthogonal matrix. \n\nFor an $n$-vertex graph $G$ with adjacency matrix $A$, the \\emph{walk matrix} of $G$ is \\begin{equation*}\nW(G):=[e,Ae,\\ldots,A^{n-1}e],\n\\end{equation*} where $e$ is the all-ones vector. A graph is \\emph{controllable} \\cite{godsil} if $W(G)$ is nonsingular. We remark that the set of controllable graphs is \\emph{closed} under generalized cospectrality, that is, any graph generalized cospectral with a controllable graph must be controllable. A key observation of Wang \\cite{wang2006} is that, when restricted to controllable graphs,   the regular orthogonal matrix connecting the adjacency matrices of two generalized cospectral graphs $G$ and $H$ is unique and rational. \n\nWe are mainly concerned with controllable graphs in this paper. We use $\\RO_n(\\mathbb{Q})$ to denote the group of rational regular orthogonal matrices of order $n$. For a controllable graph $G$ with adjacency matrix $A$, we write\n\\begin{equation*}\n\t\\mathcal{Q}(G)=\\{Q\\in \\RO_n(\\mathbb{Q})\\colon\\, Q^\\mathsf{T} AQ \\text{~is a $(0,1)$-matrix} \\}.\n\t\\end{equation*}\nIt is easy to show that any $(0,1)$-matrix of the form $Q^\\mathsf{T} A Q$ must be the adjacency matrix of some graph (see \\cite[Remark 1]{qiu2023}). Let $\\mathcal{C}(G)$ be the set of all graphs that are generalized cospectral with $G$. As there exists a one-to-one correspondence between $\\mathcal{C}(G)$ and $\\mathcal{Q}(G)$, the problem of determining the set $\\mathcal{C}(G)$ can naturally be  reduced to that of determining the set $\\mathcal{Q}(G)$. For example, $\\mathcal{Q}(G)$ always contains all permutation matrices of order $n$ which  correspond to graphs that are isomorphic to $G$.  Moreover, if $\\mathcal{Q}(G)$ contains a non-permutation matrix, then  $\\mathcal{C}(G)$ contains a graph that is not isomorphic to $G$, i.e., $G$ is not DGS.  The following notion introduced in \\cite{wang2006} is crucial to investigate the set $\\mathcal{Q}(G)$.\n\\begin{definition}[\\cite{wang2006}] \\normalfont\n Let $Q$ be a rational matrix. The \\emph{level} of $Q$, denoted by $\\ell(Q)$, is the\nsmallest positive integer $k$ such that $kQ$ is an integral matrix.\n\\end{definition}\nLet \n\\begin{equation*}\nL(G)=\\lcm\\{\\ell(Q)\\colon\\,Q\\in \\mathcal{Q}(G)\\},\n\\end{equation*}the least common multiple of all levels  $\\ell(Q)$ for $Q\\in \\mathcal{Q}(G)$. Clearly, if $L(G)=1$ then $\\mathcal{Q}(G)$ contains only permutation matrices which implies that $G$ is DGS.   A theorem of Wang \\cite{wang2013EJC,wang2017JCTB} states that if the integer $2^{-\\lfloor\\frac{n}{2}\\rfloor}\\det W(G)$ is  odd and square-free, then   $L(G)=1$ and hence $G$ is DGS. Wang proves this theorem by excluding the possibilities of $p\\mid L(G)$ for odd primes $p$ and for $p=2$ separately.\n\\begin{proposition}[\\cite{wang2013EJC}]\\label{oddcase}\n\tIf $p$ is an odd prime with $p^2\\nmid \\det W(G)$ then $p\\nmid L(G)$.\n\\end{proposition}\n\\begin{proposition}[\\cite{wang2017JCTB}]\\label{evencase}\n\tIf $2^{-\\lfloor\\frac{n}{2}\\rfloor}$ is odd then $L(G)$ is odd.\n\\end{proposition}\nRecently, Qiu et al.~\\cite{qiu2023} improved both Proposition \\ref{oddcase} and Proposition \\ref{evencase} using a new and much  simpler argument. The current paper is a further improvement of the result in \\cite{qiu2023}, but only for the odd prime case.  The result of  Qiu et al.~\\cite{qiu2023} for the odd prime case is the following.\n\\begin{proposition}[\\cite{qiu2023}]\\label{qiuodd} If $p$ is an odd prime with $\\rank_p W(G)=n-1$ then $L(G)\\mid \\frac{\\det W(G)}{p}$.\n\\end{proposition}\nFor a prime $p$ and a nonzero integer $m$, we use $v_p(m)$ to denote the \\emph{$p$-adic valuation} \\cite{gouvea} of $m$, that is, the maximum nonnegative integer $k$ such that $p^k$ divides $m$. Note that the conclusion of Proposition  \\ref{qiuodd} can be rewritten as the inequality $v_p(L(G))\\le v_p(\\det W(G))-1$. The main aim of this paper is to give a better upper bound of $v_p(L(G))$ under the same assumption of Proposition \\ref{qiuodd}.", "context": "For a graph $G$, the \\emph{spectrum} of $G$ is the multiset of all eigenvalues of its adjacency matrix. Two graphs are \\emph{cospectral} if they share the same spectrum. We say two graphs $G$ and $H$ are \\emph{generalized cospectral} if (i) $G$ is cospectral with $H$ and (ii) $\\overline{G}$ is cospectral with $\\overline{H}$, where $\\overline{G}$ denotes the complement of $G$. A graph $G$ is \\emph{determined by its generalized spectrum} (DGS) if any graph generalized cospectral with $G$ is isomorphic to $G$. An orthogonal matrix $Q$ is \\emph{regular} if the sum of each row is 1. A classic result of Johnson and Newman \\cite{johnson} states that two graphs $G$ and $H$ are generalized cospectral if and only if their adjacency matrices are similar via a regular orthogonal matrix.\n\nFor an $n$-vertex graph $G$ with adjacency matrix $A$, the \\emph{walk matrix} of $G$ is \\begin{equation*}\nW(G):=[e,Ae,\\ldots,A^{n-1}e],\n\\end{equation*} where $e$ is the all-ones vector. A graph is \\emph{controllable} \\cite{godsil} if $W(G)$ is nonsingular. We remark that the set of controllable graphs is \\emph{closed} under generalized cospectrality, that is, any graph generalized cospectral with a controllable graph must be controllable. A key observation of Wang \\cite{wang2006} is that, when restricted to controllable graphs, the regular orthogonal matrix connecting the adjacency matrices of two generalized cospectral graphs $G$ and $H$ is unique and rational.\n\nWe are mainly concerned with controllable graphs in this paper. We use $\\RO_n(\\mathbb{Q})$ to denote the group of rational regular orthogonal matrices of order $n$. For a controllable graph $G$ with adjacency matrix $A$, we write\n\\begin{equation*}\n \\mathcal{Q}(G)=\\{Q\\in \\RO_n(\\mathbb{Q})\\colon\\, Q^\\mathsf{T} AQ \\text{~is a $(0,1)$-matrix} \\}.\n \\end{equation*}\nIt is easy to show that any $(0,1)$-matrix of the form $Q^\\mathsf{T} A Q$ must be the adjacency matrix of some graph (see \\cite[Remark 1]{qiu2023}). Let $\\mathcal{C}(G)$ be the set of all graphs that are generalized cospectral with $G$. As there exists a one-to-one correspondence between $\\mathcal{C}(G)$ and $\\mathcal{Q}(G)$, the problem of determining the set $\\mathcal{C}(G)$ can naturally be reduced to that of determining the set $\\mathcal{Q}(G)$. For example, $\\mathcal{Q}(G)$ always contains all permutation matrices of order $n$ which correspond to graphs that are isomorphic to $G$. Moreover, if $\\mathcal{Q}(G)$ contains a non-permutation matrix, then $\\mathcal{C}(G)$ contains a graph that is not isomorphic to $G$, i.e., $G$ is not DGS. The following notion introduced in \\cite{wang2006} is crucial to investigate the set $\\mathcal{Q}(G)$.\n\\begin{definition}[\\cite{wang2006}] \\normalfont\n Let $Q$ be a rational matrix. The \\emph{level} of $Q$, denoted by $\\ell(Q)$, is the\nsmallest positive integer $k$ such that $kQ$ is an integral matrix.\n\\end{definition}\nLet \n\\begin{equation*}\nL(G)=\\lcm\\{\\ell(Q)\\colon\\,Q\\in \\mathcal{Q}(G)\\},\n\\end{equation*}the least common multiple of all levels $\\ell(Q)$ for $Q\\in \\mathcal{Q}(G)$. Clearly, if $L(G)=1$ then $\\mathcal{Q}(G)$ contains only permutation matrices which implies that $G$ is DGS. A theorem of Wang \\cite{wang2013EJC,wang2017JCTB} states that if the integer $2^{-\\lfloor\\frac{n}{2}\\rfloor}\\det W(G)$ is odd and square-free, then $L(G)=1$ and hence $G$ is DGS. Wang proves this theorem by excluding the possibilities of $p\\mid L(G)$ for odd primes $p$ and for $p=2$ separately.\n\\begin{proposition}[\\cite{wang2013EJC}]\\label{oddcase}\n If $p$ is an odd prime with $p^2\\nmid \\det W(G)$ then $p\\nmid L(G)$.\n\\end{proposition}\n\\begin{proposition}[\\cite{wang2017JCTB}]\\label{evencase}\n If $2^{-\\lfloor\\frac{n}{2}\\rfloor}$ is odd then $L(G)$ is odd.\n\\end{proposition}\nRecently, Qiu et al.~\\cite{qiu2023} improved both Proposition \\ref{oddcase} and Proposition \\ref{evencase} using a new and much simpler argument. The current paper is a further improvement of the result in \\cite{qiu2023}, but only for the odd prime case. The result of Qiu et al.~\\cite{qiu2023} for the odd prime case is the following.\n\\begin{proposition}[\\cite{qiu2023}]\\label{qiuodd} If $p$ is an odd prime with $\\rank_p W(G)=n-1$ then $L(G)\\mid \\frac{\\det W(G)}{p}$.\n\\end{proposition}\nFor a prime $p$ and a nonzero integer $m$, we use $v_p(m)$ to denote the \\emph{$p$-adic valuation} \\cite{gouvea} of $m$, that is, the maximum nonnegative integer $k$ such that $p^k$ divides $m$. Note that the conclusion of Proposition \\ref{qiuodd} can be rewritten as the inequality $v_p(L(G))\\le v_p(\\det W(G))-1$. The main aim of this paper is to give a better upper bound of $v_p(L(G))$ under the same assumption of Proposition \\ref{qiuodd}.\n\n\\begin{proposition}[\\cite{wang2017JCTB}]\\label{evencase}\n\tIf $2^{-\\lfloor\\frac{n}{2}\\rfloor}$ is odd then $L(G)$ is odd.\n\\end{proposition}\n\n\\begin{proposition}[\\cite{wang2013EJC}]\\label{oddcase}\n\tIf $p$ is an odd prime with $p^2\\nmid \\det W(G)$ then $p\\nmid L(G)$.\n\\end{proposition}", "full_context": "For a graph $G$, the \\emph{spectrum} of $G$ is the multiset of all eigenvalues of its adjacency matrix. Two graphs are \\emph{cospectral} if they share the same spectrum. We say two graphs $G$ and $H$ are \\emph{generalized cospectral} if (i) $G$ is cospectral with $H$ and (ii) $\\overline{G}$ is cospectral with $\\overline{H}$, where $\\overline{G}$ denotes the complement of $G$. A graph $G$ is \\emph{determined by its generalized spectrum} (DGS) if any graph generalized cospectral with $G$ is isomorphic to $G$. An orthogonal matrix $Q$ is \\emph{regular} if the sum of each row is 1. A classic result of Johnson and Newman \\cite{johnson} states that two graphs $G$ and $H$ are generalized cospectral if and only if their adjacency matrices are similar via a regular orthogonal matrix.\n\nFor an $n$-vertex graph $G$ with adjacency matrix $A$, the \\emph{walk matrix} of $G$ is \\begin{equation*}\nW(G):=[e,Ae,\\ldots,A^{n-1}e],\n\\end{equation*} where $e$ is the all-ones vector. A graph is \\emph{controllable} \\cite{godsil} if $W(G)$ is nonsingular. We remark that the set of controllable graphs is \\emph{closed} under generalized cospectrality, that is, any graph generalized cospectral with a controllable graph must be controllable. A key observation of Wang \\cite{wang2006} is that, when restricted to controllable graphs, the regular orthogonal matrix connecting the adjacency matrices of two generalized cospectral graphs $G$ and $H$ is unique and rational.\n\nWe are mainly concerned with controllable graphs in this paper. We use $\\RO_n(\\mathbb{Q})$ to denote the group of rational regular orthogonal matrices of order $n$. For a controllable graph $G$ with adjacency matrix $A$, we write\n\\begin{equation*}\n \\mathcal{Q}(G)=\\{Q\\in \\RO_n(\\mathbb{Q})\\colon\\, Q^\\mathsf{T} AQ \\text{~is a $(0,1)$-matrix} \\}.\n \\end{equation*}\nIt is easy to show that any $(0,1)$-matrix of the form $Q^\\mathsf{T} A Q$ must be the adjacency matrix of some graph (see \\cite[Remark 1]{qiu2023}). Let $\\mathcal{C}(G)$ be the set of all graphs that are generalized cospectral with $G$. As there exists a one-to-one correspondence between $\\mathcal{C}(G)$ and $\\mathcal{Q}(G)$, the problem of determining the set $\\mathcal{C}(G)$ can naturally be reduced to that of determining the set $\\mathcal{Q}(G)$. For example, $\\mathcal{Q}(G)$ always contains all permutation matrices of order $n$ which correspond to graphs that are isomorphic to $G$. Moreover, if $\\mathcal{Q}(G)$ contains a non-permutation matrix, then $\\mathcal{C}(G)$ contains a graph that is not isomorphic to $G$, i.e., $G$ is not DGS. The following notion introduced in \\cite{wang2006} is crucial to investigate the set $\\mathcal{Q}(G)$.\n\\begin{definition}[\\cite{wang2006}] \\normalfont\n Let $Q$ be a rational matrix. The \\emph{level} of $Q$, denoted by $\\ell(Q)$, is the\nsmallest positive integer $k$ such that $kQ$ is an integral matrix.\n\\end{definition}\nLet \n\\begin{equation*}\nL(G)=\\lcm\\{\\ell(Q)\\colon\\,Q\\in \\mathcal{Q}(G)\\},\n\\end{equation*}the least common multiple of all levels $\\ell(Q)$ for $Q\\in \\mathcal{Q}(G)$. Clearly, if $L(G)=1$ then $\\mathcal{Q}(G)$ contains only permutation matrices which implies that $G$ is DGS. A theorem of Wang \\cite{wang2013EJC,wang2017JCTB} states that if the integer $2^{-\\lfloor\\frac{n}{2}\\rfloor}\\det W(G)$ is odd and square-free, then $L(G)=1$ and hence $G$ is DGS. Wang proves this theorem by excluding the possibilities of $p\\mid L(G)$ for odd primes $p$ and for $p=2$ separately.\n\\begin{proposition}[\\cite{wang2013EJC}]\\label{oddcase}\n If $p$ is an odd prime with $p^2\\nmid \\det W(G)$ then $p\\nmid L(G)$.\n\\end{proposition}\n\\begin{proposition}[\\cite{wang2017JCTB}]\\label{evencase}\n If $2^{-\\lfloor\\frac{n}{2}\\rfloor}$ is odd then $L(G)$ is odd.\n\\end{proposition}\nRecently, Qiu et al.~\\cite{qiu2023} improved both Proposition \\ref{oddcase} and Proposition \\ref{evencase} using a new and much simpler argument. The current paper is a further improvement of the result in \\cite{qiu2023}, but only for the odd prime case. The result of Qiu et al.~\\cite{qiu2023} for the odd prime case is the following.\n\\begin{proposition}[\\cite{qiu2023}]\\label{qiuodd} If $p$ is an odd prime with $\\rank_p W(G)=n-1$ then $L(G)\\mid \\frac{\\det W(G)}{p}$.\n\\end{proposition}\nFor a prime $p$ and a nonzero integer $m$, we use $v_p(m)$ to denote the \\emph{$p$-adic valuation} \\cite{gouvea} of $m$, that is, the maximum nonnegative integer $k$ such that $p^k$ divides $m$. Note that the conclusion of Proposition \\ref{qiuodd} can be rewritten as the inequality $v_p(L(G))\\le v_p(\\det W(G))-1$. The main aim of this paper is to give a better upper bound of $v_p(L(G))$ under the same assumption of Proposition \\ref{qiuodd}.\n\n\\begin{proposition}[\\cite{wang2017JCTB}]\\label{evencase}\n\tIf $2^{-\\lfloor\\frac{n}{2}\\rfloor}$ is odd then $L(G)$ is odd.\n\\end{proposition}\n\n\\begin{proposition}[\\cite{wang2013EJC}]\\label{oddcase}\n\tIf $p$ is an odd prime with $p^2\\nmid \\det W(G)$ then $p\\nmid L(G)$.\n\\end{proposition}\n\nFor an $n$-vertex graph $G$ with adjacency matrix $A$, the \\emph{walk matrix} of $G$ is \\begin{equation*}\nW(G):=[e,Ae,\\ldots,A^{n-1}e],\n\\end{equation*} where $e$ is the all-ones vector. A graph is \\emph{controllable} \\cite{godsil} if $W(G)$ is nonsingular. We remark that the set of controllable graphs is \\emph{closed} under generalized cospectrality, that is, any graph generalized cospectral with a controllable graph must be controllable. A key observation of Wang \\cite{wang2006} is that, when restricted to controllable graphs, the regular orthogonal matrix connecting the adjacency matrices of two generalized cospectral graphs $G$ and $H$ is unique and rational.\n\nThe rest of this paper is organized as follows. In Sec.~\\ref{pre}, we recall some preliminaries on Smith normal forms and modules over $\\mathbb{Z}/p^k \\mathbb{Z}$. The proof of Theorem \\ref{main} is given in Sec.~\\ref{pf}, which can be seen as a module-theoretical version of the original argument of Wang \\cite{wang2006}, combined with some improvements and simplifications developed in \\cite{wang2020}. Some direct applications of Theorem \\ref{main} are presented in Sec.~\\ref{dis}. The paper concludes with a conjecture proposing a further possible improvement to Theorem \\ref{main}.\n\\section{Preliminaries}\n\\label{pre}\nLet $R$ be a commutative principal ideal ring with identity. It is well known \\cite{stanley2016JCTA} that any matrix $M$ over $R$ has a Smith normal form (SNF); that is, there exist invertible matrices $U$ and $V$ over $R$ such that \n\\begin{equation*}\n UMV=\\begin{bmatrix}\n \\diag(d_1,\\ldots,d_r)&0\\\\\n 0&0\n \\end{bmatrix},\n\\end{equation*}\nwhere the elements $d_i$ ($i=1,\\ldots,r$) are nonzero and satisfy $d_1\\mid d_2\\mid \\cdots\\mid d_r$. The nonzero elements $d_1,\\ldots,d_r$ are called the \\emph{invariant factors} of $M$.\n\nIn the following, let $G$ be an $n$-vertex controllable graph and $p$ be an odd prime factor of $\\det W(G)$ such that $\\rank_p W(G)=n-1$. Denote $\\tau=v_p(L(G))$, i.e., $\\tau=\\max\\{v_p (\\ell(Q))\\colon\\, Q\\in \\mathcal{Q}(G)\\}$. Note that if $\\tau=0$, then Theorem \\ref{main} clearly holds. Thus, we may assume that $\\tau\\ge 1$. For simplicity, the adjacency matrix $A(G)$ and the walk matrix $W(G)$ will be denoted by $A$ and $W$, respectively, when there is no confusion.\n\n\\noindent\\textbf{Example 1}. Let $n=10$ and $G$ be the $n$-vertex graph with adjacency matrix \n\\begin{equation*}\n A={\\footnotesize\\begin{bmatrix}\n 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 \\\\\n 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 \\\\\n 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 \\\\\n 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\\\\n 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\\\\n 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 \\\\\n 0 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 \\\\\n 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\\\\n 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\\\\n \\end{bmatrix}}.\n\\end{equation*} \nIt can be verified that $2^{-\\lfloor\\frac{n}{2}\\rfloor} \\det W=3^4\\times 19$ and $\\rank_3 W=n-1$. By Theorem \\ref{main}, we know that $v_3(L(G))\\le \\frac{1}{2} v_3(\\det W)=2$. Indeed, using the program in \\cite{wang2025}, we find that, besides the permutation matrices, there exist essentially two rational regular orthogonal matrices \n\\begin{equation*}\n Q_1=\\frac{1}{3}{\\footnotesize\n \\begin{bmatrix}\n 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & -2 & 1 & 1 & 1 & 1 & 1 \\\\\n 0 & 0 & 0 & 0 & 1 & -2 & 1 & 1 & 1 & 1 \\\\\n 0 & 0 & 0 & 0 & 1 & 1 & -2 & 1 & 1 & 1 \\\\\n 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 1 & 1 & 1 & -2 & 1 & 1 \\\\\n 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & -2 & 1 \\\\\n 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & -2 \\\\\n 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n \\end{bmatrix}}\n\\end{equation*}\nand \n\\begin{equation*}\nQ_2=\\frac{1}{9}{\\footnotesize\\begin{bmatrix}\n -3 & -3 & 3 & 3 & 3 & 0 & 0 & 0 & 0 & 6 \\\\\n 3 & 3 & 6 & -3 & -3 & 0 & 0 & 0 & 0 & 3 \\\\\n 1 & 1 & 2 & 2 & 2 & -6 & 3 & 3 & 3 & -2 \\\\\n 1 & 1 & 2 & 2 & 2 & 3 & -6 & 3 & 3 & -2 \\\\\n -5 & 4 & -1 & -1 & -1 & 3 & 3 & 3 & 3 & 1 \\\\\n 3 & 3 & -3 & 6 & -3 & 0 & 0 & 0 & 0 & 3 \\\\\n 1 & 1 & 2 & 2 & 2 & 3 & 3 & -6 & 3 & -2 \\\\\n 1 & 1 & 2 & 2 & 2 & 3 & 3 & 3 & -6 & -2 \\\\\n 4 & -5 & -1 & -1 & -1 & 3 & 3 & 3 & 3 & 1 \\\\\n 3 & 3 & -3 & -3 & 6 & 0 & 0 & 0 & 0 & 3 \\\\\n\\end{bmatrix}}\n\\end{equation*}\nsuch that both $Q_1^\\mathsf{T} A Q_1$ and $Q_2^\\mathsf{T} A Q_2$ are adjacency matrices. Thus, $G$ has two generalized cospectral mates.\n\nRecently, as a significant improvement of the criterion for a graph to have at most one cospectral mate established in \\cite{wang2023ejc}, Raza et al. \\cite{raza2026} obtained the following upper bound on the number of generalized cospectral mates of a graph.\n\\begin{theorem} [\\cite{raza2026}]\\label{ngm}\n Let $G$ be a controllable graph and $d_1,\\ldots, d_n$ be the invariant factors of the walk matrix $W(G)$. Suppose $d_{\\lceil\\frac{n}{2}\\rceil}=1$ and $d_{n-1}=2$. Let $d_n=\\prod_{i=1}^k p_i^{m_i}$ be the prime factorization of $d_n$. Then $G$ has at most $\\left(\\prod_{i=1}^{k}m_i\\right)-1$ generalized cospectral mates.\n \\end{theorem}\nUsing the result of Qiu et al. \\cite{qiu2023}, it is known that, for graphs $G$ as described in Theorem \\ref{ngm}, the number $L(G)$ must be a factor of $\\prod_{i=1}^k p_i^{m_i-1}$. Thus, $\\ell(Q)$ can take at most $\\prod_{i=1}^k m_i$ different values (including 1) when $Q$ runs through $\\mathcal{Q}(G)$. A key contribution of Raza et al.~\\cite{raza2026} is to show that, under the assumptions of Theorem \\ref{ngm}, there exists at most one matrix $Q$ in $\\mathcal{Q}(G)$ for each possible level $\\ell$ (in the sense of column permutations). Noting that the matrix $Q\\in \\mathcal{Q}(G)$ with level $\\ell=1$ is the permutation matrix, the number of generalized cospectral mates of $G$ is upper bounded by the number of possible levels, excluding 1. Thus, Theorem \\ref{ngm} holds.\n\nUsing Theorem \\ref{main} for odd primes, we can obtain a better bound on $L(G)$. Indeed, let $d_n=2^{m_1}\\prod_{i=2}^kp_i^{m_i}$ be the factorization of $d_n$, where $p_2,\\ldots,p_k$ are distinct odd primes. Then for each odd prime $p_i$, we have $v_{p_i}(L(G)) \\le \\lfloor\\frac{1}{2}{m_i}\\rfloor$ by Theorem \\ref{main}. For $p=2$, we only have $v_2(L(G))\\le m_1-1$ by the result of Qiu et al.~\\cite{qiu2023}. Thus, the number of possible levels for matrices in $\\mathcal{Q}(G)$, other than 1, is at most $m_1\\left(\\prod_{i=2}^k\\left( \\lfloor\\frac{1}{2}m_i\\rfloor+1\\right)\\right)-1$. This gives an improvement of Theorem \\ref{ngm} under the same assumptions. We state it as the following theorem.\n\\begin{theorem}\n Let $G$ be a controllable graph and $d_1,\\ldots, d_n$ be the invariant factors of the walk matrix $W(G)$. Suppose $d_{\\lceil\\frac{n}{2}\\rceil}=1$ and $d_{n-1}=2$. Let $d_n=2^{m_1}\\prod_{i=2}^k p_i^{m_i}$ be the prime factorization of $d_n$, where $p_2,\\ldots,p_k$ are distinct odd primes. Then $G$ has at most $m_1\\left(\\prod_{i=2}^k\\left( \\lfloor\\frac{1}{2}m_i\\rfloor+1\\right)\\right)-1$ generalized cospectral mates.\n\\end{theorem}\nWe end this paper by proposing a conjecture that the technical assumption on $\\rank_p W(G)$ in Theorem \\ref{main} is unnecessary for the conclusion. Moreover, the upper bound in terms of the determinant may be refined using the last two invariant factors.\n \\begin{conjecture}\n Let $G$ be a controllable graph and $p$ be an odd prime factor of $\\det W(G)$. Let $d_{n-1}$ and $d_n$ be the last two invariant factors of $W(G)$. Then $v_p(L(G))\\le \\frac{1}{2}( v_p(d_{n})+v_p(d_{n-1}))$, and in particular, $v_p(L(G))\\le \\frac{1}{2} v_p(\\det W(G))$.\n \\end{conjecture}", "post_theorem_intro_text_len": 1164, "post_theorem_intro_text": "The main innovation of this paper lies in the shift of the underlying algebraic structure from the vector space over the field $\\mathbb{Z}/p\\mathbb{Z}$ (as used in \\cite{wang2013EJC,qiu2023}) to the module over the local ring $\\mathbb{Z}/p^k\\mathbb{Z}$. While working over a ring introduces the complexity of zero divisors, many basic results from linear algebra over fields still hold for the ring $\\mathbb{Z}/p^k\\mathbb{Z}$. In particular, it possesses the ``Basis Extension Property'' (Steinitz Property), which allows us to generalize arguments from linear algebra. \n\n\tThe rest of this paper is organized as follows.  In Sec.~\\ref{pre}, we recall some  preliminaries on Smith normal forms and modules over $\\mathbb{Z}/p^k \\mathbb{Z}$.  The proof of Theorem \\ref{main} is given in Sec.~\\ref{pf}, which can be seen as a module-theoretical version of the original argument of Wang \\cite{wang2006}, combined with some improvements and simplifications developed in \\cite{wang2020}.   Some direct applications of Theorem \\ref{main} are presented in Sec.~\\ref{dis}. The paper concludes with a conjecture proposing a further possible improvement to Theorem \\ref{main}.", "sketch": "The proof of Theorem~\\ref{main} is said to be \"a module-theoretical version of the original argument of Wang \\cite{wang2006}, combined with some improvements and simplifications developed in \\cite{wang2020}.\" The paper’s approach shifts \"from the vector space over the field $\\mathbb{Z}/p\\mathbb{Z}$\" to \"the module over the local ring $\\mathbb{Z}/p^k\\mathbb{Z}$\" and uses that this ring has the \"Basis Extension Property'' (Steinitz Property), which \"allows us to generalize arguments from linear algebra.\"", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{main}\n\tLet $G$ be a controllable graph and $p$ be an odd prime such that  $\\rank_p W(G)=n-1$. Then $v_p(L(G))\\le \\frac{1}{2} v_p(\\det W(G))$.", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $G$ be an $n$-vertex controllable graph with adjacency matrix $A$ and walk matrix\n\\[\nW(G)=[e,Ae,\\ldots,A^{n-1}e],\n\\]\nwhere $e$ is the all-ones vector and controllable means that $W(G)$ is nonsingular. Define\n\\[\n\\mathcal{Q}(G)=\\{Q\\in \\mathrm{RO}_n(\\mathbb{Q}) : Q^\\mathsf{T}AQ \\text{ is a }(0,1)\\text{-matrix}\\},\n\\]\nwhere $\\mathrm{RO}_n(\\mathbb{Q})$ is the set of rational regular orthogonal matrices (orthogonal matrices whose row sums are all $1$). For a rational matrix $Q$, its level $\\ell(Q)$ is the smallest positive integer $k$ such that $kQ$ is integral, and\n\\[\nL(G)=\\operatorname{lcm}\\{\\ell(Q):Q\\in \\mathcal{Q}(G)\\}.\n\\]\nFor a prime $p$, let $v_p(m)$ denote the $p$-adic valuation of a nonzero integer $m$, and let $\\operatorname{rank}_p W(G)$ denote the rank of $W(G)$ over $\\mathbb{F}_p$. If $p$ is an odd prime such that $\\operatorname{rank}_p W(G)=n-1$, which quantitative estimate holds?", "correct_choice": {"label": "A", "text": "\\[v_p(L(G))\\le \\frac{1}{2}\\,v_p(\\det W(G)).\\]"}, "choices": [{"label": "B", "text": "\\[v_p(L(G))\\le v_p(\\det W(G))-1.\\]"}, {"label": "C", "text": "\\[v_p(L(G))\\le v_p(\\det W(G)).\\]"}, {"label": "D", "text": "\\[v_p(L(G))\\le \\frac{1}{2}\\,v_p(\\det W(G))-1.\\]"}, {"label": "E", "text": "If \\(p\\) is an odd prime such that \\(\\operatorname{rank}_p W(G)=n-1\\), then \\(p\\nmid L(G)\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "half-valuation bound replaced by earlier linear-minus-one bound", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "characteristic", "tampered_component": "dropped the factor \\(\\frac12\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "characteristic", "tampered_component": "inserted an extra subtractive constant into the valuation estimate", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "collapsed rank condition to the stronger conclusion \\(p\\nmid L(G)\\)", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the bound on v_p(L(G)); it only gives definitions and hypotheses. There is no direct textual leakage of the correct choice."}, "TAS": {"score": 1, "justification": "The question is very close to a theorem-recall item: under the stated hypotheses, the correct option is essentially the theorem's exact conclusion. However, it is not completely tautological because the options include nearby variants and over/under-strengthenings."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact theorem statement from a weaker true statement, a stricter false one, and a hypothesis-broadening variant. Still, the item mainly tests precise recall of the result rather than deep generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: one is weaker but true, others reflect common failure modes such as overgeneralizing the prime condition or strengthening a non-strict inequality to a strict one."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it primarily tests precise recall of a known result rather than substantial generative reasoning."}}
{"id": "2602.14213v1", "paper_link": "http://arxiv.org/abs/2602.14213v1", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "\\label{main}\n\tLet $G$ be a controllable graph and $p$ be an odd prime such that  $\\rank_p W(G)=n-1$. Then $v_p(L(G))\\le \\frac{1}{2} v_p(\\det W(G))$.", "start_pos": 7243, "end_pos": 7420, "label": "main"}, "ref_dict": {"pf": "\\label{pf}\nWe begin with the fundamental matrix characterization of generalized cospectrality for graphs.\n\n\\begin{lemma}[\\cite{johnson,wang2006}]\n\tLet $G$ and $H$ be two graphs of the same order. Then", "evencase": "\\begin{proposition}[\\cite{wang2017JCTB}]\\label{evencase}\n\tIf $2^{-\\lfloor\\frac{n}{2}\\rfloor}$ is odd then $L(G)$ is odd.\n\\end{proposition}", "pre": "\\begin{theorem}\\label{main}\n\tLet $G$ be a controllable graph and $p$ be an odd prime such that  $\\rank_p W(G)=n-1$. Then $v_p(L(G))\\le \\frac{1}{2} v_p(\\det W(G))$.\n\\end{theorem}\n\tThe main innovation of this paper lies in the shift of the underlying algebraic structure from the vector space over the field $\\mathbb{Z}/p\\mathbb{Z}$ (as used in \\cite{wang2013EJC,qiu2023}) to the module over the local ring $\\mathbb{Z}/p^k\\mathbb{Z}$. While working over a ring introduces the complexity of zero divisors, many basic results from linear algebra over fields still hold for the ring $\\mathbb{Z}/p^k\\mathbb{Z}$. In particular, it possesses the ``Basis Extension Property'' (Steinitz Property), which allows us to generalize arguments from linear algebra. \n\n\tThe rest of this paper is organized as follows.  In Sec.~\\ref{pre}, we recall some  preliminaries on Smith normal forms and modules over $\\mathbb{Z}/p^k \\mathbb{Z}$.  The proof of Theorem \\ref{main} is given in Sec.~\\ref{pf}, which can be seen as a module-theoretical version of the original argument of Wang \\cite{wang2006}, combined with some improvements and simplifications developed in \\cite{wang2020}.   Some direct applications of Theorem \\ref{main} are presented in Sec.~\\ref{dis}. The paper concludes with a conjecture proposing a further possible improvement to Theorem \\ref{main}.\n\\section{Preliminaries}\n\\label{pre}\nLet $R$ be a commutative principal ideal ring with identity.  It is well known \\cite{stanley2016JCTA} that any matrix $M$ over $R$ has a Smith normal form (SNF); that is, there exist invertible matrices $U$ and $V$ over $R$ such that \n\\begin{equation*}\n\tUMV=\\begin{bmatrix}\n\t\t\\diag(d_1,\\ldots,d_r)&0\\\\\n\t\t0&0\n\t\\end{bmatrix},\n\\end{equation*}\nwhere the elements $d_i$ ($i=1,\\ldots,r$) are nonzero and satisfy $d_1\\mid d_2\\mid \\cdots\\mid d_r$. The nonzero elements $d_1,\\ldots,d_r$ are called the \\emph{invariant factors} of $M$.\n\nThroughout this paper, we fix an odd prime $p$ and consider three kinds of rings: $\\mathbb{Z}$, $\\mathbb{Z}/p\\mathbb{Z}$ and $\\mathbb{Z}/p^k\\mathbb{Z}$, where $k\\ge 2$. Let $\\mathbb{Z} \\to \\mathbb{Z}/p^k \\mathbb{Z} \\to \\mathbb{Z}/p\\mathbb{Z}$ be the natural projections. For a matrix $M$ over $\\mathbb{Z}$, the SNFs of the projections of $M$ are determined naturally by the SNF of $M$ over $\\mathbb{Z}$ via $p$-adic valuations. For example, suppose $M$ has the Smith normal form $\\diag(2,10,30,270)$ over $\\mathbb{Z}$. Then, over the field $\\mathbb{Z}/3\\mathbb{Z}$, the SNF of $M$ is $\\diag(1,1,0,0)$, whereas over the local ring $\\mathbb{Z}/3^2\\mathbb{Z}$, its SNF is $\\diag(1,1,3,0)$. \n\nIn this paper, we are mainly concerned with the SNFs of matrices over the local ring $\\mathbb{Z}/p^k \\mathbb{Z}$. Note that the invariant factors are unique only up to multiplication by units. To ensure uniqueness, we conventionally assume that each nonzero invariant factor takes the form $p^{c}$ for some integer $0 \\le c < k$. We formalize this in the following lemma.\n\n\\begin{lemma}\n\tFor any matrix $M$ over $\\mathbb{Z}/p^k\\mathbb{Z}$, there exist invertible matrices $U$ and $V$ over $\\mathbb{Z}/p^k\\mathbb{Z}$ such that \n\t\\begin{equation*}\n\t\tUMV = \\begin{bmatrix}\n\t\t\t\\diag(p^{c_1}, \\ldots, p^{c_r})&0\\\\\n\t\t\t0&0\n\t\t\\end{bmatrix},\n\t\\end{equation*}\n\twhere $0 \\le c_1 \\le \\cdots \\le c_r < k$.\n\\end{lemma}\nA classic result in linear algebra states that the linear system $Mx=b$ over a field has a solution if and only if $\\rank M = \\rank (M,b)$. The corresponding question over an arbitrary ring is generally more complicated, but for our purposes, the existence of the SNF gives a straightforward extension.\n\n\\begin{proposition}\\label{ssnf}\n\tThe linear system $Mx=b$ over $\\mathbb{Z}/p^k\\mathbb{Z}$ has a solution if and only if $M$ and $(M,b)$ have the same invariant factors.\n\\end{proposition} \n\n\\begin{proof}\n\tThe ``only if'' part is clear. Let $\\mathcal{C}(M)$ and $\\mathcal{C}(M,b)$ be the modules generated by the columns of $M$ and $(M,b)$, respectively. Since $M$ and $(M,b)$ have the same invariant factors, the two modules $\\mathcal{C}(M)$ and $\\mathcal{C}(M,b)$ are isomorphic. As $\\mathcal{C}(M,b)$ is finite, it cannot be isomorphic to any proper submodule. Since $\\mathcal{C}(M)$ is a submodule of $\\mathcal{C}(M,b)$, the isomorphism $\\mathcal{C}(M)\\cong \\mathcal{C}(M,b)$ implies that they must be equal. Thus, $b\\in\\mathcal{C}(M)$, i.e., $Mx=b$ has a solution. \n\\end{proof}\nThe SNF is particularly useful for determining the structure of the solution space (kernel) of a linear system. It is important to note that the kernel is not always a free module. However, under specific conditions on the invariant factors, it is.\n\n\\begin{proposition}\\label{kf}\n\tLet $M$ be an $m \\times n$ matrix over the ring $\\mathbb{Z}/p^k\\mathbb{Z}$. Let the invariant factors of $M$ be $p^{c_1}, \\ldots, p^{c_r}$ satisfying $0 \\le c_1 \\le \\dots \\le c_r < k$. Then the kernel of $M$ is given, up to isomorphism, by:\n\t\\begin{equation*}\n\t\t\\ker(M) \\cong \\bigoplus_{i=1}^r \\left( \\mathbb{Z}/p^{c_i}\\mathbb{Z} \\right) \\oplus \\left( \\mathbb{Z}/p^k \\mathbb{Z} \\right)^{n-r}.\n\t\\end{equation*}\n\tIn particular, if $c_1=\\cdots=c_r=0$, then $\\ker(M)\\cong (\\mathbb{Z}/p^k\\mathbb{Z})^{n-r}$, which is a free module of rank $n-r$.\n\\end{proposition}\n\nThe following proposition is a direct generalization of \\cite[Lemma 7]{wang2013EJC}, where the case $m=n,k=2$ was considered. The original proof in \\cite{wang2013EJC} remains valid for the general setting and is therefore omitted here.\n\n\\begin{proposition}[\\cite{wang2013EJC}]\\label{dnk}\n\tLet $M$ be an $m\\times n~(m\\ge n)$ integral matrix whose SNF is \n\t\\begin{equation*}\n\t\t\\begin{bmatrix}\n\t\t\\diag(d_1,\\ldots,d_n)\\\\\n\t\t0_{(m-n)\\times n}\n\t\t\\end{bmatrix}.\n\t\\end{equation*} Then the equation $Mz\\equiv 0\\pmod{p^{k}}$ has a solution $z\\not\\equiv 0\\pmod{p}$ if and only if $p^{k}\\mid d_n.$\n\\end{proposition}\n\nA set of vectors $v_1, \\ldots, v_m \\in (\\mathbb{Z}/p^k\\mathbb{Z})^n$ is called \\emph{linearly independent} if $\\sum_{i=1}^m c_i v_i = 0$ implies $c_i = 0$ for all $i$. It is known that $v_1,\\ldots, v_m$ are linearly independent if and only if their projections in the vector space $(\\mathbb{Z}/p\\mathbb{Z})^n$ are linearly independent (over the field $\\mathbb{Z}/p\\mathbb{Z}$). A key property of the ring $\\mathbb{Z}/p^k\\mathbb{Z}$ is the so-called Basis Extension Property (or Steinitz Property \\cite{mcdonald}), which states that any linearly independent vectors in a free $\\mathbb{Z}/p^k\\mathbb{Z}$-module $M$ can always be extended to a basis of $M$. Since we only require the finite-rank case, we formalize it in the following proposition.\n\n\\begin{proposition}\\label{bas}\n\tLet $M$ be a free submodule of rank $m$ in $(\\mathbb{Z}/p^k\\mathbb{Z})^n$. Let $v_1, \\ldots, v_k \\in M$ ($k\\le m$) be linearly independent. Then:\n\n\t\\textup{(i)} If $k=m$, the set $\\{v_1, \\ldots, v_k\\}$ constitutes a basis of $M$.\n\n\t\\textup{(ii)}  If $k< m$, there exist $m-k$ vectors $v_{k+1}, \\ldots, v_{m} \\in M$ such that $\\{v_1,\\ldots,v_m\\}$ constitutes a basis of $M$.\n\\end{proposition}\n\\section{Proof of Theorem \\ref{main}}\\label{pf}\nWe begin with the fundamental matrix characterization of generalized cospectrality for graphs.\n\n\\begin{lemma}[\\cite{johnson,wang2006}]\n\tLet $G$ and $H$ be two graphs of the same order. Then $G$ and $H$ are generalized cospectral if and only if there exists a regular orthogonal matrix $Q$ such that $Q^\\mathsf{T} A(G) Q=A(H)$. Moreover, if $G$ is controllable, then $Q^\\mathsf{T}=W(H)(W(G))^{-1}$, and hence $Q$ is unique and rational.\n\\end{lemma}\n\nIn the following, let $G$ be an $n$-vertex controllable graph and $p$ be an odd prime factor of $\\det W(G)$ such that $\\rank_p W(G)=n-1$. Denote $\\tau=v_p(L(G))$, i.e., $\\tau=\\max\\{v_p (\\ell(Q))\\colon\\, Q\\in \\mathcal{Q}(G)\\}$. Note that if $\\tau=0$, then Theorem \\ref{main} clearly holds. Thus, we may assume that $\\tau\\ge 1$. For simplicity, the adjacency matrix $A(G)$ and the walk matrix $W(G)$ will be denoted by $A$ and $W$, respectively, when there is no confusion.\n\n\\begin{lemma}\\label{fourcong}\n\tThere exist an integral vector $z_0$ with $z_0\\not\\equiv 0\\pmod{p}$ and an integer $\\lambda_0$ such that $z_0^\\mathsf{T} z_0\\equiv 0\\pmod{p^{2\\tau}}$, $z_0^\\mathsf{T} A z_0\\equiv 0\\pmod{p^{2\\tau}}$, $W^\\mathsf{T} z_0\\equiv 0\\pmod{p^\\tau}$, and $Az_0\\equiv \\lambda_0z_0\\pmod{p^\\tau}$.\n\\end{lemma}\n\n\\begin{proof}\n\tLet $Q\\in \\mathcal{Q}(G)$ be such that $v_p(\\ell(Q))=\\tau$, and let $H$ be the corresponding graph. Let $\\ell=\\ell(Q)$ and $\\hat{Q}=\\ell Q$. Then $\\hat{Q}$ is an integral matrix and $\\hat{Q}\\not\\equiv 0\\pmod{p}$. Let $z_0$ be a column of $\\hat{Q}$ such that $z_0\\not\\equiv 0\\pmod{p}$. Since $Q$ is an orthogonal matrix and $p^\\tau\\mid \\ell$, we have $\\hat{Q}^\\mathsf{T} \\hat{Q}=\\ell^2 I$, and hence $z_0^\\mathsf{T} z_0\\equiv 0\\pmod{p^{2\\tau}}$. Similarly, as $Q^\\mathsf{T} A Q$ is a $(0,1)$-matrix, we have $\\hat{Q}^\\mathsf{T} A\\hat{Q}\\equiv 0\\pmod{p^{2\\tau}}$, and hence $z_0^\\mathsf{T} A z_0\\equiv 0\\pmod{p^{2\\tau}}$. Moreover, noting that $W^\\mathsf{T} Q$ equals $(W(H))^\\mathsf{T}$, which is an integral matrix, we find that $W^\\mathsf{T} \\hat{Q}\\equiv 0\\pmod{\\ell}$, and hence $W^\\mathsf{T} z_0\\equiv 0\\pmod{p^\\tau}$.\n\n\tLet $M=\\{z\\in \\mathbb{Z}^n\\colon\\, W^\\mathsf{T} z\\equiv 0\\pmod {p^\\tau}\\}$. Since $\\rank_p W=n-1$ and $p^\\tau \\mid \\det W$, we see that, over $\\mathbb{Z}/p^\\tau \\mathbb{Z}$, the SNF of $W^\\mathsf{T}$ is $\\diag(1,1,\\ldots,1,0)$. It follows from Proposition \\ref{kf} that $M$ is a free $\\mathbb{Z}/p^\\tau \\mathbb{Z}$-module of rank one. Since $z_0\\in M$ and $z_0\\not\\equiv 0\\pmod{p}$, we see that $\\{z_0\\}$ forms a basis for $M$ over $\\mathbb{Z}/p^\\tau \\mathbb{Z}$ by Proposition \\ref{bas} (i). On the other hand, by the Cayley-Hamilton Theorem, it is easy to see that $M$ is an $A$-invariant $\\mathbb{Z}/p^\\tau \\mathbb{Z}$-submodule. This implies that there exists an integer $\\lambda_0$ such that $Az_0\\equiv \\lambda_0 z_0\\pmod{p^\\tau}$. This completes the proof.\t\n\\end{proof}\n\\begin{lemma}\\label{wy}\n\tIf $(A-\\lambda_0 I)y\\equiv s p^j z_0\\pmod{p^{j+\\tau}}$ for some integer $s$ and integer $j\\ge 0$, then\n\t\\begin{equation*}\n\t\tW^\\mathsf{T} y\\equiv (e^\\mathsf{T} y)(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T} \\pmod{p^{j+\\tau}}.\n\t\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\n\tWe proceed by induction on $k$ to show that $e^\\mathsf{T} A^k y\\equiv \\lambda_0^k e^\\mathsf{T} y\\pmod{p^{j+\\tau}}$ for $k=0,1,\\ldots,n-1$. The base case $k=0$ is trivial. Assume that the congruence holds for some integer $k$ with $0 \\le k < n-1$; we proceed to verify it for $k+1$. \n\n\tBy Lemma \\ref{fourcong}, we have $W^\\mathsf{T} z_0\\equiv 0\\pmod{p^\\tau}$, which implies that $e^\\mathsf{T} A^k z_0\\equiv 0\\pmod{p^\\tau}$. Consequently, we obtain $e^\\mathsf{T} A^k (sp^j z_0)\\equiv 0\\pmod{p^{j+\\tau}}$. By the hypothesis of this lemma, $Ay \\equiv  sp^j z_0+\\lambda_0 y \\pmod{p^{j+\\tau}}$, which implies that\n\t\\[\n\te^\\mathsf{T} A^{k+1}y \\equiv e^\\mathsf{T} A^k (sp^j z_0+\\lambda_0 y) \\equiv \\lambda_0 e^\\mathsf{T} A^k y \\equiv \\lambda_0^{k+1}e^\\mathsf{T} y \\pmod{p^{j+\\tau}}.\n\t\\]\n\tThis completes the proof.\t\n\\end{proof}\n\\begin{lemma}\\label{rA}\n\tLet $S=\\diag(f_1,\\ldots,f_n)$ be the SNF of $A-\\lambda_0 I$. Then $f_{n-2}\\not \\equiv 0\\pmod{p}$ and $f_n\\equiv 0\\pmod{p^\\tau}$.\n\\end{lemma}\n\n\\begin{proof}\n\tBy Lemma \\ref{fourcong}, we have $(A-\\lambda_0 I)z_0\\equiv 0\\pmod{p^\\tau}$ and $z_0\\not\\equiv 0\\pmod{p}$. It follows from Proposition \\ref{dnk} that $f_n\\equiv 0\\pmod{p^\\tau}$. \tIt remains to show that $f_{n-2}\\not \\equiv 0\\pmod{p}$. Suppose to the contrary that $f_{n-2}\\equiv 0\\pmod{p}$. Then $\\rank_p (A-\\lambda_0 I)\\le n-3$. \n\tLet \\begin{equation*}\n\tB=\\begin{bmatrix} A-\\lambda_0 I\\\\e^\\mathsf{T}\\end{bmatrix}.\\end{equation*}\n\tThen, we have $\\rank_p B\\le n-2$. Recall from Lemma \\ref{fourcong} that $W^\\mathsf{T} z_0 \\equiv 0 \\pmod{p}$, which implies $e^\\mathsf{T} z_0 \\equiv 0 \\pmod{p}$. Together with $(A-\\lambda_0 I)z_0 \\equiv 0 \\pmod{p}$, this yields $Bz_0 \\equiv 0 \\pmod{p}$. Since the nullity of $B$ over $\\mathbb{Z}/p\\mathbb{Z}$ is at least $n - (n-2) = 2$, the equation $Bz\\equiv 0\\pmod {p}$ must have a solution $z_1$ such that $z_0$ and $z_1$ are linearly independent over $\\mathbb{Z}/p\\mathbb{Z}$. \n\n\tNoting that $Az_1\\equiv \\lambda_0 z_1 \\pmod{p}$ and $e^\\mathsf{T} z_1\\equiv 0\\pmod{p}$, we clearly have \n\t\\[\n\te^\\mathsf{T} A^{k}z_1\\equiv \\lambda_0^k e^\\mathsf{T} z_1\\equiv 0\\pmod{p}\n\t\\]\n\tfor $k=0,1,\\ldots, n-1$. This means that $W^\\mathsf{T} z_1\\equiv 0\\pmod{p}$. By Lemma \\ref{fourcong}, we also have $W^\\mathsf{T} z_0\\equiv 0\\pmod{p}$. Therefore, $\\rank_p W^\\mathsf{T} \\le n-2$. This contradicts the assumption that $\\rank_p W = n-1$, which completes the proof.\n\\end{proof}\n\\begin{lemma} \\label{lc}\n\t\tLet $M = [A-\\lambda_0 I, z_0]$. Then $z_0^\\mathsf{T} M \\equiv 0 \\pmod{p^\\tau}$ and the SNF of $M$ is $[\\diag(I_{n-1},0),0]$. Moreover, any integral  vector $z$ satisfying $z_0^\\mathsf{T} z \\equiv 0 \\pmod{p^\\tau}$ can be expressed as a linear combination of the columns of $M$ over $\\mathbb{Z}/p^\\tau \\mathbb{Z}$.\n\t\\end{lemma}\n\t\\begin{proof} By Lemma \\ref{fourcong}, we have $(A-\\lambda_0 I)z_0\\equiv 0\\pmod{p^\\tau}$ and $z_0^\\mathsf{T} z_0\\equiv 0\\pmod{p^\\tau}$. As  $A-\\lambda_0 I$ is symmetric, we find that\n\t\t\\begin{equation}\\label{mz}\n\t\t\tM^\\mathsf{T} z_0=\\begin{bmatrix}\n\t\t\t\tA-\\lambda_0 I\\\\\n\t\t\t\tz_0^\\mathsf{T}\n\t\t\t\\end{bmatrix}z_0\\equiv 0\\pmod{p^\\tau}.\n\t\t\\end{equation} Let the SNF of $M$ be $S= [\\diag(p^{c_1},p^{c_2}, \\dots, p^{c_n}), 0]$. Clearly, the SNF of $M^\\mathsf{T}$ is $S^\\mathsf{T}$. Note that $z_0\\not\\equiv 0\\pmod{p}$. It follows from Eq.~\\eqref{mz} and Proposition \\ref{dnk}  that $p^{c_n}$ is zero over $R$. Thus, the SNF of $M$ can be simplified as \n\t\t\\begin{equation}\\label{ss}\n\t\t\tS=[\\diag(p^{c_1},\\ldots,p^{c_{n-1}},0),0].\n\t\t\\end{equation}\n\n\\noindent\\textbf{Claim}: $p^{c_{n-1}}$ is a unit in $R$, i.e., $c_{n-1}= 0$.\n\n\t\tSuppose to the contrary that $c_{n-1}\\ge 1$. Then, we have $\\rank_p S\\le n-2$, or equivalently, $\\rank_p M\\le n-2$. On the other hand, by Lemma \\ref{rA}, we see that $\\rank_p (A-\\lambda_0 I)\\ge n-2$ and hence  $\\rank_p M \\ge n-2$. Thus, we must have  $\\rank_p M=\\rank_p (A-\\lambda_0 I)=n-2$. It follows that there exists an integral vector $z_1$ such that $(A-\\lambda_0 I)z_1\\equiv z_0\\pmod{p}$.\t\n\n\t\tAs $\\rank_p  (A-\\lambda_0 I)=n-2$, $(A-\\lambda_0I)z\\equiv 0\\pmod{p}$ has two  solutions $z_2$ and $z_3$ that are linearly  independent over $\\mathbb{Z}/p\\mathbb{Z}$. Since $(A-\\lambda_0 I)z_1\\equiv z_0\\not \\equiv 0\\pmod{p}$, $z_1$ cannot be written as a linear combination of $z_2$ and $z_3$. This implies that $z_1,z_2,z_3$ are linearly independent. Consider the equation $e^\\mathsf{T}(k_1z_1+k_2z_2+k_3z_3)\\equiv 0 \\pmod{p}$ with three unknowns $k_1,k_2,k_3$. Clearly, it has at least two independent solutions over $\\mathbb{Z}/p\\mathbb{Z}$. Let $(a_1,a_2,a_3)^\\mathsf{T}$ and  $(b_1,b_2,b_3)^\\mathsf{T}$  be  two such solutions and write $\\alpha=a_1z_1+a_2z_2+a_3z_3$ and $\\beta=b_1z_1+b_2z_2+b_3z_3$. It is easy to see that $\\alpha$ and $\\beta$ are linearly independent over $\\mathbb{Z}/p\\mathbb{Z}$. Note that $(A-\\lambda_0I)\\alpha\\equiv a_1z_0$ and $e^\\mathsf{T}\\alpha\\equiv 0\\pmod{p}$. Using a similar argument as in the proof of  Lemma~\\ref{wy}, we find  that $W^\\mathsf{T}\\alpha \\equiv 0\\pmod{p}$. Also,  $W^\\mathsf{T} \\beta\\equiv 0\\pmod{p}$. Thus, we have found two linearly independent solutions of $W^\\mathsf{T} z\\equiv 0\\pmod{p}$. This contradicts the fact that $\\rank_p W^\\mathsf{T}=n-1$ and hence completes the proof of the Claim.\n\n\t\tBy the Claim, we see that Eq.~\\eqref{ss} can be further reduced to \n\t\t\\begin{equation}\\label{s3}\n\t\t\tS=[\\diag(I_{n-1},0),0].\n\t\t\\end{equation}\n\t\tLet $\\mathcal{C}(M)$ be the $R$-module generated by the columns of $M$ and $N$ be the module $\\{z\\in R^n\\colon\\, z_0^\\mathsf{T} z=0\\}$. By Eq.~\\eqref{s3}, we know that $\\mathcal{C}(M)$ is isomorphic to the free module $R^{n-1}$. Since $z_0\\not\\equiv 0\\pmod{p}$, we find that $N$ is also isomorphic to $R^{n-1}$. Thus, the two $R$-modules $\\mathcal{C}(M)$ and $N$ are isomorphic. Since $z_0^\\mathsf{T} M=0$ over $R$, we know that $\\mathcal{C}(M)$ is a submodule of $N$. As $R$ is a finite ring, we must have $\\mathcal{C}(M)=N$, which completes the proof of Lemma \\ref{lc}.\n\t\\end{proof}\n\n\\begin{lemma}\\label{ez}\n\tThere exists an  integral vector $z_1$ such that $e^\\mathsf{T} z_1\\not\\equiv 0\\pmod {p}$ and $(A-\\lambda_0 I)z_1\\equiv p^cz_0\\pmod{p^\\tau}$ for some $c\\in \\{0,1,\\ldots,\\tau\\}$.\n\\end{lemma}\n\\begin{proof}\n\tLet $R=\\mathbb{Z}/p^\\tau \\mathbb{Z}$.  By Lemma \\ref{rA}, we know that the SNF of $A-\\lambda_0 I$ is\n\t\\begin{equation}\\label{smA}\n\t\tS=\\diag(1,1,\\ldots,1,p^c,0) \\text{~for some~} c\\in\\{0,1,\\ldots,\\tau\\}.\n\t\\end{equation}\nLet $M_i=[A-\\lambda_0 I, p^i z_0]$ for $i\\in\\{0,1,\\ldots,c\\}$. \n\n\\noindent\\textbf{Claim 1}: The SNF of $M_i$ is $[\\diag(1,1,\\ldots,1,p^i,0),0]$ for $i\\in\\{0,1,\\ldots,c\\}$.\n\nLet $S_i$ be the SNF of $M_i$. Noting that $\\rank_p M_i\\ge \\rank_p (A-\\lambda_0 I)\\ge n-2$, we conclude that $S_i$ must have the form\n\\begin{equation}\\label{si}\n\tS_i=[\\diag(1,\\ldots,1,p^{t_{i}},p^{t'_{i}}),0].\n\\end{equation}\nSince $ M_i^\\mathsf{T} z_0\\equiv 0\\pmod{p^\\tau}$ and $z_0\\not\\equiv 0\\pmod{p}$, Proposition \\ref{dnk} indicates that $p^{t'_i}$ is the zero element in $R$. Thus, Eq.~\\eqref{si} can be simplified as\n\\begin{equation}\\label{st}\n\tS_i=[\\diag(1,\\ldots,1,p^{t_{i}},0),0].\n\\end{equation}\nWe need to show that $t_i=i$ for each $i\\in\\{0,1,\\ldots,c\\}$. Clearly, $t_0=0$ by Lemma \\ref{lc}. Thus, it suffices to show that the sequence $\\{t_i\\}_{0\\le i\\le c}$ satisfies \n\\begin{equation}\\label{tc}\n\tt_c=c\n\\end{equation}\nand \n\\begin{equation}\\label{icr}\n\tt_i\\le t_{i+1}\\le t_{i}+1\\text{~for~} 0\\le i<c.\n\\end{equation}\nNow regard $A-\\lambda_0 I$ and $M_i$ as matrices over $\\mathbb{Z}$. Let $\\Delta$ and $\\Delta^{(i)}$ be the $(n-1)$-th determinantal factors of $A-\\lambda_{0}I$ and $M_i$, respectively. Note that Eq.~\\eqref{tc} holds for the case $c=\\tau$. Indeed, if $c=\\tau$ then $M_c=[A-\\lambda_0 I,p^\\tau z_0]=[A-\\lambda_0 I,0]$ over $R$, which implies that the SNF of $M_c$ is $S_c=[S,0]$, i.e., $t_c=c$ as desired. Now assume $c<\\tau$. From Eq.~\\eqref{smA}, we know that $v_p(\\Delta)=c$. Since $M_c$ contains $A-\\lambda_0 I$ as a submatrix, we must have $v_p(\\Delta^{(c)})\\le v_p(\\Delta)$ and hence  $v_p(\\Delta^{(c)})\\le c$. Let $D$ be any nonzero $(n-1)$-th minor of $M_c$. If $D$ does not include the last column of $M_c$, then $D$ is also a minor of $A-\\lambda_0 I$ and hence $v_p(D)\\ge v_p(\\Delta)\\ge c$. Otherwise,  each entry of the last column of $D$ is a multiple of $p^c$ and hence $v_p(D)\\ge c$. Thus, we always have $v_p(D)\\ge c$, which implies  $v_p(\\Delta^{(c)})\\ge c$. This means $v_p(\\Delta^{(c)})= c$. But by Eq.~\\eqref{st} for the case $i=c$, we know that $v_p(\\Delta^{(c)})=t_c$ and hence $t_c=c$. This proves Eq.~\\eqref{tc}.\n\nWe proceed to verify Eq.~\\eqref{icr}. We may assume $c>0$ since otherwise we have nothing to show. Let $D^{(i)}$ and $D^{(i+1)}$ be any two corresponding $(n-1)$-th minors of $M_i$ and $M_{i+1}$, respectively. By the constructions of $M_i$ and $M_{i+1}$, we have either $D^{(i+1)}=D^{(i)}$ or $D^{(i+1)}=pD^{(i)}$. It follows that $v_p(\\Delta^{(i)})\\le v_p(\\Delta^{(i+1)}) \\le v_p(\\Delta^{(i)}) +1$, i.e., $t_i\\le t_{i+1}\\le t_i+1$. Thus,  Claim 1 follows.\n\nTo complete the proof of Lemma \\ref{ez}, we consider the following three cases:\n\n\\noindent\\emph{Case 1}:  $c=0$. Since $A-\\lambda_0 I$ and $M_0=[A-\\lambda_0 I,z_0]$ have the same invariant factors over $R$, Proposition \\ref{ssnf} implies that there exists an integral vector $z_1$ such that  \\begin{equation}\\label{as}\n(A-\\lambda_0 I)z_1\\equiv z_0\\pmod{p^\\tau}.\n\\end{equation} By Lemma \\ref{wy}, we have \n\t\\begin{equation*}\n\tW^\\mathsf{T} z_1\\equiv e^\\mathsf{T} z_1(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T} \\pmod{p^{\\tau}}.\n\\end{equation*}\nSuppose to the contrary that $e^\\mathsf{T} z_1\\equiv 0\\pmod {p}$. Then we have $W^\\mathsf{T} z_1\\equiv 0\\pmod{p}$.  Since $\\rank_p W=n-1$ and $z_0$ is a nonzero (over $\\mathbb{Z}/p\\mathbb{Z}$) solution of $W^\\mathsf{T} z\\equiv 0\\pmod{p}$, we conclude that $z_1\\equiv kz_0\\pmod{p}$ for some integer $k$. But this would imply \n\\begin{equation*}\n(A-\\lambda_0 I) z_1\\equiv k(A-\\lambda_0 I) z_0\\equiv 0\\pmod{p},\n\\end{equation*}\nwhich, combining with Eq.~\\eqref{as}, leads to $z_0\\equiv 0\\pmod{p}$. This is a contradiction and hence we must have  $e^\\mathsf{T} z_1\\not\\equiv 0\\pmod {p}$.\n\n\\noindent\\emph{Case 2}: $c=\\tau$. As the SNF of $A-\\lambda_0 I$ is $S=\\diag(I_{n-1},0,0)$, we know that the kernel of $(A-\\lambda_0 I)$ is a free $R$-module of rank $2$ by Proposition \\ref{kf}. Let $K=\\ker (A-\\lambda_0 I)$ be the kernel. Since $z_0\\in K$ and $z_0\\not\\equiv 0\\pmod{p}$, Proposition \\ref{bas} (ii) implies that there exists an integer vector $z_1$ such that $\\{z_0,z_1\\}$ (over $R$) constitutes a basis of the free module $K$. In particular, $(A-\\lambda_0 I)z_1\\equiv 0\\pmod{p^\\tau}$. Using Lemma \\ref{wy}, we conclude that \n\t\\begin{equation*}\n\tW^\\mathsf{T} z_1\\equiv e^\\mathsf{T} z_1(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T} \\pmod{p^{\\tau}}.\n\\end{equation*}\nIf $e^\\mathsf{T} z_1\\equiv 0\\pmod{p}$ then we would have \t$W^\\mathsf{T} z_1\\equiv 0\\pmod{p}$. But since $z_0$ and $z_1$ are independent over the field $\\mathbb{Z}/p\\mathbb{Z}$, we must have $\\rank_p W^\\mathsf{T}\\le n-2$. This is a contradiction and hence $e^\\mathsf{T} z_1\\not\\equiv 0\\pmod{p}$.\n\n\\noindent{\\emph{Case 3}}:  $0<c<\\tau$. By Claim 1, we see that $A-\\lambda_0 I$ and $M_c=[A-\\lambda_0 I, p^c z_0]$ have the same invariant factors. By Proposition \\ref{ssnf}, there exists an integral vector $z_1$ such that \n\\begin{equation}\\label{az1}\n\t(A-\\lambda_0 I)z_1\\equiv p^c z_0\\pmod{p^\\tau}.\n\\end{equation} \n\n\\noindent\\textbf{Claim 2}: $z_1$ and $z_0$ are linearly independent over $\\mathbb{Z}/p\\mathbb{Z}$.\n\nSuppose to the contrary that $z_1$ and $z_0$ are linearly dependent over $\\mathbb{Z}/p\\mathbb{Z}$. As $z_0\\not\\equiv 0\\pmod{p}$, there exist an integer $k$ and an integral vector $\\tilde{z}_1$ such that $z_1=kz_0+p\\tilde{z}_1$. Noting that $(A-\\lambda_0 I)z_0\\equiv 0\\pmod{p^\\tau}$, we obtain from Eq.~\\eqref{az1} that $(A-\\lambda_0 I)p\\tilde{z}_1\\equiv p^c z_0\\pmod{p^\\tau}$, i.e., \n\\begin{equation}\\label{az1s}\n\t(A-\\lambda_0 I)\\tilde{z}_1\\equiv p^{c-1} z_0\\pmod{p^{\\tau-1}}.\n\t\\end{equation}\nLet $R'=\\mathbb{Z}/p^{\\tau-1}\\mathbb{Z}$. By Eq.~\\eqref{az1s}, we see that, over $R'$, the two matrices $A-\\lambda_0 I$ and $M_{c-1}=[A-\\lambda_0 I, p^{c-1}z_0]$ have the same invariant factors. Since $c\\le \\tau-1$, we see that $p^c$ and $p^{c-1}$ are not associates over $R'$. Thus, Claim 1 means that $A-\\lambda_0 I$ and $M_{c-1}=[A-\\lambda_0 I, p^{c-1}z_0]$ cannot have the same invariant factors. This contradiction completes the proof of Claim 2. \n\nBy Eq.~\\eqref{az1} and Lemma \\ref{wy}, we have \n\\begin{equation*}\n\tW^\\mathsf{T} z_1\\equiv e^\\mathsf{T} z_1(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T}\\pmod{p^\\tau}.\n\\end{equation*}\nBy Claim 2 and the same argument as in Case 2, we must have $e^\\mathsf{T} z_1\\not\\equiv 0\\pmod{p}$. \n\nCombining the three cases,  Lemma \\ref{ez} follows.\n\\end{proof}\nNow we are in a position to present a proof of Theorem \\ref{main}.\n\n\\begin{proof}[Proof of Theorem \\ref{main}]  Let $\\tau=v_p(L(G))$. Let $z_0$ and $\\lambda_0$ be the integral vector and the integer as described in  Lemma \\ref{fourcong}. Then we have \n\\begin{equation*}\n\tz_0^\\mathsf{T}(A-\\lambda_0 I)z_0\\equiv 0\\pmod{p^{2\\tau}}\n\\end{equation*}\nand hence\n\\begin{equation*}\n\tz_0^\\mathsf{T}\\frac{(A-\\lambda_0 I)z_0}{p^\\tau}\\equiv 0\\pmod{p^{\\tau}}.\n\\end{equation*}\nIt follows from Lemma \\ref{lc} that \n\\begin{equation}\\label{aps}\n\\frac{(A-\\lambda_0 I)z_0}{p^\\tau}\\equiv (A-\\lambda_0 I)y+s z_0\\pmod{p^{\\tau}}\n\\end{equation}\nfor some $y\\in \\mathbb{Z}^{n}$ and $s\\in \\mathbb{Z}$.\nMultiplying both sides of Eq.~\\eqref{aps} by $p^\\tau$ and rearranging the terms, we obtain\n\\begin{equation*}\n\t(A-\\lambda_0 I)(z_0-p^\\tau y)\\equiv sp^\\tau z_0\\pmod{p^{2\\tau}}.\n\\end{equation*}\nConsequently, by Lemma \\ref{wy}, we have\n\\begin{equation}\\label{wp}\n\tW^\\mathsf{T} (z_0-p^\\tau y)\\equiv e^\\mathsf{T} (z_0-p^\\tau y)(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T}\\pmod{p^{2\\tau}}.\n\\end{equation}\nBy Lemma \\ref{ez}, there exist an integral vector $z_1$ and an integer $c\\in\\{0,1,\\ldots,\\tau\\}$ such that $e^\\mathsf{T} z_1\\not\\equiv 0\\pmod{p}$ and\n \\begin{equation}\\label{apta}\n(A-\\lambda_0 I)z_1\\equiv p^c z_0\\pmod{p^\\tau}.\n\\end{equation} As $e^\\mathsf{T} (z_0-p^\\tau y)\\equiv e^\\mathsf{T} z_0\\equiv 0\\pmod{p^\\tau}$, we see that ${p^{-\\tau}}e^\\mathsf{T} (z_0-p^\\tau y) $ is an integer. Note that $e^\\mathsf{T} z_1$ is a unit in $\\mathbb{Z}/p^{\\tau}\\mathbb{Z}$. Thus, there exists some integer $g$ such that \n\\begin{equation}\\label{ep}\n\t\\frac{e^\\mathsf{T} (z_0-p^\\tau y)}{p^\\tau}\\equiv g e^\\mathsf{T} z_1\\pmod{p^{\\tau}}.\n\\end{equation}\nWe know from Eq.~\\eqref{apta} and Lemma \\ref{wy} that \n\\begin{equation*}\n\tW^\\mathsf{T} z_1\\equiv e^\\mathsf{T} z_1(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T} \\pmod{p^\\tau}.\n\\end{equation*}\nThis, together with Eq.~\\eqref{ep}, leads to \n\\begin{equation*}\n\tW^\\mathsf{T} (g z_1)\\equiv g e^\\mathsf{T} z_1(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T}\\equiv \t\\frac{e^\\mathsf{T} (z_0-p^\\tau y)}{p^\\tau}(1,\\lambda_0,\\ldots,\\lambda_0^{n-1})^\\mathsf{T} \\pmod{p^\\tau}.\n\\end{equation*}\nMultiplying both sides by $p^\\tau$ and using Eq.~\\eqref{wp}, we find that \n\\begin{equation*}\nW^\\mathsf{T} (gp^\\tau z_1) \\equiv W^\\mathsf{T} (z_0-p^\\tau y)\\pmod{p^{2\\tau}},\n\\end{equation*}\ni.e., \n\\begin{equation*}\nW^\\mathsf{T} (z_0-p^\\tau y-gp^\\tau z_1) \\equiv 0\\pmod{p^{2\\tau}}.\n\\end{equation*}\nAs $z_0-p^\\tau y-g p^\\tau z_1\\equiv z_0\\not\\equiv 0\\pmod{p}$, Proposition \\ref{dnk} implies that $d_n\\equiv 0\\pmod{p^{2\\tau}}$, where $d_n$ is the $n$-th invariant factor of the integral matrix $W^\\mathsf{T}$. As $\\rank_p W=n-1$, we have $v_p(\\det W) =v_p(d_n)$ and hence $v_p(\\det W)\\ge 2\\tau$. This completes the proof of Theorem \\ref{main}.\n\\end{proof}\n\\section{Applications and future work}\\label{dis}\nAs a natural extension of the arithmetic criterion of DGS-graphs established in \\cite{wang2017JCTB}, Wang et al. \\cite{wang2023ejc} defined, for each odd prime $p$,  a family of graphs \n \\begin{equation*}\n\\mathcal{F}_{n,p}=\\{\\text{$n$-vertex graphs~} G\\colon\\, 2^{-\\lfloor n/2 \\rfloor}\\det W =p^2b\\text{~and~} \\rank_p W=n-1\\},\n \\end{equation*}\nwhere $b$ is odd, square-free and $p\\nmid b$. It was shown that each graph in $\\mathcal{F}_{n,p}$ has at most one generalized cospectral mate \\cite{wang2023ejc}. Using Theorem \\ref{main}, we can show the same conclusion for a larger family of graphs:\n \\begin{equation*}\n \\widetilde{\\mathcal{F}}_{n,p}=\\{\\text{$n$-vertex graphs~} G\\colon\\, 2^{-\\lfloor n/2 \\rfloor}\\det W \\in\\{p^2b, p^3b\\} \\text{~and~} \\rank_p W=n-1\\},\n \\end{equation*}\n where $p$ and $b$ satisfy the same restrictions. Indeed, let $G\\in \\widetilde{\\mathcal{F}}_{n,p}$ and  $Q$ be any matrix in $\\mathcal{Q}(G)$. Then, by Propositions \\ref{oddcase} and \\ref{evencase}, the only possible prime factor of $\\ell(Q)$ is $p$. It follows from Theorem \\ref{main} that\n \\begin{equation*}\nv_p(\\ell(Q))\\le \\frac{1}{2}v_p(\\det W)\\le 3/2,\n \\end{equation*}\n i.e., $v_p(\\ell(Q))\\in\\{0,1\\}$. In other words, $Q$ is either a permutation matrix, or a matrix with level exactly $p$. Now, the original argument in \\cite{wang2023ejc} shows that $G$ has at most one generalized cospectral mate; refer to \\cite{wang2023ejc} for details.\n\n The following small example illustrates the optimality of the family $\\widetilde{\\mathcal{F}}_{n,p}$ in the sense that $G$ may have two or more generalized cospectral mates if the $v_p(\\det W)\\ge 4$ for some odd prime $p$.\n\n \\noindent\\textbf{Example 1}. Let $n=10$ and $G$ be the $n$-vertex graph with adjacency matrix \n\\begin{equation*}\n\tA={\\footnotesize\\begin{bmatrix}\n\t\t0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 \\\\\n\t1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 \\\\\n\t0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 \\\\\n\t1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\\\\n\t1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\\\\n\t0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\\\\n\t1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 \\\\\n\t0 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 \\\\\n\t0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\\\\n\t1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\\\\n\t\\end{bmatrix}}.\n\\end{equation*} \nIt can be verified  that $2^{-\\lfloor\\frac{n}{2}\\rfloor} \\det W=3^4\\times 19$ and $\\rank_3 W=n-1$. By Theorem \\ref{main}, we know that $v_3(L(G))\\le \\frac{1}{2} v_3(\\det W)=2$. Indeed, using the program in \\cite{wang2025}, we find that, besides the permutation matrices,  there exist essentially two rational regular orthogonal matrices \n\\begin{equation*}\n\tQ_1=\\frac{1}{3}{\\footnotesize\n\t\\begin{bmatrix}\n\t\t3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\t\t0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\t\t0 & 0 & 0 & 0 & -2 & 1 & 1 & 1 & 1 & 1 \\\\\n\t\t0 & 0 & 0 & 0 & 1 & -2 & 1 & 1 & 1 & 1 \\\\\n\t\t0 & 0 & 0 & 0 & 1 & 1 & -2 & 1 & 1 & 1 \\\\\n\t\t0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\t\t0 & 0 & 0 & 0 & 1 & 1 & 1 & -2 & 1 & 1 \\\\\n\t\t0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & -2 & 1 \\\\\n\t\t0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & -2 \\\\\n\t\t0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\t\\end{bmatrix}}\n\\end{equation*}\nand \n\\begin{equation*}\nQ_2=\\frac{1}{9}{\\footnotesize\\begin{bmatrix}\n\t\t-3 & -3 & 3 & 3 & 3 & 0 & 0 & 0 & 0 & 6 \\\\\n\t\t3 & 3 & 6 & -3 & -3 & 0 & 0 & 0 & 0 & 3 \\\\\n\t\t1 & 1 & 2 & 2 & 2 & -6 & 3 & 3 & 3 & -2 \\\\\n\t\t1 & 1 & 2 & 2 & 2 & 3 & -6 & 3 & 3 & -2 \\\\\n\t\t-5 & 4 & -1 & -1 & -1 & 3 & 3 & 3 & 3 & 1 \\\\\n\t\t3 & 3 & -3 & 6 & -3 & 0 & 0 & 0 & 0 & 3 \\\\\n\t\t1 & 1 & 2 & 2 & 2 & 3 & 3 & -6 & 3 & -2 \\\\\n\t\t1 & 1 & 2 & 2 & 2 & 3 & 3 & 3 & -6 & -2 \\\\\n\t\t4 & -5 & -1 & -1 & -1 & 3 & 3 & 3 & 3 & 1 \\\\\n\t\t3 & 3 & -3 & -3 & 6 & 0 & 0 & 0 & 0 & 3 \\\\\n\\end{bmatrix}}\n\\end{equation*}\nsuch that both $Q_1^\\mathsf{T} A Q_1$ and $Q_2^\\mathsf{T} A Q_2$ are adjacency matrices. Thus, $G$ has two generalized cospectral mates.\n\nRecently, as a significant improvement of the criterion for a graph to have at most one cospectral mate established in \\cite{wang2023ejc}, Raza et al. \\cite{raza2026} obtained the following upper bound on the number of generalized cospectral mates of a graph.\n\\begin{theorem} [\\cite{raza2026}]\\label{ngm}\n\tLet $G$ be a controllable graph and $d_1,\\ldots, d_n$ be the invariant factors of the walk matrix $W(G)$. Suppose $d_{\\lceil\\frac{n}{2}\\rceil}=1$ and $d_{n-1}=2$. Let $d_n=\\prod_{i=1}^k p_i^{m_i}$ be the prime factorization of $d_n$. Then $G$ has at most $\\left(\\prod_{i=1}^{k}m_i\\right)-1$ generalized cospectral mates.\n\t\\end{theorem}", "dis": "\\label{dis}\nAs a natural extension of the arithmetic criterion of DGS-graphs established in \\cite{wang2017JCTB}, Wang et al. \\cite{wang2023ejc} defined, for each odd prime $p$,  a family of graphs \n \\", "main": "\\begin{theorem}\\label{main}\n\tLet $G$ be a controllable graph and $p$ be an odd prime such that  $\\rank_p W(G)=n-1$. Then $v_p(L(G))\\le \\frac{1}{2} v_p(\\det W(G))$.\n\\end{theorem}", "oddcase": "\\begin{proposition}[\\cite{wang2013EJC}]\\label{oddcase}\n\tIf $p$ is an odd prime with $p^2\\nmid \\det W(G)$ then $p\\nmid L(G)$.\n\\end{proposition}", "qiuodd": "\\begin{proposition}[\\cite{qiu2023}]\\label{qiuodd} If $p$ is an odd prime with $\\rank_p W(G)=n-1$ then $L(G)\\mid \\frac{\\det W(G)}{p}$.\n\\end{proposition}"}, "pre_theorem_intro_text_len": 4698, "pre_theorem_intro_text": "For a graph $G$, the \\emph{spectrum} of $G$ is the multiset of all eigenvalues of its adjacency matrix. Two graphs are \\emph{cospectral} if they share the same spectrum.   We say two graphs $G$ and $H$ are \\emph{generalized cospectral} if (i) $G$ is cospectral with $H$ and (ii) $\\overline{G}$ is cospectral with $\\overline{H}$, where $\\overline{G}$ denotes the complement of $G$. A graph $G$ is \\emph{determined by its generalized spectrum} (DGS)  if any graph generalized cospectral with $G$ is isomorphic to $G$. An orthogonal matrix $Q$ is \\emph{regular} if the sum of each row is 1.  A classic result of Johnson and Newman \\cite{johnson} states that two graphs $G$ and $H$ are generalized cospectral if and only if their adjacency matrices are similar via a regular orthogonal matrix. \n\nFor an $n$-vertex graph $G$ with adjacency matrix $A$, the \\emph{walk matrix} of $G$ is \\begin{equation*}\nW(G):=[e,Ae,\\ldots,A^{n-1}e],\n\\end{equation*} where $e$ is the all-ones vector. A graph is \\emph{controllable} \\cite{godsil} if $W(G)$ is nonsingular. We remark that the set of controllable graphs is \\emph{closed} under generalized cospectrality, that is, any graph generalized cospectral with a controllable graph must be controllable. A key observation of Wang \\cite{wang2006} is that, when restricted to controllable graphs,   the regular orthogonal matrix connecting the adjacency matrices of two generalized cospectral graphs $G$ and $H$ is unique and rational. \n\nWe are mainly concerned with controllable graphs in this paper. We use $\\RO_n(\\mathbb{Q})$ to denote the group of rational regular orthogonal matrices of order $n$. For a controllable graph $G$ with adjacency matrix $A$, we write\n\\begin{equation*}\n\t\\mathcal{Q}(G)=\\{Q\\in \\RO_n(\\mathbb{Q})\\colon\\, Q^\\mathsf{T} AQ \\text{~is a $(0,1)$-matrix} \\}.\n\t\\end{equation*}\nIt is easy to show that any $(0,1)$-matrix of the form $Q^\\mathsf{T} A Q$ must be the adjacency matrix of some graph (see \\cite[Remark 1]{qiu2023}). Let $\\mathcal{C}(G)$ be the set of all graphs that are generalized cospectral with $G$. As there exists a one-to-one correspondence between $\\mathcal{C}(G)$ and $\\mathcal{Q}(G)$, the problem of determining the set $\\mathcal{C}(G)$ can naturally be  reduced to that of determining the set $\\mathcal{Q}(G)$. For example, $\\mathcal{Q}(G)$ always contains all permutation matrices of order $n$ which  correspond to graphs that are isomorphic to $G$.  Moreover, if $\\mathcal{Q}(G)$ contains a non-permutation matrix, then  $\\mathcal{C}(G)$ contains a graph that is not isomorphic to $G$, i.e., $G$ is not DGS.  The following notion introduced in \\cite{wang2006} is crucial to investigate the set $\\mathcal{Q}(G)$.\n\\begin{definition}[\\cite{wang2006}] \\normalfont\n Let $Q$ be a rational matrix. The \\emph{level} of $Q$, denoted by $\\ell(Q)$, is the\nsmallest positive integer $k$ such that $kQ$ is an integral matrix.\n\\end{definition}\nLet \n\\begin{equation*}\nL(G)=\\lcm\\{\\ell(Q)\\colon\\,Q\\in \\mathcal{Q}(G)\\},\n\\end{equation*}the least common multiple of all levels  $\\ell(Q)$ for $Q\\in \\mathcal{Q}(G)$. Clearly, if $L(G)=1$ then $\\mathcal{Q}(G)$ contains only permutation matrices which implies that $G$ is DGS.   A theorem of Wang \\cite{wang2013EJC,wang2017JCTB} states that if the integer $2^{-\\lfloor\\frac{n}{2}\\rfloor}\\det W(G)$ is  odd and square-free, then   $L(G)=1$ and hence $G$ is DGS. Wang proves this theorem by excluding the possibilities of $p\\mid L(G)$ for odd primes $p$ and for $p=2$ separately.\n\\begin{proposition}[\\cite{wang2013EJC}]\\label{oddcase}\n\tIf $p$ is an odd prime with $p^2\\nmid \\det W(G)$ then $p\\nmid L(G)$.\n\\end{proposition}\n\\begin{proposition}[\\cite{wang2017JCTB}]\\label{evencase}\n\tIf $2^{-\\lfloor\\frac{n}{2}\\rfloor}$ is odd then $L(G)$ is odd.\n\\end{proposition}\nRecently, Qiu et al.~\\cite{qiu2023} improved both Proposition \\ref{oddcase} and Proposition \\ref{evencase} using a new and much  simpler argument. The current paper is a further improvement of the result in \\cite{qiu2023}, but only for the odd prime case.  The result of  Qiu et al.~\\cite{qiu2023} for the odd prime case is the following.\n\\begin{proposition}[\\cite{qiu2023}]\\label{qiuodd} If $p$ is an odd prime with $\\rank_p W(G)=n-1$ then $L(G)\\mid \\frac{\\det W(G)}{p}$.\n\\end{proposition}\nFor a prime $p$ and a nonzero integer $m$, we use $v_p(m)$ to denote the \\emph{$p$-adic valuation} \\cite{gouvea} of $m$, that is, the maximum nonnegative integer $k$ such that $p^k$ divides $m$. Note that the conclusion of Proposition  \\ref{qiuodd} can be rewritten as the inequality $v_p(L(G))\\le v_p(\\det W(G))-1$. The main aim of this paper is to give a better upper bound of $v_p(L(G))$ under the same assumption of Proposition \\ref{qiuodd}.", "context": "For a graph $G$, the \\emph{spectrum} of $G$ is the multiset of all eigenvalues of its adjacency matrix. Two graphs are \\emph{cospectral} if they share the same spectrum. We say two graphs $G$ and $H$ are \\emph{generalized cospectral} if (i) $G$ is cospectral with $H$ and (ii) $\\overline{G}$ is cospectral with $\\overline{H}$, where $\\overline{G}$ denotes the complement of $G$. A graph $G$ is \\emph{determined by its generalized spectrum} (DGS) if any graph generalized cospectral with $G$ is isomorphic to $G$. An orthogonal matrix $Q$ is \\emph{regular} if the sum of each row is 1. A classic result of Johnson and Newman \\cite{johnson} states that two graphs $G$ and $H$ are generalized cospectral if and only if their adjacency matrices are similar via a regular orthogonal matrix.\n\nFor an $n$-vertex graph $G$ with adjacency matrix $A$, the \\emph{walk matrix} of $G$ is \\begin{equation*}\nW(G):=[e,Ae,\\ldots,A^{n-1}e],\n\\end{equation*} where $e$ is the all-ones vector. A graph is \\emph{controllable} \\cite{godsil} if $W(G)$ is nonsingular. We remark that the set of controllable graphs is \\emph{closed} under generalized cospectrality, that is, any graph generalized cospectral with a controllable graph must be controllable. A key observation of Wang \\cite{wang2006} is that, when restricted to controllable graphs, the regular orthogonal matrix connecting the adjacency matrices of two generalized cospectral graphs $G$ and $H$ is unique and rational.\n\nWe are mainly concerned with controllable graphs in this paper. We use $\\RO_n(\\mathbb{Q})$ to denote the group of rational regular orthogonal matrices of order $n$. For a controllable graph $G$ with adjacency matrix $A$, we write\n\\begin{equation*}\n \\mathcal{Q}(G)=\\{Q\\in \\RO_n(\\mathbb{Q})\\colon\\, Q^\\mathsf{T} AQ \\text{~is a $(0,1)$-matrix} \\}.\n \\end{equation*}\nIt is easy to show that any $(0,1)$-matrix of the form $Q^\\mathsf{T} A Q$ must be the adjacency matrix of some graph (see \\cite[Remark 1]{qiu2023}). Let $\\mathcal{C}(G)$ be the set of all graphs that are generalized cospectral with $G$. As there exists a one-to-one correspondence between $\\mathcal{C}(G)$ and $\\mathcal{Q}(G)$, the problem of determining the set $\\mathcal{C}(G)$ can naturally be reduced to that of determining the set $\\mathcal{Q}(G)$. For example, $\\mathcal{Q}(G)$ always contains all permutation matrices of order $n$ which correspond to graphs that are isomorphic to $G$. Moreover, if $\\mathcal{Q}(G)$ contains a non-permutation matrix, then $\\mathcal{C}(G)$ contains a graph that is not isomorphic to $G$, i.e., $G$ is not DGS. The following notion introduced in \\cite{wang2006} is crucial to investigate the set $\\mathcal{Q}(G)$.\n\\begin{definition}[\\cite{wang2006}] \\normalfont\n Let $Q$ be a rational matrix. The \\emph{level} of $Q$, denoted by $\\ell(Q)$, is the\nsmallest positive integer $k$ such that $kQ$ is an integral matrix.\n\\end{definition}\nLet \n\\begin{equation*}\nL(G)=\\lcm\\{\\ell(Q)\\colon\\,Q\\in \\mathcal{Q}(G)\\},\n\\end{equation*}the least common multiple of all levels $\\ell(Q)$ for $Q\\in \\mathcal{Q}(G)$. Clearly, if $L(G)=1$ then $\\mathcal{Q}(G)$ contains only permutation matrices which implies that $G$ is DGS. A theorem of Wang \\cite{wang2013EJC,wang2017JCTB} states that if the integer $2^{-\\lfloor\\frac{n}{2}\\rfloor}\\det W(G)$ is odd and square-free, then $L(G)=1$ and hence $G$ is DGS. Wang proves this theorem by excluding the possibilities of $p\\mid L(G)$ for odd primes $p$ and for $p=2$ separately.\n\\begin{proposition}[\\cite{wang2013EJC}]\\label{oddcase}\n If $p$ is an odd prime with $p^2\\nmid \\det W(G)$ then $p\\nmid L(G)$.\n\\end{proposition}\n\\begin{proposition}[\\cite{wang2017JCTB}]\\label{evencase}\n If $2^{-\\lfloor\\frac{n}{2}\\rfloor}$ is odd then $L(G)$ is odd.\n\\end{proposition}\nRecently, Qiu et al.~\\cite{qiu2023} improved both Proposition \\ref{oddcase} and Proposition \\ref{evencase} using a new and much simpler argument. The current paper is a further improvement of the result in \\cite{qiu2023}, but only for the odd prime case. The result of Qiu et al.~\\cite{qiu2023} for the odd prime case is the following.\n\\begin{proposition}[\\cite{qiu2023}]\\label{qiuodd} If $p$ is an odd prime with $\\rank_p W(G)=n-1$ then $L(G)\\mid \\frac{\\det W(G)}{p}$.\n\\end{proposition}\nFor a prime $p$ and a nonzero integer $m$, we use $v_p(m)$ to denote the \\emph{$p$-adic valuation} \\cite{gouvea} of $m$, that is, the maximum nonnegative integer $k$ such that $p^k$ divides $m$. Note that the conclusion of Proposition \\ref{qiuodd} can be rewritten as the inequality $v_p(L(G))\\le v_p(\\det W(G))-1$. The main aim of this paper is to give a better upper bound of $v_p(L(G))$ under the same assumption of Proposition \\ref{qiuodd}.\n\n\\begin{proposition}[\\cite{wang2017JCTB}]\\label{evencase}\n\tIf $2^{-\\lfloor\\frac{n}{2}\\rfloor}$ is odd then $L(G)$ is odd.\n\\end{proposition}\n\n\\begin{proposition}[\\cite{wang2013EJC}]\\label{oddcase}\n\tIf $p$ is an odd prime with $p^2\\nmid \\det W(G)$ then $p\\nmid L(G)$.\n\\end{proposition}", "full_context": "For a graph $G$, the \\emph{spectrum} of $G$ is the multiset of all eigenvalues of its adjacency matrix. Two graphs are \\emph{cospectral} if they share the same spectrum. We say two graphs $G$ and $H$ are \\emph{generalized cospectral} if (i) $G$ is cospectral with $H$ and (ii) $\\overline{G}$ is cospectral with $\\overline{H}$, where $\\overline{G}$ denotes the complement of $G$. A graph $G$ is \\emph{determined by its generalized spectrum} (DGS) if any graph generalized cospectral with $G$ is isomorphic to $G$. An orthogonal matrix $Q$ is \\emph{regular} if the sum of each row is 1. A classic result of Johnson and Newman \\cite{johnson} states that two graphs $G$ and $H$ are generalized cospectral if and only if their adjacency matrices are similar via a regular orthogonal matrix.\n\nFor an $n$-vertex graph $G$ with adjacency matrix $A$, the \\emph{walk matrix} of $G$ is \\begin{equation*}\nW(G):=[e,Ae,\\ldots,A^{n-1}e],\n\\end{equation*} where $e$ is the all-ones vector. A graph is \\emph{controllable} \\cite{godsil} if $W(G)$ is nonsingular. We remark that the set of controllable graphs is \\emph{closed} under generalized cospectrality, that is, any graph generalized cospectral with a controllable graph must be controllable. A key observation of Wang \\cite{wang2006} is that, when restricted to controllable graphs, the regular orthogonal matrix connecting the adjacency matrices of two generalized cospectral graphs $G$ and $H$ is unique and rational.\n\nWe are mainly concerned with controllable graphs in this paper. We use $\\RO_n(\\mathbb{Q})$ to denote the group of rational regular orthogonal matrices of order $n$. For a controllable graph $G$ with adjacency matrix $A$, we write\n\\begin{equation*}\n \\mathcal{Q}(G)=\\{Q\\in \\RO_n(\\mathbb{Q})\\colon\\, Q^\\mathsf{T} AQ \\text{~is a $(0,1)$-matrix} \\}.\n \\end{equation*}\nIt is easy to show that any $(0,1)$-matrix of the form $Q^\\mathsf{T} A Q$ must be the adjacency matrix of some graph (see \\cite[Remark 1]{qiu2023}). Let $\\mathcal{C}(G)$ be the set of all graphs that are generalized cospectral with $G$. As there exists a one-to-one correspondence between $\\mathcal{C}(G)$ and $\\mathcal{Q}(G)$, the problem of determining the set $\\mathcal{C}(G)$ can naturally be reduced to that of determining the set $\\mathcal{Q}(G)$. For example, $\\mathcal{Q}(G)$ always contains all permutation matrices of order $n$ which correspond to graphs that are isomorphic to $G$. Moreover, if $\\mathcal{Q}(G)$ contains a non-permutation matrix, then $\\mathcal{C}(G)$ contains a graph that is not isomorphic to $G$, i.e., $G$ is not DGS. The following notion introduced in \\cite{wang2006} is crucial to investigate the set $\\mathcal{Q}(G)$.\n\\begin{definition}[\\cite{wang2006}] \\normalfont\n Let $Q$ be a rational matrix. The \\emph{level} of $Q$, denoted by $\\ell(Q)$, is the\nsmallest positive integer $k$ such that $kQ$ is an integral matrix.\n\\end{definition}\nLet \n\\begin{equation*}\nL(G)=\\lcm\\{\\ell(Q)\\colon\\,Q\\in \\mathcal{Q}(G)\\},\n\\end{equation*}the least common multiple of all levels $\\ell(Q)$ for $Q\\in \\mathcal{Q}(G)$. Clearly, if $L(G)=1$ then $\\mathcal{Q}(G)$ contains only permutation matrices which implies that $G$ is DGS. A theorem of Wang \\cite{wang2013EJC,wang2017JCTB} states that if the integer $2^{-\\lfloor\\frac{n}{2}\\rfloor}\\det W(G)$ is odd and square-free, then $L(G)=1$ and hence $G$ is DGS. Wang proves this theorem by excluding the possibilities of $p\\mid L(G)$ for odd primes $p$ and for $p=2$ separately.\n\\begin{proposition}[\\cite{wang2013EJC}]\\label{oddcase}\n If $p$ is an odd prime with $p^2\\nmid \\det W(G)$ then $p\\nmid L(G)$.\n\\end{proposition}\n\\begin{proposition}[\\cite{wang2017JCTB}]\\label{evencase}\n If $2^{-\\lfloor\\frac{n}{2}\\rfloor}$ is odd then $L(G)$ is odd.\n\\end{proposition}\nRecently, Qiu et al.~\\cite{qiu2023} improved both Proposition \\ref{oddcase} and Proposition \\ref{evencase} using a new and much simpler argument. The current paper is a further improvement of the result in \\cite{qiu2023}, but only for the odd prime case. The result of Qiu et al.~\\cite{qiu2023} for the odd prime case is the following.\n\\begin{proposition}[\\cite{qiu2023}]\\label{qiuodd} If $p$ is an odd prime with $\\rank_p W(G)=n-1$ then $L(G)\\mid \\frac{\\det W(G)}{p}$.\n\\end{proposition}\nFor a prime $p$ and a nonzero integer $m$, we use $v_p(m)$ to denote the \\emph{$p$-adic valuation} \\cite{gouvea} of $m$, that is, the maximum nonnegative integer $k$ such that $p^k$ divides $m$. Note that the conclusion of Proposition \\ref{qiuodd} can be rewritten as the inequality $v_p(L(G))\\le v_p(\\det W(G))-1$. The main aim of this paper is to give a better upper bound of $v_p(L(G))$ under the same assumption of Proposition \\ref{qiuodd}.\n\n\\begin{proposition}[\\cite{wang2017JCTB}]\\label{evencase}\n\tIf $2^{-\\lfloor\\frac{n}{2}\\rfloor}$ is odd then $L(G)$ is odd.\n\\end{proposition}\n\n\\begin{proposition}[\\cite{wang2013EJC}]\\label{oddcase}\n\tIf $p$ is an odd prime with $p^2\\nmid \\det W(G)$ then $p\\nmid L(G)$.\n\\end{proposition}\n\nFor an $n$-vertex graph $G$ with adjacency matrix $A$, the \\emph{walk matrix} of $G$ is \\begin{equation*}\nW(G):=[e,Ae,\\ldots,A^{n-1}e],\n\\end{equation*} where $e$ is the all-ones vector. A graph is \\emph{controllable} \\cite{godsil} if $W(G)$ is nonsingular. We remark that the set of controllable graphs is \\emph{closed} under generalized cospectrality, that is, any graph generalized cospectral with a controllable graph must be controllable. A key observation of Wang \\cite{wang2006} is that, when restricted to controllable graphs, the regular orthogonal matrix connecting the adjacency matrices of two generalized cospectral graphs $G$ and $H$ is unique and rational.\n\nThe rest of this paper is organized as follows. In Sec.~\\ref{pre}, we recall some preliminaries on Smith normal forms and modules over $\\mathbb{Z}/p^k \\mathbb{Z}$. The proof of Theorem \\ref{main} is given in Sec.~\\ref{pf}, which can be seen as a module-theoretical version of the original argument of Wang \\cite{wang2006}, combined with some improvements and simplifications developed in \\cite{wang2020}. Some direct applications of Theorem \\ref{main} are presented in Sec.~\\ref{dis}. The paper concludes with a conjecture proposing a further possible improvement to Theorem \\ref{main}.\n\\section{Preliminaries}\n\\label{pre}\nLet $R$ be a commutative principal ideal ring with identity. It is well known \\cite{stanley2016JCTA} that any matrix $M$ over $R$ has a Smith normal form (SNF); that is, there exist invertible matrices $U$ and $V$ over $R$ such that \n\\begin{equation*}\n UMV=\\begin{bmatrix}\n \\diag(d_1,\\ldots,d_r)&0\\\\\n 0&0\n \\end{bmatrix},\n\\end{equation*}\nwhere the elements $d_i$ ($i=1,\\ldots,r$) are nonzero and satisfy $d_1\\mid d_2\\mid \\cdots\\mid d_r$. The nonzero elements $d_1,\\ldots,d_r$ are called the \\emph{invariant factors} of $M$.\n\nIn the following, let $G$ be an $n$-vertex controllable graph and $p$ be an odd prime factor of $\\det W(G)$ such that $\\rank_p W(G)=n-1$. Denote $\\tau=v_p(L(G))$, i.e., $\\tau=\\max\\{v_p (\\ell(Q))\\colon\\, Q\\in \\mathcal{Q}(G)\\}$. Note that if $\\tau=0$, then Theorem \\ref{main} clearly holds. Thus, we may assume that $\\tau\\ge 1$. For simplicity, the adjacency matrix $A(G)$ and the walk matrix $W(G)$ will be denoted by $A$ and $W$, respectively, when there is no confusion.\n\n\\noindent\\textbf{Example 1}. Let $n=10$ and $G$ be the $n$-vertex graph with adjacency matrix \n\\begin{equation*}\n A={\\footnotesize\\begin{bmatrix}\n 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 \\\\\n 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 \\\\\n 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 \\\\\n 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\\\\n 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\\\\n 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 \\\\\n 0 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 \\\\\n 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\\\\n 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\\\\n \\end{bmatrix}}.\n\\end{equation*} \nIt can be verified that $2^{-\\lfloor\\frac{n}{2}\\rfloor} \\det W=3^4\\times 19$ and $\\rank_3 W=n-1$. By Theorem \\ref{main}, we know that $v_3(L(G))\\le \\frac{1}{2} v_3(\\det W)=2$. Indeed, using the program in \\cite{wang2025}, we find that, besides the permutation matrices, there exist essentially two rational regular orthogonal matrices \n\\begin{equation*}\n Q_1=\\frac{1}{3}{\\footnotesize\n \\begin{bmatrix}\n 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & -2 & 1 & 1 & 1 & 1 & 1 \\\\\n 0 & 0 & 0 & 0 & 1 & -2 & 1 & 1 & 1 & 1 \\\\\n 0 & 0 & 0 & 0 & 1 & 1 & -2 & 1 & 1 & 1 \\\\\n 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 1 & 1 & 1 & -2 & 1 & 1 \\\\\n 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & -2 & 1 \\\\\n 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & -2 \\\\\n 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n \\end{bmatrix}}\n\\end{equation*}\nand \n\\begin{equation*}\nQ_2=\\frac{1}{9}{\\footnotesize\\begin{bmatrix}\n -3 & -3 & 3 & 3 & 3 & 0 & 0 & 0 & 0 & 6 \\\\\n 3 & 3 & 6 & -3 & -3 & 0 & 0 & 0 & 0 & 3 \\\\\n 1 & 1 & 2 & 2 & 2 & -6 & 3 & 3 & 3 & -2 \\\\\n 1 & 1 & 2 & 2 & 2 & 3 & -6 & 3 & 3 & -2 \\\\\n -5 & 4 & -1 & -1 & -1 & 3 & 3 & 3 & 3 & 1 \\\\\n 3 & 3 & -3 & 6 & -3 & 0 & 0 & 0 & 0 & 3 \\\\\n 1 & 1 & 2 & 2 & 2 & 3 & 3 & -6 & 3 & -2 \\\\\n 1 & 1 & 2 & 2 & 2 & 3 & 3 & 3 & -6 & -2 \\\\\n 4 & -5 & -1 & -1 & -1 & 3 & 3 & 3 & 3 & 1 \\\\\n 3 & 3 & -3 & -3 & 6 & 0 & 0 & 0 & 0 & 3 \\\\\n\\end{bmatrix}}\n\\end{equation*}\nsuch that both $Q_1^\\mathsf{T} A Q_1$ and $Q_2^\\mathsf{T} A Q_2$ are adjacency matrices. Thus, $G$ has two generalized cospectral mates.\n\nRecently, as a significant improvement of the criterion for a graph to have at most one cospectral mate established in \\cite{wang2023ejc}, Raza et al. \\cite{raza2026} obtained the following upper bound on the number of generalized cospectral mates of a graph.\n\\begin{theorem} [\\cite{raza2026}]\\label{ngm}\n Let $G$ be a controllable graph and $d_1,\\ldots, d_n$ be the invariant factors of the walk matrix $W(G)$. Suppose $d_{\\lceil\\frac{n}{2}\\rceil}=1$ and $d_{n-1}=2$. Let $d_n=\\prod_{i=1}^k p_i^{m_i}$ be the prime factorization of $d_n$. Then $G$ has at most $\\left(\\prod_{i=1}^{k}m_i\\right)-1$ generalized cospectral mates.\n \\end{theorem}\nUsing the result of Qiu et al. \\cite{qiu2023}, it is known that, for graphs $G$ as described in Theorem \\ref{ngm}, the number $L(G)$ must be a factor of $\\prod_{i=1}^k p_i^{m_i-1}$. Thus, $\\ell(Q)$ can take at most $\\prod_{i=1}^k m_i$ different values (including 1) when $Q$ runs through $\\mathcal{Q}(G)$. A key contribution of Raza et al.~\\cite{raza2026} is to show that, under the assumptions of Theorem \\ref{ngm}, there exists at most one matrix $Q$ in $\\mathcal{Q}(G)$ for each possible level $\\ell$ (in the sense of column permutations). Noting that the matrix $Q\\in \\mathcal{Q}(G)$ with level $\\ell=1$ is the permutation matrix, the number of generalized cospectral mates of $G$ is upper bounded by the number of possible levels, excluding 1. Thus, Theorem \\ref{ngm} holds.\n\nUsing Theorem \\ref{main} for odd primes, we can obtain a better bound on $L(G)$. Indeed, let $d_n=2^{m_1}\\prod_{i=2}^kp_i^{m_i}$ be the factorization of $d_n$, where $p_2,\\ldots,p_k$ are distinct odd primes. Then for each odd prime $p_i$, we have $v_{p_i}(L(G)) \\le \\lfloor\\frac{1}{2}{m_i}\\rfloor$ by Theorem \\ref{main}. For $p=2$, we only have $v_2(L(G))\\le m_1-1$ by the result of Qiu et al.~\\cite{qiu2023}. Thus, the number of possible levels for matrices in $\\mathcal{Q}(G)$, other than 1, is at most $m_1\\left(\\prod_{i=2}^k\\left( \\lfloor\\frac{1}{2}m_i\\rfloor+1\\right)\\right)-1$. This gives an improvement of Theorem \\ref{ngm} under the same assumptions. We state it as the following theorem.\n\\begin{theorem}\n Let $G$ be a controllable graph and $d_1,\\ldots, d_n$ be the invariant factors of the walk matrix $W(G)$. Suppose $d_{\\lceil\\frac{n}{2}\\rceil}=1$ and $d_{n-1}=2$. Let $d_n=2^{m_1}\\prod_{i=2}^k p_i^{m_i}$ be the prime factorization of $d_n$, where $p_2,\\ldots,p_k$ are distinct odd primes. Then $G$ has at most $m_1\\left(\\prod_{i=2}^k\\left( \\lfloor\\frac{1}{2}m_i\\rfloor+1\\right)\\right)-1$ generalized cospectral mates.\n\\end{theorem}\nWe end this paper by proposing a conjecture that the technical assumption on $\\rank_p W(G)$ in Theorem \\ref{main} is unnecessary for the conclusion. Moreover, the upper bound in terms of the determinant may be refined using the last two invariant factors.\n \\begin{conjecture}\n Let $G$ be a controllable graph and $p$ be an odd prime factor of $\\det W(G)$. Let $d_{n-1}$ and $d_n$ be the last two invariant factors of $W(G)$. Then $v_p(L(G))\\le \\frac{1}{2}( v_p(d_{n})+v_p(d_{n-1}))$, and in particular, $v_p(L(G))\\le \\frac{1}{2} v_p(\\det W(G))$.\n \\end{conjecture}", "post_theorem_intro_text_len": 1164, "post_theorem_intro_text": "The main innovation of this paper lies in the shift of the underlying algebraic structure from the vector space over the field $\\mathbb{Z}/p\\mathbb{Z}$ (as used in \\cite{wang2013EJC,qiu2023}) to the module over the local ring $\\mathbb{Z}/p^k\\mathbb{Z}$. While working over a ring introduces the complexity of zero divisors, many basic results from linear algebra over fields still hold for the ring $\\mathbb{Z}/p^k\\mathbb{Z}$. In particular, it possesses the ``Basis Extension Property'' (Steinitz Property), which allows us to generalize arguments from linear algebra. \n\n\tThe rest of this paper is organized as follows.  In Sec.~\\ref{pre}, we recall some  preliminaries on Smith normal forms and modules over $\\mathbb{Z}/p^k \\mathbb{Z}$.  The proof of Theorem \\ref{main} is given in Sec.~\\ref{pf}, which can be seen as a module-theoretical version of the original argument of Wang \\cite{wang2006}, combined with some improvements and simplifications developed in \\cite{wang2020}.   Some direct applications of Theorem \\ref{main} are presented in Sec.~\\ref{dis}. The paper concludes with a conjecture proposing a further possible improvement to Theorem \\ref{main}.", "sketch": "The proof of Theorem~\\ref{main} is said to be \"a module-theoretical version of the original argument of Wang \\cite{wang2006}, combined with some improvements and simplifications developed in \\cite{wang2020}.\" The paper’s approach shifts \"from the vector space over the field $\\mathbb{Z}/p\\mathbb{Z}$\" to \"the module over the local ring $\\mathbb{Z}/p^k\\mathbb{Z}$\" and uses that this ring has the \"Basis Extension Property'' (Steinitz Property), which \"allows us to generalize arguments from linear algebra.\"", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{main}\n\tLet $G$ be a controllable graph and $p$ be an odd prime such that  $\\rank_p W(G)=n-1$. Then $v_p(L(G))\\le \\frac{1}{2} v_p(\\det W(G))$.", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $G$ be an $n$-vertex controllable graph with adjacency matrix $A$ and walk matrix\n\\[\nW(G)=[e,Ae,\\ldots,A^{n-1}e],\n\\]\nwhere $e$ is the all-ones vector and controllable means that $W(G)$ is nonsingular. Define\n\\[\n\\mathcal{Q}(G)=\\{Q\\in \\mathrm{RO}_n(\\mathbb{Q}) : Q^\\mathsf{T}AQ \\text{ is a }(0,1)\\text{-matrix}\\},\n\\]\nwhere $\\mathrm{RO}_n(\\mathbb{Q})$ is the set of rational regular orthogonal matrices (orthogonal matrices whose row sums are all $1$). For a rational matrix $Q$, its level $\\ell(Q)$ is the smallest positive integer $k$ such that $kQ$ is integral, and\n\\[\nL(G)=\\operatorname{lcm}\\{\\ell(Q):Q\\in \\mathcal{Q}(G)\\}.\n\\]\nFor a prime $p$, let $v_p(m)$ denote the $p$-adic valuation of a nonzero integer $m$, and let $\\operatorname{rank}_p W(G)$ denote the rank of $W(G)$ over $\\mathbb{F}_p$. If $p$ is an odd prime such that $\\operatorname{rank}_p W(G)=n-1$, which quantitative estimate holds?", "correct_choice": {"label": "A", "text": "\\[v_p(L(G))\\le \\frac{1}{2}\\,v_p(\\det W(G)).\\]"}, "choices": [{"label": "B", "text": "\\[v_p(L(G))\\le v_p(\\det W(G))-1.\\]"}, {"label": "C", "text": "\\[v_p(L(G))\\le v_p(\\det W(G)).\\]"}, {"label": "D", "text": "\\[v_p(L(G))\\le \\frac{1}{2}\\,v_p(\\det W(G))-1.\\]"}, {"label": "E", "text": "If \\(p\\) is an odd prime such that \\(\\operatorname{rank}_p W(G)=n-1\\), then \\(p\\nmid L(G)\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "half-valuation bound replaced by earlier linear-minus-one bound", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "characteristic", "tampered_component": "dropped the factor \\(\\frac12\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "characteristic", "tampered_component": "inserted an extra subtractive constant into the valuation estimate", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "collapsed rank condition to the stronger conclusion \\(p\\nmid L(G)\\)", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct bound. It provides the setup and hypotheses, but the conclusion itself is not leaked beyond asking which estimate is valid."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: under the exact stated hypotheses, the student is asked to identify the theorem’s quantitative conclusion. It is very close to a restatement rather than a new application."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the choices include a weaker true statement, overly strong false statements, and nearby quantitative variants. However, the item mostly tests recall/recognition of the precise bound rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically aligned with common errors: dropping the factor 1/2, strengthening to p∤L(G), or inserting an extra -1. They are distinct and target realistic theorem-confusion failure modes."}, "total_score": 5, "overall_assessment": "Well-constructed distractors and no answer leakage, but the question is largely theorem recognition and is close to tautological rather than genuinely generative."}}
{"id": "2602.14266v1", "paper_link": "http://arxiv.org/abs/2602.14266v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "[Center Theorem]\\label{center}\nA canonical maximal admissible weighted center\nintersects the nc locus $X^{\\mathrm{nc}}(\\mathcal{I})$\nonly if it is entirely contained in it.", "start_pos": 22234, "end_pos": 22435, "label": "center"}, "ref_dict": {"thm:embedded-NC": "\\begin{theorem}[Embedded NC-preserving resolution]\n\\label{thm:embedded-NC}\nLet $(X,E)$ be a smooth irreducible variety, or more generally a smooth\nirreducible Deligne-Mumford stack, over a field of characteristic~$0$,\nequipped with an SNC divisor $E$, and let\n$Y \\subset X$ be a closed subscheme (respectively, a closed substack).\n\nThere exists a canonical sequence of weighted blow-ups\ndefined by the invariant $\\inv_{(X,E)}(Y)$.\n\\[\n   \\Pi: X_N \\longrightarrow \\cdots \\longrightarrow X_1\n   \\longrightarrow X_0 = X\n\\]\nwith maximal admissible centers disjoint from the normal crossings locus\n$X_*^{\\mathrm{nc}}$, such that:\n\n\\begin{enumerate}\n\\item Each center coincides with the locus where the invariant\n$\\inv_{(X,E)}(Y)$ attains its maximal value outside $X_*^{\\mathrm{nc}}$.\n\n\\item The invariant strictly decreases after each blow-up away from\nthe NC locus. In particular, the process terminates.\n\n\\item The morphism $\\Pi$ restricts to an isomorphism over\n$X_*^{\\mathrm{nc}}$.\n\n\\item The strict transform $\\Pi^s(Y)\\subset X_N$ has normal crossings\nadapted to the SNC divisor $E' \\cup D$, where\n$D$ is the exceptional divisor and $E'=\\Pi^s(E)$.\n\\end{enumerate}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 4264, "pre_theorem_intro_text": "Hironaka's theorem (1964) shows that every algebraic variety over a field\nof characteristic zero admits a resolution of singularities, and that the\nexceptional divisor can be arranged to have simple normal crossings (snc).\nSubsequent functorial algorithms refined this statement by producing\nsimplified canonical resolutions compatible with smooth morphisms.\n\nHowever, classical desingularization procedures do not distinguish between\ngenuinely singular points and those that already possess normal crossings\nstructure. From both geometric and birational perspectives this distinction\nis essential: normal crossings singularities behave, in many respects,\nas terminal objects of desingularization theory.\nThey arise naturally as boundary strata in moduli problems\n(e.g.\\ stable curves, KSBA compactifications),\nin Hodge theory (semistable reduction and degenerations),\nand in birational geometry (log pairs $(X,D)$ in the Minimal Model Program).\nBlowing them up further often obscures intrinsic structure rather than\nsimplifying it.\n\nThe guiding principle of this work is that the normal crossings locus\nshould be fixed by a canonical desingularization procedure.\nWe therefore seek a resolution algorithm that removes all singularities\noutside the normal crossings locus while leaving that locus untouched.\nThis leads to a notion of \\emph{resolution up to normal crossings}.\nResolution except for SNC singularities was established in\n\\cite{BM12,BLM12,BDMP14,BBL22},\nwhere the desingularization invariant is refined so as to avoid blowing up\nsnc points and to isolate minimal residual singularities.\nIn that setting the splitting of components is already visible\nZariski-locally.\n\nThe problem treated here is fundamentally different.\nFor nc singularities the splitting of branches may occur only after\nan \\'etale or Galois extension.\nThus the nc condition is intrinsically non-local.\nThis forces the introduction of stack-theoretic methods and more flexible\nbirational transformations, such as weighted blow-ups.\nMore generally, resolution except for \\emph{locally toric}\nsingularities was obtained in~\\cite{W22t} using logarithmic and combinatorial methods,\nbut the \\'etale-locally toric case requires new structural tools.\n\nThe purpose of this article is to construct a canonical\nresolution algorithm that removes all singularities outside the nc locus\nwhile preserving every pre-existing nc structure.\nThe method is based on weighted blow-ups adapted to the combinatorial\ntype of the singularity.\nApplied canonically, these transformations simplify several components\nsimultaneously and guide the resolution process in a controlled manner.\n\n{\\it While completing this proof, I became aware of a parallel paper in preparation by\nA.~Belotto~da~Silva and E.~Bierstone addressing, in particular, the same result as Theorem \\ref{thm:embedded-NC} in the case of  reduced normal crossings and in a non-SNC divisorial setting for exceptional divisors, \nusing different techniques~\\cite{BB26}.}\n\n Their approach combines formal splitting\ntechniques of Newton--Puiseux type with a combinatorial analysis of circulant\nsingularities near the normal crossings locus, together with weighted blow-ups.\n\nOur construction provides a functorial resolution that avoids the reduced or non-reduced nc locus and is governed\nby the following principle.\n\n\\subsection{Center Theorem}\n\nLet $X^{\\mathrm{nc}}(\\mathcal{I})$ denote the locus of points where an ideal\n$\\mathcal{I}$ (reduced or not) has at most normal crossings singularities\non a smooth variety $X$.\n\nThe NC-preserving algorithm rests on two ingredients.\n\nFirst, we use the weighted blow-up resolution framework of\nMcQuillan and Abramovich-Temkin-W\\l odarczyk\n\\cite{ATW-weighted,Marzo-McQuillan},\ntogether with its logarithmic refinement developed in\n\\cite{W22,W23}.\nThe centers are chosen as canonical maximal admissible centers\nsupported on the maximal locus of the logarithmic invariant\n$\\mathrm{inv}_{(X,E)}$, introduced in  \\cite{W22,W23}, which refines the invariant of~\\cite{ATW-weighted}.\n\nSecond and this is the key structural property we show that such\ncanonical weighted centers are rigid with respect to the nc locus.\nUnlike classical smooth centers, they cannot intersect\n$X^{\\mathrm{nc}}(\\mathcal{I})$ partially.", "context": "Hironaka's theorem (1964) shows that every algebraic variety over a field\nof characteristic zero admits a resolution of singularities, and that the\nexceptional divisor can be arranged to have simple normal crossings (snc).\nSubsequent functorial algorithms refined this statement by producing\nsimplified canonical resolutions compatible with smooth morphisms.\n\nThe guiding principle of this work is that the normal crossings locus\nshould be fixed by a canonical desingularization procedure.\nWe therefore seek a resolution algorithm that removes all singularities\noutside the normal crossings locus while leaving that locus untouched.\nThis leads to a notion of \\emph{resolution up to normal crossings}.\nResolution except for SNC singularities was established in\n\\cite{BM12,BLM12,BDMP14,BBL22},\nwhere the desingularization invariant is refined so as to avoid blowing up\nsnc points and to isolate minimal residual singularities.\nIn that setting the splitting of components is already visible\nZariski-locally.\n\nThe purpose of this article is to construct a canonical\nresolution algorithm that removes all singularities outside the nc locus\nwhile preserving every pre-existing nc structure.\nThe method is based on weighted blow-ups adapted to the combinatorial\ntype of the singularity.\nApplied canonically, these transformations simplify several components\nsimultaneously and guide the resolution process in a controlled manner.\n\nLet $X^{\\mathrm{nc}}(\\mathcal{I})$ denote the locus of points where an ideal\n$\\mathcal{I}$ (reduced or not) has at most normal crossings singularities\non a smooth variety $X$.\n\nFirst, we use the weighted blow-up resolution framework of\nMcQuillan and Abramovich-Temkin-W\\l odarczyk\n\\cite{ATW-weighted,Marzo-McQuillan},\ntogether with its logarithmic refinement developed in\n\\cite{W22,W23}.\nThe centers are chosen as canonical maximal admissible centers\nsupported on the maximal locus of the logarithmic invariant\n$\\mathrm{inv}_{(X,E)}$, introduced in \\cite{W22,W23}, which refines the invariant of~\\cite{ATW-weighted}.\n\nSecond and this is the key structural property we show that such\ncanonical weighted centers are rigid with respect to the nc locus.\nUnlike classical smooth centers, they cannot intersect\n$X^{\\mathrm{nc}}(\\mathcal{I})$ partially.", "full_context": "Hironaka's theorem (1964) shows that every algebraic variety over a field\nof characteristic zero admits a resolution of singularities, and that the\nexceptional divisor can be arranged to have simple normal crossings (snc).\nSubsequent functorial algorithms refined this statement by producing\nsimplified canonical resolutions compatible with smooth morphisms.\n\nThe guiding principle of this work is that the normal crossings locus\nshould be fixed by a canonical desingularization procedure.\nWe therefore seek a resolution algorithm that removes all singularities\noutside the normal crossings locus while leaving that locus untouched.\nThis leads to a notion of \\emph{resolution up to normal crossings}.\nResolution except for SNC singularities was established in\n\\cite{BM12,BLM12,BDMP14,BBL22},\nwhere the desingularization invariant is refined so as to avoid blowing up\nsnc points and to isolate minimal residual singularities.\nIn that setting the splitting of components is already visible\nZariski-locally.\n\nThe purpose of this article is to construct a canonical\nresolution algorithm that removes all singularities outside the nc locus\nwhile preserving every pre-existing nc structure.\nThe method is based on weighted blow-ups adapted to the combinatorial\ntype of the singularity.\nApplied canonically, these transformations simplify several components\nsimultaneously and guide the resolution process in a controlled manner.\n\nLet $X^{\\mathrm{nc}}(\\mathcal{I})$ denote the locus of points where an ideal\n$\\mathcal{I}$ (reduced or not) has at most normal crossings singularities\non a smooth variety $X$.\n\nFirst, we use the weighted blow-up resolution framework of\nMcQuillan and Abramovich-Temkin-W\\l odarczyk\n\\cite{ATW-weighted,Marzo-McQuillan},\ntogether with its logarithmic refinement developed in\n\\cite{W22,W23}.\nThe centers are chosen as canonical maximal admissible centers\nsupported on the maximal locus of the logarithmic invariant\n$\\mathrm{inv}_{(X,E)}$, introduced in \\cite{W22,W23}, which refines the invariant of~\\cite{ATW-weighted}.\n\nSecond and this is the key structural property we show that such\ncanonical weighted centers are rigid with respect to the nc locus.\nUnlike classical smooth centers, they cannot intersect\n$X^{\\mathrm{nc}}(\\mathcal{I})$ partially.\n\nFirst, we use the weighted blow-up resolution framework of\nMcQuillan and Abramovich-Temkin-W\\l odarczyk\n\\cite{ATW-weighted,Marzo-McQuillan},\ntogether with its logarithmic refinement developed in\n\\cite{W22,W23}.\nThe centers are chosen as canonical maximal admissible centers\nsupported on the maximal locus of the logarithmic invariant\n$\\mathrm{inv}_{(X,E)}$, introduced in \\cite{W22,W23}, which refines the invariant of~\\cite{ATW-weighted}.\n\nSecond and this is the key structural property we show that such\ncanonical weighted centers are rigid with respect to the nc locus.\nUnlike classical smooth centers, they cannot intersect\n$X^{\\mathrm{nc}}(\\mathcal{I})$ partially.\n\nThis rigidity forces the algorithm to preserve normal crossings:\ncenters chosen outside the nc locus remain disjoint from it.\nThe proof of the theorem forms the main technical part of the paper.\n\\subsection{Algorithmic Rule}\n\nThe algorithm proceeds as follows:\n\\begin{itemize}\n \\item The nc locus $X^{\\mathrm{nc}}(\\mathcal{I})$ is open.\n \\item At each step, one blows up a canonical admissible center associated with the\n maximum of the invariant $\\mathrm{inv}$ on the closed complement\n $X \\setminus X^{\\mathrm{nc}}(\\mathcal{I})$. By the Center Theorem, the associated\n center is disjoint from the nc locus.\n \\item The process stops when every point lies in the nc locus, so\n that $X = X^{\\mathrm{nc}}(\\mathcal{I})$.\n\\end{itemize}\n\nThere exists a canonical semi-continuous invariant\n\\[\n\\inv_{(X,E)}(\\mathcal{I}) : X \\to \\Sigma,\n\\]\nwith values in a well-ordered set (see Section~\\ref{sec:invariant},\nDefinition~\\ref{cntr}),\nand a canonical sequence of weighted blow-ups\n\\[\n \\Pi\\colon X_N \\longrightarrow \\cdots \\longrightarrow X_1\n \\longrightarrow X_0 = X\n\\]\nwith smooth centers disjoint from the normal crossings locus\n$X^{\\mathrm{nc}}_*$ such that:\n\nThere exists a canonical sequence of weighted blow-ups\ndefined by the invariant $\\inv_{(X,E)}(Y)$.\n\\[\n \\Pi: X_N \\longrightarrow \\cdots \\longrightarrow X_1\n \\longrightarrow X_0 = X\n\\]\nwith maximal admissible centers disjoint from the normal crossings locus\n$X_*^{\\mathrm{nc}}$, such that:\n\n\\begin{remark}\n\\label{rem:inv-compat-QIDL}\nLet \n\\[\n\\mathcal{A}_J\n=\n\\mathcal{O}_{X,p}\n\\bigl[x_1 t^{1/a_1},\\ldots,x_k t^{1/a_k}\\bigr]^{\\mathrm{int}}\n=\n\\bigoplus_{q\\in\\QQ_{\\ge0}} (\\mathcal{A}_J)_q t^q\n\\]\nbe a weighted center and let \n\\(\nJ=(x_1^{a_1},\\ldots,x_k^{a_k})\n\\)\nbe the corresponding $\\QQ$-ideal.\nThen the degree $1$ component satisfies\n$\n(\\mathcal{A}_J)_1 = J_{X,p}.\n$\nHence\n\\[\n\\mathcal{I}_p\\, t \\subset \\mathcal{A}_{J,p}\n\\quad\\Longleftrightarrow\\quad\n\\mathcal{I}_p \\subset J_{X,p}.\n\\]\nTherefore the definition of $\\inv_p(\\mathcal{I})$ via weighted centers\ncoincides with the previous $\\QQ$-ideal definition.\n\\end{remark}\n\nIn general, the process yields principalization of $\\mathcal{I}$:\nits total transform becomes a locally monomial ideal supported on an SNC divisor.\n\\end{theorem}\\subsection{Invariant associated with NC singularities}\n\\begin{lemma} \\label{invar} Let $\\cI$ be a Normal Crossing of codimension $r+1$ at $p\\in X$ so that in an \\'etale neighborhood $\\cI$ admits a presentation\n\\[\n\\cI = (u_1, \\ldots, u_r, m),\n\\qquad \nm = u_{r+1}^{a_{r+1}} \\cdots u_k^{a_k},\n\\]\nwhere $u_1, \\ldots, u_k$ form a partial system of local parameters and \n$a_{r+1} + \\cdots + a_k = d$, where $u_1,\\ldots,u_{r+t}$ are free and $u_{r+t+1},\\ldots,u_{r}$, are divisorial. \nThen the maximal admissile center at $p\\in X$ is given by $$\\cJ=(u_1,\\ldots,u_{r},u^d_{r+1},\\ldots,u^d_{r+t},u^d_{r+t+1},\\ldots,u^d_{k}),$$ and the invariant is given by $$\\inv_p(\\cI)=\\inv(\\cJ)=(\\underbrace{1, \\ldots, 1}_{r \\text{ times}}, \\underbrace{d, \\ldots, d}_{t \\text{ times}},\\underbrace{d_+, \\ldots, d_+}_{k-r-t \\text{ times}})$$\n\\end{lemma}\n\\begin{proof} We follow the algorithm from \\cite[Section 3.3]{ W22}. We compute the invariant in the given \\'etale neighborhood. Consider the algebra $R_1=\\cO_X[\\cI t]$.\nIts order is equal $$\\ord_p(R_1)=\\ord_p(\\cI)=1,$$ with the tangent ideal $$T(\\cI)=\\cD^{1-1}(\\cI)=\\cD^0(\\cI)=\\cI,$$ and a graded multiple maximal contact $\\overline{u}_1t$ where $$\\overline{u}_1:=(u_1,\\ldots,u_{r}).$$\nThen $$R_2:=C_{\\overline{u}_1t}(R_1)=(mt)$$ is generated by the only coefficient $m$ with respect to presenataion in $\\overline{u}_1$. Thus $$\\ord_p(R_2)=\\ord_p(mt)=\\ord_p(m)=d,$$ In this case multiple maximal contact is given by \n$\\overline{u}_2t^{1/d}$, where $$\\overline{u}_2:=(u_{r+1},\\ldots,u_k)\\subset T(m)=\\cD^{d-1}(m).$$ Note that\n$R_3=C_{\\overline{u}_2t^{1/d}}(R_2)=0$. This determines the Rees algebra of the center $$\\cA_{\\cJ}=\\cO_X[\\overline{u}_1t,\\overline{u}_2t^{1/d}],$$ associated with $\\QQ$-ideal center $$\\cJ=(\\overline{u}_1,\\overline{u}_2^d)=(u_1,\\ldots,u_r,u^d_{r+1},\\ldots,u^d_k)$$ and the invariant $$\\inv_p(\\cI)=\\inv(\\cJ)=(\\underbrace{1, \\ldots, 1}_{r \\text{ times}},, \\underbrace{d, \\ldots, d}_{t \\text{ times}},\\underbrace{d_+, \\ldots, d_+}_{k-r-t \\text{ times}}).$$", "post_theorem_intro_text_len": 3988, "post_theorem_intro_text": "This rigidity forces the algorithm to preserve normal crossings:\ncenters chosen outside the nc locus remain disjoint from it.\nThe proof of the theorem forms the main technical part of the paper.\n\\subsection{Algorithmic Rule}\n\nThe algorithm proceeds as follows:\n\\begin{itemize}\n  \\item The nc locus $X^{\\mathrm{nc}}(\\mathcal{I})$ is open.\n  \\item At each step, one blows up a canonical admissible center associated with the\n  maximum of the invariant $\\mathrm{inv}$ on the closed complement\n  $X \\smallsetminus X^{\\mathrm{nc}}(\\mathcal{I})$. By the Center Theorem, the associated\n  center is disjoint from the nc locus.\n  \\item The process stops when every point lies in the nc locus, so\n  that $X = X^{\\mathrm{nc}}(\\mathcal{I})$.\n\\end{itemize}\n\n\\subsection{Galois groups, weights, and stacks}\n\nThe obstruction to eliminating nc singularities without modifying the nc locus\nis already visible in classical examples, most notably the pinch point\n(Whitney umbrella), as explained by Koll\\'ar\n\\cite[Paragraph 8]{Kollar}; see also\n\\cite[Corollary 3.6.10]{Fujino} and \\cite[Example 1.7]{BM12}.\nAt a nonzero point of the axis, the pinch point has two analytic branches\nwhich are interchanged when one loops around the origin.\nAny birational morphism that is an isomorphism over the generic point\nof the axis preserves this monodromy.\nHence the singularity cannot be eliminated without blowing up the axis itself.\n\nThis example reflects a general feature of nc singularities.\nAlthough they are locally defined by monomial equations,\ntheir branches may split only after passing to a finite Galois extension.\nIn particular, a normal crossings (nc) singularity becomes\nsimple normal crossings (snc) on a finite Galois cover,\nand the original nc structure is recovered as the quotient by a finite\ngroup acting by permutation of branches.\n\n\\medskip\n\n\\noindent\n\\textbf{Structural principle.}\nNormal crossings singularities are \\'etale-locally simple normal crossings.\nEquivalently, nc singularities are quotients of snc singularities\nby finite Galois groups permuting their components.\n\nThus nc geometry is inherently equivariant geometry.\nSmooth blow-ups performed equivariantly on the splitting cover\ndescend to weighted blow-ups on the base,\nwith weights encoding the stabilizer and ramification data\nof the finite group action.\n\nSince these stabilizers are intrinsic to the nc structure,\nthe natural ambient category is that of Deligne-Mumford stacks:\nthe stack quotient records the finite isotropy groups,\nwhile the coarse space corresponds to the weighted transform.\nStacks therefore arise canonically in nc-preserving resolution.\n\n\\medskip\n\nTo make this structure canonical, we develop a Galois theory\nof multivariate splitting forms.\n\nLet\n\\[\nF(x_1,\\ldots,x_k)=\\prod_i \\ell_i^{a_i}\\in K[x_1,\\ldots,x_k]\n\\]\nbe a homogeneous form whose linear factors $\\ell_i$\nsplit only after a finite extension.\nWe associate to $F$ a canonical splitting field $K_F$\nwith Galois group\n\\[\nG_F=\\mathrm{Gal}(K_F/K),\n\\]\nwhere $K$ is the fraction field of a ring $R$,\nand denote by $R_F$ the integral closure of $R$ in $K_F$.\nThe morphism\n\\[\n\\operatorname{Spec} R_F[x_1,\\ldots,x_k]\\longrightarrow \\operatorname{Spec} R[x_1,\\ldots,x_k]\n\\]\nencodes the splitting behavior of $F$.\nHere $R$ is the coordinate ring of the canonical center,\nand $x_1,\\ldots,x_k$ generate its defining ideal.\n\nWe prove that the locus where $F$ has distinct linear factors\ncoincides with the \\'etale locus of this morphism.\nThus the nc locus is precisely the locus where the associated\nGalois cover is unramified.\nOn the splitting cover the initial form splits and the singularity becomes formally simple normal crossings along the canonical center; equivariant smooth centers resolving the ramification locus descend to canonical weighted centers downstairs.\n\nThis Galois-weight-stack correspondence is the structural input\nin the proof of the Center Theorem and explains\nwhy canonical weighted centers cannot partially intersect\nthe nc locus.", "sketch": "This rigidity forces the algorithm to preserve normal crossings: centers chosen outside the nc locus remain disjoint from it (the proof “forms the main technical part of the paper”). The “structural input in the proof of the Center Theorem” is that nc singularities are “\\'etale-locally simple normal crossings,” i.e. “quotients of snc singularities by finite Galois groups permuting their components,” so nc geometry is “inherently equivariant geometry.” One develops a Galois theory of “multivariate splitting forms” by associating to a homogeneous form $F=\\prod_i \\ell_i^{a_i}$ a canonical splitting field $K_F$ with Galois group $G_F$ and integral closure $R_F$, and considers the morphism $\\Spec R_F[x]\\to \\Spec R[x]$ encoding splitting. The key identification is: “the locus where $F$ has distinct linear factors coincides with the \\'etale locus of this morphism,” hence “the nc locus is precisely the locus where the associated Galois cover is unramified.” On the splitting cover “the initial form splits and the singularity becomes formally simple normal crossings along the canonical center;” then “equivariant smooth centers resolving the ramification locus descend to canonical weighted centers downstairs.” This Galois–weight–stack correspondence “explains why canonical weighted centers cannot partially intersect the nc locus,” yielding the Center Theorem’s conclusion that a canonical maximal admissible weighted center meets $X^{\\mathrm{nc}}(\\mathcal I)$ only if it is contained in it.", "expanded_sketch": "This rigidity forces the algorithm to preserve normal crossings: centers chosen outside the nc locus remain disjoint from it (the proof “forms the main technical part of the paper”). The “structural input in the proof of the Center Theorem” is that nc singularities are “\\'etale-locally simple normal crossings,” i.e. “quotients of snc singularities by finite Galois groups permuting their components,” so nc geometry is “inherently equivariant geometry.” One develops a Galois theory of “multivariate splitting forms” by associating to a homogeneous form $F=\\prod_i \\ell_i^{a_i}$ a canonical splitting field $K_F$ with Galois group $G_F$ and integral closure $R_F$, and considers the morphism $\\Spec R_F[x]\\to \\Spec R[x]$ encoding splitting. The key identification is: “the locus where $F$ has distinct linear factors coincides with the \\'etale locus of this morphism,” hence “the nc locus is precisely the locus where the associated Galois cover is unramified.” On the splitting cover “the initial form splits and the singularity becomes formally simple normal crossings along the canonical center;” then “equivariant smooth centers resolving the ramification locus descend to canonical weighted centers downstairs.” This Galois–weight–stack correspondence “explains why canonical weighted centers cannot partially intersect the nc locus,” yielding the conclusion needed in establishing the main theorem: a canonical maximal admissible weighted center meets $X^{\\mathrm{nc}}(\\mathcal I)$ only if it is contained in it.", "expanded_theorem": "[Center Theorem]\\label{center}\nA canonical maximal admissible weighted center\nintersects the nc locus $X^{\\mathrm{nc}}(\\mathcal{I})$\nonly if it is entirely contained in it.,", "theorem_type": ["Implication"], "mcq": {"question": "Let $X$ be a smooth variety, let $\\mathcal I$ be an ideal sheaf on $X$, and let $X^{\\mathrm{nc}}(\\mathcal I)$ denote the locus of points of $X$ where $\\mathcal I$ has at most normal crossings singularities. Consider a canonical maximal admissible weighted center, meaning a weighted blow-up center chosen canonically by the resolution procedure and maximal among admissible centers. Which statement about its relation to the normal crossings locus is valid?", "correct_choice": {"label": "A", "text": "A canonical maximal admissible weighted center can intersect $X^{\\mathrm{nc}}(\\mathcal I)$ only if the entire center is contained in $X^{\\mathrm{nc}}(\\mathcal I)$; equivalently, it cannot meet the normal crossings locus only partially."}, "choices": [{"label": "B", "text": "A canonical maximal admissible weighted center may intersect $X^{\\mathrm{nc}}(\\mathcal I)$ in a proper closed subset, but whenever this happens the intersection is itself a union of strata of the normal crossings locus."}, {"label": "C", "text": "If a canonical maximal admissible weighted center is disjoint from $X^{\\mathrm{nc}}(\\mathcal I)$, then it does not meet the normal crossings locus."}, {"label": "D", "text": "A canonical maximal admissible weighted center intersects $X^{\\mathrm{nc}}(\\mathcal I)$ only if, after passing to an étale neighborhood of each intersection point, the induced center is entirely contained in the pullback of $X^{\\mathrm{nc}}(\\mathcal I)$."}, {"label": "E", "text": "A canonical maximal admissible weighted center intersects $X^{\\mathrm{nc}}(\\mathcal I)$ only if its reduced support is entirely contained in $X^{\\mathrm{nc}}(\\mathcal I)$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "global rigidity against partial intersection", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "drops the nontrivial implication from meeting to containment and keeps only the disjointness case", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "etale_local", "tampered_component": "global descent from étale-local containment", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "scheme-theoretic center versus reduced support", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state or strongly hint at the correct conclusion. It asks generally for the valid relation between the canonical maximal admissible weighted center and the normal crossings locus."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: the correct option appears to state the key claim essentially directly. However, the presence of nearby variants (weaker, local, and scheme-theoretic reformulations) means it is not a pure tautological restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the strongest valid statement from subtle alternatives, especially against the weaker true distractor and the etale-local/global confusion. Still, the item mainly tests recognition of the theorem rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful. They target common failure modes: weakening the implication, confusing local with global containment, allowing partial intersection via strata, and conflating the scheme-theoretic center with its reduced support."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it leans more toward theorem recognition than genuine generative reasoning and is somewhat close to a direct statement of the underlying result."}}
{"id": "2602.14366v1", "paper_link": "http://arxiv.org/abs/2602.14366v1", "theorems_cnt": 2, "theorem": {"env_name": "theoA", "content": "\\label{thm:A}\n    Let $G$ be a finite group and $P\\in\\Syl_3(G)$. Then $|P:\\Phi(P)|=9$ if the principal $3$-block $B_0(G)$ contains exactly $6$ or $9$ $\\sigma$-invariant characters of degree coprime to $3$.", "start_pos": 10147, "end_pos": 10377, "label": "thm:A"}, "ref_dict": {"rem:thm B": "\\begin{rem}\\label{rem:thm B}\n    The hypothesis on the Sylow subgroups of $G/N$ being cyclic is only used in the second-to-last paragraph of the proof of Theorem \\ref{thm:ppalblock gam}, to apply \\cite[Thm. 3.4]{Nav04}. In general, we could assume only that, for every block $B$ of maximal defect of $G/N$ with Brauer correspondent $b$, there is a bijection\n    $\\Omega_B:\\irrtau(B)\\rightarrow\\irrtau(b)$. Moreover, if one assumes $\\Omega_B(\\psi)(1)\\leq \\psi(1)$ for all $\\psi\\in\\irrtau(B)$ and all blocks $B$ of maximal defect, then the proof of Theorem \\ref{thm:ppalblock gam} gives a bijection $\\Omega:\\Irr_{0,\\tau}(B_0(G))\\rightarrow\\Irr_{0,\\tau}(B_0(\\norm G P))$ with $\\Omega(\\chi)(1)\\leq\\chi(1)$. The existence of such bijections has been proposed recently in \\cite{Gia25}.\n\\end{rem}", "thm:B": "\\begin{theoA}\\label{thm:B}\nLet $G$ be a finite group of order divisible by $p$, let $N\\normal G$ be an abelian $p$-group and assume $G/N$ has cyclic Sylow $p$-subgroups and let $\\tau\\in\\mathcal{H}$ be a $p$-power order element. Then $\\tau$ fixes the same number of height-zero characters in $\\Irr(B_0(G))$ as in $\\Irr(B_0(\\norm G P))$, where $P\\in{\\rm Syl}_p(G)$.\n\\end{theoA}", "thm:A": "\\begin{theoA}\\label{thm:A}\n    Let $G$ be a finite group and $P\\in\\Syl_3(G)$. Then $|P:\\Phi(P)|=9$ if the principal $3$-block $B_0(G)$ contains exactly $6$ or $9$ $\\sigma$-invariant characters of degree coprime to $3$.\n\\end{theoA}"}, "pre_theorem_intro_text_len": 913, "pre_theorem_intro_text": "In Problem 12 of his famous list of problems from 1963, Brauer asked what information could be gleaned about a Sylow $p$-subgroup $P$ of a finite group $G$ from its character table. In particular, the question of whether the number of generators of $P$ could be obtained in this way has been studied in recent years, see e.g. \\cite{Riz-Sch-Val20, Nav-Riz-Sch-Val20, Mor-Sam23, Va23}. Many questions in this realm seem linked to the action of the Galois automorphism $\\sigma\\in{\\rm Gal}(\\mathbb{Q}^{ab}/\\mathbb{Q})$ that fixes $p'$-roots of unity and sends $p$-power roots of unity to their $1+p$ power. For any cyclotomic extension $\\mathbb{Q}_n:=\\mathbb{Q}(e^{2\\pi i/n})$ of $\\mathbb{Q}$, we keep the notation $\\sigma$ for the restriction of this automorphism to $\\mathbb{Q}_n$.\nHere, we contribute to this line of problems by proving the following direction of the main conjecture from \\cite{Nav-Riz-Sch-Val20}.", "context": "In Problem 12 of his famous list of problems from 1963, Brauer asked what information could be gleaned about a Sylow $p$-subgroup $P$ of a finite group $G$ from its character table. In particular, the question of whether the number of generators of $P$ could be obtained in this way has been studied in recent years, see e.g. \\cite{Riz-Sch-Val20, Nav-Riz-Sch-Val20, Mor-Sam23, Va23}. Many questions in this realm seem linked to the action of the Galois automorphism $\\sigma\\in{\\rm Gal}(\\mathbb{Q}^{ab}/\\mathbb{Q})$ that fixes $p'$-roots of unity and sends $p$-power roots of unity to their $1+p$ power. For any cyclotomic extension $\\mathbb{Q}_n:=\\mathbb{Q}(e^{2\\pi i/n})$ of $\\mathbb{Q}$, we keep the notation $\\sigma$ for the restriction of this automorphism to $\\mathbb{Q}_n$.\nHere, we contribute to this line of problems by proving the following direction of the main conjecture from \\cite{Nav-Riz-Sch-Val20}.", "full_context": "In Problem 12 of his famous list of problems from 1963, Brauer asked what information could be gleaned about a Sylow $p$-subgroup $P$ of a finite group $G$ from its character table. In particular, the question of whether the number of generators of $P$ could be obtained in this way has been studied in recent years, see e.g. \\cite{Riz-Sch-Val20, Nav-Riz-Sch-Val20, Mor-Sam23, Va23}. Many questions in this realm seem linked to the action of the Galois automorphism $\\sigma\\in{\\rm Gal}(\\mathbb{Q}^{ab}/\\mathbb{Q})$ that fixes $p'$-roots of unity and sends $p$-power roots of unity to their $1+p$ power. For any cyclotomic extension $\\mathbb{Q}_n:=\\mathbb{Q}(e^{2\\pi i/n})$ of $\\mathbb{Q}$, we keep the notation $\\sigma$ for the restriction of this automorphism to $\\mathbb{Q}_n$.\nHere, we contribute to this line of problems by proving the following direction of the main conjecture from \\cite{Nav-Riz-Sch-Val20}.\n\n\\section{Introduction}\n\nAs noted in \\cite{Nav-Riz-Sch-Val20}, Theorem \\ref{thm:A} (and its converse) would follow from the Alperin--McKay--Navarro conjecture \\cite[Conj.~B]{Nav04}, or even the more restrictive version \\cite[Conj.~D]{Isa-Nav02}. Hence, this result gives further evidence for these elusive conjectures. We remark that although the block-free version \\cite[Conj.~C]{Isa-Nav02} was recently established in \\cite{RSF25}, the blockwise version (even for principal blocks) is much further from completion and has not yet been reduced to a problem on simple groups. For this reason, finding additional evidence of this blockwise version remains pertinent.\n\nWe begin by collecting some basic facts about principal blocks. Throughout, given a prime $p$, $B_0(G)$ denotes the principal $p$-block of the finite group $G$. We denote by $\\Irr(B_0(G))$ the set of irreducible characters in $B_0(G)$ and by $\\Irr_0(B_0(G))$ the subset of those with height zero (that is, $p'$-degree). The set of $\\sigma$-invariant characters in $\\Irr_0(B_0(G))$ will be denoted $\\irrsigma(B_0(G))$, and we set $\\ksigma(B_0(G)):=|\\irrsigma(B_0(G))|$.\n\n\\begin{lem}\\label{lem:kernel of sigmas}\nLet $G$ be a finite group with $\\oh{p'}G=1$ and let $P\\in\\Syl_p(G)$. Then\n$$K=\\bigcap_{\\chi\\in\\irrsigma(B_0(G))}\\ker\\chi\\leq \\Phi(P).$$\n\\end{lem}\n\\begin{proof}\n Let $\\lambda\\in\\Irr(P/\\Phi(P))$. Write $P\\cent G P=P\\times X$ and let $\\psi=\\lambda\\times 1_X\\in\\irrsigma(B_0(P\\cent G P))$. By Lemma \\ref{lem:3.7MMSV} we have that $\\lambda^G$ contains a constituent $\\chi\\in\\irrsigma(B_0(G))$. By Frobenius reciprocity, $\\chi_{P\\cent G P}$ contains $\\lambda\\times 1_X$ so $\\chi_P$ contains $\\lambda_P$. Moreover $\\chi_{P\\cap K}=\\chi(1)1_{P\\cap K}$ contains $\\lambda_{P\\cap K}$ which shows that $\\ker{\\lambda}$ contains $P\\cap K$. Therefore $$P\\cap K\\sbs\\bigcap_{\\lambda\\in\\Irr(P/\\Phi(P))}\\ker\\lambda=\\Phi(P).$$\nBy Tate's theorem \\cite[Cor.~6.14]{Nav18}, this implies that $K$ is $p$-nilpotent. Then if $K$ is not a $p$-group, $\\oh{p'}K>1$, which contradicts $\\oh{p'}G=1$. Therefore $K$ is a normal $p$-subgroup so $K=P\\cap K\\sbs\\Phi(P)$, as desired.\n\\end{proof}\n\n\\begin{pro}\\label{pro:noelia relative sigma} Let $G$ be a finite group, $p\\in\\{2,3\\}$, and $P\\in\\Syl_p(G)$. Let $N\\normal G$ and assume $p$ divides $|G:N|$. Suppose that $\\theta\\in\\Irr(B_0(N))$ is $P\\times\\langle \\sigma\\rangle$-invariant. Then $p$ divides $ \\ksigma(B_0(G)|\\theta)$. In particular, if $\\theta$ extends to $\\hat\\theta\\in\\irrsigma(B_0(PN))$ then $\\ksigma(B_0(G)|\\theta)\\geq p$.\n\\end{pro}\n\\begin{proof}\n Arguing as in \\cite[Thm.~2.7]{Gia-Riz-Sch-Val24} we get that\n $$\\sum_{\\chi\\in\\Irr_0(B_0(G)|\\theta)}\\chi(1)^2\\equiv 0\\mod p.$$\n\n\\begin{pro}\\label{thm:ppalblock gam2}\n Let $N\\normal G$ be perfect and let $P\\in\\Syl_p(G)$. Assume every $P$-invariant character $\\theta\\in\\irrsigma(B_0(N))$ that extends to a $\\sigma$-invariant character of $PN$ also extends to a $\\sigma$-invariant character of $B_0(G_\\theta)$ and assume $G/N$ has cyclic Sylow $p$-subgroups. Then $|\\irrsigma(B_0(G))|=|\\irrsigma(B_0(N\\norm G P))|$.\n\\end{pro}\n\\begin{proof}\nWe may argue as in the proof of Theorem \\ref{thm:ppalblock gam} that every $\\chi\\in\\irrsigma(B_0(G))$ lies over a $P$-invariant $\\theta\\in\\Irr(B_0(N))$. Since $\\theta$ has a number of $G$-conjugates not divisible by $p$ and $\\langle \\sigma \\rangle$ acts on the set of $G$-conjugates of $\\theta$, it follows that $\\theta$ is also $\\sigma$-invariant. Arguing again as in Theorem \\ref{thm:ppalblock gam} we conclude that $\\chi\\in\\Irr_{0}(B_0(G))$ is $\\sigma$-invariant if and only if its Clifford correspondent $\\psi\\in\\Irr_{0}(B_0(G_\\theta))$ is $\\sigma$-invariant.\n\n\\begin{thm}\n Let $G$ be a finite group, let $B_0(G)$ be the principal 3-block of $G$, and let $P\\in\\syl 3 G$. Suppose that $k_{0,\\sigma}(B_0(G))\\in\\{6,9\\}$. Then $|P:\\Phi(P)|=9.$\n\\end{thm}\n\\begin{proof}\n We argue by induction on $|G|$. Since $\\irrsigma(B_0(G))=\\irrsigma(B_0(G/\\oh{p'}G))$ by \\cite[Thm.~9.9]{Nav98}, we may assume $\\oh{p'}G=1$. We can also assume that $P$ is not cyclic by \\cite[Thm.~A]{Riz-Sch-Val20}.\n\n\\begin{thm}\\label{thm:simple group conditions}\nLet $S$ be a nonabelian simple group of order divisible by $3$ and let $X\\in\\Syl_3(\\Aut(S))$. \n\\begin{enumerate}\n \\item One of the following holds.\n \\begin{enumerate}\n \\item There exist $1_S\\neq\\theta_1,\\theta_2\\in\\irrsigma(B_0(S))$ nonconjugate in $\\Aut(S)$ and invariant under $X$, or\n \\item There is an $X$-invariant $1_S\\neq \\theta\\in\\irrsigma(B_0(S))$ that extends to some $\\sigma$-invariant character in $B_0(T)$ for all $S\\leq T\\leq \\Aut(S)_\\theta$.\n \\end{enumerate}\n \\item The set of degrees of characters in $\\irrsigma(B_0(S))$ has size at least $3$.\n \\item If $S$ has noncyclic Sylow $3$-subgroups, then there exist $1_S\\neq \\theta_1,\\theta_2,\\theta_3\\in\\irrsigma(B_0(S))$ nonconjugate in $\\Aut(S)$. \n\\end{enumerate}\n\\end{thm}\n\n\\begin{theoA}\\label{thm:A}\n    Let $G$ be a finite group and $P\\in\\Syl_3(G)$. Then $|P:\\Phi(P)|=9$ if the principal $3$-block $B_0(G)$ contains exactly $6$ or $9$ $\\sigma$-invariant characters of degree coprime to $3$.\n\\end{theoA}", "post_theorem_intro_text_len": 3118, "post_theorem_intro_text": "As noted in \\cite{Nav-Riz-Sch-Val20}, Theorem \\ref{thm:A} (and its converse) would follow from the Alperin--McKay--Navarro conjecture \\cite[Conj.~B]{Nav04}, or even the more restrictive version  \\cite[Conj.~D]{Isa-Nav02}. Hence, this result gives further evidence for these elusive conjectures. We remark that although the block-free version \\cite[Conj.~C]{Isa-Nav02} was recently established in \\cite{RSF25}, the blockwise version (even for principal blocks) is much further from completion and has not yet been reduced to a problem on simple groups. For this reason, finding additional evidence of this blockwise version remains pertinent.  \n\nTheorem \\ref{thm:A} can be thought of simultaneously as an extension of the main result of \\cite{Nav-Riz-Sch-Val20}, which addressed the analogous question for $p=2$, and of the main result of \\cite{Gia-Riz-Sch-Val24}, which showed that $[P:P']=9$ if and only if $B_0(G)$ contains exactly 6 or 9 irreducible characters of degree coprime to $3$.\n\nOur proof of Theorem \\ref{thm:A} uses the recent work of Ketchum \\cite{eden}, which shows that the statement of Theorem \\ref{thm:A} (and its converse) hold for almost simple groups. It also uses a reduction theorem and additional results on almost simple groups analogous to those in \\cite{Gia-Riz-Sch-Val24}. Moreover, the reduction relies on beautiful work of R. Brauer \\cite{Bra76} on the structure of groups having cyclic Sylow $p$-subgroups.\n\nAs mentioned above, \\cite[Conj.~C]{Isa-Nav02} has been completed in \\cite{RSF25}. Our next main theorem considers the principal block version, which is part of \\cite[Conj.~D]{Isa-Nav02}, and is a key step towards the proof of Theorem \\ref{thm:A}. We believe it may be of independent interest. Let $\\mathcal{H}\\leq {\\rm Gal}(\\mathbb{Q}^{ab}/\\mathbb{Q})$ be the Galois group considered in \\cite{Nav04}.\n\n\\begin{theoA}\\label{thm:B}\nLet $G$ be a finite group of order divisible by $p$, let $N\\triangleleft\\, G$ be an abelian $p$-group and assume $G/N$ has cyclic Sylow $p$-subgroups and let $\\tau\\in\\mathcal{H}$ be a $p$-power order element. Then $\\tau$ fixes the same number of height-zero characters in $\\operatorname{Irr}(B_0(G))$ as in $\\operatorname{Irr}(B_0(\\norm G P))$, where $P\\in{\\rm Syl}_p(G)$.\n\\end{theoA}\n\nIn particular, \\cite[Conj.~D]{Isa-Nav02} holds for principal blocks of groups satisfying the hypotheses of Theorem \\ref{thm:B}. In fact, as explained in Remark \\ref{rem:thm B}, the same result holds by assuming only that every block of maximal defect of $G/N$ satisfies \\cite[Conj.~D]{Isa-Nav02}, or \\cite[Conj.~B]{Nav04}. (See also \\cite[Prop.~5.4]{Mar-Mar-Sch-Val24} for another related, but restrictive, case.)\n\nThe paper is structured as follows: Section \\ref{sec:aux} contains preliminary lemmas and results, as well as a Galois version of \\cite[Thm.~B]{Gia-Riz-Sch-Val24}. Theorem \\ref{thm:B} is proved in Section \\ref{sec:thm B}. Theorem \\ref{thm:A} is reduced to simple groups in Section \\ref{sec:reduction} and finally completed in Section \\ref{sec:simples}, where we also provide some evidence for the version of Theorem \\ref{thm:A} for arbitrary blocks.", "sketch": "Our proof of Theorem~\\ref{thm:A} uses the recent work of Ketchum \\cite{eden}, which shows that the statement of Theorem~\\ref{thm:A} (and its converse) hold for almost simple groups. It also uses a reduction theorem and additional results on almost simple groups analogous to those in \\cite{Gia-Riz-Sch-Val24}. Moreover, the reduction relies on beautiful work of R.\\ Brauer \\cite{Bra76} on the structure of groups having cyclic Sylow $p$-subgroups.\n\nA key step toward Theorem~\\ref{thm:A} is the “next main theorem” Theorem~\\ref{thm:B}, a principal block version (part of \\cite[Conj.~D]{Isa-Nav02}). The overall structure is: prove Theorem~\\ref{thm:B} (Section~\\ref{sec:thm B}); then reduce Theorem~\\ref{thm:A} to simple groups (Section~\\ref{sec:reduction}); and finally complete the proof in the simple/almost simple case (Section~\\ref{sec:simples}).", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:A}\n    Let $G$ be a finite group and $P\\in\\Syl_3(G)$. Then $|P:\\Phi(P)|=9$ if the principal $3$-block $B_0(G)$ contains exactly $6$ or $9$ $\\sigma$-invariant characters of degree coprime to $3$.", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $G$ be a finite group, let $P\\in\\Syl_3(G)$ be a Sylow $3$-subgroup, and let $B_0(G)$ denote the principal $3$-block of $G$. Write $\\Irr_0(B_0(G))$ for the irreducible characters in $B_0(G)$ of degree coprime to $3$ (height zero). Let $\\sigma$ be the Galois automorphism that fixes all roots of unity of order prime to $3$ and sends each root of unity of $3$-power order to its fourth power; call a character in $\\Irr_0(B_0(G))$ $\\sigma$-invariant if it is fixed by $\\sigma$. If $B_0(G)$ contains exactly $6$ or exactly $9$ such $\\sigma$-invariant characters, which quantitative statement about the Sylow $3$-subgroup holds?", "correct_choice": {"label": "A", "text": "The Frattini quotient of $P$ has order $9$; equivalently, $|P:\\Phi(P)|=9$."}, "choices": [{"label": "B", "text": "The Sylow $3$-subgroup $P$ is elementary abelian of order $9$; equivalently, $P\\cong C_3\\times C_3$."}, {"label": "C", "text": "The Frattini quotient of $P$ has order at most $9$; equivalently, $|P:\\Phi(P)|\\le 9$."}, {"label": "D", "text": "The Frattini quotient of $P$ has order $9$ whenever the principal $3$-block $B_0(G)$ contains at most $9$ $\\sigma$-invariant characters of degree coprime to $3$."}, {"label": "E", "text": "The Frattini quotient of $P$ has order $9$ if and only if the principal $3$-block $B_0(G)$ contains exactly $6$ or exactly $9$ $\\sigma$-invariant characters of degree coprime to $3$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "conclusion weakened from Frattini quotient size to full group structure", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "replaced exact value $|P:\\Phi(P)|=9$ by upper bound $|P:\\Phi(P)|\\le 9$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "exact count set $\\{6,9\\}$ replaced by broad bound \"at most $9$\"", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "one-way implication upgraded to biconditional", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the conclusion. It gives the hypotheses and asks for the resulting property of the Sylow 3-subgroup, so the answer is not leaked directly."}, "TAS": {"score": 1, "justification": "This is essentially a theorem-recall item: the stem states the hypotheses of a specific result and asks for its conclusion. The presence of nearby variants prevents complete tautology, but it is still close to a restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact conclusion from stronger or overgeneralized alternatives, but the item mostly tests recall of a known implication rather than substantial mathematical generation or derivation."}, "DQS": {"score": 1, "justification": "Several distractors are plausible mathematical confusions (exact quotient size vs full structure, implication vs biconditional, exact count vs upper bound). However, choice C is a weaker true statement, which makes the item potentially ambiguous and weakens distractor quality."}, "total_score": 5, "overall_assessment": "Moderately good theorem-recall MCQ with no major answer leakage, but it is close to a restated theorem and is weakened by a distractor that appears genuinely true as a weaker consequence."}}
{"id": "2602.14368v1", "paper_link": "http://arxiv.org/abs/2602.14368v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "[Local density of $\\mathcal{S}$ in short intervals]\\label{thm:main-elements}\nFix $\\varepsilon>0$. Let $\\theta$ satisfy $2/15+\\varepsilon<\\theta<1-\\delta_0$ for some absolute constant $\\delta_0>0$, and set $h=X^{\\theta}$.\nThen for all but $O_{\\varepsilon}\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)$ integers $x\\in[X,2X]$, one has\n\\[\n\\#\\left(\\mathcal{S}\\cap(x,x+h]\\right)\\ \\asymp\\ h.\n\\]\nThe implied constants depend at most on $\\lambda$, $\\varepsilon$ and $\\theta$.", "start_pos": 7019, "end_pos": 7514, "label": "thm:main-elements"}, "ref_dict": {"thm:erdos-short-interval": "\\begin{theorem}[Large multiplicities in short intervals]\\label{thm:erdos-short-interval}\n\\begin{enumerate}[label=(\\roman*),leftmargin=*]\n\\item There exist constants $c_1,c_2>0$ such that for $X$ sufficiently large there exists $n\\in\\left(X,X+X\\exp(-c_1\\sqrt{\\log X})\\right]$ with\n\\[\n  f_{\\mathrm{Rom}}(n)\\ \\geqslant\\ c_2\\,\\log_2 n.\n\\]\n\\item Let $0<\\theta<1$ and set $h:=X^{\\theta}$. There exist constants $c_1(\\theta),c_2(\\theta)>0$ such that, for $X$ sufficiently large, for at least $c_1(\\theta)\\,X$ integers $x\\in(X,2X]$ there exists $n\\in(x,x+h]$ with\n\\[\n  f_{\\mathrm{Rom}}(n)\\ \\geqslant\\ c_2(\\theta)\\,\\log_2 n.\n\\]\n\\end{enumerate}\n\\end{theorem}", "eq:reciprocal-sum": "\\begin{equation}\\label{eq:reciprocal-sum}\n\\frac{1}{r_1}+\\cdots+\\frac{1}{r_{s-1}}<1\\leqslant \\frac{1}{r_1}+\\cdots+\\frac{1}{r_s}.\n\\end{equation}", "thm:main-elements": "\\begin{theorem}[Local density of $\\cS$ in short intervals]\\label{thm:main-elements}\nFix $\\varepsilon>0$. Let $\\theta$ satisfy $2/15+\\varepsilon<\\theta<1-\\delta_0$ for some absolute constant $\\delta_0>0$, and set $h=X^{\\theta}$.\nThen for all but $O_{\\varepsilon}\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)$ integers $x\\in[X,2X]$, one has\n\\[\n\\#\\left(\\cS\\cap(x,x+h]\\right)\\ \\asymp\\ h.\n\\]\nThe implied constants depend at most on $\\lambda$, $\\varepsilon$ and $\\theta$.\n\\end{theorem}", "thm:main-rep": "\\begin{theorem}[Typical number of representations]\\label{thm:main-rep}\nFix $\\varepsilon>0$. Let $\\theta$ satisfy $2/15+\\varepsilon<\\theta<1-\\delta_0$ for some absolute constant $\\delta_0>0$, and set $h=X^{\\theta}$.\nThen for all but $O_{\\varepsilon}\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)$ integers $x\\in[X,2X]$, one has\n\\[\nR(x)\\ \\asymp\\ h.\n\\]\nThe implied constants depend at most on $\\lambda$, $\\varepsilon$ and $\\theta$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 3708, "pre_theorem_intro_text": "Let $\\mathbb{P}$ denote the set of prime numbers.\nRomanoff proved that the sumset $\\mathbb{P}+\\{2^k:k\\geqslant 0\\}$ has positive lower density~\\cite{Romanoff1934}, following earlier questions of de~Polignac~\\cite{dePolignac1849a,dePolignac1849b}.\nErd\\H{o}s initiated a systematic study of multiplicities and related variants~\\cite{Erdos1950} (see also van~der~Corput~\\cite{vanDerCorput1950}), and quantitative bounds for Romanoff-type representation functions have been developed more recently, for instance by Chen and Ding~\\cite{ChenDing2023}.\nSee also Elsholtz and Schlage-Puchta~\\cite{ElsholtzSchlagePuchta2018} for explicit numerical lower bounds on Romanov's constant.\n\nIn this paper we study a short-interval refinement for the structured additive set $\\cA_{\\lambda}$ introduced by Ding and Zhai~\\cite{DingZhai}.\nFor $h=X^{\\theta}$ in the range of \\Cref{thm:main-elements,thm:main-rep}, we show that for all but a stretched-exponentially small exceptional set of $x\\in[X,2X]$ the interval $(x,x+h]$ contains $\\asymp h$ elements of the sumset $\\mathbb{P}+\\cA_{\\lambda}(X)$, and the number of representations in such windows is also $\\asymp h$.\nIn the model case of the Romanoff representation function \n$$ f_{\\mathrm{Rom}}(n):=\\#\\left\\{(p,k)\\in\\mathbb{P}\\times\\mathbb N_0:\\ p+2^k=n\\right\\}, $$\nwe additionally record a short-interval Erd\\H{o}s-type statement for a positive proportion of windows, of an integer with unusually large pointwise multiplicity. We now set up the notation and state the main results.\n\nLet $X\\geqslant 3$ and set $h=X^{\\theta}$ with $\\theta\\in(0,1)$.\nFix an integer $s\\geqslant 2$ and positive real numbers $r_1,\\dots,r_s>1$ such that\n\\begin{equation}\\label{eq:reciprocal-sum}\n\\frac{1}{r_1}+\\cdots+\\frac{1}{r_{s-1}}<1\\leqslant \\frac{1}{r_1}+\\cdots+\\frac{1}{r_s}.\n\\end{equation}\nLet $\\lambda>0$ be defined by\n\\begin{equation}\\label{eq:lambda-balance}\n\\frac{1}{r_1}+\\cdots+\\frac{1}{r_{s-1}}+\\frac{1}{\\lambda r_s}\\ =\\ 1,\n\\end{equation}\nso that in particular $\\lambda\\geqslant 1$ by \\eqref{eq:reciprocal-sum}.\nDefine the (Ding--Zhai~\\cite{DingZhai}) additive set\n\\begin{equation}\\label{eq:def-A}\n\\cA_{\\lambda}\\ :=\\ \\left\\{2^{\\lfloor k_1^{r_1}\\rfloor}+\\cdots+2^{\\lfloor k_{s-1}^{r_{s-1}}\\rfloor}+2^{\\lfloor \\lfloor k_s^{\\lambda}\\rfloor^{r_s}\\rfloor}\\ :\\ k_1,\\dots,k_s\\in\\mathbb{N}\\right\\}\\subset\\mathbb{N}.\n\\end{equation}\nFor $X\\geqslant 3$ we truncate\n\\[\n\\cA_{\\lambda}(X)\\ :=\\ \\cA_{\\lambda}\\cap\\left[1,2X\\right],\n\\qquad\n\\mathcal{S}\\ :=\\ \\mathbb{P}+\\cA_{\\lambda}(X).\n\\]\nFor real $x\\geqslant 1$ we denote by\n\\[\nA_{\\lambda}(x)\\ :=\\ \\#\\left(\\cA_{\\lambda}\\cap\\left[1,x\\right]\\right)\n\\]\nthe associated counting function. The bound $A_{\\lambda}(x)\\asymp\\log x$ holds for all $x\\geqslant 3$ (see Ding--Zhai~\\cite{DingZhai}, ultimately due to Chen--Xu~\\cite{ChenXu2024}).\nIn particular,\n\\begin{equation}\\label{eq:sizeA}\n\\#\\cA_{\\lambda}(X)\\ \\asymp\\ \\log X.\n\\end{equation}\n\nIn what follows, the parameters $s,r_1,\\dots,r_s$ (and hence $\\lambda$ and the set $\\cA_{\\lambda}$) are fixed, and all implied constants are allowed to depend on this data.\n\nDefine the representation function\n\\begin{equation}\\label{eq:def-f}\nf(n)\\ :=\\ \\sum_{a\\in\\cA_{\\lambda}(X)} \\1_{\\mathbb{P}}(n-a),\n\\end{equation}\nso that $f(n)$ counts representations $n=p+a$ with $p\\in\\mathbb{P}$ and $a\\in\\cA_{\\lambda}(X)$.\nFor real $x$ define the window sums\n\\begin{equation}\\label{eq:def-R}\nR(x)\\ :=\\ \\sum_{x<n\\leqslant x+h} f(n),\n\\qquad\nQ(x)\\ :=\\ \\sum_{x<n\\leqslant x+h} f(n)^2,\n\\end{equation}\nand the corresponding count of \\emph{elements} of the sumset $\\mathcal{S}$,\n\\begin{equation}\\label{eq:def-Sx}\nS(x)\\ :=\\ \\#\\left(\\mathcal{S}\\cap(x,x+h]\\right)\\ =\\ \\sum_{x<n\\leqslant x+h}\\1_{f(n)\\geqslant 1}.\n\\end{equation}\n\n\\medskip", "context": "Let $\\mathbb{P}$ denote the set of prime numbers.\nRomanoff proved that the sumset $\\mathbb{P}+\\{2^k:k\\geqslant 0\\}$ has positive lower density~\\cite{Romanoff1934}, following earlier questions of de~Polignac~\\cite{dePolignac1849a,dePolignac1849b}.\nErd\\H{o}s initiated a systematic study of multiplicities and related variants~\\cite{Erdos1950} (see also van~der~Corput~\\cite{vanDerCorput1950}), and quantitative bounds for Romanoff-type representation functions have been developed more recently, for instance by Chen and Ding~\\cite{ChenDing2023}.\nSee also Elsholtz and Schlage-Puchta~\\cite{ElsholtzSchlagePuchta2018} for explicit numerical lower bounds on Romanov's constant.\n\nIn this paper we study a short-interval refinement for the structured additive set $\\cA_{\\lambda}$ introduced by Ding and Zhai~\\cite{DingZhai}.\nFor $h=X^{\\theta}$ in the range of \\Cref{thm:main-elements,thm:main-rep}, we show that for all but a stretched-exponentially small exceptional set of $x\\in[X,2X]$ the interval $(x,x+h]$ contains $\\asymp h$ elements of the sumset $\\mathbb{P}+\\cA_{\\lambda}(X)$, and the number of representations in such windows is also $\\asymp h$.\nIn the model case of the Romanoff representation function \n$$ f_{\\mathrm{Rom}}(n):=\\#\\left\\{(p,k)\\in\\mathbb{P}\\times\\mathbb N_0:\\ p+2^k=n\\right\\}, $$\nwe additionally record a short-interval Erd\\H{o}s-type statement for a positive proportion of windows, of an integer with unusually large pointwise multiplicity. We now set up the notation and state the main results.\n\nLet $X\\geqslant 3$ and set $h=X^{\\theta}$ with $\\theta\\in(0,1)$.\nFix an integer $s\\geqslant 2$ and positive real numbers $r_1,\\dots,r_s>1$ such that\n\\begin{equation}\\label{eq:reciprocal-sum}\n\\frac{1}{r_1}+\\cdots+\\frac{1}{r_{s-1}}<1\\leqslant \\frac{1}{r_1}+\\cdots+\\frac{1}{r_s}.\n\\end{equation}\nLet $\\lambda>0$ be defined by\n\\begin{equation}\\label{eq:lambda-balance}\n\\frac{1}{r_1}+\\cdots+\\frac{1}{r_{s-1}}+\\frac{1}{\\lambda r_s}\\ =\\ 1,\n\\end{equation}\nso that in particular $\\lambda\\geqslant 1$ by \\eqref{eq:reciprocal-sum}.\nDefine the (Ding--Zhai~\\cite{DingZhai}) additive set\n\\begin{equation}\\label{eq:def-A}\n\\cA_{\\lambda}\\ :=\\ \\left\\{2^{\\lfloor k_1^{r_1}\\rfloor}+\\cdots+2^{\\lfloor k_{s-1}^{r_{s-1}}\\rfloor}+2^{\\lfloor \\lfloor k_s^{\\lambda}\\rfloor^{r_s}\\rfloor}\\ :\\ k_1,\\dots,k_s\\in\\mathbb{N}\\right\\}\\subset\\mathbb{N}.\n\\end{equation}\nFor $X\\geqslant 3$ we truncate\n\\[\n\\cA_{\\lambda}(X)\\ :=\\ \\cA_{\\lambda}\\cap\\left[1,2X\\right],\n\\qquad\n\\mathcal{S}\\ :=\\ \\mathbb{P}+\\cA_{\\lambda}(X).\n\\]\nFor real $x\\geqslant 1$ we denote by\n\\[\nA_{\\lambda}(x)\\ :=\\ \\#\\left(\\cA_{\\lambda}\\cap\\left[1,x\\right]\\right)\n\\]\nthe associated counting function. The bound $A_{\\lambda}(x)\\asymp\\log x$ holds for all $x\\geqslant 3$ (see Ding--Zhai~\\cite{DingZhai}, ultimately due to Chen--Xu~\\cite{ChenXu2024}).\nIn particular,\n\\begin{equation}\\label{eq:sizeA}\n\\#\\cA_{\\lambda}(X)\\ \\asymp\\ \\log X.\n\\end{equation}\n\nIn what follows, the parameters $s,r_1,\\dots,r_s$ (and hence $\\lambda$ and the set $\\cA_{\\lambda}$) are fixed, and all implied constants are allowed to depend on this data.\n\nDefine the representation function\n\\begin{equation}\\label{eq:def-f}\nf(n)\\ :=\\ \\sum_{a\\in\\cA_{\\lambda}(X)} \\1_{\\mathbb{P}}(n-a),\n\\end{equation}\nso that $f(n)$ counts representations $n=p+a$ with $p\\in\\mathbb{P}$ and $a\\in\\cA_{\\lambda}(X)$.\nFor real $x$ define the window sums\n\\begin{equation}\\label{eq:def-R}\nR(x)\\ :=\\ \\sum_{x<n\\leqslant x+h} f(n),\n\\qquad\nQ(x)\\ :=\\ \\sum_{x<n\\leqslant x+h} f(n)^2,\n\\end{equation}\nand the corresponding count of \\emph{elements} of the sumset $\\mathcal{S}$,\n\\begin{equation}\\label{eq:def-Sx}\nS(x)\\ :=\\ \\#\\left(\\mathcal{S}\\cap(x,x+h]\\right)\\ =\\ \\sum_{x<n\\leqslant x+h}\\1_{f(n)\\geqslant 1}.\n\\end{equation}\n\n\\medskip", "full_context": "Let $\\mathbb{P}$ denote the set of prime numbers.\nRomanoff proved that the sumset $\\mathbb{P}+\\{2^k:k\\geqslant 0\\}$ has positive lower density~\\cite{Romanoff1934}, following earlier questions of de~Polignac~\\cite{dePolignac1849a,dePolignac1849b}.\nErd\\H{o}s initiated a systematic study of multiplicities and related variants~\\cite{Erdos1950} (see also van~der~Corput~\\cite{vanDerCorput1950}), and quantitative bounds for Romanoff-type representation functions have been developed more recently, for instance by Chen and Ding~\\cite{ChenDing2023}.\nSee also Elsholtz and Schlage-Puchta~\\cite{ElsholtzSchlagePuchta2018} for explicit numerical lower bounds on Romanov's constant.\n\nIn this paper we study a short-interval refinement for the structured additive set $\\cA_{\\lambda}$ introduced by Ding and Zhai~\\cite{DingZhai}.\nFor $h=X^{\\theta}$ in the range of \\Cref{thm:main-elements,thm:main-rep}, we show that for all but a stretched-exponentially small exceptional set of $x\\in[X,2X]$ the interval $(x,x+h]$ contains $\\asymp h$ elements of the sumset $\\mathbb{P}+\\cA_{\\lambda}(X)$, and the number of representations in such windows is also $\\asymp h$.\nIn the model case of the Romanoff representation function \n$$ f_{\\mathrm{Rom}}(n):=\\#\\left\\{(p,k)\\in\\mathbb{P}\\times\\mathbb N_0:\\ p+2^k=n\\right\\}, $$\nwe additionally record a short-interval Erd\\H{o}s-type statement for a positive proportion of windows, of an integer with unusually large pointwise multiplicity. We now set up the notation and state the main results.\n\nLet $X\\geqslant 3$ and set $h=X^{\\theta}$ with $\\theta\\in(0,1)$.\nFix an integer $s\\geqslant 2$ and positive real numbers $r_1,\\dots,r_s>1$ such that\n\\begin{equation}\\label{eq:reciprocal-sum}\n\\frac{1}{r_1}+\\cdots+\\frac{1}{r_{s-1}}<1\\leqslant \\frac{1}{r_1}+\\cdots+\\frac{1}{r_s}.\n\\end{equation}\nLet $\\lambda>0$ be defined by\n\\begin{equation}\\label{eq:lambda-balance}\n\\frac{1}{r_1}+\\cdots+\\frac{1}{r_{s-1}}+\\frac{1}{\\lambda r_s}\\ =\\ 1,\n\\end{equation}\nso that in particular $\\lambda\\geqslant 1$ by \\eqref{eq:reciprocal-sum}.\nDefine the (Ding--Zhai~\\cite{DingZhai}) additive set\n\\begin{equation}\\label{eq:def-A}\n\\cA_{\\lambda}\\ :=\\ \\left\\{2^{\\lfloor k_1^{r_1}\\rfloor}+\\cdots+2^{\\lfloor k_{s-1}^{r_{s-1}}\\rfloor}+2^{\\lfloor \\lfloor k_s^{\\lambda}\\rfloor^{r_s}\\rfloor}\\ :\\ k_1,\\dots,k_s\\in\\mathbb{N}\\right\\}\\subset\\mathbb{N}.\n\\end{equation}\nFor $X\\geqslant 3$ we truncate\n\\[\n\\cA_{\\lambda}(X)\\ :=\\ \\cA_{\\lambda}\\cap\\left[1,2X\\right],\n\\qquad\n\\mathcal{S}\\ :=\\ \\mathbb{P}+\\cA_{\\lambda}(X).\n\\]\nFor real $x\\geqslant 1$ we denote by\n\\[\nA_{\\lambda}(x)\\ :=\\ \\#\\left(\\cA_{\\lambda}\\cap\\left[1,x\\right]\\right)\n\\]\nthe associated counting function. The bound $A_{\\lambda}(x)\\asymp\\log x$ holds for all $x\\geqslant 3$ (see Ding--Zhai~\\cite{DingZhai}, ultimately due to Chen--Xu~\\cite{ChenXu2024}).\nIn particular,\n\\begin{equation}\\label{eq:sizeA}\n\\#\\cA_{\\lambda}(X)\\ \\asymp\\ \\log X.\n\\end{equation}\n\nIn what follows, the parameters $s,r_1,\\dots,r_s$ (and hence $\\lambda$ and the set $\\cA_{\\lambda}$) are fixed, and all implied constants are allowed to depend on this data.\n\nDefine the representation function\n\\begin{equation}\\label{eq:def-f}\nf(n)\\ :=\\ \\sum_{a\\in\\cA_{\\lambda}(X)} \\1_{\\mathbb{P}}(n-a),\n\\end{equation}\nso that $f(n)$ counts representations $n=p+a$ with $p\\in\\mathbb{P}$ and $a\\in\\cA_{\\lambda}(X)$.\nFor real $x$ define the window sums\n\\begin{equation}\\label{eq:def-R}\nR(x)\\ :=\\ \\sum_{x<n\\leqslant x+h} f(n),\n\\qquad\nQ(x)\\ :=\\ \\sum_{x<n\\leqslant x+h} f(n)^2,\n\\end{equation}\nand the corresponding count of \\emph{elements} of the sumset $\\mathcal{S}$,\n\\begin{equation}\\label{eq:def-Sx}\nS(x)\\ :=\\ \\#\\left(\\mathcal{S}\\cap(x,x+h]\\right)\\ =\\ \\sum_{x<n\\leqslant x+h}\\1_{f(n)\\geqslant 1}.\n\\end{equation}\n\n\\medskip\n\n\\begin{abstract}\nLet $X$ be large and let $\\PP$ denote the set of primes. Fix positive real parameters $r_1,\\dots,r_s$ and a parameter $\\lambda>0$ determined by a balancing relation, and let $\\cA_\\lambda(X)\\subset[1,2X]$ be the associated lacunary set generated by sums of powers of $2$ with polynomially growing exponents. Set $\\cS_\\lambda:=\\PP+\\cA_\\lambda(X)$.\nFix $\\varepsilon>0$, choose $\\theta$ with $2/15+\\varepsilon<\\theta<1-\\delta_0$, where $\\delta_0>0$ is an absolute constant, and set $h=X^{\\theta}$.\nWe prove that for all but $O_{\\varepsilon}\\!\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)$ values of $x\\in[X,2X]$, the short interval $(x,x+h]$ contains $\\asymp_{\\varepsilon} h$ integers of the form $p+a$ with $p\\in\\PP$ and $a\\in\\cA_\\lambda(X)$.\n\\end{abstract}\n\n\\medskip\n\nTheorem~\\ref{thm:main-elements} is the conclusion we ultimately care about: it asserts that the\nsumset $\\cS$ meets almost all intervals of length $h$ with the expected order of magnitude.\nTo access it through Cauchy--Schwarz, it is convenient to also record the corresponding\nfirst-moment statement for the representation function $f$.\n\n\\begin{theorem}[Typical number of representations]\\label{thm:main-rep}\nFix $\\varepsilon>0$. Let $\\theta$ satisfy $2/15+\\varepsilon<\\theta<1-\\delta_0$ for some absolute constant $\\delta_0>0$, and set $h=X^{\\theta}$.\nThen for all but $O_{\\varepsilon}\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)$ integers $x\\in[X,2X]$, one has\n\\[\nR(x)\\ \\asymp\\ h.\n\\]\nThe implied constants depend at most on $\\lambda$, $\\varepsilon$ and $\\theta$.\n\\end{theorem}\n\n\\begin{lemma}[Guth--Maynard, almost-all short intervals]\\label{lem:GM}\nFix $\\varepsilon>0$. There exists an absolute constant $\\delta_0>0$ such that the following holds. Let $X\\geqslant 3$ and let $y$ satisfy\n\\[\nX^{2/15+\\varepsilon}\\ \\leqslant\\ y\\ \\leqslant\\ X^{1-\\delta_0}.\n\\]\nThen there exists a constant $c=c_{\\varepsilon}>0$ such that for all but\n$O_{\\varepsilon}\\left(X\\exp\\left(-c(\\log X)^{1/4}\\right)\\right)$ integers $t\\in[X,2X]$ one has\n\\[\n\\pi(t+y)-\\pi(t)\\ =\\ \\frac{y}{\\log t}\\ +\\ O_{\\varepsilon}\\left(y\\exp\\left(-c(\\log X)^{1/4}\\right)\\right).\n\\]\nThis follows from \\cite[Cor.~1.4]{GuthMaynard2024}; the corollary is stated for real $x$, and restricting to integer $t$ gives the same exceptional-set bound.\n\\end{lemma}\n\n\\begin{proposition}[First moment lower bound]\\label{prop:first-moment}\nFix $\\varepsilon>0$ and let $\\delta_0>0$ be as in \\cref{lem:GM}. Let $\\theta$ satisfy $2/15+\\varepsilon\\leqslant\\theta<1-\\delta_0$ and set $h=X^{\\theta}$. Then there exists a set\n$\\mathcal{E}\\subset[X,2X]\\cap\\Z$ with\n\\[\n\\#\\mathcal{E}\\ \\ll_{\\varepsilon,\\lambda}\\ X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\n\\]\nsuch that\n\\[\nR(x)\\ \\gg\\ h\n\\]\nfor all $x\\in([X,2X]\\cap\\Z)\\setminus\\mathcal{E}$.\n\\end{proposition}\n\nWe apply \\cref{lem:GM} twice: first with the scale parameter $X' := X/2$ to control $\\pi(t+h)-\\pi(t)$ for $t\\in[X/2,X]$, and then with the scale parameter $X' := X$ to control it for $t\\in[X,2X]$. Since $\\theta\\geqslant 2/15+\\varepsilon$, for $X$ sufficiently large we have $(X/2)^{2/15+\\varepsilon/2}\\leqslant h$. Moreover, since $\\theta<1-\\delta_0$ with $\\delta_0>0$ fixed, for $X$ sufficiently large we also have $h=X^{\\theta}\\leqslant (X/2)^{1-\\delta_0}$. Consequently the hypotheses of \\cref{lem:GM} apply (with $\\varepsilon$ replaced by $\\varepsilon/2$) on both dyadic blocks. We therefore obtain an exceptional set $\\mathcal{E}_a\\subset[X,2X]\\cap\\Z$ with\n\\[\n\\#\\mathcal{E}_a\\ \\ll_{\\varepsilon}\\ X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\n\\]\nsuch that for all $x\\in([X,2X]\\cap\\Z)\\setminus\\mathcal{E}_a$ we have\n\\begin{equation}\\label{eq:GM-shifted}\n\\pi(x+h-a)-\\pi(x-a)\\ =\\ \\frac{h}{\\log(x-a)}\\ +\\ O_{\\varepsilon}\\left(h\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right).\n\\end{equation}\n\nFor $x\\in([X,2X]\\cap\\Z)\\setminus\\mathcal{E}$, summing \\eqref{eq:GM-shifted} over $a\\in\\cA_{\\lambda}(X;h)$ yields\n\\[\n\\begin{aligned}\nR(x)\n&\\geqslant\\ \\sum_{a\\in\\cA_{\\lambda}(X;h)}\\left(\\pi(x+h-a)-\\pi(x-a)\\right)\n\\\\\n&=\\ h\\sum_{a\\in\\cA_{\\lambda}(X;h)}\\frac{1}{\\log(x-a)}\\ +\\ O_{\\varepsilon}\\left(A_h h\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right).\n\\end{aligned}\n\\]\nSince $x-a\\asymp X$ uniformly for $x\\in[X,2X]$ and $a\\leqslant h/2\\leqslant X/2$, the main term satisfies\n\\[\n\\sum_{a\\in\\cA_{\\lambda}(X;h)}\\frac{1}{\\log(x-a)}\\ \\asymp\\ \\frac{A_h}{\\log X}\\ \\asymp\\ \\frac{\\log h}{\\log X},\n\\]\nand the error term is negligible. Since $h=X^{\\theta}$, we have $\\log h=\\theta \\log X$, hence $R(x)\\gg h$ for all $x\\notin\\mathcal{E}$.\n\\end{proof}\n\n\\begin{theorem}[Local density of $\\cS$ in short intervals]\\label{thm:main-elements}\nFix $\\varepsilon>0$. Let $\\theta$ satisfy $2/15+\\varepsilon<\\theta<1-\\delta_0$ for some absolute constant $\\delta_0>0$, and set $h=X^{\\theta}$.\nThen for all but $O_{\\varepsilon}\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)$ integers $x\\in[X,2X]$, one has\n\\[\n\\#\\left(\\cS\\cap(x,x+h]\\right)\\ \\asymp\\ h.\n\\]\nThe implied constants depend at most on $\\lambda$, $\\varepsilon$ and $\\theta$.\n\\end{theorem}", "post_theorem_intro_text_len": 2129, "post_theorem_intro_text": "Theorem~\\ref{thm:main-elements} is the conclusion we ultimately care about: it asserts that the\nsumset $\\mathcal{S}$ meets almost all intervals of length $h$ with the expected order of magnitude.\nTo access it through Cauchy--Schwarz, it is convenient to also record the corresponding\nfirst-moment statement for the representation function $f$.\n\n\\begin{theorem}[Typical number of representations]\\label{thm:main-rep}\nFix $\\varepsilon>0$. Let $\\theta$ satisfy $2/15+\\varepsilon<\\theta<1-\\delta_0$ for some absolute constant $\\delta_0>0$, and set $h=X^{\\theta}$.\nThen for all but $O_{\\varepsilon}\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)$ integers $x\\in[X,2X]$, one has\n\\[\nR(x)\\ \\asymp\\ h.\n\\]\nThe implied constants depend at most on $\\lambda$, $\\varepsilon$ and $\\theta$.\n\\end{theorem}\n\n\\medskip\n\n\\noindent\\emph{Strategy.} The pointwise upper bound $R(x)\\ll h$ follows from Brun--Titchmarsh together with $\\#\\cA_{\\lambda}(X)\\asymp\\log X$.\nThe almost-all lower bound $R(x)\\gg h$ is obtained by restricting to $a\\leqslant h/2$ and applying the Guth--Maynard~\\cite[Cor.~1.4]{GuthMaynard2024} asymptotic for primes in almost all short intervals (with a stretched-exponential exceptional set).\nTo pass from representations to elements we combine Cauchy--Schwarz, which relates $R(x)$, $S(x)$ and the local second moment $Q(x)$, with a uniform bound $Q(x)\\ll h$.\nThe latter is proved using Selberg's upper-bound sieve for prime pairs together with Chen--Ding--Xu--Zhai's estimate controlling the contribution of small prime divisors in differences $a_1-a_2$ with $a_1,a_2\\in\\cA_{\\lambda}(X)$.\n\n\\medskip\n\n\\noindent\\emph{Organization of the paper.} Section~2 establishes the first-moment estimate for the local representation count $R(x)$ in almost all short intervals. Section~3 proves a uniform local mean-square bound, leading to the required second-moment control. In Section~4 we combine these inputs with Cauchy--Schwarz to deduce Theorems~\\ref{thm:main-elements} and~\\ref{thm:main-rep}. Finally, Section~5 proves Theorem~\\ref{thm:erdos-short-interval} on large short-interval multiplicities in the Romanoff model.", "sketch": "“To access [Theorem~\\ref{thm:main-elements}] through Cauchy--Schwarz,” the paper first proves a first-moment statement for the representation function $f$, namely Theorem~\\ref{thm:main-rep} that $R(x)\\asymp h$ for almost all $x\\in[X,2X]$. For this, the “pointwise upper bound $R(x)\\ll h$ follows from Brun--Titchmarsh together with $\\#\\cA_{\\lambda}(X)\\asymp\\log X$,” while the “almost-all lower bound $R(x)\\gg h$ is obtained by restricting to $a\\leqslant h/2$ and applying the Guth--Maynard ... asymptotic for primes in almost all short intervals (with a stretched-exponential exceptional set).”\n\n“To pass from representations to elements” (i.e. to deduce Theorem~\\ref{thm:main-elements}), they “combine Cauchy--Schwarz, which relates $R(x)$, $S(x)$ and the local second moment $Q(x)$, with a uniform bound $Q(x)\\ll h$.” This uniform second-moment bound “is proved using Selberg's upper-bound sieve for prime pairs together with Chen--Ding--Xu--Zhai's estimate controlling the contribution of small prime divisors in differences $a_1-a_2$ with $a_1,a_2\\in\\cA_{\\lambda}(X)$.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "[Local density of $\\mathcal{S}$ in short intervals]\\label{thm:main-elements}\nFix $\\varepsilon>0$. Let $\\theta$ satisfy $2/15+\\varepsilon<\\theta<1-\\delta_0$ for some absolute constant $\\delta_0>0$, and set $h=X^{\\theta}$.\nThen for all but $O_{\\varepsilon}\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)$ integers $x\\in[X,2X]$, one has\n\\[\n\\#\\left(\\mathcal{S}\\cap(x,x+h]\\right)\\ \\asymp\\ h.\n\\]\nThe implied constants depend at most on $\\lambda$, $\\varepsilon$ and $\\theta$.", "theorem_type": ["Universal", "Asymptotic or Limit"], "mcq": {"question": "Let \\(\\mathbb P\\) denote the prime numbers. Fix an integer \\(s\\ge 2\\) and real numbers \\(r_1,\\dots,r_s>1\\) such that\n\\[\n\\frac{1}{r_1}+\\cdots+\\frac{1}{r_{s-1}}<1\\le \\frac{1}{r_1}+\\cdots+\\frac{1}{r_s},\n\\]\nand define \\(\\lambda>0\\) by\n\\[\n\\frac{1}{r_1}+\\cdots+\\frac{1}{r_{s-1}}+\\frac{1}{\\lambda r_s}=1.\n\\]\nSet\n\\[\n\\mathcal A_\\lambda:=\\left\\{2^{\\lfloor k_1^{r_1}\\rfloor}+\\cdots+2^{\\lfloor k_{s-1}^{r_{s-1}}\\rfloor}+2^{\\lfloor \\lfloor k_s^{\\lambda}\\rfloor^{r_s}\\rfloor}: k_1,\\dots,k_s\\in\\mathbb N\\right\\},\n\\]\n\\[\n\\mathcal A_\\lambda(X):=\\mathcal A_\\lambda\\cap[1,2X],\\qquad \\mathcal S:=\\mathbb P+\\mathcal A_\\lambda(X).\n\\]\nFix \\(\\varepsilon>0\\), choose \\(\\theta\\) with \\(2/15+\\varepsilon<\\theta<1-\\delta_0\\), where \\(\\delta_0>0\\) is an absolute constant, and set \\(h=X^{\\theta}\\). Which statement holds for all but \\(O_{\\varepsilon}\\!\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)\\) integers \\(x\\in[X,2X]\\cap\\mathbb Z\\)?", "correct_choice": {"label": "A", "text": "\\[\\#\\bigl(\\mathcal S\\cap(x,x+h]\\bigr)\\asymp h,\\]\nthat is, for all but that exceptional set of integers \\(x\\), the interval \\((x,x+h]\\) contains on the order of \\(h\\) integers of the form \\(p+a\\) with \\(p\\in\\mathbb P\\) and \\(a\\in\\mathcal A_\\lambda(X)\\); the implied constants depend at most on \\(\\lambda\\), \\(\\varepsilon\\), and \\(\\theta\\)."}, "choices": [{"label": "B", "text": "\\[\\#\\bigl(\\mathcal S\\cap(x,x+h]\\bigr)\\asymp h,\\]\nfor every integer \\(x\\in[X,2X]\\cap\\mathbb Z\\); equivalently, there is no exceptional set, and the implied constants depend at most on \\(\\lambda\\), \\(\\varepsilon\\), and \\(\\theta\\)."}, {"label": "C", "text": "\\[\\#\\bigl(\\mathcal S\\cap(x,x+h]\\bigr)\\gg h,\\]\nfor all but \\(O_{\\varepsilon}\\!\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)\\) integers \\(x\\in[X,2X]\\cap\\mathbb Z\\); the implied constant depends at most on \\(\\lambda\\), \\(\\varepsilon\\), and \\(\\theta\\)."}, {"label": "D", "text": "\\[\\#\\bigl(\\mathcal S\\cap(x,x+h]\\bigr)\\asymp h,\\]\nfor all but \\(O_{\\varepsilon}\\!\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)\\) real numbers \\(x\\in[X,2X]\\); the implied constants depend at most on \\(\\lambda\\), \\(\\varepsilon\\), and \\(\\theta\\)."}, {"label": "E", "text": "\\[\\#\\bigl(\\mathcal S\\cap(x,x+h]\\bigr)=h+O_{\\varepsilon}\\!\\left(h\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right),\\]\nfor all but \\(O_{\\varepsilon}\\!\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)\\) integers \\(x\\in[X,2X]\\cap\\mathbb Z\\), with implied constants depending at most on \\(\\lambda\\), \\(\\varepsilon\\), and \\(\\theta\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "presence_of_exceptional_set", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped_upper_bound_in_asymp", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "integer_domain_for_x", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "order_of_magnitude_replaced_by_pointwise_asymptotic", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It sets up the hypotheses and asks for the resulting asymptotic statement, without giving away the exceptional-set size or the exact comparability conclusion."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the stem presents the full hypotheses of a specific result and asks which asymptotic conclusion holds. The correct answer is the theorem statement almost verbatim rather than a genuinely derived consequence."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the student must distinguish the exact strength of the result from stronger, weaker, or mis-scaled alternatives. However, the task is still mainly recognition of the precise theorem statement rather than substantive mathematical generation."}, "DQS": {"score": 2, "justification": "The distractors are strong: one is an overstrong uniform version, one is a weaker true-looking statement, one gives the wrong local density scale, and one replaces an almost-all estimate with a sharp pointwise asymptotic. These reflect realistic failure modes in reading asymptotic theorems."}, "total_score": 5, "overall_assessment": "A solid theorem-discrimination MCQ with good distractors and no direct answer leakage, but it is largely a restatement/recognition question rather than a generative reasoning task."}}
{"id": "2602.14368v1", "paper_link": "http://arxiv.org/abs/2602.14368v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "[Local density of $\\mathcal{S}$ in short intervals]\\label{thm:main-elements}\nFix $\\varepsilon>0$. Let $\\theta$ satisfy $2/15+\\varepsilon<\\theta<1-\\delta_0$ for some absolute constant $\\delta_0>0$, and set $h=X^{\\theta}$.\nThen for all but $O_{\\varepsilon}\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)$ integers $x\\in[X,2X]$, one has\n\\[\n\\#\\left(\\mathcal{S}\\cap(x,x+h]\\right)\\ \\asymp\\ h.\n\\]\nThe implied constants depend at most on $\\lambda$, $\\varepsilon$ and $\\theta$.", "start_pos": 7019, "end_pos": 7514, "label": "thm:main-elements"}, "ref_dict": {"thm:erdos-short-interval": "\\begin{theorem}[Large multiplicities in short intervals]\\label{thm:erdos-short-interval}\n\\begin{enumerate}[label=(\\roman*),leftmargin=*]\n\\item There exist constants $c_1,c_2>0$ such that for $X$ sufficiently large there exists $n\\in\\left(X,X+X\\exp(-c_1\\sqrt{\\log X})\\right]$ with\n\\[\n  f_{\\mathrm{Rom}}(n)\\ \\geqslant\\ c_2\\,\\log_2 n.\n\\]\n\\item Let $0<\\theta<1$ and set $h:=X^{\\theta}$. There exist constants $c_1(\\theta),c_2(\\theta)>0$ such that, for $X$ sufficiently large, for at least $c_1(\\theta)\\,X$ integers $x\\in(X,2X]$ there exists $n\\in(x,x+h]$ with\n\\[\n  f_{\\mathrm{Rom}}(n)\\ \\geqslant\\ c_2(\\theta)\\,\\log_2 n.\n\\]\n\\end{enumerate}\n\\end{theorem}", "eq:reciprocal-sum": "\\begin{equation}\\label{eq:reciprocal-sum}\n\\frac{1}{r_1}+\\cdots+\\frac{1}{r_{s-1}}<1\\leqslant \\frac{1}{r_1}+\\cdots+\\frac{1}{r_s}.\n\\end{equation}", "thm:main-elements": "\\begin{theorem}[Local density of $\\cS$ in short intervals]\\label{thm:main-elements}\nFix $\\varepsilon>0$. Let $\\theta$ satisfy $2/15+\\varepsilon<\\theta<1-\\delta_0$ for some absolute constant $\\delta_0>0$, and set $h=X^{\\theta}$.\nThen for all but $O_{\\varepsilon}\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)$ integers $x\\in[X,2X]$, one has\n\\[\n\\#\\left(\\cS\\cap(x,x+h]\\right)\\ \\asymp\\ h.\n\\]\nThe implied constants depend at most on $\\lambda$, $\\varepsilon$ and $\\theta$.\n\\end{theorem}", "thm:main-rep": "\\begin{theorem}[Typical number of representations]\\label{thm:main-rep}\nFix $\\varepsilon>0$. Let $\\theta$ satisfy $2/15+\\varepsilon<\\theta<1-\\delta_0$ for some absolute constant $\\delta_0>0$, and set $h=X^{\\theta}$.\nThen for all but $O_{\\varepsilon}\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)$ integers $x\\in[X,2X]$, one has\n\\[\nR(x)\\ \\asymp\\ h.\n\\]\nThe implied constants depend at most on $\\lambda$, $\\varepsilon$ and $\\theta$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 3708, "pre_theorem_intro_text": "Let $\\mathbb{P}$ denote the set of prime numbers.\nRomanoff proved that the sumset $\\mathbb{P}+\\{2^k:k\\geqslant 0\\}$ has positive lower density~\\cite{Romanoff1934}, following earlier questions of de~Polignac~\\cite{dePolignac1849a,dePolignac1849b}.\nErd\\H{o}s initiated a systematic study of multiplicities and related variants~\\cite{Erdos1950} (see also van~der~Corput~\\cite{vanDerCorput1950}), and quantitative bounds for Romanoff-type representation functions have been developed more recently, for instance by Chen and Ding~\\cite{ChenDing2023}.\nSee also Elsholtz and Schlage-Puchta~\\cite{ElsholtzSchlagePuchta2018} for explicit numerical lower bounds on Romanov's constant.\n\nIn this paper we study a short-interval refinement for the structured additive set $\\cA_{\\lambda}$ introduced by Ding and Zhai~\\cite{DingZhai}.\nFor $h=X^{\\theta}$ in the range of \\Cref{thm:main-elements,thm:main-rep}, we show that for all but a stretched-exponentially small exceptional set of $x\\in[X,2X]$ the interval $(x,x+h]$ contains $\\asymp h$ elements of the sumset $\\mathbb{P}+\\cA_{\\lambda}(X)$, and the number of representations in such windows is also $\\asymp h$.\nIn the model case of the Romanoff representation function \n$$ f_{\\mathrm{Rom}}(n):=\\#\\left\\{(p,k)\\in\\mathbb{P}\\times\\mathbb N_0:\\ p+2^k=n\\right\\}, $$\nwe additionally record a short-interval Erd\\H{o}s-type statement for a positive proportion of windows, of an integer with unusually large pointwise multiplicity. We now set up the notation and state the main results.\n\nLet $X\\geqslant 3$ and set $h=X^{\\theta}$ with $\\theta\\in(0,1)$.\nFix an integer $s\\geqslant 2$ and positive real numbers $r_1,\\dots,r_s>1$ such that\n\\begin{equation}\\label{eq:reciprocal-sum}\n\\frac{1}{r_1}+\\cdots+\\frac{1}{r_{s-1}}<1\\leqslant \\frac{1}{r_1}+\\cdots+\\frac{1}{r_s}.\n\\end{equation}\nLet $\\lambda>0$ be defined by\n\\begin{equation}\\label{eq:lambda-balance}\n\\frac{1}{r_1}+\\cdots+\\frac{1}{r_{s-1}}+\\frac{1}{\\lambda r_s}\\ =\\ 1,\n\\end{equation}\nso that in particular $\\lambda\\geqslant 1$ by \\eqref{eq:reciprocal-sum}.\nDefine the (Ding--Zhai~\\cite{DingZhai}) additive set\n\\begin{equation}\\label{eq:def-A}\n\\cA_{\\lambda}\\ :=\\ \\left\\{2^{\\lfloor k_1^{r_1}\\rfloor}+\\cdots+2^{\\lfloor k_{s-1}^{r_{s-1}}\\rfloor}+2^{\\lfloor \\lfloor k_s^{\\lambda}\\rfloor^{r_s}\\rfloor}\\ :\\ k_1,\\dots,k_s\\in\\mathbb{N}\\right\\}\\subset\\mathbb{N}.\n\\end{equation}\nFor $X\\geqslant 3$ we truncate\n\\[\n\\cA_{\\lambda}(X)\\ :=\\ \\cA_{\\lambda}\\cap\\left[1,2X\\right],\n\\qquad\n\\mathcal{S}\\ :=\\ \\mathbb{P}+\\cA_{\\lambda}(X).\n\\]\nFor real $x\\geqslant 1$ we denote by\n\\[\nA_{\\lambda}(x)\\ :=\\ \\#\\left(\\cA_{\\lambda}\\cap\\left[1,x\\right]\\right)\n\\]\nthe associated counting function. The bound $A_{\\lambda}(x)\\asymp\\log x$ holds for all $x\\geqslant 3$ (see Ding--Zhai~\\cite{DingZhai}, ultimately due to Chen--Xu~\\cite{ChenXu2024}).\nIn particular,\n\\begin{equation}\\label{eq:sizeA}\n\\#\\cA_{\\lambda}(X)\\ \\asymp\\ \\log X.\n\\end{equation}\n\nIn what follows, the parameters $s,r_1,\\dots,r_s$ (and hence $\\lambda$ and the set $\\cA_{\\lambda}$) are fixed, and all implied constants are allowed to depend on this data.\n\nDefine the representation function\n\\begin{equation}\\label{eq:def-f}\nf(n)\\ :=\\ \\sum_{a\\in\\cA_{\\lambda}(X)} \\1_{\\mathbb{P}}(n-a),\n\\end{equation}\nso that $f(n)$ counts representations $n=p+a$ with $p\\in\\mathbb{P}$ and $a\\in\\cA_{\\lambda}(X)$.\nFor real $x$ define the window sums\n\\begin{equation}\\label{eq:def-R}\nR(x)\\ :=\\ \\sum_{x<n\\leqslant x+h} f(n),\n\\qquad\nQ(x)\\ :=\\ \\sum_{x<n\\leqslant x+h} f(n)^2,\n\\end{equation}\nand the corresponding count of \\emph{elements} of the sumset $\\mathcal{S}$,\n\\begin{equation}\\label{eq:def-Sx}\nS(x)\\ :=\\ \\#\\left(\\mathcal{S}\\cap(x,x+h]\\right)\\ =\\ \\sum_{x<n\\leqslant x+h}\\1_{f(n)\\geqslant 1}.\n\\end{equation}\n\n\\medskip", "context": "Let $\\mathbb{P}$ denote the set of prime numbers.\nRomanoff proved that the sumset $\\mathbb{P}+\\{2^k:k\\geqslant 0\\}$ has positive lower density~\\cite{Romanoff1934}, following earlier questions of de~Polignac~\\cite{dePolignac1849a,dePolignac1849b}.\nErd\\H{o}s initiated a systematic study of multiplicities and related variants~\\cite{Erdos1950} (see also van~der~Corput~\\cite{vanDerCorput1950}), and quantitative bounds for Romanoff-type representation functions have been developed more recently, for instance by Chen and Ding~\\cite{ChenDing2023}.\nSee also Elsholtz and Schlage-Puchta~\\cite{ElsholtzSchlagePuchta2018} for explicit numerical lower bounds on Romanov's constant.\n\nIn this paper we study a short-interval refinement for the structured additive set $\\cA_{\\lambda}$ introduced by Ding and Zhai~\\cite{DingZhai}.\nFor $h=X^{\\theta}$ in the range of \\Cref{thm:main-elements,thm:main-rep}, we show that for all but a stretched-exponentially small exceptional set of $x\\in[X,2X]$ the interval $(x,x+h]$ contains $\\asymp h$ elements of the sumset $\\mathbb{P}+\\cA_{\\lambda}(X)$, and the number of representations in such windows is also $\\asymp h$.\nIn the model case of the Romanoff representation function \n$$ f_{\\mathrm{Rom}}(n):=\\#\\left\\{(p,k)\\in\\mathbb{P}\\times\\mathbb N_0:\\ p+2^k=n\\right\\}, $$\nwe additionally record a short-interval Erd\\H{o}s-type statement for a positive proportion of windows, of an integer with unusually large pointwise multiplicity. We now set up the notation and state the main results.\n\nLet $X\\geqslant 3$ and set $h=X^{\\theta}$ with $\\theta\\in(0,1)$.\nFix an integer $s\\geqslant 2$ and positive real numbers $r_1,\\dots,r_s>1$ such that\n\\begin{equation}\\label{eq:reciprocal-sum}\n\\frac{1}{r_1}+\\cdots+\\frac{1}{r_{s-1}}<1\\leqslant \\frac{1}{r_1}+\\cdots+\\frac{1}{r_s}.\n\\end{equation}\nLet $\\lambda>0$ be defined by\n\\begin{equation}\\label{eq:lambda-balance}\n\\frac{1}{r_1}+\\cdots+\\frac{1}{r_{s-1}}+\\frac{1}{\\lambda r_s}\\ =\\ 1,\n\\end{equation}\nso that in particular $\\lambda\\geqslant 1$ by \\eqref{eq:reciprocal-sum}.\nDefine the (Ding--Zhai~\\cite{DingZhai}) additive set\n\\begin{equation}\\label{eq:def-A}\n\\cA_{\\lambda}\\ :=\\ \\left\\{2^{\\lfloor k_1^{r_1}\\rfloor}+\\cdots+2^{\\lfloor k_{s-1}^{r_{s-1}}\\rfloor}+2^{\\lfloor \\lfloor k_s^{\\lambda}\\rfloor^{r_s}\\rfloor}\\ :\\ k_1,\\dots,k_s\\in\\mathbb{N}\\right\\}\\subset\\mathbb{N}.\n\\end{equation}\nFor $X\\geqslant 3$ we truncate\n\\[\n\\cA_{\\lambda}(X)\\ :=\\ \\cA_{\\lambda}\\cap\\left[1,2X\\right],\n\\qquad\n\\mathcal{S}\\ :=\\ \\mathbb{P}+\\cA_{\\lambda}(X).\n\\]\nFor real $x\\geqslant 1$ we denote by\n\\[\nA_{\\lambda}(x)\\ :=\\ \\#\\left(\\cA_{\\lambda}\\cap\\left[1,x\\right]\\right)\n\\]\nthe associated counting function. The bound $A_{\\lambda}(x)\\asymp\\log x$ holds for all $x\\geqslant 3$ (see Ding--Zhai~\\cite{DingZhai}, ultimately due to Chen--Xu~\\cite{ChenXu2024}).\nIn particular,\n\\begin{equation}\\label{eq:sizeA}\n\\#\\cA_{\\lambda}(X)\\ \\asymp\\ \\log X.\n\\end{equation}\n\nIn what follows, the parameters $s,r_1,\\dots,r_s$ (and hence $\\lambda$ and the set $\\cA_{\\lambda}$) are fixed, and all implied constants are allowed to depend on this data.\n\nDefine the representation function\n\\begin{equation}\\label{eq:def-f}\nf(n)\\ :=\\ \\sum_{a\\in\\cA_{\\lambda}(X)} \\1_{\\mathbb{P}}(n-a),\n\\end{equation}\nso that $f(n)$ counts representations $n=p+a$ with $p\\in\\mathbb{P}$ and $a\\in\\cA_{\\lambda}(X)$.\nFor real $x$ define the window sums\n\\begin{equation}\\label{eq:def-R}\nR(x)\\ :=\\ \\sum_{x<n\\leqslant x+h} f(n),\n\\qquad\nQ(x)\\ :=\\ \\sum_{x<n\\leqslant x+h} f(n)^2,\n\\end{equation}\nand the corresponding count of \\emph{elements} of the sumset $\\mathcal{S}$,\n\\begin{equation}\\label{eq:def-Sx}\nS(x)\\ :=\\ \\#\\left(\\mathcal{S}\\cap(x,x+h]\\right)\\ =\\ \\sum_{x<n\\leqslant x+h}\\1_{f(n)\\geqslant 1}.\n\\end{equation}\n\n\\medskip", "full_context": "Let $\\mathbb{P}$ denote the set of prime numbers.\nRomanoff proved that the sumset $\\mathbb{P}+\\{2^k:k\\geqslant 0\\}$ has positive lower density~\\cite{Romanoff1934}, following earlier questions of de~Polignac~\\cite{dePolignac1849a,dePolignac1849b}.\nErd\\H{o}s initiated a systematic study of multiplicities and related variants~\\cite{Erdos1950} (see also van~der~Corput~\\cite{vanDerCorput1950}), and quantitative bounds for Romanoff-type representation functions have been developed more recently, for instance by Chen and Ding~\\cite{ChenDing2023}.\nSee also Elsholtz and Schlage-Puchta~\\cite{ElsholtzSchlagePuchta2018} for explicit numerical lower bounds on Romanov's constant.\n\nIn this paper we study a short-interval refinement for the structured additive set $\\cA_{\\lambda}$ introduced by Ding and Zhai~\\cite{DingZhai}.\nFor $h=X^{\\theta}$ in the range of \\Cref{thm:main-elements,thm:main-rep}, we show that for all but a stretched-exponentially small exceptional set of $x\\in[X,2X]$ the interval $(x,x+h]$ contains $\\asymp h$ elements of the sumset $\\mathbb{P}+\\cA_{\\lambda}(X)$, and the number of representations in such windows is also $\\asymp h$.\nIn the model case of the Romanoff representation function \n$$ f_{\\mathrm{Rom}}(n):=\\#\\left\\{(p,k)\\in\\mathbb{P}\\times\\mathbb N_0:\\ p+2^k=n\\right\\}, $$\nwe additionally record a short-interval Erd\\H{o}s-type statement for a positive proportion of windows, of an integer with unusually large pointwise multiplicity. We now set up the notation and state the main results.\n\nLet $X\\geqslant 3$ and set $h=X^{\\theta}$ with $\\theta\\in(0,1)$.\nFix an integer $s\\geqslant 2$ and positive real numbers $r_1,\\dots,r_s>1$ such that\n\\begin{equation}\\label{eq:reciprocal-sum}\n\\frac{1}{r_1}+\\cdots+\\frac{1}{r_{s-1}}<1\\leqslant \\frac{1}{r_1}+\\cdots+\\frac{1}{r_s}.\n\\end{equation}\nLet $\\lambda>0$ be defined by\n\\begin{equation}\\label{eq:lambda-balance}\n\\frac{1}{r_1}+\\cdots+\\frac{1}{r_{s-1}}+\\frac{1}{\\lambda r_s}\\ =\\ 1,\n\\end{equation}\nso that in particular $\\lambda\\geqslant 1$ by \\eqref{eq:reciprocal-sum}.\nDefine the (Ding--Zhai~\\cite{DingZhai}) additive set\n\\begin{equation}\\label{eq:def-A}\n\\cA_{\\lambda}\\ :=\\ \\left\\{2^{\\lfloor k_1^{r_1}\\rfloor}+\\cdots+2^{\\lfloor k_{s-1}^{r_{s-1}}\\rfloor}+2^{\\lfloor \\lfloor k_s^{\\lambda}\\rfloor^{r_s}\\rfloor}\\ :\\ k_1,\\dots,k_s\\in\\mathbb{N}\\right\\}\\subset\\mathbb{N}.\n\\end{equation}\nFor $X\\geqslant 3$ we truncate\n\\[\n\\cA_{\\lambda}(X)\\ :=\\ \\cA_{\\lambda}\\cap\\left[1,2X\\right],\n\\qquad\n\\mathcal{S}\\ :=\\ \\mathbb{P}+\\cA_{\\lambda}(X).\n\\]\nFor real $x\\geqslant 1$ we denote by\n\\[\nA_{\\lambda}(x)\\ :=\\ \\#\\left(\\cA_{\\lambda}\\cap\\left[1,x\\right]\\right)\n\\]\nthe associated counting function. The bound $A_{\\lambda}(x)\\asymp\\log x$ holds for all $x\\geqslant 3$ (see Ding--Zhai~\\cite{DingZhai}, ultimately due to Chen--Xu~\\cite{ChenXu2024}).\nIn particular,\n\\begin{equation}\\label{eq:sizeA}\n\\#\\cA_{\\lambda}(X)\\ \\asymp\\ \\log X.\n\\end{equation}\n\nIn what follows, the parameters $s,r_1,\\dots,r_s$ (and hence $\\lambda$ and the set $\\cA_{\\lambda}$) are fixed, and all implied constants are allowed to depend on this data.\n\nDefine the representation function\n\\begin{equation}\\label{eq:def-f}\nf(n)\\ :=\\ \\sum_{a\\in\\cA_{\\lambda}(X)} \\1_{\\mathbb{P}}(n-a),\n\\end{equation}\nso that $f(n)$ counts representations $n=p+a$ with $p\\in\\mathbb{P}$ and $a\\in\\cA_{\\lambda}(X)$.\nFor real $x$ define the window sums\n\\begin{equation}\\label{eq:def-R}\nR(x)\\ :=\\ \\sum_{x<n\\leqslant x+h} f(n),\n\\qquad\nQ(x)\\ :=\\ \\sum_{x<n\\leqslant x+h} f(n)^2,\n\\end{equation}\nand the corresponding count of \\emph{elements} of the sumset $\\mathcal{S}$,\n\\begin{equation}\\label{eq:def-Sx}\nS(x)\\ :=\\ \\#\\left(\\mathcal{S}\\cap(x,x+h]\\right)\\ =\\ \\sum_{x<n\\leqslant x+h}\\1_{f(n)\\geqslant 1}.\n\\end{equation}\n\n\\medskip\n\n\\begin{abstract}\nLet $X$ be large and let $\\PP$ denote the set of primes. Fix positive real parameters $r_1,\\dots,r_s$ and a parameter $\\lambda>0$ determined by a balancing relation, and let $\\cA_\\lambda(X)\\subset[1,2X]$ be the associated lacunary set generated by sums of powers of $2$ with polynomially growing exponents. Set $\\cS_\\lambda:=\\PP+\\cA_\\lambda(X)$.\nFix $\\varepsilon>0$, choose $\\theta$ with $2/15+\\varepsilon<\\theta<1-\\delta_0$, where $\\delta_0>0$ is an absolute constant, and set $h=X^{\\theta}$.\nWe prove that for all but $O_{\\varepsilon}\\!\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)$ values of $x\\in[X,2X]$, the short interval $(x,x+h]$ contains $\\asymp_{\\varepsilon} h$ integers of the form $p+a$ with $p\\in\\PP$ and $a\\in\\cA_\\lambda(X)$.\n\\end{abstract}\n\n\\medskip\n\nTheorem~\\ref{thm:main-elements} is the conclusion we ultimately care about: it asserts that the\nsumset $\\cS$ meets almost all intervals of length $h$ with the expected order of magnitude.\nTo access it through Cauchy--Schwarz, it is convenient to also record the corresponding\nfirst-moment statement for the representation function $f$.\n\n\\begin{theorem}[Typical number of representations]\\label{thm:main-rep}\nFix $\\varepsilon>0$. Let $\\theta$ satisfy $2/15+\\varepsilon<\\theta<1-\\delta_0$ for some absolute constant $\\delta_0>0$, and set $h=X^{\\theta}$.\nThen for all but $O_{\\varepsilon}\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)$ integers $x\\in[X,2X]$, one has\n\\[\nR(x)\\ \\asymp\\ h.\n\\]\nThe implied constants depend at most on $\\lambda$, $\\varepsilon$ and $\\theta$.\n\\end{theorem}\n\n\\begin{lemma}[Guth--Maynard, almost-all short intervals]\\label{lem:GM}\nFix $\\varepsilon>0$. There exists an absolute constant $\\delta_0>0$ such that the following holds. Let $X\\geqslant 3$ and let $y$ satisfy\n\\[\nX^{2/15+\\varepsilon}\\ \\leqslant\\ y\\ \\leqslant\\ X^{1-\\delta_0}.\n\\]\nThen there exists a constant $c=c_{\\varepsilon}>0$ such that for all but\n$O_{\\varepsilon}\\left(X\\exp\\left(-c(\\log X)^{1/4}\\right)\\right)$ integers $t\\in[X,2X]$ one has\n\\[\n\\pi(t+y)-\\pi(t)\\ =\\ \\frac{y}{\\log t}\\ +\\ O_{\\varepsilon}\\left(y\\exp\\left(-c(\\log X)^{1/4}\\right)\\right).\n\\]\nThis follows from \\cite[Cor.~1.4]{GuthMaynard2024}; the corollary is stated for real $x$, and restricting to integer $t$ gives the same exceptional-set bound.\n\\end{lemma}\n\n\\begin{proposition}[First moment lower bound]\\label{prop:first-moment}\nFix $\\varepsilon>0$ and let $\\delta_0>0$ be as in \\cref{lem:GM}. Let $\\theta$ satisfy $2/15+\\varepsilon\\leqslant\\theta<1-\\delta_0$ and set $h=X^{\\theta}$. Then there exists a set\n$\\mathcal{E}\\subset[X,2X]\\cap\\Z$ with\n\\[\n\\#\\mathcal{E}\\ \\ll_{\\varepsilon,\\lambda}\\ X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\n\\]\nsuch that\n\\[\nR(x)\\ \\gg\\ h\n\\]\nfor all $x\\in([X,2X]\\cap\\Z)\\setminus\\mathcal{E}$.\n\\end{proposition}\n\nWe apply \\cref{lem:GM} twice: first with the scale parameter $X' := X/2$ to control $\\pi(t+h)-\\pi(t)$ for $t\\in[X/2,X]$, and then with the scale parameter $X' := X$ to control it for $t\\in[X,2X]$. Since $\\theta\\geqslant 2/15+\\varepsilon$, for $X$ sufficiently large we have $(X/2)^{2/15+\\varepsilon/2}\\leqslant h$. Moreover, since $\\theta<1-\\delta_0$ with $\\delta_0>0$ fixed, for $X$ sufficiently large we also have $h=X^{\\theta}\\leqslant (X/2)^{1-\\delta_0}$. Consequently the hypotheses of \\cref{lem:GM} apply (with $\\varepsilon$ replaced by $\\varepsilon/2$) on both dyadic blocks. We therefore obtain an exceptional set $\\mathcal{E}_a\\subset[X,2X]\\cap\\Z$ with\n\\[\n\\#\\mathcal{E}_a\\ \\ll_{\\varepsilon}\\ X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\n\\]\nsuch that for all $x\\in([X,2X]\\cap\\Z)\\setminus\\mathcal{E}_a$ we have\n\\begin{equation}\\label{eq:GM-shifted}\n\\pi(x+h-a)-\\pi(x-a)\\ =\\ \\frac{h}{\\log(x-a)}\\ +\\ O_{\\varepsilon}\\left(h\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right).\n\\end{equation}\n\nFor $x\\in([X,2X]\\cap\\Z)\\setminus\\mathcal{E}$, summing \\eqref{eq:GM-shifted} over $a\\in\\cA_{\\lambda}(X;h)$ yields\n\\[\n\\begin{aligned}\nR(x)\n&\\geqslant\\ \\sum_{a\\in\\cA_{\\lambda}(X;h)}\\left(\\pi(x+h-a)-\\pi(x-a)\\right)\n\\\\\n&=\\ h\\sum_{a\\in\\cA_{\\lambda}(X;h)}\\frac{1}{\\log(x-a)}\\ +\\ O_{\\varepsilon}\\left(A_h h\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right).\n\\end{aligned}\n\\]\nSince $x-a\\asymp X$ uniformly for $x\\in[X,2X]$ and $a\\leqslant h/2\\leqslant X/2$, the main term satisfies\n\\[\n\\sum_{a\\in\\cA_{\\lambda}(X;h)}\\frac{1}{\\log(x-a)}\\ \\asymp\\ \\frac{A_h}{\\log X}\\ \\asymp\\ \\frac{\\log h}{\\log X},\n\\]\nand the error term is negligible. Since $h=X^{\\theta}$, we have $\\log h=\\theta \\log X$, hence $R(x)\\gg h$ for all $x\\notin\\mathcal{E}$.\n\\end{proof}\n\n\\begin{theorem}[Local density of $\\cS$ in short intervals]\\label{thm:main-elements}\nFix $\\varepsilon>0$. Let $\\theta$ satisfy $2/15+\\varepsilon<\\theta<1-\\delta_0$ for some absolute constant $\\delta_0>0$, and set $h=X^{\\theta}$.\nThen for all but $O_{\\varepsilon}\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)$ integers $x\\in[X,2X]$, one has\n\\[\n\\#\\left(\\cS\\cap(x,x+h]\\right)\\ \\asymp\\ h.\n\\]\nThe implied constants depend at most on $\\lambda$, $\\varepsilon$ and $\\theta$.\n\\end{theorem}", "post_theorem_intro_text_len": 2129, "post_theorem_intro_text": "Theorem~\\ref{thm:main-elements} is the conclusion we ultimately care about: it asserts that the\nsumset $\\mathcal{S}$ meets almost all intervals of length $h$ with the expected order of magnitude.\nTo access it through Cauchy--Schwarz, it is convenient to also record the corresponding\nfirst-moment statement for the representation function $f$.\n\n\\begin{theorem}[Typical number of representations]\\label{thm:main-rep}\nFix $\\varepsilon>0$. Let $\\theta$ satisfy $2/15+\\varepsilon<\\theta<1-\\delta_0$ for some absolute constant $\\delta_0>0$, and set $h=X^{\\theta}$.\nThen for all but $O_{\\varepsilon}\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)$ integers $x\\in[X,2X]$, one has\n\\[\nR(x)\\ \\asymp\\ h.\n\\]\nThe implied constants depend at most on $\\lambda$, $\\varepsilon$ and $\\theta$.\n\\end{theorem}\n\n\\medskip\n\n\\noindent\\emph{Strategy.} The pointwise upper bound $R(x)\\ll h$ follows from Brun--Titchmarsh together with $\\#\\cA_{\\lambda}(X)\\asymp\\log X$.\nThe almost-all lower bound $R(x)\\gg h$ is obtained by restricting to $a\\leqslant h/2$ and applying the Guth--Maynard~\\cite[Cor.~1.4]{GuthMaynard2024} asymptotic for primes in almost all short intervals (with a stretched-exponential exceptional set).\nTo pass from representations to elements we combine Cauchy--Schwarz, which relates $R(x)$, $S(x)$ and the local second moment $Q(x)$, with a uniform bound $Q(x)\\ll h$.\nThe latter is proved using Selberg's upper-bound sieve for prime pairs together with Chen--Ding--Xu--Zhai's estimate controlling the contribution of small prime divisors in differences $a_1-a_2$ with $a_1,a_2\\in\\cA_{\\lambda}(X)$.\n\n\\medskip\n\n\\noindent\\emph{Organization of the paper.} Section~2 establishes the first-moment estimate for the local representation count $R(x)$ in almost all short intervals. Section~3 proves a uniform local mean-square bound, leading to the required second-moment control. In Section~4 we combine these inputs with Cauchy--Schwarz to deduce Theorems~\\ref{thm:main-elements} and~\\ref{thm:main-rep}. Finally, Section~5 proves Theorem~\\ref{thm:erdos-short-interval} on large short-interval multiplicities in the Romanoff model.", "sketch": "“To access [Theorem~\\ref{thm:main-elements}] through Cauchy--Schwarz,” the paper first proves a first-moment statement for the representation function $f$, namely Theorem~\\ref{thm:main-rep} that $R(x)\\asymp h$ for almost all $x\\in[X,2X]$. For this, the “pointwise upper bound $R(x)\\ll h$ follows from Brun--Titchmarsh together with $\\#\\cA_{\\lambda}(X)\\asymp\\log X$,” while the “almost-all lower bound $R(x)\\gg h$ is obtained by restricting to $a\\leqslant h/2$ and applying the Guth--Maynard ... asymptotic for primes in almost all short intervals (with a stretched-exponential exceptional set).”\n\n“To pass from representations to elements” (i.e. to deduce Theorem~\\ref{thm:main-elements}), they “combine Cauchy--Schwarz, which relates $R(x)$, $S(x)$ and the local second moment $Q(x)$, with a uniform bound $Q(x)\\ll h$.” This uniform second-moment bound “is proved using Selberg's upper-bound sieve for prime pairs together with Chen--Ding--Xu--Zhai's estimate controlling the contribution of small prime divisors in differences $a_1-a_2$ with $a_1,a_2\\in\\cA_{\\lambda}(X)$.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "[Local density of $\\mathcal{S}$ in short intervals]\\label{thm:main-elements}\nFix $\\varepsilon>0$. Let $\\theta$ satisfy $2/15+\\varepsilon<\\theta<1-\\delta_0$ for some absolute constant $\\delta_0>0$, and set $h=X^{\\theta}$.\nThen for all but $O_{\\varepsilon}\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)$ integers $x\\in[X,2X]$, one has\n\\[\n\\#\\left(\\mathcal{S}\\cap(x,x+h]\\right)\\ \\asymp\\ h.\n\\]\nThe implied constants depend at most on $\\lambda$, $\\varepsilon$ and $\\theta$.", "theorem_type": ["Universal", "Asymptotic or Limit"], "mcq": {"question": "Let \\(\\mathbb P\\) denote the prime numbers. Fix an integer \\(s\\ge 2\\) and real numbers \\(r_1,\\dots,r_s>1\\) such that\n\\[\n\\frac{1}{r_1}+\\cdots+\\frac{1}{r_{s-1}}<1\\le \\frac{1}{r_1}+\\cdots+\\frac{1}{r_s},\n\\]\nand define \\(\\lambda>0\\) by\n\\[\n\\frac{1}{r_1}+\\cdots+\\frac{1}{r_{s-1}}+\\frac{1}{\\lambda r_s}=1.\n\\]\nSet\n\\[\n\\mathcal A_\\lambda:=\\left\\{2^{\\lfloor k_1^{r_1}\\rfloor}+\\cdots+2^{\\lfloor k_{s-1}^{r_{s-1}}\\rfloor}+2^{\\lfloor \\lfloor k_s^{\\lambda}\\rfloor^{r_s}\\rfloor}: k_1,\\dots,k_s\\in\\mathbb N\\right\\},\n\\]\n\\[\n\\mathcal A_\\lambda(X):=\\mathcal A_\\lambda\\cap[1,2X],\\qquad \\mathcal S:=\\mathbb P+\\mathcal A_\\lambda(X).\n\\]\nFix \\(\\varepsilon>0\\), choose \\(\\theta\\) with \\(2/15+\\varepsilon<\\theta<1-\\delta_0\\), where \\(\\delta_0>0\\) is an absolute constant, and set \\(h=X^{\\theta}\\). Which statement holds for all but \\(O_{\\varepsilon}\\!\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)\\) integers \\(x\\in[X,2X]\\cap\\mathbb Z\\)?", "correct_choice": {"label": "A", "text": "\\[\\#\\bigl(\\mathcal S\\cap(x,x+h]\\bigr)\\asymp h,\\]\nthat is, for all but that exceptional set of integers \\(x\\), the interval \\((x,x+h]\\) contains on the order of \\(h\\) integers of the form \\(p+a\\) with \\(p\\in\\mathbb P\\) and \\(a\\in\\mathcal A_\\lambda(X)\\); the implied constants depend at most on \\(\\lambda\\), \\(\\varepsilon\\), and \\(\\theta\\)."}, "choices": [{"label": "B", "text": "\\[\\#\\bigl(\\mathcal S\\cap(x,x+h]\\bigr)\\asymp h,\\]\nfor every integer \\(x\\in[X,2X]\\cap\\mathbb Z\\); equivalently, there is no exceptional set, and the implied constants depend at most on \\(\\lambda\\), \\(\\varepsilon\\), and \\(\\theta\\)."}, {"label": "C", "text": "\\[\\#\\bigl(\\mathcal S\\cap(x,x+h]\\bigr)\\gg h,\\]\nfor all but \\(O_{\\varepsilon}\\!\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)\\) integers \\(x\\in[X,2X]\\cap\\mathbb Z\\); the implied constant depends at most on \\(\\lambda\\), \\(\\varepsilon\\), and \\(\\theta\\)."}, {"label": "D", "text": "\\[\\#\\bigl(\\mathcal S\\cap(x,x+h]\\bigr)\\asymp h,\\]\nfor all but \\(O_{\\varepsilon}\\!\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)\\) real numbers \\(x\\in[X,2X]\\); the implied constants depend at most on \\(\\lambda\\), \\(\\varepsilon\\), and \\(\\theta\\)."}, {"label": "E", "text": "\\[\\#\\bigl(\\mathcal S\\cap(x,x+h]\\bigr)=h+O_{\\varepsilon}\\!\\left(h\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right),\\]\nfor all but \\(O_{\\varepsilon}\\!\\left(X\\exp\\left(-c_{\\varepsilon}(\\log X)^{1/4}\\right)\\right)\\) integers \\(x\\in[X,2X]\\cap\\mathbb Z\\), with implied constants depending at most on \\(\\lambda\\), \\(\\varepsilon\\), and \\(\\theta\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "presence_of_exceptional_set", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped_upper_bound_in_asymp", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "integer_domain_for_x", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "order_of_magnitude_replaced_by_pointwise_asymptotic", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It states the hypotheses and asks for the valid conclusion; the correct answer is not leaked verbatim outside the answer choices."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the hypotheses are given and the correct choice is the theorem’s conclusion with only nearby variants as alternatives. It does not materially go beyond restating the result."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in distinguishing the exact strength of the conclusion from stronger or weaker variants (exceptional set, asymp vs. lower bound, integers vs. reals). However, it mainly tests precise recall of the theorem statement rather than substantial mathematical generation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: one removes the exceptional set, one weakens asymp to a one-sided bound, one changes the domain from integers to reals, and one strengthens to a sharp asymptotic. These reflect common failure modes in parsing analytic number theory statements."}, "total_score": 5, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no direct answer leakage, but it is largely a restatement/recall question rather than a genuinely generative reasoning task."}}
{"id": "2602.14658v1", "paper_link": "http://arxiv.org/abs/2602.14658v1", "theorems_cnt": 5, "theorem": {"env_name": "Thm", "content": "[Gradient regularity in Morrey spaces]\\label{MainThm}\nLet $u \\in W^{1,p}(\\Omega)$ be a local minimizer of  the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin\\eqref{MainF}, and assume that\n$\\partial E$ is a hypersurface of class $C^1$.\nSuppose that assumptions \\emph{(H1)}--\\emph{(H2)} are satisfied where $2<p<\\frac{2n}{n-2}$.\nThen, for every $0 \\leq \\lambda < n$,\n\\[\n\\nabla u \\in L^{{p},\\lambda}_{\\mathrm{loc}}(\\Omega).\n\\]\nMoreover, for every $\\Omega' \\Subset \\Omega$, there exists a constant\n$C = C\\big(n, p, \\nu, L, \\beta, E, \\operatorname{diam}(\\Omega),\n\\operatorname{dist}(\\Omega', \\partial\\Omega)\\big) > 0$\nsuch that\n\\[\n\\|\\nabla u\\|_{L^{{p},\\lambda}(\\Omega')} \\leq C.\n\\]", "start_pos": 8951, "end_pos": 9651, "label": "MainThm"}, "ref_dict": {"FJNL": "\\begin{Prop}\\label{FJNL}\nLet $u$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin \\eqref{MainF} under the assumption of Theorem~\\ref{MainThm}.\nThen there exists a constant $0 < \\tau_0 < 1$ such that the following holds:\nfor every $\\tau \\in (0,\\tau_0)$ there exists $\\varepsilon_0 = \\varepsilon_0(\\tau) > 0$\nwith the property that, if $B_r(x_0) \\Subset \\Omega$ and one of the following\nconditions is satisfied,\n\\begin{enumerate}[label=(\\roman*)]\n\\item $|E \\cap B_r(x_0)| < \\varepsilon_0 |B_r|$,\n\\item $|B_r(x_0) \\setminus E| < \\varepsilon_0 |B_r|$,\n\\item there exists a half-space $H$ such that\n\\[\n\\frac{|(E \\Delta H) \\cap B_r(x_0)|}{|B_r|} < \\varepsilon_0,\n\\]\n\\end{enumerate}\nthen, for every $0 < \\delta < n$, the estimate\n\\[\n\\int_{B_{\\tau r}(x_0)} |\\nabla u|^p \\, dx\n\\leq\nC_0 \\tau^{n-\\delta}\n\\left(\n\\int_{B_r(x_0)} |\\nabla u|^p \\, dx + r^n\n\\right)\n\\]\nholds, where the constant $C_0$ depends only on\n$n, \\nu, L, \\alpha,\\beta, \\delta$ and $\\|\\nabla u\\|_{L^2(\\Omega)}$.\n\\end{Prop}", "MainF": "\\begin{equation}\\label{MainF}\n\\mathcal{F}(u;E) := \\int_{\\Omega} \\sigma_E(x)\\, F(\\nabla u)\\, dx,\n\\end{equation}", "MainThm": "\\begin{Thm}[Gradient regularity in Morrey spaces]\\label{MainThm}\nLet $u \\in W^{1,p}(\\Omega)$ be a local minimizer of  the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin\\eqref{MainF}, and assume that\n$\\partial E$ is a hypersurface of class $C^1$.\nSuppose that assumptions \\emph{(H1)}--\\emph{(H2)} are satisfied where $2<p<\\frac{2n}{n-2}$.\nThen, for every $0 \\leq \\lambda < n$,\n\\[\n\\nabla u \\in L^{{p},\\lambda}_{\\mathrm{loc}}(\\Omega).\n\\]\nMoreover, for every $\\Omega' \\Subset \\Omega$, there exists a constant\n$C = C\\big(n, p, \\nu, L, \\beta, E, \\operatorname{diam}(\\Omega),\n\\operatorname{dist}(\\Omega', \\partial\\Omega)\\big) > 0$\nsuch that\n\\[\n\\|\\nabla u\\|_{L^{{p},\\lambda}(\\Omega')} \\leq C.\n\\]\n\\end{Thm}"}, "pre_theorem_intro_text_len": 5502, "pre_theorem_intro_text": "Transmission problems arise naturally in the study of diffusive processes in\nheterogeneous media, where the governing laws change abruptly across fixed interfaces.\nTypical applications include elasticity theory, composite materials, and conductivity\nphenomena. This kind of problems appear also in the study of thermal insulation of bodies under sources and prescribed boundary conditions (see for example \\cite{AC,DPNST})\nFrom a mathematical perspective, such problems are characterised by the presence of\npiecewise-defined operators or energy densities, together with suitable transmission\nconditions prescribed along the separating interface.\nThe first systematic investigation of transmission problems in elasticity theory dates\nback to the seminal work of M.~Picone in the 1950s (see \\cite{Pico}).\n\nIn this paper, we investigate a class of \\emph{variational transmission problems}\nassociated with nonlinear integral functionals exhibiting discontinuous coefficients\nacross a prescribed interface.\nOur focus is on the regularity properties of local minimizers, with particular attention\nto the behaviour of the gradient near the interface.\n\nLet $\\Omega \\subset \\mathbb{R}^n$ be a bounded domain and let $E \\subset \\Omega$ be\na measurable subset.\nWe assume that the interface $\\partial E$ is a hypersurface of class $C^1$.\nWe fix two constants $0 < \\alpha < \\beta < +\\infty$ and define\n\\[\n\\sigma_E := \\beta \\mathbbm{1}_E + \\alpha \\mathbbm{1}_{E^c},\n\\]\nwhere $\\mathbbm{1}_E$ denotes the characteristic function of the set $E$ and $E^c$ is the complementary of $E$.\nWe consider the integral functional\n\\begin{equation}\\label{MainF}\n\\mathcal{F}(u;E) := \\int_{\\Omega} \\sigma_E(x)\\, F(\\nabla u)\\, dx,\n\\end{equation}\ndefined for functions $u \\in W^{1,p}(\\Omega)$.\n\nThe integrand $F \\colon \\mathbb{R}^n \\to \\mathbb{R}$ is assumed to belong to\n$C^2(\\mathbb{R}^n)$ and to satisfy standard $p$-growth and monotonicity conditions:\nfor all $\\xi, \\eta \\in \\mathbb{R}^n$,\n\\begin{equation}\n\\label{Monotonicity}\\tag{H1}\n\\langle \\nabla F(\\xi) - \\nabla F(\\eta), \\xi - \\eta \\rangle\n\\geq \\nu \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|^2,\n\\end{equation}\n\\begin{equation}\n\\label{Growth}\\tag{H2}\n|\\nabla F(\\xi) - \\nabla F(\\eta)|\n\\leq L \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|,\n\\end{equation}\nfor some constants $\\nu, L > 0$, $\\mu \\in (0,1]$, and $p > 2$.\n\nLocal minimizers of the functional $\\mathcal{F}$ are understood in the usual\nvariational sense.\n\n\\begin{Def}\nLet $u \\in W^{1,p}(\\Omega)$.\nWe say that $u$ is a \\emph{local minimizer} of $\\mathcal{F}(\\cdot;\\Omega)$ in $\\Omega$\nif, for every ball $B_r(x_0) \\subset \\Omega$ and every\n$\\phi \\in W^{1,p}_0(B_r(x_0))$, one has\n\\[\n\\mathcal{F}(u;B_r(x_0)) \\leq \\mathcal{F}(u + \\phi;B_r(x_0)).\n\\]\n\\end{Def}\n\nBefore stating our main result, let us place it in the context of the existing literature.\nIn the linear case $p = 2$, transmission problems with piecewise constant coefficients\nhave been extensively studied, and gradient regularity across sufficiently smooth\ninterfaces is by now classical (see, e.g.,~\\cite{GT,LV,Grisvard}).\nSuch results follow from the elliptic regularity theory for divergence-form operators\nwith discontinuous coefficients (see for instance\n\\cite{Evans, GT, LM, Grisvard} and \\cite[Theorem~7.53]{AFP}.\nIn this setting, local $W^{1,2}$- and H\\\"older-type regularity for the gradient of minimizers is well\nunderstood under mild geometric assumptions on the interface.\n\nMore recently, fine regularity properties for linear transmission problems across\n$C^{1,\\alpha}$ interfaces have been obtained by Caffarelli, Soria-Carro, and Stinga\n\\cite{CSS}.\n\nBy contrast, the nonlinear case $p \\neq 2$ remains far less developed.\nTo the best of our knowledge, the only available regularity results for nonlinear\ntransmission problems concern equations subject to \\emph{additive} transmission\nconditions on the interface.\nIn particular, in the recent work \\cite{BPU}, BMO regularity of the gradient is obtained\nfor degenerate quasilinear equations under transmission conditions of the form\n\\[\ng(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n-\ng(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n= f \\quad \\text{on } \\partial E.\n\\]\n\nIn contrast, the transmission condition naturally associated with the variational\nfunctional \\eqref{MainF} is of \\emph{multiplicative} type, namely\n\\[\ng_1(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n=\ng_2(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n\\quad \\text{on } \\partial E,\n\\]\nwhich arises intrinsically from the Euler--Lagrange equations and expresses the\ncontinuity of the nonlinear flux across the interface. We stress that this multiplicative transmission condition is intrinsic to the\nvariational structure of the functional \\eqref{MainF}, as it arises directly from\nthe Euler--Lagrange equations and cannot be prescribed independently.\nThe presence of a discontinuous coefficient combined with nonlinear growth prevents\na direct application of standard techniques such as freezing arguments or difference\nquotient estimates across the interface.\nTo the best of our knowledge, no Morrey-space regularity results for gradients of local\nminimizers are available in this genuinely nonlinear variational transmission setting.\n\nDenoting by $L^{{p},\\lambda}$ the classical Morrey spaces, we can now state the main\nresult of this paper, which establishes local Morrey regularity for the gradient of\nlocal minimizers under minimal geometric assumptions on the interface.", "context": "Let $\\Omega \\subset \\mathbb{R}^n$ be a bounded domain and let $E \\subset \\Omega$ be\na measurable subset.\nWe assume that the interface $\\partial E$ is a hypersurface of class $C^1$.\nWe fix two constants $0 < \\alpha < \\beta < +\\infty$ and define\n\\[\n\\sigma_E := \\beta \\mathbbm{1}_E + \\alpha \\mathbbm{1}_{E^c},\n\\]\nwhere $\\mathbbm{1}_E$ denotes the characteristic function of the set $E$ and $E^c$ is the complementary of $E$.\nWe consider the integral functional\n\\begin{equation}\\label{MainF}\n\\mathcal{F}(u;E) := \\int_{\\Omega} \\sigma_E(x)\\, F(\\nabla u)\\, dx,\n\\end{equation}\ndefined for functions $u \\in W^{1,p}(\\Omega)$.\n\nThe integrand $F \\colon \\mathbb{R}^n \\to \\mathbb{R}$ is assumed to belong to\n$C^2(\\mathbb{R}^n)$ and to satisfy standard $p$-growth and monotonicity conditions:\nfor all $\\xi, \\eta \\in \\mathbb{R}^n$,\n\\begin{equation}\n\\label{Monotonicity}\\tag{H1}\n\\langle \\nabla F(\\xi) - \\nabla F(\\eta), \\xi - \\eta \\rangle\n\\geq \\nu \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|^2,\n\\end{equation}\n\\begin{equation}\n\\label{Growth}\\tag{H2}\n|\\nabla F(\\xi) - \\nabla F(\\eta)|\n\\leq L \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|,\n\\end{equation}\nfor some constants $\\nu, L > 0$, $\\mu \\in (0,1]$, and $p > 2$.\n\n\\begin{Def}\nLet $u \\in W^{1,p}(\\Omega)$.\nWe say that $u$ is a \\emph{local minimizer} of $\\mathcal{F}(\\cdot;\\Omega)$ in $\\Omega$\nif, for every ball $B_r(x_0) \\subset \\Omega$ and every\n$\\phi \\in W^{1,p}_0(B_r(x_0))$, one has\n\\[\n\\mathcal{F}(u;B_r(x_0)) \\leq \\mathcal{F}(u + \\phi;B_r(x_0)).\n\\]\n\\end{Def}\n\nBy contrast, the nonlinear case $p \\neq 2$ remains far less developed.\nTo the best of our knowledge, the only available regularity results for nonlinear\ntransmission problems concern equations subject to \\emph{additive} transmission\nconditions on the interface.\nIn particular, in the recent work \\cite{BPU}, BMO regularity of the gradient is obtained\nfor degenerate quasilinear equations under transmission conditions of the form\n\\[\ng(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n-\ng(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n= f \\quad \\text{on } \\partial E.\n\\]\n\nIn contrast, the transmission condition naturally associated with the variational\nfunctional \\eqref{MainF} is of \\emph{multiplicative} type, namely\n\\[\ng_1(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n=\ng_2(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n\\quad \\text{on } \\partial E,\n\\]\nwhich arises intrinsically from the Euler--Lagrange equations and expresses the\ncontinuity of the nonlinear flux across the interface. We stress that this multiplicative transmission condition is intrinsic to the\nvariational structure of the functional \\eqref{MainF}, as it arises directly from\nthe Euler--Lagrange equations and cannot be prescribed independently.\nThe presence of a discontinuous coefficient combined with nonlinear growth prevents\na direct application of standard techniques such as freezing arguments or difference\nquotient estimates across the interface.\nTo the best of our knowledge, no Morrey-space regularity results for gradients of local\nminimizers are available in this genuinely nonlinear variational transmission setting.\n\nDenoting by $L^{{p},\\lambda}$ the classical Morrey spaces, we can now state the main\nresult of this paper, which establishes local Morrey regularity for the gradient of\nlocal minimizers under minimal geometric assumptions on the interface.", "full_context": "Let $\\Omega \\subset \\mathbb{R}^n$ be a bounded domain and let $E \\subset \\Omega$ be\na measurable subset.\nWe assume that the interface $\\partial E$ is a hypersurface of class $C^1$.\nWe fix two constants $0 < \\alpha < \\beta < +\\infty$ and define\n\\[\n\\sigma_E := \\beta \\mathbbm{1}_E + \\alpha \\mathbbm{1}_{E^c},\n\\]\nwhere $\\mathbbm{1}_E$ denotes the characteristic function of the set $E$ and $E^c$ is the complementary of $E$.\nWe consider the integral functional\n\\begin{equation}\\label{MainF}\n\\mathcal{F}(u;E) := \\int_{\\Omega} \\sigma_E(x)\\, F(\\nabla u)\\, dx,\n\\end{equation}\ndefined for functions $u \\in W^{1,p}(\\Omega)$.\n\nThe integrand $F \\colon \\mathbb{R}^n \\to \\mathbb{R}$ is assumed to belong to\n$C^2(\\mathbb{R}^n)$ and to satisfy standard $p$-growth and monotonicity conditions:\nfor all $\\xi, \\eta \\in \\mathbb{R}^n$,\n\\begin{equation}\n\\label{Monotonicity}\\tag{H1}\n\\langle \\nabla F(\\xi) - \\nabla F(\\eta), \\xi - \\eta \\rangle\n\\geq \\nu \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|^2,\n\\end{equation}\n\\begin{equation}\n\\label{Growth}\\tag{H2}\n|\\nabla F(\\xi) - \\nabla F(\\eta)|\n\\leq L \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|,\n\\end{equation}\nfor some constants $\\nu, L > 0$, $\\mu \\in (0,1]$, and $p > 2$.\n\n\\begin{Def}\nLet $u \\in W^{1,p}(\\Omega)$.\nWe say that $u$ is a \\emph{local minimizer} of $\\mathcal{F}(\\cdot;\\Omega)$ in $\\Omega$\nif, for every ball $B_r(x_0) \\subset \\Omega$ and every\n$\\phi \\in W^{1,p}_0(B_r(x_0))$, one has\n\\[\n\\mathcal{F}(u;B_r(x_0)) \\leq \\mathcal{F}(u + \\phi;B_r(x_0)).\n\\]\n\\end{Def}\n\nBy contrast, the nonlinear case $p \\neq 2$ remains far less developed.\nTo the best of our knowledge, the only available regularity results for nonlinear\ntransmission problems concern equations subject to \\emph{additive} transmission\nconditions on the interface.\nIn particular, in the recent work \\cite{BPU}, BMO regularity of the gradient is obtained\nfor degenerate quasilinear equations under transmission conditions of the form\n\\[\ng(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n-\ng(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n= f \\quad \\text{on } \\partial E.\n\\]\n\nIn contrast, the transmission condition naturally associated with the variational\nfunctional \\eqref{MainF} is of \\emph{multiplicative} type, namely\n\\[\ng_1(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n=\ng_2(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n\\quad \\text{on } \\partial E,\n\\]\nwhich arises intrinsically from the Euler--Lagrange equations and expresses the\ncontinuity of the nonlinear flux across the interface. We stress that this multiplicative transmission condition is intrinsic to the\nvariational structure of the functional \\eqref{MainF}, as it arises directly from\nthe Euler--Lagrange equations and cannot be prescribed independently.\nThe presence of a discontinuous coefficient combined with nonlinear growth prevents\na direct application of standard techniques such as freezing arguments or difference\nquotient estimates across the interface.\nTo the best of our knowledge, no Morrey-space regularity results for gradients of local\nminimizers are available in this genuinely nonlinear variational transmission setting.\n\nDenoting by $L^{{p},\\lambda}$ the classical Morrey spaces, we can now state the main\nresult of this paper, which establishes local Morrey regularity for the gradient of\nlocal minimizers under minimal geometric assumptions on the interface.\n\n\\begin{Prop}\\label{FJNL}\nLet $u$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin \\eqref{MainF} under the assumption of Theorem~\\ref{MainThm}.\nThen there exists a constant $0 < \\tau_0 < 1$ such that the following holds:\nfor every $\\tau \\in (0,\\tau_0)$ there exists $\\varepsilon_0 = \\varepsilon_0(\\tau) > 0$\nwith the property that, if $B_r(x_0) \\Subset \\Omega$ and one of the following\nconditions is satisfied,\n\\begin{enumerate}[label=(\\roman*)]\n\\item $|E \\cap B_r(x_0)| < \\varepsilon_0 |B_r|$,\n\\item $|B_r(x_0) \\setminus E| < \\varepsilon_0 |B_r|$,\n\\item there exists a half-space $H$ such that\n\\[\n\\frac{|(E \\Delta H) \\cap B_r(x_0)|}{|B_r|} < \\varepsilon_0,\n\\]\n\\end{enumerate}\nthen, for every $0 < \\delta < n$, the estimate\n\\[\n\\int_{B_{\\tau r}(x_0)} |\\nabla u|^p \\, dx\n\\leq\nC_0 \\tau^{n-\\delta}\n\\left(\n\\int_{B_r(x_0)} |\\nabla u|^p \\, dx + r^n\n\\right)\n\\]\nholds, where the constant $C_0$ depends only on\n$n, \\nu, L, \\alpha,\\beta, \\delta$ and $\\|\\nabla u\\|_{L^2(\\Omega)}$.\n\\end{Prop}\n\\begin{Rem}\n\\label{Reg}\n For the proof of Theorem \\ref{MainThm}, it would be sufficient to state the proposition only in case (i). We nevertheless include the other two cases for the sake of completeness.\n\\end{Rem}\n\n\\begin{Lem}[Standard $p$-growth bounds]\\label{lem:p-growth}\nAssume \\((H1)\\)--\\((H2)\\). Then there exists a constant $C=C(n,p,\\nu,L)>0$ such that for all $\\xi\\in\\R^n$\n\\[\nC^{-1}(\\mu^2+|\\xi|^2)^{\\frac p2}-C\\mu^p \\le F(\\xi)\\le C(\\mu^2+|\\xi|^2)^{\\frac p2}+C\\mu^p .\n\\]\n\\end{Lem}\nThe following results on higher integrability and local Hölder continuity\nof minimizers are classical. Since their proofs are standard, we omit them and refer the reader to\n\\cite{Giu,GiaMar}.\n{\n\\begin{Thm}[Higher integrability]\n\\label{HigherInt}\nLet $u \\in W^{1,p}(\\Omega)$ be a local minimizer of the functional \\eqref{MainF}.\nThere exist $s > 1$ and $C = C(n,p,\\nu,L,\\beta)$ such that for any ball\n$B_{2r}(x_{0})\\subset\\subset\\Omega$\n\\begin{equation}\n \\int_{B_r(x_0)}|\\nabla u|^{sp}\\,dx\\leq C\\bigg(\\int_{B_{2r}(x_0)}\\big(|\\nabla u|^p+\\mu^p\\big)\\,dx\n \\bigg)^s.\n\\end{equation}\n\\end{Thm}\n\\begin{Thm}[H\\\"older continuity]Let $u \\in W^{1,p}(\\Omega)$ be a local minimizer of the functional \\eqref{MainF}. Then\n\\begin{enumerate}[label=(\\roman*)]\n\\item For any open set $\\Omega'\\Subset \\Omega$ the quantity $\\norm{u}_{L^{\\infty}(\\Omega')}$ is bounded by a constant $C = C(n,p,\\nu,L,\\beta)\\norm{u}_{L^2(\\Omega)}$.\n\\item u is locally H\\\"older continuous in $\\Omega$.\n\\end{enumerate}\n\\end{Thm}\n}\n\n\\begin{Thm}\\label{DS}\nLet $u$ be a minimizer of $\\mathcal{F}$ in $B_r(x_0)$. Then $\\partial_i u\\in W^{1,2}_{loc}(B_{r})$ for every $i=1\\dots n-1$. Moreover the following properties hold:\n\\begin{enumerate}[label=\\roman*)]\n\\item $u \\in W^{2,2}_{\\mathrm{loc}}(B_r^+)$ e $u \\in W^{2,2}_{\\mathrm{loc}}(B_r^-)$;\n\\item for every $x_0 \\in \\Gamma_r$ and $R>0$ such that $B_{4R}(x_0)\\subset B_r$, it holds that\n\\begin{align}\n\\int_{B_{R/2}} V^{p-2}\\,|\\nabla\\nabla' u|^2 \\,dx\n&\\leq \\frac{C}{R^2}\\int_{B_{2R}} V^p \\,dx, \\label{eq:tangential-estimate}\n\\end{align}\nwhere $C=C\\!\\left(\\tfrac{L}{\\nu},\\,n,\\,p,\\,\\alpha,\\,\\beta,\\mu\\right)$ is a positive constant.\n\\end{enumerate}\n\\end{Thm}\n\n\\section{Proof of Proposition 1.3}\n\\begin{proof}\nFix $\\tau\\in(0,\\tau_0)$ (to be chosen small only depending on $n,p,\\nu,L,\\beta/\\alpha$) and assume without loss of generality that $x_0=0$.\nWe prove that there exists $\\e_0=\\e_0(\\tau)>0$ such that if $B_r(x_0)\\Subset\\Om$ and one of {\\rm(i)}--{\\rm(iii)}\nholds, then for every $0<\\delta<n$, then\n\\begin{equation}\\label{eq:prop13-goal}\n\\int_{B_{\\tau r}} |\\nabla u|^p\\, dx\n\\leq\nC_0\\tau^{n-\\delta}\\bigg( \\int_{B_r} |\\nabla u|^p \\, dx\n+r^n\\bigg),\n\\end{equation}\nwhere $C_0=C_0(n,p,\\nu,L,\\beta,\\delta,\\|\\nabla u\\|_{L^2(\\Om)})>0$.\\\\\n\\indent \n\\noindent\\textbf{Step 1: the flat comparison in case (iii).}\nAssume that (iii) holds true. Up to a rigid motion we may assume that $H=\\{x_n>0\\}$ and set\n\\[\nE_H:=H\\cap B_{r},\\quad \\sigma_H:=\\beta\\mathbbm 1_{E_H}+\\alpha\\mathbbm 1_{B_r\\setminus E_H}.\n\\]\nLet $u_H\\in u+W^{1,p}_0(B_\\frac{r}{2})$ be the unique minimizer of the functional\n\\[\n\\mathcal F_H(w;B_\\frac{r}{2}):=\\int_{B_\\frac{r}{2}}\\sigma_H(x)\\,F(\\nabla w)\\,dx .\n\\]\nSubtracting the Euler-Lagrange equation satisfied by $u$ from the one satisfied by $u_H$ and testing them with $u-u_H$, we get\n\\begin{align}\n \\int_{B_{\\frac{r}{2}}} \\sigma_E\n\\big(DF(\\nabla u)-DF(\\nabla u_H)\\big)\n\\cdot(\\nabla u-\\nabla u_H)\\,dx\n& =\n\\int_{B_{\\frac{r}{2}}} (\\sigma_H-\\sigma_E)\\,\nDF(\\nabla u_H)\\cdot(\\nabla u-\\nabla u_H)\\,dx\n\\end{align}\nUsing assumptions \\eqref{Monotonicity} and \\eqref{Growth} and H\\\"older's inequality, for every $\\varepsilon>0$ it holds that\n\\begin{align}\n \\int_{B_{\\frac{r}{2}}}|\\D u_H-\\D u|^p\\,dx\n & \\leq C\\int_{(E\\Delta E_H)\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D u_H|^2\\big)^{\\frac{p-2}{2}}|\\D u-\\D u_H|\\,dx\\\\\n& \\leq C\\bigg(\\frac{1}{\\varepsilon}\\int_{(E\\Delta E_H)\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D u_H|^2\\big)^{\\frac{p}{2}}\\,dx+\\varepsilon\\int_{B_\\frac{r}{2}}|\\D u-\\D u_H|^p\\,dx \\bigg).\n\\end{align}\nChoosing $\\varepsilon$ sufficiently small, we get \n\\begin{align}\n \\int_{B_{\\frac{r}{2}}}|\\D u_H-\\D u|^p\\,dx\n& \\leq C\\int_{(E\\Delta E_H)\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D u_H|^2\\big)^{\\frac{p}{2}}\\,dx.\n\\end{align}\nThus, the minimality of $u_H$ with respect to $u$, H\\\"older's inequality and Theorem \\ref{HigherInt} yield\n\\begin{align}\n\\label{eqq1}\n\\int_{B_{\\frac{r}{2}}}|\\D u_H-\\D u|^p\\,dx& \\leq \\int_{(E\\Delta E_H)\\cap B_\\frac{r}{2}} (1+|\\nabla u|^2)^{\\frac p2}\\,dx\\\\\n& \\le \\bigg(\\frac{|(E\\Delta E_H)\\cap B_r|}{|B_r|}\\bigg)^{1-\\frac1s}|B_r|\\bigg(\\dashint_{B_r}(1+|\\nabla u|^2)^{\\frac{sp}{2}}\\,dx\\bigg)^{\\frac1s}\\\\\n& \\le C\\,\\e_0^{1-\\frac1s}\\int_{B_r}(1+|\\nabla u|^2)^{\\frac p2}\\,dx,\n\\end{align}\nwhere we have used assumption \\emph{iii)} and $C=C(n,p,\\beta/\\alpha,\\nu/L)$ is a positive constant.\n\n\\medskip\n\\noindent\\textbf{Step 3: the one-phase cases (i) and (ii).}\nAssume (i): $|E\\cap B_r|<\\e_0|B_r|$ (the case (ii) is analogous). Let us denote by $v\\in u+W^{1,p}_0(B_\\frac{r}{2})$ be the unique minimizer of the functional\n\\[\n\\mathcal G(w):=\\int_{B_\\frac{r}{2}}\\,F(\\nabla w)\\,dx .\n\\]\nWe can proceed as in step $(iii)$ subtracting the Euler-Lagrange equation satisfied by $u$ from the one satisfied by $v$ and testing them with $u-v$, getting\n\\begin{align}\n \\int_{B_{\\frac{r}{2}}} \\sigma_E\n\\big(DF(\\nabla u)-DF(\\nabla v)\\big)\n\\cdot(\\nabla u-\\nabla v)\\,dx\n& =\n\\int_{B_{\\frac{r}{2}}} (\\alpha-\\sigma_E)\\,\nDF(\\nabla v)\\cdot(\\nabla u-\\nabla v)\\,dx\n\\end{align}\nUsing assumptions \\eqref{Monotonicity} and \\eqref{Growth} and H\\\"older's inequality, for every $\\varepsilon>0$ it holds that\n\\begin{align}\n \\int_{B_{\\frac{r}{2}}}|\\D v -\\D u|^p\\,dx\n & \\leq C\\int_{E\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D v|^2\\big)^{\\frac{p-2}{2}}|\\D u-\\D v|\\,dx\\\\\n& \\leq C\\bigg(\\frac{1}{\\varepsilon}\\int_{E\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D v|^2\\big)^{\\frac{p}{2}}\\,dx+\\varepsilon\\int_{B_\\frac{r}{2}}|\\D u-\\D v|^p\\,dx \\bigg).\n\\end{align}\nChoosing $\\varepsilon$ sufficiently small, we get \n\\begin{align}\n \\int_{B_{\\frac{r}{2}}}|\\D v-\\D u|^p\\,dx\n& \\leq C\\int_{E\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D v|^2\\big)^{\\frac{p}{2}}\\,dx.\n\\end{align}\nUsing the minimality of $v$ compared with $u$ and Theorem\n\\ref{HigherInt} we deduce,\n\\begin{align}\n\\label{eqq11}\n\\int_{B_{\\frac{r}{2}}}|\\D v-\\D u|^p\\,dx& \\leq \\int_{E\\cap B_{\\frac{r}{2}}} (1+|\\nabla u|^2)^{\\frac p2}\\,dx\\\\\n& \\le \\bigg(\\frac{E\\cap B_{r}}{|B_r|}\\bigg)^{1-\\frac1s}|B_r|\\bigg(\\dashint_{B_r}(1+|\\nabla u|^2)^{\\frac{sp}{2}}\\,dx\\bigg)^{\\frac1s}\\\\\n& \\le C\\,\\e_0^{1-\\frac1s}\\int_{B_r}(1+|\\nabla u|^2)^{\\frac p2}\\,dx,\n\\end{align}\nwhere we have used assumption \\emph{i)} and $C=C(n,p,\\beta/\\alpha,\\nu/L)$ is a positive constant. Thereafter we can argue as in the previous step using the classical decay estimate for minimizers of regular integrals in the Calculus of Variations\n\\[\n\\int_{B_{\\tau \\rho}}|\\nabla v|^p\\,dx\n\\le C\\tau^{n}\\int_{B_{ \\rho}}|\\nabla u|^p\\,dx,\n\\]\nfor every $\\rho\\leq \\frac{r}{2}$.\n\\end{proof}", "post_theorem_intro_text_len": 3397, "post_theorem_intro_text": "\\begin{Rem}\nWithin the present variational transmission framework, Morrey regularity represents\nthe natural level of regularity that can be expected for the gradient of local\nminimizers, in view of the discontinuous coefficients and the intrinsic transmission\ncondition.\n\\end{Rem}\nThe proof of Theorem~\\ref{MainThm} is obtained as a consequence of the following\ndecay estimate for the energy.\n\n\\begin{Prop}\\label{FJNL}\nLet $u$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin \\eqref{MainF} under the assumption of Theorem~\\ref{MainThm}.\nThen there exists a constant $0 < \\tau_0 < 1$ such that the following holds:\nfor every $\\tau \\in (0,\\tau_0)$ there exists $\\varepsilon_0 = \\varepsilon_0(\\tau) > 0$\nwith the property that, if $B_r(x_0) \\Subset \\Omega$ and one of the following\nconditions is satisfied,\n\\begin{enumerate}[label=(\\roman*)]\n\\item $|E \\cap B_r(x_0)| < \\varepsilon_0 |B_r|$,\n\\item $|B_r(x_0) \\setminus E| < \\varepsilon_0 |B_r|$,\n\\item there exists a half-space $H$ such that\n\\[\n\\frac{|(E \\Delta H) \\cap B_r(x_0)|}{|B_r|} < \\varepsilon_0,\n\\]\n\\end{enumerate}\nthen, for every $0 < \\delta < n$, the estimate\n\\[\n\\int_{B_{\\tau r}(x_0)} |\\nabla u|^p \\, dx\n\\leq\nC_0 \\tau^{n-\\delta}\n\\left(\n\\int_{B_r(x_0)} |\\nabla u|^p \\, dx + r^n\n\\right)\n\\]\nholds, where the constant $C_0$ depends only on\n$n, \\nu, L, \\alpha,\\beta, \\delta$ and $\\|\\nabla u\\|_{L^2(\\Omega)}$.\n\\end{Prop}\n\\begin{Rem}\n\\label{Reg}\n    For the proof of Theorem \\ref{MainThm}, it would be sufficient to state the proposition only in case (i). We nevertheless include the other two cases for the sake of completeness.\n\\end{Rem}\n\nWe give an outline of the proof of Proposition~\\ref{FJNL}. It relies on decay estimates for the gradient\nnear the interface, first established in a flat configuration, i.e.\\ when the interface coincides\nwith a hyperplane.\nSection~3 is devoted to this ``flat case''.\n\nThe starting point is a local boundedness estimate for the tangential gradient\n$\\nabla' u$, where $\\nabla'=(\\partial_{x_1},\\dots,\\partial_{x_{n-1}})$ denotes the vector of derivatives parallel to\nthe interface.\nThis is proved in Proposition~3.2 via a difference-quotient argument with test functions involving\nonly tangential increments.\nA key ingredient is a Moser iteration applied to\n$Z:=|\\Delta_{i,h}u|^{m/p}$.\nDuring the estimates, the full gradient $\\nabla u$ enters through the structure conditions\n\\emph{(H1)}--\\emph{(H2)} and is treated as a weight; this is the origin of the restriction\n$2<p<\\frac{2n}{n-2}$.\n\nThe bound obtained in Proposition~3.2 is preliminary, since the decay exponent in the estimates is not optimal and\nthe constant may depend on $\\mu$.\nTo obtain the scale-invariant Lipschitz estimate for $\\nabla' u$ stated in Proposition~3.4,\nwe perform a rescaling argument, which requires uniformity with respect to $\\mu$.\nThis uniformity is proved in Lemma~3.3, exploiting the boundedness of $\\nabla' u$ and a Poincar\\'e-type\ninequality for $Z:=|\\partial_i u|^{m/p}$, $i=1,\\dots,n-1$, which is an admissible Sobolev function by\nTheorem~3.1.\n\nOnce the tangential Lipschitz bound is available, we prove the Morrey estimate in the flat case in\nTheorem~3.5 by a comparison argument (Section~3.2).\nFinally, the general case follows from a standard flattening/approximation procedure using the $C^1$\nregularity of the interface, along the lines of the quadratic theory (see, e.g.,~\\cite{E,FJ}).", "sketch": "The proof of Theorem~\\ref{MainThm} is obtained as a consequence of the decay estimate for the energy stated in Proposition~\\ref{FJNL}.\n\nAn outline is given for the proof of Proposition~\\ref{FJNL}:\n\\begin{itemize}\n\\item The argument \\emph{relies on decay estimates for the gradient near the interface}, first established in a \\emph{flat configuration} (interface coincides with a hyperplane), treated in Section~3.\n\\item Starting point: a \\emph{local boundedness estimate for the tangential gradient} $\\nabla' u$ (derivatives parallel to the interface). This is proved in Proposition~3.2 via a \\emph{difference-quotient argument} with test functions involving only tangential increments.\n\\item Key ingredient: a \\emph{Moser iteration} applied to $Z:=|\\Delta_{i,h}u|^{m/p}$. In the estimates, the full gradient $\\nabla u$ appears through (H1)--(H2) and is \\emph{treated as a weight}; this is cited as the origin of the restriction $2<p<\\frac{2n}{n-2}$.\n\\item Since the Proposition~3.2 bound has a non-optimal decay exponent and constants that may depend on $\\mu$, one obtains the \\emph{scale-invariant Lipschitz estimate for $\\nabla' u$} (Proposition~3.4) by a \\emph{rescaling argument} requiring \\emph{uniformity with respect to $\\mu$}. This uniformity is proved in Lemma~3.3, using boundedness of $\\nabla' u$ and a \\emph{Poincar\\'e-type inequality} for $Z:=|\\partial_i u|^{m/p}$, $i=1,\\dots,n-1$, which is admissible by Theorem~3.1.\n\\item With the tangential Lipschitz bound, the \\emph{Morrey estimate in the flat case} is proved in Theorem~3.5 by a \\emph{comparison argument} (Section~3.2).\n\\item The \\emph{general case} then follows by a \\emph{standard flattening/approximation procedure} using the $C^1$ regularity of the interface, along the lines of the quadratic theory.\n\\end{itemize}\n\nAdditionally, Remark~\\ref{Reg} notes that for proving Theorem~\\ref{MainThm} it would be sufficient to have Proposition~\\ref{FJNL} only in case (i), though cases (ii)--(iii) are included for completeness.", "expanded_sketch": "The proof of the main theorem is obtained as a consequence of the decay estimate for the energy stated in the following proposition.\n\n\\begin{Prop}\\label{FJNL}\nLet $u$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin \\eqref{MainF} under the assumption of Theorem~\\ref{MainThm}.\nThen there exists a constant $0 < \\tau_0 < 1$ such that the following holds:\nfor every $\\tau \\in (0,\\tau_0)$ there exists $\\varepsilon_0 = \\varepsilon_0(\\tau) > 0$\nwith the property that, if $B_r(x_0) \\Subset \\Omega$ and one of the following\nconditions is satisfied,\n\\begin{enumerate}[label=(\\roman*)]\n\\item $|E \\cap B_r(x_0)| < \\varepsilon_0 |B_r|$,\n\\item $|B_r(x_0) \\setminus E| < \\varepsilon_0 |B_r|$,\n\\item there exists a half-space $H$ such that\n\\[\n\\frac{|(E \\Delta H) \\cap B_r(x_0)|}{|B_r|} < \\varepsilon_0,\n\\]\n\\end{enumerate}\nthen, for every $0 < \\delta < n$, the estimate\n\\[\n\\int_{B_{\\tau r}(x_0)} |\\nabla u|^p \\, dx\n\\leq\nC_0 \\tau^{n-\\delta}\n\\left(\n\\int_{B_r(x_0)} |\\nabla u|^p \\, dx + r^n\n\\right)\n\\]\nholds, where the constant $C_0$ depends only on\n$n, \\nu, L, \\alpha,\\beta, \\delta$ and $\\|\\nabla u\\|_{L^2(\\Omega)}$.\n\\end{Prop}\n\nAn outline is given for the proof of Proposition~\\ref{FJNL}:\n\\begin{itemize}\n\\item The argument \\emph{relies on decay estimates for the gradient near the interface}, first established in a \\emph{flat configuration} (interface coincides with a hyperplane), treated next.\n\\item Starting point: a \\emph{local boundedness estimate for the tangential gradient} $\\nabla' u$ (derivatives parallel to the interface). This is proved in Proposition~3.2 via a \\emph{difference-quotient argument} with test functions involving only tangential increments.\n\\item Key ingredient: a \\emph{Moser iteration} applied to $Z:=|\\Delta_{i,h}u|^{m/p}$. In the estimates, the full gradient $\\nabla u$ appears through (H1)--(H2) and is \\emph{treated as a weight}; this is cited as the origin of the restriction $2<p<\\frac{2n}{n-2}$.\n\\item Since the Proposition~3.2 bound has a non-optimal decay exponent and constants that may depend on $\\mu$, one obtains the \\emph{scale-invariant Lipschitz estimate for $\\nabla' u$} (Proposition~3.4) by a \\emph{rescaling argument} requiring \\emph{uniformity with respect to $\\mu$}. This uniformity is proved in Lemma~3.3, using boundedness of $\\nabla' u$ and a \\emph{Poincar\\'e-type inequality} for $Z:=|\\partial_i u|^{m/p}$, $i=1,\\dots,n-1$, which is admissible by Theorem~3.1.\n\\item With the tangential Lipschitz bound, the \\emph{Morrey estimate in the flat case} is proved in Theorem~3.5 by a \\emph{comparison argument}.\n\\item The \\emph{general case} then follows by a \\emph{standard flattening/approximation procedure} using the $C^1$ regularity of the interface, along the lines of the quadratic theory.\n\\end{itemize}\n\nAdditionally, Remark~\\ref{Reg} notes that for proving the main theorem it would be sufficient to have Proposition~\\ref{FJNL} only in case (i), though cases (ii)--(iii) are included for completeness.", "expanded_theorem": "[Gradient regularity in Morrey spaces]\\label{MainThm}\nLet $u \\in W^{1,p}(\\Omega)$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined in\n\\begin{equation}\\label{MainF}\n\\mathcal{F}(u;E) := \\int_{\\Omega} \\sigma_E(x)\\, F(\\nabla u)\\, dx,\n\\end{equation}\nand assume that $\\partial E$ is a hypersurface of class $C^1$.\nSuppose that assumptions \\emph{(H1)}--\\emph{(H2)} are satisfied where $2<p<\\frac{2n}{n-2}$.\nThen, for every $0 \\leq \\lambda < n$,\n\\[\n\\nabla u \\in L^{{p},\\lambda}_{\\mathrm{loc}}(\\Omega).\n\\]\nMoreover, for every $\\Omega' \\Subset \\Omega$, there exists a constant\n$C = C\\big(n, p, \\nu, L, \\beta, E, \\operatorname{diam}(\\Omega),\n\\operatorname{dist}(\\Omega', \\partial\\Omega)\\big) > 0$\nsuch that\n\\[\n\\|\\nabla u\\|_{L^{{p},\\lambda}(\\Omega')} \\leq C.\n\\]", "theorem_type": ["Implication", "Existential–Universal"], "mcq": {"question": "Let \\(\\Omega\\subset\\mathbb{R}^n\\) be a bounded domain, let \\(E\\subset\\Omega\\) be measurable with \\(\\partial E\\) a \\(C^1\\) hypersurface, and fix constants \\(0<\\alpha<\\beta<\\infty\\). Define \\(\\sigma_E:=\\beta\\mathbbm{1}_E+\\alpha\\mathbbm{1}_{E^c}\\) and, for \\(v\\in W^{1,p}(\\Omega)\\),\n\\[\\mathcal F(v;E):=\\int_{\\Omega}\\sigma_E(x)F(\\nabla v)\\,dx.\\]\nAssume \\(F\\in C^2(\\mathbb{R}^n)\\) and that for some \\(\\nu,L>0\\), \\(\\mu\\in(0,1]\\), and every \\(\\xi,\\eta\\in\\mathbb{R}^n\\),\n\\[\\langle \\nabla F(\\xi)-\\nabla F(\\eta),\\xi-\\eta\\rangle\\ge \\nu\\big(\\mu^2+|\\xi|^2+|\\eta|^2\\big)^{\\frac{p-2}{2}}|\\xi-\\eta|^2,\\]\n\\[|\\nabla F(\\xi)-\\nabla F(\\eta)|\\le L\\big(\\mu^2+|\\xi|^2+|\\eta|^2\\big)^{\\frac{p-2}{2}}|\\xi-\\eta|,\\]\nwith \\(2<p<\\frac{2n}{n-2}\\). Suppose \\(u\\in W^{1,p}(\\Omega)\\) is a local minimizer in the sense that for every ball \\(B_r(x_0)\\subset\\Omega\\) and every \\(\\phi\\in W_0^{1,p}(B_r(x_0))\\),\n\\[\\mathcal F(u;B_r(x_0))\\le \\mathcal F(u+\\phi;B_r(x_0)).\\]\nWhich conclusion about \\(\\nabla u\\) holds? Here \\(L^{p,\\lambda}(\\Omega')\\) is the classical Morrey space, for instance with norm \\(\\|f\\|_{L^{p,\\lambda}(\\Omega')}:=\\sup_{B_\\rho(x)\\subset\\Omega'}\\big(\\rho^{-\\lambda}\\int_{B_\\rho(x)}|f|^p\\,dy\\big)^{1/p}\\).", "correct_choice": {"label": "A", "text": "For every \\(0\\le \\lambda<n\\), one has \\(\\nabla u\\in L^{p,\\lambda}_{\\mathrm{loc}}(\\Omega)\\). Moreover, for every \\(\\Omega'\\Subset\\Omega\\), there exists a constant \\(C=C\\big(n,p,\\nu,L,\\beta,E,\\operatorname{diam}(\\Omega),\\operatorname{dist}(\\Omega',\\partial\\Omega)\\big)>0\\) such that\n\\[\\|\\nabla u\\|_{L^{p,\\lambda}(\\Omega')}\\le C.\\]"}, "choices": [{"label": "B", "text": "For every \\(0\\le \\lambda<n\\), one has \\(\\nabla u\\in L^{p,\\lambda}_{\\mathrm{loc}}(\\Omega)\\). Moreover, for every \\(\\Omega'\\Subset\\Omega\\), there exists a constant \\(C=C\\big(n,p,\\nu,L,\\alpha,\\beta,E,\\operatorname{diam}(\\Omega),\\operatorname{dist}(\\Omega',\\partial\\Omega)\\big)>0\\), independent of \\(\\lambda\\), such that\n\\[\\|\\nabla u\\|_{L^{p,\\lambda}(\\Omega')}\\le C.\\]"}, {"label": "C", "text": "One has \\(\\nabla u\\in L^{p,\\lambda}_{\\mathrm{loc}}(\\Omega)\\) for at least one exponent \\(\\lambda\\in[0,n)\\)."}, {"label": "D", "text": "For every \\(0\\le \\lambda<n\\), one has \\(\\nabla u\\in L^{p,\\lambda}_{\\mathrm{loc}}(\\Omega)\\). Moreover, for every \\(\\Omega'\\Subset\\Omega\\), there exists a constant \\(C=C\\big(n,p,\\nu,L,\\beta,E,\\operatorname{diam}(\\Omega),\\operatorname{dist}(\\Omega',\\partial\\Omega)\\big)>0\\) such that\n\\[\\|\\nabla u\\|_{L^{p,\\lambda}(\\Omega')}\\le C\\,\\|\\nabla u\\|_{L^{p}(\\Omega)}.\\]"}, {"label": "E", "text": "There exists \\(\\lambda_0\\in(0,n)\\), depending only on \\(n,p,\\nu,L,\\alpha,\\beta\\), such that \\(\\nabla u\\in L^{p,\\lambda_0}_{\\mathrm{loc}}(\\Omega)\\). Moreover, for every \\(\\Omega'\\Subset\\Omega\\), there exists a constant \\(C=C\\big(n,p,\\nu,L,\\beta,E,\\operatorname{diam}(\\Omega),\\operatorname{dist}(\\Omega',\\partial\\Omega)\\big)>0\\) with\n\\[\\|\\nabla u\\|_{L^{p,\\lambda_0}(\\Omega')}\\le C.\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "uniformity", "tampered_component": "dependence on Morrey exponent in the local bound", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "universal quantifier over all \\(\\lambda<n\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "absolute local Morrey bound replaced by a bound proportional to \\(\\|\\nabla u\\|_{L^p(\\Omega)}\\)", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "full range \\(0\\le \\lambda<n\\) weakened to existence of a single universal \\(\\lambda_0\\)", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem states the hypotheses in detail but does not reveal the conclusion. The correct option is not explicitly or implicitly signaled by wording in the prompt."}, "TAS": {"score": 1, "justification": "This is largely a theorem-recall item: the setup mirrors a regularity theorem and asks for its conclusion. However, it is not a pure restatement because the options vary in quantifiers, endpoint cases, and parameter dependence."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to reject overly strong or improperly generalized statements (such as including λ = n, removing dependence on E, or extending to all p > 2). Still, the item mainly tests recognition of the precise theorem statement rather than deep derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: weakening the quantifier, adding the forbidden endpoint λ = n, claiming uniformity in E, and overextending the p-range. They are distinct and nontrivial."}, "total_score": 6, "overall_assessment": "A solid theorem-identification MCQ with strong distractors and no answer leakage, but it leans more toward precise recall of a known result than toward genuinely generative mathematical reasoning."}}
{"id": "2602.14658v1", "paper_link": "http://arxiv.org/abs/2602.14658v1", "theorems_cnt": 5, "theorem": {"env_name": "Thm", "content": "[Gradient regularity in Morrey spaces]\\label{MainThm}\nLet $u \\in W^{1,p}(\\Omega)$ be a local minimizer of  the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin\\eqref{MainF}, and assume that\n$\\partial E$ is a hypersurface of class $C^1$.\nSuppose that assumptions \\emph{(H1)}--\\emph{(H2)} are satisfied where $2<p<\\frac{2n}{n-2}$.\nThen, for every $0 \\leq \\lambda < n$,\n\\[\n\\nabla u \\in L^{{p},\\lambda}_{\\mathrm{loc}}(\\Omega).\n\\]\nMoreover, for every $\\Omega' \\Subset \\Omega$, there exists a constant\n$C = C\\big(n, p, \\nu, L, \\beta, E, \\operatorname{diam}(\\Omega),\n\\operatorname{dist}(\\Omega', \\partial\\Omega)\\big) > 0$\nsuch that\n\\[\n\\|\\nabla u\\|_{L^{{p},\\lambda}(\\Omega')} \\leq C.\n\\]", "start_pos": 8951, "end_pos": 9651, "label": "MainThm"}, "ref_dict": {"FJNL": "\\begin{Prop}\\label{FJNL}\nLet $u$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin \\eqref{MainF} under the assumption of Theorem~\\ref{MainThm}.\nThen there exists a constant $0 < \\tau_0 < 1$ such that the following holds:\nfor every $\\tau \\in (0,\\tau_0)$ there exists $\\varepsilon_0 = \\varepsilon_0(\\tau) > 0$\nwith the property that, if $B_r(x_0) \\Subset \\Omega$ and one of the following\nconditions is satisfied,\n\\begin{enumerate}[label=(\\roman*)]\n\\item $|E \\cap B_r(x_0)| < \\varepsilon_0 |B_r|$,\n\\item $|B_r(x_0) \\setminus E| < \\varepsilon_0 |B_r|$,\n\\item there exists a half-space $H$ such that\n\\[\n\\frac{|(E \\Delta H) \\cap B_r(x_0)|}{|B_r|} < \\varepsilon_0,\n\\]\n\\end{enumerate}\nthen, for every $0 < \\delta < n$, the estimate\n\\[\n\\int_{B_{\\tau r}(x_0)} |\\nabla u|^p \\, dx\n\\leq\nC_0 \\tau^{n-\\delta}\n\\left(\n\\int_{B_r(x_0)} |\\nabla u|^p \\, dx + r^n\n\\right)\n\\]\nholds, where the constant $C_0$ depends only on\n$n, \\nu, L, \\alpha,\\beta, \\delta$ and $\\|\\nabla u\\|_{L^2(\\Omega)}$.\n\\end{Prop}", "MainF": "\\begin{equation}\\label{MainF}\n\\mathcal{F}(u;E) := \\int_{\\Omega} \\sigma_E(x)\\, F(\\nabla u)\\, dx,\n\\end{equation}", "MainThm": "\\begin{Thm}[Gradient regularity in Morrey spaces]\\label{MainThm}\nLet $u \\in W^{1,p}(\\Omega)$ be a local minimizer of  the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin\\eqref{MainF}, and assume that\n$\\partial E$ is a hypersurface of class $C^1$.\nSuppose that assumptions \\emph{(H1)}--\\emph{(H2)} are satisfied where $2<p<\\frac{2n}{n-2}$.\nThen, for every $0 \\leq \\lambda < n$,\n\\[\n\\nabla u \\in L^{{p},\\lambda}_{\\mathrm{loc}}(\\Omega).\n\\]\nMoreover, for every $\\Omega' \\Subset \\Omega$, there exists a constant\n$C = C\\big(n, p, \\nu, L, \\beta, E, \\operatorname{diam}(\\Omega),\n\\operatorname{dist}(\\Omega', \\partial\\Omega)\\big) > 0$\nsuch that\n\\[\n\\|\\nabla u\\|_{L^{{p},\\lambda}(\\Omega')} \\leq C.\n\\]\n\\end{Thm}"}, "pre_theorem_intro_text_len": 5502, "pre_theorem_intro_text": "Transmission problems arise naturally in the study of diffusive processes in\nheterogeneous media, where the governing laws change abruptly across fixed interfaces.\nTypical applications include elasticity theory, composite materials, and conductivity\nphenomena. This kind of problems appear also in the study of thermal insulation of bodies under sources and prescribed boundary conditions (see for example \\cite{AC,DPNST})\nFrom a mathematical perspective, such problems are characterised by the presence of\npiecewise-defined operators or energy densities, together with suitable transmission\nconditions prescribed along the separating interface.\nThe first systematic investigation of transmission problems in elasticity theory dates\nback to the seminal work of M.~Picone in the 1950s (see \\cite{Pico}).\n\nIn this paper, we investigate a class of \\emph{variational transmission problems}\nassociated with nonlinear integral functionals exhibiting discontinuous coefficients\nacross a prescribed interface.\nOur focus is on the regularity properties of local minimizers, with particular attention\nto the behaviour of the gradient near the interface.\n\nLet $\\Omega \\subset \\mathbb{R}^n$ be a bounded domain and let $E \\subset \\Omega$ be\na measurable subset.\nWe assume that the interface $\\partial E$ is a hypersurface of class $C^1$.\nWe fix two constants $0 < \\alpha < \\beta < +\\infty$ and define\n\\[\n\\sigma_E := \\beta \\mathbbm{1}_E + \\alpha \\mathbbm{1}_{E^c},\n\\]\nwhere $\\mathbbm{1}_E$ denotes the characteristic function of the set $E$ and $E^c$ is the complementary of $E$.\nWe consider the integral functional\n\\begin{equation}\\label{MainF}\n\\mathcal{F}(u;E) := \\int_{\\Omega} \\sigma_E(x)\\, F(\\nabla u)\\, dx,\n\\end{equation}\ndefined for functions $u \\in W^{1,p}(\\Omega)$.\n\nThe integrand $F \\colon \\mathbb{R}^n \\to \\mathbb{R}$ is assumed to belong to\n$C^2(\\mathbb{R}^n)$ and to satisfy standard $p$-growth and monotonicity conditions:\nfor all $\\xi, \\eta \\in \\mathbb{R}^n$,\n\\begin{equation}\n\\label{Monotonicity}\\tag{H1}\n\\langle \\nabla F(\\xi) - \\nabla F(\\eta), \\xi - \\eta \\rangle\n\\geq \\nu \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|^2,\n\\end{equation}\n\\begin{equation}\n\\label{Growth}\\tag{H2}\n|\\nabla F(\\xi) - \\nabla F(\\eta)|\n\\leq L \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|,\n\\end{equation}\nfor some constants $\\nu, L > 0$, $\\mu \\in (0,1]$, and $p > 2$.\n\nLocal minimizers of the functional $\\mathcal{F}$ are understood in the usual\nvariational sense.\n\n\\begin{Def}\nLet $u \\in W^{1,p}(\\Omega)$.\nWe say that $u$ is a \\emph{local minimizer} of $\\mathcal{F}(\\cdot;\\Omega)$ in $\\Omega$\nif, for every ball $B_r(x_0) \\subset \\Omega$ and every\n$\\phi \\in W^{1,p}_0(B_r(x_0))$, one has\n\\[\n\\mathcal{F}(u;B_r(x_0)) \\leq \\mathcal{F}(u + \\phi;B_r(x_0)).\n\\]\n\\end{Def}\n\nBefore stating our main result, let us place it in the context of the existing literature.\nIn the linear case $p = 2$, transmission problems with piecewise constant coefficients\nhave been extensively studied, and gradient regularity across sufficiently smooth\ninterfaces is by now classical (see, e.g.,~\\cite{GT,LV,Grisvard}).\nSuch results follow from the elliptic regularity theory for divergence-form operators\nwith discontinuous coefficients (see for instance\n\\cite{Evans, GT, LM, Grisvard} and \\cite[Theorem~7.53]{AFP}.\nIn this setting, local $W^{1,2}$- and H\\\"older-type regularity for the gradient of minimizers is well\nunderstood under mild geometric assumptions on the interface.\n\nMore recently, fine regularity properties for linear transmission problems across\n$C^{1,\\alpha}$ interfaces have been obtained by Caffarelli, Soria-Carro, and Stinga\n\\cite{CSS}.\n\nBy contrast, the nonlinear case $p \\neq 2$ remains far less developed.\nTo the best of our knowledge, the only available regularity results for nonlinear\ntransmission problems concern equations subject to \\emph{additive} transmission\nconditions on the interface.\nIn particular, in the recent work \\cite{BPU}, BMO regularity of the gradient is obtained\nfor degenerate quasilinear equations under transmission conditions of the form\n\\[\ng(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n-\ng(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n= f \\quad \\text{on } \\partial E.\n\\]\n\nIn contrast, the transmission condition naturally associated with the variational\nfunctional \\eqref{MainF} is of \\emph{multiplicative} type, namely\n\\[\ng_1(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n=\ng_2(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n\\quad \\text{on } \\partial E,\n\\]\nwhich arises intrinsically from the Euler--Lagrange equations and expresses the\ncontinuity of the nonlinear flux across the interface. We stress that this multiplicative transmission condition is intrinsic to the\nvariational structure of the functional \\eqref{MainF}, as it arises directly from\nthe Euler--Lagrange equations and cannot be prescribed independently.\nThe presence of a discontinuous coefficient combined with nonlinear growth prevents\na direct application of standard techniques such as freezing arguments or difference\nquotient estimates across the interface.\nTo the best of our knowledge, no Morrey-space regularity results for gradients of local\nminimizers are available in this genuinely nonlinear variational transmission setting.\n\nDenoting by $L^{{p},\\lambda}$ the classical Morrey spaces, we can now state the main\nresult of this paper, which establishes local Morrey regularity for the gradient of\nlocal minimizers under minimal geometric assumptions on the interface.", "context": "Let $\\Omega \\subset \\mathbb{R}^n$ be a bounded domain and let $E \\subset \\Omega$ be\na measurable subset.\nWe assume that the interface $\\partial E$ is a hypersurface of class $C^1$.\nWe fix two constants $0 < \\alpha < \\beta < +\\infty$ and define\n\\[\n\\sigma_E := \\beta \\mathbbm{1}_E + \\alpha \\mathbbm{1}_{E^c},\n\\]\nwhere $\\mathbbm{1}_E$ denotes the characteristic function of the set $E$ and $E^c$ is the complementary of $E$.\nWe consider the integral functional\n\\begin{equation}\\label{MainF}\n\\mathcal{F}(u;E) := \\int_{\\Omega} \\sigma_E(x)\\, F(\\nabla u)\\, dx,\n\\end{equation}\ndefined for functions $u \\in W^{1,p}(\\Omega)$.\n\nThe integrand $F \\colon \\mathbb{R}^n \\to \\mathbb{R}$ is assumed to belong to\n$C^2(\\mathbb{R}^n)$ and to satisfy standard $p$-growth and monotonicity conditions:\nfor all $\\xi, \\eta \\in \\mathbb{R}^n$,\n\\begin{equation}\n\\label{Monotonicity}\\tag{H1}\n\\langle \\nabla F(\\xi) - \\nabla F(\\eta), \\xi - \\eta \\rangle\n\\geq \\nu \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|^2,\n\\end{equation}\n\\begin{equation}\n\\label{Growth}\\tag{H2}\n|\\nabla F(\\xi) - \\nabla F(\\eta)|\n\\leq L \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|,\n\\end{equation}\nfor some constants $\\nu, L > 0$, $\\mu \\in (0,1]$, and $p > 2$.\n\n\\begin{Def}\nLet $u \\in W^{1,p}(\\Omega)$.\nWe say that $u$ is a \\emph{local minimizer} of $\\mathcal{F}(\\cdot;\\Omega)$ in $\\Omega$\nif, for every ball $B_r(x_0) \\subset \\Omega$ and every\n$\\phi \\in W^{1,p}_0(B_r(x_0))$, one has\n\\[\n\\mathcal{F}(u;B_r(x_0)) \\leq \\mathcal{F}(u + \\phi;B_r(x_0)).\n\\]\n\\end{Def}\n\nBy contrast, the nonlinear case $p \\neq 2$ remains far less developed.\nTo the best of our knowledge, the only available regularity results for nonlinear\ntransmission problems concern equations subject to \\emph{additive} transmission\nconditions on the interface.\nIn particular, in the recent work \\cite{BPU}, BMO regularity of the gradient is obtained\nfor degenerate quasilinear equations under transmission conditions of the form\n\\[\ng(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n-\ng(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n= f \\quad \\text{on } \\partial E.\n\\]\n\nIn contrast, the transmission condition naturally associated with the variational\nfunctional \\eqref{MainF} is of \\emph{multiplicative} type, namely\n\\[\ng_1(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n=\ng_2(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n\\quad \\text{on } \\partial E,\n\\]\nwhich arises intrinsically from the Euler--Lagrange equations and expresses the\ncontinuity of the nonlinear flux across the interface. We stress that this multiplicative transmission condition is intrinsic to the\nvariational structure of the functional \\eqref{MainF}, as it arises directly from\nthe Euler--Lagrange equations and cannot be prescribed independently.\nThe presence of a discontinuous coefficient combined with nonlinear growth prevents\na direct application of standard techniques such as freezing arguments or difference\nquotient estimates across the interface.\nTo the best of our knowledge, no Morrey-space regularity results for gradients of local\nminimizers are available in this genuinely nonlinear variational transmission setting.\n\nDenoting by $L^{{p},\\lambda}$ the classical Morrey spaces, we can now state the main\nresult of this paper, which establishes local Morrey regularity for the gradient of\nlocal minimizers under minimal geometric assumptions on the interface.", "full_context": "Let $\\Omega \\subset \\mathbb{R}^n$ be a bounded domain and let $E \\subset \\Omega$ be\na measurable subset.\nWe assume that the interface $\\partial E$ is a hypersurface of class $C^1$.\nWe fix two constants $0 < \\alpha < \\beta < +\\infty$ and define\n\\[\n\\sigma_E := \\beta \\mathbbm{1}_E + \\alpha \\mathbbm{1}_{E^c},\n\\]\nwhere $\\mathbbm{1}_E$ denotes the characteristic function of the set $E$ and $E^c$ is the complementary of $E$.\nWe consider the integral functional\n\\begin{equation}\\label{MainF}\n\\mathcal{F}(u;E) := \\int_{\\Omega} \\sigma_E(x)\\, F(\\nabla u)\\, dx,\n\\end{equation}\ndefined for functions $u \\in W^{1,p}(\\Omega)$.\n\nThe integrand $F \\colon \\mathbb{R}^n \\to \\mathbb{R}$ is assumed to belong to\n$C^2(\\mathbb{R}^n)$ and to satisfy standard $p$-growth and monotonicity conditions:\nfor all $\\xi, \\eta \\in \\mathbb{R}^n$,\n\\begin{equation}\n\\label{Monotonicity}\\tag{H1}\n\\langle \\nabla F(\\xi) - \\nabla F(\\eta), \\xi - \\eta \\rangle\n\\geq \\nu \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|^2,\n\\end{equation}\n\\begin{equation}\n\\label{Growth}\\tag{H2}\n|\\nabla F(\\xi) - \\nabla F(\\eta)|\n\\leq L \\big( \\mu^2 + |\\xi|^2 + |\\eta|^2 \\big)^{\\frac{p-2}{2}} |\\xi - \\eta|,\n\\end{equation}\nfor some constants $\\nu, L > 0$, $\\mu \\in (0,1]$, and $p > 2$.\n\n\\begin{Def}\nLet $u \\in W^{1,p}(\\Omega)$.\nWe say that $u$ is a \\emph{local minimizer} of $\\mathcal{F}(\\cdot;\\Omega)$ in $\\Omega$\nif, for every ball $B_r(x_0) \\subset \\Omega$ and every\n$\\phi \\in W^{1,p}_0(B_r(x_0))$, one has\n\\[\n\\mathcal{F}(u;B_r(x_0)) \\leq \\mathcal{F}(u + \\phi;B_r(x_0)).\n\\]\n\\end{Def}\n\nBy contrast, the nonlinear case $p \\neq 2$ remains far less developed.\nTo the best of our knowledge, the only available regularity results for nonlinear\ntransmission problems concern equations subject to \\emph{additive} transmission\nconditions on the interface.\nIn particular, in the recent work \\cite{BPU}, BMO regularity of the gradient is obtained\nfor degenerate quasilinear equations under transmission conditions of the form\n\\[\ng(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n-\ng(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n= f \\quad \\text{on } \\partial E.\n\\]\n\nIn contrast, the transmission condition naturally associated with the variational\nfunctional \\eqref{MainF} is of \\emph{multiplicative} type, namely\n\\[\ng_1(|\\nabla u_1|)\\frac{\\partial u_1}{\\partial \\nu}\n=\ng_2(|\\nabla u_2|)\\frac{\\partial u_2}{\\partial \\nu}\n\\quad \\text{on } \\partial E,\n\\]\nwhich arises intrinsically from the Euler--Lagrange equations and expresses the\ncontinuity of the nonlinear flux across the interface. We stress that this multiplicative transmission condition is intrinsic to the\nvariational structure of the functional \\eqref{MainF}, as it arises directly from\nthe Euler--Lagrange equations and cannot be prescribed independently.\nThe presence of a discontinuous coefficient combined with nonlinear growth prevents\na direct application of standard techniques such as freezing arguments or difference\nquotient estimates across the interface.\nTo the best of our knowledge, no Morrey-space regularity results for gradients of local\nminimizers are available in this genuinely nonlinear variational transmission setting.\n\nDenoting by $L^{{p},\\lambda}$ the classical Morrey spaces, we can now state the main\nresult of this paper, which establishes local Morrey regularity for the gradient of\nlocal minimizers under minimal geometric assumptions on the interface.\n\n\\begin{Prop}\\label{FJNL}\nLet $u$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin \\eqref{MainF} under the assumption of Theorem~\\ref{MainThm}.\nThen there exists a constant $0 < \\tau_0 < 1$ such that the following holds:\nfor every $\\tau \\in (0,\\tau_0)$ there exists $\\varepsilon_0 = \\varepsilon_0(\\tau) > 0$\nwith the property that, if $B_r(x_0) \\Subset \\Omega$ and one of the following\nconditions is satisfied,\n\\begin{enumerate}[label=(\\roman*)]\n\\item $|E \\cap B_r(x_0)| < \\varepsilon_0 |B_r|$,\n\\item $|B_r(x_0) \\setminus E| < \\varepsilon_0 |B_r|$,\n\\item there exists a half-space $H$ such that\n\\[\n\\frac{|(E \\Delta H) \\cap B_r(x_0)|}{|B_r|} < \\varepsilon_0,\n\\]\n\\end{enumerate}\nthen, for every $0 < \\delta < n$, the estimate\n\\[\n\\int_{B_{\\tau r}(x_0)} |\\nabla u|^p \\, dx\n\\leq\nC_0 \\tau^{n-\\delta}\n\\left(\n\\int_{B_r(x_0)} |\\nabla u|^p \\, dx + r^n\n\\right)\n\\]\nholds, where the constant $C_0$ depends only on\n$n, \\nu, L, \\alpha,\\beta, \\delta$ and $\\|\\nabla u\\|_{L^2(\\Omega)}$.\n\\end{Prop}\n\\begin{Rem}\n\\label{Reg}\n For the proof of Theorem \\ref{MainThm}, it would be sufficient to state the proposition only in case (i). We nevertheless include the other two cases for the sake of completeness.\n\\end{Rem}\n\n\\begin{Lem}[Standard $p$-growth bounds]\\label{lem:p-growth}\nAssume \\((H1)\\)--\\((H2)\\). Then there exists a constant $C=C(n,p,\\nu,L)>0$ such that for all $\\xi\\in\\R^n$\n\\[\nC^{-1}(\\mu^2+|\\xi|^2)^{\\frac p2}-C\\mu^p \\le F(\\xi)\\le C(\\mu^2+|\\xi|^2)^{\\frac p2}+C\\mu^p .\n\\]\n\\end{Lem}\nThe following results on higher integrability and local Hölder continuity\nof minimizers are classical. Since their proofs are standard, we omit them and refer the reader to\n\\cite{Giu,GiaMar}.\n{\n\\begin{Thm}[Higher integrability]\n\\label{HigherInt}\nLet $u \\in W^{1,p}(\\Omega)$ be a local minimizer of the functional \\eqref{MainF}.\nThere exist $s > 1$ and $C = C(n,p,\\nu,L,\\beta)$ such that for any ball\n$B_{2r}(x_{0})\\subset\\subset\\Omega$\n\\begin{equation}\n \\int_{B_r(x_0)}|\\nabla u|^{sp}\\,dx\\leq C\\bigg(\\int_{B_{2r}(x_0)}\\big(|\\nabla u|^p+\\mu^p\\big)\\,dx\n \\bigg)^s.\n\\end{equation}\n\\end{Thm}\n\\begin{Thm}[H\\\"older continuity]Let $u \\in W^{1,p}(\\Omega)$ be a local minimizer of the functional \\eqref{MainF}. Then\n\\begin{enumerate}[label=(\\roman*)]\n\\item For any open set $\\Omega'\\Subset \\Omega$ the quantity $\\norm{u}_{L^{\\infty}(\\Omega')}$ is bounded by a constant $C = C(n,p,\\nu,L,\\beta)\\norm{u}_{L^2(\\Omega)}$.\n\\item u is locally H\\\"older continuous in $\\Omega$.\n\\end{enumerate}\n\\end{Thm}\n}\n\n\\begin{Thm}\\label{DS}\nLet $u$ be a minimizer of $\\mathcal{F}$ in $B_r(x_0)$. Then $\\partial_i u\\in W^{1,2}_{loc}(B_{r})$ for every $i=1\\dots n-1$. Moreover the following properties hold:\n\\begin{enumerate}[label=\\roman*)]\n\\item $u \\in W^{2,2}_{\\mathrm{loc}}(B_r^+)$ e $u \\in W^{2,2}_{\\mathrm{loc}}(B_r^-)$;\n\\item for every $x_0 \\in \\Gamma_r$ and $R>0$ such that $B_{4R}(x_0)\\subset B_r$, it holds that\n\\begin{align}\n\\int_{B_{R/2}} V^{p-2}\\,|\\nabla\\nabla' u|^2 \\,dx\n&\\leq \\frac{C}{R^2}\\int_{B_{2R}} V^p \\,dx, \\label{eq:tangential-estimate}\n\\end{align}\nwhere $C=C\\!\\left(\\tfrac{L}{\\nu},\\,n,\\,p,\\,\\alpha,\\,\\beta,\\mu\\right)$ is a positive constant.\n\\end{enumerate}\n\\end{Thm}\n\n\\section{Proof of Proposition 1.3}\n\\begin{proof}\nFix $\\tau\\in(0,\\tau_0)$ (to be chosen small only depending on $n,p,\\nu,L,\\beta/\\alpha$) and assume without loss of generality that $x_0=0$.\nWe prove that there exists $\\e_0=\\e_0(\\tau)>0$ such that if $B_r(x_0)\\Subset\\Om$ and one of {\\rm(i)}--{\\rm(iii)}\nholds, then for every $0<\\delta<n$, then\n\\begin{equation}\\label{eq:prop13-goal}\n\\int_{B_{\\tau r}} |\\nabla u|^p\\, dx\n\\leq\nC_0\\tau^{n-\\delta}\\bigg( \\int_{B_r} |\\nabla u|^p \\, dx\n+r^n\\bigg),\n\\end{equation}\nwhere $C_0=C_0(n,p,\\nu,L,\\beta,\\delta,\\|\\nabla u\\|_{L^2(\\Om)})>0$.\\\\\n\\indent \n\\noindent\\textbf{Step 1: the flat comparison in case (iii).}\nAssume that (iii) holds true. Up to a rigid motion we may assume that $H=\\{x_n>0\\}$ and set\n\\[\nE_H:=H\\cap B_{r},\\quad \\sigma_H:=\\beta\\mathbbm 1_{E_H}+\\alpha\\mathbbm 1_{B_r\\setminus E_H}.\n\\]\nLet $u_H\\in u+W^{1,p}_0(B_\\frac{r}{2})$ be the unique minimizer of the functional\n\\[\n\\mathcal F_H(w;B_\\frac{r}{2}):=\\int_{B_\\frac{r}{2}}\\sigma_H(x)\\,F(\\nabla w)\\,dx .\n\\]\nSubtracting the Euler-Lagrange equation satisfied by $u$ from the one satisfied by $u_H$ and testing them with $u-u_H$, we get\n\\begin{align}\n \\int_{B_{\\frac{r}{2}}} \\sigma_E\n\\big(DF(\\nabla u)-DF(\\nabla u_H)\\big)\n\\cdot(\\nabla u-\\nabla u_H)\\,dx\n& =\n\\int_{B_{\\frac{r}{2}}} (\\sigma_H-\\sigma_E)\\,\nDF(\\nabla u_H)\\cdot(\\nabla u-\\nabla u_H)\\,dx\n\\end{align}\nUsing assumptions \\eqref{Monotonicity} and \\eqref{Growth} and H\\\"older's inequality, for every $\\varepsilon>0$ it holds that\n\\begin{align}\n \\int_{B_{\\frac{r}{2}}}|\\D u_H-\\D u|^p\\,dx\n & \\leq C\\int_{(E\\Delta E_H)\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D u_H|^2\\big)^{\\frac{p-2}{2}}|\\D u-\\D u_H|\\,dx\\\\\n& \\leq C\\bigg(\\frac{1}{\\varepsilon}\\int_{(E\\Delta E_H)\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D u_H|^2\\big)^{\\frac{p}{2}}\\,dx+\\varepsilon\\int_{B_\\frac{r}{2}}|\\D u-\\D u_H|^p\\,dx \\bigg).\n\\end{align}\nChoosing $\\varepsilon$ sufficiently small, we get \n\\begin{align}\n \\int_{B_{\\frac{r}{2}}}|\\D u_H-\\D u|^p\\,dx\n& \\leq C\\int_{(E\\Delta E_H)\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D u_H|^2\\big)^{\\frac{p}{2}}\\,dx.\n\\end{align}\nThus, the minimality of $u_H$ with respect to $u$, H\\\"older's inequality and Theorem \\ref{HigherInt} yield\n\\begin{align}\n\\label{eqq1}\n\\int_{B_{\\frac{r}{2}}}|\\D u_H-\\D u|^p\\,dx& \\leq \\int_{(E\\Delta E_H)\\cap B_\\frac{r}{2}} (1+|\\nabla u|^2)^{\\frac p2}\\,dx\\\\\n& \\le \\bigg(\\frac{|(E\\Delta E_H)\\cap B_r|}{|B_r|}\\bigg)^{1-\\frac1s}|B_r|\\bigg(\\dashint_{B_r}(1+|\\nabla u|^2)^{\\frac{sp}{2}}\\,dx\\bigg)^{\\frac1s}\\\\\n& \\le C\\,\\e_0^{1-\\frac1s}\\int_{B_r}(1+|\\nabla u|^2)^{\\frac p2}\\,dx,\n\\end{align}\nwhere we have used assumption \\emph{iii)} and $C=C(n,p,\\beta/\\alpha,\\nu/L)$ is a positive constant.\n\n\\medskip\n\\noindent\\textbf{Step 3: the one-phase cases (i) and (ii).}\nAssume (i): $|E\\cap B_r|<\\e_0|B_r|$ (the case (ii) is analogous). Let us denote by $v\\in u+W^{1,p}_0(B_\\frac{r}{2})$ be the unique minimizer of the functional\n\\[\n\\mathcal G(w):=\\int_{B_\\frac{r}{2}}\\,F(\\nabla w)\\,dx .\n\\]\nWe can proceed as in step $(iii)$ subtracting the Euler-Lagrange equation satisfied by $u$ from the one satisfied by $v$ and testing them with $u-v$, getting\n\\begin{align}\n \\int_{B_{\\frac{r}{2}}} \\sigma_E\n\\big(DF(\\nabla u)-DF(\\nabla v)\\big)\n\\cdot(\\nabla u-\\nabla v)\\,dx\n& =\n\\int_{B_{\\frac{r}{2}}} (\\alpha-\\sigma_E)\\,\nDF(\\nabla v)\\cdot(\\nabla u-\\nabla v)\\,dx\n\\end{align}\nUsing assumptions \\eqref{Monotonicity} and \\eqref{Growth} and H\\\"older's inequality, for every $\\varepsilon>0$ it holds that\n\\begin{align}\n \\int_{B_{\\frac{r}{2}}}|\\D v -\\D u|^p\\,dx\n & \\leq C\\int_{E\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D v|^2\\big)^{\\frac{p-2}{2}}|\\D u-\\D v|\\,dx\\\\\n& \\leq C\\bigg(\\frac{1}{\\varepsilon}\\int_{E\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D v|^2\\big)^{\\frac{p}{2}}\\,dx+\\varepsilon\\int_{B_\\frac{r}{2}}|\\D u-\\D v|^p\\,dx \\bigg).\n\\end{align}\nChoosing $\\varepsilon$ sufficiently small, we get \n\\begin{align}\n \\int_{B_{\\frac{r}{2}}}|\\D v-\\D u|^p\\,dx\n& \\leq C\\int_{E\\cap B_{\\frac{r}{2}}}\\big(\\mu^2+|\\D v|^2\\big)^{\\frac{p}{2}}\\,dx.\n\\end{align}\nUsing the minimality of $v$ compared with $u$ and Theorem\n\\ref{HigherInt} we deduce,\n\\begin{align}\n\\label{eqq11}\n\\int_{B_{\\frac{r}{2}}}|\\D v-\\D u|^p\\,dx& \\leq \\int_{E\\cap B_{\\frac{r}{2}}} (1+|\\nabla u|^2)^{\\frac p2}\\,dx\\\\\n& \\le \\bigg(\\frac{E\\cap B_{r}}{|B_r|}\\bigg)^{1-\\frac1s}|B_r|\\bigg(\\dashint_{B_r}(1+|\\nabla u|^2)^{\\frac{sp}{2}}\\,dx\\bigg)^{\\frac1s}\\\\\n& \\le C\\,\\e_0^{1-\\frac1s}\\int_{B_r}(1+|\\nabla u|^2)^{\\frac p2}\\,dx,\n\\end{align}\nwhere we have used assumption \\emph{i)} and $C=C(n,p,\\beta/\\alpha,\\nu/L)$ is a positive constant. Thereafter we can argue as in the previous step using the classical decay estimate for minimizers of regular integrals in the Calculus of Variations\n\\[\n\\int_{B_{\\tau \\rho}}|\\nabla v|^p\\,dx\n\\le C\\tau^{n}\\int_{B_{ \\rho}}|\\nabla u|^p\\,dx,\n\\]\nfor every $\\rho\\leq \\frac{r}{2}$.\n\\end{proof}", "post_theorem_intro_text_len": 3397, "post_theorem_intro_text": "\\begin{Rem}\nWithin the present variational transmission framework, Morrey regularity represents\nthe natural level of regularity that can be expected for the gradient of local\nminimizers, in view of the discontinuous coefficients and the intrinsic transmission\ncondition.\n\\end{Rem}\nThe proof of Theorem~\\ref{MainThm} is obtained as a consequence of the following\ndecay estimate for the energy.\n\n\\begin{Prop}\\label{FJNL}\nLet $u$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin \\eqref{MainF} under the assumption of Theorem~\\ref{MainThm}.\nThen there exists a constant $0 < \\tau_0 < 1$ such that the following holds:\nfor every $\\tau \\in (0,\\tau_0)$ there exists $\\varepsilon_0 = \\varepsilon_0(\\tau) > 0$\nwith the property that, if $B_r(x_0) \\Subset \\Omega$ and one of the following\nconditions is satisfied,\n\\begin{enumerate}[label=(\\roman*)]\n\\item $|E \\cap B_r(x_0)| < \\varepsilon_0 |B_r|$,\n\\item $|B_r(x_0) \\setminus E| < \\varepsilon_0 |B_r|$,\n\\item there exists a half-space $H$ such that\n\\[\n\\frac{|(E \\Delta H) \\cap B_r(x_0)|}{|B_r|} < \\varepsilon_0,\n\\]\n\\end{enumerate}\nthen, for every $0 < \\delta < n$, the estimate\n\\[\n\\int_{B_{\\tau r}(x_0)} |\\nabla u|^p \\, dx\n\\leq\nC_0 \\tau^{n-\\delta}\n\\left(\n\\int_{B_r(x_0)} |\\nabla u|^p \\, dx + r^n\n\\right)\n\\]\nholds, where the constant $C_0$ depends only on\n$n, \\nu, L, \\alpha,\\beta, \\delta$ and $\\|\\nabla u\\|_{L^2(\\Omega)}$.\n\\end{Prop}\n\\begin{Rem}\n\\label{Reg}\n    For the proof of Theorem \\ref{MainThm}, it would be sufficient to state the proposition only in case (i). We nevertheless include the other two cases for the sake of completeness.\n\\end{Rem}\n\nWe give an outline of the proof of Proposition~\\ref{FJNL}. It relies on decay estimates for the gradient\nnear the interface, first established in a flat configuration, i.e.\\ when the interface coincides\nwith a hyperplane.\nSection~3 is devoted to this ``flat case''.\n\nThe starting point is a local boundedness estimate for the tangential gradient\n$\\nabla' u$, where $\\nabla'=(\\partial_{x_1},\\dots,\\partial_{x_{n-1}})$ denotes the vector of derivatives parallel to\nthe interface.\nThis is proved in Proposition~3.2 via a difference-quotient argument with test functions involving\nonly tangential increments.\nA key ingredient is a Moser iteration applied to\n$Z:=|\\Delta_{i,h}u|^{m/p}$.\nDuring the estimates, the full gradient $\\nabla u$ enters through the structure conditions\n\\emph{(H1)}--\\emph{(H2)} and is treated as a weight; this is the origin of the restriction\n$2<p<\\frac{2n}{n-2}$.\n\nThe bound obtained in Proposition~3.2 is preliminary, since the decay exponent in the estimates is not optimal and\nthe constant may depend on $\\mu$.\nTo obtain the scale-invariant Lipschitz estimate for $\\nabla' u$ stated in Proposition~3.4,\nwe perform a rescaling argument, which requires uniformity with respect to $\\mu$.\nThis uniformity is proved in Lemma~3.3, exploiting the boundedness of $\\nabla' u$ and a Poincar\\'e-type\ninequality for $Z:=|\\partial_i u|^{m/p}$, $i=1,\\dots,n-1$, which is an admissible Sobolev function by\nTheorem~3.1.\n\nOnce the tangential Lipschitz bound is available, we prove the Morrey estimate in the flat case in\nTheorem~3.5 by a comparison argument (Section~3.2).\nFinally, the general case follows from a standard flattening/approximation procedure using the $C^1$\nregularity of the interface, along the lines of the quadratic theory (see, e.g.,~\\cite{E,FJ}).", "sketch": "The proof of Theorem~\\ref{MainThm} is obtained as a consequence of the decay estimate for the energy stated in Proposition~\\ref{FJNL}.\n\nAn outline is given for the proof of Proposition~\\ref{FJNL}:\n\\begin{itemize}\n\\item The argument \\emph{relies on decay estimates for the gradient near the interface}, first established in a \\emph{flat configuration} (interface coincides with a hyperplane), treated in Section~3.\n\\item Starting point: a \\emph{local boundedness estimate for the tangential gradient} $\\nabla' u$ (derivatives parallel to the interface). This is proved in Proposition~3.2 via a \\emph{difference-quotient argument} with test functions involving only tangential increments.\n\\item Key ingredient: a \\emph{Moser iteration} applied to $Z:=|\\Delta_{i,h}u|^{m/p}$. In the estimates, the full gradient $\\nabla u$ appears through (H1)--(H2) and is \\emph{treated as a weight}; this is cited as the origin of the restriction $2<p<\\frac{2n}{n-2}$.\n\\item Since the Proposition~3.2 bound has a non-optimal decay exponent and constants that may depend on $\\mu$, one obtains the \\emph{scale-invariant Lipschitz estimate for $\\nabla' u$} (Proposition~3.4) by a \\emph{rescaling argument} requiring \\emph{uniformity with respect to $\\mu$}. This uniformity is proved in Lemma~3.3, using boundedness of $\\nabla' u$ and a \\emph{Poincar\\'e-type inequality} for $Z:=|\\partial_i u|^{m/p}$, $i=1,\\dots,n-1$, which is admissible by Theorem~3.1.\n\\item With the tangential Lipschitz bound, the \\emph{Morrey estimate in the flat case} is proved in Theorem~3.5 by a \\emph{comparison argument} (Section~3.2).\n\\item The \\emph{general case} then follows by a \\emph{standard flattening/approximation procedure} using the $C^1$ regularity of the interface, along the lines of the quadratic theory.\n\\end{itemize}\n\nAdditionally, Remark~\\ref{Reg} notes that for proving Theorem~\\ref{MainThm} it would be sufficient to have Proposition~\\ref{FJNL} only in case (i), though cases (ii)--(iii) are included for completeness.", "expanded_sketch": "The proof of the main theorem is obtained as a consequence of the decay estimate for the energy stated in the following proposition.\n\n\\begin{Prop}\\label{FJNL}\nLet $u$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined\nin \\eqref{MainF} under the assumption of Theorem~\\ref{MainThm}.\nThen there exists a constant $0 < \\tau_0 < 1$ such that the following holds:\nfor every $\\tau \\in (0,\\tau_0)$ there exists $\\varepsilon_0 = \\varepsilon_0(\\tau) > 0$\nwith the property that, if $B_r(x_0) \\Subset \\Omega$ and one of the following\nconditions is satisfied,\n\\begin{enumerate}[label=(\\roman*)]\n\\item $|E \\cap B_r(x_0)| < \\varepsilon_0 |B_r|$,\n\\item $|B_r(x_0) \\setminus E| < \\varepsilon_0 |B_r|$,\n\\item there exists a half-space $H$ such that\n\\[\n\\frac{|(E \\Delta H) \\cap B_r(x_0)|}{|B_r|} < \\varepsilon_0,\n\\]\n\\end{enumerate}\nthen, for every $0 < \\delta < n$, the estimate\n\\[\n\\int_{B_{\\tau r}(x_0)} |\\nabla u|^p \\, dx\n\\leq\nC_0 \\tau^{n-\\delta}\n\\left(\n\\int_{B_r(x_0)} |\\nabla u|^p \\, dx + r^n\n\\right)\n\\]\nholds, where the constant $C_0$ depends only on\n$n, \\nu, L, \\alpha,\\beta, \\delta$ and $\\|\\nabla u\\|_{L^2(\\Omega)}$.\n\\end{Prop}\n\nAn outline is given for the proof of Proposition~\\ref{FJNL}:\n\\begin{itemize}\n\\item The argument \\emph{relies on decay estimates for the gradient near the interface}, first established in a \\emph{flat configuration} (interface coincides with a hyperplane), treated next.\n\\item Starting point: a \\emph{local boundedness estimate for the tangential gradient} $\\nabla' u$ (derivatives parallel to the interface). This is proved in Proposition~3.2 via a \\emph{difference-quotient argument} with test functions involving only tangential increments.\n\\item Key ingredient: a \\emph{Moser iteration} applied to $Z:=|\\Delta_{i,h}u|^{m/p}$. In the estimates, the full gradient $\\nabla u$ appears through (H1)--(H2) and is \\emph{treated as a weight}; this is cited as the origin of the restriction $2<p<\\frac{2n}{n-2}$.\n\\item Since the Proposition~3.2 bound has a non-optimal decay exponent and constants that may depend on $\\mu$, one obtains the \\emph{scale-invariant Lipschitz estimate for $\\nabla' u$} (Proposition~3.4) by a \\emph{rescaling argument} requiring \\emph{uniformity with respect to $\\mu$}. This uniformity is proved in Lemma~3.3, using boundedness of $\\nabla' u$ and a \\emph{Poincar\\'e-type inequality} for $Z:=|\\partial_i u|^{m/p}$, $i=1,\\dots,n-1$, which is admissible by Theorem~3.1.\n\\item With the tangential Lipschitz bound, the \\emph{Morrey estimate in the flat case} is proved in Theorem~3.5 by a \\emph{comparison argument}.\n\\item The \\emph{general case} then follows by a \\emph{standard flattening/approximation procedure} using the $C^1$ regularity of the interface, along the lines of the quadratic theory.\n\\end{itemize}\n\nAdditionally, Remark~\\ref{Reg} notes that for proving the main theorem it would be sufficient to have Proposition~\\ref{FJNL} only in case (i), though cases (ii)--(iii) are included for completeness.", "expanded_theorem": "[Gradient regularity in Morrey spaces]\\label{MainThm}\nLet $u \\in W^{1,p}(\\Omega)$ be a local minimizer of the functional $\\mathcal{F}(\\cdot;\\Omega)$ defined in\n\\begin{equation}\\label{MainF}\n\\mathcal{F}(u;E) := \\int_{\\Omega} \\sigma_E(x)\\, F(\\nabla u)\\, dx,\n\\end{equation}\nand assume that $\\partial E$ is a hypersurface of class $C^1$.\nSuppose that assumptions \\emph{(H1)}--\\emph{(H2)} are satisfied where $2<p<\\frac{2n}{n-2}$.\nThen, for every $0 \\leq \\lambda < n$,\n\\[\n\\nabla u \\in L^{{p},\\lambda}_{\\mathrm{loc}}(\\Omega).\n\\]\nMoreover, for every $\\Omega' \\Subset \\Omega$, there exists a constant\n$C = C\\big(n, p, \\nu, L, \\beta, E, \\operatorname{diam}(\\Omega),\n\\operatorname{dist}(\\Omega', \\partial\\Omega)\\big) > 0$\nsuch that\n\\[\n\\|\\nabla u\\|_{L^{{p},\\lambda}(\\Omega')} \\leq C.\n\\]", "theorem_type": ["Implication", "Existential–Universal"], "mcq": {"question": "Let \\(\\Omega\\subset\\mathbb{R}^n\\) be a bounded domain, let \\(E\\subset\\Omega\\) be measurable with \\(\\partial E\\) a \\(C^1\\) hypersurface, and fix constants \\(0<\\alpha<\\beta<\\infty\\). Define \\(\\sigma_E:=\\beta\\mathbbm{1}_E+\\alpha\\mathbbm{1}_{E^c}\\) and, for \\(v\\in W^{1,p}(\\Omega)\\),\n\\[\\mathcal F(v;E):=\\int_{\\Omega}\\sigma_E(x)F(\\nabla v)\\,dx.\\]\nAssume \\(F\\in C^2(\\mathbb{R}^n)\\) and that for some \\(\\nu,L>0\\), \\(\\mu\\in(0,1]\\), and every \\(\\xi,\\eta\\in\\mathbb{R}^n\\),\n\\[\\langle \\nabla F(\\xi)-\\nabla F(\\eta),\\xi-\\eta\\rangle\\ge \\nu\\big(\\mu^2+|\\xi|^2+|\\eta|^2\\big)^{\\frac{p-2}{2}}|\\xi-\\eta|^2,\\]\n\\[|\\nabla F(\\xi)-\\nabla F(\\eta)|\\le L\\big(\\mu^2+|\\xi|^2+|\\eta|^2\\big)^{\\frac{p-2}{2}}|\\xi-\\eta|,\\]\nwith \\(2<p<\\frac{2n}{n-2}\\). Suppose \\(u\\in W^{1,p}(\\Omega)\\) is a local minimizer in the sense that for every ball \\(B_r(x_0)\\subset\\Omega\\) and every \\(\\phi\\in W_0^{1,p}(B_r(x_0))\\),\n\\[\\mathcal F(u;B_r(x_0))\\le \\mathcal F(u+\\phi;B_r(x_0)).\\]\nWhich conclusion about \\(\\nabla u\\) holds? Here \\(L^{p,\\lambda}(\\Omega')\\) is the classical Morrey space, for instance with norm \\(\\|f\\|_{L^{p,\\lambda}(\\Omega')}:=\\sup_{B_\\rho(x)\\subset\\Omega'}\\big(\\rho^{-\\lambda}\\int_{B_\\rho(x)}|f|^p\\,dy\\big)^{1/p}\\).", "correct_choice": {"label": "A", "text": "For every \\(0\\le \\lambda<n\\), one has \\(\\nabla u\\in L^{p,\\lambda}_{\\mathrm{loc}}(\\Omega)\\). Moreover, for every \\(\\Omega'\\Subset\\Omega\\), there exists a constant \\(C=C\\big(n,p,\\nu,L,\\beta,E,\\operatorname{diam}(\\Omega),\\operatorname{dist}(\\Omega',\\partial\\Omega)\\big)>0\\) such that\n\\[\\|\\nabla u\\|_{L^{p,\\lambda}(\\Omega')}\\le C.\\]"}, "choices": [{"label": "B", "text": "For every \\(0\\le \\lambda<n\\), one has \\(\\nabla u\\in L^{p,\\lambda}_{\\mathrm{loc}}(\\Omega)\\). Moreover, for every \\(\\Omega'\\Subset\\Omega\\), there exists a constant \\(C=C\\big(n,p,\\nu,L,\\alpha,\\beta,E,\\operatorname{diam}(\\Omega),\\operatorname{dist}(\\Omega',\\partial\\Omega)\\big)>0\\), independent of \\(\\lambda\\), such that\n\\[\\|\\nabla u\\|_{L^{p,\\lambda}(\\Omega')}\\le C.\\]"}, {"label": "C", "text": "One has \\(\\nabla u\\in L^{p,\\lambda}_{\\mathrm{loc}}(\\Omega)\\) for at least one exponent \\(\\lambda\\in[0,n)\\)."}, {"label": "D", "text": "For every \\(0\\le \\lambda<n\\), one has \\(\\nabla u\\in L^{p,\\lambda}_{\\mathrm{loc}}(\\Omega)\\). Moreover, for every \\(\\Omega'\\Subset\\Omega\\), there exists a constant \\(C=C\\big(n,p,\\nu,L,\\beta,E,\\operatorname{diam}(\\Omega),\\operatorname{dist}(\\Omega',\\partial\\Omega)\\big)>0\\) such that\n\\[\\|\\nabla u\\|_{L^{p,\\lambda}(\\Omega')}\\le C\\,\\|\\nabla u\\|_{L^{p}(\\Omega)}.\\]"}, {"label": "E", "text": "There exists \\(\\lambda_0\\in(0,n)\\), depending only on \\(n,p,\\nu,L,\\alpha,\\beta\\), such that \\(\\nabla u\\in L^{p,\\lambda_0}_{\\mathrm{loc}}(\\Omega)\\). Moreover, for every \\(\\Omega'\\Subset\\Omega\\), there exists a constant \\(C=C\\big(n,p,\\nu,L,\\beta,E,\\operatorname{diam}(\\Omega),\\operatorname{dist}(\\Omega',\\partial\\Omega)\\big)>0\\) with\n\\[\\|\\nabla u\\|_{L^{p,\\lambda_0}(\\Omega')}\\le C.\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "uniformity", "tampered_component": "dependence on Morrey exponent in the local bound", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "universal quantifier over all \\(\\lambda<n\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "absolute local Morrey bound replaced by a bound proportional to \\(\\|\\nabla u\\|_{L^p(\\Omega)}\\)", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "full range \\(0\\le \\lambda<n\\) weakened to existence of a single universal \\(\\lambda_0\\)", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option, and no choice is singled out by wording. Although the hypotheses strongly signal a specific regularity theorem, the answer is not leaked directly."}, "TAS": {"score": 1, "justification": "This is very close to a theorem-recall item: the stem lists the full assumptions and asks for the corresponding conclusion. However, it is not a pure tautology because the options differ in quantifiers, dependence of constants, and strength of the Morrey regularity claim."}, "GPS": {"score": 1, "justification": "The item requires some reasoning about the strength of conclusions and the role of parameter dependence, especially to distinguish the full-for-all-lambda statement from weaker or tampered variants. Still, it mainly tests recognition/recall of a known theorem rather than deep derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they vary universal quantifiers, dependence on lambda, presence of an extra global norm factor, and weakening to existence of one exponent. These reflect realistic failure modes in reading regularity statements."}, "total_score": 6, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no direct answer leakage, but it is still fairly close to a restatement of a known result and only moderately tests generative reasoning."}}
{"id": "2602.14809v2", "paper_link": "http://arxiv.org/abs/2602.14809v2", "theorems_cnt": 3, "theorem": {"env_name": "mainthm", "content": "[\\ref{Purehomogenous},~\\ref{prp:MainNonSimp}]\\label{prp:MainThm1}\n    Let $X$ be a compact metric space, and let $A$ be a pure \\ca{}. Assume additionally that\n    \\begin{enumerate}\n        \\item $A$ is simple; or \n        \\item every quotient of $A$ is stably finite.\n    \\end{enumerate}\n\n    Then, $C(X,A)$ is pure.", "start_pos": 8129, "end_pos": 8474, "label": "prp:MainThm1"}, "ref_dict": {"pgr:GGP": "\\begin{pgr}[The Global Glimm Property]\\label{pgr:GGP}\n    A \\ca{} $A$ is said to satisfy the \\emph{Global Glimm Property} if for every $a\\in A_+$ and $\\varepsilon>0$ there exists a square-zero element $r\\in\\overline{aAa}$ such that $(a-\\varepsilon)_+\\in \\overline{ArA}$. This property was originally introduced by Kirchberg and R\\o{}rdam in their study of non-simple purely infinite \\ca{s} \\cite[Definition~4.12]{KirRor02InfNonSimpleCalgAbsOInfty}, but has recently found various and deep applications in the context of dimension reduction phenomena; see, for example, \\cite{ThiVil23Glimm,Vil25:IntroGGP}.\n\n    As shown in \\cite[Theorem~3.6]{ThiVil23Glimm}, the property can be characterized in terms of a divisibility condition on the Cuntz semigroup: $A$ has the Global Glimm Property if and only if $\\Cu (A)$ is $(2,\\omega )$-divisible, that is, for  every pair $x',x\\in\\Cu (A)$ such that $x'\\ll x$, there exists $y\\in\\Cu (A)$ and $n\\in\\N$ such that $2y\\leq x$ and $x'\\leq ny$.\n\\end{pgr}", "prp:CXAPureIfGGP": "\\begin{theorem}\\label{prp:CXAPureIfGGP}\n    Let $X$ be a finite-dimensional compact metric space and let $A$ be a pure \\ca{}. Then, $C(X,A)$ is pure if and only if $C(X,A)$ has the Global Glimm Property.\n\\end{theorem}", "DpureRSHA": "\\begin{prop}\\label{DpureRSHA}\nLet $D$ be a unital simple pure \\ca. Let $R_D$ be a recursive subhomogeneous algebra over $D$ and let $R$ be any recursive homogeneous algebra over $\\mathbb{C}$.\nThen both $R_D$ and $R \\otimes D$ are pure.\n\\end{prop}", "RSHA": "\\begin{defn}[{\\cite[Definition~3.2]{archey2020structure}}]\\label{RSHA}\nFor a simple unital \\ca{} $D$, the class $\\mathcal{R}$ of \n{\\emph{recursive subhomogeneous algebras over $D$}} is\nthe smallest class of \\ca{s} closed under isomorphism such that:\n\\begin{enumerate}\n\\item\\label{9223_RSHA_1}\n$C (X, M_{n} (D)) \\in \\mathcal{R}$ whenever $X$ is a compact Hausdorff space and $n \\geq 1$.\n\\item\\label{9223_RSHA_2}\nAny pullback of the form \n\\begin{align*}\n& B \\oplus_{C (X^{(0)}, \\, M_{n} (D)), \\, \\ph, \\, \\rh} C (X, M_{n} (D))\n\\\\\n& \\hspace*{3em} {\\mbox{}}\n  = \\bset{ (b, f) \\in B \\oplus C (X, M_{n} (D)) \\colon\n        \\ph (b) = f |_{X^{(0)}} }\n\\end{align*}\nis in $\\mathcal{R}$ whenever $B \\in \\mathcal{R}$, $X$ is compact Hausdorff,\n$n \\geq 1$, $X^{(0)} \\subseteq X$ is closed (possibly empty),\n$\\ph \\colon B \\to C (X^{(0)}, \\, M_{n} (D))$\nis any unital $^*$-homomorphism\n(the zero \\hm{} if $X^{(0)}$ is empty), and\n$\\rho \\colon C (X, M_{n} (D)) \\to C (X^{(0)}, \\, M_{n} (D))$ is the\nrestriction $^*$-homomorphism.\n\\end{enumerate}\n\\end{defn}", "Purehomogenous": "\\begin{cor}\\label{Purehomogenous}\n    Let $A$ be a simple pure \\ca{}, and let $X$ be a compact metric space. Then, $C(X,A)$ is pure.\n\\end{cor}", "prp:MainThm2": "\\begin{mainthm}[\\ref{prp:ASHtenPure}]\\label{prp:MainThm2}\n    Let $A$ be a pure \\ca{} satisfying (1) or (2) in Theorem \\ref{prp:MainThm1}, and let $B$ be a unital separable ASH-algebra. Then, $A\\otimes B$ is pure.\n\\end{mainthm}", "prp:MainCor1": "\\begin{maincor}\\label{prp:MainCor1}\n Let $G$ be a countable acylindrically hyperbolic group, and let $H$ be a virtually abelian group. Then, $C_r^*(G\\times H)$ is pure. In particular, it has strict comparison.\n\\end{maincor}", "prp:MainNonSimp": "\\begin{theorem}\\label{prp:MainNonSimp}\n    Let $X$ be a compact metric space, and let $A$ be a pure \\ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, $C(X,A)$ is pure.\n\\end{theorem}", "def_almost_divisibility": "\\begin{defn}\\label{def_almost_divisibility}\n    Let $A$ be a \\ca{} and $n\\in\\N$. We say that $A$ (and its Cuntz semigroup $\\Cu (A)$) is \\emph{$n$-almost divisible} if for any pair $x',x\\in\\mathrm{Cu}(A)$ such that $x'\\ll x$ and any $N\\in \\N$, there exists $y\\in \\mathrm{Cu}(A)$ such that \n    \\[\n     N y\\leq x\\quad{\\mbox{and}}\\quad x'\\leq (N+1)(n+1)y.\n    \\]\n\n    $A$ is said to be \\emph{almost divisible} if it is $0$-almost divisible.\n\\end{defn}", "robert_analogy_2": "\\begin{cor}\\label{robert_analogy_2}\n    Let $X$ be a finite-dimensional compact metric space, and let $A$ be an almost divisible \\ca{}. Then, there exists $M\\in\\N$ such that, for every $N\\geq 1$ and every pair $x'\\ll x$ in $\\Cu (C(X,A))$, there exists $y\\in\\Cu (C(X,A))$ such that\n    \\[\n        x'\\ll Ny\\ll Mx.\n    \\]\n\\end{cor}", "qst:VillZstab": "\\begin{qst}\\label{qst:VillZstab}\n    Let $B$ be a Villadsen algebra of the first kind without strict comparison (for example, Toms' \\cite{Tom08ClassificationNuclear}). Is $B\\otimes C_r^*(\\mathbb{F}_2)$ $\\mathcal{Z}$-stable?\n\\end{qst}", "prp:ASHtenPure": "\\begin{theorem}\\label{prp:ASHtenPure}\n   Let $B$ be a unital separable ASH-algebra and $A$ be simple and pure. Then, $B\\otimes A$ is pure.\n\\end{theorem}", "comp_c(x,A)": "\\begin{prop}\\label{comp_c(x,A)}\n    Let $X$ be a compact metric space of covering dimension $m$, and let $A$ be a \\ca{} with strict comparison. Then $C(X, A)$ has $m$-comparison.\n\\end{prop}", "prp:MainThm1": "\\begin{mainthm}[\\ref{Purehomogenous},~\\ref{prp:MainNonSimp}]\\label{prp:MainThm1}\n    Let $X$ be a compact metric space, and let $A$ be a pure \\ca{}. Assume additionally that\n    \\begin{enumerate}\n        \\item $A$ is simple; or \n        \\item every quotient of $A$ is stably finite.\n    \\end{enumerate}\n\n    Then, $C(X,A)$ is pure.\n\\end{mainthm}", "prp:Gen_DimRed": "\\begin{theorem}\\label{prp:Gen_DimRed}\n    Let $A$ be a \\ca{}, and let $m,M\\in\\mathbb{N}$. Assume that $A$ has $m$-comparison and that, for every $N\\geq 1$ and every pair of elements $x', x\\in \\Cu (A)$ such that $x'\\ll x$, there exists $y\\in\\mathrm{Cu} (A)$ satisfying \n    \\[\n        x'\\ll Ny\\ll M x.\n    \\]\n\n    Then, the following are equivalent:\n    \\begin{itemize}\n        \\item[(i)] $A$ has the Global Glimm Property;\n        \\item[(ii)] there exists $L\\in\\N$ such that, for every pair $x',x\\in\\Cu (A)$ with $x'\\ll x$, there exist $y_0,y_1\\in\\mathrm{Cu} (A)$ satisfying\n        \\[\n        y_0+y_1\\leq x,\\quad\\text{and}\\quad \n        x'\\ll Ly_0, Ly_1;\n        \\]\n        \\item[(iii)] $A$ is pure.\n    \\end{itemize}\n\\end{theorem}", "dfn:MComp": "\\begin{defn}[{\\cite[Definition~2.1]{Win12NuclDimZstable}}]\\label{dfn:MComp}\n    Let $A$ be a \\ca{} and let $n\\in\\N$. We say that $A$ (and its Cuntz semigroup $\\Cu (A)$) has the \\emph{$n$-comparison} property if $x \\leq_s y_j$ for $x, y_j \\in \\Cu (A)$ and $j = 0, 1, \\ldots, n$, implies $x\\leq \\sum_{j=0}^n y_j$.\n\n    $\\Cu (A)$ is said to be \\emph{almost unperforated} if it has $0$-comparison.\n\\end{defn}", "rem:StCompAlmUnperf": "\\begin{rem}\\label{rem:StCompAlmUnperf}\n    Let $A$ be a \\ca{}. For any $[0,\\infty]$-valued lower semi-continuous $2$-quasitrace $\\tau$ on $A$ (which extends to $A\\otimes\\mathcal{K}$ and is denoted by the same symbol), the associated \\emph{dimension function} $d_\\tau\\colon \\Cu (A)\\to [0,\\infty]$ is defined by $d_\\tau ([a])=\\lim_n\\tau (a^{1/n})$.\n\n    As shown in \\cite[Proposition~6.2]{elliott2011cone}, $\\Cu(A)$ is almost unperforated (in the sense described above) if and only if, for every pair $a,b\\in  (A\\otimes\\mathcal{K})_+$ such that $a$ is in the ideal generated by $b$ and $d_\\tau ([a])<d_\\tau([b])$ whenever $d_\\tau ([b])=1$, one has $[a]\\leq [b]$ in $\\Cu (A)$.\n\n    This property is known as \\emph{strict comparison} (of positive elements by $[0,\\infty]$-valued lower semi-continuous $2$-quasitraces). Note that, if $A$ is simple, $A$ has strict comparison if and only if, for every pair of non-zero elements $a,b\\in  (A\\otimes\\mathcal{K})_+$ such that $d_\\tau ([a])<d_\\tau([b])$ for every normalized quasitrace $\\tau$, one has $[a]\\leq [b]$ in $\\Cu (A)$. We refer the reader to \\cite[Paragraph~3.4]{AntPerThiVil24arX:PureCAlgs} for an in-depth introduction.\n\\end{rem}", "prp:CXNoQuot": "\\begin{prop}\\label{prp:CXNoQuot}\n    Let $X$ be a finite-dimensional compact metric space, and let $A$ be a pure \\ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, there exists $L\\in\\N$ such that, for every pair $x',x\\in\\Cu (C(X,A))$ with $x'\\ll x$, there exist $y_0,y_1\\in\\mathrm{Cu} (C(X,A))$ satisfying\n        \\[\n        y_0+y_1\\leq x,\\quad\\text{and}\\quad \n        x'\\ll Ly_0, Ly_1.\n        \\]\n\\end{prop}"}, "pre_theorem_intro_text_len": 4095, "pre_theorem_intro_text": "The properties of \\emph{strict comparison} and \\emph{almost divisibility} play a fundamental role in the modern structure theory of $\\mathrm{C}^*$-algebras. As defined by Winter \\cite{Win12NuclDimZstable}, $\\mathrm{C}^*$-algebras that satisfy both properties are termed \\emph{pure}. In the simple finite setting, such algebras should be regarded as the correct analogues of II$_1$ factors; roughly speaking, the defining properties of pureness correspond to two key features of projections in II$_1$factors, adapted to the setting of positive elements ---a necessary adaptation given that many $\\mathrm{C}^*$-algebras contain few or no nontrivial projections.\n\nJust as the unique trace on a II$_1$ factor $\\mathcal{N}$ determines the order of projections, \\emph{strict comparison} \\cite{Bla88Comparison} asserts that tracial state data determines when one positive operator can be approximately compressed to another (Remark \\ref{rem:StCompAlmUnperf}). Similarly, \\emph{almost divisibility} (Definition \\ref{def_almost_divisibility}) serves as the analogue to the fact that projections in $\\mathcal{N}$ can be divided arbitrarily. Whilst these two properties are automatic for II$_1$ factors, they can fail for simple $\\mathrm{C}^*$-algebras \\cite{Tom08ClassificationNuclear,Vil99SRSimpleCa}. Consequently, determining which algebras satisfy these properties ---and, in particular, developing mechanisms to deduce pureness--- has been a major focus in the field for the past few years \\cite{AGKEP25,AntPerThiVil24arX:PureCAlgs,Rob25Self}. Examples of pure $\\mathrm{C}^*$-algebras can be found both in the nuclear and non nuclear case, and include $\\mathcal{Z}$-stable algebras, purely infinite algebras \\cite{kirchberg2000non}, von Neumann algebras with trivial type I summand, and reduced (twisted) group $\\mathrm{C}^*$-algebras of acylindrically hyperbolic groups \\cite{FloLisCobPag:PureTwi,RauThiVil:StCompTwi}.\n\nIn the realm of simple nuclear $\\mathrm{C}^*$-algebras, pureness features prominently in the Toms-Winter conjecture \\cite{CasEviTikWhiWin21NucDimSimple,Ror04StableRealRankZ,Win12NuclDimZstable} and serves as a crucial regularity condition in the Elliott classification program \\cite{Whi23ICM,Win18ICM}. More generally, the notion is expected to play an important role in the classification of $^*$-homomorphisms \\cite{CarGabSchTikWhi23arX:ClassifHom1,Sza26:UniqThmKK}, potentially by weakening the requirement of $\\mathcal{Z}$-stability on the codomain to that of pureness. In fact, as shown in \\cite[Theorem~7.3.11]{antoine2018tensor}, one may think of pureness as $\\mathcal{Z}$-stability at the level of the Cuntz semigroup.\n\nOf the two defining properties of pureness, strict comparison has received the most attention in recent years, finding deep connections with the theory of group $\\mathrm{C}^*$-algebras \\cite{AGKEP25,Oza25:Proxim}, time-frequency analysis \\cite{BedEnsvVelt22,EnsThiVil25Criteria}, and the recent negative resolution to Tarski's problem \\cite{KES:NegativeTarski}. However, despite its importance, strict comparison is often difficult to verify directly. For this reason, attention has shifted towards the stronger, more rigid property of pureness, which is particularly susceptible to \\emph{dimension reduction phenomena}. Specifically, it was recently proved in \\cite{AntPerThiVil24arX:PureCAlgs} that any $(m, n)$-pure \\ca{} ---a formal `quantized' weakening of pureness--- is automatically pure. This result allows one to deduce pureness (and thus strict comparison) in situations where direct verification was previously out of reach. It has also paved the way for new permanence results; for instance, it was established in \\cite{perera2025extensions} that an extension of pure $\\mathrm{C}^*$-algebras is itself pure, mirroring the corresponding property for $\\mathcal{Z}$-stability \\cite[Theorem~4.3]{TomWin07ssa}. An important question in this line of inquiry, raised at a number of conferences, is whether a minimal tensor product $B \\otimes A$ is pure whenever $A$ is. In the present paper, we investigate this question for the case $B = C(X)$.", "context": "The properties of \\emph{strict comparison} and \\emph{almost divisibility} play a fundamental role in the modern structure theory of $\\mathrm{C}^*$-algebras. As defined by Winter \\cite{Win12NuclDimZstable}, $\\mathrm{C}^*$-algebras that satisfy both properties are termed \\emph{pure}. In the simple finite setting, such algebras should be regarded as the correct analogues of II$_1$ factors; roughly speaking, the defining properties of pureness correspond to two key features of projections in II$_1$factors, adapted to the setting of positive elements ---a necessary adaptation given that many $\\mathrm{C}^*$-algebras contain few or no nontrivial projections.\n\nJust as the unique trace on a II$_1$ factor $\\mathcal{N}$ determines the order of projections, \\emph{strict comparison} \\cite{Bla88Comparison} asserts that tracial state data determines when one positive operator can be approximately compressed to another (Remark \\ref{rem:StCompAlmUnperf}). Similarly, \\emph{almost divisibility} (Definition \\ref{def_almost_divisibility}) serves as the analogue to the fact that projections in $\\mathcal{N}$ can be divided arbitrarily. Whilst these two properties are automatic for II$_1$ factors, they can fail for simple $\\mathrm{C}^*$-algebras \\cite{Tom08ClassificationNuclear,Vil99SRSimpleCa}. Consequently, determining which algebras satisfy these properties ---and, in particular, developing mechanisms to deduce pureness--- has been a major focus in the field for the past few years \\cite{AGKEP25,AntPerThiVil24arX:PureCAlgs,Rob25Self}. Examples of pure $\\mathrm{C}^*$-algebras can be found both in the nuclear and non nuclear case, and include $\\mathcal{Z}$-stable algebras, purely infinite algebras \\cite{kirchberg2000non}, von Neumann algebras with trivial type I summand, and reduced (twisted) group $\\mathrm{C}^*$-algebras of acylindrically hyperbolic groups \\cite{FloLisCobPag:PureTwi,RauThiVil:StCompTwi}.\n\nIn the realm of simple nuclear $\\mathrm{C}^*$-algebras, pureness features prominently in the Toms-Winter conjecture \\cite{CasEviTikWhiWin21NucDimSimple,Ror04StableRealRankZ,Win12NuclDimZstable} and serves as a crucial regularity condition in the Elliott classification program \\cite{Whi23ICM,Win18ICM}. More generally, the notion is expected to play an important role in the classification of $^*$-homomorphisms \\cite{CarGabSchTikWhi23arX:ClassifHom1,Sza26:UniqThmKK}, potentially by weakening the requirement of $\\mathcal{Z}$-stability on the codomain to that of pureness. In fact, as shown in \\cite[Theorem~7.3.11]{antoine2018tensor}, one may think of pureness as $\\mathcal{Z}$-stability at the level of the Cuntz semigroup.\n\nOf the two defining properties of pureness, strict comparison has received the most attention in recent years, finding deep connections with the theory of group $\\mathrm{C}^*$-algebras \\cite{AGKEP25,Oza25:Proxim}, time-frequency analysis \\cite{BedEnsvVelt22,EnsThiVil25Criteria}, and the recent negative resolution to Tarski's problem \\cite{KES:NegativeTarski}. However, despite its importance, strict comparison is often difficult to verify directly. For this reason, attention has shifted towards the stronger, more rigid property of pureness, which is particularly susceptible to \\emph{dimension reduction phenomena}. Specifically, it was recently proved in \\cite{AntPerThiVil24arX:PureCAlgs} that any $(m, n)$-pure \\ca{} ---a formal `quantized' weakening of pureness--- is automatically pure. This result allows one to deduce pureness (and thus strict comparison) in situations where direct verification was previously out of reach. It has also paved the way for new permanence results; for instance, it was established in \\cite{perera2025extensions} that an extension of pure $\\mathrm{C}^*$-algebras is itself pure, mirroring the corresponding property for $\\mathcal{Z}$-stability \\cite[Theorem~4.3]{TomWin07ssa}. An important question in this line of inquiry, raised at a number of conferences, is whether a minimal tensor product $B \\otimes A$ is pure whenever $A$ is. In the present paper, we investigate this question for the case $B = C(X)$.", "full_context": "The properties of \\emph{strict comparison} and \\emph{almost divisibility} play a fundamental role in the modern structure theory of $\\mathrm{C}^*$-algebras. As defined by Winter \\cite{Win12NuclDimZstable}, $\\mathrm{C}^*$-algebras that satisfy both properties are termed \\emph{pure}. In the simple finite setting, such algebras should be regarded as the correct analogues of II$_1$ factors; roughly speaking, the defining properties of pureness correspond to two key features of projections in II$_1$factors, adapted to the setting of positive elements ---a necessary adaptation given that many $\\mathrm{C}^*$-algebras contain few or no nontrivial projections.\n\nJust as the unique trace on a II$_1$ factor $\\mathcal{N}$ determines the order of projections, \\emph{strict comparison} \\cite{Bla88Comparison} asserts that tracial state data determines when one positive operator can be approximately compressed to another (Remark \\ref{rem:StCompAlmUnperf}). Similarly, \\emph{almost divisibility} (Definition \\ref{def_almost_divisibility}) serves as the analogue to the fact that projections in $\\mathcal{N}$ can be divided arbitrarily. Whilst these two properties are automatic for II$_1$ factors, they can fail for simple $\\mathrm{C}^*$-algebras \\cite{Tom08ClassificationNuclear,Vil99SRSimpleCa}. Consequently, determining which algebras satisfy these properties ---and, in particular, developing mechanisms to deduce pureness--- has been a major focus in the field for the past few years \\cite{AGKEP25,AntPerThiVil24arX:PureCAlgs,Rob25Self}. Examples of pure $\\mathrm{C}^*$-algebras can be found both in the nuclear and non nuclear case, and include $\\mathcal{Z}$-stable algebras, purely infinite algebras \\cite{kirchberg2000non}, von Neumann algebras with trivial type I summand, and reduced (twisted) group $\\mathrm{C}^*$-algebras of acylindrically hyperbolic groups \\cite{FloLisCobPag:PureTwi,RauThiVil:StCompTwi}.\n\nIn the realm of simple nuclear $\\mathrm{C}^*$-algebras, pureness features prominently in the Toms-Winter conjecture \\cite{CasEviTikWhiWin21NucDimSimple,Ror04StableRealRankZ,Win12NuclDimZstable} and serves as a crucial regularity condition in the Elliott classification program \\cite{Whi23ICM,Win18ICM}. More generally, the notion is expected to play an important role in the classification of $^*$-homomorphisms \\cite{CarGabSchTikWhi23arX:ClassifHom1,Sza26:UniqThmKK}, potentially by weakening the requirement of $\\mathcal{Z}$-stability on the codomain to that of pureness. In fact, as shown in \\cite[Theorem~7.3.11]{antoine2018tensor}, one may think of pureness as $\\mathcal{Z}$-stability at the level of the Cuntz semigroup.\n\nOf the two defining properties of pureness, strict comparison has received the most attention in recent years, finding deep connections with the theory of group $\\mathrm{C}^*$-algebras \\cite{AGKEP25,Oza25:Proxim}, time-frequency analysis \\cite{BedEnsvVelt22,EnsThiVil25Criteria}, and the recent negative resolution to Tarski's problem \\cite{KES:NegativeTarski}. However, despite its importance, strict comparison is often difficult to verify directly. For this reason, attention has shifted towards the stronger, more rigid property of pureness, which is particularly susceptible to \\emph{dimension reduction phenomena}. Specifically, it was recently proved in \\cite{AntPerThiVil24arX:PureCAlgs} that any $(m, n)$-pure \\ca{} ---a formal `quantized' weakening of pureness--- is automatically pure. This result allows one to deduce pureness (and thus strict comparison) in situations where direct verification was previously out of reach. It has also paved the way for new permanence results; for instance, it was established in \\cite{perera2025extensions} that an extension of pure $\\mathrm{C}^*$-algebras is itself pure, mirroring the corresponding property for $\\mathcal{Z}$-stability \\cite[Theorem~4.3]{TomWin07ssa}. An important question in this line of inquiry, raised at a number of conferences, is whether a minimal tensor product $B \\otimes A$ is pure whenever $A$ is. In the present paper, we investigate this question for the case $B = C(X)$.\n\n\\begin{document}\n\\begin{abstract}\n Let $X$ be a compact metric space, and let $A$ be a pure \\ca{}. We show that $C(X,A)$ is pure whenever\n \\begin{enumerate}\n \\item $A$ is simple; or \n \\item every quotient of $A$ is stably finite (e.g., $A$ has stable rank one).\n \\end{enumerate}\n\nOf the two defining properties of pureness, strict comparison has received the most attention in recent years, finding deep connections with the theory of group \\ca{s} \\cite{AGKEP25,Oza25:Proxim}, time-frequency analysis \\cite{BedEnsvVelt22,EnsThiVil25Criteria}, and the recent negative resolution to Tarski's problem \\cite{KES:NegativeTarski}. However, despite its importance, strict comparison is often difficult to verify directly. For this reason, attention has shifted towards the stronger, more rigid property of pureness, which is particularly susceptible to \\emph{dimension reduction phenomena}. Specifically, it was recently proved in \\cite{AntPerThiVil24arX:PureCAlgs} that any $(m, n)$-pure \\ca{} ---a formal `quantized' weakening of pureness--- is automatically pure. This result allows one to deduce pureness (and thus strict comparison) in situations where direct verification was previously out of reach. It has also paved the way for new permanence results; for instance, it was established in \\cite{perera2025extensions} that an extension of pure \\ca{s} is itself pure, mirroring the corresponding property for $\\mathcal{Z}$-stability \\cite[Theorem~4.3]{TomWin07ssa}. An important question in this line of inquiry, raised at a number of conferences, is whether a minimal tensor product $B \\otimes A$ is pure whenever $A$ is. In the present paper, we investigate this question for the case $B = C(X)$.\n\nBefore outlining the proof strategy and the challenges involved in establishing Theorem \\ref{prp:MainThm1}, we highlight some important consequences. First, combining Theorem \\ref{prp:MainThm1} with several permanence properties of pureness, we obtain the following application:\n\n\\begin{cor}\\label{robert_analogy_2}\n Let $X$ be a finite-dimensional compact metric space, and let $A$ be an almost divisible \\ca{}. Then, there exists $M\\in\\N$ such that, for every $N\\geq 1$ and every pair $x'\\ll x$ in $\\Cu (C(X,A))$, there exists $y\\in\\Cu (C(X,A))$ such that\n \\[\n x'\\ll Ny\\ll Mx.\n \\]\n\\end{cor}\n\\begin{proof}\n Set $M=2(\\dim (X)+1)^2$ and let $N\\in\\N$. If $N=1$, any $y\\in\\Cu (A)$ such that $x'\\ll y\\ll x$ satisfies the required condition. Thus, we may assume $N>1$.\n As in the previous proof, we may also assume that $A$ is stable. Find $w\\in \\Cu (C(X,A))$ such that $x'\\ll w\\ll x$, and let $w=[f]$ with $f\\in C(X,A)_+$. By the definition of compact containment, there exists $\\varepsilon>0$ such that $x'\\ll [(f-\\varepsilon)_+]$. Now apply Proposition \\ref{robert_analogy} to $f$, $\\varepsilon$ and $N-1$ to find $y\\in \\Cu (C(X,A))$ such that \n \\[\n [(f-\\varepsilon)_+]\\leq Ny\\quad\\text{and}\\quad (N-1)y\\leq (\\dim(X)+1)^2[f].\n \\]\n\n\\begin{theorem}\\label{prp:CXAPureIfGGP}\n Let $X$ be a finite-dimensional compact metric space and let $A$ be a pure \\ca{}. Then, $C(X,A)$ is pure if and only if $C(X,A)$ has the Global Glimm Property.\n\\end{theorem}\n\\begin{proof}\n Proposition \\ref{comp_c(x,A)} shows that $C(X,A)$ has $m$-comparison for $m=\\dim (X)$. Further, it follows from Corollary \\ref{robert_analogy_2} that $C(X,A)$ satisfies the divisibility condition in Theorem \\ref{prp:Gen_DimRed}. Thus, Theorem \\ref{prp:Gen_DimRed} gives the desired equivalence.\n\\end{proof}\n\n\\begin{cor}\\label{Purehomogenous}\n Let $A$ be a simple pure \\ca{}, and let $X$ be a compact metric space. Then, $C(X,A)$ is pure.\n\\end{cor}\n\\begin{proof}\n If $X$ is finite-dimensional, $C(X,A)$ has the Global Glimm Property by \\cite[Theorem~4.3]{BlaKir04GlimmHalving}. Thus, the result follows from Theorem \\ref{prp:CXAPureIfGGP}.\n\n\\begin{prop}\\label{prp:CXNoQuot}\n Let $X$ be a finite-dimensional compact metric space, and let $A$ be a pure \\ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, there exists $L\\in\\N$ such that, for every pair $x',x\\in\\Cu (C(X,A))$ with $x'\\ll x$, there exist $y_0,y_1\\in\\mathrm{Cu} (C(X,A))$ satisfying\n \\[\n y_0+y_1\\leq x,\\quad\\text{and}\\quad \n x'\\ll Ly_0, Ly_1.\n \\]\n\\end{prop}\n\\begin{proof}\n We may assume without loss of generality that $A$ is stable, and that there exists $\\varepsilon>0$ such that that $x=[f]$ and $x'=[(f-\\varepsilon)_+]$ for some $f\\in C(X,A)_+$. We adapt the argument from \\cite[Proposition~3.4]{RobTik17NucDimNonSimple}. However, distinct technical difficulties arise in the present context that demand additional care. For this reason, we present the full proof in detail:\n\n\\begin{theorem}\\label{prp:MainNonSimp}\n Let $X$ be a compact metric space, and let $A$ be a pure \\ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, $C(X,A)$ is pure.\n\\end{theorem}\n\\begin{proof}\n If $X$ is finite-dimensional, the result follows directly from Proposition \\ref{prp:CXNoQuot} together with Theorem \\ref{prp:Gen_DimRed} (together with Proposition \\ref{comp_c(x,A)} and Corollary \\ref{robert_analogy_2}).\n\n\\begin{cor}\\label{Purehomogenous}\n    Let $A$ be a simple pure \\ca{}, and let $X$ be a compact metric space. Then, $C(X,A)$ is pure.\n\\end{cor}\n\n\\begin{prop}\\label{comp_c(x,A)}\n    Let $X$ be a compact metric space of covering dimension $m$, and let $A$ be a \\ca{} with strict comparison. Then $C(X, A)$ has $m$-comparison.\n\\end{prop}\n\n\\begin{theorem}\\label{prp:CXAPureIfGGP}\n    Let $X$ be a finite-dimensional compact metric space and let $A$ be a pure \\ca{}. Then, $C(X,A)$ is pure if and only if $C(X,A)$ has the Global Glimm Property.\n\\end{theorem}\n\n\\begin{prop}\\label{prp:CXNoQuot}\n    Let $X$ be a finite-dimensional compact metric space, and let $A$ be a pure \\ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, there exists $L\\in\\N$ such that, for every pair $x',x\\in\\Cu (C(X,A))$ with $x'\\ll x$, there exist $y_0,y_1\\in\\mathrm{Cu} (C(X,A))$ satisfying\n        \\[\n        y_0+y_1\\leq x,\\quad\\text{and}\\quad \n        x'\\ll Ly_0, Ly_1.\n        \\]\n\\end{prop}\n\n\\begin{theorem}\\label{prp:Gen_DimRed}\n    Let $A$ be a \\ca{}, and let $m,M\\in\\mathbb{N}$. Assume that $A$ has $m$-comparison and that, for every $N\\geq 1$ and every pair of elements $x', x\\in \\Cu (A)$ such that $x'\\ll x$, there exists $y\\in\\mathrm{Cu} (A)$ satisfying \n    \\[\n        x'\\ll Ny\\ll M x.\n    \\]\n\n    Then, the following are equivalent:\n    \\begin{itemize}\n        \\item[(i)] $A$ has the Global Glimm Property;\n        \\item[(ii)] there exists $L\\in\\N$ such that, for every pair $x',x\\in\\Cu (A)$ with $x'\\ll x$, there exist $y_0,y_1\\in\\mathrm{Cu} (A)$ satisfying\n        \\[\n        y_0+y_1\\leq x,\\quad\\text{and}\\quad \n        x'\\ll Ly_0, Ly_1;\n        \\]\n        \\item[(iii)] $A$ is pure.\n    \\end{itemize}\n\\end{theorem}\n\n\\begin{theorem}\\label{prp:MainNonSimp}\n    Let $X$ be a compact metric space, and let $A$ be a pure \\ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, $C(X,A)$ is pure.\n\\end{theorem}\n\n\\begin{mainthm}[\\ref{Purehomogenous},~\\ref{prp:MainNonSimp}]\\label{prp:MainThm1}\n    Let $X$ be a compact metric space, and let $A$ be a pure \\ca{}. Assume additionally that\n    \\begin{enumerate}\n        \\item $A$ is simple; or \n        \\item every quotient of $A$ is stably finite.\n    \\end{enumerate}\n\n    Then, $C(X,A)$ is pure.\n\\end{mainthm}\n\n\\begin{cor}\\label{robert_analogy_2}\n    Let $X$ be a finite-dimensional compact metric space, and let $A$ be an almost divisible \\ca{}. Then, there exists $M\\in\\N$ such that, for every $N\\geq 1$ and every pair $x'\\ll x$ in $\\Cu (C(X,A))$, there exists $y\\in\\Cu (C(X,A))$ such that\n    \\[\n        x'\\ll Ny\\ll Mx.\n    \\]\n\\end{cor}", "post_theorem_intro_text_len": 6354, "post_theorem_intro_text": "Before outlining the proof strategy and the challenges involved in establishing Theorem \\ref{prp:MainThm1}, we highlight some important consequences. First, combining Theorem \\ref{prp:MainThm1} with several permanence properties of pureness, we obtain the following application:\n\n\\begin{mainthm}[\\ref{prp:ASHtenPure}]\\label{prp:MainThm2}\n    Let $A$ be a pure \\ca{} satisfying (1) or (2) in Theorem \\ref{prp:MainThm1}, and let $B$ be a unital separable ASH-algebra. Then, $A\\otimes B$ is pure.\n\\end{mainthm}\n\nAs a particular example of this result, Theorem \\ref{prp:MainThm2} shows that the tensor product of any Villadsen algebra of type I with the reduced group \\ca{} of any free group is pure. It is currently unknown whether these specific examples are $\\mathcal{Z}$-stable (Question \\ref{qst:VillZstab}), highlighting the utility of pureness as a regularity invariant. In fact, Theorem \\ref{prp:MainThm2} is slightly more general: As we prove in Theorem \\ref{DpureRSHA}, any recursive subhomogeneous \\ca{} over $A$ \\cite[Definition~3.2]{archey2020structure} (Definition \\ref{RSHA}) is pure whenever $A$ is.\n\nA further application of Theorem \\ref{prp:MainThm2} arises in the study of determining which nonamenable groups have a reduced group \\ca{} with strict comparison. Since $C_r^*(G)$ is a direct sum of simple pure $\\mathrm{C}^*$-algebras whenever $G$ is countable and acylindrically hyperbolic \\cite[Proof of Theorem~C]{FloLisCobPag:PureTwi}, and $C_r^*(H)$ is subhomogeneous whenever $H$ is virtually abelian, we get:\n\n\\begin{maincor}\\label{prp:MainCor1}\n Let $G$ be a countable acylindrically hyperbolic group, and let $H$ be a virtually abelian group. Then, $C_r^*(G\\times H)$ is pure. In particular, it has strict comparison.\n\\end{maincor}\n\nIn addition to significantly enlarging the known class of pure $\\mathrm{C}^*$-algebras, Theorems \\ref{prp:MainThm1} and \\ref{prp:MainThm2} shed new light on the \\emph{Global Glimm Problem}, a major open problem in the field \\cite[Problem~LXXIII]{SchTikWhi99arX:Prob}. The problem asks whether the \\emph{Global Glimm Property} ---a generalization of non-elementariness for non-simple $\\mathrm{C}^*$-algebras \\cite{KirRor02InfNonSimpleCalgAbsOInfty} which we recall in Paragraph \\ref{pgr:GGP}--- is equivalent to having no nonzero elementary ideals of quotients, and has been solved affirmatively in numerous cases (see \\cite[Section~4]{Vil25:IntroGGP} for an overview). While it was previously known that $C(X, A)$ possesses the Global Glimm Property whenever $X$ is finite-dimensional and $A$ is simple and has the property \\cite[Theorem~4.3]{BlaKir04GlimmHalving}, this was not known for pure $\\mathrm{C}^*$-algebras $A$ satisfying condition (2) of Theorem \\ref{prp:MainThm1}, nor for the tensor products appearing in Theorem \\ref{prp:MainThm2}. As we explain below, this is not a consequence of our results; rather, it is a necessary intermediate step in our proof of pureness.\n\nApart from the applications presented in this paper, Theorem \\ref{prp:MainThm1} has further consequences. In particular, this result will be used in forthcoming work of the first named author with Evington, Hua, Schafhauser, and White, which will study \\emph{$K$-stability} for pure $\\mathrm{C}^*$-algebras satisfying conditions as in Theorem~\\ref{prp:MainThm1}. It has been shown in \\cite{lin2025strict} that finite simple pure $\\mathrm{C}^*$-algebras have stable rank one and are therefore $K_1$-bijective. The future work will focus on the stronger property of $K$-stability for this class.\n\n\\subsection*{Strategy of the proof} Given a \\ca{} $A$ with strict comparison, it is well-known that $C(X,A)$ need not have comparison. Topological obstructions, often manifesting as non-trivial vector bundles, can impede comparison even in simple settings ---for example, when $A=\\mathbb{C}$ and $X=\\mathbb{T}^4$ \\cite{Vil98SimpleCaPerforation}. Thus, one of the challenges in establishing Theorem \\ref{prp:MainThm1} is that strict comparison and almost divisibility cannot be verified separately.\n\nOur strategy relies on a multi-step dimension reduction argument. First, we restrict our attention to the case where $X$ is finite-dimensional. In Sections \\ref{sec:CompCuA} and \\ref{sec:DivCuA}, we prove that if $X$ is a compact metric space of dimension $m$ and $A$ is pure, the algebra $C(X, A)$ satisfies a weakening of $(m,n)$-pureness, with $n$ depending only on $m$. Concretely, we see that $C(X,A)$ has $m$-comparison (Proposition \\ref{comp_c(x,A)}) and satisfies a \\emph{weak} version of $n$-almost divisibility (Corollary \\ref{robert_analogy_2}); see Definitions \\ref{dfn:MComp} and \\ref{def_almost_divisibility} for details.\n\nAs Corollary \\ref{robert_analogy_2} does not establish $n$-almost divisibility for any $n$, the rigidity results from \\cite{AntPerThiVil24arX:PureCAlgs} cannot be applied directly. Instead, building on the techniques from \\cite{AntPerThiVil24arX:PureCAlgs}, we prove a new general reduction result (Theorem \\ref{prp:Gen_DimRed}), which we then use in Theorem \\ref{prp:CXAPureIfGGP} to deduce that $C(X, A)$ is pure if and only if it possesses the Global Glimm Property. Relying on the fact that all these properties pass to inductive limits, we are able to dispense with the assumption of finite dimensionality on $X$. Thus, the proof of Theorem \\ref{prp:MainThm1} reduces to proving the Global Glimm Property for $C(X,A)$. For $\\mathrm{C}^*$-algebras satisfying condition (1), this follows from \\cite[Theorem~4.3]{BlaKir04GlimmHalving}; for those satisfying (2), the result is new and is proven in Proposition \\ref{prp:CXNoQuot}.\n\n\\subsection*{Acknowledgments} This paper started during EV's visit to the University of Oxford during the \\emph{Mini Course: Topological Phenomena in the Cuntz semigroup}, which was given by Andrew Toms and organized by Stuart White.  We are grateful to both, as well as to the University of Oxford and St John’s College, for their kind hospitality. We would further like to thank Stuart White for his helpful feedback on the paper, and Hannes Thiel for pointing out Corollary \\ref{prp:MainCor1}. AS is  grateful to N. Christopher Phillips for valuable discussions concerning pureness of $\\mathrm{C}^*$-algebras and for feedback on earlier drafts of this paper. AS also thanks Julian Buck, Dawn Archey, and Javad Mohammadkarimi for insightful discussions.", "sketch": "The proof of Theorem \\ref{prp:MainThm1} addresses the issue that, although $A$ has strict comparison, it is \"well-known that $C(X,A)$ need not have comparison\" due to \"topological obstructions\" (e.g., non-trivial vector bundles), so \"strict comparison and almost divisibility cannot be verified separately.\"\n\nThe strategy is a \"multi-step dimension reduction argument\":\n\\begin{itemize}\n\\item First reduce to $X$ finite-dimensional: for $\\dim(X)=m$ and $A$ pure, in Sections \\ref{sec:CompCuA} and \\ref{sec:DivCuA} they show $C(X,A)$ satisfies a weakening of $(m,n)$-pureness (with $n$ depending only on $m$), namely that $C(X,A)$ has \"$m$-comparison\" (Proposition \\ref{comp_c(x,A)}) and a \"\\emph{weak} version of $n$-almost divisibility\" (Corollary \\ref{robert_analogy_2}).\n\\item Since Corollary \\ref{robert_analogy_2} \"does not establish $n$-almost divisibility for any $n$\", the rigidity results of \\cite{AntPerThiVil24arX:PureCAlgs} \"cannot be applied directly.\" Instead, \"building on the techniques\" there, they prove \"a new general reduction result\" (Theorem \\ref{prp:Gen_DimRed}).\n\\item Use this to show (Theorem \\ref{prp:CXAPureIfGGP}) that $C(X,A)$ is pure \"if and only if it possesses the Global Glimm Property.\"\n\\item \"Relying on the fact that all these properties pass to inductive limits,\" remove the finite-dimensionality hypothesis on $X$.\n\\item Conclude that proving Theorem \\ref{prp:MainThm1} reduces to proving the Global Glimm Property for $C(X,A)$: under condition (1) this follows from \\cite[Theorem~4.3]{BlaKir04GlimmHalving}, while under condition (2) it is new and is proved in Proposition \\ref{prp:CXNoQuot}.\n\\end{itemize}", "expanded_sketch": "To prove the main theorem, the proof addresses the issue that, although $A$ has strict comparison, it is \"well-known that $C(X,A)$ need not have comparison\" due to \"topological obstructions\" (e.g., non-trivial vector bundles), so \"strict comparison and almost divisibility cannot be verified separately.\"\n\nThe strategy is a \"multi-step dimension reduction argument\":\n\\begin{itemize}\n\\item First reduce to $X$ finite-dimensional: for $\\dim(X)=m$ and $A$ pure, they show that $C(X,A)$ satisfies a weakening of $(m,n)$-pureness (with $n$ depending only on $m$), namely the following.\n\n\\begin{prop}\\label{comp_c(x,A)}\n    Let $X$ be a compact metric space of covering dimension $m$, and let $A$ be a \\ca{} with strict comparison. Then $C(X, A)$ has $m$-comparison.\n\\end{prop}\n\nMoreover, they prove the following weak form of almost divisibility.\n\n\\begin{cor}\\label{robert_analogy_2}\n    Let $X$ be a finite-dimensional compact metric space, and let $A$ be an almost divisible \\ca{}. Then, there exists $M\\in\\N$ such that, for every $N\\geq 1$ and every pair $x'\\ll x$ in $\\Cu (C(X,A))$, there exists $y\\in\\Cu (C(X,A))$ such that\n    \\[\n        x'\\ll Ny\\ll Mx.\n    \\]\n\\end{cor}\n\n\\item Since the corollary above \"does not establish $n$-almost divisibility for any $n$\", the rigidity results of AntPerThiVil24arX:PureCAlgs \"cannot be applied directly.\" Instead, \"building on the techniques\" there, they prove the following new general reduction result.\n\n\\begin{theorem}\\label{prp:Gen_DimRed}\n    Let $A$ be a \\ca{}, and let $m,M\\in\\mathbb{N}$. Assume that $A$ has $m$-comparison and that, for every $N\\geq 1$ and every pair of elements $x', x\\in \\Cu (A)$ such that $x'\\ll x$, there exists $y\\in\\mathrm{Cu} (A)$ satisfying \n    \\[\n        x'\\ll Ny\\ll M x.\n    \\]\n\n    Then, the following are equivalent:\n    \\begin{itemize}\n        \\item[(i)] $A$ has the Global Glimm Property;\n        \\item[(ii)] there exists $L\\in\\N$ such that, for every pair $x',x\\in\\Cu (A)$ with $x'\\ll x$, there exist $y_0,y_1\\in\\mathrm{Cu} (A)$ satisfying\n        \\[\n        y_0+y_1\\leq x,\\quad\\text{and}\\quad \n        x'\\ll Ly_0, Ly_1;\n        \\]\n        \\item[(iii)] $A$ is pure.\n    \\end{itemize}\n\\end{theorem}\n\n\\item Use this to show the following characterization of purity for $C(X,A)$ in terms of the Global Glimm Property.\n\n\\begin{theorem}\\label{prp:CXAPureIfGGP}\n    Let $X$ be a finite-dimensional compact metric space and let $A$ be a pure \\ca{}. Then, $C(X,A)$ is pure if and only if $C(X,A)$ has the Global Glimm Property.\n\\end{theorem}\n\n\\item \"Relying on the fact that all these properties pass to inductive limits,\" remove the finite-dimensionality hypothesis on $X$.\n\n\\item Conclude that proving the main theorem reduces to proving the Global Glimm Property for $C(X,A)$: under condition (1) this follows from \\cite[Theorem~4.3]{BlaKir04GlimmHalving}, while under condition (2) it is new and is proved in the following proposition.\n\n\\begin{prop}\\label{prp:CXNoQuot}\n    Let $X$ be a finite-dimensional compact metric space, and let $A$ be a pure \\ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, there exists $L\\in\\N$ such that, for every pair $x',x\\in\\Cu (C(X,A))$ with $x'\\ll x$, there exist $y_0,y_1\\in\\mathrm{Cu} (C(X,A))$ satisfying\n        \\[\n        y_0+y_1\\leq x,\\quad\\text{and}\\quad \n        x'\\ll Ly_0, Ly_1.\n        \\]\n\\end{prop}\n\\end{itemize}", "expanded_theorem": "\\begin{cor}\\label{Purehomogenous}\n    Let $A$ be a simple pure \\ca{}, and let $X$ be a compact metric space. Then, $C(X,A)$ is pure.\n\\end{cor} To prove the main theorem, we use the following result.\n\n\\begin{theorem}\\label{prp:MainNonSimp}\n    Let $X$ be a compact metric space, and let $A$ be a pure \\ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, $C(X,A)$ is pure.\n\\end{theorem}\n\n\\label{prp:MainThm1}\n    Let $X$ be a compact metric space, and let $A$ be a pure \\ca{}. Assume additionally that\n    \\begin{enumerate}\n        \\item $A$ is simple; or \n        \\item every quotient of $A$ is stably finite.\n    \\end{enumerate}\n\n    Then, $C(X,A)$ is pure.", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let \\(X\\) be a compact metric space, and let \\(A\\) be a pure \\(\\mathrm{C}^*\\)-algebra, where \"pure\" means that \\(A\\) has strict comparison and almost divisibility. Assume in addition that either \\(A\\) is simple, or every quotient \\(\\mathrm{C}^*\\)-algebra of \\(A\\) is stably finite. Here \\(C(X,A)\\) denotes the \\(\\mathrm{C}^*\\)-algebra of continuous \\(A\\)-valued functions on \\(X\\). Which statement holds for every such pair \\((X,A)\\)?", "correct_choice": {"label": "A", "text": "\\(C(X,A)\\) is pure; equivalently, the \\(\\mathrm{C}^*\\)-algebra of continuous \\(A\\)-valued functions on \\(X\\) has strict comparison and almost divisibility."}, "choices": [{"label": "B", "text": "\\(C(X,A)\\) has strict comparison, but it need not be almost divisible in general; thus one can only conclude a comparison property for \\(C(X,A)\\), not pureness."}, {"label": "C", "text": "\\(C(X,A)\\) has strict comparison."}, {"label": "D", "text": "\\(C(X,A)\\) is pure whenever \\(X\\) is finite-dimensional; for arbitrary compact metric \\(X\\), the conclusion can fail even if \\(A\\) is simple or every quotient of \\(A\\) is stably finite."}, {"label": "E", "text": "\\(C(X,A)\\) is pure provided both that \\(A\\) is simple and that every quotient \\(\\mathrm{C}^*\\)-algebra of \\(A\\) is stably finite; assuming only one of these conditions is not sufficient."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "dimension-reduction needed to recover almost divisibility jointly with comparison", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped almost divisibility/pureness and kept only strict comparison", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "removal of finite-dimensionality via inductive limits", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "logical disjunction of hypotheses (simple OR all quotients stably finite)", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the conclusion, and no option is textually echoed in a way that gives away the answer. It presents hypotheses and asks for the resulting conclusion, so there is no direct answer leakage."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the hypotheses are stated in full and the correct choice is the theorem's conclusion. It is very close to a restatement rather than a novel inference task."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the student must distinguish the strongest valid conclusion from weaker or unnecessarily qualified alternatives. However, the item mainly tests recognition/recall of a known permanence result rather than substantive generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target common confusions: mistaking strict comparison for purity, adding an unnecessary finite-dimensional hypothesis, or selecting a weaker true statement instead of the strongest correct one. They are distinct and reasonably well designed."}, "total_score": 5, "overall_assessment": "A solid theorem-recall MCQ with good distractors, but it is highly tautological and only moderately probes reasoning."}}
{"id": "2602.14809v2", "paper_link": "http://arxiv.org/abs/2602.14809v2", "theorems_cnt": 3, "theorem": {"env_name": "mainthm", "content": "[\\ref{Purehomogenous},~\\ref{prp:MainNonSimp}]\\label{prp:MainThm1}\n    Let $X$ be a compact metric space, and let $A$ be a pure \\ca{}. Assume additionally that\n    \\begin{enumerate}\n        \\item $A$ is simple; or \n        \\item every quotient of $A$ is stably finite.\n    \\end{enumerate}\n\n    Then, $C(X,A)$ is pure.", "start_pos": 8129, "end_pos": 8474, "label": "prp:MainThm1"}, "ref_dict": {"pgr:GGP": "\\begin{pgr}[The Global Glimm Property]\\label{pgr:GGP}\n    A \\ca{} $A$ is said to satisfy the \\emph{Global Glimm Property} if for every $a\\in A_+$ and $\\varepsilon>0$ there exists a square-zero element $r\\in\\overline{aAa}$ such that $(a-\\varepsilon)_+\\in \\overline{ArA}$. This property was originally introduced by Kirchberg and R\\o{}rdam in their study of non-simple purely infinite \\ca{s} \\cite[Definition~4.12]{KirRor02InfNonSimpleCalgAbsOInfty}, but has recently found various and deep applications in the context of dimension reduction phenomena; see, for example, \\cite{ThiVil23Glimm,Vil25:IntroGGP}.\n\n    As shown in \\cite[Theorem~3.6]{ThiVil23Glimm}, the property can be characterized in terms of a divisibility condition on the Cuntz semigroup: $A$ has the Global Glimm Property if and only if $\\Cu (A)$ is $(2,\\omega )$-divisible, that is, for  every pair $x',x\\in\\Cu (A)$ such that $x'\\ll x$, there exists $y\\in\\Cu (A)$ and $n\\in\\N$ such that $2y\\leq x$ and $x'\\leq ny$.\n\\end{pgr}", "prp:CXAPureIfGGP": "\\begin{theorem}\\label{prp:CXAPureIfGGP}\n    Let $X$ be a finite-dimensional compact metric space and let $A$ be a pure \\ca{}. Then, $C(X,A)$ is pure if and only if $C(X,A)$ has the Global Glimm Property.\n\\end{theorem}", "DpureRSHA": "\\begin{prop}\\label{DpureRSHA}\nLet $D$ be a unital simple pure \\ca. Let $R_D$ be a recursive subhomogeneous algebra over $D$ and let $R$ be any recursive homogeneous algebra over $\\mathbb{C}$.\nThen both $R_D$ and $R \\otimes D$ are pure.\n\\end{prop}", "RSHA": "\\begin{defn}[{\\cite[Definition~3.2]{archey2020structure}}]\\label{RSHA}\nFor a simple unital \\ca{} $D$, the class $\\mathcal{R}$ of \n{\\emph{recursive subhomogeneous algebras over $D$}} is\nthe smallest class of \\ca{s} closed under isomorphism such that:\n\\begin{enumerate}\n\\item\\label{9223_RSHA_1}\n$C (X, M_{n} (D)) \\in \\mathcal{R}$ whenever $X$ is a compact Hausdorff space and $n \\geq 1$.\n\\item\\label{9223_RSHA_2}\nAny pullback of the form \n\\begin{align*}\n& B \\oplus_{C (X^{(0)}, \\, M_{n} (D)), \\, \\ph, \\, \\rh} C (X, M_{n} (D))\n\\\\\n& \\hspace*{3em} {\\mbox{}}\n  = \\bset{ (b, f) \\in B \\oplus C (X, M_{n} (D)) \\colon\n        \\ph (b) = f |_{X^{(0)}} }\n\\end{align*}\nis in $\\mathcal{R}$ whenever $B \\in \\mathcal{R}$, $X$ is compact Hausdorff,\n$n \\geq 1$, $X^{(0)} \\subseteq X$ is closed (possibly empty),\n$\\ph \\colon B \\to C (X^{(0)}, \\, M_{n} (D))$\nis any unital $^*$-homomorphism\n(the zero \\hm{} if $X^{(0)}$ is empty), and\n$\\rho \\colon C (X, M_{n} (D)) \\to C (X^{(0)}, \\, M_{n} (D))$ is the\nrestriction $^*$-homomorphism.\n\\end{enumerate}\n\\end{defn}", "Purehomogenous": "\\begin{cor}\\label{Purehomogenous}\n    Let $A$ be a simple pure \\ca{}, and let $X$ be a compact metric space. Then, $C(X,A)$ is pure.\n\\end{cor}", "prp:MainThm2": "\\begin{mainthm}[\\ref{prp:ASHtenPure}]\\label{prp:MainThm2}\n    Let $A$ be a pure \\ca{} satisfying (1) or (2) in Theorem \\ref{prp:MainThm1}, and let $B$ be a unital separable ASH-algebra. Then, $A\\otimes B$ is pure.\n\\end{mainthm}", "prp:MainCor1": "\\begin{maincor}\\label{prp:MainCor1}\n Let $G$ be a countable acylindrically hyperbolic group, and let $H$ be a virtually abelian group. Then, $C_r^*(G\\times H)$ is pure. In particular, it has strict comparison.\n\\end{maincor}", "prp:MainNonSimp": "\\begin{theorem}\\label{prp:MainNonSimp}\n    Let $X$ be a compact metric space, and let $A$ be a pure \\ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, $C(X,A)$ is pure.\n\\end{theorem}", "def_almost_divisibility": "\\begin{defn}\\label{def_almost_divisibility}\n    Let $A$ be a \\ca{} and $n\\in\\N$. We say that $A$ (and its Cuntz semigroup $\\Cu (A)$) is \\emph{$n$-almost divisible} if for any pair $x',x\\in\\mathrm{Cu}(A)$ such that $x'\\ll x$ and any $N\\in \\N$, there exists $y\\in \\mathrm{Cu}(A)$ such that \n    \\[\n     N y\\leq x\\quad{\\mbox{and}}\\quad x'\\leq (N+1)(n+1)y.\n    \\]\n\n    $A$ is said to be \\emph{almost divisible} if it is $0$-almost divisible.\n\\end{defn}", "robert_analogy_2": "\\begin{cor}\\label{robert_analogy_2}\n    Let $X$ be a finite-dimensional compact metric space, and let $A$ be an almost divisible \\ca{}. Then, there exists $M\\in\\N$ such that, for every $N\\geq 1$ and every pair $x'\\ll x$ in $\\Cu (C(X,A))$, there exists $y\\in\\Cu (C(X,A))$ such that\n    \\[\n        x'\\ll Ny\\ll Mx.\n    \\]\n\\end{cor}", "qst:VillZstab": "\\begin{qst}\\label{qst:VillZstab}\n    Let $B$ be a Villadsen algebra of the first kind without strict comparison (for example, Toms' \\cite{Tom08ClassificationNuclear}). Is $B\\otimes C_r^*(\\mathbb{F}_2)$ $\\mathcal{Z}$-stable?\n\\end{qst}", "prp:ASHtenPure": "\\begin{theorem}\\label{prp:ASHtenPure}\n   Let $B$ be a unital separable ASH-algebra and $A$ be simple and pure. Then, $B\\otimes A$ is pure.\n\\end{theorem}", "comp_c(x,A)": "\\begin{prop}\\label{comp_c(x,A)}\n    Let $X$ be a compact metric space of covering dimension $m$, and let $A$ be a \\ca{} with strict comparison. Then $C(X, A)$ has $m$-comparison.\n\\end{prop}", "prp:MainThm1": "\\begin{mainthm}[\\ref{Purehomogenous},~\\ref{prp:MainNonSimp}]\\label{prp:MainThm1}\n    Let $X$ be a compact metric space, and let $A$ be a pure \\ca{}. Assume additionally that\n    \\begin{enumerate}\n        \\item $A$ is simple; or \n        \\item every quotient of $A$ is stably finite.\n    \\end{enumerate}\n\n    Then, $C(X,A)$ is pure.\n\\end{mainthm}", "prp:Gen_DimRed": "\\begin{theorem}\\label{prp:Gen_DimRed}\n    Let $A$ be a \\ca{}, and let $m,M\\in\\mathbb{N}$. Assume that $A$ has $m$-comparison and that, for every $N\\geq 1$ and every pair of elements $x', x\\in \\Cu (A)$ such that $x'\\ll x$, there exists $y\\in\\mathrm{Cu} (A)$ satisfying \n    \\[\n        x'\\ll Ny\\ll M x.\n    \\]\n\n    Then, the following are equivalent:\n    \\begin{itemize}\n        \\item[(i)] $A$ has the Global Glimm Property;\n        \\item[(ii)] there exists $L\\in\\N$ such that, for every pair $x',x\\in\\Cu (A)$ with $x'\\ll x$, there exist $y_0,y_1\\in\\mathrm{Cu} (A)$ satisfying\n        \\[\n        y_0+y_1\\leq x,\\quad\\text{and}\\quad \n        x'\\ll Ly_0, Ly_1;\n        \\]\n        \\item[(iii)] $A$ is pure.\n    \\end{itemize}\n\\end{theorem}", "dfn:MComp": "\\begin{defn}[{\\cite[Definition~2.1]{Win12NuclDimZstable}}]\\label{dfn:MComp}\n    Let $A$ be a \\ca{} and let $n\\in\\N$. We say that $A$ (and its Cuntz semigroup $\\Cu (A)$) has the \\emph{$n$-comparison} property if $x \\leq_s y_j$ for $x, y_j \\in \\Cu (A)$ and $j = 0, 1, \\ldots, n$, implies $x\\leq \\sum_{j=0}^n y_j$.\n\n    $\\Cu (A)$ is said to be \\emph{almost unperforated} if it has $0$-comparison.\n\\end{defn}", "rem:StCompAlmUnperf": "\\begin{rem}\\label{rem:StCompAlmUnperf}\n    Let $A$ be a \\ca{}. For any $[0,\\infty]$-valued lower semi-continuous $2$-quasitrace $\\tau$ on $A$ (which extends to $A\\otimes\\mathcal{K}$ and is denoted by the same symbol), the associated \\emph{dimension function} $d_\\tau\\colon \\Cu (A)\\to [0,\\infty]$ is defined by $d_\\tau ([a])=\\lim_n\\tau (a^{1/n})$.\n\n    As shown in \\cite[Proposition~6.2]{elliott2011cone}, $\\Cu(A)$ is almost unperforated (in the sense described above) if and only if, for every pair $a,b\\in  (A\\otimes\\mathcal{K})_+$ such that $a$ is in the ideal generated by $b$ and $d_\\tau ([a])<d_\\tau([b])$ whenever $d_\\tau ([b])=1$, one has $[a]\\leq [b]$ in $\\Cu (A)$.\n\n    This property is known as \\emph{strict comparison} (of positive elements by $[0,\\infty]$-valued lower semi-continuous $2$-quasitraces). Note that, if $A$ is simple, $A$ has strict comparison if and only if, for every pair of non-zero elements $a,b\\in  (A\\otimes\\mathcal{K})_+$ such that $d_\\tau ([a])<d_\\tau([b])$ for every normalized quasitrace $\\tau$, one has $[a]\\leq [b]$ in $\\Cu (A)$. We refer the reader to \\cite[Paragraph~3.4]{AntPerThiVil24arX:PureCAlgs} for an in-depth introduction.\n\\end{rem}", "prp:CXNoQuot": "\\begin{prop}\\label{prp:CXNoQuot}\n    Let $X$ be a finite-dimensional compact metric space, and let $A$ be a pure \\ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, there exists $L\\in\\N$ such that, for every pair $x',x\\in\\Cu (C(X,A))$ with $x'\\ll x$, there exist $y_0,y_1\\in\\mathrm{Cu} (C(X,A))$ satisfying\n        \\[\n        y_0+y_1\\leq x,\\quad\\text{and}\\quad \n        x'\\ll Ly_0, Ly_1.\n        \\]\n\\end{prop}"}, "pre_theorem_intro_text_len": 4095, "pre_theorem_intro_text": "The properties of \\emph{strict comparison} and \\emph{almost divisibility} play a fundamental role in the modern structure theory of $\\mathrm{C}^*$-algebras. As defined by Winter \\cite{Win12NuclDimZstable}, $\\mathrm{C}^*$-algebras that satisfy both properties are termed \\emph{pure}. In the simple finite setting, such algebras should be regarded as the correct analogues of II$_1$ factors; roughly speaking, the defining properties of pureness correspond to two key features of projections in II$_1$factors, adapted to the setting of positive elements ---a necessary adaptation given that many $\\mathrm{C}^*$-algebras contain few or no nontrivial projections.\n\nJust as the unique trace on a II$_1$ factor $\\mathcal{N}$ determines the order of projections, \\emph{strict comparison} \\cite{Bla88Comparison} asserts that tracial state data determines when one positive operator can be approximately compressed to another (Remark \\ref{rem:StCompAlmUnperf}). Similarly, \\emph{almost divisibility} (Definition \\ref{def_almost_divisibility}) serves as the analogue to the fact that projections in $\\mathcal{N}$ can be divided arbitrarily. Whilst these two properties are automatic for II$_1$ factors, they can fail for simple $\\mathrm{C}^*$-algebras \\cite{Tom08ClassificationNuclear,Vil99SRSimpleCa}. Consequently, determining which algebras satisfy these properties ---and, in particular, developing mechanisms to deduce pureness--- has been a major focus in the field for the past few years \\cite{AGKEP25,AntPerThiVil24arX:PureCAlgs,Rob25Self}. Examples of pure $\\mathrm{C}^*$-algebras can be found both in the nuclear and non nuclear case, and include $\\mathcal{Z}$-stable algebras, purely infinite algebras \\cite{kirchberg2000non}, von Neumann algebras with trivial type I summand, and reduced (twisted) group $\\mathrm{C}^*$-algebras of acylindrically hyperbolic groups \\cite{FloLisCobPag:PureTwi,RauThiVil:StCompTwi}.\n\nIn the realm of simple nuclear $\\mathrm{C}^*$-algebras, pureness features prominently in the Toms-Winter conjecture \\cite{CasEviTikWhiWin21NucDimSimple,Ror04StableRealRankZ,Win12NuclDimZstable} and serves as a crucial regularity condition in the Elliott classification program \\cite{Whi23ICM,Win18ICM}. More generally, the notion is expected to play an important role in the classification of $^*$-homomorphisms \\cite{CarGabSchTikWhi23arX:ClassifHom1,Sza26:UniqThmKK}, potentially by weakening the requirement of $\\mathcal{Z}$-stability on the codomain to that of pureness. In fact, as shown in \\cite[Theorem~7.3.11]{antoine2018tensor}, one may think of pureness as $\\mathcal{Z}$-stability at the level of the Cuntz semigroup.\n\nOf the two defining properties of pureness, strict comparison has received the most attention in recent years, finding deep connections with the theory of group $\\mathrm{C}^*$-algebras \\cite{AGKEP25,Oza25:Proxim}, time-frequency analysis \\cite{BedEnsvVelt22,EnsThiVil25Criteria}, and the recent negative resolution to Tarski's problem \\cite{KES:NegativeTarski}. However, despite its importance, strict comparison is often difficult to verify directly. For this reason, attention has shifted towards the stronger, more rigid property of pureness, which is particularly susceptible to \\emph{dimension reduction phenomena}. Specifically, it was recently proved in \\cite{AntPerThiVil24arX:PureCAlgs} that any $(m, n)$-pure \\ca{} ---a formal `quantized' weakening of pureness--- is automatically pure. This result allows one to deduce pureness (and thus strict comparison) in situations where direct verification was previously out of reach. It has also paved the way for new permanence results; for instance, it was established in \\cite{perera2025extensions} that an extension of pure $\\mathrm{C}^*$-algebras is itself pure, mirroring the corresponding property for $\\mathcal{Z}$-stability \\cite[Theorem~4.3]{TomWin07ssa}. An important question in this line of inquiry, raised at a number of conferences, is whether a minimal tensor product $B \\otimes A$ is pure whenever $A$ is. In the present paper, we investigate this question for the case $B = C(X)$.", "context": "The properties of \\emph{strict comparison} and \\emph{almost divisibility} play a fundamental role in the modern structure theory of $\\mathrm{C}^*$-algebras. As defined by Winter \\cite{Win12NuclDimZstable}, $\\mathrm{C}^*$-algebras that satisfy both properties are termed \\emph{pure}. In the simple finite setting, such algebras should be regarded as the correct analogues of II$_1$ factors; roughly speaking, the defining properties of pureness correspond to two key features of projections in II$_1$factors, adapted to the setting of positive elements ---a necessary adaptation given that many $\\mathrm{C}^*$-algebras contain few or no nontrivial projections.\n\nJust as the unique trace on a II$_1$ factor $\\mathcal{N}$ determines the order of projections, \\emph{strict comparison} \\cite{Bla88Comparison} asserts that tracial state data determines when one positive operator can be approximately compressed to another (Remark \\ref{rem:StCompAlmUnperf}). Similarly, \\emph{almost divisibility} (Definition \\ref{def_almost_divisibility}) serves as the analogue to the fact that projections in $\\mathcal{N}$ can be divided arbitrarily. Whilst these two properties are automatic for II$_1$ factors, they can fail for simple $\\mathrm{C}^*$-algebras \\cite{Tom08ClassificationNuclear,Vil99SRSimpleCa}. Consequently, determining which algebras satisfy these properties ---and, in particular, developing mechanisms to deduce pureness--- has been a major focus in the field for the past few years \\cite{AGKEP25,AntPerThiVil24arX:PureCAlgs,Rob25Self}. Examples of pure $\\mathrm{C}^*$-algebras can be found both in the nuclear and non nuclear case, and include $\\mathcal{Z}$-stable algebras, purely infinite algebras \\cite{kirchberg2000non}, von Neumann algebras with trivial type I summand, and reduced (twisted) group $\\mathrm{C}^*$-algebras of acylindrically hyperbolic groups \\cite{FloLisCobPag:PureTwi,RauThiVil:StCompTwi}.\n\nIn the realm of simple nuclear $\\mathrm{C}^*$-algebras, pureness features prominently in the Toms-Winter conjecture \\cite{CasEviTikWhiWin21NucDimSimple,Ror04StableRealRankZ,Win12NuclDimZstable} and serves as a crucial regularity condition in the Elliott classification program \\cite{Whi23ICM,Win18ICM}. More generally, the notion is expected to play an important role in the classification of $^*$-homomorphisms \\cite{CarGabSchTikWhi23arX:ClassifHom1,Sza26:UniqThmKK}, potentially by weakening the requirement of $\\mathcal{Z}$-stability on the codomain to that of pureness. In fact, as shown in \\cite[Theorem~7.3.11]{antoine2018tensor}, one may think of pureness as $\\mathcal{Z}$-stability at the level of the Cuntz semigroup.\n\nOf the two defining properties of pureness, strict comparison has received the most attention in recent years, finding deep connections with the theory of group $\\mathrm{C}^*$-algebras \\cite{AGKEP25,Oza25:Proxim}, time-frequency analysis \\cite{BedEnsvVelt22,EnsThiVil25Criteria}, and the recent negative resolution to Tarski's problem \\cite{KES:NegativeTarski}. However, despite its importance, strict comparison is often difficult to verify directly. For this reason, attention has shifted towards the stronger, more rigid property of pureness, which is particularly susceptible to \\emph{dimension reduction phenomena}. Specifically, it was recently proved in \\cite{AntPerThiVil24arX:PureCAlgs} that any $(m, n)$-pure \\ca{} ---a formal `quantized' weakening of pureness--- is automatically pure. This result allows one to deduce pureness (and thus strict comparison) in situations where direct verification was previously out of reach. It has also paved the way for new permanence results; for instance, it was established in \\cite{perera2025extensions} that an extension of pure $\\mathrm{C}^*$-algebras is itself pure, mirroring the corresponding property for $\\mathcal{Z}$-stability \\cite[Theorem~4.3]{TomWin07ssa}. An important question in this line of inquiry, raised at a number of conferences, is whether a minimal tensor product $B \\otimes A$ is pure whenever $A$ is. In the present paper, we investigate this question for the case $B = C(X)$.", "full_context": "The properties of \\emph{strict comparison} and \\emph{almost divisibility} play a fundamental role in the modern structure theory of $\\mathrm{C}^*$-algebras. As defined by Winter \\cite{Win12NuclDimZstable}, $\\mathrm{C}^*$-algebras that satisfy both properties are termed \\emph{pure}. In the simple finite setting, such algebras should be regarded as the correct analogues of II$_1$ factors; roughly speaking, the defining properties of pureness correspond to two key features of projections in II$_1$factors, adapted to the setting of positive elements ---a necessary adaptation given that many $\\mathrm{C}^*$-algebras contain few or no nontrivial projections.\n\nJust as the unique trace on a II$_1$ factor $\\mathcal{N}$ determines the order of projections, \\emph{strict comparison} \\cite{Bla88Comparison} asserts that tracial state data determines when one positive operator can be approximately compressed to another (Remark \\ref{rem:StCompAlmUnperf}). Similarly, \\emph{almost divisibility} (Definition \\ref{def_almost_divisibility}) serves as the analogue to the fact that projections in $\\mathcal{N}$ can be divided arbitrarily. Whilst these two properties are automatic for II$_1$ factors, they can fail for simple $\\mathrm{C}^*$-algebras \\cite{Tom08ClassificationNuclear,Vil99SRSimpleCa}. Consequently, determining which algebras satisfy these properties ---and, in particular, developing mechanisms to deduce pureness--- has been a major focus in the field for the past few years \\cite{AGKEP25,AntPerThiVil24arX:PureCAlgs,Rob25Self}. Examples of pure $\\mathrm{C}^*$-algebras can be found both in the nuclear and non nuclear case, and include $\\mathcal{Z}$-stable algebras, purely infinite algebras \\cite{kirchberg2000non}, von Neumann algebras with trivial type I summand, and reduced (twisted) group $\\mathrm{C}^*$-algebras of acylindrically hyperbolic groups \\cite{FloLisCobPag:PureTwi,RauThiVil:StCompTwi}.\n\nIn the realm of simple nuclear $\\mathrm{C}^*$-algebras, pureness features prominently in the Toms-Winter conjecture \\cite{CasEviTikWhiWin21NucDimSimple,Ror04StableRealRankZ,Win12NuclDimZstable} and serves as a crucial regularity condition in the Elliott classification program \\cite{Whi23ICM,Win18ICM}. More generally, the notion is expected to play an important role in the classification of $^*$-homomorphisms \\cite{CarGabSchTikWhi23arX:ClassifHom1,Sza26:UniqThmKK}, potentially by weakening the requirement of $\\mathcal{Z}$-stability on the codomain to that of pureness. In fact, as shown in \\cite[Theorem~7.3.11]{antoine2018tensor}, one may think of pureness as $\\mathcal{Z}$-stability at the level of the Cuntz semigroup.\n\nOf the two defining properties of pureness, strict comparison has received the most attention in recent years, finding deep connections with the theory of group $\\mathrm{C}^*$-algebras \\cite{AGKEP25,Oza25:Proxim}, time-frequency analysis \\cite{BedEnsvVelt22,EnsThiVil25Criteria}, and the recent negative resolution to Tarski's problem \\cite{KES:NegativeTarski}. However, despite its importance, strict comparison is often difficult to verify directly. For this reason, attention has shifted towards the stronger, more rigid property of pureness, which is particularly susceptible to \\emph{dimension reduction phenomena}. Specifically, it was recently proved in \\cite{AntPerThiVil24arX:PureCAlgs} that any $(m, n)$-pure \\ca{} ---a formal `quantized' weakening of pureness--- is automatically pure. This result allows one to deduce pureness (and thus strict comparison) in situations where direct verification was previously out of reach. It has also paved the way for new permanence results; for instance, it was established in \\cite{perera2025extensions} that an extension of pure $\\mathrm{C}^*$-algebras is itself pure, mirroring the corresponding property for $\\mathcal{Z}$-stability \\cite[Theorem~4.3]{TomWin07ssa}. An important question in this line of inquiry, raised at a number of conferences, is whether a minimal tensor product $B \\otimes A$ is pure whenever $A$ is. In the present paper, we investigate this question for the case $B = C(X)$.\n\n\\begin{document}\n\\begin{abstract}\n Let $X$ be a compact metric space, and let $A$ be a pure \\ca{}. We show that $C(X,A)$ is pure whenever\n \\begin{enumerate}\n \\item $A$ is simple; or \n \\item every quotient of $A$ is stably finite (e.g., $A$ has stable rank one).\n \\end{enumerate}\n\nOf the two defining properties of pureness, strict comparison has received the most attention in recent years, finding deep connections with the theory of group \\ca{s} \\cite{AGKEP25,Oza25:Proxim}, time-frequency analysis \\cite{BedEnsvVelt22,EnsThiVil25Criteria}, and the recent negative resolution to Tarski's problem \\cite{KES:NegativeTarski}. However, despite its importance, strict comparison is often difficult to verify directly. For this reason, attention has shifted towards the stronger, more rigid property of pureness, which is particularly susceptible to \\emph{dimension reduction phenomena}. Specifically, it was recently proved in \\cite{AntPerThiVil24arX:PureCAlgs} that any $(m, n)$-pure \\ca{} ---a formal `quantized' weakening of pureness--- is automatically pure. This result allows one to deduce pureness (and thus strict comparison) in situations where direct verification was previously out of reach. It has also paved the way for new permanence results; for instance, it was established in \\cite{perera2025extensions} that an extension of pure \\ca{s} is itself pure, mirroring the corresponding property for $\\mathcal{Z}$-stability \\cite[Theorem~4.3]{TomWin07ssa}. An important question in this line of inquiry, raised at a number of conferences, is whether a minimal tensor product $B \\otimes A$ is pure whenever $A$ is. In the present paper, we investigate this question for the case $B = C(X)$.\n\nBefore outlining the proof strategy and the challenges involved in establishing Theorem \\ref{prp:MainThm1}, we highlight some important consequences. First, combining Theorem \\ref{prp:MainThm1} with several permanence properties of pureness, we obtain the following application:\n\n\\begin{cor}\\label{robert_analogy_2}\n Let $X$ be a finite-dimensional compact metric space, and let $A$ be an almost divisible \\ca{}. Then, there exists $M\\in\\N$ such that, for every $N\\geq 1$ and every pair $x'\\ll x$ in $\\Cu (C(X,A))$, there exists $y\\in\\Cu (C(X,A))$ such that\n \\[\n x'\\ll Ny\\ll Mx.\n \\]\n\\end{cor}\n\\begin{proof}\n Set $M=2(\\dim (X)+1)^2$ and let $N\\in\\N$. If $N=1$, any $y\\in\\Cu (A)$ such that $x'\\ll y\\ll x$ satisfies the required condition. Thus, we may assume $N>1$.\n As in the previous proof, we may also assume that $A$ is stable. Find $w\\in \\Cu (C(X,A))$ such that $x'\\ll w\\ll x$, and let $w=[f]$ with $f\\in C(X,A)_+$. By the definition of compact containment, there exists $\\varepsilon>0$ such that $x'\\ll [(f-\\varepsilon)_+]$. Now apply Proposition \\ref{robert_analogy} to $f$, $\\varepsilon$ and $N-1$ to find $y\\in \\Cu (C(X,A))$ such that \n \\[\n [(f-\\varepsilon)_+]\\leq Ny\\quad\\text{and}\\quad (N-1)y\\leq (\\dim(X)+1)^2[f].\n \\]\n\n\\begin{theorem}\\label{prp:CXAPureIfGGP}\n Let $X$ be a finite-dimensional compact metric space and let $A$ be a pure \\ca{}. Then, $C(X,A)$ is pure if and only if $C(X,A)$ has the Global Glimm Property.\n\\end{theorem}\n\\begin{proof}\n Proposition \\ref{comp_c(x,A)} shows that $C(X,A)$ has $m$-comparison for $m=\\dim (X)$. Further, it follows from Corollary \\ref{robert_analogy_2} that $C(X,A)$ satisfies the divisibility condition in Theorem \\ref{prp:Gen_DimRed}. Thus, Theorem \\ref{prp:Gen_DimRed} gives the desired equivalence.\n\\end{proof}\n\n\\begin{cor}\\label{Purehomogenous}\n Let $A$ be a simple pure \\ca{}, and let $X$ be a compact metric space. Then, $C(X,A)$ is pure.\n\\end{cor}\n\\begin{proof}\n If $X$ is finite-dimensional, $C(X,A)$ has the Global Glimm Property by \\cite[Theorem~4.3]{BlaKir04GlimmHalving}. Thus, the result follows from Theorem \\ref{prp:CXAPureIfGGP}.\n\n\\begin{prop}\\label{prp:CXNoQuot}\n Let $X$ be a finite-dimensional compact metric space, and let $A$ be a pure \\ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, there exists $L\\in\\N$ such that, for every pair $x',x\\in\\Cu (C(X,A))$ with $x'\\ll x$, there exist $y_0,y_1\\in\\mathrm{Cu} (C(X,A))$ satisfying\n \\[\n y_0+y_1\\leq x,\\quad\\text{and}\\quad \n x'\\ll Ly_0, Ly_1.\n \\]\n\\end{prop}\n\\begin{proof}\n We may assume without loss of generality that $A$ is stable, and that there exists $\\varepsilon>0$ such that that $x=[f]$ and $x'=[(f-\\varepsilon)_+]$ for some $f\\in C(X,A)_+$. We adapt the argument from \\cite[Proposition~3.4]{RobTik17NucDimNonSimple}. However, distinct technical difficulties arise in the present context that demand additional care. For this reason, we present the full proof in detail:\n\n\\begin{theorem}\\label{prp:MainNonSimp}\n Let $X$ be a compact metric space, and let $A$ be a pure \\ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, $C(X,A)$ is pure.\n\\end{theorem}\n\\begin{proof}\n If $X$ is finite-dimensional, the result follows directly from Proposition \\ref{prp:CXNoQuot} together with Theorem \\ref{prp:Gen_DimRed} (together with Proposition \\ref{comp_c(x,A)} and Corollary \\ref{robert_analogy_2}).\n\n\\begin{cor}\\label{Purehomogenous}\n    Let $A$ be a simple pure \\ca{}, and let $X$ be a compact metric space. Then, $C(X,A)$ is pure.\n\\end{cor}\n\n\\begin{prop}\\label{comp_c(x,A)}\n    Let $X$ be a compact metric space of covering dimension $m$, and let $A$ be a \\ca{} with strict comparison. Then $C(X, A)$ has $m$-comparison.\n\\end{prop}\n\n\\begin{theorem}\\label{prp:CXAPureIfGGP}\n    Let $X$ be a finite-dimensional compact metric space and let $A$ be a pure \\ca{}. Then, $C(X,A)$ is pure if and only if $C(X,A)$ has the Global Glimm Property.\n\\end{theorem}\n\n\\begin{prop}\\label{prp:CXNoQuot}\n    Let $X$ be a finite-dimensional compact metric space, and let $A$ be a pure \\ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, there exists $L\\in\\N$ such that, for every pair $x',x\\in\\Cu (C(X,A))$ with $x'\\ll x$, there exist $y_0,y_1\\in\\mathrm{Cu} (C(X,A))$ satisfying\n        \\[\n        y_0+y_1\\leq x,\\quad\\text{and}\\quad \n        x'\\ll Ly_0, Ly_1.\n        \\]\n\\end{prop}\n\n\\begin{theorem}\\label{prp:Gen_DimRed}\n    Let $A$ be a \\ca{}, and let $m,M\\in\\mathbb{N}$. Assume that $A$ has $m$-comparison and that, for every $N\\geq 1$ and every pair of elements $x', x\\in \\Cu (A)$ such that $x'\\ll x$, there exists $y\\in\\mathrm{Cu} (A)$ satisfying \n    \\[\n        x'\\ll Ny\\ll M x.\n    \\]\n\n    Then, the following are equivalent:\n    \\begin{itemize}\n        \\item[(i)] $A$ has the Global Glimm Property;\n        \\item[(ii)] there exists $L\\in\\N$ such that, for every pair $x',x\\in\\Cu (A)$ with $x'\\ll x$, there exist $y_0,y_1\\in\\mathrm{Cu} (A)$ satisfying\n        \\[\n        y_0+y_1\\leq x,\\quad\\text{and}\\quad \n        x'\\ll Ly_0, Ly_1;\n        \\]\n        \\item[(iii)] $A$ is pure.\n    \\end{itemize}\n\\end{theorem}\n\n\\begin{theorem}\\label{prp:MainNonSimp}\n    Let $X$ be a compact metric space, and let $A$ be a pure \\ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, $C(X,A)$ is pure.\n\\end{theorem}\n\n\\begin{mainthm}[\\ref{Purehomogenous},~\\ref{prp:MainNonSimp}]\\label{prp:MainThm1}\n    Let $X$ be a compact metric space, and let $A$ be a pure \\ca{}. Assume additionally that\n    \\begin{enumerate}\n        \\item $A$ is simple; or \n        \\item every quotient of $A$ is stably finite.\n    \\end{enumerate}\n\n    Then, $C(X,A)$ is pure.\n\\end{mainthm}\n\n\\begin{cor}\\label{robert_analogy_2}\n    Let $X$ be a finite-dimensional compact metric space, and let $A$ be an almost divisible \\ca{}. Then, there exists $M\\in\\N$ such that, for every $N\\geq 1$ and every pair $x'\\ll x$ in $\\Cu (C(X,A))$, there exists $y\\in\\Cu (C(X,A))$ such that\n    \\[\n        x'\\ll Ny\\ll Mx.\n    \\]\n\\end{cor}", "post_theorem_intro_text_len": 6354, "post_theorem_intro_text": "Before outlining the proof strategy and the challenges involved in establishing Theorem \\ref{prp:MainThm1}, we highlight some important consequences. First, combining Theorem \\ref{prp:MainThm1} with several permanence properties of pureness, we obtain the following application:\n\n\\begin{mainthm}[\\ref{prp:ASHtenPure}]\\label{prp:MainThm2}\n    Let $A$ be a pure \\ca{} satisfying (1) or (2) in Theorem \\ref{prp:MainThm1}, and let $B$ be a unital separable ASH-algebra. Then, $A\\otimes B$ is pure.\n\\end{mainthm}\n\nAs a particular example of this result, Theorem \\ref{prp:MainThm2} shows that the tensor product of any Villadsen algebra of type I with the reduced group \\ca{} of any free group is pure. It is currently unknown whether these specific examples are $\\mathcal{Z}$-stable (Question \\ref{qst:VillZstab}), highlighting the utility of pureness as a regularity invariant. In fact, Theorem \\ref{prp:MainThm2} is slightly more general: As we prove in Theorem \\ref{DpureRSHA}, any recursive subhomogeneous \\ca{} over $A$ \\cite[Definition~3.2]{archey2020structure} (Definition \\ref{RSHA}) is pure whenever $A$ is.\n\nA further application of Theorem \\ref{prp:MainThm2} arises in the study of determining which nonamenable groups have a reduced group \\ca{} with strict comparison. Since $C_r^*(G)$ is a direct sum of simple pure $\\mathrm{C}^*$-algebras whenever $G$ is countable and acylindrically hyperbolic \\cite[Proof of Theorem~C]{FloLisCobPag:PureTwi}, and $C_r^*(H)$ is subhomogeneous whenever $H$ is virtually abelian, we get:\n\n\\begin{maincor}\\label{prp:MainCor1}\n Let $G$ be a countable acylindrically hyperbolic group, and let $H$ be a virtually abelian group. Then, $C_r^*(G\\times H)$ is pure. In particular, it has strict comparison.\n\\end{maincor}\n\nIn addition to significantly enlarging the known class of pure $\\mathrm{C}^*$-algebras, Theorems \\ref{prp:MainThm1} and \\ref{prp:MainThm2} shed new light on the \\emph{Global Glimm Problem}, a major open problem in the field \\cite[Problem~LXXIII]{SchTikWhi99arX:Prob}. The problem asks whether the \\emph{Global Glimm Property} ---a generalization of non-elementariness for non-simple $\\mathrm{C}^*$-algebras \\cite{KirRor02InfNonSimpleCalgAbsOInfty} which we recall in Paragraph \\ref{pgr:GGP}--- is equivalent to having no nonzero elementary ideals of quotients, and has been solved affirmatively in numerous cases (see \\cite[Section~4]{Vil25:IntroGGP} for an overview). While it was previously known that $C(X, A)$ possesses the Global Glimm Property whenever $X$ is finite-dimensional and $A$ is simple and has the property \\cite[Theorem~4.3]{BlaKir04GlimmHalving}, this was not known for pure $\\mathrm{C}^*$-algebras $A$ satisfying condition (2) of Theorem \\ref{prp:MainThm1}, nor for the tensor products appearing in Theorem \\ref{prp:MainThm2}. As we explain below, this is not a consequence of our results; rather, it is a necessary intermediate step in our proof of pureness.\n\nApart from the applications presented in this paper, Theorem \\ref{prp:MainThm1} has further consequences. In particular, this result will be used in forthcoming work of the first named author with Evington, Hua, Schafhauser, and White, which will study \\emph{$K$-stability} for pure $\\mathrm{C}^*$-algebras satisfying conditions as in Theorem~\\ref{prp:MainThm1}. It has been shown in \\cite{lin2025strict} that finite simple pure $\\mathrm{C}^*$-algebras have stable rank one and are therefore $K_1$-bijective. The future work will focus on the stronger property of $K$-stability for this class.\n\n\\subsection*{Strategy of the proof} Given a \\ca{} $A$ with strict comparison, it is well-known that $C(X,A)$ need not have comparison. Topological obstructions, often manifesting as non-trivial vector bundles, can impede comparison even in simple settings ---for example, when $A=\\mathbb{C}$ and $X=\\mathbb{T}^4$ \\cite{Vil98SimpleCaPerforation}. Thus, one of the challenges in establishing Theorem \\ref{prp:MainThm1} is that strict comparison and almost divisibility cannot be verified separately.\n\nOur strategy relies on a multi-step dimension reduction argument. First, we restrict our attention to the case where $X$ is finite-dimensional. In Sections \\ref{sec:CompCuA} and \\ref{sec:DivCuA}, we prove that if $X$ is a compact metric space of dimension $m$ and $A$ is pure, the algebra $C(X, A)$ satisfies a weakening of $(m,n)$-pureness, with $n$ depending only on $m$. Concretely, we see that $C(X,A)$ has $m$-comparison (Proposition \\ref{comp_c(x,A)}) and satisfies a \\emph{weak} version of $n$-almost divisibility (Corollary \\ref{robert_analogy_2}); see Definitions \\ref{dfn:MComp} and \\ref{def_almost_divisibility} for details.\n\nAs Corollary \\ref{robert_analogy_2} does not establish $n$-almost divisibility for any $n$, the rigidity results from \\cite{AntPerThiVil24arX:PureCAlgs} cannot be applied directly. Instead, building on the techniques from \\cite{AntPerThiVil24arX:PureCAlgs}, we prove a new general reduction result (Theorem \\ref{prp:Gen_DimRed}), which we then use in Theorem \\ref{prp:CXAPureIfGGP} to deduce that $C(X, A)$ is pure if and only if it possesses the Global Glimm Property. Relying on the fact that all these properties pass to inductive limits, we are able to dispense with the assumption of finite dimensionality on $X$. Thus, the proof of Theorem \\ref{prp:MainThm1} reduces to proving the Global Glimm Property for $C(X,A)$. For $\\mathrm{C}^*$-algebras satisfying condition (1), this follows from \\cite[Theorem~4.3]{BlaKir04GlimmHalving}; for those satisfying (2), the result is new and is proven in Proposition \\ref{prp:CXNoQuot}.\n\n\\subsection*{Acknowledgments} This paper started during EV's visit to the University of Oxford during the \\emph{Mini Course: Topological Phenomena in the Cuntz semigroup}, which was given by Andrew Toms and organized by Stuart White.  We are grateful to both, as well as to the University of Oxford and St John’s College, for their kind hospitality. We would further like to thank Stuart White for his helpful feedback on the paper, and Hannes Thiel for pointing out Corollary \\ref{prp:MainCor1}. AS is  grateful to N. Christopher Phillips for valuable discussions concerning pureness of $\\mathrm{C}^*$-algebras and for feedback on earlier drafts of this paper. AS also thanks Julian Buck, Dawn Archey, and Javad Mohammadkarimi for insightful discussions.", "sketch": "The proof of Theorem \\ref{prp:MainThm1} addresses the issue that, although $A$ has strict comparison, it is \"well-known that $C(X,A)$ need not have comparison\" due to \"topological obstructions\" (e.g., non-trivial vector bundles), so \"strict comparison and almost divisibility cannot be verified separately.\"\n\nThe strategy is a \"multi-step dimension reduction argument\":\n\\begin{itemize}\n\\item First reduce to $X$ finite-dimensional: for $\\dim(X)=m$ and $A$ pure, in Sections \\ref{sec:CompCuA} and \\ref{sec:DivCuA} they show $C(X,A)$ satisfies a weakening of $(m,n)$-pureness (with $n$ depending only on $m$), namely that $C(X,A)$ has \"$m$-comparison\" (Proposition \\ref{comp_c(x,A)}) and a \"\\emph{weak} version of $n$-almost divisibility\" (Corollary \\ref{robert_analogy_2}).\n\\item Since Corollary \\ref{robert_analogy_2} \"does not establish $n$-almost divisibility for any $n$\", the rigidity results of \\cite{AntPerThiVil24arX:PureCAlgs} \"cannot be applied directly.\" Instead, \"building on the techniques\" there, they prove \"a new general reduction result\" (Theorem \\ref{prp:Gen_DimRed}).\n\\item Use this to show (Theorem \\ref{prp:CXAPureIfGGP}) that $C(X,A)$ is pure \"if and only if it possesses the Global Glimm Property.\"\n\\item \"Relying on the fact that all these properties pass to inductive limits,\" remove the finite-dimensionality hypothesis on $X$.\n\\item Conclude that proving Theorem \\ref{prp:MainThm1} reduces to proving the Global Glimm Property for $C(X,A)$: under condition (1) this follows from \\cite[Theorem~4.3]{BlaKir04GlimmHalving}, while under condition (2) it is new and is proved in Proposition \\ref{prp:CXNoQuot}.\n\\end{itemize}", "expanded_sketch": "To prove the main theorem, the proof addresses the issue that, although $A$ has strict comparison, it is \"well-known that $C(X,A)$ need not have comparison\" due to \"topological obstructions\" (e.g., non-trivial vector bundles), so \"strict comparison and almost divisibility cannot be verified separately.\"\n\nThe strategy is a \"multi-step dimension reduction argument\":\n\\begin{itemize}\n\\item First reduce to $X$ finite-dimensional: for $\\dim(X)=m$ and $A$ pure, they show that $C(X,A)$ satisfies a weakening of $(m,n)$-pureness (with $n$ depending only on $m$), namely the following.\n\n\\begin{prop}\\label{comp_c(x,A)}\n    Let $X$ be a compact metric space of covering dimension $m$, and let $A$ be a \\ca{} with strict comparison. Then $C(X, A)$ has $m$-comparison.\n\\end{prop}\n\nMoreover, they prove the following weak form of almost divisibility.\n\n\\begin{cor}\\label{robert_analogy_2}\n    Let $X$ be a finite-dimensional compact metric space, and let $A$ be an almost divisible \\ca{}. Then, there exists $M\\in\\N$ such that, for every $N\\geq 1$ and every pair $x'\\ll x$ in $\\Cu (C(X,A))$, there exists $y\\in\\Cu (C(X,A))$ such that\n    \\[\n        x'\\ll Ny\\ll Mx.\n    \\]\n\\end{cor}\n\n\\item Since the corollary above \"does not establish $n$-almost divisibility for any $n$\", the rigidity results of AntPerThiVil24arX:PureCAlgs \"cannot be applied directly.\" Instead, \"building on the techniques\" there, they prove the following new general reduction result.\n\n\\begin{theorem}\\label{prp:Gen_DimRed}\n    Let $A$ be a \\ca{}, and let $m,M\\in\\mathbb{N}$. Assume that $A$ has $m$-comparison and that, for every $N\\geq 1$ and every pair of elements $x', x\\in \\Cu (A)$ such that $x'\\ll x$, there exists $y\\in\\mathrm{Cu} (A)$ satisfying \n    \\[\n        x'\\ll Ny\\ll M x.\n    \\]\n\n    Then, the following are equivalent:\n    \\begin{itemize}\n        \\item[(i)] $A$ has the Global Glimm Property;\n        \\item[(ii)] there exists $L\\in\\N$ such that, for every pair $x',x\\in\\Cu (A)$ with $x'\\ll x$, there exist $y_0,y_1\\in\\mathrm{Cu} (A)$ satisfying\n        \\[\n        y_0+y_1\\leq x,\\quad\\text{and}\\quad \n        x'\\ll Ly_0, Ly_1;\n        \\]\n        \\item[(iii)] $A$ is pure.\n    \\end{itemize}\n\\end{theorem}\n\n\\item Use this to show the following characterization of purity for $C(X,A)$ in terms of the Global Glimm Property.\n\n\\begin{theorem}\\label{prp:CXAPureIfGGP}\n    Let $X$ be a finite-dimensional compact metric space and let $A$ be a pure \\ca{}. Then, $C(X,A)$ is pure if and only if $C(X,A)$ has the Global Glimm Property.\n\\end{theorem}\n\n\\item \"Relying on the fact that all these properties pass to inductive limits,\" remove the finite-dimensionality hypothesis on $X$.\n\n\\item Conclude that proving the main theorem reduces to proving the Global Glimm Property for $C(X,A)$: under condition (1) this follows from \\cite[Theorem~4.3]{BlaKir04GlimmHalving}, while under condition (2) it is new and is proved in the following proposition.\n\n\\begin{prop}\\label{prp:CXNoQuot}\n    Let $X$ be a finite-dimensional compact metric space, and let $A$ be a pure \\ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, there exists $L\\in\\N$ such that, for every pair $x',x\\in\\Cu (C(X,A))$ with $x'\\ll x$, there exist $y_0,y_1\\in\\mathrm{Cu} (C(X,A))$ satisfying\n        \\[\n        y_0+y_1\\leq x,\\quad\\text{and}\\quad \n        x'\\ll Ly_0, Ly_1.\n        \\]\n\\end{prop}\n\\end{itemize}", "expanded_theorem": "\\begin{cor}\\label{Purehomogenous}\n    Let $A$ be a simple pure \\ca{}, and let $X$ be a compact metric space. Then, $C(X,A)$ is pure.\n\\end{cor} To prove the main theorem, we use the following result.\n\n\\begin{theorem}\\label{prp:MainNonSimp}\n    Let $X$ be a compact metric space, and let $A$ be a pure \\ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, $C(X,A)$ is pure.\n\\end{theorem}\n\n\\label{prp:MainThm1}\n    Let $X$ be a compact metric space, and let $A$ be a pure \\ca{}. Assume additionally that\n    \\begin{enumerate}\n        \\item $A$ is simple; or \n        \\item every quotient of $A$ is stably finite.\n    \\end{enumerate}\n\n    Then, $C(X,A)$ is pure.", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let \\(X\\) be a compact metric space, and let \\(A\\) be a pure \\(\\mathrm{C}^*\\)-algebra, where \"pure\" means that \\(A\\) has strict comparison and almost divisibility. Assume in addition that either \\(A\\) is simple, or every quotient \\(\\mathrm{C}^*\\)-algebra of \\(A\\) is stably finite. Here \\(C(X,A)\\) denotes the \\(\\mathrm{C}^*\\)-algebra of continuous \\(A\\)-valued functions on \\(X\\). Which statement holds for every such pair \\((X,A)\\)?", "correct_choice": {"label": "A", "text": "\\(C(X,A)\\) is pure; equivalently, the \\(\\mathrm{C}^*\\)-algebra of continuous \\(A\\)-valued functions on \\(X\\) has strict comparison and almost divisibility."}, "choices": [{"label": "B", "text": "\\(C(X,A)\\) has strict comparison, but it need not be almost divisible in general; thus one can only conclude a comparison property for \\(C(X,A)\\), not pureness."}, {"label": "C", "text": "\\(C(X,A)\\) has strict comparison."}, {"label": "D", "text": "\\(C(X,A)\\) is pure whenever \\(X\\) is finite-dimensional; for arbitrary compact metric \\(X\\), the conclusion can fail even if \\(A\\) is simple or every quotient of \\(A\\) is stably finite."}, {"label": "E", "text": "\\(C(X,A)\\) is pure provided both that \\(A\\) is simple and that every quotient \\(\\mathrm{C}^*\\)-algebra of \\(A\\) is stably finite; assuming only one of these conditions is not sufficient."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "dimension-reduction needed to recover almost divisibility jointly with comparison", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped almost divisibility/pureness and kept only strict comparison", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "removal of finite-dimensionality via inductive limits", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "logical disjunction of hypotheses (simple OR all quotients stably finite)", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the conclusion. It states the hypotheses and asks which consequence follows, so there is no direct answer leakage, though it is clearly theorem-shaped."}, "TAS": {"score": 0, "justification": "This is essentially a direct restatement/application of a permanence theorem: under the listed hypotheses, the correct choice repeats the theorem’s conclusion that C(X,A) is pure."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because one distractor (C) is a weaker true-looking statement, so the test-taker must identify the strongest valid conclusion. Still, the item mainly checks recall of the theorem rather than deeper generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one weakens the conclusion to strict comparison only, one inserts an unnecessary finite-dimensionality restriction, and one incorrectly strengthens the hypothesis from an OR to an AND. These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A reasonably well-constructed theorem-recall MCQ with strong distractors, but it is largely tautological because it asks for the exact conclusion of the stated result rather than eliciting substantial generative reasoning."}}
{"id": "2602.14864v1", "paper_link": "http://arxiv.org/abs/2602.14864v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "(Theorem \\ref{KresThm}, Proposition  \\ref{finiterepprop}, Theorem \\ref{egvaluethm})\n(1) The representation \n$\\tau|_M$ is\nmultiplicity-free if and only if \n$(K, \\tau)$ is in the list in Theorem \\ref{KresThm}.\n\n(2) Let $(K, \\tau)$ be as\nin (1) acting on $V_\\tau$. For any finite-dimensional irreducible representation\n$(W_\\lambda, G^{\\mathbb{C}})$ there is up to scalar at most one $G$-equivariant map into $C^\\infty(G/K, V_\\tau)$.\n\n(3)\nLet $(K, \\tau)\n$ be as in (1), $\\sigma$\nan irreducible representation in  $\\tau|_M$\nand \n$(W_\\lambda, G^{\\mathbb C})$ \nas\nin (2). Let $D$ \nbe an invariant\ndifferential operator\non $C^\\infty(G/K, V_\\tau)$.\nWrite   the eigenvalues \nof  $D$ on $W_\\lambda$\nas $\\omega(D)(\\sigma_{\\lambda},-\\lambda|_{\\mathfrak{a}}\n)$\nwhich can be extended uniquely to a polynomial of $\n\\lambda|_{\\mathfrak{a}}\n\\in \n(\\mathfrak a^{\\mathbb C})'\n$.\nThen the eigenvalue\n$\\omega(D)$ has following\ninvariance \nproperty,\n$$\\omega(D)(\\sigma,\\kappa + \\rho_{\\mathfrak{a}}) = \\omega(D)(s \\cdot \\sigma,s \\cdot \\kappa + \\rho_{\\mathfrak{a}}), \\ \\forall \\ \\kappa \\in \n(\\mathfrak a^{\\mathbb C})', s\\in W.$$", "start_pos": 5241, "end_pos": 6365, "label": null}, "ref_dict": {"KresThm": "\\begin{theorem}\n\\label{KresThm}\nLet $(G, K)$ be an irreducible Hermitian symmetric pair\nand\n $M = C_K(\\mathfrak{a})$\n be the centralizer of the\n Cartan subalgebra \n $\\mathfrak{a}\\subset \\mathfrak p$ in $K$. \nThe complete list of $(K, \\tau)$ with $\\tau|_M$ being multiplicity-free is given as follows:\n\n\\begin{enumerate}\n    \\item Type A, \n    $G = SU(r+b,r), \n    K = S(U(r+b) \\times U(r))$; irreducible representations of $U(n)$ with highest\n    weights $\\mu$\n    will be denoted by \n    $\\pi^{\\mu}_n$ as  in \\cite[Theorem 5.5.22]{good}. \n\\begin{enumerate}\n    \\item $r=1$. $\\tau$ is arbitrary.\n\\item  $r=2$. $\\tau= \\pi^{\\mu}_{2+b} \\otimes \\pi^{\\nu}_2$ where $\\pi_{2+b}^{\\mu}$ is\na character and $\\pi_2^{\\nu}$ is any representations, or $\\pi_2^{\\nu}$ is \na character and\n$$\n\\mu = (\\overbrace{l_1+l_2,\\cdots, l_1+l_2}^{{j }},\\overbrace{l_2,\\cdots,l_2}^{{b+2-j }}).\n$$\n\n\\item  $r \\geq 3$. $\\tau= \\pi^{\\mu}_{2+b} \\otimes \\pi^{\\nu}_2$ where $\\pi_{r+b}^{\\mu}$\nor $\\pi_r^{\\nu}$ is a character and the other must be the symmetric\ntensor powers\n$S^m(\\mathbb C^{p})$ \nor  an exterior power\n$\n\\wedge^j \\mathbb C^p$ when restricted to $SL(p)$\n($p=r+b$ or $p=r$\naccordingly).\n\\end{enumerate} \n\n\\item Type C,  $G=Sp(n,\\R)$, $K = U(n)$. $\\tau$ is a character \nor its tensor product\nwith  the exterior power representations\n$\n\\wedge^j \\mathbb C^n$. The same holds for the double cover $(G,K) = (Mp(n,\\R),\\widehat{U(n)})$.\n\n\\item Type D,  $G = SO^*(2n), K = U(n)$. $\\tau$ \nis a character\nor its tensor product\nwith  the symmetric tensor\npowers\n$S^m(\\mathbb C^{p})$  or their duals.\n\n\\item Type BD,  $G=SO_0(n,2)$, \n $K = SO(n) \\times SO(2)$, $n>4$;\nirreducible \nrepresentations\nof $SO(n)$\nwill be denoted\nby \n$\\pi_{n}^{\\mu}$\nas in \\cite[Theorem 19.22]{FulHar}.\n $\\tau=\\pi_{n}^{0}\\otimes \\pi_{2}^\\nu$ \n if  $n$ is odd, i.e., $\\tau$ is a character; for $n$ even $\\tau=\n \\pi_{n}^{\\mu}$ \n or  its \n tensor product\n with a character\n$\\tau=\\pi_{n}^{\\mu}\n\\otimes \\pi_{2}^\\nu$,  \n where $\\mu=(a,\\dots,a,\\pm a)$ for some $a\\ge 0$.\nIf $(G,K) = (Spin_0(n,2), \nSpin(n) \\times SO(2)/\\Z_2)$, where $Spin_0(n,2)$ is the double cover of $SO_0(n,2)$\nthen up to a character $\\tau$ is the spin representation $(\\frac{1}{2}, \\dots, \\frac{1}{2})$ when $n$ is odd, and\n$\\tau= \\pi_{n}^{\\mu}$, $\\mu=(a,\\dots,a,\\pm a)$ for some integer $a\\ge 0$ a half-integer\nwhen $n$ is even.\n\n\\item Type $E_6$, $G=E_{6(-14)}, K = Spin(10) \\times U(1) / \\Gamma$, where $\\Gamma$ is a finite subgroup.\n$\\tau=\\pi_{\\omega_1} \\otimes \\chi_n$ for  $n \\equiv 1 \\mod 4$, or $\\pi_{\\omega_2} \\otimes \\chi_n$ for $n \\equiv 3 \\mod 4$. Here $\\omega_1$ and $\\omega_2$ are the positive and negative spin representations of $Spin(10)$ and $U(1) = \\{e^{it \\frac{Z}{2}} \\ | \\ t \\in \\R \\}$.\n\n\\item Type $E_7$,  $G=E_{7(-25)}, K = E_6  U(1)$.\n$\\tau$ is a character.\n\n\\end{enumerate} \n\\end{theorem}", "finiterepprop": "\\begin{proposition}\n\\label{finiterepprop}\n(1) Let $(W_{\\lambda}, G^{\\C})$ be a highest weight representation of $G^{\\C}$ and its Lie algebra $\\mathfrak{g}^{\\C}$. For the multiplicities we have $[V_{\\tau} : W_{\\lambda}|_K] \\leq 1$, and if $V_{\\tau} \\subseteq W_{\\lambda}|_K$ we have $[W_{\\lambda} : I_{\\sigma,-\\lambda|_{\\af} - \\rho_{\\af}}^0 ] \\leq 1$, viewed as Lie algebra representations.  \n\n(2)\nLet $(W_\\lambda, G ^{\\C})$ be\na finite-dimensional\nirreducible representation\nsuch that \n$V_{\\tau} \\subseteq W_{\\lambda}|_K$. Up to  constants there is a unique $G$-equivariant map\n$$F : W_{\\lambda} \\rightarrow C^{\\infty}(G/K,V_{\\tau})$$\ndefined by\n$$F(w)(g) = P_{\\tau} (g^{-1} \\cdot w), g\\in G, w\\in W,$$\nwhere $P_{\\tau}$ is the projection onto $V_{\\tau}$. Moreover, if we normalize $J_{\\lambda}$ so that for all $v \\in V_{\\tau}$ we have $J_{\\lambda}(v) = f_v$, where $f_v(kan) =  a^{\\lambda|_{\\mathfrak{a}}} P_{\\sigma} \\tau(k)^{-1} v$,\nthen there is a factorization\nof $F$ \nas a product $G$-equivariant maps,\n$$\nF=\nS_{\\sigma, -\\lambda|_{\\mathfrak{a}} - \\rho_{\\mathfrak{a}}}\nJ_{\\lambda}: (W_\\lambda, G^{\\C})\\to I_{\\sigma, -\\lambda|_{\\mathfrak{a}} - \\rho_{\\mathfrak{a}}}^0\n\\to C^\\infty(G/K,V_{\\tau}).\n$$\n\\end{proposition}", "egvaluethm": "\\begin{theorem}\n\\label{egvaluethm}\nAssume the irreducible finite-dimensional  representation $V_\\tau$  \nof $K$ is  multiplicity-free  when  restricted to  $M$.\n\n(1) For any $M$-irreducible subrepresentation $(\\sigma,U_\\sigma)$ in $\\tau|_M$,  $f \\in I_{\\sigma,\\nu}^0$ and $D \\in \\mathcal{D}^G(G/K, V_{\\tau})$, we have $DS_{\\sigma, \\nu}(f) = \\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}})S_{\\sigma, \\nu}(f)$ for a constant $\\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}})$.\nFurthermore, we have the invariance property:\n$$\\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}}) = \\omega(D)(s \\cdot \\sigma,s \\cdot \\nu + \\rho_{\\mathfrak{a}}), \\ \\forall \\ s\\in W.$$\n\n(2) Let $(W_\\lambda, G^{\\C})$\nbe a  finite-dimensional\nrepresentation of $\nG^{\\C}$ containing $V_\\tau$.  Let $D \\in \\mathcal{D}^G(G/K,V_{\\tau})$, then $D$ acts on \n$F(W_\\lambda)$\nas the constant\n$$\nDF(w) = \\omega(D)(\\sigma_{\\lambda},-\\lambda|_{\\mathfrak{a}})F(w), \n\\quad  \\ \\forall \\ w \\in W_\\lambda.\n$$\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2244, "pre_theorem_intro_text": "In the present paper we shall study invariant differential operators on homogeneous vector bundles over Hermitian symmetric domains, their eigenfunctions and  eigenvalues. \nLet $G$ be a\nnon-compact semisimple \nHermitian Lie group and $K$ its maximal compact subgroup. Let\n$V_\\tau$\nbe a finite-dimensional irreducible \nrepresentation of  $K$ and $C^\\infty(G/K,V_{\\tau})$ \nbe the space of smooth sections \nof the homogeneous vector \nbundle \n$G\\times_{(K, \\tau)} V_\\tau$\ndefined by $V_\\tau$.\nThe algebraic properties\nof \nthe ring\n$\\mathcal D^G(G/K,V_{\\tau})$ \nof invariant\ndifferential operators on\n$C^\\infty(G/K,V_{\\tau})$ is of fundamental interest, both in analysis on symmetric spaces \n and in the study of universal\n enveloping algebras \\cite{deitmar, helg, ricc, Sh}.\nWhen $(K, \\tau)$\nis a character, i.e., a one-dimensional\nrepresentation, the ring is commutative, and \nShimura has constructed\na linear basis \n$\\{\\mathcal L_{\\mu}\\}$\nconsisting of formally positive \noperators\nby using the Schmid decomposition\nof the symmetric tensor\nalgebra\n$S(\\mathfrak p^+)$\nunder $K$, where \n$\\mathfrak p^+$\nis the holomorphic tangent space of $G/K$\nat $o=K\\in G/K$.\nFor general \n$V_\\tau$ it is proved by Deitmar\n\\cite{deitmar} \nthat \n$\\mathcal D^G(G/K, V_\\tau)$\nis commutative\nif and only if \nthe restriction  $\\tau|_M$\nto $M$ is multiplicity-free, where $M$\nis the centralizer in $K$ of the real Cartan group of $G$. \nIn the present paper we shall\nclassify all such representations\n$(K, \\tau)$. \nWe construct \n eigenfunctions of invariant\ndifferential operators\nusing the Szegö transform and\nwe study the invariance\nproperties of the eigenvalues\nof invariant differential\noperators. This question can be posed for any symmetric space, we study the Hermitian case because there are canonical constructions of invariant differential operators related to the Hua-Kostant-Schmid decomposition \\cite{Sh} of invariant differential operator. It is an interesting problem to further study the eigenvalue problem for those operators;\nsee  \\cite{SZ}\nfor the case of one-dimensional representations $V_\\tau$ of $K$. \n\n\\subsection{Main\nresults and methods}\n\nThe main result of this paper\nis the following theorem; see\nbelow for the exact notation\nand definitions.", "context": "In the present paper we shall study invariant differential operators on homogeneous vector bundles over Hermitian symmetric domains, their eigenfunctions and eigenvalues. \nLet $G$ be a\nnon-compact semisimple \nHermitian Lie group and $K$ its maximal compact subgroup. Let\n$V_\\tau$\nbe a finite-dimensional irreducible \nrepresentation of $K$ and $C^\\infty(G/K,V_{\\tau})$ \nbe the space of smooth sections \nof the homogeneous vector \nbundle \n$G\\times_{(K, \\tau)} V_\\tau$\ndefined by $V_\\tau$.\nThe algebraic properties\nof \nthe ring\n$\\mathcal D^G(G/K,V_{\\tau})$ \nof invariant\ndifferential operators on\n$C^\\infty(G/K,V_{\\tau})$ is of fundamental interest, both in analysis on symmetric spaces \n and in the study of universal\n enveloping algebras \\cite{deitmar, helg, ricc, Sh}.\nWhen $(K, \\tau)$\nis a character, i.e., a one-dimensional\nrepresentation, the ring is commutative, and \nShimura has constructed\na linear basis \n$\\{\\mathcal L_{\\mu}\\}$\nconsisting of formally positive \noperators\nby using the Schmid decomposition\nof the symmetric tensor\nalgebra\n$S(\\mathfrak p^+)$\nunder $K$, where \n$\\mathfrak p^+$\nis the holomorphic tangent space of $G/K$\nat $o=K\\in G/K$.\nFor general \n$V_\\tau$ it is proved by Deitmar\n\\cite{deitmar} \nthat \n$\\mathcal D^G(G/K, V_\\tau)$\nis commutative\nif and only if \nthe restriction $\\tau|_M$\nto $M$ is multiplicity-free, where $M$\nis the centralizer in $K$ of the real Cartan group of $G$. \nIn the present paper we shall\nclassify all such representations\n$(K, \\tau)$. \nWe construct \n eigenfunctions of invariant\ndifferential operators\nusing the Szegö transform and\nwe study the invariance\nproperties of the eigenvalues\nof invariant differential\noperators. This question can be posed for any symmetric space, we study the Hermitian case because there are canonical constructions of invariant differential operators related to the Hua-Kostant-Schmid decomposition \\cite{Sh} of invariant differential operator. It is an interesting problem to further study the eigenvalue problem for those operators;\nsee \\cite{SZ}\nfor the case of one-dimensional representations $V_\\tau$ of $K$.\n\n\\subsection{Main\nresults and methods}\n\nThe main result of this paper\nis the following theorem; see\nbelow for the exact notation\nand definitions.", "full_context": "In the present paper we shall study invariant differential operators on homogeneous vector bundles over Hermitian symmetric domains, their eigenfunctions and eigenvalues. \nLet $G$ be a\nnon-compact semisimple \nHermitian Lie group and $K$ its maximal compact subgroup. Let\n$V_\\tau$\nbe a finite-dimensional irreducible \nrepresentation of $K$ and $C^\\infty(G/K,V_{\\tau})$ \nbe the space of smooth sections \nof the homogeneous vector \nbundle \n$G\\times_{(K, \\tau)} V_\\tau$\ndefined by $V_\\tau$.\nThe algebraic properties\nof \nthe ring\n$\\mathcal D^G(G/K,V_{\\tau})$ \nof invariant\ndifferential operators on\n$C^\\infty(G/K,V_{\\tau})$ is of fundamental interest, both in analysis on symmetric spaces \n and in the study of universal\n enveloping algebras \\cite{deitmar, helg, ricc, Sh}.\nWhen $(K, \\tau)$\nis a character, i.e., a one-dimensional\nrepresentation, the ring is commutative, and \nShimura has constructed\na linear basis \n$\\{\\mathcal L_{\\mu}\\}$\nconsisting of formally positive \noperators\nby using the Schmid decomposition\nof the symmetric tensor\nalgebra\n$S(\\mathfrak p^+)$\nunder $K$, where \n$\\mathfrak p^+$\nis the holomorphic tangent space of $G/K$\nat $o=K\\in G/K$.\nFor general \n$V_\\tau$ it is proved by Deitmar\n\\cite{deitmar} \nthat \n$\\mathcal D^G(G/K, V_\\tau)$\nis commutative\nif and only if \nthe restriction $\\tau|_M$\nto $M$ is multiplicity-free, where $M$\nis the centralizer in $K$ of the real Cartan group of $G$. \nIn the present paper we shall\nclassify all such representations\n$(K, \\tau)$. \nWe construct \n eigenfunctions of invariant\ndifferential operators\nusing the Szegö transform and\nwe study the invariance\nproperties of the eigenvalues\nof invariant differential\noperators. This question can be posed for any symmetric space, we study the Hermitian case because there are canonical constructions of invariant differential operators related to the Hua-Kostant-Schmid decomposition \\cite{Sh} of invariant differential operator. It is an interesting problem to further study the eigenvalue problem for those operators;\nsee \\cite{SZ}\nfor the case of one-dimensional representations $V_\\tau$ of $K$.\n\n\\subsection{Main\nresults and methods}\n\nThe main result of this paper\nis the following theorem; see\nbelow for the exact notation\nand definitions.\n\nIn the present paper we shall study invariant differential operators on homogeneous vector bundles over Hermitian symmetric domains, their eigenfunctions and eigenvalues. \nLet $G$ be a\nnon-compact semisimple \nHermitian Lie group and $K$ its maximal compact subgroup. Let\n$V_\\tau$\nbe a finite-dimensional irreducible \nrepresentation of $K$ and $C^\\infty(G/K,V_{\\tau})$ \nbe the space of smooth sections \nof the homogeneous vector \nbundle \n$G\\times_{(K, \\tau)} V_\\tau$\ndefined by $V_\\tau$.\nThe algebraic properties\nof \nthe ring\n$\\mathcal D^G(G/K,V_{\\tau})$ \nof invariant\ndifferential operators on\n$C^\\infty(G/K,V_{\\tau})$ is of fundamental interest, both in analysis on symmetric spaces \n and in the study of universal\n enveloping algebras \\cite{deitmar, helg, ricc, Sh}.\nWhen $(K, \\tau)$\nis a character, i.e., a one-dimensional\nrepresentation, the ring is commutative, and \nShimura has constructed\na linear basis \n$\\{\\mathcal L_{\\mu}\\}$\nconsisting of formally positive \noperators\nby using the Schmid decomposition\nof the symmetric tensor\nalgebra\n$S(\\mathfrak p^+)$\nunder $K$, where \n$\\mathfrak p^+$\nis the holomorphic tangent space of $G/K$\nat $o=K\\in G/K$.\nFor general \n$V_\\tau$ it is proved by Deitmar\n\\cite{deitmar} \nthat \n$\\mathcal D^G(G/K, V_\\tau)$\nis commutative\nif and only if \nthe restriction $\\tau|_M$\nto $M$ is multiplicity-free, where $M$\nis the centralizer in $K$ of the real Cartan group of $G$. \nIn the present paper we shall\nclassify all such representations\n$(K, \\tau)$. \nWe construct \n eigenfunctions of invariant\ndifferential operators\nusing the Szegö transform and\nwe study the invariance\nproperties of the eigenvalues\nof invariant differential\noperators. This question can be posed for any symmetric space, we study the Hermitian case because there are canonical constructions of invariant differential operators related to the Hua-Kostant-Schmid decomposition \\cite{Sh} of invariant differential operator. It is an interesting problem to further study the eigenvalue problem for those operators;\nsee \\cite{SZ}\nfor the case of one-dimensional representations $V_\\tau$ of $K$.\n\nThe main result of this paper\nis the following theorem; see\nbelow for the exact notation\nand definitions.\n\nThe proof of the \nclassification\nis through a case by case\ncomputation\nand is technically involved.\nWe shall repeatedly use the classical branching rules in\n\\cite{good},\nthe classifications\nof weight-free\nrepresentations\nin \\cite{Howe}\nand multiplicity-free results in\n\\cite{St}.\nThe proof \nfor the uniqueness\nof $F: W_\\lambda \\to \nC^\\infty(G/K, V_\\tau)\n$ is somewhat\nnatural,\nit is done through\nthe realization\n$J: W_\\lambda \n\\to Ind_{MAN}^G(\\sigma\\otimes \\alpha\\otimes 1)$\nof finite-dimensional\nrepresentations $W_\\lambda$\nin the induced representation\nin $Ind_{MAN}^G(\\sigma\\otimes \\alpha\\otimes 1)$,\na factorization\nof $F=SJ$\nas a product of $J$\nand the Poisson-Szeg\\\"o{}\ntransform, and the multiplicity-free property of $V_\\tau$\nunder $M$. The result\non the invariance\nof eigenvalues\nof differential\noperators is \na consequence of a\ngeneral result of Lepowsky \\cite{L}\nand the factorization.\n\n(2)\nLet $(W_\\lambda, G ^{\\C})$ be\na finite-dimensional\nirreducible representation\nsuch that \n$V_{\\tau} \\subseteq W_{\\lambda}|_K$. Up to constants there is a unique $G$-equivariant map\n$$F : W_{\\lambda} \\rightarrow C^{\\infty}(G/K,V_{\\tau})$$\ndefined by\n$$F(w)(g) = P_{\\tau} (g^{-1} \\cdot w), g\\in G, w\\in W,$$\nwhere $P_{\\tau}$ is the projection onto $V_{\\tau}$. Moreover, if we normalize $J_{\\lambda}$ so that for all $v \\in V_{\\tau}$ we have $J_{\\lambda}(v) = f_v$, where $f_v(kan) = a^{\\lambda|_{\\mathfrak{a}}} P_{\\sigma} \\tau(k)^{-1} v$,\nthen there is a factorization\nof $F$ \nas a product $G$-equivariant maps,\n$$\nF=\nS_{\\sigma, -\\lambda|_{\\mathfrak{a}} - \\rho_{\\mathfrak{a}}}\nJ_{\\lambda}: (W_\\lambda, G^{\\C})\\to I_{\\sigma, -\\lambda|_{\\mathfrak{a}} - \\rho_{\\mathfrak{a}}}^0\n\\to C^\\infty(G/K,V_{\\tau}).\n$$\n\\end{proposition}\n\nWe want to prove invariance properties on the eigenvalues of differential operators. These invariance properties have been used in \\cite{SZ} to compute the eigenvalues of Shimura differential operators for one-dimensional representations $\\tau$. We first state some facts derived from \\cite[Section 3]{L}. Note that from the Iwasawa decomposition for the Lie algebra we have the decomposition \n$$U(\\mathfrak g^{\\mathbb C}\n)\n=U(\\mathfrak a^{\\mathbb C}) U(\\mathfrak k^{\\mathbb C})\\oplus\n\\mathfrak n^{\\mathbb C}\nU(\\mathfrak g^{\\mathbb C}).$$\nLet \n$$\n\\Omega:U( \\mathfrak g^{\\mathbb C})\\rightarrow U( \\mathfrak a^{\\mathbb C}) U(\\mathfrak k^{\\mathbb C})\n$$\nbe the correponding\nprojection. We give $U(\\mathfrak a^{\\mathbb C}) U(\\mathfrak k^{\\mathbb C})$ an algebra structure by identifying it with the algebra $U(\\mathfrak a^{\\mathbb C}) \\otimes U(\\mathfrak k^{\\mathbb C})$ and regard $\\Omega$ as a map to $U( \\mathfrak a^{\\mathbb C}) \\otimes U(\\mathfrak k^{\\mathbb C})$. Then $\\Omega(uv)=\\Omega(v)\\Omega(u), $ for any $u \\in U(\\mathfrak g^{\\mathbb C}), v \\in (U(\\mathfrak g^{\\mathbb C})^{K}$ and $$\\Omega(U(\\mathfrak g^{\\mathbb C})^{M})\\subseteq U( \\mathfrak a^{\\mathbb C}) \\otimes U(\\mathfrak k^{\\mathbb C})^{M}.$$\nLet $T:U(\\mathfrak k^{\\mathbb C})\\rightarrow U(\n\\mathfrak k^{\\mathbb C})$ be the canonical anti-automorphism of $U(\\mathfrak k^{\\mathbb C})$ defined by :\n$$T(1)=1, T(x)=-x, T(xy)=T(y)T(x).$$\nPut $$\\Omega_{\\tau}=(1\\otimes \\tau) \\circ (1 \\otimes T)\\circ \\Omega$$\nThen $ \\Omega_{\\tau}:U(\\mathfrak g^{\\mathbb C})^{K}\\rightarrow \nU(\n\\mathfrak a^{\\mathbb C}\n)\\otimes \\mbox{End}_{M}V_{\\tau}$ is an algebra homomorphism.\nAny $\\lambda \\in \\mathfrak (a^{\\mathbb C})'$\ncan be extended to an evaluation on \n$\nU(\\mathfrak a^{\\mathbb C})$, which we also denote by $\\lambda$.\nLet\n$$\\Omega_{\\tau, \\lambda}=(\\lambda\\otimes 1)\\circ \\Omega_{\\tau}:U(\\mathfrak g^{\\mathbb C})^{K}\\rightarrow \\mbox{End}_{M}V_{\\tau}.$$\n\n\\begin{lemma}\n\\label{egvalue}\nFor $v \\in V_{\\tau}$ let $f_{v, \\lambda}\\in C^\\infty(G/K, V_\\tau)$\nbe defined by $f(x) = \\Psi_{\\tau, \\lambda}(x) v, x\\in G$,\nwhere\n$\\Psi_{\\tau, \\lambda}\n$ is defined in (\\ref{Psi}). For any $D \\in U(\\mathfrak{g}^{\\C})^{K}\n\\subset\n(\nU(\\mathfrak{g}^{\\C}\\otimes End(V_\\tau))^{K}\n$\nrealized \nas differential\noperator on $C^\\infty(G/K, V_\\tau)$\nvia $\\nabla(D)$ in\n(\\ref{diffopUnAlg})\nwe have $$(\\nabla(D)f_{v, \\lambda})(x)= \\Psi_{\\tau, \\lambda}(x) \\Omega_{\n\\tau, -\\lambda+\\rho_{\\mathfrak{a}}\n}(D)v, \\ \\forall \\ x \\in G.$$\n \\end{lemma}\n\n(1) For any $M$-irreducible subrepresentation $(\\sigma,U_\\sigma)$ in $\\tau|_M$, $f \\in I_{\\sigma,\\nu}^0$ and $D \\in \\mathcal{D}^G(G/K, V_{\\tau})$, we have $DS_{\\sigma, \\nu}(f) = \\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}})S_{\\sigma, \\nu}(f)$ for a constant $\\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}})$.\nFurthermore, we have the invariance property:\n$$\\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}}) = \\omega(D)(s \\cdot \\sigma,s \\cdot \\nu + \\rho_{\\mathfrak{a}}), \\ \\forall \\ s\\in W.$$\n\n\\begin{proof}(1) Recall that any invariant differential operator $D \\in \\mathcal{D}^G(G/K, V_{\\tau})$ can be represented by an element of $U(\\mathfrak{g}^{\\mathbb C})^K$ via\nthe natural\nmap (\\ref{diffopUnAlg});\nsee also \\cite[Proposition 2.1]{Sh},\nand \n$\\Omega_{\\nu,\\tau}(D)$ \nis well-defined using\nthe representative in $U(\\mathfrak{g}^{\\C})$.\nAs $\\tau|_M$ is multiplicity-free and $\\Omega_{-\\nu + \\rho_{\\mathfrak{a}}}(D)|_{U_{\\sigma}} \\in \\mbox{End}_{M}(U_\\sigma) = \\C I_{U_{\\sigma}}$, by Lemma \\ref{egvalue},\nwe have $D.S_{\\sigma, \\nu}(f) =\\Omega_{-\\nu + \\rho_{\\mathfrak{a}},U_{\\sigma}}(D)S_{\\sigma, \\nu}(f) = \\omega(D)(\\sigma, \\nu + \\rho_{\\mathfrak{a}}) S_{\\sigma,\\nu}(f)$.\nThe invariance property follows directly from\n\\cite[Theorem 9.8(2)]{L}.", "post_theorem_intro_text_len": 5239, "post_theorem_intro_text": "The proof of the \nclassification\nis through a case by case\ncomputation\nand is technically involved.\nWe shall repeatedly use the classical branching rules in\n\\cite{good},\nthe classifications\nof weight-free\nrepresentations\nin \\cite{Howe}\nand multiplicity-free results in\n\\cite{St}.\nThe proof \nfor the uniqueness\nof $F: W_\\lambda \\to \nC^\\infty(G/K, V_\\tau)\n$  is somewhat\nnatural,\nit is done through\nthe realization\n$J: W_\\lambda \n\\to Ind_{MAN}^G(\\sigma\\otimes \\alpha\\otimes 1)$\nof finite-dimensional\nrepresentations $W_\\lambda$\nin the induced representation\nin $Ind_{MAN}^G(\\sigma\\otimes \\alpha\\otimes 1)$,\na factorization\nof $F=SJ$\nas a product of $J$\nand the Poisson-Szeg\\\"o{}\ntransform, and  the multiplicity-free property of $V_\\tau$\nunder $M$. The result\non the invariance\nof eigenvalues\nof  differential\noperators is \na consequence of a\ngeneral result of  Lepowsky \\cite{L}\nand the factorization.\n\n\\subsection{Related\nresults and questions}\n\nWhen  $ \\tau$ is a character Shimura \\cite{Sh} \nhas constructed\na system of $r$ generators\nfor the ring $\\mathcal D^G(G/K, V_\\tau)$ \nand proposed several\nquestions on determining\nthe domains of positivity\nfor the eigenvalues\nof the generators\nand of the \nwhole linear basis\n$\\{\\mathcal L_{\\mu}\\}$\nof formally positive\noperators. A partial\nanswer to these\nquestions has been obtained\nin \\cite{SZ, Zhang}. Here invariance of eigenvalues under the Weyl group was used. \nWe may also pose similar\nquestions as above for $(K, \\tau)$\nin the list of our classification.  \nThis is related to the antihomomorphism $\\Omega:U(\\mathfrak{g})^{K}\\rightarrow  U( \\mathfrak a^{\\mathbb C}) \\otimes U(\\mathfrak k^{\\mathbb C})^{M}$ \nin Section \\ref{sect-4}; see \\cite{ACT1, ACT2}\nfor the case\nof real rank \none groups.\n\nThe computation\nof eigenvalues\nof invariant \ndifferential\noperators is closely\nrelated to the \nproblem of  characterizing\nfinite-dimensional representations\nof $G$ containing\na fixed $K$-type $\\tau$.\nWhen $\\tau$\nis a character there\nis the Cartan-Helgason\ntheorem and its generalization\nby Schlichtkrull \\cite{schlichtkrull}. On\nthe analytic side there is the explicit Plancherel formula for\nthe $L^2(G/K, V_\\tau)$\nby Shimeno \\cite{Shimeno}.\nIt might be possible\nto find a characterization\nof the finite-dimensional\n$G$-representations\ncontaining $(\\tau, K)$\nand to find\nthe discrete\nseries $(\\pi, G)$\ncontained in \n$L^2(G/K, V_\\tau)$,\nin this case \nthey appear with multiplicity\nat most one (see e.g. \\cite[Prop. 2.2, (40)]{Campo},\n\\cite{ricc} or \\cite[Vol. II, Proposition 6.1.1.6]{war}).\n\n\\subsection{List of notation and\nsymbols}\n\n\\begin{enumerate}\n\\item $G/K$: Hermitian\nsymmetric space.\n\n\\item \n$\n\\mathfrak g=\\mathfrak\nk +\\mathfrak p$,\n$\\mathfrak g^{\\mathbb C}=\n\\mathfrak p^{-}\n+\\mathfrak\nk^{\\mathbb C}\n+\\mathfrak p^+\n$: Cartan and\nHarish-Chandra decompositions of the Lie algebra $\\mathfrak g$ of $G$ and its\ncomplexification $\n\\mathfrak g^{\\mathbb C}$.\n\n\\item $\\mathfrak t^{\\mathbb C}\n=\\mathfrak\n(t^-)^{\\mathbb C} \n\\oplus \n\\mathfrak (t^+)^{\\mathbb C} \n\\subset \n\\mathfrak\nk^{\\mathbb C}$:\nCartan\nsubalgebra of $\\mathfrak{k}^{\\mathbb C}$, and\n$\\gamma_r>\\cdots >\\gamma_1$ are the\nHarish-Chandra strongly\nortogonal roots\nwith $\\gamma_j|_{\n\\mathfrak t_{\\mathbb C}^+}=0 $ and \n $\\gamma_r$ being\n the highest non-compact\n root.\n\n\\item \n $\\mathfrak a\n \\subset\\mathfrak p$:\n maximal abelian \nsubspace (Cartan subspace) of $\\mathfrak p$;\n$c: \\mathfrak a^{\\mathbb C}\n\\to \\mathfrak (t^{-})^{\\mathbb C}\n$ is then the Cayley transform and\n$\\alpha_j = \\gamma_j\\circ c$ is the\nCayley transform\nof the Harish-Chandra\northogonal roots $\\gamma_j$\nto $\\mathfrak a^{\\mathbb C}$. Furthermore,\n$M\\subset K, M'\\subset K$, $W=W(\\mathfrak g, \\mathfrak a)$ are respectively the centralizer\nand normalizer of $\\mathfrak a$\nin $K$, and the Weyl group. Also, $\\mathfrak h^{\\mathbb C}=\n\\mathfrak a^{\\mathbb C}\n+\\mathfrak (t^{+})^{\\mathbb C}$.\n\n\\item \n$(W_{\\lambda}, G^{\\mathbb{C}}),\n(V_\\tau, K),\n(U_\\sigma, M)$:\n Finite-dimensional\nirreducible\nrepresentations\nof the respective\ngroups, the highest\nweight $\\lambda$\nbeing defined\non the Cartan subalgebra $\\mathfrak{h}^{\\mathbb{C}}$ of $\\mathfrak{g}^{\\mathbb{C}}$, and\nhighest weight of\n$\\tau$ being defined\non the Cartan subalgebra $\\mathfrak{t}^{\\mathbb{C}}$.\n\n\\item $G=NAK$, \n$g=n(g)e^{H(g)}k(g)$: the Iwasawa decomposition \nof $G$ with $n(g)\\in N, H(g)\\in \\mathfrak{a}, k(g)\\in K$ and $g \\in G$.\n\n\\item $P=MAN$:\nminimal \nparabolic subgroup of $G$.\n\n\\item $I_{\\sigma, \\nu}=Ind_{P}^G(\\sigma\\otimes e^{\\nu}\\otimes 1)$\n: normalized\ninduced representation\nof $G$ from $P$,\nthe corresponding\nrepresentation \nof $\\mathfrak g^{\\mathbb C}$ on the subspace of $K$-finite vectors will\nalso be written as\n$I_{\\sigma, \\nu}^0$.\n\n\\item $C^\\infty(G/K,V_{\\tau})$: \nspace of\nsmooth sections\nof the vector bundle\n$G\\times_K V_\\tau$\nover $G/K$\ndefined by\nthe representation $(K, \\tau)$. We let $\\mathcal D^G(G/K,V_{\\tau})$ then be the ring \nof $G$-invariant\ndifferential operators on \n$C^\\infty(G/K,V_{\\tau})$.\n\n\\end{enumerate}\n\n\\subsection*{Acknowledgments}\nWe would like to thank Pavle Pandzic for informing \nus that some of our results\non the classifications\nof $(V_\\tau, \\tau)$\nhave also been obtained\nearlier by his joint work with \nSoo-Teck Lee and for some further discussions.", "sketch": "The proof breaks into three parts corresponding to (1)--(3).\n\n(1) For the classification, “the proof of the classification is through a case by case computation and is technically involved,” repeatedly using “the classical branching rules in \\cite{good}, the classifications of weight-free representations in \\cite{Howe} and multiplicity-free results in \\cite{St}.”\n\n(2) For uniqueness of the $G$-equivariant map $F:W_\\lambda\\to C^\\infty(G/K,V_\\tau)$, the argument “is done through the realization $J:W_\\lambda\\to \\Ind_{MAN}^G(\\sigma\\otimes\\alpha\\otimes 1)$ of finite-dimensional representations $W_\\lambda$ in the induced representation,” together with “a factorization of $F=SJ$ as a product of $J$ and the Poisson-Szeg\\\"o{} transform,” and using “the multiplicity-free property of $V_\\tau$ under $M$.”\n\n(3) The Weyl-group invariance of eigenvalues of invariant differential operators “is a consequence of a general result of Lepowsky \\cite{L} and the factorization.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "(\\begin{theorem}\n\\label{KresThm}\nLet $(G, K)$ be an irreducible Hermitian symmetric pair\nand\n $M = C_K(\\mathfrak{a})$\n be the centralizer of the\n Cartan subalgebra \n $\\mathfrak{a}\\subset \\mathfrak p$ in $K$. \nThe complete list of $(K, \\tau)$ with $\\tau|_M$ being multiplicity-free is given as follows:\n\n\\begin{enumerate}\n    \\item Type A, \n    $G = SU(r+b,r), \n    K = S(U(r+b) \\times U(r))$; irreducible representations of $U(n)$ with highest\n    weights $\\mu$\n    will be denoted by \n    $\\pi^{\\mu}_n$ as  in \\cite[Theorem 5.5.22]{good}. \n\\begin{enumerate}\n    \\item $r=1$. $\\tau$ is arbitrary.\n\\item  $r=2$. $\\tau= \\pi^{\\mu}_{2+b} \\otimes \\pi^{\\nu}_2$ where $\\pi_{2+b}^{\\mu}$ is\na character and $\\pi_2^{\\nu}$ is any representations, or $\\pi_2^{\\nu}$ is \na character and\n$$\n\\mu = (\\overbrace{l_1+l_2,\\cdots, l_1+l_2}^{{j }},\\overbrace{l_2,\\cdots,l_2}^{{b+2-j }}).\n$$\n\n\\item  $r \\geq 3$. $\\tau= \\pi^{\\mu}_{2+b} \\otimes \\pi^{\\nu}_2$ where $\\pi_{r+b}^{\\mu}$\nor $\\pi_r^{\\nu}$ is a character and the other must be the symmetric\ntensor powers\n$S^m(\\mathbb C^{p})$ \nor  an exterior power\n$\n\\wedge^j \\mathbb C^p$ when restricted to $SL(p)$\n($p=r+b$ or $p=r$\naccordingly).\n\\end{enumerate} \n\n\\item Type C,  $G=Sp(n,\\R)$, $K = U(n)$. $\\tau$ is a character \nor its tensor product\nwith  the exterior power representations\n$\n\\wedge^j \\mathbb C^n$. The same holds for the double cover $(G,K) = (Mp(n,\\R),\\widehat{U(n)})$.\n\n\\item Type D,  $G = SO^*(2n), K = U(n)$. $\\tau$ \nis a character\nor its tensor product\nwith  the symmetric tensor\npowers\n$S^m(\\mathbb C^{p})$  or their duals.\n\n\\item Type BD,  $G=SO_0(n,2)$, \n $K = SO(n) \\times SO(2)$, $n>4$;\nirreducible \nrepresentations\nof $SO(n)$\nwill be denoted\nby \n$\\pi_{n}^{\\mu}$\nas in \\cite[Theorem 19.22]{FulHar}.\n $\\tau=\\pi_{n}^{0}\\otimes \\pi_{2}^\\nu$ \n if  $n$ is odd, i.e., $\\tau$ is a character; for $n$ even $\\tau=\n \\pi_{n}^{\\mu}$ \n or  its \n tensor product\n with a character\n$\\tau=\\pi_{n}^{\\mu}\n\\otimes \\pi_{2}^\\nu$,  \n where $\\mu=(a,\\dots,a,\\pm a)$ for some $a\\ge 0$.\nIf $(G,K) = (Spin_0(n,2), \nSpin(n) \\times SO(2)/\\Z_2)$, where $Spin_0(n,2)$ is the double cover of $SO_0(n,2)$\nthen up to a character $\\tau$ is the spin representation $(\\frac{1}{2}, \\dots, \\frac{1}{2})$ when $n$ is odd, and\n$\\tau= \\pi_{n}^{\\mu}$, $\\mu=(a,\\dots,a,\\pm a)$ for some integer $a\\ge 0$ a half-integer\nwhen $n$ is even.\n\n\\item Type $E_6$, $G=E_{6(-14)}, K = Spin(10) \\times U(1) / \\Gamma$, where $\\Gamma$ is a finite subgroup.\n$\\tau=\\pi_{\\omega_1} \\otimes \\chi_n$ for  $n \\equiv 1 \\mod 4$, or $\\pi_{\\omega_2} \\otimes \\chi_n$ for $n \\equiv 3 \\mod 4$. Here $\\omega_1$ and $\\omega_2$ are the positive and negative spin representations of $Spin(10)$ and $U(1) = \\{e^{it \\frac{Z}{2}} \\ | \\ t \\in \\R \\}$.\n\n\\item Type $E_7$,  $G=E_{7(-25)}, K = E_6  U(1)$.\n$\\tau$ is a character.\n\n\\end{enumerate} \n\\end{theorem},\n\\begin{proposition}\n\\label{finiterepprop}\n(1) Let $(W_{\\lambda}, G^{\\C})$ be a highest weight representation of $G^{\\C}$ and its Lie algebra $\\mathfrak{g}^{\\C}$. For the multiplicities we have $[V_{\\tau} : W_{\\lambda}|_K] \\leq 1$, and if $V_{\\tau} \\subseteq W_{\\lambda}|_K$ we have $[W_{\\lambda} : I_{\\sigma,-\\lambda|_{\\af} - \\rho_{\\af}}^0 ] \\leq 1$, viewed as Lie algebra representations.  \n\n(2)\nLet $(W_\\lambda, G ^{\\C})$ be\na finite-dimensional\nirreducible representation\nsuch that \n$V_{\\tau} \\subseteq W_{\\lambda}|_K$. Up to  constants there is a unique $G$-equivariant map\n$$F : W_{\\lambda} \\rightarrow C^{\\infty}(G/K,V_{\\tau})$$\ndefined by\n$$F(w)(g) = P_{\\tau} (g^{-1} \\cdot w), g\\in G, w\\in W,$$\nwhere $P_{\\tau}$ is the projection onto $V_{\\tau}$. Moreover, if we normalize $J_{\\lambda}$ so that for all $v \\in V_{\\tau}$ we have $J_{\\lambda}(v) = f_v$, where $f_v(kan) =  a^{\\lambda|_{\\mathfrak{a}}} P_{\\sigma} \\tau(k)^{-1} v$,\nthen there is a factorization\nof $F$ \nas a product $G$-equivariant maps,\n$$\nF=\nS_{\\sigma, -\\lambda|_{\\mathfrak{a}} - \\rho_{\\mathfrak{a}}}\nJ_{\\lambda}: (W_\\lambda, G^{\\C})\\to I_{\\sigma, -\\lambda|_{\\mathfrak{a}} - \\rho_{\\mathfrak{a}}}^0\n\\to C^\\infty(G/K,V_{\\tau}).\n$$\n\\end{proposition},\n\\begin{theorem}\n\\label{egvaluethm}\nAssume the irreducible finite-dimensional  representation $V_\\tau$  \nof $K$ is  multiplicity-free  when  restricted to  $M$.\n\n(1) For any $M$-irreducible subrepresentation $(\\sigma,U_\\sigma)$ in $\\tau|_M$,  $f \\in I_{\\sigma,\\nu}^0$ and $D \\in \\mathcal{D}^G(G/K, V_{\\tau})$, we have $DS_{\\sigma, \\nu}(f) = \\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}})S_{\\sigma, \\nu}(f)$ for a constant $\\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}})$.\nFurthermore, we have the invariance property:\n$$\\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}}) = \\omega(D)(s \\cdot \\sigma,s \\cdot \\nu + \\rho_{\\mathfrak{a}}), \\ \\forall \\ s\\in W.$$\n\n(2) Let $(W_\\lambda, G^{\\C})$\nbe a  finite-dimensional\nrepresentation of $\nG^{\\C}$ containing $V_\\tau$.  Let $D \\in \\mathcal{D}^G(G/K,V_{\\tau})$, then $D$ acts on \n$F(W_\\lambda)$\nas the constant\n$$\nDF(w) = \\omega(D)(\\sigma_{\\lambda},-\\lambda|_{\\mathfrak{a}})F(w), \n\\quad  \\ \\forall \\ w \\in W_\\lambda.\n$$\n\\end{theorem})\n(1) The representation \n$\\tau|_M$ is\nmultiplicity-free if and only if \n$(K, \\tau)$ is in the list above.\n\n(2) Let $(K, \\tau)$ be as\nin (1) acting on $V_\\tau$. For any finite-dimensional irreducible representation\n$(W_\\lambda, G^{\\mathbb{C}})$ there is up to scalar at most one $G$-equivariant map into $C^\\infty(G/K, V_\\tau)$.\n\n(3)\nLet $(K, \\tau)\n$ be as in (1), $\\sigma$\nan irreducible representation in  $\\tau|_M$\nand \n$(W_\\lambda, G^{\\mathbb C})$ \nas\nin (2). Let $D$ \nbe an invariant\ndifferential operator\non $C^\\infty(G/K, V_\\tau)$.\nWrite   the eigenvalues \nof  $D$ on $W_\\lambda$\nas $\\omega(D)(\\sigma_{\\lambda},-\\lambda|_{\\mathfrak{a}}\n)$\nwhich can be extended uniquely to a polynomial of $\n\\lambda|_{\\mathfrak{a}}\n\\in \n(\\mathfrak a^{\\mathbb C})'\n$.\nThen the eigenvalue\n$\\omega(D)$ has following\ninvariance \nproperty,\n$$\\omega(D)(\\sigma,\\kappa + \\rho_{\\mathfrak{a}}) = \\omega(D)(s \\cdot \\sigma,s \\cdot \\kappa + \\rho_{\\mathfrak{a}}), \\ \\forall \\ \\kappa \\in \n(\\mathfrak a^{\\mathbb C})', s\\in W.$$\n", "theorem_type": ["Biconditional or Equivalence", "Universal"], "mcq": {"question": "Let (G,K) be an irreducible Hermitian symmetric pair with Cartan decomposition \\(\\mathfrak g=\\mathfrak k\\oplus\\mathfrak p\\), let \\(\\mathfrak a\\subset \\mathfrak p\\) be a Cartan subalgebra, and let \\(M=C_K(\\mathfrak a)\\). Let \\(\\tau\\) be an irreducible finite-dimensional representation of \\(K\\) on \\(V_\\tau\\), let \\(C^\\infty(G/K,V_\\tau)\\) be the smooth sections of the homogeneous bundle \\(G\\times_{(K,\\tau)}V_\\tau\\), and let \\(\\mathcal D^G(G/K,V_\\tau)\\) be the algebra of \\(G\\)-invariant differential operators on that space. Write \\(W\\) for the Weyl group of \\((\\mathfrak g,\\mathfrak a)\\) and \\(\\rho_{\\mathfrak a}\\) for the half-sum of positive restricted roots. In type A, \\(\\pi_n^\\mu\\) denotes the irreducible \\(U(n)\\)-representation of highest weight \\(\\mu\\); in type BD, \\(\\pi_n^\\mu\\) denotes the irreducible \\(SO(n)\\)-representation of highest weight \\(\\mu\\); in type \\(E_6\\), \\(\\chi_n\\) denotes the weight-\\(n\\) character of \\(U(1)\\), and \\(\\omega_1,\\omega_2\\) are the positive and negative half-spin representations of \\(\\mathrm{Spin}(10)\\). Which statement holds for every such \\((G,K,\\tau)\\)?", "correct_choice": {"label": "A", "text": "The restriction \\(\\tau|_M\\) is multiplicity-free if and only if \\((K,\\tau)\\) is one of the following: (1) Type A: \\(G=SU(r+b,r)\\), \\(K=S(U(r+b)\\times U(r))\\): (a) if \\(r=1\\), then \\(\\tau\\) is arbitrary; (b) if \\(r=2\\), then \\(\\tau=\\pi_{b+2}^{\\mu}\\otimes \\pi_2^{\\nu}\\), where either \\(\\pi_{b+2}^{\\mu}\\) is a character and \\(\\pi_2^{\\nu}\\) is arbitrary, or \\(\\pi_2^{\\nu}\\) is a character and \\(\\mu=(\\overbrace{l_1+l_2,\\dots,l_1+l_2}^{j},\\overbrace{l_2,\\dots,l_2}^{b+2-j})\\); (c) if \\(r\\ge 3\\), then \\(\\tau=\\pi_{r+b}^{\\mu}\\otimes \\pi_r^{\\nu}\\), where one of \\(\\pi_{r+b}^{\\mu}\\) or \\(\\pi_r^{\\nu}\\) is a character and the other, upon restriction to \\(SL(p)\\) (with \\(p=r+b\\) or \\(p=r\\), respectively), is either a symmetric power \\(S^m(\\mathbb C^p)\\) or an exterior power \\(\\wedge^j\\mathbb C^p\\); (2) Type C: \\(G=Sp(n,\\mathbb R)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with an exterior power \\(\\wedge^j\\mathbb C^n\\); the same holds for the double cover \\((G,K)=(Mp(n,\\mathbb R),\\widehat{U(n)})\\); (3) Type D: \\(G=SO^*(2n)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with a symmetric tensor power \\(S^m(\\mathbb C^n)\\) or its dual; (4) Type BD: \\(G=SO_0(n,2)\\), \\(K=SO(n)\\times SO(2)\\), \\(n>4\\); if \\(n\\) is odd, then \\(\\tau=\\pi_n^0\\otimes \\pi_2^\\nu\\), i.e. \\(\\tau\\) is a character; if \\(n\\) is even, then \\(\\tau=\\pi_n^\\mu\\) or \\(\\tau=\\pi_n^\\mu\\otimes \\pi_2^\\nu\\), where \\(\\mu=(a,\\dots,a,\\pm a)\\) for some \\(a\\ge 0\\); for the double cover \\((G,K)=(\\mathrm{Spin}_0(n,2),\\,\\mathrm{Spin}(n)\\times SO(2)/\\mathbb Z_2)\\), up to a character \\(\\tau\\) is the spin representation \\((\\tfrac12,\\dots,\\tfrac12)\\) when \\(n\\) is odd, and when \\(n\\) is even one has \\(\\tau=\\pi_n^\\mu\\) with \\(\\mu=(a,\\dots,a,\\pm a)\\) for some nonnegative integral or half-integral \\(a\\); (5) Type \\(E_6\\): \\(G=E_{6(-14)}\\), \\(K=\\mathrm{Spin}(10)\\times U(1)/\\Gamma\\), and \\(\\tau=\\pi_{\\omega_1}\\otimes \\chi_n\\) for \\(n\\equiv 1\\pmod 4\\), or \\(\\tau=\\pi_{\\omega_2}\\otimes \\chi_n\\) for \\(n\\equiv 3\\pmod 4\\); (6) Type \\(E_7\\): \\(G=E_{7(-25)}\\), \\(K=E_6U(1)\\), and \\(\\tau\\) is a character. Moreover, for every \\((K,\\tau)\\) in this list and every finite-dimensional irreducible representation \\((W_\\lambda,G^{\\mathbb C})\\), there is, up to scalar, at most one \\(G\\)-equivariant map \\(W_\\lambda\\to C^\\infty(G/K,V_\\tau)\\). Finally, if \\(\\sigma\\) is an irreducible summand of \\(\\tau|_M\\), \\(D\\in \\mathcal D^G(G/K,V_\\tau)\\), and the eigenvalue of \\(D\\) on \\(W_\\lambda\\) is written as \\(\\omega(D)(\\sigma_\\lambda,-\\lambda|_{\\mathfrak a})\\) and extended uniquely to a polynomial in \\(\\lambda|_{\\mathfrak a}\\in (\\mathfrak a^{\\mathbb C})'\\), then \\(\\omega(D)\\) satisfies the Weyl-group invariance property \\(\\omega(D)(\\sigma,\\kappa+\\rho_{\\mathfrak a})=\\omega(D)(s\\cdot \\sigma,s\\cdot \\kappa+\\rho_{\\mathfrak a})\\) for all \\(\\kappa\\in (\\mathfrak a^{\\mathbb C})'\\) and all \\(s\\in W\\)."}, "choices": [{"label": "B", "text": "The restriction \\(\\tau|_M\\) is multiplicity-free if and only if \\((K,\\tau)\\) is one of the following: (1) Type A: \\(G=SU(r+b,r)\\), \\(K=S(U(r+b)\\times U(r))\\): (a) if \\(r=1\\), then \\(\\tau\\) is arbitrary; (b) if \\(r=2\\), then \\(\\tau=\\pi_{b+2}^{\\mu}\\otimes \\pi_2^{\\nu}\\), where either \\(\\pi_{b+2}^{\\mu}\\) is a character and \\(\\pi_2^{\\nu}\\) is arbitrary, or \\(\\pi_2^{\\nu}\\) is a character and \\(\\mu=(\\overbrace{l_1+l_2,\\dots,l_1+l_2}^{j},\\overbrace{l_2,\\dots,l_2}^{b+2-j})\\); (c) if \\(r\\ge 3\\), then \\(\\tau=\\pi_{r+b}^{\\mu}\\otimes \\pi_r^{\\nu}\\), where one of \\(\\pi_{r+b}^{\\mu}\\) or \\(\\pi_r^{\\nu}\\) is a character and the other, upon restriction to \\(SL(p)\\) (with \\(p=r+b\\) or \\(p=r\\), respectively), is either a symmetric power \\(S^m(\\mathbb C^p)\\) or an exterior power \\(\\wedge^j\\mathbb C^p\\); (2) Type C: \\(G=Sp(n,\\mathbb R)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with an exterior power \\(\\wedge^j\\mathbb C^n\\); the same holds for the double cover \\((G,K)=(Mp(n,\\mathbb R),\\widehat{U(n)})\\); (3) Type D: \\(G=SO^*(2n)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with a symmetric tensor power \\(S^m(\\mathbb C^n)\\), an exterior power \\(\\wedge^j\\mathbb C^n\\), or their duals; (4) Type BD: \\(G=SO_0(n,2)\\), \\(K=SO(n)\\times SO(2)\\), \\(n>4\\); if \\(n\\) is odd, then \\(\\tau=\\pi_n^0\\otimes \\pi_2^\\nu\\), i.e. \\(\\tau\\) is a character; if \\(n\\) is even, then \\(\\tau=\\pi_n^\\mu\\) or \\(\\tau=\\pi_n^\\mu\\otimes \\pi_2^\\nu\\), where \\(\\mu=(a,\\dots,a,\\pm a)\\) for some \\(a\\ge 0\\); for the double cover \\((G,K)=(\\mathrm{Spin}_0(n,2),\\,\\mathrm{Spin}(n)\\times SO(2)/\\mathbb Z_2)\\), up to a character \\(\\tau\\) is the spin representation \\((\\tfrac12,\\dots,\\tfrac12)\\) when \\(n\\) is odd, and when \\(n\\) is even one has \\(\\tau=\\pi_n^\\mu\\) with \\(\\mu=(a,\\dots,a,\\pm a)\\) for some nonnegative integral or half-integral \\(a\\); (5) Type \\(E_6\\): \\(G=E_{6(-14)}\\), \\(K=\\mathrm{Spin}(10)\\times U(1)/\\Gamma\\), and \\(\\tau=\\pi_{\\omega_1}\\otimes \\chi_n\\) for \\(n\\equiv 1\\pmod 4\\), or \\(\\tau=\\pi_{\\omega_2}\\otimes \\chi_n\\) for \\(n\\equiv 3\\pmod 4\\); (6) Type \\(E_7\\): \\(G=E_{7(-25)}\\), \\(K=E_6U(1)\\), and \\(\\tau\\) is a character. Moreover, for every \\((K,\\tau)\\) in this list and every finite-dimensional irreducible representation \\((W_\\lambda,G^{\\mathbb C})\\), there exists a unique \\(G\\)-equivariant map \\(W_\\lambda\\to C^\\infty(G/K,V_\\tau)\\). Finally, if \\(\\sigma\\) is an irreducible summand of \\(\\tau|_M\\), \\(D\\in \\mathcal D^G(G/K,V_\\tau)\\), and the eigenvalue of \\(D\\) on \\(W_\\lambda\\) is written as \\(\\omega(D)(\\sigma_\\lambda,-\\lambda|_{\\mathfrak a})\\) and extended uniquely to a polynomial in \\(\\lambda|_{\\mathfrak a}\\in (\\mathfrak a^{\\mathbb C})'\\), then \\(\\omega(D)\\) satisfies the Weyl-group invariance property \\(\\omega(D)(\\sigma,\\kappa+\\rho_{\\mathfrak a})=\\omega(D)(s\\cdot \\sigma,s\\cdot \\kappa+\\rho_{\\mathfrak a})\\) for all \\(\\kappa\\in (\\mathfrak a^{\\mathbb C})'\\) and all \\(s\\in W\\)."}, {"label": "C", "text": "If \\(\\tau|_M\\) is multiplicity-free, then \\((K,\\tau)\\) is one of the representations in the classification listed above. Moreover, for every such \\((K,\\tau)\\) and every finite-dimensional irreducible representation \\((W_\\lambda,G^{\\mathbb C})\\), there is, up to scalar, at most one \\(G\\)-equivariant map \\(W_\\lambda\\to C^\\infty(G/K,V_\\tau)\\). Finally, if \\(\\sigma\\) is an irreducible summand of \\(\\tau|_M\\), \\(D\\in \\mathcal D^G(G/K,V_\\tau)\\), and the eigenvalue of \\(D\\) on \\(W_\\lambda\\) is written as \\(\\omega(D)(\\sigma_\\lambda,-\\lambda|_{\\mathfrak a})\\) and extended uniquely to a polynomial in \\(\\lambda|_{\\mathfrak a}\\in (\\mathfrak a^{\\mathbb C})'\\), then \\(\\omega(D)(\\sigma,\\kappa+\\rho_{\\mathfrak a})=\\omega(D)(s\\cdot \\sigma,s\\cdot \\kappa+\\rho_{\\mathfrak a})\\) for all \\(\\kappa\\in (\\mathfrak a^{\\mathbb C})'\\) and all \\(s\\in W\\)."}, {"label": "D", "text": "The restriction \\(\\tau|_M\\) is multiplicity-free if and only if \\((K,\\tau)\\) is one of the following: (1) Type A: \\(G=SU(r+b,r)\\), \\(K=S(U(r+b)\\times U(r))\\): (a) if \\(r=1\\), then \\(\\tau\\) is arbitrary; (b) if \\(r=2\\), then \\(\\tau=\\pi_{b+2}^{\\mu}\\otimes \\pi_2^{\\nu}\\), where either \\(\\pi_{b+2}^{\\mu}\\) is a character and \\(\\pi_2^{\\nu}\\) is arbitrary, or \\(\\pi_2^{\\nu}\\) is a character and \\(\\mu=(\\overbrace{l_1+l_2,\\dots,l_1+l_2}^{j},\\overbrace{l_2,\\dots,l_2}^{b+2-j})\\); (c) if \\(r\\ge 3\\), then \\(\\tau=\\pi_{r+b}^{\\mu}\\otimes \\pi_r^{\\nu}\\), where one of \\(\\pi_{r+b}^{\\mu}\\) or \\(\\pi_r^{\\nu}\\) is a character and the other, upon restriction to \\(SL(p)\\) (with \\(p=r+b\\) or \\(p=r\\), respectively), is either a symmetric power \\(S^m(\\mathbb C^p)\\) or an exterior power \\(\\wedge^j\\mathbb C^p\\); (2) Type C: \\(G=Sp(n,\\mathbb R)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with an exterior power \\(\\wedge^j\\mathbb C^n\\); the same holds for the double cover \\((G,K)=(Mp(n,\\mathbb R),\\widehat{U(n)})\\); (3) Type D: \\(G=SO^*(2n)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with a symmetric tensor power \\(S^m(\\mathbb C^n)\\) or its dual; (4) Type BD: \\(G=SO_0(n,2)\\), \\(K=SO(n)\\times SO(2)\\), \\(n\\ge 4\\); if \\(n\\) is odd, then \\(\\tau=\\pi_n^0\\otimes \\pi_2^\\nu\\), i.e. \\(\\tau\\) is a character; if \\(n\\) is even, then \\(\\tau=\\pi_n^\\mu\\) or \\(\\tau=\\pi_n^\\mu\\otimes \\pi_2^\\nu\\), where \\(\\mu=(a,\\dots,a,\\pm a)\\) for some \\(a\\ge 0\\); for the double cover \\((G,K)=(\\mathrm{Spin}_0(n,2),\\,\\mathrm{Spin}(n)\\times SO(2)/\\mathbb Z_2)\\), up to a character \\(\\tau\\) is the spin representation \\((\\tfrac12,\\dots,\\tfrac12)\\) when \\(n\\) is odd, and when \\(n\\) is even one has \\(\\tau=\\pi_n^\\mu\\) with \\(\\mu=(a,\\dots,a,\\pm a)\\) for some nonnegative integral or half-integral \\(a\\); (5) Type \\(E_6\\): \\(G=E_{6(-14)}\\), \\(K=\\mathrm{Spin}(10)\\times U(1)/\\Gamma\\), and \\(\\tau=\\pi_{\\omega_1}\\otimes \\chi_n\\) for \\(n\\equiv 1\\pmod 4\\), or \\(\\tau=\\pi_{\\omega_2}\\otimes \\chi_n\\) for \\(n\\equiv 3\\pmod 4\\); (6) Type \\(E_7\\): \\(G=E_{7(-25)}\\), \\(K=E_6U(1)\\), and \\(\\tau\\) is a character. Moreover, for every \\((K,\\tau)\\) in this list and every finite-dimensional irreducible representation \\((W_\\lambda,G^{\\mathbb C})\\), there is, up to scalar, at most one \\(G\\)-equivariant map \\(W_\\lambda\\to C^\\infty(G/K,V_\\tau)\\). Finally, if \\(\\sigma\\) is an irreducible summand of \\(\\tau|_M\\), \\(D\\in \\mathcal D^G(G/K,V_\\tau)\\), and the eigenvalue of \\(D\\) on \\(W_\\lambda\\) is written as \\(\\omega(D)(\\sigma_\\lambda,-\\lambda|_{\\mathfrak a})\\) and extended uniquely to a polynomial in \\(\\lambda|_{\\mathfrak a}\\in (\\mathfrak a^{\\mathbb C})'\\), then \\(\\omega(D)\\) satisfies the Weyl-group invariance property \\(\\omega(D)(\\sigma,\\kappa+\\rho_{\\mathfrak a})=\\omega(D)(s\\cdot \\sigma,s\\cdot \\kappa+\\rho_{\\mathfrak a})\\) for all \\(\\kappa\\in (\\mathfrak a^{\\mathbb C})'\\) and all \\(s\\in W\\)."}, {"label": "E", "text": "The restriction \\(\\tau|_M\\) is multiplicity-free if and only if \\((K,\\tau)\\) is one of the following: (1) Type A: \\(G=SU(r+b,r)\\), \\(K=S(U(r+b)\\times U(r))\\): (a) if \\(r=1\\), then \\(\\tau\\) is arbitrary; (b) if \\(r=2\\), then \\(\\tau=\\pi_{b+2}^{\\mu}\\otimes \\pi_2^{\\nu}\\), where either \\(\\pi_{b+2}^{\\mu}\\) is a character and \\(\\pi_2^{\\nu}\\) is arbitrary, or \\(\\pi_2^{\\nu}\\) is a character and \\(\\mu=(\\overbrace{l_1+l_2,\\dots,l_1+l_2}^{j},\\overbrace{l_2,\\dots,l_2}^{b+2-j})\\); (c) if \\(r\\ge 3\\), then \\(\\tau=\\pi_{r+b}^{\\mu}\\otimes \\pi_r^{\\nu}\\), where one of \\(\\pi_{r+b}^{\\mu}\\) or \\(\\pi_r^{\\nu}\\) is a character and the other, upon restriction to \\(SL(p)\\) (with \\(p=r+b\\) or \\(p=r\\), respectively), is either a symmetric power \\(S^m(\\mathbb C^p)\\) or an exterior power \\(\\wedge^j\\mathbb C^p\\); (2) Type C: \\(G=Sp(n,\\mathbb R)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with an exterior power \\(\\wedge^j\\mathbb C^n\\); the same holds for the double cover \\((G,K)=(Mp(n,\\mathbb R),\\widehat{U(n)})\\); (3) Type D: \\(G=SO^*(2n)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with a symmetric tensor power \\(S^m(\\mathbb C^n)\\) or its dual; (4) Type BD: \\(G=SO_0(n,2)\\), \\(K=SO(n)\\times SO(2)\\), \\(n>4\\); if \\(n\\) is odd, then \\(\\tau=\\pi_n^0\\otimes \\pi_2^\\nu\\), i.e. \\(\\tau\\) is a character; if \\(n\\) is even, then \\(\\tau=\\pi_n^\\mu\\) or \\(\\tau=\\pi_n^\\mu\\otimes \\pi_2^\\nu\\), where \\(\\mu=(a,\\dots,a,\\pm a)\\) for some \\(a\\ge 0\\); for the double cover \\((G,K)=(\\mathrm{Spin}_0(n,2),\\,\\mathrm{Spin}(n)\\times SO(2)/\\mathbb Z_2)\\), up to a character \\(\\tau\\) is the spin representation \\((\\tfrac12,\\dots,\\tfrac12)\\) when \\(n\\) is odd, and when \\(n\\) is even one has \\(\\tau=\\pi_n^\\mu\\) with \\(\\mu=(a,\\dots,a,\\pm a)\\) for some nonnegative integral or half-integral \\(a\\); (5) Type \\(E_6\\): \\(G=E_{6(-14)}\\), \\(K=\\mathrm{Spin}(10)\\times U(1)/\\Gamma\\), and \\(\\tau=\\pi_{\\omega_1}\\otimes \\chi_n\\) for \\(n\\equiv 1\\pmod 4\\), or \\(\\tau=\\pi_{\\omega_2}\\otimes \\chi_n\\) for \\(n\\equiv 3\\pmod 4\\); (6) Type \\(E_7\\): \\(G=E_{7(-25)}\\), \\(K=E_6U(1)\\), and \\(\\tau\\) is a character. Moreover, for every \\((K,\\tau)\\) in this list and every finite-dimensional irreducible representation \\((W_\\lambda,G^{\\mathbb C})\\), there is, up to scalar, at most one \\(G\\)-equivariant map \\(W_\\lambda\\to C^\\infty(G/K,V_\\tau)\\). Finally, if \\(\\sigma\\) is an irreducible summand of \\(\\tau|_M\\), \\(D\\in \\mathcal D^G(G/K,V_\\tau)\\), and the eigenvalue of \\(D\\) on \\(W_\\lambda\\) is written as \\(\\omega(D)(\\sigma_\\lambda,-\\lambda|_{\\mathfrak a})\\) and extended uniquely to a polynomial in \\(\\lambda|_{\\mathfrak a}\\in (\\mathfrak a^{\\mathbb C})'\\), then \\(\\omega(D)\\) satisfies the stronger invariance property \\(\\omega(D)(\\sigma,\\kappa)=\\omega(D)(s\\cdot \\sigma,s\\cdot \\kappa)\\) for all \\(\\kappa\\in (\\mathfrak a^{\\mathbb C})'\\) and all \\(s\\in W\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "uniqueness up to scalar replaced by absolute existence and uniqueness", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "case_split", "tampered_component": "dropped the converse direction in the classification equivalence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "type BD boundary changed from n>4 to n\\ge 4", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "Weyl-invariance shift by \\rho_{\\mathfrak a} removed", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the classification or directly hint at the correct option. It only asks for an equivalent condition, so the correct answer is not leaked by the wording itself."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the correct choice is the exact classification statement equivalent to multiplicity-freeness. The question mostly asks the test-taker to recognize the theorem's precise wording rather than infer a new conclusion."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in checking subtle boundary cases and comparing nearly identical classification lists, but the task is mainly precise recall/verification of a known theorem rather than genuine generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong: they introduce plausible alterations such as weakened implication, incorrect boundary ranges, overextended representation families, and confusion with a related operator-algebra criterion. These reflect realistic mathematical failure modes."}, "total_score": 5, "overall_assessment": "A technically strong but theorem-recall-heavy MCQ. It avoids answer leakage and uses high-quality distractors, but it is close to a direct restatement of a classification theorem and only moderately tests reasoning."}}
{"id": "2602.14864v1", "paper_link": "http://arxiv.org/abs/2602.14864v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "(Theorem \\ref{KresThm}, Proposition  \\ref{finiterepprop}, Theorem \\ref{egvaluethm})\n(1) The representation \n$\\tau|_M$ is\nmultiplicity-free if and only if \n$(K, \\tau)$ is in the list in Theorem \\ref{KresThm}.\n\n(2) Let $(K, \\tau)$ be as\nin (1) acting on $V_\\tau$. For any finite-dimensional irreducible representation\n$(W_\\lambda, G^{\\mathbb{C}})$ there is up to scalar at most one $G$-equivariant map into $C^\\infty(G/K, V_\\tau)$.\n\n(3)\nLet $(K, \\tau)\n$ be as in (1), $\\sigma$\nan irreducible representation in  $\\tau|_M$\nand \n$(W_\\lambda, G^{\\mathbb C})$ \nas\nin (2). Let $D$ \nbe an invariant\ndifferential operator\non $C^\\infty(G/K, V_\\tau)$.\nWrite   the eigenvalues \nof  $D$ on $W_\\lambda$\nas $\\omega(D)(\\sigma_{\\lambda},-\\lambda|_{\\mathfrak{a}}\n)$\nwhich can be extended uniquely to a polynomial of $\n\\lambda|_{\\mathfrak{a}}\n\\in \n(\\mathfrak a^{\\mathbb C})'\n$.\nThen the eigenvalue\n$\\omega(D)$ has following\ninvariance \nproperty,\n$$\\omega(D)(\\sigma,\\kappa + \\rho_{\\mathfrak{a}}) = \\omega(D)(s \\cdot \\sigma,s \\cdot \\kappa + \\rho_{\\mathfrak{a}}), \\ \\forall \\ \\kappa \\in \n(\\mathfrak a^{\\mathbb C})', s\\in W.$$", "start_pos": 5241, "end_pos": 6365, "label": null}, "ref_dict": {"KresThm": "\\begin{theorem}\n\\label{KresThm}\nLet $(G, K)$ be an irreducible Hermitian symmetric pair\nand\n $M = C_K(\\mathfrak{a})$\n be the centralizer of the\n Cartan subalgebra \n $\\mathfrak{a}\\subset \\mathfrak p$ in $K$. \nThe complete list of $(K, \\tau)$ with $\\tau|_M$ being multiplicity-free is given as follows:\n\n\\begin{enumerate}\n    \\item Type A, \n    $G = SU(r+b,r), \n    K = S(U(r+b) \\times U(r))$; irreducible representations of $U(n)$ with highest\n    weights $\\mu$\n    will be denoted by \n    $\\pi^{\\mu}_n$ as  in \\cite[Theorem 5.5.22]{good}. \n\\begin{enumerate}\n    \\item $r=1$. $\\tau$ is arbitrary.\n\\item  $r=2$. $\\tau= \\pi^{\\mu}_{2+b} \\otimes \\pi^{\\nu}_2$ where $\\pi_{2+b}^{\\mu}$ is\na character and $\\pi_2^{\\nu}$ is any representations, or $\\pi_2^{\\nu}$ is \na character and\n$$\n\\mu = (\\overbrace{l_1+l_2,\\cdots, l_1+l_2}^{{j }},\\overbrace{l_2,\\cdots,l_2}^{{b+2-j }}).\n$$\n\n\\item  $r \\geq 3$. $\\tau= \\pi^{\\mu}_{2+b} \\otimes \\pi^{\\nu}_2$ where $\\pi_{r+b}^{\\mu}$\nor $\\pi_r^{\\nu}$ is a character and the other must be the symmetric\ntensor powers\n$S^m(\\mathbb C^{p})$ \nor  an exterior power\n$\n\\wedge^j \\mathbb C^p$ when restricted to $SL(p)$\n($p=r+b$ or $p=r$\naccordingly).\n\\end{enumerate} \n\n\\item Type C,  $G=Sp(n,\\R)$, $K = U(n)$. $\\tau$ is a character \nor its tensor product\nwith  the exterior power representations\n$\n\\wedge^j \\mathbb C^n$. The same holds for the double cover $(G,K) = (Mp(n,\\R),\\widehat{U(n)})$.\n\n\\item Type D,  $G = SO^*(2n), K = U(n)$. $\\tau$ \nis a character\nor its tensor product\nwith  the symmetric tensor\npowers\n$S^m(\\mathbb C^{p})$  or their duals.\n\n\\item Type BD,  $G=SO_0(n,2)$, \n $K = SO(n) \\times SO(2)$, $n>4$;\nirreducible \nrepresentations\nof $SO(n)$\nwill be denoted\nby \n$\\pi_{n}^{\\mu}$\nas in \\cite[Theorem 19.22]{FulHar}.\n $\\tau=\\pi_{n}^{0}\\otimes \\pi_{2}^\\nu$ \n if  $n$ is odd, i.e., $\\tau$ is a character; for $n$ even $\\tau=\n \\pi_{n}^{\\mu}$ \n or  its \n tensor product\n with a character\n$\\tau=\\pi_{n}^{\\mu}\n\\otimes \\pi_{2}^\\nu$,  \n where $\\mu=(a,\\dots,a,\\pm a)$ for some $a\\ge 0$.\nIf $(G,K) = (Spin_0(n,2), \nSpin(n) \\times SO(2)/\\Z_2)$, where $Spin_0(n,2)$ is the double cover of $SO_0(n,2)$\nthen up to a character $\\tau$ is the spin representation $(\\frac{1}{2}, \\dots, \\frac{1}{2})$ when $n$ is odd, and\n$\\tau= \\pi_{n}^{\\mu}$, $\\mu=(a,\\dots,a,\\pm a)$ for some integer $a\\ge 0$ a half-integer\nwhen $n$ is even.\n\n\\item Type $E_6$, $G=E_{6(-14)}, K = Spin(10) \\times U(1) / \\Gamma$, where $\\Gamma$ is a finite subgroup.\n$\\tau=\\pi_{\\omega_1} \\otimes \\chi_n$ for  $n \\equiv 1 \\mod 4$, or $\\pi_{\\omega_2} \\otimes \\chi_n$ for $n \\equiv 3 \\mod 4$. Here $\\omega_1$ and $\\omega_2$ are the positive and negative spin representations of $Spin(10)$ and $U(1) = \\{e^{it \\frac{Z}{2}} \\ | \\ t \\in \\R \\}$.\n\n\\item Type $E_7$,  $G=E_{7(-25)}, K = E_6  U(1)$.\n$\\tau$ is a character.\n\n\\end{enumerate} \n\\end{theorem}", "finiterepprop": "\\begin{proposition}\n\\label{finiterepprop}\n(1) Let $(W_{\\lambda}, G^{\\C})$ be a highest weight representation of $G^{\\C}$ and its Lie algebra $\\mathfrak{g}^{\\C}$. For the multiplicities we have $[V_{\\tau} : W_{\\lambda}|_K] \\leq 1$, and if $V_{\\tau} \\subseteq W_{\\lambda}|_K$ we have $[W_{\\lambda} : I_{\\sigma,-\\lambda|_{\\af} - \\rho_{\\af}}^0 ] \\leq 1$, viewed as Lie algebra representations.  \n\n(2)\nLet $(W_\\lambda, G ^{\\C})$ be\na finite-dimensional\nirreducible representation\nsuch that \n$V_{\\tau} \\subseteq W_{\\lambda}|_K$. Up to  constants there is a unique $G$-equivariant map\n$$F : W_{\\lambda} \\rightarrow C^{\\infty}(G/K,V_{\\tau})$$\ndefined by\n$$F(w)(g) = P_{\\tau} (g^{-1} \\cdot w), g\\in G, w\\in W,$$\nwhere $P_{\\tau}$ is the projection onto $V_{\\tau}$. Moreover, if we normalize $J_{\\lambda}$ so that for all $v \\in V_{\\tau}$ we have $J_{\\lambda}(v) = f_v$, where $f_v(kan) =  a^{\\lambda|_{\\mathfrak{a}}} P_{\\sigma} \\tau(k)^{-1} v$,\nthen there is a factorization\nof $F$ \nas a product $G$-equivariant maps,\n$$\nF=\nS_{\\sigma, -\\lambda|_{\\mathfrak{a}} - \\rho_{\\mathfrak{a}}}\nJ_{\\lambda}: (W_\\lambda, G^{\\C})\\to I_{\\sigma, -\\lambda|_{\\mathfrak{a}} - \\rho_{\\mathfrak{a}}}^0\n\\to C^\\infty(G/K,V_{\\tau}).\n$$\n\\end{proposition}", "egvaluethm": "\\begin{theorem}\n\\label{egvaluethm}\nAssume the irreducible finite-dimensional  representation $V_\\tau$  \nof $K$ is  multiplicity-free  when  restricted to  $M$.\n\n(1) For any $M$-irreducible subrepresentation $(\\sigma,U_\\sigma)$ in $\\tau|_M$,  $f \\in I_{\\sigma,\\nu}^0$ and $D \\in \\mathcal{D}^G(G/K, V_{\\tau})$, we have $DS_{\\sigma, \\nu}(f) = \\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}})S_{\\sigma, \\nu}(f)$ for a constant $\\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}})$.\nFurthermore, we have the invariance property:\n$$\\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}}) = \\omega(D)(s \\cdot \\sigma,s \\cdot \\nu + \\rho_{\\mathfrak{a}}), \\ \\forall \\ s\\in W.$$\n\n(2) Let $(W_\\lambda, G^{\\C})$\nbe a  finite-dimensional\nrepresentation of $\nG^{\\C}$ containing $V_\\tau$.  Let $D \\in \\mathcal{D}^G(G/K,V_{\\tau})$, then $D$ acts on \n$F(W_\\lambda)$\nas the constant\n$$\nDF(w) = \\omega(D)(\\sigma_{\\lambda},-\\lambda|_{\\mathfrak{a}})F(w), \n\\quad  \\ \\forall \\ w \\in W_\\lambda.\n$$\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2244, "pre_theorem_intro_text": "In the present paper we shall study invariant differential operators on homogeneous vector bundles over Hermitian symmetric domains, their eigenfunctions and  eigenvalues. \nLet $G$ be a\nnon-compact semisimple \nHermitian Lie group and $K$ its maximal compact subgroup. Let\n$V_\\tau$\nbe a finite-dimensional irreducible \nrepresentation of  $K$ and $C^\\infty(G/K,V_{\\tau})$ \nbe the space of smooth sections \nof the homogeneous vector \nbundle \n$G\\times_{(K, \\tau)} V_\\tau$\ndefined by $V_\\tau$.\nThe algebraic properties\nof \nthe ring\n$\\mathcal D^G(G/K,V_{\\tau})$ \nof invariant\ndifferential operators on\n$C^\\infty(G/K,V_{\\tau})$ is of fundamental interest, both in analysis on symmetric spaces \n and in the study of universal\n enveloping algebras \\cite{deitmar, helg, ricc, Sh}.\nWhen $(K, \\tau)$\nis a character, i.e., a one-dimensional\nrepresentation, the ring is commutative, and \nShimura has constructed\na linear basis \n$\\{\\mathcal L_{\\mu}\\}$\nconsisting of formally positive \noperators\nby using the Schmid decomposition\nof the symmetric tensor\nalgebra\n$S(\\mathfrak p^+)$\nunder $K$, where \n$\\mathfrak p^+$\nis the holomorphic tangent space of $G/K$\nat $o=K\\in G/K$.\nFor general \n$V_\\tau$ it is proved by Deitmar\n\\cite{deitmar} \nthat \n$\\mathcal D^G(G/K, V_\\tau)$\nis commutative\nif and only if \nthe restriction  $\\tau|_M$\nto $M$ is multiplicity-free, where $M$\nis the centralizer in $K$ of the real Cartan group of $G$. \nIn the present paper we shall\nclassify all such representations\n$(K, \\tau)$. \nWe construct \n eigenfunctions of invariant\ndifferential operators\nusing the Szegö transform and\nwe study the invariance\nproperties of the eigenvalues\nof invariant differential\noperators. This question can be posed for any symmetric space, we study the Hermitian case because there are canonical constructions of invariant differential operators related to the Hua-Kostant-Schmid decomposition \\cite{Sh} of invariant differential operator. It is an interesting problem to further study the eigenvalue problem for those operators;\nsee  \\cite{SZ}\nfor the case of one-dimensional representations $V_\\tau$ of $K$. \n\n\\subsection{Main\nresults and methods}\n\nThe main result of this paper\nis the following theorem; see\nbelow for the exact notation\nand definitions.", "context": "In the present paper we shall study invariant differential operators on homogeneous vector bundles over Hermitian symmetric domains, their eigenfunctions and eigenvalues. \nLet $G$ be a\nnon-compact semisimple \nHermitian Lie group and $K$ its maximal compact subgroup. Let\n$V_\\tau$\nbe a finite-dimensional irreducible \nrepresentation of $K$ and $C^\\infty(G/K,V_{\\tau})$ \nbe the space of smooth sections \nof the homogeneous vector \nbundle \n$G\\times_{(K, \\tau)} V_\\tau$\ndefined by $V_\\tau$.\nThe algebraic properties\nof \nthe ring\n$\\mathcal D^G(G/K,V_{\\tau})$ \nof invariant\ndifferential operators on\n$C^\\infty(G/K,V_{\\tau})$ is of fundamental interest, both in analysis on symmetric spaces \n and in the study of universal\n enveloping algebras \\cite{deitmar, helg, ricc, Sh}.\nWhen $(K, \\tau)$\nis a character, i.e., a one-dimensional\nrepresentation, the ring is commutative, and \nShimura has constructed\na linear basis \n$\\{\\mathcal L_{\\mu}\\}$\nconsisting of formally positive \noperators\nby using the Schmid decomposition\nof the symmetric tensor\nalgebra\n$S(\\mathfrak p^+)$\nunder $K$, where \n$\\mathfrak p^+$\nis the holomorphic tangent space of $G/K$\nat $o=K\\in G/K$.\nFor general \n$V_\\tau$ it is proved by Deitmar\n\\cite{deitmar} \nthat \n$\\mathcal D^G(G/K, V_\\tau)$\nis commutative\nif and only if \nthe restriction $\\tau|_M$\nto $M$ is multiplicity-free, where $M$\nis the centralizer in $K$ of the real Cartan group of $G$. \nIn the present paper we shall\nclassify all such representations\n$(K, \\tau)$. \nWe construct \n eigenfunctions of invariant\ndifferential operators\nusing the Szegö transform and\nwe study the invariance\nproperties of the eigenvalues\nof invariant differential\noperators. This question can be posed for any symmetric space, we study the Hermitian case because there are canonical constructions of invariant differential operators related to the Hua-Kostant-Schmid decomposition \\cite{Sh} of invariant differential operator. It is an interesting problem to further study the eigenvalue problem for those operators;\nsee \\cite{SZ}\nfor the case of one-dimensional representations $V_\\tau$ of $K$.\n\n\\subsection{Main\nresults and methods}\n\nThe main result of this paper\nis the following theorem; see\nbelow for the exact notation\nand definitions.", "full_context": "In the present paper we shall study invariant differential operators on homogeneous vector bundles over Hermitian symmetric domains, their eigenfunctions and eigenvalues. \nLet $G$ be a\nnon-compact semisimple \nHermitian Lie group and $K$ its maximal compact subgroup. Let\n$V_\\tau$\nbe a finite-dimensional irreducible \nrepresentation of $K$ and $C^\\infty(G/K,V_{\\tau})$ \nbe the space of smooth sections \nof the homogeneous vector \nbundle \n$G\\times_{(K, \\tau)} V_\\tau$\ndefined by $V_\\tau$.\nThe algebraic properties\nof \nthe ring\n$\\mathcal D^G(G/K,V_{\\tau})$ \nof invariant\ndifferential operators on\n$C^\\infty(G/K,V_{\\tau})$ is of fundamental interest, both in analysis on symmetric spaces \n and in the study of universal\n enveloping algebras \\cite{deitmar, helg, ricc, Sh}.\nWhen $(K, \\tau)$\nis a character, i.e., a one-dimensional\nrepresentation, the ring is commutative, and \nShimura has constructed\na linear basis \n$\\{\\mathcal L_{\\mu}\\}$\nconsisting of formally positive \noperators\nby using the Schmid decomposition\nof the symmetric tensor\nalgebra\n$S(\\mathfrak p^+)$\nunder $K$, where \n$\\mathfrak p^+$\nis the holomorphic tangent space of $G/K$\nat $o=K\\in G/K$.\nFor general \n$V_\\tau$ it is proved by Deitmar\n\\cite{deitmar} \nthat \n$\\mathcal D^G(G/K, V_\\tau)$\nis commutative\nif and only if \nthe restriction $\\tau|_M$\nto $M$ is multiplicity-free, where $M$\nis the centralizer in $K$ of the real Cartan group of $G$. \nIn the present paper we shall\nclassify all such representations\n$(K, \\tau)$. \nWe construct \n eigenfunctions of invariant\ndifferential operators\nusing the Szegö transform and\nwe study the invariance\nproperties of the eigenvalues\nof invariant differential\noperators. This question can be posed for any symmetric space, we study the Hermitian case because there are canonical constructions of invariant differential operators related to the Hua-Kostant-Schmid decomposition \\cite{Sh} of invariant differential operator. It is an interesting problem to further study the eigenvalue problem for those operators;\nsee \\cite{SZ}\nfor the case of one-dimensional representations $V_\\tau$ of $K$.\n\n\\subsection{Main\nresults and methods}\n\nThe main result of this paper\nis the following theorem; see\nbelow for the exact notation\nand definitions.\n\nIn the present paper we shall study invariant differential operators on homogeneous vector bundles over Hermitian symmetric domains, their eigenfunctions and eigenvalues. \nLet $G$ be a\nnon-compact semisimple \nHermitian Lie group and $K$ its maximal compact subgroup. Let\n$V_\\tau$\nbe a finite-dimensional irreducible \nrepresentation of $K$ and $C^\\infty(G/K,V_{\\tau})$ \nbe the space of smooth sections \nof the homogeneous vector \nbundle \n$G\\times_{(K, \\tau)} V_\\tau$\ndefined by $V_\\tau$.\nThe algebraic properties\nof \nthe ring\n$\\mathcal D^G(G/K,V_{\\tau})$ \nof invariant\ndifferential operators on\n$C^\\infty(G/K,V_{\\tau})$ is of fundamental interest, both in analysis on symmetric spaces \n and in the study of universal\n enveloping algebras \\cite{deitmar, helg, ricc, Sh}.\nWhen $(K, \\tau)$\nis a character, i.e., a one-dimensional\nrepresentation, the ring is commutative, and \nShimura has constructed\na linear basis \n$\\{\\mathcal L_{\\mu}\\}$\nconsisting of formally positive \noperators\nby using the Schmid decomposition\nof the symmetric tensor\nalgebra\n$S(\\mathfrak p^+)$\nunder $K$, where \n$\\mathfrak p^+$\nis the holomorphic tangent space of $G/K$\nat $o=K\\in G/K$.\nFor general \n$V_\\tau$ it is proved by Deitmar\n\\cite{deitmar} \nthat \n$\\mathcal D^G(G/K, V_\\tau)$\nis commutative\nif and only if \nthe restriction $\\tau|_M$\nto $M$ is multiplicity-free, where $M$\nis the centralizer in $K$ of the real Cartan group of $G$. \nIn the present paper we shall\nclassify all such representations\n$(K, \\tau)$. \nWe construct \n eigenfunctions of invariant\ndifferential operators\nusing the Szegö transform and\nwe study the invariance\nproperties of the eigenvalues\nof invariant differential\noperators. This question can be posed for any symmetric space, we study the Hermitian case because there are canonical constructions of invariant differential operators related to the Hua-Kostant-Schmid decomposition \\cite{Sh} of invariant differential operator. It is an interesting problem to further study the eigenvalue problem for those operators;\nsee \\cite{SZ}\nfor the case of one-dimensional representations $V_\\tau$ of $K$.\n\nThe main result of this paper\nis the following theorem; see\nbelow for the exact notation\nand definitions.\n\nThe proof of the \nclassification\nis through a case by case\ncomputation\nand is technically involved.\nWe shall repeatedly use the classical branching rules in\n\\cite{good},\nthe classifications\nof weight-free\nrepresentations\nin \\cite{Howe}\nand multiplicity-free results in\n\\cite{St}.\nThe proof \nfor the uniqueness\nof $F: W_\\lambda \\to \nC^\\infty(G/K, V_\\tau)\n$ is somewhat\nnatural,\nit is done through\nthe realization\n$J: W_\\lambda \n\\to Ind_{MAN}^G(\\sigma\\otimes \\alpha\\otimes 1)$\nof finite-dimensional\nrepresentations $W_\\lambda$\nin the induced representation\nin $Ind_{MAN}^G(\\sigma\\otimes \\alpha\\otimes 1)$,\na factorization\nof $F=SJ$\nas a product of $J$\nand the Poisson-Szeg\\\"o{}\ntransform, and the multiplicity-free property of $V_\\tau$\nunder $M$. The result\non the invariance\nof eigenvalues\nof differential\noperators is \na consequence of a\ngeneral result of Lepowsky \\cite{L}\nand the factorization.\n\n(2)\nLet $(W_\\lambda, G ^{\\C})$ be\na finite-dimensional\nirreducible representation\nsuch that \n$V_{\\tau} \\subseteq W_{\\lambda}|_K$. Up to constants there is a unique $G$-equivariant map\n$$F : W_{\\lambda} \\rightarrow C^{\\infty}(G/K,V_{\\tau})$$\ndefined by\n$$F(w)(g) = P_{\\tau} (g^{-1} \\cdot w), g\\in G, w\\in W,$$\nwhere $P_{\\tau}$ is the projection onto $V_{\\tau}$. Moreover, if we normalize $J_{\\lambda}$ so that for all $v \\in V_{\\tau}$ we have $J_{\\lambda}(v) = f_v$, where $f_v(kan) = a^{\\lambda|_{\\mathfrak{a}}} P_{\\sigma} \\tau(k)^{-1} v$,\nthen there is a factorization\nof $F$ \nas a product $G$-equivariant maps,\n$$\nF=\nS_{\\sigma, -\\lambda|_{\\mathfrak{a}} - \\rho_{\\mathfrak{a}}}\nJ_{\\lambda}: (W_\\lambda, G^{\\C})\\to I_{\\sigma, -\\lambda|_{\\mathfrak{a}} - \\rho_{\\mathfrak{a}}}^0\n\\to C^\\infty(G/K,V_{\\tau}).\n$$\n\\end{proposition}\n\nWe want to prove invariance properties on the eigenvalues of differential operators. These invariance properties have been used in \\cite{SZ} to compute the eigenvalues of Shimura differential operators for one-dimensional representations $\\tau$. We first state some facts derived from \\cite[Section 3]{L}. Note that from the Iwasawa decomposition for the Lie algebra we have the decomposition \n$$U(\\mathfrak g^{\\mathbb C}\n)\n=U(\\mathfrak a^{\\mathbb C}) U(\\mathfrak k^{\\mathbb C})\\oplus\n\\mathfrak n^{\\mathbb C}\nU(\\mathfrak g^{\\mathbb C}).$$\nLet \n$$\n\\Omega:U( \\mathfrak g^{\\mathbb C})\\rightarrow U( \\mathfrak a^{\\mathbb C}) U(\\mathfrak k^{\\mathbb C})\n$$\nbe the correponding\nprojection. We give $U(\\mathfrak a^{\\mathbb C}) U(\\mathfrak k^{\\mathbb C})$ an algebra structure by identifying it with the algebra $U(\\mathfrak a^{\\mathbb C}) \\otimes U(\\mathfrak k^{\\mathbb C})$ and regard $\\Omega$ as a map to $U( \\mathfrak a^{\\mathbb C}) \\otimes U(\\mathfrak k^{\\mathbb C})$. Then $\\Omega(uv)=\\Omega(v)\\Omega(u), $ for any $u \\in U(\\mathfrak g^{\\mathbb C}), v \\in (U(\\mathfrak g^{\\mathbb C})^{K}$ and $$\\Omega(U(\\mathfrak g^{\\mathbb C})^{M})\\subseteq U( \\mathfrak a^{\\mathbb C}) \\otimes U(\\mathfrak k^{\\mathbb C})^{M}.$$\nLet $T:U(\\mathfrak k^{\\mathbb C})\\rightarrow U(\n\\mathfrak k^{\\mathbb C})$ be the canonical anti-automorphism of $U(\\mathfrak k^{\\mathbb C})$ defined by :\n$$T(1)=1, T(x)=-x, T(xy)=T(y)T(x).$$\nPut $$\\Omega_{\\tau}=(1\\otimes \\tau) \\circ (1 \\otimes T)\\circ \\Omega$$\nThen $ \\Omega_{\\tau}:U(\\mathfrak g^{\\mathbb C})^{K}\\rightarrow \nU(\n\\mathfrak a^{\\mathbb C}\n)\\otimes \\mbox{End}_{M}V_{\\tau}$ is an algebra homomorphism.\nAny $\\lambda \\in \\mathfrak (a^{\\mathbb C})'$\ncan be extended to an evaluation on \n$\nU(\\mathfrak a^{\\mathbb C})$, which we also denote by $\\lambda$.\nLet\n$$\\Omega_{\\tau, \\lambda}=(\\lambda\\otimes 1)\\circ \\Omega_{\\tau}:U(\\mathfrak g^{\\mathbb C})^{K}\\rightarrow \\mbox{End}_{M}V_{\\tau}.$$\n\n\\begin{lemma}\n\\label{egvalue}\nFor $v \\in V_{\\tau}$ let $f_{v, \\lambda}\\in C^\\infty(G/K, V_\\tau)$\nbe defined by $f(x) = \\Psi_{\\tau, \\lambda}(x) v, x\\in G$,\nwhere\n$\\Psi_{\\tau, \\lambda}\n$ is defined in (\\ref{Psi}). For any $D \\in U(\\mathfrak{g}^{\\C})^{K}\n\\subset\n(\nU(\\mathfrak{g}^{\\C}\\otimes End(V_\\tau))^{K}\n$\nrealized \nas differential\noperator on $C^\\infty(G/K, V_\\tau)$\nvia $\\nabla(D)$ in\n(\\ref{diffopUnAlg})\nwe have $$(\\nabla(D)f_{v, \\lambda})(x)= \\Psi_{\\tau, \\lambda}(x) \\Omega_{\n\\tau, -\\lambda+\\rho_{\\mathfrak{a}}\n}(D)v, \\ \\forall \\ x \\in G.$$\n \\end{lemma}\n\n(1) For any $M$-irreducible subrepresentation $(\\sigma,U_\\sigma)$ in $\\tau|_M$, $f \\in I_{\\sigma,\\nu}^0$ and $D \\in \\mathcal{D}^G(G/K, V_{\\tau})$, we have $DS_{\\sigma, \\nu}(f) = \\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}})S_{\\sigma, \\nu}(f)$ for a constant $\\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}})$.\nFurthermore, we have the invariance property:\n$$\\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}}) = \\omega(D)(s \\cdot \\sigma,s \\cdot \\nu + \\rho_{\\mathfrak{a}}), \\ \\forall \\ s\\in W.$$\n\n\\begin{proof}(1) Recall that any invariant differential operator $D \\in \\mathcal{D}^G(G/K, V_{\\tau})$ can be represented by an element of $U(\\mathfrak{g}^{\\mathbb C})^K$ via\nthe natural\nmap (\\ref{diffopUnAlg});\nsee also \\cite[Proposition 2.1]{Sh},\nand \n$\\Omega_{\\nu,\\tau}(D)$ \nis well-defined using\nthe representative in $U(\\mathfrak{g}^{\\C})$.\nAs $\\tau|_M$ is multiplicity-free and $\\Omega_{-\\nu + \\rho_{\\mathfrak{a}}}(D)|_{U_{\\sigma}} \\in \\mbox{End}_{M}(U_\\sigma) = \\C I_{U_{\\sigma}}$, by Lemma \\ref{egvalue},\nwe have $D.S_{\\sigma, \\nu}(f) =\\Omega_{-\\nu + \\rho_{\\mathfrak{a}},U_{\\sigma}}(D)S_{\\sigma, \\nu}(f) = \\omega(D)(\\sigma, \\nu + \\rho_{\\mathfrak{a}}) S_{\\sigma,\\nu}(f)$.\nThe invariance property follows directly from\n\\cite[Theorem 9.8(2)]{L}.", "post_theorem_intro_text_len": 5239, "post_theorem_intro_text": "The proof of the \nclassification\nis through a case by case\ncomputation\nand is technically involved.\nWe shall repeatedly use the classical branching rules in\n\\cite{good},\nthe classifications\nof weight-free\nrepresentations\nin \\cite{Howe}\nand multiplicity-free results in\n\\cite{St}.\nThe proof \nfor the uniqueness\nof $F: W_\\lambda \\to \nC^\\infty(G/K, V_\\tau)\n$  is somewhat\nnatural,\nit is done through\nthe realization\n$J: W_\\lambda \n\\to Ind_{MAN}^G(\\sigma\\otimes \\alpha\\otimes 1)$\nof finite-dimensional\nrepresentations $W_\\lambda$\nin the induced representation\nin $Ind_{MAN}^G(\\sigma\\otimes \\alpha\\otimes 1)$,\na factorization\nof $F=SJ$\nas a product of $J$\nand the Poisson-Szeg\\\"o{}\ntransform, and  the multiplicity-free property of $V_\\tau$\nunder $M$. The result\non the invariance\nof eigenvalues\nof  differential\noperators is \na consequence of a\ngeneral result of  Lepowsky \\cite{L}\nand the factorization.\n\n\\subsection{Related\nresults and questions}\n\nWhen  $ \\tau$ is a character Shimura \\cite{Sh} \nhas constructed\na system of $r$ generators\nfor the ring $\\mathcal D^G(G/K, V_\\tau)$ \nand proposed several\nquestions on determining\nthe domains of positivity\nfor the eigenvalues\nof the generators\nand of the \nwhole linear basis\n$\\{\\mathcal L_{\\mu}\\}$\nof formally positive\noperators. A partial\nanswer to these\nquestions has been obtained\nin \\cite{SZ, Zhang}. Here invariance of eigenvalues under the Weyl group was used. \nWe may also pose similar\nquestions as above for $(K, \\tau)$\nin the list of our classification.  \nThis is related to the antihomomorphism $\\Omega:U(\\mathfrak{g})^{K}\\rightarrow  U( \\mathfrak a^{\\mathbb C}) \\otimes U(\\mathfrak k^{\\mathbb C})^{M}$ \nin Section \\ref{sect-4}; see \\cite{ACT1, ACT2}\nfor the case\nof real rank \none groups.\n\nThe computation\nof eigenvalues\nof invariant \ndifferential\noperators is closely\nrelated to the \nproblem of  characterizing\nfinite-dimensional representations\nof $G$ containing\na fixed $K$-type $\\tau$.\nWhen $\\tau$\nis a character there\nis the Cartan-Helgason\ntheorem and its generalization\nby Schlichtkrull \\cite{schlichtkrull}. On\nthe analytic side there is the explicit Plancherel formula for\nthe $L^2(G/K, V_\\tau)$\nby Shimeno \\cite{Shimeno}.\nIt might be possible\nto find a characterization\nof the finite-dimensional\n$G$-representations\ncontaining $(\\tau, K)$\nand to find\nthe discrete\nseries $(\\pi, G)$\ncontained in \n$L^2(G/K, V_\\tau)$,\nin this case \nthey appear with multiplicity\nat most one (see e.g. \\cite[Prop. 2.2, (40)]{Campo},\n\\cite{ricc} or \\cite[Vol. II, Proposition 6.1.1.6]{war}).\n\n\\subsection{List of notation and\nsymbols}\n\n\\begin{enumerate}\n\\item $G/K$: Hermitian\nsymmetric space.\n\n\\item \n$\n\\mathfrak g=\\mathfrak\nk +\\mathfrak p$,\n$\\mathfrak g^{\\mathbb C}=\n\\mathfrak p^{-}\n+\\mathfrak\nk^{\\mathbb C}\n+\\mathfrak p^+\n$: Cartan and\nHarish-Chandra decompositions of the Lie algebra $\\mathfrak g$ of $G$ and its\ncomplexification $\n\\mathfrak g^{\\mathbb C}$.\n\n\\item $\\mathfrak t^{\\mathbb C}\n=\\mathfrak\n(t^-)^{\\mathbb C} \n\\oplus \n\\mathfrak (t^+)^{\\mathbb C} \n\\subset \n\\mathfrak\nk^{\\mathbb C}$:\nCartan\nsubalgebra of $\\mathfrak{k}^{\\mathbb C}$, and\n$\\gamma_r>\\cdots >\\gamma_1$ are the\nHarish-Chandra strongly\nortogonal roots\nwith $\\gamma_j|_{\n\\mathfrak t_{\\mathbb C}^+}=0 $ and \n $\\gamma_r$ being\n the highest non-compact\n root.\n\n\\item \n $\\mathfrak a\n \\subset\\mathfrak p$:\n maximal abelian \nsubspace (Cartan subspace) of $\\mathfrak p$;\n$c: \\mathfrak a^{\\mathbb C}\n\\to \\mathfrak (t^{-})^{\\mathbb C}\n$ is then the Cayley transform and\n$\\alpha_j = \\gamma_j\\circ c$ is the\nCayley transform\nof the Harish-Chandra\northogonal roots $\\gamma_j$\nto $\\mathfrak a^{\\mathbb C}$. Furthermore,\n$M\\subset K, M'\\subset K$, $W=W(\\mathfrak g, \\mathfrak a)$ are respectively the centralizer\nand normalizer of $\\mathfrak a$\nin $K$, and the Weyl group. Also, $\\mathfrak h^{\\mathbb C}=\n\\mathfrak a^{\\mathbb C}\n+\\mathfrak (t^{+})^{\\mathbb C}$.\n\n\\item \n$(W_{\\lambda}, G^{\\mathbb{C}}),\n(V_\\tau, K),\n(U_\\sigma, M)$:\n Finite-dimensional\nirreducible\nrepresentations\nof the respective\ngroups, the highest\nweight $\\lambda$\nbeing defined\non the Cartan subalgebra $\\mathfrak{h}^{\\mathbb{C}}$ of $\\mathfrak{g}^{\\mathbb{C}}$, and\nhighest weight of\n$\\tau$ being defined\non the Cartan subalgebra $\\mathfrak{t}^{\\mathbb{C}}$.\n\n\\item $G=NAK$, \n$g=n(g)e^{H(g)}k(g)$: the Iwasawa decomposition \nof $G$ with $n(g)\\in N, H(g)\\in \\mathfrak{a}, k(g)\\in K$ and $g \\in G$.\n\n\\item $P=MAN$:\nminimal \nparabolic subgroup of $G$.\n\n\\item $I_{\\sigma, \\nu}=Ind_{P}^G(\\sigma\\otimes e^{\\nu}\\otimes 1)$\n: normalized\ninduced representation\nof $G$ from $P$,\nthe corresponding\nrepresentation \nof $\\mathfrak g^{\\mathbb C}$ on the subspace of $K$-finite vectors will\nalso be written as\n$I_{\\sigma, \\nu}^0$.\n\n\\item $C^\\infty(G/K,V_{\\tau})$: \nspace of\nsmooth sections\nof the vector bundle\n$G\\times_K V_\\tau$\nover $G/K$\ndefined by\nthe representation $(K, \\tau)$. We let $\\mathcal D^G(G/K,V_{\\tau})$ then be the ring \nof $G$-invariant\ndifferential operators on \n$C^\\infty(G/K,V_{\\tau})$.\n\n\\end{enumerate}\n\n\\subsection*{Acknowledgments}\nWe would like to thank Pavle Pandzic for informing \nus that some of our results\non the classifications\nof $(V_\\tau, \\tau)$\nhave also been obtained\nearlier by his joint work with \nSoo-Teck Lee and for some further discussions.", "sketch": "The proof breaks into three parts corresponding to (1)--(3).\n\n(1) For the classification, “the proof of the classification is through a case by case computation and is technically involved,” repeatedly using “the classical branching rules in \\cite{good}, the classifications of weight-free representations in \\cite{Howe} and multiplicity-free results in \\cite{St}.”\n\n(2) For uniqueness of the $G$-equivariant map $F:W_\\lambda\\to C^\\infty(G/K,V_\\tau)$, the argument “is done through the realization $J:W_\\lambda\\to \\Ind_{MAN}^G(\\sigma\\otimes\\alpha\\otimes 1)$ of finite-dimensional representations $W_\\lambda$ in the induced representation,” together with “a factorization of $F=SJ$ as a product of $J$ and the Poisson-Szeg\\\"o{} transform,” and using “the multiplicity-free property of $V_\\tau$ under $M$.”\n\n(3) The Weyl-group invariance of eigenvalues of invariant differential operators “is a consequence of a general result of Lepowsky \\cite{L} and the factorization.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "(\\begin{theorem}\n\\label{KresThm}\nLet $(G, K)$ be an irreducible Hermitian symmetric pair\nand\n $M = C_K(\\mathfrak{a})$\n be the centralizer of the\n Cartan subalgebra \n $\\mathfrak{a}\\subset \\mathfrak p$ in $K$. \nThe complete list of $(K, \\tau)$ with $\\tau|_M$ being multiplicity-free is given as follows:\n\n\\begin{enumerate}\n    \\item Type A, \n    $G = SU(r+b,r), \n    K = S(U(r+b) \\times U(r))$; irreducible representations of $U(n)$ with highest\n    weights $\\mu$\n    will be denoted by \n    $\\pi^{\\mu}_n$ as  in \\cite[Theorem 5.5.22]{good}. \n\\begin{enumerate}\n    \\item $r=1$. $\\tau$ is arbitrary.\n\\item  $r=2$. $\\tau= \\pi^{\\mu}_{2+b} \\otimes \\pi^{\\nu}_2$ where $\\pi_{2+b}^{\\mu}$ is\na character and $\\pi_2^{\\nu}$ is any representations, or $\\pi_2^{\\nu}$ is \na character and\n$$\n\\mu = (\\overbrace{l_1+l_2,\\cdots, l_1+l_2}^{{j }},\\overbrace{l_2,\\cdots,l_2}^{{b+2-j }}).\n$$\n\n\\item  $r \\geq 3$. $\\tau= \\pi^{\\mu}_{2+b} \\otimes \\pi^{\\nu}_2$ where $\\pi_{r+b}^{\\mu}$\nor $\\pi_r^{\\nu}$ is a character and the other must be the symmetric\ntensor powers\n$S^m(\\mathbb C^{p})$ \nor  an exterior power\n$\n\\wedge^j \\mathbb C^p$ when restricted to $SL(p)$\n($p=r+b$ or $p=r$\naccordingly).\n\\end{enumerate} \n\n\\item Type C,  $G=Sp(n,\\R)$, $K = U(n)$. $\\tau$ is a character \nor its tensor product\nwith  the exterior power representations\n$\n\\wedge^j \\mathbb C^n$. The same holds for the double cover $(G,K) = (Mp(n,\\R),\\widehat{U(n)})$.\n\n\\item Type D,  $G = SO^*(2n), K = U(n)$. $\\tau$ \nis a character\nor its tensor product\nwith  the symmetric tensor\npowers\n$S^m(\\mathbb C^{p})$  or their duals.\n\n\\item Type BD,  $G=SO_0(n,2)$, \n $K = SO(n) \\times SO(2)$, $n>4$;\nirreducible \nrepresentations\nof $SO(n)$\nwill be denoted\nby \n$\\pi_{n}^{\\mu}$\nas in \\cite[Theorem 19.22]{FulHar}.\n $\\tau=\\pi_{n}^{0}\\otimes \\pi_{2}^\\nu$ \n if  $n$ is odd, i.e., $\\tau$ is a character; for $n$ even $\\tau=\n \\pi_{n}^{\\mu}$ \n or  its \n tensor product\n with a character\n$\\tau=\\pi_{n}^{\\mu}\n\\otimes \\pi_{2}^\\nu$,  \n where $\\mu=(a,\\dots,a,\\pm a)$ for some $a\\ge 0$.\nIf $(G,K) = (Spin_0(n,2), \nSpin(n) \\times SO(2)/\\Z_2)$, where $Spin_0(n,2)$ is the double cover of $SO_0(n,2)$\nthen up to a character $\\tau$ is the spin representation $(\\frac{1}{2}, \\dots, \\frac{1}{2})$ when $n$ is odd, and\n$\\tau= \\pi_{n}^{\\mu}$, $\\mu=(a,\\dots,a,\\pm a)$ for some integer $a\\ge 0$ a half-integer\nwhen $n$ is even.\n\n\\item Type $E_6$, $G=E_{6(-14)}, K = Spin(10) \\times U(1) / \\Gamma$, where $\\Gamma$ is a finite subgroup.\n$\\tau=\\pi_{\\omega_1} \\otimes \\chi_n$ for  $n \\equiv 1 \\mod 4$, or $\\pi_{\\omega_2} \\otimes \\chi_n$ for $n \\equiv 3 \\mod 4$. Here $\\omega_1$ and $\\omega_2$ are the positive and negative spin representations of $Spin(10)$ and $U(1) = \\{e^{it \\frac{Z}{2}} \\ | \\ t \\in \\R \\}$.\n\n\\item Type $E_7$,  $G=E_{7(-25)}, K = E_6  U(1)$.\n$\\tau$ is a character.\n\n\\end{enumerate} \n\\end{theorem},\n\\begin{proposition}\n\\label{finiterepprop}\n(1) Let $(W_{\\lambda}, G^{\\C})$ be a highest weight representation of $G^{\\C}$ and its Lie algebra $\\mathfrak{g}^{\\C}$. For the multiplicities we have $[V_{\\tau} : W_{\\lambda}|_K] \\leq 1$, and if $V_{\\tau} \\subseteq W_{\\lambda}|_K$ we have $[W_{\\lambda} : I_{\\sigma,-\\lambda|_{\\af} - \\rho_{\\af}}^0 ] \\leq 1$, viewed as Lie algebra representations.  \n\n(2)\nLet $(W_\\lambda, G ^{\\C})$ be\na finite-dimensional\nirreducible representation\nsuch that \n$V_{\\tau} \\subseteq W_{\\lambda}|_K$. Up to  constants there is a unique $G$-equivariant map\n$$F : W_{\\lambda} \\rightarrow C^{\\infty}(G/K,V_{\\tau})$$\ndefined by\n$$F(w)(g) = P_{\\tau} (g^{-1} \\cdot w), g\\in G, w\\in W,$$\nwhere $P_{\\tau}$ is the projection onto $V_{\\tau}$. Moreover, if we normalize $J_{\\lambda}$ so that for all $v \\in V_{\\tau}$ we have $J_{\\lambda}(v) = f_v$, where $f_v(kan) =  a^{\\lambda|_{\\mathfrak{a}}} P_{\\sigma} \\tau(k)^{-1} v$,\nthen there is a factorization\nof $F$ \nas a product $G$-equivariant maps,\n$$\nF=\nS_{\\sigma, -\\lambda|_{\\mathfrak{a}} - \\rho_{\\mathfrak{a}}}\nJ_{\\lambda}: (W_\\lambda, G^{\\C})\\to I_{\\sigma, -\\lambda|_{\\mathfrak{a}} - \\rho_{\\mathfrak{a}}}^0\n\\to C^\\infty(G/K,V_{\\tau}).\n$$\n\\end{proposition},\n\\begin{theorem}\n\\label{egvaluethm}\nAssume the irreducible finite-dimensional  representation $V_\\tau$  \nof $K$ is  multiplicity-free  when  restricted to  $M$.\n\n(1) For any $M$-irreducible subrepresentation $(\\sigma,U_\\sigma)$ in $\\tau|_M$,  $f \\in I_{\\sigma,\\nu}^0$ and $D \\in \\mathcal{D}^G(G/K, V_{\\tau})$, we have $DS_{\\sigma, \\nu}(f) = \\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}})S_{\\sigma, \\nu}(f)$ for a constant $\\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}})$.\nFurthermore, we have the invariance property:\n$$\\omega(D)(\\sigma,\\nu + \\rho_{\\mathfrak{a}}) = \\omega(D)(s \\cdot \\sigma,s \\cdot \\nu + \\rho_{\\mathfrak{a}}), \\ \\forall \\ s\\in W.$$\n\n(2) Let $(W_\\lambda, G^{\\C})$\nbe a  finite-dimensional\nrepresentation of $\nG^{\\C}$ containing $V_\\tau$.  Let $D \\in \\mathcal{D}^G(G/K,V_{\\tau})$, then $D$ acts on \n$F(W_\\lambda)$\nas the constant\n$$\nDF(w) = \\omega(D)(\\sigma_{\\lambda},-\\lambda|_{\\mathfrak{a}})F(w), \n\\quad  \\ \\forall \\ w \\in W_\\lambda.\n$$\n\\end{theorem})\n(1) The representation \n$\\tau|_M$ is\nmultiplicity-free if and only if \n$(K, \\tau)$ is in the list above.\n\n(2) Let $(K, \\tau)$ be as\nin (1) acting on $V_\\tau$. For any finite-dimensional irreducible representation\n$(W_\\lambda, G^{\\mathbb{C}})$ there is up to scalar at most one $G$-equivariant map into $C^\\infty(G/K, V_\\tau)$.\n\n(3)\nLet $(K, \\tau)\n$ be as in (1), $\\sigma$\nan irreducible representation in  $\\tau|_M$\nand \n$(W_\\lambda, G^{\\mathbb C})$ \nas\nin (2). Let $D$ \nbe an invariant\ndifferential operator\non $C^\\infty(G/K, V_\\tau)$.\nWrite   the eigenvalues \nof  $D$ on $W_\\lambda$\nas $\\omega(D)(\\sigma_{\\lambda},-\\lambda|_{\\mathfrak{a}}\n)$\nwhich can be extended uniquely to a polynomial of $\n\\lambda|_{\\mathfrak{a}}\n\\in \n(\\mathfrak a^{\\mathbb C})'\n$.\nThen the eigenvalue\n$\\omega(D)$ has following\ninvariance \nproperty,\n$$\\omega(D)(\\sigma,\\kappa + \\rho_{\\mathfrak{a}}) = \\omega(D)(s \\cdot \\sigma,s \\cdot \\kappa + \\rho_{\\mathfrak{a}}), \\ \\forall \\ \\kappa \\in \n(\\mathfrak a^{\\mathbb C})', s\\in W.$$\n", "theorem_type": ["Biconditional or Equivalence", "Universal"], "mcq": {"question": "Let (G,K) be an irreducible Hermitian symmetric pair with Cartan decomposition \\(\\mathfrak g=\\mathfrak k\\oplus\\mathfrak p\\), let \\(\\mathfrak a\\subset \\mathfrak p\\) be a Cartan subalgebra, and let \\(M=C_K(\\mathfrak a)\\). Let \\(\\tau\\) be an irreducible finite-dimensional representation of \\(K\\) on \\(V_\\tau\\), let \\(C^\\infty(G/K,V_\\tau)\\) be the smooth sections of the homogeneous bundle \\(G\\times_{(K,\\tau)}V_\\tau\\), and let \\(\\mathcal D^G(G/K,V_\\tau)\\) be the algebra of \\(G\\)-invariant differential operators on that space. Write \\(W\\) for the Weyl group of \\((\\mathfrak g,\\mathfrak a)\\) and \\(\\rho_{\\mathfrak a}\\) for the half-sum of positive restricted roots. In type A, \\(\\pi_n^\\mu\\) denotes the irreducible \\(U(n)\\)-representation of highest weight \\(\\mu\\); in type BD, \\(\\pi_n^\\mu\\) denotes the irreducible \\(SO(n)\\)-representation of highest weight \\(\\mu\\); in type \\(E_6\\), \\(\\chi_n\\) denotes the weight-\\(n\\) character of \\(U(1)\\), and \\(\\omega_1,\\omega_2\\) are the positive and negative half-spin representations of \\(\\mathrm{Spin}(10)\\). Which statement holds for every such \\((G,K,\\tau)\\)?", "correct_choice": {"label": "A", "text": "The restriction \\(\\tau|_M\\) is multiplicity-free if and only if \\((K,\\tau)\\) is one of the following: (1) Type A: \\(G=SU(r+b,r)\\), \\(K=S(U(r+b)\\times U(r))\\): (a) if \\(r=1\\), then \\(\\tau\\) is arbitrary; (b) if \\(r=2\\), then \\(\\tau=\\pi_{b+2}^{\\mu}\\otimes \\pi_2^{\\nu}\\), where either \\(\\pi_{b+2}^{\\mu}\\) is a character and \\(\\pi_2^{\\nu}\\) is arbitrary, or \\(\\pi_2^{\\nu}\\) is a character and \\(\\mu=(\\overbrace{l_1+l_2,\\dots,l_1+l_2}^{j},\\overbrace{l_2,\\dots,l_2}^{b+2-j})\\); (c) if \\(r\\ge 3\\), then \\(\\tau=\\pi_{r+b}^{\\mu}\\otimes \\pi_r^{\\nu}\\), where one of \\(\\pi_{r+b}^{\\mu}\\) or \\(\\pi_r^{\\nu}\\) is a character and the other, upon restriction to \\(SL(p)\\) (with \\(p=r+b\\) or \\(p=r\\), respectively), is either a symmetric power \\(S^m(\\mathbb C^p)\\) or an exterior power \\(\\wedge^j\\mathbb C^p\\); (2) Type C: \\(G=Sp(n,\\mathbb R)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with an exterior power \\(\\wedge^j\\mathbb C^n\\); the same holds for the double cover \\((G,K)=(Mp(n,\\mathbb R),\\widehat{U(n)})\\); (3) Type D: \\(G=SO^*(2n)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with a symmetric tensor power \\(S^m(\\mathbb C^n)\\) or its dual; (4) Type BD: \\(G=SO_0(n,2)\\), \\(K=SO(n)\\times SO(2)\\), \\(n>4\\); if \\(n\\) is odd, then \\(\\tau=\\pi_n^0\\otimes \\pi_2^\\nu\\), i.e. \\(\\tau\\) is a character; if \\(n\\) is even, then \\(\\tau=\\pi_n^\\mu\\) or \\(\\tau=\\pi_n^\\mu\\otimes \\pi_2^\\nu\\), where \\(\\mu=(a,\\dots,a,\\pm a)\\) for some \\(a\\ge 0\\); for the double cover \\((G,K)=(\\mathrm{Spin}_0(n,2),\\,\\mathrm{Spin}(n)\\times SO(2)/\\mathbb Z_2)\\), up to a character \\(\\tau\\) is the spin representation \\((\\tfrac12,\\dots,\\tfrac12)\\) when \\(n\\) is odd, and when \\(n\\) is even one has \\(\\tau=\\pi_n^\\mu\\) with \\(\\mu=(a,\\dots,a,\\pm a)\\) for some nonnegative integral or half-integral \\(a\\); (5) Type \\(E_6\\): \\(G=E_{6(-14)}\\), \\(K=\\mathrm{Spin}(10)\\times U(1)/\\Gamma\\), and \\(\\tau=\\pi_{\\omega_1}\\otimes \\chi_n\\) for \\(n\\equiv 1\\pmod 4\\), or \\(\\tau=\\pi_{\\omega_2}\\otimes \\chi_n\\) for \\(n\\equiv 3\\pmod 4\\); (6) Type \\(E_7\\): \\(G=E_{7(-25)}\\), \\(K=E_6U(1)\\), and \\(\\tau\\) is a character. Moreover, for every \\((K,\\tau)\\) in this list and every finite-dimensional irreducible representation \\((W_\\lambda,G^{\\mathbb C})\\), there is, up to scalar, at most one \\(G\\)-equivariant map \\(W_\\lambda\\to C^\\infty(G/K,V_\\tau)\\). Finally, if \\(\\sigma\\) is an irreducible summand of \\(\\tau|_M\\), \\(D\\in \\mathcal D^G(G/K,V_\\tau)\\), and the eigenvalue of \\(D\\) on \\(W_\\lambda\\) is written as \\(\\omega(D)(\\sigma_\\lambda,-\\lambda|_{\\mathfrak a})\\) and extended uniquely to a polynomial in \\(\\lambda|_{\\mathfrak a}\\in (\\mathfrak a^{\\mathbb C})'\\), then \\(\\omega(D)\\) satisfies the Weyl-group invariance property \\(\\omega(D)(\\sigma,\\kappa+\\rho_{\\mathfrak a})=\\omega(D)(s\\cdot \\sigma,s\\cdot \\kappa+\\rho_{\\mathfrak a})\\) for all \\(\\kappa\\in (\\mathfrak a^{\\mathbb C})'\\) and all \\(s\\in W\\)."}, "choices": [{"label": "B", "text": "The restriction \\(\\tau|_M\\) is multiplicity-free if and only if \\((K,\\tau)\\) is one of the following: (1) Type A: \\(G=SU(r+b,r)\\), \\(K=S(U(r+b)\\times U(r))\\): (a) if \\(r=1\\), then \\(\\tau\\) is arbitrary; (b) if \\(r=2\\), then \\(\\tau=\\pi_{b+2}^{\\mu}\\otimes \\pi_2^{\\nu}\\), where either \\(\\pi_{b+2}^{\\mu}\\) is a character and \\(\\pi_2^{\\nu}\\) is arbitrary, or \\(\\pi_2^{\\nu}\\) is a character and \\(\\mu=(\\overbrace{l_1+l_2,\\dots,l_1+l_2}^{j},\\overbrace{l_2,\\dots,l_2}^{b+2-j})\\); (c) if \\(r\\ge 3\\), then \\(\\tau=\\pi_{r+b}^{\\mu}\\otimes \\pi_r^{\\nu}\\), where one of \\(\\pi_{r+b}^{\\mu}\\) or \\(\\pi_r^{\\nu}\\) is a character and the other, upon restriction to \\(SL(p)\\) (with \\(p=r+b\\) or \\(p=r\\), respectively), is either a symmetric power \\(S^m(\\mathbb C^p)\\) or an exterior power \\(\\wedge^j\\mathbb C^p\\); (2) Type C: \\(G=Sp(n,\\mathbb R)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with an exterior power \\(\\wedge^j\\mathbb C^n\\); the same holds for the double cover \\((G,K)=(Mp(n,\\mathbb R),\\widehat{U(n)})\\); (3) Type D: \\(G=SO^*(2n)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with a symmetric tensor power \\(S^m(\\mathbb C^n)\\), an exterior power \\(\\wedge^j\\mathbb C^n\\), or their duals; (4) Type BD: \\(G=SO_0(n,2)\\), \\(K=SO(n)\\times SO(2)\\), \\(n>4\\); if \\(n\\) is odd, then \\(\\tau=\\pi_n^0\\otimes \\pi_2^\\nu\\), i.e. \\(\\tau\\) is a character; if \\(n\\) is even, then \\(\\tau=\\pi_n^\\mu\\) or \\(\\tau=\\pi_n^\\mu\\otimes \\pi_2^\\nu\\), where \\(\\mu=(a,\\dots,a,\\pm a)\\) for some \\(a\\ge 0\\); for the double cover \\((G,K)=(\\mathrm{Spin}_0(n,2),\\,\\mathrm{Spin}(n)\\times SO(2)/\\mathbb Z_2)\\), up to a character \\(\\tau\\) is the spin representation \\((\\tfrac12,\\dots,\\tfrac12)\\) when \\(n\\) is odd, and when \\(n\\) is even one has \\(\\tau=\\pi_n^\\mu\\) with \\(\\mu=(a,\\dots,a,\\pm a)\\) for some nonnegative integral or half-integral \\(a\\); (5) Type \\(E_6\\): \\(G=E_{6(-14)}\\), \\(K=\\mathrm{Spin}(10)\\times U(1)/\\Gamma\\), and \\(\\tau=\\pi_{\\omega_1}\\otimes \\chi_n\\) for \\(n\\equiv 1\\pmod 4\\), or \\(\\tau=\\pi_{\\omega_2}\\otimes \\chi_n\\) for \\(n\\equiv 3\\pmod 4\\); (6) Type \\(E_7\\): \\(G=E_{7(-25)}\\), \\(K=E_6U(1)\\), and \\(\\tau\\) is a character. Moreover, for every \\((K,\\tau)\\) in this list and every finite-dimensional irreducible representation \\((W_\\lambda,G^{\\mathbb C})\\), there exists a unique \\(G\\)-equivariant map \\(W_\\lambda\\to C^\\infty(G/K,V_\\tau)\\). Finally, if \\(\\sigma\\) is an irreducible summand of \\(\\tau|_M\\), \\(D\\in \\mathcal D^G(G/K,V_\\tau)\\), and the eigenvalue of \\(D\\) on \\(W_\\lambda\\) is written as \\(\\omega(D)(\\sigma_\\lambda,-\\lambda|_{\\mathfrak a})\\) and extended uniquely to a polynomial in \\(\\lambda|_{\\mathfrak a}\\in (\\mathfrak a^{\\mathbb C})'\\), then \\(\\omega(D)\\) satisfies the Weyl-group invariance property \\(\\omega(D)(\\sigma,\\kappa+\\rho_{\\mathfrak a})=\\omega(D)(s\\cdot \\sigma,s\\cdot \\kappa+\\rho_{\\mathfrak a})\\) for all \\(\\kappa\\in (\\mathfrak a^{\\mathbb C})'\\) and all \\(s\\in W\\)."}, {"label": "C", "text": "If \\(\\tau|_M\\) is multiplicity-free, then \\((K,\\tau)\\) is one of the representations in the classification listed above. Moreover, for every such \\((K,\\tau)\\) and every finite-dimensional irreducible representation \\((W_\\lambda,G^{\\mathbb C})\\), there is, up to scalar, at most one \\(G\\)-equivariant map \\(W_\\lambda\\to C^\\infty(G/K,V_\\tau)\\). Finally, if \\(\\sigma\\) is an irreducible summand of \\(\\tau|_M\\), \\(D\\in \\mathcal D^G(G/K,V_\\tau)\\), and the eigenvalue of \\(D\\) on \\(W_\\lambda\\) is written as \\(\\omega(D)(\\sigma_\\lambda,-\\lambda|_{\\mathfrak a})\\) and extended uniquely to a polynomial in \\(\\lambda|_{\\mathfrak a}\\in (\\mathfrak a^{\\mathbb C})'\\), then \\(\\omega(D)(\\sigma,\\kappa+\\rho_{\\mathfrak a})=\\omega(D)(s\\cdot \\sigma,s\\cdot \\kappa+\\rho_{\\mathfrak a})\\) for all \\(\\kappa\\in (\\mathfrak a^{\\mathbb C})'\\) and all \\(s\\in W\\)."}, {"label": "D", "text": "The restriction \\(\\tau|_M\\) is multiplicity-free if and only if \\((K,\\tau)\\) is one of the following: (1) Type A: \\(G=SU(r+b,r)\\), \\(K=S(U(r+b)\\times U(r))\\): (a) if \\(r=1\\), then \\(\\tau\\) is arbitrary; (b) if \\(r=2\\), then \\(\\tau=\\pi_{b+2}^{\\mu}\\otimes \\pi_2^{\\nu}\\), where either \\(\\pi_{b+2}^{\\mu}\\) is a character and \\(\\pi_2^{\\nu}\\) is arbitrary, or \\(\\pi_2^{\\nu}\\) is a character and \\(\\mu=(\\overbrace{l_1+l_2,\\dots,l_1+l_2}^{j},\\overbrace{l_2,\\dots,l_2}^{b+2-j})\\); (c) if \\(r\\ge 3\\), then \\(\\tau=\\pi_{r+b}^{\\mu}\\otimes \\pi_r^{\\nu}\\), where one of \\(\\pi_{r+b}^{\\mu}\\) or \\(\\pi_r^{\\nu}\\) is a character and the other, upon restriction to \\(SL(p)\\) (with \\(p=r+b\\) or \\(p=r\\), respectively), is either a symmetric power \\(S^m(\\mathbb C^p)\\) or an exterior power \\(\\wedge^j\\mathbb C^p\\); (2) Type C: \\(G=Sp(n,\\mathbb R)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with an exterior power \\(\\wedge^j\\mathbb C^n\\); the same holds for the double cover \\((G,K)=(Mp(n,\\mathbb R),\\widehat{U(n)})\\); (3) Type D: \\(G=SO^*(2n)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with a symmetric tensor power \\(S^m(\\mathbb C^n)\\) or its dual; (4) Type BD: \\(G=SO_0(n,2)\\), \\(K=SO(n)\\times SO(2)\\), \\(n\\ge 4\\); if \\(n\\) is odd, then \\(\\tau=\\pi_n^0\\otimes \\pi_2^\\nu\\), i.e. \\(\\tau\\) is a character; if \\(n\\) is even, then \\(\\tau=\\pi_n^\\mu\\) or \\(\\tau=\\pi_n^\\mu\\otimes \\pi_2^\\nu\\), where \\(\\mu=(a,\\dots,a,\\pm a)\\) for some \\(a\\ge 0\\); for the double cover \\((G,K)=(\\mathrm{Spin}_0(n,2),\\,\\mathrm{Spin}(n)\\times SO(2)/\\mathbb Z_2)\\), up to a character \\(\\tau\\) is the spin representation \\((\\tfrac12,\\dots,\\tfrac12)\\) when \\(n\\) is odd, and when \\(n\\) is even one has \\(\\tau=\\pi_n^\\mu\\) with \\(\\mu=(a,\\dots,a,\\pm a)\\) for some nonnegative integral or half-integral \\(a\\); (5) Type \\(E_6\\): \\(G=E_{6(-14)}\\), \\(K=\\mathrm{Spin}(10)\\times U(1)/\\Gamma\\), and \\(\\tau=\\pi_{\\omega_1}\\otimes \\chi_n\\) for \\(n\\equiv 1\\pmod 4\\), or \\(\\tau=\\pi_{\\omega_2}\\otimes \\chi_n\\) for \\(n\\equiv 3\\pmod 4\\); (6) Type \\(E_7\\): \\(G=E_{7(-25)}\\), \\(K=E_6U(1)\\), and \\(\\tau\\) is a character. Moreover, for every \\((K,\\tau)\\) in this list and every finite-dimensional irreducible representation \\((W_\\lambda,G^{\\mathbb C})\\), there is, up to scalar, at most one \\(G\\)-equivariant map \\(W_\\lambda\\to C^\\infty(G/K,V_\\tau)\\). Finally, if \\(\\sigma\\) is an irreducible summand of \\(\\tau|_M\\), \\(D\\in \\mathcal D^G(G/K,V_\\tau)\\), and the eigenvalue of \\(D\\) on \\(W_\\lambda\\) is written as \\(\\omega(D)(\\sigma_\\lambda,-\\lambda|_{\\mathfrak a})\\) and extended uniquely to a polynomial in \\(\\lambda|_{\\mathfrak a}\\in (\\mathfrak a^{\\mathbb C})'\\), then \\(\\omega(D)\\) satisfies the Weyl-group invariance property \\(\\omega(D)(\\sigma,\\kappa+\\rho_{\\mathfrak a})=\\omega(D)(s\\cdot \\sigma,s\\cdot \\kappa+\\rho_{\\mathfrak a})\\) for all \\(\\kappa\\in (\\mathfrak a^{\\mathbb C})'\\) and all \\(s\\in W\\)."}, {"label": "E", "text": "The restriction \\(\\tau|_M\\) is multiplicity-free if and only if \\((K,\\tau)\\) is one of the following: (1) Type A: \\(G=SU(r+b,r)\\), \\(K=S(U(r+b)\\times U(r))\\): (a) if \\(r=1\\), then \\(\\tau\\) is arbitrary; (b) if \\(r=2\\), then \\(\\tau=\\pi_{b+2}^{\\mu}\\otimes \\pi_2^{\\nu}\\), where either \\(\\pi_{b+2}^{\\mu}\\) is a character and \\(\\pi_2^{\\nu}\\) is arbitrary, or \\(\\pi_2^{\\nu}\\) is a character and \\(\\mu=(\\overbrace{l_1+l_2,\\dots,l_1+l_2}^{j},\\overbrace{l_2,\\dots,l_2}^{b+2-j})\\); (c) if \\(r\\ge 3\\), then \\(\\tau=\\pi_{r+b}^{\\mu}\\otimes \\pi_r^{\\nu}\\), where one of \\(\\pi_{r+b}^{\\mu}\\) or \\(\\pi_r^{\\nu}\\) is a character and the other, upon restriction to \\(SL(p)\\) (with \\(p=r+b\\) or \\(p=r\\), respectively), is either a symmetric power \\(S^m(\\mathbb C^p)\\) or an exterior power \\(\\wedge^j\\mathbb C^p\\); (2) Type C: \\(G=Sp(n,\\mathbb R)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with an exterior power \\(\\wedge^j\\mathbb C^n\\); the same holds for the double cover \\((G,K)=(Mp(n,\\mathbb R),\\widehat{U(n)})\\); (3) Type D: \\(G=SO^*(2n)\\), \\(K=U(n)\\), and \\(\\tau\\) is a character or its tensor product with a symmetric tensor power \\(S^m(\\mathbb C^n)\\) or its dual; (4) Type BD: \\(G=SO_0(n,2)\\), \\(K=SO(n)\\times SO(2)\\), \\(n>4\\); if \\(n\\) is odd, then \\(\\tau=\\pi_n^0\\otimes \\pi_2^\\nu\\), i.e. \\(\\tau\\) is a character; if \\(n\\) is even, then \\(\\tau=\\pi_n^\\mu\\) or \\(\\tau=\\pi_n^\\mu\\otimes \\pi_2^\\nu\\), where \\(\\mu=(a,\\dots,a,\\pm a)\\) for some \\(a\\ge 0\\); for the double cover \\((G,K)=(\\mathrm{Spin}_0(n,2),\\,\\mathrm{Spin}(n)\\times SO(2)/\\mathbb Z_2)\\), up to a character \\(\\tau\\) is the spin representation \\((\\tfrac12,\\dots,\\tfrac12)\\) when \\(n\\) is odd, and when \\(n\\) is even one has \\(\\tau=\\pi_n^\\mu\\) with \\(\\mu=(a,\\dots,a,\\pm a)\\) for some nonnegative integral or half-integral \\(a\\); (5) Type \\(E_6\\): \\(G=E_{6(-14)}\\), \\(K=\\mathrm{Spin}(10)\\times U(1)/\\Gamma\\), and \\(\\tau=\\pi_{\\omega_1}\\otimes \\chi_n\\) for \\(n\\equiv 1\\pmod 4\\), or \\(\\tau=\\pi_{\\omega_2}\\otimes \\chi_n\\) for \\(n\\equiv 3\\pmod 4\\); (6) Type \\(E_7\\): \\(G=E_{7(-25)}\\), \\(K=E_6U(1)\\), and \\(\\tau\\) is a character. Moreover, for every \\((K,\\tau)\\) in this list and every finite-dimensional irreducible representation \\((W_\\lambda,G^{\\mathbb C})\\), there is, up to scalar, at most one \\(G\\)-equivariant map \\(W_\\lambda\\to C^\\infty(G/K,V_\\tau)\\). Finally, if \\(\\sigma\\) is an irreducible summand of \\(\\tau|_M\\), \\(D\\in \\mathcal D^G(G/K,V_\\tau)\\), and the eigenvalue of \\(D\\) on \\(W_\\lambda\\) is written as \\(\\omega(D)(\\sigma_\\lambda,-\\lambda|_{\\mathfrak a})\\) and extended uniquely to a polynomial in \\(\\lambda|_{\\mathfrak a}\\in (\\mathfrak a^{\\mathbb C})'\\), then \\(\\omega(D)\\) satisfies the stronger invariance property \\(\\omega(D)(\\sigma,\\kappa)=\\omega(D)(s\\cdot \\sigma,s\\cdot \\kappa)\\) for all \\(\\kappa\\in (\\mathfrak a^{\\mathbb C})'\\) and all \\(s\\in W\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "uniqueness up to scalar replaced by absolute existence and uniqueness", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "case_split", "tampered_component": "dropped the converse direction in the classification equivalence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "type BD boundary changed from n>4 to n\\ge 4", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "Weyl-invariance shift by \\rho_{\\mathfrak a} removed", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem introduces notation and context only; it does not state the classification, uniqueness, or Weyl-invariance conclusion. There is no explicit or trivial cue pointing to choice A."}, "TAS": {"score": 1, "justification": "The item is not a literal restatement of the stem, but it is essentially a theorem-recognition question: the student is asked to identify the exact global statement of a classification theorem from near-verbatim variants."}, "GPS": {"score": 1, "justification": "The question requires careful comparison of subtle logical and technical differences across options, but it mainly tests precise recall/discrimination rather than generative mathematical reasoning from the setup."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they alter converse direction, uniqueness strength, boundary cases, and the rho-shift in Weyl invariance. These are distinct and aligned with realistic failure modes."}, "total_score": 6, "overall_assessment": "A strong theorem-discrimination MCQ with excellent distractors and little answer leakage, but it leans more toward recognition of an exact classification statement than genuine generative reasoning."}}
{"id": "2602.14931v1", "paper_link": "http://arxiv.org/abs/2602.14931v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{thm:char}\n\nLet $\\lambda$ be a partition with $n$ parts, and let \n$r_i = \\lambda_i - \\lambda_{i+1}$ for $1 \\le i < n$, and \n$r_n = \\lambda_n$. \nThus $r_i$ denotes the number of columns of height $i$ in $\\lambda$.\n\nDefine\n\\[\na_i = \\left\\lfloor \\frac{r_i}{2} \\right\\rfloor,\n\\qquad\nb_i = \\left\\lceil \\frac{r_i}{2} \\right\\rceil,\n\\]\nor vice versa.\n\nAssume Conjecture~\\ref{conj:hankel} holds, so that every minimal matrix is Hankel. \nLet $M=(m_{i,j})$ be a minimal Hankel matrix of shape $\\lambda$, \nand write $m_{i,j}=s_{i+j}$. Then\n\\[\ns_i = a_{i-1}\n\\quad \\text{and} \\quad\ns_{2n-i} = b_{i-1}\n\\qquad\n\\text{for } 2 \\le i \\le n.\n\\]", "start_pos": 6856, "end_pos": 7507, "label": "thm:char"}, "ref_dict": {"conj:hankel": "\\begin{conj}\\label{conj:hankel}\nLet $\\lambda$ be a partition with $n$ parts, and let \n$M=(m_{i,j})_{1\\le i,j \\le n} \\in \\mathcal{M}_\\lambda$ \nbe a minimal matrix of shape $\\lambda$. Then $M$ is symmetric, that is,\n\\[\nm_{i,j} = m_{j,i} \\quad \\text{for all } 1 \\le i,j \\le n.\n\\]\nMoreover, every minimal matrix of shape $\\lambda$ is Hankel: there exists a sequence of integers \n\\[\ns_2, s_3, \\dots, s_{2n}\n\\]\nsuch that\n\\[\nm_{i,j} = s_{i+j}\n\\quad \\text{for all } 1 \\le i,j \\le n.\n\\]\n\n\\end{conj}"}, "pre_theorem_intro_text_len": 3487, "pre_theorem_intro_text": "\\label{sec:introPASEP}\n\nA permutation has many statistics associated with it. One of the most fundamental is the length of its longest increasing subsequence. Robinson~\\cite{Robinson} introduced the insertion algorithm in the study of representations of the symmetric group. Decades later, Schensted~\\cite{Schensted} independently rediscovered it and connected it to the problem of longest increasing and decreasing subsequences. The resulting correspondence, now known as the Robinson-Schensted algorithm, establishes a bijection between a permutation of $N$ elements and a pair of standard Young tableaux of the same shape, where the shape is a partition of $N$.\n\nMoreover, this partition can also be characterised in terms of increasing subsequences of the permutation, via Greene's theorem~\\cite{Greene}. By a slight abuse of notation, we will refer to this partition as the shape of the permutation $\\omega$. \n\nKnuth later generalised the Robinson-Schensted algorithm to associate a nonnegative integer matrix with a pair of semi-standard Young tableaux. This extension is known as the Robinson-Schensted-Knuth (RSK) algorithm. In this setting, the shape of a matrix is defined to be the common shape of the resulting pair of semistandard Young tableaux.\n\nAnother classic statistic of a permutation is its inversion number. An inversion of a permutation $\\omega$ is a pair of entries $(\\omega_i, \\omega_j)$ that appear in descending order, that is, indices $i<j$ such that $\\omega_i>\\omega_j$.\n\nHohlweg~\\cite{Hohlweg} studied the set of permutations of a fixed shape $\\lambda$, denoted $\\mathcal{S}_\\lambda$, and characterised those with the minimum number of inversions. He also obtained an explicit formula for this minimum number. Han~\\cite{Han} later provided a combinatorial proof of Hohlweg's result using Viennot's geometric construction involving shadow line representation of a permutation~\\cite{Sagan}.\n\nIn this paper, we consider the corresponding set of matrices $\\mathcal{M}_\\lambda$ and aim to generalise Hohlweg's results to this setting. Since a permutation of $N$ elements can be represented as a $0$-$1$ matrix of size $N \\times N$, the minimal permutations characterised by Hohlweg and Han belong to $\\mathcal{M}_\\lambda$, and are in fact minimal in this larger set as well. Our focus, however, is on the non-trivial case in which the size of the matrices in $\\mathcal{M}_\\lambda$ is as small as possible. This minimal size is equal to the number of parts of the partition $\\lambda$.\n\nWe therefore fix a partition $\\lambda$ with $n$ parts and restrict our attention to matrices in $M_\\lambda$ of size $n \\times n$. Our goal is to study the inversion statistic on this set, and in particular to determine the minimum number of inversions among matrices of shape $\\lambda$.  A matrix $M \\in \\mathcal{M}_\\lambda$ that attains this minimum will be called a \\textit{minimal matrix of shape} $\\lambda$. This matrix need not be unique.\n\nThis leads us to the following conjecture.\n\n\\begin{conj}\\label{conj:hankel}\nLet $\\lambda$ be a partition with $n$ parts, and let \n$M=(m_{i,j})_{1\\le i,j \\le n} \\in \\mathcal{M}_\\lambda$ \nbe a minimal matrix of shape $\\lambda$. Then $M$ is symmetric, that is,\n\\[\nm_{i,j} = m_{j,i} \\quad \\text{for all } 1 \\le i,j \\le n.\n\\]\nMoreover, every minimal matrix of shape $\\lambda$ is Hankel: there exists a sequence of integers \n\\[\ns_2, s_3, \\dots, s_{2n}\n\\]\nsuch that\n\\[\nm_{i,j} = s_{i+j}\n\\quad \\text{for all } 1 \\le i,j \\le n.\n\\]\n\n\\end{conj}", "context": "A permutation has many statistics associated with it. One of the most fundamental is the length of its longest increasing subsequence. Robinson~\\cite{Robinson} introduced the insertion algorithm in the study of representations of the symmetric group. Decades later, Schensted~\\cite{Schensted} independently rediscovered it and connected it to the problem of longest increasing and decreasing subsequences. The resulting correspondence, now known as the Robinson-Schensted algorithm, establishes a bijection between a permutation of $N$ elements and a pair of standard Young tableaux of the same shape, where the shape is a partition of $N$.\n\nHohlweg~\\cite{Hohlweg} studied the set of permutations of a fixed shape $\\lambda$, denoted $\\mathcal{S}_\\lambda$, and characterised those with the minimum number of inversions. He also obtained an explicit formula for this minimum number. Han~\\cite{Han} later provided a combinatorial proof of Hohlweg's result using Viennot's geometric construction involving shadow line representation of a permutation~\\cite{Sagan}.\n\nIn this paper, we consider the corresponding set of matrices $\\mathcal{M}_\\lambda$ and aim to generalise Hohlweg's results to this setting. Since a permutation of $N$ elements can be represented as a $0$-$1$ matrix of size $N \\times N$, the minimal permutations characterised by Hohlweg and Han belong to $\\mathcal{M}_\\lambda$, and are in fact minimal in this larger set as well. Our focus, however, is on the non-trivial case in which the size of the matrices in $\\mathcal{M}_\\lambda$ is as small as possible. This minimal size is equal to the number of parts of the partition $\\lambda$.\n\nWe therefore fix a partition $\\lambda$ with $n$ parts and restrict our attention to matrices in $M_\\lambda$ of size $n \\times n$. Our goal is to study the inversion statistic on this set, and in particular to determine the minimum number of inversions among matrices of shape $\\lambda$. A matrix $M \\in \\mathcal{M}_\\lambda$ that attains this minimum will be called a \\textit{minimal matrix of shape} $\\lambda$. This matrix need not be unique.\n\n\\begin{conj}\\label{conj:hankel}\nLet $\\lambda$ be a partition with $n$ parts, and let \n$M=(m_{i,j})_{1\\le i,j \\le n} \\in \\mathcal{M}_\\lambda$ \nbe a minimal matrix of shape $\\lambda$. Then $M$ is symmetric, that is,\n\\[\nm_{i,j} = m_{j,i} \\quad \\text{for all } 1 \\le i,j \\le n.\n\\]\nMoreover, every minimal matrix of shape $\\lambda$ is Hankel: there exists a sequence of integers \n\\[\ns_2, s_3, \\dots, s_{2n}\n\\]\nsuch that\n\\[\nm_{i,j} = s_{i+j}\n\\quad \\text{for all } 1 \\le i,j \\le n.\n\\]\n\n\\end{conj}", "full_context": "A permutation has many statistics associated with it. One of the most fundamental is the length of its longest increasing subsequence. Robinson~\\cite{Robinson} introduced the insertion algorithm in the study of representations of the symmetric group. Decades later, Schensted~\\cite{Schensted} independently rediscovered it and connected it to the problem of longest increasing and decreasing subsequences. The resulting correspondence, now known as the Robinson-Schensted algorithm, establishes a bijection between a permutation of $N$ elements and a pair of standard Young tableaux of the same shape, where the shape is a partition of $N$.\n\nHohlweg~\\cite{Hohlweg} studied the set of permutations of a fixed shape $\\lambda$, denoted $\\mathcal{S}_\\lambda$, and characterised those with the minimum number of inversions. He also obtained an explicit formula for this minimum number. Han~\\cite{Han} later provided a combinatorial proof of Hohlweg's result using Viennot's geometric construction involving shadow line representation of a permutation~\\cite{Sagan}.\n\nIn this paper, we consider the corresponding set of matrices $\\mathcal{M}_\\lambda$ and aim to generalise Hohlweg's results to this setting. Since a permutation of $N$ elements can be represented as a $0$-$1$ matrix of size $N \\times N$, the minimal permutations characterised by Hohlweg and Han belong to $\\mathcal{M}_\\lambda$, and are in fact minimal in this larger set as well. Our focus, however, is on the non-trivial case in which the size of the matrices in $\\mathcal{M}_\\lambda$ is as small as possible. This minimal size is equal to the number of parts of the partition $\\lambda$.\n\nWe therefore fix a partition $\\lambda$ with $n$ parts and restrict our attention to matrices in $M_\\lambda$ of size $n \\times n$. Our goal is to study the inversion statistic on this set, and in particular to determine the minimum number of inversions among matrices of shape $\\lambda$. A matrix $M \\in \\mathcal{M}_\\lambda$ that attains this minimum will be called a \\textit{minimal matrix of shape} $\\lambda$. This matrix need not be unique.\n\n\\begin{conj}\\label{conj:hankel}\nLet $\\lambda$ be a partition with $n$ parts, and let \n$M=(m_{i,j})_{1\\le i,j \\le n} \\in \\mathcal{M}_\\lambda$ \nbe a minimal matrix of shape $\\lambda$. Then $M$ is symmetric, that is,\n\\[\nm_{i,j} = m_{j,i} \\quad \\text{for all } 1 \\le i,j \\le n.\n\\]\nMoreover, every minimal matrix of shape $\\lambda$ is Hankel: there exists a sequence of integers \n\\[\ns_2, s_3, \\dots, s_{2n}\n\\]\nsuch that\n\\[\nm_{i,j} = s_{i+j}\n\\quad \\text{for all } 1 \\le i,j \\le n.\n\\]\n\n\\end{conj}\n\n\\begin{conj}\\label{conj:hankel}\nLet $\\lambda$ be a partition with $n$ parts, and let \n$M=(m_{i,j})_{1\\le i,j \\le n} \\in \\mathcal{M}_\\lambda$ \nbe a minimal matrix of shape $\\lambda$. Then $M$ is symmetric, that is,\n\\[\nm_{i,j} = m_{j,i} \\quad \\text{for all } 1 \\le i,j \\le n.\n\\]\nMoreover, every minimal matrix of shape $\\lambda$ is Hankel: there exists a sequence of integers \n\\[\ns_2, s_3, \\dots, s_{2n}\n\\]\nsuch that\n\\[\nm_{i,j} = s_{i+j}\n\\quad \\text{for all } 1 \\le i,j \\le n.\n\\]\n\n\\end{conj}\n\nDefine\n\\[\na_i = \\left\\lfloor \\frac{r_i}{2} \\right\\rfloor,\n\\qquad\nb_i = \\left\\lceil \\frac{r_i}{2} \\right\\rceil,\n\\]\nor vice versa.\n\nAssume Conjecture~\\ref{conj:hankel} holds, so that every minimal matrix is Hankel. \nLet $M=(m_{i,j})$ be a minimal Hankel matrix of shape $\\lambda$, \nand write $m_{i,j}=s_{i+j}$. Then\n\\[\ns_i = a_{i-1}\n\\quad \\text{and} \\quad\ns_{2n-i} = b_{i-1}\n\\qquad\n\\text{for } 2 \\le i \\le n.\n\\]\n\nThe conjectured form of the parameters $s_i$ reflects the Hankel structure imposed on minimal matrices. Since the multiplicities $r_i$ describe the number of columns of height $i$ in the Young diagram of $\\lambda$, they encode how the weight of a matrix (sum of all the entries) must be distributed across anti-diagonals to obtain the prescribed shape.\n\nIn this case, the matrix has the form\n\\[\n \\begin{bmatrix}\n a_1 & a_2 & a_3 & a_4 & s_6 \\\\\n a_2 & a_3 & a_4 & s_6 & b_1 \\\\\n a_3 & a_4 & s_6 & b_1 & b_2 \\\\\n a_4 & s_6 & b_1 & b_2 & b_3 \\\\\n s_6 & b_1 & b_2 & b_3 & b_4\n\\end{bmatrix},\\]\nwhere for $1 \\le i \\le 4$ we have \n\\[\n\\{a_i,b_i\\}\n=\n\\left\\{\n\\left\\lfloor \\frac{r_i}{2} \\right\\rfloor,\n\\left\\lceil \\frac{r_i}{2} \\right\\rceil\n\\right\\},\n\\]\nand the central anti-diagonal entry $s_6$ corresponds to $r_5=\\lambda_5$.\n\nAssuming Conjecture~\\ref{conj:hankel}, we now prove theorem~\\ref{thm:char}.\n\\begin{proof}[Proof of Theorem~\\ref{thm:char}]\nConsider a minimal matrix $M$. Assuming Conjecture~\\ref{conj:hankel}, $M$ is a Hankel matrix with associated anti-diagonal parameters \n\\[\n(s_2, s_3, \\dots, s_{2n}).\n\\]\nThus $m_{i,j} = s_{i+j}$.\nAny right-down path in $M$ from $(1,1)$ to $(n,n)$ intersects \neach anti-diagonal exactly once. Therefore, every such path has \nthe same weight,\n\\[\ns_2 + s_3 + \\cdots + s_{2n}.\n\\]\nConsequently, a longest increasing subsequence of $M$ has length $s_2 + s_3 + \\cdots + s_{2n}$. Because of the Hankel structure of the matrix, a longest $2$-increasing subsequence in $M$ is indeed the union of a longest increasing subsequence with the second longest subsequence obtained after deleting the entries from this LIS.\nMore generally, a longest $k$-increasing subsequence of $M$ corresponds to a union of $k$ non-intersecting maximal right-down paths. Applying Greene’s theorem, we obtain the system\n\\[\n\\begin{aligned}\ns_2 + s_3 + s_4\\cdots + s_{2n} &= \\lambda_1, \\\\\ns_3 + s_4 + \\cdots + s_{2n-1} &= \\lambda_2, \\\\\n&\\ \\ \\vdots \\\\\ns_{n} + s_{n+1} + s_{n+2} &= \\lambda_{n-1}, \\\\\ns_{n+1} &= \\lambda_n.\n\\end{aligned}\n\\]\nSolving this system, we get \\[s_{k+1}+s_{2n-k+1}=\\lambda_k-\\lambda_{k+1}.\\]\nThe quantity on the right-hand side is exactly equal to $r_k$, which is the number of columns of height $k$ in $\\lambda$. Looking at the inversions number, the formula becomes a sum of quadratic expressions in $s_k$'s for $k \\in \\{3,4,\\ldots,2n-1\\}$. Inversions are minimised when they are as close to $\\frac{r_k}{2}$ as possible. But since $s_k$ is an integer for all $k$, we have\n\n\\begin{thm}\nLet $\\lambda$ be a partition and let $\\lambda'$ denote its conjugate. \nThe minimum number of inversions among words of shape $\\lambda$ is given by\n\\[ \\sum_{i \\ge 1}\n\\left( \\left\\lfloor \\frac{i}{2} \\right\\rfloor + 1 \\right)\n\\binom{\\lambda'_i}{2}\n\\;+\\;\n\\sum_{\\substack{i \\ge 1 \\\\ i \\text{ odd}}}\n\\;\\sum_{\\substack{j \\ge 2 \\\\ j \\text{ even}}}\n\\binom{\\lambda'_i + \\lambda'_j - n}{2},\n\\]\nwhere $n = \\ell(\\lambda)$ is the number of parts of $\\lambda$.\n\\end{thm}", "post_theorem_intro_text_len": 2149, "post_theorem_intro_text": "The conjectured form of the parameters $s_i$ reflects the Hankel structure imposed on minimal matrices. Since the multiplicities $r_i$ describe the number of columns of height $i$ in the Young diagram of $\\lambda$, they encode how the weight of a matrix (sum of all the entries) must be distributed across anti-diagonals to obtain the prescribed shape.\n\nUnder the fixed shape restriction, contributions to the inversion number coming from opposite anti-diagonals are naturally paired, because of the anti-diagonals being of the same length. Minimising inversions, therefore, suggests distributing the total mass $r_i$ as evenly as possible between the corresponding positions $s_{i+1}$ and $s_{2n-i+1}$. This leads to the balanced floor–ceiling splitting, which heuristically minimises the inversion count.\nTo illustrate the conjectured structure, suppose that $\\lambda$ has $5$ parts. If $M$ is a minimal matrix of shape $\\lambda$, then according to Conjecture~\\ref{conj:hankel}, it is Hankel. Hence $M=(m_{i,j})$ satisfies $m_{i,j}=s_{i+j}$, where the sequence \n$(s_2,s_3,\\dots,s_{10})$ is determined by the splitting of the column \nmultiplicities $r_i$.\n\nIn this case, the matrix has the form\n\\[\n \\begin{bmatrix}\n a_1 & a_2 & a_3 & a_4 & s_6 \\\\\n a_2 & a_3 & a_4 & s_6 & b_1 \\\\\n a_3 & a_4 & s_6 & b_1 & b_2 \\\\\n a_4 & s_6 & b_1 & b_2 & b_3 \\\\\n s_6 & b_1 & b_2 & b_3 & b_4\n\\end{bmatrix},\\]\nwhere for $1 \\le i \\le 4$ we have \n\\[\n\\{a_i,b_i\\}\n=\n\\left\\{\n\\left\\lfloor \\frac{r_i}{2} \\right\\rfloor,\n\\left\\lceil \\frac{r_i}{2} \\right\\rceil\n\\right\\},\n\\]\nand the central anti-diagonal entry $s_6$ corresponds to $r_5=\\lambda_5$.\n\nThe paper is organised as follows. In Section~\\ref{sec:prelim}, we recall the necessary background on the Robinson–Schensted–Knuth correspondence, inversion statistics, and related combinatorial tools. In Section~\\ref{sec:structure}, we introduce the class of matrices $\\mathcal{M}_{\\lambda}$ and establish the properties that will be used throughout the paper. Section~\\ref{sec:main} contains the proof of our main results, including the determination of the minimal inversion number and the characterisation of minimal matrices.", "sketch": "The introduction gives a heuristic argument for Theorem~\\ref{thm:char}: the parameters $s_i$ should reflect the Hankel structure of minimal matrices, and since the multiplicities $r_i$ (columns of height $i$) “encode how the weight of a matrix … must be distributed across anti-diagonals,” one considers how to place this mass on Hankel anti-diagonals. Under the fixed shape restriction, “contributions to the inversion number coming from opposite anti-diagonals are naturally paired,” so “minimising inversions … suggests distributing the total mass $r_i$ as evenly as possible between the corresponding positions $s_{i+1}$ and $s_{2n-i+1}$,” yielding the “balanced floor–ceiling splitting” $\\{a_i,b_i\\}=\\{\\lfloor r_i/2\\rfloor,\\lceil r_i/2\\rceil\\}$, which “heuristically minimises the inversion count.” The $n=5$ example illustrates this by writing the Hankel matrix $m_{i,j}=s_{i+j}$ with entries arranged so opposite anti-diagonals take values $a_i$ and $b_i$, and the central anti-diagonal entry $s_6$ corresponding to $r_5=\\lambda_5$.", "expanded_sketch": "The introduction gives a heuristic argument for the main theorem: the parameters $s_i$ should reflect the Hankel structure of minimal matrices, and since the multiplicities $r_i$ (columns of height $i$) “encode how the weight of a matrix … must be distributed across anti-diagonals,” one considers how to place this mass on Hankel anti-diagonals. Under the fixed shape restriction, “contributions to the inversion number coming from opposite anti-diagonals are naturally paired,” so “minimising inversions … suggests distributing the total mass $r_i$ as evenly as possible between the corresponding positions $s_{i+1}$ and $s_{2n-i+1}$,” yielding the “balanced floor–ceiling splitting” $\\{a_i,b_i\\}=\\{\\lfloor r_i/2\\rfloor,\\lceil r_i/2\\rceil\\}$, which “heuristically minimises the inversion count.” The $n=5$ example illustrates this by writing the Hankel matrix $m_{i,j}=s_{i+j}$ with entries arranged so opposite anti-diagonals take values $a_i$ and $b_i$, and the central anti-diagonal entry $s_6$ corresponding to $r_5=\\lambda_5$.", "expanded_theorem": "\\label{thm:char}\n\nLet $\\lambda$ be a partition with $n$ parts, and let \n$r_i = \\lambda_i - \\lambda_{i+1}$ for $1 \\le i < n$, and \n$r_n = \\lambda_n$. \nThus $r_i$ denotes the number of columns of height $i$ in $\\lambda$.\n\nDefine\n\\[\na_i = \\left\\lfloor \\frac{r_i}{2} \\right\\rfloor,\n\\qquad\nb_i = \\left\\lceil \\frac{r_i}{2} \\right\\rceil,\n\\]\nor vice versa.\n\nAssume the following conjecture holds.\n\\begin{conj}\\label{conj:hankel}\nLet $\\lambda$ be a partition with $n$ parts, and let \n$M=(m_{i,j})_{1\\le i,j \\le n} \\in \\mathcal{M}_\\lambda$ \nbe a minimal matrix of shape $\\lambda$. Then $M$ is symmetric, that is,\n\\[\nm_{i,j} = m_{j,i} \\quad \\text{for all } 1 \\le i,j \\le n.\n\\]\nMoreover, every minimal matrix of shape $\\lambda$ is Hankel: there exists a sequence of integers \n\\[\ns_2, s_3, \\dots, s_{2n}\n\\]\nsuch that\n\\[\nm_{i,j} = s_{i+j}\n\\quad \\text{for all } 1 \\le i,j \\le n.\n\\]\n\n\\end{conj}\nThen every minimal matrix is Hankel. \nLet $M=(m_{i,j})$ be a minimal Hankel matrix of shape $\\lambda$, \nand write $m_{i,j}=s_{i+j}$. Then\n\\[\ns_i = a_{i-1}\n\\quad \\text{and} \\quad\ns_{2n-i} = b_{i-1}\n\\qquad\n\\text{for } 2 \\le i \\le n.\n\\],", "theorem_type": ["Implication", "Classification or Bijection"], "mcq": {"question": "Let \\(\\lambda=(\\lambda_1,\\dots,\\lambda_n)\\) be a partition with \\(n\\) parts. Define\n\\[\nr_i=\\lambda_i-\\lambda_{i+1}\\quad(1\\le i<n),\\qquad r_n=\\lambda_n,\n\\]\nso that \\(r_i\\) is the number of columns of height \\(i\\) in \\(\\lambda\\). For each \\(i\\), let \\(a_i\\) and \\(b_i\\) be the two numbers\n\\[\n\\left\\lfloor \\frac{r_i}{2}\\right\\rfloor,\\qquad \\left\\lceil \\frac{r_i}{2}\\right\\rceil\n\\]\nin either order. Let \\(\\mathcal M_\\lambda\\) denote the set of matrices of shape \\(\\lambda\\), and call a matrix in \\(\\mathcal M_\\lambda\\) minimal if it attains the minimum inversion number among matrices in \\(\\mathcal M_\\lambda\\). Assume that every minimal matrix \\(M=(m_{i,j})_{1\\le i,j\\le n}\\in\\mathcal M_\\lambda\\) is symmetric and Hankel, meaning that \\(m_{i,j}=m_{j,i}\\) for all \\(i,j\\) and there exist integers \\(s_2,s_3,\\dots,s_{2n}\\) such that\n\\[\nm_{i,j}=s_{i+j}\\qquad(1\\le i,j\\le n).\n\\]\nIf \\(M\\) is a minimal Hankel matrix of shape \\(\\lambda\\) and \\(m_{i,j}=s_{i+j}\\), which statement about the anti-diagonal parameters \\(s_k\\) holds?", "correct_choice": {"label": "A", "text": "For every \\(2\\le i\\le n\\),\n\\[\ns_i=a_{i-1}\\qquad\\text{and}\\qquad s_{2n-i}=b_{i-1},\n\\]\nwhere for each index \\(j\\), the pair \\((a_j,b_j)\\) is \\(\\bigl(\\lfloor r_j/2\\rfloor,\\lceil r_j/2\\rceil\\bigr)\\) or \\(\\bigl(\\lceil r_j/2\\rceil,\\lfloor r_j/2\\rfloor\\bigr)\\)."}, "choices": [{"label": "B", "text": "For every \\(2\\le i\\le n\\),\n\\[\ns_i=a_{i-1}\\qquad\\text{and}\\qquad s_{2n-i+1}=b_{i-1},\n\\]\nwhere for each index \\(j\\), the pair \\((a_j,b_j)\\) is \\(\\bigl(\\lfloor r_j/2\\rfloor,\\lceil r_j/2\\rceil\\bigr)\\) or \\(\\bigl(\\lceil r_j/2\\rceil,\\lfloor r_j/2\\rfloor\\bigr)\\)."}, {"label": "C", "text": "For every \\(2\\le i\\le n\\), the two anti-diagonal parameters \\(s_i\\) and \\(s_{2n-i}\\) are exactly the two numbers\n\\[\n\\left\\lfloor \\frac{r_{i-1}}{2}\\right\\rfloor,\n\\qquad\n\\left\\lceil \\frac{r_{i-1}}{2}\\right\\rceil\n\\]\nin some order."}, {"label": "D", "text": "For every \\(2\\le i\\le n\\),\n\\[\ns_i=\\left\\lfloor \\frac{r_{i-1}}{2}\\right\\rfloor\\qquad\\text{and}\\qquad s_{2n-i}=\\left\\lceil \\frac{r_{i-1}}{2}\\right\\rceil,\n\\]\nso the smaller half always occurs on the lower-index anti-diagonal and the larger half on the opposite one."}, {"label": "E", "text": "For every \\(2\\le i\\le n\\),\n\\[\ns_i+s_{2n-i}=r_i,\n\\]\nand moreover each of \\(s_i\\) and \\(s_{2n-i}\\) is as close as possible to \\(r_i/2\\), i.e. the pair \\((s_i,s_{2n-i})\\) is \\(\\bigl(\\lfloor r_i/2\\rfloor,\\lceil r_i/2\\rceil\\bigr)\\) or \\(\\bigl(\\lceil r_i/2\\rceil,\\lfloor r_i/2\\rfloor\\bigr)\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "opposite anti-diagonal index pairing", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "dropped the assignment of which opposite anti-diagonal gets which half", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "orientation of floor/ceiling split made fixed rather than up to either order", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "wrong multiplicity index in the balanced split relation", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the conclusion in choice A. It introduces the notation and hypotheses, but the exact anti-diagonal assignment \\(s_i=a_{i-1}\\), \\(s_{2n-i}=b_{i-1}\\) is not leaked."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-conclusion recall item: the stem gives the full setup and asks which statement must hold. The correct option is the theorem’s precise conclusion rather than a genuinely new inference."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact statement from nearby variants (index shift, reversed orientation, weaker true statement), but the item mainly tests recall of the precise result rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and well targeted: one has an indexing error, one is a weaker true statement, one reverses the floor/ceiling assignment, and one misuses the relevant \\(r_i\\)-dependence."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-recall MCQ with strong distractors and no answer leakage, but it is largely tautological and only moderately tests reasoning."}}
{"id": "2602.15140v1", "paper_link": "http://arxiv.org/abs/2602.15140v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{mainThm}\n    Let $(\\Gamma, \\Delta)$ be a Zamolodchikov periodic $n\\times n$ $B$-matrix with Coxeter numbers $h_{\\Gamma}, h_{\\Delta}$.\n    Let $T(t)$ be the $T$-system associated to $(\\Gamma, \\Delta)$.\n    Then for all $i\\in [n]$,\n    \\begin{align*}\n        T_i(t + h_{\\Gamma} + h_{\\Delta}) = T_{\\sigma(i)}(t)\n    \\end{align*}\n    for some automorphism $\\sigma\\in S_n$ of $(\\Gamma, \\Delta)$ of order at most two.\n    When $h_{\\Gamma} + h_{\\Delta}$ is even, this permutation is bipartite color preserving.\n    Otherwise, it is color reversing.", "start_pos": 10774, "end_pos": 11352, "label": "mainThm"}, "ref_dict": {"prelims": "\\begin{theorem}\n\\label{mainThm}\n    Let $(\\Gamma, \\Delta)$ be a Zamolodchikov periodic $n\\times n$ $B$-matrix with Coxeter numbers $h_{\\Gamma}, h_{\\Delta}$.\n    Let $T(t)$ be the $T$-system associated to $(\\Gamma, \\Delta)$.\n    Then for all $i\\in [n]$,\n    \\begin{align*}\n        T_i(t + h_{\\Gamma} + h_{\\Delta}) = T_{\\sigma(i)}(t)\n    \\end{align*}\n    for some automorphism $\\sigma\\in S_n$ of $(\\Gamma, \\Delta)$ of order at most two.\n    When $h_{\\Gamma} + h_{\\Delta}$ is even, this permutation is bipartite color preserving.\n    Otherwise, it is color reversing.\n\\end{theorem}\n\nIn Section \\ref{prelims}, we introduce cluster algebras, maximal green sequences, and bipartite dynamics of cluster algebras, known as \\textit{$T$-systems} and \\textit{$Y$-systems}.\nIn Section \\ref{sec:maximalGreen}, we prove that for all Zamolodchikov periodic $B$-matrices, the bipartite belt is a maximal green sequence.\nIn Section \\ref{sec:half-periodicity}, we prove the form at the half-period is a permutation of the cluster variables, and observe when it gives the identity permutation.\nIn Section \\ref{sec:coloredMutations}, we observe that in the case of single Dynkin diagrams under a certain global tropical ordering, half-periodicity yields a bijection between almost positive roots and colored tropical mutations, as conjectured in our earlier work \\cite{chin}.\n\n\\section{Preliminaries}\n\\label{prelims}\n\n\\subsection{Introduction to cluster algebras with coefficients}\n\nAn $n\\times n$ matrix $B$ is \\textit{skew-symmetrizable} if there exists some $c\\in \\mathbb{R}_{>0}^n$ such that $\\forall i, j\\in [n]$, $c_ib_{ij} = -c_jb_{ji}$.\nIn other words, a matrix $B$ is skew-symmetrizable if there exists some nonzero scaling of the rows $c\\in \\mathbb{R}_{>0}^n$ that makes the matrix skew-symmetric.\n\n\\begin{definition}\n    Let $I$ be a finite index set.\n    \\begin{itemize}\n        \\item[(1)]\n            A \\textit{semifield} is an abelian multiplicative group $(\\mathbb{P}, \\oplus, \\cdot)$ equipped with addition $\\oplus$ which is commutative, associative, and distributive with respect to multiplication in $\\mathbb{P}$.\n        \\item[(2)]\n            For an $I$-tuple of variables $y = (y_i)_{i\\in I}$, the \\textit{universal semifield} $\\mathbb{Q}_{sf}(y)$ consists of all rational functions $P(y) / Q(y)$, where $P(y)$ and $Q(y)$ are nonzero polynomials in the $y_i$'s over $\\mathbb{Z}_{>0}$, equipped with the usual operations of multiplication and addition.\n        \\item[(3)]\n            For an $I$-tuple of variables $y = (y_i)_{i\\in I}$, the \\textit{tropical semifield} Trop($y$) is the abelian multiplicative group freely generated by the variables $(y_i)_{i\\in I}$, equipped with addition $\\oplus$ defined by\n            $$\\prod_i y_i^{a_i} \\oplus \\prod_i y_i^{b_i} = \\prod_i y_i^{\\min(a_i, b_i)}.$$\n    \\end{itemize}\n\\end{definition}\n\n\\begin{definition}\n    A \\textit{labeled seed} in field $\\mathcal{F}$ is a triple $(x, y, B)$ where\n    \\begin{itemize}\n        \\item \n            $B = (b_{ij})$ is an $n\\times n$ skew-symmetrizable integer matrix,\n        \\item \n            $x = (x_1, \\dots, x_n)$ is an $n$-tuple of elements in $\\mathcal{F}$ forming a \\textit{free generating set}; that is, $x_1, \\dots, x_n$ are algebraically independent over $\\mathbb{QP}$, and $\\mathcal{F} = \\mathbb{QP}(x_1, \\dots, x_n)$,\n        \\item \n            $y = (y_1, \\dots, y_n)$ is an $n$-tuple of algebraically independent elements in $\\mathbb{P}$.\n    \\end{itemize}\n\\end{definition}\nWe use the following notation:\n\\begin{itemize}\n    \\item \n        $x$ is the (labeled) \\textit{cluster} of this seed $(x, B)$,\n    \\item \n        the elements $x_1, \\dots, x_n$ are its \\textit{cluster variables},\n    \\item \n        the elements $y_1, \\dots, y_n$ are its \\textit{frozen variables} (or \\textit{coefficients}),\n    \\item \n        $B$ is the \\textit{exchange matrix}, also known as the $B$-matrix.\n\\end{itemize}\nWhen the coefficients lie in the tropical semifield Trop($y$), they are called \\textit{principal coefficients}.\n\n\\begin{definition}\n    Let $(x, y, B)$ be a labeled seed. \n    Let $k\\in [n]$.\n    The \\textit{seed mutation} $\\mu_k$ in direction $k$ transforms $(x, y, B)$ into the new labeled seed $\\mu_k(x, y, B) = (x', y', B')$ defined as follows:\n    \\begin{itemize}\n        \\item \n            The cluster $x' = (x_1', \\dots, x_n')$ is given by $x_j' = x_j$ for all $j\\neq k$, whereas $x_k'$ is determined by the \\textit{exchange relation}\n            \\begin{equation}\n            \\label{mutationX}\n                x_kx_k' = \\frac{y_{k}}{1 \\oplus y_k}\\prod_{b_{ik} > 0} x_i^{|b_{ik}|} + \\frac{1}{1\\oplus y_k}\\prod_{b_{ik} < 0} x_i^{|b_{ik}|}.\n            \\end{equation}\n        \\item \n            The coefficients $y' = (y_1', \\dots, y_n')$ are given by\n            \\begin{equation}\n            \\label{mutationY}\n                y_i' = \\begin{cases}\n                    y_i^{-1} & i = k,\\\\\n                    y_iy_k^{\\max(0, b_{ki})}(1\\oplus y_k)^{-b_{ki}} & i\\neq k.\n                \\end{cases}\n            \\end{equation}\n        \\item  \n            The exchange matrix $B' = (b_{ij}')$ is given by\n            $$b_{ij}' = \\begin{cases}\n                -b_{ij} & \\text{if }i = k \\text{ or } j = k,\\\\\n                b_{ij} + b_{ik}b_{kj} & \\text{if } b_{ik}, b_{kj} > 0,\\\\\n                b_{ij} - b_{ik}b_{kj} & \\text{if } b_{ik}, b_{kj} < 0,\\\\\n                b_{ij} & \\text{otherwise.}\n            \\end{cases}$$\n    \\end{itemize}\n\\end{definition}\n\n\\begin{definition}\n    Let $(x, y, B)$ be a seed.\n    Let $\\mathcal{Y}$ be the union of all $\\{\\frac{y_i}{1\\oplus y_i}, \\frac{1}{1\\oplus y_i}\\}$ for every coefficient $y_i$ obtained from any sequence of mutations of this seed.\n    Let $\\mathcal{X}$ be the set of all cluster variables obtained from any sequence of mutations of this seed.\n    The \\textit{cluster algebra} $\\mathcal{A}$ generated by the seed $(x, y, B)$ is the $\\mathbb{ZP}$-subalgebra of the ambient field $\\mathcal{F}$ generated by all cluster variables: $\\mathcal{A} = \\mathbb{ZP}[\\mathcal{X}]$.\n\\end{definition}\n\nIn general, $\\mathcal{X}$ is an infinite set. When $\\mathcal{X}$ is finite, we say the cluster algebra $\\mathcal{A}$ is of \\textit{finite type}.\nThe following result introduces an interesting connection between root systems and cluster algebras of finite type.\n\n\\begin{theorem}[\\cite{fomin}]\n\\label{fominBijection}\n    Let $(B, x, y)$ be a seed of a finite type cluster algebra of type $\\Lambda$.\n    There is a unique bijection between the almost positive roots of $\\Lambda$ and the cluster variables of the cluster algebra generated by $(B, x, y)$, which sends $\\alpha\\mapsto x[\\alpha]$, where\n    \\[x[\\alpha] = \\frac{P_{\\alpha}(x)}{x^{\\alpha}}.\\]\n    Here, $P_{\\alpha}(x)$ is a polynomial over $\\mathbb{Z}\\mathbb{P}$ with nonzero constant term.\n    Moreover, every coefficient of $P_{\\alpha}$ is a polynomial in the elements of $\\mathcal{Y}$ with positive integer coefficients.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2686, "pre_theorem_intro_text": "\\subsection{Zamolodchikov periodicity}\n\nCluster algebras were first introduced in \\cite{fomin1} by Fomin and Zelevinsky with applications to dual canonical bases, total positivity in semisimple Lie groups, and Zamolodchikov periodicity for $Y$-systems of finite root systems.\nIn their fourth paper on cluster algebras, Fomin--Zelevinsky introduced what is known as the \\textit{bipartite belt}, a sequence of bipartite mutations defined below whose exchange relations form a discrete dynamical system they called a \\textit{generalized $Y$-system} \\cite{fomin4}.\n\nAn $n\\times n$ matrix $B$ is \\textit{bipartite} if there exists a coloring $\\epsilon: [n]\\rightarrow \\{\\circ, \\bullet\\}$ of $[n]$ such that for all $i, j$ of the same color, $b_{ij} = b_{ji} = 0$.\nIt is well known that mutation is a local move, so mutations $\\mu_i$ and $\\mu_j$ commute if $b_{ij} = b_{ji} = 0$.\nThus in a bipartite matrix, we may define the \\textit{bipartite mutations} $\\mu_{\\circ}$ and  $\\mu_{\\bullet}$ as the combined mutation at all white vertices and all black vertices, respectively.\nA bipartite matrix $B$ is \\textit{recurrent} if $\\mu_{\\circ}\\mu_{\\bullet}(B) = \\mu_{\\circ}(-B) = B$. \nThe sequence $\\mu_{\\circ}\\mu_{\\bullet}\\mu_{\\circ}\\mu_{\\bullet}\\dots$ is known as the \\textit{bipartite belt}.\n\nA bipartite recurrent matrix $B$ is \\textit{Zamolodchikov periodic} if the bipartite belt is periodic.\nZamolodchikov periodicity was first observed by Zamolodchikov in his study \\cite{zamolodchikov} of thermodynamic Bethe ansatz, initially focusing on the simply-laced Dynkin diagrams $A_n, D_n$, and $E_n$.\nIn \\cite{fomin4}, it was proved that the bipartite belt associated with a generalized Cartan matrix is periodic exactly when the matrix is of finite type.\nThe conjecture was further proved for all tensor products $\\Gamma\\otimes \\Delta$ of Dynkin diagrams by Keller \\cite{keller} and later in a different way by Inoue--Iyama--Keller--Kuniba--Nakanishi \\cite{inoue, inoue2}.\nIn our previous work \\cite{chin}, we fully classified all Zamolodchikov periodic cluster algebras.\n\nIn particular, the period is a divisor of $2(h + h')$, where $h, h'$ are the Coxeter numbers associated to the bipartite recurrent $B$-matrix.\nThe behavior after $(h + h')$ mutations is known as \\textit{half-periodicity}, and was conjectured in \\cite{kuniba} for $Y$-systems and in \\cite{inoue2} for $T$-systems of finite type Cartan matrices.\nIn \\cite{inoue}, half-periodicity was proved for tensor products of two simply-laced Dynkin diagrams.\nIn this paper, we prove half-periodicity for every Zamolodchikov periodic cluster algebra.\n\n\\subsection{The main theorem}\nThe following theorem is the main result of this paper.", "context": "\\subsection{Zamolodchikov periodicity}\n\nCluster algebras were first introduced in \\cite{fomin1} by Fomin and Zelevinsky with applications to dual canonical bases, total positivity in semisimple Lie groups, and Zamolodchikov periodicity for $Y$-systems of finite root systems.\nIn their fourth paper on cluster algebras, Fomin--Zelevinsky introduced what is known as the \\textit{bipartite belt}, a sequence of bipartite mutations defined below whose exchange relations form a discrete dynamical system they called a \\textit{generalized $Y$-system} \\cite{fomin4}.\n\nAn $n\\times n$ matrix $B$ is \\textit{bipartite} if there exists a coloring $\\epsilon: [n]\\rightarrow \\{\\circ, \\bullet\\}$ of $[n]$ such that for all $i, j$ of the same color, $b_{ij} = b_{ji} = 0$.\nIt is well known that mutation is a local move, so mutations $\\mu_i$ and $\\mu_j$ commute if $b_{ij} = b_{ji} = 0$.\nThus in a bipartite matrix, we may define the \\textit{bipartite mutations} $\\mu_{\\circ}$ and $\\mu_{\\bullet}$ as the combined mutation at all white vertices and all black vertices, respectively.\nA bipartite matrix $B$ is \\textit{recurrent} if $\\mu_{\\circ}\\mu_{\\bullet}(B) = \\mu_{\\circ}(-B) = B$. \nThe sequence $\\mu_{\\circ}\\mu_{\\bullet}\\mu_{\\circ}\\mu_{\\bullet}\\dots$ is known as the \\textit{bipartite belt}.\n\nA bipartite recurrent matrix $B$ is \\textit{Zamolodchikov periodic} if the bipartite belt is periodic.\nZamolodchikov periodicity was first observed by Zamolodchikov in his study \\cite{zamolodchikov} of thermodynamic Bethe ansatz, initially focusing on the simply-laced Dynkin diagrams $A_n, D_n$, and $E_n$.\nIn \\cite{fomin4}, it was proved that the bipartite belt associated with a generalized Cartan matrix is periodic exactly when the matrix is of finite type.\nThe conjecture was further proved for all tensor products $\\Gamma\\otimes \\Delta$ of Dynkin diagrams by Keller \\cite{keller} and later in a different way by Inoue--Iyama--Keller--Kuniba--Nakanishi \\cite{inoue, inoue2}.\nIn our previous work \\cite{chin}, we fully classified all Zamolodchikov periodic cluster algebras.\n\nIn particular, the period is a divisor of $2(h + h')$, where $h, h'$ are the Coxeter numbers associated to the bipartite recurrent $B$-matrix.\nThe behavior after $(h + h')$ mutations is known as \\textit{half-periodicity}, and was conjectured in \\cite{kuniba} for $Y$-systems and in \\cite{inoue2} for $T$-systems of finite type Cartan matrices.\nIn \\cite{inoue}, half-periodicity was proved for tensor products of two simply-laced Dynkin diagrams.\nIn this paper, we prove half-periodicity for every Zamolodchikov periodic cluster algebra.\n\n\\subsection{The main theorem}\nThe following theorem is the main result of this paper.", "full_context": "\\subsection{Zamolodchikov periodicity}\n\nCluster algebras were first introduced in \\cite{fomin1} by Fomin and Zelevinsky with applications to dual canonical bases, total positivity in semisimple Lie groups, and Zamolodchikov periodicity for $Y$-systems of finite root systems.\nIn their fourth paper on cluster algebras, Fomin--Zelevinsky introduced what is known as the \\textit{bipartite belt}, a sequence of bipartite mutations defined below whose exchange relations form a discrete dynamical system they called a \\textit{generalized $Y$-system} \\cite{fomin4}.\n\nAn $n\\times n$ matrix $B$ is \\textit{bipartite} if there exists a coloring $\\epsilon: [n]\\rightarrow \\{\\circ, \\bullet\\}$ of $[n]$ such that for all $i, j$ of the same color, $b_{ij} = b_{ji} = 0$.\nIt is well known that mutation is a local move, so mutations $\\mu_i$ and $\\mu_j$ commute if $b_{ij} = b_{ji} = 0$.\nThus in a bipartite matrix, we may define the \\textit{bipartite mutations} $\\mu_{\\circ}$ and $\\mu_{\\bullet}$ as the combined mutation at all white vertices and all black vertices, respectively.\nA bipartite matrix $B$ is \\textit{recurrent} if $\\mu_{\\circ}\\mu_{\\bullet}(B) = \\mu_{\\circ}(-B) = B$. \nThe sequence $\\mu_{\\circ}\\mu_{\\bullet}\\mu_{\\circ}\\mu_{\\bullet}\\dots$ is known as the \\textit{bipartite belt}.\n\nA bipartite recurrent matrix $B$ is \\textit{Zamolodchikov periodic} if the bipartite belt is periodic.\nZamolodchikov periodicity was first observed by Zamolodchikov in his study \\cite{zamolodchikov} of thermodynamic Bethe ansatz, initially focusing on the simply-laced Dynkin diagrams $A_n, D_n$, and $E_n$.\nIn \\cite{fomin4}, it was proved that the bipartite belt associated with a generalized Cartan matrix is periodic exactly when the matrix is of finite type.\nThe conjecture was further proved for all tensor products $\\Gamma\\otimes \\Delta$ of Dynkin diagrams by Keller \\cite{keller} and later in a different way by Inoue--Iyama--Keller--Kuniba--Nakanishi \\cite{inoue, inoue2}.\nIn our previous work \\cite{chin}, we fully classified all Zamolodchikov periodic cluster algebras.\n\nIn particular, the period is a divisor of $2(h + h')$, where $h, h'$ are the Coxeter numbers associated to the bipartite recurrent $B$-matrix.\nThe behavior after $(h + h')$ mutations is known as \\textit{half-periodicity}, and was conjectured in \\cite{kuniba} for $Y$-systems and in \\cite{inoue2} for $T$-systems of finite type Cartan matrices.\nIn \\cite{inoue}, half-periodicity was proved for tensor products of two simply-laced Dynkin diagrams.\nIn this paper, we prove half-periodicity for every Zamolodchikov periodic cluster algebra.\n\n\\subsection{The main theorem}\nThe following theorem is the main result of this paper.\n\n\\subsection{The main theorem}\nThe following theorem is the main result of this paper.\n\nIn Section \\ref{prelims}, we introduce cluster algebras, maximal green sequences, and bipartite dynamics of cluster algebras, known as \\textit{$T$-systems} and \\textit{$Y$-systems}.\nIn Section \\ref{sec:maximalGreen}, we prove that for all Zamolodchikov periodic $B$-matrices, the bipartite belt is a maximal green sequence.\nIn Section \\ref{sec:half-periodicity}, we prove the form at the half-period is a permutation of the cluster variables, and observe when it gives the identity permutation.\nIn Section \\ref{sec:coloredMutations}, we observe that in the case of single Dynkin diagrams under a certain global tropical ordering, half-periodicity yields a bijection between almost positive roots and colored tropical mutations, as conjectured in our earlier work \\cite{chin}.\n\nAt each time step, when $\\epsilon_k = \\circ$, $T_k(t+1)$ and $Y_k(t+1)$ are only dependent on black indices ($\\epsilon_i = \\bullet$).\nSimilarly when $\\epsilon_k = \\bullet$, $T_k(t+1)$ and $Y_k(t+1)$ are only dependent on white indices ($\\epsilon_i = \\circ$).\nBecause of this, both systems associated with $B$ split into two independent components.\nFor our purposes, we consider only one component.\nFrom now on, assume that the two systems are defined only for $k\\in [n]$ and $t\\in \\mathbb{Z}$ such that\n\\begin{equation}\n\\label{splitTsystem}\n \\epsilon_k = \\circ\\quad\\text{and}\\quad t\\equiv 0\\hspace{-2mm}\\pmod{2}\\qquad \\text{or} \\qquad \\epsilon_k = \\bullet\\quad \\text{and} \\quad t\\equiv 1\\hspace{-2mm}\\pmod{2}.\n\\end{equation}\nBoth systems are set to the following initial conditions:\n\\begin{align*}\n T_k(0) = Y_k(0) = x_k\\quad \\text{and}\\quad T_k(1) = Y_k(1) = x_k, \\quad\\text{for all } k \\text{ satisfying } (\\ref{splitTsystem}).\n\\end{align*}\nNotice that the $T$-system relations are exactly the same as the exchange relations (\\ref{mutationX}) for cluster mutation in a bipartite recurrent setting, when specialized at $y_i = 1$. In particular, this initialization will result in the same sequence of cluster variables as the mutation sequence $\\mu_{\\circ}\\mu_{\\bullet}\\mu_{\\circ}\\mu_{\\bullet}\\dots$ acting on the initial seed $(B, x)$.\nWe say the $T$-system or $Y$-system is \\textit{periodic} if there exists some positive integer $N$ such that $T_k(t + 2N) = T_k(t)$ for all $k\\in[n], t\\in\\mathbb{Z}$. \n\\begin{remark}\nIn particular, a bipartite recurrent matrix $B$ is Zamolodchikov periodic if its associated $T$-system or $Y$-system is periodic.\nThe original formulation of Zamolodchikov periodicity was in terms of $Y$-systems.\n\\end{remark}\n\n\\begin{proposition}\n\\label{maximalGreen}\n Let $B = (\\Gamma, \\Delta)$ be a Zamolodchikov periodic $B$-matrix with associated Coxeter numbers $h_{\\Gamma}, h_{\\Delta}$. \n Let $\\epsilon: [n]\\rightarrow \\{\\circ, \\bullet\\}$ be a bipartition on $(\\Gamma, \\Delta)$ such that white vertices are sinks in $\\Gamma$ and sources in $\\Delta$, and black vertices are sources in $\\Gamma$ and sinks in $\\Delta$.\n Then, $\\mu_{\\circ}\\mu_{\\bullet}\\mu_{\\circ}\\cdots$ ($h_{\\Gamma}$ factors) and $\\mu_{\\bullet}\\mu_{\\circ}\\mu_{\\bullet}\\cdots$ ($h_{\\Delta}$ factors) are maximal green sequences for the framed matrix $\\hat{B}$.\n\\end{proposition}\n\nThe following proposition states Theorem \\ref{mainThm} for simply-laced types.\n\\begin{proposition}\n\\label{ADE}\n Let $(\\Gamma, \\Delta)$ be an ADE bigraph with Coxeter numbers $h_{\\Gamma}, h_{\\Delta}$.\n Let $T(t)$ be the $T$-system associated to $(\\Gamma, \\Delta)$.\n Then for all $i\\in [n]$,\n \\begin{align*}\n T_i(t + h_{\\Gamma} + h_{\\Delta}) = T_{\\sigma(i)}(t)\n \\end{align*}\n for some automorphism $\\sigma\\in S_n$ of $(\\Gamma, \\Delta)$ of order at most two.\n When $h_{\\Gamma} + h_{\\Delta}$ is even, this permutation is bipartite color preserving.\n Otherwise, it is color reversing.\n\\end{proposition}\n\\begin{proof}\n Let the initial seed be our coframed matrix $\\check{B}$, and let $\\hat{y}_i$ denote the tropical evaluation of the coefficient $y_i$ (in the tropical semifield Trop($y$)).\n As noted in Remark \\ref{remark1}, applying $\\mu_{\\circ}\\mu_{\\bullet}\\cdots$ ($h_{\\Gamma} + h_{\\Delta}$ factors) gives us a frozen isomorphism, or a permutation $\\sigma$ of the $C$-matrix.\n Moreover by the separation formula (\\ref{separation}), $c$-vectors are exactly the exponent vectors for principal coefficients, the tropical evaluation of coefficients, so we have $\\hat{y}_i(t) = \\hat{y}_{\\sigma(i)}$, $\\forall i\\in [n]$.\n Thus by the periodicity theorem (\\ref{periodicity}), the cluster variables also exhibit the same permutation $\\sigma$ of variables at time $N = h_{\\Gamma} + h_{\\Delta}$.\n\n\\begin{lemma}\n\\label{tropicalT}\n Let $B$ be a bipartite recurrent $B$-matrix.\n Let $\\sigma\\in S_n$ be a bipartite color preserving automorphism of $B$, and let $N\\in \\mathbb{Z}_{>0}$ be an even integer. \n The following are equivalent:\n \\begin{itemize}\n \\item[(1)]\n $\\boldsymbol{t}^{\\delta_j}_i(t + N) = \\boldsymbol{t}^{\\delta_j}_{\\sigma(i)}(t)$ for all $i, j\\in [n], t\\in \\mathbb{Z}$.\n \\item[(2)]\n $\\boldsymbol{t}^{\\lambda}_i(t + N) = \\boldsymbol{t}^{\\lambda}_{\\sigma(i)}(t)$ for all $i\\in [n], t\\in \\mathbb{Z}$, for any initial labeling $\\lambda\\in \\mathbb{R}^n$.\n \\item[(3)]\n There exists a labeling $c\\in\\mathbb{R}_{>0}^n$ such that $T_i(t + N) = c_{\\sigma(i)}T_{\\sigma(i)}(t)$ for all $i\\in [n], t\\in\\mathbb{Z}$.\n \\end{itemize}\n\\end{lemma}\n\\begin{proof}\n\n\\begin{lemma}\n\\label{transposeCommutes}\n Let $\\sigma\\in S_n$ be a bipartite color preserving automorphism of $B$, and let $N\\in \\mathbb{Z}_{>0}$ be an even integer. \n For any initial labeling $\\lambda\\in\\mathbb{R}^n$, let the tropical $T$-system of $B$ satisfy\n $$\\boldsymbol{t}^{\\lambda}_i(t + N) = \\boldsymbol{t}^{\\lambda}_{\\sigma(i)}(t), \\text{ for all } i\\in [n], t\\in \\mathbb{Z}.$$\n Then, the tropical $T$-system of the Langland dual $-B^{\\top}$ satisfies the same relations.\n\\end{lemma}\n\n\\section{Colored mutations}\n\\label{sec:coloredMutations}\nLet $B$ be a Zamolodchikov periodic $B$-matrix, and let $\\boldsymbol{t}_i^{\\lambda}$ be its corresponding tropical $T$-system with initial labeling $\\lambda\\in \\mathbb{R}^n$.\nRecall the form for tropical mutation at index $k\\in [n]$:\n\\begin{align*}\n \\boldsymbol{t}_k^{\\lambda}(t + 1) + \\boldsymbol{t}_k^{\\lambda}(t - 1) = \\max\\left(\\sum_i\\Gamma_{ik}\\boldsymbol{t}_i^{\\lambda}(t), \\sum_j \\Delta_{jk}\\boldsymbol{t}_j^{\\lambda}(t)\\right).\n\\end{align*}\nFor a generic initial labeling $\\lambda\\in\\mathbb{R}^n$, we can color this mutation either red or blue, depending on the evaluation of the max function. \nIf the maximum is $\\sum_i\\Gamma_{ik}\\boldsymbol{t}_i^{\\lambda}(t)$, then we call this a \\textit{$\\Gamma$-mutation}, and color it red.\nOtherwise if the maximum is $\\sum_j\\Delta_{jk}\\boldsymbol{t}_j^{\\lambda}(t)$, then we call this a \\textit{$\\Delta$-mutation}, and color it blue.", "post_theorem_intro_text_len": 775, "post_theorem_intro_text": "In Section \\ref{prelims}, we introduce cluster algebras, maximal green sequences, and bipartite dynamics of cluster algebras, known as \\textit{$T$-systems} and \\textit{$Y$-systems}.\nIn Section \\ref{sec:maximalGreen}, we prove that for all Zamolodchikov periodic $B$-matrices, the bipartite belt is a maximal green sequence.\nIn Section \\ref{sec:half-periodicity}, we prove the form at the half-period is a permutation of the cluster variables, and observe when it gives the identity permutation.\nIn Section \\ref{sec:coloredMutations}, we observe that in the case of single Dynkin diagrams under a certain global tropical ordering, half-periodicity yields a bijection between almost positive roots and colored tropical mutations, as conjectured in our earlier work \\cite{chin}.", "sketch": "The post-theorem introduction does not explicitly outline a proof of Theorem~\\ref{mainThm}, but it indicates the supporting steps across sections: Section~\\ref{prelims} introduces \"cluster algebras, maximal green sequences, and bipartite dynamics of cluster algebras, known as \\textit{$T$-systems} and \\textit{$Y$-systems}.\" Section~\\ref{sec:maximalGreen} proves that \"for all Zamolodchikov periodic $B$-matrices, the bipartite belt is a maximal green sequence.\" Section~\\ref{sec:half-periodicity} proves that \"the form at the half-period is a permutation of the cluster variables,\" and notes when this permutation is the identity. Section~\\ref{sec:coloredMutations} makes an additional observation (for single Dynkin diagrams under a certain global tropical ordering) relating half-periodicity to a bijection between almost positive roots and colored tropical mutations.", "expanded_sketch": "The post-theorem introduction does not explicitly outline a proof of the main theorem, but it indicates the supporting steps across sections: Next, we introduce cluster algebras, maximal green sequences, and bipartite dynamics of cluster algebras, known as \\textit{$T$-systems} and \\textit{$Y$-systems}. Next we prove that for all Zamolodchikov periodic $B$-matrices, the bipartite belt is a maximal green sequence. Next we prove that the form at the half-period is a permutation of the cluster variables, and note when this permutation is the identity. Finally, we make an additional observation (for single Dynkin diagrams under a certain global tropical ordering) relating half-periodicity to a bijection between almost positive roots and colored tropical mutations.", "expanded_theorem": "\\label{mainThm}\n    Let $(\\Gamma, \\Delta)$ be a Zamolodchikov periodic $n\\times n$ $B$-matrix with Coxeter numbers $h_{\\Gamma}, h_{\\Delta}$.\n    Let $T(t)$ be the $T$-system associated to $(\\Gamma, \\Delta)$.\n    Then for all $i\\in [n]$,\n    \\begin{align*}\n        T_i(t + h_{\\Gamma} + h_{\\Delta}) = T_{\\sigma(i)}(t)\n    \\end{align*}\n    for some automorphism $\\sigma\\in S_n$ of $(\\Gamma, \\Delta)$ of order at most two.\n    When $h_{\\Gamma} + h_{\\Delta}$ is even, this permutation is bipartite color preserving.\n    Otherwise, it is color reversing.", "theorem_type": ["Universal", "Existential–Universal"], "mcq": {"question": "Let (Γ, Δ) be an n×n Zamolodchikov periodic B-matrix, meaning a bipartite recurrent matrix B with a bipartite coloring ε:[n]→{○, ●} such that the simultaneous mutations μ○ and μ● at all white and black vertices satisfy μ○μ●(B)=B, and the alternating bipartite belt μ○μ●μ○μ●⋯ is periodic. Let h_Γ and h_Δ be its Coxeter numbers, and let T(t)={T_i(t)} be the associated T-system. An automorphism of (Γ, Δ) means a permutation σ∈S_n preserving the pair (Γ, Δ); it is bipartite color preserving if it preserves the coloring ε, and color reversing if it swaps the two colors. Which statement holds for every such (Γ, Δ) and associated T-system?", "correct_choice": {"label": "A", "text": "There exists an automorphism σ∈S_n of (Γ, Δ) with order at most 2 such that T_i(t + h_Γ + h_Δ) = T_{σ(i)}(t) for every i∈[n] and every time t for which the T-system terms are defined. Moreover, if h_Γ + h_Δ is even, then σ is bipartite color preserving; if h_Γ + h_Δ is odd, then σ is color reversing."}, "choices": [{"label": "B", "text": "There exists an automorphism \\(\\sigma\\in S_n\\) of \\((\\Gamma,\\Delta)\\) with order at most \\(2\\) such that \\(T_i(t + h_\\Gamma + h_\\Delta) = T_{\\sigma(i)}(t)\\) for every \\(i\\in[n]\\) and every time \\(t\\) for which the \\(T\\)-system terms are defined. Moreover, if \\(h_\\Gamma + h_\\Delta\\) is even, then \\(\\sigma\\) is color reversing; if \\(h_\\Gamma + h_\\Delta\\) is odd, then \\(\\sigma\\) is bipartite color preserving."}, {"label": "C", "text": "There exists an automorphism \\(\\sigma\\in S_n\\) of \\((\\Gamma,\\Delta)\\) such that \\(T_i(t + h_\\Gamma + h_\\Delta) = T_{\\sigma(i)}(t)\\) for every \\(i\\in[n]\\) and every time \\(t\\) for which the \\(T\\)-system terms are defined."}, {"label": "D", "text": "There exists a bipartite color preserving automorphism \\(\\sigma\\in S_n\\) of \\((\\Gamma,\\Delta)\\) with order at most \\(2\\) such that \\(T_i(t + h_\\Gamma + h_\\Delta) = T_{\\sigma(i)}(t)\\) for every \\(i\\in[n]\\) and every time \\(t\\) for which the \\(T\\)-system terms are defined."}, {"label": "E", "text": "There exists an automorphism \\(\\sigma\\in S_n\\) of \\((\\Gamma,\\Delta)\\) with order at most \\(2\\) such that \\(T_i(t + h_\\Gamma + h_\\Delta) = T_{\\sigma(i)}(t)\\) for every \\(i\\in[n]\\) and for some time \\(t\\) for which the \\(T\\)-system terms are defined. Moreover, if \\(h_\\Gamma + h_\\Delta\\) is even, then \\(\\sigma\\) is bipartite color preserving; if \\(h_\\Gamma + h_\\Delta\\) is odd, then \\(\\sigma\\) is color reversing."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "case_split", "tampered_component": "parity-to-color-action correspondence", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "case_split", "tampered_component": "dropped order-at-most-two and parity/color conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "odd-parity color-reversing case", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "uniformity in time t", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem sets up definitions and asks which conclusion holds, but it does not explicitly reveal the correct conclusion or give a strong hint toward choice A over the others."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem states the hypotheses and asks for the exact conclusion. The correct option is basically the theorem statement itself rather than an application or derived consequence."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in distinguishing close variants involving parity, color behavior, order-at-most-two, and existential vs. universal quantification. However, the item primarily tests precise recall of the theorem, not substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong: they are mathematically nearby statements produced by plausible modifications of the theorem (dropping conditions, changing parity behavior, strengthening existential to universal, or forcing color preservation). These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A solid recall/discrimination MCQ with no answer leakage and high-quality distractors, but it is mostly a direct theorem-identification question rather than a genuinely generative reasoning task."}}
{"id": "2602.15243v1", "paper_link": "http://arxiv.org/abs/2602.15243v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "If~$M$ and~$N$ are two upset-decomposable modules of pointwise dimension at most~$r$, then\n\\[\nd_P(M,N) \\leq d_B(M,N) \\leq (2r-1)d_P(M,N).\n\\]", "start_pos": 8489, "end_pos": 8659, "label": null}, "ref_dict": {"theorem_main": "\\begin{theorem}\\label{theorem_main}\n       Let~$M$ and~$N$ be pfd upset-decomposable modules with ~$r$ =~$\\operatorname{supdim}$~$M < \\infty$ then\n       \\[d_B(M,N) \\leq (2r-1)\\,d_P(M,N).\\]\n   \\end{theorem}", "theorem_3": "\\begin{theorem}\\label{theorem_3}\n       Let~$M$ and~$N$ be pfd upset-decomposable modules then\n       \\[d_P(M,N) \\leq d_B(M,N).\\]\n   \\end{theorem}", "theorem_1": "\\begin{theorem}\\label{theorem_1}\nLet~$\\alpha \\geq 0$ and let~$M = \\bigoplus_{l=1}^r M_l$ be an upset-decomposable module. Then\n\\[\n\\pi_{l_1}(\\operatorname{Pru}_\\alpha(M)) = \\bigcap_{l \\in JM_{l_1}} M_l(-\\alpha),\n\\]\nwhere~$JM_{l_1}$ is as defined above.\n\\end{theorem}", "ex_2": "\\begin{example}\\label{ex_2}\nLet~$\\alpha>0$ and~$r \\in \\mathbb{N^*}$. And let\n$\nM = \\bigoplus_{i=1}^r M_i \\text{ and } \nN = \\bigoplus_{i=1}^r N_i,\n$ be two~$d$-parameter upset-decomposable modules\nwith supports\n\\[\n\\operatorname{supp} (M_i) = \\{(x_1,\\dots,x_d) \\mid (x_1,\\dots,x_d) \\ge (2i\\alpha,\\dots,2i\\alpha)\\}\\]\n\\[\n\\operatorname{supp} (N_i) = \\{(x_1,\\dots,x_d) \\mid (x_1,\\dots,x_d) \\ge \\left( (2r+1)\\alpha,\\dots,(2r+1)\\alpha \\right)\\}.\n\\]\nThen one can check that \n\\[\n\\operatorname{Pru}_{\\alpha}(M) \\cong N,\n\\]\nso~$\\operatorname{Pru}_{\\alpha}(M)$ is an~$\\infty$-refinement of~$N$. \nMoreover, for every~$\\delta>0$, \n\\[\n\\operatorname{Pru}_{\\alpha+\\delta} (M) \\cong \\operatorname{Pru}_{\\delta} (N) \\cong N(-\\delta),\n\\] \nso we also have~$\\operatorname{Pru}_{\\alpha+\\delta}(M)$ as an~$\\infty$-refinement of~$N(-\\delta)$.\nWe can also easily check that~$\\forall \\varepsilon < \\alpha, \\operatorname{Pru}_{\\varepsilon}(M)$ is not an~$\\infty$-refinement of~$N$.\nSimilarly,~$N$ is an~$\\infty$-refinement of~$M$, and for every~$\\delta>0$,~$\\operatorname{Pru}_{\\delta}(N)$ is an~$\\infty$-refinement of~$\\operatorname{Pru}_{\\delta}(M)$.\\\\\nFrom these observations we obtain that\n\\[\nd_P(M,N) = \\alpha.\n\\]\nFurthermore, since all~$N_i$ have the same support, it is easy to check that\n\\[\nd_B(M,N) = d_I(M,N) = (2r-1)\\alpha,\n\\]\nshowing that the upper bound in Theorem~\\ref{theorem_main} can indeed be attained ~$\\forall \\alpha>0$,~$\\forall d \\in \\mathbb{N}^*$ and~$\\forall r \\in \\mathbb{N}^*$. In particular, the constant~$2r-1$ is optimal, and the inequality cannot be strengthened neither for the bottleneck distance~$d_B$ nor for the interleaving distance~$d_I$.\n\n\\end{example}", "def_pruning": "\\begin{definition}\\label{def_pruning}\n\\emph{(Pruning)} Let~$M$ be a module. Let~$I$ be the largest submodule of~$M$ such that for any morphism~$f: M \\to M(2\\varepsilon)$, we have~$f(I) \\subseteq M_{0 \\to 2\\varepsilon}(I)$. \\\\\nLet~$K$ be the smallest submodule of~$I$ such that for any morphism~$f: M \\to M(2\\varepsilon)$, it holds that~$I^{-1}_{0 \\to 2\\varepsilon}(f(K)) \\subseteq K$.\\\\\nWe define the \\emph{$\\varepsilon$-pruning pair} of~$M$ to be the pair~$(I, K)$, and the \\emph{$\\varepsilon$-pruning} of~$M$ to be \n\\[\n\\Pru_\\varepsilon(M) := (I/K)(-\\varepsilon).\n\\]\n\\textit{For example} we have that the~$0$-pruning pair of a module~$M$ is~$(M, 0)$, and the~$0$-pruning of~$M$ is \n\\[\n\\Pru_0(M) = (M/0)(0) \\cong M.\n\\]\n\\end{definition}", "def_pruning_distance": "\\begin{definition}\n\\label{def_pruning_distance}\nThe \\emph{pruning distance}~$d_P$ is defined by\n\\begin{align*}\nd_P(M,N) = \\inf\\{&\\varepsilon\\geq 0\\mid \\forall \\delta\\geq 0,\\\\ &\\Pru_{\\varepsilon+\\delta}(M) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(N) \\text{ and }\\\\ &\\Pru_{\\varepsilon+\\delta}(N) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(M)\\}\n\\end{align*}\nfor any pfd modules~$M$ and~$N$.\n\\end{definition}"}, "pre_theorem_intro_text_len": 4197, "pre_theorem_intro_text": "\\markboth{1. Introduction}{} \n\nDeveloped initially for data analysis, topological persistence provides invariants of filtered topological spaces through computation of their homology. This method produces algebraic objects called \\emph{persistence modules}, which are formally functors from the ordered set~$(\\mathbb{R},\\leq)$ to the category of vector spaces. A central result of the theory ensures that these modules decompose into simple elements associated with intervals when their vector spaces are finite-dimensional. The collection of intervals appearing in such a decomposition, the \\emph{persistence barcode}, completely describes the original persistence module up to isomorphism and, therefore, contains fine information about the initial topological filtration.\n\nThe remarkable isometry theorem \\cite{chazal_isometry} guarantees that the algebraically defined \\emph{interleaving distance}~$d_I$ between modules coincides with the combinatorial \\emph{bottleneck distance}~$d_B$ between their barcodes. The latter is defined through persistence diagrams and can be formulated as a minimum-cost bottleneck matching problem on a complete bipartite graph \\cite{Edelsbrunner_Harer}, which can be solved in polynomial time \\cite{Kerber_Morozov}. In particular, this theorem shows that the interleaving distance, while stable with respect to distance on data by construction, is also computable in polynomial time.\n\nWhen dealing with multi-parameter data, it is natural to consider filtrations depending on several parameters instead of a single one, thus creating a \\emph{multi-parameter persistence module}, that is, persistence modules indexed by~$\\mathbb{R}^n$ endowed with the product order. However, although the interleaving distance and the bottleneck distance are both generalized to the multiparameter setting, the isometry theorem no longer holds: multiparameter modules are not described in a stable way by their indecomposables. So, the bottleneck distance fails to satisfy the stability property in the multi-parameter case, making its usage useless. Henceforth, the computational advantage of the interleaving distance disappears in the multi-parameter setting, in particular, computing the interleaving distance for multi-parameter persistence modules has been proven to be NP-hard \\cite{bjerkevik_Botnan_kerber}. \n\nBjerkevik introduced the \\emph{$\\varepsilon$-pruning} (Definition~\\ref{def_pruning}) of a module in \\cite{bjerkevik_stabilizing}. The~$\\varepsilon$-pruning of~$M$ is an approximation of~$M$ which is particularly decomposable while discarding features below scale~$\\varepsilon$. This notion of pruning allowed Bjerkevik to introduce the pruning distance~$d_P$ (Definition~\\ref{def_pruning_distance}) which is related to both the interleaving distance and the decompositions. Therefore, it is unclear whether its computation is NP-hard or polynomial time computable. Bjerkevik conjectured that computing prunings can be done in polynomial time, and observes that, in the examples used to establish NP-hardness of computing the interleaving distance in \\cite{bjerkevik_botnan}, the pruning distance appears to be computable in polynomial time, suggesting the possibility of a general polynomial time algorithm.\n\nImportantly, Bjerkevik conjectured that the pruning distance is stable, and even Lipschitz equivalent to the interleaving distance, on persistence modules of a bounded pointwise dimension. More precisely, if~$M$ and~$N$ are persistence modules on an arbitrary poset with pointwise dimension smaller than a positive integer~$r$,\n\\[\nd_P(M,N) \\leq d_I(M,N) \\leq 2r\\,d_P(M,N).\n\\]\n\nIn this paper, we prove that the pruning distance is Lipschitz equivalent to the bottleneck distance for the class of upset-decomposable persistence modules, that is, modules that decompose as direct sums of interval persistence modules supported on subsets~$U \\subset \\mathbb{R}^n$ which are upward closed under the product order. Upset-decomposable modules are an important family of multi-parameter persistence modules that we study in detail in Section~\\ref{sec:pruning-upset}. Our main results, Theorem~\\ref{theorem_main} and Theorem~\\ref{theorem_3}, show the following", "context": "Developed initially for data analysis, topological persistence provides invariants of filtered topological spaces through computation of their homology. This method produces algebraic objects called \\emph{persistence modules}, which are formally functors from the ordered set~$(\\mathbb{R},\\leq)$ to the category of vector spaces. A central result of the theory ensures that these modules decompose into simple elements associated with intervals when their vector spaces are finite-dimensional. The collection of intervals appearing in such a decomposition, the \\emph{persistence barcode}, completely describes the original persistence module up to isomorphism and, therefore, contains fine information about the initial topological filtration.\n\nThe remarkable isometry theorem \\cite{chazal_isometry} guarantees that the algebraically defined \\emph{interleaving distance}~$d_I$ between modules coincides with the combinatorial \\emph{bottleneck distance}~$d_B$ between their barcodes. The latter is defined through persistence diagrams and can be formulated as a minimum-cost bottleneck matching problem on a complete bipartite graph \\cite{Edelsbrunner_Harer}, which can be solved in polynomial time \\cite{Kerber_Morozov}. In particular, this theorem shows that the interleaving distance, while stable with respect to distance on data by construction, is also computable in polynomial time.\n\nWhen dealing with multi-parameter data, it is natural to consider filtrations depending on several parameters instead of a single one, thus creating a \\emph{multi-parameter persistence module}, that is, persistence modules indexed by~$\\mathbb{R}^n$ endowed with the product order. However, although the interleaving distance and the bottleneck distance are both generalized to the multiparameter setting, the isometry theorem no longer holds: multiparameter modules are not described in a stable way by their indecomposables. So, the bottleneck distance fails to satisfy the stability property in the multi-parameter case, making its usage useless. Henceforth, the computational advantage of the interleaving distance disappears in the multi-parameter setting, in particular, computing the interleaving distance for multi-parameter persistence modules has been proven to be NP-hard \\cite{bjerkevik_Botnan_kerber}.\n\nBjerkevik introduced the \\emph{$\\varepsilon$-pruning} (Definition~\\ref{def_pruning}) of a module in \\cite{bjerkevik_stabilizing}. The~$\\varepsilon$-pruning of~$M$ is an approximation of~$M$ which is particularly decomposable while discarding features below scale~$\\varepsilon$. This notion of pruning allowed Bjerkevik to introduce the pruning distance~$d_P$ (Definition~\\ref{def_pruning_distance}) which is related to both the interleaving distance and the decompositions. Therefore, it is unclear whether its computation is NP-hard or polynomial time computable. Bjerkevik conjectured that computing prunings can be done in polynomial time, and observes that, in the examples used to establish NP-hardness of computing the interleaving distance in \\cite{bjerkevik_botnan}, the pruning distance appears to be computable in polynomial time, suggesting the possibility of a general polynomial time algorithm.\n\nImportantly, Bjerkevik conjectured that the pruning distance is stable, and even Lipschitz equivalent to the interleaving distance, on persistence modules of a bounded pointwise dimension. More precisely, if~$M$ and~$N$ are persistence modules on an arbitrary poset with pointwise dimension smaller than a positive integer~$r$,\n\\[\nd_P(M,N) \\leq d_I(M,N) \\leq 2r\\,d_P(M,N).\n\\]\n\nIn this paper, we prove that the pruning distance is Lipschitz equivalent to the bottleneck distance for the class of upset-decomposable persistence modules, that is, modules that decompose as direct sums of interval persistence modules supported on subsets~$U \\subset \\mathbb{R}^n$ which are upward closed under the product order. Upset-decomposable modules are an important family of multi-parameter persistence modules that we study in detail in Section~\\ref{sec:pruning-upset}. Our main results, Theorem~\\ref{theorem_main} and Theorem~\\ref{theorem_3}, show the following\n\n\\begin{definition}\\label{def_pruning}\n\\emph{(Pruning)} Let~$M$ be a module. Let~$I$ be the largest submodule of~$M$ such that for any morphism~$f: M \\to M(2\\varepsilon)$, we have~$f(I) \\subseteq M_{0 \\to 2\\varepsilon}(I)$. \\\\\nLet~$K$ be the smallest submodule of~$I$ such that for any morphism~$f: M \\to M(2\\varepsilon)$, it holds that~$I^{-1}_{0 \\to 2\\varepsilon}(f(K)) \\subseteq K$.\\\\\nWe define the \\emph{$\\varepsilon$-pruning pair} of~$M$ to be the pair~$(I, K)$, and the \\emph{$\\varepsilon$-pruning} of~$M$ to be \n\\[\n\\Pru_\\varepsilon(M) := (I/K)(-\\varepsilon).\n\\]\n\\textit{For example} we have that the~$0$-pruning pair of a module~$M$ is~$(M, 0)$, and the~$0$-pruning of~$M$ is \n\\[\n\\Pru_0(M) = (M/0)(0) \\cong M.\n\\]\n\\end{definition}\n\n\\begin{definition}\n\\label{def_pruning_distance}\nThe \\emph{pruning distance}~$d_P$ is defined by\n\\begin{align*}\nd_P(M,N) = \\inf\\{&\\varepsilon\\geq 0\\mid \\forall \\delta\\geq 0,\\\\ &\\Pru_{\\varepsilon+\\delta}(M) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(N) \\text{ and }\\\\ &\\Pru_{\\varepsilon+\\delta}(N) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(M)\\}\n\\end{align*}\nfor any pfd modules~$M$ and~$N$.\n\\end{definition}\n\n\\begin{theorem}\\label{theorem_3}\n       Let~$M$ and~$N$ be pfd upset-decomposable modules then\n       \\[d_P(M,N) \\leq d_B(M,N).\\]\n   \\end{theorem}\n\n\\begin{theorem}\\label{theorem_main}\n       Let~$M$ and~$N$ be pfd upset-decomposable modules with ~$r$ =~$\\operatorname{supdim}$~$M < \\infty$ then\n       \\[d_B(M,N) \\leq (2r-1)\\,d_P(M,N).\\]\n   \\end{theorem}", "full_context": "Developed initially for data analysis, topological persistence provides invariants of filtered topological spaces through computation of their homology. This method produces algebraic objects called \\emph{persistence modules}, which are formally functors from the ordered set~$(\\mathbb{R},\\leq)$ to the category of vector spaces. A central result of the theory ensures that these modules decompose into simple elements associated with intervals when their vector spaces are finite-dimensional. The collection of intervals appearing in such a decomposition, the \\emph{persistence barcode}, completely describes the original persistence module up to isomorphism and, therefore, contains fine information about the initial topological filtration.\n\nThe remarkable isometry theorem \\cite{chazal_isometry} guarantees that the algebraically defined \\emph{interleaving distance}~$d_I$ between modules coincides with the combinatorial \\emph{bottleneck distance}~$d_B$ between their barcodes. The latter is defined through persistence diagrams and can be formulated as a minimum-cost bottleneck matching problem on a complete bipartite graph \\cite{Edelsbrunner_Harer}, which can be solved in polynomial time \\cite{Kerber_Morozov}. In particular, this theorem shows that the interleaving distance, while stable with respect to distance on data by construction, is also computable in polynomial time.\n\nWhen dealing with multi-parameter data, it is natural to consider filtrations depending on several parameters instead of a single one, thus creating a \\emph{multi-parameter persistence module}, that is, persistence modules indexed by~$\\mathbb{R}^n$ endowed with the product order. However, although the interleaving distance and the bottleneck distance are both generalized to the multiparameter setting, the isometry theorem no longer holds: multiparameter modules are not described in a stable way by their indecomposables. So, the bottleneck distance fails to satisfy the stability property in the multi-parameter case, making its usage useless. Henceforth, the computational advantage of the interleaving distance disappears in the multi-parameter setting, in particular, computing the interleaving distance for multi-parameter persistence modules has been proven to be NP-hard \\cite{bjerkevik_Botnan_kerber}.\n\nBjerkevik introduced the \\emph{$\\varepsilon$-pruning} (Definition~\\ref{def_pruning}) of a module in \\cite{bjerkevik_stabilizing}. The~$\\varepsilon$-pruning of~$M$ is an approximation of~$M$ which is particularly decomposable while discarding features below scale~$\\varepsilon$. This notion of pruning allowed Bjerkevik to introduce the pruning distance~$d_P$ (Definition~\\ref{def_pruning_distance}) which is related to both the interleaving distance and the decompositions. Therefore, it is unclear whether its computation is NP-hard or polynomial time computable. Bjerkevik conjectured that computing prunings can be done in polynomial time, and observes that, in the examples used to establish NP-hardness of computing the interleaving distance in \\cite{bjerkevik_botnan}, the pruning distance appears to be computable in polynomial time, suggesting the possibility of a general polynomial time algorithm.\n\nImportantly, Bjerkevik conjectured that the pruning distance is stable, and even Lipschitz equivalent to the interleaving distance, on persistence modules of a bounded pointwise dimension. More precisely, if~$M$ and~$N$ are persistence modules on an arbitrary poset with pointwise dimension smaller than a positive integer~$r$,\n\\[\nd_P(M,N) \\leq d_I(M,N) \\leq 2r\\,d_P(M,N).\n\\]\n\nIn this paper, we prove that the pruning distance is Lipschitz equivalent to the bottleneck distance for the class of upset-decomposable persistence modules, that is, modules that decompose as direct sums of interval persistence modules supported on subsets~$U \\subset \\mathbb{R}^n$ which are upward closed under the product order. Upset-decomposable modules are an important family of multi-parameter persistence modules that we study in detail in Section~\\ref{sec:pruning-upset}. Our main results, Theorem~\\ref{theorem_main} and Theorem~\\ref{theorem_3}, show the following\n\n\\begin{definition}\\label{def_pruning}\n\\emph{(Pruning)} Let~$M$ be a module. Let~$I$ be the largest submodule of~$M$ such that for any morphism~$f: M \\to M(2\\varepsilon)$, we have~$f(I) \\subseteq M_{0 \\to 2\\varepsilon}(I)$. \\\\\nLet~$K$ be the smallest submodule of~$I$ such that for any morphism~$f: M \\to M(2\\varepsilon)$, it holds that~$I^{-1}_{0 \\to 2\\varepsilon}(f(K)) \\subseteq K$.\\\\\nWe define the \\emph{$\\varepsilon$-pruning pair} of~$M$ to be the pair~$(I, K)$, and the \\emph{$\\varepsilon$-pruning} of~$M$ to be \n\\[\n\\Pru_\\varepsilon(M) := (I/K)(-\\varepsilon).\n\\]\n\\textit{For example} we have that the~$0$-pruning pair of a module~$M$ is~$(M, 0)$, and the~$0$-pruning of~$M$ is \n\\[\n\\Pru_0(M) = (M/0)(0) \\cong M.\n\\]\n\\end{definition}\n\n\\begin{definition}\n\\label{def_pruning_distance}\nThe \\emph{pruning distance}~$d_P$ is defined by\n\\begin{align*}\nd_P(M,N) = \\inf\\{&\\varepsilon\\geq 0\\mid \\forall \\delta\\geq 0,\\\\ &\\Pru_{\\varepsilon+\\delta}(M) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(N) \\text{ and }\\\\ &\\Pru_{\\varepsilon+\\delta}(N) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(M)\\}\n\\end{align*}\nfor any pfd modules~$M$ and~$N$.\n\\end{definition}\n\n\\begin{theorem}\\label{theorem_3}\n       Let~$M$ and~$N$ be pfd upset-decomposable modules then\n       \\[d_P(M,N) \\leq d_B(M,N).\\]\n   \\end{theorem}\n\n\\begin{theorem}\\label{theorem_main}\n       Let~$M$ and~$N$ be pfd upset-decomposable modules with ~$r$ =~$\\operatorname{supdim}$~$M < \\infty$ then\n       \\[d_B(M,N) \\leq (2r-1)\\,d_P(M,N).\\]\n   \\end{theorem}\n\nImportantly, Bjerkevik conjectured that the pruning distance is stable, and even Lipschitz equivalent to the interleaving distance, on persistence modules of a bounded pointwise dimension. More precisely, if~$M$ and~$N$ are persistence modules on an arbitrary poset with pointwise dimension smaller than a positive integer~$r$,\n\\[\nd_P(M,N) \\leq d_I(M,N) \\leq 2r\\,d_P(M,N).\n\\]\n\nOur approach is based on an explicit computation of the pruning of upset-decomposable persistence modules, carried out in Theorem~\\ref{theorem_1}. To compute the pruning, we associate a directed graph to an upset-decomposable module, thereby translate the problem into computing the reachable indices via the directed paths in the graph, which yields a purely combinatorial procedure.\n\n\\begin{theorem}\\label{theorem_3}\n Let~$M$ and~$N$ be pfd upset-decomposable modules then\n \\[d_P(M,N) \\leq d_B(M,N).\\]\n \\end{theorem}\n\\begin{proof}\nLet~$r = \\operatorname{supdim}$~$M$, if~$\\operatorname{supdim}$~$N \\neq r$ then both distances~$= \\infty$ so nothing to prove.\\\\\nSuppose that~$\\operatorname{supdim}$~$N = r$ i.e~$M=\\bigoplus_{l=1}^r M_l$ and~$N=\\bigoplus_{l=1}^r N_l.$ Let~$\\alpha \\geq 0$, such that there exists a matching between~$M$ and~$N$. This means that there exists a permutation~$\\sigma$ of~$\\{1,\\dots,r\\}$ such that \n\\[\nM_l \\subseteq N_{\\sigma(l)}(\\alpha) \\quad \\text{and} \\quad N_l \\subseteq M_{\\sigma^{-1}(l)}(\\alpha), \\quad \\forall l \\in \\{1,\\dots,r\\}.\n\\]\nDefine~$N' = \\bigoplus_{l=1}^r N'_l$ where~$N'_l = N_{\\sigma(l)}$, then~$N' \\cong N$. We get that\n \\begin{equation}\nM_l \\subseteq N'_l(\\alpha) \\quad \\text{and} \\quad N'_l \\subseteq M_l(\\alpha), \\quad \\forall l \\in \\{1,\\dots,r\\}.\n \\tag{4}\n\\label{eq:4}\n\\end{equation}\nFrom now on, to simplify the notation,~$N$ refers to~$N'$.\\\\\nWe want to show that~$\\forall \\delta\\geq 0, \\Pru_{\\alpha+\\delta}(M) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(N)$ and~$\\Pru_{\\alpha+\\delta}(N)$ is an~$\\infty\\text{-refinement of } \\Pru_\\delta(M).$ We prove only the first statement; the second follows by symmetry. By Lemma~\\ref{lem_infinit_refinement} it is sufficient to show that \n\\[\n\\forall \\delta\\geq 0, \\, \\forall l \\in \\{1,\\dots,r\\}, \\quad \\pi_{l}(\\operatorname{Pru}_{\\alpha+\\delta}(M)) \\subseteq \\pi_{l}(\\operatorname{Pru}_{\\delta}(N)).\n\\]\nFix~$l_1 \\in \\{1,\\dots,r\\}$ and~$\\delta \\geq 0$. Let~$l \\in JN_{l_1}$ in the directed graph~$G(N,\\delta)$, then there exists a path\n\\[\nP:\\; N_{l_1}=N_{l'_0}\\to N_{l'_1}\\to\\cdots\\to N_{l'_{c-1}}\\to N_{l'_{c}}=N_{l},\n\\]\nfor some integer~$c$.\nBy the construction of~$G(N,\\delta)$ each arrow gives a containment shifted by~$2\\delta$, hence\n\\[\nN_{l'_0} \\subseteq N_{l'_1}(2\\delta), N_{l'_1} \\subseteq N_{l'_2}(2\\delta), \\cdots , N_{l'_{c-1}} \\subseteq N_{l'_{c}}(2\\delta).\n\\]\nBy~\\eqref{eq:4} we get that \n\\[\nM_{l'_i} \\subseteq N_{l'_i}(\\alpha) \\subseteq N_{l'_{i+1}}(\\alpha+2\\delta) \\subseteq M_{l'_{i+1}}(2\\alpha+2\\delta), \\quad \\forall i \\in \\{0,\\dots,c-1\\}.\n\\]\nSo we get that there exists a path \n\\[\nP':\\; M_{l_1}=M_{l'_0}\\to M_{l'_1}\\to\\cdots\\to M_{l'_{c-1}}\\to M_{l'_{c}}=M_{l},\n\\]\nin the directed graph~$G(M,\\alpha+\\delta)$. Therefore~$l \\in JM_{l_1}$ and \n\\[\nJN_{l_1} \\subseteq JM_{l_1}.\n\\]\nTherefore, by Theorem~\\ref{theorem_1} and~\\eqref{eq:4} we obtain\n\\[\\pi_{l_1}(\\operatorname{Pru}_{\\alpha+\\delta}(M)) = \\bigcap_{l \\in JM_{l_1}} M_l(-\\delta)(-\\alpha) \\subseteq \\bigcap_{l \\in JN_{l_1}} N_l(-\\delta)=\\pi_{l_1}(\\operatorname{Pru}_\\delta(N)).\\]\nSince~$l_1$ and~$\\delta$ were arbitrary,\n\\[\n\\forall \\delta \\geq 0, \\forall l \\in \\{1,\\dots,r\\}, \\quad \\pi_l(\\operatorname{Pru}_{\\alpha+\\delta}(M)) \\subseteq \\pi_l(\\operatorname{Pru}_\\delta(N)).\n\\]\nThus~$\\forall \\delta\\geq 0, \\Pru_{\\alpha+\\delta}(M) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(N)$, and we conclude that \\[d_P(M,N) \\leq d_B(M,N).\\]\n\\end{proof}\n\nImportantly, Bjerkevik conjectured that the pruning distance is stable, and even Lipschitz equivalent to the interleaving distance, on persistence modules of a bounded pointwise dimension. More precisely, if~$M$ and~$N$ are persistence modules on an arbitrary poset with pointwise dimension smaller than a positive integer~$r$,\n\\[\nd_P(M,N) \\leq d_I(M,N) \\leq 2r\\,d_P(M,N).\n\\]\n\n\\begin{theorem}\\label{theorem_3}\n Let~$M$ and~$N$ be pfd upset-decomposable modules then\n \\[d_P(M,N) \\leq d_B(M,N).\\]\n \\end{theorem}\n\\begin{proof}\nLet~$r = \\operatorname{supdim}$~$M$, if~$\\operatorname{supdim}$~$N \\neq r$ then both distances~$= \\infty$ so nothing to prove.\\\\\nSuppose that~$\\operatorname{supdim}$~$N = r$ i.e~$M=\\bigoplus_{l=1}^r M_l$ and~$N=\\bigoplus_{l=1}^r N_l.$ Let~$\\alpha \\geq 0$, such that there exists a matching between~$M$ and~$N$. This means that there exists a permutation~$\\sigma$ of~$\\{1,\\dots,r\\}$ such that \n\\[\nM_l \\subseteq N_{\\sigma(l)}(\\alpha) \\quad \\text{and} \\quad N_l \\subseteq M_{\\sigma^{-1}(l)}(\\alpha), \\quad \\forall l \\in \\{1,\\dots,r\\}.\n\\]\nDefine~$N' = \\bigoplus_{l=1}^r N'_l$ where~$N'_l = N_{\\sigma(l)}$, then~$N' \\cong N$. We get that\n \\begin{equation}\nM_l \\subseteq N'_l(\\alpha) \\quad \\text{and} \\quad N'_l \\subseteq M_l(\\alpha), \\quad \\forall l \\in \\{1,\\dots,r\\}.\n \\tag{4}\n\\label{eq:4}\n\\end{equation}\nFrom now on, to simplify the notation,~$N$ refers to~$N'$.\\\\\nWe want to show that~$\\forall \\delta\\geq 0, \\Pru_{\\alpha+\\delta}(M) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(N)$ and~$\\Pru_{\\alpha+\\delta}(N)$ is an~$\\infty\\text{-refinement of } \\Pru_\\delta(M).$ We prove only the first statement; the second follows by symmetry. By Lemma~\\ref{lem_infinit_refinement} it is sufficient to show that \n\\[\n\\forall \\delta\\geq 0, \\, \\forall l \\in \\{1,\\dots,r\\}, \\quad \\pi_{l}(\\operatorname{Pru}_{\\alpha+\\delta}(M)) \\subseteq \\pi_{l}(\\operatorname{Pru}_{\\delta}(N)).\n\\]\nFix~$l_1 \\in \\{1,\\dots,r\\}$ and~$\\delta \\geq 0$. Let~$l \\in JN_{l_1}$ in the directed graph~$G(N,\\delta)$, then there exists a path\n\\[\nP:\\; N_{l_1}=N_{l'_0}\\to N_{l'_1}\\to\\cdots\\to N_{l'_{c-1}}\\to N_{l'_{c}}=N_{l},\n\\]\nfor some integer~$c$.\nBy the construction of~$G(N,\\delta)$ each arrow gives a containment shifted by~$2\\delta$, hence\n\\[\nN_{l'_0} \\subseteq N_{l'_1}(2\\delta), N_{l'_1} \\subseteq N_{l'_2}(2\\delta), \\cdots , N_{l'_{c-1}} \\subseteq N_{l'_{c}}(2\\delta).\n\\]\nBy~\\eqref{eq:4} we get that \n\\[\nM_{l'_i} \\subseteq N_{l'_i}(\\alpha) \\subseteq N_{l'_{i+1}}(\\alpha+2\\delta) \\subseteq M_{l'_{i+1}}(2\\alpha+2\\delta), \\quad \\forall i \\in \\{0,\\dots,c-1\\}.\n\\]\nSo we get that there exists a path \n\\[\nP':\\; M_{l_1}=M_{l'_0}\\to M_{l'_1}\\to\\cdots\\to M_{l'_{c-1}}\\to M_{l'_{c}}=M_{l},\n\\]\nin the directed graph~$G(M,\\alpha+\\delta)$. Therefore~$l \\in JM_{l_1}$ and \n\\[\nJN_{l_1} \\subseteq JM_{l_1}.\n\\]\nTherefore, by Theorem~\\ref{theorem_1} and~\\eqref{eq:4} we obtain\n\\[\\pi_{l_1}(\\operatorname{Pru}_{\\alpha+\\delta}(M)) = \\bigcap_{l \\in JM_{l_1}} M_l(-\\delta)(-\\alpha) \\subseteq \\bigcap_{l \\in JN_{l_1}} N_l(-\\delta)=\\pi_{l_1}(\\operatorname{Pru}_\\delta(N)).\\]\nSince~$l_1$ and~$\\delta$ were arbitrary,\n\\[\n\\forall \\delta \\geq 0, \\forall l \\in \\{1,\\dots,r\\}, \\quad \\pi_l(\\operatorname{Pru}_{\\alpha+\\delta}(M)) \\subseteq \\pi_l(\\operatorname{Pru}_\\delta(N)).\n\\]\nThus~$\\forall \\delta\\geq 0, \\Pru_{\\alpha+\\delta}(M) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(N)$, and we conclude that \\[d_P(M,N) \\leq d_B(M,N).\\]\n\\end{proof}", "post_theorem_intro_text_len": 966, "post_theorem_intro_text": "This result improves the right-hand inequality in Bjerkevik’s conjecture by reducing the constant from~$2r$ to~$2r-1$, and strengthens it by replacing the interleaving distance with the bottleneck distance, which dominates the interleaving distance. Moreover, in Example~\\ref{ex_2} we show that the constant~$2r-1$ is optimal on the class of upset-decomposable persistence modules, for  both the interleaving distance and the bottleneck distance. The left-hand inequality follows from our analysis and is weaker than the conjectured inequality involving the interleaving distance.\n\nOur approach is based on an explicit computation of the pruning of upset-decomposable persistence modules, carried out in Theorem~\\ref{theorem_1}. To compute the pruning, we associate a directed graph to an upset-decomposable module, thereby translate the problem into computing the reachable indices via the directed paths in the graph, which yields a purely combinatorial procedure.", "sketch": "Our approach is based on an explicit computation of the pruning of upset-decomposable persistence modules, carried out in Theorem~\\ref{theorem_1}. To compute the pruning, we associate a directed graph to an upset-decomposable module, thereby translate the problem into computing the reachable indices via the directed paths in the graph, which yields a purely combinatorial procedure.", "expanded_sketch": "Our approach is based on an explicit computation of the pruning of upset-decomposable persistence modules, carried out in the following theorem. \\begin{theorem}\\label{theorem_1}\nLet~$\\alpha \\geq 0$ and let~$M = \\bigoplus_{l=1}^r M_l$ be an upset-decomposable module. Then\n\\[\n\\pi_{l_1}(\\operatorname{Pru}_\\alpha(M)) = \\bigcap_{l \\in JM_{l_1}} M_l(-\\alpha),\n\\]\nwhere~$JM_{l_1}$ is as defined above.\n\\end{theorem}\nTo compute the pruning, we associate a directed graph to an upset-decomposable module, thereby translate the problem into computing the reachable indices via the directed paths in the graph, which yields a purely combinatorial procedure.", "expanded_theorem": "If~$M$ and~$N$ are two upset-decomposable modules of pointwise dimension at most~$r$, then\n\\[\nd_P(M,N) \\leq d_B(M,N) \\leq (2r-1)d_P(M,N).\n\\],", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $M$ and $N$ be persistence modules indexed by $(\\mathbb{R}^n,\\leq)$ with the product order. Assume that both are upset-decomposable, meaning that each can be written as a direct sum of interval persistence modules supported on subsets $U\\subseteq \\mathbb{R}^n$ that are upward closed under the product order, and assume their pointwise dimensions are bounded by $r$ (i.e. at every parameter value, the corresponding vector space has dimension at most $r$). If $d_P$ denotes the pruning distance (defined via the $\\varepsilon$-prunings $\\operatorname{Pru}_\\varepsilon$) and $d_B$ denotes the bottleneck distance for these decompositions, which quantitative estimate holds?", "correct_choice": {"label": "A", "text": "For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has\n\\[\nd_P(M,N) \\le d_B(M,N) \\le (2r-1)\\, d_P(M,N).\n\\]"}, "choices": [{"label": "B", "text": "For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has\n\\[\nd_P(M,N) \\le d_B(M,N) \\le 2r\\, d_P(M,N).\n\\]"}, {"label": "C", "text": "For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has\n\\[\nd_P(M,N) \\le d_B(M,N).\n\\]"}, {"label": "D", "text": "For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has\n\\[\nd_B(M,N) \\le d_P(M,N) \\le (2r-1)\\, d_B(M,N).\n\\]"}, {"label": "E", "text": "For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has\n\\[\nd_P(M,N) \\le d_B(M,N) \\le (2^r-1)\\, d_P(M,N).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "counting_estimate", "tampered_component": "sharp constant from reachable-index/counting argument", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "geometric_construction", "tampered_component": "dropped the upper Lipschitz bound $d_B\\le(2r-1)d_P$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "direction of comparison between pruning and bottleneck distances", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "linear dependence on pointwise dimension $r$ replaced by exponential dependence", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It states the hypotheses and asks which distance comparison is valid, without embedding the sharp constant or direction of the inequalities."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem lists the exact hypotheses and asks for the corresponding conclusion. It closely mirrors a standard theorem statement rather than testing transfer or application."}, "GPS": {"score": 1, "justification": "There is some pressure to distinguish a sharp equivalence from weaker, reversed, or mis-constant variants, but the item mainly rewards memory of the theorem rather than substantial derivation or generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one has a slightly wrong constant, one gives only a weaker true statement, one incorrectly removes dependence on r, and one reverses the inequality direction. These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A technically well-constructed but theorem-recall-heavy MCQ. It avoids direct answer leakage and uses strong distractors, yet it is largely tautological and only moderately tests reasoning."}}
{"id": "2602.15243v1", "paper_link": "http://arxiv.org/abs/2602.15243v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "If~$M$ and~$N$ are two upset-decomposable modules of pointwise dimension at most~$r$, then\n\\[\nd_P(M,N) \\leq d_B(M,N) \\leq (2r-1)d_P(M,N).\n\\]", "start_pos": 8489, "end_pos": 8659, "label": null}, "ref_dict": {"theorem_main": "\\begin{theorem}\\label{theorem_main}\n       Let~$M$ and~$N$ be pfd upset-decomposable modules with ~$r$ =~$\\operatorname{supdim}$~$M < \\infty$ then\n       \\[d_B(M,N) \\leq (2r-1)\\,d_P(M,N).\\]\n   \\end{theorem}", "theorem_3": "\\begin{theorem}\\label{theorem_3}\n       Let~$M$ and~$N$ be pfd upset-decomposable modules then\n       \\[d_P(M,N) \\leq d_B(M,N).\\]\n   \\end{theorem}", "theorem_1": "\\begin{theorem}\\label{theorem_1}\nLet~$\\alpha \\geq 0$ and let~$M = \\bigoplus_{l=1}^r M_l$ be an upset-decomposable module. Then\n\\[\n\\pi_{l_1}(\\operatorname{Pru}_\\alpha(M)) = \\bigcap_{l \\in JM_{l_1}} M_l(-\\alpha),\n\\]\nwhere~$JM_{l_1}$ is as defined above.\n\\end{theorem}", "ex_2": "\\begin{example}\\label{ex_2}\nLet~$\\alpha>0$ and~$r \\in \\mathbb{N^*}$. And let\n$\nM = \\bigoplus_{i=1}^r M_i \\text{ and } \nN = \\bigoplus_{i=1}^r N_i,\n$ be two~$d$-parameter upset-decomposable modules\nwith supports\n\\[\n\\operatorname{supp} (M_i) = \\{(x_1,\\dots,x_d) \\mid (x_1,\\dots,x_d) \\ge (2i\\alpha,\\dots,2i\\alpha)\\}\\]\n\\[\n\\operatorname{supp} (N_i) = \\{(x_1,\\dots,x_d) \\mid (x_1,\\dots,x_d) \\ge \\left( (2r+1)\\alpha,\\dots,(2r+1)\\alpha \\right)\\}.\n\\]\nThen one can check that \n\\[\n\\operatorname{Pru}_{\\alpha}(M) \\cong N,\n\\]\nso~$\\operatorname{Pru}_{\\alpha}(M)$ is an~$\\infty$-refinement of~$N$. \nMoreover, for every~$\\delta>0$, \n\\[\n\\operatorname{Pru}_{\\alpha+\\delta} (M) \\cong \\operatorname{Pru}_{\\delta} (N) \\cong N(-\\delta),\n\\] \nso we also have~$\\operatorname{Pru}_{\\alpha+\\delta}(M)$ as an~$\\infty$-refinement of~$N(-\\delta)$.\nWe can also easily check that~$\\forall \\varepsilon < \\alpha, \\operatorname{Pru}_{\\varepsilon}(M)$ is not an~$\\infty$-refinement of~$N$.\nSimilarly,~$N$ is an~$\\infty$-refinement of~$M$, and for every~$\\delta>0$,~$\\operatorname{Pru}_{\\delta}(N)$ is an~$\\infty$-refinement of~$\\operatorname{Pru}_{\\delta}(M)$.\\\\\nFrom these observations we obtain that\n\\[\nd_P(M,N) = \\alpha.\n\\]\nFurthermore, since all~$N_i$ have the same support, it is easy to check that\n\\[\nd_B(M,N) = d_I(M,N) = (2r-1)\\alpha,\n\\]\nshowing that the upper bound in Theorem~\\ref{theorem_main} can indeed be attained ~$\\forall \\alpha>0$,~$\\forall d \\in \\mathbb{N}^*$ and~$\\forall r \\in \\mathbb{N}^*$. In particular, the constant~$2r-1$ is optimal, and the inequality cannot be strengthened neither for the bottleneck distance~$d_B$ nor for the interleaving distance~$d_I$.\n\n\\end{example}", "def_pruning": "\\begin{definition}\\label{def_pruning}\n\\emph{(Pruning)} Let~$M$ be a module. Let~$I$ be the largest submodule of~$M$ such that for any morphism~$f: M \\to M(2\\varepsilon)$, we have~$f(I) \\subseteq M_{0 \\to 2\\varepsilon}(I)$. \\\\\nLet~$K$ be the smallest submodule of~$I$ such that for any morphism~$f: M \\to M(2\\varepsilon)$, it holds that~$I^{-1}_{0 \\to 2\\varepsilon}(f(K)) \\subseteq K$.\\\\\nWe define the \\emph{$\\varepsilon$-pruning pair} of~$M$ to be the pair~$(I, K)$, and the \\emph{$\\varepsilon$-pruning} of~$M$ to be \n\\[\n\\Pru_\\varepsilon(M) := (I/K)(-\\varepsilon).\n\\]\n\\textit{For example} we have that the~$0$-pruning pair of a module~$M$ is~$(M, 0)$, and the~$0$-pruning of~$M$ is \n\\[\n\\Pru_0(M) = (M/0)(0) \\cong M.\n\\]\n\\end{definition}", "def_pruning_distance": "\\begin{definition}\n\\label{def_pruning_distance}\nThe \\emph{pruning distance}~$d_P$ is defined by\n\\begin{align*}\nd_P(M,N) = \\inf\\{&\\varepsilon\\geq 0\\mid \\forall \\delta\\geq 0,\\\\ &\\Pru_{\\varepsilon+\\delta}(M) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(N) \\text{ and }\\\\ &\\Pru_{\\varepsilon+\\delta}(N) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(M)\\}\n\\end{align*}\nfor any pfd modules~$M$ and~$N$.\n\\end{definition}"}, "pre_theorem_intro_text_len": 4197, "pre_theorem_intro_text": "\\markboth{1. Introduction}{} \n\nDeveloped initially for data analysis, topological persistence provides invariants of filtered topological spaces through computation of their homology. This method produces algebraic objects called \\emph{persistence modules}, which are formally functors from the ordered set~$(\\mathbb{R},\\leq)$ to the category of vector spaces. A central result of the theory ensures that these modules decompose into simple elements associated with intervals when their vector spaces are finite-dimensional. The collection of intervals appearing in such a decomposition, the \\emph{persistence barcode}, completely describes the original persistence module up to isomorphism and, therefore, contains fine information about the initial topological filtration.\n\nThe remarkable isometry theorem \\cite{chazal_isometry} guarantees that the algebraically defined \\emph{interleaving distance}~$d_I$ between modules coincides with the combinatorial \\emph{bottleneck distance}~$d_B$ between their barcodes. The latter is defined through persistence diagrams and can be formulated as a minimum-cost bottleneck matching problem on a complete bipartite graph \\cite{Edelsbrunner_Harer}, which can be solved in polynomial time \\cite{Kerber_Morozov}. In particular, this theorem shows that the interleaving distance, while stable with respect to distance on data by construction, is also computable in polynomial time.\n\nWhen dealing with multi-parameter data, it is natural to consider filtrations depending on several parameters instead of a single one, thus creating a \\emph{multi-parameter persistence module}, that is, persistence modules indexed by~$\\mathbb{R}^n$ endowed with the product order. However, although the interleaving distance and the bottleneck distance are both generalized to the multiparameter setting, the isometry theorem no longer holds: multiparameter modules are not described in a stable way by their indecomposables. So, the bottleneck distance fails to satisfy the stability property in the multi-parameter case, making its usage useless. Henceforth, the computational advantage of the interleaving distance disappears in the multi-parameter setting, in particular, computing the interleaving distance for multi-parameter persistence modules has been proven to be NP-hard \\cite{bjerkevik_Botnan_kerber}. \n\nBjerkevik introduced the \\emph{$\\varepsilon$-pruning} (Definition~\\ref{def_pruning}) of a module in \\cite{bjerkevik_stabilizing}. The~$\\varepsilon$-pruning of~$M$ is an approximation of~$M$ which is particularly decomposable while discarding features below scale~$\\varepsilon$. This notion of pruning allowed Bjerkevik to introduce the pruning distance~$d_P$ (Definition~\\ref{def_pruning_distance}) which is related to both the interleaving distance and the decompositions. Therefore, it is unclear whether its computation is NP-hard or polynomial time computable. Bjerkevik conjectured that computing prunings can be done in polynomial time, and observes that, in the examples used to establish NP-hardness of computing the interleaving distance in \\cite{bjerkevik_botnan}, the pruning distance appears to be computable in polynomial time, suggesting the possibility of a general polynomial time algorithm.\n\nImportantly, Bjerkevik conjectured that the pruning distance is stable, and even Lipschitz equivalent to the interleaving distance, on persistence modules of a bounded pointwise dimension. More precisely, if~$M$ and~$N$ are persistence modules on an arbitrary poset with pointwise dimension smaller than a positive integer~$r$,\n\\[\nd_P(M,N) \\leq d_I(M,N) \\leq 2r\\,d_P(M,N).\n\\]\n\nIn this paper, we prove that the pruning distance is Lipschitz equivalent to the bottleneck distance for the class of upset-decomposable persistence modules, that is, modules that decompose as direct sums of interval persistence modules supported on subsets~$U \\subset \\mathbb{R}^n$ which are upward closed under the product order. Upset-decomposable modules are an important family of multi-parameter persistence modules that we study in detail in Section~\\ref{sec:pruning-upset}. Our main results, Theorem~\\ref{theorem_main} and Theorem~\\ref{theorem_3}, show the following", "context": "Developed initially for data analysis, topological persistence provides invariants of filtered topological spaces through computation of their homology. This method produces algebraic objects called \\emph{persistence modules}, which are formally functors from the ordered set~$(\\mathbb{R},\\leq)$ to the category of vector spaces. A central result of the theory ensures that these modules decompose into simple elements associated with intervals when their vector spaces are finite-dimensional. The collection of intervals appearing in such a decomposition, the \\emph{persistence barcode}, completely describes the original persistence module up to isomorphism and, therefore, contains fine information about the initial topological filtration.\n\nThe remarkable isometry theorem \\cite{chazal_isometry} guarantees that the algebraically defined \\emph{interleaving distance}~$d_I$ between modules coincides with the combinatorial \\emph{bottleneck distance}~$d_B$ between their barcodes. The latter is defined through persistence diagrams and can be formulated as a minimum-cost bottleneck matching problem on a complete bipartite graph \\cite{Edelsbrunner_Harer}, which can be solved in polynomial time \\cite{Kerber_Morozov}. In particular, this theorem shows that the interleaving distance, while stable with respect to distance on data by construction, is also computable in polynomial time.\n\nWhen dealing with multi-parameter data, it is natural to consider filtrations depending on several parameters instead of a single one, thus creating a \\emph{multi-parameter persistence module}, that is, persistence modules indexed by~$\\mathbb{R}^n$ endowed with the product order. However, although the interleaving distance and the bottleneck distance are both generalized to the multiparameter setting, the isometry theorem no longer holds: multiparameter modules are not described in a stable way by their indecomposables. So, the bottleneck distance fails to satisfy the stability property in the multi-parameter case, making its usage useless. Henceforth, the computational advantage of the interleaving distance disappears in the multi-parameter setting, in particular, computing the interleaving distance for multi-parameter persistence modules has been proven to be NP-hard \\cite{bjerkevik_Botnan_kerber}.\n\nBjerkevik introduced the \\emph{$\\varepsilon$-pruning} (Definition~\\ref{def_pruning}) of a module in \\cite{bjerkevik_stabilizing}. The~$\\varepsilon$-pruning of~$M$ is an approximation of~$M$ which is particularly decomposable while discarding features below scale~$\\varepsilon$. This notion of pruning allowed Bjerkevik to introduce the pruning distance~$d_P$ (Definition~\\ref{def_pruning_distance}) which is related to both the interleaving distance and the decompositions. Therefore, it is unclear whether its computation is NP-hard or polynomial time computable. Bjerkevik conjectured that computing prunings can be done in polynomial time, and observes that, in the examples used to establish NP-hardness of computing the interleaving distance in \\cite{bjerkevik_botnan}, the pruning distance appears to be computable in polynomial time, suggesting the possibility of a general polynomial time algorithm.\n\nImportantly, Bjerkevik conjectured that the pruning distance is stable, and even Lipschitz equivalent to the interleaving distance, on persistence modules of a bounded pointwise dimension. More precisely, if~$M$ and~$N$ are persistence modules on an arbitrary poset with pointwise dimension smaller than a positive integer~$r$,\n\\[\nd_P(M,N) \\leq d_I(M,N) \\leq 2r\\,d_P(M,N).\n\\]\n\nIn this paper, we prove that the pruning distance is Lipschitz equivalent to the bottleneck distance for the class of upset-decomposable persistence modules, that is, modules that decompose as direct sums of interval persistence modules supported on subsets~$U \\subset \\mathbb{R}^n$ which are upward closed under the product order. Upset-decomposable modules are an important family of multi-parameter persistence modules that we study in detail in Section~\\ref{sec:pruning-upset}. Our main results, Theorem~\\ref{theorem_main} and Theorem~\\ref{theorem_3}, show the following\n\n\\begin{definition}\\label{def_pruning}\n\\emph{(Pruning)} Let~$M$ be a module. Let~$I$ be the largest submodule of~$M$ such that for any morphism~$f: M \\to M(2\\varepsilon)$, we have~$f(I) \\subseteq M_{0 \\to 2\\varepsilon}(I)$. \\\\\nLet~$K$ be the smallest submodule of~$I$ such that for any morphism~$f: M \\to M(2\\varepsilon)$, it holds that~$I^{-1}_{0 \\to 2\\varepsilon}(f(K)) \\subseteq K$.\\\\\nWe define the \\emph{$\\varepsilon$-pruning pair} of~$M$ to be the pair~$(I, K)$, and the \\emph{$\\varepsilon$-pruning} of~$M$ to be \n\\[\n\\Pru_\\varepsilon(M) := (I/K)(-\\varepsilon).\n\\]\n\\textit{For example} we have that the~$0$-pruning pair of a module~$M$ is~$(M, 0)$, and the~$0$-pruning of~$M$ is \n\\[\n\\Pru_0(M) = (M/0)(0) \\cong M.\n\\]\n\\end{definition}\n\n\\begin{definition}\n\\label{def_pruning_distance}\nThe \\emph{pruning distance}~$d_P$ is defined by\n\\begin{align*}\nd_P(M,N) = \\inf\\{&\\varepsilon\\geq 0\\mid \\forall \\delta\\geq 0,\\\\ &\\Pru_{\\varepsilon+\\delta}(M) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(N) \\text{ and }\\\\ &\\Pru_{\\varepsilon+\\delta}(N) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(M)\\}\n\\end{align*}\nfor any pfd modules~$M$ and~$N$.\n\\end{definition}\n\n\\begin{theorem}\\label{theorem_3}\n       Let~$M$ and~$N$ be pfd upset-decomposable modules then\n       \\[d_P(M,N) \\leq d_B(M,N).\\]\n   \\end{theorem}\n\n\\begin{theorem}\\label{theorem_main}\n       Let~$M$ and~$N$ be pfd upset-decomposable modules with ~$r$ =~$\\operatorname{supdim}$~$M < \\infty$ then\n       \\[d_B(M,N) \\leq (2r-1)\\,d_P(M,N).\\]\n   \\end{theorem}", "full_context": "Developed initially for data analysis, topological persistence provides invariants of filtered topological spaces through computation of their homology. This method produces algebraic objects called \\emph{persistence modules}, which are formally functors from the ordered set~$(\\mathbb{R},\\leq)$ to the category of vector spaces. A central result of the theory ensures that these modules decompose into simple elements associated with intervals when their vector spaces are finite-dimensional. The collection of intervals appearing in such a decomposition, the \\emph{persistence barcode}, completely describes the original persistence module up to isomorphism and, therefore, contains fine information about the initial topological filtration.\n\nThe remarkable isometry theorem \\cite{chazal_isometry} guarantees that the algebraically defined \\emph{interleaving distance}~$d_I$ between modules coincides with the combinatorial \\emph{bottleneck distance}~$d_B$ between their barcodes. The latter is defined through persistence diagrams and can be formulated as a minimum-cost bottleneck matching problem on a complete bipartite graph \\cite{Edelsbrunner_Harer}, which can be solved in polynomial time \\cite{Kerber_Morozov}. In particular, this theorem shows that the interleaving distance, while stable with respect to distance on data by construction, is also computable in polynomial time.\n\nWhen dealing with multi-parameter data, it is natural to consider filtrations depending on several parameters instead of a single one, thus creating a \\emph{multi-parameter persistence module}, that is, persistence modules indexed by~$\\mathbb{R}^n$ endowed with the product order. However, although the interleaving distance and the bottleneck distance are both generalized to the multiparameter setting, the isometry theorem no longer holds: multiparameter modules are not described in a stable way by their indecomposables. So, the bottleneck distance fails to satisfy the stability property in the multi-parameter case, making its usage useless. Henceforth, the computational advantage of the interleaving distance disappears in the multi-parameter setting, in particular, computing the interleaving distance for multi-parameter persistence modules has been proven to be NP-hard \\cite{bjerkevik_Botnan_kerber}.\n\nBjerkevik introduced the \\emph{$\\varepsilon$-pruning} (Definition~\\ref{def_pruning}) of a module in \\cite{bjerkevik_stabilizing}. The~$\\varepsilon$-pruning of~$M$ is an approximation of~$M$ which is particularly decomposable while discarding features below scale~$\\varepsilon$. This notion of pruning allowed Bjerkevik to introduce the pruning distance~$d_P$ (Definition~\\ref{def_pruning_distance}) which is related to both the interleaving distance and the decompositions. Therefore, it is unclear whether its computation is NP-hard or polynomial time computable. Bjerkevik conjectured that computing prunings can be done in polynomial time, and observes that, in the examples used to establish NP-hardness of computing the interleaving distance in \\cite{bjerkevik_botnan}, the pruning distance appears to be computable in polynomial time, suggesting the possibility of a general polynomial time algorithm.\n\nImportantly, Bjerkevik conjectured that the pruning distance is stable, and even Lipschitz equivalent to the interleaving distance, on persistence modules of a bounded pointwise dimension. More precisely, if~$M$ and~$N$ are persistence modules on an arbitrary poset with pointwise dimension smaller than a positive integer~$r$,\n\\[\nd_P(M,N) \\leq d_I(M,N) \\leq 2r\\,d_P(M,N).\n\\]\n\nIn this paper, we prove that the pruning distance is Lipschitz equivalent to the bottleneck distance for the class of upset-decomposable persistence modules, that is, modules that decompose as direct sums of interval persistence modules supported on subsets~$U \\subset \\mathbb{R}^n$ which are upward closed under the product order. Upset-decomposable modules are an important family of multi-parameter persistence modules that we study in detail in Section~\\ref{sec:pruning-upset}. Our main results, Theorem~\\ref{theorem_main} and Theorem~\\ref{theorem_3}, show the following\n\n\\begin{definition}\\label{def_pruning}\n\\emph{(Pruning)} Let~$M$ be a module. Let~$I$ be the largest submodule of~$M$ such that for any morphism~$f: M \\to M(2\\varepsilon)$, we have~$f(I) \\subseteq M_{0 \\to 2\\varepsilon}(I)$. \\\\\nLet~$K$ be the smallest submodule of~$I$ such that for any morphism~$f: M \\to M(2\\varepsilon)$, it holds that~$I^{-1}_{0 \\to 2\\varepsilon}(f(K)) \\subseteq K$.\\\\\nWe define the \\emph{$\\varepsilon$-pruning pair} of~$M$ to be the pair~$(I, K)$, and the \\emph{$\\varepsilon$-pruning} of~$M$ to be \n\\[\n\\Pru_\\varepsilon(M) := (I/K)(-\\varepsilon).\n\\]\n\\textit{For example} we have that the~$0$-pruning pair of a module~$M$ is~$(M, 0)$, and the~$0$-pruning of~$M$ is \n\\[\n\\Pru_0(M) = (M/0)(0) \\cong M.\n\\]\n\\end{definition}\n\n\\begin{definition}\n\\label{def_pruning_distance}\nThe \\emph{pruning distance}~$d_P$ is defined by\n\\begin{align*}\nd_P(M,N) = \\inf\\{&\\varepsilon\\geq 0\\mid \\forall \\delta\\geq 0,\\\\ &\\Pru_{\\varepsilon+\\delta}(M) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(N) \\text{ and }\\\\ &\\Pru_{\\varepsilon+\\delta}(N) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(M)\\}\n\\end{align*}\nfor any pfd modules~$M$ and~$N$.\n\\end{definition}\n\n\\begin{theorem}\\label{theorem_3}\n       Let~$M$ and~$N$ be pfd upset-decomposable modules then\n       \\[d_P(M,N) \\leq d_B(M,N).\\]\n   \\end{theorem}\n\n\\begin{theorem}\\label{theorem_main}\n       Let~$M$ and~$N$ be pfd upset-decomposable modules with ~$r$ =~$\\operatorname{supdim}$~$M < \\infty$ then\n       \\[d_B(M,N) \\leq (2r-1)\\,d_P(M,N).\\]\n   \\end{theorem}\n\nImportantly, Bjerkevik conjectured that the pruning distance is stable, and even Lipschitz equivalent to the interleaving distance, on persistence modules of a bounded pointwise dimension. More precisely, if~$M$ and~$N$ are persistence modules on an arbitrary poset with pointwise dimension smaller than a positive integer~$r$,\n\\[\nd_P(M,N) \\leq d_I(M,N) \\leq 2r\\,d_P(M,N).\n\\]\n\nOur approach is based on an explicit computation of the pruning of upset-decomposable persistence modules, carried out in Theorem~\\ref{theorem_1}. To compute the pruning, we associate a directed graph to an upset-decomposable module, thereby translate the problem into computing the reachable indices via the directed paths in the graph, which yields a purely combinatorial procedure.\n\n\\begin{theorem}\\label{theorem_3}\n Let~$M$ and~$N$ be pfd upset-decomposable modules then\n \\[d_P(M,N) \\leq d_B(M,N).\\]\n \\end{theorem}\n\\begin{proof}\nLet~$r = \\operatorname{supdim}$~$M$, if~$\\operatorname{supdim}$~$N \\neq r$ then both distances~$= \\infty$ so nothing to prove.\\\\\nSuppose that~$\\operatorname{supdim}$~$N = r$ i.e~$M=\\bigoplus_{l=1}^r M_l$ and~$N=\\bigoplus_{l=1}^r N_l.$ Let~$\\alpha \\geq 0$, such that there exists a matching between~$M$ and~$N$. This means that there exists a permutation~$\\sigma$ of~$\\{1,\\dots,r\\}$ such that \n\\[\nM_l \\subseteq N_{\\sigma(l)}(\\alpha) \\quad \\text{and} \\quad N_l \\subseteq M_{\\sigma^{-1}(l)}(\\alpha), \\quad \\forall l \\in \\{1,\\dots,r\\}.\n\\]\nDefine~$N' = \\bigoplus_{l=1}^r N'_l$ where~$N'_l = N_{\\sigma(l)}$, then~$N' \\cong N$. We get that\n \\begin{equation}\nM_l \\subseteq N'_l(\\alpha) \\quad \\text{and} \\quad N'_l \\subseteq M_l(\\alpha), \\quad \\forall l \\in \\{1,\\dots,r\\}.\n \\tag{4}\n\\label{eq:4}\n\\end{equation}\nFrom now on, to simplify the notation,~$N$ refers to~$N'$.\\\\\nWe want to show that~$\\forall \\delta\\geq 0, \\Pru_{\\alpha+\\delta}(M) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(N)$ and~$\\Pru_{\\alpha+\\delta}(N)$ is an~$\\infty\\text{-refinement of } \\Pru_\\delta(M).$ We prove only the first statement; the second follows by symmetry. By Lemma~\\ref{lem_infinit_refinement} it is sufficient to show that \n\\[\n\\forall \\delta\\geq 0, \\, \\forall l \\in \\{1,\\dots,r\\}, \\quad \\pi_{l}(\\operatorname{Pru}_{\\alpha+\\delta}(M)) \\subseteq \\pi_{l}(\\operatorname{Pru}_{\\delta}(N)).\n\\]\nFix~$l_1 \\in \\{1,\\dots,r\\}$ and~$\\delta \\geq 0$. Let~$l \\in JN_{l_1}$ in the directed graph~$G(N,\\delta)$, then there exists a path\n\\[\nP:\\; N_{l_1}=N_{l'_0}\\to N_{l'_1}\\to\\cdots\\to N_{l'_{c-1}}\\to N_{l'_{c}}=N_{l},\n\\]\nfor some integer~$c$.\nBy the construction of~$G(N,\\delta)$ each arrow gives a containment shifted by~$2\\delta$, hence\n\\[\nN_{l'_0} \\subseteq N_{l'_1}(2\\delta), N_{l'_1} \\subseteq N_{l'_2}(2\\delta), \\cdots , N_{l'_{c-1}} \\subseteq N_{l'_{c}}(2\\delta).\n\\]\nBy~\\eqref{eq:4} we get that \n\\[\nM_{l'_i} \\subseteq N_{l'_i}(\\alpha) \\subseteq N_{l'_{i+1}}(\\alpha+2\\delta) \\subseteq M_{l'_{i+1}}(2\\alpha+2\\delta), \\quad \\forall i \\in \\{0,\\dots,c-1\\}.\n\\]\nSo we get that there exists a path \n\\[\nP':\\; M_{l_1}=M_{l'_0}\\to M_{l'_1}\\to\\cdots\\to M_{l'_{c-1}}\\to M_{l'_{c}}=M_{l},\n\\]\nin the directed graph~$G(M,\\alpha+\\delta)$. Therefore~$l \\in JM_{l_1}$ and \n\\[\nJN_{l_1} \\subseteq JM_{l_1}.\n\\]\nTherefore, by Theorem~\\ref{theorem_1} and~\\eqref{eq:4} we obtain\n\\[\\pi_{l_1}(\\operatorname{Pru}_{\\alpha+\\delta}(M)) = \\bigcap_{l \\in JM_{l_1}} M_l(-\\delta)(-\\alpha) \\subseteq \\bigcap_{l \\in JN_{l_1}} N_l(-\\delta)=\\pi_{l_1}(\\operatorname{Pru}_\\delta(N)).\\]\nSince~$l_1$ and~$\\delta$ were arbitrary,\n\\[\n\\forall \\delta \\geq 0, \\forall l \\in \\{1,\\dots,r\\}, \\quad \\pi_l(\\operatorname{Pru}_{\\alpha+\\delta}(M)) \\subseteq \\pi_l(\\operatorname{Pru}_\\delta(N)).\n\\]\nThus~$\\forall \\delta\\geq 0, \\Pru_{\\alpha+\\delta}(M) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(N)$, and we conclude that \\[d_P(M,N) \\leq d_B(M,N).\\]\n\\end{proof}\n\nImportantly, Bjerkevik conjectured that the pruning distance is stable, and even Lipschitz equivalent to the interleaving distance, on persistence modules of a bounded pointwise dimension. More precisely, if~$M$ and~$N$ are persistence modules on an arbitrary poset with pointwise dimension smaller than a positive integer~$r$,\n\\[\nd_P(M,N) \\leq d_I(M,N) \\leq 2r\\,d_P(M,N).\n\\]\n\n\\begin{theorem}\\label{theorem_3}\n Let~$M$ and~$N$ be pfd upset-decomposable modules then\n \\[d_P(M,N) \\leq d_B(M,N).\\]\n \\end{theorem}\n\\begin{proof}\nLet~$r = \\operatorname{supdim}$~$M$, if~$\\operatorname{supdim}$~$N \\neq r$ then both distances~$= \\infty$ so nothing to prove.\\\\\nSuppose that~$\\operatorname{supdim}$~$N = r$ i.e~$M=\\bigoplus_{l=1}^r M_l$ and~$N=\\bigoplus_{l=1}^r N_l.$ Let~$\\alpha \\geq 0$, such that there exists a matching between~$M$ and~$N$. This means that there exists a permutation~$\\sigma$ of~$\\{1,\\dots,r\\}$ such that \n\\[\nM_l \\subseteq N_{\\sigma(l)}(\\alpha) \\quad \\text{and} \\quad N_l \\subseteq M_{\\sigma^{-1}(l)}(\\alpha), \\quad \\forall l \\in \\{1,\\dots,r\\}.\n\\]\nDefine~$N' = \\bigoplus_{l=1}^r N'_l$ where~$N'_l = N_{\\sigma(l)}$, then~$N' \\cong N$. We get that\n \\begin{equation}\nM_l \\subseteq N'_l(\\alpha) \\quad \\text{and} \\quad N'_l \\subseteq M_l(\\alpha), \\quad \\forall l \\in \\{1,\\dots,r\\}.\n \\tag{4}\n\\label{eq:4}\n\\end{equation}\nFrom now on, to simplify the notation,~$N$ refers to~$N'$.\\\\\nWe want to show that~$\\forall \\delta\\geq 0, \\Pru_{\\alpha+\\delta}(M) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(N)$ and~$\\Pru_{\\alpha+\\delta}(N)$ is an~$\\infty\\text{-refinement of } \\Pru_\\delta(M).$ We prove only the first statement; the second follows by symmetry. By Lemma~\\ref{lem_infinit_refinement} it is sufficient to show that \n\\[\n\\forall \\delta\\geq 0, \\, \\forall l \\in \\{1,\\dots,r\\}, \\quad \\pi_{l}(\\operatorname{Pru}_{\\alpha+\\delta}(M)) \\subseteq \\pi_{l}(\\operatorname{Pru}_{\\delta}(N)).\n\\]\nFix~$l_1 \\in \\{1,\\dots,r\\}$ and~$\\delta \\geq 0$. Let~$l \\in JN_{l_1}$ in the directed graph~$G(N,\\delta)$, then there exists a path\n\\[\nP:\\; N_{l_1}=N_{l'_0}\\to N_{l'_1}\\to\\cdots\\to N_{l'_{c-1}}\\to N_{l'_{c}}=N_{l},\n\\]\nfor some integer~$c$.\nBy the construction of~$G(N,\\delta)$ each arrow gives a containment shifted by~$2\\delta$, hence\n\\[\nN_{l'_0} \\subseteq N_{l'_1}(2\\delta), N_{l'_1} \\subseteq N_{l'_2}(2\\delta), \\cdots , N_{l'_{c-1}} \\subseteq N_{l'_{c}}(2\\delta).\n\\]\nBy~\\eqref{eq:4} we get that \n\\[\nM_{l'_i} \\subseteq N_{l'_i}(\\alpha) \\subseteq N_{l'_{i+1}}(\\alpha+2\\delta) \\subseteq M_{l'_{i+1}}(2\\alpha+2\\delta), \\quad \\forall i \\in \\{0,\\dots,c-1\\}.\n\\]\nSo we get that there exists a path \n\\[\nP':\\; M_{l_1}=M_{l'_0}\\to M_{l'_1}\\to\\cdots\\to M_{l'_{c-1}}\\to M_{l'_{c}}=M_{l},\n\\]\nin the directed graph~$G(M,\\alpha+\\delta)$. Therefore~$l \\in JM_{l_1}$ and \n\\[\nJN_{l_1} \\subseteq JM_{l_1}.\n\\]\nTherefore, by Theorem~\\ref{theorem_1} and~\\eqref{eq:4} we obtain\n\\[\\pi_{l_1}(\\operatorname{Pru}_{\\alpha+\\delta}(M)) = \\bigcap_{l \\in JM_{l_1}} M_l(-\\delta)(-\\alpha) \\subseteq \\bigcap_{l \\in JN_{l_1}} N_l(-\\delta)=\\pi_{l_1}(\\operatorname{Pru}_\\delta(N)).\\]\nSince~$l_1$ and~$\\delta$ were arbitrary,\n\\[\n\\forall \\delta \\geq 0, \\forall l \\in \\{1,\\dots,r\\}, \\quad \\pi_l(\\operatorname{Pru}_{\\alpha+\\delta}(M)) \\subseteq \\pi_l(\\operatorname{Pru}_\\delta(N)).\n\\]\nThus~$\\forall \\delta\\geq 0, \\Pru_{\\alpha+\\delta}(M) \\text{ is an } \\infty\\text{-refinement of } \\Pru_\\delta(N)$, and we conclude that \\[d_P(M,N) \\leq d_B(M,N).\\]\n\\end{proof}", "post_theorem_intro_text_len": 966, "post_theorem_intro_text": "This result improves the right-hand inequality in Bjerkevik’s conjecture by reducing the constant from~$2r$ to~$2r-1$, and strengthens it by replacing the interleaving distance with the bottleneck distance, which dominates the interleaving distance. Moreover, in Example~\\ref{ex_2} we show that the constant~$2r-1$ is optimal on the class of upset-decomposable persistence modules, for  both the interleaving distance and the bottleneck distance. The left-hand inequality follows from our analysis and is weaker than the conjectured inequality involving the interleaving distance.\n\nOur approach is based on an explicit computation of the pruning of upset-decomposable persistence modules, carried out in Theorem~\\ref{theorem_1}. To compute the pruning, we associate a directed graph to an upset-decomposable module, thereby translate the problem into computing the reachable indices via the directed paths in the graph, which yields a purely combinatorial procedure.", "sketch": "Our approach is based on an explicit computation of the pruning of upset-decomposable persistence modules, carried out in Theorem~\\ref{theorem_1}. To compute the pruning, we associate a directed graph to an upset-decomposable module, thereby translate the problem into computing the reachable indices via the directed paths in the graph, which yields a purely combinatorial procedure.", "expanded_sketch": "Our approach is based on an explicit computation of the pruning of upset-decomposable persistence modules, carried out in the following theorem. \\begin{theorem}\\label{theorem_1}\nLet~$\\alpha \\geq 0$ and let~$M = \\bigoplus_{l=1}^r M_l$ be an upset-decomposable module. Then\n\\[\n\\pi_{l_1}(\\operatorname{Pru}_\\alpha(M)) = \\bigcap_{l \\in JM_{l_1}} M_l(-\\alpha),\n\\]\nwhere~$JM_{l_1}$ is as defined above.\n\\end{theorem}\nTo compute the pruning, we associate a directed graph to an upset-decomposable module, thereby translate the problem into computing the reachable indices via the directed paths in the graph, which yields a purely combinatorial procedure.", "expanded_theorem": "If~$M$ and~$N$ are two upset-decomposable modules of pointwise dimension at most~$r$, then\n\\[\nd_P(M,N) \\leq d_B(M,N) \\leq (2r-1)d_P(M,N).\n\\],", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $M$ and $N$ be persistence modules indexed by $(\\mathbb{R}^n,\\leq)$ with the product order. Assume that both are upset-decomposable, meaning that each can be written as a direct sum of interval persistence modules supported on subsets $U\\subseteq \\mathbb{R}^n$ that are upward closed under the product order, and assume their pointwise dimensions are bounded by $r$ (i.e. at every parameter value, the corresponding vector space has dimension at most $r$). If $d_P$ denotes the pruning distance (defined via the $\\varepsilon$-prunings $\\operatorname{Pru}_\\varepsilon$) and $d_B$ denotes the bottleneck distance for these decompositions, which quantitative estimate holds?", "correct_choice": {"label": "A", "text": "For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has\n\\[\nd_P(M,N) \\le d_B(M,N) \\le (2r-1)\\, d_P(M,N).\n\\]"}, "choices": [{"label": "B", "text": "For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has\n\\[\nd_P(M,N) \\le d_B(M,N) \\le 2r\\, d_P(M,N).\n\\]"}, {"label": "C", "text": "For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has\n\\[\nd_P(M,N) \\le d_B(M,N).\n\\]"}, {"label": "D", "text": "For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has\n\\[\nd_B(M,N) \\le d_P(M,N) \\le (2r-1)\\, d_B(M,N).\n\\]"}, {"label": "E", "text": "For any two such upset-decomposable modules $M$ and $N$ of pointwise dimension at most $r$, one has\n\\[\nd_P(M,N) \\le d_B(M,N) \\le (2^r-1)\\, d_P(M,N).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "counting_estimate", "tampered_component": "sharp constant from reachable-index/counting argument", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "geometric_construction", "tampered_component": "dropped the upper Lipschitz bound $d_B\\le(2r-1)d_P$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "direction of comparison between pruning and bottleneck distances", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "linear dependence on pointwise dimension $r$ replaced by exponential dependence", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct inequality or constant. It specifies the setup and asks which estimate holds, so there is no direct answer leakage beyond the theorem context itself."}, "TAS": {"score": 0, "justification": "This is essentially a direct recall of a specific theorem statement: under the listed hypotheses, identify the exact quantitative bound. It does not substantially transform the underlying result into a new inference task."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the student must distinguish the sharp bound from a weaker true statement and several near-miss variants. However, the task is still primarily theorem recognition/recall rather than genuine derivation."}, "DQS": {"score": 2, "justification": "The distractors are strong: one is a close constant perturbation, one is a weaker true statement, one reverses the comparison, and one introduces an implausibly stronger growth rate. These reflect realistic mathematical confusions."}, "total_score": 5, "overall_assessment": "A mathematically well-constructed recall-style MCQ with strong distractors, but it is close to a theorem restatement and only moderately tests generative reasoning."}}
{"id": "2602.16048v3", "paper_link": "http://arxiv.org/abs/2602.16048v3", "theorems_cnt": 5, "theorem": {"env_name": "mainthm", "content": "\\label{thm:chordarc}\n    Let $3\\leq n\\leq 5$, \\(\\Sigma^n \\subset \\mathbb{R}^{n+1}\\) be a two-sided, embedded minimal disk with \\(\\sup_{\\Sigma}|A| < \\infty\\).\n    Then there exists a constant \\(c = c\\bigl(n,\\sup_{\\Sigma}|A|\\bigr) > 0\\) with the following property: for every \\(x\\in \\Sigma\\) and every \\(R>1\\) such that the intrinsic ball \\(\\mathcal{B}_{R}(x)\\subset \\Sigma\\), the connected component \\(\\Sigma_{x, R}\\) of the Euclidean ball \\(B_{R}(x)\\cap \\Sigma\\) containing \\(x\\) satisfies\n    \\[\n        \\Sigma_{x, R}\\subset \\mathcal{B}_{cR^n}(x).\n    \\]\n    In particular, intrinsic distances are quantitatively controlled by extrinsic distances on \\(\\Sigma_{x, R}\\), with constants depending only on the dimension and the curvature bound.", "start_pos": 17233, "end_pos": 18003, "label": "thm:chordarc"}, "ref_dict": {"thm:unbounded": "\\begin{mainthm}\\label{thm:unbounded}\nLet \\(\\Sigma^{n} \\subset \\mathbb{R}^{n+1}\\) be a complete, embedded minimal hypersurface and assume that \\(\\sup_{\\Sigma}|A|^2 < \\infty\\). Then \\(\\Sigma\\) is unbounded.\n\\end{mainthm}", "con:1": "\\begin{con}\\label{con:1}\nLet $n\\geq 2$. A complete minimal hypersurface $\\Sigma^{n}\\subset \\mathbb{R}^{n+1}$ must be unbounded.\n\\end{con}", "thm:nonproper": "\\begin{mainthm}\\label{thm:nonproper}\nFor every \\(n\\ge 3\\), there exists a complete, embedded, minimal hypersurface \\(\\Sigma_\\infty^{n}\\subset \\mathbb{R}^{n+1}\\) that is not proper.\n\\end{mainthm}", "thm:chordarc": "\\begin{mainthm}\\label{thm:chordarc}\n    Let $3\\leq n\\leq 5$, \\(\\Sigma^n \\subset \\mathbb{R}^{n+1}\\) be a two-sided, embedded minimal disk with \\(\\sup_{\\Sigma}|A| < \\infty\\).\n    Then there exists a constant \\(c = c\\bigl(n,\\sup_{\\Sigma}|A|\\bigr) > 0\\) with the following property: for every \\(x\\in \\Sigma\\) and every \\(R>1\\) such that the intrinsic ball \\(\\mathcal{B}_{R}(x)\\subset \\Sigma\\), the connected component \\(\\Sigma_{x, R}\\) of the Euclidean ball \\(B_{R}(x)\\cap \\Sigma\\) containing \\(x\\) satisfies\n    \\[\n        \\Sigma_{x, R}\\subset \\mathcal{B}_{cR^n}(x).\n    \\]\n    In particular, intrinsic distances are quantitatively controlled by extrinsic distances on \\(\\Sigma_{x, R}\\), with constants depending only on the dimension and the curvature bound.\n\\end{mainthm}"}, "pre_theorem_intro_text_len": 1171, "pre_theorem_intro_text": "In 1965, E. Calabi \\cite{Ca} proposed the following conjecture:\n\\begin{con}\\label{con:1}\nLet $n\\geq 2$. A complete minimal hypersurface $\\Sigma^{n}\\subset \\mathbb{R}^{n+1}$ must be unbounded.\n\\end{con}\nWhen $n=2$, Nadirashvili \\cite{Na1} constructed a complete minimal immersion of a disk into the unit ball, thus disproving the conjecture for minimal immersions in $\\mathbb{R}^3$. Conjecture \\ref{con:1} also appeared in \\cite[pg.~212]{chern1966geometry} and was later highlighted in Yau’s survey \\cite[pg.~360]{Ya2}, where, referring to Nadirashvili's construction, Yau asked what geometric properties such examples must satisfy. In particular, Yau asked whether any such bounded minimal surface can be embedded. This was addressed in \\cite{CM10}, where Colding--Minicozzi proved the Calabi--Yau conjectures by showing that any complete embedded minimal surface in $\\mathbb{R}^3$ with finite topology is proper and therefore unbounded. It is therefore natural to ask what remains true in higher dimensions, that is, for $n\\ge 3$.\n\nOur first main result proves Conjecture \\ref{con:1} under a uniform curvature bound, via a chord--arc estimate for embedded minimal disks:", "context": "In 1965, E. Calabi \\cite{Ca} proposed the following conjecture:\n\\begin{con}\\label{con:1}\nLet $n\\geq 2$. A complete minimal hypersurface $\\Sigma^{n}\\subset \\mathbb{R}^{n+1}$ must be unbounded.\n\\end{con}\nWhen $n=2$, Nadirashvili \\cite{Na1} constructed a complete minimal immersion of a disk into the unit ball, thus disproving the conjecture for minimal immersions in $\\mathbb{R}^3$. Conjecture \\ref{con:1} also appeared in \\cite[pg.~212]{chern1966geometry} and was later highlighted in Yau’s survey \\cite[pg.~360]{Ya2}, where, referring to Nadirashvili's construction, Yau asked what geometric properties such examples must satisfy. In particular, Yau asked whether any such bounded minimal surface can be embedded. This was addressed in \\cite{CM10}, where Colding--Minicozzi proved the Calabi--Yau conjectures by showing that any complete embedded minimal surface in $\\mathbb{R}^3$ with finite topology is proper and therefore unbounded. It is therefore natural to ask what remains true in higher dimensions, that is, for $n\\ge 3$.\n\nOur first main result proves Conjecture \\ref{con:1} under a uniform curvature bound, via a chord--arc estimate for embedded minimal disks:", "full_context": "In 1965, E. Calabi \\cite{Ca} proposed the following conjecture:\n\\begin{con}\\label{con:1}\nLet $n\\geq 2$. A complete minimal hypersurface $\\Sigma^{n}\\subset \\mathbb{R}^{n+1}$ must be unbounded.\n\\end{con}\nWhen $n=2$, Nadirashvili \\cite{Na1} constructed a complete minimal immersion of a disk into the unit ball, thus disproving the conjecture for minimal immersions in $\\mathbb{R}^3$. Conjecture \\ref{con:1} also appeared in \\cite[pg.~212]{chern1966geometry} and was later highlighted in Yau’s survey \\cite[pg.~360]{Ya2}, where, referring to Nadirashvili's construction, Yau asked what geometric properties such examples must satisfy. In particular, Yau asked whether any such bounded minimal surface can be embedded. This was addressed in \\cite{CM10}, where Colding--Minicozzi proved the Calabi--Yau conjectures by showing that any complete embedded minimal surface in $\\mathbb{R}^3$ with finite topology is proper and therefore unbounded. It is therefore natural to ask what remains true in higher dimensions, that is, for $n\\ge 3$.\n\nOur first main result proves Conjecture \\ref{con:1} under a uniform curvature bound, via a chord--arc estimate for embedded minimal disks:\n\nLet \\(\\Sigma \\subset \\bbR^n\\) be an embedded minimal hypersurface with bounded curvature, i.e. \\(\\sup_\\Sigma |A| < \\infty\\), and let \\(x \\in \\Sigma\\). For \\(R > 0\\), we let \\(B_R(x)\\) denote an extrinsic ball of radius \\(R\\) in \\(\\bbR^{n+1}\\), and we let \\(\\calB_R(x)\\) denote an intrinsic ball of radius \\(R\\) in \\(\\Sigma\\). We denote by \\(\\Sigma_{x, R}\\) the component of \\(\\Sigma \\cap B_R(x)\\) containing \\(x\\).\nFurthermore, since we assume $\\Sigma$ to have bounded curvature, we can find a graphical radius as follows:\n\\begin{lem}\\label{graphical radius and chord arc for graphical radius}\n Let \\(\\Sigma^{n} \\subset \\mathbb{R}^{n+1}\\) be an embedded minimal surface with bounded curvature \\(\\sup_{\\Sigma}|A| \\le C_A\\). Then there is a number \\(0 < R_{\\Sigma} < 1/C_A\\) such that for all \\(x \\in \\Sigma\\) and \\(R \\le R_{\\Sigma}\\), the following properties hold:\n \\begin{enumerate}\n \\item The intrinsic ball \\(\\mathcal{B}_{R}(x)\\) is graphical.\n \\item \\(\\calB_R(x)\\) can be written as a graph of a function \\(u: \\Omega \\subset T_x\\Sigma \\rightarrow \\mathbb{R}^{n + 1}\\) such that \\(|\\nabla u| < 1\\) and \\(|\\nabla^2 u| < c(C_A)\\).\n \\item There exists a \\(\\delta_c > 0\\) depending only on dimension and the curvature bound \\(C_A\\) such that\n \\begin{equation}\\label{eq: graphical weak chord arc}\n \\Sigma_{x, \\delta_cR} \\subset \\mathcal{B}_{R/2}(x).\n \\end{equation}\n \\end{enumerate}\n Importantly, the number \\(R_\\Sigma\\) depends only on the curvature bound; in particular, this means that \\(\\calB_{R_\\Sigma}(x)\\) may contain points in \\(\\partial \\Sigma\\).\n\nThe final ingredient needed is based on the recent work on Stable Bernstein Conjecture by \\cite{chodosh2023stable,CL, catino2024two,chodosh2024stable,mazet2024stable} and its extension to the $\\delta$-stable case by \\cite{hong2024delta,cheng2025complete}.\n\\begin{thm}\\label{thm:delta-stable-flat}\nFor $3 \\leq n \\leq 5$ and $\\delta\\in (0,1-\\delta_1(n))$, an $n$-dimensional complete two-sided $\\delta$--stable minimal hypersurface $\\Sigma^n \\subset \\mathbb{R}^{n+1}$ is a hyperplane, where $\\delta_1(3)=3 / 8$, $\\delta_1(4)=2 / 3$ and $\\delta_1(5)=21 / 22$.\n\\end{thm}\nBy a standard blow up argument, Theorem \\ref{thm:delta-stable-flat} implies the following curvature estimate,\n\\begin{cor}\\label{cor:curvature-estimate}\nLet $3 \\leq n \\leq 5$ and \\(\\delta \\in (0, 1-\\delta_1(n))\\) as in Theorem \\ref{thm:delta-stable-flat}. There exists \\(C(\\delta, n) < \\infty\\) such that if \\(\\Sigma^n \\subset \\mathbb{R}^{n+1}\\) is a two-sided, \\(\\delta\\)-stable minimal immersion, then\n\\[|A_M|d(p, \\partial M) \\le C(\\delta, n).\\]\n\\end{cor}\nWe now ready to give a proof of Theorem \\ref{thm:chordarc}.\n\\begin{proof}[Proof of Theorem \\ref{thm:chordarc}]\nFor clarity, we will rename \\(R\\) in the theorem statement to \\(R_0\\) in this proof. Because \\(\\Sigma\\) is an embedded disk, \\(\\Sigma_{x, R_0}\\) is also an embedded disk by the maximum principle and the monotonicity of topology (cf. \\cite{CM1}). Suppose that \\(\\Sigma_{x, R_0}\\) were not contained in an intrinsic ball \\(\\mathcal{B}_{2NR_{\\Sigma}}(x)\\) for any integer \\(N\\), where $R_\\Sigma>0$ is defined in Lemma \\ref{graphical radius and chord arc for graphical radius}. Then we can choose points \\(z_i \\in \\partial \\mathcal{B}_{2R_{\\Sigma}i} \\cap \\Sigma_{x, R_0}\\) for every \\(i\\), as well as a curve \\(\\sigma \\subset \\Sigma_{x, R_0}\\) which passes through each \\(z_i\\). This sequence is contained in the ball \\(B_{R_0}\\), and thus is Cauchy. Choose \\(z_{i_1}, z_{i_2} \\in \\{z_i\\}\\) with\n\\[|z_{i_1} - z_{i_2}| < \\veps < \\veps_0\\]\nwhere \\(\\veps_0\\) is produced from Lemma \\ref{lem: lemma 2.11 in CM}. For \\(r_{\\Sigma} < R_\\Sigma/2\\) the intrinsic balls \\(\\mathcal{B}_{r_{\\Sigma}}(z_{i_j})\\) are graphical and disjoint. Reducing \\(r_\\Sigma\\) until it is less than the radius \\(r_0\\) from Lemma \\ref{lem: lemma 2.11 in CM}, the balls \\(\\mathcal{B}_{r_{\\Sigma}}(z_{i_1})\\) and \\(\\mathcal{B}_{r_{\\Sigma}}(z_{i_2})\\) are both graphs over \\(T_{z_{i_1}}\\Sigma\\), and it is possible to define \\(u: \\mathcal{B}_{r_{\\Sigma}}(z_{i_1}) \\rightarrow \\mathbb{R}\\) such that \\(x + u(x)\\mathbf{n}_{\\Sigma}(x) \\in \\calB_{r_\\Sigma}(z_{i_2})\\) for \\(x \\in \\mathcal{B}_{r_{\\Sigma}}(z_{i_1})\\) with \\(|\\nabla u| + |u||A| < 1\\).\n\nBy our original construction, the surfaces \\(\\Sigma_i\\) lie above the plane \\(\\{x_n = 0\\}\\), because before the rotation of \\(\\mathbb{R}^n\\) they were in the interior of \\(B_{R_0}(0)\\). By the maximum principle, the graph of \\(u\\) must lie entirely in the plane \\(\\{x_n = 0\\}\\) or go below the plane \\(\\{x_n = 0\\}\\), since the graph is a minimal surface and \\(u(0) = 0\\). But then it follows that some \\(\\Sigma_i\\) leaves the ball \\(B_{R_0}\\), a contradiction. This completes the proof.\n\\end{proof}\nFinally, while we cannot prove that a complete embedded minimal hypersurface $\\Sigma^n\\subset \\R^{n+1}$ with bounded curvature is proper, we observe that $\\bar{\\Sigma}$ is a minimal lamination of $\\R^{n+1}$. Recall that a minimal lamination of a smooth manifold $M^{n+1}$ is defined as follows (cf. Appendix B of \\cite{CM6})\n\\begin{defn}[Minimal lamination]\\label{def:minimal-lamination}\nLet $M^{n+1}$ be an $(n+1)$-dimensional smooth manifold.\nA \\emph{codimension-one lamination} of $M$ is a collection $\\mathcal L$ of smooth, disjoint $n$-dimensional submanifolds of $M$ (called \\emph{leaves})\nsuch that:\n\\begin{enumerate}\n \\item The union $\\bigcup_{\\Lambda\\in\\mathcal L}\\Lambda$ is a closed subset of $M$.\n \\item For every $x\\in M$ there exists an open neighborhood $U\\ni x$ and a coordinate chart\n $\\Phi:U\\to \\Phi(U)\\subset \\mathbb R^{n+1}$ with the property that, in these coordinates,\n the leaves of $\\mathcal L$ pass through $\\Phi(U)$ in slices of the form $(\\mathbb R^n\\times\\{0\\})\\cap \\Phi(U)$.\n\\end{enumerate}\nA \\emph{minimal lamination} is a lamination whose leaves are minimal. A \\emph{limit leaf} is a leaf $\\Lambda$ of $\\calL$ that is contained in the closure of $\\calL\\setminus \\Lam$ (cf. Definition 4.3.2 in \\cite{zhou2024geometry}).\n\\end{defn}\nFurthermore, by the recent resolution of the Stable Bernstein Conjecture in $\\R^{n+1}$ for $3\\leq n \\leq 5$ we can now show that the limit leaves of $\\bar{\\Sigma}$ are flat, which yields the following trichotomy:\n\\begin{lem}\nLet $3\\leq n\\leq 5$. Suppose $\\Sigma^n\\subset \\R^{n+1}$ is a complete connected embedded minimal surface in $\\mathbb{R}^{n+1}$ with $|A_\\Sigma|\\leq C_A <\\infty$. Then \\(\\overline{\\Sigma}\\) is a minimal lamination, and one of the following holds:\n\\begin{enumerate}\n \\item $\\Sigma$ is properly embedded in $\\mathbb{R}^{n+1}$,\n \\item $\\Sigma$ is properly embedded in an open halfspace of $\\mathbb{R}^{n+1}$ with limit set the boundary plane of this halfspace, or\n \\item $\\Sigma$ is properly embedded in an open slab of $\\mathbb{R}^{n+1}$ with limit set consisting of the boundary planes.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nThe proof follows from the same argument as in the proof of Lemma 1.1 in \\cite{meeks2005uniqueness} where the authors, using the bound on the curvature, first show that $\\bar{\\Sigma}$ is a minimal lamination of $\\R^{n+1}$. Next, by considering the universal cover $\\hat{\\Lam}$ of any limit leaf $\\Lam$ one can show that $\\hat{\\Lam}$ is stable since any compact domain $\\hat{D}$ of $\\hat{\\Lam}$ is a limit of disjoint minimal domains, which in turn implies that $\\hat{D}$ is stable by a contradiction argument. Therefore, $\\Lam$ is stable, which by the works of \\cite{chodosh2023stable,CL, catino2024two,chodosh2024stable,mazet2024stable} implies that is $\\Lam$ is a plane, thus completing the proof of the Lemma.\n\\end{proof}", "post_theorem_intro_text_len": 5049, "post_theorem_intro_text": "The second part of the paper addresses the stronger version of the Calabi--Yau conjecture in higher dimensions, namely properness. While properness holds for complete embedded minimal surfaces in \\(\\mathbb{R}^3\\) by \\cite{CM10}, we show that the natural higher-dimensional analogue fails in general:\n\\begin{mainthm}\\label{thm:nonproper}\nFor every \\(n\\ge 3\\), there exists a complete, embedded, minimal hypersurface \\(\\Sigma_\\infty^{n}\\subset \\mathbb{R}^{n+1}\\) that is not proper.\n\\end{mainthm}\n\nThe restrictions on the dimensions in Theorem \\ref{thm:chordarc} arise from curvature estimates coming from the recent resolution of the Stable Bernstein conjecture in dimensions $3\\leq n \\leq 5$. Since Theorem \\ref{thm:chordarc} implies unboundedness for complete embedded minimal hypersurfaces with bounded curvature in dimensions $3\\leq n\\leq 5$, we also include, for completeness, a proof of unboundedness in all dimensions $n\\geq 3$. While this fact seems well known to experts (cf.\\ \\cite{Carlotto_2016} and the proof of Theorem 2 in \\cite{rosenberg2001intersection}), we have not found a detailed argument in the literature that applies uniformly to all dimensions.\n\\begin{mainthm}\\label{thm:unbounded}\nLet \\(\\Sigma^{n} \\subset \\mathbb{R}^{n+1}\\) be a complete, embedded minimal hypersurface and assume that \\(\\sup_{\\Sigma}|A|^2 < \\infty\\). Then \\(\\Sigma\\) is unbounded.\n\\end{mainthm}\n\nExtending the above result to show properness for complete embedded minimal hypersurfaces with bounded curvature seems nontrivial, in part due to the failure of the half-space theorem. We expect that with suitable topological assumptions one could get around these issues. However, we can show that for a complete, improperly embedded minimal surface \\(\\Sigma\\), the closure \\(\\bar \\Sigma\\) is a minimal lamination, as defined in \\cite{CM6} Appendix B with planar limit leaves in ambient dimensions $3\\leq n \\leq 5.$\n\nAt a conceptual level, the proofs of Theorem \\ref{thm:chordarc} and Theorem \\ref{thm:unbounded} follow the strategy of \\cite{CM10} in the bounded-curvature regime: when two embedded minimal sheets are very close but disjoint, their separation function satisfies a uniformly elliptic equation whose coefficients are controlled by \\(|A|\\); a Harnack inequality then yields quantitative control on how fast the sheets can separate.\n\nTheorem \\ref{thm:nonproper} is inspired by the work of Fakhi and Pacard \\cite{Fakhi-Pacard} and Kapouleas \\cite{MR1601434}. Fahki and Pacard construct complete, immersed minimal surfaces in \\(\\mathbb{R}^{n+1}\\), \\(n \\ge 3\\), with finite total curvature and arbitrarily many ends, and Kapouleas produces complete, embedded minimal surfaces in \\(\\mathbb{R}^3\\) with finite total curvature and arbitrarily many ends. These results have also been generalized by Coutant in \\cite{coutant2012deformation} to higher dimensions where, amongst other things, the author constructs examples of complete, embedded minimal hypersurfaces in \\(\\mathbb{R}^{n+1}\\) with finite total curvature and arbitrarily many ends. In our work, we modify Fahki and Pacard's construction to produce embedded examples:\n\\begin{mainthm}\\label{thm:k-ends}\nFor \\(n\\ge 3\\) and \\(k \\ge 1\\), there exists a complete, embedded, minimal hypersurface \\(\\Sigma_k^{n}\\subset \\mathbb{R}^{n+1}\\) that has finite total curvature and \\(k\\) ends.\n\\end{mainthm}\nNotably, both Kapouleas \\cite{MR1601434} and Fahki-Pacard \\cite{Fakhi-Pacard} produce their examples by gluing catenoidal necks. We proceed similarly by iteratively gluing catenoids to form a sequence of surfaces \\(\\Sigma_k^n\\) with \\(k\\)-ends. To prove Theorem \\ref{thm:nonproper}, we will construct such a sequence \\(\\Sigma_k^n\\) that converges locally smoothly to a hypersurface \\(\\Sigma_\\infty^n\\) that is complete, embedded, and improper. To our knowledge, this gives the first example of an improperly embedded minimal hypersurface in $\\mathbb{R}^{n+1}$. The constructed hypersurface has unbounded curvature, infinite genus, and infinite ends. In particular, it shows that the properness conjecture fails for embedded minimal hypersurfaces of infinite topology in higher dimensions. By contrast, in the surface case $n=2$, the results of \\cite{CM10, meeks2021embedded} indicate that properness for embedded minimal surfaces should hold under the additional assumption of finite genus and therefore it would be interesting to find an analogue of Theorem \\ref{thm:nonproper} when $n=2.$\n\nFinally, we direct the reader to the survey \\cite{MR4415896} by Breiner, Kapouleas, and Kleene for an overview on the state of gluing constructions for constant mean curvature surfaces.\n\n\\subsection{Acknowledgments}\nThe authors would like to thank their advisors Tobias Colding and William Minicozzi for many helpful discussions and encouragement. We are also grateful for stimulating discussions with Otis Chodosh, David Jersion and Peter McGrath. The first author acknowledges support from the Simons Dissertation Fellowship and the second author acknowledges support from the National Science Foundation.", "sketch": "At a conceptual level, the proofs of Theorem \\ref{thm:chordarc} (and Theorem \\ref{thm:unbounded}) “follow the strategy of \\cite{CM10} in the bounded-curvature regime”: when two embedded minimal sheets are “very close but disjoint,” their “separation function satisfies a uniformly elliptic equation whose coefficients are controlled by \\(|A|\\)”; then “a Harnack inequality” gives “quantitative control on how fast the sheets can separate.”", "expanded_sketch": "At a conceptual level, the proof of the main theorem (and the following theorem) “follow the strategy of \\cite{CM10} in the bounded-curvature regime”: when two embedded minimal sheets are “very close but disjoint,” their “separation function satisfies a uniformly elliptic equation whose coefficients are controlled by \\(|A|\\)”; then “a Harnack inequality” gives “quantitative control on how fast the sheets can separate.” We first prove the following theorem.\n\n\\begin{mainthm}\\label{thm:unbounded}\nLet \\(\\Sigma^{n} \\subset \\mathbb{R}^{n+1}\\) be a complete, embedded minimal hypersurface and assume that \\(\\sup_{\\Sigma}|A|^2 < \\infty\\). Then \\(\\Sigma\\) is unbounded.\n\\end{mainthm}", "expanded_theorem": "\\label{thm:chordarc}\n    Let $3\\leq n\\leq 5$, \\(\\Sigma^n \\subset \\mathbb{R}^{n+1}\\) be a two-sided, embedded minimal disk with \\(\\sup_{\\Sigma}|A| < \\infty\\).\n    Then there exists a constant \\(c = c\\bigl(n,\\sup_{\\Sigma}|A|\\bigr) > 0\\) with the following property: for every \\(x\\in \\Sigma\\) and every \\(R>1\\) such that the intrinsic ball \\(\\mathcal{B}_{R}(x)\\subset \\Sigma\\), the connected component \\(\\Sigma_{x, R}\\) of the Euclidean ball \\(B_{R}(x)\\cap \\Sigma\\) containing \\(x\\) satisfies\n    \\[\n        \\Sigma_{x, R}\\subset \\mathcal{B}_{cR^n}(x).\n    \\]\n    In particular, intrinsic distances are quantitatively controlled by extrinsic distances on \\(\\Sigma_{x, R}\\), with constants depending only on the dimension and the curvature bound.", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let $3\\le n\\le 5$, and let $\\Sigma^n\\subset \\mathbb{R}^{n+1}$ be a two-sided, embedded minimal disk with bounded curvature, i.e. $\\sup_{\\Sigma}|A|<\\infty$. For $x\\in \\Sigma$, let $\\mathcal B_R(x)$ denote the intrinsic ball of radius $R$ in $\\Sigma$ (with respect to the induced metric), let $B_R(x)$ denote the Euclidean ball of radius $R$ in $\\mathbb{R}^{n+1}$, and let $\\Sigma_{x,R}$ be the connected component of $B_R(x)\\cap \\Sigma$ that contains $x$. Which of the following quantitative estimates holds for every $x\\in\\Sigma$ and every $R>1$ such that $\\mathcal B_R(x)\\subset \\Sigma$?", "correct_choice": {"label": "A", "text": "There exists a constant $c=c\\bigl(n,\\sup_{\\Sigma}|A|\\bigr)>0$ such that for every $x\\in\\Sigma$ and every $R>1$ with $\\mathcal B_R(x)\\subset \\Sigma$, one has\n\\[\n\\Sigma_{x,R}\\subset \\mathcal B_{cR^n}(x).\n\\]\nEquivalently, on each such component $\\Sigma_{x,R}$, intrinsic distance is quantitatively controlled by extrinsic distance, with constants depending only on $n$ and the curvature bound."}, "choices": [{"label": "B", "text": "There exists a constant $c=c\\bigl(n,\\sup_{\\Sigma}|A|\\bigr)>0$ such that for every $x\\in\\Sigma$ and every $R>1$ with $\\mathcal B_R(x)\\subset \\Sigma$, one has\n\\[\n\\Sigma_{x,R}\\subset \\mathcal B_{cR}(x).\n\\]\nEquivalently, on each such component $\\Sigma_{x,R}$, intrinsic distance is quantitatively controlled linearly by extrinsic distance, with constants depending only on $n$ and the curvature bound."}, {"label": "C", "text": "There exists a constant $c=c\\bigl(n,\\sup_{\\Sigma}|A|\\bigr)>0$ such that for every $x\\in\\Sigma$ and every $R>1$ with $\\mathcal B_R(x)\\subset \\Sigma$, one has\n\\[\n\\Sigma_{x,R}\\subset \\mathcal B_{cR^{n+1}}(x).\n\\]"}, {"label": "D", "text": "For every $x\\in\\Sigma$ and every $R>1$ with $\\mathcal B_R(x)\\subset \\Sigma$, there exists a constant $c=c\\bigl(x,R,n,\\sup_{\\Sigma}|A|\\bigr)>0$ such that\n\\[\n\\Sigma_{x,R}\\subset \\mathcal B_{cR^n}(x).\n\\]\nEquivalently, intrinsic distance is quantitatively controlled by extrinsic distance on each $\\Sigma_{x,R}$, but the controlling constant may depend on the center and scale."}, {"label": "E", "text": "There exists a constant $c=c\\bigl(n,\\sup_{\\Sigma}|A|\\bigr)>0$ such that for every $x\\in\\Sigma$ and every $R>1$ with $\\mathcal B_R(x)\\subset \\Sigma$, the entire intersection with the Euclidean ball satisfies\n\\[\nB_R(x)\\cap \\Sigma\\subset \\mathcal B_{cR^n}(x).\n\\]\nEquivalently, every point of $B_R(x)\\cap \\Sigma$ has intrinsic distance at most $cR^n$ from $x$, with constants depending only on $n$ and the curvature bound."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "polynomial_growth_exponent_n", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "sharp_exponent_n_replaced_by_larger_power", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "uniform_dependence_of_constant", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "connected_component_restriction", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal choice A, nor does it contain a giveaway phrase uniquely matching the correct option. It asks for the valid uniform existence statement, so the answer must be inferred from the options."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the correct option states the target conclusion almost verbatim, with the task being to recognize the precise formulation rather than derive a new consequence."}, "GPS": {"score": 1, "justification": "Some reasoning is required to track quantifiers, uniformity of the constant, the radius threshold, and the growth exponent. However, the main burden is careful recognition of the exact theorem statement rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they vary the radius range, uniformity in the constant, direction of containment, and the exponent. These reflect realistic failure modes in parsing geometric estimates."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-identification MCQ with strong distractors and no obvious answer leakage, but it scores poorly on tautology avoidance because it mainly tests recognition of the exact statement rather than genuine derivational reasoning."}}
{"id": "2602.16329v1", "paper_link": "http://arxiv.org/abs/2602.16329v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{MainThm}\n\t\t$(T_t^{(2)})_{t\\geq 0}$ is hypercontractive. Moreover, the optimal time $t_p$ satisfies\n\t\t\\[\n\t\t\\widetilde{c}(\\beta) (p - 1) \\leq e^{2\\tau t_p} \\leq \\widetilde{C}(\\beta) (p - 1), \\quad \\forall\\, 2 < p < \\infty,\n\t\t\\]\n\t\twhere $\\widetilde{c}(\\beta)$ and $\\widetilde{C}(\\beta)$ are positive constants depending only on $\\beta$.", "start_pos": 13047, "end_pos": 13381, "label": "MainThm"}, "ref_dict": {"MainThm": "\\begin{thm}\\label{MainThm}\n\t\t$(T_t^{(2)})_{t\\geq 0}$ is hypercontractive. Moreover, the optimal time $t_p$ satisfies\n\t\t\\[\n\t\t\\wt{c}(\\beta) (p - 1) \\leq e^{2\\tau t_p} \\leq \\wt{C}(\\beta) (p - 1), \\quad \\forall\\, 2 < p < \\infty,\n\t\t\\]\n\t\twhere $\\wt{c}(\\beta)$ and $\\wt{C}(\\beta)$ are positive constants depending only on $\\beta$.\n\t\\end{thm}", "Main-Lem": "\\begin{thm}\\label{Main-Lem}\n\t\tLet $f_{k, n, m}$ be defined as before. For all $k \\geq 1$, $-k \\leq m \\leq k$, there exists a constant $C(\\beta)$ such that\n\t\t\\begin{equation}\n\t\t\t\\left( \\sum_{n=0}^\\8  e^{-n\\beta}| f_{k, n, m} |^p  \\right)^{1/p}  \\leq e^{m\\beta(1/2p-1/4)} \\mk{ C(\\beta)p }^{ k/2 } \\left( \\sum_{n=0}^\\8 e^{-n\\beta} | f_{k, n, m} |^2  \\right)^{1/2}.\n\t\t\\end{equation}\n\t\\end{thm}", "gap": "\\begin{equation}\\label{gap}\n\t\t\\tau=\\min\\{\\tau_1, \\tau_2\\}>0. \n\t\\end{equation}", "Prop2.1": "\\begin{proposition}\\label{Prop2.1}\n\t\t$T_t^{(2)}$ is hypercontractive if for any $2<p<\\8$, there exists a constant $c_p>0$ such that for any $x \\in L_k$, $k\\in \\nat$,\n\t\t\\begin{align}\\label{KL}\n\t\t\t\\left\\| \\rho^{1/(2p) - 1/4} x \\rho^{1/(2p) - 1/4}\\right\\|_{p} \\leq c_p^k \\left\\|  x  \\right\\|_{2}.   \n\t\t\\end{align}\n\t\tMoreover, the optimal time $t_p$ satisfies\n\t\t\\begin{equation}\\label{optimal-1}\n\t\t\te^{2\\tau t_p} \\leq 2c_p^2, \\quad \\forall \\, 2<p<\\8.\n\t\t\\end{equation}\n\t\\end{proposition}", "MainThm2": "\\begin{corollary}\\label{MainThm2}\n\t\tFor $2<p<\\8$, the optimal time $t'_p$ satisfies\n\t\t\\[\n\t\t\\wt{c}_1(\\beta) (p - 1) \\leq e^{2\\tau t'_p} \\leq \\wt{C}_1(\\beta) (p - 1)^2,\n\t\t\\]\n\t\twhere $\\wt{c}_1(\\beta)$ and $\\wt{C}_1(\\beta)$ are positive constants depending only on $\\beta$.\n\t\\end{corollary}"}, "pre_theorem_intro_text_len": 7176, "pre_theorem_intro_text": "\\label{Introduction}\n\tIn the celebrated papers \\cite{EN1966,EN1973}, Nelson showed that the classical Ornstein-Uhlenbeck semigroup is hypercontractive and utilized such property to establish the existence and uniqueness of the ground state of a special semigroup which arises from the constructive quantum field theory. Since then, hypercontractivity becomes a useful and powerful tool in quantum field theory, quantum statistical mechanics and quantum information. \n\n\tGross discovered the equivalence of log-Sobolev inequalities and hypercontractivity, and also gave a new proof of the hypercontractivity for the classical Ornstein-Uhlenbeck semigroup in the remarkable work \\cite{LG1975}. This surprising observation led to the application of hypercontractivity in numerous fields, such as Boolean analysis, concentration inequalities, geometric inequalities, statistical physics, among others. We refer to \\cite{LG1993,BGL2013} for more information.\n\n\tQuantum Markov semigroups, which are a generalization of classical Markov semigroups, are a fundamental tool to describe open quantum systems. In \\cite{CFL2000}, Cipriani \\emph{et al.} rigorously proved that a specific unbounded Lindblad-type operator generates a quantum Markov semigroup by virtue of the remarkable quantum semigroup theory established in \\cite{C1992,C1997}. Such unbounded Lindblad-type operator has been extensively studied in quantum optics models of masers and lasers \\cite{D1976,FRS1994}. The quantum Markov semigroup appearing in \\cite{CFL2000}, which is now called quantum Ornstein-Uhlenbeck semigroup, has also been shown to be hypercontractive in the seminal work \\cite{CS2008}.\n\n\tThe need to construct quantum Markov semigroups on von Neumann algebras, which are symmetric with respect to a nontracial state, is clear for various applications to open quantum systems, quantum statistical mechanics and quantum information (see \\cite{KP} for more information). In the remarkable paper \\cite{KP}, Ko \\emph{et al.} constructed a family of quantum Ornstein-Uhlenbeck semigroups, which can be seen as a generalization of the quantum Ornstein-Uhlenbeck semigroup appearing in \\cite{CFL2000, D1976,FRS1994}.\n\n\tIn this paper, we aim to study hypercontractivities of the quantum Ornstein-Uhlenbeck semigroups established in \\cite{KP}. At first, we introduce such quantum Ornstein-Uhlenbeck semigroups. To this end, we present Weyl operators and CCR. \n\n\tLet $H=\\ell_2(\\mathbb{N})$, where $\\mathbb{N}=\\{0,1,2, \\cdots\\}$ is the set of all natural numbers. Let $\\{e_n\\}_{n=0}^\\infty$ be the canonical orthonormal basis of $H$. Then we define the creation and annihilation operators $a^*$ and $a$ as follows\n\t\\[\n\ta^* e_n = \\sqrt{n+1} \\, e_{n+1}, \\quad a e_n = \\sqrt{n} \\, e_{n-1}, \\quad (e_{-1} := 0).\n\t\\]\n\tIt is clear that they satisfy the canonical commutation relation (CCR)\n\t\\[\n\t[a, a^*] = 1,\n\t\\]\n\twhere $1$ denotes the identity operator by a slight abuse of notation. The number operator $N := a^* a$ acts as\n\t\\[\n\tN e_n = n \\cdot e_n.\n\t\\]\n\tThe position and momentum operators are defined as\n\t\\[\n\tQ = \\frac{1}{\\sqrt{2}}(a + a^*), \\quad P = \\frac{1}{\\sqrt{2}i}(a - a^*),\n\t\\]\n\tand so\n\t\\[\n\t[Q, P] = i \\cdot 1.\n\t\\]\n\tFor any $z \\in \\mathbb{C}$, define the Weyl operators\n\t\\[\n\tW(z) = e^{\\frac {i}{\\sqrt{2}}(z a^* + \\bar{z} a)} = e^{i ( \\Re z \\cdot Q + \\Im z \\cdot P ) }.\n\t\\]\n\tThese operators satisfy the Weyl relation\n\t\\[\n\tW(z)^*=W(-z), \\quad \\text{and} \\quad \\forall z, w\\in\\mathbb{C}, \\ W(z) W(w) = e^{-\\frac{i}{2} \\Im (\\bar{z} w)} W(z + w).\n\t\\]\n\tLet $\\mathcal{A}$ be the $C^*$-algebra generated by all Weyl operators. It is well-known that $\\mathcal{A}\\subsetneqq B(H)$ and $\\mathcal{A}{''}=B(H)$. Let $\\omega$ be the normal faithful state on $B(H)$, which is defined by\n\t$$ \\omega(W(z))=e^{-\\frac{|z|^2}{4} (1+e^{-\\beta})(1-e^{-\\beta})^{-1}}, \\quad \\forall \\, z\\in \\mathbb{C}  $$\n\twhere $\\beta>0$ is a fixed inverse temperature. Note that $e^{-\\beta N}$ is a trace class operator in $B(H)$. Let \n\t$$ \\rho=\\dfrac{e^{-\\beta N}}{\\text{Tr}(e^{-\\beta N})}=(1-e^{-\\beta})e^{-\\beta N}. $$\n\tIndeed, $\\omega$ is a Gibbs state, and for any $x\\in B(H)$,\n\t$$  \\omega(x)=\\text{Tr}(\\rho x).  $$\n\tWe refer the interested reader to \\cite{BR} for more details. Now we introduce the quantum Ornstein-Uhlenbeck semigroups constructed in \\cite{KP}. Let the parameters $\\alpha_1, \\alpha_2, \\alpha_3 \\in \\mathbb{R}$ satisfy the following relations\n\t\\begin{align*}\n\t\t& \\frac{1}{2}\\left(1+\\alpha_2^2\\right) \\sinh (\\beta / 2)=-\\alpha_1 \\cosh (\\beta / 2), \\\\\n\t\t& \\frac{1}{2}\\left(\\alpha_1^2+\\alpha_3^2\\right) \\sinh (\\beta / 2)=\\alpha_2 \\alpha_3 \\cosh (\\beta / 2).\n\t\\end{align*}\n\tLet $G$ be the elliptic operator on $B(H)$ given by\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\tG(A)=\\frac{\\gamma}{2}\\left(1+\\alpha_2^2\\right)[P,[P, A]]&+\\frac{\\gamma}{2}\\left(\\alpha_1^2+\\alpha_3^2\\right)[Q,[Q, A]]-i \\gamma \\alpha_1(Q[P, A]+[P, A] Q)\\\\\n\t\t\t&-i \\gamma \\alpha_2 \\alpha_3(P[Q, A]+[Q, A] P), \\quad \\quad \\forall \\, A\\in B(H),\n\t\t\\end{aligned}\n\t\\end{equation*}\n\twhere $\\gamma=\\left(1+\\alpha_2^2\\right)^{-1}$ is the normalized constant. Since $P$ and $Q$ are unbounded operators affiliated to $\\mathcal{M}$, the above is a formal expression.\n\n\tFor $0<p\\leq \\infty$, let $S_p$ be the Schatten $p$-class of $B(H)$ with the usual Schatten $p$-norm denoted by $\\|\\cdot\\|_p$. We use Kosaki's definition \\cite{Ko1984} for noncommutative $L_p$ spaces. \n\tLet $L_p(\\rho)$ be the closure of all elements in $\\rho^{1/2}B(H)\\rho^{1/2}$ with respect to the following norm\n\t$$  \\| \\rho^{1/2} x \\rho^{1/2} \\|_{L_p(\\rho)}:= \\| \\rho^{1/2p} x \\rho^{1/2p}\\|_p, \\quad  \\forall \\, x\\in B(H). $$\n\tDefine $\\mathcal{G}^{(2)}$ on $S_2$ by\n\t$$  \\mathcal{G}^{(2)}(\\rho^{1/4}x\\rho^{1/4})=\\rho^{1/4}G(x)\\rho^{1/4}, \\quad \\forall \\, x\\in B(H).  $$\n\tAccording to \\cite{KP}, $P_t= e^{-t\\mathcal{G}^{(2)}}$ ($\\forall \\, t\\geq 0$) is a symmetric semigroup on $S_2$. For any $x\\in B(H)$, define\n\t$$  \\rho^{1/4} T_t^{(\\infty)}(x) \\rho^{1/4}= P_t(\\rho^{1/4}x\\rho^{1/4}).  $$\n\tThen $T_t^{(\\infty)}$ is an ergodic quantum Markov semigroup on $S_\\infty$, and $\\omega$ is the unique invariant state associated with $T_t^{(\\infty)}$. See \\cite{KP} for more details. \n\n\tFor $1\\leq p<\\infty$, denote by $T_t^{(p)}: L_p(\\rho)\\rightarrow L_p(\\rho)$ the usual restriction of $T_t^{(\\infty)}$ to $L_p(\\rho)$, by using the same way as in \\cite{GL1995}. More explicitly, for $p=2$, if $x\\in B(H)$, then\n\t$$  T_t^{(2)}(\\rho^{1/2}x\\rho^{1/2})=\\rho^{1/4}P_t(\\rho^{1/4}x\\rho^{1/4})\\rho^{1/4}. $$\n\tDenote by $\\tau$ the spectral gap of $T_t^{(2)}$, which is defined in \\eqref{gap}.\n\n\tWe present the definition of hypercontractivity. Let $(T_t)_{t\\geq 0}$ be an ergodic quantum Markov semigroup with the invariant state induced by $\\rho$. Then $(T_t)_{t\\geq 0}$ is said to be hypercontractive if for any $2<p<\\infty$, there exists $t_p>0$ such that for any $t \\geq t_p$, and for any $ x\\in B(H) $ with $\\omega(x)=0$,\n\t\\begin{equation}\\label{hypercontractivity}\n\t\t\\| T_t(\\rho^{1/2}x\\rho^{1/2}) \\|_{L_p(\\rho)} \\leq \\|\\rho^{1/2}x\\rho^{1/2}\\|_{L_2(\\rho)}.\n\t\\end{equation}\n\tMoreover, the least $t_p$ is called the optimal time of $T_t$ with respect to the parameter $p$.\n\n\tOur result is the following theorem.", "context": "\\label{Introduction}\n In the celebrated papers \\cite{EN1966,EN1973}, Nelson showed that the classical Ornstein-Uhlenbeck semigroup is hypercontractive and utilized such property to establish the existence and uniqueness of the ground state of a special semigroup which arises from the constructive quantum field theory. Since then, hypercontractivity becomes a useful and powerful tool in quantum field theory, quantum statistical mechanics and quantum information.\n\nLet $H=\\ell_2(\\mathbb{N})$, where $\\mathbb{N}=\\{0,1,2, \\cdots\\}$ is the set of all natural numbers. Let $\\{e_n\\}_{n=0}^\\infty$ be the canonical orthonormal basis of $H$. Then we define the creation and annihilation operators $a^*$ and $a$ as follows\n \\[\n a^* e_n = \\sqrt{n+1} \\, e_{n+1}, \\quad a e_n = \\sqrt{n} \\, e_{n-1}, \\quad (e_{-1} := 0).\n \\]\n It is clear that they satisfy the canonical commutation relation (CCR)\n \\[\n [a, a^*] = 1,\n \\]\n where $1$ denotes the identity operator by a slight abuse of notation. The number operator $N := a^* a$ acts as\n \\[\n N e_n = n \\cdot e_n.\n \\]\n The position and momentum operators are defined as\n \\[\n Q = \\frac{1}{\\sqrt{2}}(a + a^*), \\quad P = \\frac{1}{\\sqrt{2}i}(a - a^*),\n \\]\n and so\n \\[\n [Q, P] = i \\cdot 1.\n \\]\n For any $z \\in \\mathbb{C}$, define the Weyl operators\n \\[\n W(z) = e^{\\frac {i}{\\sqrt{2}}(z a^* + \\bar{z} a)} = e^{i ( \\Re z \\cdot Q + \\Im z \\cdot P ) }.\n \\]\n These operators satisfy the Weyl relation\n \\[\n W(z)^*=W(-z), \\quad \\text{and} \\quad \\forall z, w\\in\\mathbb{C}, \\ W(z) W(w) = e^{-\\frac{i}{2} \\Im (\\bar{z} w)} W(z + w).\n \\]\n Let $\\mathcal{A}$ be the $C^*$-algebra generated by all Weyl operators. It is well-known that $\\mathcal{A}\\subsetneqq B(H)$ and $\\mathcal{A}{''}=B(H)$. Let $\\omega$ be the normal faithful state on $B(H)$, which is defined by\n $$ \\omega(W(z))=e^{-\\frac{|z|^2}{4} (1+e^{-\\beta})(1-e^{-\\beta})^{-1}}, \\quad \\forall \\, z\\in \\mathbb{C} $$\n where $\\beta>0$ is a fixed inverse temperature. Note that $e^{-\\beta N}$ is a trace class operator in $B(H)$. Let \n $$ \\rho=\\dfrac{e^{-\\beta N}}{\\text{Tr}(e^{-\\beta N})}=(1-e^{-\\beta})e^{-\\beta N}. $$\n Indeed, $\\omega$ is a Gibbs state, and for any $x\\in B(H)$,\n $$ \\omega(x)=\\text{Tr}(\\rho x). $$\n We refer the interested reader to \\cite{BR} for more details. Now we introduce the quantum Ornstein-Uhlenbeck semigroups constructed in \\cite{KP}. Let the parameters $\\alpha_1, \\alpha_2, \\alpha_3 \\in \\mathbb{R}$ satisfy the following relations\n \\begin{align*}\n & \\frac{1}{2}\\left(1+\\alpha_2^2\\right) \\sinh (\\beta / 2)=-\\alpha_1 \\cosh (\\beta / 2), \\\\\n & \\frac{1}{2}\\left(\\alpha_1^2+\\alpha_3^2\\right) \\sinh (\\beta / 2)=\\alpha_2 \\alpha_3 \\cosh (\\beta / 2).\n \\end{align*}\n Let $G$ be the elliptic operator on $B(H)$ given by\n \\begin{equation*}\n \\begin{aligned}\n G(A)=\\frac{\\gamma}{2}\\left(1+\\alpha_2^2\\right)[P,[P, A]]&+\\frac{\\gamma}{2}\\left(\\alpha_1^2+\\alpha_3^2\\right)[Q,[Q, A]]-i \\gamma \\alpha_1(Q[P, A]+[P, A] Q)\\\\\n &-i \\gamma \\alpha_2 \\alpha_3(P[Q, A]+[Q, A] P), \\quad \\quad \\forall \\, A\\in B(H),\n \\end{aligned}\n \\end{equation*}\n where $\\gamma=\\left(1+\\alpha_2^2\\right)^{-1}$ is the normalized constant. Since $P$ and $Q$ are unbounded operators affiliated to $\\mathcal{M}$, the above is a formal expression.\n\nFor $0<p\\leq \\infty$, let $S_p$ be the Schatten $p$-class of $B(H)$ with the usual Schatten $p$-norm denoted by $\\|\\cdot\\|_p$. We use Kosaki's definition \\cite{Ko1984} for noncommutative $L_p$ spaces. \n Let $L_p(\\rho)$ be the closure of all elements in $\\rho^{1/2}B(H)\\rho^{1/2}$ with respect to the following norm\n $$ \\| \\rho^{1/2} x \\rho^{1/2} \\|_{L_p(\\rho)}:= \\| \\rho^{1/2p} x \\rho^{1/2p}\\|_p, \\quad \\forall \\, x\\in B(H). $$\n Define $\\mathcal{G}^{(2)}$ on $S_2$ by\n $$ \\mathcal{G}^{(2)}(\\rho^{1/4}x\\rho^{1/4})=\\rho^{1/4}G(x)\\rho^{1/4}, \\quad \\forall \\, x\\in B(H). $$\n According to \\cite{KP}, $P_t= e^{-t\\mathcal{G}^{(2)}}$ ($\\forall \\, t\\geq 0$) is a symmetric semigroup on $S_2$. For any $x\\in B(H)$, define\n $$ \\rho^{1/4} T_t^{(\\infty)}(x) \\rho^{1/4}= P_t(\\rho^{1/4}x\\rho^{1/4}). $$\n Then $T_t^{(\\infty)}$ is an ergodic quantum Markov semigroup on $S_\\infty$, and $\\omega$ is the unique invariant state associated with $T_t^{(\\infty)}$. See \\cite{KP} for more details.\n\nFor $1\\leq p<\\infty$, denote by $T_t^{(p)}: L_p(\\rho)\\rightarrow L_p(\\rho)$ the usual restriction of $T_t^{(\\infty)}$ to $L_p(\\rho)$, by using the same way as in \\cite{GL1995}. More explicitly, for $p=2$, if $x\\in B(H)$, then\n $$ T_t^{(2)}(\\rho^{1/2}x\\rho^{1/2})=\\rho^{1/4}P_t(\\rho^{1/4}x\\rho^{1/4})\\rho^{1/4}. $$\n Denote by $\\tau$ the spectral gap of $T_t^{(2)}$, which is defined in \\eqref{gap}.\n\nWe present the definition of hypercontractivity. Let $(T_t)_{t\\geq 0}$ be an ergodic quantum Markov semigroup with the invariant state induced by $\\rho$. Then $(T_t)_{t\\geq 0}$ is said to be hypercontractive if for any $2<p<\\infty$, there exists $t_p>0$ such that for any $t \\geq t_p$, and for any $ x\\in B(H) $ with $\\omega(x)=0$,\n \\begin{equation}\\label{hypercontractivity}\n \\| T_t(\\rho^{1/2}x\\rho^{1/2}) \\|_{L_p(\\rho)} \\leq \\|\\rho^{1/2}x\\rho^{1/2}\\|_{L_2(\\rho)}.\n \\end{equation}\n Moreover, the least $t_p$ is called the optimal time of $T_t$ with respect to the parameter $p$.\n\nOur result is the following theorem.\n\n\\begin{equation}\\label{gap}\n\t\t\\tau=\\min\\{\\tau_1, \\tau_2\\}>0. \n\t\\end{equation}", "full_context": "\\label{Introduction}\n In the celebrated papers \\cite{EN1966,EN1973}, Nelson showed that the classical Ornstein-Uhlenbeck semigroup is hypercontractive and utilized such property to establish the existence and uniqueness of the ground state of a special semigroup which arises from the constructive quantum field theory. Since then, hypercontractivity becomes a useful and powerful tool in quantum field theory, quantum statistical mechanics and quantum information.\n\nLet $H=\\ell_2(\\mathbb{N})$, where $\\mathbb{N}=\\{0,1,2, \\cdots\\}$ is the set of all natural numbers. Let $\\{e_n\\}_{n=0}^\\infty$ be the canonical orthonormal basis of $H$. Then we define the creation and annihilation operators $a^*$ and $a$ as follows\n \\[\n a^* e_n = \\sqrt{n+1} \\, e_{n+1}, \\quad a e_n = \\sqrt{n} \\, e_{n-1}, \\quad (e_{-1} := 0).\n \\]\n It is clear that they satisfy the canonical commutation relation (CCR)\n \\[\n [a, a^*] = 1,\n \\]\n where $1$ denotes the identity operator by a slight abuse of notation. The number operator $N := a^* a$ acts as\n \\[\n N e_n = n \\cdot e_n.\n \\]\n The position and momentum operators are defined as\n \\[\n Q = \\frac{1}{\\sqrt{2}}(a + a^*), \\quad P = \\frac{1}{\\sqrt{2}i}(a - a^*),\n \\]\n and so\n \\[\n [Q, P] = i \\cdot 1.\n \\]\n For any $z \\in \\mathbb{C}$, define the Weyl operators\n \\[\n W(z) = e^{\\frac {i}{\\sqrt{2}}(z a^* + \\bar{z} a)} = e^{i ( \\Re z \\cdot Q + \\Im z \\cdot P ) }.\n \\]\n These operators satisfy the Weyl relation\n \\[\n W(z)^*=W(-z), \\quad \\text{and} \\quad \\forall z, w\\in\\mathbb{C}, \\ W(z) W(w) = e^{-\\frac{i}{2} \\Im (\\bar{z} w)} W(z + w).\n \\]\n Let $\\mathcal{A}$ be the $C^*$-algebra generated by all Weyl operators. It is well-known that $\\mathcal{A}\\subsetneqq B(H)$ and $\\mathcal{A}{''}=B(H)$. Let $\\omega$ be the normal faithful state on $B(H)$, which is defined by\n $$ \\omega(W(z))=e^{-\\frac{|z|^2}{4} (1+e^{-\\beta})(1-e^{-\\beta})^{-1}}, \\quad \\forall \\, z\\in \\mathbb{C} $$\n where $\\beta>0$ is a fixed inverse temperature. Note that $e^{-\\beta N}$ is a trace class operator in $B(H)$. Let \n $$ \\rho=\\dfrac{e^{-\\beta N}}{\\text{Tr}(e^{-\\beta N})}=(1-e^{-\\beta})e^{-\\beta N}. $$\n Indeed, $\\omega$ is a Gibbs state, and for any $x\\in B(H)$,\n $$ \\omega(x)=\\text{Tr}(\\rho x). $$\n We refer the interested reader to \\cite{BR} for more details. Now we introduce the quantum Ornstein-Uhlenbeck semigroups constructed in \\cite{KP}. Let the parameters $\\alpha_1, \\alpha_2, \\alpha_3 \\in \\mathbb{R}$ satisfy the following relations\n \\begin{align*}\n & \\frac{1}{2}\\left(1+\\alpha_2^2\\right) \\sinh (\\beta / 2)=-\\alpha_1 \\cosh (\\beta / 2), \\\\\n & \\frac{1}{2}\\left(\\alpha_1^2+\\alpha_3^2\\right) \\sinh (\\beta / 2)=\\alpha_2 \\alpha_3 \\cosh (\\beta / 2).\n \\end{align*}\n Let $G$ be the elliptic operator on $B(H)$ given by\n \\begin{equation*}\n \\begin{aligned}\n G(A)=\\frac{\\gamma}{2}\\left(1+\\alpha_2^2\\right)[P,[P, A]]&+\\frac{\\gamma}{2}\\left(\\alpha_1^2+\\alpha_3^2\\right)[Q,[Q, A]]-i \\gamma \\alpha_1(Q[P, A]+[P, A] Q)\\\\\n &-i \\gamma \\alpha_2 \\alpha_3(P[Q, A]+[Q, A] P), \\quad \\quad \\forall \\, A\\in B(H),\n \\end{aligned}\n \\end{equation*}\n where $\\gamma=\\left(1+\\alpha_2^2\\right)^{-1}$ is the normalized constant. Since $P$ and $Q$ are unbounded operators affiliated to $\\mathcal{M}$, the above is a formal expression.\n\nFor $0<p\\leq \\infty$, let $S_p$ be the Schatten $p$-class of $B(H)$ with the usual Schatten $p$-norm denoted by $\\|\\cdot\\|_p$. We use Kosaki's definition \\cite{Ko1984} for noncommutative $L_p$ spaces. \n Let $L_p(\\rho)$ be the closure of all elements in $\\rho^{1/2}B(H)\\rho^{1/2}$ with respect to the following norm\n $$ \\| \\rho^{1/2} x \\rho^{1/2} \\|_{L_p(\\rho)}:= \\| \\rho^{1/2p} x \\rho^{1/2p}\\|_p, \\quad \\forall \\, x\\in B(H). $$\n Define $\\mathcal{G}^{(2)}$ on $S_2$ by\n $$ \\mathcal{G}^{(2)}(\\rho^{1/4}x\\rho^{1/4})=\\rho^{1/4}G(x)\\rho^{1/4}, \\quad \\forall \\, x\\in B(H). $$\n According to \\cite{KP}, $P_t= e^{-t\\mathcal{G}^{(2)}}$ ($\\forall \\, t\\geq 0$) is a symmetric semigroup on $S_2$. For any $x\\in B(H)$, define\n $$ \\rho^{1/4} T_t^{(\\infty)}(x) \\rho^{1/4}= P_t(\\rho^{1/4}x\\rho^{1/4}). $$\n Then $T_t^{(\\infty)}$ is an ergodic quantum Markov semigroup on $S_\\infty$, and $\\omega$ is the unique invariant state associated with $T_t^{(\\infty)}$. See \\cite{KP} for more details.\n\nFor $1\\leq p<\\infty$, denote by $T_t^{(p)}: L_p(\\rho)\\rightarrow L_p(\\rho)$ the usual restriction of $T_t^{(\\infty)}$ to $L_p(\\rho)$, by using the same way as in \\cite{GL1995}. More explicitly, for $p=2$, if $x\\in B(H)$, then\n $$ T_t^{(2)}(\\rho^{1/2}x\\rho^{1/2})=\\rho^{1/4}P_t(\\rho^{1/4}x\\rho^{1/4})\\rho^{1/4}. $$\n Denote by $\\tau$ the spectral gap of $T_t^{(2)}$, which is defined in \\eqref{gap}.\n\nWe present the definition of hypercontractivity. Let $(T_t)_{t\\geq 0}$ be an ergodic quantum Markov semigroup with the invariant state induced by $\\rho$. Then $(T_t)_{t\\geq 0}$ is said to be hypercontractive if for any $2<p<\\infty$, there exists $t_p>0$ such that for any $t \\geq t_p$, and for any $ x\\in B(H) $ with $\\omega(x)=0$,\n \\begin{equation}\\label{hypercontractivity}\n \\| T_t(\\rho^{1/2}x\\rho^{1/2}) \\|_{L_p(\\rho)} \\leq \\|\\rho^{1/2}x\\rho^{1/2}\\|_{L_2(\\rho)}.\n \\end{equation}\n Moreover, the least $t_p$ is called the optimal time of $T_t$ with respect to the parameter $p$.\n\nOur result is the following theorem.\n\n\\begin{equation}\\label{gap}\n\t\t\\tau=\\min\\{\\tau_1, \\tau_2\\}>0. \n\t\\end{equation}\n\nOur result is the following theorem.\n\nWe can also obtain some estimate of the optimal time of $(T_t^{(2)})_{t\\geq 0}$ for all $ x\\in B(H) $ without the restriction $\\omega(x)=0$ with the help of Theorem \\ref{MainThm}. For any $2<p<\\8$, let $t_p'>0$ be the least constant such that for any $t \\geq t_p'$, and for any $ x\\in B(H) $,\n \\begin{equation*}\n \\| T_t^{(2)}(\\rho^{1/2}x\\rho^{1/2}) \\|_{L_p(\\rho)} \\leq \\|\\rho^{1/2}x\\rho^{1/2}\\|_{L_2(\\rho)}.\n \\end{equation*}\n\n\\begin{corollary}\\label{MainThm2}\n For $2<p<\\8$, the optimal time $t'_p$ satisfies\n \\[\n \\wt{c}_1(\\beta) (p - 1) \\leq e^{2\\tau t'_p} \\leq \\wt{C}_1(\\beta) (p - 1)^2,\n \\]\n where $\\wt{c}_1(\\beta)$ and $\\wt{C}_1(\\beta)$ are positive constants depending only on $\\beta$.\n \\end{corollary}\n\n\\begin{proposition}\\label{Prop2.1}\n $T_t^{(2)}$ is hypercontractive if for any $2<p<\\8$, there exists a constant $c_p>0$ such that for any $x \\in L_k$, $k\\in \\nat$,\n \\begin{align}\\label{KL}\n \\left\\| \\rho^{1/(2p) - 1/4} x \\rho^{1/(2p) - 1/4}\\right\\|_{p} \\leq c_p^k \\left\\| x \\right\\|_{2}. \n \\end{align}\n Moreover, the optimal time $t_p$ satisfies\n \\begin{equation}\\label{optimal-1}\n e^{2\\tau t_p} \\leq 2c_p^2, \\quad \\forall \\, 2<p<\\8.\n \\end{equation}\n \\end{proposition}\n\n\\section{Proof of Theorem \\ref{Main-Lem}}\\label{Section3}\n In this section, we will show Theorem \\ref{Main-Lem}, which finishes the proof of Theorem \\ref{MainThm}. To begin with, we introduce Meixiner polynomials, which are the main ingredient for our proof of Theorem \\ref{Main-Lem}. Recall that $p\\geq 2$.\n \\subsection{Orthogonal polynomials}\n We present a special family of Meixiner polynomials (or Laguerre polynomials), which is orthogonal with respect to the weight $w(n) = e^{-n\\beta}$ ($n\\in \\nat$). For any $k \\geq 0$, define\n \\begin{equation}\n L_k(n) = e^{-k\\beta}\\sum_{j = 0}^k (-1)^{j} (e^{\\beta} - 1)^j \\binom{k}{j} \\binom{n}{j}, \\quad \\forall \\, n\\in\\nat.\n \\end{equation}\n We have the following proposition, see \\cite[Chapter 9]{koekoek2010}.\n \\begin{proposition}\\label{Orthogonal}\n The family $\\lk{ L_0, L_1, \\cdots, L_n, \\cdots }$ is orthogonal with respect to the weight $w(n) = e^{-n \\beta}$. More precisely, for any $m, l\\in \\nat$\n \\begin{equation}\\label{Orthogonal-eq1}\n \\sum_{n = 0}^\\infty e^{-n \\beta} L_m(n) L_l(n) = \\frac{ e^{-m\\beta} }{ 1 - e^{-\\beta} } \\delta_{ml}, \n \\end{equation} \n where $\\delta_{ml}$ is the Kronecker delta function. \n \\end{proposition}\n\nWe restate Theorem \\ref{MainThm} on the estimate of the optimal time $t_p$.\n \\begin{thm}\\label{Optimal}\n Let $p\\geq 2$. There exist positive constants $\\wt{c}(\\beta)$, $\\wt{C}(\\beta)$ such that,\n \\[\n \\wt{c}(\\beta)(p-1) \\leq e^{2\\tau t_p} \\leq \\wt{C}(\\beta) (p - 1).\n \\]\n More precisely,\n \\[\n \\wt{c}(\\beta) = \\frac{ \\min\\lk{\\beta^{-2}, \\beta^{-1}} }{ 36e}, \\quad \\wt{C}(\\beta) = 24 (1 - e^{-\\beta})^{-1}C(\\beta),\n \\]\n where $C(\\beta)$ is the constant in Theorem \\ref{Main-Lem}.\n \\end{thm}\n\nNote that one has\n \\begin{equation}\\label{Lower-1}\n \\e \\| e^{-\\tau t_p} \\rho^{1/(2p) - 1/4} A_1^*\\xi_0 \\rho^{1/(2p) - 1/4} \\|_p = \\| T_{t_p}^{(2)}( \\rho^{1/2} x \\rho^{1/2} ) \\|_{L_p(\\rho)} \\leq \\| \\rho^{1/2} x \\rho^{1/2} \\|_{L_2(\\rho)} = \\e.\n \\end{equation}\n By definition, we can write $A_1^*\\xi_0$ as a matrix of the form\n \\begin{align*}\n A_1^*\\xi_0 & = \\frac{1}{\\sqrt{2}} (D_1^* - D_2^*)\\rho^{1/2} \\\\\n & = \\frac{ \\sk{ 2 \\sinh(\\beta/2) }^{-1/2} }{\\sqrt{2}} \\sk{ e^{\\beta/4} - e^{-3\\beta/4} } \\sk{ a^*\\rho^{1/2} + \\rho^{1/2}a } \\\\\n & = \\frac{\\sk{ 2\\sinh(\\beta/2) }^{-1/2}}{\\sqrt{2}} \\sk{ e^{\\beta/4} - e^{-3\\beta/4} } \\sum_{n = 0}^\\infty \\frac{ e^{-n \\beta/2}\\sqrt{n + 1} }{ \\sk{ 1 - e^{-\\beta} }^{1/2} } \\sk{ e_n \\otimes e_{n + 1} + e_{n + 1} \\otimes e_n } .\n \\end{align*}\n For convenience, we denote\n \\[\n \\quad Z = e^{-\\tau t_p} \\rho^{1/(2p) - 1/4} A_1^*\\xi_0 \\rho^{1/(2p) - 1/4}.\n \\]\n Hence\n $$ Z=e^{-\\tau t_p} \\frac{ \\sk{ 2 \\sinh(\\beta/2) }^{1/2-1/p} }{\\sqrt{2}} \\sum_{n = 0}^\\infty \\sqrt{n + 1} e^{-n\\beta/p } \\sk{ e_n \\otimes e_{n + 1} + e_{n + 1} \\otimes e_n }. $$\n However, form \\eqref{Lower-1} and by \\eqref{Schatten-norm},\n \\begin{align}\\label{Z-p-norm}\n 1 & \\geq \\| Z \\|_p \\geq e^{-\\tau t_p} \\frac{ 2^{1/p}\\sk{ 2 \\sinh(\\beta/2) }^{1/2-1/p} }{3\\sqrt{2}} \\sk{ \\sum_{n = 0}^\\infty (n + 1)^{p/2} e^{-n\\beta } }^{1/p} \\notag\\\\\n & \\geq e^{-\\tau t_p} \\frac{ 2^{1/p}\\sk{ 2 \\sinh(\\beta/2) }^{1/2-1/p} }{3\\sqrt{2}} \\beta^{-(1/2 + 1/p)} \\sk{ \\frac{ \\pi p}{4} }^{1/(2p)} \\sk{ \\frac{p}{2e} }^{1/2} \\notag\\\\\n & \\geq e^{-\\tau t_p} \\frac{ \\min\\lk{ \\beta^{-1}, \\beta^{-1/2} } }{6 \\sqrt{e}} p^{1/2}.\n \\end{align}\n This implies\n $$ e^{\\tau t_p} \\geq \\frac{ \\min\\lk{ \\beta^{-1}, \\beta^{-1/2} } }{6 \\sqrt{e}} p^{1/2}. $$\n Hence, the proof is completed.\n\n\\begin{proof}[Proof of Corollary \\ref{MainThm2}]\n It suffices to show\n $$ e^{2\\tau t'_p} \\leq \\wt{C}_1(\\beta) (p - 1)^2. $$\n We use the same notation with Proposition \\ref{Prop2.1}. Take an arbitrary $y\\in B(H)$ where $\\omega(y)$ might be non-zero. Now applying the Ball-Carlen-Lieb convexity inequality (see \\cite{BCL1994,OZ1999,RX2016}), one has for all $p \\geq 2$,\n \\[\n \\| \\rho^{1/2} y \\rho^{1/2} \\|_{L_p(\\rho)}^2 \\leq |\\omega(y)|^2 + (p - 1) \\| \\rho^{1/2} (y - \\omega(y)) \\rho^{1/2} \\|_{L_p(\\rho)}^2.\n \\]\n Then we obtain\n \\begin{align*}\n \\| T_t^{(2)}( \\rho^{1/2} y \\rho^{1/2} ) \\|_{L_p(\\rho)} & \\leq \\sk{ | \\omega(y) |^2 + (p - 1) \\| T_t^{(2)}\\sk{ \\rho^{1/2} (y - \\omega(y) ) \\rho^{1/2} } \\|_{L_p(\\rho)}^2 }^{1/2} \\\\\n & \\leq \\sk{ | \\omega(y) |^2 + (p - 1) \\frac{ e^{-2\\tau t} c_p^2 }{ 1 - e^{-2\\tau t}c_p^2 } \\| \\rho^{1/2} ( y - \\omega(y) ) \\rho^{1/2} \\|_{L_2(\\rho)}^2 }^{1/2} \\\\\n & \\leq \\sk{ | \\omega(y) |^2 + \\| \\rho^{1/2} ( y - \\omega(y) ) \\rho^{1/2} \\|_{L_2(\\rho)}^2 }^{1/2} = \\| \\rho^{1/2} y \\rho^{1/2} \\|_{L_2(\\rho)},\n \\end{align*}\n where we require\n \\[\n (p - 1)\\frac{ e^{-2\\tau t} c_p^2 }{ 1 - e^{-2\\tau t}c_p^2 } \\leq 1 \\quad \\text{i.e.} \\quad e^{2\\tau t} \\geq pc_p^2.\n \\]\n Hence, the optimal time $t_p'$ satisfies\n \\[\n e^{2\\tau t_p'} \\leq p c_p^2.\n \\]\n We complete the proof.\n \\end{proof}\n\n\\begin{thm}\\label{Main-Lem}\n\t\tLet $f_{k, n, m}$ be defined as before. For all $k \\geq 1$, $-k \\leq m \\leq k$, there exists a constant $C(\\beta)$ such that\n\t\t\\begin{equation}\n\t\t\t\\left( \\sum_{n=0}^\\8  e^{-n\\beta}| f_{k, n, m} |^p  \\right)^{1/p}  \\leq e^{m\\beta(1/2p-1/4)} \\mk{ C(\\beta)p }^{ k/2 } \\left( \\sum_{n=0}^\\8 e^{-n\\beta} | f_{k, n, m} |^2  \\right)^{1/2}.\n\t\t\\end{equation}\n\t\\end{thm}\n\n\\begin{thm}\\label{MainThm}\n\t\t$(T_t^{(2)})_{t\\geq 0}$ is hypercontractive. Moreover, the optimal time $t_p$ satisfies\n\t\t\\[\n\t\t\\wt{c}(\\beta) (p - 1) \\leq e^{2\\tau t_p} \\leq \\wt{C}(\\beta) (p - 1), \\quad \\forall\\, 2 < p < \\infty,\n\t\t\\]\n\t\twhere $\\wt{c}(\\beta)$ and $\\wt{C}(\\beta)$ are positive constants depending only on $\\beta$.\n\t\\end{thm}\n\n\\begin{corollary}\\label{MainThm2}\n\t\tFor $2<p<\\8$, the optimal time $t'_p$ satisfies\n\t\t\\[\n\t\t\\wt{c}_1(\\beta) (p - 1) \\leq e^{2\\tau t'_p} \\leq \\wt{C}_1(\\beta) (p - 1)^2,\n\t\t\\]\n\t\twhere $\\wt{c}_1(\\beta)$ and $\\wt{C}_1(\\beta)$ are positive constants depending only on $\\beta$.\n\t\\end{corollary}\n\n\\begin{proposition}\\label{Prop2.1}\n\t\t$T_t^{(2)}$ is hypercontractive if for any $2<p<\\8$, there exists a constant $c_p>0$ such that for any $x \\in L_k$, $k\\in \\nat$,\n\t\t\\begin{align}\\label{KL}\n\t\t\t\\left\\| \\rho^{1/(2p) - 1/4} x \\rho^{1/(2p) - 1/4}\\right\\|_{p} \\leq c_p^k \\left\\|  x  \\right\\|_{2}.   \n\t\t\\end{align}\n\t\tMoreover, the optimal time $t_p$ satisfies\n\t\t\\begin{equation}\\label{optimal-1}\n\t\t\te^{2\\tau t_p} \\leq 2c_p^2, \\quad \\forall \\, 2<p<\\8.\n\t\t\\end{equation}\n\t\\end{proposition}", "post_theorem_intro_text_len": 4650, "post_theorem_intro_text": "We can also obtain some estimate of the optimal time of $(T_t^{(2)})_{t\\geq 0}$ for all $ x\\in B(H) $ without the restriction $\\omega(x)=0$ with the help of Theorem \\ref{MainThm}. For any $2<p<\\infty$, let $t_p'>0$ be the least constant such that for any $t \\geq t_p'$, and for any $ x\\in B(H) $,\n\t\\begin{equation*}\n\t\t\\| T_t^{(2)}(\\rho^{1/2}x\\rho^{1/2}) \\|_{L_p(\\rho)} \\leq \\|\\rho^{1/2}x\\rho^{1/2}\\|_{L_2(\\rho)}.\n\t\\end{equation*}\n\n\t\\begin{corollary}\\label{MainThm2}\n\t\tFor $2<p<\\infty$, the optimal time $t'_p$ satisfies\n\t\t\\[\n\t\t\\widetilde{c}_1(\\beta) (p - 1) \\leq e^{2\\tau t'_p} \\leq \\widetilde{C}_1(\\beta) (p - 1)^2,\n\t\t\\]\n\t\twhere $\\widetilde{c}_1(\\beta)$ and $\\widetilde{C}_1(\\beta)$ are positive constants depending only on $\\beta$.\n\t\\end{corollary}\n\n\tSeveral comments are in order. Theorem \\ref{MainThm} implies that $(p-1)^{1/2}$ is the optimal order of $e^{\\tau t_p}$ up to a constant depending only on $\\beta$. Our proof of Theorem \\ref{MainThm} relies heavily on Meixiner polynomials. To the best of our knowledge, this is perhaps the first time that Meixiner polynomials are used to study quantum Ornstein-Uhlenbeck semigroups. Therefore, the proof of Theorem \\ref{MainThm} is different from the previous ones. Throughout this paper, we do not care about the constants $\\widetilde{c}(\\beta)$ and $\\widetilde{C}(\\beta)$ as we are only concerned with the order $(p-1)^{1/2}$. In addition, we would like to mention that at the time of this writing, we do not know the optimal order of the optimal time $t_p'$. We come up with a conjecture on $t_p'$ at the end of this paper.\n\n\t$(T_t^{(2)})_{t\\geq 0}$ is called quantum Ornstein-Uhlenbeck semigroups, but it does not behave like the classcial Ornstein-Uhlenbeck semigroup. Indeed, for the classcial Ornstein-Uhlenbeck semigroup \\cite{EN1973} or the Ornstein-Uhlenbeck semigroup in the mixed $q$-Gaussian setting \\cite{CL1993,BI1997}, or even for the Ornstein-Uhlenbeck semigroup on the \\(q\\)-Araki--Woods algebras \\cite{LR2011}, the optimal time has been already calculated explicitly or estimated. Besides, they are all ultracontractive \\cite{BM2021}. This is because the constant $c_p$ appearing in Proposition \\ref{Prop2.1} is always a universal constant which does not depend on $p$ for all Ornstein-Uhlenbeck semigroups mentioned above except the quantum Ornstein-Uhlenbeck semigroups considered in this paper. We will see that for quantum Ornstein-Uhlenbeck semigroups, the constant $c_p$ appearing in Proposition \\ref{Prop2.1} does depend on $p$, which is also the main difficulty in the estimate of $t_p$ and $t'_p$.\n\n\tIn particular, when $\\alpha_1=-\\alpha_2$ and $\\alpha_3=1$, the quantum Ornstein-Uhlenbeck semigroup $T_t^{(2)}$ is as same as in that in \\cite{CFL2000}. On the one hand, in this special case, it has already been shown in \\cite{CS2008} that $T_t^{(2)}$ is hypercontractive. The authors in \\cite{CS2008} employed the birth and death processes in an ingenious way to show the hypercontractivity of $T_t^{(2)}$. However, they only showed that $T_t^{(2)}$ is contractive from $L_2(\\rho)$ to $L_4(\\rho)$ once $t$ is larger than some fixed time. They did not consider directly the contractivity of $T_t^{(2)}$ from $L_2(\\rho)$ to $L_p(\\rho)$ for $2\\leq p\\neq 4<\\infty$. It seems that their method does not yield the best estimate of optimal time when $p$ is very large. On the other hand, the best constants of $p$-log-Sobolev inequalities of $T_t^{(2)}$ have been calculated explicitly in the remarkable papers by Beigi and Rahimi-Keshari \\cite{BK2024} for $1\\leq p\\leq 2$ and by Carlen and Maas \\cite{CJ2017} for $p=1$ respectively. In \\cite{BK2024}, Beigi \\emph{et al.} also gave an estimate of the optimal time, but their coefficient of $\\ln(p-1)$ for the optimal time is larger than ours given in Theorem \\ref{MainThm} or Corollary \\ref{MainThm2}. This also reveals that in general, the best constants of log-Sobolev inequalities cannot yield the optimal time, which is quite different from the case of the classical Ornstein-Uhlenbeck semigroup. In addition, we would like to emphasize that the quantum Ornstein-Uhlenbeck semigroups which we consider in this paper are much more general than that in \\cite{CFL2000}.\n\n\tThe paper is organized as follows. In Section \\ref{Section2}, we present more details on the eigenvalues and corresponding eigenvectors of $T_t^{(2)}$. We also reduce the proof of Theorem \\ref{MainThm} to that of Theorem \\ref{Main-Lem}. Section \\ref{Section3} is devoted to the proof of Theorem \\ref{Main-Lem}. In Section \\ref{Section4} and Section \\ref{Section5}, we will give the estimate of the optimal time $t_p$ and $t'_p$ respectively. \n\n\t\\bigskip", "sketch": "Our proof of Theorem~\\ref{MainThm} relies heavily on Meixiner polynomials; to the best of our knowledge, this is perhaps the first time that Meixiner polynomials are used to study quantum Ornstein--Uhlenbeck semigroups, and therefore the proof of Theorem~\\ref{MainThm} is different from the previous ones. In Section~\\ref{Section2}, we present more details on the eigenvalues and corresponding eigenvectors of $T_t^{(2)}$, and we also reduce the proof of Theorem~\\ref{MainThm} to that of Theorem~\\ref{Main-Lem}. Section~\\ref{Section3} is devoted to the proof of Theorem~\\ref{Main-Lem}. In Section~\\ref{Section4} we give the estimate of the optimal time $t_p$ (yielding the order $(p-1)^{1/2}$ for $e^{\\tau t_p}$ up to constants depending only on $\\beta$).", "expanded_sketch": "Our proof of the main theorem relies heavily on Meixiner polynomials; to the best of our knowledge, this is perhaps the first time that Meixiner polynomials are used to study quantum Ornstein--Uhlenbeck semigroups, and therefore the proof of the main theorem is different from the previous ones. Next, we present more details on the eigenvalues and corresponding eigenvectors of $T_t^{(2)}$, and we also reduce the proof of the main theorem to the following theorem. \n\n\\begin{thm}\\label{Main-Lem}\n\t\tLet $f_{k, n, m}$ be defined as before. For all $k \\geq 1$, $-k \\leq m \\leq k$, there exists a constant $C(\\beta)$ such that\n\t\t\\begin{equation}\n\t\t\t\\left( \\sum_{n=0}^\\8  e^{-n\\beta}| f_{k, n, m} |^p  \\right)^{1/p}  \\leq e^{m\\beta(1/2p-1/4)} \\mk{ C(\\beta)p }^{ k/2 } \\left( \\sum_{n=0}^\\8 e^{-n\\beta} | f_{k, n, m} |^2  \\right)^{1/2}.\n\t\t\\end{equation}\n\t\\end{thm}\n\nWe then prove this theorem. Finally, we give the estimate of the optimal time $t_p$ (yielding the order $(p-1)^{1/2}$ for $e^{\\tau t_p}$ up to constants depending only on $\\beta$).", "expanded_theorem": "\\label{MainThm}\n\t\t$(T_t^{(2)})_{t\\geq 0}$ is hypercontractive. Moreover, the optimal time $t_p$ satisfies\n\t\t\\[\n\t\t\\widetilde{c}(\\beta) (p - 1) \\leq e^{2\\tau t_p} \\leq \\widetilde{C}(\\beta) (p - 1), \\quad \\forall\\, 2 < p < \\infty,\n\t\t\\]\n\t\twhere $\\widetilde{c}(\\beta)$ and $\\widetilde{C}(\\beta)$ are positive constants depending only on $\\beta$.", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Fix an inverse temperature $\\beta>0$. Let $H=\\ell_2(\\mathbb N)$, let $N=a^*a$ be the number operator, and let\n\\[\n\\rho=(1-e^{-\\beta})e^{-\\beta N},\\qquad \\omega(x)=\\operatorname{Tr}(\\rho x).\n\\]\nLet $\\alpha_1,\\alpha_2,\\alpha_3\\in\\mathbb R$ satisfy\n\\[\n\\tfrac12(1+\\alpha_2^2)\\sinh(\\beta/2)=-\\alpha_1\\cosh(\\beta/2),\n\\]\n\\[\n\\tfrac12(\\alpha_1^2+\\alpha_3^2)\\sinh(\\beta/2)=\\alpha_2\\alpha_3\\cosh(\\beta/2).\n\\]\nUsing these parameters, form the corresponding quantum Ornstein-Uhlenbeck semigroup $\\bigl(T_t^{(2)}\\bigr)_{t\\ge 0}$ on $L_2(\\rho)$, where $T_t^{(2)}$ is induced from $P_t=e^{-t\\mathcal G^{(2)}}$ via\n\\[\n\\mathcal G^{(2)}(\\rho^{1/4}x\\rho^{1/4})=\\rho^{1/4}G(x)\\rho^{1/4}.\n\\]\nLet $\\tau$ denote the spectral gap of $T_t^{(2)}$. For $2<p<\\infty$, call $\\bigl(T_t^{(2)}\\bigr)_{t\\ge0}$ hypercontractive if there exists $t_p>0$ such that for all $t\\ge t_p$ and all $x\\in B(H)$ with $\\omega(x)=0$,\n\\[\n\\|T_t^{(2)}(\\rho^{1/2}x\\rho^{1/2})\\|_{L_p(\\rho)}\\le \\|\\rho^{1/2}x\\rho^{1/2}\\|_{L_2(\\rho)}.\n\\]\nWhen this holds, let $t_p$ be the optimal such time. Which statement holds for this semigroup for every $2<p<\\infty$?", "correct_choice": {"label": "A", "text": "The semigroup $\\bigl(T_t^{(2)}\\bigr)_{t\\ge 0}$ is hypercontractive, and there exist positive constants $\\widetilde c(\\beta)$ and $\\widetilde C(\\beta)$, depending only on $\\beta$, such that for every $2<p<\\infty$,\n\\[\n\\widetilde c(\\beta)(p-1)\\le e^{2\\tau t_p}\\le \\widetilde C(\\beta)(p-1).\n\\]"}, "choices": [{"label": "B", "text": "The semigroup $\\bigl(T_t^{(2)}\\bigr)_{t\\ge 0}$ is hypercontractive, and there exist positive constants $\\widetilde c(\\beta)$ and $\\widetilde C(\\beta)$, depending only on $\\beta$, such that for every $2<p<\\infty$,\n\\[\n\\widetilde c(\\beta)(p-1)^2\\le e^{2\\tau t_p}\\le \\widetilde C(\\beta)(p-1)^2.\n\\]"}, {"label": "C", "text": "The semigroup $\\bigl(T_t^{(2)}\\bigr)_{t\\ge 0}$ is hypercontractive, and for each fixed $p\\in(2,\\infty)$ there exist positive constants $c_{p,\\beta}$ and $C_{p,\\beta}$ such that\n\\[\nc_{p,\\beta}\\le e^{2\\tau t_p}\\le C_{p,\\beta}.\n\\]"}, {"label": "D", "text": "The semigroup $\\bigl(T_t^{(2)}\\bigr)_{t\\ge 0}$ is hypercontractive, and there exist positive constants $\\widetilde c$ and $\\widetilde C$, independent of both $p$ and $\\beta$, such that for every $2<p<\\infty$,\n\\[\n\\widetilde c(p-1)\\le e^{2\\tau t_p}\\le \\widetilde C(p-1).\n\\]"}, {"label": "E", "text": "The semigroup $\\bigl(T_t^{(2)}\\bigr)_{t\\ge 0}$ is hypercontractive, and there exist positive constants $\\widetilde c(\\beta)$ and $\\widetilde C(\\beta)$, depending only on $\\beta$, such that for every $2<p<\\infty$,\n\\[\n\\widetilde c(\\beta)(p-1)\\le e^{\\tau t_p}\\le \\widetilde C(\\beta)(p-1).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "counting_estimate", "tampered_component": "linear-in-(p-1) growth of e^{2\\tau t_p}", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "uniform linear bounds with constants depending only on \\beta", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "dependence of constants on \\beta", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "exponentiated quantity is e^{2\\tau t_p}, not e^{\\tau t_p}", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct estimate explicitly or through obvious phrasing. It sets up the semigroup and asks for the valid quantitative bound without signaling which dependence on p or beta is correct."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: it asks which quantitative estimate holds for the optimal hypercontractive time. Although the choices introduce meaningful variants, the task is still largely to recognize the theorem’s stated conclusion."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish p from p-1, one-sided from two-sided bounds, and beta-dependent from beta-independent constants. However, the question mainly tests recognition of the known result rather than generating a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: they vary asymptotic order, remove a lower bound, or alter parameter dependence. These reflect realistic failure modes rather than arbitrary wrong answers."}, "total_score": 6, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it only moderately tests genuine generative reasoning because it is close to direct recall of a specific quantitative result."}}
{"id": "2602.16361v1", "paper_link": "http://arxiv.org/abs/2602.16361v1", "theorems_cnt": 2, "theorem": {"env_name": "Theorem", "content": "\\label{thm:PrefMain} Let $(W,S)$ be a Coxeter system. \n\\begin{enumerate}\n\\item The language $\\Pref_T$ of reduced words for reflection-prefixes is regular.\n\\item The language of palindromic reduced words is:\n$$\n\\operatorname{Pal}=\\{s_1\\cdots  s_k s_{k+1}  s_{k}\\cdots  s_1 \\in S^*\\mid  s_1\\cdots  s_k s_{k+1}\\in\\Pref_T\\}.\n$$\n\\item The following Poincar\\'e series is rational:\n$$\n\\Pref_T(q)=\\sum_{\\bold w \\in \\Pref_T} q^{\\ell(w)}=\\sum_{k\\geq 0} \\pal_kq^k,\n$$\nwhere $\\pal_k$ is the number of palindromic reduced words of length $k$.\n\n\\item The Poincar\\'e series $\\operatorname{Pal}(q)=q\\Pref_T(q^2)$ is rational.\n\\end{enumerate}", "start_pos": 10997, "end_pos": 11626, "label": "thm:PrefMain"}, "ref_dict": {"prop:Pref1": "\\begin{Proposition}\\label{prop:Pref1} Let $t\\in T$ with corresponding positive root $\\alpha_t$. Let  $w\\in W$ and assume there is $r\\in D_R(w)$ such that $t=wrw^{-1}$ and $\\ell(t)=2\\ell(w)-1$. Then $D_R(w)=\\{r\\}$. Moreover: \n \\begin{enumerate}\n\n\\item if $w=s_1\\cdots s_k s_{k+1}$ is a reduced word for $w$, then $s_{k+1}=r$ and $t=s_1\\cdots s_k r s_k\\cdots s_1$ is a palindromic reduced word for $t$;\n\\item $\\alpha_t=w(\\alpha_r)= s_1\\cdots s_k(\\alpha_r)$ and  $\\dep(\\alpha_t)=k=\\ell(w)-1$.\n\\end{enumerate}\n\\end{Proposition}", "se:affine": "\\begin{Corollary}\\label{cor:Refregular}\nThe language $\\Pref_T$ is regular and the generating function \n\\begin{align*}\n\\sum_{u\\in \\Pref_T} q^{|u|}& =\\sum_{t\\in T} |\\{s_1 \\cdots s_k s_{k+1} s_k\\cdots s_1 \\in \\Pref_T \\mid t=s_1 \\cdots s_k s_{k+1}s_k\\cdots s_1\\}| \\, q^{\\ell(t)}\\\\\n&=\\sum_{k\\in\\mathbb{N}} a_k q^k\n\\end{align*}\nis rational, where $a_k$ is the number of saturated chains of length $k$ in the root poset of $W$. In particular, $\\RPal(q)=q\\Pref_T(q^2)$ is rational.\n\\end{Corollary}\n\n\\section{Poincare series of affine Coxeter groups}\\label{se:affine}\nIn this section we provide a direct solution to Stembridge’s problem by providing an explicit description of the Poincaré series for affine Coxeter groups. This formula is based on a detailed description of the Hasse diagram of the associated root poset.\n\n\\subsection{Crystallographic root systems}\n\nWhen one works with an affine Coxeter system $(W,S)$, with underlying finite Weyl group $(W_0,S_0)$, it is customary to use its associated crystallographic root system in a positive semi-definite quadratic space $(V,B)$. In this case, the classical geometric representation (see \\S\\ref{ss:InversionSets}) is the unique geometric representation, and  the simple system $\\Delta$ is a basis of $V$. The crystallographic condition allows us to consider root systems with non-necessarily unitary roots. We briefly recall this construction below. \n\nLet $\\Phi_0$ be a reduced, irreducible, crystallographic root system for the finite\nWeyl group $W_0$, realized in a Euclidean space $(V_0,B_0)$ with corresponding set of simple roots $\\Delta_0=\\{\\alpha_s\\mid s\\in S_0\\}$ and positive roots $\\Phi_0^+$ (see for instance \\cite{bour,Hum}). Then for $s,t\\in S_0$, we have:\n\\[\nB_0(\\alpha_s, \\alpha_t) = \n\\begin{cases} \n-\\|\\alpha_s\\| \\|\\alpha_t\\| \\cos \\left(\\frac{\\pi}{m_{st}}\\right) & \\text{if } m_{st} < \\infty, \\\\\n-1 & \\text{if } m_{st} = \\infty.\n\\end{cases}\n\\]\nThe quadratic space $(V,B)$ is obtained as follows:  $V = V_0 \\oplus \\mathbb{R}\\delta$, and $B$ is the extension of  $B_0$ by setting the radical of $B$ to be equal to  $Q = \\mathbb{R}\\delta$. Let $\\omega$ denote the highest root of $\\Phi_0$; the crystallographic affine simple system is $\\Delta = \\Delta_0 \\cup \\{ \\delta - \\omega \\}$ with root system \n$\\Phi = \\{ \\alpha + k\\delta \\mid \\alpha \\in \\Phi_0,\\; k \\in \\mathbb{Z} \\}$.\nThe set of positive roots is:\n\\[\\Phi^+=\n\\{\\,  k\\delta + \\alpha,\\;  (k+1)\\delta -\\alpha \\mid \\alpha \\in \\Phi_0^+,\\; k \\in \\mathbb{N} \\,\\}=(\\mathbb{N}\\delta+\\Phi^+_0)\\sqcup (\\mathbb{N}^*\\delta-\\Phi^+_0).\n\\]\nIt follows that $\\Phi$ is a {\\em crystallographic (infinite) root system} for the affine Weyl group $(W,S)$, where $S=S_0 \\cup \\{s_{\\delta-\\omega}\\}$.\n\n\\subsection{The root poset and dominance order in the affine case} Structural results for unitary root systems, specifically those determined by the sign of the bilinear form $B$, generally extend to crystallographic systems. However, properties that depend on the specific numerical values of the form, such as dominance and long edges, must be precisely reformulated for crystallographic root systems; see~\\cite[Example 3.9]{DH} for more details. \n\nFrom the crystallographic root system, $\\Phi$, one obtains a {\\em unitary} root system in the sense of \\S\\ref{ss:InversionSets} by normalizing each root via the map $\\alpha\\mapsto \\frac{\\alpha}{\\|\\alpha\\|}$. Let $\\alpha,\\beta\\in\\Phi$, then:     \n$$\nB\\left(\\frac{\\alpha}{\\|\\alpha\\|},\\frac{\\beta}{\\|\\beta\\|}\\right) = \\frac{B(\\alpha,\\beta)}{\\|\\alpha\\|\\|\\beta\\|}.\n$$\nTherefore, the condition $B(\\alpha',\\beta')\\leq -1$ for unitary roots $\\alpha',\\beta'$, which appear in Definition~\\ref{def:long} or Proposition~\\ref{prop:inf-depth}, becomes:  $$B(\\alpha,\\beta)\\leq -\\|\\alpha\\|\\|\\beta\\|.$$ \nFor this reason Definition~\\ref{def:long} in the crystallographic setting becomes the following.\n\n\\begin{Definition}[Long and short edges in crystallographic root system] A covering $\\alpha\\lhd s(\\alpha)$ in the root poset $(\\Phi^+,\\leq)$ is \\textit{short} if $-\\|\\alpha\\| \\|\\alpha_s\\| < B(\\alpha, \\alpha_s)<0$. Otherwise we say that the covering is \\textit{long}.\n\\end{Definition}\n\nThe Hasse diagram of the root poset is studied in more details in the subsequent section. \n\n\\smallskip\nIn the affine case, the Hasse diagram of the dominance order is the union of $2|\\Phi_0^+|$ total ordered chains. Indeed, the only dominance relations in  $\\Phi$ are of the form\n$$\n k\\delta +\\alpha \\;\\leq_d\\;  l\\delta+\\alpha,\n\\quad\n\\text{or} \\quad \n (k+1)\\delta -\\alpha\\;\\leq_d\\;  (l+1)\\delta - \\alpha,\n$$\nwhere $k\\leq l$ in $\\mathbb N$ and $\\alpha\\in\\Phi_0^+$; see~\\cite[Example 3.9]{DH}. \n\nAn interesting consequence of these dominance relations is that for $\\alpha\\in\\Phi_0^+$ and $k\\in\\mathbb N$ we have:\n$$\n\\dep_\\infty(k\\delta+\\alpha)=k\\quad\\textrm{and}\\quad\\dep_\\infty((k+1)\\delta-\\alpha)=k,\n$$\nsee~\\cite[Example 3.9, Eq.($\\diamond$)]{DH}. In particular, the set of $k$-small roots (see Section \\ref{ss:mSmallRoots}) for $k\\in\\mathbb N$ is in the affine case:\n\\begin{equation}\n\\label{eq:mSmall}\n\\Sigma_k=\\left(k\\delta+\\Phi^+_0\\right)\\sqcup \\left((k+1)\\delta-\\Phi_0^+\\right).\n\\end{equation}\n\n\\begin{Example}\nLet $(W,S)$ be a Coxeter system of type $\\widetilde{A}_2$, \n\n\\begin{center}\n\\begin{tikzpicture}\n\t[scale=2,\n\t q/.style={teal,line join=round},\n\t racine/.style={blue},\n\t racinesimple/.style={blue},\n\t racinedih/.style={blue},\n\t sommet/.style={inner sep=2pt,circle,draw=black,fill=blue!40,thick,anchor=base},\n\t rotate=0]\n \\tikzstyle{every node}=[font=\\small]\n\\def\\grosseursimple{0.025}\n\\coordinate (ancre) at (0,3);\n\n\\node[sommet,label=above:$s_{\\delta-\\omega}$] (a2) at ($(ancre)+(0.25,0.4)$) {};\n\\node[sommet,label=below right : $s_2$] (a3) at ($(ancre)+(0.5,0)$) {} edge[thick] node[auto,swap,right] {}(a2) ;\n\\node[sommet,label=below left:$s_1$] (a4) at (ancre) {} edge[thick] node[auto,swap,below] {} (a3) edge[thick] node[auto,swap,left] {} (a2);\n\\end{tikzpicture}\n\\end{center}\n\\noindent\nthen the underlying finite Coxeter system $(W_0,S_0)$ is of type $A_2$. Consider $S=\\{s_1, s_2, s_{\\delta-\\omega}\\}$, where $s_1,s_2\\in S_0$ and $\\omega=\\alpha_1+\\alpha_2$ is the highest root of $W_0$. The finite positive root system is $\\Phi^+_0=\\{\\alpha_1, \\alpha_2, \\alpha_1+\\alpha_2\\}$ and the affine simple system is $\\Delta=\\{\\alpha_1, \\alpha_2, \\delta-\\omega\\}$. The set of $k$-small roots for $k\\in\\mathbb N$ is:\n\\begin{eqnarray*}\n\\Sigma_k &=& \\{\\alpha_1 + k\\delta, \\alpha_2 + k\\delta, \\alpha_1+\\alpha_2+k\\delta\\}\\\\\n    &&\\sqcup\\  \\{-\\alpha_1+(k+1)\\delta, -\\alpha_2+(k+1)\\delta, -(\\alpha_1+\\alpha_2)+(k+1)\\delta\\}. \n\\end{eqnarray*}\nThe affine positive root system is:\n$$\n\\Phi^+=\\bigsqcup_{m\\in\\mathbb N} \\Sigma_m.\n$$\nThe Hasse diagram of $(\\Phi^+,\\leq)$ is depicted in Figure \\ref{fig:rootposetA2}, up to depth equal to 3, where each color of the edges corresponds to the simple reflection that is applied: red for $s_1$, yellow for $s_2$ and green for $s_3:=s_{\\delta -\\omega}$. Moreover, the dashed edges correspond to the long coverings.\n\\end{Example}\n\n\\begin{figure}[ht]\n    \\centering    \\includegraphics[width=0.9\\linewidth]{rootposetA22.pdf}\n    \\caption{Hasse diagram of the root poset of type $\\widetilde{A}_2$ up to depth~3, with $\\Sigma=\\Sigma_0$ and $\\Sigma_1$ that correspond to the blocks connected with non-dotted edges.}\n    \\label{fig:rootposetA2}\n\\end{figure}\n\n\\begin{Example}\n    Let $(W,S)$ be a Coxeter system of type $\\widetilde{B}_3$ with the following Coxeter graph\n    \\begin{center}\n\\begin{tikzpicture}\n\t[scale=2,\n\t q/.style={teal,line join=round},\n\t racine/.style={blue},\n\t racinesimple/.style={blue},\n\t racinedih/.style={blue},\n\t sommet/.style={inner sep=2pt,circle,draw=black,fill=blue!40,thick,anchor=base},\n\t rotate=0]\n \\tikzstyle{every node}=[font=\\small]\n\\def\\grosseursimple{0.025}\n\\coordinate (ancre) at (0,3);\n\n\\node[sommet,label=right:$s_3$] (a2) at ($(ancre)+(0.8,0.25)$) {};\n\\node[sommet,label=above :$s_2$] (a3) at ($(ancre)+(0.5,0)$) {} edge[thick] node[auto,swap,right] {}(a2) ;\n\\node[sommet,label=above:$s_1$] (a4) at (ancre) {} edge[thick] node[auto,swap,below] {$4$} (a3);\n\\node[sommet,label=right :$s_{\\delta-\\omega}$] (a5) at ($(ancre)+(0.8,-0.25)$) {} edge[thick] node[auto,swap,right] {}(a3) ;\n\\end{tikzpicture}\n\\end{center}\n\n\\noindent    \n    In this case we have $\\|\\alpha_1\\|=1$ and $\\|\\alpha_2\\|=\\|\\alpha_3\\|=\\|\\omega\\|=\\sqrt{2}$, where $\\omega=2\\alpha_1+2\\alpha_2+\\alpha_3$. For instance, $B(\\alpha_2,\\delta-\\omega)=-B(\\alpha_2,\\omega)=-1>-\\|\\alpha_2\\|\\|\\omega\\|=-2$, and $B(\\delta-\\alpha_3, \\alpha_3)=-\\|\\alpha_3\\|^2=-2$. Hence the covering $\\alpha_2\\lhd s_{\\delta-\\omega}(\\alpha_2)$ is short, while $\\delta-\\alpha_3\\lhd s_3(\\delta-\\alpha_3)=\\delta+\\alpha_3$ is long. The Hasse diagram of the root poset  restricted to $\\Sigma=\\Sigma_0$ of type $\\widetilde{B}_3$ is in Figure \\ref{fig:root-poset-b3}.\n\\end{Example}\n\n\\subsection{Poset isomorphisms within the root poset}\n\nLet $(W,S)$ be an affine Coxeter system with underlying finite system $(W_0,S_0)$ and crystallographic root system $\\Phi_0$. Since $\\Phi_0$ is irreducible, we have two cases for the decomposition of $\\Phi_0$ into $W_0$-orbits:\n\\begin{itemize}\n    \\item $\\Phi_0=\\mo_1$ is a single orbit if $(W,S)$ is simply laced;\n    \\item $\\Phi_0=\\mo_1\\sqcup\\mo_2$ if $(W,S)$ is of type $\\tilde B_n$, $\\tilde C_n$, $\\tilde G_2$ or $\\tilde F_4$.\n\\end{itemize}\nFor simplification, we write:\n$$\n\\Phi_0^+=\\bigsqcup_{i=1}^h \\mo_i^+,\n$$ \nwhere $\\mo_i^+=\\mo_i \\cap \\Phi_0^+$ and $1\\leq i\\leq h$ ($h=1,2$). We also assume that $\\mo_1^+$ is the $W_0$-orbit containing the highest root $\\omega$. \n\n\\smallskip\n\nLet $\\mo$ be a $W_0$-orbit. For $k\\in\\mathbb N$, the set of $k$-small roots in $\\mo$ is:\n\\begin{equation}\n\\Sigma_{\\mo,k} =(k\\delta+\\mo^+)\\sqcup \\left((k+1)\\delta-\\mo^+\\right) =\\{k\\delta + \\alpha, (k+1)\\delta - \\alpha \\mid \\alpha \\in \\mo^+\\}.\n\\end{equation}\nMoreover, by Eq~(\\ref{eq:mSmall}), we have $\\Sigma_k=\\sqcup_{i=1}^h\\Sigma_{\\mo_i,k}$, where $h=1,2$. Set \n$$\n\\Phi_{\\mo}=\\bigsqcup_{k\\in \\mathbb N} \\Sigma_{\\mo,k}.\n$$\nSo $\\Phi^+=\\sqcup_{i=1}^h\\Phi_{\\mo_i}$, where $h=1,2$. \n\nThe Hasse diagrams of the subposet $(\\Phi_{\\mo},\\leq)$  exhibit strong symmetry properties that are fundamental to prove the main result of this section. \n\n\\subsection*{Symmetry at the level of small roots} We denote for simplification, the set of small root in $\\mo$ by:\n$$\n\\Sigma_{\\mo}=\\Sigma_{\\mo,0}=\\mo^+\\sqcup (\\delta-\\mo^+).\n$$\nRecall that $\\Sigma$ is an order ideal in the root poset (see~\\cite[Corollary~4.7.7]{BB}) and any cover relation within $\\Sigma$ is short (see~\\cite[Lemma~4.7.5 and Theorem~4.7.6]{BB}).\n\n\\begin{Theorem}\\label{th:iso1} Let $\\mo$ be a $W_0$-orbit. \nThe map $\\phi: \\mo^+ \\to \\delta - \\mo^+$ defined by $\\phi(\\alpha) = \\delta - \\alpha$ is an isomorphism of posets between $(\\mo^+, \\leq)$ and the opposite poset $(\\delta - \\mo^+, \\leq)^{op}$: for any $\\alpha, \\beta \\in \\mo^+$, we have $\\alpha \\leq \\beta$ if and only of  $\\delta - \\beta \\leq \\delta - \\alpha$. Moreover all covering relations in $(\\mo^+,\\leq)$ and in $(\\delta - \\mo^+, \\leq)$ are short.\n\\end{Theorem}\n\n\\begin{proof}\nSince a poset isomorphism is uniquely determined by the preservation (or reversal) of covering relations, it suffices to show that $\\alpha \\lhd \\beta$ in $\\mo^+$ if and only if $\\delta - \\beta \\lhd \\delta - \\alpha$ in $\\delta - \\mo^+$. Recall that for any $\\alpha, \\beta \\in \\Phi^+$, a covering $\\alpha \\lhd \\beta$ exists if and only if there is a simple reflection $s \\in S$ such that $\\beta = s(\\alpha)$ and $B(\\alpha, \\alpha_s) < 0$. We examine the two types of simple reflections in the affine Weyl group $W$.\n\n\\begin{itemize}\n    \\item[(i)] Let $s \\in S_0$ be a finite reflection. For a finite simple root $\\alpha_s$, we have:\n    \\[ B(\\delta - \\alpha, \\alpha_s) = B(\\delta, \\alpha_s) - B(\\alpha, \\alpha_s) = -B(\\alpha, \\alpha_s). \\]\n    It follows that $B(\\alpha, \\alpha_s) < 0$ if and only if $B(\\delta - \\alpha, \\alpha_s) > 0$. Thus, $\\alpha \\lhd s(\\alpha)$ if and only if  $s(\\delta - \\alpha) = \\delta - s(\\alpha) \\lhd \\delta - \\alpha$.\n\n    \\item[(ii)] Let $s_{\\delta-\\omega}$ be the affine reflection. Similarly, for the affine simple root $\\delta - \\omega$, the property of the radical implies:\n    \\[ B(\\delta - \\alpha, \\delta - \\omega) = B(\\delta, \\delta - \\omega) - B(\\alpha, \\delta - \\omega) = -B(\\alpha, \\delta - \\omega). \\]\n    Again, the sign of the bilinear form is reversed: $B(\\alpha, \\delta - \\omega) < 0$ if and only if $B(\\delta - \\alpha, \\delta - \\omega) > 0$. This ensures that $\\alpha \\lhd s_{\\delta-\\omega}(\\alpha)$ if and only if $s_{\\delta-\\omega}(\\delta - \\alpha) = \\delta - s_{\\delta-\\omega}(\\alpha) \\lhd \\delta - \\alpha$.\n\\end{itemize}\n\nNow, let $\\alpha \\lhd s(\\alpha)$ be a covering in $\\mo^+$ for some $s \\in S$. The type of this cover depends on the value $B(\\alpha, \\alpha_s)$. Under the map $\\phi$, this corresponds to the cover $\\delta-s(\\alpha) \\lhd \\delta -\\alpha$. By the properties of the reflection $s$ and of the radical, we have:\n\\[\nB(\\delta - s(\\alpha), \\alpha_s) = -B(s(\\alpha), \\alpha_s) = B(\\alpha, \\alpha_s).\n\\]\nSince  $B(\\cdot, \\alpha_s)$ is identical in both case, the condition for being short or long in Definition \\ref{def:long} is preserved. Since $\\mo^+$ and $\\delta-\\mo^+$ decomposes the set of small roots~$\\Sigma$, so they are all short. \n\\end{proof}\n\n\\begin{Example}\n    If $(W,S)$ is of type $\\widetilde{B}_3$, then $\\Phi_0^+=\\mo_1^+\\sqcup\\mo_2^+$, where $\\mo_1^+=\\{\\alpha_1, s_2(\\alpha_1), s_3s_2(\\alpha_1)\\}$ and $\\mo_2^+=\\{\\alpha_2, \\alpha_3, s_1(\\alpha_2), s_3(\\alpha_2), s_1s_3(\\alpha_2), s_2s_1s_3(\\alpha_2)\\}$. The root poset restricted to $\\Sigma_{\\mo_1}\\sqcup\\Sigma_{\\mo_2}$ is depicted in Figure \\ref{fig:root-poset-b3}.\n\\end{Example}\n\n\\begin{figure}[hbtp]\n\\centering\n\\includegraphics[scale=.75]{rootposetB3.pdf}\n\\caption{Root poset of type $\\widetilde{B}_3$ restricted to $\\Sigma_{\\mo_1}\\sqcup \\Sigma_{\\mo_2}$. The portion of poset in the red lines is isomorphic to opposite in the black lines, as stated in Theorem \\ref{th:iso1}.}\n\\label{fig:root-poset-b3}\n\\end{figure}\n\n\\subsection*{Symmetry of the root poset restricted to a $W_0$- orbit}\nFollowing \\cite[Ch. VI, \\S 1]{bour}, for $\\alpha, \\beta\\in \\Phi^+$, we set \n$$n(\\alpha,\\beta):=2\\frac{B(\\alpha,\\beta)}{B(\\beta,\\beta)},$$  \nso that $s_\\beta(\\alpha)=\\alpha - n(\\alpha,\\beta)\\beta.$ By \\cite[Ch. VI, \\S 1, Proposition~8]{bour}, if $\\alpha\\neq \\beta$ and $\\|\\alpha\\|\\leq \\|\\beta\\|$, then $n(\\alpha,\\beta) \\in \\{0,\\pm 1\\}$. It follows that $n(\\alpha, \\omega) \\in \\{0,\\pm 1\\}$ when $\\alpha\\neq \\omega$, since $\\|\\alpha\\|\\leq \\|\\omega\\|$ for all positive roots $\\alpha$. Finally, note that $n(k\\delta+\\alpha, \\beta)=n(\\alpha, \\beta)=n(\\alpha, m\\delta+\\beta)$ and $n(k\\delta-\\alpha, \\beta)=-n(\\alpha, \\beta)=n(\\alpha, m\\delta-\\beta)$ for $k,m\\in\\mathbb{N}^*$.\n\\smallskip\n\n We describe now the covering relations inside the sets $\\Sigma_{\\mo,k}$. Let $\\alpha \\in \\Phi_0^+$ and $k \\in \\mathbb{N}$.\n\n\\begin{enumerate}\n\\item[(1)]\nThere are no covering relations of the form\n\\[\n(k+1)\\delta - \\alpha \\lhd s_{\\delta-\\omega}\\big((k+1)\\delta - \\alpha\\big).\n\\]\nIndeed, if such a covering existed, then\n$0 > B\\big((k+1)\\delta - \\alpha,\\, \\delta - \\omega\\big) = B(\\alpha,\\omega)$,\nso that $n(\\alpha,\\omega) = -1$. It follows that\n\\[\ns_{\\delta-\\omega}(\\alpha)\n= \\alpha - n(\\alpha,\\delta-\\omega)(\\delta-\\omega)\n= \\alpha + n(\\alpha,\\omega)(\\delta-\\omega)\n= \\alpha - \\delta + \\omega.\n\\]\nThus, $s_{\\delta-\\omega}\\big((k+1)\\delta - \\alpha\\big)\n= (k+2)\\delta - (\\alpha + \\omega)\n\\notin \\Sigma_{\\mo,k},$ a contradiction.\n\\smallskip\n\n\\item[(2)]\nAll covering relations in $\\Sigma_{\\mo,k}$ are of one of the following three types:\n\\begin{itemize}\n\\item[(i)]\n$k\\delta + \\alpha \\lhd s(k\\delta + \\alpha) = k\\delta + s(\\alpha),\n\\qquad \\text{when } B(\\alpha,\\alpha_s) < 0;\n$\n\n\\item[(ii)]\n$\n(k+1)\\delta - \\alpha \\lhd s\\big((k+1)\\delta - \\alpha\\big)\n= (k+1)\\delta - s(\\alpha),\n$\nwhen $B(\\alpha,\\alpha_s) > 0$ and $\\alpha \\neq \\alpha_s$.\nNote that if $\\alpha = \\alpha_s$, then\n$(k+1)\\delta - s(\\alpha) = (k+1)\\delta + \\alpha_s \\in \\Sigma_{\\mo,k+1}$.\n\n\\item[(iii)]\n$\nk\\delta + \\alpha \\lhd s_{\\delta-\\omega}(k\\delta + \\alpha)\n= k\\delta + s_{\\delta-\\omega}(\\alpha),\n$\nwhen $B(\\alpha,\\delta-\\omega) < 0$ and $\\alpha \\neq \\omega$.\nOtherwise, $s_{\\delta-\\omega}(k\\delta + \\omega) = (k+2)\\delta - \\omega\n\\in \\Sigma_{\\mo,k+1}.$\n\\end{itemize}\n\\end{enumerate}\n\nWe state now the second symmetry of the root poset.\n\n\\begin{Theorem}\\label{th:iso2} Let $\\mo$ be a $W_0$-orbit. There is a poset isomorphism $$(\\Sigma_{\\mo,k}, \\leq) \\cong (\\Sigma_{\\mo}, \\leq)$$ for all $k\\in\\mathbb{N}$. Moreover, all covering in $\\Sigma_{\\mo,k}$ are short.\n\\end{Theorem}\n\n\\begin{proof}\nThe map $\\pi_k : \\Sigma_{\\mo,k} \\longrightarrow \\Sigma_{\\mo}$ defined by\n\\[\n\\pi_k(k\\delta+\\alpha)=\\alpha,\n\\qquad\n\\pi_k((k+1)\\delta-\\alpha)=\\delta-\\alpha,\n\\]\nfor $\\alpha\\in\\Phi_0^+$ is a bijection. By the discussion preceding the theorem, we have three types of coverings in $\\Sigma_{\\mo,k}$.\n\\smallskip\n\nA covering $k\\delta+\\alpha \\lhd k\\delta+s(\\alpha)$ of type (i) occurs if and only if\n$0>B(k\\delta+\\alpha,\\alpha_s)=B(\\alpha,\\alpha_s)$,\nwhich is equivalent to $\\alpha \\lhd s(\\alpha)$ being a covering in $\\Sigma_{\\mo}$. \n\\smallskip\n\nA covering $(k+1)\\delta-\\alpha \\lhd (k+1)\\delta-s(\\alpha)$ of type (ii), occurs if and only if\n$ 0>B((k+1)\\delta-\\alpha,\\alpha_s)=-B(\\alpha,\\alpha_s)$,\nwith $\\alpha\\neq\\alpha_s$, which is equivalent to\n$\\delta-\\alpha \\lhd \\delta-s(\\alpha)$\nbeing a covering in $\\Sigma_{\\mo}$.\n\\smallskip\n\nA covering $k\\delta+\\alpha \\lhd s_{\\delta-\\omega}(k\\delta+\\alpha)$ of type (iii)\noccurs if and only if $0>B(k\\delta+\\alpha,\\delta-\\omega)=B(\\alpha,\\delta-\\omega)$, with $\\alpha\\neq\\omega$, which is equivalent to $\\alpha \\lhd s_{\\delta-\\omega}(\\alpha)\n$ being a covering in $\\Sigma_{\\mo}$. \n\\smallskip\n\nThus $\\pi_k$ preserves and reflects covering relations and is therefore a poset isomorphism. The final statement follows from Theorem~\\ref{th:iso1} since all covering are short in $\\Sigma_\\mo$.\n\\end{proof}\n\n\\begin{Example} In the Figure~\\ref{fig:root-poset-b3-iso2} below, the the first three slices of the Hasse diagram of the root poset of type $\\widetilde{B}_3$ are represented. The first slice coincides with the poset shown in Figure \\ref{fig:root-poset-b3}, although the order relations are drawn differently. This alternative depiction makes the isomorphisms described in Theorem \\ref{th:iso2} more apparent.\n\\end{Example}\n\n\\begin{figure}[hbtp]\n\\centering\n\\includegraphics[scale=.7]{periodicB3str.pdf}\n\\caption{}\n\\label{fig:root-poset-b3-iso2}\n\\end{figure}\n\nWe end this discussion by showing that a covering $\\alpha\\lhd \\beta$ with $\\alpha\\in \\Sigma_{\\mo,k}$ and $\\beta\\in \\Sigma_{\\mo,k+1}$ is long.\n\n\\begin{Proposition}[Covering relations]\\label{prop:long-cover}\\\n\\begin{itemize}\n    \\item[(i)]  For all $k\\in\\mathbb{N}, s\\in S_0$, $(k+1)\\delta-\\alpha_s\\lhd s((k+1)\\delta - \\alpha_s)$ is a long covering, where $(k+1)\\delta-\\alpha_s\\in \\Sigma_{\\mo,k}$ and $s((k+1)\\delta - \\alpha_s)\\in \\Sigma_{\\mo, k+1}$.\n    \\item[(ii)] For all $k\\in\\mathbb{N}$, $k\\delta +\\omega\\lhd s_{\\delta-\\omega}(k\\delta + \\omega)$ is a long covering, where $k\\delta +\\omega\\in \\Sigma_{\\mo,k}$ and $s_{\\delta-\\omega}(k\\delta + \\omega)\\in \\Sigma_{\\mo,k+1}$. \n\\end{itemize}\n\n\\end{Proposition}\n\n\\begin{proof}\n    Consider the  $(k+1)\\delta-\\alpha_s\\lhd s((k+1)\\delta - \\alpha_s)$, then $B((k+1)\\delta-\\alpha_s, \\alpha_s)=-\\|\\alpha_s\\|^2$, hence the covering is long, and $s((k+1)\\delta - \\alpha_s)=(k+1)\\delta+\\alpha_s \\in \\Sigma_{\\mo,k+1}$. Analogously, if we consider the covering $k\\delta +\\omega\\lhd s_{\\delta-\\omega}(k\\delta + \\omega)$, $B(k\\delta +\\omega, \\delta-\\omega)=-\\|\\omega\\|^2$, so the covering is long. In particular, $s_{\\delta -\\omega}(k\\delta +\\omega)=(k+2)\\delta -\\omega \\in \\Sigma_{\\mo,k+1}$. \n    \\end{proof}\n\n\\subsection{An explicit formula for the generating function of depth} We are ready now to provide a closed formula for the generating function of the depth of positive roots in $\\Phi$, using the restriction of the root poset to the sets $\\Sigma_\\mo$ and $\\Phi_\\mo$ introduced in the previous section.\n\\medskip\n\nLet $\\mo \\subseteq \\Phi_0$ be $W_0$-orbit. Define \n$$\nM_{\\mo}=\\max\\{\\dep(\\beta) \\mid \\beta \\in \\Sigma_{\\mo}\\}.\n$$\nWe show that the depth of a positive root depends only on $M_{\\mo}$ and on\nthe depth of a root in $\\mo^+$.\n\n\\begin{Corollary}\\label{cor:maxdep}\n    Let $\\alpha\\in \\mo^+$ and $k\\in \\mathbb{N}$, then :\n    \\begin{enumerate}\n        \\item $\\dep(k\\delta + \\alpha)=\\dep(\\alpha)+k(M_\\mo +1)$;\n        \\item $\\dep((k+1)\\delta -\\alpha) = \\dep(\\delta-\\alpha) +k(M_\\mo +1)$.\n    \\end{enumerate}  \n\\end{Corollary}", "th:iso1": "\\begin{Theorem}\\label{th:iso1} Let $\\mo$ be a $W_0$-orbit. \nThe map $\\phi: \\mo^+ \\to \\delta - \\mo^+$ defined by $\\phi(\\alpha) = \\delta - \\alpha$ is an isomorphism of posets between $(\\mo^+, \\leq)$ and the opposite poset $(\\delta - \\mo^+, \\leq)^{op}$: for any $\\alpha, \\beta \\in \\mo^+$, we have $\\alpha \\leq \\beta$ if and only of  $\\delta - \\beta \\leq \\delta - \\alpha$. Moreover all covering relations in $(\\mo^+,\\leq)$ and in $(\\delta - \\mo^+, \\leq)$ are short.\n\\end{Theorem}", "prop:PrefDom": "\\begin{Proposition}\\label{prop:PrefDom} Let $\\beta \\in \\Phi^+$ and let $t=s_\\beta$ be the corresponding reflection. For any $t$-prefix $p_t$ we have:  \n\\begin{eqnarray*}\n\\dom(\\beta) & = & \\{ \\alpha\\in \\Phi(p_t) \\ \\mid\\  B(\\alpha,\\beta)\\geq 1\\}\\\\\n&=&\\{\\beta\\}\\sqcup \\{ \\alpha_s\\in \\Phi(p_t) \\ \\mid\\  |\\langle s,t\\rangle| =\\infty\\}.\n\\end{eqnarray*}\nIn particular, $\\dep_\\infty (\\beta)=|\\{ \\alpha_s\\in \\Phi(p_t) \\ \\mid\\ |\\langle s,t\\rangle| =\\infty\\}|$.\n\\end{Proposition}", "se:reflectionprefix": "\\begin{proof} Since $\\pi_m(w)\\leq_R w$, we have $\\Sigma_m(\\pi_m(w))\\subseteq \\Sigma_m(w)$. The converse follows from  \\cite[Theorem 6.4]{DFHM24} in which the authors prove that the set \n$$\n\\{g\\in W\\mid \\Sigma_m(g)=\\Sigma_m(w)\\}\n$$\nhas a unique element of minimal length $x\\in L_m$ and that $x\\leq_R w$. So $\\Sigma_m(w)=\\Sigma_m(x)$. But $x\\leq_R\\pi_m(x)$ by definition. Therefore  $ \\Sigma_m(w)=\\Sigma_m(x)\\subseteq \\Sigma_m(\\pi_m(w))$.\n\\end{proof}\n\n\\section{Reflection-prefixes and palindromic reduced words of reflections}\\label{se:reflectionprefix}\n\nIn this section, we  first define and state properties of {\\em reflection-prefixes} in relation to the root poset, to the dominance order and to the canonical generators of maximal dihedral subgroups.  Then we produce a family of automata built from the finite Garside shadows $L_m$ ($m\\in\\mathbb N)$  that recognize the language of reflection-prefixes. \n\n\\subsection{Reflection-prefixes}\\label{ss:reflectionprefixes}\n\n Let $t\\in T$ be a reflection of $(W,S)$. It is well known that $\\ell(t)=2k+1$ for some $k\\in\\mathbb N$ and that  if $t=s_1\\dots s_k s_{k+1} s_{k+2} \\dots s_{2k+1}$ is a reduced word for $t$, the word $s_1 \\dots s_k s_{k+1} s_k\\dots s_1$ is a {\\em palindromic reduced word} for $t$; see for instance \\cite[Proposition~2.3]{DFHM24}.  \n\n The following proposition shows in particular that $t$ is uniquely determined by the prefix $s_1\\cdots s_ks_{k+1}$ of any of its palindromic reduced words. \n\n \\begin{Proposition}\\label{prop:Pref1} Let $t\\in T$ with corresponding positive root $\\alpha_t$. Let  $w\\in W$ and assume there is $r\\in D_R(w)$ such that $t=wrw^{-1}$ and $\\ell(t)=2\\ell(w)-1$. Then $D_R(w)=\\{r\\}$. Moreover: \n \\begin{enumerate}\n\n\\item if $w=s_1\\cdots s_k s_{k+1}$ is a reduced word for $w$, then $s_{k+1}=r$ and $t=s_1\\cdots s_k r s_k\\cdots s_1$ is a palindromic reduced word for $t$;\n\\item $\\alpha_t=w(\\alpha_r)= s_1\\cdots s_k(\\alpha_r)$ and  $\\dep(\\alpha_t)=k=\\ell(w)-1$.\n\\end{enumerate}\n\\end{Proposition}\n\\begin{proof}   Let $w=s_1\\cdots s_k r$ be a reduced word.  We show that $D_R(w)=\\{r\\}$. By contradiction, assume that $|D_R(w)|\\geq 2$. So there exists  a $s\\in D_R(w)$ with $s\\not = r$; set $I=\\{s,r\\}\\subseteq D_R(w)$.  By Lemma~\\ref{lem:Descents},  we know that $W_I$ is finite and  there is $u\\in X_I$ such that $w=u w_{\\circ,I}$ is a reduced product. So $\\ell(w)=\\ell(u)+m$, where $m=m_{sr}\\geq 2$. Now, observe that  $w_{\\circ,I}rw_{\\circ,I}\\in I$, since the longest element of a finite Coxeter system acts by conjugation on the set of simple reflections. Therefore \n$$\nt= s_1\\cdots s_k s_{k+1} s_k\\cdots s_1 =s_1\\cdots s_k r s_k\\cdots s_1=w r w^{-1}=uw_{\\circ,I}rw_{\\circ,I}u^{-1},\n$$\nimplying that $\\ell(t)\\leq 2\\ell(u)+1=2\\ell(w)-2m+1<2\\ell(w)-1=\\ell(t)$, a contradiction. The remaining claims follow from the uniqueness of $r$, the definition of the depth of roots and from the equalities $\\ell(t)=2\\dep(\\alpha_t)+1=2\\ell(w)-1$.\n\\end{proof}", "prop:PrefRoot": "\\begin{Proposition}\\label{prop:PrefRoot} Let $t\\in T$ and $\\alpha_t\\in \\Phi^+$ be the corresponding positive root. \n\\begin{enumerate}\n\\item Let  $\\alpha_{r}\\lhd s_k(\\alpha_{r})\\lhd\\dots\\lhd s_1\\cdots s_{k-1} s_k(\\alpha_r)=\\alpha_t$ be a saturated chain in $(\\Phi^+,\\leq)$, where $r\\in S$. Then $p_t=s_1\\dots s_kr$ is a reduced word for some $t$-prefix $p_t$. \n\n\\item Let $p_t$ be a  $t$-prefix and fix a reduced word $p_t=s_1\\cdots s_k r $, with $r$ the unique right-descent. Then $\\alpha_{r}\\lhd s_k(\\alpha_{r})\\lhd\\cdots\\lhd s_1\\cdots s_{k-1} s_k(\\alpha_r)=\\alpha_t$ is a saturated chain in $(\\Phi^+,\\leq)$.\n\\end{enumerate}\n\\end{Proposition}", "th:main": "\\begin{Theorem}\\label{th:main}\n    The generating function of the depth of positive roots is of the form\n    $$\\Phi^+(q)=\\frac{P(q)}{1-q^M}$$\n    where $M=\\lcm(M_\\mo+1\\mid \\mo\\ \\mbox{ $W_0$-orbit})$ and $P$ is palindromic.\n\\end{Theorem}", "th:iso2": "\\begin{Theorem}\\label{th:iso2} Let $\\mo$ be a $W_0$-orbit. There is a poset isomorphism $$(\\Sigma_{\\mo,k}, \\leq) \\cong (\\Sigma_{\\mo}, \\leq)$$ for all $k\\in\\mathbb{N}$. Moreover, all covering in $\\Sigma_{\\mo,k}$ are short.\n\\end{Theorem}", "se:openproblems": "\\label{se:openproblems}\n\nWe propose here some natural problems arising from this article.\n\n\\begin{Problem}\n    Is the language of palindromic reduced word (or lexicographically ordered palindromic red", "prop:algo": "\\begin{Theorem}\\label{prop:algo} Let $r,t\\in T$ with $r\\not = t$ and let $W'=\\langle r,t\\rangle$ be the dihedral reflection subgroup generated by $r$ and $t$. Let $p_r$ be a $r$-prefix and $p_t$ be a $t$-prefix.\n\\begin{enumerate}\n\\item If $|\\Phi_{W'}(p_r)|=|\\Phi_{W'}(p_t)|=1$, then:\n$$\n\\Delta_{W'}=\\{\\alpha_r,\\alpha_t\\}\\quad\\textrm{and}\\quad\\chi(W')=\\{r,t\\}.\n$$\n\\item If   $|\\Phi_{W'}(p_r)|>1$ (resp. $|\\Phi_{W'}(p_t)|>1$), then:\n $$\n \\Delta_{W'}=\\{\\alpha_1,\\alpha_2\\}\\quad\\textrm{and}\\quad\\chi(W')=\\{s_1,s_2\\},\n $$\n where $\\alpha_1$ is the unique root with the smallest depth in $\\Phi(p_r)\\cap \\Phi_{W'}$ (resp. $\\Phi(p_t)\\cap \\Phi_{W'}$) and $s_1(\\alpha_2)$ is the unique root with smallest depth in $\\Phi(p_r)\\cap \\Phi_{W'}\\setminus\\{\\alpha_1\\}$ (resp. $\\Phi(p_t)\\cap \\Phi_{W'}\\setminus\\{\\alpha_1\\}$). \\qed\n\\end{enumerate}\n \\end{Theorem}", "cor:Refregular": "\\begin{Corollary}\\label{cor:Refregular}\nThe language $\\Pref_T$ is regular and the generating function \n\\begin{align*}\n\\sum_{u\\in \\Pref_T} q^{|u|}& =\\sum_{t\\in T} |\\{s_1 \\cdots s_k s_{k+1} s_k\\cdots s_1 \\in \\Pref_T \\mid t=s_1 \\cdots s_k s_{k+1}s_k\\cdots s_1\\}| \\, q^{\\ell(t)}\\\\\n&=\\sum_{k\\in\\mathbb{N}} a_k q^k\n\\end{align*}\nis rational, where $a_k$ is the number of saturated chains of length $k$ in the root poset of $W$. In particular, $\\RPal(q)=q\\Pref_T(q^2)$ is rational.\n\\end{Corollary}", "ss:reflectionprefixes": "\\begin{proof} Since $\\pi_m(w)\\leq_R w$, we have $\\Sigma_m(\\pi_m(w))\\subseteq \\Sigma_m(w)$. The converse follows from  \\cite[Theorem 6.4]{DFHM24} in which the authors prove that the set \n$$\n\\{g\\in W\\mid \\Sigma_m(g)=\\Sigma_m(w)\\}\n$$\nhas a unique element of minimal length $x\\in L_m$ and that $x\\leq_R w$. So $\\Sigma_m(w)=\\Sigma_m(x)$. But $x\\leq_R\\pi_m(x)$ by definition. Therefore  $ \\Sigma_m(w)=\\Sigma_m(x)\\subseteq \\Sigma_m(\\pi_m(w))$.\n\\end{proof}\n\n\\section{Reflection-prefixes and palindromic reduced words of reflections}\\label{se:reflectionprefix}\n\nIn this section, we  first define and state properties of {\\em reflection-prefixes} in relation to the root poset, to the dominance order and to the canonical generators of maximal dihedral subgroups.  Then we produce a family of automata built from the finite Garside shadows $L_m$ ($m\\in\\mathbb N)$  that recognize the language of reflection-prefixes. \n\n\\subsection{Reflection-prefixes}\\label{ss:reflectionprefixes}\n\n Let $t\\in T$ be a reflection of $(W,S)$. It is well known that $\\ell(t)=2k+1$ for some $k\\in\\mathbb N$ and that  if $t=s_1\\dots s_k s_{k+1} s_{k+2} \\dots s_{2k+1}$ is a reduced word for $t$, the word $s_1 \\dots s_k s_{k+1} s_k\\dots s_1$ is a {\\em palindromic reduced word} for $t$; see for instance \\cite[Proposition~2.3]{DFHM24}.  \n\n The following proposition shows in particular that $t$ is uniquely determined by the prefix $s_1\\cdots s_ks_{k+1}$ of any of its palindromic reduced words. \n\n \\begin{Proposition}\\label{prop:Pref1} Let $t\\in T$ with corresponding positive root $\\alpha_t$. Let  $w\\in W$ and assume there is $r\\in D_R(w)$ such that $t=wrw^{-1}$ and $\\ell(t)=2\\ell(w)-1$. Then $D_R(w)=\\{r\\}$. Moreover: \n \\begin{enumerate}\n\n\\item if $w=s_1\\cdots s_k s_{k+1}$ is a reduced word for $w$, then $s_{k+1}=r$ and $t=s_1\\cdots s_k r s_k\\cdots s_1$ is a palindromic reduced word for $t$;\n\\item $\\alpha_t=w(\\alpha_r)= s_1\\cdots s_k(\\alpha_r)$ and  $\\dep(\\alpha_t)=k=\\ell(w)-1$.\n\\end{enumerate}\n\\end{Proposition}\n\\begin{proof}   Let $w=s_1\\cdots s_k r$ be a reduced word.  We show that $D_R(w)=\\{r\\}$. By contradiction, assume that $|D_R(w)|\\geq 2$. So there exists  a $s\\in D_R(w)$ with $s\\not = r$; set $I=\\{s,r\\}\\subseteq D_R(w)$.  By Lemma~\\ref{lem:Descents},  we know that $W_I$ is finite and  there is $u\\in X_I$ such that $w=u w_{\\circ,I}$ is a reduced product. So $\\ell(w)=\\ell(u)+m$, where $m=m_{sr}\\geq 2$. Now, observe that  $w_{\\circ,I}rw_{\\circ,I}\\in I$, since the longest element of a finite Coxeter system acts by conjugation on the set of simple reflections. Therefore \n$$\nt= s_1\\cdots s_k s_{k+1} s_k\\cdots s_1 =s_1\\cdots s_k r s_k\\cdots s_1=w r w^{-1}=uw_{\\circ,I}rw_{\\circ,I}u^{-1},\n$$\nimplying that $\\ell(t)\\leq 2\\ell(u)+1=2\\ell(w)-2m+1<2\\ell(w)-1=\\ell(t)$, a contradiction. The remaining claims follow from the uniqueness of $r$, the definition of the depth of roots and from the equalities $\\ell(t)=2\\dep(\\alpha_t)+1=2\\ell(w)-1$.\n\\end{proof}"}, "pre_theorem_intro_text_len": 6807, "pre_theorem_intro_text": "Let $(W,S)$ be a Coxeter system, with $S$ finite. We denote by $S^*$ the free monoid on the alphabet $S$. To distinguish between a word and its corresponding group element, we denote a word in $S^*$ using bold letters, $\\bold w =  s_1 \\cdots  s_k$, while the resulting product in $W$ is denoted by $w=s_1\\cdots s_k$. Let $w\\in W$, a word $\\bold w= s_1 \\cdots  s_k\\in S^*$ is a {\\em reduced word for $w$} if  $w=s_1\\cdots s_k$  and $k$ is minimal for this property; in this case the {\\em length} of $w$ is $\\ell(w)=k$. The identity $e\\in W$ is represented by the empty word, and $\\ell(e)=0$. We denote the set of all reduced words of $w\\in W$ by $\\Red(w)$, and the set of all reduced words in $W$ by $\\Red=\\bigsqcup_{w\\in W} \\Red(w)$, where $\\sqcup$ denotes the disjoint union.\nFinally, for any subset $A\\subseteq W$, the {\\em Poincar\\'e series of $A$} is the formal power series\n$A(q):=\\sum_{w\\in A} q^{\\ell(w)}.$\n\\smallskip\n\nIn this article, we study two combinatorial problems regarding the {\\em set of reflections} $T:=\\bigcup_{w\\in W} wSw^{-1}$ of a Coxeter group $W$:\n\\begin{enumerate}\n    \\item Is the language of {\\em palindromic reduced words (for the reflections)} regular?\n    \\item Are there explicit and elegant formulas for the Poincar\\'e series of the set $T$ of reflections ?\n\\end{enumerate}\nOur main contributions to those questions are the following: \n\\begin{itemize}\n\\item We introduce the notion of  {\\em reflection-prefixes}, a class of elements in $W$ arising naturally from  palindromic reduced words of reflections, and study their properties in relation to the {\\em root poset}, the {\\em dominance order on roots}  and {\\em dihedral reflection subgroups}.\n\\item For any Coxeter system, we show that the {\\em language of reduced words for reflection-prefixes}, $\\Pref_T$, is regular. This is achieved using the family of $m$-canonical automata associated with $m$-Shi arrangements (see~\\cite[\\S3.4]{HNW} and \\cite{DFHM24}), which provide a family of finite deterministic automata recognizing $\\Pref_T$. As a consequence, we show that the generating function  of palindromic reduced words is rational.\n\\item In the case of affine Coxeter groups, we derive a simple expression for the Poincar\\'e series of the set of reflections in terms of rational fractions. This formula is obtained from some symmetries of the Hasse diagram of the root poset.\n\\end{itemize}\n\nWe discuss now some history and motivations that lead to this article. \nIf $W$ is finite, $A(q)$ is clearly a polynomial for any $A\\subseteq W$. A natural question in combinatorics of infinite Coxeter systems is to classify for which subsets $A$ the Poincar\\'e series $A(q)$ is {\\em rational}, that is, can be written as a ratio of two polynomials in $q$. It is well-known that $W(q)$ is rational, see for instance~\\cite[Corollary~7.1.8]{BB}. Furthermore, an explicit recursive formula for $W(q)$ in term of standard parabolic subgroups is provided in~\\cite[Proposition~7.1.7]{BB}. \n\nA further natural direction is the study of generating functions of words in $S^*$ in relation to the Coxeter system $(W,S)$.  More precisely, given the canonical projection $\\pi:S^* \\rightarrow W$ sending a word $\\bold w= s_1 \\cdots  s_k$ to the element $w=s_1\\cdots s_k$,\nit is natural to  consider the {\\em ‘‘lifted'' Poincar\\'e series} of a subset $B\\subseteq S^*$ relative to $(W,S)$:\n$$\nB(q)=\\sum_{\\bold w\\in B} q^{\\ell(w)},\n$$\nwhere the notation ${\\ell(w)}$ is understood to be $\\ell(\\pi({\\bf w}))$. \n\nA longstanding open problem in combinatorics of Coxeter groups is the enumeration of $|\\Red(w)|$; for a detailed discussion see~\\cite[p.123]{BB}, and for various partial results, see~\\cite{Eriksson,Hart}.  However, if we consider the set of all elements of a given length, we know that the numbers $r_k$ of reduced words of length $k\\in \\mathbb N$ is enumerated by  the following Poincar\\'e series\n$$\n\\Red(q)=\\sum_{\\bold w\\in \\Red} q^{\\ell(w)}=\\sum_{k\\in\\mathbb N} r_kq^k,\n$$\nwhich is known to be rational. The proof follows from the existence of a finite deterministic automaton that recognizes the language $\\Red$; see for instance~\\cite[Theorem~4.9.1]{BB} for more details. \n\n\\smallskip\n\nAs reported by Brenti~\\cite{B1}, Stembridge proposed the following problem during an open problem session at the Mathematical Sciences Research Institute at Berkeley in 1997: Is it true that the Poincar\\'e series\n$$\nT(q)=\\sum_{t\\in T} q^{\\ell(t)}\\quad \\textrm{is rational?}\n$$ \n\n\\textbf{}Since reflections are known to admit palindromic reduced words, a natural approach to this question is to construct a finite automaton that recognizes exactly one palindromic reduced word for each reflection. The existence of such a regular language would imply that the associated generating function is rational. However, the property of being regular is not generally preserved under the palindromic constraint. If a language is regular, it is well-known that the sublanguage of its palindromic words is context-free but not necessarily regular; for instance if $S=\\{a,b\\}$ then $S^*$ is regular but a standard application of the {\\em pumping lemma} shows that the sublanguage of its palindroms is not.  Indeed, the regularity of $\\operatorname{Pal}$, {\\em the language of all palindromic reduced words}, which are necessarily reduced words for reflections, remains an open question. In a recent article, Mili\\'cevi\\'c ~\\cite{Mili} provides palindromic reduced words for all reflections in finite Weyl groups, but highlights the challenge of finding a general algorithm applicable to any Coxeter group. To overcome these challenges, we shift to the language of reflection-prefixes which we define below. \n\n\\subsection*{The language of reflection-prefixes}  \n\nWe recall that the {\\em (right) weak order} $(W,\\leq_R)$ is the poset defined by $u\\leq_R w$ if there exists a reduced word for $u\\in W$ that is a prefix for a reduced word for $w\\in W$. Equivalently, $u\\leq_R w$ if and only if $\\ell(w)=\\ell(u^{-1}w)+\\ell(u)$.\n\nIt is well-known that every reflection has an odd length. Let $t\\in T$ be a reflection of length $\\ell(t)=2k+1$, we define a {\\em $t$-prefix} to be any element $p_t\\in W$ such that $p_t\\leq_R t$ and $\\ell(p_t)=k+1$.\n  It turns out that $ s_1\\cdots s_k s_{k+1}  s_{k}\\cdots  s_1$ is a palindromic reduced word for $t$  if and only if $s_1\\cdots  s_k s_{k+1}$ is a reduced word for a $t$-prefix; see \\S\\ref{ss:reflectionprefixes}. \n\nReflection-prefixes turn out to be a useful combinatorial framework to study the root poset (Proposition~\\ref{prop:PrefRoot}), the dominance order (Proposition~\\ref{prop:PrefDom}) and to find the canonical generators of dihedral reflection subgroups (Theorem~\\ref{prop:algo}).  Our main results concerning reflection-prefixes and their associated languages are summarized in the following theorem.", "context": "Let $(W,S)$ be a Coxeter system, with $S$ finite. We denote by $S^*$ the free monoid on the alphabet $S$. To distinguish between a word and its corresponding group element, we denote a word in $S^*$ using bold letters, $\\bold w = s_1 \\cdots s_k$, while the resulting product in $W$ is denoted by $w=s_1\\cdots s_k$. Let $w\\in W$, a word $\\bold w= s_1 \\cdots s_k\\in S^*$ is a {\\em reduced word for $w$} if $w=s_1\\cdots s_k$ and $k$ is minimal for this property; in this case the {\\em length} of $w$ is $\\ell(w)=k$. The identity $e\\in W$ is represented by the empty word, and $\\ell(e)=0$. We denote the set of all reduced words of $w\\in W$ by $\\Red(w)$, and the set of all reduced words in $W$ by $\\Red=\\bigsqcup_{w\\in W} \\Red(w)$, where $\\sqcup$ denotes the disjoint union.\nFinally, for any subset $A\\subseteq W$, the {\\em Poincar\\'e series of $A$} is the formal power series\n$A(q):=\\sum_{w\\in A} q^{\\ell(w)}.$\n\\smallskip\n\nIn this article, we study two combinatorial problems regarding the {\\em set of reflections} $T:=\\bigcup_{w\\in W} wSw^{-1}$ of a Coxeter group $W$:\n\\begin{enumerate}\n \\item Is the language of {\\em palindromic reduced words (for the reflections)} regular?\n \\item Are there explicit and elegant formulas for the Poincar\\'e series of the set $T$ of reflections ?\n\\end{enumerate}\nOur main contributions to those questions are the following: \n\\begin{itemize}\n\\item We introduce the notion of {\\em reflection-prefixes}, a class of elements in $W$ arising naturally from palindromic reduced words of reflections, and study their properties in relation to the {\\em root poset}, the {\\em dominance order on roots} and {\\em dihedral reflection subgroups}.\n\\item For any Coxeter system, we show that the {\\em language of reduced words for reflection-prefixes}, $\\Pref_T$, is regular. This is achieved using the family of $m$-canonical automata associated with $m$-Shi arrangements (see~\\cite[\\S3.4]{HNW} and \\cite{DFHM24}), which provide a family of finite deterministic automata recognizing $\\Pref_T$. As a consequence, we show that the generating function of palindromic reduced words is rational.\n\\item In the case of affine Coxeter groups, we derive a simple expression for the Poincar\\'e series of the set of reflections in terms of rational fractions. This formula is obtained from some symmetries of the Hasse diagram of the root poset.\n\\end{itemize}\n\nA further natural direction is the study of generating functions of words in $S^*$ in relation to the Coxeter system $(W,S)$. More precisely, given the canonical projection $\\pi:S^* \\rightarrow W$ sending a word $\\bold w= s_1 \\cdots s_k$ to the element $w=s_1\\cdots s_k$,\nit is natural to consider the {\\em ‘‘lifted'' Poincar\\'e series} of a subset $B\\subseteq S^*$ relative to $(W,S)$:\n$$\nB(q)=\\sum_{\\bold w\\in B} q^{\\ell(w)},\n$$\nwhere the notation ${\\ell(w)}$ is understood to be $\\ell(\\pi({\\bf w}))$.\n\nA longstanding open problem in combinatorics of Coxeter groups is the enumeration of $|\\Red(w)|$; for a detailed discussion see~\\cite[p.123]{BB}, and for various partial results, see~\\cite{Eriksson,Hart}. However, if we consider the set of all elements of a given length, we know that the numbers $r_k$ of reduced words of length $k\\in \\mathbb N$ is enumerated by the following Poincar\\'e series\n$$\n\\Red(q)=\\sum_{\\bold w\\in \\Red} q^{\\ell(w)}=\\sum_{k\\in\\mathbb N} r_kq^k,\n$$\nwhich is known to be rational. The proof follows from the existence of a finite deterministic automaton that recognizes the language $\\Red$; see for instance~\\cite[Theorem~4.9.1]{BB} for more details.\n\nIt is well-known that every reflection has an odd length. Let $t\\in T$ be a reflection of length $\\ell(t)=2k+1$, we define a {\\em $t$-prefix} to be any element $p_t\\in W$ such that $p_t\\leq_R t$ and $\\ell(p_t)=k+1$.\n It turns out that $ s_1\\cdots s_k s_{k+1} s_{k}\\cdots s_1$ is a palindromic reduced word for $t$ if and only if $s_1\\cdots s_k s_{k+1}$ is a reduced word for a $t$-prefix; see \\S\\ref{ss:reflectionprefixes}.\n\nReflection-prefixes turn out to be a useful combinatorial framework to study the root poset (Proposition~\\ref{prop:PrefRoot}), the dominance order (Proposition~\\ref{prop:PrefDom}) and to find the canonical generators of dihedral reflection subgroups (Theorem~\\ref{prop:algo}). Our main results concerning reflection-prefixes and their associated languages are summarized in the following theorem.", "full_context": "Let $(W,S)$ be a Coxeter system, with $S$ finite. We denote by $S^*$ the free monoid on the alphabet $S$. To distinguish between a word and its corresponding group element, we denote a word in $S^*$ using bold letters, $\\bold w = s_1 \\cdots s_k$, while the resulting product in $W$ is denoted by $w=s_1\\cdots s_k$. Let $w\\in W$, a word $\\bold w= s_1 \\cdots s_k\\in S^*$ is a {\\em reduced word for $w$} if $w=s_1\\cdots s_k$ and $k$ is minimal for this property; in this case the {\\em length} of $w$ is $\\ell(w)=k$. The identity $e\\in W$ is represented by the empty word, and $\\ell(e)=0$. We denote the set of all reduced words of $w\\in W$ by $\\Red(w)$, and the set of all reduced words in $W$ by $\\Red=\\bigsqcup_{w\\in W} \\Red(w)$, where $\\sqcup$ denotes the disjoint union.\nFinally, for any subset $A\\subseteq W$, the {\\em Poincar\\'e series of $A$} is the formal power series\n$A(q):=\\sum_{w\\in A} q^{\\ell(w)}.$\n\\smallskip\n\nIn this article, we study two combinatorial problems regarding the {\\em set of reflections} $T:=\\bigcup_{w\\in W} wSw^{-1}$ of a Coxeter group $W$:\n\\begin{enumerate}\n \\item Is the language of {\\em palindromic reduced words (for the reflections)} regular?\n \\item Are there explicit and elegant formulas for the Poincar\\'e series of the set $T$ of reflections ?\n\\end{enumerate}\nOur main contributions to those questions are the following: \n\\begin{itemize}\n\\item We introduce the notion of {\\em reflection-prefixes}, a class of elements in $W$ arising naturally from palindromic reduced words of reflections, and study their properties in relation to the {\\em root poset}, the {\\em dominance order on roots} and {\\em dihedral reflection subgroups}.\n\\item For any Coxeter system, we show that the {\\em language of reduced words for reflection-prefixes}, $\\Pref_T$, is regular. This is achieved using the family of $m$-canonical automata associated with $m$-Shi arrangements (see~\\cite[\\S3.4]{HNW} and \\cite{DFHM24}), which provide a family of finite deterministic automata recognizing $\\Pref_T$. As a consequence, we show that the generating function of palindromic reduced words is rational.\n\\item In the case of affine Coxeter groups, we derive a simple expression for the Poincar\\'e series of the set of reflections in terms of rational fractions. This formula is obtained from some symmetries of the Hasse diagram of the root poset.\n\\end{itemize}\n\nA further natural direction is the study of generating functions of words in $S^*$ in relation to the Coxeter system $(W,S)$. More precisely, given the canonical projection $\\pi:S^* \\rightarrow W$ sending a word $\\bold w= s_1 \\cdots s_k$ to the element $w=s_1\\cdots s_k$,\nit is natural to consider the {\\em ‘‘lifted'' Poincar\\'e series} of a subset $B\\subseteq S^*$ relative to $(W,S)$:\n$$\nB(q)=\\sum_{\\bold w\\in B} q^{\\ell(w)},\n$$\nwhere the notation ${\\ell(w)}$ is understood to be $\\ell(\\pi({\\bf w}))$.\n\nA longstanding open problem in combinatorics of Coxeter groups is the enumeration of $|\\Red(w)|$; for a detailed discussion see~\\cite[p.123]{BB}, and for various partial results, see~\\cite{Eriksson,Hart}. However, if we consider the set of all elements of a given length, we know that the numbers $r_k$ of reduced words of length $k\\in \\mathbb N$ is enumerated by the following Poincar\\'e series\n$$\n\\Red(q)=\\sum_{\\bold w\\in \\Red} q^{\\ell(w)}=\\sum_{k\\in\\mathbb N} r_kq^k,\n$$\nwhich is known to be rational. The proof follows from the existence of a finite deterministic automaton that recognizes the language $\\Red$; see for instance~\\cite[Theorem~4.9.1]{BB} for more details.\n\nIt is well-known that every reflection has an odd length. Let $t\\in T$ be a reflection of length $\\ell(t)=2k+1$, we define a {\\em $t$-prefix} to be any element $p_t\\in W$ such that $p_t\\leq_R t$ and $\\ell(p_t)=k+1$.\n It turns out that $ s_1\\cdots s_k s_{k+1} s_{k}\\cdots s_1$ is a palindromic reduced word for $t$ if and only if $s_1\\cdots s_k s_{k+1}$ is a reduced word for a $t$-prefix; see \\S\\ref{ss:reflectionprefixes}.\n\nReflection-prefixes turn out to be a useful combinatorial framework to study the root poset (Proposition~\\ref{prop:PrefRoot}), the dominance order (Proposition~\\ref{prop:PrefDom}) and to find the canonical generators of dihedral reflection subgroups (Theorem~\\ref{prop:algo}). Our main results concerning reflection-prefixes and their associated languages are summarized in the following theorem.\n\nLet $(W,S)$ be a Coxeter system, with $S$ finite. We denote by $S^*$ the free monoid on the alphabet $S$. To distinguish between a word and its corresponding group element, we denote a word in $S^*$ using bold letters, $\\bold w = s_1 \\cdots s_k$, while the resulting product in $W$ is denoted by $w=s_1\\cdots s_k$. Let $w\\in W$, a word $\\bold w= s_1 \\cdots s_k\\in S^*$ is a {\\em reduced word for $w$} if $w=s_1\\cdots s_k$ and $k$ is minimal for this property; in this case the {\\em length} of $w$ is $\\ell(w)=k$. The identity $e\\in W$ is represented by the empty word, and $\\ell(e)=0$. We denote the set of all reduced words of $w\\in W$ by $\\Red(w)$, and the set of all reduced words in $W$ by $\\Red=\\bigsqcup_{w\\in W} \\Red(w)$, where $\\sqcup$ denotes the disjoint union.\nFinally, for any subset $A\\subseteq W$, the {\\em Poincar\\'e series of $A$} is the formal power series\n$A(q):=\\sum_{w\\in A} q^{\\ell(w)}.$\n\\smallskip\n\nIn this article, we study two combinatorial problems regarding the {\\em set of reflections} $T:=\\bigcup_{w\\in W} wSw^{-1}$ of a Coxeter group $W$:\n\\begin{enumerate}\n \\item Is the language of {\\em palindromic reduced words (for the reflections)} regular?\n \\item Are there explicit and elegant formulas for the Poincar\\'e series of the set $T$ of reflections ?\n\\end{enumerate}\nOur main contributions to those questions are the following: \n\\begin{itemize}\n\\item We introduce the notion of {\\em reflection-prefixes}, a class of elements in $W$ arising naturally from palindromic reduced words of reflections, and study their properties in relation to the {\\em root poset}, the {\\em dominance order on roots} and {\\em dihedral reflection subgroups}.\n\\item For any Coxeter system, we show that the {\\em language of reduced words for reflection-prefixes}, $\\Pref_T$, is regular. This is achieved using the family of $m$-canonical automata associated with $m$-Shi arrangements (see~\\cite[\\S3.4]{HNW} and \\cite{DFHM24}), which provide a family of finite deterministic automata recognizing $\\Pref_T$. As a consequence, we show that the generating function of palindromic reduced words is rational.\n\\item In the case of affine Coxeter groups, we derive a simple expression for the Poincar\\'e series of the set of reflections in terms of rational fractions. This formula is obtained from some symmetries of the Hasse diagram of the root poset.\n\\end{itemize}\n\nA longstanding open problem in combinatorics of Coxeter groups is the enumeration of $|\\Red(w)|$; for a detailed discussion see~\\cite[p.123]{BB}, and for various partial results, see~\\cite{Eriksson,Hart}. However, if we consider the set of all elements of a given length, we know that the numbers $r_k$ of reduced words of length $k\\in \\mathbb N$ is enumerated by the following Poincar\\'e series\n$$\n\\Red(q)=\\sum_{\\bold w\\in \\Red} q^{\\ell(w)}=\\sum_{k\\in\\mathbb N} r_kq^k,\n$$\nwhich is known to be rational. The proof follows from the existence of a finite deterministic automaton that recognizes the language $\\Red$; see for instance~\\cite[Theorem~4.9.1]{BB} for more details.\n\nReflection-prefixes turn out to be a useful combinatorial framework to study the root poset (Proposition~\\ref{prop:PrefRoot}), the dominance order (Proposition~\\ref{prop:PrefDom}) and to find the canonical generators of dihedral reflection subgroups (Theorem~\\ref{prop:algo}). Our main results concerning reflection-prefixes and their associated languages are summarized in the following theorem.\n\nThe first assertion, the regularity of $\\Pref_T$, is established in Corollary~\\ref{cor:Refregular}. The proof relies on a construction of automata derived from the family of Garside shadow automata associated with $m$-Shi arrangements $(m \\in \\mathbb{N})$; for further details, see~\\cite{HNW, DFHM24}. The second assertion follows from Proposition~\\ref{prop:Pref1}, while the third and fourth are consequence of the first two.\n\n\\item if $w=s_1\\cdots s_k s_{k+1}$ is a reduced word for $w$, then $s_{k+1}=r$ and $t=s_1\\cdots s_k r s_k\\cdots s_1$ is a palindromic reduced word for $t$;\n\\item $\\alpha_t=w(\\alpha_r)= s_1\\cdots s_k(\\alpha_r)$ and $\\dep(\\alpha_t)=k=\\ell(w)-1$.\n\\end{enumerate}\n\\end{Proposition}\n\\begin{proof} Let $w=s_1\\cdots s_k r$ be a reduced word. We show that $D_R(w)=\\{r\\}$. By contradiction, assume that $|D_R(w)|\\geq 2$. So there exists a $s\\in D_R(w)$ with $s\\not = r$; set $I=\\{s,r\\}\\subseteq D_R(w)$. By Lemma~\\ref{lem:Descents}, we know that $W_I$ is finite and there is $u\\in X_I$ such that $w=u w_{\\circ,I}$ is a reduced product. So $\\ell(w)=\\ell(u)+m$, where $m=m_{sr}\\geq 2$. Now, observe that $w_{\\circ,I}rw_{\\circ,I}\\in I$, since the longest element of a finite Coxeter system acts by conjugation on the set of simple reflections. Therefore \n$$\nt= s_1\\cdots s_k s_{k+1} s_k\\cdots s_1 =s_1\\cdots s_k r s_k\\cdots s_1=w r w^{-1}=uw_{\\circ,I}rw_{\\circ,I}u^{-1},\n$$\nimplying that $\\ell(t)\\leq 2\\ell(u)+1=2\\ell(w)-2m+1<2\\ell(w)-1=\\ell(t)$, a contradiction. The remaining claims follow from the uniqueness of $r$, the definition of the depth of roots and from the equalities $\\ell(t)=2\\dep(\\alpha_t)+1=2\\ell(w)-1$.\n\\end{proof}\n\n\\begin{Example}[Universal Coxeter system]\n Let $(U_n, S)$ be the Universal Coxeter system of rank $n$, that is $U_n=\\langle S\\mid s^2=e \\ \\mbox{for all $s \\in S$} \\rangle$. Every element has a unique reduced word, so the language of palindromic reduced words for $(U_n,S)$ is given by $\\RPal= \\{s_1\\cdots s_{i-1}s_is_{i-1}\\cdots s_1 \\mid s_j\\in S \\mbox{ and } s_j\\neq s_{j+1}\\}$. Furthermore, the language of reduced words for reflection-prefixes coincides with the set of all reduced words in $W$, that is, $\\Pref_T=\\Red$.\n\\end{Example}\n\n\\begin{Corollary}\\label{cor:Refregular}\nThe language $\\Pref_T$ is regular and the generating function \n\\begin{align*}\n\\sum_{u\\in \\Pref_T} q^{|u|}& =\\sum_{t\\in T} |\\{s_1 \\cdots s_k s_{k+1} s_k\\cdots s_1 \\in \\Pref_T \\mid t=s_1 \\cdots s_k s_{k+1}s_k\\cdots s_1\\}| \\, q^{\\ell(t)}\\\\\n&=\\sum_{k\\in\\mathbb{N}} a_k q^k\n\\end{align*}\nis rational, where $a_k$ is the number of saturated chains of length $k$ in the root poset of $W$. In particular, $\\RPal(q)=q\\Pref_T(q^2)$ is rational.\n\\end{Corollary}", "post_theorem_intro_text_len": 3614, "post_theorem_intro_text": "The first assertion, the regularity of $\\Pref_T$, is established in Corollary~\\ref{cor:Refregular}. The proof relies on a construction of automata derived from the family of Garside shadow automata associated with $m$-Shi arrangements $(m \\in \\mathbb{N})$; for further details, see~\\cite{HNW, DFHM24}. The second assertion follows from Proposition~\\ref{prop:Pref1}, while the third and fourth are consequence of the first two.\n\n\\subsection*{Closed formulas for $T(q)$ in affine Coxeter systems} Remarkably, a solution to Stembridge's problem was provided in 1999 by de~Man~\\cite{DM}. However, the author seemed unaware of Stembridge's question and the combinatorics community was not aware of the solution. This oversight may be partly due to the title of de~Man's article {\\em ``The generating function for the number of roots of a Coxeter group''}, which only address counting reflections by length at the end of the article in \\cite[\\S5.2]{DM}. His solution is based on the Brink--Howlett automaton~\\cite{BH}  that recognizes the language of {\\em lexicographically ordered reduced word}, that is, each element $w$ has a unique lexicographically reduced word in $\\Red(w)$.  \n\nMore precisely, in \\cite[Theorem 4.1]{DM}, de Man shows that the generating function  $\\Phi^+(q)$ of the depth of positive roots is a sum of rational functions that are obtained from a subset of states of the Brink-Howlett automaton and from the enumeration of the vertices of a family of graphs parameterized by $T$. So the generating function $\\Phi^+(q)$ is rational. Since $T(q)=q\\, \\Phi^+(q^2)$, the Poincar\\'e series $T(q)$  is rational as well. However, the Brink--Howlett automaton contains a lot of states, most of them ``dead states'', which make this automaton difficult to use in practice. For instance, in affine type $\\widetilde{A}_2$, the number of states of the Brink--Howlett automaton is bounded by $2^6$; de Man provides some example of $T(q)$ using a computer to handle his formula.  \n\n In this article, we give, in the case of affine Coxeter systems, an alternative solution to Stembridge's problem by providing a simple formula for  $T(q)$. \n\n\\begin{Theorem}\\label{thm:Main1} Let $(W,S)$ be an affine Coxeter system. Then \n$$\nT(q):=\\sum_{t\\in T}q^{\\ell(t)}=q\\frac{P(q^2)}{1-q^{2M}},\n$$ \nwhere $M\\in \\mathbb{N}$ and $P(q)$ is a palindromic polynomial.\n\\end{Theorem}\n\nThe proof of the above theorem relies on two fundamental symmetries of the root poset on $\\Phi^+$ (Theorems \\ref{th:iso1} and \\ref{th:iso2}). These properties yield an explicit formula for $M$ and $P(q)$ (Theorem \\ref{th:main})), from which the above result follows.\n\n\\subsection*{Plan of the article} The article is organized as follows. In~\\S\\ref{sec:root-poset}, we recall some needed facts about the combinatorics of words and roots and, in particular, about two partial orders on the root system of a Coxeter system: the root poset and the dominance order, which were introduced by Brink and Howlett~\\cite{BH} in their work to show that Coxeter systems are automatic. In~\\S\\ref{se:reflectionprefix}, we discuss reflection-prefix, palindromic reduced words for reflections and automata and, in \\S\\ref{se:affine},  we give the enumeration of reflections by length in affine types. Finally, we propose some open problems in \\S\\ref{se:openproblems}.\n\n\\subsection*{Acknowledgements}  CH warmly thanks Francesco Brenti, James Parkinson and Christophe Reutenauer for very instructive discussions. \nThe authors are particularly indebted to Matthew Dyer for bringing de Man’s article~\\cite{DM} to their attention during the early stages of this project.", "sketch": "For Theorem~\\ref{thm:PrefMain}: the first assertion (regularity of $\\Pref_T$) is proved in Corollary~\\ref{cor:Refregular}, and the proof \"relies on a construction of automata derived from the family of Garside shadow automata associated with $m$-Shi arrangements $(m\\in\\mathbb{N})$\" (with details in~\\cite{HNW, DFHM24}). The second assertion \"follows from Proposition~\\ref{prop:Pref1}\\.\" The third and fourth assertions are stated to be \"consequence of the first two.\"", "expanded_sketch": "For Theorem~\\ref{thm:PrefMain}: the first assertion (regularity of $\\Pref_T$) is proved in\n\\begin{Corollary}\\label{cor:Refregular}\nThe language $\\Pref_T$ is regular and the generating function \n\\begin{align*}\n\\sum_{u\\in \\Pref_T} q^{|u|}& =\\sum_{t\\in T} |\\{s_1 \\cdots s_k s_{k+1} s_k\\cdots s_1 \\in \\Pref_T \\mid t=s_1 \\cdots s_k s_{k+1}s_k\\cdots s_1\\}| \\, q^{\\ell(t)}\\\\\n&=\\sum_{k\\in\\mathbb{N}} a_k q^k\n\\end{align*}\nis rational, where $a_k$ is the number of saturated chains of length $k$ in the root poset of $W$. In particular, $\\RPal(q)=q\\Pref_T(q^2)$ is rational.\n\\end{Corollary}\nand the proof relies on a construction of automata derived from the family of Garside shadow automata associated with $m$-Shi arrangements $(m\\in\\mathbb{N})$ (with details in~\\cite{HNW, DFHM24}). The second assertion follows from\n\\begin{Proposition}\\label{prop:Pref1} Let $t\\in T$ with corresponding positive root $\\alpha_t$. Let  $w\\in W$ and assume there is $r\\in D_R(w)$ such that $t=wrw^{-1}$ and $\\ell(t)=2\\ell(w)-1$. Then $D_R(w)=\\{r\\}$. Moreover: \n \\begin{enumerate}\n\n\\item if $w=s_1\\cdots s_k s_{k+1}$ is a reduced word for $w$, then $s_{k+1}=r$ and $t=s_1\\cdots s_k r s_k\\cdots s_1$ is a palindromic reduced word for $t$;\n\\item $\\alpha_t=w(\\alpha_r)= s_1\\cdots s_k(\\alpha_r)$ and  $\\dep(\\alpha_t)=k=\\ell(w)-1$.\n\\end{enumerate}\n\\end{Proposition}\nThe third and fourth assertions are stated to be consequence of the first two.", "expanded_theorem": "\\label{thm:PrefMain} Let $(W,S)$ be a Coxeter system. \n\\begin{enumerate}\n\\item The language $\\Pref_T$ of reduced words for reflection-prefixes is regular.\n\\item The language of palindromic reduced words is:\n$$\n\\operatorname{Pal}=\\{s_1\\cdots  s_k s_{k+1}  s_{k}\\cdots  s_1 \\in S^*\\mid  s_1\\cdots  s_k s_{k+1}\\in\\Pref_T\\}.\n$$\n\\item The following Poincar\\'e series is rational:\n$$\n\\Pref_T(q)=\\sum_{\\bold w \\in \\Pref_T} q^{\\ell(w)}=\\sum_{k\\geq 0} \\pal_kq^k,\n$$\nwhere $\\pal_k$ is the number of palindromic reduced words of length $k$.\n\n\\item The Poincar\\'e series $\\operatorname{Pal}(q)=q\\Pref_T(q^2)$ is rational.\n\\end{enumerate}", "theorem_type": ["Universal", "Classification or Bijection"], "mcq": {"question": "Let (W,S) be a Coxeter system with S finite, let S* be the free monoid on S, and let π: S* → W be the canonical projection. A word s1⋯sk ∈ S* is reduced if k is minimal among words representing π(s1⋯sk), and then ℓ(π(s1⋯sk)) = k. Let T = ⋃_{w∈W} wSw^{-1} be the set of reflections. For u,v ∈ W, write u ≤_R v for the right weak order, i.e. ℓ(v) = ℓ(u) + ℓ(u^{-1}v). Since every reflection t ∈ T has odd length, if ℓ(t)=2k+1 then a t-prefix is an element p_t ∈ W such that p_t ≤_R t and ℓ(p_t)=k+1. Let Pref_T ⊆ S* be the set of reduced words of all reflection-prefixes, and let Pal be the language of palindromic reduced words for reflections. For B ⊆ S*, define B(q)=∑_{w∈B} q^{ℓ(π(w))}. Which statement holds for every such Coxeter system?", "correct_choice": {"label": "A", "text": "The language Pref_T is regular; moreover, the palindromic reduced words are exactly\nPal = {s1⋯sk s_{k+1} s_k⋯s1 ∈ S* | s1⋯sk s_{k+1} ∈ Pref_T}.\nIn addition, the Poincaré series\nPref_T(q) = ∑_{w∈Pref_T} q^{ℓ(π(w))} = ∑_{k≥0} pal_k q^k\nis rational, where pal_k is the number of palindromic reduced words of length k; consequently,\nPal(q) = q Pref_T(q^2)\nis rational."}, "choices": [{"label": "B", "text": "The language Pref_T is regular; moreover, the palindromic reduced words are exactly\nPal = {s1⋯sk s_{k+1} s_k⋯s1 ∈ S* | s1⋯sk ∈ Pref_T}.\nIn addition, the Poincaré series\nPref_T(q) = ∑_{w∈Pref_T} q^{ℓ(π(w))} = ∑_{k≥0} pal_k q^k\nis rational, where pal_k is the number of palindromic reduced words of length k; consequently,\nPal(q) = q Pref_T(q^2)\nis rational."}, {"label": "C", "text": "The language Pref_T is regular. Moreover, the Poincaré series\nPal(q)=\\sum_{w\\in \\mathrm{Pal}} q^{\\ell(\\pi(w))}\nis rational."}, {"label": "D", "text": "The language Pref_T is regular; moreover, the palindromic reduced words are exactly\nPal = {s1⋯sk s_{k+1} s_k⋯s1 ∈ S* | s1⋯sk s_{k+1} ∈ Pref_T}.\nIn addition, the Poincaré series\nPref_T(q) = ∑_{w∈Pref_T} q^{ℓ(π(w))} = ∑_{k≥0} pal_k q^k\nis rational, where pal_k is the number of palindromic reduced words of length k; consequently,\nPal(q) = Pref_T(q^2)\nis rational."}, {"label": "E", "text": "The language Pref_T is regular; moreover, every palindromic reduced word has the form\ns1⋯sk s_{k+1} s_k⋯s1 with s1⋯sk s_{k+1} ∈ Pref_T, and every such word lies in Pal.\nIn addition, there is a single finite deterministic automaton, independent of any auxiliary parameter, that recognizes Pref_T for every Coxeter system; consequently the series Pref_T(q) and Pal(q) are rational."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "trace_identity", "tampered_component": "full-prefix length k+1 in the characterization of palindromes", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped the exact characterization of Pal and the explicit identity Pal(q)=q Pref_T(q^2)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "trace_identity", "tampered_component": "odd-length shift encoded by Pal(q)=q Pref_T(q^2)", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "dependence on the family of m-Shi/Garside shadow automata rather than one parameter-free universal automaton", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem provides definitions and asks for the correct full statement, but it does not explicitly reveal the correct option. There is only a mild implicit cue in the phrase 'complete description,' which helps rule out the weaker true option."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: it asks for the exact characterization of palindromic reduced words and the associated regularity/generating-function conclusions. The task is very close to restating the target result."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish subtle variants (off-by-one prefix condition, missing factor of q, weaker one-way implication, extraneous uniform automaton claim). However, the item mainly tests recognition of the exact theorem rather than generating a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: boundary-length confusion, omission of a converse, generating-series substitution error, and an overstrong uniformity claim. They are distinct and nontrivial."}, "total_score": 5, "overall_assessment": "A solid theorem-recall MCQ with strong distractors and little answer leakage, but it is close to a direct restatement of the result and only moderately tests genuine generative reasoning."}}
{"id": "2602.16361v1", "paper_link": "http://arxiv.org/abs/2602.16361v1", "theorems_cnt": 2, "theorem": {"env_name": "Theorem", "content": "\\label{thm:PrefMain} Let $(W,S)$ be a Coxeter system. \n\\begin{enumerate}\n\\item The language $\\Pref_T$ of reduced words for reflection-prefixes is regular.\n\\item The language of palindromic reduced words is:\n$$\n\\operatorname{Pal}=\\{s_1\\cdots  s_k s_{k+1}  s_{k}\\cdots  s_1 \\in S^*\\mid  s_1\\cdots  s_k s_{k+1}\\in\\Pref_T\\}.\n$$\n\\item The following Poincar\\'e series is rational:\n$$\n\\Pref_T(q)=\\sum_{\\bold w \\in \\Pref_T} q^{\\ell(w)}=\\sum_{k\\geq 0} \\pal_kq^k,\n$$\nwhere $\\pal_k$ is the number of palindromic reduced words of length $k$.\n\n\\item The Poincar\\'e series $\\operatorname{Pal}(q)=q\\Pref_T(q^2)$ is rational.\n\\end{enumerate}", "start_pos": 10997, "end_pos": 11626, "label": "thm:PrefMain"}, "ref_dict": {"prop:Pref1": "\\begin{Proposition}\\label{prop:Pref1} Let $t\\in T$ with corresponding positive root $\\alpha_t$. Let  $w\\in W$ and assume there is $r\\in D_R(w)$ such that $t=wrw^{-1}$ and $\\ell(t)=2\\ell(w)-1$. Then $D_R(w)=\\{r\\}$. Moreover: \n \\begin{enumerate}\n\n\\item if $w=s_1\\cdots s_k s_{k+1}$ is a reduced word for $w$, then $s_{k+1}=r$ and $t=s_1\\cdots s_k r s_k\\cdots s_1$ is a palindromic reduced word for $t$;\n\\item $\\alpha_t=w(\\alpha_r)= s_1\\cdots s_k(\\alpha_r)$ and  $\\dep(\\alpha_t)=k=\\ell(w)-1$.\n\\end{enumerate}\n\\end{Proposition}", "se:affine": "\\begin{Corollary}\\label{cor:Refregular}\nThe language $\\Pref_T$ is regular and the generating function \n\\begin{align*}\n\\sum_{u\\in \\Pref_T} q^{|u|}& =\\sum_{t\\in T} |\\{s_1 \\cdots s_k s_{k+1} s_k\\cdots s_1 \\in \\Pref_T \\mid t=s_1 \\cdots s_k s_{k+1}s_k\\cdots s_1\\}| \\, q^{\\ell(t)}\\\\\n&=\\sum_{k\\in\\mathbb{N}} a_k q^k\n\\end{align*}\nis rational, where $a_k$ is the number of saturated chains of length $k$ in the root poset of $W$. In particular, $\\RPal(q)=q\\Pref_T(q^2)$ is rational.\n\\end{Corollary}\n\n\\section{Poincare series of affine Coxeter groups}\\label{se:affine}\nIn this section we provide a direct solution to Stembridge’s problem by providing an explicit description of the Poincaré series for affine Coxeter groups. This formula is based on a detailed description of the Hasse diagram of the associated root poset.\n\n\\subsection{Crystallographic root systems}\n\nWhen one works with an affine Coxeter system $(W,S)$, with underlying finite Weyl group $(W_0,S_0)$, it is customary to use its associated crystallographic root system in a positive semi-definite quadratic space $(V,B)$. In this case, the classical geometric representation (see \\S\\ref{ss:InversionSets}) is the unique geometric representation, and  the simple system $\\Delta$ is a basis of $V$. The crystallographic condition allows us to consider root systems with non-necessarily unitary roots. We briefly recall this construction below. \n\nLet $\\Phi_0$ be a reduced, irreducible, crystallographic root system for the finite\nWeyl group $W_0$, realized in a Euclidean space $(V_0,B_0)$ with corresponding set of simple roots $\\Delta_0=\\{\\alpha_s\\mid s\\in S_0\\}$ and positive roots $\\Phi_0^+$ (see for instance \\cite{bour,Hum}). Then for $s,t\\in S_0$, we have:\n\\[\nB_0(\\alpha_s, \\alpha_t) = \n\\begin{cases} \n-\\|\\alpha_s\\| \\|\\alpha_t\\| \\cos \\left(\\frac{\\pi}{m_{st}}\\right) & \\text{if } m_{st} < \\infty, \\\\\n-1 & \\text{if } m_{st} = \\infty.\n\\end{cases}\n\\]\nThe quadratic space $(V,B)$ is obtained as follows:  $V = V_0 \\oplus \\mathbb{R}\\delta$, and $B$ is the extension of  $B_0$ by setting the radical of $B$ to be equal to  $Q = \\mathbb{R}\\delta$. Let $\\omega$ denote the highest root of $\\Phi_0$; the crystallographic affine simple system is $\\Delta = \\Delta_0 \\cup \\{ \\delta - \\omega \\}$ with root system \n$\\Phi = \\{ \\alpha + k\\delta \\mid \\alpha \\in \\Phi_0,\\; k \\in \\mathbb{Z} \\}$.\nThe set of positive roots is:\n\\[\\Phi^+=\n\\{\\,  k\\delta + \\alpha,\\;  (k+1)\\delta -\\alpha \\mid \\alpha \\in \\Phi_0^+,\\; k \\in \\mathbb{N} \\,\\}=(\\mathbb{N}\\delta+\\Phi^+_0)\\sqcup (\\mathbb{N}^*\\delta-\\Phi^+_0).\n\\]\nIt follows that $\\Phi$ is a {\\em crystallographic (infinite) root system} for the affine Weyl group $(W,S)$, where $S=S_0 \\cup \\{s_{\\delta-\\omega}\\}$.\n\n\\subsection{The root poset and dominance order in the affine case} Structural results for unitary root systems, specifically those determined by the sign of the bilinear form $B$, generally extend to crystallographic systems. However, properties that depend on the specific numerical values of the form, such as dominance and long edges, must be precisely reformulated for crystallographic root systems; see~\\cite[Example 3.9]{DH} for more details. \n\nFrom the crystallographic root system, $\\Phi$, one obtains a {\\em unitary} root system in the sense of \\S\\ref{ss:InversionSets} by normalizing each root via the map $\\alpha\\mapsto \\frac{\\alpha}{\\|\\alpha\\|}$. Let $\\alpha,\\beta\\in\\Phi$, then:     \n$$\nB\\left(\\frac{\\alpha}{\\|\\alpha\\|},\\frac{\\beta}{\\|\\beta\\|}\\right) = \\frac{B(\\alpha,\\beta)}{\\|\\alpha\\|\\|\\beta\\|}.\n$$\nTherefore, the condition $B(\\alpha',\\beta')\\leq -1$ for unitary roots $\\alpha',\\beta'$, which appear in Definition~\\ref{def:long} or Proposition~\\ref{prop:inf-depth}, becomes:  $$B(\\alpha,\\beta)\\leq -\\|\\alpha\\|\\|\\beta\\|.$$ \nFor this reason Definition~\\ref{def:long} in the crystallographic setting becomes the following.\n\n\\begin{Definition}[Long and short edges in crystallographic root system] A covering $\\alpha\\lhd s(\\alpha)$ in the root poset $(\\Phi^+,\\leq)$ is \\textit{short} if $-\\|\\alpha\\| \\|\\alpha_s\\| < B(\\alpha, \\alpha_s)<0$. Otherwise we say that the covering is \\textit{long}.\n\\end{Definition}\n\nThe Hasse diagram of the root poset is studied in more details in the subsequent section. \n\n\\smallskip\nIn the affine case, the Hasse diagram of the dominance order is the union of $2|\\Phi_0^+|$ total ordered chains. Indeed, the only dominance relations in  $\\Phi$ are of the form\n$$\n k\\delta +\\alpha \\;\\leq_d\\;  l\\delta+\\alpha,\n\\quad\n\\text{or} \\quad \n (k+1)\\delta -\\alpha\\;\\leq_d\\;  (l+1)\\delta - \\alpha,\n$$\nwhere $k\\leq l$ in $\\mathbb N$ and $\\alpha\\in\\Phi_0^+$; see~\\cite[Example 3.9]{DH}. \n\nAn interesting consequence of these dominance relations is that for $\\alpha\\in\\Phi_0^+$ and $k\\in\\mathbb N$ we have:\n$$\n\\dep_\\infty(k\\delta+\\alpha)=k\\quad\\textrm{and}\\quad\\dep_\\infty((k+1)\\delta-\\alpha)=k,\n$$\nsee~\\cite[Example 3.9, Eq.($\\diamond$)]{DH}. In particular, the set of $k$-small roots (see Section \\ref{ss:mSmallRoots}) for $k\\in\\mathbb N$ is in the affine case:\n\\begin{equation}\n\\label{eq:mSmall}\n\\Sigma_k=\\left(k\\delta+\\Phi^+_0\\right)\\sqcup \\left((k+1)\\delta-\\Phi_0^+\\right).\n\\end{equation}\n\n\\begin{Example}\nLet $(W,S)$ be a Coxeter system of type $\\widetilde{A}_2$, \n\n\\begin{center}\n\\begin{tikzpicture}\n\t[scale=2,\n\t q/.style={teal,line join=round},\n\t racine/.style={blue},\n\t racinesimple/.style={blue},\n\t racinedih/.style={blue},\n\t sommet/.style={inner sep=2pt,circle,draw=black,fill=blue!40,thick,anchor=base},\n\t rotate=0]\n \\tikzstyle{every node}=[font=\\small]\n\\def\\grosseursimple{0.025}\n\\coordinate (ancre) at (0,3);\n\n\\node[sommet,label=above:$s_{\\delta-\\omega}$] (a2) at ($(ancre)+(0.25,0.4)$) {};\n\\node[sommet,label=below right : $s_2$] (a3) at ($(ancre)+(0.5,0)$) {} edge[thick] node[auto,swap,right] {}(a2) ;\n\\node[sommet,label=below left:$s_1$] (a4) at (ancre) {} edge[thick] node[auto,swap,below] {} (a3) edge[thick] node[auto,swap,left] {} (a2);\n\\end{tikzpicture}\n\\end{center}\n\\noindent\nthen the underlying finite Coxeter system $(W_0,S_0)$ is of type $A_2$. Consider $S=\\{s_1, s_2, s_{\\delta-\\omega}\\}$, where $s_1,s_2\\in S_0$ and $\\omega=\\alpha_1+\\alpha_2$ is the highest root of $W_0$. The finite positive root system is $\\Phi^+_0=\\{\\alpha_1, \\alpha_2, \\alpha_1+\\alpha_2\\}$ and the affine simple system is $\\Delta=\\{\\alpha_1, \\alpha_2, \\delta-\\omega\\}$. The set of $k$-small roots for $k\\in\\mathbb N$ is:\n\\begin{eqnarray*}\n\\Sigma_k &=& \\{\\alpha_1 + k\\delta, \\alpha_2 + k\\delta, \\alpha_1+\\alpha_2+k\\delta\\}\\\\\n    &&\\sqcup\\  \\{-\\alpha_1+(k+1)\\delta, -\\alpha_2+(k+1)\\delta, -(\\alpha_1+\\alpha_2)+(k+1)\\delta\\}. \n\\end{eqnarray*}\nThe affine positive root system is:\n$$\n\\Phi^+=\\bigsqcup_{m\\in\\mathbb N} \\Sigma_m.\n$$\nThe Hasse diagram of $(\\Phi^+,\\leq)$ is depicted in Figure \\ref{fig:rootposetA2}, up to depth equal to 3, where each color of the edges corresponds to the simple reflection that is applied: red for $s_1$, yellow for $s_2$ and green for $s_3:=s_{\\delta -\\omega}$. Moreover, the dashed edges correspond to the long coverings.\n\\end{Example}\n\n\\begin{figure}[ht]\n    \\centering    \\includegraphics[width=0.9\\linewidth]{rootposetA22.pdf}\n    \\caption{Hasse diagram of the root poset of type $\\widetilde{A}_2$ up to depth~3, with $\\Sigma=\\Sigma_0$ and $\\Sigma_1$ that correspond to the blocks connected with non-dotted edges.}\n    \\label{fig:rootposetA2}\n\\end{figure}\n\n\\begin{Example}\n    Let $(W,S)$ be a Coxeter system of type $\\widetilde{B}_3$ with the following Coxeter graph\n    \\begin{center}\n\\begin{tikzpicture}\n\t[scale=2,\n\t q/.style={teal,line join=round},\n\t racine/.style={blue},\n\t racinesimple/.style={blue},\n\t racinedih/.style={blue},\n\t sommet/.style={inner sep=2pt,circle,draw=black,fill=blue!40,thick,anchor=base},\n\t rotate=0]\n \\tikzstyle{every node}=[font=\\small]\n\\def\\grosseursimple{0.025}\n\\coordinate (ancre) at (0,3);\n\n\\node[sommet,label=right:$s_3$] (a2) at ($(ancre)+(0.8,0.25)$) {};\n\\node[sommet,label=above :$s_2$] (a3) at ($(ancre)+(0.5,0)$) {} edge[thick] node[auto,swap,right] {}(a2) ;\n\\node[sommet,label=above:$s_1$] (a4) at (ancre) {} edge[thick] node[auto,swap,below] {$4$} (a3);\n\\node[sommet,label=right :$s_{\\delta-\\omega}$] (a5) at ($(ancre)+(0.8,-0.25)$) {} edge[thick] node[auto,swap,right] {}(a3) ;\n\\end{tikzpicture}\n\\end{center}\n\n\\noindent    \n    In this case we have $\\|\\alpha_1\\|=1$ and $\\|\\alpha_2\\|=\\|\\alpha_3\\|=\\|\\omega\\|=\\sqrt{2}$, where $\\omega=2\\alpha_1+2\\alpha_2+\\alpha_3$. For instance, $B(\\alpha_2,\\delta-\\omega)=-B(\\alpha_2,\\omega)=-1>-\\|\\alpha_2\\|\\|\\omega\\|=-2$, and $B(\\delta-\\alpha_3, \\alpha_3)=-\\|\\alpha_3\\|^2=-2$. Hence the covering $\\alpha_2\\lhd s_{\\delta-\\omega}(\\alpha_2)$ is short, while $\\delta-\\alpha_3\\lhd s_3(\\delta-\\alpha_3)=\\delta+\\alpha_3$ is long. The Hasse diagram of the root poset  restricted to $\\Sigma=\\Sigma_0$ of type $\\widetilde{B}_3$ is in Figure \\ref{fig:root-poset-b3}.\n\\end{Example}\n\n\\subsection{Poset isomorphisms within the root poset}\n\nLet $(W,S)$ be an affine Coxeter system with underlying finite system $(W_0,S_0)$ and crystallographic root system $\\Phi_0$. Since $\\Phi_0$ is irreducible, we have two cases for the decomposition of $\\Phi_0$ into $W_0$-orbits:\n\\begin{itemize}\n    \\item $\\Phi_0=\\mo_1$ is a single orbit if $(W,S)$ is simply laced;\n    \\item $\\Phi_0=\\mo_1\\sqcup\\mo_2$ if $(W,S)$ is of type $\\tilde B_n$, $\\tilde C_n$, $\\tilde G_2$ or $\\tilde F_4$.\n\\end{itemize}\nFor simplification, we write:\n$$\n\\Phi_0^+=\\bigsqcup_{i=1}^h \\mo_i^+,\n$$ \nwhere $\\mo_i^+=\\mo_i \\cap \\Phi_0^+$ and $1\\leq i\\leq h$ ($h=1,2$). We also assume that $\\mo_1^+$ is the $W_0$-orbit containing the highest root $\\omega$. \n\n\\smallskip\n\nLet $\\mo$ be a $W_0$-orbit. For $k\\in\\mathbb N$, the set of $k$-small roots in $\\mo$ is:\n\\begin{equation}\n\\Sigma_{\\mo,k} =(k\\delta+\\mo^+)\\sqcup \\left((k+1)\\delta-\\mo^+\\right) =\\{k\\delta + \\alpha, (k+1)\\delta - \\alpha \\mid \\alpha \\in \\mo^+\\}.\n\\end{equation}\nMoreover, by Eq~(\\ref{eq:mSmall}), we have $\\Sigma_k=\\sqcup_{i=1}^h\\Sigma_{\\mo_i,k}$, where $h=1,2$. Set \n$$\n\\Phi_{\\mo}=\\bigsqcup_{k\\in \\mathbb N} \\Sigma_{\\mo,k}.\n$$\nSo $\\Phi^+=\\sqcup_{i=1}^h\\Phi_{\\mo_i}$, where $h=1,2$. \n\nThe Hasse diagrams of the subposet $(\\Phi_{\\mo},\\leq)$  exhibit strong symmetry properties that are fundamental to prove the main result of this section. \n\n\\subsection*{Symmetry at the level of small roots} We denote for simplification, the set of small root in $\\mo$ by:\n$$\n\\Sigma_{\\mo}=\\Sigma_{\\mo,0}=\\mo^+\\sqcup (\\delta-\\mo^+).\n$$\nRecall that $\\Sigma$ is an order ideal in the root poset (see~\\cite[Corollary~4.7.7]{BB}) and any cover relation within $\\Sigma$ is short (see~\\cite[Lemma~4.7.5 and Theorem~4.7.6]{BB}).\n\n\\begin{Theorem}\\label{th:iso1} Let $\\mo$ be a $W_0$-orbit. \nThe map $\\phi: \\mo^+ \\to \\delta - \\mo^+$ defined by $\\phi(\\alpha) = \\delta - \\alpha$ is an isomorphism of posets between $(\\mo^+, \\leq)$ and the opposite poset $(\\delta - \\mo^+, \\leq)^{op}$: for any $\\alpha, \\beta \\in \\mo^+$, we have $\\alpha \\leq \\beta$ if and only of  $\\delta - \\beta \\leq \\delta - \\alpha$. Moreover all covering relations in $(\\mo^+,\\leq)$ and in $(\\delta - \\mo^+, \\leq)$ are short.\n\\end{Theorem}\n\n\\begin{proof}\nSince a poset isomorphism is uniquely determined by the preservation (or reversal) of covering relations, it suffices to show that $\\alpha \\lhd \\beta$ in $\\mo^+$ if and only if $\\delta - \\beta \\lhd \\delta - \\alpha$ in $\\delta - \\mo^+$. Recall that for any $\\alpha, \\beta \\in \\Phi^+$, a covering $\\alpha \\lhd \\beta$ exists if and only if there is a simple reflection $s \\in S$ such that $\\beta = s(\\alpha)$ and $B(\\alpha, \\alpha_s) < 0$. We examine the two types of simple reflections in the affine Weyl group $W$.\n\n\\begin{itemize}\n    \\item[(i)] Let $s \\in S_0$ be a finite reflection. For a finite simple root $\\alpha_s$, we have:\n    \\[ B(\\delta - \\alpha, \\alpha_s) = B(\\delta, \\alpha_s) - B(\\alpha, \\alpha_s) = -B(\\alpha, \\alpha_s). \\]\n    It follows that $B(\\alpha, \\alpha_s) < 0$ if and only if $B(\\delta - \\alpha, \\alpha_s) > 0$. Thus, $\\alpha \\lhd s(\\alpha)$ if and only if  $s(\\delta - \\alpha) = \\delta - s(\\alpha) \\lhd \\delta - \\alpha$.\n\n    \\item[(ii)] Let $s_{\\delta-\\omega}$ be the affine reflection. Similarly, for the affine simple root $\\delta - \\omega$, the property of the radical implies:\n    \\[ B(\\delta - \\alpha, \\delta - \\omega) = B(\\delta, \\delta - \\omega) - B(\\alpha, \\delta - \\omega) = -B(\\alpha, \\delta - \\omega). \\]\n    Again, the sign of the bilinear form is reversed: $B(\\alpha, \\delta - \\omega) < 0$ if and only if $B(\\delta - \\alpha, \\delta - \\omega) > 0$. This ensures that $\\alpha \\lhd s_{\\delta-\\omega}(\\alpha)$ if and only if $s_{\\delta-\\omega}(\\delta - \\alpha) = \\delta - s_{\\delta-\\omega}(\\alpha) \\lhd \\delta - \\alpha$.\n\\end{itemize}\n\nNow, let $\\alpha \\lhd s(\\alpha)$ be a covering in $\\mo^+$ for some $s \\in S$. The type of this cover depends on the value $B(\\alpha, \\alpha_s)$. Under the map $\\phi$, this corresponds to the cover $\\delta-s(\\alpha) \\lhd \\delta -\\alpha$. By the properties of the reflection $s$ and of the radical, we have:\n\\[\nB(\\delta - s(\\alpha), \\alpha_s) = -B(s(\\alpha), \\alpha_s) = B(\\alpha, \\alpha_s).\n\\]\nSince  $B(\\cdot, \\alpha_s)$ is identical in both case, the condition for being short or long in Definition \\ref{def:long} is preserved. Since $\\mo^+$ and $\\delta-\\mo^+$ decomposes the set of small roots~$\\Sigma$, so they are all short. \n\\end{proof}\n\n\\begin{Example}\n    If $(W,S)$ is of type $\\widetilde{B}_3$, then $\\Phi_0^+=\\mo_1^+\\sqcup\\mo_2^+$, where $\\mo_1^+=\\{\\alpha_1, s_2(\\alpha_1), s_3s_2(\\alpha_1)\\}$ and $\\mo_2^+=\\{\\alpha_2, \\alpha_3, s_1(\\alpha_2), s_3(\\alpha_2), s_1s_3(\\alpha_2), s_2s_1s_3(\\alpha_2)\\}$. The root poset restricted to $\\Sigma_{\\mo_1}\\sqcup\\Sigma_{\\mo_2}$ is depicted in Figure \\ref{fig:root-poset-b3}.\n\\end{Example}\n\n\\begin{figure}[hbtp]\n\\centering\n\\includegraphics[scale=.75]{rootposetB3.pdf}\n\\caption{Root poset of type $\\widetilde{B}_3$ restricted to $\\Sigma_{\\mo_1}\\sqcup \\Sigma_{\\mo_2}$. The portion of poset in the red lines is isomorphic to opposite in the black lines, as stated in Theorem \\ref{th:iso1}.}\n\\label{fig:root-poset-b3}\n\\end{figure}\n\n\\subsection*{Symmetry of the root poset restricted to a $W_0$- orbit}\nFollowing \\cite[Ch. VI, \\S 1]{bour}, for $\\alpha, \\beta\\in \\Phi^+$, we set \n$$n(\\alpha,\\beta):=2\\frac{B(\\alpha,\\beta)}{B(\\beta,\\beta)},$$  \nso that $s_\\beta(\\alpha)=\\alpha - n(\\alpha,\\beta)\\beta.$ By \\cite[Ch. VI, \\S 1, Proposition~8]{bour}, if $\\alpha\\neq \\beta$ and $\\|\\alpha\\|\\leq \\|\\beta\\|$, then $n(\\alpha,\\beta) \\in \\{0,\\pm 1\\}$. It follows that $n(\\alpha, \\omega) \\in \\{0,\\pm 1\\}$ when $\\alpha\\neq \\omega$, since $\\|\\alpha\\|\\leq \\|\\omega\\|$ for all positive roots $\\alpha$. Finally, note that $n(k\\delta+\\alpha, \\beta)=n(\\alpha, \\beta)=n(\\alpha, m\\delta+\\beta)$ and $n(k\\delta-\\alpha, \\beta)=-n(\\alpha, \\beta)=n(\\alpha, m\\delta-\\beta)$ for $k,m\\in\\mathbb{N}^*$.\n\\smallskip\n\n We describe now the covering relations inside the sets $\\Sigma_{\\mo,k}$. Let $\\alpha \\in \\Phi_0^+$ and $k \\in \\mathbb{N}$.\n\n\\begin{enumerate}\n\\item[(1)]\nThere are no covering relations of the form\n\\[\n(k+1)\\delta - \\alpha \\lhd s_{\\delta-\\omega}\\big((k+1)\\delta - \\alpha\\big).\n\\]\nIndeed, if such a covering existed, then\n$0 > B\\big((k+1)\\delta - \\alpha,\\, \\delta - \\omega\\big) = B(\\alpha,\\omega)$,\nso that $n(\\alpha,\\omega) = -1$. It follows that\n\\[\ns_{\\delta-\\omega}(\\alpha)\n= \\alpha - n(\\alpha,\\delta-\\omega)(\\delta-\\omega)\n= \\alpha + n(\\alpha,\\omega)(\\delta-\\omega)\n= \\alpha - \\delta + \\omega.\n\\]\nThus, $s_{\\delta-\\omega}\\big((k+1)\\delta - \\alpha\\big)\n= (k+2)\\delta - (\\alpha + \\omega)\n\\notin \\Sigma_{\\mo,k},$ a contradiction.\n\\smallskip\n\n\\item[(2)]\nAll covering relations in $\\Sigma_{\\mo,k}$ are of one of the following three types:\n\\begin{itemize}\n\\item[(i)]\n$k\\delta + \\alpha \\lhd s(k\\delta + \\alpha) = k\\delta + s(\\alpha),\n\\qquad \\text{when } B(\\alpha,\\alpha_s) < 0;\n$\n\n\\item[(ii)]\n$\n(k+1)\\delta - \\alpha \\lhd s\\big((k+1)\\delta - \\alpha\\big)\n= (k+1)\\delta - s(\\alpha),\n$\nwhen $B(\\alpha,\\alpha_s) > 0$ and $\\alpha \\neq \\alpha_s$.\nNote that if $\\alpha = \\alpha_s$, then\n$(k+1)\\delta - s(\\alpha) = (k+1)\\delta + \\alpha_s \\in \\Sigma_{\\mo,k+1}$.\n\n\\item[(iii)]\n$\nk\\delta + \\alpha \\lhd s_{\\delta-\\omega}(k\\delta + \\alpha)\n= k\\delta + s_{\\delta-\\omega}(\\alpha),\n$\nwhen $B(\\alpha,\\delta-\\omega) < 0$ and $\\alpha \\neq \\omega$.\nOtherwise, $s_{\\delta-\\omega}(k\\delta + \\omega) = (k+2)\\delta - \\omega\n\\in \\Sigma_{\\mo,k+1}.$\n\\end{itemize}\n\\end{enumerate}\n\nWe state now the second symmetry of the root poset.\n\n\\begin{Theorem}\\label{th:iso2} Let $\\mo$ be a $W_0$-orbit. There is a poset isomorphism $$(\\Sigma_{\\mo,k}, \\leq) \\cong (\\Sigma_{\\mo}, \\leq)$$ for all $k\\in\\mathbb{N}$. Moreover, all covering in $\\Sigma_{\\mo,k}$ are short.\n\\end{Theorem}\n\n\\begin{proof}\nThe map $\\pi_k : \\Sigma_{\\mo,k} \\longrightarrow \\Sigma_{\\mo}$ defined by\n\\[\n\\pi_k(k\\delta+\\alpha)=\\alpha,\n\\qquad\n\\pi_k((k+1)\\delta-\\alpha)=\\delta-\\alpha,\n\\]\nfor $\\alpha\\in\\Phi_0^+$ is a bijection. By the discussion preceding the theorem, we have three types of coverings in $\\Sigma_{\\mo,k}$.\n\\smallskip\n\nA covering $k\\delta+\\alpha \\lhd k\\delta+s(\\alpha)$ of type (i) occurs if and only if\n$0>B(k\\delta+\\alpha,\\alpha_s)=B(\\alpha,\\alpha_s)$,\nwhich is equivalent to $\\alpha \\lhd s(\\alpha)$ being a covering in $\\Sigma_{\\mo}$. \n\\smallskip\n\nA covering $(k+1)\\delta-\\alpha \\lhd (k+1)\\delta-s(\\alpha)$ of type (ii), occurs if and only if\n$ 0>B((k+1)\\delta-\\alpha,\\alpha_s)=-B(\\alpha,\\alpha_s)$,\nwith $\\alpha\\neq\\alpha_s$, which is equivalent to\n$\\delta-\\alpha \\lhd \\delta-s(\\alpha)$\nbeing a covering in $\\Sigma_{\\mo}$.\n\\smallskip\n\nA covering $k\\delta+\\alpha \\lhd s_{\\delta-\\omega}(k\\delta+\\alpha)$ of type (iii)\noccurs if and only if $0>B(k\\delta+\\alpha,\\delta-\\omega)=B(\\alpha,\\delta-\\omega)$, with $\\alpha\\neq\\omega$, which is equivalent to $\\alpha \\lhd s_{\\delta-\\omega}(\\alpha)\n$ being a covering in $\\Sigma_{\\mo}$. \n\\smallskip\n\nThus $\\pi_k$ preserves and reflects covering relations and is therefore a poset isomorphism. The final statement follows from Theorem~\\ref{th:iso1} since all covering are short in $\\Sigma_\\mo$.\n\\end{proof}\n\n\\begin{Example} In the Figure~\\ref{fig:root-poset-b3-iso2} below, the the first three slices of the Hasse diagram of the root poset of type $\\widetilde{B}_3$ are represented. The first slice coincides with the poset shown in Figure \\ref{fig:root-poset-b3}, although the order relations are drawn differently. This alternative depiction makes the isomorphisms described in Theorem \\ref{th:iso2} more apparent.\n\\end{Example}\n\n\\begin{figure}[hbtp]\n\\centering\n\\includegraphics[scale=.7]{periodicB3str.pdf}\n\\caption{}\n\\label{fig:root-poset-b3-iso2}\n\\end{figure}\n\nWe end this discussion by showing that a covering $\\alpha\\lhd \\beta$ with $\\alpha\\in \\Sigma_{\\mo,k}$ and $\\beta\\in \\Sigma_{\\mo,k+1}$ is long.\n\n\\begin{Proposition}[Covering relations]\\label{prop:long-cover}\\\n\\begin{itemize}\n    \\item[(i)]  For all $k\\in\\mathbb{N}, s\\in S_0$, $(k+1)\\delta-\\alpha_s\\lhd s((k+1)\\delta - \\alpha_s)$ is a long covering, where $(k+1)\\delta-\\alpha_s\\in \\Sigma_{\\mo,k}$ and $s((k+1)\\delta - \\alpha_s)\\in \\Sigma_{\\mo, k+1}$.\n    \\item[(ii)] For all $k\\in\\mathbb{N}$, $k\\delta +\\omega\\lhd s_{\\delta-\\omega}(k\\delta + \\omega)$ is a long covering, where $k\\delta +\\omega\\in \\Sigma_{\\mo,k}$ and $s_{\\delta-\\omega}(k\\delta + \\omega)\\in \\Sigma_{\\mo,k+1}$. \n\\end{itemize}\n\n\\end{Proposition}\n\n\\begin{proof}\n    Consider the  $(k+1)\\delta-\\alpha_s\\lhd s((k+1)\\delta - \\alpha_s)$, then $B((k+1)\\delta-\\alpha_s, \\alpha_s)=-\\|\\alpha_s\\|^2$, hence the covering is long, and $s((k+1)\\delta - \\alpha_s)=(k+1)\\delta+\\alpha_s \\in \\Sigma_{\\mo,k+1}$. Analogously, if we consider the covering $k\\delta +\\omega\\lhd s_{\\delta-\\omega}(k\\delta + \\omega)$, $B(k\\delta +\\omega, \\delta-\\omega)=-\\|\\omega\\|^2$, so the covering is long. In particular, $s_{\\delta -\\omega}(k\\delta +\\omega)=(k+2)\\delta -\\omega \\in \\Sigma_{\\mo,k+1}$. \n    \\end{proof}\n\n\\subsection{An explicit formula for the generating function of depth} We are ready now to provide a closed formula for the generating function of the depth of positive roots in $\\Phi$, using the restriction of the root poset to the sets $\\Sigma_\\mo$ and $\\Phi_\\mo$ introduced in the previous section.\n\\medskip\n\nLet $\\mo \\subseteq \\Phi_0$ be $W_0$-orbit. Define \n$$\nM_{\\mo}=\\max\\{\\dep(\\beta) \\mid \\beta \\in \\Sigma_{\\mo}\\}.\n$$\nWe show that the depth of a positive root depends only on $M_{\\mo}$ and on\nthe depth of a root in $\\mo^+$.\n\n\\begin{Corollary}\\label{cor:maxdep}\n    Let $\\alpha\\in \\mo^+$ and $k\\in \\mathbb{N}$, then :\n    \\begin{enumerate}\n        \\item $\\dep(k\\delta + \\alpha)=\\dep(\\alpha)+k(M_\\mo +1)$;\n        \\item $\\dep((k+1)\\delta -\\alpha) = \\dep(\\delta-\\alpha) +k(M_\\mo +1)$.\n    \\end{enumerate}  \n\\end{Corollary}", "th:iso1": "\\begin{Theorem}\\label{th:iso1} Let $\\mo$ be a $W_0$-orbit. \nThe map $\\phi: \\mo^+ \\to \\delta - \\mo^+$ defined by $\\phi(\\alpha) = \\delta - \\alpha$ is an isomorphism of posets between $(\\mo^+, \\leq)$ and the opposite poset $(\\delta - \\mo^+, \\leq)^{op}$: for any $\\alpha, \\beta \\in \\mo^+$, we have $\\alpha \\leq \\beta$ if and only of  $\\delta - \\beta \\leq \\delta - \\alpha$. Moreover all covering relations in $(\\mo^+,\\leq)$ and in $(\\delta - \\mo^+, \\leq)$ are short.\n\\end{Theorem}", "prop:PrefDom": "\\begin{Proposition}\\label{prop:PrefDom} Let $\\beta \\in \\Phi^+$ and let $t=s_\\beta$ be the corresponding reflection. For any $t$-prefix $p_t$ we have:  \n\\begin{eqnarray*}\n\\dom(\\beta) & = & \\{ \\alpha\\in \\Phi(p_t) \\ \\mid\\  B(\\alpha,\\beta)\\geq 1\\}\\\\\n&=&\\{\\beta\\}\\sqcup \\{ \\alpha_s\\in \\Phi(p_t) \\ \\mid\\  |\\langle s,t\\rangle| =\\infty\\}.\n\\end{eqnarray*}\nIn particular, $\\dep_\\infty (\\beta)=|\\{ \\alpha_s\\in \\Phi(p_t) \\ \\mid\\ |\\langle s,t\\rangle| =\\infty\\}|$.\n\\end{Proposition}", "se:reflectionprefix": "\\begin{proof} Since $\\pi_m(w)\\leq_R w$, we have $\\Sigma_m(\\pi_m(w))\\subseteq \\Sigma_m(w)$. The converse follows from  \\cite[Theorem 6.4]{DFHM24} in which the authors prove that the set \n$$\n\\{g\\in W\\mid \\Sigma_m(g)=\\Sigma_m(w)\\}\n$$\nhas a unique element of minimal length $x\\in L_m$ and that $x\\leq_R w$. So $\\Sigma_m(w)=\\Sigma_m(x)$. But $x\\leq_R\\pi_m(x)$ by definition. Therefore  $ \\Sigma_m(w)=\\Sigma_m(x)\\subseteq \\Sigma_m(\\pi_m(w))$.\n\\end{proof}\n\n\\section{Reflection-prefixes and palindromic reduced words of reflections}\\label{se:reflectionprefix}\n\nIn this section, we  first define and state properties of {\\em reflection-prefixes} in relation to the root poset, to the dominance order and to the canonical generators of maximal dihedral subgroups.  Then we produce a family of automata built from the finite Garside shadows $L_m$ ($m\\in\\mathbb N)$  that recognize the language of reflection-prefixes. \n\n\\subsection{Reflection-prefixes}\\label{ss:reflectionprefixes}\n\n Let $t\\in T$ be a reflection of $(W,S)$. It is well known that $\\ell(t)=2k+1$ for some $k\\in\\mathbb N$ and that  if $t=s_1\\dots s_k s_{k+1} s_{k+2} \\dots s_{2k+1}$ is a reduced word for $t$, the word $s_1 \\dots s_k s_{k+1} s_k\\dots s_1$ is a {\\em palindromic reduced word} for $t$; see for instance \\cite[Proposition~2.3]{DFHM24}.  \n\n The following proposition shows in particular that $t$ is uniquely determined by the prefix $s_1\\cdots s_ks_{k+1}$ of any of its palindromic reduced words. \n\n \\begin{Proposition}\\label{prop:Pref1} Let $t\\in T$ with corresponding positive root $\\alpha_t$. Let  $w\\in W$ and assume there is $r\\in D_R(w)$ such that $t=wrw^{-1}$ and $\\ell(t)=2\\ell(w)-1$. Then $D_R(w)=\\{r\\}$. Moreover: \n \\begin{enumerate}\n\n\\item if $w=s_1\\cdots s_k s_{k+1}$ is a reduced word for $w$, then $s_{k+1}=r$ and $t=s_1\\cdots s_k r s_k\\cdots s_1$ is a palindromic reduced word for $t$;\n\\item $\\alpha_t=w(\\alpha_r)= s_1\\cdots s_k(\\alpha_r)$ and  $\\dep(\\alpha_t)=k=\\ell(w)-1$.\n\\end{enumerate}\n\\end{Proposition}\n\\begin{proof}   Let $w=s_1\\cdots s_k r$ be a reduced word.  We show that $D_R(w)=\\{r\\}$. By contradiction, assume that $|D_R(w)|\\geq 2$. So there exists  a $s\\in D_R(w)$ with $s\\not = r$; set $I=\\{s,r\\}\\subseteq D_R(w)$.  By Lemma~\\ref{lem:Descents},  we know that $W_I$ is finite and  there is $u\\in X_I$ such that $w=u w_{\\circ,I}$ is a reduced product. So $\\ell(w)=\\ell(u)+m$, where $m=m_{sr}\\geq 2$. Now, observe that  $w_{\\circ,I}rw_{\\circ,I}\\in I$, since the longest element of a finite Coxeter system acts by conjugation on the set of simple reflections. Therefore \n$$\nt= s_1\\cdots s_k s_{k+1} s_k\\cdots s_1 =s_1\\cdots s_k r s_k\\cdots s_1=w r w^{-1}=uw_{\\circ,I}rw_{\\circ,I}u^{-1},\n$$\nimplying that $\\ell(t)\\leq 2\\ell(u)+1=2\\ell(w)-2m+1<2\\ell(w)-1=\\ell(t)$, a contradiction. The remaining claims follow from the uniqueness of $r$, the definition of the depth of roots and from the equalities $\\ell(t)=2\\dep(\\alpha_t)+1=2\\ell(w)-1$.\n\\end{proof}", "prop:PrefRoot": "\\begin{Proposition}\\label{prop:PrefRoot} Let $t\\in T$ and $\\alpha_t\\in \\Phi^+$ be the corresponding positive root. \n\\begin{enumerate}\n\\item Let  $\\alpha_{r}\\lhd s_k(\\alpha_{r})\\lhd\\dots\\lhd s_1\\cdots s_{k-1} s_k(\\alpha_r)=\\alpha_t$ be a saturated chain in $(\\Phi^+,\\leq)$, where $r\\in S$. Then $p_t=s_1\\dots s_kr$ is a reduced word for some $t$-prefix $p_t$. \n\n\\item Let $p_t$ be a  $t$-prefix and fix a reduced word $p_t=s_1\\cdots s_k r $, with $r$ the unique right-descent. Then $\\alpha_{r}\\lhd s_k(\\alpha_{r})\\lhd\\cdots\\lhd s_1\\cdots s_{k-1} s_k(\\alpha_r)=\\alpha_t$ is a saturated chain in $(\\Phi^+,\\leq)$.\n\\end{enumerate}\n\\end{Proposition}", "th:main": "\\begin{Theorem}\\label{th:main}\n    The generating function of the depth of positive roots is of the form\n    $$\\Phi^+(q)=\\frac{P(q)}{1-q^M}$$\n    where $M=\\lcm(M_\\mo+1\\mid \\mo\\ \\mbox{ $W_0$-orbit})$ and $P$ is palindromic.\n\\end{Theorem}", "th:iso2": "\\begin{Theorem}\\label{th:iso2} Let $\\mo$ be a $W_0$-orbit. There is a poset isomorphism $$(\\Sigma_{\\mo,k}, \\leq) \\cong (\\Sigma_{\\mo}, \\leq)$$ for all $k\\in\\mathbb{N}$. Moreover, all covering in $\\Sigma_{\\mo,k}$ are short.\n\\end{Theorem}", "se:openproblems": "\\label{se:openproblems}\n\nWe propose here some natural problems arising from this article.\n\n\\begin{Problem}\n    Is the language of palindromic reduced word (or lexicographically ordered palindromic red", "prop:algo": "\\begin{Theorem}\\label{prop:algo} Let $r,t\\in T$ with $r\\not = t$ and let $W'=\\langle r,t\\rangle$ be the dihedral reflection subgroup generated by $r$ and $t$. Let $p_r$ be a $r$-prefix and $p_t$ be a $t$-prefix.\n\\begin{enumerate}\n\\item If $|\\Phi_{W'}(p_r)|=|\\Phi_{W'}(p_t)|=1$, then:\n$$\n\\Delta_{W'}=\\{\\alpha_r,\\alpha_t\\}\\quad\\textrm{and}\\quad\\chi(W')=\\{r,t\\}.\n$$\n\\item If   $|\\Phi_{W'}(p_r)|>1$ (resp. $|\\Phi_{W'}(p_t)|>1$), then:\n $$\n \\Delta_{W'}=\\{\\alpha_1,\\alpha_2\\}\\quad\\textrm{and}\\quad\\chi(W')=\\{s_1,s_2\\},\n $$\n where $\\alpha_1$ is the unique root with the smallest depth in $\\Phi(p_r)\\cap \\Phi_{W'}$ (resp. $\\Phi(p_t)\\cap \\Phi_{W'}$) and $s_1(\\alpha_2)$ is the unique root with smallest depth in $\\Phi(p_r)\\cap \\Phi_{W'}\\setminus\\{\\alpha_1\\}$ (resp. $\\Phi(p_t)\\cap \\Phi_{W'}\\setminus\\{\\alpha_1\\}$). \\qed\n\\end{enumerate}\n \\end{Theorem}", "cor:Refregular": "\\begin{Corollary}\\label{cor:Refregular}\nThe language $\\Pref_T$ is regular and the generating function \n\\begin{align*}\n\\sum_{u\\in \\Pref_T} q^{|u|}& =\\sum_{t\\in T} |\\{s_1 \\cdots s_k s_{k+1} s_k\\cdots s_1 \\in \\Pref_T \\mid t=s_1 \\cdots s_k s_{k+1}s_k\\cdots s_1\\}| \\, q^{\\ell(t)}\\\\\n&=\\sum_{k\\in\\mathbb{N}} a_k q^k\n\\end{align*}\nis rational, where $a_k$ is the number of saturated chains of length $k$ in the root poset of $W$. In particular, $\\RPal(q)=q\\Pref_T(q^2)$ is rational.\n\\end{Corollary}", "ss:reflectionprefixes": "\\begin{proof} Since $\\pi_m(w)\\leq_R w$, we have $\\Sigma_m(\\pi_m(w))\\subseteq \\Sigma_m(w)$. The converse follows from  \\cite[Theorem 6.4]{DFHM24} in which the authors prove that the set \n$$\n\\{g\\in W\\mid \\Sigma_m(g)=\\Sigma_m(w)\\}\n$$\nhas a unique element of minimal length $x\\in L_m$ and that $x\\leq_R w$. So $\\Sigma_m(w)=\\Sigma_m(x)$. But $x\\leq_R\\pi_m(x)$ by definition. Therefore  $ \\Sigma_m(w)=\\Sigma_m(x)\\subseteq \\Sigma_m(\\pi_m(w))$.\n\\end{proof}\n\n\\section{Reflection-prefixes and palindromic reduced words of reflections}\\label{se:reflectionprefix}\n\nIn this section, we  first define and state properties of {\\em reflection-prefixes} in relation to the root poset, to the dominance order and to the canonical generators of maximal dihedral subgroups.  Then we produce a family of automata built from the finite Garside shadows $L_m$ ($m\\in\\mathbb N)$  that recognize the language of reflection-prefixes. \n\n\\subsection{Reflection-prefixes}\\label{ss:reflectionprefixes}\n\n Let $t\\in T$ be a reflection of $(W,S)$. It is well known that $\\ell(t)=2k+1$ for some $k\\in\\mathbb N$ and that  if $t=s_1\\dots s_k s_{k+1} s_{k+2} \\dots s_{2k+1}$ is a reduced word for $t$, the word $s_1 \\dots s_k s_{k+1} s_k\\dots s_1$ is a {\\em palindromic reduced word} for $t$; see for instance \\cite[Proposition~2.3]{DFHM24}.  \n\n The following proposition shows in particular that $t$ is uniquely determined by the prefix $s_1\\cdots s_ks_{k+1}$ of any of its palindromic reduced words. \n\n \\begin{Proposition}\\label{prop:Pref1} Let $t\\in T$ with corresponding positive root $\\alpha_t$. Let  $w\\in W$ and assume there is $r\\in D_R(w)$ such that $t=wrw^{-1}$ and $\\ell(t)=2\\ell(w)-1$. Then $D_R(w)=\\{r\\}$. Moreover: \n \\begin{enumerate}\n\n\\item if $w=s_1\\cdots s_k s_{k+1}$ is a reduced word for $w$, then $s_{k+1}=r$ and $t=s_1\\cdots s_k r s_k\\cdots s_1$ is a palindromic reduced word for $t$;\n\\item $\\alpha_t=w(\\alpha_r)= s_1\\cdots s_k(\\alpha_r)$ and  $\\dep(\\alpha_t)=k=\\ell(w)-1$.\n\\end{enumerate}\n\\end{Proposition}\n\\begin{proof}   Let $w=s_1\\cdots s_k r$ be a reduced word.  We show that $D_R(w)=\\{r\\}$. By contradiction, assume that $|D_R(w)|\\geq 2$. So there exists  a $s\\in D_R(w)$ with $s\\not = r$; set $I=\\{s,r\\}\\subseteq D_R(w)$.  By Lemma~\\ref{lem:Descents},  we know that $W_I$ is finite and  there is $u\\in X_I$ such that $w=u w_{\\circ,I}$ is a reduced product. So $\\ell(w)=\\ell(u)+m$, where $m=m_{sr}\\geq 2$. Now, observe that  $w_{\\circ,I}rw_{\\circ,I}\\in I$, since the longest element of a finite Coxeter system acts by conjugation on the set of simple reflections. Therefore \n$$\nt= s_1\\cdots s_k s_{k+1} s_k\\cdots s_1 =s_1\\cdots s_k r s_k\\cdots s_1=w r w^{-1}=uw_{\\circ,I}rw_{\\circ,I}u^{-1},\n$$\nimplying that $\\ell(t)\\leq 2\\ell(u)+1=2\\ell(w)-2m+1<2\\ell(w)-1=\\ell(t)$, a contradiction. The remaining claims follow from the uniqueness of $r$, the definition of the depth of roots and from the equalities $\\ell(t)=2\\dep(\\alpha_t)+1=2\\ell(w)-1$.\n\\end{proof}"}, "pre_theorem_intro_text_len": 6807, "pre_theorem_intro_text": "Let $(W,S)$ be a Coxeter system, with $S$ finite. We denote by $S^*$ the free monoid on the alphabet $S$. To distinguish between a word and its corresponding group element, we denote a word in $S^*$ using bold letters, $\\bold w =  s_1 \\cdots  s_k$, while the resulting product in $W$ is denoted by $w=s_1\\cdots s_k$. Let $w\\in W$, a word $\\bold w= s_1 \\cdots  s_k\\in S^*$ is a {\\em reduced word for $w$} if  $w=s_1\\cdots s_k$  and $k$ is minimal for this property; in this case the {\\em length} of $w$ is $\\ell(w)=k$. The identity $e\\in W$ is represented by the empty word, and $\\ell(e)=0$. We denote the set of all reduced words of $w\\in W$ by $\\Red(w)$, and the set of all reduced words in $W$ by $\\Red=\\bigsqcup_{w\\in W} \\Red(w)$, where $\\sqcup$ denotes the disjoint union.\nFinally, for any subset $A\\subseteq W$, the {\\em Poincar\\'e series of $A$} is the formal power series\n$A(q):=\\sum_{w\\in A} q^{\\ell(w)}.$\n\\smallskip\n\nIn this article, we study two combinatorial problems regarding the {\\em set of reflections} $T:=\\bigcup_{w\\in W} wSw^{-1}$ of a Coxeter group $W$:\n\\begin{enumerate}\n    \\item Is the language of {\\em palindromic reduced words (for the reflections)} regular?\n    \\item Are there explicit and elegant formulas for the Poincar\\'e series of the set $T$ of reflections ?\n\\end{enumerate}\nOur main contributions to those questions are the following: \n\\begin{itemize}\n\\item We introduce the notion of  {\\em reflection-prefixes}, a class of elements in $W$ arising naturally from  palindromic reduced words of reflections, and study their properties in relation to the {\\em root poset}, the {\\em dominance order on roots}  and {\\em dihedral reflection subgroups}.\n\\item For any Coxeter system, we show that the {\\em language of reduced words for reflection-prefixes}, $\\Pref_T$, is regular. This is achieved using the family of $m$-canonical automata associated with $m$-Shi arrangements (see~\\cite[\\S3.4]{HNW} and \\cite{DFHM24}), which provide a family of finite deterministic automata recognizing $\\Pref_T$. As a consequence, we show that the generating function  of palindromic reduced words is rational.\n\\item In the case of affine Coxeter groups, we derive a simple expression for the Poincar\\'e series of the set of reflections in terms of rational fractions. This formula is obtained from some symmetries of the Hasse diagram of the root poset.\n\\end{itemize}\n\nWe discuss now some history and motivations that lead to this article. \nIf $W$ is finite, $A(q)$ is clearly a polynomial for any $A\\subseteq W$. A natural question in combinatorics of infinite Coxeter systems is to classify for which subsets $A$ the Poincar\\'e series $A(q)$ is {\\em rational}, that is, can be written as a ratio of two polynomials in $q$. It is well-known that $W(q)$ is rational, see for instance~\\cite[Corollary~7.1.8]{BB}. Furthermore, an explicit recursive formula for $W(q)$ in term of standard parabolic subgroups is provided in~\\cite[Proposition~7.1.7]{BB}. \n\nA further natural direction is the study of generating functions of words in $S^*$ in relation to the Coxeter system $(W,S)$.  More precisely, given the canonical projection $\\pi:S^* \\rightarrow W$ sending a word $\\bold w= s_1 \\cdots  s_k$ to the element $w=s_1\\cdots s_k$,\nit is natural to  consider the {\\em ‘‘lifted'' Poincar\\'e series} of a subset $B\\subseteq S^*$ relative to $(W,S)$:\n$$\nB(q)=\\sum_{\\bold w\\in B} q^{\\ell(w)},\n$$\nwhere the notation ${\\ell(w)}$ is understood to be $\\ell(\\pi({\\bf w}))$. \n\nA longstanding open problem in combinatorics of Coxeter groups is the enumeration of $|\\Red(w)|$; for a detailed discussion see~\\cite[p.123]{BB}, and for various partial results, see~\\cite{Eriksson,Hart}.  However, if we consider the set of all elements of a given length, we know that the numbers $r_k$ of reduced words of length $k\\in \\mathbb N$ is enumerated by  the following Poincar\\'e series\n$$\n\\Red(q)=\\sum_{\\bold w\\in \\Red} q^{\\ell(w)}=\\sum_{k\\in\\mathbb N} r_kq^k,\n$$\nwhich is known to be rational. The proof follows from the existence of a finite deterministic automaton that recognizes the language $\\Red$; see for instance~\\cite[Theorem~4.9.1]{BB} for more details. \n\n\\smallskip\n\nAs reported by Brenti~\\cite{B1}, Stembridge proposed the following problem during an open problem session at the Mathematical Sciences Research Institute at Berkeley in 1997: Is it true that the Poincar\\'e series\n$$\nT(q)=\\sum_{t\\in T} q^{\\ell(t)}\\quad \\textrm{is rational?}\n$$ \n\n\\textbf{}Since reflections are known to admit palindromic reduced words, a natural approach to this question is to construct a finite automaton that recognizes exactly one palindromic reduced word for each reflection. The existence of such a regular language would imply that the associated generating function is rational. However, the property of being regular is not generally preserved under the palindromic constraint. If a language is regular, it is well-known that the sublanguage of its palindromic words is context-free but not necessarily regular; for instance if $S=\\{a,b\\}$ then $S^*$ is regular but a standard application of the {\\em pumping lemma} shows that the sublanguage of its palindroms is not.  Indeed, the regularity of $\\operatorname{Pal}$, {\\em the language of all palindromic reduced words}, which are necessarily reduced words for reflections, remains an open question. In a recent article, Mili\\'cevi\\'c ~\\cite{Mili} provides palindromic reduced words for all reflections in finite Weyl groups, but highlights the challenge of finding a general algorithm applicable to any Coxeter group. To overcome these challenges, we shift to the language of reflection-prefixes which we define below. \n\n\\subsection*{The language of reflection-prefixes}  \n\nWe recall that the {\\em (right) weak order} $(W,\\leq_R)$ is the poset defined by $u\\leq_R w$ if there exists a reduced word for $u\\in W$ that is a prefix for a reduced word for $w\\in W$. Equivalently, $u\\leq_R w$ if and only if $\\ell(w)=\\ell(u^{-1}w)+\\ell(u)$.\n\nIt is well-known that every reflection has an odd length. Let $t\\in T$ be a reflection of length $\\ell(t)=2k+1$, we define a {\\em $t$-prefix} to be any element $p_t\\in W$ such that $p_t\\leq_R t$ and $\\ell(p_t)=k+1$.\n  It turns out that $ s_1\\cdots s_k s_{k+1}  s_{k}\\cdots  s_1$ is a palindromic reduced word for $t$  if and only if $s_1\\cdots  s_k s_{k+1}$ is a reduced word for a $t$-prefix; see \\S\\ref{ss:reflectionprefixes}. \n\nReflection-prefixes turn out to be a useful combinatorial framework to study the root poset (Proposition~\\ref{prop:PrefRoot}), the dominance order (Proposition~\\ref{prop:PrefDom}) and to find the canonical generators of dihedral reflection subgroups (Theorem~\\ref{prop:algo}).  Our main results concerning reflection-prefixes and their associated languages are summarized in the following theorem.", "context": "Let $(W,S)$ be a Coxeter system, with $S$ finite. We denote by $S^*$ the free monoid on the alphabet $S$. To distinguish between a word and its corresponding group element, we denote a word in $S^*$ using bold letters, $\\bold w = s_1 \\cdots s_k$, while the resulting product in $W$ is denoted by $w=s_1\\cdots s_k$. Let $w\\in W$, a word $\\bold w= s_1 \\cdots s_k\\in S^*$ is a {\\em reduced word for $w$} if $w=s_1\\cdots s_k$ and $k$ is minimal for this property; in this case the {\\em length} of $w$ is $\\ell(w)=k$. The identity $e\\in W$ is represented by the empty word, and $\\ell(e)=0$. We denote the set of all reduced words of $w\\in W$ by $\\Red(w)$, and the set of all reduced words in $W$ by $\\Red=\\bigsqcup_{w\\in W} \\Red(w)$, where $\\sqcup$ denotes the disjoint union.\nFinally, for any subset $A\\subseteq W$, the {\\em Poincar\\'e series of $A$} is the formal power series\n$A(q):=\\sum_{w\\in A} q^{\\ell(w)}.$\n\\smallskip\n\nIn this article, we study two combinatorial problems regarding the {\\em set of reflections} $T:=\\bigcup_{w\\in W} wSw^{-1}$ of a Coxeter group $W$:\n\\begin{enumerate}\n \\item Is the language of {\\em palindromic reduced words (for the reflections)} regular?\n \\item Are there explicit and elegant formulas for the Poincar\\'e series of the set $T$ of reflections ?\n\\end{enumerate}\nOur main contributions to those questions are the following: \n\\begin{itemize}\n\\item We introduce the notion of {\\em reflection-prefixes}, a class of elements in $W$ arising naturally from palindromic reduced words of reflections, and study their properties in relation to the {\\em root poset}, the {\\em dominance order on roots} and {\\em dihedral reflection subgroups}.\n\\item For any Coxeter system, we show that the {\\em language of reduced words for reflection-prefixes}, $\\Pref_T$, is regular. This is achieved using the family of $m$-canonical automata associated with $m$-Shi arrangements (see~\\cite[\\S3.4]{HNW} and \\cite{DFHM24}), which provide a family of finite deterministic automata recognizing $\\Pref_T$. As a consequence, we show that the generating function of palindromic reduced words is rational.\n\\item In the case of affine Coxeter groups, we derive a simple expression for the Poincar\\'e series of the set of reflections in terms of rational fractions. This formula is obtained from some symmetries of the Hasse diagram of the root poset.\n\\end{itemize}\n\nA further natural direction is the study of generating functions of words in $S^*$ in relation to the Coxeter system $(W,S)$. More precisely, given the canonical projection $\\pi:S^* \\rightarrow W$ sending a word $\\bold w= s_1 \\cdots s_k$ to the element $w=s_1\\cdots s_k$,\nit is natural to consider the {\\em ‘‘lifted'' Poincar\\'e series} of a subset $B\\subseteq S^*$ relative to $(W,S)$:\n$$\nB(q)=\\sum_{\\bold w\\in B} q^{\\ell(w)},\n$$\nwhere the notation ${\\ell(w)}$ is understood to be $\\ell(\\pi({\\bf w}))$.\n\nA longstanding open problem in combinatorics of Coxeter groups is the enumeration of $|\\Red(w)|$; for a detailed discussion see~\\cite[p.123]{BB}, and for various partial results, see~\\cite{Eriksson,Hart}. However, if we consider the set of all elements of a given length, we know that the numbers $r_k$ of reduced words of length $k\\in \\mathbb N$ is enumerated by the following Poincar\\'e series\n$$\n\\Red(q)=\\sum_{\\bold w\\in \\Red} q^{\\ell(w)}=\\sum_{k\\in\\mathbb N} r_kq^k,\n$$\nwhich is known to be rational. The proof follows from the existence of a finite deterministic automaton that recognizes the language $\\Red$; see for instance~\\cite[Theorem~4.9.1]{BB} for more details.\n\nIt is well-known that every reflection has an odd length. Let $t\\in T$ be a reflection of length $\\ell(t)=2k+1$, we define a {\\em $t$-prefix} to be any element $p_t\\in W$ such that $p_t\\leq_R t$ and $\\ell(p_t)=k+1$.\n It turns out that $ s_1\\cdots s_k s_{k+1} s_{k}\\cdots s_1$ is a palindromic reduced word for $t$ if and only if $s_1\\cdots s_k s_{k+1}$ is a reduced word for a $t$-prefix; see \\S\\ref{ss:reflectionprefixes}.\n\nReflection-prefixes turn out to be a useful combinatorial framework to study the root poset (Proposition~\\ref{prop:PrefRoot}), the dominance order (Proposition~\\ref{prop:PrefDom}) and to find the canonical generators of dihedral reflection subgroups (Theorem~\\ref{prop:algo}). Our main results concerning reflection-prefixes and their associated languages are summarized in the following theorem.", "full_context": "Let $(W,S)$ be a Coxeter system, with $S$ finite. We denote by $S^*$ the free monoid on the alphabet $S$. To distinguish between a word and its corresponding group element, we denote a word in $S^*$ using bold letters, $\\bold w = s_1 \\cdots s_k$, while the resulting product in $W$ is denoted by $w=s_1\\cdots s_k$. Let $w\\in W$, a word $\\bold w= s_1 \\cdots s_k\\in S^*$ is a {\\em reduced word for $w$} if $w=s_1\\cdots s_k$ and $k$ is minimal for this property; in this case the {\\em length} of $w$ is $\\ell(w)=k$. The identity $e\\in W$ is represented by the empty word, and $\\ell(e)=0$. We denote the set of all reduced words of $w\\in W$ by $\\Red(w)$, and the set of all reduced words in $W$ by $\\Red=\\bigsqcup_{w\\in W} \\Red(w)$, where $\\sqcup$ denotes the disjoint union.\nFinally, for any subset $A\\subseteq W$, the {\\em Poincar\\'e series of $A$} is the formal power series\n$A(q):=\\sum_{w\\in A} q^{\\ell(w)}.$\n\\smallskip\n\nIn this article, we study two combinatorial problems regarding the {\\em set of reflections} $T:=\\bigcup_{w\\in W} wSw^{-1}$ of a Coxeter group $W$:\n\\begin{enumerate}\n \\item Is the language of {\\em palindromic reduced words (for the reflections)} regular?\n \\item Are there explicit and elegant formulas for the Poincar\\'e series of the set $T$ of reflections ?\n\\end{enumerate}\nOur main contributions to those questions are the following: \n\\begin{itemize}\n\\item We introduce the notion of {\\em reflection-prefixes}, a class of elements in $W$ arising naturally from palindromic reduced words of reflections, and study their properties in relation to the {\\em root poset}, the {\\em dominance order on roots} and {\\em dihedral reflection subgroups}.\n\\item For any Coxeter system, we show that the {\\em language of reduced words for reflection-prefixes}, $\\Pref_T$, is regular. This is achieved using the family of $m$-canonical automata associated with $m$-Shi arrangements (see~\\cite[\\S3.4]{HNW} and \\cite{DFHM24}), which provide a family of finite deterministic automata recognizing $\\Pref_T$. As a consequence, we show that the generating function of palindromic reduced words is rational.\n\\item In the case of affine Coxeter groups, we derive a simple expression for the Poincar\\'e series of the set of reflections in terms of rational fractions. This formula is obtained from some symmetries of the Hasse diagram of the root poset.\n\\end{itemize}\n\nA further natural direction is the study of generating functions of words in $S^*$ in relation to the Coxeter system $(W,S)$. More precisely, given the canonical projection $\\pi:S^* \\rightarrow W$ sending a word $\\bold w= s_1 \\cdots s_k$ to the element $w=s_1\\cdots s_k$,\nit is natural to consider the {\\em ‘‘lifted'' Poincar\\'e series} of a subset $B\\subseteq S^*$ relative to $(W,S)$:\n$$\nB(q)=\\sum_{\\bold w\\in B} q^{\\ell(w)},\n$$\nwhere the notation ${\\ell(w)}$ is understood to be $\\ell(\\pi({\\bf w}))$.\n\nA longstanding open problem in combinatorics of Coxeter groups is the enumeration of $|\\Red(w)|$; for a detailed discussion see~\\cite[p.123]{BB}, and for various partial results, see~\\cite{Eriksson,Hart}. However, if we consider the set of all elements of a given length, we know that the numbers $r_k$ of reduced words of length $k\\in \\mathbb N$ is enumerated by the following Poincar\\'e series\n$$\n\\Red(q)=\\sum_{\\bold w\\in \\Red} q^{\\ell(w)}=\\sum_{k\\in\\mathbb N} r_kq^k,\n$$\nwhich is known to be rational. The proof follows from the existence of a finite deterministic automaton that recognizes the language $\\Red$; see for instance~\\cite[Theorem~4.9.1]{BB} for more details.\n\nIt is well-known that every reflection has an odd length. Let $t\\in T$ be a reflection of length $\\ell(t)=2k+1$, we define a {\\em $t$-prefix} to be any element $p_t\\in W$ such that $p_t\\leq_R t$ and $\\ell(p_t)=k+1$.\n It turns out that $ s_1\\cdots s_k s_{k+1} s_{k}\\cdots s_1$ is a palindromic reduced word for $t$ if and only if $s_1\\cdots s_k s_{k+1}$ is a reduced word for a $t$-prefix; see \\S\\ref{ss:reflectionprefixes}.\n\nReflection-prefixes turn out to be a useful combinatorial framework to study the root poset (Proposition~\\ref{prop:PrefRoot}), the dominance order (Proposition~\\ref{prop:PrefDom}) and to find the canonical generators of dihedral reflection subgroups (Theorem~\\ref{prop:algo}). Our main results concerning reflection-prefixes and their associated languages are summarized in the following theorem.\n\nLet $(W,S)$ be a Coxeter system, with $S$ finite. We denote by $S^*$ the free monoid on the alphabet $S$. To distinguish between a word and its corresponding group element, we denote a word in $S^*$ using bold letters, $\\bold w = s_1 \\cdots s_k$, while the resulting product in $W$ is denoted by $w=s_1\\cdots s_k$. Let $w\\in W$, a word $\\bold w= s_1 \\cdots s_k\\in S^*$ is a {\\em reduced word for $w$} if $w=s_1\\cdots s_k$ and $k$ is minimal for this property; in this case the {\\em length} of $w$ is $\\ell(w)=k$. The identity $e\\in W$ is represented by the empty word, and $\\ell(e)=0$. We denote the set of all reduced words of $w\\in W$ by $\\Red(w)$, and the set of all reduced words in $W$ by $\\Red=\\bigsqcup_{w\\in W} \\Red(w)$, where $\\sqcup$ denotes the disjoint union.\nFinally, for any subset $A\\subseteq W$, the {\\em Poincar\\'e series of $A$} is the formal power series\n$A(q):=\\sum_{w\\in A} q^{\\ell(w)}.$\n\\smallskip\n\nIn this article, we study two combinatorial problems regarding the {\\em set of reflections} $T:=\\bigcup_{w\\in W} wSw^{-1}$ of a Coxeter group $W$:\n\\begin{enumerate}\n \\item Is the language of {\\em palindromic reduced words (for the reflections)} regular?\n \\item Are there explicit and elegant formulas for the Poincar\\'e series of the set $T$ of reflections ?\n\\end{enumerate}\nOur main contributions to those questions are the following: \n\\begin{itemize}\n\\item We introduce the notion of {\\em reflection-prefixes}, a class of elements in $W$ arising naturally from palindromic reduced words of reflections, and study their properties in relation to the {\\em root poset}, the {\\em dominance order on roots} and {\\em dihedral reflection subgroups}.\n\\item For any Coxeter system, we show that the {\\em language of reduced words for reflection-prefixes}, $\\Pref_T$, is regular. This is achieved using the family of $m$-canonical automata associated with $m$-Shi arrangements (see~\\cite[\\S3.4]{HNW} and \\cite{DFHM24}), which provide a family of finite deterministic automata recognizing $\\Pref_T$. As a consequence, we show that the generating function of palindromic reduced words is rational.\n\\item In the case of affine Coxeter groups, we derive a simple expression for the Poincar\\'e series of the set of reflections in terms of rational fractions. This formula is obtained from some symmetries of the Hasse diagram of the root poset.\n\\end{itemize}\n\nA longstanding open problem in combinatorics of Coxeter groups is the enumeration of $|\\Red(w)|$; for a detailed discussion see~\\cite[p.123]{BB}, and for various partial results, see~\\cite{Eriksson,Hart}. However, if we consider the set of all elements of a given length, we know that the numbers $r_k$ of reduced words of length $k\\in \\mathbb N$ is enumerated by the following Poincar\\'e series\n$$\n\\Red(q)=\\sum_{\\bold w\\in \\Red} q^{\\ell(w)}=\\sum_{k\\in\\mathbb N} r_kq^k,\n$$\nwhich is known to be rational. The proof follows from the existence of a finite deterministic automaton that recognizes the language $\\Red$; see for instance~\\cite[Theorem~4.9.1]{BB} for more details.\n\nReflection-prefixes turn out to be a useful combinatorial framework to study the root poset (Proposition~\\ref{prop:PrefRoot}), the dominance order (Proposition~\\ref{prop:PrefDom}) and to find the canonical generators of dihedral reflection subgroups (Theorem~\\ref{prop:algo}). Our main results concerning reflection-prefixes and their associated languages are summarized in the following theorem.\n\nThe first assertion, the regularity of $\\Pref_T$, is established in Corollary~\\ref{cor:Refregular}. The proof relies on a construction of automata derived from the family of Garside shadow automata associated with $m$-Shi arrangements $(m \\in \\mathbb{N})$; for further details, see~\\cite{HNW, DFHM24}. The second assertion follows from Proposition~\\ref{prop:Pref1}, while the third and fourth are consequence of the first two.\n\n\\item if $w=s_1\\cdots s_k s_{k+1}$ is a reduced word for $w$, then $s_{k+1}=r$ and $t=s_1\\cdots s_k r s_k\\cdots s_1$ is a palindromic reduced word for $t$;\n\\item $\\alpha_t=w(\\alpha_r)= s_1\\cdots s_k(\\alpha_r)$ and $\\dep(\\alpha_t)=k=\\ell(w)-1$.\n\\end{enumerate}\n\\end{Proposition}\n\\begin{proof} Let $w=s_1\\cdots s_k r$ be a reduced word. We show that $D_R(w)=\\{r\\}$. By contradiction, assume that $|D_R(w)|\\geq 2$. So there exists a $s\\in D_R(w)$ with $s\\not = r$; set $I=\\{s,r\\}\\subseteq D_R(w)$. By Lemma~\\ref{lem:Descents}, we know that $W_I$ is finite and there is $u\\in X_I$ such that $w=u w_{\\circ,I}$ is a reduced product. So $\\ell(w)=\\ell(u)+m$, where $m=m_{sr}\\geq 2$. Now, observe that $w_{\\circ,I}rw_{\\circ,I}\\in I$, since the longest element of a finite Coxeter system acts by conjugation on the set of simple reflections. Therefore \n$$\nt= s_1\\cdots s_k s_{k+1} s_k\\cdots s_1 =s_1\\cdots s_k r s_k\\cdots s_1=w r w^{-1}=uw_{\\circ,I}rw_{\\circ,I}u^{-1},\n$$\nimplying that $\\ell(t)\\leq 2\\ell(u)+1=2\\ell(w)-2m+1<2\\ell(w)-1=\\ell(t)$, a contradiction. The remaining claims follow from the uniqueness of $r$, the definition of the depth of roots and from the equalities $\\ell(t)=2\\dep(\\alpha_t)+1=2\\ell(w)-1$.\n\\end{proof}\n\n\\begin{Example}[Universal Coxeter system]\n Let $(U_n, S)$ be the Universal Coxeter system of rank $n$, that is $U_n=\\langle S\\mid s^2=e \\ \\mbox{for all $s \\in S$} \\rangle$. Every element has a unique reduced word, so the language of palindromic reduced words for $(U_n,S)$ is given by $\\RPal= \\{s_1\\cdots s_{i-1}s_is_{i-1}\\cdots s_1 \\mid s_j\\in S \\mbox{ and } s_j\\neq s_{j+1}\\}$. Furthermore, the language of reduced words for reflection-prefixes coincides with the set of all reduced words in $W$, that is, $\\Pref_T=\\Red$.\n\\end{Example}\n\n\\begin{Corollary}\\label{cor:Refregular}\nThe language $\\Pref_T$ is regular and the generating function \n\\begin{align*}\n\\sum_{u\\in \\Pref_T} q^{|u|}& =\\sum_{t\\in T} |\\{s_1 \\cdots s_k s_{k+1} s_k\\cdots s_1 \\in \\Pref_T \\mid t=s_1 \\cdots s_k s_{k+1}s_k\\cdots s_1\\}| \\, q^{\\ell(t)}\\\\\n&=\\sum_{k\\in\\mathbb{N}} a_k q^k\n\\end{align*}\nis rational, where $a_k$ is the number of saturated chains of length $k$ in the root poset of $W$. In particular, $\\RPal(q)=q\\Pref_T(q^2)$ is rational.\n\\end{Corollary}", "post_theorem_intro_text_len": 3614, "post_theorem_intro_text": "The first assertion, the regularity of $\\Pref_T$, is established in Corollary~\\ref{cor:Refregular}. The proof relies on a construction of automata derived from the family of Garside shadow automata associated with $m$-Shi arrangements $(m \\in \\mathbb{N})$; for further details, see~\\cite{HNW, DFHM24}. The second assertion follows from Proposition~\\ref{prop:Pref1}, while the third and fourth are consequence of the first two.\n\n\\subsection*{Closed formulas for $T(q)$ in affine Coxeter systems} Remarkably, a solution to Stembridge's problem was provided in 1999 by de~Man~\\cite{DM}. However, the author seemed unaware of Stembridge's question and the combinatorics community was not aware of the solution. This oversight may be partly due to the title of de~Man's article {\\em ``The generating function for the number of roots of a Coxeter group''}, which only address counting reflections by length at the end of the article in \\cite[\\S5.2]{DM}. His solution is based on the Brink--Howlett automaton~\\cite{BH}  that recognizes the language of {\\em lexicographically ordered reduced word}, that is, each element $w$ has a unique lexicographically reduced word in $\\Red(w)$.  \n\nMore precisely, in \\cite[Theorem 4.1]{DM}, de Man shows that the generating function  $\\Phi^+(q)$ of the depth of positive roots is a sum of rational functions that are obtained from a subset of states of the Brink-Howlett automaton and from the enumeration of the vertices of a family of graphs parameterized by $T$. So the generating function $\\Phi^+(q)$ is rational. Since $T(q)=q\\, \\Phi^+(q^2)$, the Poincar\\'e series $T(q)$  is rational as well. However, the Brink--Howlett automaton contains a lot of states, most of them ``dead states'', which make this automaton difficult to use in practice. For instance, in affine type $\\widetilde{A}_2$, the number of states of the Brink--Howlett automaton is bounded by $2^6$; de Man provides some example of $T(q)$ using a computer to handle his formula.  \n\n In this article, we give, in the case of affine Coxeter systems, an alternative solution to Stembridge's problem by providing a simple formula for  $T(q)$. \n\n\\begin{Theorem}\\label{thm:Main1} Let $(W,S)$ be an affine Coxeter system. Then \n$$\nT(q):=\\sum_{t\\in T}q^{\\ell(t)}=q\\frac{P(q^2)}{1-q^{2M}},\n$$ \nwhere $M\\in \\mathbb{N}$ and $P(q)$ is a palindromic polynomial.\n\\end{Theorem}\n\nThe proof of the above theorem relies on two fundamental symmetries of the root poset on $\\Phi^+$ (Theorems \\ref{th:iso1} and \\ref{th:iso2}). These properties yield an explicit formula for $M$ and $P(q)$ (Theorem \\ref{th:main})), from which the above result follows.\n\n\\subsection*{Plan of the article} The article is organized as follows. In~\\S\\ref{sec:root-poset}, we recall some needed facts about the combinatorics of words and roots and, in particular, about two partial orders on the root system of a Coxeter system: the root poset and the dominance order, which were introduced by Brink and Howlett~\\cite{BH} in their work to show that Coxeter systems are automatic. In~\\S\\ref{se:reflectionprefix}, we discuss reflection-prefix, palindromic reduced words for reflections and automata and, in \\S\\ref{se:affine},  we give the enumeration of reflections by length in affine types. Finally, we propose some open problems in \\S\\ref{se:openproblems}.\n\n\\subsection*{Acknowledgements}  CH warmly thanks Francesco Brenti, James Parkinson and Christophe Reutenauer for very instructive discussions. \nThe authors are particularly indebted to Matthew Dyer for bringing de Man’s article~\\cite{DM} to their attention during the early stages of this project.", "sketch": "For Theorem~\\ref{thm:PrefMain}: the first assertion (regularity of $\\Pref_T$) is proved in Corollary~\\ref{cor:Refregular}, and the proof \"relies on a construction of automata derived from the family of Garside shadow automata associated with $m$-Shi arrangements $(m\\in\\mathbb{N})$\" (with details in~\\cite{HNW, DFHM24}). The second assertion \"follows from Proposition~\\ref{prop:Pref1}\\.\" The third and fourth assertions are stated to be \"consequence of the first two.\"", "expanded_sketch": "For Theorem~\\ref{thm:PrefMain}: the first assertion (regularity of $\\Pref_T$) is proved in\n\\begin{Corollary}\\label{cor:Refregular}\nThe language $\\Pref_T$ is regular and the generating function \n\\begin{align*}\n\\sum_{u\\in \\Pref_T} q^{|u|}& =\\sum_{t\\in T} |\\{s_1 \\cdots s_k s_{k+1} s_k\\cdots s_1 \\in \\Pref_T \\mid t=s_1 \\cdots s_k s_{k+1}s_k\\cdots s_1\\}| \\, q^{\\ell(t)}\\\\\n&=\\sum_{k\\in\\mathbb{N}} a_k q^k\n\\end{align*}\nis rational, where $a_k$ is the number of saturated chains of length $k$ in the root poset of $W$. In particular, $\\RPal(q)=q\\Pref_T(q^2)$ is rational.\n\\end{Corollary}\nand the proof relies on a construction of automata derived from the family of Garside shadow automata associated with $m$-Shi arrangements $(m\\in\\mathbb{N})$ (with details in~\\cite{HNW, DFHM24}). The second assertion follows from\n\\begin{Proposition}\\label{prop:Pref1} Let $t\\in T$ with corresponding positive root $\\alpha_t$. Let  $w\\in W$ and assume there is $r\\in D_R(w)$ such that $t=wrw^{-1}$ and $\\ell(t)=2\\ell(w)-1$. Then $D_R(w)=\\{r\\}$. Moreover: \n \\begin{enumerate}\n\n\\item if $w=s_1\\cdots s_k s_{k+1}$ is a reduced word for $w$, then $s_{k+1}=r$ and $t=s_1\\cdots s_k r s_k\\cdots s_1$ is a palindromic reduced word for $t$;\n\\item $\\alpha_t=w(\\alpha_r)= s_1\\cdots s_k(\\alpha_r)$ and  $\\dep(\\alpha_t)=k=\\ell(w)-1$.\n\\end{enumerate}\n\\end{Proposition}\nThe third and fourth assertions are stated to be consequence of the first two.", "expanded_theorem": "\\label{thm:PrefMain} Let $(W,S)$ be a Coxeter system. \n\\begin{enumerate}\n\\item The language $\\Pref_T$ of reduced words for reflection-prefixes is regular.\n\\item The language of palindromic reduced words is:\n$$\n\\operatorname{Pal}=\\{s_1\\cdots  s_k s_{k+1}  s_{k}\\cdots  s_1 \\in S^*\\mid  s_1\\cdots  s_k s_{k+1}\\in\\Pref_T\\}.\n$$\n\\item The following Poincar\\'e series is rational:\n$$\n\\Pref_T(q)=\\sum_{\\bold w \\in \\Pref_T} q^{\\ell(w)}=\\sum_{k\\geq 0} \\pal_kq^k,\n$$\nwhere $\\pal_k$ is the number of palindromic reduced words of length $k$.\n\n\\item The Poincar\\'e series $\\operatorname{Pal}(q)=q\\Pref_T(q^2)$ is rational.\n\\end{enumerate}", "theorem_type": ["Universal", "Classification or Bijection"], "mcq": {"question": "Let (W,S) be a Coxeter system with S finite, let S* be the free monoid on S, and let π: S* → W be the canonical projection. A word s1⋯sk ∈ S* is reduced if k is minimal among words representing π(s1⋯sk), and then ℓ(π(s1⋯sk)) = k. Let T = ⋃_{w∈W} wSw^{-1} be the set of reflections. For u,v ∈ W, write u ≤_R v for the right weak order, i.e. ℓ(v) = ℓ(u) + ℓ(u^{-1}v). Since every reflection t ∈ T has odd length, if ℓ(t)=2k+1 then a t-prefix is an element p_t ∈ W such that p_t ≤_R t and ℓ(p_t)=k+1. Let Pref_T ⊆ S* be the set of reduced words of all reflection-prefixes, and let Pal be the language of palindromic reduced words for reflections. For B ⊆ S*, define B(q)=∑_{w∈B} q^{ℓ(π(w))}. Which statement holds for every such Coxeter system?", "correct_choice": {"label": "A", "text": "The language Pref_T is regular; moreover, the palindromic reduced words are exactly\nPal = {s1⋯sk s_{k+1} s_k⋯s1 ∈ S* | s1⋯sk s_{k+1} ∈ Pref_T}.\nIn addition, the Poincaré series\nPref_T(q) = ∑_{w∈Pref_T} q^{ℓ(π(w))} = ∑_{k≥0} pal_k q^k\nis rational, where pal_k is the number of palindromic reduced words of length k; consequently,\nPal(q) = q Pref_T(q^2)\nis rational."}, "choices": [{"label": "B", "text": "The language Pref_T is regular; moreover, the palindromic reduced words are exactly\nPal = {s1⋯sk s_{k+1} s_k⋯s1 ∈ S* | s1⋯sk ∈ Pref_T}.\nIn addition, the Poincaré series\nPref_T(q) = ∑_{w∈Pref_T} q^{ℓ(π(w))} = ∑_{k≥0} pal_k q^k\nis rational, where pal_k is the number of palindromic reduced words of length k; consequently,\nPal(q) = q Pref_T(q^2)\nis rational."}, {"label": "C", "text": "The language Pref_T is regular. Moreover, the Poincaré series\nPal(q)=\\sum_{w\\in \\mathrm{Pal}} q^{\\ell(\\pi(w))}\nis rational."}, {"label": "D", "text": "The language Pref_T is regular; moreover, the palindromic reduced words are exactly\nPal = {s1⋯sk s_{k+1} s_k⋯s1 ∈ S* | s1⋯sk s_{k+1} ∈ Pref_T}.\nIn addition, the Poincaré series\nPref_T(q) = ∑_{w∈Pref_T} q^{ℓ(π(w))} = ∑_{k≥0} pal_k q^k\nis rational, where pal_k is the number of palindromic reduced words of length k; consequently,\nPal(q) = Pref_T(q^2)\nis rational."}, {"label": "E", "text": "The language Pref_T is regular; moreover, every palindromic reduced word has the form\ns1⋯sk s_{k+1} s_k⋯s1 with s1⋯sk s_{k+1} ∈ Pref_T, and every such word lies in Pal.\nIn addition, there is a single finite deterministic automaton, independent of any auxiliary parameter, that recognizes Pref_T for every Coxeter system; consequently the series Pref_T(q) and Pal(q) are rational."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "trace_identity", "tampered_component": "full-prefix length k+1 in the characterization of palindromes", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped the exact characterization of Pal and the explicit identity Pal(q)=q Pref_T(q^2)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "trace_identity", "tampered_component": "odd-length shift encoded by Pal(q)=q Pref_T(q^2)", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "dependence on the family of m-Shi/Garside shadow automata rather than one parameter-free universal automaton", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem provides definitions and notation but does not explicitly reveal the correct statement. There is no direct answer leakage, though the setup narrows the topic to a specific theorem."}, "TAS": {"score": 0, "justification": "The item is essentially a theorem-identification question: the correct option states the full theorem almost verbatim, while the others are small perturbations of it. This makes it close to a direct restatement rather than a genuinely independent conclusion-selection task."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to detect subtle errors such as the off-by-one prefix condition and the missing factor of q in Pal(q)=q Pref_T(q^2). However, the task mainly rewards recall or careful comparison of theorem-level statements rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: weakening the conclusion, confusing the prefix condition, dropping the odd-length shift factor, and overclaiming automaton uniformity. They are distinct and well aligned with likely misconceptions."}, "total_score": 5, "overall_assessment": "A technically well-crafted theorem-recognition MCQ with strong distractors, but it is close to a restatement of the target result and only moderately tests genuine generative reasoning."}}
{"id": "2602.16506v1", "paper_link": "http://arxiv.org/abs/2602.16506v1", "theorems_cnt": 5, "theorem": {"env_name": "Thm", "content": "\\label{thm1.1}\n\tAssume $N\\ge1$, $m\\ge2$, and \\textbf{\\textup{$(A_1)$--$(A_3)$}}. Then for $\\varepsilon>0$, \n\t\\eqref{eq1.1} admits a sequence of sign-changing solutions\n\t\\[\n\t\\mathbf{u}^{(k)}_\\varepsilon=(u^{(k)}_{\\varepsilon,1},\\dots,u^{(k)}_{\\varepsilon,m})\\in H, ~ k\\in\\mathbb{N}^*,\n\t\\]\n\tsatisfying\n\t\\[\n\t0<J_\\varepsilon\\bigl(\\mathbf{u}^{(1)}_\\varepsilon\\bigr)\n\t< \\cdots < J_\\varepsilon\\bigl(\\mathbf{u}^{(k)}_\\varepsilon\\bigr)\n    < J_\\varepsilon\\bigl(\\mathbf{u}^{(k+1)}_\\varepsilon\\bigr)\n\t< \\cdots, ~\\text{and}~ \\lim\\limits_{k\\to\\infty}\n\tJ_\\varepsilon\\bigl(\\mathbf{u}^{(k)}_\\varepsilon\\bigr)=+\\infty.\n\t\\]", "start_pos": 10852, "end_pos": 11478, "label": "thm1.1"}, "ref_dict": {"thm1.1": "\\begin{Thm}\\label{thm1.1}\n\tAssume $N\\ge1$, $m\\ge2$, and \\textbf{\\textup{$(A_1)$--$(A_3)$}}. Then for $\\varepsilon>0$, \n\t\\eqref{eq1.1} admits a sequence of sign-changing solutions\n\t\\[\n\t\\mathbf{u}^{(k)}_\\varepsilon=(u^{(k)}_{\\varepsilon,1},\\dots,u^{(k)}_{\\varepsilon,m})\\in H, ~ k\\in\\mathbb{N}^*,\n\t\\]\n\tsatisfying\n\t\\[\n\t0<J_\\varepsilon\\bigl(\\mathbf{u}^{(1)}_\\varepsilon\\bigr)\n\t< \\cdots < J_\\varepsilon\\bigl(\\mathbf{u}^{(k)}_\\varepsilon\\bigr)\n    < J_\\varepsilon\\bigl(\\mathbf{u}^{(k+1)}_\\varepsilon\\bigr)\n\t< \\cdots, ~\\text{and}~ \\lim\\limits_{k\\to\\infty}\n\tJ_\\varepsilon\\bigl(\\mathbf{u}^{(k)}_\\varepsilon\\bigr)=+\\infty.\n\t\\]\n\\end{Thm}", "eq1.1": "\\begin{equation}\\label{eq1.1}\n\t\\begin{cases}\n\t\t-\\Delta u_i + u_i\n\t\t= \\mu_i Q_\\varepsilon(x-y_i) |u_i|^{2p-2}u_i\n\t\t+ \\sum_{j\\neq i} \\lambda_{ij} |u_j|^{p} |u_i|^{p-2}u_i, \\\\[6pt]\n\t\tu_i \\in H^1(\\mathbb{R}^N),\\ i=1,\\dots,m.\n\t\\end{cases}\t\n\\end{equation}", "eq1.2": "\\begin{equation}\\label{eq1.2}\n\t\\begin{cases}\n\t-\\Delta U_i + U_i\n\t= \\mu_i Q(-y_i)\\,|U_i|^{2p-2}U_i\n\t+ \\sum_{j\\neq i}\\lambda_{ij}|U_j|^p|U_i|^{p-2}U_i,\\\\[6pt]\n\t U_i\\in H^1(\\mathbb{R}^N),\\ i=1,\\dots,m.\n\\end{cases}\n\\end{equation}", "thm1.2": "\\begin{Thm}\\label{thm1.2}\n\tAssume $N\\ge1$, $m\\ge2$, and \\textup{(A$_1$)--(A$_3$)}.\n\tLet $\\{\\varepsilon_n\\}\\subset(0,+\\infty)$ be a sequence with $\\varepsilon_n\\to 0$.\n\tFor each fixed $k\\in\\mathbb{N}^*$, let $\\{\\mathbf{u}^{(k)}_{\\varepsilon_n}\\}$ be the sequence of\n\tsign-changing solutions given by Theorem~\\ref{thm1.1}.\n\tThen there exist $m$ sequences $\\{x^{(k)}_{\\varepsilon_n,i}\\}\\subset\\mathbb{R}^N$ and a sign-changing solution\n\t$\\mathbf{U}^{(k)}=(U^{(k)}_1,\\dots,U^{(k)}_m)\\in H$ of the limiting system \\eqref{eq1.2}\n\tsuch that, up to a subsequence,\n\t\\begin{itemize}\n\t\t\\item[(i)]\n\t\t$\\varepsilon_n x_{\\varepsilon_n,i}^{(k)}\\to 0$ as $n\\to\\infty$ for $i=1,\\dots,m$.\n\tIn particular, for every $R>0$,\n\t\t\\[\n\t\tQ\\bigl(\\varepsilon_n(\\,\\cdot+x_{\\varepsilon_n,i}^{(k)})-y_i\\bigr)\\to Q(-y_i)\n\t\t~\\text{uniformly in}~B_R(0),~i=1,\\dots,m.\n\t\t\\]\n\n\t\t\\item[(ii)]\n\t\t$u^{(k)}_{\\varepsilon_n,i}(\\,\\cdot + x^{(k)}_{\\varepsilon_n,i})\n\t\t\\longrightarrow U^{(k)}_i$ in $H^1(\\mathbb{R}^N)$ as $n\\to\\infty$ for  $i=1,\\dots,m$. \n\n\t\\item[(iii)]\n\tfor every $q\\in[1,\\infty)$,\n\t\\[\n\t\\lim_{R\\to+\\infty}\\,\\limsup_{n\\to\\infty}\n\t\\int_{\\mathbb{R}^N\\setminus B_{R/{\\varepsilon_n}}(0)}\n\t|u^{(k)}_{\\varepsilon_n, i}(x)|^q\\,dx = 0,\n\t~ i=1,\\dots,m,\n\t\\]\n\tand for $q=\\infty$,\n\t\\[\n\t\\lim_{R\\to+\\infty}\\,\\limsup_{n\\to\\infty}\n\t\\|u^{(k)}_{\\varepsilon_n, i}\\|_{L^\\infty(\\mathbb{R}^N\\setminus B_{R/{\\varepsilon_n}}(0))} = 0,\n\t~ i=1,\\dots,m.\n\t\\]\n\n\t\t\\item[(iv)]\n\t\t$\\{\\mathbf{U}^{(k)}\\}$ are pairwise distinct and satisfy\n\t\t\\[\n\t\tJ_0\\big(\\mathbf{U}^{(1)}\\big) < J_0\\big(\\mathbf{U}^{(2)}\\big) < \\cdots \\to +\\infty.\n\t\t\\]\n\t\\end{itemize}\n\\end{Thm}"}, "pre_theorem_intro_text_len": 8231, "pre_theorem_intro_text": "\\label{introduction}\n\\par \nIn this paper we consider the following competitive Schr\\\"odinger system\n\\begin{equation}\\label{eq1.1}\n\t\\begin{cases}\n\t\t-\\Delta u_i + u_i\n\t\t= \\mu_i Q_\\varepsilon(x-y_i) |u_i|^{2p-2}u_i\n\t\t+ \\sum_{j\\neq i} \\lambda_{ij} |u_j|^{p} |u_i|^{p-2}u_i, \\\\[6pt]\n\t\tu_i \\in H^1(\\mathbb{R}^N),\\ i=1,\\dots,m.\n\t\\end{cases}\t\n\\end{equation}\nwhere $N\\ge1$, $m\\ge2$. Nonlinear Schr\\\"odinger systems with competing interactions have been extensively studied in recent years, motivated both by physical models such as Bose--Einstein\ncondensates and nonlinear optics, and by the rich mathematical structure of coupled\nelliptic equations.\nAmong the various phenomena that may occur in this context, a\nparticularly interesting situation arises when the nonlinear potentials are localized in\nsmall regions that shrink to isolated points as a small parameter $\\varepsilon>0$ tends to zero. In this case,\nsolutions may concentrate around finitely many prescribed points and exhibit a delicate\ninteraction between the different components.\n\\par\n We now state the structural assumptions that will be used throughout the paper. They are standard in this context and reflect the subcritical regime and the competitive nature of the system.\n\\begin{itemize}\n  \\item [$(A_1)$] $1<p<\\frac{2^*}{2}$ if $N\\ge3$, and $p>1$ if $N=1,2$,\nwhere $2^*=\\frac{2N}{N-2}$ if $N\\ge3$.\n  \\item [$(A_2)$] $\\mu_i>0$, $i=1,\\dots,m$, $\\lambda_{ij}=\\lambda_{ji}<0$, for all $i\\neq j$.\n\\item[$(A_3)$]\nLet $Q_\\varepsilon(x-y_i):=Q(\\varepsilon x-y_i)$, where\n$Q\\in C(\\mathbb{R}^N)$ is nonnegative, $\\operatorname{supp}(Q)$ is bounded, and there exist exactly pairwise distinct points $y_1,\\dots,y_m\\in\\mathbb{R}^N$ such that\n\\[\nQ(-y_i)=\\|Q\\|_\\infty=\\max\\limits_{x\\in\\mathbb{R}^N}Q(x)~\\text{for all }i=1,\\dots,m.\n\\]\n\\end{itemize}\n\\par\nIn this setting, the solutions $u_i$ may be interpreted as standing wave profiles of\ndifferent species which are attracted to regions where $Q_\\varepsilon$ is\npositive and repelled from their complement. The assumption $\\lambda_{ij}<0$ means that\ndistinct components repel each other, which in turn favors spatial segregation, so the system \\eqref{eq1.1} is competitive. \n\\begin{Exam}\\label{exam1.1}\nThere are functions satisfying $(A_3)$.\nAssume $N\\ge2$ and fix $m$ pairwise distinct points $y_1,\\dots,y_m\\in\\mathbb R^N$.\nSet\n\\[\nd:=\\min_{i\\neq j}|y_i-y_j|>0,\n~\nr:=\\frac{d}{4}.\n\\]\nLet $\\rho\\in C_0^\\infty(\\mathbb R^N)$ be the standard compactly supported bump\n\\[\n\\rho(x):=\n\\begin{cases}\n\\exp\\!\\bigl(-\\frac{1}{1-|x|^2}\\bigr), & |x|<1,\\\\[2mm]\n0, & |x|\\ge 1,\n\\end{cases}\n~\n\\text{and define}~\n\\phi(x):=\\frac{\\rho(x)}{\\rho(0)} .\n\\]\nThen $\\phi\\in C_0^\\infty(\\mathbb R^N)$, $\\phi\\ge0$, $\\operatorname{supp}(\\phi)\\subset \\overline{B_1(0)}$,\n$\\phi(0)=1$, and $\\phi(x)<1$ for all $x\\neq0$.\n\nFor each $i=1,\\dots,m$ define\n\\[\n\\phi_i(x):=\\phi\\!\\left(\\frac{x+y_i}{r}\\right),\n~ x\\in\\mathbb R^N,\n\\]\nand set\n\\[\nQ(x):=\\max_{1\\le i\\le m}\\phi_i(x),~ x\\in\\mathbb R^N.\n\\]\nThen $Q$ satisfies $(A_3)$.\n\\end{Exam}\n\\par\nIn the scalar case, a fundamental contribution is due to Ackermann and Szulkin~\\cite{AckermannSzulkin2013},\nwho showed that when the positive region of the nonlinear coefficient collapses to isolated\npoints, every nontrivial solution concentrates at one of these cores, and ground states\nselect a single core without splitting their mass.  In this framework, the sign structure of the nonlinearity is the sole driver of localization, without any periodicity or\nsymmetry assumption on the linear part. Since then, their approach has been extended\nin various directions, including problems on the whole space and coupled systems;\nsee, e.g.,~\\cite{ClappHernandezSantamariaSaldana2025,ClappSaldanaSzulkin2025,ZhongZou2014}\nand the references therein. We point out that these works focus primarily on positive solutions or\nthe least energy solutions, for which the Mountain Pass Theorem (\\cite{Willem1996}) and the\nmaximum principles are available. Very recently, in \\cite{ClappSaldanaSzulkin2025}, Clapp, Salda\\~na and Szulkin studied the system \\eqref{eq1.1} with a single or multiple shrinking domains, the existence of nonnegative least energy solution and concentration behavior were obtained as $\\varepsilon\\to0$. Furthermore, the\nauthors characterized the limit profile and described how the components either decouple\nor remain coupled depending on the geometry of the attraction centers. Related results\nfor scalar equations, including concentration of semiclassical states and the description\nof their limit profiles, were obtained in earlier works such as\n\\cite{AckermannSzulkin2013,FangWang2020}, and see \n\\cite{TavaresTerracini2012} and the references therein for phase separation phenomena of competitive systems. We refer \\cite{Clapp-Pistoia-Saldana-2026-JMPA} to the existence of concentrating positive solutions via a Lyapunov Schmidt reduction strategy.\n\\par\nHowever, much less is known about sign-changing solutions for systems of the form\n\\eqref{eq1.1}. Even for the scalar equation, constructing nodal solutions requires a refined variational approach based on nodal Nehari sets and careful control of the positive and negative parts of the solutions. As far as we known, there are only two papers \\cite{ClappHernandezSantamariaSaldana2025,Clapp-Pistoia-Saldana-2026-JMPA} concerned with this topic. In \\cite{ClappHernandezSantamariaSaldana2025}, Clapp, Hern\\'andez-Santamar\\'ia and Salda\\~na obtained the existence and concentration of  nodal solutions via the nodal Nehari manifold method,  and characterized the symmetries and the polynomial decay of the least-energy nodal limiting profiles. In \\cite{Clapp-Pistoia-Saldana-2026-JMPA}, Clapp, Pistoia and Salda\\~na established the existence of concentrating nodal solutions via a Lyapunov Schmidt reduction method.\nBut for systems, the situation is substantially more\ndelicate, since one has to keep track simultaneously of the sign structure of each\ncomponent and of the competitive couplings between different components. It seems that there is no work concerned with the existence and concentration of sign-changing solutions of the system \\eqref{eq1.1}.\n\\par\nFirst, we state the meaning of sign-changing solutions of \\eqref{eq1.1}.\n\\begin{Def}\\label{def1.1}\nA solution $\\mathbf{u}=(u_1,u_2,\\cdots,u_m)$ of \\eqref{eq1.1} is called \\emph{sign-changing} if for each $i=1,\\dots,m$, both the positive part $u_i^+=\\max\\{u_i,0\\}$ and the negative part $u_i^-=\\min\\{u_i,0\\}$ are nonzero in $H^1(\\mathbb{R}^N)$, i.e., $\\|u_i^\\pm\\|_{H^1}>0$.\n\\end{Def}\n\\par\nThe purpose of\nthis paper is to develop a variational framework to seek for infinitely many sign-changing solutions of \\eqref{eq1.1}, and to analyze their concentration behavior as $\\varepsilon\\to 0$. \nThe main technical difficulties arise from the sign-changing nature of the solutions. \nFirst, the natural Nehari manifold associated with nonnegative solutions does\nnot capture directly sign changing solutions, so for elements in the Nehari manifold, one has to impose constraints separately on the\npositive and negative parts of each component. This leads to a nonlinear and\nnonconvex constraint set on which the direct minimization is not available.  \nSecond, the competitive couplings $\\lambda_{ij}<0$ mix the components in a nontrivial way, and one must show that on the nodal constraint the energy still controls the\n$H^{1}$-norm uniformly, so that Palais-Smale sequences are bounded.  \nThird, compactness issues are more severe in the nodal setting, one has to\nprevent vanishing of some sign components and to rule out loss of mass at\ninfinity while keeping track of the nodal structure. So it is interesting to seek for sign-changing solutions of \\eqref{eq1.1} since the above difficulties prevent one to use techniques based on the maximum principles, the comparison arguments, or the monotonicity methods. \nThese obstacles make it necessary to develop a variational strategy that is tailored to\nthe nodal structure of the problem and that remains effective in the presence of\ncompetitive couplings. \n\\par\nWe are now in a position to state the main results. Our first theorem\nshows that, for small $\\varepsilon>0$, system~\\eqref{eq1.1} admits a sequence of sign-changing solutions with unbounded energies.", "context": "For each $i=1,\\dots,m$ define\n\\[\n\\phi_i(x):=\\phi\\!\\left(\\frac{x+y_i}{r}\\right),\n~ x\\in\\mathbb R^N,\n\\]\nand set\n\\[\nQ(x):=\\max_{1\\le i\\le m}\\phi_i(x),~ x\\in\\mathbb R^N.\n\\]\nThen $Q$ satisfies $(A_3)$.\n\\end{Exam}\n\\par\nIn the scalar case, a fundamental contribution is due to Ackermann and Szulkin~\\cite{AckermannSzulkin2013},\nwho showed that when the positive region of the nonlinear coefficient collapses to isolated\npoints, every nontrivial solution concentrates at one of these cores, and ground states\nselect a single core without splitting their mass. In this framework, the sign structure of the nonlinearity is the sole driver of localization, without any periodicity or\nsymmetry assumption on the linear part. Since then, their approach has been extended\nin various directions, including problems on the whole space and coupled systems;\nsee, e.g.,~\\cite{ClappHernandezSantamariaSaldana2025,ClappSaldanaSzulkin2025,ZhongZou2014}\nand the references therein. We point out that these works focus primarily on positive solutions or\nthe least energy solutions, for which the Mountain Pass Theorem (\\cite{Willem1996}) and the\nmaximum principles are available. Very recently, in \\cite{ClappSaldanaSzulkin2025}, Clapp, Salda\\~na and Szulkin studied the system \\eqref{eq1.1} with a single or multiple shrinking domains, the existence of nonnegative least energy solution and concentration behavior were obtained as $\\varepsilon\\to0$. Furthermore, the\nauthors characterized the limit profile and described how the components either decouple\nor remain coupled depending on the geometry of the attraction centers. Related results\nfor scalar equations, including concentration of semiclassical states and the description\nof their limit profiles, were obtained in earlier works such as\n\\cite{AckermannSzulkin2013,FangWang2020}, and see \n\\cite{TavaresTerracini2012} and the references therein for phase separation phenomena of competitive systems. We refer \\cite{Clapp-Pistoia-Saldana-2026-JMPA} to the existence of concentrating positive solutions via a Lyapunov Schmidt reduction strategy.\n\\par\nHowever, much less is known about sign-changing solutions for systems of the form\n\\eqref{eq1.1}. Even for the scalar equation, constructing nodal solutions requires a refined variational approach based on nodal Nehari sets and careful control of the positive and negative parts of the solutions. As far as we known, there are only two papers \\cite{ClappHernandezSantamariaSaldana2025,Clapp-Pistoia-Saldana-2026-JMPA} concerned with this topic. In \\cite{ClappHernandezSantamariaSaldana2025}, Clapp, Hern\\'andez-Santamar\\'ia and Salda\\~na obtained the existence and concentration of nodal solutions via the nodal Nehari manifold method, and characterized the symmetries and the polynomial decay of the least-energy nodal limiting profiles. In \\cite{Clapp-Pistoia-Saldana-2026-JMPA}, Clapp, Pistoia and Salda\\~na established the existence of concentrating nodal solutions via a Lyapunov Schmidt reduction method.\nBut for systems, the situation is substantially more\ndelicate, since one has to keep track simultaneously of the sign structure of each\ncomponent and of the competitive couplings between different components. It seems that there is no work concerned with the existence and concentration of sign-changing solutions of the system \\eqref{eq1.1}.\n\\par\nFirst, we state the meaning of sign-changing solutions of \\eqref{eq1.1}.\n\\begin{Def}\\label{def1.1}\nA solution $\\mathbf{u}=(u_1,u_2,\\cdots,u_m)$ of \\eqref{eq1.1} is called \\emph{sign-changing} if for each $i=1,\\dots,m$, both the positive part $u_i^+=\\max\\{u_i,0\\}$ and the negative part $u_i^-=\\min\\{u_i,0\\}$ are nonzero in $H^1(\\mathbb{R}^N)$, i.e., $\\|u_i^\\pm\\|_{H^1}>0$.\n\\end{Def}\n\\par\nThe purpose of\nthis paper is to develop a variational framework to seek for infinitely many sign-changing solutions of \\eqref{eq1.1}, and to analyze their concentration behavior as $\\varepsilon\\to 0$. \nThe main technical difficulties arise from the sign-changing nature of the solutions. \nFirst, the natural Nehari manifold associated with nonnegative solutions does\nnot capture directly sign changing solutions, so for elements in the Nehari manifold, one has to impose constraints separately on the\npositive and negative parts of each component. This leads to a nonlinear and\nnonconvex constraint set on which the direct minimization is not available. \nSecond, the competitive couplings $\\lambda_{ij}<0$ mix the components in a nontrivial way, and one must show that on the nodal constraint the energy still controls the\n$H^{1}$-norm uniformly, so that Palais-Smale sequences are bounded. \nThird, compactness issues are more severe in the nodal setting, one has to\nprevent vanishing of some sign components and to rule out loss of mass at\ninfinity while keeping track of the nodal structure. So it is interesting to seek for sign-changing solutions of \\eqref{eq1.1} since the above difficulties prevent one to use techniques based on the maximum principles, the comparison arguments, or the monotonicity methods. \nThese obstacles make it necessary to develop a variational strategy that is tailored to\nthe nodal structure of the problem and that remains effective in the presence of\ncompetitive couplings. \n\\par\nWe are now in a position to state the main results. Our first theorem\nshows that, for small $\\varepsilon>0$, system~\\eqref{eq1.1} admits a sequence of sign-changing solutions with unbounded energies.", "full_context": "For each $i=1,\\dots,m$ define\n\\[\n\\phi_i(x):=\\phi\\!\\left(\\frac{x+y_i}{r}\\right),\n~ x\\in\\mathbb R^N,\n\\]\nand set\n\\[\nQ(x):=\\max_{1\\le i\\le m}\\phi_i(x),~ x\\in\\mathbb R^N.\n\\]\nThen $Q$ satisfies $(A_3)$.\n\\end{Exam}\n\\par\nIn the scalar case, a fundamental contribution is due to Ackermann and Szulkin~\\cite{AckermannSzulkin2013},\nwho showed that when the positive region of the nonlinear coefficient collapses to isolated\npoints, every nontrivial solution concentrates at one of these cores, and ground states\nselect a single core without splitting their mass. In this framework, the sign structure of the nonlinearity is the sole driver of localization, without any periodicity or\nsymmetry assumption on the linear part. Since then, their approach has been extended\nin various directions, including problems on the whole space and coupled systems;\nsee, e.g.,~\\cite{ClappHernandezSantamariaSaldana2025,ClappSaldanaSzulkin2025,ZhongZou2014}\nand the references therein. We point out that these works focus primarily on positive solutions or\nthe least energy solutions, for which the Mountain Pass Theorem (\\cite{Willem1996}) and the\nmaximum principles are available. Very recently, in \\cite{ClappSaldanaSzulkin2025}, Clapp, Salda\\~na and Szulkin studied the system \\eqref{eq1.1} with a single or multiple shrinking domains, the existence of nonnegative least energy solution and concentration behavior were obtained as $\\varepsilon\\to0$. Furthermore, the\nauthors characterized the limit profile and described how the components either decouple\nor remain coupled depending on the geometry of the attraction centers. Related results\nfor scalar equations, including concentration of semiclassical states and the description\nof their limit profiles, were obtained in earlier works such as\n\\cite{AckermannSzulkin2013,FangWang2020}, and see \n\\cite{TavaresTerracini2012} and the references therein for phase separation phenomena of competitive systems. We refer \\cite{Clapp-Pistoia-Saldana-2026-JMPA} to the existence of concentrating positive solutions via a Lyapunov Schmidt reduction strategy.\n\\par\nHowever, much less is known about sign-changing solutions for systems of the form\n\\eqref{eq1.1}. Even for the scalar equation, constructing nodal solutions requires a refined variational approach based on nodal Nehari sets and careful control of the positive and negative parts of the solutions. As far as we known, there are only two papers \\cite{ClappHernandezSantamariaSaldana2025,Clapp-Pistoia-Saldana-2026-JMPA} concerned with this topic. In \\cite{ClappHernandezSantamariaSaldana2025}, Clapp, Hern\\'andez-Santamar\\'ia and Salda\\~na obtained the existence and concentration of nodal solutions via the nodal Nehari manifold method, and characterized the symmetries and the polynomial decay of the least-energy nodal limiting profiles. In \\cite{Clapp-Pistoia-Saldana-2026-JMPA}, Clapp, Pistoia and Salda\\~na established the existence of concentrating nodal solutions via a Lyapunov Schmidt reduction method.\nBut for systems, the situation is substantially more\ndelicate, since one has to keep track simultaneously of the sign structure of each\ncomponent and of the competitive couplings between different components. It seems that there is no work concerned with the existence and concentration of sign-changing solutions of the system \\eqref{eq1.1}.\n\\par\nFirst, we state the meaning of sign-changing solutions of \\eqref{eq1.1}.\n\\begin{Def}\\label{def1.1}\nA solution $\\mathbf{u}=(u_1,u_2,\\cdots,u_m)$ of \\eqref{eq1.1} is called \\emph{sign-changing} if for each $i=1,\\dots,m$, both the positive part $u_i^+=\\max\\{u_i,0\\}$ and the negative part $u_i^-=\\min\\{u_i,0\\}$ are nonzero in $H^1(\\mathbb{R}^N)$, i.e., $\\|u_i^\\pm\\|_{H^1}>0$.\n\\end{Def}\n\\par\nThe purpose of\nthis paper is to develop a variational framework to seek for infinitely many sign-changing solutions of \\eqref{eq1.1}, and to analyze their concentration behavior as $\\varepsilon\\to 0$. \nThe main technical difficulties arise from the sign-changing nature of the solutions. \nFirst, the natural Nehari manifold associated with nonnegative solutions does\nnot capture directly sign changing solutions, so for elements in the Nehari manifold, one has to impose constraints separately on the\npositive and negative parts of each component. This leads to a nonlinear and\nnonconvex constraint set on which the direct minimization is not available. \nSecond, the competitive couplings $\\lambda_{ij}<0$ mix the components in a nontrivial way, and one must show that on the nodal constraint the energy still controls the\n$H^{1}$-norm uniformly, so that Palais-Smale sequences are bounded. \nThird, compactness issues are more severe in the nodal setting, one has to\nprevent vanishing of some sign components and to rule out loss of mass at\ninfinity while keeping track of the nodal structure. So it is interesting to seek for sign-changing solutions of \\eqref{eq1.1} since the above difficulties prevent one to use techniques based on the maximum principles, the comparison arguments, or the monotonicity methods. \nThese obstacles make it necessary to develop a variational strategy that is tailored to\nthe nodal structure of the problem and that remains effective in the presence of\ncompetitive couplings. \n\\par\nWe are now in a position to state the main results. Our first theorem\nshows that, for small $\\varepsilon>0$, system~\\eqref{eq1.1} admits a sequence of sign-changing solutions with unbounded energies.\n\n\\medskip\n\nBy \\eqref{eq2.3} for $\\mathbf{u}_{\\varepsilon_n}$ and make a change of variables we obtain\n\\begin{equation}\\label{eq5.20}\n\\begin{aligned}\n\\|\\tilde{u}_{j,n}^\\sigma\\|_{H^1}^2 &= \\mu_j \\int_{\\mathbb{R}^N} Q_{\\varepsilon_n}(x + x_{i,n} - y_j) |\\tilde{u}_{j,n}^\\sigma|^{2p} \\, dx \n+ \\sum_{k \\neq j} \\lambda_{jk} \\int_{\\mathbb{R}^N} |\\tilde{u}_{k,n}|^p \n|\\tilde{u}_{j,n}^\\sigma|^p \\, dx\\\\\n&\\le \\mu_j \\int_{\\mathbb{R}^N} Q_{\\varepsilon_n}(x + x_{i,n} - y_j) |\\tilde{u}_{j,n}^\\sigma|^{2p} \\, dx,\n\\end{aligned}\n\\end{equation}\nsince $\\lambda_{jk} < 0$.\n\\par\nFor any $M>0$, it follows from $\\tilde{u}_{j,n}^\\sigma \\to 0$ in $L^{2p}(B_M(0))$ and $|Q({\\varepsilon_n}(x + x_{i,n} - y_j)| \\leq \\|Q\\|_\\infty$ that \n\\begin{equation}\\label{eq5.21}\n\\int_{B_M(0)} Q_{\\varepsilon_n}(x + x_{i,n} - y_j) |\\tilde{u}_{j,n}^\\sigma|^{2p} \\, dx \\to 0.\n\\end{equation}\nOn the other hand, by the standard Agmon-type estimates (see \\cite{Agmon1982}), there exist $C,\\alpha>0$ independent of $n$, such that\n\\[\n|\\tilde{u}_{j,n}(x)|\\leq C e^{-\\alpha|x|} \\quad \\text{for all } x\\in\\mathbb{R}^N.\n\\]\nConsequently,\n\\[\n\\int_{|x|>M} Q_{\\epsilon_n}(x+x_{i,n}-y_j)|\\tilde{u}_{j,n}^\\sigma|^{2p}dx \n\\leq \\|Q\\|_\\infty \\int_{|x|>M} |\\tilde{u}_{j,n}^\\sigma|^{2p}dx \n\\leq \\|Q\\|_\\infty\\, C^{2p} \\int_{|x|>M} e^{-2p\\alpha|x|}dx.\n\\]\nFor any $\\eta>0$, choose $M=M(\\eta)>0$ such that\n\\[\n\\int_{|x|>M} e^{-2p\\alpha|x|}dx < \\eta,\n\\]\njointly with \\eqref{eq5.20} and \\eqref{eq5.21}, implies that $\\|\\tilde{u}_{j,n}^\\sigma\\|_{H^1}\\to0$. This contradicts \\eqref{eq5.19}.\n Therefore, \n$U_j^\\sigma \\neq 0$ for every $j$ and $\\sigma$, i.e., $\\mathbf{U} \\in \\mathcal{A}$.\n\\par\nWe claim that the sequence $\\{\\varepsilon_n x_{i,n}\\}$ is bounded.\n Assume by contradiction that, up to a subsequence, $|\\varepsilon_n x_{i,n}|\\to\\infty$.\n Fix $\\varphi\\in C_c^\\infty(\\R^N)$, and set $K:=\\operatorname{supp}\\varphi$.\n Since $\\mathbf u_{\\varepsilon_n}$ is a weak solution of the system, changing variables gives\n \\begin{align*}\n \\int_{\\R^N}\\bigl(\\nabla \\tilde u_{\\ell,n}\\cdot\\nabla\\varphi+\\tilde u_{\\ell,n}\\varphi\\bigr)\\,dx\n &=\n \\mu_\\ell\\int_{\\R^N} Q\\bigl(\\varepsilon_n(x+x_{i,n})-y_\\ell\\bigr)\\,\n |\\tilde u_{\\ell,n}|^{2p-2}\\tilde u_{\\ell,n}\\,\\varphi\\,dx \\\\\n &~\n +\\sum_{j\\neq \\ell}\\lambda_{\\ell j}\\int_{\\R^N}\n |\\tilde u_{j,n}|^{p}|\\tilde u_{\\ell,n}|^{p-2}\\tilde u_{\\ell,n}\\,\\varphi\\,dx .\n \\end{align*}\n By $(A_3)$, $\\operatorname{supp}(Q)\\subset B_{R_Q}(0)$ for some $R_Q>0$.\n Since $K$ is bounded and $|\\varepsilon_n x_{i,n}|\\to\\infty$, for all large $n$ we have\n \\[\n \\bigl|\\varepsilon_n(x+x_{i,n})-y_\\ell\\bigr|\n \\ge |\\varepsilon_n x_{i,n}|-|y_\\ell|-\\varepsilon_n|x|>R_Q\n ~\\text{for all }x\\in K,\n \\]\n hence $Q(\\varepsilon_n(x+x_{i,n})-y_\\ell)\\equiv0$ on $K$ for all large $n$.\n Letting $n\\to\\infty$ and using $\\tilde{\\mathbf u}_n\\to\\mathbf U$ in $L^{2p}_{\\rm loc}(\\R^N)^m$, we obtain\n \\[\n \\int_{\\R^N}\\bigl(\\nabla U_\\ell\\cdot\\nabla\\varphi+U_\\ell\\varphi\\bigr)\\,dx\n =\n \\sum_{j\\neq \\ell}\\lambda_{\\ell j}\\int_{\\R^N}|U_j|^{p}|U_\\ell|^{p-2}U_\\ell\\,\\varphi\\,dx,\n ~\\forall\\,\\varphi\\in C_c^\\infty(\\R^N).\n \\]\nChoose $\\chi_R\\in C_c^\\infty(\\R^N)$ such that $0\\le \\chi_R\\le 1$, $\\chi_R\\equiv 1$ on $B_R(0)$,\n$\\operatorname{supp}(\\chi_R)\\subset B_{2R}(0)$ and $|\\nabla\\chi_R|\\le C/R$.\nTesting the above identity with $\\varphi=\\chi_R^2U_\\ell$ and using $\\lambda_{\\ell j}<0$, we obtain\n\\[\n\\int_{\\R^N}\\Bigl(\\nabla U_\\ell\\cdot\\nabla(\\chi_R^2U_\\ell)+\\chi_R^2|U_\\ell|^2\\Bigr)\\,dx\n=\\sum_{j\\neq \\ell}\\lambda_{\\ell j}\\int_{\\R^N}\\chi_R^2|U_j|^p|U_\\ell|^p\\,dx\\le 0.\n\\]\nMoreover, by the product rule,\n\\[\n\\int_{\\R^N}\\nabla U_\\ell\\cdot\\nabla(\\chi_R^2U_\\ell)\\,dx\n=\\int_{\\R^N}\\Bigl(|\\nabla(\\chi_RU_\\ell)|^2-|U_\\ell|^2|\\nabla\\chi_R|^2\\Bigr)\\,dx.\n\\]\nConsequently,\n\\[\n\\int_{\\R^N}\\Bigl(|\\nabla(\\chi_RU_\\ell)|^2+\\chi_R^2|U_\\ell|^2\\Bigr)\\,dx\n\\le \\int_{\\R^N}|U_\\ell|^2|\\nabla\\chi_R|^2\\,dx.\n\\]\nSince $|\\nabla\\chi_R|\\le C/R$ and $\\operatorname{supp}(\\nabla\\chi_R)\\subset B_{2R}(0)\\setminus B_R(0)$, we have\n\\[\n\\int_{\\R^N}|U_\\ell|^2|\\nabla\\chi_R|^2\\,dx\n\\le \\frac{C}{R^2}\\int_{B_{2R}(0)\\setminus B_R(0)}|U_\\ell|^2\\,dx\\to 0\n\\]\nas $R\\to\\infty$, which implies\n\\[\n\\int_{\\R^N}\\bigl(|\\nabla U_\\ell|^2+|U_\\ell|^2\\bigr)\\,dx=0,\n\\]\ni.e., $U_\\ell= 0$, which contradicts $U_\\ell\\neq 0$. Consequently, passing to a subsequence, there exists $\\xi_i\\in\\R^N$ such that\n \\begin{equation}\\label{eq5.22}\n \\varepsilon_n x_{i,n}\\to \\xi_i~\\text{as}~n\\to\\infty.\n \\end{equation}\n\\par \n Next, we show that $\\xi_i = 0$ by contradiction. Suppose that $\\xi_i \\neq 0$. \n\\par\nWe first claim that\n\\begin{equation}\\label{eq5.23}\n Q(\\xi_i - y_\\ell)<Q(-y_\\ell)~\\text{for some}~\\ell\\in\\{1,\\cdots,m\\}.\n\\end{equation} \nSuppose to the contrary that $Q(\\xi_i-y_\\ell)=Q(-y_\\ell)=\\|Q\\|_\\infty$ for all $\\ell$. Then, by $(A_3)$, for each $\\ell$, there exists$j_\\ell\\in{1,\\dots,m}$ such that $\\xi_i-y_\\ell=-y_{j_\\ell}$. If $j_\\ell=\\ell$ for some $\\ell$, then $\\xi_i=0$, contradicting $\\xi_i\\neq0$. Hence $j_\\ell\\neq\\ell$ for all $\\ell$. Define a map $\\sigma:\\{1,\\dots,m\\}\\to\\{1,\\dots,m\\}$ by $\\sigma(\\ell)=j_\\ell$. Since the points $y_1,\\dots,y_m$ are pairwise distinct, $\\sigma$ is a permutation of $\\{1,\\dots,m\\}$ without fixed points. Moreover, we have\n\\begin{equation*}\ny_{\\sigma(\\ell)}=y_\\ell-\\xi_i,~\\text{for each}~\\ell.\n\\end{equation*}\nIterating this relation gives $y_{\\sigma^k(\\ell)}=y_\\ell-k\\xi_i$ for any positive integer $k$. Because the set $\\{y_1,\\dots,y_m\\}$ is finite, there exist $k_1\\neq k_2$ such that $y_{\\sigma^{k_1}(\\ell)}=y_{\\sigma^{k_2}(\\ell)}$, which implies $(k_1-k_2)\\xi_i=0$ and hence $\\xi_i=0$, a contradiction. Therefore, \\eqref{eq5.23} must hold.\n\\par\nObserve that $\\mathbf{U}$ is a weak solution of\n\\begin{equation}\\label{eq5.24}\n-\\Delta U_\\ell + U_\\ell = \\mu_\\ell Q(\\xi_i - y_\\ell) |U_\\ell|^{2p-2}U_\\ell + \\sum_{j\\neq\\ell}\\lambda_{\\ell j}|U_j|^p |U_\\ell|^{p-2}U_\\ell, ~ \\ell=1,\\dots,m .\n\\end{equation}\nDenote by $J_\\xi$ the energy functional associated with \\eqref{eq5.24} and recall $J_0$ be the original limiting functional with coefficients $Q(-y_\\ell)$\n\\begin{equation*}\n-\\Delta U_\\ell + U_\\ell = \\mu_\\ell Q(- y_\\ell) |U_\\ell|^{2p-2}U_\\ell + \\sum_{j\\neq\\ell}\\lambda_{\\ell j}|U_j|^p |U_\\ell|^{p-2}U_\\ell, ~ \\ell=1,\\dots,m .\n\\end{equation*}\n\\par\nFrom Lemma \\ref{Lem5.3} we have $c_{\\varepsilon_n,k} \\to c_{0,k}$. The concentration-compactness principle together with the Br\\'ezis--Lieb lemma yield\n\\begin{equation}\\label{eq5.25}\nc_{0,k} = \\lim_{n\\to\\infty} J_{\\varepsilon_n}(\\mathbf{u}_{\\varepsilon_n}) \\ge J_\\xi(\\mathbf{U}).\n\\end{equation}\n\\par\nBy \\eqref{eq5.23} and $U_\\ell \\neq 0$, we obtain\n\\begin{equation}\\label{eq5.26}\n\\begin{aligned}\nJ_\\xi(\\mathbf{U}) - J_0(\\mathbf{U}) &= \\frac{1}{2p}\\sum_{\\ell=1}^m \\mu_\\ell\\bigl[Q(-y_\\ell) - Q(\\xi_i - y_\\ell)\\bigr] \\int_{\\mathbb{R}^N} |U_\\ell|^{2p}\\\\\n&\\ge \\frac{1}{2p}\\mu_\\ell\\bigl[Q(-y_\\ell) - Q(\\xi_i-y_\\ell)\\bigr] \\int |U_\\ell|^{2p} > 0.\n\\end{aligned}\n\\end{equation}", "post_theorem_intro_text_len": 6513, "post_theorem_intro_text": "\\par\nTo treat sign-changing solutions of competitive Schr\\\"odinger systems with shrinking nonlinear potentials, we develop a variational framework tailored to the nodal setting of \\eqref{eq1.1}, inspired by the ideas in\n\\cite{AckermannSzulkin2013,ClappSaldanaSzulkin2025} but adapted to the present\nsystem. We work in the product space $H:=(H^1(\\mathbb{R}^N))^m$ endowed with the natural norm, and we consider the energy functional associated with\n\\eqref{eq1.1} $J_\\varepsilon:H\\to\\mathbb{R}$. Our first step is to introduce a fully sign-changing Nehari set\n$\\mathcal{N}^{\\mathrm{sc}}_\\varepsilon$ (see \\S \\ref{sec:framework}), by imposing Nehari type constraints\nseparately on the positive and negative parts of each component. On a suitable open\nsubset $\\mathcal{A}$ of $H$, rather than on a\nlinear subspace, we define a nodal projection\n\\[\nm_\\varepsilon:\\mathcal{A}\\longrightarrow\n\\mathcal{N}^{\\mathrm{sc}}_\\varepsilon,\n\\]\nand we use it to construct an even reduced functional\n$\\Psi_\\varepsilon:=J_\\varepsilon\\circ m_\\varepsilon$ on\n$\\mathcal{A}$. The symmetry of $\\Psi_\\varepsilon$ allows us to apply\nKrasnosel'skii genus theory and to implement a symmetric minimax scheme, which provide infinitely many critical values and\n\tyield a sequence of nodal solutions with unbounded energies.\nThis feature is essential for applying genus theory in a genuinely nodal\nsetting and allows us to recover a symmetric minimax structure despite the strong\nnonlinearity of the constraints. The proof of Theorem~\\ref{thm1.1} will be carried out in Section~\\ref{sec:existence}.\n\n\\medskip\n\nThe second result in this paper concerns the asymptotic behavior of the above sign-changing solutions as $\\varepsilon\\to 0^+$. After a suitable rescaling, the potentials $Q_\\varepsilon$ converge to the constants $Q(-y_i)$, and one is naturally led to the limiting autonomous system\n\\begin{equation}\\label{eq1.2}\n\t\\begin{cases}\n\t-\\Delta U_i + U_i\n\t= \\mu_i Q(-y_i)\\,|U_i|^{2p-2}U_i\n\t+ \\sum_{j\\neq i}\\lambda_{ij}|U_j|^p|U_i|^{p-2}U_i,\\\\[6pt]\n\t U_i\\in H^1(\\mathbb{R}^N),\\ i=1,\\dots,m.\n\\end{cases}\n\\end{equation}\nThe energy functional associated with\n\\eqref{eq1.2} is denoted by $J_0(\\mathbf{u}): H\\to\\mathbb{R}$.\n\\begin{Thm}\\label{thm1.2}\n\tAssume $N\\ge1$, $m\\ge2$, and \\textup{(A$_1$)--(A$_3$)}.\n\tLet $\\{\\varepsilon_n\\}\\subset(0,+\\infty)$ be a sequence with $\\varepsilon_n\\to 0$.\n\tFor each fixed $k\\in\\mathbb{N}^*$, let $\\{\\mathbf{u}^{(k)}_{\\varepsilon_n}\\}$ be the sequence of\n\tsign-changing solutions given by Theorem~\\ref{thm1.1}.\n\tThen there exist $m$ sequences $\\{x^{(k)}_{\\varepsilon_n,i}\\}\\subset\\mathbb{R}^N$ and a sign-changing solution\n\t$\\mathbf{U}^{(k)}=(U^{(k)}_1,\\dots,U^{(k)}_m)\\in H$ of the limiting system \\eqref{eq1.2}\n\tsuch that, up to a subsequence,\n\t\\begin{itemize}\n\t\t\\item[(i)]\n\t\t$\\varepsilon_n x_{\\varepsilon_n,i}^{(k)}\\to 0$ as $n\\to\\infty$ for $i=1,\\dots,m$.\n\tIn particular, for every $R>0$,\n\t\t\\[\n\t\tQ\\bigl(\\varepsilon_n(\\,\\cdot+x_{\\varepsilon_n,i}^{(k)})-y_i\\bigr)\\to Q(-y_i)\n\t\t~\\text{uniformly in}~B_R(0),~i=1,\\dots,m.\n\t\t\\]\n\n\t\t\\item[(ii)]\n\t\t$u^{(k)}_{\\varepsilon_n,i}(\\,\\cdot + x^{(k)}_{\\varepsilon_n,i})\n\t\t\\longrightarrow U^{(k)}_i$ in $H^1(\\mathbb{R}^N)$ as $n\\to\\infty$ for  $i=1,\\dots,m$. \n\n\t\\item[(iii)]\n\tfor every $q\\in[1,\\infty)$,\n\t\\[\n\t\\lim_{R\\to+\\infty}\\,\\limsup_{n\\to\\infty}\n\t\\int_{\\mathbb{R}^N\\setminus B_{R/{\\varepsilon_n}}(0)}\n\t|u^{(k)}_{\\varepsilon_n, i}(x)|^q\\,dx = 0,\n\t~ i=1,\\dots,m,\n\t\\]\n\tand for $q=\\infty$,\n\t\\[\n\t\\lim_{R\\to+\\infty}\\,\\limsup_{n\\to\\infty}\n\t\\|u^{(k)}_{\\varepsilon_n, i}\\|_{L^\\infty(\\mathbb{R}^N\\setminus B_{R/{\\varepsilon_n}}(0))} = 0,\n\t~ i=1,\\dots,m.\n\t\\]\n\n\t\t\\item[(iv)]\n\t\t$\\{\\mathbf{U}^{(k)}\\}$ are pairwise distinct and satisfy\n\t\t\\[\n\t\tJ_0\\big(\\mathbf{U}^{(1)}\\big) < J_0\\big(\\mathbf{U}^{(2)}\\big) < \\cdots \\to +\\infty.\n\t\t\\]\n\t\\end{itemize}\n\\end{Thm}\n\n\\begin{Rem}\\label{rem1.2}\n\tTheorem~\\ref{thm1.2} extends the concentration results for nonnegative solutions obtained in\n\t\\cite{ClappSaldanaSzulkin2025} to the sign-changing case.\n\tAssertions \\textup{(i)}--\\textup{(iii)} show that, after suitable translations, each component concentrates\n\taround the origin, while assertion \\textup{(iv)} reveals a new phenomenon: the limiting autonomous system\n\tinherits infinitely many sign-changing solutions with unbounded energies, arising from the genus-based\n\tconstruction for the original problem. \n\\end{Rem}\n\nThe proof of Theorem~\\ref{thm1.2} is based on a concentration--compactness argument for suitable\ntranslations of sign-changing solutions $\\{\\mathbf u_n\\}$, which yields a nontrivial limiting profile solving\n\\eqref{eq1.2} and the strong convergence in (ii) via a Br\\'ezis--Lieb type energy\ndecomposition.\nThe decay property in (iii) follows from localized estimates with cut--off functions, interpolation,\nand Agmon-type exponential decay bounds; the case $q=\\infty$ is obtained by a translation argument\ncombined with local elliptic regularity and the compact support of $Q$.\nIt is carried out in Section~\\ref{sec:concentration}.\n\nThe paper is organized as follows.\nIn Section~\\ref{sec:framework} we introduce the variational setting, the fully nodal Nehari set and the reduced functional, and state our assumptions on the nonlinear potentials.\nIn Section~\\ref{sec:ps} we establish a compactness result for the reduced functional $\\Psi_{\\varepsilon}$, which will be crucial in the construction of sign-changing solutions.\nSection~\\ref{sec:existence} is devoted to a genus-based minimax scheme on $\\mathcal N_{\\varepsilon}^{\\mathrm{sc}}$, and hence obtain infinitely many sign-changing critical points. In Section~\\ref{sec:concentration} we prove the concentration result Theorem~\\ref{thm1.2}, including the\ndescription of the limiting profiles for the semiclassical states.\n\\medskip\n\n\\textbf{Notation.}\n\\begin{itemize}\n\t\\item Let $w^+(x):=\\max\\{w(x),0\\}$ and $w^-(x):=\\min\\{w(x),0\\}$ be the positive and negative parts of $w$, respectively. \n\t\\item For $r>0$ and $x\\in\\mathbb{R}^N$, $B_r(x)$ is the open ball centered at $x$ with radius $r$.\n    \\item Boldface $\\mathbf{u}$ denote a $\\mathbb{R}^m$ vector-value function, $u_i$ denotes the $i$-th component function of $\\mathbf{u}$. \n    \\item \"$\\to$\" denotes the strong convergence and \"$\\rightharpoonup$\" denotes the weak convergence.\n    \\item For $k\\in\\mathbb{Z}^N$, we denote by $k*\\mathbf u$ the integer translation of $\\mathbf u$:\n    \\[\n    (k*\\mathbf u)(x):=\\mathbf u(x-k),~ x\\in\\mathbb{R}^N .\n    \\]\n    \\item For a function space $E$, $E^m$ denotes the product space $\\underbrace{E\\times\\cdots\\times E}_{m}$.\n  \\end{itemize}", "sketch": "To prove Theorem~\\ref{thm1.1}, the authors “develop a variational framework tailored to the nodal setting of \\eqref{eq1.1}” in the product space $H:=(H^1(\\mathbb{R}^N))^m$ and consider the associated energy functional $J_\\varepsilon:H\\to\\mathbb{R}$. The “first step is to introduce a fully sign-changing Nehari set $\\mathcal{N}^{\\mathrm{sc}}_\\varepsilon$ … by imposing Nehari type constraints separately on the positive and negative parts of each component.” On “a suitable open subset $\\mathcal{A}$ of $H$, rather than on a linear subspace,” they define a nodal projection $m_\\varepsilon:\\mathcal{A}\\to\\mathcal{N}^{\\mathrm{sc}}_\\varepsilon$ and use it to construct “an even reduced functional $\\Psi_\\varepsilon:=J_\\varepsilon\\circ m_\\varepsilon$ on $\\mathcal{A}$.” Since “the symmetry of $\\Psi_\\varepsilon$ allows us to apply Krasnosel'skii genus theory,” they “implement a symmetric minimax scheme,” which “provide[s] infinitely many critical values” and “yield[s] a sequence of nodal solutions with unbounded energies,” i.e., the desired infinitely many sign-changing solutions with increasing energy levels. The proof is said to be “carried out in Section~\\ref{sec:existence},” with compactness for $\\Psi_\\varepsilon$ established earlier as “crucial in the construction of sign-changing solutions.”", "expanded_sketch": "To prove the main theorem, the authors “develop a variational framework tailored to the nodal setting of\n\\begin{equation}\\label{eq1.1}\n\t\\begin{cases}\n\t\t-\\Delta u_i + u_i\n\t\t= \\mu_i Q_\\varepsilon(x-y_i) |u_i|^{2p-2}u_i\n\t\t+ \\sum_{j\\neq i} \\lambda_{ij} |u_j|^{p} |u_i|^{p-2}u_i, \\\\[6pt]\n\t\tu_i \\in H^1(\\mathbb{R}^N),\\ i=1,\\dots,m.\n\t\\end{cases}\t\n\\end{equation}\n” in the product space $H:=(H^1(\\mathbb{R}^N))^m$ and consider the associated energy functional $J_\\varepsilon:H\\to\\mathbb{R}$. The “first step is to introduce a fully sign-changing Nehari set $\\mathcal{N}^{\\mathrm{sc}}_\\varepsilon$ … by imposing Nehari type constraints separately on the positive and negative parts of each component.” On “a suitable open subset $\\mathcal{A}$ of $H$, rather than on a linear subspace,” they define a nodal projection $m_\\varepsilon:\\mathcal{A}\\to\\mathcal{N}^{\\mathrm{sc}}_\\varepsilon$ and use it to construct “an even reduced functional $\\Psi_\\varepsilon:=J_\\varepsilon\\circ m_\\varepsilon$ on $\\mathcal{A}$.” Since “the symmetry of $\\Psi_\\varepsilon$ allows us to apply Krasnosel'skii genus theory,” they “implement a symmetric minimax scheme,” which “provide[s] infinitely many critical values” and “yield[s] a sequence of nodal solutions with unbounded energies,” i.e., the desired infinitely many sign-changing solutions with increasing energy levels. The proof is said to be “carried out later,” with compactness for $\\Psi_\\varepsilon$ established earlier as “crucial in the construction of sign-changing solutions.”", "expanded_theorem": "\\label{thm1.1}\n\tAssume $N\\ge1$, $m\\ge2$, and \\textbf{\\textup{$(A_1)$--$(A_3)$}}. Then for $\\varepsilon>0$, \n\t\\begin{equation}\\label{eq1.1}\n\t\\begin{cases}\n\t\t-\\Delta u_i + u_i\n\t\t= \\mu_i Q_\\varepsilon(x-y_i) |u_i|^{2p-2}u_i\n\t\t+ \\sum_{j\\neq i} \\lambda_{ij} |u_j|^{p} |u_i|^{p-2}u_i, \\\\[6pt]\n\t\tu_i \\in H^1(\\mathbb{R}^N),\\ i=1,\\dots,m.\n\t\\end{cases}\t\n\\end{equation}\n\tadmits a sequence of sign-changing solutions\n\t\\[\n\t\\mathbf{u}^{(k)}_\\varepsilon=(u^{(k)}_{\\varepsilon,1},\\dots,u^{(k)}_{\\varepsilon,m})\\in H, ~ k\\in\\mathbb{N}^*,\n\t\\]\n\tsatisfying\n\t\\[\n\t0<J_\\varepsilon\\bigl(\\mathbf{u}^{(1)}_\\varepsilon\\bigr)\n\t< \\cdots < J_\\varepsilon\\bigl(\\mathbf{u}^{(k)}_\\varepsilon\\bigr)\n    < J_\\varepsilon\\bigl(\\mathbf{u}^{(k+1)}_\\varepsilon\\bigr)\n\t< \\cdots, ~\\text{and}~ \\lim\\limits_{k\\to\\infty}\n\tJ_\\varepsilon\\bigl(\\mathbf{u}^{(k)}_\\varepsilon\\bigr)=+\\infty.\n\t\\]", "theorem_type": ["Existence", "Asymptotic or Limit"], "mcq": {"question": "Let \n\\[\nH:=[H^1(\\mathbb R^N)]^m,\n\\]\nand consider, for a fixed \\(\\varepsilon>0\\), the system\n\\[\n\\begin{cases}\n-\\Delta u_i+u_i\n=\\mu_i Q_\\varepsilon(x-y_i)|u_i|^{2p-2}u_i\n+\\displaystyle\\sum_{j\\ne i}\\lambda_{ij}|u_j|^p|u_i|^{p-2}u_i,\\\\[4pt]\nu_i\\in H^1(\\mathbb R^N),\\quad i=1,\\dots,m.\n\\end{cases}\n\\]\nA solution \\(\\mathbf u=(u_1,\\dots,u_m)\\) is called sign-changing if for every \\(i=1,\\dots,m\\), both\n\\[\nu_i^+=\\max\\{u_i,0\\},\\qquad u_i^-=\\min\\{u_i,0\\}\n\\]\nare nonzero in \\(H^1(\\mathbb R^N)\\). Assume \\(N\\ge 1\\), \\(m\\ge 2\\), and that the data satisfy \\((A_1)\\)–\\((A_3)\\). Which existence statement holds for this system?", "correct_choice": {"label": "A", "text": "For every \\(\\varepsilon>0\\), the system admits a sequence of sign-changing solutions\n\\[\n\\mathbf u^{(k)}_\\varepsilon=(u^{(k)}_{\\varepsilon,1},\\dots,u^{(k)}_{\\varepsilon,m})\\in H,\\qquad k\\in\\mathbb N^*,\n\\]\nsuch that\n\\[\n0<J_\\varepsilon(\\mathbf u^{(1)}_\\varepsilon)<\\cdots<J_\\varepsilon(\\mathbf u^{(k)}_\\varepsilon)<J_\\varepsilon(\\mathbf u^{(k+1)}_\\varepsilon)<\\cdots,\n\\]\nand\n\\[\n\\lim_{k\\to\\infty}J_\\varepsilon(\\mathbf u^{(k)}_\\varepsilon)=+\\infty.\n\\]"}, "choices": [{"label": "B", "text": "For every \\(\\varepsilon>0\\), the system admits a sign-changing ground state \\(\\mathbf u_\\varepsilon\\in H\\) such that\n\\[\nJ_\\varepsilon(\\mathbf u_\\varepsilon)=\\inf\\bigl\\{J_\\varepsilon(\\mathbf u):\\mathbf u\\in H\\setminus\\{0\\}\\text{ is sign-changing}\\bigr\\},\n\\]\nand every sign-changing solution of least energy is obtained in this way."}, {"label": "C", "text": "For every \\(\\varepsilon>0\\), the system admits at least one sign-changing solution \\(\\mathbf u_\\varepsilon\\in H\\) with\n\\[\nJ_\\varepsilon(\\mathbf u_\\varepsilon)>0.\n\\]"}, {"label": "D", "text": "There exists \\(\\varepsilon_0>0\\) such that for every \\(0<\\varepsilon<\\varepsilon_0\\), the system admits a sequence of sign-changing solutions\n\\[\n\\mathbf u^{(k)}_\\varepsilon=(u^{(k)}_{\\varepsilon,1},\\dots,u^{(k)}_{\\varepsilon,m})\\in H,\\qquad k\\in\\mathbb N^*,\n\\]\nsatisfying\n\\[\n0<J_\\varepsilon(\\mathbf u^{(1)}_\\varepsilon)<\\cdots<J_\\varepsilon(\\mathbf u^{(k)}_\\varepsilon)<J_\\varepsilon(\\mathbf u^{(k+1)}_\\varepsilon)<\\cdots,\n\\]\nand\n\\[\n\\lim_{k\\to\\infty}J_\\varepsilon(\\mathbf u^{(k)}_\\varepsilon)=+\\infty.\n\\]"}, {"label": "E", "text": "For every \\(\\varepsilon>0\\), the system admits a sequence of nontrivial solutions\n\\[\n\\mathbf u^{(k)}_\\varepsilon=(u^{(k)}_{\\varepsilon,1},\\dots,u^{(k)}_{\\varepsilon,m})\\in H,\\qquad k\\in\\mathbb N^*,\n\\]\nsuch that each component \\(u^{(k)}_{\\varepsilon,i}\\) changes sign, the energies satisfy\n\\[\n0<J_\\varepsilon(\\mathbf u^{(1)}_\\varepsilon)\\le \\cdots\\le J_\\varepsilon(\\mathbf u^{(k)}_varepsilon)\\le J_\\varepsilon(\\mathbf u^{(k+1)}_\\varepsilon)\\le\\cdots,\n\\]\nand the set \\(\\{J_\\varepsilon(\\mathbf u^{(k)}_\\varepsilon):k\\in\\mathbb N^*\\}\\) is unbounded above."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "genus-minimax gives multiplicity, not least-energy ground state characterization", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped infinite multiplicity and strict energy ladder", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "quantifier in epsilon changed from all positive epsilon to only sufficiently small epsilon", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "strictly increasing divergent critical values weakened to merely nondecreasing unbounded energies", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only the PDE system, assumptions, and the definition of sign-changing; it does not explicitly reveal the theorem's conclusion or uniquely point to choice A."}, "TAS": {"score": 1, "justification": "This is essentially a theorem-identification item: the task is to recognize the exact existence statement among nearby variants. It is not a pure restatement in the stem, but it is still very close to selecting the theorem's formal conclusion."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to compare quantifiers and strength of conclusions (all ε vs small ε, one solution vs infinitely many, strict energy ladder vs weaker unboundedness), but the item mainly tests precise recall rather than deeper mathematical generation."}, "DQS": {"score": 1, "justification": "Most distractors are plausible theorem-level variants, but choice C is a weaker statement that is also logically true if A is true. Since the stem asks which statement holds, this creates ambiguity and weakens distractor quality."}, "total_score": 5, "overall_assessment": "A moderately good theorem-recall MCQ with no real answer leakage, but it is close to theorem restatement and is weakened by an ambiguous weaker-true distractor."}}
{"id": "2602.16692v1", "paper_link": "http://arxiv.org/abs/2602.16692v1", "theorems_cnt": 7, "theorem": {"env_name": "thmA", "content": "[\\cite{wagner}]\n\t\\label{wagner-thm}\n\tIf a graph $G$ is maximal $K_5$-minor-free, then $G$ can be built by a sequence of $2$-sums and \n\t$3$-sums, without deleting edges, starting from maximal planar graphs and copies of $M_8$.", "start_pos": 13282, "end_pos": 13532, "label": "wagner-thm"}, "ref_dict": {"conj1": "\\begin{conj}\\label{conj1}\n\tIf $G$ is $K_5$-minor-free and $\\MM$ is a correspondence 6-cover, then $G$ has 6 disjoint \n\t$\\MM$-colorings.  That is, $\\chisc(\\K_5)=6$.\n\\end{conj}"}, "pre_theorem_intro_text_len": 9594, "pre_theorem_intro_text": "\\subsection{Definitions and Examples}\nCorrespondence coloring generalizes list coloring, and was introduced to resolve a longstanding open \nquestion \\cite{DP} on the list-chromatic number of certain planar graphs.  Recall that a list assignment \n$L$ for a graph $G$ gives to each vertex $v$ of $G$ a set $L(v)$ of allowable colors.  An \\emph{$L$-coloring} \nis a proper coloring $\\vph$ such that $\\vph(v)\\in L(v)$ for all vertices $v$.  If $|L(v)|=t$ for all $v$, \nthen $L$ is called a $t$-assignment and an \\emph{$L$-packing}\nconsists of $L$-colorings $\\vph_1,\\ldots,\\vph_t$ such that $\\vph_i(v)\\ne \\vph_j(v)$ for all vertices $v$\nand all distinct $i,j\\in[t]$.  That is, the colors $\\vph_i(v)$ partition $L(v)$ for each vertex $v$.\n\n\tFor a graph $G$, a \\emph{correspondence $t$-cover} $\\MM$ assigns \n\tto each $v\\in V(G)$ a set of allowable colors $\\{1_v,\\ldots,t_v\\}$ and to each edge $vw\\in E(G)$\n\ta matching between $\\{1_v,\\ldots,t_v\\}$ and $\\{1_w,\\ldots,t_w\\}$. An $\\MM$-coloring $\\vph$\n\tpicks for each vertex $v$ a color $\\vph(v)$ (from the set $\\{1_v,\\ldots,t_v\\}$) such that for\n\teach edge $vw\\in E(G)$ the colors $\\vph(v),\\vph(w)$ are not matched to each other.\n\t(It is straightforward to verify that correspondence coloring generalizes list coloring.)\n\tTwo $\\MM$-colorings $\\vph_1,\\vph_2$ of $G$ are \\emph{disjoint} if $\\vph_1(v)\\ne\\vph_2(v)$ for all \n\t$v\\in V(G)$.  An $\\MM$-packing for $G$ consists of disjoint $\\MM$-colorings $\\vph_1,\\ldots, \\vph_t$.\n\n\tThe \\emph{list-chromatic number}, $\\chil(G)$, and \\emph{list packing number},\n\t$\\chisl(G)$, are the minimum values of $t$ such that for every $t$-assignment $G$ admits,\n\trespectively, a list coloring and a list packing.  Analogously, for correspondence $t$-covers, we \n\tdefine the \\emph{correspondence chromatic number}, $\\chic(G)$, and the \\emph{correspondence packing \n\tnumber}, $\\chisc(G)$. List packing and correspondence packing were introduced in~\\cite{CCvBDK}, where\n\tthe authors discussed connections with other areas of discrete math.  These parameters have since \n\tbeen studied in numerous papers~\\cite{BCK25,CCvB,CCvBDK2,CCvBZ,CH23,CDKH24,CSr,KMMP22,KM25,M23}.\n\n\tAs a warmup with these definitions, we provide a few examples.  Note that always $\\chi(G)\\le \\chil(G)\n\t\\le \\chic(G)\\le \\chisc(G)$; and also $\\chil(G)\\le \\chisl(G)\\le \\chisc(G)$.  For a graph class $\\G$\n\tand a graph parameter $f$, let $f(\\G):=\\max_{G\\in \\G}f(G)$.  Let $\\F$ denote the class of forests,\n\t$\\C$ the class of cycles, and $\\C_e$ the class of cycles of even length.\n\tSo $2=\\chi(\\F)=\\chil(\\F)=\\chisl(\\F)=\\chic(\\F)=\\chisc(\\F)$.  The lower bound is trivial, and the \n\tupper bound follows from \\Cref{ext-lem} below.  It is well-known (and easy to check) that \n\t$\\chil(\\C_e) = 2 < 3 = \\chil(\\C)$.  But the reader may find the following more surpising:\n\t$\\chisl(\\C_e)=\\chisl(\\C)=3$, $\\chic(\\C_e)=\\chic(\\C)=3$, and $\\chisc(\\C_e)=\\chisc(\\C)=4$. \n\tIn the first case, the upper bound requires a bit of work.  But in the second and third cases, \n\tthe upper bounds hold because the graphs are $2$-degenerate; in \n\tparticular, the last of these follows from \\Cref{ext-lem} below.  The lower bounds on both\n\t$\\chic(\\C_e)$ and $\\chisc(\\C_e)$ come from constructing a cover that violates a necessary\n\tparity-type invariant needed to find the desired coloring or packing.\n\tFor more details on these lower bounds, see \\cite{CCvBDK2} or the introduction of~\\cite{CSr}.\n\n\\subsection{Main Result}\nThe authors of \\cite{CCvBDK} proposed the problem of determining $\\chisc(\\G)$ for various graph classes \n$\\G$.  Let $\\K_r$ denote the class of all graphs that are $K_r$-minor-free.  It is easy to check that \n$\\chisc(\\K_2)=1$, $\\chisc(\\K_3)=2$, and $\\chisc(\\K_4)=4$; the final upper bound follows from Lemma~\\ref{ext-lem}\nbelow because all graphs in $\\K_4$ are 2-degenerate.  So the first open case is $\\chisc(\\K_5)$.\nThe best known upper bound is $\\chisc(\\K_5)\\le 8$.  This follows from \\cite[Theorem~1.7]{CCvBZ} because every\n$K_5$-minor-free graph has maximum average degree less than 6.  We cautiously believe that this upper bound \ncan be improved as follows.\n\n\\begin{conj}\\label{conj1}\n\tIf $G$ is $K_5$-minor-free and $\\MM$ is a correspondence 6-cover, then $G$ has 6 disjoint \n\t$\\MM$-colorings.  That is, $\\chisc(\\K_5)=6$.\n\\end{conj}\n\nDvo\\v{r}\\'{a}k, Norin, and Postle~\\cite{DNP19} called a graph \\emph{weighted $\\epsilon$-flexible} with \nrespect to a list-assignment $L$ if there exists a probability distribution on the $L$-colorings $\\phi$ \nof $G$ such that $\\mathbb{P}(\\phi(v)=c) \\ge \\epsilon$ for every vertex $v$ and color $c\\in L(v)$.\nIn other words, weighted $\\epsilon$-flexibility ensures that at each vertex each color has a decent chance \nto appear in the random list-coloring.  Dvo\\v{r}\\'{a}k, Norin, and Postle  proved that there is an absolute \nconstant $\\epsilon>0$ such that all planar graphs are weighted $\\epsilon$-flexible with respect to every \n$7$-assignment, and they posed as an open problem to show the same for $6$-assignments or even \n$5$-assignments.  While this problem remains wide open, for $7$-assignments the value of $\\epsilon$ has been \nimproved considerably from $7^{-36}$ to $2^{-7}$ by~\\cite{KMMP22} and~\\cite{CCvB}.\nAs detailed in~\\cite{CCvBZ} and~\\cite{KMMP22}, every graph $G$ is weighted $\\frac{1}{\\chisl(G)}$-flexible \nwith respect to every $\\chisl(G)$-assignment. Thus a proof of Conjecture~\\ref{conj1}  will make significant \nprogress: it will imply that every planar graph (which is $K_5$-minor-free) is weighted \n$\\frac{1}{6}$-flexible with respect to every $6$-assignment.\n\\bigskip\n\nThe main goal of this note is to prove the following result, in support of the above conjecture.\n\n\\begin{mainthm}\n\t\\label{thm1}\n\tLet $G$ be a $K_5$-minor-free graph.  (i) For every correspondence 6-cover $\\MM$ of $G$, \n\tthere exist disjoint $\\MM$-colorings $\\vph_1,\\vph_2,\\vph_3$.  That is, for all $v\\in V(G)$ \n\tthe colors $\\vph_1(v),\\vph_2(v),\\vph_3(v)$ are pairwise distinct.\n\t(ii) But the analogous statement is false if we replace 6-cover with 5-cover.\n\\end{mainthm}\n\nThe fact that $K_5$-minor-free graphs are $5$-degenerate easily implies that they have list-chromatic number \nat most $6$, which is  close to the optimal bound $5$. However, for the correspondence packing number this \napproach performs quite poorly.  As discussed above, $\\chisc(\\C) \\le 4$ due to the $2$-degeneracy of cycles.\nMore generally, by \\Cref{ext-lem} we have $\\chisc(G)\\leq 2d$ for every $d$-degenerate graph $G$, and this is \nsharp \\cite[Prop. 24]{CCvBDK} for every $d \\ge 1$, due to large complete bipartite graphs. Therefore a vanilla \ndegeneracy argument can merely yield $\\chisc(\\K_5) \\le  10$, while it is known that $\\chisc(\\K_5)\\le 8$. This \nindicates that establishing Conjecture~\\ref{conj1} is nontrivial. On the lower bound side, note that the \nsecond part of our Main Theorem implies that $\\chisc(\\K_5)\\ge 6$.\n\\bigskip\n\n\\subsection{Proof overview}\nWhen constructing disjoint list colorings and correspondence colorings, it is often convenient \nto proceed by induction.  Given a graph $G$, a correspondence $t$-cover $\\MM$ of $G$\nand disjoint partial $\\MM$-colorings $\\vph^0_1,\\ldots,\\vph^0_s$, for each vertex $v$\nthat is not yet colored we seek a color $\\vph_j(v)$ to extend $\\vph^0_j$ to $v$.\nWe write $L_j(v)$ for the subset of $[t]$ that is not yet forbidden from use as \n$\\vph_j(v)$, due to the colors already used on $N(v)$ and the matchings $\\MM$.\nThe following lemma, from~\\cite{CCvBDK}, is generally useful for handling vertices of low degree.\nIts proof is instructive, so we include it.\n\n\\begin{lemA}[\\cite{CCvBDK}]\n\t\\label[lem]{ext-lem}\n\tLet $G$ be a graph with a correspondence $t$-cover $\\MM$ and fix $v\\in V(G)$.\n\tIf $d(v)\\le t/2$, then any disjoint $\\MM$-colorings $\\vph_1',\\ldots,\\vph_s'$\n\tof $G-v$ extend to disjoint $\\MM$-colorings $\\vph_1,\\ldots,\\vph_s$ of $G$.\n\\end{lemA}\n\\begin{proof}\n\tWe build an auxiliary bipartite graph $\\B$ with $1,\\ldots,t$ as the vertices of the first\n\tpart, and $\\vph'_1,\\ldots,\\vph'_s$ as the vertices of the second part, and $i\\vph'_j\\in E(\\B)$\n\tif $i\\in L_j(v)$.  To extend the $\\MM$-colorings $\\vph_j'$, we seek a matching $M_{\\B}$\n\tsaturating the second part.  For this we use Hall's Theorem.\n\n\tNote that $d_{\\B}(i)\\ge s-d(v)$ and $d_{\\B}(\\vph'_j)\\ge t-d(v)$, for all $i\\in[t]$\n\tand $j\\in[s]$.  Consider $S\\subseteq\\{\\vph_1',\\ldots,\\vph_s'\\}$.  Clearly \n\t$|N_{\\B}(S)|\\ge t-d(v)\\ge t/2$.  So if $|S|\\le t/2$, then $|N_{\\B}(S)|\\ge |S|$\n\tas desired.  Otherwise, $|S|>t/2\\ge d(v)$. So $|S|+d_{\\B}(i) \\ge |S|+s-d(v) > s$ because\n\t$|S|> d(v)$.  So by Pigeonhole $i\\in N_{\\B}(S)$ for all $i\\in [t]$.  That is, $|N_{\\B}(S)| = \n\tt \\ge |S|$, as desired.  Thus, by Hall's Theorem we have the desired matching $M_{\\B}$ in $\\B$.\n\\end{proof}\n\nBy definition, each matching in $\\MM$ need not be perfect.  But to prove the results in this paper, it\nsuffices to consider the case that all matchings are perfect; if not, then we can grow them greedily. \n(In what follows, we implicitly assume that all matchings in $\\MM$ are perfect.)\n\nAn \\emph{$r$-sum} of graphs $G_1$ and $G_2$ is formed from the disjoint union $G_1+G_2$ by specifying \nan $r$-clique $x_1,\\ldots,x_r$ in $G_1$ and an $r$-clique $y_1,\\ldots,y_r$ in $G_2$, identifying $x_i$ with \n$y_i$ for each $i\\in[r]$, and possibly deleting some edges of the resulting $r$-clique.  The \\emph{Wagner \ngraph} $M_8$ (also called the \\emph{M\\\"{o}bius 8-ladder}) is formed from the $8$-cycle $z_1\\cdots z_8$ by \nadding the 4 chords $z_iz_{i+4}$ for each $i\\in [4]$.\nWagner~\\cite{wagner} characterized the class of maximal $K_5$-minor-free graphs as follows.", "context": "\\subsection{Main Result}\nThe authors of \\cite{CCvBDK} proposed the problem of determining $\\chisc(\\G)$ for various graph classes \n$\\G$. Let $\\K_r$ denote the class of all graphs that are $K_r$-minor-free. It is easy to check that \n$\\chisc(\\K_2)=1$, $\\chisc(\\K_3)=2$, and $\\chisc(\\K_4)=4$; the final upper bound follows from Lemma~\\ref{ext-lem}\nbelow because all graphs in $\\K_4$ are 2-degenerate. So the first open case is $\\chisc(\\K_5)$.\nThe best known upper bound is $\\chisc(\\K_5)\\le 8$. This follows from \\cite[Theorem~1.7]{CCvBZ} because every\n$K_5$-minor-free graph has maximum average degree less than 6. We cautiously believe that this upper bound \ncan be improved as follows.\n\nDvo\\v{r}\\'{a}k, Norin, and Postle~\\cite{DNP19} called a graph \\emph{weighted $\\epsilon$-flexible} with \nrespect to a list-assignment $L$ if there exists a probability distribution on the $L$-colorings $\\phi$ \nof $G$ such that $\\mathbb{P}(\\phi(v)=c) \\ge \\epsilon$ for every vertex $v$ and color $c\\in L(v)$.\nIn other words, weighted $\\epsilon$-flexibility ensures that at each vertex each color has a decent chance \nto appear in the random list-coloring. Dvo\\v{r}\\'{a}k, Norin, and Postle proved that there is an absolute \nconstant $\\epsilon>0$ such that all planar graphs are weighted $\\epsilon$-flexible with respect to every \n$7$-assignment, and they posed as an open problem to show the same for $6$-assignments or even \n$5$-assignments. While this problem remains wide open, for $7$-assignments the value of $\\epsilon$ has been \nimproved considerably from $7^{-36}$ to $2^{-7}$ by~\\cite{KMMP22} and~\\cite{CCvB}.\nAs detailed in~\\cite{CCvBZ} and~\\cite{KMMP22}, every graph $G$ is weighted $\\frac{1}{\\chisl(G)}$-flexible \nwith respect to every $\\chisl(G)$-assignment. Thus a proof of Conjecture~\\ref{conj1} will make significant \nprogress: it will imply that every planar graph (which is $K_5$-minor-free) is weighted \n$\\frac{1}{6}$-flexible with respect to every $6$-assignment.\n\\bigskip\n\nThe fact that $K_5$-minor-free graphs are $5$-degenerate easily implies that they have list-chromatic number \nat most $6$, which is close to the optimal bound $5$. However, for the correspondence packing number this \napproach performs quite poorly. As discussed above, $\\chisc(\\C) \\le 4$ due to the $2$-degeneracy of cycles.\nMore generally, by \\Cref{ext-lem} we have $\\chisc(G)\\leq 2d$ for every $d$-degenerate graph $G$, and this is \nsharp \\cite[Prop. 24]{CCvBDK} for every $d \\ge 1$, due to large complete bipartite graphs. Therefore a vanilla \ndegeneracy argument can merely yield $\\chisc(\\K_5) \\le 10$, while it is known that $\\chisc(\\K_5)\\le 8$. This \nindicates that establishing Conjecture~\\ref{conj1} is nontrivial. On the lower bound side, note that the \nsecond part of our Main Theorem implies that $\\chisc(\\K_5)\\ge 6$.\n\\bigskip\n\n\\begin{lemA}[\\cite{CCvBDK}]\n \\label[lem]{ext-lem}\n Let $G$ be a graph with a correspondence $t$-cover $\\MM$ and fix $v\\in V(G)$.\n If $d(v)\\le t/2$, then any disjoint $\\MM$-colorings $\\vph_1',\\ldots,\\vph_s'$\n of $G-v$ extend to disjoint $\\MM$-colorings $\\vph_1,\\ldots,\\vph_s$ of $G$.\n\\end{lemA}\n\\begin{proof}\n We build an auxiliary bipartite graph $\\B$ with $1,\\ldots,t$ as the vertices of the first\n part, and $\\vph'_1,\\ldots,\\vph'_s$ as the vertices of the second part, and $i\\vph'_j\\in E(\\B)$\n if $i\\in L_j(v)$. To extend the $\\MM$-colorings $\\vph_j'$, we seek a matching $M_{\\B}$\n saturating the second part. For this we use Hall's Theorem.\n\nBy definition, each matching in $\\MM$ need not be perfect. But to prove the results in this paper, it\nsuffices to consider the case that all matchings are perfect; if not, then we can grow them greedily. \n(In what follows, we implicitly assume that all matchings in $\\MM$ are perfect.)\n\nAn \\emph{$r$-sum} of graphs $G_1$ and $G_2$ is formed from the disjoint union $G_1+G_2$ by specifying \nan $r$-clique $x_1,\\ldots,x_r$ in $G_1$ and an $r$-clique $y_1,\\ldots,y_r$ in $G_2$, identifying $x_i$ with \n$y_i$ for each $i\\in[r]$, and possibly deleting some edges of the resulting $r$-clique. The \\emph{Wagner \ngraph} $M_8$ (also called the \\emph{M\\\"{o}bius 8-ladder}) is formed from the $8$-cycle $z_1\\cdots z_8$ by \nadding the 4 chords $z_iz_{i+4}$ for each $i\\in [4]$.\nWagner~\\cite{wagner} characterized the class of maximal $K_5$-minor-free graphs as follows.\n\n\\begin{conj}\\label{conj1}\n\tIf $G$ is $K_5$-minor-free and $\\MM$ is a correspondence 6-cover, then $G$ has 6 disjoint \n\t$\\MM$-colorings.  That is, $\\chisc(\\K_5)=6$.\n\\end{conj}", "full_context": "\\subsection{Main Result}\nThe authors of \\cite{CCvBDK} proposed the problem of determining $\\chisc(\\G)$ for various graph classes \n$\\G$. Let $\\K_r$ denote the class of all graphs that are $K_r$-minor-free. It is easy to check that \n$\\chisc(\\K_2)=1$, $\\chisc(\\K_3)=2$, and $\\chisc(\\K_4)=4$; the final upper bound follows from Lemma~\\ref{ext-lem}\nbelow because all graphs in $\\K_4$ are 2-degenerate. So the first open case is $\\chisc(\\K_5)$.\nThe best known upper bound is $\\chisc(\\K_5)\\le 8$. This follows from \\cite[Theorem~1.7]{CCvBZ} because every\n$K_5$-minor-free graph has maximum average degree less than 6. We cautiously believe that this upper bound \ncan be improved as follows.\n\nDvo\\v{r}\\'{a}k, Norin, and Postle~\\cite{DNP19} called a graph \\emph{weighted $\\epsilon$-flexible} with \nrespect to a list-assignment $L$ if there exists a probability distribution on the $L$-colorings $\\phi$ \nof $G$ such that $\\mathbb{P}(\\phi(v)=c) \\ge \\epsilon$ for every vertex $v$ and color $c\\in L(v)$.\nIn other words, weighted $\\epsilon$-flexibility ensures that at each vertex each color has a decent chance \nto appear in the random list-coloring. Dvo\\v{r}\\'{a}k, Norin, and Postle proved that there is an absolute \nconstant $\\epsilon>0$ such that all planar graphs are weighted $\\epsilon$-flexible with respect to every \n$7$-assignment, and they posed as an open problem to show the same for $6$-assignments or even \n$5$-assignments. While this problem remains wide open, for $7$-assignments the value of $\\epsilon$ has been \nimproved considerably from $7^{-36}$ to $2^{-7}$ by~\\cite{KMMP22} and~\\cite{CCvB}.\nAs detailed in~\\cite{CCvBZ} and~\\cite{KMMP22}, every graph $G$ is weighted $\\frac{1}{\\chisl(G)}$-flexible \nwith respect to every $\\chisl(G)$-assignment. Thus a proof of Conjecture~\\ref{conj1} will make significant \nprogress: it will imply that every planar graph (which is $K_5$-minor-free) is weighted \n$\\frac{1}{6}$-flexible with respect to every $6$-assignment.\n\\bigskip\n\nThe fact that $K_5$-minor-free graphs are $5$-degenerate easily implies that they have list-chromatic number \nat most $6$, which is close to the optimal bound $5$. However, for the correspondence packing number this \napproach performs quite poorly. As discussed above, $\\chisc(\\C) \\le 4$ due to the $2$-degeneracy of cycles.\nMore generally, by \\Cref{ext-lem} we have $\\chisc(G)\\leq 2d$ for every $d$-degenerate graph $G$, and this is \nsharp \\cite[Prop. 24]{CCvBDK} for every $d \\ge 1$, due to large complete bipartite graphs. Therefore a vanilla \ndegeneracy argument can merely yield $\\chisc(\\K_5) \\le 10$, while it is known that $\\chisc(\\K_5)\\le 8$. This \nindicates that establishing Conjecture~\\ref{conj1} is nontrivial. On the lower bound side, note that the \nsecond part of our Main Theorem implies that $\\chisc(\\K_5)\\ge 6$.\n\\bigskip\n\n\\begin{lemA}[\\cite{CCvBDK}]\n \\label[lem]{ext-lem}\n Let $G$ be a graph with a correspondence $t$-cover $\\MM$ and fix $v\\in V(G)$.\n If $d(v)\\le t/2$, then any disjoint $\\MM$-colorings $\\vph_1',\\ldots,\\vph_s'$\n of $G-v$ extend to disjoint $\\MM$-colorings $\\vph_1,\\ldots,\\vph_s$ of $G$.\n\\end{lemA}\n\\begin{proof}\n We build an auxiliary bipartite graph $\\B$ with $1,\\ldots,t$ as the vertices of the first\n part, and $\\vph'_1,\\ldots,\\vph'_s$ as the vertices of the second part, and $i\\vph'_j\\in E(\\B)$\n if $i\\in L_j(v)$. To extend the $\\MM$-colorings $\\vph_j'$, we seek a matching $M_{\\B}$\n saturating the second part. For this we use Hall's Theorem.\n\nBy definition, each matching in $\\MM$ need not be perfect. But to prove the results in this paper, it\nsuffices to consider the case that all matchings are perfect; if not, then we can grow them greedily. \n(In what follows, we implicitly assume that all matchings in $\\MM$ are perfect.)\n\nAn \\emph{$r$-sum} of graphs $G_1$ and $G_2$ is formed from the disjoint union $G_1+G_2$ by specifying \nan $r$-clique $x_1,\\ldots,x_r$ in $G_1$ and an $r$-clique $y_1,\\ldots,y_r$ in $G_2$, identifying $x_i$ with \n$y_i$ for each $i\\in[r]$, and possibly deleting some edges of the resulting $r$-clique. The \\emph{Wagner \ngraph} $M_8$ (also called the \\emph{M\\\"{o}bius 8-ladder}) is formed from the $8$-cycle $z_1\\cdots z_8$ by \nadding the 4 chords $z_iz_{i+4}$ for each $i\\in [4]$.\nWagner~\\cite{wagner} characterized the class of maximal $K_5$-minor-free graphs as follows.\n\n\\begin{conj}\\label{conj1}\n\tIf $G$ is $K_5$-minor-free and $\\MM$ is a correspondence 6-cover, then $G$ has 6 disjoint \n\t$\\MM$-colorings.  That is, $\\chisc(\\K_5)=6$.\n\\end{conj}\n\n\\subsection{Main Result}\nThe authors of \\cite{CCvBDK} proposed the problem of determining $\\chisc(\\G)$ for various graph classes \n$\\G$. Let $\\K_r$ denote the class of all graphs that are $K_r$-minor-free. It is easy to check that \n$\\chisc(\\K_2)=1$, $\\chisc(\\K_3)=2$, and $\\chisc(\\K_4)=4$; the final upper bound follows from Lemma~\\ref{ext-lem}\nbelow because all graphs in $\\K_4$ are 2-degenerate. So the first open case is $\\chisc(\\K_5)$.\nThe best known upper bound is $\\chisc(\\K_5)\\le 8$. This follows from \\cite[Theorem~1.7]{CCvBZ} because every\n$K_5$-minor-free graph has maximum average degree less than 6. We cautiously believe that this upper bound \ncan be improved as follows.\n\nDvo\\v{r}\\'{a}k, Norin, and Postle~\\cite{DNP19} called a graph \\emph{weighted $\\epsilon$-flexible} with \nrespect to a list-assignment $L$ if there exists a probability distribution on the $L$-colorings $\\phi$ \nof $G$ such that $\\mathbb{P}(\\phi(v)=c) \\ge \\epsilon$ for every vertex $v$ and color $c\\in L(v)$.\nIn other words, weighted $\\epsilon$-flexibility ensures that at each vertex each color has a decent chance \nto appear in the random list-coloring. Dvo\\v{r}\\'{a}k, Norin, and Postle proved that there is an absolute \nconstant $\\epsilon>0$ such that all planar graphs are weighted $\\epsilon$-flexible with respect to every \n$7$-assignment, and they posed as an open problem to show the same for $6$-assignments or even \n$5$-assignments. While this problem remains wide open, for $7$-assignments the value of $\\epsilon$ has been \nimproved considerably from $7^{-36}$ to $2^{-7}$ by~\\cite{KMMP22} and~\\cite{CCvB}.\nAs detailed in~\\cite{CCvBZ} and~\\cite{KMMP22}, every graph $G$ is weighted $\\frac{1}{\\chisl(G)}$-flexible \nwith respect to every $\\chisl(G)$-assignment. Thus a proof of Conjecture~\\ref{conj1} will make significant \nprogress: it will imply that every planar graph (which is $K_5$-minor-free) is weighted \n$\\frac{1}{6}$-flexible with respect to every $6$-assignment.\n\\bigskip\n\n\\begin{lemA}[\\cite{CCvBDK}]\n \\label[lem]{ext-lem}\n Let $G$ be a graph with a correspondence $t$-cover $\\MM$ and fix $v\\in V(G)$.\n If $d(v)\\le t/2$, then any disjoint $\\MM$-colorings $\\vph_1',\\ldots,\\vph_s'$\n of $G-v$ extend to disjoint $\\MM$-colorings $\\vph_1,\\ldots,\\vph_s$ of $G$.\n\\end{lemA}\n\\begin{proof}\n We build an auxiliary bipartite graph $\\B$ with $1,\\ldots,t$ as the vertices of the first\n part, and $\\vph'_1,\\ldots,\\vph'_s$ as the vertices of the second part, and $i\\vph'_j\\in E(\\B)$\n if $i\\in L_j(v)$. To extend the $\\MM$-colorings $\\vph_j'$, we seek a matching $M_{\\B}$\n saturating the second part. For this we use Hall's Theorem.\n\nAn \\emph{$r$-sum} of graphs $G_1$ and $G_2$ is formed from the disjoint union $G_1+G_2$ by specifying \nan $r$-clique $x_1,\\ldots,x_r$ in $G_1$ and an $r$-clique $y_1,\\ldots,y_r$ in $G_2$, identifying $x_i$ with \n$y_i$ for each $i\\in[r]$, and possibly deleting some edges of the resulting $r$-clique. The \\emph{Wagner \ngraph} $M_8$ (also called the \\emph{M\\\"{o}bius 8-ladder}) is formed from the $8$-cycle $z_1\\cdots z_8$ by \nadding the 4 chords $z_iz_{i+4}$ for each $i\\in [4]$.\nWagner~\\cite{wagner} characterized the class of maximal $K_5$-minor-free graphs as follows.\n\nSo to prove our Main Theorem, we would like to reduce to the case of planar graphs and copies of\n $M_8$. However, we must ensure that our disjoint $\\MM$-colorings $\\vph_1,\\vph_2,\\vph_3$\n agree on the copies of $K_2$ and $K_3$ where we perform our 2-sums and 3-sums. This motivates the\n following slightly stronger lemmas. We handle the copies of $M_8$ first, even for 6 disjoint\n $\\MM$-colorings, since this step is easy.\n\nIn this section we present proofs of our results. We first prove part (i) of our Main Theorem, by using \n\\Cref{M8-lem}, \\Cref{planar-lem}, and Theorem~\\ref{wagner-thm} (Wagner's characterization of \n$K_5$-minor-free graphs). That is, given a $K_5$-minor-free graph \n$G$ and a correspondence 6-cover $\\MM$, we show how to construct disjoint $\\MM$-colorings \n$\\vph_1,\\vph_2,\\vph_3$. Next, we prove \\Cref{planar-lem} using our Key Lemma; and after this we prove our \nKey Lemma via a ``Thomassen-style'' induction proof. Finally, we conclude this section by proving part (ii) \nof our Main Theorem. That is, we construct a $K_5$-minor-free graph $G$ and a correspondence 6-cover \n$\\MM$ of $G$ that do not admit 3 disjoint $\\MM$-colorings.\n\nBy \\Cref{wagner-thm}, $G$ can be constructed by a sequence of 2-sums and 3-sums starting from\n copies of $M_8$ and planar graphs. Furthermore, by possibly reordering the sequence, we can assume\n that for each 2-sum or 3-sum one of the graphs being summed is itself either a copy of $M_8$ or a \n planar graph. Fix such a construction sequence, and let $q$ be the sum of its\n numbers of 2-sums and 3-sums. We proceed by induction on $q$. If $q=0$, then $G$ is planar or a \n copy of $M_8$, so we are done by \\Cref{M8-lem} or by \\Cref{planar-lem}. To be precise, in the latter\n case, we should first fix a copy $C$ of $K_3$ and disjoint $\\MM$-colorings of $C$. But this\n is easy by \\Cref{ext-lem}. This completes the base case, $q=0$.\n\nNow assume that $q\\ge 1$. So $G$ is formed from a 2-sum or 3-sum of graphs $G_1$ and $G_2$,\n where $G_2$ is planar or a copy of $M_8$ and $G_1$ is formed from a sequence of 2-sums and\n 3-sums of length $q-1$. By the induction hypothesis, $G_1$ has\n disjoint $\\MM$-colorings $\\vph_1',\\vph_2',\\vph_3'$. Let the copy of $K_2$ or $K_3$ in $G_2$, \n that is involved in the sum to form $G$, inherit from the corresponding vertices in $G_1$ three\n disjoint $\\MM$-colorings $\\vph_1^0,\\vph_2^0,\\vph_3^0$. By \\Cref{M8-lem} or \\Cref{planar-lem}, \n we can extend $\\vph_1^0,\\vph_2^0,\\vph_3^0$ to disjoint $\\MM$-colorings $\\vph_1'',\\vph_2'', \n \\vph_3''$ of $G_2$. (If $G_2$ is planar and only a $K_2$ is precolored, then we first extend \n the precoloring to a $K_3$, via \\Cref{ext-lem}, before invoking \\Cref{planar-lem}; this is possible \n because $G_2$ is \\emph{maximal planar}, so every $K_2$ is contained in a $K_3$.) Since \n $\\vph_j'$ and $\\vph_j''$ agree on the vertices of $V(G_1)\\cap V(G_2)$, for all $j\\in[3]$, \n together they give our disjoint $\\MM$-colorings $\\vph_1,\\vph_2,\\vph_3$ of $G$.\n\\end{proof}\n\n\\begin{conj}\\label{conj1}\n\tIf $G$ is $K_5$-minor-free and $\\MM$ is a correspondence 6-cover, then $G$ has 6 disjoint \n\t$\\MM$-colorings.  That is, $\\chisc(\\K_5)=6$.\n\\end{conj}", "post_theorem_intro_text_len": 2788, "post_theorem_intro_text": "So to prove our Main Theorem, we would like to reduce to the case of planar graphs and copies of\n\t$M_8$.  However, we must ensure that our disjoint $\\MM$-colorings $\\vph_1,\\vph_2,\\vph_3$\n\tagree on the copies of $K_2$ and $K_3$ where we perform our 2-sums and 3-sums.  This motivates the\n\tfollowing slightly stronger lemmas.  We handle the copies of $M_8$ first, even for 6 disjoint\n\t$\\MM$-colorings, since this step is easy.\n\n\\begin{lem}\n\t\\label[lem]{M8-lem}\n\tGiven a copy of $M_8$ and a correspondence 6-cover $\\MM$ of $M_8$, there exist disjoint\n\t$\\MM$-colorings $\\vph_1,\\ldots,\\vph_6$.  Furthermore, if $\\vph_1^0,\\ldots,\\vph_6^0$ are\n\tdisjoint $\\MM$-colorings of a specified $K_2$ in $M_8$, then we can choose \n\t$\\vph_1,\\ldots,\\vph_6$ to extend $\\vph_1^0,\\ldots,\\vph_6^0$.\n\\end{lem}\n\\begin{proof}\n\tWe use induction on the number $n'$ of uncolored vertices.  The base case, $n'=0$, is trivial.\n\tAnd the induction step holds, because $M_8$ is 3-degenerate, by Lemma~\\ref{ext-lem} with $s:=6$\n\tand $t:=6$.\n\\end{proof}\n\nNow we consider planar graphs.\n\n\\begin{lem}\n\t\\label[lem]{planar-lem}\n\tLet $G$ be a planar graph with a 6-correspondence cover $\\MM$.  If $C$ is a $K_3$ in $G$,\n\tthen each choice $\\vph_1^0,\\vph_2^0, \\vph_3^0$ of disjoint $\\MM$-colorings of $C$ extends to \n\tdisjoint $\\MM$-colorings $\\vph_1,\\vph_2,\\vph_3$ of $G$.\n\\end{lem}\n\nThe key to the proof of Lemma~\\ref{planar-lem} is proving a still stronger statement, given below in the Key Lemma, that more easily facilitates \nproof by induction.  Our technique is known as a ``Thomassen-style'' proof, due to the strikingly short and \nelegant proof of Carsten Thomassen~\\cite{Thomassen-5choosable} that all planar graphs are 5-choosable.  This \nmethod has since been applied many times~\\cite{CCKPSZ, GZ, KT, LWZZ, thomassen-grotzsch2, WL, \nzhu-4choosable}.  For a unified study of numerous such examples, see~\\cite[Chapter 11]{GCM}.\n\n\\begin{keylem}\n\tLet $G$ be a plane near-triangulation, with a correspondence 6-cover $\\MM$.  Denote the vertices \n\ton the outer face of $G$, in clockwise order, by $w_1,\\ldots,w_n$.  There exist disjoint \n\t$\\MM$-colorings $\\vph_1,\\vph_2,\\vph_3$ that extend $\\vph_1^0,\\vph_2^0,\\vph_3^0$ and that have \n\t$\\vph_j(v)\\in L_j(v)$ for all $j\\in[3]$ if $\\vph_1^0,\\vph_2^0,\\vph_3^0$ and $L_1,L_2,L_3$ \n\tsatisfy the following conditions\n\t\\begin{enumerate}\n\t\t\\item[(1)] $L_1(v)=L_2(v)=L_3(v)=[6]$ \n\t\t\tfor each vertex $v$ not on the outer face.\n\t\t\\item[(2)] For each vertex $w_i$ with $i\\in\\{3,\\ldots,n\\}$ there exist distinct colors \n\t\t\t$c_1,\\ldots,c_6\\in [6]$ with \n\t\t\t$L_1(w_i)=\\{c_1,c_2,c_3,c_4\\}$ and\n\t\t\t$L_2(w_i)=\\{c_1,c_2,c_5,c_6\\}$ and\n\t\t\t$L_3(w_i)=\\{c_3,c_4,c_5,c_6\\}$.\n\t\t\\item[(3)] \n\t\t\t$\\vph_1^0,\\vph_2^0,\\vph_3^0$ are disjoint $\\MM$-colorings of $G[\\{w_1,w_2\\}]$.\n\t\\end{enumerate}\n\\end{keylem}", "sketch": "To prove the Main Theorem, the authors “would like to reduce to the case of planar graphs and copies of $M_8$” using Wagner’s decomposition, but they note they “must ensure that [the] disjoint $\\MM$-colorings $\\vph_1,\\vph_2,\\vph_3$ agree on the copies of $K_2$ and $K_3$ where we perform our 2-sums and 3-sums,” which “motivates the following slightly stronger lemmas.” They “handle the copies of $M_8$ first, even for 6 disjoint $\\MM$-colorings, since this step is easy”: for $M_8$ they “use induction on the number $n'$ of uncolored vertices,” with base case “$n'=0$” trivial, and the induction step because “$M_8$ is 3-degenerate, by Lemma~\\ref{ext-lem} with $s:=6$ and $t:=6$,” yielding disjoint colorings that can be chosen to extend prescribed disjoint colorings on a specified $K_2$.\n\nFor planar graphs, they aim to prove Lemma~\\ref{planar-lem} (extending prescribed disjoint $\\MM$-colorings on a $K_3$), and state that “the key to the proof of Lemma~\\ref{planar-lem} is proving a still stronger statement, given below in the Key Lemma, that more easily facilitates proof by induction,” using a “Thomassen-style” inductive method on plane near-triangulations with specified outer-face structure and list conditions.", "expanded_sketch": "To prove the Main Theorem, the authors “would like to reduce to the case of planar graphs and copies of $M_8$” using Wagner’s decomposition, but they note they “must ensure that [the] disjoint $\\MM$-colorings $\\vph_1,\\vph_2,\\vph_3$ agree on the copies of $K_2$ and $K_3$ where we perform our 2-sums and 3-sums,” which “motivates the following slightly stronger lemmas.” They “handle the copies of $M_8$ first, even for 6 disjoint $\\MM$-colorings, since this step is easy”: for $M_8$ they “use induction on the number $n'$ of uncolored vertices,” with base case “$n'=0$” trivial, and the induction step because “$M_8$ is 3-degenerate, by Lemma~\\ref{ext-lem} with $s:=6$ and $t:=6$,” yielding disjoint colorings that can be chosen to extend prescribed disjoint colorings on a specified $K_2$.\n\nFor planar graphs, they aim to prove Lemma~\\ref{planar-lem} (extending prescribed disjoint $\\MM$-colorings on a $K_3$), and state that “the key to the proof of Lemma~\\ref{planar-lem} is proving a still stronger statement, given below in the Key Lemma, that more easily facilitates proof by induction,” using a “Thomassen-style” inductive method on plane near-triangulations with specified outer-face structure and list conditions.", "expanded_theorem": "[\\cite{wagner}]\n\t\\label{wagner-thm}\n\tIf a graph $G$ is maximal $K_5$-minor-free, then $G$ can be built by a sequence of $2$-sums and \n\t$3$-sums, without deleting edges, starting from maximal planar graphs and copies of $M_8$.", "theorem_type": ["Implication", "Algorithmic or Constructive"], "mcq": {"question": "Let $G$ be a maximal $K_5$-minor-free graph, meaning that $G$ has no $K_5$ minor and adding any missing edge to $G$ creates a $K_5$ minor. An $r$-sum of graphs $G_1$ and $G_2$ is obtained from their disjoint union by choosing an $r$-clique $x_1,\\dots,x_r$ in $G_1$ and an $r$-clique $y_1,\\dots,y_r$ in $G_2$, identifying $x_i$ with $y_i$ for each $i\\in[r]$, and in general one may optionally delete edges of the resulting identified $r$-clique. The Wagner graph $M_8$ is the graph formed from the cycle $z_1z_2\\cdots z_8z_1$ by adding the four chords $z_iz_{i+4}$ for $i\\in[4]$. Under these assumptions, which structural statement about $G$ holds?", "correct_choice": {"label": "A", "text": "$G$ can be built from maximal planar graphs and copies of $M_8$ by a sequence of $2$-sums and $3$-sums, with no deletion of edges from the identified clique at any step."}, "choices": [{"label": "B", "text": "$G$ can be built from planar graphs and copies of $M_8$ by a sequence of $2$-sums and $3$-sums, where at each step one may delete some edges of the identified clique."}, {"label": "C", "text": "$G$ can be built from maximal planar graphs and copies of $M_8$ by a sequence of $2$-sums and $3$-sums."}, {"label": "D", "text": "$G$ can be built from maximal planar graphs and copies of $M_8$ by a sequence of $1$-sums, $2$-sums, and $3$-sums, with no deletion of edges from the identified clique at any step."}, {"label": "E", "text": "$G$ can be built from maximal planar graphs and copies of $M_8$ by a sequence of $2$-sums and $3$-sums so that, after each sum, the identified clique separates the two previously constructed pieces."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "no-edge-deletion requirement in Wagner decomposition", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "geometric_construction", "tampered_component": "dropped the clause forbidding deletion of edges in the identified clique", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "allowed sum sizes beyond the stated $2$- and $3$-sum range", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "imposed an extra separator/nesting condition not present in the decomposition theorem", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the decomposition conclusion or the key clause 'with no deletion of edges from the identified clique.' It provides definitions and asks for the correct structural theorem, so there is no direct answer leakage."}, "TAS": {"score": 0, "justification": "This is essentially a recall of a specific decomposition theorem for maximal K5-minor-free graphs. The question asks which theorem statement holds, making it very close to a direct restatement rather than a novel application."}, "GPS": {"score": 1, "justification": "Some discrimination is required because the choices differ in subtle but important ways: maximal planar vs. planar, allowing edge deletion vs. forbidding it, 2/3-sums vs. also 1-sums, and an extra separator condition. However, the task is still mainly theorem recognition rather than substantial generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful. They target realistic failure modes: accepting a weaker true statement, overgeneralizing the allowed sums, confusing the edge-deletion convention, or adding an unsupported separator condition."}, "total_score": 5, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it is largely tautological and tests precise recall more than genuine generative reasoning."}}
{"id": "2602.17191v1", "paper_link": "http://arxiv.org/abs/2602.17191v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{thm:main}\n  Let $\\dim X =2$. There is a unique $\\hat T\\in\\PD{X}{\\mathbb{R}^2}$ such that $\\hat T(B_X)\\supset B_{\\mathbb{R}^2}$ and\n  \\begin{equation}\n    \\label{eq:tm=d2}\n    \\|\\hat T\\| = d_2(X).\n  \\end{equation}\n  $\\hat T$ is characterised by the following property:\n\n  There are $x_i\\in S_X$, $i=1,2$, with $x_1\\neq \\pm x_2$, and $y_i\\in S_X$, $i=1,2$, with $y_1\\neq \\pm y_2$, such that:\n  \\begin{equation}\n    \\label{eq:x-distance}\n    |\\hat T x_i| = \\|\\hat T\\|,\\quad i=1,2,\n  \\end{equation}\n  \\begin{equation}\n    \\label{eq:y-contact}\n    |\\hat T y_i| = 1,\\quad i=1,2,\n  \\end{equation}\n  \\begin{equation}\n    \\label{eq:x-between-y}\n    y_1\\in\\mathrm{cone}\\, \\{x_1,x_2\\},\\quad y_2\\in\\mathrm{cone}\\,\\{-x_1,x_2\\}.\n  \\end{equation}", "start_pos": 3774, "end_pos": 4472, "label": "thm:main"}, "ref_dict": {"eq:def-bmd": "\\begin{equation}\n  \\label{eq:def-bmd}\n  d(X,Y) := \\inf \\{\\|T\\|\\cdot\\|T^{-1}\\|:\\ T:X\\to Y\\ \\text{isomorphism}\\},\n\\end{equation}", "eq:d2-inclusion": "\\begin{equation}\n  \\label{eq:d2-inclusion}\n  d_2(X) = \\min \\{\\|T\\|:\\ T:X\\to \\R^2,\\ T(B_X)\\supset B_{\\R^2}\\},\n\\end{equation}", "thm:main": "\\begin{thm}\n  \\label{thm:main}\n  Let $\\dim X =2$. There is a unique $\\T\\in\\PD{X}{\\R^2}$ such that $\\T(B_X)\\supset B_{\\R^2}$ and\n  \\begin{equation}\n    \\label{eq:tm=d2}\n    \\|\\T\\| = d_2(X).\n  \\end{equation}\n  $\\T$ is characterised by the following property:\n\n  There are $x_i\\in S_X$, $i=1,2$, with $x_1\\neq \\pm x_2$, and $y_i\\in S_X$, $i=1,2$, with $y_1\\neq \\pm y_2$, such that:\n  \\begin{equation}\n    \\label{eq:x-distance}\n    |\\T x_i| = \\|\\T\\|,\\quad i=1,2,\n  \\end{equation}\n  \\begin{equation}\n    \\label{eq:y-contact}\n    |\\T y_i| = 1,\\quad i=1,2,\n  \\end{equation}\n  \\begin{equation}\n    \\label{eq:x-between-y}\n    y_1\\in\\cone \\{x_1,x_2\\},\\quad y_2\\in\\cone\\{-x_1,x_2\\}.\n  \\end{equation}\n\\end{thm}"}, "pre_theorem_intro_text_len": 2242, "pre_theorem_intro_text": "Recall that the Banach-Mazur distance between the isomorphic normed spaces $X$ and $Y$ is denoted by \n\\begin{equation}\n  \\label{eq:def-bmd}\n  d(X,Y) := \\inf \\{\\|T\\|\\cdot\\|T^{-1}\\|:\\ T:X\\to Y\\ \\text{isomorphism}\\},\n\\end{equation}\nfor details see e.g.\\ \\cite[Section 6.1.1]{pietsch}.\n\nIf $X$ and $Y$ are of one and the same finite dimension, then the minimum in \\eqref{eq:def-bmd} is attained.\n\nAn important characteristic of a finite dimensional space is its distance to the Euclidean space. Here we study this characteristic in the simplest case: the plane.\n\nSo, for a 2-dimensional normed space $(X,\\|\\cdot\\|)$\n$$\n  d_2(X) := d(X,\\mathbb{R}^2),\n$$\nwhere $\\mathbb{R}^2$ is considered with the standard Euclidean norm $|\\cdot|$. It is easy to check that\n\\begin{equation}\n  \\label{eq:d2-inclusion}\n  d_2(X) = \\min \\{\\|T\\|:\\ T:X\\to \\mathbb{R}^2,\\ T(B_X)\\supset B_{\\mathbb{R}^2}\\},\n\\end{equation}\nwhere $B_X$ and $S_X$ stand for the unit ball and the unit sphere of $X$ respectively. Indeed, $T(B_X)\\supset B_{\\mathbb{R}^2}$ is equivalent to $\\|T^{-1}\\| \\le 1$, so we have that $d_2(X)$ is at most the right-hand side of \\eqref{eq:d2-inclusion}. On the other hand, multiplying the operator $T$ by a real constant does not change $\\|T\\|\\cdot\\|T^{-1}\\|$, so we may assume that $\\|T^{-1}\\| = 1$ in \\eqref{eq:def-bmd}.\n\nUsing \\textit{polar decomposition}, we can reduce \\eqref{eq:d2-inclusion}. Note that if $U$ is in $\\mathbf{O}(2)$ -- the group of isometries of $\\mathbb{R}^2$, that is, $U^{-1} = U^t$ if considered as a matrix -- then $\\|UT\\| = \\max_{x\\in S_X} |UTx| = \\max_{x\\in S_X} |Tx| = \\|T\\|$, and similarly $\\|T^{-1}U^{-1}\\| = \\|T^{-1}\\|$, so we can quotient by $\\mathbf{O}(2)$. Denote by $\\PD{X}{\\mathbb{R}^2}$ the operators from $X$ to $\\mathbb{R}^2$ that are positive definite symmetric matrices in any basis. By \\cite[Theorem~7.3.1, p.449]{hj} each isomorphism $T:X\\to\\mathbb{R}^2$ can be represented as $T=UT_1$, where $U\\in\\mathbf{O}(2)$ and $T_1\\in\\PD{X}{\\mathbb{R}^2}$. Therefore,\n\\begin{equation}\n  \\label{eq:d2-reduction}\n  d_2(X) = \\min \\{\\|T\\|:\\ T\\in\\PD{X}{\\mathbb{R}^2},\\ T(B_X)\\supset B_{\\mathbb{R}^2}\\}.\n\\end{equation}\n\nWe prove in a new way the following characterisation, which is essentially contained in \\cite{ader,mory}.", "context": "Recall that the Banach-Mazur distance between the isomorphic normed spaces $X$ and $Y$ is denoted by \n\\begin{equation}\n \\label{eq:def-bmd}\n d(X,Y) := \\inf \\{\\|T\\|\\cdot\\|T^{-1}\\|:\\ T:X\\to Y\\ \\text{isomorphism}\\},\n\\end{equation}\nfor details see e.g.\\ \\cite[Section 6.1.1]{pietsch}.\n\nIf $X$ and $Y$ are of one and the same finite dimension, then the minimum in \\eqref{eq:def-bmd} is attained.\n\nAn important characteristic of a finite dimensional space is its distance to the Euclidean space. Here we study this characteristic in the simplest case: the plane.\n\nSo, for a 2-dimensional normed space $(X,\\|\\cdot\\|)$\n$$\n d_2(X) := d(X,\\mathbb{R}^2),\n$$\nwhere $\\mathbb{R}^2$ is considered with the standard Euclidean norm $|\\cdot|$. It is easy to check that\n\\begin{equation}\n \\label{eq:d2-inclusion}\n d_2(X) = \\min \\{\\|T\\|:\\ T:X\\to \\mathbb{R}^2,\\ T(B_X)\\supset B_{\\mathbb{R}^2}\\},\n\\end{equation}\nwhere $B_X$ and $S_X$ stand for the unit ball and the unit sphere of $X$ respectively. Indeed, $T(B_X)\\supset B_{\\mathbb{R}^2}$ is equivalent to $\\|T^{-1}\\| \\le 1$, so we have that $d_2(X)$ is at most the right-hand side of \\eqref{eq:d2-inclusion}. On the other hand, multiplying the operator $T$ by a real constant does not change $\\|T\\|\\cdot\\|T^{-1}\\|$, so we may assume that $\\|T^{-1}\\| = 1$ in \\eqref{eq:def-bmd}.\n\nUsing \\textit{polar decomposition}, we can reduce \\eqref{eq:d2-inclusion}. Note that if $U$ is in $\\mathbf{O}(2)$ -- the group of isometries of $\\mathbb{R}^2$, that is, $U^{-1} = U^t$ if considered as a matrix -- then $\\|UT\\| = \\max_{x\\in S_X} |UTx| = \\max_{x\\in S_X} |Tx| = \\|T\\|$, and similarly $\\|T^{-1}U^{-1}\\| = \\|T^{-1}\\|$, so we can quotient by $\\mathbf{O}(2)$. Denote by $\\PD{X}{\\mathbb{R}^2}$ the operators from $X$ to $\\mathbb{R}^2$ that are positive definite symmetric matrices in any basis. By \\cite[Theorem~7.3.1, p.449]{hj} each isomorphism $T:X\\to\\mathbb{R}^2$ can be represented as $T=UT_1$, where $U\\in\\mathbf{O}(2)$ and $T_1\\in\\PD{X}{\\mathbb{R}^2}$. Therefore,\n\\begin{equation}\n \\label{eq:d2-reduction}\n d_2(X) = \\min \\{\\|T\\|:\\ T\\in\\PD{X}{\\mathbb{R}^2},\\ T(B_X)\\supset B_{\\mathbb{R}^2}\\}.\n\\end{equation}\n\nWe prove in a new way the following characterisation, which is essentially contained in \\cite{ader,mory}.\n\n\\begin{equation}\n  \\label{eq:d2-inclusion}\n  d_2(X) = \\min \\{\\|T\\|:\\ T:X\\to \\R^2,\\ T(B_X)\\supset B_{\\R^2}\\},\n\\end{equation}\n\n\\begin{equation}\n  \\label{eq:def-bmd}\n  d(X,Y) := \\inf \\{\\|T\\|\\cdot\\|T^{-1}\\|:\\ T:X\\to Y\\ \\text{isomorphism}\\},\n\\end{equation}", "full_context": "Recall that the Banach-Mazur distance between the isomorphic normed spaces $X$ and $Y$ is denoted by \n\\begin{equation}\n \\label{eq:def-bmd}\n d(X,Y) := \\inf \\{\\|T\\|\\cdot\\|T^{-1}\\|:\\ T:X\\to Y\\ \\text{isomorphism}\\},\n\\end{equation}\nfor details see e.g.\\ \\cite[Section 6.1.1]{pietsch}.\n\nIf $X$ and $Y$ are of one and the same finite dimension, then the minimum in \\eqref{eq:def-bmd} is attained.\n\nAn important characteristic of a finite dimensional space is its distance to the Euclidean space. Here we study this characteristic in the simplest case: the plane.\n\nSo, for a 2-dimensional normed space $(X,\\|\\cdot\\|)$\n$$\n d_2(X) := d(X,\\mathbb{R}^2),\n$$\nwhere $\\mathbb{R}^2$ is considered with the standard Euclidean norm $|\\cdot|$. It is easy to check that\n\\begin{equation}\n \\label{eq:d2-inclusion}\n d_2(X) = \\min \\{\\|T\\|:\\ T:X\\to \\mathbb{R}^2,\\ T(B_X)\\supset B_{\\mathbb{R}^2}\\},\n\\end{equation}\nwhere $B_X$ and $S_X$ stand for the unit ball and the unit sphere of $X$ respectively. Indeed, $T(B_X)\\supset B_{\\mathbb{R}^2}$ is equivalent to $\\|T^{-1}\\| \\le 1$, so we have that $d_2(X)$ is at most the right-hand side of \\eqref{eq:d2-inclusion}. On the other hand, multiplying the operator $T$ by a real constant does not change $\\|T\\|\\cdot\\|T^{-1}\\|$, so we may assume that $\\|T^{-1}\\| = 1$ in \\eqref{eq:def-bmd}.\n\nUsing \\textit{polar decomposition}, we can reduce \\eqref{eq:d2-inclusion}. Note that if $U$ is in $\\mathbf{O}(2)$ -- the group of isometries of $\\mathbb{R}^2$, that is, $U^{-1} = U^t$ if considered as a matrix -- then $\\|UT\\| = \\max_{x\\in S_X} |UTx| = \\max_{x\\in S_X} |Tx| = \\|T\\|$, and similarly $\\|T^{-1}U^{-1}\\| = \\|T^{-1}\\|$, so we can quotient by $\\mathbf{O}(2)$. Denote by $\\PD{X}{\\mathbb{R}^2}$ the operators from $X$ to $\\mathbb{R}^2$ that are positive definite symmetric matrices in any basis. By \\cite[Theorem~7.3.1, p.449]{hj} each isomorphism $T:X\\to\\mathbb{R}^2$ can be represented as $T=UT_1$, where $U\\in\\mathbf{O}(2)$ and $T_1\\in\\PD{X}{\\mathbb{R}^2}$. Therefore,\n\\begin{equation}\n \\label{eq:d2-reduction}\n d_2(X) = \\min \\{\\|T\\|:\\ T\\in\\PD{X}{\\mathbb{R}^2},\\ T(B_X)\\supset B_{\\mathbb{R}^2}\\}.\n\\end{equation}\n\nWe prove in a new way the following characterisation, which is essentially contained in \\cite{ader,mory}.\n\n\\begin{equation}\n  \\label{eq:d2-inclusion}\n  d_2(X) = \\min \\{\\|T\\|:\\ T:X\\to \\R^2,\\ T(B_X)\\supset B_{\\R^2}\\},\n\\end{equation}\n\n\\begin{equation}\n  \\label{eq:def-bmd}\n  d(X,Y) := \\inf \\{\\|T\\|\\cdot\\|T^{-1}\\|:\\ T:X\\to Y\\ \\text{isomorphism}\\},\n\\end{equation}\n\nAs an illustration, consider the following picture.\n\nThere are $u_i,v_i\\in S_{\\R^2}$, $i=1,2$, with $u_1\\neq \\pm u_2$, $v_1\\neq \\pm v_2$, $v_1\\in\\cone \\{u_1,u_2\\}$, $v_2\\in\\cone \\{-u_1,u_2\\}$, and\n \\begin{equation}\n \\label{eq:equiv-alternance}\n \\|\\tilde Tu_i\\| = 1/\\|\\tilde T^{-1}\\|,\\quad \\|\\tilde T v_i\\| = 1,\\quad i=1,2.\n \\end{equation}\n \\end{pro}\n Let a co-ordinate system be fixed in $X$ in which\n \\begin{equation}\n \\label{eq:r-*}\n S_X = \\{r(\\varphi)(\\cos\\varphi,\\sin\\varphi):\\ \\varphi\\in\\R\\},\n \\end{equation}\n for a strictly positive, $\\pi$-periodic and continuous $r:\\R\\to\\R^+$.\n\nIn these fixed coordinates each $T\\in\\PD{\\R^2}{X}$ is identified with a matrix\n $$\n T = \\begin{pmatrix} \n a & b \\\\ b & c \n \\end{pmatrix},\\quad a,c>0,\\quad \\det T = ac-b^2>0.\n $$\n Obviously,\n $$\n x \\in T(S_{\\mathbb{R}^2}) \\iff |T^{-1}x| = 1 \\iff x^t T^{-2} x = 1.\n $$\n That is, $T(S_{\\mathbb{R}^2})$ is the ellipse:\n $$\n a_1 x_1^2 + 2 b_1 x_1x_2 + c_1 x_2^2 = 1,\n $$\n where\n $$\n \\begin{pmatrix} a_1 & b_1 \\\\ b_1 & c_1 \\end{pmatrix} = T^{-2},\n $$\n so $a_1,c_1 > 0$ and $b_1^2 < a_1c_1$. In polar co-ordinates then $T(S_{\\mathbb{R}^2}) = \\{\\rho(\\varphi)(\\cos\\varphi,\\sin\\varphi):\\ \\varphi\\in\\R\\}$, where\n \\begin{equation}\n \\label{eq:rho-trig}\n \\rho^{-2}(\\varphi) = a_2 + b_2\\cos 2\\varphi + c_2\\sin 2\\varphi,\n \\end{equation}\n with $a_2 = (a_1+c_1)/2$, $b_2 = (a_1-c_1)/2$ and $c_2 = b_1$, so\n \\begin{equation}\n \\label{eq:ellipse-cond}\n a_2 > 0,\\quad a_2^2 > b_2^2 + c_2^2.\n \\end{equation}\n Usually this is expressed in terms of $\\cos2(\\varphi-\\theta)$, where $\\theta$ is the tilt angle. Let $\\mathcal E$ be the family of all ellipses in polar co-ordinates, that is, all curves in the form \\eqref{eq:rho-trig} for all values of the parameters $a_2,b_2,c_2$ which satisfy \\eqref{eq:ellipse-cond}. It is clear that the cone \\eqref{eq:ellipse-cond} parametrises one-to-one the ellipses, as well as $\\PD{\\R^2}{X}$, which we will use when establishing the uniqueness part of our main result. Let\n \\begin{equation}\n \\label{eq:C-def}\n \\mathcal{C} := \\{\\log \\rho:\\ \\rho\\in\\mathcal{E}\\}.\n \\end{equation}\n Let also, see \\eqref{eq:r-*},\n $$\n f := \\log r.\n $$\n Since $\\|T\\| \\le 1$ is equivalent to $T(S_{\\mathbb{R}^2})\\subset B_X$, which in polar co-ordinates translates to\n $$\n \\rho \\le r \\iff g := \\log \\rho \\le f,\n $$\n the constraint in \\eqref{eq:t-1-prob} becomes $g\\in\\mathcal{C}$, $g \\le f$. For the objective, since $\\rho(\\varphi)(\\cos\\varphi,\\sin\\varphi) \\in T(S_{\\R^2})$, we have $|T^{-1}(\\rho(\\varphi)(\\cos\\varphi,\\sin\\varphi))| = 1$, hence $|T^{-1}(\\cos\\varphi,\\sin\\varphi)| = 1/\\rho(\\varphi)$. For $x = r(\\varphi)(\\cos\\varphi,\\sin\\varphi) \\in S_X$,\n $$\n |T^{-1}x| = \\frac{r(\\varphi)}{\\rho(\\varphi)} = \\exp (f(\\varphi)-g(\\varphi)),\n $$\n and the problem \\eqref{eq:t-1-prob} is equivalent to\n $$\n \\begin{cases}\n \\max_\\varphi \\exp (f(\\varphi)-g(\\varphi)) \\to \\min\\\\\n g\\in\\mathcal{C},\\quad g \\le f.\n \\end{cases}\n $$\n Since the exponential is strictly monotonically increasing, Proposition~\\ref{prop:equiv} is equivalent to Proposition~\\ref{prop:C-cal-equiv} below. There, for a $\\pi$-periodic function $h$\n $$\n \\|h\\|_\\infty := \\max_\\varphi |h(\\varphi)|.\n $$\n \\begin{pro}\n \\label{prop:C-cal-equiv}\n The optimisation problem\n \\begin{equation}\n \\label{eq:C-cal-equiv}\n \\begin{cases}\n \\|f - g\\|_\\infty \\to \\min\\\\\n g\\in\\mathcal{C},\\quad g \\le f\n \\end{cases}\n \\end{equation}\n has a unique solution $\\bar g$ characterised by the following property:\n\nWe are ready to prove the uniqueness part of our main result.\n \\begin{pro}\n \\label{prop:unical}\n Let $\\bar g\\in \\mathcal{C}$. Assume that there exist points $\\varphi_i,\\psi_i$, $i=1,2$, satisfying \\eqref{eq:alternance-pts} and such that\n \\begin{equation}\n \\label{eq:alternance-pts-values-2}\n f(\\varphi_i) - \\bar g(\\varphi_i) = \\bar g(\\psi_i) - f(\\psi_i) =\\|f-\\bar g\\|_\\infty,\\quad i=1,2.\n \\end{equation}\n Then $\\bar g$ is the unique solution to \\eqref{eq:Ch}.\n \\end{pro}\n \\begin{proof}\n Let $\\hat g \\in \\mathcal{C}$ be such that\n \\begin{equation}\n \\label{eq:51}\n \\|f-\\hat g\\|_\\infty \\le \\|f - \\bar g\\|_\\infty.\n \\end{equation}\n If we can prove that from this follows $\\hat g = \\bar g$ then obviously $\\bar g$ is a solution to \\eqref{eq:Ch} and, moreover, unique.\n\nAssume that\n \\begin{equation}\n \\label{eq:71}\n \\bar g \\neq \\hat g.\n \\end{equation} \n Let $h := \\bar g - \\hat g$. We can write $h(\\varphi_i) = (f-\\hat g) (\\varphi_i) - (f-\\bar g) (\\varphi_i)$. By \\eqref{eq:alternance-pts-values-2} we have $(f-\\bar g)(\\varphi_i) = \\|f-\\bar g\\|_\\infty$, so $h(\\varphi_i) = (f-\\hat g)(\\varphi_i) - \\|f-\\bar g\\|_\\infty \\le \\|f-\\hat g\\|_\\infty - \\|f-\\bar g\\|_\\infty \\le 0$ by \\eqref{eq:51}. Thus $h(\\varphi_i) \\le 0$.\n Similarly, $h(\\psi_i) = (f-\\hat g)(\\psi_i) + (\\bar g -f)(\\psi_i) = (f-\\hat g)(\\psi_i) + \\|f-\\bar g\\|_\\infty \\ge \\|f-\\bar g\\|_\\infty - \\|f-\\hat g\\|_\\infty \\ge 0$. We have shown that\n \\begin{equation}\n \\label{eq:72}\n h(\\varphi_i) \\le 0 \\le h(\\psi_i),\\quad i=1,2.\n \\end{equation}\n Consequently, there exist $\\xi_i\\in[\\varphi_i,\\psi_i]$, $i=1,2$ (in particular, $\\xi_1<\\xi_2$) such that $h(\\xi_i) = 0$ for $i=1,2$. If there were a third point in $[\\varphi_1,\\varphi_1+\\pi)$ at which $h=0$, then Proposition~\\ref{prop:elipses-3-pts-equial} would have implied $\\bar g = \\hat g$, contradicting \\eqref{eq:71}, so\n \\begin{equation}\n \\label{eq:73}\n h(\\varphi) \\neq 0,\\quad \\forall \\varphi \\in [\\varphi_1,\\varphi_1+\\pi) \\setminus \\{\\xi_1,\\xi_2\\}.\n \\end{equation}\n Also, if $h'(\\xi_i) = 0$ for some $i=1,2$, then again Proposition~\\ref{prop:elipses-3-pts-equial} implies $\\bar g = \\hat g$, so\n \\begin{equation}\n \\label{eq:74}\n h'(\\xi_i) \\neq 0,\\quad i=1,2.\n \\end{equation}\n From \\eqref{eq:73} and $\\xi_1\\le\\psi_1<\\varphi_2\\le\\xi_2$ it follows that $h(\\varphi) \\neq 0$ for all $\\varphi\\in(\\psi_1,\\varphi_2)$. Since $h$ is continuous, it has a constant sign on $(\\psi_1,\\varphi_2)$, so there are two possibilities:\n\nThus \\eqref{eq:contra} is verified, but it contradicts \\eqref{eq:alternance-*}. Therefore, there is $\\varphi_2\\in\\Phi\\cap(\\psi_1,\\psi_2)$ and $\\varphi_i,\\psi_i$ satisfy \\eqref{eq:alternance-pts} and \\eqref{eq:alternance-pts-values-2}.\n \\end{proof}\n In view of the work so far the following completes our study.\n \\begin{pro}\n \\label{prop:final}\n The problem \\eqref{eq:Ch} has a solution.\n \\end{pro}\n \\begin{proof}\n Here it is easier to work with the standard form of the ellipses in polar co-ordinates. Take $(a,b,c)\\in K$, that is, $a>\\sqrt{b^2+c^2}$, and set \n $$\n b':= \\sqrt{b^2+c^2},\\quad b = b'\\cos 2\\theta,\\ c = b'\\sin 2\\theta.\n $$\n Then\n $$\n \\mathfrak{R}(a,b,c) = a + b'\\cos 2(\\varphi-\\theta).\n $$\n $K$ translates to\n $$\n K' := \\{(a,b',\\theta):\\ a>b'\\ge0,\\ \\theta\\in[0,\\pi]\\},\n $$\n and\n $$\n \\mathcal{C} = \\{-(1/2)\\log(a + b'\\cos 2(\\varphi-\\theta)) : (a,b',\\theta) \\in K'\\}.\n $$\n Let $(a_n,b_n',\\theta_n)\\in K'$ be such that for\n $$\n g_n(\\varphi) := -\\frac{1}{2}\\log(a_n + b_n'\\cos 2(\\varphi-\\theta_n))\n $$\n we have\n $$\n \\lim_{n\\to\\infty} \\|f-g_n\\|_\\infty = \\mathfrak{d}.\n $$\n By taking a subsequence we may assume that $\\theta_n\\to\\theta$, as $n\\to\\infty$. Since\n $$\n g_n(\\theta_n+\\pi/4) = -\\frac{1}{2}\\log a_n,\n $$\n and $\\|g_n\\|_\\infty < \\|f\\|_\\infty + \\mathfrak{d} + 1$ for large enough $n$'s, the sequence $\\{\\log a_n\\}_{n=1}^\\infty$ is bounded, so there is $r>0$ such that\n $$\n r \\le a_n \\le 1/r,\\quad\\forall n\\in\\mathbb{N}.\n $$\n Therefore, we may assume that $a_n\\to a >0$, as $n\\to\\infty$. We also assume that $b_n'\\to b'\\le a$, as $n\\to\\infty$. If $b'=a$ then\n $$\n g_n(\\theta_n+\\pi/2) = -\\frac{1}{2}\\log(a_n - b_n') \\to \\infty,\\text{ as }n\\to\\infty,\n $$\n which is a contradiction. So, $(a,b',\\theta)\\in K'$ and by continuity it follows that the corresponding $g\\in\\mathcal{C}$ solves \\eqref{eq:Ch}.\n \\end{proof}", "post_theorem_intro_text_len": 4849, "post_theorem_intro_text": "Here\n$$\n  \\mathrm{cone}\\, \\{x,y\\} := \\{\\alpha x + \\beta y:\\ \\alpha,\\beta \\ge 0\\}.\n$$\nNote that the condition characterising the optimal operator depends only on the operator itself and not on $d_2(X)$. This is to be expected, because the problem suitably stated is convex. Not strictly convex, however, so the uniqueness is not immediate.\n\nAs an illustration, consider the following picture.\n\n\\begin{center}\n  \\begin{tikzpicture}\n    \\draw[color=gray] (0,0) circle [radius = 2cm];\n    \\coordinate (y1) at (86:2);\n    \\coordinate (y1m) at (180+86:2);\nlldraw (y1) circle (1pt);\n    \\node[above] at (y1) {$\\hat Ty_2$};\n\n    \\coordinate (y2) at (26:2);\n    \\coordinate (y2m) at (180+26:2);\nlldraw (y2) circle (1pt);\n    \\node[right] at (y2) {$\\hat Ty_1$};\n\n    \\coordinate (x1) at (56:4/1.73205080757);\n    \\coordinate (x1m) at (180+56:4/1.73205080757);\nlldraw (x1) circle (1pt);\n    \\node[right, yshift=1.6mm] at (x1) {$\\hat Tx_2$};\n\n    \\draw[thick] (y1) -- (x1) -- (y2);\n    \\draw[thick] (y1m) -- (x1m) -- (y2m);\n\n    \\coordinate (a) at (-12:2);\n    \\coordinate (am) at (180-12:2);\n    \\coordinate (b) at (-42:4/1.73205080757);\n    \\coordinate (bm) at (180-42:4/1.73205080757);\n\n    \\draw[thick] (a) -- (b);\n    \\draw[thick] (am) -- (bm);\n    \\draw[thick] (y2) arc(26:-12:2);\n    \\draw[thick] (y2m) arc(180+26:180-12:2);\n\n    \\coordinate (u) at (272:2);\n    \\coordinate (um) at (92:2);\n    \\draw[thick] (y1m) arc(266:272:2);\n    \\draw[thick] (y1) arc(86:92:2);\n\n    \\coordinate (v) at (302:4/1.73205080757);\n    \\draw[thick] (u) -- (v);\n    \\coordinate (vm) at (122:4/1.73205080757);\n    \\draw[thick] (um) -- (vm);\n\n    \\draw[thick] (v) arc(302:318:4/1.73205080757);\n    \\draw[thick] (vm) arc(122:138:4/1.73205080757);\n\n    \\coordinate (x2) at (310:4/1.73205080757);\nlldraw (x2) circle (1pt);\n    \\node[right] at (x2) {$\\hat T x_1$};\n  \\end{tikzpicture}\n\\end{center}\n\nFrom it we can also see one important, but immediate, corollary. Since one of the angles between $\\hat T y_1$ and $\\hat T y_2$, and between $\\hat T y_2$ and $-\\hat T y_1$ (say the one between $\\hat T y_1$ and $\\hat T y_2$ as on the picture) will be $\\le \\pi/2$; and since $\\hat T x_2$ will be within the angle between the tangents to the unit circle at $\\hat T y_1$ and $\\hat T y_2$, due to convexity, the distance from $\\hat T x_2$ to zero, that is $d_2(X)$, is at most $\\sqrt{2}$. Therefore,\n$$\n  d_2(X) \\le \\sqrt{2}.\n$$\n\nNote that Theorem~\\ref{thm:main} easily follows from\n\\begin{principle}[\\cite{ader}]\n  For the unique $\\hat T$ there is no angle such that all points $x\\in S_X$ with $|\\hat T x| = \\|\\hat T\\|$ are inside this angle and all points $y\\in S_X$ with $|\\hat T y| = 1$ are outside.\n\\end{principle}\nHere, an angle is\n$$\n  \\mathrm{cone}\\, \\{a,b\\} \\cup \\mathrm{cone}\\, \\{-a,-b\\}\n$$\nfor some $a,b\\in\\mathbb{R}^2$.\n\nThe original work \\cite{ader} states that there should be at most $4$ points realising Ader's Non-separation Principle, and for these $4$ points it is then clear that the alternance described in Theorem~\\ref{thm:main} must be in place. By Maurey's result \\cite{mory}, see also \\cite{gru-ko}, the ellipse of minimal Banach-Mazur distance is unique, which yields the uniqueness part of Theorem~\\ref{thm:main}.\n\nObviously, the problem can be re-stated more geometrically like: find the minimal $k\\ge1$ such that there is an ellipse inscribed in $B_X$ whose homothetic image with ratio $k$ is circumscribed about $B_X$. We will actually be using this form below, but we mention it here to point out the relationship to the celebrated L\\\"owner--John ellipse, see e.g.\\ \\cite{john}, that is, the ellipse of maximal area inscribed in $B_X$. The L\\\"owner--John ellipsoid is extensively used for estimating the Banach-Mazur distance to Euclidean space, probably because Ader's result \\cite{ader} had apparently been forgotten until Grundbacher and Kobos recently unearthed it, see \\cite{gru-ko}.\n\nHowever, if $B_X$ is not symmetric the estimate through the L\\\"owner--John ellipsoid can grow progressively worse with the dimension. Consider $\\mathbb{R}^n$ and let $B_{X_n}$ be the convex hull of $B_{\\mathbb{R}^n}$ and $\\sqrt{n}e$, where $e\\in S_{\\mathbb{R}^n}$. Then $B_{\\mathbb{R}^n}$ is the L\\\"owner--John ellipsoid for $B_{X_n}$, see \\cite{rakov} for a nice and simple proof, so it gives only the generic estimate $\\sqrt{n}$. On the other hand, by the rotational symmetry of $B_{X_n}$ around the axis through $e$, $d(X_n,\\mathbb{R}^n)\\le\\sqrt{3}$.\n\nOur interest in the specifically two-dimensional case comes from our previous works \\cite{iv-tr,iv-pa-tr,nik1,nik2}. Of course, similar problems can also be considered in different metrics, like e.g.\\ the Hausdorff, see \\cite{kenderov,gru-ke,ke-ki}.\n\nIn conclusion, note that the convexity of $B_X$ will not be used at all in the proofs below, which is not surprising, as they go through Chebyshev approximation.", "sketch": "A proof route for Theorem~\\ref{thm:main} is indicated via Ader’s Non-separation Principle. The text states that Theorem~\\ref{thm:main} “easily follows from” the following: for the unique optimal $\\hat T$, “there is no angle such that all points $x\\in S_X$ with $|\\hat T x|=\\|\\hat T\\|$ are inside this angle and all points $y\\in S_X$ with $|\\hat T y|=1$ are outside,” where an “angle” is $\\mathrm{cone}\\{a,b\\}\\cup\\mathrm{cone}\\{-a,-b\\}$. The original work \\cite{ader} asserts there are “at most $4$ points” realizing this non-separation; “for these $4$ points it is then clear that the alternance described in Theorem~\\ref{thm:main} must be in place,” yielding the configuration of two norming points $x_i$ with $|\\hat T x_i|=\\|\\hat T\\|$ and two contact points $y_i$ with $|\\hat T y_i|=1$ arranged in the stated cone relations. The uniqueness part is attributed to Maurey’s result (see \\cite{mory,gru-ko}) that “the ellipse of minimal Banach–Mazur distance is unique,” which “yields the uniqueness part of Theorem~\\ref{thm:main}.”", "expanded_sketch": "A proof route for the main theorem is indicated via Ader’s Non-separation Principle. The text states that the main theorem “easily follows from” the following: for the unique optimal $\\hat T$, “there is no angle such that all points $x\\in S_X$ with $|\\hat T x|=\\|\\hat T\\|$ are inside this angle and all points $y\\in S_X$ with $|\\hat T y|=1$ are outside,” where an “angle” is $\\mathrm{cone}\\{a,b\\}\\cup\\mathrm{cone}\\{-a,-b\\}$. The original work \\cite{ader} asserts there are “at most $4$ points” realizing this non-separation; “for these $4$ points it is then clear that the alternance described in the main theorem must be in place,” yielding the configuration of two norming points $x_i$ with $|\\hat T x_i|=\\|\\hat T\\|$ and two contact points $y_i$ with $|\\hat T y_i|=1$ arranged in the stated cone relations. The uniqueness part is attributed to Maurey’s result (see \\cite{mory,gru-ko}) that “the ellipse of minimal Banach–Mazur distance is unique,” which “yields the uniqueness part of the main theorem.”", "expanded_theorem": "\\label{thm:main}\n  Let $\\dim X =2$. There is a unique $\\hat T\\in\\PD{X}{\\mathbb{R}^2}$ such that $\\hat T(B_X)\\supset B_{\\mathbb{R}^2}$ and\n  \\begin{equation}\n    \\label{eq:tm=d2}\n    \\|\\hat T\\| = d_2(X).\n  \\end{equation}\n  $\\hat T$ is characterised by the following property:\n\n  There are $x_i\\in S_X$, $i=1,2$, with $x_1\\neq \\pm x_2$, and $y_i\\in S_X$, $i=1,2$, with $y_1\\neq \\pm y_2$, such that:\n  \\begin{equation}\n    \\label{eq:x-distance}\n    |\\hat T x_i| = \\|\\hat T\\|,\\quad i=1,2,\n  \\end{equation}\n  \\begin{equation}\n    \\label{eq:y-contact}\n    |\\hat T y_i| = 1,\\quad i=1,2,\n  \\end{equation}\n  \\begin{equation}\n    \\label{eq:x-between-y}\n    y_1\\in\\mathrm{cone}\\, \\{x_1,x_2\\},\\quad y_2\\in\\mathrm{cone}\\,\\{-x_1,x_2\\}.\n  \\end{equation}", "theorem_type": ["Uniqueness", "Existence"], "mcq": {"question": "Let $(X,\\|\\cdot\\|)$ be a 2-dimensional normed space, with unit ball $B_X$ and unit sphere $S_X$. Let $\\mathbb{R}^2$ carry the standard Euclidean norm $|\\cdot|$, with unit ball $B_{\\mathbb{R}^2}$. Define\n\\[\n d_2(X):=d(X,\\mathbb{R}^2),\\qquad d(X,\\mathbb{R}^2):=\\inf\\{\\|T\\|\\,\\|T^{-1}\\|:T:X\\to\\mathbb{R}^2\\text{ is an isomorphism}\\},\n\\]\nand equivalently\n\\[\n d_2(X)=\\min\\{\\|T\\|:T:X\\to\\mathbb{R}^2,\\ T(B_X)\\supset B_{\\mathbb{R}^2}\\}.\n\\]\nLet $\\PD{X}{\\mathbb{R}^2}$ denote the class of linear maps $T:X\\to\\mathbb{R}^2$ that are represented by a positive definite symmetric $2\\times2$ matrix in coordinates, and for vectors $u,v$ write\n\\[\n\\mathrm{cone}\\{u,v\\}:=\\{\\alpha u+\\beta v:\\alpha,\\beta\\ge 0\\}.\n\\]\nWhich statement holds about existence and uniqueness of an operator in $\\PD{X}{\\mathbb{R}^2}$ attaining $d_2(X)$?", "correct_choice": {"label": "A", "text": "There exists a unique operator $\\hat T\\in\\PD{X}{\\mathbb{R}^2}$ such that $\\hat T(B_X)\\supset B_{\\mathbb{R}^2}$ and $\\|\\hat T\\|=d_2(X)$. Moreover, this unique operator is characterized by the existence of points $x_i\\in S_X$ and $y_i\\in S_X$ for $i=1,2$ such that $x_1\\neq \\pm x_2$, $y_1\\neq \\pm y_2$,\n\\[\n|\\hat T x_i|=\\|\\hat T\\|\\quad (i=1,2),\n\\]\n\\[\n|\\hat T y_i|=1\\quad (i=1,2),\n\\]\nand\n\\[\ny_1\\in \\mathrm{cone}\\{x_1,x_2\\},\\qquad y_2\\in \\mathrm{cone}\\{-x_1,x_2\\}.\n\\]"}, "choices": [{"label": "B", "text": "There exists a unique operator $\\hat T\\in\\PD{X}{\\mathbb{R}^2}$ such that $\\hat T(B_X)\\supset B_{\\mathbb{R}^2}$ and $\\|\\hat T\\|=d_2(X)$. Moreover, this unique operator is characterized by the existence of points $x_i\\in S_X$ and $y_i\\in S_X$ for $i=1,2$ such that $x_1\\neq \\pm x_2$, $y_1\\neq \\pm y_2$,\n\\[\n|\\hat T x_i|=\\|\\hat T\\|\\quad (i=1,2),\n\\]\n\\[\n|\\hat T y_i|=1\\quad (i=1,2),\n\\]\nand\n\\[\ny_1\\in \\mathrm{cone}\\{x_1,x_2\\},\\qquad y_2\\in \\mathrm{cone}\\{x_1,-x_2\\}.\n\\]"}, {"label": "C", "text": "There exists an operator $\\hat T\\in\\PD{X}{\\mathbb{R}^2}$ such that $\\hat T(B_X)\\supset B_{\\mathbb{R}^2}$ and $\\|\\hat T\\|=d_2(X)$."}, {"label": "D", "text": "There exists a unique operator $\\hat T\\in\\PD{X}{\\mathbb{R}^2}$ such that $\\hat T(B_X)\\supset B_{\\mathbb{R}^2}$ and $\\|\\hat T\\|=d_2(X)$. Moreover, this unique operator is characterized by the following stronger property: for every choice of points $x_i\\in S_X$ and $y_i\\in S_X$ for $i=1,2$ satisfying $x_1\\neq \\pm x_2$, $y_1\\neq \\pm y_2$,\n\\[\n|\\hat T x_i|=\\|\\hat T\\|\\quad (i=1,2),\n\\]\n\\[\n|\\hat T y_i|=1\\quad (i=1,2),\n\\]\none necessarily has\n\\[\ny_1\\in \\mathrm{cone}\\{x_1,x_2\\},\\qquad y_2\\in \\mathrm{cone}\\{-x_1,x_2\\}.\n\\]"}, {"label": "E", "text": "There exists an operator $\\hat T\\in\\PD{X}{\\mathbb{R}^2}$ such that $\\hat T(B_X)= B_{\\mathbb{R}^2}$ and $\\|\\hat T\\|=d_2(X)$. This operator is unique up to composition on the left with an orthogonal map in $\\mathbf O(2)$, and it is characterized by the existence of points $x_i\\in S_X$ and $y_i\\in S_X$ for $i=1,2$ such that $x_1\\neq \\pm x_2$, $y_1\\neq \\pm y_2$,\n\\[\n|\\hat T x_i|=\\|\\hat T\\|\\quad (i=1,2),\n\\]\n\\[\n|\\hat T y_i|=1\\quad (i=1,2),\n\\]\nand\n\\[\ny_1\\in \\mathrm{cone}\\{x_1,x_2\\},\\qquad y_2\\in \\mathrm{cone}\\{-x_1,x_2\\}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "alternating cone placement of the second contact point", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped uniqueness and the four-point cone characterization", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "existential four-point alternance replaced by universal characterization", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "containment condition $\\hat T(B_X)\\supset B_{\\mathbb R^2}$ replaced by equality and uniqueness weakened to orthogonal ambiguity", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem introduces notation and asks which theorem-style statement is correct, but it does not reveal the correct option explicitly or give a decisive hint toward A."}, "TAS": {"score": 0, "justification": "The item is essentially a recognition task for the exact theorem statement about existence/uniqueness and its characterization, so it functions largely as a restatement rather than a problem built from premises."}, "GPS": {"score": 1, "justification": "There is some logical comparison required among subtle variants (existential vs universal, uniqueness vs mere existence, containment vs equality, cone sign pattern), but the question mainly tests recall/discrimination of a known result rather than generating a conclusion from mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and well targeted: one is weaker-but-true, others introduce natural overstrengthening or geometric sign errors, and they reflect realistic theorem-misreading failure modes."}, "total_score": 5, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it is highly tautological and only moderately probes reasoning."}}
{"id": "2602.17863v1", "paper_link": "http://arxiv.org/abs/2602.17863v1", "theorems_cnt": 2, "theorem": {"env_name": "corollary", "content": "For $3$-manifolds obtained by Dehn surgery on a $(1,1)$ knot in $S^3$, the implications\n\\[\n(a)\\Leftrightarrow(c)\\quad\\text{and}\\quad (a)\\Leftarrow(b)\n\\]\nhold in the L-space conjecture.", "start_pos": 5518, "end_pos": 5736, "label": null}, "ref_dict": {"alg:brick-curve": "\\begin{algorithm}[!htbp]\n\\caption{Construction of the curves}\n\\label{alg:brick-curve}\n\\begin{algorithmic}[1]\n\\Require A positive braid diagram $D\\subset S^1\\times I$ and its associated brick diagram\n\\Ensure Simple closed curves satisfying Properties~\\ref{G1}--\\ref{G8}\n\n\\State Choose a non-vertex point $q$ in the $i$-th row lying on a component of $L$ which is not the unknot, with $i$ minimal.\n\\State $A\\gets q$, $B\\gets q$, $\\gamma\\gets\\{q\\}$, $\\Gamma\\gets\\varnothing$\n\\While{$d_{\\mathrm{up}}(B,A)=2$}\n    \\State \\Call{StepBelow}{$A,B,\\gamma$}\n\\EndWhile\n\\State \\Call{CrossBrickStep}{$A,B,\\gamma$} \\label{line:CrossBrickStep}\n\\While{$A$ is not on the last row}\n    \\If{$d_{\\mathrm{up}}(B,A)\\ge2$}\n        \\State \\Call{StepBelow}{$A,B,\\gamma$}\n    \\Else\n        \\State \\Call{CloseAndStore}{$\\gamma,\\Gamma,[B,A]$} \\label{line:close-1}\n        \\State \\Call{RestartBelow}{$A,B,\\gamma$} \\label{line:restart-below}\n    \\EndIf\n\\EndWhile\n\\State \\Call{CloseAndStore}{$\\gamma,\\Gamma,[B,A]$}\\label{line:close-2}\n\\If{$q$ is not on the first row}\n    \\State $A\\gets q$, $B\\gets q$\n    \\State \\Call{RestartAbove}{$A,B,\\gamma$}\\label{line:restart-above1}\n    \\While{$A$ is not on the first row}\n        \\If{$d_{\\mathrm{up}}(A,B)\\ge2$}\n            \\State \\Call{StepAbove}{$A,B,\\gamma$}\n        \\Else\n            \\State \\Call{CloseAndStore}{$\\gamma,\\Gamma,[A,B]$}\\label{line:close-3}\n            \\State \\Call{RestartAbove}{$A,B,\\gamma$}\\label{line:restart-above2}\n        \\EndIf\n    \\EndWhile\n    \\State \\Call{CloseAndStore}{$\\gamma,\\Gamma,[A,B]$}\\label{line:close-4}\n\\EndIf\n\n\\State \\Return $\\Gamma$\n\\end{algorithmic}\n\n\\end{algorithm}", "G8": "\\begin{enumerate}[label=(G\\arabic*)]\n    \\item\\label{G1} Each curve $\\gamma_i$ is transverse to every vertical line and hence intersects each vertical line $\\{x\\}\\times I$ once.\n    \\item\\label{G2} Each curve $\\gamma_i$ is composed of two arcs: one arc connects the two vertices of a bigon in the complement of $\\tau$, while the other arc lies on the train track $\\tau$.\n    \\item\\label{G3} For each vertical line $\\{x\\}\\times I$ with $x\\in S^1$, the intersection point\n    $\\gamma_i \\cap (\\{x\\}\\times I)$ lies above or coincides with\n    $\\gamma_{i+1} \\cap (\\{x\\}\\times I)$ for all $i=1,\\ldots,k-1$.\n    \\item\\label{G4} The bigons associated to the curves $\\gamma_1,\\gamma_2, \\ldots, \\gamma_k$ are pairwise distinct.\n    \\item\\label{G5} There exists a point $p\\in \\gamma_1$ such that $p$ lies on an under-type arc of $\\tau$, and there is no bigon directly above $p$.\n    \\item\\label{G6} There exists a point $p\\in \\gamma_k$ such that $p$ lies on an under-type arc of $\\tau$, and there is no bigon directly below $p$.\n    \\item\\label{G7} For each $1\\le i\\le k-1$, there exists a vertical segment connecting a point $p_1 \\in \\gamma_i$ to a point $p_2 \\in \\gamma_{i+1}$ whose interior is disjoint from $\\tau$, such that one of the points $p_1,p_2$ lies on an under-type arc of $\\tau$ and the other does not lie on $\\tau$.\n    \\item\\label{G8} For each under-arc of $D$, there exists a point $p\\in \\tau$ lying on an under-type arc of $\\tau$ arising from that under-arc, such that $p\\notin \\gamma_i$ for all $1\\le i\\le k$.\n\\end{enumerate}", "G1": "\\begin{enumerate}[label=(G\\arabic*)]\n    \\item\\label{G1} Each curve $\\gamma_i$ is transverse to every vertical line and hence intersects each vertical line $\\{x\\}\\times I$ once.\n    \\item\\label{G2} Each curve $\\gamma_i$ is composed of two arcs: one arc connects the two vertices of a bigon in the complement of $\\tau$, while the other arc lies on the train track $\\tau$.\n    \\item\\label{G3} For each vertical line $\\{x\\}\\times I$ with $x\\in S^1$, the intersection point\n    $\\gamma_i \\cap (\\{x\\}\\times I)$ lies above or coincides with\n    $\\gamma_{i+1} \\cap (\\{x\\}\\times I)$ for all $i=1,\\ldots,k-1$.\n    \\item\\label{G4} The bigons associated to the curves $\\gamma_1,\\gamma_2, \\ldots, \\gamma_k$ are pairwise distinct.\n    \\item\\label{G5} There exists a point $p\\in \\gamma_1$ such that $p$ lies on an under-type arc of $\\tau$, and there is no bigon directly above $p$.\n    \\item\\label{G6} There exists a point $p\\in \\gamma_k$ such that $p$ lies on an under-type arc of $\\tau$, and there is no bigon directly below $p$.\n    \\item\\label{G7} For each $1\\le i\\le k-1$, there exists a vertical segment connecting a point $p_1 \\in \\gamma_i$ to a point $p_2 \\in \\gamma_{i+1}$ whose interior is disjoint from $\\tau$, such that one of the points $p_1,p_2$ lies on an under-type arc of $\\tau$ and the other does not lie on $\\tau$.\n    \\item\\label{G8} For each under-arc of $D$, there exists a point $p\\in \\tau$ lying on an under-type arc of $\\tau$ arising from that under-arc, such that $p\\notin \\gamma_i$ for all $1\\le i\\le k$.\n\\end{enumerate}"}, "pre_theorem_intro_text_len": 3300, "pre_theorem_intro_text": "A central theme in $3$-manifold topology is the relationship between taut foliations, left-orderability, and Heegaard Floer homology. The L-space conjecture \\cite{boyer2013spaces,juhasz2015survey} predicts that the following are equivalent for an irreducible rational homology $3$-sphere $Y$:\n\\begin{enumerate}[label=(\\alph*)]\n    \\item $Y$ is a non-L-space;\n    \\item $\\pi_1(Y)$ is left-orderable;\n    \\item $Y$ admits a co-oriented taut foliation.\n\\end{enumerate}\nIn general, only the implication (c)$\\Rightarrow$(a) \\cite{ozsvath2004holomorphic,bowden2016approximating,kazez2017c0} is known.\n\nDehn surgery on knots in $S^3$ provides a natural source of both L-spaces and non-L-spaces. For a nontrivial knot $K\\subset S^3$, the set of slopes yielding L-spaces \\cite{kronheimer2007monopoles,rasmussen2017floer} is known to be one of the following three possibilities:\n\\begin{enumerate}\n\\item $[2g(K)-1,\\infty)$, in which case $K$ is called a positive L-space knot;\n\\item $(-\\infty,-2g(K)+1]$, in which case $K$ is called a negative L-space knot;\n\\item the empty set, in which case $K$ is called a non-L-space knot.\n\\end{enumerate}\nHere $g(K)$ denotes the Seifert genus of $K$. Thus, the L-space interval is completely determined once we know which case $K$ falls into. In contrast, the corresponding interval for taut foliations or left-orderability is much less understood.\n\nIn this paper, we study the existence of taut foliations on non-L-spaces obtained by Dehn surgeries on L-space knots in $S^3$.\n\nMost known positive L-space knots in $S^3$ arise as positive braid closures. For example, among the $632$ positive L-space knots in the SnapPy census with at most $9$ tetrahedra, $631$ are positive braid closures \\cite{baker2024census}, and the remaining example $o9\\_30634$ is the only known exception. Moreover, every positive $(1,1)$ L-space knot in $S^3$ is a positive braid closure \\cite{nie2021explicit}. \n\nWhen a link is represented by a braid closure, it is not straightforward to determine whether it is a knot or a multi-component link. For this reason, we work in the more general setting of links rather than restricting ourselves to knots. Our main theorem constructs taut foliations on the complement of a non-split positive braid closure in $S^3$ with at least one nontrivial component\\footnote{It is not sufficient to merely assume $L$ is nontrivial. The Hopf link provides a counterexample.}.\n\n\\begin{namedtheorem}[Main Theorem]\nSuppose $L$ is the closure of a non-split positive braid in $S^3$ with components $L_1,\\ldots,L_n$, and assume that at least one component is not the unknot. Then the Dehn surgery along a multislope $(s_1,\\ldots,s_n)\\in\\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for all $i=1,\\ldots,n$ yields a $3$-manifold admitting a co-oriented taut foliation.\n\\end{namedtheorem}\n\nAs an immediate corollary, the resulting manifolds are non-L-spaces.\n\nLyu proved that $(1,1)$ non-L-space knots are persistently foliar \\cite{lyu2025persistent}. It is also known that positive $(1,1)$ L-space knots in $S^3$ are positive braid closures, and that the L-spaces obtained by Dehn surgeries on $(1,1)$ knots have non-left-orderable fundamental groups \\cite{nie2021explicit,Li2024TautFoliations}. Combining these results with our main theorem yields the following corollary.", "context": "A central theme in $3$-manifold topology is the relationship between taut foliations, left-orderability, and Heegaard Floer homology. The L-space conjecture \\cite{boyer2013spaces,juhasz2015survey} predicts that the following are equivalent for an irreducible rational homology $3$-sphere $Y$:\n\\begin{enumerate}[label=(\\alph*)]\n \\item $Y$ is a non-L-space;\n \\item $\\pi_1(Y)$ is left-orderable;\n \\item $Y$ admits a co-oriented taut foliation.\n\\end{enumerate}\nIn general, only the implication (c)$\\Rightarrow$(a) \\cite{ozsvath2004holomorphic,bowden2016approximating,kazez2017c0} is known.\n\nDehn surgery on knots in $S^3$ provides a natural source of both L-spaces and non-L-spaces. For a nontrivial knot $K\\subset S^3$, the set of slopes yielding L-spaces \\cite{kronheimer2007monopoles,rasmussen2017floer} is known to be one of the following three possibilities:\n\\begin{enumerate}\n\\item $[2g(K)-1,\\infty)$, in which case $K$ is called a positive L-space knot;\n\\item $(-\\infty,-2g(K)+1]$, in which case $K$ is called a negative L-space knot;\n\\item the empty set, in which case $K$ is called a non-L-space knot.\n\\end{enumerate}\nHere $g(K)$ denotes the Seifert genus of $K$. Thus, the L-space interval is completely determined once we know which case $K$ falls into. In contrast, the corresponding interval for taut foliations or left-orderability is much less understood.\n\nMost known positive L-space knots in $S^3$ arise as positive braid closures. For example, among the $632$ positive L-space knots in the SnapPy census with at most $9$ tetrahedra, $631$ are positive braid closures \\cite{baker2024census}, and the remaining example $o9\\_30634$ is the only known exception. Moreover, every positive $(1,1)$ L-space knot in $S^3$ is a positive braid closure \\cite{nie2021explicit}.\n\n\\begin{namedtheorem}[Main Theorem]\nSuppose $L$ is the closure of a non-split positive braid in $S^3$ with components $L_1,\\ldots,L_n$, and assume that at least one component is not the unknot. Then the Dehn surgery along a multislope $(s_1,\\ldots,s_n)\\in\\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for all $i=1,\\ldots,n$ yields a $3$-manifold admitting a co-oriented taut foliation.\n\\end{namedtheorem}\n\nAs an immediate corollary, the resulting manifolds are non-L-spaces.\n\nLyu proved that $(1,1)$ non-L-space knots are persistently foliar \\cite{lyu2025persistent}. It is also known that positive $(1,1)$ L-space knots in $S^3$ are positive braid closures, and that the L-spaces obtained by Dehn surgeries on $(1,1)$ knots have non-left-orderable fundamental groups \\cite{nie2021explicit,Li2024TautFoliations}. Combining these results with our main theorem yields the following corollary.", "full_context": "A central theme in $3$-manifold topology is the relationship between taut foliations, left-orderability, and Heegaard Floer homology. The L-space conjecture \\cite{boyer2013spaces,juhasz2015survey} predicts that the following are equivalent for an irreducible rational homology $3$-sphere $Y$:\n\\begin{enumerate}[label=(\\alph*)]\n \\item $Y$ is a non-L-space;\n \\item $\\pi_1(Y)$ is left-orderable;\n \\item $Y$ admits a co-oriented taut foliation.\n\\end{enumerate}\nIn general, only the implication (c)$\\Rightarrow$(a) \\cite{ozsvath2004holomorphic,bowden2016approximating,kazez2017c0} is known.\n\nDehn surgery on knots in $S^3$ provides a natural source of both L-spaces and non-L-spaces. For a nontrivial knot $K\\subset S^3$, the set of slopes yielding L-spaces \\cite{kronheimer2007monopoles,rasmussen2017floer} is known to be one of the following three possibilities:\n\\begin{enumerate}\n\\item $[2g(K)-1,\\infty)$, in which case $K$ is called a positive L-space knot;\n\\item $(-\\infty,-2g(K)+1]$, in which case $K$ is called a negative L-space knot;\n\\item the empty set, in which case $K$ is called a non-L-space knot.\n\\end{enumerate}\nHere $g(K)$ denotes the Seifert genus of $K$. Thus, the L-space interval is completely determined once we know which case $K$ falls into. In contrast, the corresponding interval for taut foliations or left-orderability is much less understood.\n\nMost known positive L-space knots in $S^3$ arise as positive braid closures. For example, among the $632$ positive L-space knots in the SnapPy census with at most $9$ tetrahedra, $631$ are positive braid closures \\cite{baker2024census}, and the remaining example $o9\\_30634$ is the only known exception. Moreover, every positive $(1,1)$ L-space knot in $S^3$ is a positive braid closure \\cite{nie2021explicit}.\n\n\\begin{namedtheorem}[Main Theorem]\nSuppose $L$ is the closure of a non-split positive braid in $S^3$ with components $L_1,\\ldots,L_n$, and assume that at least one component is not the unknot. Then the Dehn surgery along a multislope $(s_1,\\ldots,s_n)\\in\\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for all $i=1,\\ldots,n$ yields a $3$-manifold admitting a co-oriented taut foliation.\n\\end{namedtheorem}\n\nAs an immediate corollary, the resulting manifolds are non-L-spaces.\n\nLyu proved that $(1,1)$ non-L-space knots are persistently foliar \\cite{lyu2025persistent}. It is also known that positive $(1,1)$ L-space knots in $S^3$ are positive braid closures, and that the L-spaces obtained by Dehn surgeries on $(1,1)$ knots have non-left-orderable fundamental groups \\cite{nie2021explicit,Li2024TautFoliations}. Combining these results with our main theorem yields the following corollary.\n\n\\begin{abstract}\nWe study taut foliations on the complements of non-split positive braid closures in $S^3$. If $L$ is such a link with components $L_1,\\ldots,L_n$ and at least one component is not the unknot, then the Dehn surgery along a multislope $(s_1,\\ldots,s_n)\\in\\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for $i=1,2,\\ldots, n$ yields a non-L-space that admits a co-oriented taut foliation.\n\\end{abstract}\n\\maketitle\n\\section{Introduction}\nA central theme in $3$-manifold topology is the relationship between taut foliations, left-orderability, and Heegaard Floer homology. The L-space conjecture \\cite{boyer2013spaces,juhasz2015survey} predicts that the following are equivalent for an irreducible rational homology $3$-sphere $Y$:\n\\begin{enumerate}[label=(\\alph*)]\n \\item $Y$ is a non-L-space;\n \\item $\\pi_1(Y)$ is left-orderable;\n \\item $Y$ admits a co-oriented taut foliation.\n\\end{enumerate}\nIn general, only the implication (c)$\\Rightarrow$(a) \\cite{ozsvath2004holomorphic,bowden2016approximating,kazez2017c0} is known.\n\nDehn surgery on knots in $S^3$ provides a natural source of both L-spaces and non-L-spaces. For a nontrivial knot $K\\subset S^3$, the set of slopes yielding L-spaces \\cite{kronheimer2007monopoles,rasmussen2017floer} is known to be one of the following three possibilities:\n\\begin{enumerate}\n\\item $[2g(K)-1,\\infty)$, in which case $K$ is called a positive L-space knot;\n\\item $(-\\infty,-2g(K)+1]$, in which case $K$ is called a negative L-space knot;\n\\item the empty set, in which case $K$ is called a non-L-space knot.\n\\end{enumerate}\nHere $g(K)$ denotes the Seifert genus of $K$. Thus, the L-space interval is completely determined once we know which case $K$ falls into. In contrast, the corresponding interval for taut foliations or left-orderability is much less understood.\n\nIn this paper, we study the existence of taut foliations on non-L-spaces obtained by Dehn surgeries on L-space knots in $S^3$.\n\n\\begin{namedtheorem}[Main Theorem]\nSuppose $L$ is the closure of a non-split positive braid in $S^3$ with components $L_1,\\ldots,L_n$, and assume that at least one component is not the unknot. Then the Dehn surgery along a multislope $(s_1,\\ldots,s_n)\\in\\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for all $i=1,\\ldots,n$ yields a $3$-manifold admitting a co-oriented taut foliation.\n\\end{namedtheorem}\n\nLyu proved that $(1,1)$ non-L-space knots are persistently foliar \\cite{lyu2025persistent}. It is also known that positive $(1,1)$ L-space knots in $S^3$ are positive braid closures, and that the L-spaces obtained by Dehn surgeries on $(1,1)$ knots have non-left-orderable fundamental groups \\cite{nie2021explicit,Li2024TautFoliations}. Combining these results with our main theorem yields the following corollary.\n\nThe remaining implication $(a)\\Rightarrow(b)$ is still open, even for the $(-2,3,7)$-pretzel knot \\cite{varvarezos2021representations}.\n\nNext, define a homotopy\n\\[\nf \\colon \\tau \\times [0,3] \\times [0,1] \\longrightarrow \\tau \\times [0,3]\n\\quad\\text{by}\\quad\nf(p,t,x) := \\bigl(p,\\max(t,3x)\\bigr).\n\\]\nThis homotopy gives a deformation retraction from\n\\[\n\\bigl\\{ (p,t)\\in \\tau\\times[0,3] : t \\ge h(p) \\bigr\\}\n\\]\nonto $\\tau \\times \\{3\\}$.\n\n\\begin{theorem}\nSuppose $L$ is the closure of a non-split positive braid in $S^3$ with components $L_1,\\ldots,L_n$, and assume that at least one component is not the unknot. Then the Dehn surgery along a multislope $(s_1,\\ldots,s_n)\\in\\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for all $i=1,\\ldots,n$ yields a $3$-manifold admitting a co-oriented taut foliation.\n\\end{theorem}\n\\begin{proof}\n Since $L$ is non-split and non-trivial, $S^3\\setminus \\operatorname{int}(\\nu(L))$ is an irreducible, orientable $3$-manifold whose boundary is a union of incompressible tori. By \\Cref{laminar}, $B'$ is a laminar branched surface properly embedded in $S^3 \\setminus \\operatorname{int}(\\nu(L))$. By \\Cref{bigon}, $\\partial \\nu(L)\\setminus B'$ is a union of bigons. By \\Cref{realizable-slopes}, any multislope $(s_1,\\ldots,s_n)\\in\\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for all $i=1,\\ldots,n$ is realized by the boundary train track $B'\\cap \\partial \\nu(L)$.", "post_theorem_intro_text_len": 2468, "post_theorem_intro_text": "The remaining implication $(a)\\Rightarrow(b)$ is still open, even for the $(-2,3,7)$-pretzel knot \\cite{varvarezos2021representations}. \n\nOur construction differs from previous approaches. Most existing methods begin with a Seifert surface and then apply sutured manifold decompositions and splittings to produce a laminar branched surface. This has been the standard approach for over two decades \\cite{roberts2000taut,roberts2001taut,li2003boundary,nakae20072,li2014taut,krishna2020taut,santoro2024spaces,krishna2025taut}.\n\nInstead, we first pinch the Seifert surface to obtain a simpler branched surface, and then perform splittings. From the theory of laminar branched surfaces \\cite{li2002laminar}, this intermediate step is not logically necessary. However, for explicit constructions and algorithmic implementation, simplifying the branched surface provides substantial advantages.\n\nIn Li's general framework, finding suitable splittings may require searching an exponentially large space of possibilities. In contrast, we design a greedy splitting algorithm of linear complexity (Algorithm~\\ref{alg:brick-curve}) that produces a laminar branched surface from the simplified model.\n\nIn Krishna's work \\cite{krishna2020taut,krishna2025taut} on taut foliations in braid closure complements, the branched surface is constructed by specifying coorientations for the product disks in Gabai's disk decomposition. This is equivalent to choosing curves $\\gamma_1,\\gamma_2,\\ldots,\\gamma_k$ in Section~\\ref{sec:curve} with the requirement that each $\\gamma_i$ lies in the closed region between the $i$-th row and the $(i+1)$-st row. Our construction removes this restriction, which allows greater flexibility and leads to a more general result.\n\nThe paper is organized as follows. Section~\\ref{sec:2} constructs the branched surface $B$. Section~\\ref{sec:3} splits $B$ into a laminar branched surface $B'$, assuming the existence of simple closed curves \\(\\gamma_1, \\gamma_2, \\ldots, \\gamma_k\\) satisfying Properties~\\ref{G1}--\\ref{G8}. Section~\\ref{sec:curve} describes the algorithm producing the curves $\\gamma_1,\\dots,\\gamma_k$ used in the splitting. Section~\\ref{sec:5} analyzes the boundary train track. Section~\\ref{sec:6} combines these ingredients to prove the main theorem.\n\\subsection*{Acknowledgements.}\nThe author is grateful to Qingfeng Lyu for many stimulating and ongoing discussions, and to Jacob Rasmussen for valuable advice and insightful discussions.", "sketch": "The post-theorem introduction does not give a proof sketch for the stated implications in the theorem. It only notes that the remaining implication $(a)\\Rightarrow(b)$ is open, and then describes the authors' overall construction/organization: they \"first pinch the Seifert surface to obtain a simpler branched surface, and then perform splittings,\" use a \"greedy splitting algorithm of linear complexity (Algorithm~\\ref{alg:brick-curve})\" to produce a laminar branched surface, and outline sections: Section~\\ref{sec:2} constructs $B$, Section~\\ref{sec:3} splits $B$ into laminar $B'$ assuming curves $\\gamma_1,\\dots,\\gamma_k$ satisfying Properties~\\ref{G1}--\\ref{G8}, Section~\\ref{sec:curve} produces these curves, Section~\\ref{sec:5} analyzes the boundary train track, and Section~\\ref{sec:6} \"combines these ingredients to prove the main theorem.\" No step-by-step argument establishing $(a)\\Leftrightarrow(c)$ and $(a)\\Leftarrow(b)$ is stated here.", "expanded_sketch": "The post-theorem introduction does not give a proof sketch for the stated implications in the theorem. It only notes that the remaining implication $(a)\\Rightarrow(b)$ is open, and then describes the authors' overall construction/organization: they ``first pinch the Seifert surface to obtain a simpler branched surface, and then perform splittings,'' use a ``greedy splitting algorithm of linear complexity'' given by\n\n\\begin{algorithm}[!htbp]\n\\caption{Construction of the curves}\n\\label{alg:brick-curve}\n\\begin{algorithmic}[1]\n\\Require A positive braid diagram $D\\subset S^1\\times I$ and its associated brick diagram\n\\Ensure Simple closed curves satisfying Properties~\\ref{G1}--\\ref{G8}\n\n\\State Choose a non-vertex point $q$ in the $i$-th row lying on a component of $L$ which is not the unknot, with $i$ minimal.\n\\State $A\\gets q$, $B\\gets q$, $\\gamma\\gets\\{q\\}$, $\\Gamma\\gets\\varnothing$\n\\While{$d_{\\mathrm{up}}(B,A)=2$}\n    \\State \\Call{StepBelow}{$A,B,\\gamma$}\n\\EndWhile\n\\State \\Call{CrossBrickStep}{$A,B,\\gamma$} \\label{line:CrossBrickStep}\n\\While{$A$ is not on the last row}\n    \\If{$d_{\\mathrm{up}}(B,A)\\ge2$}\n        \\State \\Call{StepBelow}{$A,B,\\gamma$}\n    \\Else\n        \\State \\Call{CloseAndStore}{$\\gamma,\\Gamma,[B,A]$} \\label{line:close-1}\n        \\State \\Call{RestartBelow}{$A,B,\\gamma$} \\label{line:restart-below}\n    \\EndIf\n\\EndWhile\n\\State \\Call{CloseAndStore}{$\\gamma,\\Gamma,[B,A]$}\\label{line:close-2}\n\\If{$q$ is not on the first row}\n    \\State $A\\gets q$, $B\\gets q$\n    \\State \\Call{RestartAbove}{$A,B,\\gamma$}\\label{line:restart-above1}\n    \\While{$A$ is not on the first row}\n        \\If{$d_{\\mathrm{up}}(A,B)\\ge2$}\n            \\State \\Call{StepAbove}{$A,B,\\gamma$}\n        \\Else\n            \\State \\Call{CloseAndStore}{$\\gamma,\\Gamma,[A,B]$}\\label{line:close-3}\n            \\State \\Call{RestartAbove}{$A,B,\\gamma$}\\label{line:restart-above2}\n        \\EndIf\n    \\EndWhile\n    \\State \\Call{CloseAndStore}{$\\gamma,\\Gamma,[A,B]$}\\label{line:close-4}\n\\EndIf\n\n\\State \\Return $\\Gamma$\n\\end{algorithmic}\n\n\\end{algorithm}\n\nto produce a laminar branched surface, and outlines the later sections: next they construct $B$; then they split $B$ into laminar $B'$ assuming curves $\\gamma_1,\\dots,\\gamma_k$ satisfying the following properties:\n\n\\begin{enumerate}[label=(G\\arabic*)]\n    \\item\\label{G1} Each curve $\\gamma_i$ is transverse to every vertical line and hence intersects each vertical line $\\{x\\}\\times I$ once.\n    \\item\\label{G2} Each curve $\\gamma_i$ is composed of two arcs: one arc connects the two vertices of a bigon in the complement of $\\tau$, while the other arc lies on the train track $\\tau$.\n    \\item\\label{G3} For each vertical line $\\{x\\}\\times I$ with $x\\in S^1$, the intersection point\n    $\\gamma_i \\cap (\\{x\\}\\times I)$ lies above or coincides with\n    $\\gamma_{i+1} \\cap (\\{x\\}\\times I)$ for all $i=1,\\ldots,k-1$.\n    \\item\\label{G4} The bigons associated to the curves $\\gamma_1,\\gamma_2, \\ldots, \\gamma_k$ are pairwise distinct.\n    \\item\\label{G5} There exists a point $p\\in \\gamma_1$ such that $p$ lies on an under-type arc of $\\tau$, and there is no bigon directly above $p$.\n    \\item\\label{G6} There exists a point $p\\in \\gamma_k$ such that $p$ lies on an under-type arc of $\\tau$, and there is no bigon directly below $p$.\n    \\item\\label{G7} For each $1\\le i\\le k-1$, there exists a vertical segment connecting a point $p_1 \\in \\gamma_i$ to a point $p_2 \\in \\gamma_{i+1}$ whose interior is disjoint from $\\tau$, such that one of the points $p_1,p_2$ lies on an under-type arc of $\\tau$ and the other does not lie on $\\tau$.\n    \\item\\label{G8} For each under-arc of $D$, there exists a point $p\\in \\tau$ lying on an under-type arc of $\\tau$ arising from that under-arc, such that $p\\notin \\gamma_i$ for all $1\\le i\\le k$.\n\\end{enumerate}\n\nAfter that they produce these curves; then they analyze the boundary train track; and finally they ``combine these ingredients to prove the main theorem.'' No step-by-step argument establishing $(a)\\Leftrightarrow(c)$ and $(a)\\Leftarrow(b)$ is stated here.", "expanded_theorem": "For $3$-manifolds obtained by Dehn surgery on a $(1,1)$ knot in $S^3$, the implications\n\\[\n(a)\\Leftrightarrow(c)\\quad\\text{and}\\quad (a)\\Leftarrow(b)\n\\]\nhold in the L-space conjecture.", "theorem_type": ["Implication", "Biconditional or Equivalence"], "mcq": {"question": "Let \\(Y\\) be a 3-manifold obtained by Dehn surgery on a \\((1,1)\\) knot in \\(S^3\\), where a \\((1,1)\\) knot means a knot admitting a genus-one Heegaard splitting of \\(S^3\\) in which the knot meets each solid torus in a single boundary-parallel arc. In the notation of the L-space conjecture, let\n\\((a)\\) mean that \\(Y\\) is a non-L-space,\n\\((b)\\) mean that \\(\\pi_1(Y)\\) is left-orderable, and\n\\((c)\\) mean that \\(Y\\) admits a co-oriented taut foliation.\nWhich statement about these conditions holds for all such surgery manifolds \\(Y\\)?", "correct_choice": {"label": "A", "text": "For every 3-manifold \\(Y\\) obtained by Dehn surgery on a \\((1,1)\\) knot in \\(S^3\\), conditions \\((a)\\) and \\((c)\\) are equivalent, and condition \\((b)\\) implies \\((a)\\); that is, \\(Y\\) is a non-L-space if and only if \\(Y\\) admits a co-oriented taut foliation, and if \\(\\pi_1(Y)\\) is left-orderable then \\(Y\\) is a non-L-space."}, "choices": [{"label": "B", "text": "For every 3-manifold \\(Y\\) obtained by Dehn surgery on a \\((1,1)\\) knot in \\(S^3\\), conditions \\((a)\\) and \\((b)\\) are equivalent, and condition \\((c)\\) implies \\((a)\\); that is, \\(Y\\) is a non-L-space if and only if \\(\\pi_1(Y)\\) is left-orderable, and if \\(Y\\) admits a co-oriented taut foliation then \\(Y\\) is a non-L-space."}, {"label": "C", "text": "For every 3-manifold \\(Y\\) obtained by Dehn surgery on a \\((1,1)\\) knot in \\(S^3\\), if \\(Y\\) admits a co-oriented taut foliation, then \\(Y\\) is a non-L-space; that is, condition \\((c)\\) implies condition \\((a)\\)."}, {"label": "D", "text": "For every 3-manifold \\(Y\\) obtained by Dehn surgery on a \\((1,1)\\) knot in \\(S^3\\), conditions \\((a)\\), \\((b)\\), and \\((c)\\) are all equivalent; that is, \\(Y\\) is a non-L-space if and only if \\(\\pi_1(Y)\\) is left-orderable if and only if \\(Y\\) admits a co-oriented taut foliation."}, {"label": "E", "text": "For every 3-manifold \\(Y\\) obtained by Dehn surgery on a \\((1,1)\\) knot in \\(S^3\\), condition \\((a)\\) implies condition \\((b)\\), and condition \\((a)\\) is equivalent to condition \\((c)\\); that is, if \\(Y\\) is a non-L-space then \\(\\pi_1(Y)\\) is left-orderable, and \\(Y\\) is a non-L-space if and only if \\(Y\\) admits a co-oriented taut foliation."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "which implication involving left-orderability is established", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "drops the converse (a)⇒(c) and the separate implication (b)⇒(a)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "open implication (a)⇒(b)", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "reverses the one-way implication between non-L-space and left-orderability", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem only defines the three properties and asks for the correct implication pattern; it does not reveal or strongly hint at which implications are true."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the task is to identify the published implication structure among (a), (b), and (c) for this class of manifolds, rather than derive a new conclusion from setup-specific data."}, "GPS": {"score": 1, "justification": "There is some pressure to distinguish subtle logical variants (equivalence vs one-way implication, stronger vs weaker claims), but the problem mainly tests recall/recognition of the theorem rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and well targeted: they include a stronger conjectural equivalence, a weaker true statement, and incorrect reversals/replacements of implication directions that match common confusion about the L-space conjecture."}, "total_score": 5, "overall_assessment": "A solid recall-based MCQ with strong distractors and no answer leakage, but it is close to a direct restatement of a theorem and only moderately tests reasoning."}}
{"id": "2602.17863v1", "paper_link": "http://arxiv.org/abs/2602.17863v1", "theorems_cnt": 2, "theorem": {"env_name": "corollary", "content": "For $3$-manifolds obtained by Dehn surgery on a $(1,1)$ knot in $S^3$, the implications\n\\[\n(a)\\Leftrightarrow(c)\\quad\\text{and}\\quad (a)\\Leftarrow(b)\n\\]\nhold in the L-space conjecture.", "start_pos": 5518, "end_pos": 5736, "label": null}, "ref_dict": {"alg:brick-curve": "\\begin{algorithm}[!htbp]\n\\caption{Construction of the curves}\n\\label{alg:brick-curve}\n\\begin{algorithmic}[1]\n\\Require A positive braid diagram $D\\subset S^1\\times I$ and its associated brick diagram\n\\Ensure Simple closed curves satisfying Properties~\\ref{G1}--\\ref{G8}\n\n\\State Choose a non-vertex point $q$ in the $i$-th row lying on a component of $L$ which is not the unknot, with $i$ minimal.\n\\State $A\\gets q$, $B\\gets q$, $\\gamma\\gets\\{q\\}$, $\\Gamma\\gets\\varnothing$\n\\While{$d_{\\mathrm{up}}(B,A)=2$}\n    \\State \\Call{StepBelow}{$A,B,\\gamma$}\n\\EndWhile\n\\State \\Call{CrossBrickStep}{$A,B,\\gamma$} \\label{line:CrossBrickStep}\n\\While{$A$ is not on the last row}\n    \\If{$d_{\\mathrm{up}}(B,A)\\ge2$}\n        \\State \\Call{StepBelow}{$A,B,\\gamma$}\n    \\Else\n        \\State \\Call{CloseAndStore}{$\\gamma,\\Gamma,[B,A]$} \\label{line:close-1}\n        \\State \\Call{RestartBelow}{$A,B,\\gamma$} \\label{line:restart-below}\n    \\EndIf\n\\EndWhile\n\\State \\Call{CloseAndStore}{$\\gamma,\\Gamma,[B,A]$}\\label{line:close-2}\n\\If{$q$ is not on the first row}\n    \\State $A\\gets q$, $B\\gets q$\n    \\State \\Call{RestartAbove}{$A,B,\\gamma$}\\label{line:restart-above1}\n    \\While{$A$ is not on the first row}\n        \\If{$d_{\\mathrm{up}}(A,B)\\ge2$}\n            \\State \\Call{StepAbove}{$A,B,\\gamma$}\n        \\Else\n            \\State \\Call{CloseAndStore}{$\\gamma,\\Gamma,[A,B]$}\\label{line:close-3}\n            \\State \\Call{RestartAbove}{$A,B,\\gamma$}\\label{line:restart-above2}\n        \\EndIf\n    \\EndWhile\n    \\State \\Call{CloseAndStore}{$\\gamma,\\Gamma,[A,B]$}\\label{line:close-4}\n\\EndIf\n\n\\State \\Return $\\Gamma$\n\\end{algorithmic}\n\n\\end{algorithm}", "G8": "\\begin{enumerate}[label=(G\\arabic*)]\n    \\item\\label{G1} Each curve $\\gamma_i$ is transverse to every vertical line and hence intersects each vertical line $\\{x\\}\\times I$ once.\n    \\item\\label{G2} Each curve $\\gamma_i$ is composed of two arcs: one arc connects the two vertices of a bigon in the complement of $\\tau$, while the other arc lies on the train track $\\tau$.\n    \\item\\label{G3} For each vertical line $\\{x\\}\\times I$ with $x\\in S^1$, the intersection point\n    $\\gamma_i \\cap (\\{x\\}\\times I)$ lies above or coincides with\n    $\\gamma_{i+1} \\cap (\\{x\\}\\times I)$ for all $i=1,\\ldots,k-1$.\n    \\item\\label{G4} The bigons associated to the curves $\\gamma_1,\\gamma_2, \\ldots, \\gamma_k$ are pairwise distinct.\n    \\item\\label{G5} There exists a point $p\\in \\gamma_1$ such that $p$ lies on an under-type arc of $\\tau$, and there is no bigon directly above $p$.\n    \\item\\label{G6} There exists a point $p\\in \\gamma_k$ such that $p$ lies on an under-type arc of $\\tau$, and there is no bigon directly below $p$.\n    \\item\\label{G7} For each $1\\le i\\le k-1$, there exists a vertical segment connecting a point $p_1 \\in \\gamma_i$ to a point $p_2 \\in \\gamma_{i+1}$ whose interior is disjoint from $\\tau$, such that one of the points $p_1,p_2$ lies on an under-type arc of $\\tau$ and the other does not lie on $\\tau$.\n    \\item\\label{G8} For each under-arc of $D$, there exists a point $p\\in \\tau$ lying on an under-type arc of $\\tau$ arising from that under-arc, such that $p\\notin \\gamma_i$ for all $1\\le i\\le k$.\n\\end{enumerate}", "G1": "\\begin{enumerate}[label=(G\\arabic*)]\n    \\item\\label{G1} Each curve $\\gamma_i$ is transverse to every vertical line and hence intersects each vertical line $\\{x\\}\\times I$ once.\n    \\item\\label{G2} Each curve $\\gamma_i$ is composed of two arcs: one arc connects the two vertices of a bigon in the complement of $\\tau$, while the other arc lies on the train track $\\tau$.\n    \\item\\label{G3} For each vertical line $\\{x\\}\\times I$ with $x\\in S^1$, the intersection point\n    $\\gamma_i \\cap (\\{x\\}\\times I)$ lies above or coincides with\n    $\\gamma_{i+1} \\cap (\\{x\\}\\times I)$ for all $i=1,\\ldots,k-1$.\n    \\item\\label{G4} The bigons associated to the curves $\\gamma_1,\\gamma_2, \\ldots, \\gamma_k$ are pairwise distinct.\n    \\item\\label{G5} There exists a point $p\\in \\gamma_1$ such that $p$ lies on an under-type arc of $\\tau$, and there is no bigon directly above $p$.\n    \\item\\label{G6} There exists a point $p\\in \\gamma_k$ such that $p$ lies on an under-type arc of $\\tau$, and there is no bigon directly below $p$.\n    \\item\\label{G7} For each $1\\le i\\le k-1$, there exists a vertical segment connecting a point $p_1 \\in \\gamma_i$ to a point $p_2 \\in \\gamma_{i+1}$ whose interior is disjoint from $\\tau$, such that one of the points $p_1,p_2$ lies on an under-type arc of $\\tau$ and the other does not lie on $\\tau$.\n    \\item\\label{G8} For each under-arc of $D$, there exists a point $p\\in \\tau$ lying on an under-type arc of $\\tau$ arising from that under-arc, such that $p\\notin \\gamma_i$ for all $1\\le i\\le k$.\n\\end{enumerate}"}, "pre_theorem_intro_text_len": 3300, "pre_theorem_intro_text": "A central theme in $3$-manifold topology is the relationship between taut foliations, left-orderability, and Heegaard Floer homology. The L-space conjecture \\cite{boyer2013spaces,juhasz2015survey} predicts that the following are equivalent for an irreducible rational homology $3$-sphere $Y$:\n\\begin{enumerate}[label=(\\alph*)]\n    \\item $Y$ is a non-L-space;\n    \\item $\\pi_1(Y)$ is left-orderable;\n    \\item $Y$ admits a co-oriented taut foliation.\n\\end{enumerate}\nIn general, only the implication (c)$\\Rightarrow$(a) \\cite{ozsvath2004holomorphic,bowden2016approximating,kazez2017c0} is known.\n\nDehn surgery on knots in $S^3$ provides a natural source of both L-spaces and non-L-spaces. For a nontrivial knot $K\\subset S^3$, the set of slopes yielding L-spaces \\cite{kronheimer2007monopoles,rasmussen2017floer} is known to be one of the following three possibilities:\n\\begin{enumerate}\n\\item $[2g(K)-1,\\infty)$, in which case $K$ is called a positive L-space knot;\n\\item $(-\\infty,-2g(K)+1]$, in which case $K$ is called a negative L-space knot;\n\\item the empty set, in which case $K$ is called a non-L-space knot.\n\\end{enumerate}\nHere $g(K)$ denotes the Seifert genus of $K$. Thus, the L-space interval is completely determined once we know which case $K$ falls into. In contrast, the corresponding interval for taut foliations or left-orderability is much less understood.\n\nIn this paper, we study the existence of taut foliations on non-L-spaces obtained by Dehn surgeries on L-space knots in $S^3$.\n\nMost known positive L-space knots in $S^3$ arise as positive braid closures. For example, among the $632$ positive L-space knots in the SnapPy census with at most $9$ tetrahedra, $631$ are positive braid closures \\cite{baker2024census}, and the remaining example $o9\\_30634$ is the only known exception. Moreover, every positive $(1,1)$ L-space knot in $S^3$ is a positive braid closure \\cite{nie2021explicit}. \n\nWhen a link is represented by a braid closure, it is not straightforward to determine whether it is a knot or a multi-component link. For this reason, we work in the more general setting of links rather than restricting ourselves to knots. Our main theorem constructs taut foliations on the complement of a non-split positive braid closure in $S^3$ with at least one nontrivial component\\footnote{It is not sufficient to merely assume $L$ is nontrivial. The Hopf link provides a counterexample.}.\n\n\\begin{namedtheorem}[Main Theorem]\nSuppose $L$ is the closure of a non-split positive braid in $S^3$ with components $L_1,\\ldots,L_n$, and assume that at least one component is not the unknot. Then the Dehn surgery along a multislope $(s_1,\\ldots,s_n)\\in\\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for all $i=1,\\ldots,n$ yields a $3$-manifold admitting a co-oriented taut foliation.\n\\end{namedtheorem}\n\nAs an immediate corollary, the resulting manifolds are non-L-spaces.\n\nLyu proved that $(1,1)$ non-L-space knots are persistently foliar \\cite{lyu2025persistent}. It is also known that positive $(1,1)$ L-space knots in $S^3$ are positive braid closures, and that the L-spaces obtained by Dehn surgeries on $(1,1)$ knots have non-left-orderable fundamental groups \\cite{nie2021explicit,Li2024TautFoliations}. Combining these results with our main theorem yields the following corollary.", "context": "A central theme in $3$-manifold topology is the relationship between taut foliations, left-orderability, and Heegaard Floer homology. The L-space conjecture \\cite{boyer2013spaces,juhasz2015survey} predicts that the following are equivalent for an irreducible rational homology $3$-sphere $Y$:\n\\begin{enumerate}[label=(\\alph*)]\n \\item $Y$ is a non-L-space;\n \\item $\\pi_1(Y)$ is left-orderable;\n \\item $Y$ admits a co-oriented taut foliation.\n\\end{enumerate}\nIn general, only the implication (c)$\\Rightarrow$(a) \\cite{ozsvath2004holomorphic,bowden2016approximating,kazez2017c0} is known.\n\nDehn surgery on knots in $S^3$ provides a natural source of both L-spaces and non-L-spaces. For a nontrivial knot $K\\subset S^3$, the set of slopes yielding L-spaces \\cite{kronheimer2007monopoles,rasmussen2017floer} is known to be one of the following three possibilities:\n\\begin{enumerate}\n\\item $[2g(K)-1,\\infty)$, in which case $K$ is called a positive L-space knot;\n\\item $(-\\infty,-2g(K)+1]$, in which case $K$ is called a negative L-space knot;\n\\item the empty set, in which case $K$ is called a non-L-space knot.\n\\end{enumerate}\nHere $g(K)$ denotes the Seifert genus of $K$. Thus, the L-space interval is completely determined once we know which case $K$ falls into. In contrast, the corresponding interval for taut foliations or left-orderability is much less understood.\n\nMost known positive L-space knots in $S^3$ arise as positive braid closures. For example, among the $632$ positive L-space knots in the SnapPy census with at most $9$ tetrahedra, $631$ are positive braid closures \\cite{baker2024census}, and the remaining example $o9\\_30634$ is the only known exception. Moreover, every positive $(1,1)$ L-space knot in $S^3$ is a positive braid closure \\cite{nie2021explicit}.\n\n\\begin{namedtheorem}[Main Theorem]\nSuppose $L$ is the closure of a non-split positive braid in $S^3$ with components $L_1,\\ldots,L_n$, and assume that at least one component is not the unknot. Then the Dehn surgery along a multislope $(s_1,\\ldots,s_n)\\in\\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for all $i=1,\\ldots,n$ yields a $3$-manifold admitting a co-oriented taut foliation.\n\\end{namedtheorem}\n\nAs an immediate corollary, the resulting manifolds are non-L-spaces.\n\nLyu proved that $(1,1)$ non-L-space knots are persistently foliar \\cite{lyu2025persistent}. It is also known that positive $(1,1)$ L-space knots in $S^3$ are positive braid closures, and that the L-spaces obtained by Dehn surgeries on $(1,1)$ knots have non-left-orderable fundamental groups \\cite{nie2021explicit,Li2024TautFoliations}. Combining these results with our main theorem yields the following corollary.", "full_context": "A central theme in $3$-manifold topology is the relationship between taut foliations, left-orderability, and Heegaard Floer homology. The L-space conjecture \\cite{boyer2013spaces,juhasz2015survey} predicts that the following are equivalent for an irreducible rational homology $3$-sphere $Y$:\n\\begin{enumerate}[label=(\\alph*)]\n \\item $Y$ is a non-L-space;\n \\item $\\pi_1(Y)$ is left-orderable;\n \\item $Y$ admits a co-oriented taut foliation.\n\\end{enumerate}\nIn general, only the implication (c)$\\Rightarrow$(a) \\cite{ozsvath2004holomorphic,bowden2016approximating,kazez2017c0} is known.\n\nDehn surgery on knots in $S^3$ provides a natural source of both L-spaces and non-L-spaces. For a nontrivial knot $K\\subset S^3$, the set of slopes yielding L-spaces \\cite{kronheimer2007monopoles,rasmussen2017floer} is known to be one of the following three possibilities:\n\\begin{enumerate}\n\\item $[2g(K)-1,\\infty)$, in which case $K$ is called a positive L-space knot;\n\\item $(-\\infty,-2g(K)+1]$, in which case $K$ is called a negative L-space knot;\n\\item the empty set, in which case $K$ is called a non-L-space knot.\n\\end{enumerate}\nHere $g(K)$ denotes the Seifert genus of $K$. Thus, the L-space interval is completely determined once we know which case $K$ falls into. In contrast, the corresponding interval for taut foliations or left-orderability is much less understood.\n\nMost known positive L-space knots in $S^3$ arise as positive braid closures. For example, among the $632$ positive L-space knots in the SnapPy census with at most $9$ tetrahedra, $631$ are positive braid closures \\cite{baker2024census}, and the remaining example $o9\\_30634$ is the only known exception. Moreover, every positive $(1,1)$ L-space knot in $S^3$ is a positive braid closure \\cite{nie2021explicit}.\n\n\\begin{namedtheorem}[Main Theorem]\nSuppose $L$ is the closure of a non-split positive braid in $S^3$ with components $L_1,\\ldots,L_n$, and assume that at least one component is not the unknot. Then the Dehn surgery along a multislope $(s_1,\\ldots,s_n)\\in\\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for all $i=1,\\ldots,n$ yields a $3$-manifold admitting a co-oriented taut foliation.\n\\end{namedtheorem}\n\nAs an immediate corollary, the resulting manifolds are non-L-spaces.\n\nLyu proved that $(1,1)$ non-L-space knots are persistently foliar \\cite{lyu2025persistent}. It is also known that positive $(1,1)$ L-space knots in $S^3$ are positive braid closures, and that the L-spaces obtained by Dehn surgeries on $(1,1)$ knots have non-left-orderable fundamental groups \\cite{nie2021explicit,Li2024TautFoliations}. Combining these results with our main theorem yields the following corollary.\n\n\\begin{abstract}\nWe study taut foliations on the complements of non-split positive braid closures in $S^3$. If $L$ is such a link with components $L_1,\\ldots,L_n$ and at least one component is not the unknot, then the Dehn surgery along a multislope $(s_1,\\ldots,s_n)\\in\\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for $i=1,2,\\ldots, n$ yields a non-L-space that admits a co-oriented taut foliation.\n\\end{abstract}\n\\maketitle\n\\section{Introduction}\nA central theme in $3$-manifold topology is the relationship between taut foliations, left-orderability, and Heegaard Floer homology. The L-space conjecture \\cite{boyer2013spaces,juhasz2015survey} predicts that the following are equivalent for an irreducible rational homology $3$-sphere $Y$:\n\\begin{enumerate}[label=(\\alph*)]\n \\item $Y$ is a non-L-space;\n \\item $\\pi_1(Y)$ is left-orderable;\n \\item $Y$ admits a co-oriented taut foliation.\n\\end{enumerate}\nIn general, only the implication (c)$\\Rightarrow$(a) \\cite{ozsvath2004holomorphic,bowden2016approximating,kazez2017c0} is known.\n\nDehn surgery on knots in $S^3$ provides a natural source of both L-spaces and non-L-spaces. For a nontrivial knot $K\\subset S^3$, the set of slopes yielding L-spaces \\cite{kronheimer2007monopoles,rasmussen2017floer} is known to be one of the following three possibilities:\n\\begin{enumerate}\n\\item $[2g(K)-1,\\infty)$, in which case $K$ is called a positive L-space knot;\n\\item $(-\\infty,-2g(K)+1]$, in which case $K$ is called a negative L-space knot;\n\\item the empty set, in which case $K$ is called a non-L-space knot.\n\\end{enumerate}\nHere $g(K)$ denotes the Seifert genus of $K$. Thus, the L-space interval is completely determined once we know which case $K$ falls into. In contrast, the corresponding interval for taut foliations or left-orderability is much less understood.\n\nIn this paper, we study the existence of taut foliations on non-L-spaces obtained by Dehn surgeries on L-space knots in $S^3$.\n\n\\begin{namedtheorem}[Main Theorem]\nSuppose $L$ is the closure of a non-split positive braid in $S^3$ with components $L_1,\\ldots,L_n$, and assume that at least one component is not the unknot. Then the Dehn surgery along a multislope $(s_1,\\ldots,s_n)\\in\\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for all $i=1,\\ldots,n$ yields a $3$-manifold admitting a co-oriented taut foliation.\n\\end{namedtheorem}\n\nLyu proved that $(1,1)$ non-L-space knots are persistently foliar \\cite{lyu2025persistent}. It is also known that positive $(1,1)$ L-space knots in $S^3$ are positive braid closures, and that the L-spaces obtained by Dehn surgeries on $(1,1)$ knots have non-left-orderable fundamental groups \\cite{nie2021explicit,Li2024TautFoliations}. Combining these results with our main theorem yields the following corollary.\n\nThe remaining implication $(a)\\Rightarrow(b)$ is still open, even for the $(-2,3,7)$-pretzel knot \\cite{varvarezos2021representations}.\n\nNext, define a homotopy\n\\[\nf \\colon \\tau \\times [0,3] \\times [0,1] \\longrightarrow \\tau \\times [0,3]\n\\quad\\text{by}\\quad\nf(p,t,x) := \\bigl(p,\\max(t,3x)\\bigr).\n\\]\nThis homotopy gives a deformation retraction from\n\\[\n\\bigl\\{ (p,t)\\in \\tau\\times[0,3] : t \\ge h(p) \\bigr\\}\n\\]\nonto $\\tau \\times \\{3\\}$.\n\n\\begin{theorem}\nSuppose $L$ is the closure of a non-split positive braid in $S^3$ with components $L_1,\\ldots,L_n$, and assume that at least one component is not the unknot. Then the Dehn surgery along a multislope $(s_1,\\ldots,s_n)\\in\\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for all $i=1,\\ldots,n$ yields a $3$-manifold admitting a co-oriented taut foliation.\n\\end{theorem}\n\\begin{proof}\n Since $L$ is non-split and non-trivial, $S^3\\setminus \\operatorname{int}(\\nu(L))$ is an irreducible, orientable $3$-manifold whose boundary is a union of incompressible tori. By \\Cref{laminar}, $B'$ is a laminar branched surface properly embedded in $S^3 \\setminus \\operatorname{int}(\\nu(L))$. By \\Cref{bigon}, $\\partial \\nu(L)\\setminus B'$ is a union of bigons. By \\Cref{realizable-slopes}, any multislope $(s_1,\\ldots,s_n)\\in\\mathbb{Q}^n$ satisfying $s_i<2g(L_i)-1$ for all $i=1,\\ldots,n$ is realized by the boundary train track $B'\\cap \\partial \\nu(L)$.", "post_theorem_intro_text_len": 2468, "post_theorem_intro_text": "The remaining implication $(a)\\Rightarrow(b)$ is still open, even for the $(-2,3,7)$-pretzel knot \\cite{varvarezos2021representations}. \n\nOur construction differs from previous approaches. Most existing methods begin with a Seifert surface and then apply sutured manifold decompositions and splittings to produce a laminar branched surface. This has been the standard approach for over two decades \\cite{roberts2000taut,roberts2001taut,li2003boundary,nakae20072,li2014taut,krishna2020taut,santoro2024spaces,krishna2025taut}.\n\nInstead, we first pinch the Seifert surface to obtain a simpler branched surface, and then perform splittings. From the theory of laminar branched surfaces \\cite{li2002laminar}, this intermediate step is not logically necessary. However, for explicit constructions and algorithmic implementation, simplifying the branched surface provides substantial advantages.\n\nIn Li's general framework, finding suitable splittings may require searching an exponentially large space of possibilities. In contrast, we design a greedy splitting algorithm of linear complexity (Algorithm~\\ref{alg:brick-curve}) that produces a laminar branched surface from the simplified model.\n\nIn Krishna's work \\cite{krishna2020taut,krishna2025taut} on taut foliations in braid closure complements, the branched surface is constructed by specifying coorientations for the product disks in Gabai's disk decomposition. This is equivalent to choosing curves $\\gamma_1,\\gamma_2,\\ldots,\\gamma_k$ in Section~\\ref{sec:curve} with the requirement that each $\\gamma_i$ lies in the closed region between the $i$-th row and the $(i+1)$-st row. Our construction removes this restriction, which allows greater flexibility and leads to a more general result.\n\nThe paper is organized as follows. Section~\\ref{sec:2} constructs the branched surface $B$. Section~\\ref{sec:3} splits $B$ into a laminar branched surface $B'$, assuming the existence of simple closed curves \\(\\gamma_1, \\gamma_2, \\ldots, \\gamma_k\\) satisfying Properties~\\ref{G1}--\\ref{G8}. Section~\\ref{sec:curve} describes the algorithm producing the curves $\\gamma_1,\\dots,\\gamma_k$ used in the splitting. Section~\\ref{sec:5} analyzes the boundary train track. Section~\\ref{sec:6} combines these ingredients to prove the main theorem.\n\\subsection*{Acknowledgements.}\nThe author is grateful to Qingfeng Lyu for many stimulating and ongoing discussions, and to Jacob Rasmussen for valuable advice and insightful discussions.", "sketch": "The post-theorem introduction does not give a proof sketch for the stated implications in the theorem. It only notes that the remaining implication $(a)\\Rightarrow(b)$ is open, and then describes the authors' overall construction/organization: they \"first pinch the Seifert surface to obtain a simpler branched surface, and then perform splittings,\" use a \"greedy splitting algorithm of linear complexity (Algorithm~\\ref{alg:brick-curve})\" to produce a laminar branched surface, and outline sections: Section~\\ref{sec:2} constructs $B$, Section~\\ref{sec:3} splits $B$ into laminar $B'$ assuming curves $\\gamma_1,\\dots,\\gamma_k$ satisfying Properties~\\ref{G1}--\\ref{G8}, Section~\\ref{sec:curve} produces these curves, Section~\\ref{sec:5} analyzes the boundary train track, and Section~\\ref{sec:6} \"combines these ingredients to prove the main theorem.\" No step-by-step argument establishing $(a)\\Leftrightarrow(c)$ and $(a)\\Leftarrow(b)$ is stated here.", "expanded_sketch": "The post-theorem introduction does not give a proof sketch for the stated implications in the theorem. It only notes that the remaining implication $(a)\\Rightarrow(b)$ is open, and then describes the authors' overall construction/organization: they ``first pinch the Seifert surface to obtain a simpler branched surface, and then perform splittings,'' use a ``greedy splitting algorithm of linear complexity'' given by\n\n\\begin{algorithm}[!htbp]\n\\caption{Construction of the curves}\n\\label{alg:brick-curve}\n\\begin{algorithmic}[1]\n\\Require A positive braid diagram $D\\subset S^1\\times I$ and its associated brick diagram\n\\Ensure Simple closed curves satisfying Properties~\\ref{G1}--\\ref{G8}\n\n\\State Choose a non-vertex point $q$ in the $i$-th row lying on a component of $L$ which is not the unknot, with $i$ minimal.\n\\State $A\\gets q$, $B\\gets q$, $\\gamma\\gets\\{q\\}$, $\\Gamma\\gets\\varnothing$\n\\While{$d_{\\mathrm{up}}(B,A)=2$}\n    \\State \\Call{StepBelow}{$A,B,\\gamma$}\n\\EndWhile\n\\State \\Call{CrossBrickStep}{$A,B,\\gamma$} \\label{line:CrossBrickStep}\n\\While{$A$ is not on the last row}\n    \\If{$d_{\\mathrm{up}}(B,A)\\ge2$}\n        \\State \\Call{StepBelow}{$A,B,\\gamma$}\n    \\Else\n        \\State \\Call{CloseAndStore}{$\\gamma,\\Gamma,[B,A]$} \\label{line:close-1}\n        \\State \\Call{RestartBelow}{$A,B,\\gamma$} \\label{line:restart-below}\n    \\EndIf\n\\EndWhile\n\\State \\Call{CloseAndStore}{$\\gamma,\\Gamma,[B,A]$}\\label{line:close-2}\n\\If{$q$ is not on the first row}\n    \\State $A\\gets q$, $B\\gets q$\n    \\State \\Call{RestartAbove}{$A,B,\\gamma$}\\label{line:restart-above1}\n    \\While{$A$ is not on the first row}\n        \\If{$d_{\\mathrm{up}}(A,B)\\ge2$}\n            \\State \\Call{StepAbove}{$A,B,\\gamma$}\n        \\Else\n            \\State \\Call{CloseAndStore}{$\\gamma,\\Gamma,[A,B]$}\\label{line:close-3}\n            \\State \\Call{RestartAbove}{$A,B,\\gamma$}\\label{line:restart-above2}\n        \\EndIf\n    \\EndWhile\n    \\State \\Call{CloseAndStore}{$\\gamma,\\Gamma,[A,B]$}\\label{line:close-4}\n\\EndIf\n\n\\State \\Return $\\Gamma$\n\\end{algorithmic}\n\n\\end{algorithm}\n\nto produce a laminar branched surface, and outlines the later sections: next they construct $B$; then they split $B$ into laminar $B'$ assuming curves $\\gamma_1,\\dots,\\gamma_k$ satisfying the following properties:\n\n\\begin{enumerate}[label=(G\\arabic*)]\n    \\item\\label{G1} Each curve $\\gamma_i$ is transverse to every vertical line and hence intersects each vertical line $\\{x\\}\\times I$ once.\n    \\item\\label{G2} Each curve $\\gamma_i$ is composed of two arcs: one arc connects the two vertices of a bigon in the complement of $\\tau$, while the other arc lies on the train track $\\tau$.\n    \\item\\label{G3} For each vertical line $\\{x\\}\\times I$ with $x\\in S^1$, the intersection point\n    $\\gamma_i \\cap (\\{x\\}\\times I)$ lies above or coincides with\n    $\\gamma_{i+1} \\cap (\\{x\\}\\times I)$ for all $i=1,\\ldots,k-1$.\n    \\item\\label{G4} The bigons associated to the curves $\\gamma_1,\\gamma_2, \\ldots, \\gamma_k$ are pairwise distinct.\n    \\item\\label{G5} There exists a point $p\\in \\gamma_1$ such that $p$ lies on an under-type arc of $\\tau$, and there is no bigon directly above $p$.\n    \\item\\label{G6} There exists a point $p\\in \\gamma_k$ such that $p$ lies on an under-type arc of $\\tau$, and there is no bigon directly below $p$.\n    \\item\\label{G7} For each $1\\le i\\le k-1$, there exists a vertical segment connecting a point $p_1 \\in \\gamma_i$ to a point $p_2 \\in \\gamma_{i+1}$ whose interior is disjoint from $\\tau$, such that one of the points $p_1,p_2$ lies on an under-type arc of $\\tau$ and the other does not lie on $\\tau$.\n    \\item\\label{G8} For each under-arc of $D$, there exists a point $p\\in \\tau$ lying on an under-type arc of $\\tau$ arising from that under-arc, such that $p\\notin \\gamma_i$ for all $1\\le i\\le k$.\n\\end{enumerate}\n\nAfter that they produce these curves; then they analyze the boundary train track; and finally they ``combine these ingredients to prove the main theorem.'' No step-by-step argument establishing $(a)\\Leftrightarrow(c)$ and $(a)\\Leftarrow(b)$ is stated here.", "expanded_theorem": "For $3$-manifolds obtained by Dehn surgery on a $(1,1)$ knot in $S^3$, the implications\n\\[\n(a)\\Leftrightarrow(c)\\quad\\text{and}\\quad (a)\\Leftarrow(b)\n\\]\nhold in the L-space conjecture.", "theorem_type": ["Implication", "Biconditional or Equivalence"], "mcq": {"question": "Let \\(Y\\) be a 3-manifold obtained by Dehn surgery on a \\((1,1)\\) knot in \\(S^3\\), where a \\((1,1)\\) knot means a knot admitting a genus-one Heegaard splitting of \\(S^3\\) in which the knot meets each solid torus in a single boundary-parallel arc. In the notation of the L-space conjecture, let\n\\((a)\\) mean that \\(Y\\) is a non-L-space,\n\\((b)\\) mean that \\(\\pi_1(Y)\\) is left-orderable, and\n\\((c)\\) mean that \\(Y\\) admits a co-oriented taut foliation.\nWhich statement about these conditions holds for all such surgery manifolds \\(Y\\)?", "correct_choice": {"label": "A", "text": "For every 3-manifold \\(Y\\) obtained by Dehn surgery on a \\((1,1)\\) knot in \\(S^3\\), conditions \\((a)\\) and \\((c)\\) are equivalent, and condition \\((b)\\) implies \\((a)\\); that is, \\(Y\\) is a non-L-space if and only if \\(Y\\) admits a co-oriented taut foliation, and if \\(\\pi_1(Y)\\) is left-orderable then \\(Y\\) is a non-L-space."}, "choices": [{"label": "B", "text": "For every 3-manifold \\(Y\\) obtained by Dehn surgery on a \\((1,1)\\) knot in \\(S^3\\), conditions \\((a)\\) and \\((b)\\) are equivalent, and condition \\((c)\\) implies \\((a)\\); that is, \\(Y\\) is a non-L-space if and only if \\(\\pi_1(Y)\\) is left-orderable, and if \\(Y\\) admits a co-oriented taut foliation then \\(Y\\) is a non-L-space."}, {"label": "C", "text": "For every 3-manifold \\(Y\\) obtained by Dehn surgery on a \\((1,1)\\) knot in \\(S^3\\), if \\(Y\\) admits a co-oriented taut foliation, then \\(Y\\) is a non-L-space; that is, condition \\((c)\\) implies condition \\((a)\\)."}, {"label": "D", "text": "For every 3-manifold \\(Y\\) obtained by Dehn surgery on a \\((1,1)\\) knot in \\(S^3\\), conditions \\((a)\\), \\((b)\\), and \\((c)\\) are all equivalent; that is, \\(Y\\) is a non-L-space if and only if \\(\\pi_1(Y)\\) is left-orderable if and only if \\(Y\\) admits a co-oriented taut foliation."}, {"label": "E", "text": "For every 3-manifold \\(Y\\) obtained by Dehn surgery on a \\((1,1)\\) knot in \\(S^3\\), condition \\((a)\\) implies condition \\((b)\\), and condition \\((a)\\) is equivalent to condition \\((c)\\); that is, if \\(Y\\) is a non-L-space then \\(\\pi_1(Y)\\) is left-orderable, and \\(Y\\) is a non-L-space if and only if \\(Y\\) admits a co-oriented taut foliation."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "which implication involving left-orderability is established", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "drops the converse (a)⇒(c) and the separate implication (b)⇒(a)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "open implication (a)⇒(b)", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "reverses the one-way implication between non-L-space and left-orderability", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the three conditions neutrally and asks which relationship holds; it does not explicitly state or strongly hint at the correct implication pattern. There are no obvious wording cues in the stem that single out choice A."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: it asks for the precise logical relationship among (a), (b), and (c) for this class of manifolds. However, it is not a pure restatement, since the options offer competing strengthenings, weakenings, and reversals of the theorem."}, "GPS": {"score": 1, "justification": "The question requires some reasoning about logical strength and known versus open implications, especially to reject the weaker true statement C and the overstrong D/E. Still, success mainly depends on recalling the exact theorem rather than generating a novel argument."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they test common confusions about converse implications, full equivalence versus one-way implications, and weaker true statements versus strongest true statements. They are distinct and well-targeted."}, "total_score": 6, "overall_assessment": "A solid MCQ with strong distractors and little answer leakage, but it is primarily a precise theorem-identification item rather than a deeply generative reasoning question."}}
{"id": "2602.18612v1", "paper_link": "http://arxiv.org/abs/2602.18612v1", "theorems_cnt": 1, "theorem": {"env_name": "conjecture", "content": "[Song--Wang {\\cite{wang2}}]\\label{conj}\nThere exists, for each level $\\ell$, a combinatorial bijection\n\\begin{equation}\n\\label{e:psi}\n\\psi_\\ell : \\mathrm{ParMat}_{\\nu,\\mu}^{\\flat} \\longrightarrow \n\\bigsqcup_{\\lambda \\vdash_\\ell n}\n\\mathrm{SST}_\\ell(\\lambda,\\mu)\\times \\mathrm{SST}_\\ell(\\lambda,\\nu),\n\\end{equation}\nwhich extends the level $\\ell-1$ bijection and recovers the classical RSK correspondence when $\\ell=1$.", "start_pos": 25931, "end_pos": 26384, "label": "conj"}, "ref_dict": {}, "pre_theorem_intro_text_len": 2515, "pre_theorem_intro_text": "The Robinson--Schensted--Knuth correspondence is a bijection between (bi)words or nonnegative integer matrices and pairs of Young tableaux of the same shape, with applications to symmetric functions, permutation statistics such as longest increasing subsequences, and the representation theory of the symmetric and general linear groups. With the development of web categories, representation theories traditionally governed by Young tableaux have been reformulated in diagrammatic terms that incorporate additional structure, such as multiple components, block decompositions, and restrictions on how labels may appear. In this setting, the relevant combinatorial data consist not of a single word or matrix but of several interrelated pieces, and the resulting tableau data record this multi-component structure. This viewpoint naturally raises the question of whether classical correspondences such as RSK admit higher--level analogues that organize such data in a way compatible with the additional structure.\n\nIn \\cite{beeo}, the authors introduced the \\emph{Schur category}, whose path algebra recovers certain subalgebras of Schur algebras, alongside a category \\emph{Web}, a $\\mathfrak{gl}_{\\infty}$--analogue of the $\\mathfrak{sl}_n$ web categories. They showed that these categories are isomorphic, and that both contain the (degenerate) Hecke category via fully faithful functors. Building on this framework, Song and Wang \\cite{wang1,wang2} defined \\emph{affine Schur} and \\emph{affine web} categories. In contrast to the finite case, the affine web category embeds as a full subcategory of the affine Schur category rather than being equivalent to it. Their work further introduced \\emph{cyclotomic} quotients of these categories, which retain embeddings of the cyclotomic Hecke category and exhibit rich combinatorial structure.\n\nA key result of \\cite{wang2} identifies the path algebra of the cyclotomic Schur category with the cyclotomic Schur algebra of Dipper--James--Mathas \\cite{djm}. Song and Wang describe two natural bases for this algebra. The first is indexed by pairs of tableaux and corresponds, under this identification, to the cellular basis of the cyclotomic Schur algebra. The second is a family of basis elements indexed by enriched block matrices arising from elementary ``chicken--foot'' diagrams in the cyclotomic web category. The coexistence of these two parametrizations naturally suggests a bijection relating block-matrix data to pairs of tableaux in the cyclotomic setting.", "context": "The Robinson--Schensted--Knuth correspondence is a bijection between (bi)words or nonnegative integer matrices and pairs of Young tableaux of the same shape, with applications to symmetric functions, permutation statistics such as longest increasing subsequences, and the representation theory of the symmetric and general linear groups. With the development of web categories, representation theories traditionally governed by Young tableaux have been reformulated in diagrammatic terms that incorporate additional structure, such as multiple components, block decompositions, and restrictions on how labels may appear. In this setting, the relevant combinatorial data consist not of a single word or matrix but of several interrelated pieces, and the resulting tableau data record this multi-component structure. This viewpoint naturally raises the question of whether classical correspondences such as RSK admit higher--level analogues that organize such data in a way compatible with the additional structure.\n\nIn \\cite{beeo}, the authors introduced the \\emph{Schur category}, whose path algebra recovers certain subalgebras of Schur algebras, alongside a category \\emph{Web}, a $\\mathfrak{gl}_{\\infty}$--analogue of the $\\mathfrak{sl}_n$ web categories. They showed that these categories are isomorphic, and that both contain the (degenerate) Hecke category via fully faithful functors. Building on this framework, Song and Wang \\cite{wang1,wang2} defined \\emph{affine Schur} and \\emph{affine web} categories. In contrast to the finite case, the affine web category embeds as a full subcategory of the affine Schur category rather than being equivalent to it. Their work further introduced \\emph{cyclotomic} quotients of these categories, which retain embeddings of the cyclotomic Hecke category and exhibit rich combinatorial structure.\n\nA key result of \\cite{wang2} identifies the path algebra of the cyclotomic Schur category with the cyclotomic Schur algebra of Dipper--James--Mathas \\cite{djm}. Song and Wang describe two natural bases for this algebra. The first is indexed by pairs of tableaux and corresponds, under this identification, to the cellular basis of the cyclotomic Schur algebra. The second is a family of basis elements indexed by enriched block matrices arising from elementary ``chicken--foot'' diagrams in the cyclotomic web category. The coexistence of these two parametrizations naturally suggests a bijection relating block-matrix data to pairs of tableaux in the cyclotomic setting.", "full_context": "The Robinson--Schensted--Knuth correspondence is a bijection between (bi)words or nonnegative integer matrices and pairs of Young tableaux of the same shape, with applications to symmetric functions, permutation statistics such as longest increasing subsequences, and the representation theory of the symmetric and general linear groups. With the development of web categories, representation theories traditionally governed by Young tableaux have been reformulated in diagrammatic terms that incorporate additional structure, such as multiple components, block decompositions, and restrictions on how labels may appear. In this setting, the relevant combinatorial data consist not of a single word or matrix but of several interrelated pieces, and the resulting tableau data record this multi-component structure. This viewpoint naturally raises the question of whether classical correspondences such as RSK admit higher--level analogues that organize such data in a way compatible with the additional structure.\n\nIn \\cite{beeo}, the authors introduced the \\emph{Schur category}, whose path algebra recovers certain subalgebras of Schur algebras, alongside a category \\emph{Web}, a $\\mathfrak{gl}_{\\infty}$--analogue of the $\\mathfrak{sl}_n$ web categories. They showed that these categories are isomorphic, and that both contain the (degenerate) Hecke category via fully faithful functors. Building on this framework, Song and Wang \\cite{wang1,wang2} defined \\emph{affine Schur} and \\emph{affine web} categories. In contrast to the finite case, the affine web category embeds as a full subcategory of the affine Schur category rather than being equivalent to it. Their work further introduced \\emph{cyclotomic} quotients of these categories, which retain embeddings of the cyclotomic Hecke category and exhibit rich combinatorial structure.\n\nA key result of \\cite{wang2} identifies the path algebra of the cyclotomic Schur category with the cyclotomic Schur algebra of Dipper--James--Mathas \\cite{djm}. Song and Wang describe two natural bases for this algebra. The first is indexed by pairs of tableaux and corresponds, under this identification, to the cellular basis of the cyclotomic Schur algebra. The second is a family of basis elements indexed by enriched block matrices arising from elementary ``chicken--foot'' diagrams in the cyclotomic web category. The coexistence of these two parametrizations naturally suggests a bijection relating block-matrix data to pairs of tableaux in the cyclotomic setting.\n\nA key result of \\cite{wang2} identifies the path algebra of the cyclotomic Schur category with the cyclotomic Schur algebra of Dipper--James--Mathas \\cite{djm}. Song and Wang describe two natural bases for this algebra. The first is indexed by pairs of tableaux and corresponds, under this identification, to the cellular basis of the cyclotomic Schur algebra. The second is a family of basis elements indexed by enriched block matrices arising from elementary ``chicken--foot'' diagrams in the cyclotomic web category. The coexistence of these two parametrizations naturally suggests a bijection relating block-matrix data to pairs of tableaux in the cyclotomic setting.\n\nWe confirm this conjecture by giving an explicit bijection at level~$\\ell$ that uses only the classical RSK correspondence. The key step is to recast the combinatorial parameters in the set $\\mathrm{ParMat}^{\\flat}$ as \\emph{flagged block composition matrices}. These matrices encode, in a transparent way, collections of biwords subject to the level-dependent labeling restrictions. With this reformulation in hand, the bijection is obtained by applying the classical RSK correspondence independently in each component and then assembling the outputs as semistandard multitableaux of common shape.\n\n\\begin{theorem}\\label{classicalRSK}[Robinson--Schensted--Knuth \\cite{RSK}]\nThere is a bijection\n\\[\n\\mathrm{BW}(\\mathbb N)\n\\longleftrightarrow\n\\coprod_{\\lambda\\vdash n}\n\\mathrm{SST}(\\mathbb N,\\lambda)\n\\times\n\\mathrm{SST}(\\mathbb N,\\lambda).\n\\]\nFor biword $w$ corresponding to $(P,Q)$, \n\\(\n\\mathrm{wt}(P)=\\text{content of } w_{\\mathrm{bot}}\\) \nand\n\\(\\mathrm{wt}(Q)=\\text{content of } w_{\\mathrm{top}}.\n\\)\n\\end{theorem}\n\nEach biword $w\\in \\mathrm{BW}(\\mathbb N)$ naturally determines an\n$\\mathbb N$-matrix $A$ by setting $a_{ij}$ equal to the multiplicity of the\nbiletter $\\begin{pmatrix} i \\\\ j \\end{pmatrix}$ in $w$.\nUnder this correspondence,\n\\[\n\\mathrm{row}(A)_i=\\#\\{\\text{occurrences of } i \\text{ in } w_{\\mathrm{top}}\\}\n\\quad\\text{and}\\quad\n\\mathrm{col}(A)_j=\\#\\{\\text{occurrences of } j \\text{ in } w_{\\mathrm{bot}}\\}.\n\\]\nThus the RSK correspondence may equivalently be written as a bijection\n\\begin{equation}\n\\label{e rskmatrix}\n \\left\\{\\NN\\mathrm{-matrices}\\,\\,A\\setmid\\begin{tabular}{c}$\\mathrm{row}(A)=\\nu$\\\\$\\mathrm{col}(A)=\\mu$\\end{tabular}\\right\\}\\longleftrightarrow\\coprod_{\\lambda\\vdash n}\\text{\\rm SST}(\\NN,\\lambda,\\mu)\\times\\text{\\rm SST}(\\NN,\\lambda,\\nu).\n \\end{equation}\n\nThe cyclotomic web category is connected in \\cite{wang2} to a combinatorial model called \n(elementary) chicken foot diagrams. \nThe diagrammatic parameters indexing the cyclotomic web basis may be encoded by certain partition-enriched block matrices. These matrices are packaged into a set\n\\[\n\\mathrm{ParMat}^{\\flat}_{\\nu,\\mu},\n\\]\nindexed by multicompositions $\\nu,\\mu \\vDash_\\ell n$ which record the prescribed row and column data of the underlying block matrix.\nInformally, an element of $\\mathrm{ParMat}^{\\flat}_{\\nu,\\mu}$ consists of:\n\\begin{itemize}\n\\item an $\\ell\\times\\ell$ block matrix of nonnegative integers whose block row and column sums are fixed by $\\nu$ and $\\mu$, and\n\\item a choice of a partition in each matrix entry, subject to natural size and length constraints determined by the block position.\n\\end{itemize}\n\n\\begin{lemma}\\label{lem:parmat-to-bcm}\nFix $\\mu,\\nu \\vDash_\\ell n$. There is a bijection\n\\begin{align*}\n\\mathrm{ParMat}^{\\flat}_{\\nu,\\mu}&\\longleftrightarrow\\text{BCM}_\\ell(\\nu,\\mu),\\\\\n(A,P)&\\mapsto B,\n\\end{align*}\nsatisfying $a_{ij}^{(pq)}=\\left|b^{(pq)}_{ij}\\right|$ and $\\mathrm{length}\\left(\\eta_{ij}^{(pq)}\\right)=\\mathrm{length}\\left(b_{ij}^{(pq)}\\right)-1$.\n\\end{lemma}\n\\begin{proof}\nThere is a natural, length-respecting bijection between compositions of $k$ and partitions with parts at most $k$: if $\\lambda$ is such a partition,\nthe associated composition $c$ is given by\n\\(\nc_t=\\lambda_{t-1}-\\lambda_t,\n\\)\nwith the convention $\\lambda_0=k$ and $\\lambda_t=0$ for $t>\\ell(\\lambda)$.\nThe inverse map recovers $\\lambda$ from $c$ by taking partial sums\n$\\lambda_t=\\sum_{s>t} c_s$.\n\n\\begin{example}\nFor $\\ell=3$, $\\mu=(212,3,23)$ and $\\nu=(13,3,222)$, the flagged compositional\nblock matrix $B$ from Example~\\ref{ex:BCM-with-B}, written with \na letter-coordinate key, is sent to the flagged multi biword of \nExample~\\ref{uvword}\\\\\n\\begin{tabular}{c | c c | c | c c c}\n\\color{violet}\\text{key:}& \\color{violet}$1_1$ & \\color{violet}$2_1$ & \\color{violet}$1_2$ & \\color{violet}$1_3$ & \\color{violet}$2_3$ & \\color{violet}$3_3$\\\\\n\\hline\n{\\color{violet}$1_1$}&$\\varnothing$ & $2$ & $\\varnothing$ & $\\varnothing$ & $\\varnothing$ & $\\varnothing$\\\\\n{\\color{violet}$2_1$}&$\\varnothing$ & $\\varnothing$ & $\\varnothing$ & $1$ & $\\varnothing$ & $\\varnothing$\\\\\n{\\color{violet}$3_1$}&$1$ & $\\varnothing$ & $\\varnothing$ & $\\varnothing$ & $\\varnothing$ & $1$\\\\\n\\hline\n{\\color{violet}$1_2$}&$\\varnothing$ & $\\varnothing$ & $03$ & $\\varnothing$ & $\\varnothing$ & $\\varnothing$\\\\\n\\hline\n{\\color{violet}$1_3$}&$\\varnothing$ & $\\varnothing$ & $\\varnothing$ & $001$ & $\\varnothing$ & $01$\\\\\n{\\color{violet}$2_3$}&$\\varnothing$ & $1$ & $\\varnothing$ & $\\varnothing$ & $011$ & $\\varnothing$\\\\\n\\end{tabular}\n$\\longleftrightarrow$\n$\\left(\\begin{pmatrix}1_1\\ 1_1\\ 2_1 \\ 3_1 \\ 3_1\\ 2_3 \\\\ 2_1\\ 2_1 \\ 1_3 \\ 1_1 \\ 3_3\\ 2_1\\end{pmatrix},\n\\begin{pmatrix}1_2\\ 1_2\\ 1_2 \\ 1_3 \\ 2_3 \\\\ 1_2\\ 1_2 \\ 1_2 \\ 3_3\\ 2_3\\end{pmatrix}\\,,\n\\begin{pmatrix}1_3\\ 2_3\\\\ 1_3\\ 2_3\\end{pmatrix}\\right)$.\n\\end{example}\n\\begin{lemma}\\label{lem:biwords-to-tableaux}\nFix $\\mu,\\nu \\models_\\ell n$. There is a bijection\n\\[\nBW_\\ell(\\nu,\\mu)\\;\n\\longleftrightarrow\\coprod_{\\lambda\\vdash_\\ell n}\\text{\\rm SST}_\\ell(\\lambda,\\mu)\\times\\text{\\rm SST}_\\ell(\\lambda,\\nu),\n\\]\nobtained by applying the classical Robinson--Schensted--Knuth correspondence\ncomponentwise, after an order-preserving relabeling of the $\\mu$-- and\n$\\nu$--alphabets.\n\\end{lemma}\n\\begin{proof}\nLet $w=(w^{(1)},\\dots,w^{(\\ell)})\\in BW_\\ell(\\nu,\\mu)$.\nConstruct the pair $(P,Q)$ of semistandard multitableaux componentwise using the usual RSK correspondence of Thm~\\ref{classicalRSK}, where\nwe set $(P^{(i)},Q^{(i)})=\\text{RSK}(w^{(i)})$, but with the modification that the (totally ordered) $\\mu$- and $\\nu$-alphabets serve in place of the usual letters $[n]$.\n\n\\section{The correspondence}\n\\begin{theorem}\nFix $\\mu,\\nu \\models_\\ell n$. Applying the classical RSK correspondence of Thm~\\ref{classicalRSK}\ncomponentwise defines a bijection\n\\begin{equation}\n\\text{ParMat}^\\flat_{\\nu,\\mu}\n\\longleftrightarrow\\coprod_{\\lambda\\vdash_\\ell n}\\text{\\rm SST}_\\ell(\\lambda,\\mu)\\times\\text{\\rm SST}_\\ell(\\lambda,\\nu).\n\\end{equation}\nWhen restricted to multipartitions satisfying\n$\\mu^{(\\ell)}=\\nu^{(\\ell)}=\\lambda^{(\\ell)}=\\varnothing$, this bijection agrees with the\nlevel--$(\\ell-1)$ correspondence. For $\\ell=1$, it reduces to the usual RSK\ncorrespondence.\n\\end{theorem}\n\\begin{proof}\nWe first use Lemma~\\ref{lem:parmat-to-bcm} to replace the domain with \n$\\text{\\rm BCM}_\\ell(\\nu,\\mu)$. From there,\nLemma~\\ref{lem:BCM-to-biwords} takes flagged block composition matrices bijectively to\nflagged biwords. Lemma~\\ref{lem:biwords-to-tableaux} then applies\nclassical RSK componentwise to obtain the desired correspondence. The\nspecialization and restriction properties follow directly from the definitions.\n\\end{proof}", "post_theorem_intro_text_len": 1808, "post_theorem_intro_text": "We confirm this conjecture by giving an explicit bijection at level~$\\ell$ that uses only the classical RSK correspondence. The key step is to recast the combinatorial parameters in the set $\\mathrm{ParMat}^{\\flat}$  as \\emph{flagged block composition matrices}. These matrices encode, in a transparent way, collections of biwords subject to the level-dependent labeling restrictions. With this reformulation in hand, the bijection is obtained by applying the classical RSK correspondence independently in each component and then assembling the outputs as semistandard multitableaux of common shape.\n\nOur approach proceeds in three steps. First, we reinterpret the $\\mathrm{ParMat}^{\\flat}$-indexed data in terms of the {flagged block composition matrices}, which present the same information in a form amenable to combinatorial manipulation. Second, we establish a natural bijection between these matrices and collections of \\emph{flagged biwords}, extending the classical correspondence between nonnegative integer matrices and biwords. Finally, we apply the classical RSK correspondence componentwise (after a simple relabeling of the alphabets) to obtain pairs of semistandard multitableaux of common shape.\n\nThe resulting correspondence is compatible with restriction to lower levels, specializes to the usual RSK correspondence when $\\ell=1$, and reflects the multipartition data inherent in the cyclotomic setting. In particular, it provides a direct combinatorial explanation for the relationship between the tableau--indexed and diagrammatic bases of the cyclotomic Schur category.\n\n\\noindent {\\bf Acknowledgement.} We thank Linliang Song and Weiqiang Wang for formulating the initiating conjecture. We are especially grateful to Weiqiang Wang for personally recommending this combinatorial problem.", "sketch": "We “confirm this conjecture by giving an explicit bijection at level~$\\ell$ that uses only the classical RSK correspondence.” The “key step is to recast the combinatorial parameters in the set $\\mathrm{ParMat}^{\\flat}$ as \\emph{flagged block composition matrices},” which “encode, in a transparent way, collections of biwords subject to the level-dependent labeling restrictions.”\n\nThe approach “proceeds in three steps.” (1) “Reinterpret the $\\mathrm{ParMat}^{\\flat}$-indexed data in terms of the flagged block composition matrices,” putting the data “in a form amenable to combinatorial manipulation.” (2) “Establish a natural bijection between these matrices and collections of \\emph{flagged biwords}, extending the classical correspondence between nonnegative integer matrices and biwords.” (3) “Apply the classical RSK correspondence componentwise (after a simple relabeling of the alphabets) to obtain pairs of semistandard multitableaux of common shape,” i.e. “apply the classical RSK correspondence independently in each component and then assemble the outputs as semistandard multitableaux of common shape.”\n\nThe resulting correspondence is stated to be “compatible with restriction to lower levels” and to “specialize[] to the usual RSK correspondence when $\\ell=1$,” matching the properties required for the bijection $\\psi_\\ell$ in Theorem~\\ref{conj}.", "expanded_sketch": "We “confirm this conjecture by giving an explicit bijection at level~$\\ell$ that uses only the classical RSK correspondence.” The “key step is to recast the combinatorial parameters in the set $\\mathrm{ParMat}^{\\flat}$ as \\emph{flagged block composition matrices},” which “encode, in a transparent way, collections of biwords subject to the level-dependent labeling restrictions.”\n\nThe approach “proceeds in three steps.” (1) “Reinterpret the $\\mathrm{ParMat}^{\\flat}$-indexed data in terms of the flagged block composition matrices,” putting the data “in a form amenable to combinatorial manipulation.” (2) “Establish a natural bijection between these matrices and collections of \\emph{flagged biwords}, extending the classical correspondence between nonnegative integer matrices and biwords.” (3) “Apply the classical RSK correspondence componentwise (after a simple relabeling of the alphabets) to obtain pairs of semistandard multitableaux of common shape,” i.e. “apply the classical RSK correspondence independently in each component and then assemble the outputs as semistandard multitableaux of common shape.”\n\nThe resulting correspondence is stated to be “compatible with restriction to lower levels” and to “specialize[] to the usual RSK correspondence when $\\ell=1$,” matching the properties required for the bijection $\\psi_\\ell$ in establishing the main theorem.", "expanded_theorem": "[Song--Wang {\\cite{wang2}}]\\label{conj}\nThere exists, for each level $\\ell$, a combinatorial bijection\n\\begin{equation}\n\\label{e:psi}\n\\psi_\\ell : \\mathrm{ParMat}_{\\nu,\\mu}^{\\flat} \\longrightarrow \n\\bigsqcup_{\\lambda \\vdash_\\ell n}\n\\mathrm{SST}_\\ell(\\lambda,\\mu)\\times \\mathrm{SST}_\\ell(\\lambda,\\nu),\n\\end{equation}\nwhich extends the level $\\ell-1$ bijection and recovers the classical RSK correspondence when $\\ell=1$.", "theorem_type": ["Classification or Bijection", "Universal–Existential"], "mcq": {"question": "Fix a level \\(\\ell\\) and \\(\\ell\\)-compositions \\(\\mu,\\nu\\models_\\ell n\\). Let \\(\\mathrm{ParMat}^{\\flat}_{\\nu,\\mu}\\) denote the paper’s flagged block-matrix/combinatorial matrix data with parameters \\((\\nu,\\mu)\\), and for an \\(\\ell\\)-multipartition \\(\\lambda\\vdash_\\ell n\\), let \\(\\mathrm{SST}_\\ell(\\lambda,\\mu)\\) and \\(\\mathrm{SST}_\\ell(\\lambda,\\nu)\\) be the sets of semistandard multitableaux of shape \\(\\lambda\\) and weights \\(\\mu\\) and \\(\\nu\\), respectively. Which statement holds for these objects?", "correct_choice": {"label": "A", "text": "For each level \\(\\ell\\), there exists a combinatorial bijection\n\\[\n\\psi_\\ell:\\mathrm{ParMat}^{\\flat}_{\\nu,\\mu}\\longrightarrow \\bigsqcup_{\\lambda\\vdash_\\ell n}\\mathrm{SST}_\\ell(\\lambda,\\mu)\\times \\mathrm{SST}_\\ell(\\lambda,\\nu),\n\\]\nwhich extends the level \\(\\ell-1\\) bijection and recovers the classical Robinson--Schensted--Knuth correspondence when \\(\\ell=1\\)."}, "choices": [{"label": "B", "text": "For each level \\(\\ell\\), there exists a combinatorial bijection\n\\[\n\\psi_\\ell:\\mathrm{ParMat}^{\\flat}_{\\nu,\\mu}\\longrightarrow \\bigsqcup_{\\lambda\\vdash_\\ell n}\\mathrm{SST}_\\ell(\\lambda,\\mu)\\times \\mathrm{SST}_\\ell(\\lambda,\\nu),\n\\]\nwhich extends the level \\(\\ell-1\\) bijection and, for \\(\\ell=1\\), agrees with the classical Robinson--Schensted--Knuth correspondence only after restricting to ordinary partitions \\(\\lambda\\) with a single nonempty component."}, {"label": "C", "text": "For each level \\(\\ell\\), there exists a combinatorial bijection\n\\[\n\\psi_\\ell:\\mathrm{ParMat}^{\\flat}_{\\nu,\\mu}\\longrightarrow \\bigsqcup_{\\lambda\\vdash_\\ell n}\\mathrm{SST}_\\ell(\\lambda,\\mu)\\times \\mathrm{SST}_\\ell(\\lambda,\\nu).\n\\]"}, {"label": "D", "text": "For each level \\(\\ell\\), there exists a combinatorial bijection\n\\[\n\\psi_\\ell:\\mathrm{ParMat}^{\\flat}_{\\nu,\\mu}\\longrightarrow \\bigsqcup_{\\lambda\\vdash_\\ell n}\\mathrm{SST}_\\ell(\\lambda,\\mu)\\times \\mathrm{SST}_\\ell(\\lambda,\\nu),\n\\]\nwhich recovers the classical Robinson--Schensted--Knuth correspondence when \\(\\ell=1\\), and whose restriction to multipartitions satisfying \\(\\mu^{(\\ell)}=\\nu^{(\\ell)}=\\lambda^{(\\ell)}=\\varnothing\\) is canonically the level \\(\\ell\\) correspondence."}, {"label": "E", "text": "For each level \\(\\ell\\), there exists a combinatorial bijection\n\\[\n\\psi_\\ell:\\mathrm{ParMat}^{\\flat}_{\\nu,\\mu}\\longrightarrow \\bigsqcup_{\\lambda\\vdash_\\ell n}\\mathrm{SST}_\\ell(\\lambda,\\nu)\\times \\mathrm{SST}_\\ell(\\lambda,\\mu),\n\\]\nwhich extends the level \\(\\ell-1\\) bijection and recovers the classical Robinson--Schensted--Knuth correspondence when \\(\\ell=1\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "exact \\ell=1 specialization to classical RSK", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped compatibility with lower levels and \\ell=1 specialization", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "lower-level restriction identifies with level \\ell-1, not level \\ell", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "componentwise RSK preserves ordered pairing of \\mu- and \\nu-weight tableau factors", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It names the source and target-type objects involved in the theorem, but the exact bijective conclusion and its precise qualifiers still have to be identified from the choices."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: it asks for the explicit family appearing in the stated bijection, together with the extension and \\(\\ell=1\\) compatibility. The correct option is basically the theorem statement restated."}, "GPS": {"score": 1, "justification": "There is some pressure to distinguish the exact strongest statement from nearby variants (subset vs bijection, wrong weight pairing, extra canonicity claim, misuse of classical RSK), but the task is mostly precise recall rather than genuinely generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted. They reflect common failure modes: weakening bijectivity, confusing how RSK is applied, altering the weight data, or adding an unjustified independence/canonicity claim."}, "total_score": 5, "overall_assessment": "A reasonably well-crafted theorem-recall MCQ with strong distractors and little answer leakage, but it is highly close to a direct restatement and only modestly tests generative reasoning."}}
{"id": "2602.18632v1", "paper_link": "http://arxiv.org/abs/2602.18632v1", "theorems_cnt": 1, "theorem": {"env_name": "problem", "content": "[\\cite{Cho}]\\label{problem:Cho}\n    For $\\eta  \\leqslant  \\theta$, give a new definition of the plactic skew Schur $P$-function $\\sPlacticSchur_{\\theta/\\eta}$\n so that it belongs to the ring generated by plactic Schur $P$-functions and it describes\nthe multiplicities of the ordinary $P_\\lambda$ in the expansion of $P_{\\theta/\\eta}$ in a nice way.", "start_pos": 8284, "end_pos": 8656, "label": "problem:Cho"}, "ref_dict": {"thm:mixedRectification": "\\begin{theorem}\\label{thm:mixedRectification}\n        Let $T$ be an arbitrary shifted Young tableau and $\\unpr y$ a high letter. Then performing the mixed insertion of $\\unpr y$ on $T$ results in the tableau \n        $\\mathrm{rect}_\\mix \\left( T\\oplus \\unpr y \\right)$.\n    \\end{theorem}", "problem:Cho": "\\begin{problem}[\\cite{Cho}]\\label{problem:Cho}\n    For $\\eta  \\leq  \\theta$, give a new definition of the plactic skew Schur $P$-function $\\sPlacticSchur_{\\theta/\\eta}$\n so that it belongs to the ring generated by plactic Schur $P$-functions and it describes\nthe multiplicities of the ordinary $P_\\lambda$ in the expansion of $P_{\\theta/\\eta}$ in a nice way.\n\\end{problem}"}, "pre_theorem_intro_text_len": 3774, "pre_theorem_intro_text": "The \\emph{Schur functions} $s_\\lambda$ are a basis of the ring of symmetric functions and their structure coefficients $c_{\\lambda,\\mu}^\\nu$, appearing in the expansions\n\\[\ns_\\lambda \\cdot s_\\mu  = \\sum_\\nu c_{\\lambda,\\mu}^\\nu s_\\nu,\n\\]\nare the \\emph{Littlewood--Richardson coefficients}. These objects play important roles in the linear representation theory of symmetric groups (where $s_\\lambda$ represents a \\emph{Specht module}), in the linear representation theory of general linear groups (where $s_\\lambda$ represents a \\emph{Schur module}), and in the Schubert calculus of Grassmannians (where $s_\\lambda$ represents a \\emph{Schubert cycle}). The Littlewood--Richardson coefficients may be studied combinatorially through the interlocking machinery of \\emph{jeu de taquin} on \\emph{Young tableaux}, the \\emph{Robinson--Schensted--Knuth} (RSK) insertion algorithm, and the \\emph{plactic monoid}. For exposition of these now-standard ideas, see, for example, the textbooks \\cite{Fulton:YT,Manivel}.\n\nThe ideas discussed so far may all be considered ``type A.'' In other classical types, the corresponding theories are more complicated and the relations less well developed. The analogues of the usual Schur functions are the \\emph{Schur $P$-functions} and \\emph{Schur $Q$-functions}, which play similar roles for the projective representation theory of symmetric groups as well as for the Schubert calculus of Lagrangian and maximal orthogonal Grassmannians.\n\nIn this setting, \\emph{Sagan--Worley} \\cite{Sagan,Worley} insertion is an analogue of RSK that is closely connected to a \\emph{Sagan--Worley} jeu de taquin. This Sagan--Worley theory does not appear to have a corresponding analogue of the plactic monoid. On the other hand, \\emph{mixed} insertion \\cite{Haiman} is a different analogue of RSK insertion with a corresponding \\emph{shifted plactic monoid} \\cite{Serrano}. This latter theory has not been known to have a corresponding jeu de taquin. We provide\\footnote{The results of this paper were first announced in the extended abstract \\cite{EstupinanSalamanca.Pechenik:FPSAC}.} such a jeu de taquin in \\cref{sec:mixedJeudeTaquin}. Our new jeu de taquin computes mixed insertions, despite lacking some of the properties of ordinary jeu de taquin, such as confluence. There is precedent for useful theories of jeu de taquin without confluence, such as the $K$-theoretic jeu de taquin theories of \\cite{Thomas.Yong:K, Pechenik.Yong} and we hope that our mixed jeu de taquin will lead to similar such developments.\n\nIn type A, the combinatorics of Schur functions can realized through \\emph{plactic skew Schur functions}, which plactically encode both jeu de taquin rectification and RSK insertion. This realization yields an algebraic perspective on these combinatorial theories, as well as on the Littlewood--Richardson coefficients. Serrano \\cite[p.~379]{Serrano} introduced a notion of \\emph{plactic skew Schur $P$-functions}, which he conjectured \\cite[Conjecture~2.12]{Serrano} (see also, \\cite[Conjecture~7.8]{Cho}) would play an analogous role in other classical types. If true, this would have given an algebraic substitute for the missing jeu de taquin associated to mixed insertion and to the shifted plactic monoid. Unfortunately, Serrano's conjecture was not correct. Precisely, Cho \\cite[\\textsection 7]{Cho} gave explicit examples to show that Serrano's plactic skew Schur $P$-functions do not lie in the desired ring and hence they cannot serve the desired role in substituting for jeu de taquin or in shedding algebraic light on structure coefficients. Having disproved Serrano's conjecture, Cho asks in \\cite[Open Problem~7.12(1)]{Cho} for a new definition that would accomplish Serrano's goals. More precisely, she sets the following task:", "context": "The \\emph{Schur functions} $s_\\lambda$ are a basis of the ring of symmetric functions and their structure coefficients $c_{\\lambda,\\mu}^\\nu$, appearing in the expansions\n\\[\ns_\\lambda \\cdot s_\\mu = \\sum_\\nu c_{\\lambda,\\mu}^\\nu s_\\nu,\n\\]\nare the \\emph{Littlewood--Richardson coefficients}. These objects play important roles in the linear representation theory of symmetric groups (where $s_\\lambda$ represents a \\emph{Specht module}), in the linear representation theory of general linear groups (where $s_\\lambda$ represents a \\emph{Schur module}), and in the Schubert calculus of Grassmannians (where $s_\\lambda$ represents a \\emph{Schubert cycle}). The Littlewood--Richardson coefficients may be studied combinatorially through the interlocking machinery of \\emph{jeu de taquin} on \\emph{Young tableaux}, the \\emph{Robinson--Schensted--Knuth} (RSK) insertion algorithm, and the \\emph{plactic monoid}. For exposition of these now-standard ideas, see, for example, the textbooks \\cite{Fulton:YT,Manivel}.\n\nThe ideas discussed so far may all be considered ``type A.'' In other classical types, the corresponding theories are more complicated and the relations less well developed. The analogues of the usual Schur functions are the \\emph{Schur $P$-functions} and \\emph{Schur $Q$-functions}, which play similar roles for the projective representation theory of symmetric groups as well as for the Schubert calculus of Lagrangian and maximal orthogonal Grassmannians.\n\nIn this setting, \\emph{Sagan--Worley} \\cite{Sagan,Worley} insertion is an analogue of RSK that is closely connected to a \\emph{Sagan--Worley} jeu de taquin. This Sagan--Worley theory does not appear to have a corresponding analogue of the plactic monoid. On the other hand, \\emph{mixed} insertion \\cite{Haiman} is a different analogue of RSK insertion with a corresponding \\emph{shifted plactic monoid} \\cite{Serrano}. This latter theory has not been known to have a corresponding jeu de taquin. We provide\\footnote{The results of this paper were first announced in the extended abstract \\cite{EstupinanSalamanca.Pechenik:FPSAC}.} such a jeu de taquin in \\cref{sec:mixedJeudeTaquin}. Our new jeu de taquin computes mixed insertions, despite lacking some of the properties of ordinary jeu de taquin, such as confluence. There is precedent for useful theories of jeu de taquin without confluence, such as the $K$-theoretic jeu de taquin theories of \\cite{Thomas.Yong:K, Pechenik.Yong} and we hope that our mixed jeu de taquin will lead to similar such developments.\n\nIn type A, the combinatorics of Schur functions can realized through \\emph{plactic skew Schur functions}, which plactically encode both jeu de taquin rectification and RSK insertion. This realization yields an algebraic perspective on these combinatorial theories, as well as on the Littlewood--Richardson coefficients. Serrano \\cite[p.~379]{Serrano} introduced a notion of \\emph{plactic skew Schur $P$-functions}, which he conjectured \\cite[Conjecture~2.12]{Serrano} (see also, \\cite[Conjecture~7.8]{Cho}) would play an analogous role in other classical types. If true, this would have given an algebraic substitute for the missing jeu de taquin associated to mixed insertion and to the shifted plactic monoid. Unfortunately, Serrano's conjecture was not correct. Precisely, Cho \\cite[\\textsection 7]{Cho} gave explicit examples to show that Serrano's plactic skew Schur $P$-functions do not lie in the desired ring and hence they cannot serve the desired role in substituting for jeu de taquin or in shedding algebraic light on structure coefficients. Having disproved Serrano's conjecture, Cho asks in \\cite[Open Problem~7.12(1)]{Cho} for a new definition that would accomplish Serrano's goals. More precisely, she sets the following task:", "full_context": "The \\emph{Schur functions} $s_\\lambda$ are a basis of the ring of symmetric functions and their structure coefficients $c_{\\lambda,\\mu}^\\nu$, appearing in the expansions\n\\[\ns_\\lambda \\cdot s_\\mu = \\sum_\\nu c_{\\lambda,\\mu}^\\nu s_\\nu,\n\\]\nare the \\emph{Littlewood--Richardson coefficients}. These objects play important roles in the linear representation theory of symmetric groups (where $s_\\lambda$ represents a \\emph{Specht module}), in the linear representation theory of general linear groups (where $s_\\lambda$ represents a \\emph{Schur module}), and in the Schubert calculus of Grassmannians (where $s_\\lambda$ represents a \\emph{Schubert cycle}). The Littlewood--Richardson coefficients may be studied combinatorially through the interlocking machinery of \\emph{jeu de taquin} on \\emph{Young tableaux}, the \\emph{Robinson--Schensted--Knuth} (RSK) insertion algorithm, and the \\emph{plactic monoid}. For exposition of these now-standard ideas, see, for example, the textbooks \\cite{Fulton:YT,Manivel}.\n\nThe ideas discussed so far may all be considered ``type A.'' In other classical types, the corresponding theories are more complicated and the relations less well developed. The analogues of the usual Schur functions are the \\emph{Schur $P$-functions} and \\emph{Schur $Q$-functions}, which play similar roles for the projective representation theory of symmetric groups as well as for the Schubert calculus of Lagrangian and maximal orthogonal Grassmannians.\n\nIn this setting, \\emph{Sagan--Worley} \\cite{Sagan,Worley} insertion is an analogue of RSK that is closely connected to a \\emph{Sagan--Worley} jeu de taquin. This Sagan--Worley theory does not appear to have a corresponding analogue of the plactic monoid. On the other hand, \\emph{mixed} insertion \\cite{Haiman} is a different analogue of RSK insertion with a corresponding \\emph{shifted plactic monoid} \\cite{Serrano}. This latter theory has not been known to have a corresponding jeu de taquin. We provide\\footnote{The results of this paper were first announced in the extended abstract \\cite{EstupinanSalamanca.Pechenik:FPSAC}.} such a jeu de taquin in \\cref{sec:mixedJeudeTaquin}. Our new jeu de taquin computes mixed insertions, despite lacking some of the properties of ordinary jeu de taquin, such as confluence. There is precedent for useful theories of jeu de taquin without confluence, such as the $K$-theoretic jeu de taquin theories of \\cite{Thomas.Yong:K, Pechenik.Yong} and we hope that our mixed jeu de taquin will lead to similar such developments.\n\nIn type A, the combinatorics of Schur functions can realized through \\emph{plactic skew Schur functions}, which plactically encode both jeu de taquin rectification and RSK insertion. This realization yields an algebraic perspective on these combinatorial theories, as well as on the Littlewood--Richardson coefficients. Serrano \\cite[p.~379]{Serrano} introduced a notion of \\emph{plactic skew Schur $P$-functions}, which he conjectured \\cite[Conjecture~2.12]{Serrano} (see also, \\cite[Conjecture~7.8]{Cho}) would play an analogous role in other classical types. If true, this would have given an algebraic substitute for the missing jeu de taquin associated to mixed insertion and to the shifted plactic monoid. Unfortunately, Serrano's conjecture was not correct. Precisely, Cho \\cite[\\textsection 7]{Cho} gave explicit examples to show that Serrano's plactic skew Schur $P$-functions do not lie in the desired ring and hence they cannot serve the desired role in substituting for jeu de taquin or in shedding algebraic light on structure coefficients. Having disproved Serrano's conjecture, Cho asks in \\cite[Open Problem~7.12(1)]{Cho} for a new definition that would accomplish Serrano's goals. More precisely, she sets the following task:\n\nThe \\emph{Schur functions} $s_\\lambda$ are a basis of the ring of symmetric functions and their structure coefficients $c_{\\lambda,\\mu}^\\nu$, appearing in the expansions\n\\[\ns_\\lambda \\cdot s_\\mu = \\sum_\\nu c_{\\lambda,\\mu}^\\nu s_\\nu,\n\\]\nare the \\emph{Littlewood--Richardson coefficients}. These objects play important roles in the linear representation theory of symmetric groups (where $s_\\lambda$ represents a \\emph{Specht module}), in the linear representation theory of general linear groups (where $s_\\lambda$ represents a \\emph{Schur module}), and in the Schubert calculus of Grassmannians (where $s_\\lambda$ represents a \\emph{Schubert cycle}). The Littlewood--Richardson coefficients may be studied combinatorially through the interlocking machinery of \\emph{jeu de taquin} on \\emph{Young tableaux}, the \\emph{Robinson--Schensted--Knuth} (RSK) insertion algorithm, and the \\emph{plactic monoid}. For exposition of these now-standard ideas, see, for example, the textbooks \\cite{Fulton:YT,Manivel}.\n\nIn this setting, \\emph{Sagan--Worley} \\cite{Sagan,Worley} insertion is an analogue of RSK that is closely connected to a \\emph{Sagan--Worley} jeu de taquin. This Sagan--Worley theory does not appear to have a corresponding analogue of the plactic monoid. On the other hand, \\emph{mixed} insertion \\cite{Haiman} is a different analogue of RSK insertion with a corresponding \\emph{shifted plactic monoid} \\cite{Serrano}. This latter theory has not been known to have a corresponding jeu de taquin. We provide\\footnote{The results of this paper were first announced in the extended abstract \\cite{EstupinanSalamanca.Pechenik:FPSAC}.} such a jeu de taquin in \\cref{sec:mixedJeudeTaquin}. Our new jeu de taquin computes mixed insertions, despite lacking some of the properties of ordinary jeu de taquin, such as confluence. There is precedent for useful theories of jeu de taquin without confluence, such as the $K$-theoretic jeu de taquin theories of \\cite{Thomas.Yong:K, Pechenik.Yong} and we hope that our mixed jeu de taquin will lead to similar such developments.\n\n\\medskip\n{\\bf This paper is structured as follows.} In \\cref{sec:background}, we recall background and give necessary new definitions. Our treatment differs notationally from standard approaches in ways that we believe are clarifying. In \\cref{sec:mixedJeudeTaquin}, we introduce mixed jeu de taquin and prove various properties of it. In particular, we show in \\cref{thm:mixedRectification} that it computes Haiman's mixed insertions. \\cref{sec:Cho} gives a new definition of plactic skew Schur $P$-functions $\\sPlacticSchur_{\\theta/\\eta}$ and shows that these give a solution to \\cref{problem:Cho}.\n\nA \\newword{hook subword} of a word $w$ in alphabet $\\cN$ is a subword $d\\cdot i$ such that $d$ is strictly decreasing and $i$ is weakly increasing. If $w$ is a hook subword of itself, we say $w$ is a \\newword{hook word}. For example, $4211688$ is a hook word.\n Let $\\lambda$ be a strict partition. Denote by $\\hookword(\\lambda)$ the set of words $w=w_1\\ldots w_k$, where each $w_i$ is a hook word of length $\\lambda_{\\ell(\\lambda)-i+1}$ and such that, for all $i>1$, $w_i$ is a longest hook subword of $w_{i-1}w_i$. \n Then the \\newword{shifted free Schur function} of shape $\\lambda$ is the formal power series in noncommuting variables\n\\[\\sFreeSchur_\\lambda \\coloneqq \\sum_{ w\\: : \\: w\\in \\hookword(\\lambda) } w \\in \\Z\\npowerser{x_1, x_2, \\dots}.\\]\n(Shifted free Schur functions were first introduced by Serrano \\cite{Serrano} using a definition based on \\emph{mixed reading words} for certain tableaux; our definition, first stated in \\cite{EstupinanSalamanca.Pechenik}, is different from his, but is equivalent.) Note that shifted free Schur functions are not being defined in the skew setting; this fact is the crux of Cho's open problem.\n\nThe \\newword{shifted plactic monoid} \\cite{Serrano} is the quotient $\\sPlactic$ of the free monoid on $\\cN$ by the eight families of relations\n \\begin{align*}\n &abdc\\sim adbc \\quad \\text{for all } a\\leq b \\leq c<d; \\\\\n &acdb\\sim acbd \\quad \\text{for all } a\\leq b < c \\leq d; \\\\ \n &dacb\\sim adcb \\quad \\text{for all } a\\leq b < c <d; \\\\ \n &badc\\sim bdac \\quad \\text{for all } a< b \\leq c<d; \\\\ \n &cbda\\sim cdba \\quad \\text{for all } a< b <c \\leq d; \\\\ \n &dbca\\sim bdca \\quad \\text{for all } a< b \\leq c<d; \\\\ \n &bcda\\sim bcad \\quad \\text{for all } a< b \\leq c\\leq d; \\\\ \n &cadb\\sim cdab \\quad \\text{for all } a\\leq b < c\\leq d. \n \\end{align*}\nThe \\newword{shifted plactic Schur function} $\\sPlacticSchur_\\lambda$ is the image of the shifted free Schur function $\\sFreeSchur_\\lambda$ under the projection map identifying monomials whose sequences of variable subscripts are equivalent in the shifted plactic monoid. Then $\\sPlacticSchur_\\lambda \\in \\Z \\llbracket \\sPlactic \\rrbracket$, the quotient of $\\Z\\npowerser{x_1, x_2, \\dots}$ by the shifted plactic relations on variables. By further projecting to the ring in commuting variables, we recover the ordinary Schur $P$-functions.\n\n\\begin{definition}\\label{def:skewplacticS}\nThe \\newword{skew plactic Schur $P$-function} is\n \\[ \\sPlacticSchur_{\\nu/\\mu}\\coloneqq \\frac{1}{2^{\\diag(\\nu/\\mu)}} \\sum_{T\\in \\shSSYTprime(\\nu/\\mu)} [F\\circ \\rectify(T)]_\\sPlactic \\in \\Z \\llbracket \\sPlactic \\rrbracket, \\]\nwhere $F$ is the map that raises all diagonal positions, $\\rectify(\\cdot)$ signifies modified Sagan--Worley rectification, and $\\diag(\\nu/\\mu)$ denotes the number of diagonal positions in the skew shape $\\nu/\\mu.$ (Note that $\\diag(\\nu / \\mu) = \\ell(\\nu) - \\ell(\\mu)$.)\n\\end{definition}\n\nWe need some preliminary results clarifying Sagan--Worley's rectification on skew semistandard tableaux which may contain low entries on the diagonal. While it is well known that rectifying shifted standard skew tableaux to a shifted standard target yields $b_{\\lambda,\\mu}^\\nu$ tableaux regardless of the target chosen, the analogous result is not as well explained in the literature for the more general case. Essentially, \\cref{thm:generalizedSaganWorley} and \\cref{lem:WorleyRectCondition} can be found in \\cite{Sagan} and \\cite{Worley}, respectively; but the proof in \\cite{Worley} is difficult to retrieve and it seems that \\cref{lem:WorleyRectCondition} is not used for our same purposes in \\cite{Sagan}. These results are also stated in \\cite[Proposition~7.3]{Cho} without proof. For the sake of completeness, we include proofs of those results below and use them to show that our definition of shifted plactic skew Schur $P$-functions satisfies Cho's requirements.\n\nThe following shows that our definition solves Cho's Open Problem:\n\\begin{thm}\n Let $\\mu<\\nu$ be strict partitions. Then $\\sPlacticSchur_{\\nu/\\mu} \\in \\mathbb{Q}[\\sPlacticSchur_\\lambda]_\\lambda$ and\n \\[\\sPlacticSchur_{\\nu/\\mu} = \\sum_{\\lambda} \\frac{2^{\\ell(\\lambda)}}{2^{\\diag(\\nu/\\mu)}} b_{\\lambda,\\mu}^\\nu \\sPlacticSchur_\\lambda. \\]\n In particular, the expansion coefficients of $\\sPlacticSchur_{\\nu/\\mu}$ in the plactic Schur $P$-basis are equal to the expansion coefficients of the ordinary skew Schur $P$-function $P_{\\nu/\\mu}$ in the ordinary Schur $P$-basis. \n\\end{thm}\n\\begin{proof}\n By \\cref{def:skewplacticS}, we need to prove that for any semistandard tableau $S \\in \\shSSYTprime(\\lambda)$, the number of skew tableaux $T \\in \\shSSYTprime(\\nu/\\mu)$ with $[F\\circ \\rectify(T)]_\\sPlactic = [F(S)]_\\sPlactic$ is equal to $2^{\\ell(\\lambda)}b_{\\lambda,\\mu}^\\nu$. Noting that $b_{\\lambda,\\mu}^\\nu$ is the number of skew tableaux $T \\in \\shSSYTprime(\\nu/\\mu)$ rectifying to any semistandard tableau $U \\in \\shSSYTprime(\\lambda)$ by \\cref{thm:generalizedSaganWorley}, it suffices to establish that there are $2^{\\ell(\\lambda)}$ semistandard tableaux $\\hat{S} \\in \\shSSYTprime(\\lambda)$ with $F(\\hat{S})=F(S).$", "post_theorem_intro_text_len": 934, "post_theorem_intro_text": "We solve this problem. Interestingly, the key tool of our solution is the Sagan--Worley jeu de taquin theory. By combining this jeu de taquin with shifted plactic theory, we obtain plactic skew Schur $P$-functions that lie in the correct ring and that govern the expansion multiplicities of ordinary Schur $P$-functions, as desired.\n\n\\medskip\n{\\bf This paper is structured as follows.} In \\cref{sec:background}, we recall background and give necessary new definitions. Our treatment differs notationally from standard approaches in ways that we believe are clarifying. In \\cref{sec:mixedJeudeTaquin}, we introduce mixed jeu de taquin and prove various properties of it. In particular, we show in \\cref{thm:mixedRectification} that it computes Haiman's mixed insertions. \\cref{sec:Cho} gives a new definition of plactic skew Schur $P$-functions $\\sPlacticSchur_{\\theta/\\eta}$ and shows that these give a solution to \\cref{problem:Cho}.", "sketch": "To solve \\cref{problem:Cho}, the paper uses “the Sagan--Worley jeu de taquin theory” as the “key tool.” “By combining this jeu de taquin with shifted plactic theory,” it obtains plactic skew Schur $P$-functions that “lie in the correct ring” and “govern the expansion multiplicities of ordinary Schur $P$-functions, as desired.” The structure indicated is: background/definitions (\\cref{sec:background}); introduction of mixed jeu de taquin and its properties (\\cref{sec:mixedJeudeTaquin}), including showing it computes Haiman’s mixed insertions via \\cref{thm:mixedRectification}; then \\cref{sec:Cho} “gives a new definition” of $\\sPlacticSchur_{\\theta/\\eta}$ and shows these solve \\cref{problem:Cho}.", "expanded_sketch": "To solve the main problem, the paper uses “the Sagan--Worley jeu de taquin theory” as the “key tool.” “By combining this jeu de taquin with shifted plactic theory,” it obtains plactic skew Schur $P$-functions that “lie in the correct ring” and “govern the expansion multiplicities of ordinary Schur $P$-functions, as desired.” The structure indicated is: background/definitions (\\cref{sec:background}); introduction of mixed jeu de taquin and its properties (\\cref{sec:mixedJeudeTaquin}), including showing it computes Haiman’s mixed insertions via the following theorem. \n\n\\begin{theorem}\\label{thm:mixedRectification}\n        Let $T$ be an arbitrary shifted Young tableau and $\\unpr y$ a high letter. Then performing the mixed insertion of $\\unpr y$ on $T$ results in the tableau \n        $\\mathrm{rect}_\\mix \\left( T\\oplus \\unpr y \\right)$.\n    \\end{theorem}\n\nThen \\cref{sec:Cho} “gives a new definition” of $\\sPlacticSchur_{\\theta/\\eta}$ and shows these establish the main problem.", "expanded_theorem": "[Cho]\n    For $\\eta  \\leqslant  \\theta$, give a new definition of the plactic skew Schur $P$-function $\\sPlacticSchur_{\\theta/\\eta}$\n so that it belongs to the ring generated by plactic Schur $P$-functions and it describes\nthe multiplicities of the ordinary $P_\\lambda$ in the expansion of $P_{\\theta/\\eta}$ in a nice way.", "theorem_type": ["Algorithmic or Constructive", "Existence"], "mcq": {"question": "Let \\(\\eta \\leq \\theta\\) be strict partitions, so that \\(\\theta/\\eta\\) is a shifted skew shape. Here \\(P_{\\theta/\\eta}\\) denotes the ordinary skew Schur \\(P\\)-function, and the plactic Schur \\(P\\)-functions are the generators of the corresponding plactic ring. Which statement gives the required property of a new definition of the plactic skew Schur \\(P\\)-function \\(\\sPlacticSchur_{\\theta/\\eta}\\)?", "correct_choice": {"label": "A", "text": "One seeks a definition of \\(\\sPlacticSchur_{\\theta/\\eta}\\) such that \\(\\sPlacticSchur_{\\theta/\\eta}\\) lies in the ring generated by the plactic Schur \\(P\\)-functions and, moreover, it describes in a natural way the multiplicities of the ordinary \\(P_\\lambda\\) appearing in the expansion of \\(P_{\\theta/\\eta}\\)."}, "choices": [{"label": "B", "text": "One seeks a definition of \\(\\sPlacticSchur_{\\theta/\\eta}\\) such that \\(\\sPlacticSchur_{\\theta/\\eta}\\) itself is a single plactic Schur \\(P\\)-function, and, moreover, its coefficients directly equal the Littlewood--Richardson-type structure constants for the product of plactic Schur \\(P\\)-functions."}, {"label": "C", "text": "One seeks a definition of \\(\\sPlacticSchur_{\\theta/\\eta}\\) such that \\(\\sPlacticSchur_{\\theta/\\eta}\\) lies in the ring generated by the plactic Schur \\(P\\)-functions."}, {"label": "D", "text": "One seeks a definition of \\(\\sPlacticSchur_{\\theta/\\eta}\\) such that \\(\\sPlacticSchur_{\\theta/\\eta}\\) lies in the ring generated by the plactic Schur \\(P\\)-functions and, moreover, it computes in a natural way the multiplicities of the ordinary \\(P_\\lambda\\) appearing in the product expansion of \\(P_\\theta P_\\eta\\)."}, {"label": "E", "text": "One seeks a definition of \\(\\sPlacticSchur_{\\theta/\\eta}\\) such that, for every pair of strict partitions \\(\\eta \\leq \\theta\\), there is a canonical expression of \\(\\sPlacticSchur_{\\theta/\\eta}\\) as a polynomial with nonnegative integer coefficients in the plactic Schur \\(P\\)-functions, and these coefficients coincide with the multiplicities of the ordinary \\(P_\\lambda\\) in \\(P_{\\theta/\\eta}\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "membership-in-generated-ring vs single-basis-element / expansion-multiplicity role", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the requirement about describing multiplicities of ordinary \\(P_\\lambda\\) in \\(P_{\\theta/\\eta}\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "skew-expansion multiplicities replaced by product-expansion multiplicities", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "unjustified strengthening to canonical polynomial with nonnegative integer coefficients", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives background terminology but does not state the target property itself. The correct answer is not explicitly revealed, and the alternatives are close enough that the answer is not leaked trivially."}, "TAS": {"score": 0, "justification": "The item mainly asks the test-taker to identify the precise stated property of the new definition, which is essentially theorem/definition recall rather than a nontrivial mathematical conclusion derived from premises."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to separate the exact claim from weaker, stronger, or confused variants (especially C, D, and E), but the task is primarily recognition of the intended statement rather than genuine generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and well-targeted: one is too weak, one confuses skew-expansion with product-expansion, one overstrengthens the claim, and one changes the object entirely. These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A reasonably well-constructed recall/discrimination MCQ with strong distractors and little answer leakage, but it is largely a statement-identification item rather than a genuinely generative reasoning question."}}
{"id": "2602.18656v1", "paper_link": "http://arxiv.org/abs/2602.18656v1", "theorems_cnt": 1, "theorem": {"env_name": "example", "content": "\\label{BernoulliExample}\nLet $X = (X_1, X_2, ..., X_5)$ be iid Bernoulli($\\theta$) random variables with joint probability mass function $p_\\theta(x) = \\theta^{T(\\mathbf{x})}(1-\\theta)^{5-T(\\mathbf{x})}$ where $T(x) = \\sum_ix_i$, and consider testing null hypothesis $H_0:\\theta = 0.5$ against alternative hypothesis $H_1:\\theta = 0.8$.", "start_pos": 4175, "end_pos": 4540, "label": "BernoulliExample"}, "ref_dict": {"BernoulliTestFunction": "\\begin{equation}\\label{BernoulliTestFunction}\n\\phi(x) = \\left\\{\\begin{array}{rcl} 1 & \\mbox{if} &T(x) > 4 \\\\\n 0.44 & \\mbox{if} & T(x) = 4  \\\\\n 0 & \\mbox{if}& T(x) < 4.\n \\end{array}\\right.\n\\end{equation}", "BernoulliExample": "\\begin{example}\\label{BernoulliExample}\nLet $X = (X_1, X_2, ..., X_5)$ be iid Bernoulli($\\theta$) random variables with joint probability mass function $p_\\theta(x) = \\theta^{T(\\mathbf{x})}(1-\\theta)^{5-T(\\mathbf{x})}$ where $T(x) = \\sum_ix_i$, and consider testing null hypothesis $H_0:\\theta = 0.5$ against alternative hypothesis $H_1:\\theta = 0.8$.\n\\end{example}"}, "pre_theorem_intro_text_len": 1526, "pre_theorem_intro_text": "In statistical hypothesis testing the objective is to decide if a null hypothesis $H_0$, which specifies a distribution or statistical model for random data $X$, should be retained or rejected in favor of alternative hypothesis $H_1$ based on realization $X = x$. The decision can be denoted by $\\delta(x)\\in\\{0,1\\}$, where $\\delta(x) =$ 0 (1) means that $H_0$ is retained (rejected).  The Neyman-Pearson (NP) paradigm stipulates that $\\delta$ should  maximize power subject to the constraint that the type 1 error rate is not more than some prespecified level $\\alpha$. \nFormally, $\\delta$ should maximize $E_1[\\delta(X)]$ subject to the constraint that $E_0[\\delta(X)]\\leq \\alpha$, where $E_j$ denotes an expectation taken under $H_j$.   For example, in the simple versus simple setting  $H_0:X\\sim P_0$ is tested against $H_1:X\\sim P_1$, and the most powerful level $\\alpha$ decision rule is $\\delta(x) = I(\\lambda(x)\\geq c)$ where $I(\\cdot)$ is the indicator function, $\\lambda(x) = \\frac{p_1(x)}{p_0(x)}$ is the likelihood ratio statistic, $p_j$ is the probability mass function for $P_j$ under $H_j$, and $c$ is chosen so that $E_0[\\delta(X)] = \\alpha$ \\citep{Neyman1933}. The corresponding $p$-value can be computed via $p(x) = \\Pr_0\\{\\lambda(X)\\geq \\lambda(x)\\}$.  \n\nHowever, if $X$ has a discrete distribution, it is not generally possible to define $\\delta(X)$ such that $E_0[\\delta(X)]=\\alpha$, and several different types of $p$-values can be considered.  Let us consider a simple example to facilitate discussion.", "context": "In statistical hypothesis testing the objective is to decide if a null hypothesis $H_0$, which specifies a distribution or statistical model for random data $X$, should be retained or rejected in favor of alternative hypothesis $H_1$ based on realization $X = x$. The decision can be denoted by $\\delta(x)\\in\\{0,1\\}$, where $\\delta(x) =$ 0 (1) means that $H_0$ is retained (rejected). The Neyman-Pearson (NP) paradigm stipulates that $\\delta$ should maximize power subject to the constraint that the type 1 error rate is not more than some prespecified level $\\alpha$. \nFormally, $\\delta$ should maximize $E_1[\\delta(X)]$ subject to the constraint that $E_0[\\delta(X)]\\leq \\alpha$, where $E_j$ denotes an expectation taken under $H_j$. For example, in the simple versus simple setting $H_0:X\\sim P_0$ is tested against $H_1:X\\sim P_1$, and the most powerful level $\\alpha$ decision rule is $\\delta(x) = I(\\lambda(x)\\geq c)$ where $I(\\cdot)$ is the indicator function, $\\lambda(x) = \\frac{p_1(x)}{p_0(x)}$ is the likelihood ratio statistic, $p_j$ is the probability mass function for $P_j$ under $H_j$, and $c$ is chosen so that $E_0[\\delta(X)] = \\alpha$ \\citep{Neyman1933}. The corresponding $p$-value can be computed via $p(x) = \\Pr_0\\{\\lambda(X)\\geq \\lambda(x)\\}$.\n\nHowever, if $X$ has a discrete distribution, it is not generally possible to define $\\delta(X)$ such that $E_0[\\delta(X)]=\\alpha$, and several different types of $p$-values can be considered. Let us consider a simple example to facilitate discussion.", "full_context": "In statistical hypothesis testing the objective is to decide if a null hypothesis $H_0$, which specifies a distribution or statistical model for random data $X$, should be retained or rejected in favor of alternative hypothesis $H_1$ based on realization $X = x$. The decision can be denoted by $\\delta(x)\\in\\{0,1\\}$, where $\\delta(x) =$ 0 (1) means that $H_0$ is retained (rejected). The Neyman-Pearson (NP) paradigm stipulates that $\\delta$ should maximize power subject to the constraint that the type 1 error rate is not more than some prespecified level $\\alpha$. \nFormally, $\\delta$ should maximize $E_1[\\delta(X)]$ subject to the constraint that $E_0[\\delta(X)]\\leq \\alpha$, where $E_j$ denotes an expectation taken under $H_j$. For example, in the simple versus simple setting $H_0:X\\sim P_0$ is tested against $H_1:X\\sim P_1$, and the most powerful level $\\alpha$ decision rule is $\\delta(x) = I(\\lambda(x)\\geq c)$ where $I(\\cdot)$ is the indicator function, $\\lambda(x) = \\frac{p_1(x)}{p_0(x)}$ is the likelihood ratio statistic, $p_j$ is the probability mass function for $P_j$ under $H_j$, and $c$ is chosen so that $E_0[\\delta(X)] = \\alpha$ \\citep{Neyman1933}. The corresponding $p$-value can be computed via $p(x) = \\Pr_0\\{\\lambda(X)\\geq \\lambda(x)\\}$.\n\nHowever, if $X$ has a discrete distribution, it is not generally possible to define $\\delta(X)$ such that $E_0[\\delta(X)]=\\alpha$, and several different types of $p$-values can be considered. Let us consider a simple example to facilitate discussion.\n\nLet us pause to consider a concrete example. Table \\ref{ExampleTable} provides an illustration of MD natural $p$-values that agree with the natural $p$-values from the usual LRT in Example \\ref{BernoulliTestFunction}. \n\\begin{table}[ht]\n\\caption{\\label{ExampleTable} The first 5 columns list an $x \\in \\mathcal{X}$ in Example \\ref{BernoulliExample}. The following columns contain $p_0(x)$, $p_1(x)$, likelihood ratio $\\lambda(x)$, a ranking $R(x)$ that agrees with $\\lambda(x)$, and natural MD $p$-values based on $R(x)$ computed $P^{MD-\\lambda}(x,1) = \\Pr_0\\{R(X)\\leq R(x)\\}$ and natural $p$-value based on the likelihood ratio statistic $P^\\lambda(x,1) = \\Pr_0\\{\\lambda(X)\\geq \\lambda(x)\\}$}\n\\begin{tabular}{cccccccccll}\n$x_1$&$x_2$&$x_3$&$x_4$&$x_5$&$p_0(x)$ & $p_1(x)$ & $\\lambda(x)$ & $R(x)$ & $P^{MD}(x,1)$& $P^\\lambda(x,1)$\\\\ \\hline\n1 & 1 & 1& 1& 1 & 0.03125 & 0.32768 & 10.4857 & 1 & 0.03125 & 0.03125\\\\\n0 & 1 & 1& 1& 1 & 0.03125 & 0.08192 & 2.62144 & 2 & 0.06250 & 0.1875 \\\\\n1 & 0 & 1& 1& 1 & 0.03125 & 0.08192 & 2.62144 & 3 & 0.09375& 0.1875\\\\\n1 & 1 & 0& 1& 1 & 0.03125 & 0.08192 & 2.62144 & 4 & 0.125& 0.1875\\\\\n1 & 1 & 1& 0& 1 & 0.03125 & 0.08192 & 2.62144 & 5 & 0.15625& 0.1875 \\\\\n1 & 1 & 1& 1& 0 & 0.03125 & 0.08192 & 2.62144 & 6 & 0.1875& 0.1875\\\\\n0 & 0 & 1& 1& 1 & 0.03125 & 0.02048 & 0.65536 & 7 & 0.21875&0.5\\\\\n1 & 0 & 0& 1& 1 & 0.03125 & 0.02048 & 0.65536 & 8 & 0.25&0.5\\\\ \n\\end{tabular}\n\\end{table}\nObserve that $R$ agrees with $\\lambda$ since $\\lambda(x)\\geq\\lambda(y)$ implies $R(x)\\leq R(y)$ for each $x$,$y$ listed. We further observe that $R$ is one-to-one since each $x$ is assigned a unique ranking for the support points depicted. Hence, the resulting $p$-value $P^{MD-\\lambda}(x,1)$ is minimally discrete. Less technically, the terminology ``minimal'' emphasizes that among all possible other natural $p$-values based on the likelihood ratio test, this $p$-value has as many support points as possible (32 in this example), and hence minimizes discreteness effect. The p-value based on the LRT is not minimal since it only has 6 support points when 32 are possible.\n\nNext let us compare MR-$T$ $p$-values to randomized $p$-values based on $T$ that need not be MR. Claim (C5) stipulates that both $p$-values are uniformly distributed under the null hypothesis. Claim (C6) states that if $T$ is a sufficient statistic, then both $p$-value statistics are equal in distribution whether the null hypothesis is true or false. This is perhaps not surprising given that most powerful size $\\alpha$ decision rules are not unique in discrete testing (recall the Introduction) and both tests are size $\\alpha$ tests based on sufficient statistics. However, the MR $p$-value dominates its non-MR counterpart in that the generation of $u$ has ``minimal impact'' on $p(x,u)$. In particular, Claim (C7) states that the variance attributable to the auxiliary $U$ is smaller for the MR-$T$ $p$-value if $T$ is sufficient. Consequently, variation from replication to replication that is attributable to generation of $U$ is minimized. In this sense, we may say that the replicability of $p^{MR-T}(X,U)$ is greater. \n\\begin{theorem}[MR-$T$ versus $T$: Randomized p-values and decision functions] \\label{Thm2}\nLet decision functions and p-values be defined as Definitions \\ref{T} and \\ref{MDT} with $U$ being uniformly distributed over the unit interval and independent of $X$. Then claim (C5) below is true. If additionally $T$ is a sufficient statistic, then claim (C6) and (C7) are true: \n\\begin{enumerate}\n\\item[(C5)] $E_0[\\delta_\\alpha^{MR-T}(X,U)] = E_0[\\delta_\\alpha^{T}(X,U)] = \\alpha$ $\\forall \\alpha\\in[0,1]$ and hence \\\\\n$Pr_0\\{\\Pr(P^{MR-T}(X,U)\\leq t\\} = \\Pr_0\\{P^{T}(X,U)\\leq t\\} = t$ $\\forall t \\in [0,1]$,\\\\\n\\item[(C6)] $E_\\theta[\\delta_\\alpha^{MR-T}(X,U)] = E_\\theta[\\delta_\\alpha^T(X,U)]$ $\\forall \\alpha\\in[0,1],\\theta\\in\\Theta$ \\\\\nand hence $Pr_\\theta\\{\\Pr(P^{MR-T}(X,U)\\leq t\\} = \\Pr_\\theta\\{P^{T}(X,U)\\leq t\\}$ $\\forall t\\in[0,1], \\theta\\in\\Theta$, \\\\\n\\item[(C7)] $Var_\\theta(P^{MD-T}(X,U)|X)\\leq Var_\\theta(P^{T}(X,U)|X)$ $\\forall \\theta\\in\\Theta$.\n\\end{enumerate}\n\\end{theorem}\n\nA brief discussion regarding the potential impact of results here is warranted. First consider multiple hypothesis testing. A multiple hypothesis testing (MTP) procedure tests null hypotheses $H_{01},H_{02},..., H_{0M}$ with $p$-values $P_1, P_2, ..., P_M$, which are often assumed to be uniformly distributed under the null hypotheses. Formally, they are defined $\\delta_i = I(P_i\\leq t)$ for $i = 1, 2, ..., M$ and some $t\\in[0,1]$. For example, a Bonferroni procedure chooses $t = \\alpha/M$ while the \\cite{Benjamini1995} procedure for False Discovery Rate control chooses $\\hat{t} = \\sup\\{s:\\frac{s}{\\max\\{\\sum_i I(P_i\\leq s),1\\}}\\leq \\alpha\\}$ (with the supremum of the empty set defined to be 0), which can be shown to converge to some constant $t$ as $M\\rightarrow \\infty$ with probability 1 under weak dependence (cf. \\cite{Genovese2002,Storey2004, Habiger2015}). While procedures are often studied under the assumption that $\\Pr_0(P_i\\leq t) = t$, this assumption can be relaxed to $\\Pr_0(P_i\\leq t)\\geq t$. Claim (C4) in Theorem \\ref{Thm1} ensures that, if opting for natural $p$-values, the MD-natural $p$-values are still valid since they satisfy the later relaxed assumption. Claims (C1) and (C3) give that they result in more power. If opting for randomized $p$-values then both MR $p$-values and their non-minimal counterparts are uniformly distributed under the null hypotheses. If randomized $p$-values are functions of sufficient statistics, then we should not anticipate any more or less power if opting for their MR counterparts (claim (C6)), as the Neyman Pearson Lemma suggests. However, claim (C7) stipulates that fewer discoveries will rely on the generation of $u$ if utilizing the MR $p$-values. To see this, consider Example \\ref{BernoulliTestFunction} and consider implementing a level $\\alpha = 0.05$ test. In this example, the decision to reject or retain $H_0$ will depend upon $U$ with probability $\\Pr\\{T(X) = 4\\}$, which is 5 times more likely than its MR counterpart that only utilizes $u$ for one $x\\in [T(x)=4]$.\n\n\\section{False Discovery Rate Methods and MD $p$-values}\nNext consider testing null hypotheses $H_m:\\theta_m = 0$ with data $X_m\\in\\mathcal{X}_m$, where $\\mathcal{X}_m$ is the countable support of $X_m$, for $m = 1, 2, ..., M$. A size $\\alpha$ test function and its corresponding decision function and $p$-value are defined as in Section 2 but indexed by $m$. Specifically $\\phi_{\\alpha,m}(x_m)$ is nondecreasing and right continuous in $\\alpha$ and satisfies $E_{0}[\\phi_{\\alpha,m}(X_m)] = \\alpha$, where $E_0$ denotes an expectation taken over $X_m$ under $H:\\theta_m = 0$. Likewise, $\\delta_{\\alpha,m}(x_m,u_m) = I(u_m\\leq \\phi_{\\alpha,m}(x_m))$ and $P_m(x_m, u_m) = \\inf\\{\\alpha:\\delta_{\\alpha,m}(x_m,u_m) = 1\\}$. We shall write $P_m = P_m(x_m,u_m)$ for short whenever possible in this section and let $P$ denote a collection of $p$-values.", "post_theorem_intro_text_len": 6694, "post_theorem_intro_text": "\\noindent Observe that $\\lambda(x) = p_{1}(x)/p_{0}(x)$ is increasing in $T(x)$ so the likelihood ratio test rejects $H_0$ for large values of $T(x)$. However, observe $T(x)$ takes values in $\\{0,1,...,5\\}$ and that $\\Pr_0\\{T(X) \\geq 5\\} = 0.03125$ but $\\Pr_0\\{T(X)\\geq 4\\} = 0.1875$.  Hence, it is not possible to define a $\\delta$, i.e. choose $c$, such that $E_0[\\delta(X)] = \\Pr_0\\{T(X)\\geq c\\} = \\alpha$ if, for example, $\\alpha = 0.1$.  Randomized testing \\citep{Pearson1950, Tocher1950} would be necessary. In Example \\ref{BernoulliExample}, a most powerful level $0.1$ test function is \n\\begin{equation}\\label{BernoulliTestFunction}\n\\phi(x) = \\left\\{\\begin{array}{rcl} 1 & \\mbox{if} &T(x) > 4 \\\\\n 0.44 & \\mbox{if} & T(x) = 4  \\\\\n 0 & \\mbox{if}& T(x) < 4.\n \\end{array}\\right.\n\\end{equation}\nwhere $\\phi(x)$ represents the rejection probability. See, for example, \\cite{Lehmann1993}. The randomized test, when implemented, can be written as a randomized decision rule $\\delta(x,u) = I(u\\leq \\phi(x))$ where $u$ is a realization from a standard uniform distribution. The corresponding randomized $p$-value can be computed $p(x,u) =  \\Pr_{0}\\{T(X)>T(x)\\}+ u\\Pr_{0}\\{T(X) = T(x)\\}$.  See, for example, \\cite{Pena2011, Habiger2011}.   In practice, we may also report the corresponding natural $p$-value: $p(x,1)=\\Pr_{0}\\{T(X)\\geq T(x)\\}$ (cf. \\cite{Casella2002}), mid-p-value \\citep{Lan61}: $p(x,1/2) = \\Pr_{0}\\{T(X)>T(x)\\}+ 1/2\\Pr_{0}\\{T(X) = T(x)\\}$, or even the fuzzy $p$-value \\citep{GeyMee05}, which in this example amounts to reporting the interval $[p(x,0), p(x,1)]$.  The curious reader is referred to \\cite{Agr07, Wells2010} for comparison of approaches in traditional single null hypothesis testing settings. More recent research studies the impact of the employing mid-p-values, natural p-values, or randomized in downstream methods. \n\nFirst let us consider randomized $p$-value literature. \\cite{Habiger2011, Dickhaus2013} provided conditions that ensure randomized $p$-values are uniformly distributed under the null hypotheses so that the \\cite{Benjamini1995} multiple hypothesis testing procedure, and other adaptive procedures (eg. \\cite{Storey2004}) for false discovery rate (FDR) control are valid.  \\cite{Ochieng2024} combined randomization methods in \\cite{Hoang2022,Hoang2022b}  with aforementioned randomization methods to develop improved 2 step procedures for multiple testing of composite null hypotheses with discrete data. \\cite{Habiger2015, Dai2019} demonstrated that randomized p-values can be used to develop more powerful multiple testing procedures even if (adjusted) randomized $p$-values are not ultimately reported, say as in the fuzzy BH procedure in \\cite{Kulinskaya2009}. Of course, randomized $p$-values may be more variable than their non-randomized counterparts, which can be manifested as decreasing replication probability. That is, the probability of realizing the same randomized p-value in a replicated experiment is 0 if $u$ is regenerated. \n\nOther research is aimed at establishing conservative (or less conservative) behavior for mid and natural p-values.  For example, \\cite{Doehler2018, Chen2020, Chen2020b} provide upper bounds for the FDR when the BH procedure is applied to mid-p-values.   In meta analysis, \\cite{Rubin-Delanchy2018} provide bounds for the distribution of Fisher's test statistic $T= -2 \\sum_i \\log(P_i)$ under the null hypothesis when $P_i$ represents a mid-p-value. The idea builds upon \\cite{Hwang2001}, where it was shown that the mid-p-value is stochastically less than or equal to a uniform distribution in the convex order \\citep{Shaked2007} under the null hypothesis.  \\cite{Wang2024} also used stochastic ordering to calibrate mid-p-values into valid p-values, valid E-values \\citep{Vovk2021} and Bayes Factors. Of course, established inequalities need not be sharp. \n\nIdeally inequalities for downstream methods that utilize mid-p-values or natural p-values would be as sharp as possible and, if opting for randomized $p$-values, generated uniform variates would add minimal variation.  This motivates our minimality principle, which ensures that the aforementioned detrimental impacts when dealing with the ``discreteness'' of the data are minimized.  Specifically, we develop \\textit{minimally discrete} (MD) mid-p-values,  MD natural $p$-values, and \\textit{minimally randomized} (MR) p-values that dominate their traditional mid-$p$-value, natural $p$-value and randomized $p$-value counterparts, respectively. Specifically, it is shown that inequalities based on the usual stochastic order for natural $p$-values and inequalities based on the convex order for mid-$p$-values are sharpened.  Likewise, variation attributable to independeintly generated auxiliary variates for randomized $p$-values is smaller.  \n\nOur implication that ``less discrete'' and ``less randomized'' p-values are available is bold and therefore warrants a brief discussion. Reconsider Example \\ref{BernoulliExample} and let us compare \\begin{equation}\\label{AnotherMPTest}\n\\phi^*(x) = \\left\\{\\begin{array}{rcl} 1 & \\mbox{if} & x\\in \\{ (1,1,1,1,1), (1,1,1,1,0),  (0,1,1,1,1)\\}  \\\\ \n 0.2 & \\mbox{if} & x=(1,1,0,1,1)  \\\\ \n 0 & & \\textrm{otherwise}\n \\end{array}\\right.\n\\end{equation}\nto the most powerful test $\\phi(x)$ in \\eqref{BernoulliTestFunction}. A curious reader may verify that the type 1 error rates are $E_0[\\phi(X)] = E_0[\\phi^*(X)] = 0.1$ and that the powers are $E_1[\\phi(X)] = E_1[\\phi^*(X)] = 0.5079$. Clearly $\\phi^*$ and $\\phi$ are two distinct most powerful level $0.1$ tests. In particular, both tests automatically reject $H_0$ when $T(x)>4$ and automatically retain $H_0$ when $T(x)<4$.  The difference is that $\\phi(x)$ rejects $H_0$ with probability 0.44 for all five $x\\in [T(x)=4]$ while $\\phi^*(x)$ only randomly rejects $H_0$ for one $x\\in [T(x)=4]$.  It automatically rejects $H_0$ if $x = (1,1,1,1,0)$ or $x= (0,1,1,1,1)$ and automatically retains $H_0$ when $x=(1,0,1,1,1)$ or $x = (1,1,1,0,1)$.  The key realization is that a most powerful test function can be defined ``arbitrarily on the set $[\\lambda(x) = k]$'' \\citep{Lehmann1997} as long as the resulting test function has expectation equal to $\\alpha$ under $H_0$.  While mathematical statistics texts (cf. \\cite{Cox1974, Lehmann1997}) acknowledge this point, further development is cited as beyond scope: in testing a single null hypothesis $\\phi$ is sometimes viewed as a mathematical mechanism for illustrating the NP Lemma rather than a practical method.  However, as noted previously, p-values derived from test functions like $\\phi$ are increasingly utilized as inputs in a downstream method.  In this setting, further research is warranted.", "sketch": "In Example \\ref{BernoulliExample}, the argument is that the likelihood ratio \\(\\lambda(x)=p_1(x)/p_0(x)\\) is increasing in \\(T(x)\\), so the likelihood ratio test rejects \\(H_0\\) for large values of \\(T(x)\\). Since \\(T(x)\\in\\{0,1,\\dots,5\\}\\) is discrete, one cannot generally choose a cutoff \\(c\\) such that \\(E_0[\\delta(X)]=\\Pr_0\\{T(X)\\ge c\\}=\\alpha\\) (e.g. \\(\\Pr_0\\{T(X)\\ge5\\}=0.03125\\) but \\(\\Pr_0\\{T(X)\\ge4\\}=0.1875\\), so \\(\\alpha=0.1\\) cannot be matched by a nonrandomized cutoff). Therefore randomized testing is necessary, and a most powerful level \\(0.1\\) test is obtained by rejecting with probability 1 when \\(T(x)>4\\), with probability \\(0.44\\) when \\(T(x)=4\\), and with probability 0 when \\(T(x)<4\\), i.e. \\eqref{BernoulliTestFunction}. The text further notes that the randomized test can be implemented as \\(\\delta(x,u)=I(u\\le \\phi(x))\\) with \\(u\\sim \\mathrm{Unif}(0,1)\\), yielding a randomized \\(p\\)-value \\(p(x,u)=\\Pr_0\\{T(X)>T(x)\\}+u\\Pr_0\\{T(X)=T(x)\\}\\). It is also observed that most powerful tests need not be unique: a different most powerful level \\(0.1\\) test \\(\\phi^*\\) in \\eqref{AnotherMPTest} has the same type I error and power as \\(\\phi\\), illustrating that one may define a most powerful test function \"arbitrarily on the set \\([\\lambda(x)=k]\\)\" as long as its expectation under \\(H_0\\) equals \\(\\alpha\\).", "expanded_sketch": "In the following example.\n\n\\begin{example}\\label{BernoulliExample}\nLet $X = (X_1, X_2, ..., X_5)$ be iid Bernoulli($\\theta$) random variables with joint probability mass function $p_\\theta(x) = \\theta^{T(\\mathbf{x})}(1-\\theta)^{5-T(\\mathbf{x})}$ where $T(x) = \\sum_ix_i$, and consider testing null hypothesis $H_0:\\theta = 0.5$ against alternative hypothesis $H_1:\\theta = 0.8$.\n\\end{example}\n\nthe argument is that the likelihood ratio \\(\\lambda(x)=p_1(x)/p_0(x)\\) is increasing in \\(T(x)\\), so the likelihood ratio test rejects \\(H_0\\) for large values of \\(T(x)\\). Since \\(T(x)\\in\\{0,1,\\dots,5\\}\\) is discrete, one cannot generally choose a cutoff \\(c\\) such that \\(E_0[\\delta(X)]=\\Pr_0\\{T(X)\\ge c\\}=\\alpha\\) (e.g. \\(\\Pr_0\\{T(X)\\ge5\\}=0.03125\\) but \\(\\Pr_0\\{T(X)\\ge4\\}=0.1875\\), so \\(\\alpha=0.1\\) cannot be matched by a nonrandomized cutoff). Therefore randomized testing is necessary, and a most powerful level \\(0.1\\) test is obtained by rejecting with probability 1 when \\(T(x)>4\\), with probability \\(0.44\\) when \\(T(x)=4\\), and with probability 0 when \\(T(x)<4\\), i.e.\n\\begin{equation}\\label{BernoulliTestFunction}\n\\phi(x) = \\left\\{\\begin{array}{rcl} 1 & \\mbox{if} &T(x) > 4 \\\\\n 0.44 & \\mbox{if} & T(x) = 4  \\\\\n 0 & \\mbox{if}& T(x) < 4.\n \\end{array}\\right.\n\\end{equation}\nThe text further notes that the randomized test can be implemented as \\(\\delta(x,u)=I(u\\le \\phi(x))\\) with \\(u\\sim \\mathrm{Unif}(0,1)\\), yielding a randomized \\(p\\)-value \\(p(x,u)=\\Pr_0\\{T(X)>T(x)\\}+u\\Pr_0\\{T(X)=T(x)\\}\\). It is also observed that most powerful tests need not be unique: a different most powerful level \\(0.1\\) test \\(\\phi^*\\) in \\eqref{AnotherMPTest} has the same type I error and power as \\(\\phi\\), illustrating that one may define a most powerful test function \"arbitrarily on the set \\([\\lambda(x)=k]\\)\" as long as its expectation under \\(H_0\\) equals \\(\\alpha\\).", "expanded_theorem": "\\label{BernoulliExample}\nLet $X = (X_1, X_2, ..., X_5)$ be iid Bernoulli($\\theta$) random variables with joint probability mass function $p_\\theta(x) = \\theta^{T(\\mathbf{x})}(1-\\theta)^{5-T(\\mathbf{x})}$ where $T(x) = \\sum_ix_i$, and consider testing null hypothesis $H_0:\\theta = 0.5$ against alternative hypothesis $H_1:\\theta = 0.8$.", "theorem_type": ["Universal"], "mcq": {"question": "Consider a hypothesis-testing setup in which \\(X=(X_1,X_2,\\ldots,X_5)\\) consists of five independent and identically distributed Bernoulli\\((\\theta)\\) random variables. For \\(x=(x_1,\\ldots,x_5)\\), define \\(T(x)=\\sum_{i=1}^5 x_i\\), so the joint probability mass function is \\(p_\\theta(x)=\\theta^{T(x)}(1-\\theta)^{5-T(x)}\\). Which hypothesis test is being considered in this model?", "correct_choice": {"label": "A", "text": "The null hypothesis is \\(H_0:\\theta=0.5\\) and the alternative hypothesis is \\(H_1:\\theta=0.8\\)."}, "choices": [{"label": "B", "text": "The null hypothesis is \\(H_0:\\theta=0.8\\) and the alternative hypothesis is \\(H_1:\\theta=0.5\\)."}, {"label": "C", "text": "The test compares two simple hypotheses for the Bernoulli parameter \\(\\theta\\), with \\(H_0:\\theta=0.5\\) against a fixed alternative \\(H_1:\\theta\\neq 0.5\\)."}, {"label": "D", "text": "The null hypothesis is \\(H_0:\\theta\\le 0.5\\) and the alternative hypothesis is \\(H_1:\\theta>0.5\\)."}, {"label": "E", "text": "The null hypothesis is \\(H_0:\\theta=0.5\\) and the alternative hypothesis is \\(H_1:\\theta\\ge 0.8\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "direction_of_simple_vs_simple_pair", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "fixed_simple_alternative_0.8_replaced_by_composite_alternative", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "simple_null_and_simple_alternative_replaced_by_one_sided_composite_test", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "point_alternative_0.8_replaced_by_range_alternative", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem states only the Bernoulli sampling model and sufficient statistic; it does not explicitly or implicitly reveal the specific null and alternative hypotheses keyed in choice A."}, "TAS": {"score": 2, "justification": "This is not a direct restatement of a theorem or definition. The item asks for identification of a testing setup rather than merely rephrasing a stated conclusion."}, "GPS": {"score": 0, "justification": "There is essentially no valid generative reasoning path from the stem to the keyed answer, because the hypotheses are not determined by the model description alone. The item tests guessing or hidden-context recall rather than mathematical reasoning."}, "DQS": {"score": 1, "justification": "The distractors are mathematically plausible and reflect common confusions (reversing null/alternative, simple vs. composite alternatives, one-sided formulations). However, because the stem does not specify the test, multiple options remain equally plausible, weakening distractor effectiveness."}, "total_score": 5, "overall_assessment": "The question avoids answer leakage and is not tautological, but it is fundamentally underdetermined: the stem gives only the Bernoulli model, not the actual hypotheses. As a result, it does not genuinely test reasoning, and the keyed answer is unsupported by the provided information."}}
{"id": "2602.18719v1", "paper_link": "http://arxiv.org/abs/2602.18719v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\nLet $(D,\\mu)$ be a measure space, and let $a=(a_1,\\hdots,a_m)^\\top$\nand $b= (b_k)_{k\\in \\mathbb I}^\\top$ be families of square-integrable functions on $D$,\nwhere $\\mathbb I$ is at most countable.\nAssume that $I:=\\int_D a(x)a(x)^*d\\mu(x)$ and $J:=\\int_D b(x)b(x)^*d\\mu(x)$ are positive definite, and that $J$\nhas finite trace and effective dimension\n\\[\n\\Tr(J) \\,=\\, \\sum_{k\\in \\mathbb I} \\|b_k\\|_{L_2(D,\\mu)}^2 \\,<\\,\\infty\n\\quad\\text{and}\\quad M:=\\frac{\\Tr(J)}{\\lambda_{\\max}(J)}.\n\\]\nThen, for any\n$n\\geq m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^{n} w_i\\, \na(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m-1}{n}}\\right)^2\n\\lambda_{\\min}(I)\n\\]\nand\n\\[\n\\lambda_{\\max}\\left(\\sum_{i=1}^{n} w_i\\, b(x_i)b(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{M-1}{n}}\\right)^2\n\\lambda_{\\max}(J).\n\\]", "start_pos": 5360, "end_pos": 6265, "label": "thm:main"}, "ref_dict": {"cor:sampling-numbers-H": "\\begin{coro}\\label{cor:sampling-numbers-H}\nLet $(D,\\mu)$ be a measure space \nand $H$ a reproducing kernel Hilbert space \nsatisfying \\eqref{cond:continuous}, \\eqref{cond:injective} and \\eqref{eq:finite-trace}.\nThen, for all $m\\in \\N$ and $n\\geq m$,\n\\[\ng_n(B_H,L_2) \n\\,\\leq\\, \\left(1+\\frac{1}{1-r}\\right)\n\\left(\\sigma_m + \\sqrt{\\frac{1}{n}\\sum_{k> m} \\sigma_k^2} \\right),\n\\]\nwhere $r=\\sqrt\\frac{m-1}{n}$ and $\\sigma_0 \\geq \\sigma_1 \\geq \\hdots$ are the singular values\nof the embedding $H\\hookrightarrow L_2$.\n\\end{coro}", "coro:intro-equal": "\\begin{prop}\n\\label{coro:intro-equal}\nLet $(D,\\mu)$ be a probability space, and \n$V_m\\subset L_2(D,\\mu)\\cap B(D)$ with $\\dim(V_m)=m$. \nFor $n\\geq m$, there exist points $x_1,\\hdots,x_{2n}\\in D$ such that,\nfor all $f\\in V_m$ and $2\\leq p\\leq\\infty$, \n\\[\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)\n\\cdot \\|f\\|_p \n\\;\\leq\\; \n\\,m^{\\frac12-\\frac1p}\\cdot\n\\sqrt{\\frac{1}{n}\\sum_{i=1}^{2n} |f(x_i)|^2}.\n\\]\nIn the case $p\\in\\{2,\\infty\\}$, we can take $n$ points instead of $2n$ points.\\\\\nIn the case $n=m$, the factor $1-\\sqrt{\\frac{m}{n}}$ can be replaced by $\\frac{1}{2m}$.\n\\end{prop}", "cor:recovery": "\\begin{coro}\\label{cor:recovery}\nLet $(D,\\mu)$ be a measure space. \nLet $H$ be a reproducing kernel Hilbert space satisfying conditions \\eqref{cond:continuous}, \\eqref{cond:injective} and \\eqref{eq:finite-trace}.\nLet $V_m \\subset H$ be an $m$-dimensional subspace and\nlet $n\\geq m$.\nThen, there are points $x_1,\\hdots,x_n \\in D$\nand weights $w_1,\\hdots,w_n>0$\nsuch that, for any $f\\in H$, the weighted least squares approximation~\\eqref{def:wls} with $y_i=f(x_i)$ satisfies\n\\[\n\\big\\|f- \\tilde f\\big\\|_2 \n\\,\\leq\\, \n\\left(1+\\frac{1+s}{1-r}\\right)\\sqrt{\\lambda_m} \\,\\big\\|f-P_m f\\big\\|_H,\n\\]\nwhere\n\\[\n\\lambda_m:=\\sup_{f \\in H_m} \\frac{\\Vert f \\Vert_2^2}{\\Vert f \\Vert_{H}^2},\n\\qquad\nr:=\\sqrt\\frac{m-1}{n}\n\\quad\\text{and}\\quad\ns:=\\sqrt{\\frac{\\Tr(K_m)-\\lambda_m}{n\\,\\lambda_m}}.\n\\] \nHere, $H_m$ is the $H$-orthogonal complement of $V_m$,\n$P_m$ is the $H$-orthogonal projection onto $V_m$,\nand $K_m$ is the reproducing kernel of $(H_m,\\|\\cdot\\|_H)$.\n\\end{coro}", "rk:weight_control": "\\begin{prop}\n\\label{rk:weight_control}\nLet the assumptions of Corollary~\\ref{coro:main} hold.\nIf $(D,\\mu)$ is a probability space and $\\int f\\,d\\mu=0$ for all $f\\in H$,\nthen for $n\\geq m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ \nsuch that \\eqref{eq-low-coro-main} holds, \\eqref{eq-up-coro-main} holds with $M$ instead of $M-1$, and \n\\[\n\\sum_{i=1}^n w_i \\leq \\left(1+\\sqrt\\frac{M}{n}\\right)^2.\n\\]\n\\end{prop}", "coro:noisy-recovery": "\\begin{coro}\n\\label{coro:noisy-recovery}\nAssume that $(D,\\mu)$ is a probability space, that $H$ satisfies conditions \\eqref{cond:continuous}, \\eqref{cond:injective} and \\eqref{eq:finite-trace}, and that $V_m \\subset H$ has dimension $m$. \nMoreover, assume that \n$V_m$ contains the Riesz representer of the integral, i.e., \n\\[\nh=\\int K(x,\\cdot)\\,d\\mu(x) \\,\\in V_m.\n\\]\nThen, for $n \\ge m$, \nthere are points\n$x_1,\\hdots,x_n \\in D$\nand weights $w_1,\\hdots,w_n>0$\nsuch that, for any $f\\in H$ and $e\\in \\C^n$, the least squares approximation \\eqref{def:wls} on $V_m$ with noisy measurements $y_i=f(x_i)+e_i$ satisfies\n\\begin{equation}\n\\label{eq-prop-noisy}\n\\big\\|f- \\tilde f\\big\\|_2 \n\\,\\leq\\, \n\\frac{1+\\tilde s}{1-r}\\, \\Big(2\\,\\sqrt{\\lambda_m} \\,\\big\\|f-P_m f\\big\\|_H\n+\\,\\Vert e \\Vert_{\\infty}\\Big),\n\\end{equation}\nwith $\\lambda_m$, $K_m$, $P_m$ and $r$ as in Corollary~\\ref{cor:recovery}, \nand $\\tilde s=\\sqrt{\\frac{\\Tr(K_m)}{n\\,\\lambda_m}}$.\n\\end{coro}", "thm:main": "\\begin{theorem}\n\\label{thm:main}\nLet $(D,\\mu)$ be a measure space, and let $a=(a_1,\\hdots,a_m)^\\top$\nand $b= (b_k)_{k\\in \\mathbb I}^\\top$ be families of square-integrable functions on $D$,\nwhere $\\mathbb I$ is at most countable.\nAssume that $I:=\\int_D a(x)a(x)^*d\\mu(x)$ and $J:=\\int_D b(x)b(x)^*d\\mu(x)$ are positive definite, and that $J$\nhas finite trace and effective dimension\n\\[\n\\Tr(J) \\,=\\, \\sum_{k\\in \\mathbb I} \\|b_k\\|_{L_2(D,\\mu)}^2 \\,<\\,\\infty\n\\quad\\text{and}\\quad M:=\\frac{\\Tr(J)}{\\lambda_{\\max}(J)}.\n\\]\nThen, for any\n$n\\geq m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^{n} w_i\\, \na(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m-1}{n}}\\right)^2\n\\lambda_{\\min}(I)\n\\]\nand\n\\[\n\\lambda_{\\max}\\left(\\sum_{i=1}^{n} w_i\\, b(x_i)b(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{M-1}{n}}\\right)^2\n\\lambda_{\\max}(J).\n\\]\n\\end{theorem}", "fig:graph-results": "\\begin{figure}[H]\n\\label{fig:graph-results}\n\\centering\n\\begin{tikzcd}\n\\rm Prop.~\\ref{thm:BSS}\n&\\rm Prop.~\\ref{prop:BDM} \\arrow[l]\n&\\rm Thm.~\\ref{thm:main} \\arrow[l] \\arrow[ld] \\arrow[rd]\n&\\rm Prop.~\\ref{prop:KW} \\arrow[d] \\\\\n\\rm Prop.~\\ref{rk:weight_control} \\arrow[d]\n&\\rm Cor.~\\ref{coro:main} \\arrow[l] \\arrow[d] \\arrow[rr]\n& &\\rm Prop.~\\ref{coro:intro-equal} \\arrow[d] \\\\\n\\rm Cor.~\\ref{coro:noisy-recovery}\n&\\rm Cor.~\\ref{cor:recovery} \\arrow[l] \\arrow[r]\n&\\rm Cor.~\\ref{cor:sampling-numbers-H}\n&\\rm Cor.~\\ref{cor:recovery-unif}\n\\end{tikzcd}\n\\caption{Summary of the results of Sections~\\ref{sec:intro}, \\ref{sec:discretization} and \\ref{sec:sampling}, with arrows denoting implications. The first line contains linear algebra results, the second deals with norm discretization and the third with sampling recovery. The lower left block details the reproducing kernel Hilbert space (RKHS) setting, with the second column directly applying Thm.~\\ref{thm:main}, while the first column adds a control on the weights, and the third looks at sampling numbers. Lastly, the fourth column is focused on the $L_p$-setting, in which all weights are equal.}\n\\end{figure}", "ex:mixed-smoothness": "\\begin{example}\n\\label{ex:mixed-smoothness}\n    An example, where Theorem~\\ref{thm:constructive} could be applied\n    is the approximation on Sobolev spaces $H^\\alpha_{\\rm mix}$\n    of $d$-variate periodic functions with mixed smoothness $\\alpha>1/2$.\n    These can be defined by \n    \\[\n     H^\\alpha_{\\rm mix} =\n     \\left\\{ f \\in C([0,1]^d) \\colon \\Vert f \\Vert_{\\alpha}^2 := \\sum_{ k \\in \\Z^d} w_{ k}^{2\\alpha} |\\hat{f}( k)|^2 < \\infty \\right\\},\n    \\]\n    where $\\hat{f}( k)$ are the usual Fourier coefficients and $w_{ k} = \\prod_{i=1}^d \\max\\{1,|k_i|\\}$. \n    For integer $\\alpha$, they can also be defined as spaces of functions possessing (weak) derivatives \n    in $L_2$ up to mixed order $\\alpha$,\n    see, e.g., \\cite[Sec.\\,2.1]{KSU} for a few details,\n    and \\cite{DTU} for a comprehensive treatment of dominating mixed smoothness.\n    If we choose the functions $\\eta_k$ as the trigonometric monomials\n    with the frequencies $k \\in \\Z^d$ ordered according to their hyperbolic distance $w_{k}$ to the origin\n    and $\\alpha_0>1/2$,\n    the algorithm from Theorem~\\ref{thm:constructive} satisfies\n    \\begin{equation}\\label{eq:order-mix}\n     \\Vert f - \\tilde f \\Vert_2 \\,\\lesssim\\, m^{-\\alpha} \n     (\\log m)^{\\alpha(d-1)} \n     \\cdot \\Vert f \\Vert_{\\alpha},\n    \\end{equation}\n    for any smoothness $\\alpha\\geq \\alpha_0$.\n\n    We present this example since\n    for the class $H^\\alpha_{\\rm mix}$,\n    no (polynomial time) construction was previously known to give the optimal order~\\eqref{eq:order-mix}.\n    Only the existence of such algorithms was known due to~\\cite{DKU}.\n    Previous constructions that come close to the optimal order include sparse grids (Smolyak's algorithm), see~\\cite{SU},\n    random points~\\cite{KU1},\n    subsamples of random points~\\cite{BSU}\n    and generated sets (lattices with a non-integer generating vector), see~\\cite{CNW}.\n    \\end{example}", "thm:BSS": "\\begin{prop}[\\cite{BSS}, Theorem 3.1]\n\\label{thm:BSS}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$\nbe an orthonormal family in $L_2(D,\\mu)$.\nThen, for any $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^n w_i\\, a(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\lambda_{\\max}\\left(\\sum_{i=1}^n w_ia(x_i)a(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{m}{n}}\\right)^2.\n\\]\n\\end{prop}", "prop:BDM": "\\begin{prop}[\\cite{BDM14}, Lemma 13]\n\\label{prop:BDM}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$ and $b=(b_1,\\dots,b_M)^\\top$\nbe orthonormal families in $L_2(D,\\mu)$.\nThen for any integer $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^{n} w_i\\, \na(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\,\n\\lambda_{\\max}\\left(\\sum_{i=1}^{n} w_i\\, b(x_i)b(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{M}{n}}\\right)^2.\n\\]\n\\end{prop}", "coro:main": "\\begin{coro} \\label{coro:main}\nLet $(D,\\mu)$ be a measure space, \n$V_m\\subset L_2(D,\\mu)$ be an $m$-dimensional space of functions and let $H$ be \na reproducing kernel Hilbert space satisfying conditions \\eqref{cond:continuous}, \\eqref{cond:injective} and \\eqref{eq:finite-trace}. \nFor $n\\geq m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\begin{equation}\n\\label{eq-low-coro-main}\n\\left(1-\\sqrt{\\frac{m-1}{n}}\\right)\\,\\|f\\|_2\n\\,\\leq\\,\n\\sqrt{\\sum_{i=1}^n w_i|f(x_i)|^2}\n\\qquad \\text{for all } f \\in V_m\n\\end{equation}\nand \n\\begin{equation}\n\\label{eq-up-coro-main}\n\\sqrt{\\sum_{i=1}^n w_i|g(x_i)|^2}\n\\,\\leq\\,\n\\left(1 +\\sqrt{\\frac{M-1}{n}} \\right)\\, \\sqrt\\lambda\\,\n\\left\\| g \\right\\|_H\n\\qquad \\text{for all } g \\in H,\n\\end{equation}\nwhere $M=\\Tr(K)/\\lambda$ is the ``effective dimension'' of $H$ in $L_2(D,\\mu)$.\n\\end{coro}", "subsec:numerics": "\\label{subsec:numerics}\n\nWe now discuss some numerical examples\nfor the sampling points and weights \nfrom Theorem~\\ref{thm:constructive}.\nFigures~\\ref{fig:trig-blocked}--\\ref{fig:legendre-new} show th", "alg:abstract-construction": "\\label{alg:abstract-construction}\n\\end{algorithm}\n\nRegarding the first issue, one option is to find a finite subset of $D$ in which at least one point satisfies the condition of line~\\ref{state:test}.", "prop:KW": "\\begin{prop}[\\cite{KW}]\\label{prop:KW}\nLet $D$ be a set and $V_m$ be an $m$-dimen\\-sion\\-al subspace of~$B(D)$. \nFor every $\\varepsilon>0$, \nthere exists a finitely supported measure $\\mu_\\eps$ on~$D$,\nsuch that \n\\[\n \\|f\\|_{\\infty} \\,\\leq\\, \\sqrt{m+\\varepsilon} \\cdot \\Vert f \\Vert_{L_2(D,\\mu_\\eps)}. \n\\]  \n\\end{prop}", "thm:constructive": "\\begin{theorem}\n\\label{thm:constructive}\nLet $(D,\\mu)$ be a probability space, $(\\eta_k)_{k\\in\\II}$ an $L_2$-orthonormal system and $(I_\\ell)_{\\ell\\in\\N}$ nested index sets with $|I_\\ell|\\le\\ell$ that satisfy condition~\\eqref{eq:assum-uniform-bound}, $m\\geq 2$, $n=2 m$, $\\alpha_0>\\theta$ and $N=\\lceil m^{\\alpha_0/(\\alpha_0-\\theta)}\\rceil$.\nAssume that the constant function $1$ belongs to $\\{\\eta_k,\\;k\\in I_m\\}$. Algorithm~\\ref{alg:abstract-construction} can be implemented, with $a$ and $b$ defined in \\eqref{eq:a-b-constructive}, in time\n\\[\nO\\big((c+Nn)n^2\\mathcal G\\big),\n\\]\nwhere $c$ is the cost of drawing one point from the Christoffel density, and ${\\mathcal G\\in\\N}$ is a shifted geometric random variable of parameter $1/2$. It yields points $x_1,\\dots,x_n\\in D$ and weights $w_1,\\dots,w_n>0$ which satisfy the following.\nIf $f\\in L_2(D,\\mu)$ with\n\\[\n\\|f-f_{I_\\ell}\\|_2\\leq C_f\\, \\ell^{-\\alpha}\\log^\\beta(\\ell)\n\\]\nfor some $\\alpha\\geq \\alpha_0$ and $\\beta\\geq 0$, and if\n$f(x_i)=\\lim_{\\ell\\to\\infty}f_{I_\\ell}(x_i)$ for $1\\leq i\\leq n$,\nthe weighted least-squares approximation \\eqref{def:wls}\nfrom these values satisfies\n\\[\n\\|f-\\tilde f\\|_2\\leq C_f\\,C\\, m^{-\\alpha}\\log^\\beta(m),\n\\]\nwhere $C$ depends on $\\theta$, $\\alpha_0$, $\\alpha$, $\\beta$ and $C_\\eta$. \n\\end{theorem}", "cor:recovery-unif": "\\begin{coro}\\label{cor:recovery-unif}\nLet $(D,\\mu)$ be a probability space, $V_m\\subset L_2(D,\\mu)\\cap B(D)$ with $\\dim(V_m)=m$, \nand $2\\le p\\le\\infty$. \nFor all $n> m$,\nthere are points $x_1,\\hdots,x_{2n} \\in D$\nsuch that the plain least squares \napproximation~\\eqref{def:pls} \nwith $y_i = f(x_i) + e_i$\nsatisfies\n\\[\n\\big\\|f- \\tilde f\\big\\|_p \n\\,\\leq\\, \\left(1 + \\sqrt{2}\\cdot\\frac{m^{1/2-1/p}}{1-\\sqrt{m/n}}\\right) \\Big( \\min_{g\\in V_m} \\,\\big\\|f-g\\big\\|_\\infty + \\Vert e \\Vert_\\infty \\Big)\n\\]\nfor all $f\\in B(D)$ and any noise $e_1,\\hdots,e_n \\in \\C$.\nFor $p\\in\\{2,\\infty\\}$, we can use $n$ instead of $2n$ points, remove the factor $\\sqrt{2}$, \nand it suffices to have $V_m \\subset L_2$ or $V_m \\subset B(D)$, respectively.\n\\end{coro}"}, "pre_theorem_intro_text_len": 3025, "pre_theorem_intro_text": "\\label{sec:intro}\nIn recent years, much\nprogress has been made in norm discretization and sampling recovery, first\nby considering concentration inequalities for random points,\nthen by applying a sample sparsification using \\cite{MSS} or \\cite{BSS}. \nThis follows the trend observed in other fields, including graph sparsification, frame discretization, matrix\nsketching, subspace selection, and more generally randomized numerical linear algebra.\n\nHere, we prove a generalization of the result from~\\cite{BSS} by Batson, Spielman, and Srivastava (BSS), and expand on its implications\nfor the problems of norm discretization and sampling recovery. \nThe original result from~\\cite{BSS} can be stated as follows:\n\n\\begin{prop}[\\cite{BSS}, Theorem 3.1]\n\\label{thm:BSS}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$\nbe an orthonormal family in $L_2(D,\\mu)$.\nThen, for any $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^n w_i\\, a(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\lambda_{\\max}\\left(\\sum_{i=1}^n w_ia(x_i)a(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{m}{n}}\\right)^2.\n\\]\n\\end{prop}\n\nHere and in the following, we denote by \n$\\lambda_{\\min}(A)$ and $\\lambda_{\\max}(A)$ the smallest and largest eigenvalues of $A$.\nThe points and weights from Proposition~\\ref{thm:BSS} can be constructed in polynomial time.\n\nIn \\cite{BSS}, this theorem is applied to the problem of graph sparsification, see also \\cite{HGN+24,SS11,ST11}.\nFor our applications, we want to generalize the result in three ways: by allowing general measure spaces $(D,\\mu)$; by decoupling the lower and upper frame bounds with two different\nfamilies of $L_2$-functions; and by making the upper frame bound independent of the dimension, thus allowing to consider very large and even infinite-dimensional families. \n\nThe first two kinds of generalizations are already available in the literature. General probability spaces are treated, for instance, in \\cite[Theorem~6.4]{DPSTT}.\nThe second kind of generalization was achieved in \\cite{BDM14}.\nThe result of \\cite{BDM14}, called dual set spectral sparsification, can be reformulated as follows.\n\n\\begin{prop}[\\cite{BDM14}, Lemma 13]\n\\label{prop:BDM}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$ and $b=(b_1,\\dots,b_M)^\\top$\nbe orthonormal families in $L_2(D,\\mu)$.\nThen for any integer $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^{n} w_i\\, \na(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\,\n\\lambda_{\\max}\\left(\\sum_{i=1}^{n} w_i\\, b(x_i)b(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{M}{n}}\\right)^2.\n\\]\n\\end{prop}\n\nOur main contribution, stated below, is the third kind of generalization. Namely, we replace the dimension $M$ by an effective dimension in the\nupper frame bound.\n\\smallskip", "context": "\\label{sec:intro}\nIn recent years, much\nprogress has been made in norm discretization and sampling recovery, first\nby considering concentration inequalities for random points,\nthen by applying a sample sparsification using \\cite{MSS} or \\cite{BSS}. \nThis follows the trend observed in other fields, including graph sparsification, frame discretization, matrix\nsketching, subspace selection, and more generally randomized numerical linear algebra.\n\n\\begin{prop}[\\cite{BSS}, Theorem 3.1]\n\\label{thm:BSS}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$\nbe an orthonormal family in $L_2(D,\\mu)$.\nThen, for any $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^n w_i\\, a(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\lambda_{\\max}\\left(\\sum_{i=1}^n w_ia(x_i)a(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{m}{n}}\\right)^2.\n\\]\n\\end{prop}\n\nHere and in the following, we denote by \n$\\lambda_{\\min}(A)$ and $\\lambda_{\\max}(A)$ the smallest and largest eigenvalues of $A$.\nThe points and weights from Proposition~\\ref{thm:BSS} can be constructed in polynomial time.\n\nIn \\cite{BSS}, this theorem is applied to the problem of graph sparsification, see also \\cite{HGN+24,SS11,ST11}.\nFor our applications, we want to generalize the result in three ways: by allowing general measure spaces $(D,\\mu)$; by decoupling the lower and upper frame bounds with two different\nfamilies of $L_2$-functions; and by making the upper frame bound independent of the dimension, thus allowing to consider very large and even infinite-dimensional families.\n\n\\begin{prop}[\\cite{BDM14}, Lemma 13]\n\\label{prop:BDM}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$ and $b=(b_1,\\dots,b_M)^\\top$\nbe orthonormal families in $L_2(D,\\mu)$.\nThen for any integer $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^{n} w_i\\, \na(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\,\n\\lambda_{\\max}\\left(\\sum_{i=1}^{n} w_i\\, b(x_i)b(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{M}{n}}\\right)^2.\n\\]\n\\end{prop}\n\nOur main contribution, stated below, is the third kind of generalization. Namely, we replace the dimension $M$ by an effective dimension in the\nupper frame bound.\n\\smallskip", "full_context": "\\label{sec:intro}\nIn recent years, much\nprogress has been made in norm discretization and sampling recovery, first\nby considering concentration inequalities for random points,\nthen by applying a sample sparsification using \\cite{MSS} or \\cite{BSS}. \nThis follows the trend observed in other fields, including graph sparsification, frame discretization, matrix\nsketching, subspace selection, and more generally randomized numerical linear algebra.\n\n\\begin{prop}[\\cite{BSS}, Theorem 3.1]\n\\label{thm:BSS}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$\nbe an orthonormal family in $L_2(D,\\mu)$.\nThen, for any $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^n w_i\\, a(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\lambda_{\\max}\\left(\\sum_{i=1}^n w_ia(x_i)a(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{m}{n}}\\right)^2.\n\\]\n\\end{prop}\n\nHere and in the following, we denote by \n$\\lambda_{\\min}(A)$ and $\\lambda_{\\max}(A)$ the smallest and largest eigenvalues of $A$.\nThe points and weights from Proposition~\\ref{thm:BSS} can be constructed in polynomial time.\n\nIn \\cite{BSS}, this theorem is applied to the problem of graph sparsification, see also \\cite{HGN+24,SS11,ST11}.\nFor our applications, we want to generalize the result in three ways: by allowing general measure spaces $(D,\\mu)$; by decoupling the lower and upper frame bounds with two different\nfamilies of $L_2$-functions; and by making the upper frame bound independent of the dimension, thus allowing to consider very large and even infinite-dimensional families.\n\n\\begin{prop}[\\cite{BDM14}, Lemma 13]\n\\label{prop:BDM}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$ and $b=(b_1,\\dots,b_M)^\\top$\nbe orthonormal families in $L_2(D,\\mu)$.\nThen for any integer $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^{n} w_i\\, \na(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\,\n\\lambda_{\\max}\\left(\\sum_{i=1}^{n} w_i\\, b(x_i)b(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{M}{n}}\\right)^2.\n\\]\n\\end{prop}\n\nOur main contribution, stated below, is the third kind of generalization. Namely, we replace the dimension $M$ by an effective dimension in the\nupper frame bound.\n\\smallskip\n\n\\begin{prop}[\\cite{BDM14}, Lemma 13]\n\\label{prop:BDM}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$ and $b=(b_1,\\dots,b_M)^\\top$\nbe orthonormal families in $L_2(D,\\mu)$.\nThen for any integer $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^{n} w_i\\, \na(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\,\n\\lambda_{\\max}\\left(\\sum_{i=1}^{n} w_i\\, b(x_i)b(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{M}{n}}\\right)^2.\n\\]\n\\end{prop}\n\nOur main contribution, stated below, is the third kind of generalization. Namely, we replace the dimension $M$ by an effective dimension in the\nupper frame bound.\n\\smallskip\n\n\\smallskip\n\n\\begin{coro} \\label{coro:main}\nLet $(D,\\mu)$ be a measure space, \n$V_m\\subset L_2(D,\\mu)$ be an $m$-dimensional space of functions and let $H$ be \na reproducing kernel Hilbert space satisfying conditions \\eqref{cond:continuous}, \\eqref{cond:injective} and \\eqref{eq:finite-trace}. \nFor $n\\geq m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\begin{equation}\n\\label{eq-low-coro-main}\n\\left(1-\\sqrt{\\frac{m-1}{n}}\\right)\\,\\|f\\|_2\n\\,\\leq\\,\n\\sqrt{\\sum_{i=1}^n w_i|f(x_i)|^2}\n\\qquad \\text{for all } f \\in V_m\n\\end{equation}\nand \n\\begin{equation}\n\\label{eq-up-coro-main}\n\\sqrt{\\sum_{i=1}^n w_i|g(x_i)|^2}\n\\,\\leq\\,\n\\left(1 +\\sqrt{\\frac{M-1}{n}} \\right)\\, \\sqrt\\lambda\\,\n\\left\\| g \\right\\|_H\n\\qquad \\text{for all } g \\in H,\n\\end{equation}\nwhere $M=\\Tr(K)/\\lambda$ is the ``effective dimension'' of $H$ in $L_2(D,\\mu)$.\n\\end{coro}\n\n\\begin{coro}\\label{cor:recovery}\nLet $(D,\\mu)$ be a measure space. \nLet $H$ be a reproducing kernel Hilbert space satisfying conditions \\eqref{cond:continuous}, \\eqref{cond:injective} and \\eqref{eq:finite-trace}.\nLet $V_m \\subset H$ be an $m$-dimensional subspace and\nlet $n\\geq m$.\nThen, there are points $x_1,\\hdots,x_n \\in D$\nand weights $w_1,\\hdots,w_n>0$\nsuch that, for any $f\\in H$, the weighted least squares approximation~\\eqref{def:wls} with $y_i=f(x_i)$ satisfies\n\\[\n\\big\\|f- \\tilde f\\big\\|_2 \n\\,\\leq\\, \n\\left(1+\\frac{1+s}{1-r}\\right)\\sqrt{\\lambda_m} \\,\\big\\|f-P_m f\\big\\|_H,\n\\]\nwhere\n\\[\n\\lambda_m:=\\sup_{f \\in H_m} \\frac{\\Vert f \\Vert_2^2}{\\Vert f \\Vert_{H}^2},\n\\qquad\nr:=\\sqrt\\frac{m-1}{n}\n\\quad\\text{and}\\quad\ns:=\\sqrt{\\frac{\\Tr(K_m)-\\lambda_m}{n\\,\\lambda_m}}.\n\\] \nHere, $H_m$ is the $H$-orthogonal complement of $V_m$,\n$P_m$ is the $H$-orthogonal projection onto $V_m$,\nand $K_m$ is the reproducing kernel of $(H_m,\\|\\cdot\\|_H)$.\n\\end{coro}\n\n\\begin{lemma}\n\\label{lem:accept_proba}\nIf $x$ is drawn according to the Christoffel measure $\\rho$ and the increment $\\delta=\\frac{1-r}{n}$ is replaced by\n\\[\n\\delta_\\epsilon=\\frac{1-r-\\epsilon}{n}\n\\]\nfor some parameter $\\epsilon < (1-r)^2$ in the computation of $L_A^\\delta$ and the updates $A\\leftarrow A'$ (but not in the initialization $A_0$),\nthen the acceptance probability satisfies\n\\[\n\\mathbb P\\left(L_A^{\\delta_\\eps}(a(x)) \\ge U_B^\\zeta(b(x))\\right) \\ge \\frac{\\epsilon}{m}.\n\\]\nIn addition, any points $x_1,\\dots x_n$ returned by our algorithm for the modified value of $\\delta$ satisfy the upper bound \\eqref{eq:UB} while the lower bound \\eqref{eq:LB} becomes\n\\[\n\\lambda_{\\min}\\left(\\sum_{i=1}^{n} w_ia(x_i)a(x_i)^*\\right)\n\\geq \\Big(n\\delta_\\epsilon+\\frac{\\delta}{r}-\\frac{m\\delta}{r}\\,\\Big)= (1-r)^2-\\epsilon.\n\\]\n\\end{lemma}\nIn the sequel, we will take $\\epsilon=\\frac{(1-r)^2}{2}$, which changes \\eqref{eq:LB} by a factor 2.\n\\begin{proof}\nWe start with $A \\in \\mathcal P_m$ and $B\\in \\mathcal P$ such that \\eqref{ineq-lemboth} holds for the original $\\delta=\\frac{1-r}{n}$.\nLet\n\\[\n\\Delta(x):= \\frac{L_A^{\\delta_\\epsilon}(a(x))- U_B^\\zeta(b(x))}{\\sum_{k=1}^m|a_k(x)|^2}.\n\\]\nWe now consider a random variable $x\\sim \\rho$.\nAccording to Lemma~\\ref{lemma:pos_measure},\nthe expectation of $\\Delta(x)$ is \n\\begin{align*}\n\\E(\\Delta(x))\n=\\int_D \\frac{L_A^{\\delta_\\epsilon}(a(x))-U_B^\\zeta(b(x))}m \\, d\\mu(x)\n\\geq \\frac{1}{m}\\left(\\frac{1}{\\delta_\\epsilon}-\\Phi(A)- \\frac{1}{\\zeta}-\\Psi(B)\\right).\n\\end{align*}\nThanks to condition \\eqref{ineq-lemboth}, \nwe get\n\\[\n\\E(\\Delta(x))\n\\geq \\frac{1}{m}\\left(\\frac{1}{\\delta_\\epsilon}-\\frac{1}{\\delta}\\right)=\\frac{\\epsilon}{m \\,\\delta_\\epsilon(1-r)}.\n\\]\nOn the other hand, $\\Delta(x)$ is almost surely bounded by\n\\begin{align*}\n\\Delta(x)\n\\leq \\sup_{\\substack{a\\in\\C^m \n}} \\frac{L_A^{\\delta_\\eps}(a)}{\\sum_{k=1}^m|a_k|^2} \n\\leq \\frac{\\lambda_{\\max}(Z^2)}{\\Tr(Z)-\\Tr(Y)}\\leq \\frac{\\Tr(Z^2)}{\\delta_\\epsilon\\Tr(YZ)}\\leq \\frac{1}{\\delta_\\epsilon(1-r)},\n\\end{align*}\nwhere $Y$ and $Z$ are from Section~\\ref{sec:main-proof} with $\\delta_\\eps$ in place of $\\delta$. Here,\nthe third inequality follows from $Z-Y=\\delta_\\epsilon YZ$\nand the last inequality comes from the fact that\n\\[\n\\Tr(Z^2)-\\Tr(YZ)=\\delta_\\epsilon \\Tr(Y Z^2)\\leq \\delta_\\epsilon\\Tr(Y)\\Tr(Z^2)\\leq r\\Tr(Z^2),\n\\]\nby a Cauchy-Schwarz inequality~\\eqref{property:CS} between $Y^{1/2}$ and $Y^{1/2}Z$ and\nsince $\\Tr(Y)=\\Phi(A)\\leq r/\\delta_\\epsilon$. In the end, we obtain an acceptance probability\n\\[\n\\mathbb P(\\Delta(x)\\geq 0)\n\\geq \\frac{\\int_D \\Delta\\, d\\rho}{\\sup_{x\\in D} \\Delta(x)}\\geq \\frac{\\epsilon}m.\n\\]\nNote that, by Lemma~\\ref{lemboth}, choosing $x_i$ with $\\Delta(x_i) \\ge 0$ and $w_i$ accordingly, we obtain that the potentials do not increase \nwhen switching to the updates \n$A':=A-\\delta_\\eps I+wa(x_i)a(x_i)^*$ and \n$ B':=B+\\zeta J-wb(x_i)b(x_i)^*$ and\nso \ncondition~\\eqref{ineq-lemboth} remains true with the original $\\delta$ also for the updated operators.\n\\end{proof}\n\n\\begin{example}\n\\label{ex:christoffel}\nFor each $m\\in\\N$, consider $D=[0,m]$ with the Lebesgue measure $\\mu$\nand define a RKHS $H$ on $D$ by its kernel $K(x,y)=\\sum_{k\\in \\II} \\sigma_k^2 b_k(x) b_k(y)$, where the functions\n\\[\nb_k := \\left\\{\\begin{array}{cl}\n\\mathbbm 1_{[k-1,k]} &\\text{ for } 1\\leq k\\leq m,\\\\\n\\cos(2\\pi (k-m)x) \\mathbbm 1_{[0,1]} &\\text{ for } k>m,\\\\\n\\sin(2\\pi (m-k)x) \\mathbbm 1_{[0,1]} &\\text{ for } k<-m,\\\\\n\\end{array}\\right.\n\\]\nare orthonormal in $L_2(D)$,\nand $\\sigma_k:=\\frac1{|k|}$ for $k\\in\\II:=\\Z\\setminus\\{-m,\\dots,0\\}$. Let also $V_m=\\Span\\{b_k \\colon 1\\leq k\\leq m\\}$. From~\\eqref{eq:n-is-2m}, we get\n\\[\n g_{2m}^{\\rm lin}(B_H,L_2) \\,\\leq\\, \\frac{10}m.\n\\]\nOn the other hand,\nlet $A_n$ be an approximation that uses information from at\nmost $n$ samples $x_1,\\hdots,x_n$.\nLet $\\{y_1,\\hdots,y_{n'}\\} = \\{x_1,\\hdots,x_n\\} \\cap [0,1]$\nbe the samples in $[0,1]$. Observing that $A_n(-f)=A_n(f)$ if $f(x_1)=\\dots f(x_n)=0$,\n\\[\n \\sup_{\\Vert f \\Vert_H \\leq 1} \\Vert f - A_n(f) \\Vert_{L_2([0,m])}\n \\geq \\sup_{\\substack{\\Vert f \\Vert_H \\leq 1 \\\\ f(x_i)=0}} \\Vert f \\Vert_{L_2([0,m])}\n \\geq \\sup_{\\substack{\\Vert f \\Vert_H \\leq 1 \\\\ f(y_i)=0}} \\Vert f \\Vert_{L_2([0,1])}.\n \\]\nAs $\\|f\\|_{L_2([0,1])}\\geq \\big|\\int_0^1f\\big|$, the lower bound from \\cite[Corollary~2]{KV} with $\\lambda_0=1$ and $\\lambda_k=\\sigma_{|k|+m}^2$ for $k\\neq 0$ gives\n \\[\n\\sup_{\\substack{\\Vert f \\Vert_H \\leq 1 \\\\ f(y_i)=0}} \\Vert f \\Vert_{L_2([0,1])}\n \\geq \\sqrt{\\min\\bigg\\{\\frac{\\lambda_0}2, \\ \\frac{1}{8n'} \\sum_{k> 4n'} \\lambda_k\\bigg\\}}\n \\geq \\frac{1}{8\\max(n',\\sqrt{mn'})}.\n\\]\nAs a consequence, the approximation $A_n$ can only attain an optimal error if $n' \\gtrsim m$.\nAs the Christoffel density of $V_m$ is uniform on $[0,m]$,\nif $x_1,\\hdots,x_n$ are drawn i.i.d. from that density, \nwe must have $n \\gtrsim m^2$ with high probability. In summary, any \nalgorithm achieving an error of optimal order \nmust use $\\Omega(m^2)$ Christoffel samples.\n\\end{example}", "post_theorem_intro_text_len": 4424, "post_theorem_intro_text": "\\smallskip\n\nOne can retrieve Proposition~\\ref{prop:BDM} from Theorem~\\ref{thm:main} by taking $\\mathbb I=\\{1,\\dots,M\\}$ and assuming that $a$ and $b$ are orthonormal families. Indeed, in that case, we see that $I$ and $J$ are the $m\\times m$ and $M\\times M$ identity matrices. In turn, Proposition~\\ref{prop:BDM} implies Proposition~\\ref{thm:BSS} by taking $M=m$ and $b=a$.\n\n\\begin{remark}\nThe factor $m-1$ instead of $m$ in the lower bound is a minor improvement\nalready appearing in \\cite{ChkifaDolbeault24}. It allows to treat the case $n=m$, with the bound\n\\[\n1-\\sqrt{\\frac{m-1}{m}}>\\frac{1}{2m}.\n\\]\nIn the same way, the factor $M-1$ instead of $M$ \nin the upper bound is most interesting when $J$ has rank one, where we recover \\cite[Lemma~14]{BDM14}.\nThis case will be used in Corollary~\\ref{coro:intro-equal} to obtain discretizations with equal weights. We will also leverage these improvements in Propositions~\\ref{rk:weight_control} and Corollary~\\ref{coro:noisy-recovery}, when considering families $a$ and $b$ containing one more function.\n\\end{remark}\n\n\\begin{remark}\nThroughout the article, $L_2(D,\\mu)$ and its subspaces are assumed to consist of complex-valued functions, not equivalence classes, which is needed for discretization. \nMoreover, it is possible to extend our results to Hilbert-valued functions, by combining them with \\cite[Theorem 2.1]{BartelDung24}.\n\\end{remark}\n\nThe proof of Theorem~\\ref{thm:main}, see Section~\\ref{sec:proof}, \nbuilds upon the original proof of Proposition~\\ref{thm:BSS},  \nand the points and weights can be constructed in a quite similar way, see Algorithm~\\ref{alg:abstract-construction} in Section~\\ref{sec:implementation}.\nAlthough the proof of Theorem~\\ref{thm:main} is self-contained, \nwe recommend reading the proof of \\cite[Theorem~3.1]{BSS} first, since it comes with a very nice physical intuition.\n\nBefore we come to the proof, we first discuss several applications of this linear algebra result.\nIn Section~\\ref{sec:discretization}, we discuss its implications to the problem of norm discretization.\nIn Section~\\ref{sec:sampling}, we use these discretization results to obtain results on the problem of sampling recovery\nand on the error of (weighted) least squares algorithms. The relation between these results is illustrated in Figure~\\ref{fig:graph-results}.\n\n\\begin{figure}[H]\n\\label{fig:graph-results}\n\\centering\n\\begin{tikzcd}\n\\rm Prop.~\\ref{thm:BSS}\n&\\rm Prop.~\\ref{prop:BDM} \\arrow[l]\n&\\rm Thm.~\\ref{thm:main} \\arrow[l] \\arrow[ld] \\arrow[rd]\n&\\rm Prop.~\\ref{prop:KW} \\arrow[d] \\\\\n\\rm Prop.~\\ref{rk:weight_control} \\arrow[d]\n&\\rm Cor.~\\ref{coro:main} \\arrow[l] \\arrow[d] \\arrow[rr]\n& &\\rm Prop.~\\ref{coro:intro-equal} \\arrow[d] \\\\\n\\rm Cor.~\\ref{coro:noisy-recovery}\n&\\rm Cor.~\\ref{cor:recovery} \\arrow[l] \\arrow[r]\n&\\rm Cor.~\\ref{cor:sampling-numbers-H}\n&\\rm Cor.~\\ref{cor:recovery-unif}\n\\end{tikzcd}\n\\caption{Summary of the results of Sections~\\ref{sec:intro}, \\ref{sec:discretization} and \\ref{sec:sampling}, with arrows denoting implications. The first line contains linear algebra results, the second deals with norm discretization and the third with sampling recovery. The lower left block details the reproducing kernel Hilbert space (RKHS) setting, with the second column directly applying Thm.~\\ref{thm:main}, while the first column adds a control on the weights, and the third looks at sampling numbers. Lastly, the fourth column is focused on the $L_p$-setting, in which all weights are equal.}\n\\end{figure}\n\nIn fact, it is quite interesting that many of the recent advances in \nthe areas of both norm discretization and sampling recovery\ncan be derived solely on the basis of Theorem~\\ref{thm:main}.\nPrevious results used the non-constructive Kadison-Singer theorem from \\cite{MSS},\nsee, e.g., \\cite{CohenDolbeault,DKU,NSU},\nwhich now becomes unnecessary.\nThis does not only simplify the proofs of those results\nbut also much improves the absolute constants.\nMoreover, the least squares algorithms based on Theorem~\\ref{thm:main} \nare implementable for various examples where \nan implementable construction of an optimal algorithm \nhas previously been unknown\n(e.g., for function spaces of mixed smoothness),\nsee Theorem~\\ref{thm:constructive} and Example~\\ref{ex:mixed-smoothness}.\nA discussion of the practical implementation, along with numerical illustrations, is provided in Sections~\\ref{sec:implementation} and~\\ref{subsec:numerics}.", "sketch": "The proof of Theorem~\\ref{thm:main} (see Section~\\ref{sec:proof}) is said to “builds upon the original proof of Proposition~\\ref{thm:BSS},” and the “points and weights can be constructed in a quite similar way,” referring to Algorithm~\\ref{alg:abstract-construction} in Section~\\ref{sec:implementation}. Although the proof is “self-contained,” the authors recommend reading the proof of \\cite[Theorem~3.1]{BSS} first “since it comes with a very nice physical intuition.”", "expanded_sketch": "The proof of the main theorem (see Section~\\ref{sec:proof}) is said to “builds upon the original proof of \\begin{prop}[\\cite{BSS}, Theorem 3.1]\n\\label{thm:BSS}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$\nbe an orthonormal family in $L_2(D,\\mu)$.\nThen, for any $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^n w_i\\, a(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\lambda_{\\max}\\left(\\sum_{i=1}^n w_ia(x_i)a(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{m}{n}}\\right)^2.\n\\]\n\\end{prop},” and the “points and weights can be constructed in a quite similar way,” referring to Algorithm~\\ref{alg:abstract-construction} in Section~\\ref{sec:implementation}:\n\\[\n\\label{alg:abstract-construction}\n\\end{algorithm}\n\nRegarding the first issue, one option is to find a finite subset of $D$ in which at least one point satisfies the condition of line~\\ref{state:test}.\n\\]\nAlthough the proof is “self-contained,” the authors recommend reading the proof of \\cite{BSS}, Theorem~3.1 first “since it comes with a very nice physical intuition.”", "expanded_theorem": "\\label{thm:main}\nLet $(D,\\mu)$ be a measure space, and let $a=(a_1,\\hdots,a_m)^\\top$\nand $b= (b_k)_{k\\in \\mathbb I}^\\top$ be families of square-integrable functions on $D$,\nwhere $\\mathbb I$ is at most countable.\nAssume that $I:=\\int_D a(x)a(x)^*d\\mu(x)$ and $J:=\\int_D b(x)b(x)^*d\\mu(x)$ are positive definite, and that $J$\nhas finite trace and effective dimension\n\\[\n\\Tr(J) \\,=\\, \\sum_{k\\in \\mathbb I} \\|b_k\\|_{L_2(D,\\mu)}^2 \\,<\\,\\infty\n\\quad\\text{and}\\quad M:=\\frac{\\Tr(J)}{\\lambda_{\\max}(J)}.\n\\]\nThen, for any\n$n\\geq m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^{n} w_i\\, \na(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m-1}{n}}\\right)^2\n\\lambda_{\\min}(I)\n\\]\nand\n\\[\n\\lambda_{\\max}\\left(\\sum_{i=1}^{n} w_i\\, b(x_i)b(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{M-1}{n}}\\right)^2\n\\lambda_{\\max}(J).\n\\]", "theorem_type": ["Existence", "Inequality or Bound"], "mcq": {"question": "Let $(D,\\mu)$ be a measure space. Let $a=(a_1,\\dots,a_m)^\\top$ and $b=(b_k)_{k\\in\\mathbb I}^\\top$ be families of square-integrable functions on $D$, where $\\mathbb I$ is at most countable. Define\n\\[\nI:=\\int_D a(x)a(x)^*\\,d\\mu(x),\\qquad J:=\\int_D b(x)b(x)^*\\,d\\mu(x),\n\\]\nand assume that both $I$ and $J$ are positive definite. Assume also that $J$ has finite trace\n\\[\n\\operatorname{Tr}(J)=\\sum_{k\\in\\mathbb I}\\|b_k\\|_{L_2(D,\\mu)}^2<\\infty,\n\\]\nand define its effective dimension by\n\\[\nM:=\\frac{\\operatorname{Tr}(J)}{\\lambda_{\\max}(J)}.\n\\]\nHere $\\lambda_{\\min}(\\cdot)$ and $\\lambda_{\\max}(\\cdot)$ denote the smallest and largest eigenvalues. For a given integer $n\\ge m$, which quantitative estimate is valid?", "correct_choice": {"label": "A", "text": "There exist points $x_1,\\dots,x_n\\in D$ and weights $w_1,\\dots,w_n>0$ such that\n\\[\n\\lambda_{\\min}\\!\\left(\\sum_{i=1}^n w_i\\,a(x_i)a(x_i)^*\\right)\n\\ge \\left(1-\\sqrt{\\frac{m-1}{n}}\\right)^2\\lambda_{\\min}(I)\n\\]\nand\n\\[\n\\lambda_{\\max}\\!\\left(\\sum_{i=1}^n w_i\\,b(x_i)b(x_i)^*\\right)\n\\le \\left(1+\\sqrt{\\frac{M-1}{n}}\\right)^2\\lambda_{\\max}(J).\n\\]"}, "choices": [{"label": "B", "text": "There exist points $x_1,\\dots,x_n\\in D$ and weights $w_1,\\dots,w_n>0$ such that\n\\[\n\\lambda_{\\min}\\!\\left(\\sum_{i=1}^n w_i\\,a(x_i)a(x_i)^*\\right)\n\\ge \\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\\lambda_{\\min}(I)\n\\]\nand\n\\[\n\\lambda_{\\max}\\!\\left(\\sum_{i=1}^n w_i\\,b(x_i)b(x_i)^*\\right)\n\\le \\left(1+\\sqrt{\\frac{M}{n}}\\right)^2\\lambda_{\\max}(J).\n\\]"}, {"label": "C", "text": "There exist points $x_1,\\dots,x_n\\in D$ and weights $w_1,\\dots,w_n>0$ such that\n\\[\n\\lambda_{\\min}\\!\\left(\\sum_{i=1}^n w_i\\,a(x_i)a(x_i)^*\\right)\n\\ge \\left(1-\\sqrt{\\frac{m-1}{n}}\\right)^2\\lambda_{\\min}(I).\n\\]"}, {"label": "D", "text": "There exist points $x_1,\\dots,x_n\\in D$ and weights $w_1,\\dots,w_n>0$, depending only on $n$, $m$, and $M$, such that\n\\[\n\\lambda_{\\min}\\!\\left(\\sum_{i=1}^n w_i\\,a(x_i)a(x_i)^*\\right)\n\\ge \\left(1-\\sqrt{\\frac{m-1}{n}}\\right)^2\\lambda_{\\min}(I)\n\\]\nand\n\\[\n\\lambda_{\\max}\\!\\left(\\sum_{i=1}^n w_i\\,b(x_i)b(x_i)^*\\right)\n\\le \\left(1+\\sqrt{\\frac{M-1}{n}}\\right)^2\\lambda_{\\max}(J).\n\\]"}, {"label": "E", "text": "For every choice of points $x_1,\\dots,x_n\\in D$, there exist weights $w_1,\\dots,w_n>0$ such that\n\\[\n\\lambda_{\\min}\\!\\left(\\sum_{i=1}^n w_i\\,a(x_i)a(x_i)^*\\right)\n\\ge \\left(1-\\sqrt{\\frac{m-1}{n}}\\right)^2\\lambda_{\\min}(I)\n\\]\nand\n\\[\n\\lambda_{\\max}\\!\\left(\\sum_{i=1}^n w_i\\,b(x_i)b(x_i)^*\\right)\n\\le \\left(1+\\sqrt{\\frac{M-1}{n}}\\right)^2\\lambda_{\\max}(J).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "sharp m-1 and M-1 boundary terms", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "upper eigenvalue estimate for the $b$-family", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "dependence of selected points and weights on the actual families $a,b$ and operators $I,J$", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "existential choice of sample points produced by the construction", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It provides assumptions and definitions, but the exact two-sided eigenvalue conclusion in A is not stated in the prompt itself."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the question asks which existence statement holds under the listed assumptions, and choice A is the theorem’s conclusion almost verbatim rather than an application or derivation."}, "GPS": {"score": 1, "justification": "There is some reasoning required to distinguish the exact sharp form from nearby variants, especially because one distractor is a weaker true statement and others involve plausible parameter confusions. However, the main task is still recognition of the theorem statement rather than substantial generation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: replacing sharp constants by nearby ones, dropping one half of the conclusion, confusing effective dimension with trace, and mixing up λ_max(J) with Tr(J)."}, "total_score": 5, "overall_assessment": "A technically well-constructed recall-style MCQ with strong distractors, but it is largely a direct restatement of a theorem rather than a genuinely generative reasoning task."}}
{"id": "2602.18719v1", "paper_link": "http://arxiv.org/abs/2602.18719v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\nLet $(D,\\mu)$ be a measure space, and let $a=(a_1,\\hdots,a_m)^\\top$\nand $b= (b_k)_{k\\in \\mathbb I}^\\top$ be families of square-integrable functions on $D$,\nwhere $\\mathbb I$ is at most countable.\nAssume that $I:=\\int_D a(x)a(x)^*d\\mu(x)$ and $J:=\\int_D b(x)b(x)^*d\\mu(x)$ are positive definite, and that $J$\nhas finite trace and effective dimension\n\\[\n\\Tr(J) \\,=\\, \\sum_{k\\in \\mathbb I} \\|b_k\\|_{L_2(D,\\mu)}^2 \\,<\\,\\infty\n\\quad\\text{and}\\quad M:=\\frac{\\Tr(J)}{\\lambda_{\\max}(J)}.\n\\]\nThen, for any\n$n\\geq m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^{n} w_i\\, \na(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m-1}{n}}\\right)^2\n\\lambda_{\\min}(I)\n\\]\nand\n\\[\n\\lambda_{\\max}\\left(\\sum_{i=1}^{n} w_i\\, b(x_i)b(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{M-1}{n}}\\right)^2\n\\lambda_{\\max}(J).\n\\]", "start_pos": 5360, "end_pos": 6265, "label": "thm:main"}, "ref_dict": {"cor:sampling-numbers-H": "\\begin{coro}\\label{cor:sampling-numbers-H}\nLet $(D,\\mu)$ be a measure space \nand $H$ a reproducing kernel Hilbert space \nsatisfying \\eqref{cond:continuous}, \\eqref{cond:injective} and \\eqref{eq:finite-trace}.\nThen, for all $m\\in \\N$ and $n\\geq m$,\n\\[\ng_n(B_H,L_2) \n\\,\\leq\\, \\left(1+\\frac{1}{1-r}\\right)\n\\left(\\sigma_m + \\sqrt{\\frac{1}{n}\\sum_{k> m} \\sigma_k^2} \\right),\n\\]\nwhere $r=\\sqrt\\frac{m-1}{n}$ and $\\sigma_0 \\geq \\sigma_1 \\geq \\hdots$ are the singular values\nof the embedding $H\\hookrightarrow L_2$.\n\\end{coro}", "coro:intro-equal": "\\begin{prop}\n\\label{coro:intro-equal}\nLet $(D,\\mu)$ be a probability space, and \n$V_m\\subset L_2(D,\\mu)\\cap B(D)$ with $\\dim(V_m)=m$. \nFor $n\\geq m$, there exist points $x_1,\\hdots,x_{2n}\\in D$ such that,\nfor all $f\\in V_m$ and $2\\leq p\\leq\\infty$, \n\\[\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)\n\\cdot \\|f\\|_p \n\\;\\leq\\; \n\\,m^{\\frac12-\\frac1p}\\cdot\n\\sqrt{\\frac{1}{n}\\sum_{i=1}^{2n} |f(x_i)|^2}.\n\\]\nIn the case $p\\in\\{2,\\infty\\}$, we can take $n$ points instead of $2n$ points.\\\\\nIn the case $n=m$, the factor $1-\\sqrt{\\frac{m}{n}}$ can be replaced by $\\frac{1}{2m}$.\n\\end{prop}", "cor:recovery": "\\begin{coro}\\label{cor:recovery}\nLet $(D,\\mu)$ be a measure space. \nLet $H$ be a reproducing kernel Hilbert space satisfying conditions \\eqref{cond:continuous}, \\eqref{cond:injective} and \\eqref{eq:finite-trace}.\nLet $V_m \\subset H$ be an $m$-dimensional subspace and\nlet $n\\geq m$.\nThen, there are points $x_1,\\hdots,x_n \\in D$\nand weights $w_1,\\hdots,w_n>0$\nsuch that, for any $f\\in H$, the weighted least squares approximation~\\eqref{def:wls} with $y_i=f(x_i)$ satisfies\n\\[\n\\big\\|f- \\tilde f\\big\\|_2 \n\\,\\leq\\, \n\\left(1+\\frac{1+s}{1-r}\\right)\\sqrt{\\lambda_m} \\,\\big\\|f-P_m f\\big\\|_H,\n\\]\nwhere\n\\[\n\\lambda_m:=\\sup_{f \\in H_m} \\frac{\\Vert f \\Vert_2^2}{\\Vert f \\Vert_{H}^2},\n\\qquad\nr:=\\sqrt\\frac{m-1}{n}\n\\quad\\text{and}\\quad\ns:=\\sqrt{\\frac{\\Tr(K_m)-\\lambda_m}{n\\,\\lambda_m}}.\n\\] \nHere, $H_m$ is the $H$-orthogonal complement of $V_m$,\n$P_m$ is the $H$-orthogonal projection onto $V_m$,\nand $K_m$ is the reproducing kernel of $(H_m,\\|\\cdot\\|_H)$.\n\\end{coro}", "rk:weight_control": "\\begin{prop}\n\\label{rk:weight_control}\nLet the assumptions of Corollary~\\ref{coro:main} hold.\nIf $(D,\\mu)$ is a probability space and $\\int f\\,d\\mu=0$ for all $f\\in H$,\nthen for $n\\geq m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ \nsuch that \\eqref{eq-low-coro-main} holds, \\eqref{eq-up-coro-main} holds with $M$ instead of $M-1$, and \n\\[\n\\sum_{i=1}^n w_i \\leq \\left(1+\\sqrt\\frac{M}{n}\\right)^2.\n\\]\n\\end{prop}", "coro:noisy-recovery": "\\begin{coro}\n\\label{coro:noisy-recovery}\nAssume that $(D,\\mu)$ is a probability space, that $H$ satisfies conditions \\eqref{cond:continuous}, \\eqref{cond:injective} and \\eqref{eq:finite-trace}, and that $V_m \\subset H$ has dimension $m$. \nMoreover, assume that \n$V_m$ contains the Riesz representer of the integral, i.e., \n\\[\nh=\\int K(x,\\cdot)\\,d\\mu(x) \\,\\in V_m.\n\\]\nThen, for $n \\ge m$, \nthere are points\n$x_1,\\hdots,x_n \\in D$\nand weights $w_1,\\hdots,w_n>0$\nsuch that, for any $f\\in H$ and $e\\in \\C^n$, the least squares approximation \\eqref{def:wls} on $V_m$ with noisy measurements $y_i=f(x_i)+e_i$ satisfies\n\\begin{equation}\n\\label{eq-prop-noisy}\n\\big\\|f- \\tilde f\\big\\|_2 \n\\,\\leq\\, \n\\frac{1+\\tilde s}{1-r}\\, \\Big(2\\,\\sqrt{\\lambda_m} \\,\\big\\|f-P_m f\\big\\|_H\n+\\,\\Vert e \\Vert_{\\infty}\\Big),\n\\end{equation}\nwith $\\lambda_m$, $K_m$, $P_m$ and $r$ as in Corollary~\\ref{cor:recovery}, \nand $\\tilde s=\\sqrt{\\frac{\\Tr(K_m)}{n\\,\\lambda_m}}$.\n\\end{coro}", "thm:main": "\\begin{theorem}\n\\label{thm:main}\nLet $(D,\\mu)$ be a measure space, and let $a=(a_1,\\hdots,a_m)^\\top$\nand $b= (b_k)_{k\\in \\mathbb I}^\\top$ be families of square-integrable functions on $D$,\nwhere $\\mathbb I$ is at most countable.\nAssume that $I:=\\int_D a(x)a(x)^*d\\mu(x)$ and $J:=\\int_D b(x)b(x)^*d\\mu(x)$ are positive definite, and that $J$\nhas finite trace and effective dimension\n\\[\n\\Tr(J) \\,=\\, \\sum_{k\\in \\mathbb I} \\|b_k\\|_{L_2(D,\\mu)}^2 \\,<\\,\\infty\n\\quad\\text{and}\\quad M:=\\frac{\\Tr(J)}{\\lambda_{\\max}(J)}.\n\\]\nThen, for any\n$n\\geq m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^{n} w_i\\, \na(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m-1}{n}}\\right)^2\n\\lambda_{\\min}(I)\n\\]\nand\n\\[\n\\lambda_{\\max}\\left(\\sum_{i=1}^{n} w_i\\, b(x_i)b(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{M-1}{n}}\\right)^2\n\\lambda_{\\max}(J).\n\\]\n\\end{theorem}", "fig:graph-results": "\\begin{figure}[H]\n\\label{fig:graph-results}\n\\centering\n\\begin{tikzcd}\n\\rm Prop.~\\ref{thm:BSS}\n&\\rm Prop.~\\ref{prop:BDM} \\arrow[l]\n&\\rm Thm.~\\ref{thm:main} \\arrow[l] \\arrow[ld] \\arrow[rd]\n&\\rm Prop.~\\ref{prop:KW} \\arrow[d] \\\\\n\\rm Prop.~\\ref{rk:weight_control} \\arrow[d]\n&\\rm Cor.~\\ref{coro:main} \\arrow[l] \\arrow[d] \\arrow[rr]\n& &\\rm Prop.~\\ref{coro:intro-equal} \\arrow[d] \\\\\n\\rm Cor.~\\ref{coro:noisy-recovery}\n&\\rm Cor.~\\ref{cor:recovery} \\arrow[l] \\arrow[r]\n&\\rm Cor.~\\ref{cor:sampling-numbers-H}\n&\\rm Cor.~\\ref{cor:recovery-unif}\n\\end{tikzcd}\n\\caption{Summary of the results of Sections~\\ref{sec:intro}, \\ref{sec:discretization} and \\ref{sec:sampling}, with arrows denoting implications. The first line contains linear algebra results, the second deals with norm discretization and the third with sampling recovery. The lower left block details the reproducing kernel Hilbert space (RKHS) setting, with the second column directly applying Thm.~\\ref{thm:main}, while the first column adds a control on the weights, and the third looks at sampling numbers. Lastly, the fourth column is focused on the $L_p$-setting, in which all weights are equal.}\n\\end{figure}", "ex:mixed-smoothness": "\\begin{example}\n\\label{ex:mixed-smoothness}\n    An example, where Theorem~\\ref{thm:constructive} could be applied\n    is the approximation on Sobolev spaces $H^\\alpha_{\\rm mix}$\n    of $d$-variate periodic functions with mixed smoothness $\\alpha>1/2$.\n    These can be defined by \n    \\[\n     H^\\alpha_{\\rm mix} =\n     \\left\\{ f \\in C([0,1]^d) \\colon \\Vert f \\Vert_{\\alpha}^2 := \\sum_{ k \\in \\Z^d} w_{ k}^{2\\alpha} |\\hat{f}( k)|^2 < \\infty \\right\\},\n    \\]\n    where $\\hat{f}( k)$ are the usual Fourier coefficients and $w_{ k} = \\prod_{i=1}^d \\max\\{1,|k_i|\\}$. \n    For integer $\\alpha$, they can also be defined as spaces of functions possessing (weak) derivatives \n    in $L_2$ up to mixed order $\\alpha$,\n    see, e.g., \\cite[Sec.\\,2.1]{KSU} for a few details,\n    and \\cite{DTU} for a comprehensive treatment of dominating mixed smoothness.\n    If we choose the functions $\\eta_k$ as the trigonometric monomials\n    with the frequencies $k \\in \\Z^d$ ordered according to their hyperbolic distance $w_{k}$ to the origin\n    and $\\alpha_0>1/2$,\n    the algorithm from Theorem~\\ref{thm:constructive} satisfies\n    \\begin{equation}\\label{eq:order-mix}\n     \\Vert f - \\tilde f \\Vert_2 \\,\\lesssim\\, m^{-\\alpha} \n     (\\log m)^{\\alpha(d-1)} \n     \\cdot \\Vert f \\Vert_{\\alpha},\n    \\end{equation}\n    for any smoothness $\\alpha\\geq \\alpha_0$.\n\n    We present this example since\n    for the class $H^\\alpha_{\\rm mix}$,\n    no (polynomial time) construction was previously known to give the optimal order~\\eqref{eq:order-mix}.\n    Only the existence of such algorithms was known due to~\\cite{DKU}.\n    Previous constructions that come close to the optimal order include sparse grids (Smolyak's algorithm), see~\\cite{SU},\n    random points~\\cite{KU1},\n    subsamples of random points~\\cite{BSU}\n    and generated sets (lattices with a non-integer generating vector), see~\\cite{CNW}.\n    \\end{example}", "thm:BSS": "\\begin{prop}[\\cite{BSS}, Theorem 3.1]\n\\label{thm:BSS}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$\nbe an orthonormal family in $L_2(D,\\mu)$.\nThen, for any $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^n w_i\\, a(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\lambda_{\\max}\\left(\\sum_{i=1}^n w_ia(x_i)a(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{m}{n}}\\right)^2.\n\\]\n\\end{prop}", "prop:BDM": "\\begin{prop}[\\cite{BDM14}, Lemma 13]\n\\label{prop:BDM}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$ and $b=(b_1,\\dots,b_M)^\\top$\nbe orthonormal families in $L_2(D,\\mu)$.\nThen for any integer $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^{n} w_i\\, \na(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\,\n\\lambda_{\\max}\\left(\\sum_{i=1}^{n} w_i\\, b(x_i)b(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{M}{n}}\\right)^2.\n\\]\n\\end{prop}", "coro:main": "\\begin{coro} \\label{coro:main}\nLet $(D,\\mu)$ be a measure space, \n$V_m\\subset L_2(D,\\mu)$ be an $m$-dimensional space of functions and let $H$ be \na reproducing kernel Hilbert space satisfying conditions \\eqref{cond:continuous}, \\eqref{cond:injective} and \\eqref{eq:finite-trace}. \nFor $n\\geq m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\begin{equation}\n\\label{eq-low-coro-main}\n\\left(1-\\sqrt{\\frac{m-1}{n}}\\right)\\,\\|f\\|_2\n\\,\\leq\\,\n\\sqrt{\\sum_{i=1}^n w_i|f(x_i)|^2}\n\\qquad \\text{for all } f \\in V_m\n\\end{equation}\nand \n\\begin{equation}\n\\label{eq-up-coro-main}\n\\sqrt{\\sum_{i=1}^n w_i|g(x_i)|^2}\n\\,\\leq\\,\n\\left(1 +\\sqrt{\\frac{M-1}{n}} \\right)\\, \\sqrt\\lambda\\,\n\\left\\| g \\right\\|_H\n\\qquad \\text{for all } g \\in H,\n\\end{equation}\nwhere $M=\\Tr(K)/\\lambda$ is the ``effective dimension'' of $H$ in $L_2(D,\\mu)$.\n\\end{coro}", "subsec:numerics": "\\label{subsec:numerics}\n\nWe now discuss some numerical examples\nfor the sampling points and weights \nfrom Theorem~\\ref{thm:constructive}.\nFigures~\\ref{fig:trig-blocked}--\\ref{fig:legendre-new} show th", "alg:abstract-construction": "\\label{alg:abstract-construction}\n\\end{algorithm}\n\nRegarding the first issue, one option is to find a finite subset of $D$ in which at least one point satisfies the condition of line~\\ref{state:test}.", "prop:KW": "\\begin{prop}[\\cite{KW}]\\label{prop:KW}\nLet $D$ be a set and $V_m$ be an $m$-dimen\\-sion\\-al subspace of~$B(D)$. \nFor every $\\varepsilon>0$, \nthere exists a finitely supported measure $\\mu_\\eps$ on~$D$,\nsuch that \n\\[\n \\|f\\|_{\\infty} \\,\\leq\\, \\sqrt{m+\\varepsilon} \\cdot \\Vert f \\Vert_{L_2(D,\\mu_\\eps)}. \n\\]  \n\\end{prop}", "thm:constructive": "\\begin{theorem}\n\\label{thm:constructive}\nLet $(D,\\mu)$ be a probability space, $(\\eta_k)_{k\\in\\II}$ an $L_2$-orthonormal system and $(I_\\ell)_{\\ell\\in\\N}$ nested index sets with $|I_\\ell|\\le\\ell$ that satisfy condition~\\eqref{eq:assum-uniform-bound}, $m\\geq 2$, $n=2 m$, $\\alpha_0>\\theta$ and $N=\\lceil m^{\\alpha_0/(\\alpha_0-\\theta)}\\rceil$.\nAssume that the constant function $1$ belongs to $\\{\\eta_k,\\;k\\in I_m\\}$. Algorithm~\\ref{alg:abstract-construction} can be implemented, with $a$ and $b$ defined in \\eqref{eq:a-b-constructive}, in time\n\\[\nO\\big((c+Nn)n^2\\mathcal G\\big),\n\\]\nwhere $c$ is the cost of drawing one point from the Christoffel density, and ${\\mathcal G\\in\\N}$ is a shifted geometric random variable of parameter $1/2$. It yields points $x_1,\\dots,x_n\\in D$ and weights $w_1,\\dots,w_n>0$ which satisfy the following.\nIf $f\\in L_2(D,\\mu)$ with\n\\[\n\\|f-f_{I_\\ell}\\|_2\\leq C_f\\, \\ell^{-\\alpha}\\log^\\beta(\\ell)\n\\]\nfor some $\\alpha\\geq \\alpha_0$ and $\\beta\\geq 0$, and if\n$f(x_i)=\\lim_{\\ell\\to\\infty}f_{I_\\ell}(x_i)$ for $1\\leq i\\leq n$,\nthe weighted least-squares approximation \\eqref{def:wls}\nfrom these values satisfies\n\\[\n\\|f-\\tilde f\\|_2\\leq C_f\\,C\\, m^{-\\alpha}\\log^\\beta(m),\n\\]\nwhere $C$ depends on $\\theta$, $\\alpha_0$, $\\alpha$, $\\beta$ and $C_\\eta$. \n\\end{theorem}", "cor:recovery-unif": "\\begin{coro}\\label{cor:recovery-unif}\nLet $(D,\\mu)$ be a probability space, $V_m\\subset L_2(D,\\mu)\\cap B(D)$ with $\\dim(V_m)=m$, \nand $2\\le p\\le\\infty$. \nFor all $n> m$,\nthere are points $x_1,\\hdots,x_{2n} \\in D$\nsuch that the plain least squares \napproximation~\\eqref{def:pls} \nwith $y_i = f(x_i) + e_i$\nsatisfies\n\\[\n\\big\\|f- \\tilde f\\big\\|_p \n\\,\\leq\\, \\left(1 + \\sqrt{2}\\cdot\\frac{m^{1/2-1/p}}{1-\\sqrt{m/n}}\\right) \\Big( \\min_{g\\in V_m} \\,\\big\\|f-g\\big\\|_\\infty + \\Vert e \\Vert_\\infty \\Big)\n\\]\nfor all $f\\in B(D)$ and any noise $e_1,\\hdots,e_n \\in \\C$.\nFor $p\\in\\{2,\\infty\\}$, we can use $n$ instead of $2n$ points, remove the factor $\\sqrt{2}$, \nand it suffices to have $V_m \\subset L_2$ or $V_m \\subset B(D)$, respectively.\n\\end{coro}"}, "pre_theorem_intro_text_len": 3025, "pre_theorem_intro_text": "\\label{sec:intro}\nIn recent years, much\nprogress has been made in norm discretization and sampling recovery, first\nby considering concentration inequalities for random points,\nthen by applying a sample sparsification using \\cite{MSS} or \\cite{BSS}. \nThis follows the trend observed in other fields, including graph sparsification, frame discretization, matrix\nsketching, subspace selection, and more generally randomized numerical linear algebra.\n\nHere, we prove a generalization of the result from~\\cite{BSS} by Batson, Spielman, and Srivastava (BSS), and expand on its implications\nfor the problems of norm discretization and sampling recovery. \nThe original result from~\\cite{BSS} can be stated as follows:\n\n\\begin{prop}[\\cite{BSS}, Theorem 3.1]\n\\label{thm:BSS}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$\nbe an orthonormal family in $L_2(D,\\mu)$.\nThen, for any $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^n w_i\\, a(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\lambda_{\\max}\\left(\\sum_{i=1}^n w_ia(x_i)a(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{m}{n}}\\right)^2.\n\\]\n\\end{prop}\n\nHere and in the following, we denote by \n$\\lambda_{\\min}(A)$ and $\\lambda_{\\max}(A)$ the smallest and largest eigenvalues of $A$.\nThe points and weights from Proposition~\\ref{thm:BSS} can be constructed in polynomial time.\n\nIn \\cite{BSS}, this theorem is applied to the problem of graph sparsification, see also \\cite{HGN+24,SS11,ST11}.\nFor our applications, we want to generalize the result in three ways: by allowing general measure spaces $(D,\\mu)$; by decoupling the lower and upper frame bounds with two different\nfamilies of $L_2$-functions; and by making the upper frame bound independent of the dimension, thus allowing to consider very large and even infinite-dimensional families. \n\nThe first two kinds of generalizations are already available in the literature. General probability spaces are treated, for instance, in \\cite[Theorem~6.4]{DPSTT}.\nThe second kind of generalization was achieved in \\cite{BDM14}.\nThe result of \\cite{BDM14}, called dual set spectral sparsification, can be reformulated as follows.\n\n\\begin{prop}[\\cite{BDM14}, Lemma 13]\n\\label{prop:BDM}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$ and $b=(b_1,\\dots,b_M)^\\top$\nbe orthonormal families in $L_2(D,\\mu)$.\nThen for any integer $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^{n} w_i\\, \na(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\,\n\\lambda_{\\max}\\left(\\sum_{i=1}^{n} w_i\\, b(x_i)b(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{M}{n}}\\right)^2.\n\\]\n\\end{prop}\n\nOur main contribution, stated below, is the third kind of generalization. Namely, we replace the dimension $M$ by an effective dimension in the\nupper frame bound.\n\\smallskip", "context": "\\label{sec:intro}\nIn recent years, much\nprogress has been made in norm discretization and sampling recovery, first\nby considering concentration inequalities for random points,\nthen by applying a sample sparsification using \\cite{MSS} or \\cite{BSS}. \nThis follows the trend observed in other fields, including graph sparsification, frame discretization, matrix\nsketching, subspace selection, and more generally randomized numerical linear algebra.\n\n\\begin{prop}[\\cite{BSS}, Theorem 3.1]\n\\label{thm:BSS}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$\nbe an orthonormal family in $L_2(D,\\mu)$.\nThen, for any $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^n w_i\\, a(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\lambda_{\\max}\\left(\\sum_{i=1}^n w_ia(x_i)a(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{m}{n}}\\right)^2.\n\\]\n\\end{prop}\n\nHere and in the following, we denote by \n$\\lambda_{\\min}(A)$ and $\\lambda_{\\max}(A)$ the smallest and largest eigenvalues of $A$.\nThe points and weights from Proposition~\\ref{thm:BSS} can be constructed in polynomial time.\n\nIn \\cite{BSS}, this theorem is applied to the problem of graph sparsification, see also \\cite{HGN+24,SS11,ST11}.\nFor our applications, we want to generalize the result in three ways: by allowing general measure spaces $(D,\\mu)$; by decoupling the lower and upper frame bounds with two different\nfamilies of $L_2$-functions; and by making the upper frame bound independent of the dimension, thus allowing to consider very large and even infinite-dimensional families.\n\n\\begin{prop}[\\cite{BDM14}, Lemma 13]\n\\label{prop:BDM}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$ and $b=(b_1,\\dots,b_M)^\\top$\nbe orthonormal families in $L_2(D,\\mu)$.\nThen for any integer $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^{n} w_i\\, \na(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\,\n\\lambda_{\\max}\\left(\\sum_{i=1}^{n} w_i\\, b(x_i)b(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{M}{n}}\\right)^2.\n\\]\n\\end{prop}\n\nOur main contribution, stated below, is the third kind of generalization. Namely, we replace the dimension $M$ by an effective dimension in the\nupper frame bound.\n\\smallskip", "full_context": "\\label{sec:intro}\nIn recent years, much\nprogress has been made in norm discretization and sampling recovery, first\nby considering concentration inequalities for random points,\nthen by applying a sample sparsification using \\cite{MSS} or \\cite{BSS}. \nThis follows the trend observed in other fields, including graph sparsification, frame discretization, matrix\nsketching, subspace selection, and more generally randomized numerical linear algebra.\n\n\\begin{prop}[\\cite{BSS}, Theorem 3.1]\n\\label{thm:BSS}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$\nbe an orthonormal family in $L_2(D,\\mu)$.\nThen, for any $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^n w_i\\, a(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\lambda_{\\max}\\left(\\sum_{i=1}^n w_ia(x_i)a(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{m}{n}}\\right)^2.\n\\]\n\\end{prop}\n\nHere and in the following, we denote by \n$\\lambda_{\\min}(A)$ and $\\lambda_{\\max}(A)$ the smallest and largest eigenvalues of $A$.\nThe points and weights from Proposition~\\ref{thm:BSS} can be constructed in polynomial time.\n\nIn \\cite{BSS}, this theorem is applied to the problem of graph sparsification, see also \\cite{HGN+24,SS11,ST11}.\nFor our applications, we want to generalize the result in three ways: by allowing general measure spaces $(D,\\mu)$; by decoupling the lower and upper frame bounds with two different\nfamilies of $L_2$-functions; and by making the upper frame bound independent of the dimension, thus allowing to consider very large and even infinite-dimensional families.\n\n\\begin{prop}[\\cite{BDM14}, Lemma 13]\n\\label{prop:BDM}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$ and $b=(b_1,\\dots,b_M)^\\top$\nbe orthonormal families in $L_2(D,\\mu)$.\nThen for any integer $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^{n} w_i\\, \na(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\,\n\\lambda_{\\max}\\left(\\sum_{i=1}^{n} w_i\\, b(x_i)b(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{M}{n}}\\right)^2.\n\\]\n\\end{prop}\n\nOur main contribution, stated below, is the third kind of generalization. Namely, we replace the dimension $M$ by an effective dimension in the\nupper frame bound.\n\\smallskip\n\n\\begin{prop}[\\cite{BDM14}, Lemma 13]\n\\label{prop:BDM}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$ and $b=(b_1,\\dots,b_M)^\\top$\nbe orthonormal families in $L_2(D,\\mu)$.\nThen for any integer $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^{n} w_i\\, \na(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\,\n\\lambda_{\\max}\\left(\\sum_{i=1}^{n} w_i\\, b(x_i)b(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{M}{n}}\\right)^2.\n\\]\n\\end{prop}\n\nOur main contribution, stated below, is the third kind of generalization. Namely, we replace the dimension $M$ by an effective dimension in the\nupper frame bound.\n\\smallskip\n\n\\smallskip\n\n\\begin{coro} \\label{coro:main}\nLet $(D,\\mu)$ be a measure space, \n$V_m\\subset L_2(D,\\mu)$ be an $m$-dimensional space of functions and let $H$ be \na reproducing kernel Hilbert space satisfying conditions \\eqref{cond:continuous}, \\eqref{cond:injective} and \\eqref{eq:finite-trace}. \nFor $n\\geq m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\begin{equation}\n\\label{eq-low-coro-main}\n\\left(1-\\sqrt{\\frac{m-1}{n}}\\right)\\,\\|f\\|_2\n\\,\\leq\\,\n\\sqrt{\\sum_{i=1}^n w_i|f(x_i)|^2}\n\\qquad \\text{for all } f \\in V_m\n\\end{equation}\nand \n\\begin{equation}\n\\label{eq-up-coro-main}\n\\sqrt{\\sum_{i=1}^n w_i|g(x_i)|^2}\n\\,\\leq\\,\n\\left(1 +\\sqrt{\\frac{M-1}{n}} \\right)\\, \\sqrt\\lambda\\,\n\\left\\| g \\right\\|_H\n\\qquad \\text{for all } g \\in H,\n\\end{equation}\nwhere $M=\\Tr(K)/\\lambda$ is the ``effective dimension'' of $H$ in $L_2(D,\\mu)$.\n\\end{coro}\n\n\\begin{coro}\\label{cor:recovery}\nLet $(D,\\mu)$ be a measure space. \nLet $H$ be a reproducing kernel Hilbert space satisfying conditions \\eqref{cond:continuous}, \\eqref{cond:injective} and \\eqref{eq:finite-trace}.\nLet $V_m \\subset H$ be an $m$-dimensional subspace and\nlet $n\\geq m$.\nThen, there are points $x_1,\\hdots,x_n \\in D$\nand weights $w_1,\\hdots,w_n>0$\nsuch that, for any $f\\in H$, the weighted least squares approximation~\\eqref{def:wls} with $y_i=f(x_i)$ satisfies\n\\[\n\\big\\|f- \\tilde f\\big\\|_2 \n\\,\\leq\\, \n\\left(1+\\frac{1+s}{1-r}\\right)\\sqrt{\\lambda_m} \\,\\big\\|f-P_m f\\big\\|_H,\n\\]\nwhere\n\\[\n\\lambda_m:=\\sup_{f \\in H_m} \\frac{\\Vert f \\Vert_2^2}{\\Vert f \\Vert_{H}^2},\n\\qquad\nr:=\\sqrt\\frac{m-1}{n}\n\\quad\\text{and}\\quad\ns:=\\sqrt{\\frac{\\Tr(K_m)-\\lambda_m}{n\\,\\lambda_m}}.\n\\] \nHere, $H_m$ is the $H$-orthogonal complement of $V_m$,\n$P_m$ is the $H$-orthogonal projection onto $V_m$,\nand $K_m$ is the reproducing kernel of $(H_m,\\|\\cdot\\|_H)$.\n\\end{coro}\n\n\\begin{lemma}\n\\label{lem:accept_proba}\nIf $x$ is drawn according to the Christoffel measure $\\rho$ and the increment $\\delta=\\frac{1-r}{n}$ is replaced by\n\\[\n\\delta_\\epsilon=\\frac{1-r-\\epsilon}{n}\n\\]\nfor some parameter $\\epsilon < (1-r)^2$ in the computation of $L_A^\\delta$ and the updates $A\\leftarrow A'$ (but not in the initialization $A_0$),\nthen the acceptance probability satisfies\n\\[\n\\mathbb P\\left(L_A^{\\delta_\\eps}(a(x)) \\ge U_B^\\zeta(b(x))\\right) \\ge \\frac{\\epsilon}{m}.\n\\]\nIn addition, any points $x_1,\\dots x_n$ returned by our algorithm for the modified value of $\\delta$ satisfy the upper bound \\eqref{eq:UB} while the lower bound \\eqref{eq:LB} becomes\n\\[\n\\lambda_{\\min}\\left(\\sum_{i=1}^{n} w_ia(x_i)a(x_i)^*\\right)\n\\geq \\Big(n\\delta_\\epsilon+\\frac{\\delta}{r}-\\frac{m\\delta}{r}\\,\\Big)= (1-r)^2-\\epsilon.\n\\]\n\\end{lemma}\nIn the sequel, we will take $\\epsilon=\\frac{(1-r)^2}{2}$, which changes \\eqref{eq:LB} by a factor 2.\n\\begin{proof}\nWe start with $A \\in \\mathcal P_m$ and $B\\in \\mathcal P$ such that \\eqref{ineq-lemboth} holds for the original $\\delta=\\frac{1-r}{n}$.\nLet\n\\[\n\\Delta(x):= \\frac{L_A^{\\delta_\\epsilon}(a(x))- U_B^\\zeta(b(x))}{\\sum_{k=1}^m|a_k(x)|^2}.\n\\]\nWe now consider a random variable $x\\sim \\rho$.\nAccording to Lemma~\\ref{lemma:pos_measure},\nthe expectation of $\\Delta(x)$ is \n\\begin{align*}\n\\E(\\Delta(x))\n=\\int_D \\frac{L_A^{\\delta_\\epsilon}(a(x))-U_B^\\zeta(b(x))}m \\, d\\mu(x)\n\\geq \\frac{1}{m}\\left(\\frac{1}{\\delta_\\epsilon}-\\Phi(A)- \\frac{1}{\\zeta}-\\Psi(B)\\right).\n\\end{align*}\nThanks to condition \\eqref{ineq-lemboth}, \nwe get\n\\[\n\\E(\\Delta(x))\n\\geq \\frac{1}{m}\\left(\\frac{1}{\\delta_\\epsilon}-\\frac{1}{\\delta}\\right)=\\frac{\\epsilon}{m \\,\\delta_\\epsilon(1-r)}.\n\\]\nOn the other hand, $\\Delta(x)$ is almost surely bounded by\n\\begin{align*}\n\\Delta(x)\n\\leq \\sup_{\\substack{a\\in\\C^m \n}} \\frac{L_A^{\\delta_\\eps}(a)}{\\sum_{k=1}^m|a_k|^2} \n\\leq \\frac{\\lambda_{\\max}(Z^2)}{\\Tr(Z)-\\Tr(Y)}\\leq \\frac{\\Tr(Z^2)}{\\delta_\\epsilon\\Tr(YZ)}\\leq \\frac{1}{\\delta_\\epsilon(1-r)},\n\\end{align*}\nwhere $Y$ and $Z$ are from Section~\\ref{sec:main-proof} with $\\delta_\\eps$ in place of $\\delta$. Here,\nthe third inequality follows from $Z-Y=\\delta_\\epsilon YZ$\nand the last inequality comes from the fact that\n\\[\n\\Tr(Z^2)-\\Tr(YZ)=\\delta_\\epsilon \\Tr(Y Z^2)\\leq \\delta_\\epsilon\\Tr(Y)\\Tr(Z^2)\\leq r\\Tr(Z^2),\n\\]\nby a Cauchy-Schwarz inequality~\\eqref{property:CS} between $Y^{1/2}$ and $Y^{1/2}Z$ and\nsince $\\Tr(Y)=\\Phi(A)\\leq r/\\delta_\\epsilon$. In the end, we obtain an acceptance probability\n\\[\n\\mathbb P(\\Delta(x)\\geq 0)\n\\geq \\frac{\\int_D \\Delta\\, d\\rho}{\\sup_{x\\in D} \\Delta(x)}\\geq \\frac{\\epsilon}m.\n\\]\nNote that, by Lemma~\\ref{lemboth}, choosing $x_i$ with $\\Delta(x_i) \\ge 0$ and $w_i$ accordingly, we obtain that the potentials do not increase \nwhen switching to the updates \n$A':=A-\\delta_\\eps I+wa(x_i)a(x_i)^*$ and \n$ B':=B+\\zeta J-wb(x_i)b(x_i)^*$ and\nso \ncondition~\\eqref{ineq-lemboth} remains true with the original $\\delta$ also for the updated operators.\n\\end{proof}\n\n\\begin{example}\n\\label{ex:christoffel}\nFor each $m\\in\\N$, consider $D=[0,m]$ with the Lebesgue measure $\\mu$\nand define a RKHS $H$ on $D$ by its kernel $K(x,y)=\\sum_{k\\in \\II} \\sigma_k^2 b_k(x) b_k(y)$, where the functions\n\\[\nb_k := \\left\\{\\begin{array}{cl}\n\\mathbbm 1_{[k-1,k]} &\\text{ for } 1\\leq k\\leq m,\\\\\n\\cos(2\\pi (k-m)x) \\mathbbm 1_{[0,1]} &\\text{ for } k>m,\\\\\n\\sin(2\\pi (m-k)x) \\mathbbm 1_{[0,1]} &\\text{ for } k<-m,\\\\\n\\end{array}\\right.\n\\]\nare orthonormal in $L_2(D)$,\nand $\\sigma_k:=\\frac1{|k|}$ for $k\\in\\II:=\\Z\\setminus\\{-m,\\dots,0\\}$. Let also $V_m=\\Span\\{b_k \\colon 1\\leq k\\leq m\\}$. From~\\eqref{eq:n-is-2m}, we get\n\\[\n g_{2m}^{\\rm lin}(B_H,L_2) \\,\\leq\\, \\frac{10}m.\n\\]\nOn the other hand,\nlet $A_n$ be an approximation that uses information from at\nmost $n$ samples $x_1,\\hdots,x_n$.\nLet $\\{y_1,\\hdots,y_{n'}\\} = \\{x_1,\\hdots,x_n\\} \\cap [0,1]$\nbe the samples in $[0,1]$. Observing that $A_n(-f)=A_n(f)$ if $f(x_1)=\\dots f(x_n)=0$,\n\\[\n \\sup_{\\Vert f \\Vert_H \\leq 1} \\Vert f - A_n(f) \\Vert_{L_2([0,m])}\n \\geq \\sup_{\\substack{\\Vert f \\Vert_H \\leq 1 \\\\ f(x_i)=0}} \\Vert f \\Vert_{L_2([0,m])}\n \\geq \\sup_{\\substack{\\Vert f \\Vert_H \\leq 1 \\\\ f(y_i)=0}} \\Vert f \\Vert_{L_2([0,1])}.\n \\]\nAs $\\|f\\|_{L_2([0,1])}\\geq \\big|\\int_0^1f\\big|$, the lower bound from \\cite[Corollary~2]{KV} with $\\lambda_0=1$ and $\\lambda_k=\\sigma_{|k|+m}^2$ for $k\\neq 0$ gives\n \\[\n\\sup_{\\substack{\\Vert f \\Vert_H \\leq 1 \\\\ f(y_i)=0}} \\Vert f \\Vert_{L_2([0,1])}\n \\geq \\sqrt{\\min\\bigg\\{\\frac{\\lambda_0}2, \\ \\frac{1}{8n'} \\sum_{k> 4n'} \\lambda_k\\bigg\\}}\n \\geq \\frac{1}{8\\max(n',\\sqrt{mn'})}.\n\\]\nAs a consequence, the approximation $A_n$ can only attain an optimal error if $n' \\gtrsim m$.\nAs the Christoffel density of $V_m$ is uniform on $[0,m]$,\nif $x_1,\\hdots,x_n$ are drawn i.i.d. from that density, \nwe must have $n \\gtrsim m^2$ with high probability. In summary, any \nalgorithm achieving an error of optimal order \nmust use $\\Omega(m^2)$ Christoffel samples.\n\\end{example}", "post_theorem_intro_text_len": 4424, "post_theorem_intro_text": "\\smallskip\n\nOne can retrieve Proposition~\\ref{prop:BDM} from Theorem~\\ref{thm:main} by taking $\\mathbb I=\\{1,\\dots,M\\}$ and assuming that $a$ and $b$ are orthonormal families. Indeed, in that case, we see that $I$ and $J$ are the $m\\times m$ and $M\\times M$ identity matrices. In turn, Proposition~\\ref{prop:BDM} implies Proposition~\\ref{thm:BSS} by taking $M=m$ and $b=a$.\n\n\\begin{remark}\nThe factor $m-1$ instead of $m$ in the lower bound is a minor improvement\nalready appearing in \\cite{ChkifaDolbeault24}. It allows to treat the case $n=m$, with the bound\n\\[\n1-\\sqrt{\\frac{m-1}{m}}>\\frac{1}{2m}.\n\\]\nIn the same way, the factor $M-1$ instead of $M$ \nin the upper bound is most interesting when $J$ has rank one, where we recover \\cite[Lemma~14]{BDM14}.\nThis case will be used in Corollary~\\ref{coro:intro-equal} to obtain discretizations with equal weights. We will also leverage these improvements in Propositions~\\ref{rk:weight_control} and Corollary~\\ref{coro:noisy-recovery}, when considering families $a$ and $b$ containing one more function.\n\\end{remark}\n\n\\begin{remark}\nThroughout the article, $L_2(D,\\mu)$ and its subspaces are assumed to consist of complex-valued functions, not equivalence classes, which is needed for discretization. \nMoreover, it is possible to extend our results to Hilbert-valued functions, by combining them with \\cite[Theorem 2.1]{BartelDung24}.\n\\end{remark}\n\nThe proof of Theorem~\\ref{thm:main}, see Section~\\ref{sec:proof}, \nbuilds upon the original proof of Proposition~\\ref{thm:BSS},  \nand the points and weights can be constructed in a quite similar way, see Algorithm~\\ref{alg:abstract-construction} in Section~\\ref{sec:implementation}.\nAlthough the proof of Theorem~\\ref{thm:main} is self-contained, \nwe recommend reading the proof of \\cite[Theorem~3.1]{BSS} first, since it comes with a very nice physical intuition.\n\nBefore we come to the proof, we first discuss several applications of this linear algebra result.\nIn Section~\\ref{sec:discretization}, we discuss its implications to the problem of norm discretization.\nIn Section~\\ref{sec:sampling}, we use these discretization results to obtain results on the problem of sampling recovery\nand on the error of (weighted) least squares algorithms. The relation between these results is illustrated in Figure~\\ref{fig:graph-results}.\n\n\\begin{figure}[H]\n\\label{fig:graph-results}\n\\centering\n\\begin{tikzcd}\n\\rm Prop.~\\ref{thm:BSS}\n&\\rm Prop.~\\ref{prop:BDM} \\arrow[l]\n&\\rm Thm.~\\ref{thm:main} \\arrow[l] \\arrow[ld] \\arrow[rd]\n&\\rm Prop.~\\ref{prop:KW} \\arrow[d] \\\\\n\\rm Prop.~\\ref{rk:weight_control} \\arrow[d]\n&\\rm Cor.~\\ref{coro:main} \\arrow[l] \\arrow[d] \\arrow[rr]\n& &\\rm Prop.~\\ref{coro:intro-equal} \\arrow[d] \\\\\n\\rm Cor.~\\ref{coro:noisy-recovery}\n&\\rm Cor.~\\ref{cor:recovery} \\arrow[l] \\arrow[r]\n&\\rm Cor.~\\ref{cor:sampling-numbers-H}\n&\\rm Cor.~\\ref{cor:recovery-unif}\n\\end{tikzcd}\n\\caption{Summary of the results of Sections~\\ref{sec:intro}, \\ref{sec:discretization} and \\ref{sec:sampling}, with arrows denoting implications. The first line contains linear algebra results, the second deals with norm discretization and the third with sampling recovery. The lower left block details the reproducing kernel Hilbert space (RKHS) setting, with the second column directly applying Thm.~\\ref{thm:main}, while the first column adds a control on the weights, and the third looks at sampling numbers. Lastly, the fourth column is focused on the $L_p$-setting, in which all weights are equal.}\n\\end{figure}\n\nIn fact, it is quite interesting that many of the recent advances in \nthe areas of both norm discretization and sampling recovery\ncan be derived solely on the basis of Theorem~\\ref{thm:main}.\nPrevious results used the non-constructive Kadison-Singer theorem from \\cite{MSS},\nsee, e.g., \\cite{CohenDolbeault,DKU,NSU},\nwhich now becomes unnecessary.\nThis does not only simplify the proofs of those results\nbut also much improves the absolute constants.\nMoreover, the least squares algorithms based on Theorem~\\ref{thm:main} \nare implementable for various examples where \nan implementable construction of an optimal algorithm \nhas previously been unknown\n(e.g., for function spaces of mixed smoothness),\nsee Theorem~\\ref{thm:constructive} and Example~\\ref{ex:mixed-smoothness}.\nA discussion of the practical implementation, along with numerical illustrations, is provided in Sections~\\ref{sec:implementation} and~\\ref{subsec:numerics}.", "sketch": "The proof of Theorem~\\ref{thm:main} (see Section~\\ref{sec:proof}) is said to “builds upon the original proof of Proposition~\\ref{thm:BSS},” and the “points and weights can be constructed in a quite similar way,” referring to Algorithm~\\ref{alg:abstract-construction} in Section~\\ref{sec:implementation}. Although the proof is “self-contained,” the authors recommend reading the proof of \\cite[Theorem~3.1]{BSS} first “since it comes with a very nice physical intuition.”", "expanded_sketch": "The proof of the main theorem (see Section~\\ref{sec:proof}) is said to “builds upon the original proof of \\begin{prop}[\\cite{BSS}, Theorem 3.1]\n\\label{thm:BSS}\nLet $D$ be a finite set and let $\\mu$ be the uniform distribution on $D$. Let $a=(a_1,\\hdots,a_m)^\\top$\nbe an orthonormal family in $L_2(D,\\mu)$.\nThen, for any $n>m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^n w_i\\, a(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\n\\]\nand\n\\[\n\\lambda_{\\max}\\left(\\sum_{i=1}^n w_ia(x_i)a(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{m}{n}}\\right)^2.\n\\]\n\\end{prop},” and the “points and weights can be constructed in a quite similar way,” referring to Algorithm~\\ref{alg:abstract-construction} in Section~\\ref{sec:implementation}:\n\\[\n\\label{alg:abstract-construction}\n\\end{algorithm}\n\nRegarding the first issue, one option is to find a finite subset of $D$ in which at least one point satisfies the condition of line~\\ref{state:test}.\n\\]\nAlthough the proof is “self-contained,” the authors recommend reading the proof of \\cite{BSS}, Theorem~3.1 first “since it comes with a very nice physical intuition.”", "expanded_theorem": "\\label{thm:main}\nLet $(D,\\mu)$ be a measure space, and let $a=(a_1,\\hdots,a_m)^\\top$\nand $b= (b_k)_{k\\in \\mathbb I}^\\top$ be families of square-integrable functions on $D$,\nwhere $\\mathbb I$ is at most countable.\nAssume that $I:=\\int_D a(x)a(x)^*d\\mu(x)$ and $J:=\\int_D b(x)b(x)^*d\\mu(x)$ are positive definite, and that $J$\nhas finite trace and effective dimension\n\\[\n\\Tr(J) \\,=\\, \\sum_{k\\in \\mathbb I} \\|b_k\\|_{L_2(D,\\mu)}^2 \\,<\\,\\infty\n\\quad\\text{and}\\quad M:=\\frac{\\Tr(J)}{\\lambda_{\\max}(J)}.\n\\]\nThen, for any\n$n\\geq m$, there exist points $x_1,\\dots,x_n\\in D$ and weights ${w_1,\\dots,w_n>0}$ such that\n\\[\n\\lambda_{\\min}\\left( \\sum_{i=1}^{n} w_i\\, \na(x_i)a(x_i)^* \\right) \\geq\n\\left(1-\\sqrt{\\frac{m-1}{n}}\\right)^2\n\\lambda_{\\min}(I)\n\\]\nand\n\\[\n\\lambda_{\\max}\\left(\\sum_{i=1}^{n} w_i\\, b(x_i)b(x_i)^* \\right) \\leq \\left(1+\\sqrt{\\frac{M-1}{n}}\\right)^2\n\\lambda_{\\max}(J).\n\\]", "theorem_type": ["Existence", "Inequality or Bound"], "mcq": {"question": "Let $(D,\\mu)$ be a measure space. Let $a=(a_1,\\dots,a_m)^\\top$ and $b=(b_k)_{k\\in\\mathbb I}^\\top$ be families of square-integrable functions on $D$, where $\\mathbb I$ is at most countable. Define\n\\[\nI:=\\int_D a(x)a(x)^*\\,d\\mu(x),\\qquad J:=\\int_D b(x)b(x)^*\\,d\\mu(x),\n\\]\nand assume that both $I$ and $J$ are positive definite. Assume also that $J$ has finite trace\n\\[\n\\operatorname{Tr}(J)=\\sum_{k\\in\\mathbb I}\\|b_k\\|_{L_2(D,\\mu)}^2<\\infty,\n\\]\nand define its effective dimension by\n\\[\nM:=\\frac{\\operatorname{Tr}(J)}{\\lambda_{\\max}(J)}.\n\\]\nHere $\\lambda_{\\min}(\\cdot)$ and $\\lambda_{\\max}(\\cdot)$ denote the smallest and largest eigenvalues. For a given integer $n\\ge m$, which quantitative estimate is valid?", "correct_choice": {"label": "A", "text": "There exist points $x_1,\\dots,x_n\\in D$ and weights $w_1,\\dots,w_n>0$ such that\n\\[\n\\lambda_{\\min}\\!\\left(\\sum_{i=1}^n w_i\\,a(x_i)a(x_i)^*\\right)\n\\ge \\left(1-\\sqrt{\\frac{m-1}{n}}\\right)^2\\lambda_{\\min}(I)\n\\]\nand\n\\[\n\\lambda_{\\max}\\!\\left(\\sum_{i=1}^n w_i\\,b(x_i)b(x_i)^*\\right)\n\\le \\left(1+\\sqrt{\\frac{M-1}{n}}\\right)^2\\lambda_{\\max}(J).\n\\]"}, "choices": [{"label": "B", "text": "There exist points $x_1,\\dots,x_n\\in D$ and weights $w_1,\\dots,w_n>0$ such that\n\\[\n\\lambda_{\\min}\\!\\left(\\sum_{i=1}^n w_i\\,a(x_i)a(x_i)^*\\right)\n\\ge \\left(1-\\sqrt{\\frac{m}{n}}\\right)^2\\lambda_{\\min}(I)\n\\]\nand\n\\[\n\\lambda_{\\max}\\!\\left(\\sum_{i=1}^n w_i\\,b(x_i)b(x_i)^*\\right)\n\\le \\left(1+\\sqrt{\\frac{M}{n}}\\right)^2\\lambda_{\\max}(J).\n\\]"}, {"label": "C", "text": "There exist points $x_1,\\dots,x_n\\in D$ and weights $w_1,\\dots,w_n>0$ such that\n\\[\n\\lambda_{\\min}\\!\\left(\\sum_{i=1}^n w_i\\,a(x_i)a(x_i)^*\\right)\n\\ge \\left(1-\\sqrt{\\frac{m-1}{n}}\\right)^2\\lambda_{\\min}(I).\n\\]"}, {"label": "D", "text": "There exist points $x_1,\\dots,x_n\\in D$ and weights $w_1,\\dots,w_n>0$, depending only on $n$, $m$, and $M$, such that\n\\[\n\\lambda_{\\min}\\!\\left(\\sum_{i=1}^n w_i\\,a(x_i)a(x_i)^*\\right)\n\\ge \\left(1-\\sqrt{\\frac{m-1}{n}}\\right)^2\\lambda_{\\min}(I)\n\\]\nand\n\\[\n\\lambda_{\\max}\\!\\left(\\sum_{i=1}^n w_i\\,b(x_i)b(x_i)^*\\right)\n\\le \\left(1+\\sqrt{\\frac{M-1}{n}}\\right)^2\\lambda_{\\max}(J).\n\\]"}, {"label": "E", "text": "For every choice of points $x_1,\\dots,x_n\\in D$, there exist weights $w_1,\\dots,w_n>0$ such that\n\\[\n\\lambda_{\\min}\\!\\left(\\sum_{i=1}^n w_i\\,a(x_i)a(x_i)^*\\right)\n\\ge \\left(1-\\sqrt{\\frac{m-1}{n}}\\right)^2\\lambda_{\\min}(I)\n\\]\nand\n\\[\n\\lambda_{\\max}\\!\\left(\\sum_{i=1}^n w_i\\,b(x_i)b(x_i)^*\\right)\n\\le \\left(1+\\sqrt{\\frac{M-1}{n}}\\right)^2\\lambda_{\\max}(J).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "sharp m-1 and M-1 boundary terms", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "upper eigenvalue estimate for the $b$-family", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "dependence of selected points and weights on the actual families $a,b$ and operators $I,J$", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "existential choice of sample points produced by the construction", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly or implicitly reveal option A. It gives the hypotheses and asks for the valid estimate, but no wording singles out the exact constants or quantifiers in the correct choice."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem presents the full setup and asks which conclusion is valid. The correct option is basically the exact theorem statement rather than a conclusion derived from independent reasoning."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in checking sharp constants, completeness of the conclusion, and quantifier dependence across the options. However, the task is still mostly recognition of the exact theorem statement rather than substantive mathematical generation."}, "DQS": {"score": 2, "justification": "The distractors are strong: B alters sharp boundary terms, C gives a weaker partial conclusion, D mishandles dependence/quantifiers, and E improperly strengthens the existential claim to a universal one. These are plausible and mathematically meaningful failure modes."}, "total_score": 5, "overall_assessment": "A solid recall/discrimination MCQ with strong distractors and no answer leakage, but it is largely tautological because it mainly tests recognition of the theorem’s exact statement rather than deeper generative reasoning."}}
{"id": "2602.18784v1", "paper_link": "http://arxiv.org/abs/2602.18784v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:single}\n   Let the infection rate be $\\alpha$ and the recovery rate be $\\mu$ for a single-type SIR process on $GW (p)$. At time 0, only the root is infected. The probability of survival is greater than 0 if and only if   $   \\alpha/\\mu > 1/{(m-1)}   $, where $m$ is the mean of $p$. Thus, if $\\mu$ is fixed, then the critical value for $\\alpha$ is  $\\mu/(m-1)$.", "start_pos": 13144, "end_pos": 13549, "label": "thm:single"}, "ref_dict": {"rates table": "\\begin{table}[h]\n       \\label{rates table}\n\\begin{center}\n\\begin{tabular}{c|c|c|c}\n\\hline\n& original infection rate   & increased infection rate & recovery rate\\\\\n\\hline \ndisease $A$ &  $\\alpha_1$  & $\\beta_1\\in [\\alpha_1,\\infty]$ & $\\mu_1$\\\\\n\\hline \ndisease $B$ &  $\\alpha_2$  & $\\beta_2\\in [\\alpha_2, \\infty]$  & $\\mu_2$\\\\\n\\hline \n\\end{tabular}\n\\end{center}\n\\caption{Parameters for the two-type SIR process.}\n    \\end{table}", "thm:single": "\\begin{theorem}\\label{thm:single}\n   Let the infection rate be $\\alpha$ and the recovery rate be $\\mu$ for a single-type SIR process on $GW (p)$. At time 0, only the root is infected. The probability of survival is greater than 0 if and only if   $   \\alpha/\\mu > 1/{(m-1)}   $, where $m$ is the mean of $p$. Thus, if $\\mu$ is fixed, then the critical value for $\\alpha$ is  $\\mu/(m-1)$. \n   \\end{theorem}", "eq:ode": "\\begin{equation}    \\label{eq:ode}\n   \\begin{cases}\n      s'(t)=-2\\alpha s(t)x(t),  \\\\  \n  q'(t)=(\\alpha s(t)-C \\alpha q(t))x(t), \\\\\n  x'(t)=(\\alpha s(t)+C \\alpha q(t))x(t)-x(t). \n    \\end{cases}\n\\end{equation}", "rmk:c=infty": "\\begin{remark}\\label{rmk:c=infty}\nIn the special case of equal infection/recovery rates for both diseases (denoted by $\\alpha,\\beta=C\\alpha$ and $\\mu$)  with $C=\\infty$,  the conclusion of    Theorem \\ref{general}  is consistent with  the prediction made in \\cite{grassberger2016phase} regarding infinite tree graphs, where the authors claimed that (for $C=\\infty$), \n$$\n\\mathbb{P}(\\mbox{infinitely many vertices are infected with both diseases})>0\\, \\mbox{ if and only if }\\, \\frac{\\alpha}{\\mu}>\\frac{1}{m-1}. \n$$ See also the discussions above Table       \\ref{rates table}.\n\\end{remark}", "general": "\\begin{theorem}\n    \\label{general}\nConsider the two-type SIR process on $GW (p)$ with parameters $\\alpha_i$, $\\beta_i$, $\\mu_i$, $i=1,2$. At time $0$, the root is infected with both disease $A$ and disease $B$. All other vertices are susceptible to both $A$ and $B$. Let $m$ be the mean of $p$. If\n    \\begin{equation}\\label{eq:assump}\n      \\max\\left\\{\\frac{\\alpha_1}{\\mu_1},\\frac{\\alpha_2}{\\mu_2}\\right\\}\\leq \\frac{1}{m-1}\n      \\textnormal {,}\n    \\end{equation}\nthen the probability of survival is $0$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 9852, "pre_theorem_intro_text": "SIR models and their variants have been widely used to predict and control the disease spreading process.  In the standard single-type SIR process based on a graph, each node can be in one of three states: S (representing `susceptible'), I (representing `infected') and R (representing `recovered'). Vertices in state I try to infect their neighbors at rate $\\alpha$, independently across each edge, and recover (turn into state R) at rate $\\mu$. Alternatively, in terms of species evolution, one might imagine that each individual gives birth after a random amount of time distributed as Exp($\\alpha$)  and dies after time distributed as Exp($\\mu$), where all the exponential random variables are independent. A closely related model is the SIS model, also called the contact process, where a vertex turns into state S instead of R when it recovers.\n\nResearchers have also been fascinated by the behavior of epidemic models that incorporate interactions (competition or cooperation) among multiple species.  Two-type SIS models with competitions\nfor space on the $d$-dimensional integer lattice $\\mathbb{Z}^d$\nwere investigated in \\cite{MounTford2019,Neuhauser1992,Stover2020}.\nFor the case of equal death rates,  Neuhauser \\cite{Neuhauser1992} proved that the species with the smaller birth rate dies out locally, while \nMountford,  Pantoja, and Valesin \\cite{MounTford2019}  proved that the winner takes over a ball whose radius grows linearly over time.\nNeuhauser conjectured that for general death rates, the species with the higher birth/death ratio wins the competition, which has been verified by Stover \\cite{Stover2020} for certain cases but still remains largely open.\nFor cooperative interactions, Durrett and Yao \\cite{DurrettYao2020} considered a symbiotic contact process on $\\mathbb{Z}^d$\nwhere the presence of one species can reduce the death rate of the other type at the same site. It was proved that strong symbiosis can lower the critical value, but the general case (especially weak  symbiosis) remains unclear.\n\nIn another direction, Lanchier and Neuhauser\n\\cite{lanchier2010stochastic,lanchier2006stochastic} studied stochastic models with hosts and symbionts. In these models, each host can be infected with a symbiont, while a few species of hosts compete against each other.  Durrett and Lanchier \\cite{durrett2008coexistence} studied another case where only one of two species of hosts can be infected with one type of symbionts. In these papers, conditions for survival and coexistence have been studied, and the main results describe the long-term behavior of the models under certain conditions.\nA few years later, Lanchier and Zhang \\cite{lanchier2016some} studied the ``stacked contact process'', and Ma \\cite{ma2022complete} studied the ``two-level contact process''. In these two models, there are uninfected hosts and infected hosts. The results focused on the phase transition and the limiting distribution of the models.   \n\n\tMotivated by a series of works by Ghanbarnejad and coauthors\n    \\cite{cai2015avalanche, chen2013outbreaks,grassberger2016phase,zarei2019exact},\n    we consider a \\emph{two-type cooperative SIR process} on a network.  Suppose that two diseases, $A$ and $B$, spread simultaneously on the network, each type acting as an SIR process. Vertices in the graph can be infected by both types. Thus, there are nine possible states for each vertex,\n    \\begin{equation}\\label{eq:statespace}\n            S\\text{, }A \\text{, } B \\text{, } a \\text{, } b \\text{, } Ab \\text{, } aB \\text{, } AB\\text{ and }ab.\n    \\end{equation}\n    Here, capital $A$ and $B$ indicate active infection, and  lower case $a$ and $b$ indicate recovery from $A$ and $B$. We assume that the two diseases interact in a cooperative way. \n    When a vertex $x$  is susceptible to both diseases, then it can be infected with $A$ at rate $\\alpha_1$. On the other hand, \n    if  $x$ has been infected with $B$ before (regardless of whether it has recovered from $B$ or not), then $x$ acquires  $A$ at a higher rate  $\\beta_1 \\in [\\alpha_1, \\infty]$.\n    Note that an infinite rate means that an infection transmits immediately.  The constant $C _ 1:= \\beta _1 / \\alpha _1 \\in [1,\\infty]$ is called the \\emph{cooperativity coefficient} for  $A$. Likewise, a vertex $x$ can be infected by $B$ at a higher rate $\\beta _2 \\geq \\alpha _2$ if it has been infected with $A$ before. The cooperativity coefficient of $B$ is $C _ 2 = \\beta _2 / \\alpha _2$. Our definition of the two-type SIR process simulates the situation in which the immune system weakens because of a prior infection.\n\nChen et al.\\@ \\cite{chen2013outbreaks} analyzed a deterministic version of this model with mean-field methods, involving a homogeneously mixed population of infinitely many agents. \nIn total, there are nine ordinary differential equations for the evolution of all states. Let $[z](t)$ be the fraction of agents in state $z$ at time $t$.\nUnder the assumption of equal infection rates for both diseases, and letting the recovery rate be 1, \\cite{chen2013outbreaks} reduced the system of nine equations to three:\n\\begin{equation}    \\label{eq:ode}\n   \\begin{cases}\n      s'(t)=-2\\alpha s(t)x(t),  \\\\  \n  q'(t)=(\\alpha s(t)-C \\alpha q(t))x(t), \\\\\n  x'(t)=(\\alpha s(t)+C \\alpha q(t))x(t)-x(t). \n    \\end{cases}\n\\end{equation}\nIn these equations, $s(t)=[S](t)$ is the fraction of susceptible agents, \n$$x(t)=[A](t)+[AB](t)+[Ab](t)=[B](t)+[AB](t)+[aB](t)$$\nis the fraction of agents that are actively infected  by disease $A$  (or $B$) at time $t$, and $$q(t)=[A](t)+[a](t)=[B](t)+[b](t)$$ accounts for the agents which have  an infection history of only one disease. The initial condition is set to $[A](0)=[B](0)=\\epsilon/2$, $[S] (0) = 1 - \\epsilon$ and the fractions of all other states are $0$. Hence, $s(0)=1-\\epsilon$ and $q(0)=x(0)=\\epsilon/2$.   \n\nDenote the final epidemic size of the system of equations \\eqref{eq:ode} by $$\\mathcal{R}(\\alpha, C, \\epsilon) = 1-\\lim_{t\\to\\infty} s(t).$$\nZarei et al.\\@  \\cite{zarei2019exact} found\n$\\mathcal R(\\alpha, C, \\epsilon)=1-s(0)\\exp(-2\\alpha T_0)$, where $T_0$ is given by\n$$\n\\inf\\left\\{t>0: t+s(0)\\exp(-2\\alpha t)+q(0)\\exp(-C\\beta t)-\\frac{ s(0)}{C-2}(\\exp(-C\\alpha t)-\\exp(-2\\alpha t) )=1  \\right\\}\n$$\nif $C\\neq 2$.\nNote that this expression already implies the criticality at $C=2$.\nLet\n$$\n\\mathcal{R}_*(\\alpha, C) = \\lim_{\\epsilon\\to 0} \\mathcal R(\\alpha, C, \\epsilon)\n$$\nwhen the limit exists (otherwise, one may replace the limit with upper limit). Based on numerical experiments and some non-rigorous arguments,  \\cite{chen2013outbreaks, zarei2019exact} claimed the following:\n\\begin{enumerate}[label = (\\roman*)]\n\t\\item If $C \\leq 2$, then $\\mathcal R(\\alpha, C, \\epsilon)$ is continuous in $\\alpha$. Moreover, $\\mathcal R_*(\\alpha, C)=0$ for $\\alpha \\leq 1$, and behaves like $\\alpha-1$ ($C<2$) or $\\sqrt{\\alpha-1}$ ($C=2$) for $\\alpha$ close to 1.\n\t\\item If $C>2$, then $\\mathcal R(\\alpha, C, \\epsilon)$  is discontinuous in $\\alpha$ at some $\\alpha_0=\\alpha_0(C,\\epsilon)$, and\n\\begin{equation}\\label{eq:discont}\n    \\lim_{\\epsilon \\to 0} \\mathcal R(\\alpha_0-, C, \\epsilon)=0 \\quad \\textnormal{ and } \\quad \n\\lim_{\\epsilon \\to 0} \\mathcal R(\\alpha_0+, C, \\epsilon)>0. \n\\end{equation}\n\\item For $C>2$,  the critical infection rate $\\alpha_0(C, \\epsilon)=1-\\sqrt{(C-2)\\epsilon}+O(\\epsilon)$ as $\\epsilon\\to 0$.\n\\item The quantity $\\mathcal R_*(\\alpha, C)\\equiv 0$ for $\\alpha \\leq 1/2$, and  $$\\lim_{\\epsilon \\to 0}\\lim_{C\\to\\infty}R(\\alpha, C,\\epsilon)  \\sim \\alpha-\\frac{1}{2}\\quad  \\textnormal{ as } \\alpha \\to 1/2 \\textnormal{.}$$\n\\end{enumerate}\n\nIt is generally expected that\nmean-field ODEs may give good approximations for the stochastic particle systems  on the complete graph.\n In real-world networks, nontrivial spatial structures are often present.  Grassberger et al.\\@ \\cite{grassberger2016phase} conducted simulations for the two-type SIR model on  Erd\\H{o}s-R\\'enyi graphs and the integer lattice $\\mathbb{Z}^d$. They argued that \n Erd\\H{o}s-R\\'enyi graphs have a discontinuous \n phase transition for the fraction of eventually infected vertices when the cooperativity is sufficiently strong,  as predicted by the mean-field ODEs. For the case of $\\mathbb{Z}^d$ with $d\\geq 2$, simulations in  \\cite{grassberger2016phase}  showed that the two-type model shares similar critical and near-critical features  with the  single-type SIR process if and only if $d\\leq 3$. \n \\cite{grassberger2016phase}  also considered   the near-critical behavior of the two-type SIR model on trees. In particular,  \\cite{grassberger2016phase}  suggested that the critical value (for infinitely many vertices to be infected with both diseases) remains the same as the single-type model for $C=\\infty$. See also Remark \\ref{rmk:c=infty} below.  \n\n In this paper, we consider the two-type SIR process on Galton-Watson trees with cooperative interactions  defined before. The rates are not necessarily the same for $A$ and $B$, and there are six parameters in total, shown in Table \\ref{rates table}.\n\n    \\begin{table}[h]\n       \\label{rates table}\n\\begin{center}\n\\begin{tabular}{c|c|c|c}\n\\hline\n& original infection rate   & increased infection rate & recovery rate\\\\\n\\hline \ndisease $A$ &  $\\alpha_1$  & $\\beta_1\\in [\\alpha_1,\\infty]$ & $\\mu_1$\\\\\n\\hline \ndisease $B$ &  $\\alpha_2$  & $\\beta_2\\in [\\alpha_2, \\infty]$  & $\\mu_2$\\\\\n\\hline \n\\end{tabular}\n\\end{center}\n\\caption{Parameters for the two-type SIR process.}\n    \\end{table}\n\nLet $GW (p)$ be the Galton-Watson tree with the offspring distribution $p$. Let  $m \\in (1,\\infty)$ be the finite mean of $p$. (We let $m>1$ so that $GW(p)$ itself is supercritical.) Consider the single-type SIR process on $GW(p)$. We say that it \\textit{survives} if for all $t\\geq 0$, there is at least one infected vertex at time $t$. The following result is classic.", "context": "Motivated by a series of works by Ghanbarnejad and coauthors\n \\cite{cai2015avalanche, chen2013outbreaks,grassberger2016phase,zarei2019exact},\n we consider a \\emph{two-type cooperative SIR process} on a network. Suppose that two diseases, $A$ and $B$, spread simultaneously on the network, each type acting as an SIR process. Vertices in the graph can be infected by both types. Thus, there are nine possible states for each vertex,\n \\begin{equation}\\label{eq:statespace}\n S\\text{, }A \\text{, } B \\text{, } a \\text{, } b \\text{, } Ab \\text{, } aB \\text{, } AB\\text{ and }ab.\n \\end{equation}\n Here, capital $A$ and $B$ indicate active infection, and lower case $a$ and $b$ indicate recovery from $A$ and $B$. We assume that the two diseases interact in a cooperative way. \n When a vertex $x$ is susceptible to both diseases, then it can be infected with $A$ at rate $\\alpha_1$. On the other hand, \n if $x$ has been infected with $B$ before (regardless of whether it has recovered from $B$ or not), then $x$ acquires $A$ at a higher rate $\\beta_1 \\in [\\alpha_1, \\infty]$.\n Note that an infinite rate means that an infection transmits immediately. The constant $C _ 1:= \\beta _1 / \\alpha _1 \\in [1,\\infty]$ is called the \\emph{cooperativity coefficient} for $A$. Likewise, a vertex $x$ can be infected by $B$ at a higher rate $\\beta _2 \\geq \\alpha _2$ if it has been infected with $A$ before. The cooperativity coefficient of $B$ is $C _ 2 = \\beta _2 / \\alpha _2$. Our definition of the two-type SIR process simulates the situation in which the immune system weakens because of a prior infection.\n\nChen et al.\\@ \\cite{chen2013outbreaks} analyzed a deterministic version of this model with mean-field methods, involving a homogeneously mixed population of infinitely many agents. \nIn total, there are nine ordinary differential equations for the evolution of all states. Let $[z](t)$ be the fraction of agents in state $z$ at time $t$.\nUnder the assumption of equal infection rates for both diseases, and letting the recovery rate be 1, \\cite{chen2013outbreaks} reduced the system of nine equations to three:\n\\begin{equation} \\label{eq:ode}\n \\begin{cases}\n s'(t)=-2\\alpha s(t)x(t), \\\\ \n q'(t)=(\\alpha s(t)-C \\alpha q(t))x(t), \\\\\n x'(t)=(\\alpha s(t)+C \\alpha q(t))x(t)-x(t). \n \\end{cases}\n\\end{equation}\nIn these equations, $s(t)=[S](t)$ is the fraction of susceptible agents, \n$$x(t)=[A](t)+[AB](t)+[Ab](t)=[B](t)+[AB](t)+[aB](t)$$\nis the fraction of agents that are actively infected by disease $A$ (or $B$) at time $t$, and $$q(t)=[A](t)+[a](t)=[B](t)+[b](t)$$ accounts for the agents which have an infection history of only one disease. The initial condition is set to $[A](0)=[B](0)=\\epsilon/2$, $[S] (0) = 1 - \\epsilon$ and the fractions of all other states are $0$. Hence, $s(0)=1-\\epsilon$ and $q(0)=x(0)=\\epsilon/2$.\n\nDenote the final epidemic size of the system of equations \\eqref{eq:ode} by $$\\mathcal{R}(\\alpha, C, \\epsilon) = 1-\\lim_{t\\to\\infty} s(t).$$\nZarei et al.\\@ \\cite{zarei2019exact} found\n$\\mathcal R(\\alpha, C, \\epsilon)=1-s(0)\\exp(-2\\alpha T_0)$, where $T_0$ is given by\n$$\n\\inf\\left\\{t>0: t+s(0)\\exp(-2\\alpha t)+q(0)\\exp(-C\\beta t)-\\frac{ s(0)}{C-2}(\\exp(-C\\alpha t)-\\exp(-2\\alpha t) )=1 \\right\\}\n$$\nif $C\\neq 2$.\nNote that this expression already implies the criticality at $C=2$.\nLet\n$$\n\\mathcal{R}_*(\\alpha, C) = \\lim_{\\epsilon\\to 0} \\mathcal R(\\alpha, C, \\epsilon)\n$$\nwhen the limit exists (otherwise, one may replace the limit with upper limit). Based on numerical experiments and some non-rigorous arguments, \\cite{chen2013outbreaks, zarei2019exact} claimed the following:\n\\begin{enumerate}[label = (\\roman*)]\n \\item If $C \\leq 2$, then $\\mathcal R(\\alpha, C, \\epsilon)$ is continuous in $\\alpha$. Moreover, $\\mathcal R_*(\\alpha, C)=0$ for $\\alpha \\leq 1$, and behaves like $\\alpha-1$ ($C<2$) or $\\sqrt{\\alpha-1}$ ($C=2$) for $\\alpha$ close to 1.\n \\item If $C>2$, then $\\mathcal R(\\alpha, C, \\epsilon)$ is discontinuous in $\\alpha$ at some $\\alpha_0=\\alpha_0(C,\\epsilon)$, and\n\\begin{equation}\\label{eq:discont}\n \\lim_{\\epsilon \\to 0} \\mathcal R(\\alpha_0-, C, \\epsilon)=0 \\quad \\textnormal{ and } \\quad \n\\lim_{\\epsilon \\to 0} \\mathcal R(\\alpha_0+, C, \\epsilon)>0. \n\\end{equation}\n\\item For $C>2$, the critical infection rate $\\alpha_0(C, \\epsilon)=1-\\sqrt{(C-2)\\epsilon}+O(\\epsilon)$ as $\\epsilon\\to 0$.\n\\item The quantity $\\mathcal R_*(\\alpha, C)\\equiv 0$ for $\\alpha \\leq 1/2$, and $$\\lim_{\\epsilon \\to 0}\\lim_{C\\to\\infty}R(\\alpha, C,\\epsilon) \\sim \\alpha-\\frac{1}{2}\\quad \\textnormal{ as } \\alpha \\to 1/2 \\textnormal{.}$$\n\\end{enumerate}\n\n\\begin{table}[h]\n \\label{rates table}\n\\begin{center}\n\\begin{tabular}{c|c|c|c}\n\\hline\n& original infection rate & increased infection rate & recovery rate\\\\\n\\hline \ndisease $A$ & $\\alpha_1$ & $\\beta_1\\in [\\alpha_1,\\infty]$ & $\\mu_1$\\\\\n\\hline \ndisease $B$ & $\\alpha_2$ & $\\beta_2\\in [\\alpha_2, \\infty]$ & $\\mu_2$\\\\\n\\hline \n\\end{tabular}\n\\end{center}\n\\caption{Parameters for the two-type SIR process.}\n \\end{table}\n\nLet $GW (p)$ be the Galton-Watson tree with the offspring distribution $p$. Let $m \\in (1,\\infty)$ be the finite mean of $p$. (We let $m>1$ so that $GW(p)$ itself is supercritical.) Consider the single-type SIR process on $GW(p)$. We say that it \\textit{survives} if for all $t\\geq 0$, there is at least one infected vertex at time $t$. The following result is classic.", "full_context": "Motivated by a series of works by Ghanbarnejad and coauthors\n \\cite{cai2015avalanche, chen2013outbreaks,grassberger2016phase,zarei2019exact},\n we consider a \\emph{two-type cooperative SIR process} on a network. Suppose that two diseases, $A$ and $B$, spread simultaneously on the network, each type acting as an SIR process. Vertices in the graph can be infected by both types. Thus, there are nine possible states for each vertex,\n \\begin{equation}\\label{eq:statespace}\n S\\text{, }A \\text{, } B \\text{, } a \\text{, } b \\text{, } Ab \\text{, } aB \\text{, } AB\\text{ and }ab.\n \\end{equation}\n Here, capital $A$ and $B$ indicate active infection, and lower case $a$ and $b$ indicate recovery from $A$ and $B$. We assume that the two diseases interact in a cooperative way. \n When a vertex $x$ is susceptible to both diseases, then it can be infected with $A$ at rate $\\alpha_1$. On the other hand, \n if $x$ has been infected with $B$ before (regardless of whether it has recovered from $B$ or not), then $x$ acquires $A$ at a higher rate $\\beta_1 \\in [\\alpha_1, \\infty]$.\n Note that an infinite rate means that an infection transmits immediately. The constant $C _ 1:= \\beta _1 / \\alpha _1 \\in [1,\\infty]$ is called the \\emph{cooperativity coefficient} for $A$. Likewise, a vertex $x$ can be infected by $B$ at a higher rate $\\beta _2 \\geq \\alpha _2$ if it has been infected with $A$ before. The cooperativity coefficient of $B$ is $C _ 2 = \\beta _2 / \\alpha _2$. Our definition of the two-type SIR process simulates the situation in which the immune system weakens because of a prior infection.\n\nChen et al.\\@ \\cite{chen2013outbreaks} analyzed a deterministic version of this model with mean-field methods, involving a homogeneously mixed population of infinitely many agents. \nIn total, there are nine ordinary differential equations for the evolution of all states. Let $[z](t)$ be the fraction of agents in state $z$ at time $t$.\nUnder the assumption of equal infection rates for both diseases, and letting the recovery rate be 1, \\cite{chen2013outbreaks} reduced the system of nine equations to three:\n\\begin{equation} \\label{eq:ode}\n \\begin{cases}\n s'(t)=-2\\alpha s(t)x(t), \\\\ \n q'(t)=(\\alpha s(t)-C \\alpha q(t))x(t), \\\\\n x'(t)=(\\alpha s(t)+C \\alpha q(t))x(t)-x(t). \n \\end{cases}\n\\end{equation}\nIn these equations, $s(t)=[S](t)$ is the fraction of susceptible agents, \n$$x(t)=[A](t)+[AB](t)+[Ab](t)=[B](t)+[AB](t)+[aB](t)$$\nis the fraction of agents that are actively infected by disease $A$ (or $B$) at time $t$, and $$q(t)=[A](t)+[a](t)=[B](t)+[b](t)$$ accounts for the agents which have an infection history of only one disease. The initial condition is set to $[A](0)=[B](0)=\\epsilon/2$, $[S] (0) = 1 - \\epsilon$ and the fractions of all other states are $0$. Hence, $s(0)=1-\\epsilon$ and $q(0)=x(0)=\\epsilon/2$.\n\nDenote the final epidemic size of the system of equations \\eqref{eq:ode} by $$\\mathcal{R}(\\alpha, C, \\epsilon) = 1-\\lim_{t\\to\\infty} s(t).$$\nZarei et al.\\@ \\cite{zarei2019exact} found\n$\\mathcal R(\\alpha, C, \\epsilon)=1-s(0)\\exp(-2\\alpha T_0)$, where $T_0$ is given by\n$$\n\\inf\\left\\{t>0: t+s(0)\\exp(-2\\alpha t)+q(0)\\exp(-C\\beta t)-\\frac{ s(0)}{C-2}(\\exp(-C\\alpha t)-\\exp(-2\\alpha t) )=1 \\right\\}\n$$\nif $C\\neq 2$.\nNote that this expression already implies the criticality at $C=2$.\nLet\n$$\n\\mathcal{R}_*(\\alpha, C) = \\lim_{\\epsilon\\to 0} \\mathcal R(\\alpha, C, \\epsilon)\n$$\nwhen the limit exists (otherwise, one may replace the limit with upper limit). Based on numerical experiments and some non-rigorous arguments, \\cite{chen2013outbreaks, zarei2019exact} claimed the following:\n\\begin{enumerate}[label = (\\roman*)]\n \\item If $C \\leq 2$, then $\\mathcal R(\\alpha, C, \\epsilon)$ is continuous in $\\alpha$. Moreover, $\\mathcal R_*(\\alpha, C)=0$ for $\\alpha \\leq 1$, and behaves like $\\alpha-1$ ($C<2$) or $\\sqrt{\\alpha-1}$ ($C=2$) for $\\alpha$ close to 1.\n \\item If $C>2$, then $\\mathcal R(\\alpha, C, \\epsilon)$ is discontinuous in $\\alpha$ at some $\\alpha_0=\\alpha_0(C,\\epsilon)$, and\n\\begin{equation}\\label{eq:discont}\n \\lim_{\\epsilon \\to 0} \\mathcal R(\\alpha_0-, C, \\epsilon)=0 \\quad \\textnormal{ and } \\quad \n\\lim_{\\epsilon \\to 0} \\mathcal R(\\alpha_0+, C, \\epsilon)>0. \n\\end{equation}\n\\item For $C>2$, the critical infection rate $\\alpha_0(C, \\epsilon)=1-\\sqrt{(C-2)\\epsilon}+O(\\epsilon)$ as $\\epsilon\\to 0$.\n\\item The quantity $\\mathcal R_*(\\alpha, C)\\equiv 0$ for $\\alpha \\leq 1/2$, and $$\\lim_{\\epsilon \\to 0}\\lim_{C\\to\\infty}R(\\alpha, C,\\epsilon) \\sim \\alpha-\\frac{1}{2}\\quad \\textnormal{ as } \\alpha \\to 1/2 \\textnormal{.}$$\n\\end{enumerate}\n\n\\begin{table}[h]\n \\label{rates table}\n\\begin{center}\n\\begin{tabular}{c|c|c|c}\n\\hline\n& original infection rate & increased infection rate & recovery rate\\\\\n\\hline \ndisease $A$ & $\\alpha_1$ & $\\beta_1\\in [\\alpha_1,\\infty]$ & $\\mu_1$\\\\\n\\hline \ndisease $B$ & $\\alpha_2$ & $\\beta_2\\in [\\alpha_2, \\infty]$ & $\\mu_2$\\\\\n\\hline \n\\end{tabular}\n\\end{center}\n\\caption{Parameters for the two-type SIR process.}\n \\end{table}\n\nLet $GW (p)$ be the Galton-Watson tree with the offspring distribution $p$. Let $m \\in (1,\\infty)$ be the finite mean of $p$. (We let $m>1$ so that $GW(p)$ itself is supercritical.) Consider the single-type SIR process on $GW(p)$. We say that it \\textit{survives} if for all $t\\geq 0$, there is at least one infected vertex at time $t$. The following result is classic.\n\n\\begin{table}[h]\n \\label{rates table}\n\\begin{center}\n\\begin{tabular}{c|c|c|c}\n\\hline\n& original infection rate & increased infection rate & recovery rate\\\\\n\\hline \ndisease $A$ & $\\alpha_1$ & $\\beta_1\\in [\\alpha_1,\\infty]$ & $\\mu_1$\\\\\n\\hline \ndisease $B$ & $\\alpha_2$ & $\\beta_2\\in [\\alpha_2, \\infty]$ & $\\mu_2$\\\\\n\\hline \n\\end{tabular}\n\\end{center}\n\\caption{Parameters for the two-type SIR process.}\n \\end{table}\n\n\\begin{theorem}\n \\label{general}\nConsider the two-type SIR process on $GW (p)$ with parameters $\\alpha_i$, $\\beta_i$, $\\mu_i$, $i=1,2$. At time $0$, the root is infected with both disease $A$ and disease $B$. All other vertices are susceptible to both $A$ and $B$. Let $m$ be the mean of $p$. If\n \\begin{equation}\\label{eq:assump}\n \\max\\left\\{\\frac{\\alpha_1}{\\mu_1},\\frac{\\alpha_2}{\\mu_2}\\right\\}\\leq \\frac{1}{m-1}\n \\textnormal {,}\n \\end{equation}\nthen the probability of survival is $0$.\n\\end{theorem} \n\\begin{remark}\\label{rmk:c=infty}\nIn the special case of equal infection/recovery rates for both diseases (denoted by $\\alpha,\\beta=C\\alpha$ and $\\mu$) with $C=\\infty$, the conclusion of Theorem \\ref{general} is consistent with the prediction made in \\cite{grassberger2016phase} regarding infinite tree graphs, where the authors claimed that (for $C=\\infty$), \n$$\n\\mathbb{P}(\\mbox{infinitely many vertices are infected with both diseases})>0\\, \\mbox{ if and only if }\\, \\frac{\\alpha}{\\mu}>\\frac{1}{m-1}. \n$$ See also the discussions above Table \\ref{rates table}.\n\\end{remark}\nCombining Theorems \\ref{thm:single} and \\ref{general}, the two-type SIR process survives with a positive probability if and only if\n $$\n \\max\\left\\{\\frac{\\alpha_1}{\\mu_1},\\frac{\\alpha_2}{\\mu_2}\\right\\}> \\frac{1}{m-1}.\n $$\n Since the Galton-Watson tree and its variants arise as local limits of some random graph models such as the configuration model and the {\\ER} graph, our result may be useful for studying the two-type SIR processes on these random graphs. We leave further investigations as future work.\n\nThough Theorem \\ref{thm:single} is more or less well-known, here we give a proof for the sake of completeness and also to illustrate the basic idea that will also be used in the proof of Theorem \\ref{general}. We couple the Galton-Watson tree with an epidemic process by \n revealing the number of children of a vertex after it becomes infected. At time 0 only the root $o$ is infected and \n the only available information about the tree is the degree of $o$. Since $o$ infects each of its children at rate $\\alpha$ and recovers at rate $\\mu$, the probability that any given child is infected before $o$ recovers is equal to $\\alpha/(\\alpha+\\mu)$ by standard properties of the exponential distribution. \n For any vertex $x$,\n we let $N_x$ be the number of children of $x$ that are eventually infected by $x$. Since the degree $D_o$ of $o$ is distributed according to $p$, we see that the mean of $N_o$ is \n \\begin{equation}\\label{eq:edo}\n \\E\\left( \\frac{\\alpha}{\\alpha + \\mu} D_o\\right)=\\frac{m\\alpha}{\\alpha+ \\mu}.\n \\end{equation}\n Denote the distribution of $N_o$ by $q$. \n Since the number of children of any given vertex has the same distribution $p$, we see that the number of infected children of any infected vertex must have the distribution $q$, and must be independent of its ancestors.\n\nLet $Y_n$ be the number of infected vertices in the $n$-th generation of $GW (p)$. Then\n$Y_n$ forms a branching process with the initial value $Y_0=1$ and branching distribution $q$, whose \nmean is equal to $m\\alpha/(\\alpha+ \\mu)$.\n Standard results on the branching process imply that \n \\begin{equation}\\label{eq:thm}\n \\P(Y_n>0, \\forall n)>0 \\Leftrightarrow \n \\frac{\\alpha}{\\alpha+ \\mu}m>1 \\Leftrightarrow \\alpha>\\frac{ \\mu}{m-1}\n \\textnormal {,}\n \\end{equation}\nwhich proves Theorem \\ref{thm:single}.\n\nLet $\\mathcal{X}_n$ be the set of vertices in the $n$-th generation of $GW (p)$ that eventually receive both infections, and let $X_n=\\abs{\\mathcal{X}_n}$, for all $n \\geq 0$. Note that the root is the single vertex in the zeroth generation. Let $\\mathcal{F}_n$ be the $\\sigma$-algebra generated by the number of children of all $x$ up to the $ (n-1)$-th generation and all $\\tau_A(x)$, $\\tau_B(x)$, for $x$ up to the $n$-th generation. By Lemma \\ref{t}, \n\\begin{equation*}\n \\E(X_{n+1}|\\mathcal{F}_n)\\leq X_n\\textnormal{ a.s., for }n\\geq 0.\n\\end{equation*}\nThus $\\{X_n,n\\geq 0\\}$ is a non-negative integer-valued supermartingale. Hence $X_n$ must converge almost surely to some limit $X$. Moreover, there exists a constant $c>0$ such that for all $k,n\\geq 1$,\n\\begin{equation}\\label{xn=0}\n\\P(X_{n+1}=0|X_n=k)\\geq c^k,\n\\end{equation}\nby considering the event where all vertices in $\\mathcal{X}_n$ recover before infecting any child. Let $\\hat{\\Omega}$ be the event\n $$\n \\left\\{ \\lim_{n\\to\\infty} X_n=0 \\right\\}=\\{X_n=0\\textnormal { for some }n\\}.\n $$\nBy Levy's 0-1 Law \\cite[Theorem 4.6.9] {durrett2019probability} and equation \\eqref{xn=0},\n \\begin{equation*}\n \\P ( \\mathbf{1}_{\\hat{\\Omega}} \\mid \\mathcal{F}_n) \\to 1=1_{\\hat{\\Omega}} \\text{ a.s.}\n \\end{equation*}\nThus, $\\P(\\hat{\\Omega})=1$ and $X=0$ a.s. Now define\n\\begin{equation*}\n N=\\inf\\{n: X_n=0\\}.\n\\end{equation*}\nAll vertices $z$ in the $N$-th generation of $GW (p)$ are either infected by one type of disease, or never infected. Since the subtree of $GW (p)$ re-rooted at $z$ has the same distribution as $GW (p)$, we can apply Theorem \\ref{thm:single} to deduce that the number of infected vertices in that subtree is almost surely finite. Thus, we conclude that the two-type SIR process \nsurvives with probability 0.\n\\end{proof}\n\n\\begin{itemize}\n \\item \\textbf{Rigorous analysis for the ODE system \\eqref{eq:ode}.} It would be interesting to inspect the claims made by physicists \\cite{chen2013outbreaks} regarding properties of final epidemic size $\\mathcal R(\\alpha, C, \\epsilon)$, particularly the discontinuous transition \\eqref{eq:discont}. One can also study whether the mean-field equations approximate the true dynamics on the complete graph. \n \\item \\textbf{Survival of the weaker species in the asymmetric case.} If one species is (sub)critical while the other is supercritical, can the weaker species survive with positive probability when the cooperation is strong? See simulations (Figures \\ref{sim growth} and \\ref{sim_surv}) and discussions below.\n \\item \\textbf{Effects of the graph structure on the two-type SIR process.} As mentioned in the introduction, the simulations in \\cite{grassberger2016phase} found that the structure of the underlying graph (particularly the existence of loops in the graph) has a major impact on the two-type SIR process thereon. We do expect a qualitatively different situation for the survival probability if the Galton-Watson tree is replaced with the integer lattice.\n \\item \\textbf{Different cooperativity mechanism.}\n One can also consider the cooperativity mechanism as in \\cite{DurrettYao2020}, where the recovery rates decrease from $\\mu_i$ to $\\hat{\\mu}_i$ if a node has two infections. By comparison with two independent single-type SIR processes, it can be shown that the critical value for survival matches the single-type model if $\\hat{\\mu}_i$ is sufficiently close to $\\mu_i$. However, the case where $\\hat{\\mu}_i$ is small remains unknown.\n \\item \\textbf{Cooperative SIS models.} If we consider the SIS dynamics on a Galton-Watson tree, by an oriented percolation argument, the critical value becomes smaller if the cooperation is sufficiently strong. As indicated by \\cite{DurrettYao2020}, the general case may be rather challenging. \n\\end{itemize}", "post_theorem_intro_text_len": 2338, "post_theorem_intro_text": "Given Theorem \\ref{thm:single}, it is natural to probe whether the critical value gets smaller in the cooperative SIR process. We say that the two-type SIR process survives if for all $t\\geq 0$, there is at least one infected vertex (with $A$ or $B$) at time $t$. According to the following Theorem \\ref{general}, the answer is no.  \n\n\\begin{theorem}\n    \\label{general}\nConsider the two-type SIR process on $GW (p)$ with parameters $\\alpha_i$, $\\beta_i$, $\\mu_i$, $i=1,2$. At time $0$, the root is infected with both disease $A$ and disease $B$. All other vertices are susceptible to both $A$ and $B$. Let $m$ be the mean of $p$. If\n    \\begin{equation}\\label{eq:assump}\n      \\max\\left\\{\\frac{\\alpha_1}{\\mu_1},\\frac{\\alpha_2}{\\mu_2}\\right\\}\\leq \\frac{1}{m-1}\n      \\textnormal {,}\n    \\end{equation}\nthen the probability of survival is $0$.\n\\end{theorem} \n\\begin{remark}\\label{rmk:c=infty}\nIn the special case of equal infection/recovery rates for both diseases (denoted by $\\alpha,\\beta=C\\alpha$ and $\\mu$)  with $C=\\infty$,  the conclusion of    Theorem \\ref{general}  is consistent with  the prediction made in \\cite{grassberger2016phase} regarding infinite tree graphs, where the authors claimed that (for $C=\\infty$), \n$$\n\\mathbb{P}(\\mbox{infinitely many vertices are infected with both diseases})>0\\, \\mbox{ if and only if }\\, \\frac{\\alpha}{\\mu}>\\frac{1}{m-1}. \n$$ See also the discussions above Table       \\ref{rates table}.\n\\end{remark}\nCombining Theorems \\ref{thm:single} and \\ref{general}, the two-type SIR process survives with a positive probability if and only if\n  $$\n      \\max\\left\\{\\frac{\\alpha_1}{\\mu_1},\\frac{\\alpha_2}{\\mu_2}\\right\\}> \\frac{1}{m-1}.\n  $$\n Since the Galton-Watson tree and its variants arise as local limits of some random graph models such as the configuration model and the {Erd\\H{o}s-R\\'enyi} graph, our result may be useful for studying the two-type SIR processes on these random graphs. We leave further investigations as  future work.   \n\nIn the remainder of this paper, we prove\nTheorems \\ref{thm:single} and  \\ref{general}  in Section \\ref{sec:single} and Section \\ref{sec:two}, respectively, by coupling the Galton-Watson tree with the epidemic process. Section \\ref{sec:conclusion} summarizes the main findings of this paper and states many open problems as well as some simulation results.", "sketch": "In the remainder of this paper, we prove Theorems \\ref{thm:single} and \\ref{general} in Section \\ref{sec:single} and Section \\ref{sec:two}, respectively, \\emph{by coupling the Galton-Watson tree with the epidemic process}.", "expanded_sketch": "In the remainder of this paper, we prove the main theorem and we first prove the following theorem.\n\n\\begin{theorem}\n    \\label{general}\nConsider the two-type SIR process on $GW (p)$ with parameters $\\alpha_i$, $\\beta_i$, $\\mu_i$, $i=1,2$. At time $0$, the root is infected with both disease $A$ and disease $B$. All other vertices are susceptible to both $A$ and $B$. Let $m$ be the mean of $p$. If\n    \\begin{equation}\\label{eq:assump}\n      \\max\\left\\{\\frac{\\alpha_1}{\\mu_1},\\frac{\\alpha_2}{\\mu_2}\\right\\}\\leq \\frac{1}{m-1}\n      \\textnormal {,}\n    \\end{equation}\nthen the probability of survival is $0$.\n\\end{theorem}\n\nWe do so \\emph{by coupling the Galton-Watson tree with the epidemic process}.", "expanded_theorem": "\\label{thm:single}\n   Let the infection rate be $\\alpha$ and the recovery rate be $\\mu$ for a single-type SIR process on $GW (p)$. At time 0, only the root is infected. The probability of survival is greater than 0 if and only if   $   \\alpha/\\mu > 1/{(m-1)}   $, where $m$ is the mean of $p$. Thus, if $\\mu$ is fixed, then the critical value for $\\alpha$ is  $\\mu/(m-1)$.", "theorem_type": ["Biconditional or Equivalence", "Existence"], "mcq": {"question": "Let \\(GW(p)\\) be a Galton--Watson tree with offspring distribution \\(p\\) and mean \\(m\\). Consider a single-type SIR process on \\(GW(p)\\) with infection rate \\(\\alpha\\) and recovery rate \\(\\mu\\), started at time \\(0\\) with only the root infected. Here, “survival” means that the infection does not die out. Which statement holds about when survival has positive probability?", "correct_choice": {"label": "A", "text": "The survival probability is greater than \\(0\\) if and only if \\(\\alpha/\\mu > 1/(m-1)\\). Equivalently, if \\(\\mu\\) is fixed, then the critical value of the infection rate is \\(\\mu/(m-1)\\)."}, "choices": [{"label": "B", "text": "The survival probability is greater than \\(0\\) if and only if \\(\\alpha/\\mu \\ge 1/(m-1)\\). Equivalently, if \\(\\mu\\) is fixed, then the critical value of the infection rate is attained already at \\(\\alpha=\\mu/(m-1)\\)."}, {"label": "C", "text": "If \\(\\alpha/\\mu > 1/(m-1)\\), then the survival probability is greater than \\(0\\). Equivalently, if \\(\\mu\\) is fixed and \\(\\alpha>\\mu/(m-1)\\), then survival has positive probability."}, {"label": "D", "text": "The survival probability is greater than \\(0\\) if and only if \\(\\alpha/\\mu > 1/m\\). Equivalently, if \\(\\mu\\) is fixed, then the critical value of the infection rate is \\(\\mu/m\\)."}, {"label": "E", "text": "The survival probability is greater than \\(0\\) whenever the mean offspring satisfies \\(m\\,\\alpha/\\mu > 1\\), and in particular the critical value for \\(\\alpha\\) is \\(\\mu/m\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "counting_estimate", "tampered_component": "strictness_of_threshold", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "only_if_direction", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "effective_branching_factor_m_minus_1", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "root_vs_nonroot_branching_count", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state or strongly hint at the threshold formula; it only asks for the correct survival criterion. The correct answer is not leaked directly."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question, since the correct option essentially states the exact threshold result. However, it is not a pure tautology because the alternatives vary in strictness, logical strength, and the branching-factor term."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact 'if and only if' statement from the weaker true implication and from the common confusion between m and m-1. Still, the item mainly tests recognition/recall of the threshold rather than deep derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: B tests boundary strictness, C is a weaker-but-true statement, and D/E reflect a natural m versus m-1 branching-factor confusion. These align well with common failure modes."}, "total_score": 6, "overall_assessment": "A solid MCQ with strong distractors and no answer leakage, but it leans more toward theorem recognition than genuinely generative reasoning."}}
{"id": "2602.18953v1", "paper_link": "http://arxiv.org/abs/2602.18953v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{main_thm} If $0<p<3/4$ then there exists positive constants $C_1$, $C_2$, $C_3$ and $C_4$ such that, for all $N \\ge 2$ and $t \\geq 1$, the following bound holds:\n\n    \\begin{equation*}\n        C_1 \\exp \\left( -C_2 \\frac{t}{N^2} \\right) \\leq \\mathbb{P} \\left( \\tau_N > t \\right) \\le C_3 \\exp \\left(-C_4\\frac{t}{N^2}\\right).\n    \\end{equation*}", "start_pos": 8988, "end_pos": 9358, "label": "main_thm"}, "ref_dict": {"main_thm": "\\begin{theorem}\\label{main_thm} If $0<p<3/4$ then there exists positive constants $C_1$, $C_2$, $C_3$ and $C_4$ such that, for all $N \\ge 2$ and $t \\geq 1$, the following bound holds:\n\n    \\begin{equation*}\n        C_1 \\exp \\left( -C_2 \\frac{t}{N^2} \\right) \\leq \\P \\left( \\tau_N > t \\right) \\le C_3 \\exp \\left(-C_4\\frac{t}{N^2}\\right).\n    \\end{equation*}\n\\end{theorem}", "prop_1": "\\begin{proposition}\\label{prop_1}\n    If $0<p<3/4$ then there exists a constant $\\theta$ such that: $$\\E[\\tau_N] \\underset{N \\rightarrow\\infty}{\\sim} \\theta N^2. $$\n\\end{proposition}"}, "pre_theorem_intro_text_len": 6539, "pre_theorem_intro_text": "The gambler’s ruin may be the oldest problem in probability theory, dating back to a letter from Blaise Pascal to Pierre de Fermat in 1656. In the classical gambler's ruin, a gambler is betting on the outcome of a possibly biased coin, winning or loosing one dollar at each step, with probability $p$ and $1-p$ respectively, independently of what happened up to the current step. Assume that the gambler starts with fortune $N$ and stop either when the totality of its fortune has been lost, or when it has doubled. For simplicity we adopt the representation in which $S_n$ represents the gambler's gains (when $S_n \\geq 0$) or losses (when $S_n \\leq 0$) with $S_0=0$. Thus $(S_n)_{n \\ge 0}$ is a Simple Random Walk (SRW) on the one-dimensional integer lattice, starting at the origin. Moreover we denote the game's duration by $\\sigma_N = \\inf\\left\\{n\\ge 1: \\left|S_n\\right| = N \\right\\}$. One of the main object of interest is the expected duration of the game, and, in this standard (Markovian and time-homogeneous) setting the problem has long been fully and analytically solved. Classical results\\footnote{See for example \\cite{feller}, pages 348 and 349.} tell us that $\\mathbb{E}[\\sigma_N]$ is of quadratic order in the symmetric case, while it is of linear order in the asymmetric case. More precisely, if $p=1/2$, then $\\mathbb{E} \\left[\\sigma_N \\right] = N^2$, otherwise: $$\\mathbb{E} \\left[\\sigma_N \\right] = \\frac{N}{2p - 1} \\left( \\frac{1 - \\left( \\frac{1-p}{p}\\right)^N}{1 + \\left( \\frac{1-p}{p}\\right)^N}\\right).$$\nThe proofs of these closed-form formulas are highly dependent on the Markovian and time-homogeneous nature of the dynamic. In the present work, we develop an approach allowing the analysis of a case in which the random walk has a past-dependent dynamic. Namely, we consider the problem above when the classical simple random walk is replaced by the elephant's random walk.\n\n\\vspace{0.2 cm}\n\nThe Elephant's Random Walk (ERW) is a discrete time stochastic process introduced by Schütz and Trimper \\cite{schutz2004elephants} as a paradigmatic example of past-dependent random walks. It is both quantitatively tractable and qualitatively rich---exhibiting in particular a phase transition from diffusive to super-diffusive behavior---and has therefore recently become the standard choice for testing classical results outside the traditional Markovian setting, when memory is allowed to play a role. It can be defined as follows. Let $(Z_n)_{n \\geq 0}$ be a stochastic chain with value in $\\mathbb{Z}$, making jumps of unit size---either upward or downward---at each time step. It can be defined, like classical random walks, as the partial sum of a sequence $(X_n)_{n \\geq 0}$ of random variables: $$Z_n = \\sum_{k=0}^n X_k,$$ with the crucial difference that the $(X_n)_{n \\geq 0}$ are not independent. Conventionally $X_0 = 0$, and the first jump leads to either $X_1 = 1$ or $X_1 = -1$, with probabilities that are unimportant for our purposes, since in either cases $|Z_1|=1$. Then, for any $n \\geq 1$, the jump at time $n+1$ is given by:\n$$X_{n+1} = \\begin{cases} X_{K(n)} &\\text{ with probability } p,\\\\\n-X_{K(n)} &\\text{ with probability } 1-p,\n\\end{cases}$$\nwhere $(K(n))_{n \\geq 1}$ is a sequence of independent random variables, $K(n)$ having uniform distribution on $\\{1, \\ldots, n\\}$. In words, at each time step, the ERW selects a step uniformly at random from its entire past and repeats it with probability $p$, or takes the opposite step with probability $1 - p$. The parameter $p$ is often referred to as the \\textit{memory} of the elephant: when $p$ is close to $1$, the dependence on the past tends to persist over a long period, whereas it fades rapidly when $p$ is close to $0$.\n\n\\vspace{0.2 cm}\n\nMuch effort has been done during the last two decades to understand the effect of the memory of the elephant on its asymptotic behavior, as compared to classical Markovian random walks, with a particular focus on laws of large numbers, central limit theorems and functional limit theorems---see for example \\cite{baur2016elephant, bercu, coletticlt, coletticlt2}. Results on hitting times have been obtained, but mostly from the view point of \\textit{return times} and recurrence/transience---see \\cite{bertoinZeros, colettiRec}. Surprisingly, such natural questions as the gambler's ruin problem---in other words, the question of \\textit{escape times}---has, to the best of our knowledge, never been addressed for the ERW, even though it was briefly mentioned in \\cite{stadtmuller} and \\cite{pozdnyakov}.\n\n\\vspace{0.2 cm}\n\nThe question is whether or not the classical behavior mentioned at the beginning of this section remains when $(S_n)_{n \\ge 0}$ is replaced by $(Z_n)_{n \\ge 0}$. For concreteness, one can imagine a gambling game in which the coin is replaced by Pólya's urn\\footnote{See \\cite{baur2016elephant} for details on the connection between ERW and Pólya's urn.}. More precisely, suppose an urn contains initially a unique ball, which is either green or red. Then at any time $n \\geq 1$ a ball is drawn at random (uniformly) from the urn, and then put back in with another ball, which has either the same color or the other color, with probability $p$ and $1-p$ respectively. Then replacing $(S_n)_{n \\geq 0}$ by $(Z_n)_{n \\geq 0}$ is the same as saying that the gambler is betting on the outcome of the urn sampling, and win or loose one dollar at time $n$ depending on whether the ball drawn at time $n$ is green or not. Define:\n\n\\begin{equation*}\n    \\tau_N = \\inf\\left\\{k\\ge 1: |Z_k| = N\\right\\}.\n\\end{equation*}\n\n\\vspace{0.2 cm}\n\nIt is well-known (see \\cite{schutz2004elephants}) that the behavior of the ERW critically depends on the value of $p$, with a phase transition at $3/4$. $(Z_n)_{n \\geq 0}$ is said to be in the \\textit{diffusive regime} when $p < 3/4$, and in the \\textit{super-diffusive} regime when $p > 3/4$. We prove that, in the diffusive regime, the expectation of the game duration exhibits similar quadratic behavior as in the classical symmetric gambler's ruin for big $N$. More precisely, we prove the following.\n\n\\begin{proposition}\\label{prop_1}\n    If $0<p<3/4$ then there exists a constant $\\theta$ such that: $$\\mathbb{E}[\\tau_N] \\underset{N \\rightarrow\\infty}{\\sim} \\theta N^2. $$\n\\end{proposition}\n\n\\vspace{0.2 cm}\n\nThe above result is obtained as a corollary of a much stronger one, which might actually be considered as the main contribution of this work. Namely we obtain tight bounds on the tail of the game's duration of the elephant's ruin.", "context": "The gambler’s ruin may be the oldest problem in probability theory, dating back to a letter from Blaise Pascal to Pierre de Fermat in 1656. In the classical gambler's ruin, a gambler is betting on the outcome of a possibly biased coin, winning or loosing one dollar at each step, with probability $p$ and $1-p$ respectively, independently of what happened up to the current step. Assume that the gambler starts with fortune $N$ and stop either when the totality of its fortune has been lost, or when it has doubled. For simplicity we adopt the representation in which $S_n$ represents the gambler's gains (when $S_n \\geq 0$) or losses (when $S_n \\leq 0$) with $S_0=0$. Thus $(S_n)_{n \\ge 0}$ is a Simple Random Walk (SRW) on the one-dimensional integer lattice, starting at the origin. Moreover we denote the game's duration by $\\sigma_N = \\inf\\left\\{n\\ge 1: \\left|S_n\\right| = N \\right\\}$. One of the main object of interest is the expected duration of the game, and, in this standard (Markovian and time-homogeneous) setting the problem has long been fully and analytically solved. Classical results\\footnote{See for example \\cite{feller}, pages 348 and 349.} tell us that $\\mathbb{E}[\\sigma_N]$ is of quadratic order in the symmetric case, while it is of linear order in the asymmetric case. More precisely, if $p=1/2$, then $\\mathbb{E} \\left[\\sigma_N \\right] = N^2$, otherwise: $$\\mathbb{E} \\left[\\sigma_N \\right] = \\frac{N}{2p - 1} \\left( \\frac{1 - \\left( \\frac{1-p}{p}\\right)^N}{1 + \\left( \\frac{1-p}{p}\\right)^N}\\right).$$\nThe proofs of these closed-form formulas are highly dependent on the Markovian and time-homogeneous nature of the dynamic. In the present work, we develop an approach allowing the analysis of a case in which the random walk has a past-dependent dynamic. Namely, we consider the problem above when the classical simple random walk is replaced by the elephant's random walk.\n\nThe Elephant's Random Walk (ERW) is a discrete time stochastic process introduced by Schütz and Trimper \\cite{schutz2004elephants} as a paradigmatic example of past-dependent random walks. It is both quantitatively tractable and qualitatively rich---exhibiting in particular a phase transition from diffusive to super-diffusive behavior---and has therefore recently become the standard choice for testing classical results outside the traditional Markovian setting, when memory is allowed to play a role. It can be defined as follows. Let $(Z_n)_{n \\geq 0}$ be a stochastic chain with value in $\\mathbb{Z}$, making jumps of unit size---either upward or downward---at each time step. It can be defined, like classical random walks, as the partial sum of a sequence $(X_n)_{n \\geq 0}$ of random variables: $$Z_n = \\sum_{k=0}^n X_k,$$ with the crucial difference that the $(X_n)_{n \\geq 0}$ are not independent. Conventionally $X_0 = 0$, and the first jump leads to either $X_1 = 1$ or $X_1 = -1$, with probabilities that are unimportant for our purposes, since in either cases $|Z_1|=1$. Then, for any $n \\geq 1$, the jump at time $n+1$ is given by:\n$$X_{n+1} = \\begin{cases} X_{K(n)} &\\text{ with probability } p,\\\\\n-X_{K(n)} &\\text{ with probability } 1-p,\n\\end{cases}$$\nwhere $(K(n))_{n \\geq 1}$ is a sequence of independent random variables, $K(n)$ having uniform distribution on $\\{1, \\ldots, n\\}$. In words, at each time step, the ERW selects a step uniformly at random from its entire past and repeats it with probability $p$, or takes the opposite step with probability $1 - p$. The parameter $p$ is often referred to as the \\textit{memory} of the elephant: when $p$ is close to $1$, the dependence on the past tends to persist over a long period, whereas it fades rapidly when $p$ is close to $0$.\n\n\\begin{equation*}\n \\tau_N = \\inf\\left\\{k\\ge 1: |Z_k| = N\\right\\}.\n\\end{equation*}\n\n\\begin{proposition}\\label{prop_1}\n If $0<p<3/4$ then there exists a constant $\\theta$ such that: $$\\mathbb{E}[\\tau_N] \\underset{N \\rightarrow\\infty}{\\sim} \\theta N^2. $$\n\\end{proposition}\n\n\\vspace{0.2 cm}\n\nThe above result is obtained as a corollary of a much stronger one, which might actually be considered as the main contribution of this work. Namely we obtain tight bounds on the tail of the game's duration of the elephant's ruin.", "full_context": "The gambler’s ruin may be the oldest problem in probability theory, dating back to a letter from Blaise Pascal to Pierre de Fermat in 1656. In the classical gambler's ruin, a gambler is betting on the outcome of a possibly biased coin, winning or loosing one dollar at each step, with probability $p$ and $1-p$ respectively, independently of what happened up to the current step. Assume that the gambler starts with fortune $N$ and stop either when the totality of its fortune has been lost, or when it has doubled. For simplicity we adopt the representation in which $S_n$ represents the gambler's gains (when $S_n \\geq 0$) or losses (when $S_n \\leq 0$) with $S_0=0$. Thus $(S_n)_{n \\ge 0}$ is a Simple Random Walk (SRW) on the one-dimensional integer lattice, starting at the origin. Moreover we denote the game's duration by $\\sigma_N = \\inf\\left\\{n\\ge 1: \\left|S_n\\right| = N \\right\\}$. One of the main object of interest is the expected duration of the game, and, in this standard (Markovian and time-homogeneous) setting the problem has long been fully and analytically solved. Classical results\\footnote{See for example \\cite{feller}, pages 348 and 349.} tell us that $\\mathbb{E}[\\sigma_N]$ is of quadratic order in the symmetric case, while it is of linear order in the asymmetric case. More precisely, if $p=1/2$, then $\\mathbb{E} \\left[\\sigma_N \\right] = N^2$, otherwise: $$\\mathbb{E} \\left[\\sigma_N \\right] = \\frac{N}{2p - 1} \\left( \\frac{1 - \\left( \\frac{1-p}{p}\\right)^N}{1 + \\left( \\frac{1-p}{p}\\right)^N}\\right).$$\nThe proofs of these closed-form formulas are highly dependent on the Markovian and time-homogeneous nature of the dynamic. In the present work, we develop an approach allowing the analysis of a case in which the random walk has a past-dependent dynamic. Namely, we consider the problem above when the classical simple random walk is replaced by the elephant's random walk.\n\nThe Elephant's Random Walk (ERW) is a discrete time stochastic process introduced by Schütz and Trimper \\cite{schutz2004elephants} as a paradigmatic example of past-dependent random walks. It is both quantitatively tractable and qualitatively rich---exhibiting in particular a phase transition from diffusive to super-diffusive behavior---and has therefore recently become the standard choice for testing classical results outside the traditional Markovian setting, when memory is allowed to play a role. It can be defined as follows. Let $(Z_n)_{n \\geq 0}$ be a stochastic chain with value in $\\mathbb{Z}$, making jumps of unit size---either upward or downward---at each time step. It can be defined, like classical random walks, as the partial sum of a sequence $(X_n)_{n \\geq 0}$ of random variables: $$Z_n = \\sum_{k=0}^n X_k,$$ with the crucial difference that the $(X_n)_{n \\geq 0}$ are not independent. Conventionally $X_0 = 0$, and the first jump leads to either $X_1 = 1$ or $X_1 = -1$, with probabilities that are unimportant for our purposes, since in either cases $|Z_1|=1$. Then, for any $n \\geq 1$, the jump at time $n+1$ is given by:\n$$X_{n+1} = \\begin{cases} X_{K(n)} &\\text{ with probability } p,\\\\\n-X_{K(n)} &\\text{ with probability } 1-p,\n\\end{cases}$$\nwhere $(K(n))_{n \\geq 1}$ is a sequence of independent random variables, $K(n)$ having uniform distribution on $\\{1, \\ldots, n\\}$. In words, at each time step, the ERW selects a step uniformly at random from its entire past and repeats it with probability $p$, or takes the opposite step with probability $1 - p$. The parameter $p$ is often referred to as the \\textit{memory} of the elephant: when $p$ is close to $1$, the dependence on the past tends to persist over a long period, whereas it fades rapidly when $p$ is close to $0$.\n\n\\begin{equation*}\n \\tau_N = \\inf\\left\\{k\\ge 1: |Z_k| = N\\right\\}.\n\\end{equation*}\n\n\\begin{proposition}\\label{prop_1}\n If $0<p<3/4$ then there exists a constant $\\theta$ such that: $$\\mathbb{E}[\\tau_N] \\underset{N \\rightarrow\\infty}{\\sim} \\theta N^2. $$\n\\end{proposition}\n\n\\vspace{0.2 cm}\n\nThe above result is obtained as a corollary of a much stronger one, which might actually be considered as the main contribution of this work. Namely we obtain tight bounds on the tail of the game's duration of the elephant's ruin.\n\nThe above result is obtained as a corollary of a much stronger one, which might actually be considered as the main contribution of this work. Namely we obtain tight bounds on the tail of the game's duration of the elephant's ruin.\n\n\\begin{remark}\n Even though formally the result above is stated only for $0 < p < 3/4$, the upper bound actually holds for all $0 < p < 1$. Moreover, some of the constants above may depend on $p$.\n\\end{remark}\n\n\\begin{lemma}\\label{lemma:sym} There exists positive absolute constants $C_1$, $C_2$, $C_3$ and $C_4$ such that \n\\begin{equation*}\n C_1 \\exp \\left(-C_2\\frac{t}{N^2}\\right) \\le \\mathbb{P}(\\sigma_N > t) \\le C_3 \\exp \\left(-C_4\\frac{t}{N^2}\\right),\n\\end{equation*}\nfor all $N\\ge 2$ and $t \\ge 1$. \n\\end{lemma}\n\n\\begin{enumerate}[(i)]\n \\item \\textbf{Case $\\bm{p \\ge 1/2}$.}\n Let $\\tilde{S}_0 = 0$ and, for $k\\ge 1$, define recursively $\\xi_k$ and $\\tilde{S}_k$ by:\n \\begin{equation*}\n \\xi_{k} = \\mathbbm{1}_{\\left\\{\\tilde{S}_{k-1} = 0 \\right\\}} + \\mathbbm{1}_{\\left\\{\\tilde{S}_{k-1} \\neq 0 \\right\\}} \\left( \\mathbbm{1}_{\\left\\{U_{k} < \\frac{1}{2} \\right\\}} - \\mathbbm{1}_{\\left\\{U_{k} \\ge \\frac{1}{2}\\right\\}}\\right), \n \\end{equation*}\n and\n \\begin{equation}\\label{srw-coupling}\n \\tilde{S}_{k} = \\sum_{j=1}^{k}\\xi_j.\n \\end{equation}\n Then $\\tilde{S}_k$ has the same distribution as $|S_k|$, where $(S_k)_{k \\ge 0}$ is still the symmetric simple random walk defined in the previous section. Moreover, $(\\tilde{S}_k)_{k\\ge 0}$ and $(\\tilde{Z}_{k} )_{k\\ge 0}$ are constructed in such a way that the following holds:\n \\begin{equation}\\label{enq_1}\n \\tilde{Z}_k \\ge \\tilde{S}_k, \\, \\text{ for all } k\\ge 0.\n \\end{equation}\n Notice that the only way inequality \\eqref{enq_1} could be broken would be if at some point $\\tilde{Z}_k = 1$ and $\\tilde{S}_k = 0$ (and $\\tilde{Z}_{k}$ jumps to the left between $k$ and $k+1$). Fortunately this never happens since $\\tilde{Z}_k$ and $\\tilde{S}_k$ have the same parity. Below, we deliberately abuse notation, redefining $\\tau_N$ and $\\sigma_N$ in terms of $(\\tilde{S}_k)_{k\\ge 0}$ and $(\\tilde{Z}_{k} )_{k\\ge 0}$. By inequality \\eqref{enq_1} and Lemma \\ref{lemma:sym}, for all $N\\ge 1$ and $t\\ge 1$, one indeed has:\n\n\\begin{equation}\\label{enq_2}\n \\tilde{S}_k - 2\\sum_{j=1}^k\\mathbbm{1}_{\\left\\{\\frac{1}{2} + \\frac{(2p-1)\\tilde{Z}_{j +dN^{2}}}{j + dN^2}\\le U_{j + dN^{2}} < \\frac{1}{2}\\right\\}} \\le \\tilde{Z}_{k + dN^{2}} , \\, \\text{ for all } k\\ge 0.\n \\end{equation}\n Let $c > 12 + d$. For any $x < N$ we have\n \\begin{align*}\\label{eq:boundonethird+}\n \\P_x \\left(\\tau_N > cN^2\\right) &= \\P_x\\left(\\tau_N > cN^{2}, \\sigma_{2N} > 3(2N)^2\\right) + \\P_x \\left(\\tau_N > cN^{2}, \\sigma_{2N} \\le 3(2N)^2)\\right)\\\\\n &\\le \\P_x \\left(\\sigma_{2N} > 3(2N)^2\\right) + \\P_x \\left(\\tau_N > cN^{2}, \\sigma_{2N} \\le 3(2N)^2\\right)\n \\end{align*}\n But $\\P_x \\left(\\sigma_{2N} > 3(2N)^2 \\right) \\leq 1/3$ by Markov inequality (see inequality $(\\ref{eq:markov_sym})$ in the proof of Lemma \\ref{lemma:sym}), hence:\n \\begin{equation}\\label{eq:boundonethird+}\n \\P_x \\left(\\tau_N > cN^2 \\right) \\le \\frac{1}{3} + \\P_x \\left(\\tau_N > cN^{2}, \\, \\sigma_{2N} \\le 3(2N)^2 \\right).\n \\end{equation}\n Now observe that $\\tau_N > cN^{2}$ means that the elephant has not reached $N$ yet at time $c N^2$ while $\\sigma_{2N} \\le 3(2N)^2$ means that the symmetric random walk reached $2N$ in the meantime, even though it was exactly at the same point $12N^2$ time-steps earlier (or more). Now assume that both hold. Then $\\tilde{Z}_k < N$ for any $k \\le dN^2 + \\sigma_{2N}$, where $dN^2 + \\sigma_{2N}$ represents the time at which $(\\tilde{S}_k)_{k \\geq 0}$ reaches $2N$, but on the $(\\tilde{Z}_k)_{k \\geq 0}$ time-scale. Then the inequality $\\tilde{Z}_{dN^2 + \\sigma_{2N}}< N$ together with \\eqref{enq_2} implies that \n $$2N - 2 \\sum_{j=1}^{\\sigma_{2N}}\\mathbbm{1}_{\\big\\{\\frac{1}{2} + \\frac{(2p-1)\\tilde{Z}_{j + dN^{2}}}{j + dN^{2}}\\le U_{j + dN^{2}} < \\frac{1}{2}\\big\\}} < N.$$\n Moreover, since $\\sigma_{2N} \\le 3(2N)^2$ and $\\tilde{Z}_{j + dN^2} < N$ for any of the indices $j$ in the sum above, the above inequality implies that\n $$ 2N - 2\\sum_{j=1}^{3 (2 N)^2}\\mathbbm{1}_{\\{\\frac{1}{2} + \\frac{(2p - 1)}{dN}\\le U_{j + dN^{2}} < \\frac{1}{2} \\} } < N.$$\n\n\\begin{equation*}\n \\big\\{ \\tau_N > cN^{2}, \\sigma_{2N} \\le 3(2N)^2 \\big\\} \\subset \\Big\\{B_N > \\frac{N}{2} \\Big\\},\n \\end{equation*}\n where: $$B_N = \\sum_{j=1}^{12 N^2}\\mathbbm{1}_{\\left\\{\\frac{1}{2} + \\frac{(2p - 1)}{dN}\\le U_{j + dN^{2}} < \\frac{1}{2} \\right\\}} \\sim \\text{Binomial}\\left(12 N^2, \\, \\frac{1-2p}{dN}\\right).$$ \n Now, by letting $d > \\lceil 24(1-2p)\\rceil$, we have:\n \\begin{equation}\\label{eq_2}\n \\mathbb{E}[B_N] = \\frac{12(1 - 2p)N}{d} < \\frac{N}{2},\n \\end{equation}\n and thus\n \\begin{equation*}\n \\P\\Big(B_N > \\frac{N}{2}\\Big) \\le \\frac{1}{2}.\n \\end{equation*}\n Hence, for our choice of $d$, it holds that:\n \\begin{equation} \\label{eq:onehalfpart}\n \\P_x \\left(\\tau_N > cN^{2}, \\sigma_{2N} \\le 3(2N)^2 \\right) \\le \\frac{1}{2}.\n \\end{equation}\n Finally, inequality \\eqref{eq:onehalfpart} together with inequality \\eqref{eq:boundonethird+} leads to:\n \\begin{equation*}\n \\P_x \\left(\\tau_N > cN^2 \\right) \\le \\frac{5}{6}.\n \\end{equation*}\n Then the same line of reasoning as in the end of the proof of Lemma \\ref{lemma:sym} allows us to conclude:\n\nNotice that, since ${\\left\\{U_j < p \\right\\} = \\left\\{U_j < \\frac{1}{2} + \\frac{2p-1}{2} \\right\\}}$, one has $\\tilde{Z}_k \\le \\hat{S}_k$, for all $k\\ge 0$. Therefore, if \n\\begin{equation*}\n \\hat{\\sigma}_N = \\inf\\{k\\ge 1: \\hat{S}_k = N\\},\n\\end{equation*}\nthen, for all $N \\ge 2$ and $t\\ge 1$, one has: \n\\begin{equation*}\n \\mathbb{P}(\\tau_N > t) \\ge \\mathbb{P}(\\hat{\\sigma}_N > t).\n\\end{equation*}\nObserving that $\\{U_1 > p , \\, \\ldots , \\, U_t > p\\}\\subset \\{\\hat{\\sigma}_N > t\\}$, we obtain the following---very crude, but good enough---bound:\n\\begin{equation*}\n \\mathbb{P}(\\hat{\\sigma}_N > t) \\ge (1-p)^t = e^{\\log(1 - p) t} \\ge e^{-\\log \\left(\\frac{1}{1-p}\\right)C_A^2 \\frac{t}{N^2}},\n\\end{equation*}\nfor all $t\\ge 1$ and $2\\le N\\le C_A$. Hence taking $C_2 = \\max \\left\\{\\log \\left(\\frac{1}{1-p}\\right)C_A^2, \\tilde{C}_2 \\right \\}$, we can indeed conclude:\n\\begin{equation*}\n \\mathbbm{P}(\\tau_N \\ge t) \\ge C_1e^{-C_2\\frac{t}{N^2}},\n\\end{equation*}\nfor all $N\\ge 2$ and $t\\ge 1$.\n\\end{enumerate}\n\nWe start with the upper bound and then proceed to the lower bound. Fix some $c > 1$. For any $x \\in \\llbracket -N, N \\rrbracket$, a classical martingale argument (see for example Theorem 4.8.7 in \\cite{durrett2019probability}) gives us:\n \\begin{equation*}\n \\E_x \\left[\\sigma_N \\right] = (N-x)(x+N) \\le N^2.\n \\end{equation*}\n Hence, by Markov inequality:\n \\begin{equation}\\label{eq:markov_sym}\n \\P_x \\left(\\sigma_N > cN^2 \\right) \\le \\frac{\\E_x \\left[ \\sigma_N \\right]}{cN^2}\\le \\frac{1}{c}.\n \\end{equation}\n For $t \\geq 0$, it follows that:\n \\begin{equation*}\n \\P(\\sigma_N > t) \\leq \\P \\left(\\sigma_N \\ge \\left\\lfloor \\frac{t}{cN^2}\\right\\rfloor cN^2 \\right) \\le \\left(\\frac{1}{c}\\right)^{\\left\\lfloor \\frac{t}{cN^2}\\right\\rfloor} \\le \\left(\\frac{1}{c}\\right)^{\\frac{t}{cN^2} - 1} = C_3e^{-C_4\\frac{t}{N^2}},\n \\end{equation*}\n with $C_3 = c$ and $C_4 = \\log(c)/c$. Notice that, for $t \\geq cN^2$, the second inequality above is a consequence of \\eqref{eq:markov_sym} together with the strong Markov property, whereas it is trivial for $t < c N^2$, since in that case $\\left(\\frac{1}{c}\\right)^{\\left\\lfloor \\frac{t}{cN^2} \\right\\rfloor} = 1$.", "post_theorem_intro_text_len": 2482, "post_theorem_intro_text": "\\begin{remark}\n    Even though formally the result above is stated only for $0 < p < 3/4$, the upper bound actually holds for all $0 < p < 1$. Moreover, some of the constants above may depend on $p$.\n\\end{remark}\n\nThe main idea behind the proof of Theorem \\ref{main_thm} is to first establish analogous bounds for the SRW, and then demonstrate via couplings that, on a quadratic time scale, the ERW does not deviate excessively from the SRW. These couplings rely on a well-known time-inhomogeneous Markovian description of the ERW. Given a sequence of i.i.d. random variables with uniform distribution on $(0,1)$, we can construct both the ERW and the SRW on the same probability space. Each walk determines its next jump based on the value of the current uniform random variable relative to a specified threshold. For the SRW this threshold is always $1/2$, while for the ERW it depends on both time and the walk’s current state—though it is always either greater or less than $1/2$, depending on whether $p>1/2$ or not. This construction yields joint realizations in which the ERW either dominates or is dominated by the SRW, according to the value of $p$. By leveraging the SRW's bounds, this approach already resolves half of the cases. To address the remaining cases and thus complete the proof of Theorem \\ref{main_thm}, one then needs to control the distance between the SRW and the ERW on the appropriate time scale. Specifically, for $t = cN^2$, we identify some scaling constant $A$ such that, whenever the ERW or the SRW reaches the boundaries of $[-AN, AN]$, the other one has vanishingly small probability of not having reached $\\{-N,N\\}$ yet. This allows us prove Theorem \\ref{main_thm} on the quadratic time scale $t = cN^2$. Then, by partitioning the interval $[0, t]$ into sub-intervals of length $t/cN^2$, we extend the result to any $t \\geq 1$. Finally, Proposition \\ref{prop_1} follows directly from the functional limit theorem in \\cite{baur2016elephant} and uniform integrability.\n\n\\vspace{0.2 cm}\n\nThe paper is organized as follows. In Section 2 we derive a well-known (time inhomogeneous) Markovian description of the ERW allowing a coupling with the simple random walks. In Section 3 we prove the exponential bounds on the tail of the simple random walk. In Section 4 we define a coupling between ERW and the classical random walks and prove our main result, Theorem \\ref{main_thm}. Finally, in Section 5, we prove Proposition \\ref{prop_1}.\n\n\\vspace{0.2 cm}", "sketch": "The proof of Theorem \\ref{main_thm} proceeds as follows:\n\n- First, “establish analogous bounds for the SRW, and then demonstrate via couplings that, on a quadratic time scale, the ERW does not deviate excessively from the SRW.”\n\n- The couplings use “a well-known time-inhomogeneous Markovian description of the ERW.” Using i.i.d. uniforms on $(0,1)$, “we can construct both the ERW and the SRW on the same probability space,” where each walk’s next jump is determined by comparing the current uniform to a threshold: for SRW the threshold is always $1/2$, while for ERW it “depends on both time and the walk’s current state.” This produces joint realizations where “the ERW either dominates or is dominated by the SRW, according to the value of $p$,” and “by leveraging the SRW's bounds, this approach already resolves half of the cases.”\n\n- For the remaining cases, “one then needs to control the distance between the SRW and the ERW on the appropriate time scale.” Concretely, for $t=cN^2$, choose a scaling constant $A$ such that whenever one walk hits the boundary of $[-AN,AN]$, “the other one has vanishingly small probability of not having reached $\\{-N,N\\}$ yet.” This yields Theorem \\ref{main_thm} “on the quadratic time scale $t=cN^2$.”\n\n- To pass from $t=cN^2$ to general $t\\ge 1$, “partitioning the interval $[0,t]$ into sub-intervals of length $t/cN^2$,” thereby extending the bound to “any $t\\ge 1$.”", "expanded_sketch": "The proof of the main theorem proceeds as follows:\n\n- First, “establish analogous bounds for the SRW, and then demonstrate via couplings that, on a quadratic time scale, the ERW does not deviate excessively from the SRW.”\n\n- The couplings use “a well-known time-inhomogeneous Markovian description of the ERW.” Using i.i.d. uniforms on $(0,1)$, “we can construct both the ERW and the SRW on the same probability space,” where each walk’s next jump is determined by comparing the current uniform to a threshold: for SRW the threshold is always $1/2$, while for ERW it “depends on both time and the walk’s current state.” This produces joint realizations where “the ERW either dominates or is dominated by the SRW, according to the value of $p$,” and “by leveraging the SRW's bounds, this approach already resolves half of the cases.”\n\n- For the remaining cases, “one then needs to control the distance between the SRW and the ERW on the appropriate time scale.” Concretely, for $t=cN^2$, choose a scaling constant $A$ such that whenever one walk hits the boundary of $[-AN,AN]$, “the other one has vanishingly small probability of not having reached $\\{-N,N\\}$ yet.” This yields the main theorem “on the quadratic time scale $t=cN^2$.”\n\n- To pass from $t=cN^2$ to general $t\\ge 1$, “partitioning the interval $[0,t]$ into sub-intervals of length $t/cN^2$,” thereby extending the bound to “any $t\\ge 1$.”", "expanded_theorem": "\\label{main_thm} If $0<p<3/4$ then there exists positive constants $C_1$, $C_2$, $C_3$ and $C_4$ such that, for all $N \\ge 2$ and $t \\geq 1$, the following bound holds:\n\n    \\begin{equation*}\n        C_1 \\exp \\left( -C_2 \\frac{t}{N^2} \\right) \\leq \\mathbb{P} \\left( \\tau_N > t \\right) \\le C_3 \\exp \\left(-C_4\\frac{t}{N^2}\\right).\n    \\end{equation*}", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Consider the elephant random walk $(Z_n)_{n\\ge 0}$ on $\\mathbb Z$, defined by $Z_n=\\sum_{k=0}^n X_k$ with $X_0=0$, the first step $X_1\\in\\{-1,1\\}$, and for each $n\\ge 1$, after choosing $K(n)$ uniformly from $\\{1,\\dots,n\\}$ independently of the past,\n\\[\nX_{n+1}=\\begin{cases}\nX_{K(n)} & \\text{with probability } p,\\\\\n- X_{K(n)} & \\text{with probability } 1-p.\n\\end{cases}\n\\]\nFor $N\\ge 2$, define the hitting time of the two-sided boundary $\\{\\pm N\\}$ by\n\\[\n\\tau_N=\\inf\\{k\\ge 1: |Z_k|=N\\}.\n\\]\nAssume $0<p<3/4$. Which quantitative estimate holds for the tail of $\\tau_N$ for all $N\\ge 2$ and $t\\ge 1$?", "correct_choice": {"label": "A", "text": "There exist positive constants $C_1,C_2,C_3,C_4$ such that, for every $N\\ge 2$ and $t\\ge 1$,\n\\[\nC_1\\exp\\!\\left(-C_2\\frac{t}{N^2}\\right)\\le \\mathbb P(\\tau_N>t)\\le C_3\\exp\\!\\left(-C_4\\frac{t}{N^2}\\right).\n\\]"}, "choices": [{"label": "B", "text": "There exist positive constants $C_1,C_2,C_3,C_4$ such that, for every $N\\ge 2$ and $t\\ge 1$,\n\\[\nC_1\\exp\\!\\left(-C_2\\frac{t}{N}\\right)\\le \\mathbb P(\\tau_N>t)\\le C_3\\exp\\!\\left(-C_4\\frac{t}{N}\\right).\n\\]"}, {"label": "C", "text": "There exist positive constants $C_3,C_4$ such that, for every $N\\ge 2$ and $t\\ge 1$,\n\\[\n\\mathbb P(\\tau_N>t)\\le C_3\\exp\\!\\left(-C_4\\frac{t}{N^2}\\right).\n\\]"}, {"label": "D", "text": "For every $N\\ge 2$ there exist positive constants $C_1(N),C_2(N),C_3(N),C_4(N)$ such that, for every $t\\ge 1$,\n\\[\nC_1(N)\\exp\\!\\left(-C_2(N)\\frac{t}{N^2}\\right)\\le \\mathbb P(\\tau_N>t)\\le C_3(N)\\exp\\!\\left(-C_4(N)\\frac{t}{N^2}\\right).\n\\]"}, {"label": "E", "text": "There exist positive constants $C_1,C_2,C_3,C_4$ such that, for every $N\\ge 2$ and $t\\ge 1$,\n\\[\nC_1\\exp\\!\\left(-C_2\\frac{t}{N^2}\\right)\\le \\mathbb P(\\tau_N=t)\\le C_3\\exp\\!\\left(-C_4\\frac{t}{N^2}\\right).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "quadratic_time_scale", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "lower_tail_bound_removed", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "uniform_constants_in_N", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "tail_event_replaced_by_point_mass", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives the model and parameter regime but does not state or strongly hint at the correct tail estimate. The correct answer is not leaked explicitly or by obvious wording cues."}, "TAS": {"score": 1, "justification": "The item is essentially asking for the theorem-level conclusion for the subcritical regime, so it is close to a restatement of a known result. However, the alternatives vary in scaling, quantifier strength, and whether one asks for a tail or point mass, so it is not a pure verbatim restatement."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure: a strong respondent must distinguish the correct diffusive scale N^2, uniform-in-N constants, and two-sided tail bounds from weaker or incorrect variants. But the item mostly tests recognition/recall of the theorem, and the presence of another true statement (C) weakens the need for a uniquely reasoned choice."}, "DQS": {"score": 1, "justification": "Most distractors are plausible and target meaningful failure modes: wrong time scale (B), loss of uniformity in N (D), and confusing tail probability with point mass (E). However, C is a weaker statement that is still true, so the distractor set is not fully clean and creates ambiguity."}, "total_score": 5, "overall_assessment": "Good on avoiding answer leakage, with mathematically meaningful alternatives, but only moderate as an assessment item: it is close to theorem recall and is flawed by ambiguity because choice C also holds as written."}}
{"id": "2602.18953v1", "paper_link": "http://arxiv.org/abs/2602.18953v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{main_thm} If $0<p<3/4$ then there exists positive constants $C_1$, $C_2$, $C_3$ and $C_4$ such that, for all $N \\ge 2$ and $t \\geq 1$, the following bound holds:\n\n    \\begin{equation*}\n        C_1 \\exp \\left( -C_2 \\frac{t}{N^2} \\right) \\leq \\mathbb{P} \\left( \\tau_N > t \\right) \\le C_3 \\exp \\left(-C_4\\frac{t}{N^2}\\right).\n    \\end{equation*}", "start_pos": 8988, "end_pos": 9358, "label": "main_thm"}, "ref_dict": {"main_thm": "\\begin{theorem}\\label{main_thm} If $0<p<3/4$ then there exists positive constants $C_1$, $C_2$, $C_3$ and $C_4$ such that, for all $N \\ge 2$ and $t \\geq 1$, the following bound holds:\n\n    \\begin{equation*}\n        C_1 \\exp \\left( -C_2 \\frac{t}{N^2} \\right) \\leq \\P \\left( \\tau_N > t \\right) \\le C_3 \\exp \\left(-C_4\\frac{t}{N^2}\\right).\n    \\end{equation*}\n\\end{theorem}", "prop_1": "\\begin{proposition}\\label{prop_1}\n    If $0<p<3/4$ then there exists a constant $\\theta$ such that: $$\\E[\\tau_N] \\underset{N \\rightarrow\\infty}{\\sim} \\theta N^2. $$\n\\end{proposition}"}, "pre_theorem_intro_text_len": 6539, "pre_theorem_intro_text": "The gambler’s ruin may be the oldest problem in probability theory, dating back to a letter from Blaise Pascal to Pierre de Fermat in 1656. In the classical gambler's ruin, a gambler is betting on the outcome of a possibly biased coin, winning or loosing one dollar at each step, with probability $p$ and $1-p$ respectively, independently of what happened up to the current step. Assume that the gambler starts with fortune $N$ and stop either when the totality of its fortune has been lost, or when it has doubled. For simplicity we adopt the representation in which $S_n$ represents the gambler's gains (when $S_n \\geq 0$) or losses (when $S_n \\leq 0$) with $S_0=0$. Thus $(S_n)_{n \\ge 0}$ is a Simple Random Walk (SRW) on the one-dimensional integer lattice, starting at the origin. Moreover we denote the game's duration by $\\sigma_N = \\inf\\left\\{n\\ge 1: \\left|S_n\\right| = N \\right\\}$. One of the main object of interest is the expected duration of the game, and, in this standard (Markovian and time-homogeneous) setting the problem has long been fully and analytically solved. Classical results\\footnote{See for example \\cite{feller}, pages 348 and 349.} tell us that $\\mathbb{E}[\\sigma_N]$ is of quadratic order in the symmetric case, while it is of linear order in the asymmetric case. More precisely, if $p=1/2$, then $\\mathbb{E} \\left[\\sigma_N \\right] = N^2$, otherwise: $$\\mathbb{E} \\left[\\sigma_N \\right] = \\frac{N}{2p - 1} \\left( \\frac{1 - \\left( \\frac{1-p}{p}\\right)^N}{1 + \\left( \\frac{1-p}{p}\\right)^N}\\right).$$\nThe proofs of these closed-form formulas are highly dependent on the Markovian and time-homogeneous nature of the dynamic. In the present work, we develop an approach allowing the analysis of a case in which the random walk has a past-dependent dynamic. Namely, we consider the problem above when the classical simple random walk is replaced by the elephant's random walk.\n\n\\vspace{0.2 cm}\n\nThe Elephant's Random Walk (ERW) is a discrete time stochastic process introduced by Schütz and Trimper \\cite{schutz2004elephants} as a paradigmatic example of past-dependent random walks. It is both quantitatively tractable and qualitatively rich---exhibiting in particular a phase transition from diffusive to super-diffusive behavior---and has therefore recently become the standard choice for testing classical results outside the traditional Markovian setting, when memory is allowed to play a role. It can be defined as follows. Let $(Z_n)_{n \\geq 0}$ be a stochastic chain with value in $\\mathbb{Z}$, making jumps of unit size---either upward or downward---at each time step. It can be defined, like classical random walks, as the partial sum of a sequence $(X_n)_{n \\geq 0}$ of random variables: $$Z_n = \\sum_{k=0}^n X_k,$$ with the crucial difference that the $(X_n)_{n \\geq 0}$ are not independent. Conventionally $X_0 = 0$, and the first jump leads to either $X_1 = 1$ or $X_1 = -1$, with probabilities that are unimportant for our purposes, since in either cases $|Z_1|=1$. Then, for any $n \\geq 1$, the jump at time $n+1$ is given by:\n$$X_{n+1} = \\begin{cases} X_{K(n)} &\\text{ with probability } p,\\\\\n-X_{K(n)} &\\text{ with probability } 1-p,\n\\end{cases}$$\nwhere $(K(n))_{n \\geq 1}$ is a sequence of independent random variables, $K(n)$ having uniform distribution on $\\{1, \\ldots, n\\}$. In words, at each time step, the ERW selects a step uniformly at random from its entire past and repeats it with probability $p$, or takes the opposite step with probability $1 - p$. The parameter $p$ is often referred to as the \\textit{memory} of the elephant: when $p$ is close to $1$, the dependence on the past tends to persist over a long period, whereas it fades rapidly when $p$ is close to $0$.\n\n\\vspace{0.2 cm}\n\nMuch effort has been done during the last two decades to understand the effect of the memory of the elephant on its asymptotic behavior, as compared to classical Markovian random walks, with a particular focus on laws of large numbers, central limit theorems and functional limit theorems---see for example \\cite{baur2016elephant, bercu, coletticlt, coletticlt2}. Results on hitting times have been obtained, but mostly from the view point of \\textit{return times} and recurrence/transience---see \\cite{bertoinZeros, colettiRec}. Surprisingly, such natural questions as the gambler's ruin problem---in other words, the question of \\textit{escape times}---has, to the best of our knowledge, never been addressed for the ERW, even though it was briefly mentioned in \\cite{stadtmuller} and \\cite{pozdnyakov}.\n\n\\vspace{0.2 cm}\n\nThe question is whether or not the classical behavior mentioned at the beginning of this section remains when $(S_n)_{n \\ge 0}$ is replaced by $(Z_n)_{n \\ge 0}$. For concreteness, one can imagine a gambling game in which the coin is replaced by Pólya's urn\\footnote{See \\cite{baur2016elephant} for details on the connection between ERW and Pólya's urn.}. More precisely, suppose an urn contains initially a unique ball, which is either green or red. Then at any time $n \\geq 1$ a ball is drawn at random (uniformly) from the urn, and then put back in with another ball, which has either the same color or the other color, with probability $p$ and $1-p$ respectively. Then replacing $(S_n)_{n \\geq 0}$ by $(Z_n)_{n \\geq 0}$ is the same as saying that the gambler is betting on the outcome of the urn sampling, and win or loose one dollar at time $n$ depending on whether the ball drawn at time $n$ is green or not. Define:\n\n\\begin{equation*}\n    \\tau_N = \\inf\\left\\{k\\ge 1: |Z_k| = N\\right\\}.\n\\end{equation*}\n\n\\vspace{0.2 cm}\n\nIt is well-known (see \\cite{schutz2004elephants}) that the behavior of the ERW critically depends on the value of $p$, with a phase transition at $3/4$. $(Z_n)_{n \\geq 0}$ is said to be in the \\textit{diffusive regime} when $p < 3/4$, and in the \\textit{super-diffusive} regime when $p > 3/4$. We prove that, in the diffusive regime, the expectation of the game duration exhibits similar quadratic behavior as in the classical symmetric gambler's ruin for big $N$. More precisely, we prove the following.\n\n\\begin{proposition}\\label{prop_1}\n    If $0<p<3/4$ then there exists a constant $\\theta$ such that: $$\\mathbb{E}[\\tau_N] \\underset{N \\rightarrow\\infty}{\\sim} \\theta N^2. $$\n\\end{proposition}\n\n\\vspace{0.2 cm}\n\nThe above result is obtained as a corollary of a much stronger one, which might actually be considered as the main contribution of this work. Namely we obtain tight bounds on the tail of the game's duration of the elephant's ruin.", "context": "The gambler’s ruin may be the oldest problem in probability theory, dating back to a letter from Blaise Pascal to Pierre de Fermat in 1656. In the classical gambler's ruin, a gambler is betting on the outcome of a possibly biased coin, winning or loosing one dollar at each step, with probability $p$ and $1-p$ respectively, independently of what happened up to the current step. Assume that the gambler starts with fortune $N$ and stop either when the totality of its fortune has been lost, or when it has doubled. For simplicity we adopt the representation in which $S_n$ represents the gambler's gains (when $S_n \\geq 0$) or losses (when $S_n \\leq 0$) with $S_0=0$. Thus $(S_n)_{n \\ge 0}$ is a Simple Random Walk (SRW) on the one-dimensional integer lattice, starting at the origin. Moreover we denote the game's duration by $\\sigma_N = \\inf\\left\\{n\\ge 1: \\left|S_n\\right| = N \\right\\}$. One of the main object of interest is the expected duration of the game, and, in this standard (Markovian and time-homogeneous) setting the problem has long been fully and analytically solved. Classical results\\footnote{See for example \\cite{feller}, pages 348 and 349.} tell us that $\\mathbb{E}[\\sigma_N]$ is of quadratic order in the symmetric case, while it is of linear order in the asymmetric case. More precisely, if $p=1/2$, then $\\mathbb{E} \\left[\\sigma_N \\right] = N^2$, otherwise: $$\\mathbb{E} \\left[\\sigma_N \\right] = \\frac{N}{2p - 1} \\left( \\frac{1 - \\left( \\frac{1-p}{p}\\right)^N}{1 + \\left( \\frac{1-p}{p}\\right)^N}\\right).$$\nThe proofs of these closed-form formulas are highly dependent on the Markovian and time-homogeneous nature of the dynamic. In the present work, we develop an approach allowing the analysis of a case in which the random walk has a past-dependent dynamic. Namely, we consider the problem above when the classical simple random walk is replaced by the elephant's random walk.\n\nThe Elephant's Random Walk (ERW) is a discrete time stochastic process introduced by Schütz and Trimper \\cite{schutz2004elephants} as a paradigmatic example of past-dependent random walks. It is both quantitatively tractable and qualitatively rich---exhibiting in particular a phase transition from diffusive to super-diffusive behavior---and has therefore recently become the standard choice for testing classical results outside the traditional Markovian setting, when memory is allowed to play a role. It can be defined as follows. Let $(Z_n)_{n \\geq 0}$ be a stochastic chain with value in $\\mathbb{Z}$, making jumps of unit size---either upward or downward---at each time step. It can be defined, like classical random walks, as the partial sum of a sequence $(X_n)_{n \\geq 0}$ of random variables: $$Z_n = \\sum_{k=0}^n X_k,$$ with the crucial difference that the $(X_n)_{n \\geq 0}$ are not independent. Conventionally $X_0 = 0$, and the first jump leads to either $X_1 = 1$ or $X_1 = -1$, with probabilities that are unimportant for our purposes, since in either cases $|Z_1|=1$. Then, for any $n \\geq 1$, the jump at time $n+1$ is given by:\n$$X_{n+1} = \\begin{cases} X_{K(n)} &\\text{ with probability } p,\\\\\n-X_{K(n)} &\\text{ with probability } 1-p,\n\\end{cases}$$\nwhere $(K(n))_{n \\geq 1}$ is a sequence of independent random variables, $K(n)$ having uniform distribution on $\\{1, \\ldots, n\\}$. In words, at each time step, the ERW selects a step uniformly at random from its entire past and repeats it with probability $p$, or takes the opposite step with probability $1 - p$. The parameter $p$ is often referred to as the \\textit{memory} of the elephant: when $p$ is close to $1$, the dependence on the past tends to persist over a long period, whereas it fades rapidly when $p$ is close to $0$.\n\n\\begin{equation*}\n \\tau_N = \\inf\\left\\{k\\ge 1: |Z_k| = N\\right\\}.\n\\end{equation*}\n\n\\begin{proposition}\\label{prop_1}\n If $0<p<3/4$ then there exists a constant $\\theta$ such that: $$\\mathbb{E}[\\tau_N] \\underset{N \\rightarrow\\infty}{\\sim} \\theta N^2. $$\n\\end{proposition}\n\n\\vspace{0.2 cm}\n\nThe above result is obtained as a corollary of a much stronger one, which might actually be considered as the main contribution of this work. Namely we obtain tight bounds on the tail of the game's duration of the elephant's ruin.", "full_context": "The gambler’s ruin may be the oldest problem in probability theory, dating back to a letter from Blaise Pascal to Pierre de Fermat in 1656. In the classical gambler's ruin, a gambler is betting on the outcome of a possibly biased coin, winning or loosing one dollar at each step, with probability $p$ and $1-p$ respectively, independently of what happened up to the current step. Assume that the gambler starts with fortune $N$ and stop either when the totality of its fortune has been lost, or when it has doubled. For simplicity we adopt the representation in which $S_n$ represents the gambler's gains (when $S_n \\geq 0$) or losses (when $S_n \\leq 0$) with $S_0=0$. Thus $(S_n)_{n \\ge 0}$ is a Simple Random Walk (SRW) on the one-dimensional integer lattice, starting at the origin. Moreover we denote the game's duration by $\\sigma_N = \\inf\\left\\{n\\ge 1: \\left|S_n\\right| = N \\right\\}$. One of the main object of interest is the expected duration of the game, and, in this standard (Markovian and time-homogeneous) setting the problem has long been fully and analytically solved. Classical results\\footnote{See for example \\cite{feller}, pages 348 and 349.} tell us that $\\mathbb{E}[\\sigma_N]$ is of quadratic order in the symmetric case, while it is of linear order in the asymmetric case. More precisely, if $p=1/2$, then $\\mathbb{E} \\left[\\sigma_N \\right] = N^2$, otherwise: $$\\mathbb{E} \\left[\\sigma_N \\right] = \\frac{N}{2p - 1} \\left( \\frac{1 - \\left( \\frac{1-p}{p}\\right)^N}{1 + \\left( \\frac{1-p}{p}\\right)^N}\\right).$$\nThe proofs of these closed-form formulas are highly dependent on the Markovian and time-homogeneous nature of the dynamic. In the present work, we develop an approach allowing the analysis of a case in which the random walk has a past-dependent dynamic. Namely, we consider the problem above when the classical simple random walk is replaced by the elephant's random walk.\n\nThe Elephant's Random Walk (ERW) is a discrete time stochastic process introduced by Schütz and Trimper \\cite{schutz2004elephants} as a paradigmatic example of past-dependent random walks. It is both quantitatively tractable and qualitatively rich---exhibiting in particular a phase transition from diffusive to super-diffusive behavior---and has therefore recently become the standard choice for testing classical results outside the traditional Markovian setting, when memory is allowed to play a role. It can be defined as follows. Let $(Z_n)_{n \\geq 0}$ be a stochastic chain with value in $\\mathbb{Z}$, making jumps of unit size---either upward or downward---at each time step. It can be defined, like classical random walks, as the partial sum of a sequence $(X_n)_{n \\geq 0}$ of random variables: $$Z_n = \\sum_{k=0}^n X_k,$$ with the crucial difference that the $(X_n)_{n \\geq 0}$ are not independent. Conventionally $X_0 = 0$, and the first jump leads to either $X_1 = 1$ or $X_1 = -1$, with probabilities that are unimportant for our purposes, since in either cases $|Z_1|=1$. Then, for any $n \\geq 1$, the jump at time $n+1$ is given by:\n$$X_{n+1} = \\begin{cases} X_{K(n)} &\\text{ with probability } p,\\\\\n-X_{K(n)} &\\text{ with probability } 1-p,\n\\end{cases}$$\nwhere $(K(n))_{n \\geq 1}$ is a sequence of independent random variables, $K(n)$ having uniform distribution on $\\{1, \\ldots, n\\}$. In words, at each time step, the ERW selects a step uniformly at random from its entire past and repeats it with probability $p$, or takes the opposite step with probability $1 - p$. The parameter $p$ is often referred to as the \\textit{memory} of the elephant: when $p$ is close to $1$, the dependence on the past tends to persist over a long period, whereas it fades rapidly when $p$ is close to $0$.\n\n\\begin{equation*}\n \\tau_N = \\inf\\left\\{k\\ge 1: |Z_k| = N\\right\\}.\n\\end{equation*}\n\n\\begin{proposition}\\label{prop_1}\n If $0<p<3/4$ then there exists a constant $\\theta$ such that: $$\\mathbb{E}[\\tau_N] \\underset{N \\rightarrow\\infty}{\\sim} \\theta N^2. $$\n\\end{proposition}\n\n\\vspace{0.2 cm}\n\nThe above result is obtained as a corollary of a much stronger one, which might actually be considered as the main contribution of this work. Namely we obtain tight bounds on the tail of the game's duration of the elephant's ruin.\n\nThe above result is obtained as a corollary of a much stronger one, which might actually be considered as the main contribution of this work. Namely we obtain tight bounds on the tail of the game's duration of the elephant's ruin.\n\n\\begin{remark}\n Even though formally the result above is stated only for $0 < p < 3/4$, the upper bound actually holds for all $0 < p < 1$. Moreover, some of the constants above may depend on $p$.\n\\end{remark}\n\n\\begin{lemma}\\label{lemma:sym} There exists positive absolute constants $C_1$, $C_2$, $C_3$ and $C_4$ such that \n\\begin{equation*}\n C_1 \\exp \\left(-C_2\\frac{t}{N^2}\\right) \\le \\mathbb{P}(\\sigma_N > t) \\le C_3 \\exp \\left(-C_4\\frac{t}{N^2}\\right),\n\\end{equation*}\nfor all $N\\ge 2$ and $t \\ge 1$. \n\\end{lemma}\n\n\\begin{enumerate}[(i)]\n \\item \\textbf{Case $\\bm{p \\ge 1/2}$.}\n Let $\\tilde{S}_0 = 0$ and, for $k\\ge 1$, define recursively $\\xi_k$ and $\\tilde{S}_k$ by:\n \\begin{equation*}\n \\xi_{k} = \\mathbbm{1}_{\\left\\{\\tilde{S}_{k-1} = 0 \\right\\}} + \\mathbbm{1}_{\\left\\{\\tilde{S}_{k-1} \\neq 0 \\right\\}} \\left( \\mathbbm{1}_{\\left\\{U_{k} < \\frac{1}{2} \\right\\}} - \\mathbbm{1}_{\\left\\{U_{k} \\ge \\frac{1}{2}\\right\\}}\\right), \n \\end{equation*}\n and\n \\begin{equation}\\label{srw-coupling}\n \\tilde{S}_{k} = \\sum_{j=1}^{k}\\xi_j.\n \\end{equation}\n Then $\\tilde{S}_k$ has the same distribution as $|S_k|$, where $(S_k)_{k \\ge 0}$ is still the symmetric simple random walk defined in the previous section. Moreover, $(\\tilde{S}_k)_{k\\ge 0}$ and $(\\tilde{Z}_{k} )_{k\\ge 0}$ are constructed in such a way that the following holds:\n \\begin{equation}\\label{enq_1}\n \\tilde{Z}_k \\ge \\tilde{S}_k, \\, \\text{ for all } k\\ge 0.\n \\end{equation}\n Notice that the only way inequality \\eqref{enq_1} could be broken would be if at some point $\\tilde{Z}_k = 1$ and $\\tilde{S}_k = 0$ (and $\\tilde{Z}_{k}$ jumps to the left between $k$ and $k+1$). Fortunately this never happens since $\\tilde{Z}_k$ and $\\tilde{S}_k$ have the same parity. Below, we deliberately abuse notation, redefining $\\tau_N$ and $\\sigma_N$ in terms of $(\\tilde{S}_k)_{k\\ge 0}$ and $(\\tilde{Z}_{k} )_{k\\ge 0}$. By inequality \\eqref{enq_1} and Lemma \\ref{lemma:sym}, for all $N\\ge 1$ and $t\\ge 1$, one indeed has:\n\n\\begin{equation}\\label{enq_2}\n \\tilde{S}_k - 2\\sum_{j=1}^k\\mathbbm{1}_{\\left\\{\\frac{1}{2} + \\frac{(2p-1)\\tilde{Z}_{j +dN^{2}}}{j + dN^2}\\le U_{j + dN^{2}} < \\frac{1}{2}\\right\\}} \\le \\tilde{Z}_{k + dN^{2}} , \\, \\text{ for all } k\\ge 0.\n \\end{equation}\n Let $c > 12 + d$. For any $x < N$ we have\n \\begin{align*}\\label{eq:boundonethird+}\n \\P_x \\left(\\tau_N > cN^2\\right) &= \\P_x\\left(\\tau_N > cN^{2}, \\sigma_{2N} > 3(2N)^2\\right) + \\P_x \\left(\\tau_N > cN^{2}, \\sigma_{2N} \\le 3(2N)^2)\\right)\\\\\n &\\le \\P_x \\left(\\sigma_{2N} > 3(2N)^2\\right) + \\P_x \\left(\\tau_N > cN^{2}, \\sigma_{2N} \\le 3(2N)^2\\right)\n \\end{align*}\n But $\\P_x \\left(\\sigma_{2N} > 3(2N)^2 \\right) \\leq 1/3$ by Markov inequality (see inequality $(\\ref{eq:markov_sym})$ in the proof of Lemma \\ref{lemma:sym}), hence:\n \\begin{equation}\\label{eq:boundonethird+}\n \\P_x \\left(\\tau_N > cN^2 \\right) \\le \\frac{1}{3} + \\P_x \\left(\\tau_N > cN^{2}, \\, \\sigma_{2N} \\le 3(2N)^2 \\right).\n \\end{equation}\n Now observe that $\\tau_N > cN^{2}$ means that the elephant has not reached $N$ yet at time $c N^2$ while $\\sigma_{2N} \\le 3(2N)^2$ means that the symmetric random walk reached $2N$ in the meantime, even though it was exactly at the same point $12N^2$ time-steps earlier (or more). Now assume that both hold. Then $\\tilde{Z}_k < N$ for any $k \\le dN^2 + \\sigma_{2N}$, where $dN^2 + \\sigma_{2N}$ represents the time at which $(\\tilde{S}_k)_{k \\geq 0}$ reaches $2N$, but on the $(\\tilde{Z}_k)_{k \\geq 0}$ time-scale. Then the inequality $\\tilde{Z}_{dN^2 + \\sigma_{2N}}< N$ together with \\eqref{enq_2} implies that \n $$2N - 2 \\sum_{j=1}^{\\sigma_{2N}}\\mathbbm{1}_{\\big\\{\\frac{1}{2} + \\frac{(2p-1)\\tilde{Z}_{j + dN^{2}}}{j + dN^{2}}\\le U_{j + dN^{2}} < \\frac{1}{2}\\big\\}} < N.$$\n Moreover, since $\\sigma_{2N} \\le 3(2N)^2$ and $\\tilde{Z}_{j + dN^2} < N$ for any of the indices $j$ in the sum above, the above inequality implies that\n $$ 2N - 2\\sum_{j=1}^{3 (2 N)^2}\\mathbbm{1}_{\\{\\frac{1}{2} + \\frac{(2p - 1)}{dN}\\le U_{j + dN^{2}} < \\frac{1}{2} \\} } < N.$$\n\n\\begin{equation*}\n \\big\\{ \\tau_N > cN^{2}, \\sigma_{2N} \\le 3(2N)^2 \\big\\} \\subset \\Big\\{B_N > \\frac{N}{2} \\Big\\},\n \\end{equation*}\n where: $$B_N = \\sum_{j=1}^{12 N^2}\\mathbbm{1}_{\\left\\{\\frac{1}{2} + \\frac{(2p - 1)}{dN}\\le U_{j + dN^{2}} < \\frac{1}{2} \\right\\}} \\sim \\text{Binomial}\\left(12 N^2, \\, \\frac{1-2p}{dN}\\right).$$ \n Now, by letting $d > \\lceil 24(1-2p)\\rceil$, we have:\n \\begin{equation}\\label{eq_2}\n \\mathbb{E}[B_N] = \\frac{12(1 - 2p)N}{d} < \\frac{N}{2},\n \\end{equation}\n and thus\n \\begin{equation*}\n \\P\\Big(B_N > \\frac{N}{2}\\Big) \\le \\frac{1}{2}.\n \\end{equation*}\n Hence, for our choice of $d$, it holds that:\n \\begin{equation} \\label{eq:onehalfpart}\n \\P_x \\left(\\tau_N > cN^{2}, \\sigma_{2N} \\le 3(2N)^2 \\right) \\le \\frac{1}{2}.\n \\end{equation}\n Finally, inequality \\eqref{eq:onehalfpart} together with inequality \\eqref{eq:boundonethird+} leads to:\n \\begin{equation*}\n \\P_x \\left(\\tau_N > cN^2 \\right) \\le \\frac{5}{6}.\n \\end{equation*}\n Then the same line of reasoning as in the end of the proof of Lemma \\ref{lemma:sym} allows us to conclude:\n\nNotice that, since ${\\left\\{U_j < p \\right\\} = \\left\\{U_j < \\frac{1}{2} + \\frac{2p-1}{2} \\right\\}}$, one has $\\tilde{Z}_k \\le \\hat{S}_k$, for all $k\\ge 0$. Therefore, if \n\\begin{equation*}\n \\hat{\\sigma}_N = \\inf\\{k\\ge 1: \\hat{S}_k = N\\},\n\\end{equation*}\nthen, for all $N \\ge 2$ and $t\\ge 1$, one has: \n\\begin{equation*}\n \\mathbb{P}(\\tau_N > t) \\ge \\mathbb{P}(\\hat{\\sigma}_N > t).\n\\end{equation*}\nObserving that $\\{U_1 > p , \\, \\ldots , \\, U_t > p\\}\\subset \\{\\hat{\\sigma}_N > t\\}$, we obtain the following---very crude, but good enough---bound:\n\\begin{equation*}\n \\mathbb{P}(\\hat{\\sigma}_N > t) \\ge (1-p)^t = e^{\\log(1 - p) t} \\ge e^{-\\log \\left(\\frac{1}{1-p}\\right)C_A^2 \\frac{t}{N^2}},\n\\end{equation*}\nfor all $t\\ge 1$ and $2\\le N\\le C_A$. Hence taking $C_2 = \\max \\left\\{\\log \\left(\\frac{1}{1-p}\\right)C_A^2, \\tilde{C}_2 \\right \\}$, we can indeed conclude:\n\\begin{equation*}\n \\mathbbm{P}(\\tau_N \\ge t) \\ge C_1e^{-C_2\\frac{t}{N^2}},\n\\end{equation*}\nfor all $N\\ge 2$ and $t\\ge 1$.\n\\end{enumerate}\n\nWe start with the upper bound and then proceed to the lower bound. Fix some $c > 1$. For any $x \\in \\llbracket -N, N \\rrbracket$, a classical martingale argument (see for example Theorem 4.8.7 in \\cite{durrett2019probability}) gives us:\n \\begin{equation*}\n \\E_x \\left[\\sigma_N \\right] = (N-x)(x+N) \\le N^2.\n \\end{equation*}\n Hence, by Markov inequality:\n \\begin{equation}\\label{eq:markov_sym}\n \\P_x \\left(\\sigma_N > cN^2 \\right) \\le \\frac{\\E_x \\left[ \\sigma_N \\right]}{cN^2}\\le \\frac{1}{c}.\n \\end{equation}\n For $t \\geq 0$, it follows that:\n \\begin{equation*}\n \\P(\\sigma_N > t) \\leq \\P \\left(\\sigma_N \\ge \\left\\lfloor \\frac{t}{cN^2}\\right\\rfloor cN^2 \\right) \\le \\left(\\frac{1}{c}\\right)^{\\left\\lfloor \\frac{t}{cN^2}\\right\\rfloor} \\le \\left(\\frac{1}{c}\\right)^{\\frac{t}{cN^2} - 1} = C_3e^{-C_4\\frac{t}{N^2}},\n \\end{equation*}\n with $C_3 = c$ and $C_4 = \\log(c)/c$. Notice that, for $t \\geq cN^2$, the second inequality above is a consequence of \\eqref{eq:markov_sym} together with the strong Markov property, whereas it is trivial for $t < c N^2$, since in that case $\\left(\\frac{1}{c}\\right)^{\\left\\lfloor \\frac{t}{cN^2} \\right\\rfloor} = 1$.", "post_theorem_intro_text_len": 2482, "post_theorem_intro_text": "\\begin{remark}\n    Even though formally the result above is stated only for $0 < p < 3/4$, the upper bound actually holds for all $0 < p < 1$. Moreover, some of the constants above may depend on $p$.\n\\end{remark}\n\nThe main idea behind the proof of Theorem \\ref{main_thm} is to first establish analogous bounds for the SRW, and then demonstrate via couplings that, on a quadratic time scale, the ERW does not deviate excessively from the SRW. These couplings rely on a well-known time-inhomogeneous Markovian description of the ERW. Given a sequence of i.i.d. random variables with uniform distribution on $(0,1)$, we can construct both the ERW and the SRW on the same probability space. Each walk determines its next jump based on the value of the current uniform random variable relative to a specified threshold. For the SRW this threshold is always $1/2$, while for the ERW it depends on both time and the walk’s current state—though it is always either greater or less than $1/2$, depending on whether $p>1/2$ or not. This construction yields joint realizations in which the ERW either dominates or is dominated by the SRW, according to the value of $p$. By leveraging the SRW's bounds, this approach already resolves half of the cases. To address the remaining cases and thus complete the proof of Theorem \\ref{main_thm}, one then needs to control the distance between the SRW and the ERW on the appropriate time scale. Specifically, for $t = cN^2$, we identify some scaling constant $A$ such that, whenever the ERW or the SRW reaches the boundaries of $[-AN, AN]$, the other one has vanishingly small probability of not having reached $\\{-N,N\\}$ yet. This allows us prove Theorem \\ref{main_thm} on the quadratic time scale $t = cN^2$. Then, by partitioning the interval $[0, t]$ into sub-intervals of length $t/cN^2$, we extend the result to any $t \\geq 1$. Finally, Proposition \\ref{prop_1} follows directly from the functional limit theorem in \\cite{baur2016elephant} and uniform integrability.\n\n\\vspace{0.2 cm}\n\nThe paper is organized as follows. In Section 2 we derive a well-known (time inhomogeneous) Markovian description of the ERW allowing a coupling with the simple random walks. In Section 3 we prove the exponential bounds on the tail of the simple random walk. In Section 4 we define a coupling between ERW and the classical random walks and prove our main result, Theorem \\ref{main_thm}. Finally, in Section 5, we prove Proposition \\ref{prop_1}.\n\n\\vspace{0.2 cm}", "sketch": "The proof of Theorem \\ref{main_thm} proceeds as follows:\n\n- First, “establish analogous bounds for the SRW, and then demonstrate via couplings that, on a quadratic time scale, the ERW does not deviate excessively from the SRW.”\n\n- The couplings use “a well-known time-inhomogeneous Markovian description of the ERW.” Using i.i.d. uniforms on $(0,1)$, “we can construct both the ERW and the SRW on the same probability space,” where each walk’s next jump is determined by comparing the current uniform to a threshold: for SRW the threshold is always $1/2$, while for ERW it “depends on both time and the walk’s current state.” This produces joint realizations where “the ERW either dominates or is dominated by the SRW, according to the value of $p$,” and “by leveraging the SRW's bounds, this approach already resolves half of the cases.”\n\n- For the remaining cases, “one then needs to control the distance between the SRW and the ERW on the appropriate time scale.” Concretely, for $t=cN^2$, choose a scaling constant $A$ such that whenever one walk hits the boundary of $[-AN,AN]$, “the other one has vanishingly small probability of not having reached $\\{-N,N\\}$ yet.” This yields Theorem \\ref{main_thm} “on the quadratic time scale $t=cN^2$.”\n\n- To pass from $t=cN^2$ to general $t\\ge 1$, “partitioning the interval $[0,t]$ into sub-intervals of length $t/cN^2$,” thereby extending the bound to “any $t\\ge 1$.”", "expanded_sketch": "The proof of the main theorem proceeds as follows:\n\n- First, “establish analogous bounds for the SRW, and then demonstrate via couplings that, on a quadratic time scale, the ERW does not deviate excessively from the SRW.”\n\n- The couplings use “a well-known time-inhomogeneous Markovian description of the ERW.” Using i.i.d. uniforms on $(0,1)$, “we can construct both the ERW and the SRW on the same probability space,” where each walk’s next jump is determined by comparing the current uniform to a threshold: for SRW the threshold is always $1/2$, while for ERW it “depends on both time and the walk’s current state.” This produces joint realizations where “the ERW either dominates or is dominated by the SRW, according to the value of $p$,” and “by leveraging the SRW's bounds, this approach already resolves half of the cases.”\n\n- For the remaining cases, “one then needs to control the distance between the SRW and the ERW on the appropriate time scale.” Concretely, for $t=cN^2$, choose a scaling constant $A$ such that whenever one walk hits the boundary of $[-AN,AN]$, “the other one has vanishingly small probability of not having reached $\\{-N,N\\}$ yet.” This yields the main theorem “on the quadratic time scale $t=cN^2$.”\n\n- To pass from $t=cN^2$ to general $t\\ge 1$, “partitioning the interval $[0,t]$ into sub-intervals of length $t/cN^2$,” thereby extending the bound to “any $t\\ge 1$.”", "expanded_theorem": "\\label{main_thm} If $0<p<3/4$ then there exists positive constants $C_1$, $C_2$, $C_3$ and $C_4$ such that, for all $N \\ge 2$ and $t \\geq 1$, the following bound holds:\n\n    \\begin{equation*}\n        C_1 \\exp \\left( -C_2 \\frac{t}{N^2} \\right) \\leq \\mathbb{P} \\left( \\tau_N > t \\right) \\le C_3 \\exp \\left(-C_4\\frac{t}{N^2}\\right).\n    \\end{equation*}", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Consider the elephant random walk $(Z_n)_{n\\ge 0}$ on $\\mathbb Z$, defined by $Z_n=\\sum_{k=0}^n X_k$ with $X_0=0$, the first step $X_1\\in\\{-1,1\\}$, and for each $n\\ge 1$, after choosing $K(n)$ uniformly from $\\{1,\\dots,n\\}$ independently of the past,\n\\[\nX_{n+1}=\\begin{cases}\nX_{K(n)} & \\text{with probability } p,\\\\\n- X_{K(n)} & \\text{with probability } 1-p.\n\\end{cases}\n\\]\nFor $N\\ge 2$, define the hitting time of the two-sided boundary $\\{\\pm N\\}$ by\n\\[\n\\tau_N=\\inf\\{k\\ge 1: |Z_k|=N\\}.\n\\]\nAssume $0<p<3/4$. Which quantitative estimate holds for the tail of $\\tau_N$ for all $N\\ge 2$ and $t\\ge 1$?", "correct_choice": {"label": "A", "text": "There exist positive constants $C_1,C_2,C_3,C_4$ such that, for every $N\\ge 2$ and $t\\ge 1$,\n\\[\nC_1\\exp\\!\\left(-C_2\\frac{t}{N^2}\\right)\\le \\mathbb P(\\tau_N>t)\\le C_3\\exp\\!\\left(-C_4\\frac{t}{N^2}\\right).\n\\]"}, "choices": [{"label": "B", "text": "There exist positive constants $C_1,C_2,C_3,C_4$ such that, for every $N\\ge 2$ and $t\\ge 1$,\n\\[\nC_1\\exp\\!\\left(-C_2\\frac{t}{N}\\right)\\le \\mathbb P(\\tau_N>t)\\le C_3\\exp\\!\\left(-C_4\\frac{t}{N}\\right).\n\\]"}, {"label": "C", "text": "There exist positive constants $C_3,C_4$ such that, for every $N\\ge 2$ and $t\\ge 1$,\n\\[\n\\mathbb P(\\tau_N>t)\\le C_3\\exp\\!\\left(-C_4\\frac{t}{N^2}\\right).\n\\]"}, {"label": "D", "text": "For every $N\\ge 2$ there exist positive constants $C_1(N),C_2(N),C_3(N),C_4(N)$ such that, for every $t\\ge 1$,\n\\[\nC_1(N)\\exp\\!\\left(-C_2(N)\\frac{t}{N^2}\\right)\\le \\mathbb P(\\tau_N>t)\\le C_3(N)\\exp\\!\\left(-C_4(N)\\frac{t}{N^2}\\right).\n\\]"}, {"label": "E", "text": "There exist positive constants $C_1,C_2,C_3,C_4$ such that, for every $N\\ge 2$ and $t\\ge 1$,\n\\[\nC_1\\exp\\!\\left(-C_2\\frac{t}{N^2}\\right)\\le \\mathbb P(\\tau_N=t)\\le C_3\\exp\\!\\left(-C_4\\frac{t}{N^2}\\right).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "quadratic_time_scale", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "lower_tail_bound_removed", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "uniform_constants_in_N", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "tail_event_replaced_by_point_mass", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It specifies the model and asks for the valid tail estimate, without wording that uniquely points to choice A."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: it essentially asks for the precise quantitative conclusion for a stated regime. There is some selection among variants, but the task is still largely identifying the theorem statement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish diffusive scaling (N^2) from incorrect scales, uniform versus N-dependent constants, and tail probability versus point mass. However, the main demand is recognition of the known result rather than deeper derivation."}, "DQS": {"score": 1, "justification": "Most distractors are plausible and mathematically meaningful: wrong time scale, weakened one-sided bound, quantifier error, and tail/point-mass confusion. But choice C is also true if A is true, so the item is not cleanly single-answer as written."}, "total_score": 5, "overall_assessment": "A mathematically sophisticated but somewhat theorem-recall-style MCQ. Its main weakness is ambiguity: a weaker true statement appears among the distractors, reducing clarity and single-answer validity."}}
{"id": "2602.18969v1", "paper_link": "http://arxiv.org/abs/2602.18969v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "The Prym maps \n$$\n\\mathcal{P}^{iso}_3 :  \\cRH_3^{iso} \\rightarrow  \\cA_6^{(1,1,1,2,2,4)} \\qquad \\mathcal{P}^{ni}_3 :  \\cRH_3^{ni} \\rightarrow  \\cA_6^{(1,1,1,1,4,4)}\n$$\nare injective on each irreducible component of the source moduli space.", "start_pos": 7298, "end_pos": 7529, "label": null}, "ref_dict": {"4components": "\\begin{lem}\\label{4components}\nThe moduli space $\\cRH_3^{V_4}$ consists of four irreducible components, corresponding to the configuration of the Weierstrass points in the generators $ \\eta, \\ \\xi$ as shown \nin I.1, I.2, II.1 and II.2. Moreover, the degree of the forgetful \nmap  $\\cRH_3^{V_4} \\ra \\cH_3$ on each component is 56, 280, 105, 210, respectively. \n\\end{lem}", "autoofP": "\\begin{prop}\\label{autoofP}\nLet $(P, \\Xi)$ be an element in the image of the Prym map \n$\\cP^{iso}_3$, respectively in $\\cP^{ni}_3$ Case I.2.  Then it holds \n$$\nB:=\\{\\psi \\in \\Aut (P,\\Xi) \\ : \\ \\psi^2=id, \\ \n\\psi_{|K(\\Xi)}= \\pm id \\} = \\langle \\sigma, \\tau, j, -1 \\rangle \\simeq \\ZZ_2^4 \n$$\nwhere $j$ is any lift of the hyperelliptic involution from $H$.\n\\end{prop}", "FullKlein": "\\begin{thm}\\label{FullKlein}\n    The Klein Prym maps $$\n\\cP^{iso}_3 :  \\cR_3^{iso} \\ra  \\cA_6^{(1,1,1,2,2,4)} \\qquad \\cP^{ni}_3 :  \\cR_3^{ni} \\ra  \\cA_6^{(1,1,1,1,4,4)}\n$$ are generically finite. Moreover they are of degree $1$ on images of coverings of hyperelliptic curves of types $I.2, II.1, II.2$.\n\\end{thm}"}, "pre_theorem_intro_text_len": 2700, "pre_theorem_intro_text": "Let $H$ be a hyperelliptic smooth projective curve over $\\mathbb C$ of genus 3. We consider Klein coverings $f:\\widetilde{C} \\rightarrow H$, that is, \n$\\widetilde{C}$ admits an action by a group of fixed point free automorphisms $V_4$ isomorphic to $\\ZZ_2 \\times \\ZZ_2$ such that $H = \\widetilde{C} / V_4$. \nOne can associate to such a covering a polarised abelian sixfold in the following way. Let $\\Nm_f : J\\widetilde{C} \\rightarrow JH$ be the norm map, sending the divisor class $[\\sum_i n_ip_i]$ to $[\\sum_i n_if(p_i)]$,\nwith $p_i \\in \\widetilde{C}$ and $n_i\\in \\mathbb Z$. The Prym variety $P(f)$ of the covering $f$ is defined as the connected component of 0 of the kernel of $\\Nm_f$. It is an abelian subvariety of $J\\widetilde{C}$ of dimension the difference of the genera of the curves, which in this case is 6 and the restriction of the principal polarisation on $J\\widetilde{C}$ induces a non-principal one on $P(f)$. \n\nPrym varieties of étale double covers have been extensively studied in the past and more recently it has been proven that \nthe Prym map is injective for double covers  ramified in at least six points (\\cite{NO22}). For étale cyclic coverings over a hyperelliptic curve, the (generic) injectivity \nof the Prym has been shown for infinitely many degrees (\\cite{NOS24}, \n\\cite{NOPS25}). There are also new results on the fiber of the Prym map for non-cyclic coverings in low genera (\\cite{S25}).    \n\nKlein coverings can be defined from the curve $H$ by choosing a subgroup \n$\\langle \\eta, \\xi\\rangle $ of order four of the group $JH[2]$ of 2-torsion points in $JH$.  \nWe distinguish two types of Klein coverings depending whether the subgroup is isotropic or not with respect to the Weil pairing on  $JH[2]$. \nLet $\\cRH_3^{iso}$, respectively  $\\cRH_3^{ni}$, denote the moduli space parametrising isotropic, respectively  non-isotropic, Klein \ncoverings on hyperelliptic curves of genus 3. \nEach of these moduli spaces consists of two irreducible components (see Lemma \\ref{4components}). \n\nConsider now the Prym map that sends $[f:\\widetilde{C} \\rightarrow H] \\in \\cRH_3^{V_4}=\\cRH_3^{iso} \\sqcup \\cRH_3^{ni} $ to the \npolarised Prym variety $(P,\\Xi)$. In this article we continue the study of the injectivity of the Prym map for Klein  \ncoverings of genus $3$ curves, which was started in \\cite{BO24}, where we prove the injectivity on the locus of hyperelliptic coverings $f:\\widetilde{C} \\rightarrow H $ (i.e. $\\widetilde{C}$ is also hyperelliptic) \ncorresponding to the case I.1 below.  Let us denote by\n$\\cA_g^{\\delta}$ the moduli space of polarised abelian varieties of dimension $g$ with polarisation type $\\delta$.\nThe main theorem of the paper is the following.", "context": "Let $H$ be a hyperelliptic smooth projective curve over $\\mathbb C$ of genus 3. We consider Klein coverings $f:\\widetilde{C} \\rightarrow H$, that is, \n$\\widetilde{C}$ admits an action by a group of fixed point free automorphisms $V_4$ isomorphic to $\\ZZ_2 \\times \\ZZ_2$ such that $H = \\widetilde{C} / V_4$. \nOne can associate to such a covering a polarised abelian sixfold in the following way. Let $\\Nm_f : J\\widetilde{C} \\rightarrow JH$ be the norm map, sending the divisor class $[\\sum_i n_ip_i]$ to $[\\sum_i n_if(p_i)]$,\nwith $p_i \\in \\widetilde{C}$ and $n_i\\in \\mathbb Z$. The Prym variety $P(f)$ of the covering $f$ is defined as the connected component of 0 of the kernel of $\\Nm_f$. It is an abelian subvariety of $J\\widetilde{C}$ of dimension the difference of the genera of the curves, which in this case is 6 and the restriction of the principal polarisation on $J\\widetilde{C}$ induces a non-principal one on $P(f)$.\n\nPrym varieties of étale double covers have been extensively studied in the past and more recently it has been proven that \nthe Prym map is injective for double covers ramified in at least six points (\\cite{NO22}). For étale cyclic coverings over a hyperelliptic curve, the (generic) injectivity \nof the Prym has been shown for infinitely many degrees (\\cite{NOS24}, \n\\cite{NOPS25}). There are also new results on the fiber of the Prym map for non-cyclic coverings in low genera (\\cite{S25}).\n\nKlein coverings can be defined from the curve $H$ by choosing a subgroup \n$\\langle \\eta, \\xi\\rangle $ of order four of the group $JH[2]$ of 2-torsion points in $JH$. \nWe distinguish two types of Klein coverings depending whether the subgroup is isotropic or not with respect to the Weil pairing on $JH[2]$. \nLet $\\cRH_3^{iso}$, respectively $\\cRH_3^{ni}$, denote the moduli space parametrising isotropic, respectively non-isotropic, Klein \ncoverings on hyperelliptic curves of genus 3. \nEach of these moduli spaces consists of two irreducible components (see Lemma \\ref{4components}).\n\nConsider now the Prym map that sends $[f:\\widetilde{C} \\rightarrow H] \\in \\cRH_3^{V_4}=\\cRH_3^{iso} \\sqcup \\cRH_3^{ni} $ to the \npolarised Prym variety $(P,\\Xi)$. In this article we continue the study of the injectivity of the Prym map for Klein \ncoverings of genus $3$ curves, which was started in \\cite{BO24}, where we prove the injectivity on the locus of hyperelliptic coverings $f:\\widetilde{C} \\rightarrow H $ (i.e. $\\widetilde{C}$ is also hyperelliptic) \ncorresponding to the case I.1 below. Let us denote by\n$\\cA_g^{\\delta}$ the moduli space of polarised abelian varieties of dimension $g$ with polarisation type $\\delta$.\nThe main theorem of the paper is the following.\n\n\\begin{lem}\\label{4components}\nThe moduli space $\\cRH_3^{V_4}$ consists of four irreducible components, corresponding to the configuration of the Weierstrass points in the generators $ \\eta, \\ \\xi$ as shown \nin I.1, I.2, II.1 and II.2. Moreover, the degree of the forgetful \nmap  $\\cRH_3^{V_4} \\ra \\cH_3$ on each component is 56, 280, 105, 210, respectively. \n\\end{lem}", "full_context": "Let $H$ be a hyperelliptic smooth projective curve over $\\mathbb C$ of genus 3. We consider Klein coverings $f:\\widetilde{C} \\rightarrow H$, that is, \n$\\widetilde{C}$ admits an action by a group of fixed point free automorphisms $V_4$ isomorphic to $\\ZZ_2 \\times \\ZZ_2$ such that $H = \\widetilde{C} / V_4$. \nOne can associate to such a covering a polarised abelian sixfold in the following way. Let $\\Nm_f : J\\widetilde{C} \\rightarrow JH$ be the norm map, sending the divisor class $[\\sum_i n_ip_i]$ to $[\\sum_i n_if(p_i)]$,\nwith $p_i \\in \\widetilde{C}$ and $n_i\\in \\mathbb Z$. The Prym variety $P(f)$ of the covering $f$ is defined as the connected component of 0 of the kernel of $\\Nm_f$. It is an abelian subvariety of $J\\widetilde{C}$ of dimension the difference of the genera of the curves, which in this case is 6 and the restriction of the principal polarisation on $J\\widetilde{C}$ induces a non-principal one on $P(f)$.\n\nPrym varieties of étale double covers have been extensively studied in the past and more recently it has been proven that \nthe Prym map is injective for double covers ramified in at least six points (\\cite{NO22}). For étale cyclic coverings over a hyperelliptic curve, the (generic) injectivity \nof the Prym has been shown for infinitely many degrees (\\cite{NOS24}, \n\\cite{NOPS25}). There are also new results on the fiber of the Prym map for non-cyclic coverings in low genera (\\cite{S25}).\n\nKlein coverings can be defined from the curve $H$ by choosing a subgroup \n$\\langle \\eta, \\xi\\rangle $ of order four of the group $JH[2]$ of 2-torsion points in $JH$. \nWe distinguish two types of Klein coverings depending whether the subgroup is isotropic or not with respect to the Weil pairing on $JH[2]$. \nLet $\\cRH_3^{iso}$, respectively $\\cRH_3^{ni}$, denote the moduli space parametrising isotropic, respectively non-isotropic, Klein \ncoverings on hyperelliptic curves of genus 3. \nEach of these moduli spaces consists of two irreducible components (see Lemma \\ref{4components}).\n\nConsider now the Prym map that sends $[f:\\widetilde{C} \\rightarrow H] \\in \\cRH_3^{V_4}=\\cRH_3^{iso} \\sqcup \\cRH_3^{ni} $ to the \npolarised Prym variety $(P,\\Xi)$. In this article we continue the study of the injectivity of the Prym map for Klein \ncoverings of genus $3$ curves, which was started in \\cite{BO24}, where we prove the injectivity on the locus of hyperelliptic coverings $f:\\widetilde{C} \\rightarrow H $ (i.e. $\\widetilde{C}$ is also hyperelliptic) \ncorresponding to the case I.1 below. Let us denote by\n$\\cA_g^{\\delta}$ the moduli space of polarised abelian varieties of dimension $g$ with polarisation type $\\delta$.\nThe main theorem of the paper is the following.\n\n\\begin{lem}\\label{4components}\nThe moduli space $\\cRH_3^{V_4}$ consists of four irreducible components, corresponding to the configuration of the Weierstrass points in the generators $ \\eta, \\ \\xi$ as shown \nin I.1, I.2, II.1 and II.2. Moreover, the degree of the forgetful \nmap  $\\cRH_3^{V_4} \\ra \\cH_3$ on each component is 56, 280, 105, 210, respectively. \n\\end{lem}\n\nKlein coverings can be defined from the curve $H$ by choosing a subgroup \n$\\langle \\eta, \\xi\\rangle $ of order four of the group $JH[2]$ of 2-torsion points in $JH$. \nWe distinguish two types of Klein coverings depending whether the subgroup is isotropic or not with respect to the Weil pairing on $JH[2]$. \nLet $\\cRH_3^{iso}$, respectively $\\cRH_3^{ni}$, denote the moduli space parametrising isotropic, respectively non-isotropic, Klein \ncoverings on hyperelliptic curves of genus 3. \nEach of these moduli spaces consists of two irreducible components (see Lemma \\ref{4components}).\n\nConsider now the Prym map that sends $[f:\\tC \\ra H] \\in \\cRH_3^{V_4}=\\cRH_3^{iso} \\sqcup \\cRH_3^{ni} $ to the \npolarised Prym variety $(P,\\Xi)$. In this article we continue the study of the injectivity of the Prym map for Klein \ncoverings of genus $3$ curves, which was started in \\cite{BO24}, where we prove the injectivity on the locus of hyperelliptic coverings $f:\\tC \\ra H $ (i.e. $\\tC$ is also hyperelliptic) \ncorresponding to the case I.1 below. Let us denote by\n$\\cA_g^{\\delta}$ the moduli space of polarised abelian varieties of dimension $g$ with polarisation type $\\delta$.\nThe main theorem of the paper is the following.\n\nActually, the injectivity also holds on the entire moduli space $\\cRH_3^{V_4}$ since the dimensions of the factors appearing in the isotypical decomposition of the associated Prym variety, depends on the type of the Klein covering and it is different for each irreducible component.\n\n\\begin{thm}\\label{thmI.2}\nThe Prym map \n$\\cP^{ni}_3 : \\cRH_3^{ni} \\ra \\cA_6^{(1,1,1,1,4,4)}$ is injective on the irreducible component consisting of Klein coverings of type I.2.\n\n\\begin{thm}\nThe Prym map \n$\\cP^{iso}_3 : \\cRH_3^{iso} \\ra \\cA_6^{(1,1,1,2,2,4)}$ is injective on the irreducible component consisting of Klein coverings of type II.1.\n\\end{thm}\n\\begin{proof}\nLet $(P, \\Xi)$ be the image of a Klein covering of type $II.1$ under $\\cP^{iso}_3$. First note that, by Proposition \\ref{autoofP}, $(P, \\Xi)$ determines the group of involutions $B\\simeq \\ZZ^4_2$ on $P$.\nOne then look at the fixed loci of the involutions together with the restricted polarisation types to distinguish $j,j\\s,j\\tau,j\\s\\tau$ and $JC_j,JC_{j\\s},JC_{j\\tau},JC_{j\\s\\tau}$ respectively. Now, one chooses any two of these, say $j$ and $j\\s$ and by Torelli Theorem one recovers $C_j$ and $C_{j\\s}$.\nBy Lemma \\ref{intersection_Jacobians_II.1} and Lemma \\ref{key_lemma} we obtain the maps $C_j\\to E$ and $C_{j\\s}\\to E$, so we have the following fibered product.\n\n\\begin{thm}\nThe Prym map \n$\\cP^{iso}_3 : \\cRH_3^{iso} \\ra \\cA_6^{(1,1,1,2,2,4)}$ is injective on the irreducible component consisting of Klein coverings of type II.2.\n\\end{thm}\n\\begin{proof}\nLet $(P, \\Xi)$ be an element on the image of $\\cP^{ni}_3$. Then the group $\\langle \\sigma, \\tau,j, -1 \\rangle \\simeq \\ZZ_2^4$ acts on $P$ and according to Proposition \\ref{isotyp_II.2}, this action induces the isotypical decomposition \n$$\nP = P^*(C_{\\s}/H) \\boxplus P^*(C_{\\tau}/H) \\boxplus E_j \\boxplus F\n$$\nwith a distinguished elliptic curve $F$ of exponent $(4)$. \nMoreover, by construction \n$$\nP^*(C_{\\s}/H)+F \\sim (JC_{j\\s}, 2\\Theta_{C_{j\\s}}), \\quad P^*(C_{\\tau}/H)+F \\sim (JC_{j\\tau}, 2\\Theta_{C_{j\\tau}}).\n$$\nSo the isotypical components determine uniquely the \npolarised Jacobians called $JC_{j\\s}$ and $JD_{j\\tau}$ and\nby Torelli Theorem the genus 3 curves $C_{j\\s}$ and $ C_{j\\tau}$ are also determined. Since $F$ is embedded in the Jacobians by Lemma \\ref{key_lemma}, we consider the fibered product \n\\begin{equation}\\label{fibred_product_II.2}\n\\xymatrix@R=.9cm@C=1cm{\n \\widetilde{C_{j\\s} \\times_{F} C_{j\\tau}}\\ar[d]_{2:1} \\ar[r]^{ \\ 2:1} & C_{j\\s} \\ar[d]^{h_\\s} \\\\\nC_{j\\tau} \\ar[r]_{h_\\tau} & F \n}\n\\end{equation}\n\n\\begin{thm}\\label{FullKlein}\n The Klein Prym maps $$\n\\cP^{iso}_3 : \\cR_3^{iso} \\ra \\cA_6^{(1,1,1,2,2,4)} \\qquad \\cP^{ni}_3 : \\cR_3^{ni} \\ra \\cA_6^{(1,1,1,1,4,4)}\n$$ are generically finite. Moreover they are of degree $1$ on images of coverings of hyperelliptic curves of types $I.2, II.1, II.2$.\n\\end{thm}\n\\begin{proof}\n Let us consider the non-isotropic case. By Theorems \\ref{thmI.1} and \\ref{thmI.2} the Prym map ${\\cP^{ni}_3}_{|\\cRH_3^{ni}}$ is injective on each component. Moreover, images of respective components are disjoint as the isotypical decomposition yields components of different dimensions. Hence, the image of the Prym map, being an irreducible variety and containing $2$ disjoint subvarieties of dimension $5$ has to be of dimension at least $6=\\dim(\\cR_3^{ni})$. This shows that the map is generically finite.\n\nThe above theorem is an evidence to a positive answer of the following question.\n\\begin{question}\nAre the following Prym maps\n$$\n\\cP^{iso}_3 : \\cR_3^{iso} \\ra \\cA_6^{(1,1,1,2,2,4)} \\qquad \\cP^{ni}_3 : \\cR_3^{ni} \\ra \\cA_6^{(1,1,1,1,4,4)}\n$$\n injective?\n\\end{question}\n\n\\begin{lem}\\label{4components}\nThe moduli space $\\cRH_3^{V_4}$ consists of four irreducible components, corresponding to the configuration of the Weierstrass points in the generators $ \\eta, \\ \\xi$ as shown \nin I.1, I.2, II.1 and II.2. Moreover, the degree of the forgetful \nmap  $\\cRH_3^{V_4} \\ra \\cH_3$ on each component is 56, 280, 105, 210, respectively. \n\\end{lem}\n\n\\begin{prop}\\label{autoofP}\nLet $(P, \\Xi)$ be an element in the image of the Prym map \n$\\cP^{iso}_3$, respectively in $\\cP^{ni}_3$ Case I.2.  Then it holds \n$$\nB:=\\{\\psi \\in \\Aut (P,\\Xi) \\ : \\ \\psi^2=id, \\ \n\\psi_{|K(\\Xi)}= \\pm id \\} = \\langle \\sigma, \\tau, j, -1 \\rangle \\simeq \\ZZ_2^4 \n$$\nwhere $j$ is any lift of the hyperelliptic involution from $H$.\n\\end{prop}", "post_theorem_intro_text_len": 2121, "post_theorem_intro_text": "Actually, the injectivity also holds on the entire moduli space $\\cRH_3^{V_4}$ since the dimensions of the factors appearing in the isotypical decomposition of the associated Prym variety,  depends on the type of the Klein covering and it is different for each irreducible component. \n\nThe general idea of the proof, with some variations in each case, is the following. A Prym variety $(P,\\Xi)$ in the image of the Prym map determines a group of automorphisms of $P$ isomorphic to \n$\\ZZ_2^4$ (see Proposition \\ref{autoofP}). This gives us the isotypical decomposition of $P$ and one can identify the Jacobians by means of the type of the restricted polarisation and using the action of the involutions on the kernel of the polarisation.  With this information we can reconstruct the quotients curves and finally recover $\\widetilde{C}$ as a fibered product.  \n\nAs an application, we can show the following theorem for Klein coverings over {\\it any} curve of genus 3, see Theorem \\ref{FullKlein}.\n\\begin{thm}\n    The Klein Prym maps $$\n\\mathcal{P}^{iso}_3 :  \\cR_3^{iso} \\rightarrow  \\cA_6^{(1,1,1,2,2,4)} \\qquad \\mathcal{P}^{ni}_3 :  \\cR_3^{ni} \\rightarrow  \\cA_6^{(1,1,1,1,4,4)}\n$$ are generically finite. Moreover they are of degree $1$ on images of coverings of hyperelliptic curves of types $I.2, II.1, II.2$.\n\\end{thm}\n\nThe structure of the paper is as follows. In Section 1, we state basic definitions and key lemmas and show that the moduli space have 4 components, called I.1, I.2, II.1, II.2. Each one of the Sections 2,\\ 3,\\ 4 and 5 is devoted to one of these cases and to the proof of the injectivity of the corresponding  Prym map. Finally, Section 6 is devoted to Klein coverings of any genus 3 curve.\n\n\\subsection*{Acknowledgments}\nThe authors would like to thank Anatoli Shatsila for finding a reference that shortened considerably one of the proofs. The first author has been supported by the Polish National Science Centre project number 2024/54/E/ST1/00330. Some results of the paper were obtained during his visit to Humboldt University in Berlin. He would like to thank the university for hospitality.", "sketch": "Actually, the injectivity also holds on the entire moduli space $\\cRH_3^{V_4}$ since the dimensions of the factors appearing in the isotypical decomposition of the associated Prym variety depends on the type of the Klein covering and it is different for each irreducible component.\n\nThe general idea of the proof (``with some variations in each case'') is:\n\\begin{itemize}\n\\item A Prym variety $(P,\\Xi)$ in the image of the Prym map determines a group of automorphisms of $P$ isomorphic to $\\ZZ_2^4$ (Proposition \\ref{autoofP}).\n\\item This yields the isotypical decomposition of $P$.\n\\item ``One can identify the Jacobians\" using the type of the restricted polarisation and the action of the involutions on the kernel of the polarisation.\n\\item From this data, ``we can reconstruct the quotients curves\" and ``finally recover $\\widetilde{C}$ as a fibered product.\"\n\\end{itemize}", "expanded_sketch": "Actually, the injectivity also holds on the entire moduli space $\\cRH_3^{V_4}$ since the dimensions of the factors appearing in the isotypical decomposition of the associated Prym variety depends on the type of the Klein covering and it is different for each irreducible component.\n\nThe general idea of the proof (``with some variations in each case'') is:\n\\begin{itemize}\n\\item A Prym variety $(P,\\Xi)$ in the image of the Prym map determines a group of automorphisms of $P$ isomorphic to $\\ZZ_2^4$. More precisely:\n\\begin{prop}\\label{autoofP}\nLet $(P, \\Xi)$ be an element in the image of the Prym map \n$\\cP^{iso}_3$, respectively in $\\cP^{ni}_3$ Case I.2.  Then it holds \n$$\nB:=\\{\\psi \\in \\Aut (P,\\Xi) \\ : \\ \\psi^2=id, \\ \n\\psi_{|K(\\Xi)}= \\pm id \\} = \\langle \\sigma, \\tau, j, -1 \\rangle \\simeq \\ZZ_2^4 \n$$\nwhere $j$ is any lift of the hyperelliptic involution from $H$.\n\\end{prop}\n\\item This yields the isotypical decomposition of $P$.\n\\item ``One can identify the Jacobians\" using the type of the restricted polarisation and the action of the involutions on the kernel of the polarisation.\n\\item From this data, ``we can reconstruct the quotients curves\" and ``finally recover $\\widetilde{C}$ as a fibered product.\"\n\\end{itemize}", "expanded_theorem": "The Prym maps \n$$\n\\mathcal{P}^{iso}_3 :  \\cRH_3^{iso} \\rightarrow  \\cA_6^{(1,1,1,2,2,4)} \\qquad \\mathcal{P}^{ni}_3 :  \\cRH_3^{ni} \\rightarrow  \\cA_6^{(1,1,1,1,4,4)}\n$$\nare injective on each irreducible component of the source moduli space.", "theorem_type": ["Universal", "Implication"], "mcq": {"question": "Let \\(\\cRH_3^{iso}\\) (respectively \\(\\cRH_3^{ni}\\)) be the moduli space of Klein coverings \\(f:\\widetilde C\\to H\\) of hyperelliptic smooth projective curves \\(H\\) of genus \\(3\\) over \\(\\mathbb C\\), where \\(\\widetilde C\\) admits a fixed-point-free action of \\(V_4\\simeq \\mathbb Z_2\\times \\mathbb Z_2\\) with quotient \\(H=\\widetilde C/V_4\\), and where the associated order-four subgroup \\(\\langle \\eta,\\xi\\rangle\\subset JH[2]\\) is isotropic (respectively non-isotropic) with respect to the Weil pairing. For such a covering, let \\(P(f)\\) be the Prym variety, namely the connected component of \\(0\\) in \\(\\ker(\\operatorname{Nm}_f:J\\widetilde C\\to JH)\\), endowed with the induced polarization. Consider the Prym maps\n\\[\n\\mathcal P^{iso}_3: \\cRH_3^{iso}\\to \\cA_6^{(1,1,1,2,2,4)},\\qquad\n\\mathcal P^{ni}_3: \\cRH_3^{ni}\\to \\cA_6^{(1,1,1,1,4,4)},\n\\]\nwhere \\(\\cA_6^{\\delta}\\) denotes the moduli space of polarized abelian sixfolds of polarization type \\(\\delta\\). Which statement holds for every irreducible component of the source moduli spaces?", "correct_choice": {"label": "A", "text": "For both maps, the restriction to any irreducible component of the source is injective; that is, on each irreducible component of \\(\\cRH_3^{iso}\\) and on each irreducible component of \\(\\cRH_3^{ni}\\), two Klein coverings with the same polarized Prym variety represent the same point of that component."}, "choices": [{"label": "B", "text": "For both maps, the Prym variety determines the Klein covering globally on the whole source moduli space; that is, \\(\\mathcal P^{iso}_3\\) is injective on all of \\(\\cRH_3^{iso}\\) and \\(\\mathcal P^{ni}_3\\) is injective on all of \\(\\cRH_3^{ni}\\), so two Klein coverings with the same polarized Prym variety necessarily coincide even if they lie in different irreducible components."}, {"label": "C", "text": "For both maps, the restriction to each irreducible component has finite fibers; equivalently, on every irreducible component of \\(\\cRH_3^{iso}\\) and of \\(\\cRH_3^{ni}\\), only finitely many Klein coverings can have the same polarized Prym variety."}, {"label": "D", "text": "For both maps, there is a single reconstruction procedure from the polarized Prym variety that is uniform across all irreducible components, in the sense that the same isotypical decomposition and the same pair of quotient Jacobians recover the Klein covering in every isotropic and non-isotropic case."}, {"label": "E", "text": "For both maps, the polarized Prym variety determines the underlying hyperelliptic base curve \\(H\\) on each irreducible component, but not necessarily the full Klein covering; in other words, equality of polarized Prym varieties forces the same quotient curve \\(H\\), although the corresponding order-four subgroup \\(\\langle\\eta,\\xi\\rangle\\subset JH[2]\\) may still vary."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "case_split", "tampered_component": "componentwise injectivity versus global injectivity across distinct irreducible components", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped injective and kept finite fibers on each irreducible component", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "case-dependent isotypical decomposition and reconstruction data", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "reconstruction of the full covering from quotient curves and fibered product, not merely the base curve", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the Prym maps and asks which conclusion holds, but it does not explicitly or implicitly reveal that the correct property is injectivity on each irreducible component."}, "TAS": {"score": 1, "justification": "The item is largely a theorem-identification question: it asks for the precise valid statement about the Prym maps among nearby variants. This is not a pure restatement, since the options include stronger and weaker alternatives, but it remains close to recalling the theorem's formulation."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish full injectivity, componentwise injectivity, generic finiteness, and generically injective behavior. However, the task mainly tests precise theorem recall rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they vary the quantifiers and strength of the conclusion in ways that reflect common confusion between injective, generically injective, and generically finite, as well as componentwise versus global statements."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise recall of a known result than toward deep generative reasoning."}}
{"id": "2602.19175v1", "paper_link": "http://arxiv.org/abs/2602.19175v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm0.1}\n\t\tLet $(X,{\\mathrm d},\\mathfrak m)$ be an ${\\rm RCD}(K,N)$ metric measure space. Let $S\\subseteq X$ be a John domain and $Y\\subseteq X$ be compact with $\\mathfrak m(Y)>0$. Let $\\rho\\in \\mathcal{P}(S)$  be with $a_1\\mathfrak m\\restr{S} \\leq \\rho\\leq a_2\\mathfrak m\\restr{S}$ for some positive constants $a_1,a_2$. Then there exists a constant $C>0$, depending on $K,N,a_1,a_2, S, \\mathop{\\rm diam}\\nolimits(S\\cup Y)$, such that for any $\\mu,\\nu\\in\\mathcal{P}(Y)$, \n\t\t\\begin{equation}\n\t\t\t\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}\\leq C W_1^\\frac{1}{2}(\\mu,\\nu),\n\t\t\\end{equation}\n\t\twhere $\\phi_\\mu$ and $\\phi_\\nu$ are the Kantorovich potentials from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively.", "start_pos": 11594, "end_pos": 12265, "label": "thm0.1"}, "ref_dict": {"A": "\\begin{appendices}\n\t\\section{Poincar\\'e inequality: local to global}\\label{A}\n\t\t\\begin{lemma}\\label{p3.10}\n\tLet $x_0\\in S$, $0<r_0\\leq1$ and  $B=B(x_0,r_0)\\subseteq S$,  $\\rho_B=\\frac{\\rho\\restr{B}}{\\rho(B)}$.  Then the following strong local $(1,1)$-Poincar\\'e inequality holds for $f\\in \\Lip(B,  \\d)$:\n\t\t\t\\begin{equation}\n\t\t\t\\int_{B}|f(x)-\\E_{\\rho_B}(f)|\\,\\d\\rho_B(x)\\leq C_4 r_0\\int_B |\\nabla_x f|\\,\\d\\rho_B,\n\t\t\\end{equation}\n\t\twhere $C_4$ depends on $K,N,a_1,a_2$.\n\t\\end{lemma}\n\n\t\\begin{proof}\n\t\tBy \\cite[Chapter 9]{zbMATH01474795} and \\cite[Remark 3.3]{zbMATH06043352},  $\\mm$ satisfies the  $(1,1)$-Poincar\\'e inequality:\n\t\t\\begin{equation}\n\t\t\t\\int_{B} |f-\\E_{\\mm_B}(f)|\\,\\d\\mm\\leq c_1 r_0 \\int_B |\\nabla_x f|\\,\\d\\mm,\n\t\t\\end{equation}\n\t\twhere $c_1$ depends on $K,N$, $\\mm_B=\\frac{\\mm\\restr{B}}{\\mm(B)}$. Since $a_1\\mm \\leq \\rho\\leq a_2 \\mm$, we have \n\t\t\\begin{equation}\n\t\t\t\\begin{aligned}\n\t\t\t\t\\int_{B} |f-\\E_{\\rho_B}(f)|\\,\\d\\rho\\leq &\\int_{B} |f-\\E_{\\mm_B}(f)|\\,\\d\\rho+\\int_{B} |\\E_{\\mm_B}(f)-\\E_{\\rho_B}(f)|\\,\\d\\rho\\\\\n\t\t\t\t\\leq & 2a_2\\int_{B} |f-\\E_{\\mm_B}(f)|\\,\\d\\mm\\\\\n\t\t\t\t\\leq &2a_2 c_1 r_0 \\int_B |\\nabla_x f|\\,\\d\\mm\n\t\t\t\t\\leq  2\\frac{a_2}{a_1}c_1 r_0 \\int_B |\\nabla_x f|\\,\\d\\rho,\\\\\n\t\t\t\\end{aligned}\n\t\t\\end{equation}\n\twhich is the thesis.\n\t\\end{proof}\n\n\tNext we prove an $L^1$-variant of the gluing Lemma in \\cite[Lemma 3.3]{arXiv:2411.04908},   following the same Boman-chain decomposition,  utilizing the doubling property directly instead of the maximal function estimates. \n\n\t\t\\begin{definition}[Boman chain condition]\\label{Boman chain condition}\n\t\tWe say that a probability measure $\\rho$ on an open set $S$ of a metric space satisfies  \\emph{Boman chain condition} with parameters \\(E, F,G>1\\) if there is a covering \\(\\mathcal{F}\\) of $S$ by open balls  $B \\in \\mathcal{F}$ such that:\n\n\t\t\\begin{enumerate}\n\t\t\t\t\\item For any \\(x \\in S\\),\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\sum_{B \\in \\mathcal{F}} \\chi_{2B}(x) \\leq E \\chi_{S}(x).\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\\item For some fixed ball \\(B_0\\) in \\(\\mathcal{F}\\), called the \\emph{central ball}, and for every \\(B \\in \\mathcal{F}\\), there exists a chain \\(B_0, B_1, \\ldots, B_N = B\\) of distinct balls from \\(\\mathcal{F}\\) such that\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\tB \\subset F B_j,~~~\\forall j \\in \\{0, \\ldots, N-1\\}.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\\item Consecutive balls of the above chain overlap quantitatively:\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\rho(B_j \\cap B_{j+1}) \\geq G^{-1} \\max(\\rho(B_j), \\rho(B_{j+1})),~~~\\forall j \\in \\{0, \\ldots, N-1\\}.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\\end{enumerate}\n\t\\end{definition}\n\n\t\\begin{lemma}\\label{3.1}\n\tFor any $f\\in L^1(\\rho)$,   it holds that\n\t\\begin{equation}\\label{A.5}\n\t\t\\int_S |f(x)-\\E_{\\rho}(f)|\\,\\d\\rho(x)\\leq C_5 \\sum_{B\\in\\mathcal{F}}\\rho(B)\\int_B |f(x)-\\E_{\\rho_B}(f)|\\,\\d\\rho_B(x),\n\t\\end{equation}\n\twhere $C_5$ depends on $K,N,\\diam(S),S$.\n\t\\end{lemma}\n\n    \\begin{proof} \n    \tSince $S$ is a John domain, and $\\rho$ is a doubling measure on $S$, by  \\cite[Proposition 3.7]{arXiv:2504.05412}, $\\rho$ satisfies the Boman chain condition. Hence there exists a covering $\\mathcal{F}$ of $S$ satisfying Definition \\ref{Boman chain condition}.\n\n    \tFor the central ball $B_0$, note that\n    \t\\begin{equation}\\label{A.2}\n    \t\t\\begin{aligned}\n    \t\t\t\\int_S |f(x)-\\E_{\\rho}(f)|\\,\\d\\rho(x)\n    \t\t\t\\leq & \\int_S |f(x)-\\E_{\\rho_{B_0}}(f)|\\,\\d\\rho(x)+\\int_S |\\E_{\\rho_{B_0}}(f)-\\E_{\\rho}(f)|\\,\\d\\rho(x)\\\\\n    \t\t\t\\leq & 2\\int_S |f(x)-\\E_{\\rho_{B_0}}(f)|\\,\\d\\rho(x).\n    \t\t\\end{aligned}\n    \t\\end{equation}\n\n   For  $B\\in\\mathcal{F}$,  \tdenote $a_B=\\int_B |f-\\E_{\\rho_{B}}(f)|\\,\\d\\rho=\\rho(B)\\int_B |f-\\E_{\\rho_{B}}(f)|\\,\\d\\rho_B$. Since $\\mathcal{F}$ is a covering of $S$, we have\n    \t\\begin{equation}\\label{A.3}\n    \t\t\\begin{aligned}\n    \t\t\t&\\int_S |f(x)-\\E_{\\rho_{B_0}}(f)|\\,\\d\\rho(x)\n    \t\t\t\\leq  \\sum_{B \\in \\mathcal{F}} \\int_B |f(x)-\\E_{\\rho_{B_0}}(f)|\\,\\d\\rho(x)\\\\\n    \t\t\t\\leq & \\sum_{B \\in \\mathcal{F}} \\left(\\int_B |f(x)-\\E_{\\rho_{B}}(f)|\\,\\d\\rho(x)+\\int_B |\\E_{\\rho_{B}}(f)-\\E_{\\rho_{B_0}}(f)|\\,\\d\\rho(x)\\right)\\\\\n    \t\t\t\\leq &  \\sum_{B \\in \\mathcal{F}} \\left(a_B+\\rho(B) |\\E_{\\rho_{B}}(f)-\\E_{\\rho_{B_0}}(f)|\\right).\n    \t\t\\end{aligned}\n    \t\\end{equation}\n\n    \tFor any $B\\in\\mathcal{F}$, by Boman chain condition, there exists a chain $B_0, B_1,\\dots, B_N=B$ of distinct balls from \\(\\mathcal{F}\\), such that for any \\(j \\in \\{0, \\ldots, N-1\\}\\),\n    \t\\begin{equation}\n    \t\t\\begin{aligned}\n    \t\t|\\E_{\\rho_{B_j}}(f)-\\E_{\\rho_{B_{j+1}}}(f)|= &\\left|\\frac{1}{\\rho(B_j\\cap B_{j+1})}\\int_{B_j\\cap B_{j+1}}\\left(\\E_{\\rho_{B_j}}(f)-\\E_{\\rho_{B_{j+1}}}(f)\\right)\\,\\d\\rho\\right|\\\\\n    \t\t\\leq &\\frac{1}{\\rho(B_j\\cap B_{j+1})}\\int_{B_j\\cap B_{j+1}}\\left|\\E_{\\rho_{B_j}}(f)-\\E_{\\rho_{B_{j+1}}}(f)\\right|\\,\\d\\rho\\\\\n    \t\t\\leq &\\frac{1}{\\rho(B_j\\cap B_{j+1})}\\left(\\int_{B_j\\cap B_{j+1}}\\left|f-\\E_{\\rho_{B_j}}(f)\\right|+\\left|f-\\E_{\\rho_{B_{j+1}}}(f)\\right|\\,\\d\\rho\\right)\\\\\n    \t\t\\leq & \\frac{a_{B_j}+a_{B_{j+1}}}{\\rho(B_j\\cap B_{j+1})}\n    \t\t\\overset{*}{\\leq}  G \\left(\\frac{a_{B_j}}{\\rho(B_j)}+\\frac{a_{B_{j+1}}}{\\rho(B_{j+1})}\\right),\n    \t\t\\end{aligned}\n    \t\\end{equation}\n    \twhere $(*)$ follows from the quantitative chain overlap of Boman chain condition $3$.\n\n    \t Thus, we have\n    \t \\begin{equation}\n    \t |\\E_{\\rho_{B}}(f)-\\E_{\\rho_{B_0}}(f)|\\leq \\sum_{j=0}^{N-1}|\\E_{\\rho_{B_j}}(f)-\\E_{\\rho_{B_{j+1}}}(f)|\\leq 2G \\sum_{j=0}^{N}\\frac{a_{B_j}}{\\rho(B_j)}\\overset{*}{\\leq} 2G \\sum_{B\\subset F \\bar{B}}\\frac{a_{\\bar{B}}}{\\rho(\\bar{B})},\n    \t \\end{equation}\n    \t where $ \\sum_{B\\subset F \\bar{B}}$ means that the sum runs over all  $\\bar{B}\\in\\mathcal{F}$ satisfying $B\\subset F \\bar{B}$, and $(*)$ follows from the Boman chain condition $2$.\n    \t Then by Fubini--Tonelli theorem,  \n    \t \\begin{equation}\\label{A.6}\n    \t \t\\begin{aligned}\n    \t \t \t  \\sum_{B \\in \\mathcal{F}} \\rho(B) |\\E_{\\rho_{B}}(f)-\\E_{\\rho_{B_0}}(f)|\n    \t \t \t  \\leq & 2G\\sum_{B \\in \\mathcal{F}} \\rho(B) \\sum_{B\\subset F \\overline{B}}\\frac{a_{\\overline{B}}}{\\rho(\\overline{B})}\n    \t \t \t  \\leq 2G\\sum_{\\bar{B} \\in \\mathcal{F}} \\frac{a_{\\bar{B}}}{\\rho(\\bar{B})}\\sum_{B\\subset F \\bar{B}}\\rho(B).\n    \t \t\\end{aligned}\n    \t \\end{equation}\n\nBy Boman chain condition $1$ and  the doubling property of $\\rho$, we have  \n    \t \\begin{equation}\\label{aa}\n       \\sum_{B\\subset F \\bar{B}}\\rho(B)\\leq E\\rho(F\\bar{B})\\leq E\\beta^2F^\\frac{\\log \\beta}{\\log 2}\\rho(\\bar{B}),\n    \t \\end{equation}\nwhere  $\\beta=\\beta(K,N,\\diam(S))$ is the doubling constant.\n\n    \t Combining \\eqref{A.2}, \\eqref{A.3},  \\eqref{A.6} and \\eqref{aa}, we obtain\n    \t \\begin{equation}\\label{A_A}\n    \t \t\\begin{aligned}\n    \t \t\t \\int_S |f(x)-\\E_{\\rho}(f)|\\,\\d\\rho(x)\\leq & 2 \\sum_{B \\in \\mathcal{F}} \\left(a_B+\\rho(B) |\\E_{\\rho_{B}}(f)-\\E_{\\rho_{B_0}}(f)|\\right)\\\\\n    \t \t\t \\leq &2(1+2\\beta^2 EF^\\frac{\\log \\beta}{\\log 2}G) \\sum_{B \\in \\mathcal{F}} \\rho(B)\\int_B |f-\\E_{\\rho_{B}}(f)|\\,\\d\\rho_B,\n    \t \t\\end{aligned}\n    \t \\end{equation}\n  which is the thesis.\n    \\end{proof}\n\n\t\\end{appendices}", "thm0.1": "\\begin{theorem}\\label{thm0.1}\n\t\tLet $(X,\\d,\\mm)$ be an ${\\rm RCD}(K,N)$ metric measure space. Let $S\\subseteq X$ be a John domain and $Y\\subseteq X$ be compact with $\\mm(Y)>0$. Let $\\rho\\in \\mathcal{P}(S)$  be with $a_1\\mm\\restr{S} \\leq \\rho\\leq a_2\\mm\\restr{S}$ for some positive constants $a_1,a_2$. Then there exists a constant $C>0$, depending on $K,N,a_1,a_2, S, \\diam(S\\cup Y)$, such that for any $\\mu,\\nu\\in\\mathcal{P}(Y)$, \n\t\t\\begin{equation}\n\t\t\t\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}\\leq C W_1^\\frac{1}{2}(\\mu,\\nu),\n\t\t\\end{equation}\n\t\twhere $\\phi_\\mu$ and $\\phi_\\nu$ are the Kantorovich potentials from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively.\n\t\\end{theorem}"}, "pre_theorem_intro_text_len": 6021, "pre_theorem_intro_text": "\\subsection{Motivation and setting}\n\t\\paragraph{Optimal transport.}Optimal transport, initiated by Monge  \\cite{monge1781} and reframed by  Kantorovich \\cite{zbMATH03099866},  seeks the most efficient way to redistribute mass between two probability distributions.   Precisely, given two probability measures $\\rho$ and $\\mu$ defined on Polish spaces $X$ and $Y$ respectively, and a  cost function $c:X\\times Y\\rightarrow \\mathbb{R}$, the Monge optimal transport problem aims to find a minimizer  $T$,   called an optimal transport map,   of the following optimization problem among all measurable maps $H: X \\to Y$ pushing $\\rho$ forward to $\\mu$:\n\t\\begin{equation}\\label{2.1}\n\t\t\\int_X c(x, T(x))  \\,{\\mathrm d}\\rho(x)=\\inf_{H_\\sharp \\rho = \\mu}\\int_X c(x, H(x))  \\,{\\mathrm d}\\rho(x), \\tag{MP}\n\t\\end{equation}\n The   Monge problem  prevents mass splitting, sometimes making the problem ill-posed.  Its relaxation,  called Kantorovich problem, optimizes over joint couplings instead of deterministic maps. A fundamental result known as Kantorovich duality, asserts that the Kantorovich problem is equivalent to the  following dual problem:\n\t\\begin{equation}\n\t\t\\sup_{ \\phi(x) + \\psi(y) \\leq c(x, y)} \\left\\{ \\int_X \\phi(x) \\, {\\mathrm d}\\rho(x) + \\int_Y \\psi(y) \\, {\\mathrm d}\\mu(y)  \\right\\}.\\tag{KD}\n\t\\end{equation}\n\tThe optimal functions $\\phi,\\psi$,  which always exist under standard assumptions,   are called  Kantorovich potentials \\cite{AG-U, V-O}.  \n\n\tIn the Euclidean space,   for $c(x,y)=\\frac{1}{2}|x-y|^2$, Brenier's landmark result \\cite{brenier1991polar} established that, for absolutely continuous source measures, the optimal transport map is unique and takes the form $T(x)=\\nabla u(x)$ for a convex function $u$. This was later extended to Riemannian manifolds by McCann \\cite{mccann2001polar}, who showed that for $c(x,y)=\\frac{1}{2}{\\mathrm d}^2(x,y)$, the optimal transport map is given almost everywhere by $T(x)=\\exp_x(-\\nabla\\varphi(x))$ for a  Kantorovich potential $\\varphi$.\n\n\t A fundamental question in both theoretical and applied contexts of optimal transport is the quantitative stability of optimal transport maps and Kantorovich potentials under perturbations of the target measure (see Letrouit's lecture note \\cite{LetrouitLecture} and the references therein).   Based on recent breakthroughs in the quantitative stability of optimal transport  on  Euclidean spaces \\cite{zbMATH07794624, arXiv:2411.04908},    on boundaries of convex bodies \\cite{Kitagawa12,  Kitagawa25}  and  on Riemannian manifolds  \\cite{arXiv:2504.05412},     \nKitagawa--Letrouit--M\\'erigot   \\cite[\\S 1.2]{arXiv:2504.05412} conjecture that:\n\t\\begin{center}\n\t\t\\emph{the quantitative stability results are also true in more general metric measure spaces with synthetic curvature bounds.}\n\t\\end{center}\n\n\t\\paragraph{General setting.}\n\tIn this paper, we confirm the conjecture of Kitagawa-- Letrouit--M\\'erigot for the following spaces:\n\t\\begin{itemize}\n\t\t\\item  RCD$(K,N)$  spaces (metric measure spaces with synthetic Ricci curvature lower bound):  we prove quantitative $L^1$-stability of Kantorovich potentials with respect to Wasserstein perturbations of the target measure;\n\t\t\\item Alexandrov spaces with (sectional) curvature lower bound:   we establish stability estimates for optimal transport maps in terms of the $L^2$-norm of the potential difference.\n\t\\end{itemize}\n\n\tAn Alexandrov space is a  geodesic  space of finite Hausdorff dimension and of curvature bounded from below (cf.  \\cite{zbMATH01626771}).   An ${\\rm RCD}(K,N)$ space  is  a metric measure spaces verifying the synthetic Riemannian curvature-dimension condition  \\cite{zbMATH05049052,zbMATH05578758, zbMATH06303881, G-O}.   An $n$-dimensional Alexandrov space with curvature bounded from below by $k$,  equipped with its $n$-Hausdorff measure,   is an ${\\rm RCD}(k(n-1), n)$ space  \\cite{zbMATH06032507,zbMATH05962558}.\n\t\t After \\cite{brenier1991polar, mccann2001polar},   Gigli--Rajala--Sturm \\cite[Theorem 1.1]{ gigli2016optimal}  proved the existence and uniqueness of the optimal transport map on an $\\rm{RCD}(K,N)$ space $(X,{\\mathrm d},\\mathfrak m)$,  for the quadratic cost $c(x,y)=\\frac{1}{2}{\\mathrm d}^2(x,y)$ and the source measure $\\rho\\ll \\mathfrak m$. \n\n\t  John domains,    named after  F. John \\cite{zbMATH03166766},   encompass many cases of interest, such as  bounded Lipschitz domains, bounded domains satisfying a cone condition and certain fractal domains (see \\cite{zbMATH00858510} for more discussions). \n\t\\begin{definition}[John domain]\n\t\tA bounded open subset $S$ of a metric space is called a \\emph{John domain} if there is a distinguished point $x_0 \\in S$ and a constant $\\eta > 0$ such that, for every $x \\in S$, there is a rectifiable curve $\\gamma \\colon [0, \\ell(\\gamma)] \\to S$ parametrized by arc length, such that $\\gamma(0) = x$, $\\gamma(\\ell(\\gamma)) = x_0$, and\n\t\t\\begin{equation*}\n\t\t\t{\\mathrm d}(\\gamma(t), S^c) \\geq \\eta t,~~~\\forall t \\in [0, \\ell(\\gamma)],\n\t\t\\end{equation*}\n\t\twhere $S^c$ denotes the complement of $S$.\n\t\\end{definition}\n\tIn  \\cite[Theorem 1.9]{arXiv:2411.04908}, it has been shown that when the source measure $\\rho$ is the uniform density on some non-John domain  $S\\subset \\mathbb{R}^n$, then no quantitative stability estimates of the form $\\|\\phi_\\mu-\\phi_\\nu\\|_{L^2(\\rho)}\\leq C W_p^\\alpha(\\mu,\\nu)$ \n\tcan hold for any $C, \\alpha, p>0$. \n\tMoreover, examples in \\cite{letrouit2025unstableoptimaltransportmaps} indicates that both unboundedness of the density and openness of    $S$ may cause  instability  of optimal transport maps.\n\n\t In this paper,  we always assume that the Kantorovich potential  $\\phi_\\mu$ from $\\rho$ to $\\mu$ satisfies $\\E_\\rho(\\phi_\\mu)=\\int_S\\phi_\\mu\\,{\\mathrm d}\\rho=0$.  Together with the uniqueness of the optimal transport map, such  $\\phi_\\mu$ is  unique and thus makes sense to study its stability.\n\n\t\\subsection{Main results}\n\tOur main theorem concerns the quantitative  $L^1$ stability of Kantorovich potentials on ${\\rm RCD}(K,N)$  spaces.", "context": "\\subsection{Motivation and setting}\n \\paragraph{Optimal transport.}Optimal transport, initiated by Monge \\cite{monge1781} and reframed by Kantorovich \\cite{zbMATH03099866}, seeks the most efficient way to redistribute mass between two probability distributions. Precisely, given two probability measures $\\rho$ and $\\mu$ defined on Polish spaces $X$ and $Y$ respectively, and a cost function $c:X\\times Y\\rightarrow \\mathbb{R}$, the Monge optimal transport problem aims to find a minimizer $T$, called an optimal transport map, of the following optimization problem among all measurable maps $H: X \\to Y$ pushing $\\rho$ forward to $\\mu$:\n \\begin{equation}\\label{2.1}\n \\int_X c(x, T(x)) \\,{\\mathrm d}\\rho(x)=\\inf_{H_\\sharp \\rho = \\mu}\\int_X c(x, H(x)) \\,{\\mathrm d}\\rho(x), \\tag{MP}\n \\end{equation}\n The Monge problem prevents mass splitting, sometimes making the problem ill-posed. Its relaxation, called Kantorovich problem, optimizes over joint couplings instead of deterministic maps. A fundamental result known as Kantorovich duality, asserts that the Kantorovich problem is equivalent to the following dual problem:\n \\begin{equation}\n \\sup_{ \\phi(x) + \\psi(y) \\leq c(x, y)} \\left\\{ \\int_X \\phi(x) \\, {\\mathrm d}\\rho(x) + \\int_Y \\psi(y) \\, {\\mathrm d}\\mu(y) \\right\\}.\\tag{KD}\n \\end{equation}\n The optimal functions $\\phi,\\psi$, which always exist under standard assumptions, are called Kantorovich potentials \\cite{AG-U, V-O}.\n\nA fundamental question in both theoretical and applied contexts of optimal transport is the quantitative stability of optimal transport maps and Kantorovich potentials under perturbations of the target measure (see Letrouit's lecture note \\cite{LetrouitLecture} and the references therein). Based on recent breakthroughs in the quantitative stability of optimal transport on Euclidean spaces \\cite{zbMATH07794624, arXiv:2411.04908}, on boundaries of convex bodies \\cite{Kitagawa12, Kitagawa25} and on Riemannian manifolds \\cite{arXiv:2504.05412}, \nKitagawa--Letrouit--M\\'erigot \\cite[\\S 1.2]{arXiv:2504.05412} conjecture that:\n \\begin{center}\n \\emph{the quantitative stability results are also true in more general metric measure spaces with synthetic curvature bounds.}\n \\end{center}\n\nAn Alexandrov space is a geodesic space of finite Hausdorff dimension and of curvature bounded from below (cf. \\cite{zbMATH01626771}). An ${\\rm RCD}(K,N)$ space is a metric measure spaces verifying the synthetic Riemannian curvature-dimension condition \\cite{zbMATH05049052,zbMATH05578758, zbMATH06303881, G-O}. An $n$-dimensional Alexandrov space with curvature bounded from below by $k$, equipped with its $n$-Hausdorff measure, is an ${\\rm RCD}(k(n-1), n)$ space \\cite{zbMATH06032507,zbMATH05962558}.\n After \\cite{brenier1991polar, mccann2001polar}, Gigli--Rajala--Sturm \\cite[Theorem 1.1]{ gigli2016optimal} proved the existence and uniqueness of the optimal transport map on an $\\rm{RCD}(K,N)$ space $(X,{\\mathrm d},\\mathfrak m)$, for the quadratic cost $c(x,y)=\\frac{1}{2}{\\mathrm d}^2(x,y)$ and the source measure $\\rho\\ll \\mathfrak m$.\n\nJohn domains, named after F. John \\cite{zbMATH03166766}, encompass many cases of interest, such as bounded Lipschitz domains, bounded domains satisfying a cone condition and certain fractal domains (see \\cite{zbMATH00858510} for more discussions). \n \\begin{definition}[John domain]\n A bounded open subset $S$ of a metric space is called a \\emph{John domain} if there is a distinguished point $x_0 \\in S$ and a constant $\\eta > 0$ such that, for every $x \\in S$, there is a rectifiable curve $\\gamma \\colon [0, \\ell(\\gamma)] \\to S$ parametrized by arc length, such that $\\gamma(0) = x$, $\\gamma(\\ell(\\gamma)) = x_0$, and\n \\begin{equation*}\n {\\mathrm d}(\\gamma(t), S^c) \\geq \\eta t,~~~\\forall t \\in [0, \\ell(\\gamma)],\n \\end{equation*}\n where $S^c$ denotes the complement of $S$.\n \\end{definition}\n In \\cite[Theorem 1.9]{arXiv:2411.04908}, it has been shown that when the source measure $\\rho$ is the uniform density on some non-John domain $S\\subset \\mathbb{R}^n$, then no quantitative stability estimates of the form $\\|\\phi_\\mu-\\phi_\\nu\\|_{L^2(\\rho)}\\leq C W_p^\\alpha(\\mu,\\nu)$ \n can hold for any $C, \\alpha, p>0$. \n Moreover, examples in \\cite{letrouit2025unstableoptimaltransportmaps} indicates that both unboundedness of the density and openness of $S$ may cause instability of optimal transport maps.\n\nIn this paper, we always assume that the Kantorovich potential $\\phi_\\mu$ from $\\rho$ to $\\mu$ satisfies $\\E_\\rho(\\phi_\\mu)=\\int_S\\phi_\\mu\\,{\\mathrm d}\\rho=0$. Together with the uniqueness of the optimal transport map, such $\\phi_\\mu$ is unique and thus makes sense to study its stability.\n\n\\subsection{Main results}\n Our main theorem concerns the quantitative $L^1$ stability of Kantorovich potentials on ${\\rm RCD}(K,N)$ spaces.", "full_context": "\\subsection{Motivation and setting}\n \\paragraph{Optimal transport.}Optimal transport, initiated by Monge \\cite{monge1781} and reframed by Kantorovich \\cite{zbMATH03099866}, seeks the most efficient way to redistribute mass between two probability distributions. Precisely, given two probability measures $\\rho$ and $\\mu$ defined on Polish spaces $X$ and $Y$ respectively, and a cost function $c:X\\times Y\\rightarrow \\mathbb{R}$, the Monge optimal transport problem aims to find a minimizer $T$, called an optimal transport map, of the following optimization problem among all measurable maps $H: X \\to Y$ pushing $\\rho$ forward to $\\mu$:\n \\begin{equation}\\label{2.1}\n \\int_X c(x, T(x)) \\,{\\mathrm d}\\rho(x)=\\inf_{H_\\sharp \\rho = \\mu}\\int_X c(x, H(x)) \\,{\\mathrm d}\\rho(x), \\tag{MP}\n \\end{equation}\n The Monge problem prevents mass splitting, sometimes making the problem ill-posed. Its relaxation, called Kantorovich problem, optimizes over joint couplings instead of deterministic maps. A fundamental result known as Kantorovich duality, asserts that the Kantorovich problem is equivalent to the following dual problem:\n \\begin{equation}\n \\sup_{ \\phi(x) + \\psi(y) \\leq c(x, y)} \\left\\{ \\int_X \\phi(x) \\, {\\mathrm d}\\rho(x) + \\int_Y \\psi(y) \\, {\\mathrm d}\\mu(y) \\right\\}.\\tag{KD}\n \\end{equation}\n The optimal functions $\\phi,\\psi$, which always exist under standard assumptions, are called Kantorovich potentials \\cite{AG-U, V-O}.\n\nA fundamental question in both theoretical and applied contexts of optimal transport is the quantitative stability of optimal transport maps and Kantorovich potentials under perturbations of the target measure (see Letrouit's lecture note \\cite{LetrouitLecture} and the references therein). Based on recent breakthroughs in the quantitative stability of optimal transport on Euclidean spaces \\cite{zbMATH07794624, arXiv:2411.04908}, on boundaries of convex bodies \\cite{Kitagawa12, Kitagawa25} and on Riemannian manifolds \\cite{arXiv:2504.05412}, \nKitagawa--Letrouit--M\\'erigot \\cite[\\S 1.2]{arXiv:2504.05412} conjecture that:\n \\begin{center}\n \\emph{the quantitative stability results are also true in more general metric measure spaces with synthetic curvature bounds.}\n \\end{center}\n\nAn Alexandrov space is a geodesic space of finite Hausdorff dimension and of curvature bounded from below (cf. \\cite{zbMATH01626771}). An ${\\rm RCD}(K,N)$ space is a metric measure spaces verifying the synthetic Riemannian curvature-dimension condition \\cite{zbMATH05049052,zbMATH05578758, zbMATH06303881, G-O}. An $n$-dimensional Alexandrov space with curvature bounded from below by $k$, equipped with its $n$-Hausdorff measure, is an ${\\rm RCD}(k(n-1), n)$ space \\cite{zbMATH06032507,zbMATH05962558}.\n After \\cite{brenier1991polar, mccann2001polar}, Gigli--Rajala--Sturm \\cite[Theorem 1.1]{ gigli2016optimal} proved the existence and uniqueness of the optimal transport map on an $\\rm{RCD}(K,N)$ space $(X,{\\mathrm d},\\mathfrak m)$, for the quadratic cost $c(x,y)=\\frac{1}{2}{\\mathrm d}^2(x,y)$ and the source measure $\\rho\\ll \\mathfrak m$.\n\nJohn domains, named after F. John \\cite{zbMATH03166766}, encompass many cases of interest, such as bounded Lipschitz domains, bounded domains satisfying a cone condition and certain fractal domains (see \\cite{zbMATH00858510} for more discussions). \n \\begin{definition}[John domain]\n A bounded open subset $S$ of a metric space is called a \\emph{John domain} if there is a distinguished point $x_0 \\in S$ and a constant $\\eta > 0$ such that, for every $x \\in S$, there is a rectifiable curve $\\gamma \\colon [0, \\ell(\\gamma)] \\to S$ parametrized by arc length, such that $\\gamma(0) = x$, $\\gamma(\\ell(\\gamma)) = x_0$, and\n \\begin{equation*}\n {\\mathrm d}(\\gamma(t), S^c) \\geq \\eta t,~~~\\forall t \\in [0, \\ell(\\gamma)],\n \\end{equation*}\n where $S^c$ denotes the complement of $S$.\n \\end{definition}\n In \\cite[Theorem 1.9]{arXiv:2411.04908}, it has been shown that when the source measure $\\rho$ is the uniform density on some non-John domain $S\\subset \\mathbb{R}^n$, then no quantitative stability estimates of the form $\\|\\phi_\\mu-\\phi_\\nu\\|_{L^2(\\rho)}\\leq C W_p^\\alpha(\\mu,\\nu)$ \n can hold for any $C, \\alpha, p>0$. \n Moreover, examples in \\cite{letrouit2025unstableoptimaltransportmaps} indicates that both unboundedness of the density and openness of $S$ may cause instability of optimal transport maps.\n\nIn this paper, we always assume that the Kantorovich potential $\\phi_\\mu$ from $\\rho$ to $\\mu$ satisfies $\\E_\\rho(\\phi_\\mu)=\\int_S\\phi_\\mu\\,{\\mathrm d}\\rho=0$. Together with the uniqueness of the optimal transport map, such $\\phi_\\mu$ is unique and thus makes sense to study its stability.\n\n\\subsection{Main results}\n Our main theorem concerns the quantitative $L^1$ stability of Kantorovich potentials on ${\\rm RCD}(K,N)$ spaces.\n\nJohn domains, named after F. John \\cite{zbMATH03166766}, encompass many cases of interest, such as bounded Lipschitz domains, bounded domains satisfying a cone condition and certain fractal domains (see \\cite{zbMATH00858510} for more discussions). \n \\begin{definition}[John domain]\n A bounded open subset $S$ of a metric space is called a \\emph{John domain} if there is a distinguished point $x_0 \\in S$ and a constant $\\eta > 0$ such that, for every $x \\in S$, there is a rectifiable curve $\\gamma \\colon [0, \\ell(\\gamma)] \\to S$ parametrized by arc length, such that $\\gamma(0) = x$, $\\gamma(\\ell(\\gamma)) = x_0$, and\n \\begin{equation*}\n \\d(\\gamma(t), S^c) \\geq \\eta t,~~~\\forall t \\in [0, \\ell(\\gamma)],\n \\end{equation*}\n where $S^c$ denotes the complement of $S$.\n \\end{definition}\n In \\cite[Theorem 1.9]{arXiv:2411.04908}, it has been shown that when the source measure $\\rho$ is the uniform density on some non-John domain $S\\subset \\R^n$, then no quantitative stability estimates of the form $\\|\\phi_\\mu-\\phi_\\nu\\|_{L^2(\\rho)}\\leq C W_p^\\alpha(\\mu,\\nu)$ \n can hold for any $C, \\alpha, p>0$. \n Moreover, examples in \\cite{letrouit2025unstableoptimaltransportmaps} indicates that both unboundedness of the density and openness of $S$ may cause instability of optimal transport maps.\n\n\\subsection{Main results}\n Our main theorem concerns the quantitative $L^1$ stability of Kantorovich potentials on ${\\rm RCD}(K,N)$ spaces.\n\nIf $(S, \\d, \\mm)$ is a compact $\\rcdkn$ space, it supports a global $(1,1)$-Poincar\\'e inequality (cf.\\cite{zbMATH01474795, zbMATH06043352}) and we can remove the John domain assumption:\n \\begin{corollary}\\label{cor}\n Let $(X,\\d,\\mm)$ be a compact ${\\rm RCD}(K,N)$ metric measure space. Let $\\rho\\in \\mathcal{P}(X)$ be with $a_1\\mm \\leq \\rho\\leq a_2\\mm$ for some positive constants $a_1,a_2$. Then the conclusion of Theorem \\ref{thm0.1} holds.\n \\end{corollary}\n\nAdapting the strategy of \\cite{arXiv:2504.05412}, we can also prove the stability of optimal transport maps on Alexandrov spaces.\n \\begin{theorem}\\label{thm0.2}\n Let $(X, \\d)$ be an $n$-dimensional Alexandrov space, $\\mm$ be the $n$-Hausdorff measure. Under the same assumption for $S, Y$ and $\\rho$ as in Theorem \\ref{thm0.1}, and if $S$ additionally has finite perimeter, then there exists a constant $C>0$, depending on $k,n,a_1, a_2, \\diam(S\\cup Y), {\\rm{Per}}(S), S$, such that for any $\\mu,\\nu\\in\\mathcal{P}_2(Y)$, we have\n \\begin{equation}\n \\int_S \\d^2(T_\\mu(x), T_\\nu(x))\\,\\d \\rho(x) \\leq CW_1^{1/6}(\\mu, \\nu),\n \\end{equation}\n where $T_\\mu$ and $T_\\nu$ are the optimal transport maps from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively.\n \\end{theorem}\n\n\\begin{proposition}\\label{2.11}\n It holds that\n \\begin{equation}\n \\int_{S}\\big| \\E_{\\mu_x^t[\\psi_*]}(v)-\\E_{\\mu^t[\\psi_*]}(v))\\big|\\,\\d\\rho(x)\\leq \\frac{\\kappa}{\\sqrt{t}}\\left(\\int_S {\\rm{Var}}_{ \\mu_x^t[\\psi_*]}(v)\\,\\d\\rho(x)\\right)^\\frac{1}{2},\n \\end{equation}\n where $\\mu^t[\\psi_*]$ is defined as in \\eqref{mu} and $\\kappa$ depends on $K,N,\\Lambda_\\psi,a_1,a_2, S$.\n \\end{proposition}\n \\begin{proof}\n Since $a_1\\mm\\leq \\rho\\leq a_2\\mm$, by \\cite[Corollary 2.4]{zbMATH05049052}, $\\rho$ is a doubling measure. Moreover, \n since $S$ is a John domain, by \\cite[Proposition 3.7]{arXiv:2504.05412}, $\\rho$ satisfies the Boman chain condition (see Definition \\ref{Boman chain condition}) and we can choose a covering $\\mathcal{F}$, such that for any $B\\in \\mathcal{F}$, $r_B\\leq 1$. Then \n \\begin{equation}\n \\begin{aligned}\n &\\int_{S}\\big| \\E_{\\mu_x^t[\\psi_*]}(v)-\\E_{\\mu^t[\\psi_*]}(v))\\big|\\,\\d\\rho(x)\\\\\n \\mathop{\\leq}^{\\text {Lemma \\ref{3.1}}} &C_5 \\sum_{B\\in\\mathcal{F}}\\rho(B)\\int_B \\big| \\E_{\\mu_x^t[\\psi_*]}(v)-{\\rho(B)}^{-1}\\E_{\\mu^t[\\psi_*]}(v))\\big|\\,\\d\\rho_B(x)\\\\\n \\mathop{\\leq }^{\\text{Lemma \\ref{p3.10}}}&C_5 \\sum_{B\\in\\mathcal{F}}\\rho(B)C_4 r_B\\int_B \\big|\\nabla_x \\E_{\\mu_x^t[\\psi_*]}(v)\\big|\\,\\d\\rho_B(x)\\\\\n \\mathop{\\leq}^{\\text{Proposition \\ref{3.9}} }& \\frac{C_3C_4C_5}{\\sqrt{t}}\\sum_{B\\in\\mathcal{F}}\\sqrt{\\mm(B_{2r_B})}\\sqrt{r_B}\\left(\\int_B {\\rm{Var}}_{ \\mu_x^t[\\psi_*]}(v)\\,\\d\\rho(x)\\right)^\\frac{1}{2}\\\\\n \\mathop{\\leq}^{\\text{H\\\"{o}lder}~~}&\\frac{C_3C_4C_5}{\\sqrt{t}}\\left(\\sum_{B\\in\\mathcal{F}}\\mm(B_{2r_B})\\right)^\\frac{1}{2}\\left(\\sum_{B\\in\\mathcal{F}}\\int_B {\\rm{Var}}_{ \\mu_x^t[\\psi_*]}(v)\\,\\d\\rho(x)\\right)^\\frac{1}{2}\\\\\n \\overset{\\text{Boman chain~}}{\\leq}&\\frac{\\kappa}{\\sqrt{t}}\\left(\\int_S {\\rm{Var}}_{ \\mu_x^t[\\psi_*]}(v)\\,\\d\\rho(x)\\right)^\\frac{1}{2}\n \\end{aligned}\n \\end{equation}\n which is the thesis.\n \\end{proof}\n\n\\begin{theorem}\\label{thm2}\n Let $(X, \\d)$ be an Alexandrov space with no boundary. Then under the same assumptions of Theorem \\ref{thm0.2}, there exists a constant $\\bar{C}>0$, depending on $k,n,a_1, a_2, \\diam(S\\cup Y), {\\rm{Per}}(S), S$, such that for any $\\mu,\\nu\\in\\mathcal{P}_2(Y)$, we have \n \\begin{equation}\n \\int_S |\\nabla\\phi_\\mu(x)-\\nabla\\phi_\\nu(x)|^2\\,\\d\\rho(x)\\leq \\bar{C} \\left(\\int_S|\\phi_\\mu(x)-\\phi_\\nu(x)|^2\\,\\d\\rho(x)\\right)^\\frac{1}{3},\n \\end{equation}\n where $\\phi_\\mu$ and $\\phi_\\nu$ are the Kantorovich potentials from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively.\n \\end{theorem}\n\n\\begin{lemma}[\\cite{zbMATH07794624}, Lemma 5.1]\\label{l0.18}\n Let $I\\subseteq \\R$ be a compact segment and let $u,v: I\\rightarrow\\R$ be two convex functions such that $|u'|$ and $|v'|$ (defined a.e. on $I$) are uniformly bounded on $I$. Then\n \\begin{equation*}\n \\|u'-v'\\|_{L^2(I)}^2\\leq 8 (\\|u'\\|_{L^\\infty(I)}+\\|v'\\|_{L^\\infty(I)})^\\frac{4}{3}\\|u-v\\|_{L^2(I)}^\\frac{2}{3}.\n \\end{equation*}\n \\end{lemma}\n Therefore, applying Lemma \\ref{l0.18} to the functions $\\zeta|s|^2-u_\\mu^{(x,v)}$ and $\\zeta|s|^2-u_\\nu^{(x,v)}$ on each compact segment $[\\alpha_i(x,v),\\beta_i(x,v)]$ and using H\\\"{o}lder inequality, we obtain\n \\begin{equation}\\label{0.55}\n \\begin{aligned}\n &\\int_S |\\nabla\\phi_\\mu(x)-\\nabla\\phi_\\nu(x)|^2\\,\\d\\rho(x)\\\\\n \\mathop{\\leq}^{\\eqref{0.54}}& \\frac{a_2}{c_nd_n}\\int_{{\\rm S}X}\\sum_{i\\in I_S(x,v)}\\int_{\\alpha_i(x,v)}^{\\beta_i(x,v)}\\langle\\nabla\\phi_\\mu(b_s(x,v))-\\nabla\\phi_\\nu(b_s(x,v)),t_s(x,v)\\rangle^2\\,\\d s \\,\\d\\mm_S(x,v)\\\\\n \\leq&C_1 \\int_{{\\rm S}X}\\sum_{i\\in I_S(x,v)}\\left(\\int_{\\alpha_i(x,v)}^{\\beta_i(x,v)}|\\phi_\\mu(b_s(x,v))-\\phi_\\nu(b_s(x,v))|^2\\,\\d s\\right)^\\frac{1}{3}\\,\\d\\mm_S(x,v)\\\\\n \\leq&C_1 \\int_{{\\rm S}X}\\left(\\# I_S\\right)^\\frac{2}{3}\\left(\\int_0^1|\\phi_\\mu(b_s)-\\phi_\\nu(b_s)|^2\\chi_S(b_s)\\,\\d s\\right)^\\frac{1}{3}\\d\\mm_S\\\\\n \\leq&C_1 \\left(\\int_{{\\rm S}X}\\# I_S\\,\\d\\mm_S\\right)^\\frac{2}{3}\\left(\\int_{{\\rm S}X}\\int_0^1|\\phi_\\mu(b_s)-\\phi_\\nu(b_s)|^2\\chi_S(b_s)\\,\\d s \\,\\d\\mm_S\\right)^\\frac{1}{3}\\\\\n \\leq&C_1 \\left(\\frac{c_n}{a_1}\\right)^\\frac{1}{3}\\left(\\int_{{\\rm S}X}\\# I_S\\,\\d\\mm_S\\right)^\\frac{2}{3}\\left(\\int_S|\\phi_\\mu-\\phi_\\nu|^2\\,\\d\\rho\\right)^\\frac{1}{3},\n \\end{aligned}\n \\end{equation}\n where $C_1$ depends on $k,n,a_2, \\diam(S\\cup Y)$, and the last inequality follows from the invariance of $\\mm_S$ under the geodesic flow.\n\nLet $\\phi_\\mu$ and $\\phi_\\nu$ be the Kantorovich potentials from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively. From \\cite{zbMATH05349267,zbMATH06032507} we know that $\\nabla \\phi_\\mu(x), \\nabla \\phi_\\nu(x) \\in {\\rm{T}}_xX$ and\n $T_\\mu(x)= \\exp_x(-\\nabla \\phi_\\mu(x))$, $T_\\nu(x)= \\exp_x(-\\nabla \\phi_\\nu(x))$ are well-defined for almost every $x\\in S$. By triangle comparison condition (cf. \\cite{zbMATH05342782}), it holds\n \\begin{equation}\\label{exp}\n \\d \\big(T_\\mu(x), T_\\nu(x) \\big) \\leq C_2|\\nabla \\phi_\\mu(x)-\\nabla \\phi_\\nu(x)|,\n \\end{equation}\n for some constant $C_2>0$ which depends on $k, \\diam(S\\cup Y)$. \n \\medskip", "post_theorem_intro_text_len": 3816, "post_theorem_intro_text": "If $(S,  {\\mathrm d},  \\mathfrak m)$  is a compact ${\\rm RCD}(K, N)$ space,   it  supports a global $(1,1)$-Poincar\\'e inequality (cf.\\cite{zbMATH01474795, zbMATH06043352}) and we can remove the John domain assumption:\n\t\\begin{corollary}\\label{cor}\n\t\tLet $(X,{\\mathrm d},\\mathfrak m)$ be a compact ${\\rm RCD}(K,N)$ metric measure space.  Let $\\rho\\in \\mathcal{P}(X)$ be with $a_1\\mathfrak m \\leq \\rho\\leq a_2\\mathfrak m$ for some positive constants $a_1,a_2$. Then the conclusion of Theorem \\ref{thm0.1} holds.\n\t\\end{corollary}\n\n\tAdapting the  strategy of \\cite{arXiv:2504.05412}, we can also prove the stability of optimal transport maps on Alexandrov spaces.\n\t\t\\begin{theorem}\\label{thm0.2}\n\t\tLet $(X,  {\\mathrm d})$ be an $n$-dimensional Alexandrov space, $\\mathfrak m$ be the $n$-Hausdorff measure. Under the same assumption for $S, Y$ and $\\rho$ as in Theorem \\ref{thm0.1}, and if $S$ additionally has finite perimeter, then there exists a constant $C>0$, depending on $k,n,a_1, a_2, \\mathop{\\rm diam}\\nolimits(S\\cup Y), {\\rm{Per}}(S), S$, such that for any $\\mu,\\nu\\in\\mathcal{P}_2(Y)$, we have\n\t\t\\begin{equation}\n\t\t\t\\int_S {\\mathrm d}^2(T_\\mu(x), T_\\nu(x))\\,{\\mathrm d} \\rho(x) \\leq CW_1^{1/6}(\\mu, \\nu),\n\t\t\\end{equation}\n\t\twhere $T_\\mu$ and $T_\\nu$ are the optimal transport maps from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively.\n\t\\end{theorem}\n\n\t\\subsection{Strategy: heat kernel-regularized \\texorpdfstring{$c$}{c}-transform}\n\nMotivated by   regularized $c$-transforms  using  Gibbs kernels $e^{-c(x, y)/\\varepsilon}$ \\cite{arXiv:2504.05412},  entropic optimal transport \\cite{GTJEMS, GTT25}  and Varadhan's formula\n\\[\n\\lim_{t \\to 0} -t \\log p_{t/2}(x, y) = \\frac{1}{2} {\\mathrm d}^2(x, y)=c(x,y),\n\\]\n  we make use of {heat kernel-regularized $c$-transform} (Hopf--Cole transform):\n\\[\n\\mathop{\\rm Lip}\\nolimits(X, {\\mathrm d}) \\ni \\psi  \\mapsto \\Phi_t[\\psi](x) = -t \\log \\int_X e^{\\frac{\\psi(y)}{t}} p_{t/2}(x, y) \\, {\\mathrm d}\\mathfrak m(y).\n\\]\nThis approach  allows us to bypass the low regularity of Kantorovich potentials in the non-smooth setting. \n\nWe define the heat kernel regularized Kantorovich functional as\n\t\\begin{equation*}\n\t\t\\K_t[\\varphi]:=\\int_S \\Phi_t[\\varphi]\\,{\\mathrm d}\\rho.\n\t\\end{equation*}\nSimilar to \\cite{arXiv:2504.05412},  the key in our proof is the strong concavity of the functional $\\K_t$.\nTo achieve this,  we  first derive a local strong concavity estimate using heat kernel estimate,    then globalize the estimate on the support of the source measure using a Boman chain argument for John domains.\n\nUnlike the regularized $c$-transform used in \\cite{arXiv:2504.05412},  the existence of the boundary of $Y$ may lead to the failure in our heat kernel regularization argument.  This possibility is ruled out by using the measure concentration property of the heat kernel and by making a careful choice of Lipschitz extension.\n\n\\medskip\n\n\t\\noindent {\\bf Organization.}  This paper is structured as follows. In Section~\\ref{sec3}, we prove quantitative stability of Kantorovich potentials on $\\mathrm{RCD}(K,N)$ spaces. In Section \\ref{sec4}, we establish stability of optimal transport maps on Alexandrov spaces. The Appendix \\ref{A} contains technical lemmas about Poincar\\'e inequalities.\n\n\t\\bigskip\n\n\\noindent {\\bf Declaration.} The authors declare no conflict of interest and that the manuscript has no associated data.\n\n\\medskip\n\n\\noindent {\\bf Acknowledgement.} This work is supported by the Ministry of Science \\& Technology of China (2021YFA1000900, 2021YFA1002200),  National Natural Science Foundation of China  (12201596)   and Shandong Provincial Natural Science Foundation (ZR2025QB05).  The authors  thank Nicola Gigli, Jun Kitagawa, Nan Li,  Cyril Letrouit, Quentin M\\'erigot, Luca Tamanini for  helpful discussions and  suggestions on the bibliography.", "sketch": "Motivated by “regularized $c$-transforms … entropic optimal transport … and Varadhan's formula”, the proof of Theorem~\\ref{thm0.1} uses a **heat kernel-regularized $c$-transform** (Hopf--Cole transform)\n\\[\n\\psi\\mapsto \\Phi_t[\\psi](x)=-t\\log\\int_X e^{\\frac{\\psi(y)}{t}}\\,p_{t/2}(x,y)\\,\\mathrm d\\mathfrak m(y),\n\\]\nwhich “allows us to bypass the low regularity of Kantorovich potentials in the non-smooth setting.” One then defines the “heat kernel regularized Kantorovich functional”\n\\[\n\\mathcal K_t[\\varphi]:=\\int_S \\Phi_t[\\varphi]\\,\\mathrm d\\rho.\n\\]\n“Similar to \\cite{arXiv:2504.05412}, the key in our proof is the strong concavity of the functional $\\mathcal K_t$.” To obtain this strong concavity, they “first derive a local strong concavity estimate using heat kernel estimate, then globalize the estimate on the support of the source measure using a Boman chain argument for John domains.”\n\nThey also note a technical issue specific to this regularization: “the existence of the boundary of $Y$ may lead to the failure in our heat kernel regularization argument,” and this is “ruled out by using the measure concentration property of the heat kernel and by making a careful choice of Lipschitz extension.”", "expanded_sketch": "Motivated by “regularized $c$-transforms … entropic optimal transport … and Varadhan's formula”, in establishing the main theorem, the proof uses a **heat kernel-regularized $c$-transform** (Hopf--Cole transform)\n\\[\n\\psi\\mapsto \\Phi_t[\\psi](x)=-t\\log\\int_X e^{\\frac{\\psi(y)}{t}}\\,p_{t/2}(x,y)\\,\\mathrm d\\mathfrak m(y),\n\\]\nwhich “allows us to bypass the low regularity of Kantorovich potentials in the non-smooth setting.” One then defines the “heat kernel regularized Kantorovich functional”\n\\[\n\\mathcal K_t[\\varphi]:=\\int_S \\Phi_t[\\varphi]\\,\\mathrm d\\rho.\n\\]\n“Similar to arXiv:2504.05412, the key in our proof is the strong concavity of the functional $\\mathcal K_t$.” To obtain this strong concavity, they “first derive a local strong concavity estimate using heat kernel estimate, then globalize the estimate on the support of the source measure using a Boman chain argument for John domains.”\n\nThey also note a technical issue specific to this regularization: “the existence of the boundary of $Y$ may lead to the failure in our heat kernel regularization argument,” and this is “ruled out by using the measure concentration property of the heat kernel and by making a careful choice of Lipschitz extension.”,", "expanded_theorem": "\\label{thm0.1}\n\t\tLet $(X,{\\mathrm d},\\mathfrak m)$ be an ${\\rm RCD}(K,N)$ metric measure space. Let $S\\subseteq X$ be a John domain and $Y\\subseteq X$ be compact with $\\mathfrak m(Y)>0$. Let $\\rho\\in \\mathcal{P}(S)$  be with $a_1\\mathfrak m\\restr{S} \\leq \\rho\\leq a_2\\mathfrak m\\restr{S}$ for some positive constants $a_1,a_2$. Then there exists a constant $C>0$, depending on $K,N,a_1,a_2, S, \\mathop{\\rm diam}\\nolimits(S\\cup Y)$, such that for any $\\mu,\\nu\\in\\mathcal{P}(Y)$, \n\t\t\\begin{equation}\n\t\t\t\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}\\leq C W_1^\\frac{1}{2}(\\mu,\\nu),\n\t\t\\end{equation}\n\t\twhere $\\phi_\\mu$ and $\\phi_\\nu$ are the Kantorovich potentials from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively.,", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let \\((X,{\\mathrm d},\\mathfrak m)\\) be an \\({\\rm RCD}(K,N)\\) metric measure space. Let \\(S\\subseteq X\\) be a John domain, meaning that \\(S\\) is a bounded open set for which there exist \\(x_0\\in S\\) and \\(\\eta>0\\) such that every \\(x\\in S\\) can be joined to \\(x_0\\) by a rectifiable curve \\(\\gamma:[0,\\ell(\\gamma)]\\to S\\), parametrized by arc length, with \\(\\gamma(0)=x\\), \\(\\gamma(\\ell(\\gamma))=x_0\\), and \\({\\mathrm d}(\\gamma(t),X\\setminus S)\\ge \\eta t\\) for all \\(t\\in[0,\\ell(\\gamma)]\\). Let \\(Y\\subseteq X\\) be compact with \\(\\mathfrak m(Y)>0\\). Assume \\(\\rho\\in \\mathcal P(S)\\) satisfies \\(a_1\\,\\mathfrak m\\!\\restriction_S \\le \\rho \\le a_2\\,\\mathfrak m\\!\\restriction_S\\) for some constants \\(a_1,a_2>0\\). For each \\(\\mu\\in\\mathcal P(Y)\\), let \\(\\phi_\\mu\\) denote a Kantorovich potential for the quadratic cost \\(c(x,y)=\\tfrac12{\\mathrm d}(x,y)^2\\) from \\(\\rho\\) to \\(\\mu\\), normalized by \\(\\int_S \\phi_\\mu\\,{\\mathrm d}\\rho=0\\). Also let\n\\[\nW_1(\\mu,\\nu):=\\inf_{\\pi\\in\\Pi(\\mu,\\nu)}\\int_{Y\\times Y}{\\mathrm d}(y,z)\\,{\\mathrm d}\\pi(y,z)\n\\]\nbe the 1-Wasserstein distance. Which quantitative estimate holds for all \\(\\mu,\\nu\\in\\mathcal P(Y)\\)?", "correct_choice": {"label": "A", "text": "There exists a constant \\(C>0\\), depending only on \\(K,N,a_1,a_2,S\\), and \\(\\operatorname{diam}(S\\cup Y)\\), such that for every \\(\\mu,\\nu\\in\\mathcal P(Y)\\),\n\\[\n\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}=\\int_S |\\phi_\\mu-\\phi_\\nu|\\,{\\mathrm d}\\rho \\le C\\,W_1(\\mu,\\nu)^{1/2}.\n\\]"}, "choices": [{"label": "B", "text": "There exists a constant \\(C>0\\), depending only on \\(K,N,a_1,a_2,S\\), and \\(\\operatorname{diam}(S\\cup Y)\\), such that for every \\(\\mu,\\nu\\in\\mathcal P(Y)\\),\n\\[\n\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}=\\int_S |\\phi_\\mu-\\phi_\\nu|\\,{\\mathrm d}\\rho \\le C\\,W_1(\\mu,\\nu).\n\\]"}, {"label": "C", "text": "There exists a constant \\(C>0\\), depending only on \\(K,N,a_1,a_2,S\\), and \\(\\operatorname{diam}(S\\cup Y)\\), such that for every \\(\\mu,\\nu\\in\\mathcal P(Y)\\),\n\\[\n\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}=\\int_S |\\phi_\\mu-\\phi_\\nu|\\,{\\mathrm d}\\rho \\le C.\n\\]"}, {"label": "D", "text": "There exists a constant \\(C>0\\), depending only on \\(K,N,a_1,a_2\\), such that for every John domain \\(S\\subseteq X\\), every compact \\(Y\\subseteq X\\) with \\(\\mathfrak m(Y)>0\\), and every \\(\\mu,\\nu\\in\\mathcal P(Y)\\),\n\\[\n\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}=\\int_S |\\phi_\\mu-\\phi_\\nu|\\,{\\mathrm d}\\rho \\le C\\,W_1(\\mu,\\nu)^{1/2}.\n\\]"}, {"label": "E", "text": "There exists a constant \\(C>0\\), depending only on \\(K,N,a_1,a_2,S\\), and \\(\\operatorname{diam}(S\\cup Y)\\), such that for every \\(\\mu,\\nu\\in\\mathcal P(Y)\\),\n\\[\n\\|\\phi_\\mu-\\phi_\\nu\\|_{L^\\infty(S)}\\le C\\,W_1(\\mu,\\nu)^{1/2}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "sharp_holder_exponent_one_half", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "wasserstein_rate_factor", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "dependence_on_domain_geometry_and_diameter", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "L1_control_replaced_by_Linfty_control", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem states hypotheses and asks for the resulting uniform stability estimate, but it does not explicitly reveal the correct exponent, norm, or quantifier structure. There is no direct answer leakage beyond the standard setup."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: it essentially asks for the exact conclusion under a detailed set of assumptions. The answer is not literally restated in the stem, but the task is mainly to recognize the theorem's precise formulation."}, "GPS": {"score": 1, "justification": "Some reasoning is required to distinguish the sharp statement from nearby variants involving a stronger Lipschitz rate, weaker boundedness-only claim, dependence of constants on the measures, or an incorrect L^infty norm. However, this is mostly discriminating among theorem variants rather than generating a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target common errors: replacing the sharp 1/2 exponent by 1, dropping the Wasserstein dependence, weakening uniformity via target-dependent constants, and strengthening L^1 to L^infty. They are distinct and well aligned with likely failure modes."}, "total_score": 6, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no obvious leakage, but it mainly tests precise recall/recognition of a known result rather than deeper generative reasoning."}}
{"id": "2602.19175v1", "paper_link": "http://arxiv.org/abs/2602.19175v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm0.1}\n\t\tLet $(X,{\\mathrm d},\\mathfrak m)$ be an ${\\rm RCD}(K,N)$ metric measure space. Let $S\\subseteq X$ be a John domain and $Y\\subseteq X$ be compact with $\\mathfrak m(Y)>0$. Let $\\rho\\in \\mathcal{P}(S)$  be with $a_1\\mathfrak m\\restr{S} \\leq \\rho\\leq a_2\\mathfrak m\\restr{S}$ for some positive constants $a_1,a_2$. Then there exists a constant $C>0$, depending on $K,N,a_1,a_2, S, \\mathop{\\rm diam}\\nolimits(S\\cup Y)$, such that for any $\\mu,\\nu\\in\\mathcal{P}(Y)$, \n\t\t\\begin{equation}\n\t\t\t\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}\\leq C W_1^\\frac{1}{2}(\\mu,\\nu),\n\t\t\\end{equation}\n\t\twhere $\\phi_\\mu$ and $\\phi_\\nu$ are the Kantorovich potentials from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively.", "start_pos": 11594, "end_pos": 12265, "label": "thm0.1"}, "ref_dict": {"A": "\\begin{appendices}\n\t\\section{Poincar\\'e inequality: local to global}\\label{A}\n\t\t\\begin{lemma}\\label{p3.10}\n\tLet $x_0\\in S$, $0<r_0\\leq1$ and  $B=B(x_0,r_0)\\subseteq S$,  $\\rho_B=\\frac{\\rho\\restr{B}}{\\rho(B)}$.  Then the following strong local $(1,1)$-Poincar\\'e inequality holds for $f\\in \\Lip(B,  \\d)$:\n\t\t\t\\begin{equation}\n\t\t\t\\int_{B}|f(x)-\\E_{\\rho_B}(f)|\\,\\d\\rho_B(x)\\leq C_4 r_0\\int_B |\\nabla_x f|\\,\\d\\rho_B,\n\t\t\\end{equation}\n\t\twhere $C_4$ depends on $K,N,a_1,a_2$.\n\t\\end{lemma}\n\n\t\\begin{proof}\n\t\tBy \\cite[Chapter 9]{zbMATH01474795} and \\cite[Remark 3.3]{zbMATH06043352},  $\\mm$ satisfies the  $(1,1)$-Poincar\\'e inequality:\n\t\t\\begin{equation}\n\t\t\t\\int_{B} |f-\\E_{\\mm_B}(f)|\\,\\d\\mm\\leq c_1 r_0 \\int_B |\\nabla_x f|\\,\\d\\mm,\n\t\t\\end{equation}\n\t\twhere $c_1$ depends on $K,N$, $\\mm_B=\\frac{\\mm\\restr{B}}{\\mm(B)}$. Since $a_1\\mm \\leq \\rho\\leq a_2 \\mm$, we have \n\t\t\\begin{equation}\n\t\t\t\\begin{aligned}\n\t\t\t\t\\int_{B} |f-\\E_{\\rho_B}(f)|\\,\\d\\rho\\leq &\\int_{B} |f-\\E_{\\mm_B}(f)|\\,\\d\\rho+\\int_{B} |\\E_{\\mm_B}(f)-\\E_{\\rho_B}(f)|\\,\\d\\rho\\\\\n\t\t\t\t\\leq & 2a_2\\int_{B} |f-\\E_{\\mm_B}(f)|\\,\\d\\mm\\\\\n\t\t\t\t\\leq &2a_2 c_1 r_0 \\int_B |\\nabla_x f|\\,\\d\\mm\n\t\t\t\t\\leq  2\\frac{a_2}{a_1}c_1 r_0 \\int_B |\\nabla_x f|\\,\\d\\rho,\\\\\n\t\t\t\\end{aligned}\n\t\t\\end{equation}\n\twhich is the thesis.\n\t\\end{proof}\n\n\tNext we prove an $L^1$-variant of the gluing Lemma in \\cite[Lemma 3.3]{arXiv:2411.04908},   following the same Boman-chain decomposition,  utilizing the doubling property directly instead of the maximal function estimates. \n\n\t\t\\begin{definition}[Boman chain condition]\\label{Boman chain condition}\n\t\tWe say that a probability measure $\\rho$ on an open set $S$ of a metric space satisfies  \\emph{Boman chain condition} with parameters \\(E, F,G>1\\) if there is a covering \\(\\mathcal{F}\\) of $S$ by open balls  $B \\in \\mathcal{F}$ such that:\n\n\t\t\\begin{enumerate}\n\t\t\t\t\\item For any \\(x \\in S\\),\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\sum_{B \\in \\mathcal{F}} \\chi_{2B}(x) \\leq E \\chi_{S}(x).\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\\item For some fixed ball \\(B_0\\) in \\(\\mathcal{F}\\), called the \\emph{central ball}, and for every \\(B \\in \\mathcal{F}\\), there exists a chain \\(B_0, B_1, \\ldots, B_N = B\\) of distinct balls from \\(\\mathcal{F}\\) such that\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\tB \\subset F B_j,~~~\\forall j \\in \\{0, \\ldots, N-1\\}.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\\item Consecutive balls of the above chain overlap quantitatively:\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\rho(B_j \\cap B_{j+1}) \\geq G^{-1} \\max(\\rho(B_j), \\rho(B_{j+1})),~~~\\forall j \\in \\{0, \\ldots, N-1\\}.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\\end{enumerate}\n\t\\end{definition}\n\n\t\\begin{lemma}\\label{3.1}\n\tFor any $f\\in L^1(\\rho)$,   it holds that\n\t\\begin{equation}\\label{A.5}\n\t\t\\int_S |f(x)-\\E_{\\rho}(f)|\\,\\d\\rho(x)\\leq C_5 \\sum_{B\\in\\mathcal{F}}\\rho(B)\\int_B |f(x)-\\E_{\\rho_B}(f)|\\,\\d\\rho_B(x),\n\t\\end{equation}\n\twhere $C_5$ depends on $K,N,\\diam(S),S$.\n\t\\end{lemma}\n\n    \\begin{proof} \n    \tSince $S$ is a John domain, and $\\rho$ is a doubling measure on $S$, by  \\cite[Proposition 3.7]{arXiv:2504.05412}, $\\rho$ satisfies the Boman chain condition. Hence there exists a covering $\\mathcal{F}$ of $S$ satisfying Definition \\ref{Boman chain condition}.\n\n    \tFor the central ball $B_0$, note that\n    \t\\begin{equation}\\label{A.2}\n    \t\t\\begin{aligned}\n    \t\t\t\\int_S |f(x)-\\E_{\\rho}(f)|\\,\\d\\rho(x)\n    \t\t\t\\leq & \\int_S |f(x)-\\E_{\\rho_{B_0}}(f)|\\,\\d\\rho(x)+\\int_S |\\E_{\\rho_{B_0}}(f)-\\E_{\\rho}(f)|\\,\\d\\rho(x)\\\\\n    \t\t\t\\leq & 2\\int_S |f(x)-\\E_{\\rho_{B_0}}(f)|\\,\\d\\rho(x).\n    \t\t\\end{aligned}\n    \t\\end{equation}\n\n   For  $B\\in\\mathcal{F}$,  \tdenote $a_B=\\int_B |f-\\E_{\\rho_{B}}(f)|\\,\\d\\rho=\\rho(B)\\int_B |f-\\E_{\\rho_{B}}(f)|\\,\\d\\rho_B$. Since $\\mathcal{F}$ is a covering of $S$, we have\n    \t\\begin{equation}\\label{A.3}\n    \t\t\\begin{aligned}\n    \t\t\t&\\int_S |f(x)-\\E_{\\rho_{B_0}}(f)|\\,\\d\\rho(x)\n    \t\t\t\\leq  \\sum_{B \\in \\mathcal{F}} \\int_B |f(x)-\\E_{\\rho_{B_0}}(f)|\\,\\d\\rho(x)\\\\\n    \t\t\t\\leq & \\sum_{B \\in \\mathcal{F}} \\left(\\int_B |f(x)-\\E_{\\rho_{B}}(f)|\\,\\d\\rho(x)+\\int_B |\\E_{\\rho_{B}}(f)-\\E_{\\rho_{B_0}}(f)|\\,\\d\\rho(x)\\right)\\\\\n    \t\t\t\\leq &  \\sum_{B \\in \\mathcal{F}} \\left(a_B+\\rho(B) |\\E_{\\rho_{B}}(f)-\\E_{\\rho_{B_0}}(f)|\\right).\n    \t\t\\end{aligned}\n    \t\\end{equation}\n\n    \tFor any $B\\in\\mathcal{F}$, by Boman chain condition, there exists a chain $B_0, B_1,\\dots, B_N=B$ of distinct balls from \\(\\mathcal{F}\\), such that for any \\(j \\in \\{0, \\ldots, N-1\\}\\),\n    \t\\begin{equation}\n    \t\t\\begin{aligned}\n    \t\t|\\E_{\\rho_{B_j}}(f)-\\E_{\\rho_{B_{j+1}}}(f)|= &\\left|\\frac{1}{\\rho(B_j\\cap B_{j+1})}\\int_{B_j\\cap B_{j+1}}\\left(\\E_{\\rho_{B_j}}(f)-\\E_{\\rho_{B_{j+1}}}(f)\\right)\\,\\d\\rho\\right|\\\\\n    \t\t\\leq &\\frac{1}{\\rho(B_j\\cap B_{j+1})}\\int_{B_j\\cap B_{j+1}}\\left|\\E_{\\rho_{B_j}}(f)-\\E_{\\rho_{B_{j+1}}}(f)\\right|\\,\\d\\rho\\\\\n    \t\t\\leq &\\frac{1}{\\rho(B_j\\cap B_{j+1})}\\left(\\int_{B_j\\cap B_{j+1}}\\left|f-\\E_{\\rho_{B_j}}(f)\\right|+\\left|f-\\E_{\\rho_{B_{j+1}}}(f)\\right|\\,\\d\\rho\\right)\\\\\n    \t\t\\leq & \\frac{a_{B_j}+a_{B_{j+1}}}{\\rho(B_j\\cap B_{j+1})}\n    \t\t\\overset{*}{\\leq}  G \\left(\\frac{a_{B_j}}{\\rho(B_j)}+\\frac{a_{B_{j+1}}}{\\rho(B_{j+1})}\\right),\n    \t\t\\end{aligned}\n    \t\\end{equation}\n    \twhere $(*)$ follows from the quantitative chain overlap of Boman chain condition $3$.\n\n    \t Thus, we have\n    \t \\begin{equation}\n    \t |\\E_{\\rho_{B}}(f)-\\E_{\\rho_{B_0}}(f)|\\leq \\sum_{j=0}^{N-1}|\\E_{\\rho_{B_j}}(f)-\\E_{\\rho_{B_{j+1}}}(f)|\\leq 2G \\sum_{j=0}^{N}\\frac{a_{B_j}}{\\rho(B_j)}\\overset{*}{\\leq} 2G \\sum_{B\\subset F \\bar{B}}\\frac{a_{\\bar{B}}}{\\rho(\\bar{B})},\n    \t \\end{equation}\n    \t where $ \\sum_{B\\subset F \\bar{B}}$ means that the sum runs over all  $\\bar{B}\\in\\mathcal{F}$ satisfying $B\\subset F \\bar{B}$, and $(*)$ follows from the Boman chain condition $2$.\n    \t Then by Fubini--Tonelli theorem,  \n    \t \\begin{equation}\\label{A.6}\n    \t \t\\begin{aligned}\n    \t \t \t  \\sum_{B \\in \\mathcal{F}} \\rho(B) |\\E_{\\rho_{B}}(f)-\\E_{\\rho_{B_0}}(f)|\n    \t \t \t  \\leq & 2G\\sum_{B \\in \\mathcal{F}} \\rho(B) \\sum_{B\\subset F \\overline{B}}\\frac{a_{\\overline{B}}}{\\rho(\\overline{B})}\n    \t \t \t  \\leq 2G\\sum_{\\bar{B} \\in \\mathcal{F}} \\frac{a_{\\bar{B}}}{\\rho(\\bar{B})}\\sum_{B\\subset F \\bar{B}}\\rho(B).\n    \t \t\\end{aligned}\n    \t \\end{equation}\n\nBy Boman chain condition $1$ and  the doubling property of $\\rho$, we have  \n    \t \\begin{equation}\\label{aa}\n       \\sum_{B\\subset F \\bar{B}}\\rho(B)\\leq E\\rho(F\\bar{B})\\leq E\\beta^2F^\\frac{\\log \\beta}{\\log 2}\\rho(\\bar{B}),\n    \t \\end{equation}\nwhere  $\\beta=\\beta(K,N,\\diam(S))$ is the doubling constant.\n\n    \t Combining \\eqref{A.2}, \\eqref{A.3},  \\eqref{A.6} and \\eqref{aa}, we obtain\n    \t \\begin{equation}\\label{A_A}\n    \t \t\\begin{aligned}\n    \t \t\t \\int_S |f(x)-\\E_{\\rho}(f)|\\,\\d\\rho(x)\\leq & 2 \\sum_{B \\in \\mathcal{F}} \\left(a_B+\\rho(B) |\\E_{\\rho_{B}}(f)-\\E_{\\rho_{B_0}}(f)|\\right)\\\\\n    \t \t\t \\leq &2(1+2\\beta^2 EF^\\frac{\\log \\beta}{\\log 2}G) \\sum_{B \\in \\mathcal{F}} \\rho(B)\\int_B |f-\\E_{\\rho_{B}}(f)|\\,\\d\\rho_B,\n    \t \t\\end{aligned}\n    \t \\end{equation}\n  which is the thesis.\n    \\end{proof}\n\n\t\\end{appendices}", "thm0.1": "\\begin{theorem}\\label{thm0.1}\n\t\tLet $(X,\\d,\\mm)$ be an ${\\rm RCD}(K,N)$ metric measure space. Let $S\\subseteq X$ be a John domain and $Y\\subseteq X$ be compact with $\\mm(Y)>0$. Let $\\rho\\in \\mathcal{P}(S)$  be with $a_1\\mm\\restr{S} \\leq \\rho\\leq a_2\\mm\\restr{S}$ for some positive constants $a_1,a_2$. Then there exists a constant $C>0$, depending on $K,N,a_1,a_2, S, \\diam(S\\cup Y)$, such that for any $\\mu,\\nu\\in\\mathcal{P}(Y)$, \n\t\t\\begin{equation}\n\t\t\t\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}\\leq C W_1^\\frac{1}{2}(\\mu,\\nu),\n\t\t\\end{equation}\n\t\twhere $\\phi_\\mu$ and $\\phi_\\nu$ are the Kantorovich potentials from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively.\n\t\\end{theorem}"}, "pre_theorem_intro_text_len": 6021, "pre_theorem_intro_text": "\\subsection{Motivation and setting}\n\t\\paragraph{Optimal transport.}Optimal transport, initiated by Monge  \\cite{monge1781} and reframed by  Kantorovich \\cite{zbMATH03099866},  seeks the most efficient way to redistribute mass between two probability distributions.   Precisely, given two probability measures $\\rho$ and $\\mu$ defined on Polish spaces $X$ and $Y$ respectively, and a  cost function $c:X\\times Y\\rightarrow \\mathbb{R}$, the Monge optimal transport problem aims to find a minimizer  $T$,   called an optimal transport map,   of the following optimization problem among all measurable maps $H: X \\to Y$ pushing $\\rho$ forward to $\\mu$:\n\t\\begin{equation}\\label{2.1}\n\t\t\\int_X c(x, T(x))  \\,{\\mathrm d}\\rho(x)=\\inf_{H_\\sharp \\rho = \\mu}\\int_X c(x, H(x))  \\,{\\mathrm d}\\rho(x), \\tag{MP}\n\t\\end{equation}\n The   Monge problem  prevents mass splitting, sometimes making the problem ill-posed.  Its relaxation,  called Kantorovich problem, optimizes over joint couplings instead of deterministic maps. A fundamental result known as Kantorovich duality, asserts that the Kantorovich problem is equivalent to the  following dual problem:\n\t\\begin{equation}\n\t\t\\sup_{ \\phi(x) + \\psi(y) \\leq c(x, y)} \\left\\{ \\int_X \\phi(x) \\, {\\mathrm d}\\rho(x) + \\int_Y \\psi(y) \\, {\\mathrm d}\\mu(y)  \\right\\}.\\tag{KD}\n\t\\end{equation}\n\tThe optimal functions $\\phi,\\psi$,  which always exist under standard assumptions,   are called  Kantorovich potentials \\cite{AG-U, V-O}.  \n\n\tIn the Euclidean space,   for $c(x,y)=\\frac{1}{2}|x-y|^2$, Brenier's landmark result \\cite{brenier1991polar} established that, for absolutely continuous source measures, the optimal transport map is unique and takes the form $T(x)=\\nabla u(x)$ for a convex function $u$. This was later extended to Riemannian manifolds by McCann \\cite{mccann2001polar}, who showed that for $c(x,y)=\\frac{1}{2}{\\mathrm d}^2(x,y)$, the optimal transport map is given almost everywhere by $T(x)=\\exp_x(-\\nabla\\varphi(x))$ for a  Kantorovich potential $\\varphi$.\n\n\t A fundamental question in both theoretical and applied contexts of optimal transport is the quantitative stability of optimal transport maps and Kantorovich potentials under perturbations of the target measure (see Letrouit's lecture note \\cite{LetrouitLecture} and the references therein).   Based on recent breakthroughs in the quantitative stability of optimal transport  on  Euclidean spaces \\cite{zbMATH07794624, arXiv:2411.04908},    on boundaries of convex bodies \\cite{Kitagawa12,  Kitagawa25}  and  on Riemannian manifolds  \\cite{arXiv:2504.05412},     \nKitagawa--Letrouit--M\\'erigot   \\cite[\\S 1.2]{arXiv:2504.05412} conjecture that:\n\t\\begin{center}\n\t\t\\emph{the quantitative stability results are also true in more general metric measure spaces with synthetic curvature bounds.}\n\t\\end{center}\n\n\t\\paragraph{General setting.}\n\tIn this paper, we confirm the conjecture of Kitagawa-- Letrouit--M\\'erigot for the following spaces:\n\t\\begin{itemize}\n\t\t\\item  RCD$(K,N)$  spaces (metric measure spaces with synthetic Ricci curvature lower bound):  we prove quantitative $L^1$-stability of Kantorovich potentials with respect to Wasserstein perturbations of the target measure;\n\t\t\\item Alexandrov spaces with (sectional) curvature lower bound:   we establish stability estimates for optimal transport maps in terms of the $L^2$-norm of the potential difference.\n\t\\end{itemize}\n\n\tAn Alexandrov space is a  geodesic  space of finite Hausdorff dimension and of curvature bounded from below (cf.  \\cite{zbMATH01626771}).   An ${\\rm RCD}(K,N)$ space  is  a metric measure spaces verifying the synthetic Riemannian curvature-dimension condition  \\cite{zbMATH05049052,zbMATH05578758, zbMATH06303881, G-O}.   An $n$-dimensional Alexandrov space with curvature bounded from below by $k$,  equipped with its $n$-Hausdorff measure,   is an ${\\rm RCD}(k(n-1), n)$ space  \\cite{zbMATH06032507,zbMATH05962558}.\n\t\t After \\cite{brenier1991polar, mccann2001polar},   Gigli--Rajala--Sturm \\cite[Theorem 1.1]{ gigli2016optimal}  proved the existence and uniqueness of the optimal transport map on an $\\rm{RCD}(K,N)$ space $(X,{\\mathrm d},\\mathfrak m)$,  for the quadratic cost $c(x,y)=\\frac{1}{2}{\\mathrm d}^2(x,y)$ and the source measure $\\rho\\ll \\mathfrak m$. \n\n\t  John domains,    named after  F. John \\cite{zbMATH03166766},   encompass many cases of interest, such as  bounded Lipschitz domains, bounded domains satisfying a cone condition and certain fractal domains (see \\cite{zbMATH00858510} for more discussions). \n\t\\begin{definition}[John domain]\n\t\tA bounded open subset $S$ of a metric space is called a \\emph{John domain} if there is a distinguished point $x_0 \\in S$ and a constant $\\eta > 0$ such that, for every $x \\in S$, there is a rectifiable curve $\\gamma \\colon [0, \\ell(\\gamma)] \\to S$ parametrized by arc length, such that $\\gamma(0) = x$, $\\gamma(\\ell(\\gamma)) = x_0$, and\n\t\t\\begin{equation*}\n\t\t\t{\\mathrm d}(\\gamma(t), S^c) \\geq \\eta t,~~~\\forall t \\in [0, \\ell(\\gamma)],\n\t\t\\end{equation*}\n\t\twhere $S^c$ denotes the complement of $S$.\n\t\\end{definition}\n\tIn  \\cite[Theorem 1.9]{arXiv:2411.04908}, it has been shown that when the source measure $\\rho$ is the uniform density on some non-John domain  $S\\subset \\mathbb{R}^n$, then no quantitative stability estimates of the form $\\|\\phi_\\mu-\\phi_\\nu\\|_{L^2(\\rho)}\\leq C W_p^\\alpha(\\mu,\\nu)$ \n\tcan hold for any $C, \\alpha, p>0$. \n\tMoreover, examples in \\cite{letrouit2025unstableoptimaltransportmaps} indicates that both unboundedness of the density and openness of    $S$ may cause  instability  of optimal transport maps.\n\n\t In this paper,  we always assume that the Kantorovich potential  $\\phi_\\mu$ from $\\rho$ to $\\mu$ satisfies $\\E_\\rho(\\phi_\\mu)=\\int_S\\phi_\\mu\\,{\\mathrm d}\\rho=0$.  Together with the uniqueness of the optimal transport map, such  $\\phi_\\mu$ is  unique and thus makes sense to study its stability.\n\n\t\\subsection{Main results}\n\tOur main theorem concerns the quantitative  $L^1$ stability of Kantorovich potentials on ${\\rm RCD}(K,N)$  spaces.", "context": "\\subsection{Motivation and setting}\n \\paragraph{Optimal transport.}Optimal transport, initiated by Monge \\cite{monge1781} and reframed by Kantorovich \\cite{zbMATH03099866}, seeks the most efficient way to redistribute mass between two probability distributions. Precisely, given two probability measures $\\rho$ and $\\mu$ defined on Polish spaces $X$ and $Y$ respectively, and a cost function $c:X\\times Y\\rightarrow \\mathbb{R}$, the Monge optimal transport problem aims to find a minimizer $T$, called an optimal transport map, of the following optimization problem among all measurable maps $H: X \\to Y$ pushing $\\rho$ forward to $\\mu$:\n \\begin{equation}\\label{2.1}\n \\int_X c(x, T(x)) \\,{\\mathrm d}\\rho(x)=\\inf_{H_\\sharp \\rho = \\mu}\\int_X c(x, H(x)) \\,{\\mathrm d}\\rho(x), \\tag{MP}\n \\end{equation}\n The Monge problem prevents mass splitting, sometimes making the problem ill-posed. Its relaxation, called Kantorovich problem, optimizes over joint couplings instead of deterministic maps. A fundamental result known as Kantorovich duality, asserts that the Kantorovich problem is equivalent to the following dual problem:\n \\begin{equation}\n \\sup_{ \\phi(x) + \\psi(y) \\leq c(x, y)} \\left\\{ \\int_X \\phi(x) \\, {\\mathrm d}\\rho(x) + \\int_Y \\psi(y) \\, {\\mathrm d}\\mu(y) \\right\\}.\\tag{KD}\n \\end{equation}\n The optimal functions $\\phi,\\psi$, which always exist under standard assumptions, are called Kantorovich potentials \\cite{AG-U, V-O}.\n\nA fundamental question in both theoretical and applied contexts of optimal transport is the quantitative stability of optimal transport maps and Kantorovich potentials under perturbations of the target measure (see Letrouit's lecture note \\cite{LetrouitLecture} and the references therein). Based on recent breakthroughs in the quantitative stability of optimal transport on Euclidean spaces \\cite{zbMATH07794624, arXiv:2411.04908}, on boundaries of convex bodies \\cite{Kitagawa12, Kitagawa25} and on Riemannian manifolds \\cite{arXiv:2504.05412}, \nKitagawa--Letrouit--M\\'erigot \\cite[\\S 1.2]{arXiv:2504.05412} conjecture that:\n \\begin{center}\n \\emph{the quantitative stability results are also true in more general metric measure spaces with synthetic curvature bounds.}\n \\end{center}\n\nAn Alexandrov space is a geodesic space of finite Hausdorff dimension and of curvature bounded from below (cf. \\cite{zbMATH01626771}). An ${\\rm RCD}(K,N)$ space is a metric measure spaces verifying the synthetic Riemannian curvature-dimension condition \\cite{zbMATH05049052,zbMATH05578758, zbMATH06303881, G-O}. An $n$-dimensional Alexandrov space with curvature bounded from below by $k$, equipped with its $n$-Hausdorff measure, is an ${\\rm RCD}(k(n-1), n)$ space \\cite{zbMATH06032507,zbMATH05962558}.\n After \\cite{brenier1991polar, mccann2001polar}, Gigli--Rajala--Sturm \\cite[Theorem 1.1]{ gigli2016optimal} proved the existence and uniqueness of the optimal transport map on an $\\rm{RCD}(K,N)$ space $(X,{\\mathrm d},\\mathfrak m)$, for the quadratic cost $c(x,y)=\\frac{1}{2}{\\mathrm d}^2(x,y)$ and the source measure $\\rho\\ll \\mathfrak m$.\n\nJohn domains, named after F. John \\cite{zbMATH03166766}, encompass many cases of interest, such as bounded Lipschitz domains, bounded domains satisfying a cone condition and certain fractal domains (see \\cite{zbMATH00858510} for more discussions). \n \\begin{definition}[John domain]\n A bounded open subset $S$ of a metric space is called a \\emph{John domain} if there is a distinguished point $x_0 \\in S$ and a constant $\\eta > 0$ such that, for every $x \\in S$, there is a rectifiable curve $\\gamma \\colon [0, \\ell(\\gamma)] \\to S$ parametrized by arc length, such that $\\gamma(0) = x$, $\\gamma(\\ell(\\gamma)) = x_0$, and\n \\begin{equation*}\n {\\mathrm d}(\\gamma(t), S^c) \\geq \\eta t,~~~\\forall t \\in [0, \\ell(\\gamma)],\n \\end{equation*}\n where $S^c$ denotes the complement of $S$.\n \\end{definition}\n In \\cite[Theorem 1.9]{arXiv:2411.04908}, it has been shown that when the source measure $\\rho$ is the uniform density on some non-John domain $S\\subset \\mathbb{R}^n$, then no quantitative stability estimates of the form $\\|\\phi_\\mu-\\phi_\\nu\\|_{L^2(\\rho)}\\leq C W_p^\\alpha(\\mu,\\nu)$ \n can hold for any $C, \\alpha, p>0$. \n Moreover, examples in \\cite{letrouit2025unstableoptimaltransportmaps} indicates that both unboundedness of the density and openness of $S$ may cause instability of optimal transport maps.\n\nIn this paper, we always assume that the Kantorovich potential $\\phi_\\mu$ from $\\rho$ to $\\mu$ satisfies $\\E_\\rho(\\phi_\\mu)=\\int_S\\phi_\\mu\\,{\\mathrm d}\\rho=0$. Together with the uniqueness of the optimal transport map, such $\\phi_\\mu$ is unique and thus makes sense to study its stability.\n\n\\subsection{Main results}\n Our main theorem concerns the quantitative $L^1$ stability of Kantorovich potentials on ${\\rm RCD}(K,N)$ spaces.", "full_context": "\\subsection{Motivation and setting}\n \\paragraph{Optimal transport.}Optimal transport, initiated by Monge \\cite{monge1781} and reframed by Kantorovich \\cite{zbMATH03099866}, seeks the most efficient way to redistribute mass between two probability distributions. Precisely, given two probability measures $\\rho$ and $\\mu$ defined on Polish spaces $X$ and $Y$ respectively, and a cost function $c:X\\times Y\\rightarrow \\mathbb{R}$, the Monge optimal transport problem aims to find a minimizer $T$, called an optimal transport map, of the following optimization problem among all measurable maps $H: X \\to Y$ pushing $\\rho$ forward to $\\mu$:\n \\begin{equation}\\label{2.1}\n \\int_X c(x, T(x)) \\,{\\mathrm d}\\rho(x)=\\inf_{H_\\sharp \\rho = \\mu}\\int_X c(x, H(x)) \\,{\\mathrm d}\\rho(x), \\tag{MP}\n \\end{equation}\n The Monge problem prevents mass splitting, sometimes making the problem ill-posed. Its relaxation, called Kantorovich problem, optimizes over joint couplings instead of deterministic maps. A fundamental result known as Kantorovich duality, asserts that the Kantorovich problem is equivalent to the following dual problem:\n \\begin{equation}\n \\sup_{ \\phi(x) + \\psi(y) \\leq c(x, y)} \\left\\{ \\int_X \\phi(x) \\, {\\mathrm d}\\rho(x) + \\int_Y \\psi(y) \\, {\\mathrm d}\\mu(y) \\right\\}.\\tag{KD}\n \\end{equation}\n The optimal functions $\\phi,\\psi$, which always exist under standard assumptions, are called Kantorovich potentials \\cite{AG-U, V-O}.\n\nA fundamental question in both theoretical and applied contexts of optimal transport is the quantitative stability of optimal transport maps and Kantorovich potentials under perturbations of the target measure (see Letrouit's lecture note \\cite{LetrouitLecture} and the references therein). Based on recent breakthroughs in the quantitative stability of optimal transport on Euclidean spaces \\cite{zbMATH07794624, arXiv:2411.04908}, on boundaries of convex bodies \\cite{Kitagawa12, Kitagawa25} and on Riemannian manifolds \\cite{arXiv:2504.05412}, \nKitagawa--Letrouit--M\\'erigot \\cite[\\S 1.2]{arXiv:2504.05412} conjecture that:\n \\begin{center}\n \\emph{the quantitative stability results are also true in more general metric measure spaces with synthetic curvature bounds.}\n \\end{center}\n\nAn Alexandrov space is a geodesic space of finite Hausdorff dimension and of curvature bounded from below (cf. \\cite{zbMATH01626771}). An ${\\rm RCD}(K,N)$ space is a metric measure spaces verifying the synthetic Riemannian curvature-dimension condition \\cite{zbMATH05049052,zbMATH05578758, zbMATH06303881, G-O}. An $n$-dimensional Alexandrov space with curvature bounded from below by $k$, equipped with its $n$-Hausdorff measure, is an ${\\rm RCD}(k(n-1), n)$ space \\cite{zbMATH06032507,zbMATH05962558}.\n After \\cite{brenier1991polar, mccann2001polar}, Gigli--Rajala--Sturm \\cite[Theorem 1.1]{ gigli2016optimal} proved the existence and uniqueness of the optimal transport map on an $\\rm{RCD}(K,N)$ space $(X,{\\mathrm d},\\mathfrak m)$, for the quadratic cost $c(x,y)=\\frac{1}{2}{\\mathrm d}^2(x,y)$ and the source measure $\\rho\\ll \\mathfrak m$.\n\nJohn domains, named after F. John \\cite{zbMATH03166766}, encompass many cases of interest, such as bounded Lipschitz domains, bounded domains satisfying a cone condition and certain fractal domains (see \\cite{zbMATH00858510} for more discussions). \n \\begin{definition}[John domain]\n A bounded open subset $S$ of a metric space is called a \\emph{John domain} if there is a distinguished point $x_0 \\in S$ and a constant $\\eta > 0$ such that, for every $x \\in S$, there is a rectifiable curve $\\gamma \\colon [0, \\ell(\\gamma)] \\to S$ parametrized by arc length, such that $\\gamma(0) = x$, $\\gamma(\\ell(\\gamma)) = x_0$, and\n \\begin{equation*}\n {\\mathrm d}(\\gamma(t), S^c) \\geq \\eta t,~~~\\forall t \\in [0, \\ell(\\gamma)],\n \\end{equation*}\n where $S^c$ denotes the complement of $S$.\n \\end{definition}\n In \\cite[Theorem 1.9]{arXiv:2411.04908}, it has been shown that when the source measure $\\rho$ is the uniform density on some non-John domain $S\\subset \\mathbb{R}^n$, then no quantitative stability estimates of the form $\\|\\phi_\\mu-\\phi_\\nu\\|_{L^2(\\rho)}\\leq C W_p^\\alpha(\\mu,\\nu)$ \n can hold for any $C, \\alpha, p>0$. \n Moreover, examples in \\cite{letrouit2025unstableoptimaltransportmaps} indicates that both unboundedness of the density and openness of $S$ may cause instability of optimal transport maps.\n\nIn this paper, we always assume that the Kantorovich potential $\\phi_\\mu$ from $\\rho$ to $\\mu$ satisfies $\\E_\\rho(\\phi_\\mu)=\\int_S\\phi_\\mu\\,{\\mathrm d}\\rho=0$. Together with the uniqueness of the optimal transport map, such $\\phi_\\mu$ is unique and thus makes sense to study its stability.\n\n\\subsection{Main results}\n Our main theorem concerns the quantitative $L^1$ stability of Kantorovich potentials on ${\\rm RCD}(K,N)$ spaces.\n\nJohn domains, named after F. John \\cite{zbMATH03166766}, encompass many cases of interest, such as bounded Lipschitz domains, bounded domains satisfying a cone condition and certain fractal domains (see \\cite{zbMATH00858510} for more discussions). \n \\begin{definition}[John domain]\n A bounded open subset $S$ of a metric space is called a \\emph{John domain} if there is a distinguished point $x_0 \\in S$ and a constant $\\eta > 0$ such that, for every $x \\in S$, there is a rectifiable curve $\\gamma \\colon [0, \\ell(\\gamma)] \\to S$ parametrized by arc length, such that $\\gamma(0) = x$, $\\gamma(\\ell(\\gamma)) = x_0$, and\n \\begin{equation*}\n \\d(\\gamma(t), S^c) \\geq \\eta t,~~~\\forall t \\in [0, \\ell(\\gamma)],\n \\end{equation*}\n where $S^c$ denotes the complement of $S$.\n \\end{definition}\n In \\cite[Theorem 1.9]{arXiv:2411.04908}, it has been shown that when the source measure $\\rho$ is the uniform density on some non-John domain $S\\subset \\R^n$, then no quantitative stability estimates of the form $\\|\\phi_\\mu-\\phi_\\nu\\|_{L^2(\\rho)}\\leq C W_p^\\alpha(\\mu,\\nu)$ \n can hold for any $C, \\alpha, p>0$. \n Moreover, examples in \\cite{letrouit2025unstableoptimaltransportmaps} indicates that both unboundedness of the density and openness of $S$ may cause instability of optimal transport maps.\n\n\\subsection{Main results}\n Our main theorem concerns the quantitative $L^1$ stability of Kantorovich potentials on ${\\rm RCD}(K,N)$ spaces.\n\nIf $(S, \\d, \\mm)$ is a compact $\\rcdkn$ space, it supports a global $(1,1)$-Poincar\\'e inequality (cf.\\cite{zbMATH01474795, zbMATH06043352}) and we can remove the John domain assumption:\n \\begin{corollary}\\label{cor}\n Let $(X,\\d,\\mm)$ be a compact ${\\rm RCD}(K,N)$ metric measure space. Let $\\rho\\in \\mathcal{P}(X)$ be with $a_1\\mm \\leq \\rho\\leq a_2\\mm$ for some positive constants $a_1,a_2$. Then the conclusion of Theorem \\ref{thm0.1} holds.\n \\end{corollary}\n\nAdapting the strategy of \\cite{arXiv:2504.05412}, we can also prove the stability of optimal transport maps on Alexandrov spaces.\n \\begin{theorem}\\label{thm0.2}\n Let $(X, \\d)$ be an $n$-dimensional Alexandrov space, $\\mm$ be the $n$-Hausdorff measure. Under the same assumption for $S, Y$ and $\\rho$ as in Theorem \\ref{thm0.1}, and if $S$ additionally has finite perimeter, then there exists a constant $C>0$, depending on $k,n,a_1, a_2, \\diam(S\\cup Y), {\\rm{Per}}(S), S$, such that for any $\\mu,\\nu\\in\\mathcal{P}_2(Y)$, we have\n \\begin{equation}\n \\int_S \\d^2(T_\\mu(x), T_\\nu(x))\\,\\d \\rho(x) \\leq CW_1^{1/6}(\\mu, \\nu),\n \\end{equation}\n where $T_\\mu$ and $T_\\nu$ are the optimal transport maps from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively.\n \\end{theorem}\n\n\\begin{proposition}\\label{2.11}\n It holds that\n \\begin{equation}\n \\int_{S}\\big| \\E_{\\mu_x^t[\\psi_*]}(v)-\\E_{\\mu^t[\\psi_*]}(v))\\big|\\,\\d\\rho(x)\\leq \\frac{\\kappa}{\\sqrt{t}}\\left(\\int_S {\\rm{Var}}_{ \\mu_x^t[\\psi_*]}(v)\\,\\d\\rho(x)\\right)^\\frac{1}{2},\n \\end{equation}\n where $\\mu^t[\\psi_*]$ is defined as in \\eqref{mu} and $\\kappa$ depends on $K,N,\\Lambda_\\psi,a_1,a_2, S$.\n \\end{proposition}\n \\begin{proof}\n Since $a_1\\mm\\leq \\rho\\leq a_2\\mm$, by \\cite[Corollary 2.4]{zbMATH05049052}, $\\rho$ is a doubling measure. Moreover, \n since $S$ is a John domain, by \\cite[Proposition 3.7]{arXiv:2504.05412}, $\\rho$ satisfies the Boman chain condition (see Definition \\ref{Boman chain condition}) and we can choose a covering $\\mathcal{F}$, such that for any $B\\in \\mathcal{F}$, $r_B\\leq 1$. Then \n \\begin{equation}\n \\begin{aligned}\n &\\int_{S}\\big| \\E_{\\mu_x^t[\\psi_*]}(v)-\\E_{\\mu^t[\\psi_*]}(v))\\big|\\,\\d\\rho(x)\\\\\n \\mathop{\\leq}^{\\text {Lemma \\ref{3.1}}} &C_5 \\sum_{B\\in\\mathcal{F}}\\rho(B)\\int_B \\big| \\E_{\\mu_x^t[\\psi_*]}(v)-{\\rho(B)}^{-1}\\E_{\\mu^t[\\psi_*]}(v))\\big|\\,\\d\\rho_B(x)\\\\\n \\mathop{\\leq }^{\\text{Lemma \\ref{p3.10}}}&C_5 \\sum_{B\\in\\mathcal{F}}\\rho(B)C_4 r_B\\int_B \\big|\\nabla_x \\E_{\\mu_x^t[\\psi_*]}(v)\\big|\\,\\d\\rho_B(x)\\\\\n \\mathop{\\leq}^{\\text{Proposition \\ref{3.9}} }& \\frac{C_3C_4C_5}{\\sqrt{t}}\\sum_{B\\in\\mathcal{F}}\\sqrt{\\mm(B_{2r_B})}\\sqrt{r_B}\\left(\\int_B {\\rm{Var}}_{ \\mu_x^t[\\psi_*]}(v)\\,\\d\\rho(x)\\right)^\\frac{1}{2}\\\\\n \\mathop{\\leq}^{\\text{H\\\"{o}lder}~~}&\\frac{C_3C_4C_5}{\\sqrt{t}}\\left(\\sum_{B\\in\\mathcal{F}}\\mm(B_{2r_B})\\right)^\\frac{1}{2}\\left(\\sum_{B\\in\\mathcal{F}}\\int_B {\\rm{Var}}_{ \\mu_x^t[\\psi_*]}(v)\\,\\d\\rho(x)\\right)^\\frac{1}{2}\\\\\n \\overset{\\text{Boman chain~}}{\\leq}&\\frac{\\kappa}{\\sqrt{t}}\\left(\\int_S {\\rm{Var}}_{ \\mu_x^t[\\psi_*]}(v)\\,\\d\\rho(x)\\right)^\\frac{1}{2}\n \\end{aligned}\n \\end{equation}\n which is the thesis.\n \\end{proof}\n\n\\begin{theorem}\\label{thm2}\n Let $(X, \\d)$ be an Alexandrov space with no boundary. Then under the same assumptions of Theorem \\ref{thm0.2}, there exists a constant $\\bar{C}>0$, depending on $k,n,a_1, a_2, \\diam(S\\cup Y), {\\rm{Per}}(S), S$, such that for any $\\mu,\\nu\\in\\mathcal{P}_2(Y)$, we have \n \\begin{equation}\n \\int_S |\\nabla\\phi_\\mu(x)-\\nabla\\phi_\\nu(x)|^2\\,\\d\\rho(x)\\leq \\bar{C} \\left(\\int_S|\\phi_\\mu(x)-\\phi_\\nu(x)|^2\\,\\d\\rho(x)\\right)^\\frac{1}{3},\n \\end{equation}\n where $\\phi_\\mu$ and $\\phi_\\nu$ are the Kantorovich potentials from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively.\n \\end{theorem}\n\n\\begin{lemma}[\\cite{zbMATH07794624}, Lemma 5.1]\\label{l0.18}\n Let $I\\subseteq \\R$ be a compact segment and let $u,v: I\\rightarrow\\R$ be two convex functions such that $|u'|$ and $|v'|$ (defined a.e. on $I$) are uniformly bounded on $I$. Then\n \\begin{equation*}\n \\|u'-v'\\|_{L^2(I)}^2\\leq 8 (\\|u'\\|_{L^\\infty(I)}+\\|v'\\|_{L^\\infty(I)})^\\frac{4}{3}\\|u-v\\|_{L^2(I)}^\\frac{2}{3}.\n \\end{equation*}\n \\end{lemma}\n Therefore, applying Lemma \\ref{l0.18} to the functions $\\zeta|s|^2-u_\\mu^{(x,v)}$ and $\\zeta|s|^2-u_\\nu^{(x,v)}$ on each compact segment $[\\alpha_i(x,v),\\beta_i(x,v)]$ and using H\\\"{o}lder inequality, we obtain\n \\begin{equation}\\label{0.55}\n \\begin{aligned}\n &\\int_S |\\nabla\\phi_\\mu(x)-\\nabla\\phi_\\nu(x)|^2\\,\\d\\rho(x)\\\\\n \\mathop{\\leq}^{\\eqref{0.54}}& \\frac{a_2}{c_nd_n}\\int_{{\\rm S}X}\\sum_{i\\in I_S(x,v)}\\int_{\\alpha_i(x,v)}^{\\beta_i(x,v)}\\langle\\nabla\\phi_\\mu(b_s(x,v))-\\nabla\\phi_\\nu(b_s(x,v)),t_s(x,v)\\rangle^2\\,\\d s \\,\\d\\mm_S(x,v)\\\\\n \\leq&C_1 \\int_{{\\rm S}X}\\sum_{i\\in I_S(x,v)}\\left(\\int_{\\alpha_i(x,v)}^{\\beta_i(x,v)}|\\phi_\\mu(b_s(x,v))-\\phi_\\nu(b_s(x,v))|^2\\,\\d s\\right)^\\frac{1}{3}\\,\\d\\mm_S(x,v)\\\\\n \\leq&C_1 \\int_{{\\rm S}X}\\left(\\# I_S\\right)^\\frac{2}{3}\\left(\\int_0^1|\\phi_\\mu(b_s)-\\phi_\\nu(b_s)|^2\\chi_S(b_s)\\,\\d s\\right)^\\frac{1}{3}\\d\\mm_S\\\\\n \\leq&C_1 \\left(\\int_{{\\rm S}X}\\# I_S\\,\\d\\mm_S\\right)^\\frac{2}{3}\\left(\\int_{{\\rm S}X}\\int_0^1|\\phi_\\mu(b_s)-\\phi_\\nu(b_s)|^2\\chi_S(b_s)\\,\\d s \\,\\d\\mm_S\\right)^\\frac{1}{3}\\\\\n \\leq&C_1 \\left(\\frac{c_n}{a_1}\\right)^\\frac{1}{3}\\left(\\int_{{\\rm S}X}\\# I_S\\,\\d\\mm_S\\right)^\\frac{2}{3}\\left(\\int_S|\\phi_\\mu-\\phi_\\nu|^2\\,\\d\\rho\\right)^\\frac{1}{3},\n \\end{aligned}\n \\end{equation}\n where $C_1$ depends on $k,n,a_2, \\diam(S\\cup Y)$, and the last inequality follows from the invariance of $\\mm_S$ under the geodesic flow.\n\nLet $\\phi_\\mu$ and $\\phi_\\nu$ be the Kantorovich potentials from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively. From \\cite{zbMATH05349267,zbMATH06032507} we know that $\\nabla \\phi_\\mu(x), \\nabla \\phi_\\nu(x) \\in {\\rm{T}}_xX$ and\n $T_\\mu(x)= \\exp_x(-\\nabla \\phi_\\mu(x))$, $T_\\nu(x)= \\exp_x(-\\nabla \\phi_\\nu(x))$ are well-defined for almost every $x\\in S$. By triangle comparison condition (cf. \\cite{zbMATH05342782}), it holds\n \\begin{equation}\\label{exp}\n \\d \\big(T_\\mu(x), T_\\nu(x) \\big) \\leq C_2|\\nabla \\phi_\\mu(x)-\\nabla \\phi_\\nu(x)|,\n \\end{equation}\n for some constant $C_2>0$ which depends on $k, \\diam(S\\cup Y)$. \n \\medskip", "post_theorem_intro_text_len": 3816, "post_theorem_intro_text": "If $(S,  {\\mathrm d},  \\mathfrak m)$  is a compact ${\\rm RCD}(K, N)$ space,   it  supports a global $(1,1)$-Poincar\\'e inequality (cf.\\cite{zbMATH01474795, zbMATH06043352}) and we can remove the John domain assumption:\n\t\\begin{corollary}\\label{cor}\n\t\tLet $(X,{\\mathrm d},\\mathfrak m)$ be a compact ${\\rm RCD}(K,N)$ metric measure space.  Let $\\rho\\in \\mathcal{P}(X)$ be with $a_1\\mathfrak m \\leq \\rho\\leq a_2\\mathfrak m$ for some positive constants $a_1,a_2$. Then the conclusion of Theorem \\ref{thm0.1} holds.\n\t\\end{corollary}\n\n\tAdapting the  strategy of \\cite{arXiv:2504.05412}, we can also prove the stability of optimal transport maps on Alexandrov spaces.\n\t\t\\begin{theorem}\\label{thm0.2}\n\t\tLet $(X,  {\\mathrm d})$ be an $n$-dimensional Alexandrov space, $\\mathfrak m$ be the $n$-Hausdorff measure. Under the same assumption for $S, Y$ and $\\rho$ as in Theorem \\ref{thm0.1}, and if $S$ additionally has finite perimeter, then there exists a constant $C>0$, depending on $k,n,a_1, a_2, \\mathop{\\rm diam}\\nolimits(S\\cup Y), {\\rm{Per}}(S), S$, such that for any $\\mu,\\nu\\in\\mathcal{P}_2(Y)$, we have\n\t\t\\begin{equation}\n\t\t\t\\int_S {\\mathrm d}^2(T_\\mu(x), T_\\nu(x))\\,{\\mathrm d} \\rho(x) \\leq CW_1^{1/6}(\\mu, \\nu),\n\t\t\\end{equation}\n\t\twhere $T_\\mu$ and $T_\\nu$ are the optimal transport maps from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively.\n\t\\end{theorem}\n\n\t\\subsection{Strategy: heat kernel-regularized \\texorpdfstring{$c$}{c}-transform}\n\nMotivated by   regularized $c$-transforms  using  Gibbs kernels $e^{-c(x, y)/\\varepsilon}$ \\cite{arXiv:2504.05412},  entropic optimal transport \\cite{GTJEMS, GTT25}  and Varadhan's formula\n\\[\n\\lim_{t \\to 0} -t \\log p_{t/2}(x, y) = \\frac{1}{2} {\\mathrm d}^2(x, y)=c(x,y),\n\\]\n  we make use of {heat kernel-regularized $c$-transform} (Hopf--Cole transform):\n\\[\n\\mathop{\\rm Lip}\\nolimits(X, {\\mathrm d}) \\ni \\psi  \\mapsto \\Phi_t[\\psi](x) = -t \\log \\int_X e^{\\frac{\\psi(y)}{t}} p_{t/2}(x, y) \\, {\\mathrm d}\\mathfrak m(y).\n\\]\nThis approach  allows us to bypass the low regularity of Kantorovich potentials in the non-smooth setting. \n\nWe define the heat kernel regularized Kantorovich functional as\n\t\\begin{equation*}\n\t\t\\K_t[\\varphi]:=\\int_S \\Phi_t[\\varphi]\\,{\\mathrm d}\\rho.\n\t\\end{equation*}\nSimilar to \\cite{arXiv:2504.05412},  the key in our proof is the strong concavity of the functional $\\K_t$.\nTo achieve this,  we  first derive a local strong concavity estimate using heat kernel estimate,    then globalize the estimate on the support of the source measure using a Boman chain argument for John domains.\n\nUnlike the regularized $c$-transform used in \\cite{arXiv:2504.05412},  the existence of the boundary of $Y$ may lead to the failure in our heat kernel regularization argument.  This possibility is ruled out by using the measure concentration property of the heat kernel and by making a careful choice of Lipschitz extension.\n\n\\medskip\n\n\t\\noindent {\\bf Organization.}  This paper is structured as follows. In Section~\\ref{sec3}, we prove quantitative stability of Kantorovich potentials on $\\mathrm{RCD}(K,N)$ spaces. In Section \\ref{sec4}, we establish stability of optimal transport maps on Alexandrov spaces. The Appendix \\ref{A} contains technical lemmas about Poincar\\'e inequalities.\n\n\t\\bigskip\n\n\\noindent {\\bf Declaration.} The authors declare no conflict of interest and that the manuscript has no associated data.\n\n\\medskip\n\n\\noindent {\\bf Acknowledgement.} This work is supported by the Ministry of Science \\& Technology of China (2021YFA1000900, 2021YFA1002200),  National Natural Science Foundation of China  (12201596)   and Shandong Provincial Natural Science Foundation (ZR2025QB05).  The authors  thank Nicola Gigli, Jun Kitagawa, Nan Li,  Cyril Letrouit, Quentin M\\'erigot, Luca Tamanini for  helpful discussions and  suggestions on the bibliography.", "sketch": "Motivated by “regularized $c$-transforms … entropic optimal transport … and Varadhan's formula”, the proof of Theorem~\\ref{thm0.1} uses a **heat kernel-regularized $c$-transform** (Hopf--Cole transform)\n\\[\n\\psi\\mapsto \\Phi_t[\\psi](x)=-t\\log\\int_X e^{\\frac{\\psi(y)}{t}}\\,p_{t/2}(x,y)\\,\\mathrm d\\mathfrak m(y),\n\\]\nwhich “allows us to bypass the low regularity of Kantorovich potentials in the non-smooth setting.” One then defines the “heat kernel regularized Kantorovich functional”\n\\[\n\\mathcal K_t[\\varphi]:=\\int_S \\Phi_t[\\varphi]\\,\\mathrm d\\rho.\n\\]\n“Similar to \\cite{arXiv:2504.05412}, the key in our proof is the strong concavity of the functional $\\mathcal K_t$.” To obtain this strong concavity, they “first derive a local strong concavity estimate using heat kernel estimate, then globalize the estimate on the support of the source measure using a Boman chain argument for John domains.”\n\nThey also note a technical issue specific to this regularization: “the existence of the boundary of $Y$ may lead to the failure in our heat kernel regularization argument,” and this is “ruled out by using the measure concentration property of the heat kernel and by making a careful choice of Lipschitz extension.”", "expanded_sketch": "Motivated by “regularized $c$-transforms … entropic optimal transport … and Varadhan's formula”, in establishing the main theorem, the proof uses a **heat kernel-regularized $c$-transform** (Hopf--Cole transform)\n\\[\n\\psi\\mapsto \\Phi_t[\\psi](x)=-t\\log\\int_X e^{\\frac{\\psi(y)}{t}}\\,p_{t/2}(x,y)\\,\\mathrm d\\mathfrak m(y),\n\\]\nwhich “allows us to bypass the low regularity of Kantorovich potentials in the non-smooth setting.” One then defines the “heat kernel regularized Kantorovich functional”\n\\[\n\\mathcal K_t[\\varphi]:=\\int_S \\Phi_t[\\varphi]\\,\\mathrm d\\rho.\n\\]\n“Similar to arXiv:2504.05412, the key in our proof is the strong concavity of the functional $\\mathcal K_t$.” To obtain this strong concavity, they “first derive a local strong concavity estimate using heat kernel estimate, then globalize the estimate on the support of the source measure using a Boman chain argument for John domains.”\n\nThey also note a technical issue specific to this regularization: “the existence of the boundary of $Y$ may lead to the failure in our heat kernel regularization argument,” and this is “ruled out by using the measure concentration property of the heat kernel and by making a careful choice of Lipschitz extension.”,", "expanded_theorem": "\\label{thm0.1}\n\t\tLet $(X,{\\mathrm d},\\mathfrak m)$ be an ${\\rm RCD}(K,N)$ metric measure space. Let $S\\subseteq X$ be a John domain and $Y\\subseteq X$ be compact with $\\mathfrak m(Y)>0$. Let $\\rho\\in \\mathcal{P}(S)$  be with $a_1\\mathfrak m\\restr{S} \\leq \\rho\\leq a_2\\mathfrak m\\restr{S}$ for some positive constants $a_1,a_2$. Then there exists a constant $C>0$, depending on $K,N,a_1,a_2, S, \\mathop{\\rm diam}\\nolimits(S\\cup Y)$, such that for any $\\mu,\\nu\\in\\mathcal{P}(Y)$, \n\t\t\\begin{equation}\n\t\t\t\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}\\leq C W_1^\\frac{1}{2}(\\mu,\\nu),\n\t\t\\end{equation}\n\t\twhere $\\phi_\\mu$ and $\\phi_\\nu$ are the Kantorovich potentials from $\\rho$ to $\\mu$ and $\\rho$ to $\\nu$ respectively.,", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let \\((X,{\\mathrm d},\\mathfrak m)\\) be an \\({\\rm RCD}(K,N)\\) metric measure space. Let \\(S\\subseteq X\\) be a John domain, meaning that \\(S\\) is a bounded open set for which there exist \\(x_0\\in S\\) and \\(\\eta>0\\) such that every \\(x\\in S\\) can be joined to \\(x_0\\) by a rectifiable curve \\(\\gamma:[0,\\ell(\\gamma)]\\to S\\), parametrized by arc length, with \\(\\gamma(0)=x\\), \\(\\gamma(\\ell(\\gamma))=x_0\\), and \\({\\mathrm d}(\\gamma(t),X\\setminus S)\\ge \\eta t\\) for all \\(t\\in[0,\\ell(\\gamma)]\\). Let \\(Y\\subseteq X\\) be compact with \\(\\mathfrak m(Y)>0\\). Assume \\(\\rho\\in \\mathcal P(S)\\) satisfies \\(a_1\\,\\mathfrak m\\!\\restriction_S \\le \\rho \\le a_2\\,\\mathfrak m\\!\\restriction_S\\) for some constants \\(a_1,a_2>0\\). For each \\(\\mu\\in\\mathcal P(Y)\\), let \\(\\phi_\\mu\\) denote a Kantorovich potential for the quadratic cost \\(c(x,y)=\\tfrac12{\\mathrm d}(x,y)^2\\) from \\(\\rho\\) to \\(\\mu\\), normalized by \\(\\int_S \\phi_\\mu\\,{\\mathrm d}\\rho=0\\). Also let\n\\[\nW_1(\\mu,\\nu):=\\inf_{\\pi\\in\\Pi(\\mu,\\nu)}\\int_{Y\\times Y}{\\mathrm d}(y,z)\\,{\\mathrm d}\\pi(y,z)\n\\]\nbe the 1-Wasserstein distance. Which quantitative estimate holds for all \\(\\mu,\\nu\\in\\mathcal P(Y)\\)?", "correct_choice": {"label": "A", "text": "There exists a constant \\(C>0\\), depending only on \\(K,N,a_1,a_2,S\\), and \\(\\operatorname{diam}(S\\cup Y)\\), such that for every \\(\\mu,\\nu\\in\\mathcal P(Y)\\),\n\\[\n\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}=\\int_S |\\phi_\\mu-\\phi_\\nu|\\,{\\mathrm d}\\rho \\le C\\,W_1(\\mu,\\nu)^{1/2}.\n\\]"}, "choices": [{"label": "B", "text": "There exists a constant \\(C>0\\), depending only on \\(K,N,a_1,a_2,S\\), and \\(\\operatorname{diam}(S\\cup Y)\\), such that for every \\(\\mu,\\nu\\in\\mathcal P(Y)\\),\n\\[\n\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}=\\int_S |\\phi_\\mu-\\phi_\\nu|\\,{\\mathrm d}\\rho \\le C\\,W_1(\\mu,\\nu).\n\\]"}, {"label": "C", "text": "There exists a constant \\(C>0\\), depending only on \\(K,N,a_1,a_2,S\\), and \\(\\operatorname{diam}(S\\cup Y)\\), such that for every \\(\\mu,\\nu\\in\\mathcal P(Y)\\),\n\\[\n\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}=\\int_S |\\phi_\\mu-\\phi_\\nu|\\,{\\mathrm d}\\rho \\le C.\n\\]"}, {"label": "D", "text": "There exists a constant \\(C>0\\), depending only on \\(K,N,a_1,a_2\\), such that for every John domain \\(S\\subseteq X\\), every compact \\(Y\\subseteq X\\) with \\(\\mathfrak m(Y)>0\\), and every \\(\\mu,\\nu\\in\\mathcal P(Y)\\),\n\\[\n\\|\\phi_\\mu-\\phi_\\nu\\|_{L^1(\\rho)}=\\int_S |\\phi_\\mu-\\phi_\\nu|\\,{\\mathrm d}\\rho \\le C\\,W_1(\\mu,\\nu)^{1/2}.\n\\]"}, {"label": "E", "text": "There exists a constant \\(C>0\\), depending only on \\(K,N,a_1,a_2,S\\), and \\(\\operatorname{diam}(S\\cup Y)\\), such that for every \\(\\mu,\\nu\\in\\mathcal P(Y)\\),\n\\[\n\\|\\phi_\\mu-\\phi_\\nu\\|_{L^\\infty(S)}\\le C\\,W_1(\\mu,\\nu)^{1/2}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "sharp_holder_exponent_one_half", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "wasserstein_rate_factor", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "dependence_on_domain_geometry_and_diameter", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "L1_control_replaced_by_Linfty_control", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the target estimate or reveal the correct option. It gives the hypotheses and asks for the valid conclusion, without obvious verbal cues favoring A."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem lays out the full hypotheses and asks which estimate holds. The correct answer is basically the theorem's conclusion rather than a genuinely new inference from competing principles."}, "GPS": {"score": 1, "justification": "Some discrimination is required among exponent, norm, and parameter-dependence choices, especially between A and the plausible stronger/weaker variants. However, the task is primarily recognition/recall of the sharp statement, not substantial generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically meaningful and target common failure modes: overstrong Lipschitz rate (B), weaker but non-sharp bound (C), incorrect uniformity in constants (D), and an unjustified upgrade from L1 to L∞ control (E)."}, "total_score": 5, "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and little answer leakage, but it is mostly a direct restatement of a known result rather than a deep reasoning question."}}
{"id": "2602.19195v1", "paper_link": "http://arxiv.org/abs/2602.19195v1", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "\\label{theorem1.01}\n(Zagier \\cite{[Z-AM]}) For non-negative integers $a$ and $b$,\n\\begin{align}\nH(a,b)=2\\sum_{k=1}^{a+b+1}(-1)^kc_{a,b}\\zeta(2k+1)\n\\zeta(\\{2\\}^{a+b+1-k}),\n\\end{align}\nwhere\n$$c_{a,b}:=\\binom{2k}{2a+2}-\\Big(1-\\frac{1}{2^{2k}}\\Big)\\binom{2k}{2b+1}.$$", "start_pos": 5505, "end_pos": 5798, "label": "theorem1.01"}, "ref_dict": {"eq1.1": "\\begin{align}\\label{eq1.1}\nK(a,b)=\\sum_{k=1}^{a+b+1}(-1)^{k-1}d_{a,b}K(a+b-k+1)\\zeta(2k+1),\n\\end{align}", "theorem1.01": "\\begin{theorem}\\label{theorem1.01}\n(Zagier \\cite{[Z-AM]}) For non-negative integers $a$ and $b$,\n\\begin{align}\nH(a,b)=2\\sum_{k=1}^{a+b+1}(-1)^kc_{a,b}\\zeta(2k+1)\n\\zeta(\\{2\\}^{a+b+1-k}),\n\\end{align}\nwhere\n$$c_{a,b}:=\\binom{2k}{2a+2}-\\Big(1-\\frac{1}{2^{2k}}\\Big)\\binom{2k}{2b+1}.$$\n\\end{theorem}", "theorem1.2": "\\begin{theorem}\\label{theorem1.2}\nFor nonnegative integers $a$ and $b$, we have $T(a,b)=\\hat L(a,b)$, where\n\\begin{align*}\n\\hat L(a,b):=\\frac{-2}{(2a+1)!}\\left(\\frac{\\pi}{2}\\right)^{2a+2b+2}\n\\sum_{n=0}^{\\infty}\\frac{\\zeta(2n)}{(2n+2a+1)(2n+2a+2)\\cdots(2n+2a+2b+2)2^{2n}}.\n\\end{align*}\n\\end{theorem}", "con1": "\\begin{conjecture}\\label{con1}\nFor nonnegative integers $a$ and $b$, we have\n\\begin{align}\nH(a,b)=\\frac{-4\\pi^{2a+2b+2}}{(2a+2)!}\\sum_{n=0}^{\\infty}\n\\frac{\\zeta(2n)}{(2n+2a+2)(2n+2a+3)\\cdots(2n+2a+2b+3)2^{2n}}.\n\\end{align}\n\\end{conjecture}", "con2": "\\begin{conjecture}\\label{con2}\nFor nonnegative integers $a$ and $b$, we have\n\\begin{align}\nT(a,b)=\\frac{-2}{(2a+1)!}\\left(\\frac{\\pi}{2}\\right)^{2a+2b+2}\n\\sum_{n=0}^{\\infty}\\frac{\\zeta(2n)}{(2n+2a+1)(2n+2a+2)\\cdots(2n+2a+2b+2)2^{2n}}.\n\\end{align}\n\\end{conjecture}", "theorem1.1": "\\begin{theorem}\\label{theorem1.1}\nFor nonnegative integers $a$ and $b$, we have $H(a,b)=L(a,b)$, where\n\\begin{align*}\nL(a,b):=\\frac{-4\\pi^{2a+2b+2}}{(2a+2)!}\\sum_{n=0}^{\\infty}\n\\frac{\\zeta(2n)}{(2n+2a+2)(2n+2a+3)\\cdots(2n+2a+2b+3)2^{2n}}.\n\\end{align*}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 1592, "pre_theorem_intro_text": "The systematic study of multiple zeta values (MZVs) started\nin the early 1990s with the work of Hoffman \\cite{[H-PJM]}\nand Zagier \\cite{[Z-FECM]}. Let $r\\ge 1$ be an integer. For any multiple index\n$\\mathbf{s}=(s_1,s_2,\\cdots,s_r)\\in \\mathbb{Z}_{\\geq 1}^r$ with\n$s_r\\textgreater 1$, the {\\it multiple zeta value (MZV for short)}\nis defined by\n\\begin{align*}\n\\zeta(s_1,s_2,\\cdots,s_r):=\\sum_{1\\leq k_1<k_2<\\cdots<k_r}\n\\frac{1}{k_1^{s_1}k_2^{s_2}\\cdots k_r^{s_r}}.\n\\end{align*}\nThe {\\it weight} of MZV is defined to be the number\n$|\\mathbf{s}|:=s_1+\\cdots+s_r$, and its {\\it depth} is defined\nby $l(\\mathbf{s}):=r$. For brevity, we let $\\{a_1,\\cdots,a_k\\}^d$\ndenote the sequence formed by repeating the sequence\n$\\{a_1,\\cdots,a_k\\}$ exactly $d$ times. The simplest precise\nevaluations of MZVs and MZSVs are given by\n\\begin{eqnarray*}\n\\zeta(\\{2\\}^d)=\\frac{\\pi^{2d}}{(2d+1)!}.\n\\end{eqnarray*}\n\nA central problem in MZV is to investigate the algebraic structure\nof the $\\mathbb{Q}$-vector space $\\mathfrak{Z}_k$ spanned by MZVs\nof weight $k$. In this direction, Hoffman \\cite{[H-JA]} proposed a\nconjecture that stats that every MZV of weight $k$ can be expressed as a\n$\\mathbb{Q}$-linear combination of those MZVs whose entries are only\n$2$'s and $3$'s. This conjecture was proved by Brown \\cite{[B-ANN]}\nusing motivic method. In particular, Brown showed that\n$H(a,b):=\\zeta(\\{2\\}^a,3,\\{2\\}^b)$ can be expressed as a $\\mathbb{Q}$-linear\ncombination of the products $\\pi^{2m}\\zeta(2n+1)$ with $m+n=a+b+1$.\nIn 2012, Zagier \\cite{[Z-AM]} found surprisingly an elegant and explicit\nformula as follows.", "context": "The systematic study of multiple zeta values (MZVs) started\nin the early 1990s with the work of Hoffman \\cite{[H-PJM]}\nand Zagier \\cite{[Z-FECM]}. Let $r\\ge 1$ be an integer. For any multiple index\n$\\mathbf{s}=(s_1,s_2,\\cdots,s_r)\\in \\mathbb{Z}_{\\geq 1}^r$ with\n$s_r\\textgreater 1$, the {\\it multiple zeta value (MZV for short)}\nis defined by\n\\begin{align*}\n\\zeta(s_1,s_2,\\cdots,s_r):=\\sum_{1\\leq k_1<k_2<\\cdots<k_r}\n\\frac{1}{k_1^{s_1}k_2^{s_2}\\cdots k_r^{s_r}}.\n\\end{align*}\nThe {\\it weight} of MZV is defined to be the number\n$|\\mathbf{s}|:=s_1+\\cdots+s_r$, and its {\\it depth} is defined\nby $l(\\mathbf{s}):=r$. For brevity, we let $\\{a_1,\\cdots,a_k\\}^d$\ndenote the sequence formed by repeating the sequence\n$\\{a_1,\\cdots,a_k\\}$ exactly $d$ times. The simplest precise\nevaluations of MZVs and MZSVs are given by\n\\begin{eqnarray*}\n\\zeta(\\{2\\}^d)=\\frac{\\pi^{2d}}{(2d+1)!}.\n\\end{eqnarray*}\n\nA central problem in MZV is to investigate the algebraic structure\nof the $\\mathbb{Q}$-vector space $\\mathfrak{Z}_k$ spanned by MZVs\nof weight $k$. In this direction, Hoffman \\cite{[H-JA]} proposed a\nconjecture that stats that every MZV of weight $k$ can be expressed as a\n$\\mathbb{Q}$-linear combination of those MZVs whose entries are only\n$2$'s and $3$'s. This conjecture was proved by Brown \\cite{[B-ANN]}\nusing motivic method. In particular, Brown showed that\n$H(a,b):=\\zeta(\\{2\\}^a,3,\\{2\\}^b)$ can be expressed as a $\\mathbb{Q}$-linear\ncombination of the products $\\pi^{2m}\\zeta(2n+1)$ with $m+n=a+b+1$.\nIn 2012, Zagier \\cite{[Z-AM]} found surprisingly an elegant and explicit\nformula as follows.", "full_context": "The systematic study of multiple zeta values (MZVs) started\nin the early 1990s with the work of Hoffman \\cite{[H-PJM]}\nand Zagier \\cite{[Z-FECM]}. Let $r\\ge 1$ be an integer. For any multiple index\n$\\mathbf{s}=(s_1,s_2,\\cdots,s_r)\\in \\mathbb{Z}_{\\geq 1}^r$ with\n$s_r\\textgreater 1$, the {\\it multiple zeta value (MZV for short)}\nis defined by\n\\begin{align*}\n\\zeta(s_1,s_2,\\cdots,s_r):=\\sum_{1\\leq k_1<k_2<\\cdots<k_r}\n\\frac{1}{k_1^{s_1}k_2^{s_2}\\cdots k_r^{s_r}}.\n\\end{align*}\nThe {\\it weight} of MZV is defined to be the number\n$|\\mathbf{s}|:=s_1+\\cdots+s_r$, and its {\\it depth} is defined\nby $l(\\mathbf{s}):=r$. For brevity, we let $\\{a_1,\\cdots,a_k\\}^d$\ndenote the sequence formed by repeating the sequence\n$\\{a_1,\\cdots,a_k\\}$ exactly $d$ times. The simplest precise\nevaluations of MZVs and MZSVs are given by\n\\begin{eqnarray*}\n\\zeta(\\{2\\}^d)=\\frac{\\pi^{2d}}{(2d+1)!}.\n\\end{eqnarray*}\n\nA central problem in MZV is to investigate the algebraic structure\nof the $\\mathbb{Q}$-vector space $\\mathfrak{Z}_k$ spanned by MZVs\nof weight $k$. In this direction, Hoffman \\cite{[H-JA]} proposed a\nconjecture that stats that every MZV of weight $k$ can be expressed as a\n$\\mathbb{Q}$-linear combination of those MZVs whose entries are only\n$2$'s and $3$'s. This conjecture was proved by Brown \\cite{[B-ANN]}\nusing motivic method. In particular, Brown showed that\n$H(a,b):=\\zeta(\\{2\\}^a,3,\\{2\\}^b)$ can be expressed as a $\\mathbb{Q}$-linear\ncombination of the products $\\pi^{2m}\\zeta(2n+1)$ with $m+n=a+b+1$.\nIn 2012, Zagier \\cite{[Z-AM]} found surprisingly an elegant and explicit\nformula as follows.\n\n\\section{Introduction}\nThe systematic study of multiple zeta values (MZVs) started\nin the early 1990s with the work of Hoffman \\cite{[H-PJM]}\nand Zagier \\cite{[Z-FECM]}. Let $r\\ge 1$ be an integer. For any multiple index\n$\\mathbf{s}=(s_1,s_2,\\cdots,s_r)\\in \\mathbb{Z}_{\\geq 1}^r$ with\n$s_r\\textgreater 1$, the {\\it multiple zeta value (MZV for short)}\nis defined by\n\\begin{align*}\n\\zeta(s_1,s_2,\\cdots,s_r):=\\sum_{1\\leq k_1<k_2<\\cdots<k_r}\n\\frac{1}{k_1^{s_1}k_2^{s_2}\\cdots k_r^{s_r}}.\n\\end{align*}\nThe {\\it weight} of MZV is defined to be the number\n$|\\mathbf{s}|:=s_1+\\cdots+s_r$, and its {\\it depth} is defined\nby $l(\\mathbf{s}):=r$. For brevity, we let $\\{a_1,\\cdots,a_k\\}^d$\ndenote the sequence formed by repeating the sequence\n$\\{a_1,\\cdots,a_k\\}$ exactly $d$ times. The simplest precise\nevaluations of MZVs and MZSVs are given by\n\\begin{eqnarray*}\n\\zeta(\\{2\\}^d)=\\frac{\\pi^{2d}}{(2d+1)!}.\n\\end{eqnarray*}\n\nA central problem in MZV is to investigate the algebraic structure\nof the $\\mathbb{Q}$-vector space $\\mathfrak{Z}_k$ spanned by MZVs\nof weight $k$. In this direction, Hoffman \\cite{[H-JA]} proposed a\nconjecture that stats that every MZV of weight $k$ can be expressed as a\n$\\mathbb{Q}$-linear combination of those MZVs whose entries are only\n$2$'s and $3$'s. This conjecture was proved by Brown \\cite{[B-ANN]}\nusing motivic method. In particular, Brown showed that\n$H(a,b):=\\zeta(\\{2\\}^a,3,\\{2\\}^b)$ can be expressed as a $\\mathbb{Q}$-linear\ncombination of the products $\\pi^{2m}\\zeta(2n+1)$ with $m+n=a+b+1$.\nIn 2012, Zagier \\cite{[Z-AM]} found surprisingly an elegant and explicit\nformula as follows.\n\nIn a similar way, Hoffman \\cite{[H-CNTP]} defined the multiple\n$t$-values (``odd variant'' of MZVs). For any multiple index\n$\\mathbf{s}=(s_1,s_2,...,s_r)\\in \\mathbb{Z}_{\\geq 1}^r$, the\n{\\it multiple $t$-value } is defined by\n\\begin{align*}\nt(s_1,s_2,...,s_r):=\\sum_{1\\leq k_1<k_2<...<k_r}\n\\frac{1}{(2k_1-1)^{s_1}(2k_2-1)^{s_2}...(2k_r-1)^{s_r}},\n\\end{align*}\nwhere if the index is empty, then we define the value $t(\\emptyset):=1$.\nAs we stated above, first let us recall the the analogue formula\nfor the simplest multiple zeta value (\\cite{[MK-PSPM]},\\cite{[Z-FM]})\n\\begin{align*}\nt(\\mathop{\\underbrace{2,2,...,2}}_{n})=\\frac{\\pi^{2n}}{2^{2n}(2n)!}.\n\\end{align*}\n\nBy the method of establishing the generating functions, Murakami \\cite{[M-MA]}\nobtained an equivalent version of Theorem \\ref{theorem1.01}. Define\n$$\\widetilde{t}(k_1,k_2,...,k_r):=2^{|k|}t(k_1,k_2,...,k_r)$$\nand set\n$$K(a,b):=\\widetilde{t}(\\{2\\}^a,3,\\{2\\}^b),K(n)=\\widetilde{t}(\\{2\\}^n)$$\nfor any $a,b,n\\in \\mathbb{N}_{\\geq 0}$. Then for all integers $a,b\\geq 0$,\nMurakami \\cite{[M-MA]} provided the following formula\n\\begin{align}\\label{eq1.1}\nK(a,b)=\\sum_{k=1}^{a+b+1}(-1)^{k-1}d_{a,b}K(a+b-k+1)\\zeta(2k+1),\n\\end{align}\nwhere\n$$d_{a,b}:=\\binom{2k}{2a+1}+\\binom{2k}{2b+1}\\Big(1-\\frac{1}{2^{2k}}\\Big).$$\nMoreover, let $T(a,b):=t(\\{2\\}^a,3,\\{2\\}^b)$. Then \\eqref{eq1.1} is equivalent to\n\\begin{align}\nT(a,b)=2\\sum_{k=1}^{r+s+1}(-1)^{k-1}d_{a,b}\\frac{1}{2^{2k}}\\zeta(2k+1)\nt(\\{2\\}^{a+b-k+1}),\n\\end{align}\n\n\\begin{conjecture}\\label{con2}\nFor nonnegative integers $a$ and $b$, we have\n\\begin{align}\nT(a,b)=\\frac{-2}{(2a+1)!}\\left(\\frac{\\pi}{2}\\right)^{2a+2b+2}\n\\sum_{n=0}^{\\infty}\\frac{\\zeta(2n)}{(2n+2a+1)(2n+2a+2)\\cdots(2n+2a+2b+2)2^{2n}}.\n\\end{align}\n\\end{conjecture}\n\\noindent Notice that Conjectures 1.2 and 1.3 are shown in \\cite{[L-MZ]}\nto be true when $b=0$. But for the general case $b>0$, these two conjectures\nare still kept open so far.\n\n\\begin{theorem}\\label{theorem1.1}\nFor nonnegative integers $a$ and $b$, we have $H(a,b)=L(a,b)$, where\n\\begin{align*}\nL(a,b):=\\frac{-4\\pi^{2a+2b+2}}{(2a+2)!}\\sum_{n=0}^{\\infty}\n\\frac{\\zeta(2n)}{(2n+2a+2)(2n+2a+3)\\cdots(2n+2a+2b+3)2^{2n}}.\n\\end{align*}\n\\end{theorem}\n\n\\begin{theorem}\\label{theorem1.2}\nFor nonnegative integers $a$ and $b$, we have $T(a,b)=\\hat L(a,b)$, where\n\\begin{align*}\n\\hat L(a,b):=\\frac{-2}{(2a+1)!}\\left(\\frac{\\pi}{2}\\right)^{2a+2b+2}\n\\sum_{n=0}^{\\infty}\\frac{\\zeta(2n)}{(2n+2a+1)(2n+2a+2)\\cdots(2n+2a+2b+2)2^{2n}}.\n\\end{align*}\n\\end{theorem}\n\nFirst, we consider the integral\n$$\\int_{0}^{\\frac{\\pi}{2}} x^{2a + 2}\\left(1 - \\frac{2}{\\pi} x\\right)^{2b + 1} \\cot(x)\\mathrm{d}x.$$\nBy using the well-known identity (see, for example, Theorem 1.2.4 of \\cite{[AAR]})\n\\begin{align}\\label{eq3.1'}\nx\\cot(x)=-2\\sum_{n=0}^{\\infty}\\frac{\\zeta(2n)}{\\pi^{2n}}x^{2n},\n\\end{align}\nwe derive that\n\\begin{align}\\label{eq3.2}\n&\\int_{0}^{\\frac{\\pi}{2}}x^{2a+2}\\Big(1-\\frac{2}{\\pi} x\\Big)^{2b+1}\n\\cot(x)\\mathrm{d}x\\nonumber\\\\\n=&-2\\int_{0}^{\\frac{\\pi}{2}}x^{2a+1}\\Big(1-\\frac{2}{\\pi} x\\Big)^{2b+1}\n\\sum_{n=0}^{\\infty}\\frac{\\zeta(2n)}{\\pi^{2n}}x^{2n} \\mathrm{d}x \\nonumber\\\\\n=&-2\\sum_{n=0}^{\\infty}\\frac{\\zeta(2n)}{\\pi^{2n}}\n\\int_{0}^{\\frac{\\pi}{2}}x^{2n+2a+1}\\Big(1-\\frac{2}{\\pi} x\\Big)^{2b+1}\\mathrm{d}x.\n\\end{align}\n\n\\begin{theorem}\\label{theorem1.01}\n(Zagier \\cite{[Z-AM]}) For non-negative integers $a$ and $b$,\n\\begin{align}\nH(a,b)=2\\sum_{k=1}^{a+b+1}(-1)^kc_{a,b}\\zeta(2k+1)\n\\zeta(\\{2\\}^{a+b+1-k}),\n\\end{align}\nwhere\n$$c_{a,b}:=\\binom{2k}{2a+2}-\\Big(1-\\frac{1}{2^{2k}}\\Big)\\binom{2k}{2b+1}.$$\n\\end{theorem}", "post_theorem_intro_text_len": 4374, "post_theorem_intro_text": "\\noindent The proof of Theorem \\ref{theorem1.01} is given by computing the associated\ngenerating functions of both sides in a closed form, and then showing\nthey are entire functions of exponential growth that agree at\nsufficiently many points to force their equality. Later on, his proof\nwas simplified by Li \\cite{[L-MRL]}. In 2017, Hessami Pilehrood\nand Hessami Pilehrood \\cite{[HHT-JMAA]} gave an alternative proof of Theorem\n\\ref{theorem1.01}.\n\nIn a similar way, Hoffman \\cite{[H-CNTP]} defined the multiple\n$t$-values  (``odd variant'' of MZVs). For any multiple index\n$\\mathbf{s}=(s_1,s_2,...,s_r)\\in \\mathbb{Z}_{\\geq 1}^r$, the\n{\\it multiple $t$-value } is defined by\n\\begin{align*}\nt(s_1,s_2,...,s_r):=\\sum_{1\\leq k_1<k_2<...<k_r}\n\\frac{1}{(2k_1-1)^{s_1}(2k_2-1)^{s_2}...(2k_r-1)^{s_r}},\n\\end{align*}\nwhere if the index is empty, then we define the value $t(\\emptyset):=1$.\nAs we stated above, first let us recall the the analogue formula\nfor the simplest multiple zeta value (\\cite{[MK-PSPM]},\\cite{[Z-FM]})\n\\begin{align*}\nt(\\mathop{\\underbrace{2,2,...,2}}_{n})=\\frac{\\pi^{2n}}{2^{2n}(2n)!}.\n\\end{align*}\n\nBy the method of establishing the generating functions, Murakami \\cite{[M-MA]}\nobtained an equivalent version of Theorem \\ref{theorem1.01}. Define\n$$\\widetilde{t}(k_1,k_2,...,k_r):=2^{|k|}t(k_1,k_2,...,k_r)$$\nand set\n$$K(a,b):=\\widetilde{t}(\\{2\\}^a,3,\\{2\\}^b),K(n)=\\widetilde{t}(\\{2\\}^n)$$\nfor any $a,b,n\\in \\mathbb{N}_{\\geq 0}$. Then for all integers $a,b\\geq 0$,\nMurakami \\cite{[M-MA]} provided the following formula\n\\begin{align}\\label{eq1.1}\nK(a,b)=\\sum_{k=1}^{a+b+1}(-1)^{k-1}d_{a,b}K(a+b-k+1)\\zeta(2k+1),\n\\end{align}\nwhere\n$$d_{a,b}:=\\binom{2k}{2a+1}+\\binom{2k}{2b+1}\\Big(1-\\frac{1}{2^{2k}}\\Big).$$\nMoreover, let $T(a,b):=t(\\{2\\}^a,3,\\{2\\}^b)$. Then \\eqref{eq1.1} is equivalent to\n\\begin{align}\nT(a,b)=2\\sum_{k=1}^{r+s+1}(-1)^{k-1}d_{a,b}\\frac{1}{2^{2k}}\\zeta(2k+1)\nt(\\{2\\}^{a+b-k+1}),\n\\end{align}\n\nIn a different approach, Lupu \\cite{[L-MZ]} provided elementary proofs of\nZagier's formula for the special case $b=0$, and also established a Zagier-type\nformula for multiple $t$-values when $b=0$. For the general case $b>0$, Lupu\n\\cite{[L-MZ]} proposed the following two conjectures.\n\n\\begin{conjecture}\\label{con1}\nFor nonnegative integers $a$ and $b$, we have\n\\begin{align}\nH(a,b)=\\frac{-4\\pi^{2a+2b+2}}{(2a+2)!}\\sum_{n=0}^{\\infty}\n\\frac{\\zeta(2n)}{(2n+2a+2)(2n+2a+3)\\cdots(2n+2a+2b+3)2^{2n}}.\n\\end{align}\n\\end{conjecture}\n\n\\begin{conjecture}\\label{con2}\nFor nonnegative integers $a$ and $b$, we have\n\\begin{align}\nT(a,b)=\\frac{-2}{(2a+1)!}\\left(\\frac{\\pi}{2}\\right)^{2a+2b+2}\n\\sum_{n=0}^{\\infty}\\frac{\\zeta(2n)}{(2n+2a+1)(2n+2a+2)\\cdots(2n+2a+2b+2)2^{2n}}.\n\\end{align}\n\\end{conjecture}\n\\noindent Notice that Conjectures 1.2 and 1.3 are shown in \\cite{[L-MZ]}\nto be true when $b=0$. But for the general case $b>0$, these two conjectures\nare still kept open so far.\n\nIn 2026, Lai, Lupu, and Orr \\cite{[LLO-PAMS]} gave elementary and direct proofs\nfor both $H(a, b)$ and $T(a, b)$. In fact, they \\cite{[LLO-PAMS]} first provided\nan integral expression for $H(a, b)$, and then showed that this integral expression\nis equal to Zagier's result. The same proof method was applied to $T(a, b)$.\n\nIn this paper, our main goal is to investigate Conjectures \\ref{con1} and \\ref{con2}.\nWe will prove that both conjectures of Lupu are true. Actually, by employing the integral\nexpressions presented in \\cite{[LLO-PAMS]} for $H(a,b)$ and $T(a,b)$ and using\nthe properties of Beta and Gamma functions, we show the following main results\nof this paper.\n\n\\begin{theorem}\\label{theorem1.1}\nFor nonnegative integers $a$ and $b$, we have $H(a,b)=L(a,b)$, where\n\\begin{align*}\nL(a,b):=\\frac{-4\\pi^{2a+2b+2}}{(2a+2)!}\\sum_{n=0}^{\\infty}\n\\frac{\\zeta(2n)}{(2n+2a+2)(2n+2a+3)\\cdots(2n+2a+2b+3)2^{2n}}.\n\\end{align*}\n\\end{theorem}\n\n\\begin{theorem}\\label{theorem1.2}\nFor nonnegative integers $a$ and $b$, we have $T(a,b)=\\hat L(a,b)$, where\n\\begin{align*}\n\\hat L(a,b):=\\frac{-2}{(2a+1)!}\\left(\\frac{\\pi}{2}\\right)^{2a+2b+2}\n\\sum_{n=0}^{\\infty}\\frac{\\zeta(2n)}{(2n+2a+1)(2n+2a+2)\\cdots(2n+2a+2b+2)2^{2n}}.\n\\end{align*}\n\\end{theorem}\n\nThis paper is organized as follows. In Section 2, we supply several preliminary lemmas\nthat are needed in the proofs of our main results. Sections 3 and 4 are devoted to\nthe proofs of Theorems \\ref{theorem1.1} and \\ref{theorem1.2}, respectively.", "sketch": "The post-theorem introduction sketches Zagier's strategy for proving Theorem~\\ref{theorem1.01}: compute the associated generating functions of both sides \"in a closed form,\" then show these generating functions are \"entire functions of exponential growth\" and that they \"agree at sufficiently many points to force their equality.\" It is also noted that \"his proof was simplified by Li\" and that Hessami Pilehrood and Hessami Pilehrood gave \"an alternative proof\" of Theorem~\\ref{theorem1.01}.", "expanded_sketch": "The post-theorem introduction sketches Zagier's strategy for proving the main theorem: compute the associated generating functions of both sides \"in a closed form,\" then show these generating functions are \"entire functions of exponential growth\" and that they \"agree at sufficiently many points to force their equality.\" It is also noted that \"his proof was simplified by Li\" and that Hessami Pilehrood and Hessami Pilehrood gave \"an alternative proof\" of the main theorem.", "expanded_theorem": "\\label{theorem1.01}\n(Zagier \\cite{[Z-AM]}) For non-negative integers $a$ and $b$,\n\\begin{align}\nH(a,b)=2\\sum_{k=1}^{a+b+1}(-1)^kc_{a,b}\\zeta(2k+1)\n\\zeta(\\{2\\}^{a+b+1-k}),\n\\end{align}\nwhere\n$$c_{a,b}:=\\binom{2k}{2a+2}-\\Big(1-\\frac{1}{2^{2k}}\\Big)\\binom{2k}{2b+1}.$$", "theorem_type": ["Universal", "Classification or Bijection"], "mcq": {"question": "Let \\(a,b\\in \\mathbb Z_{\\ge 0}\\). For a multiple index \\((s_1,\\dots,s_r)\\) with \\(s_r>1\\), define the multiple zeta value\n\\[\n\\zeta(s_1,\\dots,s_r)=\\sum_{1\\le k_1<\\cdots<k_r}\\frac{1}{k_1^{s_1}\\cdots k_r^{s_r}}.\n\\]\nLet \\(\\{2\\}^m\\) denote the index consisting of \\(m\\) copies of \\(2\\), and define\n\\[\nH(a,b):=\\zeta(\\{2\\}^a,3,\\{2\\}^b).\n\\]\nWhich statement holds for every such pair \\((a,b)\\)?", "correct_choice": {"label": "A", "text": "For all nonnegative integers \\(a\\) and \\(b\\),\n\\[\nH(a,b)=2\\sum_{k=1}^{a+b+1}(-1)^k\\left[\\binom{2k}{2a+2}-\\left(1-\\frac{1}{2^{2k}}\\right)\\binom{2k}{2b+1}\\right]\\zeta(2k+1)\\,\\zeta(\\{2\\}^{a+b+1-k}).\n\\]\nEquivalently, in the \\(k\\)-th summand the coefficient is \\(c_{a,b}(k)=\\binom{2k}{2a+2}-\\left(1-2^{-2k}\\right)\\binom{2k}{2b+1}\\)."}, "choices": [{"label": "B", "text": "For all nonnegative integers \\(a\\) and \\(b\\),\n\\[\nH(a,b)=2\\sum_{k=0}^{a+b+1}(-1)^k\\left[\\binom{2k}{2a+2}-\\left(1-\\frac{1}{2^{2k}}\\right)\\binom{2k}{2b+1}\\right]\\zeta(2k+1)\\,\\zeta(\\{2\\}^{a+b+1-k}).\n\\]\nEquivalently, the same coefficient \\(c_{a,b}(k)=\\binom{2k}{2a+2}-\\left(1-2^{-2k}\\right)\\binom{2k}{2b+1}\\) is used, but the sum starts at \\(k=0\\)."}, {"label": "C", "text": "For all nonnegative integers \\(a\\) and \\(b\\), the value \\(H(a,b)\\) is a \\(\\mathbb{Q}\\)-linear combination of products of the form\n\\[\n\\zeta(2k+1)\\,\\zeta(\\{2\\}^{a+b+1-k}) \\qquad (1\\le k\\le a+b+1).\n\\]\nIn particular, \\(H(a,b)\\) can be written as a \\(\\mathbb{Q}\\)-linear combination of \\(\\pi^{2m}\\zeta(2n+1)\\) with \\(m+n=a+b+1\\)."}, {"label": "D", "text": "For all nonnegative integers \\(a\\) and \\(b\\),\n\\[\nH(a,b)=2\\sum_{k=1}^{a+b+1}(-1)^k\\left[\\binom{2k}{2a+2}-\\left(1-\\frac{1}{2^{2k}}\\right)\\binom{2k}{2b+1}\\right]\\zeta(2k+1)\\,\\zeta(\\{2\\}^{a+b-k}).\n\\]\nEquivalently, the coefficient is \\(c_{a,b}(k)=\\binom{2k}{2a+2}-\\left(1-2^{-2k}\\right)\\binom{2k}{2b+1}\\), but the depth-\\((a+b-k)\\) block of twos appears in the second zeta factor."}, {"label": "E", "text": "For all nonnegative integers \\(a\\) and \\(b\\),\n\\[\nH(a,b)=2\\sum_{k=1}^{a+b+1}(-1)^k\\left[\\binom{2k}{2a+2}-\\binom{2k}{2b+1}\\right]\\zeta(2k+1)\\,\\zeta(\\{2\\}^{a+b+1-k}).\n\\]\nEquivalently, the coefficient in the \\(k\\)-th summand is \\(c_{a,b}(k)=\\binom{2k}{2a+2}-\\binom{2k}{2b+1}\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "summation lower bound", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "explicit coefficient formula and exact closed-form identity", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "weight-balancing index in the second zeta factor", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "essential factor \\((1-2^{-2k})\\) in the coefficient", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem only defines the notation and asks which statement is universally true; it does not reveal the specific identity, coefficients, or range of summation."}, "TAS": {"score": 1, "justification": "The item is largely a recognition task for the precise theorem statement about \u001dH(a,b)\u001d, though it does include competing nearby alternatives rather than simply restating the theorem in the stem."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure: one can reject distractors by checking weight balance, the lower summation bound, and the missing factor \\((1-2^{-2k})\\). However, success still depends mostly on recalling the exact identity rather than generating a conclusion from premises."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic errors: wrong summation boundary, weaker true statement, weight/index mismatch, and omission of an essential coefficient factor."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it mainly tests precise theorem recognition rather than deeper generative mathematical reasoning."}}
{"id": "2602.19456v1", "paper_link": "http://arxiv.org/abs/2602.19456v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm 1}\nLet $\\Omega\\subset \\mathbb{R}^m$ be a bounded  Lipschitz domain symmetric about the origin. Suppose $h(r)\\in C^2([0, +\\infty))$ satisfies  $h'(r)\\ge 0$ and $h''(r)\\ge 0$ for $r\\in (0, +\\infty)$. Let  $B \\subset \\mathbb{R}^m$  be the  origin-centered  round ball with the same $\\gamma_h$-volume as $\\Omega$, namely $ \\int_{\\Omega} e^{h(|x|)}\\, dx= \\int_{B} e^{h(|x|)}\\, dx$. Denote by $\\sigma_1(B; \\gamma_h)$ the first nonzero Steklov eigenvalue of \\eqref{eq 1.4} on $B$, and $\\l_{2,\\alpha}(\\Omega; \\gamma_h)$ the second Robin eigenvalue of \\eqref{eq 1.1} on $\\Omega$.  If  $\\alpha \\in [-\\sigma_1(B; \\gamma_h), 0]$, then \n\\begin{align}\\label{eq 1.6}\n\\l_{2,\\alpha}(\\Omega; \\gamma_h) \\le \\l_{2,\\alpha}(B; \\gamma_h).\n\\end{align}\nEquality holds  if and only if \n$\\Omega=B$.", "start_pos": 9013, "end_pos": 9793, "label": "thm 1"}, "ref_dict": {"eq 1.4": "\\begin{align}\\label{eq 1.4}\n\\begin{cases}\n      -\\div (e^{h(|x|)}\\nabla u)=0 \\quad & x\\in \\Omega,\\\\\n        \\frac{\\p u}{\\p \\nu}=\\sigma u, \\quad & x\\in \\p \\Omega.\n\\end{cases}    \n\\end{align}", "eq 1.6": "\\begin{align}\\label{eq 1.6}\n\\l_{2,\\a}(\\Omega; \\gamma_h) \\le \\l_{2,\\a}(B; \\gamma_h).\n\\end{align}", "cor": "\\begin{corollary}\\label{cor}\nUnder the hypotheses of Theorem \\ref{thm 1}, we have \n\\begin{align*}\n\\sigma_1(\\Omega; \\gamma_h)\\le \\sigma_1(B; \\gamma_h).\n\\end{align*}\nEquality holds if and only if $\\Omega$ is isometric to the round ball $B$.\n\\end{corollary}", "eq 3.5": "\\begin{align}\\label{eq 3.5}\n        \\int_\\Omega v_i(x)u(x)   \\, d\\gamma_h=0, \\quad i=1,2,\\cdots, m.\n    \\end{align}", "eq 1.1": "\\begin{align}\\label{eq 1.1}\n    \\begin{cases}\n        -\\div (e^{h(|x|)}\\nabla u)=\\l e^{h(|x|)} u, \\quad & x\\in \\Omega,\\\\\n        \\frac{\\p u}{\\p \\nu}+\\a u=0, \\quad & x\\in \\p \\Omega,\n    \\end{cases}\n\\end{align}", "thm 1": "\\begin{theorem}\\label{thm 1}\nLet $\\Omega\\subset \\R^m$ be a bounded  Lipschitz domain symmetric about the origin. Suppose $h(r)\\in C^2([0, +\\infty))$ satisfies  $h'(r)\\ge 0$ and $h''(r)\\ge 0$ for $r\\in (0, +\\infty)$. Let  $B \\subset \\R^m$  be the  origin-centered  round ball with the same $\\gamma_h$-volume as $\\Omega$, namely $ \\int_{\\Omega} e^{h(|x|)}\\, dx= \\int_{B} e^{h(|x|)}\\, dx$. Denote by $\\sigma_1(B; \\gamma_h)$ the first nonzero Steklov eigenvalue of \\eqref{eq 1.4} on $B$, and $\\l_{2,\\a}(\\Omega; \\gamma_h)$ the second Robin eigenvalue of \\eqref{eq 1.1} on $\\Omega$.  If  $\\a \\in [-\\sigma_1(B; \\gamma_h), 0]$, then \n\\begin{align}\\label{eq 1.6}\n\\l_{2,\\a}(\\Omega; \\gamma_h) \\le \\l_{2,\\a}(B; \\gamma_h).\n\\end{align}\nEquality holds  if and only if \n$\\Omega=B$. \n\\end{theorem}"}, "pre_theorem_intro_text_len": 4883, "pre_theorem_intro_text": "The classical Szeg\\\"o-Weinberger inequality \\cite{Sz54, Wei56} states that the ball uniquely maximizes the second (first nonzero) Neumann eigenvalue of Laplacian  among all bounded domains of the same volume in Euclidean space.   Using the arguments of  Szeg\\\"o and Weinberger, analogous inequalities for the second Neumann eigenvalues have been established in various settings:  Ashbaugh-Benguria \\cite{AB95} extended the  the inequality to bounded domains in hemisphere and in hyperbolic space; Chiacchio and di Blasio \\cite{CB12} obtained a sharp isopermetric inequality for the first nonzero Neumann eigenvalue for bounded domians symmetric about the origin in Gauss space. \nMoreover, it is expected (cf. \\cite[Problem 4.41]{Hen17}) that for a range of negative Robin parameters, the second Robin eigenvalue should be maximized by  a round ball. This expectation has recently been confirmed by Freitas and Laugesen via the Szeg\\\"o \\cite{FL20} and the Weinberger  \\cite{FL21} approaches. \nFor further results on Szeg\\\"o-Weinberger type inequalities, we refer to \\cite{AS96,  As99, BBC20, BCB16,  LL23, LWW22, LWW23, MW24, Wang19, XW23} and references therein.\n\nEigenvalue optimization problems for the weighted Laplacian have also attracted  considerable attention. For instance,  Chiacchio and di Blasio \\cite{CB12} proved that, among all origin-symmetric regions with  fixed Gaussian volume, the ball maximizes the first nonzero Neumann eigenvalue; Brock, Francesco and di Blasio \\cite{BCB16} obtained  optimal  Szeg\\\"o-Weinberger type inequalities under a weighted measure.\nAdditional results on weighted eigenvalue problems can be found in \\cite{BCHT13, BCKT16, BCB16, CM24, CG22, GW26} and the references therein.\n\nInspired by the work of Brock, Francesco and di Blasio \\cite{BCB16} on the weighted Laplace operator and that of Freitas and Laugesen  \\cite{FL21} on the second Robin eigenvalue, we study the shape optimization problem for the second Robin eigenvalue of the following class of problems\n\\begin{align}\\label{eq 1.1}\n    \\begin{cases}\n        -\\operatorname{div} (e^{h(|x|)}\\nabla u)=\\lambda e^{h(|x|)} u, \\quad & x\\in \\Omega,\\\\\n        \\frac{\\partial u}{\\partial \\nu}+\\alpha u=0, \\quad & x\\in \\partial \\Omega,\n    \\end{cases}\n\\end{align}\nwhere $\\alpha\\in \\mathbb{R}$. Here and in the sequel, $\\Omega$ denotes a bounded connected  domain in $\\mathbb{R}^m$ with Lipchitz boundary, $\\nu$ is the outward normal to $\\partial \\Omega$, and $h(r)$ is a $C^2$ function on $[0, +\\infty)$.  We define the weighted measure associated with $h$ by \n$$\nd\\gamma_h=e^{h(|x|)}\\,dx, \\qquad  x\\in \\mathbb{R}^m,\n$$\nand denote by $W^{1,2}(\\Omega; \\gamma_h)$   the weighted Sobolev space equipped with the norm\n\\begin{align*}\n||u||_{W^{1,2}(\\Omega;\\gamma_h)}:=\\left(\\int_\\Omega u^2 \\, d\\gamma_h\\right)^{1/2}+ \\left(\\int_\\Omega |\\nabla u|^2 \\, d\\gamma_h\\right)^{1/2}. \n\\end{align*}\n\nIt is well known that the problem \\eqref{eq 1.1} has discrete  spectrum. We  denote its eigenvalue by $\\l_{k, \\alpha}(\\Omega; \\gamma_h)$ for $k=1,2, \\cdots$, which satisfy\n  \\begin{align*}\n       \\l_{1, \\alpha}(\\Omega;\\gamma_h) < \\l_{2, \\alpha}(\\Omega;\\gamma_h)  \\leq \\l_{3, \\alpha}(\\Omega;\\gamma_h) \\leq \\cdots \\to +\\infty,\n    \\end{align*}\nwith each eigenvalue repeated according to its multiplicity. \nBy standard spectral theory for self-adjoint compact operators,  the first two eigenvalues of \\eqref{eq 1.1} admit the variational characterizations: \n\\begin{align}\n     \\lambda_{1,\\alpha}(\\Omega; \\gamma_h)=\\inf\\{\\frac{\\int_{\\Omega}|\\nabla u|^{2}d\\gamma_h+\\alpha\\int_{\\partial \\Omega}u^{2} e^{h}\\, dA}{\\int_{\\Omega}u^{2}d\\gamma_h}: u\\in W^{1,2}(\\Omega;\\gamma_h)\\setminus\\{0\\}\\},\n\\end{align}\nwhere $dA$ is  the\ninduced measure on $\\partial \\Omega$;\nand\n\\begin{align}\\label{eq 1.3}\n    \\lambda_{2,\\alpha}(\\Omega;\\gamma_h)=\\inf\\{\\frac{\\int_{\\Omega}|\\nabla u|^{2}d\\gamma_h+\\alpha\\int_{\\partial \\Omega}u^{2} e^{h}dA}{\\int_{\\Omega}u^{2}d\\gamma_h}: u\\in W^{1,2}(\\Omega;\\gamma_h)\\setminus\\{0\\}, \\int_{\\Omega}uu_{1}d\\gamma_h=0\\},\n\\end{align}\nwhere $u_1(x)$ is an eigenfunction corresponding to $\\l_{1, \\alpha}(\\Omega;\\gamma_h)$. \n\nTo state our main results, we introduce the  following Steklov  eigenvalue problem on $\\Omega$ \n\\begin{align}\\label{eq 1.4}\n\\begin{cases}\n      -\\operatorname{div} (e^{h(|x|)}\\nabla u)=0 \\quad & x\\in \\Omega,\\\\\n        \\frac{\\partial u}{\\partial \\nu}=\\sigma u, \\quad & x\\in \\partial \\Omega.\n\\end{cases}    \n\\end{align}\nThe first nonzero Steklov eigenvalue can be characterized  variationally by\n\\begin{align}\n\\sigma_1(\\Omega; \\gamma_h)=\\inf\\left\\{ \\frac{\\int_{\\Omega} |\\nabla u|^2 \\, d\\gamma_h}{\\int_{\\partial \\Omega}  u^2 e^{h}\\, dA} : u\\in W^{1,2}(\\Omega; \\gamma_h )\\setminus\\{0\\},\\;\\; \\int_{\\partial \\Omega} ue^{h(|x|)}\\ dA=0 \\right\\}.\n\\end{align}\n In this paper,  we establish the following optimal isoperimetric inequality for the second Robin eigenvalue of \\eqref{eq 1.1}.", "context": "The classical Szeg\\\"o-Weinberger inequality \\cite{Sz54, Wei56} states that the ball uniquely maximizes the second (first nonzero) Neumann eigenvalue of Laplacian among all bounded domains of the same volume in Euclidean space. Using the arguments of Szeg\\\"o and Weinberger, analogous inequalities for the second Neumann eigenvalues have been established in various settings: Ashbaugh-Benguria \\cite{AB95} extended the the inequality to bounded domains in hemisphere and in hyperbolic space; Chiacchio and di Blasio \\cite{CB12} obtained a sharp isopermetric inequality for the first nonzero Neumann eigenvalue for bounded domians symmetric about the origin in Gauss space. \nMoreover, it is expected (cf. \\cite[Problem 4.41]{Hen17}) that for a range of negative Robin parameters, the second Robin eigenvalue should be maximized by a round ball. This expectation has recently been confirmed by Freitas and Laugesen via the Szeg\\\"o \\cite{FL20} and the Weinberger \\cite{FL21} approaches. \nFor further results on Szeg\\\"o-Weinberger type inequalities, we refer to \\cite{AS96, As99, BBC20, BCB16, LL23, LWW22, LWW23, MW24, Wang19, XW23} and references therein.\n\nEigenvalue optimization problems for the weighted Laplacian have also attracted considerable attention. For instance, Chiacchio and di Blasio \\cite{CB12} proved that, among all origin-symmetric regions with fixed Gaussian volume, the ball maximizes the first nonzero Neumann eigenvalue; Brock, Francesco and di Blasio \\cite{BCB16} obtained optimal Szeg\\\"o-Weinberger type inequalities under a weighted measure.\nAdditional results on weighted eigenvalue problems can be found in \\cite{BCHT13, BCKT16, BCB16, CM24, CG22, GW26} and the references therein.\n\nInspired by the work of Brock, Francesco and di Blasio \\cite{BCB16} on the weighted Laplace operator and that of Freitas and Laugesen \\cite{FL21} on the second Robin eigenvalue, we study the shape optimization problem for the second Robin eigenvalue of the following class of problems\n\\begin{align}\\label{eq 1.1}\n \\begin{cases}\n -\\operatorname{div} (e^{h(|x|)}\\nabla u)=\\lambda e^{h(|x|)} u, \\quad & x\\in \\Omega,\\\\\n \\frac{\\partial u}{\\partial \\nu}+\\alpha u=0, \\quad & x\\in \\partial \\Omega,\n \\end{cases}\n\\end{align}\nwhere $\\alpha\\in \\mathbb{R}$. Here and in the sequel, $\\Omega$ denotes a bounded connected domain in $\\mathbb{R}^m$ with Lipchitz boundary, $\\nu$ is the outward normal to $\\partial \\Omega$, and $h(r)$ is a $C^2$ function on $[0, +\\infty)$. We define the weighted measure associated with $h$ by \n$$\nd\\gamma_h=e^{h(|x|)}\\,dx, \\qquad x\\in \\mathbb{R}^m,\n$$\nand denote by $W^{1,2}(\\Omega; \\gamma_h)$ the weighted Sobolev space equipped with the norm\n\\begin{align*}\n||u||_{W^{1,2}(\\Omega;\\gamma_h)}:=\\left(\\int_\\Omega u^2 \\, d\\gamma_h\\right)^{1/2}+ \\left(\\int_\\Omega |\\nabla u|^2 \\, d\\gamma_h\\right)^{1/2}. \n\\end{align*}\n\nIt is well known that the problem \\eqref{eq 1.1} has discrete spectrum. We denote its eigenvalue by $\\l_{k, \\alpha}(\\Omega; \\gamma_h)$ for $k=1,2, \\cdots$, which satisfy\n \\begin{align*}\n \\l_{1, \\alpha}(\\Omega;\\gamma_h) < \\l_{2, \\alpha}(\\Omega;\\gamma_h) \\leq \\l_{3, \\alpha}(\\Omega;\\gamma_h) \\leq \\cdots \\to +\\infty,\n \\end{align*}\nwith each eigenvalue repeated according to its multiplicity. \nBy standard spectral theory for self-adjoint compact operators, the first two eigenvalues of \\eqref{eq 1.1} admit the variational characterizations: \n\\begin{align}\n \\lambda_{1,\\alpha}(\\Omega; \\gamma_h)=\\inf\\{\\frac{\\int_{\\Omega}|\\nabla u|^{2}d\\gamma_h+\\alpha\\int_{\\partial \\Omega}u^{2} e^{h}\\, dA}{\\int_{\\Omega}u^{2}d\\gamma_h}: u\\in W^{1,2}(\\Omega;\\gamma_h)\\setminus\\{0\\}\\},\n\\end{align}\nwhere $dA$ is the\ninduced measure on $\\partial \\Omega$;\nand\n\\begin{align}\\label{eq 1.3}\n \\lambda_{2,\\alpha}(\\Omega;\\gamma_h)=\\inf\\{\\frac{\\int_{\\Omega}|\\nabla u|^{2}d\\gamma_h+\\alpha\\int_{\\partial \\Omega}u^{2} e^{h}dA}{\\int_{\\Omega}u^{2}d\\gamma_h}: u\\in W^{1,2}(\\Omega;\\gamma_h)\\setminus\\{0\\}, \\int_{\\Omega}uu_{1}d\\gamma_h=0\\},\n\\end{align}\nwhere $u_1(x)$ is an eigenfunction corresponding to $\\l_{1, \\alpha}(\\Omega;\\gamma_h)$.\n\nTo state our main results, we introduce the following Steklov eigenvalue problem on $\\Omega$ \n\\begin{align}\\label{eq 1.4}\n\\begin{cases}\n -\\operatorname{div} (e^{h(|x|)}\\nabla u)=0 \\quad & x\\in \\Omega,\\\\\n \\frac{\\partial u}{\\partial \\nu}=\\sigma u, \\quad & x\\in \\partial \\Omega.\n\\end{cases} \n\\end{align}\nThe first nonzero Steklov eigenvalue can be characterized variationally by\n\\begin{align}\n\\sigma_1(\\Omega; \\gamma_h)=\\inf\\left\\{ \\frac{\\int_{\\Omega} |\\nabla u|^2 \\, d\\gamma_h}{\\int_{\\partial \\Omega} u^2 e^{h}\\, dA} : u\\in W^{1,2}(\\Omega; \\gamma_h )\\setminus\\{0\\},\\;\\; \\int_{\\partial \\Omega} ue^{h(|x|)}\\ dA=0 \\right\\}.\n\\end{align}\n In this paper, we establish the following optimal isoperimetric inequality for the second Robin eigenvalue of \\eqref{eq 1.1}.\n\n\\begin{align}\\label{eq 1.1}\n    \\begin{cases}\n        -\\div (e^{h(|x|)}\\nabla u)=\\l e^{h(|x|)} u, \\quad & x\\in \\Omega,\\\\\n        \\frac{\\p u}{\\p \\nu}+\\a u=0, \\quad & x\\in \\p \\Omega,\n    \\end{cases}\n\\end{align}\n\n\\begin{align}\\label{eq 1.4}\n\\begin{cases}\n      -\\div (e^{h(|x|)}\\nabla u)=0 \\quad & x\\in \\Omega,\\\\\n        \\frac{\\p u}{\\p \\nu}=\\sigma u, \\quad & x\\in \\p \\Omega.\n\\end{cases}    \n\\end{align}", "full_context": "The classical Szeg\\\"o-Weinberger inequality \\cite{Sz54, Wei56} states that the ball uniquely maximizes the second (first nonzero) Neumann eigenvalue of Laplacian among all bounded domains of the same volume in Euclidean space. Using the arguments of Szeg\\\"o and Weinberger, analogous inequalities for the second Neumann eigenvalues have been established in various settings: Ashbaugh-Benguria \\cite{AB95} extended the the inequality to bounded domains in hemisphere and in hyperbolic space; Chiacchio and di Blasio \\cite{CB12} obtained a sharp isopermetric inequality for the first nonzero Neumann eigenvalue for bounded domians symmetric about the origin in Gauss space. \nMoreover, it is expected (cf. \\cite[Problem 4.41]{Hen17}) that for a range of negative Robin parameters, the second Robin eigenvalue should be maximized by a round ball. This expectation has recently been confirmed by Freitas and Laugesen via the Szeg\\\"o \\cite{FL20} and the Weinberger \\cite{FL21} approaches. \nFor further results on Szeg\\\"o-Weinberger type inequalities, we refer to \\cite{AS96, As99, BBC20, BCB16, LL23, LWW22, LWW23, MW24, Wang19, XW23} and references therein.\n\nEigenvalue optimization problems for the weighted Laplacian have also attracted considerable attention. For instance, Chiacchio and di Blasio \\cite{CB12} proved that, among all origin-symmetric regions with fixed Gaussian volume, the ball maximizes the first nonzero Neumann eigenvalue; Brock, Francesco and di Blasio \\cite{BCB16} obtained optimal Szeg\\\"o-Weinberger type inequalities under a weighted measure.\nAdditional results on weighted eigenvalue problems can be found in \\cite{BCHT13, BCKT16, BCB16, CM24, CG22, GW26} and the references therein.\n\nInspired by the work of Brock, Francesco and di Blasio \\cite{BCB16} on the weighted Laplace operator and that of Freitas and Laugesen \\cite{FL21} on the second Robin eigenvalue, we study the shape optimization problem for the second Robin eigenvalue of the following class of problems\n\\begin{align}\\label{eq 1.1}\n \\begin{cases}\n -\\operatorname{div} (e^{h(|x|)}\\nabla u)=\\lambda e^{h(|x|)} u, \\quad & x\\in \\Omega,\\\\\n \\frac{\\partial u}{\\partial \\nu}+\\alpha u=0, \\quad & x\\in \\partial \\Omega,\n \\end{cases}\n\\end{align}\nwhere $\\alpha\\in \\mathbb{R}$. Here and in the sequel, $\\Omega$ denotes a bounded connected domain in $\\mathbb{R}^m$ with Lipchitz boundary, $\\nu$ is the outward normal to $\\partial \\Omega$, and $h(r)$ is a $C^2$ function on $[0, +\\infty)$. We define the weighted measure associated with $h$ by \n$$\nd\\gamma_h=e^{h(|x|)}\\,dx, \\qquad x\\in \\mathbb{R}^m,\n$$\nand denote by $W^{1,2}(\\Omega; \\gamma_h)$ the weighted Sobolev space equipped with the norm\n\\begin{align*}\n||u||_{W^{1,2}(\\Omega;\\gamma_h)}:=\\left(\\int_\\Omega u^2 \\, d\\gamma_h\\right)^{1/2}+ \\left(\\int_\\Omega |\\nabla u|^2 \\, d\\gamma_h\\right)^{1/2}. \n\\end{align*}\n\nIt is well known that the problem \\eqref{eq 1.1} has discrete spectrum. We denote its eigenvalue by $\\l_{k, \\alpha}(\\Omega; \\gamma_h)$ for $k=1,2, \\cdots$, which satisfy\n \\begin{align*}\n \\l_{1, \\alpha}(\\Omega;\\gamma_h) < \\l_{2, \\alpha}(\\Omega;\\gamma_h) \\leq \\l_{3, \\alpha}(\\Omega;\\gamma_h) \\leq \\cdots \\to +\\infty,\n \\end{align*}\nwith each eigenvalue repeated according to its multiplicity. \nBy standard spectral theory for self-adjoint compact operators, the first two eigenvalues of \\eqref{eq 1.1} admit the variational characterizations: \n\\begin{align}\n \\lambda_{1,\\alpha}(\\Omega; \\gamma_h)=\\inf\\{\\frac{\\int_{\\Omega}|\\nabla u|^{2}d\\gamma_h+\\alpha\\int_{\\partial \\Omega}u^{2} e^{h}\\, dA}{\\int_{\\Omega}u^{2}d\\gamma_h}: u\\in W^{1,2}(\\Omega;\\gamma_h)\\setminus\\{0\\}\\},\n\\end{align}\nwhere $dA$ is the\ninduced measure on $\\partial \\Omega$;\nand\n\\begin{align}\\label{eq 1.3}\n \\lambda_{2,\\alpha}(\\Omega;\\gamma_h)=\\inf\\{\\frac{\\int_{\\Omega}|\\nabla u|^{2}d\\gamma_h+\\alpha\\int_{\\partial \\Omega}u^{2} e^{h}dA}{\\int_{\\Omega}u^{2}d\\gamma_h}: u\\in W^{1,2}(\\Omega;\\gamma_h)\\setminus\\{0\\}, \\int_{\\Omega}uu_{1}d\\gamma_h=0\\},\n\\end{align}\nwhere $u_1(x)$ is an eigenfunction corresponding to $\\l_{1, \\alpha}(\\Omega;\\gamma_h)$.\n\nTo state our main results, we introduce the following Steklov eigenvalue problem on $\\Omega$ \n\\begin{align}\\label{eq 1.4}\n\\begin{cases}\n -\\operatorname{div} (e^{h(|x|)}\\nabla u)=0 \\quad & x\\in \\Omega,\\\\\n \\frac{\\partial u}{\\partial \\nu}=\\sigma u, \\quad & x\\in \\partial \\Omega.\n\\end{cases} \n\\end{align}\nThe first nonzero Steklov eigenvalue can be characterized variationally by\n\\begin{align}\n\\sigma_1(\\Omega; \\gamma_h)=\\inf\\left\\{ \\frac{\\int_{\\Omega} |\\nabla u|^2 \\, d\\gamma_h}{\\int_{\\partial \\Omega} u^2 e^{h}\\, dA} : u\\in W^{1,2}(\\Omega; \\gamma_h )\\setminus\\{0\\},\\;\\; \\int_{\\partial \\Omega} ue^{h(|x|)}\\ dA=0 \\right\\}.\n\\end{align}\n In this paper, we establish the following optimal isoperimetric inequality for the second Robin eigenvalue of \\eqref{eq 1.1}.\n\n\\begin{align}\\label{eq 1.1}\n    \\begin{cases}\n        -\\div (e^{h(|x|)}\\nabla u)=\\l e^{h(|x|)} u, \\quad & x\\in \\Omega,\\\\\n        \\frac{\\p u}{\\p \\nu}+\\a u=0, \\quad & x\\in \\p \\Omega,\n    \\end{cases}\n\\end{align}\n\n\\begin{align}\\label{eq 1.4}\n\\begin{cases}\n      -\\div (e^{h(|x|)}\\nabla u)=0 \\quad & x\\in \\Omega,\\\\\n        \\frac{\\p u}{\\p \\nu}=\\sigma u, \\quad & x\\in \\p \\Omega.\n\\end{cases}    \n\\end{align}\n\nIt is well known that the problem \\eqref{eq 1.1} has discrete spectrum. We denote its eigenvalue by $\\l_{k, \\a}(\\Omega; \\gamma_h)$ for $k=1,2, \\cdots$, which satisfy\n \\begin{align*}\n \\l_{1, \\a}(\\Omega;\\gamma_h) < \\l_{2, \\a}(\\Omega;\\gamma_h) \\leq \\l_{3, \\a}(\\Omega;\\gamma_h) \\leq \\cdots \\to +\\infty,\n \\end{align*}\nwith each eigenvalue repeated according to its multiplicity. \nBy standard spectral theory for self-adjoint compact operators, the first two eigenvalues of \\eqref{eq 1.1} admit the variational characterizations: \n\\begin{align}\n \\lambda_{1,\\alpha}(\\Omega; \\gamma_h)=\\inf\\{\\frac{\\int_{\\Omega}|\\nabla u|^{2}d\\gamma_h+\\alpha\\int_{\\partial \\Omega}u^{2} e^{h}\\, dA}{\\int_{\\Omega}u^{2}d\\gamma_h}: u\\in W^{1,2}(\\Omega;\\gamma_h)\\setminus\\{0\\}\\},\n\\end{align}\nwhere $dA$ is the\ninduced measure on $\\p \\Omega$;\nand\n\\begin{align}\\label{eq 1.3}\n \\lambda_{2,\\alpha}(\\Omega;\\gamma_h)=\\inf\\{\\frac{\\int_{\\Omega}|\\nabla u|^{2}d\\gamma_h+\\alpha\\int_{\\partial \\Omega}u^{2} e^{h}dA}{\\int_{\\Omega}u^{2}d\\gamma_h}: u\\in W^{1,2}(\\Omega;\\gamma_h)\\setminus\\{0\\}, \\int_{\\Omega}uu_{1}d\\gamma_h=0\\},\n\\end{align}\nwhere $u_1(x)$ is an eigenfunction corresponding to $\\l_{1, \\a}(\\Omega;\\gamma_h)$.\n\nBy taking $\\a=-\\sigma_1(B;\\gamma_h)$, Theorem \\ref{thm 1} yields the following Brock–Weinstock inequality, which was proved by Brock and Chiacchio \\cite[Theorems 1.4 and 1.5]{BC25} and by Mao and Zhang \\cite[Corollary 1.2]{MZ24}. \n\\begin{corollary}\\label{cor}\nUnder the hypotheses of Theorem \\ref{thm 1}, we have \n\\begin{align*}\n\\sigma_1(\\Omega; \\gamma_h)\\le \\sigma_1(B; \\gamma_h).\n\\end{align*}\nEquality holds if and only if $\\Omega$ is isometric to the round ball $B$.\n\\end{corollary}\n\n\\begin{proposition} \\label{prop 2.4}\nSuppose $h'(r)\\ge 0$, and $\\a\\ge -\\sigma_1(B(R); \\gamma_h)$. Then\n\\begin{align*}\n \\lambda_{2,\\alpha}(B(R); \\gamma_h)\\geq 0. \n\\end{align*}\n\\end{proposition}\n\\begin{proof}\n Let $g(r)$ be as defined in Proposition \\ref{prop 2.3}. \n Since \n \\begin{align*}\n \\int_{\\partial B(R)}g(r)\\frac{x_{i}}{r}e^{h(r)}dA=0, 1\\leq i\\leq m,\n \\end{align*}\n the functions $u_{i}=g(r)\\frac{x_{i}}{r}, 1\\leq i\\leq m,$ are admissible test functions for $\\sigma_{1}(B(R), \\gamma_h)$. Therefore,\n \\begin{align}\\label{eq 2.22}\n \\sum_{i=1}^{m}\\int_{B(R)}|\\nabla u_{i}|^{2}\\, d\\gamma_h\\geq \\sigma_1(B(R); \\gamma_h) \\sum_{i=1}^{m}\\int_{\\partial B(R)}|u_{i}|^{2}e^{h(r)}dA,\n \\end{align}\n where $dA$ is the induced measure on $\\p B$.\nUsing \n \\begin{align*}\n \\sum_{i=1}^{m}|\\nabla u_{i}|^{2}=g'(r)^{2}+\\frac{m-1}{r^{2}}g(r)^{2}.\n \\end{align*}\n and the fact $h(r)\\le h(R)$ for $r\\in[0, R]$, we estimate from \\ref{eq 2.22} \n \\begin{align}\\label{eq 2.23}\n 0&\\leq \\sum_{i=1}^{m}\\int_{B(R)}|\\nabla u_{i}|^{2}e^{h(r)}dx- \\sigma_1(B(R); \\gamma_h)\\sum_{i=1}^{m}\\int_{\\partial B(R)}|u_{i}|^{2}e^{h(r)}dA\\nonumber\\\\\n &\\leq \\omega_{m-1}\\int_{0}^{R}\\left(g'(r)^{2}+\\frac{m-1}{r^{2}}g^{2}\\right)e^{h(r)}r^{m-1}dr+\\alpha \\omega_{m-1} g(R)^{2}e^{h(R)}R^{m-1},\n \\end{align}\n where $\\omega_{m-1}$ denotes the surface area of the unit sphere $\\mathbb{S}^{m-1}$ in $\\R^m$.\nUsing integration by parts and \\eqref{eq 2.11}, we compute \n\\begin{align*}\n &\\int_{0}^{R}\\left(g'(r)^{2}+\\frac{m-1}{r^{2}}g^{2}\\right)e^{h(r)}r^{m-1}dr\\\\\n =\\quad &gg'e^{h(r)}r^{m-1}\\Big|_{0}^{R}-\\int_{0}^{R}g(g'e^{h(r)}r^{m-1})'dr+\\int_{0}^{R}\\frac{m-1}{r^{2}}g^{2}e^{h(r)}r^{m-1}dr\\\\\n =\\quad &g(R)g'(R)e^{h(R)}R^{m-1}-\\int_{0}^{R}g(g''+(\\frac{m-1}{r}+h')g')e^{h(r)}r^{m-1}dr\\\\\n &+\\int_{0}^{R}\\frac{m-1}{r^{2}}g^{2}e^{h(r)}r^{m-1}dr\\\\\n =&g(R)g'(R)e^{h(R)}R^{m-1}+\\int_{0}^{R}g^{2}(\\lambda_{2,\\alpha}(B(R);\\gamma_h)-\\frac{m-1}{r^{2}})e^{h(r)}r^{m-1}dr\\\\\n &+\\int_{0}^{R}\\frac{m-1}{r^{2}}g^{2}e^{h(r)}r^{m-1}dr.\n\\end{align*}\nSubstituting this equality into inequality \\eqref{eq 2.23} yields\n \\begin{align*}\n 0\\leq& \\lambda_{2,\\alpha}(B(R);\\gamma_h)\\int_{0}^{R}g^{2}(r)e^{h(r)}r^{m-1}dr+g(R)e^{h(R)}R^{m-1}(g'(R)+\\alpha g(R))\\\\\n =&\\lambda_{2,\\alpha}(B(R);\\gamma_h)\\int_{0}^{R}g^{2}(r)e^{h(r)}r^{m-1}\\, dr,\n \\end{align*}\nwhich implies $\\lambda_{2,\\alpha}(B(R);\\gamma_h)\\geq 0$.\n\\end{proof}\n\nFrom now on, we assume $\\Omega\\subset \\R^m$ is a bounded Lipschitz domain symmetric about the origin, and $B\\subset \\R^m$ is the origin-centered round bound with the same $\\gamma_h$ voulme as $\\Omega$. Let $R$ be the radius of $B$. We extend the function $g$ defined in \\eqref{eq 2.11} by\n\\begin{equation}\\label{eq 3.1}\ng(r)=g(R) e^{-\\a(r-R)}\\text{\\quad for \\quad} r\\ge R.\n\\end{equation}\nBy definition, $g$ is continuously differentiable on $(0,\\infty)$. If $\\a\\le 0$, then by Proposition \\ref{prop 2.2}, $g$ is increasing on $(0,\\infty)$. In the sequel, $\\sigma_1(B; \\gamma_h)$ denotes the first nonzero Steklov eigenvalue of $B$ as defined in \\eqref{eq 2.2}.\n\n{\\bf Case 2.} $r\\ge R$. For $r\\ge R$, \n\\begin{align*}\n F(r)=(-\\alpha^{2}+\\frac{m-1}{r^{2}}+\\alpha\\frac{m-1}{r}+\\alpha h^{'})g^{2}(r),\n\\end{align*}\nand a direct computation yields\n\\begin{align*}\n F'(r)=(2\\alpha^{3}+\\alpha h''-2\\alpha^{2}h'-2\\alpha^{2}\\frac{m-1}{r}-3\\alpha \\frac{m-1}{r^{2}}-2\\frac{m-1}{r^{3}})g^{2}(r).\n\\end{align*}\nSince $\\alpha\\leq 0$, $h'\\geq 0$ and $h^{''}\\geq 0$, we obtain\n\\begin{align*}\n F'(r)\\le& \\left(-2\\alpha^{2}\\frac{m-1}{r}-3\\alpha \\frac{m-1}{r^{2}}-2\\frac{m-1}{r^{3}}\\right) g(r)^2\\\\\n =&-\\frac{m-1}{r}(2(\\alpha+\\frac{3}{4r})^{2}+\\frac{7}{8}\\frac{1}{r^{2}})g(r)^2<0,\n\\end{align*}\nproving that $F(r)$ is monotonically decreasing on $(0, R)$.\n\\end{proof}\n The main idea in proving Theorem \\ref{thm 1} is to construct trial functions for $\\l_{2, \\a}$ (see \\eqref{eq 1.3}) using techniques introduced by Weinberger. We first recall a monotonicity lemma for weighted symmetrization.\n\\begin{lemma}\\label{Lemma 3.2}\n Let $\\Omega\\subset \\R^m$ be a bounded set, and let $B$ be the origin-centered round ball with the same $\\gamma_h$ volume as $\\Omega$. If $f(r)$ is monotonically decreasing on $[0,+\\infty)$, then \n \\begin{align*}\n \\int_\\Omega f(|x|)\\, d\\gamma_m\\le \\int_B f(|x|)\\, d\\gamma_m.\n \\end{align*}\n Equality holds if and only if $\\Omega=B$.\n\\end{lemma}\n\\begin{proof}\n Lemma 3.2 follows directly from Hardy-Littlewood inequality; see the proof of inequality (4.24) in \\cite[Page 213]{BCB16}.\n\\end{proof}\n\nNow we now proceed to prove Theorem \\ref{thm 1}.\n\\begin{proof}\n Let $\\Omega$ be a bounded Lipschitz domain symmetric about the origin in $\\R^m$, and let $B\\subset \\R^m$ be the origin-centered round ball of radius $R$ satisfying \n $$\n\\int_B \\, d\\gamma_h =\\int_\\Omega \\, d\\gamma_h.\n $$\nDefine $g(r)$ for $r\\in [0, +\\infty)$ as in \\eqref{eq 3.1}, and for each $1\\le i\\le m$ set\n\\begin{align*}\n v_i(x)=g(|x|)\\frac{x_i}{|x|}, \n\\end{align*}\nwhere $x=(x_1, x_2, \\cdots, x_m)\\in \\R^m$.", "post_theorem_intro_text_len": 2156, "post_theorem_intro_text": "By taking $\\alpha=0$, Theorem \\ref{thm 1} implies that the ball centered at the\norigin is the unique set maximizer the second Neumann eigenvalue $\\mu_2(\\Omega;\\gamma_h)$ among all bounded Lipschitz domains $\\Omega$ in $\\mathbb{R}^m$ with prescribed $\\gamma_h$-measure and symmetric about the origin. This result was previously  proved by Brock, Chiacchio, and di Blasio (Theorem 1.2 of \\cite{BCB16}). Note that Theorem \\ref{thm 1} applies in particular  to $\\l_{2, \\alpha}(\\Omega; e^{|x|^2/2})$. Hence, our result provides information about  eigenvalues of the problem\n$$\n-\\Delta u-x\\cdot \\nabla u=\\lambda u\n$$\nwhich has been widely studied in literature (see, e.g. \\cite{BCM12, BMP13}). \nThe symmetry assumption on $\\Omega$ in Theorem \\ref{thm 1}\n ensures the validity of  the orthogonality conditions \\eqref{eq 3.5}, as the weighted measure $e^{h(|x|)}$ is  radially\n symmetric about the origin.\nIt remains an interesting question that whether \\eqref{eq 1.6} \nholds in the case of the Gauss measure $e^{-|x|^2/2}dx$.   \n\nBy taking $\\alpha=-\\sigma_1(B;\\gamma_h)$, Theorem \\ref{thm 1} yields the following Brock–Weinstock inequality, which was proved by Brock and Chiacchio \\cite[Theorems 1.4 and 1.5]{BC25} and by Mao and Zhang \\cite[Corollary 1.2]{MZ24}. \n\\begin{corollary}\\label{cor}\nUnder the hypotheses of Theorem \\ref{thm 1}, we have \n\\begin{align*}\n\\sigma_1(\\Omega; \\gamma_h)\\le \\sigma_1(B; \\gamma_h).\n\\end{align*}\nEquality holds if and only if $\\Omega$ is isometric to the round ball $B$.\n\\end{corollary}\n\nThe main technique in the proof of Theorem \\ref{thm 1} relies on Weinberger’s trick, in which  test functions are constructed using eigenfunctions on a round ball to estimate eigenvalues. The main difficulties in the proof include analyzing the properties of the second eigenfunction of \\eqref{eq 1.1} on the round ball  and establishing the required monotonicity to apply Weinberger’s method. The rest of the paper is organized as follows.\nIn Section \\ref{sect2}, we study properties of the Robin and Steklov eigenvalue  problems for balls in the weighted space. In Section  \\ref{sect3}, we prove Theorem \\ref{thm 1} and Corollary \\ref{cor}.", "sketch": "The proof of Theorem~\\ref{thm 1} \"relies on Weinberger’s trick, in which test functions are constructed using eigenfunctions on a round ball to estimate eigenvalues.\" The \"main difficulties\" are \"analyzing the properties of the second eigenfunction of \\eqref{eq 1.1} on the round ball\" and \"establishing the required monotonicity to apply Weinberger’s method.\" The symmetry assumption on $\\Omega$ \"ensures the validity of the orthogonality conditions \\eqref{eq 3.5}, as the weighted measure $e^{h(|x|)}$ is radially symmetric about the origin.\"", "expanded_sketch": "The proof of the main theorem relies on Weinberger’s trick, in which test functions are constructed using eigenfunctions on a round ball to estimate eigenvalues. The main difficulties are analyzing the properties of the second eigenfunction of\n\\begin{align}\\label{eq 1.1}\n    \\begin{cases}\n        -\\div (e^{h(|x|)}\\nabla u)=\\l e^{h(|x|)} u, \\quad & x\\in \\Omega,\\\\\n        \\frac{\\p u}{\\p \\nu}+\\a u=0, \\quad & x\\in \\p \\Omega,\n    \\end{cases}\n\\end{align}\non the round ball and establishing the required monotonicity to apply Weinberger’s method. The symmetry assumption on $\\Omega$ ensures the validity of the orthogonality conditions\n\\begin{align}\\label{eq 3.5}\n        \\int_\\Omega v_i(x)u(x)   \\, d\\gamma_h=0, \\quad i=1,2,\\cdots, m.\n    \\end{align}\nas the weighted measure $e^{h(|x|)}$ is radially symmetric about the origin.", "expanded_theorem": "\\label{thm 1}\nLet $\\Omega\\subset \\mathbb{R}^m$ be a bounded  Lipschitz domain symmetric about the origin. Suppose $h(r)\\in C^2([0, +\\infty))$ satisfies  $h'(r)\\ge 0$ and $h''(r)\\ge 0$ for $r\\in (0, +\\infty)$. Let  $B \\subset \\mathbb{R}^m$  be the  origin-centered  round ball with the same $\\gamma_h$-volume as $\\Omega$, namely $ \\int_{\\Omega} e^{h(|x|)}\\, dx= \\int_{B} e^{h(|x|)}\\, dx$. Denote by $\\sigma_1(B; \\gamma_h)$ the first nonzero Steklov eigenvalue of\n\\begin{align}\\label{eq 1.4}\n\\begin{cases}\n      -\\div (e^{h(|x|)}\\nabla u)=0 \\quad & x\\in \\Omega,\\\\\n        \\frac{\\p u}{\\p \\nu}=\\sigma u, \\quad & x\\in \\p \\Omega.\n\\end{cases}    \n\\end{align}\non $B$, and $\\l_{2,\\alpha}(\\Omega; \\gamma_h)$ the second Robin eigenvalue of\n\\begin{align}\\label{eq 1.1}\n    \\begin{cases}\n        -\\div (e^{h(|x|)}\\nabla u)=\\l e^{h(|x|)} u, \\quad & x\\in \\Omega,\\\\\n        \\frac{\\p u}{\\p \\nu}+\\a u=0, \\quad & x\\in \\p \\Omega,\n    \\end{cases}\n\\end{align}\non $\\Omega$.  If  $\\alpha \\in [-\\sigma_1(B; \\gamma_h), 0]$, then \n\\begin{align}\\label{eq 1.6}\n\\l_{2,\\alpha}(\\Omega; \\gamma_h) \\le \\l_{2,\\alpha}(B; \\gamma_h).\n\\end{align}\nEquality holds  if and only if \n$\\Omega=B$.,", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let Ω ⊂ R^m be a bounded Lipschitz domain symmetric about the origin, and let h ∈ C^2([0,∞)) satisfy h'(r) ≥ 0 and h''(r) ≥ 0 for all r > 0. Define the weighted measure dγ_h = e^{h(|x|)} dx. Let B ⊂ R^m be the origin-centered round ball with the same γ_h-volume as Ω, i.e. ∫_Ω e^{h(|x|)} dx = ∫_B e^{h(|x|)} dx. For a domain D, let λ_{2,α}(D; γ_h) denote the second eigenvalue of the weighted Robin problem -div(e^{h(|x|)}∇u) = λ e^{h(|x|)}u in D, with boundary condition ∂u/∂ν + αu = 0 on ∂D. Let σ_1(B; γ_h) denote the first nonzero Steklov eigenvalue on B for -div(e^{h(|x|)}∇u) = 0 in B, with ∂u/∂ν = σu on ∂B. If α ∈ [-σ_1(B; γ_h), 0], which quantitative estimate holds?", "correct_choice": {"label": "A", "text": "One has λ_{2,α}(Ω; γ_h) ≤ λ_{2,α}(B; γ_h), and equality holds if and only if Ω = B."}, "choices": [{"label": "B", "text": "One has \\(\\lambda_{2,\\alpha}(\\Omega; \\gamma_h) \\le \\lambda_{2,\\alpha}(B; \\gamma_h)\\) for every \\(\\alpha \\le 0\\), and equality holds if and only if \\(\\Omega = B\\)."}, {"label": "C", "text": "One has \\(\\lambda_{2,\\alpha}(\\Omega; \\gamma_h) \\le \\lambda_{2,\\alpha}(B; \\gamma_h)\\)."}, {"label": "D", "text": "One has \\(\\lambda_{2,\\alpha}(\\Omega; \\gamma_h) \\le \\lambda_{2,\\alpha}(B; \\gamma_h)\\) whenever \\(\\alpha \\in [-\\sigma_1(\\Omega; \\gamma_h),0]\\), and equality holds if and only if \\(\\Omega = B\\)."}, {"label": "E", "text": "One has \\(\\lambda_{2,\\alpha}(\\Omega; \\gamma_h) \\ge \\lambda_{2,\\alpha}(B; \\gamma_h)\\), and equality holds if and only if \\(\\Omega = B\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "sharp range for \\alpha tied to monotonicity/Steklov threshold", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the equality characterization", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence of the threshold on B rather than on \\Omega", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "extremal direction from Weinberger trial-function argument", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It states the hypotheses and asks for the valid conclusion, without giving away the inequality direction or the precise equality condition."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem lists the exact hypotheses and asks for the corresponding conclusion. The correct choice is basically the theorem statement itself."}, "GPS": {"score": 1, "justification": "Some reasoning or careful discrimination is needed because the options vary in subtle but important ways (range of α, equality characterization, comparison direction). However, the task is closer to recognition of the precise theorem than to genuine derivation."}, "DQS": {"score": 2, "justification": "The distractors are strong and plausible: one overextends the α-range, one gives only a weaker true conclusion, one swaps the Steklov threshold to Ω, and one reverses the inequality. These reflect realistic mathematical confusions."}, "total_score": 5, "overall_assessment": "A solid MCQ with no answer leakage and high-quality distractors, but it mainly tests precise theorem recall/recognition rather than generative reasoning."}}
{"id": "2602.19456v1", "paper_link": "http://arxiv.org/abs/2602.19456v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm 1}\nLet $\\Omega\\subset \\mathbb{R}^m$ be a bounded  Lipschitz domain symmetric about the origin. Suppose $h(r)\\in C^2([0, +\\infty))$ satisfies  $h'(r)\\ge 0$ and $h''(r)\\ge 0$ for $r\\in (0, +\\infty)$. Let  $B \\subset \\mathbb{R}^m$  be the  origin-centered  round ball with the same $\\gamma_h$-volume as $\\Omega$, namely $ \\int_{\\Omega} e^{h(|x|)}\\, dx= \\int_{B} e^{h(|x|)}\\, dx$. Denote by $\\sigma_1(B; \\gamma_h)$ the first nonzero Steklov eigenvalue of \\eqref{eq 1.4} on $B$, and $\\l_{2,\\alpha}(\\Omega; \\gamma_h)$ the second Robin eigenvalue of \\eqref{eq 1.1} on $\\Omega$.  If  $\\alpha \\in [-\\sigma_1(B; \\gamma_h), 0]$, then \n\\begin{align}\\label{eq 1.6}\n\\l_{2,\\alpha}(\\Omega; \\gamma_h) \\le \\l_{2,\\alpha}(B; \\gamma_h).\n\\end{align}\nEquality holds  if and only if \n$\\Omega=B$.", "start_pos": 9013, "end_pos": 9793, "label": "thm 1"}, "ref_dict": {"eq 1.4": "\\begin{align}\\label{eq 1.4}\n\\begin{cases}\n      -\\div (e^{h(|x|)}\\nabla u)=0 \\quad & x\\in \\Omega,\\\\\n        \\frac{\\p u}{\\p \\nu}=\\sigma u, \\quad & x\\in \\p \\Omega.\n\\end{cases}    \n\\end{align}", "eq 1.6": "\\begin{align}\\label{eq 1.6}\n\\l_{2,\\a}(\\Omega; \\gamma_h) \\le \\l_{2,\\a}(B; \\gamma_h).\n\\end{align}", "cor": "\\begin{corollary}\\label{cor}\nUnder the hypotheses of Theorem \\ref{thm 1}, we have \n\\begin{align*}\n\\sigma_1(\\Omega; \\gamma_h)\\le \\sigma_1(B; \\gamma_h).\n\\end{align*}\nEquality holds if and only if $\\Omega$ is isometric to the round ball $B$.\n\\end{corollary}", "eq 3.5": "\\begin{align}\\label{eq 3.5}\n        \\int_\\Omega v_i(x)u(x)   \\, d\\gamma_h=0, \\quad i=1,2,\\cdots, m.\n    \\end{align}", "eq 1.1": "\\begin{align}\\label{eq 1.1}\n    \\begin{cases}\n        -\\div (e^{h(|x|)}\\nabla u)=\\l e^{h(|x|)} u, \\quad & x\\in \\Omega,\\\\\n        \\frac{\\p u}{\\p \\nu}+\\a u=0, \\quad & x\\in \\p \\Omega,\n    \\end{cases}\n\\end{align}", "thm 1": "\\begin{theorem}\\label{thm 1}\nLet $\\Omega\\subset \\R^m$ be a bounded  Lipschitz domain symmetric about the origin. Suppose $h(r)\\in C^2([0, +\\infty))$ satisfies  $h'(r)\\ge 0$ and $h''(r)\\ge 0$ for $r\\in (0, +\\infty)$. Let  $B \\subset \\R^m$  be the  origin-centered  round ball with the same $\\gamma_h$-volume as $\\Omega$, namely $ \\int_{\\Omega} e^{h(|x|)}\\, dx= \\int_{B} e^{h(|x|)}\\, dx$. Denote by $\\sigma_1(B; \\gamma_h)$ the first nonzero Steklov eigenvalue of \\eqref{eq 1.4} on $B$, and $\\l_{2,\\a}(\\Omega; \\gamma_h)$ the second Robin eigenvalue of \\eqref{eq 1.1} on $\\Omega$.  If  $\\a \\in [-\\sigma_1(B; \\gamma_h), 0]$, then \n\\begin{align}\\label{eq 1.6}\n\\l_{2,\\a}(\\Omega; \\gamma_h) \\le \\l_{2,\\a}(B; \\gamma_h).\n\\end{align}\nEquality holds  if and only if \n$\\Omega=B$. \n\\end{theorem}"}, "pre_theorem_intro_text_len": 4883, "pre_theorem_intro_text": "The classical Szeg\\\"o-Weinberger inequality \\cite{Sz54, Wei56} states that the ball uniquely maximizes the second (first nonzero) Neumann eigenvalue of Laplacian  among all bounded domains of the same volume in Euclidean space.   Using the arguments of  Szeg\\\"o and Weinberger, analogous inequalities for the second Neumann eigenvalues have been established in various settings:  Ashbaugh-Benguria \\cite{AB95} extended the  the inequality to bounded domains in hemisphere and in hyperbolic space; Chiacchio and di Blasio \\cite{CB12} obtained a sharp isopermetric inequality for the first nonzero Neumann eigenvalue for bounded domians symmetric about the origin in Gauss space. \nMoreover, it is expected (cf. \\cite[Problem 4.41]{Hen17}) that for a range of negative Robin parameters, the second Robin eigenvalue should be maximized by  a round ball. This expectation has recently been confirmed by Freitas and Laugesen via the Szeg\\\"o \\cite{FL20} and the Weinberger  \\cite{FL21} approaches. \nFor further results on Szeg\\\"o-Weinberger type inequalities, we refer to \\cite{AS96,  As99, BBC20, BCB16,  LL23, LWW22, LWW23, MW24, Wang19, XW23} and references therein.\n\nEigenvalue optimization problems for the weighted Laplacian have also attracted  considerable attention. For instance,  Chiacchio and di Blasio \\cite{CB12} proved that, among all origin-symmetric regions with  fixed Gaussian volume, the ball maximizes the first nonzero Neumann eigenvalue; Brock, Francesco and di Blasio \\cite{BCB16} obtained  optimal  Szeg\\\"o-Weinberger type inequalities under a weighted measure.\nAdditional results on weighted eigenvalue problems can be found in \\cite{BCHT13, BCKT16, BCB16, CM24, CG22, GW26} and the references therein.\n\nInspired by the work of Brock, Francesco and di Blasio \\cite{BCB16} on the weighted Laplace operator and that of Freitas and Laugesen  \\cite{FL21} on the second Robin eigenvalue, we study the shape optimization problem for the second Robin eigenvalue of the following class of problems\n\\begin{align}\\label{eq 1.1}\n    \\begin{cases}\n        -\\operatorname{div} (e^{h(|x|)}\\nabla u)=\\lambda e^{h(|x|)} u, \\quad & x\\in \\Omega,\\\\\n        \\frac{\\partial u}{\\partial \\nu}+\\alpha u=0, \\quad & x\\in \\partial \\Omega,\n    \\end{cases}\n\\end{align}\nwhere $\\alpha\\in \\mathbb{R}$. Here and in the sequel, $\\Omega$ denotes a bounded connected  domain in $\\mathbb{R}^m$ with Lipchitz boundary, $\\nu$ is the outward normal to $\\partial \\Omega$, and $h(r)$ is a $C^2$ function on $[0, +\\infty)$.  We define the weighted measure associated with $h$ by \n$$\nd\\gamma_h=e^{h(|x|)}\\,dx, \\qquad  x\\in \\mathbb{R}^m,\n$$\nand denote by $W^{1,2}(\\Omega; \\gamma_h)$   the weighted Sobolev space equipped with the norm\n\\begin{align*}\n||u||_{W^{1,2}(\\Omega;\\gamma_h)}:=\\left(\\int_\\Omega u^2 \\, d\\gamma_h\\right)^{1/2}+ \\left(\\int_\\Omega |\\nabla u|^2 \\, d\\gamma_h\\right)^{1/2}. \n\\end{align*}\n\nIt is well known that the problem \\eqref{eq 1.1} has discrete  spectrum. We  denote its eigenvalue by $\\l_{k, \\alpha}(\\Omega; \\gamma_h)$ for $k=1,2, \\cdots$, which satisfy\n  \\begin{align*}\n       \\l_{1, \\alpha}(\\Omega;\\gamma_h) < \\l_{2, \\alpha}(\\Omega;\\gamma_h)  \\leq \\l_{3, \\alpha}(\\Omega;\\gamma_h) \\leq \\cdots \\to +\\infty,\n    \\end{align*}\nwith each eigenvalue repeated according to its multiplicity. \nBy standard spectral theory for self-adjoint compact operators,  the first two eigenvalues of \\eqref{eq 1.1} admit the variational characterizations: \n\\begin{align}\n     \\lambda_{1,\\alpha}(\\Omega; \\gamma_h)=\\inf\\{\\frac{\\int_{\\Omega}|\\nabla u|^{2}d\\gamma_h+\\alpha\\int_{\\partial \\Omega}u^{2} e^{h}\\, dA}{\\int_{\\Omega}u^{2}d\\gamma_h}: u\\in W^{1,2}(\\Omega;\\gamma_h)\\setminus\\{0\\}\\},\n\\end{align}\nwhere $dA$ is  the\ninduced measure on $\\partial \\Omega$;\nand\n\\begin{align}\\label{eq 1.3}\n    \\lambda_{2,\\alpha}(\\Omega;\\gamma_h)=\\inf\\{\\frac{\\int_{\\Omega}|\\nabla u|^{2}d\\gamma_h+\\alpha\\int_{\\partial \\Omega}u^{2} e^{h}dA}{\\int_{\\Omega}u^{2}d\\gamma_h}: u\\in W^{1,2}(\\Omega;\\gamma_h)\\setminus\\{0\\}, \\int_{\\Omega}uu_{1}d\\gamma_h=0\\},\n\\end{align}\nwhere $u_1(x)$ is an eigenfunction corresponding to $\\l_{1, \\alpha}(\\Omega;\\gamma_h)$. \n\nTo state our main results, we introduce the  following Steklov  eigenvalue problem on $\\Omega$ \n\\begin{align}\\label{eq 1.4}\n\\begin{cases}\n      -\\operatorname{div} (e^{h(|x|)}\\nabla u)=0 \\quad & x\\in \\Omega,\\\\\n        \\frac{\\partial u}{\\partial \\nu}=\\sigma u, \\quad & x\\in \\partial \\Omega.\n\\end{cases}    \n\\end{align}\nThe first nonzero Steklov eigenvalue can be characterized  variationally by\n\\begin{align}\n\\sigma_1(\\Omega; \\gamma_h)=\\inf\\left\\{ \\frac{\\int_{\\Omega} |\\nabla u|^2 \\, d\\gamma_h}{\\int_{\\partial \\Omega}  u^2 e^{h}\\, dA} : u\\in W^{1,2}(\\Omega; \\gamma_h )\\setminus\\{0\\},\\;\\; \\int_{\\partial \\Omega} ue^{h(|x|)}\\ dA=0 \\right\\}.\n\\end{align}\n In this paper,  we establish the following optimal isoperimetric inequality for the second Robin eigenvalue of \\eqref{eq 1.1}.", "context": "The classical Szeg\\\"o-Weinberger inequality \\cite{Sz54, Wei56} states that the ball uniquely maximizes the second (first nonzero) Neumann eigenvalue of Laplacian among all bounded domains of the same volume in Euclidean space. Using the arguments of Szeg\\\"o and Weinberger, analogous inequalities for the second Neumann eigenvalues have been established in various settings: Ashbaugh-Benguria \\cite{AB95} extended the the inequality to bounded domains in hemisphere and in hyperbolic space; Chiacchio and di Blasio \\cite{CB12} obtained a sharp isopermetric inequality for the first nonzero Neumann eigenvalue for bounded domians symmetric about the origin in Gauss space. \nMoreover, it is expected (cf. \\cite[Problem 4.41]{Hen17}) that for a range of negative Robin parameters, the second Robin eigenvalue should be maximized by a round ball. This expectation has recently been confirmed by Freitas and Laugesen via the Szeg\\\"o \\cite{FL20} and the Weinberger \\cite{FL21} approaches. \nFor further results on Szeg\\\"o-Weinberger type inequalities, we refer to \\cite{AS96, As99, BBC20, BCB16, LL23, LWW22, LWW23, MW24, Wang19, XW23} and references therein.\n\nEigenvalue optimization problems for the weighted Laplacian have also attracted considerable attention. For instance, Chiacchio and di Blasio \\cite{CB12} proved that, among all origin-symmetric regions with fixed Gaussian volume, the ball maximizes the first nonzero Neumann eigenvalue; Brock, Francesco and di Blasio \\cite{BCB16} obtained optimal Szeg\\\"o-Weinberger type inequalities under a weighted measure.\nAdditional results on weighted eigenvalue problems can be found in \\cite{BCHT13, BCKT16, BCB16, CM24, CG22, GW26} and the references therein.\n\nInspired by the work of Brock, Francesco and di Blasio \\cite{BCB16} on the weighted Laplace operator and that of Freitas and Laugesen \\cite{FL21} on the second Robin eigenvalue, we study the shape optimization problem for the second Robin eigenvalue of the following class of problems\n\\begin{align}\\label{eq 1.1}\n \\begin{cases}\n -\\operatorname{div} (e^{h(|x|)}\\nabla u)=\\lambda e^{h(|x|)} u, \\quad & x\\in \\Omega,\\\\\n \\frac{\\partial u}{\\partial \\nu}+\\alpha u=0, \\quad & x\\in \\partial \\Omega,\n \\end{cases}\n\\end{align}\nwhere $\\alpha\\in \\mathbb{R}$. Here and in the sequel, $\\Omega$ denotes a bounded connected domain in $\\mathbb{R}^m$ with Lipchitz boundary, $\\nu$ is the outward normal to $\\partial \\Omega$, and $h(r)$ is a $C^2$ function on $[0, +\\infty)$. We define the weighted measure associated with $h$ by \n$$\nd\\gamma_h=e^{h(|x|)}\\,dx, \\qquad x\\in \\mathbb{R}^m,\n$$\nand denote by $W^{1,2}(\\Omega; \\gamma_h)$ the weighted Sobolev space equipped with the norm\n\\begin{align*}\n||u||_{W^{1,2}(\\Omega;\\gamma_h)}:=\\left(\\int_\\Omega u^2 \\, d\\gamma_h\\right)^{1/2}+ \\left(\\int_\\Omega |\\nabla u|^2 \\, d\\gamma_h\\right)^{1/2}. \n\\end{align*}\n\nIt is well known that the problem \\eqref{eq 1.1} has discrete spectrum. We denote its eigenvalue by $\\l_{k, \\alpha}(\\Omega; \\gamma_h)$ for $k=1,2, \\cdots$, which satisfy\n \\begin{align*}\n \\l_{1, \\alpha}(\\Omega;\\gamma_h) < \\l_{2, \\alpha}(\\Omega;\\gamma_h) \\leq \\l_{3, \\alpha}(\\Omega;\\gamma_h) \\leq \\cdots \\to +\\infty,\n \\end{align*}\nwith each eigenvalue repeated according to its multiplicity. \nBy standard spectral theory for self-adjoint compact operators, the first two eigenvalues of \\eqref{eq 1.1} admit the variational characterizations: \n\\begin{align}\n \\lambda_{1,\\alpha}(\\Omega; \\gamma_h)=\\inf\\{\\frac{\\int_{\\Omega}|\\nabla u|^{2}d\\gamma_h+\\alpha\\int_{\\partial \\Omega}u^{2} e^{h}\\, dA}{\\int_{\\Omega}u^{2}d\\gamma_h}: u\\in W^{1,2}(\\Omega;\\gamma_h)\\setminus\\{0\\}\\},\n\\end{align}\nwhere $dA$ is the\ninduced measure on $\\partial \\Omega$;\nand\n\\begin{align}\\label{eq 1.3}\n \\lambda_{2,\\alpha}(\\Omega;\\gamma_h)=\\inf\\{\\frac{\\int_{\\Omega}|\\nabla u|^{2}d\\gamma_h+\\alpha\\int_{\\partial \\Omega}u^{2} e^{h}dA}{\\int_{\\Omega}u^{2}d\\gamma_h}: u\\in W^{1,2}(\\Omega;\\gamma_h)\\setminus\\{0\\}, \\int_{\\Omega}uu_{1}d\\gamma_h=0\\},\n\\end{align}\nwhere $u_1(x)$ is an eigenfunction corresponding to $\\l_{1, \\alpha}(\\Omega;\\gamma_h)$.\n\nTo state our main results, we introduce the following Steklov eigenvalue problem on $\\Omega$ \n\\begin{align}\\label{eq 1.4}\n\\begin{cases}\n -\\operatorname{div} (e^{h(|x|)}\\nabla u)=0 \\quad & x\\in \\Omega,\\\\\n \\frac{\\partial u}{\\partial \\nu}=\\sigma u, \\quad & x\\in \\partial \\Omega.\n\\end{cases} \n\\end{align}\nThe first nonzero Steklov eigenvalue can be characterized variationally by\n\\begin{align}\n\\sigma_1(\\Omega; \\gamma_h)=\\inf\\left\\{ \\frac{\\int_{\\Omega} |\\nabla u|^2 \\, d\\gamma_h}{\\int_{\\partial \\Omega} u^2 e^{h}\\, dA} : u\\in W^{1,2}(\\Omega; \\gamma_h )\\setminus\\{0\\},\\;\\; \\int_{\\partial \\Omega} ue^{h(|x|)}\\ dA=0 \\right\\}.\n\\end{align}\n In this paper, we establish the following optimal isoperimetric inequality for the second Robin eigenvalue of \\eqref{eq 1.1}.\n\n\\begin{align}\\label{eq 1.1}\n    \\begin{cases}\n        -\\div (e^{h(|x|)}\\nabla u)=\\l e^{h(|x|)} u, \\quad & x\\in \\Omega,\\\\\n        \\frac{\\p u}{\\p \\nu}+\\a u=0, \\quad & x\\in \\p \\Omega,\n    \\end{cases}\n\\end{align}\n\n\\begin{align}\\label{eq 1.4}\n\\begin{cases}\n      -\\div (e^{h(|x|)}\\nabla u)=0 \\quad & x\\in \\Omega,\\\\\n        \\frac{\\p u}{\\p \\nu}=\\sigma u, \\quad & x\\in \\p \\Omega.\n\\end{cases}    \n\\end{align}", "full_context": "The classical Szeg\\\"o-Weinberger inequality \\cite{Sz54, Wei56} states that the ball uniquely maximizes the second (first nonzero) Neumann eigenvalue of Laplacian among all bounded domains of the same volume in Euclidean space. Using the arguments of Szeg\\\"o and Weinberger, analogous inequalities for the second Neumann eigenvalues have been established in various settings: Ashbaugh-Benguria \\cite{AB95} extended the the inequality to bounded domains in hemisphere and in hyperbolic space; Chiacchio and di Blasio \\cite{CB12} obtained a sharp isopermetric inequality for the first nonzero Neumann eigenvalue for bounded domians symmetric about the origin in Gauss space. \nMoreover, it is expected (cf. \\cite[Problem 4.41]{Hen17}) that for a range of negative Robin parameters, the second Robin eigenvalue should be maximized by a round ball. This expectation has recently been confirmed by Freitas and Laugesen via the Szeg\\\"o \\cite{FL20} and the Weinberger \\cite{FL21} approaches. \nFor further results on Szeg\\\"o-Weinberger type inequalities, we refer to \\cite{AS96, As99, BBC20, BCB16, LL23, LWW22, LWW23, MW24, Wang19, XW23} and references therein.\n\nEigenvalue optimization problems for the weighted Laplacian have also attracted considerable attention. For instance, Chiacchio and di Blasio \\cite{CB12} proved that, among all origin-symmetric regions with fixed Gaussian volume, the ball maximizes the first nonzero Neumann eigenvalue; Brock, Francesco and di Blasio \\cite{BCB16} obtained optimal Szeg\\\"o-Weinberger type inequalities under a weighted measure.\nAdditional results on weighted eigenvalue problems can be found in \\cite{BCHT13, BCKT16, BCB16, CM24, CG22, GW26} and the references therein.\n\nInspired by the work of Brock, Francesco and di Blasio \\cite{BCB16} on the weighted Laplace operator and that of Freitas and Laugesen \\cite{FL21} on the second Robin eigenvalue, we study the shape optimization problem for the second Robin eigenvalue of the following class of problems\n\\begin{align}\\label{eq 1.1}\n \\begin{cases}\n -\\operatorname{div} (e^{h(|x|)}\\nabla u)=\\lambda e^{h(|x|)} u, \\quad & x\\in \\Omega,\\\\\n \\frac{\\partial u}{\\partial \\nu}+\\alpha u=0, \\quad & x\\in \\partial \\Omega,\n \\end{cases}\n\\end{align}\nwhere $\\alpha\\in \\mathbb{R}$. Here and in the sequel, $\\Omega$ denotes a bounded connected domain in $\\mathbb{R}^m$ with Lipchitz boundary, $\\nu$ is the outward normal to $\\partial \\Omega$, and $h(r)$ is a $C^2$ function on $[0, +\\infty)$. We define the weighted measure associated with $h$ by \n$$\nd\\gamma_h=e^{h(|x|)}\\,dx, \\qquad x\\in \\mathbb{R}^m,\n$$\nand denote by $W^{1,2}(\\Omega; \\gamma_h)$ the weighted Sobolev space equipped with the norm\n\\begin{align*}\n||u||_{W^{1,2}(\\Omega;\\gamma_h)}:=\\left(\\int_\\Omega u^2 \\, d\\gamma_h\\right)^{1/2}+ \\left(\\int_\\Omega |\\nabla u|^2 \\, d\\gamma_h\\right)^{1/2}. \n\\end{align*}\n\nIt is well known that the problem \\eqref{eq 1.1} has discrete spectrum. We denote its eigenvalue by $\\l_{k, \\alpha}(\\Omega; \\gamma_h)$ for $k=1,2, \\cdots$, which satisfy\n \\begin{align*}\n \\l_{1, \\alpha}(\\Omega;\\gamma_h) < \\l_{2, \\alpha}(\\Omega;\\gamma_h) \\leq \\l_{3, \\alpha}(\\Omega;\\gamma_h) \\leq \\cdots \\to +\\infty,\n \\end{align*}\nwith each eigenvalue repeated according to its multiplicity. \nBy standard spectral theory for self-adjoint compact operators, the first two eigenvalues of \\eqref{eq 1.1} admit the variational characterizations: \n\\begin{align}\n \\lambda_{1,\\alpha}(\\Omega; \\gamma_h)=\\inf\\{\\frac{\\int_{\\Omega}|\\nabla u|^{2}d\\gamma_h+\\alpha\\int_{\\partial \\Omega}u^{2} e^{h}\\, dA}{\\int_{\\Omega}u^{2}d\\gamma_h}: u\\in W^{1,2}(\\Omega;\\gamma_h)\\setminus\\{0\\}\\},\n\\end{align}\nwhere $dA$ is the\ninduced measure on $\\partial \\Omega$;\nand\n\\begin{align}\\label{eq 1.3}\n \\lambda_{2,\\alpha}(\\Omega;\\gamma_h)=\\inf\\{\\frac{\\int_{\\Omega}|\\nabla u|^{2}d\\gamma_h+\\alpha\\int_{\\partial \\Omega}u^{2} e^{h}dA}{\\int_{\\Omega}u^{2}d\\gamma_h}: u\\in W^{1,2}(\\Omega;\\gamma_h)\\setminus\\{0\\}, \\int_{\\Omega}uu_{1}d\\gamma_h=0\\},\n\\end{align}\nwhere $u_1(x)$ is an eigenfunction corresponding to $\\l_{1, \\alpha}(\\Omega;\\gamma_h)$.\n\nTo state our main results, we introduce the following Steklov eigenvalue problem on $\\Omega$ \n\\begin{align}\\label{eq 1.4}\n\\begin{cases}\n -\\operatorname{div} (e^{h(|x|)}\\nabla u)=0 \\quad & x\\in \\Omega,\\\\\n \\frac{\\partial u}{\\partial \\nu}=\\sigma u, \\quad & x\\in \\partial \\Omega.\n\\end{cases} \n\\end{align}\nThe first nonzero Steklov eigenvalue can be characterized variationally by\n\\begin{align}\n\\sigma_1(\\Omega; \\gamma_h)=\\inf\\left\\{ \\frac{\\int_{\\Omega} |\\nabla u|^2 \\, d\\gamma_h}{\\int_{\\partial \\Omega} u^2 e^{h}\\, dA} : u\\in W^{1,2}(\\Omega; \\gamma_h )\\setminus\\{0\\},\\;\\; \\int_{\\partial \\Omega} ue^{h(|x|)}\\ dA=0 \\right\\}.\n\\end{align}\n In this paper, we establish the following optimal isoperimetric inequality for the second Robin eigenvalue of \\eqref{eq 1.1}.\n\n\\begin{align}\\label{eq 1.1}\n    \\begin{cases}\n        -\\div (e^{h(|x|)}\\nabla u)=\\l e^{h(|x|)} u, \\quad & x\\in \\Omega,\\\\\n        \\frac{\\p u}{\\p \\nu}+\\a u=0, \\quad & x\\in \\p \\Omega,\n    \\end{cases}\n\\end{align}\n\n\\begin{align}\\label{eq 1.4}\n\\begin{cases}\n      -\\div (e^{h(|x|)}\\nabla u)=0 \\quad & x\\in \\Omega,\\\\\n        \\frac{\\p u}{\\p \\nu}=\\sigma u, \\quad & x\\in \\p \\Omega.\n\\end{cases}    \n\\end{align}\n\nIt is well known that the problem \\eqref{eq 1.1} has discrete spectrum. We denote its eigenvalue by $\\l_{k, \\a}(\\Omega; \\gamma_h)$ for $k=1,2, \\cdots$, which satisfy\n \\begin{align*}\n \\l_{1, \\a}(\\Omega;\\gamma_h) < \\l_{2, \\a}(\\Omega;\\gamma_h) \\leq \\l_{3, \\a}(\\Omega;\\gamma_h) \\leq \\cdots \\to +\\infty,\n \\end{align*}\nwith each eigenvalue repeated according to its multiplicity. \nBy standard spectral theory for self-adjoint compact operators, the first two eigenvalues of \\eqref{eq 1.1} admit the variational characterizations: \n\\begin{align}\n \\lambda_{1,\\alpha}(\\Omega; \\gamma_h)=\\inf\\{\\frac{\\int_{\\Omega}|\\nabla u|^{2}d\\gamma_h+\\alpha\\int_{\\partial \\Omega}u^{2} e^{h}\\, dA}{\\int_{\\Omega}u^{2}d\\gamma_h}: u\\in W^{1,2}(\\Omega;\\gamma_h)\\setminus\\{0\\}\\},\n\\end{align}\nwhere $dA$ is the\ninduced measure on $\\p \\Omega$;\nand\n\\begin{align}\\label{eq 1.3}\n \\lambda_{2,\\alpha}(\\Omega;\\gamma_h)=\\inf\\{\\frac{\\int_{\\Omega}|\\nabla u|^{2}d\\gamma_h+\\alpha\\int_{\\partial \\Omega}u^{2} e^{h}dA}{\\int_{\\Omega}u^{2}d\\gamma_h}: u\\in W^{1,2}(\\Omega;\\gamma_h)\\setminus\\{0\\}, \\int_{\\Omega}uu_{1}d\\gamma_h=0\\},\n\\end{align}\nwhere $u_1(x)$ is an eigenfunction corresponding to $\\l_{1, \\a}(\\Omega;\\gamma_h)$.\n\nBy taking $\\a=-\\sigma_1(B;\\gamma_h)$, Theorem \\ref{thm 1} yields the following Brock–Weinstock inequality, which was proved by Brock and Chiacchio \\cite[Theorems 1.4 and 1.5]{BC25} and by Mao and Zhang \\cite[Corollary 1.2]{MZ24}. \n\\begin{corollary}\\label{cor}\nUnder the hypotheses of Theorem \\ref{thm 1}, we have \n\\begin{align*}\n\\sigma_1(\\Omega; \\gamma_h)\\le \\sigma_1(B; \\gamma_h).\n\\end{align*}\nEquality holds if and only if $\\Omega$ is isometric to the round ball $B$.\n\\end{corollary}\n\n\\begin{proposition} \\label{prop 2.4}\nSuppose $h'(r)\\ge 0$, and $\\a\\ge -\\sigma_1(B(R); \\gamma_h)$. Then\n\\begin{align*}\n \\lambda_{2,\\alpha}(B(R); \\gamma_h)\\geq 0. \n\\end{align*}\n\\end{proposition}\n\\begin{proof}\n Let $g(r)$ be as defined in Proposition \\ref{prop 2.3}. \n Since \n \\begin{align*}\n \\int_{\\partial B(R)}g(r)\\frac{x_{i}}{r}e^{h(r)}dA=0, 1\\leq i\\leq m,\n \\end{align*}\n the functions $u_{i}=g(r)\\frac{x_{i}}{r}, 1\\leq i\\leq m,$ are admissible test functions for $\\sigma_{1}(B(R), \\gamma_h)$. Therefore,\n \\begin{align}\\label{eq 2.22}\n \\sum_{i=1}^{m}\\int_{B(R)}|\\nabla u_{i}|^{2}\\, d\\gamma_h\\geq \\sigma_1(B(R); \\gamma_h) \\sum_{i=1}^{m}\\int_{\\partial B(R)}|u_{i}|^{2}e^{h(r)}dA,\n \\end{align}\n where $dA$ is the induced measure on $\\p B$.\nUsing \n \\begin{align*}\n \\sum_{i=1}^{m}|\\nabla u_{i}|^{2}=g'(r)^{2}+\\frac{m-1}{r^{2}}g(r)^{2}.\n \\end{align*}\n and the fact $h(r)\\le h(R)$ for $r\\in[0, R]$, we estimate from \\ref{eq 2.22} \n \\begin{align}\\label{eq 2.23}\n 0&\\leq \\sum_{i=1}^{m}\\int_{B(R)}|\\nabla u_{i}|^{2}e^{h(r)}dx- \\sigma_1(B(R); \\gamma_h)\\sum_{i=1}^{m}\\int_{\\partial B(R)}|u_{i}|^{2}e^{h(r)}dA\\nonumber\\\\\n &\\leq \\omega_{m-1}\\int_{0}^{R}\\left(g'(r)^{2}+\\frac{m-1}{r^{2}}g^{2}\\right)e^{h(r)}r^{m-1}dr+\\alpha \\omega_{m-1} g(R)^{2}e^{h(R)}R^{m-1},\n \\end{align}\n where $\\omega_{m-1}$ denotes the surface area of the unit sphere $\\mathbb{S}^{m-1}$ in $\\R^m$.\nUsing integration by parts and \\eqref{eq 2.11}, we compute \n\\begin{align*}\n &\\int_{0}^{R}\\left(g'(r)^{2}+\\frac{m-1}{r^{2}}g^{2}\\right)e^{h(r)}r^{m-1}dr\\\\\n =\\quad &gg'e^{h(r)}r^{m-1}\\Big|_{0}^{R}-\\int_{0}^{R}g(g'e^{h(r)}r^{m-1})'dr+\\int_{0}^{R}\\frac{m-1}{r^{2}}g^{2}e^{h(r)}r^{m-1}dr\\\\\n =\\quad &g(R)g'(R)e^{h(R)}R^{m-1}-\\int_{0}^{R}g(g''+(\\frac{m-1}{r}+h')g')e^{h(r)}r^{m-1}dr\\\\\n &+\\int_{0}^{R}\\frac{m-1}{r^{2}}g^{2}e^{h(r)}r^{m-1}dr\\\\\n =&g(R)g'(R)e^{h(R)}R^{m-1}+\\int_{0}^{R}g^{2}(\\lambda_{2,\\alpha}(B(R);\\gamma_h)-\\frac{m-1}{r^{2}})e^{h(r)}r^{m-1}dr\\\\\n &+\\int_{0}^{R}\\frac{m-1}{r^{2}}g^{2}e^{h(r)}r^{m-1}dr.\n\\end{align*}\nSubstituting this equality into inequality \\eqref{eq 2.23} yields\n \\begin{align*}\n 0\\leq& \\lambda_{2,\\alpha}(B(R);\\gamma_h)\\int_{0}^{R}g^{2}(r)e^{h(r)}r^{m-1}dr+g(R)e^{h(R)}R^{m-1}(g'(R)+\\alpha g(R))\\\\\n =&\\lambda_{2,\\alpha}(B(R);\\gamma_h)\\int_{0}^{R}g^{2}(r)e^{h(r)}r^{m-1}\\, dr,\n \\end{align*}\nwhich implies $\\lambda_{2,\\alpha}(B(R);\\gamma_h)\\geq 0$.\n\\end{proof}\n\nFrom now on, we assume $\\Omega\\subset \\R^m$ is a bounded Lipschitz domain symmetric about the origin, and $B\\subset \\R^m$ is the origin-centered round bound with the same $\\gamma_h$ voulme as $\\Omega$. Let $R$ be the radius of $B$. We extend the function $g$ defined in \\eqref{eq 2.11} by\n\\begin{equation}\\label{eq 3.1}\ng(r)=g(R) e^{-\\a(r-R)}\\text{\\quad for \\quad} r\\ge R.\n\\end{equation}\nBy definition, $g$ is continuously differentiable on $(0,\\infty)$. If $\\a\\le 0$, then by Proposition \\ref{prop 2.2}, $g$ is increasing on $(0,\\infty)$. In the sequel, $\\sigma_1(B; \\gamma_h)$ denotes the first nonzero Steklov eigenvalue of $B$ as defined in \\eqref{eq 2.2}.\n\n{\\bf Case 2.} $r\\ge R$. For $r\\ge R$, \n\\begin{align*}\n F(r)=(-\\alpha^{2}+\\frac{m-1}{r^{2}}+\\alpha\\frac{m-1}{r}+\\alpha h^{'})g^{2}(r),\n\\end{align*}\nand a direct computation yields\n\\begin{align*}\n F'(r)=(2\\alpha^{3}+\\alpha h''-2\\alpha^{2}h'-2\\alpha^{2}\\frac{m-1}{r}-3\\alpha \\frac{m-1}{r^{2}}-2\\frac{m-1}{r^{3}})g^{2}(r).\n\\end{align*}\nSince $\\alpha\\leq 0$, $h'\\geq 0$ and $h^{''}\\geq 0$, we obtain\n\\begin{align*}\n F'(r)\\le& \\left(-2\\alpha^{2}\\frac{m-1}{r}-3\\alpha \\frac{m-1}{r^{2}}-2\\frac{m-1}{r^{3}}\\right) g(r)^2\\\\\n =&-\\frac{m-1}{r}(2(\\alpha+\\frac{3}{4r})^{2}+\\frac{7}{8}\\frac{1}{r^{2}})g(r)^2<0,\n\\end{align*}\nproving that $F(r)$ is monotonically decreasing on $(0, R)$.\n\\end{proof}\n The main idea in proving Theorem \\ref{thm 1} is to construct trial functions for $\\l_{2, \\a}$ (see \\eqref{eq 1.3}) using techniques introduced by Weinberger. We first recall a monotonicity lemma for weighted symmetrization.\n\\begin{lemma}\\label{Lemma 3.2}\n Let $\\Omega\\subset \\R^m$ be a bounded set, and let $B$ be the origin-centered round ball with the same $\\gamma_h$ volume as $\\Omega$. If $f(r)$ is monotonically decreasing on $[0,+\\infty)$, then \n \\begin{align*}\n \\int_\\Omega f(|x|)\\, d\\gamma_m\\le \\int_B f(|x|)\\, d\\gamma_m.\n \\end{align*}\n Equality holds if and only if $\\Omega=B$.\n\\end{lemma}\n\\begin{proof}\n Lemma 3.2 follows directly from Hardy-Littlewood inequality; see the proof of inequality (4.24) in \\cite[Page 213]{BCB16}.\n\\end{proof}\n\nNow we now proceed to prove Theorem \\ref{thm 1}.\n\\begin{proof}\n Let $\\Omega$ be a bounded Lipschitz domain symmetric about the origin in $\\R^m$, and let $B\\subset \\R^m$ be the origin-centered round ball of radius $R$ satisfying \n $$\n\\int_B \\, d\\gamma_h =\\int_\\Omega \\, d\\gamma_h.\n $$\nDefine $g(r)$ for $r\\in [0, +\\infty)$ as in \\eqref{eq 3.1}, and for each $1\\le i\\le m$ set\n\\begin{align*}\n v_i(x)=g(|x|)\\frac{x_i}{|x|}, \n\\end{align*}\nwhere $x=(x_1, x_2, \\cdots, x_m)\\in \\R^m$.", "post_theorem_intro_text_len": 2156, "post_theorem_intro_text": "By taking $\\alpha=0$, Theorem \\ref{thm 1} implies that the ball centered at the\norigin is the unique set maximizer the second Neumann eigenvalue $\\mu_2(\\Omega;\\gamma_h)$ among all bounded Lipschitz domains $\\Omega$ in $\\mathbb{R}^m$ with prescribed $\\gamma_h$-measure and symmetric about the origin. This result was previously  proved by Brock, Chiacchio, and di Blasio (Theorem 1.2 of \\cite{BCB16}). Note that Theorem \\ref{thm 1} applies in particular  to $\\l_{2, \\alpha}(\\Omega; e^{|x|^2/2})$. Hence, our result provides information about  eigenvalues of the problem\n$$\n-\\Delta u-x\\cdot \\nabla u=\\lambda u\n$$\nwhich has been widely studied in literature (see, e.g. \\cite{BCM12, BMP13}). \nThe symmetry assumption on $\\Omega$ in Theorem \\ref{thm 1}\n ensures the validity of  the orthogonality conditions \\eqref{eq 3.5}, as the weighted measure $e^{h(|x|)}$ is  radially\n symmetric about the origin.\nIt remains an interesting question that whether \\eqref{eq 1.6} \nholds in the case of the Gauss measure $e^{-|x|^2/2}dx$.   \n\nBy taking $\\alpha=-\\sigma_1(B;\\gamma_h)$, Theorem \\ref{thm 1} yields the following Brock–Weinstock inequality, which was proved by Brock and Chiacchio \\cite[Theorems 1.4 and 1.5]{BC25} and by Mao and Zhang \\cite[Corollary 1.2]{MZ24}. \n\\begin{corollary}\\label{cor}\nUnder the hypotheses of Theorem \\ref{thm 1}, we have \n\\begin{align*}\n\\sigma_1(\\Omega; \\gamma_h)\\le \\sigma_1(B; \\gamma_h).\n\\end{align*}\nEquality holds if and only if $\\Omega$ is isometric to the round ball $B$.\n\\end{corollary}\n\nThe main technique in the proof of Theorem \\ref{thm 1} relies on Weinberger’s trick, in which  test functions are constructed using eigenfunctions on a round ball to estimate eigenvalues. The main difficulties in the proof include analyzing the properties of the second eigenfunction of \\eqref{eq 1.1} on the round ball  and establishing the required monotonicity to apply Weinberger’s method. The rest of the paper is organized as follows.\nIn Section \\ref{sect2}, we study properties of the Robin and Steklov eigenvalue  problems for balls in the weighted space. In Section  \\ref{sect3}, we prove Theorem \\ref{thm 1} and Corollary \\ref{cor}.", "sketch": "The proof of Theorem~\\ref{thm 1} \"relies on Weinberger’s trick, in which test functions are constructed using eigenfunctions on a round ball to estimate eigenvalues.\" The \"main difficulties\" are \"analyzing the properties of the second eigenfunction of \\eqref{eq 1.1} on the round ball\" and \"establishing the required monotonicity to apply Weinberger’s method.\" The symmetry assumption on $\\Omega$ \"ensures the validity of the orthogonality conditions \\eqref{eq 3.5}, as the weighted measure $e^{h(|x|)}$ is radially symmetric about the origin.\"", "expanded_sketch": "The proof of the main theorem relies on Weinberger’s trick, in which test functions are constructed using eigenfunctions on a round ball to estimate eigenvalues. The main difficulties are analyzing the properties of the second eigenfunction of\n\\begin{align}\\label{eq 1.1}\n    \\begin{cases}\n        -\\div (e^{h(|x|)}\\nabla u)=\\l e^{h(|x|)} u, \\quad & x\\in \\Omega,\\\\\n        \\frac{\\p u}{\\p \\nu}+\\a u=0, \\quad & x\\in \\p \\Omega,\n    \\end{cases}\n\\end{align}\non the round ball and establishing the required monotonicity to apply Weinberger’s method. The symmetry assumption on $\\Omega$ ensures the validity of the orthogonality conditions\n\\begin{align}\\label{eq 3.5}\n        \\int_\\Omega v_i(x)u(x)   \\, d\\gamma_h=0, \\quad i=1,2,\\cdots, m.\n    \\end{align}\nas the weighted measure $e^{h(|x|)}$ is radially symmetric about the origin.", "expanded_theorem": "\\label{thm 1}\nLet $\\Omega\\subset \\mathbb{R}^m$ be a bounded  Lipschitz domain symmetric about the origin. Suppose $h(r)\\in C^2([0, +\\infty))$ satisfies  $h'(r)\\ge 0$ and $h''(r)\\ge 0$ for $r\\in (0, +\\infty)$. Let  $B \\subset \\mathbb{R}^m$  be the  origin-centered  round ball with the same $\\gamma_h$-volume as $\\Omega$, namely $ \\int_{\\Omega} e^{h(|x|)}\\, dx= \\int_{B} e^{h(|x|)}\\, dx$. Denote by $\\sigma_1(B; \\gamma_h)$ the first nonzero Steklov eigenvalue of\n\\begin{align}\\label{eq 1.4}\n\\begin{cases}\n      -\\div (e^{h(|x|)}\\nabla u)=0 \\quad & x\\in \\Omega,\\\\\n        \\frac{\\p u}{\\p \\nu}=\\sigma u, \\quad & x\\in \\p \\Omega.\n\\end{cases}    \n\\end{align}\non $B$, and $\\l_{2,\\alpha}(\\Omega; \\gamma_h)$ the second Robin eigenvalue of\n\\begin{align}\\label{eq 1.1}\n    \\begin{cases}\n        -\\div (e^{h(|x|)}\\nabla u)=\\l e^{h(|x|)} u, \\quad & x\\in \\Omega,\\\\\n        \\frac{\\p u}{\\p \\nu}+\\a u=0, \\quad & x\\in \\p \\Omega,\n    \\end{cases}\n\\end{align}\non $\\Omega$.  If  $\\alpha \\in [-\\sigma_1(B; \\gamma_h), 0]$, then \n\\begin{align}\\label{eq 1.6}\n\\l_{2,\\alpha}(\\Omega; \\gamma_h) \\le \\l_{2,\\alpha}(B; \\gamma_h).\n\\end{align}\nEquality holds  if and only if \n$\\Omega=B$.,", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let Ω ⊂ R^m be a bounded Lipschitz domain symmetric about the origin, and let h ∈ C^2([0,∞)) satisfy h'(r) ≥ 0 and h''(r) ≥ 0 for all r > 0. Define the weighted measure dγ_h = e^{h(|x|)} dx. Let B ⊂ R^m be the origin-centered round ball with the same γ_h-volume as Ω, i.e. ∫_Ω e^{h(|x|)} dx = ∫_B e^{h(|x|)} dx. For a domain D, let λ_{2,α}(D; γ_h) denote the second eigenvalue of the weighted Robin problem -div(e^{h(|x|)}∇u) = λ e^{h(|x|)}u in D, with boundary condition ∂u/∂ν + αu = 0 on ∂D. Let σ_1(B; γ_h) denote the first nonzero Steklov eigenvalue on B for -div(e^{h(|x|)}∇u) = 0 in B, with ∂u/∂ν = σu on ∂B. If α ∈ [-σ_1(B; γ_h), 0], which quantitative estimate holds?", "correct_choice": {"label": "A", "text": "One has λ_{2,α}(Ω; γ_h) ≤ λ_{2,α}(B; γ_h), and equality holds if and only if Ω = B."}, "choices": [{"label": "B", "text": "One has \\(\\lambda_{2,\\alpha}(\\Omega; \\gamma_h) \\le \\lambda_{2,\\alpha}(B; \\gamma_h)\\) for every \\(\\alpha \\le 0\\), and equality holds if and only if \\(\\Omega = B\\)."}, {"label": "C", "text": "One has \\(\\lambda_{2,\\alpha}(\\Omega; \\gamma_h) \\le \\lambda_{2,\\alpha}(B; \\gamma_h)\\)."}, {"label": "D", "text": "One has \\(\\lambda_{2,\\alpha}(\\Omega; \\gamma_h) \\le \\lambda_{2,\\alpha}(B; \\gamma_h)\\) whenever \\(\\alpha \\in [-\\sigma_1(\\Omega; \\gamma_h),0]\\), and equality holds if and only if \\(\\Omega = B\\)."}, {"label": "E", "text": "One has \\(\\lambda_{2,\\alpha}(\\Omega; \\gamma_h) \\ge \\lambda_{2,\\alpha}(B; \\gamma_h)\\), and equality holds if and only if \\(\\Omega = B\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "sharp range for \\alpha tied to monotonicity/Steklov threshold", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the equality characterization", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence of the threshold on B rather than on \\Omega", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "extremal direction from Weinberger trial-function argument", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the conclusion directly; it only states the hypotheses and asks which estimate is valid. The correct choice is not leaked by wording, though the setup strongly suggests a known extremal theorem."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem reproduces the full hypothesis set and asks for the exact conclusion. The correct option is basically the theorem statement itself."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact sharp statement from nearby variants: stronger range in B, weaker conclusion in C, wrong threshold domain in D, and reversed inequality in E. But it mainly tests precise recall rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong and mathematically meaningful: overgeneralized parameter range, omission of the equality characterization, wrong dependence on the Steklov threshold, and reversed extremal direction. These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A solid theorem-discrimination MCQ with good distractors and no answer leakage, but it is largely a direct restatement of a known result rather than a deep reasoning question."}}
{"id": "2602.19511v2", "paper_link": "http://arxiv.org/abs/2602.19511v2", "theorems_cnt": 2, "theorem": {"env_name": "Theorem", "content": "\\label{ILR2n} \nLet $n$ be a positive integer, and let $K$ be a simplicial $n$-complex that is $N_1^{(n)}$, $N_2^{(n)}$ or $N_3^{(n)}$. Then, for every embedding $f$ of $K$ into ${\\mathbb R}^{2n}$, there exists an element $\\lambda$ in $\\Lambda^{n-1,n}(K)$ such that ${\\rm lk}_{2}(f(\\lambda)) = 1$.", "start_pos": 6546, "end_pos": 6872, "label": "ILR2n"}, "ref_dict": {"VKF": "\\begin{Theorem}{\\rm (van Kampen \\cite{VK33}, Flores \\cite{Flores32})}\\label{VKF} \nLet $n$ be a positive integer, and let $K$ be a simplicial $n$-complex that is $\\sigma_{2n+2}^{n}$ or $[3]^{*n+1}$. Then for every generic immersion $\\varphi$ of $K$ into ${\\mathbb R}^{2n}$, the following holds: \n\\begin{eqnarray*}\n\\sum_{\\substack{s,s'\\in\\varDelta^{n}(K)\\\\ s\\cap s'=\\varnothing}}l(\\varphi(s),\\varphi(s'))\\equiv 1\\pmod{2}. \n\\end{eqnarray*}\n\\end{Theorem}", "ILR2n": "\\begin{Theorem}\\label{ILR2n} \nLet $n$ be a positive integer, and let $K$ be a simplicial $n$-complex that is $N_1^{(n)}$, $N_2^{(n)}$ or $N_3^{(n)}$. Then, for every embedding $f$ of $K$ into ${\\mathbb R}^{2n}$, there exists an element $\\lambda$ in $\\Lambda^{n-1,n}(K)$ such that ${\\rm lk}_{2}(f(\\lambda)) = 1$. \n\\end{Theorem}", "min": "\\begin{Proposition}\\label{min}\nLet $N$ be a proper subcomplex of $N_1^{(n)}$, $N_2^{(n)}$ or $N_3^{(n)}$. Then there exists an embedding $f:N\\to {\\mathbb R}^{2n}$ such that ${\\rm lk}_{2}(f(\\lambda))= 0$ for any element $\\lambda$ in $\\Lambda^{n-1,n}(N)$.\n\\end{Proposition}", "nonsp": "\\begin{center}\n\\scalebox{0.525}{\\includegraphics*{nonsp}}\n\\caption{$K_1 \\sqcup K_4$, $K_1 \\sqcup K_{2,3}$ and $K_{1,1,3}$}\n\\label{nonsp}\n\\end{center}"}, "pre_theorem_intro_text_len": 4297, "pre_theorem_intro_text": "Throughout this paper, we work in the piecewise linear category. For the fundamentals of piecewise-linear topology, we refer the reader to \\cite{H69}, \\cite{RS82}. Let $K$ be a finite simplicial $n$-complex, which we identify with its polyhedron in this context. For an integer $k\\le n$, let $\\varDelta^{k}(K)$ denote the set of all $k$-simplices in $K$. Then the subcomplex $\\bigcup_{l\\le k}\\varDelta^{l}(K)$ of $K$ is called the {\\it $k$-skeleton} of $K$ and denoted by $K^{k}$. Let $f$ be an embedding of $K$ into $\\mathbb{R}^{m}$. Let $\\Lambda^{p,q}(K)$ be the set of all unordered pairs $\\{\\gamma, \\gamma'\\}$ of mutually disjoint subcomplexes of $K$ such that $\\gamma$ is homeomorphic to the $p$-sphere and $\\gamma'$ is homeomorphic to the $q$-sphere. We identify any pair $(\\gamma,\\gamma')$ in $\\Lambda^{p,q}(K)$ with the disjoint union $\\gamma\\sqcup \\gamma'$. Then, for any $\\lambda = \\gamma \\sqcup \\gamma' \\in \\Lambda^{p,q}(K)$, the image $f(\\lambda) = f(\\gamma) \\sqcup f(\\gamma')$ forms a two-component link in $f(K)$ consisting of a $p$-sphere and a $q$-sphere. If $p+q = m-1$, the \\textit{$\\mathbb{Z}_2$-linking number} ${\\rm lk}_2(L) = {\\rm lk}_2(S^p, S^q) = {\\rm lk}_2(S^q, S^p) \\in \\mathbb{Z}_2$ of a two-component link $L = S^p \\sqcup S^q$ in ${\\mathbb R}^m$ is well-defined (cf.\\ \\cite{ST80}). If $m=2n+1$, there are known examples of simplicial $n$-complexes with the following property: every embedding into $\\mathbb{R}^{2n+1}$ necessarily contains a two-component link of $n$-spheres with $\\mathbb{Z}_2$-linking number $1$. Such examples were constructed by Segal--Spie\\.{z} \\cite{SeSp92}, Lovasz--Schrijver \\cite{LS98}, Skopenkov \\cite{Sk03}, Taniyama \\cite{T00}, and more recently by the author \\cite{N26}. In particular, the case of $n=1$ is famous as the Conway--Gordon--Sachs theorem (\\cite{CG83}, \\cite{S84}), and such a simplicial $1$-complex (i.e., a graph) is said to be {\\it intrinsically linked}. \n\nOur purpose in this paper is to present an example of a minimal simplicial $n$-complex that possesses such an intrinsic linking property with respect to embeddings into $\\mathbb{R}^{2n}$. Let us introduce three simplicial $n$-complexes $N_{1}^{(n)}$, $N_2^{(n)}$ and $N_3^{(n)}$ as follows. First, let $\\sigma_{m}=|a_{0}a_{1}\\cdots a_{m}|$ be an $m$-simplex whose vertices are  $a_{0},a_{1},\\ldots,a_{m}$. This can be naturally regarded as a simplicial $m$-complex consisting of itself and all its faces. In this setting, the $k$-skeleton of $\\sigma_{m}$ is standardly denoted by $\\sigma_{m}^{k}$ for $k\\le m$. Then we define \n\\begin{eqnarray*}\nN_{1}^{(n)} = \\sigma_{2n+2}^{n}\\setminus \\{|a_{0}a_{j_{1}}\\cdots a_{j_{n}}|\\ |\\ 1\\le j_{1}<j_{2}<\\cdots<j_{n}\\le 2n+2\\}. \n\\end{eqnarray*}\nIn other words, $N_{1}^{(n)}$ is the subcomplex of $\\sigma_{2n+2}^{n}$ obtained by removing all $n$-simplices that contain the vertex $a_0$. Next, for a positive integer $k$, let $[k]^{*m}$ denote the $m$-fold join of $k$ points. Explicitly, we set\n\\begin{eqnarray*}\n[k]^{*m} = \n\\{a_{0}^{0},a_{1}^{0},\\ldots,a_{k-1}^{0}\\} * \\{a_{0}^{1},a_{1}^{1},\\ldots,a_{k-1}^{1}\\} * \\cdots * \\{a_{0}^{n},a_{1}^{n},\\ldots,a_{k-1}^{n}\\}. \n\\end{eqnarray*}\nNote that $[k]^{*n+1}$ can naturally be regarded as a simplicial $n$-complex. Then we define \n\\begin{eqnarray*}\nN_{2}^{(n)} = [3]^{*n+1}\\setminus \\{|a_{0}^{0}a_{j_{1}}^{1}\\cdots a_{j_{n}}^{n}|\\ |\\ j_{1},j_{2},\\ldots,j_{n}\\in\\{0,1,2\\}\\}. \n\\end{eqnarray*}\nIn other words, $N_{2}^{(n)}$ is the subcomplex of $[3]^{*n+1}$ obtained by removing all $n$-simplices that contain the vertex $a_{0}^{0}$. Finally, we define \n\\begin{eqnarray*}\nN_{3}^{(n)} = \\sigma_{2n+2}^{n}\\setminus \\varDelta^{n}(\\sigma_{n+1}). \n\\end{eqnarray*}\nIn other words, $N_{3}^{(n)}$ is the subcomplex of $\\sigma_{2n+2}^{n}$ obtained by removing all $n$-simplices contained in $\\sigma_{n+1}=|a_{0}a_{1}\\cdots a_{n+1}|$. As we will see in the next section, it is well known that neither $\\sigma_{2n+2}^{n}$ nor $[3]^{*n+1}$ cannot be embedded in ${\\mathbb R}^{2n}$ (Theorem \\ref{VKF}). On the other hand, it is also known that any proper subcomplex of $\\sigma_{2n+2}^{n}$ and of $[3]^{*n+1}$ can be embedded in $\\mathbb{R}^{2n}$ (Gr\\\"{u}nbaum \\cite{Grun69}). Therefore, each of $N_1^{(n)}$, $N_2^{(n)}$ and $N_3^{(n)}$ is embeddable in $\\mathbb{R}^{2n}$. Then we have the following:", "context": "Throughout this paper, we work in the piecewise linear category. For the fundamentals of piecewise-linear topology, we refer the reader to \\cite{H69}, \\cite{RS82}. Let $K$ be a finite simplicial $n$-complex, which we identify with its polyhedron in this context. For an integer $k\\le n$, let $\\varDelta^{k}(K)$ denote the set of all $k$-simplices in $K$. Then the subcomplex $\\bigcup_{l\\le k}\\varDelta^{l}(K)$ of $K$ is called the {\\it $k$-skeleton} of $K$ and denoted by $K^{k}$. Let $f$ be an embedding of $K$ into $\\mathbb{R}^{m}$. Let $\\Lambda^{p,q}(K)$ be the set of all unordered pairs $\\{\\gamma, \\gamma'\\}$ of mutually disjoint subcomplexes of $K$ such that $\\gamma$ is homeomorphic to the $p$-sphere and $\\gamma'$ is homeomorphic to the $q$-sphere. We identify any pair $(\\gamma,\\gamma')$ in $\\Lambda^{p,q}(K)$ with the disjoint union $\\gamma\\sqcup \\gamma'$. Then, for any $\\lambda = \\gamma \\sqcup \\gamma' \\in \\Lambda^{p,q}(K)$, the image $f(\\lambda) = f(\\gamma) \\sqcup f(\\gamma')$ forms a two-component link in $f(K)$ consisting of a $p$-sphere and a $q$-sphere. If $p+q = m-1$, the \\textit{$\\mathbb{Z}_2$-linking number} ${\\rm lk}_2(L) = {\\rm lk}_2(S^p, S^q) = {\\rm lk}_2(S^q, S^p) \\in \\mathbb{Z}_2$ of a two-component link $L = S^p \\sqcup S^q$ in ${\\mathbb R}^m$ is well-defined (cf.\\ \\cite{ST80}). If $m=2n+1$, there are known examples of simplicial $n$-complexes with the following property: every embedding into $\\mathbb{R}^{2n+1}$ necessarily contains a two-component link of $n$-spheres with $\\mathbb{Z}_2$-linking number $1$. Such examples were constructed by Segal--Spie\\.{z} \\cite{SeSp92}, Lovasz--Schrijver \\cite{LS98}, Skopenkov \\cite{Sk03}, Taniyama \\cite{T00}, and more recently by the author \\cite{N26}. In particular, the case of $n=1$ is famous as the Conway--Gordon--Sachs theorem (\\cite{CG83}, \\cite{S84}), and such a simplicial $1$-complex (i.e., a graph) is said to be {\\it intrinsically linked}.\n\nOur purpose in this paper is to present an example of a minimal simplicial $n$-complex that possesses such an intrinsic linking property with respect to embeddings into $\\mathbb{R}^{2n}$. Let us introduce three simplicial $n$-complexes $N_{1}^{(n)}$, $N_2^{(n)}$ and $N_3^{(n)}$ as follows. First, let $\\sigma_{m}=|a_{0}a_{1}\\cdots a_{m}|$ be an $m$-simplex whose vertices are $a_{0},a_{1},\\ldots,a_{m}$. This can be naturally regarded as a simplicial $m$-complex consisting of itself and all its faces. In this setting, the $k$-skeleton of $\\sigma_{m}$ is standardly denoted by $\\sigma_{m}^{k}$ for $k\\le m$. Then we define \n\\begin{eqnarray*}\nN_{1}^{(n)} = \\sigma_{2n+2}^{n}\\setminus \\{|a_{0}a_{j_{1}}\\cdots a_{j_{n}}|\\ |\\ 1\\le j_{1}<j_{2}<\\cdots<j_{n}\\le 2n+2\\}. \n\\end{eqnarray*}\nIn other words, $N_{1}^{(n)}$ is the subcomplex of $\\sigma_{2n+2}^{n}$ obtained by removing all $n$-simplices that contain the vertex $a_0$. Next, for a positive integer $k$, let $[k]^{*m}$ denote the $m$-fold join of $k$ points. Explicitly, we set\n\\begin{eqnarray*}\n[k]^{*m} = \n\\{a_{0}^{0},a_{1}^{0},\\ldots,a_{k-1}^{0}\\} * \\{a_{0}^{1},a_{1}^{1},\\ldots,a_{k-1}^{1}\\} * \\cdots * \\{a_{0}^{n},a_{1}^{n},\\ldots,a_{k-1}^{n}\\}. \n\\end{eqnarray*}\nNote that $[k]^{*n+1}$ can naturally be regarded as a simplicial $n$-complex. Then we define \n\\begin{eqnarray*}\nN_{2}^{(n)} = [3]^{*n+1}\\setminus \\{|a_{0}^{0}a_{j_{1}}^{1}\\cdots a_{j_{n}}^{n}|\\ |\\ j_{1},j_{2},\\ldots,j_{n}\\in\\{0,1,2\\}\\}. \n\\end{eqnarray*}\nIn other words, $N_{2}^{(n)}$ is the subcomplex of $[3]^{*n+1}$ obtained by removing all $n$-simplices that contain the vertex $a_{0}^{0}$. Finally, we define \n\\begin{eqnarray*}\nN_{3}^{(n)} = \\sigma_{2n+2}^{n}\\setminus \\varDelta^{n}(\\sigma_{n+1}). \n\\end{eqnarray*}\nIn other words, $N_{3}^{(n)}$ is the subcomplex of $\\sigma_{2n+2}^{n}$ obtained by removing all $n$-simplices contained in $\\sigma_{n+1}=|a_{0}a_{1}\\cdots a_{n+1}|$. As we will see in the next section, it is well known that neither $\\sigma_{2n+2}^{n}$ nor $[3]^{*n+1}$ cannot be embedded in ${\\mathbb R}^{2n}$ (Theorem \\ref{VKF}). On the other hand, it is also known that any proper subcomplex of $\\sigma_{2n+2}^{n}$ and of $[3]^{*n+1}$ can be embedded in $\\mathbb{R}^{2n}$ (Gr\\\"{u}nbaum \\cite{Grun69}). Therefore, each of $N_1^{(n)}$, $N_2^{(n)}$ and $N_3^{(n)}$ is embeddable in $\\mathbb{R}^{2n}$. Then we have the following:\n\n\\begin{Theorem}{\\rm (van Kampen \\cite{VK33}, Flores \\cite{Flores32})}\\label{VKF} \nLet $n$ be a positive integer, and let $K$ be a simplicial $n$-complex that is $\\sigma_{2n+2}^{n}$ or $[3]^{*n+1}$. Then for every generic immersion $\\varphi$ of $K$ into ${\\mathbb R}^{2n}$, the following holds: \n\\begin{eqnarray*}\n\\sum_{\\substack{s,s'\\in\\varDelta^{n}(K)\\\\ s\\cap s'=\\varnothing}}l(\\varphi(s),\\varphi(s'))\\equiv 1\\pmod{2}. \n\\end{eqnarray*}\n\\end{Theorem}", "full_context": "Throughout this paper, we work in the piecewise linear category. For the fundamentals of piecewise-linear topology, we refer the reader to \\cite{H69}, \\cite{RS82}. Let $K$ be a finite simplicial $n$-complex, which we identify with its polyhedron in this context. For an integer $k\\le n$, let $\\varDelta^{k}(K)$ denote the set of all $k$-simplices in $K$. Then the subcomplex $\\bigcup_{l\\le k}\\varDelta^{l}(K)$ of $K$ is called the {\\it $k$-skeleton} of $K$ and denoted by $K^{k}$. Let $f$ be an embedding of $K$ into $\\mathbb{R}^{m}$. Let $\\Lambda^{p,q}(K)$ be the set of all unordered pairs $\\{\\gamma, \\gamma'\\}$ of mutually disjoint subcomplexes of $K$ such that $\\gamma$ is homeomorphic to the $p$-sphere and $\\gamma'$ is homeomorphic to the $q$-sphere. We identify any pair $(\\gamma,\\gamma')$ in $\\Lambda^{p,q}(K)$ with the disjoint union $\\gamma\\sqcup \\gamma'$. Then, for any $\\lambda = \\gamma \\sqcup \\gamma' \\in \\Lambda^{p,q}(K)$, the image $f(\\lambda) = f(\\gamma) \\sqcup f(\\gamma')$ forms a two-component link in $f(K)$ consisting of a $p$-sphere and a $q$-sphere. If $p+q = m-1$, the \\textit{$\\mathbb{Z}_2$-linking number} ${\\rm lk}_2(L) = {\\rm lk}_2(S^p, S^q) = {\\rm lk}_2(S^q, S^p) \\in \\mathbb{Z}_2$ of a two-component link $L = S^p \\sqcup S^q$ in ${\\mathbb R}^m$ is well-defined (cf.\\ \\cite{ST80}). If $m=2n+1$, there are known examples of simplicial $n$-complexes with the following property: every embedding into $\\mathbb{R}^{2n+1}$ necessarily contains a two-component link of $n$-spheres with $\\mathbb{Z}_2$-linking number $1$. Such examples were constructed by Segal--Spie\\.{z} \\cite{SeSp92}, Lovasz--Schrijver \\cite{LS98}, Skopenkov \\cite{Sk03}, Taniyama \\cite{T00}, and more recently by the author \\cite{N26}. In particular, the case of $n=1$ is famous as the Conway--Gordon--Sachs theorem (\\cite{CG83}, \\cite{S84}), and such a simplicial $1$-complex (i.e., a graph) is said to be {\\it intrinsically linked}.\n\nOur purpose in this paper is to present an example of a minimal simplicial $n$-complex that possesses such an intrinsic linking property with respect to embeddings into $\\mathbb{R}^{2n}$. Let us introduce three simplicial $n$-complexes $N_{1}^{(n)}$, $N_2^{(n)}$ and $N_3^{(n)}$ as follows. First, let $\\sigma_{m}=|a_{0}a_{1}\\cdots a_{m}|$ be an $m$-simplex whose vertices are $a_{0},a_{1},\\ldots,a_{m}$. This can be naturally regarded as a simplicial $m$-complex consisting of itself and all its faces. In this setting, the $k$-skeleton of $\\sigma_{m}$ is standardly denoted by $\\sigma_{m}^{k}$ for $k\\le m$. Then we define \n\\begin{eqnarray*}\nN_{1}^{(n)} = \\sigma_{2n+2}^{n}\\setminus \\{|a_{0}a_{j_{1}}\\cdots a_{j_{n}}|\\ |\\ 1\\le j_{1}<j_{2}<\\cdots<j_{n}\\le 2n+2\\}. \n\\end{eqnarray*}\nIn other words, $N_{1}^{(n)}$ is the subcomplex of $\\sigma_{2n+2}^{n}$ obtained by removing all $n$-simplices that contain the vertex $a_0$. Next, for a positive integer $k$, let $[k]^{*m}$ denote the $m$-fold join of $k$ points. Explicitly, we set\n\\begin{eqnarray*}\n[k]^{*m} = \n\\{a_{0}^{0},a_{1}^{0},\\ldots,a_{k-1}^{0}\\} * \\{a_{0}^{1},a_{1}^{1},\\ldots,a_{k-1}^{1}\\} * \\cdots * \\{a_{0}^{n},a_{1}^{n},\\ldots,a_{k-1}^{n}\\}. \n\\end{eqnarray*}\nNote that $[k]^{*n+1}$ can naturally be regarded as a simplicial $n$-complex. Then we define \n\\begin{eqnarray*}\nN_{2}^{(n)} = [3]^{*n+1}\\setminus \\{|a_{0}^{0}a_{j_{1}}^{1}\\cdots a_{j_{n}}^{n}|\\ |\\ j_{1},j_{2},\\ldots,j_{n}\\in\\{0,1,2\\}\\}. \n\\end{eqnarray*}\nIn other words, $N_{2}^{(n)}$ is the subcomplex of $[3]^{*n+1}$ obtained by removing all $n$-simplices that contain the vertex $a_{0}^{0}$. Finally, we define \n\\begin{eqnarray*}\nN_{3}^{(n)} = \\sigma_{2n+2}^{n}\\setminus \\varDelta^{n}(\\sigma_{n+1}). \n\\end{eqnarray*}\nIn other words, $N_{3}^{(n)}$ is the subcomplex of $\\sigma_{2n+2}^{n}$ obtained by removing all $n$-simplices contained in $\\sigma_{n+1}=|a_{0}a_{1}\\cdots a_{n+1}|$. As we will see in the next section, it is well known that neither $\\sigma_{2n+2}^{n}$ nor $[3]^{*n+1}$ cannot be embedded in ${\\mathbb R}^{2n}$ (Theorem \\ref{VKF}). On the other hand, it is also known that any proper subcomplex of $\\sigma_{2n+2}^{n}$ and of $[3]^{*n+1}$ can be embedded in $\\mathbb{R}^{2n}$ (Gr\\\"{u}nbaum \\cite{Grun69}). Therefore, each of $N_1^{(n)}$, $N_2^{(n)}$ and $N_3^{(n)}$ is embeddable in $\\mathbb{R}^{2n}$. Then we have the following:\n\n\\begin{Theorem}{\\rm (van Kampen \\cite{VK33}, Flores \\cite{Flores32})}\\label{VKF} \nLet $n$ be a positive integer, and let $K$ be a simplicial $n$-complex that is $\\sigma_{2n+2}^{n}$ or $[3]^{*n+1}$. Then for every generic immersion $\\varphi$ of $K$ into ${\\mathbb R}^{2n}$, the following holds: \n\\begin{eqnarray*}\n\\sum_{\\substack{s,s'\\in\\varDelta^{n}(K)\\\\ s\\cap s'=\\varnothing}}l(\\varphi(s),\\varphi(s'))\\equiv 1\\pmod{2}. \n\\end{eqnarray*}\n\\end{Theorem}\n\nTherefore, $N_1^{(n)}$, $N_2^{(n)}$ and $N_3^{(n)}$ are simplicial $n$-complexes that are `intrinsically linked' with respect to embeddings into ${\\mathbb R}^{2n}$. We note that the simplicial $n$-complex $\\sigma_{2n+2}^{n}\\setminus\\{|a_{0}a_{1}\\cdots a_{n}|\\}$ has already been shown to satisfy this property (Freedman--Krushkal \\cite[Proposition 4.1]{FK14}; see also Freedman--Krushkal--Teichner \\cite[Lemma 6]{FKT94}). However, both our $N_{1}^{(n)}$ and $N_{3}^{(n)}$ are proper subcomplexes of this complex. Moreover, the complexes $N_{1}^{(n)}$, $N_{2}^{(n)}$ and $N_{3}^{(n)}$ are minimal in the following sense among all simplicial $n$-complexes possessing this property.\n\n\\begin{Proposition}\\label{min}\nLet $N$ be a proper subcomplex of $N_1^{(n)}$, $N_2^{(n)}$ or $N_3^{(n)}$. Then there exists an embedding $f:N\\to {\\mathbb R}^{2n}$ such that ${\\rm lk}_{2}(f(\\lambda))= 0$ for any element $\\lambda$ in $\\Lambda^{n-1,n}(N)$.\n\\end{Proposition}\n\n\\begin{Remark}\\label{nsp}\nA graph $G$ is said to be {\\it non-separating planar} if there exists an embedding $f: G \\to \\mathbb{R}^2$ such that, for every cycle $\\gamma$ in $G$, all vertices of $G$ not lying on $\\gamma$ belong to the same connected component of $\\mathbb{R}^2 \\setminus f(\\gamma)$. Equivalently, $G$ is not non-separating planar if and only if for every embedding $f:G\\to \\mathbb{R}^2$, the image $f(G)$ contains a two-component link consisting of a $0$-sphere and a $1$-sphere with $\\mathbb{Z}_2$-linking number $1$. Dehkordi--Farr proved in \\cite{DF} that a graph is non-separating planar if and only if it contains no subgraph homeomorphic to $K_1 \\sqcup K_4$, $K_1 \\sqcup K_{2,3}$, or $K_{1,1,3}$ as illustrated in Fig. \\ref{nonsp}. They further applied this characterization to the study of linkless embeddings of graphs in $\\mathbb{R}^3$. For related results, see Fowler--Li--Pavelescu \\cite{FLP23}, Pavelescu--Pavelescu \\cite{PP24}. Note that $N_1^{(1)}$ is isomorphic to $K_1 \\sqcup K_4$, $N_2^{(1)}$ is isomorphic to $K_1 \\sqcup K_{2,3}$, and $N_3^{(1)}$ is isomorphic to $K_{1,1,3}$. Thus, our simplicial $n$-complexes $N_{1}^{(n)}$, $N_{2}^{(n)}$ and $N_{3}^{(n)}$ may be regarded as higher-dimensional analogues of the minimal subgraphs that prevent a graph from being non-separating planar.\n\\end{Remark}\n\n\\begin{Theorem}{\\rm (van Kampen \\cite{VK33}, Flores \\cite{Flores32})}\\label{VKF} \nLet $n$ be a positive integer, and let $K$ be a simplicial $n$-complex that is $\\sigma_{2n+2}^{n}$ or $[3]^{*n+1}$. Then for every generic immersion $\\varphi$ of $K$ into ${\\mathbb R}^{2n}$, the following holds: \n\\begin{eqnarray*}\n\\sum_{\\substack{s,s'\\in\\varDelta^{n}(K)\\\\ s\\cap s'=\\varnothing}}l(\\varphi(s),\\varphi(s'))\\equiv 1\\pmod{2}. \n\\end{eqnarray*}\n\\end{Theorem}\n\n\\begin{proof}[Proof of Theorem \\ref{ILR2n}]\nWe first consider the case $K = N_1^{(n)}$. In what follows, for an $m$-simplex $s$, we will also denote its $(m-1)$-skeleton $s^{m-1}$ by $\\partial s$. We define collections $\\Gamma^{n-1}$ and $\\Gamma^n$ of $(n-1)$ and $n$-subcomplexes of $N_1^{(n)}$, respectively, as follows: \n\\begin{eqnarray*}\n\\Gamma^{n-1} &=& \\left\\{\\partial s \\mid s\\in \\varDelta^{n}(\\sigma_{2n+2}^{n}),\\,a_{0}\\in s\\right\\},\\\\\n\\Gamma^{n} &=& \\left\\{\\partial t \\mid t\\in \\varDelta^{n+1}(\\sigma_{2n+2}),\\,a_{0}\\notin t\\right\\}. \n\\end{eqnarray*}\nThen every $\\lambda \\in \\Lambda^{n-1,n}(N_1^{(n)})$ is the disjoint union of an element $\\gamma$ of $\\Gamma^{n-1}$ and an element $\\gamma'$ of $\\Gamma^n$, see Fig. \\ref{N_1_link} for $n=2$. Let $f:N_{1}^{(n)}\\to {\\mathbb R}^{2n}$ be an embedding. This can be extended to a generic immersion $\\hat{f}:\\sigma_{2n+2}^{n} \\to \\mathbb{R}^{2n}$ such that, for any two $n$-simplices $s,s' \\in \\varDelta^n(\\sigma_{2n+2}^{n})$ with $a_0 \\in s$ and $a_0 \\notin s'$, the images $\\hat{f}(s)$ and $\\hat{f}(s')$ intersect transversely at finitely many double points, all of which occur in the interiors of both $n$-simplices. Note that for any $\\lambda = \\gamma \\sqcup \\gamma' \\in \\Lambda^{n-1,n}(N_1^{(n)})$ with $\\gamma = \\partial s$, $s \\in \\varDelta^n(\\sigma_{2n+2}^n)$ and $a_0 \\in s$, we have \n\\begin{eqnarray}\\label{lkdef}\n{\\rm lk}_{2}(f(\\gamma),f(\\gamma'))\n\\equiv \\sum_{s'\\in \\varDelta^{n}(\\gamma')}l(\\hat{f}(s),\\hat{f}(s'))\\pmod{2}.\n\\end{eqnarray}\nOn the other hand, for any two disjoint $n$-simplices $s, s' \\in \\varDelta^n(\\sigma_{2n+2}^n)$ with $a_0 \\in s$ and $a_0 \\notin s'$, there uniquely exists an element $\\lambda = \\gamma \\sqcup \\gamma' \\in \\Lambda^{n-1,n}(N_1^{(n)})$ such that $\\gamma = \\partial s$ and $\\gamma' \\supset s'$. Then, by (\\ref{lkdef}) and Theorem \\ref{VKF}, we obtain \n\\begin{eqnarray*}\\label{lksum}\n\\sum_{\\lambda\\in \\Lambda^{n-1,n}(N_{1}^{(n)})}{\\rm lk}_{2}(f(\\lambda))\n&=& \\sum_{\\substack{\\gamma\\sqcup\\gamma'\\in \\Lambda^{n-1,n}(N_{1}^{(n)}) \\\\ \\gamma\\in \\Gamma^{n-1},\\,\\gamma'\\in\\Gamma^{n}}}{\\rm lk}_{2}(f(\\gamma),f(\\gamma'))\\\\\n&\\equiv& \\sum_{\\substack{\\gamma\\sqcup\\gamma'\\in \\Lambda^{n-1,n}(N_{1}^{(n)}) \\\\ \\gamma=\\partial s,\\,s\\in \\varDelta^{n}(\\sigma_{2n+2}^{n}),\\,a_{0}\\in s}}\n\\bigg(\\sum_{s'\\in \\varDelta^{n}(\\gamma')}l(\\hat{f}(s),\\hat{f}(s'))\\bigg)\\nonumber\\\\\n&=& \\sum_{\\substack{s,\\,s'\\in \\varDelta^{n}(\\sigma_{2n+2}^{n}) \\\\ s\\cap s'=\\varnothing}} \nl(\\hat{f}(s),\\hat{f}(s'))\\nonumber\\\\\n&\\equiv& 1\\pmod{2}. \\nonumber\n\\end{eqnarray*}\nThus, there exists some $\\lambda$ in $\\Lambda^{n-1,n}(N_{1}^{(n)})$ such that ${\\rm lk}_{2}(f(\\lambda))=1$.\n\nNext, we consider the case $K = N_{2}^{(n)}$. We define collections $\\Gamma^{n-1}$ and $\\Gamma^n$ of $(n-1)$ and $n$-subcomplexes of $N_{2}^{(n)}$, respectively, as follows: \n\\begin{eqnarray*}\n\\Gamma^{n-1} &=& \\left\\{\\partial s \\mid s\\in \\varDelta^{n}([3]^{*n+1}),\\,a_{0}^{0}\\in s\\right\\},\\\\\n\\Gamma^{n} &=& \\left\\{\\left\\{a_{1}^{0},a_{2}^{0}\\right\\}*\\left\\{a_{i_{1}}^{1},a_{j_{1}}^{1}\\right\\}*\\cdots*\\left\\{a_{i_{n}}^{n},a_{j_{n}}^{n}\\right\\} \\mid 0\\le i_{q}<j_{q}\\le 2,\\,q\\ge 1\\right\\}. \n\\end{eqnarray*}\nNote that the $(n+1)$-fold join of two points is homeomorphic to the $n$-sphere. Then every $\\lambda \\in \\Lambda^{n-1,n}(N_{2}^{(n)})$ is the disjoint union of an element $\\gamma$ of $\\Gamma^{n-1}$ and an element $\\gamma'$ of $\\Gamma^n$, see Fig. \\ref{N_2_link} for $n=2$. Let $f:N_{2}^{(n)}\\to {\\mathbb R}^{2n}$ be an embedding. This can be extended to a generic immersion $\\hat{f}:[3]^{*n+1} \\to \\mathbb{R}^{2n}$ such that, for any two $n$-simplices $s,s' \\in \\varDelta^n([3]^{*n+1})$ with $a_{0}^{0} \\in s$ and $a_{0}^{0} \\notin s'$, the images $\\hat{f}(s)$ and $\\hat{f}(s')$ intersect transversely at finitely many double points, all of which occur in the interiors of both $n$-simplices. Then, in a similar way as in the case of $N_{1}^{(n)}$, we have \n\\begin{eqnarray*}\\label{lksum2}\n\\sum_{\\lambda\\in \\Lambda^{n-1,n}(N_{2}^{(n)})}{\\rm lk}_{2}(f(\\lambda))\n&=& \\sum_{\\substack{\\gamma\\sqcup\\gamma'\\in \\Lambda^{n-1,n}(N_{2}^{(n)}) \\\\ \\gamma\\in \\Gamma^{n-1},\\,\\gamma'\\in\\Gamma^{n}}}{\\rm lk}_{2}(f(\\gamma),f(\\gamma'))\\\\\n&\\equiv& \\sum_{\\substack{\\gamma\\sqcup\\gamma'\\in \\Lambda^{n-1,n}(N_{2}^{(n)}) \\\\ \\gamma=\\partial s,\\,s\\in \\varDelta^{n}([3]^{*n+1}),\\,a_{0}^{0}\\in s}}\n\\bigg(\\sum_{s'\\in \\varDelta^{n}(\\gamma')}l(\\hat{f}(s),\\hat{f}(s'))\\bigg)\\nonumber\\\\\n&=& \\sum_{\\substack{s,\\,s'\\in \\varDelta^{n}([3]^{*n+1}) \\\\ s\\cap s'=\\varnothing}} \nl(\\hat{f}(s),\\hat{f}(s'))\\nonumber\\\\\n&\\equiv& 1\\pmod{2}. \\nonumber\n\\end{eqnarray*}\nThus the result follows.", "post_theorem_intro_text_len": 2891, "post_theorem_intro_text": "Therefore, $N_1^{(n)}$, $N_2^{(n)}$ and $N_3^{(n)}$ are simplicial $n$-complexes that are `intrinsically linked' with respect to embeddings into ${\\mathbb R}^{2n}$. We note that the simplicial $n$-complex $\\sigma_{2n+2}^{n}\\setminus\\{|a_{0}a_{1}\\cdots a_{n}|\\}$ has already been shown to satisfy this property (Freedman--Krushkal \\cite[Proposition 4.1]{FK14}; see also Freedman--Krushkal--Teichner \\cite[Lemma 6]{FKT94}). However, both our $N_{1}^{(n)}$ and $N_{3}^{(n)}$ are proper subcomplexes of this complex. Moreover, the complexes $N_{1}^{(n)}$, $N_{2}^{(n)}$ and $N_{3}^{(n)}$ are minimal in the following sense among all simplicial $n$-complexes possessing this property.\n\n\\begin{Proposition}\\label{min}\nLet $N$ be a proper subcomplex of $N_1^{(n)}$, $N_2^{(n)}$ or $N_3^{(n)}$. Then there exists an embedding $f:N\\to {\\mathbb R}^{2n}$ such that ${\\rm lk}_{2}(f(\\lambda))= 0$ for any element $\\lambda$ in $\\Lambda^{n-1,n}(N)$.\n\\end{Proposition}\n\nA central role in the proof of Theorem \\ref{ILR2n} is played by the van Kampen--Flores theorem. In the next section, we recall this theorem and present proofs of Theorem \\ref{ILR2n} and Proposition \\ref{min} using it. The van Kampen--Flores theorem is also used to construct examples of simplicial $n$-complexes that are intrinsically linked with respect to embeddings into $\\mathbb{R}^{2n+1}$, see Nikkuni \\cite{N26}. \n\n\\begin{Remark}\\label{nsp}\nA graph $G$ is said to be {\\it non-separating planar} if there exists an embedding $f: G \\to \\mathbb{R}^2$ such that, for every cycle $\\gamma$ in $G$, all vertices of $G$ not lying on $\\gamma$ belong to the same connected component of $\\mathbb{R}^2 \\setminus f(\\gamma)$. Equivalently, $G$ is not non-separating planar if and only if for every embedding $f:G\\to \\mathbb{R}^2$, the image $f(G)$ contains a two-component link consisting of a $0$-sphere and a $1$-sphere with $\\mathbb{Z}_2$-linking number $1$. Dehkordi--Farr proved in \\cite{DF} that a graph is non-separating planar if and only if it contains no subgraph homeomorphic to $K_1 \\sqcup K_4$, $K_1 \\sqcup K_{2,3}$, or $K_{1,1,3}$ as illustrated in Fig. \\ref{nonsp}. They further applied this characterization to the study of linkless embeddings of graphs in $\\mathbb{R}^3$. For related results, see Fowler--Li--Pavelescu \\cite{FLP23}, Pavelescu--Pavelescu \\cite{PP24}. Note that $N_1^{(1)}$ is isomorphic to $K_1 \\sqcup K_4$, $N_2^{(1)}$ is isomorphic to $K_1 \\sqcup K_{2,3}$, and $N_3^{(1)}$ is isomorphic to $K_{1,1,3}$. Thus, our simplicial $n$-complexes $N_{1}^{(n)}$, $N_{2}^{(n)}$ and $N_{3}^{(n)}$ may be regarded as higher-dimensional analogues of the minimal subgraphs that prevent a graph from being non-separating planar.\n\\end{Remark}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\scalebox{0.525}{\\includegraphics*{nonsp}}\n\\caption{$K_1 \\sqcup K_4$, $K_1 \\sqcup K_{2,3}$ and $K_{1,1,3}$}\n\\label{nonsp}\n\\end{center}\n\\end{figure}", "sketch": "A proof sketch is not given here beyond indicating the main tool: “A central role in the proof of Theorem \\ref{ILR2n} is played by the van Kampen--Flores theorem.” The authors say that “In the next section, we recall this theorem and present proofs of Theorem \\ref{ILR2n} and Proposition \\ref{min} using it.”", "expanded_sketch": "A proof sketch is not given here beyond indicating the main tool: “A central role in the proof of the main theorem is played by the van Kampen--Flores theorem.” The authors say that “Next we recall this theorem and present proofs of the main theorem and \\begin{Proposition}\\label{min}\nLet $N$ be a proper subcomplex of $N_1^{(n)}$, $N_2^{(n)}$ or $N_3^{(n)}$. Then there exists an embedding $f:N\\to {\\mathbb R}^{2n}$ such that ${\\rm lk}_{2}(f(\\lambda))= 0$ for any element $\\lambda$ in $\\Lambda^{n-1,n}(N)$.\n\\end{Proposition} using it.”", "expanded_theorem": "\\label{ILR2n} \nLet $n$ be a positive integer, and let $K$ be a simplicial $n$-complex that is $N_1^{(n)}$, $N_2^{(n)}$ or $N_3^{(n)}$. Then, for every embedding $f$ of $K$ into ${\\mathbb R}^{2n}$, there exists an element $\\lambda$ in $\\Lambda^{n-1,n}(K)$ such that ${\\rm lk}_{2}(f(\\lambda)) = 1$.", "theorem_type": ["Universal–Existential", "Implication"], "mcq": {"question": "Let $n$ be a positive integer. Write $\\sigma_m=|a_0a_1\\cdots a_m|$ for an $m$-simplex, and let $\\sigma_m^n$ denote its $n$-skeleton. Define three simplicial $n$-complexes by\n\\[\nN_1^{(n)}=\\sigma_{2n+2}^n\\setminus\\{\\,|a_0a_{j_1}\\cdots a_{j_n}|\\mid 1\\le j_1<\\cdots<j_n\\le 2n+2\\,\\},\n\\]\nso $N_1^{(n)}$ is obtained from $\\sigma_{2n+2}^n$ by removing all $n$-simplices containing $a_0$,\n\\[\nN_2^{(n)}=[3]^{*n+1}\\setminus\\{\\,|a_0^0a_{j_1}^1\\cdots a_{j_n}^n|\\mid j_1,\\dots,j_n\\in\\{0,1,2\\}\\,\\},\n\\]\nwhere $[3]^{*n+1}=\\{a_0^0,a_1^0,a_2^0\\}*\\{a_0^1,a_1^1,a_2^1\\}*\\cdots*\\{a_0^n,a_1^n,a_2^n\\}$ is the simplicial join of $n+1$ three-point sets, and\n\\[\nN_3^{(n)}=\\sigma_{2n+2}^n\\setminus \\Delta^n(\\sigma_{n+1}),\n\\]\nso $N_3^{(n)}$ is obtained from $\\sigma_{2n+2}^n$ by removing all $n$-simplices contained in the face $\\sigma_{n+1}=|a_0a_1\\cdots a_{n+1}|$.\n\nFor a simplicial complex $K$, let $\\Lambda^{n-1,n}(K)$ be the set of unordered pairs $\\lambda=\\gamma\\sqcup\\gamma'$ of mutually disjoint subcomplexes of $K$ such that $\\gamma$ is homeomorphic to $S^{n-1}$ and $\\gamma'$ is homeomorphic to $S^n$. If $f:K\\to\\mathbb{R}^{2n}$ is an embedding, then $f(\\lambda)=f(\\gamma)\\sqcup f(\\gamma')$ is a two-component link in $\\mathbb{R}^{2n}$, and ${\\rm lk}_2(f(\\lambda))\\in\\mathbb Z_2$ denotes its mod-2 linking number.\n\nSuppose $K$ is one of $N_1^{(n)}$, $N_2^{(n)}$, or $N_3^{(n)}$. Which conclusion about embeddings $f:K\\to\\mathbb{R}^{2n}$ holds?", "correct_choice": {"label": "A", "text": "For every embedding $f:K\\to\\mathbb{R}^{2n}$, there exists some $\\lambda\\in\\Lambda^{n-1,n}(K)$ such that ${\\rm lk}_2(f(\\lambda))=1$; equivalently, $f(K)$ contains disjoint embedded spheres of dimensions $n-1$ and $n$ whose mod-2 linking number is $1$."}, "choices": [{"label": "B", "text": "For every embedding $f:K\\to\\mathbb{R}^{2n}$, one has ${\\rm lk}_2(f(\\lambda))=1$ for every $\\lambda\\in\\Lambda^{n-1,n}(K)$; equivalently, every pair of disjoint subcomplexes of types $S^{n-1}$ and $S^n$ occurring in $K$ forms a mod-2 linked pair in $\\mathbb{R}^{2n}$."}, {"label": "C", "text": "For every embedding $f:K\\to\\mathbb{R}^{2n}$, there exists some $\\lambda\\in\\Lambda^{n-1,n}(K)$ such that ${\\rm lk}_2(f(\\lambda))\\neq 0$."}, {"label": "D", "text": "There exists an embedding $f:K\\to\\mathbb{R}^{2n}$ such that for every $\\lambda\\in\\Lambda^{n-1,n}(K)$, one has ${\\rm lk}_2(f(\\lambda))=1$; equivalently, each of $N_1^{(n)},N_2^{(n)},N_3^{(n)}$ admits a realization in $\\mathbb{R}^{2n}$ in which all disjoint $S^{n-1}\\sqcup S^n$ pairs are mod-2 linked."}, {"label": "E", "text": "For every embedding $f:K\\to\\mathbb{R}^{2n}$, there exist disjoint subcomplexes $\\lambda=\\gamma\\sqcup\\gamma'\\in\\Lambda^{n,n}(K)$ such that ${\\rm lk}_2(f(\\lambda))=1$; equivalently, $f(K)$ always contains a two-component link of two embedded $n$-spheres with mod-2 linking number $1$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "counting_estimate", "tampered_component": "odd-total-sum-implies-existence-not-universality", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "replace explicit value 1 in \\mathbb Z_2 by nonzero", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "quantifier-order on embedding versus linked pairs", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "sphere-dimension type (n-1,n) replaced by (n,n)", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct conclusion. It defines the complexes and the mod-2 linking setup, but does not directly signal which quantifier pattern or sphere-dimension pattern is correct."}, "TAS": {"score": 2, "justification": "The item is not a pure restatement: the options vary meaningfully by quantifiers, universality versus existence, and sphere dimensions. A student must distinguish among competing conclusions rather than recognize a verbatim theorem statement."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the choices differ by subtle logical strength. However, this is weakened substantially because choice C is effectively equivalent to A: since the linking number is stated to lie in Z_2, saying it is '!= 0' is the same as saying it equals 1."}, "DQS": {"score": 0, "justification": "The distractors are fatally compromised by choice C, which is not genuinely false or weaker in this mod-2 setting and duplicates the intended correct answer. Although B, D, and E are plausible mathematical traps, the A/C ambiguity makes the distractor set poor overall."}, "total_score": 5, "overall_assessment": "Conceptually sophisticated and mostly free of stem-level leakage, but the MCQ is flawed because A and C are both correct in Z_2. This ambiguity seriously weakens its validity as a multiple-choice item."}}
{"id": "2602.19529v1", "paper_link": "http://arxiv.org/abs/2602.19529v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{geometricallyFiniteNearOrbit}\n    Suppose that a group $G$ admits a non-elementary proper action on a proper Gromov hyperbolic space $X$. Then the action is geometrically infinite if and only if there exists an escaping sequence of hyperbolic elements in $G$.", "start_pos": 4505, "end_pos": 4800, "label": "geometricallyFiniteNearOrbit"}, "ref_dict": {"geometricallyFiniteNonconicalCountable": "\\begin{theorem}\\label{geometricallyFiniteNonconicalCountable}\n    Suppose  a group $G$ admits a non-elementary proper action on a proper Gromov hyperbolic space $X$. Then the action is geometrically infinite if and only if the set of non-conical points is uncountable.\n\\end{theorem}", "geometricallyFiniteNearOrbit": "\\begin{theorem}\\label{geometricallyFiniteNearOrbit}\n    Suppose that a group $G$ admits a non-elementary proper action on a proper Gromov hyperbolic space $X$. Then the action is geometrically infinite if and only if there exists an escaping sequence of hyperbolic elements in $G$.\n\\end{theorem}", "preliminary": "\\label{preliminary}\nIn this section we introduce the preliminaries we shall use on convergence group actions (cf. \\cite{tukia1994convergence, bowditch1999convergence}) and hyperbolic spaces (cf. \\cite", "producingHyperbolicElements": "\\begin{lemma}\\label{producingHyperbolicElements}\n    Let the action of $G$ on $M$ be a non-elementary convergence group action.\n    If there exist an element $f \\in G$ and an open set $U \\subset M$ such that $f\\overline{U} \\subsetneqq U$, then $f$ is hyperbolic.\n\\end{lemma}"}, "pre_theorem_intro_text_len": 2043, "pre_theorem_intro_text": "Let $X$ be a proper Gromov hyperbolic space, on which a countable group $G$ acts properly by isometries. The action naturally extends to a convergence group action on the Gromov boundary $\\partial X$, which decomposes $\\partial X$ into  the \\textit{limit set} $\\Lambda G$ and the  domain of discontinuity. Let $\\Pi \\subset \\Lambda G$ be the set of parabolic points.  The action of $G$ on $X$ is called \\textit{geometrically finite} if there exists a sufficiently separated $G$-invariant family of horoballs $\\{B_p:p\\in \\Pi\\}$  so that the action on the complement $\\mathrm{CH}(\\Lambda G)\\setminus \\bigcup_{p\\in \\Pi} B_p$ is cocompact, where $CH(\\Lambda G)$ denotes the weak convex hull of $\\Lambda G$ in $X$.  In this case, $\\Lambda G\\setminus \\Pi$ consists precisely of \\textit{conical points}. We refer to Section \\ref{preliminary} for precise definitions.\n\nHistorically, the notion of geometric finiteness originated in the theory of Kleinian groups (see \\cite{BeaMas,Marden,thurstonnotes}) and has since been extended to the much broader class of groups considered here in a series of works \\cite{Bow3,Bow4,bowditch2012relatively}. Since the formal introduction in \\cite{Gromov}, this class of groups has been intensively studied in geometric group theory under the name of \\textit{relatively hyperbolic groups}, encompassing many examples of geometric and algebraic origin. Several equivalent characterizations of relative hyperbolicity are then proposed in the literature, generalizing and going well beyond those known for Kleinian groups; see, for example, \\cite{Farb, bowditch2012relatively, Osin, DruSapir, Ge1}, as well as \\cite{Hru} for an overview and references therein. \n\nThe aim of this paper is to supplement these descriptions with two new characterizations in the present general setting.  \nLet $\\{g_n\\}_{n\\ge 1}$ be a sequence of hyperbolic elements in $G$. We say that $\\{g_n\\}_{n\\ge 1}$ is \\textit{escaping} if for any point $o$ and any $R>0$, the $R$-neighborhood of $Go$ intersects the axes of only finitely many $g_n$.", "context": "Let $X$ be a proper Gromov hyperbolic space, on which a countable group $G$ acts properly by isometries. The action naturally extends to a convergence group action on the Gromov boundary $\\partial X$, which decomposes $\\partial X$ into the \\textit{limit set} $\\Lambda G$ and the domain of discontinuity. Let $\\Pi \\subset \\Lambda G$ be the set of parabolic points. The action of $G$ on $X$ is called \\textit{geometrically finite} if there exists a sufficiently separated $G$-invariant family of horoballs $\\{B_p:p\\in \\Pi\\}$ so that the action on the complement $\\mathrm{CH}(\\Lambda G)\\setminus \\bigcup_{p\\in \\Pi} B_p$ is cocompact, where $CH(\\Lambda G)$ denotes the weak convex hull of $\\Lambda G$ in $X$. In this case, $\\Lambda G\\setminus \\Pi$ consists precisely of \\textit{conical points}. We refer to Section \\ref{preliminary} for precise definitions.\n\nHistorically, the notion of geometric finiteness originated in the theory of Kleinian groups (see \\cite{BeaMas,Marden,thurstonnotes}) and has since been extended to the much broader class of groups considered here in a series of works \\cite{Bow3,Bow4,bowditch2012relatively}. Since the formal introduction in \\cite{Gromov}, this class of groups has been intensively studied in geometric group theory under the name of \\textit{relatively hyperbolic groups}, encompassing many examples of geometric and algebraic origin. Several equivalent characterizations of relative hyperbolicity are then proposed in the literature, generalizing and going well beyond those known for Kleinian groups; see, for example, \\cite{Farb, bowditch2012relatively, Osin, DruSapir, Ge1}, as well as \\cite{Hru} for an overview and references therein.\n\nThe aim of this paper is to supplement these descriptions with two new characterizations in the present general setting. \nLet $\\{g_n\\}_{n\\ge 1}$ be a sequence of hyperbolic elements in $G$. We say that $\\{g_n\\}_{n\\ge 1}$ is \\textit{escaping} if for any point $o$ and any $R>0$, the $R$-neighborhood of $Go$ intersects the axes of only finitely many $g_n$.\n\n\\label{preliminary}\nIn this section we introduce the preliminaries we shall use on convergence group actions (cf. \\cite{tukia1994convergence, bowditch1999convergence}) and hyperbolic spaces (cf. \\cite", "full_context": "Let $X$ be a proper Gromov hyperbolic space, on which a countable group $G$ acts properly by isometries. The action naturally extends to a convergence group action on the Gromov boundary $\\partial X$, which decomposes $\\partial X$ into the \\textit{limit set} $\\Lambda G$ and the domain of discontinuity. Let $\\Pi \\subset \\Lambda G$ be the set of parabolic points. The action of $G$ on $X$ is called \\textit{geometrically finite} if there exists a sufficiently separated $G$-invariant family of horoballs $\\{B_p:p\\in \\Pi\\}$ so that the action on the complement $\\mathrm{CH}(\\Lambda G)\\setminus \\bigcup_{p\\in \\Pi} B_p$ is cocompact, where $CH(\\Lambda G)$ denotes the weak convex hull of $\\Lambda G$ in $X$. In this case, $\\Lambda G\\setminus \\Pi$ consists precisely of \\textit{conical points}. We refer to Section \\ref{preliminary} for precise definitions.\n\nHistorically, the notion of geometric finiteness originated in the theory of Kleinian groups (see \\cite{BeaMas,Marden,thurstonnotes}) and has since been extended to the much broader class of groups considered here in a series of works \\cite{Bow3,Bow4,bowditch2012relatively}. Since the formal introduction in \\cite{Gromov}, this class of groups has been intensively studied in geometric group theory under the name of \\textit{relatively hyperbolic groups}, encompassing many examples of geometric and algebraic origin. Several equivalent characterizations of relative hyperbolicity are then proposed in the literature, generalizing and going well beyond those known for Kleinian groups; see, for example, \\cite{Farb, bowditch2012relatively, Osin, DruSapir, Ge1}, as well as \\cite{Hru} for an overview and references therein.\n\nThe aim of this paper is to supplement these descriptions with two new characterizations in the present general setting. \nLet $\\{g_n\\}_{n\\ge 1}$ be a sequence of hyperbolic elements in $G$. We say that $\\{g_n\\}_{n\\ge 1}$ is \\textit{escaping} if for any point $o$ and any $R>0$, the $R$-neighborhood of $Go$ intersects the axes of only finitely many $g_n$.\n\n\\label{preliminary}\nIn this section we introduce the preliminaries we shall use on convergence group actions (cf. \\cite{tukia1994convergence, bowditch1999convergence}) and hyperbolic spaces (cf. \\cite\n\nLet $X$ be a proper Gromov hyperbolic space, on which a countable group $G$ acts properly by isometries. The action naturally extends to a convergence group action on the Gromov boundary $\\partial X$, which decomposes $\\partial X$ into the \\textit{limit set} $\\Lambda G$ and the domain of discontinuity. Let $\\Pi \\subset \\Lambda G$ be the set of parabolic points. The action of $G$ on $X$ is called \\textit{geometrically finite} if there exists a sufficiently separated $G$-invariant family of horoballs $\\{B_p:p\\in \\Pi\\}$ so that the action on the complement $\\mathrm{CH}(\\Lambda G)\\setminus \\bigcup_{p\\in \\Pi} B_p$ is cocompact, where $CH(\\Lambda G)$ denotes the weak convex hull of $\\Lambda G$ in $X$. In this case, $\\Lambda G\\setminus \\Pi$ consists precisely of \\textit{conical points}. We refer to Section \\ref{preliminary} for precise definitions.\n\nThe aim of this paper is to supplement these descriptions with two new characterizations in the present general setting. \nLet $\\{g_n\\}_{n\\ge 1}$ be a sequence of hyperbolic elements in $G$. We say that $\\{g_n\\}_{n\\ge 1}$ is \\textit{escaping} if for any point $o$ and any $R>0$, the $R$-neighborhood of $Go$ intersects the axes of only finitely many $g_n$.\n\nThe corresponding result for Kleinian groups was proved by Bonahon \\cite{bonahon-bouts} and was recently generalized to pinched negatively curved manifolds by Kapovich and Liu \\cite{KL19}. In contrast to these works, our approach relies only on elementary arguments in hyperbolic geometry and does not invoke the Margulis lemma.\n\n\\begin{theorem}\\label{geometricallyFiniteNonconicalCountable}\n Suppose a group $G$ admits a non-elementary proper action on a proper Gromov hyperbolic space $X$. Then the action is geometrically infinite if and only if the set of non-conical points is uncountable.\n\\end{theorem}\n\nFinally, let us conclude the Introduction with an open question. \nBy a theorem of Yaman \\cite{Yaman} (and of Bowditch \\cite{Bow98} in the cocompact case), geometrically finite actions on Gromov hyperbolic spaces can be recovered from the dynamics on the boundary. More precisely, if a convergence group action is geometrically finite (i.e., its limit set consists only of conical points and bounded parabolic points) then the group admits a geometrically finite action on some proper Gromov hyperbolic space. Moreover, the induced action on the boundary coincides with the original convergence group action. The following question asks whether the assumption of bounded parabolic points could be weakend as follows.\n\n\\begin{question}\\label{geometricallyFiniteConvergenceGroup}\n Suppose a discrete group $G$ admits a convergence group action on a compact metrizable space with countably many non-conical limit points. Does $G$ admit a geometrically finite action on some Gromov hyperbolic space?\n\\end{question}\n\nTogether with the standing assumption that the action is non-elementary (which ensures the existence of independent hyperbolic elements), we have a finiteness statement about conjugacy classes.\n\\begin{lemma}\\label{stableTranslationLength}\n For any $r > 0$, there are only finitely many conjugacy classes of hyperbolic elements in $G$ with translation length less than $r$.\n\\end{lemma}\n\\begin{proof}\n Let $r > 0$, and let $g \\in G$ be a hyperbolic element with $\\tau(g) < r$. By our assumption, there exists $h \\in G$ and an axis $\\gamma$ of $g$ such that $d(h o, \\gamma) \\le R_0$. Thus, we can choose a point $x \\in \\gamma$ satisfying $d(h o, x) \\le R_0$. Applying the triangle inequality and using Lemma~\\ref{translationDistance}, we estimate:\n $$\n d(h o, g h o) \\le d(h o, x) + d(x, g x) + d(g x, g h o) \\le R_0 + \\tau(g) + D_1 + R_0 < 2R_0 + D_1 + r.\n $$\n This yields $d(o, h^{-1} g h \\, o) = d(h o, g h o) < 2R_0 + D_1 + r$.\n Since the action of $G$ on $X$ is proper, the set $\\{ k \\in G : d(o, k o) < 2R_0 + D_1 + r \\}$ is finite. Therefore, the set of conjugates $h^{-1}gh$ belongs to this finite set. Consequently, there are only finitely many distinct conjugacy classes of such elements $g$.\n\\end{proof}\n\nThroughout this section, let $G$ be a discrete group that admits a non-elementary, geometrically infinite, proper action on a proper $\\delta$-hyperbolic space $X$ for some $\\delta\\ge 0$, and fix a basepoint $o\\in X$. \nBy Lemma~\\ref{localQuasiGeodesic}, there exist constants $L>16\\delta$, $\\lambda \\ge 1$, and $C\\ge 0$ such that every $L$-local $(3, 2D_0)$-quasi-geodesic in $X$ is a $(\\lambda, C)$-quasi-geodesic.\nBy Lemma~\\ref{MorseLemma}, there exists a constant $B>0$ such that $d_{\\text{Haus}}(p, [p_-, p_+])\\le B$ for every $(\\lambda, C)$-quasi-geodesic $p$.\n\n\\label{preliminary}\nIn this section we introduce the preliminaries we shall use on convergence group actions (cf. \\cite{tukia1994convergence, bowditch1999convergence}) and hyperbolic spaces (cf. \\cite", "post_theorem_intro_text_len": 5175, "post_theorem_intro_text": "The corresponding result for Kleinian groups was proved by Bonahon \\cite{bonahon-bouts} and was recently generalized to pinched negatively curved manifolds by Kapovich and Liu \\cite{KL19}. In contrast to these works, our approach relies only on elementary arguments in hyperbolic geometry and does not invoke the Margulis lemma.\n\nFollowing the strategy of Bishop \\cite{Bishop96} for Kleinian groups, we derive the second characterization of geometric infiniteness  from Theorem \\ref{geometricallyFiniteNearOrbit}. This refines the result of Beardon and Maskit \\cite{BeaMas} in terms of non-conical limit sets. \n\n\\begin{theorem}\\label{geometricallyFiniteNonconicalCountable}\n    Suppose  a group $G$ admits a non-elementary proper action on a proper Gromov hyperbolic space $X$. Then the action is geometrically infinite if and only if the set of non-conical points is uncountable.\n\\end{theorem}\n\nWe remark that  the negatively pinched  manifold case was due to  Kapovich and Liu   \\cite{KL20} after their work \\cite{KL19} on Theorem \\ref{geometricallyFiniteNearOrbit}. To put into perspective, for finitely generated geometrically infinite Kleinian groups, the non‑conical limit set has   maximal Hausdorff dimension whose precise value was computed by Bishop–Jones \\cite{BJ97bddgeo}, subsequent contributions \\cite{KL20}, and recently completed in \\cite{mj2025hausdorff} in full general case. The question of whether Theorem~\\ref{geometricallyFiniteNonconicalCountable} holds in this level of generality arose naturally in the course of that study.\n\nFinally, let us conclude the Introduction with an open question.  \nBy a theorem of Yaman \\cite{Yaman} (and of Bowditch \\cite{Bow98} in the cocompact case), geometrically finite actions on Gromov hyperbolic spaces can be recovered from the dynamics on the boundary. More precisely, if a convergence group action is geometrically finite (i.e., its limit set consists only of conical points and bounded parabolic points) then the group admits a geometrically finite action on some proper Gromov hyperbolic space. Moreover, the induced action on the boundary coincides with the original convergence group action. The following question asks whether the assumption of bounded parabolic points could be weakend as follows. \n\n\\begin{question}\\label{geometricallyFiniteConvergenceGroup}\n    Suppose a  discrete group  $G$ admits a convergence group action on a compact metrizable space with  countably many non-conical limit points. Does $G$ admit a geometrically finite action on some Gromov hyperbolic space?\n\\end{question}\n\n\\begin{remark}\n\n\\begin{enumerate}\n\\item\nThis is related to a question of Kapovich \\cite[Problem 7]{Kap05} asking whether every convergence group action can be realized as the boundary action of a proper action on some Gromov hyperbolic space. If so, then a positive answer would follow from Theorem~\\ref{geometricallyFiniteNonconicalCountable} together with Yaman’s theorem.\nMoreover, in \\cite{sun2019dynamicalcharacterizationacylindricallyhyperbolic}, Bin Sun proved that the action constructed in \\cite{Bow98} on a hyperbolic space contains WPD (i.e. weak proper discontinuity) elements.  \n\n\\item The assumption that the action is a convergence group action is necessary. It is easy to give examples of \\emph{non-relatively hyperbolic} groups $G$ which admit a minimal action on a compact metriziable space $M$ so that all but {countably many points} $\\xi\\in M$ are \\emph{conical points} in the usual sense.\n\\begin{itemize}\n    \\item  there exist a sequence $\\{g_n\\}$ in $G$ and distinct points $a, b \\in M$ such that $g_n x \\to b$ locally uniformly for all $x \\in M \\setminus \\{\\xi\\}$, while $g_n \\xi \\to a$.  \n\\end{itemize}\nIndeed, any non-elementary action on a locally infinite tree $T$ gives raise to the desired action on the compact space $M=\\partial T\\cup T^0$, the union of the ends of the tree and the vertex set. We equip $M$ with Bowditch topology  (since $T$ is a fine graph in sense of \\cite{bowditch2012relatively}). It is easy to check that the ends of the tree are conical points in the above sense.\n\\end{enumerate}\n\n\\end{remark}\n\\subsection*{Structure of the paper}\nSection 2 reviews the necessary preliminaries on convergence group actions and $\\delta$-hyperbolic spaces. A key technical lemma (Lemma~\\ref{producingHyperbolicElements}) for producing hyperbolic elements under a dynamical condition is presented.\nIn Section 3, assuming the condition in Theorem~\\ref{geometricallyFiniteNearOrbit}, we construct geometric objects analogous to those in a cusp-uniform action. We prove that every point in the limit set $\\Lambda G$ is either conical or bounded parabolic, thereby establishing the theorem.\nIn Section 4, we largely follow the strategy of \\cite{Bishop96} to construct uncountably many non-conical limit points under the hypothesis of geometric infiniteness. The proof uses the tool of $L$-local quasi-geodesics to build the required paths, thus proving Theorem~\\ref{geometricallyFiniteNonconicalCountable}.\n\n\\subsection*{Acknowledgments}\nThe second-named author is grateful to Mahan MJ  for insightful discussions about  Theorem \\ref{geometricallyFiniteNonconicalCountable}.", "sketch": "A proof sketch for Theorem~\\ref{geometricallyFiniteNearOrbit} is indicated in the “Structure of the paper” paragraph: in Section~3, “assuming the condition in Theorem~\\ref{geometricallyFiniteNearOrbit}, we construct geometric objects analogous to those in a cusp-uniform action” and then “prove that every point in the limit set $\\Lambda G$ is either conical or bounded parabolic, thereby establishing the theorem.” The introduction also notes that the approach “relies only on elementary arguments in hyperbolic geometry and does not invoke the Margulis lemma,” and that a “key technical lemma (Lemma~\\ref{producingHyperbolicElements}) for producing hyperbolic elements under a dynamical condition is presented” in Section~2.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{geometricallyFiniteNearOrbit}\n    Suppose that a group $G$ admits a non-elementary proper action on a proper Gromov hyperbolic space $X$. Then the action is geometrically infinite if and only if there exists an escaping sequence of hyperbolic elements in $G$.", "theorem_type": ["Biconditional or Equivalence"], "mcq": {"question": "Let a group $G$ act non-elementarily and properly by isometries on a proper Gromov hyperbolic space $X$. Write $\\Lambda G\\subset \\partial X$ for the limit set, $\\Pi\\subset \\Lambda G$ for the parabolic points, and $CH(\\Lambda G)$ for the weak convex hull of $\\Lambda G$. The action is called geometrically finite if there exists a sufficiently separated $G$-invariant family of horoballs $\\{B_p:p\\in \\Pi\\}$ such that the action on $CH(\\Lambda G)\\setminus \\bigcup_{p\\in \\Pi} B_p$ is cocompact; geometrically infinite means not geometrically finite. A sequence $\\{g_n\\}_{n\\ge 1}$ of hyperbolic elements of $G$ is called escaping if, for every $o\\in X$ and every $R>0$, the $R$-neighborhood of the orbit $Go$ intersects the axes of only finitely many $g_n$. Which of the following statements is equivalent to the action being geometrically infinite?", "correct_choice": {"label": "A", "text": "There exists an escaping sequence of hyperbolic elements in $G$."}, "choices": [{"label": "B", "text": "For every sequence $\\{g_n\\}_{n\\ge 1}$ of hyperbolic elements in $G$ whose conjugacy classes are pairwise distinct, the sequence is escaping."}, {"label": "C", "text": "There exists a sequence $\\{g_n\\}_{n\\ge 1}$ of hyperbolic elements in $G$ such that, for some $o\\in X$ and some $R>0$, the $R$-neighborhood of the orbit $Go$ intersects the axes of only finitely many $g_n$."}, {"label": "D", "text": "There exists a sequence $\\{g_n\\}_{n\\ge 1}$ of hyperbolic elements in $G$ such that, for every $o\\in X$, there exists $R>0$ for which the $R$-neighborhood of the orbit $Go$ intersects the axes of only finitely many $g_n$."}, {"label": "E", "text": "There exists an escaping sequence $\\{g_n\\}_{n\\ge 1}$ of loxodromic isometries of $X$, not necessarily contained in $G$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "existential escaping sequence replaced by universal claim over distinct conjugacy classes", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the quantifier requirement 'for every $o$ and every $R>0$'", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "quantifier order on $R$ weakened from 'for every $R>0$' to 'there exists $R>0$'", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "sequence no longer required to lie in the acting group $G$", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It gives the definitions of geometrically finite/infinite and escaping, but does not itself state the equivalence with escaping sequences."}, "TAS": {"score": 0, "justification": "The item is essentially a direct theorem-recall question: it asks which statement is equivalent to geometrical infiniteness, and the correct choice is the theorem statement itself with no substantive reformulation."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to track quantifiers and group-membership conditions across the distractors, especially to reject weakened or strengthened variants. However, the intended answer is still mainly identifiable by recognizing the exact theorem statement rather than generating a deeper conclusion."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: overstrengthening to a universal claim, weakening quantifiers, changing quantifier order, and dropping the requirement that elements lie in G."}, "total_score": 5, "overall_assessment": "A solid theorem-recall MCQ with strong distractors, but it is highly tautological: the correct answer is essentially the exact equivalence being tested, so it offers only moderate generative reasoning."}}
{"id": "2602.19537v1", "paper_link": "http://arxiv.org/abs/2602.19537v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "Suppose that $\\S$ is compact, spacelike (mean) convex and acausal, and $F: \\S^n \\times I \\to \\R^{n+1,k}$ is a solution to MCF, and for $n = 1$ we require the curve to have turning number (degree of the Gauss map) $1$ , then noncollapsing is preserved.", "start_pos": 7012, "end_pos": 7298, "label": null}, "ref_dict": {"noncollapsing section": "\\begin{proposition}[Short time existence]\\label{short time existence MCF}\n    Suppose that $\\S$ is compact, spacelike-convex and let $F_0: \\S^n \\to \\R^{n+1,k}$ be a smooth immersion, then there exists $\\e > 0$ and a smooth solution $F(\\cdot,t): \\S^n \\to \\R^{n+1,k}$ to MCF defined for $t\\in [0, \\e)$ satisfying $F(\\cdot,0) = F_0$.\n\\end{proposition}\n\n\\section{Noncollapsing}\\label{noncollapsing section}\n\nThe result holds for a more general class of submanifolds, which is an analogue of mean-convex hypersurfaces. \n\n\\begin{definition}\n    A spacelike submanifold $\\S^n \\subset \\R^{n+1,k}$ is \\emph{spacelike-mean-convex} if its mean curvature is everywhere spacelike and non-vanishing.\n\\end{definition}\n\nFirst, we show that spacelike-mean-convexity is preserved, assuming that the induced metric stays spacelike, this condition is implied by noncollapsing so we omit the assumption for results in this section. \n\\begin{proposition}\n    Suppose that $\\S$ is compact, spacelike-mean-convex and $F: \\S^n \\times I \\to \\R^{n+1,k}$ is a solution to MCF, then spacelike-mean-convexity is preserved along the flow. \n\\end{proposition}"}, "pre_theorem_intro_text_len": 5140, "pre_theorem_intro_text": "The evolution of hypersurfaces by their mean curvature has been studied extensively since Huisken \\cite{huisken84}, who proved that closed embedded convex hypersurfaces in Euclidean space contract to round points under mean curvature flow. \nMore recently, the mean curvature flow of submanifolds with higher codimension has been studied. The first author and Baker \\cite{andrewsbaker2010} proved an analogous result to that of \\cite{huisken84} in any codimension of Euclidean background under a quadratic curvature pinching condition $|h|^2 \\leq c |H|^2$ on the initial data, where $h$ is the second fundamental form, $H$ is the mean curvature, and $c$ is a dimensional constant. Since then, Baker and Nguyen \\cite{bakernguyen2023} have explored the flow in the case where the background is a sphere of arbitrary codimension, where the sharp quartic pinching condition for the sphere is found in \\cite{vogiatzi2024sharpquarticpinchingmean}. Other directions in high codimension have been explored, including ancient solutions \\cite{lynchnguyen2021, Risa_2018}, singularity formation \\cite{vogiatzi2023singularitymodelshighcodimension}, flow in complex projective space \\cite{vogiatzi2023meancurvatureflowhigh}, and flow in pseudo-Euclidean space of submanifolds with maximal spacelike dimension \\cite{Lambert_2019}. \n\nIn this paper, we consider the mean curvature flow (MCF) of embedded spacelike submanifolds under the assumption that the normal subspace at each point has signature $(1,k)$, so that the codimension may be arbitrary but no two spacelike normal vectors are orthogonal.  This implies that the cone of strictly spacelike normal vectors at any chosen point separates into two connected components.  In addition, we impose the condition of \\emph{spacelike convexity}, which is the condition that the acceleration vector along any geodesic of the submanifold is strictly spacelike, or equivalently that the second fundamental form $h(v,v)$ is strictly spacelike for any non-zero tangent vector $v$.  At each point, the values of $h(v,v)$ must lie in the same connected component of the spacelike normal cone, which we call the \\emph{inward normal cone}.\n Spacelike-convex submanifolds are a natural analogue of convex hypersurfaces in Euclidean space:  For example, the orthogonal projection of such a submanifold onto any maximal spacelike subspace $\\R^{n+1,0} \\subset \\R^{n+1,k}$ is locally convex, and the sectional curvature of the induced metric is positive. \n More interestingly for our purposes, the condition of spacelike convexity leads to a  natural pinching condition for the second fundamental form: There exists $\\a > 0$ such that $h(v,v) - \\a H$ is inward spacelike for all unit $v\\in T\\S$, and similarly there exists $\\b > 0$ such that $h(v,v) - \\b H$ lies in the outward normal cone.  We call the largest $\\a$ and smallest $\\b$ for which this holds the inward and outward pinching ratios respectively.  Our first result is that such pinching is preserved under the flow. \n\n\\begin{proposition}\n    Suppose that $\\S$ is compact and $F: \\S^n \\times I \\to \\R^{n+1,k}$ is a solution to MCF with $F(.,0)$ spacelike and spacelike-convex. Then the inward pinching ratio $\\a$ is non-decreasing in $t$, and the outward pinching ratio $\\b$ is non-increasing in $t$.\n\\end{proposition}\n\nIn \\cite{huisken84}, pinching of the form $h(v,v)\\geq \\alpha Hg(v,v)$ is preserved under the flow and can be improved via Stampaccia iteration, eventually leading to Huisken's theorem. Such an approach may be possible in our setting, but we instead use an alternative approach using non-collapsing estimates, which has the advantage of giving direct control on the global geometry of the evolving submanifolds. Sheng and Wang \\cite{shengwang} proved that embedded mean-convex solutions of mean curvature flow in Euclidean space satisfy a non-collapsing estimate:  For any point on the hypersurface there is a touching enclosed ball with curvature comparable to the mean curvature at that point.  In \\cite{andrews2011noncollapsingmeanconvexmeancurvature}, the first author found a proof of such an estimate using a direct maximum principle argument, which also provides a corresponding result for balls touching from the exterior.  This argument enables a very direct proof of the result of \\cite{huisken84}.  In this paper in Section \\ref{noncollapsing section} we prove a version of noncollapsing for space-like convex submanifolds (the first high-codimension setting where such a result is known). This non-collapsing result holds also for the more general class of acausal submanifolds which are spacelike-mean-convex (that is, the mean curvature is strictly spacelike at each point).  We say that an acausal spacelike (mean) convex embedding $F: \\S^n \\to \\R^{n+1,k}$ is (interior) noncollapsed with parameter $\\delta$ if, for any $x\\in \\S$, the pseudoball $\\left\\{|p-F(x)-\\delta\\frac{H(x)}{|H(x)|^2}|< \\frac{\\delta}{|H(x)|}\\right\\}$ has $F(\\Sigma)$ in its exterior, touching at $F(x)$.  There is a similar definition of exterior noncollapsing. Our second result is that the flow preserves noncollapsing (\\cref{noncollapsing section}).", "context": "The evolution of hypersurfaces by their mean curvature has been studied extensively since Huisken \\cite{huisken84}, who proved that closed embedded convex hypersurfaces in Euclidean space contract to round points under mean curvature flow. \nMore recently, the mean curvature flow of submanifolds with higher codimension has been studied. The first author and Baker \\cite{andrewsbaker2010} proved an analogous result to that of \\cite{huisken84} in any codimension of Euclidean background under a quadratic curvature pinching condition $|h|^2 \\leq c |H|^2$ on the initial data, where $h$ is the second fundamental form, $H$ is the mean curvature, and $c$ is a dimensional constant. Since then, Baker and Nguyen \\cite{bakernguyen2023} have explored the flow in the case where the background is a sphere of arbitrary codimension, where the sharp quartic pinching condition for the sphere is found in \\cite{vogiatzi2024sharpquarticpinchingmean}. Other directions in high codimension have been explored, including ancient solutions \\cite{lynchnguyen2021, Risa_2018}, singularity formation \\cite{vogiatzi2023singularitymodelshighcodimension}, flow in complex projective space \\cite{vogiatzi2023meancurvatureflowhigh}, and flow in pseudo-Euclidean space of submanifolds with maximal spacelike dimension \\cite{Lambert_2019}.\n\nIn this paper, we consider the mean curvature flow (MCF) of embedded spacelike submanifolds under the assumption that the normal subspace at each point has signature $(1,k)$, so that the codimension may be arbitrary but no two spacelike normal vectors are orthogonal. This implies that the cone of strictly spacelike normal vectors at any chosen point separates into two connected components. In addition, we impose the condition of \\emph{spacelike convexity}, which is the condition that the acceleration vector along any geodesic of the submanifold is strictly spacelike, or equivalently that the second fundamental form $h(v,v)$ is strictly spacelike for any non-zero tangent vector $v$. At each point, the values of $h(v,v)$ must lie in the same connected component of the spacelike normal cone, which we call the \\emph{inward normal cone}.\n Spacelike-convex submanifolds are a natural analogue of convex hypersurfaces in Euclidean space: For example, the orthogonal projection of such a submanifold onto any maximal spacelike subspace $\\R^{n+1,0} \\subset \\R^{n+1,k}$ is locally convex, and the sectional curvature of the induced metric is positive. \n More interestingly for our purposes, the condition of spacelike convexity leads to a natural pinching condition for the second fundamental form: There exists $\\a > 0$ such that $h(v,v) - \\a H$ is inward spacelike for all unit $v\\in T\\S$, and similarly there exists $\\b > 0$ such that $h(v,v) - \\b H$ lies in the outward normal cone. We call the largest $\\a$ and smallest $\\b$ for which this holds the inward and outward pinching ratios respectively. Our first result is that such pinching is preserved under the flow.\n\n\\begin{proposition}\n Suppose that $\\S$ is compact and $F: \\S^n \\times I \\to \\R^{n+1,k}$ is a solution to MCF with $F(.,0)$ spacelike and spacelike-convex. Then the inward pinching ratio $\\a$ is non-decreasing in $t$, and the outward pinching ratio $\\b$ is non-increasing in $t$.\n\\end{proposition}\n\nIn \\cite{huisken84}, pinching of the form $h(v,v)\\geq \\alpha Hg(v,v)$ is preserved under the flow and can be improved via Stampaccia iteration, eventually leading to Huisken's theorem. Such an approach may be possible in our setting, but we instead use an alternative approach using non-collapsing estimates, which has the advantage of giving direct control on the global geometry of the evolving submanifolds. Sheng and Wang \\cite{shengwang} proved that embedded mean-convex solutions of mean curvature flow in Euclidean space satisfy a non-collapsing estimate: For any point on the hypersurface there is a touching enclosed ball with curvature comparable to the mean curvature at that point. In \\cite{andrews2011noncollapsingmeanconvexmeancurvature}, the first author found a proof of such an estimate using a direct maximum principle argument, which also provides a corresponding result for balls touching from the exterior. This argument enables a very direct proof of the result of \\cite{huisken84}. In this paper in Section \\ref{noncollapsing section} we prove a version of noncollapsing for space-like convex submanifolds (the first high-codimension setting where such a result is known). This non-collapsing result holds also for the more general class of acausal submanifolds which are spacelike-mean-convex (that is, the mean curvature is strictly spacelike at each point). We say that an acausal spacelike (mean) convex embedding $F: \\S^n \\to \\R^{n+1,k}$ is (interior) noncollapsed with parameter $\\delta$ if, for any $x\\in \\S$, the pseudoball $\\left\\{|p-F(x)-\\delta\\frac{H(x)}{|H(x)|^2}|< \\frac{\\delta}{|H(x)|}\\right\\}$ has $F(\\Sigma)$ in its exterior, touching at $F(x)$. There is a similar definition of exterior noncollapsing. Our second result is that the flow preserves noncollapsing (\\cref{noncollapsing section}).", "full_context": "The evolution of hypersurfaces by their mean curvature has been studied extensively since Huisken \\cite{huisken84}, who proved that closed embedded convex hypersurfaces in Euclidean space contract to round points under mean curvature flow. \nMore recently, the mean curvature flow of submanifolds with higher codimension has been studied. The first author and Baker \\cite{andrewsbaker2010} proved an analogous result to that of \\cite{huisken84} in any codimension of Euclidean background under a quadratic curvature pinching condition $|h|^2 \\leq c |H|^2$ on the initial data, where $h$ is the second fundamental form, $H$ is the mean curvature, and $c$ is a dimensional constant. Since then, Baker and Nguyen \\cite{bakernguyen2023} have explored the flow in the case where the background is a sphere of arbitrary codimension, where the sharp quartic pinching condition for the sphere is found in \\cite{vogiatzi2024sharpquarticpinchingmean}. Other directions in high codimension have been explored, including ancient solutions \\cite{lynchnguyen2021, Risa_2018}, singularity formation \\cite{vogiatzi2023singularitymodelshighcodimension}, flow in complex projective space \\cite{vogiatzi2023meancurvatureflowhigh}, and flow in pseudo-Euclidean space of submanifolds with maximal spacelike dimension \\cite{Lambert_2019}.\n\nIn this paper, we consider the mean curvature flow (MCF) of embedded spacelike submanifolds under the assumption that the normal subspace at each point has signature $(1,k)$, so that the codimension may be arbitrary but no two spacelike normal vectors are orthogonal. This implies that the cone of strictly spacelike normal vectors at any chosen point separates into two connected components. In addition, we impose the condition of \\emph{spacelike convexity}, which is the condition that the acceleration vector along any geodesic of the submanifold is strictly spacelike, or equivalently that the second fundamental form $h(v,v)$ is strictly spacelike for any non-zero tangent vector $v$. At each point, the values of $h(v,v)$ must lie in the same connected component of the spacelike normal cone, which we call the \\emph{inward normal cone}.\n Spacelike-convex submanifolds are a natural analogue of convex hypersurfaces in Euclidean space: For example, the orthogonal projection of such a submanifold onto any maximal spacelike subspace $\\R^{n+1,0} \\subset \\R^{n+1,k}$ is locally convex, and the sectional curvature of the induced metric is positive. \n More interestingly for our purposes, the condition of spacelike convexity leads to a natural pinching condition for the second fundamental form: There exists $\\a > 0$ such that $h(v,v) - \\a H$ is inward spacelike for all unit $v\\in T\\S$, and similarly there exists $\\b > 0$ such that $h(v,v) - \\b H$ lies in the outward normal cone. We call the largest $\\a$ and smallest $\\b$ for which this holds the inward and outward pinching ratios respectively. Our first result is that such pinching is preserved under the flow.\n\n\\begin{proposition}\n Suppose that $\\S$ is compact and $F: \\S^n \\times I \\to \\R^{n+1,k}$ is a solution to MCF with $F(.,0)$ spacelike and spacelike-convex. Then the inward pinching ratio $\\a$ is non-decreasing in $t$, and the outward pinching ratio $\\b$ is non-increasing in $t$.\n\\end{proposition}\n\nIn \\cite{huisken84}, pinching of the form $h(v,v)\\geq \\alpha Hg(v,v)$ is preserved under the flow and can be improved via Stampaccia iteration, eventually leading to Huisken's theorem. Such an approach may be possible in our setting, but we instead use an alternative approach using non-collapsing estimates, which has the advantage of giving direct control on the global geometry of the evolving submanifolds. Sheng and Wang \\cite{shengwang} proved that embedded mean-convex solutions of mean curvature flow in Euclidean space satisfy a non-collapsing estimate: For any point on the hypersurface there is a touching enclosed ball with curvature comparable to the mean curvature at that point. In \\cite{andrews2011noncollapsingmeanconvexmeancurvature}, the first author found a proof of such an estimate using a direct maximum principle argument, which also provides a corresponding result for balls touching from the exterior. This argument enables a very direct proof of the result of \\cite{huisken84}. In this paper in Section \\ref{noncollapsing section} we prove a version of noncollapsing for space-like convex submanifolds (the first high-codimension setting where such a result is known). This non-collapsing result holds also for the more general class of acausal submanifolds which are spacelike-mean-convex (that is, the mean curvature is strictly spacelike at each point). We say that an acausal spacelike (mean) convex embedding $F: \\S^n \\to \\R^{n+1,k}$ is (interior) noncollapsed with parameter $\\delta$ if, for any $x\\in \\S$, the pseudoball $\\left\\{|p-F(x)-\\delta\\frac{H(x)}{|H(x)|^2}|< \\frac{\\delta}{|H(x)|}\\right\\}$ has $F(\\Sigma)$ in its exterior, touching at $F(x)$. There is a similar definition of exterior noncollapsing. Our second result is that the flow preserves noncollapsing (\\cref{noncollapsing section}).\n\n\\begin{proposition}\n Suppose that $\\S$ is compact and $F: \\S^n \\times I \\to \\R^{n+1,k}$ is a solution to MCF with $F(.,0)$ spacelike and spacelike-convex. Then the inward pinching ratio $\\a$ is non-decreasing in $t$, and the outward pinching ratio $\\b$ is non-increasing in $t$.\n\\end{proposition}\n\nIn \\cite{huisken84}, pinching of the form $h(v,v)\\geq \\alpha Hg(v,v)$ is preserved under the flow and can be improved via Stampaccia iteration, eventually leading to Huisken's theorem. Such an approach may be possible in our setting, but we instead use an alternative approach using non-collapsing estimates, which has the advantage of giving direct control on the global geometry of the evolving submanifolds. Sheng and Wang \\cite{shengwang} proved that embedded mean-convex solutions of mean curvature flow in Euclidean space satisfy a non-collapsing estimate: For any point on the hypersurface there is a touching enclosed ball with curvature comparable to the mean curvature at that point. In \\cite{andrews2011noncollapsingmeanconvexmeancurvature}, the first author found a proof of such an estimate using a direct maximum principle argument, which also provides a corresponding result for balls touching from the exterior. This argument enables a very direct proof of the result of \\cite{huisken84}. In this paper in Section \\ref{noncollapsing section} we prove a version of noncollapsing for space-like convex submanifolds (the first high-codimension setting where such a result is known). This non-collapsing result holds also for the more general class of acausal submanifolds which are spacelike-mean-convex (that is, the mean curvature is strictly spacelike at each point). We say that an acausal spacelike (mean) convex embedding $F: \\S^n \\to \\R^{n+1,k}$ is (interior) noncollapsed with parameter $\\delta$ if, for any $x\\in \\S$, the pseudoball $\\left\\{|p-F(x)-\\delta\\frac{H(x)}{|H(x)|^2}|< \\frac{\\delta}{|H(x)|}\\right\\}$ has $F(\\Sigma)$ in its exterior, touching at $F(x)$. There is a similar definition of exterior noncollapsing. Our second result is that the flow preserves noncollapsing (\\cref{noncollapsing section}).\n\nThe assumption of turning number $1$ is carried out throughout the paper. The Euclidean inner product leads to a notion of long time existence, hence a subsequence convergence result that a suitably rescaled flow admits a subsequence that converges smoothly to a sphere in some maximal spacelike subspace $\\R^{n+1,0} \\subset \\R^{n+1,k}$. The approach to full convergence is motivated by first author's approach on Gauss curvature flow with the idea of considering normalised flow invariant under affine transformations \\cite{andrews96} and showing that the original flow is stable near the final shape \\cite[Chapter 16]{ben2020}. Since the rescaled flow consists of a Lorentz transform at each subsequent time, we consider the normalised flow that fixes the volume, centre of mass and the maximal spacelike subspace that has the minimum projected volume. This leads to a stability argument which proves our final result: an analogue of Huisken's theorem.\n\n\\begin{proposition}[spacelike-convexity is preserved]\\label{spacelike-convexity is preserved}\n Suppose that $\\S$ is compact, spacelike-convex and $F: \\S^n \\times I \\to \\R^{n+1,k}$ is a solution to MCF, then spacelike-convexity is preserved along the flow. \n\\end{proposition}\n\n\\begin{proposition}[Pinching is preserved]\\label{pinching preserved prop}\n Suppose that $\\S$ is compact, spacelike-convex and $F: \\S^n \\times I \\to \\R^{n+1,k}$ is a solution to MCF, then there exists some $\\a > 0$ such that $h(v,v) - \\a H$ is spacelike for all unit $v\\in T\\S$ at all times. \n\\end{proposition}\n\nFirst, we show that spacelike-mean-convexity is preserved, assuming that the induced metric stays spacelike, this condition is implied by noncollapsing so we omit the assumption for results in this section. \n\\begin{proposition}\n Suppose that $\\S$ is compact, spacelike-mean-convex and $F: \\S^n \\times I \\to \\R^{n+1,k}$ is a solution to MCF, then spacelike-mean-convexity is preserved along the flow. \n\\end{proposition}\n\n\\begin{theorem}\\label{acausal spacelike-mean-convex submanifold preserves noncollapsing}\n Suppose $F: \\S^n \\times [0, T) \\to \\R^{n+1,k}$ is a spacelike-convex, compact and embedded solution to MCF, if $\\S\\times \\{0\\}$ is $\\d_-$-exterior noncollapsed and $\\d_+$-interior noncollapsed, then\n \\begin{equation*}\n \\overline{Q}(x, \\d_-, t) \\leq 0, \\qquad \\underline{Q}(x,\\d_+, t) \\geq 0\n \\end{equation*}\n for all $t\\in [0, T)$. \n\\end{theorem}\n\n\\begin{corollary}\\label{spacelike-mean-convex data stays spacelike coro}\n Suppose that $\\S$ is compact, spacelike-mean-convex and $F: \\S^n \\times I \\to \\R^{n+1,k}$ is a solution to MCF, then $\\S_t$ stays spacelike under the flow. \n\\end{corollary}\n\n\\begin{proposition}[Short time existence]\\label{short time existence MCF}\n    Suppose that $\\S$ is compact, spacelike-convex and let $F_0: \\S^n \\to \\R^{n+1,k}$ be a smooth immersion, then there exists $\\e > 0$ and a smooth solution $F(\\cdot,t): \\S^n \\to \\R^{n+1,k}$ to MCF defined for $t\\in [0, \\e)$ satisfying $F(\\cdot,0) = F_0$.\n\\end{proposition}\n\n\\section{Noncollapsing}\\label{noncollapsing section}\n\nThe result holds for a more general class of submanifolds, which is an analogue of mean-convex hypersurfaces. \n\n\\begin{definition}\n    A spacelike submanifold $\\S^n \\subset \\R^{n+1,k}$ is \\emph{spacelike-mean-convex} if its mean curvature is everywhere spacelike and non-vanishing.\n\\end{definition}\n\nFirst, we show that spacelike-mean-convexity is preserved, assuming that the induced metric stays spacelike, this condition is implied by noncollapsing so we omit the assumption for results in this section. \n\\begin{proposition}\n    Suppose that $\\S$ is compact, spacelike-mean-convex and $F: \\S^n \\times I \\to \\R^{n+1,k}$ is a solution to MCF, then spacelike-mean-convexity is preserved along the flow. \n\\end{proposition}", "post_theorem_intro_text_len": 1521, "post_theorem_intro_text": "The assumption of turning number $1$ is carried out throughout the paper. The Euclidean inner product leads to a notion of long time existence, hence a subsequence convergence result that a suitably rescaled flow admits a subsequence that converges smoothly to a sphere in some maximal spacelike subspace $\\R^{n+1,0} \\subset \\R^{n+1,k}$. The approach to full convergence is motivated by first author's approach on Gauss curvature flow with the idea of considering normalised flow invariant under affine transformations \\cite{andrews96} and showing that the original flow is stable near the final shape \\cite[Chapter 16]{ben2020}. Since the rescaled flow consists of a Lorentz transform at each subsequent time, we consider the normalised flow that fixes the volume, centre of mass and the maximal spacelike subspace that has the minimum projected volume. This leads to a stability argument which proves our final result: an analogue of Huisken's theorem. \n\n\\begin{theorem}\n    Let $F: \\S^n \\times [0, T) \\to \\R^{n+1,k}$ be a maximal solution to MCF, if $F(\\cdot, 0)$ is a spacelike-convex embedding, then $F(\\cdot, t)$ is a spacelike-convex embedding for all $0 < t < T $, $F_t$ converges uniformly to a point as $t \\to T$. Moreover, a suitably rescaled flow $\\Tilde{F}(\\cdot, \\tau)$ exists for all $\\tau \\in [0, \\infty)$ and converges in smooth topology to a smooth embedding whose image coincides with $S^n \\subset \\R^{n+1,0}$ of $\\R^{n+1,k}$. \n\\end{theorem}\n\nFor $n = 1$, this is an analogue of Gage--Hamilton theorem.", "sketch": "The post-theorem introduction does not sketch a proof of the stated noncollapsing preservation result. It instead outlines an approach toward a later “final result” (an analogue of Huisken's theorem): using the Euclidean inner product to obtain “long time existence” and a “subsequence convergence result” for a suitably rescaled flow converging smoothly to a sphere in a maximal spacelike subspace; then, to upgrade subsequence convergence to full convergence, it follows an approach “motivated by” work on Gauss curvature flow by considering a “normalised flow invariant under affine transformations” and proving the “original flow is stable near the final shape.” Because the rescaled flow involves “a Lorentz transform at each subsequent time,” it instead considers a normalised flow that “fixes the volume, centre of mass and the maximal spacelike subspace that has the minimum projected volume,” leading to “a stability argument” establishing the Huisken-type convergence.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "Suppose that $\\S$ is compact, spacelike (mean) convex and acausal, and $F: \\S^n \\times I \\to \\R^{n+1,k}$ is a solution to MCF, and for $n = 1$ we require the curve to have turning number (degree of the Gauss map) $1$ , then noncollapsing is preserved.,", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let $F:\\Sigma^n\\times I\\to\\mathbb{R}^{n+1,k}$ be a solution of mean curvature flow $(\\partial_tF=H)$ for a compact acausal spacelike submanifold, where acausal means that no two points of the submanifold are timelike-separated in the ambient pseudo-Euclidean space $\\mathbb{R}^{n+1,k}$. Assume the submanifold is spacelike (mean) convex: either it is spacelike-convex, meaning $h(v,v)$ is strictly spacelike for every nonzero tangent vector $v$, or it is spacelike-mean-convex, meaning the mean curvature vector $H$ is everywhere spacelike and nonzero. If $n=1$, also assume the curve has turning number (degree of the Gauss map) $1$. For $\\delta>0$, say a time-slice is $\\delta$-interior noncollapsed if for every $x\\in\\Sigma$ the pseudoball\n$$\\left\\{p:\\left|p-F(x)-\\delta\\frac{H(x)}{|H(x)|^2}\\right|<\\frac{\\delta}{|H(x)|}\\right\\}$$\ntouches the submanifold at $F(x)$ and the image lies outside it; define $\\delta$-exterior noncollapsed analogously using the corresponding exterior touching pseudoball. Which statement holds for every such flow?", "correct_choice": {"label": "A", "text": "Noncollapsing is preserved along the flow: if the initial slice is noncollapsed (in particular, if it is $\\delta_+$-interior noncollapsed and/or $\\delta_-$-exterior noncollapsed), then every later time-slice remains noncollapsed with the same corresponding noncollapsing parameter(s) for all times in $I$."}, "choices": [{"label": "B", "text": "Noncollapsing improves along the flow: if the initial slice is noncollapsed, then for every later time-slice there exists a parameter strictly larger than the initial interior noncollapsing parameter and strictly larger than the initial exterior noncollapsing parameter for which the corresponding interior and exterior noncollapsing inequalities hold throughout $I$."}, {"label": "C", "text": "One-sided noncollapsing is preserved along the flow: if the initial slice is $\\delta_+$-interior noncollapsed, then every later time-slice remains $\\delta_+$-interior noncollapsed for all times in $I$; likewise, if the initial slice is $\\delta_-$-exterior noncollapsed, then every later time-slice remains $\\delta_-$-exterior noncollapsed."}, {"label": "D", "text": "Noncollapsing holds automatically along the flow from the geometric assumptions alone: every compact acausal spacelike (mean) convex solution, with the turning-number-$1$ hypothesis when $n=1$, is both $\\delta_+$-interior noncollapsed and $\\delta_-$-exterior noncollapsed for some positive constants $\\delta_+,\\delta_-$ depending only on $n$ and $k$, for all times in $I$, without any initial noncollapsing assumption."}, {"label": "E", "text": "Noncollapsing is preserved along the flow provided one strengthens the initial hypothesis to two-sided noncollapsing: if the initial slice is simultaneously $\\delta$-interior noncollapsed and $\\delta$-exterior noncollapsed for the same parameter $\\delta>0$, then every later time-slice remains both $\\delta$-interior and $\\delta$-exterior noncollapsed for all times in $I$; however, preservation need not hold for only one of the two sides separately."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "preserved_same_parameter_vs_strict_improvement", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_full_noncollapsed_to_one_sided_preservation", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "initial_noncollapsing_hypothesis_and_dependence_of_constants", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "independence_of_interior_and_exterior_preservation", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct conclusion. It states the hypotheses and definitions, but the exact preservation statement in choice A is not directly given in the prompt."}, "TAS": {"score": 0, "justification": "The item is essentially a direct theorem-recall question: the stem lists the assumptions of a preservation theorem and asks for the conclusion. Choice A is basically the theorem statement itself."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ by subtle quantifier and strength changes (one-sided vs two-sided, same parameter vs weakened parameter). However, the task is mostly recognition of the exact theorem rather than genuine derivation or synthesis."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically relevant: they reflect common misreadings such as requiring both sides initially, allowing parameter degradation, or preserving only one side. They are distinct and well targeted."}, "total_score": 5, "overall_assessment": "A solid recognition-style theorem question with strong distractors, but it is highly tautological and only moderately tests reasoning rather than generative understanding."}}
{"id": "2602.19801v1", "paper_link": "http://arxiv.org/abs/2602.19801v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thmmain}\nGiven $v_{0}\\in H^3(\\mathcal O)$, $\\sigma_{0} \\in H^2(\\mathcal O)$, and $p_{0}\\in H^3(\\mathbb T^2),$ such that\n$$\n\\sigma_{0}\\geq\\underline\\sigma, \\quad p_{0}\\geq\\underline p,\\quad \\partial_zv_{0}|_{z=0,1}=0,\\quad \\partial_z\\sigma_{0}|_{z=0,1}=0,\n$$\nfor two positive numbers $\\underline\\sigma$ and $\\underline p$.\nThen, there is a positive time $\\mathcal T_0$,\ndepending only on $\\gamma$, $\\nu$, $\\mu$, $\\lambda$, $\\underline\\sigma$, $\\underline p$, and\n  $\\|v_{0}\\|_{H^3}^2+\\|\\sigma_{0}\\|_{H^2}^2+\\|p_0\\|_{H^3}^2$, such that system (\\ref{EQv0})--(\\ref{EQp0}),\n  subject to (\\ref{BC1})--(\\ref{IC}), has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$,\n  satisfying\n  \\begin{eqnarray*}\n  &&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\geq0.5\\underline\\sigma,\\quad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\geq0.5\\underline p,\\\\\n    &&v\\in C([0,\\mathcal T_0]; H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0; H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0; H^4(\\mathcal O)),\\\\\n    &&\\sigma\\in C([0,\\mathcal T_0]; H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0; H^2(\\mathcal O)),\\quad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\\\\n    &&\\partial_tv\\in L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\quad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0; H^1(\\mathcal O)),\\\\\n    &&p\\in C([0,\\mathcal T_0]; H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0; H^3(\\mathbb T^2)),\\quad\\partial_tp\\in  L^2(0,\\mathcal T_0; H^2(\\mathbb T^2)).\n  \\end{eqnarray*}", "start_pos": 14353, "end_pos": 15876, "label": "thmmain"}, "ref_dict": {"EQp0": "\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n  &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n  \\label{EQs0}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}", "EQp00": "\\begin{eqnarray}\n \\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)},\\label{EQp00}\\\\\n\\tilde v\\cdot\\nabla_hp+\\gamma p(\\text{div}_h\\tilde v+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)\\widetilde{Q(\\nabla v)}. \\label{EQw000}\n\\end{eqnarray}", "IC": "\\begin{equation}\n  \\label{IC}\n  (v, \\sigma, p)|_{t=0}=(v_{0}, \\sigma_{0}, p_{0}).\n\\end{equation}", "EQv00": "\\begin{equation}\n\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma \\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma \\nabla_h\\text{div}_hv. \\label{EQv00}\n\\end{equation}", "EQw00": "\\begin{eqnarray}\\label{eq:dz-w}\n\t\t \\dz w &=&-\\text{div}_h\\tilde v+\\frac{\\gamma-1}{\\gamma p}\\left(\\kappa\\partial_z^2\\theta+\\widetilde{Q(\\nabla v)}\\right)-\\frac{1}{\\gamma p}\\tilde v\\cdot\\nabla_hp\\nonumber\\\\\n&=&\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}\\partial_{zz} \\sigma -\\text{div}_h\\tilde v-\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right)\\nonumber\\\\\n&=&\\nu\\partial_z^2\\sigma-\\phi(v,p),\\label{EQw00}\n\\end{eqnarray}", "phi": "\\begin{eqnarray}\n  \\phi(v,p):=\\text{div}_h\\tilde v+\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right).\\label{phi}\n\\end{eqnarray}", "Q": "\\begin{eqnarray}\n\t&\\mathbb S_h = \\mu (\\nablah v + \\nablah^\\top v) + \\lambda \\dvh v \\mathbb I_2, \\quad\n\tp = R \\rho \\theta, \\nonumber\\\\\n  &Q(\\nabla v)=\\mathbb S_h:\\nabla_hv+\\mu|\\partial_zv|^2,\\label{Q}\n\\end{eqnarray}", "eq:dz-w": "\\begin{eqnarray}\\label{eq:dz-w}\n\t\t \\dz w &=&-\\text{div}_h\\tilde v+\\frac{\\gamma-1}{\\gamma p}\\left(\\kappa\\partial_z^2\\theta+\\widetilde{Q(\\nabla v)}\\right)-\\frac{1}{\\gamma p}\\tilde v\\cdot\\nabla_hp\\nonumber\\\\\n&=&\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}\\partial_{zz} \\sigma -\\text{div}_h\\tilde v-\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right)\\nonumber\\\\\n&=&\\nu\\partial_z^2\\sigma-\\phi(v,p),\\label{EQw00}\n\\end{eqnarray}", "EQs00": "\\begin{equation}\\label{EQs00}\n\t\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma.\n\\end{equation}", "BC1": "\\begin{eqnarray}\n  &v, \\sigma, \\mbox{ and } p \\mbox{ are periodic in }x, y, \\label{BC1}\\\\\n  &\\partial_zv|_{z=0,1}=0, \\quad\\partial_z\\sigma|_{z=0,1}=0. \\label{BC2}\n\\end{eqnarray}", "EQv": "\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv}\\\\\n  &w=\\nu\\partial_zv-\\int_0^z\\phi(v,p)dz',\\label{EQw}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma+\\epsilon\\Delta_h\\sigma,\n  \\label{EQs}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=\\epsilon\\Delta_hp+(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp}\n\\end{eqnarray}", "EQw000": "\\begin{eqnarray}\n \\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)},\\label{EQp00}\\\\\n\\tilde v\\cdot\\nabla_hp+\\gamma p(\\text{div}_h\\tilde v+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)\\widetilde{Q(\\nabla v)}. \\label{EQw000}\n\\end{eqnarray}", "eq:CPE": "\\begin{equation}\\label{eq:CPE}\n\t\\begin{cases}\n\t\t\\dt \\rho + \\dvh (\\rho v)  + \\dz (\\rho w) = 0, \\\\\n\t\t\\dt (\\rho v) + \\dvh(\\rho v\\otimes v) + \\dz(\\rho w v) + \\nablah p = \\dvh \\mathbb S_h + \\mu \\partial_{zz} v, \\\\\n\t\t\\dz p = 0, \\\\\n\t\t\\dfrac{1}{\\gamma-1}\\bigl(\\dt p + \\dvh(p v) + \\dz (pw) \\bigr) + p(\\dvh v + \\dz w) = \\dz (\\kappa \\dz \\theta )+Q(\\nabla v),\n\t\\end{cases}\n\\end{equation}", "EQw00'": "\\begin{equation}\n  w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz'.\\label{EQw00'}\n\\end{equation}", "EQv0": "\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n  &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n  \\label{EQs0}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}", "EQs0": "\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n  &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n  \\label{EQs0}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}", "EQp": "\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv}\\\\\n  &w=\\nu\\partial_zv-\\int_0^z\\phi(v,p)dz',\\label{EQw}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma+\\epsilon\\Delta_h\\sigma,\n  \\label{EQs}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=\\epsilon\\Delta_hp+(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp}\n\\end{eqnarray}", "thmmain": "\\begin{theorem}\\label{thmmain}\nGiven $v_{0}\\in H^3(\\mathcal O)$, $\\sigma_{0} \\in H^2(\\mathcal O)$, and $p_{0}\\in H^3(\\mathbb T^2),$ such that\n$$\n\\sigma_{0}\\geq\\underline\\sigma, \\quad p_{0}\\geq\\underline p,\\quad \\partial_zv_{0}|_{z=0,1}=0,\\quad \\partial_z\\sigma_{0}|_{z=0,1}=0,\n$$\nfor two positive numbers $\\underline\\sigma$ and $\\underline p$.\nThen, there is a positive time $\\mathcal T_0$,\ndepending only on $\\gamma$, $\\nu$, $\\mu$, $\\lambda$, $\\underline\\sigma$, $\\underline p$, and\n  $\\|v_{0}\\|_{H^3}^2+\\|\\sigma_{0}\\|_{H^2}^2+\\|p_0\\|_{H^3}^2$, such that system (\\ref{EQv0})--(\\ref{EQp0}),\n  subject to (\\ref{BC1})--(\\ref{IC}), has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$,\n  satisfying\n  \\begin{eqnarray*}\n  &&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\geq0.5\\underline\\sigma,\\quad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\geq0.5\\underline p,\\\\\n    &&v\\in C([0,\\mathcal T_0]; H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0; H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0; H^4(\\mathcal O)),\\\\\n    &&\\sigma\\in C([0,\\mathcal T_0]; H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0; H^2(\\mathcal O)),\\quad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\\\\n    &&\\partial_tv\\in L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\quad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0; H^1(\\mathcal O)),\\\\\n    &&p\\in C([0,\\mathcal T_0]; H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0; H^3(\\mathbb T^2)),\\quad\\partial_tp\\in  L^2(0,\\mathcal T_0; H^2(\\mathbb T^2)).\n  \\end{eqnarray*}\n\\end{theorem}", "bc:CPE": "\\begin{equation}\\label{bc:CPE}\n\t\\dz \\theta\\big|_{z=0,1} = 0, \\quad \\dz v\\big|_{z=0,1} = 0, \\quad w\\big|_{z=0,1} = 0.\n\\end{equation}"}, "pre_theorem_intro_text_len": 10452, "pre_theorem_intro_text": "The compressible primitive equations, formally obtained as the hydrostatic approximation of the compressible Navier-Stokes equations by replacing the vertical momentum balance equation with the hydrostatic balance\nequation, are the fundamental equations for modern meteorological study (see, e.g., \\cite[Chapter 4]{Richardson1965}). In particular, it is the starting point of many large scale\nmodels in the theoretical investigations and practical weather predictions (see, e.g., \\cite{Lions1992,JLLions1992,Washington2005}). In fact, such an approximation is reliable and useful in the following sense: (i)\nthe vertical scale of the atmosphere is significantly smaller than the planetary horizontal scale; (ii) the balance of gravity and pressure dominates the dynamic in the vertical direction; (iii) the vertical velocity is\nusually hard to observe in reality. Rigorous justifications of the hydrostatic approximation of hydrodynamic equations can be found in \\cite{Li2017,Azerad2001} (see also \\cite{FugigaHHK,FuGiHiHuKaW1,FuruGigaKas,LiTitiYuan}) for incompressible flows and \\cite{Liu2019} for\nisentropic compressible flows.\n\nIn this work, our goal is to investigate the fundamental problem of the well-posedness of solutions to the compressible primitive equations with non-trivial entropy. In particular, we consider the non-isentropic compressible primitive equations with only vertical diffusion for the temperature:\n\\begin{equation}\\label{eq:CPE}\n\t\\begin{cases}\n\t\t\\partial_t \\rho + \\mathrm{div}_{h}\\, (\\rho v)  + \\partial_z (\\rho w) = 0, \\\\\n\t\t\\partial_t (\\rho v) + \\mathrm{div}_{h}\\,(\\rho v\\otimes v) + \\partial_z(\\rho w v) + \\nabla_{h} p = \\mathrm{div}_{h}\\, \\mathbb S_h + \\mu \\partial_{zz} v, \\\\\n\t\t\\partial_z p = 0, \\\\\n\t\t\\dfrac{1}{\\gamma-1}\\bigl(\\partial_t p + \\mathrm{div}_{h}\\,(p v) + \\partial_z (pw) \\bigr) + p(\\mathrm{div}_{h}\\, v + \\partial_z w) = \\partial_z (\\kappa \\partial_z \\theta )+Q(\\nabla v),\n\t\\end{cases}\n\\end{equation}\nwith\n\\begin{eqnarray}\n\t&\\mathbb S_h = \\mu (\\nabla_{h} v + \\nabla_{h}^\\top v) + \\lambda \\mathrm{div}_{h}\\, v \\mathbb I_2, \\quad\n\tp = R \\rho \\theta, \\nonumber\\\\\n  &Q(\\nabla v)=\\mathbb S_h:\\nabla_hv+\\mu|\\partial_zv|^2,\\label{Q}\n\\end{eqnarray}\nwith constant coefficients $\\gamma$, $\\mu$, $\\lambda$, $\\kappa,$ and $R$ satisfying $\\gamma > 1$, $\\mu+\\lambda>0$, $\\mu>0$, $\\kappa>0$, and $R>0$.\nHere, $ \\rho, v, w $, and $ p $ represent the density, the horizontal velocity, vertical velocity, and pressure, respectively. $ \\mathrm{div}_{h}\\,, \\nabla_{h} $, and $\\Delta_{h} $ are, and will be, the divergence, gradient, and Laplace operators in the horizontal variables $ (x,y) $, respectively.\nWe investigate system \\eqref{eq:CPE} in the horizontally periodic channel\n$$ \\mathcal O:= \\mathbb T^2 \\times (0,1) = \\lbrace (x,y,z) | (x,y) \\in \\mathbb T^2, z \\in (0,1) \\rbrace,\n$$\nand consider the following\nboundary conditions:\n\\begin{equation}\\label{bc:CPE}\n\t\\partial_z \\theta\\big|_{z=0,1} = 0, \\quad \\partial_z v\\big|_{z=0,1} = 0, \\quad w\\big|_{z=0,1} = 0.\n\\end{equation}\n\nThe isentropic compressible primitive equations have been investigated by the last two authors in \\cite{LT2018a} for local strong solutions and \\cite{LT2018b} for global weak solutions. The existence of global weak solutions is also studied by \\cite{Wang2020}, independently. See also, \\cite{Jiu2018,Ersoy2011a,Ersoy2012,Gatapov2005}.\n\nOn the other hand, the incompressible primitive equations have been the subject of intensive mathematical research since the introduction\nby Lions, Temam, and Wang in \\cite{Lions1992,JLLions1992}. For instance, Guill\\'en-Gonz\\'alez, Masmoudi, and Rodr\\'iguez-Bellido in\n\\cite{GuillenGonzalez2001} study the local existence of strong solutions and global existence of strong solutions with small initial data. In\n\\cite{HuTemamZiane2003}, the authors address the global existence of strong solutions in a domain with small depth. The first breakthrough\nconcerning the global well-posedness of incompressible primitive equations is obtained by Cao and Titi in \\cite{Cao2007}. See also,\n\\cite{Cao2014a,Cao2014b,Cao2016,Cao2016a,CaoLiTiti2020,CaoLiTitiWang2024,Boling,MHATK,MHTKa,GMK,IKMZ,Li2017a,LiYuan2022} and the references therein for\nrelated literatures; in particular, global well-posedness of strong solutions was proved in \\cite{Cao2016,Cao2016a,CaoLiTiti2020,CaoLiTitiWang2024} to the primitive equations with only horizontal viscosity.\n\nWe have introduced the PE diagram in \\cite{LT2018LowMach1}, which concerns the low Mach number limit and the small aspect ratio (between the vertical and horizontal scales) limit. In \\cite{LT2018LowMach1,LT2018LowMach2}, we also establish the low Mach number limit for the isentropic compressible primitive equations. However, the counterpart study of the compressible Navier-Stokes-Fourier equations (see, e.g., \\cite{Alazard2006}) indicates that the PE diagram might be completely different for non-isentropic flows, due to the additional heat conductivity. We leave such a subject to future study.\n\nFor interested readers, we refer to \\cite{Temam1977,Lions1996,Lions1998,Feireisl2004,Feireisl2009a} for the study of hydrodynamic equations.\n\nOne can easily see that, compared to the Navier-Stokes equations, the evolutionary equation for vertical velocity is missing in system\n\\eqref{eq:CPE}. In fact, this is one of the main challenges in the study of the compressible primitive equations. In order to have a better\nunderstanding of the system, we derive a representation of vertical velocity in terms of horizontal velocity, density, and pressure.\nConsequently, we will reformulate system \\eqref{eq:CPE} to an equivalent one.\n\nUse \\subeqref{eq:CPE}{1} to rewrite \\subeqref{eq:CPE}{2} as\n$$\n\\rho(\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv)+\\nabla_hp=\\mu\\Delta v+(\\mu+\\lambda)\\nabla_h\\text{div}_hv\n$$\nand define\n\\begin{equation}\\label{def:rho-1}\n\t\\sigma := \\frac1\\rho.\n\\end{equation}\nThen, it is clear that\n\\begin{equation}\n\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma \\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma \\nabla_h\\text{div}_hv. \\label{EQv00}\n\\end{equation}\n\nNote that \\subeqref{eq:CPE}{3} implies that $p$ is independent of the vertical variable $z$ and one can use \\subeqref{eq:CPE}{3} to rewrite \\subeqref{eq:CPE}{4} as\n\\begin{equation*}\n \\partial_tp+v\\cdot\\nabla_hp+\\gamma p(\\text{div}_hv+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)Q(\\nabla v).\n\\end{equation*}\nSeparating the $z$-average part and the fluctuation part of the above equation, and recalling the boundary conditions $\\partial_z\\theta|_{z=0,1}=w|_{z=0,1}=0$, one obtains that\n\\begin{eqnarray}\n \\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)},\\label{EQp00}\\\\\n\\tilde v\\cdot\\nabla_hp+\\gamma p(\\text{div}_h\\tilde v+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)\\widetilde{Q(\\nabla v)}. \\label{EQw000}\n\\end{eqnarray}\nHere, for a function $f$, we use\n\\begin{equation}\\label{TBf}\n\\bar f:=\\int_0^1f dz\\quad\\mbox{and}\\quad  \\tilde f:=f-\\bar f,\n\\end{equation}\nto represent the vertical integral (equivalently, average) and the vertical fluctuation of the quantity considered, respectively.\n\nSince $\\theta = R^{-1} \\rho^{-1} p=R^{-1}\\sigma p,$\nit follows from \\eqref{EQw000} and \\subeqref{eq:CPE}{3} that\n\\begin{eqnarray}\\label{eq:dz-w}\n\t\t \\partial_z w &=&-\\text{div}_h\\tilde v+\\frac{\\gamma-1}{\\gamma p}\\left(\\kappa\\partial_z^2\\theta+\\widetilde{Q(\\nabla v)}\\right)-\\frac{1}{\\gamma p}\\tilde v\\cdot\\nabla_hp\\nonumber\\\\\n&=&\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}\\partial_{zz} \\sigma -\\text{div}_h\\tilde v-\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right)\\nonumber\\\\\n&=&\\nu\\partial_z^2\\sigma-\\phi(v,p),\\label{EQw00}\n\\end{eqnarray}\nwhere $\\nu=\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}$ and\n\\begin{eqnarray}\n  \\phi(v,p):=\\text{div}_h\\tilde v+\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right).\\label{phi}\n\\end{eqnarray}\nRecalling that $p$ is independent of $z$, it holds that $\\partial_z\\sigma=R\\partial_z(p^{-1}\\theta)= R p^{-1}\\partial_z\\theta$. Then, the boundary condition \\eqref{bc:CPE} implies\n\\begin{equation}\\label{bc:rf-CPE}\n\\partial_z \\sigma\\big|_{z=0,1} = 0.\n\\end{equation}\nThanks to this and recalling that $w|_{z=0}$, it follows from \\eqref{EQw00} that\n\\begin{equation}\n  w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz'.\\label{EQw00'}\n\\end{equation}\n\nIt follows from \\subeqref{eq:CPE}{1} that\n\\begin{equation*}\\label{eq:rho-1}\n\t\\partial_t \\rho^{-1} + v \\cdot \\nabla_{h} \\rho^{-1} - \\rho^{-1} \\mathrm{div}_{h}\\, v = \\rho^{-2} \\partial_z (\\rho w) = \\rho^{-1} \\partial_z w - w \\partial_z \\rho^{-1},\n\\end{equation*}\nfrom which, using \\eqref{eq:dz-w} and recalling $\\sigma=\\frac1\\rho$, one gets\n\\begin{equation}\\label{EQs00}\n\t\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma.\n\\end{equation}\n\nNow, collecting \\eqref{EQv00}, \\eqref{EQp00}, \\eqref{EQw00'}, \\eqref{EQs00}, we have the following reformulated system which is equivalent to \\eqref{eq:CPE}\n\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n  &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n  \\label{EQs0}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\nwhere $\\phi(v,p)$ and $Q(\\nabla v)$ are expressed as in (\\ref{phi}) and (\\ref{Q}), respectively. The boundary conditions read as\n\\begin{eqnarray}\n  &v, \\sigma, \\mbox{ and } p \\mbox{ are periodic in }x, y, \\label{BC1}\\\\\n  &\\partial_zv|_{z=0,1}=0, \\quad\\partial_z\\sigma|_{z=0,1}=0. \\label{BC2}\n\\end{eqnarray}\nThe initial condition is\n\\begin{equation}\n  \\label{IC}\n  (v, \\sigma, p)|_{t=0}=(v_{0}, \\sigma_{0}, p_{0}).\n\\end{equation}\n\nBefore stating the main result, we recall some standard notations. For positive integer $k$ and $q\\in[1,\\infty]$, $L^q(\\mathcal O)$ and\n$L^q(\\mathbb T^2)$ are the Lebesgue spaces,\n$W^{k,q}(\\mathcal O)$ and $W^{k,q}(\\mathbb T^2)$ are the Sobolev spaces. If $q=2$, we use $H^k$ instead of $W^{k,2}$. For simplicity,\nwe will use $\\|\\cdot\\|_q$ to denote $L^q(\\mathcal O)$ or $L^q(\\mathbb T^2)$ norms of the corresponding function,\nif the meaning is clear from the context.\n\nWe are now in the position to state the main result of this paper.", "context": "In this work, our goal is to investigate the fundamental problem of the well-posedness of solutions to the compressible primitive equations with non-trivial entropy. In particular, we consider the non-isentropic compressible primitive equations with only vertical diffusion for the temperature:\n\\begin{equation}\\label{eq:CPE}\n \\begin{cases}\n \\partial_t \\rho + \\mathrm{div}_{h}\\, (\\rho v) + \\partial_z (\\rho w) = 0, \\\\\n \\partial_t (\\rho v) + \\mathrm{div}_{h}\\,(\\rho v\\otimes v) + \\partial_z(\\rho w v) + \\nabla_{h} p = \\mathrm{div}_{h}\\, \\mathbb S_h + \\mu \\partial_{zz} v, \\\\\n \\partial_z p = 0, \\\\\n \\dfrac{1}{\\gamma-1}\\bigl(\\partial_t p + \\mathrm{div}_{h}\\,(p v) + \\partial_z (pw) \\bigr) + p(\\mathrm{div}_{h}\\, v + \\partial_z w) = \\partial_z (\\kappa \\partial_z \\theta )+Q(\\nabla v),\n \\end{cases}\n\\end{equation}\nwith\n\\begin{eqnarray}\n &\\mathbb S_h = \\mu (\\nabla_{h} v + \\nabla_{h}^\\top v) + \\lambda \\mathrm{div}_{h}\\, v \\mathbb I_2, \\quad\n p = R \\rho \\theta, \\nonumber\\\\\n &Q(\\nabla v)=\\mathbb S_h:\\nabla_hv+\\mu|\\partial_zv|^2,\\label{Q}\n\\end{eqnarray}\nwith constant coefficients $\\gamma$, $\\mu$, $\\lambda$, $\\kappa,$ and $R$ satisfying $\\gamma > 1$, $\\mu+\\lambda>0$, $\\mu>0$, $\\kappa>0$, and $R>0$.\nHere, $ \\rho, v, w $, and $ p $ represent the density, the horizontal velocity, vertical velocity, and pressure, respectively. $ \\mathrm{div}_{h}\\,, \\nabla_{h} $, and $\\Delta_{h} $ are, and will be, the divergence, gradient, and Laplace operators in the horizontal variables $ (x,y) $, respectively.\nWe investigate system \\eqref{eq:CPE} in the horizontally periodic channel\n$$ \\mathcal O:= \\mathbb T^2 \\times (0,1) = \\lbrace (x,y,z) | (x,y) \\in \\mathbb T^2, z \\in (0,1) \\rbrace,\n$$\nand consider the following\nboundary conditions:\n\\begin{equation}\\label{bc:CPE}\n \\partial_z \\theta\\big|_{z=0,1} = 0, \\quad \\partial_z v\\big|_{z=0,1} = 0, \\quad w\\big|_{z=0,1} = 0.\n\\end{equation}\n\nNote that \\subeqref{eq:CPE}{3} implies that $p$ is independent of the vertical variable $z$ and one can use \\subeqref{eq:CPE}{3} to rewrite \\subeqref{eq:CPE}{4} as\n\\begin{equation*}\n \\partial_tp+v\\cdot\\nabla_hp+\\gamma p(\\text{div}_hv+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)Q(\\nabla v).\n\\end{equation*}\nSeparating the $z$-average part and the fluctuation part of the above equation, and recalling the boundary conditions $\\partial_z\\theta|_{z=0,1}=w|_{z=0,1}=0$, one obtains that\n\\begin{eqnarray}\n \\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)},\\label{EQp00}\\\\\n\\tilde v\\cdot\\nabla_hp+\\gamma p(\\text{div}_h\\tilde v+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)\\widetilde{Q(\\nabla v)}. \\label{EQw000}\n\\end{eqnarray}\nHere, for a function $f$, we use\n\\begin{equation}\\label{TBf}\n\\bar f:=\\int_0^1f dz\\quad\\mbox{and}\\quad \\tilde f:=f-\\bar f,\n\\end{equation}\nto represent the vertical integral (equivalently, average) and the vertical fluctuation of the quantity considered, respectively.\n\nSince $\\theta = R^{-1} \\rho^{-1} p=R^{-1}\\sigma p,$\nit follows from \\eqref{EQw000} and \\subeqref{eq:CPE}{3} that\n\\begin{eqnarray}\\label{eq:dz-w}\n \\partial_z w &=&-\\text{div}_h\\tilde v+\\frac{\\gamma-1}{\\gamma p}\\left(\\kappa\\partial_z^2\\theta+\\widetilde{Q(\\nabla v)}\\right)-\\frac{1}{\\gamma p}\\tilde v\\cdot\\nabla_hp\\nonumber\\\\\n&=&\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}\\partial_{zz} \\sigma -\\text{div}_h\\tilde v-\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right)\\nonumber\\\\\n&=&\\nu\\partial_z^2\\sigma-\\phi(v,p),\\label{EQw00}\n\\end{eqnarray}\nwhere $\\nu=\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}$ and\n\\begin{eqnarray}\n \\phi(v,p):=\\text{div}_h\\tilde v+\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right).\\label{phi}\n\\end{eqnarray}\nRecalling that $p$ is independent of $z$, it holds that $\\partial_z\\sigma=R\\partial_z(p^{-1}\\theta)= R p^{-1}\\partial_z\\theta$. Then, the boundary condition \\eqref{bc:CPE} implies\n\\begin{equation}\\label{bc:rf-CPE}\n\\partial_z \\sigma\\big|_{z=0,1} = 0.\n\\end{equation}\nThanks to this and recalling that $w|_{z=0}$, it follows from \\eqref{EQw00} that\n\\begin{equation}\n w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz'.\\label{EQw00'}\n\\end{equation}\n\nNow, collecting \\eqref{EQv00}, \\eqref{EQp00}, \\eqref{EQw00'}, \\eqref{EQs00}, we have the following reformulated system which is equivalent to \\eqref{eq:CPE}\n\\begin{eqnarray}\n &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n \\label{EQs0}\\\\\n &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\nwhere $\\phi(v,p)$ and $Q(\\nabla v)$ are expressed as in (\\ref{phi}) and (\\ref{Q}), respectively. The boundary conditions read as\n\\begin{eqnarray}\n &v, \\sigma, \\mbox{ and } p \\mbox{ are periodic in }x, y, \\label{BC1}\\\\\n &\\partial_zv|_{z=0,1}=0, \\quad\\partial_z\\sigma|_{z=0,1}=0. \\label{BC2}\n\\end{eqnarray}\nThe initial condition is\n\\begin{equation}\n \\label{IC}\n (v, \\sigma, p)|_{t=0}=(v_{0}, \\sigma_{0}, p_{0}).\n\\end{equation}\n\nBefore stating the main result, we recall some standard notations. For positive integer $k$ and $q\\in[1,\\infty]$, $L^q(\\mathcal O)$ and\n$L^q(\\mathbb T^2)$ are the Lebesgue spaces,\n$W^{k,q}(\\mathcal O)$ and $W^{k,q}(\\mathbb T^2)$ are the Sobolev spaces. If $q=2$, we use $H^k$ instead of $W^{k,2}$. For simplicity,\nwe will use $\\|\\cdot\\|_q$ to denote $L^q(\\mathcal O)$ or $L^q(\\mathbb T^2)$ norms of the corresponding function,\nif the meaning is clear from the context.\n\nWe are now in the position to state the main result of this paper.", "full_context": "In this work, our goal is to investigate the fundamental problem of the well-posedness of solutions to the compressible primitive equations with non-trivial entropy. In particular, we consider the non-isentropic compressible primitive equations with only vertical diffusion for the temperature:\n\\begin{equation}\\label{eq:CPE}\n \\begin{cases}\n \\partial_t \\rho + \\mathrm{div}_{h}\\, (\\rho v) + \\partial_z (\\rho w) = 0, \\\\\n \\partial_t (\\rho v) + \\mathrm{div}_{h}\\,(\\rho v\\otimes v) + \\partial_z(\\rho w v) + \\nabla_{h} p = \\mathrm{div}_{h}\\, \\mathbb S_h + \\mu \\partial_{zz} v, \\\\\n \\partial_z p = 0, \\\\\n \\dfrac{1}{\\gamma-1}\\bigl(\\partial_t p + \\mathrm{div}_{h}\\,(p v) + \\partial_z (pw) \\bigr) + p(\\mathrm{div}_{h}\\, v + \\partial_z w) = \\partial_z (\\kappa \\partial_z \\theta )+Q(\\nabla v),\n \\end{cases}\n\\end{equation}\nwith\n\\begin{eqnarray}\n &\\mathbb S_h = \\mu (\\nabla_{h} v + \\nabla_{h}^\\top v) + \\lambda \\mathrm{div}_{h}\\, v \\mathbb I_2, \\quad\n p = R \\rho \\theta, \\nonumber\\\\\n &Q(\\nabla v)=\\mathbb S_h:\\nabla_hv+\\mu|\\partial_zv|^2,\\label{Q}\n\\end{eqnarray}\nwith constant coefficients $\\gamma$, $\\mu$, $\\lambda$, $\\kappa,$ and $R$ satisfying $\\gamma > 1$, $\\mu+\\lambda>0$, $\\mu>0$, $\\kappa>0$, and $R>0$.\nHere, $ \\rho, v, w $, and $ p $ represent the density, the horizontal velocity, vertical velocity, and pressure, respectively. $ \\mathrm{div}_{h}\\,, \\nabla_{h} $, and $\\Delta_{h} $ are, and will be, the divergence, gradient, and Laplace operators in the horizontal variables $ (x,y) $, respectively.\nWe investigate system \\eqref{eq:CPE} in the horizontally periodic channel\n$$ \\mathcal O:= \\mathbb T^2 \\times (0,1) = \\lbrace (x,y,z) | (x,y) \\in \\mathbb T^2, z \\in (0,1) \\rbrace,\n$$\nand consider the following\nboundary conditions:\n\\begin{equation}\\label{bc:CPE}\n \\partial_z \\theta\\big|_{z=0,1} = 0, \\quad \\partial_z v\\big|_{z=0,1} = 0, \\quad w\\big|_{z=0,1} = 0.\n\\end{equation}\n\nNote that \\subeqref{eq:CPE}{3} implies that $p$ is independent of the vertical variable $z$ and one can use \\subeqref{eq:CPE}{3} to rewrite \\subeqref{eq:CPE}{4} as\n\\begin{equation*}\n \\partial_tp+v\\cdot\\nabla_hp+\\gamma p(\\text{div}_hv+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)Q(\\nabla v).\n\\end{equation*}\nSeparating the $z$-average part and the fluctuation part of the above equation, and recalling the boundary conditions $\\partial_z\\theta|_{z=0,1}=w|_{z=0,1}=0$, one obtains that\n\\begin{eqnarray}\n \\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)},\\label{EQp00}\\\\\n\\tilde v\\cdot\\nabla_hp+\\gamma p(\\text{div}_h\\tilde v+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)\\widetilde{Q(\\nabla v)}. \\label{EQw000}\n\\end{eqnarray}\nHere, for a function $f$, we use\n\\begin{equation}\\label{TBf}\n\\bar f:=\\int_0^1f dz\\quad\\mbox{and}\\quad \\tilde f:=f-\\bar f,\n\\end{equation}\nto represent the vertical integral (equivalently, average) and the vertical fluctuation of the quantity considered, respectively.\n\nSince $\\theta = R^{-1} \\rho^{-1} p=R^{-1}\\sigma p,$\nit follows from \\eqref{EQw000} and \\subeqref{eq:CPE}{3} that\n\\begin{eqnarray}\\label{eq:dz-w}\n \\partial_z w &=&-\\text{div}_h\\tilde v+\\frac{\\gamma-1}{\\gamma p}\\left(\\kappa\\partial_z^2\\theta+\\widetilde{Q(\\nabla v)}\\right)-\\frac{1}{\\gamma p}\\tilde v\\cdot\\nabla_hp\\nonumber\\\\\n&=&\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}\\partial_{zz} \\sigma -\\text{div}_h\\tilde v-\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right)\\nonumber\\\\\n&=&\\nu\\partial_z^2\\sigma-\\phi(v,p),\\label{EQw00}\n\\end{eqnarray}\nwhere $\\nu=\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}$ and\n\\begin{eqnarray}\n \\phi(v,p):=\\text{div}_h\\tilde v+\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right).\\label{phi}\n\\end{eqnarray}\nRecalling that $p$ is independent of $z$, it holds that $\\partial_z\\sigma=R\\partial_z(p^{-1}\\theta)= R p^{-1}\\partial_z\\theta$. Then, the boundary condition \\eqref{bc:CPE} implies\n\\begin{equation}\\label{bc:rf-CPE}\n\\partial_z \\sigma\\big|_{z=0,1} = 0.\n\\end{equation}\nThanks to this and recalling that $w|_{z=0}$, it follows from \\eqref{EQw00} that\n\\begin{equation}\n w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz'.\\label{EQw00'}\n\\end{equation}\n\nNow, collecting \\eqref{EQv00}, \\eqref{EQp00}, \\eqref{EQw00'}, \\eqref{EQs00}, we have the following reformulated system which is equivalent to \\eqref{eq:CPE}\n\\begin{eqnarray}\n &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n \\label{EQs0}\\\\\n &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\nwhere $\\phi(v,p)$ and $Q(\\nabla v)$ are expressed as in (\\ref{phi}) and (\\ref{Q}), respectively. The boundary conditions read as\n\\begin{eqnarray}\n &v, \\sigma, \\mbox{ and } p \\mbox{ are periodic in }x, y, \\label{BC1}\\\\\n &\\partial_zv|_{z=0,1}=0, \\quad\\partial_z\\sigma|_{z=0,1}=0. \\label{BC2}\n\\end{eqnarray}\nThe initial condition is\n\\begin{equation}\n \\label{IC}\n (v, \\sigma, p)|_{t=0}=(v_{0}, \\sigma_{0}, p_{0}).\n\\end{equation}\n\nBefore stating the main result, we recall some standard notations. For positive integer $k$ and $q\\in[1,\\infty]$, $L^q(\\mathcal O)$ and\n$L^q(\\mathbb T^2)$ are the Lebesgue spaces,\n$W^{k,q}(\\mathcal O)$ and $W^{k,q}(\\mathbb T^2)$ are the Sobolev spaces. If $q=2$, we use $H^k$ instead of $W^{k,2}$. For simplicity,\nwe will use $\\|\\cdot\\|_q$ to denote $L^q(\\mathcal O)$ or $L^q(\\mathbb T^2)$ norms of the corresponding function,\nif the meaning is clear from the context.\n\nWe are now in the position to state the main result of this paper.\n\nWe are now in the position to state the main result of this paper.\n\n\\begin{proposition}\n \\label{PROP-APR}\nGiven $(v_{0}, \\sigma_{0}, p_{0})$ satisfying (\\ref{ASSUMIC1})--(\\ref{ASSUMIC2}). Let $X_0$ be an arbitrary constant such that\n$\\|v_{0}\\|_{H^3}^2+\\|\\sigma_{0}\\|_{H^2}^2+\\|p_0\\|_{H^3}^2\\leq X_0$. Then, there is a positive time $\\mathcal T_0$,\ndepending only on $\\gamma$, $\\nu$, $\\mu$, $\\lambda$, $\\underline\\sigma$, $\\underline p$, and\n $X_0$, such that system (\\ref{EQv})--(\\ref{EQp}),\n subject to (\\ref{BC1})--(\\ref{IC}), has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$,\n satisfying\n \\begin{eqnarray*}\n &&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\geq0.5\\underline\\sigma,\\quad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\geq0.5\\underline p,\\\\\n &&\\sup_{0\\leq t\\leq\\mathcal T_0}(\\|v\\|_{H^3}^2+\\|\\sigma\\|_{H^2}^2+\\|p\\|_{H^3}^2)(t)+\\epsilon\\int_0^{\\mathcal T_0}(\\|\\nabla_h\\sigma\\|_{H^2}^2+\\|\\nabla_hp\\|_{H^3}^2)dt\\nonumber\\\\\n &&\\, \\, ~~~+\\int_0^{\\mathcal T_0}(\\|v\\|_{H^4}^2+\\|\\partial_z\\sigma\\|_{H^2}^2+\\|\\partial_tv\\|_{H^2}^2+\\|\\partial_t\\sigma\\|_{H^1}^2+\\|\\partial_tp\\|_{H^2}^2)dt \\leq K_0,\n \\end{eqnarray*}\n for a positive constant $K_0$ depending only on $\\gamma$, $\\nu$, $\\mu$, $\\lambda$, $\\underline\\sigma$, $\\underline p$, and\n $X_0$.\n\\end{proposition}\n\nBy Proposition \\ref{PROP-APR}, there is a positive time $\\mathcal T_0$ depending only on $\\gamma,$ $\\nu,$ $\\mu$, $\\lambda$,\n$\\underline\\sigma$, $\\underline p$, and $X_0$, such that for any $\\epsilon\\in(0,\\epsilon_0)$, system \\eqref{EQv}--\\eqref{EQp}, subject to \\eqref{BC1}--\\eqref{IC}, has a unique solution $(v_\\epsilon,\n\\sigma_\\epsilon, p_\\epsilon)$ on $\\mathcal O\\times(0,\\mathcal T_0)$,\nwith initial data $(v_{0,\\epsilon}, \\sigma_{0,\\epsilon}, p_{0,\\epsilon})$, satisfying\n \\begin{eqnarray}\n &&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma_\\epsilon\\geq0.5\\underline\\sigma,\\quad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p_\\epsilon\\geq0.5\\underline p,\\label{APRLBsp}\\\\\n &&\\sup_{0\\leq t\\leq\\mathcal T_0}(\\|v_\\epsilon\\|_{H^3}^2+\\|\\sigma_\\epsilon\\|_{H^2}^2+\\|p_\\epsilon\\|_{H^3}^2)(t)\n +\\epsilon\\int_0^{\\mathcal T_0}(\\|\\nabla_h\\sigma_\\epsilon\\|_{H^2}^2+\\|\\nabla_hp_\\epsilon\\|_{H^3}^2)dt\\nonumber\\\\\n &&\\, \\, ~~~+\\int_0^{\\mathcal T_0}(\\|v_\\epsilon\\|_{H^4}^2+\\|\\partial_z\\sigma_\\epsilon\\|_{H^2}^2+\\|\\partial_tv_\\epsilon\\|_{H^2}^2+\\|\\partial_t\\sigma_\\epsilon\n \\|_{H^1}^2+\\|\\partial_tp_\\epsilon\\|_{H^2}^2)dt \\leq K_0,\\label{APRHk}\n \\end{eqnarray}\n for a positive constant $K_0$ depending only on $\\gamma$, $\\nu$, $\\mu$, $\\lambda$, $\\underline\\sigma$, $\\underline p$, and\n $X_0$.\n\nThanks to \\eqref{APRHk}, by the Banach-Alaoglu theorem, and using Cantor's diagonal argument,\n there is a subsequence $(v_{\\epsilon_n}, \\sigma_{\\epsilon_n}, p_{\\epsilon_n})$ and a triple $(v,\\sigma,p)$,\n such that\n \\begin{eqnarray}\n &&v_{\\epsilon_n}\\stackrel{\\ast}{\\rightharpoonup}v,\\quad\\mbox{in }L^\\infty(0,\\mathcal T_0; H^3(\\mathcal O)),\\label{WeaC1}\\\\\n &&\\sigma_{\\epsilon_n}\\stackrel{\\ast}{\\rightharpoonup}\\sigma,\\quad\\mbox{in }L^\\infty(0,\\mathcal T_0; H^2(\\mathcal O)),\\label{WeaC2}\\\\\n &&p_{\\epsilon_n}\\stackrel{\\ast}{\\rightharpoonup}p,\\quad\\mbox{in }L^\\infty(0,\\mathcal T_0; H^3(\\mathbb T^2)),\\label{WeaC3}\\\\\n &&v_{\\epsilon_n}{\\rightharpoonup}v,\\quad\\mbox{in }L^2(0,\\mathcal T_0; H^4(\\mathcal O)),\\label{WeaC4}\\\\\n &&\\partial_z\\sigma_{\\epsilon_n} {\\rightharpoonup}\\partial_z\\sigma,\\quad\\mbox{in }L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\label{WeaC5}\\\\\n &&\\partial_tv_{\\epsilon_n} {\\rightharpoonup}\\partial_tv,\\quad\\mbox{in }L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\label{WeaC6}\\\\\n &&\\partial_t\\sigma_{\\epsilon_n} {\\rightharpoonup}\\partial_t\\sigma,\\quad\\mbox{in }L^2(0,\\mathcal T_0; H^1(\\mathcal O)),\\label{WeaC7}\\\\\n &&\\partial_tp_{\\epsilon_n} {\\rightharpoonup}\\partial_tp,\\quad\\mbox{in }L^2(0,\\mathcal T_0; H^2(\\mathbb T^2)),\\label{WeaC8}\n \\end{eqnarray}\n where $\\stackrel{\\ast}{\\rightharpoonup}$ and $\\rightharpoonup$ represent the weak-* and weak convergence, respectively, in the corresponding spaces. With the aid of the above convergence, noticing that $H^3(\\mathcal O)\\hookrightarrow\\hookrightarrow H^2(\\mathcal O)$, $H^2(\\mathcal O)\\hookrightarrow\\hookrightarrow H^1(\\mathcal O)\\cap C(\\overline{\\mathcal O})$, and $H^3(\\mathbb T^2)\\hookrightarrow\\hookrightarrow H^2(\\mathbb T^2)\\cap C(\\mathbb T^2)$,\n it follows from the Aubin-Lions lemma that\n \\begin{eqnarray}\n &&v_{\\epsilon_n} \\rightarrow v,\\quad\\mbox{in }C([0,\\mathcal T_0]; H^2(\\mathcal O)),\\label{StrC1}\\\\\n &&\\sigma_{\\epsilon_n} \\rightarrow\\sigma,\\quad\\mbox{in }C([0,\\mathcal T_0]; H^1(\\mathcal O)\\cap C(\\overline{\\mathcal O})),\\label{StrC2}\\\\\n &&\\partial_z\\sigma_{\\epsilon_n} \\rightarrow\\partial_z\\sigma,\\quad\\mbox{in }L^2(0,\\mathcal T_0; H^1(\\mathcal O)),\\\\\n &&p_{\\epsilon_n} \\rightarrow p,\\quad\\mbox{in }C([0,\\mathcal T_0]; H^2(\\mathbb T^2)\\cap C(\\mathbb T^2)).\\label{StrC3}\n \\end{eqnarray}\n These imply that $(v,\\sigma,p)$ has initial data $(v_0,\\sigma_0,p_0)$, enjoys all regularities stated in Theorem \\ref{thmmain}, and fulfills the boundary conditions \\eqref{BC1} and \\eqref{BC2}.\n Besides, by \\eqref{StrC2} and \\eqref{StrC3}, it follows from \\eqref{APRLBsp} that\n \\begin{equation*}\n \\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma \\geq0.5\\underline\\sigma,\\quad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p \\geq0.5\\underline p.\n \\end{equation*}\n By \\eqref{WeaC1}--\\eqref{StrC3}, one can take the limit as $n\\rightarrow\\infty$ to \\eqref{EQv}--\\eqref{EQp} to show that $(v,\\sigma,p)$ satisfies system \\eqref{EQv0}--\\eqref{EQp0} in the sense of distribution and further pointwisely, by the regularities of $(v,\\sigma,p)$. Therefore, $(v,\\sigma,p)$ is a strong solution\n to system \\eqref{EQv}--\\eqref{EQp}, subject to \\eqref{BC1}--\\eqref{IC}, on $\\mathcal O\\times(0,\\mathcal T_0)$.\n\n\\section{Appendix B: local existence of the regularized system}\n\\label{APPENDIXB}\nDenote\n\\begin{eqnarray*}\n N_1(v,\\sigma,p):=-[(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp],\\label{N1}\\\\\n N_2(v,\\sigma,p):=-[v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)],\\label{N2}\\\\\n N_3(v,p):=-(\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v)+(\\gamma-1)\\overline{Q(\\nabla v)},\\label{N3}\n\\end{eqnarray*}\nwhere $w, \\phi(v,p), \\bar f,$ and $Q(\\nabla v)$ are expressed, respectively, by\\eqref{EQw00'}, \\eqref{phi}, \\eqref{TBf}, and \\eqref{Q}.\nRecall $\\mathcal O=\\mathbb T^2\\times(0,1)$. Define a solution mapping\n\\begin{equation}\\label{DEFF}\n(v,\\sigma,p)\\mapsto(V,\\Sigma,P)=:\\mathfrak{F}(v,\\sigma,p),\n\\end{equation}\nwhere $(V,\\Sigma,P)$ is the unique solution to the following linear system\n\\begin{eqnarray}\n \\partial_tV-\\mu\\sigma\\Delta V-(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hV&=&N_1(v,\\sigma,p),\\label{AEQV}\\\\\n \\partial_t\\Sigma-\\nu\\sigma\\partial_z^2\\Sigma-\\epsilon\\Delta_h\\Sigma&=&N_2(v,\\sigma,p),\\label{AEQSIGMA}\\\\\n \\partial_tP-\\epsilon\\Delta_hP&=&N_3(v,p), \\label{AEQP}\n\\end{eqnarray}\nin $\\mathcal O\\times(0,T)$, subject to\n\\begin{eqnarray}\n &V,\\Sigma, P\\text{ are periodic in }x,y, \\quad\\partial_zV|_{z=0,1}=0,\\quad \\partial_z\\Sigma|_{z=0,1}=0,\\label{ABC}\\\\\n &(V,\\Sigma,P)|_{t=0}=(v_0,\\sigma_0,p_0),\\label{AIC}\n\\end{eqnarray}\nwhere $(v_0,\\sigma_0)\\in H^3(\\mathcal O)$, $p_0\\in H^3(\\mathbb T^2),$ and\n\\begin{equation*}\n \\partial_zv_0|_{z=0,1}=0,\\quad \\partial_z\\sigma_0|_{z=0,1}=0,\\quad\\sigma_0\\geq\\underline\\sigma, \\quad p_0\\geq\\underline p,\\label{A1}\n\\end{equation*}\nfor two positive numbers $\\underline\\sigma$ and $\\underline p$.", "post_theorem_intro_text_len": 2475, "post_theorem_intro_text": "Some comments on the proof of Theorem \\ref{thmmain} are presented as follows.\nRegularity assumption $\\sigma_0\\in H^2(\\mathcal O)$ comes from the following observation: for the toy model\n$$\n\\partial_t\\sigma=\\nu\\sigma\\partial_z^2\\sigma,\n$$\nin order that $H^k(\\mathcal O)$ regularity propagates, one needs $k\\geq2$.\nFortunately, such $H^2(\\mathcal O)$ regularity on $\\sigma_0$ is still sufficient for the more complicated equation \\eqref{EQs0}.\nWhile for the aim of getting $H^2(\\mathcal O)$ estimate on $\\sigma$, one may apply $\\Delta$ to \\eqref{EQs0} and, as a result, the\nquantity $\\Delta\\phi(v,p)$ and further, recalling the expression of $\\phi(v,p)$, $\\nabla_h\\Delta_hp$ will be encountered.\nIn order to get information on $\\nabla_h\\Delta_hp$, by equation \\eqref{EQp0}, one requires $p_0\\in H^3(\\mathbb T^2)$ and moreover\nthe quantity $\\nabla_h^4v$ will be encountered. Due to this and by \\eqref{EQv0}, one has to assume $v_0\\in H^3(\\mathcal O)$.\nThe existence of solutions to system \\eqref{EQv0}--\\eqref{EQp0}, subject to \\eqref{BC1}--\\eqref{IC}, is established by parabolic\nregularization argument. A regularized system is introduced by adding the regularizing dissipative terms $\\epsilon\\Delta_h\\sigma$ and $\\epsilon\\Delta_hp$\nto \\eqref{EQs0} and \\eqref{EQp0}, respectively, see \\eqref{EQv}--\\eqref{EQp}, below. Such a regularized system is a semi-linear\nparabolic system. The main effort of the proof is then to carry out\nsuitable $\\epsilon$-independent {\\it a priori} estimates and the local existence\nfollows passing to the limit $\\epsilon\\rightarrow0$.\nContinuous dependence on the initial data, which in particular implies\nthe uniqueness, is proved by performing energy estimates to the difference between two solutions. We employ $L^2$ energy estimate to the\nsubtracted $\\sigma$ equation which contains a term involving $\\nabla_hp$. This, due to the absence of dissipation\nin the pressure equation, forces us to perform $H^1(\\mathbb T^2)$ type energy estimate to the subtracted pressure equation and\nfurther $H^1(\\mathcal O)$ energy to the subtracted velocity equations.\n\nThe rest of this paper is arranged as follows: in the next section, Section \\ref{sec3}, we consider the $\\epsilon$-regularized system\nand perform suitable a priori estimates; the main result, Theorem \\ref{thmmain}, is proved in Section \\ref{sec4}; some commutator estimates are proved in Appendix A; local existence and uniqueness of the $\\epsilon$-regularized system is proved in Appendix B.", "sketch": "For Theorem~\\ref{thmmain}, the authors explain the proof strategy and why the initial regularities are chosen.\n\n- **Why $\\sigma_0\\in H^2(\\mathcal O)$:** From the toy model $\\partial_t\\sigma=\\nu\\sigma\\partial_z^2\\sigma$, “in order that $H^k(\\mathcal O)$ regularity propagates, one needs $k\\ge 2$.” This $H^2$ assumption is said to be “still sufficient for the more complicated equation \\eqref{EQs0}.”\n\n- **Why $p_0\\in H^3(\\mathbb T^2)$ and $v_0\\in H^3(\\mathcal O)$:** To get an “$H^2(\\mathcal O)$ estimate on $\\sigma$,” one applies “$\\Delta$ to \\eqref{EQs0},” which leads to “$\\Delta\\phi(v,p)$” and hence to “$\\nabla_h\\Delta_h p$.” To control $\\nabla_h\\Delta_h p$ using \\eqref{EQp0}, one needs “$p_0\\in H^3(\\mathbb T^2)$,” and in the process “the quantity $\\nabla_h^4 v$ will be encountered,” which (via \\eqref{EQv0}) “has to assume $v_0\\in H^3(\\mathcal O)$.”\n\n- **Existence via parabolic regularization and $\\epsilon$-independent estimates:** Existence is established by a “parabolic regularization argument.” They introduce a regularized system by adding “the regularizing dissipative terms $\\epsilon\\Delta_h\\sigma$ and $\\epsilon\\Delta_h p$ to \\eqref{EQs0} and \\eqref{EQp0}, respectively.” The resulting system is “a semi-linear parabolic system.” The “main effort” is to derive “suitable $\\epsilon$-independent {\\it a priori} estimates,” and then “the local existence follows passing to the limit $\\epsilon\\to 0$.”\n\n- **Uniqueness/continuous dependence by energy estimates on differences:** “Continuous dependence on the initial data, which in particular implies the uniqueness,” is proved by “performing energy estimates to the difference between two solutions.” Specifically, they use an “$L^2$ energy estimate to the subtracted $\\sigma$ equation” but this equation has “a term involving $\\nabla_h p$.” Because of “the absence of dissipation in the pressure equation,” they are “force[d]” to do an “$H^1(\\mathbb T^2)$ type energy estimate to the subtracted pressure equation” and “further $H^1(\\mathcal O)$ energy to the subtracted velocity equations.”", "expanded_sketch": "For the main theorem, the authors explain the proof strategy and why the initial regularities are chosen.\n\n- **Why $\\sigma_0\\in H^2(\\mathcal O)$:** From the toy model $\\partial_t\\sigma=\\nu\\sigma\\partial_z^2\\sigma$, “in order that $H^k(\\mathcal O)$ regularity propagates, one needs $k\\ge 2$.” This $H^2$ assumption is said to be “still sufficient for the more complicated equation\n\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n  &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n  \\label{EQs0}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\n.”\n\n- **Why $p_0\\in H^3(\\mathbb T^2)$ and $v_0\\in H^3(\\mathcal O)$:** To get an “$H^2(\\mathcal O)$ estimate on $\\sigma$,” one applies “$\\Delta$ to” the $\\sigma$-equation in\n\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n  &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n  \\label{EQs0}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\nwhich leads to “$\\Delta\\phi(v,p)$” and hence to “$\\nabla_h\\Delta_h p$.” To control $\\nabla_h\\Delta_h p$ using the pressure equation in the same system (i.e. the $p$-equation displayed above), one needs “$p_0\\in H^3(\\mathbb T^2)$,” and in the process “the quantity $\\nabla_h^4 v$ will be encountered,” which (via the velocity equation in the displayed system above) “has to assume $v_0\\in H^3(\\mathcal O)$.”\n\n- **Existence via parabolic regularization and $\\epsilon$-independent estimates:** Existence is established by a “parabolic regularization argument.” They introduce a regularized system by adding “the regularizing dissipative terms $\\epsilon\\Delta_h\\sigma$ and $\\epsilon\\Delta_h p$” to the $\\sigma$- and $p$-equations in\n\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n  &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n  \\label{EQs0}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\nrespectively. The resulting system is “a semi-linear parabolic system.” The “main effort” is to derive “suitable $\\epsilon$-independent {\\it a priori} estimates,” and then “the local existence follows passing to the limit $\\epsilon\\to 0$.”\n\n- **Uniqueness/continuous dependence by energy estimates on differences:** “Continuous dependence on the initial data, which in particular implies the uniqueness,” is proved by “performing energy estimates to the difference between two solutions.” Specifically, they use an “$L^2$ energy estimate to the subtracted $\\sigma$ equation” but this equation has “a term involving $\\nabla_h p$.” Because of “the absence of dissipation in the pressure equation” (namely, in the $p$-equation in the displayed system above), they are “force[d]” to do an “$H^1(\\mathbb T^2)$ type energy estimate to the subtracted pressure equation” and “further $H^1(\\mathcal O)$ energy to the subtracted velocity equations.”", "expanded_theorem": "\\label{thmmain}\nGiven $v_{0}\\in H^3(\\mathcal O)$, $\\sigma_{0} \\in H^2(\\mathcal O)$, and $p_{0}\\in H^3(\\mathbb T^2),$ such that\n$$\n\\sigma_{0}\\geq\\underline\\sigma, \\quad p_{0}\\geq\\underline p,\\quad \\partial_zv_{0}|_{z=0,1}=0,\\quad \\partial_z\\sigma_{0}|_{z=0,1}=0,\n$$\nfor two positive numbers $\\underline\\sigma$ and $\\underline p$.\nThen, there is a positive time $\\mathcal T_0$,\ndepending only on $\\gamma$, $\\nu$, $\\mu$, $\\lambda$, $\\underline\\sigma$, $\\underline p$, and\n  $\\|v_{0}\\|_{H^3}^2+\\|\\sigma_{0}\\|_{H^2}^2+\\|p_0\\|_{H^3}^2$, such that the system\n\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n  &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n  \\label{EQs0}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\n  subject to\n\\begin{eqnarray}\n  &v, \\sigma, \\mbox{ and } p \\mbox{ are periodic in }x, y, \\label{BC1}\\\\\n  &\\partial_zv|_{z=0,1}=0, \\quad\\partial_z\\sigma|_{z=0,1}=0. \\label{BC2}\n\\end{eqnarray}\nand\n\\begin{equation}\n  \\label{IC}\n  (v, \\sigma, p)|_{t=0}=(v_{0}, \\sigma_{0}, p_{0}).\n\\end{equation}\nhas a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$,\n  satisfying\n  \\begin{eqnarray*}\n  &&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\geq0.5\\underline\\sigma,\\quad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\geq0.5\\underline p,\\\\\n    &&v\\in C([0,\\mathcal T_0]; H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0; H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0; H^4(\\mathcal O)),\\\\\n    &&\\sigma\\in C([0,\\mathcal T_0]; H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0; H^2(\\mathcal O)),\\quad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\\\\n    &&\\partial_tv\\in L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\quad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0; H^1(\\mathcal O)),\\\\\n    &&p\\in C([0,\\mathcal T_0]; H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0; H^3(\\mathbb T^2)),\\quad\\partial_tp\\in  L^2(0,\\mathcal T_0; H^2(\\mathbb T^2)).\n  \\end{eqnarray*},", "theorem_type": ["Uniqueness", "Existence"], "mcq": {"question": "Let $\\mathcal O=\\mathbb T^2\\times(0,1)$. For a function $f(x,y,z)$, define\n$$\\bar f(x,y):=\\int_0^1 f(x,y,z)\\,dz,\\qquad \\tilde f:=f-\\bar f.$$ \nLet\n$$Q(\\nabla v):=\\bigl[\\mu(\\nabla_h v+\\nabla_h^{\\top}v)+\\lambda\\,\\mathrm{div}_h v\\,\\mathbb I_2\\bigr]:\\nabla_h v+\\mu|\\partial_z v|^2,$$\n$$\\phi(v,p):=\\mathrm{div}_h\\tilde v+\\frac{1}{\\gamma p}\\Bigl(\\tilde v\\cdot\\nabla_h p-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\Bigr),$$\n$$w=\\nu\\partial_z\\sigma-\\int_0^z \\phi(v,p)(x,y,z',t)\\,dz'.$$\nAssume\n$$v_0\\in H^3(\\mathcal O),\\qquad \\sigma_0\\in H^2(\\mathcal O),\\qquad p_0\\in H^3(\\mathbb T^2),$$\nand for some positive constants $\\underline\\sigma,\\underline p$,\n$$\\sigma_0\\ge \\underline\\sigma,\\qquad p_0\\ge \\underline p,\\qquad \\partial_z v_0|_{z=0,1}=0,\\qquad \\partial_z\\sigma_0|_{z=0,1}=0.$$ \nWhich statement holds for the problem of finding $v,\\sigma$ on $\\mathcal O\\times(0,T)$ and $p$ on $\\mathbb T^2\\times(0,T)$ such that\n\\begin{eqnarray*}\n&&\\partial_t v+(v\\cdot\\nabla_h)v+w\\partial_z v+\\sigma\\nabla_h p=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\mathrm{div}_h v,\\\\\n&&\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma\\bigl(\\phi(v,p)-\\mathrm{div}_h v\\bigr)=\\nu\\sigma\\partial_z^2\\sigma,\\\\\n&&\\partial_t p+\\bar v\\cdot\\nabla_h p+\\gamma p\\,\\mathrm{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)},\n\\end{eqnarray*}\nwith $v,\\sigma,p$ periodic in $(x,y)$, boundary conditions $\\partial_z v|_{z=0,1}=0$ and $\\partial_z\\sigma|_{z=0,1}=0$, and initial data $(v,\\sigma,p)|_{t=0}=(v_0,\\sigma_0,p_0)$?", "correct_choice": {"label": "A", "text": "There exists a time $\\mathcal T_0>0$, depending only on $\\gamma,\\nu,\\mu,\\lambda,\\underline\\sigma,\\underline p$, and $\\|v_0\\|_{H^3}^2+\\|\\sigma_0\\|_{H^2}^2+\\|p_0\\|_{H^3}^2$, such that the above system has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$ satisfying\n\\begin{eqnarray*}\n&&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\ge 0.5\\,\\underline\\sigma,\\qquad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\ge 0.5\\,\\underline p,\\\\\n&&v\\in C([0,\\mathcal T_0];H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0;H^4(\\mathcal O)),\\\\\n&&\\sigma\\in C([0,\\mathcal T_0];H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\\\\n&&\\partial_t v\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0;H^1(\\mathcal O)),\\\\\n&&p\\in C([0,\\mathcal T_0];H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathbb T^2)),\\qquad \\partial_t p\\in L^2(0,\\mathcal T_0;H^2(\\mathbb T^2)).\n\\end{eqnarray*}"}, "choices": [{"label": "B", "text": "There exists a time $\\mathcal T_0>0$, depending only on $\\gamma,\\nu,\\mu,\\lambda,\\underline\\sigma,\\underline p$, and $\\|v_0\\|_{H^3}^2+\\|\\sigma_0\\|_{H^1}^2+\\|p_0\\|_{H^2}^2$, such that the above system has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$ satisfying\n\\begin{eqnarray*}\n&&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\ge 0.5\\,\\underline\\sigma,\\qquad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\ge 0.5\\,\\underline p,\\\\\n&&v\\in C([0,\\mathcal T_0];H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0;H^4(\\mathcal O)),\\\\\n&&\\sigma\\in C([0,\\mathcal T_0];L^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^1(\\mathcal O)),\\qquad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0;H^1(\\mathcal O)),\\\\\n&&\\partial_t v\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0;L^2(\\mathcal O)),\\\\\n&&p\\in C([0,\\mathcal T_0];H^1(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0;H^2(\\mathbb T^2)),\\qquad \\partial_t p\\in L^2(0,\\mathcal T_0;H^1(\\mathbb T^2)).\n\\end{eqnarray*}"}, {"label": "C", "text": "There exists a time $\\mathcal T_0>0$, depending only on $\\gamma,\\nu,\\mu,\\lambda,\\underline\\sigma,\\underline p$, and $\\|v_0\\|_{H^3}^2+\\|\\sigma_0\\|_{H^2}^2+\\|p_0\\|_{H^3}^2$, such that the above system has at least one local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$ satisfying\n\\begin{eqnarray*}\n&&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\ge 0.5\\,\\underline\\sigma,\\qquad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\ge 0.5\\,\\underline p,\\\\\n&&v\\in C([0,\\mathcal T_0];H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0;H^4(\\mathcal O)),\\\\\n&&\\sigma\\in C([0,\\mathcal T_0];H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\\\\n&&\\partial_t v\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0;H^1(\\mathcal O)),\\\\\n&&p\\in C([0,\\mathcal T_0];H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathbb T^2)),\\qquad \\partial_t p\\in L^2(0,\\mathcal T_0;H^2(\\mathbb T^2)).\n\\end{eqnarray*}"}, {"label": "D", "text": "There exists a universal time $\\mathcal T_0>0$, depending only on $\\gamma,\\nu,\\mu,\\lambda,\\underline\\sigma,$ and $\\underline p$ but independent of $\\|v_0\\|_{H^3},\\|\\sigma_0\\|_{H^2},\\|p_0\\|_{H^3}$, such that the above system has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$ satisfying\n\\begin{eqnarray*}\n&&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\ge 0.5\\,\\underline\\sigma,\\qquad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\ge 0.5\\,\\underline p,\\\\\n&&v\\in C([0,\\mathcal T_0];H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0;H^4(\\mathcal O)),\\\\\n&&\\sigma\\in C([0,\\mathcal T_0];H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\\\\n&&\\partial_t v\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0;H^1(\\mathcal O)),\\\\\n&&p\\in C([0,\\mathcal T_0];H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathbb T^2)),\\qquad \\partial_t p\\in L^2(0,\\mathcal T_0;H^2(\\mathbb T^2)).\n\\end{eqnarray*}"}, {"label": "E", "text": "There exists a time $\\mathcal T_0>0$, depending only on $\\gamma,\\nu,\\mu,\\lambda,\\underline\\sigma,\\underline p$, and $\\|v_0\\|_{H^3}^2+\\|\\sigma_0\\|_{H^2}^2+\\|p_0\\|_{H^3}^2$, such that the above system has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$ satisfying\n\\begin{eqnarray*}\n&&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\ge \\underline\\sigma,\\qquad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\ge \\underline p,\\\\\n&&v\\in C([0,\\mathcal T_0];H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0;H^4(\\mathcal O)),\\\\\n&&\\sigma\\in C([0,\\mathcal T_0];H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\\\\n&&\\partial_t v\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0;H^1(\\mathcal O)),\\\\\n&&p\\in C([0,\\mathcal T_0];H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathbb T^2)),\\qquad \\partial_t p\\in L^2(0,\\mathcal T_0;H^2(\\mathbb T^2)).\n\\end{eqnarray*}"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "minimal initial regularity for $\\sigma_0$ and $p_0$", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped uniqueness while keeping the same existence/regularity conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "dependence of lifespan on Sobolev norms of initial data", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "strict preservation of the full lower bounds instead of the halved bounds", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the correct existence theorem. It presents the PDE system and assumptions, but the reader must still distinguish among uniqueness, regularity, and lifespan-dependence claims."}, "TAS": {"score": 1, "justification": "This is close to a theorem-identification item: the task is essentially to pick the correct local well-posedness statement for the displayed system. The alternatives introduce some variation, but the question still largely tests recognition of the theorem rather than a genuinely independent conclusion."}, "GPS": {"score": 1, "justification": "Moderate reasoning is needed to assess which statement is most plausible, especially regarding uniqueness, top-order time continuity, and dependence on positivity lower bounds. However, it does not strongly force derivation or synthesis beyond theorem-level matching."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: weakened initial regularity, loss of uniqueness, incorrect parameter dependence, and an overly strong regularity upgrade. They are distinct and well aligned with common PDE theorem confusions."}, "total_score": 6, "overall_assessment": "A solid but theorem-recognition-heavy MCQ: little answer leakage and strong distractors, but only moderate generative reasoning and limited avoidance of theorem-restatement structure."}}
{"id": "2602.19801v1", "paper_link": "http://arxiv.org/abs/2602.19801v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thmmain}\nGiven $v_{0}\\in H^3(\\mathcal O)$, $\\sigma_{0} \\in H^2(\\mathcal O)$, and $p_{0}\\in H^3(\\mathbb T^2),$ such that\n$$\n\\sigma_{0}\\geq\\underline\\sigma, \\quad p_{0}\\geq\\underline p,\\quad \\partial_zv_{0}|_{z=0,1}=0,\\quad \\partial_z\\sigma_{0}|_{z=0,1}=0,\n$$\nfor two positive numbers $\\underline\\sigma$ and $\\underline p$.\nThen, there is a positive time $\\mathcal T_0$,\ndepending only on $\\gamma$, $\\nu$, $\\mu$, $\\lambda$, $\\underline\\sigma$, $\\underline p$, and\n  $\\|v_{0}\\|_{H^3}^2+\\|\\sigma_{0}\\|_{H^2}^2+\\|p_0\\|_{H^3}^2$, such that system (\\ref{EQv0})--(\\ref{EQp0}),\n  subject to (\\ref{BC1})--(\\ref{IC}), has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$,\n  satisfying\n  \\begin{eqnarray*}\n  &&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\geq0.5\\underline\\sigma,\\quad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\geq0.5\\underline p,\\\\\n    &&v\\in C([0,\\mathcal T_0]; H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0; H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0; H^4(\\mathcal O)),\\\\\n    &&\\sigma\\in C([0,\\mathcal T_0]; H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0; H^2(\\mathcal O)),\\quad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\\\\n    &&\\partial_tv\\in L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\quad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0; H^1(\\mathcal O)),\\\\\n    &&p\\in C([0,\\mathcal T_0]; H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0; H^3(\\mathbb T^2)),\\quad\\partial_tp\\in  L^2(0,\\mathcal T_0; H^2(\\mathbb T^2)).\n  \\end{eqnarray*}", "start_pos": 14353, "end_pos": 15876, "label": "thmmain"}, "ref_dict": {"EQp0": "\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n  &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n  \\label{EQs0}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}", "EQp00": "\\begin{eqnarray}\n \\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)},\\label{EQp00}\\\\\n\\tilde v\\cdot\\nabla_hp+\\gamma p(\\text{div}_h\\tilde v+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)\\widetilde{Q(\\nabla v)}. \\label{EQw000}\n\\end{eqnarray}", "IC": "\\begin{equation}\n  \\label{IC}\n  (v, \\sigma, p)|_{t=0}=(v_{0}, \\sigma_{0}, p_{0}).\n\\end{equation}", "EQv00": "\\begin{equation}\n\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma \\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma \\nabla_h\\text{div}_hv. \\label{EQv00}\n\\end{equation}", "EQw00": "\\begin{eqnarray}\\label{eq:dz-w}\n\t\t \\dz w &=&-\\text{div}_h\\tilde v+\\frac{\\gamma-1}{\\gamma p}\\left(\\kappa\\partial_z^2\\theta+\\widetilde{Q(\\nabla v)}\\right)-\\frac{1}{\\gamma p}\\tilde v\\cdot\\nabla_hp\\nonumber\\\\\n&=&\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}\\partial_{zz} \\sigma -\\text{div}_h\\tilde v-\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right)\\nonumber\\\\\n&=&\\nu\\partial_z^2\\sigma-\\phi(v,p),\\label{EQw00}\n\\end{eqnarray}", "phi": "\\begin{eqnarray}\n  \\phi(v,p):=\\text{div}_h\\tilde v+\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right).\\label{phi}\n\\end{eqnarray}", "Q": "\\begin{eqnarray}\n\t&\\mathbb S_h = \\mu (\\nablah v + \\nablah^\\top v) + \\lambda \\dvh v \\mathbb I_2, \\quad\n\tp = R \\rho \\theta, \\nonumber\\\\\n  &Q(\\nabla v)=\\mathbb S_h:\\nabla_hv+\\mu|\\partial_zv|^2,\\label{Q}\n\\end{eqnarray}", "eq:dz-w": "\\begin{eqnarray}\\label{eq:dz-w}\n\t\t \\dz w &=&-\\text{div}_h\\tilde v+\\frac{\\gamma-1}{\\gamma p}\\left(\\kappa\\partial_z^2\\theta+\\widetilde{Q(\\nabla v)}\\right)-\\frac{1}{\\gamma p}\\tilde v\\cdot\\nabla_hp\\nonumber\\\\\n&=&\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}\\partial_{zz} \\sigma -\\text{div}_h\\tilde v-\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right)\\nonumber\\\\\n&=&\\nu\\partial_z^2\\sigma-\\phi(v,p),\\label{EQw00}\n\\end{eqnarray}", "EQs00": "\\begin{equation}\\label{EQs00}\n\t\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma.\n\\end{equation}", "BC1": "\\begin{eqnarray}\n  &v, \\sigma, \\mbox{ and } p \\mbox{ are periodic in }x, y, \\label{BC1}\\\\\n  &\\partial_zv|_{z=0,1}=0, \\quad\\partial_z\\sigma|_{z=0,1}=0. \\label{BC2}\n\\end{eqnarray}", "EQv": "\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv}\\\\\n  &w=\\nu\\partial_zv-\\int_0^z\\phi(v,p)dz',\\label{EQw}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma+\\epsilon\\Delta_h\\sigma,\n  \\label{EQs}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=\\epsilon\\Delta_hp+(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp}\n\\end{eqnarray}", "EQw000": "\\begin{eqnarray}\n \\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)},\\label{EQp00}\\\\\n\\tilde v\\cdot\\nabla_hp+\\gamma p(\\text{div}_h\\tilde v+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)\\widetilde{Q(\\nabla v)}. \\label{EQw000}\n\\end{eqnarray}", "eq:CPE": "\\begin{equation}\\label{eq:CPE}\n\t\\begin{cases}\n\t\t\\dt \\rho + \\dvh (\\rho v)  + \\dz (\\rho w) = 0, \\\\\n\t\t\\dt (\\rho v) + \\dvh(\\rho v\\otimes v) + \\dz(\\rho w v) + \\nablah p = \\dvh \\mathbb S_h + \\mu \\partial_{zz} v, \\\\\n\t\t\\dz p = 0, \\\\\n\t\t\\dfrac{1}{\\gamma-1}\\bigl(\\dt p + \\dvh(p v) + \\dz (pw) \\bigr) + p(\\dvh v + \\dz w) = \\dz (\\kappa \\dz \\theta )+Q(\\nabla v),\n\t\\end{cases}\n\\end{equation}", "EQw00'": "\\begin{equation}\n  w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz'.\\label{EQw00'}\n\\end{equation}", "EQv0": "\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n  &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n  \\label{EQs0}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}", "EQs0": "\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n  &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n  \\label{EQs0}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}", "EQp": "\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv}\\\\\n  &w=\\nu\\partial_zv-\\int_0^z\\phi(v,p)dz',\\label{EQw}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma+\\epsilon\\Delta_h\\sigma,\n  \\label{EQs}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=\\epsilon\\Delta_hp+(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp}\n\\end{eqnarray}", "thmmain": "\\begin{theorem}\\label{thmmain}\nGiven $v_{0}\\in H^3(\\mathcal O)$, $\\sigma_{0} \\in H^2(\\mathcal O)$, and $p_{0}\\in H^3(\\mathbb T^2),$ such that\n$$\n\\sigma_{0}\\geq\\underline\\sigma, \\quad p_{0}\\geq\\underline p,\\quad \\partial_zv_{0}|_{z=0,1}=0,\\quad \\partial_z\\sigma_{0}|_{z=0,1}=0,\n$$\nfor two positive numbers $\\underline\\sigma$ and $\\underline p$.\nThen, there is a positive time $\\mathcal T_0$,\ndepending only on $\\gamma$, $\\nu$, $\\mu$, $\\lambda$, $\\underline\\sigma$, $\\underline p$, and\n  $\\|v_{0}\\|_{H^3}^2+\\|\\sigma_{0}\\|_{H^2}^2+\\|p_0\\|_{H^3}^2$, such that system (\\ref{EQv0})--(\\ref{EQp0}),\n  subject to (\\ref{BC1})--(\\ref{IC}), has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$,\n  satisfying\n  \\begin{eqnarray*}\n  &&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\geq0.5\\underline\\sigma,\\quad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\geq0.5\\underline p,\\\\\n    &&v\\in C([0,\\mathcal T_0]; H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0; H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0; H^4(\\mathcal O)),\\\\\n    &&\\sigma\\in C([0,\\mathcal T_0]; H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0; H^2(\\mathcal O)),\\quad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\\\\n    &&\\partial_tv\\in L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\quad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0; H^1(\\mathcal O)),\\\\\n    &&p\\in C([0,\\mathcal T_0]; H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0; H^3(\\mathbb T^2)),\\quad\\partial_tp\\in  L^2(0,\\mathcal T_0; H^2(\\mathbb T^2)).\n  \\end{eqnarray*}\n\\end{theorem}", "bc:CPE": "\\begin{equation}\\label{bc:CPE}\n\t\\dz \\theta\\big|_{z=0,1} = 0, \\quad \\dz v\\big|_{z=0,1} = 0, \\quad w\\big|_{z=0,1} = 0.\n\\end{equation}"}, "pre_theorem_intro_text_len": 10452, "pre_theorem_intro_text": "The compressible primitive equations, formally obtained as the hydrostatic approximation of the compressible Navier-Stokes equations by replacing the vertical momentum balance equation with the hydrostatic balance\nequation, are the fundamental equations for modern meteorological study (see, e.g., \\cite[Chapter 4]{Richardson1965}). In particular, it is the starting point of many large scale\nmodels in the theoretical investigations and practical weather predictions (see, e.g., \\cite{Lions1992,JLLions1992,Washington2005}). In fact, such an approximation is reliable and useful in the following sense: (i)\nthe vertical scale of the atmosphere is significantly smaller than the planetary horizontal scale; (ii) the balance of gravity and pressure dominates the dynamic in the vertical direction; (iii) the vertical velocity is\nusually hard to observe in reality. Rigorous justifications of the hydrostatic approximation of hydrodynamic equations can be found in \\cite{Li2017,Azerad2001} (see also \\cite{FugigaHHK,FuGiHiHuKaW1,FuruGigaKas,LiTitiYuan}) for incompressible flows and \\cite{Liu2019} for\nisentropic compressible flows.\n\nIn this work, our goal is to investigate the fundamental problem of the well-posedness of solutions to the compressible primitive equations with non-trivial entropy. In particular, we consider the non-isentropic compressible primitive equations with only vertical diffusion for the temperature:\n\\begin{equation}\\label{eq:CPE}\n\t\\begin{cases}\n\t\t\\partial_t \\rho + \\mathrm{div}_{h}\\, (\\rho v)  + \\partial_z (\\rho w) = 0, \\\\\n\t\t\\partial_t (\\rho v) + \\mathrm{div}_{h}\\,(\\rho v\\otimes v) + \\partial_z(\\rho w v) + \\nabla_{h} p = \\mathrm{div}_{h}\\, \\mathbb S_h + \\mu \\partial_{zz} v, \\\\\n\t\t\\partial_z p = 0, \\\\\n\t\t\\dfrac{1}{\\gamma-1}\\bigl(\\partial_t p + \\mathrm{div}_{h}\\,(p v) + \\partial_z (pw) \\bigr) + p(\\mathrm{div}_{h}\\, v + \\partial_z w) = \\partial_z (\\kappa \\partial_z \\theta )+Q(\\nabla v),\n\t\\end{cases}\n\\end{equation}\nwith\n\\begin{eqnarray}\n\t&\\mathbb S_h = \\mu (\\nabla_{h} v + \\nabla_{h}^\\top v) + \\lambda \\mathrm{div}_{h}\\, v \\mathbb I_2, \\quad\n\tp = R \\rho \\theta, \\nonumber\\\\\n  &Q(\\nabla v)=\\mathbb S_h:\\nabla_hv+\\mu|\\partial_zv|^2,\\label{Q}\n\\end{eqnarray}\nwith constant coefficients $\\gamma$, $\\mu$, $\\lambda$, $\\kappa,$ and $R$ satisfying $\\gamma > 1$, $\\mu+\\lambda>0$, $\\mu>0$, $\\kappa>0$, and $R>0$.\nHere, $ \\rho, v, w $, and $ p $ represent the density, the horizontal velocity, vertical velocity, and pressure, respectively. $ \\mathrm{div}_{h}\\,, \\nabla_{h} $, and $\\Delta_{h} $ are, and will be, the divergence, gradient, and Laplace operators in the horizontal variables $ (x,y) $, respectively.\nWe investigate system \\eqref{eq:CPE} in the horizontally periodic channel\n$$ \\mathcal O:= \\mathbb T^2 \\times (0,1) = \\lbrace (x,y,z) | (x,y) \\in \\mathbb T^2, z \\in (0,1) \\rbrace,\n$$\nand consider the following\nboundary conditions:\n\\begin{equation}\\label{bc:CPE}\n\t\\partial_z \\theta\\big|_{z=0,1} = 0, \\quad \\partial_z v\\big|_{z=0,1} = 0, \\quad w\\big|_{z=0,1} = 0.\n\\end{equation}\n\nThe isentropic compressible primitive equations have been investigated by the last two authors in \\cite{LT2018a} for local strong solutions and \\cite{LT2018b} for global weak solutions. The existence of global weak solutions is also studied by \\cite{Wang2020}, independently. See also, \\cite{Jiu2018,Ersoy2011a,Ersoy2012,Gatapov2005}.\n\nOn the other hand, the incompressible primitive equations have been the subject of intensive mathematical research since the introduction\nby Lions, Temam, and Wang in \\cite{Lions1992,JLLions1992}. For instance, Guill\\'en-Gonz\\'alez, Masmoudi, and Rodr\\'iguez-Bellido in\n\\cite{GuillenGonzalez2001} study the local existence of strong solutions and global existence of strong solutions with small initial data. In\n\\cite{HuTemamZiane2003}, the authors address the global existence of strong solutions in a domain with small depth. The first breakthrough\nconcerning the global well-posedness of incompressible primitive equations is obtained by Cao and Titi in \\cite{Cao2007}. See also,\n\\cite{Cao2014a,Cao2014b,Cao2016,Cao2016a,CaoLiTiti2020,CaoLiTitiWang2024,Boling,MHATK,MHTKa,GMK,IKMZ,Li2017a,LiYuan2022} and the references therein for\nrelated literatures; in particular, global well-posedness of strong solutions was proved in \\cite{Cao2016,Cao2016a,CaoLiTiti2020,CaoLiTitiWang2024} to the primitive equations with only horizontal viscosity.\n\nWe have introduced the PE diagram in \\cite{LT2018LowMach1}, which concerns the low Mach number limit and the small aspect ratio (between the vertical and horizontal scales) limit. In \\cite{LT2018LowMach1,LT2018LowMach2}, we also establish the low Mach number limit for the isentropic compressible primitive equations. However, the counterpart study of the compressible Navier-Stokes-Fourier equations (see, e.g., \\cite{Alazard2006}) indicates that the PE diagram might be completely different for non-isentropic flows, due to the additional heat conductivity. We leave such a subject to future study.\n\nFor interested readers, we refer to \\cite{Temam1977,Lions1996,Lions1998,Feireisl2004,Feireisl2009a} for the study of hydrodynamic equations.\n\nOne can easily see that, compared to the Navier-Stokes equations, the evolutionary equation for vertical velocity is missing in system\n\\eqref{eq:CPE}. In fact, this is one of the main challenges in the study of the compressible primitive equations. In order to have a better\nunderstanding of the system, we derive a representation of vertical velocity in terms of horizontal velocity, density, and pressure.\nConsequently, we will reformulate system \\eqref{eq:CPE} to an equivalent one.\n\nUse \\subeqref{eq:CPE}{1} to rewrite \\subeqref{eq:CPE}{2} as\n$$\n\\rho(\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv)+\\nabla_hp=\\mu\\Delta v+(\\mu+\\lambda)\\nabla_h\\text{div}_hv\n$$\nand define\n\\begin{equation}\\label{def:rho-1}\n\t\\sigma := \\frac1\\rho.\n\\end{equation}\nThen, it is clear that\n\\begin{equation}\n\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma \\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma \\nabla_h\\text{div}_hv. \\label{EQv00}\n\\end{equation}\n\nNote that \\subeqref{eq:CPE}{3} implies that $p$ is independent of the vertical variable $z$ and one can use \\subeqref{eq:CPE}{3} to rewrite \\subeqref{eq:CPE}{4} as\n\\begin{equation*}\n \\partial_tp+v\\cdot\\nabla_hp+\\gamma p(\\text{div}_hv+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)Q(\\nabla v).\n\\end{equation*}\nSeparating the $z$-average part and the fluctuation part of the above equation, and recalling the boundary conditions $\\partial_z\\theta|_{z=0,1}=w|_{z=0,1}=0$, one obtains that\n\\begin{eqnarray}\n \\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)},\\label{EQp00}\\\\\n\\tilde v\\cdot\\nabla_hp+\\gamma p(\\text{div}_h\\tilde v+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)\\widetilde{Q(\\nabla v)}. \\label{EQw000}\n\\end{eqnarray}\nHere, for a function $f$, we use\n\\begin{equation}\\label{TBf}\n\\bar f:=\\int_0^1f dz\\quad\\mbox{and}\\quad  \\tilde f:=f-\\bar f,\n\\end{equation}\nto represent the vertical integral (equivalently, average) and the vertical fluctuation of the quantity considered, respectively.\n\nSince $\\theta = R^{-1} \\rho^{-1} p=R^{-1}\\sigma p,$\nit follows from \\eqref{EQw000} and \\subeqref{eq:CPE}{3} that\n\\begin{eqnarray}\\label{eq:dz-w}\n\t\t \\partial_z w &=&-\\text{div}_h\\tilde v+\\frac{\\gamma-1}{\\gamma p}\\left(\\kappa\\partial_z^2\\theta+\\widetilde{Q(\\nabla v)}\\right)-\\frac{1}{\\gamma p}\\tilde v\\cdot\\nabla_hp\\nonumber\\\\\n&=&\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}\\partial_{zz} \\sigma -\\text{div}_h\\tilde v-\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right)\\nonumber\\\\\n&=&\\nu\\partial_z^2\\sigma-\\phi(v,p),\\label{EQw00}\n\\end{eqnarray}\nwhere $\\nu=\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}$ and\n\\begin{eqnarray}\n  \\phi(v,p):=\\text{div}_h\\tilde v+\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right).\\label{phi}\n\\end{eqnarray}\nRecalling that $p$ is independent of $z$, it holds that $\\partial_z\\sigma=R\\partial_z(p^{-1}\\theta)= R p^{-1}\\partial_z\\theta$. Then, the boundary condition \\eqref{bc:CPE} implies\n\\begin{equation}\\label{bc:rf-CPE}\n\\partial_z \\sigma\\big|_{z=0,1} = 0.\n\\end{equation}\nThanks to this and recalling that $w|_{z=0}$, it follows from \\eqref{EQw00} that\n\\begin{equation}\n  w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz'.\\label{EQw00'}\n\\end{equation}\n\nIt follows from \\subeqref{eq:CPE}{1} that\n\\begin{equation*}\\label{eq:rho-1}\n\t\\partial_t \\rho^{-1} + v \\cdot \\nabla_{h} \\rho^{-1} - \\rho^{-1} \\mathrm{div}_{h}\\, v = \\rho^{-2} \\partial_z (\\rho w) = \\rho^{-1} \\partial_z w - w \\partial_z \\rho^{-1},\n\\end{equation*}\nfrom which, using \\eqref{eq:dz-w} and recalling $\\sigma=\\frac1\\rho$, one gets\n\\begin{equation}\\label{EQs00}\n\t\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma.\n\\end{equation}\n\nNow, collecting \\eqref{EQv00}, \\eqref{EQp00}, \\eqref{EQw00'}, \\eqref{EQs00}, we have the following reformulated system which is equivalent to \\eqref{eq:CPE}\n\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n  &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n  \\label{EQs0}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\nwhere $\\phi(v,p)$ and $Q(\\nabla v)$ are expressed as in (\\ref{phi}) and (\\ref{Q}), respectively. The boundary conditions read as\n\\begin{eqnarray}\n  &v, \\sigma, \\mbox{ and } p \\mbox{ are periodic in }x, y, \\label{BC1}\\\\\n  &\\partial_zv|_{z=0,1}=0, \\quad\\partial_z\\sigma|_{z=0,1}=0. \\label{BC2}\n\\end{eqnarray}\nThe initial condition is\n\\begin{equation}\n  \\label{IC}\n  (v, \\sigma, p)|_{t=0}=(v_{0}, \\sigma_{0}, p_{0}).\n\\end{equation}\n\nBefore stating the main result, we recall some standard notations. For positive integer $k$ and $q\\in[1,\\infty]$, $L^q(\\mathcal O)$ and\n$L^q(\\mathbb T^2)$ are the Lebesgue spaces,\n$W^{k,q}(\\mathcal O)$ and $W^{k,q}(\\mathbb T^2)$ are the Sobolev spaces. If $q=2$, we use $H^k$ instead of $W^{k,2}$. For simplicity,\nwe will use $\\|\\cdot\\|_q$ to denote $L^q(\\mathcal O)$ or $L^q(\\mathbb T^2)$ norms of the corresponding function,\nif the meaning is clear from the context.\n\nWe are now in the position to state the main result of this paper.", "context": "In this work, our goal is to investigate the fundamental problem of the well-posedness of solutions to the compressible primitive equations with non-trivial entropy. In particular, we consider the non-isentropic compressible primitive equations with only vertical diffusion for the temperature:\n\\begin{equation}\\label{eq:CPE}\n \\begin{cases}\n \\partial_t \\rho + \\mathrm{div}_{h}\\, (\\rho v) + \\partial_z (\\rho w) = 0, \\\\\n \\partial_t (\\rho v) + \\mathrm{div}_{h}\\,(\\rho v\\otimes v) + \\partial_z(\\rho w v) + \\nabla_{h} p = \\mathrm{div}_{h}\\, \\mathbb S_h + \\mu \\partial_{zz} v, \\\\\n \\partial_z p = 0, \\\\\n \\dfrac{1}{\\gamma-1}\\bigl(\\partial_t p + \\mathrm{div}_{h}\\,(p v) + \\partial_z (pw) \\bigr) + p(\\mathrm{div}_{h}\\, v + \\partial_z w) = \\partial_z (\\kappa \\partial_z \\theta )+Q(\\nabla v),\n \\end{cases}\n\\end{equation}\nwith\n\\begin{eqnarray}\n &\\mathbb S_h = \\mu (\\nabla_{h} v + \\nabla_{h}^\\top v) + \\lambda \\mathrm{div}_{h}\\, v \\mathbb I_2, \\quad\n p = R \\rho \\theta, \\nonumber\\\\\n &Q(\\nabla v)=\\mathbb S_h:\\nabla_hv+\\mu|\\partial_zv|^2,\\label{Q}\n\\end{eqnarray}\nwith constant coefficients $\\gamma$, $\\mu$, $\\lambda$, $\\kappa,$ and $R$ satisfying $\\gamma > 1$, $\\mu+\\lambda>0$, $\\mu>0$, $\\kappa>0$, and $R>0$.\nHere, $ \\rho, v, w $, and $ p $ represent the density, the horizontal velocity, vertical velocity, and pressure, respectively. $ \\mathrm{div}_{h}\\,, \\nabla_{h} $, and $\\Delta_{h} $ are, and will be, the divergence, gradient, and Laplace operators in the horizontal variables $ (x,y) $, respectively.\nWe investigate system \\eqref{eq:CPE} in the horizontally periodic channel\n$$ \\mathcal O:= \\mathbb T^2 \\times (0,1) = \\lbrace (x,y,z) | (x,y) \\in \\mathbb T^2, z \\in (0,1) \\rbrace,\n$$\nand consider the following\nboundary conditions:\n\\begin{equation}\\label{bc:CPE}\n \\partial_z \\theta\\big|_{z=0,1} = 0, \\quad \\partial_z v\\big|_{z=0,1} = 0, \\quad w\\big|_{z=0,1} = 0.\n\\end{equation}\n\nNote that \\subeqref{eq:CPE}{3} implies that $p$ is independent of the vertical variable $z$ and one can use \\subeqref{eq:CPE}{3} to rewrite \\subeqref{eq:CPE}{4} as\n\\begin{equation*}\n \\partial_tp+v\\cdot\\nabla_hp+\\gamma p(\\text{div}_hv+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)Q(\\nabla v).\n\\end{equation*}\nSeparating the $z$-average part and the fluctuation part of the above equation, and recalling the boundary conditions $\\partial_z\\theta|_{z=0,1}=w|_{z=0,1}=0$, one obtains that\n\\begin{eqnarray}\n \\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)},\\label{EQp00}\\\\\n\\tilde v\\cdot\\nabla_hp+\\gamma p(\\text{div}_h\\tilde v+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)\\widetilde{Q(\\nabla v)}. \\label{EQw000}\n\\end{eqnarray}\nHere, for a function $f$, we use\n\\begin{equation}\\label{TBf}\n\\bar f:=\\int_0^1f dz\\quad\\mbox{and}\\quad \\tilde f:=f-\\bar f,\n\\end{equation}\nto represent the vertical integral (equivalently, average) and the vertical fluctuation of the quantity considered, respectively.\n\nSince $\\theta = R^{-1} \\rho^{-1} p=R^{-1}\\sigma p,$\nit follows from \\eqref{EQw000} and \\subeqref{eq:CPE}{3} that\n\\begin{eqnarray}\\label{eq:dz-w}\n \\partial_z w &=&-\\text{div}_h\\tilde v+\\frac{\\gamma-1}{\\gamma p}\\left(\\kappa\\partial_z^2\\theta+\\widetilde{Q(\\nabla v)}\\right)-\\frac{1}{\\gamma p}\\tilde v\\cdot\\nabla_hp\\nonumber\\\\\n&=&\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}\\partial_{zz} \\sigma -\\text{div}_h\\tilde v-\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right)\\nonumber\\\\\n&=&\\nu\\partial_z^2\\sigma-\\phi(v,p),\\label{EQw00}\n\\end{eqnarray}\nwhere $\\nu=\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}$ and\n\\begin{eqnarray}\n \\phi(v,p):=\\text{div}_h\\tilde v+\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right).\\label{phi}\n\\end{eqnarray}\nRecalling that $p$ is independent of $z$, it holds that $\\partial_z\\sigma=R\\partial_z(p^{-1}\\theta)= R p^{-1}\\partial_z\\theta$. Then, the boundary condition \\eqref{bc:CPE} implies\n\\begin{equation}\\label{bc:rf-CPE}\n\\partial_z \\sigma\\big|_{z=0,1} = 0.\n\\end{equation}\nThanks to this and recalling that $w|_{z=0}$, it follows from \\eqref{EQw00} that\n\\begin{equation}\n w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz'.\\label{EQw00'}\n\\end{equation}\n\nNow, collecting \\eqref{EQv00}, \\eqref{EQp00}, \\eqref{EQw00'}, \\eqref{EQs00}, we have the following reformulated system which is equivalent to \\eqref{eq:CPE}\n\\begin{eqnarray}\n &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n \\label{EQs0}\\\\\n &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\nwhere $\\phi(v,p)$ and $Q(\\nabla v)$ are expressed as in (\\ref{phi}) and (\\ref{Q}), respectively. The boundary conditions read as\n\\begin{eqnarray}\n &v, \\sigma, \\mbox{ and } p \\mbox{ are periodic in }x, y, \\label{BC1}\\\\\n &\\partial_zv|_{z=0,1}=0, \\quad\\partial_z\\sigma|_{z=0,1}=0. \\label{BC2}\n\\end{eqnarray}\nThe initial condition is\n\\begin{equation}\n \\label{IC}\n (v, \\sigma, p)|_{t=0}=(v_{0}, \\sigma_{0}, p_{0}).\n\\end{equation}\n\nBefore stating the main result, we recall some standard notations. For positive integer $k$ and $q\\in[1,\\infty]$, $L^q(\\mathcal O)$ and\n$L^q(\\mathbb T^2)$ are the Lebesgue spaces,\n$W^{k,q}(\\mathcal O)$ and $W^{k,q}(\\mathbb T^2)$ are the Sobolev spaces. If $q=2$, we use $H^k$ instead of $W^{k,2}$. For simplicity,\nwe will use $\\|\\cdot\\|_q$ to denote $L^q(\\mathcal O)$ or $L^q(\\mathbb T^2)$ norms of the corresponding function,\nif the meaning is clear from the context.\n\nWe are now in the position to state the main result of this paper.", "full_context": "In this work, our goal is to investigate the fundamental problem of the well-posedness of solutions to the compressible primitive equations with non-trivial entropy. In particular, we consider the non-isentropic compressible primitive equations with only vertical diffusion for the temperature:\n\\begin{equation}\\label{eq:CPE}\n \\begin{cases}\n \\partial_t \\rho + \\mathrm{div}_{h}\\, (\\rho v) + \\partial_z (\\rho w) = 0, \\\\\n \\partial_t (\\rho v) + \\mathrm{div}_{h}\\,(\\rho v\\otimes v) + \\partial_z(\\rho w v) + \\nabla_{h} p = \\mathrm{div}_{h}\\, \\mathbb S_h + \\mu \\partial_{zz} v, \\\\\n \\partial_z p = 0, \\\\\n \\dfrac{1}{\\gamma-1}\\bigl(\\partial_t p + \\mathrm{div}_{h}\\,(p v) + \\partial_z (pw) \\bigr) + p(\\mathrm{div}_{h}\\, v + \\partial_z w) = \\partial_z (\\kappa \\partial_z \\theta )+Q(\\nabla v),\n \\end{cases}\n\\end{equation}\nwith\n\\begin{eqnarray}\n &\\mathbb S_h = \\mu (\\nabla_{h} v + \\nabla_{h}^\\top v) + \\lambda \\mathrm{div}_{h}\\, v \\mathbb I_2, \\quad\n p = R \\rho \\theta, \\nonumber\\\\\n &Q(\\nabla v)=\\mathbb S_h:\\nabla_hv+\\mu|\\partial_zv|^2,\\label{Q}\n\\end{eqnarray}\nwith constant coefficients $\\gamma$, $\\mu$, $\\lambda$, $\\kappa,$ and $R$ satisfying $\\gamma > 1$, $\\mu+\\lambda>0$, $\\mu>0$, $\\kappa>0$, and $R>0$.\nHere, $ \\rho, v, w $, and $ p $ represent the density, the horizontal velocity, vertical velocity, and pressure, respectively. $ \\mathrm{div}_{h}\\,, \\nabla_{h} $, and $\\Delta_{h} $ are, and will be, the divergence, gradient, and Laplace operators in the horizontal variables $ (x,y) $, respectively.\nWe investigate system \\eqref{eq:CPE} in the horizontally periodic channel\n$$ \\mathcal O:= \\mathbb T^2 \\times (0,1) = \\lbrace (x,y,z) | (x,y) \\in \\mathbb T^2, z \\in (0,1) \\rbrace,\n$$\nand consider the following\nboundary conditions:\n\\begin{equation}\\label{bc:CPE}\n \\partial_z \\theta\\big|_{z=0,1} = 0, \\quad \\partial_z v\\big|_{z=0,1} = 0, \\quad w\\big|_{z=0,1} = 0.\n\\end{equation}\n\nNote that \\subeqref{eq:CPE}{3} implies that $p$ is independent of the vertical variable $z$ and one can use \\subeqref{eq:CPE}{3} to rewrite \\subeqref{eq:CPE}{4} as\n\\begin{equation*}\n \\partial_tp+v\\cdot\\nabla_hp+\\gamma p(\\text{div}_hv+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)Q(\\nabla v).\n\\end{equation*}\nSeparating the $z$-average part and the fluctuation part of the above equation, and recalling the boundary conditions $\\partial_z\\theta|_{z=0,1}=w|_{z=0,1}=0$, one obtains that\n\\begin{eqnarray}\n \\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)},\\label{EQp00}\\\\\n\\tilde v\\cdot\\nabla_hp+\\gamma p(\\text{div}_h\\tilde v+\\partial_zw)=(\\gamma-1)\\kappa\\partial_z^2\\theta+(\\gamma-1)\\widetilde{Q(\\nabla v)}. \\label{EQw000}\n\\end{eqnarray}\nHere, for a function $f$, we use\n\\begin{equation}\\label{TBf}\n\\bar f:=\\int_0^1f dz\\quad\\mbox{and}\\quad \\tilde f:=f-\\bar f,\n\\end{equation}\nto represent the vertical integral (equivalently, average) and the vertical fluctuation of the quantity considered, respectively.\n\nSince $\\theta = R^{-1} \\rho^{-1} p=R^{-1}\\sigma p,$\nit follows from \\eqref{EQw000} and \\subeqref{eq:CPE}{3} that\n\\begin{eqnarray}\\label{eq:dz-w}\n \\partial_z w &=&-\\text{div}_h\\tilde v+\\frac{\\gamma-1}{\\gamma p}\\left(\\kappa\\partial_z^2\\theta+\\widetilde{Q(\\nabla v)}\\right)-\\frac{1}{\\gamma p}\\tilde v\\cdot\\nabla_hp\\nonumber\\\\\n&=&\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}\\partial_{zz} \\sigma -\\text{div}_h\\tilde v-\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right)\\nonumber\\\\\n&=&\\nu\\partial_z^2\\sigma-\\phi(v,p),\\label{EQw00}\n\\end{eqnarray}\nwhere $\\nu=\\dfrac{(\\gamma-1)\\kappa}{\\gamma R}$ and\n\\begin{eqnarray}\n \\phi(v,p):=\\text{div}_h\\tilde v+\\frac{1}{\\gamma p}\\left(\\tilde v\\cdot\\nabla_hp-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\right).\\label{phi}\n\\end{eqnarray}\nRecalling that $p$ is independent of $z$, it holds that $\\partial_z\\sigma=R\\partial_z(p^{-1}\\theta)= R p^{-1}\\partial_z\\theta$. Then, the boundary condition \\eqref{bc:CPE} implies\n\\begin{equation}\\label{bc:rf-CPE}\n\\partial_z \\sigma\\big|_{z=0,1} = 0.\n\\end{equation}\nThanks to this and recalling that $w|_{z=0}$, it follows from \\eqref{EQw00} that\n\\begin{equation}\n w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz'.\\label{EQw00'}\n\\end{equation}\n\nNow, collecting \\eqref{EQv00}, \\eqref{EQp00}, \\eqref{EQw00'}, \\eqref{EQs00}, we have the following reformulated system which is equivalent to \\eqref{eq:CPE}\n\\begin{eqnarray}\n &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n \\label{EQs0}\\\\\n &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\nwhere $\\phi(v,p)$ and $Q(\\nabla v)$ are expressed as in (\\ref{phi}) and (\\ref{Q}), respectively. The boundary conditions read as\n\\begin{eqnarray}\n &v, \\sigma, \\mbox{ and } p \\mbox{ are periodic in }x, y, \\label{BC1}\\\\\n &\\partial_zv|_{z=0,1}=0, \\quad\\partial_z\\sigma|_{z=0,1}=0. \\label{BC2}\n\\end{eqnarray}\nThe initial condition is\n\\begin{equation}\n \\label{IC}\n (v, \\sigma, p)|_{t=0}=(v_{0}, \\sigma_{0}, p_{0}).\n\\end{equation}\n\nBefore stating the main result, we recall some standard notations. For positive integer $k$ and $q\\in[1,\\infty]$, $L^q(\\mathcal O)$ and\n$L^q(\\mathbb T^2)$ are the Lebesgue spaces,\n$W^{k,q}(\\mathcal O)$ and $W^{k,q}(\\mathbb T^2)$ are the Sobolev spaces. If $q=2$, we use $H^k$ instead of $W^{k,2}$. For simplicity,\nwe will use $\\|\\cdot\\|_q$ to denote $L^q(\\mathcal O)$ or $L^q(\\mathbb T^2)$ norms of the corresponding function,\nif the meaning is clear from the context.\n\nWe are now in the position to state the main result of this paper.\n\nWe are now in the position to state the main result of this paper.\n\n\\begin{proposition}\n \\label{PROP-APR}\nGiven $(v_{0}, \\sigma_{0}, p_{0})$ satisfying (\\ref{ASSUMIC1})--(\\ref{ASSUMIC2}). Let $X_0$ be an arbitrary constant such that\n$\\|v_{0}\\|_{H^3}^2+\\|\\sigma_{0}\\|_{H^2}^2+\\|p_0\\|_{H^3}^2\\leq X_0$. Then, there is a positive time $\\mathcal T_0$,\ndepending only on $\\gamma$, $\\nu$, $\\mu$, $\\lambda$, $\\underline\\sigma$, $\\underline p$, and\n $X_0$, such that system (\\ref{EQv})--(\\ref{EQp}),\n subject to (\\ref{BC1})--(\\ref{IC}), has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$,\n satisfying\n \\begin{eqnarray*}\n &&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\geq0.5\\underline\\sigma,\\quad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\geq0.5\\underline p,\\\\\n &&\\sup_{0\\leq t\\leq\\mathcal T_0}(\\|v\\|_{H^3}^2+\\|\\sigma\\|_{H^2}^2+\\|p\\|_{H^3}^2)(t)+\\epsilon\\int_0^{\\mathcal T_0}(\\|\\nabla_h\\sigma\\|_{H^2}^2+\\|\\nabla_hp\\|_{H^3}^2)dt\\nonumber\\\\\n &&\\, \\, ~~~+\\int_0^{\\mathcal T_0}(\\|v\\|_{H^4}^2+\\|\\partial_z\\sigma\\|_{H^2}^2+\\|\\partial_tv\\|_{H^2}^2+\\|\\partial_t\\sigma\\|_{H^1}^2+\\|\\partial_tp\\|_{H^2}^2)dt \\leq K_0,\n \\end{eqnarray*}\n for a positive constant $K_0$ depending only on $\\gamma$, $\\nu$, $\\mu$, $\\lambda$, $\\underline\\sigma$, $\\underline p$, and\n $X_0$.\n\\end{proposition}\n\nBy Proposition \\ref{PROP-APR}, there is a positive time $\\mathcal T_0$ depending only on $\\gamma,$ $\\nu,$ $\\mu$, $\\lambda$,\n$\\underline\\sigma$, $\\underline p$, and $X_0$, such that for any $\\epsilon\\in(0,\\epsilon_0)$, system \\eqref{EQv}--\\eqref{EQp}, subject to \\eqref{BC1}--\\eqref{IC}, has a unique solution $(v_\\epsilon,\n\\sigma_\\epsilon, p_\\epsilon)$ on $\\mathcal O\\times(0,\\mathcal T_0)$,\nwith initial data $(v_{0,\\epsilon}, \\sigma_{0,\\epsilon}, p_{0,\\epsilon})$, satisfying\n \\begin{eqnarray}\n &&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma_\\epsilon\\geq0.5\\underline\\sigma,\\quad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p_\\epsilon\\geq0.5\\underline p,\\label{APRLBsp}\\\\\n &&\\sup_{0\\leq t\\leq\\mathcal T_0}(\\|v_\\epsilon\\|_{H^3}^2+\\|\\sigma_\\epsilon\\|_{H^2}^2+\\|p_\\epsilon\\|_{H^3}^2)(t)\n +\\epsilon\\int_0^{\\mathcal T_0}(\\|\\nabla_h\\sigma_\\epsilon\\|_{H^2}^2+\\|\\nabla_hp_\\epsilon\\|_{H^3}^2)dt\\nonumber\\\\\n &&\\, \\, ~~~+\\int_0^{\\mathcal T_0}(\\|v_\\epsilon\\|_{H^4}^2+\\|\\partial_z\\sigma_\\epsilon\\|_{H^2}^2+\\|\\partial_tv_\\epsilon\\|_{H^2}^2+\\|\\partial_t\\sigma_\\epsilon\n \\|_{H^1}^2+\\|\\partial_tp_\\epsilon\\|_{H^2}^2)dt \\leq K_0,\\label{APRHk}\n \\end{eqnarray}\n for a positive constant $K_0$ depending only on $\\gamma$, $\\nu$, $\\mu$, $\\lambda$, $\\underline\\sigma$, $\\underline p$, and\n $X_0$.\n\nThanks to \\eqref{APRHk}, by the Banach-Alaoglu theorem, and using Cantor's diagonal argument,\n there is a subsequence $(v_{\\epsilon_n}, \\sigma_{\\epsilon_n}, p_{\\epsilon_n})$ and a triple $(v,\\sigma,p)$,\n such that\n \\begin{eqnarray}\n &&v_{\\epsilon_n}\\stackrel{\\ast}{\\rightharpoonup}v,\\quad\\mbox{in }L^\\infty(0,\\mathcal T_0; H^3(\\mathcal O)),\\label{WeaC1}\\\\\n &&\\sigma_{\\epsilon_n}\\stackrel{\\ast}{\\rightharpoonup}\\sigma,\\quad\\mbox{in }L^\\infty(0,\\mathcal T_0; H^2(\\mathcal O)),\\label{WeaC2}\\\\\n &&p_{\\epsilon_n}\\stackrel{\\ast}{\\rightharpoonup}p,\\quad\\mbox{in }L^\\infty(0,\\mathcal T_0; H^3(\\mathbb T^2)),\\label{WeaC3}\\\\\n &&v_{\\epsilon_n}{\\rightharpoonup}v,\\quad\\mbox{in }L^2(0,\\mathcal T_0; H^4(\\mathcal O)),\\label{WeaC4}\\\\\n &&\\partial_z\\sigma_{\\epsilon_n} {\\rightharpoonup}\\partial_z\\sigma,\\quad\\mbox{in }L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\label{WeaC5}\\\\\n &&\\partial_tv_{\\epsilon_n} {\\rightharpoonup}\\partial_tv,\\quad\\mbox{in }L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\label{WeaC6}\\\\\n &&\\partial_t\\sigma_{\\epsilon_n} {\\rightharpoonup}\\partial_t\\sigma,\\quad\\mbox{in }L^2(0,\\mathcal T_0; H^1(\\mathcal O)),\\label{WeaC7}\\\\\n &&\\partial_tp_{\\epsilon_n} {\\rightharpoonup}\\partial_tp,\\quad\\mbox{in }L^2(0,\\mathcal T_0; H^2(\\mathbb T^2)),\\label{WeaC8}\n \\end{eqnarray}\n where $\\stackrel{\\ast}{\\rightharpoonup}$ and $\\rightharpoonup$ represent the weak-* and weak convergence, respectively, in the corresponding spaces. With the aid of the above convergence, noticing that $H^3(\\mathcal O)\\hookrightarrow\\hookrightarrow H^2(\\mathcal O)$, $H^2(\\mathcal O)\\hookrightarrow\\hookrightarrow H^1(\\mathcal O)\\cap C(\\overline{\\mathcal O})$, and $H^3(\\mathbb T^2)\\hookrightarrow\\hookrightarrow H^2(\\mathbb T^2)\\cap C(\\mathbb T^2)$,\n it follows from the Aubin-Lions lemma that\n \\begin{eqnarray}\n &&v_{\\epsilon_n} \\rightarrow v,\\quad\\mbox{in }C([0,\\mathcal T_0]; H^2(\\mathcal O)),\\label{StrC1}\\\\\n &&\\sigma_{\\epsilon_n} \\rightarrow\\sigma,\\quad\\mbox{in }C([0,\\mathcal T_0]; H^1(\\mathcal O)\\cap C(\\overline{\\mathcal O})),\\label{StrC2}\\\\\n &&\\partial_z\\sigma_{\\epsilon_n} \\rightarrow\\partial_z\\sigma,\\quad\\mbox{in }L^2(0,\\mathcal T_0; H^1(\\mathcal O)),\\\\\n &&p_{\\epsilon_n} \\rightarrow p,\\quad\\mbox{in }C([0,\\mathcal T_0]; H^2(\\mathbb T^2)\\cap C(\\mathbb T^2)).\\label{StrC3}\n \\end{eqnarray}\n These imply that $(v,\\sigma,p)$ has initial data $(v_0,\\sigma_0,p_0)$, enjoys all regularities stated in Theorem \\ref{thmmain}, and fulfills the boundary conditions \\eqref{BC1} and \\eqref{BC2}.\n Besides, by \\eqref{StrC2} and \\eqref{StrC3}, it follows from \\eqref{APRLBsp} that\n \\begin{equation*}\n \\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma \\geq0.5\\underline\\sigma,\\quad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p \\geq0.5\\underline p.\n \\end{equation*}\n By \\eqref{WeaC1}--\\eqref{StrC3}, one can take the limit as $n\\rightarrow\\infty$ to \\eqref{EQv}--\\eqref{EQp} to show that $(v,\\sigma,p)$ satisfies system \\eqref{EQv0}--\\eqref{EQp0} in the sense of distribution and further pointwisely, by the regularities of $(v,\\sigma,p)$. Therefore, $(v,\\sigma,p)$ is a strong solution\n to system \\eqref{EQv}--\\eqref{EQp}, subject to \\eqref{BC1}--\\eqref{IC}, on $\\mathcal O\\times(0,\\mathcal T_0)$.\n\n\\section{Appendix B: local existence of the regularized system}\n\\label{APPENDIXB}\nDenote\n\\begin{eqnarray*}\n N_1(v,\\sigma,p):=-[(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp],\\label{N1}\\\\\n N_2(v,\\sigma,p):=-[v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)],\\label{N2}\\\\\n N_3(v,p):=-(\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v)+(\\gamma-1)\\overline{Q(\\nabla v)},\\label{N3}\n\\end{eqnarray*}\nwhere $w, \\phi(v,p), \\bar f,$ and $Q(\\nabla v)$ are expressed, respectively, by\\eqref{EQw00'}, \\eqref{phi}, \\eqref{TBf}, and \\eqref{Q}.\nRecall $\\mathcal O=\\mathbb T^2\\times(0,1)$. Define a solution mapping\n\\begin{equation}\\label{DEFF}\n(v,\\sigma,p)\\mapsto(V,\\Sigma,P)=:\\mathfrak{F}(v,\\sigma,p),\n\\end{equation}\nwhere $(V,\\Sigma,P)$ is the unique solution to the following linear system\n\\begin{eqnarray}\n \\partial_tV-\\mu\\sigma\\Delta V-(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hV&=&N_1(v,\\sigma,p),\\label{AEQV}\\\\\n \\partial_t\\Sigma-\\nu\\sigma\\partial_z^2\\Sigma-\\epsilon\\Delta_h\\Sigma&=&N_2(v,\\sigma,p),\\label{AEQSIGMA}\\\\\n \\partial_tP-\\epsilon\\Delta_hP&=&N_3(v,p), \\label{AEQP}\n\\end{eqnarray}\nin $\\mathcal O\\times(0,T)$, subject to\n\\begin{eqnarray}\n &V,\\Sigma, P\\text{ are periodic in }x,y, \\quad\\partial_zV|_{z=0,1}=0,\\quad \\partial_z\\Sigma|_{z=0,1}=0,\\label{ABC}\\\\\n &(V,\\Sigma,P)|_{t=0}=(v_0,\\sigma_0,p_0),\\label{AIC}\n\\end{eqnarray}\nwhere $(v_0,\\sigma_0)\\in H^3(\\mathcal O)$, $p_0\\in H^3(\\mathbb T^2),$ and\n\\begin{equation*}\n \\partial_zv_0|_{z=0,1}=0,\\quad \\partial_z\\sigma_0|_{z=0,1}=0,\\quad\\sigma_0\\geq\\underline\\sigma, \\quad p_0\\geq\\underline p,\\label{A1}\n\\end{equation*}\nfor two positive numbers $\\underline\\sigma$ and $\\underline p$.", "post_theorem_intro_text_len": 2475, "post_theorem_intro_text": "Some comments on the proof of Theorem \\ref{thmmain} are presented as follows.\nRegularity assumption $\\sigma_0\\in H^2(\\mathcal O)$ comes from the following observation: for the toy model\n$$\n\\partial_t\\sigma=\\nu\\sigma\\partial_z^2\\sigma,\n$$\nin order that $H^k(\\mathcal O)$ regularity propagates, one needs $k\\geq2$.\nFortunately, such $H^2(\\mathcal O)$ regularity on $\\sigma_0$ is still sufficient for the more complicated equation \\eqref{EQs0}.\nWhile for the aim of getting $H^2(\\mathcal O)$ estimate on $\\sigma$, one may apply $\\Delta$ to \\eqref{EQs0} and, as a result, the\nquantity $\\Delta\\phi(v,p)$ and further, recalling the expression of $\\phi(v,p)$, $\\nabla_h\\Delta_hp$ will be encountered.\nIn order to get information on $\\nabla_h\\Delta_hp$, by equation \\eqref{EQp0}, one requires $p_0\\in H^3(\\mathbb T^2)$ and moreover\nthe quantity $\\nabla_h^4v$ will be encountered. Due to this and by \\eqref{EQv0}, one has to assume $v_0\\in H^3(\\mathcal O)$.\nThe existence of solutions to system \\eqref{EQv0}--\\eqref{EQp0}, subject to \\eqref{BC1}--\\eqref{IC}, is established by parabolic\nregularization argument. A regularized system is introduced by adding the regularizing dissipative terms $\\epsilon\\Delta_h\\sigma$ and $\\epsilon\\Delta_hp$\nto \\eqref{EQs0} and \\eqref{EQp0}, respectively, see \\eqref{EQv}--\\eqref{EQp}, below. Such a regularized system is a semi-linear\nparabolic system. The main effort of the proof is then to carry out\nsuitable $\\epsilon$-independent {\\it a priori} estimates and the local existence\nfollows passing to the limit $\\epsilon\\rightarrow0$.\nContinuous dependence on the initial data, which in particular implies\nthe uniqueness, is proved by performing energy estimates to the difference between two solutions. We employ $L^2$ energy estimate to the\nsubtracted $\\sigma$ equation which contains a term involving $\\nabla_hp$. This, due to the absence of dissipation\nin the pressure equation, forces us to perform $H^1(\\mathbb T^2)$ type energy estimate to the subtracted pressure equation and\nfurther $H^1(\\mathcal O)$ energy to the subtracted velocity equations.\n\nThe rest of this paper is arranged as follows: in the next section, Section \\ref{sec3}, we consider the $\\epsilon$-regularized system\nand perform suitable a priori estimates; the main result, Theorem \\ref{thmmain}, is proved in Section \\ref{sec4}; some commutator estimates are proved in Appendix A; local existence and uniqueness of the $\\epsilon$-regularized system is proved in Appendix B.", "sketch": "For Theorem~\\ref{thmmain}, the authors explain the proof strategy and why the initial regularities are chosen.\n\n- **Why $\\sigma_0\\in H^2(\\mathcal O)$:** From the toy model $\\partial_t\\sigma=\\nu\\sigma\\partial_z^2\\sigma$, “in order that $H^k(\\mathcal O)$ regularity propagates, one needs $k\\ge 2$.” This $H^2$ assumption is said to be “still sufficient for the more complicated equation \\eqref{EQs0}.”\n\n- **Why $p_0\\in H^3(\\mathbb T^2)$ and $v_0\\in H^3(\\mathcal O)$:** To get an “$H^2(\\mathcal O)$ estimate on $\\sigma$,” one applies “$\\Delta$ to \\eqref{EQs0},” which leads to “$\\Delta\\phi(v,p)$” and hence to “$\\nabla_h\\Delta_h p$.” To control $\\nabla_h\\Delta_h p$ using \\eqref{EQp0}, one needs “$p_0\\in H^3(\\mathbb T^2)$,” and in the process “the quantity $\\nabla_h^4 v$ will be encountered,” which (via \\eqref{EQv0}) “has to assume $v_0\\in H^3(\\mathcal O)$.”\n\n- **Existence via parabolic regularization and $\\epsilon$-independent estimates:** Existence is established by a “parabolic regularization argument.” They introduce a regularized system by adding “the regularizing dissipative terms $\\epsilon\\Delta_h\\sigma$ and $\\epsilon\\Delta_h p$ to \\eqref{EQs0} and \\eqref{EQp0}, respectively.” The resulting system is “a semi-linear parabolic system.” The “main effort” is to derive “suitable $\\epsilon$-independent {\\it a priori} estimates,” and then “the local existence follows passing to the limit $\\epsilon\\to 0$.”\n\n- **Uniqueness/continuous dependence by energy estimates on differences:** “Continuous dependence on the initial data, which in particular implies the uniqueness,” is proved by “performing energy estimates to the difference between two solutions.” Specifically, they use an “$L^2$ energy estimate to the subtracted $\\sigma$ equation” but this equation has “a term involving $\\nabla_h p$.” Because of “the absence of dissipation in the pressure equation,” they are “force[d]” to do an “$H^1(\\mathbb T^2)$ type energy estimate to the subtracted pressure equation” and “further $H^1(\\mathcal O)$ energy to the subtracted velocity equations.”", "expanded_sketch": "For the main theorem, the authors explain the proof strategy and why the initial regularities are chosen.\n\n- **Why $\\sigma_0\\in H^2(\\mathcal O)$:** From the toy model $\\partial_t\\sigma=\\nu\\sigma\\partial_z^2\\sigma$, “in order that $H^k(\\mathcal O)$ regularity propagates, one needs $k\\ge 2$.” This $H^2$ assumption is said to be “still sufficient for the more complicated equation\n\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n  &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n  \\label{EQs0}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\n.”\n\n- **Why $p_0\\in H^3(\\mathbb T^2)$ and $v_0\\in H^3(\\mathcal O)$:** To get an “$H^2(\\mathcal O)$ estimate on $\\sigma$,” one applies “$\\Delta$ to” the $\\sigma$-equation in\n\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n  &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n  \\label{EQs0}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\nwhich leads to “$\\Delta\\phi(v,p)$” and hence to “$\\nabla_h\\Delta_h p$.” To control $\\nabla_h\\Delta_h p$ using the pressure equation in the same system (i.e. the $p$-equation displayed above), one needs “$p_0\\in H^3(\\mathbb T^2)$,” and in the process “the quantity $\\nabla_h^4 v$ will be encountered,” which (via the velocity equation in the displayed system above) “has to assume $v_0\\in H^3(\\mathcal O)$.”\n\n- **Existence via parabolic regularization and $\\epsilon$-independent estimates:** Existence is established by a “parabolic regularization argument.” They introduce a regularized system by adding “the regularizing dissipative terms $\\epsilon\\Delta_h\\sigma$ and $\\epsilon\\Delta_h p$” to the $\\sigma$- and $p$-equations in\n\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n  &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n  \\label{EQs0}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\nrespectively. The resulting system is “a semi-linear parabolic system.” The “main effort” is to derive “suitable $\\epsilon$-independent {\\it a priori} estimates,” and then “the local existence follows passing to the limit $\\epsilon\\to 0$.”\n\n- **Uniqueness/continuous dependence by energy estimates on differences:** “Continuous dependence on the initial data, which in particular implies the uniqueness,” is proved by “performing energy estimates to the difference between two solutions.” Specifically, they use an “$L^2$ energy estimate to the subtracted $\\sigma$ equation” but this equation has “a term involving $\\nabla_h p$.” Because of “the absence of dissipation in the pressure equation” (namely, in the $p$-equation in the displayed system above), they are “force[d]” to do an “$H^1(\\mathbb T^2)$ type energy estimate to the subtracted pressure equation” and “further $H^1(\\mathcal O)$ energy to the subtracted velocity equations.”", "expanded_theorem": "\\label{thmmain}\nGiven $v_{0}\\in H^3(\\mathcal O)$, $\\sigma_{0} \\in H^2(\\mathcal O)$, and $p_{0}\\in H^3(\\mathbb T^2),$ such that\n$$\n\\sigma_{0}\\geq\\underline\\sigma, \\quad p_{0}\\geq\\underline p,\\quad \\partial_zv_{0}|_{z=0,1}=0,\\quad \\partial_z\\sigma_{0}|_{z=0,1}=0,\n$$\nfor two positive numbers $\\underline\\sigma$ and $\\underline p$.\nThen, there is a positive time $\\mathcal T_0$,\ndepending only on $\\gamma$, $\\nu$, $\\mu$, $\\lambda$, $\\underline\\sigma$, $\\underline p$, and\n  $\\|v_{0}\\|_{H^3}^2+\\|\\sigma_{0}\\|_{H^2}^2+\\|p_0\\|_{H^3}^2$, such that the system\n\\begin{eqnarray}\n  &\\partial_tv+(v\\cdot\\nabla_h)v+w\\partial_zv+\\sigma\\nabla_hp=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\text{div}_hv,\\label{EQv0}\\\\\n  &w=\\nu\\partial_z\\sigma-\\int_0^z\\phi(v,p)dz',\\label{EQw0}\\\\\n  &\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma(\\phi(v,p)-\\text{div}_hv)=\\nu\\sigma\\partial_z^2\\sigma,\n  \\label{EQs0}\\\\\n  &\\partial_tp+\\bar v\\cdot\\nabla_hp+\\gamma p\\text{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)}, \\label{EQp0}\n\\end{eqnarray}\n  subject to\n\\begin{eqnarray}\n  &v, \\sigma, \\mbox{ and } p \\mbox{ are periodic in }x, y, \\label{BC1}\\\\\n  &\\partial_zv|_{z=0,1}=0, \\quad\\partial_z\\sigma|_{z=0,1}=0. \\label{BC2}\n\\end{eqnarray}\nand\n\\begin{equation}\n  \\label{IC}\n  (v, \\sigma, p)|_{t=0}=(v_{0}, \\sigma_{0}, p_{0}).\n\\end{equation}\nhas a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$,\n  satisfying\n  \\begin{eqnarray*}\n  &&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\geq0.5\\underline\\sigma,\\quad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\geq0.5\\underline p,\\\\\n    &&v\\in C([0,\\mathcal T_0]; H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0; H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0; H^4(\\mathcal O)),\\\\\n    &&\\sigma\\in C([0,\\mathcal T_0]; H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0; H^2(\\mathcal O)),\\quad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\\\\n    &&\\partial_tv\\in L^2(0,\\mathcal T_0; H^2(\\mathcal O)),\\quad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0; H^1(\\mathcal O)),\\\\\n    &&p\\in C([0,\\mathcal T_0]; H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0; H^3(\\mathbb T^2)),\\quad\\partial_tp\\in  L^2(0,\\mathcal T_0; H^2(\\mathbb T^2)).\n  \\end{eqnarray*},", "theorem_type": ["Uniqueness", "Existence"], "mcq": {"question": "Let $\\mathcal O=\\mathbb T^2\\times(0,1)$. For a function $f(x,y,z)$, define\n$$\\bar f(x,y):=\\int_0^1 f(x,y,z)\\,dz,\\qquad \\tilde f:=f-\\bar f.$$ \nLet\n$$Q(\\nabla v):=\\bigl[\\mu(\\nabla_h v+\\nabla_h^{\\top}v)+\\lambda\\,\\mathrm{div}_h v\\,\\mathbb I_2\\bigr]:\\nabla_h v+\\mu|\\partial_z v|^2,$$\n$$\\phi(v,p):=\\mathrm{div}_h\\tilde v+\\frac{1}{\\gamma p}\\Bigl(\\tilde v\\cdot\\nabla_h p-(\\gamma-1)\\widetilde{Q(\\nabla v)}\\Bigr),$$\n$$w=\\nu\\partial_z\\sigma-\\int_0^z \\phi(v,p)(x,y,z',t)\\,dz'.$$\nAssume\n$$v_0\\in H^3(\\mathcal O),\\qquad \\sigma_0\\in H^2(\\mathcal O),\\qquad p_0\\in H^3(\\mathbb T^2),$$\nand for some positive constants $\\underline\\sigma,\\underline p$,\n$$\\sigma_0\\ge \\underline\\sigma,\\qquad p_0\\ge \\underline p,\\qquad \\partial_z v_0|_{z=0,1}=0,\\qquad \\partial_z\\sigma_0|_{z=0,1}=0.$$ \nWhich statement holds for the problem of finding $v,\\sigma$ on $\\mathcal O\\times(0,T)$ and $p$ on $\\mathbb T^2\\times(0,T)$ such that\n\\begin{eqnarray*}\n&&\\partial_t v+(v\\cdot\\nabla_h)v+w\\partial_z v+\\sigma\\nabla_h p=\\mu\\sigma\\Delta v+(\\mu+\\lambda)\\sigma\\nabla_h\\mathrm{div}_h v,\\\\\n&&\\partial_t\\sigma+v\\cdot\\nabla_h\\sigma+w\\partial_z\\sigma+\\sigma\\bigl(\\phi(v,p)-\\mathrm{div}_h v\\bigr)=\\nu\\sigma\\partial_z^2\\sigma,\\\\\n&&\\partial_t p+\\bar v\\cdot\\nabla_h p+\\gamma p\\,\\mathrm{div}_h\\bar v=(\\gamma-1)\\overline{Q(\\nabla v)},\n\\end{eqnarray*}\nwith $v,\\sigma,p$ periodic in $(x,y)$, boundary conditions $\\partial_z v|_{z=0,1}=0$ and $\\partial_z\\sigma|_{z=0,1}=0$, and initial data $(v,\\sigma,p)|_{t=0}=(v_0,\\sigma_0,p_0)$?", "correct_choice": {"label": "A", "text": "There exists a time $\\mathcal T_0>0$, depending only on $\\gamma,\\nu,\\mu,\\lambda,\\underline\\sigma,\\underline p$, and $\\|v_0\\|_{H^3}^2+\\|\\sigma_0\\|_{H^2}^2+\\|p_0\\|_{H^3}^2$, such that the above system has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$ satisfying\n\\begin{eqnarray*}\n&&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\ge 0.5\\,\\underline\\sigma,\\qquad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\ge 0.5\\,\\underline p,\\\\\n&&v\\in C([0,\\mathcal T_0];H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0;H^4(\\mathcal O)),\\\\\n&&\\sigma\\in C([0,\\mathcal T_0];H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\\\\n&&\\partial_t v\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0;H^1(\\mathcal O)),\\\\\n&&p\\in C([0,\\mathcal T_0];H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathbb T^2)),\\qquad \\partial_t p\\in L^2(0,\\mathcal T_0;H^2(\\mathbb T^2)).\n\\end{eqnarray*}"}, "choices": [{"label": "B", "text": "There exists a time $\\mathcal T_0>0$, depending only on $\\gamma,\\nu,\\mu,\\lambda,\\underline\\sigma,\\underline p$, and $\\|v_0\\|_{H^3}^2+\\|\\sigma_0\\|_{H^1}^2+\\|p_0\\|_{H^2}^2$, such that the above system has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$ satisfying\n\\begin{eqnarray*}\n&&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\ge 0.5\\,\\underline\\sigma,\\qquad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\ge 0.5\\,\\underline p,\\\\\n&&v\\in C([0,\\mathcal T_0];H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0;H^4(\\mathcal O)),\\\\\n&&\\sigma\\in C([0,\\mathcal T_0];L^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^1(\\mathcal O)),\\qquad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0;H^1(\\mathcal O)),\\\\\n&&\\partial_t v\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0;L^2(\\mathcal O)),\\\\\n&&p\\in C([0,\\mathcal T_0];H^1(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0;H^2(\\mathbb T^2)),\\qquad \\partial_t p\\in L^2(0,\\mathcal T_0;H^1(\\mathbb T^2)).\n\\end{eqnarray*}"}, {"label": "C", "text": "There exists a time $\\mathcal T_0>0$, depending only on $\\gamma,\\nu,\\mu,\\lambda,\\underline\\sigma,\\underline p$, and $\\|v_0\\|_{H^3}^2+\\|\\sigma_0\\|_{H^2}^2+\\|p_0\\|_{H^3}^2$, such that the above system has at least one local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$ satisfying\n\\begin{eqnarray*}\n&&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\ge 0.5\\,\\underline\\sigma,\\qquad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\ge 0.5\\,\\underline p,\\\\\n&&v\\in C([0,\\mathcal T_0];H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0;H^4(\\mathcal O)),\\\\\n&&\\sigma\\in C([0,\\mathcal T_0];H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\\\\n&&\\partial_t v\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0;H^1(\\mathcal O)),\\\\\n&&p\\in C([0,\\mathcal T_0];H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathbb T^2)),\\qquad \\partial_t p\\in L^2(0,\\mathcal T_0;H^2(\\mathbb T^2)).\n\\end{eqnarray*}"}, {"label": "D", "text": "There exists a universal time $\\mathcal T_0>0$, depending only on $\\gamma,\\nu,\\mu,\\lambda,\\underline\\sigma,$ and $\\underline p$ but independent of $\\|v_0\\|_{H^3},\\|\\sigma_0\\|_{H^2},\\|p_0\\|_{H^3}$, such that the above system has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$ satisfying\n\\begin{eqnarray*}\n&&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\ge 0.5\\,\\underline\\sigma,\\qquad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\ge 0.5\\,\\underline p,\\\\\n&&v\\in C([0,\\mathcal T_0];H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0;H^4(\\mathcal O)),\\\\\n&&\\sigma\\in C([0,\\mathcal T_0];H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\\\\n&&\\partial_t v\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0;H^1(\\mathcal O)),\\\\\n&&p\\in C([0,\\mathcal T_0];H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathbb T^2)),\\qquad \\partial_t p\\in L^2(0,\\mathcal T_0;H^2(\\mathbb T^2)).\n\\end{eqnarray*}"}, {"label": "E", "text": "There exists a time $\\mathcal T_0>0$, depending only on $\\gamma,\\nu,\\mu,\\lambda,\\underline\\sigma,\\underline p$, and $\\|v_0\\|_{H^3}^2+\\|\\sigma_0\\|_{H^2}^2+\\|p_0\\|_{H^3}^2$, such that the above system has a unique local solution $(v,\\sigma,p)$ on $\\mathcal O\\times(0,\\mathcal T_0)$ satisfying\n\\begin{eqnarray*}\n&&\\inf_{(x,y,z,t)\\in\\mathcal O\\times[0,\\mathcal T_0]}\\sigma\\ge \\underline\\sigma,\\qquad \\inf_{(x,y,t)\\in\\mathbb T^2\\times[0,\\mathcal T_0]}p\\ge \\underline p,\\\\\n&&v\\in C([0,\\mathcal T_0];H^2(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathcal O))\\cap L^2(0,\\mathcal T_0;H^4(\\mathcal O)),\\\\\n&&\\sigma\\in C([0,\\mathcal T_0];H^1(\\mathcal O))\\cap L^\\infty(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_z\\sigma\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\\\\n&&\\partial_t v\\in L^2(0,\\mathcal T_0;H^2(\\mathcal O)),\\qquad \\partial_t\\sigma\\in L^2(0,\\mathcal T_0;H^1(\\mathcal O)),\\\\\n&&p\\in C([0,\\mathcal T_0];H^2(\\mathbb T^2))\\cap L^\\infty(0,\\mathcal T_0;H^3(\\mathbb T^2)),\\qquad \\partial_t p\\in L^2(0,\\mathcal T_0;H^2(\\mathbb T^2)).\n\\end{eqnarray*}"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "minimal initial regularity for $\\sigma_0$ and $p_0$", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped uniqueness while keeping the same existence/regularity conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "dependence of lifespan on Sobolev norms of initial data", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "strict preservation of the full lower bounds instead of the halved bounds", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It sets up the PDE system and assumptions, but the exact conclusion must still be selected from several close variants."}, "TAS": {"score": 0, "justification": "This is essentially a direct recognition task for the precise local well-posedness theorem statement. The correct answer is the theorem itself, only contrasted against slightly modified versions."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in subtle but meaningful ways: uniqueness vs. existence, lifespan dependence, lower-bound propagation, and Sobolev regularity. However, the task is still mostly theorem recall/verification rather than generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful. They target common failure modes: weakened regularity assumptions, dropping uniqueness, incorrect lifespan dependence, and overstating positivity preservation."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-recognition MCQ with strong distractors, but it is largely a restatement of a known result and only moderately tests reasoning."}}
{"id": "2602.19882v1", "paper_link": "http://arxiv.org/abs/2602.19882v1", "theorems_cnt": 6, "theorem": {"env_name": "theorem", "content": "\\label{th-main}\nFor each positive integer $k$, a finite graph is $k$-set-homogeneous if and only if it is $k$-homogeneous.\nMoreover, { if $\\Ga$ is a $k$-homogeneous graph with $k\\geqslant 2$, then} one of the following holds:\n\\begin{itemize}\n\\item [{\\rm(1)}] $k\\geqslant 5$, and $\\Ga$ is one of { $\\C_5 (\\cong\\overline{\\C_5})$, $\\K_3\\square\\K_3 (\\cong \\overline{\\K_3\\square\\K_3})$, $n\\K_m$, and $\\K_{n[m]}$; furthermore, $\\Ga$ is homogeneous.}\n\n{ \\item[(2)] $\\Ga$ is either the Schl\\\"{a}fli graph on $27$ vertices or its complement (the two non-trivial orbital graphs of $\\mathrm{PSU}_4(2)$ acting on the isotropic lines); furthermore, $\\Ga$ is $4$-homogeneous but not $5$-homogeneous.\n\n\\item[(3)] $\\Ga$ is either $\\K_n\\square\\K_n$ with $n\\geqslant4$, or a graph from Theorem~\\ref{th:3sethomo}(1)--(4), or the complement of any such graph; furthermore, $\\Ga$ is $3$-homogeneous but not $4$-homogeneous.\n\n\\item[(4)] $\\Ga$ is an orbital graph of a primitive permutation group of rank $3$ which is not covered by parts (1)--(3); furthermore, $\\Ga$ is $2$-homogeneous but not $3$-homogeneous. }\n\n\\end{itemize}", "start_pos": 11163, "end_pos": 12292, "label": "th-main"}, "ref_dict": {"th:2sethomo": "\\begin{theorem}\\label{th:2sethomo}\nLet $\\Ga=(V,E)$ be a finite  $(G,2)$-set-homogeneous graph, where $G\\leq\\Aut(\\Ga)$.\nThen one of the following holds:\n\\begin{itemize}\n\\item[{\\rm(1)}] $\\Ga\\cong\\K_n$ or $n\\K_1$, and $G$ is a $2$-homogeneous permutation group on $V$.\n\n\\item[{\\rm(2)}] $\\Ga\\cong\\K_{m[b]}$ or $m\\K_b$, and  $G\\leqslant X\\wr Y$,\nwhere $X,Y$ are $2$-homogeneous permutation groups of degree $b,m$, respectively.\n\n\\item[{\\rm(3)}] $G$ is primitive of rank $3$, and $\\Ga$ is $(G,2)$-homogeneous.\n\n\\item[{\\rm(4)}] $G\\cong\\PSU_3(3)$ with vertex stabiliser $\\PSL_3(2)$, $\\Ga$ is of order $36$ and\nvalency $14$ or $21$, but $\\Aut(\\Ga)=\\PSU_3(3).2$ is $2$-homogeneous on $\\Ga$.\n\n\\item[(5)] $\\Ga$ is a Paley graph of order $p^r$, where $p\\equiv5 \\pmod{8}$ and $r$ is odd, and\n$G\\leq\\AGammaL_1(p^r)$ is $2$-set-homogeneous on $\\Ga$.\n\\end{itemize}\nIn particular, all $2$-set-homogeneous graphs are $2$-homogeneous.\n\\end{theorem}", "set:3sethom": "\\begin{array}{cllllc}\n\\hline\n\\text{Line}&S& S_\\alpha & \\text{degree} &\\text{subdegrees}& \\text{rank of }\\N_{{\\rm Sym}(\\Om)}(S) \\\\\n\\hline\n48 &  \\mathrm{P\\Omega}_7(3)  &  \\PSp_6(2)  & 3159 &  1,288,630,2240  &  4 \\\\\n49 &  \\mathrm{P\\Omega}^{+}_8(2)  &  \\A_9  & 960 &  1,84,315,560  &  4 \\\\\n50 & \\mathrm{P\\Omega}^{+}_8(3)  &  \\mathrm{P\\Omega}^{+}_8(2)  &  28431  &  1,960,960,960,3150,22400  &  4 \\\\\n\\hline\n\\end{array}\n\\]\nIt implies that all orbitals of $G$ are self-paired.\n\\medskip\n\n\\f{\\bf Case~5.}\\ The Exceptional Group of Lie Type.\\medskip\n\nIn this case, $(S, S_\\alpha)$ is one of the pairs in Lines 51--68 of \\cite[Table~I]{Cuypers}.\n\n \\vspace{0.1cm}\n\\underline{Lines  51--54, 58--59, 61, 63, 65, 67--68}:\\  These lines are impossible by Lemma~\\ref{lm:parabolic}.\n\n \\vspace{0.1cm}\n\\underline{Lines 55--56}:\\\nNow $S=G_2(q)$ and $S_\\alpha=\\mathrm{SU}_3(q).2$ or $\\SL_3(q).2$.\nBy \\cite{Liebeck-2-closures}, all orbitals of $S$ are self-paired.\n\n \\vspace{0.1cm}\n\\underline{Line  57}:\\\nNow $S=G_2(4)$ and $S_\\alpha=\\mathrm{J}_2$, which  gives  rise to a group of rank $3$ by~\\cite{Liebeck-Saxl}.\n\n \\vspace{0.1cm}\n\\underline{Line  60}:\\\nNow $S={}^2F_4(2)'$ and $S_\\alpha=\\PSL_3(2).2$. By computation in Magma~\\cite{BCP},  $S$ has rank $4$ with subdegrees $1$, $312$, $351$ and $936$.\n\n \\vspace{0.1cm}\n\\underline{Line  62}:\\\nNow $S=F_4(2)$ and $S_\\alpha=\\PSp_8(2)$. From Atlas~\\cite{Conway-Atlas} we see that $G=S$.\n By~\\cite[Lemma~8.6~(i)]{Cuypers}, $G$ would have  rank $5$.\n\n \\vspace{0.1cm}\n\\underline{Line 64}:\\\nNow $S={}^2E_6(2)$ and $S_\\alpha=F_4(2)$. By \\cite[Lemma~8.6(ii)]{Cuypers}, $G$ has rank $4$. In this case, $S_\\alpha$ is the centraliser of the out-involution $\\s$ of $S$. Let $G={}^2E_6(2){:}\\lg\\s\\rg$ and $C=F_4(2)\\times\\lg\\s\\rg$. Set $\\Om=\\{\\s^g \\mid g\\in G\\}$. Now let $H\\leq G$ s.t. $H=\\PSU_6(2){:}\\lg\\s\\rg$. Then $C_H(\\s)=\\PSp_6(2)\\times\\lg\\s\\rg$. Take $x,y\\in\\Om$ such that $x,y$ are in the same orbit of $C$. Then $\\s x$ and $\\s y$ are conjugate, and so they have the same order. Let $X=\\{\\s^h\\ |\\ h\\in H\\}$. Then $H$ is primitive on $X$ with rank $4$ and all orbitals are self-paired. By Magma~\\cite{BCP}, we can obtain that $|\\{o (\\s x)\\ |\\ x\\in X\\}|=4$, and so each orbital of $H$ on $X$ is contained in exactly one orbital of $G$ on $\\Om$. This implies that all orbitals of $G$ are also self-paired.\n(We are grateful to Prof. Frank L${\\rm \\ddot{u}}$beck who kindly helps us to obtain the subdegrees (namely, $1,69615, 6336512, 16707600$) of this group by using GAP. )\n\n \\vspace{0.1cm}\n\\underline{Line  66}:\\\nNow $S=E_6(2)$ and $S_\\alpha=F_4(2)$, and $G$ contains a graph automorphism. By \\cite[Lemma~8.7]{Cuypers}, $G$ would have rank $5$.\n\\hfill\\qed\n\nNow combining Propositions~\\ref{prop-sporadic}, \\ref{prop-alt} and \\ref{prop-sim-lie}, the proof of Theorem~\\ref{rank-4} is complete.\n\n\\section{Proof of Theorems~\\ref{th:2sethomo} and \\ref{trans-rank-4}}\\label{sec6:proof-main-theorems}\n\n\\f{\\bf Proof of Theorem~\\ref{th:2sethomo}.}\\ Let $\\Ga=(V,E)$ be a $(G,2)$-set-homogeneous graph.\nSuppose that $\\Ga$ is neither complete nor empty.\nIf $G$ is imprimitive on the vertex set $V$, then by Lemma~\\ref{imp},\n$\\Ga\\cong m\\K_b$ or $\\K_{m[b]}$ for some positive integers $m$ and $ b$. Then Theorem~\\ref{th:2sethomo}~(2) holds.\n\nAssume that $G$ is primitive on $V$. By Lemma~\\ref{rank<=5}, $G$ is of rank at most 5.\n\nIf $G$ is of rank $3$, then $\\Ga$ is $(G,2)$-homogeneous and all candidates for $G$ are listed in \\cite{Bannai,Kantor,Liebeck-affine,Liebeck-Saxl}. Then Theorem~\\ref{th:2sethomo}~(3)  holds.\n\nIf $G$ is of rank $4$, then exactly one of the orbitals is self-paired. By Corollary~\\ref{reduction}, $G$ is affine or almost simple. By Theorem~\\ref{affine-rank-4}, $G$ is not affine. If $G$ is almost simple, then by Theorem~\\ref{rank-4}, $G\\cong\\PSU_3(3)$ with stabiliser $\\PSL_3(2)$, and the subdegrees are $1,7,7$ and $21$. Then $\\Ga$ is either the orbital graph of $G$ of order $36$ and of valency $21$, or the complement of this graph. Furthermore, by Magma~\\cite{BCP}, $\\Aut(\\Ga)={\\rm P\\Gamma U}_3(3)$ is $2$-homogeneous on $\\Ga$. This proves part (4) of our theorem.\n\nIf $G$ is of rank 5, then none of the non-trivial orbitals of $G$ is self-paired. From Proposition~\\ref{AGL_1(p^r)} we obtain part (5) of our theorem. \\hfill\\qed\n\n\\medskip\n\\f{\\bf Proof of Theorem~\\ref{trans-rank-4}.}\\ Combining Proposition~\\ref{th-rank-4}, Lemma~\\ref{non-HA-AS}, Theorem~\\ref{affine-rank-4} and Theorem~\\ref{rank-4}, we obtain the proof of this theorem.\\hfill\\qed\n\n\\section{Finite $3$-set-homogeneous graphs}\\label{set:3sethom}\n\nIn this section, we shall classify finite $3$-set-homogeneous graphs, and then prove Theorem~\\ref{th:3sethomo}.\n\n\\subsection{Preliminaries}\\label{subsec:3reduce}\n\\subsubsection{A reduction}\nA finite graph $\\Ga$ is \\emph{strongly regular} with parameters $(v, k, \\lambda, \\mu)$ if it has $v$ vertices and is regular with valency $k$ where $0<k<v-1$, and if any two adjacent vertices have $\\ld=\\ld(\\Ga)$ common neighbours and any two non-adjacent vertices have $\\mu=\\mu(\\Ga)$ common neighbours. Note that a strongly regular graph is neither complete nor edgeless.\n\n\\begin{lem}\\label{lem:prim-3ch-is-srg}\nLet $\\Ga $ be a finite $(G, 3)$-set-homogeneous graph. Then one of the following holds.\n\\begin{enumerate}\n\\item[{\\rm(1)}] $\\Ga \\cong\\K_n$ or $n\\K_1$;\n\n\\item[{\\rm(2)}]  $\\Ga \\cong \\K_{m[b]}$ or $m\\K_b$ where $m,b\\geq 2$;\n\n\\item [{\\rm (3)}]\\ $G$ is primitive on $V(\\Ga)$ of rank $3$, and $\\Ga$ is strongly regular of diameter $2$. Furthermore, if $\\Ga$ is not the pentagon, then either $\\Ga$ or its complement $\\overline{\\Ga}$ has girth $3$.\n\\end{enumerate}\n\\end{lem}\n\n\\proof By \\cite[Theorem~1.1 \\& Corollary~1.2]{Zhou-EJC2021}, $\\Ga$ is $(G,2)$-homogeneous, and then by Theorem~\\ref{th:2sethomo}, we conclude that either parts (1) or (2) hold, or $G$ is primitive on $V(\\Ga)$ of rank $3$.\n\nAssume that neither part (1) nor part (2) happens. Then $\\Ga$ has diameter $2$ and $G$ is arc-transitive on both $\\Ga$ and the complement $\\overline{\\Ga}$ of $\\Ga$. This implies that $\\Ga$ is strongly regular. The first assertion of the part (3) is proved. If $\\Ga$ has valency $2$, then since $\\Ga$ has diameter $2$, it follows that $\\Ga$ is the pentagon. If $\\Ga$ has valency at least $3$, then either $\\Ga$ or its complement $\\overline{\\Ga}$ has girth $3$. \\hfill\\qed\n\nBy Lemma~\\ref{lem:prim-3ch-is-srg}, to prove Theorem~\\ref{th:3sethomo}, it suffices to classify those $(G,3)$-set-homogeneous graphs $\\Ga$ such that $G$ is primitive on $V(\\Ga)$ of rank $3$. The following result can be obtained by the fundamental O'Nan-Scott theorem (see \\cite[\\textsection{1}]{Liebeck-ONan-Scott}).\n\n\\begin{theorem}\\label{th:rank3groups}\nLet $G$ be a primitive group of rank  $3$.\nThen one of the following holds:\n\\begin{itemize}\n\\item[\\rm (1)]  $T\\times T\\lhd G\\leqslant H\\wr\\S_2$, where $H$ is a $2$-transitive permutation group of degree $n$ and $\\soc(H)=T$ is a nonabelian simple group, and $G$ has production action;\n\\item[\\rm (2)]  $G$ is an affine group;\n\\item[\\rm (3)]  $G$ is an almost simple group.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{lemma}\\label{lm:CaseI}\nSuppose that $\\Ga$ is a finite $(G,3)$-set-homogeneous graph such that $G$ satisfies Theorem~{\\rm\\ref{th:rank3groups}~(1)}.\nThen $\\Ga$ is isomorphic to $\\K_n\\square\\K_n$ or its complement $\\K_n\\times\\K_n$, and $\\Ga$ is $3$-homogeneous.\n\\end{lemma}\n\n\\proof Without loss of generality, assume that $H$ is a $2$-transitive permutation group on the set $\\Del =\\{1,2,\\dots, n\\}$. Then $V(\\Ga)=\\Del \\times\\Del $. Take $\\a=(1, 1)\\in V(\\Ga)$. Then the three orbits of $G$ are\n\\[\\Ome_0=\\{\\a\\}, \\Ome_1=\\{(1, i), (i,1)\\mid i\\in\\Delta\\setminus\\{1\\}\\}, \\Ome_2=\\{(i,j)\\mid i, j\\in\\Del \\setminus\\{1\\}\\}.\\]\nSo $\\Ga(\\a)=\\Ome_1$ or $\\Ome_2$. It is easy to see that $\\Ga\\cong\\K_n\\square\\K_n$ or $\\K_n\\times\\K_n$. By \\cite[Corollary~1.2]{Cameron-3-hom}, $\\Ga$ is $3$-homogeneous.\\hfill\\qed\n\nBy Lemmas~\\ref{lem:prim-3ch-is-srg} and \\ref{lm:CaseI}, from now on we will adopt the following hypothesis.\n\n\\begin{hypothesis}\\label{hyp}\nLet $\\Ga=(V, E)$ be a finite $(G,3)$-set-homogeneous graph satisfying the following:\n\\begin{itemize}\n  \\item $G\\leq\\Aut(\\Ga)$ is primitive on $V$ with rank $3$,\n  \\item $\\soc(G)$ is either abelian or nonabelian simple,\n  \\item $\\Ga$ has girth $3$ and diameter $2$.\n\\end{itemize}\n\\end{hypothesis}\n\nWe will deal with the case where $\\soc(G)$ is abelian in Subsection~\\ref{subsec:affine}, and the case where $\\soc(G)$ is nonabelian simple in Subsection~\\ref{subsec:simple}.\n\n\\subsubsection{Some useful lemmas}\n\nThe first two lemmas concern the arc-stabilisers of $(G,3)$-set-homogeneous graphs. Let $s$ be a positive integer and let $\\Sig$ be a graph. For a subgroup $H$ of $\\Aut(\\Sig)$, we say that $\\Sig$ is {\\em $(H, s)$-connected-set-homogeneous} if any two isomorphic connected induced subgraphs of $\\Sigma$ on at most $s$ vertices are equivalent under $H$. Clearly, every $(H,s)$-set-homogeneous graph is $(H,s)$-connected-set-homogeneous.\nFor any vertex $v$ of a graph $\\Sig$ of diameter at least two, let $\\Sig(v)$ be the vertices of $\\Sig$ adjacent to $v$, and let $\\Sig_2(v)$ be the vertices of $\\Sig$ at distance $2$ from $v$.\n\n\\begin{lemma}{\\rm \\cite[Theorem~1.3~(3)-(4)]{Zhou-EJC2021}}\\label{rank}\nLet $\\Sig$ be a finite $(H, 3)$-connected-set-homogeneous non-complete graph of girth $3$ with $H\\leq\\Aut(\\Sig)$. Then for every edge $\\{\\a, \\b\\}$ of $\\Sig$, we have the following:\n\\begin{enumerate}\n  \\item [{\\rm (1)}]\\ $H_{\\a\\b}$ has $s$ orbits on $\\Sig(\\a)\\cap \\Sig(\\b)$ with equal size, where $s=1, 2, 3$ or $6$.\n  \\item [{\\rm (2)}]\\ $H_{\\a\\b}$ has $t$ orbits on $\\Sig(\\a)-(\\Sig(\\a)\\cap \\Sig(\\b))\\cup\\{\\b\\}$ with equal size, where $t=1$ or $ 2$.\n      \\end{enumerate}\n\\end{lemma}\n\n\\begin{lemma}\\label{lm:Gab10orbits}\nUnder Hypothesis~{\\rm \\ref{hyp}}, let $\\{\\alpha,\\beta\\}\\in E$. Then $G_{\\alpha\\beta}$ has at most $12$ orbits on $V(\\mathit{\\Ga})\\setminus\\{\\alpha,\\beta\\}$.\n\\end{lemma}\n\n\\begin{proof}By Hypothesis~{\\rm \\ref{hyp}}, $\\Ga$ is a $(G,3)$-set-homogeneous graph. Furthermore, $\\Ga$ has diameter $2$ and girth $3$. Note that $V(\\mathit{\\Ga})\\setminus\\{\\alpha,\\beta\\}$ is partitioned into the following four subsets:\n\\[\\Ga(\\a)\\cap\\Ga(\\b), \\Ga(\\a)\\cap\\Ga_2(\\b), \\Ga_2(\\a)\\cap\\Ga(\\b), \\Ga_2(\\a)\\cap\\Ga_2(\\b).\\]\nMoreover, $G_{\\a\\b}$ setwise fixes each of the four subsets.\n\nSince every $(G,3)$-set-homogeneous graph is $(G,3)$-connected-set-homogeneous, by Lemma~\\ref{rank} $G_{\\a\\b}$ has at most $6$ orbits on $\\Ga(\\a)\\cap\\Ga(\\b)$, and has at most $2$ orbits on $\\Ga(\\a)\\cap\\Ga_2(\\b)$ and on $\\Ga_2(\\a)\\cap\\Ga(\\b)$.\n\nTo complete the proof, it suffices to prove that $G_{\\a\\b}$ has at most two orbits on $\\Ga_2(\\a)\\cap\\Ga_2(\\b)$. To see this, we may assume that $G_{\\a\\b}$ is intransitive on $\\Ga(\\a)\\cap\\Ga_2(\\b)$. Then there exist $\\d_1, \\d_2\\in\\Ga_2(\\a)\\cap\\Ga_2(\\b)$ such that the orbits $\\d_1^{G_{\\a\\b}}\\neq \\d_2^{G_{\\a\\b}}$. Then $\\d_1,\\d_2$ are not adjacent to $\\a$ and $\\b$. Since $\\Ga$ is $(G,3)$-set-homogeneous, there exists a $g\\in G$ such that $\\d_1^g=\\d_2$, and $\\a^g=\\b$ and $\\b^g=\\a$. For any $\\d\\in \\Ga_2(\\a)\\cap\\Ga_2(\\b)$, also since $\\Ga$ is $(G,3)$-set-homogeneous, there exists $h\\in G_{\\{\\a,\\b\\}}$ such that $\\d^h=\\d_1$. If $h\\in G_{\\a\\b}$, then $\\d\\in \\d_1^{G_{\\a\\b}}$. If $h\\notin G_{\\a\\b}$, then $h$ swaps $\\a$ and $\\b$ and so $hg\\in G_{\\a\\b}$. Furthermore, $\\d^{hg}=\\d_1^g=\\d_2$, and hence $\\d\\in \\d_2^{G_{\\a\\b}}$. Thus, $\\Ga_2(\\a)\\cap\\Ga_2(\\b)=\\d_1^{G_{\\a\\b}}\\cup \\d_2^{G_{\\a\\b}}$.\nThe proof is completed.\n\\hfill\\qed\n\\end{proof}\n\nThe next lemma gives some basic properties about strongly regular graphs, see \\cite[\\textsection{1.1.1-1.1.2}]{BrouwerSRG}.\n\n\\begin{lem}\\label{lem:drg-basic}\nLet $\\Ga$ be a strongly regular graph with parameters $(v, k, \\lambda, \\mu)$. Then the following hold:\n\\begin{enumerate}\n  \\item [{\\rm (1)}]\\ The complement $\\overline{\\Ga}$ of $\\Ga$ is strongly regular with parameters $(v, \\overline{k}, \\overline{\\ld}, \\overline{\\mu})$, where $\\overline{k}=v-k-1$, $\\overline{\\ld}=v-2k+\\mu-2$ and $\\overline{\\mu}=v-2k+\\lambda$;\n  \\item [{\\rm (2)}]\\ $0\\leq \\ld\\leq k-1$ and $0\\leq\\mu\\leq k$;\n  \\item [{\\rm (3)}]\\ $k(k-1-\\lambda)=\\mu(v-1-k)$.\n\\end{enumerate}\n\\end{lem}\n\nLet $\\mathbf{P}_{3}$ be a path of length $2$. For a graph $\\Ga$, let\n\\[\n\\begin{array}{l}\n\\Ome_1(\\Ga):=\\{U\\subseteq V(\\Ga) \\mid [U]\\cong \\K_1\\cup\\K_2 \\},\\\\\n\\Ome_2(\\Ga):=\\{U\\subseteq V(\\Ga) \\mid [U]\\cong \\mathbf{P}_{3}\\},\\\\\n\\Ome_3(\\Ga):=\\{U\\subseteq V(\\Ga) \\mid [U]\\cong \\K_3\\},\\\\\n\\Ome_4(\\Ga):=\\{U\\subseteq V(\\Ga) \\mid [U]\\cong 3\\K_1\\}.\\\\\n\\end{array}", "set:proof-of-main": "\\label{set:proof-of-main}\n\nBy definition, both 1-homogeneous graphs and 1-set-homogeneous graphs are exactly the vertex-transitive graphs. Assume now that $k\\geq 2$. It is obvious by definition that e", "th-main": "\\begin{theorem}\\label{th-main}\nFor each positive integer $k$, a finite graph is $k$-set-homogeneous if and only if it is $k$-homogeneous.\nMoreover, { if $\\Ga$ is a $k$-homogeneous graph with $k\\geq 2$, then} one of the following holds:\n\\begin{itemize}\n\\item [{\\rm(1)}] $k\\geqslant 5$, and $\\Ga$ is one of { $\\C_5 (\\cong\\overline{\\C_5})$, $\\K_3\\square\\K_3 (\\cong \\overline{\\K_3\\square\\K_3})$, $n\\K_m$, and $\\K_{n[m]}$; furthermore, $\\Ga$ is homogeneous.}\n\n{ \\item[(2)] $\\Ga$ is either the Schl\\\"{a}fli graph on $27$ vertices or its complement (the two non-trivial orbital graphs of $\\mathrm{PSU}_4(2)$ acting on the isotropic lines); furthermore, $\\Ga$ is $4$-homogeneous but not $5$-homogeneous.\n\n\\item[(3)] $\\Ga$ is either $\\K_n\\square\\K_n$ with $n\\geqslant4$, or a graph from Theorem~\\ref{th:3sethomo}(1)--(4), or the complement of any such graph; furthermore, $\\Ga$ is $3$-homogeneous but not $4$-homogeneous.\n\n\\item[(4)] $\\Ga$ is an orbital graph of a primitive permutation group of rank $3$ which is not covered by parts (1)--(3); furthermore, $\\Ga$ is $2$-homogeneous but not $3$-homogeneous. }\n\n\\end{itemize}\n\\end{theorem}", "th:3sethomo": "\\begin{theorem}\\label{th:3sethomo}\nLet $\\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\\Ga$ is isomorphic either to one of the graphs: $\\C_5$, $\\K_n$, $\\K_{m[b]}$, $\\K_n\\square\\K_n$, or to one of the following graphs:\n\\begin{itemize}\n\\item[{\\rm(1)}] the affine polar graph $\\mathrm{VO}_{2m}^{\\epsilon}(2)$ with $m\\geq 1$ and $\\epsilon \\in \\{+,-\\}$;\n\n\\item[{\\rm(2)}]  the elliptic orthogonal graph $\\mathrm{\\Gamma}(\\mathrm{O}^-_6(q))$ with $q$ a prime power;\n\n\\item[{\\rm(3)}] an orbital graph of the Higman-Sims group $\\mathrm{HS}$ acting on $100$ points;\n\n\\item[{\\rm(4)}] an orbital graph of the McLaughlin group $\\mathrm{McL}$ acting on $275$ points.\n\\end{itemize}\nIn particular, all $3$-set-homogeneous graphs are $3$-homogeneous.\n\\end{theorem}", "trans-rank-4": "\\begin{theorem}\\label{trans-rank-4}\nLet $G$ be a primitive permutation group of rank $4$. Then either all orbitals of $G$ are self-paired, or $G=\\PSU_3(3)$ acting on $36$ points.\n\\end{theorem}", "subsec:2reduce": "\\begin{itemize}\n\\item[{\\rm(1)}] the affine polar graph $\\mathrm{VO}_{2m}^{\\epsilon}(2)$ with $m\\geq 1$ and $\\epsilon \\in \\{+,-\\}$;\n\n\\item[{\\rm(2)}]  the elliptic orthogonal graph $\\mathrm{\\Gamma}(\\mathrm{O}^-_6(q))$ with $q$ a prime power;\n\n\\item[{\\rm(3)}] an orbital graph of the Higman-Sims group $\\mathrm{HS}$ acting on $100$ points;\n\n\\item[{\\rm(4)}] an orbital graph of the McLaughlin group $\\mathrm{McL}$ acting on $275$ points.\n\\end{itemize}\nIn particular, all $3$-set-homogeneous graphs are $3$-homogeneous.\n\\end{theorem}\n\nNote that all graphs in Theorem~\\ref{th:3sethomo}~(1)--(4) are actually $3$-homogeneous graphs, which {are known} in the literature.\n\nThe rest of the paper is organised as follows. In Sections~\\ref{subsec:2reduce}--\\ref{sec6:proof-main-theorems}, we deal with finite $(G, 2)$-set-homogeneous graphs.\nIn Section~\\ref{sec6:proof-main-theorems}, we prove Theorem~\\ref{th:2sethomo} and Theorem~\\ref{trans-rank-4}.\nIn Section~\\ref{set:3sethom}, we deal with finite $(G, 3)$-set-homogeneous graphs, and prove Theorem~\\ref{th:3sethomo}.\nIn Section~\\ref{set:proof-of-main}, we prove Theorem~\\ref{th-main}.\n\n\\section{A reduction}\\label{subsec:2reduce}\nIn this section, we shall reduce the proof of Theorem~\\ref{th:2sethomo} to two cases: $G$ is a primitive permutation group of rank $5$ { and odd order}, and $G$ is a primitive permutation group of rank $4$ which has exactly one self-paired orbital. The former case will be treated in Section~\\ref{subset:prime-rank5}, and the latter case will be treated in Sections~\\ref{subsec:affineprimitive-rank4} and \\ref{subsec:almostsimple-rank4}. Based on these, Theorem~\\ref{th:2sethomo} and Theorem~\\ref{trans-rank-4} will be proved in Section~\\ref{sec6:proof-main-theorems}.\n\nWe first introduce several terms of permutation groups.\nLet $\\Ome$ be a finite nonempty set. Denote by $\\Sym(\\Ome)$ the symmetric group on $\\Ome$. Let $G\\leqslant\\Sym(\\Ome)$ be transitive.\nLet $\\Ome\\times\\Ome=\\{(u,v)\\mid u,v\\in \\Ome\\}$.\nAn orbit $(u,v)^G$ of $G$ on $\\Ome\\times\\Ome$ is called an {\\it orbital} of $G$ on $\\Ome$, and\n$(v,u)^G$ is called the {\\it paired orbital} of $(u,v)^G$.\nAn orbital  $(u,v)^G$ is called {\\it self-paired} if it equals $(v,u)^G$.\nThe orbital $(u,u)^G$ is called {\\it trivial}, and the others are non-trivial. To an orbital $E$ we\nassociate the digraph with vertex set $\\Ome$ and arc set $E$, called the {\\em orbital digraph} for\n$E$, denoted by the pair $(\\Ome, E)$. For a given point $u$, each orbital $E=(u,v)^G$ corresponds to an orbit of $G_u$ on $\\Ome$, namely, $\\{v\\ |\\ (u,v)\\in E\\}$. The orbits of $G_u$ on $\\Om$ are called {\\em suborbits} of $G$, and their lengths are called the {\\em subdegrees} of $G$.\nFor a set $\\Ome$, let $\\Ome^{\\{2\\}}$ be the set of 2-subsets of $\\Ome$, namely, $\\Ome^{\\{2\\}}=\\{\\{u,v\\}\\mid u\\not=v\\in \\Ome\\}$,  and let $\\Ome^{(2)}=\\{(u,v)\\mid u\\not=v\\in \\Ome\\}$.\n\n\\begin{lemma}\\label{rank<=5}\nLet $\\Ga=(V,E)$ be a $(G, 2)$-set-homogeneous graph, where $G\\leq\\Aut(\\Ga)$.\nThen $G$ is a permutation group on $V$  of rank at most $5$, and\none of the following holds:\n\\begin{itemize}\n\\item[{\\rm(1)}] $G$ is of rank $2$, and $\\Ga$ is a complete graph or an empty graph.\n\n\\item[{\\rm(2)}] $G$ is of rank $3$, and {either $\\Ga$ is a complete graph or an empty graph, or $\\Ga$ is $(G, 2)$-homogeneous}.\n\n\\item[{\\rm(3)}] $G$ is of rank $4$, and exactly one of the $3$ non-trivial orbitals is self-paired.\n\n\\item[{\\rm(4)}] $G$ is of rank $5$, and none of the $4$ non-trivial orbitals is self-paired.\n\n\\end{itemize}", "Def-homog": "\\begin{definition}\\label{Def-homog}\n{\\rm\nLet $\\Ga=(V,E)$ be a graph, $G\\leqslant\\Aut(\\Ga)$, and  $k$ a positive integer.\n\\begin{itemize}\n\\item[(1)] If each isomorphism between any pair of isomorphic induced subgraphs of $\\Ga$ of order at most $k$\nextends to an automorphism of $\\Ga$ { contained} in $G$, then $\\Ga$ is called a {\\it $(G, k)$-homogeneous} graph.\n(The {\\it order} of a graph is the number of vertices.)\n\n\\item[(2)] If any two isomorphic induced subgraphs of $\\Ga$ of order at most $k$ are equivalent under $G$, then $\\Ga$ is called a {\\it $(G,k)$-set-homogeneous} graph.\n\n\\item[(3)] {When $G=\\Aut(\\Ga)$, a $(G, k)$-homogeneous or $(G, k)$-set-homogeneous graph is simply called {\\em $k$-homogeneous} or {\\em $k$-set-homogeneous}, respectively.}\n\\end{itemize}\n}\n\\end{definition}", "rank-4": "\\begin{theorem}\\label{rank-4}\nLet $G$ be an almost simple primitive rank $4$ permutation group on a set $\\Om$ with a unique non-self-paired orbital.\nThen $G=\\PSU_3(3)$ with stabiliser $\\PSL_3(2)$, and the subdegrees are $1$, $7$, $7$ and $21$. Moreover, $\\mathbf{N}_{\\mathrm{Sym}(\\mathit{\\Omega})}(G)=\\mathrm{P\\Gamma U}_3(3)$ is a primitive rank $3$ group with subdegrees $1$, $14$ and $21$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 5005, "pre_theorem_intro_text": "The main purpose of this paper is to solve the long-standing open problem:\n\n\\vskip0.1in\n\\f{\\bf Problem A.} {\\rm {Classify the finite graphs in which, for a given positive integer $k$, any two isomorphic induced subgraphs of order at most $k$ are equivalent under the automorphism group.}}\n\\vskip0.1in\n\nBefore proceeding, we give some background on this topic.\nThe strongest possible symmetry condition that one can impose {on a graph is {\\it homogeneity}.}\nA graph is called {\\em homogeneous} if every isomorphism between any pair of isomorphic induced subgraphs extends to an automorphism of the graph. The study of graphs with {certain homogeneity-type properties} was initiated by Sheehan~{\\cite{Sheehan1972,Sheehan1973,Sheehan1973a,Sheehan1974}},\nand Gardiner~\\cite{Gardiner1976} in the 1970s.\n{It turns out that} the only finite homogeneous graphs are the following `trivial graphs':\n\\[\\C_5,\\ \\K_3\\square\\K_3,\\ n\\K_m,\\ \\K_{n[m]}.\\]\nA graph $\\Ga$ is called {\\em set-homogeneous} if isomorphic subgraphs of $\\Ga$ are equivalent under $\\Aut(\\Ga)$.\n{In 1978, Ronse~\\cite{Ronse}} proved that every finite set-homogeneous graph is homogeneous.\n{Slightly weaker versions of homogeneity have been studied in the literature} (see Definition~\\ref{Def-homog} for instance) and also {have been extended} to other mathematical structures. Over the past 50 years, { a fair amount of work has been done on objects with certain homogeneity-type properties. The reader may refer to the excellent survey paper of Macpherson~\\cite{Macpherson-survey} for an overview.}\n\n{One of these weaker conditions is as follows.}\n\n\\begin{definition}\\label{Def-homog}\n{\\rm\nLet $\\Ga=(V,E)$ be a graph, $G\\leqslant\\Aut(\\Ga)$, and  $k$ a positive integer.\n\\begin{itemize}\n\\item[(1)] If each isomorphism between any pair of isomorphic induced subgraphs of $\\Ga$ of order at most $k$\nextends to an automorphism of $\\Ga$ { contained} in $G$, then $\\Ga$ is called a {\\it $(G, k)$-homogeneous} graph.\n(The {\\it order} of a graph is the number of vertices.)\n\n\\item[(2)] If any two isomorphic induced subgraphs of $\\Ga$ of order at most $k$ are equivalent under $G$, then $\\Ga$ is called a {\\it $(G,k)$-set-homogeneous} graph.\n\n\\item[(3)] {When $G=\\Aut(\\Ga)$, a $(G, k)$-homogeneous or $(G, k)$-set-homogeneous graph is simply called {\\em $k$-homogeneous} or {\\em $k$-set-homogeneous}, respectively.}\n\\end{itemize}\n}\n\\end{definition}\n\nA major theme in the area has been the following classification problem.\n\n\\vskip0.1in\n\\f{\\bf Problem B.}\n{\\rm Classify $k$-homogeneous graphs and $k$-set-homogeneous graphs.}\n\\vskip0.1in\n\nThis paper completely solves the problem, and particularly shows that a finite graph is $k$-homogeneous if and only if it is $k$-set-homogeneous.\n\n{The study of Problem~B has lasted for over half a century, dating back to Sheehan and Gardiner's classification of finite homogeneous graphs in the 1970s mentioned above.} Later, Lachlan and Woodrow~\\cite{Lachlan-Woodrow} obtained a classification of infinite countable homogeneous graphs. For more results regarding homogeneous graphs, we refer the reader to \\cite{Enomoto} and \\cite{Gray,Gray-Macpherson,GMPR,HH,Henson}.\n\nBy definition, $1$-homogeneous graphs and $1$-set-homogeneous graphs are exactly vertex-transitive graphs.\nIt is infeasible to obtain { an explicit list of vertex-transitive graphs.}\nHowever, { an explicit classification has been achieved for}\nfinite $k$-homogeneous graphs with $k\\geqslant 2$ in a collection of papers. Cameron~\\cite{Cameron-6-transitive} proved that the list of finite $5$-homogeneous graphs is the same as that for homogeneous graphs;\nBuczak~\\cite{Buczak} classified finite $4$-homogeneous graphs; and Cameron and Macpherson~\\cite{Cameron-3-hom} classified finite $3$-homogeneous graphs.\n{We observe} that a graph $\\Ga$ is $2$-homogeneous if and only if $\\Ga$ and its complement $\\ov\\Ga$ are both arc-transitive. {Thus, a finite $2$-homogeneous graph is a complete multipartite graph or its complement, or an orbital graph of a primitive permutation group of rank $3$.\nNote that primitive permutation groups of rank $3$ have been classified in \\cite{Bannai,Kantor,Liebeck-affine,Liebeck-Saxl}.}\n\nThe main purpose of this paper is to solve Problem~B by proving that every $k$-set-homogeneous graph is $k$-homogeneous for $k\\geqslant 2$, { so that $k$-set-homogeneous graphs are explicitly classified, as described in the previous paragraph.}\nTo state {our main theorem}, we fix some graph notation. Let $m$ and $n$ be positive integers. We denote the complete graph with $n$ vertices by $\\K_n$, the cycle with $n$ vertices by $\\C_n$, the complete multipartite graph with $n$ parts of\nsize $m$ by $\\K_{n[m]}$, and its complement (the disjoint union of $n$ copies of $\\K_m$) by $n\\K_m$. Let $\\K_n\\square\\K_n$ and $\\K_n\\times\\K_n$ be the Cartesian product and direct product of two $\\K_n$'s, respectively. For a graph $\\Ga$, we denote by $\\ov\\Ga$ the {\\it complement} of $\\Ga$. Note that $\\overline{\\K_n\\square\\K_n}=\\K_n\\times\\K_n$.", "context": "\\vskip0.1in\n\\f{\\bf Problem A.} {\\rm {Classify the finite graphs in which, for a given positive integer $k$, any two isomorphic induced subgraphs of order at most $k$ are equivalent under the automorphism group.}}\n\\vskip0.1in\n\nBefore proceeding, we give some background on this topic.\nThe strongest possible symmetry condition that one can impose {on a graph is {\\it homogeneity}.}\nA graph is called {\\em homogeneous} if every isomorphism between any pair of isomorphic induced subgraphs extends to an automorphism of the graph. The study of graphs with {certain homogeneity-type properties} was initiated by Sheehan~{\\cite{Sheehan1972,Sheehan1973,Sheehan1973a,Sheehan1974}},\nand Gardiner~\\cite{Gardiner1976} in the 1970s.\n{It turns out that} the only finite homogeneous graphs are the following `trivial graphs':\n\\[\\C_5,\\ \\K_3\\square\\K_3,\\ n\\K_m,\\ \\K_{n[m]}.\\]\nA graph $\\Ga$ is called {\\em set-homogeneous} if isomorphic subgraphs of $\\Ga$ are equivalent under $\\Aut(\\Ga)$.\n{In 1978, Ronse~\\cite{Ronse}} proved that every finite set-homogeneous graph is homogeneous.\n{Slightly weaker versions of homogeneity have been studied in the literature} (see Definition~\\ref{Def-homog} for instance) and also {have been extended} to other mathematical structures. Over the past 50 years, { a fair amount of work has been done on objects with certain homogeneity-type properties. The reader may refer to the excellent survey paper of Macpherson~\\cite{Macpherson-survey} for an overview.}\n\n\\begin{definition}\\label{Def-homog}\n{\\rm\nLet $\\Ga=(V,E)$ be a graph, $G\\leqslant\\Aut(\\Ga)$, and $k$ a positive integer.\n\\begin{itemize}\n\\item[(1)] If each isomorphism between any pair of isomorphic induced subgraphs of $\\Ga$ of order at most $k$\nextends to an automorphism of $\\Ga$ { contained} in $G$, then $\\Ga$ is called a {\\it $(G, k)$-homogeneous} graph.\n(The {\\it order} of a graph is the number of vertices.)\n\n\\item[(3)] {When $G=\\Aut(\\Ga)$, a $(G, k)$-homogeneous or $(G, k)$-set-homogeneous graph is simply called {\\em $k$-homogeneous} or {\\em $k$-set-homogeneous}, respectively.}\n\\end{itemize}\n}\n\\end{definition}\n\nBy definition, $1$-homogeneous graphs and $1$-set-homogeneous graphs are exactly vertex-transitive graphs.\nIt is infeasible to obtain { an explicit list of vertex-transitive graphs.}\nHowever, { an explicit classification has been achieved for}\nfinite $k$-homogeneous graphs with $k\\geqslant 2$ in a collection of papers. Cameron~\\cite{Cameron-6-transitive} proved that the list of finite $5$-homogeneous graphs is the same as that for homogeneous graphs;\nBuczak~\\cite{Buczak} classified finite $4$-homogeneous graphs; and Cameron and Macpherson~\\cite{Cameron-3-hom} classified finite $3$-homogeneous graphs.\n{We observe} that a graph $\\Ga$ is $2$-homogeneous if and only if $\\Ga$ and its complement $\\ov\\Ga$ are both arc-transitive. {Thus, a finite $2$-homogeneous graph is a complete multipartite graph or its complement, or an orbital graph of a primitive permutation group of rank $3$.\nNote that primitive permutation groups of rank $3$ have been classified in \\cite{Bannai,Kantor,Liebeck-affine,Liebeck-Saxl}.}\n\nThe main purpose of this paper is to solve Problem~B by proving that every $k$-set-homogeneous graph is $k$-homogeneous for $k\\geqslant 2$, { so that $k$-set-homogeneous graphs are explicitly classified, as described in the previous paragraph.}\nTo state {our main theorem}, we fix some graph notation. Let $m$ and $n$ be positive integers. We denote the complete graph with $n$ vertices by $\\K_n$, the cycle with $n$ vertices by $\\C_n$, the complete multipartite graph with $n$ parts of\nsize $m$ by $\\K_{n[m]}$, and its complement (the disjoint union of $n$ copies of $\\K_m$) by $n\\K_m$. Let $\\K_n\\square\\K_n$ and $\\K_n\\times\\K_n$ be the Cartesian product and direct product of two $\\K_n$'s, respectively. For a graph $\\Ga$, we denote by $\\ov\\Ga$ the {\\it complement} of $\\Ga$. Note that $\\overline{\\K_n\\square\\K_n}=\\K_n\\times\\K_n$.\n\n\\begin{definition}\\label{Def-homog}\n{\\rm\nLet $\\Ga=(V,E)$ be a graph, $G\\leqslant\\Aut(\\Ga)$, and  $k$ a positive integer.\n\\begin{itemize}\n\\item[(1)] If each isomorphism between any pair of isomorphic induced subgraphs of $\\Ga$ of order at most $k$\nextends to an automorphism of $\\Ga$ { contained} in $G$, then $\\Ga$ is called a {\\it $(G, k)$-homogeneous} graph.\n(The {\\it order} of a graph is the number of vertices.)\n\n\\item[(2)] If any two isomorphic induced subgraphs of $\\Ga$ of order at most $k$ are equivalent under $G$, then $\\Ga$ is called a {\\it $(G,k)$-set-homogeneous} graph.\n\n\\item[(3)] {When $G=\\Aut(\\Ga)$, a $(G, k)$-homogeneous or $(G, k)$-set-homogeneous graph is simply called {\\em $k$-homogeneous} or {\\em $k$-set-homogeneous}, respectively.}\n\\end{itemize}\n}\n\\end{definition}\n\n\\begin{theorem}\\label{th:3sethomo}\nLet $\\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\\Ga$ is isomorphic either to one of the graphs: $\\C_5$, $\\K_n$, $\\K_{m[b]}$, $\\K_n\\square\\K_n$, or to one of the following graphs:\n\\begin{itemize}\n\\item[{\\rm(1)}] the affine polar graph $\\mathrm{VO}_{2m}^{\\epsilon}(2)$ with $m\\geq 1$ and $\\epsilon \\in \\{+,-\\}$;\n\n\\item[{\\rm(2)}]  the elliptic orthogonal graph $\\mathrm{\\Gamma}(\\mathrm{O}^-_6(q))$ with $q$ a prime power;\n\n\\item[{\\rm(3)}] an orbital graph of the Higman-Sims group $\\mathrm{HS}$ acting on $100$ points;\n\n\\item[{\\rm(4)}] an orbital graph of the McLaughlin group $\\mathrm{McL}$ acting on $275$ points.\n\\end{itemize}\nIn particular, all $3$-set-homogeneous graphs are $3$-homogeneous.\n\\end{theorem}", "full_context": "\\vskip0.1in\n\\f{\\bf Problem A.} {\\rm {Classify the finite graphs in which, for a given positive integer $k$, any two isomorphic induced subgraphs of order at most $k$ are equivalent under the automorphism group.}}\n\\vskip0.1in\n\nBefore proceeding, we give some background on this topic.\nThe strongest possible symmetry condition that one can impose {on a graph is {\\it homogeneity}.}\nA graph is called {\\em homogeneous} if every isomorphism between any pair of isomorphic induced subgraphs extends to an automorphism of the graph. The study of graphs with {certain homogeneity-type properties} was initiated by Sheehan~{\\cite{Sheehan1972,Sheehan1973,Sheehan1973a,Sheehan1974}},\nand Gardiner~\\cite{Gardiner1976} in the 1970s.\n{It turns out that} the only finite homogeneous graphs are the following `trivial graphs':\n\\[\\C_5,\\ \\K_3\\square\\K_3,\\ n\\K_m,\\ \\K_{n[m]}.\\]\nA graph $\\Ga$ is called {\\em set-homogeneous} if isomorphic subgraphs of $\\Ga$ are equivalent under $\\Aut(\\Ga)$.\n{In 1978, Ronse~\\cite{Ronse}} proved that every finite set-homogeneous graph is homogeneous.\n{Slightly weaker versions of homogeneity have been studied in the literature} (see Definition~\\ref{Def-homog} for instance) and also {have been extended} to other mathematical structures. Over the past 50 years, { a fair amount of work has been done on objects with certain homogeneity-type properties. The reader may refer to the excellent survey paper of Macpherson~\\cite{Macpherson-survey} for an overview.}\n\n\\begin{definition}\\label{Def-homog}\n{\\rm\nLet $\\Ga=(V,E)$ be a graph, $G\\leqslant\\Aut(\\Ga)$, and $k$ a positive integer.\n\\begin{itemize}\n\\item[(1)] If each isomorphism between any pair of isomorphic induced subgraphs of $\\Ga$ of order at most $k$\nextends to an automorphism of $\\Ga$ { contained} in $G$, then $\\Ga$ is called a {\\it $(G, k)$-homogeneous} graph.\n(The {\\it order} of a graph is the number of vertices.)\n\n\\item[(3)] {When $G=\\Aut(\\Ga)$, a $(G, k)$-homogeneous or $(G, k)$-set-homogeneous graph is simply called {\\em $k$-homogeneous} or {\\em $k$-set-homogeneous}, respectively.}\n\\end{itemize}\n}\n\\end{definition}\n\nBy definition, $1$-homogeneous graphs and $1$-set-homogeneous graphs are exactly vertex-transitive graphs.\nIt is infeasible to obtain { an explicit list of vertex-transitive graphs.}\nHowever, { an explicit classification has been achieved for}\nfinite $k$-homogeneous graphs with $k\\geqslant 2$ in a collection of papers. Cameron~\\cite{Cameron-6-transitive} proved that the list of finite $5$-homogeneous graphs is the same as that for homogeneous graphs;\nBuczak~\\cite{Buczak} classified finite $4$-homogeneous graphs; and Cameron and Macpherson~\\cite{Cameron-3-hom} classified finite $3$-homogeneous graphs.\n{We observe} that a graph $\\Ga$ is $2$-homogeneous if and only if $\\Ga$ and its complement $\\ov\\Ga$ are both arc-transitive. {Thus, a finite $2$-homogeneous graph is a complete multipartite graph or its complement, or an orbital graph of a primitive permutation group of rank $3$.\nNote that primitive permutation groups of rank $3$ have been classified in \\cite{Bannai,Kantor,Liebeck-affine,Liebeck-Saxl}.}\n\nThe main purpose of this paper is to solve Problem~B by proving that every $k$-set-homogeneous graph is $k$-homogeneous for $k\\geqslant 2$, { so that $k$-set-homogeneous graphs are explicitly classified, as described in the previous paragraph.}\nTo state {our main theorem}, we fix some graph notation. Let $m$ and $n$ be positive integers. We denote the complete graph with $n$ vertices by $\\K_n$, the cycle with $n$ vertices by $\\C_n$, the complete multipartite graph with $n$ parts of\nsize $m$ by $\\K_{n[m]}$, and its complement (the disjoint union of $n$ copies of $\\K_m$) by $n\\K_m$. Let $\\K_n\\square\\K_n$ and $\\K_n\\times\\K_n$ be the Cartesian product and direct product of two $\\K_n$'s, respectively. For a graph $\\Ga$, we denote by $\\ov\\Ga$ the {\\it complement} of $\\Ga$. Note that $\\overline{\\K_n\\square\\K_n}=\\K_n\\times\\K_n$.\n\n\\begin{definition}\\label{Def-homog}\n{\\rm\nLet $\\Ga=(V,E)$ be a graph, $G\\leqslant\\Aut(\\Ga)$, and  $k$ a positive integer.\n\\begin{itemize}\n\\item[(1)] If each isomorphism between any pair of isomorphic induced subgraphs of $\\Ga$ of order at most $k$\nextends to an automorphism of $\\Ga$ { contained} in $G$, then $\\Ga$ is called a {\\it $(G, k)$-homogeneous} graph.\n(The {\\it order} of a graph is the number of vertices.)\n\n\\item[(2)] If any two isomorphic induced subgraphs of $\\Ga$ of order at most $k$ are equivalent under $G$, then $\\Ga$ is called a {\\it $(G,k)$-set-homogeneous} graph.\n\n\\item[(3)] {When $G=\\Aut(\\Ga)$, a $(G, k)$-homogeneous or $(G, k)$-set-homogeneous graph is simply called {\\em $k$-homogeneous} or {\\em $k$-set-homogeneous}, respectively.}\n\\end{itemize}\n}\n\\end{definition}\n\n\\begin{theorem}\\label{th:3sethomo}\nLet $\\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\\Ga$ is isomorphic either to one of the graphs: $\\C_5$, $\\K_n$, $\\K_{m[b]}$, $\\K_n\\square\\K_n$, or to one of the following graphs:\n\\begin{itemize}\n\\item[{\\rm(1)}] the affine polar graph $\\mathrm{VO}_{2m}^{\\epsilon}(2)$ with $m\\geq 1$ and $\\epsilon \\in \\{+,-\\}$;\n\n\\item[{\\rm(2)}]  the elliptic orthogonal graph $\\mathrm{\\Gamma}(\\mathrm{O}^-_6(q))$ with $q$ a prime power;\n\n\\item[{\\rm(3)}] an orbital graph of the Higman-Sims group $\\mathrm{HS}$ acting on $100$ points;\n\n\\item[{\\rm(4)}] an orbital graph of the McLaughlin group $\\mathrm{McL}$ acting on $275$ points.\n\\end{itemize}\nIn particular, all $3$-set-homogeneous graphs are $3$-homogeneous.\n\\end{theorem}\n\n\\begin{abstract}\nA classification is given of finite $k$-set-homogeneous graphs for $k\\geqslant 2$, leading to a striking result that each finite $k$-set-homogeneous graph is $k$-homogeneous. It shows that $3$-set-homogeneous graphs are rare, consisting of the following graphs and their complements: $\\C_5$, $\\K_n\\square\\K_n$, $n\\K_m$, the Schl\\\"{a}fli graph of order 27, the Higman-Sims graph, the MaLaughlin graph, {affine polar graphs, and elliptic orthogonal graphs}. As an ingredient for the proof, it is shown that all orbitals in a primitive permutation group of rank $4$ are self-paired, except for $\\PSU_3(3)$ acting on 36 points.\n\\bigskip\n\nThe main purpose of this paper is to solve Problem~B by proving that every $k$-set-homogeneous graph is $k$-homogeneous for $k\\geqslant 2$, { so that $k$-set-homogeneous graphs are explicitly classified, as described in the previous paragraph.}\nTo state {our main theorem}, we fix some graph notation. Let $m$ and $n$ be positive integers. We denote the complete graph with $n$ vertices by $\\K_n$, the cycle with $n$ vertices by $\\C_n$, the complete multipartite graph with $n$ parts of\nsize $m$ by $\\K_{n[m]}$, and its complement (the disjoint union of $n$ copies of $\\K_m$) by $n\\K_m$. Let $\\K_n\\square\\K_n$ and $\\K_n\\times\\K_n$ be the Cartesian product and direct product of two $\\K_n$'s, respectively. For a graph $\\Ga$, we denote by $\\ov\\Ga$ the {\\it complement} of $\\Ga$. Note that $\\overline{\\K_n\\square\\K_n}=\\K_n\\times\\K_n$.\n\n{\\noindent{\\bf Remark on Theorem~\\ref{th-main}.}\\ For $k\\geq2$, all $k$-homogeneous graphs have already been classified in the literature, { as mentioned above}, and so all graphs in Theorem~\\ref{th-main}~(1)--(4) are known.}\\medskip\n\n\\begin{theorem}\\label{th:2sethomo}\nLet $\\Ga=(V,E)$ be a finite $(G,2)$-set-homogeneous graph, where $G\\leq\\Aut(\\Ga)$.\nThen one of the following holds:\n\\begin{itemize}\n\\item[{\\rm(1)}] $\\Ga\\cong\\K_n$ or $n\\K_1$, and $G$ is a $2$-homogeneous permutation group on $V$.\n\n\\begin{theorem}\\label{th:3sethomo}\nLet $\\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\\Ga$ is isomorphic either to one of the graphs: $\\C_5$, $\\K_n$, $\\K_{m[b]}$, $\\K_n\\square\\K_n$, or to one of the following graphs:\n\\begin{itemize}\n\\item[{\\rm(1)}] the affine polar graph $\\mathrm{VO}_{2m}^{\\epsilon}(2)$ with $m\\geq 1$ and $\\epsilon \\in \\{+,-\\}$;\n\n\\begin{theorem}[\\cite{Liebeck-affine}]\\label{subdegree}\nLet $X$ be a finite primitive affine permutation group of rank $3$ and of degree $p^d$, with socle $V$, where $V\\cong\\mz_p^d$ for some prime $p$, and let $X_0$ be the stabiliser of the zero vector in $V$. Then $X_0$ belongs to one of the following classes $($and, conversely, each of the possibilities listed below does give rise to a rank $3$ affine group$)$.\n\\begin{itemize}\n\\item[\\rm (A)] Infinite classes. These are\n\\begin{itemize}\n\\item[\\rm (1)] $X_0\\leq \\mathrm{\\Gamma L}_1(p^d)$: all possibilities for $X_0$ are determined in~\\cite[\\textsection{3}]{Foulser-Kallaher-solvable-rank-3};\n\\item[\\rm (2)] $X_0$ imprimitive: $X_0$ stabilises a pair $\\{V_1,V_2 \\}$ of subspaces of $V$ where $V=V_1 \\oplus V_2$ and $\\dim(V_1)=\\dim(V_2)$; moreover, $(X_0)_{V_i}$ is transitive on $V_i\\setminus 0$ for $i=1$, $2$ (and hence $ X_0$ is determined by Hering's Theorem~\\cite{Hering});\n\\item[\\rm (3)] tensor product case: for some $a$, $q$ with $q^a= p^d$, consider $V$ as a vector space $\\GF(q)^a$ of dimension $a$ over $\\GF(q)$; then $X_0$ stabilises a decomposition of $\\GF(q)^a$ as a tensor product $V_1 \\otimes V_2$ where $\\dim_{\\GF(q)}(V_1)=2$; moreover, $(X_0)^{V_2}\\vartriangleright \\SL(V_2)$, or $(X_0)^{V_2}=\\A_7<\\SL_4(2)$ $($and $p=q=2$, $d=a=8)$, or $\\dim_{\\GF(q)}(V_2)\\leq 3$;\n\\item[\\rm (4)] $X_0\\vartriangleright \\SL_a(q) $ and $p^d=q^{2a}$;\n\\item[\\rm (5)] $X_0\\vartriangleright \\SL_2(q) $ and $p^d=q^{6}$;\n\\item[\\rm (6)] $X_0\\vartriangleright \\mathrm{SU}_a(q) $ and $p^d=q^{2a}$;\n\\item[\\rm (7)] $X_0\\vartriangleright \\Omega^{\\pm}_{2a}(q)$ $($and if $q$ is odd, $X_0$ contains an automorphism interchanging the two orbits of $\\Omega^{\\pm}_{2a}(q)$ on nonsingular $1$-spaces$)$;\n\\item[\\rm (8)] $X_0\\vartriangleright \\SL_5(q) $ and $p^d=q^{10}$ $($from the action of $\\SL_5(q)$ on $\\bigwedge^2( \\GF(q) ^5)$, the exterior square of $\\GF(q)^5)$;\n\n\\begin{lemma}\\label{lem:class:b-c}\nIf $G$ satisfies Theorem~{\\rm\\ref{subdegree}} {\\rm(B)--(C)}, then one of the following holds:\n\\begin{enumerate} [\\rm (1)]\n\\item $\\Ga\\cong \\mathrm{VO}^{-}_6(2)$ or $\\overline{\\mathrm{VO}^{-}_6(2)}$, arising from the case where $p^d=2^6$ and $G_0\\leq \\N_{\\GL_6(2)}(3^{1+2})$ in {\\rm(B)};\n\\item $\\Ga\\cong \\K_9\\square\\K_9$ or $\\K_9\\times\\K_9$, arising from the case where $p^d=3^4$ and $G_0\\leq \\N_{\\GL_4(3)}(2^{1+4}) $ in {\\rm(B)};\n\\item $\\Ga\\cong \\mathrm{VO}^{+}_8(2)$ or $\\overline{\\mathrm{VO}^{+}_8(2)}$, arising from the case where $p^d=2^8$ and $\\soc(G_0/\\Z(G_0))=\\A_9<\\mathrm{P\\Omega}^{+}_8(2)$ in {\\rm(C)}.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof} Assume that $G$ is of type {\\rm(A1)}. Then $G\\leq \\mathrm{A\\Gamma L}_1(p^d)$. Let $\\{\\alpha, \\beta\\}$ be an edge of $\\Ga$.\nThen $|V(\\Ga)|=p^d$ and $G_{\\alpha\\beta}\\leq \\ZZ_d$. Furthermore, $G_{\\a\\b}$ is semiregular on $V(\\Ga) \\setminus \\{\\alpha, \\beta\\}$.\nFrom Lemma~\\ref{lm:Gab10orbits} it follows that $G_{\\alpha\\beta}$ has at most $12$ orbits on $V(\\Ga) \\setminus \\{\\alpha, \\beta\\}$.\nTherefore, $G_{\\alpha\\beta}$ has one orbit of length at least $(|V(\\Ga)|-2)/12$.\nIt follows that $|G_{\\alpha\\beta}|\\geq (|V(\\Ga)|-2)/12$, which means\n\\[ p^d-2 \\leq 12d.\\]\nThis implies that\n\\[\np^d\\in\\{2^i (1\\leq i\\leq6), 3^j (1\\leq j\\leq 3), 5^k, 7^k, 11^k (1\\leq k\\leq 2), 13\\}.\n\\]\nSince $\\Ga$ has girth $3$ and diameter $2$ by Hypothesis~\\ref{hyp}, it follows that $|V(\\Ga)|=p^d\\geq7$. For all of the remaining possible values of $p^d$, we make use of Magma~\\cite{BCP} to construct all primitive rank $3$ subgroups $G$ of $\\mathrm{A\\Gamma L}_1(p^d)$, and then for each rank $3$ graph arising from $G$, we can further check if $|\\Om_i(\\Ga)|$ divides $|G|$ for $i=1,2,3,4$, where $\\Om_i(\\Ga)$ is given in Lemma~\\ref{lm:Gaorder}. We find that the only possibility is $p^d=3^2$, and $\\Ga\\cong \\mathrm{Paley}(9)\\cong\\K_3\\square\\K_3$. \\hfill\\qed\n\\end{proof}", "post_theorem_intro_text_len": 7099, "post_theorem_intro_text": "{\\noindent{\\bf Remark on Theorem~\\ref{th-main}.}\\ For $k\\geq2$, all $k$-homogeneous graphs have already been classified in the literature, { as mentioned above}, and so all graphs in Theorem~\\ref{th-main}~(1)--(4) are known.}\\medskip\n\nBy definition, a $k$-homogeneous graph is $k$-set-homogeneous.\nThe converse of this statement is clearly true for $k=1$.\n{In the case where $k\\ge4$,}\n{Devillers~\\cite{Devillers-2002paper}} proved that every finite $5$-set-homogeneous graph is set-homogeneous, and that a $4$-set-homogeneous graph is either homogeneous or one of the Schl\\\"{a}fli graph on $27$ vertices and its complement, which are $4$-homogeneous (see, for example, \\cite[Proposition~3.2.1]{Devillers}).\n{So, to prove Theorem~\\ref{th-main}, it suffices to show that every $k$-set-homogeneous graph is also $k$-homogeneous for $k=2$ and 3.}\n\nBefore stating the results, we introduce some families of graphs.\n\n\\begin{definition}\\label{Def-graph-families}\n{\\rm \\begin{itemize}\n\\item[\\rm (1)]  Let ${\\rm GF}(q)$ be a field of order a prime power $q$ such that $q\\equiv 1\\ (\\mod 4)$, and let $\\o$ be a fixed primitive { element} of ${\\rm GF}(q)$.\nLet $S=\\{\\o^2, \\o^4, \\ldots, \\o^{q-3}, \\o^{q-1}\\}$. The {\\it Paley graph} of order $q$, denoted by {\\rm Paley}$(q)$, is the graph with vertex set ${\\rm GF}(q)$ { where} two vertices $x,y$ are adjacent if and only if $x-y\\in S$.\n\n\\item[\\rm (2)]  For a positive integer $n$, the {\\it Hamming graph}, denoted by $\\mathrm{H}(2,n)$, is the graph with vertex set {$\\{(i, j) \\mid 1\\leqslant i, j\\leqslant n\\}$}, and for every vertex $(i, j)$, its neighbour set is\n\\[\n\\{ (i,j')\\mid j' \\in \\{1,2,\\dots,n \\}\\setminus \\{j\\}\\} \\cup \\{ (i',j)\\mid i' \\in \\{1,2,\\dots,n \\}\\setminus \\{i\\}\\}.\n\\]\nNote that $\\mathrm{H}(2,n)$ is just the Cartesian product $\\K_n\\square\\K_n$.\n\n\\item[\\rm (3)]  For $\\epsilon \\in \\{+,-\\}$, a positive integer $m$ and a prime power $q$, the {\\it affine polar graph}, denoted by $\\mathrm{VO}_{2m}^{\\epsilon}(q)$, is a graph with vertex set $\\GF(q)^{2m}$ equipped with a nondegenerate quadratic form $Q$ of type $\\epsilon$, and two vertices $\\alpha$ and $\\alpha'$ being adjacent if and only if $Q(\\alpha-\\alpha')=0$ (see \\cite[\\textsection{3.3.1}]{BrouwerSRG}). Since a nondegenerate $2$-dimensional orthogonal vector space of type $-$ contains no singular vector, it follows that $\\mathrm{VO}_{2}^{-}(q)\\cong q^2\\K_1$. As remarked in~\\cite[\\textsection{3.3.1}]{BrouwerSRG}, we see that $\\mathrm{VO}_{2}^{+}(q)\\cong \\K_q\\square\\K_q=\\mathrm{H}(2,q)$.\n\n\\item[\\rm (4)] For a prime power $q$, the {\\it elliptic orthogonal graph}, denoted by $\\mathrm{\\Gamma}(\\mathrm{O}^{-}_6(q))$ is a graph with vertex set consisting of  all singular $1$-subspaces of the orthogonal space on {$\\GF(q)^{6}$ of type $-$},\n    and two vertices $\\alpha=\\langle w \\rangle$ and $\\alpha'=\\langle w' \\rangle$ being adjacent if and only if $w\\perp w'$ (see \\cite[\\textsection{2.6.3}]{BrouwerSRG}). The Schl\\\"{a}fli graph is the complement of the elliptic orthogonal graph $\\mathrm{\\Gamma}(\\mathrm{O}^{-}_6(2))$ (see \\cite[Table~11.8]{BrouwerSRG}).\n\\end{itemize}}\n\\end{definition}\n\nWe recall that a transitive permutation group $G$ on a set $\\Ome$ is {\\it $2$-homogeneous} if\nall $2$-subsets of $\\Ome$ are equivalent under $G$.\nIn the next theorem, a classification of finite $2$-set-homogeneous graphs is given.\n\n\\begin{theorem}\\label{th:2sethomo}\nLet $\\Ga=(V,E)$ be a finite  $(G,2)$-set-homogeneous graph, where $G\\leqslant\\Aut(\\Ga)$.\nThen one of the following holds:\n\\begin{itemize}\n\\item[{\\rm(1)}] $\\Ga\\cong\\K_n$ or $n\\K_1$, and $G$ is a $2$-homogeneous permutation group on $V$.\n\n\\item[{\\rm(2)}] $\\Ga\\cong\\K_{m[b]}$ or $m\\K_b$, and  $G\\leqslant X\\wr Y$,\nwhere $X,Y$ are $2$-homogeneous permutation groups of degree $b,m$, respectively.\n\n\\item[{\\rm(3)}] $G$ is primitive of rank $3$, and $\\Ga$ is $(G,2)$-homogeneous.\n\n\\item[{\\rm(4)}] $G\\cong\\PSU_3(3)$ with vertex stabiliser $\\PSL_3(2)$, $\\Ga$ is of order $36$ and\nvalency $14$ or $21$, but $\\Aut(\\Ga)=\\PSU_3(3).2$ is $2$-homogeneous on $\\Ga$.\n\n\\item[(5)] $\\Ga$ is a Paley graph of order $p^r$, where $p\\equiv5 \\pmod{8}$ and $r$ is odd, and\n$G\\leqslant\\AGammaL_1(p^r)$ is $2$-set-homogeneous on $\\Ga$.\n\\end{itemize}\nIn particular, all $2$-set-homogeneous graphs are $2$-homogeneous.\n\\end{theorem}\n\n{It is worth mentioning} that all graphs in Theorem~\\ref{th:2sethomo}~(1)--(5) are $2$-homogeneous and already known in the literature.\n{Since} a graph $\\Ga$ is $(G, 2)$-set-homogeneous if and only if both $\\Ga$ and $\\ov\\Ga$ are $G$-edge-transitive, { our} main approach for proving the above theorem is to {analyse} permutation groups of rank at most 5. It is well-known that all orbitals of a permutation group of rank 3 and {of even order} are self-paired. The following result shows that, except for a single group, all orbitals of a primitive permutation group of rank 4 are self-paired.\n\n\\begin{theorem}\\label{trans-rank-4}\nLet $G$ be a primitive permutation group of rank $4$. Then either all orbitals of $G$ are self-paired, or $G=\\PSU_3(3)$ acting on $36$ points.\n\\end{theorem}\n\n\\begin{remark}\n{\\rm\\begin{itemize}\n\\item[\\rm (a)] More detailed information for the single group $\\PSU_3(3)$ and its orbitals is given\nin Theorem~\\ref{rank-4}.\n\\item[\\rm (b)] There are infinitely many imprimitive permutation groups of rank $4$ with exactly one non-trivial self-paired orbital.\nHere is an example. Let $X$ be a $2$-homogeneous but not $2$-transitive permutation group, and let\n$Y$ be a $2$-transitive permutation group. Then $X\\wr Y$ is an imprimitive permutation group of rank $4$ with exactly one non-trivial self-paired orbital.\n\\end{itemize}}\n\\end{remark}\n\nOur last theorem gives a classification of finite $3$-set-homogeneous graphs.\n\n\\begin{theorem}\\label{th:3sethomo}\nLet $\\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\\Ga$ is isomorphic either to one of the graphs: $\\C_5$, $\\K_n$, $\\K_{m[b]}$, $\\K_n\\square\\K_n$, or to one of the following graphs:\n\\begin{itemize}\n\\item[{\\rm(1)}] the affine polar graph $\\mathrm{VO}_{2m}^{\\epsilon}(2)$ with $m\\geqslant 1$ and $\\epsilon \\in \\{+,-\\}$;\n\n\\item[{\\rm(2)}]  the elliptic orthogonal graph $\\mathrm{\\Gamma}(\\mathrm{O}^-_6(q))$ with $q$ a prime power;\n\n\\item[{\\rm(3)}] an orbital graph of the Higman-Sims group $\\mathrm{HS}$ acting on $100$ points;\n\n\\item[{\\rm(4)}] an orbital graph of the McLaughlin group $\\mathrm{McL}$ acting on $275$ points.\n\\end{itemize}\nIn particular, all $3$-set-homogeneous graphs are $3$-homogeneous.\n\\end{theorem}\n\nNote that all graphs in Theorem~\\ref{th:3sethomo}~(1)--(4) are actually $3$-homogeneous graphs, which {are known} in the literature.\n\nThe rest of the paper is organised as follows. In Sections~\\ref{subsec:2reduce}--\\ref{sec6:proof-main-theorems}, we deal with finite $(G, 2)$-set-homogeneous graphs.\nIn Section~\\ref{sec6:proof-main-theorems}, we prove Theorem~\\ref{th:2sethomo} and Theorem~\\ref{trans-rank-4}.\nIn Section~\\ref{set:3sethom}, we deal with finite $(G, 3)$-set-homogeneous graphs, and prove Theorem~\\ref{th:3sethomo}.\nIn Section~\\ref{set:proof-of-main}, we prove Theorem~\\ref{th-main}.", "sketch": "By definition, a $k$-homogeneous graph is $k$-set-homogeneous. For $k=1$ the converse is clearly true. For $k\\ge4$, Devillers~\\cite{Devillers-2002paper} proved that every finite $5$-set-homogeneous graph is set-homogeneous, and that a $4$-set-homogeneous graph is either homogeneous or one of the Schl\\\"{a}fli graph on $27$ vertices and its complement (which are $4$-homogeneous). Hence, to prove Theorem~\\ref{th-main}, it suffices to show that every $k$-set-homogeneous graph is also $k$-homogeneous for $k=2$ and $3$.\n\nThis is done by classifying finite $2$-set-homogeneous graphs (Theorem~\\ref{th:2sethomo}, yielding in particular that all $2$-set-homogeneous graphs are $2$-homogeneous) and finite $3$-set-homogeneous graphs (Theorem~\\ref{th:3sethomo}, yielding in particular that all $3$-set-homogeneous graphs are $3$-homogeneous), and then concluding Theorem~\\ref{th-main} in Section~\\ref{set:proof-of-main}.", "expanded_sketch": "By definition, a $k$-homogeneous graph is $k$-set-homogeneous. For $k=1$ the converse is clearly true. For $k\\ge4$, Devillers~\\cite{Devillers-2002paper} proved that every finite $5$-set-homogeneous graph is set-homogeneous, and that a $4$-set-homogeneous graph is either homogeneous or one of the Schl\\\"{a}fli graph on $27$ vertices and its complement (which are $4$-homogeneous). Hence, to prove the main theorem, it suffices to show that every $k$-set-homogeneous graph is also $k$-homogeneous for $k=2$ and $3$.\n\nWe first prove the following theorem.\n\\begin{theorem}\\label{th:2sethomo}\nLet $\\Ga=(V,E)$ be a finite  $(G,2)$-set-homogeneous graph, where $G\\leq\\Aut(\\Ga)$.\nThen one of the following holds:\n\\begin{itemize}\n\\item[{\\rm(1)}] $\\Ga\\cong\\K_n$ or $n\\K_1$, and $G$ is a $2$-homogeneous permutation group on $V$.\n\n\\item[{\\rm(2)}] $\\Ga\\cong\\K_{m[b]}$ or $m\\K_b$, and  $G\\leqslant X\\wr Y$,\nwhere $X,Y$ are $2$-homogeneous permutation groups of degree $b,m$, respectively.\n\n\\item[{\\rm(3)}] $G$ is primitive of rank $3$, and $\\Ga$ is $(G,2)$-homogeneous.\n\n\\item[{\\rm(4)}] $G\\cong\\PSU_3(3)$ with vertex stabiliser $\\PSL_3(2)$, $\\Ga$ is of order $36$ and\nvalency $14$ or $21$, but $\\Aut(\\Ga)=\\PSU_3(3).2$ is $2$-homogeneous on $\\Ga$.\n\n\\item[(5)] $\\Ga$ is a Paley graph of order $p^r$, where $p\\equiv5 \\pmod{8}$ and $r$ is odd, and\n$G\\leq\\AGammaL_1(p^r)$ is $2$-set-homogeneous on $\\Ga$.\n\\end{itemize}\nIn particular, all $2$-set-homogeneous graphs are $2$-homogeneous.\n\\end{theorem}\n\nWe next prove the following theorem.\n\\begin{theorem}\\label{th:3sethomo}\nLet $\\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\\Ga$ is isomorphic either to one of the graphs: $\\C_5$, $\\K_n$, $\\K_{m[b]}$, $\\K_n\\square\\K_n$, or to one of the following graphs:\n\\begin{itemize}\n\\item[{\\rm(1)}] the affine polar graph $\\mathrm{VO}_{2m}^{\\epsilon}(2)$ with $m\\geq 1$ and $\\epsilon \\in \\{+,-\\}$;\n\n\\item[{\\rm(2)}]  the elliptic orthogonal graph $\\mathrm{\\Gamma}(\\mathrm{O}^-_6(q))$ with $q$ a prime power;\n\n\\item[{\\rm(3)}] an orbital graph of the Higman-Sims group $\\mathrm{HS}$ acting on $100$ points;\n\n\\item[{\\rm(4)}] an orbital graph of the McLaughlin group $\\mathrm{McL}$ acting on $275$ points.\n\\end{itemize}\nIn particular, all $3$-set-homogeneous graphs are $3$-homogeneous.\n\\end{theorem}\n\nFinally, we conclude the proof of the main theorem as follows:\n\\label{set:proof-of-main}\n\nBy definition, both 1-homogeneous graphs and 1-set-homogeneous graphs are exactly the vertex-transitive graphs. Assume now that $k\\geq 2$. It is obvious by definition that e", "expanded_theorem": "\\label{th-main}\nFor each positive integer $k$, a finite graph is $k$-set-homogeneous if and only if it is $k$-homogeneous.\nMoreover, { if $\\Ga$ is a $k$-homogeneous graph with $k\\geqslant 2$, then} one of the following holds:\n\\begin{itemize}\n\\item [{\\rm(1)}] $k\\geqslant 5$, and $\\Ga$ is one of { $\\C_5 (\\cong\\overline{\\C_5})$, $\\K_3\\square\\K_3 (\\cong \\overline{\\K_3\\square\\K_3})$, $n\\K_m$, and $\\K_{n[m]}$; furthermore, $\\Ga$ is homogeneous.}\n\n{ \\item[(2)] $\\Ga$ is either the Schl\\\"{a}fli graph on $27$ vertices or its complement (the two non-trivial orbital graphs of $\\mathrm{PSU}_4(2)$ acting on the isotropic lines); furthermore, $\\Ga$ is $4$-homogeneous but not $5$-homogeneous.\n\n\\item[(3)] $\\Ga$ is either $\\K_n\\square\\K_n$ with $n\\geqslant4$, or a graph from the following theorem (items (1)--(4)), or the complement of any such graph; furthermore, $\\Ga$ is $3$-homogeneous but not $4$-homogeneous.\n\nWe first prove the following theorem.\n\\begin{theorem}\\label{th:3sethomo}\nLet $\\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\\Ga$ is isomorphic either to one of the graphs: $\\C_5$, $\\K_n$, $\\K_{m[b]}$, $\\K_n\\square\\K_n$, or to one of the following graphs:\n\\begin{itemize}\n\\item[{\\rm(1)}] the affine polar graph $\\mathrm{VO}_{2m}^{\\epsilon}(2)$ with $m\\geq 1$ and $\\epsilon \\in \\{+,-\\}$;\n\n\\item[{\\rm(2)}]  the elliptic orthogonal graph $\\mathrm{\\Gamma}(\\mathrm{O}^-_6(q))$ with $q$ a prime power;\n\n\\item[{\\rm(3)}] an orbital graph of the Higman-Sims group $\\mathrm{HS}$ acting on $100$ points;\n\n\\item[{\\rm(4)}] an orbital graph of the McLaughlin group $\\mathrm{McL}$ acting on $275$ points.\n\\end{itemize}\nIn particular, all $3$-set-homogeneous graphs are $3$-homogeneous.\n\\end{theorem}\n\n\\item[(4)] $\\Ga$ is an orbital graph of a primitive permutation group of rank $3$ which is not covered by parts (1)--(3); furthermore, $\\Ga$ is $2$-homogeneous but not $3$-homogeneous. }\n\n\\end{itemize}", "theorem_type": ["Biconditional or Equivalence", "Classification or Bijection"], "mcq": {"question": "Let be a finite graph, and let k be a positive integer. Call k-set-homogeneous if any two isomorphic induced subgraphs of with at most k vertices lie in the same orbit of Aut(). Call k-homogeneous if every isomorphism between any two isomorphic induced subgraphs of with at most k vertices extends to an automorphism of . Here nK_m denotes the disjoint union of n copies of K_m, K_{n[m]} the complete multipartite graph with n parts of size m, K_n\\square K_n the Cartesian product of two copies of K_n, and an orbital graph is a graph whose edge set is an orbit of a permutation group on unordered pairs of vertices. Which statement gives the complete classification of finite k-set-homogeneous graphs?", "correct_choice": {"label": "A", "text": "For every positive integer k, a finite graph is k-set-homogeneous if and only if it is k-homogeneous. Moreover, if is k-homogeneous with k\\ge 2, then one of the following holds: (1) k\\ge 5, and is one of C_5, K_3\\square K_3, nK_m, or K_{n[m]}; in fact is homogeneous. (2) is the Schl\\\"afli graph on 27 vertices or its complement; then is 4-homogeneous but not 5-homogeneous. (3) is K_n\\square K_n for some n\\ge 4, or its complement, or one of the affine polar graphs VO_{2m}^{\\epsilon}(2) with m\\ge 1 and \\epsilon\\in\\{+,-\\}, the elliptic orthogonal graphs \\Gamma(O^-_6(q)) with q a prime power, an orbital graph of the Higman-Sims group HS on 100 points, or an orbital graph of the McLaughlin group McL on 275 points, or the complement of one of these; then is 3-homogeneous but not 4-homogeneous. (4) Otherwise, is an orbital graph of a primitive permutation group of rank 3 not already covered by (1)-(3); then is 2-homogeneous but not 3-homogeneous."}, "choices": [{"label": "B", "text": "For every positive integer k, a finite graph is k-set-homogeneous if and only if it is k-homogeneous. Moreover, if \\Gamma is k-homogeneous with k\\ge 2, then one of the following holds: (1) k\\ge 5, and \\Gamma is one of C_5, K_3\\square K_3, nK_m, or K_{n[m]}; in fact \\Gamma is homogeneous. (2) \\Gamma is the Schl\\\"afli graph on 27 vertices or its complement; then \\Gamma is 4-homogeneous but not 5-homogeneous. (3) \\Gamma is K_n\\square K_n for some n\\ge 3, or its complement, or one of the affine polar graphs VO_{2m}^{\\epsilon}(2) with m\\ge 1 and \\epsilon\\in\\{+,-\\}, the elliptic orthogonal graphs \\Gamma(O^-_6(q)) with q a prime power, an orbital graph of the Higman-Sims group HS on 100 points, or an orbital graph of the McLaughlin group McL on 275 points, or the complement of one of these; then \\Gamma is 3-homogeneous but not 4-homogeneous. (4) Otherwise, \\Gamma is an orbital graph of a primitive permutation group of rank 3 not already covered by (1)-(3); then \\Gamma is 2-homogeneous but not 3-homogeneous."}, {"label": "C", "text": "For every positive integer k, every finite k-homogeneous graph is k-set-homogeneous. Moreover, every finite k-set-homogeneous graph belongs to one of the four families listed in (1)-(4) of the classification for k-homogeneous graphs."}, {"label": "D", "text": "For every positive integer k, a finite graph is k-set-homogeneous if and only if it is k-homogeneous. Moreover, if \\Gamma is k-homogeneous with k\\ge 2, then one of the following holds: (1) k\\ge 5, and \\Gamma is one of C_5, K_3\\square K_3, nK_m, or K_{n[m]}; in fact \\Gamma is homogeneous. (2) \\Gamma is the Schl\\\"afli graph on 27 vertices or its complement; then \\Gamma is 4-homogeneous but not 5-homogeneous. (3) \\Gamma is K_n\\square K_n for some n\\ge 4, or its complement, or one of the affine polar graphs VO_{2m}^{\\epsilon}(2) with m\\ge 1 and \\epsilon\\in\\{+,-\\}, the elliptic orthogonal graphs \\Gamma(O^-_6(q)) with q a prime power, an orbital graph of the Higman-Sims group HS on 100 points, or an orbital graph of the McLaughlin group McL on 275 points, or the complement of one of these; then \\Gamma is 3-homogeneous but not 5-homogeneous. (4) Otherwise, \\Gamma is an orbital graph of a primitive permutation group of rank 3 not already covered by (1)-(3); then \\Gamma is 2-homogeneous but not 3-homogeneous."}, {"label": "E", "text": "For every positive integer k, a finite graph is k-set-homogeneous if and only if it is k-homogeneous. Moreover, if \\Gamma is k-homogeneous with k\\ge 2, then one of the following holds: (1) k\\ge 5, and \\Gamma is one of C_5, K_3\\square K_3, nK_m, or K_{n[m]}; in fact \\Gamma is homogeneous. (2) \\Gamma is the Schl\\\"afli graph on 27 vertices or its complement; then \\Gamma is 4-homogeneous but not 5-homogeneous. (3) \\Gamma is K_n\\square K_n for some n\\ge 4, or one of the affine polar graphs VO_{2m}^{\\epsilon}(2) with m\\ge 1 and \\epsilon\\in\\{+,-\\}, the elliptic orthogonal graphs \\Gamma(O^-_6(q)) with q a prime power, an orbital graph of the Higman-Sims group HS on 100 points, or an orbital graph of the McLaughlin group McL on 275 points; then \\Gamma is 3-homogeneous but not 4-homogeneous. (4) Otherwise, \\Gamma is an orbital graph of a primitive permutation group of rank 3 not already covered by (1)-(3), or the complement of such a graph; then \\Gamma is 2-homogeneous but not 3-homogeneous."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "case_split", "tampered_component": "threshold n\\ge 4 for K_n\\square K_n in the 3-homogeneous-but-not-4-homogeneous case", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "case_split", "tampered_component": "dropped the converse direction and the per-case sharp homogeneity level conclusions", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "sharp exclusion '3-homogeneous but not 4-homogeneous' replaced by 'not 5-homogeneous'", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "complement closure moved from part (3) to part (4), omitting complements in the 3-homogeneous family", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the notions involved but does not reveal the classification or explicitly signal the correct option. There is no direct answer leakage beyond standard terminology."}, "TAS": {"score": 1, "justification": "The item asks for an equivalent statement of a theorem, so it is not a pure restatement of the stem, but it is still fundamentally theorem-recognition rather than a genuinely independent conclusion drawn from new data."}, "GPS": {"score": 0, "justification": "The question mainly tests recall of a precise classification statement and the ability to detect small textual perturbations. It does not require substantive mathematical generation or multi-step reasoning from the definitions."}, "DQS": {"score": 2, "justification": "The distractors are plausible and well-targeted: one is a weaker true statement, while others alter thresholds, homogeneity levels, or complement-closure details. These reflect realistic mathematical failure modes."}, "total_score": 5, "overall_assessment": "A technically well-constructed recall/checking item with strong distractors and no leakage, but it has limited generative reasoning value and leans heavily on theorem memorization."}}
{"id": "2602.19882v1", "paper_link": "http://arxiv.org/abs/2602.19882v1", "theorems_cnt": 6, "theorem": {"env_name": "theorem", "content": "\\label{th-main}\nFor each positive integer $k$, a finite graph is $k$-set-homogeneous if and only if it is $k$-homogeneous.\nMoreover, { if $\\Ga$ is a $k$-homogeneous graph with $k\\geqslant 2$, then} one of the following holds:\n\\begin{itemize}\n\\item [{\\rm(1)}] $k\\geqslant 5$, and $\\Ga$ is one of { $\\C_5 (\\cong\\overline{\\C_5})$, $\\K_3\\square\\K_3 (\\cong \\overline{\\K_3\\square\\K_3})$, $n\\K_m$, and $\\K_{n[m]}$; furthermore, $\\Ga$ is homogeneous.}\n\n{ \\item[(2)] $\\Ga$ is either the Schl\\\"{a}fli graph on $27$ vertices or its complement (the two non-trivial orbital graphs of $\\mathrm{PSU}_4(2)$ acting on the isotropic lines); furthermore, $\\Ga$ is $4$-homogeneous but not $5$-homogeneous.\n\n\\item[(3)] $\\Ga$ is either $\\K_n\\square\\K_n$ with $n\\geqslant4$, or a graph from Theorem~\\ref{th:3sethomo}(1)--(4), or the complement of any such graph; furthermore, $\\Ga$ is $3$-homogeneous but not $4$-homogeneous.\n\n\\item[(4)] $\\Ga$ is an orbital graph of a primitive permutation group of rank $3$ which is not covered by parts (1)--(3); furthermore, $\\Ga$ is $2$-homogeneous but not $3$-homogeneous. }\n\n\\end{itemize}", "start_pos": 11163, "end_pos": 12292, "label": "th-main"}, "ref_dict": {"th:2sethomo": "\\begin{theorem}\\label{th:2sethomo}\nLet $\\Ga=(V,E)$ be a finite  $(G,2)$-set-homogeneous graph, where $G\\leq\\Aut(\\Ga)$.\nThen one of the following holds:\n\\begin{itemize}\n\\item[{\\rm(1)}] $\\Ga\\cong\\K_n$ or $n\\K_1$, and $G$ is a $2$-homogeneous permutation group on $V$.\n\n\\item[{\\rm(2)}] $\\Ga\\cong\\K_{m[b]}$ or $m\\K_b$, and  $G\\leqslant X\\wr Y$,\nwhere $X,Y$ are $2$-homogeneous permutation groups of degree $b,m$, respectively.\n\n\\item[{\\rm(3)}] $G$ is primitive of rank $3$, and $\\Ga$ is $(G,2)$-homogeneous.\n\n\\item[{\\rm(4)}] $G\\cong\\PSU_3(3)$ with vertex stabiliser $\\PSL_3(2)$, $\\Ga$ is of order $36$ and\nvalency $14$ or $21$, but $\\Aut(\\Ga)=\\PSU_3(3).2$ is $2$-homogeneous on $\\Ga$.\n\n\\item[(5)] $\\Ga$ is a Paley graph of order $p^r$, where $p\\equiv5 \\pmod{8}$ and $r$ is odd, and\n$G\\leq\\AGammaL_1(p^r)$ is $2$-set-homogeneous on $\\Ga$.\n\\end{itemize}\nIn particular, all $2$-set-homogeneous graphs are $2$-homogeneous.\n\\end{theorem}", "set:3sethom": "\\begin{array}{cllllc}\n\\hline\n\\text{Line}&S& S_\\alpha & \\text{degree} &\\text{subdegrees}& \\text{rank of }\\N_{{\\rm Sym}(\\Om)}(S) \\\\\n\\hline\n48 &  \\mathrm{P\\Omega}_7(3)  &  \\PSp_6(2)  & 3159 &  1,288,630,2240  &  4 \\\\\n49 &  \\mathrm{P\\Omega}^{+}_8(2)  &  \\A_9  & 960 &  1,84,315,560  &  4 \\\\\n50 & \\mathrm{P\\Omega}^{+}_8(3)  &  \\mathrm{P\\Omega}^{+}_8(2)  &  28431  &  1,960,960,960,3150,22400  &  4 \\\\\n\\hline\n\\end{array}\n\\]\nIt implies that all orbitals of $G$ are self-paired.\n\\medskip\n\n\\f{\\bf Case~5.}\\ The Exceptional Group of Lie Type.\\medskip\n\nIn this case, $(S, S_\\alpha)$ is one of the pairs in Lines 51--68 of \\cite[Table~I]{Cuypers}.\n\n \\vspace{0.1cm}\n\\underline{Lines  51--54, 58--59, 61, 63, 65, 67--68}:\\  These lines are impossible by Lemma~\\ref{lm:parabolic}.\n\n \\vspace{0.1cm}\n\\underline{Lines 55--56}:\\\nNow $S=G_2(q)$ and $S_\\alpha=\\mathrm{SU}_3(q).2$ or $\\SL_3(q).2$.\nBy \\cite{Liebeck-2-closures}, all orbitals of $S$ are self-paired.\n\n \\vspace{0.1cm}\n\\underline{Line  57}:\\\nNow $S=G_2(4)$ and $S_\\alpha=\\mathrm{J}_2$, which  gives  rise to a group of rank $3$ by~\\cite{Liebeck-Saxl}.\n\n \\vspace{0.1cm}\n\\underline{Line  60}:\\\nNow $S={}^2F_4(2)'$ and $S_\\alpha=\\PSL_3(2).2$. By computation in Magma~\\cite{BCP},  $S$ has rank $4$ with subdegrees $1$, $312$, $351$ and $936$.\n\n \\vspace{0.1cm}\n\\underline{Line  62}:\\\nNow $S=F_4(2)$ and $S_\\alpha=\\PSp_8(2)$. From Atlas~\\cite{Conway-Atlas} we see that $G=S$.\n By~\\cite[Lemma~8.6~(i)]{Cuypers}, $G$ would have  rank $5$.\n\n \\vspace{0.1cm}\n\\underline{Line 64}:\\\nNow $S={}^2E_6(2)$ and $S_\\alpha=F_4(2)$. By \\cite[Lemma~8.6(ii)]{Cuypers}, $G$ has rank $4$. In this case, $S_\\alpha$ is the centraliser of the out-involution $\\s$ of $S$. Let $G={}^2E_6(2){:}\\lg\\s\\rg$ and $C=F_4(2)\\times\\lg\\s\\rg$. Set $\\Om=\\{\\s^g \\mid g\\in G\\}$. Now let $H\\leq G$ s.t. $H=\\PSU_6(2){:}\\lg\\s\\rg$. Then $C_H(\\s)=\\PSp_6(2)\\times\\lg\\s\\rg$. Take $x,y\\in\\Om$ such that $x,y$ are in the same orbit of $C$. Then $\\s x$ and $\\s y$ are conjugate, and so they have the same order. Let $X=\\{\\s^h\\ |\\ h\\in H\\}$. Then $H$ is primitive on $X$ with rank $4$ and all orbitals are self-paired. By Magma~\\cite{BCP}, we can obtain that $|\\{o (\\s x)\\ |\\ x\\in X\\}|=4$, and so each orbital of $H$ on $X$ is contained in exactly one orbital of $G$ on $\\Om$. This implies that all orbitals of $G$ are also self-paired.\n(We are grateful to Prof. Frank L${\\rm \\ddot{u}}$beck who kindly helps us to obtain the subdegrees (namely, $1,69615, 6336512, 16707600$) of this group by using GAP. )\n\n \\vspace{0.1cm}\n\\underline{Line  66}:\\\nNow $S=E_6(2)$ and $S_\\alpha=F_4(2)$, and $G$ contains a graph automorphism. By \\cite[Lemma~8.7]{Cuypers}, $G$ would have rank $5$.\n\\hfill\\qed\n\nNow combining Propositions~\\ref{prop-sporadic}, \\ref{prop-alt} and \\ref{prop-sim-lie}, the proof of Theorem~\\ref{rank-4} is complete.\n\n\\section{Proof of Theorems~\\ref{th:2sethomo} and \\ref{trans-rank-4}}\\label{sec6:proof-main-theorems}\n\n\\f{\\bf Proof of Theorem~\\ref{th:2sethomo}.}\\ Let $\\Ga=(V,E)$ be a $(G,2)$-set-homogeneous graph.\nSuppose that $\\Ga$ is neither complete nor empty.\nIf $G$ is imprimitive on the vertex set $V$, then by Lemma~\\ref{imp},\n$\\Ga\\cong m\\K_b$ or $\\K_{m[b]}$ for some positive integers $m$ and $ b$. Then Theorem~\\ref{th:2sethomo}~(2) holds.\n\nAssume that $G$ is primitive on $V$. By Lemma~\\ref{rank<=5}, $G$ is of rank at most 5.\n\nIf $G$ is of rank $3$, then $\\Ga$ is $(G,2)$-homogeneous and all candidates for $G$ are listed in \\cite{Bannai,Kantor,Liebeck-affine,Liebeck-Saxl}. Then Theorem~\\ref{th:2sethomo}~(3)  holds.\n\nIf $G$ is of rank $4$, then exactly one of the orbitals is self-paired. By Corollary~\\ref{reduction}, $G$ is affine or almost simple. By Theorem~\\ref{affine-rank-4}, $G$ is not affine. If $G$ is almost simple, then by Theorem~\\ref{rank-4}, $G\\cong\\PSU_3(3)$ with stabiliser $\\PSL_3(2)$, and the subdegrees are $1,7,7$ and $21$. Then $\\Ga$ is either the orbital graph of $G$ of order $36$ and of valency $21$, or the complement of this graph. Furthermore, by Magma~\\cite{BCP}, $\\Aut(\\Ga)={\\rm P\\Gamma U}_3(3)$ is $2$-homogeneous on $\\Ga$. This proves part (4) of our theorem.\n\nIf $G$ is of rank 5, then none of the non-trivial orbitals of $G$ is self-paired. From Proposition~\\ref{AGL_1(p^r)} we obtain part (5) of our theorem. \\hfill\\qed\n\n\\medskip\n\\f{\\bf Proof of Theorem~\\ref{trans-rank-4}.}\\ Combining Proposition~\\ref{th-rank-4}, Lemma~\\ref{non-HA-AS}, Theorem~\\ref{affine-rank-4} and Theorem~\\ref{rank-4}, we obtain the proof of this theorem.\\hfill\\qed\n\n\\section{Finite $3$-set-homogeneous graphs}\\label{set:3sethom}\n\nIn this section, we shall classify finite $3$-set-homogeneous graphs, and then prove Theorem~\\ref{th:3sethomo}.\n\n\\subsection{Preliminaries}\\label{subsec:3reduce}\n\\subsubsection{A reduction}\nA finite graph $\\Ga$ is \\emph{strongly regular} with parameters $(v, k, \\lambda, \\mu)$ if it has $v$ vertices and is regular with valency $k$ where $0<k<v-1$, and if any two adjacent vertices have $\\ld=\\ld(\\Ga)$ common neighbours and any two non-adjacent vertices have $\\mu=\\mu(\\Ga)$ common neighbours. Note that a strongly regular graph is neither complete nor edgeless.\n\n\\begin{lem}\\label{lem:prim-3ch-is-srg}\nLet $\\Ga $ be a finite $(G, 3)$-set-homogeneous graph. Then one of the following holds.\n\\begin{enumerate}\n\\item[{\\rm(1)}] $\\Ga \\cong\\K_n$ or $n\\K_1$;\n\n\\item[{\\rm(2)}]  $\\Ga \\cong \\K_{m[b]}$ or $m\\K_b$ where $m,b\\geq 2$;\n\n\\item [{\\rm (3)}]\\ $G$ is primitive on $V(\\Ga)$ of rank $3$, and $\\Ga$ is strongly regular of diameter $2$. Furthermore, if $\\Ga$ is not the pentagon, then either $\\Ga$ or its complement $\\overline{\\Ga}$ has girth $3$.\n\\end{enumerate}\n\\end{lem}\n\n\\proof By \\cite[Theorem~1.1 \\& Corollary~1.2]{Zhou-EJC2021}, $\\Ga$ is $(G,2)$-homogeneous, and then by Theorem~\\ref{th:2sethomo}, we conclude that either parts (1) or (2) hold, or $G$ is primitive on $V(\\Ga)$ of rank $3$.\n\nAssume that neither part (1) nor part (2) happens. Then $\\Ga$ has diameter $2$ and $G$ is arc-transitive on both $\\Ga$ and the complement $\\overline{\\Ga}$ of $\\Ga$. This implies that $\\Ga$ is strongly regular. The first assertion of the part (3) is proved. If $\\Ga$ has valency $2$, then since $\\Ga$ has diameter $2$, it follows that $\\Ga$ is the pentagon. If $\\Ga$ has valency at least $3$, then either $\\Ga$ or its complement $\\overline{\\Ga}$ has girth $3$. \\hfill\\qed\n\nBy Lemma~\\ref{lem:prim-3ch-is-srg}, to prove Theorem~\\ref{th:3sethomo}, it suffices to classify those $(G,3)$-set-homogeneous graphs $\\Ga$ such that $G$ is primitive on $V(\\Ga)$ of rank $3$. The following result can be obtained by the fundamental O'Nan-Scott theorem (see \\cite[\\textsection{1}]{Liebeck-ONan-Scott}).\n\n\\begin{theorem}\\label{th:rank3groups}\nLet $G$ be a primitive group of rank  $3$.\nThen one of the following holds:\n\\begin{itemize}\n\\item[\\rm (1)]  $T\\times T\\lhd G\\leqslant H\\wr\\S_2$, where $H$ is a $2$-transitive permutation group of degree $n$ and $\\soc(H)=T$ is a nonabelian simple group, and $G$ has production action;\n\\item[\\rm (2)]  $G$ is an affine group;\n\\item[\\rm (3)]  $G$ is an almost simple group.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{lemma}\\label{lm:CaseI}\nSuppose that $\\Ga$ is a finite $(G,3)$-set-homogeneous graph such that $G$ satisfies Theorem~{\\rm\\ref{th:rank3groups}~(1)}.\nThen $\\Ga$ is isomorphic to $\\K_n\\square\\K_n$ or its complement $\\K_n\\times\\K_n$, and $\\Ga$ is $3$-homogeneous.\n\\end{lemma}\n\n\\proof Without loss of generality, assume that $H$ is a $2$-transitive permutation group on the set $\\Del =\\{1,2,\\dots, n\\}$. Then $V(\\Ga)=\\Del \\times\\Del $. Take $\\a=(1, 1)\\in V(\\Ga)$. Then the three orbits of $G$ are\n\\[\\Ome_0=\\{\\a\\}, \\Ome_1=\\{(1, i), (i,1)\\mid i\\in\\Delta\\setminus\\{1\\}\\}, \\Ome_2=\\{(i,j)\\mid i, j\\in\\Del \\setminus\\{1\\}\\}.\\]\nSo $\\Ga(\\a)=\\Ome_1$ or $\\Ome_2$. It is easy to see that $\\Ga\\cong\\K_n\\square\\K_n$ or $\\K_n\\times\\K_n$. By \\cite[Corollary~1.2]{Cameron-3-hom}, $\\Ga$ is $3$-homogeneous.\\hfill\\qed\n\nBy Lemmas~\\ref{lem:prim-3ch-is-srg} and \\ref{lm:CaseI}, from now on we will adopt the following hypothesis.\n\n\\begin{hypothesis}\\label{hyp}\nLet $\\Ga=(V, E)$ be a finite $(G,3)$-set-homogeneous graph satisfying the following:\n\\begin{itemize}\n  \\item $G\\leq\\Aut(\\Ga)$ is primitive on $V$ with rank $3$,\n  \\item $\\soc(G)$ is either abelian or nonabelian simple,\n  \\item $\\Ga$ has girth $3$ and diameter $2$.\n\\end{itemize}\n\\end{hypothesis}\n\nWe will deal with the case where $\\soc(G)$ is abelian in Subsection~\\ref{subsec:affine}, and the case where $\\soc(G)$ is nonabelian simple in Subsection~\\ref{subsec:simple}.\n\n\\subsubsection{Some useful lemmas}\n\nThe first two lemmas concern the arc-stabilisers of $(G,3)$-set-homogeneous graphs. Let $s$ be a positive integer and let $\\Sig$ be a graph. For a subgroup $H$ of $\\Aut(\\Sig)$, we say that $\\Sig$ is {\\em $(H, s)$-connected-set-homogeneous} if any two isomorphic connected induced subgraphs of $\\Sigma$ on at most $s$ vertices are equivalent under $H$. Clearly, every $(H,s)$-set-homogeneous graph is $(H,s)$-connected-set-homogeneous.\nFor any vertex $v$ of a graph $\\Sig$ of diameter at least two, let $\\Sig(v)$ be the vertices of $\\Sig$ adjacent to $v$, and let $\\Sig_2(v)$ be the vertices of $\\Sig$ at distance $2$ from $v$.\n\n\\begin{lemma}{\\rm \\cite[Theorem~1.3~(3)-(4)]{Zhou-EJC2021}}\\label{rank}\nLet $\\Sig$ be a finite $(H, 3)$-connected-set-homogeneous non-complete graph of girth $3$ with $H\\leq\\Aut(\\Sig)$. Then for every edge $\\{\\a, \\b\\}$ of $\\Sig$, we have the following:\n\\begin{enumerate}\n  \\item [{\\rm (1)}]\\ $H_{\\a\\b}$ has $s$ orbits on $\\Sig(\\a)\\cap \\Sig(\\b)$ with equal size, where $s=1, 2, 3$ or $6$.\n  \\item [{\\rm (2)}]\\ $H_{\\a\\b}$ has $t$ orbits on $\\Sig(\\a)-(\\Sig(\\a)\\cap \\Sig(\\b))\\cup\\{\\b\\}$ with equal size, where $t=1$ or $ 2$.\n      \\end{enumerate}\n\\end{lemma}\n\n\\begin{lemma}\\label{lm:Gab10orbits}\nUnder Hypothesis~{\\rm \\ref{hyp}}, let $\\{\\alpha,\\beta\\}\\in E$. Then $G_{\\alpha\\beta}$ has at most $12$ orbits on $V(\\mathit{\\Ga})\\setminus\\{\\alpha,\\beta\\}$.\n\\end{lemma}\n\n\\begin{proof}By Hypothesis~{\\rm \\ref{hyp}}, $\\Ga$ is a $(G,3)$-set-homogeneous graph. Furthermore, $\\Ga$ has diameter $2$ and girth $3$. Note that $V(\\mathit{\\Ga})\\setminus\\{\\alpha,\\beta\\}$ is partitioned into the following four subsets:\n\\[\\Ga(\\a)\\cap\\Ga(\\b), \\Ga(\\a)\\cap\\Ga_2(\\b), \\Ga_2(\\a)\\cap\\Ga(\\b), \\Ga_2(\\a)\\cap\\Ga_2(\\b).\\]\nMoreover, $G_{\\a\\b}$ setwise fixes each of the four subsets.\n\nSince every $(G,3)$-set-homogeneous graph is $(G,3)$-connected-set-homogeneous, by Lemma~\\ref{rank} $G_{\\a\\b}$ has at most $6$ orbits on $\\Ga(\\a)\\cap\\Ga(\\b)$, and has at most $2$ orbits on $\\Ga(\\a)\\cap\\Ga_2(\\b)$ and on $\\Ga_2(\\a)\\cap\\Ga(\\b)$.\n\nTo complete the proof, it suffices to prove that $G_{\\a\\b}$ has at most two orbits on $\\Ga_2(\\a)\\cap\\Ga_2(\\b)$. To see this, we may assume that $G_{\\a\\b}$ is intransitive on $\\Ga(\\a)\\cap\\Ga_2(\\b)$. Then there exist $\\d_1, \\d_2\\in\\Ga_2(\\a)\\cap\\Ga_2(\\b)$ such that the orbits $\\d_1^{G_{\\a\\b}}\\neq \\d_2^{G_{\\a\\b}}$. Then $\\d_1,\\d_2$ are not adjacent to $\\a$ and $\\b$. Since $\\Ga$ is $(G,3)$-set-homogeneous, there exists a $g\\in G$ such that $\\d_1^g=\\d_2$, and $\\a^g=\\b$ and $\\b^g=\\a$. For any $\\d\\in \\Ga_2(\\a)\\cap\\Ga_2(\\b)$, also since $\\Ga$ is $(G,3)$-set-homogeneous, there exists $h\\in G_{\\{\\a,\\b\\}}$ such that $\\d^h=\\d_1$. If $h\\in G_{\\a\\b}$, then $\\d\\in \\d_1^{G_{\\a\\b}}$. If $h\\notin G_{\\a\\b}$, then $h$ swaps $\\a$ and $\\b$ and so $hg\\in G_{\\a\\b}$. Furthermore, $\\d^{hg}=\\d_1^g=\\d_2$, and hence $\\d\\in \\d_2^{G_{\\a\\b}}$. Thus, $\\Ga_2(\\a)\\cap\\Ga_2(\\b)=\\d_1^{G_{\\a\\b}}\\cup \\d_2^{G_{\\a\\b}}$.\nThe proof is completed.\n\\hfill\\qed\n\\end{proof}\n\nThe next lemma gives some basic properties about strongly regular graphs, see \\cite[\\textsection{1.1.1-1.1.2}]{BrouwerSRG}.\n\n\\begin{lem}\\label{lem:drg-basic}\nLet $\\Ga$ be a strongly regular graph with parameters $(v, k, \\lambda, \\mu)$. Then the following hold:\n\\begin{enumerate}\n  \\item [{\\rm (1)}]\\ The complement $\\overline{\\Ga}$ of $\\Ga$ is strongly regular with parameters $(v, \\overline{k}, \\overline{\\ld}, \\overline{\\mu})$, where $\\overline{k}=v-k-1$, $\\overline{\\ld}=v-2k+\\mu-2$ and $\\overline{\\mu}=v-2k+\\lambda$;\n  \\item [{\\rm (2)}]\\ $0\\leq \\ld\\leq k-1$ and $0\\leq\\mu\\leq k$;\n  \\item [{\\rm (3)}]\\ $k(k-1-\\lambda)=\\mu(v-1-k)$.\n\\end{enumerate}\n\\end{lem}\n\nLet $\\mathbf{P}_{3}$ be a path of length $2$. For a graph $\\Ga$, let\n\\[\n\\begin{array}{l}\n\\Ome_1(\\Ga):=\\{U\\subseteq V(\\Ga) \\mid [U]\\cong \\K_1\\cup\\K_2 \\},\\\\\n\\Ome_2(\\Ga):=\\{U\\subseteq V(\\Ga) \\mid [U]\\cong \\mathbf{P}_{3}\\},\\\\\n\\Ome_3(\\Ga):=\\{U\\subseteq V(\\Ga) \\mid [U]\\cong \\K_3\\},\\\\\n\\Ome_4(\\Ga):=\\{U\\subseteq V(\\Ga) \\mid [U]\\cong 3\\K_1\\}.\\\\\n\\end{array}", "set:proof-of-main": "\\label{set:proof-of-main}\n\nBy definition, both 1-homogeneous graphs and 1-set-homogeneous graphs are exactly the vertex-transitive graphs. Assume now that $k\\geq 2$. It is obvious by definition that e", "th-main": "\\begin{theorem}\\label{th-main}\nFor each positive integer $k$, a finite graph is $k$-set-homogeneous if and only if it is $k$-homogeneous.\nMoreover, { if $\\Ga$ is a $k$-homogeneous graph with $k\\geq 2$, then} one of the following holds:\n\\begin{itemize}\n\\item [{\\rm(1)}] $k\\geqslant 5$, and $\\Ga$ is one of { $\\C_5 (\\cong\\overline{\\C_5})$, $\\K_3\\square\\K_3 (\\cong \\overline{\\K_3\\square\\K_3})$, $n\\K_m$, and $\\K_{n[m]}$; furthermore, $\\Ga$ is homogeneous.}\n\n{ \\item[(2)] $\\Ga$ is either the Schl\\\"{a}fli graph on $27$ vertices or its complement (the two non-trivial orbital graphs of $\\mathrm{PSU}_4(2)$ acting on the isotropic lines); furthermore, $\\Ga$ is $4$-homogeneous but not $5$-homogeneous.\n\n\\item[(3)] $\\Ga$ is either $\\K_n\\square\\K_n$ with $n\\geqslant4$, or a graph from Theorem~\\ref{th:3sethomo}(1)--(4), or the complement of any such graph; furthermore, $\\Ga$ is $3$-homogeneous but not $4$-homogeneous.\n\n\\item[(4)] $\\Ga$ is an orbital graph of a primitive permutation group of rank $3$ which is not covered by parts (1)--(3); furthermore, $\\Ga$ is $2$-homogeneous but not $3$-homogeneous. }\n\n\\end{itemize}\n\\end{theorem}", "th:3sethomo": "\\begin{theorem}\\label{th:3sethomo}\nLet $\\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\\Ga$ is isomorphic either to one of the graphs: $\\C_5$, $\\K_n$, $\\K_{m[b]}$, $\\K_n\\square\\K_n$, or to one of the following graphs:\n\\begin{itemize}\n\\item[{\\rm(1)}] the affine polar graph $\\mathrm{VO}_{2m}^{\\epsilon}(2)$ with $m\\geq 1$ and $\\epsilon \\in \\{+,-\\}$;\n\n\\item[{\\rm(2)}]  the elliptic orthogonal graph $\\mathrm{\\Gamma}(\\mathrm{O}^-_6(q))$ with $q$ a prime power;\n\n\\item[{\\rm(3)}] an orbital graph of the Higman-Sims group $\\mathrm{HS}$ acting on $100$ points;\n\n\\item[{\\rm(4)}] an orbital graph of the McLaughlin group $\\mathrm{McL}$ acting on $275$ points.\n\\end{itemize}\nIn particular, all $3$-set-homogeneous graphs are $3$-homogeneous.\n\\end{theorem}", "trans-rank-4": "\\begin{theorem}\\label{trans-rank-4}\nLet $G$ be a primitive permutation group of rank $4$. Then either all orbitals of $G$ are self-paired, or $G=\\PSU_3(3)$ acting on $36$ points.\n\\end{theorem}", "subsec:2reduce": "\\begin{itemize}\n\\item[{\\rm(1)}] the affine polar graph $\\mathrm{VO}_{2m}^{\\epsilon}(2)$ with $m\\geq 1$ and $\\epsilon \\in \\{+,-\\}$;\n\n\\item[{\\rm(2)}]  the elliptic orthogonal graph $\\mathrm{\\Gamma}(\\mathrm{O}^-_6(q))$ with $q$ a prime power;\n\n\\item[{\\rm(3)}] an orbital graph of the Higman-Sims group $\\mathrm{HS}$ acting on $100$ points;\n\n\\item[{\\rm(4)}] an orbital graph of the McLaughlin group $\\mathrm{McL}$ acting on $275$ points.\n\\end{itemize}\nIn particular, all $3$-set-homogeneous graphs are $3$-homogeneous.\n\\end{theorem}\n\nNote that all graphs in Theorem~\\ref{th:3sethomo}~(1)--(4) are actually $3$-homogeneous graphs, which {are known} in the literature.\n\nThe rest of the paper is organised as follows. In Sections~\\ref{subsec:2reduce}--\\ref{sec6:proof-main-theorems}, we deal with finite $(G, 2)$-set-homogeneous graphs.\nIn Section~\\ref{sec6:proof-main-theorems}, we prove Theorem~\\ref{th:2sethomo} and Theorem~\\ref{trans-rank-4}.\nIn Section~\\ref{set:3sethom}, we deal with finite $(G, 3)$-set-homogeneous graphs, and prove Theorem~\\ref{th:3sethomo}.\nIn Section~\\ref{set:proof-of-main}, we prove Theorem~\\ref{th-main}.\n\n\\section{A reduction}\\label{subsec:2reduce}\nIn this section, we shall reduce the proof of Theorem~\\ref{th:2sethomo} to two cases: $G$ is a primitive permutation group of rank $5$ { and odd order}, and $G$ is a primitive permutation group of rank $4$ which has exactly one self-paired orbital. The former case will be treated in Section~\\ref{subset:prime-rank5}, and the latter case will be treated in Sections~\\ref{subsec:affineprimitive-rank4} and \\ref{subsec:almostsimple-rank4}. Based on these, Theorem~\\ref{th:2sethomo} and Theorem~\\ref{trans-rank-4} will be proved in Section~\\ref{sec6:proof-main-theorems}.\n\nWe first introduce several terms of permutation groups.\nLet $\\Ome$ be a finite nonempty set. Denote by $\\Sym(\\Ome)$ the symmetric group on $\\Ome$. Let $G\\leqslant\\Sym(\\Ome)$ be transitive.\nLet $\\Ome\\times\\Ome=\\{(u,v)\\mid u,v\\in \\Ome\\}$.\nAn orbit $(u,v)^G$ of $G$ on $\\Ome\\times\\Ome$ is called an {\\it orbital} of $G$ on $\\Ome$, and\n$(v,u)^G$ is called the {\\it paired orbital} of $(u,v)^G$.\nAn orbital  $(u,v)^G$ is called {\\it self-paired} if it equals $(v,u)^G$.\nThe orbital $(u,u)^G$ is called {\\it trivial}, and the others are non-trivial. To an orbital $E$ we\nassociate the digraph with vertex set $\\Ome$ and arc set $E$, called the {\\em orbital digraph} for\n$E$, denoted by the pair $(\\Ome, E)$. For a given point $u$, each orbital $E=(u,v)^G$ corresponds to an orbit of $G_u$ on $\\Ome$, namely, $\\{v\\ |\\ (u,v)\\in E\\}$. The orbits of $G_u$ on $\\Om$ are called {\\em suborbits} of $G$, and their lengths are called the {\\em subdegrees} of $G$.\nFor a set $\\Ome$, let $\\Ome^{\\{2\\}}$ be the set of 2-subsets of $\\Ome$, namely, $\\Ome^{\\{2\\}}=\\{\\{u,v\\}\\mid u\\not=v\\in \\Ome\\}$,  and let $\\Ome^{(2)}=\\{(u,v)\\mid u\\not=v\\in \\Ome\\}$.\n\n\\begin{lemma}\\label{rank<=5}\nLet $\\Ga=(V,E)$ be a $(G, 2)$-set-homogeneous graph, where $G\\leq\\Aut(\\Ga)$.\nThen $G$ is a permutation group on $V$  of rank at most $5$, and\none of the following holds:\n\\begin{itemize}\n\\item[{\\rm(1)}] $G$ is of rank $2$, and $\\Ga$ is a complete graph or an empty graph.\n\n\\item[{\\rm(2)}] $G$ is of rank $3$, and {either $\\Ga$ is a complete graph or an empty graph, or $\\Ga$ is $(G, 2)$-homogeneous}.\n\n\\item[{\\rm(3)}] $G$ is of rank $4$, and exactly one of the $3$ non-trivial orbitals is self-paired.\n\n\\item[{\\rm(4)}] $G$ is of rank $5$, and none of the $4$ non-trivial orbitals is self-paired.\n\n\\end{itemize}", "Def-homog": "\\begin{definition}\\label{Def-homog}\n{\\rm\nLet $\\Ga=(V,E)$ be a graph, $G\\leqslant\\Aut(\\Ga)$, and  $k$ a positive integer.\n\\begin{itemize}\n\\item[(1)] If each isomorphism between any pair of isomorphic induced subgraphs of $\\Ga$ of order at most $k$\nextends to an automorphism of $\\Ga$ { contained} in $G$, then $\\Ga$ is called a {\\it $(G, k)$-homogeneous} graph.\n(The {\\it order} of a graph is the number of vertices.)\n\n\\item[(2)] If any two isomorphic induced subgraphs of $\\Ga$ of order at most $k$ are equivalent under $G$, then $\\Ga$ is called a {\\it $(G,k)$-set-homogeneous} graph.\n\n\\item[(3)] {When $G=\\Aut(\\Ga)$, a $(G, k)$-homogeneous or $(G, k)$-set-homogeneous graph is simply called {\\em $k$-homogeneous} or {\\em $k$-set-homogeneous}, respectively.}\n\\end{itemize}\n}\n\\end{definition}", "rank-4": "\\begin{theorem}\\label{rank-4}\nLet $G$ be an almost simple primitive rank $4$ permutation group on a set $\\Om$ with a unique non-self-paired orbital.\nThen $G=\\PSU_3(3)$ with stabiliser $\\PSL_3(2)$, and the subdegrees are $1$, $7$, $7$ and $21$. Moreover, $\\mathbf{N}_{\\mathrm{Sym}(\\mathit{\\Omega})}(G)=\\mathrm{P\\Gamma U}_3(3)$ is a primitive rank $3$ group with subdegrees $1$, $14$ and $21$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 5005, "pre_theorem_intro_text": "The main purpose of this paper is to solve the long-standing open problem:\n\n\\vskip0.1in\n\\f{\\bf Problem A.} {\\rm {Classify the finite graphs in which, for a given positive integer $k$, any two isomorphic induced subgraphs of order at most $k$ are equivalent under the automorphism group.}}\n\\vskip0.1in\n\nBefore proceeding, we give some background on this topic.\nThe strongest possible symmetry condition that one can impose {on a graph is {\\it homogeneity}.}\nA graph is called {\\em homogeneous} if every isomorphism between any pair of isomorphic induced subgraphs extends to an automorphism of the graph. The study of graphs with {certain homogeneity-type properties} was initiated by Sheehan~{\\cite{Sheehan1972,Sheehan1973,Sheehan1973a,Sheehan1974}},\nand Gardiner~\\cite{Gardiner1976} in the 1970s.\n{It turns out that} the only finite homogeneous graphs are the following `trivial graphs':\n\\[\\C_5,\\ \\K_3\\square\\K_3,\\ n\\K_m,\\ \\K_{n[m]}.\\]\nA graph $\\Ga$ is called {\\em set-homogeneous} if isomorphic subgraphs of $\\Ga$ are equivalent under $\\Aut(\\Ga)$.\n{In 1978, Ronse~\\cite{Ronse}} proved that every finite set-homogeneous graph is homogeneous.\n{Slightly weaker versions of homogeneity have been studied in the literature} (see Definition~\\ref{Def-homog} for instance) and also {have been extended} to other mathematical structures. Over the past 50 years, { a fair amount of work has been done on objects with certain homogeneity-type properties. The reader may refer to the excellent survey paper of Macpherson~\\cite{Macpherson-survey} for an overview.}\n\n{One of these weaker conditions is as follows.}\n\n\\begin{definition}\\label{Def-homog}\n{\\rm\nLet $\\Ga=(V,E)$ be a graph, $G\\leqslant\\Aut(\\Ga)$, and  $k$ a positive integer.\n\\begin{itemize}\n\\item[(1)] If each isomorphism between any pair of isomorphic induced subgraphs of $\\Ga$ of order at most $k$\nextends to an automorphism of $\\Ga$ { contained} in $G$, then $\\Ga$ is called a {\\it $(G, k)$-homogeneous} graph.\n(The {\\it order} of a graph is the number of vertices.)\n\n\\item[(2)] If any two isomorphic induced subgraphs of $\\Ga$ of order at most $k$ are equivalent under $G$, then $\\Ga$ is called a {\\it $(G,k)$-set-homogeneous} graph.\n\n\\item[(3)] {When $G=\\Aut(\\Ga)$, a $(G, k)$-homogeneous or $(G, k)$-set-homogeneous graph is simply called {\\em $k$-homogeneous} or {\\em $k$-set-homogeneous}, respectively.}\n\\end{itemize}\n}\n\\end{definition}\n\nA major theme in the area has been the following classification problem.\n\n\\vskip0.1in\n\\f{\\bf Problem B.}\n{\\rm Classify $k$-homogeneous graphs and $k$-set-homogeneous graphs.}\n\\vskip0.1in\n\nThis paper completely solves the problem, and particularly shows that a finite graph is $k$-homogeneous if and only if it is $k$-set-homogeneous.\n\n{The study of Problem~B has lasted for over half a century, dating back to Sheehan and Gardiner's classification of finite homogeneous graphs in the 1970s mentioned above.} Later, Lachlan and Woodrow~\\cite{Lachlan-Woodrow} obtained a classification of infinite countable homogeneous graphs. For more results regarding homogeneous graphs, we refer the reader to \\cite{Enomoto} and \\cite{Gray,Gray-Macpherson,GMPR,HH,Henson}.\n\nBy definition, $1$-homogeneous graphs and $1$-set-homogeneous graphs are exactly vertex-transitive graphs.\nIt is infeasible to obtain { an explicit list of vertex-transitive graphs.}\nHowever, { an explicit classification has been achieved for}\nfinite $k$-homogeneous graphs with $k\\geqslant 2$ in a collection of papers. Cameron~\\cite{Cameron-6-transitive} proved that the list of finite $5$-homogeneous graphs is the same as that for homogeneous graphs;\nBuczak~\\cite{Buczak} classified finite $4$-homogeneous graphs; and Cameron and Macpherson~\\cite{Cameron-3-hom} classified finite $3$-homogeneous graphs.\n{We observe} that a graph $\\Ga$ is $2$-homogeneous if and only if $\\Ga$ and its complement $\\ov\\Ga$ are both arc-transitive. {Thus, a finite $2$-homogeneous graph is a complete multipartite graph or its complement, or an orbital graph of a primitive permutation group of rank $3$.\nNote that primitive permutation groups of rank $3$ have been classified in \\cite{Bannai,Kantor,Liebeck-affine,Liebeck-Saxl}.}\n\nThe main purpose of this paper is to solve Problem~B by proving that every $k$-set-homogeneous graph is $k$-homogeneous for $k\\geqslant 2$, { so that $k$-set-homogeneous graphs are explicitly classified, as described in the previous paragraph.}\nTo state {our main theorem}, we fix some graph notation. Let $m$ and $n$ be positive integers. We denote the complete graph with $n$ vertices by $\\K_n$, the cycle with $n$ vertices by $\\C_n$, the complete multipartite graph with $n$ parts of\nsize $m$ by $\\K_{n[m]}$, and its complement (the disjoint union of $n$ copies of $\\K_m$) by $n\\K_m$. Let $\\K_n\\square\\K_n$ and $\\K_n\\times\\K_n$ be the Cartesian product and direct product of two $\\K_n$'s, respectively. For a graph $\\Ga$, we denote by $\\ov\\Ga$ the {\\it complement} of $\\Ga$. Note that $\\overline{\\K_n\\square\\K_n}=\\K_n\\times\\K_n$.", "context": "\\vskip0.1in\n\\f{\\bf Problem A.} {\\rm {Classify the finite graphs in which, for a given positive integer $k$, any two isomorphic induced subgraphs of order at most $k$ are equivalent under the automorphism group.}}\n\\vskip0.1in\n\nBefore proceeding, we give some background on this topic.\nThe strongest possible symmetry condition that one can impose {on a graph is {\\it homogeneity}.}\nA graph is called {\\em homogeneous} if every isomorphism between any pair of isomorphic induced subgraphs extends to an automorphism of the graph. The study of graphs with {certain homogeneity-type properties} was initiated by Sheehan~{\\cite{Sheehan1972,Sheehan1973,Sheehan1973a,Sheehan1974}},\nand Gardiner~\\cite{Gardiner1976} in the 1970s.\n{It turns out that} the only finite homogeneous graphs are the following `trivial graphs':\n\\[\\C_5,\\ \\K_3\\square\\K_3,\\ n\\K_m,\\ \\K_{n[m]}.\\]\nA graph $\\Ga$ is called {\\em set-homogeneous} if isomorphic subgraphs of $\\Ga$ are equivalent under $\\Aut(\\Ga)$.\n{In 1978, Ronse~\\cite{Ronse}} proved that every finite set-homogeneous graph is homogeneous.\n{Slightly weaker versions of homogeneity have been studied in the literature} (see Definition~\\ref{Def-homog} for instance) and also {have been extended} to other mathematical structures. Over the past 50 years, { a fair amount of work has been done on objects with certain homogeneity-type properties. The reader may refer to the excellent survey paper of Macpherson~\\cite{Macpherson-survey} for an overview.}\n\n\\begin{definition}\\label{Def-homog}\n{\\rm\nLet $\\Ga=(V,E)$ be a graph, $G\\leqslant\\Aut(\\Ga)$, and $k$ a positive integer.\n\\begin{itemize}\n\\item[(1)] If each isomorphism between any pair of isomorphic induced subgraphs of $\\Ga$ of order at most $k$\nextends to an automorphism of $\\Ga$ { contained} in $G$, then $\\Ga$ is called a {\\it $(G, k)$-homogeneous} graph.\n(The {\\it order} of a graph is the number of vertices.)\n\n\\item[(3)] {When $G=\\Aut(\\Ga)$, a $(G, k)$-homogeneous or $(G, k)$-set-homogeneous graph is simply called {\\em $k$-homogeneous} or {\\em $k$-set-homogeneous}, respectively.}\n\\end{itemize}\n}\n\\end{definition}\n\nBy definition, $1$-homogeneous graphs and $1$-set-homogeneous graphs are exactly vertex-transitive graphs.\nIt is infeasible to obtain { an explicit list of vertex-transitive graphs.}\nHowever, { an explicit classification has been achieved for}\nfinite $k$-homogeneous graphs with $k\\geqslant 2$ in a collection of papers. Cameron~\\cite{Cameron-6-transitive} proved that the list of finite $5$-homogeneous graphs is the same as that for homogeneous graphs;\nBuczak~\\cite{Buczak} classified finite $4$-homogeneous graphs; and Cameron and Macpherson~\\cite{Cameron-3-hom} classified finite $3$-homogeneous graphs.\n{We observe} that a graph $\\Ga$ is $2$-homogeneous if and only if $\\Ga$ and its complement $\\ov\\Ga$ are both arc-transitive. {Thus, a finite $2$-homogeneous graph is a complete multipartite graph or its complement, or an orbital graph of a primitive permutation group of rank $3$.\nNote that primitive permutation groups of rank $3$ have been classified in \\cite{Bannai,Kantor,Liebeck-affine,Liebeck-Saxl}.}\n\nThe main purpose of this paper is to solve Problem~B by proving that every $k$-set-homogeneous graph is $k$-homogeneous for $k\\geqslant 2$, { so that $k$-set-homogeneous graphs are explicitly classified, as described in the previous paragraph.}\nTo state {our main theorem}, we fix some graph notation. Let $m$ and $n$ be positive integers. We denote the complete graph with $n$ vertices by $\\K_n$, the cycle with $n$ vertices by $\\C_n$, the complete multipartite graph with $n$ parts of\nsize $m$ by $\\K_{n[m]}$, and its complement (the disjoint union of $n$ copies of $\\K_m$) by $n\\K_m$. Let $\\K_n\\square\\K_n$ and $\\K_n\\times\\K_n$ be the Cartesian product and direct product of two $\\K_n$'s, respectively. For a graph $\\Ga$, we denote by $\\ov\\Ga$ the {\\it complement} of $\\Ga$. Note that $\\overline{\\K_n\\square\\K_n}=\\K_n\\times\\K_n$.\n\n\\begin{definition}\\label{Def-homog}\n{\\rm\nLet $\\Ga=(V,E)$ be a graph, $G\\leqslant\\Aut(\\Ga)$, and  $k$ a positive integer.\n\\begin{itemize}\n\\item[(1)] If each isomorphism between any pair of isomorphic induced subgraphs of $\\Ga$ of order at most $k$\nextends to an automorphism of $\\Ga$ { contained} in $G$, then $\\Ga$ is called a {\\it $(G, k)$-homogeneous} graph.\n(The {\\it order} of a graph is the number of vertices.)\n\n\\item[(2)] If any two isomorphic induced subgraphs of $\\Ga$ of order at most $k$ are equivalent under $G$, then $\\Ga$ is called a {\\it $(G,k)$-set-homogeneous} graph.\n\n\\item[(3)] {When $G=\\Aut(\\Ga)$, a $(G, k)$-homogeneous or $(G, k)$-set-homogeneous graph is simply called {\\em $k$-homogeneous} or {\\em $k$-set-homogeneous}, respectively.}\n\\end{itemize}\n}\n\\end{definition}\n\n\\begin{theorem}\\label{th:3sethomo}\nLet $\\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\\Ga$ is isomorphic either to one of the graphs: $\\C_5$, $\\K_n$, $\\K_{m[b]}$, $\\K_n\\square\\K_n$, or to one of the following graphs:\n\\begin{itemize}\n\\item[{\\rm(1)}] the affine polar graph $\\mathrm{VO}_{2m}^{\\epsilon}(2)$ with $m\\geq 1$ and $\\epsilon \\in \\{+,-\\}$;\n\n\\item[{\\rm(2)}]  the elliptic orthogonal graph $\\mathrm{\\Gamma}(\\mathrm{O}^-_6(q))$ with $q$ a prime power;\n\n\\item[{\\rm(3)}] an orbital graph of the Higman-Sims group $\\mathrm{HS}$ acting on $100$ points;\n\n\\item[{\\rm(4)}] an orbital graph of the McLaughlin group $\\mathrm{McL}$ acting on $275$ points.\n\\end{itemize}\nIn particular, all $3$-set-homogeneous graphs are $3$-homogeneous.\n\\end{theorem}", "full_context": "\\vskip0.1in\n\\f{\\bf Problem A.} {\\rm {Classify the finite graphs in which, for a given positive integer $k$, any two isomorphic induced subgraphs of order at most $k$ are equivalent under the automorphism group.}}\n\\vskip0.1in\n\nBefore proceeding, we give some background on this topic.\nThe strongest possible symmetry condition that one can impose {on a graph is {\\it homogeneity}.}\nA graph is called {\\em homogeneous} if every isomorphism between any pair of isomorphic induced subgraphs extends to an automorphism of the graph. The study of graphs with {certain homogeneity-type properties} was initiated by Sheehan~{\\cite{Sheehan1972,Sheehan1973,Sheehan1973a,Sheehan1974}},\nand Gardiner~\\cite{Gardiner1976} in the 1970s.\n{It turns out that} the only finite homogeneous graphs are the following `trivial graphs':\n\\[\\C_5,\\ \\K_3\\square\\K_3,\\ n\\K_m,\\ \\K_{n[m]}.\\]\nA graph $\\Ga$ is called {\\em set-homogeneous} if isomorphic subgraphs of $\\Ga$ are equivalent under $\\Aut(\\Ga)$.\n{In 1978, Ronse~\\cite{Ronse}} proved that every finite set-homogeneous graph is homogeneous.\n{Slightly weaker versions of homogeneity have been studied in the literature} (see Definition~\\ref{Def-homog} for instance) and also {have been extended} to other mathematical structures. Over the past 50 years, { a fair amount of work has been done on objects with certain homogeneity-type properties. The reader may refer to the excellent survey paper of Macpherson~\\cite{Macpherson-survey} for an overview.}\n\n\\begin{definition}\\label{Def-homog}\n{\\rm\nLet $\\Ga=(V,E)$ be a graph, $G\\leqslant\\Aut(\\Ga)$, and $k$ a positive integer.\n\\begin{itemize}\n\\item[(1)] If each isomorphism between any pair of isomorphic induced subgraphs of $\\Ga$ of order at most $k$\nextends to an automorphism of $\\Ga$ { contained} in $G$, then $\\Ga$ is called a {\\it $(G, k)$-homogeneous} graph.\n(The {\\it order} of a graph is the number of vertices.)\n\n\\item[(3)] {When $G=\\Aut(\\Ga)$, a $(G, k)$-homogeneous or $(G, k)$-set-homogeneous graph is simply called {\\em $k$-homogeneous} or {\\em $k$-set-homogeneous}, respectively.}\n\\end{itemize}\n}\n\\end{definition}\n\nBy definition, $1$-homogeneous graphs and $1$-set-homogeneous graphs are exactly vertex-transitive graphs.\nIt is infeasible to obtain { an explicit list of vertex-transitive graphs.}\nHowever, { an explicit classification has been achieved for}\nfinite $k$-homogeneous graphs with $k\\geqslant 2$ in a collection of papers. Cameron~\\cite{Cameron-6-transitive} proved that the list of finite $5$-homogeneous graphs is the same as that for homogeneous graphs;\nBuczak~\\cite{Buczak} classified finite $4$-homogeneous graphs; and Cameron and Macpherson~\\cite{Cameron-3-hom} classified finite $3$-homogeneous graphs.\n{We observe} that a graph $\\Ga$ is $2$-homogeneous if and only if $\\Ga$ and its complement $\\ov\\Ga$ are both arc-transitive. {Thus, a finite $2$-homogeneous graph is a complete multipartite graph or its complement, or an orbital graph of a primitive permutation group of rank $3$.\nNote that primitive permutation groups of rank $3$ have been classified in \\cite{Bannai,Kantor,Liebeck-affine,Liebeck-Saxl}.}\n\nThe main purpose of this paper is to solve Problem~B by proving that every $k$-set-homogeneous graph is $k$-homogeneous for $k\\geqslant 2$, { so that $k$-set-homogeneous graphs are explicitly classified, as described in the previous paragraph.}\nTo state {our main theorem}, we fix some graph notation. Let $m$ and $n$ be positive integers. We denote the complete graph with $n$ vertices by $\\K_n$, the cycle with $n$ vertices by $\\C_n$, the complete multipartite graph with $n$ parts of\nsize $m$ by $\\K_{n[m]}$, and its complement (the disjoint union of $n$ copies of $\\K_m$) by $n\\K_m$. Let $\\K_n\\square\\K_n$ and $\\K_n\\times\\K_n$ be the Cartesian product and direct product of two $\\K_n$'s, respectively. For a graph $\\Ga$, we denote by $\\ov\\Ga$ the {\\it complement} of $\\Ga$. Note that $\\overline{\\K_n\\square\\K_n}=\\K_n\\times\\K_n$.\n\n\\begin{definition}\\label{Def-homog}\n{\\rm\nLet $\\Ga=(V,E)$ be a graph, $G\\leqslant\\Aut(\\Ga)$, and  $k$ a positive integer.\n\\begin{itemize}\n\\item[(1)] If each isomorphism between any pair of isomorphic induced subgraphs of $\\Ga$ of order at most $k$\nextends to an automorphism of $\\Ga$ { contained} in $G$, then $\\Ga$ is called a {\\it $(G, k)$-homogeneous} graph.\n(The {\\it order} of a graph is the number of vertices.)\n\n\\item[(2)] If any two isomorphic induced subgraphs of $\\Ga$ of order at most $k$ are equivalent under $G$, then $\\Ga$ is called a {\\it $(G,k)$-set-homogeneous} graph.\n\n\\item[(3)] {When $G=\\Aut(\\Ga)$, a $(G, k)$-homogeneous or $(G, k)$-set-homogeneous graph is simply called {\\em $k$-homogeneous} or {\\em $k$-set-homogeneous}, respectively.}\n\\end{itemize}\n}\n\\end{definition}\n\n\\begin{theorem}\\label{th:3sethomo}\nLet $\\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\\Ga$ is isomorphic either to one of the graphs: $\\C_5$, $\\K_n$, $\\K_{m[b]}$, $\\K_n\\square\\K_n$, or to one of the following graphs:\n\\begin{itemize}\n\\item[{\\rm(1)}] the affine polar graph $\\mathrm{VO}_{2m}^{\\epsilon}(2)$ with $m\\geq 1$ and $\\epsilon \\in \\{+,-\\}$;\n\n\\item[{\\rm(2)}]  the elliptic orthogonal graph $\\mathrm{\\Gamma}(\\mathrm{O}^-_6(q))$ with $q$ a prime power;\n\n\\item[{\\rm(3)}] an orbital graph of the Higman-Sims group $\\mathrm{HS}$ acting on $100$ points;\n\n\\item[{\\rm(4)}] an orbital graph of the McLaughlin group $\\mathrm{McL}$ acting on $275$ points.\n\\end{itemize}\nIn particular, all $3$-set-homogeneous graphs are $3$-homogeneous.\n\\end{theorem}\n\n\\begin{abstract}\nA classification is given of finite $k$-set-homogeneous graphs for $k\\geqslant 2$, leading to a striking result that each finite $k$-set-homogeneous graph is $k$-homogeneous. It shows that $3$-set-homogeneous graphs are rare, consisting of the following graphs and their complements: $\\C_5$, $\\K_n\\square\\K_n$, $n\\K_m$, the Schl\\\"{a}fli graph of order 27, the Higman-Sims graph, the MaLaughlin graph, {affine polar graphs, and elliptic orthogonal graphs}. As an ingredient for the proof, it is shown that all orbitals in a primitive permutation group of rank $4$ are self-paired, except for $\\PSU_3(3)$ acting on 36 points.\n\\bigskip\n\nThe main purpose of this paper is to solve Problem~B by proving that every $k$-set-homogeneous graph is $k$-homogeneous for $k\\geqslant 2$, { so that $k$-set-homogeneous graphs are explicitly classified, as described in the previous paragraph.}\nTo state {our main theorem}, we fix some graph notation. Let $m$ and $n$ be positive integers. We denote the complete graph with $n$ vertices by $\\K_n$, the cycle with $n$ vertices by $\\C_n$, the complete multipartite graph with $n$ parts of\nsize $m$ by $\\K_{n[m]}$, and its complement (the disjoint union of $n$ copies of $\\K_m$) by $n\\K_m$. Let $\\K_n\\square\\K_n$ and $\\K_n\\times\\K_n$ be the Cartesian product and direct product of two $\\K_n$'s, respectively. For a graph $\\Ga$, we denote by $\\ov\\Ga$ the {\\it complement} of $\\Ga$. Note that $\\overline{\\K_n\\square\\K_n}=\\K_n\\times\\K_n$.\n\n{\\noindent{\\bf Remark on Theorem~\\ref{th-main}.}\\ For $k\\geq2$, all $k$-homogeneous graphs have already been classified in the literature, { as mentioned above}, and so all graphs in Theorem~\\ref{th-main}~(1)--(4) are known.}\\medskip\n\n\\begin{theorem}\\label{th:2sethomo}\nLet $\\Ga=(V,E)$ be a finite $(G,2)$-set-homogeneous graph, where $G\\leq\\Aut(\\Ga)$.\nThen one of the following holds:\n\\begin{itemize}\n\\item[{\\rm(1)}] $\\Ga\\cong\\K_n$ or $n\\K_1$, and $G$ is a $2$-homogeneous permutation group on $V$.\n\n\\begin{theorem}\\label{th:3sethomo}\nLet $\\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\\Ga$ is isomorphic either to one of the graphs: $\\C_5$, $\\K_n$, $\\K_{m[b]}$, $\\K_n\\square\\K_n$, or to one of the following graphs:\n\\begin{itemize}\n\\item[{\\rm(1)}] the affine polar graph $\\mathrm{VO}_{2m}^{\\epsilon}(2)$ with $m\\geq 1$ and $\\epsilon \\in \\{+,-\\}$;\n\n\\begin{theorem}[\\cite{Liebeck-affine}]\\label{subdegree}\nLet $X$ be a finite primitive affine permutation group of rank $3$ and of degree $p^d$, with socle $V$, where $V\\cong\\mz_p^d$ for some prime $p$, and let $X_0$ be the stabiliser of the zero vector in $V$. Then $X_0$ belongs to one of the following classes $($and, conversely, each of the possibilities listed below does give rise to a rank $3$ affine group$)$.\n\\begin{itemize}\n\\item[\\rm (A)] Infinite classes. These are\n\\begin{itemize}\n\\item[\\rm (1)] $X_0\\leq \\mathrm{\\Gamma L}_1(p^d)$: all possibilities for $X_0$ are determined in~\\cite[\\textsection{3}]{Foulser-Kallaher-solvable-rank-3};\n\\item[\\rm (2)] $X_0$ imprimitive: $X_0$ stabilises a pair $\\{V_1,V_2 \\}$ of subspaces of $V$ where $V=V_1 \\oplus V_2$ and $\\dim(V_1)=\\dim(V_2)$; moreover, $(X_0)_{V_i}$ is transitive on $V_i\\setminus 0$ for $i=1$, $2$ (and hence $ X_0$ is determined by Hering's Theorem~\\cite{Hering});\n\\item[\\rm (3)] tensor product case: for some $a$, $q$ with $q^a= p^d$, consider $V$ as a vector space $\\GF(q)^a$ of dimension $a$ over $\\GF(q)$; then $X_0$ stabilises a decomposition of $\\GF(q)^a$ as a tensor product $V_1 \\otimes V_2$ where $\\dim_{\\GF(q)}(V_1)=2$; moreover, $(X_0)^{V_2}\\vartriangleright \\SL(V_2)$, or $(X_0)^{V_2}=\\A_7<\\SL_4(2)$ $($and $p=q=2$, $d=a=8)$, or $\\dim_{\\GF(q)}(V_2)\\leq 3$;\n\\item[\\rm (4)] $X_0\\vartriangleright \\SL_a(q) $ and $p^d=q^{2a}$;\n\\item[\\rm (5)] $X_0\\vartriangleright \\SL_2(q) $ and $p^d=q^{6}$;\n\\item[\\rm (6)] $X_0\\vartriangleright \\mathrm{SU}_a(q) $ and $p^d=q^{2a}$;\n\\item[\\rm (7)] $X_0\\vartriangleright \\Omega^{\\pm}_{2a}(q)$ $($and if $q$ is odd, $X_0$ contains an automorphism interchanging the two orbits of $\\Omega^{\\pm}_{2a}(q)$ on nonsingular $1$-spaces$)$;\n\\item[\\rm (8)] $X_0\\vartriangleright \\SL_5(q) $ and $p^d=q^{10}$ $($from the action of $\\SL_5(q)$ on $\\bigwedge^2( \\GF(q) ^5)$, the exterior square of $\\GF(q)^5)$;\n\n\\begin{lemma}\\label{lem:class:b-c}\nIf $G$ satisfies Theorem~{\\rm\\ref{subdegree}} {\\rm(B)--(C)}, then one of the following holds:\n\\begin{enumerate} [\\rm (1)]\n\\item $\\Ga\\cong \\mathrm{VO}^{-}_6(2)$ or $\\overline{\\mathrm{VO}^{-}_6(2)}$, arising from the case where $p^d=2^6$ and $G_0\\leq \\N_{\\GL_6(2)}(3^{1+2})$ in {\\rm(B)};\n\\item $\\Ga\\cong \\K_9\\square\\K_9$ or $\\K_9\\times\\K_9$, arising from the case where $p^d=3^4$ and $G_0\\leq \\N_{\\GL_4(3)}(2^{1+4}) $ in {\\rm(B)};\n\\item $\\Ga\\cong \\mathrm{VO}^{+}_8(2)$ or $\\overline{\\mathrm{VO}^{+}_8(2)}$, arising from the case where $p^d=2^8$ and $\\soc(G_0/\\Z(G_0))=\\A_9<\\mathrm{P\\Omega}^{+}_8(2)$ in {\\rm(C)}.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof} Assume that $G$ is of type {\\rm(A1)}. Then $G\\leq \\mathrm{A\\Gamma L}_1(p^d)$. Let $\\{\\alpha, \\beta\\}$ be an edge of $\\Ga$.\nThen $|V(\\Ga)|=p^d$ and $G_{\\alpha\\beta}\\leq \\ZZ_d$. Furthermore, $G_{\\a\\b}$ is semiregular on $V(\\Ga) \\setminus \\{\\alpha, \\beta\\}$.\nFrom Lemma~\\ref{lm:Gab10orbits} it follows that $G_{\\alpha\\beta}$ has at most $12$ orbits on $V(\\Ga) \\setminus \\{\\alpha, \\beta\\}$.\nTherefore, $G_{\\alpha\\beta}$ has one orbit of length at least $(|V(\\Ga)|-2)/12$.\nIt follows that $|G_{\\alpha\\beta}|\\geq (|V(\\Ga)|-2)/12$, which means\n\\[ p^d-2 \\leq 12d.\\]\nThis implies that\n\\[\np^d\\in\\{2^i (1\\leq i\\leq6), 3^j (1\\leq j\\leq 3), 5^k, 7^k, 11^k (1\\leq k\\leq 2), 13\\}.\n\\]\nSince $\\Ga$ has girth $3$ and diameter $2$ by Hypothesis~\\ref{hyp}, it follows that $|V(\\Ga)|=p^d\\geq7$. For all of the remaining possible values of $p^d$, we make use of Magma~\\cite{BCP} to construct all primitive rank $3$ subgroups $G$ of $\\mathrm{A\\Gamma L}_1(p^d)$, and then for each rank $3$ graph arising from $G$, we can further check if $|\\Om_i(\\Ga)|$ divides $|G|$ for $i=1,2,3,4$, where $\\Om_i(\\Ga)$ is given in Lemma~\\ref{lm:Gaorder}. We find that the only possibility is $p^d=3^2$, and $\\Ga\\cong \\mathrm{Paley}(9)\\cong\\K_3\\square\\K_3$. \\hfill\\qed\n\\end{proof}", "post_theorem_intro_text_len": 7099, "post_theorem_intro_text": "{\\noindent{\\bf Remark on Theorem~\\ref{th-main}.}\\ For $k\\geq2$, all $k$-homogeneous graphs have already been classified in the literature, { as mentioned above}, and so all graphs in Theorem~\\ref{th-main}~(1)--(4) are known.}\\medskip\n\nBy definition, a $k$-homogeneous graph is $k$-set-homogeneous.\nThe converse of this statement is clearly true for $k=1$.\n{In the case where $k\\ge4$,}\n{Devillers~\\cite{Devillers-2002paper}} proved that every finite $5$-set-homogeneous graph is set-homogeneous, and that a $4$-set-homogeneous graph is either homogeneous or one of the Schl\\\"{a}fli graph on $27$ vertices and its complement, which are $4$-homogeneous (see, for example, \\cite[Proposition~3.2.1]{Devillers}).\n{So, to prove Theorem~\\ref{th-main}, it suffices to show that every $k$-set-homogeneous graph is also $k$-homogeneous for $k=2$ and 3.}\n\nBefore stating the results, we introduce some families of graphs.\n\n\\begin{definition}\\label{Def-graph-families}\n{\\rm \\begin{itemize}\n\\item[\\rm (1)]  Let ${\\rm GF}(q)$ be a field of order a prime power $q$ such that $q\\equiv 1\\ (\\mod 4)$, and let $\\o$ be a fixed primitive { element} of ${\\rm GF}(q)$.\nLet $S=\\{\\o^2, \\o^4, \\ldots, \\o^{q-3}, \\o^{q-1}\\}$. The {\\it Paley graph} of order $q$, denoted by {\\rm Paley}$(q)$, is the graph with vertex set ${\\rm GF}(q)$ { where} two vertices $x,y$ are adjacent if and only if $x-y\\in S$.\n\n\\item[\\rm (2)]  For a positive integer $n$, the {\\it Hamming graph}, denoted by $\\mathrm{H}(2,n)$, is the graph with vertex set {$\\{(i, j) \\mid 1\\leqslant i, j\\leqslant n\\}$}, and for every vertex $(i, j)$, its neighbour set is\n\\[\n\\{ (i,j')\\mid j' \\in \\{1,2,\\dots,n \\}\\setminus \\{j\\}\\} \\cup \\{ (i',j)\\mid i' \\in \\{1,2,\\dots,n \\}\\setminus \\{i\\}\\}.\n\\]\nNote that $\\mathrm{H}(2,n)$ is just the Cartesian product $\\K_n\\square\\K_n$.\n\n\\item[\\rm (3)]  For $\\epsilon \\in \\{+,-\\}$, a positive integer $m$ and a prime power $q$, the {\\it affine polar graph}, denoted by $\\mathrm{VO}_{2m}^{\\epsilon}(q)$, is a graph with vertex set $\\GF(q)^{2m}$ equipped with a nondegenerate quadratic form $Q$ of type $\\epsilon$, and two vertices $\\alpha$ and $\\alpha'$ being adjacent if and only if $Q(\\alpha-\\alpha')=0$ (see \\cite[\\textsection{3.3.1}]{BrouwerSRG}). Since a nondegenerate $2$-dimensional orthogonal vector space of type $-$ contains no singular vector, it follows that $\\mathrm{VO}_{2}^{-}(q)\\cong q^2\\K_1$. As remarked in~\\cite[\\textsection{3.3.1}]{BrouwerSRG}, we see that $\\mathrm{VO}_{2}^{+}(q)\\cong \\K_q\\square\\K_q=\\mathrm{H}(2,q)$.\n\n\\item[\\rm (4)] For a prime power $q$, the {\\it elliptic orthogonal graph}, denoted by $\\mathrm{\\Gamma}(\\mathrm{O}^{-}_6(q))$ is a graph with vertex set consisting of  all singular $1$-subspaces of the orthogonal space on {$\\GF(q)^{6}$ of type $-$},\n    and two vertices $\\alpha=\\langle w \\rangle$ and $\\alpha'=\\langle w' \\rangle$ being adjacent if and only if $w\\perp w'$ (see \\cite[\\textsection{2.6.3}]{BrouwerSRG}). The Schl\\\"{a}fli graph is the complement of the elliptic orthogonal graph $\\mathrm{\\Gamma}(\\mathrm{O}^{-}_6(2))$ (see \\cite[Table~11.8]{BrouwerSRG}).\n\\end{itemize}}\n\\end{definition}\n\nWe recall that a transitive permutation group $G$ on a set $\\Ome$ is {\\it $2$-homogeneous} if\nall $2$-subsets of $\\Ome$ are equivalent under $G$.\nIn the next theorem, a classification of finite $2$-set-homogeneous graphs is given.\n\n\\begin{theorem}\\label{th:2sethomo}\nLet $\\Ga=(V,E)$ be a finite  $(G,2)$-set-homogeneous graph, where $G\\leqslant\\Aut(\\Ga)$.\nThen one of the following holds:\n\\begin{itemize}\n\\item[{\\rm(1)}] $\\Ga\\cong\\K_n$ or $n\\K_1$, and $G$ is a $2$-homogeneous permutation group on $V$.\n\n\\item[{\\rm(2)}] $\\Ga\\cong\\K_{m[b]}$ or $m\\K_b$, and  $G\\leqslant X\\wr Y$,\nwhere $X,Y$ are $2$-homogeneous permutation groups of degree $b,m$, respectively.\n\n\\item[{\\rm(3)}] $G$ is primitive of rank $3$, and $\\Ga$ is $(G,2)$-homogeneous.\n\n\\item[{\\rm(4)}] $G\\cong\\PSU_3(3)$ with vertex stabiliser $\\PSL_3(2)$, $\\Ga$ is of order $36$ and\nvalency $14$ or $21$, but $\\Aut(\\Ga)=\\PSU_3(3).2$ is $2$-homogeneous on $\\Ga$.\n\n\\item[(5)] $\\Ga$ is a Paley graph of order $p^r$, where $p\\equiv5 \\pmod{8}$ and $r$ is odd, and\n$G\\leqslant\\AGammaL_1(p^r)$ is $2$-set-homogeneous on $\\Ga$.\n\\end{itemize}\nIn particular, all $2$-set-homogeneous graphs are $2$-homogeneous.\n\\end{theorem}\n\n{It is worth mentioning} that all graphs in Theorem~\\ref{th:2sethomo}~(1)--(5) are $2$-homogeneous and already known in the literature.\n{Since} a graph $\\Ga$ is $(G, 2)$-set-homogeneous if and only if both $\\Ga$ and $\\ov\\Ga$ are $G$-edge-transitive, { our} main approach for proving the above theorem is to {analyse} permutation groups of rank at most 5. It is well-known that all orbitals of a permutation group of rank 3 and {of even order} are self-paired. The following result shows that, except for a single group, all orbitals of a primitive permutation group of rank 4 are self-paired.\n\n\\begin{theorem}\\label{trans-rank-4}\nLet $G$ be a primitive permutation group of rank $4$. Then either all orbitals of $G$ are self-paired, or $G=\\PSU_3(3)$ acting on $36$ points.\n\\end{theorem}\n\n\\begin{remark}\n{\\rm\\begin{itemize}\n\\item[\\rm (a)] More detailed information for the single group $\\PSU_3(3)$ and its orbitals is given\nin Theorem~\\ref{rank-4}.\n\\item[\\rm (b)] There are infinitely many imprimitive permutation groups of rank $4$ with exactly one non-trivial self-paired orbital.\nHere is an example. Let $X$ be a $2$-homogeneous but not $2$-transitive permutation group, and let\n$Y$ be a $2$-transitive permutation group. Then $X\\wr Y$ is an imprimitive permutation group of rank $4$ with exactly one non-trivial self-paired orbital.\n\\end{itemize}}\n\\end{remark}\n\nOur last theorem gives a classification of finite $3$-set-homogeneous graphs.\n\n\\begin{theorem}\\label{th:3sethomo}\nLet $\\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\\Ga$ is isomorphic either to one of the graphs: $\\C_5$, $\\K_n$, $\\K_{m[b]}$, $\\K_n\\square\\K_n$, or to one of the following graphs:\n\\begin{itemize}\n\\item[{\\rm(1)}] the affine polar graph $\\mathrm{VO}_{2m}^{\\epsilon}(2)$ with $m\\geqslant 1$ and $\\epsilon \\in \\{+,-\\}$;\n\n\\item[{\\rm(2)}]  the elliptic orthogonal graph $\\mathrm{\\Gamma}(\\mathrm{O}^-_6(q))$ with $q$ a prime power;\n\n\\item[{\\rm(3)}] an orbital graph of the Higman-Sims group $\\mathrm{HS}$ acting on $100$ points;\n\n\\item[{\\rm(4)}] an orbital graph of the McLaughlin group $\\mathrm{McL}$ acting on $275$ points.\n\\end{itemize}\nIn particular, all $3$-set-homogeneous graphs are $3$-homogeneous.\n\\end{theorem}\n\nNote that all graphs in Theorem~\\ref{th:3sethomo}~(1)--(4) are actually $3$-homogeneous graphs, which {are known} in the literature.\n\nThe rest of the paper is organised as follows. In Sections~\\ref{subsec:2reduce}--\\ref{sec6:proof-main-theorems}, we deal with finite $(G, 2)$-set-homogeneous graphs.\nIn Section~\\ref{sec6:proof-main-theorems}, we prove Theorem~\\ref{th:2sethomo} and Theorem~\\ref{trans-rank-4}.\nIn Section~\\ref{set:3sethom}, we deal with finite $(G, 3)$-set-homogeneous graphs, and prove Theorem~\\ref{th:3sethomo}.\nIn Section~\\ref{set:proof-of-main}, we prove Theorem~\\ref{th-main}.", "sketch": "By definition, a $k$-homogeneous graph is $k$-set-homogeneous. For $k=1$ the converse is clearly true. For $k\\ge4$, Devillers~\\cite{Devillers-2002paper} proved that every finite $5$-set-homogeneous graph is set-homogeneous, and that a $4$-set-homogeneous graph is either homogeneous or one of the Schl\\\"{a}fli graph on $27$ vertices and its complement (which are $4$-homogeneous). Hence, to prove Theorem~\\ref{th-main}, it suffices to show that every $k$-set-homogeneous graph is also $k$-homogeneous for $k=2$ and $3$.\n\nThis is done by classifying finite $2$-set-homogeneous graphs (Theorem~\\ref{th:2sethomo}, yielding in particular that all $2$-set-homogeneous graphs are $2$-homogeneous) and finite $3$-set-homogeneous graphs (Theorem~\\ref{th:3sethomo}, yielding in particular that all $3$-set-homogeneous graphs are $3$-homogeneous), and then concluding Theorem~\\ref{th-main} in Section~\\ref{set:proof-of-main}.", "expanded_sketch": "By definition, a $k$-homogeneous graph is $k$-set-homogeneous. For $k=1$ the converse is clearly true. For $k\\ge4$, Devillers~\\cite{Devillers-2002paper} proved that every finite $5$-set-homogeneous graph is set-homogeneous, and that a $4$-set-homogeneous graph is either homogeneous or one of the Schl\\\"{a}fli graph on $27$ vertices and its complement (which are $4$-homogeneous). Hence, to prove the main theorem, it suffices to show that every $k$-set-homogeneous graph is also $k$-homogeneous for $k=2$ and $3$.\n\nWe first prove the following theorem.\n\\begin{theorem}\\label{th:2sethomo}\nLet $\\Ga=(V,E)$ be a finite  $(G,2)$-set-homogeneous graph, where $G\\leq\\Aut(\\Ga)$.\nThen one of the following holds:\n\\begin{itemize}\n\\item[{\\rm(1)}] $\\Ga\\cong\\K_n$ or $n\\K_1$, and $G$ is a $2$-homogeneous permutation group on $V$.\n\n\\item[{\\rm(2)}] $\\Ga\\cong\\K_{m[b]}$ or $m\\K_b$, and  $G\\leqslant X\\wr Y$,\nwhere $X,Y$ are $2$-homogeneous permutation groups of degree $b,m$, respectively.\n\n\\item[{\\rm(3)}] $G$ is primitive of rank $3$, and $\\Ga$ is $(G,2)$-homogeneous.\n\n\\item[{\\rm(4)}] $G\\cong\\PSU_3(3)$ with vertex stabiliser $\\PSL_3(2)$, $\\Ga$ is of order $36$ and\nvalency $14$ or $21$, but $\\Aut(\\Ga)=\\PSU_3(3).2$ is $2$-homogeneous on $\\Ga$.\n\n\\item[(5)] $\\Ga$ is a Paley graph of order $p^r$, where $p\\equiv5 \\pmod{8}$ and $r$ is odd, and\n$G\\leq\\AGammaL_1(p^r)$ is $2$-set-homogeneous on $\\Ga$.\n\\end{itemize}\nIn particular, all $2$-set-homogeneous graphs are $2$-homogeneous.\n\\end{theorem}\n\nWe next prove the following theorem.\n\\begin{theorem}\\label{th:3sethomo}\nLet $\\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\\Ga$ is isomorphic either to one of the graphs: $\\C_5$, $\\K_n$, $\\K_{m[b]}$, $\\K_n\\square\\K_n$, or to one of the following graphs:\n\\begin{itemize}\n\\item[{\\rm(1)}] the affine polar graph $\\mathrm{VO}_{2m}^{\\epsilon}(2)$ with $m\\geq 1$ and $\\epsilon \\in \\{+,-\\}$;\n\n\\item[{\\rm(2)}]  the elliptic orthogonal graph $\\mathrm{\\Gamma}(\\mathrm{O}^-_6(q))$ with $q$ a prime power;\n\n\\item[{\\rm(3)}] an orbital graph of the Higman-Sims group $\\mathrm{HS}$ acting on $100$ points;\n\n\\item[{\\rm(4)}] an orbital graph of the McLaughlin group $\\mathrm{McL}$ acting on $275$ points.\n\\end{itemize}\nIn particular, all $3$-set-homogeneous graphs are $3$-homogeneous.\n\\end{theorem}\n\nFinally, we conclude the proof of the main theorem as follows:\n\\label{set:proof-of-main}\n\nBy definition, both 1-homogeneous graphs and 1-set-homogeneous graphs are exactly the vertex-transitive graphs. Assume now that $k\\geq 2$. It is obvious by definition that e", "expanded_theorem": "\\label{th-main}\nFor each positive integer $k$, a finite graph is $k$-set-homogeneous if and only if it is $k$-homogeneous.\nMoreover, { if $\\Ga$ is a $k$-homogeneous graph with $k\\geqslant 2$, then} one of the following holds:\n\\begin{itemize}\n\\item [{\\rm(1)}] $k\\geqslant 5$, and $\\Ga$ is one of { $\\C_5 (\\cong\\overline{\\C_5})$, $\\K_3\\square\\K_3 (\\cong \\overline{\\K_3\\square\\K_3})$, $n\\K_m$, and $\\K_{n[m]}$; furthermore, $\\Ga$ is homogeneous.}\n\n{ \\item[(2)] $\\Ga$ is either the Schl\\\"{a}fli graph on $27$ vertices or its complement (the two non-trivial orbital graphs of $\\mathrm{PSU}_4(2)$ acting on the isotropic lines); furthermore, $\\Ga$ is $4$-homogeneous but not $5$-homogeneous.\n\n\\item[(3)] $\\Ga$ is either $\\K_n\\square\\K_n$ with $n\\geqslant4$, or a graph from the following theorem (items (1)--(4)), or the complement of any such graph; furthermore, $\\Ga$ is $3$-homogeneous but not $4$-homogeneous.\n\nWe first prove the following theorem.\n\\begin{theorem}\\label{th:3sethomo}\nLet $\\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\\Ga$ is isomorphic either to one of the graphs: $\\C_5$, $\\K_n$, $\\K_{m[b]}$, $\\K_n\\square\\K_n$, or to one of the following graphs:\n\\begin{itemize}\n\\item[{\\rm(1)}] the affine polar graph $\\mathrm{VO}_{2m}^{\\epsilon}(2)$ with $m\\geq 1$ and $\\epsilon \\in \\{+,-\\}$;\n\n\\item[{\\rm(2)}]  the elliptic orthogonal graph $\\mathrm{\\Gamma}(\\mathrm{O}^-_6(q))$ with $q$ a prime power;\n\n\\item[{\\rm(3)}] an orbital graph of the Higman-Sims group $\\mathrm{HS}$ acting on $100$ points;\n\n\\item[{\\rm(4)}] an orbital graph of the McLaughlin group $\\mathrm{McL}$ acting on $275$ points.\n\\end{itemize}\nIn particular, all $3$-set-homogeneous graphs are $3$-homogeneous.\n\\end{theorem}\n\n\\item[(4)] $\\Ga$ is an orbital graph of a primitive permutation group of rank $3$ which is not covered by parts (1)--(3); furthermore, $\\Ga$ is $2$-homogeneous but not $3$-homogeneous. }\n\n\\end{itemize}", "theorem_type": ["Biconditional or Equivalence", "Classification or Bijection"], "mcq": {"question": "Let be a finite graph, and let k be a positive integer. Call k-set-homogeneous if any two isomorphic induced subgraphs of with at most k vertices lie in the same orbit of Aut(). Call k-homogeneous if every isomorphism between any two isomorphic induced subgraphs of with at most k vertices extends to an automorphism of . Here nK_m denotes the disjoint union of n copies of K_m, K_{n[m]} the complete multipartite graph with n parts of size m, K_n\\square K_n the Cartesian product of two copies of K_n, and an orbital graph is a graph whose edge set is an orbit of a permutation group on unordered pairs of vertices. Which statement gives the complete classification of finite k-set-homogeneous graphs?", "correct_choice": {"label": "A", "text": "For every positive integer k, a finite graph is k-set-homogeneous if and only if it is k-homogeneous. Moreover, if is k-homogeneous with k\\ge 2, then one of the following holds: (1) k\\ge 5, and is one of C_5, K_3\\square K_3, nK_m, or K_{n[m]}; in fact is homogeneous. (2) is the Schl\\\"afli graph on 27 vertices or its complement; then is 4-homogeneous but not 5-homogeneous. (3) is K_n\\square K_n for some n\\ge 4, or its complement, or one of the affine polar graphs VO_{2m}^{\\epsilon}(2) with m\\ge 1 and \\epsilon\\in\\{+,-\\}, the elliptic orthogonal graphs \\Gamma(O^-_6(q)) with q a prime power, an orbital graph of the Higman-Sims group HS on 100 points, or an orbital graph of the McLaughlin group McL on 275 points, or the complement of one of these; then is 3-homogeneous but not 4-homogeneous. (4) Otherwise, is an orbital graph of a primitive permutation group of rank 3 not already covered by (1)-(3); then is 2-homogeneous but not 3-homogeneous."}, "choices": [{"label": "B", "text": "For every positive integer k, a finite graph is k-set-homogeneous if and only if it is k-homogeneous. Moreover, if \\Gamma is k-homogeneous with k\\ge 2, then one of the following holds: (1) k\\ge 5, and \\Gamma is one of C_5, K_3\\square K_3, nK_m, or K_{n[m]}; in fact \\Gamma is homogeneous. (2) \\Gamma is the Schl\\\"afli graph on 27 vertices or its complement; then \\Gamma is 4-homogeneous but not 5-homogeneous. (3) \\Gamma is K_n\\square K_n for some n\\ge 3, or its complement, or one of the affine polar graphs VO_{2m}^{\\epsilon}(2) with m\\ge 1 and \\epsilon\\in\\{+,-\\}, the elliptic orthogonal graphs \\Gamma(O^-_6(q)) with q a prime power, an orbital graph of the Higman-Sims group HS on 100 points, or an orbital graph of the McLaughlin group McL on 275 points, or the complement of one of these; then \\Gamma is 3-homogeneous but not 4-homogeneous. (4) Otherwise, \\Gamma is an orbital graph of a primitive permutation group of rank 3 not already covered by (1)-(3); then \\Gamma is 2-homogeneous but not 3-homogeneous."}, {"label": "C", "text": "For every positive integer k, every finite k-homogeneous graph is k-set-homogeneous. Moreover, every finite k-set-homogeneous graph belongs to one of the four families listed in (1)-(4) of the classification for k-homogeneous graphs."}, {"label": "D", "text": "For every positive integer k, a finite graph is k-set-homogeneous if and only if it is k-homogeneous. Moreover, if \\Gamma is k-homogeneous with k\\ge 2, then one of the following holds: (1) k\\ge 5, and \\Gamma is one of C_5, K_3\\square K_3, nK_m, or K_{n[m]}; in fact \\Gamma is homogeneous. (2) \\Gamma is the Schl\\\"afli graph on 27 vertices or its complement; then \\Gamma is 4-homogeneous but not 5-homogeneous. (3) \\Gamma is K_n\\square K_n for some n\\ge 4, or its complement, or one of the affine polar graphs VO_{2m}^{\\epsilon}(2) with m\\ge 1 and \\epsilon\\in\\{+,-\\}, the elliptic orthogonal graphs \\Gamma(O^-_6(q)) with q a prime power, an orbital graph of the Higman-Sims group HS on 100 points, or an orbital graph of the McLaughlin group McL on 275 points, or the complement of one of these; then \\Gamma is 3-homogeneous but not 5-homogeneous. (4) Otherwise, \\Gamma is an orbital graph of a primitive permutation group of rank 3 not already covered by (1)-(3); then \\Gamma is 2-homogeneous but not 3-homogeneous."}, {"label": "E", "text": "For every positive integer k, a finite graph is k-set-homogeneous if and only if it is k-homogeneous. Moreover, if \\Gamma is k-homogeneous with k\\ge 2, then one of the following holds: (1) k\\ge 5, and \\Gamma is one of C_5, K_3\\square K_3, nK_m, or K_{n[m]}; in fact \\Gamma is homogeneous. (2) \\Gamma is the Schl\\\"afli graph on 27 vertices or its complement; then \\Gamma is 4-homogeneous but not 5-homogeneous. (3) \\Gamma is K_n\\square K_n for some n\\ge 4, or one of the affine polar graphs VO_{2m}^{\\epsilon}(2) with m\\ge 1 and \\epsilon\\in\\{+,-\\}, the elliptic orthogonal graphs \\Gamma(O^-_6(q)) with q a prime power, an orbital graph of the Higman-Sims group HS on 100 points, or an orbital graph of the McLaughlin group McL on 275 points; then \\Gamma is 3-homogeneous but not 4-homogeneous. (4) Otherwise, \\Gamma is an orbital graph of a primitive permutation group of rank 3 not already covered by (1)-(3), or the complement of such a graph; then \\Gamma is 2-homogeneous but not 3-homogeneous."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "case_split", "tampered_component": "threshold n\\ge 4 for K_n\\square K_n in the 3-homogeneous-but-not-4-homogeneous case", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "case_split", "tampered_component": "dropped the converse direction and the per-case sharp homogeneity level conclusions", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "sharp exclusion '3-homogeneous but not 4-homogeneous' replaced by 'not 5-homogeneous'", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "complement closure moved from part (3) to part (4), omitting complements in the 3-homogeneous family", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct option. It introduces definitions and notation, then asks for the classification; the correct answer is not explicitly or implicitly singled out."}, "TAS": {"score": 0, "justification": "The item mainly asks the test-taker to recognize the exact classification theorem statement. This is essentially a theorem-recall/restatement question rather than a problem requiring selection among genuinely competing mathematical conclusions."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the distractors are subtle near-miss variants involving boundary cases, converse strength, and complement closure. However, the task is still primarily precise recall/comparison of theorem statements, not generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong: they are plausible, mathematically close to the correct theorem, and exploit realistic failure modes such as weakened converses, altered thresholds, and incorrect sharp homogeneity levels."}, "total_score": 5, "overall_assessment": "A solid theorem-recognition MCQ with excellent distractors and no answer leakage, but it is weak on tautology avoidance and only moderately tests genuine generative reasoning."}}
{"id": "2602.19913v1", "paper_link": "http://arxiv.org/abs/2602.19913v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{main}In the setting of Conjecture \\ref{con}, we have that as $t\\to\\infty$ the flow $(X,\\omega(t))$ converges in the Gromov-Hausdorff topology to $(Y,d_{\\mathrm{can}})$. In particular, Conjecture \\ref{con} holds.", "start_pos": 8806, "end_pos": 9053, "label": "main"}, "ref_dict": {"eq--KRF": "\\begin{equation}\\label{eq--KRF}\n\\left\\{\n                \\begin{aligned}\n                  &\\frac{\\de}{\\de t}\\omega(t)=-\\Ric(\\omega(t))-\\omega(t),\\\\\n                  &\\omega(0)=\\omega_0,\n                \\end{aligned}\n              \\right.\n\\end{equation}", "main": "\\begin{theorem}\\label{main}In the setting of Conjecture \\ref{con}, we have that as $t\\to\\infty$ the flow $(X,\\omega(t))$ converges in the Gromov-Hausdorff topology to $(Y,d_{\\mathrm{can}})$. In particular, Conjecture \\ref{con} holds.\n\\end{theorem}", "con": "\\begin{conjecture}\\label{con}\nLet $(X^n,\\omega_0)$ be a compact K\\\"ahler manifold with $K_X$ semiample, and let $\\omega(t)$ be the solution of the K\\\"ahler-Ricci flow \\eqref{eq--KRF}. Then as $t\\to\\infty$, $(X,\\omega(t))$ converges in the Gromov-Hausdorff topology to the metric completion of $(Y^\\circ,\\omega_{\\rm can})$. This is a compact metric space homeomorphic to the canonical model $Y$.\n\\end{conjecture}"}, "pre_theorem_intro_text_len": 6073, "pre_theorem_intro_text": "Let $(X^n,\\omega_0)$ be a compact K\\\"ahler manifold, and let $\\omega(t)$ be the solution of the normalized K\\\"ahler-Ricci flow\n\\begin{equation}\\label{eq--KRF}\n\\left\\{\n                \\begin{aligned}\n                  &\\frac{\\partial}{\\partial t}\\omega(t)=-\\mathrm{Ric}(\\omega(t))-\\omega(t),\\\\\n                  &\\omega(0)=\\omega_0,\n                \\end{aligned}\n              \\right.\n\\end{equation}\nstarting at $\\omega_0$. We are interested in the case when the solution $\\omega(t)$ is {\\em immortal}, i.e. it exists for all $t\\geqslant 0$. By a result of Tian-Zhang \\cite{TiZh}, this happens if and only if the canonical bundle $K_X$ is nef, i.e. $c_1(K_X)$ is a limit of K\\\"ahler classes. We are interested in the behavior of the metrics $\\omega(t)$ as $t\\to\\infty$, and refer to the second-named author's survey on this topic \\cite{To}.\n\nThe Abundance Conjecture in birational geometry, and its natural extension to K\\\"ahler manifolds, predicts that if $K_X$ is nef then $K_X$ is semiample, which means that $\\ell K_X$ is base-point free for some $\\ell\\geqslant 1$. Abundance is currently known for $n\\leqslant 3$, and we will assume from now on that $K_X$ is indeed semiample.\n\nIn this case, the Kodaira dimension $m:=\\kappa(X)$ satisfies $0\\leqslant m\\leqslant n$, and the linear system $|\\ell K_X|$ for $\\ell$ sufficiently divisible gives a fiber space (the Iitaka fibration of $X$)\n\\begin{equation}\nf:X\\to Y\\subset\\mathbb{P}^N \\text{ with } \\ell K_X=f^*(\\mathcal{O}_{\\mathbb{P}^N}(1)|_Y),\n\\end{equation}\n onto a normal projective variety $Y$ of dimension $\\dim Y=m$. This implies that the generic fibers of $f$ are Calabi-Yau $(n-m)$-folds. The variety $Y=\\mathrm{Proj}\\oplus_{k\\geqslant 0}H^0(X,kK_X)$ is the canonical model of $X$.\n\nWe define $D\\subset Y$ as the closed subvariety given by the union of the singularities of $Y$ together with the discriminant locus of $f$, so that if we call $X^\\circ:=X\\backslash f^{-1}(D)$ and $Y^\\circ:=Y\\backslash D$, then $f:X^\\circ\\to Y^\\circ$ is a proper holomorphic submersion with connected $(n-m)$-dimensional Calabi-Yau fibers.\n\nIn the case when $m=0$, we have that $Y$ is a point, $D=\\emptyset$, $X$ itself is Calabi-Yau, and the behavior of the flow has been fully understood since the 1980s \\cite{Cao}: the flow \\eqref{eq--KRF} shrinks the metric smoothly to zero as $t\\to\\infty$ (in particular, $(X,\\omega(t))$ converges to a point in the Gromov-Hausdorff topology), and reparametrizing the flow to have constant volume, it converges smoothly to a Ricci-flat K\\\"ahler metric on $X$.\n\nThe case $m=n$ is also special, because in this case the flow is volume non-collapsed, in the sense that $\\mathrm{Vol}(X,\\omega(t))\\geqslant c>0$ for all $t\\geqslant 0$. In this case $K_X$ is nef and big, and $X$ is known as a minimal model of general type. In this case, Tsuji \\cite{Ts} and Tian-Zhang \\cite{TiZh} proved that the flow converges smoothly on compact subsets of $X^\\circ$ to a negative K\\\"ahler-Einstein metric $\\omega_{\\rm can}$ on $Y^\\circ$ (pulled back via $f:X^\\circ\\to Y^\\circ,$ which is an isomorphism in this case).\n\nThe remaining cases when $0<m<n$ turned out to be vastly more complicated, due to the fact that the flow is now volume collapsing as $t\\to\\infty$. This setup was first considered by Song-Tian in \\cite{ST0,ST}. In this case, after much work by many people (see \\cite{To} for references), it was recently shown by Hein and the first and second-named authors \\cite{HLT} that as $t\\to\\infty$, we have $\\omega(t)\\to f^*\\omega_{\\rm can}$ smoothly on $X^\\circ$, where $\\omega_{\\can}$ is a twisted K\\\"ahler-Einstein metric on $Y^\\circ$, constructed by Song-Tian \\cite{ST}.\n\nIn this paper we are interested in the global limiting behavior of $(X,\\omega(t))$ as $t\\to\\infty$, in the Gromov-Hausdorff topology. The following basic conjecture is due to Song-Tian \\cite[P.651]{ST0}, \\cite[Conjectures 6.2 and 6.3]{ST3}, and it fits into their picture of analytic minimal model program:\n\\begin{conjecture}\\label{con}\nLet $(X^n,\\omega_0)$ be a compact K\\\"ahler manifold with $K_X$ semiample, and let $\\omega(t)$ be the solution of the K\\\"ahler-Ricci flow \\eqref{eq--KRF}. Then as $t\\to\\infty$, $(X,\\omega(t))$ converges in the Gromov-Hausdorff topology to the metric completion of $(Y^\\circ,\\omega_{\\rm can})$. This is a compact metric space homeomorphic to the canonical model $Y$.\n\\end{conjecture}\n\nThis conjecture has received much attention since it was first posed in 2006. As mentioned above, we now know \\cite{HLT} that $\\omega(t)$ converges locally smoothly to $f^*\\omega_{\\rm can}$ on $X^\\circ$. However, obtaining reasonable estimates for the flow near $f^{-1}(D)=X\\backslash X^\\circ$ has proved to be very challenging.\n\nThe case $m=n$ was studied first, and the conjecture is known in low dimensions by Guo-Song-Weinkove \\cite{GSW} and Tian-Zhang \\cite{TiZ} or under the additional assumption of a uniform Ricci lower bound by Guo \\cite{Gu}. When $m=n$, Conjecture \\ref{con} was settled by Wang \\cite{Wang}. The harder case when $0<m<n$ has also been much studied. In particular, Conjecture \\ref{con} was proved by Song-Tian-Zhang \\cite{STZ} when $m=1$ and the generic fibers of $f$ are tori, and by Li and the second-named author \\cite{LT} in arbitrary dimensions assuming that $Y$ is smooth and the divisorial components of $D$ have simple normal crossings (which always holds when $m=1$).\n\nLet us remark that the uniform diameter bound and existence of subsequential Gromov-Hausdorff limits of $(X,\\omega(t))$ was only achieved recently in \\cite{JS}, and even more generally in \\cite{GPSS, GPSS2,guedjT,vu2024} assuming only that $K_X$ is nef. Lastly, Sz\\'ekelyhidi \\cite{Sz} has very recently shown that in the setting of Conjecture \\ref{con}, the metric completion of $(Y^\\circ,\\omega_{\\rm can})$ is homeomorphic to $Y$, is a non-collapsed $\\mathrm{RCD}(-1,2m)$-space, and inside this metric space, $Y\\backslash Y^\\circ$ has real Hausdorff codimension at least $2$. We will denote this metric completion by $(Y,d_{\\rm can})$.\n\nOur main result finally resolves Conjecture \\ref{con} in general:", "context": "Let $(X^n,\\omega_0)$ be a compact K\\\"ahler manifold, and let $\\omega(t)$ be the solution of the normalized K\\\"ahler-Ricci flow\n\\begin{equation}\\label{eq--KRF}\n\\left\\{\n \\begin{aligned}\n &\\frac{\\partial}{\\partial t}\\omega(t)=-\\mathrm{Ric}(\\omega(t))-\\omega(t),\\\\\n &\\omega(0)=\\omega_0,\n \\end{aligned}\n \\right.\n\\end{equation}\nstarting at $\\omega_0$. We are interested in the case when the solution $\\omega(t)$ is {\\em immortal}, i.e. it exists for all $t\\geqslant 0$. By a result of Tian-Zhang \\cite{TiZh}, this happens if and only if the canonical bundle $K_X$ is nef, i.e. $c_1(K_X)$ is a limit of K\\\"ahler classes. We are interested in the behavior of the metrics $\\omega(t)$ as $t\\to\\infty$, and refer to the second-named author's survey on this topic \\cite{To}.\n\nIn the case when $m=0$, we have that $Y$ is a point, $D=\\emptyset$, $X$ itself is Calabi-Yau, and the behavior of the flow has been fully understood since the 1980s \\cite{Cao}: the flow \\eqref{eq--KRF} shrinks the metric smoothly to zero as $t\\to\\infty$ (in particular, $(X,\\omega(t))$ converges to a point in the Gromov-Hausdorff topology), and reparametrizing the flow to have constant volume, it converges smoothly to a Ricci-flat K\\\"ahler metric on $X$.\n\nThe case $m=n$ is also special, because in this case the flow is volume non-collapsed, in the sense that $\\mathrm{Vol}(X,\\omega(t))\\geqslant c>0$ for all $t\\geqslant 0$. In this case $K_X$ is nef and big, and $X$ is known as a minimal model of general type. In this case, Tsuji \\cite{Ts} and Tian-Zhang \\cite{TiZh} proved that the flow converges smoothly on compact subsets of $X^\\circ$ to a negative K\\\"ahler-Einstein metric $\\omega_{\\rm can}$ on $Y^\\circ$ (pulled back via $f:X^\\circ\\to Y^\\circ,$ which is an isomorphism in this case).\n\nIn this paper we are interested in the global limiting behavior of $(X,\\omega(t))$ as $t\\to\\infty$, in the Gromov-Hausdorff topology. The following basic conjecture is due to Song-Tian \\cite[P.651]{ST0}, \\cite[Conjectures 6.2 and 6.3]{ST3}, and it fits into their picture of analytic minimal model program:\n\\begin{conjecture}\\label{con}\nLet $(X^n,\\omega_0)$ be a compact K\\\"ahler manifold with $K_X$ semiample, and let $\\omega(t)$ be the solution of the K\\\"ahler-Ricci flow \\eqref{eq--KRF}. Then as $t\\to\\infty$, $(X,\\omega(t))$ converges in the Gromov-Hausdorff topology to the metric completion of $(Y^\\circ,\\omega_{\\rm can})$. This is a compact metric space homeomorphic to the canonical model $Y$.\n\\end{conjecture}\n\nLet us remark that the uniform diameter bound and existence of subsequential Gromov-Hausdorff limits of $(X,\\omega(t))$ was only achieved recently in \\cite{JS}, and even more generally in \\cite{GPSS, GPSS2,guedjT,vu2024} assuming only that $K_X$ is nef. Lastly, Sz\\'ekelyhidi \\cite{Sz} has very recently shown that in the setting of Conjecture \\ref{con}, the metric completion of $(Y^\\circ,\\omega_{\\rm can})$ is homeomorphic to $Y$, is a non-collapsed $\\mathrm{RCD}(-1,2m)$-space, and inside this metric space, $Y\\backslash Y^\\circ$ has real Hausdorff codimension at least $2$. We will denote this metric completion by $(Y,d_{\\rm can})$.\n\nOur main result finally resolves Conjecture \\ref{con} in general:\n\n\\begin{equation}\\label{eq--KRF}\n\\left\\{\n                \\begin{aligned}\n                  &\\frac{\\de}{\\de t}\\omega(t)=-\\Ric(\\omega(t))-\\omega(t),\\\\\n                  &\\omega(0)=\\omega_0,\n                \\end{aligned}\n              \\right.\n\\end{equation}", "full_context": "Let $(X^n,\\omega_0)$ be a compact K\\\"ahler manifold, and let $\\omega(t)$ be the solution of the normalized K\\\"ahler-Ricci flow\n\\begin{equation}\\label{eq--KRF}\n\\left\\{\n \\begin{aligned}\n &\\frac{\\partial}{\\partial t}\\omega(t)=-\\mathrm{Ric}(\\omega(t))-\\omega(t),\\\\\n &\\omega(0)=\\omega_0,\n \\end{aligned}\n \\right.\n\\end{equation}\nstarting at $\\omega_0$. We are interested in the case when the solution $\\omega(t)$ is {\\em immortal}, i.e. it exists for all $t\\geqslant 0$. By a result of Tian-Zhang \\cite{TiZh}, this happens if and only if the canonical bundle $K_X$ is nef, i.e. $c_1(K_X)$ is a limit of K\\\"ahler classes. We are interested in the behavior of the metrics $\\omega(t)$ as $t\\to\\infty$, and refer to the second-named author's survey on this topic \\cite{To}.\n\nIn the case when $m=0$, we have that $Y$ is a point, $D=\\emptyset$, $X$ itself is Calabi-Yau, and the behavior of the flow has been fully understood since the 1980s \\cite{Cao}: the flow \\eqref{eq--KRF} shrinks the metric smoothly to zero as $t\\to\\infty$ (in particular, $(X,\\omega(t))$ converges to a point in the Gromov-Hausdorff topology), and reparametrizing the flow to have constant volume, it converges smoothly to a Ricci-flat K\\\"ahler metric on $X$.\n\nThe case $m=n$ is also special, because in this case the flow is volume non-collapsed, in the sense that $\\mathrm{Vol}(X,\\omega(t))\\geqslant c>0$ for all $t\\geqslant 0$. In this case $K_X$ is nef and big, and $X$ is known as a minimal model of general type. In this case, Tsuji \\cite{Ts} and Tian-Zhang \\cite{TiZh} proved that the flow converges smoothly on compact subsets of $X^\\circ$ to a negative K\\\"ahler-Einstein metric $\\omega_{\\rm can}$ on $Y^\\circ$ (pulled back via $f:X^\\circ\\to Y^\\circ,$ which is an isomorphism in this case).\n\nIn this paper we are interested in the global limiting behavior of $(X,\\omega(t))$ as $t\\to\\infty$, in the Gromov-Hausdorff topology. The following basic conjecture is due to Song-Tian \\cite[P.651]{ST0}, \\cite[Conjectures 6.2 and 6.3]{ST3}, and it fits into their picture of analytic minimal model program:\n\\begin{conjecture}\\label{con}\nLet $(X^n,\\omega_0)$ be a compact K\\\"ahler manifold with $K_X$ semiample, and let $\\omega(t)$ be the solution of the K\\\"ahler-Ricci flow \\eqref{eq--KRF}. Then as $t\\to\\infty$, $(X,\\omega(t))$ converges in the Gromov-Hausdorff topology to the metric completion of $(Y^\\circ,\\omega_{\\rm can})$. This is a compact metric space homeomorphic to the canonical model $Y$.\n\\end{conjecture}\n\nLet us remark that the uniform diameter bound and existence of subsequential Gromov-Hausdorff limits of $(X,\\omega(t))$ was only achieved recently in \\cite{JS}, and even more generally in \\cite{GPSS, GPSS2,guedjT,vu2024} assuming only that $K_X$ is nef. Lastly, Sz\\'ekelyhidi \\cite{Sz} has very recently shown that in the setting of Conjecture \\ref{con}, the metric completion of $(Y^\\circ,\\omega_{\\rm can})$ is homeomorphic to $Y$, is a non-collapsed $\\mathrm{RCD}(-1,2m)$-space, and inside this metric space, $Y\\backslash Y^\\circ$ has real Hausdorff codimension at least $2$. We will denote this metric completion by $(Y,d_{\\rm can})$.\n\nOur main result finally resolves Conjecture \\ref{con} in general:\n\n\\begin{equation}\\label{eq--KRF}\n\\left\\{\n                \\begin{aligned}\n                  &\\frac{\\de}{\\de t}\\omega(t)=-\\Ric(\\omega(t))-\\omega(t),\\\\\n                  &\\omega(0)=\\omega_0,\n                \\end{aligned}\n              \\right.\n\\end{equation}\n\n\\section{Introduction}\nLet $(X^n,\\omega_0)$ be a compact K\\\"ahler manifold, and let $\\omega(t)$ be the solution of the normalized K\\\"ahler-Ricci flow\n\\begin{equation}\\label{eq--KRF}\n\\left\\{\n \\begin{aligned}\n &\\frac{\\de}{\\de t}\\omega(t)=-\\Ric(\\omega(t))-\\omega(t),\\\\\n &\\omega(0)=\\omega_0,\n \\end{aligned}\n \\right.\n\\end{equation}\nstarting at $\\omega_0$. We are interested in the case when the solution $\\omega(t)$ is {\\em immortal}, i.e. it exists for all $t\\geq 0$. By a result of Tian-Zhang \\cite{TiZh}, this happens if and only if the canonical bundle $K_X$ is nef, i.e. $c_1(K_X)$ is a limit of K\\\"ahler classes. We are interested in the behavior of the metrics $\\omega(t)$ as $t\\to\\infty$, and refer to the second-named author's survey on this topic \\cite{To}.\n\nIn the case when $m=0$, we have that $Y$ is a point, $D=\\emptyset$, $X$ itself is Calabi-Yau, and the behavior of the flow has been fully understood since the 1980s \\cite{Cao}: the flow \\eqref{eq--KRF} shrinks the metric smoothly to zero as $t\\to\\infty$ (in particular, $(X,\\omega(t))$ converges to a point in the Gromov-Hausdorff topology), and reparametrizing the flow to have constant volume, it converges smoothly to a Ricci-flat K\\\"ahler metric on $X$.\n\nIn this paper we are interested in the global limiting behavior of $(X,\\omega(t))$ as $t\\to\\infty$, in the Gromov-Hausdorff topology. The following basic conjecture is due to Song-Tian \\cite[P.651]{ST0}, \\cite[Conjectures 6.2 and 6.3]{ST3}, and it fits into their picture of analytic minimal model program:\n\\begin{conjecture}\\label{con}\nLet $(X^n,\\omega_0)$ be a compact K\\\"ahler manifold with $K_X$ semiample, and let $\\omega(t)$ be the solution of the K\\\"ahler-Ricci flow \\eqref{eq--KRF}. Then as $t\\to\\infty$, $(X,\\omega(t))$ converges in the Gromov-Hausdorff topology to the metric completion of $(Y^\\circ,\\omega_{\\rm can})$. This is a compact metric space homeomorphic to the canonical model $Y$.\n\\end{conjecture}\n\nOur main result finally resolves Conjecture \\ref{con} in general:\n\nIn particular, in the case when $m=n$, our arguments are different from those in \\cite{Wang}, so we obtain a new proof of that case. For general $m$, our proof follows a strategy introduced in \\cite{LT}, where a key input is Perelman's monotonicity of the reduced volume, viewed as a parabolic analogue of the Bishop-Gromov volume comparison for Riemannian metrics with Ricci curvature bounded below. This allows for some control over how much time minimizing $\\mathcal{L}$-geodesics spend in a neighborhood of the ``bad region'' $X\\backslash X^\\circ=f^{-1}(D)$. However, to push this strategy to the end, \\cite{LT} had to use a result from \\cite{GTZ2} which shows that when $Y$ is smooth and the divisorial components of $D$ have simple normal crossings, $\\omega_{\\rm can}$ is quasi-isometric to a {\\em conical K\\\"ahler metric} near these components, up to a small logarithmic error.\n\n\\subsection{Reduction to an upper of $d_{\\can}$ by $d_t$}In the following, let $d_t$ denote the distance function on $X$ defined by the metric $\\omega(t)$. To prove Theorem \\ref{main} we need to show that $(X,d_t)$ converges in the Gromov-Hausdorff topology to $(Y,d_{\\rm can})$. Recall that we have the analytic subvariety $D\\subset Y$ so that $Y^\\circ=Y\\backslash D$ is smooth and $f$ is a submersion over $Y^\\circ$. Given $\\ve>0$ we define\n\\begin{equation}\n V_{\\epsilon}:=\\{y\\in Y\\mid d_{\\can}(D,y)<\\epsilon\\}, \\quad \\tilde V_{\\epsilon}:=f^{-1}(V_{\\epsilon}).\n\\end{equation}and let $(Y\\setminus V_{\\epsilon},d_{\\can})$ denote the restriction of the metric $d_{\\can}$ from $Y$ to $Y\\setminus V_{\\epsilon}$, and similarly let $(X\\setminus \\tilde V_{\\epsilon},d_t)$ be restriction of $d_t$ from $X$ to $X\\setminus \\tilde V_{\\epsilon}$.\n\n\\begin{lemma}\\label{lem--dcan bound by dt}\n Suppose the following is true: given any small $\\ve>0$ and two points $p, q\\in X\\setminus \\tilde V_{\\epsilon}$, we have\n \\begin{equation}\\label{eq--to prove}\n d_{\\can}(f(p),f(q))\\leq d_t(p,q)+\\Psi(t^{-1}|\\epsilon).\n\\end{equation}\nIn this case, we then have\n\\begin{equation}\\label{agogna}\n \\lim_{t\\rightarrow\\infty}d_{\\gh}((X,d_t),(Y,d_{\\can}))=0\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\n Thanks to Lemma \\ref{lem--gh close for smooth exhaustion}, to prove \\eqref{agogna} is enough to show that\n \\begin{equation}\\label{eq--intermediate}\n d_{\\gh}((X\\setminus \\tilde V_{\\epsilon},d_t), (Y\\setminus V_{\\epsilon},d_{\\can}))=\\Psi(t^{-1}| \\epsilon).\n \\end{equation} To show this, we will show that the fibration map\n \\begin{equation}\n f:X\\setminus \\tilde V_{\\epsilon}\\rightarrow Y\\setminus V_{\\epsilon}\n \\end{equation}is the Gromov-Hausdorff approximation that we want.\nIndeed, by Lemma \\ref{lem--convexity on y}, and the $C^0_{\\loc}$-convergence of $\\omega(t)$ to $\\omega_{\\can}$ on $f^{-1}(Y^{\\circ})$ in \\eqref{conv}, we know that for any $p,q\\in X\\setminus \\tilde V_{\\epsilon}$, we have\n \\begin{equation}\n d_t(p,q)\\leq d_{\\can}(f(p),f(q))+\\Psi(t^{-1}|\\epsilon).\n \\end{equation}\nCombining this with \\eqref{eq--to prove} then gives\n \\begin{equation}\n \\left|d_t(p,q)-d_{\\can}(f(p),f(q))\\right|=\\Psi(t^{-1}|\\epsilon),\n \\end{equation}\nand since clearly $f$ is surjective, we obtain the desired Gromov-Hausdorff approximation, which shows \\eqref{eq--intermediate}.\n\\end{proof}\n\nFollowing the argument in \\cite{LT}, we now show that Proposition \\ref{prop--main to prove} implies Theorem \\ref{main}:\n\\begin{proof}[Proof of Theorem \\ref{main}, assuming Proposition \\ref{prop--main to prove}]\nThanks to Lemma \\ref{lem--dcan bound by dt}, it suffices to show that \\eqref{eq--to prove} holds. For this, we first observe the following easy upper bound of the $\\mathcal L$-distance. By Lemma \\ref{lem--convexity on y}, given $p,q\\in X\\setminus \\tilde V_{\\epsilon}$, we can find a curve $\\gamma$ contained in $Y\\setminus V_{\\epsilon'}$ connecting $f(p)$ and $f(q)$ such that\n\\begin{equation}\n \\mathrm{length}_{\\omega_{\\can}}(\\gamma)\\leq d_{\\can}(f(p),f(q))+\\Psi(\\epsilon'|\\epsilon).\n\\end{equation}\nThen using the fiber bundle structure of $f$ and the $C^0_{\\loc}$ convergence of $\\omega(t)$, arguing as in \\cite[Proof of Lemma 9.1]{TWY2}, we can construct a curve $\\tilde \\gamma$ contained in $X^\\circ$, connecting $p$ and $q$, such that\n\\begin{equation}\\label{eq--length upper bound}\n \\mathrm{length}_{\\omega(T)}(\\tilde \\gamma)\\leq d_{\\can}(f(p),f(q))+\\Psi(\\epsilon'|\\epsilon)+\\Psi(T^{-1}|\\epsilon').\n\\end{equation}Note that this curve $\\tilde\\gamma$ depends on the parameter $\\epsilon'$ and for simplicity of notations, in the following, we omit the dependence on $\\epsilon'$.\n\n\\begin{equation}\\label{eq--KRF}\n\\left\\{\n                \\begin{aligned}\n                  &\\frac{\\de}{\\de t}\\omega(t)=-\\Ric(\\omega(t))-\\omega(t),\\\\\n                  &\\omega(0)=\\omega_0,\n                \\end{aligned}\n              \\right.\n\\end{equation}\n\n\\begin{theorem}\\label{main}In the setting of Conjecture \\ref{con}, we have that as $t\\to\\infty$ the flow $(X,\\omega(t))$ converges in the Gromov-Hausdorff topology to $(Y,d_{\\mathrm{can}})$. In particular, Conjecture \\ref{con} holds.\n\\end{theorem}", "post_theorem_intro_text_len": 3163, "post_theorem_intro_text": "In particular, in the case when $m=n$, our arguments are different from those in \\cite{Wang}, so we obtain a new proof of that case. For general $m$, our proof follows a strategy introduced in \\cite{LT}, where a key input is Perelman's monotonicity of the reduced volume, viewed as a parabolic analogue of the Bishop-Gromov volume comparison for Riemannian metrics with Ricci curvature bounded below. This allows for some control over how much time minimizing $\\mathcal{L}$-geodesics spend in a neighborhood of the ``bad region'' $X\\backslash X^\\circ=f^{-1}(D)$. However, to push this strategy to the end, \\cite{LT} had to use a result from \\cite{GTZ2} which shows that when $Y$ is smooth and the divisorial components of $D$ have simple normal crossings, $\\omega_{\\rm can}$ is quasi-isometric to a {\\em conical K\\\"ahler metric} near these components, up to a small logarithmic error.\n\nGiven that no such conical description is available for general $Y$, in this paper we refine the analysis in \\cite{LT} to establish the existence of $\\mathcal{L}$-geodesics that almost avoid a small neighborhood of $f^{-1}(D)$ (or more precisely, they visit this neighborhood a controlled number of times for a controlled length of time). After carefully choosing appropriate neighborhoods of $f^{-1}(D)$ (here the RCD property of $(Y,d_{\\rm can})$ is crucial), and combining this almost-avoidance property with a H\\\"older estimate for $d_{\\can}$, we complete the proof of Theorem \\ref{main}.\n\nThe paper is organized as follows. In Section \\ref{sec--pre}, we collect various results from the literature concerning estimates for $(X,\\omega(t))$ and $(Y^{\\circ},\\omega_{\\can})$. In Section \\ref{sec-holder}, we establish a H\\\"older estimate for the potential of $\\omega_{\\can}$ following \\cite{GKSS}, and then derive a H\\\"older estimate for $d_{\\can}$ using \\cite{Li}. In Section \\ref{sec--reduction}, following \\cite{LT}, we reduce the main theorem to an estimate asserting that the $d_{\\can}$-distance can be bounded above by the $\\mathcal{L}$-distance.\nSection \\ref{sec--main} contains the core of the paper. We first prove a general almost-avoidance principle, roughly stating that for suitably small neighborhoods of $f^{-1}(D)$, a generic $\\mathcal{L}$-geodesic cannot intersect such neighborhoods too many times. Then we carefully choose the small neighborhoods of $f^{-1}(D)$ to apply the almost-avoidance principle to, and we use the RCD property of $(Y,d_{\\can})$ to establish an upper volume bound for these neighborhoods.\n Finally combining the almost-avoidance principle with the H\\\"older estimate obtained in Section \\ref{sec-holder}, we complete the proof.\n\n\\subsection*{Acknowledgements} We are grateful to G\\'abor Sz\\'ekelyhidi for very useful discussions, and to Shouhei Honda, Wenshuai Jiang, Xiaochun Rong, and Song Sun for email communications. The third-named author thanks Yifan Guo for discussions about Poincar\\'e inequalities on minimal surfaces.  The first-named author is supported by Hong Kong RGC grants No. 14300623 and No. 14304225, and an Asian Young Scientist Fellowship. The second-named author was partially supported by NSF grant DMS-2404599.", "sketch": "For general $m$, the proof of Theorem~\\ref{main} follows the strategy of \\cite{LT}, using \"Perelman's monotonicity of the reduced volume\" to control how much time minimizing $\\mathcal{L}$-geodesics spend near the \"bad region'' $X\\backslash X^\\circ=f^{-1}(D)$. Since no conical description of $\\omega_{\\rm can}$ is available for general $Y$, the paper \"refine[s] the analysis in \\cite{LT} to establish the existence of $\\mathcal{L}$-geodesics that almost avoid a small neighborhood of $f^{-1}(D)$ (or more precisely, they visit this neighborhood a controlled number of times for a controlled length of time).\" After \"carefully choosing appropriate neighborhoods of $f^{-1}(D)$ (here the RCD property of $(Y,d_{\\rm can})$ is crucial),\" and combining this almost-avoidance property with a H\\\"older estimate for $d_{\\can}$, the proof is completed.\n\nMore structurally: (i) establish a H\\\"older estimate for the potential of $\\omega_{\\can}$ and hence for $d_{\\can}$; (ii) reduce the main theorem to an estimate that the $d_{\\can}$-distance is bounded above by the $\\mathcal{L}$-distance; (iii) prove an \"almost-avoidance principle\" that a generic $\\mathcal{L}$-geodesic cannot intersect suitably small neighborhoods of $f^{-1}(D)$ too many times; (iv) choose neighborhoods and use the RCD property of $(Y,d_{\\can})$ to get an upper volume bound for them; (v) combine the almost-avoidance principle with the H\\\"older estimate to complete the proof.", "expanded_sketch": "For general $m$, the proof of To prove the main theorem, … follows the strategy of \\cite{LT}, using \"Perelman's monotonicity of the reduced volume\" to control how much time minimizing $\\mathcal{L}$-geodesics spend near the \"bad region'' $X\\backslash X^\\circ=f^{-1}(D)$. Since no conical description of $\\omega_{\\rm can}$ is available for general $Y$, the paper \"refine[s] the analysis in \\cite{LT} to establish the existence of $\\mathcal{L}$-geodesics that almost avoid a small neighborhood of $f^{-1}(D)$ (or more precisely, they visit this neighborhood a controlled number of times for a controlled length of time).\" After \"carefully choosing appropriate neighborhoods of $f^{-1}(D)$ (here the RCD property of $(Y,d_{\\rm can})$ is crucial),\" and combining this almost-avoidance property with a H\\\"older estimate for $d_{\\can}$, the proof is completed.\n\nMore structurally: (i) establish a H\\\"older estimate for the potential of $\\omega_{\\can}$ and hence for $d_{\\can}$; (ii) reduce the main theorem to an estimate that the $d_{\\can}$-distance is bounded above by the $\\mathcal{L}$-distance; (iii) prove an \"almost-avoidance principle\" that a generic $\\mathcal{L}$-geodesic cannot intersect suitably small neighborhoods of $f^{-1}(D)$ too many times; (iv) choose neighborhoods and use the RCD property of $(Y,d_{\\can})$ to get an upper volume bound for them; (v) combine the almost-avoidance principle with the H\\\"older estimate to complete the proof.,", "expanded_theorem": "\\label{main}In the setting of \\begin{conjecture}\\label{con}\nLet $(X^n,\\omega_0)$ be a compact K\\\"ahler manifold with $K_X$ semiample, and let $\\omega(t)$ be the solution of the K\\\"ahler-Ricci flow \\eqref{eq--KRF}. Then as $t\\to\\infty$, $(X,\\omega(t))$ converges in the Gromov-Hausdorff topology to the metric completion of $(Y^\\circ,\\omega_{\\rm can})$. This is a compact metric space homeomorphic to the canonical model $Y$.\n\\end{conjecture} we have that as $t\\to\\infty$ the flow $(X,\\omega(t))$ converges in the Gromov-Hausdorff topology to $(Y,d_{\\mathrm{can}})$. In particular, the conjecture above holds.", "theorem_type": ["Asymptotic or Limit", "Implication"], "mcq": {"question": "Let $(X^n,\\omega_0)$ be a compact K\\\"ahler manifold with semiample canonical bundle $K_X$, and let $\\omega(t)$ be the solution of the normalized K\\\"ahler-Ricci flow\n\\[\n\\begin{cases}\n\\dfrac{\\partial}{\\partial t}\\omega(t)=-\\mathrm{Ric}(\\omega(t))-\\omega(t),\\\\\n\\omega(0)=\\omega_0.\n\\end{cases}\n\\]\nLet $Y$ be the canonical model of $X$, let $Y^\\circ$ denote the regular part on which the canonical metric $\\omega_{\\rm can}$ is defined, and let $(Y,d_{\\rm can})$ denote the metric completion of $(Y^\\circ,\\omega_{\\rm can})$. Under these hypotheses, which statement about the long-time behavior of the flow is valid?", "correct_choice": {"label": "A", "text": "As $t\\to\\infty$, the metric spaces $(X,\\omega(t))$ converge in the Gromov-Hausdorff topology to $(Y,d_{\\rm can})$; equivalently, they converge to the metric completion of $(Y^\\circ,\\omega_{\\rm can})$. Thus the predicted Gromov-Hausdorff limit description by the canonical model holds in this setting."}, "choices": [{"label": "B", "text": "As $t\\to\\infty$, the metric spaces $(X,\\omega(t))$ converge in the Gromov-Hausdorff topology to $(Y^\\circ,\\omega_{\\rm can})$ itself, without passing to its metric completion. Thus the singular set plays no role in the limiting metric space."}, {"label": "C", "text": "As $t\\to\\infty$, the metric spaces $(X,\\omega(t))$ admit subsequences that converge in the Gromov-Hausdorff topology to $(Y,d_{\\rm can})$. In particular, $(Y,d_{\\rm can})$ occurs as a Gromov-Hausdorff limit of the flow."}, {"label": "D", "text": "As $t\\to\\infty$, the metric spaces $(X,\\omega(t))$ converge in the Gromov-Hausdorff topology to a compact metric space homeomorphic to the canonical model $Y$, but the theorem does not identify this limit canonically with $(Y,d_{\\rm can})$ or with the metric completion of $(Y^\\circ,\\omega_{\\rm can})$."}, {"label": "E", "text": "As $t\\to\\infty$, the metric spaces $(X,\\omega(t))$ converge in the Gromov-Hausdorff topology to $(Y,d_{\\rm can})$, and moreover the convergence is smooth on all of $X$ after identifying $X$ with $Y$ through the canonical map."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "need_for_metric_completion_across_singular_set", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "full_sequence_convergence_replaced_by_subsequential_convergence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "canonical_identification_of_limit_with_(Y,d_can)", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "GH_convergence_upgraded_to_global_smooth_convergence", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It introduces the relevant objects and asks for the valid asymptotic statement, but does not itself state full Gromov-Hausdorff convergence to the completed canonical metric space."}, "TAS": {"score": 0, "justification": "The item is essentially asking for the precise statement of a known theorem. The correct choice largely restates the theorem's conclusion rather than requiring transfer of the result to a new situation."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in separating full convergence from subsequential convergence, completion versus regular part, and GH convergence versus smooth convergence. However, the main task is still theorem recall, not substantial generative reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: one is weaker-but-true sounding, one omits metric completion, one weakens canonical identification, and one overstates smoothness. These reflect common failure modes in reading geometric convergence results."}, "total_score": 5, "overall_assessment": "A solid recall-style theorem-identification MCQ with strong distractors and little answer leakage, but it is fairly tautological and only moderately tests genuine reasoning."}}
{"id": "2602.20035v1", "paper_link": "http://arxiv.org/abs/2602.20035v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "[Communication-efficient similarity search for signals]\\label{thm:jl_signals_receiver}\nFix $d\\ge 1$ and $n \\ge 8,$ and let $f_1,\\dots,f_n\\colon [0,1]^d\\to[-1,1]$ be continuous.\nA \\emph{sketching protocol} consists of an encoding map\n\\[\n\\Enc\\colon C([0,1]^d)\\to\\mathbb R^k\n\\]\nand a decoding rule\n\\[\n\\Dec\\colon \\mathbb R^k\\times \\mathbb R^k\\to\\mathbb R,\n\\]\nwhere $\\Dec(\\Enc(f_i),\\Enc(f_j))$ is interpreted as an estimate of the squared $L_2$-distance\n\\[\nD(i,j):=\\int_{[0,1]^d}\\abs{f_i(x)-f_j(x)}^2\\,\\,\\mathrm{d} x.\n\\]\n\nThen for every $\\varepsilon\\in(0,1)$ there exist $k\\le C\\,\\frac{\\ln n}{\\varepsilon^2}$ and maps $\\Enc,\\Dec$\nsuch that for all $i,j\\in[n]$,\n\\[\n\\abs{\\Dec(\\Enc(f_i),\\Enc(f_j)) - D(i,j)} \\le \\varepsilon.\n\\]\nMoreover, one may take $\\Dec(u,v)=\\left|u-v\\right|^2$ (i.e.\\ decoding is just squared Euclidean distance $\\left|\\cdot\\right|^2$) and $\\Enc$ to be linear.", "start_pos": 7434, "end_pos": 8281, "label": "thm:jl_signals_receiver"}, "ref_dict": {}, "pre_theorem_intro_text_len": 3469, "pre_theorem_intro_text": "The theorems of Helly~\\cite{helly1923mengen} and Carath\\'eodory~\\cite{caratheodory1911variabilitatsbereich} are two celebrated results that form part of the foundation of modern convexity and its adjacent areas.\nAt a basic level, these theorems reveal fundamental combinatorial and geometric properties of convex sets in finite-dimensional linear spaces and, in particular, they can be used to characterize the dimension of the ambient space.\nBoth results have given rise to extensive and largely independent lines of research; we refer to~\\cite{barany2022helly, barany2021combinatorial} for recent surveys.\n\nIn a breakthrough paper~\\cite{adiprasito2020theorems}, Adiprasito, B\\'ar\\'any, Mustafa, and Terpai obtained \\emph{no-dimensional} (or \\emph{approximate}) versions of Helly's and Carath\\'eodory's theorems in Euclidean spaces.\nThese results establish a striking connection between combinatorial convexity and approximation theory: instead of exact intersection/containment statements whose quantitative form depends on the dimension,\none gets approximate conclusions controlled by a single natural parameter.\nMotivated by potential applications to algorithmic questions (such as linear sketching and local-to-global guarantees in optimization), it is natural to ask for analogous results beyond Euclidean spaces.\n\nThe goal of this note is to bring attention to this circle of problems.\nWe formulate several open questions and record some straightforward corollaries of the no-dimensional results in Banach spaces.\nAt the same time, we make one additional tweak and ``return the dimension to the picture'': we combine no-dimensional results with a standard substitution trick from Banach space theory,\nwhich allows one to pass from $\\ell_1$-type or $\\ell_\\infty$-type geometry to an appropriately chosen $\\ell_p$-space with desired geometric properties.\n\nWe will treat the needed Banach space facts as a black box.\nFor convenience, we recall all relevant definitions and inequalities in \\Href{Section}{sec:banach_space_likbez}, in order to keep the paper self-contained.\nThen we explain in detail what we mean by no-dimensional versions of Helly's and Carath\\'eodory's theorems, survey recent results on the topic, and formulate several open problems.\nFinally, we describe a standard substitution trick and show that a number of ``local-to-global'' statements become straightforward corollaries of the no-dimensional theory.\n\n\\subsection{Applications of no-dimensional results}\n\nIn many applications \\cite{Garofalakis2013,Wang2016} one stores or transmits large objects\n(signals, images, fields), while the downstream task uses only pairwise similarity queries,\noften measured by the squared $L_2$-distance (equivalently, the $L_2$-energy).\nThe following theorem shows that, for a fixed collection of $n$ signals, one can replace each\nsignal by a short ``fingerprint'' in $\\mathbb R^k$ for sufficiently small $k$, while preserving all\npairwise squared $L_2$-distances up to additive error~$\\varepsilon$.\n\nThe result can be viewed as a weak additive variant of the celebrated Johnson--Lindenstrauss\nflattening lemma \\cite{johnson1984extensions}; see also \\cite{vershynin2018high} for modern\nexpositions. While the classical Johnson--Lindenstrauss lemma is typically proved by a random\nprojection argument, our proof is based on a deterministic construction arising from a\nno-dimensional Carath\\'eodory-type lemma. This provides a simple illustration of the method.", "context": "The theorems of Helly~\\cite{helly1923mengen} and Carath\\'eodory~\\cite{caratheodory1911variabilitatsbereich} are two celebrated results that form part of the foundation of modern convexity and its adjacent areas.\nAt a basic level, these theorems reveal fundamental combinatorial and geometric properties of convex sets in finite-dimensional linear spaces and, in particular, they can be used to characterize the dimension of the ambient space.\nBoth results have given rise to extensive and largely independent lines of research; we refer to~\\cite{barany2022helly, barany2021combinatorial} for recent surveys.\n\nIn a breakthrough paper~\\cite{adiprasito2020theorems}, Adiprasito, B\\'ar\\'any, Mustafa, and Terpai obtained \\emph{no-dimensional} (or \\emph{approximate}) versions of Helly's and Carath\\'eodory's theorems in Euclidean spaces.\nThese results establish a striking connection between combinatorial convexity and approximation theory: instead of exact intersection/containment statements whose quantitative form depends on the dimension,\none gets approximate conclusions controlled by a single natural parameter.\nMotivated by potential applications to algorithmic questions (such as linear sketching and local-to-global guarantees in optimization), it is natural to ask for analogous results beyond Euclidean spaces.\n\nThe goal of this note is to bring attention to this circle of problems.\nWe formulate several open questions and record some straightforward corollaries of the no-dimensional results in Banach spaces.\nAt the same time, we make one additional tweak and ``return the dimension to the picture'': we combine no-dimensional results with a standard substitution trick from Banach space theory,\nwhich allows one to pass from $\\ell_1$-type or $\\ell_\\infty$-type geometry to an appropriately chosen $\\ell_p$-space with desired geometric properties.\n\nWe will treat the needed Banach space facts as a black box.\nFor convenience, we recall all relevant definitions and inequalities in \\Href{Section}{sec:banach_space_likbez}, in order to keep the paper self-contained.\nThen we explain in detail what we mean by no-dimensional versions of Helly's and Carath\\'eodory's theorems, survey recent results on the topic, and formulate several open problems.\nFinally, we describe a standard substitution trick and show that a number of ``local-to-global'' statements become straightforward corollaries of the no-dimensional theory.\n\nIn many applications \\cite{Garofalakis2013,Wang2016} one stores or transmits large objects\n(signals, images, fields), while the downstream task uses only pairwise similarity queries,\noften measured by the squared $L_2$-distance (equivalently, the $L_2$-energy).\nThe following theorem shows that, for a fixed collection of $n$ signals, one can replace each\nsignal by a short ``fingerprint'' in $\\mathbb R^k$ for sufficiently small $k$, while preserving all\npairwise squared $L_2$-distances up to additive error~$\\varepsilon$.\n\nThe result can be viewed as a weak additive variant of the celebrated Johnson--Lindenstrauss\nflattening lemma \\cite{johnson1984extensions}; see also \\cite{vershynin2018high} for modern\nexpositions. While the classical Johnson--Lindenstrauss lemma is typically proved by a random\nprojection argument, our proof is based on a deterministic construction arising from a\nno-dimensional Carath\\'eodory-type lemma. This provides a simple illustration of the method.", "full_context": "The theorems of Helly~\\cite{helly1923mengen} and Carath\\'eodory~\\cite{caratheodory1911variabilitatsbereich} are two celebrated results that form part of the foundation of modern convexity and its adjacent areas.\nAt a basic level, these theorems reveal fundamental combinatorial and geometric properties of convex sets in finite-dimensional linear spaces and, in particular, they can be used to characterize the dimension of the ambient space.\nBoth results have given rise to extensive and largely independent lines of research; we refer to~\\cite{barany2022helly, barany2021combinatorial} for recent surveys.\n\nIn a breakthrough paper~\\cite{adiprasito2020theorems}, Adiprasito, B\\'ar\\'any, Mustafa, and Terpai obtained \\emph{no-dimensional} (or \\emph{approximate}) versions of Helly's and Carath\\'eodory's theorems in Euclidean spaces.\nThese results establish a striking connection between combinatorial convexity and approximation theory: instead of exact intersection/containment statements whose quantitative form depends on the dimension,\none gets approximate conclusions controlled by a single natural parameter.\nMotivated by potential applications to algorithmic questions (such as linear sketching and local-to-global guarantees in optimization), it is natural to ask for analogous results beyond Euclidean spaces.\n\nThe goal of this note is to bring attention to this circle of problems.\nWe formulate several open questions and record some straightforward corollaries of the no-dimensional results in Banach spaces.\nAt the same time, we make one additional tweak and ``return the dimension to the picture'': we combine no-dimensional results with a standard substitution trick from Banach space theory,\nwhich allows one to pass from $\\ell_1$-type or $\\ell_\\infty$-type geometry to an appropriately chosen $\\ell_p$-space with desired geometric properties.\n\nWe will treat the needed Banach space facts as a black box.\nFor convenience, we recall all relevant definitions and inequalities in \\Href{Section}{sec:banach_space_likbez}, in order to keep the paper self-contained.\nThen we explain in detail what we mean by no-dimensional versions of Helly's and Carath\\'eodory's theorems, survey recent results on the topic, and formulate several open problems.\nFinally, we describe a standard substitution trick and show that a number of ``local-to-global'' statements become straightforward corollaries of the no-dimensional theory.\n\nIn many applications \\cite{Garofalakis2013,Wang2016} one stores or transmits large objects\n(signals, images, fields), while the downstream task uses only pairwise similarity queries,\noften measured by the squared $L_2$-distance (equivalently, the $L_2$-energy).\nThe following theorem shows that, for a fixed collection of $n$ signals, one can replace each\nsignal by a short ``fingerprint'' in $\\mathbb R^k$ for sufficiently small $k$, while preserving all\npairwise squared $L_2$-distances up to additive error~$\\varepsilon$.\n\nThe result can be viewed as a weak additive variant of the celebrated Johnson--Lindenstrauss\nflattening lemma \\cite{johnson1984extensions}; see also \\cite{vershynin2018high} for modern\nexpositions. While the classical Johnson--Lindenstrauss lemma is typically proved by a random\nprojection argument, our proof is based on a deterministic construction arising from a\nno-dimensional Carath\\'eodory-type lemma. This provides a simple illustration of the method.\n\nThe result can be viewed as a weak additive variant of the celebrated Johnson--Lindenstrauss\nflattening lemma \\cite{johnson1984extensions}; see also \\cite{vershynin2018high} for modern\nexpositions. While the classical Johnson--Lindenstrauss lemma is typically proved by a random\nprojection argument, our proof is based on a deterministic construction arising from a\nno-dimensional Carath\\'eodory-type lemma. This provides a simple illustration of the method.\n\nWe emphasize that this result is much weaker than the Johnson--Lindenstrauss flattening lemma.\nFirst, we only obtain an additive approximation to the squared $L_2$-distance. Second, the range\nof the functions is a fixed interval, so the supremum norm is uniformly bounded (and we exploit\nthis in the proof). Third, the functions are defined on a domain of measure one, which is also\nimportant for our argument since we work on a probability space.\n\n\\begin{thm}[Local-to-global Chebyshev regression]\\label{thm:chebyshev_ball}\nLet $a_1,\\dots,a_m\\in [-1,1]^d$ and $b_1,\\dots,b_m\\in \\R$.\nFix $k\\in[m]$ and parameters $R\\ge 1$ and $r\\ge 0$.\nSuppose that for every subset $J\\subset[m]$ with $\\card{J}=k$ there exists a point $x_J\\in R\\ball{}_{\\ell_1^d}$ such that\n\\[\n\\max_{j\\in J}\\abs{\\iprod{a_j}{x_J}-b_j}\\ \\le\\ r.\n\\]\nThen there exists a point $x\\in eR\\ball{}_{\\ell_1^d}$ such that\n\\[\n\\max_{i\\in[m]}\\abs{\\iprod{a_i}{x}-b_i}\n\\ \\le\\\nr + 21\\,R\\,\\sqrt{\\frac{\\ln d}{k}}.\n\\]\n\\end{thm}\n\n\\begin{thm}[Quantum feasibility from local consistency]\\label{thm:quantum_psd_local_to_global}\nLet $d\\ge 3$ and $k\\in[m]$. Let $A_1,\\dots,A_m\\in\\mathcal{H}_d$ be Hermitian matrices satisfying\n\\[\n\\normsch[\\infty]{A_i}\\le 1 \\qquad\\text{for all } i\\in[m],\n\\]\nand let $b_1,\\dots,b_m\\in\\R$. Fix $t\\ge 0$.\nAssume that for every subset $J\\subset[m]$ with $\\abs{J}=k$, there exists a density matrix\n$\\rho_J\\succeq 0$ with $\\mathrm{Tr}(\\rho_J)=1$ such that\n\\[\n\\abs{\\iprod{A_j}{\\rho_J} - b_j}\n\\le t \\qquad\\text{for all } j\\in J.\n\\]\nThen there exists a density matrix $\\rho\\succeq 0$ with $\\mathrm{Tr}(\\rho)=1$ such that\n\\[\n\\abs{\\iprod{A_i}{\\rho} - b_i}\n\\ \\le\\\nt + 21\\,\\sqrt{\\frac{\\ln d}{k}},\n\\]\nfor all $i\\in[m]$.\n\\end{thm}\n\n\\begin{lem}\\label{lem:Caratheodory_sequence_l_p}\nLet $2\\le p<+\\infty$.\nThen the sequence $\\braces{21\\sqrt{\\frac{p-1}{k}}}_{k\\ge 1}$ is a Carath\\'eodory sequence for $\\ell_p$ and for $S_p$.\nMoreover, whenever the convex hull of a subset $Q$ of the unit ball $\\ball{}_X$ contains the origin,\nthere exists a greedy algorithm that selects points $x_1,\\dots,x_k\\in Q$ successively so that\n\\[\n\\norm{\\frac{x_1+\\cdots+x_k}{k}}\n\\le R_k(X)\n\\]\nat each step.\n\\end{lem}\n\n\\section{Dimension strikes back. Carath\\'eodory-type results and problems}\n\\subsection{Johnson--Lindenstrauss-type sketching}\n\\Href{Theorem}{thm:jl_signals_receiver} follows from the following result \nsince the measure of $[0,1]^d$ is one. We note that similar results were obtained in \\cite{Eskenazis2023}.\n\\begin{thm}[Johnson--Lindenstrauss sketching for bounded functions]\\label{thm:JL_for_functions}\nLet $(\\Omega,\\mu)$ be a probability space and let $f_1,\\dots,f_n\\colon \\Omega\\to[-1,1]$ be functions in $L_2(\\Omega,\\mu)$.\nFix $\\varepsilon\\in(0,1)$. Then there exists an integer\n\\[\nk \\ \\le\\ C\\,\\frac{\\ln n}{\\varepsilon^2}\n\\]\nand vectors $x_1,\\dots,x_n\\in\\R^k$ such that for all $i,j\\in[n]$,\n\\[\n\\enorm{x_i-x_j}^2- \\varepsilon \\ \\le\\ \n\\norm{f_i-f_j}_2^2 \\ \\le\\ \n\\enorm{x_i-x_j}^2 + \\varepsilon .\n\\]\nHere $C>0$ is an absolute constant.\n\\end{thm}\n\nApply \\Href{Lemma}{lem:Caratheodory_sequence_l_p} in the space $\\ell_\\infty^{ {n\\choose 2}}$ (via the substitution from \\Href{Proposition}{prp:l_p_trick_bounds})\nto obtain points $\\omega_1,\\dots,\\omega_k\\in\\Omega$ such that the empirical average\n\\[\n\\frac{1}{k}\\sum_{s=1}^k v(\\omega_s)\n\\]\napproximates $\\int_\\Omega v(\\omega)\\,d\\mu(\\omega)$ in $\\ell_\\infty^{ {n\\choose 2}}$ up to error $\\varepsilon$.\nEquivalently, for all $i<j$,\n\\[\n\\enorm{\n\\frac{1}{k}\\sum_{s=1}^k \\abs{f_i(\\omega_s)-f_j(\\omega_s)}^2\n-\n\\norm{f_i-f_j}_2^2\n}\n\\le \\varepsilon.\n\\]\nNow define\n\\[\nx_i:=\\frac{1}{\\sqrt{k}}\\big(f_i(\\omega_1),\\dots,f_i(\\omega_k)\\big)\\in\\R^k.\n\\]\nThen\n\\[\n\\enorm{x_i-x_j}^2=\\frac{1}{k}\\sum_{s=1}^k \\abs{f_i(\\omega_s)-f_j(\\omega_s)}^2,\n\\]\nand the desired inequality follows.\n\\end{proof}\n\\subsection{Positive semidefinite matrix sparsification}\n\n\\begin{conj}\\label{conj:deterministic_rudelson}\nLet $A_1,\\dots,A_n\\in \\mathcal{M}_d$ be positive semidefinite and suppose that\n\\[\n\\lambda_1A_1+\\cdots+\\lambda_nA_n=\\id_d,\n\\]\nwhere the coefficients $\\lambda_i$ are nonnegative and summing up to one.\nThen for every $\\varepsilon\\in(0,1)$ there exists a deterministic algorithm that selects indices\n$i_1,\\dots,i_k\\in[n]$ with $k\\le C(\\varepsilon)\\,d\\ln d$ and outputs positive coefficients $\\beta_1,\\dots,\\beta_k$\nsuch that\n\\[\n\\Bigl\\|\\beta_1A_{i_1}+\\cdots+\\beta_kA_{i_k}-\\id_d\\Bigr\\|_{S_\\infty}\\le \\varepsilon.\n\\]\n\\end{conj}", "post_theorem_intro_text_len": 6304, "post_theorem_intro_text": "We emphasize that this result is much weaker than the Johnson--Lindenstrauss flattening lemma.\nFirst, we only obtain an additive approximation to the squared $L_2$-distance. Second, the range\nof the functions is a fixed interval, so the supremum norm is uniformly bounded (and we exploit\nthis in the proof). Third, the functions are defined on a domain of measure one, which is also\nimportant for our argument since we work on a probability space.\n\nThe main point is that the statement admits a deterministic proof based on a no-dimensional\nCarath\\'eodory lemma.\n\nAlso, we obtain several local-to-global estimates for $\\ell_1$-minimization problems using no-dimensional Helly-type results.\nWe write $[m]:=\\braces{1,\\dots,m}$ for $m\\in\\mathbb N$, and we equip $\\mathbb R^d$ with the standard $\\ell_p$ norm $\\left\\|\\cdot\\right\\|_p$.\nWe denote by $\\ell_p^d:=(\\mathbb R^d,\\left\\|\\cdot\\right\\|_p)$ the resulting normed space and by $\\ball{}_{\\ell_p^d}$ its unit ball.\n\n\\medskip\nOne basic example is \\emph{Chebyshev regression} (also known as \\emph{minimax absolute residual fitting} or $\\ell_\\infty$-regression), the classical problem of fitting a linear model by minimizing the worst absolute residual:\n\\[\n\\min_{x\\in\\mathbb R^d}\\ \\max_{i\\in[m]} \\abs{\\iprod{a_i}{x}-b_i}.\n\\]\nIt admits an immediate linear programming formulation by introducing an auxiliary variable $t$ and enforcing the two-sided constraints\n$-t\\le \\iprod{a_i}{x}-b_i\\le t$ for all $i\\in[m]$; see, e.g., \\cite{boyd2004convex,ben2001lectures}.\nFrom an algorithmic viewpoint, this connects Chebyshev regression to the general toolbox of convex optimization and linear programming, while its geometry is governed by small active constraint sets, a perspective formalized in the LP-type framework and its randomized algorithms \\cite{clarkson1995vegas,matouvsek1992subexponential}.\nMore recently, Chebyshev/$\\ell_\\infty$ guarantees have also been studied via modern randomized linear algebra and ``sketch-and-solve'' methods, where one seeks fast approximate solutions together with explicit $\\ell_\\infty$ control \\cite{price2017fast,song2023nearly}.\nIn the present work, we pursue a different direction: using no-dimensional Helly-type results to derive \\emph{local-to-global} approximation guarantees from $k$-wise feasibility information, with explicit dimension dependence.\n\nOur first local-to-global statement shows that approximate global feasibility over the $\\ell_1$ ball follows from approximate feasibility on all $k$-subcollections of constraints, with an explicit  dependence on the dimension.\n\n\\begin{thm}[Local-to-global Chebyshev regression]\\label{thm:chebyshev_ball}\nLet $a_1,\\dots,a_m\\in [-1,1]^d$ and $b_1,\\dots,b_m\\in \\mathbb R$.\nFix $k\\in[m]$ and parameters $R\\ge 1$ and $r\\ge 0$.\nSuppose that for every subset $J\\subset[m]$ with $\\card{J}=k$ there exists a point $x_J\\in R\\ball{}_{\\ell_1^d}$ such that\n\\[\n\\max_{j\\in J}\\abs{\\iprod{a_j}{x_J}-b_j}\\ \\le\\ r.\n\\]\nThen there exists a point $x\\in eR\\ball{}_{\\ell_1^d}$ such that\n\\[\n\\max_{i\\in[m]}\\abs{\\iprod{a_i}{x}-b_i}\n\\ \\le\\\nr + 21\\,R\\,\\sqrt{\\frac{\\ln d}{k}}.\n\\]\n\\end{thm}\n\n\\medskip\nOur second application of no-dimensional Helly-type results is motivated by feasibility and consistency questions in quantum information theory.\nA prototypical instance is the \\emph{consistency of local density matrices} (also known as the \\emph{quantum marginal problem}), which asks whether a collection of local constraints can arise from a single global quantum state; this problem is known to be computationally intractable in general; it was first shown to be in QMA and QMA-complete under Turing reductions by Liu \\cite{liu2006consistency}, and subsequently proven to be QMA-complete under Karp reductions by Broadbent and Grilo \\cite{broadbent2022qma}.\nFrom a geometric viewpoint, the natural ambient spaces are Schatten classes and their convexity properties, and we refer to \\cite{aubrun2017alice} for background on the interface between asymptotic geometric analysis and quantum information.\n\nFix $d\\ge 1$ and let $\\mathbb{F}\\in\\{\\mathbb R,\\mathbb C\\}$.\nWrite $\\mathcal{M}_d:=\\mathbb{F}^{d\\times d}$ for the space of $d\\times d$ matrices over $\\mathbb{F}$.\nWe say that $A\\in\\mathcal{M}_d$ is \\emph{self-adjoint} if $A=A^\\ast$, where $A^\\ast$ denotes the transpose when $\\mathbb{F}=\\mathbb R$\nand the conjugate transpose when $\\mathbb{F}=\\mathbb C$.\nIf $\\mathbb{F}=\\mathbb C$, let\n\\[\n\\mathcal{H}_d:=\\{A\\in\\mathbb C^{d\\times d}:\\ A=A^\\ast\\}\n\\]\ndenote the  vector space of Hermitian matrices.\n\nFor $1\\le p\\le\\infty$ we write $\\normsch[p]{A}$ for the Schatten $p$-norm (the $\\ell_p$-norm of the singular value vector),\nand we equip $\\mathcal{M}_d$ with the Frobenius (Hilbert--Schmidt) inner product\n\\[\n\\iprod{A}{B}:=\\mathrm{Tr}(A^\\ast B).\n\\]\n\nA matrix $\\rho\\in\\mathcal{M}_d$ is called \\emph{positive semidefinite}, written $\\rho\\succeq 0$, if it is self-adjoint and has\nnonnegative spectrum. A \\emph{density matrix} is a positive semidefinite matrix of unit trace:\n\\[\n\\rho\\succeq 0,\n\\qquad\n\\mathrm{Tr}(\\rho)=1.\n\\]\n\nGiven self-adjoint measurement operators $A_1,\\dots,A_m\\in\\mathcal{M}_d$ and target values $b_1,\\dots,b_m\\in\\mathbb R$,\nwe consider the feasibility problem of finding a density matrix whose linear measurements $\\iprod{A_i}{\\rho}$ are simultaneously\nconsistent with $b_i$ up to a tolerance level.\nIn this work we study a \\emph{local-to-global} version: assuming that every subcollection of $k$ constraints is feasible up to error $t$,\nwe ask whether this implies the existence of a single density matrix that approximately satisfies all $m$ constraints.\n\n\\begin{thm}[Quantum feasibility from local consistency]\\label{thm:quantum_psd_local_to_global}\nLet $d\\ge 3$ and $k\\in[m]$. Let $A_1,\\dots,A_m\\in\\mathcal{H}_d$ be Hermitian matrices satisfying\n\\[\n\\normsch[\\infty]{A_i}\\le 1 \\qquad\\text{for all } i\\in[m],\n\\]\nand let $b_1,\\dots,b_m\\in\\mathbb R$. Fix $t\\ge 0$.\nAssume that for every subset $J\\subset[m]$ with $\\abs{J}=k$, there exists a density matrix\n$\\rho_J\\succeq 0$ with $\\mathrm{Tr}(\\rho_J)=1$ such that\n\\[\n\\abs{\\iprod{A_j}{\\rho_J} - b_j}\n\\le t \\qquad\\text{for all } j\\in J.\n\\]\nThen there exists a density matrix $\\rho\\succeq 0$ with $\\mathrm{Tr}(\\rho)=1$ such that\n\\[\n\\abs{\\iprod{A_i}{\\rho} - b_i}\n\\ \\le\\\nt + 21\\,\\sqrt{\\frac{\\ln d}{k}},\n\\]\nfor all $i\\in[m]$.\n\\end{thm}", "sketch": "The post-theorem introduction gives only a high-level indication of the proof method for Theorem~\\ref{thm:jl_signals_receiver}: it notes that the proof exploits that (i) the functions take values in a fixed interval so “the supremum norm is uniformly bounded (and we exploit this in the proof),” and (ii) the domain has “measure one,” which is “important for our argument since we work on a probability space.” It also states that “the statement admits a deterministic proof based on a no-dimensional Carath\\'eodory lemma.” No further steps or structure of the argument are described there.", "expanded_sketch": "The post-theorem introduction gives only a high-level indication of the proof method for the main theorem: it notes that the proof exploits that (i) the functions take values in a fixed interval so “the supremum norm is uniformly bounded (and we exploit this in the proof),” and (ii) the domain has “measure one,” which is “important for our argument since we work on a probability space.” It also states that “the statement admits a deterministic proof based on a no-dimensional Carath\\'eodory lemma.” No further steps or structure of the argument are described there.", "expanded_theorem": "[Communication-efficient similarity search for signals]\\label{thm:jl_signals_receiver}\nFix $d\\ge 1$ and $n \\ge 8,$ and let $f_1,\\dots,f_n\\colon [0,1]^d\\to[-1,1]$ be continuous.\nA \\emph{sketching protocol} consists of an encoding map\n\\[\n\\Enc\\colon C([0,1]^d)\\to\\mathbb R^k\n\\]\nand a decoding rule\n\\[\n\\Dec\\colon \\mathbb R^k\\times \\mathbb R^k\\to\\mathbb R,\n\\]\nwhere $\\Dec(\\Enc(f_i),\\Enc(f_j))$ is interpreted as an estimate of the squared $L_2$-distance\n\\[\nD(i,j):=\\int_{[0,1]^d}\\abs{f_i(x)-f_j(x)}^2\\,\\,\\mathrm{d} x.\n\\]\n\nThen for every $\\varepsilon\\in(0,1)$ there exist $k\\le C\\,\\frac{\\ln n}{\\varepsilon^2}$ and maps $\\Enc,\\Dec$\nsuch that for all $i,j\\in[n],$\n\\[\n\\abs{\\Dec(\\Enc(f_i),\\Enc(f_j)) - D(i,j)} \\le \\varepsilon.\n\\]\nMoreover, one may take $\\Dec(u,v)=\\left|u-v\\right|^2$ (i.e.\\ decoding is just squared Euclidean distance $\\left|\\cdot\\right|^2$) and $\\Enc$ to be linear.", "theorem_type": ["Existential–Universal", "Algorithmic or Constructive"], "mcq": {"question": "Let $d\\ge 1$ and $n\\ge 8$, and let $f_1,\\dots,f_n\\colon [0,1]^d\\to[-1,1]$ be continuous functions. Define the squared $L_2$-distance between two signals by\n\\[\nD(i,j):=\\int_{[0,1]^d} |f_i(x)-f_j(x)|^2\\,dx.\n\\]\nA sketching protocol consists of an encoding map $\\mathrm{Enc}\\colon C([0,1]^d)\\to\\mathbb R^k$ and a decoding rule $\\mathrm{Dec}\\colon \\mathbb R^k\\times\\mathbb R^k\\to\\mathbb R$, where $\\mathrm{Dec}(\\mathrm{Enc}(f_i),\\mathrm{Enc}(f_j))$ is used to estimate $D(i,j)$. For a given $\\varepsilon\\in(0,1)$, which statement holds for such a family of signals?", "correct_choice": {"label": "A", "text": "There exist an integer $k\\le C\\,\\frac{\\ln n}{\\varepsilon^2}$ and maps $\\mathrm{Enc}$ and $\\mathrm{Dec}$ such that for every $i,j\\in[n]$,\n\\[\n\\big|\\mathrm{Dec}(\\mathrm{Enc}(f_i),\\mathrm{Enc}(f_j)) - D(i,j)\\big|\\le \\varepsilon.\n\\]\nMoreover, one may choose $\\mathrm{Dec}(u,v)=|u-v|^2$ (the squared Euclidean distance in $\\mathbb R^k$) and choose $\\mathrm{Enc}$ to be linear."}, "choices": [{"label": "B", "text": "There exist an integer $k\\le C\\,\\frac{\\ln n}{\\varepsilon^2}$ and maps $\\mathrm{Enc}$ and $\\mathrm{Dec}$ such that for every $i,j\\in[n]$,\n\\[\n\\big|\\mathrm{Dec}(\\mathrm{Enc}(f_i),\\mathrm{Enc}(f_j)) - D(i,j)\\big|\\le \\varepsilon.\n\\]\nMoreover, one may choose $\\mathrm{Dec}(u,v)=|u-v|^2$ and choose $\\mathrm{Enc}$ to be linear, with $C>0$ an absolute constant independent of $d$, $n$, $\\varepsilon$, and of the range bound on the functions."}, {"label": "C", "text": "There exist an integer $k\\le C\\,\\frac{\\ln n}{\\varepsilon^2}$ and maps $\\mathrm{Enc}$ and $\\mathrm{Dec}$ such that for every $i,j\\in[n]$,\n\\[\n\\big|\\mathrm{Dec}(\\mathrm{Enc}(f_i),\\mathrm{Enc}(f_j)) - D(i,j)\\big|\\le \\varepsilon.\n\\]"}, {"label": "D", "text": "There exist an integer $k\\le C\\,\\frac{\\ln n}{\\varepsilon^2}$ and maps $\\mathrm{Enc}$ and $\\mathrm{Dec}$ such that for every $i,j\\in[n]$,\n\\[\n\\big|\\mathrm{Dec}(\\mathrm{Enc}(f_i),\\mathrm{Enc}(f_j)) - D(i,j)\\big|\\le \\varepsilon.\n\\]\nMoreover, one may choose $\\mathrm{Dec}(u,v)=|u-v|^2$ and choose $\\mathrm{Enc}$ to be linear even when the functions $f_1,\\dots,f_n\\colon [0,1]^d\\to\\mathbb R$ are arbitrary continuous functions, with no assumption that their values lie in $[-1,1]$."}, {"label": "E", "text": "There exist maps $\\mathrm{Enc}$ and $\\mathrm{Dec}$ such that for every $\\varepsilon\\in(0,1)$ and every $i,j\\in[n]$,\n\\[\n\\big|\\mathrm{Dec}(\\mathrm{Enc}(f_i),\\mathrm{Enc}(f_j)) - D(i,j)\\big|\\le \\varepsilon,\n\\]\nwith a sketch dimension satisfying\n\\[\nk\\le C\\,\\frac{\\ln n}{\\varepsilon},\n\\]\nand one may take $\\mathrm{Dec}(u,v)=|u-v|^2$ and $\\mathrm{Enc}$ linear."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "hidden dependence on bounded range normalization", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the additional conclusion specifying Euclidean decoding and linear encoding", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "uniform bounded-range hypothesis $f_i\\in[-1,1]$", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "the $\\varepsilon^{-2}$ dependence in the sketch dimension bound", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the setup and asks which conclusion is valid, but it does not explicitly reveal additive vs. relative error, linearity, decoder form, or quantifier dependence. There is no direct answer leakage."}, "TAS": {"score": 2, "justification": "This is not a bare restatement of a theorem in the stem. The choices differ in meaningful ways: additive vs. relative error, linear vs. arbitrary encoding, instance-dependent vs. uniform construction, and dimension scaling."}, "GPS": {"score": 1, "justification": "A solver must reason about which theorem-strengthening is justified, so there is some real mathematical pressure. However, because choice C is a weaker statement that is also true if A is true, the item does not cleanly force selection of a unique strongest conclusion unless the stem explicitly asked for the strongest statement."}, "DQS": {"score": 1, "justification": "B, D, and E are plausible theorem-variant distractors tied to common overclaim errors. But C is not a genuine distractor: it is a weaker true statement implied by A, making the item ambiguous under the wording 'Which statement holds'."}, "total_score": 6, "overall_assessment": "Good on leakage and non-tautology, but flawed as a single-answer MCQ because C is also true. It tests theorem-comparison reasoning, yet the ambiguity weakens its validity."}}
{"id": "2602.20035v1", "paper_link": "http://arxiv.org/abs/2602.20035v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "[Communication-efficient similarity search for signals]\\label{thm:jl_signals_receiver}\nFix $d\\ge 1$ and $n \\ge 8,$ and let $f_1,\\dots,f_n\\colon [0,1]^d\\to[-1,1]$ be continuous.\nA \\emph{sketching protocol} consists of an encoding map\n\\[\n\\Enc\\colon C([0,1]^d)\\to\\mathbb R^k\n\\]\nand a decoding rule\n\\[\n\\Dec\\colon \\mathbb R^k\\times \\mathbb R^k\\to\\mathbb R,\n\\]\nwhere $\\Dec(\\Enc(f_i),\\Enc(f_j))$ is interpreted as an estimate of the squared $L_2$-distance\n\\[\nD(i,j):=\\int_{[0,1]^d}\\abs{f_i(x)-f_j(x)}^2\\,\\,\\mathrm{d} x.\n\\]\n\nThen for every $\\varepsilon\\in(0,1)$ there exist $k\\le C\\,\\frac{\\ln n}{\\varepsilon^2}$ and maps $\\Enc,\\Dec$\nsuch that for all $i,j\\in[n]$,\n\\[\n\\abs{\\Dec(\\Enc(f_i),\\Enc(f_j)) - D(i,j)} \\le \\varepsilon.\n\\]\nMoreover, one may take $\\Dec(u,v)=\\left|u-v\\right|^2$ (i.e.\\ decoding is just squared Euclidean distance $\\left|\\cdot\\right|^2$) and $\\Enc$ to be linear.", "start_pos": 7434, "end_pos": 8281, "label": "thm:jl_signals_receiver"}, "ref_dict": {}, "pre_theorem_intro_text_len": 3469, "pre_theorem_intro_text": "The theorems of Helly~\\cite{helly1923mengen} and Carath\\'eodory~\\cite{caratheodory1911variabilitatsbereich} are two celebrated results that form part of the foundation of modern convexity and its adjacent areas.\nAt a basic level, these theorems reveal fundamental combinatorial and geometric properties of convex sets in finite-dimensional linear spaces and, in particular, they can be used to characterize the dimension of the ambient space.\nBoth results have given rise to extensive and largely independent lines of research; we refer to~\\cite{barany2022helly, barany2021combinatorial} for recent surveys.\n\nIn a breakthrough paper~\\cite{adiprasito2020theorems}, Adiprasito, B\\'ar\\'any, Mustafa, and Terpai obtained \\emph{no-dimensional} (or \\emph{approximate}) versions of Helly's and Carath\\'eodory's theorems in Euclidean spaces.\nThese results establish a striking connection between combinatorial convexity and approximation theory: instead of exact intersection/containment statements whose quantitative form depends on the dimension,\none gets approximate conclusions controlled by a single natural parameter.\nMotivated by potential applications to algorithmic questions (such as linear sketching and local-to-global guarantees in optimization), it is natural to ask for analogous results beyond Euclidean spaces.\n\nThe goal of this note is to bring attention to this circle of problems.\nWe formulate several open questions and record some straightforward corollaries of the no-dimensional results in Banach spaces.\nAt the same time, we make one additional tweak and ``return the dimension to the picture'': we combine no-dimensional results with a standard substitution trick from Banach space theory,\nwhich allows one to pass from $\\ell_1$-type or $\\ell_\\infty$-type geometry to an appropriately chosen $\\ell_p$-space with desired geometric properties.\n\nWe will treat the needed Banach space facts as a black box.\nFor convenience, we recall all relevant definitions and inequalities in \\Href{Section}{sec:banach_space_likbez}, in order to keep the paper self-contained.\nThen we explain in detail what we mean by no-dimensional versions of Helly's and Carath\\'eodory's theorems, survey recent results on the topic, and formulate several open problems.\nFinally, we describe a standard substitution trick and show that a number of ``local-to-global'' statements become straightforward corollaries of the no-dimensional theory.\n\n\\subsection{Applications of no-dimensional results}\n\nIn many applications \\cite{Garofalakis2013,Wang2016} one stores or transmits large objects\n(signals, images, fields), while the downstream task uses only pairwise similarity queries,\noften measured by the squared $L_2$-distance (equivalently, the $L_2$-energy).\nThe following theorem shows that, for a fixed collection of $n$ signals, one can replace each\nsignal by a short ``fingerprint'' in $\\mathbb R^k$ for sufficiently small $k$, while preserving all\npairwise squared $L_2$-distances up to additive error~$\\varepsilon$.\n\nThe result can be viewed as a weak additive variant of the celebrated Johnson--Lindenstrauss\nflattening lemma \\cite{johnson1984extensions}; see also \\cite{vershynin2018high} for modern\nexpositions. While the classical Johnson--Lindenstrauss lemma is typically proved by a random\nprojection argument, our proof is based on a deterministic construction arising from a\nno-dimensional Carath\\'eodory-type lemma. This provides a simple illustration of the method.", "context": "The theorems of Helly~\\cite{helly1923mengen} and Carath\\'eodory~\\cite{caratheodory1911variabilitatsbereich} are two celebrated results that form part of the foundation of modern convexity and its adjacent areas.\nAt a basic level, these theorems reveal fundamental combinatorial and geometric properties of convex sets in finite-dimensional linear spaces and, in particular, they can be used to characterize the dimension of the ambient space.\nBoth results have given rise to extensive and largely independent lines of research; we refer to~\\cite{barany2022helly, barany2021combinatorial} for recent surveys.\n\nIn a breakthrough paper~\\cite{adiprasito2020theorems}, Adiprasito, B\\'ar\\'any, Mustafa, and Terpai obtained \\emph{no-dimensional} (or \\emph{approximate}) versions of Helly's and Carath\\'eodory's theorems in Euclidean spaces.\nThese results establish a striking connection between combinatorial convexity and approximation theory: instead of exact intersection/containment statements whose quantitative form depends on the dimension,\none gets approximate conclusions controlled by a single natural parameter.\nMotivated by potential applications to algorithmic questions (such as linear sketching and local-to-global guarantees in optimization), it is natural to ask for analogous results beyond Euclidean spaces.\n\nThe goal of this note is to bring attention to this circle of problems.\nWe formulate several open questions and record some straightforward corollaries of the no-dimensional results in Banach spaces.\nAt the same time, we make one additional tweak and ``return the dimension to the picture'': we combine no-dimensional results with a standard substitution trick from Banach space theory,\nwhich allows one to pass from $\\ell_1$-type or $\\ell_\\infty$-type geometry to an appropriately chosen $\\ell_p$-space with desired geometric properties.\n\nWe will treat the needed Banach space facts as a black box.\nFor convenience, we recall all relevant definitions and inequalities in \\Href{Section}{sec:banach_space_likbez}, in order to keep the paper self-contained.\nThen we explain in detail what we mean by no-dimensional versions of Helly's and Carath\\'eodory's theorems, survey recent results on the topic, and formulate several open problems.\nFinally, we describe a standard substitution trick and show that a number of ``local-to-global'' statements become straightforward corollaries of the no-dimensional theory.\n\nIn many applications \\cite{Garofalakis2013,Wang2016} one stores or transmits large objects\n(signals, images, fields), while the downstream task uses only pairwise similarity queries,\noften measured by the squared $L_2$-distance (equivalently, the $L_2$-energy).\nThe following theorem shows that, for a fixed collection of $n$ signals, one can replace each\nsignal by a short ``fingerprint'' in $\\mathbb R^k$ for sufficiently small $k$, while preserving all\npairwise squared $L_2$-distances up to additive error~$\\varepsilon$.\n\nThe result can be viewed as a weak additive variant of the celebrated Johnson--Lindenstrauss\nflattening lemma \\cite{johnson1984extensions}; see also \\cite{vershynin2018high} for modern\nexpositions. While the classical Johnson--Lindenstrauss lemma is typically proved by a random\nprojection argument, our proof is based on a deterministic construction arising from a\nno-dimensional Carath\\'eodory-type lemma. This provides a simple illustration of the method.", "full_context": "The theorems of Helly~\\cite{helly1923mengen} and Carath\\'eodory~\\cite{caratheodory1911variabilitatsbereich} are two celebrated results that form part of the foundation of modern convexity and its adjacent areas.\nAt a basic level, these theorems reveal fundamental combinatorial and geometric properties of convex sets in finite-dimensional linear spaces and, in particular, they can be used to characterize the dimension of the ambient space.\nBoth results have given rise to extensive and largely independent lines of research; we refer to~\\cite{barany2022helly, barany2021combinatorial} for recent surveys.\n\nIn a breakthrough paper~\\cite{adiprasito2020theorems}, Adiprasito, B\\'ar\\'any, Mustafa, and Terpai obtained \\emph{no-dimensional} (or \\emph{approximate}) versions of Helly's and Carath\\'eodory's theorems in Euclidean spaces.\nThese results establish a striking connection between combinatorial convexity and approximation theory: instead of exact intersection/containment statements whose quantitative form depends on the dimension,\none gets approximate conclusions controlled by a single natural parameter.\nMotivated by potential applications to algorithmic questions (such as linear sketching and local-to-global guarantees in optimization), it is natural to ask for analogous results beyond Euclidean spaces.\n\nThe goal of this note is to bring attention to this circle of problems.\nWe formulate several open questions and record some straightforward corollaries of the no-dimensional results in Banach spaces.\nAt the same time, we make one additional tweak and ``return the dimension to the picture'': we combine no-dimensional results with a standard substitution trick from Banach space theory,\nwhich allows one to pass from $\\ell_1$-type or $\\ell_\\infty$-type geometry to an appropriately chosen $\\ell_p$-space with desired geometric properties.\n\nWe will treat the needed Banach space facts as a black box.\nFor convenience, we recall all relevant definitions and inequalities in \\Href{Section}{sec:banach_space_likbez}, in order to keep the paper self-contained.\nThen we explain in detail what we mean by no-dimensional versions of Helly's and Carath\\'eodory's theorems, survey recent results on the topic, and formulate several open problems.\nFinally, we describe a standard substitution trick and show that a number of ``local-to-global'' statements become straightforward corollaries of the no-dimensional theory.\n\nIn many applications \\cite{Garofalakis2013,Wang2016} one stores or transmits large objects\n(signals, images, fields), while the downstream task uses only pairwise similarity queries,\noften measured by the squared $L_2$-distance (equivalently, the $L_2$-energy).\nThe following theorem shows that, for a fixed collection of $n$ signals, one can replace each\nsignal by a short ``fingerprint'' in $\\mathbb R^k$ for sufficiently small $k$, while preserving all\npairwise squared $L_2$-distances up to additive error~$\\varepsilon$.\n\nThe result can be viewed as a weak additive variant of the celebrated Johnson--Lindenstrauss\nflattening lemma \\cite{johnson1984extensions}; see also \\cite{vershynin2018high} for modern\nexpositions. While the classical Johnson--Lindenstrauss lemma is typically proved by a random\nprojection argument, our proof is based on a deterministic construction arising from a\nno-dimensional Carath\\'eodory-type lemma. This provides a simple illustration of the method.\n\nThe result can be viewed as a weak additive variant of the celebrated Johnson--Lindenstrauss\nflattening lemma \\cite{johnson1984extensions}; see also \\cite{vershynin2018high} for modern\nexpositions. While the classical Johnson--Lindenstrauss lemma is typically proved by a random\nprojection argument, our proof is based on a deterministic construction arising from a\nno-dimensional Carath\\'eodory-type lemma. This provides a simple illustration of the method.\n\nWe emphasize that this result is much weaker than the Johnson--Lindenstrauss flattening lemma.\nFirst, we only obtain an additive approximation to the squared $L_2$-distance. Second, the range\nof the functions is a fixed interval, so the supremum norm is uniformly bounded (and we exploit\nthis in the proof). Third, the functions are defined on a domain of measure one, which is also\nimportant for our argument since we work on a probability space.\n\n\\begin{thm}[Local-to-global Chebyshev regression]\\label{thm:chebyshev_ball}\nLet $a_1,\\dots,a_m\\in [-1,1]^d$ and $b_1,\\dots,b_m\\in \\R$.\nFix $k\\in[m]$ and parameters $R\\ge 1$ and $r\\ge 0$.\nSuppose that for every subset $J\\subset[m]$ with $\\card{J}=k$ there exists a point $x_J\\in R\\ball{}_{\\ell_1^d}$ such that\n\\[\n\\max_{j\\in J}\\abs{\\iprod{a_j}{x_J}-b_j}\\ \\le\\ r.\n\\]\nThen there exists a point $x\\in eR\\ball{}_{\\ell_1^d}$ such that\n\\[\n\\max_{i\\in[m]}\\abs{\\iprod{a_i}{x}-b_i}\n\\ \\le\\\nr + 21\\,R\\,\\sqrt{\\frac{\\ln d}{k}}.\n\\]\n\\end{thm}\n\n\\begin{thm}[Quantum feasibility from local consistency]\\label{thm:quantum_psd_local_to_global}\nLet $d\\ge 3$ and $k\\in[m]$. Let $A_1,\\dots,A_m\\in\\mathcal{H}_d$ be Hermitian matrices satisfying\n\\[\n\\normsch[\\infty]{A_i}\\le 1 \\qquad\\text{for all } i\\in[m],\n\\]\nand let $b_1,\\dots,b_m\\in\\R$. Fix $t\\ge 0$.\nAssume that for every subset $J\\subset[m]$ with $\\abs{J}=k$, there exists a density matrix\n$\\rho_J\\succeq 0$ with $\\mathrm{Tr}(\\rho_J)=1$ such that\n\\[\n\\abs{\\iprod{A_j}{\\rho_J} - b_j}\n\\le t \\qquad\\text{for all } j\\in J.\n\\]\nThen there exists a density matrix $\\rho\\succeq 0$ with $\\mathrm{Tr}(\\rho)=1$ such that\n\\[\n\\abs{\\iprod{A_i}{\\rho} - b_i}\n\\ \\le\\\nt + 21\\,\\sqrt{\\frac{\\ln d}{k}},\n\\]\nfor all $i\\in[m]$.\n\\end{thm}\n\n\\begin{lem}\\label{lem:Caratheodory_sequence_l_p}\nLet $2\\le p<+\\infty$.\nThen the sequence $\\braces{21\\sqrt{\\frac{p-1}{k}}}_{k\\ge 1}$ is a Carath\\'eodory sequence for $\\ell_p$ and for $S_p$.\nMoreover, whenever the convex hull of a subset $Q$ of the unit ball $\\ball{}_X$ contains the origin,\nthere exists a greedy algorithm that selects points $x_1,\\dots,x_k\\in Q$ successively so that\n\\[\n\\norm{\\frac{x_1+\\cdots+x_k}{k}}\n\\le R_k(X)\n\\]\nat each step.\n\\end{lem}\n\n\\section{Dimension strikes back. Carath\\'eodory-type results and problems}\n\\subsection{Johnson--Lindenstrauss-type sketching}\n\\Href{Theorem}{thm:jl_signals_receiver} follows from the following result \nsince the measure of $[0,1]^d$ is one. We note that similar results were obtained in \\cite{Eskenazis2023}.\n\\begin{thm}[Johnson--Lindenstrauss sketching for bounded functions]\\label{thm:JL_for_functions}\nLet $(\\Omega,\\mu)$ be a probability space and let $f_1,\\dots,f_n\\colon \\Omega\\to[-1,1]$ be functions in $L_2(\\Omega,\\mu)$.\nFix $\\varepsilon\\in(0,1)$. Then there exists an integer\n\\[\nk \\ \\le\\ C\\,\\frac{\\ln n}{\\varepsilon^2}\n\\]\nand vectors $x_1,\\dots,x_n\\in\\R^k$ such that for all $i,j\\in[n]$,\n\\[\n\\enorm{x_i-x_j}^2- \\varepsilon \\ \\le\\ \n\\norm{f_i-f_j}_2^2 \\ \\le\\ \n\\enorm{x_i-x_j}^2 + \\varepsilon .\n\\]\nHere $C>0$ is an absolute constant.\n\\end{thm}\n\nApply \\Href{Lemma}{lem:Caratheodory_sequence_l_p} in the space $\\ell_\\infty^{ {n\\choose 2}}$ (via the substitution from \\Href{Proposition}{prp:l_p_trick_bounds})\nto obtain points $\\omega_1,\\dots,\\omega_k\\in\\Omega$ such that the empirical average\n\\[\n\\frac{1}{k}\\sum_{s=1}^k v(\\omega_s)\n\\]\napproximates $\\int_\\Omega v(\\omega)\\,d\\mu(\\omega)$ in $\\ell_\\infty^{ {n\\choose 2}}$ up to error $\\varepsilon$.\nEquivalently, for all $i<j$,\n\\[\n\\enorm{\n\\frac{1}{k}\\sum_{s=1}^k \\abs{f_i(\\omega_s)-f_j(\\omega_s)}^2\n-\n\\norm{f_i-f_j}_2^2\n}\n\\le \\varepsilon.\n\\]\nNow define\n\\[\nx_i:=\\frac{1}{\\sqrt{k}}\\big(f_i(\\omega_1),\\dots,f_i(\\omega_k)\\big)\\in\\R^k.\n\\]\nThen\n\\[\n\\enorm{x_i-x_j}^2=\\frac{1}{k}\\sum_{s=1}^k \\abs{f_i(\\omega_s)-f_j(\\omega_s)}^2,\n\\]\nand the desired inequality follows.\n\\end{proof}\n\\subsection{Positive semidefinite matrix sparsification}\n\n\\begin{conj}\\label{conj:deterministic_rudelson}\nLet $A_1,\\dots,A_n\\in \\mathcal{M}_d$ be positive semidefinite and suppose that\n\\[\n\\lambda_1A_1+\\cdots+\\lambda_nA_n=\\id_d,\n\\]\nwhere the coefficients $\\lambda_i$ are nonnegative and summing up to one.\nThen for every $\\varepsilon\\in(0,1)$ there exists a deterministic algorithm that selects indices\n$i_1,\\dots,i_k\\in[n]$ with $k\\le C(\\varepsilon)\\,d\\ln d$ and outputs positive coefficients $\\beta_1,\\dots,\\beta_k$\nsuch that\n\\[\n\\Bigl\\|\\beta_1A_{i_1}+\\cdots+\\beta_kA_{i_k}-\\id_d\\Bigr\\|_{S_\\infty}\\le \\varepsilon.\n\\]\n\\end{conj}", "post_theorem_intro_text_len": 6304, "post_theorem_intro_text": "We emphasize that this result is much weaker than the Johnson--Lindenstrauss flattening lemma.\nFirst, we only obtain an additive approximation to the squared $L_2$-distance. Second, the range\nof the functions is a fixed interval, so the supremum norm is uniformly bounded (and we exploit\nthis in the proof). Third, the functions are defined on a domain of measure one, which is also\nimportant for our argument since we work on a probability space.\n\nThe main point is that the statement admits a deterministic proof based on a no-dimensional\nCarath\\'eodory lemma.\n\nAlso, we obtain several local-to-global estimates for $\\ell_1$-minimization problems using no-dimensional Helly-type results.\nWe write $[m]:=\\braces{1,\\dots,m}$ for $m\\in\\mathbb N$, and we equip $\\mathbb R^d$ with the standard $\\ell_p$ norm $\\left\\|\\cdot\\right\\|_p$.\nWe denote by $\\ell_p^d:=(\\mathbb R^d,\\left\\|\\cdot\\right\\|_p)$ the resulting normed space and by $\\ball{}_{\\ell_p^d}$ its unit ball.\n\n\\medskip\nOne basic example is \\emph{Chebyshev regression} (also known as \\emph{minimax absolute residual fitting} or $\\ell_\\infty$-regression), the classical problem of fitting a linear model by minimizing the worst absolute residual:\n\\[\n\\min_{x\\in\\mathbb R^d}\\ \\max_{i\\in[m]} \\abs{\\iprod{a_i}{x}-b_i}.\n\\]\nIt admits an immediate linear programming formulation by introducing an auxiliary variable $t$ and enforcing the two-sided constraints\n$-t\\le \\iprod{a_i}{x}-b_i\\le t$ for all $i\\in[m]$; see, e.g., \\cite{boyd2004convex,ben2001lectures}.\nFrom an algorithmic viewpoint, this connects Chebyshev regression to the general toolbox of convex optimization and linear programming, while its geometry is governed by small active constraint sets, a perspective formalized in the LP-type framework and its randomized algorithms \\cite{clarkson1995vegas,matouvsek1992subexponential}.\nMore recently, Chebyshev/$\\ell_\\infty$ guarantees have also been studied via modern randomized linear algebra and ``sketch-and-solve'' methods, where one seeks fast approximate solutions together with explicit $\\ell_\\infty$ control \\cite{price2017fast,song2023nearly}.\nIn the present work, we pursue a different direction: using no-dimensional Helly-type results to derive \\emph{local-to-global} approximation guarantees from $k$-wise feasibility information, with explicit dimension dependence.\n\nOur first local-to-global statement shows that approximate global feasibility over the $\\ell_1$ ball follows from approximate feasibility on all $k$-subcollections of constraints, with an explicit  dependence on the dimension.\n\n\\begin{thm}[Local-to-global Chebyshev regression]\\label{thm:chebyshev_ball}\nLet $a_1,\\dots,a_m\\in [-1,1]^d$ and $b_1,\\dots,b_m\\in \\mathbb R$.\nFix $k\\in[m]$ and parameters $R\\ge 1$ and $r\\ge 0$.\nSuppose that for every subset $J\\subset[m]$ with $\\card{J}=k$ there exists a point $x_J\\in R\\ball{}_{\\ell_1^d}$ such that\n\\[\n\\max_{j\\in J}\\abs{\\iprod{a_j}{x_J}-b_j}\\ \\le\\ r.\n\\]\nThen there exists a point $x\\in eR\\ball{}_{\\ell_1^d}$ such that\n\\[\n\\max_{i\\in[m]}\\abs{\\iprod{a_i}{x}-b_i}\n\\ \\le\\\nr + 21\\,R\\,\\sqrt{\\frac{\\ln d}{k}}.\n\\]\n\\end{thm}\n\n\\medskip\nOur second application of no-dimensional Helly-type results is motivated by feasibility and consistency questions in quantum information theory.\nA prototypical instance is the \\emph{consistency of local density matrices} (also known as the \\emph{quantum marginal problem}), which asks whether a collection of local constraints can arise from a single global quantum state; this problem is known to be computationally intractable in general; it was first shown to be in QMA and QMA-complete under Turing reductions by Liu \\cite{liu2006consistency}, and subsequently proven to be QMA-complete under Karp reductions by Broadbent and Grilo \\cite{broadbent2022qma}.\nFrom a geometric viewpoint, the natural ambient spaces are Schatten classes and their convexity properties, and we refer to \\cite{aubrun2017alice} for background on the interface between asymptotic geometric analysis and quantum information.\n\nFix $d\\ge 1$ and let $\\mathbb{F}\\in\\{\\mathbb R,\\mathbb C\\}$.\nWrite $\\mathcal{M}_d:=\\mathbb{F}^{d\\times d}$ for the space of $d\\times d$ matrices over $\\mathbb{F}$.\nWe say that $A\\in\\mathcal{M}_d$ is \\emph{self-adjoint} if $A=A^\\ast$, where $A^\\ast$ denotes the transpose when $\\mathbb{F}=\\mathbb R$\nand the conjugate transpose when $\\mathbb{F}=\\mathbb C$.\nIf $\\mathbb{F}=\\mathbb C$, let\n\\[\n\\mathcal{H}_d:=\\{A\\in\\mathbb C^{d\\times d}:\\ A=A^\\ast\\}\n\\]\ndenote the  vector space of Hermitian matrices.\n\nFor $1\\le p\\le\\infty$ we write $\\normsch[p]{A}$ for the Schatten $p$-norm (the $\\ell_p$-norm of the singular value vector),\nand we equip $\\mathcal{M}_d$ with the Frobenius (Hilbert--Schmidt) inner product\n\\[\n\\iprod{A}{B}:=\\mathrm{Tr}(A^\\ast B).\n\\]\n\nA matrix $\\rho\\in\\mathcal{M}_d$ is called \\emph{positive semidefinite}, written $\\rho\\succeq 0$, if it is self-adjoint and has\nnonnegative spectrum. A \\emph{density matrix} is a positive semidefinite matrix of unit trace:\n\\[\n\\rho\\succeq 0,\n\\qquad\n\\mathrm{Tr}(\\rho)=1.\n\\]\n\nGiven self-adjoint measurement operators $A_1,\\dots,A_m\\in\\mathcal{M}_d$ and target values $b_1,\\dots,b_m\\in\\mathbb R$,\nwe consider the feasibility problem of finding a density matrix whose linear measurements $\\iprod{A_i}{\\rho}$ are simultaneously\nconsistent with $b_i$ up to a tolerance level.\nIn this work we study a \\emph{local-to-global} version: assuming that every subcollection of $k$ constraints is feasible up to error $t$,\nwe ask whether this implies the existence of a single density matrix that approximately satisfies all $m$ constraints.\n\n\\begin{thm}[Quantum feasibility from local consistency]\\label{thm:quantum_psd_local_to_global}\nLet $d\\ge 3$ and $k\\in[m]$. Let $A_1,\\dots,A_m\\in\\mathcal{H}_d$ be Hermitian matrices satisfying\n\\[\n\\normsch[\\infty]{A_i}\\le 1 \\qquad\\text{for all } i\\in[m],\n\\]\nand let $b_1,\\dots,b_m\\in\\mathbb R$. Fix $t\\ge 0$.\nAssume that for every subset $J\\subset[m]$ with $\\abs{J}=k$, there exists a density matrix\n$\\rho_J\\succeq 0$ with $\\mathrm{Tr}(\\rho_J)=1$ such that\n\\[\n\\abs{\\iprod{A_j}{\\rho_J} - b_j}\n\\le t \\qquad\\text{for all } j\\in J.\n\\]\nThen there exists a density matrix $\\rho\\succeq 0$ with $\\mathrm{Tr}(\\rho)=1$ such that\n\\[\n\\abs{\\iprod{A_i}{\\rho} - b_i}\n\\ \\le\\\nt + 21\\,\\sqrt{\\frac{\\ln d}{k}},\n\\]\nfor all $i\\in[m]$.\n\\end{thm}", "sketch": "The post-theorem introduction gives only a high-level indication of the proof method for Theorem~\\ref{thm:jl_signals_receiver}: it notes that the proof exploits that (i) the functions take values in a fixed interval so “the supremum norm is uniformly bounded (and we exploit this in the proof),” and (ii) the domain has “measure one,” which is “important for our argument since we work on a probability space.” It also states that “the statement admits a deterministic proof based on a no-dimensional Carath\\'eodory lemma.” No further steps or structure of the argument are described there.", "expanded_sketch": "The post-theorem introduction gives only a high-level indication of the proof method for the main theorem: it notes that the proof exploits that (i) the functions take values in a fixed interval so “the supremum norm is uniformly bounded (and we exploit this in the proof),” and (ii) the domain has “measure one,” which is “important for our argument since we work on a probability space.” It also states that “the statement admits a deterministic proof based on a no-dimensional Carath\\'eodory lemma.” No further steps or structure of the argument are described there.", "expanded_theorem": "[Communication-efficient similarity search for signals]\\label{thm:jl_signals_receiver}\nFix $d\\ge 1$ and $n \\ge 8,$ and let $f_1,\\dots,f_n\\colon [0,1]^d\\to[-1,1]$ be continuous.\nA \\emph{sketching protocol} consists of an encoding map\n\\[\n\\Enc\\colon C([0,1]^d)\\to\\mathbb R^k\n\\]\nand a decoding rule\n\\[\n\\Dec\\colon \\mathbb R^k\\times \\mathbb R^k\\to\\mathbb R,\n\\]\nwhere $\\Dec(\\Enc(f_i),\\Enc(f_j))$ is interpreted as an estimate of the squared $L_2$-distance\n\\[\nD(i,j):=\\int_{[0,1]^d}\\abs{f_i(x)-f_j(x)}^2\\,\\,\\mathrm{d} x.\n\\]\n\nThen for every $\\varepsilon\\in(0,1)$ there exist $k\\le C\\,\\frac{\\ln n}{\\varepsilon^2}$ and maps $\\Enc,\\Dec$\nsuch that for all $i,j\\in[n],$\n\\[\n\\abs{\\Dec(\\Enc(f_i),\\Enc(f_j)) - D(i,j)} \\le \\varepsilon.\n\\]\nMoreover, one may take $\\Dec(u,v)=\\left|u-v\\right|^2$ (i.e.\\ decoding is just squared Euclidean distance $\\left|\\cdot\\right|^2$) and $\\Enc$ to be linear.", "theorem_type": ["Existential–Universal", "Algorithmic or Constructive"], "mcq": {"question": "Let $d\\ge 1$ and $n\\ge 8$, and let $f_1,\\dots,f_n\\colon [0,1]^d\\to[-1,1]$ be continuous functions. Define the squared $L_2$-distance between two signals by\n\\[\nD(i,j):=\\int_{[0,1]^d} |f_i(x)-f_j(x)|^2\\,dx.\n\\]\nA sketching protocol consists of an encoding map $\\mathrm{Enc}\\colon C([0,1]^d)\\to\\mathbb R^k$ and a decoding rule $\\mathrm{Dec}\\colon \\mathbb R^k\\times\\mathbb R^k\\to\\mathbb R$, where $\\mathrm{Dec}(\\mathrm{Enc}(f_i),\\mathrm{Enc}(f_j))$ is used to estimate $D(i,j)$. For a given $\\varepsilon\\in(0,1)$, which statement holds for such a family of signals?", "correct_choice": {"label": "A", "text": "There exist an integer $k\\le C\\,\\frac{\\ln n}{\\varepsilon^2}$ and maps $\\mathrm{Enc}$ and $\\mathrm{Dec}$ such that for every $i,j\\in[n]$,\n\\[\n\\big|\\mathrm{Dec}(\\mathrm{Enc}(f_i),\\mathrm{Enc}(f_j)) - D(i,j)\\big|\\le \\varepsilon.\n\\]\nMoreover, one may choose $\\mathrm{Dec}(u,v)=|u-v|^2$ (the squared Euclidean distance in $\\mathbb R^k$) and choose $\\mathrm{Enc}$ to be linear."}, "choices": [{"label": "B", "text": "There exist an integer $k\\le C\\,\\frac{\\ln n}{\\varepsilon^2}$ and maps $\\mathrm{Enc}$ and $\\mathrm{Dec}$ such that for every $i,j\\in[n]$,\n\\[\n\\big|\\mathrm{Dec}(\\mathrm{Enc}(f_i),\\mathrm{Enc}(f_j)) - D(i,j)\\big|\\le \\varepsilon.\n\\]\nMoreover, one may choose $\\mathrm{Dec}(u,v)=|u-v|^2$ and choose $\\mathrm{Enc}$ to be linear, with $C>0$ an absolute constant independent of $d$, $n$, $\\varepsilon$, and of the range bound on the functions."}, {"label": "C", "text": "There exist an integer $k\\le C\\,\\frac{\\ln n}{\\varepsilon^2}$ and maps $\\mathrm{Enc}$ and $\\mathrm{Dec}$ such that for every $i,j\\in[n]$,\n\\[\n\\big|\\mathrm{Dec}(\\mathrm{Enc}(f_i),\\mathrm{Enc}(f_j)) - D(i,j)\\big|\\le \\varepsilon.\n\\]"}, {"label": "D", "text": "There exist an integer $k\\le C\\,\\frac{\\ln n}{\\varepsilon^2}$ and maps $\\mathrm{Enc}$ and $\\mathrm{Dec}$ such that for every $i,j\\in[n]$,\n\\[\n\\big|\\mathrm{Dec}(\\mathrm{Enc}(f_i),\\mathrm{Enc}(f_j)) - D(i,j)\\big|\\le \\varepsilon.\n\\]\nMoreover, one may choose $\\mathrm{Dec}(u,v)=|u-v|^2$ and choose $\\mathrm{Enc}$ to be linear even when the functions $f_1,\\dots,f_n\\colon [0,1]^d\\to\\mathbb R$ are arbitrary continuous functions, with no assumption that their values lie in $[-1,1]$."}, {"label": "E", "text": "There exist maps $\\mathrm{Enc}$ and $\\mathrm{Dec}$ such that for every $\\varepsilon\\in(0,1)$ and every $i,j\\in[n]$,\n\\[\n\\big|\\mathrm{Dec}(\\mathrm{Enc}(f_i),\\mathrm{Enc}(f_j)) - D(i,j)\\big|\\le \\varepsilon,\n\\]\nwith a sketch dimension satisfying\n\\[\nk\\le C\\,\\frac{\\ln n}{\\varepsilon},\n\\]\nand one may take $\\mathrm{Dec}(u,v)=|u-v|^2$ and $\\mathrm{Enc}$ linear."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "hidden dependence on bounded range normalization", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the additional conclusion specifying Euclidean decoding and linear encoding", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "uniform bounded-range hypothesis $f_i\\in[-1,1]$", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "the $\\varepsilon^{-2}$ dependence in the sketch dimension bound", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state or strongly hint at the full correct conclusion. It sets up the sketching problem and asks which guarantee holds, without revealing the linear-encoding/squared-Euclidean-decoding conclusion."}, "TAS": {"score": 1, "justification": "This is largely a theorem-statement recognition item: the options are mostly strengthened or weakened variants of a known embedding/sketching guarantee. It is not a pure restatement in the stem, but it still mainly tests recall of the exact theorem form."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure in comparing subtle variants (absolute-constant dependence, bounded-range hypothesis, and the epsilon exponent). However, the item does not strongly force construction or derivation, and option C is a weaker true statement, which reduces the need for identifying a uniquely strongest conclusion."}, "DQS": {"score": 1, "justification": "B, D, and E are plausible theorem-level distractors tied to common overgeneralization errors. But C is not a genuine distractor: it is a weaker true statement if A is true, so the single-correct-answer format is compromised."}, "total_score": 5, "overall_assessment": "Good on avoiding answer leakage, but only moderate as an assessment item: it is mostly theorem recall, and the presence of a weaker true option makes the MCQ structurally flawed."}}
{"id": "2602.20056v1", "paper_link": "http://arxiv.org/abs/2602.20056v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{theorem:k-dim-ds}\n   Let $k\\geq 1$, and let $\\map{\\psi}{\\mathbb{N}}{\\left[0,1/2\\right]}$ be such that $\\sum_{q=1}^\\infty \\left(\\varphi(q)\\psi(q)/q\\right)^k=\\infty$. For $\\alpha\\in \\left[0,1\\right]^k$, denote by $S_k(\\alpha,Q)$ the number of solutions $(a,q)\\in\\mathbb{Z}^k\\times \\mathbb{N}$ to the inequality:\n    \\begin{align*}\n        \\pn{q\\alpha-a}{\\infty} < \\psi(q), \\quad \\text{given } q \\leq Q,\n    \\end{align*}\n    satisfying $\\gcd(a_i, q)=1$, for $i\\in\\{1,\\dots, k\\}$, $a=(a_1,\\dots, a_k)$. Moreover, given $Q\\in \\mathbb{N}$, define:\n    \\begin{align*}\n        \\Psi_k(Q)\\coloneq \\sum_{q\\leq Q}\\left(\\frac{2\\psi(q)\\varphi(q)}{q}\\right)^k.\n    \\end{align*}\n    Then for all $\\varepsilon>0$, as $Q\\to \\infty$, we have for almost all $\\alpha$ that:\n    \\begin{align*}\n        S_k(\\alpha,Q)=\\Psi_k(Q)+O_{\\varepsilon,k}\\left( \\Psi_k(Q)^{1/2+\\varepsilon} \\right).\n    \\end{align*}", "start_pos": 10709, "end_pos": 11570, "label": "theorem:k-dim-ds"}, "ref_dict": {"theorem:k-dim-ds": "\\begin{theorem}\\label{theorem:k-dim-ds}\n   Let $k\\geq 1$, and let $\\map{\\psi}{\\N}{\\left[0,1/2\\right]}$ be such that $\\sum_{q=1}^\\infty \\left(\\varphi(q)\\psi(q)/q\\right)^k=\\infty$. For $\\alpha\\in \\left[0,1\\right]^k$, denote by $S_k(\\alpha,Q)$ the number of solutions $(a,q)\\in\\Z^k\\times \\N$ to the inequality:\n    \\begin{align*}\n        \\pn{q\\alpha-a}{\\infty} < \\psi(q), \\quad \\text{given } q \\leq Q,\n    \\end{align*}\n    satisfying $\\gcd(a_i, q)=1$, for $i\\in\\{1,\\dots, k\\}$, $a=(a_1,\\dots, a_k)$. Moreover, given $Q\\in \\N$, define:\n    \\begin{align*}\n        \\Psi_k(Q)\\coloneq \\sum_{q\\leq Q}\\left(\\frac{2\\psi(q)\\varphi(q)}{q}\\right)^k.\n    \\end{align*}\n    Then for all $\\ep>0$, as $Q\\to \\infty$, we have for almost all $\\alpha$ that:\n    \\begin{align*}\n        S_k(\\alpha,Q)=\\Psi_k(Q)+O_{\\ep,k}\\left( \\Psi_k(Q)^{1/2+\\ep} \\right).\n    \\end{align*}\n\\end{theorem}", "eq:kdim-approx-ineq": "\\begin{align}\n    \\pn{q\\alpha-a}{\\infty}< \\psi(q),\\label{eq:kdim-approx-ineq}\n\\end{align}", "theorem:k-dim-variance": "\\begin{theorem}\\label{theorem:k-dim-variance}\n    Let $k\\geq 1$ and $Q\\in \\N$, and define $S_k(\\alpha,Q)$ and $\\Psi_k(Q)$ as in Theorem~\\ref{theorem:k-dim-ds}. Then, for every fixed $\\ep>0$, we have:\n    \\begin{align*}\n         \\lint{\\left[0,1\\right]^k}{}{\\left(S_k(\\alpha, Q)-\\Psi_k(Q)\\right)^2}{\\alpha} \\leq \\Psi_k(Q) + O_{\\ep,k}(\\Psi_k(Q)^{1+\\ep}).\n     \\end{align*}\n\\end{theorem}", "eq:approx-ineq": "\\begin{align}\n    \\left\\lvert \\alpha - \\frac{p}{q}\\right\\rvert < \\frac{\\psi(q)}{q}.\\label{eq:approx-ineq}\n\\end{align}"}, "pre_theorem_intro_text_len": 6564, "pre_theorem_intro_text": "Simultaneous Diophantine approximation concerns the existence of rational approximations of multidimensional irrational points: a classical result of Dirichlet states that, given $k\\geq 1$, for every $\\alpha\\in \\mathbb{R}^k$, there are infinitely many solutions $(a,q)\\in\\mathbb{Z}^k\\times \\mathbb{N}$ to the inequality \\cite[Chapter~2]{schmidt1980diophantine}:\n\\begin{align*}\n    \\pn{q\\alpha - a}{\\infty}<q^{-1/k}.\n\\end{align*}\nMoreover, this bound is optimal up to a constant for numbers with bounded partial quotients when represented as continued fractions; this class of numbers are known as badly approximable, and contains e.g. for $k=1$ the golden ratio $\\varphi$, and $\\sqrt{2}$. However, for typical $\\alpha$, in the sense of Lebesgue measure, it turns out that this bound can be significantly improved: metric Diophantine approximation asks the question of by exactly how much. For an overview of metric Diophantine approximation (and metric number theory, in general), one should read the monograph \\cite{harman1998metricnumbertheory}. \n\nA notable foundational result of metric Diophantine approximation is Khintchine's theorem: given a function $\\map{\\psi}{\\mathbb{N}}{[0,\\infty)}$ monotonically decreasing, Khintchine's theorem states that for almost every $\\alpha\\in \\mathbb{R}^k$, there exists infinitely many pairs $(a,q)\\in\\mathbb{Z}^k\\times \\mathbb{N}$ satisfying the inequality:\n\\begin{align*}\n    \\pn{q\\alpha-a}{\\infty}< \\psi(q),\n\\end{align*}\nif and only if $\\sum_{q\\in\\mathbb{N}}(\\psi(q))^k$ diverges \\cite[Section~3]{beresnevich2009classical}. Intuitively, one can choose $\\psi(q)=1/q\\log q$ to see that on average the bound from Dirichlet's theorem can be improved by a logarithmic factor. In the case of only finitely many solutions, Khintchine's theorem is immediate from the convergence Borel-Cantelli lemma. The divergence lemma, however, can not be applied directly as there is no stochastic independence, but when $\\psi$ is monotonic, the set system under consideration behaves approximately independent in a sense that is sufficient for the statement to hold.\n\nIn 1941, Duffin and Schaeffer \\cite{duffin1941khintchinesproblem} showed that Khintchine's theorem in 1-dimension fails without the monotonicity condition, by constructing a counterexample where the series $\\sum_{q\\in \\mathbb{N}} \\psi(q)$ diverges, yet, for almost every $\\alpha$, only finitely many solutions exist. Their work relies on the multiple representations of fractions: if $p/q$ is a close approximation for $\\alpha$, then so is $np/nq$ for all $n\\in\\mathbb{N}$. As a result, if the function $\\psi$ is constructed to have values disproportionately high on integer multiples of $q$, the sum may still diverge, despite the $\\alpha$ admitting good approximations being relatively rare.\nThis led them to conjecture that a variant of Khintchine's theorem would hold when non-reduced fractions are excluded, as follows: denoting by $\\varphi$ the Euler totient function, if $\\map{\\psi}{\\mathbb{N}}{[0,\\infty)}$ is a function such that $\\sum_{q\\in\\mathbb{N}}\\psi(q)\\varphi(q)/q$ diverges, then for almost all $\\alpha\\in\\mathbb{R}$, there exists infinitely many solutions to the inequality:\n\\begin{align}\n    \\left\\lvert \\alpha - \\frac{p}{q}\\right\\rvert < \\frac{\\psi(q)}{q}.\\label{eq:approx-ineq}\n\\end{align}\nThe Duffin-Schaeffer conjecture was open for 78 years before it was proven by Koukoulopoulos and Maynard \\cite{koukoulopoulos2020duffin}. A full survey on the conjecture and an exposition of its proof can be found in \\cite{hauke2025centurymetricdiophantineapproximation}.\n\nThe work of \\cite{koukoulopoulos2020duffin}, however, left open the question of quantitative results on the number of approximations, assuming some bound on $q$. A quantitative version of Khintchine's theorem was proven by Schmidt in 1960 \\cite{schmidt1960ametricaltheorem}; one can also view \\cite[Chapter~4]{harman1998metricnumbertheory} for an overview of Schmidt's method. Schmidt-type results on the Duffin-Schaeffer conjecture have been the subject of various recent works \\cite{aistleitner2023metrictheoryofapproximations, hauke2024provingduffinschaefferconjecturegcd}, with the sharpest known being of Koukoulopoulos, Maynard and Yang \\cite{koukoulopoulos2024sharpquantitativeversionduffinschaeffer}. \nTowards this quantitative question, given $\\map{\\psi}{\\mathbb{N}}{[0,1/2]}$ and $Q\\geq 1$, we define:\n\\begin{align*}\n    \\Psi(Q) \\coloneq \\sum_{q\\leq Q}\\frac{2\\psi(q)\\varphi(q)}{q}.\n\\end{align*}\nMoreover, denote by $S(\\alpha,Q)$ the number of pairs $(p,q)\\in \\mathbb{Z} \\times \\mathbb{N}$ with $q\\leq Q,\\ \\gcd(p,q)=1$ that satisfy the inequality \\eqref{eq:approx-ineq}. The work of Koukoulopoulos, Maynard and Yang \\cite{koukoulopoulos2024sharpquantitativeversionduffinschaeffer} then states that, assuming $\\lim_{Q\\to\\infty}\\Psi(Q)=\\infty$, for every $\\varepsilon>0$ and almost all $\\alpha\\in\\mathbb{R}$, we have as $Q\\to \\infty$:\n\\begin{align*}\n    S(\\alpha,Q)=\\Psi(Q)+O_\\varepsilon\\left(\\Psi(Q)^{\\frac{1}{2}+\\varepsilon}\\right).\n\\end{align*}\n\nIn this paper, we consider a higher-dimensional generalisation of the Duffin-Schaeffer conjecture as follows: for $k\\geq 1$, given $\\map{\\psi}{\\mathbb{N}}{[0,\\infty)}$ and $\\alpha\\in \\mathbb{R}^k$, we ask whether there exists infinitely many solutions $(a,q)\\in\\mathbb{Z}^k\\times \\mathbb{N}$ satisfying the inequality:\n\\begin{align}\n    \\pn{q\\alpha-a}{\\infty}< \\psi(q),\\label{eq:kdim-approx-ineq}\n\\end{align}\nwhere $a=(a_1,\\dots,a_k)$ satisfies $\\gcd(a_i, q)=1$ for $i\\in\\{1,\\dots, k\\}$. When $k=1$, this is then exactly the question of the Duffin-Schaeffer conjecture. For $k\\geq 2$, the question was first raised by Sprind\\v{z}uk \\cite{sprindzhuk1979metric}, and then resolved by Pollington and Vaughan \\cite{pollington1990kdimensionalduffinschaeffer} as follows: assuming the notation and hypothesis above, there exists infinitely many solutions to the inequality \\eqref{eq:kdim-approx-ineq} if and only if the sum $\\sum_{q\\in\\mathbb{N}}(\\psi(q)\\varphi(q)/q)^k$ diverges.\n\nFollowing the recent progress in the 1-dimensional case, we now ask the question of quantitative results for the higher-dimensional setup of Pollington and Vaughan. By periodicity, we consider only $\\alpha \\in [0,1]^k$ and thus $0\\leq a \\leq q$, and we assume $\\psi(q)\\leq 1/2$, following \\cite{aistleitner2023metrictheoryofapproximations, koukoulopoulos2024sharpquantitativeversionduffinschaeffer}. The main theorem of this paper is the following. We state the result for all $k\\geq 1$ for completeness; our contribution, of course, is the case $k\\geq 2$.", "context": "A notable foundational result of metric Diophantine approximation is Khintchine's theorem: given a function $\\map{\\psi}{\\mathbb{N}}{[0,\\infty)}$ monotonically decreasing, Khintchine's theorem states that for almost every $\\alpha\\in \\mathbb{R}^k$, there exists infinitely many pairs $(a,q)\\in\\mathbb{Z}^k\\times \\mathbb{N}$ satisfying the inequality:\n\\begin{align*}\n \\pn{q\\alpha-a}{\\infty}< \\psi(q),\n\\end{align*}\nif and only if $\\sum_{q\\in\\mathbb{N}}(\\psi(q))^k$ diverges \\cite[Section~3]{beresnevich2009classical}. Intuitively, one can choose $\\psi(q)=1/q\\log q$ to see that on average the bound from Dirichlet's theorem can be improved by a logarithmic factor. In the case of only finitely many solutions, Khintchine's theorem is immediate from the convergence Borel-Cantelli lemma. The divergence lemma, however, can not be applied directly as there is no stochastic independence, but when $\\psi$ is monotonic, the set system under consideration behaves approximately independent in a sense that is sufficient for the statement to hold.\n\nIn 1941, Duffin and Schaeffer \\cite{duffin1941khintchinesproblem} showed that Khintchine's theorem in 1-dimension fails without the monotonicity condition, by constructing a counterexample where the series $\\sum_{q\\in \\mathbb{N}} \\psi(q)$ diverges, yet, for almost every $\\alpha$, only finitely many solutions exist. Their work relies on the multiple representations of fractions: if $p/q$ is a close approximation for $\\alpha$, then so is $np/nq$ for all $n\\in\\mathbb{N}$. As a result, if the function $\\psi$ is constructed to have values disproportionately high on integer multiples of $q$, the sum may still diverge, despite the $\\alpha$ admitting good approximations being relatively rare.\nThis led them to conjecture that a variant of Khintchine's theorem would hold when non-reduced fractions are excluded, as follows: denoting by $\\varphi$ the Euler totient function, if $\\map{\\psi}{\\mathbb{N}}{[0,\\infty)}$ is a function such that $\\sum_{q\\in\\mathbb{N}}\\psi(q)\\varphi(q)/q$ diverges, then for almost all $\\alpha\\in\\mathbb{R}$, there exists infinitely many solutions to the inequality:\n\\begin{align}\n \\left\\lvert \\alpha - \\frac{p}{q}\\right\\rvert < \\frac{\\psi(q)}{q}.\\label{eq:approx-ineq}\n\\end{align}\nThe Duffin-Schaeffer conjecture was open for 78 years before it was proven by Koukoulopoulos and Maynard \\cite{koukoulopoulos2020duffin}. A full survey on the conjecture and an exposition of its proof can be found in \\cite{hauke2025centurymetricdiophantineapproximation}.\n\nThe work of \\cite{koukoulopoulos2020duffin}, however, left open the question of quantitative results on the number of approximations, assuming some bound on $q$. A quantitative version of Khintchine's theorem was proven by Schmidt in 1960 \\cite{schmidt1960ametricaltheorem}; one can also view \\cite[Chapter~4]{harman1998metricnumbertheory} for an overview of Schmidt's method. Schmidt-type results on the Duffin-Schaeffer conjecture have been the subject of various recent works \\cite{aistleitner2023metrictheoryofapproximations, hauke2024provingduffinschaefferconjecturegcd}, with the sharpest known being of Koukoulopoulos, Maynard and Yang \\cite{koukoulopoulos2024sharpquantitativeversionduffinschaeffer}. \nTowards this quantitative question, given $\\map{\\psi}{\\mathbb{N}}{[0,1/2]}$ and $Q\\geq 1$, we define:\n\\begin{align*}\n \\Psi(Q) \\coloneq \\sum_{q\\leq Q}\\frac{2\\psi(q)\\varphi(q)}{q}.\n\\end{align*}\nMoreover, denote by $S(\\alpha,Q)$ the number of pairs $(p,q)\\in \\mathbb{Z} \\times \\mathbb{N}$ with $q\\leq Q,\\ \\gcd(p,q)=1$ that satisfy the inequality \\eqref{eq:approx-ineq}. The work of Koukoulopoulos, Maynard and Yang \\cite{koukoulopoulos2024sharpquantitativeversionduffinschaeffer} then states that, assuming $\\lim_{Q\\to\\infty}\\Psi(Q)=\\infty$, for every $\\varepsilon>0$ and almost all $\\alpha\\in\\mathbb{R}$, we have as $Q\\to \\infty$:\n\\begin{align*}\n S(\\alpha,Q)=\\Psi(Q)+O_\\varepsilon\\left(\\Psi(Q)^{\\frac{1}{2}+\\varepsilon}\\right).\n\\end{align*}\n\nIn this paper, we consider a higher-dimensional generalisation of the Duffin-Schaeffer conjecture as follows: for $k\\geq 1$, given $\\map{\\psi}{\\mathbb{N}}{[0,\\infty)}$ and $\\alpha\\in \\mathbb{R}^k$, we ask whether there exists infinitely many solutions $(a,q)\\in\\mathbb{Z}^k\\times \\mathbb{N}$ satisfying the inequality:\n\\begin{align}\n \\pn{q\\alpha-a}{\\infty}< \\psi(q),\\label{eq:kdim-approx-ineq}\n\\end{align}\nwhere $a=(a_1,\\dots,a_k)$ satisfies $\\gcd(a_i, q)=1$ for $i\\in\\{1,\\dots, k\\}$. When $k=1$, this is then exactly the question of the Duffin-Schaeffer conjecture. For $k\\geq 2$, the question was first raised by Sprind\\v{z}uk \\cite{sprindzhuk1979metric}, and then resolved by Pollington and Vaughan \\cite{pollington1990kdimensionalduffinschaeffer} as follows: assuming the notation and hypothesis above, there exists infinitely many solutions to the inequality \\eqref{eq:kdim-approx-ineq} if and only if the sum $\\sum_{q\\in\\mathbb{N}}(\\psi(q)\\varphi(q)/q)^k$ diverges.\n\nFollowing the recent progress in the 1-dimensional case, we now ask the question of quantitative results for the higher-dimensional setup of Pollington and Vaughan. By periodicity, we consider only $\\alpha \\in [0,1]^k$ and thus $0\\leq a \\leq q$, and we assume $\\psi(q)\\leq 1/2$, following \\cite{aistleitner2023metrictheoryofapproximations, koukoulopoulos2024sharpquantitativeversionduffinschaeffer}. The main theorem of this paper is the following. We state the result for all $k\\geq 1$ for completeness; our contribution, of course, is the case $k\\geq 2$.", "full_context": "A notable foundational result of metric Diophantine approximation is Khintchine's theorem: given a function $\\map{\\psi}{\\mathbb{N}}{[0,\\infty)}$ monotonically decreasing, Khintchine's theorem states that for almost every $\\alpha\\in \\mathbb{R}^k$, there exists infinitely many pairs $(a,q)\\in\\mathbb{Z}^k\\times \\mathbb{N}$ satisfying the inequality:\n\\begin{align*}\n \\pn{q\\alpha-a}{\\infty}< \\psi(q),\n\\end{align*}\nif and only if $\\sum_{q\\in\\mathbb{N}}(\\psi(q))^k$ diverges \\cite[Section~3]{beresnevich2009classical}. Intuitively, one can choose $\\psi(q)=1/q\\log q$ to see that on average the bound from Dirichlet's theorem can be improved by a logarithmic factor. In the case of only finitely many solutions, Khintchine's theorem is immediate from the convergence Borel-Cantelli lemma. The divergence lemma, however, can not be applied directly as there is no stochastic independence, but when $\\psi$ is monotonic, the set system under consideration behaves approximately independent in a sense that is sufficient for the statement to hold.\n\nIn 1941, Duffin and Schaeffer \\cite{duffin1941khintchinesproblem} showed that Khintchine's theorem in 1-dimension fails without the monotonicity condition, by constructing a counterexample where the series $\\sum_{q\\in \\mathbb{N}} \\psi(q)$ diverges, yet, for almost every $\\alpha$, only finitely many solutions exist. Their work relies on the multiple representations of fractions: if $p/q$ is a close approximation for $\\alpha$, then so is $np/nq$ for all $n\\in\\mathbb{N}$. As a result, if the function $\\psi$ is constructed to have values disproportionately high on integer multiples of $q$, the sum may still diverge, despite the $\\alpha$ admitting good approximations being relatively rare.\nThis led them to conjecture that a variant of Khintchine's theorem would hold when non-reduced fractions are excluded, as follows: denoting by $\\varphi$ the Euler totient function, if $\\map{\\psi}{\\mathbb{N}}{[0,\\infty)}$ is a function such that $\\sum_{q\\in\\mathbb{N}}\\psi(q)\\varphi(q)/q$ diverges, then for almost all $\\alpha\\in\\mathbb{R}$, there exists infinitely many solutions to the inequality:\n\\begin{align}\n \\left\\lvert \\alpha - \\frac{p}{q}\\right\\rvert < \\frac{\\psi(q)}{q}.\\label{eq:approx-ineq}\n\\end{align}\nThe Duffin-Schaeffer conjecture was open for 78 years before it was proven by Koukoulopoulos and Maynard \\cite{koukoulopoulos2020duffin}. A full survey on the conjecture and an exposition of its proof can be found in \\cite{hauke2025centurymetricdiophantineapproximation}.\n\nThe work of \\cite{koukoulopoulos2020duffin}, however, left open the question of quantitative results on the number of approximations, assuming some bound on $q$. A quantitative version of Khintchine's theorem was proven by Schmidt in 1960 \\cite{schmidt1960ametricaltheorem}; one can also view \\cite[Chapter~4]{harman1998metricnumbertheory} for an overview of Schmidt's method. Schmidt-type results on the Duffin-Schaeffer conjecture have been the subject of various recent works \\cite{aistleitner2023metrictheoryofapproximations, hauke2024provingduffinschaefferconjecturegcd}, with the sharpest known being of Koukoulopoulos, Maynard and Yang \\cite{koukoulopoulos2024sharpquantitativeversionduffinschaeffer}. \nTowards this quantitative question, given $\\map{\\psi}{\\mathbb{N}}{[0,1/2]}$ and $Q\\geq 1$, we define:\n\\begin{align*}\n \\Psi(Q) \\coloneq \\sum_{q\\leq Q}\\frac{2\\psi(q)\\varphi(q)}{q}.\n\\end{align*}\nMoreover, denote by $S(\\alpha,Q)$ the number of pairs $(p,q)\\in \\mathbb{Z} \\times \\mathbb{N}$ with $q\\leq Q,\\ \\gcd(p,q)=1$ that satisfy the inequality \\eqref{eq:approx-ineq}. The work of Koukoulopoulos, Maynard and Yang \\cite{koukoulopoulos2024sharpquantitativeversionduffinschaeffer} then states that, assuming $\\lim_{Q\\to\\infty}\\Psi(Q)=\\infty$, for every $\\varepsilon>0$ and almost all $\\alpha\\in\\mathbb{R}$, we have as $Q\\to \\infty$:\n\\begin{align*}\n S(\\alpha,Q)=\\Psi(Q)+O_\\varepsilon\\left(\\Psi(Q)^{\\frac{1}{2}+\\varepsilon}\\right).\n\\end{align*}\n\nIn this paper, we consider a higher-dimensional generalisation of the Duffin-Schaeffer conjecture as follows: for $k\\geq 1$, given $\\map{\\psi}{\\mathbb{N}}{[0,\\infty)}$ and $\\alpha\\in \\mathbb{R}^k$, we ask whether there exists infinitely many solutions $(a,q)\\in\\mathbb{Z}^k\\times \\mathbb{N}$ satisfying the inequality:\n\\begin{align}\n \\pn{q\\alpha-a}{\\infty}< \\psi(q),\\label{eq:kdim-approx-ineq}\n\\end{align}\nwhere $a=(a_1,\\dots,a_k)$ satisfies $\\gcd(a_i, q)=1$ for $i\\in\\{1,\\dots, k\\}$. When $k=1$, this is then exactly the question of the Duffin-Schaeffer conjecture. For $k\\geq 2$, the question was first raised by Sprind\\v{z}uk \\cite{sprindzhuk1979metric}, and then resolved by Pollington and Vaughan \\cite{pollington1990kdimensionalduffinschaeffer} as follows: assuming the notation and hypothesis above, there exists infinitely many solutions to the inequality \\eqref{eq:kdim-approx-ineq} if and only if the sum $\\sum_{q\\in\\mathbb{N}}(\\psi(q)\\varphi(q)/q)^k$ diverges.\n\nFollowing the recent progress in the 1-dimensional case, we now ask the question of quantitative results for the higher-dimensional setup of Pollington and Vaughan. By periodicity, we consider only $\\alpha \\in [0,1]^k$ and thus $0\\leq a \\leq q$, and we assume $\\psi(q)\\leq 1/2$, following \\cite{aistleitner2023metrictheoryofapproximations, koukoulopoulos2024sharpquantitativeversionduffinschaeffer}. The main theorem of this paper is the following. We state the result for all $k\\geq 1$ for completeness; our contribution, of course, is the case $k\\geq 2$.\n\nIn this paper, we consider a higher-dimensional generalisation of the Duffin-Schaeffer conjecture as follows: for $k\\geq 1$, given $\\map{\\psi}{\\N}{[0,\\infty)}$ and $\\alpha\\in \\R^k$, we ask whether there exists infinitely many solutions $(a,q)\\in\\Z^k\\times \\N$ satisfying the inequality:\n\\begin{align}\n \\pn{q\\alpha-a}{\\infty}< \\psi(q),\\label{eq:kdim-approx-ineq}\n\\end{align}\nwhere $a=(a_1,\\dots,a_k)$ satisfies $\\gcd(a_i, q)=1$ for $i\\in\\{1,\\dots, k\\}$. When $k=1$, this is then exactly the question of the Duffin-Schaeffer conjecture. For $k\\geq 2$, the question was first raised by Sprind\\v{z}uk \\cite{sprindzhuk1979metric}, and then resolved by Pollington and Vaughan \\cite{pollington1990kdimensionalduffinschaeffer} as follows: assuming the notation and hypothesis above, there exists infinitely many solutions to the inequality \\eqref{eq:kdim-approx-ineq} if and only if the sum $\\sum_{q\\in\\N}(\\psi(q)\\varphi(q)/q)^k$ diverges.\n\nFinally, we prove the variance estimate, and thus conclude the proof of our main result. The proof follows \\cite{koukoulopoulos2024sharpquantitativeversionduffinschaeffer} with minor modifications.\n\\begin{proof}[Proof of Theorem~\\ref{theorem:k-dim-variance}]\n Note that we have:\n \\begin{align*}\n \\lint{\\left[0,1\\right]^k}{}{\\left(S_k(\\alpha, Q)-\\Psi_k(Q)\\right)^2}{\\alpha}= \\sum_{q,r\\leq Q} \\lambda_k(\\A_q^k \\cap \\A_r^k) - \\Psi_k(Q)^2,\n \\end{align*}\n and so we will show instead that:\n \\begin{align*}\n \\sum_{q,r\\leq Q} \\lambda_k(\\A_q^k \\cap \\A_r^k) \\leq \\Psi_k(Q)^2 + \\Psi_k(Q) + O_{\\ep,k}(\\Psi_k(Q)^{1+\\ep}).\n \\end{align*}\n When $q=r$, we have simply $\\sum_{q\\leq Q}\\lambda_k(\\A_q^k) = \\Psi_k(Q)$. Thus, consider the sum of terms $q\\neq r$, and assume that $D=D(q,r)\\geq 1/2$. We consider the contributions over the following partition:\n \\begin{align*}\n \\E^{(1)} &\\coloneq\\set{(q,r)\\in \\left[1,Q\\right]^k}{L_{D^2}(q,r)\\leq \\frac{1}{D},\\ \\omega_{D^2}(q,r)\\leq \\frac{\\ep}{4k}\\log(2D)},\\\\\n \\E^{(2)} &\\coloneq\\set{(q,r)\\in \\left[1,Q\\right]^k}{L_{D^2}(q,r)> \\frac{1}{D}},\\\\\n \\E^{(3)} &\\coloneq\\set{(q,r)\\in \\left[1,Q\\right]^k}{L_{D^2}(q,r)\\leq \\frac{1}{D},\\ \\omega_{D^2}(q,r)> \\frac{\\ep}{4k}\\log(2D)}.\n \\end{align*}\n First, consider the sum over $\\E^{(1)}$. Then applying Lemma~\\ref{lem:k-dim-overlap} with $t=D^2$ gives:\n \\begin{align*}\n \\lambda_k(\\A_q^k\\cap\\A_r^k) &\\leq \\lambda_k(\\A_q^k)\\lambda_k(\\A_r^k)\\me^{2k/D}\\left(1+O\\left(\\frac{2^{\\ep\\log(2D)/4}\\log(4D)}{D}\\right)\\right)\\\\\n &\\leq \\lambda_k(\\A_q^k)\\lambda_k(\\A_r^k)\\left(1+O_k(D^{-1+\\ep/2})\\right)\n \\end{align*}\n Therefore:\n \\begin{align*}\n \\sum_{(q,r)\\in\\E^{(1)}}\\lambda_k(\\A_q^k\\cap\\A_r^k) &\\leq \\sum_{(q,r)\\in\\E^{(1)}}\\lambda_k(\\A_q^k)\\lambda_k(\\A_r^k) + O_k\\left(\\sum_{(q,r)\\in\\E^{(1)}}\\lambda_k(\\A_q^k)\\lambda_k(\\A_r^k)D^{-1+\\ep/2}\\right)\\\\\n &\\leq \\Psi_k(Q)^2+O_k\\left(\\sum_{(q,r)\\in\\E^{(1)}}\\lambda_k(\\A_q^k)\\lambda_k(\\A_r^k)D^{-1+\\ep/2}\\right).\n \\end{align*}\n Focusing on the error term, define $R\\coloneq\\sum_{(q,r)\\in\\E^{(1)}}\\lambda_k(\\A_q^k)\\lambda_k(\\A_r^k)D^{-1+\\ep/2}$, and let $I$ be the smallest non-negative integer such that $2^I\\geq \\Psi_k(Q)$. We then split $R$ via dyadic partition on $D$, writing $R_i$ as the part of the sum with $D\\in [2^{i-1},2^i)$, and $R'$ the part with $D\\geq 2^I$. Then:\n \\begin{align*}\n R'&\\leq \\Psi_k(Q)^{-1+\\ep/2}\\sum_{(q,r)\\in\\E^{(1)}}\\lambda_k(\\A_q^k)\\lambda_k(\\A_r^k) \\leq \\Psi_k(Q)^{1+\\ep/2},\n \\end{align*}\n and for $i\\in \\{0,1,\\dots,I\\}$ we have by Proposition~\\ref{proposition:overlap-sum-1} with $y=2^i$ that:\n \\begin{align*}\n R_i &\\leq (2^{i-1})^{-1+\\ep/2}\\sum_{(q,r)\\in\\E^{(1)}}\\lambda_k(\\A_q^k)\\lambda_k(\\A_r^k)\\\\\n &\\ll_{\\ep,k} (2^i)^{-1+\\ep/2}(2^i)^{1-\\ep}\\Psi_k(Q)^{1+\\ep}\\\\\n &= 2^{-i\\epsilon/2}\\Psi_k(Q)^{1+\\ep}.\n \\end{align*}\n We thus have $R \\ll_{\\ep,k}\\Psi_k(Q)^{1+\\ep}$ and so:\n \\begin{align*}\n \\sum_{(q,r)\\in\\E^{(1)}}\\lambda_k(\\A_q^k\\cap\\A_r^k) \\leq \\Psi_k(Q)^k + O_{\\ep,k}(\\Psi_k(Q)^{1+\\ep}).\n \\end{align*}\n Considering $\\E^{(2)}$, we use Lemma~\\ref{lem:pv-overlap} to say:\n \\begin{align*}\n \\sum_{(q,r)\\in\\E^{(2)}}\\lambda_k(\\A_q^k\\cap\\A_r^k) &\\ll_k \\sum_{(q,r)\\in\\E^{(2)}}\\lambda_k(\\A_q^k)\\lambda_k(\\A_r^k)\\me^{kL_D(q,r)}.\n \\end{align*}\n If $D\\in [2^{i-1},2^i)$, we have $L_{4^{i-1}}(q,r)\\geq L_{D^2}(q,r)>1/D\\geq 2^{-i}$. We then write:\n \\begin{align*}\n \\E^{(2)}=\\bigcup_{i=0}^\\infty \\bigcup_{j=i}^\\infty \\E^{(2)}_{i,j},\n \\end{align*}\n where:\n \\begin{align*}\n \\E^{(2)}_{i,i}&\\coloneq\\set{(q,r)\\in \\E^{(2)}}{D(q,r)\\in[2^{i-1},2^i),\\ L_{4^{i-1}}(q,r)\\leq 1},\\text{ and}\\\\\n \\E^{(2)}_{i,j}&\\coloneq\\set{(q,r)\\in \\E^{(2)}}{D(q,r)\\in[2^{i-1},2^i),\\ L_{4^{j-1}}(q,r)\\leq 1<L_{4^{j-2}}(q,r)},\\text{ for }j>i.\n \\end{align*}\n Then, if $(q,r)\\in\\E^{(2)}_{i,j}$, we have:\n \\begin{align*}\n L_D(q,r)\\leq L_{2^{i-1}}(q,r)\\leq \\sum_{2^{i-1}< p\\leq 4^{j-1}}\\frac{1}{p}+L_{4^{j-1}}(q,r) \\leq \\log\\frac{j+1}{i+1} + O(1),\n \\end{align*}\n and so:\n \\begin{align*}\n \\sum_{(q,r)\\in\\E^{(2)}}\\lambda_k(\\A_q^k\\cap\\A_r^k) \\ll_k \\sum_{j\\geq i\\geq 0}\\left(\\frac{j+1}{i+1}\\right)^k\\sum_{(q,r)\\in\\E^{(2)}_{i,j}}\\lambda_k(\\A_q^k)\\lambda_k(\\A_r^k).\n \\end{align*}\n Then for $i\\geq 0$, Proposition~\\ref{proposition:overlap-sum-2} with $y=2^i$, $t=\\max\\{1, 4^{i-1}\\}$, $s=2^i$ and $C=8$ gives:\n \\begin{align*}\n \\sum_{(q,r)\\in\\E^{(2)}_{i,i}}\\lambda_k(\\A_q^k)\\lambda_k(\\A_r^k) &\\ll_{\\ep,k} \\exp(-2^i)\\Psi_k(Q)^{1+\\ep}.\n \\end{align*}\n Similarly, for $j>i$, choosing $y=2^i$, $t=\\max\\{1, 4^{j-2}\\}$, $s=1$ and $C=17$ gives:\n \\begin{align*}\n \\sum_{(q,r)\\in\\E^{(2)}_{i,j}}\\lambda_k(\\A_q^k)\\lambda_k(\\A_r^k) \\ll_{\\ep,k} \\exp(-4^j)\\Psi_k(Q)^{1+\\ep}.\n \\end{align*}\n Together, this gives:\n \\begin{align*}\n \\sum_{(q,r)\\in\\E^{(2)}}\\lambda_k(\\A_q^k\\cap\\A_r^k) &\\ll_{\\ep,k} \\Psi_k(Q)^{1+\\ep}\\left(\\sum_{i\\geq 0}\\exp(-2^i) +\\sum_{j>i}\\left(\\frac{j+1}{i+1}\\right)^k\\exp(-4^j)\\right)\\\\\n &\\ll_{\\ep,k} \\Psi_k(Q)^{1+\\ep}.\n \\end{align*}\n Finally, considering $\\E^{(3)}$, we have that:\n \\begin{align*}\n L_D(q,r)\\leq \\sum_{D<p\\leq D^2}\\frac{1}{p} + L_{D^2}(q,r) \\ll \\log\\log D^2-\\log\\log D + \\frac{1}{D} \\ll 1.\n \\end{align*}\n We thus have by Lemma~\\ref{lem:pv-overlap} that:\n \\begin{align*}\n \\sum_{(q,r)\\in\\E^{(3)}}\\lambda_k(\\A_q^k\\cap\\A_r^k) &\\ll_k \\sum_{(q,r)\\in\\E^{(3)}}\\lambda_k(\\A_q^k)\\lambda_k(\\A_r^k).\n \\end{align*}\n If $D\\in [2^{i-1},2^i)$, we have:\n \\begin{align*}\n \\omega_{4^i}(q,r)\\geq \\omega_{D^2}(q,r)&> \\frac{\\ep}{4k}\\log(2D) \\geq \\frac{\\ep}{8k}\\log(4^i).\n \\end{align*}\n We can thus write:\n \\begin{align*}\n \\sum_{(q,r)\\in\\E^{(3)}}\\lambda_k(\\A_q^k\\cap\\A_r^k) &\\ll_k \\sum_{i\\geq 0}\\sum_{(q,r)\\in\\E^{(3)}_i}\\lambda_k(\\A_q^k)\\lambda_k(\\A_r^k),\n \\end{align*}\n where, for each $i\\geq 0$:\n \\begin{align*}\n \\E^{(3)}_{i}&\\coloneq\\set{(q,r)\\in \\E^{(3)}}{D(q,r)\\in[2^{i-1},2^i),\\ \\omega_{4^i}(q,r)> \\frac{\\ep}{8k}\\log(4^i)}.\n \\end{align*}\n Applying Proposition~\\ref{proposition:overlap-sum-3} with $y=2^i$, $t=4^i$, $\\kappa=\\ep/8k$ and $C=2$ gives:\n \\begin{align*}\n \\sum_{(q,r)\\in\\E^{(3)}_i}\\lambda_k(\\A_q^k)\\lambda_k(\\A_r^k) &\\ll_{\\ep,k} (4^i)^{-2}(2^i)^{1-\\ep}\\Psi_k(Q)^{1-\\ep} \\ll_\\ep 2^{-i}\\Psi_k(Q)^{1+\\ep},\n \\end{align*}\n and so:\n \\begin{align*}\n \\sum_{(q,r)\\in\\E^{(3)}}\\lambda_k(\\A_q^k\\cap\\A_r^k) &\\ll_{\\ep,k} \\sum_{i\\geq 0}2^{-i}\\Psi_k(Q)^{1+\\ep} \\ll_\\ep\\Psi_k(Q)^{1+\\ep}.\n \\end{align*}\n Putting everything together:\n \\begin{align*}\n \\sum_{q,r\\leq Q}\\lambda_k(\\A_q^k\\cap\\A_r^k) &= \\sum_{q\\leq Q}\\lambda_k(\\A_q^k) + \\sum_{i=1}^3\\sum_{(q,r)\\in\\E^{(i)}}\\lambda_k(\\A_q^k\\cap\\A_r^k)\\\\\n &\\leq \\Psi_k(Q) + \\Psi_k(Q)^2 + O_{\\ep,k}(\\Psi_k(Q)^{1+\\ep}),\n \\end{align*}\n as required.\n\\end{proof}", "post_theorem_intro_text_len": 5320, "post_theorem_intro_text": "This result establishes a quantitative estimate for the number of solutions for the $k$-dimensional Duffin-Schaeffer conjecture for all $k\\geq 2$, matching the best-known error term in the 1-dimensional case from \\cite{koukoulopoulos2024sharpquantitativeversionduffinschaeffer}. We note that, in the original, non-reduced fraction case of Khintchine, without assuming monotonicity, asymptotic formulae exist only in the case $k\\geq 3$. For $k=1,2$, we have such formulae with monotonicity; for $k=1$, we know that this monotonicity is essential, but for $k=2$, it remains an open question: see \\cite[Chapter~3]{harman1998metricnumbertheory} or \\cite[Section~5]{beresnevich2010khintchinegroshev}.\n\nThe method of proving Theorem~\\ref{theorem:k-dim-ds} and similar results in metric Diophantine approximation is a standard procedure in metric number theory, with ideas dating back to at least \\cite{cassels1950somemetricaltheorems, schmidt1964metricaltheorems}. We omit the proof, referring the reader to \\cite[Lemma~1.5]{harman1998metricnumbertheory} for the general procedure and to \\cite{koukoulopoulos2024sharpquantitativeversionduffinschaeffer} for the details in our case. The key is a variance estimate of random variables describing the number of solutions for a given $\\alpha$. From a probabilistic perspective, the 1-dimensional Duffin-Schaeffer conjecture asks whether a uniformly randomly chosen irrational $\\alpha$ is contained within the set system:\n\\begin{align*}\n    \\A_q \\coloneq \\bigcup_{\\substack{0\\leq a \\leq q,\\\\\\gcd(a,q)=1}}\\left(\\frac{a}{q}-\\frac{\\psi(q)}{q}, \\frac{a}{q} + \\frac{\\psi(q)}{q}\\right)\\cap \\left[0,1 \\right]\n\\end{align*}\nfor infinitely or finitely many $q$; for $k\\geq2$, we ask about $\\A_q^k\\coloneq \\A_q^{k-1}\\times \\A_q$. Denoting then by $\\lambda_k$ the $k$-dimensional Lebesgue measure restricted to $[0,1]^k$, we note that $\\Psi_k(Q) = \\sum_{q=1}^Q\\lambda_k(\\A_q^k)$ as:\n\\begin{align*}\n        \\lambda_k(\\A_q^k) = \\left(\\frac{2\\psi(q)\\varphi(q)}{q}\\right)^k.\n\\end{align*}\nFollowing this, one can also see that $S_k(\\alpha,Q)=\\sum_{q=1}^Q\\ind{\\A_q^k}(\\alpha)$ for every $\\alpha$. The result of Theorem~\\ref{theorem:k-dim-ds} can then be understood as describing the concentration of $S_k(\\alpha,Q)$ around its mean. As before, the result would follow immediately from the divergence Borel-Cantelli lemma with pairwise independence of the random variables $\\ind{\\A_q}$, which we do not have here. In fact, the overlap $\\lambda_k(\\A_q^k\\cap \\mathcal{A}^k_r)$ can be significantly larger than $\\lambda_k(\\A_q^k)\\lambda_k(\\A_r^k)$ for pairs $(q,r)$ where the prime factorisation of $Q$ contains a large number of small primes not dividing $r$, or vice versa. The crux of the argument involves showing that such pairs are relatively rare, and thus on average we have a form of quasi-independence. In fact, one can show that the set system moves towards pairwise independence on average as the total mass tends towards infinity. Our key result below establishes exactly this for the case $k\\geq 2$; again, stated as $k\\geq 1$ for completeness.\n\n\\begin{theorem}\\label{theorem:k-dim-variance}\n    Let $k\\geq 1$ and $Q\\in \\mathbb{N}$, and define $S_k(\\alpha,Q)$ and $\\Psi_k(Q)$ as in Theorem~\\ref{theorem:k-dim-ds}. Then, for every fixed $\\varepsilon>0$, we have:\n    \\begin{align*}\n         \\lint{\\left[0,1\\right]^k}{}{\\left(S_k(\\alpha, Q)-\\Psi_k(Q)\\right)^2}{\\alpha} \\leq \\Psi_k(Q) + O_{\\varepsilon,k}(\\Psi_k(Q)^{1+\\varepsilon}).\n     \\end{align*}\n\\end{theorem}\n\nThe rest of this paper is dedicated to constructing the proof of Theorem~\\ref{theorem:k-dim-variance}. The standard approach involves a counting argument on the prime factorisations of $q$ and $r$, followed by a sieve on the coprimality condition. In the $k$-dimensional problem, one can simply adapt the $1$-dimensional overlap estimates proven in \\cite{aistleitner2023metrictheoryofapproximations, koukoulopoulos2024sharpquantitativeversionduffinschaeffer, pollington1990kdimensionalduffinschaeffer}, as shown in Section~\\ref{sec:overlap}. From here, we construct three bilinear estimates, allowing us to bound the variance contributions of the problematic pairs of integers $(q,r)$ with many prime factors. In the $1$-dimensional case, the approach of \\cite{koukoulopoulos2020duffin, koukoulopoulos2024sharpquantitativeversionduffinschaeffer} is a technical iterative construction via GCD graphs. This approach was generalised in \\cite{hauke2024provingduffinschaefferconjecturegcd, vazquez2024almostsharpquantitativeduffinschaeffergcd}, providing a framework for similar bounds on general multiplicative functions. In Section~\\ref{sec:bilinear}, we show how this framework can be employed to construct equivalent bilinear bounds in all dimensions $k\\geq 2$. From here, the variance estimate follows with an approach adapted from \\cite{koukoulopoulos2024sharpquantitativeversionduffinschaeffer}, as shown in Section~\\ref{sec:variance}.\n\n\\subsection*{Acknowledgements}\nThe author would like to thank Manuel Hauke for introducing them to the topic and for guidance and discussion throughout the work. The work was conducted during a research stay in TU Graz, as part of the author's master's thesis in NTNU; the author would thus also like to thank the Institute of Analysis and Number Theory in TU Graz for their hospitality.", "sketch": "The proof method for Theorem~\\ref{theorem:k-dim-ds} is described as a “standard procedure in metric number theory” and is omitted (with references). The sketch given is:\n\n- Recast the counting function as a sum of indicator random variables: define the set system\n\\[\n\\A_q \\coloneq \\bigcup_{\\substack{0\\leq a \\leq q,\\\\\\gcd(a,q)=1}}\\left(\\frac{a}{q}-\\frac{\\psi(q)}{q}, \\frac{a}{q} + \\frac{\\psi(q)}{q}\\right)\\cap \\left[0,1 \\right],\n\\]\n(and for $k\\ge2$ use $\\A_q^k$), so that\n\\[\\Psi_k(Q)=\\sum_{q=1}^Q \\lambda_k(\\A_q^k), \\qquad S_k(\\alpha,Q)=\\sum_{q=1}^Q \\ind{\\A_q^k}(\\alpha).\\]\nThus Theorem~\\ref{theorem:k-dim-ds} “describ[es] the concentration of $S_k(\\alpha,Q)$ around its mean.”\n\n- Identify the main obstacle: the divergence Borel–Cantelli lemma “would follow immediately” under pairwise independence of $\\ind{\\A_q}$, but this fails because overlaps can be much larger than products:\n\\(\\lambda_k(\\A_q^k\\cap \\mathcal{A}^k_r)\\) can be “significantly larger” than \\(\\lambda_k(\\A_q^k)\\lambda_k(\\A_r^k)\\) for certain $(q,r)$ related to their prime factorizations.\n\n- State the crux: show that such highly dependent pairs are “relatively rare,” giving “on average… quasi-independence,” and in fact “the set system moves towards pairwise independence on average as the total mass tends towards infinity.”\n\n- Reduce Theorem~\\ref{theorem:k-dim-ds} to a variance bound: “The key is a variance estimate” for $S_k(\\alpha,Q)$, formalized as Theorem~\\ref{theorem:k-dim-variance}, which bounds\n\\(\\int_{[0,1]^k} (S_k(\\alpha,Q)-\\Psi_k(Q))^2\\, d\\alpha\\).\n\n- How the variance bound is proved (high level): it “involves a counting argument on the prime factorisations of $q$ and $r$, followed by a sieve on the coprimality condition,” using adapted $1$-dimensional overlap estimates, then “three bilinear estimates” to control contributions from “problematic pairs… with many prime factors,” after which “the variance estimate follows with an approach adapted from” the cited 1D work.", "expanded_sketch": "The proof method for the main theorem is described as a “standard procedure in metric number theory” and is omitted (with references). The sketch given is:\n\n- Recast the counting function as a sum of indicator random variables: define the set system\n\\[\n\\A_q \\coloneq \\bigcup_{\\substack{0\\leq a \\leq q,\\\\\\gcd(a,q)=1}}\\left(\\frac{a}{q}-\\frac{\\psi(q)}{q}, \\frac{a}{q} + \\frac{\\psi(q)}{q}\\right)\\cap \\left[0,1 \\right],\n\\]\n(and for $k\\ge2$ use $\\A_q^k$), so that\n\\[\\Psi_k(Q)=\\sum_{q=1}^Q \\lambda_k(\\A_q^k), \\qquad S_k(\\alpha,Q)=\\sum_{q=1}^Q \\ind{\\A_q^k}(\\alpha).\\]\nThus the main theorem “describ[es] the concentration of $S_k(\\alpha,Q)$ around its mean.”\n\n- Identify the main obstacle: the divergence Borel–Cantelli lemma “would follow immediately” under pairwise independence of $\\ind{\\A_q}$, but this fails because overlaps can be much larger than products:\n\\(\\lambda_k(\\A_q^k\\cap \\mathcal{A}^k_r)\\) can be “significantly larger” than \\(\\lambda_k(\\A_q^k)\\lambda_k(\\A_r^k)\\) for certain $(q,r)$ related to their prime factorizations.\n\n- State the crux: show that such highly dependent pairs are “relatively rare,” giving “on average… quasi-independence,” and in fact “the set system moves towards pairwise independence on average as the total mass tends towards infinity.”\n\n- Reduce the main theorem to a variance bound: “The key is a variance estimate” for $S_k(\\alpha,Q)$, formalized in the following result.\n\n\\begin{theorem}\\label{theorem:k-dim-variance}\n    Let $k\\geq 1$ and $Q\\in \\N$, and define $S_k(\\alpha,Q)$ and $\\Psi_k(Q)$ as in Theorem~\\ref{theorem:k-dim-ds}. Then, for every fixed $\\ep>0$, we have:\n    \\begin{align*}\n         \\lint{\\left[0,1\\right]^k}{}{\\left(S_k(\\alpha, Q)-\\Psi_k(Q)\\right)^2}{\\alpha} \\leq \\Psi_k(Q) + O_{\\ep,k}(\\Psi_k(Q)^{1+\\ep}).\n     \\end{align*}\n\\end{theorem}\n\nThis bounds\n\\(\\int_{[0,1]^k} (S_k(\\alpha,Q)-\\Psi_k(Q))^2\\, d\\alpha\\).\n\n- How the variance bound is proved (high level): it “involves a counting argument on the prime factorisations of $q$ and $r$, followed by a sieve on the coprimality condition,” using adapted $1$-dimensional overlap estimates, then “three bilinear estimates” to control contributions from “problematic pairs… with many prime factors,” after which “the variance estimate follows with an approach adapted from” the cited 1D work.", "expanded_theorem": "\\label{theorem:k-dim-ds}\n   Let $k\\geq 1$, and let $\\map{\\psi}{\\mathbb{N}}{\\left[0,1/2\\right]}$ be such that $\\sum_{q=1}^\\infty \\left(\\varphi(q)\\psi(q)/q\\right)^k=\\infty$. For $\\alpha\\in \\left[0,1\\right]^k$, denote by $S_k(\\alpha,Q)$ the number of solutions $(a,q)\\in\\mathbb{Z}^k\\times \\mathbb{N}$ to the inequality:\n    \\begin{align*}\n        \\pn{q\\alpha-a}{\\infty} < \\psi(q), \\quad \\text{given } q \\leq Q,\n    \\end{align*}\n    satisfying $\\gcd(a_i, q)=1$, for $i\\in\\{1,\\dots, k\\}$, $a=(a_1,\\dots, a_k)$. Moreover, given $Q\\in \\mathbb{N}$, define:\n    \\begin{align*}\n        \\Psi_k(Q)\\coloneq \\sum_{q\\leq Q}\\left(\\frac{2\\psi(q)\\varphi(q)}{q}\\right)^k.\n    \\end{align*}\n    Then for all $\\varepsilon>0$, as $Q\\to \\infty$, we have for almost all $\\alpha$ that:\n    \\begin{align*}\n        S_k(\\alpha,Q)=\\Psi_k(Q)+O_{\\varepsilon,k}\\left( \\Psi_k(Q)^{1/2+\\varepsilon} \\right).\n    \\end{align*}", "theorem_type": ["Asymptotic or Limit", "Inequality or Bound"], "mcq": {"question": "Let $k\\ge 1$, and let $\\psi:\\mathbb N\\to [0,1/2]$ satisfy\n\\[\n\\sum_{q=1}^\\infty \\left(\\frac{\\varphi(q)\\psi(q)}{q}\\right)^k=\\infty,\n\\]\nwhere $\\varphi$ is Euler's totient function. For $\\alpha=(\\alpha_1,\\dots,\\alpha_k)\\in[0,1]^k$ and $Q\\in\\mathbb N$, let $S_k(\\alpha,Q)$ be the number of pairs $(a,q)\\in\\mathbb Z^k\\times\\mathbb N$ with $q\\le Q$, $a=(a_1,\\dots,a_k)$, and $\\gcd(a_i,q)=1$ for each $i\\in\\{1,\\dots,k\\}$, such that\n\\[\n\\|q\\alpha-a\\|_\\infty<\\psi(q),\n\\]\nwhere $\\|x\\|_\\infty=\\max_{1\\le i\\le k}|x_i|$. Also define\n\\[\n\\Psi_k(Q):=\\sum_{q\\le Q}\\left(\\frac{2\\psi(q)\\varphi(q)}{q}\\right)^k.\n\\]\nWhich quantitative estimate holds for almost every $\\alpha\\in[0,1]^k$ as $Q\\to\\infty$?", "correct_choice": {"label": "A", "text": "For every $\\varepsilon>0$, one has\n\\[\nS_k(\\alpha,Q)=\\Psi_k(Q)+O_{\\varepsilon,k}\\bigl(\\Psi_k(Q)^{1/2+\\varepsilon}\\bigr)\n\\]\nas $Q\\to\\infty$ for almost all $\\alpha\\in[0,1]^k$."}, "choices": [{"label": "B", "text": "For every $\\varepsilon>0$, one has\n\\[\nS_k(\\alpha,Q)=\\Psi_k(Q)+O_{\\varepsilon,k}\\bigl(Q^{1/2+\\varepsilon}\\bigr)\n\\]\nas $Q\\to\\infty$ for almost all $\\alpha\\in[0,1]^k$."}, {"label": "C", "text": "For every $\\varepsilon>0$, one has\n\\[\nS_k(\\alpha,Q)=\\Psi_k(Q)+o\\bigl(\\Psi_k(Q)\\bigr)\n\\]\nas $Q\\to\\infty$ for almost all $\\alpha\\in[0,1]^k$."}, {"label": "D", "text": "There exists $C_{\\varepsilon,k}>0$ such that for every $\\alpha\\in[0,1]^k$ and all sufficiently large $Q$,\n\\[\n\\left|S_k(\\alpha,Q)-\\Psi_k(Q)\\right|\\le C_{\\varepsilon,k}\\,\\Psi_k(Q)^{1/2+\\varepsilon}.\n\\]"}, {"label": "E", "text": "For every $\\varepsilon>0$, one has\n\\[\nS_k(\\alpha,Q)=\\Psi_k(Q)+O_{\\varepsilon,k}\\bigl(\\Psi_k(Q)^{\\varepsilon}\\bigr)\n\\]\nas $Q\\to\\infty$ for almost all $\\alpha\\in[0,1]^k$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "error scaled by total mass $\\Psi_k(Q)$ rather than by the cutoff parameter $Q$", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "drop the sharp exponent $1/2+\\varepsilon$ and keep only asymptotic concentration around the mean", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "almost-everywhere conclusion replaced by a uniform-in-$\\alpha$ bound", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "variance-driven square-root fluctuation replaced by unrealistically stronger subpower error", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct estimate; it only states the hypotheses and definitions. There is no direct answer leakage, though the setup clearly points to a specific known metric theorem."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the hypotheses are stated in full and the correct option is the theorem's quantitative conclusion almost verbatim. The item tests recognition of the exact statement more than independent inference."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact error term from weaker, stronger, or quantifier-altered alternatives. However, the main task is still selecting the memorized theorem statement rather than generating a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are mathematically meaningful and target plausible failure modes: replacing Ψ_k(Q) by Q, weakening to o(Ψ_k(Q)), strengthening to uniform-in-α, or giving an unrealistically small error term. They are distinct and well-designed."}, "total_score": 5, "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no real answer leakage, but it is largely tautological because it asks for the theorem's conclusion under its exact hypotheses."}}
{"id": "2602.20086v2", "paper_link": "http://arxiv.org/abs/2602.20086v2", "theorems_cnt": 3, "theorem": {"env_name": "theoremL", "content": "\\label{thm:asboundsPoly}\n    Let $f$ be a  Rademacher or Steinhaus random multiplicative function. In the Rademacher case, assume $P\\in\\mathbb Z[x]$ is a product of at least two distinct linear factors over $\\mathbb Q$, or irreducible of degree $2$, taking infinitely many square-free values. In the Steinhaus case, assume $P(x)$ is not of the form $w(x+c)^d$ for $w\\in\\mathbb Z, c\\in\\mathbb Q$. Then almost surely we have $$\\left\\lvert\\sum_{n\\leq N} f(P(n))\\right\\rvert \\ll \\sqrt{N\\log\\log N}.$$ Moreover, almost surely there exists a sequence $N_k\\to\\infty$ such that \n    \\begin{equation*}\n        \\left\\lvert\\sum_{n\\leq N_k} f(P(n))\\right\\rvert\\gg \\sqrt{N_k\\log\\log N_k}.\n    \\end{equation*}", "start_pos": 8546, "end_pos": 9272, "label": "thm:asboundsPoly"}, "ref_dict": {"thm:asboundsPoly": "\\begin{theoremL}\\label{thm:asboundsPoly}\n    Let $f$ be a  Rademacher or Steinhaus random multiplicative function. In the Rademacher case, assume $P\\in\\mathbb Z[x]$ is a product of at least two distinct linear factors over $\\mathbb Q$, or irreducible of degree $2$, taking infinitely many square-free values. In the Steinhaus case, assume $P(x)$ is not of the form $w(x+c)^d$ for $w\\in\\mathbb Z, c\\in\\mathbb Q$. Then almost surely we have $$\\left\\lvert\\sum_{n\\leq N} f(P(n))\\right\\rvert \\ll \\sqrt{N\\log\\log N}.$$ Moreover, almost surely there exists a sequence $N_k\\to\\infty$ such that \n    \\begin{equation*}\n        \\left\\lvert\\sum_{n\\leq N_k} f(P(n))\\right\\rvert\\gg \\sqrt{N_k\\log\\log N_k}.\n    \\end{equation*}\n\\end{theoremL}", "lemma:unSmoothSquarefree": "\\begin{lemma}\\label{lemma:unSmoothSquarefree}\n    Let $N$ be large, $\\alpha>\\frac{1}{2}$, and let $N\\geq H=H(N)\\geq N^{\\frac{4\\alpha + 1}{5}}$. The number of square-free integers in the short interval $(N-H, N]$ such that $P^+(n)>N^{\\alpha}$ is \n    \\begin{equation*}\n        \\gg H+O\\left(\\frac{H}{\\log N} \\right).\n    \\end{equation*}\n\\end{lemma}"}, "pre_theorem_intro_text_len": 6200, "pre_theorem_intro_text": "Since their introduction by Wintner \\cite{Wintner1944}, random multiplicative functions have attracted a lot of attention in number theory. \n\\begin{definition}\n    A \\emph{Steinhaus} random multiplicative function $f$ is a sequence of random variables $f(1), f(2), \\ldots$ such that $f(mn)=f(m)f(n)$ for all positive integers $m$ and $n$, and for each prime $p$, $f(p)$ is uniformly distributed on the unit circle.\n\n    A \\emph{Rademacher} random multiplicative function $f$ is a sequence of random variables $f(1), f(2),$ $\\ldots$  supported on square-free integers such that $f(mn) = f(m)f(n)$ whenever $m$ and $n$ are coprime, and for each prime $p$, $f(p)$ is uniformly distributed on $\\{-1, 1\\}$. \n\\end{definition}\nAlthough random multiplicative functions are useful in modeling deterministic objects such as Dirichlet and Archimedean characters, they have become increasingly interesting to study in their own right. Building on earlier work of Hal\\'asz \\cite{Halasz1977} and Hal\\'asz and R\\'enyi \\cite{HalaszRenyi1961}, partial sums of a random multiplicative function $\\sum_{n\\leq N} f(n)$ have been extensively studied by Harper. Being a sum of random variables, it is natural to ask whether $\\frac{1}{\\sqrt N}\\sum_{n\\leq N} f(n)$ has an approximately Gaussian distribution as $N\\to\\infty$. \nHarper \\cite{Harper2013} showed that this is not the case and in fact, he later showed \\cite{Harper2020_MomentsRandomMultiplicativeI} that the typical size of these sums is of order $\\sqrt N/(\\log\\log N)^{\\frac14}$. One might therefore hope to obtain \nan interesting limiting distribution after normalizing by this factor instead. This is a delicate matter closely related to Gaussian multiplicative chaos, and has been resolved recently in the work of Gorodetsky and Wong \\cite{gorodetsky_wong_2025_limiting} in the Steinhaus case. \n\nFrom an arithmetic point of view, it is interesting to understand sums of a random multiplicative function when the summands are restricted to subsets of positive integers. This will be our focus in this paper. Examples of such restrictions that have been considered previously include: \n\\begin{itemize}\\item[(i)] integers with an atypically small number of prime factors (Hough \\cite{Hough2011}, Harper \\cite{Harper2013}), \n\\item[(ii)] integers in a short interval (Chatterjee and Soundararajan \\cite{ChatterjeeSoundararajan2012}, Soundararajan and Xu \\cite{soundararajan2023clt}, Pandey, Wang, and Xu \\cite{PandeyWangXu2024}), \n\\item[(iii)] sums of two squares in a short interval and shifted primes $p-k$ for $k\\neq 0$ (Soundararajan and Xu \\cite{soundararajan2023clt}), \n\\item[(iv)] polynomial images (Najnudel \\cite{najnudel2020consecutive}, Klurman, Shkredov, and Xu \\cite{KlurmanShkredovXu2023}, Chinis and the author \\cite{ChinisShala2025}).\n\\end{itemize} \nIn each of the aforementioned works, the distribution of the sums turns out to be approximately Gaussian, demonstrating that multiplicativity seems to interfere less with independence when the summands are restricted to a less ``multiplicatively structured'' set. In this paper, we extend the general framework of Soundararajan and Xu \\cite{soundararajan2023clt} on central limit theorems for random multiplicative functions to the setting of large fluctuations.\n\nThere has been considerable work in understanding almost sure large fluctuations for the full sum $\\sum_{n\\leq N} f(n)$ and related variants. Harper \\cite{harper2020large} showed that for any (slowly growing) function $V(N)\\to\\infty$, almost surely there is a sequence of positive integers $N_k\\to\\infty$ such that \n\\begin{equation*}\n    \\left\\lvert\\sum_{n\\leq N_k}f(n)\\right\\rvert\\gg\\sqrt{N_k}\\frac{(\\log\\log N_k)^{\\frac14}}{V(N_k)}.\n\\end{equation*}\nThis matches what one expects from the law of iterated logarithm, up to the factor $V(N)$. Proving the corresponding upper bound remains a challenge. Partial results with an upper bound of the right shape (namely $\\sqrt N$ times a power of $\\log\\log N$) have been proved by Lau, Tenenbaum and Wu \\cite{LauTenenbaumWu2012}, as well as most recently by Caich \\cite{Caich2023} with the current best known almost sure bound\n\\begin{equation*}\n    \\left\\lvert\\sum_{n\\leq N} f(n)\\right\\rvert\\ll_{\\epsilon} \\sqrt N(\\log\\log N)^{\\frac{3}{4}+\\epsilon}.\n\\end{equation*}\nIn the case when the summands are demanded to have a prime factor bigger than $\\sqrt N$, Mastrostefano \\cite{Mastrostefano2022} proved an essentially sharp almost sure upper bound with the exponent $\\frac{1}{4}+\\epsilon$ in place of $\\frac{3}4$, with the corresponding almost sure lower bound following by work of Harper \\cite{harper2020large}; see also the work of Hardy \\cite{Hardy2025LargePrime} on the distributional aspect in this case. A weighted variant $\\sum_{n\\leq N} \\frac{f(n)}{\\sqrt n}$\nhas been considered by Aymone, Heap and Zhao \\cite{aymone2021partial}, and Hardy \\cite{Hardy2023AlmostSure} in the Steinhaus case who obtained essentially sharp bounds, and Atherfold \\cite{Atherfold2025} in the Rademacher case albeit without sharp bounds.\n\nRecently, Hoban, Shah, Ismail, Verreault, and Zaman \\cite{HobanShahIsmailVerreaultZaman2025} have extended the framework of Soundararajan and Xu \\cite{soundararajan2023clt} in a different direction, namely in the function field setting. It is likely that the methods of this paper, combined with the work in \\cite{HobanShahIsmailVerreaultZaman2025}, can be used to prove analogous results over function fields.\n\nWe are now ready to state the main results of this paper. Our focus will be on two main examples: (i) polynomial images, and (ii) integers in a short interval. However, our method is rather general, so it should be able to flexibly handle several examples; see Section \\ref{section:generalSetup} for the general but rather technical results. \n\n\\subsection{Polynomial images} This case is particularly interesting due to its connection with conjectures of Chowla and Elliott \\cite{Chowla1965, Elliott1992, Elliott1994} on correlations of multiplicative functions. See the introductions of \\cite{KlurmanShkredovXu2023, ChinisShala2025} for a more detailed discussion in the setting of random multiplicative functions, where central limit theorems were proved.", "context": "Since their introduction by Wintner \\cite{Wintner1944}, random multiplicative functions have attracted a lot of attention in number theory. \n\\begin{definition}\n A \\emph{Steinhaus} random multiplicative function $f$ is a sequence of random variables $f(1), f(2), \\ldots$ such that $f(mn)=f(m)f(n)$ for all positive integers $m$ and $n$, and for each prime $p$, $f(p)$ is uniformly distributed on the unit circle.\n\nA \\emph{Rademacher} random multiplicative function $f$ is a sequence of random variables $f(1), f(2),$ $\\ldots$ supported on square-free integers such that $f(mn) = f(m)f(n)$ whenever $m$ and $n$ are coprime, and for each prime $p$, $f(p)$ is uniformly distributed on $\\{-1, 1\\}$. \n\\end{definition}\nAlthough random multiplicative functions are useful in modeling deterministic objects such as Dirichlet and Archimedean characters, they have become increasingly interesting to study in their own right. Building on earlier work of Hal\\'asz \\cite{Halasz1977} and Hal\\'asz and R\\'enyi \\cite{HalaszRenyi1961}, partial sums of a random multiplicative function $\\sum_{n\\leq N} f(n)$ have been extensively studied by Harper. Being a sum of random variables, it is natural to ask whether $\\frac{1}{\\sqrt N}\\sum_{n\\leq N} f(n)$ has an approximately Gaussian distribution as $N\\to\\infty$. \nHarper \\cite{Harper2013} showed that this is not the case and in fact, he later showed \\cite{Harper2020_MomentsRandomMultiplicativeI} that the typical size of these sums is of order $\\sqrt N/(\\log\\log N)^{\\frac14}$. One might therefore hope to obtain \nan interesting limiting distribution after normalizing by this factor instead. This is a delicate matter closely related to Gaussian multiplicative chaos, and has been resolved recently in the work of Gorodetsky and Wong \\cite{gorodetsky_wong_2025_limiting} in the Steinhaus case.\n\nFrom an arithmetic point of view, it is interesting to understand sums of a random multiplicative function when the summands are restricted to subsets of positive integers. This will be our focus in this paper. Examples of such restrictions that have been considered previously include: \n\\begin{itemize}\\item[(i)] integers with an atypically small number of prime factors (Hough \\cite{Hough2011}, Harper \\cite{Harper2013}), \n\\item[(ii)] integers in a short interval (Chatterjee and Soundararajan \\cite{ChatterjeeSoundararajan2012}, Soundararajan and Xu \\cite{soundararajan2023clt}, Pandey, Wang, and Xu \\cite{PandeyWangXu2024}), \n\\item[(iii)] sums of two squares in a short interval and shifted primes $p-k$ for $k\\neq 0$ (Soundararajan and Xu \\cite{soundararajan2023clt}), \n\\item[(iv)] polynomial images (Najnudel \\cite{najnudel2020consecutive}, Klurman, Shkredov, and Xu \\cite{KlurmanShkredovXu2023}, Chinis and the author \\cite{ChinisShala2025}).\n\\end{itemize} \nIn each of the aforementioned works, the distribution of the sums turns out to be approximately Gaussian, demonstrating that multiplicativity seems to interfere less with independence when the summands are restricted to a less ``multiplicatively structured'' set. In this paper, we extend the general framework of Soundararajan and Xu \\cite{soundararajan2023clt} on central limit theorems for random multiplicative functions to the setting of large fluctuations.\n\nThere has been considerable work in understanding almost sure large fluctuations for the full sum $\\sum_{n\\leq N} f(n)$ and related variants. Harper \\cite{harper2020large} showed that for any (slowly growing) function $V(N)\\to\\infty$, almost surely there is a sequence of positive integers $N_k\\to\\infty$ such that \n\\begin{equation*}\n \\left\\lvert\\sum_{n\\leq N_k}f(n)\\right\\rvert\\gg\\sqrt{N_k}\\frac{(\\log\\log N_k)^{\\frac14}}{V(N_k)}.\n\\end{equation*}\nThis matches what one expects from the law of iterated logarithm, up to the factor $V(N)$. Proving the corresponding upper bound remains a challenge. Partial results with an upper bound of the right shape (namely $\\sqrt N$ times a power of $\\log\\log N$) have been proved by Lau, Tenenbaum and Wu \\cite{LauTenenbaumWu2012}, as well as most recently by Caich \\cite{Caich2023} with the current best known almost sure bound\n\\begin{equation*}\n \\left\\lvert\\sum_{n\\leq N} f(n)\\right\\rvert\\ll_{\\epsilon} \\sqrt N(\\log\\log N)^{\\frac{3}{4}+\\epsilon}.\n\\end{equation*}\nIn the case when the summands are demanded to have a prime factor bigger than $\\sqrt N$, Mastrostefano \\cite{Mastrostefano2022} proved an essentially sharp almost sure upper bound with the exponent $\\frac{1}{4}+\\epsilon$ in place of $\\frac{3}4$, with the corresponding almost sure lower bound following by work of Harper \\cite{harper2020large}; see also the work of Hardy \\cite{Hardy2025LargePrime} on the distributional aspect in this case. A weighted variant $\\sum_{n\\leq N} \\frac{f(n)}{\\sqrt n}$\nhas been considered by Aymone, Heap and Zhao \\cite{aymone2021partial}, and Hardy \\cite{Hardy2023AlmostSure} in the Steinhaus case who obtained essentially sharp bounds, and Atherfold \\cite{Atherfold2025} in the Rademacher case albeit without sharp bounds.\n\nWe are now ready to state the main results of this paper. Our focus will be on two main examples: (i) polynomial images, and (ii) integers in a short interval. However, our method is rather general, so it should be able to flexibly handle several examples; see Section \\ref{section:generalSetup} for the general but rather technical results.\n\n\\subsection{Polynomial images} This case is particularly interesting due to its connection with conjectures of Chowla and Elliott \\cite{Chowla1965, Elliott1992, Elliott1994} on correlations of multiplicative functions. See the introductions of \\cite{KlurmanShkredovXu2023, ChinisShala2025} for a more detailed discussion in the setting of random multiplicative functions, where central limit theorems were proved.", "full_context": "Since their introduction by Wintner \\cite{Wintner1944}, random multiplicative functions have attracted a lot of attention in number theory. \n\\begin{definition}\n A \\emph{Steinhaus} random multiplicative function $f$ is a sequence of random variables $f(1), f(2), \\ldots$ such that $f(mn)=f(m)f(n)$ for all positive integers $m$ and $n$, and for each prime $p$, $f(p)$ is uniformly distributed on the unit circle.\n\nA \\emph{Rademacher} random multiplicative function $f$ is a sequence of random variables $f(1), f(2),$ $\\ldots$ supported on square-free integers such that $f(mn) = f(m)f(n)$ whenever $m$ and $n$ are coprime, and for each prime $p$, $f(p)$ is uniformly distributed on $\\{-1, 1\\}$. \n\\end{definition}\nAlthough random multiplicative functions are useful in modeling deterministic objects such as Dirichlet and Archimedean characters, they have become increasingly interesting to study in their own right. Building on earlier work of Hal\\'asz \\cite{Halasz1977} and Hal\\'asz and R\\'enyi \\cite{HalaszRenyi1961}, partial sums of a random multiplicative function $\\sum_{n\\leq N} f(n)$ have been extensively studied by Harper. Being a sum of random variables, it is natural to ask whether $\\frac{1}{\\sqrt N}\\sum_{n\\leq N} f(n)$ has an approximately Gaussian distribution as $N\\to\\infty$. \nHarper \\cite{Harper2013} showed that this is not the case and in fact, he later showed \\cite{Harper2020_MomentsRandomMultiplicativeI} that the typical size of these sums is of order $\\sqrt N/(\\log\\log N)^{\\frac14}$. One might therefore hope to obtain \nan interesting limiting distribution after normalizing by this factor instead. This is a delicate matter closely related to Gaussian multiplicative chaos, and has been resolved recently in the work of Gorodetsky and Wong \\cite{gorodetsky_wong_2025_limiting} in the Steinhaus case.\n\nFrom an arithmetic point of view, it is interesting to understand sums of a random multiplicative function when the summands are restricted to subsets of positive integers. This will be our focus in this paper. Examples of such restrictions that have been considered previously include: \n\\begin{itemize}\\item[(i)] integers with an atypically small number of prime factors (Hough \\cite{Hough2011}, Harper \\cite{Harper2013}), \n\\item[(ii)] integers in a short interval (Chatterjee and Soundararajan \\cite{ChatterjeeSoundararajan2012}, Soundararajan and Xu \\cite{soundararajan2023clt}, Pandey, Wang, and Xu \\cite{PandeyWangXu2024}), \n\\item[(iii)] sums of two squares in a short interval and shifted primes $p-k$ for $k\\neq 0$ (Soundararajan and Xu \\cite{soundararajan2023clt}), \n\\item[(iv)] polynomial images (Najnudel \\cite{najnudel2020consecutive}, Klurman, Shkredov, and Xu \\cite{KlurmanShkredovXu2023}, Chinis and the author \\cite{ChinisShala2025}).\n\\end{itemize} \nIn each of the aforementioned works, the distribution of the sums turns out to be approximately Gaussian, demonstrating that multiplicativity seems to interfere less with independence when the summands are restricted to a less ``multiplicatively structured'' set. In this paper, we extend the general framework of Soundararajan and Xu \\cite{soundararajan2023clt} on central limit theorems for random multiplicative functions to the setting of large fluctuations.\n\nThere has been considerable work in understanding almost sure large fluctuations for the full sum $\\sum_{n\\leq N} f(n)$ and related variants. Harper \\cite{harper2020large} showed that for any (slowly growing) function $V(N)\\to\\infty$, almost surely there is a sequence of positive integers $N_k\\to\\infty$ such that \n\\begin{equation*}\n \\left\\lvert\\sum_{n\\leq N_k}f(n)\\right\\rvert\\gg\\sqrt{N_k}\\frac{(\\log\\log N_k)^{\\frac14}}{V(N_k)}.\n\\end{equation*}\nThis matches what one expects from the law of iterated logarithm, up to the factor $V(N)$. Proving the corresponding upper bound remains a challenge. Partial results with an upper bound of the right shape (namely $\\sqrt N$ times a power of $\\log\\log N$) have been proved by Lau, Tenenbaum and Wu \\cite{LauTenenbaumWu2012}, as well as most recently by Caich \\cite{Caich2023} with the current best known almost sure bound\n\\begin{equation*}\n \\left\\lvert\\sum_{n\\leq N} f(n)\\right\\rvert\\ll_{\\epsilon} \\sqrt N(\\log\\log N)^{\\frac{3}{4}+\\epsilon}.\n\\end{equation*}\nIn the case when the summands are demanded to have a prime factor bigger than $\\sqrt N$, Mastrostefano \\cite{Mastrostefano2022} proved an essentially sharp almost sure upper bound with the exponent $\\frac{1}{4}+\\epsilon$ in place of $\\frac{3}4$, with the corresponding almost sure lower bound following by work of Harper \\cite{harper2020large}; see also the work of Hardy \\cite{Hardy2025LargePrime} on the distributional aspect in this case. A weighted variant $\\sum_{n\\leq N} \\frac{f(n)}{\\sqrt n}$\nhas been considered by Aymone, Heap and Zhao \\cite{aymone2021partial}, and Hardy \\cite{Hardy2023AlmostSure} in the Steinhaus case who obtained essentially sharp bounds, and Atherfold \\cite{Atherfold2025} in the Rademacher case albeit without sharp bounds.\n\nWe are now ready to state the main results of this paper. Our focus will be on two main examples: (i) polynomial images, and (ii) integers in a short interval. However, our method is rather general, so it should be able to flexibly handle several examples; see Section \\ref{section:generalSetup} for the general but rather technical results.\n\n\\subsection{Polynomial images} This case is particularly interesting due to its connection with conjectures of Chowla and Elliott \\cite{Chowla1965, Elliott1992, Elliott1994} on correlations of multiplicative functions. See the introductions of \\cite{KlurmanShkredovXu2023, ChinisShala2025} for a more detailed discussion in the setting of random multiplicative functions, where central limit theorems were proved.\n\nThe almost sure upper bound in Theorem \\ref{thm:asboundsPoly} is new, whereas the lower bound is an extension of results of Klurman, Shkredov, and Xu \\cite{KlurmanShkredovXu2023} in the Steinhaus case, and Chinis and the author \\cite{ChinisShala2025} in the Rademacher case, to polynomials that split as a product of linear factors over $\\mathbb Q$. Indeed, the method used in \\cite{KlurmanShkredovXu2023, ChinisShala2025} relied on the fact that for a polynomial $P$ that has an irreducible factor of degree at least two, there are many integers $n$ for which $P(n)$ has a prime factor greater than $n\\log n$.\n\n\\begin{theoremL}\\label{thm:asBoundsShort}\n Let $f$ be a Rademacher or Steinhaus random multiplicative function and suppose that $H=H(N)$ is a smooth, increasing, and concave function, such that $N^{\\frac{11}{15}}\\leq H(N)\\leq N/(\\log N)^c$\n for some constant $c>2\\log 2-1$. \n Then almost surely there exists a sequence $N_k\\to\\infty$ with corresponding $H_k = H(N_k)$ such that \n \\begin{equation*}\n \\left\\lvert\\sum_{N_k-H_k < n \\leq N_k}f(n)\\right\\rvert\\gg \\sqrt{H_k\\log\\frac{N_k}{H_k}}.\n \\end{equation*}\n\\end{theoremL}\n\n\\section{Overview of the proofs}\n\\subsection{Martingale structure} Our starting point is the observation of Harper \\cite{Harper2013} that sums of a random multiplicative function have the structure of a martingale with respect to the filtration induced by the largest prime factor, namely \n\\begin{equation*}\n \\sum_{n\\in\\mathcal A}f(n) = \\sum_{p^{\\alpha}\\leq N}f(p^{\\alpha})\\sum_{\\substack{n\\in\\mathcal A \\\\ P^+(n)=p \\\\ p^{\\alpha}\\parallel n}}f\\left(\\frac{n}{p^{\\alpha}}\\right).\n\\end{equation*}\nThis observation was utilized in almost all subsequent works on random multiplicative functions. For us, its usefulness lies in the fact that it morally reduces the understanding of $\\sum_{n\\in\\mathcal A}f(n)$ to evaluating the second and fourth moments of the sum, provided they do not blow up. More precisely, we will apply the following quantitative result from the book of Hall and Heyde \\cite{HallHeyde1980}, yielding a bound on the Kolmogorov distance between our martingale and a Gaussian. \n\\begin{prop}[\\cite{HallHeyde1980}, Theorem 3.9]\\label{prop:quantCLT}\n Let $X_1, X_2, \\ldots, X_n$ be a real square-integrable martingale difference sequence.\n Let $S_n = \\sum_{i=1}^n X_i$. Then \n \\begin{equation*}\n\\left\\lvert\\mathbb P(S_n\\leq x) - \\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^x e^{-t^2/2}\\emph{d}t\\right\\rvert\\ll \\frac{\\left(\\sum_{i=1}^n \\mathbb E\\lvert X_i\\rvert^4 + \\mathbb E\\left\\lvert\\sum_{i=1}^n X_i^2 - 1\\right\\rvert^2\\right)^{1/5}}{1+\\lvert x\\rvert^{\\frac{16}{5}}}.\n \\end{equation*}\n\\end{prop}\n\\subsection{Almost sure upper bound} Equipped with the above result, we may readily prove an almost sure upper bound for $\\sum_{n\\in\\mathcal A} f(n)$, provided our approximation is suitably strong. In order to access the tails at $x\\approx C\\sqrt{\\log k}$ which occur with probability $\\approx k^{-C^2/2}$ for the Gaussian, we will need an approximation that is at least this good. We are able to achieve this in the case of polynomial images, but not short intervals -- we will discuss this in more detail in the following subsections. Using our quantitative approximation at several spread out scales together with a slow variation property between the scales (see Proposition \\ref{prop:slowVar} and Lemma \\ref{lemma:slowVarPoly}), the Borel-Cantelli lemma establishes the almost sure upper bound.\n\n\\begin{lemma}\\label{lemma:slowVarPoly} Let $f$ be a Rademacher (or Steinhaus) random multiplicative function, and let $P\\in\\mathbb Z[x]$ be a polynomial as in Theorem \\ref{thm:asboundsPoly}. \nFor any constant $A>0$, there exists a constant $c>0$ such that for $N_l = \\lfloor e^{l^c}\\rfloor$, we have that\n \\begin{equation*}\n \\max_{N_{l}<N\\leq N_{l+1}} \\left\\lvert\\sum_{n\\leq N} f(P(n)) - \\sum_{n\\leq N_{l}} f(P(n)) \\right\\rvert \\ll \\frac{\\sqrt{N_{l+1}}}{(\\log N_{l+1})^A}\n \\end{equation*} holds almost surely for every $l\\in\\mathbb N$. \n\\end{lemma}\n\nWe finally prove the almost sure upper bound. By Lemma \\ref{lemma:slowVarPoly} with $A=1$, it suffices to show that almost surely there exists a constant $C>0$ such that for all $l\\in\\mathbb N$ we have\n \\begin{equation}\\label{eq:ubPoly}\n \\left\\lvert\\sum_{n\\leq N_l} f(P(n))\\right\\rvert\\leq C \\sqrt{N_l\\log\\log N_l},\n \\end{equation}\n where $N_l$ is as in the statement of the lemma (in the Steinhaus case, we do this separately for the real and imaginary parts). By Corollary \\ref{cor:quantCLTpoly}, the probability that the above fails for a fixed $l$ and large enough implied constant $C$ is $\\ll (\\log N_l)^{-\\frac{C^2}{2}}$. Making $C$ larger (in terms of $c$) if necessary, we have that the series of probabilities $$\\sum_{l=1}^\\infty (\\log N_l)^{-\\frac{C^2}{2}} = \\sum_{l=1}^\\infty \\frac{1}{l^{\\frac{cC^2}{2}}}<\\infty,$$ thus \\eqref{eq:ubPoly} holds almost surely by Borel-Cantelli (with a possibly even larger constant $C$ to account for the finitely many failures).\n\n\\begin{theorem}\\label{thm:localizedBoundPoly}\n Let $f$ be a Rademacher (or Steinhaus) random multiplicative function and let $P\\in\\mathbb Z[x]$ be a polynomial as in Theorem \\ref{thm:asboundsPoly}. There exists a constant $c>0$ such that for any large enough $X$, we have that \n \\begin{equation*}\n \\max_{X<N\\leq X^2} \\frac{1}{\\sqrt N} \\left\\lvert\\sum_{n\\leq N}f(P(n)) \\right\\rvert \\gg \\sqrt{\\log\\log X}\n \\end{equation*} holds with a suitably small implied constant with probability $1-O(\\exp(-(\\log X)^{c}))$. \n\\end{theorem}\n\n\\section{Proof of Theorem \\ref{thm:asBoundsShort}: Short intervals} Here we will require the more sophisticated part of Corollary \\ref{cor:normalcomparison}. Let $H$ be as in the statement of Theorem \\ref{thm:asBoundsShort} and consider $\\mathcal A_N = \\{n\\in\\mathbb N : N-H<n\\leq N\\}$. Note that $\\lvert\\mathcal A_N\\rvert\\sim \\kappa H$ for some\\footnote{In the Steinhaus case we have $\\kappa = 1$, whereas in the Rademacher case $\\kappa = \\frac{6}{\\pi^2}$.} $\\kappa>0$. Recall \n\\begin{equation*}\n S_N = \\frac{1}{\\sqrt{\\lvert\\mathcal A_N\\rvert}}\\sum_{n\\in\\mathcal A_N} \\mathfrak Tf(n)\\sim \\frac{1}{\\sqrt{\\kappa H}}\\sum_{N-H<n\\leq n }\\mathfrak Tf(n).\n\\end{equation*}\nMoreover, let $X$ be a large parameter. For some small $\\delta>0$ depending on the function $H$ and then even smaller $\\varepsilon_0>0$, find $k= (X/H(X))^{\\varepsilon_0}$ primes $l$ in the range $h(X)/2<l\\leq h(X)$, where $h(X) := (X/H(X))^{\\delta}$. We will not fix $\\delta$ or $\\varepsilon_0$ for now, as there will be various (but finitely many) places where we may have make them smaller. For these values of $l$, let $N_l = lX$ with corresponding $H_l = H(N_l)$. \nObserve that for large enough $X$ and small enough $\\delta>0$, the sets $\\mathcal A_{N_l}$ are disjoint. It is not these sets that we will apply Theorem \\ref{thm: complicatedapproxJointG} to. If we did this, the very smooth integers in the sets $\\mathcal A_{N_l}$ would prevent us from obtaining a suitably small value for $\\varepsilon_2$. We bypass this by splitting $S_{N_l}$ as \n\\begin{equation}\\label{eq:splitting1}\n \\sum_{N_l - H_l<n\\leq N_l}f(n) = \\sum_{(Xh(X))^{\\frac{2}{3}}< p}f(p)\\sum_{\\substack{N_l-H_l < n\\leq N_l \\\\ p\\mid n}}f\\left(\\frac np\\right) + \\sum_{\\substack{N_l-H_l<n\\leq N_l \\\\ P^+(n)\\leq (Xh(X))^{\\frac{2}{3}}}}f(n).\n\\end{equation}\nIf $H_1> N_1/(\\log N_1)^2$ (hence since $H$ is increasing we have $H_l \\gg Xh(X)/(\\log X)^2$ for all values of $l$), we choose some $\\varepsilon>0$ to be specified later and additionally split the first part as \n\\begin{equation}\\label{eq:splitting2}\n \\sum_{N_l - H_l<n\\leq N_l}f(n) = \\sum_{(Xh(X))^{\\frac{2}{3}}< p }f(p)\\sum_{\\substack{N_l-H_l < n\\leq N_l \\\\ p\\mid n \\\\ \\Omega(n)\\leq (1+\\varepsilon)\\log\\log X}}f\\left(\\frac np\\right) \\\\ + \\sum_{\\substack{N_l-H_l<n\\leq N_l \\\\ P^+(n)\\leq (Xh(X))^{\\frac{2}{3}} \\\\ \\text{or } \\Omega(n)> (1+\\varepsilon)\\log\\log X }}f(n).\n\\end{equation}\nWe first show that the sum over integers with $\\Omega(n)> (1+\\varepsilon)\\log\\log X$ above may be typically ignored for all values of $l$.", "post_theorem_intro_text_len": 6196, "post_theorem_intro_text": "The almost sure upper bound in Theorem \\ref{thm:asboundsPoly} is new, whereas the lower bound is an extension of results of Klurman, Shkredov, and Xu \\cite{KlurmanShkredovXu2023} in the Steinhaus case, and Chinis and the author \\cite{ChinisShala2025} in the Rademacher case, to polynomials that split as a product of linear factors over $\\mathbb Q$. Indeed, the method used in \\cite{KlurmanShkredovXu2023, ChinisShala2025} relied on the fact that for a polynomial $P$ that has an irreducible factor of degree at least two, there are many integers $n$ for which $P(n)$ has a prime factor greater than $n\\log n$. \n\nTo demonstrate how this was used, let us consider $P(n) = n^2+1$. By the aforementioned fact, one can ``pull out'' a unique large prime factor for such $n$ and still capture a good ``bulk'' of the sum for infinitely many (sparse) $N$, that is \n\\begin{equation*}\n    \\sum_{n\\leq N}f(n^2+1)\\approx \\sum_{N\\log N < p\\leq N^2}f(p)\\sum_{\\substack{n\\leq N \\\\ p\\mid n^2+1}}f\\left(\\frac{n^2+1}{p}\\right).\n\\end{equation*}\n\nUpon conditioning on the small primes, this is a weighted sum of independent random variables, which is then handled using intricate combinatorial arguments and Gaussian approximation. \n\nIndeed, the method described above cannot be directly applied to the case when $P$ is a product of linear factors, for all its prime factors are of size at most linear in $N$.\nIn this paper, we will use a different method that does not involve any combinatorial splitting of the sum or conditioning. We will rather fully utilize the martingale structure of the entire sum -- we will elaborate on this in the next section. In particular, provided enough number-theoretic input (see the work of Martin \\cite{Martin2002}), our method would also work to prove almost sure large fluctuations of $N$-smooth\\footnote{An integer is called $y$-smooth if all of the prime factors dividing it are at most $y$.} values of irreducible polynomials.\n\n\\subsection{Integers in a short interval} A central limit theorem for sums of a random multiplicative function over integers in a short interval $(N-H, N]$ was first proved by Chatterjeee and Soundararajan \\cite{ChatterjeeSoundararajan2012} when $H=o\\left(\\frac{N}{\\log N}\\right)$, using essentially only second and fourth moment information. This was later extended to $H\\leq \\frac{N}{(\\log N)^{c}}$ with $c>2\\log2-1$ by Soundararajan and Xu \\cite{soundararajan2023clt}, entering a regime of $H$ where naively the fourth moment of the sum blows up. However, this is bypassed by throwing out $0\\%$ of the integers in the short interval with atypically many prime factors, resulting in a controlled fourth moment. \n\nWe prove an almost sure lower bound in the same regime. \n\n\\begin{theoremL}\\label{thm:asBoundsShort}\n    Let $f$ be a Rademacher or Steinhaus random multiplicative function and suppose that $H=H(N)$ is a smooth, increasing, and concave function, such that $N^{\\frac{11}{15}}\\leq H(N)\\leq N/(\\log N)^c$\n    for some constant $c>2\\log 2-1$. \n    Then almost surely there exists a sequence $N_k\\to\\infty$ with corresponding $H_k = H(N_k)$ such that \n    \\begin{equation*}\n        \\left\\lvert\\sum_{N_k-H_k < n \\leq N_k}f(n)\\right\\rvert\\gg \\sqrt{H_k\\log\\frac{N_k}{H_k}}.\n    \\end{equation*}\n\\end{theoremL}\n\n\\begin{remark}\n    The fluctuations of order $\\sqrt{\\log\\frac{N}{H}}$ may seem surprisingly large at first, especially in a naive comparison with the law of iterated logarithm. However, this is explained by the fact that short sums of length $H$ decorrelate much faster, and indeed roughly $N/H$ essentially independent sums fit in a ``window'' of size $N$. In agreement with the law of iterated logarithm, the maximum of these sums has size roughly $\\sqrt{\\log\\frac{N}{H}}$. \n\\end{remark}\n\n\\begin{remark}\n    The conditions on $H$ are imposed simply for convenience -- any standard choices such as $H=N^{\\alpha}$ for $\\alpha < 1$, $H = N/\\exp\\left(\\sqrt{\\log N}\\right)$, or $H = N/(\\log N)^A$ for some constant $A>2\\log 2-1$, satisfy these conditions (for sufficiently large $N$). \n\n    We have not attempted to optimize the exponent $\\frac{11}{15}$ -- it is likely possible to bring it down to $\\frac{3}{5}$. However, the lower bound on $H$ should not be essential: for very small $H\\ll \\log N$ the large fluctuations are captured by almost sure long runs of ones, of length $\\approx \\log N$ (see the work of Erdős and Rényi \\cite{ErdosRenyi1970}), whereas if $\\log N\\ll H\\ll N^{\\frac{11}{15}}$, our method should work (with significant simplifications when $H\\ll N^{\\frac{1}{2}}$), but the restriction stems from a result on square-free smooth integers in short intervals; see Lemma \\ref{lemma:unSmoothSquarefree}. Since we are only proving a lower bound and therefore can pick convenient scales to work with, it is likely possible to circumvent this using the Matom\\\"aki--Radziwiłł machinery; see the work of Jain \\cite{jain2025smooth} and the references within. However, we have opted for simplicity, especially since our result captures the most interesting regime where there is a transition of the large fluctuations from $\\sqrt{\\log N}$ down to $\\sqrt{\\log \\log N}$.\\footnote{We thank Sarvagya Jain for discussions pertaining to this remark.}\n\\end{remark}\n\nIt would be interesting to prove the corresponding almost sure upper bound in any regime of $H$. For rather large $H$, it is likely that the sophisticated ideas in \\cite{LauTenenbaumWu2012, Mastrostefano2022, Hardy2023AlmostSure, Atherfold2025} relating the short sum $\\sum_{N-H<n\\leq N} f(n)$ to a random Euler product could be useful. It is unclear, however, whether this would result in a sharp (or almost sharp) upper bound in any regime of $H$ -- this is an ongoing investigation. \n\nUpcoming work of Harper, Soundararajan, and Xu \nestablishes a central limit theorem in the range $H = o(N)$, although with a normalization of the sum that differs from just the standard deviation. It would be interesting to determine whether our method would succeed to prove almost sure lower bounds of the right order of magnitude in this extended range. See also the work of Caich \\cite{Caich2024randomshort} for more on the intricate behavior of short sums in this regime.", "sketch": "The introduction explains that prior lower-bound methods (e.g. \\cite{KlurmanShkredovXu2023, ChinisShala2025}) for handling \\(\\sum_{n\\le N} f(P(n))\\) when \\(P\\) has an irreducible factor of degree \\(\\ge 2\\) use the fact that for many \\(n\\), \\(P(n)\\) has a prime factor \\(> n\\log n\\). For example, with \\(P(n)=n^2+1\\), one can “pull out” a unique large prime factor and still capture a “bulk” of the sum along infinitely many (sparse) \\(N\\):\n\\[\n\\sum_{n\\le N} f(n^2+1)\\approx \\sum_{N\\log N < p\\le N^2} f(p)\\sum_{\\substack{n\\le N\\\\ p\\mid n^2+1}} f\\!\\left(\\frac{n^2+1}{p}\\right).\n\\]\nAfter “conditioning on the small primes”, this becomes “a weighted sum of independent random variables”, which is then treated via “intricate combinatorial arguments and Gaussian approximation.”\n\nIt then notes this approach “cannot be directly applied” when \\(P\\) is a product of linear factors, since then “all its prime factors are of size at most linear in \\(N\\).” For Theorem~\\ref{thm:asboundsPoly}, the paper instead uses “a different method that does not involve any combinatorial splitting of the sum or conditioning” and aims to “fully utilize the martingale structure of the entire sum,” with “enough number-theoretic input (see the work of Martin \\cite{Martin2002}).”", "expanded_sketch": "The introduction explains that prior lower-bound methods (e.g. J. Klurman, I. D. Shkredov, and J. Xu, *Title not found in ref_dict* (2023), and K. Chinis and A. Shala, *Title not found in ref_dict* (2025)) for handling \\(\\sum_{n\\le N} f(P(n))\\) when \\(P\\) has an irreducible factor of degree \\(\\ge 2\\) use the fact that for many \\(n\\), \\(P(n)\\) has a prime factor \\(> n\\log n\\). For example, with \\(P(n)=n^2+1\\), one can “pull out” a unique large prime factor and still capture a “bulk” of the sum along infinitely many (sparse) \\(N\\):\n\\[\n\\sum_{n\\le N} f(n^2+1)\\approx \\sum_{N\\log N < p\\le N^2} f(p)\\sum_{\\substack{n\\le N\\\\ p\\mid n^2+1}} f\\!\\left(\\frac{n^2+1}{p}\\right).\n\\]\nAfter “conditioning on the small primes”, this becomes “a weighted sum of independent random variables”, which is then treated via “intricate combinatorial arguments and Gaussian approximation.”\n\nIt then notes this approach “cannot be directly applied” when \\(P\\) is a product of linear factors, since then “all its prime factors are of size at most linear in \\(N\\).” In establishing the main theorem, the paper instead uses “a different method that does not involve any combinatorial splitting of the sum or conditioning” and aims to “fully utilize the martingale structure of the entire sum,” with “enough number-theoretic input (see the work of Martin C. A. Martin, *Title not found in ref_dict* (2002)).”,", "expanded_theorem": "\\label{thm:asboundsPoly}\n    Let $f$ be a  Rademacher or Steinhaus random multiplicative function. In the Rademacher case, assume $P\\in\\mathbb Z[x]$ is a product of at least two distinct linear factors over $\\mathbb Q$, or irreducible of degree $2$, taking infinitely many square-free values. In the Steinhaus case, assume $P(x)$ is not of the form $w(x+c)^d$ for $w\\in\\mathbb Z, c\\in\\mathbb Q$. Then almost surely we have $$\\left\\lvert\\sum_{n\\leq N} f(P(n))\\right\\rvert \\ll \\sqrt{N\\log\\log N}.$$ Moreover, almost surely there exists a sequence $N_k\\to\\infty$ such that \n    \\begin{equation*}\n        \\left\\lvert\\sum_{n\\leq N_k} f(P(n))\\right\\rvert\\gg \\sqrt{N_k\\log\\log N_k}.\n    \\end{equation*},", "theorem_type": ["Inequality or Bound", "Asymptotic or Limit"], "mcq": {"question": "Let $f$ be either a Steinhaus or a Rademacher random multiplicative function. Here, a Steinhaus random multiplicative function satisfies $f(mn)=f(m)f(n)$ for all positive integers $m,n$, and for each prime $p$, the random variable $f(p)$ is uniformly distributed on the unit circle. A Rademacher random multiplicative function is supported on square-free integers, satisfies $f(mn)=f(m)f(n)$ whenever $(m,n)=1$, and for each prime $p$, $f(p)$ is uniformly distributed on $\\{-1,1\\}$. Let $P\\in \\mathbb Z[x]$. In the Rademacher case, assume that $P$ is either a product of at least two distinct linear factors over $\\mathbb Q$, or an irreducible polynomial of degree $2$, and that $P$ takes infinitely many square-free values. In the Steinhaus case, assume that $P(x)$ is not of the form $w(x+c)^d$ with $w\\in \\mathbb Z$ and $c\\in \\mathbb Q$. Under these assumptions, which quantitative estimate holds for the polynomially sampled sums $\\sum_{n\\le N} f(P(n))$?", "correct_choice": {"label": "A", "text": "Almost surely, one has\n$$\\left|\\sum_{n\\le N} f(P(n))\\right| \\ll \\sqrt{N\\log\\log N}.$$ \nMoreover, almost surely there exists a sequence $N_k\\to\\infty$ such that\n$$\\left|\\sum_{n\\le N_k} f(P(n))\\right| \\gg \\sqrt{N_k\\log\\log N_k}.$$"}, "choices": [{"label": "B", "text": "Almost surely, one has\n$$\\left|\\sum_{n\\le N} f(P(n))\\right| \\ll \\sqrt N\\,(\\log\\log N)^{\\frac14}.$$ \nMoreover, almost surely there exists a sequence $N_k\\to\\infty$ such that\n$$\\left|\\sum_{n\\le N_k} f(P(n))\\right| \\gg \\sqrt{N_k}\\,(\\log\\log N_k)^{\\frac14}.$$"}, {"label": "C", "text": "Almost surely, one has\n$$\\left|\\sum_{n\\le N} f(P(n))\\right| \\ll \\sqrt{N\\log\\log N}.$$"}, {"label": "D", "text": "For every sequence $N_k\\to\\infty$, almost surely one has\n$$\\left|\\sum_{n\\le N_k} f(P(n))\\right| \\gg \\sqrt{N_k\\log\\log N_k}$$\nfor all sufficiently large $k$, and in particular\n$$\\left|\\sum_{n\\le N} f(P(n))\\right| \\ll \\sqrt{N\\log\\log N}.$$"}, {"label": "E", "text": "Almost surely, one has\n$$\\left|\\sum_{n\\le N} f(P(n))\\right| \\ll \\sqrt{N\\log N}.$$ \nMoreover, almost surely there exists a sequence $N_k\\to\\infty$ such that\n$$\\left|\\sum_{n\\le N_k} f(P(n))\\right| \\gg \\sqrt{N_k\\log N_k}.$$"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "loglog_exponent_size", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "existence_of_matching_lower_bound_subsequence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "existential_subsequence_vs_uniform_eventual_lower_bound", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "correct_loglog_scale_from_martingale_method", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives hypotheses and asks for the resulting asymptotic, but it does not explicitly reveal the correct conclusion or uniquely cue choice A. The answer is not leaked beyond the general theorem setup."}, "TAS": {"score": 1, "justification": "This is essentially a theorem-recall question: the stem states a detailed set of assumptions and asks for the exact asymptotic conclusion. The options do vary in meaningful ways, so it is not a pure verbatim restatement, but it is still close to one."}, "GPS": {"score": 1, "justification": "There is some pressure to distinguish between nearby asymptotic scales and formulations, but the task mainly tests precise recall/comparison rather than genuine derivation. Moreover, because one distractor is a weaker true consequence, the item does not cleanly force identification of a unique strongest conclusion."}, "DQS": {"score": 1, "justification": "Several distractors are plausibly constructed from common theorem-distortions (wrong log-log scale, overly strong upper bound, missing absolute value). However, choice C appears to be a weaker statement implied by A, so the distractor set is not fully clean for a single-answer MCQ."}, "total_score": 5, "overall_assessment": "A technically precise but theorem-recall-heavy item with no major answer leakage. Its main weakness is ambiguity from a weaker true distractor, which reduces both generative pressure and distractor quality."}}
{"id": "2602.20086v2", "paper_link": "http://arxiv.org/abs/2602.20086v2", "theorems_cnt": 3, "theorem": {"env_name": "theoremL", "content": "\\label{thm:asboundsPoly}\n    Let $f$ be a  Rademacher or Steinhaus random multiplicative function. In the Rademacher case, assume $P\\in\\mathbb Z[x]$ is a product of at least two distinct linear factors over $\\mathbb Q$, or irreducible of degree $2$, taking infinitely many square-free values. In the Steinhaus case, assume $P(x)$ is not of the form $w(x+c)^d$ for $w\\in\\mathbb Z, c\\in\\mathbb Q$. Then almost surely we have $$\\left\\lvert\\sum_{n\\leq N} f(P(n))\\right\\rvert \\ll \\sqrt{N\\log\\log N}.$$ Moreover, almost surely there exists a sequence $N_k\\to\\infty$ such that \n    \\begin{equation*}\n        \\left\\lvert\\sum_{n\\leq N_k} f(P(n))\\right\\rvert\\gg \\sqrt{N_k\\log\\log N_k}.\n    \\end{equation*}", "start_pos": 8546, "end_pos": 9272, "label": "thm:asboundsPoly"}, "ref_dict": {"thm:asboundsPoly": "\\begin{theoremL}\\label{thm:asboundsPoly}\n    Let $f$ be a  Rademacher or Steinhaus random multiplicative function. In the Rademacher case, assume $P\\in\\mathbb Z[x]$ is a product of at least two distinct linear factors over $\\mathbb Q$, or irreducible of degree $2$, taking infinitely many square-free values. In the Steinhaus case, assume $P(x)$ is not of the form $w(x+c)^d$ for $w\\in\\mathbb Z, c\\in\\mathbb Q$. Then almost surely we have $$\\left\\lvert\\sum_{n\\leq N} f(P(n))\\right\\rvert \\ll \\sqrt{N\\log\\log N}.$$ Moreover, almost surely there exists a sequence $N_k\\to\\infty$ such that \n    \\begin{equation*}\n        \\left\\lvert\\sum_{n\\leq N_k} f(P(n))\\right\\rvert\\gg \\sqrt{N_k\\log\\log N_k}.\n    \\end{equation*}\n\\end{theoremL}", "lemma:unSmoothSquarefree": "\\begin{lemma}\\label{lemma:unSmoothSquarefree}\n    Let $N$ be large, $\\alpha>\\frac{1}{2}$, and let $N\\geq H=H(N)\\geq N^{\\frac{4\\alpha + 1}{5}}$. The number of square-free integers in the short interval $(N-H, N]$ such that $P^+(n)>N^{\\alpha}$ is \n    \\begin{equation*}\n        \\gg H+O\\left(\\frac{H}{\\log N} \\right).\n    \\end{equation*}\n\\end{lemma}"}, "pre_theorem_intro_text_len": 6200, "pre_theorem_intro_text": "Since their introduction by Wintner \\cite{Wintner1944}, random multiplicative functions have attracted a lot of attention in number theory. \n\\begin{definition}\n    A \\emph{Steinhaus} random multiplicative function $f$ is a sequence of random variables $f(1), f(2), \\ldots$ such that $f(mn)=f(m)f(n)$ for all positive integers $m$ and $n$, and for each prime $p$, $f(p)$ is uniformly distributed on the unit circle.\n\n    A \\emph{Rademacher} random multiplicative function $f$ is a sequence of random variables $f(1), f(2),$ $\\ldots$  supported on square-free integers such that $f(mn) = f(m)f(n)$ whenever $m$ and $n$ are coprime, and for each prime $p$, $f(p)$ is uniformly distributed on $\\{-1, 1\\}$. \n\\end{definition}\nAlthough random multiplicative functions are useful in modeling deterministic objects such as Dirichlet and Archimedean characters, they have become increasingly interesting to study in their own right. Building on earlier work of Hal\\'asz \\cite{Halasz1977} and Hal\\'asz and R\\'enyi \\cite{HalaszRenyi1961}, partial sums of a random multiplicative function $\\sum_{n\\leq N} f(n)$ have been extensively studied by Harper. Being a sum of random variables, it is natural to ask whether $\\frac{1}{\\sqrt N}\\sum_{n\\leq N} f(n)$ has an approximately Gaussian distribution as $N\\to\\infty$. \nHarper \\cite{Harper2013} showed that this is not the case and in fact, he later showed \\cite{Harper2020_MomentsRandomMultiplicativeI} that the typical size of these sums is of order $\\sqrt N/(\\log\\log N)^{\\frac14}$. One might therefore hope to obtain \nan interesting limiting distribution after normalizing by this factor instead. This is a delicate matter closely related to Gaussian multiplicative chaos, and has been resolved recently in the work of Gorodetsky and Wong \\cite{gorodetsky_wong_2025_limiting} in the Steinhaus case. \n\nFrom an arithmetic point of view, it is interesting to understand sums of a random multiplicative function when the summands are restricted to subsets of positive integers. This will be our focus in this paper. Examples of such restrictions that have been considered previously include: \n\\begin{itemize}\\item[(i)] integers with an atypically small number of prime factors (Hough \\cite{Hough2011}, Harper \\cite{Harper2013}), \n\\item[(ii)] integers in a short interval (Chatterjee and Soundararajan \\cite{ChatterjeeSoundararajan2012}, Soundararajan and Xu \\cite{soundararajan2023clt}, Pandey, Wang, and Xu \\cite{PandeyWangXu2024}), \n\\item[(iii)] sums of two squares in a short interval and shifted primes $p-k$ for $k\\neq 0$ (Soundararajan and Xu \\cite{soundararajan2023clt}), \n\\item[(iv)] polynomial images (Najnudel \\cite{najnudel2020consecutive}, Klurman, Shkredov, and Xu \\cite{KlurmanShkredovXu2023}, Chinis and the author \\cite{ChinisShala2025}).\n\\end{itemize} \nIn each of the aforementioned works, the distribution of the sums turns out to be approximately Gaussian, demonstrating that multiplicativity seems to interfere less with independence when the summands are restricted to a less ``multiplicatively structured'' set. In this paper, we extend the general framework of Soundararajan and Xu \\cite{soundararajan2023clt} on central limit theorems for random multiplicative functions to the setting of large fluctuations.\n\nThere has been considerable work in understanding almost sure large fluctuations for the full sum $\\sum_{n\\leq N} f(n)$ and related variants. Harper \\cite{harper2020large} showed that for any (slowly growing) function $V(N)\\to\\infty$, almost surely there is a sequence of positive integers $N_k\\to\\infty$ such that \n\\begin{equation*}\n    \\left\\lvert\\sum_{n\\leq N_k}f(n)\\right\\rvert\\gg\\sqrt{N_k}\\frac{(\\log\\log N_k)^{\\frac14}}{V(N_k)}.\n\\end{equation*}\nThis matches what one expects from the law of iterated logarithm, up to the factor $V(N)$. Proving the corresponding upper bound remains a challenge. Partial results with an upper bound of the right shape (namely $\\sqrt N$ times a power of $\\log\\log N$) have been proved by Lau, Tenenbaum and Wu \\cite{LauTenenbaumWu2012}, as well as most recently by Caich \\cite{Caich2023} with the current best known almost sure bound\n\\begin{equation*}\n    \\left\\lvert\\sum_{n\\leq N} f(n)\\right\\rvert\\ll_{\\epsilon} \\sqrt N(\\log\\log N)^{\\frac{3}{4}+\\epsilon}.\n\\end{equation*}\nIn the case when the summands are demanded to have a prime factor bigger than $\\sqrt N$, Mastrostefano \\cite{Mastrostefano2022} proved an essentially sharp almost sure upper bound with the exponent $\\frac{1}{4}+\\epsilon$ in place of $\\frac{3}4$, with the corresponding almost sure lower bound following by work of Harper \\cite{harper2020large}; see also the work of Hardy \\cite{Hardy2025LargePrime} on the distributional aspect in this case. A weighted variant $\\sum_{n\\leq N} \\frac{f(n)}{\\sqrt n}$\nhas been considered by Aymone, Heap and Zhao \\cite{aymone2021partial}, and Hardy \\cite{Hardy2023AlmostSure} in the Steinhaus case who obtained essentially sharp bounds, and Atherfold \\cite{Atherfold2025} in the Rademacher case albeit without sharp bounds.\n\nRecently, Hoban, Shah, Ismail, Verreault, and Zaman \\cite{HobanShahIsmailVerreaultZaman2025} have extended the framework of Soundararajan and Xu \\cite{soundararajan2023clt} in a different direction, namely in the function field setting. It is likely that the methods of this paper, combined with the work in \\cite{HobanShahIsmailVerreaultZaman2025}, can be used to prove analogous results over function fields.\n\nWe are now ready to state the main results of this paper. Our focus will be on two main examples: (i) polynomial images, and (ii) integers in a short interval. However, our method is rather general, so it should be able to flexibly handle several examples; see Section \\ref{section:generalSetup} for the general but rather technical results. \n\n\\subsection{Polynomial images} This case is particularly interesting due to its connection with conjectures of Chowla and Elliott \\cite{Chowla1965, Elliott1992, Elliott1994} on correlations of multiplicative functions. See the introductions of \\cite{KlurmanShkredovXu2023, ChinisShala2025} for a more detailed discussion in the setting of random multiplicative functions, where central limit theorems were proved.", "context": "Since their introduction by Wintner \\cite{Wintner1944}, random multiplicative functions have attracted a lot of attention in number theory. \n\\begin{definition}\n A \\emph{Steinhaus} random multiplicative function $f$ is a sequence of random variables $f(1), f(2), \\ldots$ such that $f(mn)=f(m)f(n)$ for all positive integers $m$ and $n$, and for each prime $p$, $f(p)$ is uniformly distributed on the unit circle.\n\nA \\emph{Rademacher} random multiplicative function $f$ is a sequence of random variables $f(1), f(2),$ $\\ldots$ supported on square-free integers such that $f(mn) = f(m)f(n)$ whenever $m$ and $n$ are coprime, and for each prime $p$, $f(p)$ is uniformly distributed on $\\{-1, 1\\}$. \n\\end{definition}\nAlthough random multiplicative functions are useful in modeling deterministic objects such as Dirichlet and Archimedean characters, they have become increasingly interesting to study in their own right. Building on earlier work of Hal\\'asz \\cite{Halasz1977} and Hal\\'asz and R\\'enyi \\cite{HalaszRenyi1961}, partial sums of a random multiplicative function $\\sum_{n\\leq N} f(n)$ have been extensively studied by Harper. Being a sum of random variables, it is natural to ask whether $\\frac{1}{\\sqrt N}\\sum_{n\\leq N} f(n)$ has an approximately Gaussian distribution as $N\\to\\infty$. \nHarper \\cite{Harper2013} showed that this is not the case and in fact, he later showed \\cite{Harper2020_MomentsRandomMultiplicativeI} that the typical size of these sums is of order $\\sqrt N/(\\log\\log N)^{\\frac14}$. One might therefore hope to obtain \nan interesting limiting distribution after normalizing by this factor instead. This is a delicate matter closely related to Gaussian multiplicative chaos, and has been resolved recently in the work of Gorodetsky and Wong \\cite{gorodetsky_wong_2025_limiting} in the Steinhaus case.\n\nFrom an arithmetic point of view, it is interesting to understand sums of a random multiplicative function when the summands are restricted to subsets of positive integers. This will be our focus in this paper. Examples of such restrictions that have been considered previously include: \n\\begin{itemize}\\item[(i)] integers with an atypically small number of prime factors (Hough \\cite{Hough2011}, Harper \\cite{Harper2013}), \n\\item[(ii)] integers in a short interval (Chatterjee and Soundararajan \\cite{ChatterjeeSoundararajan2012}, Soundararajan and Xu \\cite{soundararajan2023clt}, Pandey, Wang, and Xu \\cite{PandeyWangXu2024}), \n\\item[(iii)] sums of two squares in a short interval and shifted primes $p-k$ for $k\\neq 0$ (Soundararajan and Xu \\cite{soundararajan2023clt}), \n\\item[(iv)] polynomial images (Najnudel \\cite{najnudel2020consecutive}, Klurman, Shkredov, and Xu \\cite{KlurmanShkredovXu2023}, Chinis and the author \\cite{ChinisShala2025}).\n\\end{itemize} \nIn each of the aforementioned works, the distribution of the sums turns out to be approximately Gaussian, demonstrating that multiplicativity seems to interfere less with independence when the summands are restricted to a less ``multiplicatively structured'' set. In this paper, we extend the general framework of Soundararajan and Xu \\cite{soundararajan2023clt} on central limit theorems for random multiplicative functions to the setting of large fluctuations.\n\nThere has been considerable work in understanding almost sure large fluctuations for the full sum $\\sum_{n\\leq N} f(n)$ and related variants. Harper \\cite{harper2020large} showed that for any (slowly growing) function $V(N)\\to\\infty$, almost surely there is a sequence of positive integers $N_k\\to\\infty$ such that \n\\begin{equation*}\n \\left\\lvert\\sum_{n\\leq N_k}f(n)\\right\\rvert\\gg\\sqrt{N_k}\\frac{(\\log\\log N_k)^{\\frac14}}{V(N_k)}.\n\\end{equation*}\nThis matches what one expects from the law of iterated logarithm, up to the factor $V(N)$. Proving the corresponding upper bound remains a challenge. Partial results with an upper bound of the right shape (namely $\\sqrt N$ times a power of $\\log\\log N$) have been proved by Lau, Tenenbaum and Wu \\cite{LauTenenbaumWu2012}, as well as most recently by Caich \\cite{Caich2023} with the current best known almost sure bound\n\\begin{equation*}\n \\left\\lvert\\sum_{n\\leq N} f(n)\\right\\rvert\\ll_{\\epsilon} \\sqrt N(\\log\\log N)^{\\frac{3}{4}+\\epsilon}.\n\\end{equation*}\nIn the case when the summands are demanded to have a prime factor bigger than $\\sqrt N$, Mastrostefano \\cite{Mastrostefano2022} proved an essentially sharp almost sure upper bound with the exponent $\\frac{1}{4}+\\epsilon$ in place of $\\frac{3}4$, with the corresponding almost sure lower bound following by work of Harper \\cite{harper2020large}; see also the work of Hardy \\cite{Hardy2025LargePrime} on the distributional aspect in this case. A weighted variant $\\sum_{n\\leq N} \\frac{f(n)}{\\sqrt n}$\nhas been considered by Aymone, Heap and Zhao \\cite{aymone2021partial}, and Hardy \\cite{Hardy2023AlmostSure} in the Steinhaus case who obtained essentially sharp bounds, and Atherfold \\cite{Atherfold2025} in the Rademacher case albeit without sharp bounds.\n\nWe are now ready to state the main results of this paper. Our focus will be on two main examples: (i) polynomial images, and (ii) integers in a short interval. However, our method is rather general, so it should be able to flexibly handle several examples; see Section \\ref{section:generalSetup} for the general but rather technical results.\n\n\\subsection{Polynomial images} This case is particularly interesting due to its connection with conjectures of Chowla and Elliott \\cite{Chowla1965, Elliott1992, Elliott1994} on correlations of multiplicative functions. See the introductions of \\cite{KlurmanShkredovXu2023, ChinisShala2025} for a more detailed discussion in the setting of random multiplicative functions, where central limit theorems were proved.", "full_context": "Since their introduction by Wintner \\cite{Wintner1944}, random multiplicative functions have attracted a lot of attention in number theory. \n\\begin{definition}\n A \\emph{Steinhaus} random multiplicative function $f$ is a sequence of random variables $f(1), f(2), \\ldots$ such that $f(mn)=f(m)f(n)$ for all positive integers $m$ and $n$, and for each prime $p$, $f(p)$ is uniformly distributed on the unit circle.\n\nA \\emph{Rademacher} random multiplicative function $f$ is a sequence of random variables $f(1), f(2),$ $\\ldots$ supported on square-free integers such that $f(mn) = f(m)f(n)$ whenever $m$ and $n$ are coprime, and for each prime $p$, $f(p)$ is uniformly distributed on $\\{-1, 1\\}$. \n\\end{definition}\nAlthough random multiplicative functions are useful in modeling deterministic objects such as Dirichlet and Archimedean characters, they have become increasingly interesting to study in their own right. Building on earlier work of Hal\\'asz \\cite{Halasz1977} and Hal\\'asz and R\\'enyi \\cite{HalaszRenyi1961}, partial sums of a random multiplicative function $\\sum_{n\\leq N} f(n)$ have been extensively studied by Harper. Being a sum of random variables, it is natural to ask whether $\\frac{1}{\\sqrt N}\\sum_{n\\leq N} f(n)$ has an approximately Gaussian distribution as $N\\to\\infty$. \nHarper \\cite{Harper2013} showed that this is not the case and in fact, he later showed \\cite{Harper2020_MomentsRandomMultiplicativeI} that the typical size of these sums is of order $\\sqrt N/(\\log\\log N)^{\\frac14}$. One might therefore hope to obtain \nan interesting limiting distribution after normalizing by this factor instead. This is a delicate matter closely related to Gaussian multiplicative chaos, and has been resolved recently in the work of Gorodetsky and Wong \\cite{gorodetsky_wong_2025_limiting} in the Steinhaus case.\n\nFrom an arithmetic point of view, it is interesting to understand sums of a random multiplicative function when the summands are restricted to subsets of positive integers. This will be our focus in this paper. Examples of such restrictions that have been considered previously include: \n\\begin{itemize}\\item[(i)] integers with an atypically small number of prime factors (Hough \\cite{Hough2011}, Harper \\cite{Harper2013}), \n\\item[(ii)] integers in a short interval (Chatterjee and Soundararajan \\cite{ChatterjeeSoundararajan2012}, Soundararajan and Xu \\cite{soundararajan2023clt}, Pandey, Wang, and Xu \\cite{PandeyWangXu2024}), \n\\item[(iii)] sums of two squares in a short interval and shifted primes $p-k$ for $k\\neq 0$ (Soundararajan and Xu \\cite{soundararajan2023clt}), \n\\item[(iv)] polynomial images (Najnudel \\cite{najnudel2020consecutive}, Klurman, Shkredov, and Xu \\cite{KlurmanShkredovXu2023}, Chinis and the author \\cite{ChinisShala2025}).\n\\end{itemize} \nIn each of the aforementioned works, the distribution of the sums turns out to be approximately Gaussian, demonstrating that multiplicativity seems to interfere less with independence when the summands are restricted to a less ``multiplicatively structured'' set. In this paper, we extend the general framework of Soundararajan and Xu \\cite{soundararajan2023clt} on central limit theorems for random multiplicative functions to the setting of large fluctuations.\n\nThere has been considerable work in understanding almost sure large fluctuations for the full sum $\\sum_{n\\leq N} f(n)$ and related variants. Harper \\cite{harper2020large} showed that for any (slowly growing) function $V(N)\\to\\infty$, almost surely there is a sequence of positive integers $N_k\\to\\infty$ such that \n\\begin{equation*}\n \\left\\lvert\\sum_{n\\leq N_k}f(n)\\right\\rvert\\gg\\sqrt{N_k}\\frac{(\\log\\log N_k)^{\\frac14}}{V(N_k)}.\n\\end{equation*}\nThis matches what one expects from the law of iterated logarithm, up to the factor $V(N)$. Proving the corresponding upper bound remains a challenge. Partial results with an upper bound of the right shape (namely $\\sqrt N$ times a power of $\\log\\log N$) have been proved by Lau, Tenenbaum and Wu \\cite{LauTenenbaumWu2012}, as well as most recently by Caich \\cite{Caich2023} with the current best known almost sure bound\n\\begin{equation*}\n \\left\\lvert\\sum_{n\\leq N} f(n)\\right\\rvert\\ll_{\\epsilon} \\sqrt N(\\log\\log N)^{\\frac{3}{4}+\\epsilon}.\n\\end{equation*}\nIn the case when the summands are demanded to have a prime factor bigger than $\\sqrt N$, Mastrostefano \\cite{Mastrostefano2022} proved an essentially sharp almost sure upper bound with the exponent $\\frac{1}{4}+\\epsilon$ in place of $\\frac{3}4$, with the corresponding almost sure lower bound following by work of Harper \\cite{harper2020large}; see also the work of Hardy \\cite{Hardy2025LargePrime} on the distributional aspect in this case. A weighted variant $\\sum_{n\\leq N} \\frac{f(n)}{\\sqrt n}$\nhas been considered by Aymone, Heap and Zhao \\cite{aymone2021partial}, and Hardy \\cite{Hardy2023AlmostSure} in the Steinhaus case who obtained essentially sharp bounds, and Atherfold \\cite{Atherfold2025} in the Rademacher case albeit without sharp bounds.\n\nWe are now ready to state the main results of this paper. Our focus will be on two main examples: (i) polynomial images, and (ii) integers in a short interval. However, our method is rather general, so it should be able to flexibly handle several examples; see Section \\ref{section:generalSetup} for the general but rather technical results.\n\n\\subsection{Polynomial images} This case is particularly interesting due to its connection with conjectures of Chowla and Elliott \\cite{Chowla1965, Elliott1992, Elliott1994} on correlations of multiplicative functions. See the introductions of \\cite{KlurmanShkredovXu2023, ChinisShala2025} for a more detailed discussion in the setting of random multiplicative functions, where central limit theorems were proved.\n\nThe almost sure upper bound in Theorem \\ref{thm:asboundsPoly} is new, whereas the lower bound is an extension of results of Klurman, Shkredov, and Xu \\cite{KlurmanShkredovXu2023} in the Steinhaus case, and Chinis and the author \\cite{ChinisShala2025} in the Rademacher case, to polynomials that split as a product of linear factors over $\\mathbb Q$. Indeed, the method used in \\cite{KlurmanShkredovXu2023, ChinisShala2025} relied on the fact that for a polynomial $P$ that has an irreducible factor of degree at least two, there are many integers $n$ for which $P(n)$ has a prime factor greater than $n\\log n$.\n\n\\begin{theoremL}\\label{thm:asBoundsShort}\n Let $f$ be a Rademacher or Steinhaus random multiplicative function and suppose that $H=H(N)$ is a smooth, increasing, and concave function, such that $N^{\\frac{11}{15}}\\leq H(N)\\leq N/(\\log N)^c$\n for some constant $c>2\\log 2-1$. \n Then almost surely there exists a sequence $N_k\\to\\infty$ with corresponding $H_k = H(N_k)$ such that \n \\begin{equation*}\n \\left\\lvert\\sum_{N_k-H_k < n \\leq N_k}f(n)\\right\\rvert\\gg \\sqrt{H_k\\log\\frac{N_k}{H_k}}.\n \\end{equation*}\n\\end{theoremL}\n\n\\section{Overview of the proofs}\n\\subsection{Martingale structure} Our starting point is the observation of Harper \\cite{Harper2013} that sums of a random multiplicative function have the structure of a martingale with respect to the filtration induced by the largest prime factor, namely \n\\begin{equation*}\n \\sum_{n\\in\\mathcal A}f(n) = \\sum_{p^{\\alpha}\\leq N}f(p^{\\alpha})\\sum_{\\substack{n\\in\\mathcal A \\\\ P^+(n)=p \\\\ p^{\\alpha}\\parallel n}}f\\left(\\frac{n}{p^{\\alpha}}\\right).\n\\end{equation*}\nThis observation was utilized in almost all subsequent works on random multiplicative functions. For us, its usefulness lies in the fact that it morally reduces the understanding of $\\sum_{n\\in\\mathcal A}f(n)$ to evaluating the second and fourth moments of the sum, provided they do not blow up. More precisely, we will apply the following quantitative result from the book of Hall and Heyde \\cite{HallHeyde1980}, yielding a bound on the Kolmogorov distance between our martingale and a Gaussian. \n\\begin{prop}[\\cite{HallHeyde1980}, Theorem 3.9]\\label{prop:quantCLT}\n Let $X_1, X_2, \\ldots, X_n$ be a real square-integrable martingale difference sequence.\n Let $S_n = \\sum_{i=1}^n X_i$. Then \n \\begin{equation*}\n\\left\\lvert\\mathbb P(S_n\\leq x) - \\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^x e^{-t^2/2}\\emph{d}t\\right\\rvert\\ll \\frac{\\left(\\sum_{i=1}^n \\mathbb E\\lvert X_i\\rvert^4 + \\mathbb E\\left\\lvert\\sum_{i=1}^n X_i^2 - 1\\right\\rvert^2\\right)^{1/5}}{1+\\lvert x\\rvert^{\\frac{16}{5}}}.\n \\end{equation*}\n\\end{prop}\n\\subsection{Almost sure upper bound} Equipped with the above result, we may readily prove an almost sure upper bound for $\\sum_{n\\in\\mathcal A} f(n)$, provided our approximation is suitably strong. In order to access the tails at $x\\approx C\\sqrt{\\log k}$ which occur with probability $\\approx k^{-C^2/2}$ for the Gaussian, we will need an approximation that is at least this good. We are able to achieve this in the case of polynomial images, but not short intervals -- we will discuss this in more detail in the following subsections. Using our quantitative approximation at several spread out scales together with a slow variation property between the scales (see Proposition \\ref{prop:slowVar} and Lemma \\ref{lemma:slowVarPoly}), the Borel-Cantelli lemma establishes the almost sure upper bound.\n\n\\begin{lemma}\\label{lemma:slowVarPoly} Let $f$ be a Rademacher (or Steinhaus) random multiplicative function, and let $P\\in\\mathbb Z[x]$ be a polynomial as in Theorem \\ref{thm:asboundsPoly}. \nFor any constant $A>0$, there exists a constant $c>0$ such that for $N_l = \\lfloor e^{l^c}\\rfloor$, we have that\n \\begin{equation*}\n \\max_{N_{l}<N\\leq N_{l+1}} \\left\\lvert\\sum_{n\\leq N} f(P(n)) - \\sum_{n\\leq N_{l}} f(P(n)) \\right\\rvert \\ll \\frac{\\sqrt{N_{l+1}}}{(\\log N_{l+1})^A}\n \\end{equation*} holds almost surely for every $l\\in\\mathbb N$. \n\\end{lemma}\n\nWe finally prove the almost sure upper bound. By Lemma \\ref{lemma:slowVarPoly} with $A=1$, it suffices to show that almost surely there exists a constant $C>0$ such that for all $l\\in\\mathbb N$ we have\n \\begin{equation}\\label{eq:ubPoly}\n \\left\\lvert\\sum_{n\\leq N_l} f(P(n))\\right\\rvert\\leq C \\sqrt{N_l\\log\\log N_l},\n \\end{equation}\n where $N_l$ is as in the statement of the lemma (in the Steinhaus case, we do this separately for the real and imaginary parts). By Corollary \\ref{cor:quantCLTpoly}, the probability that the above fails for a fixed $l$ and large enough implied constant $C$ is $\\ll (\\log N_l)^{-\\frac{C^2}{2}}$. Making $C$ larger (in terms of $c$) if necessary, we have that the series of probabilities $$\\sum_{l=1}^\\infty (\\log N_l)^{-\\frac{C^2}{2}} = \\sum_{l=1}^\\infty \\frac{1}{l^{\\frac{cC^2}{2}}}<\\infty,$$ thus \\eqref{eq:ubPoly} holds almost surely by Borel-Cantelli (with a possibly even larger constant $C$ to account for the finitely many failures).\n\n\\begin{theorem}\\label{thm:localizedBoundPoly}\n Let $f$ be a Rademacher (or Steinhaus) random multiplicative function and let $P\\in\\mathbb Z[x]$ be a polynomial as in Theorem \\ref{thm:asboundsPoly}. There exists a constant $c>0$ such that for any large enough $X$, we have that \n \\begin{equation*}\n \\max_{X<N\\leq X^2} \\frac{1}{\\sqrt N} \\left\\lvert\\sum_{n\\leq N}f(P(n)) \\right\\rvert \\gg \\sqrt{\\log\\log X}\n \\end{equation*} holds with a suitably small implied constant with probability $1-O(\\exp(-(\\log X)^{c}))$. \n\\end{theorem}\n\n\\section{Proof of Theorem \\ref{thm:asBoundsShort}: Short intervals} Here we will require the more sophisticated part of Corollary \\ref{cor:normalcomparison}. Let $H$ be as in the statement of Theorem \\ref{thm:asBoundsShort} and consider $\\mathcal A_N = \\{n\\in\\mathbb N : N-H<n\\leq N\\}$. Note that $\\lvert\\mathcal A_N\\rvert\\sim \\kappa H$ for some\\footnote{In the Steinhaus case we have $\\kappa = 1$, whereas in the Rademacher case $\\kappa = \\frac{6}{\\pi^2}$.} $\\kappa>0$. Recall \n\\begin{equation*}\n S_N = \\frac{1}{\\sqrt{\\lvert\\mathcal A_N\\rvert}}\\sum_{n\\in\\mathcal A_N} \\mathfrak Tf(n)\\sim \\frac{1}{\\sqrt{\\kappa H}}\\sum_{N-H<n\\leq n }\\mathfrak Tf(n).\n\\end{equation*}\nMoreover, let $X$ be a large parameter. For some small $\\delta>0$ depending on the function $H$ and then even smaller $\\varepsilon_0>0$, find $k= (X/H(X))^{\\varepsilon_0}$ primes $l$ in the range $h(X)/2<l\\leq h(X)$, where $h(X) := (X/H(X))^{\\delta}$. We will not fix $\\delta$ or $\\varepsilon_0$ for now, as there will be various (but finitely many) places where we may have make them smaller. For these values of $l$, let $N_l = lX$ with corresponding $H_l = H(N_l)$. \nObserve that for large enough $X$ and small enough $\\delta>0$, the sets $\\mathcal A_{N_l}$ are disjoint. It is not these sets that we will apply Theorem \\ref{thm: complicatedapproxJointG} to. If we did this, the very smooth integers in the sets $\\mathcal A_{N_l}$ would prevent us from obtaining a suitably small value for $\\varepsilon_2$. We bypass this by splitting $S_{N_l}$ as \n\\begin{equation}\\label{eq:splitting1}\n \\sum_{N_l - H_l<n\\leq N_l}f(n) = \\sum_{(Xh(X))^{\\frac{2}{3}}< p}f(p)\\sum_{\\substack{N_l-H_l < n\\leq N_l \\\\ p\\mid n}}f\\left(\\frac np\\right) + \\sum_{\\substack{N_l-H_l<n\\leq N_l \\\\ P^+(n)\\leq (Xh(X))^{\\frac{2}{3}}}}f(n).\n\\end{equation}\nIf $H_1> N_1/(\\log N_1)^2$ (hence since $H$ is increasing we have $H_l \\gg Xh(X)/(\\log X)^2$ for all values of $l$), we choose some $\\varepsilon>0$ to be specified later and additionally split the first part as \n\\begin{equation}\\label{eq:splitting2}\n \\sum_{N_l - H_l<n\\leq N_l}f(n) = \\sum_{(Xh(X))^{\\frac{2}{3}}< p }f(p)\\sum_{\\substack{N_l-H_l < n\\leq N_l \\\\ p\\mid n \\\\ \\Omega(n)\\leq (1+\\varepsilon)\\log\\log X}}f\\left(\\frac np\\right) \\\\ + \\sum_{\\substack{N_l-H_l<n\\leq N_l \\\\ P^+(n)\\leq (Xh(X))^{\\frac{2}{3}} \\\\ \\text{or } \\Omega(n)> (1+\\varepsilon)\\log\\log X }}f(n).\n\\end{equation}\nWe first show that the sum over integers with $\\Omega(n)> (1+\\varepsilon)\\log\\log X$ above may be typically ignored for all values of $l$.", "post_theorem_intro_text_len": 6196, "post_theorem_intro_text": "The almost sure upper bound in Theorem \\ref{thm:asboundsPoly} is new, whereas the lower bound is an extension of results of Klurman, Shkredov, and Xu \\cite{KlurmanShkredovXu2023} in the Steinhaus case, and Chinis and the author \\cite{ChinisShala2025} in the Rademacher case, to polynomials that split as a product of linear factors over $\\mathbb Q$. Indeed, the method used in \\cite{KlurmanShkredovXu2023, ChinisShala2025} relied on the fact that for a polynomial $P$ that has an irreducible factor of degree at least two, there are many integers $n$ for which $P(n)$ has a prime factor greater than $n\\log n$. \n\nTo demonstrate how this was used, let us consider $P(n) = n^2+1$. By the aforementioned fact, one can ``pull out'' a unique large prime factor for such $n$ and still capture a good ``bulk'' of the sum for infinitely many (sparse) $N$, that is \n\\begin{equation*}\n    \\sum_{n\\leq N}f(n^2+1)\\approx \\sum_{N\\log N < p\\leq N^2}f(p)\\sum_{\\substack{n\\leq N \\\\ p\\mid n^2+1}}f\\left(\\frac{n^2+1}{p}\\right).\n\\end{equation*}\n\nUpon conditioning on the small primes, this is a weighted sum of independent random variables, which is then handled using intricate combinatorial arguments and Gaussian approximation. \n\nIndeed, the method described above cannot be directly applied to the case when $P$ is a product of linear factors, for all its prime factors are of size at most linear in $N$.\nIn this paper, we will use a different method that does not involve any combinatorial splitting of the sum or conditioning. We will rather fully utilize the martingale structure of the entire sum -- we will elaborate on this in the next section. In particular, provided enough number-theoretic input (see the work of Martin \\cite{Martin2002}), our method would also work to prove almost sure large fluctuations of $N$-smooth\\footnote{An integer is called $y$-smooth if all of the prime factors dividing it are at most $y$.} values of irreducible polynomials.\n\n\\subsection{Integers in a short interval} A central limit theorem for sums of a random multiplicative function over integers in a short interval $(N-H, N]$ was first proved by Chatterjeee and Soundararajan \\cite{ChatterjeeSoundararajan2012} when $H=o\\left(\\frac{N}{\\log N}\\right)$, using essentially only second and fourth moment information. This was later extended to $H\\leq \\frac{N}{(\\log N)^{c}}$ with $c>2\\log2-1$ by Soundararajan and Xu \\cite{soundararajan2023clt}, entering a regime of $H$ where naively the fourth moment of the sum blows up. However, this is bypassed by throwing out $0\\%$ of the integers in the short interval with atypically many prime factors, resulting in a controlled fourth moment. \n\nWe prove an almost sure lower bound in the same regime. \n\n\\begin{theoremL}\\label{thm:asBoundsShort}\n    Let $f$ be a Rademacher or Steinhaus random multiplicative function and suppose that $H=H(N)$ is a smooth, increasing, and concave function, such that $N^{\\frac{11}{15}}\\leq H(N)\\leq N/(\\log N)^c$\n    for some constant $c>2\\log 2-1$. \n    Then almost surely there exists a sequence $N_k\\to\\infty$ with corresponding $H_k = H(N_k)$ such that \n    \\begin{equation*}\n        \\left\\lvert\\sum_{N_k-H_k < n \\leq N_k}f(n)\\right\\rvert\\gg \\sqrt{H_k\\log\\frac{N_k}{H_k}}.\n    \\end{equation*}\n\\end{theoremL}\n\n\\begin{remark}\n    The fluctuations of order $\\sqrt{\\log\\frac{N}{H}}$ may seem surprisingly large at first, especially in a naive comparison with the law of iterated logarithm. However, this is explained by the fact that short sums of length $H$ decorrelate much faster, and indeed roughly $N/H$ essentially independent sums fit in a ``window'' of size $N$. In agreement with the law of iterated logarithm, the maximum of these sums has size roughly $\\sqrt{\\log\\frac{N}{H}}$. \n\\end{remark}\n\n\\begin{remark}\n    The conditions on $H$ are imposed simply for convenience -- any standard choices such as $H=N^{\\alpha}$ for $\\alpha < 1$, $H = N/\\exp\\left(\\sqrt{\\log N}\\right)$, or $H = N/(\\log N)^A$ for some constant $A>2\\log 2-1$, satisfy these conditions (for sufficiently large $N$). \n\n    We have not attempted to optimize the exponent $\\frac{11}{15}$ -- it is likely possible to bring it down to $\\frac{3}{5}$. However, the lower bound on $H$ should not be essential: for very small $H\\ll \\log N$ the large fluctuations are captured by almost sure long runs of ones, of length $\\approx \\log N$ (see the work of Erdős and Rényi \\cite{ErdosRenyi1970}), whereas if $\\log N\\ll H\\ll N^{\\frac{11}{15}}$, our method should work (with significant simplifications when $H\\ll N^{\\frac{1}{2}}$), but the restriction stems from a result on square-free smooth integers in short intervals; see Lemma \\ref{lemma:unSmoothSquarefree}. Since we are only proving a lower bound and therefore can pick convenient scales to work with, it is likely possible to circumvent this using the Matom\\\"aki--Radziwiłł machinery; see the work of Jain \\cite{jain2025smooth} and the references within. However, we have opted for simplicity, especially since our result captures the most interesting regime where there is a transition of the large fluctuations from $\\sqrt{\\log N}$ down to $\\sqrt{\\log \\log N}$.\\footnote{We thank Sarvagya Jain for discussions pertaining to this remark.}\n\\end{remark}\n\nIt would be interesting to prove the corresponding almost sure upper bound in any regime of $H$. For rather large $H$, it is likely that the sophisticated ideas in \\cite{LauTenenbaumWu2012, Mastrostefano2022, Hardy2023AlmostSure, Atherfold2025} relating the short sum $\\sum_{N-H<n\\leq N} f(n)$ to a random Euler product could be useful. It is unclear, however, whether this would result in a sharp (or almost sharp) upper bound in any regime of $H$ -- this is an ongoing investigation. \n\nUpcoming work of Harper, Soundararajan, and Xu \nestablishes a central limit theorem in the range $H = o(N)$, although with a normalization of the sum that differs from just the standard deviation. It would be interesting to determine whether our method would succeed to prove almost sure lower bounds of the right order of magnitude in this extended range. See also the work of Caich \\cite{Caich2024randomshort} for more on the intricate behavior of short sums in this regime.", "sketch": "The introduction explains that prior lower-bound methods (e.g. \\cite{KlurmanShkredovXu2023, ChinisShala2025}) for handling \\(\\sum_{n\\le N} f(P(n))\\) when \\(P\\) has an irreducible factor of degree \\(\\ge 2\\) use the fact that for many \\(n\\), \\(P(n)\\) has a prime factor \\(> n\\log n\\). For example, with \\(P(n)=n^2+1\\), one can “pull out” a unique large prime factor and still capture a “bulk” of the sum along infinitely many (sparse) \\(N\\):\n\\[\n\\sum_{n\\le N} f(n^2+1)\\approx \\sum_{N\\log N < p\\le N^2} f(p)\\sum_{\\substack{n\\le N\\\\ p\\mid n^2+1}} f\\!\\left(\\frac{n^2+1}{p}\\right).\n\\]\nAfter “conditioning on the small primes”, this becomes “a weighted sum of independent random variables”, which is then treated via “intricate combinatorial arguments and Gaussian approximation.”\n\nIt then notes this approach “cannot be directly applied” when \\(P\\) is a product of linear factors, since then “all its prime factors are of size at most linear in \\(N\\).” For Theorem~\\ref{thm:asboundsPoly}, the paper instead uses “a different method that does not involve any combinatorial splitting of the sum or conditioning” and aims to “fully utilize the martingale structure of the entire sum,” with “enough number-theoretic input (see the work of Martin \\cite{Martin2002}).”", "expanded_sketch": "The introduction explains that prior lower-bound methods (e.g. J. Klurman, I. D. Shkredov, and J. Xu, *Title not found in ref_dict* (2023), and K. Chinis and A. Shala, *Title not found in ref_dict* (2025)) for handling \\(\\sum_{n\\le N} f(P(n))\\) when \\(P\\) has an irreducible factor of degree \\(\\ge 2\\) use the fact that for many \\(n\\), \\(P(n)\\) has a prime factor \\(> n\\log n\\). For example, with \\(P(n)=n^2+1\\), one can “pull out” a unique large prime factor and still capture a “bulk” of the sum along infinitely many (sparse) \\(N\\):\n\\[\n\\sum_{n\\le N} f(n^2+1)\\approx \\sum_{N\\log N < p\\le N^2} f(p)\\sum_{\\substack{n\\le N\\\\ p\\mid n^2+1}} f\\!\\left(\\frac{n^2+1}{p}\\right).\n\\]\nAfter “conditioning on the small primes”, this becomes “a weighted sum of independent random variables”, which is then treated via “intricate combinatorial arguments and Gaussian approximation.”\n\nIt then notes this approach “cannot be directly applied” when \\(P\\) is a product of linear factors, since then “all its prime factors are of size at most linear in \\(N\\).” In establishing the main theorem, the paper instead uses “a different method that does not involve any combinatorial splitting of the sum or conditioning” and aims to “fully utilize the martingale structure of the entire sum,” with “enough number-theoretic input (see the work of Martin C. A. Martin, *Title not found in ref_dict* (2002)).”,", "expanded_theorem": "\\label{thm:asboundsPoly}\n    Let $f$ be a  Rademacher or Steinhaus random multiplicative function. In the Rademacher case, assume $P\\in\\mathbb Z[x]$ is a product of at least two distinct linear factors over $\\mathbb Q$, or irreducible of degree $2$, taking infinitely many square-free values. In the Steinhaus case, assume $P(x)$ is not of the form $w(x+c)^d$ for $w\\in\\mathbb Z, c\\in\\mathbb Q$. Then almost surely we have $$\\left\\lvert\\sum_{n\\leq N} f(P(n))\\right\\rvert \\ll \\sqrt{N\\log\\log N}.$$ Moreover, almost surely there exists a sequence $N_k\\to\\infty$ such that \n    \\begin{equation*}\n        \\left\\lvert\\sum_{n\\leq N_k} f(P(n))\\right\\rvert\\gg \\sqrt{N_k\\log\\log N_k}.\n    \\end{equation*},", "theorem_type": ["Inequality or Bound", "Asymptotic or Limit"], "mcq": {"question": "Let $f$ be either a Steinhaus or a Rademacher random multiplicative function. Here, a Steinhaus random multiplicative function satisfies $f(mn)=f(m)f(n)$ for all positive integers $m,n$, and for each prime $p$, the random variable $f(p)$ is uniformly distributed on the unit circle. A Rademacher random multiplicative function is supported on square-free integers, satisfies $f(mn)=f(m)f(n)$ whenever $(m,n)=1$, and for each prime $p$, $f(p)$ is uniformly distributed on $\\{-1,1\\}$. Let $P\\in \\mathbb Z[x]$. In the Rademacher case, assume that $P$ is either a product of at least two distinct linear factors over $\\mathbb Q$, or an irreducible polynomial of degree $2$, and that $P$ takes infinitely many square-free values. In the Steinhaus case, assume that $P(x)$ is not of the form $w(x+c)^d$ with $w\\in \\mathbb Z$ and $c\\in \\mathbb Q$. Under these assumptions, which quantitative estimate holds for the polynomially sampled sums $\\sum_{n\\le N} f(P(n))$?", "correct_choice": {"label": "A", "text": "Almost surely, one has\n$$\\left|\\sum_{n\\le N} f(P(n))\\right| \\ll \\sqrt{N\\log\\log N}.$$ \nMoreover, almost surely there exists a sequence $N_k\\to\\infty$ such that\n$$\\left|\\sum_{n\\le N_k} f(P(n))\\right| \\gg \\sqrt{N_k\\log\\log N_k}.$$"}, "choices": [{"label": "B", "text": "Almost surely, one has\n$$\\left|\\sum_{n\\le N} f(P(n))\\right| \\ll \\sqrt N\\,(\\log\\log N)^{\\frac14}.$$ \nMoreover, almost surely there exists a sequence $N_k\\to\\infty$ such that\n$$\\left|\\sum_{n\\le N_k} f(P(n))\\right| \\gg \\sqrt{N_k}\\,(\\log\\log N_k)^{\\frac14}.$$"}, {"label": "C", "text": "Almost surely, one has\n$$\\left|\\sum_{n\\le N} f(P(n))\\right| \\ll \\sqrt{N\\log\\log N}.$$"}, {"label": "D", "text": "For every sequence $N_k\\to\\infty$, almost surely one has\n$$\\left|\\sum_{n\\le N_k} f(P(n))\\right| \\gg \\sqrt{N_k\\log\\log N_k}$$\nfor all sufficiently large $k$, and in particular\n$$\\left|\\sum_{n\\le N} f(P(n))\\right| \\ll \\sqrt{N\\log\\log N}.$$"}, {"label": "E", "text": "Almost surely, one has\n$$\\left|\\sum_{n\\le N} f(P(n))\\right| \\ll \\sqrt{N\\log N}.$$ \nMoreover, almost surely there exists a sequence $N_k\\to\\infty$ such that\n$$\\left|\\sum_{n\\le N_k} f(P(n))\\right| \\gg \\sqrt{N_k\\log N_k}.$$"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "loglog_exponent_size", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "existence_of_matching_lower_bound_subsequence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "existential_subsequence_vs_uniform_eventual_lower_bound", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "correct_loglog_scale_from_martingale_method", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the conclusion or embed the correct estimate. It only gives the hypotheses and asks which quantitative bound follows."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-conclusion recall question: the stem lists the theorem's assumptions and asks for the exact estimate that holds."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the choices differ in subtle ways (log-log scale, missing lower bound, wrong quantifier strength, wrong logarithmic scale), but success is still driven mainly by recalling the precise theorem statement rather than generating a new argument."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: one is a weaker true statement, others alter the log-log exponent, quantifiers, or replace log log by log. These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A solid theorem-recall MCQ with strong distractors and no answer leakage, but it is largely tautological and only moderately tests generative reasoning."}}
{"id": "2602.20124v1", "paper_link": "http://arxiv.org/abs/2602.20124v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm:MinimalSurfaceExistence}\nFor every $p, q \\geq 1$, there exists a minimal embedding of $\\mathbb{S}^p \\times \\mathbb{S}^q \\times \\mathbb{S}^1$ in $\\mathbb{S}^{p+q+2}$.", "start_pos": 10539, "end_pos": 10717, "label": "thm:MinimalSurfaceExistence"}, "ref_dict": {"eqn:one-phase-problem": "\\begin{equation}\\label{eqn:one-phase-problem}\\tag{OP}\n        \\Delta u = 0 \\; \\text{ in } \\; \\{ u > 0 \\} \n        \\qquad \\text{and}\\qquad \n        |\\nabla u| = 1 \\; \\text{ on } \\; \\partial \\{ u > 0 \\}.\n\\end{equation}", "thm:carlotto-schulz": "\\begin{theorem}\\label{thm:carlotto-schulz}\n    For every $k \\geq 2$, there exists a smooth one-parameter family of capillary cones $\\{ \\bar{\\mathbf{C}}_{k,a} \\}_{a \\in (0,a^*_k] } \\subset \\bR^{2k+1}$ that interpolates between a free-boundary minimal cone and a homogeneous solution of the one-phase problem through cones of every angle.\n    More precisely,\n    \\begin{enumerate}[(i)]\n        \\item As $a \\downarrow 0$, the rescaled cones $\\frac{1}{a} \\bar{\\mathbf{C}}_{k,a}$ converge in $C^{\\infty}_{\\textup{loc}}$ to the graph of a homogeneous solution of the one-phase problem~\\eqref{eqn:one-phase-problem}.\n        \\item As $a \\in (0,a^*_k]$, the cones $\\bar{\\mathbf{C}}_{k,a}$ attain every capillary angle $\\theta \\in (0,\\frac{\\pi}{2}]$.\n        \\item The cone $\\bar{\\mathbf{C}}_{k,a^*_k}$ is the free-boundary cone $\\hat{\\mathbf{C}}_{2k,k,\\frac{\\pi}{2}}$, whose double has link given by a minimal embedded $\\bS^{k-1} \\times \\bS^{k-1} \\times \\bS^1 \\hookrightarrow \\bS^{2k}$.\n    \\end{enumerate}\n\\end{theorem}", "thm:MinimalSurfaceExistence": "\\begin{theorem}\\label{thm:MinimalSurfaceExistence}\nFor every $p, q \\geq 1$, there exists a minimal embedding of $\\bS^p \\times \\bS^q \\times \\bS^1$ in $\\bS^{p+q+2}$. \n\\end{theorem}", "lemma:change-of-variables": "\\begin{lemma}\\label{lemma:change-of-variables}\nFor given $(n,k)$, the function $f(t)$ solves equation~\\eqref{eqn:capillary-ODE} if and only if the function $f (\\sqrt{1-t^2})$ solves this equation for $(n,n-k)$.\nAny capillary cones produced by solving this equation with the appropriate free boundary are isometric via the involution $t \\leftrightsquigarrow \\sqrt{1-t^2}$; in particular, they form the same contact angle.\n\\end{lemma}", "lemma:bound-at-sqrt": "\\begin{lemma}\\label{lemma:bound-at-sqrt}\nAny solution $f$ of equation~\\eqref{eqn:capillary-ODE} with two zeros in $(0,1)$ satisfies $f(\\sqrt{\\frac{k-1}{n-2}}) \\leq \\frac{2}{\\sqrt{n}} \\lambda^{- \\frac{1}{2}}$.\n\\end{lemma}", "thm:capillaryConeExistence": "\\begin{theorem}\\label{thm:capillaryConeExistence}\n    For all $n \\geq 4$, $2 \\leq k \\leq n-2$, and $\\theta \\in (0,\\tfrac{\\pi}{2}]$, there exists a capillary minimal cone $\\tilde{\\mathbf{C}}_{n,k,\\theta}$ that is topologically $C(\\bS^{n-k-1}\\times \\bS^{k-1}\\times [0,1])$ with isometry group $O(n-k) \\times O(k)$. \n\n    The double of the free-boundary cone $\\tilde{\\mathbf{C}}_{n,k,\\frac{\\pi}{2}}$ is a non-isoparametric minimal hypercone $\\hat{\\mathbf{C}}_{n,k} \\subset \\bR^{n+1}$.\n    Its link $\\mathbf{S}_{n,k} \\subset \\bS^n$ is topologically $\\bS^{n-k-1} \\times \\bS^{k-1} \\times \\bS^1$, invariant under $O(n-k) \\times O(k) \\times \\bZ_2$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2723, "pre_theorem_intro_text": "We construct new minimal surfaces in the sphere as minimal embeddings $\\mathbb{S}^p \\times \\mathbb{S}^q \\times \\mathbb{S}^1 \\hookrightarrow \\mathbb{S}^{p+q+2}$, for any $p,q \\geq 1$, by doubling the links of free-boundary minimal cones in $\\mathbb{R}^{n+1}_+$.\nThese free-boundary surfaces arise as limits of families of capillary minimal surfaces constructed by solving a free boundary ODE.\nUnder suitable rescalings these cones converge, in the shallow-angle regime, to a homogeneous solution of the one-phase Bernoulli problem.\nUnlike the examples in~\\cite{FTW-1}, our families of free-boundary cones do not enjoy additional reflection symmetries and exhibit different geometric properties.\nTopologically, the links are diffeomorphic to $\\mathbb{S}^{n-k-1}\\times \\mathbb{S}^{k-1}\\times \\mathbb{S}^1$ with isometry group $O(n-k)\\times O(k) \\times \\bZ_2$, and the induced metric can be expressed as a doubly warped product over the $\\mathbb{S}^1$ factor. \nThis construction resolves problems posed by Hsiang-Lawson~\\cite{hsiang-lawson-jdg}*{Ch.~III \\S~3} and Hsiang-Hsiang~\\cite{hsiang-hsiang}*{Problem~3} on the existence of minimal submanifolds in the sphere, and advances the Hsiang-Lawson program in low cohomogeneity through a new, self-contained approach.\nOur construction also generalizes results of Carlotto-Schulz and Wang-Wang-Zhou~\\cites{carlotto-schulz, spherical-bernstein-xin-zhou} to arbitrary pairs of spherical factors.\n\nThe construction of minimal hypersurfaces in spheres $\\mathbb{S}^n$ has received great attention, leading to the development of many key techniques in differential geometry.\nThese include the Hsiang-Lawson framework for the study of minimal submanifolds with large symmetry groups~\\cites{hsiang-lawson-jdg , spherical-bernstein, spherical-bernstein-2 , ferus-karcher , free-boundary-mcgrath }, Kapouleas' gluing and desingularization methods~\\cites{doubling-clifford-torus, doubling-two-sphere, kapouleas-mcgrath-cpam , kapouleas-wiygul-annalen , choe-soret }, min-max constructions~\\cites{ willmore-conjecture , infinitely-many-MN , marques-neves-song , haslhofer-ketover , XinZhouAnnals, WangZhouFourSpheres, song , ko-min-max , spherical-bernstein-xin-zhou }, and the Allen-Cahn approach~\\cites{guaraco-phase-transitions , allen-cahn-3-manifolds}.\nSeparately, White~\\cite{white-mapping-degree} constructed complete minimal surfaces asymptotic to a given minimizing cone $\\mathbf{C}$ provided that its link is not a homology sphere. \nThe topological complexity of the links of cones is connected to their density, as proved by Ilmanen-White and Bernstein-Wang~\\cites{ilmanen-white, bernstein-wang} using techniques from mean curvature flow.\n\nOur first result presents the following:", "context": "We construct new minimal surfaces in the sphere as minimal embeddings $\\mathbb{S}^p \\times \\mathbb{S}^q \\times \\mathbb{S}^1 \\hookrightarrow \\mathbb{S}^{p+q+2}$, for any $p,q \\geq 1$, by doubling the links of free-boundary minimal cones in $\\mathbb{R}^{n+1}_+$.\nThese free-boundary surfaces arise as limits of families of capillary minimal surfaces constructed by solving a free boundary ODE.\nUnder suitable rescalings these cones converge, in the shallow-angle regime, to a homogeneous solution of the one-phase Bernoulli problem.\nUnlike the examples in~\\cite{FTW-1}, our families of free-boundary cones do not enjoy additional reflection symmetries and exhibit different geometric properties.\nTopologically, the links are diffeomorphic to $\\mathbb{S}^{n-k-1}\\times \\mathbb{S}^{k-1}\\times \\mathbb{S}^1$ with isometry group $O(n-k)\\times O(k) \\times \\bZ_2$, and the induced metric can be expressed as a doubly warped product over the $\\mathbb{S}^1$ factor. \nThis construction resolves problems posed by Hsiang-Lawson~\\cite{hsiang-lawson-jdg}*{Ch.~III \\S~3} and Hsiang-Hsiang~\\cite{hsiang-hsiang}*{Problem~3} on the existence of minimal submanifolds in the sphere, and advances the Hsiang-Lawson program in low cohomogeneity through a new, self-contained approach.\nOur construction also generalizes results of Carlotto-Schulz and Wang-Wang-Zhou~\\cites{carlotto-schulz, spherical-bernstein-xin-zhou} to arbitrary pairs of spherical factors.\n\nThe construction of minimal hypersurfaces in spheres $\\mathbb{S}^n$ has received great attention, leading to the development of many key techniques in differential geometry.\nThese include the Hsiang-Lawson framework for the study of minimal submanifolds with large symmetry groups~\\cites{hsiang-lawson-jdg , spherical-bernstein, spherical-bernstein-2 , ferus-karcher , free-boundary-mcgrath }, Kapouleas' gluing and desingularization methods~\\cites{doubling-clifford-torus, doubling-two-sphere, kapouleas-mcgrath-cpam , kapouleas-wiygul-annalen , choe-soret }, min-max constructions~\\cites{ willmore-conjecture , infinitely-many-MN , marques-neves-song , haslhofer-ketover , XinZhouAnnals, WangZhouFourSpheres, song , ko-min-max , spherical-bernstein-xin-zhou }, and the Allen-Cahn approach~\\cites{guaraco-phase-transitions , allen-cahn-3-manifolds}.\nSeparately, White~\\cite{white-mapping-degree} constructed complete minimal surfaces asymptotic to a given minimizing cone $\\mathbf{C}$ provided that its link is not a homology sphere. \nThe topological complexity of the links of cones is connected to their density, as proved by Ilmanen-White and Bernstein-Wang~\\cites{ilmanen-white, bernstein-wang} using techniques from mean curvature flow.\n\nOur first result presents the following:", "full_context": "We construct new minimal surfaces in the sphere as minimal embeddings $\\mathbb{S}^p \\times \\mathbb{S}^q \\times \\mathbb{S}^1 \\hookrightarrow \\mathbb{S}^{p+q+2}$, for any $p,q \\geq 1$, by doubling the links of free-boundary minimal cones in $\\mathbb{R}^{n+1}_+$.\nThese free-boundary surfaces arise as limits of families of capillary minimal surfaces constructed by solving a free boundary ODE.\nUnder suitable rescalings these cones converge, in the shallow-angle regime, to a homogeneous solution of the one-phase Bernoulli problem.\nUnlike the examples in~\\cite{FTW-1}, our families of free-boundary cones do not enjoy additional reflection symmetries and exhibit different geometric properties.\nTopologically, the links are diffeomorphic to $\\mathbb{S}^{n-k-1}\\times \\mathbb{S}^{k-1}\\times \\mathbb{S}^1$ with isometry group $O(n-k)\\times O(k) \\times \\bZ_2$, and the induced metric can be expressed as a doubly warped product over the $\\mathbb{S}^1$ factor. \nThis construction resolves problems posed by Hsiang-Lawson~\\cite{hsiang-lawson-jdg}*{Ch.~III \\S~3} and Hsiang-Hsiang~\\cite{hsiang-hsiang}*{Problem~3} on the existence of minimal submanifolds in the sphere, and advances the Hsiang-Lawson program in low cohomogeneity through a new, self-contained approach.\nOur construction also generalizes results of Carlotto-Schulz and Wang-Wang-Zhou~\\cites{carlotto-schulz, spherical-bernstein-xin-zhou} to arbitrary pairs of spherical factors.\n\nThe construction of minimal hypersurfaces in spheres $\\mathbb{S}^n$ has received great attention, leading to the development of many key techniques in differential geometry.\nThese include the Hsiang-Lawson framework for the study of minimal submanifolds with large symmetry groups~\\cites{hsiang-lawson-jdg , spherical-bernstein, spherical-bernstein-2 , ferus-karcher , free-boundary-mcgrath }, Kapouleas' gluing and desingularization methods~\\cites{doubling-clifford-torus, doubling-two-sphere, kapouleas-mcgrath-cpam , kapouleas-wiygul-annalen , choe-soret }, min-max constructions~\\cites{ willmore-conjecture , infinitely-many-MN , marques-neves-song , haslhofer-ketover , XinZhouAnnals, WangZhouFourSpheres, song , ko-min-max , spherical-bernstein-xin-zhou }, and the Allen-Cahn approach~\\cites{guaraco-phase-transitions , allen-cahn-3-manifolds}.\nSeparately, White~\\cite{white-mapping-degree} constructed complete minimal surfaces asymptotic to a given minimizing cone $\\mathbf{C}$ provided that its link is not a homology sphere. \nThe topological complexity of the links of cones is connected to their density, as proved by Ilmanen-White and Bernstein-Wang~\\cites{ilmanen-white, bernstein-wang} using techniques from mean curvature flow.\n\nOur first result presents the following:\n\nOur first result presents the following:\n\nTheorem~\\ref{thm:MinimalSurfaceExistence} is obtained from the study of capillary minimal cones as the other endpoint of a capillary interpolation between free-boundary minimal surfaces and the one-phase problem, first introduced in~\\cite{FTW-1}.\n\n\\begin{theorem}\\label{thm:capillaryConeExistence}\n For all $n \\geq 4$, $2 \\leq k \\leq n-2$, and $\\theta \\in (0,\\tfrac{\\pi}{2}]$, there exists a capillary minimal cone $\\tilde{\\mathbf{C}}_{n,k,\\theta}$ that is topologically $C(\\bS^{n-k-1}\\times \\bS^{k-1}\\times [0,1])$ with isometry group $O(n-k) \\times O(k)$.\n\nAs a consequence of our general technique, we extend the results of~\\cites{carlotto-schulz, spherical-bernstein-xin-zhou} to the capillary setting as follows.\n\\begin{theorem}\\label{thm:carlotto-schulz}\n For every $k \\geq 2$, there exists a smooth one-parameter family of capillary cones $\\{ \\bar{\\mathbf{C}}_{k,a} \\}_{a \\in (0,a^*_k] } \\subset \\bR^{2k+1}$ that interpolates between a free-boundary minimal cone and a homogeneous solution of the one-phase problem through cones of every angle.\n More precisely,\n \\begin{enumerate}[(i)]\n \\item As $a \\downarrow 0$, the rescaled cones $\\frac{1}{a} \\bar{\\mathbf{C}}_{k,a}$ converge in $C^{\\infty}_{\\textup{loc}}$ to the graph of a homogeneous solution of the one-phase problem~\\eqref{eqn:one-phase-problem}.\n \\item As $a \\in (0,a^*_k]$, the cones $\\bar{\\mathbf{C}}_{k,a}$ attain every capillary angle $\\theta \\in (0,\\frac{\\pi}{2}]$.\n \\item The cone $\\bar{\\mathbf{C}}_{k,a^*_k}$ is the free-boundary cone $\\hat{\\mathbf{C}}_{2k,k,\\frac{\\pi}{2}}$, whose double has link given by a minimal embedded $\\bS^{k-1} \\times \\bS^{k-1} \\times \\bS^1 \\hookrightarrow \\bS^{2k}$.\n \\end{enumerate}\n\\end{theorem}\nThis theorem provides further evidence for the importance of the capillary functional in the study of minimal submanifolds, as it provides a bridge between minimal surfaces and solutions of the one-phase problem~\\eqref{eqn:one-phase-problem}.\nThe first instance of such an interpolating family was discovered in~\\cite{FTW-1}.\n\n\\section{Construction of capillary surfaces}\\label{section:construction-capillary}\nWe adopt many conventions from~\\cite{FTW-1}.\nLet $n \\geq 4$ and $2 \\leq k \\leq n-2$, and decompose $\\bR^{n+1} = \\bR^{n-k}_x \\oplus \\bR^{k}_y \\oplus \\bR_z$.\nFor the capillary problem, we consider the upper half-space $\\bR^{n+1}_+$ with boundary $\\Pi := \\{z = 0\\} = \\partial \\{z \\geq 0 \\}$, so we assume $z \\geq 0$ throughout.\nThe $O(n-k)\\times O(k)$-action on the $(x,y)$-variables reduces the minimal surface equation to an ODE, where a natural equivariant coordinate system for the analysis is $\\rho := \\sqrt{|x|^2 + |y|^2}$ and $t := \\frac{|y|}{\\sqrt{|x|^2 + |y|^2}}$.\nAs computed in~\\cite{FTW-1}, a graphical surface $\\mathbf{C} = \\{ z = U(|x|,|y|) > 0\\}$ is a minimal cone if and only if $U = \\rho f(t)$, where $f(t)$ satisfies the ODE\n\\begin{equation}\\label{eqn:ode-star}\\tag{$\\star$}\n (1-t^2) f'' + (f - tf') + (n-2) \\left( 1 + (1-t^2) \\frac{(f')^2}{1 + f^2} \\right) (f - A_{n,k} f') = 0,\n\\end{equation}\nwhere $A_{n,k}(t) := t - \\frac{k-1}{n-2} t^{-1}$.\nThe boundary angle for $\\mathbf{C}$ at a point $t_*$ where $f(t_*) = 0$ is given by\n\\begin{equation}\\label{eqn:boundaryAngleTerm}\n \\theta = \\on{arctan} (\\sqrt{1 - t_*^2} \\, |f'(t_*)|)\n\\end{equation}\nA useful phenomenon in the study of families of capillary solutions is that, as the contact angle $\\theta \\downarrow 0$, the rescaled functions $\\frac{1}{\\tan \\theta} f_{\\theta}$ converge (subsequentially) to a multiple of a solution of the one-phase Bernoulli problem; this idea was utilized in~\\cites{improved-regularity,FTW-1}.\nFor $\\lambda \\in (0,\\infty)$ and $f$ a solution of~\\eqref{eqn:ode-star}, the rescaled function $\\frac{1}{\\sqrt{\\lambda}} f$ is a solution of the equation with parameter\n\\begin{equation}\\label{eqn:capillary-ODE} \\tag{$\\star_\\lambda$}\n (1-t^2) f'' + (f - tf') + (n-2) \\left( 1 + (1-t^2) \\frac{\\lambda (f')^2}{1 + \\lambda f^2} \\right) (f - A_{n,k} f') = 0.\n\\end{equation}\nMotivated by the above limiting behavior, we will study equation~\\eqref{eqn:capillary-ODE} uniformly in $\\lambda \\in [0,\\infty]$.\nThen, equation~\\eqref{eqn:capillary-ODE} includes the one-phase problem in the limit $\\lambda \\downarrow 0$; the resulting homogeneous solutions are further analyzed in~\\cites{FTW-1 , FTW-stability-one-phase , six-way}.\nWhen $\\lambda = 0$, the problem~\\eqref{eqn:capillary-ODE} specializes to the hypergeometric Legendre operator associated to $p(t) := t^{k-1} (1-t^2)^{\\frac{n-k}{2}}$, \n\\begin{equation}\\label{eqn:legendre-form}\n\\cL_{n,k} f = (1-t^2) f'' + (n-1) (f - t f') + (k-1) t^{-1} f' = \\tfrac{1-t^2}{p(t)} \\left[ (p f')' + (n-1) \\tfrac{p}{1-t^2} f \\right].\n\\end{equation}\nThe solutions of $\\cL_{n,k} f_0 = 0$ have the significance that $U(x,y) := \\rho f_0(t)$ is harmonic in $\\bR^n$, while the analogue of the boundary condition~\\eqref{eqn:boundaryAngleTerm} implies that $|\\nabla U| =1$ along $\\partial \\{ U > 0 \\}$.\nConsequently, $U$ is a solution of the \\textit{one-phase Bernoulli} problem\n\\begin{equation}\\label{eqn:one-phase-problem}\\tag{OP}\n \\Delta u = 0 \\; \\text{ in } \\; \\{ u > 0 \\} \n \\qquad \\text{and}\\qquad \n |\\nabla u| = 1 \\; \\text{ on } \\; \\partial \\{ u > 0 \\}.\n\\end{equation}\nThis problem is closely connected to the theory of minimal surfaces, with many constructions and theorems having direct counterparts.\nIn low dimensions, classical methods such as the Weierstrass representation and gluing techniques have led to classification results and new examples~\\cites{ entire-hairpins, traizet , jerison-kamburov, n-dim-catenoid , hines-kolesar-mcgrath}.\nWe refer the reader to~\\cites{ one-phase-simon-solomon , FTW-1 , FTW-stability-one-phase } for some recent results on the one-phase problem and its connections to the theory of minimal surfaces.\nAs in~\\cite{FTW-1}, Theorems~\\ref{thm:capillaryConeExistence} and~\\ref{thm:carlotto-schulz} deepen these connections by bridging minimal surfaces and one-phase solutions through interpolating families of capillary surfaces.\n\nThe capillary minimal cone $\\mathbf{C} = \\text{graph}(\\rho f(t))$ admits the spherical parametrization\n\\[\nF(\\rho, t , \\xi , \\eta) = \\left( \\rho \\sqrt{1-t^2} \\xi, \\rho t \\eta , \\rho f(t)\\right) , \\qquad \\xi \\in \\bS^{n-k-1}, \\; \\eta \\in \\bS^{k-1},\n\\]\nfor which $|F(\\rho, t , \\xi, \\eta)|^2 = \\rho^2 ( 1+ f(t)^2)$.\nConsequently, the link of the capillary cone is a $O(n-k) \\times O(k)$-invariant surface $\\Sigma \\subset \\bS^n_+$ inside the upper hemisphere with parametrization\n\\begin{equation}\\label{eqn:link-equation}\n \\Sigma := \\left\\{ \\left( \\frac{\\sqrt{1-t^2}}{\\sqrt{1+f^2}} \\xi, \\frac{t}{\\sqrt{1+f^2}} \\eta, \\frac{f}{\\sqrt{1+f^2}} \\right) : (\\xi, \\eta, t) \\in \\bS^{n-k-1} \\times \\bS^{k-1}\\times I_\\theta \\right\\} \\subset \\bS^n_+\n\\end{equation}\nwhere $I_{\\theta} \\subset [0,1]$ denotes the positive phase of $f$.\nFor cones of type $(ii)$, the resulting function $U = \\rho f(t)$ corresponds to a graph over a conical annulus in the base, of the form\n\\[\n\\Gamma := \\{ (x,y) : t_1 \\rho < |y| < t_2 \\rho \\}\n\\]\nwith two boundary components on $\\Pi = \\{ p_{n+1} \\geq 0 \\}$, namely the two cones $\\{ |y| = t_i \\rho \\}$.\nTopologically, the link of such a cone is diffeomorphic to $\\bS^{n-k-1} \\times \\bS^{k-1} \\times I_\\theta$.\nWe denote such a capillary cone with contact angle $\\theta$ by $\\tilde{\\mathbf{C}}_{n,k,\\theta}$, in parallel with the cones $\\mathbf{C}_{n,k,\\theta}$ of type $(i)$, constructed in~\\cite{FTW-1}*{Theorem 1.3}.\nBy construction, $\\tilde{\\mathbf{C}}_{n,k,\\theta}$ is a regular capillary cone with an isolated singularity at the origin.", "post_theorem_intro_text_len": 6616, "post_theorem_intro_text": "Theorem~\\ref{thm:MinimalSurfaceExistence} is obtained from the study of capillary minimal cones as the other endpoint of a capillary interpolation between free-boundary minimal surfaces and the one-phase problem, first introduced in~\\cite{FTW-1}.\n\n\\begin{theorem}\\label{thm:capillaryConeExistence}\n    For all $n \\geq 4$, $2 \\leq k \\leq n-2$, and $\\theta \\in (0,\\tfrac{\\pi}{2}]$, there exists a capillary minimal cone $\\tilde{\\mathbf{C}}_{n,k,\\theta}$ that is topologically $C(\\mathbb{S}^{n-k-1}\\times \\mathbb{S}^{k-1}\\times [0,1])$ with isometry group $O(n-k) \\times O(k)$. \n\n    The double of the free-boundary cone $\\tilde{\\mathbf{C}}_{n,k,\\frac{\\pi}{2}}$ is a non-isoparametric minimal hypercone $\\hat{\\mathbf{C}}_{n,k} \\subset \\mathbb{R}^{n+1}$.\n    Its link $\\mathbf{S}_{n,k} \\subset \\mathbb{S}^n$ is topologically $\\mathbb{S}^{n-k-1} \\times \\mathbb{S}^{k-1} \\times \\mathbb{S}^1$, invariant under $O(n-k) \\times O(k) \\times \\bZ_2$.\n\\end{theorem}\nThe resulting minimal submanifolds of $\\mathbb{S}^n$ are non-isoparametric, meaning that they have non-constant principal curvatures.\nIsoparametric minimal surfaces in the sphere have been classified using algebraic and analytic techniques, cf.~\\cites{cecil-chi-jensen, miyaoka, chi-four-curvatures}. \nTheorem~\\ref{thm:MinimalSurfaceExistence} is also connected to questions posed by Choe-Fraser~\\cite{choe-fraser}*{\\S~4} on the possible topology and diffeomorphism types of minimal submanifolds inside manifolds with positive Ricci curvature.\n\nA minimal surface $\\Sigma \\subset \\mathbb{S}^n$ with reflection symmetry $p_{n+1} \\leftrightsquigarrow -p_{n+1}$, restricted to the upper half-space $\\mathbb{R}^{n+1}_+$ and $\\mathbb{S}^n_+$, forms a free-boundary surface. \nConversely, since the boundary of $\\mathbb{S}^n_+$ is totally geodesic, any free-boundary surface can be doubled to produce an embedded minimal surface in the sphere.\nTherefore, when viewed as a graph, any such free-boundary minimal surface must have infinite slope along the boundary equator, making the graphical shooting approach singular in these coordinates.\nUsing a bi-orthogonal symmetry, the capillary minimal surface equation reduces to a free boundary ODE.\nFor capillary angles $\\theta \\in (0,\\tfrac{\\pi}{2})$, shooting techniques that match the contact angle are well-defined, and we show that the resulting solutions converge uniformly to an infinite-slope shot.\nThis yields the free-boundary minimal surface in Theorem~\\ref{thm:MinimalSurfaceExistence}.\n\nThe most symmetric case of Theorem~\\ref{thm:MinimalSurfaceExistence} where $n=2k$ (hence $n-k-1=k-1$), recovers the minimal hypertori produced by Carlotto-Schulz~\\cite{carlotto-schulz}.\nUsing numerical methods, Perdomo approximated the spectrum and index of the Carlotto-Schulz examples~\\cites{perdomo-spectrum, perdomo-carlotto-schulz}.\nAnalogues of this construction for constant mean curvature hypersurfaces are discussed in~\\cites{huang-wei-2022 , perdomo-navigating}.\nVery recently, Wang-Wang-Zhou~\\cite{spherical-bernstein-xin-zhou} provided an alternative construction of embedded spheres $\\mathbb{S}^3$ and hypertori $\\mathbb{S}^1 \\times \\mathbb{S}^1 \\times \\mathbb{S}^1$ inside the $4$-sphere, using equivariant min-max theory.\n\nAs a consequence of our general technique, we extend the results of~\\cites{carlotto-schulz, spherical-bernstein-xin-zhou} to the capillary setting as follows.\n\\begin{theorem}\\label{thm:carlotto-schulz}\n    For every $k \\geq 2$, there exists a smooth one-parameter family of capillary cones $\\{ \\bar{\\mathbf{C}}_{k,a} \\}_{a \\in (0,a^*_k] } \\subset \\mathbb{R}^{2k+1}$ that interpolates between a free-boundary minimal cone and a homogeneous solution of the one-phase problem through cones of every angle.\n    More precisely,\n    \\begin{enumerate}[(i)]\n        \\item As $a \\downarrow 0$, the rescaled cones $\\frac{1}{a} \\bar{\\mathbf{C}}_{k,a}$ converge in $C^{\\infty}_{\\textup{loc}}$ to the graph of a homogeneous solution of the one-phase problem~\\eqref{eqn:one-phase-problem}.\n        \\item As $a \\in (0,a^*_k]$, the cones $\\bar{\\mathbf{C}}_{k,a}$ attain every capillary angle $\\theta \\in (0,\\frac{\\pi}{2}]$.\n        \\item The cone $\\bar{\\mathbf{C}}_{k,a^*_k}$ is the free-boundary cone $\\hat{\\mathbf{C}}_{2k,k,\\frac{\\pi}{2}}$, whose double has link given by a minimal embedded $\\mathbb{S}^{k-1} \\times \\mathbb{S}^{k-1} \\times \\mathbb{S}^1 \\hookrightarrow \\mathbb{S}^{2k}$.\n    \\end{enumerate}\n\\end{theorem}\nThis theorem provides further evidence for the importance of the capillary functional in the study of minimal submanifolds, as it provides a bridge between minimal surfaces and solutions of the one-phase problem~\\eqref{eqn:one-phase-problem}.\nThe first instance of such an interpolating family was discovered in~\\cite{FTW-1}.\n\nThe general case of $n \\neq 2k$ in Theorem~\\ref{thm:capillaryConeExistence} (corresponding to $p \\neq q$ in Theorem~\\ref{thm:MinimalSurfaceExistence}) is more challenging than the case $n=2k$ ($p=q$), which enjoys an additional symmetry.\nIn the low cohomogeneity framework of Hsiang, Hsiang, and Lawson~\\cites{hsiang-lawson-jdg , hsiang-hsiang }, the existence of solutions with certain symmetries was obtained via the study of planar dynamical systems; however, their technique encounters difficulties in the current setting, as discussed in~\\cite{carlotto-schulz}.\nCarlotto-Schulz obtain their surfaces by studying a non-planar, $3 \\times 3$ ODE system.\nThe shooting method was also used in Angenent's construction of self-shrinking tori~\\cite{angenent-tori}.\n\nThe main part of our paper is devoted to developing a novel, singular shooting method for graphical solutions of the capillary problem, allowing us to study the minimal surfaces and cones in question via single-variable ODE methods.\nNotably, this provides an effective method to approximate these surfaces, which is often inaccessible through pure variational methods.\nThe existence of a minimally embedded $\\mathbb{S}^{k-1} \\times \\mathbb{S}^{k-1} \\times \\mathbb{S}^1 \\hookrightarrow \\mathbb{S}^{2k}$, connected to a homogeneous one-phase solution by a smooth interpolating family of capillary hypersurfaces, follows using only Lemma~\\ref{lemma:change-of-variables} (symmetry under $t \\leftrightsquigarrow \\sqrt{1-t^2}$) and Lemma~\\ref{lemma:bound-at-sqrt} (a uniform bound for shots reaching zero) to prove Theorem~\\ref{thm:carlotto-schulz}.\nThe constructions of Theorems~\\ref{thm:MinimalSurfaceExistence} and~\\ref{thm:capillaryConeExistence} are also connected to new $O(n-k) \\times O(k)$-invariant homogeneous one-phase solutions constructed and studied in a related paper~\\cite{six-way}.", "sketch": "Theorem~\\ref{thm:MinimalSurfaceExistence} is described as being “obtained from the study of capillary minimal cones as the other endpoint of a capillary interpolation between free-boundary minimal surfaces and the one-phase problem.” The introduction explains the proof strategy via a capillary-to-free-boundary limiting argument: using “a bi-orthogonal symmetry, the capillary minimal surface equation reduces to a free boundary ODE.” For capillary angles $\\theta\\in(0,\\tfrac{\\pi}{2})$, “shooting techniques that match the contact angle are well-defined,” and the authors “show that the resulting solutions converge uniformly to an infinite-slope shot.” Since doubling across the totally geodesic boundary of $\\mathbb{S}^n_+$ turns a free-boundary surface into an embedded minimal surface in the sphere, this limiting “infinite-slope shot” “yields the free-boundary minimal surface in Theorem~\\ref{thm:MinimalSurfaceExistence}.”", "expanded_sketch": "Theorem~\\ref{thm:MinimalSurfaceExistence} is described as being “obtained from the study of capillary minimal cones as the other endpoint of a capillary interpolation between free-boundary minimal surfaces and the one-phase problem.” The introduction explains the proof strategy via a capillary-to-free-boundary limiting argument: using “a bi-orthogonal symmetry, the capillary minimal surface equation reduces to a free boundary ODE.” For capillary angles $\\theta\\in(0,\\tfrac{\\pi}{2})$, “shooting techniques that match the contact angle are well-defined,” and the authors “show that the resulting solutions converge uniformly to an infinite-slope shot.” Since doubling across the totally geodesic boundary of $\\mathbb{S}^n_+$ turns a free-boundary surface into an embedded minimal surface in the sphere, this limiting “infinite-slope shot” is used in establishing the main theorem.", "expanded_theorem": "\\label{thm:MinimalSurfaceExistence}\nFor every $p, q \\geq 1$, there exists a minimal embedding of $\\mathbb{S}^p \\times \\mathbb{S}^q \\times \\mathbb{S}^1$ in $\\mathbb{S}^{p+q+2}$.", "theorem_type": ["Universal–Existential", "Existence"], "mcq": {"question": "Let $p,q$ be integers with $p,q\\ge 1$. Here $\\mathbb S^m$ denotes the standard round $m$-sphere, and a minimal embedding $M\\hookrightarrow \\mathbb S^N$ means a smooth embedding whose image is a minimal submanifold of $\\mathbb S^N$. Which statement holds for the product manifold $\\mathbb S^p\\times \\mathbb S^q\\times \\mathbb S^1$?", "correct_choice": {"label": "A", "text": "For every $p,q\\ge 1$, there exists a minimal embedding $\\mathbb S^p\\times \\mathbb S^q\\times \\mathbb S^1\\hookrightarrow \\mathbb S^{p+q+2}$."}, "choices": [{"label": "B", "text": "For every $p,q\\ge 1$, there exists a minimal embedding $\\mathbb S^p\\times \\mathbb S^q\\times \\mathbb S^1\\hookrightarrow \\mathbb S^{p+q+1}$."}, {"label": "C", "text": "For every $p,q\\ge 1$, there exists a smooth embedding $\\mathbb S^p\\times \\mathbb S^q\\times \\mathbb S^1\\hookrightarrow \\mathbb S^{p+q+2}$."}, {"label": "D", "text": "There exists an integer $N$ such that for every $p,q\\ge 1$, there exists a minimal embedding $\\mathbb S^p\\times \\mathbb S^q\\times \\mathbb S^1\\hookrightarrow \\mathbb S^{N}$."}, {"label": "E", "text": "For every $p,q\\ge 1$, there exists a free-boundary minimal embedding $\\mathbb S^p\\times \\mathbb S^q\\times [0,1]\\hookrightarrow \\mathbb S^{p+q+2}$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "geometric_construction", "tampered_component": "ambient-dimension-after-doubling", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "geometric_construction", "tampered_component": "minimality", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence-of-ambient-dimension-on-pq", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "doubling-free-boundary-link-vs-closed-minimal-submanifold", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly or implicitly reveal that the minimal embedding exists in dimension p+q+2; it only defines terminology and asks which statement is valid."}, "TAS": {"score": 1, "justification": "The correct option appears to closely mirror a specific theorem-level conclusion about S^p x S^q x S^1, so the item is more a recognition of the exact statement than a substantially reworked application."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure: one must compare ambient dimensions, distinguish minimal vs. smooth embedding, and reject quantifier mistakes. However, if the underlying theorem is known, the answer is immediate, and the presence of a weaker true statement reduces the need for strong generative reasoning."}, "DQS": {"score": 1, "justification": "B, D, and E are reasonably distinct and target common confusions about codimension, quantifiers, and free-boundary vs. closed minimality. But C is marked in the metadata as a weaker true statement, so it is not a valid distractor in a single-answer MCQ and creates ambiguity."}, "total_score": 5, "overall_assessment": "The item avoids answer leakage and has some mathematical discrimination, but it is weakened by being close to a theorem restatement and, more seriously, by including a weaker true option among the distractors."}}
{"id": "2602.20143v1", "paper_link": "http://arxiv.org/abs/2602.20143v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:isopery}\n    We have $\\gamma(\\alpha, n) \\leqslant \\frac{(1-\\alpha)(1-(1-\\alpha)^n)}{\\alpha n}$ for all $n\\geqslant 1$ and $\\alpha \\in (0, 1)$.", "start_pos": 5335, "end_pos": 5505, "label": "thm:isopery"}, "ref_dict": {"thm:isopery": "\\begin{theorem}\\label{thm:isopery}\n    We have $\\gamma(\\alpha, n) \\le \\frac{(1-\\alpha)(1-(1-\\alpha)^n)}{\\alpha n}$ for all $n\\ge 1$ and $\\alpha \\in (0, 1)$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2390, "pre_theorem_intro_text": "Let $\\Omega$ be a finite set and let $n\\geqslant 2$. Let $\\mu=\\mu_n$ be the uniform probability measure on $\\Omega^n$. We say that an ordered pair of words $(w, u) \\in \\Omega^n \\times \\Omega^n$ {\\em overlaps} if a final segment of $w$ coincides with an initial segment of $u$. That is, if we denote $w=(w_1, \\ldots, w_n)$, $u=(u_1, \\ldots, u_n)$ then for some $j\\in \\{1, \\ldots, n\\}$ we have $(w_{n-j+1}, \\ldots, w_n) = (u_1, \\ldots, u_j)$. Note that we in particular allow $u = w$. \nWe are interested in the following extremal question: suppose that $A, B\\subset \\Omega^n$ are sets of words such that no two words $w \\in A$ and $u \\in B$ overlap. For what pairs of densities $\\alpha, \\beta \\in (0,1)$ is it possible to have $\\mu(A) \\geqslant \\alpha$ and $\\mu(B) \\geqslant \\beta$?\n\nThere is a related question about non-overlapping codes (also known as `cross-bifix-free' codes) that has been extensively studied in the computer science literature \\cite{bernini2017gray, blackburn2015non, chee2013cross, levenshteindecoding, stanovnik2024search}. In our notation, the question is to determine the size of a largest code $A \\subset \\Omega^n$ such that no two distinct words in $A$ overlap (see \\cite{chee2013cross} for an asymptotically sharp construction). So the question we consider can be thought of as a bipartite variant of this and, to the best of our knowledge, it has not been studied before.\n\nDefine the shift map $s=s_n: \\Omega^n \\rightarrow \\Omega^{n-1}$ by\n\\[\ns(w_1, \\ldots, w_n) = (w_2, \\ldots, w_n).\n\\]\nFor a subset $A \\subset \\Omega^n$ we can define the set of words which do not overlap with $A$ as follows:\n\\[\nU= U(A) = \\Omega^n \\setminus \\bigcup_{j=0}^{n-1} s^j(A) \\times \\Omega^{j}.\n\\]\nIt is easy to see that $U(A)$ is precisely the set of all $u$ such that the pair $(w,u)$ does not overlap for all $w\\in A$. \nLet $\\gamma(\\alpha, n)$ be the largest possible measure of the set $U(A)$ over all $A\\subset \\Omega^n $ of measure $\\alpha$ and all finite sets $\\Omega$. Here we consider the uniform measure on the space $\\Omega^n$. \nNote that if $A \\subset A'$ then we have the inclusion $U(A') \\subset U(A)$. This means that \n$\\gamma(\\alpha, n)$ is a monotone decreasing function in $\\alpha$. For example, it is an easy exercise to show that $\\gamma(\\alpha,2) = \\max\\{ (1-\\alpha)^2, ~1-\\alpha^{1/2} \\}$ holds for any $\\alpha \\in (0,1)$.\n\nOur result is the following estimate.", "context": "Let $\\Omega$ be a finite set and let $n\\geqslant 2$. Let $\\mu=\\mu_n$ be the uniform probability measure on $\\Omega^n$. We say that an ordered pair of words $(w, u) \\in \\Omega^n \\times \\Omega^n$ {\\em overlaps} if a final segment of $w$ coincides with an initial segment of $u$. That is, if we denote $w=(w_1, \\ldots, w_n)$, $u=(u_1, \\ldots, u_n)$ then for some $j\\in \\{1, \\ldots, n\\}$ we have $(w_{n-j+1}, \\ldots, w_n) = (u_1, \\ldots, u_j)$. Note that we in particular allow $u = w$. \nWe are interested in the following extremal question: suppose that $A, B\\subset \\Omega^n$ are sets of words such that no two words $w \\in A$ and $u \\in B$ overlap. For what pairs of densities $\\alpha, \\beta \\in (0,1)$ is it possible to have $\\mu(A) \\geqslant \\alpha$ and $\\mu(B) \\geqslant \\beta$?\n\nThere is a related question about non-overlapping codes (also known as `cross-bifix-free' codes) that has been extensively studied in the computer science literature \\cite{bernini2017gray, blackburn2015non, chee2013cross, levenshteindecoding, stanovnik2024search}. In our notation, the question is to determine the size of a largest code $A \\subset \\Omega^n$ such that no two distinct words in $A$ overlap (see \\cite{chee2013cross} for an asymptotically sharp construction). So the question we consider can be thought of as a bipartite variant of this and, to the best of our knowledge, it has not been studied before.\n\nDefine the shift map $s=s_n: \\Omega^n \\rightarrow \\Omega^{n-1}$ by\n\\[\ns(w_1, \\ldots, w_n) = (w_2, \\ldots, w_n).\n\\]\nFor a subset $A \\subset \\Omega^n$ we can define the set of words which do not overlap with $A$ as follows:\n\\[\nU= U(A) = \\Omega^n \\setminus \\bigcup_{j=0}^{n-1} s^j(A) \\times \\Omega^{j}.\n\\]\nIt is easy to see that $U(A)$ is precisely the set of all $u$ such that the pair $(w,u)$ does not overlap for all $w\\in A$. \nLet $\\gamma(\\alpha, n)$ be the largest possible measure of the set $U(A)$ over all $A\\subset \\Omega^n $ of measure $\\alpha$ and all finite sets $\\Omega$. Here we consider the uniform measure on the space $\\Omega^n$. \nNote that if $A \\subset A'$ then we have the inclusion $U(A') \\subset U(A)$. This means that \n$\\gamma(\\alpha, n)$ is a monotone decreasing function in $\\alpha$. For example, it is an easy exercise to show that $\\gamma(\\alpha,2) = \\max\\{ (1-\\alpha)^2, ~1-\\alpha^{1/2} \\}$ holds for any $\\alpha \\in (0,1)$.\n\nOur result is the following estimate.", "full_context": "Let $\\Omega$ be a finite set and let $n\\geqslant 2$. Let $\\mu=\\mu_n$ be the uniform probability measure on $\\Omega^n$. We say that an ordered pair of words $(w, u) \\in \\Omega^n \\times \\Omega^n$ {\\em overlaps} if a final segment of $w$ coincides with an initial segment of $u$. That is, if we denote $w=(w_1, \\ldots, w_n)$, $u=(u_1, \\ldots, u_n)$ then for some $j\\in \\{1, \\ldots, n\\}$ we have $(w_{n-j+1}, \\ldots, w_n) = (u_1, \\ldots, u_j)$. Note that we in particular allow $u = w$. \nWe are interested in the following extremal question: suppose that $A, B\\subset \\Omega^n$ are sets of words such that no two words $w \\in A$ and $u \\in B$ overlap. For what pairs of densities $\\alpha, \\beta \\in (0,1)$ is it possible to have $\\mu(A) \\geqslant \\alpha$ and $\\mu(B) \\geqslant \\beta$?\n\nThere is a related question about non-overlapping codes (also known as `cross-bifix-free' codes) that has been extensively studied in the computer science literature \\cite{bernini2017gray, blackburn2015non, chee2013cross, levenshteindecoding, stanovnik2024search}. In our notation, the question is to determine the size of a largest code $A \\subset \\Omega^n$ such that no two distinct words in $A$ overlap (see \\cite{chee2013cross} for an asymptotically sharp construction). So the question we consider can be thought of as a bipartite variant of this and, to the best of our knowledge, it has not been studied before.\n\nDefine the shift map $s=s_n: \\Omega^n \\rightarrow \\Omega^{n-1}$ by\n\\[\ns(w_1, \\ldots, w_n) = (w_2, \\ldots, w_n).\n\\]\nFor a subset $A \\subset \\Omega^n$ we can define the set of words which do not overlap with $A$ as follows:\n\\[\nU= U(A) = \\Omega^n \\setminus \\bigcup_{j=0}^{n-1} s^j(A) \\times \\Omega^{j}.\n\\]\nIt is easy to see that $U(A)$ is precisely the set of all $u$ such that the pair $(w,u)$ does not overlap for all $w\\in A$. \nLet $\\gamma(\\alpha, n)$ be the largest possible measure of the set $U(A)$ over all $A\\subset \\Omega^n $ of measure $\\alpha$ and all finite sets $\\Omega$. Here we consider the uniform measure on the space $\\Omega^n$. \nNote that if $A \\subset A'$ then we have the inclusion $U(A') \\subset U(A)$. This means that \n$\\gamma(\\alpha, n)$ is a monotone decreasing function in $\\alpha$. For example, it is an easy exercise to show that $\\gamma(\\alpha,2) = \\max\\{ (1-\\alpha)^2, ~1-\\alpha^{1/2} \\}$ holds for any $\\alpha \\in (0,1)$.\n\nOur result is the following estimate.\n\nOur result is the following estimate.\n\nSo if we have a pair of non-overlapping sets $A, B\\subset \\Omega^n$ with densities $\\alpha$ and $\\beta$ then we have $\\alpha\\beta n\\le (1-\\alpha)(1-(1-\\alpha)^n) \\le 1 $.\n\nWe can interpolate between these two examples by taking $A = \\Omega^{n-k} \\times S^k$ for some $1 \\le k \\le n$.\nThen the set $U = U(A)$ is given by\n\\begin{align*}\nU = \\{ (w_1, \\ldots, w_n):~ w_1 \\not\\in S, \\quad \\{w_{j+1}, \\ldots, w_{j+k}\\}\\not\\subset S,~j=0, \\ldots, n-k \\}. \n\\end{align*}\nThe exact formula for $\\mu(U)$ is a bit complicated (it involves generalized Fibonacci numbers, see \\cite{chee2013cross}) but we can use a simple Poisson approximation inequality due to \\cite{arratia1989two} (see also \\cite{godbole1991poisson, godbole1993improved, guibas1980long}) to get a good estimate of the measure of $U$.\nDenote $p = |S|/|\\Omega|$ so that $\\alpha = p^k$. \nLet $Z_1, \\ldots, Z_{n-1} \\sim \\operatorname{Ber}(p)$ be iid Bernoulli random variables. Let $R_{n-1}$ be the length of the longest run of 1-s in the sequence $(Z_1, \\ldots, Z_{n-1})$. The measure of $U$ can then be computed in terms of $R_{n-1}$:\n\\[\n\\mu(U) = (1-p)\\Pr[ R_{n-1} < k].\n\\]\nIndeed, we can view $w_2, \\ldots, w_n \\in \\Omega$ as iid variables uniformly distributed on $\\Omega$ and select $Z_i = 1_{w_{i+1} \\in S}$. \nBy \\cite[Example 3]{arratia1989two} we have the following estimate on this probability:\n\\[\n\\left|\\Pr[ R_{n-1} < k] - e^{-\\lambda}\\right| \\le \\frac{\\lambda (2k+1)}{n-1} + 2 p^k, \\quad \\lambda = p^k ((n-2)(1-p)+1)\n\\]\nLet $k = [n\\alpha \\log(1/\\alpha) ]$. Then for $1/n \\ll \\alpha \\ll 1$ we have $p = \\alpha^{1/k} = e^{-\\frac{\\log(1/\\alpha)}{k}} = 1 - \\frac{1+o(1)}{n\\alpha}$. This gives $\\lambda = 1+o(1)$ and $\\Pr[R_{n-1} < k] = e^{-1} + o(1)$ and so we have\n\\[\n\\mu(U) = \\frac{e^{-1}+o(1)}{\\alpha n},\n\\] \nwhere $o(1)$ tends to zero as $\\min(\\alpha^{-1}, n\\alpha) \\to \\infty$. This matches the bound in Theorem \\ref{thm:isopery} for all $\\alpha \\in (1/n, 1/2)$ up to a constant factor.\n\nWe trivially have $\\beta_{1}=\\alpha_{1}$. Since $A_{j} = s(A_{j+1})$, we have the inclusion $A_{j+1} \\subset \\Omega\\times A_{j} $. We have $B_{j+1} = (B_j \\times \\Omega) \\cup A_{j+1}$ and, in particular $B_j \\times \\Omega \\subset B_{j+1}$. Together these observations imply the following chain of inequalities:\n\\[\n\\beta_n \\ge \\ldots \\ge \\beta_1 = \\alpha_1 \\ge \\ldots \\ge \\alpha_n.\n\\]\nNow let us define sets $D_j$ as follows:\n\\[\nD_j = A_j \\setminus (B_{j-1} \\times \\Omega) = B_j \\setminus (B_{j-1} \\times \\Omega),\n\\]\nwhere for $j=1$ we put $D_1 = A_1=B_1$. \nIn particular, since $B_{j-1} \\times \\Omega \\subset B_{j}$, we can write $B_j$ as a disjoint union $(B_{j-1}\\times \\Omega) \\sqcup D_j$ and\n$\\mu(D_j) = \\beta_j-\\beta_{j-1}$ for all $j=1, \\ldots, n$ (where we set $\\beta_0=0$).\nNote that we can write \n\\begin{align*}\nB_j &= D_j \\sqcup (B_{j-1}\\times \\Omega) = D_j \\sqcup (D_{j-1}\\times \\Omega) \\sqcup (B_{j-2}\\times \\Omega^2) = \\ldots \\\\\n&= D_j \\sqcup (D_{j-1}\\times \\Omega) \\sqcup (D_{j-2}\\times \\Omega^2) \\sqcup \\ldots \\sqcup (D_{1}\\times \\Omega^{j-1}) \\\\\n& = A_j \\cup \\left( (D_{j-1}\\times \\Omega) \\sqcup (D_{j-2}\\times \\Omega^2) \\sqcup \\ldots \\sqcup (D_{1}\\times \\Omega^{j-1}) \\right).\n\\end{align*}\nUsing Observation \\ref{obs1}, we have the following bounds for $i=1, \\ldots, j-1$:\n\\[\n\\mu(A_j \\cap (D_i \\times \\Omega^{j-i})) \\le \\lambda_{j, i}\\cdot \\mu(D_i) \\le \\alpha_{j-i} \\cdot \\mu(D_i)\n\\]\nSo since sets $D_i \\times \\Omega^{j-i}$ are pairwise disjoint, we obtain\n\\[\n\\mu(B_j) = \\mu(A_j) + \\sum_{i=1}^{j-1} \\mu(D_i \\times \\Omega^{j-i} \\setminus A_j) \\ge \\mu(A_j) + \\sum_{i=1}^{j-1} (1- \\alpha_{j- i}) \\mu(D_{i})\n\\]\ngiving the following relation between $\\alpha$-s and $\\beta$-s:\n\\begin{equation}\\label{eq:beta}\n \\beta_j \\ge \\alpha_j + \\sum_{i=1}^{j-1} (1-\\alpha_{j-i}) (\\beta_{i}-\\beta_{i-1}).\n\\end{equation}\nDenote $\\gamma_i = 1-\\beta_i = \\mu(U(A_i))$ and let $\\delta_i = \\alpha_{i-1}-\\alpha_i$ for $i=1, \\ldots, n$ where we put $\\alpha_0=1$ and $\\gamma_0=1$. Then (\\ref{eq:beta}) can be rewritten as follows:\n\\begin{equation}\\label{eq:gamma}\n \\gamma_j \\le \\sum_{i=0}^{j-1} \\gamma_i\\delta_{j-i}.\n\\end{equation}\nWe also have the following information about $\\gamma_i, \\delta_i$: \n\\[\n\\gamma_n \\le \\gamma_{n-1} \\le \\ldots \\le \\gamma_1 \\le \\gamma_0 = 1,\n\\]\n\\[\n\\delta_1+\\ldots+\\delta_n = 1-\\alpha, \\quad \\delta_i \\ge 0, \\quad i=1, \\ldots, n.\n\\]\nWe will use these properties to upper bound $\\gamma_n$. The idea is to use (\\ref{eq:gamma}) to compare $\\gamma$ with a random walk on $\\Z$.\n\nLet $p_i = \\frac{\\delta_i}{1-\\alpha}$ and let $Z$ be the random variable supported on $\\{1, \\ldots, n\\}$ given by the following distribution:\n\\[\n\\Pr[Z = i] = p_i.\n\\]\nLet $Z_1, \\ldots, Z_n$ be i.i.d. copies of $Z$.\n\\begin{obs}\n For every $j=0, \\ldots, n$ we have the following:\n \\begin{equation}\\label{eq:probability}\n \\gamma_j \\le \\sum_{s= 0}^n (1-\\alpha)^s \\Pr[Z_1+\\ldots+Z_s = j]. \n \\end{equation}\n\\end{obs}\n\\begin{proof}\n We prove this by induction on $j$. The base case $j=0$ is clear since only $s=0$ term contributes. Now for $j\\ge 1$ we have by (\\ref{eq:gamma}):\n \\begin{align*}\n \\gamma_j &\\le \\sum_{i=0}^{j-1} \\gamma_i \\delta_{j-i} = \n (1-\\alpha) \\sum_{i=0}^{j-1} \\gamma_i p_{j-i} \\\\\n &\\le (1-\\alpha) \\sum_{i=0}^{j-1} \\sum_{s \\ge 0} (1-\\alpha)^s \\Pr[Z_1+\\ldots+Z_s = i] \\Pr[Z_{s+1} = j-i] \\\\\n &=\\sum_{s\\ge 0} (1-\\alpha)^s\\Pr[Z_1+\\ldots+Z_s = j].\n \\end{align*}\n\\end{proof}\n\n\\begin{cor}\\label{cor:isoperimetry}\n Let $A \\subset \\Omega^n$ be a set of measure $\\alpha \\in (0,1)$. For $w \\in \\Omega^n$ define $f(w)$ to be the number of indices $j\\in \\{0, \\ldots, n-1\\}$ such that $w \\in s^j(A) \\times \\Omega^{j}$.\n Then for every integer $t \\in [1, n/4]$ we have the following level set estimate on $f$:\n \\begin{equation}\n \\mu(\\{w\\in \\Omega^n:~ f(w) \\le t\\}) \\le \\frac{8t}{\\alpha n}.\n \\end{equation}\n\\end{cor}\n\nLet $\\tilde n = [n/2t]$ and $r = n-2t \\tilde n$. By the assumption on $t$ we have $\\tilde n\\ge 1$. Consider a new alphabet $\\tilde \\Omega = \\Omega^{2t}$ and\n let $\\tilde s = \\tilde s_j: \\tilde \\Omega^{j} \\rightarrow \\tilde \\Omega^{j-1}$ denote the shift map defined on words over the alphabet $\\tilde \\Omega$. By identifying $\\tilde \\Omega^j = \\Omega^{2t j}$, we get that $\\tilde s = s^{2t}$. For $i=0, \\ldots, 2t-1$ let \n \\[\n \\tilde A_i = s^{r+i}(A) \\times \\Omega^{i} \\subset \\Omega^{2t \\tilde n} = \\tilde \\Omega^{\\tilde n}.\n \\]\n Note that $\\mu(\\tilde A_i) \\ge \\mu(A) =\\alpha$ for $i=0, \\ldots, 2t-1$. For an arbitrary subset $\\tilde A \\subset \\tilde \\Omega^{\\tilde n}$ we denote $\\tilde U(\\tilde A) = \\tilde \\Omega^{\\tilde n} \\setminus \\bigcup_{j=0}^{\\tilde n-1} \\tilde s^j(\\tilde A) \\times \\tilde \\Omega^{j}$, that is the analogue of $U(A)$ over the new alphabet.\n Let $w \\in \\Omega^n$ and denote $\\tilde w = s^r(w) \\in \\tilde\\Omega^{\\tilde n}$.\n Note that we have $\\tilde w \\not \\in \\tilde U(\\tilde A_i)$ precisely when there exists $j \\in \\{0, \\ldots, \\tilde n-1\\}$ such that $\\tilde w \\in \\tilde s^j(\\tilde A_i) \\times \\tilde \\Omega^j$. The latter is in turn equivalent to $w \\in s^{2t j + i + r}(A)$. It follows that we have\n \\[\n \\#\\{i \\in \\{0, \\ldots, 2t-1\\}: \\tilde w \\not \\in \\tilde U(\\tilde A_i)\\} \\le \\#\\{ i\\in \\{0, \\ldots, n-1\\}: w \\in s^{i}(A)\\times \\Omega^i \\} = f(w)\n \\]\n Thus, if $f(w) \\le t$ then there are at least $2t-t=t$ indices $i \\in \\{0, \\ldots, 2t-1\\}$ such that $\\tilde w \\in \\tilde U(\\tilde A_i)$. So by the union bound and Theorem \\ref{thm:isopery} applied to each $\\tilde U(\\tilde A_i)$ we have \n \\[\n t\\mu(\\{w:~f(w) \\le t\\}) \\le \\sum_{i=0}^{2t-1} \\mu(\\tilde U(\\tilde A_i)) \\le 2t \\frac{(1-\\alpha) (1 - (1-\\alpha)^{\\tilde n})}{\\alpha \\tilde n} \\le \\frac{2t}{\\alpha \\tilde n}.\n \\]\n So recalling that $\\tilde n = [n / 2t]$ we get\n \\[\n \\mu(\\{w:~f(w) \\le t\\}) \\le \\frac{2}{\\alpha [n/2t]} \\le \\frac{8t}{\\alpha n}\n \\]\n provided that $n \\ge 4t$, concluding the proof.\n\\end{proof}", "post_theorem_intro_text_len": 3064, "post_theorem_intro_text": "So if we have a pair of non-overlapping sets $A, B\\subset \\Omega^n$ with densities $\\alpha$ and $\\beta$ then we have $\\alpha\\beta  n\\leqslant (1-\\alpha)(1-(1-\\alpha)^n) \\leqslant 1 $. \n\nThe proof of Theorem \\ref{thm:isopery} is presented in the next section. The rough idea is to split the set $\\Omega^n \\setminus U(A)$ into several disjoint pieces and use inclusion-exclusion to lower bound the size of each piece. This then gives a certain recursive relationship between various densities associated with $A$ and $U(A)$ and their shifts. This relationship can be interpreted in terms of a certain random walk leading to the desired estimate.\n\nWe close this section by considering some examples essentially matching the upper bound in Theorem \\ref{thm:isopery}. Let $S \\subset \\Omega$ be an arbitrary subset.\nThen for $A = S^n$ one can check that\n\\[\nU(A) = (\\Omega\\setminus S)\\times \\Omega^{n-1}\n\\]\nand so we get\n\\[\n\\mu(U(A)) = 1-\\alpha^{1/n} = \\frac{\\log(1/\\alpha)}{n} + O\\left(\\frac{\\log^2(1/\\alpha)}{n^2}\\right).\n\\]\nThis is a good bound for $\\alpha \\in (1/2, 1)$. \n\nSimilarly, for $A = \\Omega^{n-1} \\times S$ one can check that\n\\[\nU(A) = (\\Omega\\setminus S)^n\n\\]\nso that we get\n\\[\n\\mu(U(A)) = (1-\\alpha)^n.\n\\]\nThis is a good bound for $\\alpha \\in (0, 1/n)$. \n\nWe can interpolate between these two examples by taking $A = \\Omega^{n-k} \\times S^k$ for some $1 \\leqslant k \\leqslant n$.\nThen the set $U = U(A)$ is given by\n\\begin{align*}\nU =  \\{ (w_1, \\ldots, w_n):~ w_1 \\not\\in S, \\quad \\{w_{j+1}, \\ldots, w_{j+k}\\}\\not\\subset S,~j=0, \\ldots, n-k \\}.    \n\\end{align*}\nThe exact formula for $\\mu(U)$ is a bit complicated (it involves generalized Fibonacci numbers, see \\cite{chee2013cross}) but we can use a simple Poisson approximation inequality due to \\cite{arratia1989two} (see also \\cite{godbole1991poisson, godbole1993improved, guibas1980long}) to get a good estimate of the measure of $U$.\nDenote $p = |S|/|\\Omega|$ so that $\\alpha = p^k$. \nLet $Z_1, \\ldots, Z_{n-1} \\sim \\operatorname{Ber}(p)$ be iid Bernoulli random variables. Let $R_{n-1}$ be the length of the longest run of 1-s in the sequence $(Z_1, \\ldots, Z_{n-1})$. The measure of $U$ can then be computed in terms of $R_{n-1}$:\n\\[\n\\mu(U) = (1-p)\\Pr[ R_{n-1} < k].\n\\]\nIndeed, we can view $w_2, \\ldots, w_n \\in \\Omega$ as iid variables uniformly distributed on $\\Omega$ and select $Z_i = 1_{w_{i+1} \\in S}$. \nBy \\cite[Example 3]{arratia1989two} we have the following estimate on this probability:\n\\[\n\\left|\\Pr[ R_{n-1} < k] - e^{-\\lambda}\\right| \\leqslant \\frac{\\lambda (2k+1)}{n-1} + 2 p^k, \\quad \\lambda = p^k ((n-2)(1-p)+1)\n\\]\nLet $k = [n\\alpha \\log(1/\\alpha) ]$. Then for $1/n \\ll \\alpha \\ll 1$ we have $p = \\alpha^{1/k} = e^{-\\frac{\\log(1/\\alpha)}{k}} = 1 - \\frac{1+o(1)}{n\\alpha}$. This gives $\\lambda = 1+o(1)$ and $\\Pr[R_{n-1} < k] = e^{-1} + o(1)$ and so we have\n\\[\n\\mu(U) = \\frac{e^{-1}+o(1)}{\\alpha n},\n\\] \nwhere $o(1)$ tends to zero as $\\min(\\alpha^{-1}, n\\alpha) \\to \\infty$. This matches the bound in Theorem \\ref{thm:isopery} for all $\\alpha \\in (1/n, 1/2)$ up to a constant factor.", "sketch": "The post-theorem introduction says: \"The proof of Theorem \\ref{thm:isopery} is presented in the next section. The rough idea is to split the set $\\Omega^n \\setminus U(A)$ into several disjoint pieces and use inclusion-exclusion to lower bound the size of each piece.\" This \"then gives a certain recursive relationship between various densities associated with $A$ and $U(A)$ and their shifts.\" Finally, \"[t]his relationship can be interpreted in terms of a certain random walk leading to the desired estimate.\"", "expanded_sketch": "The post-theorem introduction says: \"The proof of the main theorem is presented in the next section. The rough idea is to split the set $\\Omega^n \\setminus U(A)$ into several disjoint pieces and use inclusion-exclusion to lower bound the size of each piece.\" This \"then gives a certain recursive relationship between various densities associated with $A$ and $U(A)$ and their shifts.\" Finally, \"[t]his relationship can be interpreted in terms of a certain random walk leading to the desired estimate.\"", "expanded_theorem": "\\label{thm:isopery}\n    We have $\\gamma(\\alpha, n) \\leqslant \\frac{(1-\\alpha)(1-(1-\\alpha)^n)}{\\alpha n}$ for all $n\\geqslant 1$ and $\\alpha \\in (0, 1)$.", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "For a finite set $\\Omega$ and an integer $n\\ge 1$, let $\\mu$ be the uniform probability measure on $\\Omega^n$. Let $s: \\Omega^n\\to\\Omega^{n-1}$ be the shift map $s(w_1,\\dots,w_n)=(w_2,\\dots,w_n)$, and let $s^j$ denote its $j$-fold iterate (with $s^0$ the identity). For a set $A\\subset \\Omega^n$, define\n\\[\nU(A)=\\Omega^n\\setminus \\bigcup_{j=0}^{n-1} s^j(A)\\times \\Omega^j,\n\\]\nso $U(A)$ is the set of words in $\\Omega^n$ that do not overlap any word of $A$. For $\\alpha\\in(0,1)$, let $\\gamma(\\alpha,n)$ be the largest possible value of $\\mu(U(A))$ over all finite alphabets $\\Omega$ and all sets $A\\subset \\Omega^n$ with $\\mu(A)=\\alpha$. Which statement holds for every $n\\ge 1$ and every $\\alpha\\in(0,1)$?", "correct_choice": {"label": "A", "text": "$\\gamma(\\alpha,n)\\le \\dfrac{(1-\\alpha)\\bigl(1-(1-\\alpha)^n\\bigr)}{\\alpha n}$."}, "choices": [{"label": "B", "text": "$\\gamma(\\alpha,n)\\le \\dfrac{(1-\\alpha)\\bigl(1-(1-\\alpha)^{n-1}\\bigr)}{\\alpha n}$."}, {"label": "C", "text": "$\\gamma(\\alpha,n)\\le \\dfrac{1-\\alpha}{\\alpha n}$."}, {"label": "D", "text": "$\\gamma(\\alpha,n)\\le \\dfrac{(1-\\alpha)\\bigl(1-(1-\\alpha)^n\\bigr)}{\\alpha}$."}, {"label": "E", "text": "$\\gamma(\\alpha,n)= \\dfrac{(1-\\alpha)\\bigl(1-(1-\\alpha)^n\\bigr)}{\\alpha n}$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "counting_estimate", "tampered_component": "number_of_shift_levels", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "dropped_factor_1-(1-\\alpha)^n", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "random_walk", "tampered_component": "averaging_by_n", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "upper_bound_not_sharp_for_all_sets", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem only defines the overlap setup and the extremal quantity \u0007gamma(\u0007alpha,n); it does not explicitly reveal the bound or give a strong hint toward the exact prefactor in the correct estimate."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the task is to identify the quantitative estimate that holds, and the correct option is the theorem statement itself rather than a conclusion derived from a novel scenario."}, "GPS": {"score": 1, "justification": "There is some pressure to distinguish the sharp bound from a weaker true bound and nearby false variants, but the item mainly rewards recognition/recall of the known estimate rather than generating a substantial chain of reasoning from the definitions."}, "DQS": {"score": 2, "justification": "The distractors are mathematically meaningful and well targeted: one is a weaker true statement, one omits a key factor, one substitutes a terminal term for a cumulative term, and one incorrectly upgrades an upper bound to equality."}, "total_score": 5, "overall_assessment": "A solid multiple-choice theorem-identification item with strong distractors and no answer leakage, but it is largely a restatement/recall question rather than a genuinely generative reasoning problem."}}
{"id": "2602.20143v1", "paper_link": "http://arxiv.org/abs/2602.20143v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:isopery}\n    We have $\\gamma(\\alpha, n) \\leqslant \\frac{(1-\\alpha)(1-(1-\\alpha)^n)}{\\alpha n}$ for all $n\\geqslant 1$ and $\\alpha \\in (0, 1)$.", "start_pos": 5335, "end_pos": 5505, "label": "thm:isopery"}, "ref_dict": {"thm:isopery": "\\begin{theorem}\\label{thm:isopery}\n    We have $\\gamma(\\alpha, n) \\le \\frac{(1-\\alpha)(1-(1-\\alpha)^n)}{\\alpha n}$ for all $n\\ge 1$ and $\\alpha \\in (0, 1)$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2390, "pre_theorem_intro_text": "Let $\\Omega$ be a finite set and let $n\\geqslant 2$. Let $\\mu=\\mu_n$ be the uniform probability measure on $\\Omega^n$. We say that an ordered pair of words $(w, u) \\in \\Omega^n \\times \\Omega^n$ {\\em overlaps} if a final segment of $w$ coincides with an initial segment of $u$. That is, if we denote $w=(w_1, \\ldots, w_n)$, $u=(u_1, \\ldots, u_n)$ then for some $j\\in \\{1, \\ldots, n\\}$ we have $(w_{n-j+1}, \\ldots, w_n) = (u_1, \\ldots, u_j)$. Note that we in particular allow $u = w$. \nWe are interested in the following extremal question: suppose that $A, B\\subset \\Omega^n$ are sets of words such that no two words $w \\in A$ and $u \\in B$ overlap. For what pairs of densities $\\alpha, \\beta \\in (0,1)$ is it possible to have $\\mu(A) \\geqslant \\alpha$ and $\\mu(B) \\geqslant \\beta$?\n\nThere is a related question about non-overlapping codes (also known as `cross-bifix-free' codes) that has been extensively studied in the computer science literature \\cite{bernini2017gray, blackburn2015non, chee2013cross, levenshteindecoding, stanovnik2024search}. In our notation, the question is to determine the size of a largest code $A \\subset \\Omega^n$ such that no two distinct words in $A$ overlap (see \\cite{chee2013cross} for an asymptotically sharp construction). So the question we consider can be thought of as a bipartite variant of this and, to the best of our knowledge, it has not been studied before.\n\nDefine the shift map $s=s_n: \\Omega^n \\rightarrow \\Omega^{n-1}$ by\n\\[\ns(w_1, \\ldots, w_n) = (w_2, \\ldots, w_n).\n\\]\nFor a subset $A \\subset \\Omega^n$ we can define the set of words which do not overlap with $A$ as follows:\n\\[\nU= U(A) = \\Omega^n \\setminus \\bigcup_{j=0}^{n-1} s^j(A) \\times \\Omega^{j}.\n\\]\nIt is easy to see that $U(A)$ is precisely the set of all $u$ such that the pair $(w,u)$ does not overlap for all $w\\in A$. \nLet $\\gamma(\\alpha, n)$ be the largest possible measure of the set $U(A)$ over all $A\\subset \\Omega^n $ of measure $\\alpha$ and all finite sets $\\Omega$. Here we consider the uniform measure on the space $\\Omega^n$. \nNote that if $A \\subset A'$ then we have the inclusion $U(A') \\subset U(A)$. This means that \n$\\gamma(\\alpha, n)$ is a monotone decreasing function in $\\alpha$. For example, it is an easy exercise to show that $\\gamma(\\alpha,2) = \\max\\{ (1-\\alpha)^2, ~1-\\alpha^{1/2} \\}$ holds for any $\\alpha \\in (0,1)$.\n\nOur result is the following estimate.", "context": "Let $\\Omega$ be a finite set and let $n\\geqslant 2$. Let $\\mu=\\mu_n$ be the uniform probability measure on $\\Omega^n$. We say that an ordered pair of words $(w, u) \\in \\Omega^n \\times \\Omega^n$ {\\em overlaps} if a final segment of $w$ coincides with an initial segment of $u$. That is, if we denote $w=(w_1, \\ldots, w_n)$, $u=(u_1, \\ldots, u_n)$ then for some $j\\in \\{1, \\ldots, n\\}$ we have $(w_{n-j+1}, \\ldots, w_n) = (u_1, \\ldots, u_j)$. Note that we in particular allow $u = w$. \nWe are interested in the following extremal question: suppose that $A, B\\subset \\Omega^n$ are sets of words such that no two words $w \\in A$ and $u \\in B$ overlap. For what pairs of densities $\\alpha, \\beta \\in (0,1)$ is it possible to have $\\mu(A) \\geqslant \\alpha$ and $\\mu(B) \\geqslant \\beta$?\n\nThere is a related question about non-overlapping codes (also known as `cross-bifix-free' codes) that has been extensively studied in the computer science literature \\cite{bernini2017gray, blackburn2015non, chee2013cross, levenshteindecoding, stanovnik2024search}. In our notation, the question is to determine the size of a largest code $A \\subset \\Omega^n$ such that no two distinct words in $A$ overlap (see \\cite{chee2013cross} for an asymptotically sharp construction). So the question we consider can be thought of as a bipartite variant of this and, to the best of our knowledge, it has not been studied before.\n\nDefine the shift map $s=s_n: \\Omega^n \\rightarrow \\Omega^{n-1}$ by\n\\[\ns(w_1, \\ldots, w_n) = (w_2, \\ldots, w_n).\n\\]\nFor a subset $A \\subset \\Omega^n$ we can define the set of words which do not overlap with $A$ as follows:\n\\[\nU= U(A) = \\Omega^n \\setminus \\bigcup_{j=0}^{n-1} s^j(A) \\times \\Omega^{j}.\n\\]\nIt is easy to see that $U(A)$ is precisely the set of all $u$ such that the pair $(w,u)$ does not overlap for all $w\\in A$. \nLet $\\gamma(\\alpha, n)$ be the largest possible measure of the set $U(A)$ over all $A\\subset \\Omega^n $ of measure $\\alpha$ and all finite sets $\\Omega$. Here we consider the uniform measure on the space $\\Omega^n$. \nNote that if $A \\subset A'$ then we have the inclusion $U(A') \\subset U(A)$. This means that \n$\\gamma(\\alpha, n)$ is a monotone decreasing function in $\\alpha$. For example, it is an easy exercise to show that $\\gamma(\\alpha,2) = \\max\\{ (1-\\alpha)^2, ~1-\\alpha^{1/2} \\}$ holds for any $\\alpha \\in (0,1)$.\n\nOur result is the following estimate.", "full_context": "Let $\\Omega$ be a finite set and let $n\\geqslant 2$. Let $\\mu=\\mu_n$ be the uniform probability measure on $\\Omega^n$. We say that an ordered pair of words $(w, u) \\in \\Omega^n \\times \\Omega^n$ {\\em overlaps} if a final segment of $w$ coincides with an initial segment of $u$. That is, if we denote $w=(w_1, \\ldots, w_n)$, $u=(u_1, \\ldots, u_n)$ then for some $j\\in \\{1, \\ldots, n\\}$ we have $(w_{n-j+1}, \\ldots, w_n) = (u_1, \\ldots, u_j)$. Note that we in particular allow $u = w$. \nWe are interested in the following extremal question: suppose that $A, B\\subset \\Omega^n$ are sets of words such that no two words $w \\in A$ and $u \\in B$ overlap. For what pairs of densities $\\alpha, \\beta \\in (0,1)$ is it possible to have $\\mu(A) \\geqslant \\alpha$ and $\\mu(B) \\geqslant \\beta$?\n\nThere is a related question about non-overlapping codes (also known as `cross-bifix-free' codes) that has been extensively studied in the computer science literature \\cite{bernini2017gray, blackburn2015non, chee2013cross, levenshteindecoding, stanovnik2024search}. In our notation, the question is to determine the size of a largest code $A \\subset \\Omega^n$ such that no two distinct words in $A$ overlap (see \\cite{chee2013cross} for an asymptotically sharp construction). So the question we consider can be thought of as a bipartite variant of this and, to the best of our knowledge, it has not been studied before.\n\nDefine the shift map $s=s_n: \\Omega^n \\rightarrow \\Omega^{n-1}$ by\n\\[\ns(w_1, \\ldots, w_n) = (w_2, \\ldots, w_n).\n\\]\nFor a subset $A \\subset \\Omega^n$ we can define the set of words which do not overlap with $A$ as follows:\n\\[\nU= U(A) = \\Omega^n \\setminus \\bigcup_{j=0}^{n-1} s^j(A) \\times \\Omega^{j}.\n\\]\nIt is easy to see that $U(A)$ is precisely the set of all $u$ such that the pair $(w,u)$ does not overlap for all $w\\in A$. \nLet $\\gamma(\\alpha, n)$ be the largest possible measure of the set $U(A)$ over all $A\\subset \\Omega^n $ of measure $\\alpha$ and all finite sets $\\Omega$. Here we consider the uniform measure on the space $\\Omega^n$. \nNote that if $A \\subset A'$ then we have the inclusion $U(A') \\subset U(A)$. This means that \n$\\gamma(\\alpha, n)$ is a monotone decreasing function in $\\alpha$. For example, it is an easy exercise to show that $\\gamma(\\alpha,2) = \\max\\{ (1-\\alpha)^2, ~1-\\alpha^{1/2} \\}$ holds for any $\\alpha \\in (0,1)$.\n\nOur result is the following estimate.\n\nOur result is the following estimate.\n\nSo if we have a pair of non-overlapping sets $A, B\\subset \\Omega^n$ with densities $\\alpha$ and $\\beta$ then we have $\\alpha\\beta n\\le (1-\\alpha)(1-(1-\\alpha)^n) \\le 1 $.\n\nWe can interpolate between these two examples by taking $A = \\Omega^{n-k} \\times S^k$ for some $1 \\le k \\le n$.\nThen the set $U = U(A)$ is given by\n\\begin{align*}\nU = \\{ (w_1, \\ldots, w_n):~ w_1 \\not\\in S, \\quad \\{w_{j+1}, \\ldots, w_{j+k}\\}\\not\\subset S,~j=0, \\ldots, n-k \\}. \n\\end{align*}\nThe exact formula for $\\mu(U)$ is a bit complicated (it involves generalized Fibonacci numbers, see \\cite{chee2013cross}) but we can use a simple Poisson approximation inequality due to \\cite{arratia1989two} (see also \\cite{godbole1991poisson, godbole1993improved, guibas1980long}) to get a good estimate of the measure of $U$.\nDenote $p = |S|/|\\Omega|$ so that $\\alpha = p^k$. \nLet $Z_1, \\ldots, Z_{n-1} \\sim \\operatorname{Ber}(p)$ be iid Bernoulli random variables. Let $R_{n-1}$ be the length of the longest run of 1-s in the sequence $(Z_1, \\ldots, Z_{n-1})$. The measure of $U$ can then be computed in terms of $R_{n-1}$:\n\\[\n\\mu(U) = (1-p)\\Pr[ R_{n-1} < k].\n\\]\nIndeed, we can view $w_2, \\ldots, w_n \\in \\Omega$ as iid variables uniformly distributed on $\\Omega$ and select $Z_i = 1_{w_{i+1} \\in S}$. \nBy \\cite[Example 3]{arratia1989two} we have the following estimate on this probability:\n\\[\n\\left|\\Pr[ R_{n-1} < k] - e^{-\\lambda}\\right| \\le \\frac{\\lambda (2k+1)}{n-1} + 2 p^k, \\quad \\lambda = p^k ((n-2)(1-p)+1)\n\\]\nLet $k = [n\\alpha \\log(1/\\alpha) ]$. Then for $1/n \\ll \\alpha \\ll 1$ we have $p = \\alpha^{1/k} = e^{-\\frac{\\log(1/\\alpha)}{k}} = 1 - \\frac{1+o(1)}{n\\alpha}$. This gives $\\lambda = 1+o(1)$ and $\\Pr[R_{n-1} < k] = e^{-1} + o(1)$ and so we have\n\\[\n\\mu(U) = \\frac{e^{-1}+o(1)}{\\alpha n},\n\\] \nwhere $o(1)$ tends to zero as $\\min(\\alpha^{-1}, n\\alpha) \\to \\infty$. This matches the bound in Theorem \\ref{thm:isopery} for all $\\alpha \\in (1/n, 1/2)$ up to a constant factor.\n\nWe trivially have $\\beta_{1}=\\alpha_{1}$. Since $A_{j} = s(A_{j+1})$, we have the inclusion $A_{j+1} \\subset \\Omega\\times A_{j} $. We have $B_{j+1} = (B_j \\times \\Omega) \\cup A_{j+1}$ and, in particular $B_j \\times \\Omega \\subset B_{j+1}$. Together these observations imply the following chain of inequalities:\n\\[\n\\beta_n \\ge \\ldots \\ge \\beta_1 = \\alpha_1 \\ge \\ldots \\ge \\alpha_n.\n\\]\nNow let us define sets $D_j$ as follows:\n\\[\nD_j = A_j \\setminus (B_{j-1} \\times \\Omega) = B_j \\setminus (B_{j-1} \\times \\Omega),\n\\]\nwhere for $j=1$ we put $D_1 = A_1=B_1$. \nIn particular, since $B_{j-1} \\times \\Omega \\subset B_{j}$, we can write $B_j$ as a disjoint union $(B_{j-1}\\times \\Omega) \\sqcup D_j$ and\n$\\mu(D_j) = \\beta_j-\\beta_{j-1}$ for all $j=1, \\ldots, n$ (where we set $\\beta_0=0$).\nNote that we can write \n\\begin{align*}\nB_j &= D_j \\sqcup (B_{j-1}\\times \\Omega) = D_j \\sqcup (D_{j-1}\\times \\Omega) \\sqcup (B_{j-2}\\times \\Omega^2) = \\ldots \\\\\n&= D_j \\sqcup (D_{j-1}\\times \\Omega) \\sqcup (D_{j-2}\\times \\Omega^2) \\sqcup \\ldots \\sqcup (D_{1}\\times \\Omega^{j-1}) \\\\\n& = A_j \\cup \\left( (D_{j-1}\\times \\Omega) \\sqcup (D_{j-2}\\times \\Omega^2) \\sqcup \\ldots \\sqcup (D_{1}\\times \\Omega^{j-1}) \\right).\n\\end{align*}\nUsing Observation \\ref{obs1}, we have the following bounds for $i=1, \\ldots, j-1$:\n\\[\n\\mu(A_j \\cap (D_i \\times \\Omega^{j-i})) \\le \\lambda_{j, i}\\cdot \\mu(D_i) \\le \\alpha_{j-i} \\cdot \\mu(D_i)\n\\]\nSo since sets $D_i \\times \\Omega^{j-i}$ are pairwise disjoint, we obtain\n\\[\n\\mu(B_j) = \\mu(A_j) + \\sum_{i=1}^{j-1} \\mu(D_i \\times \\Omega^{j-i} \\setminus A_j) \\ge \\mu(A_j) + \\sum_{i=1}^{j-1} (1- \\alpha_{j- i}) \\mu(D_{i})\n\\]\ngiving the following relation between $\\alpha$-s and $\\beta$-s:\n\\begin{equation}\\label{eq:beta}\n \\beta_j \\ge \\alpha_j + \\sum_{i=1}^{j-1} (1-\\alpha_{j-i}) (\\beta_{i}-\\beta_{i-1}).\n\\end{equation}\nDenote $\\gamma_i = 1-\\beta_i = \\mu(U(A_i))$ and let $\\delta_i = \\alpha_{i-1}-\\alpha_i$ for $i=1, \\ldots, n$ where we put $\\alpha_0=1$ and $\\gamma_0=1$. Then (\\ref{eq:beta}) can be rewritten as follows:\n\\begin{equation}\\label{eq:gamma}\n \\gamma_j \\le \\sum_{i=0}^{j-1} \\gamma_i\\delta_{j-i}.\n\\end{equation}\nWe also have the following information about $\\gamma_i, \\delta_i$: \n\\[\n\\gamma_n \\le \\gamma_{n-1} \\le \\ldots \\le \\gamma_1 \\le \\gamma_0 = 1,\n\\]\n\\[\n\\delta_1+\\ldots+\\delta_n = 1-\\alpha, \\quad \\delta_i \\ge 0, \\quad i=1, \\ldots, n.\n\\]\nWe will use these properties to upper bound $\\gamma_n$. The idea is to use (\\ref{eq:gamma}) to compare $\\gamma$ with a random walk on $\\Z$.\n\nLet $p_i = \\frac{\\delta_i}{1-\\alpha}$ and let $Z$ be the random variable supported on $\\{1, \\ldots, n\\}$ given by the following distribution:\n\\[\n\\Pr[Z = i] = p_i.\n\\]\nLet $Z_1, \\ldots, Z_n$ be i.i.d. copies of $Z$.\n\\begin{obs}\n For every $j=0, \\ldots, n$ we have the following:\n \\begin{equation}\\label{eq:probability}\n \\gamma_j \\le \\sum_{s= 0}^n (1-\\alpha)^s \\Pr[Z_1+\\ldots+Z_s = j]. \n \\end{equation}\n\\end{obs}\n\\begin{proof}\n We prove this by induction on $j$. The base case $j=0$ is clear since only $s=0$ term contributes. Now for $j\\ge 1$ we have by (\\ref{eq:gamma}):\n \\begin{align*}\n \\gamma_j &\\le \\sum_{i=0}^{j-1} \\gamma_i \\delta_{j-i} = \n (1-\\alpha) \\sum_{i=0}^{j-1} \\gamma_i p_{j-i} \\\\\n &\\le (1-\\alpha) \\sum_{i=0}^{j-1} \\sum_{s \\ge 0} (1-\\alpha)^s \\Pr[Z_1+\\ldots+Z_s = i] \\Pr[Z_{s+1} = j-i] \\\\\n &=\\sum_{s\\ge 0} (1-\\alpha)^s\\Pr[Z_1+\\ldots+Z_s = j].\n \\end{align*}\n\\end{proof}\n\n\\begin{cor}\\label{cor:isoperimetry}\n Let $A \\subset \\Omega^n$ be a set of measure $\\alpha \\in (0,1)$. For $w \\in \\Omega^n$ define $f(w)$ to be the number of indices $j\\in \\{0, \\ldots, n-1\\}$ such that $w \\in s^j(A) \\times \\Omega^{j}$.\n Then for every integer $t \\in [1, n/4]$ we have the following level set estimate on $f$:\n \\begin{equation}\n \\mu(\\{w\\in \\Omega^n:~ f(w) \\le t\\}) \\le \\frac{8t}{\\alpha n}.\n \\end{equation}\n\\end{cor}\n\nLet $\\tilde n = [n/2t]$ and $r = n-2t \\tilde n$. By the assumption on $t$ we have $\\tilde n\\ge 1$. Consider a new alphabet $\\tilde \\Omega = \\Omega^{2t}$ and\n let $\\tilde s = \\tilde s_j: \\tilde \\Omega^{j} \\rightarrow \\tilde \\Omega^{j-1}$ denote the shift map defined on words over the alphabet $\\tilde \\Omega$. By identifying $\\tilde \\Omega^j = \\Omega^{2t j}$, we get that $\\tilde s = s^{2t}$. For $i=0, \\ldots, 2t-1$ let \n \\[\n \\tilde A_i = s^{r+i}(A) \\times \\Omega^{i} \\subset \\Omega^{2t \\tilde n} = \\tilde \\Omega^{\\tilde n}.\n \\]\n Note that $\\mu(\\tilde A_i) \\ge \\mu(A) =\\alpha$ for $i=0, \\ldots, 2t-1$. For an arbitrary subset $\\tilde A \\subset \\tilde \\Omega^{\\tilde n}$ we denote $\\tilde U(\\tilde A) = \\tilde \\Omega^{\\tilde n} \\setminus \\bigcup_{j=0}^{\\tilde n-1} \\tilde s^j(\\tilde A) \\times \\tilde \\Omega^{j}$, that is the analogue of $U(A)$ over the new alphabet.\n Let $w \\in \\Omega^n$ and denote $\\tilde w = s^r(w) \\in \\tilde\\Omega^{\\tilde n}$.\n Note that we have $\\tilde w \\not \\in \\tilde U(\\tilde A_i)$ precisely when there exists $j \\in \\{0, \\ldots, \\tilde n-1\\}$ such that $\\tilde w \\in \\tilde s^j(\\tilde A_i) \\times \\tilde \\Omega^j$. The latter is in turn equivalent to $w \\in s^{2t j + i + r}(A)$. It follows that we have\n \\[\n \\#\\{i \\in \\{0, \\ldots, 2t-1\\}: \\tilde w \\not \\in \\tilde U(\\tilde A_i)\\} \\le \\#\\{ i\\in \\{0, \\ldots, n-1\\}: w \\in s^{i}(A)\\times \\Omega^i \\} = f(w)\n \\]\n Thus, if $f(w) \\le t$ then there are at least $2t-t=t$ indices $i \\in \\{0, \\ldots, 2t-1\\}$ such that $\\tilde w \\in \\tilde U(\\tilde A_i)$. So by the union bound and Theorem \\ref{thm:isopery} applied to each $\\tilde U(\\tilde A_i)$ we have \n \\[\n t\\mu(\\{w:~f(w) \\le t\\}) \\le \\sum_{i=0}^{2t-1} \\mu(\\tilde U(\\tilde A_i)) \\le 2t \\frac{(1-\\alpha) (1 - (1-\\alpha)^{\\tilde n})}{\\alpha \\tilde n} \\le \\frac{2t}{\\alpha \\tilde n}.\n \\]\n So recalling that $\\tilde n = [n / 2t]$ we get\n \\[\n \\mu(\\{w:~f(w) \\le t\\}) \\le \\frac{2}{\\alpha [n/2t]} \\le \\frac{8t}{\\alpha n}\n \\]\n provided that $n \\ge 4t$, concluding the proof.\n\\end{proof}", "post_theorem_intro_text_len": 3064, "post_theorem_intro_text": "So if we have a pair of non-overlapping sets $A, B\\subset \\Omega^n$ with densities $\\alpha$ and $\\beta$ then we have $\\alpha\\beta  n\\leqslant (1-\\alpha)(1-(1-\\alpha)^n) \\leqslant 1 $. \n\nThe proof of Theorem \\ref{thm:isopery} is presented in the next section. The rough idea is to split the set $\\Omega^n \\setminus U(A)$ into several disjoint pieces and use inclusion-exclusion to lower bound the size of each piece. This then gives a certain recursive relationship between various densities associated with $A$ and $U(A)$ and their shifts. This relationship can be interpreted in terms of a certain random walk leading to the desired estimate.\n\nWe close this section by considering some examples essentially matching the upper bound in Theorem \\ref{thm:isopery}. Let $S \\subset \\Omega$ be an arbitrary subset.\nThen for $A = S^n$ one can check that\n\\[\nU(A) = (\\Omega\\setminus S)\\times \\Omega^{n-1}\n\\]\nand so we get\n\\[\n\\mu(U(A)) = 1-\\alpha^{1/n} = \\frac{\\log(1/\\alpha)}{n} + O\\left(\\frac{\\log^2(1/\\alpha)}{n^2}\\right).\n\\]\nThis is a good bound for $\\alpha \\in (1/2, 1)$. \n\nSimilarly, for $A = \\Omega^{n-1} \\times S$ one can check that\n\\[\nU(A) = (\\Omega\\setminus S)^n\n\\]\nso that we get\n\\[\n\\mu(U(A)) = (1-\\alpha)^n.\n\\]\nThis is a good bound for $\\alpha \\in (0, 1/n)$. \n\nWe can interpolate between these two examples by taking $A = \\Omega^{n-k} \\times S^k$ for some $1 \\leqslant k \\leqslant n$.\nThen the set $U = U(A)$ is given by\n\\begin{align*}\nU =  \\{ (w_1, \\ldots, w_n):~ w_1 \\not\\in S, \\quad \\{w_{j+1}, \\ldots, w_{j+k}\\}\\not\\subset S,~j=0, \\ldots, n-k \\}.    \n\\end{align*}\nThe exact formula for $\\mu(U)$ is a bit complicated (it involves generalized Fibonacci numbers, see \\cite{chee2013cross}) but we can use a simple Poisson approximation inequality due to \\cite{arratia1989two} (see also \\cite{godbole1991poisson, godbole1993improved, guibas1980long}) to get a good estimate of the measure of $U$.\nDenote $p = |S|/|\\Omega|$ so that $\\alpha = p^k$. \nLet $Z_1, \\ldots, Z_{n-1} \\sim \\operatorname{Ber}(p)$ be iid Bernoulli random variables. Let $R_{n-1}$ be the length of the longest run of 1-s in the sequence $(Z_1, \\ldots, Z_{n-1})$. The measure of $U$ can then be computed in terms of $R_{n-1}$:\n\\[\n\\mu(U) = (1-p)\\Pr[ R_{n-1} < k].\n\\]\nIndeed, we can view $w_2, \\ldots, w_n \\in \\Omega$ as iid variables uniformly distributed on $\\Omega$ and select $Z_i = 1_{w_{i+1} \\in S}$. \nBy \\cite[Example 3]{arratia1989two} we have the following estimate on this probability:\n\\[\n\\left|\\Pr[ R_{n-1} < k] - e^{-\\lambda}\\right| \\leqslant \\frac{\\lambda (2k+1)}{n-1} + 2 p^k, \\quad \\lambda = p^k ((n-2)(1-p)+1)\n\\]\nLet $k = [n\\alpha \\log(1/\\alpha) ]$. Then for $1/n \\ll \\alpha \\ll 1$ we have $p = \\alpha^{1/k} = e^{-\\frac{\\log(1/\\alpha)}{k}} = 1 - \\frac{1+o(1)}{n\\alpha}$. This gives $\\lambda = 1+o(1)$ and $\\Pr[R_{n-1} < k] = e^{-1} + o(1)$ and so we have\n\\[\n\\mu(U) = \\frac{e^{-1}+o(1)}{\\alpha n},\n\\] \nwhere $o(1)$ tends to zero as $\\min(\\alpha^{-1}, n\\alpha) \\to \\infty$. This matches the bound in Theorem \\ref{thm:isopery} for all $\\alpha \\in (1/n, 1/2)$ up to a constant factor.", "sketch": "The post-theorem introduction says: \"The proof of Theorem \\ref{thm:isopery} is presented in the next section. The rough idea is to split the set $\\Omega^n \\setminus U(A)$ into several disjoint pieces and use inclusion-exclusion to lower bound the size of each piece.\" This \"then gives a certain recursive relationship between various densities associated with $A$ and $U(A)$ and their shifts.\" Finally, \"[t]his relationship can be interpreted in terms of a certain random walk leading to the desired estimate.\"", "expanded_sketch": "The post-theorem introduction says: \"The proof of the main theorem is presented in the next section. The rough idea is to split the set $\\Omega^n \\setminus U(A)$ into several disjoint pieces and use inclusion-exclusion to lower bound the size of each piece.\" This \"then gives a certain recursive relationship between various densities associated with $A$ and $U(A)$ and their shifts.\" Finally, \"[t]his relationship can be interpreted in terms of a certain random walk leading to the desired estimate.\"", "expanded_theorem": "\\label{thm:isopery}\n    We have $\\gamma(\\alpha, n) \\leqslant \\frac{(1-\\alpha)(1-(1-\\alpha)^n)}{\\alpha n}$ for all $n\\geqslant 1$ and $\\alpha \\in (0, 1)$.", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "For a finite set $\\Omega$ and an integer $n\\ge 1$, let $\\mu$ be the uniform probability measure on $\\Omega^n$. Let $s: \\Omega^n\\to\\Omega^{n-1}$ be the shift map $s(w_1,\\dots,w_n)=(w_2,\\dots,w_n)$, and let $s^j$ denote its $j$-fold iterate (with $s^0$ the identity). For a set $A\\subset \\Omega^n$, define\n\\[\nU(A)=\\Omega^n\\setminus \\bigcup_{j=0}^{n-1} s^j(A)\\times \\Omega^j,\n\\]\nso $U(A)$ is the set of words in $\\Omega^n$ that do not overlap any word of $A$. For $\\alpha\\in(0,1)$, let $\\gamma(\\alpha,n)$ be the largest possible value of $\\mu(U(A))$ over all finite alphabets $\\Omega$ and all sets $A\\subset \\Omega^n$ with $\\mu(A)=\\alpha$. Which statement holds for every $n\\ge 1$ and every $\\alpha\\in(0,1)$?", "correct_choice": {"label": "A", "text": "$\\gamma(\\alpha,n)\\le \\dfrac{(1-\\alpha)\\bigl(1-(1-\\alpha)^n\\bigr)}{\\alpha n}$."}, "choices": [{"label": "B", "text": "$\\gamma(\\alpha,n)\\le \\dfrac{(1-\\alpha)\\bigl(1-(1-\\alpha)^{n-1}\\bigr)}{\\alpha n}$."}, {"label": "C", "text": "$\\gamma(\\alpha,n)\\le \\dfrac{1-\\alpha}{\\alpha n}$."}, {"label": "D", "text": "$\\gamma(\\alpha,n)\\le \\dfrac{(1-\\alpha)\\bigl(1-(1-\\alpha)^n\\bigr)}{\\alpha}$."}, {"label": "E", "text": "$\\gamma(\\alpha,n)= \\dfrac{(1-\\alpha)\\bigl(1-(1-\\alpha)^n\\bigr)}{\\alpha n}$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "counting_estimate", "tampered_component": "number_of_shift_levels", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "dropped_factor_1-(1-\\alpha)^n", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "random_walk", "tampered_component": "averaging_by_n", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "upper_bound_not_sharp_for_all_sets", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the objects carefully but does not reveal the bound itself or give strong clues uniquely pointing to choice A."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall format: after introducing the definitions, it asks which universal bound holds. This is not a literal restatement inside the stem, but it is largely selecting the exact theorem conclusion from nearby variants."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to compare the candidate inequalities (e.g., weaker true bound, missing factor of n, false equality), but the question mainly rewards recognition or recall of the precise bound rather than generating a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically structured: B is an off-by-one variant, C is a weaker true statement, D omits the averaging-by-n factor, and E incorrectly strengthens an upper bound to equality."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it is somewhat theorem-recall driven and only moderately tests generative mathematical reasoning."}}
{"id": "2602.20151v1", "paper_link": "http://arxiv.org/abs/2602.20151v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:main-intro}\n    Assume $\\mathcal{A}$ is symmetric and $\\beta$-stable with respect to $\\mathcal{A}^*$, that $D_{1:n+1}$ is exchangeable, and that\n\\begin{equation}\n        \\E\\left[\\ell(X_{n+1}, Y_{n+1}; \\mathcal{A}^*(D_{1:n+1})) \\right] \\leq \\alpha-\\beta.\n\\end{equation}\n    Then\n    \\begin{equation}\n        \\E\\left[\\ell(X_{n+1}, Y_{n+1}; \\mathcal{A}(D_{1:n})) \\right] \\leq \\alpha.\n    \\end{equation}", "start_pos": 4366, "end_pos": 4774, "label": "thm:main-intro"}, "ref_dict": {"thm:main-intro": "\\begin{theorem}\n    \\label{thm:main-intro}\n    Assume $\\A$ is symmetric and $\\beta$-stable with respect to $\\A^*$, that $D_{1:n+1}$ is exchangeable, and that\n\\begin{equation}\n        \\E\\left[\\ell(X_{n+1}, Y_{n+1}; \\A^*(D_{1:n+1})) \\right] \\leq \\alpha-\\beta.\n\\end{equation}\n    Then\n    \\begin{equation}\n        \\E\\left[\\ell(X_{n+1}, Y_{n+1}; \\A(D_{1:n})) \\right] \\leq \\alpha.\n    \\end{equation}\n\\end{theorem}", "eq:intro-guarantee": "\\begin{equation}\n    \\label{eq:intro-guarantee}\n    \\E\\left[ \\ell(X_{n+1}, Y_{n+1}; \\thetahat) \\right] \\leq \\alpha, \n\\end{equation}", "prop:crc-stable": "\\begin{proposition}\n    \\label{prop:crc-stable}\n    Let $d=1$, $D$ be any dataset, $\\ell$ be nonincreasing in its last argument, \n    \\begin{equation}\n        \\A(D) = \\inf\\left\\{ \\theta : \\frac{1}{|D|+1}\\sum_{(x,y) \\in D}\\ell(x,y;\\theta) \\leq \\alpha - \\frac{1}{|D|+1} \\right\\}, \n    \\end{equation}\n    and\n    \\begin{equation}\n        \\A^*(D) = \\inf\\left\\{ \\theta : \\frac{1}{|D|}\\sum_{(x,y) \\in D}\\ell(x,y;\\theta) \\leq \\alpha \\right\\}.\n    \\end{equation}\n    Then $\\A$ is $0$-stable with respect to $\\A^*$.\n\\end{proposition}"}, "pre_theorem_intro_text_len": 2990, "pre_theorem_intro_text": "Consider a sequence of random variables $D_{1:n+1} = ((X_1, Y_1), \\ldots, (X_{n+1}, Y_{n+1}))$ representing feature-label pairs with an exchangeable joint distribution.\nLet the first $n$ datapoints, $D_{1:n}$, be a known calibration set, and  last datapoint, $(X_{n+1},Y_{n+1})$, represent a test datapoint whose label is unknown.\nAlso define a bounded loss, $\\ell(x,y;\\theta) \\in [0,1]$, which is a function of a datapoint $(x,y)$ and a parameter $\\theta \\in \\R^d$.\nOur goal is to select a parameter value $\\hat{\\theta}$ using the calibration data $D_{1:n}$ to bound the expected loss on the test datapoint:\n\\begin{equation}\n    \\label{eq:intro-guarantee}\n    \\E\\left[ \\ell(X_{n+1}, Y_{n+1}; \\hat{\\theta}) \\right] \\leq \\alpha, \n\\end{equation}\nwhere $\\alpha \\in [0,1]$ is user-specified and the expectation is taken with respect to all $n+1$ datapoints.\nWe call~\\eqref{eq:intro-guarantee} a risk control guarantee.\n\nConformal risk control, a generalization of conformal prediction~\\cite{gammerman1998learning, vovk1999machine, vovk2005algorithmic,lei2013conformal,lei2018distribution} as developed in~\\cite{angelopoulos2024conformal}, handles the case where $d=1$ and $\\ell$ is monotonically nonincreasing in $\\theta$.\nConformal risk control works by setting the parameter $\\hat\\theta$ to be\n\\begin{equation}\n    \\hat\\theta = \\inf\\left\\{ \\theta : \\frac{1}{n+1}\\sum_{i=1}^{n}\\ell(X_i,Y_i;\\theta) \\leq \\alpha - \\frac{1}{n+1} \\right\\},\n\\end{equation}\nwhich is the smallest value of $\\theta$ such that the empirical risk calculated on $n+1$ datapoints is certain to be below $\\alpha$. This provides the guarantee in~\\eqref{eq:intro-guarantee} for monotonic losses, but can fail arbitrarily badly for non-monotonic losses, as shown in Proposition 1 of~\\cite{angelopoulos2024conformal}.\n\nIn this paper, we give risk control guarantees for non-monotonic losses.\nThe key insight is that the population risk of $\\hat{\\theta}$ depends on the stability of the algorithm for choosing $\\hat{\\theta}$---i.e., the change in risk when the test datapoint is added to or removed from the calibration dataset.\nMore formally, let $\\mathcal{A}$ be an algorithm mapping datasets to choices of $\\theta$ and let $D_{-i}$ denote $D_{1:n+1}$ with the $i$th element removed.\nThe parameter $\\hat{\\theta}$ is the output of $\\mathcal{A}(D_{1:n})$.\nWe say an algorithm $\\mathcal{A}$ is $\\beta$-stable with respect to a reference algorithm $\\mathcal{A}^*$ and $\\ell$ if\n\\begin{equation}\n        \\E\\left[\\frac{1}{n+1}\\sum\\limits_{i=1}^{n+1} \\ell(X_{i}, Y_{i}; \\mathcal{A}(D_{-i}))\\right] \n        \\leq \\E\\left[\\frac{1}{n+1}\\sum\\limits_{i=1}^{n+1} \\ell(X_{i}, Y_{i}; \\mathcal{A}^*(D_{1:n+1}))\\right] + \\beta.\n\\end{equation}\nThe main theorem demonstrates that if an algorithm is stable with respect to a reference algorithm that controls the risk when run on the full data, then the original algorithm also controls the risk.\nHere and throughout the paper, we assume the algorithms are symmetric (i.e. permutation-invariant).", "context": "Consider a sequence of random variables $D_{1:n+1} = ((X_1, Y_1), \\ldots, (X_{n+1}, Y_{n+1}))$ representing feature-label pairs with an exchangeable joint distribution.\nLet the first $n$ datapoints, $D_{1:n}$, be a known calibration set, and last datapoint, $(X_{n+1},Y_{n+1})$, represent a test datapoint whose label is unknown.\nAlso define a bounded loss, $\\ell(x,y;\\theta) \\in [0,1]$, which is a function of a datapoint $(x,y)$ and a parameter $\\theta \\in \\R^d$.\nOur goal is to select a parameter value $\\hat{\\theta}$ using the calibration data $D_{1:n}$ to bound the expected loss on the test datapoint:\n\\begin{equation}\n \\label{eq:intro-guarantee}\n \\E\\left[ \\ell(X_{n+1}, Y_{n+1}; \\hat{\\theta}) \\right] \\leq \\alpha, \n\\end{equation}\nwhere $\\alpha \\in [0,1]$ is user-specified and the expectation is taken with respect to all $n+1$ datapoints.\nWe call~\\eqref{eq:intro-guarantee} a risk control guarantee.\n\nConformal risk control, a generalization of conformal prediction~\\cite{gammerman1998learning, vovk1999machine, vovk2005algorithmic,lei2013conformal,lei2018distribution} as developed in~\\cite{angelopoulos2024conformal}, handles the case where $d=1$ and $\\ell$ is monotonically nonincreasing in $\\theta$.\nConformal risk control works by setting the parameter $\\hat\\theta$ to be\n\\begin{equation}\n \\hat\\theta = \\inf\\left\\{ \\theta : \\frac{1}{n+1}\\sum_{i=1}^{n}\\ell(X_i,Y_i;\\theta) \\leq \\alpha - \\frac{1}{n+1} \\right\\},\n\\end{equation}\nwhich is the smallest value of $\\theta$ such that the empirical risk calculated on $n+1$ datapoints is certain to be below $\\alpha$. This provides the guarantee in~\\eqref{eq:intro-guarantee} for monotonic losses, but can fail arbitrarily badly for non-monotonic losses, as shown in Proposition 1 of~\\cite{angelopoulos2024conformal}.\n\nIn this paper, we give risk control guarantees for non-monotonic losses.\nThe key insight is that the population risk of $\\hat{\\theta}$ depends on the stability of the algorithm for choosing $\\hat{\\theta}$---i.e., the change in risk when the test datapoint is added to or removed from the calibration dataset.\nMore formally, let $\\mathcal{A}$ be an algorithm mapping datasets to choices of $\\theta$ and let $D_{-i}$ denote $D_{1:n+1}$ with the $i$th element removed.\nThe parameter $\\hat{\\theta}$ is the output of $\\mathcal{A}(D_{1:n})$.\nWe say an algorithm $\\mathcal{A}$ is $\\beta$-stable with respect to a reference algorithm $\\mathcal{A}^*$ and $\\ell$ if\n\\begin{equation}\n \\E\\left[\\frac{1}{n+1}\\sum\\limits_{i=1}^{n+1} \\ell(X_{i}, Y_{i}; \\mathcal{A}(D_{-i}))\\right] \n \\leq \\E\\left[\\frac{1}{n+1}\\sum\\limits_{i=1}^{n+1} \\ell(X_{i}, Y_{i}; \\mathcal{A}^*(D_{1:n+1}))\\right] + \\beta.\n\\end{equation}\nThe main theorem demonstrates that if an algorithm is stable with respect to a reference algorithm that controls the risk when run on the full data, then the original algorithm also controls the risk.\nHere and throughout the paper, we assume the algorithms are symmetric (i.e. permutation-invariant).\n\n\\begin{equation}\n    \\label{eq:intro-guarantee}\n    \\E\\left[ \\ell(X_{n+1}, Y_{n+1}; \\thetahat) \\right] \\leq \\alpha, \n\\end{equation}", "full_context": "Consider a sequence of random variables $D_{1:n+1} = ((X_1, Y_1), \\ldots, (X_{n+1}, Y_{n+1}))$ representing feature-label pairs with an exchangeable joint distribution.\nLet the first $n$ datapoints, $D_{1:n}$, be a known calibration set, and last datapoint, $(X_{n+1},Y_{n+1})$, represent a test datapoint whose label is unknown.\nAlso define a bounded loss, $\\ell(x,y;\\theta) \\in [0,1]$, which is a function of a datapoint $(x,y)$ and a parameter $\\theta \\in \\R^d$.\nOur goal is to select a parameter value $\\hat{\\theta}$ using the calibration data $D_{1:n}$ to bound the expected loss on the test datapoint:\n\\begin{equation}\n \\label{eq:intro-guarantee}\n \\E\\left[ \\ell(X_{n+1}, Y_{n+1}; \\hat{\\theta}) \\right] \\leq \\alpha, \n\\end{equation}\nwhere $\\alpha \\in [0,1]$ is user-specified and the expectation is taken with respect to all $n+1$ datapoints.\nWe call~\\eqref{eq:intro-guarantee} a risk control guarantee.\n\nConformal risk control, a generalization of conformal prediction~\\cite{gammerman1998learning, vovk1999machine, vovk2005algorithmic,lei2013conformal,lei2018distribution} as developed in~\\cite{angelopoulos2024conformal}, handles the case where $d=1$ and $\\ell$ is monotonically nonincreasing in $\\theta$.\nConformal risk control works by setting the parameter $\\hat\\theta$ to be\n\\begin{equation}\n \\hat\\theta = \\inf\\left\\{ \\theta : \\frac{1}{n+1}\\sum_{i=1}^{n}\\ell(X_i,Y_i;\\theta) \\leq \\alpha - \\frac{1}{n+1} \\right\\},\n\\end{equation}\nwhich is the smallest value of $\\theta$ such that the empirical risk calculated on $n+1$ datapoints is certain to be below $\\alpha$. This provides the guarantee in~\\eqref{eq:intro-guarantee} for monotonic losses, but can fail arbitrarily badly for non-monotonic losses, as shown in Proposition 1 of~\\cite{angelopoulos2024conformal}.\n\nIn this paper, we give risk control guarantees for non-monotonic losses.\nThe key insight is that the population risk of $\\hat{\\theta}$ depends on the stability of the algorithm for choosing $\\hat{\\theta}$---i.e., the change in risk when the test datapoint is added to or removed from the calibration dataset.\nMore formally, let $\\mathcal{A}$ be an algorithm mapping datasets to choices of $\\theta$ and let $D_{-i}$ denote $D_{1:n+1}$ with the $i$th element removed.\nThe parameter $\\hat{\\theta}$ is the output of $\\mathcal{A}(D_{1:n})$.\nWe say an algorithm $\\mathcal{A}$ is $\\beta$-stable with respect to a reference algorithm $\\mathcal{A}^*$ and $\\ell$ if\n\\begin{equation}\n \\E\\left[\\frac{1}{n+1}\\sum\\limits_{i=1}^{n+1} \\ell(X_{i}, Y_{i}; \\mathcal{A}(D_{-i}))\\right] \n \\leq \\E\\left[\\frac{1}{n+1}\\sum\\limits_{i=1}^{n+1} \\ell(X_{i}, Y_{i}; \\mathcal{A}^*(D_{1:n+1}))\\right] + \\beta.\n\\end{equation}\nThe main theorem demonstrates that if an algorithm is stable with respect to a reference algorithm that controls the risk when run on the full data, then the original algorithm also controls the risk.\nHere and throughout the paper, we assume the algorithms are symmetric (i.e. permutation-invariant).\n\n\\begin{equation}\n    \\label{eq:intro-guarantee}\n    \\E\\left[ \\ell(X_{n+1}, Y_{n+1}; \\thetahat) \\right] \\leq \\alpha, \n\\end{equation}\n\n\\subsection{Conformal Risk Control is Stable for Monotonic Losses}\n\n\\subsubsection{General Bounded Losses}\n\\label{sec:general-bounded}\nTo handle general losses bounded in $[0,1]$, we will use algorithms that discretize $\\Theta$ to impose stability.\nFirst, let $\\A(D) = \\inf\\{\\theta \\in \\Theta_m = \\{0, \\tfrac{1}{m}, \\tfrac{2}{m}, \\ldots, 1\\} : \\hat{R}_D(\\theta) \\leq \\alpha\\}$ for $\\Theta = [0,1]$ and any positive integer $m$.\nThis is simply the discretized version of the previously studied algorithms.\nWe will assume nothing about the loss other than the existence of a safe solution, $\\ell(z;1)=0$ for all $z \\in \\cX\\times\\cY$.\n\\begin{proposition}\n\\label{prop:general-stability}\nLet $D_{1:n+1}$ be sampled i.i.d., $\\Theta_m=\\{0,\\tfrac1m,\\ldots,1\\}$, $\\ell(\\cdot;1)=0$, and define\n\\begin{equation}\n\\A(D)\\ :=\\ \\inf\\bigl\\{\\theta\\in\\Theta_m:\\ \\hat R_D(\\theta)\\le \\alpha\\bigr\\}.\n\\end{equation}\nThen\n\\begin{equation}\n\\E\\bigl[R(\\thetahat_{n})\\bigr]\\ \\leq\\ \\alpha+\\frac{1}{2\\sqrt{n}}\\left(\\sqrt{-W_{-1} \\left( -\\frac{1}{4n(m+1)^2}\\right)}+\\sqrt{-\\frac{1}{W_{-1} \\left( -\\frac{1}{4n(m+1)^2}\\right)}}\\right) = \\alpha + \\tilde{\\mathcal{O}}\\left(\\frac{1}{\\sqrt{n}}\\right),\n\\end{equation}\nwhere $W_{-1}$ is the $-1$st branch of the Lambert W function and $\\tilde{\\mathcal{O}}$ denotes the growth rate excluding logarithmic factors.\n\\end{proposition}\nThis proposition tells us that we can achieve risk control for general losses with the discretized algorithm up to a dominating factor of $\\tilde{\\mathcal{O}}\\left(\\tfrac{1}{\\sqrt{n}}\\right)$, since $W_{-1}$ is of logarithmic order.\n\n\\begin{proposition}\n \\label{prop:stability-selective}\n The selective classification algorithm in~\\eqref{eq:leftmost-root} is $\\beta$-stable with\n \\begin{equation}\n \\beta = \\frac{2\\max\\{\\alpha, 1-\\alpha\\}\\E[K]}{n+1}, \\text{ where } K = \\max_i|\\hat{\\jmath}_{-i} - \\hat{\\jmath}_{n+1}|.\n \\end{equation}\n\\end{proposition}\n\\begin{proof}\n By~\\eqref{eq:selective-eq-neq}, it suffices to prove that, almost surely,\n \\begin{equation}\n \\sum\\limits_{i=1}^{n+1} \\Big(\\ell(X_{i}, Y_{i}; \\A(D_{-i})) - \\ell(X_{i}, Y_{i}; \\A^*(D_{1:n+1}))\\Big) = \\sum_{i\\in [n+1]} w_i\\Delta_i\n \\leq 2\\max\\{\\alpha, 1-\\alpha\\}K,\n \\end{equation}\n where $\\Delta_i = \\ind{\\A(D_{-i}) \\geq \\hat P_i} - \\ind{\\A^*(D_{1:n+1}) \\geq \\hat P_i}$ and $w_i = \\alpha$ for $i \\in V^{=}$ and $-(1-\\alpha)$ otherwise.\n We can begin by rewriting the $\\Delta_i$ terms as \n \\begin{align}\n \\Delta_i &= \\ind{\\A(D_{-i}) \\geq \\hat{P}_i > \\A(D_{1:n+1})} - \\ind{\\A(D_{-i}) < \\hat{P}_i \\leq \\A(D_{1:n+1})} \\\\\n &= \\ind{\\hat{\\jmath}_{-i} \\leq V_i < \\hat{\\jmath}_{n+1}} - \\ind{\\hat{\\jmath}_{-i} > V_i \\geq \\hat{\\jmath}_{n+1}}.\n \\end{align}\n Note that the two events inside the indicators are mutually exclusive, and correspond to the ``crossing'' of $\\A(D_{-i})$ over $\\hat{P}_i$ in the downward or upward directions.\n Therefore,\n \\begin{equation}\n \\label{eq:pf-selective-cauchy-schwarz}\n \\sum_{i\\in [n+1]} w_i\\Delta_i \\leq \\max\\{\\alpha, 1-\\alpha\\} \\sum_{i\\in [n+1]}|\\Delta_i| \\leq \\max\\{\\alpha, 1-\\alpha\\}\\left|\\{i : \\hat{\\jmath}_{-i} \\leq V_i < \\hat{\\jmath}_{n+1} \\text{ or } \\hat{\\jmath}_{-i} > V_i \\geq \\hat{\\jmath}_{n+1}\\} \\right|.\n \\end{equation}\n Define the index sets\n\\[\nS_- \\;:=\\; \\bigl\\{i\\in[n{+}1]:\\ \\hat{\\jmath}_{-i}\\le V_i<\\hat{\\jmath}_{n+1}\\bigr\\}\n\\quad\\text{and}\\quad\nS_+ \\;:=\\; \\bigl\\{i\\in[n{+}1]:\\ \\hat{\\jmath}_{-i}> V_i\\ge \\hat{\\jmath}_{n+1}\\bigr\\}.\n\\]\nThen the set of indices that contribute nonzero terms to $\\sum_{i\\in[n+1]}|\\Delta_i|$ is $S:=S_-\\cup S_+$, and $S_-\\cap S_+=\\varnothing$ by definition.\n\n\\subsubsection{Risk Control Guarantees on the Loss}\nFirst, we show the stability of ERM on the scale of the loss.\n\\begin{proposition}\n\\label{prop:erm-stable-loss}\nLet $\\Theta=\\R^d$, $\\ell$ be convex in $\\theta$, and assume there exists $\\rho:\\cZ\\to[0,\\infty)$ such that for all $z\\in\\cZ$ and $\\theta,\\theta'\\in\\R^d$,\n\\begin{equation}\n\\label{eq:lipschitz-rho-simple}\n|\\ell(z;\\theta)-\\ell(z;\\theta')|\\ \\le\\ \\rho(z)\\,\\|\\theta-\\theta'\\|_2.\n\\end{equation}\nThen $\\A$ is $\\beta$-stable with\n\\begin{equation}\n\\label{eq:beta-simple}\n\\beta \\leq \\frac{2\\,\\E[\\rho(Z_{n+1})^2]}{\\lambda(n+1)}.\n\\end{equation}\n\\end{proposition}\nThis result can be used in conjunction with Theorem~\\ref{thm:main-intro} to give a generalization guarantee on ERM, as below.\n\\begin{corollary}\n In the setting of Proposition~\\ref{prop:erm-stable-loss},\n \\begin{equation}\n \\E\\left[ \\ell(X_{n+1}, Y_{n+1}; \\thetahat_n) + \\tfrac{\\lambda}{2}\\|\\thetahat_n\\|_2^2 \\right] \\leq R^* + \\frac{2\\,\\E[\\rho(Z_{n+1})^2]}{\\lambda(n+1)},\n \\end{equation}\n where $R^* = \\min_{\\theta \\in \\Theta} \\E[\\ell(X_{n+1},Y_{n+1};\\theta)+ \\tfrac{\\lambda}{2}\\|\\theta\\|_2^2]$.\n\\end{corollary}\nThese results represent minor variations on the canonical results in Section 5 of~\\cite{bousquet2002stability}.\n\nNext, we address stability of the expected gradient.\nBuilding towards this, we will need a multivariate notion of stability to handle gradients when $d>1$: an algorithm $\\cA$ is $\\beta$-stable with respect to $\\A^*$ and $g$ if\n\\begin{equation}\n \\E\\left[\\frac{1}{n+1}\\sum\\limits_{i=1}^{n+1} g(X_{i}, Y_{i}; \\A(D_{-i}))\\right] \n \\preceq \\E\\left[\\frac{1}{n+1}\\sum\\limits_{i=1}^{n+1} g(X_{i}, Y_{i}; \\A^*(D_{1:n+1}))\\right] + \\beta,\n\\end{equation}\nwhere $\\beta \\in \\R^d$, and $\\preceq$ represents the standard partial ordering on vectors (i.e., the inequality holds component-wise).\nTheorem~\\ref{thm:main-intro} also has an extension to $d>1$, which we present below.\n\\begin{theorem}\n \\label{thm:main-high-d}\n Assume $\\A$ is symmetric and $\\beta$-stable with respect to $\\A^*$ and $g$, that $D_{1:n+1}$ is exchangeable, and that\n\\begin{equation}\n \\E\\left[g(X_{n+1}, Y_{n+1}; \\A^*(D_{1:n+1})) \\right] \\preceq \\alpha-\\beta.\n\\end{equation}\n Then\n \\begin{equation}\n \\E\\left[g(X_{n+1}, Y_{n+1}; \\A(D_{1:n})) \\right] \\preceq \\alpha.\n \\end{equation}\n\\end{theorem}\nThe proof of this theorem is identical to that of Theorem~\\ref{thm:main-intro}, except with $\\preceq$ in place of $\\leq$.\nNow we proceed with the stability result, stated below for differentiable convex losses. The differentiability is not needed, and is only used to simplify the statement and proof of the proposition.\n\\begin{proposition}\n\\label{prop:erm-grad-stability}\nAssume $\\ell$ is a convex, differentiable function of $\\theta$ and that there exists a measurable function $\\rho : \\cZ \\to [0,\\infty)$ such that for all $\\theta,\\theta' \\in \\R^d$ and almost all $z \\in \\cZ$,\n\\begin{equation}\n\\|\\nabla\\ell(z;\\theta) - \\nabla\\ell(z;\\theta')\\|_2 \\leq \\rho(z)\\|\\theta - \\theta'\\|_2.\n\\end{equation}\nAssume also that $\\hat{R}_{1:n+1}$ is $\\mu$-strongly convex for some $\\mu \\in [0,\\infty)$ (noting that $\\mu=0$ is also allowable, so it need not be strongly convex).\nThen the regularized ERM algorithm $\\cA$ is $\\beta$-stable with respect to $\\nabla\\ell$, where\n\\begin{equation}\n\\beta = \\frac{\\E[\\rho(Z_{n+1})\\|\\nabla\\ell(Z_{n+1}; \\thetahat_n)\\|_2] + \\E\\left[\\rho(Z_{n+1})\\|\\tfrac{1}{n}\\sum_{j=1}^n\\nabla\\ell(Z_j;\\thetahat_n)\\|_2\\right]}{(\\mu + \\lambda)(n+1)} \\mathbf{1}_d.\n\\end{equation}\n\\end{proposition}\nEliding the Lipschitz constant $\\rho(Z_{n+1})$, this result tells us that ERM is stable with\n\\begin{equation}\n \\beta = \\frac{\\E[\\text{magnitude of test gradient}] + \\E[\\text{magnitude of training gradient}]}{(\\mu + \\lambda)(n+1)}.\n\\end{equation}\nUnder normal circumstances, both terms in the numerator will be constant-order, and the latter term will be near-zero (in fact, when $\\lambda=0$, it is exactly zero).\nProposition~\\ref{prop:erm-grad-stability} also holds for nondifferentiable losses, replacing every gradient in $\\beta$ with a supremum over all subgradients in the subdifferential at $\\thetahat_{n}$; the proof of this more general result is morally the same, so we omit it.\nThe assumption that $\\hat{R}_{1:n+1}$ is strongly convex is weaker than strong convexity of the loss function itself, and allows for, e.g., ordinary least squares with a full-rank design matrix, even though the individual losses may not be strongly convex.", "post_theorem_intro_text_len": 5035, "post_theorem_intro_text": "Theorem~\\ref{thm:main-intro} gives us an actionable workflow for producing risk-control algorithms.\nFirst, we try to identify a reference algorithm, $\\mathcal{A}^*$, that controls the risk when applied on the full dataset of $n+1$ datapoints.\nThen, we try to approximate that algorithm via $\\mathcal{A}$, which runs only on $n$ datapoints, and prove that the difference in risks between these two algorithms is low.\nThe proof of the main result is below. \n\\begin{proof}\nBy exchangeability and symmetry,\n\\[\n\\mathbb{E}\\!\\left[\\frac{1}{n+1}\\sum_{i=1}^{n+1}\\ell(X_i,Y_i;\\mathcal{A}^*(D_{1:n+1}))\\right]\n= \\mathbb{E}\\!\\left[\\ell(X_{n+1},Y_{n+1};\\mathcal{A}^*(D_{1:n+1}))\\right],\n\\]\nand\n\\[\n\\mathbb{E}\\!\\left[\\frac{1}{n+1}\\sum_{i=1}^{n+1}\\ell(X_i,Y_i;\\mathcal{A}(D_{-i}))\\right]\n= \\mathbb{E}\\!\\left[\\ell(X_{n+1},Y_{n+1};\\mathcal{A}(D_{1:n}))\\right].\n\\]\nBy $\\beta$-stability, $\\mathbb{E}\\!\\left[\\ell(X_{n+1},Y_{n+1};\\mathcal{A}(D_{1:n}))\\right] \\leq \\mathbb{E}\\!\\left[\\ell(X_{n+1},Y_{n+1};\\mathcal{A}^*(D_{1:n+1}))\\right] + \\beta$.\nCombining this with the assumption that $\\mathbb{E}[\\ell(X_{n+1},Y_{n+1};\\mathcal{A}^*(D_{1:n+1}))] \\leq \\alpha-\\beta$ yields the result.\n\\end{proof}\nNotice that Theorem~\\ref{thm:main-intro} applies to $\\theta$ in any space, and that boundedness of $\\ell$ was never needed in the proof of Theorem~\\ref{thm:main-intro}.\nThese concerns will come up later in determining stability bounds, building on the literature on algorithmic stability~\\cite{kearns1997algorithmic, bousquet2002stability, kutin2002almost, mukherjee2006learning, shalev-shwartz2010learnability, yu2013stability, hardt2016train, yu2017three, feldman2018generalization, bousquet2020sharper, zrnic2023post}; see Chapter 13 of~\\cite{shalev-shwartz2014understanding} for a review of this extensive field.\nThe notion of stability used herein is a form of leave-one-out stability with respect to generic algorithms $\\cA$ and $\\cA^*$; we will prove guarantees for a few different choices of algorithms depending on the setting.\n\n\\subsection{Conformal Risk Control is Stable for Monotonic Losses}\n\nHere we will show that the conformal risk control algorithm is stable for monotonic losses, and thus, satisfies the guarantee in Theorem~\\ref{thm:main-intro}.\n\n\\begin{proposition}\n    \\label{prop:crc-stable}\n    Let $d=1$, $D$ be any dataset, $\\ell$ be nonincreasing in its last argument, \n    \\begin{equation}\n        \\mathcal{A}(D) = \\inf\\left\\{ \\theta : \\frac{1}{|D|+1}\\sum_{(x,y) \\in D}\\ell(x,y;\\theta) \\leq \\alpha - \\frac{1}{|D|+1} \\right\\}, \n    \\end{equation}\n    and\n    \\begin{equation}\n        \\mathcal{A}^*(D) = \\inf\\left\\{ \\theta : \\frac{1}{|D|}\\sum_{(x,y) \\in D}\\ell(x,y;\\theta) \\leq \\alpha \\right\\}.\n    \\end{equation}\n    Then $\\mathcal{A}$ is $0$-stable with respect to $\\mathcal{A}^*$.\n\\end{proposition}\n\\begin{proof}\n    The boundedness of $\\ell$ and definition of $\\mathcal{A}$ imply that for all $i$,\n    \\begin{equation}\n        \\frac{1}{n+1}\\sum_{j=1}^{n+1}\\ell(X_j,Y_j;\\mathcal{A}(D_{-i})) \\leq \\frac{1}{n+1}\\sum_{j \\in [n+1] \\setminus i} \\ell(X_j,Y_j;\\mathcal{A}(D_{-i})) + \\frac{1}{n+1} \\leq \\alpha.\n    \\end{equation}\n    $\\mathcal{A}(D_{-i})$ therefore satisfies the constraint in the definition of $\\mathcal{A}^*(D_{1:n+1})$, and the monotonicity of the loss gives $\\mathcal{A}(D_{-i}) \\geq \\mathcal{A}^*(D_{1:n+1})$. Therefore, $\\ell(X_{i}, Y_{i}; \\mathcal{A}(D_{-i})) \\leq \\ell(X_{i}, Y_{i}; \\mathcal{A}^*(D_{1:n+1}))$, and\n    \\begin{equation}\n        \\E\\left[\\frac{1}{n+1}\\sum\\limits_{i=1}^{n+1} \\Big(\\ell(X_{i}, Y_{i}; \\mathcal{A}(D_{-i})) - \\ell(X_{i}, Y_{i}; \\mathcal{A}^*(D_{1:n+1}))\\Big)\\right] \\leq 0.\n    \\end{equation}\n    Rearranging terms gives $0$-stability.\n\\end{proof}\n\nCombining Proposition~\\ref{prop:crc-stable} with Theorem~\\ref{thm:main-intro} exactly recovers the classical conformal risk control guarantee.\n\\begin{corollary}\n    The conformal risk control algorithm $\\mathcal{A}$ defined in Proposition~\\ref{prop:crc-stable} satisfies\n    \\begin{equation}\n        \\E\\left[ \\ell(X_{n+1},Y_{n+1};\\mathcal{A}(D_{1:n})) \\right] \\leq \\alpha.\n    \\end{equation}\n\\end{corollary}\n\\begin{proof}\n    For monotonic losses, Proposition~\\ref{prop:crc-stable} shows that the conformal risk control algorithm $\\mathcal{A}$ is $0$-stable. Applying Theorem~\\ref{thm:main-intro} with $\\beta=0$ recovers the guarantee.\n\\end{proof}\n\nThe key takeaway is that the proof of conformal risk control's validity has two parts, which can be decoupled: (1) proving monotonicity implies stability, and (2) proving that stability implies risk control.\nTheorem~\\ref{thm:main-intro} tells us that (2) holds for general algorithms.\nThe remainder of the paper will be dedicated to replacing step (1) with other stable algorithms, in the context of non-monotonic losses.\nLooking forward, Section~\\ref{sec:methods} gives stability bounds for classes of non-monotonic losses and shows how to estimate these stability bounds for use in Theorem~\\ref{thm:main-intro}. Section~\\ref{sec:experiments} shows experimental validation of these methods.", "sketch": "To prove Theorem~\\ref{thm:main-intro}, the text argues:\n\n1. **Use exchangeability and symmetry to rewrite the target expectations as averages.** By exchangeability and symmetry,\n\\[\n\\mathbb{E}\\!\\left[\\frac{1}{n+1}\\sum_{i=1}^{n+1}\\ell(X_i,Y_i;\\mathcal{A}^*(D_{1:n+1}))\\right]\n= \\mathbb{E}\\!\\left[\\ell(X_{n+1},Y_{n+1};\\mathcal{A}^*(D_{1:n+1}))\\right],\n\\]\nand similarly\n\\[\n\\mathbb{E}\\!\\left[\\frac{1}{n+1}\\sum_{i=1}^{n+1}\\ell(X_i,Y_i;\\mathcal{A}(D_{-i}))\\right]\n= \\mathbb{E}\\!\\left[\\ell(X_{n+1},Y_{n+1};\\mathcal{A}(D_{1:n}))\\right].\n\\]\n\n2. **Apply $\\beta$-stability to compare the risks of $\\mathcal{A}$ and $\\mathcal{A}^*$.** The proof states: by $\\beta$-stability,\n\\[\n\\mathbb{E}\\!\\left[\\ell(X_{n+1},Y_{n+1};\\mathcal{A}(D_{1:n}))\\right] \\leq \\mathbb{E}\\!\\left[\\ell(X_{n+1},Y_{n+1};\\mathcal{A}^*(D_{1:n+1}))\\right] + \\beta.\n\\]\n\n3. **Use the assumption on $\\mathcal{A}^*$ to conclude.** Combining the above with the assumption\n\\(\\mathbb{E}[\\ell(X_{n+1},Y_{n+1};\\mathcal{A}^*(D_{1:n+1}))] \\leq \\alpha-\\beta\\)\n“yields the result,” i.e.,\n\\(\\mathbb{E}[\\ell(X_{n+1},Y_{n+1};\\mathcal{A}(D_{1:n}))] \\leq \\alpha\\).\n\nThe post-theorem discussion also notes that Theorem~\\ref{thm:main-intro} “applies to $\\theta$ in any space,” and that “boundedness of $\\ell$ was never needed in the proof of Theorem~\\ref{thm:main-intro}.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:main-intro}\n    Assume $\\mathcal{A}$ is symmetric and $\\beta$-stable with respect to $\\mathcal{A}^*$, that $D_{1:n+1}$ is exchangeable, and that\n\\begin{equation}\n        \\E\\left[\\ell(X_{n+1}, Y_{n+1}; \\mathcal{A}^*(D_{1:n+1})) \\right] \\leq \\alpha-\\beta.\n\\end{equation}\n    Then\n    \\begin{equation}\n        \\E\\left[\\ell(X_{n+1}, Y_{n+1}; \\mathcal{A}(D_{1:n})) \\right] \\leq \\alpha.\n    \\end{equation}", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $D_{1:n+1}=((X_1,Y_1),\\ldots,(X_{n+1},Y_{n+1}))$ be feature-label pairs with an exchangeable joint distribution, and let $\\ell(x,y;\\theta)$ be a loss function. For each $i$, let $D_{-i}$ denote $D_{1:n+1}$ with its $i$th datapoint removed. Suppose $\\mathcal A$ and $\\mathcal A^*$ are symmetric (permutation-invariant) algorithms mapping datasets to parameter values, and suppose $\\mathcal A$ is $\\beta$-stable with respect to $\\mathcal A^*$ and $\\ell$ in the sense that\n\\[\n\\mathbb E\\!\\left[\\frac{1}{n+1}\\sum_{i=1}^{n+1} \\ell(X_i,Y_i;\\mathcal A(D_{-i}))\\right]\n\\le\n\\mathbb E\\!\\left[\\frac{1}{n+1}\\sum_{i=1}^{n+1} \\ell(X_i,Y_i;\\mathcal A^*(D_{1:n+1}))\\right] + \\beta.\n\\]\nAssume also that\n\\[\n\\mathbb E\\bigl[\\ell(X_{n+1},Y_{n+1};\\mathcal A^*(D_{1:n+1}))\\bigr] \\le \\alpha-\\beta.\n\\]\nUnder these assumptions, which quantitative estimate holds for the test-point loss of $\\mathcal A$ trained on $D_{1:n}$?", "correct_choice": {"label": "A", "text": "\\[\n\\mathbb E\\bigl[\\ell(X_{n+1},Y_{n+1};\\mathcal A(D_{1:n}))\\bigr] \\le \\alpha.\n\\]"}, "choices": [{"label": "B", "text": "\\[\n\\mathbb E\\bigl[\\ell(X_{n+1},Y_{n+1};\\mathcal A(D_{1:n}))\\bigr] \\le \\alpha-\\beta.\n\\]"}, {"label": "C", "text": "\\[\n\\mathbb E\\bigl[\\ell(X_{n+1},Y_{n+1};\\mathcal A(D_{1:n}))\\bigr] \\le \\alpha+\\beta.\n\\]"}, {"label": "D", "text": "\\[\n\\mathbb E\\bigl[\\ell(X_{n+1},Y_{n+1};\\mathcal A(D_{1:n+1}))\\bigr] \\le \\alpha.\n\\]"}, {"label": "E", "text": "\\[\n\\mathbb E\\!\\left[\\frac{1}{n+1}\\sum_{i=1}^{n+1} \\ell(X_i,Y_i;\\mathcal A(D_{-i}))\\right] \\le \\alpha.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "final slack absorption", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the sharp conclusion threshold from \\(\\alpha\\) to the weaker bound \\(\\alpha+\\beta\\)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dataset dependence in algorithm input", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "leave-one-out average substituted for test-point conclusion", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 1, "justification": "The stem does not explicitly state the conclusion, but the assumptions strongly cue it: one bound contributes a +beta slack and the other gives alpha-beta, so alpha is implicitly suggested."}, "TAS": {"score": 1, "justification": "This is close to a direct theorem-style implication question. It is not a pure verbatim restatement, since the student must select among nearby alternatives, but it mainly asks for the standard conclusion of the stated assumptions."}, "GPS": {"score": 1, "justification": "Some reasoning is required: the solver must connect exchangeability and symmetry to the leave-one-out average and then combine the two inequalities carefully. Still, the structure of the assumptions makes the intended derivation fairly direct."}, "DQS": {"score": 2, "justification": "The distractors are mathematically meaningful and target realistic errors: forgetting slack accounting, choosing a weaker but true bound, switching the dataset argument, or dropping the comparison slack entirely."}, "total_score": 5, "overall_assessment": "A solid but theorem-proximate MCQ. The distractors are strong, but the item is somewhat guided by the setup and only moderately tests genuine generative reasoning."}}
{"id": "2602.20462v1", "paper_link": "http://arxiv.org/abs/2602.20462v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "\\label{thm:mainisoperim}\nFor all $A\\subset \\{0,1\\}^n$ with $|A|\\le \\frac12$,\n\\begin{equation}\\label{eqn:isoperimhalf}\n\\mathbf{E} \\sqrt{h_A} \\ge |A| \\sqrt{\\log_2(1/|A|)}.\n\\end{equation}", "start_pos": 2578, "end_pos": 2783, "label": "thm:mainisoperim"}, "ref_dict": {"eqn:isoperimhalf": "\\begin{equation}\\label{eqn:isoperimhalf}\n\\mathbf{E} \\sqrt{h_A} \\ge |A| \\sqrt{\\log_2(1/|A|)}.\n\\end{equation}", "thm:mainisoperim": "\\begin{thm}\\label{thm:mainisoperim}\nFor all $A\\subset \\{0,1\\}^n$ with $|A|\\le \\frac12$,\n\\begin{equation}\\label{eqn:isoperimhalf}\n\\mathbf{E} \\sqrt{h_A} \\ge |A| \\sqrt{\\log_2(1/|A|)}.\n\\end{equation}\n\\end{thm}", "thm:poincare": "\\begin{thm}\\label{thm:poincare} For all $f:\\{0,1\\}^n\\to \\{0,1\\}$,\n\\begin{equation}\\label{eqn:poincareL1}\n\\|\\nabla f\\|_1\\ge \\|f-\\mathbf{E}f\\|_1.\n\\end{equation}\nEquality is achieved when $f=\\mathbf{1}_A$ for a half-cube $A$.\n\\end{thm}", "eqn:sensitivity": "\\begin{equation}\\label{eqn:sensitivity}\n\\mathbf{E}\\sqrt{s_f} \\ge 1\n\\end{equation}", "eqn:poincareL1": "\\begin{equation}\\label{eqn:poincareL1}\n\\|\\nabla f\\|_1\\ge \\|f-\\mathbf{E}f\\|_1.\n\\end{equation}", "cor:kahnpark": "\\begin{thm}\\label{cor:kahnpark}\nLet $(A,B,W)$ be a partition of $\\{0,1\\}^n$ and assume $|A|=\\frac12$. Then\n\\[ |\\nabla(A,B)| + \\sqrt{n}\\,|W|\\ge \\tfrac12. \\]\n\\end{thm}", "eqn:isoperim2": "\\begin{equation}\\label{eqn:isoperim2}\n\\mathbf{E} \\sqrt{h_A} \\ge B_{w_1}(|A|),\n\\end{equation}"}, "pre_theorem_intro_text_len": 357, "pre_theorem_intro_text": "Let $n\\ge 1$ be an integer.\nFor $A\\subset \\{0,1\\}^n$ and $x\\in A$ let\n$h_A(x)$ denote the number of edges connecting $x$ with $\\{0,1\\}^n\\setminus A$. In other words, $h_A(x)$ is the number of single-bit flips of $x$ that leave $A$.\nIf $x\\not\\in A$, then let $h_A(x)=0$.\nHere $\\mathbf{E}f=2^{-n} \\sum_{x\\in \\{0,1\\}^n} f(x)$ and $|A|=\\mathbf{E} \\mathbf{1}_A$.", "context": "Let $n\\ge 1$ be an integer.\nFor $A\\subset \\{0,1\\}^n$ and $x\\in A$ let\n$h_A(x)$ denote the number of edges connecting $x$ with $\\{0,1\\}^n\\setminus A$. In other words, $h_A(x)$ is the number of single-bit flips of $x$ that leave $A$.\nIf $x\\not\\in A$, then let $h_A(x)=0$.\nHere $\\mathbf{E}f=2^{-n} \\sum_{x\\in \\{0,1\\}^n} f(x)$ and $|A|=\\mathbf{E} \\mathbf{1}_A$.", "full_context": "Let $n\\ge 1$ be an integer.\nFor $A\\subset \\{0,1\\}^n$ and $x\\in A$ let\n$h_A(x)$ denote the number of edges connecting $x$ with $\\{0,1\\}^n\\setminus A$. In other words, $h_A(x)$ is the number of single-bit flips of $x$ that leave $A$.\nIf $x\\not\\in A$, then let $h_A(x)=0$.\nHere $\\mathbf{E}f=2^{-n} \\sum_{x\\in \\{0,1\\}^n} f(x)$ and $|A|=\\mathbf{E} \\mathbf{1}_A$.\n\n\\section{Introduction}\nLet $n\\ge 1$ be an integer.\nFor $A\\subset \\{0,1\\}^n$ and $x\\in A$ let\n$h_A(x)$ denote the number of edges connecting $x$ with $\\{0,1\\}^n\\setminus A$. In other words, $h_A(x)$ is the number of single-bit flips of $x$ that leave $A$.\nIf $x\\not\\in A$, then let $h_A(x)=0$.\nHere $\\mathbf{E}f=2^{-n} \\sum_{x\\in \\{0,1\\}^n} f(x)$ and $|A|=\\mathbf{E} \\mathbf{1}_A$.\n\nThis inequality is sharp in two different ways: first, it is an equality when $A$ is a subcube.\nSecond, there is no corresponding dimension-free lower bound when $\\sqrt{h_A}$ is replaced by $h_A^\\beta$ for any $\\beta<\\frac12$, which can be seen by Hamming ball examples.\nThe study of $\\mathbf{E} \\sqrt{h_A}$ was first initiated by Talagrand \\cite{Tal93} who proved lower bounds such as $\\mathbf{E}\\sqrt{h_A}\\ge \\sqrt{2}|A|(1-|A|)$. This was later improved by Bobkov--G\\\"otze \\cite{BG99}, as a consequence of more general results.\n\nSharp lower bounds for $\\mathbf{E} h_A^\\beta$ for all $\\beta\\ge \\frac12$ follow from \\eqref{eqn:isoperimhalf} and H\\\"older's inequality; see e.g. \\cite[Lemma 5.1]{DIR24}.\nSuch bounds were previously proved for $\\beta\\ge 0.50057$ in \\cite{DIR24} and prior to that for $\\beta\\ge \\log_2(3/2)\\approx 0.585$ in \\cite{KP20,BIM23}.\nFor $\\beta=1$ one has Harper's classical isoperimetric inequality \\cite{Har66,Ber67,Hart76}.\nBy the Cauchy-Schwarz inequality, \\eqref{eqn:isoperimhalf} implies the following sharp strengthening of the classical isoperimetric inequality for $|A|\\le \\frac12$,\n\\begin{equation}\\label{eqn:classicalsharpening}\n\\mathbf{E} h_A \\ge \\frac{|A|}{|\\partial A|} |A|\\log_2(1/|A|),\n\\end{equation}\nwhere $\\partial A$ is the support of $h_A$, i.e. the vertex boundary of $A$.\nThis is an equality if $A$ is a subcube. It improves on the classical isoperimetric inequality because of the factor $\\frac{|A|}{|\\partial A|}\\ge 1$. Moreover, it fails when the factor is replaced by $(\\frac{|A|}{|\\partial A|})^{\\gamma}$ for any $\\gamma>1$, again by Hamming ball examples.\nSuch an inequality was observed by Beltran--Ivanisvili--Madrid \\cite[Cor. 1.2]{BIM23} for $\\gamma=\\frac1{\\log_2 (3/2)}-1\\approx 0.709$.\n\n\\subsection*{Applications}\nAs an immediate consequence of Theorem \\ref{thm:mainisoperim} we settle a conjecture of Kahn and Park \\cite[Conjecture 1.3]{KP20} on cube partitions:\n\\begin{thm}\\label{cor:kahnpark}\nLet $(A,B,W)$ be a partition of $\\{0,1\\}^n$ and assume $|A|=\\frac12$. Then\n\\[ |\\nabla(A,B)| + \\sqrt{n}\\,|W|\\ge \\tfrac12. \\]\n\\end{thm}\nHere $|W|=\\mathbf{E} \\mathbf{1}_W$ and the edge boundary measure $|\\nabla(A,B)|$ is defined by $2^{-n} \\# \\{(x,y)\\in \\mathcal{E}\\,:\\,x\\in A, y\\in B\\}$, where $\\mathcal{E}$ is the set of edges in the Hamming cube $\\{0,1\\}^n$.\nTheorem \\ref{cor:kahnpark} follows from Theorem \\ref{thm:mainisoperim} because for $|B\\cup W|=|A|=\\frac12$, we have\n\\[\\tfrac12\\le \\mathbf{E} \\sqrt{h_{B\\cup W}} \\le \\mathbf{E} (h_{B\\cup W}\\mathbf{1}_B) + \\mathbf{E} (\\sqrt{h_{B\\cup W}}\\mathbf{1}_W),\\]\nwhich is no greater than $|\\nabla(A,B)| + \\sqrt{n} |W|$ as required.\n\nOur next application is a sharp $L^1$ Poincar\\'e inequality for Boolean-valued functions.\n\\begin{thm}\\label{thm:poincare} For all $f:\\{0,1\\}^n\\to \\{0,1\\}$,\n\\begin{equation}\\label{eqn:poincareL1}\n\\|\\nabla f\\|_1\\ge \\|f-\\mathbf{E}f\\|_1.\n\\end{equation}\nEquality is achieved when $f=\\mathbf{1}_A$ for a half-cube $A$.\n\\end{thm}\nHere $|\\nabla f(x)|=\\sqrt{\\sum_{i=1}^n (\\tfrac12(f(x)-f(x\\oplus e_i)))^2}$ and $\\|f\\|_1=\\mathbf{E} |f|$. Note that it is still open to determine the sharp constant in this inequality for {\\it real-valued} $f$, but it is known that this constant lies in the interval $(\\frac2{\\pi},\\sqrt{\\frac{2}{\\pi}}]$; see \\cite{BELP08}, \\cite{ILvHV}, also \\cite{haonan1}, \\cite{Esken1} for quantum and vector-valued inequalities. In particular, \\eqref{eqn:poincareL1} shows that the lower bound improves for Boolean-valued functions, which was first observed in \\cite{BIM23}.\nTheorem \\ref{thm:poincare} does not follow directly from Theorem \\ref{thm:mainisoperim}, but from the more technical isoperimetric inequality \\eqref{eqn:isoperim2} proved below.\n\nFinally, isoperimetric inequalities for the Hamming cube are also related to a family of conjectures in information theory concerning noisy channels (Chen--Nair \\cite{CN24}). One of these is the most informative Boolean function conjecture, also known as Courtade--Kumar conjecture \\cite{KC13}.\nThe latter would be implied by a stronger conjecture, known as Hellinger conjecture \\cite{ABCJN17, CGN25}. Roughly speaking, these conjectures concern optimality questions for Boolean functions under standard noisy-channel models on $\\{0,1\\}^n$, where each bit is independently perturbed.\nIn the low-noise limit, the Hellinger conjecture would in particular imply that\n\\begin{equation}\\label{eqn:sensitivity}\n\\mathbf{E}\\sqrt{s_f} \\ge 1\n\\end{equation}\nholds for balanced Boolean functions $f:\\{0,1\\}^n\\to \\{0,1\\}$ (i.e. $\\mathbf{E}f=\\tfrac12$), where $s_f(x)$ is the sensitivity of $f$ at $x$, which is defined as the number of single-bit flips of $x$ that change the value of $f(x)$, i.e. $s_f(x)=h_A(x)+h_{A^c}(x)$.\nThis implication is well-known, but for convenience of the reader we provide details in \\S \\ref{sec:it}.\nSince $h_A$ and $h_{A^c}$ have disjoint support, $\\sqrt{s_f}=\\sqrt{h_A}+\\sqrt{h_{A^c}}$, so Theorem \\ref{thm:mainisoperim} with $|A|=\\frac12$ immediately implies \\eqref{eqn:sensitivity}.\n\nFor $0\\le x\\le y\\le 1$ and $B:[0,1]\\to [0,\\infty)$ let\n\\begin{equation}\\label{eqn:Gdef}\nG_1[B](x,y) = \\sqrt{(y-x)^{2}+B(y)^{2}} + B(x) - 2 B(\\tfrac{x+y}2),\n\\end{equation}\n\\[ G_2[B](x,y) = y-x + (\\sqrt{2}-1) B(y) + B(x) - 2 B(\\tfrac{x+y}2),\\]\nand $G = \\max(G_1, G_2).$\nKahn and Park \\cite{KP20} proved that if $B$ satisfies $B(0)=B(1)=0$ and the two-point inequality\n\\[G[B](x,y)\\ge 0\\]\nholds for all $0\\le x\\le y\\le 1$, then $\\mathbf{E} \\sqrt{h_A}\\ge B(|A|)$\nfor all $A\\subset \\{0,1\\}^n$ and $n\\ge 1$.\n\nThis reduces the proof of Theorem \\ref{thm:mainisoperim} to finding a Bellman function $B$ and verifying the two-point inequality.\nWe use the following refinement of the Bellman function used in \\cite{DIR24}.\nFor $x\\in (0,1]$ define\n\\[ L(x) = x \\sqrt{\\log_2(1/x)} \\]\nand $L(0)=0$.\nLet $Q(x)$ be the unique cubic interpolating polynomial such that $Q(0)=Q(1)=0$, $Q(\\frac12)=\\frac12$ and $Q(\\frac14)=2^{-3/2}$, i.e.\n\\[ Q(x) = \\tfrac23 x (1-x)(2^{5/2}-3+4(3-2^{3/2}) x). \\]\nThe idea of using the polynomial $Q$ goes back to Beltran--Ivanisvili--Madrid \\cite{BIM23}.\nLet $I(x)$ denote the Gaussian isoperimetric profile, i.e. the unique function on $[0,1]$ such that\n$I(0)=I(1)=0, I\\cdot I''=-1$.\nFor a parameter $w\\in (\\frac12,1]$ let\n\\[ {\\mathrm{J}}_w(x) = \\tfrac12 I(\\tfrac1{2w})^{-1} I(\\tfrac{1-x}{w}). \\]\nNote that ${\\mathrm{J}}_w(\\frac12)=\\frac12$ and ${\\mathrm{J}}_w(1)=0$.\nThen define\n\\begin{equation}\\label{eqn:bbdef}\nB_w(x) = \\left\\{ \\begin{array}{ll}\nL(x) & \\text{for } x\\in [0, \\frac14],\\\\\nQ(x)\n& \\text{for }x\\in [\\frac14, \\frac12],\\\\\n{\\mathrm{J}}_w(x) & \\text{for }x\\in [\\frac12, 1].\n\\end{array}\\right.\n\\end{equation}\nFor some motivation on the choice of such functions, see \\cite[\\S 2]{DIR24}.\nIn the following we shall fix\n\\begin{equation}\\label{eqn:w1def}\nw=w_1=\\tfrac{29}{32}\n\\end{equation}\nand write ${\\mathrm{J}}={\\mathrm{J}}_{w_1}$.\nNote that since $B_w(x)\\ge L(x)$ for $x\\in [0,\\frac12]$ (see \\cite[Lemma 5.9 (2)]{DIR24}), in order to prove Theorem \\ref{thm:mainisoperim} it suffices to prove that there exists $w$ such that\n\\begin{equation}\\label{eqn:maintwopt}\nG[B_w](x,y)\\ge 0\n\\end{equation}\nfor all $0\\le x\\le y\\le 1$.\nAs a consequence of this argument we obtain the isoperimetric inequality\n\\begin{equation}\\label{eqn:isoperim2}\n\\mathbf{E} \\sqrt{h_A} \\ge B_{w_1}(|A|),\n\\end{equation}\nvalid for all $A\\subset \\{0,1\\}^n$.", "post_theorem_intro_text_len": 5932, "post_theorem_intro_text": "This inequality is sharp in two different ways: first, it is an equality when $A$ is a subcube.\nSecond, there is no corresponding dimension-free lower bound when $\\sqrt{h_A}$ is replaced by $h_A^\\beta$ for any $\\beta<\\frac12$, which can be seen by Hamming ball examples.\nThe study of $\\mathbf{E} \\sqrt{h_A}$ was first initiated by Talagrand \\cite{Tal93} who proved lower bounds such as $\\mathbf{E}\\sqrt{h_A}\\ge \\sqrt{2}|A|(1-|A|)$. This was later improved by Bobkov--G\\\"otze \\cite{BG99}, as a consequence of more general results.\n\nSharp lower bounds for $\\mathbf{E} h_A^\\beta$ for all $\\beta\\ge \\frac12$ follow from \\eqref{eqn:isoperimhalf} and H\\\"older's inequality; see e.g. \\cite[Lemma 5.1]{DIR24}.\nSuch bounds were previously proved for $\\beta\\ge 0.50057$ in \\cite{DIR24} and prior to that for $\\beta\\ge \\log_2(3/2)\\approx 0.585$ in \\cite{KP20,BIM23}.\nFor $\\beta=1$ one has Harper's classical isoperimetric inequality \\cite{Har66,Ber67,Hart76}.\nBy the Cauchy-Schwarz inequality, \\eqref{eqn:isoperimhalf} implies the following sharp strengthening of the classical isoperimetric inequality for $|A|\\le \\frac12$,\n\\begin{equation}\\label{eqn:classicalsharpening}\n\\mathbf{E} h_A \\ge \\frac{|A|}{|\\partial A|} |A|\\log_2(1/|A|),\n\\end{equation}\nwhere $\\partial A$ is the support of $h_A$, i.e. the vertex boundary of $A$.\nThis is an equality if $A$ is a subcube. It improves on the classical isoperimetric inequality because of the factor $\\frac{|A|}{|\\partial A|}\\ge 1$. Moreover, it fails when the factor is replaced by $(\\frac{|A|}{|\\partial A|})^{\\gamma}$ for any $\\gamma>1$, again by Hamming ball examples.\nSuch an inequality was observed by Beltran--Ivanisvili--Madrid \\cite[Cor. 1.2]{BIM23} for $\\gamma=\\frac1{\\log_2 (3/2)}-1\\approx 0.709$.\n\nThe proof of Theorem \\ref{thm:mainisoperim} relies on a crucial induction argument of Kahn and Park \\cite{KP20}, which reduced proving lower bounds for $\\mathbf{E} h_A^\\beta$ to finding a certain Bellman function and verifying a two-point inequality.\nOur Bellman function is a further refinement of the function used in \\cite{DIR24}, which in turn builds on important work by Beltran--Ivanisvili--Madrid \\cite{BIM23}.\nIn the proof of the two-point inequality we draw from estimates and tools introduced in \\cite{DIR24}.\n\n\\subsection*{Applications}\nAs an immediate consequence of Theorem \\ref{thm:mainisoperim} we settle a conjecture of Kahn and Park \\cite[Conjecture 1.3]{KP20} on cube partitions:\n\\begin{thm}\\label{cor:kahnpark}\nLet $(A,B,W)$ be a partition of $\\{0,1\\}^n$ and assume $|A|=\\frac12$. Then\n\\[ |\\nabla(A,B)| + \\sqrt{n}\\,|W|\\ge \\tfrac12. \\]\n\\end{thm}\nHere $|W|=\\mathbf{E} \\mathbf{1}_W$ and the edge boundary measure $|\\nabla(A,B)|$ is defined by $2^{-n} \\# \\{(x,y)\\in \\mathcal{E}\\,:\\,x\\in A, y\\in B\\}$, where $\\mathcal{E}$ is the set of edges in the Hamming cube $\\{0,1\\}^n$.\nTheorem \\ref{cor:kahnpark} follows from Theorem \\ref{thm:mainisoperim} because for $|B\\cup W|=|A|=\\frac12$, we have\n\\[\\tfrac12\\le \\mathbf{E} \\sqrt{h_{B\\cup W}} \\le \\mathbf{E} (h_{B\\cup W}\\mathbf{1}_B) + \\mathbf{E} (\\sqrt{h_{B\\cup W}}\\mathbf{1}_W),\\]\nwhich is no greater than $|\\nabla(A,B)| + \\sqrt{n} |W|$ as required.\n\nOur next application is a sharp $L^1$ Poincar\\'e inequality for Boolean-valued functions.\n\\begin{thm}\\label{thm:poincare} For all $f:\\{0,1\\}^n\\to \\{0,1\\}$,\n\\begin{equation}\\label{eqn:poincareL1}\n\\|\\nabla f\\|_1\\ge \\|f-\\mathbf{E}f\\|_1.\n\\end{equation}\nEquality is achieved when $f=\\mathbf{1}_A$ for a half-cube $A$.\n\\end{thm}\nHere $|\\nabla f(x)|=\\sqrt{\\sum_{i=1}^n (\\tfrac12(f(x)-f(x\\oplus e_i)))^2}$ and $\\|f\\|_1=\\mathbf{E} |f|$. Note that it is still open to determine the sharp constant in this inequality for {\\it real-valued} $f$, but it is known that this constant lies in the interval $(\\frac2{\\pi},\\sqrt{\\frac{2}{\\pi}}]$; see \\cite{BELP08}, \\cite{ILvHV}, also \\cite{haonan1}, \\cite{Esken1} for quantum and vector-valued inequalities. In particular, \\eqref{eqn:poincareL1} shows that the lower bound improves for Boolean-valued functions, which was first observed in \\cite{BIM23}.\nTheorem \\ref{thm:poincare} does not follow directly from Theorem \\ref{thm:mainisoperim}, but from the more technical isoperimetric inequality \\eqref{eqn:isoperim2} proved below.\n\nFinally, isoperimetric inequalities for the Hamming cube are also related to a family of conjectures in information theory concerning noisy channels (Chen--Nair \\cite{CN24}). One of these is the most informative Boolean function conjecture, also known as Courtade--Kumar conjecture \\cite{KC13}.\nThe latter would be implied by a stronger conjecture, known as Hellinger conjecture \\cite{ABCJN17, CGN25}. Roughly speaking, these conjectures concern optimality questions for Boolean functions under standard noisy-channel models on $\\{0,1\\}^n$, where each bit is independently perturbed.\nIn the low-noise limit, the Hellinger conjecture would in particular imply that\n\\begin{equation}\\label{eqn:sensitivity}\n\\mathbf{E}\\sqrt{s_f} \\ge 1\n\\end{equation}\nholds for balanced Boolean functions $f:\\{0,1\\}^n\\to \\{0,1\\}$ (i.e. $\\mathbf{E}f=\\tfrac12$), where $s_f(x)$ is the sensitivity of $f$ at $x$, which is defined as the number of single-bit flips of $x$ that change the value of $f(x)$, i.e. $s_f(x)=h_A(x)+h_{A^c}(x)$.\nThis implication is well-known, but for convenience of the reader we provide details in \\S \\ref{sec:it}.\nSince $h_A$ and $h_{A^c}$ have disjoint support, $\\sqrt{s_f}=\\sqrt{h_A}+\\sqrt{h_{A^c}}$, so Theorem \\ref{thm:mainisoperim} with $|A|=\\frac12$ immediately implies \\eqref{eqn:sensitivity}.\n\n\\subsection*{Acknowledgments} P.D., P.I. and J.R. thank the American Institute of Mathematics (AIM) for funding our SQuaRE project. The authors were also supported in part by grants from the National Science Foundation CAREER-DMS-2152401 (P.I.), DMS-2154835 (J.R.) and by the Simons Foundation\nSFI-MPS-TSM-00013943 (P.D.), SFI-MPS-TSM-00014075 (J.R.), Simons Fellowship, and Humboldt Research Fellowship (P.I.).", "sketch": "The post-theorem introduction states that the proof of Theorem~\\ref{thm:mainisoperim} \"relies on a crucial induction argument of Kahn and Park \\cite{KP20}, which reduced proving lower bounds for $\\mathbf{E} h_A^\\beta$ to finding a certain Bellman function and verifying a two-point inequality.\" It further says: \"Our Bellman function is a further refinement of the function used in \\cite{DIR24}, which in turn builds on important work by Beltran--Ivanisvili--Madrid \\cite{BIM23}.\" Finally, it notes that \"[i]n the proof of the two-point inequality we draw from estimates and tools introduced in \\cite{DIR24}.\"", "expanded_sketch": "The post-theorem introduction states that the proof of Theorem~\\ref{thm:mainisoperim} “relies on a crucial induction argument of Kahn and Park \\cite{KP20}, which reduced proving lower bounds for $\\mathbf{E} h_A^\\beta$ to finding a certain Bellman function and verifying a two-point inequality.” It further says: “Our Bellman function is a further refinement of the function used in \\cite{DIR24}, which in turn builds on important work by Beltran--Ivanisvili--Madrid \\cite{BIM23}.” Finally, it notes that “[i]n the proof of the two-point inequality we draw from estimates and tools introduced in \\cite{DIR24}.”", "expanded_theorem": "\\label{thm:mainisoperim}\nFor all $A\\subset \\{0,1\\}^n$ with $|A|\\le \\frac12$,\n\\begin{equation}\\label{eqn:isoperimhalf}\n\\mathbf{E} \\sqrt{h_A} \\ge |A| \\sqrt{\\log_2(1/|A|)}.\n\\end{equation}", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let n\\ge 1 be an integer, and let A\\subset \\{0,1\\}^n satisfy |A|\\le \\tfrac12, where\n\\[\\mathbf{E}f=2^{-n}\\sum_{x\\in\\{0,1\\}^n} f(x),\\qquad |A|=\\mathbf{E}\\mathbf{1}_A=2^{-n}\\#A.\\]\nFor x\\in A, let h_A(x) be the number of single-bit flips of x that leave A, equivalently the number of edges from x to \\{0,1\\}^n\\setminus A; and for x\\notin A set h_A(x)=0. Which statement holds for every such set A?", "correct_choice": {"label": "A", "text": "\\[\\mathbf{E}\\sqrt{h_A}\\ge |A|\\sqrt{\\log_2(1/|A|)}.\\]"}, "choices": [{"label": "B", "text": "\\[\\mathbf{E}\\sqrt{h_A}\\ge |A|\\sqrt{\\log_2\\!\\bigl(1/(2|A|)\\bigr)}.\\]"}, {"label": "C", "text": "\\[\\mathbf{E}\\sqrt{h_A}\\ge |A|(1-|A|).\\]"}, {"label": "D", "text": "\\[\\mathbf{E}\\sqrt{h_A}\\ge |A|\\sqrt{\\log_2(1/|A|)}\\qquad\\text{for all }A\\subset\\{0,1\\}^n.\\]"}, {"label": "E", "text": "\\[\\mathbf{E}h_A\\ge |A|\\sqrt{\\log_2(1/|A|)}.\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "sharp Bellman profile normalization inside the logarithm", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "replaced the sharp logarithmic lower bound by a weaker elementary bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "hypothesis restriction |A|\\le 1/2", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "moment/order of the boundary term, replacing \\sqrt{h_A} by h_A", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the objects carefully but does not reveal the sharp inequality or otherwise point directly to choice A. There is no explicit answer cue beyond the general setup."}, "TAS": {"score": 1, "justification": "The correct option is essentially the exact target inequality, so the item does have a theorem-recognition flavor. However, it is not a pure restatement because the alternatives vary the hypothesis, normalization, and strength of conclusion in meaningful ways."}, "GPS": {"score": 2, "justification": "To select A, a student must distinguish a sharp universal bound from a weaker true statement (C), a close but incorrect logarithmic normalization (B), an overgeneralization beyond |A|<=1/2 (D), and a moment mismatch (E). This creates genuine pressure to reason about sharpness and scope."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: weakening the bound, changing the domain of validity, altering the logarithmic term, or confusing E[sqrt(h_A)] with E[h_A]. They are distinct and nontrivial."}, "total_score": 7, "overall_assessment": "A strong MCQ with no answer leakage and high-quality distractors; its main limitation is that the correct choice is still close to a direct statement of the underlying theorem, so it tests theorem discrimination more than full derivation."}}
{"id": "2602.20935v1", "paper_link": "http://arxiv.org/abs/2602.20935v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:MainIntro}\n\tFix $d\\geq 2$, $\\gamma>0$ and $0\\leq\\lambda\\leq 1$. Consider the visibility region $Z_{\\gamma,\\lambda,d}$ of $o$ in a Poisson process of $\\lambda$-geodesic hyperplanes in $\\mathbb{H}^d$ with intensity measure $\\gamma\\nu_\\lambda^o$.\n\t\\begin{itemize}\n\t\\item[(a)] Let $\\gamma_{\\mathrm{crit}}$ be as in \\eqref{eq:GammaCrit}. If $\\gamma<\\gamma_{\\mathrm{crit}}$, then $Z_{\\gamma,\\lambda,d}$ is unbounded with strictly positive probability, whereas if $\\gamma>\\gamma_{\\mathrm{crit}}$, then $Z_{\\gamma,\\lambda,d}$ is almost surely bounded. Moreover, if $d=2$, then $Z_{\\gamma_{\\mathrm{crit}},\\lambda,2}$ is almost surely bounded.\n\n\t\\item[(b)] If $\\gamma>\\gamma_{\\mathrm{crit}}$ then\n\t\\begin{equation}\\label{eq:MeanVolume}\n\t\t\\mathbb{E}{\\rm vol}_d(Z_{\\gamma,\\lambda,d})=\\mathbb{E}{\\rm vol}_d(Z_{\\gamma,0,d}) = \\pi^{\\frac{d-1}{2}} \\Gamma\\left(\\frac{d+1}{2}\\right){\\Gamma\\big({\\gamma_d^\\ast-d+1\\over 2}\\big)\\over\\Gamma\\big({\\gamma_d^\\ast+d+1\\over 2}\\big)}\n\t\\end{equation}\n\twith $\\gamma_d^\\ast := \\gamma\\,{\\Gamma({d\\over 2})\\over 2\\sqrt{\\pi}\\Gamma({d+1\\over 2})}$. On the other hand, $\\mathbb{E}{\\rm vol}_d(Z_{\\gamma,\\lambda,d})=+\\infty$ for all $\\gamma\\leq\\gamma_{\\mathrm{crit}}$.\n\t\\end{itemize}", "start_pos": 13726, "end_pos": 14962, "label": "thm:MainIntro"}, "ref_dict": {"fig: simulation": "\\label{fig: simulation}\n\t\\end{figure}\n\n\tThe case $\\lambda = 0$, corresponding to totally geodesic hyperplanes, has been studied in detail in previous works \\cite{BJST,BuehlerGusakovaRecke,BuehlerHugTh", "thm:MainIntro": "\\begin{theorem}\\label{thm:MainIntro}\n\tFix $d\\geq 2$, $\\gamma>0$ and $0\\leq\\lambda\\leq 1$. Consider the visibility region $Z_{\\gamma,\\lambda,d}$ of $o$ in a Poisson process of $\\lambda$-geodesic hyperplanes in $\\mathbb{H}^d$ with intensity measure $\\gamma\\nu_\\lambda^o$.\n\t\\begin{itemize}\n\t\\item[(a)] Let $\\gamma_{\\mathrm{crit}}$ be as in \\eqref{eq:GammaCrit}. If $\\gamma<\\gamma_{\\mathrm{crit}}$, then $Z_{\\gamma,\\lambda,d}$ is unbounded with strictly positive probability, whereas if $\\gamma>\\gamma_{\\mathrm{crit}}$, then $Z_{\\gamma,\\lambda,d}$ is almost surely bounded. Moreover, if $d=2$, then $Z_{\\gamma_{\\mathrm{crit}},\\lambda,2}$ is almost surely bounded.\n\n\t\\item[(b)] If $\\gamma>\\gamma_{\\mathrm{crit}}$ then\n\t\\begin{equation}\\label{eq:MeanVolume}\n\t\t\\mathbb{E}{\\rm vol}_d(Z_{\\gamma,\\lambda,d})=\\mathbb{E}{\\rm vol}_d(Z_{\\gamma,0,d}) = \\pi^{\\frac{d-1}{2}} \\Gamma\\left(\\frac{d+1}{2}\\right){\\Gamma\\big({\\gamma_d^\\ast-d+1\\over 2}\\big)\\over\\Gamma\\big({\\gamma_d^\\ast+d+1\\over 2}\\big)}\n\t\\end{equation}\n\twith $\\gamma_d^\\ast := \\gamma\\,{\\Gamma({d\\over 2})\\over 2\\sqrt{\\pi}\\Gamma({d+1\\over 2})}$. On the other hand, $\\mathbb{E}{\\rm vol}_d(Z_{\\gamma,\\lambda,d})=+\\infty$ for all $\\gamma\\leq\\gamma_{\\mathrm{crit}}$.\n\t\\end{itemize}\n\t\\end{theorem}", "eq:GammaCrit": "\\begin{equation}\\label{eq:GammaCrit}\n\t\\gamma_{\\mathrm{crit}}:= \\sqrt{\\pi}(d-1)^2{\\Gamma({d-1\\over 2})\\over\\Gamma({d\\over 2})}\n\t\\end{equation}", "eq:IndicatorToEulerChar": "\\begin{equation}\\label{eq:IndicatorToEulerChar}\n\t\\int_{{\\rm Hyp}_0^o}{\\bf 1}\\{H\\cap[o,x_h]\\neq\\varnothing\\}\\,\\nu_0^o(\\dint H) = \\int_{{\\rm Hyp}_0^o}\\chi(H\\cap[o,x_h])\\,\\nu_0^o(\\dint H) = {\\Gamma({d\\over 2})\\over 2\\sqrt{\\pi}\\Gamma({d+1\\over 2})}\\,h,\n\t\\end{equation}"}, "pre_theorem_intro_text_len": 7562, "pre_theorem_intro_text": "Random tessellations generated by Poisson hyperplane processes in Euclidean space form a classical and well-studied topic in stochastic geometry.  In $\\mathbb{R}^d$, a stationary Poisson hyperplane process gives rise to a mosaic of convex cells, and many aspects of its geometry have been analyzed in detail, including the cell containing the origin, see \\cite{HugSchneiderBook} and Chapter 10 of \\cite{SW08}.\n\n\tWhen passing from Euclidean space to hyperbolic space $\\mathbb{H}^d$ of constant negative curvature  $-1$, the situation becomes richer, since hyperbolic geometry admits several natural analogues of Euclidean hyperplanes. Besides totally geodesic hyperplanes, which are isometric copies of $\\mathbb{H}^{d-1}$, one encounters equidistant hypersurfaces, which are defined as the set of points at fixed distance from a given geodesic hyperplane. A further limiting case is provided by horospheres, which arise as limits of equidistant hypersurfaces whose base hyperplane recedes towards the ideal boundary of $\\mathbb{H}^d$. These three families can be unified by the notion of\n\t\\emph{$\\lambda$--geodesic hyperplanes} as introduced in \\cite{Solanes}. For $\\lambda \\in [0,1]$, a\n\t$\\lambda$--geodesic hyperplane is a complete totally umbilical hypersurface in $\\mathbb{H}^d$ with normal curvature $\\lambda $. Thus, $\\lambda = 0$ corresponds to totally geodesic hyperplanes, which have vanishing intrinsic curvature, $0 < \\lambda < 1$ yields equidistant hypersurfaces, whose curvature depends continuously on~$\\lambda$, and $\\lambda = 1$ gives horospheres.\n\n\tEvery $\\lambda$--geodesic hyperplane separates $\\mathbb{H}^d$ into two connected components.  As we are not interested in all such hypersurfaces, we introduce the space $\\mathrm{Hyp}_\\lambda^o$ of all $\\lambda$--geodesic hyperplanes for which the distinguished point\n\t$o \\in \\mathbb{H}^d$, which we refer to as the origin, does not lie on the convex side.  Let $\\nu_\\lambda$ denote the natural isometry-invariant measure on the space of all $\\lambda$-geodesic hyperplanes, as introduced in Section \\ref{sec:lambda-geodesic_hyperplanes} below,  and $\\nu_\\lambda^o$ its restriction to $\\mathrm{Hyp}_\\lambda^o$. For an intensity parameter $\\gamma > 0$, we define $\\eta_{\\gamma,\\lambda}$\n\tto be a Poisson point process on $\\mathrm{Hyp}_\\lambda^o$ with intensity measure $\\gamma \\nu_\\lambda^o$.  Each hyperplane in $\\eta_{\\gamma,\\lambda}$ acts as a random obstacle to visibility from the reference point~$o$. For a point $x \\in \\mathbb{H}^d$, we say that $x$ is visible from~$o$ if the geodesic segment $[o,x]$ connecting $o$ with $x$ does not intersect any $H \\in \\eta_{\\gamma,\\lambda}$.\n\tThis leads to the random closed set\n\t\\begin{equation}\\label{eq:DefVisibilitySet}\n\tZ_{\\gamma,\\lambda,d}\n\t:= \\{\\, x \\in \\mathbb{H}^d :\n\t[o,x] \\cap H = \\varnothing\n\t\\text{ for all } H \\in \\eta_{\\gamma,\\lambda}\n\t\\,\\},\n\t\\end{equation}\n\twhich we call the \\emph{visibility region} of~$o$. Equivalently, for $\\lambda=0$, $Z_{\\gamma,0,d}$ is the connected component of~$o$ in the complement of the union of all hyperplanes in $\\eta_{\\gamma,0}$. Simulations of Poisson processes of $\\lambda$-geodesic hyperplanes for different values of $\\lambda$ together with the corresponding visibility regions are shown in Figure \\ref{fig: simulation}.\n\n\t\t\\begin{figure}\n\t\t\\centering\n\t\t\\begin{minipage}{.3\\linewidth}\n\t\t\t\\hspace*{-0.4cm}\\includegraphics[scale=0.09]{l=0_g=2_c.pdf}\n\t\t\t\\subcaption{$\\lambda=0, \\gamma=2<\\gamma_{\\mathrm{crit}}$}\n\t\t\\end{minipage}\\quad\n\t\t\\begin{minipage}{.3\\linewidth}\n\t\t\t\\hspace*{-0.4cm}\t\\includegraphics[scale=0.09]{l=05_g=2_c.pdf}\n\t\t\t\\subcaption{$\\lambda=0.5, \\gamma=2<\\gamma_{\\mathrm{crit}}$}\n\t\t\\end{minipage}\\quad\n\t\t\\begin{minipage}{.3\\linewidth}\n\t\t\t\\hspace*{-0.4cm}\\includegraphics[scale=0.09]{l=1_g=2_c.pdf}\n\t\t\t\\subcaption{$\\lambda=1, \\gamma=2<\\gamma_{\\mathrm{crit}}$}\n\t\t\\end{minipage}\n\t\t\\begin{minipage}{.3\\linewidth}\n\t\t\t\\hspace*{-0.4cm}\\includegraphics[scale=0.09]{l=0_g=pi_c.pdf}\n\t\t\t\\subcaption{$\\lambda=0, \\gamma=\\pi=\\gamma_{\\mathrm{crit}}$}\n\t\t\\end{minipage}\\quad\n\t\t\\begin{minipage}{.3\\linewidth}\n\t\t\t\\hspace*{-0.4cm}\t\\includegraphics[scale=0.09]{l=05_g=pi_c.pdf}\n\t\t\t\\subcaption{$\\lambda=0.5, \\gamma=\\pi=\\gamma_{\\mathrm{crit}}$}\n\t\t\\end{minipage}\\quad\n\t\t\\begin{minipage}{.3\\linewidth}\n\t\t\t\\hspace*{-0.4cm}\\includegraphics[scale=0.09]{l=1_g=pi_c.pdf}\n\t\t\t\\subcaption{$\\lambda=1,\\gamma=\\pi=\\gamma_{\\mathrm{crit}}$}\n\t\t\\end{minipage}\n\t\t\\begin{minipage}{.3\\linewidth}\n\t\t\t\\hspace*{-0.4cm}\\includegraphics[scale=0.09]{l=0_g=7_c.pdf}\n\t\t\t\\subcaption{$\\lambda=0, \\gamma=7>\\gamma_{\\mathrm{crit}}$}\n\t\t\\end{minipage}\\quad\n\t\t\\begin{minipage}{.3\\linewidth}\n\t\t\t\\hspace*{-0.4cm}\t\\includegraphics[scale=0.09]{l=05_g=7_c.pdf}\n\t\t\t\\subcaption{$\\lambda=0.5, \\gamma=7>\\gamma_{\\mathrm{crit}}$}\n\t\t\\end{minipage}\\quad\n\t\t\\begin{minipage}{.3\\linewidth}\n\t\t\t\\hspace*{-0.4cm}\\includegraphics[scale=0.09]{l=1_g=7_c.pdf}\n\t\t\t\\subcaption{$\\lambda=1, \\gamma=7>\\gamma_{\\mathrm{crit}}$}\n\t\t\\end{minipage}\n\t\t\\caption{Simulation of a Poisson process of $\\lambda$-geodesic hyperplanes in $\\mathrm{Hyp}_\\lambda^o$ with the corresponding visibility set in red in the Poincar\\'e ball model. Note that for $d=2$, $\\gamma_{\\mathrm{crit}}=\\pi$.}\n\t\t\\label{fig: simulation}\n\t\\end{figure}\n\n\tThe case $\\lambda = 0$, corresponding to totally geodesic hyperplanes, has been studied in detail in previous works \\cite{BJST,BuehlerGusakovaRecke,BuehlerHugThaele,GKT22,PorretBlanc}.  These papers show that the visibility region $Z_{\\gamma,0,d}$ may be unbounded, and that its behaviour undergoes a sharp phase transition as the intensity~$\\gamma$ varies.  More precisely, there exists the critical value\n\t\\begin{equation}\\label{eq:GammaCrit}\n\t\\gamma_{\\mathrm{crit}}:= \\sqrt{\\pi}(d-1)^2{\\Gamma({d-1\\over 2})\\over\\Gamma({d\\over 2})}\n\t\\end{equation}\n\tsuch that\n\t\\begin{itemize}\n\t\t\\item if $\\gamma < \\gamma_{\\mathrm{crit}}$, then\n\t\t$Z_{\\gamma,0,d}$ is unbounded with strictly positive probability,\n\t\t\\item if $\\gamma \\geq \\gamma_{\\mathrm{crit}}$, then\n\t\t$Z_{\\gamma,0,d}$ is almost surely bounded.\n\t\\end{itemize}\n\tThe critical case $\\gamma = \\gamma_{\\mathrm{crit}}$ was treated in \\cite{GKT22,PorretBlanc} for $d=2$ and in \\cite{BuehlerGusakovaRecke} for general dimensions. Thus, for small intensities, there remains a positive chance of seeing arbitrarily far from the origin, whereas for large intensities the random hypersurfaces almost surely form a finite ``cocoon'' around~$o$.  In the latter regime, the expected hyperbolic volume of the bounded visibility region has been computed in \\cite[Theorem 7.1]{BuehlerHugThaele}, yielding an explicit expression in terms of the dimension~$d$ and the intensity parameter~$\\gamma$.\n\n\tIn the present work we investigate the same questions for general\n\t$\\lambda$-geodesic hyperplanes.  One might expect the behaviour of the visibility region to depend sensitively on the parameter $\\lambda$, since varying $\\lambda$ interpolates between different geometric objects. Surprisingly, our results show that the situation is completely rigid with respect to~$\\lambda$.  The critical intensity at which the transition from unbounded to bounded\n\tvisibility occurs is \\emph{identical} for all $\\lambda \\in [0,1]$, and in\n\tthe bounded regime the expected volume of the visibility region\n\t$Z_{\\gamma,\\lambda,d}$ agrees exactly with the expression known from the case $\\lambda = 0$.  In other words, neither the existence of infinite sightliness nor the size of the bounded visibility region depends in any way on the geometric nature of the obstructing hypersurfaces.  Our main findings are summarized in the following result.", "context": "Every $\\lambda$--geodesic hyperplane separates $\\mathbb{H}^d$ into two connected components. As we are not interested in all such hypersurfaces, we introduce the space $\\mathrm{Hyp}_\\lambda^o$ of all $\\lambda$--geodesic hyperplanes for which the distinguished point\n $o \\in \\mathbb{H}^d$, which we refer to as the origin, does not lie on the convex side. Let $\\nu_\\lambda$ denote the natural isometry-invariant measure on the space of all $\\lambda$-geodesic hyperplanes, as introduced in Section \\ref{sec:lambda-geodesic_hyperplanes} below, and $\\nu_\\lambda^o$ its restriction to $\\mathrm{Hyp}_\\lambda^o$. For an intensity parameter $\\gamma > 0$, we define $\\eta_{\\gamma,\\lambda}$\n to be a Poisson point process on $\\mathrm{Hyp}_\\lambda^o$ with intensity measure $\\gamma \\nu_\\lambda^o$. Each hyperplane in $\\eta_{\\gamma,\\lambda}$ acts as a random obstacle to visibility from the reference point~$o$. For a point $x \\in \\mathbb{H}^d$, we say that $x$ is visible from~$o$ if the geodesic segment $[o,x]$ connecting $o$ with $x$ does not intersect any $H \\in \\eta_{\\gamma,\\lambda}$.\n This leads to the random closed set\n \\begin{equation}\\label{eq:DefVisibilitySet}\n Z_{\\gamma,\\lambda,d}\n := \\{\\, x \\in \\mathbb{H}^d :\n [o,x] \\cap H = \\varnothing\n \\text{ for all } H \\in \\eta_{\\gamma,\\lambda}\n \\,\\},\n \\end{equation}\n which we call the \\emph{visibility region} of~$o$. Equivalently, for $\\lambda=0$, $Z_{\\gamma,0,d}$ is the connected component of~$o$ in the complement of the union of all hyperplanes in $\\eta_{\\gamma,0}$. Simulations of Poisson processes of $\\lambda$-geodesic hyperplanes for different values of $\\lambda$ together with the corresponding visibility regions are shown in Figure \\ref{fig: simulation}.\n\n\\begin{figure}\n \\centering\n \\begin{minipage}{.3\\linewidth}\n \\hspace*{-0.4cm}\\includegraphics[scale=0.09]{l=0_g=2_c.pdf}\n \\subcaption{$\\lambda=0, \\gamma=2<\\gamma_{\\mathrm{crit}}$}\n \\end{minipage}\\quad\n \\begin{minipage}{.3\\linewidth}\n \\hspace*{-0.4cm} \\includegraphics[scale=0.09]{l=05_g=2_c.pdf}\n \\subcaption{$\\lambda=0.5, \\gamma=2<\\gamma_{\\mathrm{crit}}$}\n \\end{minipage}\\quad\n \\begin{minipage}{.3\\linewidth}\n \\hspace*{-0.4cm}\\includegraphics[scale=0.09]{l=1_g=2_c.pdf}\n \\subcaption{$\\lambda=1, \\gamma=2<\\gamma_{\\mathrm{crit}}$}\n \\end{minipage}\n \\begin{minipage}{.3\\linewidth}\n \\hspace*{-0.4cm}\\includegraphics[scale=0.09]{l=0_g=pi_c.pdf}\n \\subcaption{$\\lambda=0, \\gamma=\\pi=\\gamma_{\\mathrm{crit}}$}\n \\end{minipage}\\quad\n \\begin{minipage}{.3\\linewidth}\n \\hspace*{-0.4cm} \\includegraphics[scale=0.09]{l=05_g=pi_c.pdf}\n \\subcaption{$\\lambda=0.5, \\gamma=\\pi=\\gamma_{\\mathrm{crit}}$}\n \\end{minipage}\\quad\n \\begin{minipage}{.3\\linewidth}\n \\hspace*{-0.4cm}\\includegraphics[scale=0.09]{l=1_g=pi_c.pdf}\n \\subcaption{$\\lambda=1,\\gamma=\\pi=\\gamma_{\\mathrm{crit}}$}\n \\end{minipage}\n \\begin{minipage}{.3\\linewidth}\n \\hspace*{-0.4cm}\\includegraphics[scale=0.09]{l=0_g=7_c.pdf}\n \\subcaption{$\\lambda=0, \\gamma=7>\\gamma_{\\mathrm{crit}}$}\n \\end{minipage}\\quad\n \\begin{minipage}{.3\\linewidth}\n \\hspace*{-0.4cm} \\includegraphics[scale=0.09]{l=05_g=7_c.pdf}\n \\subcaption{$\\lambda=0.5, \\gamma=7>\\gamma_{\\mathrm{crit}}$}\n \\end{minipage}\\quad\n \\begin{minipage}{.3\\linewidth}\n \\hspace*{-0.4cm}\\includegraphics[scale=0.09]{l=1_g=7_c.pdf}\n \\subcaption{$\\lambda=1, \\gamma=7>\\gamma_{\\mathrm{crit}}$}\n \\end{minipage}\n \\caption{Simulation of a Poisson process of $\\lambda$-geodesic hyperplanes in $\\mathrm{Hyp}_\\lambda^o$ with the corresponding visibility set in red in the Poincar\\'e ball model. Note that for $d=2$, $\\gamma_{\\mathrm{crit}}=\\pi$.}\n \\label{fig: simulation}\n \\end{figure}\n\nThe case $\\lambda = 0$, corresponding to totally geodesic hyperplanes, has been studied in detail in previous works \\cite{BJST,BuehlerGusakovaRecke,BuehlerHugThaele,GKT22,PorretBlanc}. These papers show that the visibility region $Z_{\\gamma,0,d}$ may be unbounded, and that its behaviour undergoes a sharp phase transition as the intensity~$\\gamma$ varies. More precisely, there exists the critical value\n \\begin{equation}\\label{eq:GammaCrit}\n \\gamma_{\\mathrm{crit}}:= \\sqrt{\\pi}(d-1)^2{\\Gamma({d-1\\over 2})\\over\\Gamma({d\\over 2})}\n \\end{equation}\n such that\n \\begin{itemize}\n \\item if $\\gamma < \\gamma_{\\mathrm{crit}}$, then\n $Z_{\\gamma,0,d}$ is unbounded with strictly positive probability,\n \\item if $\\gamma \\geq \\gamma_{\\mathrm{crit}}$, then\n $Z_{\\gamma,0,d}$ is almost surely bounded.\n \\end{itemize}\n The critical case $\\gamma = \\gamma_{\\mathrm{crit}}$ was treated in \\cite{GKT22,PorretBlanc} for $d=2$ and in \\cite{BuehlerGusakovaRecke} for general dimensions. Thus, for small intensities, there remains a positive chance of seeing arbitrarily far from the origin, whereas for large intensities the random hypersurfaces almost surely form a finite ``cocoon'' around~$o$. In the latter regime, the expected hyperbolic volume of the bounded visibility region has been computed in \\cite[Theorem 7.1]{BuehlerHugThaele}, yielding an explicit expression in terms of the dimension~$d$ and the intensity parameter~$\\gamma$.\n\nIn the present work we investigate the same questions for general\n $\\lambda$-geodesic hyperplanes. One might expect the behaviour of the visibility region to depend sensitively on the parameter $\\lambda$, since varying $\\lambda$ interpolates between different geometric objects. Surprisingly, our results show that the situation is completely rigid with respect to~$\\lambda$. The critical intensity at which the transition from unbounded to bounded\n visibility occurs is \\emph{identical} for all $\\lambda \\in [0,1]$, and in\n the bounded regime the expected volume of the visibility region\n $Z_{\\gamma,\\lambda,d}$ agrees exactly with the expression known from the case $\\lambda = 0$. In other words, neither the existence of infinite sightliness nor the size of the bounded visibility region depends in any way on the geometric nature of the obstructing hypersurfaces. Our main findings are summarized in the following result.", "full_context": "Every $\\lambda$--geodesic hyperplane separates $\\mathbb{H}^d$ into two connected components. As we are not interested in all such hypersurfaces, we introduce the space $\\mathrm{Hyp}_\\lambda^o$ of all $\\lambda$--geodesic hyperplanes for which the distinguished point\n $o \\in \\mathbb{H}^d$, which we refer to as the origin, does not lie on the convex side. Let $\\nu_\\lambda$ denote the natural isometry-invariant measure on the space of all $\\lambda$-geodesic hyperplanes, as introduced in Section \\ref{sec:lambda-geodesic_hyperplanes} below, and $\\nu_\\lambda^o$ its restriction to $\\mathrm{Hyp}_\\lambda^o$. For an intensity parameter $\\gamma > 0$, we define $\\eta_{\\gamma,\\lambda}$\n to be a Poisson point process on $\\mathrm{Hyp}_\\lambda^o$ with intensity measure $\\gamma \\nu_\\lambda^o$. Each hyperplane in $\\eta_{\\gamma,\\lambda}$ acts as a random obstacle to visibility from the reference point~$o$. For a point $x \\in \\mathbb{H}^d$, we say that $x$ is visible from~$o$ if the geodesic segment $[o,x]$ connecting $o$ with $x$ does not intersect any $H \\in \\eta_{\\gamma,\\lambda}$.\n This leads to the random closed set\n \\begin{equation}\\label{eq:DefVisibilitySet}\n Z_{\\gamma,\\lambda,d}\n := \\{\\, x \\in \\mathbb{H}^d :\n [o,x] \\cap H = \\varnothing\n \\text{ for all } H \\in \\eta_{\\gamma,\\lambda}\n \\,\\},\n \\end{equation}\n which we call the \\emph{visibility region} of~$o$. Equivalently, for $\\lambda=0$, $Z_{\\gamma,0,d}$ is the connected component of~$o$ in the complement of the union of all hyperplanes in $\\eta_{\\gamma,0}$. Simulations of Poisson processes of $\\lambda$-geodesic hyperplanes for different values of $\\lambda$ together with the corresponding visibility regions are shown in Figure \\ref{fig: simulation}.\n\n\\begin{figure}\n \\centering\n \\begin{minipage}{.3\\linewidth}\n \\hspace*{-0.4cm}\\includegraphics[scale=0.09]{l=0_g=2_c.pdf}\n \\subcaption{$\\lambda=0, \\gamma=2<\\gamma_{\\mathrm{crit}}$}\n \\end{minipage}\\quad\n \\begin{minipage}{.3\\linewidth}\n \\hspace*{-0.4cm} \\includegraphics[scale=0.09]{l=05_g=2_c.pdf}\n \\subcaption{$\\lambda=0.5, \\gamma=2<\\gamma_{\\mathrm{crit}}$}\n \\end{minipage}\\quad\n \\begin{minipage}{.3\\linewidth}\n \\hspace*{-0.4cm}\\includegraphics[scale=0.09]{l=1_g=2_c.pdf}\n \\subcaption{$\\lambda=1, \\gamma=2<\\gamma_{\\mathrm{crit}}$}\n \\end{minipage}\n \\begin{minipage}{.3\\linewidth}\n \\hspace*{-0.4cm}\\includegraphics[scale=0.09]{l=0_g=pi_c.pdf}\n \\subcaption{$\\lambda=0, \\gamma=\\pi=\\gamma_{\\mathrm{crit}}$}\n \\end{minipage}\\quad\n \\begin{minipage}{.3\\linewidth}\n \\hspace*{-0.4cm} \\includegraphics[scale=0.09]{l=05_g=pi_c.pdf}\n \\subcaption{$\\lambda=0.5, \\gamma=\\pi=\\gamma_{\\mathrm{crit}}$}\n \\end{minipage}\\quad\n \\begin{minipage}{.3\\linewidth}\n \\hspace*{-0.4cm}\\includegraphics[scale=0.09]{l=1_g=pi_c.pdf}\n \\subcaption{$\\lambda=1,\\gamma=\\pi=\\gamma_{\\mathrm{crit}}$}\n \\end{minipage}\n \\begin{minipage}{.3\\linewidth}\n \\hspace*{-0.4cm}\\includegraphics[scale=0.09]{l=0_g=7_c.pdf}\n \\subcaption{$\\lambda=0, \\gamma=7>\\gamma_{\\mathrm{crit}}$}\n \\end{minipage}\\quad\n \\begin{minipage}{.3\\linewidth}\n \\hspace*{-0.4cm} \\includegraphics[scale=0.09]{l=05_g=7_c.pdf}\n \\subcaption{$\\lambda=0.5, \\gamma=7>\\gamma_{\\mathrm{crit}}$}\n \\end{minipage}\\quad\n \\begin{minipage}{.3\\linewidth}\n \\hspace*{-0.4cm}\\includegraphics[scale=0.09]{l=1_g=7_c.pdf}\n \\subcaption{$\\lambda=1, \\gamma=7>\\gamma_{\\mathrm{crit}}$}\n \\end{minipage}\n \\caption{Simulation of a Poisson process of $\\lambda$-geodesic hyperplanes in $\\mathrm{Hyp}_\\lambda^o$ with the corresponding visibility set in red in the Poincar\\'e ball model. Note that for $d=2$, $\\gamma_{\\mathrm{crit}}=\\pi$.}\n \\label{fig: simulation}\n \\end{figure}\n\nThe case $\\lambda = 0$, corresponding to totally geodesic hyperplanes, has been studied in detail in previous works \\cite{BJST,BuehlerGusakovaRecke,BuehlerHugThaele,GKT22,PorretBlanc}. These papers show that the visibility region $Z_{\\gamma,0,d}$ may be unbounded, and that its behaviour undergoes a sharp phase transition as the intensity~$\\gamma$ varies. More precisely, there exists the critical value\n \\begin{equation}\\label{eq:GammaCrit}\n \\gamma_{\\mathrm{crit}}:= \\sqrt{\\pi}(d-1)^2{\\Gamma({d-1\\over 2})\\over\\Gamma({d\\over 2})}\n \\end{equation}\n such that\n \\begin{itemize}\n \\item if $\\gamma < \\gamma_{\\mathrm{crit}}$, then\n $Z_{\\gamma,0,d}$ is unbounded with strictly positive probability,\n \\item if $\\gamma \\geq \\gamma_{\\mathrm{crit}}$, then\n $Z_{\\gamma,0,d}$ is almost surely bounded.\n \\end{itemize}\n The critical case $\\gamma = \\gamma_{\\mathrm{crit}}$ was treated in \\cite{GKT22,PorretBlanc} for $d=2$ and in \\cite{BuehlerGusakovaRecke} for general dimensions. Thus, for small intensities, there remains a positive chance of seeing arbitrarily far from the origin, whereas for large intensities the random hypersurfaces almost surely form a finite ``cocoon'' around~$o$. In the latter regime, the expected hyperbolic volume of the bounded visibility region has been computed in \\cite[Theorem 7.1]{BuehlerHugThaele}, yielding an explicit expression in terms of the dimension~$d$ and the intensity parameter~$\\gamma$.\n\nIn the present work we investigate the same questions for general\n $\\lambda$-geodesic hyperplanes. One might expect the behaviour of the visibility region to depend sensitively on the parameter $\\lambda$, since varying $\\lambda$ interpolates between different geometric objects. Surprisingly, our results show that the situation is completely rigid with respect to~$\\lambda$. The critical intensity at which the transition from unbounded to bounded\n visibility occurs is \\emph{identical} for all $\\lambda \\in [0,1]$, and in\n the bounded regime the expected volume of the visibility region\n $Z_{\\gamma,\\lambda,d}$ agrees exactly with the expression known from the case $\\lambda = 0$. In other words, neither the existence of infinite sightliness nor the size of the bounded visibility region depends in any way on the geometric nature of the obstructing hypersurfaces. Our main findings are summarized in the following result.\n\n\\begin{prop}\\label{prop:gamma=gamma_crit}\n We assume the same set-up as in Theorem \\ref{thm:CoveringGenerald}. For $d=2$ and $\\gamma=\\gamma_{\\mathrm{crit}}$ it holds\n $\\displaystyle\\mathbb{P}[\\limsup_{n\\to\\infty} \\mathcal{S}^\\circ(u_n, \\varphi(r_n)) = \\mathbb{S}^{1}] = 1$. Thus, the visibility set $Z_{\\gamma_\\mathrm{crit},\\lambda,2}$ in \\eqref{eq:DefVisibilitySet} is almost surely bounded.\n \\end{prop}\n \\begin{proof}\n As before, let $0<r_1<r_2<\\dots$ be such that they form an inhomogeneous Poisson point process on $(0,1)$ with intensity function $f$ defined in \\eqref{eq:intensity_r}. Then, in the Poincar\\'e disc model a $\\lambda$-geodesic hyperplane $H(r_n,u)$ of distance $r_n$ to $o$ in direction $u\\in\\mathbb{S}^1$ covers an arc of length $2\\arccos(\\varphi(r_n))$ for $n\\in\\mathbb{N}$ (see Figure \\ref{fig: configuration_caps}). Now, by \\cite[Equation (1)]{Shepp}, Proposition \\ref{prop:gamma=gamma_crit} follows once we have shown that\n $$\\sum_{n=1}^{\\infty}\\frac{1}{n^2}e^{\\ell_1+\\ldots+\\ell_n}=\\infty,$$\n where $2\\pi\\ell_i$ is the arc length covered by $r_i$, i.e.\\ $\\ell_i=\\frac{1}{\\pi}\\arccos(\\varphi(r_i))$ for $i\\in\\mathbb{N}$. Combining \\eqref{eq: 1-phi(rn)_asymptotics}, $\\gamma_{\\mathrm{crit}}=\\pi$ and the refined $d=2$ asymptotics from Lemma~\\ref{lemma:asymptotics_rn} we can sharpen the estimate for the arc lengths $\\ell_i$.\n Indeed, for $d=2$ and $\\gamma=\\gamma_{\\mathrm{crit}}=\\pi$, Lemma~\\ref{lemma:asymptotics_rn} yields almost surely\n \\begin{equation*}\n 1-r_i=\\frac{\\pi(1+\\lambda)}{i} +O\\left(\\sqrt{\\frac{\\log\\log(i)}{i^3}}\\right),\\qquad i\\to\\infty.\n \\end{equation*}\n Together with Lemma~\\ref{lemma:asymptotics_varphi} this implies almost surely\n \\begin{equation*}\n \\delta_i:=1-\\varphi(r_i)\n =\\frac{(1-r_i)^2}{2(1+\\lambda)^2}+O\\big((1-r_i)^3\\big)\n =\\frac{\\pi^2}{2i^2}+O\\left(\\sqrt{\\frac{\\log\\log(i)}{i^5}}\\right),\n \\qquad i\\to\\infty.\n \\end{equation*}\n Using the Taylor expansions $\\arccos(1-x)=\\sqrt{2x}+O(x^{3/2})$ as $x\\downarrow 0$ and $\\sqrt{1+x}=1+\\frac{x}{2}+O(x^2)$ as $x\\downarrow 0$, we obtain almost surely\n \\[\n \\arccos(\\varphi(r_i))\n =\\arccos(1-\\delta_i)\n =\\sqrt{2\\delta_i}+O(\\delta_i^{3/2})\n =\\frac{\\pi}{i}+O\\left(\\sqrt{\\frac{\\log\\log(i)}{i^3}}\\right),\n \\qquad i\\to\\infty.\n \\]\n Consequently,\n \\begin{equation*}\n \\ell_i=\\frac{1}{\\pi}\\arccos(\\varphi(r_i))\n =\\frac{1}{i}+O\\left(\\sqrt{\\frac{\\log\\log(i)}{i^3}}\\right),\n \\qquad i\\to\\infty,\n \\end{equation*}\n almost surely. Hence, as $\\sum_{k=1}^\\infty \\sqrt{\\frac{\\log\\log(k)}{k^3}}<\\infty,$ this\n implies that there exists $c\\in\\mathbb{R}$ such that\n \\[\n \\sum_{k=1}^{n}\\ell_k\n \\ge \\sum_{k=1}^{n}\\frac{1}{k}+c\n \\ge \\ln(n)+c,\n \\]\n where we used $\\sum_{k=1}^n \\frac1k \\ge \\ln(n)$.\n Thus,\n \\begin{align*}\n \\sum_{n=1}^{\\infty}\\frac{1}{n^2}e^{\\ell_1+\\ldots+\\ell_n}\n &\\ge \\sum_{n=1}^{\\infty}\\frac{1}{n^2}e^{\\ln(n)+c}\n = \\sum_{n=1}^\\infty \\frac{e^{c}}{n}\n =\\infty,\n \\end{align*}\n which completes the proof.\n \\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:MainIntro} (b)]\n Fix $u\\in\\mathbb{S}^{d-1}$ and let $\\ell(h)$ be a geodesic segment of length $h>0$ starting at $o$ in an arbitrary direction. Then by Corollary \\ref{coro:mu(F_[o,l])} we have\n \\begin{align*}\n \\p(s_u(\\eta_{\\gamma,\\lambda})>h)&=\\p(\\eta_{\\gamma,\\lambda}([\\ell(h)]_\\lambda)=0)\n =\\exp\\big({-\\mu_{\\gamma,\\lambda}([\\ell(h)]_\\lambda)}\\big)=\\exp\\Big({-\\gamma\\,\\frac{\\Gamma(\\frac{d}{2})}{2\\sqrt{\\pi}\\,\\Gamma(\\frac{d+1}{2})}\\,h}\\Big),\n \\end{align*}\n independently of $\\lambda$.\n Hence, the random variable $s_u(\\eta_{\\gamma,\\lambda})$ is exponentially distributed with parameter $\\gamma\\,\\frac{\\Gamma(\\frac{d}{2})}{2\\sqrt{\\pi}\\,\\Gamma(\\frac{d+1}{2})}$ and by the polar decomposition of hyperbolic space we have\n \\begin{align*}\n \\mathbb{E}{\\rm vol}_d(Z_{\\gamma,\\lambda,d})\n &={2\\pi^{d/2}\\over\\Gamma({d\\over 2})}\\E\\Bigg[\\int_{\\mathbb{S}^{d-1}}\\int_0^{s_u(\\eta_{\\gamma,\\lambda})}\\sinh^{d-1}(s)\\;\\mathrm{d}s\\sigma_d(\\mathrm{d}u)\\Bigg]\\\\\n &={2\\pi^{d/2}\\over\\Gamma({d\\over 2})}\\int_{\\mathbb{S}^{d-1}}\\int_0^{\\infty}\\p(s_u(\\eta_{\\gamma,\\lambda})\\geq s)\\sinh^{d-1}(s)\\;\\mathrm{d}s\\sigma_d(\\mathrm{d}u)\\Bigg]\\\\\n &={2\\pi^{d/2}\\over\\Gamma({d\\over 2})}\\int_0^\\infty \\exp\\Big({-\\gamma\\,\\frac{\\Gamma(\\frac{d}{2})}{2\\sqrt{\\pi}\\,\\Gamma(\\frac{d+1}{2})}\\,s}\\Big)\\sinh^{d-1}(s)\\;\\mathrm{d}s,\n \\end{align*}\n where $\\sigma_d$ denotes the normalized spherical Lebesgue measure on $\\mathbb{S}^{d-1}$.\n According to Identity 3.541.1 in \\cite{GradRysz} (see also \\cite[Equation (6.9)]{BuehlerHugThaele}) it holds that\n \\begin{equation}\\label{eq:SinhExpIntegral}\n \\int_0^\\infty\\sinh^{d-1}(s)e^{-as}\\,\\mathrm{d}s=\\frac{(d-1)!}{2^d}\\frac{\\Gamma(\\frac{a-d+1}{2})}{\\Gamma(\\frac{a+d+1}{2})},\n \\end{equation}\n whenever $a>d-1$. Thus, the expected volume is finite if and only if $\\gamma\\,\\frac{\\Gamma(\\frac{d}{2})}{2\\sqrt{\\pi}\\,\\Gamma(\\frac{d+1}{2})}>d-1$ or $\\gamma>\\frac{2(d-1)\\sqrt{\\pi}\\Gamma(\\frac{d+1}{2})}{\\Gamma(\\frac{d}{2})}=(d-1)^2\\sqrt{\\pi}\\frac{\\Gamma(\\frac{d-1}{2})}{\\Gamma(\\frac{d}{2})}=\\gamma_{\\mathrm{crit}}$ and, in this case,\n $$\n \\mathbb{E}{\\rm vol}_d(Z_{\\gamma,\\lambda,d})= \\mathbb{E}{\\rm vol}_d(Z_{\\gamma,0,d}) = {2^{1-d}\\pi^{d/2}(d-1)!\\over \\Gamma({d\\over 2})}{\\Gamma\\big({\\gamma_d^\\ast-d+1\\over 2}\\big)\\over\\Gamma\\big({\\gamma_d^\\ast+d+1\\over 2}\\big)}\\quad\\text{with}\\quad \\gamma_d^\\ast := \\gamma\\,{\\Gamma({d\\over 2})\\over2\\sqrt{\\pi}\\Gamma({d+1\\over 2})}.\n $$\n To arrive at the desired expression, we simplify the factor ${2^{1-d}\\pi^{d/2}(d-1)!\\over \\Gamma({d\\over 2})}$ using the Legendre duplication formula.\n It follows that\n \\begin{align*}\n {2^{1-d}\\pi^{d/2}(d-1)!\\over \\Gamma({d\\over 2})}&=2^{1-d}\\pi^{d/2}(d-1)! \\frac{\\Gamma(\\frac{d+1}{2})}{2^{1-d}\\sqrt{\\pi}\\Gamma(d)}= \\pi^{\\frac{d-1}{2}} \\Gamma\\left(\\frac{d+1}{2}\\right).\n \\end{align*}\n Plugging this into the expression for $\\mathbb{E}{\\rm vol}_d(Z_{\\gamma,\\lambda,d})$ completes the proof.\n\\end{proof}\n\n\\begin{rema}\\rm\n The notion of $\\lambda$-geodesic hyperplanes as totally umbilical hypersurfaces of $\\mathbb{H}^d$ can be extended to the case $\\lambda>1$. In that situation, a $\\lambda$-geodesic 'hyperplane' is a hyperbolic sphere of radius\n \\[\n R_\\lambda : = \\operatorname{artanh}\\Big(\\frac{1}{\\lambda}\\Big).\n \\]\n Hence, in this regime the visibility question in presence of a Poisson process of such spheres reduces to that in a Boolean model of balls with fixed radius $R_\\lambda$, which in turn has been studied in \\cite{L96}. As for $\\lambda\\in[0,1]$, there exists a critical intensity $\\gamma_{B,\\mathrm{crit}}>0$ such that the visibility region is unbounded with positive probability if $\\gamma<\\gamma_{B,\\mathrm{crit}}$ and almost surely bounded for $\\gamma>\\gamma_{B,\\mathrm{crit}}$. However, a key difference to the case $\\lambda\\in[0,1]$ is that, for $\\lambda>1$, this critical intensity depends on $\\lambda$. By \\cite[p.\\ 447]{L96} and \\cite[Remark 6.5]{BuehlerHugThaele} it is given by $$\\gamma_{B,\\mathrm{crit}}=\\frac{(d-1)\\Gamma(\\frac{d+1}{2})}{\\pi^{(d-1)/2}\\sinh(R_\\lambda)}=\\frac{(d-1)\\Gamma(\\frac{d+1}{2})}{\\pi^{(d-1)/2}(\\lambda^2-1)^{(d-1)/2}}.$$ In the bounded phase, the expected visible volume is derived in \\cite[Theorem 6.2]{BuehlerHugThaele}. It can be expressed in the same way as the mean visible volume in \\eqref{eq:MeanVolume} by replacing $\\gamma_d^*$ by $v_d^*=\\gamma\\frac{\\pi^{(d-1)/2}}{\\Gamma(\\frac{d+1}{2})}\\sinh(R_\\lambda)=\\gamma\\frac{\\pi^{(d-1)/2}}{\\Gamma(\\frac{d+1}{2})(\\lambda^2-1)^{(d-1)/2}}$.\n\\end{rema}", "post_theorem_intro_text_len": 3947, "post_theorem_intro_text": "The proofs of the results in (a) and (b) rely on two complementary ingredients.\n\tFor the existence of a phase transition, we adapt a covering criterion due to Hoffmann--J\\o rgensen \\cite{H73}. This method parallels the approach in \\cite{GKT22} used in the case $\\lambda = 0$, and it  shows that the critical threshold does not depend on the geometric nature of the\n\thyperplanes, i.e., on $\\lambda$. The concrete computations rely on a subtle cancellation, which eliminates the $\\lambda$-dependence from the relevant density terms.\n\tThe computation of the expected volume in the bounded phase reduces to an integral-geometric question: one must determine the measure $\\nu_\\lambda^o$ of $\\lambda$--geodesic hyperplanes that intersect a fixed geodesic segment of length~$h>0$.\n\tFor $\\lambda = 0$, this quantity can be determined using the classical Crofton formula in hyperbolic space. In fact, if $x_h$ is a point with hyperbolic distance $h$ from $o$ we have\n\t\\begin{equation}\\label{eq:IndicatorToEulerChar}\n\t\\int_{{\\rm Hyp}_0^o}{\\bf 1}\\{H\\cap[o,x_h]\\neq\\varnothing\\}\\,\\nu_0^o(\\mathrm{d} H) = \\int_{{\\rm Hyp}_0^o}\\chi(H\\cap[o,x_h])\\,\\nu_0^o(\\mathrm{d} H) = {\\Gamma({d\\over 2})\\over 2\\sqrt{\\pi}\\Gamma({d+1\\over 2})}\\,h,\n\t\\end{equation}\n\twhere $\\chi$ denotes the Euler characteristic, see \\cite[Section 7]{BuehlerHugThaele}.  However, the Crofton argument breaks down for $\\lambda>0$, since a\n\t$\\lambda$--geodesic hyperplane may intersect a geodesic segment either once or twice. This implies that for all such $\\lambda$ the indicator function cannot be replaced by the Euler characteristic and it seems that no direct integral-geometric relation is available.  We overcome this  by introducing an explicit parametrization of $\\lambda$--geodesic hyperplanes and carrying out the computation directly.  The resulting expression turns out to be independent of~$\\lambda$ and again proportional to the length~$h$. This invariance is a surprising integral-geometric fact of independent interest and is ultimately responsible for the $\\lambda$--independence of the expected volume as described in Theorem \\ref{thm:MainIntro}.\n\n\tOur findings are placed within the broader context of stochastic geometry in hyperbolic space, a field that has seen significant recent activity. This area encompasses a rich variety of random structures adapted to negative curvature. A selection of recent results includes the asymptotic geometry of random polytopes \\cite{BesauThaele,FodorGruenfelder}, the investigation of the Boolean model \\cite{BJST,BuehlerHugThaele,HugLastSchulte,Tykesson} and the study of hyperbolic random graphs, which connects geometric probability with network science \\cite{BlaesiusETal,FountulakisEtal,KrioukovEtal}. The investigation of Poisson point processes and their associated Voronoi tessellations remained a central theme, with foundational works including \\cite{Isokawa}. A particular focus has recently developed around ideal Poisson--Voronoi tessellations and their applications, where the generating points reside on the boundary at infinity \\cite{Achille,AchilleEtal,BudzinskiEtal,Fraczyk}. Our work complements this diverse landscape by focusing on the geometry induced by Poisson processes of $\\lambda$-geodesic hyperplanes, uncovering a surprising universality in visibility properties.\n\n\t\\medspace\n\n\tThe paper is organized as follows. Section \\ref{sec:Background} collects the necessary background on hyperbolic geometry to make the paper self-contained. In Section \\ref{sec:Visibility}, we establish the visibility phase transition, proving Theorem \\ref{thm:MainIntro}(a). Section \\ref{sec:IntegralGeometrySegment} is dedicated to the core integral-geometric mechanism: we prove that the Crofton-type identity \\eqref{eq:IndicatorToEulerChar} holds universally for all $\\lambda\\in[0,1]$. Finally, we apply this result in Section \\ref{sec:ExpectedVolume} to derive the mean visible volume formula in Theorem \\ref{thm:MainIntro}(b).", "sketch": "The introduction states that the proofs of Theorem~\\ref{thm:MainIntro}(a) and (b) “rely on two complementary ingredients.”\n\n1. **Phase transition / (a):** “For the existence of a phase transition, we adapt a covering criterion due to Hoffmann--J\\o rgensen \\cite{H73}.” This “parallels the approach in \\cite{GKT22} used in the case $\\lambda=0$,” and “shows that the critical threshold does not depend on the geometric nature of the hyperplanes, i.e., on $\\lambda$.” The needed computations “rely on a subtle cancellation, which eliminates the $\\lambda$-dependence from the relevant density terms.”\n\n2. **Expected volume / (b):** “The computation of the expected volume in the bounded phase reduces to an integral-geometric question: one must determine the measure $\\nu_\\lambda^o$ of $\\lambda$--geodesic hyperplanes that intersect a fixed geodesic segment of length~$h>0$.” For $\\lambda=0$ this is obtained via “the classical Crofton formula,” yielding \\eqref{eq:IndicatorToEulerChar}. For $\\lambda>0$ “the Crofton argument breaks down … since a $\\lambda$--geodesic hyperplane may intersect a geodesic segment either once or twice,” so “the indicator function cannot be replaced by the Euler characteristic” and “no direct integral-geometric relation is available.” They “overcome this by introducing an explicit parametrization of $\\lambda$--geodesic hyperplanes and carrying out the computation directly.” The “resulting expression turns out to be independent of~$\\lambda$ and again proportional to the length~$h$,” and “is ultimately responsible for the $\\lambda$--independence of the expected volume as described in Theorem~\\ref{thm:MainIntro}.”\n\nFinally, they indicate the proof structure: Section~\\ref{sec:Visibility} proves Theorem~\\ref{thm:MainIntro}(a) via the phase transition argument; Section~\\ref{sec:IntegralGeometrySegment} proves the universal Crofton-type identity; Section~\\ref{sec:ExpectedVolume} applies it to derive Theorem~\\ref{thm:MainIntro}(b).", "expanded_sketch": "The introduction states that the proofs establishing the main theorem (parts (a) and (b)) “rely on two complementary ingredients.”\n\n1. **Phase transition / (a):** “For the existence of a phase transition, we adapt a covering criterion due to Hoffmann--J\\o rgensen \\cite{H73}.” This “parallels the approach in \\cite{GKT22} used in the case $\\lambda=0$,” and “shows that the critical threshold does not depend on the geometric nature of the hyperplanes, i.e., on $\\lambda$.” The needed computations “rely on a subtle cancellation, which eliminates the $\\lambda$-dependence from the relevant density terms.”\n\n2. **Expected volume / (b):** “The computation of the expected volume in the bounded phase reduces to an integral-geometric question: one must determine the measure $\\nu_\\lambda^o$ of $\\lambda$--geodesic hyperplanes that intersect a fixed geodesic segment of length~$h>0$.” For $\\lambda=0$ this is obtained via “the classical Crofton formula,” yielding\n\\begin{equation}\\label{eq:IndicatorToEulerChar}\n\t\\int_{{\\rm Hyp}_0^o}{\\bf 1}\\{H\\cap[o,x_h]\\neq\\varnothing\\}\\,\\nu_0^o(\\dint H) = \\int_{{\\rm Hyp}_0^o}\\chi(H\\cap[o,x_h])\\,\\nu_0^o(\\dint H) = {\\Gamma({d\\over 2})\\over 2\\sqrt{\\pi}\\Gamma({d+1\\over 2})}\\,h,\n\t\\end{equation}\nFor $\\lambda>0$ “the Crofton argument breaks down … since a $\\lambda$--geodesic hyperplane may intersect a geodesic segment either once or twice,” so “the indicator function cannot be replaced by the Euler characteristic” and “no direct integral-geometric relation is available.” They “overcome this by introducing an explicit parametrization of $\\lambda$--geodesic hyperplanes and carrying out the computation directly.” The “resulting expression turns out to be independent of~$\\lambda$ and again proportional to the length~$h$,” and “is ultimately responsible for the $\\lambda$--independence of the expected volume as described in establishing the main theorem.”\n\nFinally, they indicate the proof structure: Section~\\ref{sec:Visibility} proves part (a) of the main theorem via the phase transition argument; Section~\\ref{sec:IntegralGeometrySegment} proves the universal Crofton-type identity; Section~\\ref{sec:ExpectedVolume} applies it to derive part (b) of the main theorem.", "expanded_theorem": "\\label{thm:MainIntro}\n\tFix $d\\geq 2$, $\\gamma>0$ and $0\\leq\\lambda\\leq 1$. Consider the visibility region $Z_{\\gamma,\\lambda,d}$ of $o$ in a Poisson process of $\\lambda$-geodesic hyperplanes in $\\mathbb{H}^d$ with intensity measure $\\gamma\\nu_\\lambda^o$.\n\t\\begin{itemize}\n\t\\item[(a)] Let $\\gamma_{\\mathrm{crit}}$ be as in\n\t\\begin{equation}\\label{eq:GammaCrit}\n\t\\gamma_{\\mathrm{crit}}:= \\sqrt{\\pi}(d-1)^2{\\Gamma({d-1\\over 2})\\over\\Gamma({d\\over 2})}\n\t\\end{equation}\n\tIf $\\gamma<\\gamma_{\\mathrm{crit}}$, then $Z_{\\gamma,\\lambda,d}$ is unbounded with strictly positive probability, whereas if $\\gamma>\\gamma_{\\mathrm{crit}}$, then $Z_{\\gamma,\\lambda,d}$ is almost surely bounded. Moreover, if $d=2$, then $Z_{\\gamma_{\\mathrm{crit}},\\lambda,2}$ is almost surely bounded.\n\n\t\\item[(b)] If $\\gamma>\\gamma_{\\mathrm{crit}}$ then\n\t\\begin{equation}\\label{eq:MeanVolume}\n\t\t\\mathbb{E}{\\rm vol}_d(Z_{\\gamma,\\lambda,d})=\\mathbb{E}{\\rm vol}_d(Z_{\\gamma,0,d}) = \\pi^{\\frac{d-1}{2}} \\Gamma\\left(\\frac{d+1}{2}\\right){\\Gamma\\big({\\gamma_d^\\ast-d+1\\over 2}\\big)\\over\\Gamma\\big({\\gamma_d^\\ast+d+1\\over 2}\\big)}\n\t\\end{equation}\n\twith $\\gamma_d^\\ast := \\gamma\\,{\\Gamma({d\\over 2})\\over 2\\sqrt{\\pi}\\Gamma({d+1\\over 2})}$. On the other hand, $\\mathbb{E}{\\rm vol}_d(Z_{\\gamma,\\lambda,d})=+\\infty$ for all $\\gamma\\leq\\gamma_{\\mathrm{crit}}$.\n\t\\end{itemize},", "theorem_type": ["Implication", "Existential–Universal"], "mcq": {"question": "Fix $d\\ge 2$, $\\gamma>0$, and $0\\le \\lambda\\le 1$. Let $o\\in \\mathbb{H}^d$ be a distinguished origin. Let $\\mathrm{Hyp}_\\lambda^o$ be the collection of $\\lambda$-geodesic hyperplanes in $\\mathbb{H}^d$ whose convex side does not contain $o$, let $\\nu_\\lambda^o$ be the restriction of the natural isometry-invariant measure to $\\mathrm{Hyp}_\\lambda^o$, and let $\\eta_{\\gamma,\\lambda}$ be a Poisson point process on $\\mathrm{Hyp}_\\lambda^o$ with intensity measure $\\gamma\\nu_\\lambda^o$. Define the visibility region of $o$ by\n\\[\nZ_{\\gamma,\\lambda,d}:=\\{x\\in \\mathbb{H}^d:[o,x]\\cap H=\\varnothing\\text{ for all }H\\in \\eta_{\\gamma,\\lambda}\\},\n\\]\nwhere $[o,x]$ is the geodesic segment from $o$ to $x$, and let ${\\rm vol}_d$ denote hyperbolic $d$-dimensional volume. Also set\n\\[\n\\gamma_{\\mathrm{crit}}:=\\sqrt{\\pi}(d-1)^2\\frac{\\Gamma\\bigl((d-1)/2\\bigr)}{\\Gamma(d/2)}\n\\quad\\text{and}\\quad\n\\gamma_d^*:=\\gamma\\,\\frac{\\Gamma(d/2)}{2\\sqrt{\\pi}\\,\\Gamma((d+1)/2)}.\n\\]\nWhich of the following conclusions about the boundedness of $Z_{\\gamma,\\lambda,d}$ and the expectation $\\mathbb E\\,{m vol}_d(Z_{\\gamma,\\lambda,d})$ holds under these hypotheses?", "correct_choice": {"label": "A", "text": "If $\\gamma<\\gamma_{\\mathrm{crit}}$, then $Z_{\\gamma,\\lambda,d}$ is unbounded with strictly positive probability. If $\\gamma>\\gamma_{\\mathrm{crit}}$, then $Z_{\\gamma,\\lambda,d}$ is almost surely bounded. Moreover, in the critical case for $d=2$, $Z_{\\gamma_{\\mathrm{crit}},\\lambda,2}$ is almost surely bounded. In addition, if $\\gamma>\\gamma_{\\mathrm{crit}}$, then\n\\[\n\\mathbb E\\,{m vol}_d(Z_{\\gamma,\\lambda,d})=\\mathbb E\\,{m vol}_d(Z_{\\gamma,0,d})=\\pi^{\\frac{d-1}{2}}\\Gamma\\!\\left(\\frac{d+1}{2}\\right)\\frac{\\Gamma\\!\\left(\\frac{\\gamma_d^*-d+1}{2}\\right)}{\\Gamma\\!\\left(\\frac{\\gamma_d^*+d+1}{2}\\right)}.\n\\]\nOn the other hand, $\\mathbb E\\,{m vol}_d(Z_{\\gamma,\\lambda,d})=+\\infty$ for every $\\gamma\\le \\gamma_{\\mathrm{crit}}$."}, "choices": [{"label": "B", "text": "If $\\gamma<\\gamma_{\\mathrm{crit}}$, then $Z_{\\gamma,\\lambda,d}$ is unbounded with strictly positive probability. If $\\gamma\\ge \\gamma_{\\mathrm{crit}}$, then $Z_{\\gamma,\\lambda,d}$ is almost surely bounded. In addition, if $\\gamma\\ge \\gamma_{\\mathrm{crit}}$, then\n\\[\n\\mathbb E\\,{\\rm vol}_d(Z_{\\gamma,\\lambda,d})=\\mathbb E\\,{\\rm vol}_d(Z_{\\gamma,0,d})=\\pi^{\\frac{d-1}{2}}\\Gamma\\!\\left(\\frac{d+1}{2}\\right)\\frac{\\Gamma\\!\\left(\\frac{\\gamma_d^*-d+1}{2}\\right)}{\\Gamma\\!\\left(\\frac{\\gamma_d^*+d+1}{2}\\right)}.\n\\]\nOn the other hand, $\\mathbb E\\,{\\rm vol}_d(Z_{\\gamma,\\lambda,d})=+\\infty$ for every $\\gamma< \\gamma_{\\mathrm{crit}}$."}, {"label": "C", "text": "If $\\gamma<\\gamma_{\\mathrm{crit}}$, then $Z_{\\gamma,\\lambda,d}$ is unbounded with strictly positive probability. If $\\gamma>\\gamma_{\\mathrm{crit}}$, then $Z_{\\gamma,\\lambda,d}$ is almost surely bounded. Moreover, in the critical case for $d=2$, $Z_{\\gamma_{\\mathrm{crit}},\\lambda,2}$ is almost surely bounded. In addition, if $\\gamma>\\gamma_{\\mathrm{crit}}$, then\n\\[\n\\mathbb E\\,{\\rm vol}_d(Z_{\\gamma,\\lambda,d})=\\pi^{\\frac{d-1}{2}}\\Gamma\\!\\left(\\frac{d+1}{2}\\right)\\frac{\\Gamma\\!\\left(\\frac{\\gamma_d^*-d+1}{2}\\right)}{\\Gamma\\!\\left(\\frac{\\gamma_d^*+d+1}{2}\\right)}.\n\\]\nOn the other hand, $\\mathbb E\\,{\\rm vol}_d(Z_{\\gamma,\\lambda,d})=+\\infty$ for every $\\gamma\\le \\gamma_{\\mathrm{crit}}$."}, {"label": "D", "text": "If $\\gamma<\\gamma_{\\mathrm{crit}}$, then $Z_{\\gamma,\\lambda,d}$ is unbounded with strictly positive probability. If $\\gamma>\\gamma_{\\mathrm{crit}}$, then $Z_{\\gamma,\\lambda,d}$ is almost surely bounded. Moreover, in the critical case for $d=2$, $Z_{\\gamma_{\\mathrm{crit}},\\lambda,2}$ is almost surely bounded. In addition, if $\\gamma>\\gamma_{\\mathrm{crit}}$, then\n\\[\n\\mathbb E\\,{\\rm vol}_d(Z_{\\gamma,\\lambda,d})=\\pi^{\\frac{d-1}{2}}\\Gamma\\!\\left(\\frac{d+1}{2}\\right)\\frac{\\Gamma\\!\\left(\\frac{\\gamma_d^*-d+1}{2}\\right)}{\\Gamma\\!\\left(\\frac{\\gamma_d^*+d+1}{2}\\right)}.\n\\]\nOn the other hand, for every fixed $\\gamma>\\gamma_{\\mathrm{crit}}$ this expectation formula holds for each $\\lambda$, but the critical intensity governing boundedness depends on $\\lambda$."}, {"label": "E", "text": "If $\\gamma<\\gamma_{\\mathrm{crit}}$, then $Z_{\\gamma,\\lambda,d}$ is unbounded with strictly positive probability, whereas if $\\gamma>\\gamma_{\\mathrm{crit}}$, then $Z_{\\gamma,\\lambda,d}$ is almost surely bounded. Moreover, in the critical case for $d=2$, $Z_{\\gamma_{\\mathrm{crit}},\\lambda,2}$ is almost surely bounded. In addition, if $\\gamma>\\gamma_{\\mathrm{crit}}$, then\n\\[\n\\mathbb E\\,{\\rm vol}_d(Z_{\\gamma,\\lambda,d})=\\mathbb E\\,{\\rm vol}_d(Z_{\\gamma,0,d})=\\pi^{\\frac{d-1}{2}}\\Gamma\\!\\left(\\frac{d+1}{2}\\right)\\frac{\\Gamma\\!\\left(\\frac{\\gamma_d^*-d+1}{2}\\right)}{\\Gamma\\!\\left(\\frac{\\gamma_d^*+d+1}{2}\\right)}.\n\\]\nOn the other hand, $\\mathbb E\\,{\\rm vol}_d(Z_{\\gamma,\\lambda,d})<+\\infty$ for every $\\gamma\\ge \\gamma_{\\mathrm{crit}}$, and $\\mathbb E\\,{\\rm vol}_d(Z_{\\gamma,\\lambda,d})=+\\infty$ only for $\\gamma<\\gamma_{\\mathrm{crit}}$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "strict threshold for mean-volume formula and divergence at criticality", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the explicit identity $\\mathbb E\\,{\\rm vol}_d(Z_{\\gamma,\\lambda,d})=\\mathbb E\\,{\\rm vol}_d(Z_{\\gamma,0,d})$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "$\\lambda$-independence of the critical threshold", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "critical-case integrability in the gamma-ratio formula", "template_used": "uniformity_effectivity"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives definitions and parameter thresholds but does not state the boundedness/expectation conclusion itself. There is no direct or trivial leakage of the marked answer."}, "TAS": {"score": 2, "justification": "This is not a mere restatement of the stem; the respondent must distinguish among several nearby threshold statements, critical-case variants, and lambda-dependence claims."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the choices differ in subtle but mathematically meaningful ways. However, it primarily tests precise theorem recall/comparison rather than substantial generative derivation from the hypotheses."}, "DQS": {"score": 0, "justification": "Although most distractors are plausible theorem variants, choice C is a weaker statement that still appears true if A is true, so the item is not cleanly single-answer. This seriously weakens the distractor set."}, "total_score": 5, "overall_assessment": "Technically sophisticated and not leaky, but flawed as an MCQ because at least one distractor is apparently also true, making the single-correct-answer format ambiguous."}}
{"id": "2602.20938v1", "paper_link": "http://arxiv.org/abs/2602.20938v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{t1} Let $\\Omega \\subset \\mathbb{R}^2$ be a smooth bounded convex domain such that $\\mathrm{inr}(\\Omega)=r$ and $D(\\Omega)=D$. Then if $D$ is  large enough, there exists $\\lambda>0$ for which problem \\eqref{con} admits a pattern.", "start_pos": 3111, "end_pos": 3375, "label": "t1"}, "ref_dict": {"t1": "\\begin{theorem}\\label{t1} Let $\\Omega \\subset \\mathbb{R}^2$ be a smooth bounded convex domain such that $\\mathrm{inr}(\\Omega)=r$ and $D(\\Omega)=D$. Then if $D$ is  large enough, there exists $\\lambda>0$ for which problem \\eqref{con} admits a pattern. \\end{theorem}", "fig1": "\\begin{figure}[ht]\n\\centering\n\\def\\svgwidth{1\\linewidth}\n\\input{fig3.pdf_tex}\n\\caption{Convex domain\n$\\Omega \\subset \\mathbb{R}^2$ with inradius $r$ and diameter $D$.}\n\\label{fig1}\n\\end{figure}", "tmi": "\\begin{theorem}\\label{tmi}\nLet $\\Omega \\subset \\mathbb{R}^2$ be a bounded convex domain and again let\n$\nD := D(\\Omega).\n$\nThen there exists a constant $c>0$ depending only on the dimension such that\n\\[\n\\mu_1(\\Omega) \\ge \\frac{c}{D}.\n\\]\n\\end{theorem}", "con": "\\begin{equation}\\label{con}\n\\begin{cases}\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}"}, "pre_theorem_intro_text_len": 1183, "pre_theorem_intro_text": "Consider the following problem with nonlinear Neumann boundary condition\n\\begin{equation}\\label{con}\n\\begin{cases}\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}\nwhere $\\Omega \\subset \\mathbb{R}^2$ is a smooth, bounded, and convex domain, \n$\\nu$ denotes the outward unit normal vector to $\\partial\\Omega$, \n$\\lambda$ is a positive parameter, and $g$ is a bistable nonlinearity, \nhere taken in its prototypical form $g(u)=u-u^3$.\n\nA non-constant stationary solution of \\eqref{con} that is linearly stable will be called a {\\it pattern}.\n\nIn order to state our main result, we recall that the \\emph{inradius} of a set \n$\\Omega \\subset \\mathbb{R}^2$ is defined by\n\\[\n\\mathrm{inr}(\\Omega) := \\sup \\left\\{ r > 0 \\; ; \\; \\exists\\, x \\in \\Omega \n\\ \\text{such that} \\ B_r(x) \\subset \\Omega \\right\\},\n\\]\nwhere $B_r(x)$ denotes the open ball of radius $r$ centered at $x$.\n\nThe \\emph{diameter} of $\\Omega$ is defined by\n\\[\nD(\\Omega) := \\sup \\left\\{ |x-y| \\; ; \\; x,y \\in \\Omega \\right\\}.\n\\]\n\nThe main result of this work is stated as follows.", "context": "Consider the following problem with nonlinear Neumann boundary condition\n\\begin{equation}\\label{con}\n\\begin{cases}\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}\nwhere $\\Omega \\subset \\mathbb{R}^2$ is a smooth, bounded, and convex domain, \n$\\nu$ denotes the outward unit normal vector to $\\partial\\Omega$, \n$\\lambda$ is a positive parameter, and $g$ is a bistable nonlinearity, \nhere taken in its prototypical form $g(u)=u-u^3$.\n\nA non-constant stationary solution of \\eqref{con} that is linearly stable will be called a {\\it pattern}.\n\nIn order to state our main result, we recall that the \\emph{inradius} of a set \n$\\Omega \\subset \\mathbb{R}^2$ is defined by\n\\[\n\\mathrm{inr}(\\Omega) := \\sup \\left\\{ r > 0 \\; ; \\; \\exists\\, x \\in \\Omega \n\\ \\text{such that} \\ B_r(x) \\subset \\Omega \\right\\},\n\\]\nwhere $B_r(x)$ denotes the open ball of radius $r$ centered at $x$.\n\nThe \\emph{diameter} of $\\Omega$ is defined by\n\\[\nD(\\Omega) := \\sup \\left\\{ |x-y| \\; ; \\; x,y \\in \\Omega \\right\\}.\n\\]\n\nThe main result of this work is stated as follows.", "full_context": "Consider the following problem with nonlinear Neumann boundary condition\n\\begin{equation}\\label{con}\n\\begin{cases}\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}\nwhere $\\Omega \\subset \\mathbb{R}^2$ is a smooth, bounded, and convex domain, \n$\\nu$ denotes the outward unit normal vector to $\\partial\\Omega$, \n$\\lambda$ is a positive parameter, and $g$ is a bistable nonlinearity, \nhere taken in its prototypical form $g(u)=u-u^3$.\n\nA non-constant stationary solution of \\eqref{con} that is linearly stable will be called a {\\it pattern}.\n\nIn order to state our main result, we recall that the \\emph{inradius} of a set \n$\\Omega \\subset \\mathbb{R}^2$ is defined by\n\\[\n\\mathrm{inr}(\\Omega) := \\sup \\left\\{ r > 0 \\; ; \\; \\exists\\, x \\in \\Omega \n\\ \\text{such that} \\ B_r(x) \\subset \\Omega \\right\\},\n\\]\nwhere $B_r(x)$ denotes the open ball of radius $r$ centered at $x$.\n\nThe \\emph{diameter} of $\\Omega$ is defined by\n\\[\nD(\\Omega) := \\sup \\left\\{ |x-y| \\; ; \\; x,y \\in \\Omega \\right\\}.\n\\]\n\nThe main result of this work is stated as follows.\n\n\\abstract{We consider the heat equation in a smooth bounded convex domain \n$\\Omega \\subset \\mathbb{R}^2$ with nonlinear Neumann boundary condition \n$\\partial_\\nu u = \\lambda (u - u^3)$. \nStable non-constant stationary solutions do not exist when $\\Omega$ is a ball. \nWe show that this behavior is not a consequence of convexity alone. \nMore precisely, if the inradius of $\\Omega$ is fixed and its diameter is sufficiently large, then there exists $\\lambda>0$ for which the problem admits such a solution. \nThe result reveals a geometric mechanism for the emergence of stable non-constant stationary solutions in elongated convex domains.}\n\nConsider the following problem with nonlinear Neumann boundary condition\n\\begin{equation}\\label{con}\n\\begin{cases}\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}\nwhere $\\Omega \\subset \\mathbb{R}^2$ is a smooth, bounded, and convex domain, \n$\\nu$ denotes the outward unit normal vector to $\\partial\\Omega$, \n$\\lambda$ is a positive parameter, and $g$ is a bistable nonlinearity, \nhere taken in its prototypical form $g(u)=u-u^3$.\n\nThe main result of this work is stated as follows.\n\nTheorem \\ref{t1} reveals a purely geometric mechanism for pattern\nformation. While the inradius remains fixed, increasing the diameter\nforces the emergence of patterns for appropriate values\nof $\\lambda$. In other words, sufficiently elongated convex domains (see Figure \\ref{fig1})\nnecessarily admit patterns. This phenomenon is independent of symmetry assumptions and depends\nonly on the convexity and aspect ratio of the domain.\n\n\\begin{lemma}\\label{LFU}\nLet $\\Omega \\subset \\mathbb{R}^n$ be a bounded convex domain,\nand denote $D=D(\\Omega)$.\nThen there exists a constant $C=C(n)>0$ such that,\nfor every $u\\in H^1(\\Omega)$,\n\\[\n\\|u-u_{\\partial\\Omega}\\|_{L^2(\\partial\\Omega)}\n\\le\nC D^{1/2}\\|\\nabla u\\|_{L^2(\\Omega)},\n\\]\nwhere\n\\[\nu_{\\partial\\Omega}\n=\n\\frac{1}{|\\partial\\Omega|}\n\\int_{\\partial\\Omega} u\\ d\\sigma.\n\\]\n\\end{lemma}\n\n\\begin{theorem}\\label{tmi}\nLet $\\Omega \\subset \\mathbb{R}^2$ be a bounded convex domain and again let\n$\nD := D(\\Omega).\n$\nThen there exists a constant $c>0$ depending only on the dimension such that\n\\[\n\\mu_1(\\Omega) \\ge \\frac{c}{D}.\n\\]\n\\end{theorem}\n\n\\begin{lemma}\\label{le1}\nLet $\\Omega \\subset \\mathbb{R}^n$ ($n \\ge 2$) be a smooth bounded domain,\n$\\Omega_l$ and $\\Omega_r$ two disjoint subdomains of $\\Omega$\nwith smooth boundaries, and\n$S_j = \\partial\\Omega \\cap \\partial\\Omega_j$,\n$\\mathcal{H}^{n-1}(S_j) > 0$ $(j=l,r)$.\nFor $p > n$, we define the set\n\\begin{equation}\\label{L}\n\\Lambda(\\Omega)\n=\n\\left\\{\\begin{aligned}\nv \\in W^{1,p}(\\Omega) :\\quad\n&-1 \\le v(x) \\le 1,\\; x \\in \\overline{\\Omega},\\\\\n&\\int_{S_l} v \\, d\\sigma < 0,\\qquad\n\\int_{S_r} v \\, d\\sigma > 0,\\\\\n&E(v) < \\varepsilon_0 - G(1)\\mathcal{H}^{n-1}(\\partial\\Omega)\n\\end{aligned}\\right\\},\n\\end{equation}\nwhere\n\\[\n\\varepsilon_0\n=\nG(1)\n\\min\n\\left\\{\\begin{aligned}\n&\\mathcal{H}^{n-1}(S_l)\\min\\{\\lambda,\\,\\mu_1(\\Omega_l)\\},\\\\\n&\\mathcal{H}^{n-1}(S_r)\\min\\{\\lambda,\\,\\mu_1(\\Omega_r)\\}\n\\end{aligned}\\right\\},\n\\]\nand $\\mu_1(\\Omega_j)$ is the first positive eigenvalue of the Steklov\nproblem \\ref{ST} defined in $\\Omega_j$ $(j=l,r)$.\n\nFor convenience, in this section we denote a point in $\\mathbb{R}^2$ by $(x,y)$. Our objective is to prove that if $\\Omega \\subset \\mathbb{R}^2$ is smooth bounded convex domain\n with inradius $r$ and diameter $D$ (see Figure \\ref{fig1}) large enough, then one can find $\\lambda > 0$ such that\n$\n\\Lambda(\\Omega) \\neq \\emptyset.\n$, where $\\Lambda(\\Omega)$ is the set defined in \\eqref{L}.\n\n\\begin{figure}[ht]\n\\centering\n\\def\\svgwidth{1\\linewidth}\n\\input{fig3.pdf_tex}\n\\caption{Convex domain\n$\\Omega \\subset \\mathbb{R}^2$ with inradius $r$ and diameter $D$.}\n\\label{fig1}\n\\end{figure}\n\n\\begin{theorem}\\label{t1} Let $\\Omega \\subset \\mathbb{R}^2$ be a smooth bounded convex domain such that $\\mathrm{inr}(\\Omega)=r$ and $D(\\Omega)=D$. Then if $D$ is  large enough, there exists $\\lambda>0$ for which problem \\eqref{con} admits a pattern. \\end{theorem}", "post_theorem_intro_text_len": 4620, "post_theorem_intro_text": "Theorem \\ref{t1} reveals a purely geometric mechanism for pattern\nformation. While the inradius remains fixed, increasing the diameter\nforces the emergence of patterns for appropriate values\nof $\\lambda$. In other words, sufficiently elongated convex domains (see Figure \\ref{fig1})\nnecessarily admit patterns. This phenomenon is independent of symmetry assumptions and depends\nonly on the convexity and aspect ratio of the domain.\n\nFrom a biological perspective, Theorem~\\ref{t1} suggests that\ngeometric elongation of the domain can counterbalance\nthe homogenizing action of diffusion in the interior.\nWhen the diameter becomes sufficiently large compared to the inradius,\nthe global geometric structure weakens diffusive stabilization\nand allows patterns to arise.\n\nThe relation between convexity and stability of non-constant stationary solutions \nhas been a central theme in the study of semilinear parabolic equations. \nIn the classical works of Casten--Holland \\cite{CH} and Matano \\cite{matano}, the nonlinearity\nacts in the interior of the domain and homogeneous Neumann boundary\nconditions are imposed. In that setting, it was shown that in convex\ndomains patterns cannot exist under very general assumptions on the\nnonlinearity. These results reveal a strong rigidity\nphenomenon induced by convexity.\n\nIn contrast, in the present work the nonlinearity acts through the\nboundary condition. Theorem~\\ref{t1} shows that, in this setting,\nconvexity by itself does not constitute a structural obstruction\nto pattern formation.\nAlthough it is known that no pattern can exist when $\\Omega$ is a ball\n(see \\cite{NLM}), this reflects the special symmetry of the ball rather\nthan convexity per se.\n\nTo the best of our knowledge, no previous result establishes a direct\ncriterion for the existence or absence of patterns in terms of global\ngeometric quantities such as the inradius and the diameter. Conversely,\nmuch of the existing literature connects the existence of patterns to\ncurvature effects or other local geometric properties of the boundary, for instance, see \\cite{CH,SAE,FA,SN} and references therein.\n\nIt is worth mentioning that in \\cite{CJ}, through a computer-assisted analysis\ncombined with bifurcation techniques, strong evidence was provided for\nthe existence of patterns for problem \\eqref{con} in the case\n$g(u)=u-u^3$, $\\lambda>2,84083164$, and $\\Omega$ equal to the unit square,\nwhich is a convex planar domain. The authors further conjectured that the same phenomenon should persist\nfor smooth convex domains obtained by suitably rounding the corners of the square.\n\nIn addition to \\cite{CJ}, a proof in the convex case was mentioned in \\cite{CS,DP}\nas part of work in preparation. To the best of our knowledge, a\ncomplete proof has not yet appeared in the literature.\n\nIn the same framework of boundary reactions in a bidimensional domain, Del Pino {\\it et al.} \\cite{DP}\nconstructed solutions with multiple boundary transitions, addressing existence and qualitative properties, but not stability. The existence of patterns has been obtained under stronger geometric or structural assumptions. \nIn \\cite{AG,C2,C3}, existence results were proved   in special classes of non-convex bounded domains, such as dumbbell-type and perfored domains. \nMoreover, when the boundary nonlinearity depends explicitly on the spatial variable, namely $g=g(x,u)$, existence of patterns has been established in \\cite{ARRIETA} and for problems with variable diffusivity, see \\cite{ACM}.\n\nThe proof of Theorem \\ref{t1} is based on the construction of a specific set, denoted by $\\Lambda(\\Omega)$, associated with the geometry of $\\Omega$ and the positive flow generated by \\eqref{con}. The underlying strategy ensures that, whenever $\\Lambda(\\Omega)$ is nonempty, it contains the desired pattern. Thus, the core of the argument consists in proving that $\\Lambda(\\Omega)\\neq\\emptyset$. This approach appears to have been introduced in \\cite{matano} and has since been employed in different contexts, for instance in \\cite{ar1, C2}, as well as in some subsequent works \\cite{ACM}.\n\nThe present work is organized as follows. Section 2 is devoted to the preliminary results needed for the proof of Theorem \\ref{t1}. In particular, in Theorem \\ref{tmi} we establish a lower bound for the first positive Steklov eigenvalue, expressed in terms of the diameter of the domain and a constant depending only on the dimension. This estimate, which may be of independent interest in its own right, plays a key role in the proof of Theorem \\ref{t1}, carried out in Section 3. Section 4 concludes the paper with some final remarks.", "sketch": "The proof of Theorem \\ref{t1} is based on “the construction of a specific set, denoted by $\\Lambda(\\Omega)$, associated with the geometry of $\\Omega$ and the positive flow generated by \\eqref{con}.” The strategy is that “whenever $\\Lambda(\\Omega)$ is nonempty, it contains the desired pattern,” so “the core of the argument consists in proving that $\\Lambda(\\Omega)\\neq\\emptyset$.”\n\nA key ingredient used in this program is that “in Theorem \\ref{tmi} we establish a lower bound for the first positive Steklov eigenvalue, expressed in terms of the diameter of the domain and a constant depending only on the dimension,” and “this estimate … plays a key role in the proof of Theorem \\ref{t1}.”", "expanded_sketch": "In establishing the main theorem, the proof is based on “the construction of a specific set, denoted by $\\Lambda(\\Omega)$, associated with the geometry of $\\Omega$ and the positive flow generated by\n\\begin{equation}\\label{con}\n\\begin{cases}\\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}\n.” The strategy is that “whenever $\\Lambda(\\Omega)$ is nonempty, it contains the desired pattern,” so “the core of the argument consists in proving that $\\Lambda(\\Omega)\\neq\\emptyset$.”\n\nWe first prove the following theorem.\n\\begin{theorem}\\label{tmi}\nLet $\\Omega \\subset \\mathbb{R}^2$ be a bounded convex domain and again let\n$\nD := D(\\Omega).\n$\nThen there exists a constant $c>0$ depending only on the dimension such that\n\\[\n\\mu_1(\\Omega) \\ge \\frac{c}{D}.\n\\]\n\\end{theorem}\nThis estimate plays a key role in the proof of the main theorem.", "expanded_theorem": "\\label{t1} Let $\\Omega \\subset \\mathbb{R}^2$ be a smooth bounded convex domain such that $\\mathrm{inr}(\\Omega)=r$ and $D(\\Omega)=D$. Then if $D$ is  large enough, there exists $\\lambda>0$ for which the problem\n\\begin{equation}\\label{con}\n\\begin{cases}\\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}\nadmits a pattern.", "theorem_type": ["Implication", "Existence"], "mcq": {"question": "Let \\(\\Omega \\subset \\mathbb{R}^2\\) be a smooth bounded convex domain with inradius\n\\[\n\\mathrm{inr}(\\Omega):=\\sup\\{r'>0:\\exists x\\in\\Omega\\text{ such that }B_{r'}(x)\\subset \\Omega\\}=r\n\\]\nand diameter\n\\[\nD(\\Omega):=\\sup\\{|x-y|:x,y\\in\\Omega\\}=D.\n\\]\nConsider the heat equation with nonlinear Neumann boundary condition\n\\[\n\\begin{cases}\nu_t-\\Delta u=0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+,\\\\\n\\partial_\\nu u=\\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+,\\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\]\nwhere \\(\\nu\\) is the outward unit normal and \\(g(u)=u-u^3\\). A pattern means a non-constant stationary solution that is linearly stable. If \\(D\\) is sufficiently large, which conclusion about this problem is valid?", "correct_choice": {"label": "A", "text": "There exists \\(\\lambda>0\\) such that the problem admits a pattern; equivalently, for some positive \\(\\lambda\\), it has a linearly stable non-constant stationary solution."}, "choices": [{"label": "B", "text": "There exists a constant \\(\\lambda_0>0\\), depending only on the dimension, such that for every \\(\\lambda\\ge \\lambda_0\\) the problem admits a pattern whenever \\(D\\) is sufficiently large."}, {"label": "C", "text": "There exists \\(\\lambda>0\\) such that the problem admits a stationary solution that is non-constant."}, {"label": "D", "text": "For every sufficiently large diameter \\(D\\), and for every smooth bounded convex domain \\(\\Omega\\subset\\mathbb{R}^2\\) with \\(\\mathrm{inr}(\\Omega)=r\\) and \\(D(\\Omega)=D\\), the problem admits a pattern for every \\(\\lambda>0\\)."}, {"label": "E", "text": "If \\(D\\) is sufficiently large, then for every \\(\\lambda>0\\) there exists a linearly stable non-constant stationary solution of the problem."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "existential dependence of the admissible parameter \\(\\lambda\\)", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "linear stability requirement in the definition of pattern", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "quantifier order on \\(\\lambda\\): existence replaced by universality", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "parameter-selection mechanism via nonemptiness of \\(\\Lambda(\\Omega)\\) for some suitable \\(\\lambda\\)", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the correct option. It gives the PDE setup and asks for the valid existence conclusion, so the answer is not leaked directly."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: under the stated geometric hypothesis, the student must choose the matching existence claim. The quantifier and stability variations prevent it from being a pure restatement, but it is still only a mild reformulation of a theorem statement."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the choices differ in important ways: existence of some lambda vs all lambda, stable pattern vs mere non-constant stationary solution, and existence vs uniqueness. However, the task is still largely recognition of the exact theorem statement rather than substantial mathematical generation."}, "DQS": {"score": 1, "justification": "Most distractors are plausible and reflect common overstatements (uniformity in lambda, stronger quantifiers, uniqueness). However, choice C is a weaker statement that is also true if A is true, making the item potentially ambiguous and weakening distractor quality."}, "total_score": 5, "overall_assessment": "A moderately good theorem-discrimination MCQ with no clear answer leakage and reasonably plausible distractors, but it is somewhat theorem-restatement-like and is weakened by a true-but-weaker distractor that introduces ambiguity."}}
{"id": "2602.20938v1", "paper_link": "http://arxiv.org/abs/2602.20938v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{t1} Let $\\Omega \\subset \\mathbb{R}^2$ be a smooth bounded convex domain such that $\\mathrm{inr}(\\Omega)=r$ and $D(\\Omega)=D$. Then if $D$ is  large enough, there exists $\\lambda>0$ for which problem \\eqref{con} admits a pattern.", "start_pos": 3111, "end_pos": 3375, "label": "t1"}, "ref_dict": {"t1": "\\begin{theorem}\\label{t1} Let $\\Omega \\subset \\mathbb{R}^2$ be a smooth bounded convex domain such that $\\mathrm{inr}(\\Omega)=r$ and $D(\\Omega)=D$. Then if $D$ is  large enough, there exists $\\lambda>0$ for which problem \\eqref{con} admits a pattern. \\end{theorem}", "fig1": "\\begin{figure}[ht]\n\\centering\n\\def\\svgwidth{1\\linewidth}\n\\input{fig3.pdf_tex}\n\\caption{Convex domain\n$\\Omega \\subset \\mathbb{R}^2$ with inradius $r$ and diameter $D$.}\n\\label{fig1}\n\\end{figure}", "tmi": "\\begin{theorem}\\label{tmi}\nLet $\\Omega \\subset \\mathbb{R}^2$ be a bounded convex domain and again let\n$\nD := D(\\Omega).\n$\nThen there exists a constant $c>0$ depending only on the dimension such that\n\\[\n\\mu_1(\\Omega) \\ge \\frac{c}{D}.\n\\]\n\\end{theorem}", "con": "\\begin{equation}\\label{con}\n\\begin{cases}\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}"}, "pre_theorem_intro_text_len": 1183, "pre_theorem_intro_text": "Consider the following problem with nonlinear Neumann boundary condition\n\\begin{equation}\\label{con}\n\\begin{cases}\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}\nwhere $\\Omega \\subset \\mathbb{R}^2$ is a smooth, bounded, and convex domain, \n$\\nu$ denotes the outward unit normal vector to $\\partial\\Omega$, \n$\\lambda$ is a positive parameter, and $g$ is a bistable nonlinearity, \nhere taken in its prototypical form $g(u)=u-u^3$.\n\nA non-constant stationary solution of \\eqref{con} that is linearly stable will be called a {\\it pattern}.\n\nIn order to state our main result, we recall that the \\emph{inradius} of a set \n$\\Omega \\subset \\mathbb{R}^2$ is defined by\n\\[\n\\mathrm{inr}(\\Omega) := \\sup \\left\\{ r > 0 \\; ; \\; \\exists\\, x \\in \\Omega \n\\ \\text{such that} \\ B_r(x) \\subset \\Omega \\right\\},\n\\]\nwhere $B_r(x)$ denotes the open ball of radius $r$ centered at $x$.\n\nThe \\emph{diameter} of $\\Omega$ is defined by\n\\[\nD(\\Omega) := \\sup \\left\\{ |x-y| \\; ; \\; x,y \\in \\Omega \\right\\}.\n\\]\n\nThe main result of this work is stated as follows.", "context": "Consider the following problem with nonlinear Neumann boundary condition\n\\begin{equation}\\label{con}\n\\begin{cases}\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}\nwhere $\\Omega \\subset \\mathbb{R}^2$ is a smooth, bounded, and convex domain, \n$\\nu$ denotes the outward unit normal vector to $\\partial\\Omega$, \n$\\lambda$ is a positive parameter, and $g$ is a bistable nonlinearity, \nhere taken in its prototypical form $g(u)=u-u^3$.\n\nA non-constant stationary solution of \\eqref{con} that is linearly stable will be called a {\\it pattern}.\n\nIn order to state our main result, we recall that the \\emph{inradius} of a set \n$\\Omega \\subset \\mathbb{R}^2$ is defined by\n\\[\n\\mathrm{inr}(\\Omega) := \\sup \\left\\{ r > 0 \\; ; \\; \\exists\\, x \\in \\Omega \n\\ \\text{such that} \\ B_r(x) \\subset \\Omega \\right\\},\n\\]\nwhere $B_r(x)$ denotes the open ball of radius $r$ centered at $x$.\n\nThe \\emph{diameter} of $\\Omega$ is defined by\n\\[\nD(\\Omega) := \\sup \\left\\{ |x-y| \\; ; \\; x,y \\in \\Omega \\right\\}.\n\\]\n\nThe main result of this work is stated as follows.", "full_context": "Consider the following problem with nonlinear Neumann boundary condition\n\\begin{equation}\\label{con}\n\\begin{cases}\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}\nwhere $\\Omega \\subset \\mathbb{R}^2$ is a smooth, bounded, and convex domain, \n$\\nu$ denotes the outward unit normal vector to $\\partial\\Omega$, \n$\\lambda$ is a positive parameter, and $g$ is a bistable nonlinearity, \nhere taken in its prototypical form $g(u)=u-u^3$.\n\nA non-constant stationary solution of \\eqref{con} that is linearly stable will be called a {\\it pattern}.\n\nIn order to state our main result, we recall that the \\emph{inradius} of a set \n$\\Omega \\subset \\mathbb{R}^2$ is defined by\n\\[\n\\mathrm{inr}(\\Omega) := \\sup \\left\\{ r > 0 \\; ; \\; \\exists\\, x \\in \\Omega \n\\ \\text{such that} \\ B_r(x) \\subset \\Omega \\right\\},\n\\]\nwhere $B_r(x)$ denotes the open ball of radius $r$ centered at $x$.\n\nThe \\emph{diameter} of $\\Omega$ is defined by\n\\[\nD(\\Omega) := \\sup \\left\\{ |x-y| \\; ; \\; x,y \\in \\Omega \\right\\}.\n\\]\n\nThe main result of this work is stated as follows.\n\n\\abstract{We consider the heat equation in a smooth bounded convex domain \n$\\Omega \\subset \\mathbb{R}^2$ with nonlinear Neumann boundary condition \n$\\partial_\\nu u = \\lambda (u - u^3)$. \nStable non-constant stationary solutions do not exist when $\\Omega$ is a ball. \nWe show that this behavior is not a consequence of convexity alone. \nMore precisely, if the inradius of $\\Omega$ is fixed and its diameter is sufficiently large, then there exists $\\lambda>0$ for which the problem admits such a solution. \nThe result reveals a geometric mechanism for the emergence of stable non-constant stationary solutions in elongated convex domains.}\n\nConsider the following problem with nonlinear Neumann boundary condition\n\\begin{equation}\\label{con}\n\\begin{cases}\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}\nwhere $\\Omega \\subset \\mathbb{R}^2$ is a smooth, bounded, and convex domain, \n$\\nu$ denotes the outward unit normal vector to $\\partial\\Omega$, \n$\\lambda$ is a positive parameter, and $g$ is a bistable nonlinearity, \nhere taken in its prototypical form $g(u)=u-u^3$.\n\nThe main result of this work is stated as follows.\n\nTheorem \\ref{t1} reveals a purely geometric mechanism for pattern\nformation. While the inradius remains fixed, increasing the diameter\nforces the emergence of patterns for appropriate values\nof $\\lambda$. In other words, sufficiently elongated convex domains (see Figure \\ref{fig1})\nnecessarily admit patterns. This phenomenon is independent of symmetry assumptions and depends\nonly on the convexity and aspect ratio of the domain.\n\n\\begin{lemma}\\label{LFU}\nLet $\\Omega \\subset \\mathbb{R}^n$ be a bounded convex domain,\nand denote $D=D(\\Omega)$.\nThen there exists a constant $C=C(n)>0$ such that,\nfor every $u\\in H^1(\\Omega)$,\n\\[\n\\|u-u_{\\partial\\Omega}\\|_{L^2(\\partial\\Omega)}\n\\le\nC D^{1/2}\\|\\nabla u\\|_{L^2(\\Omega)},\n\\]\nwhere\n\\[\nu_{\\partial\\Omega}\n=\n\\frac{1}{|\\partial\\Omega|}\n\\int_{\\partial\\Omega} u\\ d\\sigma.\n\\]\n\\end{lemma}\n\n\\begin{theorem}\\label{tmi}\nLet $\\Omega \\subset \\mathbb{R}^2$ be a bounded convex domain and again let\n$\nD := D(\\Omega).\n$\nThen there exists a constant $c>0$ depending only on the dimension such that\n\\[\n\\mu_1(\\Omega) \\ge \\frac{c}{D}.\n\\]\n\\end{theorem}\n\n\\begin{lemma}\\label{le1}\nLet $\\Omega \\subset \\mathbb{R}^n$ ($n \\ge 2$) be a smooth bounded domain,\n$\\Omega_l$ and $\\Omega_r$ two disjoint subdomains of $\\Omega$\nwith smooth boundaries, and\n$S_j = \\partial\\Omega \\cap \\partial\\Omega_j$,\n$\\mathcal{H}^{n-1}(S_j) > 0$ $(j=l,r)$.\nFor $p > n$, we define the set\n\\begin{equation}\\label{L}\n\\Lambda(\\Omega)\n=\n\\left\\{\\begin{aligned}\nv \\in W^{1,p}(\\Omega) :\\quad\n&-1 \\le v(x) \\le 1,\\; x \\in \\overline{\\Omega},\\\\\n&\\int_{S_l} v \\, d\\sigma < 0,\\qquad\n\\int_{S_r} v \\, d\\sigma > 0,\\\\\n&E(v) < \\varepsilon_0 - G(1)\\mathcal{H}^{n-1}(\\partial\\Omega)\n\\end{aligned}\\right\\},\n\\end{equation}\nwhere\n\\[\n\\varepsilon_0\n=\nG(1)\n\\min\n\\left\\{\\begin{aligned}\n&\\mathcal{H}^{n-1}(S_l)\\min\\{\\lambda,\\,\\mu_1(\\Omega_l)\\},\\\\\n&\\mathcal{H}^{n-1}(S_r)\\min\\{\\lambda,\\,\\mu_1(\\Omega_r)\\}\n\\end{aligned}\\right\\},\n\\]\nand $\\mu_1(\\Omega_j)$ is the first positive eigenvalue of the Steklov\nproblem \\ref{ST} defined in $\\Omega_j$ $(j=l,r)$.\n\nFor convenience, in this section we denote a point in $\\mathbb{R}^2$ by $(x,y)$. Our objective is to prove that if $\\Omega \\subset \\mathbb{R}^2$ is smooth bounded convex domain\n with inradius $r$ and diameter $D$ (see Figure \\ref{fig1}) large enough, then one can find $\\lambda > 0$ such that\n$\n\\Lambda(\\Omega) \\neq \\emptyset.\n$, where $\\Lambda(\\Omega)$ is the set defined in \\eqref{L}.\n\n\\begin{figure}[ht]\n\\centering\n\\def\\svgwidth{1\\linewidth}\n\\input{fig3.pdf_tex}\n\\caption{Convex domain\n$\\Omega \\subset \\mathbb{R}^2$ with inradius $r$ and diameter $D$.}\n\\label{fig1}\n\\end{figure}\n\n\\begin{theorem}\\label{t1} Let $\\Omega \\subset \\mathbb{R}^2$ be a smooth bounded convex domain such that $\\mathrm{inr}(\\Omega)=r$ and $D(\\Omega)=D$. Then if $D$ is  large enough, there exists $\\lambda>0$ for which problem \\eqref{con} admits a pattern. \\end{theorem}", "post_theorem_intro_text_len": 4620, "post_theorem_intro_text": "Theorem \\ref{t1} reveals a purely geometric mechanism for pattern\nformation. While the inradius remains fixed, increasing the diameter\nforces the emergence of patterns for appropriate values\nof $\\lambda$. In other words, sufficiently elongated convex domains (see Figure \\ref{fig1})\nnecessarily admit patterns. This phenomenon is independent of symmetry assumptions and depends\nonly on the convexity and aspect ratio of the domain.\n\nFrom a biological perspective, Theorem~\\ref{t1} suggests that\ngeometric elongation of the domain can counterbalance\nthe homogenizing action of diffusion in the interior.\nWhen the diameter becomes sufficiently large compared to the inradius,\nthe global geometric structure weakens diffusive stabilization\nand allows patterns to arise.\n\nThe relation between convexity and stability of non-constant stationary solutions \nhas been a central theme in the study of semilinear parabolic equations. \nIn the classical works of Casten--Holland \\cite{CH} and Matano \\cite{matano}, the nonlinearity\nacts in the interior of the domain and homogeneous Neumann boundary\nconditions are imposed. In that setting, it was shown that in convex\ndomains patterns cannot exist under very general assumptions on the\nnonlinearity. These results reveal a strong rigidity\nphenomenon induced by convexity.\n\nIn contrast, in the present work the nonlinearity acts through the\nboundary condition. Theorem~\\ref{t1} shows that, in this setting,\nconvexity by itself does not constitute a structural obstruction\nto pattern formation.\nAlthough it is known that no pattern can exist when $\\Omega$ is a ball\n(see \\cite{NLM}), this reflects the special symmetry of the ball rather\nthan convexity per se.\n\nTo the best of our knowledge, no previous result establishes a direct\ncriterion for the existence or absence of patterns in terms of global\ngeometric quantities such as the inradius and the diameter. Conversely,\nmuch of the existing literature connects the existence of patterns to\ncurvature effects or other local geometric properties of the boundary, for instance, see \\cite{CH,SAE,FA,SN} and references therein.\n\nIt is worth mentioning that in \\cite{CJ}, through a computer-assisted analysis\ncombined with bifurcation techniques, strong evidence was provided for\nthe existence of patterns for problem \\eqref{con} in the case\n$g(u)=u-u^3$, $\\lambda>2,84083164$, and $\\Omega$ equal to the unit square,\nwhich is a convex planar domain. The authors further conjectured that the same phenomenon should persist\nfor smooth convex domains obtained by suitably rounding the corners of the square.\n\nIn addition to \\cite{CJ}, a proof in the convex case was mentioned in \\cite{CS,DP}\nas part of work in preparation. To the best of our knowledge, a\ncomplete proof has not yet appeared in the literature.\n\nIn the same framework of boundary reactions in a bidimensional domain, Del Pino {\\it et al.} \\cite{DP}\nconstructed solutions with multiple boundary transitions, addressing existence and qualitative properties, but not stability. The existence of patterns has been obtained under stronger geometric or structural assumptions. \nIn \\cite{AG,C2,C3}, existence results were proved   in special classes of non-convex bounded domains, such as dumbbell-type and perfored domains. \nMoreover, when the boundary nonlinearity depends explicitly on the spatial variable, namely $g=g(x,u)$, existence of patterns has been established in \\cite{ARRIETA} and for problems with variable diffusivity, see \\cite{ACM}.\n\nThe proof of Theorem \\ref{t1} is based on the construction of a specific set, denoted by $\\Lambda(\\Omega)$, associated with the geometry of $\\Omega$ and the positive flow generated by \\eqref{con}. The underlying strategy ensures that, whenever $\\Lambda(\\Omega)$ is nonempty, it contains the desired pattern. Thus, the core of the argument consists in proving that $\\Lambda(\\Omega)\\neq\\emptyset$. This approach appears to have been introduced in \\cite{matano} and has since been employed in different contexts, for instance in \\cite{ar1, C2}, as well as in some subsequent works \\cite{ACM}.\n\nThe present work is organized as follows. Section 2 is devoted to the preliminary results needed for the proof of Theorem \\ref{t1}. In particular, in Theorem \\ref{tmi} we establish a lower bound for the first positive Steklov eigenvalue, expressed in terms of the diameter of the domain and a constant depending only on the dimension. This estimate, which may be of independent interest in its own right, plays a key role in the proof of Theorem \\ref{t1}, carried out in Section 3. Section 4 concludes the paper with some final remarks.", "sketch": "The proof of Theorem \\ref{t1} is based on “the construction of a specific set, denoted by $\\Lambda(\\Omega)$, associated with the geometry of $\\Omega$ and the positive flow generated by \\eqref{con}.” The strategy is that “whenever $\\Lambda(\\Omega)$ is nonempty, it contains the desired pattern,” so “the core of the argument consists in proving that $\\Lambda(\\Omega)\\neq\\emptyset$.”\n\nA key ingredient used in this program is that “in Theorem \\ref{tmi} we establish a lower bound for the first positive Steklov eigenvalue, expressed in terms of the diameter of the domain and a constant depending only on the dimension,” and “this estimate … plays a key role in the proof of Theorem \\ref{t1}.”", "expanded_sketch": "In establishing the main theorem, the proof is based on “the construction of a specific set, denoted by $\\Lambda(\\Omega)$, associated with the geometry of $\\Omega$ and the positive flow generated by\n\\begin{equation}\\label{con}\n\\begin{cases}\\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}\n.” The strategy is that “whenever $\\Lambda(\\Omega)$ is nonempty, it contains the desired pattern,” so “the core of the argument consists in proving that $\\Lambda(\\Omega)\\neq\\emptyset$.”\n\nWe first prove the following theorem.\n\\begin{theorem}\\label{tmi}\nLet $\\Omega \\subset \\mathbb{R}^2$ be a bounded convex domain and again let\n$\nD := D(\\Omega).\n$\nThen there exists a constant $c>0$ depending only on the dimension such that\n\\[\n\\mu_1(\\Omega) \\ge \\frac{c}{D}.\n\\]\n\\end{theorem}\nThis estimate plays a key role in the proof of the main theorem.", "expanded_theorem": "\\label{t1} Let $\\Omega \\subset \\mathbb{R}^2$ be a smooth bounded convex domain such that $\\mathrm{inr}(\\Omega)=r$ and $D(\\Omega)=D$. Then if $D$ is  large enough, there exists $\\lambda>0$ for which the problem\n\\begin{equation}\\label{con}\n\\begin{cases}\\nu_t - \\Delta u = 0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+, \\\\\n\\partial_{\\nu} u = \\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+, \\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\end{equation}\nadmits a pattern.", "theorem_type": ["Implication", "Existence"], "mcq": {"question": "Let \\(\\Omega \\subset \\mathbb{R}^2\\) be a smooth bounded convex domain with inradius\n\\[\n\\mathrm{inr}(\\Omega):=\\sup\\{r'>0:\\exists x\\in\\Omega\\text{ such that }B_{r'}(x)\\subset \\Omega\\}=r\n\\]\nand diameter\n\\[\nD(\\Omega):=\\sup\\{|x-y|:x,y\\in\\Omega\\}=D.\n\\]\nConsider the heat equation with nonlinear Neumann boundary condition\n\\[\n\\begin{cases}\nu_t-\\Delta u=0, & (x,t)\\in \\Omega\\times\\mathbb{R}^+,\\\\\n\\partial_\\nu u=\\lambda g(u), & (x,t)\\in \\partial\\Omega\\times\\mathbb{R}^+,\\\\\nu(x,0)=u_0(x), & x\\in \\Omega,\n\\end{cases}\n\\]\nwhere \\(\\nu\\) is the outward unit normal and \\(g(u)=u-u^3\\). A pattern means a non-constant stationary solution that is linearly stable. If \\(D\\) is sufficiently large, which conclusion about this problem is valid?", "correct_choice": {"label": "A", "text": "There exists \\(\\lambda>0\\) such that the problem admits a pattern; equivalently, for some positive \\(\\lambda\\), it has a linearly stable non-constant stationary solution."}, "choices": [{"label": "B", "text": "There exists a constant \\(\\lambda_0>0\\), depending only on the dimension, such that for every \\(\\lambda\\ge \\lambda_0\\) the problem admits a pattern whenever \\(D\\) is sufficiently large."}, {"label": "C", "text": "There exists \\(\\lambda>0\\) such that the problem admits a stationary solution that is non-constant."}, {"label": "D", "text": "For every sufficiently large diameter \\(D\\), and for every smooth bounded convex domain \\(\\Omega\\subset\\mathbb{R}^2\\) with \\(\\mathrm{inr}(\\Omega)=r\\) and \\(D(\\Omega)=D\\), the problem admits a pattern for every \\(\\lambda>0\\)."}, {"label": "E", "text": "If \\(D\\) is sufficiently large, then for every \\(\\lambda>0\\) there exists a linearly stable non-constant stationary solution of the problem."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "existential dependence of the admissible parameter \\(\\lambda\\)", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "linear stability requirement in the definition of pattern", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "quantifier order on \\(\\lambda\\): existence replaced by universality", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "parameter-selection mechanism via nonemptiness of \\(\\Lambda(\\Omega)\\) for some suitable \\(\\lambda\\)", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state or strongly hint at the correct quantifier structure. It gives the PDE setup and definition of pattern, but the correct choice must be identified from the options."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-identification question: the correct option is essentially the precise theorem conclusion. However, it is not a pure tautology because the alternatives vary meaningfully in stability and quantifiers over λ."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish existence for some λ from stronger universal claims and from the weaker non-constant stationary conclusion. Still, the question mainly tests recognition of the exact valid statement rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target common errors: dropping linear stability, strengthening existence to universality, and asserting uniform parameter bounds. They are distinct and well aligned with likely failure modes."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it leans more toward precise theorem recall/quantifier discrimination than deep generative reasoning."}}
{"id": "2602.21070v1", "paper_link": "http://arxiv.org/abs/2602.21070v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "[Half-lift principle for sums of squares]\\label{thm:half-lift-principle}\nLet $d\\ge 1$ and set\n\\[\n  Q_d(x_1,\\dots,x_d):=x_1^2+\\cdots+x_d^2.\n\\]\nFor $n\\ge 1$ and $a\\in\\mathbb{Z}$ let\n\\[\n  N_{d,n}(a):=\\#\\{x\\in (\\mathbb{Z}/2^n\\mathbb{Z})^d: Q_d(x)\\equiv a\\pmod{2^n}\\}.\n\\]\nAssume $n\\ge 3$ and $4\\nmid a$.\nThen\n\\[\n  N_{d,n+1}(a)=2^{d-1}\\,N_{d,n}(a).\n\\]\nEquivalently, among the residue classes modulo $2^n$ satisfying $Q_d\\equiv a$, exactly half admit lifts to level $2^{n+1}$; moreover, whenever a class lifts, \\emph{all} $2^d$ lifts in its fibre are solutions (cf.\\ Lemma~\\ref{lem:fibre-invariance}).", "start_pos": 10709, "end_pos": 11308, "label": "thm:half-lift-principle"}, "ref_dict": {"rem:half-lift-hensel": "\\begin{remark}\\label{rem:half-lift-hensel}\nFor $d=3$ one can improve Step~2 by choosing a single involution direction on each parity class (as in the original ``three-squares'' half-lift lemma).\nThe minimal-index partition above is slightly less symmetric but works uniformly for all $d\\ge 1$.\nIn particular, the growth factor $2^{d-1}$ coincides with the ``smooth'' Hensel factor $p^{d-1}$ specialised to $p=2$, even though the Jacobian of the bilinear quadratic form vanishes modulo $2$.\n\\end{remark}", "tab:block-densities": "\\label{tab:block-densities}\n\\end{table}\n\n\\begin{corollary}[Dyadic vanishing thresholds in the $Q$- and $q$-normalisations]\n\\label{cor:dyadic-vanishing-thresholds}\nLet $p=2$.\nLet $L$ be an \\emph{even}", "eq:L3": "\\begin{equation}\\label{eq:L3}\n  L_3 := \\langle 2\\rangle^{\\oplus 3}\\qquad\\text{with }Q(x)=2(x_1^2+x_2^2+x_3^2)\n\\end{equation}", "lem:q-dictionary": "\\begin{lemma}[Dictionary with the $q=\\langle\\cdot,\\cdot\\rangle/2$ convention]\\label{lem:q-dictionary}\nAssume $L$ is \\emph{even}, i.e.\\ $Q(L)\\subset 2\\Ztwo$, and set $q(v):=Q(v)/2\\in\\Ztwo$.\nFor $m\\ge 1$ define\n\\[\n  r_m^{(q)}(t;L):=\\#\\{v\\in L/2^mL : q(v)\\equiv t\\!\\!\\pmod{2^m}\\}.\n\\]\nThen for $t\\in\\Ztwo$ and $n:=\\mathrm{rank}_{\\Ztwo}(L)$:\n\\begin{altenumerate}\n\\item If $t$ is odd, then $r_m(t;L)=0$ for all $m\\ge 1$.\n\\item If $t=2t'$ is even, then for every $m\\ge 2$ one has\n\\[\n  r_m(t;L)=2^{n}\\,r_{m-1}^{(q)}(t';L).\n\\]\nIn particular, $r_1(2t';L)=2^n$.\n\\end{altenumerate}\nConsequently, whenever the limits exist,\n\\[\\alpha_2(2t';L)=2\\,\\alpha_2^{(q)}(t';L),\\qquad\n  \\alpha_2^{(q)}(t';L):=\\lim_{m\\to\\infty}2^{-m(n-1)}\\,r_m^{(q)}(t';L).\\]\n\\end{lemma}", "eq:2xy": "\\begin{equation}\\label{eq:2xy}\n  2xy\\equiv s\\pmod{p^m}.\n\\end{equation}", "app:typeI": "\\label{app:typeI}\n\nWe record the square-root counts modulo $2^m$ that enter the rank-$1$ dyadic blocks.\nThese counts are classical, but we include them so that the evaluation of the base blocks can be", "thm:prime-uniform-h0": "\\begin{theorem}[Prime-uniform density for the hyperbolic plane]\n\\label{thm:prime-uniform-h0}\nLet $p$ be a prime and let $e\\ge 0$.\nLet $H_0$ denote the rank-$2$ hyperbolic plane over $\\Zp$ with bilinear form\n\\(\\langle (x,y),(x',y')\\rangle = xy'+x'y\\).\nOn the scaled lattice $p^eH_0$ consider the quadratic form\n\\[\n  q_e(v):=\\tfrac12\\langle v,v\\rangle = p^e xy\\in\\Zp.\n\\]\nFor $m\\ge 1$ and $t\\in\\Zp$ set\n\\[\n  r^{(q)}_{m}(t;p^eH_0)\n  :=\\#\\{(x,y)\\in(\\Z/p^m\\Z)^2 : p^e xy\\equiv t\\!\\!\\pmod{p^m}\\},\n\\]\nand define the associated density (when the limit exists) by\n\\[\n  \\alpha_p^{(q)}(t;p^eH_0)\n  :=\\lim_{m\\to\\infty}p^{-m}\\,r^{(q)}_{m}(t;p^eH_0).\n\\]\nLet $t\\in\\Zp\\setminus\\{0\\}$ and write $v:=v_p(t)$.\nThen\n\\[\n  \\alpha_p^{(q)}(t;p^eH_0)=\n  \\begin{cases}\n    0, & v<e,\\\\[2pt]\n    (v-e+1)(p-1)p^{e-1}, & v\\ge e.\n  \\end{cases}\n\\]\n\\smallskip\n\\noindent In particular, under the $q=\\langle\\cdot,\\cdot\\rangle/2$ normalisation, both the vanishing threshold and the density formula for the scaled hyperbolic plane are uniform for all primes $p$, including $p=2$.\n\\end{theorem}", "lem:H1": "\\begin{lemma}[Counts for the anisotropic even plane]\\label{lem:H1}\nLet $m\\ge 1$ and $t\\in\\Ztwo$.\nWrite $v:=v_2(t)$ with the convention $v=\\infty$ if $t=0$.\nThen:\n\\begin{altenumeratealpha}\n\\item If $t\\ne 0$ and $v$ is even, then $A_m(t)=0$ for all $m>v$.\n\\item If $t\\ne 0$ and $v$ is odd, then $A_m(t)=3\\cdot 2^m$ for all $m>v$.\n\\item If $t\\equiv 0\\pmod{2^m}$, then $A_m(t)=4^{\\lceil m/2\\rceil}$.\n\\end{altenumeratealpha}\n\\end{lemma}", "cor:density-hyp-scaled": "\\begin{corollary}[Scaled hyperbolic block]\\label{cor:density-hyp-scaled}\nLet $e\\ge 0$ and let $L=2^eH_0$, so that $Q(x,y)=2^{e+1}xy$.\nLet $t\\in\\Ztwo\\setminus\\{0\\}$ and set $v:=v_2(t)$.\nIf $v<e+1$ then $\\alpha_2(t;L)=0$.\nIf $v\\ge e+1$ then\n\\[\n  \\alpha_2(t;L)=2^e\\,(v-e).\n\\]\n\\end{corollary}", "eq:expansion-d": "\\begin{equation}\\label{eq:expansion-d}\n  f(\\tau_u(x))\\equiv f(x)\\pmod{2^n},\\qquad\n  f(\\tau_u(x))\\equiv f(x)+2^n(x\\cdot u)\\pmod{2^{n+1}},\n\\end{equation}", "thm:half-lift-principle": "\\begin{theorem}[Half-lift principle for sums of squares]\\label{thm:half-lift-principle}\nLet $d\\ge 1$ and set\n\\[\n  Q_d(x_1,\\dots,x_d):=x_1^2+\\cdots+x_d^2.\n\\]\nFor $n\\ge 1$ and $a\\in\\Z$ let\n\\[\n  N_{d,n}(a):=\\#\\{x\\in (\\Z/2^n\\Z)^d: Q_d(x)\\equiv a\\pmod{2^n}\\}.\n\\]\nAssume $n\\ge 3$ and $4\\nmid a$.\nThen\n\\[\n  N_{d,n+1}(a)=2^{d-1}\\,N_{d,n}(a).\n\\]\nEquivalently, among the residue classes modulo $2^n$ satisfying $Q_d\\equiv a$, exactly half admit lifts to level $2^{n+1}$; moreover, whenever a class lifts, \\emph{all} $2^d$ lifts in its fibre are solutions (cf.\\ Lemma~\\ref{lem:fibre-invariance}).\n\\end{theorem}", "lem:fibre-invariance": "\\begin{lemma}[Fibre invariance for sums of squares]\\label{lem:fibre-invariance}\nLet $d\\ge 1$ and $n\\ge 1$.\nFor any $x\\in(\\Z/2^n\\Z)^d$ and any $z\\in(\\Z/2\\Z)^d$ one has\n\\[\n  Q_d(x+2^n z)\\equiv Q_d(x)\\pmod{2^{n+1}}.\n\\]\nConsequently, for any $a\\in\\Z$ and any residue class $\\bar x\\in(\\Z/2^n\\Z)^d$, the number of lifts $\\tilde x\\in(\\Z/2^{n+1}\\Z)^d$ of $\\bar x$ satisfying\n$Q_d(\\tilde x)\\equiv a\\pmod{2^{n+1}}$\nis either $0$ or the full fibre size $2^d$.\n\\end{lemma}", "def:dyadic-base-blocks": "\\begin{definition}[Basic dyadic blocks]\\label{def:dyadic-base-blocks}\nFor the explicit computations in this paper we single out the following quadratic $\\Ztwo$-lattices; we refer to them as the \\emph{basic dyadic blocks}:\n\\begin{altenumerate}\n\\item \\textbf{Type~I (rank $1$).} The lattice $\\langle 2^a u\\rangle$ with Gram matrix $[2^a u]$, where $a\\ge 0$ and $u\\in\\Ztwo^\\times$.\n\\item \\textbf{Type~II (rank $2$, even).} The lattice $2^a H_\\varepsilon$ with Gram matrix $2^a H_\\varepsilon$, where $a\\ge 0$ and\n\\[\n  H_0:=\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}\\qquad\\text{and}\\qquad\n  H_1:=\\begin{pmatrix}2&1\\\\1&2\\end{pmatrix}.\n\\]\n\\end{altenumerate}\nOnly the square class of the unit $u$ (equivalently $u\\bmod 8$) will matter for solvability in the rank-one case.\n\\end{definition}", "lem:2xy-2": "\\begin{lemma}[Counting $2xy\\equiv s\\pmod{2^m}$ at $p=2$]\\label{lem:2xy-2}\nLet $m\\ge 1$ and $s\\in\\Z/2^m\\Z$.\nWrite $v:=v_2(s)$ with the convention $v=m$ if $s\\equiv 0\\pmod{2^m}$.\nThen:\n\\begin{altenumerate}\n\\item If $v=0$ (i.e.\\ $s$ is odd), then $M_m(s)=0$.\n\\item If $1\\le v<m$, then $M_m(s)=v\\cdot 2^m$.\n\\item If $v=m$ (i.e.\\ $s\\equiv 0\\pmod{2^m}$), then $M_m(0)=(m+1)2^m$.\n\\end{altenumerate}\n\\end{lemma}", "cor:density-hyp-scaled-odd": "\\begin{corollary}[Scaled hyperbolic plane at odd primes]\\label{cor:density-hyp-scaled-odd}\nLet $p$ be an odd prime and let $e\\ge 0$.\nLet $L=p^eH_0$, so that $Q(x,y)=2p^e xy$.\nLet $t\\in\\Zp\\setminus\\{0\\}$ and set $v:=v_p(t)$.\nIf $v<e$ then $\\alpha_p(t;L)=0$.\nIf $v\\ge e$, then\n\\[\n  \\alpha_p(t;L)=(v-e+1)(p-1)p^{e-1}.\n\\]\n\\end{corollary}", "prop:jordan": "\\begin{proposition}[Jordan decomposition]\\label{prop:jordan}\nLet $L$ be a nondegenerate quadratic lattice over $\\Ztwo$.\nThen there exists an orthogonal decomposition\n\\[\n  L \\;\\cong\\; \\perp_{a\\ge 0} 2^a L_a,\n\\]\nwith each $L_a$ a unimodular quadratic $\\Ztwo$-lattice (possibly $0$).\nMoreover, the multiset of integers $a$ together with the multiplicities $\\mathrm{rank}(L_a)$ (equivalently, the Jordan \\emph{scales} and \\emph{ranks}) is uniquely determined by $L$.\n\\end{proposition}", "lem:2xy-odd": "\\begin{lemma}[Counting $2xy\\equiv s\\pmod{p^m}$ at odd primes]\\label{lem:2xy-odd}\nLet $p$ be an odd prime, $m\\ge 1$, and $s\\in\\Z/p^m\\Z$.\nWrite $v:=v_p(s)$ with the convention $v=m$ if $s\\equiv 0\\pmod{p^m}$.\nThen:\n\\begin{altenumerate}\n\\item If $0\\le v<m$, then $M_{p,m}(s)=(v+1)(p-1)p^{m-1}$.\n\\item If $v=m$ (i.e.\\ $s\\equiv 0\\pmod{p^m}$), then $M_{p,m}(0)=(m(p-1)+p)p^{m-1}$.\n\\end{altenumerate}\n\\end{lemma}", "prop:orthogonal-convolution": "\\begin{proposition}[Orthogonal-sum convolution]\\label{prop:orthogonal-convolution}\nLet $L$ and $M$ be quadratic $\\Zp$-lattices, and equip $L\\oplus M$ with the orthogonal direct-sum quadratic form $Q_{L\\oplus M}(v,w):=Q_L(v)+Q_M(w)$.\nThen for every $m\\ge 1$ and $t\\in\\Zp$ one has the finite-level identity\n\\[\n  r_m(t;L\\oplus M)\n  \\,=\\,\n  \\sum_{s\\in \\Z/p^m\\Z} r_m(s;L)\\,r_m(t-s;M).\n\\]\n\\end{proposition}"}, "pre_theorem_intro_text_len": 6488, "pre_theorem_intro_text": "\\label{sec:intro}\n\nLocal representation densities of quadratic lattices are the local factors in many global counting problems.\nThey appear in the Siegel series and Siegel's mass formula, and they control Fourier coefficients of theta series and Eisenstein series.\nIn concrete arithmetic applications one is often forced to evaluate these local factors for specific lattices, and the dyadic prime is typically the main obstacle.\n\nAt odd primes, once the relevant solution locus is smooth modulo $p$, Hensel lifting implies that reduction maps between solution sets have constant fibre size $p^{n-1}$, and the density stabilises quickly.\nAt the dyadic prime this mechanism can break down even for very simple lattices.\nIn the bilinear-lattice convention $Q(v)=\\langle v,v\\rangle$, the linearisation of $Q-t$ at a solution $\\bar v$ is the functional $h\\mapsto 2\\langle \\bar v,h\\rangle$, which vanishes modulo~$2$.\nThus the standard Jacobian criterion yields no lifting information.\n\nEvery nondegenerate quadratic lattice over $\\Z_2$ admits a Jordan decomposition into scaled unimodular constituents (Proposition~\\ref{prop:jordan}).\nAccordingly, explicit formulas for the basic dyadic blocks are a natural starting point for dyadic computations.\nIn this paper we determine the finite-level representation counts and local densities for the basic dyadic blocks of Definition~\\ref{def:dyadic-base-blocks} (rank-$1$ Type~I blocks and the even planes $2^aH_\\varepsilon$), together with the anisotropic ternary lattice $L_3$.\n\nThe main new input is a very simple involution argument, the \\emph{half-lift}.\nFor diagonal sums of squares $Q_d=x_1^2+\\cdots+x_d^2$ and a primitive target $a$ with $4\\nmid a$, translation by $2^{n-1}u$ in a primitive direction produces a free involution on the solution set modulo $2^n$ whose orbits control lifts to level $2^{n+1}$.\nIn each orbit exactly one residue class lifts, and whenever a class lifts, all lifts in its fibre are solutions.\nThis yields the stable recursion $N_{d,n+1}(a)=2^{d-1}N_{d,n}(a)$ for $n\\ge 3$ (Theorem~\\ref{thm:half-lift-principle}).\nFor $d=3$ it leads to closed forms for the three-squares congruence counts and hence an explicit formula for $\\alpha_2(t;L_3)$.\n\n\\smallskip\nWe fix the conventions for the finite-level representation counts $r_m(t;L)$ and the local density $\\alpha_p(t;L)$ in \\S\\ref{sec:prelim}.\nAt $p=2$ and for even lattices there is a second common normalisation $q:=Q/2$; we work with $Q$ unless explicitly indicated, and Lemma~\\ref{lem:q-dictionary} gives the precise conversion.\n\nGeneral formulas for the dyadic Siegel series are known, expressed for instance in terms of Gross--Keating invariants or group-scheme methods (see for example\n\\cite{grossKeating1993,choYamauchi2020,ganYu2000,ikedaKatsurada2018,kitaokaBook}).\nThe densities computed here agree with these formulas (cf.\\ Yang's explicit expressions~\\cite{yang1998}), but this serves only as an \\emph{external consistency check} and is not used as an input to any argument.\n\nAt odd primes, the hyperbolic-plane computations of \\S\\ref{sec:hyperbolic} are classical; we include a self-contained counting argument to keep the exposition uniform and to make the prime-uniform statement of Theorem~\\ref{thm:prime-uniform-h0} transparent.\nThe genuinely new feature of this paper is dyadic: the half-lift involution of Theorem~\\ref{thm:half-lift-principle} and the resulting elementary derivation of the dyadic block densities without appealing to the full Siegel-series machinery.\n\nAll computations in the body of the paper are organised around the same elementary theme:\ndescribe the fibres of the reduction maps $L/2^{m+1}L\\to L/2^mL$ on the relevant solution sets.\nConcretely,\n\\begin{altenumerate}\n\\item for the split even plane $H_0$ this reduces to valuation-stratified counting for the bilinear congruence $2xy\\equiv t$ (see Lemmas~\\ref{lem:2xy-odd} and~\\ref{lem:2xy-2});\n\\item for the anisotropic plane $H_1$ it becomes a lifting recursion for a norm congruence in the unramified quadratic extension of~$\\Q_2$ (see Lemma~\\ref{lem:H1});\n\\item for the anisotropic ternary block $L_3$ one needs an additional involution (the half-lift) because the naive dyadic linearisation carries no information (see Theorem~\\ref{thm:half-lift-principle}).\n\\end{altenumerate}\n\nThis paper focuses on the basic dyadic blocks (Definition~\\ref{def:dyadic-base-blocks}) and the anisotropic ternary lattice $L_3$.\nOnce the basic block counts are known, representation counts for orthogonal sums reduce to finite convolutions (Proposition~\\ref{prop:orthogonal-convolution}).\nWe do not attempt closed forms for general orthogonal sums.\n\n\\smallskip\n\\noindent\nFor quick reference, Table~\\ref{tab:block-densities} in \\S\\ref{sec:densities} lists the final density values.\nReaders primarily interested in the dyadic novelty may start with \\S\\ref{sec:half-lift}.\n\\subsection{Main results}\n\nLet $p$ be a prime.\nSection~\\ref{sec:hyperbolic} treats the hyperbolic plane $H_0$ over $\\Z_p$ for every $p$ and gives explicit formulas for the congruence counts $M_{p,m}(s)$ in~\\eqref{eq:2xy}, together with their generating series and stable-range behaviour.\nSection~\\ref{sec:densities} records the resulting density formulas for $\\alpha_p(t;H_0)$ and the basic dyadic blocks.\nIn particular, after passing to the standard normalisation $q=\\langle\\cdot,\\cdot\\rangle/2$ at $p=2$ (Lemma~\\ref{lem:q-dictionary}), the density of $H_0$ and its scalings admits a single prime-uniform expression (Theorem~\\ref{thm:prime-uniform-h0}).\nThe remaining sections specialise to $p=2$ and evaluate the basic dyadic blocks.\n\nWe therefore recall the dyadic Jordan decomposition (Proposition~\\ref{prop:jordan}) and the basic dyadic blocks (Definition~\\ref{def:dyadic-base-blocks}) that will be treated explicitly.\nIn addition to the rank-$2$ even blocks, the rank-$3$ lattice\n\\begin{equation}\\label{eq:L3}\n  L_3 := \\langle 2\\rangle^{\\oplus 3}\\qquad\\text{with }Q(x)=2(x_1^2+x_2^2+x_3^2)\n\\end{equation}\nplays a distinguished role: it is the simplest anisotropic even ternary lattice over $\\Z_2$, and its representation counts amount to the classical three-squares congruence problem.\nIndeed, if $x_1^2+x_2^2+x_3^2=0$ in $\\Z_2$, then reducing modulo $8$ forces $x_1,x_2,x_3$ to be even, and iterating gives $x_1=x_2=x_3=0$; hence $L_3$ is anisotropic over $\\Q_2$.\n\nOur first main theorem is a lifting principle for diagonal sums of squares in arbitrary dimension, under the mild primitivity hypothesis $4\\nmid a$.", "context": "The main new input is a very simple involution argument, the \\emph{half-lift}.\nFor diagonal sums of squares $Q_d=x_1^2+\\cdots+x_d^2$ and a primitive target $a$ with $4\\nmid a$, translation by $2^{n-1}u$ in a primitive direction produces a free involution on the solution set modulo $2^n$ whose orbits control lifts to level $2^{n+1}$.\nIn each orbit exactly one residue class lifts, and whenever a class lifts, all lifts in its fibre are solutions.\nThis yields the stable recursion $N_{d,n+1}(a)=2^{d-1}N_{d,n}(a)$ for $n\\ge 3$ (Theorem~\\ref{thm:half-lift-principle}).\nFor $d=3$ it leads to closed forms for the three-squares congruence counts and hence an explicit formula for $\\alpha_2(t;L_3)$.\n\n\\smallskip\nWe fix the conventions for the finite-level representation counts $r_m(t;L)$ and the local density $\\alpha_p(t;L)$ in \\S\\ref{sec:prelim}.\nAt $p=2$ and for even lattices there is a second common normalisation $q:=Q/2$; we work with $Q$ unless explicitly indicated, and Lemma~\\ref{lem:q-dictionary} gives the precise conversion.\n\nAll computations in the body of the paper are organised around the same elementary theme:\ndescribe the fibres of the reduction maps $L/2^{m+1}L\\to L/2^mL$ on the relevant solution sets.\nConcretely,\n\\begin{altenumerate}\n\\item for the split even plane $H_0$ this reduces to valuation-stratified counting for the bilinear congruence $2xy\\equiv t$ (see Lemmas~\\ref{lem:2xy-odd} and~\\ref{lem:2xy-2});\n\\item for the anisotropic plane $H_1$ it becomes a lifting recursion for a norm congruence in the unramified quadratic extension of~$\\Q_2$ (see Lemma~\\ref{lem:H1});\n\\item for the anisotropic ternary block $L_3$ one needs an additional involution (the half-lift) because the naive dyadic linearisation carries no information (see Theorem~\\ref{thm:half-lift-principle}).\n\\end{altenumerate}\n\nLet $p$ be a prime.\nSection~\\ref{sec:hyperbolic} treats the hyperbolic plane $H_0$ over $\\Z_p$ for every $p$ and gives explicit formulas for the congruence counts $M_{p,m}(s)$ in~\\eqref{eq:2xy}, together with their generating series and stable-range behaviour.\nSection~\\ref{sec:densities} records the resulting density formulas for $\\alpha_p(t;H_0)$ and the basic dyadic blocks.\nIn particular, after passing to the standard normalisation $q=\\langle\\cdot,\\cdot\\rangle/2$ at $p=2$ (Lemma~\\ref{lem:q-dictionary}), the density of $H_0$ and its scalings admits a single prime-uniform expression (Theorem~\\ref{thm:prime-uniform-h0}).\nThe remaining sections specialise to $p=2$ and evaluate the basic dyadic blocks.\n\nWe therefore recall the dyadic Jordan decomposition (Proposition~\\ref{prop:jordan}) and the basic dyadic blocks (Definition~\\ref{def:dyadic-base-blocks}) that will be treated explicitly.\nIn addition to the rank-$2$ even blocks, the rank-$3$ lattice\n\\begin{equation}\\label{eq:L3}\n L_3 := \\langle 2\\rangle^{\\oplus 3}\\qquad\\text{with }Q(x)=2(x_1^2+x_2^2+x_3^2)\n\\end{equation}\nplays a distinguished role: it is the simplest anisotropic even ternary lattice over $\\Z_2$, and its representation counts amount to the classical three-squares congruence problem.\nIndeed, if $x_1^2+x_2^2+x_3^2=0$ in $\\Z_2$, then reducing modulo $8$ forces $x_1,x_2,x_3$ to be even, and iterating gives $x_1=x_2=x_3=0$; hence $L_3$ is anisotropic over $\\Q_2$.\n\nOur first main theorem is a lifting principle for diagonal sums of squares in arbitrary dimension, under the mild primitivity hypothesis $4\\nmid a$.", "full_context": "The main new input is a very simple involution argument, the \\emph{half-lift}.\nFor diagonal sums of squares $Q_d=x_1^2+\\cdots+x_d^2$ and a primitive target $a$ with $4\\nmid a$, translation by $2^{n-1}u$ in a primitive direction produces a free involution on the solution set modulo $2^n$ whose orbits control lifts to level $2^{n+1}$.\nIn each orbit exactly one residue class lifts, and whenever a class lifts, all lifts in its fibre are solutions.\nThis yields the stable recursion $N_{d,n+1}(a)=2^{d-1}N_{d,n}(a)$ for $n\\ge 3$ (Theorem~\\ref{thm:half-lift-principle}).\nFor $d=3$ it leads to closed forms for the three-squares congruence counts and hence an explicit formula for $\\alpha_2(t;L_3)$.\n\n\\smallskip\nWe fix the conventions for the finite-level representation counts $r_m(t;L)$ and the local density $\\alpha_p(t;L)$ in \\S\\ref{sec:prelim}.\nAt $p=2$ and for even lattices there is a second common normalisation $q:=Q/2$; we work with $Q$ unless explicitly indicated, and Lemma~\\ref{lem:q-dictionary} gives the precise conversion.\n\nAll computations in the body of the paper are organised around the same elementary theme:\ndescribe the fibres of the reduction maps $L/2^{m+1}L\\to L/2^mL$ on the relevant solution sets.\nConcretely,\n\\begin{altenumerate}\n\\item for the split even plane $H_0$ this reduces to valuation-stratified counting for the bilinear congruence $2xy\\equiv t$ (see Lemmas~\\ref{lem:2xy-odd} and~\\ref{lem:2xy-2});\n\\item for the anisotropic plane $H_1$ it becomes a lifting recursion for a norm congruence in the unramified quadratic extension of~$\\Q_2$ (see Lemma~\\ref{lem:H1});\n\\item for the anisotropic ternary block $L_3$ one needs an additional involution (the half-lift) because the naive dyadic linearisation carries no information (see Theorem~\\ref{thm:half-lift-principle}).\n\\end{altenumerate}\n\nLet $p$ be a prime.\nSection~\\ref{sec:hyperbolic} treats the hyperbolic plane $H_0$ over $\\Z_p$ for every $p$ and gives explicit formulas for the congruence counts $M_{p,m}(s)$ in~\\eqref{eq:2xy}, together with their generating series and stable-range behaviour.\nSection~\\ref{sec:densities} records the resulting density formulas for $\\alpha_p(t;H_0)$ and the basic dyadic blocks.\nIn particular, after passing to the standard normalisation $q=\\langle\\cdot,\\cdot\\rangle/2$ at $p=2$ (Lemma~\\ref{lem:q-dictionary}), the density of $H_0$ and its scalings admits a single prime-uniform expression (Theorem~\\ref{thm:prime-uniform-h0}).\nThe remaining sections specialise to $p=2$ and evaluate the basic dyadic blocks.\n\nWe therefore recall the dyadic Jordan decomposition (Proposition~\\ref{prop:jordan}) and the basic dyadic blocks (Definition~\\ref{def:dyadic-base-blocks}) that will be treated explicitly.\nIn addition to the rank-$2$ even blocks, the rank-$3$ lattice\n\\begin{equation}\\label{eq:L3}\n L_3 := \\langle 2\\rangle^{\\oplus 3}\\qquad\\text{with }Q(x)=2(x_1^2+x_2^2+x_3^2)\n\\end{equation}\nplays a distinguished role: it is the simplest anisotropic even ternary lattice over $\\Z_2$, and its representation counts amount to the classical three-squares congruence problem.\nIndeed, if $x_1^2+x_2^2+x_3^2=0$ in $\\Z_2$, then reducing modulo $8$ forces $x_1,x_2,x_3$ to be even, and iterating gives $x_1=x_2=x_3=0$; hence $L_3$ is anisotropic over $\\Q_2$.\n\nOur first main theorem is a lifting principle for diagonal sums of squares in arbitrary dimension, under the mild primitivity hypothesis $4\\nmid a$.\n\nThe main new input is a very simple involution argument, the \\emph{half-lift}.\nFor diagonal sums of squares $Q_d=x_1^2+\\cdots+x_d^2$ and a primitive target $a$ with $4\\nmid a$, translation by $2^{n-1}u$ in a primitive direction produces a free involution on the solution set modulo $2^n$ whose orbits control lifts to level $2^{n+1}$.\nIn each orbit exactly one residue class lifts, and whenever a class lifts, all lifts in its fibre are solutions.\nThis yields the stable recursion $N_{d,n+1}(a)=2^{d-1}N_{d,n}(a)$ for $n\\ge 3$ (Theorem~\\ref{thm:half-lift-principle}).\nFor $d=3$ it leads to closed forms for the three-squares congruence counts and hence an explicit formula for $\\alpha_2(t;L_3)$.\n\nOur first main theorem is a lifting principle for diagonal sums of squares in arbitrary dimension, under the mild primitivity hypothesis $4\\nmid a$.\n\n\\begin{remark}[Scope of the half-lift involution]\\label{rem:half-lift-scope}\nThe involution argument proving Theorem~\\ref{thm:half-lift-principle} uses the specific diagonal form\n$Q_d=x_1^2+\\cdots+x_d^2$ and the primitivity hypothesis $4\\nmid a$.\nThe primitivity is essential in general: for example, in dimension $d=4$ one checks directly that\n\\(N_{4,4}(8)=1536\\) while \\(N_{4,3}(8)=128\\), so the ratio is $12\\ne 2^{d-1}=8$.\nOn a conceptual level, two features of the proof break down once these hypotheses are relaxed:\nif $4\\mid a$ then solutions modulo $2^n$ may have all coordinates even, so the first step (which extracts an odd coordinate to choose an involution direction) can fail; and for $n=2$ the quadratic error term in the expansion \\eqref{eq:expansion-d} no longer vanishes modulo $2^{n+1}$, so the translation $x\\mapsto x+2^{n-1}u$ need not toggle $Q_d(x)$ by a clean additive $2^n$.\nWe do not claim a dimension-free half-lift recursion for arbitrary diagonal unit coefficients (or for general dyadic lattices) without further reduction.\nIn the dyadic setting, the half-lift mechanism is used here only to analyse the even ternary lattice $L_3$ in \\eqref{eq:L3};\nrank-$1$ Type~I blocks are handled separately via explicit square-root counts (Appendix~\\ref{app:typeI}).\n\\end{remark}\n\n\\begin{lemma}[Dictionary with the $q=\\langle\\cdot,\\cdot\\rangle/2$ convention]\\label{lem:q-dictionary}\nAssume $L$ is \\emph{even}, i.e.\\ $Q(L)\\subset 2\\Ztwo$, and set $q(v):=Q(v)/2\\in\\Ztwo$.\nFor $m\\ge 1$ define\n\\[\n r_m^{(q)}(t;L):=\\#\\{v\\in L/2^mL : q(v)\\equiv t\\!\\!\\pmod{2^m}\\}.\n\\]\nThen for $t\\in\\Ztwo$ and $n:=\\mathrm{rank}_{\\Ztwo}(L)$:\n\\begin{altenumerate}\n\\item If $t$ is odd, then $r_m(t;L)=0$ for all $m\\ge 1$.\n\\item If $t=2t'$ is even, then for every $m\\ge 2$ one has\n\\[\n r_m(t;L)=2^{n}\\,r_{m-1}^{(q)}(t';L).\n\\]\nIn particular, $r_1(2t';L)=2^n$.\n\\end{altenumerate}\nConsequently, whenever the limits exist,\n\\[\\alpha_2(2t';L)=2\\,\\alpha_2^{(q)}(t';L),\\qquad\n \\alpha_2^{(q)}(t';L):=\\lim_{m\\to\\infty}2^{-m(n-1)}\\,r_m^{(q)}(t';L).\\]\n\\end{lemma}\n\n\\smallskip\n\\noindent\\emph{Case $0\\le v<m$.}\nStratify by $i:=v_p(x)$.\nFor each $0\\le i\\le v$, the number of residue classes $x\\bmod p^m$ with $v_p(x)=i$ is $(p-1)p^{m-i-1}$.\nWrite $x=p^i x'$ with $x'\\in(\\Z/p^{m-i}\\Z)^\\times$.\nThen $xy\\equiv t\\pmod{p^m}$ becomes\n\\[\n x'\\,y\\equiv p^{v-i}t'\\pmod{p^{m-i}},\\qquad t':=t/p^v\\in(\\Z/p^{m-v}\\Z)^\\times.\n\\]\nAs $x'$ is a unit modulo $p^{m-i}$, there is a unique solution $y_0\\bmod p^{m-i}$.\nIt has exactly $p^i$ lifts modulo $p^m$, so for each fixed $i$ the number of pairs with $v_p(x)=i$ is\n\\[(p-1)p^{m-i-1}\\cdot p^i=(p-1)p^{m-1}.\n\\]\nSumming over $i=0,1,\\dots,v$ yields $M_{p,m}(s)=(v+1)(p-1)p^{m-1}$.\n\n\\begin{lemma}[Fibre invariance for sums of squares]\\label{lem:fibre-invariance}\nLet $d\\ge 1$ and $n\\ge 1$.\nFor any $x\\in(\\Z/2^n\\Z)^d$ and any $z\\in(\\Z/2\\Z)^d$ one has\n\\[\n Q_d(x+2^n z)\\equiv Q_d(x)\\pmod{2^{n+1}}.\n\\]\nConsequently, for any $a\\in\\Z$ and any residue class $\\bar x\\in(\\Z/2^n\\Z)^d$, the number of lifts $\\tilde x\\in(\\Z/2^{n+1}\\Z)^d$ of $\\bar x$ satisfying\n$Q_d(\\tilde x)\\equiv a\\pmod{2^{n+1}}$\nis either $0$ or the full fibre size $2^d$.\n\\end{lemma}\n\nLet\n$S_n(a):=\\{x\\in(\\Z/2^n\\Z)^d: f(x)\\equiv 0\\pmod{2^n}\\}$,\nso that $\\#S_n(a)=N_{d,n}(a)$.\nBy Lemma~\\ref{lem:fibre-invariance}, for each $\\bar x\\in(\\Z/2^n\\Z)^d$ the congruence condition modulo $2^{n+1}$ is constant across the entire fibre of its lifts.\nIn particular, if we view $f(\\bar x)\\bmod 2^{n+1}$ as the residue obtained by evaluating $f$ on any lift of $\\bar x$ to $(\\Z/2^{n+1}\\Z)^d$ (well-defined by the lemma), then\n\\[\n N_{d,n+1}(a)=2^d\\cdot\\#\\{\\bar x\\in S_n(a): f(\\bar x)\\equiv 0\\pmod{2^{n+1}}\\}.\n\\]\nWe show the subset on the right has size exactly $\\tfrac12\\#S_n(a)$.\n\n\\smallskip\n\\noindent\\emph{(ii)} Assume $b\\not\\equiv 0\\pmod{2^m}$ and write $b=2^{2j}c$ with $j\\ge 0$, $2j<m$, and $c$ odd.\nIf $x^2\\equiv 2^{2j}c\\pmod{2^m}$, then necessarily $2^j\\mid x$.\nWrite $x=2^j y$.\nDividing by $2^{2j}$ reduces the congruence to\n\\begin{equation}\\label{eq:sqroot-reduced}\n y^2\\equiv c\\pmod{2^{k}},\\qquad k:=m-2j\\ge 1.\n\\end{equation}\nThe condition \\eqref{eq:sqroot-reduced} depends only on the class of $y$ modulo $2^k$.\nConversely, each residue class $\\bar y\\in\\Z/2^k\\Z$ has exactly $2^j$ lifts to a class modulo $2^{m-j}$.\nHence the number of solutions $x\\bmod 2^m$ equals\n\\[\n 2^j\\cdot\\#\\{y\\bmod 2^k: y^2\\equiv c\\pmod{2^k}\\}.\n\\]\n\n\\label{app:typeI}\n\nWe record the square-root counts modulo $2^m$ that enter the rank-$1$ dyadic blocks.\nThese counts are classical, but we include them so that the evaluation of the base blocks can be\n\n\\begin{equation}\\label{eq:L3}\n  L_3 := \\langle 2\\rangle^{\\oplus 3}\\qquad\\text{with }Q(x)=2(x_1^2+x_2^2+x_3^2)\n\\end{equation}\n\n\\begin{equation}\\label{eq:expansion-d}\n  f(\\tau_u(x))\\equiv f(x)\\pmod{2^n},\\qquad\n  f(\\tau_u(x))\\equiv f(x)+2^n(x\\cdot u)\\pmod{2^{n+1}},\n\\end{equation}\n\n\\begin{lemma}[Fibre invariance for sums of squares]\\label{lem:fibre-invariance}\nLet $d\\ge 1$ and $n\\ge 1$.\nFor any $x\\in(\\Z/2^n\\Z)^d$ and any $z\\in(\\Z/2\\Z)^d$ one has\n\\[\n  Q_d(x+2^n z)\\equiv Q_d(x)\\pmod{2^{n+1}}.\n\\]\nConsequently, for any $a\\in\\Z$ and any residue class $\\bar x\\in(\\Z/2^n\\Z)^d$, the number of lifts $\\tilde x\\in(\\Z/2^{n+1}\\Z)^d$ of $\\bar x$ satisfying\n$Q_d(\\tilde x)\\equiv a\\pmod{2^{n+1}}$\nis either $0$ or the full fibre size $2^d$.\n\\end{lemma}\n\n\\begin{theorem}[Half-lift principle for sums of squares]\\label{thm:half-lift-principle}\nLet $d\\ge 1$ and set\n\\[\n  Q_d(x_1,\\dots,x_d):=x_1^2+\\cdots+x_d^2.\n\\]\nFor $n\\ge 1$ and $a\\in\\Z$ let\n\\[\n  N_{d,n}(a):=\\#\\{x\\in (\\Z/2^n\\Z)^d: Q_d(x)\\equiv a\\pmod{2^n}\\}.\n\\]\nAssume $n\\ge 3$ and $4\\nmid a$.\nThen\n\\[\n  N_{d,n+1}(a)=2^{d-1}\\,N_{d,n}(a).\n\\]\nEquivalently, among the residue classes modulo $2^n$ satisfying $Q_d\\equiv a$, exactly half admit lifts to level $2^{n+1}$; moreover, whenever a class lifts, \\emph{all} $2^d$ lifts in its fibre are solutions (cf.\\ Lemma~\\ref{lem:fibre-invariance}).\n\\end{theorem}", "post_theorem_intro_text_len": 4729, "post_theorem_intro_text": "\\begin{remark}[Scope of the half-lift involution]\\label{rem:half-lift-scope}\nThe involution argument proving Theorem~\\ref{thm:half-lift-principle} uses the specific diagonal form\n$Q_d=x_1^2+\\cdots+x_d^2$ and the primitivity hypothesis $4\\nmid a$.\nThe primitivity is essential in general: for example, in dimension $d=4$ one checks directly that\n\\(N_{4,4}(8)=1536\\) while \\(N_{4,3}(8)=128\\), so the ratio is $12\\ne 2^{d-1}=8$.\nOn a conceptual level, two features of the proof break down once these hypotheses are relaxed:\nif $4\\mid a$ then solutions modulo $2^n$ may have all coordinates even, so the first step (which extracts an odd coordinate to choose an involution direction) can fail; and for $n=2$ the quadratic error term in the expansion \\eqref{eq:expansion-d} no longer vanishes modulo $2^{n+1}$, so the translation $x\\mapsto x+2^{n-1}u$ need not toggle $Q_d(x)$ by a clean additive $2^n$.\nWe do not claim a dimension-free half-lift recursion for arbitrary diagonal unit coefficients (or for general dyadic lattices) without further reduction.\nIn the dyadic setting, the half-lift mechanism is used here only to analyse the even ternary lattice $L_3$ in \\eqref{eq:L3};\nrank-$1$ Type~I blocks are handled separately via explicit square-root counts (Appendix~\\ref{app:typeI}).\n\\end{remark}\n\nFor $d=3$, Theorem~\\ref{thm:half-lift-principle} combines with a simple $4$-adic descent and a base computation modulo $8$ to yield closed forms for the three-squares counts once one reaches the stable range $n\\ge 2k+3$ for $a=4^k a_0$ with $4\\nmid a_0$ (and hence an explicit formula for $\\alpha_2(t;L_3)$).\nMore generally, in the stable range the half-lift recursion has growth factor $2^{d-1}$, which is exactly the ``smooth'' Hensel factor $p^{d-1}$ specialised to $p=2$ (Remark~\\ref{rem:half-lift-hensel}).\nWe also record explicit formulas for the even rank-$2$ Jordan blocks (hyperbolic and anisotropic).\nAs a consequence we obtain explicit densities $\\alpha_2(t;B)$ for each basic dyadic block $B$ of Definition~\\ref{def:dyadic-base-blocks}.\n\n\\begin{theorem}[Prime-uniform density for the hyperbolic plane]\n\\label{thm:prime-uniform-h0}\nLet $p$ be a prime and let $e\\ge 0$.\nLet $H_0$ denote the rank-$2$ hyperbolic plane over $\\Z_p$ with bilinear form\n\\(\\langle (x,y),(x',y')\\rangle = xy'+x'y\\).\nOn the scaled lattice $p^eH_0$ consider the quadratic form\n\\[\n  q_e(v):=\\tfrac12\\langle v,v\\rangle = p^e xy\\in\\Z_p.\n\\]\nFor $m\\ge 1$ and $t\\in\\Z_p$ set\n\\[\n  r^{(q)}_{m}(t;p^eH_0)\n  :=\\#\\{(x,y)\\in(\\mathbb{Z}/p^m\\mathbb{Z})^2 : p^e xy\\equiv t\\!\\!\\pmod{p^m}\\},\n\\]\nand define the associated density (when the limit exists) by\n\\[\n  \\alpha_p^{(q)}(t;p^eH_0)\n  :=\\lim_{m\\to\\infty}p^{-m}\\,r^{(q)}_{m}(t;p^eH_0).\n\\]\nLet $t\\in\\Z_p\\setminus\\{0\\}$ and write $v:=v_p(t)$.\nThen\n\\[\n  \\alpha_p^{(q)}(t;p^eH_0)=\n  \\begin{cases}\n    0, & v<e,\\\\[2pt]\n    (v-e+1)(p-1)p^{e-1}, & v\\ge e.\n  \\end{cases}\n\\]\n\\smallskip\n\\noindent In particular, under the $q=\\langle\\cdot,\\cdot\\rangle/2$ normalisation, both the vanishing threshold and the density formula for the scaled hyperbolic plane are uniform for all primes $p$, including $p=2$.\n\\end{theorem}\n\nTheorem~\\ref{thm:prime-uniform-h0} is a formal consequence of the $Q$-normalised density formulas for the hyperbolic plane (Corollaries~\\ref{cor:density-hyp-scaled-odd} and~\\ref{cor:density-hyp-scaled}) together with the conversion Lemma~\\ref{lem:q-dictionary}; we give the short proof in \\S\\ref{sec:densities}.\n\n\\subsection*{Organisation}\nSection~\\ref{sec:prelim} fixes notation and recalls the dyadic Jordan decomposition and the basic dyadic blocks.\nSection~\\ref{sec:hyperbolic} treats the hyperbolic plane $H_0$ uniformly for all primes $p$ by explicit congruence counting, including closed forms for its stable-range counts and generating series.\nSection~\\ref{sec:anisotropic} treats the anisotropic even plane; here one can recast the counting problem as a norm-congruence in an unramified quadratic extension, where the unit locus admits a clean lifting recursion.\nSection~\\ref{sec:half-lift} proves the half-lift principle and derives closed forms for the anisotropic ternary lattice $L_3$.\nSection~\\ref{sec:densities} summarises the resulting density formulas.\nAppendix~\\ref{app:typeI} records the rank-$1$ (Type~I) dyadic block counts.\n\n\\subsection*{Computational checks}\n\nThe arguments in this paper are self-contained and elementary, and they do not rely on computation.\nAs an external consistency check, the arXiv source bundle includes ancillary files containing finite-level verifiers for the congruence counts and a Lean~4 formalisation of the half-lift involution (see \\texttt{anc/verify.py} and the directory \\texttt{anc/lean/}; build instructions are in \\texttt{anc/README\\_CODE.txt}).", "sketch": "The post-theorem introduction does not give a full step-by-step proof sketch, but it does describe the key mechanism and where it can fail. It says that the proof of Theorem~\\ref{thm:half-lift-principle} is an \\emph{“involution argument”} using the \\emph{“specific diagonal form $Q_d=x_1^2+\\cdots+x_d^2$”} together with the \\emph{“primitivity hypothesis $4\\nmid a$.”} Conceptually, it highlights two main ingredients/steps behind the involution:\n\n1. A first step that \\emph{“extracts an odd coordinate to choose an involution direction”}; this can fail when $4\\mid a$ because \\emph{“solutions modulo $2^n$ may have all coordinates even.”}\n\n2. A translation step \\emph{“$x\\mapsto x+2^{n-1}u$”} that should \\emph{“toggle $Q_d(x)$ by a clean additive $2^n$.”} This relies on the fact that for $n\\ge 3$ \\emph{“the quadratic error term in the expansion \\eqref{eq:expansion-d} … vanishes modulo $2^{n+1}$,”} whereas for $n=2$ it \\emph{“no longer vanishes modulo $2^{n+1}$,”} so the toggling may fail.", "expanded_sketch": "The post-theorem introduction does not give a full step-by-step proof sketch, but it does describe the key mechanism and where it can fail. It says that, in establishing the main theorem, the argument is an \\emph{“involution argument”} using the \\emph{“specific diagonal form $Q_d=x_1^2+\\cdots+x_d^2$”} together with the \\emph{“primitivity hypothesis $4\\nmid a$.”} Conceptually, it highlights two main ingredients/steps behind the involution:\n\n1. A first step that \\emph{“extracts an odd coordinate to choose an involution direction”}; this can fail when $4\\mid a$ because \\emph{“solutions modulo $2^n$ may have all coordinates even.”}\n\n2. A translation step \\emph{“$x\\mapsto x+2^{n-1}u$”} that should \\emph{“toggle $Q_d(x)$ by a clean additive $2^n$.”} This relies on the fact that for $n\\ge 3$ \\emph{“the quadratic error term in the expansion\n\\begin{equation}\\label{eq:expansion-d}\n  f(\\tau_u(x))\\equiv f(x)\\pmod{2^n},\\qquad\n  f(\\tau_u(x))\\equiv f(x)+2^n(x\\cdot u)\\pmod{2^{n+1}},\n\\end{equation}\n… vanishes modulo $2^{n+1}$,”} whereas for $n=2$ it \\emph{“no longer vanishes modulo $2^{n+1}$,”} so the toggling may fail.", "expanded_theorem": "[Half-lift principle for sums of squares]\\label{thm:half-lift-principle}\nLet $d\\ge 1$ and set\n\\[\n  Q_d(x_1,\\dots,x_d):=x_1^2+\\cdots+x_d^2.\n\\]\nFor $n\\ge 1$ and $a\\in\\mathbb{Z}$ let\n\\[\n  N_{d,n}(a):=\\#\\{x\\in (\\mathbb{Z}/2^n\\mathbb{Z})^d: Q_d(x)\\equiv a\\pmod{2^n}\\}.\n\\]\nAssume $n\\ge 3$ and $4\\nmid a$.\nThen\n\\[\n  N_{d,n+1}(a)=2^{d-1}\\,N_{d,n}(a).\n\\]\nEquivalently, among the residue classes modulo $2^n$ satisfying $Q_d\\equiv a$, exactly half admit lifts to level $2^{n+1}$; moreover, whenever a class lifts, \\emph{all} $2^d$ lifts in its fibre are solutions, by the following lemma.\n\n\\begin{lemma}[Fibre invariance for sums of squares]\\label{lem:fibre-invariance}\nLet $d\\ge 1$ and $n\\ge 1$.\nFor any $x\\in(\\Z/2^n\\Z)^d$ and any $z\\in(\\Z/2\\Z)^d$ one has\n\\[\n  Q_d(x+2^n z)\\equiv Q_d(x)\\pmod{2^{n+1}}.\n\\]\nConsequently, for any $a\\in\\Z$ and any residue class $\\bar x\\in(\\Z/2^n\\Z)^d$, the number of lifts $\\tilde x\\in(\\Z/2^{n+1}\\Z)^d$ of $\\bar x$ satisfying\n$Q_d(\\tilde x)\\equiv a\\pmod{2^{n+1}}$\nis either $0$ or the full fibre size $2^d$.\n\\end{lemma}", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let \\(d\\ge 1\\) and define the diagonal quadratic form\n\\[\nQ_d(x_1,\\dots,x_d)=x_1^2+\\cdots+x_d^2.\n\\]\nFor \\(n\\ge 1\\) and \\(a\\in\\mathbb Z\\), let\n\\[\nN_{d,n}(a):=\\#\\{x\\in(\\mathbb Z/2^n\\mathbb Z)^d: Q_d(x)\\equiv a\\pmod{2^n}\\}.\n\\]\nAssume \\(n\\ge 3\\) and \\(4\\nmid a\\). For a residue class \\(\\bar x\\in(\\mathbb Z/2^n\\mathbb Z)^d\\), its fibre at level \\(2^{n+1}\\) means the set of lifts \\(\\tilde x\\in(\\mathbb Z/2^{n+1}\\mathbb Z)^d\\) reducing to \\(\\bar x\\), which has size \\(2^d\\). Which quantitative statement holds?", "correct_choice": {"label": "A", "text": "One has\n\\[\nN_{d,n+1}(a)=2^{d-1}N_{d,n}(a).\n\\]\nEquivalently, among the classes \\(\\bar x\\in(\\mathbb Z/2^n\\mathbb Z)^d\\) with \\(Q_d(\\bar x)\\equiv a\\pmod{2^n}\\), exactly half admit lifts to level \\(2^{n+1}\\); and whenever such a class lifts, all \\(2^d\\) elements in its fibre are solutions modulo \\(2^{n+1}\\)."}, "choices": [{"label": "B", "text": "One has\n\\[\nN_{d,n+1}(a)=2^{d-1}N_{d,n}(a)\n\\]\nfor every \\(n\\ge 2\\) and every \\(a\\in\\mathbb Z\\) with \\(2\\nmid a\\). Equivalently, among the classes \\(\\bar x\\in(\\mathbb Z/2^n\\mathbb Z)^d\\) with \\(Q_d(\\bar x)\\equiv a\\pmod{2^n}\\), exactly half admit lifts to level \\(2^{n+1}\\); and whenever such a class lifts, all \\(2^d\\) elements in its fibre are solutions modulo \\(2^{n+1}\\)."}, {"label": "C", "text": "Whenever a class \\(\\bar x\\in(\\mathbb Z/2^n\\mathbb Z)^d\\) with \\(Q_d(\\bar x)\\equiv a\\pmod{2^n}\\) admits one lift to level \\(2^{n+1}\\), in fact all \\(2^d\\) elements in its fibre are solutions modulo \\(2^{n+1}\\)."}, {"label": "D", "text": "One has\n\\[\nN_{d,n+1}(a)=2^{d-1}N_{d,n}(a)\n\\]\nfor every \\(n\\ge 3\\) and every primitive target \\(a\\in\\mathbb Z\\). Equivalently, among the classes \\(\\bar x\\in(\\mathbb Z/2^n\\mathbb Z)^d\\) with \\(Q_d(\\bar x)\\equiv a\\pmod{2^n}\\), exactly half admit lifts to level \\(2^{n+1}\\); and whenever such a class lifts, all \\(2^d\\) elements in its fibre are solutions modulo \\(2^{n+1}\\)."}, {"label": "E", "text": "One has\n\\[\nN_{d,n+1}(a)=2^{d}N_{d,n}(a).\n\\]\nEquivalently, every class \\(\\bar x\\in(\\mathbb Z/2^n\\mathbb Z)^d\\) with \\(Q_d(\\bar x)\\equiv a\\pmod{2^n}\\) admits lifts to level \\(2^{n+1}\\), and whenever such a class lifts, all \\(2^d\\) elements in its fibre are solutions modulo \\(2^{n+1}\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "stable range n\\ge 3 replaced by n\\ge 2", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the exact-half orbit-counting conclusion and recursion, retaining only fibre invariance", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "hypothesis 4\\nmid a weakened to mere primitivity of a", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "replaced half of the classes lifting by all classes lifting", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the recursion or the 'exactly half lift' conclusion. It gives setup and definitions only, so the correct answer is not directly leaked."}, "TAS": {"score": 0, "justification": "The item is essentially asking for the exact theorem-level conclusion under the stated hypotheses. The correct option reads like a direct statement of the target result rather than a derived consequence from additional computation or context."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the choices include a weaker true statement and several close variants with altered quantifiers or lifting counts. However, the task is still mainly theorem recognition rather than substantial generation or derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic errors: wrong stability range, weakening to an inequality, confusing 'all lifts' with 'exactly one lift,' and overclaiming universal lifting."}, "total_score": 5, "overall_assessment": "A solid recognition-style MCQ with strong distractors and no direct answer leakage, but it is quite theorem-restatement-heavy and only moderately tests genuine generative reasoning."}}
{"id": "2602.21070v1", "paper_link": "http://arxiv.org/abs/2602.21070v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "[Half-lift principle for sums of squares]\\label{thm:half-lift-principle}\nLet $d\\ge 1$ and set\n\\[\n  Q_d(x_1,\\dots,x_d):=x_1^2+\\cdots+x_d^2.\n\\]\nFor $n\\ge 1$ and $a\\in\\mathbb{Z}$ let\n\\[\n  N_{d,n}(a):=\\#\\{x\\in (\\mathbb{Z}/2^n\\mathbb{Z})^d: Q_d(x)\\equiv a\\pmod{2^n}\\}.\n\\]\nAssume $n\\ge 3$ and $4\\nmid a$.\nThen\n\\[\n  N_{d,n+1}(a)=2^{d-1}\\,N_{d,n}(a).\n\\]\nEquivalently, among the residue classes modulo $2^n$ satisfying $Q_d\\equiv a$, exactly half admit lifts to level $2^{n+1}$; moreover, whenever a class lifts, \\emph{all} $2^d$ lifts in its fibre are solutions (cf.\\ Lemma~\\ref{lem:fibre-invariance}).", "start_pos": 10709, "end_pos": 11308, "label": "thm:half-lift-principle"}, "ref_dict": {"rem:half-lift-hensel": "\\begin{remark}\\label{rem:half-lift-hensel}\nFor $d=3$ one can improve Step~2 by choosing a single involution direction on each parity class (as in the original ``three-squares'' half-lift lemma).\nThe minimal-index partition above is slightly less symmetric but works uniformly for all $d\\ge 1$.\nIn particular, the growth factor $2^{d-1}$ coincides with the ``smooth'' Hensel factor $p^{d-1}$ specialised to $p=2$, even though the Jacobian of the bilinear quadratic form vanishes modulo $2$.\n\\end{remark}", "tab:block-densities": "\\label{tab:block-densities}\n\\end{table}\n\n\\begin{corollary}[Dyadic vanishing thresholds in the $Q$- and $q$-normalisations]\n\\label{cor:dyadic-vanishing-thresholds}\nLet $p=2$.\nLet $L$ be an \\emph{even}", "eq:L3": "\\begin{equation}\\label{eq:L3}\n  L_3 := \\langle 2\\rangle^{\\oplus 3}\\qquad\\text{with }Q(x)=2(x_1^2+x_2^2+x_3^2)\n\\end{equation}", "lem:q-dictionary": "\\begin{lemma}[Dictionary with the $q=\\langle\\cdot,\\cdot\\rangle/2$ convention]\\label{lem:q-dictionary}\nAssume $L$ is \\emph{even}, i.e.\\ $Q(L)\\subset 2\\Ztwo$, and set $q(v):=Q(v)/2\\in\\Ztwo$.\nFor $m\\ge 1$ define\n\\[\n  r_m^{(q)}(t;L):=\\#\\{v\\in L/2^mL : q(v)\\equiv t\\!\\!\\pmod{2^m}\\}.\n\\]\nThen for $t\\in\\Ztwo$ and $n:=\\mathrm{rank}_{\\Ztwo}(L)$:\n\\begin{altenumerate}\n\\item If $t$ is odd, then $r_m(t;L)=0$ for all $m\\ge 1$.\n\\item If $t=2t'$ is even, then for every $m\\ge 2$ one has\n\\[\n  r_m(t;L)=2^{n}\\,r_{m-1}^{(q)}(t';L).\n\\]\nIn particular, $r_1(2t';L)=2^n$.\n\\end{altenumerate}\nConsequently, whenever the limits exist,\n\\[\\alpha_2(2t';L)=2\\,\\alpha_2^{(q)}(t';L),\\qquad\n  \\alpha_2^{(q)}(t';L):=\\lim_{m\\to\\infty}2^{-m(n-1)}\\,r_m^{(q)}(t';L).\\]\n\\end{lemma}", "eq:2xy": "\\begin{equation}\\label{eq:2xy}\n  2xy\\equiv s\\pmod{p^m}.\n\\end{equation}", "app:typeI": "\\label{app:typeI}\n\nWe record the square-root counts modulo $2^m$ that enter the rank-$1$ dyadic blocks.\nThese counts are classical, but we include them so that the evaluation of the base blocks can be", "thm:prime-uniform-h0": "\\begin{theorem}[Prime-uniform density for the hyperbolic plane]\n\\label{thm:prime-uniform-h0}\nLet $p$ be a prime and let $e\\ge 0$.\nLet $H_0$ denote the rank-$2$ hyperbolic plane over $\\Zp$ with bilinear form\n\\(\\langle (x,y),(x',y')\\rangle = xy'+x'y\\).\nOn the scaled lattice $p^eH_0$ consider the quadratic form\n\\[\n  q_e(v):=\\tfrac12\\langle v,v\\rangle = p^e xy\\in\\Zp.\n\\]\nFor $m\\ge 1$ and $t\\in\\Zp$ set\n\\[\n  r^{(q)}_{m}(t;p^eH_0)\n  :=\\#\\{(x,y)\\in(\\Z/p^m\\Z)^2 : p^e xy\\equiv t\\!\\!\\pmod{p^m}\\},\n\\]\nand define the associated density (when the limit exists) by\n\\[\n  \\alpha_p^{(q)}(t;p^eH_0)\n  :=\\lim_{m\\to\\infty}p^{-m}\\,r^{(q)}_{m}(t;p^eH_0).\n\\]\nLet $t\\in\\Zp\\setminus\\{0\\}$ and write $v:=v_p(t)$.\nThen\n\\[\n  \\alpha_p^{(q)}(t;p^eH_0)=\n  \\begin{cases}\n    0, & v<e,\\\\[2pt]\n    (v-e+1)(p-1)p^{e-1}, & v\\ge e.\n  \\end{cases}\n\\]\n\\smallskip\n\\noindent In particular, under the $q=\\langle\\cdot,\\cdot\\rangle/2$ normalisation, both the vanishing threshold and the density formula for the scaled hyperbolic plane are uniform for all primes $p$, including $p=2$.\n\\end{theorem}", "lem:H1": "\\begin{lemma}[Counts for the anisotropic even plane]\\label{lem:H1}\nLet $m\\ge 1$ and $t\\in\\Ztwo$.\nWrite $v:=v_2(t)$ with the convention $v=\\infty$ if $t=0$.\nThen:\n\\begin{altenumeratealpha}\n\\item If $t\\ne 0$ and $v$ is even, then $A_m(t)=0$ for all $m>v$.\n\\item If $t\\ne 0$ and $v$ is odd, then $A_m(t)=3\\cdot 2^m$ for all $m>v$.\n\\item If $t\\equiv 0\\pmod{2^m}$, then $A_m(t)=4^{\\lceil m/2\\rceil}$.\n\\end{altenumeratealpha}\n\\end{lemma}", "cor:density-hyp-scaled": "\\begin{corollary}[Scaled hyperbolic block]\\label{cor:density-hyp-scaled}\nLet $e\\ge 0$ and let $L=2^eH_0$, so that $Q(x,y)=2^{e+1}xy$.\nLet $t\\in\\Ztwo\\setminus\\{0\\}$ and set $v:=v_2(t)$.\nIf $v<e+1$ then $\\alpha_2(t;L)=0$.\nIf $v\\ge e+1$ then\n\\[\n  \\alpha_2(t;L)=2^e\\,(v-e).\n\\]\n\\end{corollary}", "eq:expansion-d": "\\begin{equation}\\label{eq:expansion-d}\n  f(\\tau_u(x))\\equiv f(x)\\pmod{2^n},\\qquad\n  f(\\tau_u(x))\\equiv f(x)+2^n(x\\cdot u)\\pmod{2^{n+1}},\n\\end{equation}", "thm:half-lift-principle": "\\begin{theorem}[Half-lift principle for sums of squares]\\label{thm:half-lift-principle}\nLet $d\\ge 1$ and set\n\\[\n  Q_d(x_1,\\dots,x_d):=x_1^2+\\cdots+x_d^2.\n\\]\nFor $n\\ge 1$ and $a\\in\\Z$ let\n\\[\n  N_{d,n}(a):=\\#\\{x\\in (\\Z/2^n\\Z)^d: Q_d(x)\\equiv a\\pmod{2^n}\\}.\n\\]\nAssume $n\\ge 3$ and $4\\nmid a$.\nThen\n\\[\n  N_{d,n+1}(a)=2^{d-1}\\,N_{d,n}(a).\n\\]\nEquivalently, among the residue classes modulo $2^n$ satisfying $Q_d\\equiv a$, exactly half admit lifts to level $2^{n+1}$; moreover, whenever a class lifts, \\emph{all} $2^d$ lifts in its fibre are solutions (cf.\\ Lemma~\\ref{lem:fibre-invariance}).\n\\end{theorem}", "lem:fibre-invariance": "\\begin{lemma}[Fibre invariance for sums of squares]\\label{lem:fibre-invariance}\nLet $d\\ge 1$ and $n\\ge 1$.\nFor any $x\\in(\\Z/2^n\\Z)^d$ and any $z\\in(\\Z/2\\Z)^d$ one has\n\\[\n  Q_d(x+2^n z)\\equiv Q_d(x)\\pmod{2^{n+1}}.\n\\]\nConsequently, for any $a\\in\\Z$ and any residue class $\\bar x\\in(\\Z/2^n\\Z)^d$, the number of lifts $\\tilde x\\in(\\Z/2^{n+1}\\Z)^d$ of $\\bar x$ satisfying\n$Q_d(\\tilde x)\\equiv a\\pmod{2^{n+1}}$\nis either $0$ or the full fibre size $2^d$.\n\\end{lemma}", "def:dyadic-base-blocks": "\\begin{definition}[Basic dyadic blocks]\\label{def:dyadic-base-blocks}\nFor the explicit computations in this paper we single out the following quadratic $\\Ztwo$-lattices; we refer to them as the \\emph{basic dyadic blocks}:\n\\begin{altenumerate}\n\\item \\textbf{Type~I (rank $1$).} The lattice $\\langle 2^a u\\rangle$ with Gram matrix $[2^a u]$, where $a\\ge 0$ and $u\\in\\Ztwo^\\times$.\n\\item \\textbf{Type~II (rank $2$, even).} The lattice $2^a H_\\varepsilon$ with Gram matrix $2^a H_\\varepsilon$, where $a\\ge 0$ and\n\\[\n  H_0:=\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}\\qquad\\text{and}\\qquad\n  H_1:=\\begin{pmatrix}2&1\\\\1&2\\end{pmatrix}.\n\\]\n\\end{altenumerate}\nOnly the square class of the unit $u$ (equivalently $u\\bmod 8$) will matter for solvability in the rank-one case.\n\\end{definition}", "lem:2xy-2": "\\begin{lemma}[Counting $2xy\\equiv s\\pmod{2^m}$ at $p=2$]\\label{lem:2xy-2}\nLet $m\\ge 1$ and $s\\in\\Z/2^m\\Z$.\nWrite $v:=v_2(s)$ with the convention $v=m$ if $s\\equiv 0\\pmod{2^m}$.\nThen:\n\\begin{altenumerate}\n\\item If $v=0$ (i.e.\\ $s$ is odd), then $M_m(s)=0$.\n\\item If $1\\le v<m$, then $M_m(s)=v\\cdot 2^m$.\n\\item If $v=m$ (i.e.\\ $s\\equiv 0\\pmod{2^m}$), then $M_m(0)=(m+1)2^m$.\n\\end{altenumerate}\n\\end{lemma}", "cor:density-hyp-scaled-odd": "\\begin{corollary}[Scaled hyperbolic plane at odd primes]\\label{cor:density-hyp-scaled-odd}\nLet $p$ be an odd prime and let $e\\ge 0$.\nLet $L=p^eH_0$, so that $Q(x,y)=2p^e xy$.\nLet $t\\in\\Zp\\setminus\\{0\\}$ and set $v:=v_p(t)$.\nIf $v<e$ then $\\alpha_p(t;L)=0$.\nIf $v\\ge e$, then\n\\[\n  \\alpha_p(t;L)=(v-e+1)(p-1)p^{e-1}.\n\\]\n\\end{corollary}", "prop:jordan": "\\begin{proposition}[Jordan decomposition]\\label{prop:jordan}\nLet $L$ be a nondegenerate quadratic lattice over $\\Ztwo$.\nThen there exists an orthogonal decomposition\n\\[\n  L \\;\\cong\\; \\perp_{a\\ge 0} 2^a L_a,\n\\]\nwith each $L_a$ a unimodular quadratic $\\Ztwo$-lattice (possibly $0$).\nMoreover, the multiset of integers $a$ together with the multiplicities $\\mathrm{rank}(L_a)$ (equivalently, the Jordan \\emph{scales} and \\emph{ranks}) is uniquely determined by $L$.\n\\end{proposition}", "lem:2xy-odd": "\\begin{lemma}[Counting $2xy\\equiv s\\pmod{p^m}$ at odd primes]\\label{lem:2xy-odd}\nLet $p$ be an odd prime, $m\\ge 1$, and $s\\in\\Z/p^m\\Z$.\nWrite $v:=v_p(s)$ with the convention $v=m$ if $s\\equiv 0\\pmod{p^m}$.\nThen:\n\\begin{altenumerate}\n\\item If $0\\le v<m$, then $M_{p,m}(s)=(v+1)(p-1)p^{m-1}$.\n\\item If $v=m$ (i.e.\\ $s\\equiv 0\\pmod{p^m}$), then $M_{p,m}(0)=(m(p-1)+p)p^{m-1}$.\n\\end{altenumerate}\n\\end{lemma}", "prop:orthogonal-convolution": "\\begin{proposition}[Orthogonal-sum convolution]\\label{prop:orthogonal-convolution}\nLet $L$ and $M$ be quadratic $\\Zp$-lattices, and equip $L\\oplus M$ with the orthogonal direct-sum quadratic form $Q_{L\\oplus M}(v,w):=Q_L(v)+Q_M(w)$.\nThen for every $m\\ge 1$ and $t\\in\\Zp$ one has the finite-level identity\n\\[\n  r_m(t;L\\oplus M)\n  \\,=\\,\n  \\sum_{s\\in \\Z/p^m\\Z} r_m(s;L)\\,r_m(t-s;M).\n\\]\n\\end{proposition}"}, "pre_theorem_intro_text_len": 6488, "pre_theorem_intro_text": "\\label{sec:intro}\n\nLocal representation densities of quadratic lattices are the local factors in many global counting problems.\nThey appear in the Siegel series and Siegel's mass formula, and they control Fourier coefficients of theta series and Eisenstein series.\nIn concrete arithmetic applications one is often forced to evaluate these local factors for specific lattices, and the dyadic prime is typically the main obstacle.\n\nAt odd primes, once the relevant solution locus is smooth modulo $p$, Hensel lifting implies that reduction maps between solution sets have constant fibre size $p^{n-1}$, and the density stabilises quickly.\nAt the dyadic prime this mechanism can break down even for very simple lattices.\nIn the bilinear-lattice convention $Q(v)=\\langle v,v\\rangle$, the linearisation of $Q-t$ at a solution $\\bar v$ is the functional $h\\mapsto 2\\langle \\bar v,h\\rangle$, which vanishes modulo~$2$.\nThus the standard Jacobian criterion yields no lifting information.\n\nEvery nondegenerate quadratic lattice over $\\Z_2$ admits a Jordan decomposition into scaled unimodular constituents (Proposition~\\ref{prop:jordan}).\nAccordingly, explicit formulas for the basic dyadic blocks are a natural starting point for dyadic computations.\nIn this paper we determine the finite-level representation counts and local densities for the basic dyadic blocks of Definition~\\ref{def:dyadic-base-blocks} (rank-$1$ Type~I blocks and the even planes $2^aH_\\varepsilon$), together with the anisotropic ternary lattice $L_3$.\n\nThe main new input is a very simple involution argument, the \\emph{half-lift}.\nFor diagonal sums of squares $Q_d=x_1^2+\\cdots+x_d^2$ and a primitive target $a$ with $4\\nmid a$, translation by $2^{n-1}u$ in a primitive direction produces a free involution on the solution set modulo $2^n$ whose orbits control lifts to level $2^{n+1}$.\nIn each orbit exactly one residue class lifts, and whenever a class lifts, all lifts in its fibre are solutions.\nThis yields the stable recursion $N_{d,n+1}(a)=2^{d-1}N_{d,n}(a)$ for $n\\ge 3$ (Theorem~\\ref{thm:half-lift-principle}).\nFor $d=3$ it leads to closed forms for the three-squares congruence counts and hence an explicit formula for $\\alpha_2(t;L_3)$.\n\n\\smallskip\nWe fix the conventions for the finite-level representation counts $r_m(t;L)$ and the local density $\\alpha_p(t;L)$ in \\S\\ref{sec:prelim}.\nAt $p=2$ and for even lattices there is a second common normalisation $q:=Q/2$; we work with $Q$ unless explicitly indicated, and Lemma~\\ref{lem:q-dictionary} gives the precise conversion.\n\nGeneral formulas for the dyadic Siegel series are known, expressed for instance in terms of Gross--Keating invariants or group-scheme methods (see for example\n\\cite{grossKeating1993,choYamauchi2020,ganYu2000,ikedaKatsurada2018,kitaokaBook}).\nThe densities computed here agree with these formulas (cf.\\ Yang's explicit expressions~\\cite{yang1998}), but this serves only as an \\emph{external consistency check} and is not used as an input to any argument.\n\nAt odd primes, the hyperbolic-plane computations of \\S\\ref{sec:hyperbolic} are classical; we include a self-contained counting argument to keep the exposition uniform and to make the prime-uniform statement of Theorem~\\ref{thm:prime-uniform-h0} transparent.\nThe genuinely new feature of this paper is dyadic: the half-lift involution of Theorem~\\ref{thm:half-lift-principle} and the resulting elementary derivation of the dyadic block densities without appealing to the full Siegel-series machinery.\n\nAll computations in the body of the paper are organised around the same elementary theme:\ndescribe the fibres of the reduction maps $L/2^{m+1}L\\to L/2^mL$ on the relevant solution sets.\nConcretely,\n\\begin{altenumerate}\n\\item for the split even plane $H_0$ this reduces to valuation-stratified counting for the bilinear congruence $2xy\\equiv t$ (see Lemmas~\\ref{lem:2xy-odd} and~\\ref{lem:2xy-2});\n\\item for the anisotropic plane $H_1$ it becomes a lifting recursion for a norm congruence in the unramified quadratic extension of~$\\Q_2$ (see Lemma~\\ref{lem:H1});\n\\item for the anisotropic ternary block $L_3$ one needs an additional involution (the half-lift) because the naive dyadic linearisation carries no information (see Theorem~\\ref{thm:half-lift-principle}).\n\\end{altenumerate}\n\nThis paper focuses on the basic dyadic blocks (Definition~\\ref{def:dyadic-base-blocks}) and the anisotropic ternary lattice $L_3$.\nOnce the basic block counts are known, representation counts for orthogonal sums reduce to finite convolutions (Proposition~\\ref{prop:orthogonal-convolution}).\nWe do not attempt closed forms for general orthogonal sums.\n\n\\smallskip\n\\noindent\nFor quick reference, Table~\\ref{tab:block-densities} in \\S\\ref{sec:densities} lists the final density values.\nReaders primarily interested in the dyadic novelty may start with \\S\\ref{sec:half-lift}.\n\\subsection{Main results}\n\nLet $p$ be a prime.\nSection~\\ref{sec:hyperbolic} treats the hyperbolic plane $H_0$ over $\\Z_p$ for every $p$ and gives explicit formulas for the congruence counts $M_{p,m}(s)$ in~\\eqref{eq:2xy}, together with their generating series and stable-range behaviour.\nSection~\\ref{sec:densities} records the resulting density formulas for $\\alpha_p(t;H_0)$ and the basic dyadic blocks.\nIn particular, after passing to the standard normalisation $q=\\langle\\cdot,\\cdot\\rangle/2$ at $p=2$ (Lemma~\\ref{lem:q-dictionary}), the density of $H_0$ and its scalings admits a single prime-uniform expression (Theorem~\\ref{thm:prime-uniform-h0}).\nThe remaining sections specialise to $p=2$ and evaluate the basic dyadic blocks.\n\nWe therefore recall the dyadic Jordan decomposition (Proposition~\\ref{prop:jordan}) and the basic dyadic blocks (Definition~\\ref{def:dyadic-base-blocks}) that will be treated explicitly.\nIn addition to the rank-$2$ even blocks, the rank-$3$ lattice\n\\begin{equation}\\label{eq:L3}\n  L_3 := \\langle 2\\rangle^{\\oplus 3}\\qquad\\text{with }Q(x)=2(x_1^2+x_2^2+x_3^2)\n\\end{equation}\nplays a distinguished role: it is the simplest anisotropic even ternary lattice over $\\Z_2$, and its representation counts amount to the classical three-squares congruence problem.\nIndeed, if $x_1^2+x_2^2+x_3^2=0$ in $\\Z_2$, then reducing modulo $8$ forces $x_1,x_2,x_3$ to be even, and iterating gives $x_1=x_2=x_3=0$; hence $L_3$ is anisotropic over $\\Q_2$.\n\nOur first main theorem is a lifting principle for diagonal sums of squares in arbitrary dimension, under the mild primitivity hypothesis $4\\nmid a$.", "context": "The main new input is a very simple involution argument, the \\emph{half-lift}.\nFor diagonal sums of squares $Q_d=x_1^2+\\cdots+x_d^2$ and a primitive target $a$ with $4\\nmid a$, translation by $2^{n-1}u$ in a primitive direction produces a free involution on the solution set modulo $2^n$ whose orbits control lifts to level $2^{n+1}$.\nIn each orbit exactly one residue class lifts, and whenever a class lifts, all lifts in its fibre are solutions.\nThis yields the stable recursion $N_{d,n+1}(a)=2^{d-1}N_{d,n}(a)$ for $n\\ge 3$ (Theorem~\\ref{thm:half-lift-principle}).\nFor $d=3$ it leads to closed forms for the three-squares congruence counts and hence an explicit formula for $\\alpha_2(t;L_3)$.\n\n\\smallskip\nWe fix the conventions for the finite-level representation counts $r_m(t;L)$ and the local density $\\alpha_p(t;L)$ in \\S\\ref{sec:prelim}.\nAt $p=2$ and for even lattices there is a second common normalisation $q:=Q/2$; we work with $Q$ unless explicitly indicated, and Lemma~\\ref{lem:q-dictionary} gives the precise conversion.\n\nAll computations in the body of the paper are organised around the same elementary theme:\ndescribe the fibres of the reduction maps $L/2^{m+1}L\\to L/2^mL$ on the relevant solution sets.\nConcretely,\n\\begin{altenumerate}\n\\item for the split even plane $H_0$ this reduces to valuation-stratified counting for the bilinear congruence $2xy\\equiv t$ (see Lemmas~\\ref{lem:2xy-odd} and~\\ref{lem:2xy-2});\n\\item for the anisotropic plane $H_1$ it becomes a lifting recursion for a norm congruence in the unramified quadratic extension of~$\\Q_2$ (see Lemma~\\ref{lem:H1});\n\\item for the anisotropic ternary block $L_3$ one needs an additional involution (the half-lift) because the naive dyadic linearisation carries no information (see Theorem~\\ref{thm:half-lift-principle}).\n\\end{altenumerate}\n\nLet $p$ be a prime.\nSection~\\ref{sec:hyperbolic} treats the hyperbolic plane $H_0$ over $\\Z_p$ for every $p$ and gives explicit formulas for the congruence counts $M_{p,m}(s)$ in~\\eqref{eq:2xy}, together with their generating series and stable-range behaviour.\nSection~\\ref{sec:densities} records the resulting density formulas for $\\alpha_p(t;H_0)$ and the basic dyadic blocks.\nIn particular, after passing to the standard normalisation $q=\\langle\\cdot,\\cdot\\rangle/2$ at $p=2$ (Lemma~\\ref{lem:q-dictionary}), the density of $H_0$ and its scalings admits a single prime-uniform expression (Theorem~\\ref{thm:prime-uniform-h0}).\nThe remaining sections specialise to $p=2$ and evaluate the basic dyadic blocks.\n\nWe therefore recall the dyadic Jordan decomposition (Proposition~\\ref{prop:jordan}) and the basic dyadic blocks (Definition~\\ref{def:dyadic-base-blocks}) that will be treated explicitly.\nIn addition to the rank-$2$ even blocks, the rank-$3$ lattice\n\\begin{equation}\\label{eq:L3}\n L_3 := \\langle 2\\rangle^{\\oplus 3}\\qquad\\text{with }Q(x)=2(x_1^2+x_2^2+x_3^2)\n\\end{equation}\nplays a distinguished role: it is the simplest anisotropic even ternary lattice over $\\Z_2$, and its representation counts amount to the classical three-squares congruence problem.\nIndeed, if $x_1^2+x_2^2+x_3^2=0$ in $\\Z_2$, then reducing modulo $8$ forces $x_1,x_2,x_3$ to be even, and iterating gives $x_1=x_2=x_3=0$; hence $L_3$ is anisotropic over $\\Q_2$.\n\nOur first main theorem is a lifting principle for diagonal sums of squares in arbitrary dimension, under the mild primitivity hypothesis $4\\nmid a$.", "full_context": "The main new input is a very simple involution argument, the \\emph{half-lift}.\nFor diagonal sums of squares $Q_d=x_1^2+\\cdots+x_d^2$ and a primitive target $a$ with $4\\nmid a$, translation by $2^{n-1}u$ in a primitive direction produces a free involution on the solution set modulo $2^n$ whose orbits control lifts to level $2^{n+1}$.\nIn each orbit exactly one residue class lifts, and whenever a class lifts, all lifts in its fibre are solutions.\nThis yields the stable recursion $N_{d,n+1}(a)=2^{d-1}N_{d,n}(a)$ for $n\\ge 3$ (Theorem~\\ref{thm:half-lift-principle}).\nFor $d=3$ it leads to closed forms for the three-squares congruence counts and hence an explicit formula for $\\alpha_2(t;L_3)$.\n\n\\smallskip\nWe fix the conventions for the finite-level representation counts $r_m(t;L)$ and the local density $\\alpha_p(t;L)$ in \\S\\ref{sec:prelim}.\nAt $p=2$ and for even lattices there is a second common normalisation $q:=Q/2$; we work with $Q$ unless explicitly indicated, and Lemma~\\ref{lem:q-dictionary} gives the precise conversion.\n\nAll computations in the body of the paper are organised around the same elementary theme:\ndescribe the fibres of the reduction maps $L/2^{m+1}L\\to L/2^mL$ on the relevant solution sets.\nConcretely,\n\\begin{altenumerate}\n\\item for the split even plane $H_0$ this reduces to valuation-stratified counting for the bilinear congruence $2xy\\equiv t$ (see Lemmas~\\ref{lem:2xy-odd} and~\\ref{lem:2xy-2});\n\\item for the anisotropic plane $H_1$ it becomes a lifting recursion for a norm congruence in the unramified quadratic extension of~$\\Q_2$ (see Lemma~\\ref{lem:H1});\n\\item for the anisotropic ternary block $L_3$ one needs an additional involution (the half-lift) because the naive dyadic linearisation carries no information (see Theorem~\\ref{thm:half-lift-principle}).\n\\end{altenumerate}\n\nLet $p$ be a prime.\nSection~\\ref{sec:hyperbolic} treats the hyperbolic plane $H_0$ over $\\Z_p$ for every $p$ and gives explicit formulas for the congruence counts $M_{p,m}(s)$ in~\\eqref{eq:2xy}, together with their generating series and stable-range behaviour.\nSection~\\ref{sec:densities} records the resulting density formulas for $\\alpha_p(t;H_0)$ and the basic dyadic blocks.\nIn particular, after passing to the standard normalisation $q=\\langle\\cdot,\\cdot\\rangle/2$ at $p=2$ (Lemma~\\ref{lem:q-dictionary}), the density of $H_0$ and its scalings admits a single prime-uniform expression (Theorem~\\ref{thm:prime-uniform-h0}).\nThe remaining sections specialise to $p=2$ and evaluate the basic dyadic blocks.\n\nWe therefore recall the dyadic Jordan decomposition (Proposition~\\ref{prop:jordan}) and the basic dyadic blocks (Definition~\\ref{def:dyadic-base-blocks}) that will be treated explicitly.\nIn addition to the rank-$2$ even blocks, the rank-$3$ lattice\n\\begin{equation}\\label{eq:L3}\n L_3 := \\langle 2\\rangle^{\\oplus 3}\\qquad\\text{with }Q(x)=2(x_1^2+x_2^2+x_3^2)\n\\end{equation}\nplays a distinguished role: it is the simplest anisotropic even ternary lattice over $\\Z_2$, and its representation counts amount to the classical three-squares congruence problem.\nIndeed, if $x_1^2+x_2^2+x_3^2=0$ in $\\Z_2$, then reducing modulo $8$ forces $x_1,x_2,x_3$ to be even, and iterating gives $x_1=x_2=x_3=0$; hence $L_3$ is anisotropic over $\\Q_2$.\n\nOur first main theorem is a lifting principle for diagonal sums of squares in arbitrary dimension, under the mild primitivity hypothesis $4\\nmid a$.\n\nThe main new input is a very simple involution argument, the \\emph{half-lift}.\nFor diagonal sums of squares $Q_d=x_1^2+\\cdots+x_d^2$ and a primitive target $a$ with $4\\nmid a$, translation by $2^{n-1}u$ in a primitive direction produces a free involution on the solution set modulo $2^n$ whose orbits control lifts to level $2^{n+1}$.\nIn each orbit exactly one residue class lifts, and whenever a class lifts, all lifts in its fibre are solutions.\nThis yields the stable recursion $N_{d,n+1}(a)=2^{d-1}N_{d,n}(a)$ for $n\\ge 3$ (Theorem~\\ref{thm:half-lift-principle}).\nFor $d=3$ it leads to closed forms for the three-squares congruence counts and hence an explicit formula for $\\alpha_2(t;L_3)$.\n\nOur first main theorem is a lifting principle for diagonal sums of squares in arbitrary dimension, under the mild primitivity hypothesis $4\\nmid a$.\n\n\\begin{remark}[Scope of the half-lift involution]\\label{rem:half-lift-scope}\nThe involution argument proving Theorem~\\ref{thm:half-lift-principle} uses the specific diagonal form\n$Q_d=x_1^2+\\cdots+x_d^2$ and the primitivity hypothesis $4\\nmid a$.\nThe primitivity is essential in general: for example, in dimension $d=4$ one checks directly that\n\\(N_{4,4}(8)=1536\\) while \\(N_{4,3}(8)=128\\), so the ratio is $12\\ne 2^{d-1}=8$.\nOn a conceptual level, two features of the proof break down once these hypotheses are relaxed:\nif $4\\mid a$ then solutions modulo $2^n$ may have all coordinates even, so the first step (which extracts an odd coordinate to choose an involution direction) can fail; and for $n=2$ the quadratic error term in the expansion \\eqref{eq:expansion-d} no longer vanishes modulo $2^{n+1}$, so the translation $x\\mapsto x+2^{n-1}u$ need not toggle $Q_d(x)$ by a clean additive $2^n$.\nWe do not claim a dimension-free half-lift recursion for arbitrary diagonal unit coefficients (or for general dyadic lattices) without further reduction.\nIn the dyadic setting, the half-lift mechanism is used here only to analyse the even ternary lattice $L_3$ in \\eqref{eq:L3};\nrank-$1$ Type~I blocks are handled separately via explicit square-root counts (Appendix~\\ref{app:typeI}).\n\\end{remark}\n\n\\begin{lemma}[Dictionary with the $q=\\langle\\cdot,\\cdot\\rangle/2$ convention]\\label{lem:q-dictionary}\nAssume $L$ is \\emph{even}, i.e.\\ $Q(L)\\subset 2\\Ztwo$, and set $q(v):=Q(v)/2\\in\\Ztwo$.\nFor $m\\ge 1$ define\n\\[\n r_m^{(q)}(t;L):=\\#\\{v\\in L/2^mL : q(v)\\equiv t\\!\\!\\pmod{2^m}\\}.\n\\]\nThen for $t\\in\\Ztwo$ and $n:=\\mathrm{rank}_{\\Ztwo}(L)$:\n\\begin{altenumerate}\n\\item If $t$ is odd, then $r_m(t;L)=0$ for all $m\\ge 1$.\n\\item If $t=2t'$ is even, then for every $m\\ge 2$ one has\n\\[\n r_m(t;L)=2^{n}\\,r_{m-1}^{(q)}(t';L).\n\\]\nIn particular, $r_1(2t';L)=2^n$.\n\\end{altenumerate}\nConsequently, whenever the limits exist,\n\\[\\alpha_2(2t';L)=2\\,\\alpha_2^{(q)}(t';L),\\qquad\n \\alpha_2^{(q)}(t';L):=\\lim_{m\\to\\infty}2^{-m(n-1)}\\,r_m^{(q)}(t';L).\\]\n\\end{lemma}\n\n\\smallskip\n\\noindent\\emph{Case $0\\le v<m$.}\nStratify by $i:=v_p(x)$.\nFor each $0\\le i\\le v$, the number of residue classes $x\\bmod p^m$ with $v_p(x)=i$ is $(p-1)p^{m-i-1}$.\nWrite $x=p^i x'$ with $x'\\in(\\Z/p^{m-i}\\Z)^\\times$.\nThen $xy\\equiv t\\pmod{p^m}$ becomes\n\\[\n x'\\,y\\equiv p^{v-i}t'\\pmod{p^{m-i}},\\qquad t':=t/p^v\\in(\\Z/p^{m-v}\\Z)^\\times.\n\\]\nAs $x'$ is a unit modulo $p^{m-i}$, there is a unique solution $y_0\\bmod p^{m-i}$.\nIt has exactly $p^i$ lifts modulo $p^m$, so for each fixed $i$ the number of pairs with $v_p(x)=i$ is\n\\[(p-1)p^{m-i-1}\\cdot p^i=(p-1)p^{m-1}.\n\\]\nSumming over $i=0,1,\\dots,v$ yields $M_{p,m}(s)=(v+1)(p-1)p^{m-1}$.\n\n\\begin{lemma}[Fibre invariance for sums of squares]\\label{lem:fibre-invariance}\nLet $d\\ge 1$ and $n\\ge 1$.\nFor any $x\\in(\\Z/2^n\\Z)^d$ and any $z\\in(\\Z/2\\Z)^d$ one has\n\\[\n Q_d(x+2^n z)\\equiv Q_d(x)\\pmod{2^{n+1}}.\n\\]\nConsequently, for any $a\\in\\Z$ and any residue class $\\bar x\\in(\\Z/2^n\\Z)^d$, the number of lifts $\\tilde x\\in(\\Z/2^{n+1}\\Z)^d$ of $\\bar x$ satisfying\n$Q_d(\\tilde x)\\equiv a\\pmod{2^{n+1}}$\nis either $0$ or the full fibre size $2^d$.\n\\end{lemma}\n\nLet\n$S_n(a):=\\{x\\in(\\Z/2^n\\Z)^d: f(x)\\equiv 0\\pmod{2^n}\\}$,\nso that $\\#S_n(a)=N_{d,n}(a)$.\nBy Lemma~\\ref{lem:fibre-invariance}, for each $\\bar x\\in(\\Z/2^n\\Z)^d$ the congruence condition modulo $2^{n+1}$ is constant across the entire fibre of its lifts.\nIn particular, if we view $f(\\bar x)\\bmod 2^{n+1}$ as the residue obtained by evaluating $f$ on any lift of $\\bar x$ to $(\\Z/2^{n+1}\\Z)^d$ (well-defined by the lemma), then\n\\[\n N_{d,n+1}(a)=2^d\\cdot\\#\\{\\bar x\\in S_n(a): f(\\bar x)\\equiv 0\\pmod{2^{n+1}}\\}.\n\\]\nWe show the subset on the right has size exactly $\\tfrac12\\#S_n(a)$.\n\n\\smallskip\n\\noindent\\emph{(ii)} Assume $b\\not\\equiv 0\\pmod{2^m}$ and write $b=2^{2j}c$ with $j\\ge 0$, $2j<m$, and $c$ odd.\nIf $x^2\\equiv 2^{2j}c\\pmod{2^m}$, then necessarily $2^j\\mid x$.\nWrite $x=2^j y$.\nDividing by $2^{2j}$ reduces the congruence to\n\\begin{equation}\\label{eq:sqroot-reduced}\n y^2\\equiv c\\pmod{2^{k}},\\qquad k:=m-2j\\ge 1.\n\\end{equation}\nThe condition \\eqref{eq:sqroot-reduced} depends only on the class of $y$ modulo $2^k$.\nConversely, each residue class $\\bar y\\in\\Z/2^k\\Z$ has exactly $2^j$ lifts to a class modulo $2^{m-j}$.\nHence the number of solutions $x\\bmod 2^m$ equals\n\\[\n 2^j\\cdot\\#\\{y\\bmod 2^k: y^2\\equiv c\\pmod{2^k}\\}.\n\\]\n\n\\label{app:typeI}\n\nWe record the square-root counts modulo $2^m$ that enter the rank-$1$ dyadic blocks.\nThese counts are classical, but we include them so that the evaluation of the base blocks can be\n\n\\begin{equation}\\label{eq:L3}\n  L_3 := \\langle 2\\rangle^{\\oplus 3}\\qquad\\text{with }Q(x)=2(x_1^2+x_2^2+x_3^2)\n\\end{equation}\n\n\\begin{equation}\\label{eq:expansion-d}\n  f(\\tau_u(x))\\equiv f(x)\\pmod{2^n},\\qquad\n  f(\\tau_u(x))\\equiv f(x)+2^n(x\\cdot u)\\pmod{2^{n+1}},\n\\end{equation}\n\n\\begin{lemma}[Fibre invariance for sums of squares]\\label{lem:fibre-invariance}\nLet $d\\ge 1$ and $n\\ge 1$.\nFor any $x\\in(\\Z/2^n\\Z)^d$ and any $z\\in(\\Z/2\\Z)^d$ one has\n\\[\n  Q_d(x+2^n z)\\equiv Q_d(x)\\pmod{2^{n+1}}.\n\\]\nConsequently, for any $a\\in\\Z$ and any residue class $\\bar x\\in(\\Z/2^n\\Z)^d$, the number of lifts $\\tilde x\\in(\\Z/2^{n+1}\\Z)^d$ of $\\bar x$ satisfying\n$Q_d(\\tilde x)\\equiv a\\pmod{2^{n+1}}$\nis either $0$ or the full fibre size $2^d$.\n\\end{lemma}\n\n\\begin{theorem}[Half-lift principle for sums of squares]\\label{thm:half-lift-principle}\nLet $d\\ge 1$ and set\n\\[\n  Q_d(x_1,\\dots,x_d):=x_1^2+\\cdots+x_d^2.\n\\]\nFor $n\\ge 1$ and $a\\in\\Z$ let\n\\[\n  N_{d,n}(a):=\\#\\{x\\in (\\Z/2^n\\Z)^d: Q_d(x)\\equiv a\\pmod{2^n}\\}.\n\\]\nAssume $n\\ge 3$ and $4\\nmid a$.\nThen\n\\[\n  N_{d,n+1}(a)=2^{d-1}\\,N_{d,n}(a).\n\\]\nEquivalently, among the residue classes modulo $2^n$ satisfying $Q_d\\equiv a$, exactly half admit lifts to level $2^{n+1}$; moreover, whenever a class lifts, \\emph{all} $2^d$ lifts in its fibre are solutions (cf.\\ Lemma~\\ref{lem:fibre-invariance}).\n\\end{theorem}", "post_theorem_intro_text_len": 4729, "post_theorem_intro_text": "\\begin{remark}[Scope of the half-lift involution]\\label{rem:half-lift-scope}\nThe involution argument proving Theorem~\\ref{thm:half-lift-principle} uses the specific diagonal form\n$Q_d=x_1^2+\\cdots+x_d^2$ and the primitivity hypothesis $4\\nmid a$.\nThe primitivity is essential in general: for example, in dimension $d=4$ one checks directly that\n\\(N_{4,4}(8)=1536\\) while \\(N_{4,3}(8)=128\\), so the ratio is $12\\ne 2^{d-1}=8$.\nOn a conceptual level, two features of the proof break down once these hypotheses are relaxed:\nif $4\\mid a$ then solutions modulo $2^n$ may have all coordinates even, so the first step (which extracts an odd coordinate to choose an involution direction) can fail; and for $n=2$ the quadratic error term in the expansion \\eqref{eq:expansion-d} no longer vanishes modulo $2^{n+1}$, so the translation $x\\mapsto x+2^{n-1}u$ need not toggle $Q_d(x)$ by a clean additive $2^n$.\nWe do not claim a dimension-free half-lift recursion for arbitrary diagonal unit coefficients (or for general dyadic lattices) without further reduction.\nIn the dyadic setting, the half-lift mechanism is used here only to analyse the even ternary lattice $L_3$ in \\eqref{eq:L3};\nrank-$1$ Type~I blocks are handled separately via explicit square-root counts (Appendix~\\ref{app:typeI}).\n\\end{remark}\n\nFor $d=3$, Theorem~\\ref{thm:half-lift-principle} combines with a simple $4$-adic descent and a base computation modulo $8$ to yield closed forms for the three-squares counts once one reaches the stable range $n\\ge 2k+3$ for $a=4^k a_0$ with $4\\nmid a_0$ (and hence an explicit formula for $\\alpha_2(t;L_3)$).\nMore generally, in the stable range the half-lift recursion has growth factor $2^{d-1}$, which is exactly the ``smooth'' Hensel factor $p^{d-1}$ specialised to $p=2$ (Remark~\\ref{rem:half-lift-hensel}).\nWe also record explicit formulas for the even rank-$2$ Jordan blocks (hyperbolic and anisotropic).\nAs a consequence we obtain explicit densities $\\alpha_2(t;B)$ for each basic dyadic block $B$ of Definition~\\ref{def:dyadic-base-blocks}.\n\n\\begin{theorem}[Prime-uniform density for the hyperbolic plane]\n\\label{thm:prime-uniform-h0}\nLet $p$ be a prime and let $e\\ge 0$.\nLet $H_0$ denote the rank-$2$ hyperbolic plane over $\\Z_p$ with bilinear form\n\\(\\langle (x,y),(x',y')\\rangle = xy'+x'y\\).\nOn the scaled lattice $p^eH_0$ consider the quadratic form\n\\[\n  q_e(v):=\\tfrac12\\langle v,v\\rangle = p^e xy\\in\\Z_p.\n\\]\nFor $m\\ge 1$ and $t\\in\\Z_p$ set\n\\[\n  r^{(q)}_{m}(t;p^eH_0)\n  :=\\#\\{(x,y)\\in(\\mathbb{Z}/p^m\\mathbb{Z})^2 : p^e xy\\equiv t\\!\\!\\pmod{p^m}\\},\n\\]\nand define the associated density (when the limit exists) by\n\\[\n  \\alpha_p^{(q)}(t;p^eH_0)\n  :=\\lim_{m\\to\\infty}p^{-m}\\,r^{(q)}_{m}(t;p^eH_0).\n\\]\nLet $t\\in\\Z_p\\setminus\\{0\\}$ and write $v:=v_p(t)$.\nThen\n\\[\n  \\alpha_p^{(q)}(t;p^eH_0)=\n  \\begin{cases}\n    0, & v<e,\\\\[2pt]\n    (v-e+1)(p-1)p^{e-1}, & v\\ge e.\n  \\end{cases}\n\\]\n\\smallskip\n\\noindent In particular, under the $q=\\langle\\cdot,\\cdot\\rangle/2$ normalisation, both the vanishing threshold and the density formula for the scaled hyperbolic plane are uniform for all primes $p$, including $p=2$.\n\\end{theorem}\n\nTheorem~\\ref{thm:prime-uniform-h0} is a formal consequence of the $Q$-normalised density formulas for the hyperbolic plane (Corollaries~\\ref{cor:density-hyp-scaled-odd} and~\\ref{cor:density-hyp-scaled}) together with the conversion Lemma~\\ref{lem:q-dictionary}; we give the short proof in \\S\\ref{sec:densities}.\n\n\\subsection*{Organisation}\nSection~\\ref{sec:prelim} fixes notation and recalls the dyadic Jordan decomposition and the basic dyadic blocks.\nSection~\\ref{sec:hyperbolic} treats the hyperbolic plane $H_0$ uniformly for all primes $p$ by explicit congruence counting, including closed forms for its stable-range counts and generating series.\nSection~\\ref{sec:anisotropic} treats the anisotropic even plane; here one can recast the counting problem as a norm-congruence in an unramified quadratic extension, where the unit locus admits a clean lifting recursion.\nSection~\\ref{sec:half-lift} proves the half-lift principle and derives closed forms for the anisotropic ternary lattice $L_3$.\nSection~\\ref{sec:densities} summarises the resulting density formulas.\nAppendix~\\ref{app:typeI} records the rank-$1$ (Type~I) dyadic block counts.\n\n\\subsection*{Computational checks}\n\nThe arguments in this paper are self-contained and elementary, and they do not rely on computation.\nAs an external consistency check, the arXiv source bundle includes ancillary files containing finite-level verifiers for the congruence counts and a Lean~4 formalisation of the half-lift involution (see \\texttt{anc/verify.py} and the directory \\texttt{anc/lean/}; build instructions are in \\texttt{anc/README\\_CODE.txt}).", "sketch": "The post-theorem introduction does not give a full step-by-step proof sketch, but it does describe the key mechanism and where it can fail. It says that the proof of Theorem~\\ref{thm:half-lift-principle} is an \\emph{“involution argument”} using the \\emph{“specific diagonal form $Q_d=x_1^2+\\cdots+x_d^2$”} together with the \\emph{“primitivity hypothesis $4\\nmid a$.”} Conceptually, it highlights two main ingredients/steps behind the involution:\n\n1. A first step that \\emph{“extracts an odd coordinate to choose an involution direction”}; this can fail when $4\\mid a$ because \\emph{“solutions modulo $2^n$ may have all coordinates even.”}\n\n2. A translation step \\emph{“$x\\mapsto x+2^{n-1}u$”} that should \\emph{“toggle $Q_d(x)$ by a clean additive $2^n$.”} This relies on the fact that for $n\\ge 3$ \\emph{“the quadratic error term in the expansion \\eqref{eq:expansion-d} … vanishes modulo $2^{n+1}$,”} whereas for $n=2$ it \\emph{“no longer vanishes modulo $2^{n+1}$,”} so the toggling may fail.", "expanded_sketch": "The post-theorem introduction does not give a full step-by-step proof sketch, but it does describe the key mechanism and where it can fail. It says that, in establishing the main theorem, the argument is an \\emph{“involution argument”} using the \\emph{“specific diagonal form $Q_d=x_1^2+\\cdots+x_d^2$”} together with the \\emph{“primitivity hypothesis $4\\nmid a$.”} Conceptually, it highlights two main ingredients/steps behind the involution:\n\n1. A first step that \\emph{“extracts an odd coordinate to choose an involution direction”}; this can fail when $4\\mid a$ because \\emph{“solutions modulo $2^n$ may have all coordinates even.”}\n\n2. A translation step \\emph{“$x\\mapsto x+2^{n-1}u$”} that should \\emph{“toggle $Q_d(x)$ by a clean additive $2^n$.”} This relies on the fact that for $n\\ge 3$ \\emph{“the quadratic error term in the expansion\n\\begin{equation}\\label{eq:expansion-d}\n  f(\\tau_u(x))\\equiv f(x)\\pmod{2^n},\\qquad\n  f(\\tau_u(x))\\equiv f(x)+2^n(x\\cdot u)\\pmod{2^{n+1}},\n\\end{equation}\n… vanishes modulo $2^{n+1}$,”} whereas for $n=2$ it \\emph{“no longer vanishes modulo $2^{n+1}$,”} so the toggling may fail.", "expanded_theorem": "[Half-lift principle for sums of squares]\\label{thm:half-lift-principle}\nLet $d\\ge 1$ and set\n\\[\n  Q_d(x_1,\\dots,x_d):=x_1^2+\\cdots+x_d^2.\n\\]\nFor $n\\ge 1$ and $a\\in\\mathbb{Z}$ let\n\\[\n  N_{d,n}(a):=\\#\\{x\\in (\\mathbb{Z}/2^n\\mathbb{Z})^d: Q_d(x)\\equiv a\\pmod{2^n}\\}.\n\\]\nAssume $n\\ge 3$ and $4\\nmid a$.\nThen\n\\[\n  N_{d,n+1}(a)=2^{d-1}\\,N_{d,n}(a).\n\\]\nEquivalently, among the residue classes modulo $2^n$ satisfying $Q_d\\equiv a$, exactly half admit lifts to level $2^{n+1}$; moreover, whenever a class lifts, \\emph{all} $2^d$ lifts in its fibre are solutions, by the following lemma.\n\n\\begin{lemma}[Fibre invariance for sums of squares]\\label{lem:fibre-invariance}\nLet $d\\ge 1$ and $n\\ge 1$.\nFor any $x\\in(\\Z/2^n\\Z)^d$ and any $z\\in(\\Z/2\\Z)^d$ one has\n\\[\n  Q_d(x+2^n z)\\equiv Q_d(x)\\pmod{2^{n+1}}.\n\\]\nConsequently, for any $a\\in\\Z$ and any residue class $\\bar x\\in(\\Z/2^n\\Z)^d$, the number of lifts $\\tilde x\\in(\\Z/2^{n+1}\\Z)^d$ of $\\bar x$ satisfying\n$Q_d(\\tilde x)\\equiv a\\pmod{2^{n+1}}$\nis either $0$ or the full fibre size $2^d$.\n\\end{lemma}", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let \\(d\\ge 1\\) and define the diagonal quadratic form\n\\[\nQ_d(x_1,\\dots,x_d)=x_1^2+\\cdots+x_d^2.\n\\]\nFor \\(n\\ge 1\\) and \\(a\\in\\mathbb Z\\), let\n\\[\nN_{d,n}(a):=\\#\\{x\\in(\\mathbb Z/2^n\\mathbb Z)^d: Q_d(x)\\equiv a\\pmod{2^n}\\}.\n\\]\nAssume \\(n\\ge 3\\) and \\(4\\nmid a\\). For a residue class \\(\\bar x\\in(\\mathbb Z/2^n\\mathbb Z)^d\\), its fibre at level \\(2^{n+1}\\) means the set of lifts \\(\\tilde x\\in(\\mathbb Z/2^{n+1}\\mathbb Z)^d\\) reducing to \\(\\bar x\\), which has size \\(2^d\\). Which quantitative statement holds?", "correct_choice": {"label": "A", "text": "One has\n\\[\nN_{d,n+1}(a)=2^{d-1}N_{d,n}(a).\n\\]\nEquivalently, among the classes \\(\\bar x\\in(\\mathbb Z/2^n\\mathbb Z)^d\\) with \\(Q_d(\\bar x)\\equiv a\\pmod{2^n}\\), exactly half admit lifts to level \\(2^{n+1}\\); and whenever such a class lifts, all \\(2^d\\) elements in its fibre are solutions modulo \\(2^{n+1}\\)."}, "choices": [{"label": "B", "text": "One has\n\\[\nN_{d,n+1}(a)=2^{d-1}N_{d,n}(a)\n\\]\nfor every \\(n\\ge 2\\) and every \\(a\\in\\mathbb Z\\) with \\(2\\nmid a\\). Equivalently, among the classes \\(\\bar x\\in(\\mathbb Z/2^n\\mathbb Z)^d\\) with \\(Q_d(\\bar x)\\equiv a\\pmod{2^n}\\), exactly half admit lifts to level \\(2^{n+1}\\); and whenever such a class lifts, all \\(2^d\\) elements in its fibre are solutions modulo \\(2^{n+1}\\)."}, {"label": "C", "text": "Whenever a class \\(\\bar x\\in(\\mathbb Z/2^n\\mathbb Z)^d\\) with \\(Q_d(\\bar x)\\equiv a\\pmod{2^n}\\) admits one lift to level \\(2^{n+1}\\), in fact all \\(2^d\\) elements in its fibre are solutions modulo \\(2^{n+1}\\)."}, {"label": "D", "text": "One has\n\\[\nN_{d,n+1}(a)=2^{d-1}N_{d,n}(a)\n\\]\nfor every \\(n\\ge 3\\) and every primitive target \\(a\\in\\mathbb Z\\). Equivalently, among the classes \\(\\bar x\\in(\\mathbb Z/2^n\\mathbb Z)^d\\) with \\(Q_d(\\bar x)\\equiv a\\pmod{2^n}\\), exactly half admit lifts to level \\(2^{n+1}\\); and whenever such a class lifts, all \\(2^d\\) elements in its fibre are solutions modulo \\(2^{n+1}\\)."}, {"label": "E", "text": "One has\n\\[\nN_{d,n+1}(a)=2^{d}N_{d,n}(a).\n\\]\nEquivalently, every class \\(\\bar x\\in(\\mathbb Z/2^n\\mathbb Z)^d\\) with \\(Q_d(\\bar x)\\equiv a\\pmod{2^n}\\) admits lifts to level \\(2^{n+1}\\), and whenever such a class lifts, all \\(2^d\\) elements in its fibre are solutions modulo \\(2^{n+1}\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "stable range n\\ge 3 replaced by n\\ge 2", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the exact-half orbit-counting conclusion and recursion, retaining only fibre invariance", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "hypothesis 4\\nmid a weakened to mere primitivity of a", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "replaced half of the classes lifting by all classes lifting", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives definitions and hypotheses but does not state or strongly signal the exact recursion or the 'exactly half' lifting conclusion. The correct answer is not leaked."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-selection task: the correct option is essentially the full theorem statement under the given hypotheses, while the others are modified variants. This is more than a verbatim restatement, but only mildly so."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact statement from overgeneralizations in the range/hypotheses and from a weaker consequence. However, the pressure is reduced because the task mainly tests recognition of the precise theorem statement rather than deeper derivation."}, "DQS": {"score": 1, "justification": "B, D, and E are plausible theorem-perturbation distractors tied to common errors (wrong stable range, weakened hypothesis, too-strong lifting claim). But C is a weaker true statement rather than a genuine distractor, so the options do not cleanly support a single-correct-answer format."}, "total_score": 5, "overall_assessment": "Technically no answer leakage and reasonably structured near-miss distractors, but the MCQ is only moderately strong because it mostly asks for theorem recognition and includes a weaker true option, which weakens the single-answer validity."}}
{"id": "2602.21115v1", "paper_link": "http://arxiv.org/abs/2602.21115v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "\\label{thm}\nLet $0\\leq f\\in C^1([0,1))$ satisfying $F(1)=+\\infty$.\nAssume that $0\\le u\\le 1$ is a  stable radial solution to \\eqref{d-p}.\nIf $2\\le n\\le 6$, then $u$ is regular and satisfies\n$$\\frac 12 u_r(1)^2\\le F(\\|u\\|_{L^\\infty(B_1)})\\le  Cu_r(1)^2.$$\nAdditionally, if $f$ is nondecreasing, then\n $$F(\\|u\\|_{L^\\infty(B_1)})\\le C,$$\nwhere  $C>0$ is a universal constant.", "start_pos": 8818, "end_pos": 9202, "label": "thm"}, "ref_dict": {"un-bd": "\\begin{lem}\\label{un-bd}\nLet $0\\le f\\in C^1([0,1])$ be nondecreasing. Assume\nthat $0\\le u\\le 1$ is a stable radial solution to\n\\eqref{d-p}. Then\n$$|u_r(1)|\\le 2.$$\n\n\\end{lem}", "gz": "\\begin{align}\\label{gz}\n\\gz:=\\liminf_{t\\to 1^{-}}\\frac{f(t)f''(t)}{f'(t)^2}>1\n\\end{align}", "thm": "\\begin{thm}\\label{thm}\nLet $0\\leq f\\in C^1([0,1))$ satisfying $F(1)=+\\fz$.\nAssume that $0\\le u\\le 1$ is a  stable radial solution to \\eqref{d-p}.\nIf $2\\le n\\le 6$, then $u$ is regular and satisfies\n$$\\frac 12 u_r(1)^2\\le F(\\|u\\|_{L^\\fz(B_1)})\\le  Cu_r(1)^2.$$\nAdditionally, if $f$ is nondecreasing, then\n $$F(\\|u\\|_{L^\\fz(B_1)})\\le C,$$\nwhere  $C>0$ is a universal constant.\n\\end{thm}", "g-dp": "\\begin{align}\\label{g-dp}\n\\left\\{\n\\begin{aligned}\n-\\bdz u & = \\lz f(u)&\\quad{\\rm in}&\\quad B_1  \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}", "thm-2": "\\begin{thm}\\label{thm-2}\nLet $f\\in C^1([0,1))$ be nondecreasing, convex, and\n$F(1)=+\\fz$. Suppose that $0\\le u^{\\star}\\le 1$ is the extremal solution to\n\\eqref{g-dp}. Then\n$$\\|u^{\\star}\\|_{L^\\fz(B_1)}<1\\quad \\forall \\, 2\\le n\\le 6.$$\n\n\\end{thm}", "d-p": "\\begin{align}\\label{d-p}\n\\left\\{\n\\begin{aligned}\n-\\bdz u & = f(u)&\\quad{\\rm in}&\\quad B_1 \\setminus\\{ 0\\}  \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}"}, "pre_theorem_intro_text_len": 1602, "pre_theorem_intro_text": "This paper focuses on the issues associated with MEMS problem:\n\\begin{align}\\label{d-p}\n\\left\\{\n\\begin{aligned}\n-\\Delta u & = f(u)&\\quad{\\rm in}&\\quad B_1 \\setminus\\{ 0\\}  \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}\nwhere $B_1$ is the unit ball centered at the origin in ${\\mathbb R}^n$($n\\ge 2$),  $0\\le u\\le 1$ is a stable radial  solution, and $f:[0,1)\\to [0,+\\infty)$  is\n$C^1$-function which blow-up at\n$u=1$.  Throughout the paper, for each $0\\le t\\le 1$ we define $F(t):=\\int^t_0 f(s)\\,ds$ and require that $F(1)=\\int^{1}_0f(s)\\,ds=+\\infty$.\n\nRecall that a radial solution $u$ of \\eqref{d-p}  is called stable if\n$$\\int_{B_1}f'(u)\\xi^2\\,dx\\le \\int_{B_1}|\\nabla \\xi|^2\\,dx$$\nfor every $\\xi \\in C^\\infty(B_1)$ with compact support in $B_1\\backslash\\{0\\}$, where\n$f'(s)$ denotes the derivative of $f$ at the point $s$. Furthermore, if\n$\\|u\\|_{L^\\infty(B_1)}<1$, we refer to $u$ as regular in the sense of Bruera and Cabr\\'e \\cite{bc}.\nNotably, the blow-up of $f(u)$ at $u=1$ implies that the stable radial solution $u$ does not need to be regular. For instance, when $n\\ge 7$ and $f(t)=\\frac{n-1}{1-t}$, the radial function $1-|x|$ is always stable yet satisfies\n$\\|u\\|_{L^\\infty(B_1)}=1$. In contrast, for dimensions $2\\le n\\le 6$, this function is not a stable solution. For further details, see Bruera and Cabr\\'e \\cite{bc} and  Meadows \\cite{m04}.\n\nThis example naturally leads to the following question: in dimensions $2\\le n\\le 6$,  is every stable radial solution of \\eqref{d-p} regular?\n\nIn the radial case, we provide a complete answer to this question.", "context": "This paper focuses on the issues associated with MEMS problem:\n\\begin{align}\\label{d-p}\n\\left\\{\n\\begin{aligned}\n-\\Delta u & = f(u)&\\quad{\\rm in}&\\quad B_1 \\setminus\\{ 0\\} \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}\nwhere $B_1$ is the unit ball centered at the origin in ${\\mathbb R}^n$($n\\ge 2$), $0\\le u\\le 1$ is a stable radial solution, and $f:[0,1)\\to [0,+\\infty)$ is\n$C^1$-function which blow-up at\n$u=1$. Throughout the paper, for each $0\\le t\\le 1$ we define $F(t):=\\int^t_0 f(s)\\,ds$ and require that $F(1)=\\int^{1}_0f(s)\\,ds=+\\infty$.\n\nRecall that a radial solution $u$ of \\eqref{d-p} is called stable if\n$$\\int_{B_1}f'(u)\\xi^2\\,dx\\le \\int_{B_1}|\\nabla \\xi|^2\\,dx$$\nfor every $\\xi \\in C^\\infty(B_1)$ with compact support in $B_1\\backslash\\{0\\}$, where\n$f'(s)$ denotes the derivative of $f$ at the point $s$. Furthermore, if\n$\\|u\\|_{L^\\infty(B_1)}<1$, we refer to $u$ as regular in the sense of Bruera and Cabr\\'e \\cite{bc}.\nNotably, the blow-up of $f(u)$ at $u=1$ implies that the stable radial solution $u$ does not need to be regular. For instance, when $n\\ge 7$ and $f(t)=\\frac{n-1}{1-t}$, the radial function $1-|x|$ is always stable yet satisfies\n$\\|u\\|_{L^\\infty(B_1)}=1$. In contrast, for dimensions $2\\le n\\le 6$, this function is not a stable solution. For further details, see Bruera and Cabr\\'e \\cite{bc} and Meadows \\cite{m04}.\n\nThis example naturally leads to the following question: in dimensions $2\\le n\\le 6$, is every stable radial solution of \\eqref{d-p} regular?\n\nIn the radial case, we provide a complete answer to this question.\n\n\\begin{align}\\label{d-p}\n\\left\\{\n\\begin{aligned}\n-\\bdz u & = f(u)&\\quad{\\rm in}&\\quad B_1 \\setminus\\{ 0\\}  \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}", "full_context": "This paper focuses on the issues associated with MEMS problem:\n\\begin{align}\\label{d-p}\n\\left\\{\n\\begin{aligned}\n-\\Delta u & = f(u)&\\quad{\\rm in}&\\quad B_1 \\setminus\\{ 0\\} \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}\nwhere $B_1$ is the unit ball centered at the origin in ${\\mathbb R}^n$($n\\ge 2$), $0\\le u\\le 1$ is a stable radial solution, and $f:[0,1)\\to [0,+\\infty)$ is\n$C^1$-function which blow-up at\n$u=1$. Throughout the paper, for each $0\\le t\\le 1$ we define $F(t):=\\int^t_0 f(s)\\,ds$ and require that $F(1)=\\int^{1}_0f(s)\\,ds=+\\infty$.\n\nRecall that a radial solution $u$ of \\eqref{d-p} is called stable if\n$$\\int_{B_1}f'(u)\\xi^2\\,dx\\le \\int_{B_1}|\\nabla \\xi|^2\\,dx$$\nfor every $\\xi \\in C^\\infty(B_1)$ with compact support in $B_1\\backslash\\{0\\}$, where\n$f'(s)$ denotes the derivative of $f$ at the point $s$. Furthermore, if\n$\\|u\\|_{L^\\infty(B_1)}<1$, we refer to $u$ as regular in the sense of Bruera and Cabr\\'e \\cite{bc}.\nNotably, the blow-up of $f(u)$ at $u=1$ implies that the stable radial solution $u$ does not need to be regular. For instance, when $n\\ge 7$ and $f(t)=\\frac{n-1}{1-t}$, the radial function $1-|x|$ is always stable yet satisfies\n$\\|u\\|_{L^\\infty(B_1)}=1$. In contrast, for dimensions $2\\le n\\le 6$, this function is not a stable solution. For further details, see Bruera and Cabr\\'e \\cite{bc} and Meadows \\cite{m04}.\n\nThis example naturally leads to the following question: in dimensions $2\\le n\\le 6$, is every stable radial solution of \\eqref{d-p} regular?\n\nIn the radial case, we provide a complete answer to this question.\n\n\\begin{align}\\label{d-p}\n\\left\\{\n\\begin{aligned}\n-\\bdz u & = f(u)&\\quad{\\rm in}&\\quad B_1 \\setminus\\{ 0\\}  \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}\n\nRecall that a radial solution $u$ of \\eqref{d-p} is called stable if\n$$\\int_{B_1}f'(u)\\xi^2\\,dx\\le \\int_{B_1}|\\nabla \\xi|^2\\,dx$$\nfor every $\\xi \\in C^\\fz(B_1)$ with compact support in $B_1\\backslash\\{0\\}$, where\n$f'(s)$ denotes the derivative of $f$ at the point $s$. Furthermore, if\n$\\|u\\|_{L^\\fz(B_1)}<1$, we refer to $u$ as regular in the sense of Bruera and Cabr\\'e \\cite{bc}.\nNotably, the blow-up of $f(u)$ at $u=1$ implies that the stable radial solution $u$ does not need to be regular. For instance, when $n\\ge 7$ and $f(t)=\\frac{n-1}{1-t}$, the radial function $1-|x|$ is always stable yet satisfies\n$\\|u\\|_{L^\\fz(B_1)}=1$. In contrast, for dimensions $2\\le n\\le 6$, this function is not a stable solution. For further details, see Bruera and Cabr\\'e \\cite{bc} and Meadows \\cite{m04}.\n\nThis example naturally leads to the following question: in dimensions $2\\le n\\le 6$, is every stable radial solution of \\eqref{d-p} regular?\n\n(ii) When dimension\n$3\\le n\\le 6$, $F(1)=+\\infty$ is necessary for regular stable solutions to exist for certain nonlinearities $f$; counterexamples can be found in Bruera and Cabr\\'e \\cite{bc}. Indeed, Bruera and Cabr\\'e \\cite{bc}\nconstructed a singular stable solution $u(x)=1-|x|^{2/(1+p)}$ for $n\\ge 3$ and for some\n $p\\in (0,1)$ satisfying\n$$-\\bdz u=\\frac{2}{1+p}\\left(\\frac{2}{1+p}+n-2\\right)(1-u)^{-p}$$\nwith $F(1)<+\\fz$. For dimension $n=2$ (without assuming $F(1)=+\\infty$), Luo, Ye and Zhou \\cite{lyz} showed that the stable radial solution $u$ of $-\\bdz u+ c(x)\\cdot \\nabla u=f(u)$ is regular if $f$ is nonnegative, nondecreasing, and convex, where $c(x)$ is smooth vector function. Observe that when $c(x)$ is a zero vector, this reduces to the case $-\\Delta u = f(u)$.\n\nRecently, inspired by a key stability inequality established \\cite{cfrs},\nBruera and Cabr\\'e \\cite{bc} investigated the case of a nonlinearity $f(u)$\nthat blows up as $u\\to 1^{-}$. Specifically, under the following Crandall-Rabinowitz type condition:\n\\begin{align}\\label{gz}\n\\gz:=\\liminf_{t\\to 1^{-}}\\frac{f(t)f''(t)}{f'(t)^2}>1\n\\end{align}\nfor all $f\\in C^2([0,1))$ satisfying $f\\ge 0$,\n$f'\\ge 0$ and $F(1)=+\\fz$, they established the following local estimate for all stable solutions\n(not necessarily radial) up to optimal dimension:\n$$\\|u\\|_{L^{\\fz}(B_{1/2})}\\le F^{-1}(C\\|u\\|^2_{L^1(B_1)})<1\\quad\n\\forall \\, 2\\le n\\le 6,$$\nwhere constant $C$ depending only on $n$ and $\\gz$. Additionally, a forthcoming study by Figalli and Franceschini \\cite{ff} provides an explicit upper bound for the Hausdorff dimension of the singular set for stable solutions expressed in terms of $f$, $f'$, and $f''$. The result covers both globally defined nonlinearities and nonlinearities with finite blow-up.\n\nIn order to show Theorem \\ref{thm}, we recall the following important result concerning stable radial solutions, which was established by Villegas \\cite[Theorem 1.7]{v12}.\n\\begin{lem}\\label{vi-lem}\nLet $n\\geq 2$ and $f:[0,1)\\to [0,+\\fz)$ be a $C^1$ function satisfying $f\\ge 0$. Let $0\\le u\\le 1$ be a stable radial solution to\n\\eqref{d-p} in $W^{1,2}(B_1)$. Then there exists a constant $K_n$ depending only on $n$ such that\n\\begin{align}\\label{vi-le1}\n|u_r(t)|\\le K_n \\|\\nabla u\\|_{L^2(B_1\\backslash B_{1/2})}t^{-n/2+\\sqrt{n-1}+1},\n\\quad \\forall \\, 0<t<1/2.\n\\end{align}\n\\end{lem}\nBefore applying Lemma \\ref{vi-lem}, we need to verify that\n$u$ belongs to $u\\in W^{1,2}(B_1)$. For this purpose, we state the following proposition.\n\\begin{prop}\\label{w12}\nLet $n\\ge 2$ and $f:[0,1)\\to[0,+\\infty)$ be a $C^{1}$ function satisfying $f\\ge 0$. Suppose that $0\\leq u\\leq 1$ is a classical nontrivial $C^{2}$ radial solution (not necessarily stable) of $-\\bdz u = f(u) \\quad{\\rm in} \\quad B_1\\backslash\\{0\\} $. Then\n\nAs a direct consequence of Lemma \\ref{vi-lem} and Proposition \\ref{w12}, we have:\n\\begin{prop}\\label{key-prop-2}\nLet $n\\geq 2$ and $f:[0,1)\\to [0,+\\fz)$ be a $C^1$ function satisfying $f\\geq 0$. Let $0\\le u\\le 1$ be a stable radial solution to\n\\eqref{d-p}. Then there exists a constant $C_n$ depending only on $n$ such that\n\n\\noindent which completes the proof.\n\\end{proof}\nIn addition, if $f$ is nondecreasing, one can derive the universal bound\nof $u_r(1)$, which is useful for establishing an upper bound of $F(\\|u\\|_{L^\\fz(B_1)})$.\n\\begin{lem}\\label{un-bd}\nLet $0\\le f\\in C^1([0,1])$ be nondecreasing. Assume\nthat $0\\le u\\le 1$ is a stable radial solution to\n\\eqref{d-p}. Then\n$$|u_r(1)|\\le 2.$$\n\nFinally, by $u\\in C^1(\\overline B_1\\backslash B_{1/2})$, one gets\n$F(\\|u\\|_{L^\\fz(B_1)})<+\\fz$ and so that $u$ is regular. Furthermore,\nif $f$ is nondecreasing, Lemma \\ref{un-bd} tells us $|u_r(1)|\\le 2$, and thus\n$$F(\\|u\\|_{L^\\fz(B_1)})\\le M_n u_r(1)^2\\le 4M_n.$$\nSince we are considering a finite\nnumber of dimensions ($2\\le n\\le 6$), we can take $4M_n\\leq C$ in order to\nobtain a universal constant $C>0$.\nHence we finish this proof.\n\\end{proof}\nWe apply now Theorem \\ref{thm} to prove Theorem\n\\ref{thm-2}.\n\\begin{proof}[Proof of Theorem \\ref{thm-2}]\nLet $0<\\lz<\\lz^{\\star}$ and let $u_{\\lz}\\in C^2(\\overline B_1)$ be a stable radial\nsolution to \\eqref{g-dp}. It follows from Theorem \\ref{thm} that\n$$F(\\|u_{\\lz}\\|_{L^\\fz(B_1)})\\le C(n)\n\\left(\\frac{d u_{\\lz}(1)}{dr}\\right)^2.$$\nUsing Lemma 2.3 in \\cite{v12} by Villegas, one has that\n$$F(\\|u_{\\lz}\\|_{L^\\fz(B_1)})\\le C(n)\n\\left(\\frac{d u_{\\lz}(1)}{dr}\\right)^2\n\\le C(n)\\left(\\frac{d u_{\\lz}(1/2)}{dr}\\right)^2\n\\le C(n)\\int^{3/4}_{1/2}\\left(\\frac{d u_{\\lz}(t)}{dr}\\right)^2\\,dt,$$\nwhere we used the fact that $t^{2n-2}\\left(\\frac{d u_{\\lz}(t)}{dr}\\right)^2$\nis nondecreasing in the last inequality. Since\n$\\lim_{\\lz\\to \\lz^{\\star}}u_{\\lz}=u^{\\star}$ and\n$\\nabla u_{\\lz}\\to \\nabla u^{\\star}$ in $L^2(B_{3/4})$ as\n$\\lz \\to \\lz^{\\star}$, passing to the limit $\\lz\\to \\lz^{\\star}$ we conclude that\n$$F(\\|u^{\\star}\\|_{L^\\fz(B_1)})\n\\le C(n)\\int^{3/4}_{1/2}\\left(\\frac{d u^{\\star}(t)}{dr}\\right)^2\\,dt<+\\fz.$$\n\n\\begin{align}\\label{d-p}\n\\left\\{\n\\begin{aligned}\n-\\bdz u & = f(u)&\\quad{\\rm in}&\\quad B_1 \\setminus\\{ 0\\}  \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}\n\n\\begin{align}\\label{g-dp}\n\\left\\{\n\\begin{aligned}\n-\\bdz u & = \\lz f(u)&\\quad{\\rm in}&\\quad B_1  \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}\n\n\\begin{thm}\\label{thm}\nLet $0\\leq f\\in C^1([0,1))$ satisfying $F(1)=+\\fz$.\nAssume that $0\\le u\\le 1$ is a  stable radial solution to \\eqref{d-p}.\nIf $2\\le n\\le 6$, then $u$ is regular and satisfies\n$$\\frac 12 u_r(1)^2\\le F(\\|u\\|_{L^\\fz(B_1)})\\le  Cu_r(1)^2.$$\nAdditionally, if $f$ is nondecreasing, then\n $$F(\\|u\\|_{L^\\fz(B_1)})\\le C,$$\nwhere  $C>0$ is a universal constant.\n\\end{thm}\n\n\\begin{thm}\\label{thm-2}\nLet $f\\in C^1([0,1))$ be nondecreasing, convex, and\n$F(1)=+\\fz$. Suppose that $0\\le u^{\\star}\\le 1$ is the extremal solution to\n\\eqref{g-dp}. Then\n$$\\|u^{\\star}\\|_{L^\\fz(B_1)}<1\\quad \\forall \\, 2\\le n\\le 6.$$\n\n\\end{thm}\n\n\\begin{lem}\\label{un-bd}\nLet $0\\le f\\in C^1([0,1])$ be nondecreasing. Assume\nthat $0\\le u\\le 1$ is a stable radial solution to\n\\eqref{d-p}. Then\n$$|u_r(1)|\\le 2.$$\n\n\\end{lem}", "post_theorem_intro_text_len": 8000, "post_theorem_intro_text": "We now provide some remarks on the main results.\n\\begin{rem}\n\\rm\n(i) The dimension $n \\leq 6$ in Theorem \\ref{thm} is optimal: for $n \\geq 7$ and $f(t) = \\frac{n-1}{1-t}$, the function $1 - |x|$ is always stable but not regular. Additional examples can be found in Bruera and Cabr\\'e \\cite{bc}.\n\n(ii) When dimension\n$3\\le n\\le 6$, $F(1)=+\\infty$ is necessary for regular stable solutions to exist for certain nonlinearities $f$; counterexamples can be found in  Bruera and Cabr\\'e \\cite{bc}. Indeed, Bruera and Cabr\\'e \\cite{bc}\nconstructed a singular stable solution $u(x)=1-|x|^{2/(1+p)}$ for $n\\ge 3$ and for some\n $p\\in (0,1)$ satisfying\n$$-\\Delta u=\\frac{2}{1+p}\\left(\\frac{2}{1+p}+n-2\\right)(1-u)^{-p}$$\nwith $F(1)<+\\infty$. For dimension $n=2$ (without assuming $F(1)=+\\infty$), Luo, Ye and Zhou \\cite{lyz} showed that the stable radial solution $u$ of $-\\Delta u+ c(x)\\cdot \\nabla u=f(u)$ is regular if $f$ is nonnegative, nondecreasing, and convex, where $c(x)$ is smooth vector function. Observe that when $c(x)$ is a zero vector, this reduces to the case $-\\Delta u = f(u)$.\n\n(iii) In Theorem \\ref{thm} above, with slight modifications, it can be shown that the stable radial  solution is regular under the conditions $0\\le u\\le 1$, $0\\le f$  and $F(1)=+\\infty$,\nwithout requiring the boundary condition: $u=0$ on $\\partial B_1$.\n\\end{rem}\n\nLet us now review some relevant work in this direction. In the non-radial setting with a nonlinearity $f(u)$ that is unbounded as\n$u\\to +\\infty$, a pioneering study was first conducted by Crandall and Rabinowitz \\cite{cr75}\n with\n$f(t)=e^t$ and $f(t)=(1+t)^p$, for $p>1$. Subsequently, the boundedness of stable solutions\n$u$ to $-\\Delta u=f(u)$ in general domains $\\Omega$ has been extensively investigated over the past three decades; see \\cite{cc06,c10,n20,v13}. Notably, Cabr\\'e, Figalli, Ros-Oton and Serra \\cite{cfrs} made a breakthrough:\nSuppose $f$ is locally Lipschitz, nonnegative, and nondecreasing, and $u\\in W^{1,2}_0(\\Omega)$\nis stable solution of $-\\Delta u=f(u)$ in $C^3$ domain $\\Omega$, they showed that\n$$\\|u\\|_{C^{0,\\alpha}(\\overline \\Omega)}\\le C(n)\\|u\\|_{L^\\infty(\\Omega)}\\quad\n2\\le n\\le 9$$\nfor some $\\alpha=\\alpha(n)\\in (0,1)$. They also established the local H\\\"older continuity of\n$u$ for $n\\le 9$. Note that the dimension $n\\le 9$ is optimal for nonlinearities\n$f(u)$ that blow-up as $u\\to \\infty$. In addition, Cabr\\'e provided a quantitative proof of these results in \\cite{c23,c24}.\nWe also mention that, as established in\nthe work of Erneta \\cite{e23},\nthe optimal boundary regularity of $u$ in $C^3$ domains attainable already in the class of\n$C^{1,1}$-domains.\n\nRecently, inspired by a key stability inequality established \\cite{cfrs},\nBruera and Cabr\\'e \\cite{bc} investigated the case of a nonlinearity $f(u)$\nthat blows up as $u\\to 1^{-}$. Specifically, under the  following Crandall-Rabinowitz type condition:\n\\begin{align}\\label{gz}\n{\\gamma}:=\\liminf_{t\\to 1^{-}}\\frac{f(t)f''(t)}{f'(t)^2}>1\n\\end{align}\nfor all  $f\\in C^2([0,1))$ satisfying $f\\ge 0$,\n$f'\\ge 0$ and $F(1)=+\\infty$, they established the following local estimate for all stable solutions\n(not necessarily radial) up to optimal dimension:\n$$\\|u\\|_{L^{\\infty}(B_{1/2})}\\le  F^{-1}(C\\|u\\|^2_{L^1(B_1)})<1\\quad\n\\forall \\,  2\\le n\\le 6,$$\nwhere constant $C$ depending only on $n$ and ${\\gamma}$. Additionally, a forthcoming study by Figalli and Franceschini \\cite{ff} provides an explicit upper bound for the Hausdorff dimension of the singular set for stable solutions expressed in terms of $f$, $f'$, and $f''$. The result covers both globally defined nonlinearities and nonlinearities with finite blow-up.\n\nOn the other hand, for smooth domains $\\Omega$, Bruera and Cabr\\'e \\cite{bc} also derived global estimates for stable solutions\n$u\\in C^2( \\Omega)\\cap C^0(\\overline \\Omega)$  of the Dirichlet boundary value problem:\n\\begin{align*}\n\\left\\{\n\\begin{aligned}\n-\\Delta u & = f(u)&\\quad{\\rm in}&\\quad \\Omega  \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial \\Omega,\\\\\n\\end{aligned}\n\\right.\n\\end{align*}\nand showed the following:\n\\begin{itemize}\n\\item[$\\bullet$] If $n\\le 2$, then $\\|u\\|_{L^\\infty(\\Omega)}<1$ without condition \\eqref{gz}.\n\n\\item[$\\bullet$] If $3\\le n\\le 6$ and $f$ satisfies the condition \\eqref{gz}, then $\\|u\\|_{L^\\infty(\\Omega)}<1$.\n\n\\end{itemize}\nIt is worth noting that additional partial results related to this direction can be found in\n\\cite{ces,cgg,lyz,m04}. In particular, for dimension $n=2$ (without requiring $F(1)=+\\infty$), Luo, Ye and Zhou \\cite{lyz} proved that the stable radial solution $u$ of $-\\Delta u+c(x)\\cdot \\nabla u=f(u)$\n is regular provided that $f$ is nonnegative, nondecreasing, and convex.  Motivated by these findings, Bruera and Cabr\\'e \\cite{bc} proposed the following open problem:\n\\begin{center}\n{\\bf Open problem}. Does regularity of stable solutions hold up to the optimal\ndimension for singular nonlinearities under no Crandall-Rabinowitz type condition?\n\\end{center}\nIn the radial setting,\nwe provide a positive answer to this open problem in\nTheorem \\ref{thm}. However, the non-radial case remains open.\n\nTheorem \\ref{thm} has an important application to the Gelfand-type problem; see for example\nin \\cite{bv97,b03}. Precisely, let $f\\in C^1([0,1))$ satisfy $f(0)>0$ and be nondecreasing, convex, and $F(1)=+\\infty$.\n\nGiven a constant $\\lambda>0$ consider the nonlinear elliptic problem:\n\\begin{align}\\label{g-dp}\n\\left\\{\n\\begin{aligned}\n-\\Delta u & = \\lambda f(u)&\\quad{\\rm in}&\\quad B_1  \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}\nA fundamental question is to determine a  constant $\\lambda^{\\star}>0$ such that the problem\n\\eqref{g-dp} has a unique $L^1$-solution $u^{\\star}\\in L^1(B_1)$ in the sense that\n$$-\\int_{B_1}u^{\\star}\\Delta \\xi \\,dx=\n\\lambda^{\\star}\\int_{B_1}f(u^{\\star})\\xi\\,dx\\quad\\forall \\,  \\xi\\in C^{0,1}_c(B_1)$$\nSuch $L^1$-solution solution $u^{\\star}$ is called extremal solution.\nThe uniqueness and existence of extremal solution were established by\nCastorina, Esposito and Sciunzi \\cite{ces}. Moreover, they\nshowed that  for every $0<\\lambda<\\lambda^{\\star}$, there exist a stable solution\n$u_{\\lambda}\\in C^2(\\overline B_1)$ such that\n$$\\lim_{\\lambda\\to \\lambda^{\\star}}u_{\\lambda}\n=u^{\\star}.$$\nCombining this result with Theorem \\ref{thm}, we conclude that the extremal solution is always regular in dimensions $2\\le n\\le 6$.\n\n\\begin{thm}\\label{thm-2}\nLet $f\\in C^1([0,1))$ be nondecreasing, convex, and\n$F(1)=+\\infty$. Suppose that $0\\le u^{\\star}\\le 1$ is the extremal solution to\n\\eqref{g-dp}. Then\n$$\\|u^{\\star}\\|_{L^\\infty(B_1)}<1\\quad \\forall \\, 2\\le n\\le 6.$$\n\n\\end{thm}\n\nFinally, we outline the main ideas for proving the theorems.\nSince Theorem \\ref{thm-2} follows directly from Theorem \\ref{thm},\nit suffices to explain the proof strategy for Theorem \\ref{thm}.\n\nA key starting point is the following observation for general radial solutions:\n$$\nF(\\|u\\|_{L^\\infty(B_1)})= (n-1)\\int^1_0\\frac{u_r(t)^2}{t}\\,dt+\\frac{u_r(1)^2}2.\n$$\nSince $u\\in C^1(\\overline B_1\\backslash\\{0\\})$,\nthe main task is reduced to bounding the term $\\int^1_0\\frac{u_r(t)^2}{t}\\,dt$. By a key\nestimate due to Villegas \\cite[Theorem 1.7]{v12} (see also \\cite{cc06}), we deduce\n\\begin{align*}\n\\vert u_r(t)\\vert \\le C_n \\vert u_r(1)\\vert t^{-n/2+\\sqrt{n-1}+1},\n\\quad \\forall \\, 0<t\\leq 1.\n\\end{align*}\nObserve that\n$$-n/2+\\sqrt{n-1}+1>0\\quad{\\rm if \\ and\\ only\\ if}\\quad 2\\le n\\le 6.$$\nThis implies that $u_r(t)$ is H\\\"older continuous near  $t=0$ and hence\nthe behavior of\n $t^{-1}u_r(t)^2$ is like  $t^{-\\alpha}$ for some\n$\\alpha\\in (0,1)$ whenever $2\\le n\\le 6$. Therefore, the integral\n$\\int^1_0\\frac{u_r(t)^2}{t}\\,dt$ is finite, and we conclude\n\n$$F(\\|u\\|_{L^\\infty(B_1)})\\leq C_nu_r(1)^2<+\\infty.$$\nCombining this with the condition $F(1)=\\infty$, we obtain $\\|u\\|_{L^\\infty(B_1)}<1$.\nMoreover, if $f$ is nondecreasing, the universal bound on $F(\\|u\\|_{L^\\infty(B_1)})$ follows\nfrom the upper bound on $|u_r(1)|$; see Lemma \\ref{un-bd} for details.", "sketch": "To prove Theorem~\\ref{thm}, since Theorem~\\ref{thm-2} follows directly from it, the strategy starts from the identity for general radial solutions\n\\[\nF(\\|u\\|_{L^\\infty(B_1)})= (n-1)\\int^1_0\\frac{u_r(t)^2}{t}\\,dt+\\frac{u_r(1)^2}2.\n\\]\nThus (with $u\\in C^1(\\overline B_1\\backslash\\{0\\})$) the main task is to bound $\\int^1_0\\frac{u_r(t)^2}{t}\\,dt$. Using Villegas' estimate \\cite[Theorem 1.7]{v12} (see also \\cite{cc06}),\n\\[\n|u_r(t)|\\le C_n|u_r(1)|\\, t^{-n/2+\\sqrt{n-1}+1},\\qquad 0<t\\le 1.\n\\]\nSince $-n/2+\\sqrt{n-1}+1>0$ iff $2\\le n\\le 6$, this gives H\\\"older continuity of $u_r(t)$ near $t=0$; hence $t^{-1}u_r(t)^2$ behaves like $t^{-\\alpha}$ for some $\\alpha\\in(0,1)$ when $2\\le n\\le 6$. Therefore $\\int^1_0\\frac{u_r(t)^2}{t}\\,dt<\\infty$, yielding\n\\[\nF(\\|u\\|_{L^\\infty(B_1)})\\le C_n u_r(1)^2<+\\infty.\n\\]\nCombining this with $F(1)=\\infty$ gives $\\|u\\|_{L^\\infty(B_1)}<1$ (regularity). Finally, if $f$ is nondecreasing, the claimed universal bound for $F(\\|u\\|_{L^\\infty(B_1)})$ follows from an upper bound on $|u_r(1)|$ (see Lemma~\\ref{un-bd}).", "expanded_sketch": "To prove the main theorem, since\n\\begin{thm}\\label{thm-2}\nLet $f\\in C^1([0,1))$ be nondecreasing, convex, and\n$F(1)=+\\fz$. Suppose that $0\\le u^{\\star}\\le 1$ is the extremal solution to\n\\eqref{g-dp}. Then\n$$\\|u^{\\star}\\|_{L^\\fz(B_1)}<1\\quad \\forall \\, 2\\le n\\le 6.$$\n\n\\end{thm}\nfollows directly from it, the strategy starts from the identity for general radial solutions\n\\[\nF(\\|u\\|_{L^\\infty(B_1)})= (n-1)\\int^1_0\\frac{u_r(t)^2}{t}\\,dt+\\frac{u_r(1)^2}2.\n\\]\nThus (with $u\\in C^1(\\overline B_1\\backslash\\{0\\})$) the main task is to bound $\\int^1_0\\frac{u_r(t)^2}{t}\\,dt$. Using Villegas' estimate \\cite[Theorem 1.7]{v12} (see also \\cite{cc06}),\n\\[\n|u_r(t)|\\le C_n|u_r(1)|\\, t^{-n/2+\\sqrt{n-1}+1},\\qquad 0<t\\le 1.\n\\]\nSince $-n/2+\\sqrt{n-1}+1>0$ iff $2\\le n\\le 6$, this gives H\\\"older continuity of $u_r(t)$ near $t=0$; hence $t^{-1}u_r(t)^2$ behaves like $t^{-\\alpha}$ for some $\\alpha\\in(0,1)$ when $2\\le n\\le 6$. Therefore $\\int^1_0\\frac{u_r(t)^2}{t}\\,dt<\\infty$, yielding\n\\[\nF(\\|u\\|_{L^\\infty(B_1)})\\le C_n u_r(1)^2<+\\infty.\n\\]\nCombining this with $F(1)=\\infty$ gives $\\|u\\|_{L^\\infty(B_1)}<1$ (regularity). Finally, if $f$ is nondecreasing, the claimed universal bound for $F(\\|u\\|_{L^\\infty(B_1)})$ follows from an upper bound on $|u_r(1)|$. We use the following lemma.\n\\begin{lem}\\label{un-bd}\nLet $0\\le f\\in C^1([0,1])$ be nondecreasing. Assume\nthat $0\\le u\\le 1$ is a stable radial solution to\n\\eqref{d-p}. Then\n$$|u_r(1)|\\le 2.$$\n\n\\end{lem}\n", "expanded_theorem": "\\label{thm}\nLet $0\\leq f\\in C^1([0,1))$ satisfying $F(1)=+\\infty$.\nAssume that $0\\le u\\le 1$ is a  stable radial solution to\n\\begin{align}\\label{d-p}\n\\left\\{\n\\begin{aligned}\n-\\bdz u & = f(u)&\\quad{\\rm in}&\\quad B_1 \\setminus\\{ 0\\}  \\\\\nu& =0 \\quad&{\\rm on}&\\quad \\partial B_1,\\\\\n\\end{aligned}\n\\right.\n\\end{align}\nIf $2\\le n\\le 6$, then $u$ is regular and satisfies\n$$\\frac 12 u_r(1)^2\\le F(\\|u\\|_{L^\\infty(B_1)})\\le  Cu_r(1)^2.$$\nAdditionally, if $f$ is nondecreasing, then\n $$F(\\|u\\|_{L^\\infty(B_1)})\\le C,$$\nwhere  $C>0$ is a universal constant.,", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $B_1\\subset \\mathbb{R}^n$ be the unit ball, let $f:[0,1)\\to[0,\\infty)$ be a $C^1$ function, and define\n$$F(t):=\\int_0^t f(s)\\,ds.$$ \nAssume $F(1)=+\\infty$, $2\\le n\\le 6$, and that $u$ is a radial function with $0\\le u\\le 1$ solving\n$$\\begin{cases}\n-\\Delta u=f(u) & \\text{in } B_1\\setminus\\{0\\},\\\\\nu=0 & \\text{on } \\partial B_1,\n\\end{cases}$$\nwhere $u$ is stable in the sense that\n$$\\int_{B_1} f'(u)\\,\\xi^2\\,dx\\le \\int_{B_1} |\\nabla \\xi|^2\\,dx$$\nfor every $\\xi\\in C^\\infty(B_1)$ compactly supported in $B_1\\setminus\\{0\\}$. Here $u_r(1)$ denotes the radial derivative at $r=1$, and “$u$ is regular” means $\\|u\\|_{L^\\infty(B_1)}<1$. Which of the following statements is valid?", "correct_choice": {"label": "A", "text": "The solution $u$ is regular, and it satisfies\n$$\\frac12\\,u_r(1)^2\\le F\\bigl(\\|u\\|_{L^\\infty(B_1)}\\bigr)\\le C\\,u_r(1)^2.$$ \nMoreover, if $f$ is nondecreasing, then\n$$F\\bigl(\\|u\\|_{L^\\infty(B_1)}\\bigr)\\le C,$$\nfor a universal constant $C>0$."}, "choices": [{"label": "B", "text": "The solution $u$ is regular, and it satisfies\n$$\\frac12\\,u_r(1)^2\\le F\\bigl(\\|u\\|_{L^\\infty(B_1)}\\bigr)\\le C\\,u_r(1)^2.$$ \nMoreover, if $f$ is nondecreasing, then the same conclusion holds with a universal constant $C>0$ for every dimension $n\\ge 2$."}, {"label": "C", "text": "The solution $u$ is regular, and it satisfies\n$$F\\bigl(\\|u\\|_{L^\\infty(B_1)}\\bigr)\\le C\\,u_r(1)^2.$$"}, {"label": "D", "text": "The solution $u$ is regular, and it satisfies\n$$\\frac12\\,u_r(1)^2\\le F\\bigl(\\|u\\|_{L^\\infty(B_1)}\\bigr)\\le C\\,u_r(1)^2.$$ \nMoreover, if $f$ is nondecreasing, then\n$$F\\bigl(\\|u\\|_{L^\\infty(B_1)}\\bigr)\\le C,$$\nwhere $C>0$ is a constant depending only on $n$ and not on $f$ or $u$."}, {"label": "E", "text": "If $f$ is nondecreasing, then every stable radial solution $u$ satisfies\n$$\\|u\\|_{L^\\infty(B_1)}<1$$\nfor all $n\\ge 2$, and moreover\n$$\\frac12\\,u_r(1)^2\\le F\\bigl(\\|u\\|_{L^\\infty(B_1)}\\bigr)\\le C\\,u_r(1)^2.$$"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "dimension threshold from integrability exponent", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped lower bound and dropped universal nondecreasing conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "dependence of the universal constant in the monotone case", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "uses monotonicity to bypass the sharp restriction $2\\le n\\le 6$", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the conclusion in choice A, nor does it contain obvious wording cues that uniquely identify the correct option. The hypotheses are substantial, but they do not directly leak the exact form of the final estimates."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: it presents the full hypothesis set and asks for the valid conclusion. However, it is not a pure tautological restatement, since the choices differ in meaningful ways (dimension range, sharpness of inequalities, and dependence/universality of constants)."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the strongest correct conclusion from nearby variants, especially regarding the role of the dimension bound, the lower bound involving u_r(1), and universality of constants. Still, the task mainly tests recognition of the precise theorem statement rather than deeper derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target common failure modes: overextending the dimension range, weakening the conclusion, incorrectly changing constant dependence, or swapping the structure of the estimates. They are distinct and well aligned with how a learner might misremember the result."}, "total_score": 6, "overall_assessment": "A solid theorem-identification MCQ with strong distractors and little answer leakage, but it leans more toward precise recall than genuine generative mathematical reasoning."}}
{"id": "2602.21118v1", "paper_link": "http://arxiv.org/abs/2602.21118v1", "theorems_cnt": 5, "theorem": {"env_name": "main", "content": "Let $1<p<\\infty$ and let $\\Omega\\subseteq\\mathbb{R}^N$ be an open set. Let us suppose that there exists $k\\in\\mathbb{N}\\setminus\\{0\\}$ such that\n\\begin{equation}\\\n\\label{PL}\n\\lambda_{k,p}^{\\rm LS}(\\Omega)<\\mathcal{E}_p(\\Omega).\n\\end{equation}\nThen, for every $\\ell\\in\\{1,\\dots,k\\}$ the minmax value $\\lambda_{\\ell,p}^{\\rm LS}(\\Omega)$ is an eigenvalue of the Dirichlet $p-$Laplacian on $\\Omega$. \n\\par\nMoreover, we have:\n\\begin{itemize}\n\\item if $\\mathcal{E}_p(\\Omega)<+\\infty$,  for every $\\ell\\in\\{1,\\dots,k\\}$ there exist two constants $C_\\ell=C_\\ell(N, p, \\Omega)>0$ and $\\alpha_\\ell=\\alpha_\\ell(p,\\Omega)>0$ such that for every eigenfunction $u_\\ell\\in W^{1,p}_0(\\Omega)$ associated to $\\lambda_{\\ell,p}^{\\rm LS}(\\Omega)$, we have\n\\[\n|u_\\ell(x)|\\le C_\\ell\\,\\|u_\\ell\\|_{L^p(\\Omega)}\\,e^{-\\alpha_\\ell\\,|x|}\\,,\\qquad \\text{for every}\\ x\\in\\Omega;\n\\]\n\\item if $\\mathcal{E}_p(\\Omega)=+\\infty$, for every $\\ell\\in\\mathbb{N}\\setminus\\{0\\}$ and every $\\alpha>0$ there exists a constant $C_\\ell=C_\\ell(N, p,\\Omega,\\alpha)>0$ such that \n\\[\n|u_\\ell(x)|\\le C_\\ell\\,\\|u_\\ell\\|_{L^p(\\Omega)}\\,e^{-\\alpha\\,|x|}\\,,\\qquad \\text{for every}\\ x\\in\\Omega.\n\\]\n\\end{itemize}", "start_pos": 13456, "end_pos": 14636, "label": "PL"}, "ref_dict": {"defi:scp": "\\begin{defi}\n\\label{defi:scp}\nLet $V\\in L^1_{\\rm loc}(\\mathbb{R}^N)$ be a non-negative function. We say that $V$ is a {\\it strongly confining potential} if it has the following property: there exists an increasing function $\\alpha:(0,+\\infty)\\to(0,+\\infty)$ such that\n\\[\n\\lim_{R\\to +\\infty} \\alpha(R)=+\\infty,\n\\]\nand for every $R>0$ we have\n\\[\nV(x)\\ge \\alpha(R),\\qquad \\text{for a.e.}\\ x\\in \\mathbb{R}^N\\setminus B_R.\n\\]\n\\end{defi}", "coro:gradienti": "\\begin{coro}\n\\label{coro:gradienti}\nLet $1<p<\\infty$ and let $\\Omega\\subseteq\\mathbb{R}^N$ be an open set such that $\\lambda_{1,p}(\\Omega)>0$. For a constant $\\lambda$ such that \n\\[\n\\lambda_{1,p}(\\Omega)\\le \\lambda<\\mathcal E_p(\\Omega),\n\\] \nwe consider $v\\in W^{1,p}_0(\\Omega)$ a nonnegative subsolution of the equation \\eqref{subsol}.\nThen\\footnote{We use the notation $\\mathcal{C}$ for the same constant of Proposition \\ref{prop:lambdadecay}.}:\n\\begin{itemize}\n\\item if $\\mathcal{E}_p(\\Omega)<+\\infty$, there exists $M=M(p,\\lambda,\\Omega)>0$ such that \n \\begin{equation}\n\\label{decaygrad}\n\\|\\nabla v\\|_{L^p(\\Omega\\setminus B_{R+1})}\\le  M\\, e^{-\\alpha\\,R}\\,\\|v\\|_{L^p(\\Omega)}, \\qquad \\text{for every}\\ R>0,\n\\end{equation}\nwhere $\\alpha$ is the same exponent as in Proposition \\ref{prop:lambdadecay}. Moreover,  for every $0<\\beta<\\alpha$ there exists a constant  \n\\[\n\\widetilde{\\mathcal{C}}= \\widetilde{\\mathcal{C}}(\\mathcal{C}, p, \\lambda, \\beta, \\alpha-\\beta)>0,\n\\]\nsuch that \n\\begin{equation}\n\\label{decaygradiente}\n\\int_{\\Omega}\\left|\\nabla v\\right|^{p}\\,e^{\\beta p|x|}\\,dx\\le  \\widetilde{\\mathcal{C}}\\,\\|v\\|^p_{L^{p}(\\Omega)};\n\\end{equation}\n\\item if $\\mathcal{E}_p(\\Omega)=+\\infty$, for every $\\alpha>0$ there exists a constant $M=M(p, \\lambda, \\Omega,\\alpha)>0$ such that \\eqref{decaygrad} holds. Moreover,  for every $\\beta>0$ there exists a constant  \n\\[\n\\widetilde{\\mathcal{C}}= \\widetilde{\\mathcal{C}}(\\mathcal{C}, p, \\lambda, \\beta)>0,\n\\]\nsuch that \\eqref{decaygradiente} holds.\n\\end{itemize}\nIn particular, in both cases we have that \n\\[\n\\int_\\Omega |\\nabla v|^\\gamma\\,dx<+\\infty,\\qquad \\text{for every}\\ 0<\\gamma\\le p.\n\\]\n\\end{coro}", "mimmergop": "\\begin{equation}\n\\label{mimmergop}\nW^{1,p}_0(\\Omega)\\hookrightarrow L^p(\\Omega),\n\\end{equation}", "lm:delcanto": "\\begin{lm}\n\\label{lm:delcanto}\nLet $1<p<\\infty$ and let $\\Omega\\subseteq\\mathbb{R}^N$ be an open set. Then\n\\[\nW^{1,p}_0(\\Omega)\\hookrightarrow L^p(\\Omega)\\ \\text{is compact}\\qquad \\Longleftrightarrow\\qquad \\mathcal{E}_p(\\Omega)=+\\infty.\n\\]\n\\end{lm}", "prop:lambdadecay": "\\begin{prop}[Exponential decay at infinity] \n\\label{prop:lambdadecay}\nLet $1<p<\\infty$ and let $\\Omega\\subseteq\\mathbb{R}^N$ be an open set such that $\\lambda_{1,p}(\\Omega)>0$. For a constant $\\lambda$ such that \n\\[\n\\lambda_{1,p}(\\Omega)\\le \\lambda<\\mathcal E_p(\\Omega),\n\\] \nwe consider $v\\in W^{1,p}_0(\\Omega)$ a nonnegative subsolution of the equation\n\\begin{equation}\\label{subsol}\n-\\Delta_p u = \\lambda\\, u^{p-1},\\qquad \\text{in}\\ \\Omega.\n\\end{equation}\nThen:\n\\begin{itemize}\n\\item if $\\mathcal{E}_p(\\Omega)<+\\infty$,  there exist two constants $\\mathcal{C}=\\mathcal{C}(N, p, \\lambda, \\Omega)>0$ and $\\alpha=\\alpha(p, \\lambda, \\Omega)>0$ such that  $v$\nsatisfies the following decay estimate \n\\begin{equation}\n\\label{decayinfty}\n0\\le v(x)\\le \\mathcal{C}\\,\\|v\\|_{L^p(\\Omega)}\\,e^{-\\alpha\\,|x|},\\qquad \\text{for a.\\,e.}\\ x\\in\\Omega.\n\\end{equation}\nMoreover, the constant $\\alpha$ decreasingly depends on $\\lambda$ and we have \n\\[\n\\lim_{\\lambda\\nearrow \\mathcal{E}_p(\\Omega)}\\alpha=0;\n\\]\n\\item if $\\mathcal{E}_p(\\Omega)=+\\infty$, for every $\\alpha>0$ there exists a constant $\\mathcal{C}=\\mathcal{C}(N, p, \\lambda, \\Omega,\\alpha)>0$ such that \\eqref{decayinfty} holds.\n\\end{itemize}\n\\end{prop}", "minmaxLS": "\\begin{equation}\n\\label{minmaxLS}\n\\lambda_{k,p}^{\\rm LS}(\\Omega):=\\inf_{K\\in \\mathcal{W}_{k,p}(\\Omega)} \\max_{\\varphi\\in K} \\int_\\Omega |\\nabla \\varphi|^p\\,dx,\\qquad k\\in\\mathbb{N}\\setminus\\{0\\},\n\\end{equation}", "krasnodar": "\\begin{equation}\n\\label{krasnodar}\n\\gamma(A;X)=\\inf\\left\\{k\\in\\mathbb{N}\\, :\\, \\exists\\ \\text{a continuous odd map}\\ f:A\\to\\mathbb{S}^{k-1}\\right\\},\n\\end{equation}", "rem:SS": "\\begin{rem}[Fall-off]\n\\label{rem:SS}\nAs for the ``compact case'' $\\mathcal{E}_p(\\Omega)=+\\infty$, our decay estimate represents an extension to the nonlinear case of \\cite[Theorem 2]{Si2} by Simon, recalled above. The latter deals with Schr\\\"odinger operators $-\\Delta+V$ on the whole $\\mathbb{R}^N$, under the effect of a non-negative strongly confining potential $V$ (see Definition \\ref{defi:scp} below).\n\\par\nIn the ``non-compact case'' $\\mathcal{E}_p(\\Omega)<+\\infty$, our estimate is a nonlinear variation on the classical result by \\v{S}nol, previously recalled. In particular, we have that\n\\[\n\\alpha_k\\le \\dots\\le \\alpha_1,\n\\]\nwith the notation above.\nThat is, the higher the eigenvalue, the slower the exponential decay. Moreover, these exponents deteriorate (i.e. they approach $0$) as the eigenvalues approach the spectral threshold $\\mathcal{E}_p(\\Omega)$: we refer to Proposition \\ref{prop:lambdadecay} below, for this fact. \nThis behaviour is in perfect accordance with \\v{S}nol's result.\nIndeed, in \\cite[Theorem pag. 282]{Sn}, this estimate is proved for eigenfunctions of $-\\Delta+V$, corresponding to an eigenvalue $\\lambda$ below of the essential spectrum $\\mathcal{E}_V$. Also in \\cite{Sn}, the exponent $\\alpha$ dictating the fall-off depends on the difference $\\mathcal{E}_V-\\lambda$ and it goes to $0$, as this difference goes to $0$. More precisely, \\v{S}nol's exponent is given by \n\\[\n\\alpha=\\log\\left(1+\\frac{\\mathcal{E}_V-\\lambda}{K}\\right),\n\\]\nfor constant $K$ depending on the (positive) potential $V$.\n For these reasons, in what follows we will refer to the estimate of the Main Theorem as {\\it \\v{S}nol-Simon--type decay estimate}.\n\\par\nFinally, we wish to point out that our method of proof is different from those of both \\cite{Si2} and \\cite{Sn} and based on nonlinear techniques: we first obtain exponential decay in $L^p$ norm and then use an $L^\\infty-L^p$ estimate ``localized at infinity'' to transform it into a pointwise decay estimate. The decay in $L^p$ norm is obtained by combining Caccioppoli and Poincar\\'e inequalities, while the second estimate is a classical tool in the De Giorgi-Moser regularity theory for quasilinear equations.\n\\par \nIt is worth mentioning that, once we obtain decay estimates on the eigenfunctions, it is possible to exploit the equation so to obtain some decay and integrability estimates on their gradients, as well (see Corollary \\ref{coro:gradienti} below).\n\\end{rem}", "lm:giggi": "\\begin{lm}\n\\label{lm:giggi}\nLet $1<p<\\infty$ and let $\\Omega\\subseteq\\mathbb{R}^N$ be an open set such that $\\lambda_{1,p}(\\Omega)>0$. Let $X(\\Omega)$ be a Banach space with the following properties:\n\\begin{itemize}\n\\item $C^\\infty_0(\\Omega)\\subseteq X(\\Omega)$;\n\\vskip.2cm\n\\item $X(\\Omega)\\subseteq W^{1,p}_0(\\Omega)$ with continuous inclusion, i.e. there exists $C>0$ such that\n\\[\n\\|\\nabla \\varphi\\|_{L^p(\\Omega)}\\le C\\, \\|\\varphi\\|_{X(\\Omega)},\\qquad \\text{for every}\\ \\varphi\\in X(\\Omega).\n\\]\n\\end{itemize}\nWe set\n\\[\n\\mathcal{X}_{k,p}(\\Omega)=\\Big\\{K\\subseteq \\mathcal{S}_p(\\Omega)\\cap X(\\Omega)\\, :\\, K\\ \\text{symmetric and compact},\\, \\gamma(K;X(\\Omega))\\ge k\\Big\\},\n\\]\nand\n\\[\n\\mathcal{W}_{k,p}(\\Omega)=\\Big\\{K\\subseteq \\mathcal{S}_p(\\Omega)\\cap W^{1,p}_0(\\Omega)\\, :\\, K\\ \\text{symmetric and compact},\\, \\gamma(K;W^{1,p}_0(\\Omega))\\ge k\\Big\\}.\n\\]\nThen, for every $K\\in \\mathcal{W}_{k,p}(\\Omega)$ and every $\\delta>0$, there exists $K_\\delta\\in \\mathcal{X}_{k,p}(\\Omega)$ such that\n\\[\n\\max_{\\varphi\\in K_\\delta} \\int_\\Omega |\\nabla \\varphi|^p\\,dx\\le \\max_{\\varphi\\in K} \\int_\\Omega |\\nabla \\varphi|^p\\,dx+\\delta.\n\\]\n\\end{lm}", "mimmergo": "\\begin{equation}\n\\label{mimmergo}\nW^{1,2}_0(\\Omega)\\hookrightarrow L^2(\\Omega),\n\\end{equation}", "falloff": "\\begin{equation}\n\\label{falloff}\n|u(x)|\\le C\\,e^{-\\alpha\\,|x|},\\qquad \\text{for}\\ x\\in\\Omega,\n\\end{equation}", "perso": "\\begin{equation}\n\\label{perso}\n\\inf \\Big\\{\\lambda>0\\, :\\, \\lambda\\in\\sigma_{\\rm ess}(\\Omega)\\Big\\}=\\mathcal{E}(\\Omega).\n\\end{equation}"}, "pre_theorem_intro_text_len": 10848, "pre_theorem_intro_text": "\\subsection{Beneath the essential spectrum}\nLet us start by considering the spectrum of the Dirichlet-Laplacian on an open set $\\Omega\\subseteq\\mathbb{R}^N$. Whenever the embedding\n\\begin{equation}\n\\label{mimmergo}\nW^{1,2}_0(\\Omega)\\hookrightarrow L^2(\\Omega),\n\\end{equation}\nis compact, we know that this spectrum is discrete, made of a diverging sequence of positive eigenvalues $\\{\\lambda_k(\\Omega)\\}_{k\\in\\mathbb{N}\\setminus\\{0\\}}$, each one having a finite multiplicity. Characterizations of open sets for which the embedding \\eqref{mimmergo} is compact can be found for example in \\cite[Chapter 15]{Maz}.\n\\par\nMoreover, in this situation we can build a system of orthonormal (in $L^2$ sense) associated eigenfunctions $\\{u_k\\}_{k\\in\\mathbb{N}\\setminus\\{0\\}}\\subseteq W^{1,2}_0(\\Omega)$, which gives a Hilbertian basis of $L^2(\\Omega)$. Accordingly, we can characterize the eigenvalues as follows\n\\[\n\\lambda_1(\\Omega)=\\inf_{\\varphi\\in W^{1,2}_0(\\Omega)} \\left\\{\\int_\\Omega |\\nabla\\varphi|^2\\,dx\\, :\\, \\int_\\Omega |\\varphi|^2\\,dx=1\\right\\},\n\\] \nand for $k\\ge 2$\n\\[\n\\lambda_k(\\Omega)=\\inf_{\\varphi\\in W^{1,2}_0(\\Omega)} \\left\\{\\int_\\Omega |\\nabla\\varphi|^2\\,dx\\, :\\, \\int_\\Omega |\\varphi|^2\\,dx=1,\\ \\int_\\Omega \\varphi\\,u_j\\,dx=0, \\ \\text{for}\\ j=1,\\dots,k-1\\right\\}.\n\\]\nEach eigenfunction $u_k$ is a minimizer of the corresponding problem.\nSee for example \\cite[Chapter VI]{CH} or \\cite[Chapter 4, Section 8]{BraBook} for these facts.\n\\par\nWe also recall that an alternative characterization of the eigenvalues is given by the well-known {\\it Courant-Fischer formula}\n\\begin{equation}\n\\label{courant-fischer}\n\\lambda_k(\\Omega)=\\inf\\left\\{ \\max_{\\varphi\\in K\\cap \\mathcal{S}_2(\\Omega)} \\int_\\Omega|\\nabla \\varphi|^2\\,dx\\,:\\, K \\subseteq W^{1,2}_0(\\Omega)\\ \\mbox{subspace with}\\ \\dim K=k \\right\\},\n\\end{equation}\nwhere\n\\[\n\\mathcal{S}_2(\\Omega):=\\Big\\{u\\in L^2(\\Omega)\\, :\\, \\|u\\|_{L^2(\\Omega)}=1\\Big\\}.\n\\]\nHowever, for a generic open set, it may happen that the embedding \\eqref{mimmergo} fails to be compact. In this case, things become more complicated and intriguing: the spectrum of the Dirichlet-Laplacian stops being purely discrete (see \\cite[Theorem 10.1.5]{BS}) and we have a non-empty {\\it essential spectrum}, indicated by $\\sigma_{\\rm ess}(\\Omega)$. \n\\par\nWe recall that the latter can be characterized in terms of {\\it singular constrained Palais-Smale sequences} (see for example \\cite[Theorem 9.1.2]{BS} or \\cite[Theorem 7.2]{HS}). Namely, we have that \n$\\lambda\\in \\sigma_{\\rm ess}(\\Omega)$ if and only if there exists a sequence $\\{\\varphi_n\\}_{n\\in\\mathbb{N}}\\subseteq W^{1,2}_0(\\Omega)$ such that\n\\begin{enumerate}\n\\item $\\varphi_n\\in\\mathcal{S}_2(\\Omega)$, for every $n\\in\\mathbb{N}$;\n\\vskip.2cm\n\\item $\\lim\\limits_{n\\to\\infty}\\|\\nabla \\varphi_n\\|_{L^2(\\Omega)}^2=\\lambda$;\n\\vskip.2cm\n\\item we have\n\\[\n\\lim_{n\\to\\infty} \\|-\\Delta \\varphi_n-\\lambda\\,\\varphi_n\\|_{W^{-1,2}(\\Omega)}=0;\n\\]\n\\item $\\{\\varphi_n\\}_{n\\in\\mathbb{N}}$ weakly converges to $0$ in $L^2(\\Omega)$.\n\\end{enumerate}\nA celebrated result by Persson (see \\cite[Theorem 2.1]{Pe}) gives a characterization of ``geometric''  flavour to the bottom of $\\sigma_{\\rm ess}(\\Omega)$.\nMore precisely, if we introduce the {\\it Poincar\\'e constant ``at infinity''}\n\\[\n\\mathcal{E}(\\Omega):=\\sup_{R>0}\\lambda_1\\big(\\Omega\\setminus \\overline{B_R}\\big),\n\\]\nthen Persson's Theorem asserts that\n\\begin{equation}\n\\label{perso}\n\\inf \\Big\\{\\lambda>0\\, :\\, \\lambda\\in\\sigma_{\\rm ess}(\\Omega)\\Big\\}=\\mathcal{E}(\\Omega).\n\\end{equation}\nNevertheless, even when $\\sigma_{\\rm ess}(\\Omega)\\not=\\emptyset$, it may happen that some discrete eigenvalues with finite multiplicities ``pop-up'' beneath the essential spectrum, i.e. there may exist some eigenvalues $\\lambda<\\mathcal{E}(\\Omega)$. In this case, we have that these eigenvalues can still be characterized through the variational principles recalled above (see for example \\cite[Theorems 10.2.1 \\& 10.2.2]{BS}).\n\n\\par\nWe also recall that the corresponding eigenfunctions are strongly localized in space, in the sense that they exponentially ``fall-off'' at infinity.  In other words, they enjoy an estimate of the form\n\\begin{equation}\n\\label{falloff}\n|u(x)|\\le C\\,e^{-\\alpha\\,|x|},\\qquad \\text{for}\\ x\\in\\Omega,\n\\end{equation}\nfor suitable $C,\\alpha>0$. One of the very first papers in the literature proving this kind of results is \\cite{Sn} by \\v{S}nol, for Schr\\\"odinger operators of the form $-\\Delta+V$, with a non-negative potential $V$. We also wish to cite two papers \\cite{Si, Si2} by Simon, which are connected with our main result presented below. \n\\par\nNowadays, a result of the type \\eqref{falloff} is a very particular instance of decay estimates that are usually named after Agmon, following his influential work \\cite{Ag}. \nThere, one can find a systematic study of decay estimates for eigenfunctions of $-\\Delta+V$. Since then, it has become a classical subject in Spectral Theory. Many authors have contributed, by proving extensions, generalizations and refinements. Without any attempt of completeness, we mention the papers \\cite{AH2OM} and \\cite{CS}. An overview on Agmon estimates and their applications can be found in \\cite{He1,He2}.\n\n\\subsection{Eigenvalues of the $p-$Laplacian}\nThe aim of this paper is to extend to the case of the Dirichlet $p-$Laplacian the previous analysis of \n\\[\n``\\text{\\it eigenvalues lying beneath the essential spectrum\\,}''. \n\\]\nThis sentence requires some precisions: at first, what it is intended by {\\it eigenvalue} of the Dirichlet $p-$Laplacian on an open set $\\Omega\\subseteq\\mathbb{R}^N$. From the point of view of Critical Point Theory, this is readily said: these eigenvalues can be understood as critical values of the functional\n\\[\n\\varphi\\mapsto\\int_\\Omega |\\nabla \\varphi|^p\\,dx,\n\\]\nconstrained to the ``sphere''\n\\[\n\\mathcal{S}_p(\\Omega)=\\Big\\{u\\in L^p(\\Omega)\\, :\\, \\|u\\|_{L^p(\\Omega)}=1\\Big\\}.\n\\]\nThe associated critical points are the eigenfunctions, accordingly.\nThus, in other words, eigenvalues of the Dirichlet $p-$Laplacian are those numbers $\\lambda$ such that there exists $u\\in W^{1,p}_0(\\Omega)\\setminus\\{0\\}$ weakly solving\n\\[\n-\\Delta_p u=\\lambda\\,|u|^{p-2}\\,u,\\qquad \\text{in}\\ \\Omega,\n\\]\nthat is\n\\[\n\\int_\\Omega \\langle |\\nabla u|^{p-2}\\,\\nabla u,\\nabla\\varphi\\rangle\\,dx=\\lambda\\,\\int_\\Omega |u|^{p-2}\\,u\\,\\varphi,\\qquad \\text{for every}\\ \\varphi\\in C^\\infty_0(\\Omega).\n\\]\nObserve that the global infimum of the constrained Dirichlet integral\n\\[\n\\lambda_{1,p}(\\Omega):=\\inf_{\\varphi\\in W^{1,p}_0(\\Omega)} \\left\\{\\int_\\Omega |\\nabla\\varphi|^p\\,dx\\, :\\, \\int_\\Omega |\\varphi|^p\\,dx=1\\right\\},\n\\] \ncorresponds to the {\\it first eigenvalue} or {\\it ground state energy}.\n\\par\nOn the other hand, what is intended by {\\it essential spectrum} in this context is quite unclear. Nevertheless, by recalling Persson's result \\eqref{perso} for the linear case, one might bravely guess that the {\\it $L^p$ Poincar\\'e constant at infinity}\n\\[\n\\mathcal{E}_{p}(\\Omega):=\\sup_{R>0}\\lambda_{1,p}\\big(\\Omega\\setminus \\overline{B_R}\\big),\n\\]\ncould be a valid surrogate for the infimum of the essential spectrum, in this nonlinear context. We are then lead to the following questions: is it possible to produce eigenvalues of the Dirichlet $p-$Laplacian below the threshold $\\mathcal{E}_p(\\Omega)$? Do the corresponding eigenfunctions enjoy a decay estimate similar to \\eqref{falloff}? Attempting  to answer these questions, we will try to take very minimal assumptions on the open set $\\Omega$.\n\\vskip.2cm\\noindent\nLet us start with the simpler case $\\mathcal{E}_p(\\Omega)=+\\infty$: in this case, we have that the embedding\n\\begin{equation}\n\\label{mimmergop}\nW^{1,p}_0(\\Omega)\\hookrightarrow L^p(\\Omega),\n\\end{equation}\nis compact, see for example Lemma \\ref{lm:delcanto} below. In this scenario, it is well-known that we can produce a diverging sequence of positive eigenvalues by suitably mimicking the minmax Courant-Fischer procedure. This situation is very much studied in the literature, we just mention few classical references \\cite{GP, Le, Szu}, the lecture notes \\cite{Lin} and the PhD Thesis \\cite{Fra}, where one can find many more references.\n\\par\nWe briefly recall the construction of this sequence of eigenvalues:\nfor every $k\\in\\mathbb{N}\\setminus\\{0\\}$, we set\n\\[\n\\mathcal{W}_{k,p}(\\Omega)=\\Big\\{K\\subseteq \\mathcal{S}_p(\\Omega)\\cap W^{1,p}_0(\\Omega)\\, :\\, K\\ \\text{symmetric and compact},\\, \\gamma(K;W^{1,p}_0(\\Omega))\\ge k\\Big\\}.\n\\]\nHere $\\gamma(K;W^{1,p}_0(\\Omega))$ indicates the {\\it Krasnosel'ski\\u{\\i} genus} of $K$ in the norm topology of $W^{1,p}_0(\\Omega)$, see \\eqref{krasnodar} below for the definition.\nWith this class of compact sets at hand, we can define the {\\it $k-$th minmax Ljusternik-Schnirelmann value} as follows\n\\begin{equation}\n\\label{minmaxLS}\n\\lambda_{k,p}^{\\rm LS}(\\Omega):=\\inf_{K\\in \\mathcal{W}_{k,p}(\\Omega)} \\max_{\\varphi\\in K} \\int_\\Omega |\\nabla \\varphi|^p\\,dx,\\qquad k\\in\\mathbb{N}\\setminus\\{0\\},\n\\end{equation}\nand prove that this actually defines an eigenvalue, if the embedding \\eqref{mimmergop} is compact. Let us notice that, for $k=1$, the previous definition boils down $\\lambda_{1,p}(\\Omega)$, i.e. it really gives the first eigenvalue. For this reason, in what follows we will omit the superscript ``${\\rm LS}$'' in the case $k=1$. Moreover,  it is also possible to prove that, for  $k=2$, on a connected open set we have \n\\[\n\\lambda_{2,p}^{\\rm LS}(\\Omega)=\\min\\Big\\{\\lambda>\\lambda_{1,p}(\\Omega)\\,:\\, \\lambda\\ \\text{is an eigenvalue}\\Big\\},\n\\] \nsee for instance \\cite[Theorem 3.4]{JL}. \n\\begin{rem}\nIn order to appreciate these apparently pedantic remarks about $k=1$ and $k=2$, the reader should keep in mind that it is still a major open problem to know whether $\\{\\lambda_{k,p}^{\\rm LS}(\\Omega)\\}_{n\\in\\mathbb{N}\\setminus\\{0\\}}$ exhaust the whole spectrum or not, at least in the case when \\eqref{mimmergop} is compact. We also recall that it would be possible \nto define the previous minmax levels by means of another index $i$, in place of the Krasnosel'ski\\u{\\i} genus: for example, one could use the {\\it $\\mathbb{Z}_2-$cohomological index}\n\\cite{fadell_rabinowitz1977} or the {\\it Ljusternik-Schnirelman Category} \\cite[Chapter 2]{Str}.\n\\end{rem}\nWe then consider the case $\\mathcal{E}_p(\\Omega)<+\\infty$: here the embedding \\eqref{mimmergop} fails to be compact, see again Lemma \\ref{lm:delcanto} below. Nevertheless, one could still define the minmax values \\eqref{minmaxLS} and reasonably argue that whenever\n\\[\n\\lambda_{k,p}^{\\rm LS}(\\Omega)<\\mathcal{E}_p(\\Omega),\n\\]\nwe still have an eigenvalue, similarly to what happens in the linear case...\n\\subsection{Main result}\nWe will show that this is actually the case. Moreover, we will also get a \\v{S}nol-Simon--type decay estimate for the relevant eigenfunction (we refer to Remark \\ref{rem:SS} for the terminology). Namely, we prove the following", "context": "\\subsection{Eigenvalues of the $p-$Laplacian}\nThe aim of this paper is to extend to the case of the Dirichlet $p-$Laplacian the previous analysis of \n\\[\n``\\text{\\it eigenvalues lying beneath the essential spectrum\\,}''. \n\\]\nThis sentence requires some precisions: at first, what it is intended by {\\it eigenvalue} of the Dirichlet $p-$Laplacian on an open set $\\Omega\\subseteq\\mathbb{R}^N$. From the point of view of Critical Point Theory, this is readily said: these eigenvalues can be understood as critical values of the functional\n\\[\n\\varphi\\mapsto\\int_\\Omega |\\nabla \\varphi|^p\\,dx,\n\\]\nconstrained to the ``sphere''\n\\[\n\\mathcal{S}_p(\\Omega)=\\Big\\{u\\in L^p(\\Omega)\\, :\\, \\|u\\|_{L^p(\\Omega)}=1\\Big\\}.\n\\]\nThe associated critical points are the eigenfunctions, accordingly.\nThus, in other words, eigenvalues of the Dirichlet $p-$Laplacian are those numbers $\\lambda$ such that there exists $u\\in W^{1,p}_0(\\Omega)\\setminus\\{0\\}$ weakly solving\n\\[\n-\\Delta_p u=\\lambda\\,|u|^{p-2}\\,u,\\qquad \\text{in}\\ \\Omega,\n\\]\nthat is\n\\[\n\\int_\\Omega \\langle |\\nabla u|^{p-2}\\,\\nabla u,\\nabla\\varphi\\rangle\\,dx=\\lambda\\,\\int_\\Omega |u|^{p-2}\\,u\\,\\varphi,\\qquad \\text{for every}\\ \\varphi\\in C^\\infty_0(\\Omega).\n\\]\nObserve that the global infimum of the constrained Dirichlet integral\n\\[\n\\lambda_{1,p}(\\Omega):=\\inf_{\\varphi\\in W^{1,p}_0(\\Omega)} \\left\\{\\int_\\Omega |\\nabla\\varphi|^p\\,dx\\, :\\, \\int_\\Omega |\\varphi|^p\\,dx=1\\right\\},\n\\] \ncorresponds to the {\\it first eigenvalue} or {\\it ground state energy}.\n\\par\nOn the other hand, what is intended by {\\it essential spectrum} in this context is quite unclear. Nevertheless, by recalling Persson's result \\eqref{perso} for the linear case, one might bravely guess that the {\\it $L^p$ Poincar\\'e constant at infinity}\n\\[\n\\mathcal{E}_{p}(\\Omega):=\\sup_{R>0}\\lambda_{1,p}\\big(\\Omega\\setminus \\overline{B_R}\\big),\n\\]\ncould be a valid surrogate for the infimum of the essential spectrum, in this nonlinear context. We are then lead to the following questions: is it possible to produce eigenvalues of the Dirichlet $p-$Laplacian below the threshold $\\mathcal{E}_p(\\Omega)$? Do the corresponding eigenfunctions enjoy a decay estimate similar to \\eqref{falloff}? Attempting to answer these questions, we will try to take very minimal assumptions on the open set $\\Omega$.\n\\vskip.2cm\\noindent\nLet us start with the simpler case $\\mathcal{E}_p(\\Omega)=+\\infty$: in this case, we have that the embedding\n\\begin{equation}\n\\label{mimmergop}\nW^{1,p}_0(\\Omega)\\hookrightarrow L^p(\\Omega),\n\\end{equation}\nis compact, see for example Lemma \\ref{lm:delcanto} below. In this scenario, it is well-known that we can produce a diverging sequence of positive eigenvalues by suitably mimicking the minmax Courant-Fischer procedure. This situation is very much studied in the literature, we just mention few classical references \\cite{GP, Le, Szu}, the lecture notes \\cite{Lin} and the PhD Thesis \\cite{Fra}, where one can find many more references.\n\\par\nWe briefly recall the construction of this sequence of eigenvalues:\nfor every $k\\in\\mathbb{N}\\setminus\\{0\\}$, we set\n\\[\n\\mathcal{W}_{k,p}(\\Omega)=\\Big\\{K\\subseteq \\mathcal{S}_p(\\Omega)\\cap W^{1,p}_0(\\Omega)\\, :\\, K\\ \\text{symmetric and compact},\\, \\gamma(K;W^{1,p}_0(\\Omega))\\ge k\\Big\\}.\n\\]\nHere $\\gamma(K;W^{1,p}_0(\\Omega))$ indicates the {\\it Krasnosel'ski\\u{\\i} genus} of $K$ in the norm topology of $W^{1,p}_0(\\Omega)$, see \\eqref{krasnodar} below for the definition.\nWith this class of compact sets at hand, we can define the {\\it $k-$th minmax Ljusternik-Schnirelmann value} as follows\n\\begin{equation}\n\\label{minmaxLS}\n\\lambda_{k,p}^{\\rm LS}(\\Omega):=\\inf_{K\\in \\mathcal{W}_{k,p}(\\Omega)} \\max_{\\varphi\\in K} \\int_\\Omega |\\nabla \\varphi|^p\\,dx,\\qquad k\\in\\mathbb{N}\\setminus\\{0\\},\n\\end{equation}\nand prove that this actually defines an eigenvalue, if the embedding \\eqref{mimmergop} is compact. Let us notice that, for $k=1$, the previous definition boils down $\\lambda_{1,p}(\\Omega)$, i.e. it really gives the first eigenvalue. For this reason, in what follows we will omit the superscript ``${\\rm LS}$'' in the case $k=1$. Moreover, it is also possible to prove that, for $k=2$, on a connected open set we have \n\\[\n\\lambda_{2,p}^{\\rm LS}(\\Omega)=\\min\\Big\\{\\lambda>\\lambda_{1,p}(\\Omega)\\,:\\, \\lambda\\ \\text{is an eigenvalue}\\Big\\},\n\\] \nsee for instance \\cite[Theorem 3.4]{JL}. \n\\begin{rem}\nIn order to appreciate these apparently pedantic remarks about $k=1$ and $k=2$, the reader should keep in mind that it is still a major open problem to know whether $\\{\\lambda_{k,p}^{\\rm LS}(\\Omega)\\}_{n\\in\\mathbb{N}\\setminus\\{0\\}}$ exhaust the whole spectrum or not, at least in the case when \\eqref{mimmergop} is compact. We also recall that it would be possible \nto define the previous minmax levels by means of another index $i$, in place of the Krasnosel'ski\\u{\\i} genus: for example, one could use the {\\it $\\mathbb{Z}_2-$cohomological index}\n\\cite{fadell_rabinowitz1977} or the {\\it Ljusternik-Schnirelman Category} \\cite[Chapter 2]{Str}.\n\\end{rem}\nWe then consider the case $\\mathcal{E}_p(\\Omega)<+\\infty$: here the embedding \\eqref{mimmergop} fails to be compact, see again Lemma \\ref{lm:delcanto} below. Nevertheless, one could still define the minmax values \\eqref{minmaxLS} and reasonably argue that whenever\n\\[\n\\lambda_{k,p}^{\\rm LS}(\\Omega)<\\mathcal{E}_p(\\Omega),\n\\]\nwe still have an eigenvalue, similarly to what happens in the linear case...\n\\subsection{Main result}\nWe will show that this is actually the case. Moreover, we will also get a \\v{S}nol-Simon--type decay estimate for the relevant eigenfunction (we refer to Remark \\ref{rem:SS} for the terminology). Namely, we prove the following", "full_context": "\\subsection{Eigenvalues of the $p-$Laplacian}\nThe aim of this paper is to extend to the case of the Dirichlet $p-$Laplacian the previous analysis of \n\\[\n``\\text{\\it eigenvalues lying beneath the essential spectrum\\,}''. \n\\]\nThis sentence requires some precisions: at first, what it is intended by {\\it eigenvalue} of the Dirichlet $p-$Laplacian on an open set $\\Omega\\subseteq\\mathbb{R}^N$. From the point of view of Critical Point Theory, this is readily said: these eigenvalues can be understood as critical values of the functional\n\\[\n\\varphi\\mapsto\\int_\\Omega |\\nabla \\varphi|^p\\,dx,\n\\]\nconstrained to the ``sphere''\n\\[\n\\mathcal{S}_p(\\Omega)=\\Big\\{u\\in L^p(\\Omega)\\, :\\, \\|u\\|_{L^p(\\Omega)}=1\\Big\\}.\n\\]\nThe associated critical points are the eigenfunctions, accordingly.\nThus, in other words, eigenvalues of the Dirichlet $p-$Laplacian are those numbers $\\lambda$ such that there exists $u\\in W^{1,p}_0(\\Omega)\\setminus\\{0\\}$ weakly solving\n\\[\n-\\Delta_p u=\\lambda\\,|u|^{p-2}\\,u,\\qquad \\text{in}\\ \\Omega,\n\\]\nthat is\n\\[\n\\int_\\Omega \\langle |\\nabla u|^{p-2}\\,\\nabla u,\\nabla\\varphi\\rangle\\,dx=\\lambda\\,\\int_\\Omega |u|^{p-2}\\,u\\,\\varphi,\\qquad \\text{for every}\\ \\varphi\\in C^\\infty_0(\\Omega).\n\\]\nObserve that the global infimum of the constrained Dirichlet integral\n\\[\n\\lambda_{1,p}(\\Omega):=\\inf_{\\varphi\\in W^{1,p}_0(\\Omega)} \\left\\{\\int_\\Omega |\\nabla\\varphi|^p\\,dx\\, :\\, \\int_\\Omega |\\varphi|^p\\,dx=1\\right\\},\n\\] \ncorresponds to the {\\it first eigenvalue} or {\\it ground state energy}.\n\\par\nOn the other hand, what is intended by {\\it essential spectrum} in this context is quite unclear. Nevertheless, by recalling Persson's result \\eqref{perso} for the linear case, one might bravely guess that the {\\it $L^p$ Poincar\\'e constant at infinity}\n\\[\n\\mathcal{E}_{p}(\\Omega):=\\sup_{R>0}\\lambda_{1,p}\\big(\\Omega\\setminus \\overline{B_R}\\big),\n\\]\ncould be a valid surrogate for the infimum of the essential spectrum, in this nonlinear context. We are then lead to the following questions: is it possible to produce eigenvalues of the Dirichlet $p-$Laplacian below the threshold $\\mathcal{E}_p(\\Omega)$? Do the corresponding eigenfunctions enjoy a decay estimate similar to \\eqref{falloff}? Attempting to answer these questions, we will try to take very minimal assumptions on the open set $\\Omega$.\n\\vskip.2cm\\noindent\nLet us start with the simpler case $\\mathcal{E}_p(\\Omega)=+\\infty$: in this case, we have that the embedding\n\\begin{equation}\n\\label{mimmergop}\nW^{1,p}_0(\\Omega)\\hookrightarrow L^p(\\Omega),\n\\end{equation}\nis compact, see for example Lemma \\ref{lm:delcanto} below. In this scenario, it is well-known that we can produce a diverging sequence of positive eigenvalues by suitably mimicking the minmax Courant-Fischer procedure. This situation is very much studied in the literature, we just mention few classical references \\cite{GP, Le, Szu}, the lecture notes \\cite{Lin} and the PhD Thesis \\cite{Fra}, where one can find many more references.\n\\par\nWe briefly recall the construction of this sequence of eigenvalues:\nfor every $k\\in\\mathbb{N}\\setminus\\{0\\}$, we set\n\\[\n\\mathcal{W}_{k,p}(\\Omega)=\\Big\\{K\\subseteq \\mathcal{S}_p(\\Omega)\\cap W^{1,p}_0(\\Omega)\\, :\\, K\\ \\text{symmetric and compact},\\, \\gamma(K;W^{1,p}_0(\\Omega))\\ge k\\Big\\}.\n\\]\nHere $\\gamma(K;W^{1,p}_0(\\Omega))$ indicates the {\\it Krasnosel'ski\\u{\\i} genus} of $K$ in the norm topology of $W^{1,p}_0(\\Omega)$, see \\eqref{krasnodar} below for the definition.\nWith this class of compact sets at hand, we can define the {\\it $k-$th minmax Ljusternik-Schnirelmann value} as follows\n\\begin{equation}\n\\label{minmaxLS}\n\\lambda_{k,p}^{\\rm LS}(\\Omega):=\\inf_{K\\in \\mathcal{W}_{k,p}(\\Omega)} \\max_{\\varphi\\in K} \\int_\\Omega |\\nabla \\varphi|^p\\,dx,\\qquad k\\in\\mathbb{N}\\setminus\\{0\\},\n\\end{equation}\nand prove that this actually defines an eigenvalue, if the embedding \\eqref{mimmergop} is compact. Let us notice that, for $k=1$, the previous definition boils down $\\lambda_{1,p}(\\Omega)$, i.e. it really gives the first eigenvalue. For this reason, in what follows we will omit the superscript ``${\\rm LS}$'' in the case $k=1$. Moreover, it is also possible to prove that, for $k=2$, on a connected open set we have \n\\[\n\\lambda_{2,p}^{\\rm LS}(\\Omega)=\\min\\Big\\{\\lambda>\\lambda_{1,p}(\\Omega)\\,:\\, \\lambda\\ \\text{is an eigenvalue}\\Big\\},\n\\] \nsee for instance \\cite[Theorem 3.4]{JL}. \n\\begin{rem}\nIn order to appreciate these apparently pedantic remarks about $k=1$ and $k=2$, the reader should keep in mind that it is still a major open problem to know whether $\\{\\lambda_{k,p}^{\\rm LS}(\\Omega)\\}_{n\\in\\mathbb{N}\\setminus\\{0\\}}$ exhaust the whole spectrum or not, at least in the case when \\eqref{mimmergop} is compact. We also recall that it would be possible \nto define the previous minmax levels by means of another index $i$, in place of the Krasnosel'ski\\u{\\i} genus: for example, one could use the {\\it $\\mathbb{Z}_2-$cohomological index}\n\\cite{fadell_rabinowitz1977} or the {\\it Ljusternik-Schnirelman Category} \\cite[Chapter 2]{Str}.\n\\end{rem}\nWe then consider the case $\\mathcal{E}_p(\\Omega)<+\\infty$: here the embedding \\eqref{mimmergop} fails to be compact, see again Lemma \\ref{lm:delcanto} below. Nevertheless, one could still define the minmax values \\eqref{minmaxLS} and reasonably argue that whenever\n\\[\n\\lambda_{k,p}^{\\rm LS}(\\Omega)<\\mathcal{E}_p(\\Omega),\n\\]\nwe still have an eigenvalue, similarly to what happens in the linear case...\n\\subsection{Main result}\nWe will show that this is actually the case. Moreover, we will also get a \\v{S}nol-Simon--type decay estimate for the relevant eigenfunction (we refer to Remark \\ref{rem:SS} for the terminology). Namely, we prove the following\n\n\\par\nWe also recall that the corresponding eigenfunctions are strongly localized in space, in the sense that they exponentially ``fall-off'' at infinity. In other words, they enjoy an estimate of the form\n\\begin{equation}\n\\label{falloff}\n|u(x)|\\le C\\,e^{-\\alpha\\,|x|},\\qquad \\text{for}\\ x\\in\\Omega,\n\\end{equation}\nfor suitable $C,\\alpha>0$. One of the very first papers in the literature proving this kind of results is \\cite{Sn} by \\v{S}nol, for Schr\\\"odinger operators of the form $-\\Delta+V$, with a non-negative potential $V$. We also wish to cite two papers \\cite{Si, Si2} by Simon, which are connected with our main result presented below. \n\\par\nNowadays, a result of the type \\eqref{falloff} is a very particular instance of decay estimates that are usually named after Agmon, following his influential work \\cite{Ag}. \nThere, one can find a systematic study of decay estimates for eigenfunctions of $-\\Delta+V$. Since then, it has become a classical subject in Spectral Theory. Many authors have contributed, by proving extensions, generalizations and refinements. Without any attempt of completeness, we mention the papers \\cite{AH2OM} and \\cite{CS}. An overview on Agmon estimates and their applications can be found in \\cite{He1,He2}.\n\n\\subsection{Plan of the paper}\nThe paper starts with some preliminary materials contained in Section \\ref{sec:2}. The main content of Section \\ref{sec:3} is the \\v{S}nol-Simon--type estimate of Proposition \\ref{prop:lambdadecay}, which holds more generally for non-negative {\\it subsolutions} of the eigenvalue equation. As a consequence, we also obtain some decay and integrability properties for their gradients. Section \\ref{sec:4} is devoted to the ``perturbed'' eigenvalue problem described above, obtained by adding a strongly confining potential. Here, the technical Lemma \\ref{lm:giggi} may be of some independent interest. The proof of the Main Theorem is contained in Section \\ref{sec:5}, while in Section \\ref{sec:6} we provide some examples of unbounded open sets to which our result applies. The paper ends with two appendices: in particular, in Appendix \\ref{sec:B} we show that, for $p=2$, the minmax Ljusternik-Schnirelmann values below the essential spectrum coincide with the eigenvalues given by the Courant-Fischer formula.\n\n\\begin{prop}\n\\label{prop:tabaccaio}\nLet $\\Omega\\subseteq\\mathbb{R}^N$ be an open set and let us suppose that there exists $k\\in\\mathbb{N}\\setminus\\{0\\}$ such that\n\\begin{equation}\\\n\\label{PLC}\n\\lambda_{k,2}^{\\rm LS}(\\Omega)<\\mathcal{E}(\\Omega):=\\sup_{R>0} \\lambda_1(\\Omega\\setminus \\overline{B_R}).\n\\end{equation}\nThen, for every $\\ell\\in\\{1,\\dots,k\\}$ we have\n\\[\n\\lambda_{\\ell,2}^{\\rm LS}(\\Omega)=\\lambda_{\\ell}(\\Omega).\n\\]\n\\end{prop}\n\\begin{proof}\nWe can limit ourselves to the case $\\mathcal{E}(\\Omega)<+\\infty$, otherwise the result is already contained in \\cite[Theorem A.2]{BraParSqu}.\nWe may suppose that $k\\ge 2$, otherwise the claim is straightforward. We start by observing that for every vector subspace $K\\subseteq W^{1,2}_0(\\Omega)$ having dimension $\\ell$, it holds\n\\[\nK\\cap\\mathcal{S}_2(\\Omega)\\in \\mathcal{W}_{\\ell,2}(\\Omega),\n\\] \ni.e. $K\\cap\\mathcal{S}_2(\\Omega)$ is a compact symmetric subset of $W^{1,2}_0(\\Omega)$, with genus equal to $\\ell$ (see for example \\cite[Chapter II, Proposition 5.2]{Str}). This entails that \n\\[\n\\lambda_{\\ell,2}^{\\rm LS}(\\Omega)\\le \\lambda_\\ell(\\Omega).\n\\]\nIn order to prove the reverse inequality, we will use an approximation argument\\footnote{This route will avoid considering the Spectral Resolution of the Dirichlet-Laplacian on $\\Omega$ (see for example \\cite[Chapter 6]{BS}). Indeed, this last operator could have a continuous part in its spectrum, under the standing assumptions on the open set. On the contrary, the approximation argument will only need the spectral properties of operators having a discrete spectrum. The reader with a sufficiently good expertise in Spectral Theory will certainly find our argument a bit akward.}. Namely, let us consider the same operators of Proposition \\ref{prop:approssimazione}, i.e.\n\\[\n\\varphi\\mapsto-\\Delta\\varphi+V_n\\,\\varphi,\n\\]\nwith homogeneous Dirichlet boundary conditions. The quadratic form naturally associated to this operator is given by the functional $\\mathcal{G}_n$, defined in Proposition \\ref{prop:approssimazione}. By classical Spectral Theory, for every fixed $n\\in\\mathbb{N}\\setminus\\{0\\}$, the compactness of the embedding $W^{1,2}_0(\\Omega;V)\\hookrightarrow L^2(\\Omega)$ guarantees that such an operator has a discrete spectrum, made of a diverging sequence of positive eigenvalues, each one with finite multiplicity. They can be characterized through the Courant-Fischer formula\n\\[\n\\lambda_{\\ell}(\\Omega;V_n):=\\inf\\left\\{ \\max_{u\\in K\\cap \\mathcal{S}_2(\\Omega)} \\mathcal{G}_n(u)\\,:\\, K \\subseteq W^{1,2}_0(\\Omega;V)\\ \\mbox{subspace with}\\ \\dim K=\\ell \\right\\}.\n\\]\nMoreover, the infimum on the right-hand side is attained by\n\\[\nK_\\ell=\\mathrm{Vect}\\Big(\\left\\{u_{1,n},\\dots,u_{\\ell,n}\\right\\}\\Big),\n\\]\ni.e. by the vector subspace generated by the first $\\ell$ eigenfunctions of the operator. In fact, these eigenfunctions can be chosen so that \n\\[\n\\int_\\Omega u_{i,n}\\,u_{j,n}\\,dx=\\delta_{ij},\\qquad \\text{for every}\\ n\\in\\mathbb{N},\\ i,j\\in\\{1,\\dots,\\ell\\},\n\\]\nand the following characterization also holds:\n\\[\n\\lambda_{\\ell}(\\Omega;V_n)=\\min_{\\varphi\\in \\mathcal{S}_2(\\Omega)\\cap W^{1,2}_0(\\Omega;V)}\\left\\{\\mathcal{G}_n(\\varphi)\\,:\\, \\int_\\Omega \\varphi\\,u_{i,n}\\,dx=0,\\ i\\in\\{1,\\dots,\\ell-1\\}\\right\\}.\n\\]\nWe are now ready to conclude the proof: according to \\cite[Theorem A.2]{BraParSqu}, for every $\\ell\\in \\mathbb{N}\\setminus\\{0\\}$ we have\n\\[\n\\lambda_{\\ell,2}^{\\rm LS}(\\Omega;V_n)=\\lambda_{\\ell}(\\Omega;V_n)=\\int_\\Omega |\\nabla u_{\\ell,n}|^2\\,dx+\\int_\\Omega V_n\\,|u_{\\ell,n}|^2\\,dx,\n\\]\ni.e. the equality claimed in the statement holds true for the approximating operator.\nBy the proof of the Main Theorem, we know that each $\\{u_{i,n}\\}_{n\\in\\mathbb{N}}$ converges strongly in $L^2(\\Omega)$ to an eigenfunction $u_i$ of the Dirichlet-Laplacian on $\\Omega$, up to a subsequence. In particular, we still have \n\\begin{equation}\n\\label{ortaggi}\n\\int_\\Omega u_{i}\\,u_{j}\\,dx=\\delta_{ij},\\qquad \\text{for every}\\ i,j\\in\\{1,\\dots,\\ell\\}.\n\\end{equation}\nBy virtue of Proposition \\ref{prop:approssimazione} and by the lower semicontinuity of the Dirichlet integral, we thus get for every $\\ell\\in\\{1,\\dots,k\\}$\n\\[\n\\begin{split}\n\\lambda_{\\ell,2}^{\\rm LS}(\\Omega)=\\lim_{n\\to\\infty} \\lambda_{\\ell,2}^{\\rm LS}(\\Omega;V_n)\\ge\\lim_{n\\to\\infty} \\int_\\Omega |\\nabla u_{\\ell,n}|^2\\,dx\\ge \\int_\\Omega |\\nabla u_\\ell|^2\\,dx\\\\\n\\end{split}\n\\]\nOn the other hand, by defining the $\\ell-$dimensional vector subspace\n\\[\n\\overline{K}=\\mathrm{Vect}\\Big(\\left\\{u_{1},\\dots,u_{\\ell}\\right\\}\\Big),\n\\]\nwe get\n\\[\n\\begin{split}\n\\lambda_\\ell(\\Omega)\\le \\max_{\\varphi\\in \\overline{K}\\cap\\mathcal{S}_2(\\Omega)} \\int_\\Omega |\\nabla\\varphi|^2\\,dx&=\\max_{\\alpha=(\\alpha_1,\\dots,\\alpha_\\ell)\\in \\mathbb{S}^{\\ell-1}} \\int_\\Omega \\left|\\sum_{i=1}^\\ell \\alpha_i\\,\\nabla u_i\\right|^2\\,dx\\\\\n&=\\max_{\\alpha=(\\alpha_1,\\dots,\\alpha_\\ell)\\in \\mathbb{S}^{\\ell-1}} \\sum_{i=1}^\\ell |\\alpha_i|^2\\,\\int_\\Omega |\\nabla u_i|^2\\,dx\\le \\int_\\Omega |\\nabla u_\\ell|^2\\,dx,\n\\end{split}\n\\]\nthanks to the orthogonality relations \\eqref{ortaggi} and the fact that each $u_i$ is an eigenfunction. In turn, we get the reverse estimate \n\\[\n\\lambda_{\\ell,2}^{\\rm LS}(\\Omega)\\ge \\lambda_\\ell(\\Omega),\n\\]\nas desired.\n\\end{proof}", "post_theorem_intro_text_len": 7306, "post_theorem_intro_text": "The rest of this subsection is devoted to a couple of (detailed) comments on this result. We start with the exponential fall-off.\n\\begin{rem}[Fall-off]\n\\label{rem:SS}\nAs for the ``compact case'' $\\mathcal{E}_p(\\Omega)=+\\infty$, our decay estimate represents an extension to the nonlinear case of \\cite[Theorem 2]{Si2} by Simon, recalled above. The latter deals with Schr\\\"odinger operators $-\\Delta+V$ on the whole $\\mathbb{R}^N$, under the effect of a non-negative strongly confining potential $V$ (see Definition \\ref{defi:scp} below).\n\\par\nIn the ``non-compact case'' $\\mathcal{E}_p(\\Omega)<+\\infty$, our estimate is a nonlinear variation on the classical result by \\v{S}nol, previously recalled. In particular, we have that\n\\[\n\\alpha_k\\le \\dots\\le \\alpha_1,\n\\]\nwith the notation above.\nThat is, the higher the eigenvalue, the slower the exponential decay. Moreover, these exponents deteriorate (i.e. they approach $0$) as the eigenvalues approach the spectral threshold $\\mathcal{E}_p(\\Omega)$: we refer to Proposition \\ref{prop:lambdadecay} below, for this fact. \nThis behaviour is in perfect accordance with \\v{S}nol's result.\nIndeed, in \\cite[Theorem pag. 282]{Sn}, this estimate is proved for eigenfunctions of $-\\Delta+V$, corresponding to an eigenvalue $\\lambda$ below of the essential spectrum $\\mathcal{E}_V$. Also in \\cite{Sn}, the exponent $\\alpha$ dictating the fall-off depends on the difference $\\mathcal{E}_V-\\lambda$ and it goes to $0$, as this difference goes to $0$. More precisely, \\v{S}nol's exponent is given by \n\\[\n\\alpha=\\log\\left(1+\\frac{\\mathcal{E}_V-\\lambda}{K}\\right),\n\\]\nfor constant $K$ depending on the (positive) potential $V$.\n For these reasons, in what follows we will refer to the estimate of the Main Theorem as {\\it \\v{S}nol-Simon--type decay estimate}.\n\\par\nFinally, we wish to point out that our method of proof is different from those of both \\cite{Si2} and \\cite{Sn} and based on nonlinear techniques: we first obtain exponential decay in $L^p$ norm and then use an $L^\\infty-L^p$ estimate ``localized at infinity'' to transform it into a pointwise decay estimate. The decay in $L^p$ norm is obtained by combining Caccioppoli and Poincar\\'e inequalities, while the second estimate is a classical tool in the De Giorgi-Moser regularity theory for quasilinear equations.\n\\par \nIt is worth mentioning that, once we obtain decay estimates on the eigenfunctions, it is possible to exploit the equation so to obtain some decay and integrability estimates on their gradients, as well (see Corollary \\ref{coro:gradienti} below).\n\\end{rem}\nWe then briefly comment on how we prove the existence part of the Main Theorem.\n\\begin{rem}[Existence of eigenfunctions]\nThere are various ways to get existence of eigenfunctions. Of course, we can focus on the case $\\mathcal{E}_p(\\Omega)<+\\infty$, in light of what we have previously recalled. In the Hilbertian setting (i.\\,e. for $p=2$), a purely variational approach based on concentration-compactness arguments is provided by Smets, see \\cite[Theorem 4.1 $\\&$ Theorem 4.2]{Sm}, for a slightly different eigenvalue problem. His approach relies on the possibility to characterize higher eigenvalues as an infimum, possibly on the orthogonal complement of a finite dimensional vector space. This fact is true for $p=2$, but it has no counterpart for $p \\neq 2$. For this reason, it is less clear whether this approach can be applied for the full range of $p$, see for instance \\cite[Remark 4.4 (b)]{Sm}. \n\\par\nWe briefly explain our route. We use a {\\it spectral stability argument}, i.e. we prove that our minmax Ljusternik-Schnirelmann values can be approximated by the eigenvalues of a perturbed operator, having good compactness properties.\nMore precisely, we ``compactify'' the original eigenvalue problem by considering a new family of problems, each one obtained by adding a strongly confining potential $V_n$. For this family, we can prove existence of infinitely many ``perturbed'' eigenvalues, by the usual minmax procedure recalled above. Moreover, if $V_n$ tends to $0$ (in a suitable sense) as $n$ goes to $\\infty$, we can prove that these eigenvalues converge to the original minmax Ljusternik-Schnirelmann values. Then, in order to prove that those lying below $\\mathcal{E}_p(\\Omega)$ actually define an eigenvalue, we aim to infer strong convergence in $L^p$ of the ``perturbed'' eigenfunctions. Here, we crucially exploit that these ``perturbed'' eigenfunctions enjoy a uniform \\v{S}nol-Simon decay estimate, whenever they are associated to a ``perturbed'' eigenvalue which is converging to a $\\lambda_{k,p}^{\\rm LS}(\\Omega)<\\mathcal{E}_p(\\Omega)$. The uniform fall-off in turn implies an $L^p$ equi-tightness property, which permits to appeal to the classical Riesz-Fr\\'echet-Kolmogorov compactness criterion. In a nutshell, we obtain compactness through regularity. \n\\par\nIn the language of Calculus of Variations, this is essentially a $\\Gamma-$convergence--type argument in disguise, but used for higher critical points, other than the global minimum (see \\cite{Brai,Dal} for the general theory of $\\Gamma-$convergence). In doing this, we crucially borrow some ideas from the paper \\cite{CD} by Champion and De Pascale.\n\\end{rem}\n\n\\subsection{Plan of the paper}\nThe paper starts with some preliminary materials contained in Section \\ref{sec:2}. The main content of Section \\ref{sec:3} is the \\v{S}nol-Simon--type estimate of Proposition \\ref{prop:lambdadecay}, which holds more generally for non-negative {\\it subsolutions} of the eigenvalue equation. As a consequence, we also obtain some decay and integrability properties for their gradients. Section \\ref{sec:4} is devoted to the ``perturbed'' eigenvalue problem described above, obtained by adding a strongly confining potential. Here, the technical Lemma \\ref{lm:giggi} may be of some independent interest. The proof of the Main Theorem is contained in Section \\ref{sec:5}, while in Section \\ref{sec:6} we provide some examples of unbounded open sets to which our result applies. The paper ends with two appendices: in particular, in Appendix \\ref{sec:B} we show that, for $p=2$, the minmax Ljusternik-Schnirelmann values below the essential spectrum coincide with the eigenvalues given by the Courant-Fischer formula.\n\n\\begin{ack}\nWe thank Bernard Helffer for some discussions on Agmon--type estimates and for providing us a copy of his papers \\cite{He1, He2}.\nWe also thank Giovanni Franzina for some discussions on nonlinear eigenvalue problems and for a detailed reading of a preliminary draft of this paper.\n\\par\nL.\\,Briani and F.\\, Prinari are both members of the {\\it Gruppo Nazionale per l'Analisi Matematica, la Probabilit\\`a\ne le loro Applicazioni} (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). They both gratefully acknowledge the financial support of the project GNAMPA 2026  ``Problemi di ottimizzazione di forma in contesti anisotropi e non-locali\" ({\\tt  CUP E53C25002010001}).\n\\par\nL.\\,Brasco has been financially supported by the {\\it Fondo di Ateneo per la Ricerca} FAR 2024 and the {\\it Fondo per l'Incentivazione alla Ricerca Dipartimentale} FIRD 2025 of the University of Ferrara.\nThe research of L.\\,Briani has been supported by the DFG through the Emmy Noether Programme (project number 509436910).    \n\\end{ack}", "sketch": "For the \\v{S}nol-Simon--type decay estimate in Theorem~\\ref{PL}, the authors stress that their proof is “based on nonlinear techniques”: they “first obtain exponential decay in $L^p$ norm and then use an $L^\\infty-L^p$ estimate ‘localized at infinity’ to transform it into a pointwise decay estimate.” The “decay in $L^p$ norm is obtained by combining Caccioppoli and Poincar\\'e inequalities,” while the second step uses “a classical tool in the De Giorgi-Moser regularity theory for quasilinear equations.”\n\nFor the existence/eigenvalue part of Theorem~\\ref{PL}, they “use a \\emph{spectral stability argument},” i.e. “prove that [the] minmax Ljusternik-Schnirelmann values can be approximated by the eigenvalues of a perturbed operator, having good compactness properties.” Concretely, they “\\emph{compactify}” the problem by adding “a strongly confining potential $V_n$,” obtaining “infinitely many ‘perturbed’ eigenvalues, by the usual minmax procedure.” If $V_n\\to0$ suitably, these perturbed eigenvalues “converge to the original minmax Ljusternik-Schnirelmann values.” To show that limits below $\\mathcal{E}_p(\\Omega)$ yield actual eigenvalues, they seek “strong convergence in $L^p$ of the ‘perturbed’ eigenfunctions,” using that these enjoy a “uniform \\v{S}nol-Simon decay estimate” when the corresponding perturbed eigenvalues converge to some $\\lambda_{k,p}^{\\rm LS}(\\Omega)<\\mathcal{E}_p(\\Omega)$. This “uniform fall-off… implies an $L^p$ equi-tightness property,” allowing the “Riesz-Fr\\'echet-Kolmogorov compactness criterion”; “in a nutshell, we obtain compactness through regularity.” They also describe this as “essentially a $\\Gamma$-convergence--type argument in disguise, but used for higher critical points.”", "expanded_sketch": "For the \\v{S}nol-Simon--type decay estimate in Theorem~\\ref{PL}, the authors stress that their proof is “based on nonlinear techniques”: they “first obtain exponential decay in $L^p$ norm and then use an $L^\\infty-L^p$ estimate ‘localized at infinity’ to transform it into a pointwise decay estimate.” The “decay in $L^p$ norm is obtained by combining Caccioppoli and Poincar\\'e inequalities,” while the second step uses “a classical tool in the De Giorgi-Moser regularity theory for quasilinear equations.”\n\nFor the existence/eigenvalue part of the main theorem, they “use a \\emph{spectral stability argument},” i.e. “prove that [the] minmax Ljusternik-Schnirelmann values can be approximated by the eigenvalues of a perturbed operator, having good compactness properties.” Concretely, they “\\emph{compactify}” the problem by adding “a strongly confining potential $V_n$,” obtaining “infinitely many ‘perturbed’ eigenvalues, by the usual minmax procedure.” If $V_n\\to0$ suitably, these perturbed eigenvalues “converge to the original minmax Ljusternik-Schnirelmann values.” To show that limits below $\\mathcal{E}_p(\\Omega)$ yield actual eigenvalues, they seek “strong convergence in $L^p$ of the ‘perturbed’ eigenfunctions,” using that these enjoy a “uniform \\v{S}nol-Simon decay estimate” when the corresponding perturbed eigenvalues converge to some $\\lambda_{k,p}^{\\rm LS}(\\Omega)<\\mathcal{E}_p(\\Omega)$. This “uniform fall-off… implies an $L^p$ equi-tightness property,” allowing the “Riesz-Fr\\'echet-Kolmogorov compactness criterion”; “in a nutshell, we obtain compactness through regularity.” They also describe this as “essentially a $\\Gamma$-convergence--type argument in disguise, but used for higher critical points.”", "expanded_theorem": "Let $1<p<\\infty$ and let $\\Omega\\subseteq\\mathbb{R}^N$ be an open set. Let us suppose that there exists $k\\in\\mathbb{N}\\setminus\\{0\\}$ such that\n\\begin{equation}\\\n\\label{PL}\n\\lambda_{k,p}^{\\rm LS}(\\Omega)<\\mathcal{E}_p(\\Omega).\n\\end{equation}\nThen, for every $\\ell\\in\\{1,\\dots,k\\}$ the minmax value $\\lambda_{\\ell,p}^{\\rm LS}(\\Omega)$ is an eigenvalue of the Dirichlet $p-$Laplacian on $\\Omega$. \n\\par\nMoreover, we have:\n\\begin{itemize}\n\\item if $\\mathcal{E}_p(\\Omega)<+\\infty$,  for every $\\ell\\in\\{1,\\dots,k\\}$ there exist two constants $C_\\ell=C_\\ell(N, p, \\Omega)>0$ and $\\alpha_\\ell=\\alpha_\\ell(p,\\Omega)>0$ such that for every eigenfunction $u_\\ell\\in W^{1,p}_0(\\Omega)$ associated to $\\lambda_{\\ell,p}^{\\rm LS}(\\Omega)$, we have\n\\[\n|u_\\ell(x)|\\le C_\\ell\\,\\|u_\\ell\\|_{L^p(\\Omega)}\\,e^{-\\alpha_\\ell\\,|x|}\\,,\\qquad \\text{for every}\\ x\\in\\Omega;\n\\]\n\\item if $\\mathcal{E}_p(\\Omega)=+\\infty$, for every $\\ell\\in\\mathbb{N}\\setminus\\{0\\}$ and every $\\alpha>0$ there exists a constant $C_\\ell=C_\\ell(N, p,\\Omega,\\alpha)>0$ such that \n\\[\n|u_\\ell(x)|\\le C_\\ell\\,\\|u_\\ell\\|_{L^p(\\Omega)}\\,e^{-\\alpha\\,|x|}\\,,\\qquad \\text{for every}\\ x\\in\\Omega.\n\\]\n\\end{itemize}", "theorem_type": ["Implication", "Existence"], "mcq": {"question": "Let \\(1<p<\\infty\\) and let \\(\\Omega\\subseteq\\mathbb{R}^N\\) be an open set. Define\n\\[\n\\mathcal{S}_p(\\Omega)=\\{u\\in L^p(\\Omega):\\ \\|u\\|_{L^p(\\Omega)}=1\\}.\n\\]\nFor each \\(k\\in\\mathbb{N}\\setminus\\{0\\}\\), let\n\\[\n\\lambda_{k,p}^{\\rm LS}(\\Omega):=\\inf_{K\\in \\mathcal{W}_{k,p}(\\Omega)}\\ \\max_{\\varphi\\in K}\\int_\\Omega |\\nabla \\varphi|^p\\,dx,\n\\]\nwhere \\(\\mathcal{W}_{k,p}(\\Omega)\\) is the family of symmetric compact sets \\(K\\subseteq \\mathcal{S}_p(\\Omega)\\cap W^{1,p}_0(\\Omega)\\) whose Krasnosel'skii genus is at least \\(k\\). Also define\n\\[\n\\lambda_{1,p}(U):=\\inf\\left\\{\\int_U |\\nabla \\varphi|^p\\,dx:\\ \\varphi\\in W^{1,p}_0(U),\\ \\int_U |\\varphi|^p\\,dx=1\\right\\}\n\\]\nand the \\(L^p\\) Poincar\\'e constant at infinity by\n\\[\n\\mathcal{E}_p(\\Omega):=\\sup_{R>0}\\lambda_{1,p}(\\Omega\\setminus \\overline{B_R}).\n\\]\nAssume that there exists \\(k\\in\\mathbb{N}\\setminus\\{0\\}\\) such that\n\\[\n\\lambda_{k,p}^{\\rm LS}(\\Omega)<\\mathcal{E}_p(\\Omega).\n\\]\nHere \\(\\lambda\\) is an eigenvalue of the Dirichlet \\(p\\)-Laplacian on \\(\\Omega\\) if there exists \\(u\\in W^{1,p}_0(\\Omega)\\setminus\\{0\\}\\) such that \\(-\\Delta_p u=\\lambda |u|^{p-2}u\\) weakly in \\(\\Omega\\). Under these assumptions, which statement about the minmax values \\(\\lambda_{\\ell,p}^{\\rm LS}(\\Omega)\\) and their eigenfunctions holds?", "correct_choice": {"label": "A", "text": "For every \\(\\ell\\in\\{1,\\dots,k\\}\\), the minmax value \\(\\lambda_{\\ell,p}^{\\rm LS}(\\Omega)\\) is an eigenvalue of the Dirichlet \\(p\\)-Laplacian on \\(\\Omega\\). Moreover:\n\n- If \\(\\mathcal{E}_p(\\Omega)<+\\infty\\), then for every \\(\\ell\\in\\{1,\\dots,k\\}\\) there exist constants \\(C_\\ell=C_\\ell(N,p,\\Omega)>0\\) and \\(\\alpha_\\ell=\\alpha_\\ell(p,\\Omega)>0\\) such that every eigenfunction \\(u_\\ell\\in W^{1,p}_0(\\Omega)\\) associated with \\(\\lambda_{\\ell,p}^{\\rm LS}(\\Omega)\\) satisfies\n\\[\n|u_\\ell(x)|\\le C_\\ell\\,\\|u_\\ell\\|_{L^p(\\Omega)}\\,e^{-\\alpha_\\ell |x|}\\qquad\\text{for every }x\\in\\Omega.\n\\]\n- If \\(\\mathcal{E}_p(\\Omega)=+\\infty\\), then for every \\(\\ell\\in\\mathbb{N}\\setminus\\{0\\}\\) and every \\(\\alpha>0\\) there exists a constant \\(C_\\ell=C_\\ell(N,p,\\Omega,\\alpha)>0\\) such that\n\\[\n|u_\\ell(x)|\\le C_\\ell\\,\\|u_\\ell\\|_{L^p(\\Omega)}\\,e^{-\\alpha |x|}\\qquad\\text{for every }x\\in\\Omega.\n\\]"}, "choices": [{"label": "B", "text": "For every \\(\\ell\\in\\{1,\\dots,k\\}\\), the minmax value \\(\\lambda_{\\ell,p}^{\\rm LS}(\\Omega)\\) is an eigenvalue of the Dirichlet \\(p\\)-Laplacian on \\(\\Omega\\). Moreover:\n\n- If \\(\\mathcal{E}_p(\\Omega)<+\\infty\\), then for every \\(\\ell\\in\\{1,\\dots,k\\}\\) and every \\(\\alpha>0\\) there exists a constant \\(C_\\ell=C_\\ell(N,p,\\Omega,\\alpha)>0\\) such that every eigenfunction \\(u_\\ell\\in W^{1,p}_0(\\Omega)\\) associated with \\(\\lambda_{\\ell,p}^{\\rm LS}(\\Omega)\\) satisfies\n\\[\n|u_\\ell(x)|\\le C_\\ell\\,\\|u_\\ell\\|_{L^p(\\Omega)}\\,e^{-\\alpha |x|}\\qquad\\text{for every }x\\in\\Omega.\n\\]\n- If \\(\\mathcal{E}_p(\\Omega)=+\\infty\\), then for every \\(\\ell\\in\\mathbb{N}\\setminus\\{0\\}\\) there exist constants \\(C_\\ell=C_\\ell(N,p,\\Omega)>0\\) and \\(\\alpha_\\ell=\\alpha_\\ell(p,\\Omega)>0\\) such that\n\\[\n|u_\\ell(x)|\\le C_\\ell\\,\\|u_\\ell\\|_{L^p(\\Omega)}\\,e^{-\\alpha_\\ell |x|}\\qquad\\text{for every }x\\in\\Omega.\n\\]"}, {"label": "C", "text": "For every \\(\\ell\\in\\{1,\\dots,k\\}\\), the minmax value \\(\\lambda_{\\ell,p}^{\\rm LS}(\\Omega)\\) is an eigenvalue of the Dirichlet \\(p\\)-Laplacian on \\(\\Omega\\)."}, {"label": "D", "text": "For every \\(\\ell\\in\\mathbb{N}\\setminus\\{0\\}\\), the minmax value \\(\\lambda_{\\ell,p}^{\\rm LS}(\\Omega)\\) is an eigenvalue of the Dirichlet \\(p\\)-Laplacian on \\(\\Omega\\). Moreover:\n\n- If \\(\\mathcal{E}_p(\\Omega)<+\\infty\\), then for every \\(\\ell\\in\\mathbb{N}\\setminus\\{0\\}\\) there exist constants \\(C_\\ell=C_\\ell(N,p,\\Omega)>0\\) and \\(\\alpha_\\ell=\\alpha_\\ell(p,\\Omega)>0\\) such that every eigenfunction \\(u_\\ell\\in W^{1,p}_0(\\Omega)\\) associated with \\(\\lambda_{\\ell,p}^{\\rm LS}(\\Omega)\\) satisfies\n\\[\n|u_\\ell(x)|\\le C_\\ell\\,\\|u_\\ell\\|_{L^p(\\Omega)}\\,e^{-\\alpha_\\ell |x|}\\qquad\\text{for every }x\\in\\Omega.\n\\]\n- If \\(\\mathcal{E}_p(\\Omega)=+\\infty\\), then for every \\(\\ell\\in\\mathbb{N}\\setminus\\{0\\}\\) and every \\(\\alpha>0\\) there exists a constant \\(C_\\ell=C_\\ell(N,p,\\Omega,\\alpha)>0\\) such that\n\\[\n|u_\\ell(x)|\\le C_\\ell\\,\\|u_\\ell\\|_{L^p(\\Omega)}\\,e^{-\\alpha |x|}\\qquad\\text{for every }x\\in\\Omega.\n\\]"}, {"label": "E", "text": "For every \\(\\ell\\in\\{1,\\dots,k\\}\\), the minmax value \\(\\lambda_{\\ell,p}^{\\rm LS}(\\Omega)\\) is an eigenvalue of the Dirichlet \\(p\\)-Laplacian on \\(\\Omega\\). Moreover:\n\n- If \\(\\mathcal{E}_p(\\Omega)<+\\infty\\), then for every \\(\\ell\\in\\{1,\\dots,k\\}\\) there exist constants \\(C_\\ell=C_\\ell(N,p,\\Omega)>0\\) and \\(\\alpha_\\ell=\\alpha_\\ell(p,\\Omega)>0\\) such that every eigenfunction \\(u_\\ell\\in W^{1,p}_0(\\Omega)\\) associated with \\(\\lambda_{\\ell,p}^{\\rm LS}(\\Omega)\\) satisfies\n\\[\n\\|u_\\ell\\|_{L^p(\\Omega\\setminus B_R)}\\le C_\\ell\\,\\|u_\\ell\\|_{L^p(\\Omega)}\\,e^{-\\alpha_\\ell R}\\qquad\\text{for every }R>0.\n\\]\n- If \\(\\mathcal{E}_p(\\Omega)=+\\infty\\), then for every \\(\\ell\\in\\mathbb{N}\\setminus\\{0\\}\\) and every \\(\\alpha>0\\) there exists a constant \\(C_\\ell=C_\\ell(N,p,\\Omega,\\alpha)>0\\) such that\n\\[\n\\|u_\\ell\\|_{L^p(\\Omega\\setminus B_R)}\\le C_\\ell\\,\\|u_\\ell\\|_{L^p(\\Omega)}\\,e^{-\\alpha R}\\qquad\\text{for every }R>0.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "finite-vs-infinite-threshold decay quantifiers", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped the decay conclusions, keeping only eigenvalue existence for indices up to k", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "range of indices controlled by the assumption below the threshold", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "pointwise decay conclusion replaced by mere tail \\(L^p\\)-decay estimate", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct option. It gives hypotheses and asks for the valid conclusion, without obvious lexical clues pointing uniquely to A."}, "TAS": {"score": 1, "justification": "The item is very close to a theorem-recall question: the stem states the assumptions and asks which conclusion holds. The options do vary meaningful quantifiers and ranges, so it is not a pure verbatim restatement, but it remains only a mild reformulation."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to track the index range and the decay quantifiers in the cases Ep(Omega)<infinity versus Ep(Omega)=infinity. However, the task is mostly recognition of the exact theorem statement rather than genuine derivation, and the presence of a weaker true option reduces real generative pressure."}, "DQS": {"score": 1, "justification": "Several distractors are mathematically plausible and target common errors: swapping quantifiers, overextending from l<=k to all l, and weakening pointwise decay to tail L^p decay. But choice C is a weaker statement that also appears true under the stated assumptions, so the single-correct structure is compromised."}, "total_score": 5, "overall_assessment": "A mathematically sophisticated but only moderate-quality MCQ: it avoids direct leakage and has mostly strong distractor design, yet it is largely theorem-recall and is weakened by an apparently true weaker option, making the item ambiguous."}}
{"id": "2602.21304v1", "paper_link": "http://arxiv.org/abs/2602.21304v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm:intro-main}\nFor every $r\\ge 1$ there is a natural equivalence of groupoids\n\\begin{equation}\\label{eq:intro-main-equiv}\n\\Rep_r(\\G_U)\\times^{(2)}_{\\Rep_r(\\G_{\\times})} \\operatorname{Sec}(\\pi)\n\\ \\simeq\\ \\Stokes_r(X^{\\log}_n;D,\\Phi),\n\\end{equation}\ncompatible with restriction to $\\Rep_r(\\G_U)$ (forgetting the wild layer) and with restriction to $\\operatorname{Sec}(\\pi)$\n(forgetting the interior).", "start_pos": 12115, "end_pos": 12509, "label": "thm:intro-main"}, "ref_dict": {"eq:intro-global-stokes": "\\begin{equation}\\label{eq:intro-global-stokes}\n\\Stokes_r(\\Xlogn;D,\\Phi)\n:=\n\\LocSys_r(U)\\times^{(2)}_{\\LocSys_r(N^\\circ)}\n\\StokesLocal_r(N^\\circ;\\Phi).\n\\end{equation}", "eq:pushout-intro": "\\begin{equation}\\label{eq:pushout-intro}\n\\G_{\\Phi,n}\\ :=\\ \\GU \\sqcup_{\\GX} \\GC.\n\\end{equation}", "subsec:curve-model-clean": "\\begin{proof}\nThe congruence generated by $R$ forces the subgroupoid structures of $\\GU$ and $\\GC$ and imposes the strict identifications prescribed by $j_U,j_C$ on $\\GX$.\nThis is exactly the generators-and-relations model of the explicit $2$-pushout.\n\\end{proof}\n\n\\subsection{A completely explicit curve model: the explicit $2$-pushout groupoid $\\G_{\\Phi,n}$ and the forgetful map $\\pi:\\GC\\to\\CB$}\\label{subsec:curve-model-clean}\n\nThis subsection is expository and is not used later.\nIts goal is to make the explicit $2$-pushout\n\\begin{equation}\\label{eq:curve-pushout-clean}\n\\G_{\\Phi,n}\\ :=\\ \\GU\\sqcup_{\\GX}\\GC\n\\end{equation}\ncompletely concrete in complex dimension~$1$, and to clarify the comparison with the groupoids\nin \\cite[\\S3,\\S5]{GualtieriLiPym} and the Kato--Nakayama viewpoint.\n\n\\medskip\n\n\\medskip\n\n\\providecommand{\\Cech}{\\check{\\mathrm C}}\n\n\\providecommand{\\CB}{\\Cech(\\mathcal B)}\n\n\\subsubsection*{(A) Geometry on a curve and sector boxes on $X^{\\log}_n$}\n\nLet $X$ be a connected Riemann surface and let $D=\\{p_1,\\dots,p_m\\}\\subset X$ be a reduced finite divisor.\nWrite $U:=X\\setminus D$.\nFix a Kummer level $n\\ge 1$ and an irregular type $\\Phi$ along $D$ (at level~$n$).\nLet $\\tau_n:X^{\\log}_n\\to X$ be the level-$n$ Kato--Nakayama/Kummer map.\nOver $U$, $\\tau_n$ is a homeomorphism; over each $p_i$ the fiber is the level-$n$ circle of directions\n$S^1_{p_i,n}:=\\tau_n^{-1}(p_i)$.\nChoose pairwise disjoint closed discs $\\Delta_i$ around $p_i$ and set $\\Delta_i^\\times:=\\Delta_i\\setminus\\{p_i\\}$.\nLet $N_i^\\times:=\\tau_n^{-1}(\\Delta_i^\\times)\\subset X^{\\log}_n$ be the punctured collar near $p_i$.\nThe Stokes walls at $p_i$ cut $S^1_{p_i,n}$ into finitely many open arcs\n\\begin{equation}\\label{eq:arcs-clean}\nS^1_{p_i,n}\\setminus\\Sigma_{p_i}\\ =\\ \\bigsqcup_{a\\in\\ZZ/\\ell_i\\ZZ} I_{i,a},\n\\end{equation}\nand on each arc the Stokes preorder is constant.\nChoose \\emph{sector boxes}\n\\begin{equation}\\label{eq:boxes-clean}\nB_{i,a}\\ \\subset\\ N_i^\\times,\n\\qquad\nB_{i,a}\\ \\simeq\\ \\Delta_i^\\times\\times I_{i,a},\n\\qquad\nB_{i,a}\\cap B_{i,a+1}\\neq\\varnothing,\n\\end{equation}\neach contractible, and with overlaps also contractible.\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}[\n  >=Stealth,\n  font=\\scriptsize,\n  lab/.style={fill=white,inner sep=1pt},\n  vtx/.style={circle,draw,inner sep=1.2pt},\n  box/.style={draw,rounded corners,inner sep=3pt,align=center}\n]\n\n\\node  (Xbox) at (-6.0,2.35) {$X$ with $D=\\{p_1,\\dots,p_m\\}$,\\quad $U=X\\setminus D$};\n\n\\draw[thick] (-7.6,0.55) .. controls (-7.4,1.75) and (-4.6,1.75) .. (-4.4,0.55)\n             .. controls (-4.6,-0.65) and (-7.4,-0.65) .. (-7.6,0.55);\n\n\\node[vtx] (pi) at (-6.85,0.95) {$p_i$};\n\\draw[dashed,thick] (-6.85,0.95) circle (0.55);\n\\node[lab] at (-6.85,1.62) {$\\Delta_i$};\n\n\\draw[->,thick,bend left=12] (-4.4,0.55)\n  to node[lab,above,yshift=4pt] {zoom at $p_i$} (-1.70,0.9);\n\n\\node (Fbox) at (0.0,2.6) {$\\tau_n^{-1}(p_i)=S^1_{p_i,n}$};\n\n\\def\\R{1.15}\n\\coordinate (O) at (0,0.75);\n\\draw[thick] (O) circle (\\R);\n\n\\foreach \\ang in {25,115,205,295}{\n  \\draw[thick] (O) -- ({\\R*cos(\\ang)},{0.75+\\R*sin(\\ang)});\n}\n\\node[lab,anchor=west] at (1.32,0.75) {$\\Sigma_{p_i}$};\n\n\\node[lab] at ({1.48*cos(70)},{0.75+1.48*sin(70)}) {$I_{i,a}$};\n\\node[lab,xshift=-6pt] at ({1.48*cos(150)},{0.75+1.48*sin(150)}) {$I_{i,a+1}$};\n\\node[lab] at ({1.48*cos(250)},{0.75+1.48*sin(250)}) {$I_{i,a+2}$};\n\n\\node[lab,align=center] at (0,-1.05)\n{$S^1_{p_i,n}\\setminus\\Sigma_{p_i}=\\bigsqcup_{a\\in\\ZZ/\\ell_i\\ZZ} I_{i,a}$};\n\n\\draw[->,thick,bend left=10] (1.70,0.85) to node[lab,above] {thicken} (3.55,0.95);\n\n\\node (Nbox) at (5.6,2.35) {$N_i^\\times=\\tau_n^{-1}(\\Delta_i^\\times)$,\\quad $\\Delta_i^\\times=\\Delta_i\\setminus\\{p_i\\}$};\n\n\\draw[thick] (3.90,-0.10) rectangle (7.30,1.65);\n\n\\draw[->,thin] (3.95,-0.55) -- (7.25,-0.55)\n  node[lab,midway,below] {radial in $\\Delta_i^\\times$};\n\\draw[->,thin] (7.65,-0.10) -- (7.65,1.65)\n  node[lab,midway,right] {angular};\n\n\\draw[dashed,thick] (4.55,-0.10) -- (4.55,1.65);\n\\draw[dashed,thick] (5.25,-0.10) -- (5.25,1.65);\n\\draw[dashed,thick] (5.95,-0.10) -- (5.95,1.65);\n\\draw[dashed,thick] (6.65,-0.10) -- (6.65,1.65);\n\n\\node[lab] at (4.90,0.85) {$B_{i,a}$};\n\\node[lab] at (5.60,0.85) {$B_{i,a+1}$};\n\n\\node[lab,align=center] at (5.60,-1.30)\n{$B_{i,a}\\simeq \\Delta_i^\\times\\times I_{i,a}$,\\quad\n$B_{i,a}\\cap B_{i,a+1}\\neq\\varnothing$\\\\\nboxes and overlaps chosen contractible};\n\n\\end{tikzpicture}\n\\caption{At a puncture $p_i$, the level-$n$ circle of directions $S^1_{p_i,n}$ is cut by Stokes walls $\\Sigma_{p_i}$ into finitely many arcs $I_{i,a}$.\nA punctured collar $N_i^\\times$ over $\\Delta_i^\\times$ is then covered by sector boxes $B_{i,a}\\simeq \\Delta_i^\\times\\times I_{i,a}$ with adjacent overlaps.}\n\\label{fig:curve-collar-sector-boxes}\n\\end{figure}\n\nLet $\\St_\\Phi$ be the Stokes sheaf on the sector cover (as in Definition~\\ref{def:stokes-sheaf}).\nSince $B_{i,a}\\cap B_{i,a+1}$ is contractible, the group\n\\begin{equation}\\label{eq:stokes-groups-clean}\n\\St_{i;a,a+1}\\ :=\\ \\St_\\Phi(B_{i,a}\\cap B_{i,a+1})\n\\end{equation}\nis a constant (typically unipotent) subgroup of the relevant $\\Aut(\\Gr)$.\n\n\\subsubsection*{(B) The three groupoids $\\GU,\\GX,\\GC$ and the \\v{C}ech collar groupoid $\\CB$}\n\nFix once and for all the following covers.\n\n\\begin{itemize}\n\\item \\textbf{Interior cover on $U$.} Choose a good cover $\\mathcal U=\\{U_\\alpha\\}$ of $U$.\nSet $\\GU:=\\Cech(\\mathcal U)$.\n\n\\item \\textbf{Collar cover on $N^\\times:=\\bigsqcup_i N_i^\\times$.}\nSet $\\mathcal B:=\\{B_{i,a}\\}_{i,a}$ and define the \\emph{plain collar \\v{C}ech groupoid}\n\\begin{equation}\\label{eq:CB-def-clean}\n\\CB\\ :=\\ \\Cech(\\mathcal B).\n\\end{equation}\n\n\\item \\textbf{Overlap cover.} Consider the overlaps $U_\\alpha\\cap B_{i,a}\\subset U\\cap N^\\times$ and let\n$\\mathcal W:=\\{W_{\\alpha;i,a}\\}$ be any refinement by contractible opens.\nSet $\\GX:=\\Cech(\\mathcal W)$.\n\\end{itemize}\n\nNow define the \\emph{collar \\v{C}ech--Stokes groupoid} $\\GC$ as the Stokes decoration of $\\CB$:\n\\begin{equation}\\label{eq:GC-objects-clean}\n(\\GC)_0\\ :=\\ \\bigsqcup_{i,a} B_{i,a},\n\\end{equation}\nand arrows\n\\begin{equation}\\label{eq:GC-arrows-clean}\n(\\GC)_1\\ :=\\ \\bigsqcup_{i,a,a'}\\Bigl((B_{i,a}\\cap B_{i,a'})\\times \\St_\\Phi(B_{i,a}\\cap B_{i,a'})\\Bigr),\n\\end{equation}\nwhere $(x,u)$ is an arrow from the object $B_{i,a}$ to the object $B_{i,a'}$ (at the point $x$),\nand composition is\n\\begin{equation}\\label{eq:GC-comp-clean}\n(x,u')\\circ(x,u)\\ :=\\ (x,u'u).\n\\end{equation}\n\n\\medskip\n\n\\noindent\\textbf{The forgetful projection.}\nThere is a \\emph{strict} functor\n\\begin{equation}\\label{eq:pi-forgetful-clean}\n\\pi:\\GC\\longrightarrow \\CB\n\\end{equation}\ndefined by \\emph{forgetting the Stokes decoration}:\non objects it is the identity (same sector boxes), and on arrows\n\\begin{equation}\\label{eq:pi-on-arrows-clean}\n\\pi(x,u)\\ :=\\ x \\in (B_{i,a}\\cap B_{i,a'})\\ \\subset\\ (\\CB)_1.\n\\end{equation}\nSo here $\\CB$ is the \\v{C}ech groupoid of the cover $\\mathcal B$, \\emph{not} $\\CC$.\n\n\\subsubsection*{(C) The explicit $2$-pushout and its ``generators-and-relations'' meaning}\n\nThere are canonical strict functors $j_U:\\GX\\to\\GU$ and $j_C:\\GX\\to\\GC$ induced by inclusions:\non objects, $W_{\\alpha;i,a}\\subset U_\\alpha$ and $W_{\\alpha;i,a}\\subset B_{i,a}$;\non arrows, overlaps map to the corresponding \\v{C}ech arrows, and in $\\GC$ we take the\n\\emph{trivial} Stokes label $u=1$.\n\n\\begin{definition} \\label{def:curve-pushout-clean}\nDefine\n\\begin{equation}\\label{eq:pushout-clean}\n\\G_{\\Phi,n}\\ :=\\ \\GU\\sqcup_{\\GX}\\GC.\n\\end{equation}\n\\end{definition}\n\nConcretely, $\\G_{\\Phi,n}$ is obtained from the disjoint union groupoid $\\GU\\bigsqcup\\GC$\nby imposing precisely the identifications dictated by $\\GX$ (gluing the overlap charts),\nwhile keeping the Stokes-decorated arrows as genuinely new arrows.\nEquivalently, $\\G_{\\Phi,n}$ admits the standard coequalizer presentation\n\\begin{equation}\\label{eq:coeq-clean}\n\\G_{\\Phi,n}\\ \\cong\\ \\mathrm{coeq}\\Bigl(F(R)\\rightrightarrows F(E)\\Bigr),\n\\end{equation}\nwhere $E$ consists of the generating arrows coming from $\\GU$ and $\\GC$,\nand $R$ encodes only the overlap relations from $\\GX$.\n\n\\subsubsection*{(D) A finite skeleton (graph-of-groupoids picture) on curves}\n\nThe \\v{C}ech presentation is explicit but indexed by covers.\nOn a curve one can replace it by a finite skeleton without changing representation theory (Morita reduction).\n\nFix a basepoint $x_0\\in U$.\nFor each sector box $B_{i,a}$ pick a point $b_{i,a}\\in B_{i,a}$ and choose a path class\n$\\gamma_{i,a}:x_0\\to b_{i,a}$ inside $U\\cup N^\\times$ entering the collar through the sector $I_{i,a}$.\n\nDefine a small groupoid $\\G^{\\mathrm{sk}}_{\\Phi,n}$ by:\n\\begin{itemize}\n\\item objects: $\\{x_0\\}\\cup\\{b_{i,a}\\}_{i,a}$;\n\\item arrows generated by:\n  \\begin{enumerate}\n  \\item $\\pi_1(U,x_0)$ as loops at $x_0$ (ordinary monodromy),\n  \\item the spokes $\\gamma_{i,a}:x_0\\to b_{i,a}$ and formal inverses,\n  \\item for each fixed $i$ and adjacent sectors $a\\to a+1$, arrows\n  $\\sigma_{i,a}(u):b_{i,a}\\to b_{i,a+1}$ for each $u\\in \\St_{i;a,a+1}$,\n  \\end{enumerate}\n\\item relations:\n  \\begin{enumerate}\n  \\item $\\sigma_{i,a+1}(v)\\circ \\sigma_{i,a}(u)=\\sigma_{i,a}(vu)$ whenever the compositions correspond to the same overlap,\n  \\item the overlap/gluing relations expressing compatibility between the restriction of monodromy on $U$\n  and the boundary Stokes jumps (the same relations encoded by $\\GX$),\n  \\item the usual groupoid relations (identities/inverses).\n  \\end{enumerate}\n\\end{itemize}\n\n\\begin{proposition}\\label{prop:curve-skeleton-clean}\nThere is a canonical chain of Morita equivalences connecting the cover-based presenter $\\G_{\\Phi,n}$ and the finite skeletal presenter\n$\\G^{\\mathrm{sk}}_{\\Phi,n}$. In particular, they present the same classifying stack, and for every $r\\ge 1$ there is a natural equivalence\nof groupoids\n\\begin{equation}\\label{eq:repr-clean}\n\\Rep_r(\\G_{\\Phi,n})\\ \\simeq\\ \\Rep_r(\\G^{\\mathrm{sk}}_{\\Phi,n}).\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nFix the good cover $\\mathcal U=\\{U_\\alpha\\}$ of $U$, the sector-box cover $\\mathcal B=\\{B_{i,a}\\}$ of $N^\\times$, and a contractible refinement\n$\\mathcal W=\\{W_{\\alpha;i,a}\\}$ of the overlap $U\\cap N^\\times$ as in \\Cref{subsec:curve-model-clean}. The global presenter is the explicit $2$-pushout\n\\begin{equation}\\label{eq:curve-pushout-cech}\n\\G_{\\Phi,n}\\ :=\\ \\GU\\ \\sqcup_{\\GX}\\ \\GC,\n\\qquad\n\\GU=\\Cech(\\mathcal U),\\ \\GX=\\Cech(\\mathcal W),\\ \\GC=\\Cech\\text{--Stokes}(\\mathcal B).\n\\end{equation}\n\nChoose basepoints $x_\\alpha\\in U_\\alpha$ and $b_{i,a}\\in B_{i,a}$, and choose basepoints in each $W_{\\alpha;i,a}$ compatibly with the inclusions\n$W_{\\alpha;i,a}\\subset U_\\alpha$ and $W_{\\alpha;i,a}\\subset B_{i,a}$.\nSince all charts and all nonempty finite intersections are contractible, each connected overlap carries a unique homotopy class of paths between the\nchosen basepoints. Therefore each \\v{C}ech presenter admits the standard Morita reduction to a small groupoid with one object per chart and one arrow\nper nonempty overlap:\n\\begin{equation}\\label{eq:morita-reductions}\n\\GU\\ \\simeq_{\\mathrm{Morita}}\\ \\GU^{\\mathrm{bp}},\n\\qquad\n\\GX\\ \\simeq_{\\mathrm{Morita}}\\ \\GX^{\\mathrm{bp}},\n\\qquad\n\\GC\\ \\simeq_{\\mathrm{Morita}}\\ \\GC^{\\mathrm{bp}}.\n\\end{equation}\nMoreover, the refinement functors $j_U:\\GX\\to\\GU$ and $j_C:\\GX\\to\\GC$ induce functors\n$j_U^{\\mathrm{bp}}:\\GX^{\\mathrm{bp}}\\to\\GU^{\\mathrm{bp}}$ and $j_C^{\\mathrm{bp}}:\\GX^{\\mathrm{bp}}\\to\\GC^{\\mathrm{bp}}$, compatible with\n\\eqref{eq:morita-reductions}. Taking explicit $2$-pushouts (realized by Lemma~\\ref{lem:strict-pushout-is-bicategorical}) and using the $2$-pushout universal property yields a comparison functor between the two pushouts, unique up to unique isomorphism. This comparison is a Morita equivalence, hence a canonical chain\n\\begin{equation}\\label{eq:morita-chain-curve}\n\\G_{\\Phi,n}=\\GU\\sqcup_{\\GX}\\GC\\ \\simeq_{\\mathrm{Morita}}\\ \\GU^{\\mathrm{bp}}\\sqcup_{\\GX^{\\mathrm{bp}}}\\GC^{\\mathrm{bp}}\n=: \\G^{\\mathrm{bp}}_{\\Phi,n}.\n\\end{equation}\nThis passage is summarized by Diagram~\\eqref{eq:curve-morita-cube} below.\n\nFinally, by construction the reduced pushout $\\G^{\\mathrm{bp}}_{\\Phi,n}$ is the explicit finite skeleton\n$\\G^{\\mathrm{sk}}_{\\Phi,n}$ introduced in \\Cref{subsec:curve-model-clean}: it has objects $x_0$ and the sector basepoints $b_{i,a}$, and it is\ngenerated by ordinary monodromy in $U$ together with the Stokes-labelled arrows between adjacent sectors, modulo the overlap relations coming from\n$\\GX$. Thus $\\G^{\\mathrm{bp}}_{\\Phi,n}\\cong \\G^{\\mathrm{sk}}_{\\Phi,n}$. Applying $\\Rep_r(-)$, which is invariant under Morita equivalence of\npresenters, yields \\eqref{eq:repr-clean}.\n\n\\begin{equation}\\label{eq:curve-morita-cube}\n\\begin{tikzcd}[row sep=large, column sep=large]\n\\GU\n\\arrow[d, \"\\simeq_{\\mathrm{Morita}}\" description]\n\\arrow[rrr, bend left=18, \"i_U\"]\n&\n\\GX\n\\arrow[l, \"j_U\"']\n\\arrow[r, \"j_C\"]\n\\arrow[d, \"\\simeq_{\\mathrm{Morita}}\" description]\n&\n\\GC\n\\arrow[d, \"\\simeq_{\\mathrm{Morita}}\" description]\n\\arrow[r, \"i_C\"]\n&\n\\GU \\sqcup_{\\GX} \\GC\n\\arrow[d, \"\\simeq_{\\mathrm{Morita}}\" description]\n\\\\\n\\GU^{\\mathrm{bp}}\n\\arrow[rrr, bend right=18, \"i_U^{\\mathrm{bp}}\"']\n&\n\\GX^{\\mathrm{bp}}\n\\arrow[l, \"j_U^{\\mathrm{bp}}\"']\n\\arrow[r, \"j_C^{\\mathrm{bp}}\"]\n&\n\\GC^{\\mathrm{bp}}\n\\arrow[r, \"i_C^{\\mathrm{bp}}\"]\n&\n\\GU^{\\mathrm{bp}} \\sqcup_{\\GX^{\\mathrm{bp}}} \\GC^{\\mathrm{bp}}\n\\arrow[d, \"\\cong\"]\n\\\\\n&&&\n\\G^{\\mathrm{sk}}_{\\Phi,n}.\n\\end{tikzcd}\n\\end{equation}\n\\end{proof}", "thm:intro-main": "\\begin{theorem} \\label{thm:intro-main}\nFor every $r\\ge 1$ there is a natural equivalence of groupoids\n\\begin{equation}\\label{eq:intro-main-equiv}\n\\Rep_r(\\GU)\\times^{(2)}_{\\Rep_r(\\GX)} \\Sec(\\pi)\n\\ \\simeq\\ \\Stokes_r(\\Xlogn;D,\\Phi),\n\\end{equation}\ncompatible with restriction to $\\Rep_r(\\GU)$ (forgetting the wild layer) and with restriction to $\\Sec(\\pi)$\n(forgetting the interior).\n\\end{theorem}"}, "pre_theorem_intro_text_len": 6567, "pre_theorem_intro_text": "Irregular singular connections and Stokes phenomena may be viewed, at their core, as a problem of\n\\emph{topological encoding of analytic continuation near a divisor}.\nLet $X$ be a complex manifold (or a smooth complex algebraic variety), and let $D\\subset X$ be a simple normal crossings divisor.\nWrite $U:=X\\setminus D$.\nA meromorphic flat connection on $(X,D)$ restricts to $U$ as an ordinary local system, but its behavior near $D$\ninvolves additional, genuinely nonabelian gluing data: the \\emph{Stokes jumps}.\nIn dimension one this is classical (see e.g.\\ \\cite{BV89,Sabbah,Boalch2014}); in higher dimension, the same principle persists,\nwith one essential geometric change:\nthe relevant ``directions'' around $D$ vary along the strata of $D$ and must be handled functorially across charts.\nA Betti-side presenter that is useful for geometry should therefore keep track simultaneously of\n(i) the directional boundary geometry, (ii) tame/Kummer branching along all strata, and\n(iii) the wild Stokes gluing across sectorial overlaps.\n\nA standard form of the irregular Riemann--Hilbert correspondence asserts that meromorphic connections\nwith prescribed formal/irregular type are classified on the Betti side by \\emph{Stokes-filtered}\n(or Stokes-graded) local systems on a real boundary space surrounding $D$.\nAnalytically, this boundary is often realized by the real oriented blow-up along $D$,\nwhile modern formulations use enhanced or perverse-sheaf avatars of irregularity (cf.\\ \\cite{DAgnoloKashiwara,Mochizuki}).\nIn the Stokes-filtered description, the Stokes jumps are encoded by a Stokes sheaf on the boundary,\nand the classification problem becomes a torsor problem.\nWhile this framework is well-established, the Betti objects are frequently presented either analytically\nor as stacks defined by descent; in particular, the underlying \\emph{presenting groupoid} is often left implicit.\nFor applications where one wants strict gluing along covers (for instance as input to Betti-stack or Tannakian formalisms),\nit is useful to have a small and explicit strict groupoid model adapted to normal crossings strata and compatible with tame/Kummer ramification.\n\nA complementary viewpoint, especially effective on curves, is to encode wild monodromy by groupoids:\nStokes data can be organized into ``wild'' fundamental groupoids, and moduli can be described as representation groupoids\nof these presenters (see e.g.\\ \\cite{GualtieriLiPym} and the references therein).\nThis has two structural advantages: it makes cutting-and-pasting arguments transparent, and it interfaces naturally with\ngeometric structures on moduli (Poisson/symplectic forms, generators-and-relations descriptions, and related constructions).\nIn higher dimension, however, one faces an additional bookkeeping issue: the boundary geometry is naturally \\emph{logarithmic},\nand the tame/Kummer layer interacts with the Stokes layer along strata of arbitrary depth.\nThe aim of this paper is to give a single, strictly $1$-categorical, cover-based model in which the directional, tame, and wild layers\nappear together through explicit maps of small groupoids governed by universal properties (pullbacks and pushouts),\nand which admits finite skeletal models well-suited to computation (compare the \\textit{skeleton} perspective in \\cite{TeyssierSkeletons}).\n\nWe work with the logarithmic Betti realization of the pair $(X,D)$.\nLet $X^{\\log}$ denote the Kato--Nakayama space of $(X,D)$, equipped with the canonical map $\\tau:X^{\\log}\\to X$ \\cite{KatoNakayama}.\nLocally, near a point where $k$ components of $D$ meet, the fiber of $\\tau$ is a real torus $(S^1)^k$,\ncanonically recording the angular directions around each local branch.\nTo incorporate tame ramification uniformly we pass to the Kummer level $n$ and use the Kummer log cover\n$X^{\\log}_n\\to X^{\\log}$ (cf.\\ \\cite{NakayamaOgus,Ogus}); along a depth-$k$ stratum the natural deck group is $(\\mu_n)^k$.\nThis is the bookkeeping forced by multi-branch behavior once one insists on a boundary model compatible with descent.\n\nA recurring point of terminology concerns the word \\textit{representation}.\nIf $G$ is a small (topological) groupoid, we write $\\Rep_r(G)$ for the groupoid of \\emph{torsorial} data on $G$:\nprincipal $\\GL_r(\\mathbb C)^\\delta$-bundles on $G_0$ equipped with descent isomorphisms along $G_1$\n(here $\\GL_r(\\mathbb C)^\\delta$ denotes the discrete group).\nThis unpointed notion is stable under refinement and Morita equivalence, and it is the natural target for strict gluing arguments.\nWhen a basepoint and a trivialization are fixed on a connected space, one recovers the familiar $\\mathrm{Hom}(\\pi_1,-)$ description,\nbut we do not impose such auxiliary choices.\n\nFix an irregular type $\\Phi$ along $D$, and let $N^\\times\\subset X^{\\log}_n$ be a punctured collar neighborhood of the logarithmic boundary.\nWe consider the groupoid $\\LocSys_r(U)$ of rank-$r$ local systems on $U$ and the groupoid\n$\\StokesLocal_r(N^\\circ;\\Phi)$ of rank-$r$ Stokes local systems on the collar of type $\\Phi$,\ntogether with their restrictions to the overlap $N^\\circ\\subset U$.\nWe define the global Stokes object intrinsically as the $2$-fiber product\n\\begin{equation}\\label{eq:intro-global-stokes}\n\\Stokes_r(X^{\\log}_n;D,\\Phi)\n:=\n\\LocSys_r(U)\\times^{(2)}_{\\LocSys_r(N^\\circ)}\n\\StokesLocal_r(N^\\circ;\\Phi).\n\\end{equation}\nThis formalizes the basic gluing principle: an irregular object is obtained by gluing an ordinary local system on $U$\nwith a Stokes enhancement on a collar, compatibly on the overlap.\n\nOur main technical step is to present \\eqref{eq:intro-global-stokes} by an explicit \\emph{small strict groupoid}\nbuilt from \\v{C}ech data.\nChoose a good cover of $U$ and a sectorial/Stokes cover of $N^\\times$ compatible on $U\\cap N^\\times$.\nLet $\\G_U$ be the \\v{C}ech groupoid of the interior cover, let $\\G_C$ be the \\v{C}ech--Stokes collar groupoid on $N^\\times$,\nand let $\\G_{\\times}$ be the induced overlap groupoid.\nThe collar groupoid comes equipped with a canonical projection\n$\\pi:\\G_C\\to \\check{\\mathrm C}(\\mathcal B)$ to the underlying sector \\v{C}ech groupoid (forgetting the Stokes labels);\nthe correct collar moduli is the section groupoid $\\operatorname{Sec}(\\pi)$, rather than a representation groupoid of $\\G_C$.\nWe then form the explicit $2$-pushout in small groupoids\n\\begin{equation}\\label{eq:pushout-intro}\n\\G_{\\Phi,n}\\ :=\\ \\G_U \\sqcup_{\\G_{\\times}} \\G_C.\n\\end{equation}\nThe pushout is the groupoid-theoretic expression of \\textit{glue interior monodromy and boundary Stokes torsors\nalong their common restriction.}", "context": "Irregular singular connections and Stokes phenomena may be viewed, at their core, as a problem of\n\\emph{topological encoding of analytic continuation near a divisor}.\nLet $X$ be a complex manifold (or a smooth complex algebraic variety), and let $D\\subset X$ be a simple normal crossings divisor.\nWrite $U:=X\\setminus D$.\nA meromorphic flat connection on $(X,D)$ restricts to $U$ as an ordinary local system, but its behavior near $D$\ninvolves additional, genuinely nonabelian gluing data: the \\emph{Stokes jumps}.\nIn dimension one this is classical (see e.g.\\ \\cite{BV89,Sabbah,Boalch2014}); in higher dimension, the same principle persists,\nwith one essential geometric change:\nthe relevant ``directions'' around $D$ vary along the strata of $D$ and must be handled functorially across charts.\nA Betti-side presenter that is useful for geometry should therefore keep track simultaneously of\n(i) the directional boundary geometry, (ii) tame/Kummer branching along all strata, and\n(iii) the wild Stokes gluing across sectorial overlaps.\n\nA complementary viewpoint, especially effective on curves, is to encode wild monodromy by groupoids:\nStokes data can be organized into ``wild'' fundamental groupoids, and moduli can be described as representation groupoids\nof these presenters (see e.g.\\ \\cite{GualtieriLiPym} and the references therein).\nThis has two structural advantages: it makes cutting-and-pasting arguments transparent, and it interfaces naturally with\ngeometric structures on moduli (Poisson/symplectic forms, generators-and-relations descriptions, and related constructions).\nIn higher dimension, however, one faces an additional bookkeeping issue: the boundary geometry is naturally \\emph{logarithmic},\nand the tame/Kummer layer interacts with the Stokes layer along strata of arbitrary depth.\nThe aim of this paper is to give a single, strictly $1$-categorical, cover-based model in which the directional, tame, and wild layers\nappear together through explicit maps of small groupoids governed by universal properties (pullbacks and pushouts),\nand which admits finite skeletal models well-suited to computation (compare the \\textit{skeleton} perspective in \\cite{TeyssierSkeletons}).\n\nWe work with the logarithmic Betti realization of the pair $(X,D)$.\nLet $X^{\\log}$ denote the Kato--Nakayama space of $(X,D)$, equipped with the canonical map $\\tau:X^{\\log}\\to X$ \\cite{KatoNakayama}.\nLocally, near a point where $k$ components of $D$ meet, the fiber of $\\tau$ is a real torus $(S^1)^k$,\ncanonically recording the angular directions around each local branch.\nTo incorporate tame ramification uniformly we pass to the Kummer level $n$ and use the Kummer log cover\n$X^{\\log}_n\\to X^{\\log}$ (cf.\\ \\cite{NakayamaOgus,Ogus}); along a depth-$k$ stratum the natural deck group is $(\\mu_n)^k$.\nThis is the bookkeeping forced by multi-branch behavior once one insists on a boundary model compatible with descent.\n\nA recurring point of terminology concerns the word \\textit{representation}.\nIf $G$ is a small (topological) groupoid, we write $\\Rep_r(G)$ for the groupoid of \\emph{torsorial} data on $G$:\nprincipal $\\GL_r(\\mathbb C)^\\delta$-bundles on $G_0$ equipped with descent isomorphisms along $G_1$\n(here $\\GL_r(\\mathbb C)^\\delta$ denotes the discrete group).\nThis unpointed notion is stable under refinement and Morita equivalence, and it is the natural target for strict gluing arguments.\nWhen a basepoint and a trivialization are fixed on a connected space, one recovers the familiar $\\mathrm{Hom}(\\pi_1,-)$ description,\nbut we do not impose such auxiliary choices.\n\nFix an irregular type $\\Phi$ along $D$, and let $N^\\times\\subset X^{\\log}_n$ be a punctured collar neighborhood of the logarithmic boundary.\nWe consider the groupoid $\\LocSys_r(U)$ of rank-$r$ local systems on $U$ and the groupoid\n$\\StokesLocal_r(N^\\circ;\\Phi)$ of rank-$r$ Stokes local systems on the collar of type $\\Phi$,\ntogether with their restrictions to the overlap $N^\\circ\\subset U$.\nWe define the global Stokes object intrinsically as the $2$-fiber product\n\\begin{equation}\\label{eq:intro-global-stokes}\n\\Stokes_r(X^{\\log}_n;D,\\Phi)\n:=\n\\LocSys_r(U)\\times^{(2)}_{\\LocSys_r(N^\\circ)}\n\\StokesLocal_r(N^\\circ;\\Phi).\n\\end{equation}\nThis formalizes the basic gluing principle: an irregular object is obtained by gluing an ordinary local system on $U$\nwith a Stokes enhancement on a collar, compatibly on the overlap.\n\nOur main technical step is to present \\eqref{eq:intro-global-stokes} by an explicit \\emph{small strict groupoid}\nbuilt from \\v{C}ech data.\nChoose a good cover of $U$ and a sectorial/Stokes cover of $N^\\times$ compatible on $U\\cap N^\\times$.\nLet $\\G_U$ be the \\v{C}ech groupoid of the interior cover, let $\\G_C$ be the \\v{C}ech--Stokes collar groupoid on $N^\\times$,\nand let $\\G_{\\times}$ be the induced overlap groupoid.\nThe collar groupoid comes equipped with a canonical projection\n$\\pi:\\G_C\\to \\check{\\mathrm C}(\\mathcal B)$ to the underlying sector \\v{C}ech groupoid (forgetting the Stokes labels);\nthe correct collar moduli is the section groupoid $\\operatorname{Sec}(\\pi)$, rather than a representation groupoid of $\\G_C$.\nWe then form the explicit $2$-pushout in small groupoids\n\\begin{equation}\\label{eq:pushout-intro}\n\\G_{\\Phi,n}\\ :=\\ \\G_U \\sqcup_{\\G_{\\times}} \\G_C.\n\\end{equation}\nThe pushout is the groupoid-theoretic expression of \\textit{glue interior monodromy and boundary Stokes torsors\nalong their common restriction.}\n\n\\begin{equation}\\label{eq:intro-global-stokes}\n\\Stokes_r(\\Xlogn;D,\\Phi)\n:=\n\\LocSys_r(U)\\times^{(2)}_{\\LocSys_r(N^\\circ)}\n\\StokesLocal_r(N^\\circ;\\Phi).\n\\end{equation}", "full_context": "Irregular singular connections and Stokes phenomena may be viewed, at their core, as a problem of\n\\emph{topological encoding of analytic continuation near a divisor}.\nLet $X$ be a complex manifold (or a smooth complex algebraic variety), and let $D\\subset X$ be a simple normal crossings divisor.\nWrite $U:=X\\setminus D$.\nA meromorphic flat connection on $(X,D)$ restricts to $U$ as an ordinary local system, but its behavior near $D$\ninvolves additional, genuinely nonabelian gluing data: the \\emph{Stokes jumps}.\nIn dimension one this is classical (see e.g.\\ \\cite{BV89,Sabbah,Boalch2014}); in higher dimension, the same principle persists,\nwith one essential geometric change:\nthe relevant ``directions'' around $D$ vary along the strata of $D$ and must be handled functorially across charts.\nA Betti-side presenter that is useful for geometry should therefore keep track simultaneously of\n(i) the directional boundary geometry, (ii) tame/Kummer branching along all strata, and\n(iii) the wild Stokes gluing across sectorial overlaps.\n\nA complementary viewpoint, especially effective on curves, is to encode wild monodromy by groupoids:\nStokes data can be organized into ``wild'' fundamental groupoids, and moduli can be described as representation groupoids\nof these presenters (see e.g.\\ \\cite{GualtieriLiPym} and the references therein).\nThis has two structural advantages: it makes cutting-and-pasting arguments transparent, and it interfaces naturally with\ngeometric structures on moduli (Poisson/symplectic forms, generators-and-relations descriptions, and related constructions).\nIn higher dimension, however, one faces an additional bookkeeping issue: the boundary geometry is naturally \\emph{logarithmic},\nand the tame/Kummer layer interacts with the Stokes layer along strata of arbitrary depth.\nThe aim of this paper is to give a single, strictly $1$-categorical, cover-based model in which the directional, tame, and wild layers\nappear together through explicit maps of small groupoids governed by universal properties (pullbacks and pushouts),\nand which admits finite skeletal models well-suited to computation (compare the \\textit{skeleton} perspective in \\cite{TeyssierSkeletons}).\n\nWe work with the logarithmic Betti realization of the pair $(X,D)$.\nLet $X^{\\log}$ denote the Kato--Nakayama space of $(X,D)$, equipped with the canonical map $\\tau:X^{\\log}\\to X$ \\cite{KatoNakayama}.\nLocally, near a point where $k$ components of $D$ meet, the fiber of $\\tau$ is a real torus $(S^1)^k$,\ncanonically recording the angular directions around each local branch.\nTo incorporate tame ramification uniformly we pass to the Kummer level $n$ and use the Kummer log cover\n$X^{\\log}_n\\to X^{\\log}$ (cf.\\ \\cite{NakayamaOgus,Ogus}); along a depth-$k$ stratum the natural deck group is $(\\mu_n)^k$.\nThis is the bookkeeping forced by multi-branch behavior once one insists on a boundary model compatible with descent.\n\nA recurring point of terminology concerns the word \\textit{representation}.\nIf $G$ is a small (topological) groupoid, we write $\\Rep_r(G)$ for the groupoid of \\emph{torsorial} data on $G$:\nprincipal $\\GL_r(\\mathbb C)^\\delta$-bundles on $G_0$ equipped with descent isomorphisms along $G_1$\n(here $\\GL_r(\\mathbb C)^\\delta$ denotes the discrete group).\nThis unpointed notion is stable under refinement and Morita equivalence, and it is the natural target for strict gluing arguments.\nWhen a basepoint and a trivialization are fixed on a connected space, one recovers the familiar $\\mathrm{Hom}(\\pi_1,-)$ description,\nbut we do not impose such auxiliary choices.\n\nFix an irregular type $\\Phi$ along $D$, and let $N^\\times\\subset X^{\\log}_n$ be a punctured collar neighborhood of the logarithmic boundary.\nWe consider the groupoid $\\LocSys_r(U)$ of rank-$r$ local systems on $U$ and the groupoid\n$\\StokesLocal_r(N^\\circ;\\Phi)$ of rank-$r$ Stokes local systems on the collar of type $\\Phi$,\ntogether with their restrictions to the overlap $N^\\circ\\subset U$.\nWe define the global Stokes object intrinsically as the $2$-fiber product\n\\begin{equation}\\label{eq:intro-global-stokes}\n\\Stokes_r(X^{\\log}_n;D,\\Phi)\n:=\n\\LocSys_r(U)\\times^{(2)}_{\\LocSys_r(N^\\circ)}\n\\StokesLocal_r(N^\\circ;\\Phi).\n\\end{equation}\nThis formalizes the basic gluing principle: an irregular object is obtained by gluing an ordinary local system on $U$\nwith a Stokes enhancement on a collar, compatibly on the overlap.\n\nOur main technical step is to present \\eqref{eq:intro-global-stokes} by an explicit \\emph{small strict groupoid}\nbuilt from \\v{C}ech data.\nChoose a good cover of $U$ and a sectorial/Stokes cover of $N^\\times$ compatible on $U\\cap N^\\times$.\nLet $\\G_U$ be the \\v{C}ech groupoid of the interior cover, let $\\G_C$ be the \\v{C}ech--Stokes collar groupoid on $N^\\times$,\nand let $\\G_{\\times}$ be the induced overlap groupoid.\nThe collar groupoid comes equipped with a canonical projection\n$\\pi:\\G_C\\to \\check{\\mathrm C}(\\mathcal B)$ to the underlying sector \\v{C}ech groupoid (forgetting the Stokes labels);\nthe correct collar moduli is the section groupoid $\\operatorname{Sec}(\\pi)$, rather than a representation groupoid of $\\G_C$.\nWe then form the explicit $2$-pushout in small groupoids\n\\begin{equation}\\label{eq:pushout-intro}\n\\G_{\\Phi,n}\\ :=\\ \\G_U \\sqcup_{\\G_{\\times}} \\G_C.\n\\end{equation}\nThe pushout is the groupoid-theoretic expression of \\textit{glue interior monodromy and boundary Stokes torsors\nalong their common restriction.}\n\n\\begin{equation}\\label{eq:intro-global-stokes}\n\\Stokes_r(\\Xlogn;D,\\Phi)\n:=\n\\LocSys_r(U)\\times^{(2)}_{\\LocSys_r(N^\\circ)}\n\\StokesLocal_r(N^\\circ;\\Phi).\n\\end{equation}\n\nOur main technical step is to present \\eqref{eq:intro-global-stokes} by an explicit \\emph{small strict groupoid}\nbuilt from \\v{C}ech data.\nChoose a good cover of $U$ and a sectorial/Stokes cover of $N^\\times$ compatible on $U\\cap N^\\times$.\nLet $\\GU$ be the \\v{C}ech groupoid of the interior cover, let $\\GC$ be the \\v{C}ech--Stokes collar groupoid on $N^\\times$,\nand let $\\GX$ be the induced overlap groupoid.\nThe collar groupoid comes equipped with a canonical projection\n$\\pi:\\GC\\to \\Cech(\\mathcal B)$ to the underlying sector \\v{C}ech groupoid (forgetting the Stokes labels);\nthe correct collar moduli is the section groupoid $\\Sec(\\pi)$, rather than a representation groupoid of $\\GC$.\nWe then form the explicit $2$-pushout in small groupoids\n\\begin{equation}\\label{eq:pushout-intro}\n\\G_{\\Phi,n}\\ :=\\ \\GU \\sqcup_{\\GX} \\GC.\n\\end{equation}\nThe pushout is the groupoid-theoretic expression of \\textit{glue interior monodromy and boundary Stokes torsors\nalong their common restriction.}\n\nAlthough the global Stokes groupoid \\eqref{eq:intro-global-stokes} is defined intrinsically as a $2$-fiber product,\nthe main point of this paper is to provide an explicit small strict presenter for it, adapted to SNC depth and Kummer descent.\nConcretely, we:\n\\begin{itemize}\n\\item construct a strict \\v{C}ech--Stokes collar groupoid $\\GC$ and identify the correct collar moduli as $\\Sec(\\pi)$\n(rather than $\\Rep_r(\\GC)$), making the Stokes layer strictly functorial and cover-based;\n\\item build the explicit $2$-pushout presenter $\\G_{\\Phi,n}$ and prove that it computes global Stokes objects via the torsorial gluing problem\nin \\Cref{thm:intro-main};\n\\item make the tame/Kummer layer along depth strata functorial at the level of presenters, enabling explicit descent constraints\nand, in particular, a hands-on description in terms of equivariant cocycles;\n\\item provide finite skeletal models (invariant up to Morita equivalence, with contractible ambiguity) and an explicit\ngenerators-and-relations/chamber-complex description in SNC corners, solving the local presentation problem highlighted in\nProblem~12.3.\n\\end{itemize}\n\n\\begin{theorem}\\label{thm:main}\nFor every $r\\ge 1$ there is a natural equivalence of groupoids\n\\begin{equation}\\label{eq:main-equiv}\n\\Rep_r(\\GU)\\times^{(2)}_{\\Rep_r(\\GX)}\\Rep^{\\mathrm{St}}_r(\\GC)\n\\ \\simeq\\\n\\mathrm{Stokes}_r(X^{\\log}_n;D,\\Phi).\n\\end{equation}\nUnder this equivalence, the projection to $\\Rep_r(\\GU)$ corresponds to the forgetful functor\n$$\\mathrm{Stokes}_r(X^{\\log}_n;D,\\Phi)\\to\\LocSys_r(U)$$\n\\end{theorem}\n\n\\begin{proof}\nBy \\Cref{prop:rep-cech-locsys} there are natural equivalences\n\\begin{equation}\\label{eq:rep-locsys-identifications}\n\\Rep_r(\\GU)\\ \\simeq\\ \\LocSys_r(U),\n\\qquad\n\\Rep_r(\\GX)\\ \\simeq\\ \\LocSys_r(N^\\circ),\n\\end{equation}\nand by \\Cref{lem:cech-naturality} these identifications intertwine the restriction functor $$j_U^*:\\Rep_r(\\GU)\\to\\Rep_r(\\GX)$$ with the usual\nrestriction $\\LocSys_r(U)\\to\\LocSys_r(N^\\circ)$.\nOn the collar side, \\Cref{prop:RepStokesGC-eq-StokesLocal} gives a natural equivalence\n\\begin{equation}\\label{eq:collar-identification}\n\\Rep^{\\mathrm{St}}_r(\\GC)=\\Sec(\\pi)\\ \\simeq\\ \\mathrm{StokesLocal}_r(N^\\circ;\\Phi),\n\\end{equation}\nand the forgetful functor $\\Rep^{\\mathrm{St}}_r(\\GC)\\to \\Rep_r(\\GX)$ corresponds to the functor\n$\\mathrm{StokesLocal}_r(N^\\circ;\\Phi)\\to \\LocSys_r(N^\\circ)$ sending a Stokes-local object to its underlying local system.\nConsequently, replacing each term in the $2$-fiber product\n$\\Rep_r(\\GU)\\times^{(2)}_{\\Rep_r(\\GX)}\\Rep^{\\mathrm{St}}_r(\\GC)$ using \\eqref{eq:rep-locsys-identifications} and \\eqref{eq:collar-identification}\nidentifies it canonically with\n\\[\n\\LocSys_r(U)\\times^{(2)}_{\\LocSys_r(N^\\circ)}\\mathrm{StokesLocal}_r(N^\\circ;\\Phi),\n\\]\nwhich equals $\\mathrm{Stokes}_r(X^{\\log}_n;D,\\Phi)$ by Definition~\\ref{def:global-stokes}. This proves \\eqref{eq:main-equiv}.\nThe final statement follows from the fact that both sides are defined as $2$-fiber products over the restriction to $N^\\circ$, hence the projection to\n$\\Rep_r(\\GU)\\simeq \\LocSys_r(U)$ is exactly the forgetful functor.\n\\end{proof}\n\\begin{equation}\\label{eq:main-diagram}\n\\begin{tikzcd}[row sep=large, column sep=huge]\n\\Rep_r(\\GU)\\times^{(2)}_{\\Rep_r(\\GX)}\\Rep^{\\mathrm{St}}_r(\\GC)\n\\arrow[r,\"\\sim\"] \\arrow[d]\n&\n\\LocSys_r(U)\\times^{(2)}_{\\LocSys_r(N^\\circ)}\\mathrm{StokesLocal}_r(N^\\circ;\\Phi)\n\\arrow[d]\n\\\\\n\\Rep_r(\\GU) \\arrow[r,\"\\sim\"]\n&\n\\LocSys_r(U)\n\\end{tikzcd}\n\\end{equation}\n\n\\begin{proposition}\\label{prop:van-kampen-presenters}\nThere is a canonical zig-zag of Morita equivalences exhibiting the global presenter $\\G_{\\Phi,n}$ as a bicategorical pushout of\n$\\G_{\\Phi,n}^{(1)}$ and $\\G_{\\Phi,n}^{(2)}$ over $\\G_{\\Phi,n}^{(12)}$ in $\\mathbf{Grpd}$.\nConsequently, for every $r\\ge 1$ there is a natural equivalence\n\\begin{equation}\\label{eq:vk-rep-pullback}\n\\Rep^{\\mathrm{St}}_r(\\G_{\\Phi,n})\n\\ \\simeq\\\n\\Rep^{\\mathrm{St}}_r(\\G_{\\Phi,n}^{(1)})\\times^{(2)}_{\\Rep^{\\mathrm{St}}_r(\\G_{\\Phi,n}^{(12)})}\\Rep^{\\mathrm{St}}_r(\\G_{\\Phi,n}^{(2)}).\n\\end{equation}\n\\end{proposition}\n\n\\begin{proposition}\\label{prop:curve-skeleton-clean}\nThere is a canonical chain of Morita equivalences connecting the cover-based presenter $\\G_{\\Phi,n}$ and the finite skeletal presenter\n$\\G^{\\mathrm{sk}}_{\\Phi,n}$. In particular, they present the same classifying stack, and for every $r\\ge 1$ there is a natural equivalence\nof groupoids\n\\begin{equation}\\label{eq:repr-clean}\n\\Rep_r(\\G_{\\Phi,n})\\ \\simeq\\ \\Rep_r(\\G^{\\mathrm{sk}}_{\\Phi,n}).\n\\end{equation}\n\\end{proposition}\n\n\\subsubsection*{Global Stokes objects from the skeletal presenter}\n\\begin{theorem} \\label{thm:global-skeletal-pushout}\nFor every $r\\ge 1$ there is a natural equivalence of groupoids\n\\begin{equation}\\label{eq:global-skeletal-pushout}\n\\Stokes_r(X^{\\log}_n;D,\\Phi)\\ \\simeq\\\n\\Rep_r(\\GU^{\\mathrm{sk}})\\times^{(2)}_{\\Rep_r(\\GX^{\\mathrm{sk}})}\\Sec(\\pi^{\\mathrm{sk}}),\n\\end{equation}\nwhere $\\Stokes_r(X^{\\log}_n;D,\\Phi)$ is the intrinsic global Stokes groupoid from Definition~\\ref{def:global-stokes}.\nEquivalently, global Stokes objects are classified by Stokes-corrected torsorial representations of the small presenter\n$\\G_{\\Phi,n}^{\\mathrm{sk}}$.\n\\end{theorem}\n\n\\begin{theorem} \\label{thm:kummer-descent-skeletal}\nFix a depth-$k$ chart and let $G_k=(\\mu_n)^k$ be the local deck group of $q_n$.\nThen pullback along $q_n$ identifies global Stokes objects downstairs with homotopy fixed points upstairs:\n\\begin{equation}\\label{eq:kummer-descent-skeletal}\n\\Stokes_r(X^{\\log};D,\\Phi)\\ \\simeq\\ \\Stokes_r(X^{\\log}_n;D,q_n^*\\Phi)^{hG_k},\n\\end{equation}\nwhere $(-)^{hG_k}$ denotes the homotopy fixed point groupoid (Definition~\\ref{def:hfp-finite}).\nMoreover, after choosing equivariant triangulations as in Proposition~\\ref{prop:equivariant-triangulation} on the collar (and similarly on\n$U$ and on the overlap), the descent equivalence is visible on the skeletal presenter: the $G_k$-action on $\\G_{\\Phi,n}^{\\mathrm{sk}}$\ninduces a strict action on\n$$\\Rep_r(\\GU^{\\mathrm{sk}})\\times^{(2)}_{\\Rep_r(\\GX^{\\mathrm{sk}})}\\Sec(\\pi^{\\mathrm{sk}}),$$ and its homotopy fixed points compute the left-hand side of\n\\eqref{eq:kummer-descent-skeletal} under the identification of Theorem~\\ref{thm:global-skeletal-pushout}.\n\\end{theorem}\n\n\\begin{equation}\\label{eq:intro-global-stokes}\n\\Stokes_r(\\Xlogn;D,\\Phi)\n:=\n\\LocSys_r(U)\\times^{(2)}_{\\LocSys_r(N^\\circ)}\n\\StokesLocal_r(N^\\circ;\\Phi).\n\\end{equation}\n\n\\begin{theorem} \\label{thm:intro-main}\nFor every $r\\ge 1$ there is a natural equivalence of groupoids\n\\begin{equation}\\label{eq:intro-main-equiv}\n\\Rep_r(\\GU)\\times^{(2)}_{\\Rep_r(\\GX)} \\Sec(\\pi)\n\\ \\simeq\\ \\Stokes_r(\\Xlogn;D,\\Phi),\n\\end{equation}\ncompatible with restriction to $\\Rep_r(\\GU)$ (forgetting the wild layer) and with restriction to $\\Sec(\\pi)$\n(forgetting the interior).\n\\end{theorem}", "post_theorem_intro_text_len": 4032, "post_theorem_intro_text": "Although the global Stokes groupoid \\eqref{eq:intro-global-stokes} is defined intrinsically as a $2$-fiber product,\nthe main point of this paper is to provide an explicit small strict presenter for it, adapted to SNC depth and Kummer descent.\nConcretely, we:\n\\begin{itemize}\n\\item construct a strict \\v{C}ech--Stokes collar groupoid $\\G_C$ and identify the correct collar moduli as $\\operatorname{Sec}(\\pi)$\n(rather than $\\Rep_r(\\G_C)$), making the Stokes layer strictly functorial and cover-based;\n\\item build the explicit $2$-pushout presenter $\\G_{\\Phi,n}$ and prove that it computes global Stokes objects via the torsorial gluing problem\nin \\Cref{thm:intro-main};\n\\item make the tame/Kummer layer along depth strata functorial at the level of presenters, enabling explicit descent constraints\nand, in particular, a hands-on description in terms of equivariant cocycles;\n\\item provide finite skeletal models (invariant up to Morita equivalence, with contractible ambiguity) and an explicit\ngenerators-and-relations/chamber-complex description in SNC corners, solving the local presentation problem highlighted in\nProblem~12.3.\n\\end{itemize}\n\nTheorem~\\Cref{thm:intro-main} makes it possible to treat global Stokes objects using a concrete presenter:\nquestions about existence, comparison, and descent can be translated into torsors and nonabelian \\v{C}ech cocycles\non an explicit skeleton underlying $\\G_{\\Phi,n}$.\nWe record three consequences that will be used later; each is proved in the indicated section.\n\n\\begin{corollary} \\label{cor:intro-morita}\nIf one replaces the interior and collar covers by refinements, the resulting pushout groupoid $G'_{\\Phi,n}$\nis Morita equivalent to $\\G_{\\Phi,n}$; in particular the associated torsorial representation groupoids are equivalent:\n$\\Rep_r(G'_{\\Phi,n})\\simeq \\Rep_r(\\G_{\\Phi,n})$.\n\\end{corollary}\n\n\\begin{corollary} \\label{cor:intro-kummer}\nAlong a depth-$k$ stratum of $D$, the Kummer cover $X^{\\log}_n\\to X^{\\log}$ has deck group $(\\mu_n)^k$,\nand the groupoid $\\G_{\\Phi,n}$ records this tame layer functorially through the collar groupoid $\\G_C$.\nConsequently, Kummer descent constraints for Stokes objects can be expressed as strict descent conditions\non the collar data (equivalently, as equivariant cocycle constraints on a finite skeleton).\n\\end{corollary}\n\n\\begin{corollary} \\label{cor:intro-van-kampen}\nLet $X= X^{(1)}\\cup X^{(2)}$ be an open cover compatible with $D$, and assume the sectorial/Stokes data are chosen compatibly on overlaps.\nThen the presenting groupoids admit a pushout description reflecting this decomposition, so that questions about\n$\\Stokes_r(X^{\\log}_n;D,\\Phi)$ may be reduced to representations of an explicit groupoid obtained by gluing the local presenters.\n\n\\end{corollary}\n\nFor a fully explicit local model (including a finite Morita skeleton), the reader may jump directly to\n\\Cref{subsec:curve-model-clean}.\nThere we compute the presenting groupoid near a simple normal crossings corner and spell out the resulting\ngenerators-and-relations (equivalently, nonabelian cocycle) description of the Stokes data.\nThis example is included as a concrete reference point: it displays, in a single diagram, the directional layer\n($X^{\\log}$), the tame/Kummer layer (the $(\\mu_n)^k$-actions along strata), and the wild Stokes layer\n(sector overlaps and Stokes jumps), and it provides a template for later computations.\n\nAfter recalling the basic geometry of $X^{\\log}_n$ in the SNC case and fixing conventions, we define the intrinsic groupoid\n$\\Stokes_r(X^{\\log}_n;D,\\Phi)$ and establish its torsor description.\nWe then construct the interior \\v{C}ech groupoid $\\G_U$, the collar \\v{C}ech--Stokes groupoid $\\G_C$, and the overlap groupoid $\\G_{\\times}$,\nform the pushout \\eqref{eq:pushout-intro}, and prove \\Cref{thm:intro-main} in \\Cref{sec:pushout}.\nFinally, we discuss Morita invariance, Kummer descent along depth strata, and an explicit coequalizer\n(generators-and-relations) model for $\\G_{\\Phi,n}$, including the SNC computation announced above.", "sketch": "Although the global Stokes groupoid \\eqref{eq:intro-global-stokes} is defined intrinsically as a $2$-fiber product, the paper’s proof strategy for Theorem~\\Cref{thm:intro-main} is to “provide an explicit small strict presenter for it,” adapted to SNC depth and Kummer descent. Concretely, the outline given is:\n\\begin{itemize}\n\\item “construct a strict \\v{C}ech--Stokes collar groupoid $\\G_C$ and identify the correct collar moduli as $\\operatorname{Sec}(\\pi)$ (rather than $\\Rep_r(\\G_C)$), making the Stokes layer strictly functorial and cover-based;”\n\\item “build the explicit $2$-pushout presenter $\\G_{\\Phi,n}$ and prove that it computes global Stokes objects via the torsorial gluing problem in \\Cref{thm:intro-main};”\n\\item “make the tame/Kummer layer along depth strata functorial at the level of presenters, enabling explicit descent constraints and, in particular, a hands-on description in terms of equivariant cocycles;”\n\\item “provide finite skeletal models (invariant up to Morita equivalence, with contractible ambiguity) and an explicit generators-and-relations/chamber-complex description in SNC corners.”\n\\end{itemize}\nMore procedurally, the introduction says: “After recalling the basic geometry of $X^{\\log}_n$… we define the intrinsic groupoid $\\Stokes_r(X^{\\log}_n;D,\\Phi)$ and establish its torsor description. We then construct the interior \\v{C}ech groupoid $\\G_U$, the collar \\v{C}ech--Stokes groupoid $\\G_C$, and the overlap groupoid $\\G_{\\times}$, form the pushout \\eqref{eq:pushout-intro}, and prove \\Cref{thm:intro-main} in \\Cref{sec:pushout}.”", "expanded_sketch": "Although the global Stokes groupoid\n\\begin{equation}\\label{eq:intro-global-stokes}\n\\Stokes_r(\\Xlogn;D,\\Phi)\n:=\n\\LocSys_r(U)\\times^{(2)}_{\\LocSys_r(N^\\circ)}\n\\StokesLocal_r(N^\\circ;\\Phi).\n\\end{equation}\nis defined intrinsically as a $2$-fiber product, the paper’s proof strategy for the main theorem is to “provide an explicit small strict presenter for it,” adapted to SNC depth and Kummer descent. Concretely, the outline given is:\n\\begin{itemize}\n\\item “construct a strict \\v{C}ech--Stokes collar groupoid $\\G_C$ and identify the correct collar moduli as $\\operatorname{Sec}(\\pi)$ (rather than $\\Rep_r(\\G_C)$), making the Stokes layer strictly functorial and cover-based;”\n\\item “build the explicit $2$-pushout presenter $\\G_{\\Phi,n}$ and prove that it computes global Stokes objects via the torsorial gluing problem in the main theorem;”\n\\item “make the tame/Kummer layer along depth strata functorial at the level of presenters, enabling explicit descent constraints and, in particular, a hands-on description in terms of equivariant cocycles;”\n\\item “provide finite skeletal models (invariant up to Morita equivalence, with contractible ambiguity) and an explicit generators-and-relations/chamber-complex description in SNC corners.”\n\\end{itemize}\nMore procedurally, the introduction says: “After recalling the basic geometry of $X^{\\log}_n$… we define the intrinsic groupoid $\\Stokes_r(X^{\\log}_n;D,\\Phi)$ and establish its torsor description. We then construct the interior \\v{C}ech groupoid $\\G_U$, the collar \\v{C}ech--Stokes groupoid $\\G_C$, and the overlap groupoid $\\G_{\\times}$, form the pushout\n\\begin{equation}\\label{eq:pushout-intro}\n\\G_{\\Phi,n}\\ :=\\ \\GU \\sqcup_{\\GX} \\GC.\n\\end{equation}\nand prove the main theorem later.”", "expanded_theorem": "\\label{thm:intro-main}\nFor every $r\\ge 1$ there is a natural equivalence of groupoids\n\\begin{equation}\\label{eq:intro-main-equiv}\n\\Rep_r(\\G_U)\\times^{(2)}_{\\Rep_r(\\G_{\\times})} \\operatorname{Sec}(\\pi)\n\\ \\simeq\\ \\Stokes_r(X^{\\log}_n;D,\\Phi),\n\\end{equation}\ncompatible with restriction to $\\Rep_r(\\G_U)$ (forgetting the wild layer) and with restriction to $\\operatorname{Sec}(\\pi)$\n(forgetting the interior).", "theorem_type": ["Universal", "Biconditional or Equivalence"], "mcq": {"question": "Let $X$ be a complex manifold (or a smooth complex algebraic variety), let $D\\subset X$ be a simple normal crossings divisor, set $U:=X\\setminus D$, and fix an irregular type $\\Phi$ along $D$. Let $X^{\\log}_n$ be the level-$n$ Kummer log cover of the Kato--Nakayama space $X^{\\log}$. The intrinsic global Stokes groupoid is\n\\[\n\\Stokes_r(X^{\\log}_n;D,\\Phi)\n:=\n\\LocSys_r(U)\\times^{(2)}_{\\LocSys_r(N^\\circ)}\n\\StokesLocal_r(N^\\circ;\\Phi),\n\\]\nwhere $N^\\circ$ is the overlap with a collar neighborhood of the logarithmic boundary. Choose a good cover of $U$ and a compatible sectorial/Stokes cover of a punctured collar; let $\\G_U$ be the resulting interior \\v{C}ech groupoid, let $\\G_{\\times}$ be the induced overlap groupoid, and let $\\pi$ be the canonical projection from the collar \\v{C}ech--Stokes groupoid to the underlying overlap \\v{C}ech groupoid. For a small groupoid $G$, write $\\Rep_r(G)$ for the groupoid of rank-$r$ torsorial representations of $G$ (principal $\\mathrm{GL}_r(\\mathbb C)^\\delta$-bundles on objects with descent isomorphisms along arrows), and write $\\operatorname{Sec}(\\pi)$ for the paper's groupoid of sections associated with $\\pi$, equipped with its natural restriction functor to $\\Rep_r(\\G_{\\times})$. Which statement holds for every integer $r\\ge 1$?", "correct_choice": {"label": "A", "text": "There is a natural equivalence of groupoids\n\\[\n\\Rep_r(\\G_U)\\times^{(2)}_{\\Rep_r(\\G_{\\times})} \\operatorname{Sec}(\\pi)\n\\ \\simeq\\ \\Stokes_r(X^{\\log}_n;D,\\Phi),\n\\]\ncompatible with the restriction to $\\Rep_r(\\G_U)$ (forgetting the wild layer) and with the restriction to $\\operatorname{Sec}(\\pi)$ (forgetting the interior)."}, "choices": [{"label": "B", "text": "There is a natural equivalence of groupoids\n\\[\n\\Rep_r(\\G_U)\\times^{(2)}_{\\Rep_r(\\G_{\\times})} \\Rep_r(\\G_C)\n\\ \\simeq\\ \\Stokes_r(X^{\\log}_n;D,\\Phi),\n\\]\ncompatible with the restriction to $\\Rep_r(\\G_U)$ and with the restriction to $\\Rep_r(\\G_C)$."}, {"label": "C", "text": "There is a natural functor of groupoids\n\\[\n\\Rep_r(\\G_U)\\times^{(2)}_{\\Rep_r(\\G_{\\times})} \\operatorname{Sec}(\\pi)\n\\longrightarrow \\Stokes_r(X^{\\log}_n;D,\\Phi),\n\\]\ncompatible with the restriction to $\\Rep_r(\\G_U)$ and with the restriction to $\\operatorname{Sec}(\\pi)$."}, {"label": "D", "text": "There is a natural equivalence of groupoids\n\\[\n\\Rep_r(\\G_U)\\times_{\\Rep_r(\\G_{\\times})} \\operatorname{Sec}(\\pi)\n\\ \\simeq\\ \\Stokes_r(X^{\\log}_n;D,\\Phi),\n\\]\ncompatible with the restriction to $\\Rep_r(\\G_U)$ (forgetting the wild layer) and with the restriction to $\\operatorname{Sec}(\\pi)$ (forgetting the interior)."}, {"label": "E", "text": "For every $r\\ge 1$ there is a natural equivalence of groupoids\n\\[\n\\Rep_r(\\G_U)\\times^{(2)}_{\\Rep_r(\\G_{\\times})} \\operatorname{Sec}(\\pi)\n\\ \\simeq\\ \\Stokes_r(X^{\\log};D,\\Phi),\n\\]\ncompatible with the restriction to $\\Rep_r(\\G_U)$ (forgetting the wild layer) and with the restriction to $\\operatorname{Sec}(\\pi)$ (forgetting the interior)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "correct collar moduli as Sec(pi) rather than Rep_r(G_C)", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "equivalence dropped to mere natural functor", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "homotopy-coherent 2-fiber product replaced by ordinary fiber product", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "Kummer level upstairs X^{log}_n replaced by downstairs X^{log}", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not state the equivalence itself and does not explicitly reveal the correct option. It sets up the notation used in the theorem, but the key choice points—equivalence vs functor, 2-fiber product vs ordinary fiber product, the correct collar moduli, and the correct log level—are not given away."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the correct choice is the precise theorem statement, with distractors formed by small perturbations. It does not substantially move beyond restating a known result."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish subtle categorical variants and reject plausible weakenings or nearby false statements. However, the item mainly tests exact recall/recognition of the theorem rather than open-ended mathematical generation."}, "DQS": {"score": 2, "justification": "The distractors are technically plausible and target realistic failure modes: replacing Sec(pi) by ordinary collar representations, weakening equivalence to a functor, confusing 2-pullback with ordinary pullback, and using the wrong log/Kummer level."}, "total_score": 5, "overall_assessment": "Technically well-crafted with strong distractors and little direct answer leakage, but it is mostly a precise theorem-recognition question rather than a generative reasoning task."}}
{"id": "2602.21453v1", "paper_link": "http://arxiv.org/abs/2602.21453v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "{\\rm~\\cite{Rolling}}\n  \\label{conj:pak}\n  For every $ r,\\, D\\in \\mathbb{N}  $ there exist  $ C,\\,L > 0 $ such that if $ H $\n  is a graph with $ \\Delta(H) \\leq D $ and $ \\sigma(e)\\ge L\\log(|V(H^{\\sigma})|) $ for all $ e \\in E(H) $, then\n  $\\widehat{R}_r(H^{\\sigma})\\leq C|V(H^{\\sigma})| $.", "start_pos": 7081, "end_pos": 7401, "label": "conj:pak"}, "ref_dict": {"conj:pak": "\\begin{theorem}\n  {\\rm~\\cite{Rolling}}\n  \\label{conj:pak}\n  For every $ r,\\, D\\in \\mathbb{N}  $ there exist  $ C,\\,L > 0 $ such that if $ H $\n  is a graph with $ \\Delta(H) \\leq D $ and $ \\sigma(e)\\ge L\\log(|V(H^{\\sigma})|) $ for all $ e \\in E(H) $, then\n  $\\widehat{R}_r(H^{\\sigma})\\leq C|V(H^{\\sigma})| $.\n\\end{theorem}", "eq:3": "\\begin{equation}\n    \\label{eq:3}\n    \\widehat{R}_r(H^\\sigma) \\leq r^{\\,400 D\\log D} \\, n.\n  \\end{equation}", "eq:2": "\\begin{equation}\n    \\label{eq:2}\n    \\widehat{R}_r(H^\\sigma) = O\\big(r^{34}\n        D^5\\log D\\big)n.\n  \\end{equation}", "thm:odd_subd": "\\begin{theorem}{\\rm~\\cite{javadi25:_induced_long}}\n  \\label{thm:odd_subd}\n  Let $r,\\, D \\geq 2$ be integers, and let $H^\\sigma$ be a subdivision\n  of a graph~$H$ with maximum degree~$D$ with\n  $ \\sigma(e) > {2\\log_2^2r\\log_{D}n+Cr^4 2^{2r}\\log^2 D}$\n  for every $e \\in E(H)$, where $n=|V(H^\\sigma)|$ and~$C$ is a\n  suitably large absolute constant. Then\n  \\begin{equation}\n    \\label{eq:1}\n    \\widehat{R}_r(H^\\sigma) = O\\big(2^{34r}\n        r^6 \\log^5(r) D^5\\log D\\big)n.\n  \\end{equation}\n\\end{theorem}", "thm:even_subd": "\\begin{theorem}{\\rm~\\cite{javadi25:_induced_long}}\n  \\label{thm:even_subd}\n  Let $r, \\, D \\geq 2$ be integers, and let $H^\\sigma$ be a\n  subdivision of a graph~$H$ with maximum degree~$D$ with $\\sigma(e)$\n  even and larger than ${2\\log_2^2r\\log_{D}n+Cr^4 2^{2r}\\log^2 D}$ for\n  every $e \\in E(H)$, where $n=|V(H^\\sigma)|$ and~$C$ is a suitably\n  large absolute constant.  Then\n  \\begin{equation}\n    \\label{eq:2}\n    \\widehat{R}_r(H^\\sigma) = O\\big(r^{34}\n        D^5\\log D\\big)n.\n  \\end{equation}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2460, "pre_theorem_intro_text": "For given graphs $H$ and $G$, and a positive integer $r$, we say that $G$ is \\emph{Ramsey for} $H$, denoted by $G \\longrightarrow (H)_r$, if for every $r$-edge coloring of $G$ with $r$ colors, there exists a monochromatic subgraph of $G$ isomorphic to $H$.\nRamsey's pioneering work~\\cite{Ramsey} established that for any graph\n$H$ and any positive integer $r$, there exists $N\\in \\mathbb{N}$  such\nthat $K_N \\longrightarrow (H)_r$. The smallest such~$N$ is called the\n\\emph{$r$-multicolor Ramsey number} of $H$ and is denoted by $R_r(H)$.\n\nWhile this perspective focuses on minimizing the number of vertices,\nan alternative viewpoint—initiated by Erd\\H{o}s, Faudree, Rousseau,\nand Schelp~\\cite{S.R.Erd}—seeks to minimize the number of edges. This\nleads to the notion of \\emph{size-Ramsey numbers}. For a graph $H$ and\ninteger $r \\geq 2$, the \\emph{$r$-multicolor size-Ramsey number}\nof~$H$ is \n\\[\n\\widehat{R}_r(H) = \\min \\{ |E(G)| : G \\longrightarrow (H)_r \\}.\n\\]\n\nOne of the central problems in this area is to determine for which\nfamilies of graphs the size-Ramsey number grows linearly with the\nnumber of vertices. Erd\\H{o}s~\\cite{Erd_1} asked this question in the\ncase of paths and this question was addressed by Beck~\\cite{Beck_1},\nwho proved that $\\widehat{R}_2(P_n) < 900n$.  The linearity of size-Ramsey\nnumbers has since been established for various graph families:\ncycles~\\cite{Sudakov-cycle, Haxell-cycle, JKhOP, JM}, bounded-degree\ntrees~\\cite{Friedman, Haxell-tree, Xin}, powers of paths and\ncycles~\\cite{Power_path, jie}, bounded powers of bounded-degree trees\n(equivalently graphs of bounded treewidth and maximum\ndegree)~\\cite{Power_tree, treewidth, jiang13:_degree_ramsey}, and logarithmic subdivisions of bounded-degree graphs~\\cite{Rolling}. On the negative side, R\\\"{o}dl and Szemer\\'{e}di~\\cite{Rodl} constructed graphs on $n$ vertices with maximum degree $3$ and size-Ramsey number at least  $n \\log^cn$, for a small constant $c>0$. This bound has  been\nimproved to $ cne^{c\\sqrt{\\log n}} $ for some $ c > 0 $ by Tikhomirov~\\cite{Tikh}. Thus\nthe linearity of size-Ramsey numbers does not extend universally to bounded degree graphs. \n\nGiven a graph $H$ and a function $\\sigma:E(H)\\to \\mathbb{N}$, the \\emph{subdivision} $H^\\sigma$ is obtained by replacing each edge $e \\in E(H)$ with a path of length $\\sigma(e)$.\nAnswering a conjecture of \nPak~\\cite{Pak}, Dragani\\'c, Krivelevich, and Nenadov~\\cite{Rolling} proved the following theorem.", "context": "For given graphs $H$ and $G$, and a positive integer $r$, we say that $G$ is \\emph{Ramsey for} $H$, denoted by $G \\longrightarrow (H)_r$, if for every $r$-edge coloring of $G$ with $r$ colors, there exists a monochromatic subgraph of $G$ isomorphic to $H$.\nRamsey's pioneering work~\\cite{Ramsey} established that for any graph\n$H$ and any positive integer $r$, there exists $N\\in \\mathbb{N}$ such\nthat $K_N \\longrightarrow (H)_r$. The smallest such~$N$ is called the\n\\emph{$r$-multicolor Ramsey number} of $H$ and is denoted by $R_r(H)$.\n\nWhile this perspective focuses on minimizing the number of vertices,\nan alternative viewpoint—initiated by Erd\\H{o}s, Faudree, Rousseau,\nand Schelp~\\cite{S.R.Erd}—seeks to minimize the number of edges. This\nleads to the notion of \\emph{size-Ramsey numbers}. For a graph $H$ and\ninteger $r \\geq 2$, the \\emph{$r$-multicolor size-Ramsey number}\nof~$H$ is \n\\[\n\\widehat{R}_r(H) = \\min \\{ |E(G)| : G \\longrightarrow (H)_r \\}.\n\\]\n\nOne of the central problems in this area is to determine for which\nfamilies of graphs the size-Ramsey number grows linearly with the\nnumber of vertices. Erd\\H{o}s~\\cite{Erd_1} asked this question in the\ncase of paths and this question was addressed by Beck~\\cite{Beck_1},\nwho proved that $\\widehat{R}_2(P_n) < 900n$. The linearity of size-Ramsey\nnumbers has since been established for various graph families:\ncycles~\\cite{Sudakov-cycle, Haxell-cycle, JKhOP, JM}, bounded-degree\ntrees~\\cite{Friedman, Haxell-tree, Xin}, powers of paths and\ncycles~\\cite{Power_path, jie}, bounded powers of bounded-degree trees\n(equivalently graphs of bounded treewidth and maximum\ndegree)~\\cite{Power_tree, treewidth, jiang13:_degree_ramsey}, and logarithmic subdivisions of bounded-degree graphs~\\cite{Rolling}. On the negative side, R\\\"{o}dl and Szemer\\'{e}di~\\cite{Rodl} constructed graphs on $n$ vertices with maximum degree $3$ and size-Ramsey number at least $n \\log^cn$, for a small constant $c>0$. This bound has been\nimproved to $ cne^{c\\sqrt{\\log n}} $ for some $ c > 0 $ by Tikhomirov~\\cite{Tikh}. Thus\nthe linearity of size-Ramsey numbers does not extend universally to bounded degree graphs.\n\nGiven a graph $H$ and a function $\\sigma:E(H)\\to \\mathbb{N}$, the \\emph{subdivision} $H^\\sigma$ is obtained by replacing each edge $e \\in E(H)$ with a path of length $\\sigma(e)$.\nAnswering a conjecture of \nPak~\\cite{Pak}, Dragani\\'c, Krivelevich, and Nenadov~\\cite{Rolling} proved the following theorem.", "full_context": "For given graphs $H$ and $G$, and a positive integer $r$, we say that $G$ is \\emph{Ramsey for} $H$, denoted by $G \\longrightarrow (H)_r$, if for every $r$-edge coloring of $G$ with $r$ colors, there exists a monochromatic subgraph of $G$ isomorphic to $H$.\nRamsey's pioneering work~\\cite{Ramsey} established that for any graph\n$H$ and any positive integer $r$, there exists $N\\in \\mathbb{N}$ such\nthat $K_N \\longrightarrow (H)_r$. The smallest such~$N$ is called the\n\\emph{$r$-multicolor Ramsey number} of $H$ and is denoted by $R_r(H)$.\n\nWhile this perspective focuses on minimizing the number of vertices,\nan alternative viewpoint—initiated by Erd\\H{o}s, Faudree, Rousseau,\nand Schelp~\\cite{S.R.Erd}—seeks to minimize the number of edges. This\nleads to the notion of \\emph{size-Ramsey numbers}. For a graph $H$ and\ninteger $r \\geq 2$, the \\emph{$r$-multicolor size-Ramsey number}\nof~$H$ is \n\\[\n\\widehat{R}_r(H) = \\min \\{ |E(G)| : G \\longrightarrow (H)_r \\}.\n\\]\n\nOne of the central problems in this area is to determine for which\nfamilies of graphs the size-Ramsey number grows linearly with the\nnumber of vertices. Erd\\H{o}s~\\cite{Erd_1} asked this question in the\ncase of paths and this question was addressed by Beck~\\cite{Beck_1},\nwho proved that $\\widehat{R}_2(P_n) < 900n$. The linearity of size-Ramsey\nnumbers has since been established for various graph families:\ncycles~\\cite{Sudakov-cycle, Haxell-cycle, JKhOP, JM}, bounded-degree\ntrees~\\cite{Friedman, Haxell-tree, Xin}, powers of paths and\ncycles~\\cite{Power_path, jie}, bounded powers of bounded-degree trees\n(equivalently graphs of bounded treewidth and maximum\ndegree)~\\cite{Power_tree, treewidth, jiang13:_degree_ramsey}, and logarithmic subdivisions of bounded-degree graphs~\\cite{Rolling}. On the negative side, R\\\"{o}dl and Szemer\\'{e}di~\\cite{Rodl} constructed graphs on $n$ vertices with maximum degree $3$ and size-Ramsey number at least $n \\log^cn$, for a small constant $c>0$. This bound has been\nimproved to $ cne^{c\\sqrt{\\log n}} $ for some $ c > 0 $ by Tikhomirov~\\cite{Tikh}. Thus\nthe linearity of size-Ramsey numbers does not extend universally to bounded degree graphs.\n\nGiven a graph $H$ and a function $\\sigma:E(H)\\to \\mathbb{N}$, the \\emph{subdivision} $H^\\sigma$ is obtained by replacing each edge $e \\in E(H)$ with a path of length $\\sigma(e)$.\nAnswering a conjecture of \nPak~\\cite{Pak}, Dragani\\'c, Krivelevich, and Nenadov~\\cite{Rolling} proved the following theorem.\n\n\\begin{abstract}\n For a positive integer $r$, the $r$-color size-Ramsey\n number~$\\widehat{R}_r(H)$ of a graph $H$ is the minimum number of\n edges in a graph $G$ such that every $r$-edge coloring of $G$\n contains a monochromatic copy of $H$. For a graph~$H$ and a\n function $\\sigma:E(H)\\to \\mathbb{N}$, the \\emph{subdivision}\n $H^\\sigma$ is obtained by replacing every $e \\in E(H)$ with a path\n of length $\\sigma(e)$. In~\\cite{javadi25:_induced_long} it is shown\n that for all integers $r,\\, D\\geq 2 $, there exists a constant\n $c=c(r, D)$ such that for every graph $ H $ with maximum degree $D$\n if $H^{\\sigma}$ is a subdivision of~$H$ in which\n $\\sigma(e) > c \\log n $ for every $e \\in E(H)$, where\n $n=|V(H^\\sigma)|$, then\n $ \\widehat{R}_r(H^\\sigma) = O\\big(2^{34r}\n r^6 \\log^5(r) D^5\\log D\\big)n. $ We improve upon this result in the case that~$H^{\\sigma}$\n is a bipartite graph and the number of colors~$r$ is large using a\n significantly different argument, obtaining the bound\n $ \\widehat{R}_r(H^{\\sigma}) \\leq r^{400D \\log D} \\, n $.\n\\end{abstract}\n\nGiven a graph $H$ and a function $\\sigma:E(H)\\to \\mathbb{N}$, the \\emph{subdivision} $H^\\sigma$ is obtained by replacing each edge $e \\in E(H)$ with a path of length $\\sigma(e)$.\nAnswering a conjecture of \nPak~\\cite{Pak}, Dragani\\'c, Krivelevich, and Nenadov~\\cite{Rolling} proved the following theorem.\n\nThe proof of Theorem~\\ref{conj:pak} is based on Szemer\\'edi's regularity lemma and makes no attempt to optimize the constant~$C$.\n\n\\begin{theorem}{\\rm~\\cite{javadi25:_induced_long}}\n \\label{thm:odd_subd}\n Let $r,\\, D \\geq 2$ be integers, and let $H^\\sigma$ be a subdivision\n of a graph~$H$ with maximum degree~$D$ with\n $ \\sigma(e) > {2\\log_2^2r\\log_{D}n+Cr^4 2^{2r}\\log^2 D}$\n for every $e \\in E(H)$, where $n=|V(H^\\sigma)|$ and~$C$ is a\n suitably large absolute constant. Then\n \\begin{equation}\n \\label{eq:1}\n \\widehat{R}_r(H^\\sigma) = O\\big(2^{34r}\n r^6 \\log^5(r) D^5\\log D\\big)n.\n \\end{equation}\n\\end{theorem}\n\n\\begin{theorem}{\\rm~\\cite{javadi25:_induced_long}}\n \\label{thm:even_subd}\n Let $r, \\, D \\geq 2$ be integers, and let $H^\\sigma$ be a\n subdivision of a graph~$H$ with maximum degree~$D$ with $\\sigma(e)$\n even and larger than ${2\\log_2^2r\\log_{D}n+Cr^4 2^{2r}\\log^2 D}$ for\n every $e \\in E(H)$, where $n=|V(H^\\sigma)|$ and~$C$ is a suitably\n large absolute constant. Then\n \\begin{equation}\n \\label{eq:2}\n \\widehat{R}_r(H^\\sigma) = O\\big(r^{34}\n D^5\\log D\\big)n.\n \\end{equation}\n\\end{theorem}\n\n\\begin{theorem}\n \\label{thm:main}\n Let $r, \\, D\\geq 2$ be integers, and let~$H$ be a graph with maximum\n degree~$D$. Suppose~$H^\\sigma$ is a subdivision of~$H$\n with~$\\sigma(e)\\geq 2\\log_{D-1} n$ for every $e \\in E(H)$, where\n $n=|V(H^\\sigma)|$. Furthermore, suppose~$H^\\sigma$ is bipartite.\n Then\n \\begin{equation}\n \\label{eq:3}\n \\widehat{R}_r(H^\\sigma) \\leq r^{\\,400 D\\log D} \\, n.\n \\end{equation}\n\\end{theorem}\n\nIt was shown in~\\cite{JM} that there exist explicit bipartite graphs whose local edge densities mimic those expected in random bipartite graphs. The following lemma that is Lemma 2.2 in~\\cite{JM}, provides a precise quantitative formulation of this phenomenon.\n\\begin{lemma}{\\rm~\\cite{JM}}\\label{quasi}\n Let $c_1$ be a positive integer and $c_2$, $c_3$, $\\varepsilon$, $\\delta$ be positive numbers such that $c_3\\leq c_1$, $0<\\epsilon\\leq 1/2$ and $3/2\\geq \\delta >\\sqrt{6{\\log({c_1e}/{c_3})/(c_2c_3)}} $. Then there exists $n_0=n_0(c_1,c_2,c_3,\\epsilon, \\delta)$ for which the following holds for every $n\\geq n_0$.\n\nIn a complementary direction, the following lemma shows that the goodness property is also preserved when successively removing degree (at most) one vertices from the embedded graph. This is the bipartite analogue of~\\cite[Lemma~2.4]{Rolling} and we include a proof in Appendix~\\ref{app2} for completeness.\n\\begin{lemma}\\label{lem:good_removing}\nLet $n$ and $D$ be positive integers. Consider an $(n,D)$-bipartite graph $F$ and suppose that $\\phi: F \\hookrightarrow G$ is an $(n,D)$-good embedding. Assume that $F'$ is obtained from $F$ by successively deleting vertices of degree at most one. Then the restriction of $\\phi$ to $F'$ is also $(n,D)$-good.\n\\end{lemma}\nWe next recall another useful structural concept introduced in~\\cite{JM}, which allows us to pass from certain dense bipartite configurations to bipartite graphs with strong expansion.\n\\begin{definition}{\\rm~\\cite{JM}}\nLet $G=G(V_1,V_2)$ be a bipartite graph with $|V_1|=|V_2|=N$ and let $\\alpha\\in(0,1)$ be a real number. We say that $G$ is \\emph{$\\alpha$-joined} if for every pair of subsets $A\\subseteq V_1$ and $B\\subseteq V_2$ with $|A|, \\,|B|\\geq \\alpha N$, we have $e(A,B)\\neq0$.\n\\end{definition}\nThe next lemma shows that every $\\alpha$-joined bipartite graph contains a large induced subgraph with strong expansion properties. Moreover, this subgraph also supports a good embedding of a suitable null graph.\n\\begin{lemma} \\label{lem:expender_in_join} \nLet $G=G(V_1,V_2)$ be an $\\alpha$-joined bipartite graph with $0<\\alpha<1/5$ and $|V_1|=|V_2|=N$. Then there exists an induced subgraph $G'=G'(V'_1,V'_2)$ of $G$ such that for each $i\\in \\{1,2\\}$, \n \\begin{itemize}\n \\item $|V'_i|\\geq (1-\\alpha)N$,\n \\item for every $X\\subset V'_i$, with $|X|\\leq \\alpha N$, we have $|N_{G'}(X)|> (1-4\\alpha)/{(2\\alpha)} |X|$, and\n \\item for every $X\\subset V'_i$, with $|X|> \\alpha N$, we have $|N_{G'}(X)|> (1-2\\alpha)N$.\n \\end{itemize}\n In particular, $G'$ is $(6\\alpha N, (1-2\\alpha)/(6\\alpha))$-expanding. \n Moreover, there exists an embedding $\\phi: F\\hookrightarrow G'$, where $F$ is a null bipartite graph with $\\alpha N$ vertices in each part, such that $\\phi$ is a\n $(6\\alpha N, {(1-4\\alpha)}/{(6\\alpha)})$-good embedding.\n\\end{lemma}\n\\begin{proof}\n\n\\begin{theorem}\n  {\\rm~\\cite{Rolling}}\n  \\label{conj:pak}\n  For every $ r,\\, D\\in \\mathbb{N}  $ there exist  $ C,\\,L > 0 $ such that if $ H $\n  is a graph with $ \\Delta(H) \\leq D $ and $ \\sigma(e)\\ge L\\log(|V(H^{\\sigma})|) $ for all $ e \\in E(H) $, then\n  $\\widehat{R}_r(H^{\\sigma})\\leq C|V(H^{\\sigma})| $.\n\\end{theorem}", "post_theorem_intro_text_len": 3860, "post_theorem_intro_text": "The proof of Theorem~\\ref{conj:pak} is based on Szemer\\'edi's regularity lemma and makes no attempt to optimize the constant~$C$. \n\nIn a recent paper~\\cite{javadi25:_induced_long} we reprove\nTheorem~\\ref{conj:pak} with the goal of obtaining reasonable \nexplicit bounds for the constants, in particular avoiding the use of\nregularity methods. Specifically, our main result for general\nsubdivisions in~\\cite{javadi25:_induced_long} is as follows.\n\n\\begin{theorem}{\\rm~\\cite{javadi25:_induced_long}}\n  \\label{thm:odd_subd}\n  Let $r,\\, D \\geq 2$ be integers, and let $H^\\sigma$ be a subdivision\n  of a graph~$H$ with maximum degree~$D$ with\n  $ \\sigma(e) > {2\\log_2^2r\\log_{D}n+Cr^4 2^{2r}\\log^2 D}$\n  for every $e \\in E(H)$, where $n=|V(H^\\sigma)|$ and~$C$ is a\n  suitably large absolute constant. Then\n  \\begin{equation}\n    \\label{eq:1}\n    \\widehat{R}_r(H^\\sigma) = O\\big(2^{34r}\n        r^6 \\log^5(r) D^5\\log D\\big)n.\n  \\end{equation}\n\\end{theorem}\n\nIn~\\cite{javadi25:_induced_long} we also prove that if we further\nrestrict to the case that $\\sigma(e)$ is even for every $e \\in E(H)$,\nthen Theorem~\\ref{thm:odd_subd} can be significantly improved.\n\n\\begin{theorem}{\\rm~\\cite{javadi25:_induced_long}}\n  \\label{thm:even_subd}\n  Let $r, \\, D \\geq 2$ be integers, and let $H^\\sigma$ be a\n  subdivision of a graph~$H$ with maximum degree~$D$ with $\\sigma(e)$\n  even and larger than ${2\\log_2^2r\\log_{D}n+Cr^4 2^{2r}\\log^2 D}$ for\n  every $e \\in E(H)$, where $n=|V(H^\\sigma)|$ and~$C$ is a suitably\n  large absolute constant.  Then\n  \\begin{equation}\n    \\label{eq:2}\n    \\widehat{R}_r(H^\\sigma) = O\\big(r^{34}\n        D^5\\log D\\big)n.\n  \\end{equation}\n\\end{theorem}\n\nClearly, the subdivisions~$H^\\sigma$ considered in\nTheorem~\\ref{thm:even_subd} are bipartite graphs.  In this paper we\nprove that an additional improvement of Theorem~\\ref{thm:odd_subd} is\npossible when the subdivided graph is bipartite, \\textit{without}\nassuming that every~$\\sigma(e)$ is even.\n\n\\begin{theorem}\n  \\label{thm:main}\n  Let $r, \\, D\\geq 2$ be integers, and let~$H$ be a graph with maximum\n  degree~$D$.  Suppose~$H^\\sigma$ is a subdivision of~$H$\n  with~$\\sigma(e)\\geq 2\\log_{D-1} n$ for every $e \\in E(H)$, where\n  $n=|V(H^\\sigma)|$.  Furthermore, suppose~$H^\\sigma$ is bipartite.\n  Then\n  \\begin{equation}\n    \\label{eq:3}\n    \\widehat{R}_r(H^\\sigma) \\leq r^{\\,400 D\\log D} \\, n.\n  \\end{equation}\n\\end{theorem}\n\nIn general, the bound in~\\eqref{eq:3} is worse than the bound\nin~\\eqref{eq:2}, but of course~\\eqref{eq:3} applies to more general\nsubdivisions~$H^\\sigma$.  Furthermore, the method of proof employed\nhere is entirely different from the one\nin~\\cite{javadi25:_induced_long}. \n\n\\subsection*{Conventions and notation}\n\nFor a graph $ G $, we write $ V(G) $, $ E(G) $ and $ e(G) $ for the vertex set, the edge set and the number of edges of $ G $, respectively.  \nFor $ v \\in V(G) $, by $ N_{G}(v) $ we mean the set of all neighbors of $ v $ in $G$.\nFor a subset $ X\\subseteq V(G) $, we define the neighborhood of $X$ as~$ N_G(X)=\\bigcup_{x\\in X}N_G(x) $.\nIn a rooted tree $T$, the maximum distance of a vertex from the root is the height of $ T $. If a tree has only one vertex (the root), its height is zero.\nLet $ A,\\, B \\subset V(G) $ with $A\\cap B=\\emptyset$. Then $ E_G(A,B)= \\{ xy\\in E(G) : x\\in A, \\, y\\in B\\} $\nis the set of edges connecting a vertex of $ A $ to a vertex of $ B $. Also, $e_G(A,B)=|E_G(A,B)|$. A bipartite graph $ G $ with a bipartition $ (V_1, V_2) $ is denoted by $ G=G(V_1, V_2) $.\n\nGiven functions $f(n), \\, g(n)$ and $h(n)$, we write $\\Omega(h(n))=f(n)=O(g(n))$ if there exist  absolute constants $ C_1,\\,  C_2>0 $ such that $C_1h(n)\\leq f(n)\\leq C_2g(n)$ for  all sufficiently large~$n$.\n\nAll logarithms are taken to base \n$ e $, unless stated otherwise. Throughout the paper, we omit floor and ceiling symbols whenever they are not essential.", "sketch": "The post-theorem introduction only states that \"The proof of Theorem~\\ref{conj:pak} is based on Szemer\\'edi's regularity lemma and makes no attempt to optimize the constant~$C$.\" No further outline, steps, or proof structure for Theorem~\\ref{conj:pak} is given there.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "{\\rm~\\cite{Rolling}}\n  \\label{conj:pak}\n  For every $ r,\\, D\\in \\mathbb{N}  $ there exist  $ C,\\,L > 0 $ such that if $ H $\n  is a graph with $ \\Delta(H) \\leq D $ and $ \\sigma(e)\\ge L\\log(|V(H^{\\sigma})|) $ for all $ e \\in E(H) $, then\n  $\\widehat{R}_r(H^{\\sigma})\\leq C|V(H^{\\sigma})| $ .", "theorem_type": ["Universal–Existential", "Implication"], "mcq": {"question": "For a graph $J$ and a positive integer $r$, write $G\\to (J)_r$ if every $r$-edge-coloring of a graph $G$ contains a monochromatic copy of $J$, and define the $r$-color size-Ramsey number by\n\\[\n\\widehat R_r(J)=\\min\\{ |E(G)|: G\\to (J)_r\\}.\n\\]\nIf $H$ is a graph and $\\sigma:E(H)\\to \\mathbb N$, let the subdivision $H^{\\sigma}$ be the graph obtained by replacing each edge $e\\in E(H)$ with a path of length $\\sigma(e)$. For positive integers $r$ and $D$, which statement about subdivisions of graphs $H$ with maximum degree $\\Delta(H)\\le D$ is valid?", "correct_choice": {"label": "A", "text": "For every $r,D\\in \\mathbb N$, there exist constants $C,L>0$ such that whenever $H$ is a graph with $\\Delta(H)\\le D$ and $\\sigma:E(H)\\to\\mathbb N$ satisfies $\\sigma(e)\\ge L\\log\\bigl(|V(H^{\\sigma})|\\bigr)$ for every $e\\in E(H)$, the subdivision $H^{\\sigma}$ has linear $r$-color size-Ramsey number:\n\\[\n\\widehat R_r(H^{\\sigma})\\le C\\,|V(H^{\\sigma})|.\n\\]"}, "choices": [{"label": "B", "text": "For every $r,D\\in \\mathbb N$, there exist constants $C,L>0$ such that whenever $H$ is a graph with $\\Delta(H)\\le D$ and $\\sigma:E(H)\\to\\mathbb N$ satisfies $\\sigma(e)\\le L\\log\\bigl(|V(H^{\\sigma})|\\bigr)$ for every $e\\in E(H)$, the subdivision $H^{\\sigma}$ has linear $r$-color size-Ramsey number:\n\\[\n\\widehat R_r(H^{\\sigma})\\le C\\,|V(H^{\\sigma})|.\n\\]"}, {"label": "C", "text": "For every $r,D\\in \\mathbb N$, there exists a constant $C>0$ such that for every graph $H$ with $\\Delta(H)\\le D$, there is some choice of subdivision lengths $\\sigma:E(H)\\to\\mathbb N$ for which\n\\[\n\\widehat R_r(H^{\\sigma})\\le C\\,|V(H^{\\sigma})|.\n\\]"}, {"label": "D", "text": "For every $r,D\\in \\mathbb N$, there exist absolute constants $C,L>0$ such that whenever $H$ is a graph with $\\Delta(H)\\le D$ and $\\sigma:E(H)\\to\\mathbb N$ satisfies $\\sigma(e)\\ge L\\log\\bigl(|V(H^{\\sigma})|\\bigr)$ for every $e\\in E(H)$, one has\n\\[\n\\widehat R_r(H^{\\sigma})\\le C\\,|V(H^{\\sigma})|.\n\\]"}, {"label": "E", "text": "For every $r,D\\in \\mathbb N$, there exist constants $C,L>0$ such that whenever $H$ is a graph with $\\Delta(H)\\le D$ and $\\sigma:E(H)\\to\\mathbb N$ satisfies $\\sigma(e)\\ge L\\log\\bigl(|V(H^{\\sigma})|\\bigr)$ for every $e\\in E(H)$, the original graph $H$ has linear $r$-color size-Ramsey number:\n\\[\n\\widehat R_r(H)\\le C\\,|V(H)|.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "lower_bound_on_subdivision_lengths", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "universal_quantification_over_all_eligible_sigma", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "dependence_of_constants_on_r_and_D", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "target_graph_is_subdivision_not_base_graph", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives only definitions and the setup; it does not explicitly reveal the correct option. The correct answer is not signposted by wording in the stem."}, "TAS": {"score": 1, "justification": "The item is essentially asking for the exact theorem statement among nearby variants. This is more than a verbatim restatement, but it is still primarily theorem recognition rather than a genuinely new conclusion derived from the setup."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to compare quantifier strength, dependence of constants, the logarithm argument, and upper bound versus equality. However, success depends mostly on recalling or matching the theorem statement, not on substantial generative mathematical reasoning."}, "DQS": {"score": 1, "justification": "The distractors are mostly plausible and target meaningful failure modes: uniform vs parameter-dependent constants, weaker hypotheses, boundary changes, and replacing an inequality by equality. But the presence of a weaker-true option makes the distractor set problematic for a single-correct MCQ."}, "total_score": 5, "overall_assessment": "A reasonably well-disguised theorem-recognition item with no clear answer leakage, but it does not strongly test generative reasoning and is weakened by a distractor that appears to be true in a weaker form."}}
{"id": "2602.21453v1", "paper_link": "http://arxiv.org/abs/2602.21453v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "{\\rm~\\cite{Rolling}}\n  \\label{conj:pak}\n  For every $ r,\\, D\\in \\mathbb{N}  $ there exist  $ C,\\,L > 0 $ such that if $ H $\n  is a graph with $ \\Delta(H) \\leq D $ and $ \\sigma(e)\\ge L\\log(|V(H^{\\sigma})|) $ for all $ e \\in E(H) $, then\n  $\\widehat{R}_r(H^{\\sigma})\\leq C|V(H^{\\sigma})| $.", "start_pos": 7081, "end_pos": 7401, "label": "conj:pak"}, "ref_dict": {"conj:pak": "\\begin{theorem}\n  {\\rm~\\cite{Rolling}}\n  \\label{conj:pak}\n  For every $ r,\\, D\\in \\mathbb{N}  $ there exist  $ C,\\,L > 0 $ such that if $ H $\n  is a graph with $ \\Delta(H) \\leq D $ and $ \\sigma(e)\\ge L\\log(|V(H^{\\sigma})|) $ for all $ e \\in E(H) $, then\n  $\\widehat{R}_r(H^{\\sigma})\\leq C|V(H^{\\sigma})| $.\n\\end{theorem}", "eq:3": "\\begin{equation}\n    \\label{eq:3}\n    \\widehat{R}_r(H^\\sigma) \\leq r^{\\,400 D\\log D} \\, n.\n  \\end{equation}", "eq:2": "\\begin{equation}\n    \\label{eq:2}\n    \\widehat{R}_r(H^\\sigma) = O\\big(r^{34}\n        D^5\\log D\\big)n.\n  \\end{equation}", "thm:odd_subd": "\\begin{theorem}{\\rm~\\cite{javadi25:_induced_long}}\n  \\label{thm:odd_subd}\n  Let $r,\\, D \\geq 2$ be integers, and let $H^\\sigma$ be a subdivision\n  of a graph~$H$ with maximum degree~$D$ with\n  $ \\sigma(e) > {2\\log_2^2r\\log_{D}n+Cr^4 2^{2r}\\log^2 D}$\n  for every $e \\in E(H)$, where $n=|V(H^\\sigma)|$ and~$C$ is a\n  suitably large absolute constant. Then\n  \\begin{equation}\n    \\label{eq:1}\n    \\widehat{R}_r(H^\\sigma) = O\\big(2^{34r}\n        r^6 \\log^5(r) D^5\\log D\\big)n.\n  \\end{equation}\n\\end{theorem}", "thm:even_subd": "\\begin{theorem}{\\rm~\\cite{javadi25:_induced_long}}\n  \\label{thm:even_subd}\n  Let $r, \\, D \\geq 2$ be integers, and let $H^\\sigma$ be a\n  subdivision of a graph~$H$ with maximum degree~$D$ with $\\sigma(e)$\n  even and larger than ${2\\log_2^2r\\log_{D}n+Cr^4 2^{2r}\\log^2 D}$ for\n  every $e \\in E(H)$, where $n=|V(H^\\sigma)|$ and~$C$ is a suitably\n  large absolute constant.  Then\n  \\begin{equation}\n    \\label{eq:2}\n    \\widehat{R}_r(H^\\sigma) = O\\big(r^{34}\n        D^5\\log D\\big)n.\n  \\end{equation}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2460, "pre_theorem_intro_text": "For given graphs $H$ and $G$, and a positive integer $r$, we say that $G$ is \\emph{Ramsey for} $H$, denoted by $G \\longrightarrow (H)_r$, if for every $r$-edge coloring of $G$ with $r$ colors, there exists a monochromatic subgraph of $G$ isomorphic to $H$.\nRamsey's pioneering work~\\cite{Ramsey} established that for any graph\n$H$ and any positive integer $r$, there exists $N\\in \\mathbb{N}$  such\nthat $K_N \\longrightarrow (H)_r$. The smallest such~$N$ is called the\n\\emph{$r$-multicolor Ramsey number} of $H$ and is denoted by $R_r(H)$.\n\nWhile this perspective focuses on minimizing the number of vertices,\nan alternative viewpoint—initiated by Erd\\H{o}s, Faudree, Rousseau,\nand Schelp~\\cite{S.R.Erd}—seeks to minimize the number of edges. This\nleads to the notion of \\emph{size-Ramsey numbers}. For a graph $H$ and\ninteger $r \\geq 2$, the \\emph{$r$-multicolor size-Ramsey number}\nof~$H$ is \n\\[\n\\widehat{R}_r(H) = \\min \\{ |E(G)| : G \\longrightarrow (H)_r \\}.\n\\]\n\nOne of the central problems in this area is to determine for which\nfamilies of graphs the size-Ramsey number grows linearly with the\nnumber of vertices. Erd\\H{o}s~\\cite{Erd_1} asked this question in the\ncase of paths and this question was addressed by Beck~\\cite{Beck_1},\nwho proved that $\\widehat{R}_2(P_n) < 900n$.  The linearity of size-Ramsey\nnumbers has since been established for various graph families:\ncycles~\\cite{Sudakov-cycle, Haxell-cycle, JKhOP, JM}, bounded-degree\ntrees~\\cite{Friedman, Haxell-tree, Xin}, powers of paths and\ncycles~\\cite{Power_path, jie}, bounded powers of bounded-degree trees\n(equivalently graphs of bounded treewidth and maximum\ndegree)~\\cite{Power_tree, treewidth, jiang13:_degree_ramsey}, and logarithmic subdivisions of bounded-degree graphs~\\cite{Rolling}. On the negative side, R\\\"{o}dl and Szemer\\'{e}di~\\cite{Rodl} constructed graphs on $n$ vertices with maximum degree $3$ and size-Ramsey number at least  $n \\log^cn$, for a small constant $c>0$. This bound has  been\nimproved to $ cne^{c\\sqrt{\\log n}} $ for some $ c > 0 $ by Tikhomirov~\\cite{Tikh}. Thus\nthe linearity of size-Ramsey numbers does not extend universally to bounded degree graphs. \n\nGiven a graph $H$ and a function $\\sigma:E(H)\\to \\mathbb{N}$, the \\emph{subdivision} $H^\\sigma$ is obtained by replacing each edge $e \\in E(H)$ with a path of length $\\sigma(e)$.\nAnswering a conjecture of \nPak~\\cite{Pak}, Dragani\\'c, Krivelevich, and Nenadov~\\cite{Rolling} proved the following theorem.", "context": "For given graphs $H$ and $G$, and a positive integer $r$, we say that $G$ is \\emph{Ramsey for} $H$, denoted by $G \\longrightarrow (H)_r$, if for every $r$-edge coloring of $G$ with $r$ colors, there exists a monochromatic subgraph of $G$ isomorphic to $H$.\nRamsey's pioneering work~\\cite{Ramsey} established that for any graph\n$H$ and any positive integer $r$, there exists $N\\in \\mathbb{N}$ such\nthat $K_N \\longrightarrow (H)_r$. The smallest such~$N$ is called the\n\\emph{$r$-multicolor Ramsey number} of $H$ and is denoted by $R_r(H)$.\n\nWhile this perspective focuses on minimizing the number of vertices,\nan alternative viewpoint—initiated by Erd\\H{o}s, Faudree, Rousseau,\nand Schelp~\\cite{S.R.Erd}—seeks to minimize the number of edges. This\nleads to the notion of \\emph{size-Ramsey numbers}. For a graph $H$ and\ninteger $r \\geq 2$, the \\emph{$r$-multicolor size-Ramsey number}\nof~$H$ is \n\\[\n\\widehat{R}_r(H) = \\min \\{ |E(G)| : G \\longrightarrow (H)_r \\}.\n\\]\n\nOne of the central problems in this area is to determine for which\nfamilies of graphs the size-Ramsey number grows linearly with the\nnumber of vertices. Erd\\H{o}s~\\cite{Erd_1} asked this question in the\ncase of paths and this question was addressed by Beck~\\cite{Beck_1},\nwho proved that $\\widehat{R}_2(P_n) < 900n$. The linearity of size-Ramsey\nnumbers has since been established for various graph families:\ncycles~\\cite{Sudakov-cycle, Haxell-cycle, JKhOP, JM}, bounded-degree\ntrees~\\cite{Friedman, Haxell-tree, Xin}, powers of paths and\ncycles~\\cite{Power_path, jie}, bounded powers of bounded-degree trees\n(equivalently graphs of bounded treewidth and maximum\ndegree)~\\cite{Power_tree, treewidth, jiang13:_degree_ramsey}, and logarithmic subdivisions of bounded-degree graphs~\\cite{Rolling}. On the negative side, R\\\"{o}dl and Szemer\\'{e}di~\\cite{Rodl} constructed graphs on $n$ vertices with maximum degree $3$ and size-Ramsey number at least $n \\log^cn$, for a small constant $c>0$. This bound has been\nimproved to $ cne^{c\\sqrt{\\log n}} $ for some $ c > 0 $ by Tikhomirov~\\cite{Tikh}. Thus\nthe linearity of size-Ramsey numbers does not extend universally to bounded degree graphs.\n\nGiven a graph $H$ and a function $\\sigma:E(H)\\to \\mathbb{N}$, the \\emph{subdivision} $H^\\sigma$ is obtained by replacing each edge $e \\in E(H)$ with a path of length $\\sigma(e)$.\nAnswering a conjecture of \nPak~\\cite{Pak}, Dragani\\'c, Krivelevich, and Nenadov~\\cite{Rolling} proved the following theorem.", "full_context": "For given graphs $H$ and $G$, and a positive integer $r$, we say that $G$ is \\emph{Ramsey for} $H$, denoted by $G \\longrightarrow (H)_r$, if for every $r$-edge coloring of $G$ with $r$ colors, there exists a monochromatic subgraph of $G$ isomorphic to $H$.\nRamsey's pioneering work~\\cite{Ramsey} established that for any graph\n$H$ and any positive integer $r$, there exists $N\\in \\mathbb{N}$ such\nthat $K_N \\longrightarrow (H)_r$. The smallest such~$N$ is called the\n\\emph{$r$-multicolor Ramsey number} of $H$ and is denoted by $R_r(H)$.\n\nWhile this perspective focuses on minimizing the number of vertices,\nan alternative viewpoint—initiated by Erd\\H{o}s, Faudree, Rousseau,\nand Schelp~\\cite{S.R.Erd}—seeks to minimize the number of edges. This\nleads to the notion of \\emph{size-Ramsey numbers}. For a graph $H$ and\ninteger $r \\geq 2$, the \\emph{$r$-multicolor size-Ramsey number}\nof~$H$ is \n\\[\n\\widehat{R}_r(H) = \\min \\{ |E(G)| : G \\longrightarrow (H)_r \\}.\n\\]\n\nOne of the central problems in this area is to determine for which\nfamilies of graphs the size-Ramsey number grows linearly with the\nnumber of vertices. Erd\\H{o}s~\\cite{Erd_1} asked this question in the\ncase of paths and this question was addressed by Beck~\\cite{Beck_1},\nwho proved that $\\widehat{R}_2(P_n) < 900n$. The linearity of size-Ramsey\nnumbers has since been established for various graph families:\ncycles~\\cite{Sudakov-cycle, Haxell-cycle, JKhOP, JM}, bounded-degree\ntrees~\\cite{Friedman, Haxell-tree, Xin}, powers of paths and\ncycles~\\cite{Power_path, jie}, bounded powers of bounded-degree trees\n(equivalently graphs of bounded treewidth and maximum\ndegree)~\\cite{Power_tree, treewidth, jiang13:_degree_ramsey}, and logarithmic subdivisions of bounded-degree graphs~\\cite{Rolling}. On the negative side, R\\\"{o}dl and Szemer\\'{e}di~\\cite{Rodl} constructed graphs on $n$ vertices with maximum degree $3$ and size-Ramsey number at least $n \\log^cn$, for a small constant $c>0$. This bound has been\nimproved to $ cne^{c\\sqrt{\\log n}} $ for some $ c > 0 $ by Tikhomirov~\\cite{Tikh}. Thus\nthe linearity of size-Ramsey numbers does not extend universally to bounded degree graphs.\n\nGiven a graph $H$ and a function $\\sigma:E(H)\\to \\mathbb{N}$, the \\emph{subdivision} $H^\\sigma$ is obtained by replacing each edge $e \\in E(H)$ with a path of length $\\sigma(e)$.\nAnswering a conjecture of \nPak~\\cite{Pak}, Dragani\\'c, Krivelevich, and Nenadov~\\cite{Rolling} proved the following theorem.\n\n\\begin{abstract}\n For a positive integer $r$, the $r$-color size-Ramsey\n number~$\\widehat{R}_r(H)$ of a graph $H$ is the minimum number of\n edges in a graph $G$ such that every $r$-edge coloring of $G$\n contains a monochromatic copy of $H$. For a graph~$H$ and a\n function $\\sigma:E(H)\\to \\mathbb{N}$, the \\emph{subdivision}\n $H^\\sigma$ is obtained by replacing every $e \\in E(H)$ with a path\n of length $\\sigma(e)$. In~\\cite{javadi25:_induced_long} it is shown\n that for all integers $r,\\, D\\geq 2 $, there exists a constant\n $c=c(r, D)$ such that for every graph $ H $ with maximum degree $D$\n if $H^{\\sigma}$ is a subdivision of~$H$ in which\n $\\sigma(e) > c \\log n $ for every $e \\in E(H)$, where\n $n=|V(H^\\sigma)|$, then\n $ \\widehat{R}_r(H^\\sigma) = O\\big(2^{34r}\n r^6 \\log^5(r) D^5\\log D\\big)n. $ We improve upon this result in the case that~$H^{\\sigma}$\n is a bipartite graph and the number of colors~$r$ is large using a\n significantly different argument, obtaining the bound\n $ \\widehat{R}_r(H^{\\sigma}) \\leq r^{400D \\log D} \\, n $.\n\\end{abstract}\n\nGiven a graph $H$ and a function $\\sigma:E(H)\\to \\mathbb{N}$, the \\emph{subdivision} $H^\\sigma$ is obtained by replacing each edge $e \\in E(H)$ with a path of length $\\sigma(e)$.\nAnswering a conjecture of \nPak~\\cite{Pak}, Dragani\\'c, Krivelevich, and Nenadov~\\cite{Rolling} proved the following theorem.\n\nThe proof of Theorem~\\ref{conj:pak} is based on Szemer\\'edi's regularity lemma and makes no attempt to optimize the constant~$C$.\n\n\\begin{theorem}{\\rm~\\cite{javadi25:_induced_long}}\n \\label{thm:odd_subd}\n Let $r,\\, D \\geq 2$ be integers, and let $H^\\sigma$ be a subdivision\n of a graph~$H$ with maximum degree~$D$ with\n $ \\sigma(e) > {2\\log_2^2r\\log_{D}n+Cr^4 2^{2r}\\log^2 D}$\n for every $e \\in E(H)$, where $n=|V(H^\\sigma)|$ and~$C$ is a\n suitably large absolute constant. Then\n \\begin{equation}\n \\label{eq:1}\n \\widehat{R}_r(H^\\sigma) = O\\big(2^{34r}\n r^6 \\log^5(r) D^5\\log D\\big)n.\n \\end{equation}\n\\end{theorem}\n\n\\begin{theorem}{\\rm~\\cite{javadi25:_induced_long}}\n \\label{thm:even_subd}\n Let $r, \\, D \\geq 2$ be integers, and let $H^\\sigma$ be a\n subdivision of a graph~$H$ with maximum degree~$D$ with $\\sigma(e)$\n even and larger than ${2\\log_2^2r\\log_{D}n+Cr^4 2^{2r}\\log^2 D}$ for\n every $e \\in E(H)$, where $n=|V(H^\\sigma)|$ and~$C$ is a suitably\n large absolute constant. Then\n \\begin{equation}\n \\label{eq:2}\n \\widehat{R}_r(H^\\sigma) = O\\big(r^{34}\n D^5\\log D\\big)n.\n \\end{equation}\n\\end{theorem}\n\n\\begin{theorem}\n \\label{thm:main}\n Let $r, \\, D\\geq 2$ be integers, and let~$H$ be a graph with maximum\n degree~$D$. Suppose~$H^\\sigma$ is a subdivision of~$H$\n with~$\\sigma(e)\\geq 2\\log_{D-1} n$ for every $e \\in E(H)$, where\n $n=|V(H^\\sigma)|$. Furthermore, suppose~$H^\\sigma$ is bipartite.\n Then\n \\begin{equation}\n \\label{eq:3}\n \\widehat{R}_r(H^\\sigma) \\leq r^{\\,400 D\\log D} \\, n.\n \\end{equation}\n\\end{theorem}\n\nIt was shown in~\\cite{JM} that there exist explicit bipartite graphs whose local edge densities mimic those expected in random bipartite graphs. The following lemma that is Lemma 2.2 in~\\cite{JM}, provides a precise quantitative formulation of this phenomenon.\n\\begin{lemma}{\\rm~\\cite{JM}}\\label{quasi}\n Let $c_1$ be a positive integer and $c_2$, $c_3$, $\\varepsilon$, $\\delta$ be positive numbers such that $c_3\\leq c_1$, $0<\\epsilon\\leq 1/2$ and $3/2\\geq \\delta >\\sqrt{6{\\log({c_1e}/{c_3})/(c_2c_3)}} $. Then there exists $n_0=n_0(c_1,c_2,c_3,\\epsilon, \\delta)$ for which the following holds for every $n\\geq n_0$.\n\nIn a complementary direction, the following lemma shows that the goodness property is also preserved when successively removing degree (at most) one vertices from the embedded graph. This is the bipartite analogue of~\\cite[Lemma~2.4]{Rolling} and we include a proof in Appendix~\\ref{app2} for completeness.\n\\begin{lemma}\\label{lem:good_removing}\nLet $n$ and $D$ be positive integers. Consider an $(n,D)$-bipartite graph $F$ and suppose that $\\phi: F \\hookrightarrow G$ is an $(n,D)$-good embedding. Assume that $F'$ is obtained from $F$ by successively deleting vertices of degree at most one. Then the restriction of $\\phi$ to $F'$ is also $(n,D)$-good.\n\\end{lemma}\nWe next recall another useful structural concept introduced in~\\cite{JM}, which allows us to pass from certain dense bipartite configurations to bipartite graphs with strong expansion.\n\\begin{definition}{\\rm~\\cite{JM}}\nLet $G=G(V_1,V_2)$ be a bipartite graph with $|V_1|=|V_2|=N$ and let $\\alpha\\in(0,1)$ be a real number. We say that $G$ is \\emph{$\\alpha$-joined} if for every pair of subsets $A\\subseteq V_1$ and $B\\subseteq V_2$ with $|A|, \\,|B|\\geq \\alpha N$, we have $e(A,B)\\neq0$.\n\\end{definition}\nThe next lemma shows that every $\\alpha$-joined bipartite graph contains a large induced subgraph with strong expansion properties. Moreover, this subgraph also supports a good embedding of a suitable null graph.\n\\begin{lemma} \\label{lem:expender_in_join} \nLet $G=G(V_1,V_2)$ be an $\\alpha$-joined bipartite graph with $0<\\alpha<1/5$ and $|V_1|=|V_2|=N$. Then there exists an induced subgraph $G'=G'(V'_1,V'_2)$ of $G$ such that for each $i\\in \\{1,2\\}$, \n \\begin{itemize}\n \\item $|V'_i|\\geq (1-\\alpha)N$,\n \\item for every $X\\subset V'_i$, with $|X|\\leq \\alpha N$, we have $|N_{G'}(X)|> (1-4\\alpha)/{(2\\alpha)} |X|$, and\n \\item for every $X\\subset V'_i$, with $|X|> \\alpha N$, we have $|N_{G'}(X)|> (1-2\\alpha)N$.\n \\end{itemize}\n In particular, $G'$ is $(6\\alpha N, (1-2\\alpha)/(6\\alpha))$-expanding. \n Moreover, there exists an embedding $\\phi: F\\hookrightarrow G'$, where $F$ is a null bipartite graph with $\\alpha N$ vertices in each part, such that $\\phi$ is a\n $(6\\alpha N, {(1-4\\alpha)}/{(6\\alpha)})$-good embedding.\n\\end{lemma}\n\\begin{proof}\n\n\\begin{theorem}\n  {\\rm~\\cite{Rolling}}\n  \\label{conj:pak}\n  For every $ r,\\, D\\in \\mathbb{N}  $ there exist  $ C,\\,L > 0 $ such that if $ H $\n  is a graph with $ \\Delta(H) \\leq D $ and $ \\sigma(e)\\ge L\\log(|V(H^{\\sigma})|) $ for all $ e \\in E(H) $, then\n  $\\widehat{R}_r(H^{\\sigma})\\leq C|V(H^{\\sigma})| $.\n\\end{theorem}", "post_theorem_intro_text_len": 3860, "post_theorem_intro_text": "The proof of Theorem~\\ref{conj:pak} is based on Szemer\\'edi's regularity lemma and makes no attempt to optimize the constant~$C$. \n\nIn a recent paper~\\cite{javadi25:_induced_long} we reprove\nTheorem~\\ref{conj:pak} with the goal of obtaining reasonable \nexplicit bounds for the constants, in particular avoiding the use of\nregularity methods. Specifically, our main result for general\nsubdivisions in~\\cite{javadi25:_induced_long} is as follows.\n\n\\begin{theorem}{\\rm~\\cite{javadi25:_induced_long}}\n  \\label{thm:odd_subd}\n  Let $r,\\, D \\geq 2$ be integers, and let $H^\\sigma$ be a subdivision\n  of a graph~$H$ with maximum degree~$D$ with\n  $ \\sigma(e) > {2\\log_2^2r\\log_{D}n+Cr^4 2^{2r}\\log^2 D}$\n  for every $e \\in E(H)$, where $n=|V(H^\\sigma)|$ and~$C$ is a\n  suitably large absolute constant. Then\n  \\begin{equation}\n    \\label{eq:1}\n    \\widehat{R}_r(H^\\sigma) = O\\big(2^{34r}\n        r^6 \\log^5(r) D^5\\log D\\big)n.\n  \\end{equation}\n\\end{theorem}\n\nIn~\\cite{javadi25:_induced_long} we also prove that if we further\nrestrict to the case that $\\sigma(e)$ is even for every $e \\in E(H)$,\nthen Theorem~\\ref{thm:odd_subd} can be significantly improved.\n\n\\begin{theorem}{\\rm~\\cite{javadi25:_induced_long}}\n  \\label{thm:even_subd}\n  Let $r, \\, D \\geq 2$ be integers, and let $H^\\sigma$ be a\n  subdivision of a graph~$H$ with maximum degree~$D$ with $\\sigma(e)$\n  even and larger than ${2\\log_2^2r\\log_{D}n+Cr^4 2^{2r}\\log^2 D}$ for\n  every $e \\in E(H)$, where $n=|V(H^\\sigma)|$ and~$C$ is a suitably\n  large absolute constant.  Then\n  \\begin{equation}\n    \\label{eq:2}\n    \\widehat{R}_r(H^\\sigma) = O\\big(r^{34}\n        D^5\\log D\\big)n.\n  \\end{equation}\n\\end{theorem}\n\nClearly, the subdivisions~$H^\\sigma$ considered in\nTheorem~\\ref{thm:even_subd} are bipartite graphs.  In this paper we\nprove that an additional improvement of Theorem~\\ref{thm:odd_subd} is\npossible when the subdivided graph is bipartite, \\textit{without}\nassuming that every~$\\sigma(e)$ is even.\n\n\\begin{theorem}\n  \\label{thm:main}\n  Let $r, \\, D\\geq 2$ be integers, and let~$H$ be a graph with maximum\n  degree~$D$.  Suppose~$H^\\sigma$ is a subdivision of~$H$\n  with~$\\sigma(e)\\geq 2\\log_{D-1} n$ for every $e \\in E(H)$, where\n  $n=|V(H^\\sigma)|$.  Furthermore, suppose~$H^\\sigma$ is bipartite.\n  Then\n  \\begin{equation}\n    \\label{eq:3}\n    \\widehat{R}_r(H^\\sigma) \\leq r^{\\,400 D\\log D} \\, n.\n  \\end{equation}\n\\end{theorem}\n\nIn general, the bound in~\\eqref{eq:3} is worse than the bound\nin~\\eqref{eq:2}, but of course~\\eqref{eq:3} applies to more general\nsubdivisions~$H^\\sigma$.  Furthermore, the method of proof employed\nhere is entirely different from the one\nin~\\cite{javadi25:_induced_long}. \n\n\\subsection*{Conventions and notation}\n\nFor a graph $ G $, we write $ V(G) $, $ E(G) $ and $ e(G) $ for the vertex set, the edge set and the number of edges of $ G $, respectively.  \nFor $ v \\in V(G) $, by $ N_{G}(v) $ we mean the set of all neighbors of $ v $ in $G$.\nFor a subset $ X\\subseteq V(G) $, we define the neighborhood of $X$ as~$ N_G(X)=\\bigcup_{x\\in X}N_G(x) $.\nIn a rooted tree $T$, the maximum distance of a vertex from the root is the height of $ T $. If a tree has only one vertex (the root), its height is zero.\nLet $ A,\\, B \\subset V(G) $ with $A\\cap B=\\emptyset$. Then $ E_G(A,B)= \\{ xy\\in E(G) : x\\in A, \\, y\\in B\\} $\nis the set of edges connecting a vertex of $ A $ to a vertex of $ B $. Also, $e_G(A,B)=|E_G(A,B)|$. A bipartite graph $ G $ with a bipartition $ (V_1, V_2) $ is denoted by $ G=G(V_1, V_2) $.\n\nGiven functions $f(n), \\, g(n)$ and $h(n)$, we write $\\Omega(h(n))=f(n)=O(g(n))$ if there exist  absolute constants $ C_1,\\,  C_2>0 $ such that $C_1h(n)\\leq f(n)\\leq C_2g(n)$ for  all sufficiently large~$n$.\n\nAll logarithms are taken to base \n$ e $, unless stated otherwise. Throughout the paper, we omit floor and ceiling symbols whenever they are not essential.", "sketch": "The post-theorem introduction only states that \"The proof of Theorem~\\ref{conj:pak} is based on Szemer\\'edi's regularity lemma and makes no attempt to optimize the constant~$C$.\" No further outline, steps, or proof structure for Theorem~\\ref{conj:pak} is given there.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "{\\rm~\\cite{Rolling}}\n  \\label{conj:pak}\n  For every $ r,\\, D\\in \\mathbb{N}  $ there exist  $ C,\\,L > 0 $ such that if $ H $\n  is a graph with $ \\Delta(H) \\leq D $ and $ \\sigma(e)\\ge L\\log(|V(H^{\\sigma})|) $ for all $ e \\in E(H) $, then\n  $\\widehat{R}_r(H^{\\sigma})\\leq C|V(H^{\\sigma})| $ .", "theorem_type": ["Universal–Existential", "Implication"], "mcq": {"question": "For a graph $J$ and a positive integer $r$, write $G\\to (J)_r$ if every $r$-edge-coloring of a graph $G$ contains a monochromatic copy of $J$, and define the $r$-color size-Ramsey number by\n\\[\n\\widehat R_r(J)=\\min\\{ |E(G)|: G\\to (J)_r\\}.\n\\]\nIf $H$ is a graph and $\\sigma:E(H)\\to \\mathbb N$, let the subdivision $H^{\\sigma}$ be the graph obtained by replacing each edge $e\\in E(H)$ with a path of length $\\sigma(e)$. For positive integers $r$ and $D$, which statement about subdivisions of graphs $H$ with maximum degree $\\Delta(H)\\le D$ is valid?", "correct_choice": {"label": "A", "text": "For every $r,D\\in \\mathbb N$, there exist constants $C,L>0$ such that whenever $H$ is a graph with $\\Delta(H)\\le D$ and $\\sigma:E(H)\\to\\mathbb N$ satisfies $\\sigma(e)\\ge L\\log\\bigl(|V(H^{\\sigma})|\\bigr)$ for every $e\\in E(H)$, the subdivision $H^{\\sigma}$ has linear $r$-color size-Ramsey number:\n\\[\n\\widehat R_r(H^{\\sigma})\\le C\\,|V(H^{\\sigma})|.\n\\]"}, "choices": [{"label": "B", "text": "For every $r,D\\in \\mathbb N$, there exist constants $C,L>0$ such that whenever $H$ is a graph with $\\Delta(H)\\le D$ and $\\sigma:E(H)\\to\\mathbb N$ satisfies $\\sigma(e)\\le L\\log\\bigl(|V(H^{\\sigma})|\\bigr)$ for every $e\\in E(H)$, the subdivision $H^{\\sigma}$ has linear $r$-color size-Ramsey number:\n\\[\n\\widehat R_r(H^{\\sigma})\\le C\\,|V(H^{\\sigma})|.\n\\]"}, {"label": "C", "text": "For every $r,D\\in \\mathbb N$, there exists a constant $C>0$ such that for every graph $H$ with $\\Delta(H)\\le D$, there is some choice of subdivision lengths $\\sigma:E(H)\\to\\mathbb N$ for which\n\\[\n\\widehat R_r(H^{\\sigma})\\le C\\,|V(H^{\\sigma})|.\n\\]"}, {"label": "D", "text": "For every $r,D\\in \\mathbb N$, there exist absolute constants $C,L>0$ such that whenever $H$ is a graph with $\\Delta(H)\\le D$ and $\\sigma:E(H)\\to\\mathbb N$ satisfies $\\sigma(e)\\ge L\\log\\bigl(|V(H^{\\sigma})|\\bigr)$ for every $e\\in E(H)$, one has\n\\[\n\\widehat R_r(H^{\\sigma})\\le C\\,|V(H^{\\sigma})|.\n\\]"}, {"label": "E", "text": "For every $r,D\\in \\mathbb N$, there exist constants $C,L>0$ such that whenever $H$ is a graph with $\\Delta(H)\\le D$ and $\\sigma:E(H)\\to\\mathbb N$ satisfies $\\sigma(e)\\ge L\\log\\bigl(|V(H^{\\sigma})|\\bigr)$ for every $e\\in E(H)$, the original graph $H$ has linear $r$-color size-Ramsey number:\n\\[\n\\widehat R_r(H)\\le C\\,|V(H)|.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "lower_bound_on_subdivision_lengths", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "universal_quantification_over_all_eligible_sigma", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "dependence_of_constants_on_r_and_D", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "target_graph_is_subdivision_not_base_graph", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem only gives definitions and asks which statement is valid; it does not reveal the key logarithmic lower bound, the linear conclusion, or the dependence on parameters in a way that singles out choice A."}, "TAS": {"score": 2, "justification": "This is not a direct restatement of a theorem in the stem. The task is to distinguish among closely related variants with different quantifiers, parameter dependencies, and target graphs."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the solver must track subtle differences such as \u001e vs \u001c bounds, dependence of constants on r and D, and whether the conclusion concerns H or H^sigma. However, it mainly tests theorem recognition/quantifier parsing rather than deeper generative mathematical reasoning."}, "DQS": {"score": 0, "justification": "Although several distractors are plausible, choice C is a weaker statement that is also true if A is true, so the item does not have a uniquely correct option. That is a serious distractor-design flaw."}, "total_score": 5, "overall_assessment": "Good at avoiding answer leakage and not tautological, but flawed as an MCQ because one distractor is also true. It tests careful theorem-form discrimination more than genuine generative reasoning."}}
{"id": "2602.21659v1", "paper_link": "http://arxiv.org/abs/2602.21659v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "[Fundamental inequality]\\label{thm:main_intro}\nLet $F\\subset S^3$ be a closed surface. Then every knot $K\\subset S^3$ satisfies:\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$, and $\\delta(F)=g(M_1)+g(M_2)-g(F)$ is the Heegaard deficiency of $F$.", "start_pos": 2789, "end_pos": 3106, "label": "thm:main_intro"}, "ref_dict": {"thm:main_intro": "\\begin{theorem}[Fundamental inequality]\\label{thm:main_intro}\nLet $F\\subset S^3$ be a closed surface. Then every knot $K\\subset S^3$ satisfies:\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$, and $\\delta(F)=g(M_1)+g(M_2)-g(F)$ is the Heegaard deficiency of $F$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 1144, "pre_theorem_intro_text": "\\subsection{Background and Motivation}\nLow-dimensional topology quantifies knot complexity through numerical invariants. While classical invariants, such as the standard crossing number, rely on planar diagrams, these two-dimensional projections compress spatial information which does not fully reflect the global topology of the knot exterior within the ambient space $S^3$.\n\nTo capture this three-dimensional structure, we extend the notion of diagrammatic complexity to arbitrary closed surfaces. For a knot $K$ and a closed surface $F \\subset S^3$, we define the \\emph{surface crossing number} $c(K;F)$ as the minimal number of crossings among all regular diagrams of $K$ obtained by isotoping the knot into a regular neighborhood of $F$. \n\nIn this paper, we establish an inequality demonstrating that $c(K;F)$ is bounded below by the tunnel number of $K$ and the Heegaard deficiency of $F$.\n\n\\subsection{Main Results}\nThe objective of this paper is to establish a quantitative relationship between a knot's three-dimensional complexity, measured via its tunnel number, and its diagrammatic complexity on an arbitrary fixed closed surface.", "context": "\\subsection{Background and Motivation}\nLow-dimensional topology quantifies knot complexity through numerical invariants. While classical invariants, such as the standard crossing number, rely on planar diagrams, these two-dimensional projections compress spatial information which does not fully reflect the global topology of the knot exterior within the ambient space $S^3$.\n\nTo capture this three-dimensional structure, we extend the notion of diagrammatic complexity to arbitrary closed surfaces. For a knot $K$ and a closed surface $F \\subset S^3$, we define the \\emph{surface crossing number} $c(K;F)$ as the minimal number of crossings among all regular diagrams of $K$ obtained by isotoping the knot into a regular neighborhood of $F$.\n\nIn this paper, we establish an inequality demonstrating that $c(K;F)$ is bounded below by the tunnel number of $K$ and the Heegaard deficiency of $F$.\n\n\\subsection{Main Results}\nThe objective of this paper is to establish a quantitative relationship between a knot's three-dimensional complexity, measured via its tunnel number, and its diagrammatic complexity on an arbitrary fixed closed surface.", "full_context": "\\subsection{Background and Motivation}\nLow-dimensional topology quantifies knot complexity through numerical invariants. While classical invariants, such as the standard crossing number, rely on planar diagrams, these two-dimensional projections compress spatial information which does not fully reflect the global topology of the knot exterior within the ambient space $S^3$.\n\nTo capture this three-dimensional structure, we extend the notion of diagrammatic complexity to arbitrary closed surfaces. For a knot $K$ and a closed surface $F \\subset S^3$, we define the \\emph{surface crossing number} $c(K;F)$ as the minimal number of crossings among all regular diagrams of $K$ obtained by isotoping the knot into a regular neighborhood of $F$.\n\nIn this paper, we establish an inequality demonstrating that $c(K;F)$ is bounded below by the tunnel number of $K$ and the Heegaard deficiency of $F$.\n\n\\subsection{Main Results}\nThe objective of this paper is to establish a quantitative relationship between a knot's three-dimensional complexity, measured via its tunnel number, and its diagrammatic complexity on an arbitrary fixed closed surface.\n\n\\documentclass[11pt]{amsart}\n\\usepackage{amsmath,amssymb,amsthm}\n\\usepackage{hyperref}\n\\usepackage{tikz}\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{remark}[theorem]{Remark}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{question}[theorem]{Question}\n\\title[Crossing Numbers of Knots on Closed Surfaces]\n{Crossing Numbers of Knots on Closed Surfaces}\n\\author{Makoto Ozawa}\n\\address{Department of Natural Sciences, Faculty of Arts and Sciences, Komazawa University, 1-23-1 Komazawa, Setagaya-ku, Tokyo, 154-8525, Japan}\n\\email{w3c@komazawa-u.ac.jp}\n\\date{\\today}\n\\dedicatory{Dedicated to Professor Kanji Morimoto on the occasion of his retirement.}\n\\keywords{knot, surface crossing number, tunnel number, Heegaard deficiency, Heegaard splitting, surface bridge number, surface ascending number}\n\\begin{document}\n\\begin{abstract}\nLet $c(K;F)$ denote the surface crossing number of a knot $K$ with respect\nto a closed surface $F \\subset S^{3}$. We establish the lower bound\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$ and $\\delta(F)$ is the Heegaard\ndeficiency of $F$. In particular, for any fixed closed surface $F$, the\nsurface crossing number $c(K;F)$ is unbounded over all knots $K$.\nFurthermore, we construct a family of knots $K_m$ demonstrating that $c(K_m;F) = \\Theta(t(K_m))$, which shows that this lower bound is asymptotically sharp.\n\\end{abstract}\n\\maketitle\n\n\\subsection{Main Results}\nThe objective of this paper is to establish a quantitative relationship between a knot's three-dimensional complexity, measured via its tunnel number, and its diagrammatic complexity on an arbitrary fixed closed surface.\n\nThe proof relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$:\n$$\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).$$\nThis sequence demonstrates that the geometric obstruction to simplifying a knot diagram is bounded below by a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\\delta(F)$. Unlike the classical planar crossing number, $c(K;F)$ directly reflects the Heegaard-theoretic complexity of the knot.\n\nFurthermore, we prove that the linear lower bound in Theorem \\ref{thm:main_intro} is asymptotically optimal. For iterated connected sums $K_m$ of a prime knot (such as the trefoil), we obtain the linear growth estimate:\n$$c(K_m;F) \\ge 2m - 2\\delta(F) + 1.$$\nPairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F) = \\Theta(t(K_m))$. This confirms that no general lower bound of higher order exists, and that the surface crossing number grows linearly with respect to the tunnel number.\n\n\\begin{theorem}[Fundamental inequality]\\label{thm:fundamental}\nLet $F\\subset S^{3}$ be a closed separating surface, and let $K\\subset S^{3}$ be a knot. Then\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1.\n\\]\n\\end{theorem}\n\nApplying the fundamental inequality to $K_m$ yields:\n\\begin{proposition}\nFor every closed surface $F\\subset S^{3}$,\n\\[\nc(K_m;F) \\ge 2\\bigl(m-\\delta(F)\\bigr)+1 \\to \\infty \\quad \\text{as } m \\to \\infty.\n\\]\n\\end{proposition}\n\n\\section{Proof of the Main Inequality}\nIn this section, we prove Theorem \\ref{thm:fundamental} by establishing the chain\n$$\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).$$\nThroughout the proof, let $F\\subset S^{3}$ be a closed separating surface with decomposition\n$S^{3}=M_{1}\\cup_{F}M_{2}$.\n\n\\subsection{Generalization to spatial graphs}\nThe framework extends to spatial graphs. A spatial graph $G$ is a 1-dimensional CW complex embedded in $S^{3}$. The key arguments rely on the 1-dimensionality of the object: in the proof of Lemma \\ref{lem:amalgamation}, a general position argument shows that the 2-dimensional attaching disks of the 1-handles can be made disjoint from $G$. The tunnel number $t(G)$, bridge position, and ascending number extend to this setting, and one expects a lower bound of the form\n\\[\nc(G;F) \\ge 2\\bigl(t(G)-\\delta(F)\\bigr) + C_G,\n\\]\nwhere $C_G$ is a constant depending on the combinatorial structure of $G$ (such as its Euler characteristic or vertex degrees).\n\n\\begin{theorem}[Fundamental inequality]\\label{thm:main_intro}\nLet $F\\subset S^3$ be a closed surface. Then every knot $K\\subset S^3$ satisfies:\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$, and $\\delta(F)=g(M_1)+g(M_2)-g(F)$ is the Heegaard deficiency of $F$.\n\\end{theorem}", "post_theorem_intro_text_len": 2183, "post_theorem_intro_text": "The proof relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$:\n$$\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).$$\nThis sequence demonstrates that the geometric obstruction to simplifying a knot diagram is bounded below by a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\\delta(F)$. Unlike the classical planar crossing number, $c(K;F)$ directly reflects the Heegaard-theoretic complexity of the knot.\n\nFurthermore, we prove that the linear lower bound in Theorem \\ref{thm:main_intro} is asymptotically optimal. For iterated connected sums $K_m$ of a prime knot (such as the trefoil), we obtain the linear growth estimate:\n$$c(K_m;F) \\ge 2m - 2\\delta(F) + 1.$$\nPairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F) = \\Theta(t(K_m))$. This confirms that no general lower bound of higher order exists, and that the surface crossing number grows linearly with respect to the tunnel number.\n\n\\subsection{Organization of the Paper}\nThe paper is organized as follows. Section 2 recalls basic definitions and establishes the monotonicity of $c(K;F)$ under surface compression. Section 3 states the main results. Section 4 proves the fundamental inequality using the ascending number, bridge position, and the $(g,b)$-decomposition framework of \\cite{MSY}. Section 5 provides the explicit construction of the knot family $K_m$ to demonstrate the asymptotic sharpness of our lower bound. Finally, Section 6 discusses further implications and open questions. These include the hierarchy of surface invariants, their behavior under connected sums, and a comparison with the surface trunk. We outline generalizations to surfaces with boundary and spatial graphs, and remark on the necessity of the genuine manifold condition compared to singular spaces \\cite{D,G}. The section concludes with questions regarding the asymptotic behavior for prime knots, lower bounds by the Haken number, and the minimality of alternating projections on closed surfaces.", "sketch": "To prove Theorem~\\ref{thm:main_intro}, the argument “relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$”:\n\\[\n\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).\n\\]\nThis sequence yields the stated lower bound for $c(K;F)$ in terms of “a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\\delta(F)$.”\n\nFor optimality, the introduction says they “prove that the linear lower bound in Theorem~\\ref{thm:main_intro} is asymptotically optimal” by taking iterated connected sums $K_m$ of a prime knot and obtaining\n\\[\nc(K_m;F) \\ge 2m - 2\\delta(F) + 1.\n\\]\n“Pairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F)=\\Theta(t(K_m))$,” so “no general lower bound of higher order exists,” and $c(K;F)$ grows linearly with tunnel number.", "expanded_sketch": "To prove the main theorem, the argument “relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$”:\n\\[\n\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).\n\\]\nThis sequence yields the stated lower bound for $c(K;F)$ in terms of “a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\\delta(F)$.”\n\nFor optimality, the introduction says they “prove that the linear lower bound in the main theorem is asymptotically optimal” by taking iterated connected sums $K_m$ of a prime knot and obtaining\n\\[\nc(K_m;F) \\ge 2m - 2\\delta(F) + 1.\n\\]\n“Pairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F)=\\Theta(t(K_m))$,” so “no general lower bound of higher order exists,” and $c(K;F)$ grows linearly with tunnel number.", "expanded_theorem": "[Fundamental inequality]\\label{thm:main_intro}\nLet $F\\subset S^3$ be a closed surface. Then every knot $K\\subset S^3$ satisfies:\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$, and $\\delta(F)=g(M_1)+g(M_2)-g(F)$ is the Heegaard deficiency of $F$.", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let $F\\subset S^3$ be any closed surface, and write $S^3=M_1\\cup_F M_2$. Define the Heegaard deficiency of $F$ by $\\delta(F)=g(M_1)+g(M_2)-g(F)$, where $g(\\cdot)$ denotes genus. For a knot $K\\subset S^3$, let $t(K)$ be its tunnel number, and let $c(K;F)$ denote the surface crossing number of $K$ with respect to $F$, meaning the minimum number of crossings among all regular diagrams obtained by isotoping $K$ into a regular neighborhood of $F$. Which statement holds for every such closed surface $F$ and every knot $K\\subset S^3$?", "correct_choice": {"label": "A", "text": "$c(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1$."}, "choices": [{"label": "B", "text": "$c(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)$."}, {"label": "C", "text": "$c(K;F) \\ge t(K)-\\delta(F)+1$."}, {"label": "D", "text": "$c(K;F) \\ge 2\\bigl(t(K)+\\delta(F)\\bigr)+1$."}, {"label": "E", "text": "$c(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+3$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "constant-term-from-half-integer-chain", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped leading factor 2 while keeping the same dependence on $t(K)$ and $\\delta(F)$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "sign of Heegaard deficiency term", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "sharp additive constant in the final bound", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives definitions and asks for the correct universal estimate, but it does not explicitly or implicitly reveal the exact bound. The correct additive constant and the use of \u0003(F) must be inferred from the options."}, "TAS": {"score": 1, "justification": "This is largely a theorem-recall item: the task is to identify the exact published inequality among close variants. It is not a pure tautology because the choices compete on sharpness, constants, and the correct invariant, but it still mainly tests recognition of the theorem statement."}, "GPS": {"score": 1, "justification": "Some reasoning is required to compare a sharp bound, a weaker true bound, and false strengthenings or parameter substitutions. However, the item does not demand substantial derivation; it primarily rewards remembering or recognizing the exact estimate."}, "DQS": {"score": 2, "justification": "The distractors are strong: one is a stronger-but-false variant, one is a weaker true statement, one swaps in the wrong invariant, and one suggests an unjustified uniform constant. These are plausible and mathematically meaningful failure modes."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and high-quality distractors, but it mainly tests recognition of a theorem statement rather than deeper generative reasoning."}}
{"id": "2602.21659v1", "paper_link": "http://arxiv.org/abs/2602.21659v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "[Fundamental inequality]\\label{thm:main_intro}\nLet $F\\subset S^3$ be a closed surface. Then every knot $K\\subset S^3$ satisfies:\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$, and $\\delta(F)=g(M_1)+g(M_2)-g(F)$ is the Heegaard deficiency of $F$.", "start_pos": 2789, "end_pos": 3106, "label": "thm:main_intro"}, "ref_dict": {"thm:main_intro": "\\begin{theorem}[Fundamental inequality]\\label{thm:main_intro}\nLet $F\\subset S^3$ be a closed surface. Then every knot $K\\subset S^3$ satisfies:\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$, and $\\delta(F)=g(M_1)+g(M_2)-g(F)$ is the Heegaard deficiency of $F$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 1144, "pre_theorem_intro_text": "\\subsection{Background and Motivation}\nLow-dimensional topology quantifies knot complexity through numerical invariants. While classical invariants, such as the standard crossing number, rely on planar diagrams, these two-dimensional projections compress spatial information which does not fully reflect the global topology of the knot exterior within the ambient space $S^3$.\n\nTo capture this three-dimensional structure, we extend the notion of diagrammatic complexity to arbitrary closed surfaces. For a knot $K$ and a closed surface $F \\subset S^3$, we define the \\emph{surface crossing number} $c(K;F)$ as the minimal number of crossings among all regular diagrams of $K$ obtained by isotoping the knot into a regular neighborhood of $F$. \n\nIn this paper, we establish an inequality demonstrating that $c(K;F)$ is bounded below by the tunnel number of $K$ and the Heegaard deficiency of $F$.\n\n\\subsection{Main Results}\nThe objective of this paper is to establish a quantitative relationship between a knot's three-dimensional complexity, measured via its tunnel number, and its diagrammatic complexity on an arbitrary fixed closed surface.", "context": "\\subsection{Background and Motivation}\nLow-dimensional topology quantifies knot complexity through numerical invariants. While classical invariants, such as the standard crossing number, rely on planar diagrams, these two-dimensional projections compress spatial information which does not fully reflect the global topology of the knot exterior within the ambient space $S^3$.\n\nTo capture this three-dimensional structure, we extend the notion of diagrammatic complexity to arbitrary closed surfaces. For a knot $K$ and a closed surface $F \\subset S^3$, we define the \\emph{surface crossing number} $c(K;F)$ as the minimal number of crossings among all regular diagrams of $K$ obtained by isotoping the knot into a regular neighborhood of $F$.\n\nIn this paper, we establish an inequality demonstrating that $c(K;F)$ is bounded below by the tunnel number of $K$ and the Heegaard deficiency of $F$.\n\n\\subsection{Main Results}\nThe objective of this paper is to establish a quantitative relationship between a knot's three-dimensional complexity, measured via its tunnel number, and its diagrammatic complexity on an arbitrary fixed closed surface.", "full_context": "\\subsection{Background and Motivation}\nLow-dimensional topology quantifies knot complexity through numerical invariants. While classical invariants, such as the standard crossing number, rely on planar diagrams, these two-dimensional projections compress spatial information which does not fully reflect the global topology of the knot exterior within the ambient space $S^3$.\n\nTo capture this three-dimensional structure, we extend the notion of diagrammatic complexity to arbitrary closed surfaces. For a knot $K$ and a closed surface $F \\subset S^3$, we define the \\emph{surface crossing number} $c(K;F)$ as the minimal number of crossings among all regular diagrams of $K$ obtained by isotoping the knot into a regular neighborhood of $F$.\n\nIn this paper, we establish an inequality demonstrating that $c(K;F)$ is bounded below by the tunnel number of $K$ and the Heegaard deficiency of $F$.\n\n\\subsection{Main Results}\nThe objective of this paper is to establish a quantitative relationship between a knot's three-dimensional complexity, measured via its tunnel number, and its diagrammatic complexity on an arbitrary fixed closed surface.\n\n\\documentclass[11pt]{amsart}\n\\usepackage{amsmath,amssymb,amsthm}\n\\usepackage{hyperref}\n\\usepackage{tikz}\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{remark}[theorem]{Remark}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{question}[theorem]{Question}\n\\title[Crossing Numbers of Knots on Closed Surfaces]\n{Crossing Numbers of Knots on Closed Surfaces}\n\\author{Makoto Ozawa}\n\\address{Department of Natural Sciences, Faculty of Arts and Sciences, Komazawa University, 1-23-1 Komazawa, Setagaya-ku, Tokyo, 154-8525, Japan}\n\\email{w3c@komazawa-u.ac.jp}\n\\date{\\today}\n\\dedicatory{Dedicated to Professor Kanji Morimoto on the occasion of his retirement.}\n\\keywords{knot, surface crossing number, tunnel number, Heegaard deficiency, Heegaard splitting, surface bridge number, surface ascending number}\n\\begin{document}\n\\begin{abstract}\nLet $c(K;F)$ denote the surface crossing number of a knot $K$ with respect\nto a closed surface $F \\subset S^{3}$. We establish the lower bound\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$ and $\\delta(F)$ is the Heegaard\ndeficiency of $F$. In particular, for any fixed closed surface $F$, the\nsurface crossing number $c(K;F)$ is unbounded over all knots $K$.\nFurthermore, we construct a family of knots $K_m$ demonstrating that $c(K_m;F) = \\Theta(t(K_m))$, which shows that this lower bound is asymptotically sharp.\n\\end{abstract}\n\\maketitle\n\n\\subsection{Main Results}\nThe objective of this paper is to establish a quantitative relationship between a knot's three-dimensional complexity, measured via its tunnel number, and its diagrammatic complexity on an arbitrary fixed closed surface.\n\nThe proof relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$:\n$$\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).$$\nThis sequence demonstrates that the geometric obstruction to simplifying a knot diagram is bounded below by a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\\delta(F)$. Unlike the classical planar crossing number, $c(K;F)$ directly reflects the Heegaard-theoretic complexity of the knot.\n\nFurthermore, we prove that the linear lower bound in Theorem \\ref{thm:main_intro} is asymptotically optimal. For iterated connected sums $K_m$ of a prime knot (such as the trefoil), we obtain the linear growth estimate:\n$$c(K_m;F) \\ge 2m - 2\\delta(F) + 1.$$\nPairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F) = \\Theta(t(K_m))$. This confirms that no general lower bound of higher order exists, and that the surface crossing number grows linearly with respect to the tunnel number.\n\n\\begin{theorem}[Fundamental inequality]\\label{thm:fundamental}\nLet $F\\subset S^{3}$ be a closed separating surface, and let $K\\subset S^{3}$ be a knot. Then\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1.\n\\]\n\\end{theorem}\n\nApplying the fundamental inequality to $K_m$ yields:\n\\begin{proposition}\nFor every closed surface $F\\subset S^{3}$,\n\\[\nc(K_m;F) \\ge 2\\bigl(m-\\delta(F)\\bigr)+1 \\to \\infty \\quad \\text{as } m \\to \\infty.\n\\]\n\\end{proposition}\n\n\\section{Proof of the Main Inequality}\nIn this section, we prove Theorem \\ref{thm:fundamental} by establishing the chain\n$$\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).$$\nThroughout the proof, let $F\\subset S^{3}$ be a closed separating surface with decomposition\n$S^{3}=M_{1}\\cup_{F}M_{2}$.\n\n\\subsection{Generalization to spatial graphs}\nThe framework extends to spatial graphs. A spatial graph $G$ is a 1-dimensional CW complex embedded in $S^{3}$. The key arguments rely on the 1-dimensionality of the object: in the proof of Lemma \\ref{lem:amalgamation}, a general position argument shows that the 2-dimensional attaching disks of the 1-handles can be made disjoint from $G$. The tunnel number $t(G)$, bridge position, and ascending number extend to this setting, and one expects a lower bound of the form\n\\[\nc(G;F) \\ge 2\\bigl(t(G)-\\delta(F)\\bigr) + C_G,\n\\]\nwhere $C_G$ is a constant depending on the combinatorial structure of $G$ (such as its Euler characteristic or vertex degrees).\n\n\\begin{theorem}[Fundamental inequality]\\label{thm:main_intro}\nLet $F\\subset S^3$ be a closed surface. Then every knot $K\\subset S^3$ satisfies:\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$, and $\\delta(F)=g(M_1)+g(M_2)-g(F)$ is the Heegaard deficiency of $F$.\n\\end{theorem}", "post_theorem_intro_text_len": 2183, "post_theorem_intro_text": "The proof relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$:\n$$\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).$$\nThis sequence demonstrates that the geometric obstruction to simplifying a knot diagram is bounded below by a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\\delta(F)$. Unlike the classical planar crossing number, $c(K;F)$ directly reflects the Heegaard-theoretic complexity of the knot.\n\nFurthermore, we prove that the linear lower bound in Theorem \\ref{thm:main_intro} is asymptotically optimal. For iterated connected sums $K_m$ of a prime knot (such as the trefoil), we obtain the linear growth estimate:\n$$c(K_m;F) \\ge 2m - 2\\delta(F) + 1.$$\nPairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F) = \\Theta(t(K_m))$. This confirms that no general lower bound of higher order exists, and that the surface crossing number grows linearly with respect to the tunnel number.\n\n\\subsection{Organization of the Paper}\nThe paper is organized as follows. Section 2 recalls basic definitions and establishes the monotonicity of $c(K;F)$ under surface compression. Section 3 states the main results. Section 4 proves the fundamental inequality using the ascending number, bridge position, and the $(g,b)$-decomposition framework of \\cite{MSY}. Section 5 provides the explicit construction of the knot family $K_m$ to demonstrate the asymptotic sharpness of our lower bound. Finally, Section 6 discusses further implications and open questions. These include the hierarchy of surface invariants, their behavior under connected sums, and a comparison with the surface trunk. We outline generalizations to surfaces with boundary and spatial graphs, and remark on the necessity of the genuine manifold condition compared to singular spaces \\cite{D,G}. The section concludes with questions regarding the asymptotic behavior for prime knots, lower bounds by the Haken number, and the minimality of alternating projections on closed surfaces.", "sketch": "To prove Theorem~\\ref{thm:main_intro}, the argument “relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$”:\n\\[\n\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).\n\\]\nThis sequence yields the stated lower bound for $c(K;F)$ in terms of “a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\\delta(F)$.”\n\nFor optimality, the introduction says they “prove that the linear lower bound in Theorem~\\ref{thm:main_intro} is asymptotically optimal” by taking iterated connected sums $K_m$ of a prime knot and obtaining\n\\[\nc(K_m;F) \\ge 2m - 2\\delta(F) + 1.\n\\]\n“Pairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F)=\\Theta(t(K_m))$,” so “no general lower bound of higher order exists,” and $c(K;F)$ grows linearly with tunnel number.", "expanded_sketch": "To prove the main theorem, the argument “relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$”:\n\\[\n\\frac{c(K;F)-1}{2} \\ge a(K;F) \\ge b(K;F)-1 \\ge t(K)-\\delta(F).\n\\]\nThis sequence yields the stated lower bound for $c(K;F)$ in terms of “a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\\delta(F)$.”\n\nFor optimality, the introduction says they “prove that the linear lower bound in the main theorem is asymptotically optimal” by taking iterated connected sums $K_m$ of a prime knot and obtaining\n\\[\nc(K_m;F) \\ge 2m - 2\\delta(F) + 1.\n\\]\n“Pairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F)=\\Theta(t(K_m))$,” so “no general lower bound of higher order exists,” and $c(K;F)$ grows linearly with tunnel number.", "expanded_theorem": "[Fundamental inequality]\\label{thm:main_intro}\nLet $F\\subset S^3$ be a closed surface. Then every knot $K\\subset S^3$ satisfies:\n\\[\nc(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1,\n\\]\nwhere $t(K)$ is the tunnel number of $K$, and $\\delta(F)=g(M_1)+g(M_2)-g(F)$ is the Heegaard deficiency of $F$.", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let $F\\subset S^3$ be any closed surface, and write $S^3=M_1\\cup_F M_2$. Define the Heegaard deficiency of $F$ by $\\delta(F)=g(M_1)+g(M_2)-g(F)$, where $g(\\cdot)$ denotes genus. For a knot $K\\subset S^3$, let $t(K)$ be its tunnel number, and let $c(K;F)$ denote the surface crossing number of $K$ with respect to $F$, meaning the minimum number of crossings among all regular diagrams obtained by isotoping $K$ into a regular neighborhood of $F$. Which statement holds for every such closed surface $F$ and every knot $K\\subset S^3$?", "correct_choice": {"label": "A", "text": "$c(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+1$."}, "choices": [{"label": "B", "text": "$c(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)$."}, {"label": "C", "text": "$c(K;F) \\ge t(K)-\\delta(F)+1$."}, {"label": "D", "text": "$c(K;F) \\ge 2\\bigl(t(K)+\\delta(F)\\bigr)+1$."}, {"label": "E", "text": "$c(K;F) \\ge 2\\bigl(t(K)-\\delta(F)\\bigr)+3$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "constant-term-from-half-integer-chain", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped leading factor 2 while keeping the same dependence on $t(K)$ and $\\delta(F)$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "sign of Heegaard deficiency term", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "sharp additive constant in the final bound", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem only defines the quantities and asks for a universal inequality; it does not explicitly reveal the correct bound or give strong hints about the exact coefficients or constant term."}, "TAS": {"score": 1, "justification": "This is close to a theorem-statement recognition item: the task is essentially to identify the correct universal inequality. However, the nearby alternative bounds mean it is not a completely verbatim restatement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the sharp bound from weaker true or false variants, especially because the options differ by subtle coefficient/sign/constant changes. Still, the item mainly tests precise recall or recognition rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic errors: dropping the factor of 2, changing the sign of the deficiency term, or altering the additive constant. Including a weaker true statement also improves quality."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it remains primarily a theorem-recognition question rather than one that strongly tests generative mathematical reasoning."}}
{"id": "2602.21759v1", "paper_link": "http://arxiv.org/abs/2602.21759v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "[Density Theorem]\\label{thm:density Fukaya}\n  For any closed exact Lagrangian $L \\in \\Fuk(DT^*N)$ and $\\varepsilon >0$, there are points $(x_i)_{i\\in \\{1, \\ldots,l\\}}$, real numbers $(a_i)_{i\\in \\{1, \\ldots,l\\}}$ and $C \\in \\langle \\V_{(x_1,a_1)}, \\ldots, \\V_{(x_l,a_l)} \\rangle$ such that $\\gamma (L,C) < \\varepsilon$.", "start_pos": 12680, "end_pos": 13020, "label": "thm:density Fukaya"}, "ref_dict": {}, "pre_theorem_intro_text_len": 3407, "pre_theorem_intro_text": "Abouzaid proved in \\cite{Abouzaid-fiber-generation} that the wrapped Fukaya category of a cotangent bundle is generated by one cotangent fiber. In the filtered case, this can not happen: the Hamiltonian perturbation of a Lagrangian is isomorphic to the original Lagrangian only when they geometrically coincide. However the filtered Fukaya category comes with a notion of interleaving distance and Paul Biran~\\cite{Biran-V63}asked whether the iterated cones on the cotangent fibers generate a dense subcategory, in the sense that any Lagrangian is arbitrarily close to an iterated cone of cotangent fibers. We refer to \\cite{Ambrosioni-Filtered-Fukaya} for a construction of the filtered Fukaya category and~\\cite{Biran-Cornea-Zhang} for its persistence structure, which yields a notion of distance. Similar density considerations had been evoked in \\cite{Fukaya-GHdistance}.\n\nWe shall use the filtered Fukaya category constructed in~\\cite{Ambrosioni-Filtered-Fukaya}. Its objects are the compact Lagrangians so it does not contain the cotangent fibers. To make sense of the question of density of iterated cones of the fibers, we shall embed this category in its module category through the Yoneda embedding.  \n\nLet us introduce some notations. Let $N$ be a connected closed manifold and $DT^*N$ its unit ball cotangent bundle (with respect to a Riemannian metric) with Liouville form $\\lambda$.  We denote by $\\Fuk(DT^*N)$ the filtered Fukaya category whose objects are compact (exact) Lagrangian branes, that is pairs $(L, f_L)$ where $L$ is a closed exact Lagrangian contained in $DT^*N$ and $f_L$ is a primitive of $\\lambda_{\\mid L}$. Morphisms from $L$ to $L'$ are given by the Floer cochains $FC^*(L,L')$. The space of Floer cochains is filtered once we have primitives $f_L, f_{L'}$ of $\\lambda_{\\mid L}, \\lambda_{\\mid L'}$: an intersection point $x\\in L\\cap L'$ has filtration degree $f_{L'}(x)-f_L(x)$. \n\nWe let $\\Yo\\colon \\Fuk(DT^*N) \\to \\fMod(\\Fuk(DT^*N))$ be the Yoneda embedding.  Here $\\fMod(\\mathcal C)$ is the set of functors from $\\mathcal C$ to the category of filtered chain complexes, and the Yoneda morphism is given by associating to $L$ the functor $L' \\mapsto FC^*(L',L;t)$ where $FC^*(L',L;t)$ is the set of Floer chains of action greater than $t$. For simplicity we denote by $FC^*(L,L')$ the Floer chain complex with its filtration being understood, so the Yoneda embedding will send $L$ to $L' \\mapsto FC^*(L',L)$. \n\nLet $V_{(x,a)}$ be the fiber at $x$ with primitive $a$ of $\\lambda_{\\mid V_x}=0$. This Lagrangian brane $V_{(x,a)}$ does not belong to $\\Fuk(DT^*N)$ but it makes sense to define the filtered complex $FC^*(L', V_{(x,a)})$ for any $L' \\in \\Fuk(DT^*N)$ and this gives a module $\\V_{(x,a)} \\in \\fMod(\\Fuk(DT^*N))$.  The action of $FC^*(L,L')$ on $FC^*(L', V_{(x,a)})$ is given by the standard triangle product\n\\[FC^*(L,L')\\otimes FC^*(L', V_{(x,a)}) \\longrightarrow FC^*(L, V_{(x,a)})\\]\nwhich respects the filtration by \\cite{Ambrosioni-Filtered-Fukaya}. \n\nThe module category $\\fMod(\\Fuk(DT^*N))$ inherits an interleaving distance denoted by $\\gamma$.  It is also a pre-triangulated category. \n\nWe define the subcategory $\\langle \\V_{(x_1,a_1)}, \\ldots, \\V_{(x_l,a_l)} \\rangle$ of $\\fMod(\\Fuk(DT^*N))$ the subcategory generated by the $\\V_{(x_i,a_i)}$'s as the subcategory having for objects the iterated cones on the generators.\n\nNow we can state our main result.", "context": "Abouzaid proved in \\cite{Abouzaid-fiber-generation} that the wrapped Fukaya category of a cotangent bundle is generated by one cotangent fiber. In the filtered case, this can not happen: the Hamiltonian perturbation of a Lagrangian is isomorphic to the original Lagrangian only when they geometrically coincide. However the filtered Fukaya category comes with a notion of interleaving distance and Paul Biran~\\cite{Biran-V63}asked whether the iterated cones on the cotangent fibers generate a dense subcategory, in the sense that any Lagrangian is arbitrarily close to an iterated cone of cotangent fibers. We refer to \\cite{Ambrosioni-Filtered-Fukaya} for a construction of the filtered Fukaya category and~\\cite{Biran-Cornea-Zhang} for its persistence structure, which yields a notion of distance. Similar density considerations had been evoked in \\cite{Fukaya-GHdistance}.\n\nLet us introduce some notations. Let $N$ be a connected closed manifold and $DT^*N$ its unit ball cotangent bundle (with respect to a Riemannian metric) with Liouville form $\\lambda$. We denote by $\\Fuk(DT^*N)$ the filtered Fukaya category whose objects are compact (exact) Lagrangian branes, that is pairs $(L, f_L)$ where $L$ is a closed exact Lagrangian contained in $DT^*N$ and $f_L$ is a primitive of $\\lambda_{\\mid L}$. Morphisms from $L$ to $L'$ are given by the Floer cochains $FC^*(L,L')$. The space of Floer cochains is filtered once we have primitives $f_L, f_{L'}$ of $\\lambda_{\\mid L}, \\lambda_{\\mid L'}$: an intersection point $x\\in L\\cap L'$ has filtration degree $f_{L'}(x)-f_L(x)$.\n\nWe let $\\Yo\\colon \\Fuk(DT^*N) \\to \\fMod(\\Fuk(DT^*N))$ be the Yoneda embedding. Here $\\fMod(\\mathcal C)$ is the set of functors from $\\mathcal C$ to the category of filtered chain complexes, and the Yoneda morphism is given by associating to $L$ the functor $L' \\mapsto FC^*(L',L;t)$ where $FC^*(L',L;t)$ is the set of Floer chains of action greater than $t$. For simplicity we denote by $FC^*(L,L')$ the Floer chain complex with its filtration being understood, so the Yoneda embedding will send $L$ to $L' \\mapsto FC^*(L',L)$.\n\nLet $V_{(x,a)}$ be the fiber at $x$ with primitive $a$ of $\\lambda_{\\mid V_x}=0$. This Lagrangian brane $V_{(x,a)}$ does not belong to $\\Fuk(DT^*N)$ but it makes sense to define the filtered complex $FC^*(L', V_{(x,a)})$ for any $L' \\in \\Fuk(DT^*N)$ and this gives a module $\\V_{(x,a)} \\in \\fMod(\\Fuk(DT^*N))$. The action of $FC^*(L,L')$ on $FC^*(L', V_{(x,a)})$ is given by the standard triangle product\n\\[FC^*(L,L')\\otimes FC^*(L', V_{(x,a)}) \\longrightarrow FC^*(L, V_{(x,a)})\\]\nwhich respects the filtration by \\cite{Ambrosioni-Filtered-Fukaya}.\n\nWe define the subcategory $\\langle \\V_{(x_1,a_1)}, \\ldots, \\V_{(x_l,a_l)} \\rangle$ of $\\fMod(\\Fuk(DT^*N))$ the subcategory generated by the $\\V_{(x_i,a_i)}$'s as the subcategory having for objects the iterated cones on the generators.\n\nNow we can state our main result.", "full_context": "Abouzaid proved in \\cite{Abouzaid-fiber-generation} that the wrapped Fukaya category of a cotangent bundle is generated by one cotangent fiber. In the filtered case, this can not happen: the Hamiltonian perturbation of a Lagrangian is isomorphic to the original Lagrangian only when they geometrically coincide. However the filtered Fukaya category comes with a notion of interleaving distance and Paul Biran~\\cite{Biran-V63}asked whether the iterated cones on the cotangent fibers generate a dense subcategory, in the sense that any Lagrangian is arbitrarily close to an iterated cone of cotangent fibers. We refer to \\cite{Ambrosioni-Filtered-Fukaya} for a construction of the filtered Fukaya category and~\\cite{Biran-Cornea-Zhang} for its persistence structure, which yields a notion of distance. Similar density considerations had been evoked in \\cite{Fukaya-GHdistance}.\n\nLet us introduce some notations. Let $N$ be a connected closed manifold and $DT^*N$ its unit ball cotangent bundle (with respect to a Riemannian metric) with Liouville form $\\lambda$. We denote by $\\Fuk(DT^*N)$ the filtered Fukaya category whose objects are compact (exact) Lagrangian branes, that is pairs $(L, f_L)$ where $L$ is a closed exact Lagrangian contained in $DT^*N$ and $f_L$ is a primitive of $\\lambda_{\\mid L}$. Morphisms from $L$ to $L'$ are given by the Floer cochains $FC^*(L,L')$. The space of Floer cochains is filtered once we have primitives $f_L, f_{L'}$ of $\\lambda_{\\mid L}, \\lambda_{\\mid L'}$: an intersection point $x\\in L\\cap L'$ has filtration degree $f_{L'}(x)-f_L(x)$.\n\nWe let $\\Yo\\colon \\Fuk(DT^*N) \\to \\fMod(\\Fuk(DT^*N))$ be the Yoneda embedding. Here $\\fMod(\\mathcal C)$ is the set of functors from $\\mathcal C$ to the category of filtered chain complexes, and the Yoneda morphism is given by associating to $L$ the functor $L' \\mapsto FC^*(L',L;t)$ where $FC^*(L',L;t)$ is the set of Floer chains of action greater than $t$. For simplicity we denote by $FC^*(L,L')$ the Floer chain complex with its filtration being understood, so the Yoneda embedding will send $L$ to $L' \\mapsto FC^*(L',L)$.\n\nLet $V_{(x,a)}$ be the fiber at $x$ with primitive $a$ of $\\lambda_{\\mid V_x}=0$. This Lagrangian brane $V_{(x,a)}$ does not belong to $\\Fuk(DT^*N)$ but it makes sense to define the filtered complex $FC^*(L', V_{(x,a)})$ for any $L' \\in \\Fuk(DT^*N)$ and this gives a module $\\V_{(x,a)} \\in \\fMod(\\Fuk(DT^*N))$. The action of $FC^*(L,L')$ on $FC^*(L', V_{(x,a)})$ is given by the standard triangle product\n\\[FC^*(L,L')\\otimes FC^*(L', V_{(x,a)}) \\longrightarrow FC^*(L, V_{(x,a)})\\]\nwhich respects the filtration by \\cite{Ambrosioni-Filtered-Fukaya}.\n\nWe define the subcategory $\\langle \\V_{(x_1,a_1)}, \\ldots, \\V_{(x_l,a_l)} \\rangle$ of $\\fMod(\\Fuk(DT^*N))$ the subcategory generated by the $\\V_{(x_i,a_i)}$'s as the subcategory having for objects the iterated cones on the generators.\n\nNow we can state our main result.\n\nWe define the subcategory $\\langle \\V_{(x_1,a_1)}, \\ldots, \\V_{(x_l,a_l)} \\rangle$ of $\\fMod(\\Fuk(DT^*N))$ the subcategory generated by the $\\V_{(x_i,a_i)}$'s as the subcategory having for objects the iterated cones on the generators.\n\nNow we can state our main result.\n\nThe strategy of proof is to use the quantization functor constructed in~\\cite{Viterbo-Sheaves} which associates a sheaf $Q(L)$ on $N\\times\\Real$ with any closed exact Lagrangian brane $L$. In this way we embed $\\Fuk(DT^*N)$ into $\\Sh(N\\times\\Real)$, more precisely into the subcategory $\\Sh_{DT^*N}(N\\times\\Real)$ (actually the Tamarkin category $\\Tam_{DT^*N}(N)$ introduced later), formed by the sheaves with reduced microsupport contained in $DT^*N$. The category $\\Sh_{DT^*N}(N\\times\\Real)$ should be understood as a model of the wrapped Fukaya category in the filtered setting. The fiber $\\V_{(x,a)}$ corresponds to the sheaf $k_{\\{x\\} \\times [a, \\infty)}\\in \\Sh(N\\times\\Real)$, which does not belong to $\\Sh_{DT^*N}(N\\times\\Real)$. We let $P'_{DT^*N}$ be a projector which takes values in $\\Sh_{DT^*N}(N\\times\\Real)$ (it will be recalled in \\S\\ref{sec:strategyproof} - see~\\cite{Kuo-wrappedsheaves} where it is built as a wrapping functor) and we set $\\W_{(x,a)} = P'_{DT^*N}(k_{\\{x\\} \\times [a, \\infty)})$. A slight modification of~\\cite[Thm. 4.4]{Zhang-Cyclic} shows that the endomorphism algebra of $\\W_{(x,a)}$ computes the homology of length-filtered based loop spaces, which further clarifies the role of $\\Sh_{DT^*N}(N\\times\\Real)$ as ``wrapped filtered Fukaya category''. The sheaf category also inherits an interleaving distance that we denote by $\\gamma^s $.\n\nNow in the category $\\Sh_{DT^*N}(N\\times\\Real)$ we can prove a more general result: \n\\begin{thm}[Density Theorem]\\label{thm:density sheaf}\n Let $\\F \\in \\Sh_{DT^*N}(N\\times\\Real)$ such that for all $x\\in N$ the sheaf $\\F \\otimes k_{\\{x\\} \\times\\Real}$ is a $\\gamma^s$-limit of constructible sheaves on $\\Real$. Then, for any $\\varepsilon >0$, there are points $(x_i)_{i\\in \\{1, \\ldots,l\\}}$, real numbers $(a_i)_{i\\in \\{1, \\ldots,l\\}}$ and $C \\in \\langle \\W_{(x_1,a_1)}, \\ldots, \\W_{(x_l,a_l)} \\rangle$ in ${\\Sh_{DT^*N}(N\\times\\Real)}$ such that $\\gamma^s (\\F,C) < \\varepsilon$.\n\\end{thm}\n\n\\begin{rem}\nIn fact, as in \\cite{A-B-C} we prove something stronger than density, we prove what they call\n{\\it approximability}: For any $\\varepsilon>0$ we can find objects $X_1,...,X_N$ such that any object in the category is at distance at most $ \\varepsilon$ from an iterated cone of the $X_k$. Even better if we consider the set of all direct sums obtained from the $X_i$, we need only $n+1$-iterated cones (where $n$ is the dimension of the base manifold). A related notion of complexity called the interleaving Rouquier dimension will be discussed in Appendix~\\ref{sec:InterleavingRouquierdimension}. \n\\end{rem}\nOnce we have the result for sheaves, we can come back to $\\fMod(\\Fuk(DT^*N))$ using the quantization functor $Q$ since sheaf quantization of closed exact Lagrangian satisfies the assumption of the Density Theorem, due to constructibility of $Q(L) \\otimes k_{\\{x\\} \\times\\Real} \\simeq FC^*(V_{(x,0)},L)$ as sheaves on $\\Real$. In fact, the sheaf result is a little stronger in the sense that it may apply to sheaves associated with immersed Lagrangians or certain $C^0$-Lagrangians. One should also notice that the Fukaya category may depend on perturbation data; however, the sheaf category does not. Then it tells that the sheaf distance bound gives a uniform bound of Fukaya category distance for all perturbation data.\n\n\\begin{thm}[Density Theorem for sheaves]\\label{thm:density sheaves}\nLet $\\F \\in \\Tam_{DT^*N}(N\\times\\Real)$ such that for any $x\\in N$ the sheaf $\\F \\otimes k_{\\{x\\} \\times\\Real}$ is a $\\gamma$-limit of constructible sheaves on $\\Real$. Then for any $\\varepsilon >0$, there are points $(x_i)_{i\\in \\{1, \\ldots,l\\}}$, real numbers $(a_i)_{i\\in \\{1, \\ldots,l\\}}$ and $C \\in \\langle \\W_{(x_1,a_1)}, \\ldots, \\W_{(x_l,a_l)} \\rangle$ in ${\\Tam_{DT^*N}(N\\times\\Real)}$ such that $\\gamma^s (\\F,C) < \\varepsilon$.\n\\end{thm}\n\n\\begin{lem}\\label{lem:approx fibre boule}\n Let $x\\in N$, $\\varepsilon >0$ and let $Z\\subset N$ be a closed contractible subset contained in the ball with center $x$ and radius $\\varepsilon < r_{inj}(N)/3$. Then, for any $\\F \\in \\Sh(N\\times\\Real)$ such that $SS(\\F) \\subset \\{\\tau \\geq \\| p \\| \\}$, we have $\\gamma_{h^+}(\\F \\otimes k_{\\{x\\} \\times \\Real}, \\F \\otimes k_{Z \\times \\Real}) \\leq 4\\varepsilon$. \n\\end{lem}\n\\begin{proof}\n We recall that $h^+(x,p,t,\\tau) = \\max\\{ \\tau, \\| p \\| \\}$ for $\\tau \\geq 0$. Since we work on $\\Sh_{\\tau\\geq0}(N\\times\\Real)$ we can as well assume $h^+(x,p,t,\\tau) = \\max\\{ \\tau, \\| p \\| \\}$ for any $\\tau$. We first give a bound on $N^2\\times \\Real^2$ and then take the pull-back by the diagonal embedding $\\delta \\colon N\\times\\Real \\to (N\\times\\Real)^2$. To distinguish, we denote points of the second copy of $T^*(N\\times\\Real)$ with prime, and then points of $T^*(N^2\\times\\Real^2)$ are denoted by $(x,p,x',p',t,\\tau, t',\\tau')$. We will apply Lemma~\\ref{lem:distance restriction} and we look for a function $h_1$ on $T^*(N^2\\times\\Real^2)$ such that $h_1|_{\\Delta_1} = h^+ \\circ (d\\delta)^t$, where $\\Delta_1 = \\Delta_{N\\times\\Real} \\times_{N^2\\times\\Real^2} T^*(N^2\\times\\Real^2)$. We have $(d\\delta)^t(x,p,x,p',t,\\tau, t,\\tau') = (x,t, p+p', \\tau+\\tau')$ so we should extend the function $\\|p+p'\\|$ outside the diagonal. We let $\\Delta(r)$ be the neighborhood of $\\Delta_N$ formed by the pairs $(x,x')$ such that $d(x,x') < r$. For $(x,x') \\in \\Delta(r_{inj}(N))$ we identify $T_x^*N$ and $T_{x'}^*N$ through the parallel transport along the shortest geodesic between $x$ and $x'$; we can then make sense of $p+p'$. We choose a partition of unity $\\alpha, \\beta$ with $\\alpha=1$ on $\\Delta(r_{inj}(N)/2)$ and $\\beta=1$ outside $\\Delta(r_{inj}(N))$. We set $h_0(x,p,x,p',t,\\tau, t,\\tau') = \\alpha \\|p+p'\\| + \\beta \\|(p,p')\\|$. We have the inequality\n\\[\n \\|p\\| + h_0 \\geq \\|p'\\|\n \\]\n because over $\\Delta(r_{inj}(N))$ we have $\\|p\\| + h_0 \\geq \\alpha \\|p\\| + \\alpha \\|p+p'\\| + \\beta \\|p'\\| \\geq (\\alpha + \\beta)\\|p'\\| = \\|p'\\|$ and outside $\\Delta(r_{inj}(N))$ we even have $h_0 \\geq \\|p'\\|$.\n\n\\subsection{Proof of Theorem~\\ref{thm:density sheaves}}\\label{sec:proofmainthm}\nLet $0<\\varepsilon < r_{inj}(N)/3$. We pick a finite covering $N = \\bigcup_{i\\in I} U_i$ be a finite covering as in Lemma \\ref{lem:Cech} and assume moreover that all $U_i$'s are contained in a ball of radius less than $\\varepsilon$. We set $U_J^{cl} = \\bigcap_{j\\in J} \\overline{U_j}$. By Lemma \\ref{lem:Cech} $\\F$ is an $n$ steps iterated cone on $\\F \\otimes C_0[1]$, $\\F \\otimes C_1$, \\dots, $\\F \\otimes C_n[1-n]$, where $C_i = \\bigoplus_{J \\subset I, |J| = i+1} k_{U_J^{cl}\\times \\Real}$. For each $J\\subset I$ we pick $x_J$ such that $U_J^{cl} \\subset B_\\varepsilon(x_J )$. Lemma \\ref{lem:approx fibre boule} says that $\\gamma_{h^+}(\\F \\otimes k_{\\{x_J\\} \\times \\Real}, \\F \\otimes k_{U_J^{cl} \\times \\Real}) \\leq 4\\varepsilon$. We set $C'_i = \\bigoplus_{J \\subset I, |J| = i+1} k_{\\{x_J\\}\\times \\Real}$. By Lemma~\\ref{lem:distance somme} we have $\\gamma_{h^+}(\\F \\otimes C_i, \\F \\otimes C'_i) \\leq 8 \\varepsilon$.", "post_theorem_intro_text_len": 5732, "post_theorem_intro_text": "The strategy of proof is to use the quantization functor constructed in~\\cite{Viterbo-Sheaves} which associates a sheaf $Q(L)$ on $N\\times{\\mathbb R}$ with any closed exact Lagrangian brane $L$.  In this way we embed $\\Fuk(DT^*N)$ into $\\Sh(N\\times{\\mathbb R})$, more precisely into the subcategory $\\Sh_{DT^*N}(N\\times{\\mathbb R})$ (actually the Tamarkin category $\\Tam_{DT^*N}(N)$ introduced later), formed by the sheaves with reduced microsupport contained in $DT^*N$. The category $\\Sh_{DT^*N}(N\\times{\\mathbb R})$ should be understood as a model of the wrapped Fukaya category in the filtered setting. The fiber $\\V_{(x,a)}$ corresponds to the sheaf $k_{\\{x\\} \\times [a, \\infty)}\\in \\Sh(N\\times{\\mathbb R})$, which does not belong to $\\Sh_{DT^*N}(N\\times{\\mathbb R})$. We let $P'_{DT^*N}$ be a projector which takes values in $\\Sh_{DT^*N}(N\\times{\\mathbb R})$ (it will be recalled in \\S\\ref{sec:strategyproof} - see~\\cite{Kuo-wrappedsheaves} where it is built as a wrapping functor) and we set $\\W_{(x,a)} = P'_{DT^*N}(k_{\\{x\\} \\times [a, \\infty)})$. A slight modification of~\\cite[Thm. 4.4]{Zhang-Cyclic} shows that the endomorphism algebra of $\\W_{(x,a)}$ computes the homology of length-filtered based loop spaces, which further clarifies the role of $\\Sh_{DT^*N}(N\\times{\\mathbb R})$ as ``wrapped filtered Fukaya category''. The sheaf category also inherits an interleaving distance that we denote by $\\gamma^s $. \n\nNow in the category $\\Sh_{DT^*N}(N\\times{\\mathbb R})$ we can prove a more general result: \n\\begin{thm}[Density Theorem]\\label{thm:density sheaf}\n  Let ${\\mathcal F} \\in \\Sh_{DT^*N}(N\\times{\\mathbb R})$ such that for all $x\\in N$ the sheaf ${\\mathcal F} \\otimes k_{\\{x\\} \\times{\\mathbb R}}$ is a $\\gamma^s$-limit of constructible sheaves on ${\\mathbb R}$. Then, for any $\\varepsilon >0$, there are points $(x_i)_{i\\in \\{1, \\ldots,l\\}}$, real numbers $(a_i)_{i\\in \\{1, \\ldots,l\\}}$ and $C \\in \\langle \\W_{(x_1,a_1)}, \\ldots, \\W_{(x_l,a_l)} \\rangle$ in ${\\Sh_{DT^*N}(N\\times{\\mathbb R})}$ such that $\\gamma^s ({\\mathcal F},C) < \\varepsilon$.\n\\end{thm}\n\nThe idea is to use a \\v Cech resolution of the constant sheaf $k_N \\simeq C_0 \\to C_1 \\to \\cdots \\to C_m$, where $C_i = \\bigoplus_{J \\subset I, |J| = i+1} k_{U_J^{cl}}$ with $U_J^{cl} = \\bigcap_{j\\in J} \\overline{U_j}$ and the $U_j$'s are $\\varepsilon$-small balls covering $N$. Then ${\\mathcal F} = {\\mathcal F} \\otimes k_{N\\times{\\mathbb R}}$ is written as an iterated cone on the ${\\mathcal F} \\otimes k_{U_J^{cl} \\times{\\mathbb R}}$. Now ${\\mathcal F} \\otimes k_{U_J^{cl} \\times{\\mathbb R}}$ can be approximated by ${\\mathcal F} \\otimes k_{\\{x\\} \\times{\\mathbb R}}$ for some $x \\in U_J^{cl}$ and ${\\mathcal F} \\otimes k_{\\{x\\} \\times{\\mathbb R}} $ (a sheaf on a line) can be written as (in general can be approximated by) an iterated cone on the $\\W_{(x,a)}$'s by constructibility. From the argument, we know that the points $(x_i)_{i\\in \\{1, \\ldots,l\\}}$ can be taken from an a priori given dense subset of the base manifold $N$. Also, the proof may be viewed as a filtered analogue of the sectorial descent of~\\cite{G-P-S-3}, formulated in the language of sheaves.\n\n\\begin{rem}\nIn fact, as in \\cite{A-B-C} we prove something stronger than density, we prove what they call\n{\\it approximability}: For any $\\varepsilon>0$ we can find objects $X_1,...,X_N$ such that any object in the category is at distance at most $ \\varepsilon$ from an iterated cone of the $X_k$. Even better if we consider the set of all direct sums  obtained from the $X_i$, we need only $n+1$-iterated cones (where $n$ is the dimension of the base manifold). A related notion of complexity called the interleaving Rouquier dimension will be discussed in Appendix~\\ref{sec:InterleavingRouquierdimension}.  \n\\end{rem}\nOnce we have the result for sheaves, we can come back to $\\fMod(\\Fuk(DT^*N))$ using the quantization functor $Q$ since sheaf quantization of closed exact Lagrangian satisfies the assumption of the Density Theorem, due to constructibility of $Q(L) \\otimes k_{\\{x\\} \\times{\\mathbb R}} \\simeq FC^*(V_{(x,0)},L)$ as sheaves on ${\\mathbb R}$. In fact, the sheaf result is a little stronger in the sense that it may apply to sheaves associated with immersed Lagrangians or certain $C^0$-Lagrangians. One should also notice that the Fukaya category may depend on perturbation data; however, the sheaf category does not. Then it tells that the sheaf distance bound gives a uniform bound of Fukaya category distance for all perturbation data.\n\n\\medskip\n\nTo implement this plan of proof we need to be more precise on the quantization functor $Q$. In~\\cite{Viterbo-Sheaves} the quantization functor is defined on the Donaldson-Fukaya category of $T^*N$ with value in the (classical) derived category of sheaves. However the Donaldson-Fukaya category is not triangulated and we cannot state our result in this framework. For this reason we will first enhance $Q$ to a functor defined on the Fukaya category in a higher coherent way, and then we can extent $Q$ to the category of modules over Fukaya category.  Recall that the starting point in~\\cite{Viterbo-Sheaves} is to first define a presheaf whose sections on some open set $U \\times (-\\infty, a)$ is $\\lim_f FH(\\Gamma_{df}, L)$ where $f$ is a function running over the smooth functions greater than the characteristic function of $U$, rescaled by $a$.  To turn this into a functor we need some functoriality of homotopy limits. This may be true in the framework of $A_\\infty$-categories but we lack references. So we will turn our categories into $\\infty$-categories, for which the appropriate results are available in the literature. In the sequel and for simplicity, all coefficients will be in $\\mathbb Z/2 \\mathbb Z$.", "sketch": "To prove Theorem~\\ref{thm:density Fukaya} the strategy is to use the quantization functor of~\\cite{Viterbo-Sheaves}, which associates to a closed exact Lagrangian brane $L$ a sheaf $Q(L)$ on $N\\times\\mathbb R$, thereby embedding $\\Fuk(DT^*N)$ into the sheaf category $\\Sh_{DT^*N}(N\\times\\mathbb R)$ (the Tamarkin category). One introduces the objects $\\W_{(x,a)}:=P'_{DT^*N}(k_{\\{x\\}\\times[a,\\infty)})$ using a projector $P'_{DT^*N}$ landing in $\\Sh_{DT^*N}(N\\times\\mathbb R)$, and one proves first a sheaf-theoretic density statement (Theorem~\\ref{thm:density sheaf}) with respect to the sheaf interleaving distance $\\gamma^s$.\n\nThe proof idea for the sheaf density theorem is: use a \\v Cech resolution of $k_N\\simeq C_0\\to C_1\\to\\cdots\\to C_m$ where $C_i=\\bigoplus_{J\\subset I,\\ |J|=i+1} k_{U_J^{cl}}$ for an $\\varepsilon$-small ball cover $(U_j)$. Then write $\\mathcal F=\\mathcal F\\otimes k_{N\\times\\mathbb R}$ as an iterated cone on the pieces $\\mathcal F\\otimes k_{U_J^{cl}\\times\\mathbb R}$. Next, approximate $\\mathcal F\\otimes k_{U_J^{cl}\\times\\mathbb R}$ by $\\mathcal F\\otimes k_{\\{x\\}\\times\\mathbb R}$ for some $x\\in U_J^{cl}$, and (since this is a sheaf on a line) use constructibility to write (or approximate) $\\mathcal F\\otimes k_{\\{x\\}\\times\\mathbb R}$ as an iterated cone on the $\\W_{(x,a)}$'s. The argument also shows the points $x_i$ can be chosen from an a priori given dense subset of $N$.\n\nFinally, ‘once we have the result for sheaves’, one returns to $\\fMod(\\Fuk(DT^*N))$ via $Q$: sheaf quantization of closed exact Lagrangians satisfies the hypothesis of the sheaf density theorem because $Q(L)\\otimes k_{\\{x\\}\\times\\mathbb R}\\simeq FC^*(V_{(x,0)},L)$ is constructible as a sheaf on $\\mathbb R$. This transfers the approximation in $\\gamma^s$ to the desired density statement for $L$ in terms of the generators $\\V_{(x_i,a_i)}$.", "expanded_sketch": "To prove the main theorem the strategy is to use the quantization functor of~\\cite{Viterbo-Sheaves}, which associates to a closed exact Lagrangian brane $L$ a sheaf $Q(L)$ on $N\\times\\mathbb R$, thereby embedding $\\Fuk(DT^*N)$ into the sheaf category $\\Sh_{DT^*N}(N\\times\\mathbb R)$ (the Tamarkin category). One introduces the objects $\\W_{(x,a)}:=P'_{DT^*N}(k_{\\{x\\}\\times[a,\\infty)})$ using a projector $P'_{DT^*N}$ landing in $\\Sh_{DT^*N}(N\\times\\mathbb R)$, and one proves first a sheaf-theoretic density statement (Theorem~\\ref{thm:density sheaf}) with respect to the sheaf interleaving distance $\\gamma^s$.\n\nThe proof idea for the sheaf density theorem is: use a \\v Cech resolution of $k_N\\simeq C_0\\to C_1\\to\\cdots\\to C_m$ where $C_i=\\bigoplus_{J\\subset I,\\ |J|=i+1} k_{U_J^{cl}}$ for an $\\varepsilon$-small ball cover $(U_j)$. Then write $\\mathcal F=\\mathcal F\\otimes k_{N\\times\\mathbb R}$ as an iterated cone on the pieces $\\mathcal F\\otimes k_{U_J^{cl}\\times\\mathbb R}$. Next, approximate $\\mathcal F\\otimes k_{U_J^{cl}\\times\\mathbb R}$ by $\\mathcal F\\otimes k_{\\{x\\}\\times\\mathbb R}$ for some $x\\in U_J^{cl}$, and (since this is a sheaf on a line) use constructibility to write (or approximate) $\\mathcal F\\otimes k_{\\{x\\}\\times\\mathbb R}$ as an iterated cone on the $\\W_{(x,a)}$'s. The argument also shows the points $x_i$ can be chosen from an a priori given dense subset of $N$.\n\nFinally, once we have the result for sheaves, one returns to $\\fMod(\\Fuk(DT^*N))$ via $Q$: sheaf quantization of closed exact Lagrangians satisfies the hypothesis of the sheaf density theorem because $Q(L)\\otimes k_{\\{x\\}\\times\\mathbb R}\\simeq FC^*(V_{(x,0)},L)$ is constructible as a sheaf on $\\mathbb R$. This transfers the approximation in $\\gamma^s$ to the desired density statement for $L$ in terms of the generators $\\V_{(x_i,a_i)}$.", "expanded_theorem": "[Density Theorem]\\label{thm:density Fukaya}\n  For any closed exact Lagrangian $L \\in \\Fuk(DT^*N)$ and $\\varepsilon >0$, there are points $(x_i)_{i\\in \\{1, \\ldots,l\\}}$, real numbers $(a_i)_{i\\in \\{1, \\ldots,l\\}}$ and $C \\in \\langle \\V_{(x_1,a_1)}, \\ldots, \\V_{(x_l,a_l)} \\rangle$ such that $\\gamma (L,C) < \\varepsilon$., ", "theorem_type": ["Universal–Existential", "Inequality or Bound"], "mcq": {"question": "Let $N$ be a connected closed manifold, let $DT^*N$ be its unit ball cotangent bundle, and let $\\Fuk(DT^*N)$ be the filtered Fukaya category of compact exact Lagrangian branes. For $x\\in N$ and $a\\in \\mathbb R$, let $V_{(x,a)}$ denote the filtered module associated to the cotangent fiber at $x$ with primitive value $a$, and let $\\langle V_{(x_1,a_1)},\\dots,V_{(x_l,a_l)}\\rangle$ be the full subcategory of filtered modules generated from these modules by iterated cones. Using the Yoneda embedding, view any $L\\in \\Fuk(DT^*N)$ as a filtered module, and let $\\gamma$ denote the interleaving distance coming from the persistence structure on the filtered Fukaya category. Under these assumptions, which quantitative estimate holds for an arbitrary closed exact Lagrangian $L\\in \\Fuk(DT^*N)$?", "correct_choice": {"label": "A", "text": "For every closed exact Lagrangian $L\\in \\Fuk(DT^*N)$ and every $\\varepsilon>0$, there exist finitely many points $x_1,\\dots,x_l\\in N$, real numbers $a_1,\\dots,a_l\\in \\mathbb R$, and an object $C\\in \\langle V_{(x_1,a_1)},\\dots,V_{(x_l,a_l)}\\rangle$ such that $\\gamma(L,C)<\\varepsilon$."}, "choices": [{"label": "B", "text": "There exists >0 such that for every closed exact Lagrangian $L\\in \\Fuk(DT^*N)$, there exist finitely many points $x_1,\\dots,x_l\\in N$, real numbers $a_1,\\dots,a_l\\in \\mathbb R$, and an object $C\\in \\langle V_{(x_1,a_1)},\\dots,V_{(x_l,a_l)}\\rangle$ such that $\\gamma(L,C)<\\delta$."}, {"label": "C", "text": "For every closed exact Lagrangian $L\\in \\Fuk(DT^*N)$, there exist finitely many points $x_1,\\dots,x_l\\in N$, real numbers $a_1,\\dots,a_l\\in \\mathbb R$, and an object $C\\in \\langle V_{(x_1,a_1)},\\dots,V_{(x_l,a_l)}\\rangle$ such that $\\gamma(L,C)<\\infty$."}, {"label": "D", "text": "For every closed exact Lagrangian $L\\in \\Fuk(DT^*N)$ and every $\\varepsilon>0$, there exist finitely many points $x_1,\\dots,x_l\\in N$ and a single real number $a\\in \\mathbb R$ such that, if one sets $a_i=a$ for all $i$, then there is an object $C\\in \\langle V_{(x_1,a)},\\dots,V_{(x_l,a)}\\rangle$ with $\\gamma(L,C)<\\varepsilon$."}, {"label": "E", "text": "For every closed exact Lagrangian $L\\in \\Fuk(DT^*N)$ and every $\\varepsilon>0$, there exist finitely many points $x_1,\\dots,x_l\\in N$, real numbers $a_1,\\dots,a_l\\in \\mathbb R$, and an object $C\\in \\langle V_{(x_1,a_1)},\\dots,V_{(x_l,a_l)}\\rangle$ such that $L$ is quasi-isomorphic to $C$ and $\\gamma(L,C)<\\varepsilon$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "uniform_in_epsilon_accuracy", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "arbitrary_small_error_requirement", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "independent_primitive_shifts_a_i", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "approximation_vs_isomorphism", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly state the target estimate or directly reveal choice A. It introduces the setup and asks which quantitative conclusion holds, so the correct answer is not leaked verbatim from the stem."}, "TAS": {"score": 1, "justification": "The item is very close to a theorem-recall question: the correct option is essentially the sharp theorem statement with nearby quantifier perturbations. It is not a pure restatement because the choices vary meaningfully, but it remains only a mild reformulation."}, "GPS": {"score": 1, "justification": "Some reasoning is required to compare quantifiers and distinguish approximation from exact quasi-isomorphism, uniform error bounds, and shared primitive shifts. However, the task is mostly recognition of the theorem’s precise statement rather than substantial mathematical generation."}, "DQS": {"score": 1, "justification": "B, D, and E are plausible distractors tied to common theorem-reading errors, but C is a weaker statement that is also true if A is true. That makes the distractor set non-exclusive and weakens the MCQ’s validity."}, "total_score": 5, "overall_assessment": "Technically sophisticated but mostly theorem-recall. The distractors are partly well designed, yet the presence of a weaker true option (C) significantly reduces the question’s quality as a single-correct MCQ."}}
{"id": "2602.22002v1", "paper_link": "http://arxiv.org/abs/2602.22002v1", "theorems_cnt": 2, "theorem": {"env_name": "theo", "content": "\\label{marthm}\nLet $E \\subset \\mathbb{R}^2$ be a Borel or analytic set. Then\n\n(i) $\\dh \\mbox{\\rm proj}_\\theta E \\leq \\min\\{\\dh E, 1\\}$ with equality for almost all $\\theta \\in [0,\\pi)$,\n\\smallskip\n\n(ii) if $\\dh E >1$ then ${\\mathcal L} (\\mbox{\\rm proj}_\\theta E) > 0$ for almost all $\\theta \\in [0,\\pi)$.", "start_pos": 6959, "end_pos": 7286, "label": "marthm"}, "ref_dict": {"higherdim": "\\begin{theo}\\label{higherdim}\nLet $E \\subset \\mathbb{R}^n$ be a Borel or analytic set. Then\n\n(i) $\\dh\\mbox{\\rm proj}_V E \\leq \\min\\{\\dh E, m\\}$  with equality for $\\gamma_{n,m}$-almost all $V \\in G(n,m)$,\n\\smallskip\n\n(ii) if $\\dh E >m$ then ${\\mathcal L}^m (\\mbox{\\rm proj}_V E) > 0$ for $\\gamma_{n,m}$-almost all $V \\in G(n,m)$. \n\\end{theo}", "endef": "\\begin{equation}\\label{endef}\nI^s(\\mu) := \\int \\!\\! \\int \\frac{d\\mu(x)d\\mu(y)}{|x-y|^s} \n\\end{equation}", "ftend": "\\begin{equation}\\label{ftend}\nI^s(\\mu) = c_{n,s}\\int_{\\mathbb{R}^n} \\frac{|\\widehat{\\mu}(z)|^2} {|z|^{n-s}}dz \\qquad (0<s<n),\n\\end{equation}", "marthm": "\\begin{theo}\\label{marthm}\nLet $E \\subset \\mathbb{R}^2$ be a Borel or analytic set. Then\n\n(i) $\\dh \\mbox{\\rm proj}_\\theta E \\leq \\min\\{\\dh E, 1\\}$ with equality for almost all $\\theta \\in [0,\\pi)$,\n\\smallskip\n\n(ii) if $\\dh E >1$ then ${\\mathcal L} (\\mbox{\\rm proj}_\\theta E) > 0$ for almost all $\\theta \\in [0,\\pi)$.\n\\end{theo}"}, "pre_theorem_intro_text_len": 4568, "pre_theorem_intro_text": "\\setcounter{equation}{0}\n\\setcounter{theo}{0}\n\\setcounter{figure}{0}\nAt the conference {\\it Fractal Geometry and Stochastics V} held in Tabarz in 2014, I gave a survey talk entitled `Sixty years of fractal projections', a version of which, written with Jon Fraser and Xiong Jin \\cite{FFJ}, appeared in in the conference proceedings \\cite{BFZ}.  This marked the sixtieth anniversary of the publication of John Marstrand's 1954 paper \\cite{Mar} `Some fundamental geometrical properties of plane sets of fractional dimensions'  in the Proceedings of the London Mathematical Society.  For a long time the paper attracted little attention, but since the 1980s, Marstrand's projection theorems have become the prototype for  many results in fractal geometry with numerous variants and applications. This area is now more intensively researched than ever,  drawing on modern techniques from ergodic theory, CP processes, Fourier transforms, discretisation of problems and additive combinatorics, together with a lot of ingenuity.\n\nMy talk `Seventy years of fractal projections' at {\\it Fractal Geometry and Stochastics VII} held in 2024 in Chemnitz highlighted some of the very considerable progress in the area over the past 10 years. This account also surveys some of this more recent work and is a sequel to \\cite{FFJ} where many earlier results and references may be found. There are many excellent articles and books which provide other substantial overviews of projection properties, including references \\cite{Fra2,Mat3,Mat4,Mat5,Mat2,Mat6,Shm}. Also, volumes of conference proceedings, in particular of the German `Fractal Geometry and Stochastics' and the French `Fractals and Related Fields' meetings, contain many enlightening surveys on this and many other aspects of fractal geometry. \n\n\\subsection{General remarks}\nMost of this article concerns orthogonal projections of sets in the plane onto straight lines or from \n$\\mathbb{R}^n$ onto $m$-dimensional subspaces.\nIn the plane we let $L_\\theta$ be the line making  angle $\\theta \\in [0,\\pi)$ with the $x$-axis, and write $\\proj_\\theta: \\mathbb{R}^2 \\to L_\\theta$ for orthogonal projection onto $L_\\theta$, see Figure 1. By slight abuse of notation we will write ${\\mathcal L} $ for Lebesgue measure on any line $L_\\theta$, identified with $\\mathbb{R}$ in the obvious way.\n\n \\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=0.4]{projcol.png}\n\nFigure 1: Projection of a set $E$ onto a line in direction $\\theta$.\n\\end{center}\n\\end{figure}\nIn the higher dimensional setting, we write $G(n,m)$ for the Grassmanian of $m$-dimensional subspaces of $\\mathbb{R}^n$, where $1\\leq m <n$. Then $G(n,m)$ is an $m(n-m)$-dimensional compact manifold which carries a natural invariant measure $\\gamma_{n,m}$ that is locally equivalent to $m(n-m)$-dimensional Lebesgue measure. When we write `for almost all $V\\in G(n,m)$' this is taken to be with respect to $\\gamma_{n,m}$. For $V\\in G(n,m)$ let $\\proj_V : \\mathbb{R}^n\\to V$ be orthogonal projection onto  the $m$-dimensional subspace $V$, writing ${\\mathcal L}^m$ for $m$-dimensional Lebesgue measure on $V$ which we identify with $\\mathbb{R}^m$.\n\nWe make the convention that we use $\\proj_\\theta$ specifically for the planar setting and   $\\proj_V$ for the more general $G(n,m)$ context.\n\nHere our main interest is in projections of sets rather than measures, though many results for projections of sets have measure analogues, indeed many proofs concerning projections of sets depend on consideration of projections of suitable measures supported by the sets. Lower and upper Hausdorff and packing dimensions of measures can be defined in terms of local dimensions, and may all be equal for dynamically defined measures. Then often the dimension of projected measures can be expressed in terms of entropies and Lyapunov exponents. \n\nWe will assume, sometimes without mentioning it specifically,  that the subsets of $\\mathbb{R}^m$  with which we work  are all Borel or analytic, indeed for many results little is lost by assuming them to be compact. We avoid consideration of completely general sets that do not have analytic or measurable structure; they may behave strangely depending on the axioms of logic that are assumed. Nevertheless we note  that recently Lutz and Stull \\cite{LS} have used effective dimension and computational complexity to obtain some results on projections with no requirement on sets to be Borel or analytic.\n\n\\subsection{Marstrand's projection theorem}\nMarstrand's projection theorem in the plane \\cite{Mar} may be stated as follows.", "context": "\\subsection{General remarks}\nMost of this article concerns orthogonal projections of sets in the plane onto straight lines or from \n$\\mathbb{R}^n$ onto $m$-dimensional subspaces.\nIn the plane we let $L_\\theta$ be the line making angle $\\theta \\in [0,\\pi)$ with the $x$-axis, and write $\\proj_\\theta: \\mathbb{R}^2 \\to L_\\theta$ for orthogonal projection onto $L_\\theta$, see Figure 1. By slight abuse of notation we will write ${\\mathcal L} $ for Lebesgue measure on any line $L_\\theta$, identified with $\\mathbb{R}$ in the obvious way.\n\nFigure 1: Projection of a set $E$ onto a line in direction $\\theta$.\n\\end{center}\n\\end{figure}\nIn the higher dimensional setting, we write $G(n,m)$ for the Grassmanian of $m$-dimensional subspaces of $\\mathbb{R}^n$, where $1\\leq m <n$. Then $G(n,m)$ is an $m(n-m)$-dimensional compact manifold which carries a natural invariant measure $\\gamma_{n,m}$ that is locally equivalent to $m(n-m)$-dimensional Lebesgue measure. When we write `for almost all $V\\in G(n,m)$' this is taken to be with respect to $\\gamma_{n,m}$. For $V\\in G(n,m)$ let $\\proj_V : \\mathbb{R}^n\\to V$ be orthogonal projection onto the $m$-dimensional subspace $V$, writing ${\\mathcal L}^m$ for $m$-dimensional Lebesgue measure on $V$ which we identify with $\\mathbb{R}^m$.\n\nWe make the convention that we use $\\proj_\\theta$ specifically for the planar setting and $\\proj_V$ for the more general $G(n,m)$ context.\n\nHere our main interest is in projections of sets rather than measures, though many results for projections of sets have measure analogues, indeed many proofs concerning projections of sets depend on consideration of projections of suitable measures supported by the sets. Lower and upper Hausdorff and packing dimensions of measures can be defined in terms of local dimensions, and may all be equal for dynamically defined measures. Then often the dimension of projected measures can be expressed in terms of entropies and Lyapunov exponents.\n\nWe will assume, sometimes without mentioning it specifically, that the subsets of $\\mathbb{R}^m$ with which we work are all Borel or analytic, indeed for many results little is lost by assuming them to be compact. We avoid consideration of completely general sets that do not have analytic or measurable structure; they may behave strangely depending on the axioms of logic that are assumed. Nevertheless we note that recently Lutz and Stull \\cite{LS} have used effective dimension and computational complexity to obtain some results on projections with no requirement on sets to be Borel or analytic.\n\n\\subsection{Marstrand's projection theorem}\nMarstrand's projection theorem in the plane \\cite{Mar} may be stated as follows.", "full_context": "\\subsection{General remarks}\nMost of this article concerns orthogonal projections of sets in the plane onto straight lines or from \n$\\mathbb{R}^n$ onto $m$-dimensional subspaces.\nIn the plane we let $L_\\theta$ be the line making angle $\\theta \\in [0,\\pi)$ with the $x$-axis, and write $\\proj_\\theta: \\mathbb{R}^2 \\to L_\\theta$ for orthogonal projection onto $L_\\theta$, see Figure 1. By slight abuse of notation we will write ${\\mathcal L} $ for Lebesgue measure on any line $L_\\theta$, identified with $\\mathbb{R}$ in the obvious way.\n\nFigure 1: Projection of a set $E$ onto a line in direction $\\theta$.\n\\end{center}\n\\end{figure}\nIn the higher dimensional setting, we write $G(n,m)$ for the Grassmanian of $m$-dimensional subspaces of $\\mathbb{R}^n$, where $1\\leq m <n$. Then $G(n,m)$ is an $m(n-m)$-dimensional compact manifold which carries a natural invariant measure $\\gamma_{n,m}$ that is locally equivalent to $m(n-m)$-dimensional Lebesgue measure. When we write `for almost all $V\\in G(n,m)$' this is taken to be with respect to $\\gamma_{n,m}$. For $V\\in G(n,m)$ let $\\proj_V : \\mathbb{R}^n\\to V$ be orthogonal projection onto the $m$-dimensional subspace $V$, writing ${\\mathcal L}^m$ for $m$-dimensional Lebesgue measure on $V$ which we identify with $\\mathbb{R}^m$.\n\nWe make the convention that we use $\\proj_\\theta$ specifically for the planar setting and $\\proj_V$ for the more general $G(n,m)$ context.\n\nHere our main interest is in projections of sets rather than measures, though many results for projections of sets have measure analogues, indeed many proofs concerning projections of sets depend on consideration of projections of suitable measures supported by the sets. Lower and upper Hausdorff and packing dimensions of measures can be defined in terms of local dimensions, and may all be equal for dynamically defined measures. Then often the dimension of projected measures can be expressed in terms of entropies and Lyapunov exponents.\n\nWe will assume, sometimes without mentioning it specifically, that the subsets of $\\mathbb{R}^m$ with which we work are all Borel or analytic, indeed for many results little is lost by assuming them to be compact. We avoid consideration of completely general sets that do not have analytic or measurable structure; they may behave strangely depending on the axioms of logic that are assumed. Nevertheless we note that recently Lutz and Stull \\cite{LS} have used effective dimension and computational complexity to obtain some results on projections with no requirement on sets to be Borel or analytic.\n\n\\subsection{Marstrand's projection theorem}\nMarstrand's projection theorem in the plane \\cite{Mar} may be stated as follows.\n\nWe will assume, sometimes without mentioning it specifically, that the subsets of $\\mathbb{R}^m$ with which we work are all Borel or analytic, indeed for many results little is lost by assuming them to be compact. We avoid consideration of completely general sets that do not have analytic or measurable structure; they may behave strangely depending on the axioms of logic that are assumed. Nevertheless we note that recently Lutz and Stull \\cite{LS} have used effective dimension and computational complexity to obtain some results on projections with no requirement on sets to be Borel or analytic.\n\nThe natural higher dimensional analogue of Marstrand's theorem was first presented by Mattila \\cite{Mat} in 1975.\n\nBriefly, the potential-theoretic method depends on the characterisation of Hausdorff dimension in terms of energy integrals. We write $\\mathcal{M}(E)$ for the positive finite Borel measures supported on $E\\subset \\mathbb{R}^n$. For $s>0$ we write:\n\\begin{equation}\\label{endef}\nI^s(\\mu) := \\int \\!\\! \\int \\frac{d\\mu(x)d\\mu(y)}{|x-y|^s} \n\\end{equation}\nfor the $s$-{\\it energy} of the measure $\\mu\\in \\mathcal{M}(E)$. Then\n\\begin{equation*}\\label{endefhau}\n\\dh E = \\sup\\Big\\{s : \\mbox{ there exists $\\mu \\in \\mathcal{M}(E)$ such that $I^s(\\mu) < \\infty$}\\Big\\}.\n\\end{equation*}\nTheorem \\ref{marthm}(i) may then be proved by noting that for all $0<s< \\dh E$ there is a measure $\\mu\\in \\mathcal{M}(E)$ such that $I^s(\\mu) <\\infty$, and verifying that $\\int I^s(\\mu_\\theta) d\\theta \\leq c_sI^s(\\mu)$, where $\\mu_\\theta$ is the projection of the measure $\\mu$ onto the line $L_\\theta$ in direction $\\theta$, given by $\\mu_\\theta(F) = \\mu(\\proj^{-1}_\\theta F)$ for $F\\subset L_\\theta$. The higher dimensional case, Theorem \\ref{higherdim}(i), may be proved in a similar way.\n\nWhilst Marstrand's Theorem gives that the Lebesgue measure of the `exceptional set' of projection directions $\\theta$ for which $\\dh \\mbox{\\rm proj}_\\theta E < \\min\\{\\dh E, 1\\}$ has Lebesgue measure 0, often the set of exceptional directions must be somewhat smaller than this. Indeed Kaufman's potential theoretic approach \\cite{Kau} also yields an upper bound for the dimension of the exceptional set in Theorems \\ref{marthm}(i) and \\ref{higherdim}(i), namely that if $E \\subset \\mathbb{R}^n$ and $0\\leq s< \\dh E < m$ then\n\\begin{equation}\\label{Kaut}\n\\dh\\{V \\in G(n,m) : \\dh \\mbox{\\rm proj}_V E< s\\} \\leq m(n-m) -(m-s).\n\\end{equation}\n(This bound is written in this form for comparison with $m(n-m)$, the dimension of the Grassmanian $G(n,m))$.\nUsing Fourier transforms, Falconer \\cite{Fal1} in 1982 obtained bounds for the exceptional sets of an alternative form: if $0\\leq s \\leq m \\leq \\dh E$ then\n\\begin{equation}\\label{Falt}\n\\dh\\{V \\in G(n,m) : \\dh \\mbox{\\rm proj}_V E< s\\} \\leq \\max\\{m(n-m) -(\\dh E-s), 0\\},\n\\end{equation}\nand if $m \\leq \\dh E \\leq n$, \n\\begin{equation}\\label{Falt2}\n\\dh\\{V \\in G(n,m) : \\mathcal{L}^n(\\mbox{\\rm proj}_V E) =0\\} \\leq m(n-m) -(\\dh E-m);\n\\end{equation}\nthe idea here is the greater the `excess dimension' $ \\dh E - s$ or $ \\dh E - m$, the smaller the set of exceptional $V$.\nEstimates for the dimensions of exceptional sets in the spirit of \\eqref{Kaut}, where there is a direct estimate, and \\eqref{Falt}--\\eqref{Falt2}, where the bound depends on by how much the dimension of a set exceeds a threshold, have become known as `Kaufman-type' and `Falconer-type' estimates respectively.\n\n\\begin{theo}\\label{assd} \nLet $\\log 3/\\log 5 <s<1$. Then there is $\\epsilon>0$ such that\n${\\dim}_{\\rm A}\\proj_\\theta E_s=1$ for almost all $\\theta \\in (-\\epsilon, \\epsilon)$ and ${\\dim}_{\\rm A}\\proj_\\theta E_s=s<1$ for all $\\theta \\in (\\pi/4 -\\epsilon, \\pi/4+ \\epsilon)$.\n\\end{theo}\n\\noindent This example, due to Fraser and Orponen \\cite{FO}, depends on the fact that a self-similar $E\\subset \\mathbb{R}^2$ defined by contracting homotheties has\n$$\n{\\dim}_{\\rm A}\\mbox{proj}_\\theta E =\n\\left\\{\n\\begin{array}{ll}\n \\dh \\proj_\\theta E& \\text{ if } \\mathcal{H}^{\\dh {\\rm proj}_\\theta E}(\\proj_\\theta E) >0 \\\\\n 1 & \\text{ if }\\mathcal{H}^{\\dh {\\rm proj}_\\theta E}(\\proj_\\theta E)=0 \n\\end{array}\n\\right.,\n$$\nwhere $\\mathcal{H}^s$ is $s$-dimensional Hausdorff measure, so the construction is equivalent to finding a set of homotheties for which the IFS attractor $E$ has projections with positive or zero ($\\proj_\\theta E$)-dimensional Hausdorff measure in the appropriate directions.\n\n\\begin{theo}\\label{projgeneral}\nLet $P \\subset \\mathbb{R}^k$ be an open parameter set and let $E \\subset \\mathbb{R}^n$ be a Borel or analytic set. Let $\\{V(t)\\subset G(n,m): t \\in P\\}$ be a family of subspaces such that $V$ is $C^1$ with the derivative $D_t V(t)$ injective for all $t \\in P$. Then, for all $l=0,1,\\ldots,m$ and ${\\mathcal L}^k$-almost all $t \\in P$,\n$$\n\\dh\\proj_{V(t)} E \\geq \n\\left\\{\n\\begin{array}{llrl}\n \\dh E-p(l) &\\ \\mbox{if} & p(l) + l & \\leq \\dh E \\leq p(l) + l +1 \\\\\nl+1 &\\ \\mbox{if} & \\ p(l) + l +1 &\\leq \\dh E \\leq p(l+1) + l +1\n\\end{array}\n\\right. .\n$$\nMoreover, if $\\dh E > p(m-1) + m$ then ${\\mathcal L}^m ( \\proj_{V(t)} E) > 0$ for ${\\mathcal L}^k$-almost all $t \\in P$.\n\\end{theo}\n\n\\begin{theo}\\label{projline}\nLet $E \\subset \\mathbb{R}^3$ be a Borel or analytic set and let $\\theta: [0,1] \\to S^2$ be a non-degenerate family of directions. Then \n\\begin{equation}\\label{projlineeq}\n\\dh\\proj_{\\theta(t)} E = \\min\\{\\dh E, 1\\}\\quad \\text{ for almost all } t\\in [0,1],\n\\end{equation}\nand if $\\dh E >1$ then\n\\begin{equation}\\label{resmes}\n{\\mathcal L} (\\proj_{\\theta(t)} E)>0 \\quad \\text{ for almost all } t\\in [0,1].\n\\end{equation}\nConcerning exceptional directions, if $0\\leq s\\leq \\min\\{\\dh E, 1\\}$ then\n\\begin{equation}\\label{PYZ}\n \\dh\\{t : \\dh \\proj_{\\theta(t)} E <s\\} \\leq s,\n\\end{equation}\nand if $0\\leq s\\leq 1$ then \n\\begin{equation}\\label{GGM}\n \\dh \\{t: \\proj_{\\theta(t)} E <s\\} \\leq \\max\\big\\{0, 1 + {\\textstyle \\frac12} (s-\\dh E) \\big\\},\n\\end{equation}\nand\n\\begin{equation}\\label{resmes1}\n\\dh\\{t: {\\mathcal L} (\\proj_{\\theta(t)} E) = 0 \\} \\leq \\textstyle{\\frac13}(4-\\dh E).\n\\end{equation}\n\\end{theo}\n\n\\begin{theo}\\label{projplane}\nLet $E \\subset \\mathbb{R}^3$ be a Borel or analytic set and let $\\theta: [0,1] \\to S^2$ be a non-degenerate family of directions. \nThen \n\\begin{equation}\\label{projplaneeq}\n\\dh\\proj_{V_{\\theta(t)}} E = \\min\\{\\dh E, 2\\} \\quad \\text{ for almost all } t\\in [0,1].\n\\end{equation}\nMoreover, if $\\dh E >2$ then\n\\begin{equation}\\label{projlplanex}\n{\\mathcal L^2} (\\proj_{\\theta(t)} E)>0 \\quad \\text{ for almost all } t\\in [0,1].\n\\end{equation}\nFor exceptional directions, if $0\\leq s\\leq 1$ then \n\\begin{equation}\\label{GGG}\n \\dh\\{t : \\dh \\proj_{\\theta(t)} E <s\\} \\leq \\max\\{ 1 + s-\\dh E, 0 \\}.\n\\end{equation}\n\\end{theo}\n\n\\begin{theo}\\label{higherdim}\nLet $E \\subset \\mathbb{R}^n$ be a Borel or analytic set. Then\n\n(i) $\\dh\\mbox{\\rm proj}_V E \\leq \\min\\{\\dh E, m\\}$  with equality for $\\gamma_{n,m}$-almost all $V \\in G(n,m)$,\n\\smallskip\n\n(ii) if $\\dh E >m$ then ${\\mathcal L}^m (\\mbox{\\rm proj}_V E) > 0$ for $\\gamma_{n,m}$-almost all $V \\in G(n,m)$. \n\\end{theo}\n\n\\begin{theo}\\label{marthm}\nLet $E \\subset \\mathbb{R}^2$ be a Borel or analytic set. Then\n\n(i) $\\dh \\mbox{\\rm proj}_\\theta E \\leq \\min\\{\\dh E, 1\\}$ with equality for almost all $\\theta \\in [0,\\pi)$,\n\\smallskip\n\n(ii) if $\\dh E >1$ then ${\\mathcal L} (\\mbox{\\rm proj}_\\theta E) > 0$ for almost all $\\theta \\in [0,\\pi)$.\n\\end{theo}", "post_theorem_intro_text_len": 3881, "post_theorem_intro_text": "The  natural higher dimensional analogue of Marstrand's theorem was first presented by Mattila \\cite{Mat} in 1975.\n\n\\begin{theo}\\label{higherdim}\nLet $E \\subset \\mathbb{R}^n$ be a Borel or analytic set. Then\n\n(i) $\\dh\\mbox{\\rm proj}_V E \\leq \\min\\{\\dh E, m\\}$  with equality for $\\gamma_{n,m}$-almost all $V \\in G(n,m)$,\n\\smallskip\n\n(ii) if $\\dh E >m$ then ${\\mathcal L}^m (\\mbox{\\rm proj}_V E) > 0$ for $\\gamma_{n,m}$-almost all $V \\in G(n,m)$. \n\\end{theo}\n\nSince orthogonal projection is a Lipschitz map that does not increase distances between points, the inequalities in (i) of Theorems \\ref{marthm} and \\ref{higherdim}   follow easily from the definition of Hausdorff measure and dimension, but showing that equality holds for almost all $\\theta$ or $V$ is more involved. \nIn his original paper, Marstrand  \\cite{Mar} used intricate estimates involving plane geometry and measure theory, but in 1968  Kaufman \\cite{Kau} gave a new proof of Theorem \\ref{marthm}(i) using potential theory and of Theorem \\ref{marthm}(ii)  using Fourier transforms before\nMattila \\cite{Mat} used this approach in higher dimensions to obtain Theorem 1.2.\n\nBriefly, the potential-theoretic method depends on the characterisation of Hausdorff dimension in terms of energy integrals. We write $\\mathcal{M}(E)$ for the positive finite Borel measures supported on $E\\subset \\mathbb{R}^n$. For $s>0$ we write:\n\\begin{equation}\\label{endef}\nI^s(\\mu) := \\int \\!\\! \\int \\frac{d\\mu(x)d\\mu(y)}{|x-y|^s} \n\\end{equation}\nfor the $s$-{\\it energy} of the measure $\\mu\\in \\mathcal{M}(E)$. Then\n\\begin{equation*}\\label{endefhau}\n\\dh E = \\sup\\Big\\{s :  \\mbox{ there exists $\\mu \\in \\mathcal{M}(E)$ such that $I^s(\\mu) < \\infty$}\\Big\\}.\n\\end{equation*}\nTheorem \\ref{marthm}(i) may then be proved by noting that for all $0<s< \\dh E$ there is a measure $\\mu\\in \\mathcal{M}(E)$ such that $I^s(\\mu) <\\infty$,   and verifying that  $\\int I^s(\\mu_\\theta) d\\theta \\leq c_sI^s(\\mu)$, where $\\mu_\\theta$ is the projection of the measure $\\mu$ onto the line $L_\\theta$ in direction $\\theta$, given by $\\mu_\\theta(F) = \\mu( \\mbox{\\rm proj}^{-1}_\\theta F)$ for $F\\subset L_\\theta$. The higher dimensional case, Theorem \\ref{higherdim}(i), may be proved in a similar way.\n\nTheorem \\ref{marthm}(ii) may be obtained using Fourier transforms. Writing $\\widehat{\\mu}(z):= \\int e^{-2\\pi i x.z} d\\mu(x) \\  (z\\in \\mathbb{R}^n)$ for the Fourier transform of $\\mu$, applying Parseval's theorem and the convolution formulae to the energy expression in \\eqref{endef} gives \n\\begin{equation}\\label{ftend}\nI^s(\\mu) = c_{n,s}\\int_{\\mathbb{R}^n} \\frac{|\\widehat{\\mu}(z)|^2} {|z|^{n-s}}dz \\qquad (0<s<n),\n\\end{equation}\nwhere   $c_{n,s}$ depends only on $n$ and $s$, see \\cite[Theorem 3.10]{Mat2}.\nThen if $E\\subset  \\mathbb{R}^2$ and $\\dh E >1$  we may find $\\mu\\in \\mathcal{M}(E)$ and $1<s<\\dh E$ such that $I^s(\\mu) < \\infty$. Writing \\eqref{ftend} in radial coordinates this implies that $\\int \\!\\! \\int | \\widehat{\\mu_\\theta}(t)|^2 dt\\, d\\theta <\\infty$, so that for almost all $\\theta$, the projected measure $\\mu_\\theta$ on $L_\\theta$ is absolutely continuous with respect to $\\mathcal{L}$ and is a square integrable function, so in particular $E$ has support of positive $\\mathcal{L}$-measure.\n\nEnergy and Fourier methods and their generalisations and extensions have been used widely across fractal geometry and in particular in work on projections. Further examples will be encountered in this survey, but for thorough treatments see, for example, \\cite{Fal, FFJ, Mat, Mat1,Mat4,Mat5,Mat2}\n\nMarstrand's theorem is the prototype for a great deal of work in fractal geometry. If, in some setting, a dimensional property of parameterised images of a set is true for almost all parameters, perhaps for almost all $\\theta$ or $V$, we say that a `Marstrand-type theorem holds'. We will see many instances of this in the sections that follow.", "sketch": "For (i), the inequality $\\dh\\,\\mbox{\\rm proj}_\\theta E\\leq \\min\\{\\dh E,1\\}$ (and similarly $\\dh\\,\\mbox{\\rm proj}_V E\\leq \\min\\{\\dh E,m\\}$) is said to follow easily since “orthogonal projection is a Lipschitz map that does not increase distances between points”. To get the a.e. equality is “more involved”, and Kaufman’s/Mattila’s approach uses potential theory via energy integrals.\n\nPotential-theoretic step: use the characterisation\n\\[\n\\dh E = \\sup\\Big\\{s : \\exists\\,\\mu\\in\\mathcal M(E)\\text{ with }I^s(\\mu)<\\infty\\Big\\},\\qquad I^s(\\mu):=\\iint \\frac{d\\mu(x)\\,d\\mu(y)}{|x-y|^s}.\n\\]\nThen to prove Theorem~\\ref{marthm}(i), “for all $0<s<\\dh E$ there is a measure $\\mu\\in\\mathcal M(E)$ such that $I^s(\\mu)<\\infty$”, and one “verif[ies] that $\\int I^s(\\mu_\\theta)\\,d\\theta\\le c_s I^s(\\mu)$”, where $\\mu_\\theta$ is the projected measure defined by $\\mu_\\theta(F)=\\mu(\\mbox{\\rm proj}_\\theta^{-1}F)$. This yields the a.e. dimension conclusion; “The higher dimensional case, Theorem~\\ref{higherdim}(i), may be proved in a similar way.”\n\nFor (ii), the sketch uses Fourier transforms: with $\\widehat\\mu(z)=\\int e^{-2\\pi i x\\cdot z}\\,d\\mu(x)$, “applying Parseval’s theorem and the convolution formulae” gives\n\\[\nI^s(\\mu)=c_{n,s}\\int_{\\mathbb R^n}\\frac{|\\widehat\\mu(z)|^2}{|z|^{n-s}}\\,dz\\quad (0<s<n).\n\\]\nIf $E\\subset\\mathbb R^2$ and $\\dh E>1$, pick $\\mu\\in\\mathcal M(E)$ and $1<s<\\dh E$ with $I^s(\\mu)<\\infty$. “Writing \\eqref{ftend} in radial coordinates this implies that $\\iint |\\widehat{\\mu_\\theta}(t)|^2\\,dt\\,d\\theta<\\infty$,” hence for a.e. $\\theta$ the projected measure $\\mu_\\theta$ is absolutely continuous w.r.t. $\\mathcal L$ and is square-integrable, so “in particular $E$ has support of positive $\\mathcal L$-measure” (i.e. ${\\mathcal L}(\\mbox{\\rm proj}_\\theta E)>0$ for a.e. $\\theta$).", "expanded_sketch": "For (i), the inequality $\\dh\\,\\mbox{\\rm proj}_\\theta E\\leq \\min\\{\\dh E,1\\}$ (and similarly $\\dh\\,\\mbox{\\rm proj}_V E\\leq \\min\\{\\dh E,m\\}$) is said to follow easily since “orthogonal projection is a Lipschitz map that does not increase distances between points”. To get the a.e. equality is “more involved”, and Kaufman’s/Mattila’s approach uses potential theory via energy integrals.\n\nPotential-theoretic step: use the characterisation\n\\[\n\\dh E = \\sup\\Big\\{s : \\exists\\,\\mu\\in\\mathcal M(E)\\text{ with }I^s(\\mu)<\\infty\\Big\\},\\qquad I^s(\\mu):=\\iint \\frac{d\\mu(x)\\,d\\mu(y)}{|x-y|^s}.\n\\]\nThen, in establishing the main theorem, “for all $0<s<\\dh E$ there is a measure $\\mu\\in\\mathcal M(E)$ such that $I^s(\\mu)<\\infty$”, and one “verif[ies] that $\\int I^s(\\mu_\\theta)\\,d\\theta\\le c_s I^s(\\mu)$”, where $\\mu_\\theta$ is the projected measure defined by $\\mu_\\theta(F)=\\mu(\\mbox{\\rm proj}_\\theta^{-1}F)$. This yields the a.e. dimension conclusion.\n\nWe next use the following theorem.\n\\begin{theo}\\label{higherdim}\nLet $E \\subset \\mathbb{R}^n$ be a Borel or analytic set. Then\n\n(i) $\\dh\\mbox{\\rm proj}_V E \\leq \\min\\{\\dh E, m\\}$  with equality for $\\gamma_{n,m}$-almost all $V \\in G(n,m)$,\n\\smallskip\n\n(ii) if $\\dh E >m$ then ${\\mathcal L}^m (\\mbox{\\rm proj}_V E) > 0$ for $\\gamma_{n,m}$-almost all $V \\in G(n,m)$. \n\\end{theo}\n\nFor (ii), the sketch uses Fourier transforms: with $\\widehat\\mu(z)=\\int e^{-2\\pi i x\\cdot z}\\,d\\mu(x)$, “applying Parseval’s theorem and the convolution formulae” gives\n\\begin{equation}\\label{ftend}\nI^s(\\mu) = c_{n,s}\\int_{\\mathbb{R}^n} \\frac{|\\widehat{\\mu}(z)|^2} {|z|^{n-s}}dz \\qquad (0<s<n),\n\\end{equation}\nIf $E\\subset\\mathbb R^2$ and $\\dh E>1$, pick $\\mu\\in\\mathcal M(E)$ and $1<s<\\dh E$ with $I^s(\\mu)<\\infty$. “Writing the equation above in radial coordinates this implies that $\\iint |\\widehat{\\mu_\\theta}(t)|^2\\,dt\\,d\\theta<\\infty$,” hence for a.e. $\\theta$ the projected measure $\\mu_\\theta$ is absolutely continuous w.r.t. $\\mathcal L$ and is square-integrable, so “in particular $E$ has support of positive $\\mathcal L$-measure” (i.e. ${\\mathcal L}(\\mbox{\\rm proj}_\\theta E)>0$ for a.e. $\\theta$).", "expanded_theorem": "\\label{marthm}\nLet $E \\subset \\mathbb{R}^2$ be a Borel or analytic set. Then\n\n(i) $\\dh \\mbox{\\rm proj}_\\theta E \\leq \\min\\{\\dh E, 1\\}$ with equality for almost all $\\theta \\in [0,\\pi)$,\n\\smallskip\n\n(ii) if $\\dh E >1$ then ${\\mathcal L} (\\mbox{\\rm proj}_\\theta E) > 0$ for almost all $\\theta \\in [0,\\pi)$.,", "theorem_type": ["Universal", "Inequality or Bound"], "mcq": {"question": "Let $E\\subset \\mathbb{R}^2$ be a Borel or analytic set. For each angle $\\theta\\in[0,\\pi)$, let $L_\\theta$ be the line making angle $\\theta$ with the $x$-axis, and let $\\operatorname{proj}_\\theta:\\mathbb{R}^2\\to L_\\theta$ denote orthogonal projection onto $L_\\theta$. Write $\\dim_H$ for Hausdorff dimension and $\\mathcal L$ for Lebesgue measure on the line $L_\\theta$ (identified with $\\mathbb R$). Which statement holds for every such set $E$?", "correct_choice": {"label": "A", "text": "For every $\\theta\\in[0,\\pi)$, $\\dim_H(\\operatorname{proj}_\\theta E)\\le \\min\\{\\dim_H E,1\\}$, and equality holds for Lebesgue-almost every $\\theta\\in[0,\\pi)$. Moreover, if $\\dim_H E>1$, then $\\mathcal L(\\operatorname{proj}_\\theta E)>0$ for Lebesgue-almost every $\\theta\\in[0,\\pi)$."}, "choices": [{"label": "B", "text": "For every $\\theta\\in[0,\\pi)$, $\\dim_H(\\operatorname{proj}_\\theta E)\\le \\min\\{\\dim_H E,1\\}$, and equality holds for every $\\theta\\in[0,\\pi)$. Moreover, if $\\dim_H E\\ge 1$, then $\\mathcal L(\\operatorname{proj}_\\theta E)>0$ for Lebesgue-almost every $\\theta\\in[0,\\pi)$."}, {"label": "C", "text": "For Lebesgue-almost every $\\theta\\in[0,\\pi)$, $\\dim_H(\\operatorname{proj}_\\theta E)= \\min\\{\\dim_H E,1\\}$."}, {"label": "D", "text": "For Lebesgue-almost every $\\theta\\in[0,\\pi)$, $\\dim_H(\\operatorname{proj}_\\theta E)\\le \\min\\{\\dim_H E,1\\}$, and equality holds for every $\\theta\\in[0,\\pi)$. Moreover, if $\\dim_H E>1$, then there exists a set of angles of positive Lebesgue measure on which $\\mathcal L(\\operatorname{proj}_\\theta E)>0$."}, {"label": "E", "text": "For every $\\theta\\in[0,\\pi)$, $\\dim_H(\\operatorname{proj}_\\theta E)\\le \\min\\{\\dim_H E,1\\}$, and equality holds for Lebesgue-almost every $\\theta\\in[0,\\pi)$. Moreover, if $\\dim_H E>1$, then $\\mathcal L(\\operatorname{proj}_\\theta E)>0$ for every $\\theta\\in[0,\\pi)$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "strict threshold $\\dim_H E>1$ replaced by $\\dim_H E\\ge 1$ and a.e. equality upgraded to all directions", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the pointwise Lipschitz upper bound and the positive-Lebesgue-measure conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "quantifier placement on exceptional directions altered: upper bound only a.e., equality strengthened to every direction, positivity weakened from a.e. to merely positive-measure set of angles", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "almost-everywhere positivity conclusion strengthened to every direction", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly or implicitly reveal choice A; the solver must distinguish between subtle quantifier and threshold variations across the options."}, "TAS": {"score": 1, "justification": "The item is largely a recognition/recollection of Marstrand-type projection statements, but it is not a pure tautological restatement because the choices differ in meaningful ways such as 'for every' versus 'for almost every' and '>1' versus '>=1'."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to track the sharp form of the theorem and reject overstatements, but the task is still mainly theorem recall rather than substantial generative mathematical reasoning. Also, the presence of a weaker true option reduces pressure."}, "DQS": {"score": 1, "justification": "B, D, and E are plausible distractors built from common quantifier mistakes, but C is also true, making the distractor set flawed for a single-answer MCQ and weakening overall option quality."}, "total_score": 5, "overall_assessment": "A reasonably well-targeted theorem-discrimination item with good quantifier-based traps, but it is compromised as a single-answer MCQ because choice C is also true, creating ambiguity and reducing its effectiveness."}}
{"id": "2602.22167v1", "paper_link": "http://arxiv.org/abs/2602.22167v1", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "[Konyagin \\cite{Kon}]\nLet $\\varepsilon>0$ and suppose $H_i>p^{1/4+\\varepsilon}$ for all $1\\leq i\\leq n$. Then\n\\[\n\\left|\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\\right|\n\\ll_{n,\\varepsilon} p^{-\\varepsilon^2/2}|B|.\n\\]", "start_pos": 5390, "end_pos": 5631, "label": null}, "ref_dict": {"l2": "\\begin{Lemma}\\label{l2}\nWe have\n\\[\nL_{2} \\;\\ll\\; |B|^{2}(\\log p)^{n}.\n\\]\n\\end{Lemma}", "l1": "\\begin{Lemma}\\label{l1}\nWe have\n\\[\nL_{1} \\;\\ll\\; |B|^{2}\\log p.\n\\]\n\\end{Lemma}", "KT": "\\begin{Proposition}[Katz]\\label{KT}\nLet $\\chi$ be a non-trivial multiplicative character of $\\mathbb{F}_{p^n}$ and let\n$g\\in\\mathbb{F}_{p^n}$ generate the extension\n$\\mathbb{F}_{p^n}=\\mathbb{F}_p(g)$. Then, for any interval\n$I\\subseteq [1,p]\\cap\\mathbb{Z}$,\n\\[\n\\left|\\sum_{t\\in I}\\chi(g+t)\\right|\n\\le c(n)\\sqrt{p}\\log p .\n\\]\n\\end{Proposition}", "MT": "\\begin{theorem}[Main Theorem]\\label{MT}\nLet $n\\geq2$, let $\\chi$ be a nontrivial multiplicative character of $\\mathbb{F}_{p^n}$, and suppose\n\\[\n|B|\\geq p^{n(1/4+\\varepsilon)},\\qquad\nH_1\\leq H_2\\leq\\cdots\\leq H_n\\leq \\sqrt{p/2}.\n\\]\nThen\n\\[\n\\left|\\sum_{x\\in B}\\chi(x)\\right|\n\\ll |B|\\, p^{-\\varepsilon^2\n\\frac{1-\\frac{1}{2n}}{\\left(1+\\frac{1}{4n}\\right)\\left(2-\\frac{1}{2n}\\right)}}.\n\\]\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2571, "pre_theorem_intro_text": "Let $p$ be a prime, and let $\\mathbb{F}_{p^{n}}$ denote the finite field with $p^{n}$ elements. Let us define intervals of length $H_i$ in $\\mathbb{Z}$ as \n\\[ I_i=[N_i+1,N_i+H_i]\\cap \\mathbb{Z}.\\]  Fix a basis $\\{\\omega_{1},\\ldots,\\omega_{n}\\}$ of $\\mathbb{F}_{p^{n}}$ over $\\mathbb{F}_{p}$. For integers $N_i$ and $H_i$ satisfying\n\\[\n1 \\leq H_i \\leq p \\qquad (1 \\leq i \\leq n),\n\\]\nwe define the box\n\\[\nB=\\left\\{ \\sum_{i=1}^{n} x_i\\omega_i :\nN_i+1 \\leq x_i \\leq N_i+H_i,\\; 1 \\leq i \\leq n \\right\\}\n\\subseteq \\mathbb{F}_{p^{n}}.\n\\]\nand the volume of the box as \n\\[|B|=H_1\\cdot H_2 \\cdots H_n. \\]\nLet $\\chi$ be a multiplicative character on $\\mathbb{F}_{p^n}$.\n\n\\medskip\n\nWe begin with some historical background. In the case $n=1$, Burgess's bound \\cite{B1} remains the strongest result to date. It asserts that for every $\\varepsilon>0$ there exists $\\delta>0$ such that\n\\[\n\\left|\\sum_{x=N+1}^{N+H}\\chi(x)\\right|\\ll_{\\varepsilon} p^{-\\delta}H\n\\]\nwhenever $H\\geq p^{1/4+\\varepsilon}$. In subsequent work \\cite{B2}, Burgess also obtained an analogue of this bound for $n=2$ in the case of certain special bases. Karatsuba \\cite{K} later extended these ideas to general finite fields $\\mathbb{F}_{p^n}$, under the assumption that the chosen basis arises from a root of an irreducible polynomial of degree $n$ over $\\mathbb{F}_p$.\n\nFor arbitrary bases, the first nontrivial bounds were obtained by Davenport and Lewis \\cite{DL}. They studied character sums over $\\mathbb{F}_{p^n}$ of the form\n\\[\n\\sum_{x_1\\in I_1,\\ldots,x_d\\in I_n} \\chi\\!\\left(x_1\\omega_1+\\cdots+x_n\\omega_n\\right),\n\\]\nwhere $\\{\\omega_1,\\ldots,\\omega_n\\}$ is a fixed basis of $\\mathbb{F}_{p^d}$ over $\\mathbb{F}_p$ and $I_i$ are intervals in $\\mathbb{Z}$. They proved that \n\\[\n\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\n=O\\!\\left(H_1\\cdots H_n\\,p^{-\\delta(\\varepsilon)}\\right),\n\\]\nprovided that $H_i=p^{\\rho_n+\\varepsilon}$ for all $i$, where\n\\[\n\\rho_d=\\frac12-\\frac{1}{2(d+1)}.\n\\]\n\nThis result was later improved by M. C. Chang \\cite{C1}, who showed that\n\\[\n\\left|\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\\right|\n\\ll_{d,\\varepsilon} p^{-\\varepsilon^2/4}|B|,\n\\]\nwhenever $|H_i|>p^{2/5+\\varepsilon}$ for all $1\\leq i\\leq d$. In particular, this improves upon the Davenport–Lewis bound for $d\\geq5$. Chang \\cite{Cha} also obtained Burgess-strength bounds in each coordinate for $d=2$, assuming $H_1,H_2>p^{1/4+\\varepsilon}$.\n\nSubsequently, S. Konyagin generalized Chang's two-dimensional result to arbitrary finite fields of degree $n$ over $\\mathbb{F}_p$, establishing Burgess-strength estimates in every coordinate.", "context": "Let $p$ be a prime, and let $\\mathbb{F}_{p^{n}}$ denote the finite field with $p^{n}$ elements. Let us define intervals of length $H_i$ in $\\mathbb{Z}$ as \n\\[ I_i=[N_i+1,N_i+H_i]\\cap \\mathbb{Z}.\\] Fix a basis $\\{\\omega_{1},\\ldots,\\omega_{n}\\}$ of $\\mathbb{F}_{p^{n}}$ over $\\mathbb{F}_{p}$. For integers $N_i$ and $H_i$ satisfying\n\\[\n1 \\leq H_i \\leq p \\qquad (1 \\leq i \\leq n),\n\\]\nwe define the box\n\\[\nB=\\left\\{ \\sum_{i=1}^{n} x_i\\omega_i :\nN_i+1 \\leq x_i \\leq N_i+H_i,\\; 1 \\leq i \\leq n \\right\\}\n\\subseteq \\mathbb{F}_{p^{n}}.\n\\]\nand the volume of the box as \n\\[|B|=H_1\\cdot H_2 \\cdots H_n. \\]\nLet $\\chi$ be a multiplicative character on $\\mathbb{F}_{p^n}$.\n\nWe begin with some historical background. In the case $n=1$, Burgess's bound \\cite{B1} remains the strongest result to date. It asserts that for every $\\varepsilon>0$ there exists $\\delta>0$ such that\n\\[\n\\left|\\sum_{x=N+1}^{N+H}\\chi(x)\\right|\\ll_{\\varepsilon} p^{-\\delta}H\n\\]\nwhenever $H\\geq p^{1/4+\\varepsilon}$. In subsequent work \\cite{B2}, Burgess also obtained an analogue of this bound for $n=2$ in the case of certain special bases. Karatsuba \\cite{K} later extended these ideas to general finite fields $\\mathbb{F}_{p^n}$, under the assumption that the chosen basis arises from a root of an irreducible polynomial of degree $n$ over $\\mathbb{F}_p$.\n\nFor arbitrary bases, the first nontrivial bounds were obtained by Davenport and Lewis \\cite{DL}. They studied character sums over $\\mathbb{F}_{p^n}$ of the form\n\\[\n\\sum_{x_1\\in I_1,\\ldots,x_d\\in I_n} \\chi\\!\\left(x_1\\omega_1+\\cdots+x_n\\omega_n\\right),\n\\]\nwhere $\\{\\omega_1,\\ldots,\\omega_n\\}$ is a fixed basis of $\\mathbb{F}_{p^d}$ over $\\mathbb{F}_p$ and $I_i$ are intervals in $\\mathbb{Z}$. They proved that \n\\[\n\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\n=O\\!\\left(H_1\\cdots H_n\\,p^{-\\delta(\\varepsilon)}\\right),\n\\]\nprovided that $H_i=p^{\\rho_n+\\varepsilon}$ for all $i$, where\n\\[\n\\rho_d=\\frac12-\\frac{1}{2(d+1)}.\n\\]\n\nThis result was later improved by M. C. Chang \\cite{C1}, who showed that\n\\[\n\\left|\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\\right|\n\\ll_{d,\\varepsilon} p^{-\\varepsilon^2/4}|B|,\n\\]\nwhenever $|H_i|>p^{2/5+\\varepsilon}$ for all $1\\leq i\\leq d$. In particular, this improves upon the Davenport–Lewis bound for $d\\geq5$. Chang \\cite{Cha} also obtained Burgess-strength bounds in each coordinate for $d=2$, assuming $H_1,H_2>p^{1/4+\\varepsilon}$.\n\nSubsequently, S. Konyagin generalized Chang's two-dimensional result to arbitrary finite fields of degree $n$ over $\\mathbb{F}_p$, establishing Burgess-strength estimates in every coordinate.", "full_context": "Let $p$ be a prime, and let $\\mathbb{F}_{p^{n}}$ denote the finite field with $p^{n}$ elements. Let us define intervals of length $H_i$ in $\\mathbb{Z}$ as \n\\[ I_i=[N_i+1,N_i+H_i]\\cap \\mathbb{Z}.\\] Fix a basis $\\{\\omega_{1},\\ldots,\\omega_{n}\\}$ of $\\mathbb{F}_{p^{n}}$ over $\\mathbb{F}_{p}$. For integers $N_i$ and $H_i$ satisfying\n\\[\n1 \\leq H_i \\leq p \\qquad (1 \\leq i \\leq n),\n\\]\nwe define the box\n\\[\nB=\\left\\{ \\sum_{i=1}^{n} x_i\\omega_i :\nN_i+1 \\leq x_i \\leq N_i+H_i,\\; 1 \\leq i \\leq n \\right\\}\n\\subseteq \\mathbb{F}_{p^{n}}.\n\\]\nand the volume of the box as \n\\[|B|=H_1\\cdot H_2 \\cdots H_n. \\]\nLet $\\chi$ be a multiplicative character on $\\mathbb{F}_{p^n}$.\n\nWe begin with some historical background. In the case $n=1$, Burgess's bound \\cite{B1} remains the strongest result to date. It asserts that for every $\\varepsilon>0$ there exists $\\delta>0$ such that\n\\[\n\\left|\\sum_{x=N+1}^{N+H}\\chi(x)\\right|\\ll_{\\varepsilon} p^{-\\delta}H\n\\]\nwhenever $H\\geq p^{1/4+\\varepsilon}$. In subsequent work \\cite{B2}, Burgess also obtained an analogue of this bound for $n=2$ in the case of certain special bases. Karatsuba \\cite{K} later extended these ideas to general finite fields $\\mathbb{F}_{p^n}$, under the assumption that the chosen basis arises from a root of an irreducible polynomial of degree $n$ over $\\mathbb{F}_p$.\n\nFor arbitrary bases, the first nontrivial bounds were obtained by Davenport and Lewis \\cite{DL}. They studied character sums over $\\mathbb{F}_{p^n}$ of the form\n\\[\n\\sum_{x_1\\in I_1,\\ldots,x_d\\in I_n} \\chi\\!\\left(x_1\\omega_1+\\cdots+x_n\\omega_n\\right),\n\\]\nwhere $\\{\\omega_1,\\ldots,\\omega_n\\}$ is a fixed basis of $\\mathbb{F}_{p^d}$ over $\\mathbb{F}_p$ and $I_i$ are intervals in $\\mathbb{Z}$. They proved that \n\\[\n\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\n=O\\!\\left(H_1\\cdots H_n\\,p^{-\\delta(\\varepsilon)}\\right),\n\\]\nprovided that $H_i=p^{\\rho_n+\\varepsilon}$ for all $i$, where\n\\[\n\\rho_d=\\frac12-\\frac{1}{2(d+1)}.\n\\]\n\nThis result was later improved by M. C. Chang \\cite{C1}, who showed that\n\\[\n\\left|\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\\right|\n\\ll_{d,\\varepsilon} p^{-\\varepsilon^2/4}|B|,\n\\]\nwhenever $|H_i|>p^{2/5+\\varepsilon}$ for all $1\\leq i\\leq d$. In particular, this improves upon the Davenport–Lewis bound for $d\\geq5$. Chang \\cite{Cha} also obtained Burgess-strength bounds in each coordinate for $d=2$, assuming $H_1,H_2>p^{1/4+\\varepsilon}$.\n\nSubsequently, S. Konyagin generalized Chang's two-dimensional result to arbitrary finite fields of degree $n$ over $\\mathbb{F}_p$, establishing Burgess-strength estimates in every coordinate.\n\nWe begin with some historical background. In the case $n=1$, Burgess's bound \\cite{B1} remains the strongest result to date. It asserts that for every $\\varepsilon>0$ there exists $\\delta>0$ such that\n\\[\n\\left|\\sum_{x=N+1}^{N+H}\\chi(x)\\right|\\ll_{\\varepsilon} p^{-\\delta}H\n\\]\nwhenever $H\\geq p^{1/4+\\varepsilon}$. In subsequent work \\cite{B2}, Burgess also obtained an analogue of this bound for $n=2$ in the case of certain special bases. Karatsuba \\cite{K} later extended these ideas to general finite fields $\\mathbb{F}_{p^n}$, under the assumption that the chosen basis arises from a root of an irreducible polynomial of degree $n$ over $\\mathbb{F}_p$.\n\nFor arbitrary bases, the first nontrivial bounds were obtained by Davenport and Lewis \\cite{DL}. They studied character sums over $\\mathbb{F}_{p^n}$ of the form\n\\[\n\\sum_{x_1\\in I_1,\\ldots,x_d\\in I_n} \\chi\\!\\left(x_1\\omega_1+\\cdots+x_n\\omega_n\\right),\n\\]\nwhere $\\{\\omega_1,\\ldots,\\omega_n\\}$ is a fixed basis of $\\mathbb{F}_{p^d}$ over $\\mathbb{F}_p$ and $I_i$ are intervals in $\\mathbb{Z}$. They proved that \n\\[\n\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\n=O\\!\\left(H_1\\cdots H_n\\,p^{-\\delta(\\varepsilon)}\\right),\n\\]\nprovided that $H_i=p^{\\rho_n+\\varepsilon}$ for all $i$, where\n\\[\n\\rho_d=\\frac12-\\frac{1}{2(d+1)}.\n\\]\n\nThis result was later improved by M. C. Chang \\cite{C1}, who showed that\n\\[\n\\left|\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\\right|\n\\ll_{d,\\varepsilon} p^{-\\varepsilon^2/4}|B|,\n\\]\nwhenever $|H_i|>p^{2/5+\\varepsilon}$ for all $1\\leq i\\leq d$. In particular, this improves upon the Davenport–Lewis bound for $d\\geq5$. Chang \\cite{Cha} also obtained Burgess-strength bounds in each coordinate for $d=2$, assuming $H_1,H_2>p^{1/4+\\varepsilon}$.\n\nSubsequently, S. Konyagin generalized Chang's two-dimensional result to arbitrary finite fields of degree $n$ over $\\mathbb{F}_p$, establishing Burgess-strength estimates in every coordinate.\n\nWhile this theorem provides strong cancellation, it requires each interval to be of Burgess length, precluding situations in which some intervals are short while others are long. This limitation was partially overcome by M. R. Gabdullin \\cite{GB}, who generalized Konyagin's result for $n=2,3$ under a weaker product condition.\n\n\\begin{theorem}[Gabdullin \\cite{GB}]\nLet $n\\in\\{2,3\\}$, let $\\chi$ be a nontrivial multiplicative character of $\\mathbb{F}_{p^n}$, and suppose\n\\[\n|B|\\geq p^{n(1/4+\\varepsilon)}, \\qquad H_1\\leq H_2\\leq\\cdots\\leq H_n.\n\\]\nThen\n\\[\n\\left|\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\\right|\n\\ll_{n,\\varepsilon} |B| p^{-\\varepsilon^2/12}.\n\\]\n\\end{theorem}\n\n\\begin{theorem}[Main Theorem]\\label{MT}\nLet $n\\geq2$, let $\\chi$ be a nontrivial multiplicative character of $\\mathbb{F}_{p^n}$, and suppose\n\\[\n|B|\\geq p^{n(1/4+\\varepsilon)},\\qquad\nH_1\\leq H_2\\leq\\cdots\\leq H_n\\leq \\sqrt{p/2}.\n\\]\nThen\n\\[\n\\left|\\sum_{x\\in B}\\chi(x)\\right|\n\\ll |B|\\, p^{-\\varepsilon^2\n\\frac{1-\\frac{1}{2n}}{\\left(1+\\frac{1}{4n}\\right)\\left(2-\\frac{1}{2n}\\right)}}.\n\\]\n\\end{theorem}\n\nAs a consequence of our Main Theorem, we can now treat sublattices of co-dimension at least $o(n)$, while allowing each interval to have length as small as $p^{1/4+\\varepsilon}$. Moreover, we are no longer restricted to a special choice of basis; the result holds for an arbitrary basis of $\\mathbb{F}_{p^n}$ over $\\mathbb{F}_p$, in contrast with our previous work.\n\\begin{Corollary}\nLet $\\chi$ be a nontrivial multiplicative character of $\\mathbb{F}_{p^n}$, let $I_i \\subset \\mathbb{Z}$ be intervals of lengths $H_i$, and let $\\{\\omega_1,\\ldots,\\omega_n\\}$ be a basis of $\\mathbb{F}_{p^n}$ over $\\mathbb{F}_p$. \nFor every $\\varepsilon>0$, there exists $\\delta(\\varepsilon)>0$ such that\n\\[\n\\left|\\sum_{x_1\\in I_1,\\ldots,x_k\\in I_k}\n\\chi(x_1\\omega_1+\\cdots+x_k\\omega_k)\\right|\n\\ll_{n,\\varepsilon}\nH_1\\cdots H_k\\,p^{-\\delta(\\varepsilon)},\n\\]\nprovided that\n\\[\nH_i \\ge p^{\\frac{n}{k}\\left(\\frac14+\\varepsilon\\right)}\n\\quad (1\\le i\\le k),\n\\]\nwhere\n\\[\n\\delta(\\varepsilon)\n=\n\\varepsilon^2\n\\frac{1-\\frac{1}{2n}}{\\left(1+\\frac{1}{4n}\\right)\\left(2-\\frac{1}{2n}\\right)}.\n\\]\nIn particular, if $k=n-o(n)$, then the condition becomes\n$H_i\\geq p^{\\frac{n}{n-o(n)}(1/4+\\varepsilon)}$, and this exponent tends to $1/4+\\varepsilon$ as $n\\to\\infty$. Thus, in the limit of large dimension, we obtain Burgess-strength bounds in each coordinate.\n\\end{Corollary}", "post_theorem_intro_text_len": 6012, "post_theorem_intro_text": "While this theorem provides strong cancellation, it requires each interval to be of Burgess length, precluding situations in which some intervals are short while others are long. This limitation was partially overcome by M. R. Gabdullin \\cite{GB}, who generalized Konyagin's result for $n=2,3$ under a weaker product condition.\n\n\\begin{theorem}[Gabdullin \\cite{GB}]\nLet $n\\in\\{2,3\\}$, let $\\chi$ be a nontrivial multiplicative character of $\\mathbb{F}_{p^n}$, and suppose\n\\[\n|B|\\geq p^{n(1/4+\\varepsilon)}, \\qquad H_1\\leq H_2\\leq\\cdots\\leq H_n.\n\\]\nThen\n\\[\n\\left|\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\\right|\n\\ll_{n,\\varepsilon} |B| p^{-\\varepsilon^2/12}.\n\\]\n\\end{theorem}\n\nIn this paper, we generalize Gabdullin's result to arbitrary dimension $n$. Our main theorem establishes nontrivial cancellation for character sums over general boxes in $\\mathbb{F}_{p^n}$, assuming only a lower bound on the volume of the box.\n\n\\begin{theorem}[Main Theorem]\\label{MT}\nLet $n\\geq2$, let $\\chi$ be a nontrivial multiplicative character of $\\mathbb{F}_{p^n}$, and suppose\n\\[\n|B|\\geq p^{n(1/4+\\varepsilon)},\\qquad\nH_1\\leq H_2\\leq\\cdots\\leq H_n\\leq \\sqrt{p/2}.\n\\]\nThen\n\\[\n\\left|\\sum_{x\\in B}\\chi(x)\\right|\n\\ll |B|\\, p^{-\\varepsilon^2\n\\frac{1-\\frac{1}{2n}}{\\left(1+\\frac{1}{4n}\\right)\\left(2-\\frac{1}{2n}\\right)}}.\n\\]\n\\end{theorem}\n\n\\begin{remark}\nIf the extension degree $n$ is prime, the monotonicity assumption $H_1\\leq\\cdots\\leq H_n\\leq \\sqrt{p/2}$ may be removed. A sketch of the argument is given in Case 3 of the proof of Theorem \\ref{MT}.\n\\end{remark}\n\nIn a previous work \\cite{Ch}, the author studied character sums over sublattices of codimension one in $\\mathbb{F}_{p^n}$, obtaining nontrivial bounds under significantly weaker length conditions.\n\n\\begin{theorem}[Chattopadhyay \\cite{Ch}]\nLet $\\chi$ be a nontrivial multiplicative character of $\\mathbb{F}_{p^d}$, and let $\\omega$ be a generator of $\\mathbb{F}_{p^n}^\\ast$. Given $\\varepsilon>0$, there exists $\\delta(\\varepsilon)>0$ such that if $I_0,\\ldots,I_{n-2}$ are intervals of length $p^{\\rho''_n+\\varepsilon}$, where\n\\[\n\\rho''_n=\\frac{1460-1000n+\\sqrt{1000000n^2-1960000n+490000}}{960},\n\\]\nthen\n\\[\n\\sum_{x_0\\in I_0,\\ldots,x_{n-2}\\in I_{n-2}}\n\\chi(x_0+x_1\\omega+\\cdots+x_{n-2}\\omega^{n-2})\n\\ll p^{-\\delta(\\varepsilon)} |I_0|\\cdots|I_{n-2}|.\n\\]\n\\end{theorem}\n\nThe novelty of this result lied in treating character sums over sublattices of codimension one.\n\n\\medskip\n\nAs a consequence of our Main Theorem, we can now treat sublattices of co-dimension at least $o(n)$, while allowing each interval to have length as small as $p^{1/4+\\varepsilon}$. Moreover, we are no longer restricted to a special choice of basis; the result holds for an arbitrary basis of $\\mathbb{F}_{p^n}$ over $\\mathbb{F}_p$, in contrast with our previous work.\n\\begin{Corollary}\nLet $\\chi$ be a nontrivial multiplicative character of $\\mathbb{F}_{p^n}$, let $I_i \\subset \\mathbb{Z}$ be intervals of lengths $H_i$, and let $\\{\\omega_1,\\ldots,\\omega_n\\}$ be a basis of $\\mathbb{F}_{p^n}$ over $\\mathbb{F}_p$. \nFor every $\\varepsilon>0$, there exists $\\delta(\\varepsilon)>0$ such that\n\\[\n\\left|\\sum_{x_1\\in I_1,\\ldots,x_k\\in I_k}\n\\chi(x_1\\omega_1+\\cdots+x_k\\omega_k)\\right|\n\\ll_{n,\\varepsilon}\nH_1\\cdots H_k\\,p^{-\\delta(\\varepsilon)},\n\\]\nprovided that\n\\[\nH_i \\ge p^{\\frac{n}{k}\\left(\\frac14+\\varepsilon\\right)}\n\\quad (1\\le i\\le k),\n\\]\nwhere\n\\[\n\\delta(\\varepsilon)\n=\n\\varepsilon^2\n\\frac{1-\\frac{1}{2n}}{\\left(1+\\frac{1}{4n}\\right)\\left(2-\\frac{1}{2n}\\right)}.\n\\]\nIn particular, if $k=n-o(n)$, then the condition becomes\n$H_i\\geq p^{\\frac{n}{n-o(n)}(1/4+\\varepsilon)}$, and this exponent tends to $1/4+\\varepsilon$ as $n\\to\\infty$. Thus, in the limit of large dimension, we obtain Burgess-strength bounds in each coordinate.\n\\end{Corollary}\n\n\\medskip\n\nThe proof of the Main Theorem closely follows the approach of \\cite{GB}, employing techniques from \\cite{C1}, \\cite{K} and \\cite{Kon} to estimate character sums over boxes in finite fields whose side lengths are at most $\\sqrt{p/2}$. Our argument begins with the classical additive shifting technique $x\\mapsto x+yz$ introduced by Burgess \\cite{B2}. Applying H\\\"older's inequality twice to the resulting sum of shifted characters produces three terms: one is related to the multiplicative energy of the box, while the remaining terms can be bounded using Weil's bound and trivial estimates. Consequently, achieving a nontrivial saving for the character sum reduces to obtaining a bound on the multiplicative energy of a set in $\\mathbb{F}_{p^n}$ which are described in Lemma \\ref{l1} and Lemma \\ref{l2}. More specifically, we employ tools from the geometry of numbers, in particular the concepts of successive minima and Minkowski's second theorem, together with an auxiliary lemma that provides upper and lower bounds for the product of successive minimas which turns out to be an essential ingredient in our energy estimate.\\\\\n\nIn fact, we have also removed the boundedness assumption on the side lengths of the boxes in Theorem~\\ref{MT}, as described in \\textbf{Remark~1}, when the degree of the extension is prime. The key ingredient in the proof of \\textbf{Remark~1} is Theorem~\\ref{KT}, which provides square-root cancellation for the corresponding complete character sum over $\\mathbb{F}_{p}$. Under the remaining assumption that the larger side length of the box exceeds $p^{1/2+\\varepsilon}$, we obtain a non-trivial saving for the character sum along that direction. Estimating the remaining summations trivially then yields an overall non-trivial saving over the volume of the box. For further details, see \\textbf{Case~3} in the proof of the main theorem. \n\n\\medskip\n\n{\\bf Acknowledgments.} The author expresses his sincere gratitude to Professor Stephan Baier for his valuable conversations. The author thanks the Ramakrishna Mission Vivekananda Educational and Research Institute for an excellent work environment. The research of the author was supported by a CSIR Ph.D. fellowship under file number 09/0934(13170)/2022-EMR-I.", "sketch": "The proof of Theorem~\\ref{MT} “closely follows the approach of \\cite{GB},” using “techniques from \\cite{C1}, \\cite{K} and \\cite{Kon}.” It begins with “the classical additive shifting technique $x\\mapsto x+yz$ introduced by Burgess \\cite{B2}$,” applied to the character sum over a box (with side lengths “at most $\\sqrt{p/2}$”). Then, “applying H\\\"older's inequality twice to the resulting sum of shifted characters produces three terms”: one term “is related to the multiplicative energy of the box,” and “the remaining terms can be bounded using Weil's bound and trivial estimates.” Hence, getting cancellation “reduces to obtaining a bound on the multiplicative energy of a set in $\\mathbb{F}_{p^n}$,” as given in “Lemma~\\ref{l1} and Lemma~\\ref{l2}.” For the energy bound, the argument uses “tools from the geometry of numbers,” specifically “successive minima and Minkowski's second theorem,” together with “an auxiliary lemma that provides upper and lower bounds for the product of successive minimas,” described as “an essential ingredient” in the energy estimate.\n\nFor the prime-degree refinement mentioned after Theorem~\\ref{MT} (Remark~1), the boundedness on side lengths can be removed: “The key ingredient” is “Theorem~\\ref{KT}, which provides square-root cancellation for the corresponding complete character sum over $\\mathbb{F}_{p}$.” If “the larger side length of the box exceeds $p^{1/2+\\varepsilon}$,” one gets “a non-trivial saving for the character sum along that direction,” and then “estimating the remaining summations trivially” yields “an overall non-trivial saving over the volume of the box” (cf. “Case~3 in the proof of the main theorem”).", "expanded_sketch": "The proof of\n\\begin{theorem}[Main Theorem]\\label{MT}\nLet $n\\geq2$, let $\\chi$ be a nontrivial multiplicative character of $\\mathbb{F}_{p^n}$, and suppose\n\\[\n|B|\\geq p^{n(1/4+\\varepsilon)},\\qquad\nH_1\\leq H_2\\leq\\cdots\\leq H_n\\leq \\sqrt{p/2}.\n\\]\nThen\n\\[\n\\left|\\sum_{x\\in B}\\chi(x)\\right|\n\\ll |B|\\, p^{-\\varepsilon^2\n\\frac{1-\\frac{1}{2n}}{\\left(1+\\frac{1}{4n}\\right)\\left(2-\\frac{1}{2n}\\right)}}.\n\\]\n\\end{theorem}\n“closely follows the approach of \\cite{GB},” using “techniques from \\cite{C1}, \\cite{K} and \\cite{Kon}.” It begins with “the classical additive shifting technique $x\\mapsto x+yz$ introduced by Burgess \\cite{B2}$,” applied to the character sum over a box (with side lengths “at most $\\sqrt{p/2}$”). Then, “applying H\\\"older's inequality twice to the resulting sum of shifted characters produces three terms”: one term “is related to the multiplicative energy of the box,” and “the remaining terms can be bounded using Weil's bound and trivial estimates.” Hence, getting cancellation “reduces to obtaining a bound on the multiplicative energy of a set in $\\mathbb{F}_{p^n}$,” as given in the following lemmas.\n\n\\begin{Lemma}\\label{l1}\nWe have\n\\[\nL_{1} \\;\\ll\\; |B|^{2}\\log p.\n\\]\n\\end{Lemma}\n\n\\begin{Lemma}\\label{l2}\nWe have\n\\[\nL_{2} \\;\\ll\\; |B|^{2}(\\log p)^{n}.\n\\]\n\\end{Lemma}\n\nFor the energy bound, the argument uses “tools from the geometry of numbers,” specifically “successive minima and Minkowski's second theorem,” together with “an auxiliary lemma that provides upper and lower bounds for the product of successive minimas,” described as “an essential ingredient” in the energy estimate.\n\nFor the prime-degree refinement mentioned after the main theorem (Remark~1), the boundedness on side lengths can be removed: “The key ingredient” is the following proposition, which provides square-root cancellation for the corresponding complete character sum over $\\mathbb{F}_{p}$.\n\n\\begin{Proposition}[Katz]\\label{KT}\nLet $\\chi$ be a non-trivial multiplicative character of $\\mathbb{F}_{p^n}$ and let\n$g\\in\\mathbb{F}_{p^n}$ generate the extension\n$\\mathbb{F}_{p^n}=\\mathbb{F}_p(g)$. Then, for any interval\n$I\\subseteq [1,p]\\cap\\mathbb{Z}$,\n\\[\n\\left|\\sum_{t\\in I}\\chi(g+t)\\right|\n\\le c(n)\\sqrt{p}\\log p .\n\\]\n\\end{Proposition}\n\nIf “the larger side length of the box exceeds $p^{1/2+\\varepsilon}$,” one gets “a non-trivial saving for the character sum along that direction,” and then “estimating the remaining summations trivially” yields “an overall non-trivial saving over the volume of the box” (cf. “Case~3 in the proof of the main theorem”).", "expanded_theorem": "[Konyagin \\cite{Kon}]\nLet $\\varepsilon>0$ and suppose $H_i>p^{1/4+\\varepsilon}$ for all $1\\leq i\\leq n$. Then\n\\[\n\\left|\\sum_{\\mathbf{x}\\in B}\\chi(\\mathbf{x})\\right|\n\\ll_{n,\\varepsilon} p^{-\\varepsilon^2/2}|B|.\n\\],", "theorem_type": ["Inequality or Bound", "Implication"], "mcq": {"question": "Let p be a prime, let \\(\\mathbb{F}_{p^n}\\) be the finite field with \\(p^n\\) elements, and fix a basis \\(\\{\\omega_1,\\ldots,\\omega_n\\}\\) of \\(\\mathbb{F}_{p^n}\\) over \\(\\mathbb{F}_p\\). For integers \\(N_i\\) and \\(H_i\\), define intervals \\(I_i=[N_i+1,N_i+H_i]\\cap\\mathbb{Z}\\) and the box\n\\[\nB=\\left\\{\\sum_{i=1}^n x_i\\omega_i: N_i+1\\le x_i\\le N_i+H_i\\text{ for }1\\le i\\le n\\right\\}\\subseteq \\mathbb{F}_{p^n},\n\\]\nwith volume \\(|B|=H_1\\cdots H_n\\). Let \\(\\chi\\) be a nontrivial multiplicative character on \\(\\mathbb{F}_{p^n}\\). If \\(\\varepsilon>0\\) and \\(H_i>p^{1/4+\\varepsilon}\\) for every \\(1\\le i\\le n\\), which quantitative estimate holds for the character sum over \\(B\\)?", "correct_choice": {"label": "A", "text": "\\[\\left|\\sum_{x\\in B}\\chi(x)\\right|\\ll_{n,\\varepsilon} p^{-\\varepsilon^2/2}\\,|B|.\\]"}, "choices": [{"label": "B", "text": "\\[\\left|\\sum_{x\\in B}\\chi(x)\\right|\\ll_{n,\\varepsilon} p^{-\\varepsilon^2/2}\\,|B|,\\qquad \\text{provided }H_i\\ge p^{1/4+\\varepsilon}\\text{ for every }1\\le i\\le n.\\]"}, {"label": "C", "text": "\\[\\left|\\sum_{x\\in B}\\chi(x)\\right|=o_{n,\\varepsilon}(|B|)\\quad\\text{as }p\\to\\infty.\\]"}, {"label": "D", "text": "\\[\\left|\\sum_{x\\in B}\\chi(x)\\right|\\ll_{\\varepsilon} p^{-\\varepsilon^2/2}\\,|B|.\\]"}, {"label": "E", "text": "\\[\\left|\\sum_{x\\in B}\\chi(x)\\right|\\ll_{n,\\varepsilon} p^{-\\varepsilon^2}\\,|B|.\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "strict lower bound on each side length", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "explicit power saving factor p^{-\\varepsilon^2/2}", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence of the implied constant on n", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "exact exponent of the power saving", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem gives the setup and hypotheses but does not disclose the exact conclusion. The correct option is not signaled by wording in the question itself."}, "TAS": {"score": 1, "justification": "This is close to theorem recall: the task is essentially to identify the precise stated conclusion under the given hypotheses. However, it is not a verbatim restatement because the student must distinguish among nearby alternatives."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to compare exact, weaker, and stronger bounds and note subtle quantifier issues, but the item mainly tests recognition of the known result rather than genuinely generating an argument."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and well targeted: one is weaker but true, one adds unnecessary side conditions, one alters dependence of the saving exponent, and one overstates the exponent. These reflect realistic failure modes."}, "total_score": 6, "overall_assessment": "A solid theorem-identification MCQ with strong distractors and little answer leakage, but it leans more toward precise recall than toward deep generative reasoning."}}
{"id": "2602.22191v1", "paper_link": "http://arxiv.org/abs/2602.22191v1", "theorems_cnt": 2, "theorem": {"env_name": "maintheorem", "content": "\\label{thm:maintheorem}(cf. Theorem \\ref{thm:cdbound})\nLet $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\\operatorname{cd}(A,I)\\leq d-\\floor*{\\frac{d-1}{c}}.$$", "start_pos": 8062, "end_pos": 8357, "label": "thm:maintheorem"}, "ref_dict": {"thm:cdbound": "\\begin{theorem}\\label{thm:cdbound}\n    Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\\cd(A,I)\\leq d-\\floor*{\\frac{d-1}{c}}.$$\n\\end{theorem}", "lemma:p_zd_on_H_I^n": "\\begin{lemma}\\label{lemma:p_zd_on_H_I^n}\n    Let $I$, $A$, $x$, $I_1$ and $I_2$ be as in Lemma \\ref{lemma:radical(I+p)}, $M$ be an $A$-module and $n>\\max(\\cd(M,I_1),\\cd(M,I_2))$. Then $H_I^n(M)$ is $x$-power torsion, that is, every element of $H^n_I(M)$ is annihilated by some power of $x$.\n\\end{lemma}", "thm:induction": "\\begin{theorem}\\label{thm:induction}\n    Let $(A,\\mathfrak{m},k)$ be an unramified regular local ring of mixed characteristic $(0,p)$, and let $I$ be an ideal of $A$. Suppose $\\height(I)<\\height(I:(I:p))$ (for example, this holds if $p$ is a nonzerodivisor on $A/I)$.\\footnote{We use the convention that $\\height(A)=\\infty.$}\n\n    Set $c=\\bight(I)$. Let $n>c$ be an integer. Assume that for all integers $s$, with $1\\leq s\\leq c-1,$ and for all $q\\geq n-s$, $H^q_{IA_P}(A_P)=0$ for all $P\\in \\Spec(A)$ such that $I\\subseteq P$ and $\\dim(A/P)\\geq s+1.$ Then $H^q_I(A)=0$ for all $q\\geq n.$\n\\end{theorem}", "example:sharpness": "\\begin{example}\\label{example:sharpness}\n    Given an unramified regular local ring $(A,\\mathfrak{m})$ of dimension $d>0$ and a positive integer $c\\leq d$, there exists an ideal $I$ of $A$ such that $\\cd(A,I)=d-\\floor*{\\frac{d-1}{c}}.$ We construct this ideal as in \\cite{Lyu2}. \n\n    Let $N=\\floor*{\\frac{d-1}{c}}.$ Let $I_0,\\dots,I_N$ be ideals of pure height $c$ in $A$ such that $I_0+\\dots+I_N$ is $\\mathfrak{m}$-primary and let $I=I_0\\cap\\dots\\cap I_N$. Then $\\cd(A,I)=d-N.$ The proof of this statement is identical to \\cite{Lyu2}.\n\n    To resolve any conflict with the previous remark, we show that in this example, if $p\\in I$, then $\\floor*{\\frac{d-2}{c-1}}= \\floor*{\\frac{d-1}{c}}.$ Indeed, if $p\\in I,$ we have $p\\in I_j$ for all $j\\in\\{0,\\dots,N\\}$ and by Theorem \\ref{thm:Serre}, $$d=\\height(I_0+\\dots+I_N)\\leq 1+(N+1)(c-1)=Nc+c-N.$$ Let $d-1=Nc+k$, where $0\\leq k<c$. Then the above equation yields $N+k+1\\leq c$, or $N+k-1\\leq c-2$. Observe that $$d-2=N(c-1)+N+k-1,$$ and hence, $\\floor*{\\frac{d-2}{c-1}}=N=\\floor*{\\frac{d-1}{c}}$. \n\\end{example}", "thm:maintheorem": "\\begin{maintheorem}\\label{thm:maintheorem}(cf. Theorem \\ref{thm:cdbound})\nLet $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\\cd(A,I)\\leq d-\\floor*{\\frac{d-1}{c}}.$$  \n\\end{maintheorem}", "thm:indcrit": "\\begin{maintheorem}\\label{thm:indcrit}(cf. Theorem \\ref{thm:induction})\n    Let $(A,\\mathfrak{m})$ be an unramified regular local ring of mixed characteristic $(0,p)$ and $I$ be an ideal of $A$ of big height $c$. Suppose $p$ is a nonzerodivisor on $A/I.$\n\n    Let $n>c$ be an integer. Assume that for all integers $s$, with $1\\leq s\\leq c-1,$ and for all $q\\geq n-s$, $H^q_{IA_P}(A_P)=0$ for all $P\\in \\Spec(A)$ such that $I\\subseteq P$ and $\\dim(A/P)\\geq s+1.$ Then $H^q_I(A)=0$ for all $q\\geq n.$\n\\end{maintheorem}"}, "pre_theorem_intro_text_len": 3311, "pre_theorem_intro_text": "Local cohomology, introduced by Grothendieck in the early 1960s, plays a pivotal role in commutative algebra and algebraic geometry. Its vanishing behavior reflects subtle geometric and topological properties of the underlying scheme and has been studied extensively in the work of Grothendieck, Hartshorne, Faltings, and others. Understanding when local cohomology modules vanish remains a fundamental and delicate problem. \n\nLet $A$ be a commutative Noetherian ring of Krull dimension $d$ and $I$ be an ideal of $A$. The \\textit{cohomological dimension} of $I$ in $A$ is $$\\operatorname{cd}(A,I):=\\max\\{n\\:|\\:H^n_I(A)\\neq 0\\}.$$ Alternatively, $\\operatorname{cd}(A,I)$ is the least integer $n$ such that $H^q_I(M)=0$ for all $A$-modules $M$ and $q>n$ \\cite[Theorem 9.6]{24h}. As $H^q_I(M)=0$ for all $A$-modules $M$ and $q>d$ \\cite{Gr}, $\\operatorname{cd}(A,I)$ always exists and is bounded above by $d$.\n\nThe computation of better upper bounds on cohomological dimension has been of great interest over the past few decades. It is well-known that $$\\operatorname{cd}(A,I)\\leq\\operatorname{ara}(I),$$ where $\\operatorname{ara}(I)$, the $\\textit{arithmetic rank}$ of $I$, is the least number of equations required to generate $I$ up to radical. However, the arithmetic rank of an ideal is difficult to compute and can be far from the cohomological dimension \\cite{BS,Bar,JPSW,BMMP}. \n\nA complementary approach is to bound the cohomological dimension using topological invariants of the spectrum of $A/I$. The most famous result in this direction is the Hartshorne--Lichtenbaum Vanishing Theorem \\cite{Ha1}, which states that for a complete local domain $(A,\\mathfrak{m})$, we have $\\operatorname{cd}(A,I)\\leq d-1$ if and only if $I$ is not $\\mathfrak{m}$-primary. Furthermore, the Second Vanishing Theorem \\cite{Ogus,PS,HunLyu,Zhang} asserts that in equal characteristic or unramified mixed characteristic, for a complete regular local ring $(A,\\mathfrak{m})$ with separably closed residue field, we have $\\operatorname{cd}(A,I)\\leq d-2$ if and only if $\\dim(A/I)\\geq 2$ and the punctured spectrum of $A/I$ is connected. Peskine and Szpiro \\cite{PS} proved that if $(A,\\mathfrak{m)}$ is a regular local ring of equal characteristic $p$ and $A/I$ is Cohen-Macaulay, then $\\operatorname{cd}(A,I)=\\operatorname{ht}(I).$ Beyond the results discussed here, further vanishing statements have been proved in various settings \\cite{MR3078644,DaoTakagi, Burke}, though the conditions required become progressively harder to check. Ogus \\cite{Ogus}, Hartshorne and Speiser \\cite{HarSpe}, Lyubeznik \\cite{Lyu3}, Mustață and Popa \\cite{MuPo}, and Reichelt, Saito and Walther \\cite{RTW} obtained concrete formulae for cohomological dimension in equal characteristic in terms of singularity invariants, see also \\cite{BBLSZ}. However, these methods require the understanding of de Rham cohomologies, Hodge filtrations, or Frobenius structures, and in practice they can be difficult to compute.\n\nIn equal characteristic, Faltings \\cite{Faltings} established a general bound on cohomological dimension in terms of the \\textit{big height} of an ideal. The big height of an ideal is the maximum of the heights of all its minimal primes. In this article, we extend Faltings' result to the unramified mixed characteristic setting.", "context": "Local cohomology, introduced by Grothendieck in the early 1960s, plays a pivotal role in commutative algebra and algebraic geometry. Its vanishing behavior reflects subtle geometric and topological properties of the underlying scheme and has been studied extensively in the work of Grothendieck, Hartshorne, Faltings, and others. Understanding when local cohomology modules vanish remains a fundamental and delicate problem.\n\nLet $A$ be a commutative Noetherian ring of Krull dimension $d$ and $I$ be an ideal of $A$. The \\textit{cohomological dimension} of $I$ in $A$ is $$\\operatorname{cd}(A,I):=\\max\\{n\\:|\\:H^n_I(A)\\neq 0\\}.$$ Alternatively, $\\operatorname{cd}(A,I)$ is the least integer $n$ such that $H^q_I(M)=0$ for all $A$-modules $M$ and $q>n$ \\cite[Theorem 9.6]{24h}. As $H^q_I(M)=0$ for all $A$-modules $M$ and $q>d$ \\cite{Gr}, $\\operatorname{cd}(A,I)$ always exists and is bounded above by $d$.\n\nThe computation of better upper bounds on cohomological dimension has been of great interest over the past few decades. It is well-known that $$\\operatorname{cd}(A,I)\\leq\\operatorname{ara}(I),$$ where $\\operatorname{ara}(I)$, the $\\textit{arithmetic rank}$ of $I$, is the least number of equations required to generate $I$ up to radical. However, the arithmetic rank of an ideal is difficult to compute and can be far from the cohomological dimension \\cite{BS,Bar,JPSW,BMMP}.\n\nA complementary approach is to bound the cohomological dimension using topological invariants of the spectrum of $A/I$. The most famous result in this direction is the Hartshorne--Lichtenbaum Vanishing Theorem \\cite{Ha1}, which states that for a complete local domain $(A,\\mathfrak{m})$, we have $\\operatorname{cd}(A,I)\\leq d-1$ if and only if $I$ is not $\\mathfrak{m}$-primary. Furthermore, the Second Vanishing Theorem \\cite{Ogus,PS,HunLyu,Zhang} asserts that in equal characteristic or unramified mixed characteristic, for a complete regular local ring $(A,\\mathfrak{m})$ with separably closed residue field, we have $\\operatorname{cd}(A,I)\\leq d-2$ if and only if $\\dim(A/I)\\geq 2$ and the punctured spectrum of $A/I$ is connected. Peskine and Szpiro \\cite{PS} proved that if $(A,\\mathfrak{m)}$ is a regular local ring of equal characteristic $p$ and $A/I$ is Cohen-Macaulay, then $\\operatorname{cd}(A,I)=\\operatorname{ht}(I).$ Beyond the results discussed here, further vanishing statements have been proved in various settings \\cite{MR3078644,DaoTakagi, Burke}, though the conditions required become progressively harder to check. Ogus \\cite{Ogus}, Hartshorne and Speiser \\cite{HarSpe}, Lyubeznik \\cite{Lyu3}, Mustață and Popa \\cite{MuPo}, and Reichelt, Saito and Walther \\cite{RTW} obtained concrete formulae for cohomological dimension in equal characteristic in terms of singularity invariants, see also \\cite{BBLSZ}. However, these methods require the understanding of de Rham cohomologies, Hodge filtrations, or Frobenius structures, and in practice they can be difficult to compute.\n\nIn equal characteristic, Faltings \\cite{Faltings} established a general bound on cohomological dimension in terms of the \\textit{big height} of an ideal. The big height of an ideal is the maximum of the heights of all its minimal primes. In this article, we extend Faltings' result to the unramified mixed characteristic setting.\n\n\\begin{theorem}\\label{thm:cdbound}\n    Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\\cd(A,I)\\leq d-\\floor*{\\frac{d-1}{c}}.$$\n\\end{theorem}", "full_context": "Local cohomology, introduced by Grothendieck in the early 1960s, plays a pivotal role in commutative algebra and algebraic geometry. Its vanishing behavior reflects subtle geometric and topological properties of the underlying scheme and has been studied extensively in the work of Grothendieck, Hartshorne, Faltings, and others. Understanding when local cohomology modules vanish remains a fundamental and delicate problem.\n\nLet $A$ be a commutative Noetherian ring of Krull dimension $d$ and $I$ be an ideal of $A$. The \\textit{cohomological dimension} of $I$ in $A$ is $$\\operatorname{cd}(A,I):=\\max\\{n\\:|\\:H^n_I(A)\\neq 0\\}.$$ Alternatively, $\\operatorname{cd}(A,I)$ is the least integer $n$ such that $H^q_I(M)=0$ for all $A$-modules $M$ and $q>n$ \\cite[Theorem 9.6]{24h}. As $H^q_I(M)=0$ for all $A$-modules $M$ and $q>d$ \\cite{Gr}, $\\operatorname{cd}(A,I)$ always exists and is bounded above by $d$.\n\nThe computation of better upper bounds on cohomological dimension has been of great interest over the past few decades. It is well-known that $$\\operatorname{cd}(A,I)\\leq\\operatorname{ara}(I),$$ where $\\operatorname{ara}(I)$, the $\\textit{arithmetic rank}$ of $I$, is the least number of equations required to generate $I$ up to radical. However, the arithmetic rank of an ideal is difficult to compute and can be far from the cohomological dimension \\cite{BS,Bar,JPSW,BMMP}.\n\nA complementary approach is to bound the cohomological dimension using topological invariants of the spectrum of $A/I$. The most famous result in this direction is the Hartshorne--Lichtenbaum Vanishing Theorem \\cite{Ha1}, which states that for a complete local domain $(A,\\mathfrak{m})$, we have $\\operatorname{cd}(A,I)\\leq d-1$ if and only if $I$ is not $\\mathfrak{m}$-primary. Furthermore, the Second Vanishing Theorem \\cite{Ogus,PS,HunLyu,Zhang} asserts that in equal characteristic or unramified mixed characteristic, for a complete regular local ring $(A,\\mathfrak{m})$ with separably closed residue field, we have $\\operatorname{cd}(A,I)\\leq d-2$ if and only if $\\dim(A/I)\\geq 2$ and the punctured spectrum of $A/I$ is connected. Peskine and Szpiro \\cite{PS} proved that if $(A,\\mathfrak{m)}$ is a regular local ring of equal characteristic $p$ and $A/I$ is Cohen-Macaulay, then $\\operatorname{cd}(A,I)=\\operatorname{ht}(I).$ Beyond the results discussed here, further vanishing statements have been proved in various settings \\cite{MR3078644,DaoTakagi, Burke}, though the conditions required become progressively harder to check. Ogus \\cite{Ogus}, Hartshorne and Speiser \\cite{HarSpe}, Lyubeznik \\cite{Lyu3}, Mustață and Popa \\cite{MuPo}, and Reichelt, Saito and Walther \\cite{RTW} obtained concrete formulae for cohomological dimension in equal characteristic in terms of singularity invariants, see also \\cite{BBLSZ}. However, these methods require the understanding of de Rham cohomologies, Hodge filtrations, or Frobenius structures, and in practice they can be difficult to compute.\n\nIn equal characteristic, Faltings \\cite{Faltings} established a general bound on cohomological dimension in terms of the \\textit{big height} of an ideal. The big height of an ideal is the maximum of the heights of all its minimal primes. In this article, we extend Faltings' result to the unramified mixed characteristic setting.\n\n\\begin{theorem}\\label{thm:cdbound}\n    Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\\cd(A,I)\\leq d-\\floor*{\\frac{d-1}{c}}.$$\n\\end{theorem}\n\nIn equal characteristic, Faltings \\cite{Faltings} established a general bound on cohomological dimension in terms of the \\textit{big height} of an ideal. The big height of an ideal is the maximum of the heights of all its minimal primes. In this article, we extend Faltings' result to the unramified mixed characteristic setting.\n\nWe also establish an inductive criterion for the vanishing of local cohomology modules. See Theorem \\ref{thm:induction} for a more general statement than the one stated below.\n\n\\begin{theorem}\\label{thm:cdbound}\n Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\\cd(A,I)\\leq d-\\floor*{\\frac{d-1}{c}}.$$\n\\end{theorem}\n\\begin{proof} We may assume that $A$ is complete and $I$ is radical as local cohomology and big height remain unchanged. If $c=1$, $I$ is principal and $\\cd(A,I)\\leq 1$. Thus we assume $c>1.$ We also assume $c<d$ as the claimed bound is immediate if $c=d.$\n\n\\begin{corollary}[cf. {\\cite[Lemma]{Lyu2}}, {\\cite[Lemma 3.10]{HNBPW}}]\\label{cor:Lyu}\n Let $A$ be a $d$-dimensional regular local ring of equal characteristic or unramified mixed characteristic. Let $I_1,\\dots,I_N$ be ideals of $A$ of big height at most $c\\geq 1$. Then $$\\cd(A,I_1+\\dots+I_N)\\leq d-\\floor*{\\frac{d-1}{c}}+N-1.$$\n\\end{corollary}\n\\begin{proof}\n We proceed by induction on $N$. The case $N=1$ follows from \\cite[Satz 1]{Faltings} in equal characteristic and Theorem \\ref{thm:cdbound} in unramified mixed characteristic. For $N>1,$ let $I'=I_1+\\dots+I_{N-1}$, $I=I'+I_N$ and $J=I_1\\cap I_t+\\dots+I_{N-1}\\cap I_N$. Note that each $I_j\\cap I_t$ has big height at most $c$ by Remark \\ref{rmk:bight}. As $\\sqrt{J}=\\sqrt{I'\\cap I_N}$, the Mayer-Vietoris sequence for local cohomology yields $$\\cdots\\to H_J^{i-1}(A)\\to H_I^{i}(A)\\to H_{I'}^{i}(A)\\oplus H_{I_N}^{i}(A)\\to \\cdots$$ \n For $i>d-\\floor*{\\frac{d-1}{c}}+N-1$, $H_J^{i-1}(A)=H_{I'}^{i}(A)=H_{I_N}^{i}(A)=0$ by the induction hypothesis. Hence, $H_I^i(A)=0$ for $i>d-\\floor*{\\frac{d-1}{c}}+N-1$.\n\\end{proof}\n\n\\begin{theorem}[cf. {\\cite[Theorem 5.2]{HunLyu}}]\\label{thm:example}\n Let $(A,\\mathfrak{m})$ be a $d$-dimensional regular local ring of equal characteristic or unramified mixed characteristic. Let $c<d$ be a positive integer and set $N=\\floor*{\\frac{d-1}{c}}$. Let $I_1,\\dots,I_N$ be ideals of $A$ of big height at most $c$ such that $\\Spec(A/(I_1+\\dots+I_N))-\\{\\mathfrak{m}\\}$ is disconnected. Then $\\cd(A,I_1\\cap\\dots\\cap I_N)=d-N.$ \n\\end{theorem}\n\\begin{proof} \n If $(B,\\mathfrak{n})$ is a faithfully flat $A$-algebra with zero-dimensional fiber, then $\\bight(I_jB)=\\bight(I)$ (Lemma \\ref{lemma:fflat}) and $\\Spec(B/(I_1B+\\cdots+I_NB)-\\{\\mathfrak{n}\\}$ is disconnected. Hence, we may assume that $A$ has separably closed residue field.\n\n\\begin{example}[{\\cite[Example 5.4]{HunLyu}}]\n Let $(A,\\mathfrak{m})$ be a $d$-dimensional regular local ring of equal characteristic or unramified mixed characteristic. Suppose $P_1,\\dots,P_5$ are prime ideals of $A$ of height $c$ such that $\\floor*{\\frac{d-1}{c}}=2.$ Further, suppose that $P_i+P_j+P_k$ is $\\mathfrak{m}$-primary iff $$\\{i,j,k\\}\\in\\Lambda=\\{(1,3,4),(1,3,5),(1,2,5),(2,3,4),(2,4,5)\\}.$$ Let $I=P_1\\cap\\cdots\\cap P_5.$\n For instance, we may consider $A=V[[x_2,\\dots,x_9]]$, where $V$ is a discrete valuation ring with uniformizer $p$ and separably closed residue field, and $$P_1=(p,x_2,x_3,x_4), \\;P_2=(x_3,x_4,x_5,x_6),\\; P_3=(x_5,x_6,x_7,x_8),$$ $$P_4=(p,x_2,x_8,x_9) \\text{ and } P_5=(x_6,x_7,x_8,x_9).$$\n Note that $\\Lambda$ has the property that each $i$, $1\\leq i\\leq5$ appears no more than thrice and each pair $(i,j)$ appears no more than twice. This implies that we cannot split $\\{1,\\dots,5\\}$ into three distinct subsets $S_0$, $S_1$ and $S_2$ such that $P_i+P_j+P_k$ is $\\mathfrak{m}$-primary for all $i\\in S_0$, $j\\in S_1$, $k\\in S_2$. Hence, we are unable to apply the result of Example \\ref{example:sharpness}.\n\n\\begin{theorem}[{\\cite[Theorem 3.8]{HunLyu}}]\\label{thm:HunLyu_irred}\n Let $A$ be a $d$-dimensional regular local ring of equal characteristic and $I$ be a formally geometrically irreducible ideal of $A$ of big height $c$. Suppose $0<c<d.$ Then $$\\cd(A,I)\\leq d-1-\\floor*{\\frac{d-2}{c}}.$$\n\\end{theorem}\n\n\\begin{question}\\label{question:irredbound}\n Let $A$ be a $d$-dimensional regular local ring of unramified mixed characteristic and $I$ be a formally geometrically irreducible ideal of $A$ of big height $c$. Suppose $0<c<d$. Is $$\\cd(A,I)\\leq d-1-\\floor*{\\frac{d-2}{c}}?$$\n\\end{question}\n\n\\begin{example}\\label{example:sharpness}\n    Given an unramified regular local ring $(A,\\mathfrak{m})$ of dimension $d>0$ and a positive integer $c\\leq d$, there exists an ideal $I$ of $A$ such that $\\cd(A,I)=d-\\floor*{\\frac{d-1}{c}}.$ We construct this ideal as in \\cite{Lyu2}. \n\n    Let $N=\\floor*{\\frac{d-1}{c}}.$ Let $I_0,\\dots,I_N$ be ideals of pure height $c$ in $A$ such that $I_0+\\dots+I_N$ is $\\mathfrak{m}$-primary and let $I=I_0\\cap\\dots\\cap I_N$. Then $\\cd(A,I)=d-N.$ The proof of this statement is identical to \\cite{Lyu2}.\n\n    To resolve any conflict with the previous remark, we show that in this example, if $p\\in I$, then $\\floor*{\\frac{d-2}{c-1}}= \\floor*{\\frac{d-1}{c}}.$ Indeed, if $p\\in I,$ we have $p\\in I_j$ for all $j\\in\\{0,\\dots,N\\}$ and by Theorem \\ref{thm:Serre}, $$d=\\height(I_0+\\dots+I_N)\\leq 1+(N+1)(c-1)=Nc+c-N.$$ Let $d-1=Nc+k$, where $0\\leq k<c$. Then the above equation yields $N+k+1\\leq c$, or $N+k-1\\leq c-2$. Observe that $$d-2=N(c-1)+N+k-1,$$ and hence, $\\floor*{\\frac{d-2}{c-1}}=N=\\floor*{\\frac{d-1}{c}}$. \n\\end{example}\n\n\\begin{theorem}\\label{thm:cdbound}\n    Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\\cd(A,I)\\leq d-\\floor*{\\frac{d-1}{c}}.$$\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:induction}\n    Let $(A,\\mathfrak{m},k)$ be an unramified regular local ring of mixed characteristic $(0,p)$, and let $I$ be an ideal of $A$. Suppose $\\height(I)<\\height(I:(I:p))$ (for example, this holds if $p$ is a nonzerodivisor on $A/I)$.\\footnote{We use the convention that $\\height(A)=\\infty.$}\n\n    Set $c=\\bight(I)$. Let $n>c$ be an integer. Assume that for all integers $s$, with $1\\leq s\\leq c-1,$ and for all $q\\geq n-s$, $H^q_{IA_P}(A_P)=0$ for all $P\\in \\Spec(A)$ such that $I\\subseteq P$ and $\\dim(A/P)\\geq s+1.$ Then $H^q_I(A)=0$ for all $q\\geq n.$\n\\end{theorem}", "post_theorem_intro_text_len": 3988, "post_theorem_intro_text": "We also establish an inductive criterion for the vanishing of local cohomology modules. See Theorem \\ref{thm:induction} for a more general statement than the one stated below.\n\n\\begin{maintheorem}\\label{thm:indcrit}(cf. Theorem \\ref{thm:induction})\n    Let $(A,\\mathfrak{m})$ be an unramified regular local ring of mixed characteristic $(0,p)$ and $I$ be an ideal of $A$ of big height $c$. Suppose $p$ is a nonzerodivisor on $A/I.$\n\n    Let $n>c$ be an integer. Assume that for all integers $s$, with $1\\leq s\\leq c-1,$ and for all $q\\geq n-s$, $H^q_{IA_P}(A_P)=0$ for all $P\\in \\operatorname{Spec}(A)$ such that $I\\subseteq P$ and $\\dim(A/P)\\geq s+1.$ Then $H^q_I(A)=0$ for all $q\\geq n.$\n\\end{maintheorem}\n\nWe now outline the strategy for proving our results.\n\n\\begin{enumerate}\n    \\item In equal characteristic, Faltings proved a criterion \\cite[Satz 1]{Faltings} of the nature of Theorem \\ref{thm:indcrit}, and used it as an inductive tool to establish an upper bound on $\\operatorname{cd}(A,I).$ We generalize Faltings' criteria to the unramified mixed characteristic setting under the assumption that $p$ is a nonzerodivisor on $A/I$.\\footnote{This assumption can be further relaxed: see Theorem \\ref{thm:induction} for details.}\n\n    In the proof, we reduce the problem to the case of a local cohomology module supported only at the maximal ideal. We handle this remaining case by introducing an auxiliary tensor construction over the base discrete valuation ring and analyzing local cohomology with respect to the diagonal ideal. This allows us to exploit precise dimension bounds and numerical inequalities to force the collapse of a natural spectral sequence, yielding a vanishing statement in the auxiliary setting.\n    We then leverage a result of Zhou on the injective dimension of local cohomology modules in unramified mixed characteristic \\cite[Theorem 5.1]{Zhou}, together with the finiteness of Bass numbers for such modules established in \\cite{Lyu4, NB}.\n    \\item We split our proof of Theorem \\ref{thm:maintheorem} into three cases. If $p$ is a nonzerodivisor on $A/I$, we can use the inductive criterion above to prove that the local cohomology modules $H_I^n(A)$ vanish for $n$ greater than the desired bound. If $p\\in I,$ we reduce our analysis to $A/p$, a regular local ring of equal characteristic $p$, and employ the already known bound in equal characteristic.\n    \\item If $p$ is a zerodivisor on $I$, we build on the analysis of the preceding two cases to once again reduce to equal characteristic $p$. The guiding observation is that $H^n_I(A)$ is $p$-power torsion for $n$ exceeding the desired bound (Lemma \\ref{lemma:p_zd_on_H_I^n}). \n\n    At this stage, the argument must be refined to control the behavior of the big height of the ideal $I(A/p).$ A systemic analysis reveals that all cases can be resolved except when $\\operatorname{bight}(I)$ divides $\\dim(A)-1$. In this remaining case, a key input is the structural description of the critical local cohomology module $H^q_J(A/p)$ \\cite[Theorem 4.1]{Lyu1}, which allows us to reduce to the previously treated situation in which $p$ is a nonzerodivisor on $A/I$.\n\n\\end{enumerate}\n\nThe paper is organized as follows. In Section \\ref{section: InductionTheorem}, we collect the principal properties and a number of subtle features of the big height of an ideal. We then prove the inductive criteria for the vanishing of local cohomology in the unramified mixed characteristic setting (Theorem \\ref{thm:induction}). In Section \\ref{section:cdbound}, we use Theorem \\ref{thm:induction} to prove the upper bound on cohomological dimension (Theorem \\ref{thm:cdbound}). We also address the sharpness of our bound (Example \\ref{example:sharpness}) and illustrate how the bound can be used to compute the exact cohomological dimension of suitably chosen ideals. In Section \\ref{section:questions}, we collect natural questions suggested by our results, and comment on current progress and related directions.", "sketch": "To prove Theorem~\\ref{thm:maintheorem}, the authors first “generalize Faltings' criteria to the unramified mixed characteristic setting under the assumption that $p$ is a nonzerodivisor on $A/I$,” i.e. they prove an inductive vanishing criterion of the form Theorem~\\ref{thm:indcrit}. In that proof, they “reduce the problem to the case of a local cohomology module supported only at the maximal ideal,” then handle it by “introducing an auxiliary tensor construction over the base discrete valuation ring and analyzing local cohomology with respect to the diagonal ideal,” using “dimension bounds and numerical inequalities to force the collapse of a natural spectral sequence, yielding a vanishing statement in the auxiliary setting,” and finally “leverage a result of Zhou on the injective dimension of local cohomology modules… together with the finiteness of Bass numbers.”\n\nThey then “split [the] proof of Theorem~\\ref{thm:maintheorem} into three cases:”\n(1) If $p$ is a nonzerodivisor on $A/I$, they “use the inductive criterion above to prove that the local cohomology modules $H_I^n(A)$ vanish for $n$ greater than the desired bound.”\n(2) If “$p\\in I$,” they “reduce [the] analysis to $A/p$… of equal characteristic $p$, and employ the already known bound in equal characteristic.”\n(3) If $p$ is a zerodivisor on $A/I$, they “once again reduce to equal characteristic $p$,” using that “$H^n_I(A)$ is $p$-power torsion for $n$ exceeding the desired bound,” and then refine the argument to control “the behavior of the big height of the ideal $I(A/p)$”; after a “systemic analysis” they reduce remaining cases (notably when $\\operatorname{bight}(I)$ divides $\\dim(A)-1$) using “the structural description of the critical local cohomology module $H^q_J(A/p)$… which allows us to reduce to the previously treated situation in which $p$ is a nonzerodivisor on $A/I$.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:maintheorem}(To prove the main theorem, we use the following result. \\begin{theorem}\\label{thm:cdbound}\n    Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\\cd(A,I)\\leq d-\\floor*{\\frac{d-1}{c}}.$$\n\\end{theorem})\nLet $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then $$\\operatorname{cd}(A,I)\\leq d-\\floor*{\\frac{d-1}{c}}.$$, ", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let \\(A\\) be a \\(d\\)-dimensional unramified regular local ring of mixed characteristic \\((0,p)\\), and let \\(I\\subset A\\) be a nonzero proper ideal. Write \\(c=\\operatorname{bight}(I)\\), where the big height is the maximum of the heights of the minimal primes of \\(I\\). Also, let \\(\\operatorname{cd}(A,I)=\\max\\{n\\mid H_I^n(A)\\neq 0\\}\\) denote the cohomological dimension of \\(I\\) in \\(A\\). Which statement holds for every such pair \\((A,I)\\)?", "correct_choice": {"label": "A", "text": "\\(\\operatorname{cd}(A,I)\\le d-\\left\\lfloor \\frac{d-1}{c}\\right\\rfloor.\\)"}, "choices": [{"label": "B", "text": "\\(\\operatorname{cd}(A,I)\\le d-\\left\\lfloor \\frac{d}{c}\\right\\rfloor.\\)"}, {"label": "C", "text": "\\(\\operatorname{cd}(A,I)\\le d-c+1.\\)"}, {"label": "D", "text": "If \\(c<d\\), then \\(\\operatorname{cd}(A,I)\\le d-1-\\left\\lfloor \\frac{d-2}{c}\\right\\rfloor.\\)"}, {"label": "E", "text": "\\(\\operatorname{cd}(A,I)= d-\\left\\lfloor \\frac{d-1}{c}\\right\\rfloor.\\)"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "floor-expression numerator", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "replace Faltings-type floor bound by a coarser linear bound implied by it", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "imports the formally geometrically irreducible-type stronger bound without its extra hypothesis", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "upper bound promoted to equality for all ideals", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the target inequality; it only sets up notation and assumptions. There is no explicit statement of the correct bound or a giveaway phrase pointing uniquely to choice A."}, "TAS": {"score": 1, "justification": "The item is largely a theorem-recall question: it asks which estimate holds under exactly the stated hypotheses. The nearby variants create some choice pressure, but the task is still close to selecting the memorized theorem statement rather than deriving a new conclusion."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish among very similar formulas and to reject weaker/incorrect variants, but the main burden is recognition of the precise bound. It does not strongly force generative mathematical reasoning from first principles."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic confusions: changing the floor numerator, replacing big height by height, offering a weaker true bound, and using a bound from a related setting. They are distinct and nontrivial."}, "total_score": 6, "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it mainly tests precise theorem recall rather than deep generative reasoning."}}
{"id": "2602.22393v1", "paper_link": "http://arxiv.org/abs/2602.22393v1", "theorems_cnt": 3, "theorem": {"env_name": "ntheorem", "content": "\\label{thm:simply} \n  Let $S$ be a surface of infinite-type with non-planar ends. Then $\\Gamma_k(S)$ is connected and simply connected for each $k\\in \\mathbb{N}$.", "start_pos": 6068, "end_pos": 6253, "label": "thm:simply"}, "ref_dict": {"aut": "\\label{aut}\nIn this section we prove Theorem \\ref{thm:geometric} and henceforth assume that $S$ has no planar ends and boundary. We state the results proved in \\cite{irmak2004} without proof whenever", "lem:radius-zero": "\\begin{lemma}\\label{lem:radius-zero}\n  Let $p$ be a path in $\\Gamma_k(S)$ of radius $0$ with respect to some curve $a_0$. Then $p$ is null-homotopic.\n\\end{lemma}", "con": "\\begin{definition}\n  Let $S$ be an infinite-type surface with non-planar ends, $k\\in \\N$, and let $\\Gamma^0_k(S)$ be the collection of cut systems on $S$ with exactly $k$ curves. Let $\\Gamma^1_k(S)$ be the graph obtained by attaching a 1-cell between two cut systems in $\\Gamma^0_k(S)$ whenever they are related by an elementary move. The generalized Hatcher--Thurston complex, $\\Gamma_k(S)$, is obtained by attaching 2-cells to every cycle of type $1,\\ 2,$ or $3$.\n\\end{definition}\nNote that $\\Gamma^1_1(S)$ is just the Schmutz graph $ \\mathcal{G}(S)$. Since the Schmutz graph is connected and $\\text{diam}( \\mathcal{G}(S))\\leq 4$ we conclude that $\\Gamma^1_1(S)$ is connected with finite diameter. This will be useful in the proof of connectedness of $\\Gamma_k(S)$.\n\n\\section{Connectedness and Simple Connectedness of $\\Gamma_k(S)$}\\label{con}\nIn this section we prove Theorem \\ref{thm:simply}. \n\\begin{lemma}\\label{lem:common}\n  Let $v,w$ be cut systems in $\\Gamma_k(S)$ so that $v$ and $w$ have all but one curve in common. Then there is a path of length at most $4$ from $v$ to $w$ in $\\Gamma^1_1(S)$.\n\\end{lemma}\n\\begin{proof}\n  Let $v-w = \\{a\\}$ and $w-v=\\{b\\}$ and let $R$ be the surface obtained by cutting $S$ along the representatives of all the common curves, $c_1,\\cdots, c_{k-1}$, of $v$ and $w$. There is a path $a\\lr d_1\\lr\\cdots\\lr d_{n}\\lr b$ in $\\Gamma^1_1(R) = \\mathcal{G}(R)$ with $n\\leq 3$. Since $d_i$ are non-separating curves in $R$ it follows that $v_i = (d_i, c_1,\\cdots, c_{k-1})$ is a cut system. Thus, $v\\lr v_1\\lr\\cdots \\lr v_n\\lr w$ is the desired path.\n\\end{proof}\n\\begin{proof}[Proof of first half of Theorem \\ref{thm:simply}]\n  Let $v,w\\in \\Gamma^0_k(S)$. We use induction on the number of curves $\\ell = |v-v\\cap w|$. Lemma \\ref{lem:common} shows that for the base case $\\ell=1$ there is a path from $v$ to $w$. Suppose now that $v-w = \\{a_0,\\cdots, a_{\\ell-1}\\}$, $w-v=\\{b_0,\\cdots, b_{\\ell-1}\\}$, and $v\\cap w = \\{c_0,\\cdots, c_{k-\\ell-1}\\}$. Let $K\\subset S$ be a compact sub-surface containing all the curves in $v$ and $w$ and let $d$ be a non-separating curve in some component of $S-K$. Then, using induction, there is a path from $v' = (d,a_1,\\cdots,a_{\\ell-1},c_0,\\cdots,c_{k-\\ell-1})$ to $w'=(d,b_1,\\cdots,b_{\\ell-1},c_0,\\cdots,c_{k-\\ell-1})$. Using Lemma \\ref{lem:common}, there is a path from $v$ to $v'$ and $w$ to $w'$.\n\\begin{figure}[h]\n\\[\\begin{tikzcd}\n\tv && w \\\\\n\t\\\\\n\t{v'} && {w'} & {}\n\t\\arrow[squiggly, tail reversed, from=1-1, to=3-1]\n\t\\arrow[squiggly, tail reversed, from=1-3, to=3-3]\n\t\\arrow[squiggly, tail reversed, from=3-1, to=3-3]\n\\end{tikzcd}\\]\n\\caption{Constructing the path from $v$ to $w$.}\n\\end{figure}\nUsing the construction above and the fact that the length of the path in Lemma \\ref{lem:common} is at most $4$, it follows that the diameter of $\\Gamma^1_k(S)$ is at most $8k-4$.\n\\end{proof}\nIf $v,w$ are two cut systems in $\\Gamma_k(S)$ so that all of the curves in $v\\cup w$ are distinct then any path in $\\Gamma^1_k(S)$ between $v$ and $w$ has length at least $k$. This means that the diameter of $\\Gamma^1_k(S)$ has a lower bound $k$.\\\\\n\nIn order to prove that $\\Gamma_k(S)$ is simply-connected we define the following notion of radius of a path, and segments.\n\\begin{definition}\n  The radius of a path $p = v_0 \\lr \\cdots \\lr v_n$ in $\\Gamma^1_k(S)$ with respect to a non-separating curve, $a\\in v_j$ for some $j$, is defined as $\\max_{v_i}\\l(\\min_{b\\in v_i}i(a,b)\\r)$. If $a_0,\\cdots, a_n$ are non-separating curves in $S$ then $p$ is a $a_0,\\cdots,\\ a_n-$segment if $a_0,\\cdots,a_n\\in v_i$ for each vertex $v_i$ in the path $p$.\n\\end{definition}", "prop:base-case": "\\begin{proposition}\\label{prop:base-case}\n  Every closed path $p=a_0\\lr\\cdots\\lr a_{n-1}\\lr a_0$ in $\\Gamma_1(S)$ is null-homotopic.\n\\end{proposition}", "pre": "\\label{pre}\n\\textbf{Ends of a Surface.} Let $S$ be a surface of infinite-type and let $\\{K_i\\}_{i\\in \\N}$ be a strictly increasing chain of compact sets ordered by inclusion which covers $S$. Then an", "thm:geometric": "\\begin{ntheorem}\\label{thm:geometric}\n  Let $S$ be a surface of infinite-type with no planar ends and empty boundary. Then $\\text{Aut}(\\Gamma_k(S))$ is isomorphic to $\\map^*(S)$ for each $k\\in \\N$.\n\\end{ntheorem}", "thm:simply": "\\begin{ntheorem}\\label{thm:simply} \n  Let $S$ be a surface of infinite-type with non-planar ends. Then $\\Gamma_k(S)$ is connected and simply connected for each $k\\in \\N$.\n\\end{ntheorem}", "cor:iso": "\\begin{ncorollary}\\label{cor:iso}\n  Let $S,S'$ be surfaces of infinite-type with no planar ends and empty boundary. Let $\\vp:\\Gamma_k(S)\\to \\Gamma_k(S')$ be an isomorphism. Then $S$ is homeomorphic to $S'$.\n\\end{ncorollary}"}, "pre_theorem_intro_text_len": 3922, "pre_theorem_intro_text": "\\label{intro}\n\\addcontentsline{toc}{section}{Introduction}\nAll surfaces in this paper will be connected, orientable, without any punctures or marked points, and possibly with non-empty compact boundary. The mapping class group, $\\text{Map}(S)$, is the group of all orientation preserving homeomorphisms of the surface $S$ up to isotopy. The extended mapping class group, $\\text{Map}^*(S)$ is the group of all homeomorphisms of $S$ up to isotopy. When the fundamental group $\\pi_1(S)$ of the surface $S$ is finitely generated, we say that $S$ is of finite-type; otherwise $S$ is infinite-type.\\\\\n\nRecently, there has been an effort in the literature to generalize results about curve graphs and complexes from the finite-type case to the infinite-type case. In \\cite{bavard}, the authors show that the automorphism group of the curve complex, $ \\mathcal{C}(S)$, of an infinite-type surface with empty boundary is geometric, that is, every automorphism of $ \\mathcal{C}(S)$ is induced by the action of $\\text{Map}^*(S)$, generalizing the result of \\cites{ivanov, luo1999, korkmaz} for finite-type surfaces. Independently, \\cite{hernandez} also proved that the automorphism groups of the curve complex, non-separating curve complex, and the Schmutz graph are all isomorphic to each other and to $\\text{Map}^*(S)$ for surfaces of infinite-type with no planar ends and finitely many boundary components. Moreover, \\cite{hernandez, bavard} show that these complexes can differentiate infinite-type surfaces. In \\cite{branman}, the author generalizes the result that automorphism group of the pants complex is isomorphic to the extended mapping class group, a result originally proved in \\cite{margalit04} for finite-type surfaces. \\\\\n\nIn this paper, we generalize the Hatcher--Thurston complex to the setting of infinite-type surfaces with possible compact boundary and non-planar ends. The Hatcher--Thurston complex or Cut-system complex, $ \\mathcal{HT}(S)$, is a 2-dimensional CW-complex associated to surfaces of finite-type in which the vertices are maximal cut-systems of the surface, edges correspond to an elementary move between cut-systems, and 2-cells are attached to triangles, rectangles, and pentagons as described in Section \\ref{pre}. This cut-system complex was first introduced in \\cite{hatcher} to obtain a presentation for the mapping class group of finite-type surfaces. In \\cite{hatcher}, the authors use a Morse-theoretic proof to show that the complex is connected and simply connected. Later, Wajnryb gave a combinatorial proof of this in \\cite{wajnryb}. In \\cite{irmak2004}, the automorphisms of the Hatcher--Thurston complex were shown to be geometric. In this paper, the complex $ \\mathcal{HT}(S)$ is generalized to a countable collection of complexes $\\left(\\Gamma_k(S)\\right)_{k\\in \\mathbb{N}}$ whose vertices are given by cut-systems with exactly $k$ non-separating curves, as opposed to maximal cut-systems, with $1-$cells and $2-$cells defined in a similar fashion to $ \\mathcal{HT}(S)$. In case that $S$ is a finite-type surface of genus $g$, the collection $\\{\\Gamma_k(S)\\}$ is finite with $1\\leq k \\leq g$ and $\\Gamma_g(S) = \\mathcal{HT}(S)$. Precise definitions of these complexes are given in Section \\ref{pre}. Since the construction of $1$-cells and $2-$cells of the pants complex in \\cite{branman} is similar to that of the Hatcher-Thurston complex, generalization of the pants complex to infinite-type surfaces leads to similar issues with connectedness as the generalization of the Hatcher-Thurston complex. In \\cite{branman} these issues are resolved by introducing a family of new topologies on the vertex space and extending it to the enitre pants space. As will be discussed in Section \\ref{pre}, we resolve this issue, among others, by instead defining the sequence $(\\Gamma_k(S))_{k\\in \\mathbb{N}}$ of complexes discussed above. In Section \\ref{con}, we show the following result.", "context": "\\label{intro}\n\\addcontentsline{toc}{section}{Introduction}\nAll surfaces in this paper will be connected, orientable, without any punctures or marked points, and possibly with non-empty compact boundary. The mapping class group, $\\text{Map}(S)$, is the group of all orientation preserving homeomorphisms of the surface $S$ up to isotopy. The extended mapping class group, $\\text{Map}^*(S)$ is the group of all homeomorphisms of $S$ up to isotopy. When the fundamental group $\\pi_1(S)$ of the surface $S$ is finitely generated, we say that $S$ is of finite-type; otherwise $S$ is infinite-type.\\\\\n\nRecently, there has been an effort in the literature to generalize results about curve graphs and complexes from the finite-type case to the infinite-type case. In \\cite{bavard}, the authors show that the automorphism group of the curve complex, $ \\mathcal{C}(S)$, of an infinite-type surface with empty boundary is geometric, that is, every automorphism of $ \\mathcal{C}(S)$ is induced by the action of $\\text{Map}^*(S)$, generalizing the result of \\cites{ivanov, luo1999, korkmaz} for finite-type surfaces. Independently, \\cite{hernandez} also proved that the automorphism groups of the curve complex, non-separating curve complex, and the Schmutz graph are all isomorphic to each other and to $\\text{Map}^*(S)$ for surfaces of infinite-type with no planar ends and finitely many boundary components. Moreover, \\cite{hernandez, bavard} show that these complexes can differentiate infinite-type surfaces. In \\cite{branman}, the author generalizes the result that automorphism group of the pants complex is isomorphic to the extended mapping class group, a result originally proved in \\cite{margalit04} for finite-type surfaces. \\\\\n\nIn this paper, we generalize the Hatcher--Thurston complex to the setting of infinite-type surfaces with possible compact boundary and non-planar ends. The Hatcher--Thurston complex or Cut-system complex, $ \\mathcal{HT}(S)$, is a 2-dimensional CW-complex associated to surfaces of finite-type in which the vertices are maximal cut-systems of the surface, edges correspond to an elementary move between cut-systems, and 2-cells are attached to triangles, rectangles, and pentagons as described in Section \\ref{pre}. This cut-system complex was first introduced in \\cite{hatcher} to obtain a presentation for the mapping class group of finite-type surfaces. In \\cite{hatcher}, the authors use a Morse-theoretic proof to show that the complex is connected and simply connected. Later, Wajnryb gave a combinatorial proof of this in \\cite{wajnryb}. In \\cite{irmak2004}, the automorphisms of the Hatcher--Thurston complex were shown to be geometric. In this paper, the complex $ \\mathcal{HT}(S)$ is generalized to a countable collection of complexes $\\left(\\Gamma_k(S)\\right)_{k\\in \\mathbb{N}}$ whose vertices are given by cut-systems with exactly $k$ non-separating curves, as opposed to maximal cut-systems, with $1-$cells and $2-$cells defined in a similar fashion to $ \\mathcal{HT}(S)$. In case that $S$ is a finite-type surface of genus $g$, the collection $\\{\\Gamma_k(S)\\}$ is finite with $1\\leq k \\leq g$ and $\\Gamma_g(S) = \\mathcal{HT}(S)$. Precise definitions of these complexes are given in Section \\ref{pre}. Since the construction of $1$-cells and $2-$cells of the pants complex in \\cite{branman} is similar to that of the Hatcher-Thurston complex, generalization of the pants complex to infinite-type surfaces leads to similar issues with connectedness as the generalization of the Hatcher-Thurston complex. In \\cite{branman} these issues are resolved by introducing a family of new topologies on the vertex space and extending it to the enitre pants space. As will be discussed in Section \\ref{pre}, we resolve this issue, among others, by instead defining the sequence $(\\Gamma_k(S))_{k\\in \\mathbb{N}}$ of complexes discussed above. In Section \\ref{con}, we show the following result.", "full_context": "\\label{intro}\n\\addcontentsline{toc}{section}{Introduction}\nAll surfaces in this paper will be connected, orientable, without any punctures or marked points, and possibly with non-empty compact boundary. The mapping class group, $\\text{Map}(S)$, is the group of all orientation preserving homeomorphisms of the surface $S$ up to isotopy. The extended mapping class group, $\\text{Map}^*(S)$ is the group of all homeomorphisms of $S$ up to isotopy. When the fundamental group $\\pi_1(S)$ of the surface $S$ is finitely generated, we say that $S$ is of finite-type; otherwise $S$ is infinite-type.\\\\\n\nRecently, there has been an effort in the literature to generalize results about curve graphs and complexes from the finite-type case to the infinite-type case. In \\cite{bavard}, the authors show that the automorphism group of the curve complex, $ \\mathcal{C}(S)$, of an infinite-type surface with empty boundary is geometric, that is, every automorphism of $ \\mathcal{C}(S)$ is induced by the action of $\\text{Map}^*(S)$, generalizing the result of \\cites{ivanov, luo1999, korkmaz} for finite-type surfaces. Independently, \\cite{hernandez} also proved that the automorphism groups of the curve complex, non-separating curve complex, and the Schmutz graph are all isomorphic to each other and to $\\text{Map}^*(S)$ for surfaces of infinite-type with no planar ends and finitely many boundary components. Moreover, \\cite{hernandez, bavard} show that these complexes can differentiate infinite-type surfaces. In \\cite{branman}, the author generalizes the result that automorphism group of the pants complex is isomorphic to the extended mapping class group, a result originally proved in \\cite{margalit04} for finite-type surfaces. \\\\\n\nIn this paper, we generalize the Hatcher--Thurston complex to the setting of infinite-type surfaces with possible compact boundary and non-planar ends. The Hatcher--Thurston complex or Cut-system complex, $ \\mathcal{HT}(S)$, is a 2-dimensional CW-complex associated to surfaces of finite-type in which the vertices are maximal cut-systems of the surface, edges correspond to an elementary move between cut-systems, and 2-cells are attached to triangles, rectangles, and pentagons as described in Section \\ref{pre}. This cut-system complex was first introduced in \\cite{hatcher} to obtain a presentation for the mapping class group of finite-type surfaces. In \\cite{hatcher}, the authors use a Morse-theoretic proof to show that the complex is connected and simply connected. Later, Wajnryb gave a combinatorial proof of this in \\cite{wajnryb}. In \\cite{irmak2004}, the automorphisms of the Hatcher--Thurston complex were shown to be geometric. In this paper, the complex $ \\mathcal{HT}(S)$ is generalized to a countable collection of complexes $\\left(\\Gamma_k(S)\\right)_{k\\in \\mathbb{N}}$ whose vertices are given by cut-systems with exactly $k$ non-separating curves, as opposed to maximal cut-systems, with $1-$cells and $2-$cells defined in a similar fashion to $ \\mathcal{HT}(S)$. In case that $S$ is a finite-type surface of genus $g$, the collection $\\{\\Gamma_k(S)\\}$ is finite with $1\\leq k \\leq g$ and $\\Gamma_g(S) = \\mathcal{HT}(S)$. Precise definitions of these complexes are given in Section \\ref{pre}. Since the construction of $1$-cells and $2-$cells of the pants complex in \\cite{branman} is similar to that of the Hatcher-Thurston complex, generalization of the pants complex to infinite-type surfaces leads to similar issues with connectedness as the generalization of the Hatcher-Thurston complex. In \\cite{branman} these issues are resolved by introducing a family of new topologies on the vertex space and extending it to the enitre pants space. As will be discussed in Section \\ref{pre}, we resolve this issue, among others, by instead defining the sequence $(\\Gamma_k(S))_{k\\in \\mathbb{N}}$ of complexes discussed above. In Section \\ref{con}, we show the following result.\n\n\\textbf{Remark.} At multiple points throughout the proof of Theorem \\ref{thm:simply} the existence of at least one non-planar end is invoked, leading one to ask whether this is a necessary condition. In other words, is $\\Gamma_k(S)$ simply connected for all $1\\leq k \\leq g$ when $S$ is a genus $g$ surface? Since $\\Gamma_g(S) = \\mathcal{HT}(S)$ for finite-genus surfaces, as mentioned earlier, it follows that $\\Gamma_g(S)$ is simply-connected. But the proof of this, in \\cite{wajnryb, hatcher}, makes use of the maximality of the cut-systems with $g$ curves in a genus $g$ surface. To the author's knowledge, this question does not seem to be answered in the literature.\\\\\n\nNow we describe the construction of the curves $\\b_i$. Let $R$ be a subsurface of $S$ with compact boundary and at least one non-planar end so that all the curves $\\a_0,\\cdots, \\a_{n-1}$ are contained in $R$. Let $\\delta$ be a separating curve in $R$ disjoint from all $\\a_i$ so that the two surfaces, $R_1$ and $R_2$, obtained by cutting $R$ along $\\delta$ satisfy the following: $R_1$ is compact and contains all $\\a_i$ and $R_2$ is of infinite-type with a single compact boundary component corresponding to the curve $\\delta$ in $R$ and has non-planar ends. Let $\\tau_0$ and $\\tau_1$ be homotopically distinct arcs in $R_2$ with their end points lying on the boundary of $R_2$ so that $\\tau_0\\cap \\tau_1 = \\emptyset$. Since $\\delta$ and $\\a_0$ are disjoint and $i(a_0,a_1)=1$, it can be shown that there exist disjoint essential arcs, $\\sigma_0,\\ \\sigma_1$, with end points on the boundary of $R_1$ corresponding to $\\delta$ and $|\\sigma_0\\cap \\a_0| = 1$ and $|\\sigma_1\\cap \\a_1|=1$. Gluing $R_1$ and $R_2$ together along the boundary corresponding to $\\delta$ in such a way that $\\beta_0 = \\sigma_0\\cup \\tau_0$ and $\\beta_1 = \\sigma_1\\cup \\tau_1$ are closed curves. Then by construction we have $i(b_0,a_0) = i(b_1,a_1) = 1$ and that $b_i$ are non-separating. Let $\\b_2$ be a non-separating curve in $R_2$ which intersects $\\tau_0$ and $\\tau_1$ exactly once. Thus we have our desired curves $b_0,\\ b_1,$ and $b_2$.\n\\end{proof}\nThe above proposition implies that $\\Gamma_1(S)$ is simply-connected when $S$ has non-planar ends. We use this as our base case for induction on $k$, to show that $\\Gamma_k(S)$ is also simply-connected. From here on we assume the induction hypothesis: for every surface $S$ the complex $\\Gamma_j(S)$ is simply connected for all $j<k$. We prove the simple connectedness now by also inducting on radius of the path. We start by a special case of radius 0 curves.\n\\begin{proposition}\\label{prop:sp-radius-0}\n Suppose that $p = v_0 \\lr \\cdots \\lr v_{n-1}$ is a closed path in $\\Gamma_k(S)$ so that all the vertices $v_i$ contain a common curve $c$. Then $p$ is null-homotopic.\n\\end{proposition}\n\\begin{proof}\n Let $p'$ be the path in $\\Gamma_{k-1}(S)$ where the vertices are $v_i- \\{c\\}$. By the induction hypothesis, it follows that $p'$ is null-homotopic, meaning that it can be split into paths of type 1, 2, and 3. Since adding $c$ back to each vertex in $p'$ does not remove any edge connections therefore $p$ is null-homotopic.\n\\end{proof}\n\\begin{proposition}[Lemma 13, \\cite{wajnryb}]\\label{prop:hex-path}\n Let $a_0, a_1, a_2$ be disjoint non-separating curves so that:\n \\begin{enumerate}\n \\item all unions $\\a_i\\cup \\a_j$ are non-separating in $S$.\n \\item the union $\\a_0\\cup \\a_1\\cup \\a_2$ separates $S$.\n \\end{enumerate}\n Then there exists disjoint non-separating curves $b_0, b_1, b_2$ so that $i(a_i,b_i) = 1$, $i(a_{i+1}, b_{i}) = 1$ (indices modulo $3$), and $i(a_i, b_{i+1}) = 0$ for all other $i,j$. Moreover, if $(a_0,a_1,c_2,\\cdots, c_{k-1})$ is a vertex in $\\Gamma_k(S)$ then the closed path $p = v_0 \\lr \\cdots \\lr v_5 \\lr v_0$ defined as\n \\begin{align*}\n (a_0, a_1, c_2,\\cdots, c_{k-1})\\lr &(a_0, b_1,c_2,\\cdots, c_{k-1}) \\lr (a_0, a_2,c_2,\\cdots, c_{k-1}) \\lr (b_0, a_2,c_2,\\cdots, c_{k-1})\\\\ \n &\\ \\lr (a_1,a_2,c_2,\\cdots, c_{k-1})\\lr (a_1, b_2,c_2,\\cdots, c_{k-1})\\lr (a_0, a_1,c_2,\\cdots, c_{k-1})\n \\end{align*}\n is null-homotopic in $\\Gamma_k(S)$.\n\\end{proposition}\n\\begin{proof}\n Due to conditions on $a_0,\\ a_1,\\ a_2$ it follows that $S - \\bigcup_{i}\\a_i$ has exactly two connected components, $R_1$ and $R_2$, both with at least three boundary components corresponding to the curves $\\a_0,\\ \\a_1,$ and $\\a_2$. Let $\\delta_i$ be disjoint arcs in $R_1$ with endpoints on the boundary components corresponding to $\\a_i$ and $\\a_{i+1}$ (indices taken modulo 3). Similarly define disjoint arcs $\\delta'_i$ on $R_2$. Defining $\\b_i = \\delta_i\\cup \\delta'_i$ we get disjoint non-separating curves $b_0, b_1, b_2$ with the desired intersections with $a_i$. In order to prove the second part of the proposition, define $c = T_{a_1}(b_0)$. Then it follows from Proposition \\ref{prop:dehn-int} that $i(c, b_0) = i(c, a_1) = i(c,b_1) = i(c,a_0) = 1$ and $i(c,b_2) = i(c, a_2) = 0$. If $(a_0, a_1, c_2,\\cdots, c_{k-1})$ is a vertex in $\\Gamma_k(S)$ then define the vertices $w_0 = (b_1,b_2, c_2,\\cdots, c_{k-1}), w_1 = (c,b_2,c_2,\\cdots, c_{k-1}), w_2 = (c,a_2,c_2,\\cdots, c_{k-1})$. Then the path $p = v_0 \\lr \\cdots \\lr v_5 \\lr v_0$ as defined above can be split into paths of type 1, 2, and 3 as in Figure \\ref{fig:hex-path}. \n \\begin{figure}\n \\[\\begin{tikzcd}\n {v_0} && {v_1} && {v_2} && {v_3} \\\\\n \\\\\n && {w_0} && {w_2} \\\\\n &&& {w_1} \\\\\n {v_5} &&&&&& {v_4}\n \\arrow[tail reversed, from=1-1, to=1-3]\n \\arrow[tail reversed, from=1-3, to=1-5]\n \\arrow[tail reversed, from=1-5, to=1-7]\n \\arrow[tail reversed, from=1-7, to=5-7]\n \\arrow[tail reversed, from=3-3, to=1-3]\n \\arrow[tail reversed, from=3-3, to=4-4]\n \\arrow[tail reversed, from=3-3, to=5-1]\n \\arrow[tail reversed, from=3-5, to=1-5]\n \\arrow[tail reversed, from=3-5, to=1-7]\n \\arrow[tail reversed, from=3-5, to=5-7]\n \\arrow[tail reversed, from=4-4, to=3-5]\n \\arrow[tail reversed, from=4-4, to=5-1]\n \\arrow[tail reversed, from=5-1, to=1-1]\n \\arrow[tail reversed, from=5-7, to=5-1]\n\\end{tikzcd}\\]\n \\caption{Splitting the path $p$ into paths of type $1$, $2$, and $3$.}\n \\label{fig:hex-path}\n \\end{figure}\n\\end{proof}\nBefore beginning the proof of the next lemma, note that if $v$ and $w$ are vertices in $\\Gamma_k(S)$ with common curves $(a_0, \\cdots, a_m)$ then there is an $(a_0,\\cdots, a_m)-$segment connecting $v$ and $w$. This follows directly from the proof of connectedness. This fact will be repeatedly used in the proof of the next lemma.\n\\begin{lemma}\\label{lem:radius-zero}\n Let $p$ be a path in $\\Gamma_k(S)$ of radius $0$ with respect to some curve $a_0$. Then $p$ is null-homotopic.\n\\end{lemma}\n\\begin{proof}\n We prove this by induction on the number of segments in the path $p$. Suppose that $a_0\\in v_0$, where $v_0$ is a vertex in $p$. We call the maximal $a_0-$segment containing $v_0$ the first segment. From Proposition \\ref{prop:sp-radius-0} it follows that if $p$ is a $a_0-$segment then $p$ is null homotopic. Suppose this is not the case and $v_1$ is the terminal vertex of the first segment. Let $a_1\\in v_1$ be a curve distinct from $a_0$ which will serve as the common curve of the next segment, which we call the second segment. Let $v_2$ be the terminal element of second segment. If $p$ is just made up of these two segments then both $v_1$ and $v_2$ contain $a_0$ and $a_1$, thus there is a $(a_0, a_1)-$segment from $v_1$ to $v_2$, therefore implying that $p$ can be split into a $a_0-$segment and a $a_1-$segment, both of which are null-homotopic due to Proposition \\ref{prop:sp-radius-0}. Suppose now that $p$ has more than $2$ segments. Then let $a_2$ be a curve in the vertex $v$ which follows $v_2$, so that $i(a_0, a_2)=0$. Let $v_3$ be the terminal vertex in the $a_2-$segment. Now we try to reduce the number of segments.\\\\", "post_theorem_intro_text_len": 4533, "post_theorem_intro_text": "The proof of this result closely follows the combinatorial proof of \\cite{wajnryb} with alterations made when required for the infinite-type case. The proof proceeds by induction on $k$. We first show that the $1-$skeleton, $\\Gamma^1_k(S)$, is connected and that its diameter is bounded between $k$ and $8k-4$ using induction on $k$. To further prove that $\\Gamma_k(S)$ is simply connected, we introduce the notions of radius of a path in $\\Gamma_k(S)$ and segments, following the proof by Wajnryb. The remaining proof can then be divided into three steps. It is first shown in Proposition \\ref{prop:base-case} that $\\Gamma_1(S)$ is simply-connected by first showing that closed paths with radius $1$ are contractible. Note that this argument uses the assumption that $S$ has at least one non-planar end. Continuing the induction on $k$, the second step shows that any closed 0-radius path is contractible in $\\Gamma_k(S)$, see Proposition \\ref{lem:radius-zero}, by induction on the number of segments in the path. Finally, again using the result about radius $0$ paths and the assumption of a non-planar end the theorem statement follows.\\\\\n\n\\textbf{Remark.} At multiple points throughout the proof of Theorem \\ref{thm:simply} the existence of at least one non-planar end is invoked, leading one to ask whether this is a necessary condition. In other words, is $\\Gamma_k(S)$ simply connected for all $1\\leq k \\leq g$ when $S$ is a genus $g$ surface? Since $\\Gamma_g(S) = \\mathcal{HT}(S)$ for finite-genus surfaces, as mentioned earlier, it follows that $\\Gamma_g(S)$ is simply-connected. But the proof of this, in \\cite{wajnryb, hatcher}, makes use of the maximality of the cut-systems with $g$ curves in a genus $g$ surface. To the author's knowledge, this question does not seem to be answered in the literature.\\\\\n\nIn Section \\ref{aut}, we show that automorphisms of $\\Gamma_k(S)$ are geometric:\n\\begin{ntheorem}\\label{thm:geometric}\n  Let $S$ be a surface of infinite-type with no planar ends and empty boundary. Then $\\text{Aut}(\\Gamma_k(S))$ is isomorphic to $\\text{Map}^*(S)$ for each $k\\in \\mathbb{N}$.\n\\end{ntheorem}\nWe prove this by constructing an action of $\\text{Aut}(\\Gamma_k(S))$ on the Schmutz graph, $ \\mathcal{G}(S)$, as done by \\cite{irmak2004}. The proof in \\cite{irmak2004} is almost entirely transferable to the infinite-type case, but parts of it are reproduced here for the sake of completeness. As a direct corollary to Theorem \\ref{thm:geometric} we have the following:\n\\begin{ncorollary}\\label{cor:iso}\n  Let $S,S'$ be surfaces of infinite-type with no planar ends and empty boundary. Let $\\varphi:\\Gamma_k(S)\\to \\Gamma_k(S')$ be an isomorphism. Then $S$ is homeomorphic to $S'$.\n\\end{ncorollary}\nThe Hatcher--Thurston complex is an important tool in the study of mapping class group of finite-type surfaces. By studying the stabilizers of vertices, edges, and cells in $ \\mathcal{HT}(S)$ of the action of $\\text{Map}(S)$ Wajnryb gave a simple presentation of $\\text{Map}(S)$ in \\cite{wajnryb2}, which was inspired by a similar approach to the problem by Harer in \\cite{harer}. Harer in the same paper uses a simpler version of the Hatcher--Thurston complex to compute the second cohomology of the mapping class group. These methods are not directly transferable to big mapping class group as they are not finitely generated, in fact they are not even countably generated. In \\cite{patel}, the authors give topological generators for the pure mapping class group $\\text{PMap}(S)$, the subgroup of $\\text{Map}(S)$ which are end-preserving, in terms of Dehn Twists and handle shift maps. For infinite-type surfaces with countably many non-planar ends it may be possible to use the results presented in this paper to determine relations in $\\text{PMap}(S)$ between Dehn Twists and handle shift maps by applying the techniques of Wajnryb and Harer.\n\\vspace{1em}\n\\newline\n\\textbf{Outline.} In Section \\ref{pre} the main objects of this paper are defined and conventions and notations which are used throughout the paper are discussed. Section \\ref{con} is dedicated to the proof of Theorem \\ref{thm:simply} and Section \\ref{aut} to the proof of Theorem \\ref{thm:geometric} and Corollary \\ref{cor:iso}.\n\\vspace{1em}\n\\newline\n\\textbf{Acknowledgements.} The author would like to thank Sumanta Das for posing the question which led to this paper and his comments for improving the quality of the paper. The author also extends gratitude to Dan Margalit and Tyrone Ghaswala, whose comments helped improve the paper.", "sketch": "The proof of Theorem~\\ref{thm:simply} “closely follows the combinatorial proof of \\cite{wajnryb} with alterations made when required for the infinite-type case” and “proceeds by induction on $k$.” First, one shows “that the $1$-skeleton, $\\Gamma^1_k(S)$, is connected” and that “its diameter is bounded between $k$ and $8k-4$ using induction on $k$.” For simple connectedness, the proof “introduce[s] the notions of radius of a path in $\\Gamma_k(S)$ and segments, following the proof by Wajnryb,” and then splits into three steps: (1) in Proposition~\\ref{prop:base-case}, prove “$\\Gamma_1(S)$ is simply-connected by first showing that closed paths with radius $1$ are contractible,” using the assumption that “$S$ has at least one non-planar end”; (2) for general $k$, show “that any closed $0$-radius path is contractible in $\\Gamma_k(S)$, see Proposition~\\ref{lem:radius-zero}, by induction on the number of segments in the path”; (3) “finally, again using the result about radius $0$ paths and the assumption of a non-planar end the theorem statement follows.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:simply} \n  Let $S$ be a surface of infinite-type with non-planar ends. Then $\\Gamma_k(S)$ is connected and simply connected for each $k\\in \\mathbb{N}$.", "theorem_type": ["Universal"], "mcq": {"question": "Let $S$ be a connected, orientable surface with no punctures or marked points and possibly nonempty compact boundary. Assume that $S$ is of infinite type, meaning that $\\pi_1(S)$ is not finitely generated, and that $S$ has non-planar ends. For each $k\\in\\mathbb{N}$, let $\\Gamma_k(S)$ denote the generalized Hatcher--Thurston $2$-dimensional CW-complex whose vertices are cut-systems consisting of exactly $k$ non-separating curves on $S$, with $1$-cells and $2$-cells defined analogously to the Hatcher--Thurston complex. Which statement holds for every such surface $S$ and every $k\\in\\mathbb{N}$?", "correct_choice": {"label": "A", "text": "For every $k\\in\\mathbb{N}$, the complex $\\Gamma_k(S)$ is connected and simply connected."}, "choices": [{"label": "B", "text": "For every $k\\in\\mathbb{N}$, the complex $\\Gamma_k(S)$ is connected, and it is simply connected whenever $k$ is sufficiently large."}, {"label": "C", "text": "For every $k\\in\\mathbb{N}$, the complex $\\Gamma_k(S)$ is connected."}, {"label": "D", "text": "There exists $N\\in\\mathbb{N}$ such that for every $k\\ge N$, the complex $\\Gamma_k(S)$ is connected and simply connected."}, {"label": "E", "text": "For every $k\\in\\mathbb{N}$, the complex $\\Gamma_k(S)$ is path connected and contractible."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "all-k simple-connectedness", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped simple-connectedness", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "quantifier order in k", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "simple-connectedness strengthened to contractibility", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct choice or embed the conclusion in the wording. It only states the hypotheses and asks for the valid conclusion."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the correct answer appears to restate the main result under the given hypotheses, with distractors formed by weakening or strengthening that statement."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact theorem statement from nearby variants such as mere connectedness, eventual simple connectivity, or contractibility. However, the item mainly tests recognition of the precise theorem rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: one is a weaker true statement, others alter quantifiers or overstrengthen the conclusion. These reflect common failure modes in theorem selection."}, "total_score": 5, "overall_assessment": "A solid theorem-identification MCQ with strong distractors and no answer leakage, but it is largely a direct restatement of a result and only moderately tests genuine reasoning."}}
{"id": "2602.22504v1", "paper_link": "http://arxiv.org/abs/2602.22504v1", "theorems_cnt": 4, "theorem": {"env_name": "thm", "content": "\\label{main}\nLet $H$ be a split classical group over $F$, \nand let $\\phi \\in \\Phi_\\mathrm{temp}(H(F))$. \nSuppose that one of the following holds: \n\\begin{description}\n\\item[(B)]\n$H = \\SO_{2n+1}$, $p > 6n+3$ and $\\phi$ is arbitrary; \n\\item[(C)]\n$H = \\Sp_{2n}$, $p > 6n$ and $\\phi$ satisfies $(\\star)$; \n\\item[(D)]\n$H = \\SO_{2n}$, $p > 6n$ and $\\phi$ satisfies $(\\star)$. \n\\end{description}\nSet $\\psi = \\widehat\\phi$. \n\n\\begin{enumerate}\n\\item\nThere exists $\\pi \\in \\Pi_\\psi$ such that \n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi) \\in \\bar\\mathcal{N}(\\pi)^{\\max}.\n\\]\n\\item\nFor $\\pi \\in \\Pi_\\psi$ and for $\\oo_H^\\mathrm{st} \\in \\bar\\mathcal{N}(\\pi)$, \nif $\\oo_H^\\mathrm{st} \\not= d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)$, then \n\\[\n\\dim(\\oo_H^\\mathrm{st}) < \\dim(d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)).\n\\]\nHere, $\\dim(\\oo_H^\\mathrm{st})$ is the Zariski dimension of the orbit as an algebraic variety.\n\\item\nIf we further assume Hypothesis \\ref{hypo} below, then Conjecture \\ref{upper} is true for $\\phi$, i.e., \n\\[\n\\bar\\mathcal{N}(\\pi)^{\\max} \\leq d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\n\\]\nfor any $\\pi \\in \\Pi_\\psi$.\n\\end{enumerate}", "start_pos": 10589, "end_pos": 11637, "label": "main"}, "ref_dict": {"upper": "\\begin{conj}\\label{upper}\nLet $\\phi \\in \\Phi(H(F))$. \nThen for any $\\pi \\in \\Pi_\\phi$, we have\n\\[\n\\bar\\NN(\\hat\\pi)^{\\max} \\leq d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi).\n\\]\nMoreover, if $\\phi$ is tempered, \nthen the equality holds for some $\\pi \\in \\Pi_\\phi$. \n\\end{conj}", "twist": "\\begin{thm}\\label{twist}\nLet $\\phi \\colon W_F \\times \\SL_2(\\C) \\rightarrow G^\\vee$ \nbe a self-dual tempered $L$-parameter for $G(F) = \\GL_m(F)$, \nand let $\\pi_\\psi$ be the Zelevinsky dual of the irreducible tempered representation \n$\\pi_\\phi$ corresponding to $\\phi$. \nThen \n\\[\n\\bar\\NN_\\theta(\\tl\\pi_\\psi)^{\\max} = \\{d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi)\\}.\n\\]\nMoreover, $c_{\\oo_G}(\\tl\\pi_\\psi) = c_{\\oo_G'}(\\tl\\pi_\\psi)$ \nfor any rational orbits $\\oo_G$ and $\\oo_G'$ contained in $d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi)$.\n\\end{thm}", "transfer": "\\begin{thm}\\label{transfer}\nLet $\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$ be a special nilpotent orbit.\nDefine $\\oo_G^\\st \\in \\NN(\\g^\\theta(\\overline{F}))$ by \n\\[\n\\oo_G^\\st = d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st)). \n\\]\nThen for each rational orbit $\\oo_H \\subset \\oo_H^\\st$, \nthere exists a constant $\\gamma_{\\oo_H} \\in \\C$ such that \n\\[\n\\sum_{\\oo_G \\subset \\oo_G^\\st} \\hat\\mu_{\\oo_G}\n\\quad\\text{is equal to the transfer of }\\quad\n\\sum_{\\oo_H \\subset \\oo_H^\\st} \\gamma_{\\oo_H} \\hat\\mu_{\\oo_H}\n\\]\non a small enough neighborhood of $\\theta \\in G(F) \\rtimes \\theta$.\n\\end{thm}", "main": "\\begin{thm}\\label{main}\nLet $H$ be a split classical group over $F$, \nand let $\\phi \\in \\Phi_\\temp(H(F))$. \nSuppose that one of the following holds: \n\\begin{description}\n\\item[(B)]\n$H = \\SO_{2n+1}$, $p > 6n+3$ and $\\phi$ is arbitrary; \n\\item[(C)]\n$H = \\Sp_{2n}$, $p > 6n$ and $\\phi$ satisfies $(\\star)$; \n\\item[(D)]\n$H = \\SO_{2n}$, $p > 6n$ and $\\phi$ satisfies $(\\star)$. \n\\end{description}\nSet $\\psi = \\widehat\\phi$. \n\n\\begin{enumerate}\n\\item\nThere exists $\\pi \\in \\Pi_\\psi$ such that \n\\[\nd_{H^\\vee}(\\Ad(H^\\vee)N_\\phi) \\in \\bar\\NN(\\pi)^{\\max}.\n\\]\n\\item\nFor $\\pi \\in \\Pi_\\psi$ and for $\\oo_H^\\st \\in \\bar\\NN(\\pi)$, \nif $\\oo_H^\\st \\not= d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi)$, then \n\\[\n\\dim(\\oo_H^\\st) < \\dim(d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi)).\n\\]\nHere, $\\dim(\\oo_H^\\st)$ is the Zariski dimension of the orbit as an algebraic variety.\n\\item\nIf we further assume Hypothesis \\ref{hypo} below, then Conjecture \\ref{upper} is true for $\\phi$, i.e., \n\\[\n\\bar\\NN(\\pi)^{\\max} \\leq d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi)\n\\]\nfor any $\\pi \\in \\Pi_\\psi$.\n\\end{enumerate}\n\\end{thm}", "WS": "\\begin{prop}\\label{WS}\nFor $i \\in \\{1,2\\}$, let $\\oo_{H_i}^\\st \\in \\NN(\\h_i(\\overline{F}))$ be a special nilpotent orbit.\nThen \n\\[\nW(\\oo_{H_1}^\\st, \\oo_{H_2}^\\st) \n\\leq d_{H^\\vee}(\\Ad(H^\\vee)\\xi(d_{H_1}(\\oo_{H_1}^\\st), d_{H_2}(\\oo_{H_2}^\\st))). \n\\]\n\\end{prop}", "hypo": "\\begin{hyp}\\label{hypo}\nLet $\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$ be a special nilpotent orbit.\nDefine $\\oo_G^\\st \\in \\NN(\\g^\\theta(\\overline{F}))$ by \n\\[\n\\oo_G^\\st = d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st)). \n\\]\nThen for any constants $\\{c_{\\oo_H} \\;|\\; \\oo_H \\subset \\oo_H^\\st\\}$ such that \n$\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H}$ is stable, \nthere exist constants $\\{a_{\\oo_G} \\;|\\; \\oo_G \\subset \\oo_G^\\st\\}$ such that \n\\[\n\\sum_{\\oo_G \\subset \\oo_G^\\st} a_{\\oo_G} \\hat\\mu_{\\oo_G}\n\\quad\\text{is equal to the transfer of}\\quad\n\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H} \n\\]\non a small enough neighborhood of $\\theta \\in G(F) \\rtimes \\theta$.\n\\end{hyp}"}, "pre_theorem_intro_text_len": 8707, "pre_theorem_intro_text": "This work is motivated by the elusive relation \nbetween the irreducible admissible Harish-Chandra characters of a reductive $p$-adic group \nand the Langlands-Arthur parameters. \nArguably the best-known instance of this relation is the Hiraga--Ichino--Ikeda formal degree formula \nfor square-integrable characters in terms of adjoint $\\gamma$-factors \\cite{HII}. \nThe formal degree concerns the coefficient of the distribution attached to the zero nilpotent orbit \nin the Harish-Chandra--Howe local character expansion. \nAt the opposite end, \nthe largest orbits that contribute to the character expansion determine the wavefront set of the character distribution. \nThey are the focus of the present paper. \nUnder a technical assumption, which is unnecessary for $\\SO_{2n+1}$, and for $p$ large (see Theorem \\ref{main} for the precise conditions), we prove that  \nthe largest-dimensional orbits in the union of the geometric wavefront sets of the representations \nin the Aubert--Zelevinsky dual of a tempered $L$-packet for a split classical $p$-adic group \n{\\it equal} the Spaltenstein dual of the corresponding tempered nilpotent Langlands parameter. \n\\par\n\nA relation of this sort, Conjecture \\ref{upper},  was proposed in \\cite{CK,HLLS}, \nmotivated by the results of \\cite{CMBO1,CMBO2} for unipotent representations of split (and `inner-to-split') $p$-adic groups. \nThe fact that such a link should exist had already been apparent from \nthe foundational work of Adams--Barbasch--Vogan on the definition and construction of microlocal Arthur packets \\cite{ABV}. \nIn Theorem \\ref{main}, we also show that the full Conjecture \\ref{upper} follows by further assuming Hypothesis \\ref{hypo}.\n\\vskip 10pt\n\nWe fix a non-archimedean local field $F$ of characteristic zero and of residue characteristic $p \\gg 0$.\nLet $\\overline{F}$ be a fixed algebraic closure of $F$. \nIn this section, we state our main theorem (Theorem \\ref{main})\nand explain our idea for the proof. \n\n\\subsection{Wavefront set}\nLet $H$ be a connected reductive group over $F$ with the Lie algebra $\\mathfrak{h}$.\nWe denote by $\\mathcal{N}(\\mathfrak{h}(F))$ (resp.~ $\\mathcal{N}(\\mathfrak{h}(\\overline{F}))$) \nthe set of nilpotent $\\mathrm{Ad}(H(F))$-orbits (resp.~ $\\mathrm{Ad}(H(\\overline{F}))$-orbits) \nin $\\mathfrak{h}(F)$ (resp.~ $\\mathfrak{h}(\\overline{F})$).\nFor an irreducible admissible representation $\\pi$ of $H(F)$, \nwe denote its distribution character by $\\Theta_\\pi$. \nThe Harish-Chandra--Howe local character expansion (\\cite[Theorem 16.2]{HC}) states that \n\\[\n\\Theta_\\pi(\\exp(X)) = \\sum_{\\oo_H \\in \\mathcal{N}(\\mathfrak{h}(F))} c_{\\oo_H}(\\pi) \\hat\\mu_{\\oo_H}(\\exp(X))\n\\]\nfor $X \\in \\mathfrak{h}(F)$ sufficiently near to $0$, \nwhere $\\hat\\mu_{\\oo_H}$ is the Fourier transform of the orbital integral, \nand $c_{\\oo_H}(\\pi)$ is a constant. \nSet $\\mathcal{N}(\\pi) = \\{ \\oo_H \\in \\mathcal{N}(\\mathfrak{h}(F)) \\;|\\; c_{\\oo_H}(\\pi)\\not= 0\\}$ \nand let $\\bar\\mathcal{N}(\\pi)$ be the set of $\\oo_H^\\mathrm{st} \\in \\mathcal{N}(\\mathfrak{h}(\\overline{F}))$\nsuch that $\\oo_H^\\mathrm{st}$ contains a rational orbit $\\oo_H$ belonging to $\\mathcal{N}(\\pi)$.\nWe denote by $\\bar\\mathcal{N}(\\pi)^{\\max}$ the subset of $\\bar\\mathcal{N}(\\pi)$ consisting of maximal orbits\nwith respect to the closure ordering\n\\[\n\\oo_H^\\mathrm{st} \\leq \\oo_H'^\\mathrm{st} \\overset{\\text{def}}{\\iff} \\oo_H^\\mathrm{st} \\subset \\overline{\\oo_H'^\\mathrm{st}}.\n\\]\nWe call $\\bar\\mathcal{N}(\\pi)^{\\max}$ the \\emph{(geometric) wavefront set} of $\\pi$. \n\n\\subsection{$L$-parameters}\nDenote by $H^\\vee$ the complex Langlands dual group, \nand by $\\mathfrak{h}^\\vee$ the Lie algebra of $H^\\vee$. \nLet $W_F$ be the Weil group of $F$. \nAn \\emph{$L$-parameter} for $H(F)$ is a homomorphism \n\\[\n\\phi \\colon W_F \\times \\SL_2(\\mathbb{C}) \\rightarrow H^\\vee\n\\]\nsuch that $\\phi(W_F)$ consists of semisimple elements, $\\phi|_{W_F}$ is smooth and $\\phi|_{\\SL_2(\\mathbb{C})}$ is algebraic. \nIt is called \\emph{tempered} if $\\phi(W_F)$ is bounded. \nLet $\\Phi(H(F))$ be the set of $H^\\vee$-conjugacy classes of $L$-parameters for $H(F)$, \nand let $\\Phi_\\mathrm{temp}(H(F))$ be its subset consisting of tempered $L$-parameters. \nFor $\\phi \\in \\Phi(H(F))$, by the derivative of $\\phi|_{\\SL_2(\\mathbb{C})}$, \nwe have a linear map\n\\[\nd\\phi|_{\\SL_2(\\mathbb{C})} \\colon \\sl_2(\\mathbb{C}) \\rightarrow \\mathfrak{h}^\\vee.\n\\]\nWe denote by $N_{\\phi} \\in \\mathfrak{h}^\\vee$ the image of \n$\\begin{pmatrix}\n0 & 1 \\\\ 0 & 0\n\\end{pmatrix}$ under this map.\nThe $\\mathrm{Ad}(H^\\vee)$-orbit of $N_\\phi$ is denoted by $\\mathrm{Ad}(H^\\vee) N_\\phi$.\n\\par \n\nLet $\\mathcal{N}(\\mathfrak{h}^\\vee)$ be the set of nilpotent $\\mathrm{Ad}(H^\\vee)$-orbits in $\\mathfrak{h}^\\vee$.\nWe denote by \n\\[\nd_{H} \\colon \\mathcal{N}(\\mathfrak{h}(\\overline{F})) \\rightarrow \\mathcal{N}(\\mathfrak{h}^\\vee), \\quad\nd_{H^\\vee} \\colon \\mathcal{N}(\\mathfrak{h}^\\vee) \\rightarrow \\mathcal{N}(\\mathfrak{h}(\\overline{F}))\n\\]\nthe Spaltenstein duality maps (\\cite[Section 10]{Sp}). \nThe images of these maps are the sets of \\emph{special nilpotent conjugacy classes}.\nMoreover, $\\oo_H^\\mathrm{st} \\leq d_{H^\\vee}(d_H(\\oo_H^\\mathrm{st}))$, \nand the equality holds if and only if $\\oo_H^\\mathrm{st}$ is special. \nSee \\cite[Appendix A]{BV} for more details.\nAn $L$-parameter $\\phi \\in \\Phi(H(F))$ gives \na nilpotent orbit \n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee) N_\\phi) \\in \\mathcal{N}(\\mathfrak{h}(\\overline{F})). \n\\]\n\n\\subsection{$L$-packets and $A$-packets}\nNow suppose that $H$ is a split classical group over $F$. \nNamely, $H$ is a symplectic group or a special orthogonal group.\nLet $\\mathrm{Irr}(H(F))$ be the set of equivalence classes of irreducible representations of $H(F)$, \nand let $\\Irr_\\mathrm{temp}(H(F))$ be its subset consisting of tempered representations. \nThe local Langlands conjecture proven by Arthur \\cite[Theorem 2.2.1]{Ar} gives a canonical surjective map\n\\[\n\\mathrm{Irr}(H(F)) \\rightarrow \\Phi(H(F))\n\\]\nwith finite fibers, and preserving the temperedness. \nThe fiber of $\\phi \\in \\Phi(H(F))$ is denoted by $\\Pi_\\phi$ \nand is called the \\emph{$L$-packet} associated to $\\phi$.\n\\par\n\nWe say that an irreducible representation $\\pi$ of $H(F)$ is \\emph{co-tempered} \nif its Zelevinsky--Aubert dual $\\hat\\pi$ is tempered. \nFor $\\phi \\in \\Phi_\\mathrm{temp}(H(F))$, its \\emph{dual $A$-parameter} \n\\[\n\\psi = \\widehat\\phi \\colon W_F \\times \\SL_2(\\mathbb{C}) \\times \\SL_2(\\mathbb{C}) \\rightarrow H^\\vee\n\\]\nis defined by $\\psi(w,g_1,g_2) = \\phi(w,g_2)$. \nIts \\emph{$A$-packet} is given by \n\\[\n\\Pi_\\psi = \\{\\pi \\in \\mathrm{Irr}(H(F)) \\;|\\; \\hat\\pi \\in \\Pi_\\phi\\}. \n\\]\n\n\\subsection{Conjecture and the main result}\nThe following conjecture was independently formulated by Kim and the second author \\cite[Conjectures 1.1, 1.9]{CK}\nand Hazeltine--Liu--Lo--Shahidi \\cite[Conjecture 1.1]{HLLS}. \n\n\\begin{conj}\\label{upper}\nLet $\\phi \\in \\Phi(H(F))$. \nThen for any $\\pi \\in \\Pi_\\phi$, we have\n\\[\n\\bar\\mathcal{N}(\\hat\\pi)^{\\max} \\leq d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi).\n\\]\nMoreover, if $\\phi$ is tempered, \nthen the equality holds for some $\\pi \\in \\Pi_\\phi$. \n\\end{conj}\n\nThis is relevant for any connected reductive group $H$ \nprovided the local Langlands correspondence for $H(F)$ is assumed. \nThis conjecture is known in the following cases. \n\n\\begin{itemize}\n\\item\n$H=\\GL_m$, \\cite{MW}, see \\cite[Theorem 1.4]{CK}. \n\n\\item\n$\\pi$ is any depth-zero simple supercuspidal representation of a classical group (\\cite[Theorem 1.8]{CK}). \n\n\\item\n{$H$ is an inner form of a split group and} \n$\\pi$ is any unipotent representation of $H(F)$ with real infinitesimal character\n(\\cite[Theorem 1.4.1]{CMBO2}).\n\n\\item\n$H$ is an exceptional group $G_2$ (\\cite[Theorem 1.10]{CK}).\n\\end{itemize}\n\nAs explained in \\cite[Theorems 1.5, 1.8]{HLLS}, \nConjecture \\ref{upper} implies\na part of Jiang's conjecture and a part of the Generalized Shahidi Conjecture on local ABV packets \n(see \\cite[Conjectures 1.4, 1.7]{HLLS}). \n\n\\begin{rem}\nAn analogue of Conjecture \\ref{upper} for covering groups \nhas been formulated by Gao--Liu--Lo--Shahidi \\cite[Conjecture 1.2]{GLLS}, and proved for the Kazhdan--Patterson cover of $\\GL_m$ in \\cite[Theorem 1.3]{GLLS}.\n\\end{rem}\n\nIn this paper, we give a result toward Conjecture \\ref{upper} for split classical groups under some conditions. \nWe focus on the case where $\\phi$ is tempered \nsince Conjecture \\ref{upper} is reduced to this case by \\cite[Proposition 2.2]{CK} or \\cite[Theorem 1.2]{HLLS}.\nFor $\\phi \\in \\Phi_\\mathrm{temp}(H(F))$, we consider the following condition:\n\\begin{description}\n\\item[($\\star$)]\nfor any irreducible self-dual subrepresentation $\\phi_0 \\subset \\phi$, \nits determinant $\\det(\\phi_0)$ is trivial on the inertia subgroup $I_F$ of $W_F$.\n\\end{description}\nOur main result is stated as follows.", "context": "\\subsection{Wavefront set}\nLet $H$ be a connected reductive group over $F$ with the Lie algebra $\\mathfrak{h}$.\nWe denote by $\\mathcal{N}(\\mathfrak{h}(F))$ (resp.~ $\\mathcal{N}(\\mathfrak{h}(\\overline{F}))$) \nthe set of nilpotent $\\mathrm{Ad}(H(F))$-orbits (resp.~ $\\mathrm{Ad}(H(\\overline{F}))$-orbits) \nin $\\mathfrak{h}(F)$ (resp.~ $\\mathfrak{h}(\\overline{F})$).\nFor an irreducible admissible representation $\\pi$ of $H(F)$, \nwe denote its distribution character by $\\Theta_\\pi$. \nThe Harish-Chandra--Howe local character expansion (\\cite[Theorem 16.2]{HC}) states that \n\\[\n\\Theta_\\pi(\\exp(X)) = \\sum_{\\oo_H \\in \\mathcal{N}(\\mathfrak{h}(F))} c_{\\oo_H}(\\pi) \\hat\\mu_{\\oo_H}(\\exp(X))\n\\]\nfor $X \\in \\mathfrak{h}(F)$ sufficiently near to $0$, \nwhere $\\hat\\mu_{\\oo_H}$ is the Fourier transform of the orbital integral, \nand $c_{\\oo_H}(\\pi)$ is a constant. \nSet $\\mathcal{N}(\\pi) = \\{ \\oo_H \\in \\mathcal{N}(\\mathfrak{h}(F)) \\;|\\; c_{\\oo_H}(\\pi)\\not= 0\\}$ \nand let $\\bar\\mathcal{N}(\\pi)$ be the set of $\\oo_H^\\mathrm{st} \\in \\mathcal{N}(\\mathfrak{h}(\\overline{F}))$\nsuch that $\\oo_H^\\mathrm{st}$ contains a rational orbit $\\oo_H$ belonging to $\\mathcal{N}(\\pi)$.\nWe denote by $\\bar\\mathcal{N}(\\pi)^{\\max}$ the subset of $\\bar\\mathcal{N}(\\pi)$ consisting of maximal orbits\nwith respect to the closure ordering\n\\[\n\\oo_H^\\mathrm{st} \\leq \\oo_H'^\\mathrm{st} \\overset{\\text{def}}{\\iff} \\oo_H^\\mathrm{st} \\subset \\overline{\\oo_H'^\\mathrm{st}}.\n\\]\nWe call $\\bar\\mathcal{N}(\\pi)^{\\max}$ the \\emph{(geometric) wavefront set} of $\\pi$.\n\n\\subsection{$L$-parameters}\nDenote by $H^\\vee$ the complex Langlands dual group, \nand by $\\mathfrak{h}^\\vee$ the Lie algebra of $H^\\vee$. \nLet $W_F$ be the Weil group of $F$. \nAn \\emph{$L$-parameter} for $H(F)$ is a homomorphism \n\\[\n\\phi \\colon W_F \\times \\SL_2(\\mathbb{C}) \\rightarrow H^\\vee\n\\]\nsuch that $\\phi(W_F)$ consists of semisimple elements, $\\phi|_{W_F}$ is smooth and $\\phi|_{\\SL_2(\\mathbb{C})}$ is algebraic. \nIt is called \\emph{tempered} if $\\phi(W_F)$ is bounded. \nLet $\\Phi(H(F))$ be the set of $H^\\vee$-conjugacy classes of $L$-parameters for $H(F)$, \nand let $\\Phi_\\mathrm{temp}(H(F))$ be its subset consisting of tempered $L$-parameters. \nFor $\\phi \\in \\Phi(H(F))$, by the derivative of $\\phi|_{\\SL_2(\\mathbb{C})}$, \nwe have a linear map\n\\[\nd\\phi|_{\\SL_2(\\mathbb{C})} \\colon \\sl_2(\\mathbb{C}) \\rightarrow \\mathfrak{h}^\\vee.\n\\]\nWe denote by $N_{\\phi} \\in \\mathfrak{h}^\\vee$ the image of \n$\\begin{pmatrix}\n0 & 1 \\\\ 0 & 0\n\\end{pmatrix}$ under this map.\nThe $\\mathrm{Ad}(H^\\vee)$-orbit of $N_\\phi$ is denoted by $\\mathrm{Ad}(H^\\vee) N_\\phi$.\n\\par\n\nLet $\\mathcal{N}(\\mathfrak{h}^\\vee)$ be the set of nilpotent $\\mathrm{Ad}(H^\\vee)$-orbits in $\\mathfrak{h}^\\vee$.\nWe denote by \n\\[\nd_{H} \\colon \\mathcal{N}(\\mathfrak{h}(\\overline{F})) \\rightarrow \\mathcal{N}(\\mathfrak{h}^\\vee), \\quad\nd_{H^\\vee} \\colon \\mathcal{N}(\\mathfrak{h}^\\vee) \\rightarrow \\mathcal{N}(\\mathfrak{h}(\\overline{F}))\n\\]\nthe Spaltenstein duality maps (\\cite[Section 10]{Sp}). \nThe images of these maps are the sets of \\emph{special nilpotent conjugacy classes}.\nMoreover, $\\oo_H^\\mathrm{st} \\leq d_{H^\\vee}(d_H(\\oo_H^\\mathrm{st}))$, \nand the equality holds if and only if $\\oo_H^\\mathrm{st}$ is special. \nSee \\cite[Appendix A]{BV} for more details.\nAn $L$-parameter $\\phi \\in \\Phi(H(F))$ gives \na nilpotent orbit \n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee) N_\\phi) \\in \\mathcal{N}(\\mathfrak{h}(\\overline{F})). \n\\]\n\n\\begin{conj}\\label{upper}\nLet $\\phi \\in \\Phi(H(F))$. \nThen for any $\\pi \\in \\Pi_\\phi$, we have\n\\[\n\\bar\\mathcal{N}(\\hat\\pi)^{\\max} \\leq d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi).\n\\]\nMoreover, if $\\phi$ is tempered, \nthen the equality holds for some $\\pi \\in \\Pi_\\phi$. \n\\end{conj}\n\n\\begin{rem}\nAn analogue of Conjecture \\ref{upper} for covering groups \nhas been formulated by Gao--Liu--Lo--Shahidi \\cite[Conjecture 1.2]{GLLS}, and proved for the Kazhdan--Patterson cover of $\\GL_m$ in \\cite[Theorem 1.3]{GLLS}.\n\\end{rem}\n\nIn this paper, we give a result toward Conjecture \\ref{upper} for split classical groups under some conditions. \nWe focus on the case where $\\phi$ is tempered \nsince Conjecture \\ref{upper} is reduced to this case by \\cite[Proposition 2.2]{CK} or \\cite[Theorem 1.2]{HLLS}.\nFor $\\phi \\in \\Phi_\\mathrm{temp}(H(F))$, we consider the following condition:\n\\begin{description}\n\\item[($\\star$)]\nfor any irreducible self-dual subrepresentation $\\phi_0 \\subset \\phi$, \nits determinant $\\det(\\phi_0)$ is trivial on the inertia subgroup $I_F$ of $W_F$.\n\\end{description}\nOur main result is stated as follows.\n\n\\begin{hyp}\\label{hypo}\nLet $\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$ be a special nilpotent orbit.\nDefine $\\oo_G^\\st \\in \\NN(\\g^\\theta(\\overline{F}))$ by \n\\[\n\\oo_G^\\st = d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st)). \n\\]\nThen for any constants $\\{c_{\\oo_H} \\;|\\; \\oo_H \\subset \\oo_H^\\st\\}$ such that \n$\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H}$ is stable, \nthere exist constants $\\{a_{\\oo_G} \\;|\\; \\oo_G \\subset \\oo_G^\\st\\}$ such that \n\\[\n\\sum_{\\oo_G \\subset \\oo_G^\\st} a_{\\oo_G} \\hat\\mu_{\\oo_G}\n\\quad\\text{is equal to the transfer of}\\quad\n\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H} \n\\]\non a small enough neighborhood of $\\theta \\in G(F) \\rtimes \\theta$.\n\\end{hyp}\n\n\\begin{conj}\\label{upper}\nLet $\\phi \\in \\Phi(H(F))$. \nThen for any $\\pi \\in \\Pi_\\phi$, we have\n\\[\n\\bar\\NN(\\hat\\pi)^{\\max} \\leq d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi).\n\\]\nMoreover, if $\\phi$ is tempered, \nthen the equality holds for some $\\pi \\in \\Pi_\\phi$. \n\\end{conj}", "full_context": "\\subsection{Wavefront set}\nLet $H$ be a connected reductive group over $F$ with the Lie algebra $\\mathfrak{h}$.\nWe denote by $\\mathcal{N}(\\mathfrak{h}(F))$ (resp.~ $\\mathcal{N}(\\mathfrak{h}(\\overline{F}))$) \nthe set of nilpotent $\\mathrm{Ad}(H(F))$-orbits (resp.~ $\\mathrm{Ad}(H(\\overline{F}))$-orbits) \nin $\\mathfrak{h}(F)$ (resp.~ $\\mathfrak{h}(\\overline{F})$).\nFor an irreducible admissible representation $\\pi$ of $H(F)$, \nwe denote its distribution character by $\\Theta_\\pi$. \nThe Harish-Chandra--Howe local character expansion (\\cite[Theorem 16.2]{HC}) states that \n\\[\n\\Theta_\\pi(\\exp(X)) = \\sum_{\\oo_H \\in \\mathcal{N}(\\mathfrak{h}(F))} c_{\\oo_H}(\\pi) \\hat\\mu_{\\oo_H}(\\exp(X))\n\\]\nfor $X \\in \\mathfrak{h}(F)$ sufficiently near to $0$, \nwhere $\\hat\\mu_{\\oo_H}$ is the Fourier transform of the orbital integral, \nand $c_{\\oo_H}(\\pi)$ is a constant. \nSet $\\mathcal{N}(\\pi) = \\{ \\oo_H \\in \\mathcal{N}(\\mathfrak{h}(F)) \\;|\\; c_{\\oo_H}(\\pi)\\not= 0\\}$ \nand let $\\bar\\mathcal{N}(\\pi)$ be the set of $\\oo_H^\\mathrm{st} \\in \\mathcal{N}(\\mathfrak{h}(\\overline{F}))$\nsuch that $\\oo_H^\\mathrm{st}$ contains a rational orbit $\\oo_H$ belonging to $\\mathcal{N}(\\pi)$.\nWe denote by $\\bar\\mathcal{N}(\\pi)^{\\max}$ the subset of $\\bar\\mathcal{N}(\\pi)$ consisting of maximal orbits\nwith respect to the closure ordering\n\\[\n\\oo_H^\\mathrm{st} \\leq \\oo_H'^\\mathrm{st} \\overset{\\text{def}}{\\iff} \\oo_H^\\mathrm{st} \\subset \\overline{\\oo_H'^\\mathrm{st}}.\n\\]\nWe call $\\bar\\mathcal{N}(\\pi)^{\\max}$ the \\emph{(geometric) wavefront set} of $\\pi$.\n\n\\subsection{$L$-parameters}\nDenote by $H^\\vee$ the complex Langlands dual group, \nand by $\\mathfrak{h}^\\vee$ the Lie algebra of $H^\\vee$. \nLet $W_F$ be the Weil group of $F$. \nAn \\emph{$L$-parameter} for $H(F)$ is a homomorphism \n\\[\n\\phi \\colon W_F \\times \\SL_2(\\mathbb{C}) \\rightarrow H^\\vee\n\\]\nsuch that $\\phi(W_F)$ consists of semisimple elements, $\\phi|_{W_F}$ is smooth and $\\phi|_{\\SL_2(\\mathbb{C})}$ is algebraic. \nIt is called \\emph{tempered} if $\\phi(W_F)$ is bounded. \nLet $\\Phi(H(F))$ be the set of $H^\\vee$-conjugacy classes of $L$-parameters for $H(F)$, \nand let $\\Phi_\\mathrm{temp}(H(F))$ be its subset consisting of tempered $L$-parameters. \nFor $\\phi \\in \\Phi(H(F))$, by the derivative of $\\phi|_{\\SL_2(\\mathbb{C})}$, \nwe have a linear map\n\\[\nd\\phi|_{\\SL_2(\\mathbb{C})} \\colon \\sl_2(\\mathbb{C}) \\rightarrow \\mathfrak{h}^\\vee.\n\\]\nWe denote by $N_{\\phi} \\in \\mathfrak{h}^\\vee$ the image of \n$\\begin{pmatrix}\n0 & 1 \\\\ 0 & 0\n\\end{pmatrix}$ under this map.\nThe $\\mathrm{Ad}(H^\\vee)$-orbit of $N_\\phi$ is denoted by $\\mathrm{Ad}(H^\\vee) N_\\phi$.\n\\par\n\nLet $\\mathcal{N}(\\mathfrak{h}^\\vee)$ be the set of nilpotent $\\mathrm{Ad}(H^\\vee)$-orbits in $\\mathfrak{h}^\\vee$.\nWe denote by \n\\[\nd_{H} \\colon \\mathcal{N}(\\mathfrak{h}(\\overline{F})) \\rightarrow \\mathcal{N}(\\mathfrak{h}^\\vee), \\quad\nd_{H^\\vee} \\colon \\mathcal{N}(\\mathfrak{h}^\\vee) \\rightarrow \\mathcal{N}(\\mathfrak{h}(\\overline{F}))\n\\]\nthe Spaltenstein duality maps (\\cite[Section 10]{Sp}). \nThe images of these maps are the sets of \\emph{special nilpotent conjugacy classes}.\nMoreover, $\\oo_H^\\mathrm{st} \\leq d_{H^\\vee}(d_H(\\oo_H^\\mathrm{st}))$, \nand the equality holds if and only if $\\oo_H^\\mathrm{st}$ is special. \nSee \\cite[Appendix A]{BV} for more details.\nAn $L$-parameter $\\phi \\in \\Phi(H(F))$ gives \na nilpotent orbit \n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee) N_\\phi) \\in \\mathcal{N}(\\mathfrak{h}(\\overline{F})). \n\\]\n\n\\begin{conj}\\label{upper}\nLet $\\phi \\in \\Phi(H(F))$. \nThen for any $\\pi \\in \\Pi_\\phi$, we have\n\\[\n\\bar\\mathcal{N}(\\hat\\pi)^{\\max} \\leq d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi).\n\\]\nMoreover, if $\\phi$ is tempered, \nthen the equality holds for some $\\pi \\in \\Pi_\\phi$. \n\\end{conj}\n\n\\begin{rem}\nAn analogue of Conjecture \\ref{upper} for covering groups \nhas been formulated by Gao--Liu--Lo--Shahidi \\cite[Conjecture 1.2]{GLLS}, and proved for the Kazhdan--Patterson cover of $\\GL_m$ in \\cite[Theorem 1.3]{GLLS}.\n\\end{rem}\n\nIn this paper, we give a result toward Conjecture \\ref{upper} for split classical groups under some conditions. \nWe focus on the case where $\\phi$ is tempered \nsince Conjecture \\ref{upper} is reduced to this case by \\cite[Proposition 2.2]{CK} or \\cite[Theorem 1.2]{HLLS}.\nFor $\\phi \\in \\Phi_\\mathrm{temp}(H(F))$, we consider the following condition:\n\\begin{description}\n\\item[($\\star$)]\nfor any irreducible self-dual subrepresentation $\\phi_0 \\subset \\phi$, \nits determinant $\\det(\\phi_0)$ is trivial on the inertia subgroup $I_F$ of $W_F$.\n\\end{description}\nOur main result is stated as follows.\n\n\\begin{hyp}\\label{hypo}\nLet $\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$ be a special nilpotent orbit.\nDefine $\\oo_G^\\st \\in \\NN(\\g^\\theta(\\overline{F}))$ by \n\\[\n\\oo_G^\\st = d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st)). \n\\]\nThen for any constants $\\{c_{\\oo_H} \\;|\\; \\oo_H \\subset \\oo_H^\\st\\}$ such that \n$\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H}$ is stable, \nthere exist constants $\\{a_{\\oo_G} \\;|\\; \\oo_G \\subset \\oo_G^\\st\\}$ such that \n\\[\n\\sum_{\\oo_G \\subset \\oo_G^\\st} a_{\\oo_G} \\hat\\mu_{\\oo_G}\n\\quad\\text{is equal to the transfer of}\\quad\n\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H} \n\\]\non a small enough neighborhood of $\\theta \\in G(F) \\rtimes \\theta$.\n\\end{hyp}\n\n\\begin{conj}\\label{upper}\nLet $\\phi \\in \\Phi(H(F))$. \nThen for any $\\pi \\in \\Pi_\\phi$, we have\n\\[\n\\bar\\NN(\\hat\\pi)^{\\max} \\leq d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi).\n\\]\nMoreover, if $\\phi$ is tempered, \nthen the equality holds for some $\\pi \\in \\Pi_\\phi$. \n\\end{conj}\n\n\\begin{thm}\\label{twist}\nLet $\\phi \\colon W_F \\times \\SL_2(\\C) \\rightarrow G^\\vee$ \nbe a self-dual tempered $L$-parameter for $G(F) = \\GL_m(F)$, \nand let $\\pi_\\psi$ be the Zelevinsky dual of the irreducible tempered representation \n$\\pi_\\phi$ corresponding to $\\phi$. \nThen \n\\[\n\\bar\\NN_\\theta(\\tl\\pi_\\psi)^{\\max} = \\{d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi)\\}.\n\\]\nMoreover, $c_{\\oo_G}(\\tl\\pi_\\psi) = c_{\\oo_G'}(\\tl\\pi_\\psi)$ \nfor any rational orbits $\\oo_G$ and $\\oo_G'$ contained in $d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi)$.\n\\end{thm}\n\\begin{proof}\nRecall that $\\bar\\NN(\\pi_\\psi)^{\\max} = \\{d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi)\\}$ \nby M{\\oe}glin--Waldspurger \\cite{MW} as explained in \\cite[Theorem 1.4]{CK}.\nThen Varma showed in \\cite[Lemma 5.29]{V} that \nfor any $\\oo_G^\\st \\in \\bar\\NN_\\theta(\\tl\\pi_\\psi)^{\\max}$, \nthe corresponding degenerate Whittaker model is nonzero. \nHence we have\n\\[\n\\oo_G^\\st \\leq d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi).\n\\]\nOn the other hand, by Konno \\cite[Theorem 4.1]{K}\ntogether with the multiplicity one result of degenerate Whittaker models by M{\\oe}glin--Waldspurger \\cite{MW}\n(cf., see \\cite[Proposition 4.4]{K}),\nwe have\n\\[\nd_{G^\\vee}(\\Ad(G^\\vee) N_\\phi) \\in \\bar\\NN_\\theta(\\tl\\pi_\\psi)^{\\max}. \n\\]\nHence we obtain the first assertion.\nThe last assertion follows from \\cite[Theorem 4.1(2), Remark 4.2]{K}.\n\\end{proof}\n\nAs a consequence of Theorem \\ref{transfer}, we obtain the following. \n\\begin{cor}\\label{cor1}\nWe have\n\\[\nd_{H^\\vee}(\\Ad(H^\\vee) N_\\phi) \\in \\bar\\NN(\\psi).\n\\]\nMoreover, for $\\oo_H^\\st \\in \\bar\\NN(\\psi)$, \nif $\\oo_H^\\st \\not= d_{H^\\vee}(\\Ad(H^\\vee) N_\\phi)$, then \n\\[\n\\dim(\\oo_H^\\st) < \\dim(d_{H^\\vee}(\\Ad(H^\\vee) N_\\phi)).\n\\]\n\\end{cor}\n\\begin{proof}\nSet $d = \\dim(d_{H^\\vee}(\\Ad(H^\\vee) N_\\phi))$. \nAs in the proof of Theorem \\ref{transfer}, \nconsider the unique $L$-parameter $\\phi_0 \\in \\Phi_\\temp(H(F))$\nsuch that $\\phi_0$ is trivial on $W_F$ and \n\\[\n\\Ad(H^\\vee) N_{\\phi_0} = \\Ad(H^\\vee) N_\\phi. \n\\]\nSet $\\psi_0 = \\widehat\\phi_0$.\nFor a constant $e \\in \\C^\\times$\nthe transfer of $S\\Theta_\\psi - eS\\Theta_{\\psi_0}$ is \n\\[\n\\Theta_{\\tl\\pi_\\psi}-e\\Theta_{\\tl\\pi_{\\psi_0}}\n= \\sum_{\\oo_G \\in \\NN(\\g^\\theta(F))} \\left( c_{\\oo_G}(\\tl\\pi_\\psi)-e c_{\\oo_G}(\\tl\\pi_{\\psi_0}) \\right) \\hat\\mu_{\\oo_G}.\n\\]\nSince $d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi) = d_{G^\\vee}(\\Ad(G^\\vee) N_{\\phi_0})$,\nby Theorem \\ref{twist}, we can choose $e \\in \\C^\\times$ such that \nfor $\\oo_G^\\st \\in \\NN(\\g^\\theta(\\overline{F}))$, \nif $c_{\\oo_G}(\\tl\\pi_\\psi)-e c_{\\oo_G}(\\tl\\pi_{\\psi_0}) \\not= 0$ for some $\\oo_G \\subset \\oo_G^\\st$, \nthen $\\oo_G^\\st < d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi)$.\nThis together with Theorem \\ref{transfer} implies that \n\\[\n\\sum_{\\substack{\\oo_H^\\st \\in \\NN(\\h(\\overline{F})) \\\\ \\dim(\\oo_H^\\st) \\geq d}} \n\\left( \\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H}(\\psi) \\hat\\mu_{\\oo_H}\\right)\n=\ne\\sum_{\\substack{\\oo_H^\\st \\in \\NN(\\h(\\overline{F})) \\\\ \\dim(\\oo_H^\\st) \\geq d}} \n\\left( \\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H}(\\psi_0) \\hat\\mu_{\\oo_H}\\right)\n\\]\nBy \\cite[Theorem 1.4.1]{CMBO2}, \nin the right-hand side, only $\\oo_H^\\st = d_{H^\\vee}(\\Ad(H^\\vee) N_\\phi)$ can contribute. \nBy the linear independence of $\\hat\\mu_{\\oo_H}$ (\\cite[Theorem 5.11]{HC}), \nthe same holds for the left-hand side.\n\\end{proof}\n\n\\begin{prop}\\label{refine}\nAssume Hypothesis \\ref{hypo}.\nThen for any $\\oo_H^\\st \\in \\bar\\NN(\\psi)$, \nwe have \n\\[\n\\oo_H^\\st \\leq d_{H^\\vee}(\\Ad(H^\\vee) N_\\phi).\n\\]\n\\end{prop}\n\\begin{proof}\nFor each special orbit $\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$, \nlet $\\{a_{\\oo_G}(\\psi) \\;|\\; \\oo_G \\subset d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st))\\}$ \nbe the constants in Hypothesis \\ref{hypo}\ngiven by $\\{c_{\\oo_H}(\\psi) \\;|\\; \\oo_H \\subset \\oo_H^\\st\\}$.\nThen by considering the transfer of $S\\Theta_\\psi$, \nwe see that the local character expansion of $\\Theta_{\\tl\\pi_\\psi}$ is given by\n\\[\n\\Theta_{\\tl\\pi_\\psi} = \\sum_{\\oo_H^\\st} \n\\sum_{\\oo_G \\subset \\oo_G^\\st} a_{\\oo_G}(\\psi) \\hat\\mu_{\\oo_G}, \n\\]\nwhere $\\oo_H^\\st$ runs over special nilpotent orbits in $\\NN(\\h(\\overline{F}))$\nand $\\oo_G$ runs over rational orbits contained in\n$\\oo_G^\\st = d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st))$.\nIn particular, if $\\oo_H^\\st \\in \\bar\\NN(\\psi)$, \nthen $d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st)) \\in \\bar\\NN_\\theta(\\tl\\pi_\\psi)$. \nBy Theorem \\ref{twist}, we have\n\\[\nd_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st)) \\leq d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi). \n\\]\nHence $\\Ad(G^\\vee)d_{H}(\\oo_H^\\st) \\geq \\Ad(G^\\vee) N_\\phi$.\nNote that \n\\[\n\\Ad(G^\\vee)d_{H}(\\oo_H^\\st) \\cap H^\\vee = d_{H}(\\oo_H^\\st), \n\\quad\n\\Ad(G^\\vee) N_\\phi \\cap H^\\vee = \\Ad(H^\\vee) N_\\phi. \n\\]\n(Here, if $H=\\SO_{2n}$, we consider $\\O_{2n}(\\C)$-orbits actually. See Remark \\ref{OvsSO}.)\nHence $d_{H}(\\oo_H^\\st) \\geq \\Ad(H^\\vee) N_\\phi$, \nand we conclude that \n$\\oo_H^\\st \\leq d_{H^\\vee}(d_{H}(\\oo_H^\\st)) \\leq d_{H^\\vee}(\\Ad(H^\\vee) N_\\phi)$.\n\\end{proof}\n\nNow we assume that we have\n\\begin{itemize}\n\\item\n$\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$ \nwith $\\dim(\\oo_H^\\st) \\geq \\dim(d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi))$; \n\\item\n$\\oo_H \\in \\NN(\\h(F))$ with $\\oo_H \\subset \\oo_H^\\st$; \n\\item\n$\\pi_0 \\in \\Pi_\\psi$ with $\\psi = \\widehat\\phi$\n\\end{itemize}\nsuch that $c_{\\oo_H}(\\pi_0) \\not= 0$. \nThe goal is to show that $\\oo_H^\\st = d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi)$.\n\\par\n\nBy the induction hypothesis, we have \n$\\dim(\\oo_{H_i}^\\st) \\leq \\dim (d_{H_i^\\vee}(\\Ad(H_i^\\vee)N_{\\phi_i}))$ for $i \\in \\{1,2\\}$.\nHence by Proposition \\ref{WS} and Lemma \\ref{dimW}, \nwe have \n\\begin{align*}\n\\dim(\\oo_H^\\st) &= \\dim\\left(W(\\oo_{H_1}^\\st, \\oo_{H_2}^\\st)\\right)\n\\\\&\\leq \\dim\\left(\nW(d_{H_1^\\vee}(\\Ad(H_1^\\vee)N_{\\phi_1}), d_{H_2^\\vee}(\\Ad(H_2^\\vee)N_{\\phi_2}))\n\\right)\n\\\\&\\leq \n\\dim\\left(\nd_{H^\\vee}(\\Ad(H^\\vee)\\xi(\nd_{H_1}(d_{H_1^\\vee}(\\Ad(H_1^\\vee)N_{\\phi_1})), d_{H_2}(d_{H_2^\\vee}(\\Ad(H_2^\\vee)N_{\\phi_2}))\n))\n\\right)\n\\\\&\\leq\n\\dim\\left(\nd_{H^\\vee}(\\Ad(H^\\vee)\\xi(\\Ad(H_1^\\vee)N_{\\phi_1}, \\Ad(H_2^\\vee)N_{\\phi_2}))\n\\right)\n\\\\&= \n\\dim\\left(d_{H^\\vee}(\\Ad(H^\\vee)N_{\\phi})\\right).\n\\end{align*}\nHere, we used the facts \nthat $d_{H_i}(d_{H_i^\\vee}(\\Ad(H_i^\\vee)N_{\\phi_i})) \\geq \\Ad(H_i^\\vee)N_{\\phi_i}$\nand that $d_{H^\\vee}$ is an order-reversing map.\nSince we assume that $\\dim(\\oo_H^\\st) \\geq \\dim\\left(d_{H^\\vee}(\\Ad(H^\\vee)N_{\\phi})\\right)$, \nthis must be an equality.\nMoreover, it together with Lemma \\ref{dimW} implies that \n$\\oo_{H_i}^\\st = d_{H_i^\\vee}(\\Ad(H_i^\\vee)N_{\\phi_i})$ for $i \\in \\{1,2\\}$. \nThen by proposition \\ref{WS}, we have\n\\begin{align*}\n\\oo_{H^\\st} \n&\\leq d_{H^\\vee}(\\Ad(H^\\vee)\\xi(d_{H_1}(\\oo_{H_1}^\\st), d_{H_2}(\\oo_{H_2}^\\st)))\n\\\\&\\leq d_{H^\\vee}(\\Ad(H^\\vee)\\xi(\\Ad(H_1^\\vee)N_{\\phi_1}, \\Ad(H_2^\\vee)N_{\\phi_2}))\n\\\\&= d_{H^\\vee}(\\Ad(H^\\vee)N_{\\phi}).\n\\end{align*}\nSince $\\dim(\\oo_H^\\st) = \\dim\\left(d_{H^\\vee}(\\Ad(H^\\vee)N_{\\phi})\\right)$, \nwe conclude that $\\oo_H^\\st = d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi)$, \nas desired.\n\\par\n\nThe proof of Theorem \\ref{main} (3) is similar. \nSuppose that $\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$ contains a rational orbit $\\oo_H$ \nsuch that $c_{\\oo_H}(\\pi_0) \\not= 0$ for some $\\pi_0 \\in \\Pi_\\psi$. \nConsider $s \\in \\Sc_\\psi$ satisfying ($\\ast$).\nIf $s = 1$, then Proposition \\ref{refine} implies that $\\oo_H^\\st \\leq d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi)$. \nOtherwise, we can write $\\oo_H^\\st = W(\\oo_{H_1}^\\st, \\oo_{H_2}^\\st)$ as above. \nThen $\\oo_{H_i}^\\st \\leq d_{H_i^\\vee}(\\Ad(H_i^\\vee)N_{\\phi_i})$ by the induction hypothesis.\nBy Proposition \\ref{WS} and Lemma \\ref{Worder} below, we have \n\\begin{align*}\n\\oo_H^\\st &= W(\\oo_{H_1}^\\st, \\oo_{H_2}^\\st)\n\\\\&\\leq W(d_{H_1^\\vee}(\\Ad(H_1^\\vee)N_{\\phi_1}), d_{H_2^\\vee}(\\Ad(H_2^\\vee)N_{\\phi_2}))\n\\\\&\\leq \nd_{H^\\vee}(\\Ad(H^\\vee)\\xi(\nd_{H_1}(d_{H_1^\\vee}(\\Ad(H_1^\\vee)N_{\\phi_1})), d_{H_2}(d_{H_2^\\vee}(\\Ad(H_2^\\vee)N_{\\phi_2}))\n))\n\\\\&\\leq\nd_{H^\\vee}(\\Ad(H^\\vee)\\xi(\\Ad(H_1^\\vee)N_{\\phi_1}, \\Ad(H_2^\\vee)N_{\\phi_2}))\n\\\\&= \nd_{H^\\vee}(\\Ad(H^\\vee)N_{\\phi}).\n\\end{align*}\nThis completes the proof of Theorem \\ref{main}.\n\\end{proof}", "post_theorem_intro_text_len": 6376, "post_theorem_intro_text": "\\subsection{Outline of the proof}\\label{sec.outline}\nThroughout the paper, we assume that $p \\gg 0$ as in Theorem \\ref{main}.\nThe proof of Theorem \\ref{main} is similar to the proofs of the Generic Packet Conjecture \nby Konno \\cite{K} and Varma \\cite{V}. \nThe main idea is to compare local character expansions via \n\\emph{endoscopic character identities}.\n\\par\n\nFix $\\phi \\in \\Phi_\\mathrm{temp}(H(F))$ and set $\\psi = \\widehat\\phi$. \nSet $G = \\GL_m$, where $m$ is the dimension of the standard representation $H^\\vee \\rightarrow G^\\vee = \\GL_m(\\mathbb{C})$.\nBy composing with this map, we can regard $\\psi$ as an $A$-parameter for $G(F)$, \nand obtain an irreducible self-dual representation $\\pi_\\psi$ of $G(F)$. \nDefine an involution $\\theta$ on $G(F)$ by \n\\[\n\\theta(g) = J{}^tg^{-1}J^{-1}, \n\\quad\nJ = \\begin{pmatrix}\n&&&1 \\\\\n&&-1&\\\\\n&\\iddots&& \\\\\n(-1)^{m-1}&&\n\\end{pmatrix} \\in G(F).\n\\]\nWe can extend $\\pi_\\psi$ to an irreducible representation $\\tl\\pi_\\psi$ of $G \\rtimes \\langle \\theta \\rangle$.\nThe twisted character of $\\tl\\pi_\\psi$ is denoted by $\\Theta_{\\tl\\pi_\\psi}$. \n\\par\n\nLet $\\Sc_\\psi$ be the component group for $\\psi$, \nand let $\\widehat\\Sc_\\psi$ be its Pontryagin dual. \nRecall that Arthur defined a map\n\\[\n\\Pi_\\psi \\rightarrow \\widehat\\Sc_\\psi, \\;\n\\pi \\mapsto \\langle \\cdot, \\pi \\rangle_\\psi. \n\\]\nNotice that this is bijective since $\\psi = \\widehat\\phi$.\nSet $s_\\psi = \\psi(\\1_{W_F}, \\mathbf{1}, -\\mathbf{1}) = \\phi(\\1_{W_F}, -\\mathbf{1})$.\nIn this setting, \\cite[Theorem 2.2.1]{Ar} asserts that \n\\begin{description}\n\\item[(T1)]\nthe distribution\n\\[\nS\\Theta_{\\psi} = \\sum_{\\pi \\in \\Pi_\\psi} \\langle s_\\psi, \\pi \\rangle_\\psi \\Theta_\\pi\n\\]\nis stable, and $\\Theta_{\\tl\\pi_\\psi}$ is the transfer of $S\\Theta_\\psi$; \n\\item[(T2)]\nEach $s \\in \\Sc_\\psi$ with $s \\not= 1$ gives an elliptic endoscopic group $H_1 \\times H_2$ of $H$\nand an $A$-parameter $\\psi_i$ for $H_i$ \n(of the form $\\psi_i = \\widehat\\phi_i$ for some $\\phi_i \\in \\Phi_\\mathrm{temp}(H_i)$) such that \n\\[\nS\\Theta_{\\psi, s} = \\sum_{\\pi \\in \\Pi_\\psi} \\langle s \\cdot s_\\psi, \\pi \\rangle_\\psi \\Theta_\\pi\n\\]\nis the transfer of $S\\Theta_{\\psi_1} \\otimes S\\Theta_{\\psi_2}$. \n\\end{description}\n\\par\n\nThe first step of the proof of Theorem \\ref{main} is to establish an analogue for the twisted $\\GL_m$ case.\nBy the twisted local character expansion established by Clozel \\cite[Theorem 3]{C}, \nwe can expand\n\\[\n\\Theta_{\\tl\\pi_\\psi} \n= \\sum_{\\oo_G \\in \\mathcal{N}(\\mathfrak{g}^\\theta(F))} c_{\\oo_G}(\\tl\\pi_\\psi) \\hat\\mu_{\\oo_G}. \n\\]\nSee Section \\ref{sec.TLCE} below for the notations. \nWe define $\\NN_\\theta(\\tl\\pi_\\psi) \\subset \\mathcal{N}(\\mathfrak{g}^\\theta(F))$ \nand $\\bar\\NN_\\theta(\\tl\\pi_\\psi) \\subset \\mathcal{N}(\\mathfrak{g}^\\theta(\\overline{F}))$\nsimilar to $\\mathcal{N}(\\pi)$ and $\\bar\\mathcal{N}(\\pi)$, respectively. \nIn Theorem \\ref{twist} below, we will show that \n\\[\n\\bar\\NN_\\theta(\\tl\\pi_\\psi)^{\\max} = \\{d_{G^\\vee}(\\mathrm{Ad}(G^\\vee) N_\\phi)\\}.\n\\]\nThis is an immediate consequence of \nresults of Konno \\cite{K}, Varma \\cite{V}, and M{\\oe}glin--Waldspurger \\cite{MW}.\nMoreover, Konno's result \\cite[Theorem 4.1(2)]{K} says that  \nthe constant $c_{\\oo_G}(\\tl\\pi_\\psi)$ for $\\oo_G \\subset d_{G^\\vee}(\\mathrm{Ad}(G^\\vee) N_\\phi)$ \ndoes not depend on the choice of the rational orbit.\n\\par\n\nSet $\\oo_H^\\mathrm{st} = d_{H^\\vee}(\\mathrm{Ad}(H^\\vee) N_\\phi)$ and $\\oo_G^\\mathrm{st} = d_{G^\\vee}(\\mathrm{Ad}(G^\\vee) N_\\phi)$.\nThe second step of the proof of Theorem \\ref{main} is to show that \nthere exists a constant $\\gamma_{\\oo_H}$ for each rational orbit $\\oo_H \\subset \\oo_H^\\mathrm{st}$ such that \n\\[\n\\sum_{\\oo_G \\subset \\oo_G^\\mathrm{st}} \\hat\\mu_{\\oo_G}\n\\quad\n\\text{is equal to the transfer of} \n\\quad\n\\sum_{\\oo_H \\subset \\oo_H^\\mathrm{st}} \\gamma_{\\oo_H}\\hat\\mu_{\\oo_H}\n\\]\non a small enough neighborhood of $\\theta \\in G(F) \\rtimes \\theta$.\nSee Theorem \\ref{transfer} below. \nHypothesis \\ref{hypo} is an analogue of this result. \nThese claims could be proven by similar arguments to the standard case explained in the next paragraph, \nbut there is no explicit reference in the literature.\nThus we will prove Theorem \\ref{transfer} in this paper. \nThe proof uses Conjecture \\ref{upper} in the case established in \\cite{CMBO2},\nwhich requires that $H$ is split over $F$.\n\\par\n\nAn analogous result for the standard endoscopy with unramified endoscopic groups\nis known by Waldspurger \\cite{W_pave}. \nLet $H_1 \\times H_2$ be an unramified endoscopic group of $H$. \nFor a special nilpotent orbit $\\oo_{H_i}^\\mathrm{st} \\in \\mathcal{N}(\\h_i(\\overline{F}))$, \nWaldspurger explicitly defined a nilpotent orbit\n\\[\n\\oo_H^\\mathrm{st} = W(\\oo_{H_1}^\\mathrm{st}, \\oo_{H_2}^\\mathrm{st}) \\in \\mathcal{N}(\\mathfrak{h}(\\overline{F})),\n\\]\nand proved a certain transfer result in \\cite[XII.9 Th\\'eor\\`eme]{W_pave}.\nThe third step of the proof of Theorem \\ref{main} is to relate the Waldspurger map $W(\\oo_{H_1}^\\mathrm{st}, \\oo_{H_2}^\\mathrm{st})$\nand the Spaltenstein duality (Proposition \\ref{WS}). \n\\par\n\nCombining the two transfer results, we can relate the local character expansion of $S\\Theta_\\psi$ (resp.~ $S\\Theta_{\\psi,s}$)\nwith the one of $\\Theta_{\\tl\\pi_\\psi}$ (resp.~ $S\\Theta_{\\psi_1} \\otimes S\\Theta_{\\psi_2}$).\nThen by induction on $m$, \nwe can deduce Theorem \\ref{main} in Section \\ref{sec.proof}. \nTo use \\cite[XII.9 Th\\'eor\\`eme]{W_pave}, we need the condition ($\\star$) in Cases (C) and (D), \nwhich is equivalent to saying that all endoscopic groups $H_1 \\times H_2$ arising from $s \\in \\Sc_\\psi$ are unramified. \n\n\\begin{rem}\\label{OvsSO}\nThe local Langlands correspondence for $H = \\SO_{2n}$ as proven by Arthur, \nis given up to $\\O_{2n}(F)$-conjugation. \nThus, in turn, we should replace $\\mathrm{Ad}(H^\\vee)N_\\phi$ by $\\mathrm{Ad}(\\O_{2n}(\\mathbb{C}))N_\\phi$ in our results. \nWe will ignore this discrepancy since it is not an important difference for the geometric wavefront set results.\nBy replacing $\\SO_{2n}$ with $\\O_{2n}$, we can use the following properties:\n\\begin{enumerate}\n\\item\nthe distributions on $\\SO_{2n}(F)$ considered in the paper are all $\\O_{2n}(F)$-invariant;\n\\item\nthe transfer of stable distributions on $H(F)$ to $G(F) \\rtimes \\theta$ is injective;\n\\item\n$\\mathrm{Ad}(G^\\vee)\\oo_{H^\\vee} \\cap H^\\vee = \\oo_{H^\\vee}$ for $\\oo_{H^\\vee} \\in \\mathcal{N}(\\mathfrak{h}^\\vee)$.\n\\end{enumerate}\nNote that (2) and (3) hold when $H = \\SO_{2n+1}$ or $H = \\Sp_{2n}$. \n\\end{rem}", "sketch": "The proof of Theorem~\\ref{main} follows Konno~\\cite{K} and Varma~\\cite{V}: the \"main idea is to compare local character expansions via \\emph{endoscopic character identities}.\" Fix $\\phi\\in\\Phi_\\mathrm{temp}(H(F))$, set $\\psi=\\widehat\\phi$, and embed into $G=\\GL_m$ via the standard map $H^\\vee\\to G^\\vee$. This yields a self-dual $\\pi_\\psi$ of $G(F)$, extended to $\\tilde\\pi_\\psi$ on $G\\rtimes\\langle\\theta\\rangle$, whose twisted character $\\Theta_{\\tilde\\pi_\\psi}$ matches Arthur’s stable distribution\n\\[\nS\\Theta_{\\psi}=\\sum_{\\pi\\in\\Pi_\\psi}\\langle s_\\psi,\\pi\\rangle_\\psi\\,\\Theta_\\pi\n\\]\nby endoscopic character identities (Arthur’s (T1)), and similarly for $S\\Theta_{\\psi,s}$ (Arthur’s (T2)).\n\n(1) \"The first step\" is an analogue in the twisted $\\GL_m$ case: using Clozel’s twisted local character expansion\n\\[\n\\Theta_{\\tilde\\pi_\\psi}=\\sum_{\\oo_G\\in\\mathcal N(\\mathfrak g^\\theta(F))} c_{\\oo_G}(\\tilde\\pi_\\psi)\\,\\hat\\mu_{\\oo_G},\n\\]\none defines $\\bar\\mathcal N_\\theta(\\tilde\\pi_\\psi)^{\\max}$ and proves (Theorem~\\ref{twist})\n\\[\n\\bar\\mathcal N_\\theta(\\tilde\\pi_\\psi)^{\\max}=\\{d_{G^\\vee}(\\mathrm{Ad}(G^\\vee)N_\\phi)\\},\n\\]\nusing results of Konno, Varma, and M\\oe glin--Waldspurger; additionally, Konno’s result implies the constants $c_{\\oo_G}(\\tilde\\pi_\\psi)$ for $\\oo_G\\subset d_{G^\\vee}(\\mathrm{Ad}(G^\\vee)N_\\phi)$ do not depend on the rational orbit.\n\n(2) Setting $\\oo_H^{\\mathrm{st}}=d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)$ and $\\oo_G^{\\mathrm{st}}=d_{G^\\vee}(\\mathrm{Ad}(G^\\vee)N_\\phi)$, \"the second step\" is a transfer statement (Theorem~\\ref{transfer}): for each rational orbit $\\oo_H\\subset\\oo_H^{\\mathrm{st}}$ there is a constant $\\gamma_{\\oo_H}$ such that on a small neighborhood of $\\theta$,\n\\[\n\\sum_{\\oo_G\\subset\\oo_G^{\\mathrm{st}}}\\hat\\mu_{\\oo_G}\\quad\\text{equals the transfer of}\\quad\\sum_{\\oo_H\\subset\\oo_H^{\\mathrm{st}}}\\gamma_{\\oo_H}\\hat\\mu_{\\oo_H}.\n\\]\nHypothesis~\\ref{hypo} is described as an analogue of this transfer result.\n\n(3) \"The third step\" relates Waldspurger’s explicit map for standard endoscopy,\n\\[\n\\oo_H^{\\mathrm{st}}=W(\\oo_{H_1}^{\\mathrm{st}},\\oo_{H_2}^{\\mathrm{st}}),\n\\]\nto Spaltenstein duality (Proposition~\\ref{WS}).\n\nFinally, \"combining the two transfer results,\" one relates the local character expansions of $S\\Theta_\\psi$ and $S\\Theta_{\\psi,s}$ to those of $\\Theta_{\\tilde\\pi_\\psi}$ and $S\\Theta_{\\psi_1}\\otimes S\\Theta_{\\psi_2}$, and then \"by induction on $m$\" deduces Theorem~\\ref{main} (in Section~\\ref{sec.proof}). The condition $(\\star)$ in Cases (C) and (D) is needed to apply Waldspurger’s theorem, equivalently ensuring the relevant endoscopic groups $H_1\\times H_2$ are unramified.", "expanded_sketch": "The proof of Theorem~\\ref{main} follows Konno~\\cite{K} and Varma~\\cite{V}: the \"main idea is to compare local character expansions via \\emph{endoscopic character identities}.\" Fix $\\phi\\in\\Phi_\\mathrm{temp}(H(F))$, set $\\psi=\\widehat\\phi$, and embed into $G=\\GL_m$ via the standard map $H^\\vee\\to G^\\vee$. This yields a self-dual $\\pi_\\psi$ of $G(F)$, extended to $\\tilde\\pi_\\psi$ on $G\\rtimes\\langle\\theta\\rangle$, whose twisted character $\\Theta_{\\tilde\\pi_\\psi}$ matches Arthur’s stable distribution\n\\[\nS\\Theta_{\\psi}=\\sum_{\\pi\\in\\Pi_\\psi}\\langle s_\\psi,\\pi\\rangle_\\psi\\,\\Theta_\\pi\n\\]\nby endoscopic character identities (Arthur’s (T1)), and similarly for $S\\Theta_{\\psi,s}$ (Arthur’s (T2)).\n\n(1) \"The first step\" is an analogue in the twisted $\\GL_m$ case: using Clozel’s twisted local character expansion\n\\[\n\\Theta_{\\tilde\\pi_\\psi}=\\sum_{\\oo_G\\in\\mathcal N(\\mathfrak g^\\theta(F))} c_{\\oo_G}(\\tilde\\pi_\\psi)\\,\\hat\\mu_{\\oo_G},\n\\]\none defines $\\bar\\mathcal N_\\theta(\\tilde\\pi_\\psi)^{\\max}$ and proves the following theorem.\n\\begin{thm}\\label{twist}\nLet $\\phi \\colon W_F \\times \\SL_2(\\C) \\rightarrow G^\\vee$ \nbe a self-dual tempered $L$-parameter for $G(F) = \\GL_m(F)$, \nand let $\\pi_\\psi$ be the Zelevinsky dual of the irreducible tempered representation \n$\\pi_\\phi$ corresponding to $\\phi$. \nThen \n\\[\n\\bar\\NN_\\theta(\\tl\\pi_\\psi)^{\\max} = \\{d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi)\\}.\n\\]\nMoreover, $c_{\\oo_G}(\\tl\\pi_\\psi) = c_{\\oo_G'}(\\tl\\pi_\\psi)$ \nfor any rational orbits $\\oo_G$ and $\\oo_G'$ contained in $d_{G^\\vee}(\\Ad(G^\\vee) N_\\phi)$.\n\\end{thm}\nThis uses results of Konno, Varma, and M\\oe glin--Waldspurger.\n\n(2) Setting $\\oo_H^{\\mathrm{st}}=d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)$ and $\\oo_G^{\\mathrm{st}}=d_{G^\\vee}(\\mathrm{Ad}(G^\\vee)N_\\phi)$, \"the second step\" is the following transfer statement.\n\\begin{thm}\\label{transfer}\nLet $\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$ be a special nilpotent orbit.\nDefine $\\oo_G^\\st \\in \\NN(\\g^\\theta(\\overline{F}))$ by \n\\[\n\\oo_G^\\st = d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st)). \n\\]\nThen for each rational orbit $\\oo_H \\subset \\oo_H^\\st$, \nthere exists a constant $\\gamma_{\\oo_H} \\in \\C$ such that \n\\[\n\\sum_{\\oo_G \\subset \\oo_G^\\st} \\hat\\mu_{\\oo_G}\n\\quad\\text{is equal to the transfer of }\\quad\n\\sum_{\\oo_H \\subset \\oo_H^\\st} \\gamma_{\\oo_H} \\hat\\mu_{\\oo_H}\n\\]\non a small enough neighborhood of $\\theta \\in G(F) \\rtimes \\theta$.\n\\end{thm}\nThe following hypothesis is described as an analogue of this transfer result.\n\\begin{hyp}\\label{hypo}\nLet $\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$ be a special nilpotent orbit.\nDefine $\\oo_G^\\st \\in \\NN(\\g^\\theta(\\overline{F}))$ by \n\\[\n\\oo_G^\\st = d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st)). \n\\]\nThen for any constants $\\{c_{\\oo_H} \\;|\\; \\oo_H \\subset \\oo_H^\\st\\}$ such that \n$\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H}$ is stable, \nthere exist constants $\\{a_{\\oo_G} \\;|\\; \\oo_G \\subset \\oo_G^\\st\\}$ such that \n\\[\n\\sum_{\\oo_G \\subset \\oo_G^\\st} a_{\\oo_G} \\hat\\mu_{\\oo_G}\n\\quad\\text{is equal to the transfer of}\\quad\n\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H} \n\\]\non a small enough neighborhood of $\\theta \\in G(F) \\rtimes \\theta$.\n\\end{hyp}\n\n(3) \"The third step\" relates Waldspurger’s explicit map for standard endoscopy,\n\\[\n\\oo_H^{\\mathrm{st}}=W(\\oo_{H_1}^{\\mathrm{st}},\\oo_{H_2}^{\\mathrm{st}}),\n\\]\nto Spaltenstein duality using the proposition below.\n\\begin{prop}\\label{WS}\nFor $i \\in \\{1,2\\}$, let $\\oo_{H_i}^\\st \\in \\NN(\\h_i(\\overline{F}))$ be a special nilpotent orbit.\nThen \n\\[\nW(\\oo_{H_1}^\\st, \\oo_{H_2}^\\st) \n\\leq d_{H^\\vee}(\\Ad(H^\\vee)\\xi(d_{H_1}(\\oo_{H_1}^\\st), d_{H_2}(\\oo_{H_2}^\\st))). \n\\]\n\\end{prop}\n\nFinally, \"combining the two transfer results,\" one relates the local character expansions of $S\\Theta_\\psi$ and $S\\Theta_{\\psi,s}$ to those of $\\Theta_{\\tilde\\pi_\\psi}$ and $S\\Theta_{\\psi_1}\\otimes S\\Theta_{\\psi_2}$, and then \"by induction on $m$\" establishes the main theorem (later in the paper). The condition $(\\star)$ in Cases (C) and (D) is needed to apply Waldspurger’s theorem, equivalently ensuring the relevant endoscopic groups $H_1\\times H_2$ are unramified.", "expanded_theorem": "\\label{main}\nLet $H$ be a split classical group over $F$, \nand let $\\phi \\in \\Phi_\\mathrm{temp}(H(F))$. \nSuppose that one of the following holds: \n\\begin{description}\n\\item[(B)]\n$H = \\SO_{2n+1}$, $p > 6n+3$ and $\\phi$ is arbitrary; \n\\item[(C)]\n$H = \\Sp_{2n}$, $p > 6n$ and $\\phi$ satisfies $(\\star)$; \n\\item[(D)]\n$H = \\SO_{2n}$, $p > 6n$ and $\\phi$ satisfies $(\\star)$. \n\\end{description}\nSet $\\psi = \\widehat\\phi$. \n\n\\begin{enumerate}\n\\item\nThere exists $\\pi \\in \\Pi_\\psi$ such that \n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi) \\in \\bar\\mathcal{N}(\\pi)^{\\max}.\n\\]\n\\item\nFor $\\pi \\in \\Pi_\\psi$ and for $\\oo_H^\\mathrm{st} \\in \\bar\\mathcal{N}(\\pi)$, \nif $\\oo_H^\\mathrm{st} \\not= d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)$, then \n\\[\n\\dim(\\oo_H^\\mathrm{st}) < \\dim(d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)).\n\\]\nHere, $\\dim(\\oo_H^\\mathrm{st})$ is the Zariski dimension of the orbit as an algebraic variety.\n\\item\nIf we further assume the following hypothesis, then the corresponding upper-bound conjecture holds for $\\phi$, i.e., \n\\[\n\\bar\\mathcal{N}(\\pi)^{\\max} \\leq d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\n\\]\nfor any $\\pi \\in \\Pi_\\psi$.\n\n\\begin{hyp}\\label{hypo}\nLet $\\oo_H^\\st \\in \\NN(\\h(\\overline{F}))$ be a special nilpotent orbit.\nDefine $\\oo_G^\\st \\in \\NN(\\g^\\theta(\\overline{F}))$ by \n\\[\n\\oo_G^\\st = d_{G^\\vee}(\\Ad(G^\\vee)d_{H}(\\oo_H^\\st)). \n\\]\nThen for any constants $\\{c_{\\oo_H} \\;|\\; \\oo_H \\subset \\oo_H^\\st\\}$ such that \n$\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H}$ is stable, \nthere exist constants $\\{a_{\\oo_G} \\;|\\; \\oo_G \\subset \\oo_G^\\st\\}$ such that \n\\[\n\\sum_{\\oo_G \\subset \\oo_G^\\st} a_{\\oo_G} \\hat\\mu_{\\oo_G}\n\\quad\\text{is equal to the transfer of}\\quad\n\\sum_{\\oo_H \\subset \\oo_H^\\st} c_{\\oo_H} \\hat\\mu_{\\oo_H} \n\\]\non a small enough neighborhood of $\\theta \\in G(F) \\rtimes \\theta$.\n\\end{hyp}\n\n\\begin{conj}\\label{upper}\nLet $\\phi \\in \\Phi(H(F))$. \nThen for any $\\pi \\in \\Pi_\\phi$, we have\n\\[\n\\bar\\NN(\\hat\\pi)^{\\max} \\leq d_{H^\\vee}(\\Ad(H^\\vee)N_\\phi).\n\\]\nMoreover, if $\\phi$ is tempered, \nthen the equality holds for some $\\pi \\in \\Pi_\\phi$. \n\\end{conj}\n\\end{enumerate}", "theorem_type": ["Existence", "Implication"], "mcq": {"question": "Let $H$ be a split classical group over $F$ with Lie algebra $\\mathfrak h$, and let\n$\\phi: W_F\\times \\mathrm{SL}_2(\\mathbb C)\\to H^\\vee$ be a tempered $L$-parameter. Assume one of the following:\n(B) $H=\\mathrm{SO}_{2n+1}$, $p>6n+3$, and $\\phi$ is arbitrary;\n(C) $H=\\mathrm{Sp}_{2n}$, $p>6n$, and for every irreducible self-dual subrepresentation $\\phi_0\\subset \\phi$, one has $\\det(\\phi_0)|_{I_F}=1$;\n(D) $H=\\mathrm{SO}_{2n}$, $p>6n$, and for every irreducible self-dual subrepresentation $\\phi_0\\subset \\phi$, one has $\\det(\\phi_0)|_{I_F}=1$.\nSet $\\psi=\\widehat\\phi$, and let $\\Pi_\\psi$ be the packet attached to $\\psi$. Let $N_\\phi\\in \\mathfrak h^\\vee$ be the image of $\\begin{pmatrix}0&1\\\\0&0\\end{pmatrix}$ under $d\\phi|_{\\mathrm{SL}_2(\\mathbb C)}$, and let $d_H:\\mathcal N(\\mathfrak h(\\overline F))\\to \\mathcal N(\\mathfrak h^\\vee)$ and $d_{H^\\vee}:\\mathcal N(\\mathfrak h^\\vee)\\to \\mathcal N(\\mathfrak h(\\overline F))$ be the Spaltenstein duality maps. For an irreducible admissible representation $\\pi$ of $H(F)$, write\n$\\Theta_\\pi(\\exp X)=\\sum_{\\mathcal O_H} c_{\\mathcal O_H}(\\pi)\\,\\widehat\\mu_{\\mathcal O_H}(\\exp X)$\nnear $0$, define $\\mathcal N(\\pi)=\\{\\mathcal O_H: c_{\\mathcal O_H}(\\pi)\\neq 0\\}$, let $\\bar{\\mathcal N}(\\pi)$ be the set of nilpotent $\\mathrm{Ad}(H(\\overline F))$-orbits containing some rational orbit in $\\mathcal N(\\pi)$, and let $\\bar{\\mathcal N}(\\pi)^{\\max}$ be the maximal elements of $\\bar{\\mathcal N}(\\pi)$ for the closure order $\\mathcal O\\le \\mathcal O'$ iff $\\mathcal O\\subset \\overline{\\mathcal O'}$. Under these assumptions, which statement is valid?", "correct_choice": {"label": "A", "text": "All of the following hold:\n(1) There exists $\\pi\\in \\Pi_\\psi$ such that\n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\\in \\bar{\\mathcal N}(\\pi)^{\\max}.\n\\]\n(2) For every $\\pi\\in \\Pi_\\psi$ and every $\\mathcal O_H^{\\mathrm{st}}\\in \\bar{\\mathcal N}(\\pi)$, if\n$\\mathcal O_H^{\\mathrm{st}}\\neq d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)$, then\n\\[\n\\dim(\\mathcal O_H^{\\mathrm{st}})<\\dim\\bigl(d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\\bigr),\n\\]\nwhere $\\dim$ is the Zariski dimension of the orbit.\n(3) If one further assumes the following transfer hypothesis: for every special nilpotent orbit $\\mathcal O_H^{\\mathrm{st}}\\in \\mathcal N(\\mathfrak h(\\overline F))$, after defining\n\\[\n\\mathcal O_G^{\\mathrm{st}}=d_{G^\\vee}(\\mathrm{Ad}(G^\\vee)d_H(\\mathcal O_H^{\\mathrm{st}})),\n\\]\nany stable distribution of the form $\\sum_{\\mathcal O_H\\subset \\mathcal O_H^{\\mathrm{st}}} c_{\\mathcal O_H}\\widehat\\mu_{\\mathcal O_H}$ is transferred, on a sufficiently small neighborhood of $\\theta\\in G(F)\\rtimes \\theta$, to some distribution $\\sum_{\\mathcal O_G\\subset \\mathcal O_G^{\\mathrm{st}}} a_{\\mathcal O_G}\\widehat\\mu_{\\mathcal O_G}$, then for every $\\pi\\in \\Pi_\\psi$ one has\n\\[\n\\bar{\\mathcal N}(\\pi)^{\\max}\\le d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi).\n\\]"}, "choices": [{"label": "B", "text": "All of the following hold:\n(1) For every $\\pi\\in \\Pi_\\psi$ such that $d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\\in \\bar{\\mathcal N}(\\pi)$, one has\n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\\in \\bar{\\mathcal N}(\\pi)^{\\max}.\n\\]\n(2) For every $\\pi\\in \\Pi_\\psi$ and every $\\mathcal O_H^{\\mathrm{st}}\\in \\bar{\\mathcal N}(\\pi)$, if\n$\\mathcal O_H^{\\mathrm{st}}\\neq d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)$, then\n\\[\n\\dim(\\mathcal O_H^{\\mathrm{st}})\\le \\dim\\bigl(d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\\bigr),\n\\]\nwhere $\\dim$ is the Zariski dimension of the orbit.\n(3) If one further assumes the transfer hypothesis stated in the theorem, then for every $\\pi\\in \\Pi_\\psi$ one has\n\\[\n\\bar{\\mathcal N}(\\pi)\\le d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi).\n\\]"}, {"label": "C", "text": "Both of the following hold:\n(1) There exists $\\pi\\in \\Pi_\\psi$ such that\n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\\in \\bar{\\mathcal N}(\\pi)^{\\max}.\n\\]\n(2) For every $\\pi\\in \\Pi_\\psi$ and every $\\mathcal O_H^{\\mathrm{st}}\\in \\bar{\\mathcal N}(\\pi)$, if\n$\\mathcal O_H^{\\mathrm{st}}\\neq d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)$, then\n\\[\n\\dim(\\mathcal O_H^{\\mathrm{st}})<\\dim\\bigl(d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\\bigr).\n\\]"}, {"label": "D", "text": "All of the following hold:\n(1) There exists $\\pi\\in \\Pi_\\psi$ such that\n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)= \\bar{\\mathcal N}(\\pi)^{\\max}.\n\\]\n(2) For every $\\pi\\in \\Pi_\\psi$ and every $\\mathcal O_H^{\\mathrm{st}}\\in \\bar{\\mathcal N}(\\pi)$, one has\n\\[\n\\mathcal O_H^{\\mathrm{st}}\\le d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi).\n\\]\n(3) In particular, without any additional transfer hypothesis, for every $\\pi\\in \\Pi_\\psi$ one has\n\\[\n\\bar{\\mathcal N}(\\pi)^{\\max}\\le d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi).\n\\]"}, {"label": "E", "text": "All of the following hold:\n(1) There exists $\\pi\\in \\Pi_\\psi$ such that\n\\[\nd_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\\in \\bar{\\mathcal N}(\\pi)^{\\max}.\n\\]\n(2) For every $\\pi\\in \\Pi_\\psi$ and every $\\mathcal O_H^{\\mathrm{st}}\\in \\bar{\\mathcal N}(\\pi)$, if\n$\\mathcal O_H^{\\mathrm{st}}\\neq d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)$, then\n\\[\n\\dim(\\mathcal O_H^{\\mathrm{st}})<\\dim\\bigl(d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi)\\bigr).\n\\]\n(3) If one further assumes the following transfer hypothesis: for every special nilpotent orbit $\\mathcal O_H^{\\mathrm{st}}\\in \\mathcal N(\\mathfrak h(\\overline F))$, after defining\n\\[\n\\mathcal O_G^{\\mathrm{st}}=d_{G^\\vee}(\\mathrm{Ad}(G^\\vee)d_H(\\mathcal O_H^{\\mathrm{st}})),\n\\]\nthere exist constants $a_{\\mathcal O_G}$ such that the specific distribution\n\\[\n\\sum_{\\mathcal O_G\\subset \\mathcal O_G^{\\mathrm{st}}}\\widehat\\mu_{\\mathcal O_G}\n\\]\nis equal, on a sufficiently small neighborhood of $\\theta\\in G(F)\\rtimes \\theta$, to the transfer of every stable distribution of the form $\\sum_{\\mathcal O_H\\subset \\mathcal O_H^{\\mathrm{st}}} c_{\\mathcal O_H}\\widehat\\mu_{\\mathcal O_H}$; then for every $\\pi\\in \\Pi_\\psi$ one has\n\\[\n\\bar{\\mathcal N}(\\pi)^{\\max}\\le d_{H^\\vee}(\\mathrm{Ad}(H^\\vee)N_\\phi).\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "maximal-elements conclusion strengthened to all orbits and strict inequality weakened to non-strict", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the additional upper-bound conclusion under the extra transfer hypothesis", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "removed dependence on the extra transfer hypothesis and replaced existence/membership by equality of maximal set", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "quantification in the transfer hypothesis replaced by a single fixed transferred distribution independent of the stable coefficients", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem itself does not reveal the correct option; the correct statement must be identified by comparing nuanced alternatives. Although the source JSON contains an annotated correct choice, that is external metadata rather than leakage in the posed MCQ."}, "TAS": {"score": 1, "justification": "The item is very close to theorem recall: one option essentially reproduces the intended theorem, while others are slight quantifier or hypothesis modifications. It is not purely tautological, but it is only a mild reformulation rather than a genuinely new application."}, "GPS": {"score": 1, "justification": "Selecting the answer requires some careful reasoning about logical strength, strict vs. non-strict inequalities, maximal elements vs. all elements, and the role of the extra transfer hypothesis. Still, the task mainly tests precise recognition of a theorem statement rather than substantial generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are strong: they are plausible theorem-like variants involving common failure modes such as overstrengthening conclusions, weakening hypotheses, or altering quantification. They are distinct and mathematically aligned with realistic misunderstandings."}, "total_score": 6, "overall_assessment": "A well-constructed theorem-discrimination MCQ with strong distractors and no stem-level answer leakage, but it primarily tests precise recall/logical parsing of a theorem rather than deeper generative reasoning."}}
{"id": "2602.22550v1", "paper_link": "http://arxiv.org/abs/2602.22550v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm}\nAssume $1\\leq p <2$, $d\\geq 1$ and the pressure $P$ satisfies $P'(\\bar{\\rho})>0$ for $\\bar{\\rho}>0$. There is a small constant $\\delta_1>0$ such that if\n$\n\\|(n_0,\\bv_0)\\|_{\\dB_{p,1}^{\\frac{d}{p}}}^{\\ell,J_\\varepsilon}\n+\\varepsilon\\|(n_0,\\bv_0)\\|_{\\dB_{2,1}^{\\frac{d}{2}+1}}^{h,J_\\varepsilon}\n\\leq \\delta_1,\n$\nthen the Cauchy problem \\eqref{1.2}-\\eqref{1.2initial} admits a uniform global unique solution in $E_\\infty^{J_\\varepsilon}$ for all $\\varepsilon>0$.", "start_pos": 8026, "end_pos": 8525, "label": "thm"}, "ref_dict": {"1.1": "\\begin{eqnarray}\\label{1.1}\n\\left\\{\\aligned\n&\\d_t \\rho+\\div (\\rho \\bu)=0, \\\\\n&\\d_t (\\rho \\bu)+\\nabla \\cdot (\\rho \\bu\\otimes \\bu)+\\nabla P(\\rho)+\\alpha\\rho \\bu=0\n\\endaligned\\right.\n\\end{eqnarray}", "NonClassicalProLaw1": "\\begin{lemma}\\label{NonClassicalProLaw1}\nLet $0<s_1<\\frac{d}{2}$, $1\\leq p<2$. Then, we have the following inequality\n\\begin{align}\\label{newproduct}\n\\|ab\\|_{\\dB_{p,1}^{s_1}}^{\\ell,J_\\varepsilon}\n\\leq C\n\\left(\\|b\\|_{\\dB_{p,1}^{s_1}}^{\\ell,J_\\varepsilon}+ 2^{(s_1-\\frac{d}{p})J_\\varepsilon}\\|b\\|_{\\dB_{2,1}^{\\frac{d}{2}}}^{h,J_\\varepsilon} \\right)\\|a\\|_{\\dB_{2,1}^{\\frac{d}{2}}}\n\\end{align}\nwith $C$ independent on $\\varepsilon$.\n\\end{lemma}", "thm": "\\begin{theorem}\\label{thm}\nAssume $1\\leq p <2$, $d\\geq 1$ and the pressure $P$ satisfies $P'(\\bar{\\rho})>0$ for $\\bar{\\rho}>0$. There is a small constant $\\delta_1>0$ such that if\n$\n\\|(n_0,\\bv_0)\\|_{\\dB_{p,1}^{\\frac{d}{p}}}^{\\ell,J_\\varepsilon}\n+\\varepsilon\\|(n_0,\\bv_0)\\|_{\\dB_{2,1}^{\\frac{d}{2}+1}}^{h,J_\\varepsilon}\n\\leq \\delta_1,\n$\nthen the Cauchy problem \\eqref{1.2}-\\eqref{1.2initial} admits a uniform global unique solution in $E_\\infty^{J_\\varepsilon}$ for all $\\varepsilon>0$.\n\\end{theorem}", "1.2initial": "\\begin{eqnarray}\\label{1.2initial}\n (n,\\bv)|_{t=0}=(n_0, \\bv_0).\n\\end{eqnarray}", "1.2": "\\begin{eqnarray}\\label{1.2}\n\\left\\{\\aligned\n&\\d_t n + \\bv \\cdot  \\nabla n + P'(\\bar{\\rho})\\div \\bv + G(n)\\div \\bv = 0 \\\\\n&\\d_t \\bv+ \\bv\\cdot \\nabla \\bv+\\nabla n= -\\frac{1}{\\varepsilon} \\bv\n\\endaligned\\right.\n\\end{eqnarray}", "DefHLB": "\\begin{equation}\\label{DefHLB}\n\\|f\\|_{\\dB_{q_1,1}^{s_1}}^{h,J_{\\varepsilon}}\\triangleq \\sum_{j\\geq J_{\\varepsilon}}\n2^{s_1j}\\|\\ddj f\\|_{q_1}\\quad \\mbox{and}\\quad\n\\|f\\|_{\\dB_{q_2,1}^{s_2}}^{\\ell,J_{\\varepsilon}}\\triangleq \\sum_{j\\leq J_{\\varepsilon}-1}\n2^{s_2j}\\|\\ddj f\\|_{q_2}.\n\\end{equation}"}, "pre_theorem_intro_text_len": 4842, "pre_theorem_intro_text": "This work studies the compressible Euler equations with linear damping, a model describing gas flow through a porous medium where the solid matrix exerts a frictional force proportional to and opposite to the fluid momentum. The governing system is given by\n\\begin{eqnarray}\\label{1.1}\n\\left\\{\\aligned\n&\\d_t \\rho+\\mbox{\\rm div}\\;\\! (\\rho \\bu)=0, \\\\\n&\\d_t (\\rho \\bu)+\\nabla \\cdot (\\rho \\bu\\otimes \\bu)+\\nabla P(\\rho)+\\alpha\\rho \\bu=0\n\\endaligned\\right.\n\\end{eqnarray}\nfor $t\\geq0,\\ \\bm{x}\\in \\R^d(d\\geq 1)$ and a damping coefficient $\\alpha>0$\n. Here $\\rho\\in\\mathbb{R}^{+}$, $\\bu=(u^1, u^2,\\cdots, u^d)^\\top$ ($\\top$ represents transpose) denote the density and velocity of fluid flow, respectively. The pressure function $P=P(\\rho)$ is assumed to be smooth around the constant reference density $\\bar{\\rho}>0$.\n\nThere are lots of mathematical results about the existence and asymptotic behavior of system \\eqref{1.1} in sobolev space (see \\cite{HL,HuangFM2,Wangdehua,Sui,Tan3,XY} ). In critical Besov space, Fang and Xu \\cite{FXu} (with improvements in \\cite{Jiu}) studied the existence and asymptotic behavior of classical solutions. Later, Xu and Wang \\cite{XuWang} justified the relaxation convergence from \\eqref{1.1} to the porous medium equation. And also, there are some results on hyperbolic system for balance laws including \\eqref{1.1} (see \\cite{XuKawaARM14,XuKawaARM15}). Recently, Crin-Barat and Danchin \\cite{BaratDanchin,BaratDanchin2,BaratDanchin3} improved those results \\cite{XuKawaARM14,XuKawaARM15,XuWang} in the $L^2$-$L^p$ hybrid Besov spaces with $2\\leq p\\leq\\max\\{4,\\frac{2d}{d+2}\\}$, where the low frequencies are bounded in $L^p$-type spaces and the high frequencies in $L^2$-type spaces with a specific linearity assumption. And then, the first author and Xu \\cite{XuZhang} remove the assumption and to show that the results of \\cite{BaratDanchin,BaratDanchin2} hold true for the general pressure function. Very recently, Crin-Barat and Song \\cite{BSong} extend the work of \\cite{BaratDanchin,BaratDanchin2,XuZhang} to the case $p\\in[2,\\infty)$. And a natural question is how about $p<2$?\n\nIntroducing a new unknown called ``enthalpy\"\n$n(\\rho)\\triangleq \\int_1^{\\rho} \\frac{P'(s)}{s} ds.$ System \\eqref{1.1} can be rewritten as\n\\begin{eqnarray}\\label{1.2}\n\\left\\{\\aligned\n&\\d_t n + \\bv \\cdot  \\nabla n + P'(\\bar{\\rho})\\mbox{\\rm div}\\;\\! \\bv + G(n)\\mbox{\\rm div}\\;\\! \\bv = 0 \\\\\n&\\d_t \\bv+ \\bv\\cdot \\nabla \\bv+\\nabla n= -\\frac{1}{\\varepsilon} \\bv\n\\endaligned\\right.\n\\end{eqnarray}\nwith the initial data\n\\begin{eqnarray}\\label{1.2initial}\n (n,\\bv)|_{t=0}=(n_0, \\bv_0).\n\\end{eqnarray}\nHere $P'(\\bar{\\rho})>0$ for $\\bar{\\rho}>0$, $\\bar{\\rho}=P'(\\bar{\\rho})=1$ and the composite function $G(n)$ is smooth.\n\nDefine $J_\\varepsilon=-[\\log_2\\varepsilon]+k$ being the threshold between high and low frequencies with a suitable integer $k$ is to be determined later, and the space $E_T^{J_\\varepsilon}$ as follows\n\\begin{equation*}\n\\begin{split}\n&E_T^{J_\\varepsilon}=\\bigg\\{(n,\\bv)|(n^{\\ell,J_\\varepsilon},\\bv^{\\ell,J_\\varepsilon})\\in \\cC_b([0,T];\\dB_{p,1}^{\\frac{d}{p}}),\\\n\\varepsilon n^{\\ell,J_\\varepsilon}\\in \\tL_T^1(\\dB_{p,1}^{\\frac{d}{p}+2}),\\\n\\bv^{\\ell,J_\\varepsilon}\\in \\tL_T^1(\\dB_{p,1}^{\\frac{d}{p}+1}),\n\\varepsilon^{-\\frac{1}{2}} \\bv\\in \\tL_T^2(\\dB_{p,1}^{\\frac{d}{p}}),\\\\\n&\\qquad\\qquad\n(\\varepsilon n^{h,J_\\varepsilon}, \\varepsilon\\bv^{h,J_\\varepsilon})\\in \\cC_b([0,T];\\dB_{2,1}^{\\frac{d}{2}+1}),\\\n(n^{h,J_\\varepsilon},\\bv^{h,J_\\varepsilon}) \\in \\tL_T^1(\\dB_{2,1}^{\\frac{d}{2}+1}),\n\\varepsilon^{-1}\\bv+\\nabla n\\in \\tL_T^1(\\dB_{p,1}^{\\frac{d}{p}})\n\\bigg\\}.\n\\end{split}\n\\end{equation*}\nWhen $T=\\infty$ we also use $E_\\infty^{J_\\varepsilon}$ for convenience.\nFor the reader's convenience, We introduce the notations $\\|\\cdot\\|_{\\dB_{q_1,1}^s}^{h,J_{\\varepsilon}}$ and\n$\\|\\cdot\\|_{\\dB_{q_2,1}^s}^{\\ell,J_{\\varepsilon}}$ to denote Besov semi-norms with respect to the threshold $J_{\\varepsilon}$, that is,\n\\begin{equation}\\label{DefHLB}\n\\|f\\|_{\\dB_{q_1,1}^{s_1}}^{h,J_{\\varepsilon}}\\triangleq \\sum_{j\\geq J_{\\varepsilon}}\n2^{s_1j}\\|\\dot \\Delta_j f\\|_{q_1}\\quad \\mbox{and}\\quad\n\\|f\\|_{\\dB_{q_2,1}^{s_2}}^{\\ell,J_{\\varepsilon}}\\triangleq \\sum_{j\\leq J_{\\varepsilon}-1}\n2^{s_2j}\\|\\dot \\Delta_j f\\|_{q_2}.\n\\end{equation}\nIt is not difficult to deduce that for all $\\sigma_0>0$,\n\\begin{equation}\\label{HLEst}\n\\|f\\|_{\\dB_{q_1,1}^{s_1}}^{h,J_{\\varepsilon}}\n\\leq\n2^{-\\sigma_0J_{\\varepsilon}}\\|f\\|_{\\dB_{q_1,1}^{s_1+\\sigma_0}}^{h,J_{\\varepsilon}}\\quad \\mbox{and}\\quad\n\\|f\\|_{\\dB_{q_1,1}^{s_1}}^{\\ell,J_{\\varepsilon}}\n\\leq\n2^{\\sigma_0J_{\\varepsilon}}\\|f\\|_{\\dB_{q_1,1}^{s_1-\\sigma_0}}^{\\ell,J_{\\varepsilon}}.\n\\end{equation}\n\nThe main goal of this paper is to broaden the assumption on $p$ and to show that the results of \\cite{BaratDanchin,BaratDanchin2,XuZhang} hold true for the case $p<2$. Our main result is stated as follows.", "context": "This work studies the compressible Euler equations with linear damping, a model describing gas flow through a porous medium where the solid matrix exerts a frictional force proportional to and opposite to the fluid momentum. The governing system is given by\n\\begin{eqnarray}\\label{1.1}\n\\left\\{\\aligned\n&\\d_t \\rho+\\mbox{\\rm div}\\;\\! (\\rho \\bu)=0, \\\\\n&\\d_t (\\rho \\bu)+\\nabla \\cdot (\\rho \\bu\\otimes \\bu)+\\nabla P(\\rho)+\\alpha\\rho \\bu=0\n\\endaligned\\right.\n\\end{eqnarray}\nfor $t\\geq0,\\ \\bm{x}\\in \\R^d(d\\geq 1)$ and a damping coefficient $\\alpha>0$\n. Here $\\rho\\in\\mathbb{R}^{+}$, $\\bu=(u^1, u^2,\\cdots, u^d)^\\top$ ($\\top$ represents transpose) denote the density and velocity of fluid flow, respectively. The pressure function $P=P(\\rho)$ is assumed to be smooth around the constant reference density $\\bar{\\rho}>0$.\n\nThere are lots of mathematical results about the existence and asymptotic behavior of system \\eqref{1.1} in sobolev space (see \\cite{HL,HuangFM2,Wangdehua,Sui,Tan3,XY} ). In critical Besov space, Fang and Xu \\cite{FXu} (with improvements in \\cite{Jiu}) studied the existence and asymptotic behavior of classical solutions. Later, Xu and Wang \\cite{XuWang} justified the relaxation convergence from \\eqref{1.1} to the porous medium equation. And also, there are some results on hyperbolic system for balance laws including \\eqref{1.1} (see \\cite{XuKawaARM14,XuKawaARM15}). Recently, Crin-Barat and Danchin \\cite{BaratDanchin,BaratDanchin2,BaratDanchin3} improved those results \\cite{XuKawaARM14,XuKawaARM15,XuWang} in the $L^2$-$L^p$ hybrid Besov spaces with $2\\leq p\\leq\\max\\{4,\\frac{2d}{d+2}\\}$, where the low frequencies are bounded in $L^p$-type spaces and the high frequencies in $L^2$-type spaces with a specific linearity assumption. And then, the first author and Xu \\cite{XuZhang} remove the assumption and to show that the results of \\cite{BaratDanchin,BaratDanchin2} hold true for the general pressure function. Very recently, Crin-Barat and Song \\cite{BSong} extend the work of \\cite{BaratDanchin,BaratDanchin2,XuZhang} to the case $p\\in[2,\\infty)$. And a natural question is how about $p<2$?\n\nIntroducing a new unknown called ``enthalpy\"\n$n(\\rho)\\triangleq \\int_1^{\\rho} \\frac{P'(s)}{s} ds.$ System \\eqref{1.1} can be rewritten as\n\\begin{eqnarray}\\label{1.2}\n\\left\\{\\aligned\n&\\d_t n + \\bv \\cdot \\nabla n + P'(\\bar{\\rho})\\mbox{\\rm div}\\;\\! \\bv + G(n)\\mbox{\\rm div}\\;\\! \\bv = 0 \\\\\n&\\d_t \\bv+ \\bv\\cdot \\nabla \\bv+\\nabla n= -\\frac{1}{\\varepsilon} \\bv\n\\endaligned\\right.\n\\end{eqnarray}\nwith the initial data\n\\begin{eqnarray}\\label{1.2initial}\n (n,\\bv)|_{t=0}=(n_0, \\bv_0).\n\\end{eqnarray}\nHere $P'(\\bar{\\rho})>0$ for $\\bar{\\rho}>0$, $\\bar{\\rho}=P'(\\bar{\\rho})=1$ and the composite function $G(n)$ is smooth.\n\nDefine $J_\\varepsilon=-[\\log_2\\varepsilon]+k$ being the threshold between high and low frequencies with a suitable integer $k$ is to be determined later, and the space $E_T^{J_\\varepsilon}$ as follows\n\\begin{equation*}\n\\begin{split}\n&E_T^{J_\\varepsilon}=\\bigg\\{(n,\\bv)|(n^{\\ell,J_\\varepsilon},\\bv^{\\ell,J_\\varepsilon})\\in \\cC_b([0,T];\\dB_{p,1}^{\\frac{d}{p}}),\\\n\\varepsilon n^{\\ell,J_\\varepsilon}\\in \\tL_T^1(\\dB_{p,1}^{\\frac{d}{p}+2}),\\\n\\bv^{\\ell,J_\\varepsilon}\\in \\tL_T^1(\\dB_{p,1}^{\\frac{d}{p}+1}),\n\\varepsilon^{-\\frac{1}{2}} \\bv\\in \\tL_T^2(\\dB_{p,1}^{\\frac{d}{p}}),\\\\\n&\\qquad\\qquad\n(\\varepsilon n^{h,J_\\varepsilon}, \\varepsilon\\bv^{h,J_\\varepsilon})\\in \\cC_b([0,T];\\dB_{2,1}^{\\frac{d}{2}+1}),\\\n(n^{h,J_\\varepsilon},\\bv^{h,J_\\varepsilon}) \\in \\tL_T^1(\\dB_{2,1}^{\\frac{d}{2}+1}),\n\\varepsilon^{-1}\\bv+\\nabla n\\in \\tL_T^1(\\dB_{p,1}^{\\frac{d}{p}})\n\\bigg\\}.\n\\end{split}\n\\end{equation*}\nWhen $T=\\infty$ we also use $E_\\infty^{J_\\varepsilon}$ for convenience.\nFor the reader's convenience, We introduce the notations $\\|\\cdot\\|_{\\dB_{q_1,1}^s}^{h,J_{\\varepsilon}}$ and\n$\\|\\cdot\\|_{\\dB_{q_2,1}^s}^{\\ell,J_{\\varepsilon}}$ to denote Besov semi-norms with respect to the threshold $J_{\\varepsilon}$, that is,\n\\begin{equation}\\label{DefHLB}\n\\|f\\|_{\\dB_{q_1,1}^{s_1}}^{h,J_{\\varepsilon}}\\triangleq \\sum_{j\\geq J_{\\varepsilon}}\n2^{s_1j}\\|\\dot \\Delta_j f\\|_{q_1}\\quad \\mbox{and}\\quad\n\\|f\\|_{\\dB_{q_2,1}^{s_2}}^{\\ell,J_{\\varepsilon}}\\triangleq \\sum_{j\\leq J_{\\varepsilon}-1}\n2^{s_2j}\\|\\dot \\Delta_j f\\|_{q_2}.\n\\end{equation}\nIt is not difficult to deduce that for all $\\sigma_0>0$,\n\\begin{equation}\\label{HLEst}\n\\|f\\|_{\\dB_{q_1,1}^{s_1}}^{h,J_{\\varepsilon}}\n\\leq\n2^{-\\sigma_0J_{\\varepsilon}}\\|f\\|_{\\dB_{q_1,1}^{s_1+\\sigma_0}}^{h,J_{\\varepsilon}}\\quad \\mbox{and}\\quad\n\\|f\\|_{\\dB_{q_1,1}^{s_1}}^{\\ell,J_{\\varepsilon}}\n\\leq\n2^{\\sigma_0J_{\\varepsilon}}\\|f\\|_{\\dB_{q_1,1}^{s_1-\\sigma_0}}^{\\ell,J_{\\varepsilon}}.\n\\end{equation}\n\nThe main goal of this paper is to broaden the assumption on $p$ and to show that the results of \\cite{BaratDanchin,BaratDanchin2,XuZhang} hold true for the case $p<2$. Our main result is stated as follows.\n\n\\begin{eqnarray}\\label{1.1}\n\\left\\{\\aligned\n&\\d_t \\rho+\\div (\\rho \\bu)=0, \\\\\n&\\d_t (\\rho \\bu)+\\nabla \\cdot (\\rho \\bu\\otimes \\bu)+\\nabla P(\\rho)+\\alpha\\rho \\bu=0\n\\endaligned\\right.\n\\end{eqnarray}\n\n\\begin{eqnarray}\\label{1.2}\n\\left\\{\\aligned\n&\\d_t n + \\bv \\cdot  \\nabla n + P'(\\bar{\\rho})\\div \\bv + G(n)\\div \\bv = 0 \\\\\n&\\d_t \\bv+ \\bv\\cdot \\nabla \\bv+\\nabla n= -\\frac{1}{\\varepsilon} \\bv\n\\endaligned\\right.\n\\end{eqnarray}", "full_context": "This work studies the compressible Euler equations with linear damping, a model describing gas flow through a porous medium where the solid matrix exerts a frictional force proportional to and opposite to the fluid momentum. The governing system is given by\n\\begin{eqnarray}\\label{1.1}\n\\left\\{\\aligned\n&\\d_t \\rho+\\mbox{\\rm div}\\;\\! (\\rho \\bu)=0, \\\\\n&\\d_t (\\rho \\bu)+\\nabla \\cdot (\\rho \\bu\\otimes \\bu)+\\nabla P(\\rho)+\\alpha\\rho \\bu=0\n\\endaligned\\right.\n\\end{eqnarray}\nfor $t\\geq0,\\ \\bm{x}\\in \\R^d(d\\geq 1)$ and a damping coefficient $\\alpha>0$\n. Here $\\rho\\in\\mathbb{R}^{+}$, $\\bu=(u^1, u^2,\\cdots, u^d)^\\top$ ($\\top$ represents transpose) denote the density and velocity of fluid flow, respectively. The pressure function $P=P(\\rho)$ is assumed to be smooth around the constant reference density $\\bar{\\rho}>0$.\n\nThere are lots of mathematical results about the existence and asymptotic behavior of system \\eqref{1.1} in sobolev space (see \\cite{HL,HuangFM2,Wangdehua,Sui,Tan3,XY} ). In critical Besov space, Fang and Xu \\cite{FXu} (with improvements in \\cite{Jiu}) studied the existence and asymptotic behavior of classical solutions. Later, Xu and Wang \\cite{XuWang} justified the relaxation convergence from \\eqref{1.1} to the porous medium equation. And also, there are some results on hyperbolic system for balance laws including \\eqref{1.1} (see \\cite{XuKawaARM14,XuKawaARM15}). Recently, Crin-Barat and Danchin \\cite{BaratDanchin,BaratDanchin2,BaratDanchin3} improved those results \\cite{XuKawaARM14,XuKawaARM15,XuWang} in the $L^2$-$L^p$ hybrid Besov spaces with $2\\leq p\\leq\\max\\{4,\\frac{2d}{d+2}\\}$, where the low frequencies are bounded in $L^p$-type spaces and the high frequencies in $L^2$-type spaces with a specific linearity assumption. And then, the first author and Xu \\cite{XuZhang} remove the assumption and to show that the results of \\cite{BaratDanchin,BaratDanchin2} hold true for the general pressure function. Very recently, Crin-Barat and Song \\cite{BSong} extend the work of \\cite{BaratDanchin,BaratDanchin2,XuZhang} to the case $p\\in[2,\\infty)$. And a natural question is how about $p<2$?\n\nIntroducing a new unknown called ``enthalpy\"\n$n(\\rho)\\triangleq \\int_1^{\\rho} \\frac{P'(s)}{s} ds.$ System \\eqref{1.1} can be rewritten as\n\\begin{eqnarray}\\label{1.2}\n\\left\\{\\aligned\n&\\d_t n + \\bv \\cdot \\nabla n + P'(\\bar{\\rho})\\mbox{\\rm div}\\;\\! \\bv + G(n)\\mbox{\\rm div}\\;\\! \\bv = 0 \\\\\n&\\d_t \\bv+ \\bv\\cdot \\nabla \\bv+\\nabla n= -\\frac{1}{\\varepsilon} \\bv\n\\endaligned\\right.\n\\end{eqnarray}\nwith the initial data\n\\begin{eqnarray}\\label{1.2initial}\n (n,\\bv)|_{t=0}=(n_0, \\bv_0).\n\\end{eqnarray}\nHere $P'(\\bar{\\rho})>0$ for $\\bar{\\rho}>0$, $\\bar{\\rho}=P'(\\bar{\\rho})=1$ and the composite function $G(n)$ is smooth.\n\nDefine $J_\\varepsilon=-[\\log_2\\varepsilon]+k$ being the threshold between high and low frequencies with a suitable integer $k$ is to be determined later, and the space $E_T^{J_\\varepsilon}$ as follows\n\\begin{equation*}\n\\begin{split}\n&E_T^{J_\\varepsilon}=\\bigg\\{(n,\\bv)|(n^{\\ell,J_\\varepsilon},\\bv^{\\ell,J_\\varepsilon})\\in \\cC_b([0,T];\\dB_{p,1}^{\\frac{d}{p}}),\\\n\\varepsilon n^{\\ell,J_\\varepsilon}\\in \\tL_T^1(\\dB_{p,1}^{\\frac{d}{p}+2}),\\\n\\bv^{\\ell,J_\\varepsilon}\\in \\tL_T^1(\\dB_{p,1}^{\\frac{d}{p}+1}),\n\\varepsilon^{-\\frac{1}{2}} \\bv\\in \\tL_T^2(\\dB_{p,1}^{\\frac{d}{p}}),\\\\\n&\\qquad\\qquad\n(\\varepsilon n^{h,J_\\varepsilon}, \\varepsilon\\bv^{h,J_\\varepsilon})\\in \\cC_b([0,T];\\dB_{2,1}^{\\frac{d}{2}+1}),\\\n(n^{h,J_\\varepsilon},\\bv^{h,J_\\varepsilon}) \\in \\tL_T^1(\\dB_{2,1}^{\\frac{d}{2}+1}),\n\\varepsilon^{-1}\\bv+\\nabla n\\in \\tL_T^1(\\dB_{p,1}^{\\frac{d}{p}})\n\\bigg\\}.\n\\end{split}\n\\end{equation*}\nWhen $T=\\infty$ we also use $E_\\infty^{J_\\varepsilon}$ for convenience.\nFor the reader's convenience, We introduce the notations $\\|\\cdot\\|_{\\dB_{q_1,1}^s}^{h,J_{\\varepsilon}}$ and\n$\\|\\cdot\\|_{\\dB_{q_2,1}^s}^{\\ell,J_{\\varepsilon}}$ to denote Besov semi-norms with respect to the threshold $J_{\\varepsilon}$, that is,\n\\begin{equation}\\label{DefHLB}\n\\|f\\|_{\\dB_{q_1,1}^{s_1}}^{h,J_{\\varepsilon}}\\triangleq \\sum_{j\\geq J_{\\varepsilon}}\n2^{s_1j}\\|\\dot \\Delta_j f\\|_{q_1}\\quad \\mbox{and}\\quad\n\\|f\\|_{\\dB_{q_2,1}^{s_2}}^{\\ell,J_{\\varepsilon}}\\triangleq \\sum_{j\\leq J_{\\varepsilon}-1}\n2^{s_2j}\\|\\dot \\Delta_j f\\|_{q_2}.\n\\end{equation}\nIt is not difficult to deduce that for all $\\sigma_0>0$,\n\\begin{equation}\\label{HLEst}\n\\|f\\|_{\\dB_{q_1,1}^{s_1}}^{h,J_{\\varepsilon}}\n\\leq\n2^{-\\sigma_0J_{\\varepsilon}}\\|f\\|_{\\dB_{q_1,1}^{s_1+\\sigma_0}}^{h,J_{\\varepsilon}}\\quad \\mbox{and}\\quad\n\\|f\\|_{\\dB_{q_1,1}^{s_1}}^{\\ell,J_{\\varepsilon}}\n\\leq\n2^{\\sigma_0J_{\\varepsilon}}\\|f\\|_{\\dB_{q_1,1}^{s_1-\\sigma_0}}^{\\ell,J_{\\varepsilon}}.\n\\end{equation}\n\nThe main goal of this paper is to broaden the assumption on $p$ and to show that the results of \\cite{BaratDanchin,BaratDanchin2,XuZhang} hold true for the case $p<2$. Our main result is stated as follows.\n\n\\begin{eqnarray}\\label{1.1}\n\\left\\{\\aligned\n&\\d_t \\rho+\\div (\\rho \\bu)=0, \\\\\n&\\d_t (\\rho \\bu)+\\nabla \\cdot (\\rho \\bu\\otimes \\bu)+\\nabla P(\\rho)+\\alpha\\rho \\bu=0\n\\endaligned\\right.\n\\end{eqnarray}\n\n\\begin{eqnarray}\\label{1.2}\n\\left\\{\\aligned\n&\\d_t n + \\bv \\cdot  \\nabla n + P'(\\bar{\\rho})\\div \\bv + G(n)\\div \\bv = 0 \\\\\n&\\d_t \\bv+ \\bv\\cdot \\nabla \\bv+\\nabla n= -\\frac{1}{\\varepsilon} \\bv\n\\endaligned\\right.\n\\end{eqnarray}\n\nIntroducing a new unknown called ``enthalpy\"\n$n(\\rho)\\triangleq \\int_1^{\\rho} \\frac{P'(s)}{s} ds.$ System \\eqref{1.1} can be rewritten as\n\\begin{eqnarray}\\label{1.2}\n\\left\\{\\aligned\n&\\d_t n + \\bv \\cdot \\nabla n + P'(\\bar{\\rho})\\div \\bv + G(n)\\div \\bv = 0 \\\\\n&\\d_t \\bv+ \\bv\\cdot \\nabla \\bv+\\nabla n= -\\frac{1}{\\varepsilon} \\bv\n\\endaligned\\right.\n\\end{eqnarray}\nwith the initial data\n\\begin{eqnarray}\\label{1.2initial}\n (n,\\bv)|_{t=0}=(n_0, \\bv_0).\n\\end{eqnarray}\nHere $P'(\\bar{\\rho})>0$ for $\\bar{\\rho}>0$, $\\bar{\\rho}=P'(\\bar{\\rho})=1$ and the composite function $G(n)$ is smooth.\n\n\\begin{remark}\nBased on Theorem \\ref{thm}, the relaxation limit for the case $p<2$ can be performed by the same procedure as \\cite{BaratDanchin2}.\n\\end{remark}\n\n\\section{Global a priori estimates}\nIn this section, we only give the key a priori estimate for the case $p<2$, which lead to the global existence and uniqueness of solutions in $E_\\infty^{J_\\varepsilon}$. See \\cite{BShouZhang} for more details. For convenience, we use $ f\\lesssim g $ to denote that there exists a generic constant $C>0$ independent on $\\varepsilon$ such that $f\\leq C g$ in this section.\n\nFinally, by classical remainder estimates (see \\cite[Theorem 2.85]{chemin}) we can directly obtain\n\\begin{equation}\\nonumber\n\\begin{aligned}\n\\|R[a,b]\\|_{\\dB_{p,1}^{s_1}}^{\\ell,J_\\varepsilon}\n\\lesssim\n\\|b\\|_{\\dB_{\\frac{2p}{2-p},1}^{s_1-\\frac{d}{2}}}\\|a\\|_{\\dB_{2,1}^{\\frac{d}{2}}}\n\\lesssim\n\\left(\\|b\\|_{\\dB_{p,1}^{s_1}}^{\\ell,J_\\varepsilon}+ 2^{(s_1-\\frac{d}{p})J_\\varepsilon}\\|b\\|_{\\dB_{2,1}^{\\frac{d}{2}}}^{h,J_\\varepsilon} \\right)\\|a\\|_{\\dB_{2,1}^{\\frac{d}{2}}}.\n\\end{aligned}\n\\end{equation}\nAdding above three inequality together, we can finally obtain \\eqref{newproduct}.\n\\end{proof}\nFor simplicity, we define $\\cX(T):=\\|(n,\\bv)\\|_{E_T^{J\\varepsilon}}$. The proof of Theorem \\ref{thm} reduces to establishing a global-in-time a priori estimate. Specifically, we claim that if\n\\begin{equation*}\\begin{split}\n\\|n\\|_{L^\\infty}+\\|\\bv\\|_{L^\\infty} \\ll 1 \\quad \\text{on} \\quad [0,T],\n\\end{split}\n\\end{equation*}\nthen there exists a constant $C$, independent of $T$ and $\\varepsilon$, such that\n\\begin{equation}\\label{APrioriEstimate}\n\\cX(T)\\leq C\\big( \\cX(0)+\\cX^2(T)\\big) \\qquad\\text{for all } \\varepsilon>0.\n\\end{equation}\n\nNext, we will deal with the nonlinear part. For the first nonlinear term, it follows the Lemma \\ref{NonClassicalProLaw1} that\n\\begin{equation*}\n\\begin{split}\n\\|\\bv\\cdot \\nabla n\\|_{\\tL_T^1(\\dB_{p,1}^{\\frac{d}{p}})}^{\\ell,J_\\varepsilon}\n&\\lesssim\n\\left(\n\\|\\nabla n\\|_{\\tL_T^2(\\dB_{p,1}^{\\frac{d}{p}})}^{\\ell,J_\\varepsilon}\n+ \\|\\nabla n\\|_{\\tL_T^2(\\dB_{2,1}^{\\frac{d}{2}})}^{h,J_\\varepsilon}\n\\right)\n\\|\\bv\\|_{\\tL_T^2(\\dB_{2,1}^{\\frac{d}{2}})}.\n\\end{split}\n\\end{equation*}\nThen the interpolation inequalities (see Lemma 3.2 in \\cite{XuZhang}) implies\n\\begin{equation}\\label{NL1}\n\\begin{split}\n\\|\\bv\\cdot \\nabla n\\|_{\\tL_T^1(\\dB_{p,1}^{\\frac{d}{p}})}^{\\ell,J_\\varepsilon}\n&\\lesssim\n\\left(\\left(\\| n\\|_{\\tL_T^\\infty(\\dB_{p,1}^{\\frac{d}{p}})}^{\\ell,J_\\varepsilon}\n\\|n\\|_{\\tL_T^1(\\dB_{p,1}^{\\frac{d}{p}+2})}^{\\ell,J_\\varepsilon}\\right)^{\\frac{1}{2}}\n+\\left(\\|n\\|_{\\tL_T^\\infty(\\dB_{2,1}^{\\frac{d}{2}+1})}^{h,J_\\varepsilon}\n\\|n\\|_{\\tL_T^1(\\dB_{2,1}^{\\frac{d}{2}+1})}^{h,J_\\varepsilon}\\right)^{\\frac{1}{2}}\n\\right)\\\\\n&\\quad\n\\times\\left(\\|\\bv\\|_{\\tL_T^2(\\dB_{p,1}^{\\frac{d}{p}})}^{\\ell,J_\\varepsilon}\n+\\varepsilon\\left(\\|\\bv\\|_{\\tL_T^\\infty(\\dB_{2,1}^{\\frac{d}{2}+1})}^{h,J_\\varepsilon}\n\\|\\bv\\|_{\\tL_T^1(\\dB_{2,1}^{\\frac{d}{2}+1})}^{h,J_\\varepsilon}\\right)^{\\frac{1}{2}}\\right)\\\\\n&\\lesssim\n\\cX^2(T)+\\cX(T)\\varepsilon^{-\\frac{1}{2}}\\|\\bv\\|_{\\tL_T^2(\\dB_{p,1}^{\\frac{d}{p}})}^{\\ell,J_\\varepsilon}.\n\\end{split}\n\\end{equation}\nAnd similarly, for the convection term we can deduce\n\\begin{equation}\\label{NL2}\n\\begin{split}\n\\|\\bv\\cdot \\nabla \\bv\\|_{\\tL_T^1(\\dB_{p,1}^{\\frac{d}{p}})}^{\\ell,J_\\varepsilon}\n\\lesssim\n\\cX^2(T).\n\\end{split}\n\\end{equation}\nFor the last nonlinear term, the inequality \\eqref{newproduct} implies\n\\begin{equation*}\n\\begin{split}\n\\|G(n)\\div \\bv\\|_{\\tL_T^1(\\dB_{p,1}^{\\frac{d}{p}})}^{\\ell,J_\\varepsilon}\n&\\lesssim\n\\|G(n)\\|_{\\tL_T^\\infty(\\dB_{2,1}^{\\frac{d}{2}})}\n\\left(\\|\\div \\bv\\|_{\\tL_T^1(\\dB_{p,1}^{\\frac{d}{p}})}^{\\ell,J_\\varepsilon}\n+ \\|\\div \\bv\\|_{\\tL_T^1(\\dB_{2,1}^{\\frac{d}{2}})}^{h,J_\\varepsilon}\\right).\n\\end{split}\n\\end{equation*}\nBy the classical composition estimate (see \\cite{chemin}), we can obtain\n\\begin{equation}\\label{NL3}\n\\begin{split}\n\\|G(n)\\div \\bv\\|_{\\tL_T^1(\\dB_{p,1}^{\\frac{d}{p}})}^{\\ell,J_\\varepsilon}\n\\lesssim\n\\|n\\|_{\\tL_T^\\infty(\\dB_{2,1}^{\\frac{d}{2}})}\n\\left(\\|\\bv\\|_{\\tL_T^1(\\dB_{p,1}^{\\frac{d}{p}+1})}^{\\ell,J_\\varepsilon} +\\|\\bv\\|_{\\tL_T^1(\\dB_{2,1}^{\\frac{d}{2}+1})}^{h,J_\\varepsilon}\\right)\n\\lesssim\n\\cX^2(T).\n\\end{split}\n\\end{equation}\n\nInserting \\eqref{NL1}-\\eqref{NL3} into \\eqref{LowEst3}, we can obtain\n\\begin{equation*}\n\\begin{split}\n&\\|(n, \\varepsilon \\bz)\\|_{\\tL_T^\\infty(\\dot B_{p,1}^{\\frac{d}{p}})}^{\\ell,J_\\varepsilon}\n+\\bigg(\\varepsilon\\|n\\|_{\\tL_T^1(\\dot B_{p,1}^{\\frac{d}{p}+2})}^{\\ell,J_\\varepsilon}\n+\\|\\bz\\|_{\\tL_T^1(\\dot B_{p,1}^{\\frac{d}{p}})}^{\\ell,J_\\varepsilon}\\bigg)\\\\\n&\\quad\\leq C\\bigg(\n\\|(n_0, \\varepsilon \\bz_0)\\|_{\\dot B_{p,1}^{\\frac{d}{p}}}^{\\ell,J_\\varepsilon}\n+\\cX^2(T)+\\cX(T)\\varepsilon^{-\\frac{1}{2}}\n\\|\\bv\\|_{\\tL_T^2(\\dB_{p,1}^{\\frac{d}{p}})}^{\\ell,J_\\varepsilon}\\bigg) ,\n\\end{split}\n\\end{equation*}\nwhich eventually leads to\n\\begin{equation}\\label{LowEst4}\n\\begin{split}\n&\\|n\\|_{\\tL_T^\\infty(\\dot B_{p,1}^{\\frac{d}{p}})}^{\\ell,J_\\varepsilon}\n+\\|\\bv\\|_{\\tL_T^\\infty(\\dot B_{p,1}^{\\frac{d}{p}})}^{\\ell,J_\\varepsilon}\n+\\bigg(\\varepsilon\\|n\\|_{\\tL_T^1(\\dot B_{p,1}^{\\frac{d}{p}+2})}^{\\ell,J_\\varepsilon}+\\|\\varepsilon^{-1}\\bv+\\nabla n\\|_{\\tL_T^1(\\dot B_{p,1}^{\\frac{d}{p}})}^{\\ell,J_\\varepsilon}\n+\\|\\bv\\|_{\\tL_T^1(\\dot B_{p,1}^{\\frac{d}{p}+1})}^{\\ell,J_\\varepsilon}\n+\\varepsilon^{-\\frac{1}{2}}\n\\|\\bv\\|_{\\tL_T^2(\\dB_{p,1}^{\\frac{d}{p}})}^{\\ell,J_\\varepsilon}\\bigg)\\\\\n&\\quad\\lesssim\n\\cX(0)+\\cX^2(T).\n\\end{split}\n\\end{equation}\n\nSince $2^{J_\\varepsilon}\\approx 2^{k}\\varepsilon^{-1}$, by the spatial embedding\n$\\dB_{p,1}^{\\frac{d}{p}}(\\R^d)\\hookrightarrow \\dB_{2,1}^{\\frac{d}{2}}(\\R^d)\\hookrightarrow L^\\infty(\\R^d)$, the classical estimation on smooth functions (see Corollary 2.65 in \\cite{chemin}) and \\eqref{HLEst} we can obtain that\n\\begin{equation}\\label{HNonLinEst1}\n\\begin{split}\n\\varepsilon\\|(\\nabla G(n),\\nabla \\bv)\\|_{L_T^\\infty(L^\\infty)}\n\\lesssim\n\\varepsilon\\|(n, \\bv)\\|_{\\tL_T^\\infty(\\dB_{2,1}^{\\frac{d}{2}+1})}\n\\lesssim\\cX(T).\n\\end{split}\n\\end{equation}\nBy the continuity equation, it is not difficult to get\n\\begin{equation}\\label{HNonLinEst2}\n\\begin{split}\n\\varepsilon\\|\\d_t G(n)\\|_{L_T^\\infty(L^\\infty)}\n&\\lesssim\n\\varepsilon \\|\\div \\bv+\\bv \\cdot \\nabla n\n+ G(n)\\div\\bv\\|_{\\tL_T^\\infty(\\dB_{2,1}^{\\frac{d}{2}})}\\\\\n&\\lesssim\n\\varepsilon\\|\\bv\\|_{\\tL_T^\\infty(\\dB_{2,1}^{\\frac{d}{2}+1})}\n+\\|\\bv\\|_{\\tL_T^\\infty(\\dB_{2,1}^{\\frac{d}{2}})}\\varepsilon \\|\\nabla n\\|_{\\tL_T^\\infty(\\dB_{2,1}^{\\frac{d}{2}})}\\\n+\\|n\\|_{\\tL_T^\\infty(\\dB_{2,1}^{\\frac{d}{2}})}\\varepsilon \\|\\div \\bv\\|_{\\tL_T^\\infty(\\dB_{2,1}^{\\frac{d}{2}})}\\\\\n&\\lesssim\n\\cX(T)+\\cX^2(T).\n\\end{split}\n\\end{equation}", "post_theorem_intro_text_len": 1144, "post_theorem_intro_text": "\\begin{remark}\nNote that $\\dot{B}_{p,1}^{\\frac{d}{p}}\\hookrightarrow \\dot{B}_{2,1}^{\\frac{d}{2}}$ when $p<2$, that means the existence and uniqueness of solutions hold in a smaller space rather than escaping to the complement $\\dot{B}_{2,1}^{\\frac{d}{2}}\\setminus \\dot{B}_{p,1}^{\\frac{d}{p}}$. On the other hand, \\eqref{DefHLB} implies that a smaller space for $p$ corresponds to more singular and concentrated data, which naturally arises in many physical problems (such as point sources and vortex filaments).\n\\end{remark}\n\n\\begin{remark}\nBased on Theorem \\ref{thm}, the relaxation limit for the case $p<2$ can be performed by the same procedure as \\cite{BaratDanchin2}.\n\\end{remark}\n\nThe major difficulty of proof of Theorem \\ref{thm} lies in dealing with the nonlinear terms. To achieve it, a new product estimation will be developed, see Lemma \\ref{NonClassicalProLaw1} below. As a matter of fact, the new tool could be applied to investigate other systems, such as the hyperbolic-parabolic chemotaxis system (see \\cite{BaratShou2}) in $L^2-L^p$ framework and Navier-Stokes equations (see \\cite{Danchin} ) in $L^p-L^2$ framework for $p<2$.", "sketch": "The post-theorem introduction only indicates that \"[t]he major difficulty of proof of Theorem \\ref{thm} lies in dealing with the nonlinear terms\" and that, \"[t]o achieve it, a new product estimation will be developed, see Lemma \\ref{NonClassicalProLaw1} below.\" No further proof steps are described.", "expanded_sketch": "The post-theorem introduction only indicates that \"[t]he major difficulty of proof of the main theorem lies in dealing with the nonlinear terms\" and that, \"[t]o achieve it, a new product estimation will be developed,\" namely the following lemma.\n\n\\begin{lemma}\\label{NonClassicalProLaw1}\nLet $0<s_1<\\frac{d}{2}$, $1\\leq p<2$. Then, we have the following inequality\n\\begin{align}\\label{newproduct}\n\\|ab\\|_{\\dB_{p,1}^{s_1}}^{\\ell,J_\\varepsilon}\n\\leq C\n\\left(\\|b\\|_{\\dB_{p,1}^{s_1}}^{\\ell,J_\\varepsilon}+ 2^{(s_1-\\frac{d}{p})J_\\varepsilon}\\|b\\|_{\\dB_{2,1}^{\\frac{d}{2}}}^{h,J_\\varepsilon} \\right)\\|a\\|_{\\dB_{2,1}^{\\frac{d}{2}}}\n\\end{align}\nwith $C$ independent on $\\varepsilon$.\n\\end{lemma}\n\nNo further proof steps are described.", "expanded_theorem": "\\label{thm}\nAssume $1\\leq p <2$, $d\\geq 1$ and the pressure $P$ satisfies $P'(\\bar{\\rho})>0$ for $\\bar{\\rho}>0$. There is a small constant $\\delta_1>0$ such that if\n$\n\\|(n_0,\\bv_0)\\|_{\\dB_{p,1}^{\\frac{d}{p}}}^{\\ell,J_\\varepsilon}\n+\\varepsilon\\|(n_0,\\bv_0)\\|_{\\dB_{2,1}^{\\frac{d}{2}+1}}^{h,J_\\varepsilon}\n\\leq \\delta_1,\n$\nthen the Cauchy problem\n\\begin{eqnarray}\\label{1.2}\n\\left\\{\\aligned\n&\\d_t n + \\bv \\cdot  \\nabla n + P'(\\bar{\\rho})\\div \\bv + G(n)\\div \\bv = 0 \\\\\n&\\d_t \\bv+ \\bv\\cdot \\nabla \\bv+\\nabla n= -\\frac{1}{\\varepsilon} \\bv\n\\endaligned\\right.\n\\end{eqnarray}\nwith initial data\n\\begin{eqnarray}\\label{1.2initial}\n (n,\\bv)|_{t=0}=(n_0, \\bv_0).\n\\end{eqnarray}\nadmits a uniform global unique solution in $E_\\infty^{J_\\varepsilon}$ for all $\\varepsilon>0$.", "theorem_type": ["Implication", "Existence"], "mcq": {"question": "Let $1\\le p<2$, $d\\ge1$, and let the pressure law satisfy $P'(\\bar\\rho)>0$ for $\\bar\\rho>0$. For a given $\\varepsilon>0$, consider the Cauchy problem\n$$\n\\begin{cases}\n\\partial_t n+\\mathbf v\\cdot\\nabla n+P'(\\bar\\rho)\\,\\operatorname{div}\\mathbf v+G(n)\\operatorname{div}\\mathbf v=0,\\\\\n\\partial_t \\mathbf v+\\mathbf v\\cdot\\nabla \\mathbf v+\\nabla n=-\\dfrac1\\varepsilon\\mathbf v,\n\\end{cases}\n$$\nwith initial data $(n,\\mathbf v)|_{t=0}=(n_0,\\mathbf v_0)$, where $G(n)$ is the smooth function appearing in the enthalpy formulation. Let $J_\\varepsilon=-[\\log_2\\varepsilon]+k$ for a suitable integer $k$, and define the low/high-frequency Besov seminorms by\n$$\n\\|f\\|_{\\dot B_{q,1}^s}^{\\ell,J_\\varepsilon}=\\sum_{j\\le J_\\varepsilon-1}2^{sj}\\|\\dot\\Delta_j f\\|_{L^q},\n\\qquad\n\\|f\\|_{\\dot B_{q,1}^s}^{h,J_\\varepsilon}=\\sum_{j\\ge J_\\varepsilon}2^{sj}\\|\\dot\\Delta_j f\\|_{L^q}.\n$$\nWrite $n^{\\ell,J_\\varepsilon},\\mathbf v^{\\ell,J_\\varepsilon}$ and $n^{h,J_\\varepsilon},\\mathbf v^{h,J_\\varepsilon}$ for the corresponding low- and high-frequency parts. Also, $E_\\infty^{J_\\varepsilon}$ means the space obtained by taking $T=\\infty$ in\n$$\n\\begin{aligned}\nE_T^{J_\\varepsilon}=\\Big\\{(n,\\mathbf v):&\\ (n^{\\ell,J_\\varepsilon},\\mathbf v^{\\ell,J_\\varepsilon})\\in C_b([0,T];\\dot B_{p,1}^{d/p}),\\ \\varepsilon n^{\\ell,J_\\varepsilon}\\in \\widetilde L_T^1(\\dot B_{p,1}^{d/p+2}),\\\\\n&\\ \\mathbf v^{\\ell,J_\\varepsilon}\\in \\widetilde L_T^1(\\dot B_{p,1}^{d/p+1}),\\ \\varepsilon^{-1/2}\\mathbf v\\in \\widetilde L_T^2(\\dot B_{p,1}^{d/p}),\\\\\n&\\ (\\varepsilon n^{h,J_\\varepsilon},\\varepsilon\\mathbf v^{h,J_\\varepsilon})\\in C_b([0,T];\\dot B_{2,1}^{d/2+1}),\\ (n^{h,J_\\varepsilon},\\mathbf v^{h,J_\\varepsilon})\\in \\widetilde L_T^1(\\dot B_{2,1}^{d/2+1}),\\\\\n&\\ \\varepsilon^{-1}\\mathbf v+\\nabla n\\in \\widetilde L_T^1(\\dot B_{p,1}^{d/p})\\Big\\}.\n\\end{aligned}\n$$\nAssume there is a sufficiently small constant $\\delta_1>0$ such that the initial data satisfy\n$$\n\\|(n_0,\\mathbf v_0)\\|_{\\dot B_{p,1}^{d/p}}^{\\ell,J_\\varepsilon}+\\varepsilon\\|(n_0,\\mathbf v_0)\\|_{\\dot B_{2,1}^{d/2+1}}^{h,J_\\varepsilon}\\le \\delta_1.\n$$\nWhich conclusion about the solution holds under these hypotheses?", "correct_choice": {"label": "A", "text": "For every $\\varepsilon>0$, the above Cauchy problem admits a uniform global unique solution $(n,\\mathbf v)$ belonging to $E_\\infty^{J_\\varepsilon}$."}, "choices": [{"label": "B", "text": "For every $\\varepsilon>0$, the above Cauchy problem admits a uniform global unique solution $(n,\\mathbf v)$ belonging to $E_\\infty^{J_\\varepsilon}$, provided in addition that $0<\\frac{d}{p}<\\frac d2$ so that the low-frequency nonlinear estimates close."}, {"label": "C", "text": "For every $\\varepsilon>0$, the above Cauchy problem admits a global unique solution $(n,\\mathbf v)$ belonging to $E_\\infty^{J_\\varepsilon}$."}, {"label": "D", "text": "There exists $\\varepsilon_0>0$ such that for every $0<\\varepsilon\\le \\varepsilon_0$, the above Cauchy problem admits a global unique solution $(n,\\mathbf v)$ belonging to $E_\\infty^{J_\\varepsilon}$."}, {"label": "E", "text": "For every $\\varepsilon>0$, the above Cauchy problem admits a uniform global unique solution $(n,\\mathbf v)$ belonging to $E_\\infty^{J_\\varepsilon}$ whenever the smallness assumption is strengthened to\n$$\n\\|(n_0,\\mathbf v_0)\\|_{\\dot B_{p,1}^{d/p}}^{\\ell,J_\\varepsilon}+\\|(n_0,\\mathbf v_0)\\|_{\\dot B_{2,1}^{d/2+1}}^{h,J_\\varepsilon}\\le \\delta_1.\n$$"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "admissible low-frequency product range", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "uniformity in $\\varepsilon$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "for all $\\varepsilon>0$ quantifier", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "critical $\\varepsilon$-weight on high frequencies", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal choice A. It states the hypotheses in full but does not directly state the conclusion, so there is no overt answer leakage."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-recall item: the stem lays out the full assumptions and asks for the theorem's conclusion. The correct option is a near-direct restatement of the intended result rather than a genuinely new inference."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options differ in subtle ways (uniformity in ε, extra assumptions, stronger smallness, restricted ε-range). However, the task is still mostly recognition/matching of the exact theorem statement rather than substantial mathematical generation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they alter quantifiers, uniformity, admissible parameter ranges, or the ε-weighted smallness condition. These reflect realistic failure modes in reading or recalling PDE well-posedness results."}, "total_score": 5, "overall_assessment": "A technically well-constructed theorem-recognition MCQ with strong distractors, but it is largely tautological and tests recall of the exact statement more than generative reasoning."}}
{"id": "2602.22608v1", "paper_link": "http://arxiv.org/abs/2602.22608v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\nConsider any $T \\ge 1$ and $L,\\alpha>0$. Then there exist an $\\alpha$-strongly convex set $S$ and an $L$-smooth, $L$-strongly convex function $f$ in dimension $d=2(T+1)$ such that every \\texttt{FO-LMO} method applied to $(f, S)$ with $x_0 = 0$ has $x_T$ infeasible or \n\\[\nf(x_T) - \\min_{x\\in S}f(x) \\geq \\frac{1}{528}\\frac{L\\, \\mathrm{diam}(S)^2}{(T+1)^2}.\n\\]\nConsequently, for any $\\varepsilon>0$, there exist problem instances where $T\\geq \\sqrt{\\frac{L\\, \\mathrm{diam}(S)^2}{528\\varepsilon}} - 1$ iterations are required for any \\texttt{FO-LMO} method to reach a suboptimality of $f(x_T) - \\min_{x\\in S}f(x) \\leq \\varepsilon$.", "start_pos": 12690, "end_pos": 13365, "label": "thm:main"}, "ref_dict": {"thm:smooth": "\\begin{theorem}\\label{thm:smooth}\nFor any $d \\ge 1$ and $\\beta > 0$, there exist a convex $\\beta$-smooth set $S_\\beta$ and a $1$-smooth, $1$-strongly convex function $f_\\beta$ such that every \\texttt{LMO-span} method applied to $(f_\\beta, S_\\beta)$ starting from $x_0=0$ has for all $t\\leq d-1$\n    \\begin{equation}\n        f_\\beta(x_t) - \\min_{x\\in S_\\beta}f_\\beta(x) \\geq  \\max \\left(0, \\sqrt{\\frac{d-t}{2}}\\left(\\frac{\\mathrm{diam}(S_\\beta) -  2/\\beta}{\\sqrt{2}d}+\\frac{1}{\\beta\\sqrt{d}}\\right)-\\frac{1}{\\sqrt{2}\\beta}\\right)^2  . \n    \\end{equation}\n    In particular, for any fixed budget $T \\ge 1$, there exist $S_\\beta$ and $f_\\beta$ in dimension $d=2T$ such that\n    \\[\n        f_\\beta(x_T) - \\min_{x\\in S_\\beta}f_\\beta(x) \\geq \\max\\left(0,\\frac{\\mathrm{diam}(S_\\beta) - 2/\\beta}{4\\sqrt{T}} - \\frac{\\sqrt{2}-1}{2\\beta }\\right)^2 .\n    \\]\n\\end{theorem}", "thm:main": "\\begin{theorem}\\label{thm:main}\nConsider any $T \\ge 1$ and $L,\\alpha>0$. Then there exist an $\\alpha$-strongly convex set $S$ and an $L$-smooth, $L$-strongly convex function $f$ in dimension $d=2(T+1)$ such that every \\texttt{FO-LMO} method applied to $(f, S)$ with $x_0 = 0$ has $x_T$ infeasible or \n\\[\nf(x_T) - \\min_{x\\in S}f(x) \\geq \\frac{1}{528}\\frac{L\\, \\mathrm{diam}(S)^2}{(T+1)^2}.\n\\]\nConsequently, for any $\\varepsilon>0$, there exist problem instances where $T\\geq \\sqrt{\\frac{L\\, \\mathrm{diam}(S)^2}{528\\varepsilon}} - 1$ iterations are required for any \\texttt{FO-LMO} method to reach a suboptimality of $f(x_T) - \\min_{x\\in S}f(x) \\leq \\varepsilon$.\n\\end{theorem}", "thm:mixed": "\\begin{theorem}\\label{thm:mixed}\n    For any $d \\ge 1$ and $\\beta > 0$, there exist a $\\beta$-smooth, $1/(1+1/\\beta)$-strongly convex set $S_\\beta$ and a $1$-smooth, $1$-strongly convex function $f_\\beta$ such that every \\texttt{LMO-span} method applied to $(f_\\beta, S_\\beta)$ starting from $x_0 = 0$ has for all $t\\leq d-1$\n    \\begin{equation}\n        f_\\beta(x_t) - \\min_{x\\in S_\\beta}f_\\beta(x) \\geq \\max\\left(0,\\sqrt{\\frac{2}{5}\\frac{(d-t)(\\mathrm{diam}(S_\\beta) -  2/\\beta)^2}{(d+2)^3}} - \\frac{1}{\\sqrt{2}\\beta}\\right)^2  . \n    \\end{equation}\n    In particular, for any fixed budget $T \\ge 1$, there exist $S_\\beta$ and $f_\\beta$ in dimension $d=2(T+1)$ such that\n    \\[\n        f_\\beta(x_T) - \\min_{x\\in S_\\beta}f_\\beta(x) \\geq \\max\\left(0,\\frac{\\mathrm{diam}(S_\\beta) - 2/\\beta}{\\sqrt{20}(T+2)} - \\frac{1}{\\sqrt{2}\\beta }\\right)^2  .\n    \\]\n\\end{theorem}"}, "pre_theorem_intro_text_len": 8710, "pre_theorem_intro_text": "In this work, we consider convex constrained optimization problems of the form\n$$ \\begin{cases}\n\\min & f(x)\\\\\n\\mathrm{s.t.\\ } & x\\in S \\subseteq \\mathbb{R}^d.\n\\end{cases}$$\nIn particular, we consider high-dimensional problems where $d$ may be arbitrarily large.\nFrank-Wolfe methods and the broader family of ``projection-free'' algorithms using a linear minimization subroutine have found renewed interest due to their scalability. See the survey~\\cite{braun2025conditionalgradientmethods} and references therein. In a basic form, these methods iterate from an initialization $x_0$, producing for $k=0,1,\\dots$\n\\begin{align*}    \n    p_k &= \\nabla f(x_k),\\\\\n    z_{k+1} & \\in\\argmin_{x\\in S}\\langle p_k, x\\rangle,\\\\\n    x_{k+1} & \\in \\operatorname{conv}\\{x_k, z_{k+1}\\}.\n\\end{align*}\nAs examples, an exact line search implementation of Frank-Wolfe would set $x_{k+1}$ as the minimizer of $f$ on the segment $[x_k,z_{k+1}]$. Similarly, a fixed ``open-loop'' stepsize implementation would fix $x_{k+1} = \\theta_k x_k + (1-\\theta_k)z_{k+1}$ for predetermined $\\theta_k\\in [0,1]$.\nThe two key computational oracles assumed here are access to gradients of the objective $f$ to compute $p_k$ and access to a Linear Minimization Oracle (LMO) to produce $z_{k+1}$. As a shorthand, we denote $\\mathtt{LMO}_S(p) \\in \\argmin_{x\\in S}\\langle p, x\\rangle$ as an oracle providing a selection of this minimizer (potentially selected adversarially).\n\nIn this work, we provide complexity lower bounds for the family of all deterministic first-order methods using a linear minimization oracle, containing the above Frank-Wolfe methods. Our focus is on lower bounds for problem classes where the constraint set $S$ possesses structural properties like strong convexity or smoothness. Below, we formalize these classes of problems and algorithms.\n\n\\paragraph{The Families of Smooth and Strongly Convex Constraint Sets and Functions.} A problem instance is defined by a function $f\\colon \\mathbb{R}^d\\rightarrow \\mathbb{R}$ and a set $S\\subseteq\\mathbb{R}^d$ as well as an initialization $x_0$, which we will set as $x_0=0$ without loss of generality. A problem class is defined by a set of allowable $f$ and $S$ values. We consider the standard family of convex objective functions parameterized by $0\\leq\\mu\\leq L$, defined as\n\\begin{align*}\n    \\text{$f$ is $\\mu$-strongly convex if\\quad }& f(\\lambda x + (1-\\lambda)y)\\leq \\lambda f(x) + (1-\\lambda)f(y) - \\frac{\\lambda(1-\\lambda)\\mu}{2}\\|y-x\\|_2^2\\\\\n    &\\hskip7.0cm \\forall x,y\\in \\mathbb{R}^d, \\lambda\\in[0,1], \\\\\n    \\text{$f$ is $L$-smooth if\\quad } & \\|\\nabla f(y)-\\nabla f(x)\\|_2\\leq L\\|y-x\\|_2 \\quad \\forall x,y\\in \\mathbb{R}^d.\n\\end{align*}\nIf $L=\\mu$, then $f$ must be a quadratic function of the form $\\frac{L}{2}\\|x-x_\\star\\|_2^2$ up to an additive constant. Such simple quadratics will suffice for our theoretical development.\n\nWe consider compact convex sets $S$ with diameter $\\mathrm{diam}(S)=\\max_{x,y\\in S}\\|x-y\\|_2$ with at least one of the natural parallel notions of strong convexity and smoothness for constraint sets. We say\n$$ \\text{$S$ is $\\alpha$-strongly convex if\\quad } \\lambda x + (1-\\lambda)y + B\\left(0,\\ \\frac{\\lambda(1-\\lambda)\\alpha}{2}\\|y-x\\|_2^2\\right) \\subseteq S $$\nfor all $x,y\\in S, \\lambda\\in [0,1]$ where $B(0,r)=\\{x\\in\\mathbb{R}^d \\mid \\|x\\|_2\\leq r\\}$ denotes the closed ball of radius $r$. Normal vectors $n \\in N_S(x) := \\{n \\mid \\langle n, y-x\\rangle \\leq 0\\ \\forall y\\in S\\}$ at each $x\\in\\operatorname{bdry}S$ serve as the analogues of gradients. Considering any unit length normal vectors, we define\n\\begin{align*}\n    \\text{$S$ is $\\beta$-smooth if\\quad } \\|n_y - n_x\\|_2 \\leq \\beta \\|y-x\\|_2 \\quad \\forall &x,y\\in\\operatorname{bdry}S,\\\\\n    &n_y\\in \\{n\\in N_S(y) \\mid \\|n\\|_2=1\\},\\\\\n    &n_x\\in \\{n\\in N_S(x) \\mid \\|n\\|_2=1\\}.\n\\end{align*}\nNote that the above definition requires that smooth sets have a unique unit normal vector at each boundary point.\nFor a classical reference providing equivalent characterizations of the strong convexity of a set, see~\\cite{vial1982strong}. A modernized treatment of smoothness and strong convexity of sets was given by~\\cite{liu2023gauges}. Therein, parallels to the smoothness and strong convexity of functions are explored.\n\n\\paragraph{The Family of First-Order Linear Minimization Oracle Methods (\\texttt{FO-LMO}).}\nFor a given problem instance $(f,S)$ and initialization $x_0 \\in S$, a \\texttt{FO-LMO} method generates sequences of search directions $p_k\\in\\mathbb{R}^d$, linear minimization solutions $z_{k+1} = \\mathtt{LMO}_S(p_k)\\in\\argmin_{x\\in S}\\langle p_k, x\\rangle$, and iterates $x_{k+1}$. We require that $p_k$ is a deterministic function of the oracle responses $\\{(f(x_i),\\nabla f(x_i))\\}^k_{i=0}$ and $\\{z_{i}\\}^k_{i=1}$ so far and that $x_{k+1}$ is a deterministic function of the responses $\\{(f(x_i),\\nabla f(x_i))\\}^k_{i=0}$ and $\\{z_{i}\\}^{k+1}_{i=1}$. Note that we do not require that $x_{k+1}$ lie in the convex hull of $x_0$ and $\\{z_{i}\\}_{i=1}^{k+1}$. However, when outside this convex hull, the iterates may fail to be feasible.\nThis \\texttt{FO-LMO} family of methods includes, for example, the Away-step Frank–Wolfe methods~\\cite{lacoste2015global,beck2017linearly}, Pairwise Frank–Wolfe methods~\\cite{lacoste2015global,tsuji2022pairwise}, Fully Corrective Frank–Wolfe methods~\\cite{holloway1974extension,lacoste2015global}, and Blended Pairwise Frank–Wolfe methods~\\cite{braun2019blended}. \n\nIn this model, each iteration of the algorithm can make one call to a first-order oracle, returning $(f(x_k), \\nabla f(x_k))$, and one call to the LMO. Our theory then bounds iteration complexity to measure the minimum number of such pairs of oracle calls needed to reach a target suboptimality. \n\nFor the minimization of an $L$-smooth convex function $f$ over a compact convex set $S$, Frank-Wolfe methods~\\cite{frank1956algorithm,jaggi2013revisiting} are known to provide convergence rates of the following form\n$$ f(x_T) - \\min_{x\\in S}f(x) \\leq \\mathcal{O}\\left(\\frac{L\\, \\mathrm{diam}(S)^2}{T}\\right). $$\nThis provides an upper bound on the iteration complexity of computing a point with $\\varepsilon$-suboptimality (i.e., $f(x_T)-\\min_{x\\in S}f(x)\\leq\\varepsilon$) of $\\mathcal{O}(L\\, \\mathrm{diam}(S)^2/\\varepsilon)$. A matching complexity lower bound was provided by Lan~\\cite{lan2013complexity}, establishing this as the order of the optimal LMO complexity.\nGarber and Hazan~\\cite{garber2015faster} showed that given the additional structure that $f$ and $S$ are strongly convex, this rate can be accelerated to\n$$ f(x_T) - \\min_{x\\in S}f(x) \\leq \\mathcal{O}\\left(\\frac{L\\, \\mathrm{diam}(S)^2}{T^2}\\right) $$\ngiving an iteration complexity of $\\mathcal{O}(\\sqrt{L\\, \\mathrm{diam}(S)^2/\\varepsilon})$. This accelerated result poses the natural question of whether further acceleration is possible or if this complexity is the optimal strongly convex order.\n\nProviding recent progress on this question, Halbey et al.~\\cite{halbey2026lower} showed that two particular variants of Frank-Wolfe (exact line search and the short stepsize procedure) cannot improve upon this $\\mathcal{O}(1/T^2)$ rate. However, these results do not preclude the possibility of other \\texttt{FO-LMO} methods exceeding this rate.\nSimilar limitations in unconstrained minimization were overcome by the foundational work~\\cite{nemirovski1983problem}, establishing information/oracle complexity lower bounds against the whole family of gradient methods. This was accomplished by the design of hard functions whose gradient reveals only one new coordinate of information per step. This property, known as a ``zero-chain'' property, can prevent {\\it any gradient-span method} from having made substantial progress until dimension-many steps have been taken. By combining this with an adversarial ``resisting oracle'', lower bounds against {\\it all deterministic gradient methods} were achieved.\n\n\\paragraph{Our Contributions.} We extend the classical zero-chain approach to LMOs to derive complexity lower bounds on any \\texttt{FO-LMO} method over the problem classes corresponding to minimizing a quadratic $f(x)=\\frac{L}{2}\\|x-x_\\star\\|_2^2$ over structured sets, possessing either strong convexity or smoothness. Our primary result shows that over $\\alpha$-strongly convex constraints, no \\texttt{FO-LMO} method can guarantee $\\varepsilon$-suboptimality in fewer than $\\Omega(\\sqrt{L\\, \\mathrm{diam}(S)^2/\\varepsilon})$ iterations. By matching the upper bounding guarantee of~\\cite{garber2015faster}, this establishes $\\Theta(\\sqrt{L\\, \\mathrm{diam}(S)^2/\\varepsilon})$ as the optimal order of complexity for this class. Formally, we prove the following in Section~\\ref{sec:general}.", "context": "\\paragraph{The Families of Smooth and Strongly Convex Constraint Sets and Functions.} A problem instance is defined by a function $f\\colon \\mathbb{R}^d\\rightarrow \\mathbb{R}$ and a set $S\\subseteq\\mathbb{R}^d$ as well as an initialization $x_0$, which we will set as $x_0=0$ without loss of generality. A problem class is defined by a set of allowable $f$ and $S$ values. We consider the standard family of convex objective functions parameterized by $0\\leq\\mu\\leq L$, defined as\n\\begin{align*}\n \\text{$f$ is $\\mu$-strongly convex if\\quad }& f(\\lambda x + (1-\\lambda)y)\\leq \\lambda f(x) + (1-\\lambda)f(y) - \\frac{\\lambda(1-\\lambda)\\mu}{2}\\|y-x\\|_2^2\\\\\n &\\hskip7.0cm \\forall x,y\\in \\mathbb{R}^d, \\lambda\\in[0,1], \\\\\n \\text{$f$ is $L$-smooth if\\quad } & \\|\\nabla f(y)-\\nabla f(x)\\|_2\\leq L\\|y-x\\|_2 \\quad \\forall x,y\\in \\mathbb{R}^d.\n\\end{align*}\nIf $L=\\mu$, then $f$ must be a quadratic function of the form $\\frac{L}{2}\\|x-x_\\star\\|_2^2$ up to an additive constant. Such simple quadratics will suffice for our theoretical development.\n\nWe consider compact convex sets $S$ with diameter $\\mathrm{diam}(S)=\\max_{x,y\\in S}\\|x-y\\|_2$ with at least one of the natural parallel notions of strong convexity and smoothness for constraint sets. We say\n$$ \\text{$S$ is $\\alpha$-strongly convex if\\quad } \\lambda x + (1-\\lambda)y + B\\left(0,\\ \\frac{\\lambda(1-\\lambda)\\alpha}{2}\\|y-x\\|_2^2\\right) \\subseteq S $$\nfor all $x,y\\in S, \\lambda\\in [0,1]$ where $B(0,r)=\\{x\\in\\mathbb{R}^d \\mid \\|x\\|_2\\leq r\\}$ denotes the closed ball of radius $r$. Normal vectors $n \\in N_S(x) := \\{n \\mid \\langle n, y-x\\rangle \\leq 0\\ \\forall y\\in S\\}$ at each $x\\in\\operatorname{bdry}S$ serve as the analogues of gradients. Considering any unit length normal vectors, we define\n\\begin{align*}\n \\text{$S$ is $\\beta$-smooth if\\quad } \\|n_y - n_x\\|_2 \\leq \\beta \\|y-x\\|_2 \\quad \\forall &x,y\\in\\operatorname{bdry}S,\\\\\n &n_y\\in \\{n\\in N_S(y) \\mid \\|n\\|_2=1\\},\\\\\n &n_x\\in \\{n\\in N_S(x) \\mid \\|n\\|_2=1\\}.\n\\end{align*}\nNote that the above definition requires that smooth sets have a unique unit normal vector at each boundary point.\nFor a classical reference providing equivalent characterizations of the strong convexity of a set, see~\\cite{vial1982strong}. A modernized treatment of smoothness and strong convexity of sets was given by~\\cite{liu2023gauges}. Therein, parallels to the smoothness and strong convexity of functions are explored.\n\nFor the minimization of an $L$-smooth convex function $f$ over a compact convex set $S$, Frank-Wolfe methods~\\cite{frank1956algorithm,jaggi2013revisiting} are known to provide convergence rates of the following form\n$$ f(x_T) - \\min_{x\\in S}f(x) \\leq \\mathcal{O}\\left(\\frac{L\\, \\mathrm{diam}(S)^2}{T}\\right). $$\nThis provides an upper bound on the iteration complexity of computing a point with $\\varepsilon$-suboptimality (i.e., $f(x_T)-\\min_{x\\in S}f(x)\\leq\\varepsilon$) of $\\mathcal{O}(L\\, \\mathrm{diam}(S)^2/\\varepsilon)$. A matching complexity lower bound was provided by Lan~\\cite{lan2013complexity}, establishing this as the order of the optimal LMO complexity.\nGarber and Hazan~\\cite{garber2015faster} showed that given the additional structure that $f$ and $S$ are strongly convex, this rate can be accelerated to\n$$ f(x_T) - \\min_{x\\in S}f(x) \\leq \\mathcal{O}\\left(\\frac{L\\, \\mathrm{diam}(S)^2}{T^2}\\right) $$\ngiving an iteration complexity of $\\mathcal{O}(\\sqrt{L\\, \\mathrm{diam}(S)^2/\\varepsilon})$. This accelerated result poses the natural question of whether further acceleration is possible or if this complexity is the optimal strongly convex order.\n\nProviding recent progress on this question, Halbey et al.~\\cite{halbey2026lower} showed that two particular variants of Frank-Wolfe (exact line search and the short stepsize procedure) cannot improve upon this $\\mathcal{O}(1/T^2)$ rate. However, these results do not preclude the possibility of other \\texttt{FO-LMO} methods exceeding this rate.\nSimilar limitations in unconstrained minimization were overcome by the foundational work~\\cite{nemirovski1983problem}, establishing information/oracle complexity lower bounds against the whole family of gradient methods. This was accomplished by the design of hard functions whose gradient reveals only one new coordinate of information per step. This property, known as a ``zero-chain'' property, can prevent {\\it any gradient-span method} from having made substantial progress until dimension-many steps have been taken. By combining this with an adversarial ``resisting oracle'', lower bounds against {\\it all deterministic gradient methods} were achieved.\n\n\\paragraph{Our Contributions.} We extend the classical zero-chain approach to LMOs to derive complexity lower bounds on any \\texttt{FO-LMO} method over the problem classes corresponding to minimizing a quadratic $f(x)=\\frac{L}{2}\\|x-x_\\star\\|_2^2$ over structured sets, possessing either strong convexity or smoothness. Our primary result shows that over $\\alpha$-strongly convex constraints, no \\texttt{FO-LMO} method can guarantee $\\varepsilon$-suboptimality in fewer than $\\Omega(\\sqrt{L\\, \\mathrm{diam}(S)^2/\\varepsilon})$ iterations. By matching the upper bounding guarantee of~\\cite{garber2015faster}, this establishes $\\Theta(\\sqrt{L\\, \\mathrm{diam}(S)^2/\\varepsilon})$ as the optimal order of complexity for this class. Formally, we prove the following in Section~\\ref{sec:general}.", "full_context": "\\paragraph{The Families of Smooth and Strongly Convex Constraint Sets and Functions.} A problem instance is defined by a function $f\\colon \\mathbb{R}^d\\rightarrow \\mathbb{R}$ and a set $S\\subseteq\\mathbb{R}^d$ as well as an initialization $x_0$, which we will set as $x_0=0$ without loss of generality. A problem class is defined by a set of allowable $f$ and $S$ values. We consider the standard family of convex objective functions parameterized by $0\\leq\\mu\\leq L$, defined as\n\\begin{align*}\n \\text{$f$ is $\\mu$-strongly convex if\\quad }& f(\\lambda x + (1-\\lambda)y)\\leq \\lambda f(x) + (1-\\lambda)f(y) - \\frac{\\lambda(1-\\lambda)\\mu}{2}\\|y-x\\|_2^2\\\\\n &\\hskip7.0cm \\forall x,y\\in \\mathbb{R}^d, \\lambda\\in[0,1], \\\\\n \\text{$f$ is $L$-smooth if\\quad } & \\|\\nabla f(y)-\\nabla f(x)\\|_2\\leq L\\|y-x\\|_2 \\quad \\forall x,y\\in \\mathbb{R}^d.\n\\end{align*}\nIf $L=\\mu$, then $f$ must be a quadratic function of the form $\\frac{L}{2}\\|x-x_\\star\\|_2^2$ up to an additive constant. Such simple quadratics will suffice for our theoretical development.\n\nWe consider compact convex sets $S$ with diameter $\\mathrm{diam}(S)=\\max_{x,y\\in S}\\|x-y\\|_2$ with at least one of the natural parallel notions of strong convexity and smoothness for constraint sets. We say\n$$ \\text{$S$ is $\\alpha$-strongly convex if\\quad } \\lambda x + (1-\\lambda)y + B\\left(0,\\ \\frac{\\lambda(1-\\lambda)\\alpha}{2}\\|y-x\\|_2^2\\right) \\subseteq S $$\nfor all $x,y\\in S, \\lambda\\in [0,1]$ where $B(0,r)=\\{x\\in\\mathbb{R}^d \\mid \\|x\\|_2\\leq r\\}$ denotes the closed ball of radius $r$. Normal vectors $n \\in N_S(x) := \\{n \\mid \\langle n, y-x\\rangle \\leq 0\\ \\forall y\\in S\\}$ at each $x\\in\\operatorname{bdry}S$ serve as the analogues of gradients. Considering any unit length normal vectors, we define\n\\begin{align*}\n \\text{$S$ is $\\beta$-smooth if\\quad } \\|n_y - n_x\\|_2 \\leq \\beta \\|y-x\\|_2 \\quad \\forall &x,y\\in\\operatorname{bdry}S,\\\\\n &n_y\\in \\{n\\in N_S(y) \\mid \\|n\\|_2=1\\},\\\\\n &n_x\\in \\{n\\in N_S(x) \\mid \\|n\\|_2=1\\}.\n\\end{align*}\nNote that the above definition requires that smooth sets have a unique unit normal vector at each boundary point.\nFor a classical reference providing equivalent characterizations of the strong convexity of a set, see~\\cite{vial1982strong}. A modernized treatment of smoothness and strong convexity of sets was given by~\\cite{liu2023gauges}. Therein, parallels to the smoothness and strong convexity of functions are explored.\n\nFor the minimization of an $L$-smooth convex function $f$ over a compact convex set $S$, Frank-Wolfe methods~\\cite{frank1956algorithm,jaggi2013revisiting} are known to provide convergence rates of the following form\n$$ f(x_T) - \\min_{x\\in S}f(x) \\leq \\mathcal{O}\\left(\\frac{L\\, \\mathrm{diam}(S)^2}{T}\\right). $$\nThis provides an upper bound on the iteration complexity of computing a point with $\\varepsilon$-suboptimality (i.e., $f(x_T)-\\min_{x\\in S}f(x)\\leq\\varepsilon$) of $\\mathcal{O}(L\\, \\mathrm{diam}(S)^2/\\varepsilon)$. A matching complexity lower bound was provided by Lan~\\cite{lan2013complexity}, establishing this as the order of the optimal LMO complexity.\nGarber and Hazan~\\cite{garber2015faster} showed that given the additional structure that $f$ and $S$ are strongly convex, this rate can be accelerated to\n$$ f(x_T) - \\min_{x\\in S}f(x) \\leq \\mathcal{O}\\left(\\frac{L\\, \\mathrm{diam}(S)^2}{T^2}\\right) $$\ngiving an iteration complexity of $\\mathcal{O}(\\sqrt{L\\, \\mathrm{diam}(S)^2/\\varepsilon})$. This accelerated result poses the natural question of whether further acceleration is possible or if this complexity is the optimal strongly convex order.\n\nProviding recent progress on this question, Halbey et al.~\\cite{halbey2026lower} showed that two particular variants of Frank-Wolfe (exact line search and the short stepsize procedure) cannot improve upon this $\\mathcal{O}(1/T^2)$ rate. However, these results do not preclude the possibility of other \\texttt{FO-LMO} methods exceeding this rate.\nSimilar limitations in unconstrained minimization were overcome by the foundational work~\\cite{nemirovski1983problem}, establishing information/oracle complexity lower bounds against the whole family of gradient methods. This was accomplished by the design of hard functions whose gradient reveals only one new coordinate of information per step. This property, known as a ``zero-chain'' property, can prevent {\\it any gradient-span method} from having made substantial progress until dimension-many steps have been taken. By combining this with an adversarial ``resisting oracle'', lower bounds against {\\it all deterministic gradient methods} were achieved.\n\n\\paragraph{Our Contributions.} We extend the classical zero-chain approach to LMOs to derive complexity lower bounds on any \\texttt{FO-LMO} method over the problem classes corresponding to minimizing a quadratic $f(x)=\\frac{L}{2}\\|x-x_\\star\\|_2^2$ over structured sets, possessing either strong convexity or smoothness. Our primary result shows that over $\\alpha$-strongly convex constraints, no \\texttt{FO-LMO} method can guarantee $\\varepsilon$-suboptimality in fewer than $\\Omega(\\sqrt{L\\, \\mathrm{diam}(S)^2/\\varepsilon})$ iterations. By matching the upper bounding guarantee of~\\cite{garber2015faster}, this establishes $\\Theta(\\sqrt{L\\, \\mathrm{diam}(S)^2/\\varepsilon})$ as the optimal order of complexity for this class. Formally, we prove the following in Section~\\ref{sec:general}.\n\nAs a secondary result, we provide a partial generalization to $\\beta$-smooth sets. These bounds are meaningful in the regime of only modestly smooth sets, having $\\beta=\\Omega(1/\\sqrt{\\varepsilon})$. In this regime, no \\texttt{LMO-span} method (see Section~\\ref{sec:SC} for a formal definition) can improve past the optimal compact convex set complexity of $\\Theta(L\\, \\mathrm{diam}(S)^2/\\varepsilon)$ or over the optimal $\\alpha$-strongly convex set complexity of $\\Theta(\\sqrt{L\\, \\mathrm{diam}(S)^2/\\varepsilon})$. Theorems~\\ref{thm:smooth} and~\\ref{thm:mixed} formalize these limits on acceleration due to smoothness.\n\nFirst, Section~\\ref{subsec:Construction-SC} constructs our candidate hard problem instance for each dimension and verifies its validity (computing its strong convexity constant and diameter). Then Section~\\ref{subsec:zero-chain-SC} establishes a key ``zero-chain'' property of these hard instances, showing that any \\texttt{LMO-span} method applied will only discover one new coordinate per iteration. Finally, by leveraging this property, Section~\\ref{subsec:proof-SC} proves this section's main result, Theorem~\\ref{thm:SC}, showing that no method in $T$ steps can guarantee a suboptimality less than $\\Omega(L\\, \\mathrm{diam}(S)^2/T^2)$. Combined with the upper bound of~\\cite{garber2015faster}, this establishes the optimal iteration complexity for strongly convex constrained \\texttt{LMO-span} optimization as $\\Theta(\\sqrt{L\\, \\mathrm{diam}(S)^2/\\varepsilon})$.\n\nAny considered \\texttt{FO-LMO} method must set $x_T$ deterministically as a function of the history of oracle responses. Crucially, while the algorithm must fix $(x_T)_i$ for $i \\in U$, the resisting oracle remains free to select any completion of the permutation $\\pi$ over $U$. Let $\\mathcal{P}_U\\subseteq \\mathcal{P}_d$ denote all such permutations. If some completion $\\pi\\in\\mathcal{P}_U$ has $x_T\\not\\in S_\\pi$, then the adversarial oracle can select that $\\pi$, making $x_T$ infeasible and the theorem hold trivially. Hence, we can assume for all $\\pi\\in \\mathcal{P}_U$ that $x_T\\in S_\\pi$. From the strong convexity of $f$ and optimality of $x^{(\\pi)}_\\star$ guaranteeing $\\langle \\nabla f(x^{(\\pi)}_\\star), x_T - x^{(\\pi)}_\\star\\rangle\\geq 0$ since $x_T\\in S_\\pi$, the suboptimality for any $\\pi\\in\\mathcal{P}_U$ is bounded by\n\\begin{equation}\\label{eq:sc-lower-bound}\n f(x_T) - f(x_\\star^{(\\pi)}) \\ge \\frac{1}{2}\\|x_T - x_\\star^{(\\pi)}\\|_2^2 \\ge \\frac{1}{2} \\sum_{j\\in U} \\left( (x_T)_{j} - (x^{(\\pi)}_\\star)_{j} \\right)^2.\n\\end{equation}\nLet $\\pi_\\star \\in \\mathcal{P}_U$ denote the permutation choice maximizing this lower bound. Then, we have\n\\begin{align*}\n f(x_T) - f(x_\\star^{(\\pi_\\star)}) &\\geq \\frac{1}{2} \\sum_{j\\in U} \\left( (x_T)_{j} - (x^{(\\pi_\\star)}_\\star)_{j} \\right)^2\\\\\n &\\geq \\frac{1}{|\\mathcal{P}_U|}\\sum_{\\pi\\in\\mathcal{P}_U} \\left(\\frac{1}{2}\\sum_{j\\in U} \\left( (x_T)_{j} - (x^{(\\pi)}_\\star)_{j} \\right)^2\\right)\\\\\n & = \\frac{1}{2}\\sum_{j\\in U} \\frac{1}{|\\mathcal{P}_U|}\\sum_{\\pi\\in\\mathcal{P}_U} \\left( (x_T)_{j} - (x^{(\\pi)}_\\star)_{j} \\right)^2\\\\\n &\\geq \\frac{m}{2}\\mathrm{Var}\\left\\{ \\rho(w_d - w_k) \\right\\}_{k = d-m+1}^d \\\\\n &=\\frac{\\rho^2 m}{2} \\mathrm{Var}\\{w_{k}\\}_{k = d-m+1}^d\\\\\n &= \\rho^2 m C^2 \\mathrm{Var}\\{\\sqrt{k}\\}_{k = d-m+1}^d\n\\end{align*}\nwhere the first inequality is by~\\eqref{eq:sc-lower-bound}, the second inequality lower bounds this maximal $\\pi_\\star$ by the average lower bound over $\\mathcal{P}_U$, and the third inequality notes that each inner sum is minimized when $(x_T)_{j}$ is the average of $\\{\\rho(w_d - w_{d-m+1}),\\dots, \\rho(w_d - w_{d})\\}$ yielding the stated variance lower bound.\n\n\\begin{theorem}\\label{thm:smooth}\nFor any $d \\ge 1$ and $\\beta > 0$, there exist a convex $\\beta$-smooth set $S_\\beta$ and a $1$-smooth, $1$-strongly convex function $f_\\beta$ such that every \\texttt{LMO-span} method applied to $(f_\\beta, S_\\beta)$ starting from $x_0=0$ has for all $t\\leq d-1$\n \\begin{equation}\n f_\\beta(x_t) - \\min_{x\\in S_\\beta}f_\\beta(x) \\geq \\max \\left(0, \\sqrt{\\frac{d-t}{2}}\\left(\\frac{\\mathrm{diam}(S_\\beta) - 2/\\beta}{\\sqrt{2}d}+\\frac{1}{\\beta\\sqrt{d}}\\right)-\\frac{1}{\\sqrt{2}\\beta}\\right)^2 . \n \\end{equation}\n In particular, for any fixed budget $T \\ge 1$, there exist $S_\\beta$ and $f_\\beta$ in dimension $d=2T$ such that\n \\[\n f_\\beta(x_T) - \\min_{x\\in S_\\beta}f_\\beta(x) \\geq \\max\\left(0,\\frac{\\mathrm{diam}(S_\\beta) - 2/\\beta}{4\\sqrt{T}} - \\frac{\\sqrt{2}-1}{2\\beta }\\right)^2 .\n \\]\n\\end{theorem}\n\\begin{theorem}\\label{thm:mixed}\n For any $d \\ge 1$ and $\\beta > 0$, there exist a $\\beta$-smooth, $1/(1+1/\\beta)$-strongly convex set $S_\\beta$ and a $1$-smooth, $1$-strongly convex function $f_\\beta$ such that every \\texttt{LMO-span} method applied to $(f_\\beta, S_\\beta)$ starting from $x_0 = 0$ has for all $t\\leq d-1$\n \\begin{equation}\n f_\\beta(x_t) - \\min_{x\\in S_\\beta}f_\\beta(x) \\geq \\max\\left(0,\\sqrt{\\frac{2}{5}\\frac{(d-t)(\\mathrm{diam}(S_\\beta) - 2/\\beta)^2}{(d+2)^3}} - \\frac{1}{\\sqrt{2}\\beta}\\right)^2 . \n \\end{equation}\n In particular, for any fixed budget $T \\ge 1$, there exist $S_\\beta$ and $f_\\beta$ in dimension $d=2(T+1)$ such that\n \\[\n f_\\beta(x_T) - \\min_{x\\in S_\\beta}f_\\beta(x) \\geq \\max\\left(0,\\frac{\\mathrm{diam}(S_\\beta) - 2/\\beta}{\\sqrt{20}(T+2)} - \\frac{1}{\\sqrt{2}\\beta }\\right)^2 .\n \\]\n\\end{theorem}\n\\noindent In both of these theorems, the lower bounds are only meaningful in the regime of $\\beta=\\Omega(1/\\sqrt{\\varepsilon})$ as otherwise, the two-term maximums above will take value zero, making the lower bound vacuous.\n\nInductively applying Lemma~\\ref{lem:approx-chain}, the iterate $x_t \\in \\operatorname{conv}\\{x_0, z_1, \\dots, z_t\\}$ must have final $d-t$ coordinates $i\\geq t+1$ equal with value bounded by $|(x_t)_i| \\leq \\max_{1 \\le k \\le t} |\\delta_k| \\le \\frac{1}{\\beta \\sqrt{d-t+1}}$. Then the suboptimality $f_\\beta(x_t) - f_\\beta(x_\\star) = \\frac{1}{2}\\|x_t-x_\\star\\|^2_2$ is lower bounded by\n\\begin{align*}\n f_\\beta(x_t) - f_\\beta(x_\\star) &\\ge \\frac{1}{2} \\sum_{i=t+1}^d \\left((x_t)_i - \\left(\\frac{1}{d}+\\frac{1}{\\beta \\sqrt{d}}\\right)\\right)^2 \\\\\n &\\ge \\frac{d-t}{2}\\max \\left(0, \\frac{1}{d}+\\frac{1}{\\beta\\sqrt{d}}-\\frac{1}{\\beta\\sqrt{d-t+1}}\\right)^2 \\\\\n &\\ge \\max \\left(0, \\sqrt{\\frac{d-t}{2}}\\left(\\frac{\\mathrm{diam}(S_\\beta) - 2/\\beta}{\\sqrt{2}d}+\\frac{1}{\\beta\\sqrt{d}}\\right)-\\frac{1}{\\sqrt{2}\\beta}\\right)^2.\n\\end{align*}\nSpecializing to $t=T$ and $d=2T$, this provides a lower bound of\n$$ f_\\beta(x_T) - f_\\beta(x_\\star) \\geq \\max\\left(0,\\frac{\\mathrm{diam}(S_\\beta) - 2/\\beta}{4\\sqrt{T}} - \\frac{\\sqrt{2}-1}{2\\beta }\\right)^2 . $$\n\\subsection{Proof of Theorem~\\ref{thm:mixed}}\nGiven any $d>0$, consider the set $S$ previously defined in~\\eqref{eq:S-simple} and the parameter $\\nu$ defined by~\\eqref{eq:nu-def}. Fix any $\\beta > 0$ and define\n$$ S_\\beta := S + B(0,1/\\beta) . $$\nFrom Lemma~\\ref{lem:diam-SC}, it is immediate that the diameter of $S_\\beta$ is $\\mathrm{diam}(S_\\beta) = \\mathrm{diam}(S)+2/\\beta = 2C(2-\\sqrt{2}) + 2/\\beta$. Further, $S_\\beta$ is $\\beta$-smooth and $\\alpha = 1/(1+1/\\beta)$-strongly convex by the calculus rules for Minkowski sums~\\cite[Lemma 9]{liu2023gauges}.", "post_theorem_intro_text_len": 7180, "post_theorem_intro_text": "While Theorem~\\ref{thm:main} applies for any $L,\\alpha>0$, it does not allow a free selection of $\\mathrm{diam}(S)$. Rather, our constructed hard ``zero-chain'' set $S$ has $\\mathrm{diam}(S) = \\Theta(1/\\alpha d)$. One cannot arbitrarily select $\\alpha$ and $\\mathrm{diam}(S)$ as they must satisfy, for example, that $\\mathrm{diam}(S)\\leq 2/\\alpha$. In the limit where $\\mathrm{diam}(S)= 2/\\alpha$, the set $S$ is forced to be (up to translation) the ball $B(0,1/\\alpha)$. Such forced structure enables faster algorithms\\footnote{For example, when $\\mathrm{diam}(S)= 2/\\alpha$ and $S = B(0,1/\\alpha)$, an LMO can be used to explicitly compute orthogonal projections onto the feasible region $S$. Then the class of \\texttt{FO-LMO} methods includes projected gradient methods, which are known to converge linearly for smooth, strongly convex objectives.}. The development of hard instances for any selection of $\\mathrm{diam}(S) \\leq 2/\\alpha$ and resulting more nuanced complexity bounds is left as an important future direction.\n\nAs a secondary result, we provide a partial generalization to $\\beta$-smooth sets. These bounds are meaningful in the regime of only modestly smooth sets, having $\\beta=\\Omega(1/\\sqrt{\\varepsilon})$. In this regime, no \\texttt{LMO-span} method (see Section~\\ref{sec:SC} for a formal definition) can improve past the optimal compact convex set complexity of $\\Theta(L\\, \\mathrm{diam}(S)^2/\\varepsilon)$ or over the optimal $\\alpha$-strongly convex set complexity of $\\Theta(\\sqrt{L\\, \\mathrm{diam}(S)^2/\\varepsilon})$. Theorems~\\ref{thm:smooth} and~\\ref{thm:mixed} formalize these limits on acceleration due to smoothness.\n\nSince our constructions throughout use an $L$-smooth, $L$-strongly convex objective, all of our lower bounds apply against the wider class of problems with smooth, strongly convex objectives having any $0 \\leq \\mu \\leq L$. Hence, our theory highlights a fundamental difficulty of constrained optimization via LMOs: {\\it Even on perfectly conditioned objective functions, hard adversarial constraint sets constitute a barrier to linear convergence.} This stands in contrast to gradient methods with an orthogonal projection oracle where convergence is dominated by objective function conditioning.\n\n\\paragraph{Outline.} Section~\\ref{sec:SC} first derives a lower bound via a novel construction of a strongly convex feasible region that is hard for all \\texttt{FO-LMO} methods with an additional span restriction. Section~\\ref{sec:general} removes this restriction, proving Theorem~\\ref{thm:main}. This construction is the main technical innovation of our work. Section~\\ref{sec:smooth} then provides results extending lower bounds to the specialized setting of sets with modest levels of smoothness. \n\n\\subsection{Related Work}\nOur Theorem~\\ref{thm:main} differs from the previously mentioned lower bounding result of~\\cite{halbey2026lower} in two aspects, making the result complementary. In terms of algorithmic scope, our bound provides a wider guarantee, establishing a universal lower bound against all \\texttt{FO-LMO} methods. In contrast,~\\cite{halbey2026lower} provides a hard instance for two standard implementations of Frank-Wolfe, namely those fixing $p_k=\\nabla f(x_k)$ and using an exact line search or the short stepsize procedure to select $x_{k+1}$. In terms of problem class scope, our construction requires a nonsmooth feasible region and a large problem dimension $d$ (linear in the number of iterations to be run). Such a high dimensionality assumption (i.e., $d>T$) is classical and widespread in the optimization complexity literature~\\cite{nemirovski1983problem}. In contrast,~\\cite{halbey2026lower} is able to provide a hard instance using a smooth ball in $d=2$ dimensions. As a result, they provide a stronger illustration of the limitations of the two methods that their theory covers.\n\nAs mentioned above, most classical lower bounding results in unconstrained first-order optimization rely on setting the problem dimension larger than the number of iterations to be conducted. This enables ``zero-chain'' arguments where the objective function is designed to reveal only one new coordinate to the given gradient-span algorithm at each iteration. The main technical innovation of our work is the design of a hard strongly convex set where the LMO possesses a similar zero-chain property to these classical hard objective constructions. Such constructions open the possibility to extend proof techniques and insights from existing unconstrained optimization lower bounds to constrained LMO/projection-free settings.\n\nFor example, although our strongly convex convergence rate lower bound matches the upper bound of~\\cite{garber2015faster} in order, the two differ by constant factors: our lower bound has a universal constant of $1/528$ and their upper bound has $9/2$. This leaves open the question of determining exactly minimax optimal algorithms and hard problem instances. In settings of unconstrained first-order minimization, the Performance Estimation Problem (PEP) techniques pioneered by~\\cite{drori2014performance,Interpolation,Interpolation2} have provided such theory. For example, in smooth convex optimization, these facilitated the identification of the Optimized Gradient Method~\\cite{kim2016optimized} and an exactly matching lower bound~\\cite{drori2017exact}. PEP was extended to cover structured smooth and strongly convex sets by Luner and Grimmer~\\cite{luner2024performance}. As a result, future work may leverage PEP to similarly close the constant-factor gap left for \\texttt{FO-LMO} methods.\n\nAnother important property of most Frank-Wolfe methods is that their trajectory is independent of the choice of inner product used. This differs from projected-gradient methods, where the choice of inner product and notion of orthogonality for projections affects the algorithm trajectory, making good preconditioning important for practical success. The line of work~\\cite{Pena2023,Wirth2025,Wirth2026} provided Frank-Wolfe with ``affine-covariant'' convergence theory, matching the method's affine covariant nature by avoiding notions like smoothness and strong convexity defined in terms of a fixed inner product. Developing lower bounding theory in affine-covariant terms is an interesting future direction.\n\nOne can view the oracle model of an LMO as assuming first-order access to the support function $p \\mapsto \\sup\\{\\langle p, x\\rangle \\mid x\\in S\\}$. For sets with $0\\in S$, a polar transformation establishes this as dual to assuming first-order access to the Minkowski gauge $y \\mapsto \\inf\\{ \\gamma>0 \\mid y/\\gamma \\in S\\}$. This gauge model has been studied by the works~\\cite{Renegar2016,Grimmer2021-part1,Grimmer2021-part2,Zakaria2022,Lu2023} as an alternative projection-free framework. The works~\\cite{liu2023gauges} and~\\cite{samakhoana2024scalable} developed accelerated $\\mathcal{O}(1/T^2)$ convergence guarantees for gauge methods over smooth sets, complementing the accelerated LMO rates for strongly convex sets. Identifying any structural relationships between the complexity of these dual oracle models and problem settings is another interesting direction.", "sketch": "Section~\\ref{sec:SC} first derives a lower bound via a novel construction of a strongly convex feasible region that is hard for all \\texttt{FO-LMO} methods with an additional span restriction. Section~\\ref{sec:general} removes this restriction, proving Theorem~\\ref{thm:main}. This construction is the main technical innovation of our work.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:main}\nConsider any $T \\ge 1$ and $L,\\alpha>0$. Then there exist an $\\alpha$-strongly convex set $S$ and an $L$-smooth, $L$-strongly convex function $f$ in dimension $d=2(T+1)$ such that every \\texttt{FO-LMO} method applied to $(f, S)$ with $x_0 = 0$ has $x_T$ infeasible or \n\\[\nf(x_T) - \\min_{x\\in S}f(x) \\geq \\frac{1}{528}\\frac{L\\, \\mathrm{diam}(S)^2}{(T+1)^2}.\n\\]\nConsequently, for any $\\varepsilon>0$, there exist problem instances where $T\\geq \\sqrt{\\frac{L\\, \\mathrm{diam}(S)^2}{528\\varepsilon}} - 1$ iterations are required for any \\texttt{FO-LMO} method to reach a suboptimality of $f(x_T) - \\min_{x\\in S}f(x) \\leq \\varepsilon$.", "theorem_type": ["Existence", "Inequality or Bound"], "mcq": {"question": "Let \\(\\mathrm{diam}(S)=\\max_{x,y\\in S}\\|x-y\\|_2\\). A compact convex set \\(S\\subseteq\\mathbb{R}^d\\) is \\(\\alpha\\)-strongly convex if for all \\(x,y\\in S\\) and \\(\\lambda\\in[0,1]\\),\n\\[\n\\lambda x+(1-\\lambda)y+B\\!\\left(0,\\frac{\\lambda(1-\\lambda)\\alpha}{2}\\|y-x\\|_2^2\\right)\\subseteq S,\n\\]\nwhere \\(B(0,r)=\\{u\\in\\mathbb{R}^d:\\|u\\|_2\\le r\\}\\). A differentiable function \\(f\\colon\\mathbb{R}^d\\to\\mathbb{R}\\) is \\(L\\)-smooth if \\(\\|\\nabla f(y)-\\nabla f(x)\\|_2\\le L\\|y-x\\|_2\\) for all \\(x,y\\), and is \\(L\\)-strongly convex if\n\\[\nf(\\lambda x+(1-\\lambda)y)\\le \\lambda f(x)+(1-\\lambda)f(y)-\\frac{\\lambda(1-\\lambda)L}{2}\\|y-x\\|_2^2\n\\]\nfor all \\(x,y\\in\\mathbb{R}^d\\), \\(\\lambda\\in[0,1]\\). A FO-LMO method is an algorithm that uses first-order information on \\(f\\) together with a linear minimization oracle over \\(S\\), and it is started from \\(x_0=0\\). For arbitrary \\(T\\ge 1\\) and \\(L,\\alpha>0\\), which quantitative lower-bound statement holds in dimension \\(d=2(T+1)\\)?", "correct_choice": {"label": "A", "text": "There exist an \\(\\alpha\\)-strongly convex set \\(S\\subseteq\\mathbb{R}^{2(T+1)}\\) and a function \\(f\\colon\\mathbb{R}^{2(T+1)}\\to\\mathbb{R}\\) that is both \\(L\\)-smooth and \\(L\\)-strongly convex such that every FO-LMO method applied to \\((f,S)\\) with \\(x_0=0\\) either produces an infeasible iterate \\(x_T\\notin S\\) or satisfies\n\\[\nf(x_T)-\\min_{x\\in S} f(x)\\ge \\frac{1}{528}\\,\\frac{L\\,\\mathrm{diam}(S)^2}{(T+1)^2}.\n\\]\nConsequently, for every \\(\\varepsilon>0\\), there exist such problem instances for which any FO-LMO method requires at least\n\\[\n\\sqrt{\\frac{L\\,\\mathrm{diam}(S)^2}{528\\varepsilon}}-1\n\\]\niterations to obtain a point with \\(f(x_T)-\\min_{x\\in S}f(x)\\le \\varepsilon\\)."}, "choices": [{"label": "B", "text": "There exist an \\(\\alpha\\)-strongly convex set \\(S\\subseteq\\mathbb{R}^{2(T+1)}\\) and a function \\(f\\colon\\mathbb{R}^{2(T+1)}\\to\\mathbb{R}\\) that is both \\(L\\)-smooth and \\(L\\)-strongly convex such that every FO-LMO method applied to \\((f,S)\\) with \\(x_0=0\\) either produces an infeasible iterate \\(x_T\\notin S\\) or satisfies\n\\[\nf(x_T)-\\min_{x\\in S} f(x)\\ge \\frac{1}{528}\\,\\frac{L\\,\\mathrm{diam}(S)^2}{T+1}.\n\\]\nConsequently, for every \\(\\varepsilon>0\\), there exist such problem instances for which any FO-LMO method requires at least\n\\[\n\\frac{L\\,\\mathrm{diam}(S)^2}{528\\varepsilon}-1\n\\]\niterations to obtain a point with \\(f(x_T)-\\min_{x\\in S}f(x)\\le \\varepsilon\\)."}, {"label": "C", "text": "There exist an \\(\\alpha\\)-strongly convex set \\(S\\subseteq\\mathbb{R}^{2(T+1)}\\) and a function \\(f\\colon\\mathbb{R}^{2(T+1)}\\to\\mathbb{R}\\) that is both \\(L\\)-smooth and \\(L\\)-strongly convex such that every FO-LMO method applied to \\((f,S)\\) with \\(x_0=0\\) satisfies\n\\[\nf(x_T)-\\min_{x\\in S} f(x)\\ge \\frac{1}{528}\\,\\frac{L\\,\\mathrm{diam}(S)^2}{(T+1)^2}.\n\\]"}, {"label": "D", "text": "There exist an \\(\\alpha\\)-strongly convex set \\(S\\subseteq\\mathbb{R}^{2(T+1)}\\) and a function \\(f\\colon\\mathbb{R}^{2(T+1)}\\to\\mathbb{R}\\) that is both \\(L\\)-smooth and \\(L\\)-strongly convex such that every FO-LMO method applied to \\((f,S)\\) with \\(x_0=0\\) either produces an infeasible iterate \\(x_T\\notin S\\) or satisfies\n\\[\nf(x_T)-\\min_{x\\in S} f(x)\\ge \\frac{1}{528}\\,\\frac{L\\,\\mathrm{diam}(S)}{(T+1)^2}.\n\\]\nConsequently, for every \\(\\varepsilon>0\\), there exist such problem instances for which any FO-LMO method requires at least\n\\[\n\\sqrt{\\frac{L\\,\\mathrm{diam}(S)}{528\\varepsilon}}-1\n\\]\niterations to obtain a point with \\(f(x_T)-\\min_{x\\in S}f(x)\\le \\varepsilon\\)."}, {"label": "E", "text": "For every \\(\\alpha\\)-strongly convex set \\(S\\subseteq\\mathbb{R}^{2(T+1)}\\) and every function \\(f\\colon\\mathbb{R}^{2(T+1)}\\to\\mathbb{R}\\) that is both \\(L\\)-smooth and \\(L\\)-strongly convex, every FO-LMO method applied to \\((f,S)\\) with \\(x_0=0\\) either produces an infeasible iterate \\(x_T\\notin S\\) or satisfies\n\\[\nf(x_T)-\\min_{x\\in S} f(x)\\ge \\frac{1}{528}\\,\\frac{L\\,\\mathrm{diam}(S)^2}{(T+1)^2}.\n\\]\nConsequently, for every \\(\\varepsilon>0\\), any FO-LMO method requires at least\n\\[\n\\sqrt{\\frac{L\\,\\mathrm{diam}(S)^2}{528\\varepsilon}}-1\n\\]\niterations to obtain a point with \\(f(x_T)-\\min_{x\\in S}f(x)\\le \\varepsilon\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "rate_exponent_in_T", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_infeasible_iterate_alternative_and_iteration_complexity_corollary", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "diameter_squared_scaling", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "existential_instances_vs_uniform_all_instances", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct option explicitly. It gives definitions and setup, but the exact lower-bound form, quantifiers, infeasibility alternative, and iteration complexity corollary must still be identified from the choices."}, "TAS": {"score": 0, "justification": "This is essentially a theorem-identification item: the task is to recognize the exact quantitative statement of a lower-bound result. It tests recall of the theorem statement more than selection among independently reasoned mathematical conclusions."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the student must compare subtle differences in rate exponent, diameter scaling, existential vs. universal quantification, and omission of the infeasibility clause. However, the item mainly rewards precise recall rather than generating a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they alter the rate from 1/(T+1)^2 to 1/(T+1), drop a necessary caveat, change diam(S)^2 to diam(S), or switch existential to universal quantification. These reflect realistic failure modes in reading theorem statements."}, "total_score": 5, "overall_assessment": "A moderately strong MCQ with no answer leakage and high-quality distractors, but it is largely a theorem-recall/restatement question rather than a genuinely generative reasoning task."}}
{"id": "2602.22744v1", "paper_link": "http://arxiv.org/abs/2602.22744v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{thm-main11}\n    For any complex curve in a K\\\"ahler surface $M^4$, the first eigenvalue $\\Lambda_1$ of its area Jacobi operator $\\mathcal{L}$ satisfies\n    $$\\Lambda_1\\geq 2\\,\\mathfrak{Ric}.$$", "start_pos": 5914, "end_pos": 6134, "label": "thm-main11"}, "ref_dict": {"thm-main11": "\\begin{thm}\\label{thm-main11}\n    For any complex curve in a K\\\"ahler surface $M^4$, the first eigenvalue $\\Lambda_1$ of its area Jacobi operator $\\mathcal{L}$ satisfies\n    $$\\Lambda_1\\geq 2\\,\\mathfrak{Ric}.$$\n\\end{thm}", "thm-main2": "\\begin{thm}\\label{thm-main2}\n Let \\(M^{4}\\) be a K\\\"ahler-Einstein surface with Einstein constant \\(\\mathfrak{c}>0\\), and let \\(x\\colon\\Sigma \\rightarrow M^{4}\\) be a complex curve. Then \n \\begin{itemize}\n    \\item[{\\rm(1)}] the first eigenvalue $\\Lambda_1$ of its area Jacobi operator is at least $2\\mathfrak{c}$,\n    \\item[{\\rm(2)}] if $\\Sigma$ has genus $g\\leq 1$, then $\\Lambda_1=2\\mathfrak{c}$ and the corresponding eigenspace has dimension\n    \\[\n    \\mathfrak{c}\\frac{\\operatorname{Area}}{2\\pi}+1-g+\\dim H^{\\,0}(N^*_\\Sigma\\otimes K_\\Sigma^2),\n    \\]\n    where $N^*_\\Sigma$ is the dual of the normal bundle, $K_\\Sigma$ is the canonical line bundle of $\\Sigma$, and $H^{\\,0}(N^*_\\Sigma\\otimes K_\\Sigma^2)$ denotes  the space of holomorphic sections of $N^*_\\Sigma\\otimes K_\\Sigma^2$. \n\\end{itemize}\n\\end{thm}"}, "pre_theorem_intro_text_len": 4004, "pre_theorem_intro_text": "The first eigenvalue of the Laplace-Beltrami operator on a compact Riemannian manifold encodes crucial geometric information, including curvature bounds. A fundamental result in this direction is the Lichnerowicz theorem~\\cite{Lichnerowicz}, which states that if the Ricci curvature of an \\(n\\)-dimensional compact manifold satisfies \\(\\mathrm{Ric} \\ge (n-1)K\\) for some \\(K>0\\), then the first positive eigenvalue \\(\\lambda_1\\) obeys \\(\\lambda_1 \\ge nK\\). This estimate is sharp, and the Obata theorem~\\cite{Obata1962} characterizes the equality case: \\(\\lambda_1 = nK\\) holds iff the manifold is isometric to a sphere of constant sectional curvature \\(K\\). In the K\\\"ahler setting, Futaki~\\cite{Futaki} proved that the first eigenvalue of the complex Laplacian is bounded below by \\(k\\) for a compact K\\\"ahler manifold satisfying \\(\\mathrm{Ric} \\ge k\\) with \\(k>0\\). The equality case was later characterized by Tam and Yu~\\cite{Tam-Yu}.  \n\nThe area Jacobi operator $\\mathcal{L}$, which governs the second variation of the area functional for minimal submanifolds, is a second-order, self-adjoint elliptic operator. As such, it can be regarded as the extrinsic counterpart to the intrinsic Laplace–Beltrami operator. Unlike the latter, it is not positive (or even nonnegative) in general. We denote by $\\Lambda_1$ the smallest nonzero eigenvalue of $\\mathcal{L}$, referred to as the first eigenvalue. Since $\\mathcal{L}$ serves as the stability operator, $\\Lambda_1$ governs the infinitesimal behavior of the minimal submanifold under deformations. A positive $\\Lambda_1$ means the submanifold is locally minimizing, whereas a negative one indicates instability.\nIn recent decades, considerable attention has been devoted to estimating the first eigenvalue of the area Jacobi operator for minimal surfaces in various ambient spaces. In \\cite{Perdomo}, Perdomo proved that if $\\Sigma^2$ is a compact oriented minimal surface which is not totally geodesic in $\\mathbb{S}^3$, then the first eigenvalue of its area Jacobi operator satisfies $\\Lambda_1 \\leq -4$, with equality if and only if $\\Sigma^2$ is isometric to the Clifford torus.  This result sparked a series of further estimates for $\\Lambda_1$ on closed hypersurfaces of constant mean curvature or conatnt scalar curvature (see \\cite{Alias,Ambrozio,BatistaCavalcanteMelo,Batista,Bray,Chen-Cheng,Cheng,Dung,Li-Wang,Merono,Zhu} and references therein). \nMicallef and Wolfson~\\cite{MW} obtained fundamental results on the stability of minimal surfaces in 4-dimensional manifolds by considering a complex version of the second variation of area. Subsequently, for superminimal surfaces with negative spin in self-dual Einstein $4$-dimensional manifolds, Montiel and Urbano~\\cite{MU} proved that the first eigenvalue of the area Jacobi operator satisfies $\\Lambda_1\\geq-2\\mathfrak{c}$, where $\\mathfrak{c}$  denotes the Einstein constant of the ambient space, and they computed exactly the index explicitly in terms of the genus and area of the surface. \n\nThis paper focuses on another important class of codimension-2 minimal surfaces, i.e., complex curves in Kähler surfaces, which are superminimal surfaces of positive spin. It is a classical fact that such surfaces are absolutely area‑minimizing and thus stable. Consequently, their associated area Jacobi operator $\\mathcal{L}$ is a non‑negative self‑adjoint elliptic operator, exactly analogous to the Laplace–Beltrami operator on a Riemannian manifold. While the non-negativity of $\\mathcal{L}$ is well understood, obtaining sharp quantitative estimates for its spectrum in terms of the ambient curvature is a more subtle problem, and the focus of our investigation. We established a universal lower bound for the first eigenvalue $\\Lambda_1$ of $\\mathcal{L}$. To state our result, we denote by \n\\[\n\\mathfrak{Ric} \\triangleq \\inf_{\\substack{p \\in \\Sigma, \\\\ u \\in UT_p M^4}} \\operatorname{Ric}(u, u),\n\\]\nthe infimum of the Ricci curvature over all unit tangent vectors at all points of $M^4$.", "context": "The first eigenvalue of the Laplace-Beltrami operator on a compact Riemannian manifold encodes crucial geometric information, including curvature bounds. A fundamental result in this direction is the Lichnerowicz theorem~\\cite{Lichnerowicz}, which states that if the Ricci curvature of an \\(n\\)-dimensional compact manifold satisfies \\(\\mathrm{Ric} \\ge (n-1)K\\) for some \\(K>0\\), then the first positive eigenvalue \\(\\lambda_1\\) obeys \\(\\lambda_1 \\ge nK\\). This estimate is sharp, and the Obata theorem~\\cite{Obata1962} characterizes the equality case: \\(\\lambda_1 = nK\\) holds iff the manifold is isometric to a sphere of constant sectional curvature \\(K\\). In the K\\\"ahler setting, Futaki~\\cite{Futaki} proved that the first eigenvalue of the complex Laplacian is bounded below by \\(k\\) for a compact K\\\"ahler manifold satisfying \\(\\mathrm{Ric} \\ge k\\) with \\(k>0\\). The equality case was later characterized by Tam and Yu~\\cite{Tam-Yu}.\n\nThe area Jacobi operator $\\mathcal{L}$, which governs the second variation of the area functional for minimal submanifolds, is a second-order, self-adjoint elliptic operator. As such, it can be regarded as the extrinsic counterpart to the intrinsic Laplace–Beltrami operator. Unlike the latter, it is not positive (or even nonnegative) in general. We denote by $\\Lambda_1$ the smallest nonzero eigenvalue of $\\mathcal{L}$, referred to as the first eigenvalue. Since $\\mathcal{L}$ serves as the stability operator, $\\Lambda_1$ governs the infinitesimal behavior of the minimal submanifold under deformations. A positive $\\Lambda_1$ means the submanifold is locally minimizing, whereas a negative one indicates instability.\nIn recent decades, considerable attention has been devoted to estimating the first eigenvalue of the area Jacobi operator for minimal surfaces in various ambient spaces. In \\cite{Perdomo}, Perdomo proved that if $\\Sigma^2$ is a compact oriented minimal surface which is not totally geodesic in $\\mathbb{S}^3$, then the first eigenvalue of its area Jacobi operator satisfies $\\Lambda_1 \\leq -4$, with equality if and only if $\\Sigma^2$ is isometric to the Clifford torus. This result sparked a series of further estimates for $\\Lambda_1$ on closed hypersurfaces of constant mean curvature or conatnt scalar curvature (see \\cite{Alias,Ambrozio,BatistaCavalcanteMelo,Batista,Bray,Chen-Cheng,Cheng,Dung,Li-Wang,Merono,Zhu} and references therein). \nMicallef and Wolfson~\\cite{MW} obtained fundamental results on the stability of minimal surfaces in 4-dimensional manifolds by considering a complex version of the second variation of area. Subsequently, for superminimal surfaces with negative spin in self-dual Einstein $4$-dimensional manifolds, Montiel and Urbano~\\cite{MU} proved that the first eigenvalue of the area Jacobi operator satisfies $\\Lambda_1\\geq-2\\mathfrak{c}$, where $\\mathfrak{c}$ denotes the Einstein constant of the ambient space, and they computed exactly the index explicitly in terms of the genus and area of the surface.\n\nThis paper focuses on another important class of codimension-2 minimal surfaces, i.e., complex curves in Kähler surfaces, which are superminimal surfaces of positive spin. It is a classical fact that such surfaces are absolutely area‑minimizing and thus stable. Consequently, their associated area Jacobi operator $\\mathcal{L}$ is a non‑negative self‑adjoint elliptic operator, exactly analogous to the Laplace–Beltrami operator on a Riemannian manifold. While the non-negativity of $\\mathcal{L}$ is well understood, obtaining sharp quantitative estimates for its spectrum in terms of the ambient curvature is a more subtle problem, and the focus of our investigation. We established a universal lower bound for the first eigenvalue $\\Lambda_1$ of $\\mathcal{L}$. To state our result, we denote by \n\\[\n\\mathfrak{Ric} \\triangleq \\inf_{\\substack{p \\in \\Sigma, \\\\ u \\in UT_p M^4}} \\operatorname{Ric}(u, u),\n\\]\nthe infimum of the Ricci curvature over all unit tangent vectors at all points of $M^4$.", "full_context": "The first eigenvalue of the Laplace-Beltrami operator on a compact Riemannian manifold encodes crucial geometric information, including curvature bounds. A fundamental result in this direction is the Lichnerowicz theorem~\\cite{Lichnerowicz}, which states that if the Ricci curvature of an \\(n\\)-dimensional compact manifold satisfies \\(\\mathrm{Ric} \\ge (n-1)K\\) for some \\(K>0\\), then the first positive eigenvalue \\(\\lambda_1\\) obeys \\(\\lambda_1 \\ge nK\\). This estimate is sharp, and the Obata theorem~\\cite{Obata1962} characterizes the equality case: \\(\\lambda_1 = nK\\) holds iff the manifold is isometric to a sphere of constant sectional curvature \\(K\\). In the K\\\"ahler setting, Futaki~\\cite{Futaki} proved that the first eigenvalue of the complex Laplacian is bounded below by \\(k\\) for a compact K\\\"ahler manifold satisfying \\(\\mathrm{Ric} \\ge k\\) with \\(k>0\\). The equality case was later characterized by Tam and Yu~\\cite{Tam-Yu}.\n\nThe area Jacobi operator $\\mathcal{L}$, which governs the second variation of the area functional for minimal submanifolds, is a second-order, self-adjoint elliptic operator. As such, it can be regarded as the extrinsic counterpart to the intrinsic Laplace–Beltrami operator. Unlike the latter, it is not positive (or even nonnegative) in general. We denote by $\\Lambda_1$ the smallest nonzero eigenvalue of $\\mathcal{L}$, referred to as the first eigenvalue. Since $\\mathcal{L}$ serves as the stability operator, $\\Lambda_1$ governs the infinitesimal behavior of the minimal submanifold under deformations. A positive $\\Lambda_1$ means the submanifold is locally minimizing, whereas a negative one indicates instability.\nIn recent decades, considerable attention has been devoted to estimating the first eigenvalue of the area Jacobi operator for minimal surfaces in various ambient spaces. In \\cite{Perdomo}, Perdomo proved that if $\\Sigma^2$ is a compact oriented minimal surface which is not totally geodesic in $\\mathbb{S}^3$, then the first eigenvalue of its area Jacobi operator satisfies $\\Lambda_1 \\leq -4$, with equality if and only if $\\Sigma^2$ is isometric to the Clifford torus. This result sparked a series of further estimates for $\\Lambda_1$ on closed hypersurfaces of constant mean curvature or conatnt scalar curvature (see \\cite{Alias,Ambrozio,BatistaCavalcanteMelo,Batista,Bray,Chen-Cheng,Cheng,Dung,Li-Wang,Merono,Zhu} and references therein). \nMicallef and Wolfson~\\cite{MW} obtained fundamental results on the stability of minimal surfaces in 4-dimensional manifolds by considering a complex version of the second variation of area. Subsequently, for superminimal surfaces with negative spin in self-dual Einstein $4$-dimensional manifolds, Montiel and Urbano~\\cite{MU} proved that the first eigenvalue of the area Jacobi operator satisfies $\\Lambda_1\\geq-2\\mathfrak{c}$, where $\\mathfrak{c}$ denotes the Einstein constant of the ambient space, and they computed exactly the index explicitly in terms of the genus and area of the surface.\n\nThis paper focuses on another important class of codimension-2 minimal surfaces, i.e., complex curves in Kähler surfaces, which are superminimal surfaces of positive spin. It is a classical fact that such surfaces are absolutely area‑minimizing and thus stable. Consequently, their associated area Jacobi operator $\\mathcal{L}$ is a non‑negative self‑adjoint elliptic operator, exactly analogous to the Laplace–Beltrami operator on a Riemannian manifold. While the non-negativity of $\\mathcal{L}$ is well understood, obtaining sharp quantitative estimates for its spectrum in terms of the ambient curvature is a more subtle problem, and the focus of our investigation. We established a universal lower bound for the first eigenvalue $\\Lambda_1$ of $\\mathcal{L}$. To state our result, we denote by \n\\[\n\\mathfrak{Ric} \\triangleq \\inf_{\\substack{p \\in \\Sigma, \\\\ u \\in UT_p M^4}} \\operatorname{Ric}(u, u),\n\\]\nthe infimum of the Ricci curvature over all unit tangent vectors at all points of $M^4$.\n\nIf we restrict the ambient space to a Kähler–Einstein surface with positive scalar curvature, then the estimate established in Theorem \\ref{thm-main11} is sharp. In this setting, the curvature invariant $\\mathfrak{Ric}$ simplifies to the Einstein constant $\\mathfrak{c}>0$. We explore the equality case and are able to compute exactly the dimension of the first eigenspace. This is accomplished by relating it to the space of Jacobi fields of $\\mathcal{W}^+$, which corresponds precisely to the preimage of the space of holomorphic sections of a certain line bundle, and then applying the Riemann-Roch theorem in conjunction with the Dolbeault theorem. \n\\begin{thm}\\label{thm-main2}\n Let \\(M^{4}\\) be a K\\\"ahler-Einstein surface with Einstein constant \\(\\mathfrak{c}>0\\), and let \\(x\\colon\\Sigma \\rightarrow M^{4}\\) be a complex curve. Then \n \\begin{itemize}\n \\item[{\\rm(1)}] the first eigenvalue $\\Lambda_1$ of its area Jacobi operator is at least $2\\mathfrak{c}$,\n \\item[{\\rm(2)}] if $\\Sigma$ has genus $g\\leq 1$, then $\\Lambda_1=2\\mathfrak{c}$ and the corresponding eigenspace has dimension\n \\[\n \\mathfrak{c}\\frac{\\operatorname{Area}}{2\\pi}+1-g+\\dim H^{\\,0}(N^*_\\Sigma\\otimes K_\\Sigma^2),\n \\]\n where $N^*_\\Sigma$ is the dual of the normal bundle, $K_\\Sigma$ is the canonical line bundle of $\\Sigma$, and $H^{\\,0}(N^*_\\Sigma\\otimes K_\\Sigma^2)$ denotes the space of holomorphic sections of $N^*_\\Sigma\\otimes K_\\Sigma^2$. \n\\end{itemize}\n\\end{thm}\n\nNow we assume that $x$ is a complex curve, then \\eqref{eq-sfm1} yields \n$$ a \\equiv 0,~~~ b \\equiv 0,$$\nwhich implies $ x $ is a superminimal surface with positive spin. Consequently, $x$ minimizes the functional $\\mathcal{W}^+$ within its homotopy class. Without loss of generality, we assume $\\lambda\\equiv 1$ and $\\mu\\equiv 0$. Denote by $\\theta_{12}$ (resp. $\\theta_{34}$) the connection form of the tangent (resp. normal) bundle of $x$, It follows from \\eqref{eq-pull}$\\sim$\\eqref{eq-conn} that \n\\begin{equation}\\label{eq-o1100}\n x^*\\omega_{1\\bar{1}} = -i\\theta_{12}, ~~~x^*\\omega_{0\\bar{0}} = -i\\theta_{34}. \n\\end{equation}\nDifferentiating \\eqref{eq-o1100} yields the Gauss-Ricci equation\n\\begin{equation}\\label{eq-Gauss1}\n (R_{1212}+R_{1234})\\,\\theta\\wedge\\bar\\theta = -2x^*(\\Omega_{0\\bar0}+\\Omega_{1\\bar1}),\n\\end{equation}\nwhere \\(R_{1212}\\) (resp. \\(R_{1234}\\)) denotes the curvature of the tangent (resp. normal) bundle, defined by\n\\[\nd\\theta_{12} = \\frac{R_{1212}}{2i}\\,\\theta\\wedge\\bar\\theta, \\qquad \nd\\theta_{34} = \\frac{R_{1234}}{2i}\\,\\theta\\wedge\\bar\\theta.\n\\]\nNote that \n$-i(\\Omega_{0\\bar0}+ \\Omega_{1\\bar1})$ is exactly the Ricci form of $M^4$, which implies \n\\begin{equation}\\label{eq-Ric}\nx^*(\\Omega_{0\\bar0}+ \\Omega_{1\\bar1})=-\\mathrm{Ric}(e_1, \\bar e_1)\\theta\\wedge\\bar\\theta.\n\\end{equation}\nTherefore, we can also reformulate the Gauss-Ricci equation \\eqref{eq-Gauss1} as \n\\begin{equation}\\label{eq-Gauss2}\n R_{1212}+R_{1234}=2\\mathrm{Ric}(e_1,\\bar e_1).\n\\end{equation}\n\nUsing \\eqref{eq-W+3} and \\eqref{eq-lambda1}, we obtain that along the variation determined by $V$, \n\\begin{equation}\\label{eq-varW}\n \\frac{\\partial^{2}}{\\partial t^{2}} \\bigg|_{t=0} \\mathcal{W}^{+}\n\\leq 8 \\int_{\\Sigma} \\Bigl( |\\nu_{\\bar11}|^2-\\frac{1}{2}\\mathfrak{Ric}\\,|\\nu_{\\bar1}|^2\\Bigr) dA \\le \\frac{1}{2} \\int_{\\Sigma} \\bigl( \\Lambda_{1}^{2} - 2 \\Lambda_{1}\\mathfrak{Ric} \\bigr) |\\nu|^{2} dA, \n\\end{equation}\nwhere the first inequality follows from\n $$\\operatorname{Ric}(e_{1}, \\bar e_{1})=\\operatorname{Ric}(\\mathrm{Re}(e_{1}), \\mathrm{Re}(e_{1}))+\\operatorname{Ric}(\\mathrm{Im}(e_{1}), \\mathrm{Im}(e_{1}))\\geq \\frac{1}{2}\\mathfrak{Ric},$$ \nwith $\\mathrm{Re}(e_{1})$ and $\\mathrm{Im}(e_{1})$ denoting the real and imaginary parts of $e_1$. The fact that any complex curve minimizes $\\mathcal{W}^+$ implies the left-hand side of \\eqref{eq-varW} is nonnegative. Together with $\\Lambda_1>0$, this yields the lower bound \n$$\\Lambda_1\\geq 2\\,\\mathfrak{Ric}.$$\n\\end{proof}\n\\section{The proof of Theorem 2}\\label{sec5}\nWe denote by ${E}_0^{\\mathcal{W}^+}$ (resp. $E_0^{\\mathcal{A}}$) the space of Jacobi fields associated with $\\mathcal{W}^+$ (resp. $\\mathcal{A}$). It follows from \\eqref{eq-Area} and \\eqref{eq-Will} that \n\\begin{equation}\\label{eq-Jaco}\n E_0^{\\mathcal{W}^+}=\\{V=\\nu e_0+\\bar\\nu\\bar e_0 \\mid \\nu_{\\bar1\\bar1}=0\\},~~~~~~E_0^{\\mathcal{A}}=\\{V=\\nu e_0+\\bar\\nu\\bar e_0\\mid \\nu_{\\bar1}=0\\}.\n\\end{equation}\nObviously, there holds \n$$E_0^{\\mathcal{A}}\\subset E_0^{\\mathcal{W}^+}.$$\n\n{\\bf Claim 1.} {\\em If $E_0^{\\mathcal{W}^+}\\setminus E_0^{\\mathcal{A}}\\neq\\emptyset$, then $\\Lambda_1=2\\mathfrak{c}$ and $E_0^{\\mathcal{W}^+}\\setminus E_0^{\\mathcal{A}}$ is exactly the first eigenspace of $\\mathcal{L}$}. \n\\vskip 0.2cm\n\\noindent To prove this claim, first note that $\\mathcal{L}$ is a nonnegative self-adjoint operator. Hence, for each eigenvalue $\\Lambda_j$ of $\\mathcal{L}$, we can choose a basis $\\{\\mathrm{V}_{j_1}, \\mathrm{V}_{j_2}, \\cdots \\mathrm{V}_{j_{k_j}}\\}$ of the corresponding eigenspace $E_j^{\\mathcal{A}}$ such that the collection \n$$\\{\\mathrm{V}_{j_l}\\mid 1\\leq l\\leq {k_j}, j\\in \\mathbb{Z}^{\\geq 0}\\}$$\nforms a complete orthonormal system of \\( L^2(\\Gamma(N_\\Sigma)) \\). \nLet $V\\in E_0^{\\mathcal{W}^+}\\setminus E_0^{\\mathcal{A}}$ be a Jacobi field of $\\mathcal{W}^+$. Write \n\\[V = \\sum_{j=0}^{+\\infty}\\sum_{l=1}^{k_j} \\varepsilon_{j_l} \\mathrm{V}_{j_l}.\\]\nThen we have \n\\[\\mathcal{L}(V) = -\\sum_{j=1}^{+\\infty}\\sum_{l=1}^{k_j} \\Lambda_j \\varepsilon_{j_l} \\mathrm{V}_{j_l}.\\]\nand then \n\\[\\int_{\\Sigma}|\\mathcal{L}(V)|^2 dA = \\sum_{j=1}^{+\\infty}\\sum_{l=1}^{k_j} \\varepsilon_{j_l}^2\\Lambda_j^2, \\quad\\quad \\int_{\\Sigma}\\langle \\mathcal{L}(V), V \\rangle dA= -\\sum_{j=1}^{+\\infty}\\sum_{l=1}^{k_j} \\varepsilon_{j_l}^2 \\Lambda_j.\\]\nSubstituting these equalities into \\eqref{eq-W+5}, we derive \n\\[\\sum_{j=1}^{+\\infty}\\sum_{l=1}^{k_j} \\varepsilon_{j_l}^2 \\Lambda_j(\\Lambda_j - 2\\mathfrak{c}) = 0.\\]\nIt follows from \n$$\\sum_{j=1}^{+\\infty}\\sum_{l=1}^{k_j}\\varepsilon_{j_l}^2\\neq0,~~\\text{and}~~\\Lambda_{j}>\\Lambda_1\\geq 2\\mathfrak{c},~j\\geq 2$$\nthat \n$$\\Lambda_1=2c,~~\\text{and}~~\\varepsilon_{j_l}=0,~j\\geq 2,$$\nwhich implies $V$ is an eigensection associated with the first eigenvalue $\\Lambda_1=2\\mathfrak{c}$. Therefore, we obtain \n$$E_0^{\\mathcal{W}^+}\\setminus E_0^{\\mathcal{A}}\\subset E_1^{\\mathcal{A}}.$$\nWhen $\\Lambda_1=2\\mathfrak{c}$, the converse inclusion follows directly from \\eqref{eq-W+5}. Thus Claim $1$ is proved.\n\nFrom \\eqref{eq-E0W} and Claim $2$, we derive \n$$\\dim E_0^{\\mathcal{W}^+}=\\dim \\ker{\\bar\\partial_1}+\\dim \\ker{\\bar\\partial}= \\dim H^0(\\Sigma, N_\\Sigma \\otimes \\overline{K_\\Sigma})+\\dim E_0^{\\mathcal{A}}.$$\nBy Riemann-Roch Theorem, we have \n$$\\begin{aligned}\\dim H^0(\\Sigma, N_\\Sigma \\otimes \\overline{K_\\Sigma})=&\\deg(N_\\Sigma \\otimes \\overline{K_\\Sigma})+1-g+\\dim H^0(\\Sigma, N_\\Sigma^* \\otimes K_\\Sigma^2)\\\\\n=&\\,\\mathfrak{c}\\frac{\\mathrm{Area}(\\Sigma)}{2\\pi}+1-g+\\dim H^0(\\Sigma, N_\\Sigma^* \\otimes K_\\Sigma^2).\n\\end{aligned}$$\nSo when $g\\leq 1$, there holds \n$$\\dim E_0^{\\mathcal{W}^+}-\\dim E_0^{\\mathcal{A}}=\\dim H^0(\\Sigma, N_\\Sigma \\otimes \\overline{K_\\Sigma})\\geq 1.$$\nTogether with Claim 1, this yields the second conclusion of this theorem. \n\\end{proof}\n\\begin{remark}\n It follows from the proof of Claim 1 that for complex curves in K\\\"ahler-Einstein surfaces of nonpositive scalar curvature, $${E}^{\\mathcal{W}^+}_0={E}^{\\mathcal{A}}_0,$$\n i.e., the Jacobi fields of $\\mathcal{W}^+$ are exactly the same as those of $\\mathcal{A}$.\n\\end{remark}", "post_theorem_intro_text_len": 4345, "post_theorem_intro_text": "Somewhat surprisingly, the proof of this theorem relies on computing the second variation of the following  conformally invariant functional \n$$\\mathcal{W}^+(x) \\triangleq \\int_{\\Sigma} (|\\vec H|^2 + K(\\mathfrak{e}_1, \\mathfrak{e}_2, \\mathfrak{e_1}, \\mathfrak{e_2}) + K(\\mathfrak{e}_1, \\mathfrak{e}_2, \\mathfrak{e_3}, \\mathfrak{e_4}))dA,$$\nintroduced by Montiel and Urbano in \\cite{Montiel-Urbano} for closed oriented surfaces in an oriented $4$-dimensional Riemannian manifold. Here $\\vec H$ represents the mean curvature vector, $K$ denotes the ambient  curvature tensor, and $\\{\\mathfrak{e}_1, \\mathfrak{e}_2, \\mathfrak{e}_3, \\mathfrak{e}_4\\}$ is an orthonormal basis with $\\{\\mathfrak{e}_1, \\mathfrak{e}_2\\}$ tangent to the surface $x$ and $\\{\\mathfrak{e}_3, \\mathfrak{e}_4\\}$ spanning its normal space. Associated with $\\mathcal{W}^+$, Montiel and Urbano \\cite{Montiel-Urbano} also introduced  \n$$\\mathcal{W}^-(x) \\triangleq \\int_{\\Sigma} (|\\vec H|^2 + K(\\mathfrak{e}_1, \\mathfrak{e}_2, \\mathfrak{e_1}, \\mathfrak{e_2}) - K(\\mathfrak{e}_1, \\mathfrak{e}_2, \\mathfrak{e_3}, \\mathfrak{e_4}))dA.$$\nThe average of these two functionals is precisely the Willmore functional,\n$$\\mathcal{W}(x) = \\int_{\\Sigma} (|\\vec H|^2 + K_{1212}) dA,$$\nan important global conformal invariant for surfaces in general curved Riemannian manifolds that has attracted considerable attention recently (see \\cite{Marques1,Mondino,Mondino1,Wang-Xie} and the references therein). In \\cite{Montiel-Urbano}, Montiel and Urbano studied $\\mathcal{W}^+$ and $\\mathcal{W}^-$ from the perspective of twistor theory and showed that a superminimal surface with positive (resp. negative) spin minimize $\\mathcal{W}^+$ (resp. $\\mathcal{W}^-$) among its homotopy class. In \\cite{Wang-Xie}, we investigated these functionals from the viewpoint of conformal geometry and derived the following expressions, \n\\begin{equation}\\label{eq-W+}\n\\mathcal{W}^+(x) = 4 \\int_{\\Sigma} |\\psi|^2 dA + 2\\pi (\\chi + \\chi^{\\perp}),~~~\\mathcal{W}^-(x) = 4 \\int_{\\Sigma} |\\phi|^2 dA + 2\\pi (\\chi - \\chi^{\\perp}),\n\\end{equation}\nwhere \n$\\psi$ and $\\phi$ are two local conformal invariants arising from the normal-bundle-valued Hopf differential, and $\\chi$ (resp. $\\chi^{\\perp}$) denotes the Euler characteristics of the tangent bundle (resp. normal bundle). When the ambient space is a Kähler surface, complex curves are superminimal surfaces with positive spin and hence are minimizers of $\\mathcal{W}^+$.   \n\nIf we restrict the ambient space to a Kähler–Einstein surface with positive scalar curvature, then the estimate established in Theorem \\ref{thm-main11} is sharp. In this setting, the curvature invariant $\\mathfrak{Ric}$ simplifies to the Einstein constant $\\mathfrak{c}>0$. We explore the equality case and are able to compute exactly the dimension of the first eigenspace. This is accomplished by relating it to the space of Jacobi fields of $\\mathcal{W}^+$, which corresponds precisely to the preimage of the space of holomorphic sections of a certain line bundle, and then applying the Riemann-Roch theorem in conjunction with the Dolbeault theorem. \n\\begin{thm}\\label{thm-main2}\n Let \\(M^{4}\\) be a K\\\"ahler-Einstein surface with Einstein constant \\(\\mathfrak{c}>0\\), and let \\(x\\colon\\Sigma \\rightarrow M^{4}\\) be a complex curve. Then \n \\begin{itemize}\n    \\item[{\\rm(1)}] the first eigenvalue $\\Lambda_1$ of its area Jacobi operator is at least $2\\mathfrak{c}$,\n    \\item[{\\rm(2)}] if $\\Sigma$ has genus $g\\leq 1$, then $\\Lambda_1=2\\mathfrak{c}$ and the corresponding eigenspace has dimension\n    \\[\n    \\mathfrak{c}\\frac{\\operatorname{Area}}{2\\pi}+1-g+\\dim H^{\\,0}(N^*_\\Sigma\\otimes K_\\Sigma^2),\n    \\]\n    where $N^*_\\Sigma$ is the dual of the normal bundle, $K_\\Sigma$ is the canonical line bundle of $\\Sigma$, and $H^{\\,0}(N^*_\\Sigma\\otimes K_\\Sigma^2)$ denotes  the space of holomorphic sections of $N^*_\\Sigma\\otimes K_\\Sigma^2$. \n\\end{itemize}\n\\end{thm}\n\nThis paper is organized as follows. In Section~\\ref{sec2}, we review the necessary background on surface theory in  Kähler surfaces. Section~\\ref{sec3} is devoted to compute the variation of geometric quantities for a complex curve in a K\\\"ahler surface. Theorem~\\ref{thm-main11} is established in Section~\\ref{sec4}. Finally, in Section~\\ref{sec5}, we focus on the Kähler-Einstein setting and prove Theorem \\ref{thm-main2}.", "sketch": "The post-theorem introduction says that, “Somewhat surprisingly, the proof of this theorem relies on computing the second variation of the following conformally invariant functional”\n\\[\n\\mathcal{W}^+(x) \\triangleq \\int_{\\Sigma} (|\\vec H|^2 + K(\\mathfrak{e}_1, \\mathfrak{e}_2, \\mathfrak{e_1}, \\mathfrak{e_2}) + K(\\mathfrak{e}_1, \\mathfrak{e}_2, \\mathfrak{e_3}, \\mathfrak{e_4}))\\,dA.\n\\]\nIt recalls that in a K\"ahler surface “complex curves are superminimal surfaces with positive spin and hence are minimizers of $\\mathcal{W}^+$.” It then points to where the argument is carried out: “Theorem~\\ref{thm-main11} is established in Section~\\ref{sec4},” after “Section~\\ref{sec3} is devoted to compute the variation of geometric quantities for a complex curve in a K\"ahler surface.”", "expanded_sketch": "The post-theorem introduction says that, “Somewhat surprisingly, the proof of this theorem relies on computing the second variation of the following conformally invariant functional”\n\\[\n\\mathcal{W}^+(x) \\triangleq \\int_{\\Sigma} (|\\vec H|^2 + K(\\mathfrak{e}_1, \\mathfrak{e}_2, \\mathfrak{e_1}, \\mathfrak{e_2}) + K(\\mathfrak{e}_1, \\mathfrak{e}_2, \\mathfrak{e_3}, \\mathfrak{e_4}))\\,dA.\n\\]\nIt recalls that in a K\"ahler surface “complex curves are superminimal surfaces with positive spin and hence are minimizers of $\\mathcal{W}^+$.” It then points to where the argument is carried out: to prove the main theorem, the paper first devotes the next part to computing the variation of geometric quantities for a complex curve in a K\"ahler surface, and later carries out the proof.", "expanded_theorem": "\\label{thm-main11}\nFor any complex curve in a K\\\"ahler surface $M^4$, the first eigenvalue $\\Lambda_1$ of its area Jacobi operator $\\mathcal{L}$ satisfies\n$$\\Lambda_1\\geq 2\\,\\mathfrak{Ric}.$$", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let \\(M^4\\) be a K\\\"ahler surface, and let \\(x\\colon \\Sigma \\to M^4\\) be a complex curve in \\(M^4\\). Let \\(\\mathcal L\\) denote the area Jacobi operator of \\(\\Sigma\\) (the self-adjoint elliptic operator governing the second variation of area), and let \\(\\Lambda_1\\) be its first eigenvalue. Define\n\\[\n\\mathfrak{Ric}:=\\inf\\{\\operatorname{Ric}(u,u): p\\in M^4,\\ u\\in T_pM^4,\\ |u|=1\\},\n\\]\nthe infimum of the ambient Ricci curvature over all unit tangent vectors of \\(M^4\\). Which statement holds for every such complex curve?", "correct_choice": {"label": "A", "text": "The first eigenvalue of the area Jacobi operator satisfies \\(\\Lambda_1 \\ge 2\\,\\mathfrak{Ric}\\)."}, "choices": [{"label": "B", "text": "The first eigenvalue of the area Jacobi operator satisfies \\(\\Lambda_1 \\ge \\mathfrak{Ric}\\)."}, {"label": "C", "text": "The first eigenvalue of the area Jacobi operator satisfies \\(\\Lambda_1 \\ge 0\\)."}, {"label": "D", "text": "The first eigenvalue of the area Jacobi operator satisfies \\(\\Lambda_1 \\ge 2\\,\\operatorname{Ric}(u,u)\\) for every unit tangent vector \\(u\\in T_pM^4\\) at every point \\(p\\in M^4\\)."}, {"label": "E", "text": "If \\(\\mathfrak{Ric}>0\\), then the first eigenvalue of the area Jacobi operator satisfies \\(\\Lambda_1 > 2\\,\\mathfrak{Ric}\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "sharp factor 2 from second-variation identity", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "ambient-curvature-dependent lower bound dropped", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "global infimum over Ricci replaced by pointwise uniform lower bound", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "non-strict lower bound strengthened to strict inequality", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not explicitly reveal the correct estimate or its sharp constant; it only sets up notation and asks which bound holds."}, "TAS": {"score": 0, "justification": "This is essentially a direct recall of a specific theorem/result: the task is to identify the stated estimate itself rather than infer a new conclusion from premises."}, "GPS": {"score": 1, "justification": "There is some reasoning pressure because the options vary by sharp constant, quantifier strength, and weaker-vs-stronger formulations, but success still depends mainly on knowing the exact theorem statement."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they test confusion about the coefficient, a merely weaker true statement, an overly strong pointwise quantifier, and an unjustified stronger bound."}, "total_score": 5, "overall_assessment": "A mathematically well-constructed recall-style MCQ with strong distractors, but it is largely theorem restatement rather than a genuinely generative reasoning question."}}
{"id": "2602.22912v1", "paper_link": "http://arxiv.org/abs/2602.22912v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm:main:intro:ReductionToOneEnd:doubleRays}\n    The consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.", "start_pos": 7951, "end_pos": 8165, "label": "thm:main:intro:ReductionToOneEnd:doubleRays"}, "ref_dict": {"prob:ubiquity:antiraysRays": "\\begin{problem}\\label{prob:ubiquity:antiraysRays}\n    Is the disjoint union of a consistently forward and a consistently backward oriented ray ubiquitous?\n\\end{problem}", "thm:ubiquity:antiraysRays": "\\begin{thm}\\label{thm:ubiquity:antiraysRays}\n    Let $D$ be a digraph.\n    If there is a sequence $(\\cX_n)_{n\\in\\N}$ such that, for every $n\\in\\N$, the set $\\cX_n$ consists of $n$ rays and $n$ anti-rays of~$D$ that are all pairwise disjoint and such that the sequence $(\\cX_n)_{n\\in\\N}$ does not concentrate in any end of~$D$, then $D$ contains a set of infinitely many rays and infinitely many anti-rays every two of which are disjoint.\n\\end{thm}", "thm:main:intro:ReductionToOneEnd:doubleRays": "\\begin{theorem}\\label{thm:main:intro:ReductionToOneEnd:doubleRays}\n    The consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.\n\\end{theorem}", "prob:ubiquity:doubleRays": "\\begin{problem}\\cite{GKR2024}*{Problem 1.3}\\label{prob:ubiquity:doubleRays}\n    Is the consistently oriented double ray ubiquitous?\n\\end{problem}", "thm:main:intro:ReductionToOneEnd:RaysAntiRays": "\\begin{theorem}\\label{thm:main:intro:ReductionToOneEnd:RaysAntiRays}\n    The disjoint union of a consistently forward and a consistently backward oriented ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.\n\\end{theorem}", "thm:ubiquity:doubleRays": "\\begin{thm}\\label{thm:ubiquity:doubleRays}\n    Let $D$ be a digraph.\n    If there is a valid sequence $(\\cR_n)_{n\\in\\N}$ that does not concentrate in finitely many ends, then $D$ contains infinitely many pairwise disjoint double rays.\n\\end{thm}"}, "pre_theorem_intro_text_len": 2640, "pre_theorem_intro_text": "\\label{sec:intro}\n\nThe property of a graph $H$ to be \\emph{ubiquitous} describes the concept that the existence of $n$ pairwise disjoint copies for all $n\\in\\ensuremath{\\mathbb N}$ of a graph $H$ in a graph $G$ implies that $G$ already contains $\\aleph_0$ many pairwise disjoint copies of~$H$.\nOriginally, this was considered with respect to the subgraph relation and Halin showed that the ray is ubiquitous~\\cite{Halin65}*{Satz 1} and that the double ray is ubiquitous~\\cite{Halin70}*{Satz 3}.\nLater, counterexamples \\cites{A77, A02} regarding being ubiquitous were found, which even include trees.\nHence, ubiquity was also considered with respect to other relations such as the minor and topological minor relation, and we refer to \\cites{A79,A80,BEEGHPTI,BEEGHPTII,BEEGHPTIII} for ubiquity results regarding these notions.\nMany of these results leverage that finite graphs (resp.~trees) are well-quasi ordered under the minor (resp.~topological minor) relation.\n\nIt should be noted that Andreae~\\cite{A02} posed the major and still open conjecture whether all locally finite connected graphs are ubiquitous under the minor relation. In the same paper he also provided an example of an uncountable non-ubiquitous graph under the minor relation, whose construction exploits that uncountable graphs are not well-quasi ordered under minors.\n\nFurther modified questions regarding ubiquity have been raised, e.g.~where disjointness is replaced by edge-disjointness.\nIn this setting it turned out to be complicated to even verify that double rays are ubiquitous regarding the subgraph relation~\\cite{BCP15}.\n\nFor digraphs, much less is known, even for the subdigraph relation, and for the rest of this article, we shall only consider the property of being ubiquitous under that relation.\nWhile Zuther~\\cite{Z1997}*{Theorem 2.17} proved that the consistently oriented ray is ubiquitous, Gut et al.~\\cite{GKR2024}*{Theorem 1.1} proved the same for all possible orientations of the undirected ray.\nLater, Gut et al.~\\cite{GKR2025}*{Theorem 1.3} classified almost all orientations of the undirected double ray that are ubiquitous.\nThe only orientation they left out was the consistently oriented one.\nDue to that, they posed the following problem.\n\n\\begin{problem}\\cite{GKR2024}*{Problem 1.3}\\label{prob:ubiquity:doubleRays}\n    Is the consistently oriented double ray ubiquitous?\n\\end{problem}\n\nOur main result (Theorem~\\ref{thm:main:intro:ReductionToOneEnd:doubleRays}) states that it suffices to verify Problem~\\ref{prob:ubiquity:doubleRays} for one-ended digraphs.\n(We refer to Section~\\ref{sec:prelims} for the definition of ends of digraphs.)", "context": "\\label{sec:intro}\n\nThe property of a graph $H$ to be \\emph{ubiquitous} describes the concept that the existence of $n$ pairwise disjoint copies for all $n\\in\\ensuremath{\\mathbb N}$ of a graph $H$ in a graph $G$ implies that $G$ already contains $\\aleph_0$ many pairwise disjoint copies of~$H$.\nOriginally, this was considered with respect to the subgraph relation and Halin showed that the ray is ubiquitous~\\cite{Halin65}*{Satz 1} and that the double ray is ubiquitous~\\cite{Halin70}*{Satz 3}.\nLater, counterexamples \\cites{A77, A02} regarding being ubiquitous were found, which even include trees.\nHence, ubiquity was also considered with respect to other relations such as the minor and topological minor relation, and we refer to \\cites{A79,A80,BEEGHPTI,BEEGHPTII,BEEGHPTIII} for ubiquity results regarding these notions.\nMany of these results leverage that finite graphs (resp.~trees) are well-quasi ordered under the minor (resp.~topological minor) relation.\n\nIt should be noted that Andreae~\\cite{A02} posed the major and still open conjecture whether all locally finite connected graphs are ubiquitous under the minor relation. In the same paper he also provided an example of an uncountable non-ubiquitous graph under the minor relation, whose construction exploits that uncountable graphs are not well-quasi ordered under minors.\n\nFor digraphs, much less is known, even for the subdigraph relation, and for the rest of this article, we shall only consider the property of being ubiquitous under that relation.\nWhile Zuther~\\cite{Z1997}*{Theorem 2.17} proved that the consistently oriented ray is ubiquitous, Gut et al.~\\cite{GKR2024}*{Theorem 1.1} proved the same for all possible orientations of the undirected ray.\nLater, Gut et al.~\\cite{GKR2025}*{Theorem 1.3} classified almost all orientations of the undirected double ray that are ubiquitous.\nThe only orientation they left out was the consistently oriented one.\nDue to that, they posed the following problem.\n\n\\begin{problem}\\cite{GKR2024}*{Problem 1.3}\\label{prob:ubiquity:doubleRays}\n Is the consistently oriented double ray ubiquitous?\n\\end{problem}\n\nOur main result (Theorem~\\ref{thm:main:intro:ReductionToOneEnd:doubleRays}) states that it suffices to verify Problem~\\ref{prob:ubiquity:doubleRays} for one-ended digraphs.\n(We refer to Section~\\ref{sec:prelims} for the definition of ends of digraphs.)\n\n\\begin{problem}\\cite{GKR2024}*{Problem 1.3}\\label{prob:ubiquity:doubleRays}\n    Is the consistently oriented double ray ubiquitous?\n\\end{problem}\n\n\\begin{theorem}\\label{thm:main:intro:ReductionToOneEnd:doubleRays}\n    The consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.\n\\end{theorem}", "full_context": "\\label{sec:intro}\n\nThe property of a graph $H$ to be \\emph{ubiquitous} describes the concept that the existence of $n$ pairwise disjoint copies for all $n\\in\\ensuremath{\\mathbb N}$ of a graph $H$ in a graph $G$ implies that $G$ already contains $\\aleph_0$ many pairwise disjoint copies of~$H$.\nOriginally, this was considered with respect to the subgraph relation and Halin showed that the ray is ubiquitous~\\cite{Halin65}*{Satz 1} and that the double ray is ubiquitous~\\cite{Halin70}*{Satz 3}.\nLater, counterexamples \\cites{A77, A02} regarding being ubiquitous were found, which even include trees.\nHence, ubiquity was also considered with respect to other relations such as the minor and topological minor relation, and we refer to \\cites{A79,A80,BEEGHPTI,BEEGHPTII,BEEGHPTIII} for ubiquity results regarding these notions.\nMany of these results leverage that finite graphs (resp.~trees) are well-quasi ordered under the minor (resp.~topological minor) relation.\n\nIt should be noted that Andreae~\\cite{A02} posed the major and still open conjecture whether all locally finite connected graphs are ubiquitous under the minor relation. In the same paper he also provided an example of an uncountable non-ubiquitous graph under the minor relation, whose construction exploits that uncountable graphs are not well-quasi ordered under minors.\n\nFor digraphs, much less is known, even for the subdigraph relation, and for the rest of this article, we shall only consider the property of being ubiquitous under that relation.\nWhile Zuther~\\cite{Z1997}*{Theorem 2.17} proved that the consistently oriented ray is ubiquitous, Gut et al.~\\cite{GKR2024}*{Theorem 1.1} proved the same for all possible orientations of the undirected ray.\nLater, Gut et al.~\\cite{GKR2025}*{Theorem 1.3} classified almost all orientations of the undirected double ray that are ubiquitous.\nThe only orientation they left out was the consistently oriented one.\nDue to that, they posed the following problem.\n\n\\begin{problem}\\cite{GKR2024}*{Problem 1.3}\\label{prob:ubiquity:doubleRays}\n Is the consistently oriented double ray ubiquitous?\n\\end{problem}\n\nOur main result (Theorem~\\ref{thm:main:intro:ReductionToOneEnd:doubleRays}) states that it suffices to verify Problem~\\ref{prob:ubiquity:doubleRays} for one-ended digraphs.\n(We refer to Section~\\ref{sec:prelims} for the definition of ends of digraphs.)\n\n\\begin{problem}\\cite{GKR2024}*{Problem 1.3}\\label{prob:ubiquity:doubleRays}\n    Is the consistently oriented double ray ubiquitous?\n\\end{problem}\n\n\\begin{theorem}\\label{thm:main:intro:ReductionToOneEnd:doubleRays}\n    The consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.\n\\end{theorem}\n\n\\begin{abstract}\nWe prove that the consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.\nAdditionally, we prove the same equivalence for the disjoint union of a consistently forward and a consistently backward oriented ray. \nFurthermore, we discuss the connection between these two ubiquity problems.\n\\end{abstract}\n\nFor digraphs, much less is known, even for the subdigraph relation, and for the rest of this article, we shall only consider the property of being ubiquitous under that relation.\nWhile Zuther~\\cite{Z1997}*{Theorem 2.17} proved that the consistently oriented ray is ubiquitous, Gut et al.~\\cite{GKR2024}*{Theorem 1.1} proved the same for all possible orientations of the undirected ray.\nLater, Gut et al.~\\cite{GKR2025}*{Theorem 1.3} classified almost all orientations of the undirected double ray that are ubiquitous.\nThe only orientation they left out was the consistently oriented one.\nDue to that, they posed the following problem.\n\n\\begin{problem}\\cite{GKR2024}*{Problem 1.3}\\label{prob:ubiquity:doubleRays}\n Is the consistently oriented double ray ubiquitous?\n\\end{problem}\n\nOur main result (Theorem~\\ref{thm:main:intro:ReductionToOneEnd:doubleRays}) states that it suffices to verify Problem~\\ref{prob:ubiquity:doubleRays} for one-ended digraphs.\n(We refer to Section~\\ref{sec:prelims} for the definition of ends of digraphs.)\n\nAs a major step in the proof of Theorem~\\ref{thm:main:intro:ReductionToOneEnd:doubleRays}, we verify Problem~\\ref{prob:ubiquity:doubleRays} for a large, but technical, class of digraphs, see Theorem~\\ref{thm:ubiquity:doubleRays}.\n\n\\begin{problem}\\label{prob:ubiquity:antiraysRays}\n Is the disjoint union of a consistently forward and a consistently backward oriented ray ubiquitous?\n\\end{problem}\n\n\\begin{theorem}\\label{thm:main:intro:ReductionToOneEnd:RaysAntiRays}\n The disjoint union of a consistently forward and a consistently backward oriented ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:main:intro:ReductionToOneEnd:doubleRays}.]\n Let us assume that Problem~\\ref{prob:ubiquity:doubleRays} holds for the class of one-ended digraphs and let $D$ be a digraphs and $(\\cR_n)_{n\\in\\N}$ be a valid sequence in~~$D$.\n By Theorem~\\ref{thm:ubiquity:doubleRays}, we may assume that $(\\cR_n)_{n\\in\\N}$ concentrates in finitely many ends.\n Then there is a valid sequence $(\\cQ_n)_{n\\in\\N}$ induced by $(\\cR_n)_{n\\in\\N}$ such that all tails from all elements of $\\bigcup_{n\\in\\N}\\cQ_n$ are equivalent.\n By Lemma~\\ref{lem:reduction:doubleRays:oneEnd}, there exists a one-ended subdigraph $D'$ of~$D$ and a valid sequence $(\\cP_n)_{n\\in\\N}$ in~$D'$ induced by $(\\cQ_n)_{n\\in\\N}$.\n Thus, there are infinitely many pairwise disjoint double rays in~$D'$ and hence in~$D$.\n\\end{proof}\n\n\\begin{problem}\\cite{GKR2024}*{Problem 1.3}\\label{prob:ubiquity:doubleRays}\n    Is the consistently oriented double ray ubiquitous?\n\\end{problem}\n\n\\begin{theorem}\\label{thm:main:intro:ReductionToOneEnd:doubleRays}\n    The consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.\n\\end{theorem}\n\n\\begin{thm}\\label{thm:ubiquity:doubleRays}\n    Let $D$ be a digraph.\n    If there is a valid sequence $(\\cR_n)_{n\\in\\N}$ that does not concentrate in finitely many ends, then $D$ contains infinitely many pairwise disjoint double rays.\n\\end{thm}", "post_theorem_intro_text_len": 1487, "post_theorem_intro_text": "As a major step in the proof of Theorem~\\ref{thm:main:intro:ReductionToOneEnd:doubleRays}, we verify Problem~\\ref{prob:ubiquity:doubleRays} for a large, but technical, class of digraphs, see Theorem~\\ref{thm:ubiquity:doubleRays}.\n\nAdditionally, we consider the following more general version of \\cite{HH2024+}*{Problem 4.2}.\n\n\\begin{problem}\\label{prob:ubiquity:antiraysRays}\n    Is the disjoint union of a consistently forward and a consistently backward oriented ray ubiquitous?\n\\end{problem}\n\nWe also verify Problem~\\ref{prob:ubiquity:antiraysRays} for a large, but technical, class of digraphs (Theorem~\\ref{thm:ubiquity:antiraysRays}) and show that Problem~\\ref{prob:ubiquity:antiraysRays} holds if it holds for all one-ended digraphs.\n\n\\begin{theorem}\\label{thm:main:intro:ReductionToOneEnd:RaysAntiRays}\n    The disjoint union of a consistently forward and a consistently backward oriented ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.\n\\end{theorem}\n\n\\medskip\n\nThis paper is structured as follows. \nAfter introducing some terminology in Section~\\ref{sec:prelims}, we prove Theorem~\\ref{thm:main:intro:ReductionToOneEnd:RaysAntiRays} in Section~\\ref{sec:ubiquity:antiraysRays}.\nIn Section~\\ref{sec:ubiquity:DoubleRays}, we prove Theorem~\\ref{thm:main:intro:ReductionToOneEnd:doubleRays}.\nIn Section~\\ref{sec:equivalence}, we discuss connections between Problems~\\ref{prob:ubiquity:doubleRays} and \\ref{prob:ubiquity:antiraysRays}.", "sketch": "As a major step in the proof of Theorem~\\ref{thm:main:intro:ReductionToOneEnd:doubleRays}, the authors \"verify Problem~\\ref{prob:ubiquity:doubleRays} for a large, but technical, class of digraphs\" (see Theorem~\\ref{thm:ubiquity:doubleRays}).", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:main:intro:ReductionToOneEnd:doubleRays}\n    The consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.", "theorem_type": ["Biconditional or Equivalence"], "mcq": {"question": "A digraph $H$ is called ubiquitous under the subdigraph relation if, whenever a digraph $G$ contains $n$ pairwise disjoint copies of $H$ for every $n\\in\\mathbb{N}$, the digraph $G$ already contains countably infinitely many pairwise disjoint copies of $H$. Let $C$ be the consistently oriented double ray, that is, the two-way infinite directed path $\\cdots \\to v_{-1} \\to v_0 \\to v_1 \\to \\cdots$. Which of the following statements is equivalent to saying that $C$ is ubiquitous under the subdigraph relation?", "correct_choice": {"label": "A", "text": "The consistently oriented double ray $C$ is ubiquitous when restricted to one-ended digraphs; that is, for every one-ended digraph $G$ (a digraph with exactly one end), if $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\\in\\mathbb{N}$, then $G$ contains countably infinitely many pairwise disjoint copies of $C$."}, "choices": [{"label": "B", "text": "The consistently oriented double ray $C$ is ubiquitous when restricted to digraphs with at most one end; that is, for every digraph $G$ with at most one end, if $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\\in\\mathbb{N}$, then $G$ contains countably infinitely many pairwise disjoint copies of $C$."}, {"label": "C", "text": "For every one-ended digraph $G$, if $G$ contains countably infinitely many pairwise disjoint copies of $C$, then in particular $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\\in\\mathbb{N}$."}, {"label": "D", "text": "There exists a one-ended digraph $G$ such that, whenever $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\\in\\mathbb{N}$, every digraph containing $G$ as a subdigraph contains countably infinitely many pairwise disjoint copies of $C$."}, {"label": "E", "text": "The consistently oriented double ray $C$ is ubiquitous when restricted to finitely-ended digraphs; that is, for every digraph $G$ with finitely many ends, if $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\\in\\mathbb{N}$, then $G$ contains countably infinitely many pairwise disjoint copies of $C$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "one-ended_exactness_replaced_by_at_most_one_end", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "ubiquity_implication_replaced_by_trivial_converse_on_one-ended_digraphs", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "for_every_one-ended_digraph_replaced_by_there_exists_one-ended_digraph", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "restriction_class_changed_from_one-ended_to_finitely-ended", "template_used": "wildcard"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines ubiquity and the digraph C but does not explicitly or implicitly point to the correct restriction class ('one-ended digraphs'). The correct answer is not leaked by wording in the prompt."}, "TAS": {"score": 2, "justification": "The item is not a direct restatement of the definition of ubiquity. It asks for an equivalent reformulation involving a nontrivial restriction to a specific class of digraphs, so the student must distinguish among genuinely different statements."}, "GPS": {"score": 1, "justification": "The question requires some reasoning or theorem recognition to separate exact equivalence from nearby variants (at most one end, finitely many ends, converse implication, existential quantifier). However, it mainly tests precise recall/recognition rather than deep generative derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically meaningful: they vary boundary conditions, weaken/alter quantifiers, or reverse implication. These reflect common failure modes in reading equivalence statements and theorem hypotheses."}, "total_score": 7, "overall_assessment": "A strong MCQ: it avoids answer leakage, is not tautological, and uses high-quality distractors. Its main limitation is that it leans more toward theorem recognition than substantial generative reasoning."}}
{"id": "2602.23137v1", "paper_link": "http://arxiv.org/abs/2602.23137v1", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "\\label{ergodic-th}\nIf Assumption B holds, then $\\{u(t,x)\\}_{x\\in \\mathbb{R}}$ is ergodic, for any $t>0$. Consequently, by the mean ergodic theorem,\n\\[\n\\frac{1}{R}F_R(t) \\to 0 \\quad \\mbox{a.s and in $L^2(\\Omega)$}, \\quad \\mbox{as $R \\to \\infty$}.\n\\]", "start_pos": 15922, "end_pos": 16193, "label": "ergodic-th"}, "ref_dict": {"FCLT": "\\begin{theorem}[Functional Central Limit Theorem]\n\\label{FCLT}\nUnder the hypotheses of Theorem \\ref{QCLT}, for any $R>0$,\nthe process $\\{F_R(t)\\}_{t\\geq 0}$ has a $\\gamma$-H\\\"older continuous modification (denoted also $F_R$), for any $\\gamma \\in (0,\\frac{\\beta}{2})$, where $\\beta$ is given by \\eqref{def-beta}.\nMoreover,\n\\[\n\\frac{1}{R^{\\beta/2}}F_R(\\cdot) \\stackrel{d}{\\to} \\cG(\\cdot) \\quad \\mbox{in $C[0,\\infty)$ as $R \\to \\infty$},\n\\]\nwhere $\\{\\cG(t)\\}_{t\\geq 0}$ is a zero-mean Gaussian process with covariance \n$\\bE[\\cG(t) \\cG(s)]=K(t,s)$, and $K(t,s)$ given by Theorem \\ref{cov-th}. Here $\\stackrel{d}{\\to}$ denotes the convergence in distribution, and $C[0,\\infty)$ is equipped with the uniform convergence on compact sets.\n\\end{theorem}", "ergodic-th": "\\begin{theorem}\n\\label{ergodic-th}\nIf Assumption B holds, then $\\{u(t,x)\\}_{x\\in \\bR}$ is ergodic, for any $t>0$. Consequently, by the mean ergodic theorem,\n\\[\n\\frac{1}{R}F_R(t) \\to 0 \\quad \\mbox{a.s and in $L^2(\\Omega)$}, \\quad \\mbox{as $R \\to \\infty$}.\n\\]\n\\end{theorem}", "key-D": "\\begin{equation}\n\\label{key-D}\n\\|D_{r,y,z}u(t,x)\\|_p \\leq C'_{T,p,\\nu,\\k}|z| \\int_{\\bR}G_{t-r}(x-y')\\k(y-y')dy',\n\\end{equation}", "def-Mp": "\\begin{equation}\n\\label{def-Mp}\n\\cM_p(t):=\\int_0^t \\int_{\\bR} \\big|(G_{t-s}*\\k)(x)\\big|^p dx ds<\\infty \\quad \\mbox{for all} \\quad t>0,\n\\end{equation}", "Gauss-color": "\\begin{equation}\n\\label{Gauss-color}\nW_t(\\varphi)=B_t(\\varphi*\\k) \\quad \\mbox{for all $t>0$ and $\\varphi \\in \\cS(\\bR^d)$},\n\\end{equation}", "def-cov-W": "\\begin{equation}\n\\label{def-cov-W}\n\\bE[W_t(\\varphi) W_s(\\psi)]=(t \\wedge s) \\int_{\\bR^d} \\int_{\\bR^d}\\varphi(x)\\psi(y) f(x-y)dxdy,\n\\end{equation}", "appB": "\\begin{proof}\nBy direct calculation, we see that $\\tilde{f} \\star_{k}^{1} \\tilde{g} =0$ implies that $\\tilde{f} \\star_{k}^{\\ell} \\tilde{g} =0$ for all $\\ell=2,\\ldots,k$. Hence, all the terms in the product formula are zero, except those corresponding to $\\ell=0$ and $\\ell=1$:\n\\begin{align*}\nI_n(f) I_m(g) &=I_n(\\tilde{f}) I_m(\\tilde{g})=I_{n+m}(\\tilde{f} \\otimes \\tilde{g})+\n\\sum_{k=1}^{n\\wedge m}k! \\binom{n}{k}\\binom{m}{k}I_{n+m-k}(\\tilde{f} \\star_{k}^{0} \\tilde{g} ).\n\\end{align*}\nNote that $I_{n+m}(\\tilde{f} \\otimes \\tilde{g})=I_{n+m}(f \\otimes g)$ since $\\tilde{f} \\otimes \\tilde{g}$ and $f \\otimes g$ have the same symmetrization.\n\\end{proof}\n\n\\section{Inequalities for Riesz potentials}\n\\label{appB}\n\nIn this section, we include some inequalities for Riesz potentials which were used for the proof of the QCLT in the case when $\\k$ is the Riesz kernel.\n\nWe recall that the Riesz kernel of order $\\alpha \\in (0,d)$ is defined by \n\\[\nR_{d,\\alpha}(x)=C_{d,\\alpha}|x|^{-(d-\\alpha)} \\quad \\mbox{with} \\quad C_{d,\\alpha}=\\pi^{-\\frac{d}{2}}2^{-\\alpha} \\frac{\\Gamma(\\frac{d-\\alpha}{2})}{\\Gamma(\\frac{\\alpha}{2})}.\n\\]\nIts Fourier transform (in the sense of distributions) is $\\cF R_{d,\\alpha}(\\xi)=|\\xi|^{-\\alpha}$, since\n\\[\n\\int_{\\bR^d}\\varphi(x)|x|^{-\\alpha}dx=C_{d,\\alpha} \\int_{\\bR^d}\\cF \\varphi(\\xi)|\\xi|^{-(d-\\alpha)}d\\xi \\quad \n\\mbox{for any $\\varphi \\in \\cS(\\bR^d)$},\n\\]\nwhere $\\cS(\\bR^d)$ is the set of rapidly decreasing functions on $\\bR^d$.\n\n\\medskip\n\nFor the next result, we refer to Theorem 1, p.119 of \\cite{stein70}.\n\n\\begin{theorem}[Hardy-Littlewood-Sobolov inequality]\n\\label{HLS}\nFor any $\\alpha \\in (0,d)$ and $p>\\frac{d}{d-\\alpha}$, \n\\[\n\\| \\varphi *R_{d,\\alpha}\\|_{L^p(\\bR^d)} \\leq A_{d.\\alpha,p}\\|\\varphi\\|_{L^{q}(\\bR^d)} \\quad \\mbox{for any $\\varphi \\in L^q(\\bR^d)$},\n\\]\nwhere $A_{d,\\alpha,p}>0$ is a constant depending on $(d,\\alpha,p)$, and $q$ is defined by:\n\\[\n\\frac{1}{q}=\\frac{1}{p}+\\frac{\\alpha}{d}.\n\\]\n(Condition $p>\\frac{d}{d-\\alpha}$ is equivalent to $q>1$.)\n\\end{theorem}\n\nAs a consequence of Theorem \\ref{HLS}, we obtain the following lemma.\n\n\\begin{lemma}\n\\label{Hol-HLS}\nFor any $\\alpha \\in (0,d)$, $q \\in (1,\\frac{d}{\\alpha})$, $\\varphi \\in L^q(\\bR^d)$ and $\\psi \\in L^{q'}(\\bR^d)$,\n\\[\n\\left|\\int_{\\bR^d}\\int_{\\bR^d}\\varphi(x)\\psi(y) R_{d,\\alpha}(x-y) dxdy \\right| \\leq A_{d,\\alpha,p} \\|\\varphi\\|_{L^q(\\bR^d)} \\|\\psi\\|_{L^{q'}(\\bR^d)},\n\\]\nwhere $A_{d,\\alpha,p}$ is the constant  from Theorem \\ref{HLS}, and $(p,q')$ are defined by\n\\[\n\\frac{1}{p}+\\frac{1}{q}=1 \\quad \\mbox{and} \\quad \\frac{1}{q}+\\frac{1}{q'}=1+\\frac{\\alpha}{d}.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nWe apply H\\\"older's inequality, letting $\\frac{1}{p}+\\frac{1}{q}=1$, and obtain that\n\\begin{align*}\n\\left|\\int_{\\bR^d}\\int_{\\bR^d}\\varphi(x)\\psi(y) R_{d,\\alpha}(x-y) dxdy \\right|\n& =\\left| \\int_{\\bR^d}\\varphi(x) \\big(\\psi * R_{d,\\alpha} \\big) dx\\right| \\leq \\|\\varphi\\|_{L^q(\\bR^d)} \\|\\psi * R_{d,\\alpha}\\|_{L^p(\\bR^d)}.\n\\end{align*}\nNote that $q<\\frac{d}{\\alpha}$ is equivalent to $p>\\frac{d}{d-\\alpha}$.\nTherefore, by Theorem \\ref{HLS} \n\\[\n \\|\\psi * R_{d,\\alpha}\\|_{L^p(\\bR^d)} \\leq A_{d,\\alpha,p}  \\|\\psi \\|_{L^{q'}(\\bR^d)},\n\\]\nwhere $\\frac{1}{q'}=\\frac{1}{p}+\\frac{\\alpha}{d}$. Finally, we note that $\n1-\\frac{1}{q}=\\frac{1}{p}=\\frac{1}{q'}-\\frac{\\alpha}{d}$, and hence, $\\frac{1}{q}+\\frac{1}{q'}=1+\\frac{\\alpha}{d}$.\n\n\\end{proof}", "mom-u": "\\begin{equation}\n\\label{mom-u}\nK_p(T):=\\sup_{(t,x)\\in [0,T] \\times \\bR}\\|u(t,x)\\|_p<\\infty \\quad \\mbox{for all} \\quad T>0,\n\\end{equation}", "cov-th": "\\begin{theorem}[Limiting covariance]\n\\label{cov-th}\nIf Assumption B holds, then\nfor any $t,s>0$,\n\\[\n\\lim_{R \\to \\infty}\\frac{1}{R^{\\beta}}\\bE[F_R(t) F_R(s)]=K(t,s) \\quad \\mbox{is finite},\n\\] \nwhere \n\\begin{equation}\n\\label{def-beta}\n\\beta:=\\left\\{\n\\begin{array}{ll} 1 & \\mbox{if $\\k \\in L^1(\\bR)$,} \\\\\n\\alpha+1 & \\mbox{if $\\k=R_{1,\\alpha/2}$ for some $\\alpha \\in (0,1)$. }\n\\end{array} \\right.\n\\end{equation}\nIn particular, $R^{-\\beta} \\sigma_R^2(t) \\to K(t,t)$ as $R \\to \\infty$, for any $t>0$.\n\\end{theorem}", "QCLT": "\\begin{theorem}[Quantitative Central Limit Theorem]\n\\label{QCLT}\nSuppose that Assumption B holds, and there exists $p \\in (1,2]$ such that \n\\begin{equation}\n\\label{mp-m2p}\nm_{p}<\\infty \\quad \\mbox{and} \\quad m_{2p}<\\infty.\n\\end{equation}\n\nIf $\\k \\in L^1(\\bR)$, then\nfor any $t>0$,\n\\[\n{\\rm dist}\\left( \\frac{F_R(t)}{\\sigma_R(t)},Z\\right) \\leq  C_t R^{-(1-\\frac{1}{p})}\n\\]\n\nIf $\\k=R_{1,\\alpha/2}$ for some $\\alpha \\in (0,1)$, and $p>\\frac{2}{2-\\alpha}$, then\n\\[\n{\\rm dist}\\left( \\frac{F_R(t)}{\\sigma_R(t)},Z\\right) \\leq\nC_t R^{-\\e}  \\quad \\mbox{for any $\\varepsilon \\in \\big(0, 1-\\frac{1}{p}-\\frac{\\alpha}{2}\\big)$}.\n\\]\n\nHere $C_t>0$ is a constant depending on $t$, and ${\\rm dist}$ is the Fortet-Mourier distance, the 1-Wasserstein distance, or the Kolmogorov distance.\n\\end{theorem}", "mp-m2p": "\\begin{equation}\n\\label{mp-m2p}\nm_{p}<\\infty \\quad \\mbox{and} \\quad m_{2p}<\\infty.\n\\end{equation}", "def-G": "\\begin{equation}\n\\label{def-G}\nG_t(x)=\\frac{1}{2}1_{\\{|x|<t\\}} \\quad \\mbox{for all $t>0$ and $x \\in \\bR$.}\n\\end{equation}", "appA": "\\label{appA}\n\nIn this appendix section, we include some auxiliary results from Malliavin calculus which were used in the sequel. \n\\medskip\n\nLet $(Z,\\cZ,\\fm)$ be a $\\sigma$-finite measure space, and $N", "key-D2": "\\begin{equation}\n\\label{key-D2}\n\\|D_{(r_1,y_1,z_1),(r_2,y_2,z_2)}^2 u(t,x) \\|_p \\leq C_{T,p,\\nu,\\k}'' |z_1 z_2| \\int_{\\bR^2} \\widetilde{f}_2(r_1,y_1',r_2,y_2',t,x) \\k(y_1-y_1')\\k(y_2-y_2') dy_1'dy_2',\n\\end{equation}", "HAM": "\\begin{align}\n\\label{HAM}\n\t\\begin{cases}\n\t\t\\dfrac{\\partial^2 u}{\\partial^2 t} (t,x)\n\t\t=  \\dfrac{\\partial^2 u}{\\partial x^2} (t,x)+u(t,x) \\dot{X}(t,x), \\\n\t\tt>0, \\ x \\in \\bR, \\\\\n\t\tu(0,x) = 1, \\dfrac{\\partial u}{\\partial t} (0,x)=0, \\quad x \\in \\bR,\n\t\\end{cases}\n\\end{align}", "def-beta": "\\begin{equation}\n\\label{def-beta}\n\\beta:=\\left\\{\n\\begin{array}{ll} 1 & \\mbox{if $\\k \\in L^1(\\bR)$,} \\\\\n\\alpha+1 & \\mbox{if $\\k=R_{1,\\alpha/2}$ for some $\\alpha \\in (0,1)$. }\n\\end{array} \\right.\n\\end{equation}", "lem-key-Z": "\\begin{lemma}\n\\label{lem-key-Z}\nSuppose that Assumption B holds.\nLet $\\{Z(t,x);t\\geq 0,x\\in \\bR\\}$ be an adapted random field such that\n$\\{Z(t,x)\\}_{x\\in \\bR^d}$ is strictly stationary and $Z(t,x) \\in {\\rm dom}(D)$ for any $t\\geq 0$ and $x \\in \\bR$. Assume that for any $0<r<t$, $x,y \\in \\bR$ and $z \\in \\bR_0$,\n\\begin{equation}\n\\label{key-Z}\n\\|D_{r,y,z}Z(t,x)\\|_{2} \\leq C_t |z| \\int_{\\bR} G_{t-r}(x-y')\\k(y-y')dy',\n\\end{equation}\nwhere $C_t>0$ is a constant depending on $t$. Then $\\{Z(t,x)\\}_{x\\in \\bR}$ is ergodic.\n\\end{lemma}", "HLS": "\\begin{theorem}[Hardy-Littlewood-Sobolov inequality]\n\\label{HLS}\nFor any $\\alpha \\in (0,d)$ and $p>\\frac{d}{d-\\alpha}$, \n\\[\n\\| \\varphi *R_{d,\\alpha}\\|_{L^p(\\bR^d)} \\leq A_{d.\\alpha,p}\\|\\varphi\\|_{L^{q}(\\bR^d)} \\quad \\mbox{for any $\\varphi \\in L^q(\\bR^d)$},\n\\]\nwhere $A_{d,\\alpha,p}>0$ is a constant depending on $(d,\\alpha,p)$, and $q$ is defined by:\n\\[\n\\frac{1}{q}=\\frac{1}{p}+\\frac{\\alpha}{d}.\n\\]\n(Condition $p>\\frac{d}{d-\\alpha}$ is equivalent to $q>1$.)\n\\end{theorem}"}, "pre_theorem_intro_text_len": 13063, "pre_theorem_intro_text": "The theory of stochastic partial differential equations (SPDEs) has grown consistently in the last three decades, and now constitutes a central area of stochastic analysis. SPDEs model the evolution of random systems across multiple scales, ranging from microscopic particle dynamics to macroscopic continuum limits, and arise in diverse contexts, such as interface growth, disordered media, turbulence, and transport in random environments. They provide a rigorous mathematical framework for universality phenomena observed in physics. Motivated by these physical models and mathematical challenges, systematic theories for SPDEs began to emerge in the late 1980s and 1990s. Prominent developments include Walsh's \\cite{walsh86} probabilistic approach based on martingale measures and random fields, and the semigroup approach introduced by Da Prato and Zabczyk \\cite{DZ92}. Basic equations which have been studied are the stochastic heat equation and the stochastic wave equation, and their particular cases, the parabolic Anderson model (pAm), respectively the hyperbolic Anderson model (hAm). However, comprehensive solution theories for numerous other SPDEs\nremained beyond the grasp of mathematical analysis, until the early 21st century. These challenges\nstemmed from the singular nature of such equations, a singularity intricately linked to the irregularities inherent in the random data involved. A significant breakthrough arrived in 2014 when Hairer \\cite{hairer14}\nconstructed a well-posedness framework for the Kardar-Parisi-Zhang (KPZ) equation, using concepts from Lyons' rough paths theory \\cite{lyons98}. In dimension 1, the solution of the KPZ equation is directly related to the solution of (pAm\n) via the Cole-Hopf transformation.\n\n\\medskip\n\nTraditionally, SPDEs are perturbed by a space-time Gaussian white noise in dimension $d=1$, which is a zero-mean Gaussian process $\\{B_t(\\varphi);t> 0,\\varphi \\in L^2(\\mathbb{R})\\}$ with covariance:\n\\[\n\\mathbb{E}[B_t(\\varphi) B_s(\\psi)]=(t \\wedge s) \\langle \\varphi,\\psi \\rangle_{L^2(\\mathbb{R})}.\n\\]\nIn higher dimensions, Dalang introduced in \\cite{dalang99}, a spatially-homogeneous (or colored) noise, as a zero-mean Gaussian process $\\{W_t(\\varphi);t\\geq 0,\\varphi \\in \\mathcal{S}(\\mathbb{R})\\}$ with covariance:\n\\begin{equation}\n\\label{def-cov-W}\n\\mathbb{E}[W_t(\\varphi) W_s(\\psi)]=(t \\wedge s) \\int_{\\mathbb{R}^d} \\int_{\\mathbb{R}^d}\\varphi(x)\\psi(y) f(x-y)dxdy,\n\\end{equation}\nwhere $\\mathcal{S}(\\mathbb{R}^d)$ is the set of all rapidly decreasing functions on $\\mathbb{R}^d$, and $f:\\mathbb{R}^d \\to [0,\\infty]$ is the Fourier transform of a tempered measure $\\mu$. This noise can also be defined as:\n\\begin{equation}\n\\label{Gauss-color}\nW_t(\\varphi)=B_t(\\varphi*\\kappa) \\quad \\mbox{for all $t>0$ and $\\varphi \\in \\mathcal{S}(\\mathbb{R}^d)$},\n\\end{equation}\nwhere the kernel $\\kappa$ is chosen such that $k * \\widetilde{k}=f$, and $\\widetilde{k}(x):=k(-x)$ for all $x \\in \\mathbb{R}^d$.\n\n\\medskip\n\nOne of the problems which has received a lot of interest in the literature in the recent years is the study of the asymptotic behaviour as $R \\to \\infty$ of the spatial integral \n\\[\nF_R(t)=\\int_{|x|<R} \\big(u(t,x)-\\mathbb{E}[u(t,x)]\\big)dx\n\\]\nof the random-field solution $u$ of an SPDE. The major result is the {\\em Quantitative Central Limit Theorem} (QCLT), which shows that $F_R(t)/\\sqrt{{\\rm Var}\\big(F_R(t)\\big)}$ converges in distribution as $R \\to \\infty$ to $Z \\sim N(0,1)$, and gives an estimate for the rate of this convergence in the total variation distance. This was achieved for the first time in the seminal article \\cite{HNV20} for the stochastic heat equation driven by space-time Gaussian white noise, by combining tools from Malliavin calculus with Stein's method for normal approximations. Since then, this method was extended to various other models, each with its own challenges. We refer the reader to \\cite{BNZ,CKNP-delta,CKNP22,DNZ20,HNVZ,NSZ20,NXZ22,NZ20-1,NZ20-2,NZ22} for a sample of relevant references.\n\n\\medskip\n\nIn the recent article \\cite{BZ24}, the QCLT problem was considered for the first time for an SPDE, namely (hAm),\ndriven by a L\\'evy white noise, by using Poisson-Malliavin calculus tools (in this case, with respect to the underlying Poisson random measure of the noise), which have been developed for instance in \\cite{Last16,LPS16}.  On the other hand, the recent article \\cite{BJ25} has introduced a coloration procedure similar to \\eqref{Gauss-color} for the L\\'evy noise, and analyzed various SPDEs with this noise. Therefore, a natural question is whether the QCLT continues to hold for an SPDE driven by the L\\'evy colored noise of \\cite{BJ25}. We propose to address this question in the present article, by focusing on (hAm) in dimension $d=1$, as in \\cite{BZ24}.\n\n\\medskip\n\nMore precisely, we consider the (hAm): \n\\begin{align}\n\\label{HAM}\n\t\\begin{cases}\n\t\t\\dfrac{\\partial^2 u}{\\partial^2 t} (t,x)\n\t\t=  \\dfrac{\\partial^2 u}{\\partial x^2} (t,x)+u(t,x) \\dot{X}(t,x), \\\n\t\tt>0, \\ x \\in \\mathbb{R}, \\\\\n\t\tu(0,x) = 1, \\dfrac{\\partial u}{\\partial t} (0,x)=0, \\quad x \\in \\mathbb{R},\n\t\\end{cases}\n\\end{align}\nwhere $X=\\{X_t(\\varphi);t>0,\\varphi \\in \\mathcal{S}(\\mathbb{R})\\}$ is the L\\'evy colored noise, given by:\n\\[\nX_t(\\varphi)=L_t(\\varphi *\\kappa).\n\\]\nwith $L_t(\\varphi)=L(1_{[0,t]} \\varphi)$ for all $\\varphi \\in L^2(\\mathbb{R})$, and $L=\\{L(\\varphi);\\varphi \\in L^2(\\bR_{+} \\times \\mathbb{R})\\}$ being the {\\em L\\'evy white noise}:\n\\[\nL(\\varphi)=\\int_{\\bR_{+} \\times \\mathbb{R} \\times \\bR_0} \\varphi(t,x)z \\widehat{N}(dt,dx,dz) \\quad \\mbox{for all} \\quad \\varphi \\in L^2(\\bR_{+} \\times \\mathbb{R}).\n\\]\n\nHere $N$ is a Poisson random measure on $Z=\\bR_{+}\\times \\mathbb{R} \\times \\bR_0$ with intensity \n\\[\n\\mathfrak{m}(dt,dx,dz)=dt dx \\nu(dz),\n\\] \ndefined on a complete probability space $(\\Omega,\\mathcal{F},\\mathbb{P})$, and  $\\widehat{N}(A)=N(A)-\\mathfrak{m}(A)$ is the compensated version of $N$.\nThe space $\\bR_0=\\mathbb{R} \\verb2\\2 \\{0\\}$ is equipped with the distance $d(x,y)=|x^{-1}-y^{-1}|$, so that the bounded subsets of $\\bR_0$ are those that are bounded away from 0.\nWe assume that the measure $\\nu$ satisfies the following condition:\n\\[\nm_2:=\\int_{\\bR_0}|z|^2 \\nu(dz)<\\infty.\n\\]\nUnder this condition, the following isometry property holds: for any $\\varphi,\\psi \\in L^2(\\bR_{+}\\times \\mathbb{R})$,\n\\[\n\\mathbb{E}[L(\\varphi)L(\\psi)]=m_2 \\int_{\\bR_+ \\times \\mathbb{R}} \\varphi(t,x)\\psi(t,x)dtdx.\n\\]\n\nWe denote by $\\mathcal{S}'(\\mathbb{R})$ the set of tempered distributions on $\\mathbb{R}$.\nAs in \\cite{BJ25}, we assume that the kernel $\\kappa$ satisfies the following assumption:\n\n\\medskip\n\n\\noindent {\\bf Assumption A.}\n$\\kappa:\\mathbb{R} \\to [0,\\infty]$ is a continuous, symmetric, and tempered function such that:\\\\\n(a) $\\mathcal{F}\\kappa=h$ in $\\mathcal{S}'(\\mathbb{R})$ is a tempered non-negative function, and $h^2$ is tempered;\\\\\n(b) $f=\\kappa* \\widetilde{\\kappa}$ is a continuous, symmetric, and tempered function ($\\mathcal{F} f=h^2$ in $\\mathcal{S}'(\\mathbb{R}))$;\\\\\n(c)  $\\kappa(x)<\\infty$ for all $x\\not=0$, and $f(x)<\\infty$ for all $x\\not=0$.\n\n\\medskip\n\nWe will impose Assumption A throughout this article.\nUnder this assumption, the noise $X$ is well-defined, since $\\varphi*\\kappa \\in \\L^2(\\mathbb{R})$ for any $\\varphi \\in \\mathcal{S}(\\mathbb{R})$. Moreover, for any $\\varphi,\\psi \\in \\mathcal{S}(\\mathbb{R})$,\n\\[\n\\int_{\\mathbb{R}} \\int_{\\mathbb{R}}\\varphi(x)\\psi(y) f(x-y)dxdy=\\int_{\\mathbb{R}} \\mathcal{F} \\varphi(\\xi) \\overline{\\mathcal{F} \\psi(\\xi)} \\mu(d\\xi),\n\\]\nwhere $\\mathcal{F} \\varphi(\\xi)=\\int_{\\mathbb{R}}e^{-i \\xi x} \\varphi(x) dx$ is the Fourier transform of $\\varphi$, and\n\\begin{equation}\n\\label{def-mu}\n\\mu(d\\xi)=\\frac{1}{2\\pi} |\\mathcal{F} \\kappa(\\xi)|^2 d\\xi.\n\\end{equation}\n\nSince $d=1$, $\\mu$ satisfies {\\em Dalang's condition}: (see Remark 10.(b) of \\cite{dalang99})\n\\[\n\\int_{\\mathbb{R}}\\frac{1}{1+|\\xi|^2}\\mu(d\\xi)<\\infty.\n\\]\n\n\\medskip\n\nThe stochastic integral with respect to $X$ is defined using Walsh's theory \\cite{walsh86}, \nsince $X$ induces a martingale measure. This integral shares many properties with the integral defined in \\cite{dalang99} for the Gaussian colored noise, except that its moments are estimated in a different way.\n\nLet $G$ be the fundamental solution of the wave equation in dimension $d=1$, given by:\n\\begin{equation}\n\\label{def-G}\nG_t(x)=\\frac{1}{2}1_{\\{|x|<t\\}} \\quad \\mbox{for all $t>0$ and $x \\in \\mathbb{R}$.}\n\\end{equation}\n\nInspired by D'Alembert formula, we introduce the following definition. \n\n\\begin{definition}\n{\\rm A predictable process $\\{u(t,x);t\\geq 0,x\\in \\mathbb{R}\\}$ is called a {\\em (mild) solution} of \\eqref{HAM}\nif it satisfies the integral equation:\n\\begin{equation}\n\\label{def-sol}\nu(t,x)=1+\\int_0^t \\int_{\\mathbb{R}}G_{t-s}(x-y)u(s,y)X(ds,dy).\n\\end{equation}\n}\n\\end{definition}\n\nWe recall that a random field $\\{\\Phi(t,x);t\\geq 0,x\\in \\mathbb{R}\\}$ is {\\em predictable} if it is measurable with respect to the predictable $\\sigma$-field on $\\Omega \\times \\bR_{+} \\times \\mathbb{R}$, which is the minimal $\\sigma$-field with respect to which all elementary processes are measurable. An {\\em elementary process} is a linear combination of processes of the form\n\\[\n\\Phi(t,x)=Y 1_{(a,b]}(t) 1_{A}(x)\n\\]\nwhere  $0\\leq a<b$, $A \\in \\cB_b(\\mathbb{R})$ and $Y$ is $\\cF_a$-measurable. Predictable processes \n$\\{\\Phi(t,x,z);t\\geq 0,x\\in \\mathbb{R},z\\in \\bR_0\\}$ are defined similarly. Here $(\\cF_t)_{t\\geq 0}$ is the filtration induced by $N$, i.e.\n\\[\n\\cF_t=\\sigma(\\{N([0,s] \\times A \\times B); s \\in [0,t],A\\in \\cB_b(\\mathbb{R}), B \\in \\cB_b(\\bR_0)\\}) \\vee \\mathcal{N},\n\\]\nwhere $\\mathcal{N}$ is the class of $\\mathbb{P}$-null sets, and $\\mathcal{B}(\\mathbb{R})$, $\\cB_b(\\bR_0)$ are the classes of bounded Borel subsets of $\\mathbb{R}$, respectively $\\bR_0$.\n\n\\medskip\n\nBy Theorem 5.2 of \\cite{BJ25}, equation \\eqref{HAM} has a unique solution $u$, which satisfies:\n\\[\n\\sup_{(t,x)\\in [0,T] \\times \\mathbb{R}}\\mathbb{E}|u(t,x)|^2<\\infty \\quad \\mbox{for all} \\quad T>0.\n\\]\n\nBy Lemma 5.5 of \\cite{BJ25}, the process $\\{u(t,x)\\}_{x\\in \\mathbb{R}}$ is strictly stationary, for any $t>0$.\nMoreover, by Theorem 5.6 of \\cite{BJ25}, if for some $p\\geq 2$, we have\n\\begin{equation}\n\\label{def-mp}\nm_p:=\\int_{\\bR_0}|z|^p \\nu(dz)<\\infty,\n\\end{equation}\nand\n\\begin{equation}\n\\label{def-Mp}\n\\cM_p(t):=\\int_0^t \\int_{\\mathbb{R}} \\big|(G_{t-s}*\\kappa)(x)\\big|^p dx ds<\\infty \\quad \\mbox{for all} \\quad t>0,\n\\end{equation}\nthen\n\\begin{equation}\n\\label{mom-u}\nK_p(T):=\\sup_{(t,x)\\in [0,T] \\times \\mathbb{R}}\\|u(t,x)\\|_p<\\infty \\quad \\mbox{for all} \\quad T>0,\n\\end{equation}\nwhere $\\|\\cdot\\|_p$ denotes the norm in $L^p(\\Omega)$. The goal of the present paper is to complement this analysis by showing that the {\\em spatial average} of the (centered) solution:\n\\[\nF_R(t)=\\int_{-R}^{R}\\big(u(t,x)-1 \\big)dx.\n\\]\nhas asymptotic Gaussian fluctuations when $R \\to \\infty$, with a precise rate of convergence which depends on the noise.  We denote $\\sigma_R^2(t)={\\rm Var}(F_R(t))$. \n\n\\medskip\n\n\\medskip\n\nRecall that the {\\em Riesz kernel of order $\\alpha \\in (0,1)$} is given by:\n\\[\nR_{1,\\alpha}(x)=C_{1,\\alpha}|x|^{-(1-\\alpha)} \\quad \\mbox{where} \\quad C_{1,\\alpha}=\\pi^{-1/2}2^{-\\alpha}\\frac{\\Gamma(\\frac{1-\\alpha}{2})}{\\Gamma(\\frac{\\alpha}{2})}.\n\\]\nMoreover, $\\mathcal{F} R_{1,\\alpha}(\\xi)=|\\xi|^{-\\alpha}d\\xi$, and $R_{1,\\alpha}*R_{1,\\beta}=R_{1,\\alpha+\\beta}$ for any $\\alpha,\\beta>0$ with $\\alpha+\\beta<1$.\n\n\\medskip\n\nWe introduce the following assumption.\n\n\\medskip\n\n\\noindent {\\bf Assumption B.} \n(i) $\\kappa\\in L^1(\\mathbb{R})$; or (ii) $\\kappa=R_{1,\\alpha/2}$ for some $\\alpha \\in (0,1)$.\n\n\\begin{remark}\n\\label{rem-M}\n{\\rm Suppose that Assumption B holds. Applying Young's inequality if $\\kappa \\in L^1(\\mathbb{R})$, and Theorem \\ref{HLS} if $\\kappa=R_{1,\\alpha/2}$, we see that:\n\\[\n\\|G_t* \\kappa\\|_{L^p(\\mathbb{R})} \\leq\n\\left\\{\n\\begin{array}{ll} \n\\|\\kappa\\|_{L^1(\\mathbb{R})} \\|G_t\\|_{L^p(\\mathbb{R})} & \\mbox{if $\\kappa \\in L^1(\\mathbb{R})$ and $p\\geq 1$} \\\\\nA_{1,\\alpha,p}\\|G_t\\|_{L^q(\\mathbb{R})} & \\mbox{if $\\kappa=R_{1,\\alpha/2}$, $\\alpha \\in (0,1)$, $p>\\frac{2}{2-\\alpha}$ and $\\frac{1}{q}=\\frac{1}{p}+\\frac{\\alpha}{2}$}\n\\end{array} \\right.\n\\]\nSince $\\|G_t\\|_{L^p(\\mathbb{R})}^p=2^{1-p}t$ for any $p>0$, we infer that condition \\eqref{def-Mp} holds for any $p\\geq 2$. Therefore, \\eqref{mom-u} holds for any $p \\geq 2$ such that $m_p<\\infty$.\n}\n\\end{remark}\n\nWe recall the following definitions. Let $X$ and $Y$ be random variables defined on the same probability space.\nThe {\\em $1$-Wasserstein distance} between $X$ and $Y$ is:\n\\[\nd_{W}(X,Y)=\\sup_{{\\rm Lip}(h) \\leq 1} \\big|\\mathbb{E}[h(X)]-\\mathbb{E}[h(Y)]\\big|, \\quad\n\\mbox{where} \\quad\n{\\rm Lip}(h):=\\sup_{x \\not=y}\\frac{|h(x)-h(y)|}{|x-y|}.\n\\]\nThe {\\em Fortet-Mourier distance} between $X$ and $Y$ is:\n\\[\nd_{FM}(X,Y)=\\sup_{\\|h\\|_{\\infty}+{\\rm Lip}(h) \\leq 1} \\big|\\mathbb{E}[h(X)]-\\mathbb{E}[h(Y)]\\big|,\n\\]\nThe {\\em Kolmogorov distance} between $X$ and $Y$ is:\n\\[\nd_{K}(X,Y)=\\sup_{x \\in \\mathbb{E}} \\big|P(X \\leq x)-P(Y \\leq x)\\big|.\n\\]\n\nWe are now ready to state the main results of this article.", "context": "Traditionally, SPDEs are perturbed by a space-time Gaussian white noise in dimension $d=1$, which is a zero-mean Gaussian process $\\{B_t(\\varphi);t> 0,\\varphi \\in L^2(\\mathbb{R})\\}$ with covariance:\n\\[\n\\mathbb{E}[B_t(\\varphi) B_s(\\psi)]=(t \\wedge s) \\langle \\varphi,\\psi \\rangle_{L^2(\\mathbb{R})}.\n\\]\nIn higher dimensions, Dalang introduced in \\cite{dalang99}, a spatially-homogeneous (or colored) noise, as a zero-mean Gaussian process $\\{W_t(\\varphi);t\\geq 0,\\varphi \\in \\mathcal{S}(\\mathbb{R})\\}$ with covariance:\n\\begin{equation}\n\\label{def-cov-W}\n\\mathbb{E}[W_t(\\varphi) W_s(\\psi)]=(t \\wedge s) \\int_{\\mathbb{R}^d} \\int_{\\mathbb{R}^d}\\varphi(x)\\psi(y) f(x-y)dxdy,\n\\end{equation}\nwhere $\\mathcal{S}(\\mathbb{R}^d)$ is the set of all rapidly decreasing functions on $\\mathbb{R}^d$, and $f:\\mathbb{R}^d \\to [0,\\infty]$ is the Fourier transform of a tempered measure $\\mu$. This noise can also be defined as:\n\\begin{equation}\n\\label{Gauss-color}\nW_t(\\varphi)=B_t(\\varphi*\\kappa) \\quad \\mbox{for all $t>0$ and $\\varphi \\in \\mathcal{S}(\\mathbb{R}^d)$},\n\\end{equation}\nwhere the kernel $\\kappa$ is chosen such that $k * \\widetilde{k}=f$, and $\\widetilde{k}(x):=k(-x)$ for all $x \\in \\mathbb{R}^d$.\n\nLet $G$ be the fundamental solution of the wave equation in dimension $d=1$, given by:\n\\begin{equation}\n\\label{def-G}\nG_t(x)=\\frac{1}{2}1_{\\{|x|<t\\}} \\quad \\mbox{for all $t>0$ and $x \\in \\mathbb{R}$.}\n\\end{equation}\n\nBy Lemma 5.5 of \\cite{BJ25}, the process $\\{u(t,x)\\}_{x\\in \\mathbb{R}}$ is strictly stationary, for any $t>0$.\nMoreover, by Theorem 5.6 of \\cite{BJ25}, if for some $p\\geq 2$, we have\n\\begin{equation}\n\\label{def-mp}\nm_p:=\\int_{\\bR_0}|z|^p \\nu(dz)<\\infty,\n\\end{equation}\nand\n\\begin{equation}\n\\label{def-Mp}\n\\cM_p(t):=\\int_0^t \\int_{\\mathbb{R}} \\big|(G_{t-s}*\\kappa)(x)\\big|^p dx ds<\\infty \\quad \\mbox{for all} \\quad t>0,\n\\end{equation}\nthen\n\\begin{equation}\n\\label{mom-u}\nK_p(T):=\\sup_{(t,x)\\in [0,T] \\times \\mathbb{R}}\\|u(t,x)\\|_p<\\infty \\quad \\mbox{for all} \\quad T>0,\n\\end{equation}\nwhere $\\|\\cdot\\|_p$ denotes the norm in $L^p(\\Omega)$. The goal of the present paper is to complement this analysis by showing that the {\\em spatial average} of the (centered) solution:\n\\[\nF_R(t)=\\int_{-R}^{R}\\big(u(t,x)-1 \\big)dx.\n\\]\nhas asymptotic Gaussian fluctuations when $R \\to \\infty$, with a precise rate of convergence which depends on the noise. We denote $\\sigma_R^2(t)={\\rm Var}(F_R(t))$.\n\n\\begin{remark}\n\\label{rem-M}\n{\\rm Suppose that Assumption B holds. Applying Young's inequality if $\\kappa \\in L^1(\\mathbb{R})$, and Theorem \\ref{HLS} if $\\kappa=R_{1,\\alpha/2}$, we see that:\n\\[\n\\|G_t* \\kappa\\|_{L^p(\\mathbb{R})} \\leq\n\\left\\{\n\\begin{array}{ll} \n\\|\\kappa\\|_{L^1(\\mathbb{R})} \\|G_t\\|_{L^p(\\mathbb{R})} & \\mbox{if $\\kappa \\in L^1(\\mathbb{R})$ and $p\\geq 1$} \\\\\nA_{1,\\alpha,p}\\|G_t\\|_{L^q(\\mathbb{R})} & \\mbox{if $\\kappa=R_{1,\\alpha/2}$, $\\alpha \\in (0,1)$, $p>\\frac{2}{2-\\alpha}$ and $\\frac{1}{q}=\\frac{1}{p}+\\frac{\\alpha}{2}$}\n\\end{array} \\right.\n\\]\nSince $\\|G_t\\|_{L^p(\\mathbb{R})}^p=2^{1-p}t$ for any $p>0$, we infer that condition \\eqref{def-Mp} holds for any $p\\geq 2$. Therefore, \\eqref{mom-u} holds for any $p \\geq 2$ such that $m_p<\\infty$.\n}\n\\end{remark}\n\nWe recall the following definitions. Let $X$ and $Y$ be random variables defined on the same probability space.\nThe {\\em $1$-Wasserstein distance} between $X$ and $Y$ is:\n\\[\nd_{W}(X,Y)=\\sup_{{\\rm Lip}(h) \\leq 1} \\big|\\mathbb{E}[h(X)]-\\mathbb{E}[h(Y)]\\big|, \\quad\n\\mbox{where} \\quad\n{\\rm Lip}(h):=\\sup_{x \\not=y}\\frac{|h(x)-h(y)|}{|x-y|}.\n\\]\nThe {\\em Fortet-Mourier distance} between $X$ and $Y$ is:\n\\[\nd_{FM}(X,Y)=\\sup_{\\|h\\|_{\\infty}+{\\rm Lip}(h) \\leq 1} \\big|\\mathbb{E}[h(X)]-\\mathbb{E}[h(Y)]\\big|,\n\\]\nThe {\\em Kolmogorov distance} between $X$ and $Y$ is:\n\\[\nd_{K}(X,Y)=\\sup_{x \\in \\mathbb{E}} \\big|P(X \\leq x)-P(Y \\leq x)\\big|.\n\\]\n\nWe are now ready to state the main results of this article.\n\n\\begin{theorem}[Hardy-Littlewood-Sobolov inequality]\n\\label{HLS}\nFor any $\\alpha \\in (0,d)$ and $p>\\frac{d}{d-\\alpha}$, \n\\[\n\\| \\varphi *R_{d,\\alpha}\\|_{L^p(\\bR^d)} \\leq A_{d.\\alpha,p}\\|\\varphi\\|_{L^{q}(\\bR^d)} \\quad \\mbox{for any $\\varphi \\in L^q(\\bR^d)$},\n\\]\nwhere $A_{d,\\alpha,p}>0$ is a constant depending on $(d,\\alpha,p)$, and $q$ is defined by:\n\\[\n\\frac{1}{q}=\\frac{1}{p}+\\frac{\\alpha}{d}.\n\\]\n(Condition $p>\\frac{d}{d-\\alpha}$ is equivalent to $q>1$.)\n\\end{theorem}\n\n\\begin{equation}\n\\label{def-Mp}\n\\cM_p(t):=\\int_0^t \\int_{\\bR} \\big|(G_{t-s}*\\k)(x)\\big|^p dx ds<\\infty \\quad \\mbox{for all} \\quad t>0,\n\\end{equation}\n\n\\begin{equation}\n\\label{mom-u}\nK_p(T):=\\sup_{(t,x)\\in [0,T] \\times \\bR}\\|u(t,x)\\|_p<\\infty \\quad \\mbox{for all} \\quad T>0,\n\\end{equation}", "full_context": "Traditionally, SPDEs are perturbed by a space-time Gaussian white noise in dimension $d=1$, which is a zero-mean Gaussian process $\\{B_t(\\varphi);t> 0,\\varphi \\in L^2(\\mathbb{R})\\}$ with covariance:\n\\[\n\\mathbb{E}[B_t(\\varphi) B_s(\\psi)]=(t \\wedge s) \\langle \\varphi,\\psi \\rangle_{L^2(\\mathbb{R})}.\n\\]\nIn higher dimensions, Dalang introduced in \\cite{dalang99}, a spatially-homogeneous (or colored) noise, as a zero-mean Gaussian process $\\{W_t(\\varphi);t\\geq 0,\\varphi \\in \\mathcal{S}(\\mathbb{R})\\}$ with covariance:\n\\begin{equation}\n\\label{def-cov-W}\n\\mathbb{E}[W_t(\\varphi) W_s(\\psi)]=(t \\wedge s) \\int_{\\mathbb{R}^d} \\int_{\\mathbb{R}^d}\\varphi(x)\\psi(y) f(x-y)dxdy,\n\\end{equation}\nwhere $\\mathcal{S}(\\mathbb{R}^d)$ is the set of all rapidly decreasing functions on $\\mathbb{R}^d$, and $f:\\mathbb{R}^d \\to [0,\\infty]$ is the Fourier transform of a tempered measure $\\mu$. This noise can also be defined as:\n\\begin{equation}\n\\label{Gauss-color}\nW_t(\\varphi)=B_t(\\varphi*\\kappa) \\quad \\mbox{for all $t>0$ and $\\varphi \\in \\mathcal{S}(\\mathbb{R}^d)$},\n\\end{equation}\nwhere the kernel $\\kappa$ is chosen such that $k * \\widetilde{k}=f$, and $\\widetilde{k}(x):=k(-x)$ for all $x \\in \\mathbb{R}^d$.\n\nLet $G$ be the fundamental solution of the wave equation in dimension $d=1$, given by:\n\\begin{equation}\n\\label{def-G}\nG_t(x)=\\frac{1}{2}1_{\\{|x|<t\\}} \\quad \\mbox{for all $t>0$ and $x \\in \\mathbb{R}$.}\n\\end{equation}\n\nBy Lemma 5.5 of \\cite{BJ25}, the process $\\{u(t,x)\\}_{x\\in \\mathbb{R}}$ is strictly stationary, for any $t>0$.\nMoreover, by Theorem 5.6 of \\cite{BJ25}, if for some $p\\geq 2$, we have\n\\begin{equation}\n\\label{def-mp}\nm_p:=\\int_{\\bR_0}|z|^p \\nu(dz)<\\infty,\n\\end{equation}\nand\n\\begin{equation}\n\\label{def-Mp}\n\\cM_p(t):=\\int_0^t \\int_{\\mathbb{R}} \\big|(G_{t-s}*\\kappa)(x)\\big|^p dx ds<\\infty \\quad \\mbox{for all} \\quad t>0,\n\\end{equation}\nthen\n\\begin{equation}\n\\label{mom-u}\nK_p(T):=\\sup_{(t,x)\\in [0,T] \\times \\mathbb{R}}\\|u(t,x)\\|_p<\\infty \\quad \\mbox{for all} \\quad T>0,\n\\end{equation}\nwhere $\\|\\cdot\\|_p$ denotes the norm in $L^p(\\Omega)$. The goal of the present paper is to complement this analysis by showing that the {\\em spatial average} of the (centered) solution:\n\\[\nF_R(t)=\\int_{-R}^{R}\\big(u(t,x)-1 \\big)dx.\n\\]\nhas asymptotic Gaussian fluctuations when $R \\to \\infty$, with a precise rate of convergence which depends on the noise. We denote $\\sigma_R^2(t)={\\rm Var}(F_R(t))$.\n\n\\begin{remark}\n\\label{rem-M}\n{\\rm Suppose that Assumption B holds. Applying Young's inequality if $\\kappa \\in L^1(\\mathbb{R})$, and Theorem \\ref{HLS} if $\\kappa=R_{1,\\alpha/2}$, we see that:\n\\[\n\\|G_t* \\kappa\\|_{L^p(\\mathbb{R})} \\leq\n\\left\\{\n\\begin{array}{ll} \n\\|\\kappa\\|_{L^1(\\mathbb{R})} \\|G_t\\|_{L^p(\\mathbb{R})} & \\mbox{if $\\kappa \\in L^1(\\mathbb{R})$ and $p\\geq 1$} \\\\\nA_{1,\\alpha,p}\\|G_t\\|_{L^q(\\mathbb{R})} & \\mbox{if $\\kappa=R_{1,\\alpha/2}$, $\\alpha \\in (0,1)$, $p>\\frac{2}{2-\\alpha}$ and $\\frac{1}{q}=\\frac{1}{p}+\\frac{\\alpha}{2}$}\n\\end{array} \\right.\n\\]\nSince $\\|G_t\\|_{L^p(\\mathbb{R})}^p=2^{1-p}t$ for any $p>0$, we infer that condition \\eqref{def-Mp} holds for any $p\\geq 2$. Therefore, \\eqref{mom-u} holds for any $p \\geq 2$ such that $m_p<\\infty$.\n}\n\\end{remark}\n\nWe recall the following definitions. Let $X$ and $Y$ be random variables defined on the same probability space.\nThe {\\em $1$-Wasserstein distance} between $X$ and $Y$ is:\n\\[\nd_{W}(X,Y)=\\sup_{{\\rm Lip}(h) \\leq 1} \\big|\\mathbb{E}[h(X)]-\\mathbb{E}[h(Y)]\\big|, \\quad\n\\mbox{where} \\quad\n{\\rm Lip}(h):=\\sup_{x \\not=y}\\frac{|h(x)-h(y)|}{|x-y|}.\n\\]\nThe {\\em Fortet-Mourier distance} between $X$ and $Y$ is:\n\\[\nd_{FM}(X,Y)=\\sup_{\\|h\\|_{\\infty}+{\\rm Lip}(h) \\leq 1} \\big|\\mathbb{E}[h(X)]-\\mathbb{E}[h(Y)]\\big|,\n\\]\nThe {\\em Kolmogorov distance} between $X$ and $Y$ is:\n\\[\nd_{K}(X,Y)=\\sup_{x \\in \\mathbb{E}} \\big|P(X \\leq x)-P(Y \\leq x)\\big|.\n\\]\n\nWe are now ready to state the main results of this article.\n\n\\begin{theorem}[Hardy-Littlewood-Sobolov inequality]\n\\label{HLS}\nFor any $\\alpha \\in (0,d)$ and $p>\\frac{d}{d-\\alpha}$, \n\\[\n\\| \\varphi *R_{d,\\alpha}\\|_{L^p(\\bR^d)} \\leq A_{d.\\alpha,p}\\|\\varphi\\|_{L^{q}(\\bR^d)} \\quad \\mbox{for any $\\varphi \\in L^q(\\bR^d)$},\n\\]\nwhere $A_{d,\\alpha,p}>0$ is a constant depending on $(d,\\alpha,p)$, and $q$ is defined by:\n\\[\n\\frac{1}{q}=\\frac{1}{p}+\\frac{\\alpha}{d}.\n\\]\n(Condition $p>\\frac{d}{d-\\alpha}$ is equivalent to $q>1$.)\n\\end{theorem}\n\n\\begin{equation}\n\\label{def-Mp}\n\\cM_p(t):=\\int_0^t \\int_{\\bR} \\big|(G_{t-s}*\\k)(x)\\big|^p dx ds<\\infty \\quad \\mbox{for all} \\quad t>0,\n\\end{equation}\n\n\\begin{equation}\n\\label{mom-u}\nK_p(T):=\\sup_{(t,x)\\in [0,T] \\times \\bR}\\|u(t,x)\\|_p<\\infty \\quad \\mbox{for all} \\quad T>0,\n\\end{equation}\n\nBy Lemma 5.5 of \\cite{BJ25}, the process $\\{u(t,x)\\}_{x\\in \\bR}$ is strictly stationary, for any $t>0$.\nMoreover, by Theorem 5.6 of \\cite{BJ25}, if for some $p\\geq 2$, we have\n\\begin{equation}\n\\label{def-mp}\nm_p:=\\int_{\\bR_0}|z|^p \\nu(dz)<\\infty,\n\\end{equation}\nand\n\\begin{equation}\n\\label{def-Mp}\n\\cM_p(t):=\\int_0^t \\int_{\\bR} \\big|(G_{t-s}*\\k)(x)\\big|^p dx ds<\\infty \\quad \\mbox{for all} \\quad t>0,\n\\end{equation}\nthen\n\\begin{equation}\n\\label{mom-u}\nK_p(T):=\\sup_{(t,x)\\in [0,T] \\times \\bR}\\|u(t,x)\\|_p<\\infty \\quad \\mbox{for all} \\quad T>0,\n\\end{equation}\nwhere $\\|\\cdot\\|_p$ denotes the norm in $L^p(\\Omega)$. The goal of the present paper is to complement this analysis by showing that the {\\em spatial average} of the (centered) solution:\n\\[\nF_R(t)=\\int_{-R}^{R}\\big(u(t,x)-1 \\big)dx.\n\\]\nhas asymptotic Gaussian fluctuations when $R \\to \\infty$, with a precise rate of convergence which depends on the noise. We denote $\\sigma_R^2(t)={\\rm Var}(F_R(t))$.\n\n\\begin{remark}\n\\label{rem-M}\n{\\rm Suppose that Assumption B holds. Applying Young's inequality if $\\k \\in L^1(\\bR)$, and Theorem \\ref{HLS} if $\\k=R_{1,\\alpha/2}$, we see that:\n\\[\n\\|G_t* \\k\\|_{L^p(\\bR)} \\leq\n\\left\\{\n\\begin{array}{ll} \n\\|\\k\\|_{L^1(\\bR)} \\|G_t\\|_{L^p(\\bR)} & \\mbox{if $\\k \\in L^1(\\bR)$ and $p\\geq 1$} \\\\\nA_{1,\\alpha,p}\\|G_t\\|_{L^q(\\bR)} & \\mbox{if $\\k=R_{1,\\alpha/2}$, $\\alpha \\in (0,1)$, $p>\\frac{2}{2-\\alpha}$ and $\\frac{1}{q}=\\frac{1}{p}+\\frac{\\alpha}{2}$}\n\\end{array} \\right.\n\\]\nSince $\\|G_t\\|_{L^p(\\bR)}^p=2^{1-p}t$ for any $p>0$, we infer that condition \\eqref{def-Mp} holds for any $p\\geq 2$. Therefore, \\eqref{mom-u} holds for any $p \\geq 2$ such that $m_p<\\infty$.\n}\n\\end{remark}\n\nWe are now ready to state the main results of this article.\n\n\\begin{theorem}[Limiting covariance]\n\\label{cov-th}\nIf Assumption B holds, then\nfor any $t,s>0$,\n\\[\n\\lim_{R \\to \\infty}\\frac{1}{R^{\\beta}}\\bE[F_R(t) F_R(s)]=K(t,s) \\quad \\mbox{is finite},\n\\] \nwhere \n\\begin{equation}\n\\label{def-beta}\n\\beta:=\\left\\{\n\\begin{array}{ll} 1 & \\mbox{if $\\k \\in L^1(\\bR)$,} \\\\\n\\alpha+1 & \\mbox{if $\\k=R_{1,\\alpha/2}$ for some $\\alpha \\in (0,1)$. }\n\\end{array} \\right.\n\\end{equation}\nIn particular, $R^{-\\beta} \\sigma_R^2(t) \\to K(t,t)$ as $R \\to \\infty$, for any $t>0$.\n\\end{theorem}\n\n\\begin{theorem}[Quantitative Central Limit Theorem]\n\\label{QCLT}\nSuppose that Assumption B holds, and there exists $p \\in (1,2]$ such that \n\\begin{equation}\n\\label{mp-m2p}\nm_{p}<\\infty \\quad \\mbox{and} \\quad m_{2p}<\\infty.\n\\end{equation}\n\n\\begin{theorem}[Functional Central Limit Theorem]\n\\label{FCLT}\nUnder the hypotheses of Theorem \\ref{QCLT}, for any $R>0$,\nthe process $\\{F_R(t)\\}_{t\\geq 0}$ has a $\\gamma$-H\\\"older continuous modification (denoted also $F_R$), for any $\\gamma \\in (0,\\frac{\\beta}{2})$, where $\\beta$ is given by \\eqref{def-beta}.\nMoreover,\n\\[\n\\frac{1}{R^{\\beta/2}}F_R(\\cdot) \\stackrel{d}{\\to} \\cG(\\cdot) \\quad \\mbox{in $C[0,\\infty)$ as $R \\to \\infty$},\n\\]\nwhere $\\{\\cG(t)\\}_{t\\geq 0}$ is a zero-mean Gaussian process with covariance \n$\\bE[\\cG(t) \\cG(s)]=K(t,s)$, and $K(t,s)$ given by Theorem \\ref{cov-th}. Here $\\stackrel{d}{\\to}$ denotes the convergence in distribution, and $C[0,\\infty)$ is equipped with the uniform convergence on compact sets.\n\\end{theorem}\n\n$\\bullet$\nAny random variable $F \\in L^2(\\Omega)$ which is $\\cF^N$-measurable has the {\\em Poisson-chaos expansion}:\n\\begin{equation}\n\\label{Poisson-chaos}\nF=\\bE(F)+\\sum_{n\\geq 1}I_n(f_n), \\quad \\mbox{for some $f_n \\in \\cH^{\\odot n}$},\n\\end{equation}\nand the series is orthogonal in $L^2(\\Omega)$. Here $I_n$ is the multiple integral with respect to $\\widehat{N}$ and $\\cH^{\\odot n}$ is the set of symmetric functions in $\\cH^{\\otimes n}$.\nFor any $f \\in \\cH^{\\otimes n}$,\n\\[\n\\bE[I_n(f)]=0 \\quad \\mbox{and} \\quad \\bE|I_n(f)|^2=n! \\|\\widetilde{f}\\|_{\\cH^{\\otimes n}}^2,\n\\]\nwhere $\\widetilde{f}$ is the symmetrization of $f$:\n\\[\n\\widetilde{f}(\\xi_1,\\ldots,\\xi_n)=\\frac{1}{n!}\\sum_{\\rho \\in S_n}f(\\xi_{\\rho(1)},\\ldots,\\xi_{\\rho(n)}),\n\\]\n$S_n$ being the set of all permutations of $1,\\ldots,n$.\nMoreover, $I_n(f)=I_n(\\widetilde{f})$ for any $f \\in \\cH^{\\otimes n}$.\n\n{\\em Step 1.} (finite-dimensional convergence) We will prove that for any integer $m\\geq 1$ and form any $t_1,\\ldots,t_m \\in [0,T]$\n\\[\n\\Big(\\frac{1}{R^{\\beta/2}} F_R(t_1),\\ldots, \\frac{1}{R^{\\beta/2}}F_R(t_m) \\Big)\n\\stackrel{d}{\\to} \\Big(\\cG(t_1),\\ldots, \\cG(t_m) \\Big) \\quad \\mbox{as $R \\to \\infty$}.\n\\]\nBy Cram\\'er-Wold theorem, this is equivalent to showing that for any $b_1,\\ldots,b_m \\in \\bR$,\n\\[\nX_R:=\\frac{1}{R^{\\beta/2}}\\sum_{j=1}^{m}b_jF_R(t_j) \\stackrel{d}{\\to}\\sum_{j=1}^{m}b_j \\cG(t_j) \n\\quad \\mbox{as $R \\to \\infty$}.\n\\]\nUsing the same argument as on page 4215 of \\cite{BZ24}, it is enough to prove that $X_R/\\tau_R \\stackrel{d}{\\to} Z$ as $R \\to \\infty$, where $\\tau_R^2={\\rm Var}(X_R)$ and $Z \\sim N(0,1)$. By Proposition \\ref{tara}, \n\\[\nd_{W}\\left(\\frac{X_R}{\\tau_R},Z \\right) \\leq \\gamma_1+\\gamma_2+\\gamma_3,\n\\]\nwhere $\\gamma_1,\\gamma_2$ and $\\gamma_3$ are defined as in \\eqref{gamma17} with $F=X_R$. Using the same argument as in the proof of Theorem \\ref{QCLT}, we infer that \n\\[\nd_{W}\\left(\\frac{X_R}{\\tau_R},Z \\right) \\les R^{1-p} \\quad \\mbox{if $\\k \\in L^1(\\bR)$,}\n\\]\nand\n\\[\nd_{W}\\left(\\frac{X_R}{\\tau_R},Z \\right) \\les R^{-\\varepsilon} \\quad \\mbox{for any $\\varepsilon \\in \\big(0,1-\\frac{1}{p}-\\frac{\\alpha}{2}\\big)$, \\quad if $\\k =R_{1,\\alpha/2}$.}\n\\]\n\n\\begin{theorem}[Hardy-Littlewood-Sobolov inequality]\n\\label{HLS}\nFor any $\\alpha \\in (0,d)$ and $p>\\frac{d}{d-\\alpha}$, \n\\[\n\\| \\varphi *R_{d,\\alpha}\\|_{L^p(\\bR^d)} \\leq A_{d.\\alpha,p}\\|\\varphi\\|_{L^{q}(\\bR^d)} \\quad \\mbox{for any $\\varphi \\in L^q(\\bR^d)$},\n\\]\nwhere $A_{d,\\alpha,p}>0$ is a constant depending on $(d,\\alpha,p)$, and $q$ is defined by:\n\\[\n\\frac{1}{q}=\\frac{1}{p}+\\frac{\\alpha}{d}.\n\\]\n(Condition $p>\\frac{d}{d-\\alpha}$ is equivalent to $q>1$.)\n\\end{theorem}\n\n\\begin{theorem}[Quantitative Central Limit Theorem]\n\\label{QCLT}\nSuppose that Assumption B holds, and there exists $p \\in (1,2]$ such that \n\\begin{equation}\n\\label{mp-m2p}\nm_{p}<\\infty \\quad \\mbox{and} \\quad m_{2p}<\\infty.\n\\end{equation}\n\nIf $\\k \\in L^1(\\bR)$, then\nfor any $t>0$,\n\\[\n{\\rm dist}\\left( \\frac{F_R(t)}{\\sigma_R(t)},Z\\right) \\leq  C_t R^{-(1-\\frac{1}{p})}\n\\]\n\nIf $\\k=R_{1,\\alpha/2}$ for some $\\alpha \\in (0,1)$, and $p>\\frac{2}{2-\\alpha}$, then\n\\[\n{\\rm dist}\\left( \\frac{F_R(t)}{\\sigma_R(t)},Z\\right) \\leq\nC_t R^{-\\e}  \\quad \\mbox{for any $\\varepsilon \\in \\big(0, 1-\\frac{1}{p}-\\frac{\\alpha}{2}\\big)$}.\n\\]\n\nHere $C_t>0$ is a constant depending on $t$, and ${\\rm dist}$ is the Fortet-Mourier distance, the 1-Wasserstein distance, or the Kolmogorov distance.\n\\end{theorem}", "post_theorem_intro_text_len": 6581, "post_theorem_intro_text": "\\begin{theorem}[Limiting covariance]\n\\label{cov-th}\nIf Assumption B holds, then\nfor any $t,s>0$,\n\\[\n\\lim_{R \\to \\infty}\\frac{1}{R^{\\beta}}\\mathbb{E}[F_R(t) F_R(s)]=K(t,s) \\quad \\mbox{is finite},\n\\] \nwhere \n\\begin{equation}\n\\label{def-beta}\n\\beta:=\\left\\{\n\\begin{array}{ll} 1 & \\mbox{if $\\kappa \\in L^1(\\mathbb{R})$,} \\\\\n\\alpha+1 & \\mbox{if $\\kappa=R_{1,\\alpha/2}$ for some $\\alpha \\in (0,1)$. }\n\\end{array} \\right.\n\\end{equation}\nIn particular, $R^{-\\beta} \\sigma_R^2(t) \\to K(t,t)$ as $R \\to \\infty$, for any $t>0$.\n\\end{theorem}\n\n\\begin{theorem}[Quantitative Central Limit Theorem]\n\\label{QCLT}\nSuppose that Assumption B holds, and there exists $p \\in (1,2]$ such that \n\\begin{equation}\n\\label{mp-m2p}\nm_{p}<\\infty \\quad \\mbox{and} \\quad m_{2p}<\\infty.\n\\end{equation}\n\nIf $\\kappa \\in L^1(\\mathbb{R})$, then\nfor any $t>0$,\n\\[\n{\\rm dist}\\left( \\frac{F_R(t)}{\\sigma_R(t)},Z\\right) \\leq  C_t R^{-(1-\\frac{1}{p})}\n\\]\n\nIf $\\kappa=R_{1,\\alpha/2}$ for some $\\alpha \\in (0,1)$, and $p>\\frac{2}{2-\\alpha}$, then\n\\[\n{\\rm dist}\\left( \\frac{F_R(t)}{\\sigma_R(t)},Z\\right) \\leq\nC_t R^{-\\varepsilon}  \\quad \\mbox{for any $\\varepsilon \\in \\big(0, 1-\\frac{1}{p}-\\frac{\\alpha}{2}\\big)$}.\n\\]\n\nHere $C_t>0$ is a constant depending on $t$, and ${\\rm dist}$ is the Fortet-Mourier distance, the 1-Wasserstein distance, or the Kolmogorov distance.\n\\end{theorem}\n\n\\begin{theorem}[Functional Central Limit Theorem]\n\\label{FCLT}\nUnder the hypotheses of Theorem \\ref{QCLT}, for any $R>0$,\nthe process $\\{F_R(t)\\}_{t\\geq 0}$ has a $\\gamma$-H\\\"older continuous modification (denoted also $F_R$), for any $\\gamma \\in (0,\\frac{\\beta}{2})$, where $\\beta$ is given by \\eqref{def-beta}.\nMoreover,\n\\[\n\\frac{1}{R^{\\beta/2}}F_R(\\cdot) \\stackrel{d}{\\to} \\mathcal{G}(\\cdot) \\quad \\mbox{in $C[0,\\infty)$ as $R \\to \\infty$},\n\\]\nwhere $\\{\\mathcal{G}(t)\\}_{t\\geq 0}$ is a zero-mean Gaussian process with covariance \n$\\mathbb{E}[\\mathcal{G}(t) \\mathcal{G}(s)]=K(t,s)$, and $K(t,s)$ given by Theorem \\ref{cov-th}. Here $\\stackrel{d}{\\to}$ denotes the convergence in distribution, and $C[0,\\infty)$ is equipped with the uniform convergence on compact sets.\n\\end{theorem}\n\nThe proofs of these theorems are based on a key estimate  which shows that the moments of the Malliavin derivative of the solution of \\eqref{HAM} can be bounded, up to a constant, by a deterministic function (see relation \\eqref{key-D} below). Similar estimates appear in all references dedicated to QCLT, where they play a crucial role. To prove this key estimate, we proceed as in \\cite{BZ24}, developing a connection to (hAm) with delta initial velocity by using the form \\eqref{def-G} of the fundamental solution $G$.\n\nTheorem \\ref{ergodic-th} follows from the key estimate \\eqref{key-D} and a criterion which may be of independent interest (Lemma \\ref{lem-key-Z} below). \nTo prove Theorem \\ref{cov-th} we show that the covariance of the solution of \\eqref{HAM} coincides with the covariance of the solution of (hAm) driven by $\\sqrt{m_2}W$, where $W$ is the Gaussian colored noise with covariance \\eqref{def-cov-W}.\n\nTo prove Theorem \\ref{QCLT}, we apply the recent result of Trauthwein \\cite{trauthwein25}, which gives the optimal rates for the Wasserstein and Kolmogorov distances between $F/\\sqrt{{\\rm Var}(F)}$ and $Z \\sim N(0,1)$, for a centered random variable $F$ with finite variance, which is Malliavin differentiable with respect to a compensated Poisson random measure. These estimates involve the first and second order Malliavin derivatives. Therefore, to  apply the result in \\cite{trauthwein25}, we develop a similar key estimate for the second Malliavin derivative of the solution (see relation \\eqref{key-D2} below). Different techniques are used for estimating the 7 quantities $\\gamma_1,\\ldots,\\gamma_7$ which appear in the result of \\cite{trauthwein25}, which rely on Young's inequality in the case when $k$ is integrable, respectively on Hardy-Littlewood-Sobolev inequality when $\\kappa$ is the Riesz kernel. Finally, Theorem \\ref{FCLT} follows by the classical method of finite dimensional convergence and tightness.\n\n\\begin{remark}\n{\\rm a) If $\\kappa \\in L^1(\\mathbb{R})$, then $f=\\kappa * \\tilde{\\kappa} \\in L^1(\\mathbb{R})$ by Young's inequality. In this case, the variance $\\sigma_R^2(t)$ decays with rate $R$, as in the L\\'evy white noise case and the Gaussian colored noise case with integrable kernel $f$ (see Theorem 1.2 of \\cite{NZ22}). The decay rate $R^{-(1-\\frac{1}{p})}$  in the QCLT (in the $d_{W}$, $d_{FM}$ or $d_K$ distances) is the same as in the L\\'evy white noise case (see Theorem 1.1.(iii) of \\cite{BZ24}), and depends on the parameter $p$ from \\eqref{mp-m2p}. This can be explained since formally, the L\\'evy white noise can be viewed as a L\\'evy colored noise with integrable kernel $\\kappa=\\delta_0$. When $p=2$ (i.e. $m_4<\\infty)$, this rate coincides with the rate $R^{-1/2}$ obtained in the case of the Gaussian colored noise case with integrable kernel $f$, for the QCLT in the total variation distance (see Theorem 1.2 of \\cite{NZ22}).\n\n\\medskip\n\nb) If $\\kappa=R_{1,\\alpha/2}$ for some $\\alpha \\in (0,1)$, then $f =R_{1,\\alpha}$. In this case, using the parametrization $\\alpha=2H-1$ with $H \\in (\\frac{1}{2},1)$, we see that the decay rate $R^{2H}$ of $\\sigma_R^2(t)$ is the same as in the Gaussian colored noise case (see  Proposition 3.3 of \\cite{DNZ20}). In this Gaussian case, Theorem 1.1 {\\em ibid.} gives the rate $R^{-(1-H)}=R^{-\\frac{1-\\alpha}{2}}$ for the QCLT in the total variation distance, which is almost the same as the rate $R^{-\\varepsilon}$ with $\\varepsilon \\in (0,\\frac{1-\\alpha}{2})$ obtained in Theorem \\ref{QCLT} above when $p=2$ (i.e. $m_4<\\infty$).\n}\n\\end{remark}\n\nThis article is organized as follows. In Section \\ref{section-prelim}, we include some preliminaries about Poisson-Malliavin calculus and moment inequalities with respect to noises $L$ and $X$. In Sections \\ref{section-Mall1} and \\ref{section-Mall2}, we prove some key estimates for the first, respectively the second,  Malliavin derivative of the solution. In Section \\ref{section-proofs}, we present the proofs of Theorems \\ref{ergodic-th}, \\ref{cov-th}, \\ref{QCLT} and \\ref{FCLT}. Appendix \\ref{appA} contains some auxiliary results about Poisson-Malliavin calculus. Appendix \\ref{appB} presents some inequalities for Riesz potentials.\n\n\\medskip\n\nWe conclude the introduction with few words about the notation. We write $a\\lesssim b$ to indicate that $a \\leq C b$ for some positive constant $C>0$ that does not depend on $(a,b)$. \nFor any $p\\geq 1$, we denote by $\\|\\cdot \\|_p$ the norm in $L^p(\\Omega)$.", "sketch": "The post-theorem introduction outlines the following proof strategy (centered around Theorem~\\ref{ergodic-th}):\n\n- The proofs are \"based on a key estimate\" showing that \"the moments of the Malliavin derivative of the solution of \\eqref{HAM} can be bounded, up to a constant, by a deterministic function\" (relation \\eqref{key-D}).\n\n- To prove this key estimate, the authors \"proceed as in \\cite{BZ24}, developing a connection to (hAm) with delta initial velocity by using the form \\eqref{def-G} of the fundamental solution $G$.\"\n\n- Then, explicitly for the main theorem: \"Theorem \\ref{ergodic-th} follows from the key estimate \\eqref{key-D} and a criterion which may be of independent interest (Lemma \\ref{lem-key-Z} below).\"", "expanded_sketch": "The post-theorem introduction outlines the following proof strategy (centered around the main theorem):\n\n- The proofs are based on a key estimate showing that the moments of the Malliavin derivative of the solution of\n\\begin{align}\n\\label{HAM}\n\t\\begin{cases}\n\t\t\\dfrac{\\partial^2 u}{\\partial^2 t} (t,x)\n\t\t=  \\dfrac{\\partial^2 u}{\\partial x^2} (t,x)+u(t,x) \\dot{X}(t,x), \\\\\n\t\tt>0, \\ x \\in \\bR, \\\\\n\t\tu(0,x) = 1, \\dfrac{\\partial u}{\\partial t} (0,x)=0, \\quad x \\in \\bR,\n\t\\end{cases}\n\\end{align}\ncan be bounded, up to a constant, by a deterministic function, namely\n\\begin{equation}\n\\label{key-D}\n\\|D_{r,y,z}u(t,x)\\|_p \\leq C'_{T,p,\\nu,\\k}|z| \\int_{\\bR}G_{t-r}(x-y')\\k(y-y')dy',\n\\end{equation}\n\n- To prove this key estimate, the authors “proceed as in \\cite{BZ24}”, developing a connection to (hAm) with delta initial velocity by using the form\n\\begin{equation}\n\\label{def-G}\nG_t(x)=\\frac{1}{2}1_{\\{|x|<t\\}} \\quad \\mbox{for all $t>0$ and $x \\in \\bR$.}\n\\end{equation}\nof the fundamental solution $G$.\n\n- Then, in establishing the main theorem: it follows from the key estimate (the equation above) and a criterion which may be of independent interest. We first record that criterion.\n\\begin{lemma}\n\\label{lem-key-Z}\nSuppose that Assumption B holds.\nLet $\\{Z(t,x);t\\geq 0,x\\in \\bR\\}$ be an adapted random field such that\n$\\{Z(t,x)\\}_{x\\in \\bR^d}$ is strictly stationary and $Z(t,x) \\in {\\rm dom}(D)$ for any $t\\geq 0$ and $x \\in \\bR$. Assume that for any $0<r<t$, $x,y \\in \\bR$ and $z \\in \\bR_0$,\n\\begin{equation}\n\\label{key-Z}\n\\|D_{r,y,z}Z(t,x)\\|_{2} \\leq C_t |z| \\int_{\\bR} G_{t-r}(x-y')\\k(y-y')dy',\n\\end{equation}\nwhere $C_t>0$ is a constant depending on $t$. Then $\\{Z(t,x)\\}_{x\\in \\bR}$ is ergodic.\n\\end{lemma}", "expanded_theorem": "\\label{ergodic-th}\nIf Assumption B holds, then $\\{u(t,x)\\}_{x\\in \\mathbb{R}}$ is ergodic, for any $t>0$. Consequently, by the mean ergodic theorem,\n\\[\n\\frac{1}{R}F_R(t) \\to 0 \\quad \\mbox{a.s and in $L^2(\\Omega)$}, \\quad \\mbox{as $R \\to \\infty$}.\n\\]", "theorem_type": ["Implication", "Asymptotic or Limit"], "mcq": {"question": "Let \\(u(t,x)\\) be the one-dimensional random field under Assumption B, and for each fixed \\(t>0\\) define the centered spatial average over \\([-R,R]\\) by\n\\[\nF_R(t)=\\int_{-R}^{R}(u(t,x)-1)\\,dx.\n\\]\nUnder Assumption B, which statement about the spatial process \\(\\{u(t,x)\\}_{x\\in\\mathbb{R}}\\) and the asymptotic behavior of \\(F_R(t)\\) is valid for every \\(t>0\\)?", "correct_choice": {"label": "A", "text": "For every \\(t>0\\), the spatial process \\(\\{u(t,x)\\}_{x\\in\\mathbb{R}}\\) is ergodic. Consequently,\n\\[\n\\frac{1}{R}F_R(t)=\\frac{1}{R}\\int_{-R}^{R}(u(t,x)-1)\\,dx \\to 0\n\\quad \\text{as } R\\to\\infty,\n\\]\nboth almost surely and in \\(L^2(\\Omega)\\)."}, "choices": [{"label": "B", "text": "For every \\(t>0\\), the spatial process \\(\\{u(t,x)\\}_{x\\in\\mathbb{R}}\\) is mixing. Consequently,\n\\[\n\\frac{1}{R}F_R(t)=\\frac{1}{R}\\int_{-R}^{R}(u(t,x)-1)\\,dx \\to 0\n\\quad \\text{as } R\\to\\infty,\n\\]\nboth almost surely and in \\(L^2(\\Omega)\\)."}, {"label": "C", "text": "For every \\(t>0\\),\n\\[\n\\frac{1}{R}F_R(t)=\\frac{1}{R}\\int_{-R}^{R}(u(t,x)-1)\\,dx \\to 0\n\\quad \\text{as } R\\to\\infty,\n\\]\nin \\(L^2(\\Omega)\\)."}, {"label": "D", "text": "For every \\(t>0\\), the spatial process \\(\\{u(t,x)\\}_{x\\in\\mathbb{R}}\\) is ergodic. Consequently,\n\\[\n\\frac{1}{\\sqrt{R}}F_R(t)=\\frac{1}{\\sqrt{R}}\\int_{-R}^{R}(u(t,x)-1)\\,dx \\to 0\n\\quad \\text{as } R\\to\\infty,\n\\]\nboth almost surely and in \\(L^2(\\Omega)\\)."}, {"label": "E", "text": "For every \\(t>0\\), the spatial process \\(\\{u(t,x)\\}_{x\\in\\mathbb{R}}\\) is ergodic. Consequently,\n\\[\n\\frac{1}{R}F_R(t)=\\frac{1}{R}\\int_{-R}^{R}(u(t,x)-1)\\,dx \\to 0\n\\quad \\text{as } R\\to\\infty,\n\\]\nalmost surely, and the convergence is uniform in \\(t\\) on compact time intervals in \\(L^2(\\Omega)\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "ergodicity_vs_stronger_mixing", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped_ergodicity_and_a.s._convergence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "normalization_scale_1_over_R_replaced_by_1_over_sqrtR", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "pointwise_in_t_conclusion_replaced_by_uniform_on_compacts", "template_used": "quantifier_dependence"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem defines the averaged quantity and asks for the correct limiting statement, but it does not explicitly reveal ergodicity or the exact convergence mode. There is no direct answer leakage from the wording."}, "TAS": {"score": 1, "justification": "The correct option appears to be very close to a theorem-style statement under Assumption B, so the item risks being a near restatement of a known result. However, it is not completely tautological because the alternatives vary meaningfully in strength, mode of convergence, and scaling."}, "GPS": {"score": 1, "justification": "Some reasoning is required to distinguish the strongest valid conclusion from weaker true or stronger false variants, especially between A and C and among different convergence claims. Still, for someone who knows the source result, the correct answer is likely recognized rather than generated."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically well targeted: B overstates ergodicity to mixing and adds a rate, C is a weaker true statement, D adds unjustified uniformity in time, and E uses the wrong normalization scale. These reflect realistic failure modes."}, "total_score": 6, "overall_assessment": "A solid MCQ with good distractors and little answer leakage, but it remains somewhat theorem-recall driven rather than deeply generative."}}
{"id": "2602.23267v1", "paper_link": "http://arxiv.org/abs/2602.23267v1", "theorems_cnt": 2, "theorem": {"env_name": "introtheorem", "content": "\\label{introthm:tameness} \n    Suppose $\\varphi\\colon \\A\\to\\A^*$ is a primitive substitution of length $k\\geqslant 2$. Then, the following are equivalent:\n    \\begin{enumerate}\n        \\item $\\ac(X_{\\varphi})\\in \\{0,1\\}$ (with $\\ac(X_{\\varphi})=0$ corresponding to $X_{\\varphi}$ being  finite),\n        \\item $X_{\\varphi}$ is null,\n        \\item $X_{\\varphi}$ is tame.\n    \\end{enumerate}", "start_pos": 11112, "end_pos": 11534, "label": "introthm:tameness"}, "ref_dict": {"rem:inf_tiling_and_beyond_mean_eq": "\\begin{remark}\\label{rem:inf_tiling_and_beyond_mean_eq}\n    In the context of more general group actions, the authors of \\cite{BaakeGaehlerGohlke2025} study amorphic complexity—for which they suggest to use the name \\emph{orbit separation dimension}—for actions induced by primitive self-similar inflation tilings. \n    They determine it for several notable examples, including the recently discovered Hat tiling \\cite{SmithMyersKaplanGoodman-Strauss2024}.\n\n    To extend the analysis of asymptotic separation numbers beyond mean equicontinuous systems, Kasjan and Keller propose in \\cite{KasjanKeler2025} to study (Besicovitch) covering numbers of symbolic orbits of generic points. \n    Their approach applies to the wider class of symbolic systems with discrete spectrum and reduces to the study of asymptotic separation numbers in the mean equicontinuous case. \n    It allows them to distinguish between several Toeplitz subshifts associated with $\\mathcal{B}$-free systems, which were previously indistinguishable using classical topological invariants.\n\\end{remark}", "rem:Baakeresult": "\\begin{remark}\\label{rem:Baakeresult}\n    In \\cite{BaakeGaehlerGohlke2025}, the authors introduce the notion of a \\emph{discrepancy inflation} for a self-similar, nonperiodic topological tiling dynamical system $(\\mathbb{X},\\mathbb{R}^d)$ with pure-point spectrum generated by a primitive inflation, using it to establish lower and upper bounds for the amorphic complexity of $(\\mathbb{X},\\mathbb{R}^d)$. Though their work focuses on tilings (rather than the symbolic setting), it is easy to see that, for a substitution $\\varphi$ of constant length, their discrepancy inflation corresponds to our discrepancy substitution.\n\n    By translating their main result (\\cite[Thm.\\ 4.13 \\& Prop.\\ 4.14]{BaakeGaehlerGohlke2025}) to our framework, we obtain the following bounds:\n    \\[\n        \\frac{\\log k}{\\log k - \\log \\lambda_{min}} \\leq \\oac(X_{\\varphi}) \\leq \\frac{\\log k}{\\log k - \\log \\lambda_s},\n    \\]\n    where $\\lambda_s$ (resp.\\ $\\lambda_{min}$) denotes the maximal (resp.\\ minimal) growth rate of letters in the discrepancy substitution $\\varphi_s$, and $k$ is the length of the substitution. If $\\lambda_s = \\lambda_{min}$, this yields a closed-form expression for the amorphic complexity of $X_{\\varphi}$. However, in general, one typically expects that $\\lambda_{min} < \\lambda_s$; see e.g.\\ Example~\\ref{ex:sep_substitution_not_irreducible}.\n\\end{remark}", "rem:different_version_Besicovitch": "\\begin{remark}\\label{rem:different_version_Besicovitch}\n    For a fixed partition $\\mathcal{P}$ we denote by $\\fp$ the full set  \n    $\\fp = \\mathcal{P} \\ast \\mathcal{P} \\setminus \\mathcal{P}$  \n    of distinct unordered pairs of elements of $\\mathcal{P}$. With this convention,  \n    the pseudometric $D_{\\fp}$ defined in \\eqref{eq:def_of_pseudo} corresponds to  \n    the standard \\emph{Besicovitch pseudometric}\\footnote{Often, this term also  \n    refers to the pseudometric defined in \\eqref{def:Besicovitch_pseudometric}.  \n    In the symbolic context, $D_{\\fp}$ and $D_B$ are uniformly equivalent  \n    (see, for instance, the proof of Proposition 6.8 in \\cite{FuhrmannGroeger2020})  \n    but, in general, they do not need to be Lipschitz equivalent.} on a subshift.  \n    Clearly, $D_{\\fp}$ coincides with $D_\\delta$ from  \n    \\eqref{eq:besicovitch metric and density of discrepancy} for $\\delta=1$ and\n    for any $\\brt \\subseteq (\\mathcal{P} \\ast \\mathcal{P}) \\backslash \\mathcal{P}$,  \n    we have  \n    \\[\n        D_{\\brt}(x,y) \\leq D_{\\fp}(x,y),\n    \\]\n    for all $x,y \\in X$.\n\\end{remark}", "prop:denseness": "\\begin{proposition}\\label{prop:denseness}\n Let $\\A$ be an alphabet of size at least 2. In the class of minimal automatic systems over $\\A$, amorphic complexity takes a dense set of values in $[1,\\infty)$.   \n\\end{proposition}", "introthm:ac_of_automatic_systems": "\\begin{introtheorem}\\label{introthm:ac_of_automatic_systems} Let $\\varphi$ be a primitive substitution of length $k\\geq 2$.  The amorphic complexity of $X_{\\varphi}$ is given by\n\\[\\ac(X_{\\varphi})=\\frac{\\log k}{\\log k - \\log \\lambda_s},\\]\nwhere $\\lambda_s$ is the discrepancy rate of $\\varphi$.\\footnote{Assuming the usual conventions when dividing by 0 or $\\infty$ and $-\\log 0 = \\infty$.}\n\\end{introtheorem}", "introthm:tameness": "\\begin{introtheorem}\\label{introthm:tameness} \n    Suppose $\\varphi\\colon \\A\\to\\A^*$ is a primitive substitution of length $k\\geq 2$. Then, the following are equivalent:\n    \\begin{enumerate}\n        \\item $\\ac(X_{\\varphi})\\in \\{0,1\\}$ (with $\\ac(X_{\\varphi})=0$ corresponding to $X_{\\varphi}$ being  finite),\n        \\item $X_{\\varphi}$ is null,\n        \\item $X_{\\varphi}$ is tame.\n    \\end{enumerate}\n\\end{introtheorem}"}, "pre_theorem_intro_text_len": 6568, "pre_theorem_intro_text": "\\label{sec:introduction}\n\nInspired by the seminal work \\cite{BourgainFremlinTalagrand1978} of Bourgain, Fremlin, and Talagrand, K\\\"ohler introduced in \\cite{Koehler1995} the notion of \\emph{tame} dynamical systems (originally she used the term \\emph{regular}).\nLater, by adopting one of the main results of \\cite{BourgainFremlinTalagrand1978} to the dynamical context, Glasner and Megrelishvili \\cite{GlasnerMegerlishvili2006} provided the following dichotomy: the enveloping semigroup of \\emph{any} system is either ``large'' (containing the Stone--\\v{C}ech compactification of the natural numbers) or ``small'' (with its topology determined by sequential convergence), the latter case corresponding to the system being tame.\n\nA particularly useful alternative characterization of tameness is provided by the theory of topological independence \\cite{KerrLi2007}. Here, the existence or absence of large independence sets forces high or low dynamical complexity. For example, positive topological entropy of a system $T:X\\to X$ is tied to the independence of iterates  $T^n$, $n\\in\\Z$ along subsets of positive density. At a weaker level, the absence of infinite independence sets is linked to tameness, while \\emph{nullness} in the sense of Goodman \\cite{Goodman} (i.e.\\ zero topological sequence entropy) requires the lack of arbitrarily large finite independence sets, see Section \\ref{sec:tamenull} or \\cite{KerrLi2007} for precise definitions and further information.\n\nSince their introduction—and in particular in recent years—tame and null systems have been widely studied. This research has focused on characterizing their structural properties in various ways and developing more directly applicable criteria for identifying tame or null systems.\nFor more information on tame systems, see for instance \\cite{Glasner2006,GlasnerMegerlishvili2006,Huang2006,Glasner2007,Glasner2017,GlasnerMegrelishvili2018,FuhrmannKwietniak2020,FuhrmannGlasnerJagerOertel2021,GlasnerMegrelishvili2022,FuhrmannKellendonkYassawi2024,Kellendonk2025}, and for null systems, see for example \\cite{HuangLiShaoYe2003,MaassShao2007,HuangYe2009,GarciaRamos2017,FuhrmannKwietniak2020,QiuZhao2020,Leonard2025,LePavlovSchlortt2025}.\n\nAs part of these recent developments, an important focus has been on establishing concrete and computationally tractable conditions to identify tame and null systems within canonical classes of dynamical systems. \nIn this article, we provide a complete characterisation of tameness and nullness for the well-studied class of minimal automatic dynamical systems \\cite{Fogg2002,bookQueffelec} in terms of a single numerical invariant. \nSuch systems can be equivalently described as those that are conjugate to a subshift generated by a primitive, constant length substitution (see Sections \\ref{sec:substitution} and \\ref{sec:subsstsems}).\n\nThe numerical quantity we will utilize is the notion of amorphic complexity, a relatively new topological invariant introduced for the study of zero entropy systems \\cite{FuhrmannGroegerJaeger2016}. It is tailor-made for analysing minimal systems with discrete spectrum and continuous eigenfunctions \\cite{FuhrmannGroeger2020}. Beyond $\\Z$-actions, this class encompasses dynamical systems canonically induced by mathematical quasicrystals, such as Penrose and chair tilings. In recent work \\cite{FuhrmannGroegerJaegerKwietniak2023}, amorphic complexity was extended to these more general group actions; the authors of \\cite{BaakeGaehlerGohlke2025} used this to calculate amorphic complexity for some prominent examples, including the newly discovered Hat tiling \\cite{SmithMyersKaplanGoodman-Strauss2024}, see also Remarks \\ref{rem:inf_tiling_and_beyond_mean_eq} and \\ref{rem:Baakeresult}. Very recently, \\cite{Gaehler2026} provided numerical evidence that amorphic complexity, together with another finitary invariant—asymptotic composants—is strong enough to completely distinguish minimal one-dimensional inflation tilings with pure-point spectrum for a class of small ternary Pisot unit inflation factors.\nFurthermore, building on the ideas of \\cite{FuhrmannGroegerJaeger2016}, the authors of \\cite{KasjanKeler2025} were able to distinguish between several Toeplitz subshifts associated with $\\mathcal{B}$-free systems—subshifts that were previously indistinguishable using classical topological invariants.\n\nBefore stating our main results, we need to introduce some notation: we first fix an \\emph{alphabet}, i.e.\\ a finite set $\\mathcal{A}$. A \\emph{substitution} $\\varphi$ is a map that assigns to each letter $a \\in \\mathcal{A}$ a finite word in $\\mathcal{A}^*$, the set of all finite words over $\\mathcal{A}$. It is said to be of \\emph{(constant) length} $k \\geqslant 2$ if the word $\\varphi(a)$ has length $k$ for all $a \\in \\mathcal{A}$. By concatenation, $\\varphi$ naturally extends to a map on $\\mathcal{A}^*$. We call $\\varphi$ \\emph{primitive} if there exists an $n \\geqslant 1$ such that, for each $a \\in \\mathcal{A}$, the word $\\varphi^n(a)$ contains every letter of $\\mathcal{A}$. \n\nTo each substitution $\\varphi$, we can associate a canonical subset $X_{\\varphi}\\subset \\A^{\\Z}$ of the space $\\A^{\\Z}$  of all bi-infinite sequences with symbols in $\\mathcal{A}$. Specifically, $X_{\\varphi}$ is the collection of all $x \\in \\mathcal{A}^\\Z$ such that every finite word appearing in $x$ also appears in $\\varphi^n(a)$ for some $a \\in \\mathcal{A}$ and $n \\geqslant 0$. Note that there is a natural action of the \\emph{(left) shift} $T$ on $X_{\\varphi}$, defined by\n$T((x_n)_{n \\in \\Z}) = (x_{n+1})_{n \\in \\Z}$ for all $(x_n)_{n \\in \\Z} \\in X_{\\varphi}$.\nIn the following, when we say that $X_\\varphi$ is tame or null, we mean that the \\emph{dynamical system} $(X_\\varphi, T)$ is tame or null, respectively.\n\nFor the sake of simplicity, we introduce amorphic complexity here only in the symbolic context, where it admits a direct fractal geometric interpretation. The general definition is provided in Section~\\ref{sec:asymp sep numbers and ac}, along with a brief overview of some of its basic properties as a topological invariant. For a symbolic system, its \\emph{upper} and \\emph{lower amorphic complexity} can be defined simply as the lower and upper box dimension, respectively, of its maximal equicontinuous factor equipped with the Besicovitch (pseudo)metric (see Remark \\ref{rem:different_version_Besicovitch}). While these two values need not coincide in general, we will see next that for a minimal automatic system $X_\\varphi$ they do, and we denote their common value by $\\ac(X_\\varphi)$.", "context": "Inspired by the seminal work \\cite{BourgainFremlinTalagrand1978} of Bourgain, Fremlin, and Talagrand, K\\\"ohler introduced in \\cite{Koehler1995} the notion of \\emph{tame} dynamical systems (originally she used the term \\emph{regular}).\nLater, by adopting one of the main results of \\cite{BourgainFremlinTalagrand1978} to the dynamical context, Glasner and Megrelishvili \\cite{GlasnerMegerlishvili2006} provided the following dichotomy: the enveloping semigroup of \\emph{any} system is either ``large'' (containing the Stone--\\v{C}ech compactification of the natural numbers) or ``small'' (with its topology determined by sequential convergence), the latter case corresponding to the system being tame.\n\nA particularly useful alternative characterization of tameness is provided by the theory of topological independence \\cite{KerrLi2007}. Here, the existence or absence of large independence sets forces high or low dynamical complexity. For example, positive topological entropy of a system $T:X\\to X$ is tied to the independence of iterates $T^n$, $n\\in\\Z$ along subsets of positive density. At a weaker level, the absence of infinite independence sets is linked to tameness, while \\emph{nullness} in the sense of Goodman \\cite{Goodman} (i.e.\\ zero topological sequence entropy) requires the lack of arbitrarily large finite independence sets, see Section \\ref{sec:tamenull} or \\cite{KerrLi2007} for precise definitions and further information.\n\nAs part of these recent developments, an important focus has been on establishing concrete and computationally tractable conditions to identify tame and null systems within canonical classes of dynamical systems. \nIn this article, we provide a complete characterisation of tameness and nullness for the well-studied class of minimal automatic dynamical systems \\cite{Fogg2002,bookQueffelec} in terms of a single numerical invariant. \nSuch systems can be equivalently described as those that are conjugate to a subshift generated by a primitive, constant length substitution (see Sections \\ref{sec:substitution} and \\ref{sec:subsstsems}).\n\nBefore stating our main results, we need to introduce some notation: we first fix an \\emph{alphabet}, i.e.\\ a finite set $\\mathcal{A}$. A \\emph{substitution} $\\varphi$ is a map that assigns to each letter $a \\in \\mathcal{A}$ a finite word in $\\mathcal{A}^*$, the set of all finite words over $\\mathcal{A}$. It is said to be of \\emph{(constant) length} $k \\geqslant 2$ if the word $\\varphi(a)$ has length $k$ for all $a \\in \\mathcal{A}$. By concatenation, $\\varphi$ naturally extends to a map on $\\mathcal{A}^*$. We call $\\varphi$ \\emph{primitive} if there exists an $n \\geqslant 1$ such that, for each $a \\in \\mathcal{A}$, the word $\\varphi^n(a)$ contains every letter of $\\mathcal{A}$.\n\nTo each substitution $\\varphi$, we can associate a canonical subset $X_{\\varphi}\\subset \\A^{\\Z}$ of the space $\\A^{\\Z}$ of all bi-infinite sequences with symbols in $\\mathcal{A}$. Specifically, $X_{\\varphi}$ is the collection of all $x \\in \\mathcal{A}^\\Z$ such that every finite word appearing in $x$ also appears in $\\varphi^n(a)$ for some $a \\in \\mathcal{A}$ and $n \\geqslant 0$. Note that there is a natural action of the \\emph{(left) shift} $T$ on $X_{\\varphi}$, defined by\n$T((x_n)_{n \\in \\Z}) = (x_{n+1})_{n \\in \\Z}$ for all $(x_n)_{n \\in \\Z} \\in X_{\\varphi}$.\nIn the following, when we say that $X_\\varphi$ is tame or null, we mean that the \\emph{dynamical system} $(X_\\varphi, T)$ is tame or null, respectively.\n\nFor the sake of simplicity, we introduce amorphic complexity here only in the symbolic context, where it admits a direct fractal geometric interpretation. The general definition is provided in Section~\\ref{sec:asymp sep numbers and ac}, along with a brief overview of some of its basic properties as a topological invariant. For a symbolic system, its \\emph{upper} and \\emph{lower amorphic complexity} can be defined simply as the lower and upper box dimension, respectively, of its maximal equicontinuous factor equipped with the Besicovitch (pseudo)metric (see Remark \\ref{rem:different_version_Besicovitch}). While these two values need not coincide in general, we will see next that for a minimal automatic system $X_\\varphi$ they do, and we denote their common value by $\\ac(X_\\varphi)$.\n\n\\begin{remark}\\label{rem:different_version_Besicovitch}\n    For a fixed partition $\\mathcal{P}$ we denote by $\\fp$ the full set  \n    $\\fp = \\mathcal{P} \\ast \\mathcal{P} \\setminus \\mathcal{P}$  \n    of distinct unordered pairs of elements of $\\mathcal{P}$. With this convention,  \n    the pseudometric $D_{\\fp}$ defined in \\eqref{eq:def_of_pseudo} corresponds to  \n    the standard \\emph{Besicovitch pseudometric}\\footnote{Often, this term also  \n    refers to the pseudometric defined in \\eqref{def:Besicovitch_pseudometric}.  \n    In the symbolic context, $D_{\\fp}$ and $D_B$ are uniformly equivalent  \n    (see, for instance, the proof of Proposition 6.8 in \\cite{FuhrmannGroeger2020})  \n    but, in general, they do not need to be Lipschitz equivalent.} on a subshift.  \n    Clearly, $D_{\\fp}$ coincides with $D_\\delta$ from  \n    \\eqref{eq:besicovitch metric and density of discrepancy} for $\\delta=1$ and\n    for any $\\brt \\subseteq (\\mathcal{P} \\ast \\mathcal{P}) \\backslash \\mathcal{P}$,  \n    we have  \n    \\[\n        D_{\\brt}(x,y) \\leq D_{\\fp}(x,y),\n    \\]\n    for all $x,y \\in X$.\n\\end{remark}", "full_context": "Inspired by the seminal work \\cite{BourgainFremlinTalagrand1978} of Bourgain, Fremlin, and Talagrand, K\\\"ohler introduced in \\cite{Koehler1995} the notion of \\emph{tame} dynamical systems (originally she used the term \\emph{regular}).\nLater, by adopting one of the main results of \\cite{BourgainFremlinTalagrand1978} to the dynamical context, Glasner and Megrelishvili \\cite{GlasnerMegerlishvili2006} provided the following dichotomy: the enveloping semigroup of \\emph{any} system is either ``large'' (containing the Stone--\\v{C}ech compactification of the natural numbers) or ``small'' (with its topology determined by sequential convergence), the latter case corresponding to the system being tame.\n\nA particularly useful alternative characterization of tameness is provided by the theory of topological independence \\cite{KerrLi2007}. Here, the existence or absence of large independence sets forces high or low dynamical complexity. For example, positive topological entropy of a system $T:X\\to X$ is tied to the independence of iterates $T^n$, $n\\in\\Z$ along subsets of positive density. At a weaker level, the absence of infinite independence sets is linked to tameness, while \\emph{nullness} in the sense of Goodman \\cite{Goodman} (i.e.\\ zero topological sequence entropy) requires the lack of arbitrarily large finite independence sets, see Section \\ref{sec:tamenull} or \\cite{KerrLi2007} for precise definitions and further information.\n\nAs part of these recent developments, an important focus has been on establishing concrete and computationally tractable conditions to identify tame and null systems within canonical classes of dynamical systems. \nIn this article, we provide a complete characterisation of tameness and nullness for the well-studied class of minimal automatic dynamical systems \\cite{Fogg2002,bookQueffelec} in terms of a single numerical invariant. \nSuch systems can be equivalently described as those that are conjugate to a subshift generated by a primitive, constant length substitution (see Sections \\ref{sec:substitution} and \\ref{sec:subsstsems}).\n\nBefore stating our main results, we need to introduce some notation: we first fix an \\emph{alphabet}, i.e.\\ a finite set $\\mathcal{A}$. A \\emph{substitution} $\\varphi$ is a map that assigns to each letter $a \\in \\mathcal{A}$ a finite word in $\\mathcal{A}^*$, the set of all finite words over $\\mathcal{A}$. It is said to be of \\emph{(constant) length} $k \\geqslant 2$ if the word $\\varphi(a)$ has length $k$ for all $a \\in \\mathcal{A}$. By concatenation, $\\varphi$ naturally extends to a map on $\\mathcal{A}^*$. We call $\\varphi$ \\emph{primitive} if there exists an $n \\geqslant 1$ such that, for each $a \\in \\mathcal{A}$, the word $\\varphi^n(a)$ contains every letter of $\\mathcal{A}$.\n\nTo each substitution $\\varphi$, we can associate a canonical subset $X_{\\varphi}\\subset \\A^{\\Z}$ of the space $\\A^{\\Z}$ of all bi-infinite sequences with symbols in $\\mathcal{A}$. Specifically, $X_{\\varphi}$ is the collection of all $x \\in \\mathcal{A}^\\Z$ such that every finite word appearing in $x$ also appears in $\\varphi^n(a)$ for some $a \\in \\mathcal{A}$ and $n \\geqslant 0$. Note that there is a natural action of the \\emph{(left) shift} $T$ on $X_{\\varphi}$, defined by\n$T((x_n)_{n \\in \\Z}) = (x_{n+1})_{n \\in \\Z}$ for all $(x_n)_{n \\in \\Z} \\in X_{\\varphi}$.\nIn the following, when we say that $X_\\varphi$ is tame or null, we mean that the \\emph{dynamical system} $(X_\\varphi, T)$ is tame or null, respectively.\n\nFor the sake of simplicity, we introduce amorphic complexity here only in the symbolic context, where it admits a direct fractal geometric interpretation. The general definition is provided in Section~\\ref{sec:asymp sep numbers and ac}, along with a brief overview of some of its basic properties as a topological invariant. For a symbolic system, its \\emph{upper} and \\emph{lower amorphic complexity} can be defined simply as the lower and upper box dimension, respectively, of its maximal equicontinuous factor equipped with the Besicovitch (pseudo)metric (see Remark \\ref{rem:different_version_Besicovitch}). While these two values need not coincide in general, we will see next that for a minimal automatic system $X_\\varphi$ they do, and we denote their common value by $\\ac(X_\\varphi)$.\n\n\\begin{remark}\\label{rem:different_version_Besicovitch}\n    For a fixed partition $\\mathcal{P}$ we denote by $\\fp$ the full set  \n    $\\fp = \\mathcal{P} \\ast \\mathcal{P} \\setminus \\mathcal{P}$  \n    of distinct unordered pairs of elements of $\\mathcal{P}$. With this convention,  \n    the pseudometric $D_{\\fp}$ defined in \\eqref{eq:def_of_pseudo} corresponds to  \n    the standard \\emph{Besicovitch pseudometric}\\footnote{Often, this term also  \n    refers to the pseudometric defined in \\eqref{def:Besicovitch_pseudometric}.  \n    In the symbolic context, $D_{\\fp}$ and $D_B$ are uniformly equivalent  \n    (see, for instance, the proof of Proposition 6.8 in \\cite{FuhrmannGroeger2020})  \n    but, in general, they do not need to be Lipschitz equivalent.} on a subshift.  \n    Clearly, $D_{\\fp}$ coincides with $D_\\delta$ from  \n    \\eqref{eq:besicovitch metric and density of discrepancy} for $\\delta=1$ and\n    for any $\\brt \\subseteq (\\mathcal{P} \\ast \\mathcal{P}) \\backslash \\mathcal{P}$,  \n    we have  \n    \\[\n        D_{\\brt}(x,y) \\leq D_{\\fp}(x,y),\n    \\]\n    for all $x,y \\in X$.\n\\end{remark}\n\nFor the sake of simplicity, we introduce amorphic complexity here only in the symbolic context, where it admits a direct fractal geometric interpretation. The general definition is provided in Section~\\ref{sec:asymp sep numbers and ac}, along with a brief overview of some of its basic properties as a topological invariant. For a symbolic system, its \\emph{upper} and \\emph{lower amorphic complexity} can be defined simply as the lower and upper box dimension, respectively, of its maximal equicontinuous factor equipped with the Besicovitch (pseudo)metric (see Remark \\ref{rem:different_version_Besicovitch}). While these two values need not coincide in general, we will see next that for a minimal automatic system $X_\\varphi$ they do, and we denote their common value by $\\ac(X_\\varphi)$.\n\nIn particular, Theorem~\\ref{introthm:tameness} yields that a system $X_{\\varphi}$ is null iff it is tame (recall that nullness always implies tameness); to the best of our knowledge, this is a new result even in case of a binary alphabet. Furthermore, at the end of the introduction of \\cite{FuhrmannKellendonkYassawi2024}, it is pointed out that \\cite[Thm.~1.2]{FuhrmannKellendonkYassawi2024} together with \\cite[Thm.~1.1]{FuhrmannGroeger2020} yields the equivalence of (i) and (iii) in the binary case. This means that, concerning this equivalence, Theorem~\\ref{introthm:tameness} extends this observation to the full general case in the minimal setting.\n\n\\begin{lemma}\\cite[Lem.\\ 9]{DurandRigo2009}\\label{lem:growth_of_letters}\\mbox{}\n\\begin{enumerate}\n\\item\\label{lem:growth_of_letters1} For a nonnegative matrix $A$, \n\\begin{equation}\\label{eq:Mfor}\n||A^n||_1 = \\bigT (\\lambda^n n^d),\n\\end{equation} \nwhere $\\lambda$ is the dominant eigenvalue of $A$ and $d\\in\\N$.\n\\item Let $\\varphi\\colon\\mathcal{A}\\to\\mathcal{A}^{*}$ be a substitution, and let $m\\geq 1$ be such that $\\varphi^m$ is in normal primitive form. \n For each nonerasing letter $a\\in\\mathcal{A}$ there exist unique algebraic integer $\\lambda(a)\\geq 0$, integer $d(a)\\geq 0$, and $c(a)>0$ such that\n\\begin{equation}\\label{eq:growthtype}\n\\abs{\\varphi^{mn}(a)} \\sim c(a)\\cdot(mn)^{d(a)}\\cdot\\lambda(a)^{mn}.\n\\end{equation}\n If $a\\in\\A$ is erasing we put $\\lambda(a)=0$, $d(a)=1$, $c(a)=1$; in this case $\\abs{\\varphi^n(a)}=0$ for all $n$ large enough.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proposition}\\label{prop:sepnumberproperties}\n Let $\\varphi$ be a primitive substitution of length $k\\geq 2$ and let $\\lambda_s$ be its discrepancy rate.\n The following are true:\n\\begin{enumerate}\n\\item\\label{prop:sepnumberproperties1} $\\lambda_s = 0$ or $1\\leq \\lambda_s \\leq k$,\n\\item\\label{prop:sepnumberproperties2} $\\lambda_s = 0$ if and only if $X_{\\varphi}$ is finite,\n\\item\\label{prop:sepnumberproperties3} $\\lambda_s =k$ if and only if $X_{\\varphi}$ does not have discrete spectrum if and only if it is not mean equicontinuous.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\n The first claim follows from the fact that $\\lambda_s$ is the dominant eigenvalue of a nonnegative integer matrix.\n\n\\begin{lemma}\\label{lem:proportion_of_ourmetric_to_Besicovitch}\nLet $\\varphi\\colon \\A \\to \\A^*$ be a primitive substitution of length $k\\geq 2$ and height 1. Then\n\\begin{enumerate}\n\\item\\label{lem:proportion_of_ourmetric_to_Besicovitch1} $D_{\\mg}(x,y)=0$ if and only if $D_{\\mg}(\\varphi(x),\\varphi(y))=0$ for any $x,y\\in X_{\\varphi}$,\n\\item\\label{lem:proportion_of_ourmetric_to_Besicovitch2} for any $x,y\\in X_{\\varphi}$ with $D_{\\fp}(x,y)>0$ and $n$ big enough we have\n\\begin{equation}\\label{eq:proportion_of_ourmetric_to_Besicovitch}\n\\frac{D_{\\mg}(\\varphi^n(x),\\varphi^n(y))}{D_{\\fp}(\\varphi^n(x),\\varphi^n(y))}\\geq\n\\frac{D_{\\mg}(x,y)}{D_{\\fp}(x,y)}.\n\\end{equation} \n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n Property \\eqref{lem:proportion_of_ourmetric_to_Besicovitch1} is clear, since for any distinct pair of letters $\\alpha={a\\choose b}$ of not maximal growth (i.e.\\ lying in $\\A_s\\setminus \\mg$), all (pairs of) letters appearing in $\\varphi_s(\\alpha)$ are again of not maximal growth and for any distinct pair of letters $\\alpha={a\\choose b}$ of maximal growth, $\\varphi(\\alpha)$ contains at least one appearance of a (pair of) letters of maximal growth.\n\n\\begin{proposition}\\label{prop:simplekernelstructure}\n Let $\\varphi\\colon\\A\\to\\A^*$ be a primitive substitution of length $k\\geq 2$ and height 1. Let $x$ be a fixed point of $\\varphi$.\n We have \n \\begin{equation}\\label{eq:growthdm}\n \\d_m(k,x) = \\bigT (m^{d_{s}}\\lambda_s^m),\n \\end{equation} \n where $(\\lambda_s,d_{s})$ is the discrepancy type of $\\varphi$.\n Furthermore, assuming $x$ is not periodic, the following are equivalent:\n \\begin{enumerate}\n \\item\\label{prop:simplekernelstructure1} $x$ has simple $k$-kernel structure,\n \\item\\label{prop:simplekernelstructure2} $\\lambda_s=1$, \n \\item\\label{prop:simplekernelstructure3} for each nonconstant $y\\in \\Ker_k(x)$ and $m\\geq 1$, $y$ appears at most once in $y$ as $k^m$-$\\AP$,\n \\item\\label{prop:simplekernelstructure4} for each $\\mathcal{A}'\\in \\C(\\varphi)$ with $\\abs{\\A'}\\geq 2$ and each $m\\geq 1$ there exists at most one $0\\leq i<k^m$ such that $\\varphi^m_i(\\mathcal{A}')=\\mathcal{A}'$.\n\\end{enumerate} \n \\end{proposition}\n \\begin{proof}\n\n\\begin{remark}\\label{rem:graph}\n In \\cite{CovenQuasYassawi}, the authors associate with each primitive substitution $\\varphi\\colon\\A\\to\\A^*$ of length $k\\geq 2$ a certain directed graph $\\mathcal G_\\varphi$. \n If $\\varphi$ has height 1, the graph $\\mathcal G_\\varphi$ is defined as follows: its vertices are given by $\\C(\\varphi)$ and there is an oriented edge from $\\mathcal{B}$ to $\\mathcal{C}$ if $\\varphi_j(\\mathcal{B})=\\mathcal{C}$ for some $0\\leq j<k$. If $\\varphi$ does not have height 1, then $\\mathcal G_\\varphi$ is defined as the graph of its pure base.\n It is straightforward to note that Condition \\eqref{prop:simplekernelstructure4} in Proposition \\ref{prop:kernelandnullness} corresponds to the fact that $\\mathcal G_{\\varphi}$ does not contain two distinct cycles which share a common vertex. \n In \\cite[Thm.\\ 2.2]{FuhrmannKellendonkYassawi2024} it is shown that this condition precisely characterises systems $X_{\\varphi}$ that are tame. \n \\end{remark}\n\n\\begin{theorem}\\label{thm:tameness} \nLet $\\varphi\\colon \\A\\to\\A^*$ be a primitive substitution of length $k\\geq 2$ and assume that $X_{\\varphi}$ is infinite. The following are equivalent:\n\\begin{enumerate}\n\\item\\label{thm:tameness1} $\\ac(X_{\\varphi})=1$.\n\\item\\label{thm:tameness2} $X_{\\varphi}$ is null,\n\\item\\label{thm:tameness3} $X_{\\varphi}$ is tame.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}", "post_theorem_intro_text_len": 3724, "post_theorem_intro_text": "In particular, Theorem~\\ref{introthm:tameness} yields that a system $X_{\\varphi}$ is null iff it is tame (recall that nullness always implies tameness); to the best of our knowledge, this is a new result even in case of a binary alphabet. Furthermore, at the end of the introduction of \\cite{FuhrmannKellendonkYassawi2024}, it is pointed out that \\cite[Thm.~1.2]{FuhrmannKellendonkYassawi2024} together with \\cite[Thm.~1.1]{FuhrmannGroeger2020} yields the equivalence of (i) and (iii) in the binary case. This means that, concerning this equivalence, Theorem~\\ref{introthm:tameness} extends this observation to the full general case in the minimal setting.\n\nTheorem~\\ref{introthm:tameness} offers a natural interpretation of tameness and nullness in the context of minimal automatic systems. As shown in \\cite[Cor.~1.3]{FuhrmannGroeger2020}, amorphic complexity of such systems is either $0$ or lies in the interval $[1, \\infty]$. Furthermore, it is straightforward to demonstrate that amorphic complexity can attain a dense set of values in $[1, \\infty)$ in the class of minimal automatic systems over a fixed alphabet of size at least two (see Proposition~\\ref{prop:denseness}). Beyond the trivial case where amorphic complexity is $0$ (corresponding to periodic behaviour), this reveals a complexity gap between $0$ and $1$, with tame and null systems occupying exactly the minimal nontrivial complexity level of $1$.\n\nTo establish Theorem \\ref{introthm:tameness} we need to determine amorphic complexity for arbitrary minimal automatic systems.\nFor this, to each primitive substitution $\\varphi$ of constant length we associate its \\emph{discrepancy substitution} $\\varphi_s$ and define the \\emph{discrepancy rate} $\\lambda_s$ of $\\varphi$ as the dominant eigenvalue of the incidence matrix of $\\varphi_s$ (see Section \\ref{sec:discrepancysubstitution}). \nThe discrepancy substitution $\\varphi_s$ is straightforward to compute; however, in general, it is not primitive, which makes its analysis considerably more delicate.\nFor this reason, we introduce suitable dynamically defined pseudometrics that allow for a refined analysis of the discrepancy structure.\nWith this we derive a closed formula for the amorphic complexity of any minimal automatic systems, showing, in particular, that it always exists.\n\n\\begin{introtheorem}\\label{introthm:ac_of_automatic_systems} Let $\\varphi$ be a primitive substitution of length $k\\geqslant 2$.  The amorphic complexity of $X_{\\varphi}$ is given by\n\\[\\ac(X_{\\varphi})=\\frac{\\log k}{\\log k - \\log \\lambda_s},\\]\nwhere $\\lambda_s$ is the discrepancy rate of $\\varphi$.\\footnote{Assuming the usual conventions when dividing by 0 or $\\infty$ and $-\\log 0 = \\infty$.}\n\\end{introtheorem}\n\nWe note that in case of a binary alphabet, Theorem~\\ref{introthm:ac_of_automatic_systems} coincides with \\cite[Thm.~1.1]{FuhrmannGroeger2020}. Moreover, for general alphabets, \\cite{FuhrmannGroeger2020} also provides a potential algorithm to compute concrete lower and upper bounds for the lower and upper amorphic complexity of $X_\\varphi$, whereas \\cite{BaakeGaehlerGohlke2025} gives improved lower and upper bounds; see also Remark~\\ref{rem:Baakeresult}.\n\nThe paper is organized as follows. In Section~\\ref{sec:setup} we collect the necessary preliminaries. In Section~\\ref{sec:discrepancytools} we introduce the discrepancy substitution and new dynamically generated pseudometrics, which serve as the main technical tools throughout the paper. In Section~\\ref{sec:complexityofsubstitution} we prove Theorem~\\ref{introthm:ac_of_automatic_systems} concerning the formula for the amorphic complexity. Finally, in Section~\\ref{sec:tamenessandnullness} we establish Theorem~\\ref{introthm:tameness}.", "sketch": "To establish Theorem~\\ref{introthm:tameness} the authors first “need to determine amorphic complexity for arbitrary minimal automatic systems.” For this, “to each primitive substitution $\\varphi$ of constant length we associate its \\emph{discrepancy substitution} $\\varphi_s$ and define the \\emph{discrepancy rate} $\\lambda_s$ of $\\varphi$ as the dominant eigenvalue of the incidence matrix of $\\varphi_s$.” Although “the discrepancy substitution $\\varphi_s$ is straightforward to compute,” it “is not primitive,” so its analysis is “considerably more delicate.” To handle this they “introduce suitable dynamically defined pseudometrics that allow for a refined analysis of the discrepancy structure.” With these tools they “derive a closed formula for the amorphic complexity of any minimal automatic systems, showing, in particular, that it always exists,” namely Theorem~\\ref{introthm:ac_of_automatic_systems} with\n\\[\\ac(X_{\\varphi})=\\frac{\\log k}{\\log k - \\log \\lambda_s}.\\]\nFinally, “in Section~\\ref{sec:tamenessandnullness} we establish Theorem~\\ref{introthm:tameness},” i.e., the equivalence between “$\\ac(X_{\\varphi})\\in\\{0,1\\}$,” “$X_{\\varphi}$ is null,” and “$X_{\\varphi}$ is tame,” using the amorphic-complexity determination above.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{introthm:tameness} \n    Suppose $\\varphi\\colon \\A\\to\\A^*$ is a primitive substitution of length $k\\geqslant 2$. Then, the following are equivalent:\n    \\begin{enumerate}\n        \\item $\\ac(X_{\\varphi})\\in \\{0,1\\}$ (with $\\ac(X_{\\varphi})=0$ corresponding to $X_{\\varphi}$ being  finite),\n        \\item $X_{\\varphi}$ is null,\n        \\item $X_{\\varphi}$ is tame.\n    \\end{enumerate}", "theorem_type": ["Biconditional or Equivalence"], "mcq": {"question": "Let \\(\\varphi:\\mathcal A\\to\\mathcal A^*\\) be a primitive substitution of constant length \\(k\\ge 2\\) on a finite alphabet \\(\\mathcal A\\), and let \\(X_{\\varphi}\\subseteq \\mathcal A^{\\mathbb Z}\\) be the associated substitution subshift consisting of all bi-infinite sequences whose every finite block appears in some iterate \\(\\varphi^n(a)\\). Writing \\(\\operatorname{ac}(X_{\\varphi})\\) for the amorphic complexity of \\(X_{\\varphi}\\), which statement is equivalent to both of the properties \\(\"X_{\\varphi}\\) is null\" and \\(\"X_{\\varphi}\\) is tame\"?", "correct_choice": {"label": "A", "text": "\\(\\operatorname{ac}(X_{\\varphi})\\in\\{0,1\\}\\), with \\(\\operatorname{ac}(X_{\\varphi})=0\\) exactly when \\(X_{\\varphi}\\) is finite."}, "choices": [{"label": "B", "text": "\\(\\operatorname{ac}(X_{\\varphi})=1\\), with \\(\\operatorname{ac}(X_{\\varphi})=0\\) impossible for primitive substitutions of constant length."}, {"label": "C", "text": "\\(\\operatorname{ac}(X_{\\varphi})\\le 1\\)."}, {"label": "D", "text": "\\(\\operatorname{ac}(X_{\\varphi})\\in\\{0,1\\}\\), with \\(\\operatorname{ac}(X_{\\varphi})=1\\) exactly when \\(X_{\\varphi}\\) is finite."}, {"label": "E", "text": "\\(\\operatorname{ac}(X_{\\varphi})\\) is finite."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "finiteness", "tampered_component": "finite-case exclusion of ac=0", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped exact discrete range and finiteness characterization", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "reversal of the finite/ac=0 correspondence", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "discrete dichotomy replaced by mere boundedness", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the amorphic complexity characterization; it only asks for a statement equivalent to nullness and tameness. The correct answer is not stated or strongly hinted at beyond the general topic."}, "TAS": {"score": 0, "justification": "This is very close to a direct theorem-recall item: it asks for the statement equivalent to two named properties, and the correct option essentially reproduces the theorem’s exact characterization."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to reject weaker true statements like ac(X_φ) ≤ 1 and to track the finite-case exception, but the item mainly tests recognition of the known equivalence rather than substantial derivation."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one is weaker-but-true, one wrongly excludes the finite ac=0 case, one reverses the finite/ac correspondence, and one confuses exact classification with mere finiteness."}, "total_score": 5, "overall_assessment": "A solid theorem-recall MCQ with strong distractors and no answer leakage, but it is largely tautological and only moderately tests genuine generative reasoning."}}
{"id": "2602.23597v1", "paper_link": "http://arxiv.org/abs/2602.23597v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:general}\n    Let $\\beta$ be an algebraic number with $|\\beta|=1$ which is not a root of unity. Then $\\frac{\\operatorname{Arg}(\\beta)}{2\\pi} \\in \\left(-\\frac{1}{2},\\frac{1}{2}\\right]$ is a Diophantine number.", "start_pos": 8080, "end_pos": 8313, "label": "thm:general"}, "ref_dict": {"thm:general": "\\begin{theorem}\\label{thm:general}\n    Let $\\beta$ be an algebraic number with $|\\beta|=1$ which is not a root of unity. Then $\\frac{\\Arg(\\beta)}{2\\pi} \\in \\left(-\\frac{1}{2},\\frac{1}{2}\\right]$ is a Diophantine number.\n\\end{theorem}", "ineq:dioph": "\\begin{equation}\\label{ineq:dioph}\n    \\left| \\theta - \\frac{p}{q} \\right| \\ge \\frac{c}{q^\\tau}.\n\\end{equation}", "eq:main": "\\begin{equation}\\label{eq:main}\n    n \\tan \\alpha = \\tan(n \\alpha).\n\\end{equation}", "thm:3props": "\\begin{theorem}\\label{thm:3props}\n    Let $n \\ge 2$ be an integer and let $\\alpha \\in \\left( 0, \\frac{\\pi}{2} \\right)$ be a solution of \\eqref{eq:main}. Then the following statements hold.\n    \\begin{enumerate}[label=(\\alph*)]\n        \\item $\\frac{\\alpha}{2\\pi}$ is a Diophantine number.\n        \\item $\\frac{\\alpha}{2\\pi}$ is transcendental.\n        \\item $\\alpha$ is transcendental.\n    \\end{enumerate}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 5205, "pre_theorem_intro_text": "Let $\\theta$ be a real number. The {\\em irrationality exponent} of $\\theta$ is \n$$\\mu(\\theta) = \\sup\\left\\{ \\mu > 0 \\colon \\left| \\theta - \\frac{p}{q} \\right| < \\frac{1}{q^\\mu} \\text{ holds for infinitely many pairs } (p,q) \\in \\mathbb{Z} \\times \\mathbb{N} \\right\\}.$$\nWe say that $\\theta$ is a {\\em Liouville number} whenever $\\mu(\\theta) = \\infty$. On the other hand, $\\theta$ is called {\\em Diophantine} if there exist $c > 0$ and $\\tau < \\infty$ such that, for every $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$, \n\\begin{equation}\\label{ineq:dioph}\n    \\left| \\theta - \\frac{p}{q} \\right| \\ge \\frac{c}{q^\\tau}.\n\\end{equation}\nLet $\\mathcal L$ be the set of Liouville numbers and let $\\mathcal D$ be the set of Diophantine numbers.\n\nWe observe that if $\\theta = \\frac{a}{b}$ is rational, then $\\mu(\\theta) = 1$, and one may choose $\\frac{p}{q} = \\frac{a}{b}$ so that \\eqref{ineq:dioph} fails to hold. Hence, no rational number is either Diophantine or Liouville. Among irrational numbers, the sets $\\mathcal L$ and $\\mathcal D$ form a partition.\n\nA very strong result due to Roth \\cite{Roth1955} states that $\\mu(\\theta) = 2$ for every algebraic number $\\theta$. In particular, every Liouville number must be transcendental, or equivalently, every algebraic irrational number is Diophantine.\n\nArithmetic properties such as rationality or irrationality, algebraicity or transcendence,\nand Diophantine or Liouville behavior play a fundamental role in several areas of mathematics. They arise naturally in Diophantine approximation, transcendence theory, and number theory, but also have significant implications in dynamics, geometry, and analysis.\n\nIn many contexts, good arithmetic properties govern rigidity versus flexibility phenomena and the qualitative behavior of dynamical systems. In what follows, we introduce a simple trigonometric equation whose solutions exhibit\nremarkable arithmetic features and give rise to several geometric and dynamical implications,\nwhich will be discussed in the subsequent paragraphs.\n\nLet $n \\in \\mathbb{Z}$ and consider the equation \n\\begin{equation}\\label{eq:main}\n    n \\tan \\alpha = \\tan(n \\alpha).\n\\end{equation}\nIf $n\\in \\{0,1\\}$, then every $\\alpha \\in \\mathbb{R}$ is a solution. Furthermore, $\\alpha = 0$ is a solution for every $n \\in \\mathbb{Z}$. Since the tangent function is odd and $\\pi$-periodic, we may restrict attention to the case $n \\ge 2$ and $\\alpha \\in \\left(0,\\frac{\\pi}{2}\\right)$. In \\cite{Cyr2012}, Cyr proved that $\\frac{\\alpha}{2\\pi}$ is irrational.\n\nThis equation appears naturally in several topics involving plane curves and billiard dynamics. Despite its elementary appearance, its solutions encode subtle rigidity phenomena with both geometric and dynamical consequences.\n\nOne prominent context in which this equation appears is the {\\em theory of bicycle curves}. In an effort to classify noncircular bicycle curves, Tabachnikov \\cite{Tabachnikov2006} proved that the circle admits a nontrivial infinitesimal deformation as a smooth plane bicycle curve with rotation number $\\frac{\\alpha}{\\pi}$ if and only if \\eqref{eq:main} holds for some integer $n \\ge 2$. Thus, the solutions of this equation characterize the values of the rotation number for which the circle fails to be rigid in this geometric setting. In light of the Cyr theorem, it follows that the circle is rigid as a bicycle curve for any rational rotation number, except in the case $\\alpha = \\frac{\\pi}{2}$, where any curve of constant width is a bicycle curve.\n\nThe same equation arises independently in the context of {\\em mathematical billiards}. In his study of the dynamics of the billiard map inside the cross section of an infinite cylinder that floats in neutral equilibrium in any orientation, Gutkin \\cite{Gutkin2010} showed that for a regular, noncircular billiard table with boundary $\\Gamma$, the existence of a constant-angle caustic imposes the arithmetic constraint that there exist an integer $n \\ge 2$ and a real parameter $\\alpha$ satisfying \\eqref{eq:main}. Moreover, under additional assumptions, the arithmetic nature of $\\frac{\\alpha}{\\pi}$ plays a decisive role in determining whether the induced billiard dynamics on the caustic is rational or irrational.\n\nMore recently, equations of the form \\eqref{eq:main} have also appeared in the study of {\\em self-Bäcklund curves}. In this setting, the existence of non-trivial infinitesimal deformations of central conics as a self-Bäcklund is again governed by the solvability of \\eqref{eq:main} for certain values of $n \\ge 4$ (see \\cite[Section~3]{BialyBorTabachnikov2022}).\n\nThe examples discussed above show that arithmetic properties of parameters appearing in geometric and dynamical problems can strongly influence their behavior. This perspective motivates the present work, which focuses on the arithmetic properties of real numbers arising in such contexts.\n\nBefore stating the main results of this paper, we introduce some additional notation. Let $z \\in \\mathbb{C} \\setminus \\{0\\}$. We denote by $\\operatorname{Log} z = \\ln|z| + i\\operatorname{Arg}(z)$ the principal branch of the logarithm, where $\\operatorname{Arg}(z) \\in (-\\pi,\\pi]$. \nThe main results of this paper are the following.", "context": "Let $\\theta$ be a real number. The {\\em irrationality exponent} of $\\theta$ is \n$$\\mu(\\theta) = \\sup\\left\\{ \\mu > 0 \\colon \\left| \\theta - \\frac{p}{q} \\right| < \\frac{1}{q^\\mu} \\text{ holds for infinitely many pairs } (p,q) \\in \\mathbb{Z} \\times \\mathbb{N} \\right\\}.$$\nWe say that $\\theta$ is a {\\em Liouville number} whenever $\\mu(\\theta) = \\infty$. On the other hand, $\\theta$ is called {\\em Diophantine} if there exist $c > 0$ and $\\tau < \\infty$ such that, for every $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$, \n\\begin{equation}\\label{ineq:dioph}\n \\left| \\theta - \\frac{p}{q} \\right| \\ge \\frac{c}{q^\\tau}.\n\\end{equation}\nLet $\\mathcal L$ be the set of Liouville numbers and let $\\mathcal D$ be the set of Diophantine numbers.\n\nWe observe that if $\\theta = \\frac{a}{b}$ is rational, then $\\mu(\\theta) = 1$, and one may choose $\\frac{p}{q} = \\frac{a}{b}$ so that \\eqref{ineq:dioph} fails to hold. Hence, no rational number is either Diophantine or Liouville. Among irrational numbers, the sets $\\mathcal L$ and $\\mathcal D$ form a partition.\n\nLet $n \\in \\mathbb{Z}$ and consider the equation \n\\begin{equation}\\label{eq:main}\n n \\tan \\alpha = \\tan(n \\alpha).\n\\end{equation}\nIf $n\\in \\{0,1\\}$, then every $\\alpha \\in \\mathbb{R}$ is a solution. Furthermore, $\\alpha = 0$ is a solution for every $n \\in \\mathbb{Z}$. Since the tangent function is odd and $\\pi$-periodic, we may restrict attention to the case $n \\ge 2$ and $\\alpha \\in \\left(0,\\frac{\\pi}{2}\\right)$. In \\cite{Cyr2012}, Cyr proved that $\\frac{\\alpha}{2\\pi}$ is irrational.\n\nOne prominent context in which this equation appears is the {\\em theory of bicycle curves}. In an effort to classify noncircular bicycle curves, Tabachnikov \\cite{Tabachnikov2006} proved that the circle admits a nontrivial infinitesimal deformation as a smooth plane bicycle curve with rotation number $\\frac{\\alpha}{\\pi}$ if and only if \\eqref{eq:main} holds for some integer $n \\ge 2$. Thus, the solutions of this equation characterize the values of the rotation number for which the circle fails to be rigid in this geometric setting. In light of the Cyr theorem, it follows that the circle is rigid as a bicycle curve for any rational rotation number, except in the case $\\alpha = \\frac{\\pi}{2}$, where any curve of constant width is a bicycle curve.\n\nThe examples discussed above show that arithmetic properties of parameters appearing in geometric and dynamical problems can strongly influence their behavior. This perspective motivates the present work, which focuses on the arithmetic properties of real numbers arising in such contexts.\n\nBefore stating the main results of this paper, we introduce some additional notation. Let $z \\in \\mathbb{C} \\setminus \\{0\\}$. We denote by $\\operatorname{Log} z = \\ln|z| + i\\operatorname{Arg}(z)$ the principal branch of the logarithm, where $\\operatorname{Arg}(z) \\in (-\\pi,\\pi]$. \nThe main results of this paper are the following.\n\n\\begin{equation}\\label{eq:main}\n    n \\tan \\alpha = \\tan(n \\alpha).\n\\end{equation}\n\n\\begin{equation}\\label{ineq:dioph}\n    \\left| \\theta - \\frac{p}{q} \\right| \\ge \\frac{c}{q^\\tau}.\n\\end{equation}", "full_context": "Let $\\theta$ be a real number. The {\\em irrationality exponent} of $\\theta$ is \n$$\\mu(\\theta) = \\sup\\left\\{ \\mu > 0 \\colon \\left| \\theta - \\frac{p}{q} \\right| < \\frac{1}{q^\\mu} \\text{ holds for infinitely many pairs } (p,q) \\in \\mathbb{Z} \\times \\mathbb{N} \\right\\}.$$\nWe say that $\\theta$ is a {\\em Liouville number} whenever $\\mu(\\theta) = \\infty$. On the other hand, $\\theta$ is called {\\em Diophantine} if there exist $c > 0$ and $\\tau < \\infty$ such that, for every $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$, \n\\begin{equation}\\label{ineq:dioph}\n \\left| \\theta - \\frac{p}{q} \\right| \\ge \\frac{c}{q^\\tau}.\n\\end{equation}\nLet $\\mathcal L$ be the set of Liouville numbers and let $\\mathcal D$ be the set of Diophantine numbers.\n\nWe observe that if $\\theta = \\frac{a}{b}$ is rational, then $\\mu(\\theta) = 1$, and one may choose $\\frac{p}{q} = \\frac{a}{b}$ so that \\eqref{ineq:dioph} fails to hold. Hence, no rational number is either Diophantine or Liouville. Among irrational numbers, the sets $\\mathcal L$ and $\\mathcal D$ form a partition.\n\nLet $n \\in \\mathbb{Z}$ and consider the equation \n\\begin{equation}\\label{eq:main}\n n \\tan \\alpha = \\tan(n \\alpha).\n\\end{equation}\nIf $n\\in \\{0,1\\}$, then every $\\alpha \\in \\mathbb{R}$ is a solution. Furthermore, $\\alpha = 0$ is a solution for every $n \\in \\mathbb{Z}$. Since the tangent function is odd and $\\pi$-periodic, we may restrict attention to the case $n \\ge 2$ and $\\alpha \\in \\left(0,\\frac{\\pi}{2}\\right)$. In \\cite{Cyr2012}, Cyr proved that $\\frac{\\alpha}{2\\pi}$ is irrational.\n\nOne prominent context in which this equation appears is the {\\em theory of bicycle curves}. In an effort to classify noncircular bicycle curves, Tabachnikov \\cite{Tabachnikov2006} proved that the circle admits a nontrivial infinitesimal deformation as a smooth plane bicycle curve with rotation number $\\frac{\\alpha}{\\pi}$ if and only if \\eqref{eq:main} holds for some integer $n \\ge 2$. Thus, the solutions of this equation characterize the values of the rotation number for which the circle fails to be rigid in this geometric setting. In light of the Cyr theorem, it follows that the circle is rigid as a bicycle curve for any rational rotation number, except in the case $\\alpha = \\frac{\\pi}{2}$, where any curve of constant width is a bicycle curve.\n\nThe examples discussed above show that arithmetic properties of parameters appearing in geometric and dynamical problems can strongly influence their behavior. This perspective motivates the present work, which focuses on the arithmetic properties of real numbers arising in such contexts.\n\nBefore stating the main results of this paper, we introduce some additional notation. Let $z \\in \\mathbb{C} \\setminus \\{0\\}$. We denote by $\\operatorname{Log} z = \\ln|z| + i\\operatorname{Arg}(z)$ the principal branch of the logarithm, where $\\operatorname{Arg}(z) \\in (-\\pi,\\pi]$. \nThe main results of this paper are the following.\n\n\\begin{equation}\\label{eq:main}\n    n \\tan \\alpha = \\tan(n \\alpha).\n\\end{equation}\n\n\\begin{equation}\\label{ineq:dioph}\n    \\left| \\theta - \\frac{p}{q} \\right| \\ge \\frac{c}{q^\\tau}.\n\\end{equation}\n\n\\begin{abstract}\n An irrational number $\\theta$ is called Diophantine if there exist $c>0$ and $\\tau < \\infty$ such that $\\left| \\theta - \\frac{p}{q} \\right| \\ge \\frac{c}{q^\\tau}$ holds for every $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$. \n In this paper, we study Diophantine and transcendence properties of some real numbers. Using lower bounds for linear forms in logarithms, we show that if $\\beta \\in \\mathbb{C}$ is an algebraic number with $|\\beta|=1$ that is not a root of unity, then $\\frac{\\Arg(\\beta)}{2\\pi}$ is Diophantine. We also prove that if $\\beta = e^{i\\alpha}$ is algebraic, then $\\frac{\\alpha}{\\pi}$ is either rational or transcendental.\n\nLet $\\theta$ be a real number. The {\\em irrationality exponent} of $\\theta$ is \n$$\\mu(\\theta) = \\sup\\left\\{ \\mu > 0 \\colon \\left| \\theta - \\frac{p}{q} \\right| < \\frac{1}{q^\\mu} \\text{ holds for infinitely many pairs } (p,q) \\in \\mathbb{Z} \\times \\mathbb{N} \\right\\}.$$\nWe say that $\\theta$ is a {\\em Liouville number} whenever $\\mu(\\theta) = \\infty$. On the other hand, $\\theta$ is called {\\em Diophantine} if there exist $c > 0$ and $\\tau < \\infty$ such that, for every $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$, \n\\begin{equation}\\label{ineq:dioph}\n \\left| \\theta - \\frac{p}{q} \\right| \\ge \\frac{c}{q^\\tau}.\n\\end{equation}\nLet $\\mathcal L$ be the set of Liouville numbers and let $\\mathcal D$ be the set of Diophantine numbers.\n\nBefore stating the main results of this paper, we introduce some additional notation. Let $z \\in \\mathbb{C} \\setminus \\{0\\}$. We denote by $\\Log z = \\ln|z| + i\\Arg(z)$ the principal branch of the logarithm, where $\\Arg(z) \\in (-\\pi,\\pi]$. \nThe main results of this paper are the following.\n\nBy applying the previous result to the solutions of equation \\eqref{eq:main}, together with the classical theorems described in Section \\ref{sec:pre}, we obtain the following arithmetic statements.\n\n\\begin{theorem}\\label{thm:3props}\n Let $n \\ge 2$ be an integer and let $\\alpha \\in \\left( 0, \\frac{\\pi}{2} \\right)$ be a solution of \\eqref{eq:main}. Then the following statements hold.\n \\begin{enumerate}[label=(\\alph*)]\n \\item $\\frac{\\alpha}{2\\pi}$ is a Diophantine number.\n \\item $\\frac{\\alpha}{2\\pi}$ is transcendental.\n \\item $\\alpha$ is transcendental.\n \\end{enumerate}\n\\end{theorem}\n\nFor $z_1,z_2 \\in \\mathbb{C}$, $p \\in \\mathbb{Z}$ and $q \\in \\mathbb{N}$, set $L(z_1,z_2) = qz_1 - 2pz_2$ and \n$$\\Lambda(q,p) = L(\\Log(\\beta),\\Log(-1)) = q \\Log(\\beta) - 2p \\Log(-1)$$\nto be a linear form in logarithms with algebraic numbers $\\{\\beta,-1\\}$. Furthermore, we have that\n\\begin{equation}\\label{eq:Lqp}\n \\Lambda(q,p) = i\\left( q\\alpha - 2\\pi p \\right).\n\\end{equation}\nLet $\\theta = \\frac{\\Arg(\\beta)}{2\\pi} = \\frac{\\alpha}{2\\pi} \\in \\left( -\\frac{1}{2},\\frac{1}{2} \\right]$. It follows that \n\\begin{equation}\\label{eq:|Lambda|}\n |\\Lambda(q,p)| = |q\\alpha - 2\\pi p| = 2\\pi q \\left| \\theta - \\frac{p}{q} \\right|.\n\\end{equation}\n\nWe now apply Baker-Wüstholz Theorem (Theorem \\ref{thm:bw}) with $(\\alpha_1, \\alpha_2, b_1, b_2) = (\\beta, -1, q, -2p)$. Let $d = [\\mathbb{Q}(\\beta):\\mathbb{Q}]$. Since \n$$h'(L) = \\max \\left\\{ \\ln(\\max\\{q,|2p|\\}), 1\\right\\} = \\ln(\\max\\{q,|2p|,e\\}) ,$$ \nobtain that \n$$\\ln|\\Lambda(q,p)| > -C_0 \\ln(\\max\\{q,|2p|,e\\})$$\nwith $C_0 = C_0(\\beta) = C(2,d) h'(\\beta) h'(-1) \n> 0$ depending only on $\\beta$. Therefore \n$$|\\Lambda(q,p)| > (\\max\\{q,|2p|,e\\})^{-C_0}.$$\nBy \\eqref{eq:|Lambda|}, we obtain\n\\begin{equation}\\label{ineq:modulotheta}\n \\left| \\theta - \\frac{p}{q} \\right| > \\frac{1}{2\\pi} \\cdot \\frac{1}{q(\\max\\{q,|2p|,e\\})^{C_0}} \\quad \\text{ for every } p \\in \\mathbb{Z}.\n\\end{equation}\n\nNotice that $\\theta$ is irrational; otherwise $\\beta = e^{2\\pi i \\theta}$ would be a root of unity, which is a contradiction. Therefore, there exists a unique integer $p'$ closest to $q\\theta$, that is, $\\left|q\\theta - p'\\right| < \\frac{1}{2}$. Since $|p'| < q|\\theta| + \\frac{1}{2} < \\frac{q+1}{2}$, it follows that \n$$\\max\\{q,2|p'|,e\\} < 3q$$\nfor every $q \\in \\mathbb{N}$. By \\eqref{ineq:modulotheta}, we obtain\n$$\\left| \\theta - \\frac{p'}{q} \\right| > \\frac{1}{2\\pi} \\cdot \\frac{1}{q(3q)^{C_0}} = \\frac{1}{2\\pi \\cdot 3^{C_0}} \\cdot \\frac{1}{q^{C_0+1}}.$$\nLet $c = \\frac{1}{2\\pi \\cdot 3^{C_0}} > 0$ and $\\tau = C_0+1 < \\infty$. Notice that both $c$ and $\\tau$ depend only on $\\beta$, and hence only on $\\theta$. For every $p \\in \\mathbb{Z}$ and $q \\in \\mathbb{N}$, we have that $|q\\theta - p| \\ge |q\\theta - p'|$, therefore\n$$\\left| \\theta - \\frac{p}{q} \\right| \\ge \\left| \\theta - \\frac{p'}{q} \\right| \\ge \\frac{c}{q^\\tau}.$$\nThis is precisely the definition of a Diophantine number.\n\\qed\n\n\\begin{equation}\\label{eq:main}\n    n \\tan \\alpha = \\tan(n \\alpha).\n\\end{equation}", "post_theorem_intro_text_len": 1638, "post_theorem_intro_text": "By applying the previous result to the solutions of equation \\eqref{eq:main}, together with the classical theorems described in Section \\ref{sec:pre}, we obtain the following arithmetic statements.\n\n\\begin{theorem}\\label{thm:3props}\n    Let $n \\ge 2$ be an integer and let $\\alpha \\in \\left( 0, \\frac{\\pi}{2} \\right)$ be a solution of \\eqref{eq:main}. Then the following statements hold.\n    \\begin{enumerate}[label=(\\alph*)]\n        \\item $\\frac{\\alpha}{2\\pi}$ is a Diophantine number.\n        \\item $\\frac{\\alpha}{2\\pi}$ is transcendental.\n        \\item $\\alpha$ is transcendental.\n    \\end{enumerate}\n\\end{theorem}\n\nWe observe that items $(a)$ and $(b)$ of the previous theorem extend the result of Cyr~\\cite{Cyr2012}, while item $(c)$ provides the transcendence of the solution $\\alpha$ itself.\n\nThe paper is organized as follows. In Section \\ref{sec:pre}, we introduce the main tools used in the proof of Theorem \\ref{thm:general}, including several algebraic ingredients, as well as the Baker-Wüstholz Theorem, which provides a lower bound for a non-zero linear forms in logarithms, the Gelfond-Schneider Theorem and the Hermite-Lindemann Theorem, which are two remarkable results in transcendence theory. Section \\ref{sec:proof} contains the proof of the main result (Theorem \\ref{thm:general}). In Section \\ref{sec:eqtan}, we apply the main result, together with the remarkable theorems presented in Section \\ref{sec:pre}, to the solutions of equation \\eqref{eq:main} in order to prove Theorem \\ref{thm:3props}. Finally, in Section \\ref{sec:conclusion}, we summarize the results obtained in this paper and present an open problem.", "sketch": "The post-theorem introduction does not give a step-by-step proof sketch of Theorem~\\ref{thm:general}, but it does indicate the proof ingredients and where the proof is carried out: Section~\\ref{sec:pre} “introduce[s] the main tools used in the proof of Theorem~\\ref{thm:general}, including several algebraic ingredients, as well as the Baker-Wüstholz Theorem, which provides a lower bound for a non-zero linear forms in logarithms, the Gelfond-Schneider Theorem and the Hermite-Lindemann Theorem.” It then states that “Section~\\ref{sec:proof} contains the proof of the main result (Theorem~\\ref{thm:general}).”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:general}\n    Let $\\beta$ be an algebraic number with $|\\beta|=1$ which is not a root of unity. Then $\\frac{\\operatorname{Arg}(\\beta)}{2\\pi} \\in \\left(-\\frac{1}{2},\\frac{1}{2}\\right]$ is a Diophantine number.,", "theorem_type": ["Implication", "Universal"], "mcq": {"question": "Let \\(\\Arg(z)\\in(-\\pi,\\pi]\\) denote the principal argument of a nonzero complex number \\(z\\). A real number \\(\\theta\\) is called Diophantine if there exist constants \\(c>0\\) and \\(\\tau<\\infty\\) such that for every \\((p,q)\\in \\mathbb Z\\times \\mathbb N\\),\n\\[\n\\left|\\theta-\\frac pq\\right|\\ge \\frac{c}{q^{\\tau}}.\n\\]\nWhich statement holds for every algebraic number \\(\\beta\\) satisfying \\(|\\beta|=1\\) and such that \\(\\beta\\) is not a root of unity?", "correct_choice": {"label": "A", "text": "The number \\(\\dfrac{\\Arg(\\beta)}{2\\pi}\\in\\left(-\\dfrac12,\\dfrac12\\right]\\) is Diophantine; equivalently, there exist constants \\(c>0\\) and \\(\\tau<\\infty\\) such that for all \\((p,q)\\in\\mathbb Z\\times\\mathbb N\\),\n\\[\n\\left|\\frac{\\Arg(\\beta)}{2\\pi}-\\frac pq\\right|\\ge \\frac{c}{q^{\\tau}}.\n\\]"}, "choices": [{"label": "B", "text": "The number \\(\\dfrac{\\Arg(\\beta)}{2\\pi}\\in\\left(-\\dfrac12,\\dfrac12\\right]\\) is Diophantine; equivalently, there exist constants \\(c>0\\) and \\(\\tau<\\infty\\), depending only on the degree \\([\\mathbb Q(\\beta):\\mathbb Q]\\), such that for all \\((p,q)\\in\\mathbb Z\\times\\mathbb N\\),\n\\[\n\\left|\\frac{\\Arg(\\beta)}{2\\pi}-\\frac pq\\right|\\ge \\frac{c}{q^{\\tau}}.\n\\]"}, {"label": "C", "text": "The number \\(\\dfrac{\\Arg(\\beta)}{2\\pi}\\in\\left(-\\dfrac12,\\dfrac12\\right]\\) is not a rational number."}, {"label": "D", "text": "The number \\(\\dfrac{\\Arg(\\beta)}{2\\pi}\\in\\left(-\\dfrac12,\\dfrac12\\right]\\) is Liouville; equivalently, for every \\(\\tau<\\infty\\) there exist infinitely many pairs \\((p,q)\\in\\mathbb Z\\times\\mathbb N\\) such that\n\\[\n\\left|\\frac{\\Arg(\\beta)}{2\\pi}-\\frac pq\\right|< \\frac{1}{q^{\\tau}}.\n\\]"}, {"label": "E", "text": "For every algebraic number \\(\\beta\\) with \\(|\\beta|=1\\), the number \\(\\dfrac{\\Arg(\\beta)}{2\\pi}\\in\\left(-\\dfrac12,\\dfrac12\\right]\\) is Diophantine; equivalently, there exist constants \\(c>0\\) and \\(\\tau<\\infty\\) such that for all \\((p,q)\\in\\mathbb Z\\times\\mathbb N\\),\n\\[\n\\left|\\frac{\\Arg(\\beta)}{2\\pi}-\\frac pq\\right|\\ge \\frac{c}{q^{\\tau}}.\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "other", "tampered_component": "dependence_of_constants_on_beta", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped_Diophantine_conclusion_to_irrationality_only", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "Diophantine_vs_Liouville_property", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "excluded_root_of_unity_hypothesis", "template_used": "boundary_range"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct conclusion; it only supplies the definition of Diophantine and the hypothesis on algebraic numbers on the unit circle. There is no explicit answer leakage."}, "TAS": {"score": 1, "justification": "The intended correct option is essentially the theorem itself specialized to the given hypotheses, so the item is close to a direct theorem recall rather than a substantially reformulated problem. The competing options prevent it from being fully tautological, but only mildly."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish the exact Diophantine conclusion from a stronger uniform version, a weaker irrationality-only claim, and false alternatives. However, recognition of the underlying theorem is largely sufficient, so the generative demand is moderate rather than strong."}, "DQS": {"score": 2, "justification": "The distractors are mathematically meaningful and target common failure modes: confusing exact vs. uniform bounds (B), accepting a weaker true statement (C), reversing Diophantine to Liouville behavior (D), and forgetting the non-root-of-unity hypothesis (E). They are distinct and plausible."}, "total_score": 6, "overall_assessment": "Conceptually a solid theorem-recall MCQ with good distractors and no answer leakage, but it is only mildly non-tautological and does not strongly force generative reasoning. There is also a serious formatting/validity issue: the marked correct answer is labeled A, but A is missing from the displayed choices."}}
{"id": "2602.23912v1", "paper_link": "http://arxiv.org/abs/2602.23912v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm:phase-transition}\nLet $\\cC$ be a block-stable class of connected graphs, let $\\cB$ be its class of $2$-connected components whose EGF we denote by $B(y)$, and let $\\rho_B$ be the radius of convergence of $B$. Let\n\\[u_C := \\frac{1}{\\rho_B B''(\\rho_B)}.\\]\nThen the following hold. \n\\begin{itemize}\n    \\item The singular behaviour of $C^{\\bullet}(x,u)$ depends on the relative positions of $u$ and $u_C$;\n    \\item The block tree corresponding to a graph drawn according to $\\Pnu$ follows a subcritical \\BGW{} tree law if $u<u_C$, and a critical \\BGW{} tree law if $u \\ge u_C$.\n\\end{itemize}", "start_pos": 9575, "end_pos": 10204, "label": "thm:phase-transition"}, "ref_dict": {"eq:asymp-exp-B'": "\\begin{equation}\n\\label{eq:asymp-exp-B'}\nB'(y) = B'(\\rho_B) - B''(\\rho_B)(\\rho_B-y) + \\bconst (\\rho_B-y)^{3/2} + O\\((\\rho_B-y)^2\\)\n\\end{equation}", "thm:phasetransition_block_sizes": "\\begin{theorem}\\label{thm:phasetransition_block_sizes} Let $u>0$. The block-weighted graph $\\Gnu$ exhibits the following behaviours when $n\\to\\infty$.\n\\begin{description}\n    \\item[Subcritical case.] If $u<u_C$, then \\cref{th-taille-blocs-sous-crit}  holds for block sizes:\n    \\begin{align*}\n    \\frac{1}{(c(u) n)^{2/3}}&\\(\\(1-\\frac{u}{u_C}\\)n - \\LB{1}{\\Gnu}, \\(\\LB{j}{\\Gnu} : j\\geq 2\\)\\) \\\\\n    &\\hspace{170pt} \\xrightarrow[n\\to\\infty]{(d)} \\(L_1,\\(\\Delta L_{(j-1)} : j\\geq 2\\)\\)\n    \\end{align*}\n    where $(L_t)_{t\\in[0,1]}$ is a stable process of parameter $3/2$ such that $\\E{e^{-sL_1}} = e^{\\Gamma(-3/2)s^{3/2}}$ and $\\Delta L_{(1)} \\geq \\Delta L_{(2)} \\geq \\dots$ is the ranked sequence of its jumps.\n\n    \\item[Critical case.] If $u=u_C$, then \\cref{eq:size-bloc-critical} holds for the block sizes:\n    \\[\\(\\frac{\\LB{j}{\\G_{n,u_C}}}{n^{2/3}} : j \\geq 1\\) \\xrightarrow[n\\to\\infty]{(d)} \\(E_{(j)} : j \\geq 1\\),\\]\n    where the $E_{(j)}$ are as defined in \\cref{prop:degrees-Tnu}.\n    \\item[Supercritical case.] If $u>u_C$, then \\cref{eq:grandbloc} holds for the block sizes (fixing $j \\in \\mathbb{N}_0$):\n    \\[\\LB{j}{\\Gnu} = \\frac{\\ln(n)}{\\ln\\(\\frac{\\rho_B}{y(u)}\\)} - \\frac{5}{2}\\frac{\\ln(\\ln(n))}{\\ln\\(\\frac{\\rho_B}{y(u)}\\)} + O_{\\mathbb{P}}(1).\\]\n\\end{description}\n\\end{theorem}", "prop:law-TG": "\\begin{proposition}\n\\label{prop:law-TG}\nFix $u>0$, let $\\G$ be distributed according to $\\Pu$, and denote by $(\\bTT,(\\bphi_v,v\\in \\bTT))$ the decorated block tree $\\TT_\\G$, where $\\bTT$ is the undecorated block tree and $(\\bphi_v,v\\in \\bTT)$ is its family of vertex decorations. Then the law of $\\TT_\\G$ can be described as follows:\n\\begin{itemize}\n\\item $\\bTT$ follows the law $\\BGWl{\\muT^u}$;\n\\item Conditionally given $\\bTT=\\t$, the decorations $(\\bphi_{v},v\\in \\t)$ are independent random variables, and, for $v\\in\\t$, $\\bphi_{v}$ is distributed according to $\\Gamma \\Phi(C^\\bullet(\\rho(u),u),u)$ conditioned to have size $d_\\t(v)$.\n\\end{itemize}\nFor every $n\\in\\Npos$, the same statements hold under $\\Pnu$, only replacing $\\BGWl{\\muT^u}$ with $\\BGWl{\\muT^u,n}$.\n\\end{proposition}", "thm:block-sizes": "\\begin{theorem}\\label{thm:block-sizes}\nThere is a phase transition for block sizes depending on the value of $u$:\n\\begin{itemize}\n    \\item In the subcritical phase $u < u_C$, there is a largest block of size $\\Theta_\\mathbb{P}(n)$ and a second largest block of size $\\Theta_\\mathbb{P}(n^{2/3})$;\n    \\item In the critical phase $u = u_C$, there is a largest block of size $\\Theta_\\mathbb{P}(n^{2/3})$ and likewise for the next largest block sizes;\n    \\item In the supercritical phase $u > u_C$, there is a largest block of size $\\Theta_\\mathbb{P}(\\ln (n))$ and likewise for the next largest block sizes.\n\\end{itemize}\n\\end{theorem}", "thm:phase-transition": "\\begin{theorem}\\label{thm:phase-transition}\nLet $\\cC$ be a block-stable class of connected graphs, let $\\cB$ be its class of $2$-connected components whose EGF we denote by $B(y)$, and let $\\rho_B$ be the radius of convergence of $B$. Let\n\\[u_C := \\frac{1}{\\rho_B B''(\\rho_B)}.\\]\nThen the following hold. \n\\begin{itemize}\n    \\item The singular behaviour of $C^{\\bullet}(x,u)$ depends on the relative positions of $u$ and $u_C$;\n    \\item The block tree corresponding to a graph drawn according to $\\Pnu$ follows a subcritical \\BGW{} tree law if $u<u_C$, and a critical \\BGW{} tree law if $u \\ge u_C$.\n\\end{itemize}\n\\end{theorem}", "th:dvp-sing-Cbxu": "\\begin{theorem}\n\\label{th:dvp-sing-Cbxu}\nThe series $C^\\bullet(x,u)$ displays the following asymptotic behaviours when $x$ is in a $\\Delta$-domain neighbourhood of $\\rho(u)$, where we set $X := 1-\\frac{x}{\\rho(u)}$.\n\\begin{description}\n    \\item[Subcritical regime.] If $u<u_C$, then\n   \\[C^\\bullet(x, u) = \\rho_B - r(u)X  + s(u) X^{3/2} + \\Theta\\(X^2\\),\\]\n   where\n   \\[r(u) := \\rho_B\\(1-\\frac{u}{u_C}\\)^{-1}\\qquad\\text{and}\\qquad s(u) := u \\bconst \\rho_B^{5/2} \\(1-\\frac{u}{u_C}\\)^{-5/2}.\\]\n   Thus,\n   \\[[x^n] C^\\bullet(x, u) \\sim \\frac{3 s(u)}{4 \\sqrt{\\pi}} n^{-5/2} \\rho(u)^{-n}.\\]\n    \\item[Critical regime.] If $u = u_C$, then\n    \\[C^\\bullet(x, u_C) = \\rho_B - s X^{2/3} + \\Theta\\(X\\),\\]\n    where $s := u_C^{-2/3} \\bconst^{-2/3}$. Thus,\n    \\[[x^n] C^\\bullet(x, u_C) \\sim \\frac{s \\Gamma(2/3)}{\\sqrt{3}\\pi} n^{-5/3} \\rho(u_C)^{-n}.\\]\n     \\item[Supercritical regime.] If $u>u_C$, then\n    \\[C^\\bullet(x, u) = y(u) - s(u) X^{1/2} + \\Theta\\(X\\),\\]\n    where\n    \\[s(u):=\\sqrt{\\frac{2}{uB''(y(u)) + u^2 B'(y(u))^2}} = \\sqrt{\\frac{2y(u)}{1+ y(u)\\ln\\(\\frac{y(u)}{\\rho(u)}\\)^{2}}}.\\]\n    Thus,\n    \\[[x^n] C^\\bullet(x, u) \\sim \\frac{s(u)}{2\\sqrt{\\pi}} n^{-3/2} \\rho(u)^{-n}.\\]\n\\end{description}\n\\end{theorem}", "eq:Cbullet-block": "\\begin{equation}\n\\label{eq:Cbullet-block}\nC^\\bullet(x) = x \\exp(B'(C^\\bullet(x))).\n\\end{equation}", "subsec:anal-comb-block-tree": "\\begin{enumerate}\n    \\item \\emph{Block graph} $B(\\g)$ (see e.g., \\cite{harary1969}): The vertices of $B(\\g)$ are the blocks of $\\g$, and two blocks are joined by an edge in $B(\\g)$ if they overlap at a vertex. The resulting graph $B(\\g)$ is characterised by the property of being connected and that all of its blocks are cliques.\n    \\item \\emph{Block-cut graph} $T(\\g)$ (see e.g., \\cite{diestel2025graph}): The vertices of $T(\\g)$ are the blocks and the cut-vertices of $\\g$, and a cut-vertex is joined to a block by an edge in $T(\\g)$ if that vertex is contained in that block. The resulting graph $T(\\g)$ is a tree.\n\\end{enumerate}\n\nIn the present work, we use an alternative block decomposition that naturally arises from the Lagrangean form of \\eqref{eq:Cbullet-block} of the generating series $C^{\\bullet}(x)$. It originates from work of Addario-Berry \\cite{2Louigi} on planar maps (see also \\cite{Sal24}), and we give a precise algorithmic description in \\cref{subsec:anal-comb-block-tree}. We note that this construction works specifically for \\emph{rooted} labelled connected graphs, and we call its output a \\emph{decorated block tree}. In a decorated block tree, the vertices represent all the vertices of the original graph $\\g$ but each vertex is decorated by a set of blocks that contain that vertex. In \\cite{2Louigi}, the decorations were simply blocks, but in our construction the decorations are more general \\emph{sets of blocks}.\n\n\\section{Decorated block trees of connected block-stable graphs}\n\\label{sec:decorated}\nIn this section, we describe \\emph{decorated block trees}, which are a central tool in our analysis.\n\n\\subsection{Construction of decorated block tree}\n\\label{subsec:anal-comb-block-tree}\nRecalling the Lagrangean form of \\cref{eq:Cbullet-block}, we define\n\\begin{equation}\n\\label{eq:def-Phi}\n\\Phi(y) = \\sum_{k=0}^\\infty \\frac{\\phi_k}{k!} y^k \\coloneqq \\exp(B'(y)) \\in \\mathbb C [[y]],\n\\end{equation}\nwhich is precisely the exponential generating function for the class $\\Set \\circ \\mathcal{B}'$. For convenience we also use $\\Phi$ to denote  the class $\\Set \\circ \\mathcal{B}'$. It follows from \\cref{eq:Cbullet-block} that\n\\[C^\\bullet(x) = x \\Phi(C^\\bullet(x)).\\]\nAs explained in \\cite[\\S 2.2.1]{Sal24}, there is a systematic way to define a (plane) block tree from a Lagrangean decomposition scheme. Namely, each vertex of the tree is decorated with an object $\\varphi$ from the class $\\Phi$, and the subtrees attached to a vertex of the tree (corresponding to some $\\varphi$) are the trees of the rooted graphs substituted into $\\varphi$. The situation in the case of graphs is a little more involved than in the case of maps because of the labelling possibilities. Thus, we use the following notion.\n\nRemember that elements of the class $\\Phi$ are well-labelled. Therefore, for each $\\g \\in \\cC$, when we consider sets of derived blocks of $\\g$, it is always implied that the set has been relabelled with a well-labelling which is consistent with the order of the labels in $\\g$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1\\textwidth]{images/block-tree.pdf}\n\\end{center}\n\\caption{The image on the left is a connected labelled rooted planar graph $\\g$, with root highlighted in grey. The image on the right is the associated decorated block tree $\\TT_\\g$.}\n\\label{fig:block-tree-2}\n\\end{figure}\n\nNow, we give an algorithmic definition of decorated block trees. An example is provided in \\cref{fig:block-tree-2}.\n\n\\begin{definition}\nFor $\\g\\in \\cC^\\bullet$ (which for ease of presentation $\\g$ is also allowed to be a rooted derived graph, with a root and one unlabelled non-root vertex), we define the \\emph{decorated block tree $\\TT_\\g$} as the result of the following construction:\n\\begin{enumerate}\n    \\item Let $r$ be the root of $\\g$, and let $\\varphi = \\{\\b'_1, \\ldots, \\b'_\\ell\\}$ be the set of derived blocks containing $r$ \\emph{and no unlabelled vertex} in $\\g$, where the label of the vertex corresponding to $r$ is removed. Start with a root vertex $r_{\\TT}$ in $\\TT_\\g$ and decorate it with $\\varphi$.\n    \\item For $i \\in \\{1, \\ldots, \\ell\\}$, we construct a derived graph $\\g'_{\\b_i}$ by deleting $r$ from $\\g$, keeping the connected component of $\\g$ that intersects $\\b'_i$, and then adding $r$ back to this subgraph. We consistently relabel the vertices of $\\g'_{\\b_i}$ except $r$ which remains unlabelled. For every labelled vertex $v$ of $\\b'_i$, we construct the rooted derived graph $\\g_{(v)}$ by rooting $\\g_{\\b_i}'$ at $v$ (and keeping the unlabelled vertex).\n    \\item For $v$ a vertex in $\\varphi$, in increasing order of label in $\\varphi$, let $\\t_v := \\TT_{\\g_{(v)}}$ be the decorated block tree on this rooted subgraph with an unlabelled vertex. Add $\\t_v$ as the rightmost child of $r_{\\TT}$.\n\\end{enumerate}"}, "pre_theorem_intro_text_len": 6676, "pre_theorem_intro_text": "\\subsection{Motivation and background}\n\nPlanar maps and planar graphs have received much attention from enumerative and probabilistic points of view. The study of planar map enumeration started with the groundbreaking combinatorial work of Tutte, who gave a close formula for the number of rooted planar maps of size $n$ in the 1960s \\cite{tutte_1963}. Tutte's work prompted many perspectives and extensions since \\cite{ChassaingSchaeffer2004,cori_vauquelin_1981,LeGall2013,Miermont2013,Schaeffer1998}. Despite extensive studies, the asymptotic number of planar graphs were successfully determined only much more recently by Giménez and Noy \\cite{GN09} in 2009, building on work of Bender, Gao, and Wormald \\cite{bender2002number}.\n\nMany results on planar maps and planar graphs rely on applying decompositions that work in tree-like fashion, allowing combinatorial properties to be deduced. A prominent example of this theme is the \\emph{block decomposition}. For both planar maps and connected planar graphs, there is a unique way to decompose them into maximal $2$-connected submaps or subgraphs, called \\emph{blocks}  (see, e.g., \\cite{diestel2025graph, harary1969,tutte_1963}). In both cases, one can construct tree-like structures by viewing the blocks as vertices of a graph whose edges join overlapping blocks.  \n\nIn the case of planar maps, such a tree was described by Addario-Berry \\cite{2Louigi}, following Tutte's description of the block decomposition \\cite{tutte_1963}. Moreover, Addario-Berry \\cite{2Louigi} showed that this tree could be used to obtain information about \\emph{random} planar maps: Since the block tree of a uniform planar map of size $n$ follows the law of a \\emph{\\BGW{} tree}\\footnote{Also called \\emph{Galton--Watson tree} or \\emph{Bienaymé--Galton--Watson tree}, we adopt the change of notation proposed by Addario-Berry, Brandenberger, Hamdan, and Kerriou \\cite{ABBHK22}.}, whose properties are well-known, properties on the sizes of the $2$-connected components of a random map follow. It was subsequently shown that this approach fruitfully allows one to consider \\emph{block-weighted models}: instead of sampling according to a uniform law, one considers a parameter $u>0$, and samples a map of size $n$ with probability proportional to $u$ to the power of its number of blocks. (Note that $u=1$ corresponds to the usual uniform law.) Fleurat and the second author \\cite{FleuratSalvy23} showed that the block tree of such a random planar map still follows a \\BGW{} law, and used this to obtain results on the sizes of blocks of such maps as well as scaling limits. More precisely, they discovered a phase transition between the universality classes of maps (converging to the Brownian sphere) and plane trees (converging to the Brownian tree), depending on the value of $u$, obtaining precise results on block sizes in each phase.\n\nUsing an extensive list of decomposition of maps into subfamilies \\cite{airy}, the second author \\cite{Salvy23,Sal24} showed that this approach can be successfully adapted to deal with different decompositions of maps, giving a very similar phase transition between a ``map phase'' and a ``tree phase''. Albenque, Fusy, and the second author \\cite{AFS24} used this approach for a family of \\emph{decorated} maps, which are planar maps endowed with one of their spanning tree. They found a phase transition between a ``tree-rooted map phase'' and a ``tree phase''.\n\nSince this technique of using block decompositions to obtain interesting results for planar \\emph{maps} is robust, it is intriguing to extend this approach to connected planar \\emph{graphs}. The planar graph case is slightly more intricate than the planar map case because one has to take into account the labelling and cannot rely on an embedding as in the map case.\n\n\\subsection{Main results}\nThe main contributions of this paper are two-fold. First, we show that a random connected rooted labelled  graph from a block-stable class (e.g., a random connected planar graph), which is sampled according to a block-weighted law, exhibits a phase transition (see \\cref{thm:phase-transition}). To this end, we introduce a \\emph{decorated block tree} that encodes a block decomposition of such a connected graph (see \\cref{sec:decorated}). This extends the lines of previous study based on a block tree decomposition from planar \\emph{maps} to connected planar \\emph{graphs}. Because block-stable classes are very general, our results are also very general, but not necessarily precise.\n\nSecond, we show how the block tree methodology can be used to obtain more precise results. First, we use it to obtain enumeration results for the block-weighted generating function. Then we determine the distribution of block sizes of random block-weighted planar graphs and block-stable graphs of the same singular type (see \\cref{thm:block-sizes}). As a result, we obtain more precise information on block size of a uniform random connected planar graph than what was previously known in the literature \\cite{GNR13, PS11}.\n\n\\subsubsection{Phase transition in block-weighted block-stable graphs}\nA class $\\cC$ of connected graphs is said to be \\emph{block-stable} if for each graph $\\g \\in \\cC$ each of its $2$-connected components (called \\emph{blocks}) belongs to $\\cC$. Write $\\cB$ for the class of $2$-connected elements of $\\cC$. Write $B(y)$ and $C(x)$ for the corresponding exponential generating functions (EGFs), and set $C^\\bullet(x) = x C(x)$. It was shown, e.g., by Robinson \\cite[Theorem 4]{Rob70}, that\n\\begin{equation}\n\\label{eq:Cbullet-block}\nC^\\bullet(x) = x \\exp(B'(C^\\bullet(x))).\n\\end{equation}\nAs we will see in \\cref{subsec:anal-comb-block-tree}, this equation can be translated into a ``decorated'' block tree.\n\nMore precisely, let $C^\\bullet(x,u)$ be the exponential generating function for \\emph{connected rooted} graphs of $\\cC$, where $x$ marks vertices and $u$ marks blocks. Write $\\rho(u)$ for the radius of convergence of $C^\\bullet(\\cdot,u)$. For $u>0$, we consider the following \\emph{block-weighted} probability laws on $\\cC^\\bullet$: for a graph $\\g \\in \\cC$ of size $|\\g|$ with $b(\\g)$ many $2$-connected components, we set\n\\[\\pr[_{u}]{\\g} := \\frac{\\rho(u)^{|\\g|}u^{b(\\g)}}{|\\g|!C^\\bullet(\\rho(u),u)},\\qquad\\text{and}\\qquad \\pr[_{n,u}]{\\g} := \\frac{u^{b(\\g)}}{[x^n] C^\\bullet(x,u)}\\ind_{|\\g|=n}.\\]\n\nWe show that, when considering a block-weighted law on a connected rooted graphs from a block-stable classes (including connected rooted planar graphs), this block tree follows a \\BGW{} tree distribution (see \\cref{prop:law-TG}). Our first main theorem is about phase transitions in the criticality of this \\BGW{} tree.", "context": "In the case of planar maps, such a tree was described by Addario-Berry \\cite{2Louigi}, following Tutte's description of the block decomposition \\cite{tutte_1963}. Moreover, Addario-Berry \\cite{2Louigi} showed that this tree could be used to obtain information about \\emph{random} planar maps: Since the block tree of a uniform planar map of size $n$ follows the law of a \\emph{\\BGW{} tree}\\footnote{Also called \\emph{Galton--Watson tree} or \\emph{Bienaymé--Galton--Watson tree}, we adopt the change of notation proposed by Addario-Berry, Brandenberger, Hamdan, and Kerriou \\cite{ABBHK22}.}, whose properties are well-known, properties on the sizes of the $2$-connected components of a random map follow. It was subsequently shown that this approach fruitfully allows one to consider \\emph{block-weighted models}: instead of sampling according to a uniform law, one considers a parameter $u>0$, and samples a map of size $n$ with probability proportional to $u$ to the power of its number of blocks. (Note that $u=1$ corresponds to the usual uniform law.) Fleurat and the second author \\cite{FleuratSalvy23} showed that the block tree of such a random planar map still follows a \\BGW{} law, and used this to obtain results on the sizes of blocks of such maps as well as scaling limits. More precisely, they discovered a phase transition between the universality classes of maps (converging to the Brownian sphere) and plane trees (converging to the Brownian tree), depending on the value of $u$, obtaining precise results on block sizes in each phase.\n\n\\subsection{Main results}\nThe main contributions of this paper are two-fold. First, we show that a random connected rooted labelled graph from a block-stable class (e.g., a random connected planar graph), which is sampled according to a block-weighted law, exhibits a phase transition (see \\cref{thm:phase-transition}). To this end, we introduce a \\emph{decorated block tree} that encodes a block decomposition of such a connected graph (see \\cref{sec:decorated}). This extends the lines of previous study based on a block tree decomposition from planar \\emph{maps} to connected planar \\emph{graphs}. Because block-stable classes are very general, our results are also very general, but not necessarily precise.\n\nSecond, we show how the block tree methodology can be used to obtain more precise results. First, we use it to obtain enumeration results for the block-weighted generating function. Then we determine the distribution of block sizes of random block-weighted planar graphs and block-stable graphs of the same singular type (see \\cref{thm:block-sizes}). As a result, we obtain more precise information on block size of a uniform random connected planar graph than what was previously known in the literature \\cite{GNR13, PS11}.\n\n\\subsubsection{Phase transition in block-weighted block-stable graphs}\nA class $\\cC$ of connected graphs is said to be \\emph{block-stable} if for each graph $\\g \\in \\cC$ each of its $2$-connected components (called \\emph{blocks}) belongs to $\\cC$. Write $\\cB$ for the class of $2$-connected elements of $\\cC$. Write $B(y)$ and $C(x)$ for the corresponding exponential generating functions (EGFs), and set $C^\\bullet(x) = x C(x)$. It was shown, e.g., by Robinson \\cite[Theorem 4]{Rob70}, that\n\\begin{equation}\n\\label{eq:Cbullet-block}\nC^\\bullet(x) = x \\exp(B'(C^\\bullet(x))).\n\\end{equation}\nAs we will see in \\cref{subsec:anal-comb-block-tree}, this equation can be translated into a ``decorated'' block tree.\n\nMore precisely, let $C^\\bullet(x,u)$ be the exponential generating function for \\emph{connected rooted} graphs of $\\cC$, where $x$ marks vertices and $u$ marks blocks. Write $\\rho(u)$ for the radius of convergence of $C^\\bullet(\\cdot,u)$. For $u>0$, we consider the following \\emph{block-weighted} probability laws on $\\cC^\\bullet$: for a graph $\\g \\in \\cC$ of size $|\\g|$ with $b(\\g)$ many $2$-connected components, we set\n\\[\\pr[_{u}]{\\g} := \\frac{\\rho(u)^{|\\g|}u^{b(\\g)}}{|\\g|!C^\\bullet(\\rho(u),u)},\\qquad\\text{and}\\qquad \\pr[_{n,u}]{\\g} := \\frac{u^{b(\\g)}}{[x^n] C^\\bullet(x,u)}\\ind_{|\\g|=n}.\\]\n\nWe show that, when considering a block-weighted law on a connected rooted graphs from a block-stable classes (including connected rooted planar graphs), this block tree follows a \\BGW{} tree distribution (see \\cref{prop:law-TG}). Our first main theorem is about phase transitions in the criticality of this \\BGW{} tree.", "full_context": "In the case of planar maps, such a tree was described by Addario-Berry \\cite{2Louigi}, following Tutte's description of the block decomposition \\cite{tutte_1963}. Moreover, Addario-Berry \\cite{2Louigi} showed that this tree could be used to obtain information about \\emph{random} planar maps: Since the block tree of a uniform planar map of size $n$ follows the law of a \\emph{\\BGW{} tree}\\footnote{Also called \\emph{Galton--Watson tree} or \\emph{Bienaymé--Galton--Watson tree}, we adopt the change of notation proposed by Addario-Berry, Brandenberger, Hamdan, and Kerriou \\cite{ABBHK22}.}, whose properties are well-known, properties on the sizes of the $2$-connected components of a random map follow. It was subsequently shown that this approach fruitfully allows one to consider \\emph{block-weighted models}: instead of sampling according to a uniform law, one considers a parameter $u>0$, and samples a map of size $n$ with probability proportional to $u$ to the power of its number of blocks. (Note that $u=1$ corresponds to the usual uniform law.) Fleurat and the second author \\cite{FleuratSalvy23} showed that the block tree of such a random planar map still follows a \\BGW{} law, and used this to obtain results on the sizes of blocks of such maps as well as scaling limits. More precisely, they discovered a phase transition between the universality classes of maps (converging to the Brownian sphere) and plane trees (converging to the Brownian tree), depending on the value of $u$, obtaining precise results on block sizes in each phase.\n\n\\subsection{Main results}\nThe main contributions of this paper are two-fold. First, we show that a random connected rooted labelled graph from a block-stable class (e.g., a random connected planar graph), which is sampled according to a block-weighted law, exhibits a phase transition (see \\cref{thm:phase-transition}). To this end, we introduce a \\emph{decorated block tree} that encodes a block decomposition of such a connected graph (see \\cref{sec:decorated}). This extends the lines of previous study based on a block tree decomposition from planar \\emph{maps} to connected planar \\emph{graphs}. Because block-stable classes are very general, our results are also very general, but not necessarily precise.\n\nSecond, we show how the block tree methodology can be used to obtain more precise results. First, we use it to obtain enumeration results for the block-weighted generating function. Then we determine the distribution of block sizes of random block-weighted planar graphs and block-stable graphs of the same singular type (see \\cref{thm:block-sizes}). As a result, we obtain more precise information on block size of a uniform random connected planar graph than what was previously known in the literature \\cite{GNR13, PS11}.\n\n\\subsubsection{Phase transition in block-weighted block-stable graphs}\nA class $\\cC$ of connected graphs is said to be \\emph{block-stable} if for each graph $\\g \\in \\cC$ each of its $2$-connected components (called \\emph{blocks}) belongs to $\\cC$. Write $\\cB$ for the class of $2$-connected elements of $\\cC$. Write $B(y)$ and $C(x)$ for the corresponding exponential generating functions (EGFs), and set $C^\\bullet(x) = x C(x)$. It was shown, e.g., by Robinson \\cite[Theorem 4]{Rob70}, that\n\\begin{equation}\n\\label{eq:Cbullet-block}\nC^\\bullet(x) = x \\exp(B'(C^\\bullet(x))).\n\\end{equation}\nAs we will see in \\cref{subsec:anal-comb-block-tree}, this equation can be translated into a ``decorated'' block tree.\n\nMore precisely, let $C^\\bullet(x,u)$ be the exponential generating function for \\emph{connected rooted} graphs of $\\cC$, where $x$ marks vertices and $u$ marks blocks. Write $\\rho(u)$ for the radius of convergence of $C^\\bullet(\\cdot,u)$. For $u>0$, we consider the following \\emph{block-weighted} probability laws on $\\cC^\\bullet$: for a graph $\\g \\in \\cC$ of size $|\\g|$ with $b(\\g)$ many $2$-connected components, we set\n\\[\\pr[_{u}]{\\g} := \\frac{\\rho(u)^{|\\g|}u^{b(\\g)}}{|\\g|!C^\\bullet(\\rho(u),u)},\\qquad\\text{and}\\qquad \\pr[_{n,u}]{\\g} := \\frac{u^{b(\\g)}}{[x^n] C^\\bullet(x,u)}\\ind_{|\\g|=n}.\\]\n\nWe show that, when considering a block-weighted law on a connected rooted graphs from a block-stable classes (including connected rooted planar graphs), this block tree follows a \\BGW{} tree distribution (see \\cref{prop:law-TG}). Our first main theorem is about phase transitions in the criticality of this \\BGW{} tree.\n\n\\subsubsection{Phase transition in block-weighted block-stable graphs}\nA class $\\cC$ of connected graphs is said to be \\emph{block-stable} if for each graph $\\g \\in \\cC$ each of its $2$-connected components (called \\emph{blocks}) belongs to $\\cC$. Write $\\cB$ for the class of $2$-connected elements of $\\cC$. Write $B(y)$ and $C(x)$ for the corresponding exponential generating functions (EGFs), and set $C^\\bullet(x) = x C(x)$. It was shown, e.g., by Robinson \\cite[Theorem 4]{Rob70}, that\n\\begin{equation}\n\\label{eq:Cbullet-block}\nC^\\bullet(x) = x \\exp(B'(C^\\bullet(x))).\n\\end{equation}\nAs we will see in \\cref{subsec:anal-comb-block-tree}, this equation can be translated into a ``decorated'' block tree.\n\nWe show that, when considering a block-weighted law on a connected rooted graphs from a block-stable classes (including connected rooted planar graphs), this block tree follows a \\BGW{} tree distribution (see \\cref{prop:law-TG}). Our first main theorem is about phase transitions in the criticality of this \\BGW{} tree.\n\nFor a more precise version of this theorem, see \\cref{th:phase-trans-singular,thm:phasetransition_block_weighted}. For the special case of planar graphs (and similar classes), we also show that the subcritical case corresponds to a ``rooted graph phase'' while the supercritical case corresponds to a ``tree phase''. Our key proof techniques build upon the methods previously developed by Fleurat and the second author for block-weighted planar maps \\cite{FleuratSalvy23}, where analogous behaviour was exhibited.\n\n\\section{Block-weighted laws for block-stable classes}\n\\label{sec:block-weighted}\nIn this section, we study the main objects of this paper, namely \\emph{block-weighted} connected graphs. We recall that these are graphs sampled according to a block-weighted law. As explained in the introduction, let $C^\\bullet(x,u)$ be the exponential generating function for \\emph{connected rooted} graphs of $\\cC$, where $x$ marks vertices and $u$ marks blocks. Let $\\rho(u)$ denote the radius of convergence of $C^\\bullet(\\cdot,u)$. From now on, we always assume $u>0$. We consider the following \\emph{block-weighted} probability laws on $\\cC^\\bullet$: For a graph $\\g \\in \\cC^\\bullet$ with $b(\\g)$ many $2$-connected components, we set\n\\[\\pr[_{u}]{\\g} := \\frac{\\rho(u)^{|\\g|}u^{b(\\g)}}{|\\g|!C^\\bullet(\\rho(u),u)}\\qquad\\text{and}\\qquad \\pr[_{n,u}]{\\g} := \\frac{u^{b(\\g)}}{[x^n] C^\\bullet(x,u)}\\ind_{|\\g|=n}.\\]\n\n\\begin{proposition}\n\\label{prop:law-TG}\nFix $u>0$, let $\\G$ be distributed according to $\\Pu$, and denote by $(\\bTT,(\\bphi_v,v\\in \\bTT))$ the decorated block tree $\\TT_\\G$, where $\\bTT$ is the undecorated block tree and $(\\bphi_v,v\\in \\bTT)$ is its family of vertex decorations. Then the law of $\\TT_\\G$ can be described as follows:\n\\begin{itemize}\n\\item $\\bTT$ follows the law $\\BGWl{\\muT^u}$;\n\\item Conditionally given $\\bTT=\\t$, the decorations $(\\bphi_{v},v\\in \\t)$ are independent random variables, and, for $v\\in\\t$, $\\bphi_{v}$ is distributed according to $\\Gamma \\Phi(C^\\bullet(\\rho(u),u),u)$ conditioned to have size $d_\\t(v)$.\n\\end{itemize}\nFor every $n\\in\\Npos$, the same statements hold under $\\Pnu$, only replacing $\\BGWl{\\muT^u}$ with $\\BGWl{\\muT^u,n}$.\n\\end{proposition}\n\n\\begin{theorem}\\label{th:phase-trans-singular}\nLet $\\cC$ be a block-stable class of connected graphs, let $\\cB$ be its class of $2$-connected components with EGF $B(y)$, and let $\\rho_B$ denote the radius of convergence of $B$. Let\n\\[u_C := \\frac{1}{\\rho_B B''(\\rho_B)}.\\]\nThen the radius of convergence $\\rho(u)$ of the series $C^\\bullet(\\cdot,u)$ (on block-weighted rooted elements of $\\cC$) and $y(u) = C^\\bullet(\\rho(u),u)$ satisfy the following:\n\\begin{itemize}\n \\item If $u\\leq u_C$, then\n \\[\\rho(u) = \\frac{\\rho_B}{\\exp(uB'(\\rho_B))}\\qquad\\text{and}\\qquad y(u)=\\rho_B;\\]\n \\item If $u\\geq u_C$, then\n \\[\\rho(u) = \\frac{1}{uB''(y(u))\\exp(uB'(y(u)))}\\qquad\\text{and}\\qquad y(u)=\\frac{1}{uB''(y(u))}.\\]\n\\end{itemize}\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:phasetransition_block_weighted}\nLet $\\Gnu$ be a graph of size $n$ from a block stable class distributed according to $\\Pnu$. Then its block tree $\\Tnu$ follows the law $\\BGWl{\\muT^u, n}$ where $\\muT^u$ satisfies the following.\n\\begin{itemize}\n \\item If $u<u_C$, then\n \\begin{equation}\n \\label{eq:exp-muu}\n \\E{\\muT^u} = \\frac{u}{u_C}<1,\n \\end{equation} so the \\BGW{} law is subcritical;\n \\item If $u\\geq u_C$, then \\[\\E{\\muT^u} = 1,\\] so the \\BGW{} law is critical.\n\\end{itemize}\n\\end{theorem}\n\nWe want to understand the behaviour of an element of the class $\\Phi$ with a fixed size. We use the fact that the function $\\Phi_u := \\Phi(\\cdot, u)$ can be written as\n\\[\\Phi_u(y) = V(W(y)),\\]\nfor $V (\\cdot) := \\exp (\\cdot) $ and $W (\\cdot) := uB'(\\cdot)$. The \\emph{Gibbs partition} framework allows one to study the size of the elements of the class $\\cW$ (whose EGF is given by $W (\\cdot)$) in a random element $\\varphi$ drawn from a Boltzmann law on $\\Phi_u$. In particular, \\cite[Theorem 3.11]{Stu24} allows us to prove the following proposition.\n\\begin{proposition}\n\\label{prop:gibbs-partition}\nLet $\\mnu{j}$ denote the size of a largest element in the set decorating $\\vnu{j}$ in $\\Tnu$. Let $R$ be a random variable with probability generating function\n\\[P_R(z) := \\E{z^R} = \\frac{\\exp(uB'(\\rho_Bz))}{\\exp(uB'(\\rho_B))}.\\]\nThen for any $j \\in \\mathbb{N}_0$, it holds that\n\\[\\bdTnu{j} - \\mnu{j} \\xrightarrow[n\\to\\infty]{(d)} R,\\]\nso, in particular,\n\\[\\bdTnu{j} - \\mnu{j} = O_{\\mathbb{P}}(1)\\qquad\\text{as}\\quad n\\to\\infty.\\]\n\\end{proposition}\n\\begin{proof}\nThe collection of block sizes in a random decoration of a vertex of outdegree $k$ in $\\Tnu$ can be interpreted as a Gibbs partition. As described in \\cite[Theorem 3.11]{Stu24}, under certain conditions a Gibbs partition of size $k$ has a unique giant component whose size $\\bm_k$ satisfies\n\\begin{equation}\n\\label{eq:gibbs-result}\nk - \\bm_k \\xrightarrow[k\\to\\infty]{(d)} S \\coloneqq \\sum_{i=1}^{\\hat{N} - 1} X_i.\n\\end{equation}\nHere, the $X_i$'s are independent copies of a random variable $X$ whose probability generating function is\n\\[P_X(z) \\coloneqq \\mathbb{E}[z^X] = \\frac{W(\\rho_W z)}{W(\\rho_W)},\\]\nand $\\hat{N}$ is a random variable whose probability generation function is\n\\[P_{\\hat{N}}(z) \\coloneqq \\mathbb{E}[z^{\\hat{N}}] = \\frac{V'(W(\\rho_W)z)z}{V'(W(\\rho_W))}.\\]\nOne can check that the probability generating function of the random variable $S$ is\n\\[P_S(z) \\coloneqq \\mathbb{E}[z^S] = \\frac{P_{\\hat{N}}(P_X(z))}{P_X(z)} = \\frac{\\exp(uB'(\\rho_Bz))}{\\exp(uB'(\\rho_B))},\\]\nso $S$ and $R$ have the same distribution. \\Cref{prop:gibbs-partition} is a then consequence of:\n\\begin{itemize}\n \\item Stufler \\cite[Theorem 3.11]{Stu24};\n \\item For all $j\\in\\Npos$, $\\bdTnu{j}$ converges in probability to $+\\infty$;\n \\item The fact that one can sample $\\Gnu$ according to $\\Pnu$ by first sampling $\\Tnu$, then its decorations (see \\cref{prop:law-TG}).\n\\end{itemize}", "post_theorem_intro_text_len": 7161, "post_theorem_intro_text": "For a more precise version of this theorem, see  \\cref{th:phase-trans-singular,thm:phasetransition_block_weighted}. For the special case of planar graphs (and similar classes), we also show that the subcritical case corresponds to a ``rooted graph phase'' while the supercritical case corresponds to a ``tree phase''. Our key proof techniques build upon the methods previously developed by Fleurat and the second author for block-weighted planar maps \\cite{FleuratSalvy23}, where analogous behaviour was exhibited.\n\nIt is well-known that the relative positions of $\\rho_B B''(\\rho_B)$ and $1$ allow us to distinguish (sub)criticality (see, e.g., \\cite{DFKKR11}). More precisely, when $\\rho_B B''(\\rho_B) > 1$, it is known that the graphs behave ``tree-like'', which is consistent with $u_C < 1$ and the uniform case being supercritical. On the contrary, when $\\rho_B B''(\\rho_B) < 1$, the graphs behave as general planar graphs do, which is consistent with $u_C > 1$.\n\nWe note that block-weighted block-stable graph classes were already considered by Stufler \\cite[\\S5]{Stu18b} in a general setting using the framework of Gibbs partitions, but our current work has a different focus.\n\n\\subsubsection{Enumeration and block sizes in block-weighted planar graphs}\nOur next results focus on the case where $B'$ has a singularity of type $3/2$ (which encompasses planar graphs). Under this assumption, we also show how the phase transition described in \\cref{thm:phase-transition} affects the behaviour of $[x^n] C^\\bullet(x,u)$, as well as the typical sizes of blocks. The following is our main result on the former.\n\n\\begin{theorem}\\label{thm:enum} For every $u>0$, there exist $c > 0$, $c_1(u)>0$, and $c_2(u)>0$ such that:\n\\begin{itemize}\n    \\item in the subcritical phase $u < u_C$,\n    \\[[x^n] C^\\bullet(x,u) \\simn c_1(u) n^{-5/2} \\rho(u)^{-n};\\]\n    \\item in the critical phase $u = u_C$,\n    \\[[x^n] C^\\bullet(x,u) \\simn c n^{-5/3} \\rho(u_C)^{-n};\\]\n    \\item in the supercritical phase $u > u_C$,\n    \\[[x^n] C^\\bullet(x,u) \\simn c_2(u) n^{-3/2} \\rho(u)^{-n}.\\]\n\\end{itemize}\n\\end{theorem}\n\nFor a more precise version of this theorem, see \\cref{th:dvp-sing-Cbxu}. Thus, in the subcritical phase, the polynomial correction is that of rooted graphs: $n^{-5/2}$. Conversely, in the supercritical phase, the correction is that of plane trees: $n^{-3/2}$. This justifies why one can view the subcritical phase as a ``general rooted graph phase'' and the supercritical phase as a ``tree phase''.  It is worth noting that in the critical phase, i.e., when $u = u_C$, a novel asymptotic behaviour emerges, characterised by a polynomial correction of $n^{-5/3}$.\n\nThe next theorem summarises our main results on the typical block sizes in the three different phases.\n\n\\begin{theorem}\\label{thm:block-sizes}\nThere is a phase transition for block sizes depending on the value of $u$:\n\\begin{itemize}\n    \\item In the subcritical phase $u < u_C$, there is a largest block of size $\\Theta_\\mathbb{P}(n)$ and a second largest block of size $\\Theta_\\mathbb{P}(n^{2/3})$;\n    \\item In the critical phase $u = u_C$, there is a largest block of size $\\Theta_\\mathbb{P}(n^{2/3})$ and likewise for the next largest block sizes;\n    \\item In the supercritical phase $u > u_C$, there is a largest block of size $\\Theta_\\mathbb{P}(\\ln (n))$ and likewise for the next largest block sizes.\n\\end{itemize}\n\\end{theorem}\n\nAs we will see in \\cref{thm:phasetransition_block_sizes}, we also determine precise limit laws for the block sizes when appropriately scaled, in the subcritical and critical cases,  and provide a more precise description of what happens in the supercritical case. Our results extend some previous work in the literature \\cite{GNR13,PS11} from uniform to block-weighted random planar graphs. In fact, the above two results hold for any block-stable class where $B'$ is \\emph{singular with exponent $3/2$} (for a definition, see e.g., \\cite[Equation (46)]{airy}). This includes the class of planar graphs as shown by Bender, Gao, and Wormald \\cite{bender2002number} (see \\cref{eq:asymp-exp-B'} for the singular expansion of $B'$).\n\n\\subsection{Key proof techniques and related work}\n\nThe primary proof methods used in this paper build on those used for block-weighted planar maps in \\cite{FleuratSalvy23}, with some ideas originating from \\cite{2Louigi}. The Lagrangean form of \\cref{eq:Cbullet-block} implicitly gives a tree-like structure and we make it combinatorially explicit in the form of a \\emph{decorated block tree}, detailed in \\cref{subsec:anal-comb-block-tree}. Our decorated block tree looks similar to the usual notion of a block decomposition of a connected graph, but is different in that the vertices of our tree represent cut-vertices instead of blocks, and these cut-vertices are decorated by sets of blocks. We use tools from singularity analysis and results about \\BGW{} trees to pinpoint and describe a phase transition, as in \\cref{thm:phase-transition,thm:enum}.\n\nTo prove our results about typical block sizes in random block-weighted planar graphs, as in \\cref{thm:block-sizes}, we also incorporate some ideas on planar maps from \\cite{FleuratSalvy23}, but extra steps are needed due to the fact that our block tree decorations are not blocks, but \\emph{sets of blocks}, or more precisely, elements of the class $\\Set\\circ \\mathcal{B}'$. First we apply known results on degree distributions of \\BGW{} trees (by, e.g., Janson \\cite{survey-trees}) to our block tree. Then we combine this with results on Gibbs partitions by Stufler \\cite{Stu24} to deduce typical block sizes. Similar ideas were used to study block-weighted simple triangulations in \\cite[\\S4.2]{Sal24}, where the decorations were sequences of blocks.\n\nFinally, we mention some other related works. In the uniform case, Panagiotou and Steger \\cite{PS11} proved a dichotomy theorem on the behaviour of the size of a largest block in connected graph classes: For classes similar to planar graphs there is a unique largest block of linear size almost surely, whereas for classes similar to series-parallel graphs all components are at most logarithmic in size almost surely. Gim\\'enez, Noy, and Ru\\'e \\cite[Theorem 5.4]{GNR13} subsequently refined the dichotomy theorem of Panagiotou and Steger \\cite{PS11} and also showed a limiting distribution on the sizes of the largest blocks, involving the Airy distribution. As mentioned earlier, our work considers in particular planar graphs with block weights and thus generalises some of the results in the works above.\n\n\\subsection{Plan of the paper}\nIn \\cref{sec:preliminaries}, we give notation and recall usual notions of block decompositions of graphs. In \\cref{sec:decorated}, we detail the construction of the decorated block tree of a rooted graph. In \\cref{sec:block-weighted}, we introduce the block-weighted model for block-stable graph classes and obtain results about it. In \\cref{sec:block-sizes}, we focus on classes where $B'$ is singular with exponent $3/2$ (which includes planar graphs), and obtain results on enumeration and block sizes. In \\cref{sec:conclusion}, we discuss possible extensions and future work.", "sketch": "The post-theorem text does not give a step-by-step proof of Theorem~\\ref{thm:phase-transition}, but it does outline the main ingredients used to establish it. The argument is said to build on methods for block-weighted planar maps \\cite{FleuratSalvy23}. The key structural input is that “The Lagrangean form of \\cref{eq:Cbullet-block} implicitly gives a tree-like structure” which the authors “make … combinatorially explicit in the form of a \\emph{decorated block tree}.” This tree is “different in that the vertices of our tree represent cut-vertices instead of blocks, and these cut-vertices are decorated by sets of blocks.” With this explicit tree representation in hand, they “use tools from singularity analysis and results about \\BGW{} trees to pinpoint and describe a phase transition, as in \\cref{thm:phase-transition,thm:enum},” yielding the subcritical/critical dichotomy for the block tree law.\n\nThey also connect the phase transition threshold to a known (sub)criticality criterion: “It is well-known that the relative positions of $\\rho_B B''(\\rho_B)$ and $1$ allow us to distinguish (sub)criticality,” with the interpretation that when $\\rho_B B''(\\rho_B) > 1$ the graphs behave “tree-like” (consistent with $u_C<1$), while when $\\rho_B B''(\\rho_B) < 1$ they behave like “general planar graphs” (consistent with $u_C>1$).", "expanded_sketch": "The post-theorem text does not give a step-by-step proof of the main theorem, but it does outline the main ingredients used to establish it. The argument is said to build on methods for block-weighted planar maps Fleurat and Salvy, title unknown (2023). The key structural input is that “The Lagrangean form of\n\\begin{equation}\n\\label{eq:Cbullet-block}\nC^\\bullet(x) = x \\exp(B'(C^\\bullet(x))).\n\\end{equation}\nimplicitly gives a tree-like structure” which the authors “make … combinatorially explicit in the form of a \\emph{decorated block tree}.” This tree is “different in that the vertices of our tree represent cut-vertices instead of blocks, and these cut-vertices are decorated by sets of blocks.” With this explicit tree representation in hand, they “use tools from singularity analysis and results about \\BGW{} trees to pinpoint and describe a phase transition,” yielding the subcritical/critical dichotomy for the block tree law.\n\nThey also connect the phase transition threshold to a known (sub)criticality criterion: “It is well-known that the relative positions of $\\rho_B B''(\\rho_B)$ and $1$ allow us to distinguish (sub)criticality,” with the interpretation that when $\\rho_B B''(\\rho_B) > 1$ the graphs behave “tree-like” (consistent with $u_C<1$), while when $\\rho_B B''(\\rho_B) < 1$ they behave like “general planar graphs” (consistent with $u_C>1$).", "expanded_theorem": "\\label{thm:phase-transition}\nLet $\\cC$ be a block-stable class of connected graphs, let $\\cB$ be its class of $2$-connected components whose EGF we denote by $B(y)$, and let $\\rho_B$ be the radius of convergence of $B$. Let\n\\[u_C := \\frac{1}{\\rho_B B''(\\rho_B)}.\\]\nThen the following hold. \n\\begin{itemize}\n    \\item The singular behaviour of $C^{\\bullet}(x,u)$ depends on the relative positions of $u$ and $u_C$;\n    \\item The block tree corresponding to a graph drawn according to $\\Pnu$ follows a subcritical \\BGW{} tree law if $u<u_C$, and a critical \\BGW{} tree law if $u \\ge u_C$.\n\\end{itemize}", "theorem_type": ["Universal", "Classification or Bijection"], "mcq": {"question": "Let \\(\\mathcal C\\) be a block-stable class of connected labelled graphs, meaning that every 2-connected component (block) of every graph in \\(\\mathcal C\\) again belongs to \\(\\mathcal C\\). Let \\(\\mathcal B\\) be the class of 2-connected graphs in \\(\\mathcal C\\), let \\(B(y)\\) be its exponential generating function, and let \\(\\rho_B\\) be the radius of convergence of \\(B\\). Define\n\\[\nu_C:=\\frac{1}{\\rho_B B''(\\rho_B)}.\\]\nFor \\(u>0\\), let \\(C^{\\bullet}(x,u)\\) be the exponential generating function of connected rooted graphs in \\(\\mathcal C\\), where \\(x\\) marks vertices and \\(u\\) marks blocks. For each \\(n\\), let \\(\\mathbb P_{n,u}\\) be the block-weighted law on rooted connected graphs of size \\(n\\), assigning probability proportional to \\(u^{b(G)}\\), where \\(b(G)\\) is the number of blocks of \\(G\\). The associated block tree is the tree encoding the block decomposition of such a sampled graph. Which statement holds for every such class \\(\\mathcal C\\)?", "correct_choice": {"label": "A", "text": "The singular behaviour of \\(C^{\\bullet}(x,u)\\) depends on whether \\(u<\\nu_C\\) or \\(u\\ge \\nu_C\\); moreover, if a graph is sampled according to \\(\\mathbb P_{n,u}\\), then its block tree has a subcritical Bienaymé–Galton–Watson law when \\(u<\\nu_C\\), and a critical Bienaymé–Galton–Watson law when \\(u\\ge \\nu_C\\)."}, "choices": [{"label": "B", "text": "The singular behaviour of \\(C^{\\bullet}(x,u)\\) depends on whether \\(u<\\nu_C\\) or \\(u>\\nu_C\\); moreover, if a graph is sampled according to \\(\\mathbb P_{n,u}\\), then its block tree has a subcritical Bienaymé–Galton–Watson law when \\(u\\le \\nu_C\\), and a critical Bienaymé–Galton–Watson law when \\(u>\\nu_C\\)."}, {"label": "C", "text": "If a graph is sampled according to \\(\\mathbb P_{n,u}\\), then its block tree has a subcritical Bienaymé–Galton–Watson law when \\(u<\\nu_C\\), and a critical Bienaymé–Galton–Watson law when \\(u\\ge \\nu_C\\)."}, {"label": "D", "text": "The singular behaviour of \\(C^{\\bullet}(x,u)\\) depends on whether \\(u<\\nu_C\\) or \\(u\\ge \\nu_C\\); moreover, if a graph is sampled according to \\(\\mathbb P_{n,u}\\), then its block tree has a subcritical Bienaymé–Galton–Watson law whenever \\(u<\\nu_C\\), and there exists a critical Bienaymé–Galton–Watson law at \\(u=\\nu_C\\) only, while for every \\(u>\\nu_C\\) the law is no longer critical."}, {"label": "E", "text": "The singular behaviour of \\(C^{\\bullet}(x,u)\\) depends on whether \\(u<\\nu_C\\) or \\(u\\ge \\nu_C\\); moreover, if a graph is sampled according to \\(\\mathbb P_{n,u}\\), then its block tree has a critical Bienaymé–Galton–Watson law when \\(u<\\nu_C\\), and a subcritical Bienaymé–Galton–Watson law when \\(u\\ge \\nu_C\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "boundary case at the threshold \\(u=\\nu_C\\)", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "characteristic", "tampered_component": "drops the singular-behaviour clause while retaining the BGW phase-transition conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "characteristic", "tampered_component": "uniform criticality for all \\(u\\ge \\nu_C\\)", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "subcritical/critical correspondence relative to the threshold", "template_used": "property_confusion"}]}, "qa quality eval": {"ALS": {"score": 1, "justification": "The stem explicitly defines the quantity 1/(rho_B B''(rho_B)) under a different name (nu_C), and the correct option uses exactly that expression as the threshold. This does not fully reveal the whole answer, but it gives a strong hint about the critical value."}, "TAS": {"score": 1, "justification": "The item is close to a theorem-recall question: it asks for the stated phase classification associated with a previously defined threshold. However, it is not a pure tautology because the options differ on boundary inclusion, critical vs supercritical behaviour, and whether the BGW clause is complete."}, "GPS": {"score": 1, "justification": "Some reasoning is needed to distinguish subtle alternatives, especially the treatment of u = u_C and the critical/subcritical classification. Still, the question mainly tests precise recall of a known result rather than generating a conclusion from first principles."}, "DQS": {"score": 2, "justification": "The distractors are plausible and mathematically targeted: one flips the boundary case, one omits an essential clause, one substitutes supercritical for critical, and one uses the reciprocal threshold incorrectly. These reflect realistic failure modes."}, "total_score": 5, "overall_assessment": "A decent but theorem-close MCQ: distractors are strong, but the stem leaks the critical expression and the task is largely precise recall rather than genuinely generative reasoning."}}
{"id": "2602.23966v1", "paper_link": "http://arxiv.org/abs/2602.23966v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{mainthm}\nLet $X \\subset \\mathbb{P}^{n+1}\\times \\mathbb{P}^{m+1}$ be a very general irreducible smooth hypersurface of bi-degree $(a,b)$. Suppose that $a\\geq 2n+m+1$, $b\\geq 2m+n+1$, and $n,m \\geq 2 $. Then:\n\\begin{center}\n    ${ \\minimum \\Big\\{a-\\lfloor \\frac{\\sqrt{16n+9}-1}{2} \\rfloor,b-\\lfloor \\frac{\\sqrt{16m+9}-1}{2} \\rfloor \\Big\\} \\leq \\covgon(X)\\leq \\minimum \\Big\\{a-\\lfloor \\frac{\\sqrt{16n+1}-1}{2} \\rfloor,b-\\lfloor \\frac{\\sqrt{16m+1}-1}{2} \\rfloor \\Big\\} }$.\n\\end{center}\nMoreover, both sides of the inequality coincide whenever $m,n \\in \\{4\\alpha^{2}+3\\alpha, 4\\alpha^{2}+5\\alpha+1, \\alpha \\in \\mathbb{N}\\}$", "start_pos": 6193, "end_pos": 6840, "label": "mainthm"}, "ref_dict": {}, "pre_theorem_intro_text_len": 3123, "pre_theorem_intro_text": "\\end{center}\n\n\\mdseries\nA central problem in algebraic geometry is to determine the extent to which a given algebraic variety $X$ admits rationality properties—such as rationality, unirationality, or rational connectedness. Within the past few years, natural invariants which measure these rationality properties have been extensively studied. The foundational case is the one-dimensional setting. Consider $C$, an irreducible smooth curve. The gonality of $C$ is defined as the minimal degree of a morphism  $\\phi : C \\rightarrow \\mathbb{P}^{1}$ and we denote it $\\gonality(C)$. In the higher dimensional setting, various extensions of this definition exist, depending on the rationality property of interest. Consider a smooth projective variety $X$ of dimension $n$. On the one hand, the degree of irrationality of $X$ is defined as the minimal degree of a rational dominant map $f : X \\dashrightarrow \\mathbb{P}^{n}$. We denote it $\\irr(X)$.  Then, $X$ is rational if and only if $\\irr(X)=1$. This definition coincides with the gonality in the one-dimensional case. On the other hand, the covering gonality of $X$ is defined as the minimal gonality of a curve $C$ passing through a general point $x \\in X $. We denote it $\\covgon(X)$. One can notice that $X$ is uniruled if and only if $\\covgon(X)=1$. \\\\\nHistorically, the first result is due to Max Noether who showed that the gonality of a smooth plane curve $C$ of degree $d\\geq 3$ is computed by the projection from a point $x\\in C$. This result was subsequently generalized in \\cite{degreehypersurfaces} for a very general irreducible smooth hypersurface $X\\subset\\mathbb{P}^{n}$ of degree $d\\geq 2n+1$. The main result of \\cite{degreehypersurfaces} is that $\\irr(X)$ is computed by the projection from a point $x\\in X$.  More generally, these invariants are actively studied in different settings, see e.g.,  \\cite{chen2020fano}, \\cite{stapleton2020degree}, \\cite{chen2023rational}, \\cite{chen2021multiplicative}, \\cite{colombo2022degree}, \\cite{bastianelli2022covering}, \\cite{moretti2024degree}, \\cite{voisin2022fibrations} and for a global survey the reader can consult \\cite{chen2025primermeasuresirrationality}. In this note, we are interested in the covering gonality. In recent years, several results have been established in various contexts. For instance, in \\cite{martin2020conjecture},  Martin exhibits a lower bound for the covering gonality of a very general abelian variety in any dimension answering a conjecture of Voisin in \\cite{voisin2018chow}. Also, in \\cite{chen2021multiplicative}, the author gives a multiplicative lower bound for the covering gonality of certain complete intersections. Finally, in \\cite{bastianelli2019gonality} the authors compute the covering gonality of a very general irreducible smooth hypersurface $X\\subset \\mathbb{P}^{n+1}$ of degree $d\\geq 2n+2$. This work actually originated in \\cite{lopez1995curves}, \\cite{degreehypersurfaces} in which the authors investigated the lower dimensional cases. \n\nInspired by the result \\cite[Theorem 1.1]{bastianelli2019gonality}, we prove the following theorem:\n\n\\vspace{0.35cm}", "context": "\\end{center}\n\n\\mdseries\nA central problem in algebraic geometry is to determine the extent to which a given algebraic variety $X$ admits rationality properties—such as rationality, unirationality, or rational connectedness. Within the past few years, natural invariants which measure these rationality properties have been extensively studied. The foundational case is the one-dimensional setting. Consider $C$, an irreducible smooth curve. The gonality of $C$ is defined as the minimal degree of a morphism $\\phi : C \\rightarrow \\mathbb{P}^{1}$ and we denote it $\\gonality(C)$. In the higher dimensional setting, various extensions of this definition exist, depending on the rationality property of interest. Consider a smooth projective variety $X$ of dimension $n$. On the one hand, the degree of irrationality of $X$ is defined as the minimal degree of a rational dominant map $f : X \\dashrightarrow \\mathbb{P}^{n}$. We denote it $\\irr(X)$. Then, $X$ is rational if and only if $\\irr(X)=1$. This definition coincides with the gonality in the one-dimensional case. On the other hand, the covering gonality of $X$ is defined as the minimal gonality of a curve $C$ passing through a general point $x \\in X $. We denote it $\\covgon(X)$. One can notice that $X$ is uniruled if and only if $\\covgon(X)=1$. \\\\\nHistorically, the first result is due to Max Noether who showed that the gonality of a smooth plane curve $C$ of degree $d\\geq 3$ is computed by the projection from a point $x\\in C$. This result was subsequently generalized in \\cite{degreehypersurfaces} for a very general irreducible smooth hypersurface $X\\subset\\mathbb{P}^{n}$ of degree $d\\geq 2n+1$. The main result of \\cite{degreehypersurfaces} is that $\\irr(X)$ is computed by the projection from a point $x\\in X$. More generally, these invariants are actively studied in different settings, see e.g., \\cite{chen2020fano}, \\cite{stapleton2020degree}, \\cite{chen2023rational}, \\cite{chen2021multiplicative}, \\cite{colombo2022degree}, \\cite{bastianelli2022covering}, \\cite{moretti2024degree}, \\cite{voisin2022fibrations} and for a global survey the reader can consult \\cite{chen2025primermeasuresirrationality}. In this note, we are interested in the covering gonality. In recent years, several results have been established in various contexts. For instance, in \\cite{martin2020conjecture}, Martin exhibits a lower bound for the covering gonality of a very general abelian variety in any dimension answering a conjecture of Voisin in \\cite{voisin2018chow}. Also, in \\cite{chen2021multiplicative}, the author gives a multiplicative lower bound for the covering gonality of certain complete intersections. Finally, in \\cite{bastianelli2019gonality} the authors compute the covering gonality of a very general irreducible smooth hypersurface $X\\subset \\mathbb{P}^{n+1}$ of degree $d\\geq 2n+2$. This work actually originated in \\cite{lopez1995curves}, \\cite{degreehypersurfaces} in which the authors investigated the lower dimensional cases.\n\nInspired by the result \\cite[Theorem 1.1]{bastianelli2019gonality}, we prove the following theorem:\n\n\\vspace{0.35cm}", "full_context": "\\end{center}\n\n\\mdseries\nA central problem in algebraic geometry is to determine the extent to which a given algebraic variety $X$ admits rationality properties—such as rationality, unirationality, or rational connectedness. Within the past few years, natural invariants which measure these rationality properties have been extensively studied. The foundational case is the one-dimensional setting. Consider $C$, an irreducible smooth curve. The gonality of $C$ is defined as the minimal degree of a morphism $\\phi : C \\rightarrow \\mathbb{P}^{1}$ and we denote it $\\gonality(C)$. In the higher dimensional setting, various extensions of this definition exist, depending on the rationality property of interest. Consider a smooth projective variety $X$ of dimension $n$. On the one hand, the degree of irrationality of $X$ is defined as the minimal degree of a rational dominant map $f : X \\dashrightarrow \\mathbb{P}^{n}$. We denote it $\\irr(X)$. Then, $X$ is rational if and only if $\\irr(X)=1$. This definition coincides with the gonality in the one-dimensional case. On the other hand, the covering gonality of $X$ is defined as the minimal gonality of a curve $C$ passing through a general point $x \\in X $. We denote it $\\covgon(X)$. One can notice that $X$ is uniruled if and only if $\\covgon(X)=1$. \\\\\nHistorically, the first result is due to Max Noether who showed that the gonality of a smooth plane curve $C$ of degree $d\\geq 3$ is computed by the projection from a point $x\\in C$. This result was subsequently generalized in \\cite{degreehypersurfaces} for a very general irreducible smooth hypersurface $X\\subset\\mathbb{P}^{n}$ of degree $d\\geq 2n+1$. The main result of \\cite{degreehypersurfaces} is that $\\irr(X)$ is computed by the projection from a point $x\\in X$. More generally, these invariants are actively studied in different settings, see e.g., \\cite{chen2020fano}, \\cite{stapleton2020degree}, \\cite{chen2023rational}, \\cite{chen2021multiplicative}, \\cite{colombo2022degree}, \\cite{bastianelli2022covering}, \\cite{moretti2024degree}, \\cite{voisin2022fibrations} and for a global survey the reader can consult \\cite{chen2025primermeasuresirrationality}. In this note, we are interested in the covering gonality. In recent years, several results have been established in various contexts. For instance, in \\cite{martin2020conjecture}, Martin exhibits a lower bound for the covering gonality of a very general abelian variety in any dimension answering a conjecture of Voisin in \\cite{voisin2018chow}. Also, in \\cite{chen2021multiplicative}, the author gives a multiplicative lower bound for the covering gonality of certain complete intersections. Finally, in \\cite{bastianelli2019gonality} the authors compute the covering gonality of a very general irreducible smooth hypersurface $X\\subset \\mathbb{P}^{n+1}$ of degree $d\\geq 2n+2$. This work actually originated in \\cite{lopez1995curves}, \\cite{degreehypersurfaces} in which the authors investigated the lower dimensional cases.\n\nInspired by the result \\cite[Theorem 1.1]{bastianelli2019gonality}, we prove the following theorem:\n\n\\vspace{0.35cm}\n\n\\vspace{0.35cm}\n\n\\begin{thm}\nLet $X, Y \\subset \\mathbb{P}^{n+1} \\times \\mathbb{P}^{m+1}$ be very general smooth hypersurfaces \nof bi-degrees $(a_1,b_1)$ and $(a_2,b_2)$ respectively with $a_{1},a_{2} \\geq 2n+m+2$ and $b_{1},b_{2} \\geq 2m+n+2$. Then:\n\\vspace{0.5cm}\n\\begin{center}\n $\\operatorname{cov.gon}(X,Y) \\;\\geq\\; k_1 + k_2 - n - m$\n\\end{center}\nwhere : \n\\[\nk_1 = \\min\\{a_1 - n - 3, b_1 - m - 3\\}, \\quad\nk_2 = \\min\\{a_2 - n - 3, b_2 - m - 3\\}.\n\\]\n\n\\begin{prop}\n\\label{Estimate}\nLet $a\\geq 2n+m+1,b\\geq 2m+n+1$ and $B$ the congruence of lines defined of the covering family of curves of gonality $c$ as defined above. First, either $\\mathcal{B} \\subset Gr(1,n+1) \\times \\mathbb{P}^{m+1}$ or $B\\subset \\mathbb{P}^{n+1} \\times Gr(1,m+1)$ and if $B \\subset Gr(1,n+1) \\times \\mathbb{P}^{m+1}$ (resp. $\\mathcal{B}\\subset \\mathbb{P}^{n+1} \\times Gr(1,m+1)$), then : \n\\begin{center}\n $c\\geq a-n$ (resp, $c\\geq b-m$).\n\\end{center}\n\\end{prop}\n\\begin{proof}\n First, as $\\mathcal{B}$ is an irreducible subvariety of $ Gr(1,n+1)\\times \\mathbb{P}^{m+1} \\sqcup \\mathbb{P}^{n+1}\\times Gr(1,m+1)$, the irreducibility of $\\mathcal{B}$ implies that either $\\mathcal{B} \\subset Gr(1,n+1)\\times \\mathbb{P}^{m+1}$ or $\\mathcal{B} \\subset \\mathbb{P}^{n+1}\\times Gr(1,m+1)$. Suppose now that $\\mathcal{B} \\subset Gr(1,n+1)\\times \\mathbb{P}^{m+1}$ and that $c\\leq a-n-1$. Then, for $t \\in T, y\\in \\mathbb{P}^{1}$, the fibre $\\phi^{-1}(t,y)$ lie on a line $l_{(t,y)}$ of the form $(l'_{(t,y)},z) \\subset \\mathbb{P}^{n+1} \\times \\mathbb{P}^{m+1}$, where $z\\in \\mathbb{P}^{m+1}$ and $l'_{(t,y)} \\subset \\mathbb{P}^{n+1}$ is a line. Moreover, it must satisfy the Cayley-Bacharach condition with respect to $\\vert K_{X} \\vert$. But $X_{z}$ is a hypersurface of degree $a$ in $\\mathbb{P}^{n+1}$, hence $K_{X_{z}} = (a-n-2)H$ where $H$ is the hyperplane divisor of $\\mathbb{P}^{n+1}$. Then, one can produce $E' \\in \\vert K_{X_{z}} \\vert$ such that $E'$ passes through all the points in $pr_{1}(\\phi^{-1}(t,y))$ but one as there are at most $a-n-1$ points in $pr_{1}(\\phi^{-1}(t,y))$. Then, we can produce a divisor $E \\in \\vert K_{X} \\vert \\cong \\vert \\mathcal{O}(a-n-2,b-m-2)\\vert $ such that $E$ passes through any point of $\\phi^{-1}(t,y)$ but one. Indeed, consider a divisor $E'' \\in \\vert \\mathcal{O}_{\\mathbb{P}^{n+1}\\times \\mathbb{P}^{m+1}}(b-m-2) \\vert $ passing through $w\\in \\mathbb{P}^{m+1}$. Then, the divisor $E'\\boxtimes E''$ belongs to $\\vert K_{X} \\vert $ and passes through every points of $\\phi^{-1}(t,y)$ but one. This implies that the fibre does not respect the Cayley-Bacharach condition with respect to $\\vert K_{X} \\vert$, which is a contradiction. Then $c\\geq a-n$. The proof can be done the other way around to obtain that if $\\mathcal{B} \\subset \\mathbb{P}^{n+1} \\times Gr(1,m+1)$, then $c \\geq b-m$.\n\\end{proof}\n\n\\begin{prop}\n\\label{LocalisationOfCurves}\nLet $X$ be a very general and smooth hypersurface of bi-degree $(a,b)$ in $\\mathbb{P}^{n+1} \\times \\mathbb{P}^{m+1}$, such that $a \\geq 2n+m+1$ and $b\\geq 2m+n+1$. Consider a covering family of c-gonal curve over $X$ with $c \\leq k-3$. Then for each $t \\in T$ there exists $x_{t} \\in f(\\mathcal{C}_{t})$ such that $f(\\mathcal{C}_{t}) \\subset V_{x_{t}}^{a-c}:=\\{(\\ell,z) \\in Gr(1,n+1)\\times\\mathbb{P}^{m+1}, (z,\\ell) \\cdot X \\geq (a-c)x_{t}\\}$ or $f(C_{t}) \\subset {W}^{b-c}_{x_{t}} : = \\{\\ell \\in \\mathbb{P}^{n+1}\\times Gr(1,m+1), \\ell \\cdot X \\geq (b-c)x_{t}\\} $. Moreover, the gonality of the curve $f(\\mathcal{C}_{t})$ is computed by projecting from the point $x_{t}$.\n\\end{prop}\n\n\\begin{thm}[{\\cite[Theorem 4.1]{bastianelli2019gonality}}]\n\\label{Bastianelli}\nLet $X \\subset \\mathbb{P}^{n+1}$ be a very general irreducible smooth hypersurface of degree $d\\geq 2n+2$. Then: \n\\begin{center}\n $ d-\\lfloor \\frac{\\sqrt{16n+9}-1}{2} \\rfloor \\leq \\covgon(X)\\leq d-\\lfloor \\frac{\\sqrt{16n+1}-1}{2} \\rfloor $.\n\\end{center}\n\nLet $X \\subset \\mathbb{P}^{n+1}\\times \\mathbb{P}^{m+1}$ with $m,n \\geq 2$ be a very general irreducible smooth hypersurface of bi-degree $(a,b)$, such that $a \\geq 2n+m+1,b\\geq 2m+n+1$. Then: \n\\begin{center}\n ${ \\minimum \\Big\\{a-\\lfloor \\frac{\\sqrt{16n+9}-1}{2} \\rfloor,b-\\lfloor \\frac{\\sqrt{16m+9}-1}{2} \\rfloor \\Big\\} \\leq \\covgon(X)\\leq \\minimum \\Big\\{a-\\lfloor \\frac{\\sqrt{16n+1}-1}{2} \\rfloor,b-\\lfloor \\frac{\\sqrt{16m+1}-1}{2} \\rfloor \\Big\\} }$.\n\\end{center}\nMoreover, both sides of the inequality coincide whenever $m,n \\in \\{4\\alpha^{2}+3\\alpha, 4\\alpha^{2}+5\\alpha+1, \\alpha \\in \\mathbb{N}\\}$\n\\end{thm}\n\n\\begin{proof}\nAs stated throughout this work, $X$ can alternatively be seen as a family of hypersurfaces of degree $a$ over $\\mathbb{P}^{m+1}$ or as a family of hypersurfaces of degree $b$ over $\\mathbb{P}^{n+1}$. Proposition \\ref{LocalisationOfCurves} implies that $f(C_{t})$ is contained in $V_{x_{t}}^{a-c}$ for a certain $x_{t} \\in X $ or in $ W^{b-c}_{x_{t}}$. Suppose, for instance that $f(C_{t}) \\subset V_{x_{t}}^{a-c}$. More precisely, as a line that is in $V_{x_{t}}^{a-c}$ is in $Gr(1,n+1)\\times \\pi_{2}(x_{t})$, this shows that for a general $t\\in T$ we have $f(\\mathcal{C}_{t})\\subset \\pi^{-1}_{2}(\\pi_{2}(x_{t}))$, i.e. the curve is on one of the hypersurfaces of the family. This means, that computing $\\covgon(X)$ can be done fibrewise. Using Theorem \\ref{Bastianelli}, this implies that $\\covgon(X)\\geq \\minimum \\Big\\{a-\\lfloor \\frac{\\sqrt{16n+9}-1}{2} \\rfloor,b-\\lfloor \\frac{\\sqrt{16m+9}-1}{2} \\rfloor \\Big\\}$.\nMoreover, the upper bound is clear as one can construct a family of curves on $X$ by constructing it fibrewise using the family of curves mentioned in Theorem \\ref{Bastianelli}. This yields the desired upper bound. As in \\ref{Bastianelli}, both side of the inquality coincide whenever Moreover, both sides of the inequality coincide whenever $m,n \\in \\{4\\alpha^{2}+3\\alpha, 4\\alpha^{2}+5\\alpha+1, \\alpha \\in \\mathbb{N}\\}$. \n\\end{proof}\n\\begin{rmq}\nWe would like to make some comments on the assumption on the range of the bi-degree $(a,b)$ of the hypersurfaces $X$ of our theorem. Indeed, as already stated before, the assumptions $a\\geq 2m+n+1, b\\geq 2n+m+1$ are only required to get the lower bound on $\\covgon(X)$; while the upper bound is achieved as soon as $a \\geq 2n+2, b\\geq 2m+2$. This was required to carry out the computations made during the proof of the Proposition \\ref{LocalisationOfCurves}. Following \\cite{yeong2022algebraic}, these numerical assumptions are actually the ones ensuring that $X$ is algebraically hyperbolic, otherwise, Yeong shows that $X$ contains a line. As the proof relied repeatedly on the fact that $X$ does not contain any line, one would need to change the method of the proof to possibly lower the bound on $a$ and $b$.\n\\end{rmq}", "post_theorem_intro_text_len": 3533, "post_theorem_intro_text": "\\vspace{0.35cm}\nFirst, this computes the covering gonality of such hypersurfaces up to one, in the same spirit as the result of \\cite{bastianelli2019gonality}. This result admits a geometric interpretation, which lies at the core of this work. Consider $X \\subset \\mathbb{P}^{n+1} \\times \\mathbb{P}^{m+1}$ a very general smooth irreducible of bi-degree $(a,b)$ with $a \\geq 2n+m+1, b \\geq 2m+n+1$. Via the natural projection onto each factor, $X$ may alternatively be seen as a family of degree $a$ hypersurfaces in $\\mathbb{P}^{n+1}$ parametrized by $\\mathbb{P}^{m+1}$, or as a family of hypersurfaces of degree $b$ in $\\mathbb{P}^{m+1}$ parametrized by $\\mathbb{P}^{n+1}$. We prove that the covering gonality of $X$ is actually the covering gonality of a general fibre from one of these families. \\\\\nIn the final section of this paper we study the newly introduced birational invariant measure of association. This notion has been introduced by Lazarsfeld-Martin in \\cite{lazarsfeld2023measures} and the intuition is as follow: given a pair of two algebraic variety of the same dimension $n$, one can be interested by the following question: are $X$ and $Y$ birationally related? If not how can one measure that failure? In \\cite{lazarsfeld2023measures} Lazarfeld and Martin introduce the joint covering gonality of two algebraic varieties of same dimension $X$ and $Y$, that we denote $\\covgon(X,Y)$. Although less understood than in the classical setting, this invariant already satisfies interesting lower bounds. Our second main result establishes such a bound for hypersurfaces in products of projective spaces inspired by the one proven in \\cite{lazarsfeld2023measures}.\n\n\\begin{thm}\nLet $X, Y \\subset \\mathbb{P}^{n+1} \\times \\mathbb{P}^{m+1}$ be very general smooth hypersurfaces \nof bi-degrees $(a_1,b_1)$ and $(a_2,b_2)$ respectively with $a_{1},a_{2} \\geq 2n+m+2$ and $b_{1},b_{2} \\geq 2m+n+2$. Then:\n\\vspace{0.5cm}\n\\begin{center}\n   $\\operatorname{cov.gon}(X,Y) \\;\\geq\\; k_1 + k_2 - n - m$\n\\end{center}\nwhere : \n\\[\nk_1 = \\min\\{a_1 - n - 3, b_1 - m - 3\\}, \\quad\nk_2 = \\min\\{a_2 - n - 3, b_2 - m - 3\\}.\n\\]\n\n\\end{thm}\n\nConcerning the organisation of this paper it will be as follows. In the first section we recall all the needed background on the covering gonality and the classical way one has to control this invariant. The second section, much shorter is devoted to define the notion of line in a product of projective spaces which we use heavily during the proof of the first theorem in Section 3. Finally, in section 4 we explore the joint covering gonality and prove the second main theorem stated above.\n\\\\\n\n In this work, $X$ will denote a smooth irreducible  complex projective variety of  dimension $\\dim(X)$. We will say a property holds at a general point $x \\in X $ if it holds on a Zariski open subset of $X$. We will say that a property holds for a very general point in $X$ if it holds outside of countably many proper Zariski closed subsets in $X$. \n\n\\vspace{0.3cm}\n\n\\textit{Acknowledgments}: \nI would first like to express my sincere gratitude to my advisors, Damian Brotbek and Gianluca Pacienza, for their time, guidance, and valuable feedback on this work and its earlier versions. This paper owes a great deal to Francesco Bastianelli, who delivered a series of lectures at the IECL during which he generously proposed this problem. The present work is deeply inspired by his research. Finally, I warmly thank Steve Balme for his encouragement throughout the development of this work.\n\n\\begin{center}", "sketch": "We prove that the covering gonality of $X$ is actually the covering gonality of a general fibre from one of these families: via the natural projections $X\\to \\mathbb{P}^{m+1}$ and $X\\to \\mathbb{P}^{n+1}$, one can view $X$ either as “a family of degree $a$ hypersurfaces in $\\mathbb{P}^{n+1}$ parametrized by $\\mathbb{P}^{m+1}$” or as “a family of hypersurfaces of degree $b$ in $\\mathbb{P}^{m+1}$ parametrized by $\\mathbb{P}^{n+1}$,” and the argument identifies $\\covgon(X)$ with the covering gonality of a general fibre from one of these two families.", "expanded_sketch": "We prove that the covering gonality of $X$ is actually the covering gonality of a general fibre from one of these families: via the natural projections $X\\to \\mathbb{P}^{m+1}$ and $X\\to \\mathbb{P}^{n+1}$, one can view $X$ either as “a family of degree $a$ hypersurfaces in $\\mathbb{P}^{n+1}$ parametrized by $\\mathbb{P}^{m+1}$” or as “a family of hypersurfaces of degree $b$ in $\\mathbb{P}^{m+1}$ parametrized by $\\mathbb{P}^{n+1}$,” and the argument identifies $\\covgon(X)$ with the covering gonality of a general fibre from one of these two families.", "expanded_theorem": "\\label{mainthm}\nLet $X \\subset \\mathbb{P}^{n+1}\\times \\mathbb{P}^{m+1}$ be a very general irreducible smooth hypersurface of bi-degree $(a,b)$. Suppose that $a\\geq 2n+m+1$, $b\\geq 2m+n+1$, and $n,m \\geq 2 $. Then:\n\\begin{center}\n    ${ \\minimum \\Big\\{a-\\lfloor \\frac{\\sqrt{16n+9}-1}{2} \\rfloor,b-\\lfloor \\frac{\\sqrt{16m+9}-1}{2} \\rfloor \\Big\\} \\leq \\covgon(X)\\leq \\minimum \\Big\\{a-\\lfloor \\frac{\\sqrt{16n+1}-1}{2} \\rfloor,b-\\lfloor \\frac{\\sqrt{16m+1}-1}{2} \\rfloor \\Big\\} }$.\n\\end{center}\nMoreover, both sides of the inequality coincide whenever $m,n \\in \\{4\\alpha^{2}+3\\alpha, 4\\alpha^{2}+5\\alpha+1, \\alpha \\in \\mathbb{N}\\}$,", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let \\(X \\subset \\mathbb{P}^{n+1}\\times \\mathbb{P}^{m+1}\\) be a very general irreducible smooth hypersurface of bi-degree \\((a,b)\\), where \\(n,m\\ge 2\\), \\(a\\ge 2n+m+1\\), and \\(b\\ge 2m+n+1\\). Here \\(\\operatorname{cov.gon}(X)=\\covgon(X)\\) denotes the covering gonality of \\(X\\), i.e. the minimal gonality of an irreducible curve passing through a general point of \\(X\\), and the gonality of a smooth irreducible curve \\(C\\) is the minimal degree of a morphism \\(C\\to \\mathbb{P}^1\\). Which statement holds for every such hypersurface \\(X\\)?", "correct_choice": {"label": "A", "text": "\\[\\min\\!\\left\\{a-\\left\\lfloor \\frac{\\sqrt{16n+9}-1}{2}\\right\\rfloor,\\; b-\\left\\lfloor \\frac{\\sqrt{16m+9}-1}{2}\\right\\rfloor\\right\\}\\le \\covgon(X)\\le \\min\\!\\left\\{a-\\left\\lfloor \\frac{\\sqrt{16n+1}-1}{2}\\right\\rfloor,\\; b-\\left\\lfloor \\frac{\\sqrt{16m+1}-1}{2}\\right\\rfloor\\right\\}.\\] Moreover, if both \\(m\\) and \\(n\\) belong to \\(\\{4\\alpha^2+3\\alpha,\\;4\\alpha^2+5\\alpha+1\\mid \\alpha\\in\\mathbb{N}\\}\\), then the lower and upper bounds above coincide."}, "choices": [{"label": "B", "text": "\\[\\min\\!\\left\\{a-\\left\\lfloor \\frac{\\sqrt{16n+1}-1}{2}\\right\\rfloor,\\; b-\\left\\lfloor \\frac{\\sqrt{16m+1}-1}{2}\\right\\rfloor\\right\\}\\le \\covgon(X)\\le \\min\\!\\left\\{a-\\left\\lfloor \\frac{\\sqrt{16n+9}-1}{2}\\right\\rfloor,\\; b-\\left\\lfloor \\frac{\\sqrt{16m+9}-1}{2}\\right\\rfloor\\right\\}.\\] Moreover, if both \\(m\\) and \\(n\\) belong to \\(\\{4\\alpha^2+3\\alpha,\\;4\\alpha^2+5\\alpha+1\\mid \\alpha\\in\\mathbb{N}\\}\\), then the lower and upper bounds above coincide."}, {"label": "C", "text": "\\[\\covgon(X)\\ge \\min\\!\\left\\{a-\\left\\lfloor \\frac{\\sqrt{16n+9}-1}{2}\\right\\rfloor,\\; b-\\left\\lfloor \\frac{\\sqrt{16m+9}-1}{2}\\right\\rfloor\\right\\}.\\]"}, {"label": "D", "text": "\\[\\max\\!\\left\\{a-\\left\\lfloor \\frac{\\sqrt{16n+9}-1}{2}\\right\\rfloor,\\; b-\\left\\lfloor \\frac{\\sqrt{16m+9}-1}{2}\\right\\rfloor\\right\\}\\le \\covgon(X)\\le \\max\\!\\left\\{a-\\left\\lfloor \\frac{\\sqrt{16n+1}-1}{2}\\right\\rfloor,\\; b-\\left\\lfloor \\frac{\\sqrt{16m+1}-1}{2}\\right\\rfloor\\right\\}.\\] Moreover, if both \\(m\\) and \\(n\\) belong to \\(\\{4\\alpha^2+3\\alpha,\\;4\\alpha^2+5\\alpha+1\\mid \\alpha\\in\\mathbb{N}\\}\\), then the lower and upper bounds above coincide."}, {"label": "E", "text": "\\[\\min\\!\\left\\{a-\\left\\lfloor \\frac{\\sqrt{16n+9}-1}{2}\\right\\rfloor,\\; b-\\left\\lfloor \\frac{\\sqrt{16m+9}-1}{2}\\right\\rfloor\\right\\}\\le \\covgon(X)\\le \\min\\!\\left\\{a-\\left\\lfloor \\frac{\\sqrt{16n+1}-1}{2}\\right\\rfloor,\\; b-\\left\\lfloor \\frac{\\sqrt{16m+1}-1}{2}\\right\\rfloor\\right\\}.\\] Moreover, the lower and upper bounds above coincide for every such pair \\((m,n)\\) with \\(m,n\\ge 2\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "case_split", "tampered_component": "assignment of fibrewise lower versus upper bounds", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "case_split", "tampered_component": "dropped the upper bound and coincidence clause, keeping only the fibrewise lower bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "minimum over the two projection families replaced by maximum", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "restricted coincidence condition replaced by uniform coincidence for all dimensions", "template_used": "stronger_trap"}]}, "qa quality eval": {"ALS": {"score": 2, "justification": "The stem does not reveal the correct bound; it only states the hypotheses and asks for the quantitative estimate. There is no explicit answer leakage, though the setup clearly signals that a theorem-level statement is being recalled."}, "TAS": {"score": 0, "justification": "This is essentially a direct theorem-recall item: the stem gives the full hypotheses and asks for the exact conclusion. The correct option is basically the theorem statement itself rather than a derived consequence or an application."}, "GPS": {"score": 1, "justification": "There is some moderate reasoning in checking which floor-shift belongs to the lower versus upper bound, whether a min or max is appropriate, and whether the equality condition should require both indices. However, the item mainly tests recognition/recall of a precise statement rather than genuine generative mathematical reasoning."}, "DQS": {"score": 2, "justification": "The distractors are mathematically plausible and target realistic failure modes: reversing lower/upper bounds, replacing min by max, weakening the equality criterion, and offering a weaker true statement. They are distinct and nontrivial."}, "total_score": 5, "overall_assessment": "A solid theorem-recognition MCQ with no answer leakage and strong distractors, but it is largely tautological and only moderately tests reasoning rather than generation."}}